ADVANCED MATHEMATICAL THINKING
Mathematics Education Library
VOLUME 11
Managing Editor
A.J. Bishop, Cambridge, U.K.
Editorial Board
H. Bauersfeld, Bielefeld, Germany
J. Kilpatrick, Athens, U.S.A.
G. Leder, Melbourne, Australia
S. Tumau, Krakow, Poland
G. Vergnaud, Paris, France
The titles published in this series are listed at the end of this volume.
ADVANCED
MATHEMATICAL THINKING
Edited by
DAVID TALL
Science Education Department,
University of Warwick
KLUWER ACADEMIC PUBLISHERS
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TABLE OF CONTENTS
PREFACE
xiii
ACKNOWLEDGEMENTS
xvii
INTRODUCTION
CHAPTER 1 : The Psychology of Advanced Mathematical Thinking - David Tall
1.
2.
3.
4.
Cognitive considerations
1.1
Different kinds of mathematical mind
Meta-theoretical considerations
1.2
1.3
Concept image and concept definition
1.4
Cognitive development
1.5
Transition and mental reconstruction
1.6
Obstacles
1.7
Generalization and abstraction
1.8
Intuition and rigour
The growth of mathematical knowledge
2.1
The full range of advanced mathematical thinking
2.2
Building and testing theories: synthesis and analysis
2.3
Mathematical proof
Curriculum design in advanced mathematical learning
3.1
Sequencing the learning experience
3.2
Problem-solving
3.3
Proof
3.4
Differences between elementary and advanced
mathematical thinking
Looking ahead
3
4
4
6
6
7
9
9
11
13
14
14
15
16
17
17
18
19
20
20
vi
TABLE OF CONTENTS
I : THE NATURE OF
ADVANCED MATHEMATICAL THINKING
CHAPTER 2 : Advanced Mathematical Thinking Processes - Tommy Dreyfus
1.
2.
3.
4.
5.
Advanced mathematical thinking as process
Processes involved in representation
2.1
The process of representing
2.2
Switching representations and translating
2.3
Modelling
Processes involved in abstraction
3.1
Generalizing
3.2
Synthesizing
3.3
Abstracting
Relationships between representing and abstracting (in learning
processes)
A wider vista of advanced mathematical processes
CHAPTER 3 : Mathematical Creativity - Gontran Ervynck
1.
2.
3.
4.
5.
6.
7.
8.
9.
The stages of development of mathematical creativity
The structure of a mathematical theory
A tentative definition of mathematical creativity
The ingredients of mathematical creativity
The motive power of mathematical creativity
The characteristics of mathematical creativity
The results of mathematical creativity
The fallibility of mathematical creativity
Consequences in teaching advanced mathematical thinking
CHAPTER 4 : Mathematical Proof - Gila Hanna
1.
2.
3.
4.
5.
6.
Origins of the emphasis on formal proof
More recent views of mathematics
Factors in acceptance of a proof
The social process
Careful reasoning
Teaching
25
26
30
30
32
34
34
35
35
36
38
40
42
42
46
46
47
47
49
50
52
52
54
55
55
58
59
60
60
TABLE OF CONTENTS
vii
II: COGNITIVE THEORY
OF ADVANCED MATHEMATICAL THINKING
CHAPTER 5 : The Role of Definitions in the Teaching and Learning of
Mathematics - Shlomo Vinner
1.
2.
3.
4.
5.
6.
7.
8.
Definitions in mathematics and common assumptions about
Pedagogy
The cognitive situation
Concept image
Concept formation
Technical contexts
Concept image and concept definition - desirable theory and
practice
Three illustrations of common concept images
Some implications for teaching
CHAPTER 6 : The Role of Conceptual Entities and their symbols in building
AdvancedMathematicalConcepts - Guershon Harel & James Kaput
1.
2.
3.
Three roles of conceptual entities
1.1
Working-memory load
1.2a
Comprehension: the case of “uniform” and “point- wise”
operators
1.2b
Comprehension: the case of object-valued operators
1.3
Conceptual entities as aids to focus
Roles of mathematical notations
2.1
Notation and formation of cognitive
entities
2.2
Reflecting structure in elaborated notations
Summary
CHAPTER 7 : Reflective Abstraction in Advanced Mathematical Thinking Ed Dubinsky
65
65
67
68
69
69
69
73
79
82
83
84
84
86
88
88
89
91
93
95
1.Piaget’s notion of reflective abstraction
97
1.1
The importance of reflective abstraction
97
1.2
The nature of reflective abstraction
99
1.3
Examples of reflective abstraction in children’s thinking 100
1.4
Various kinds of construction in
reflective abstraction
101
2.
A theory of the development of concepts in advanced
mathematical thinking
102
...
viii
TABLE OF CONTENTS
3.
4.
2.1
Objects, processes and schemas
2.2
Constructions in advanced mathematical concepts
2.3
The organization of schemas
Genetic decompositions of three schemas
3.1
Mathematical induction
3.2
Predicate calculus
3.3
Function
Implications for education
4.1
Inadequacy of traditional teaching practices
4.2
What can be done
102
103
106
109
110
114
116
119
120
123
III : RESEARCH INTO THE TEACHING AND LEARNING
OF ADVANCED MATHEMATICAL THINKING
CHAPTER 8 : Research in Teaching and Learning Mathematics at an Advanced
Level - Aline Robert & Rolph Schwarzenberger
1.
2.
3.
Do there exist features specific to the learning of advanced
mathematics?
1.1
Social factors
1.2
Mathematical content
1.3
Assessment of students’ work
1.4
Psychological and cognitive characteristics of students
1.5
Hypotheses on student acquisition of knowledge in
advanced mathematics
1.6
Conclusion
Research on learning mathematics at the advanced level
2.1
Research into students’ acquisition of specific concepts
2.2
Research into the organization of mathematical content
at an advanced level
2.3
Research on the external environment for advanced
mathematical thinking
Conclusion
CHAPTER 9 : Functions and associated learning difficulties - Theodore
Eisenberg
1.
Historical background
2.
Deficiencies in learning theories
3.
Variables
4.
Functions, graphs and visualization
5.
Abstraction, notation, and anxiety
Representational difficulties
6.
127
128
128
128
130
131
132
133
133
134
134
136
139
140
140
142
144
145
148
151
TABLE OF CONTENTS
7.Summary
CHAPTER 10 : Limits - Bernard Cornu
1.
2.
3.
4.
5.
6.
Spontaneous conceptions and mental models
Cognitive obstacles
Epistemological obstacles in historical development
Epistemological obstacles in modem mathematics
The didactical transmission of epistemological obstacles
Towards pedagogical strategies
CHAPTER 11 : Analysis - Michèle Artigue
1.
2.
3.
4.
152
153
154
158
159
162
163
165
167
Historical background
168
1.1
Some concepts emerged early but were established late 168
1.2
Some concepts cause both enthusiasm and virulent
criticism
168
1.3
The differential/derivative conflict and its educational
repercussions
169
1.4
The non-standard analysis revival and its weak impact on
education
172
1.5
Current educational trends
173
Student conceptions
174
2.1
A cross-sectional study of the understanding of
elementary calculus in adolescents and young adults
176
2.2
A study of student conceptions of the differential, and
of the processes of differentiation and integration
180
2.2.1 The meaning and usefulness of differentials and
differential procedures
180
2.2.2 Approximation and rigour in reasoning
182
2.2.3 The role of differential elements
184
2.3
The role of education
186
Research in didactic engineering
186
"Graphic calculus"
187
3.1
3.2
Teaching integration through scientific debate
191
3.3
Didactic engineering in teaching differential equations
193
3.4
Summary
195
Conclusion and future perspectives in education
196
CHAPTER 12 : The Role of Students’Intuitions of Infinity in Teaching the
Cantorian Theory - Dina Tirosh
1.
ix
Theoretical conceptions of infinity
199
200
x
TABLE OF CONTENTS
2.
3.
4.
5.
Students’ conceptions of infinity
201
2.1
Students’ intuitive criteria for comparing infinite
quantities
203
First steps towards improving students’ intuitive understanding of
actual infinity
205
3.1
The "finite and infinite sets" learning unit
206
3.2
Raising students’ awareness of the inconsistencies in
their own thinking
206
3.3
Discussing the origins of students’ intuitions about
infinity
207
3.4
Progressing from finite to infinite sets
207
208
3.5
Stressing that it is legitimate to wonder about infinity
3.6
Emphasizing the relativity of mathematics
208
3.7
Strengthening students’ confidence in the new definitions 209
Changes in students’ understanding of actual infinity
209
Final comments
214
CHAPTER 13 : Research on Mathematical Proof - Daniel Alibert &
Michael Thomas
1.
2.
3.
4.
5.
Introduction
Students’ understanding of proofs
The structural method of proof exposition
3.1
A proof in linear style
3.2
A proof in structural style
Conjectures and proofs - the scientific debate in a mathematical
course
4.1
Generating scientific debate
4.2
An example of scientific debate
4.3
The organization of proof debates
4.4
Evaluating the role of debate
Conclusion
CHAPTER 14 : Advanced Mathematical Thinking and the Computer Ed Dubinsky and David Tall
1.
2.
3.
4.
5.
6.
215
215
216
219
221
222
224
225
226
228
229
229
231
Introduction
231
The computer in mathematical research
231
234
The computer in mathematical education - generalities
Symbolic manipulators
235
Conceptual development using a computer
237
The computer as an environment for exploration of fundamental
ideas
238
TABLE OF CONTENTS
7.
8.
Programming
The future
xi
241
243
Appendix to Chapter 14
ISETL : a computer language for advanced mathematical thinking
244
EPILOGUE
CHAPTER 15 : Reflections - David Tall
251
BIBLIOGRAPHY
261
INDEX
275
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PREFACE
Advanced Mathematical Thinking has played a central role in the development of human
civilization for over two millennia. Yet in all that time the serious study of the nature of
advanced mathematical thinking – what it is, how it functions in the minds of expert
mathematicians, how it can be encouraged and improved in the developing minds of
students – has been limited to the reflections of a few significant individuals scattered
throughout the history of mathematics. In the twentieth century the theory of mathematical
education during the compulsory years of schooling to age 16 has developed its own body
of empirical research, theory and practice. But the extensions of such theories to more
advanced levels have only occurred in the last few years.
In 1976 The International Group for the Psychology of Mathematics (known as PME)
was formed and has met annually at different venues round the world to share research
ideas. In 1985 a Working Group of PME was formed to focus on Advanced Mathematical
Thinking with a major aim of producing this volume.
The text begins with an introductory chapter on the psychology of advanced mathematical thinking, with the remaining chapters grouped under three headings:
• the nature of advanced mathematical thinking,
• cognitive theory,
and
• reviews of the progress of cognitive research into different areas of advanced
mathematics.
It is written in a style intended both for mathematicians and for mathematics educators, to
encourage an interest in the cognitive difficulties experienced by students of the former and
to extend the psychological theories of the latter through to later stages of development. We
are cognizant of the fact that it is essential to understand the nature of the thinking of
mathematical experts to see the full spectrum of mathematical growth. We therefore begin
with an introductory chapter on the psychology of advanced mathematical thinking. This
is followed by three chapters which focus on the nature of advanced mathematical thinking:
a study of the mental processes involved, the essential qualities of mathematical creativity
and the mathematician’s view of proof.
The processes prove to be subtle and complex and, sadly, few of the more advanced
processes are made available to the average student in an advanced mathematical course.
Creativity is concerned with how the subtle ideas of research are built in the mind. Proof
is how they are ordered in alogical development both to verify the nature of the relationships
and also to present them for approval to the mathematical community.
xiii
xiv
PREFACE
However, there is a huge gulf between the way in which ideas are built cognitively and
the way in which they are arranged and presented in a deductive order. This warns us that
simply presenting a mathematical theory as a sequence of definitions, theorems and proofs
(as happens in a typical university course) may show the logical structure of the
mathematics, but it fails to allow for the psychological growth of the developing human
mind.
We begin the part of the book on cognitive theory by considering the way in which
formal mathematical definitions are conceived by students and how this can be at variance
with the formal theory. As a result of mentally manipulating a (mathematical) concept an
individual develops an idiosyncratic personal concept image which is the product of
experience and mental activity. Empirical research shows how this can give rise to subtle
conflicts that can cause cognitive obstacles in the mind of the developing student and act
as a barrier to attaining the formal ideas in the theory. The next chapter looks at the mental
objects that arc the material of mathematical thought – the conceptual entities that are
manipulated in the mind during advanced mathematical thinking, and how these entities
are represented by different kinds of symbolism. The final chapter in this part considers how
these conceptual entities are formed – through the process of reflective abstraction. All
advanced mathematical concepts arc “abstract”. This chapter postulates a theory of how
these concepts start as processes which are encapsulated as mental objects that are then
available for higher level abstract thought. Such a theory can give insight into how
mathematicians develop advanced mathematical ideas, yet may fail to pass these thinking
processes on to students, and what might be done to improve the situation.
The remainder of the book is concerned with overviews of empirical research and theory
in various specific topics. First the question of the nature of advanced mathematical
thinking is addressed and how (if at all) it differs from more elementary thinking occurring
in younger children. Then there follow chapters on functions, limits, analysis, infinity,
proof, and the growing use of the computer in advanced mathematics. Each one of these
reveals a wide variety of obstacles in students’ mental imagery and often extremely limited
conceptions of formal concepts which are the unforseen consequences of the manner in
which the subject is presented to the student. A variety of more cognitively appropriate
approaches are postulated, some with empirical evidence of success. These include:
• the participation of the student in the process of mathematical thinking through
an active process of “scientific debate”, rather than passive receipt of preorganized theory,
• the direct confrontation of the student with conflict which occurs in developing
new theoretical constructs, to help them reflect on the problem and build a new,
more coherent, cognitive structure.
• the building up of appropriate intuitive foundations for the advanced mathematical concepts, through an approach which balances cognitive growth and
an appreciation of logical development.
• the use of visualization, particularly utilizing a computer, to give the student an
overall view of concepts and enabling more versatile methods of handling the
information,
PREFACE
xv
• the use of programming to cause the student to think through mathematical
processes in a way which can be encapsulated by reflective abstraction.
In all these ways we believe that empirical research into advanced thinking processes
related to complementary cognitive theory can have a significant effect in improving the
education of students at an advanced level.
In every chapter the authors have been encouraged to impress their own personalities
on their view of the phenomena, but this has been done within a framework of internal
consultation. Each participant operates from personal constructs within a context of mutual
support and constructive criticism from other authors and the final manuscript has been
recast by the editor to enable it to be read throughout as a single text rather than as a
collection of disconnected papers. This was made possible through the wonders of modem
technology, using a Macintosh SE/30 computer to enable the editor to redraft the chapters
and set the whole book as camera-ready copy.
The cognitive theory of advanced mathematical thinking is developing apace. This
study is the first step in making the broad sweep of current ideas in the advanced
mathematical education community available to a wider readership.
David Tall
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ACKNOWLEDGEMENTS
In preparing this book we acknowledge the assistance and support of a wide range of
mathematicians and mathematics educators. First we thank the International Committee of
PME for encouraging us to meet each year as a continuing Working Group of PME. Then
we thank those members of the Working Group who do not appear as authors but who have
helped in many ways – presenting papers, organizing and reporting sessions – including
Janet Duffin, Ulla Kürstein-Jensen, Miriam Amit, Anna Sfard, Janine Rogalski. Within the
group of authors, all have participated in criticizing and encouraging the writing of others.
However, it is with grateful thanks that the editor wishes to acknowledge assistance beyond
the call of duty from Tommy Dreyfus, Ted Eisenberg, Ed Dubinsky, and particularly from
Gontran Ervynek, whose enthusiasm generated the original impetus for the creation of this
book.
xvii
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INTRODUCTION
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CHAPTER 1
THE PSYCHOLOGY
OF
ADVANCED MATHEMATICAL THINKING
DAVID TALL
In the opening chapter of The Psychology of Invention in the Mathematical Field, the
mathematician Jacques Hadamard highlighted the fundamental difficulty in discussing the
nature of the psychology of advanced mathematical thinking:
... that the subject involves two disciplines, psychology and mathematics, and would require, in
order to be treated adequately, that one be both a psychologist and a mathematician. Owing to the
lack of this composite equipment, the subject has been investigated by mathematicians on the one
side, by psychologists on the other ...
(Hadamard, 1945, page 1.)
Exponents of the two disciplines are likely to view the subject in different ways – the
psychologist to extend psychological theories to thinking processes in a more complex
knowledge domain – the mathematician to seek insight into the creative thinking process,
perhaps with the hope of improving the quality of teaching or research. Although we will
consider the nature of advanced mathematical thinking from a psychological viewpoint,
our main aim will be to seek insights of value to the mathematician in his professional work
as researcher and teacher.
We begin by looking at pertinent psychological considerations which will lay the
foundations for ideas introduced not only in the remainder of the chapter, but in the book
as a whole. We then focus our attention on the full cycle of activity in advanced math–
ematical thinking: from the creative act of considering a problem context in mathematical
research that leads to the creative formulation of conjectures and on to the final stage of
refinement and proof. We postulate that many of the activities that occur in this cycle also
occur in elementary mathematical problem-solving, but the possibility of formal definition
and deduction is one factor which distinguishes advanced mathematical thinking. We will
also find that teaching undergraduate mathematics often presents the final form of the
deduced theory rather than enabling the student to participate in the full creative cycle. In
the words of Skemp (1971), current approaches to undergraduate teaching tend to give
students the product of mathematical thought rather than the process of mathematical
thinking.
Not only may current methods of presenting advanced mathematical knowledge fail to
give the full power of mathematical thinking, it also has another, equally serious,
deficiency: a logical presentation may not be appropriate for the cognitive development
of the learner. Indeed, much of the empirical theory reported in the later chapters of the book
reveals cognitive obstacles which arise as students struggle to come to terms with ideas
which challenge and contradict their current knowledge structure. Fortunately, we are also
able to report empirical evidence that appropriate sequences of learning and instruction
designed to help the student actively construct the concepts can prove highly successful.
3
4
DAVID TALL
1. COGNITIVE CONSIDERATIONS
We begin by looking, not at the logic and order of the public evidence of mathematical
thinking found in research articles and text-books, but at the way in which these coherent
relationships are built in mathematical research and implications for how this might be
implemented in teaching and learning.
1.1 DIFFERENT KINDS OF MATHEMATICAL MIND
Writing in the first decade of this century, the celebrated mathematician Henri Poincaré
asserted:
It is impossible to study the works of the great mathematicians, or even those of the lesser,
without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds
of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to
believe they have advanced only step by step, after the manner of a Vauban1 who pushes on his
trenches against the placebesieged, leaving nothing to chance. The other sort are guided by intuition
and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the
advanced guard.
(Poincaré, 1913, p. 210)
He supported his arguments by contrasting the work of various mathematicians, including
the famous German analysts, Weierstrass and Riemann, relating this to the work of
students:
Weierstrass leads everything back to the consideration of series and their analytic transformations;
to express it better, he reduces analysis to a sort of prolongation of arithmetic; you may turn through
all his books without finding a figure. Riemann, on the contrary, at once calls geometry to his aid;
each of his conceptions is an image that no one can forget, once he has caught its meaning.
... Among our students we notice the same differences; some prefer to treat their problems ‘by
analysis’, others ‘by geometry’. The first are incapable of ‘seeing in space’, the others are quickly
tired of long calculations and become perplexed.
(Poincaré, 1913, p. 212)
Of course, therearenotjust two different kinds of mathematical mind, but many. Kronecker
agreed with Weierstrass that logical proof was of paramount importance and transcended
intuitive visual arguments, but their fundamental beliefs in the nature of mathematical
concepts were very different. Weierstrass declared that “an irrational number has as real an
existence as anything else in the world of concepts”, but Kronecker was unable to accept
the actual infinity of real numbers, asserting that “God gave us the integers, the rest is the
work of man”. Based on the Weierstrassian notion of the actual infinity of real numbers,
Cantor was able to produce an infinite counting argument to show that there are strictly
“more” real numbers than algebraic numbers (solutions of polynomial equations with
integer coefficients). He therefore claimed that there exists a real non-algebraic number,
without giving an explicit method to construct one. This was anathema to Kronecker who
caused Cantor’s paper to be rejected from publication in Crelle’s Journal in 1873.
1
Sebastien de Vauban (1633-1707) was a French military engineer who revolutionized the art
of siege craft and defensive fortifications.
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
5
Such arguments about the foundations of mathematics led to the development of several
different strands of mathematical philosophy at the beginning of the twentieth century. The
intuitionist view represented by Kronecker asserted that mathematical concepts only exist
when their construction is demonstrated from the integers, the formalist view of Hilbert
affirmed that mathematics is the meaningful manipulation of meaningless marks written
on paper, whilst the logicist view of Russell, declared that mathematics consists of
deductions using the laws of logic.
Practising mathematicians tend to distance themselves from esoteric arguments and
simply get on with their work of stating and proving theorems. Thus the twentieth century
has seen the demise of Kronecker’s views and the triumph of a pragmatic mixture of
formalism and logic. It has seen the creation of a large number of formal systems based on
logical deduction from formal definitions and axioms – an approach that survived the
apparently mortal blow struck by Gödel’s incompleteness theorem, that any axiomatic
system including the integers must contain true statements that cannot be proved by a finite
sequence of steps within the system.
The textbook by Bishop (1967) on constructive analysis – which insists on algorithmic
construction proofs and disallows proof by contradiction alone – seems but an isolated
singularity in the dynamic flow of twentieth century mathematical creativity.
Nevertheless, the recent introduction of computer technology may yet see a new
renaissance in constructibility because of the way that computers manipulate data:
Computers have affected mathematics as inevitably as the development of railroads affected
patterns of land development. With computers it is possible to test hypotheses and compile data with
ease that formerly would have been accessible, if at all, only via the most sophisticated techniques.
This has affected not only the sort of questions that mathematicians work on, but the very way that
they think. One has to ask oneself which examples can be tested on a computer, a question which
forces one to consider concrete algorithms and to try to make them efficient. Because of this and
because algorithms have real-life applications of considerable importance, the development of
algorithms has become a respectable topic in its own right.
(Edwards, 1987)
The reason for raising these differences in mathematician’s perceptions is to heighten the
readers’ awareness of their own part in life’s rich tapestry, with a personal view of
mathematics that will differ in many ways from the conceptions of others. It may come as
a surprise when one first realizes that other people have radically different thinking
processes. It happened to the author when using pictures to help students visualize ideas in
mathematical analysis, at a time when he did not question the implicit belief that such an
approach was universally valid. Whilst writing a text book on complex analysis, a colleague
in the next room was engaged on a similar enterprise, yet the latter’s book had almost no
pictures at all. He only included a diagram illustrating the argument of a complex number
after a great deal of heart searching. To him a real number was an element of a complete
ordered field (satisfying specific axioms) and a complex number was an ordered pair of real
numbers. The argument of a complex number (x,y) wasdefined asareal number a such that
cos (α) =
sin (α) =
6
DAVID TALL
where sin and cos were defined by red power series. The theory did not require a
geometrical meaning. He took this hard line to make sure that his arguments were the
product of logical deduction and not dependent anywhere on geometric intuition. At the
time the author was sympathetic to this philosophical viewpoint, but considered it too
sophisticated for students. It was some considerable time later that the realization dawned
that not all students shared the geometric point of view. No one view holds universal sway.
1.2 META-THEORETICAL CONSIDERATIONS
The discussion of the preceding session is a salutary reminder that any theory of the
psychology of learning mathematics must take into account not only the growing
conceptions of the students, but the conceptions of mature mathematicians. Mathematics
is a shared culture and there are aspects which are context dependent. For example, an
analyst’s view of a differential may be very different ftom that of an applied mathematician,
and a given individual may strike up different attitudes to this concept depending on
whether it is considered in an analytic or applied context. We will see (chapter 11) that such
attitudes can cause conflicts in students too.
At a far deeper psychological level we all have subtly different ways of viewing a given
mathematical concept, depending on our previous experiences. For example, the “completeness axiom” for the real numbers is viewed by some as “filling in all the gaps between
the rational numbers to give all the points on the number line”. Such a view may imply that
there is “no room” to fit in any more numbers: the number line is now “complete”. The
“real” number line, in particular cannot contain “infinitesimals” which are smaller than any
positive rational yet not zero. But, for others, “completion” is only a technical axiom to
adjoin the limit points of cauchy sequences of rational numbers. In this case it is perfectly
possible to embed the real numbers in a variety of larger number fields, which include
infiitesimals and infinite numbers. It is this view which leads to the modern infinitesimal
theory of “non-standard analysis”. The latter idea, however, is anathema to many
mathematicians, including Cantor, who denied the existence of infinitesimals on the
grounds that it was not possible to calculate the reciprocal of an infiite number in his theory
of cardinal infinities. Even today many mathematicians are troubled by the infinitesimal
ideas of non-standard analysis; they may not deny its logic, but they sense a deep-seated
psychological unease as to its validity.
Thus any theory of the psychology of mathematical thinking must be seen in the wider
context of human mental and cultural activity. There is not one true, absolute way of
thinking about mathematics, but diverse culturally developed ways of thinking in which
various aspects are relative to the context.
1.3 CONCEPT IMAGE AND CONCEPT DEFINITION
In Tall & Vinner (1981). the distinction is made between the individual’s way of thinking
of a concept and its formal definition, thus distinguishing between mathematics as a mental
activity and mathematics as a formal system. This theory applies to expert mathematicians
as well as developing students:
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
7
The human brain is not a purely logical entity. The complex manner in which it functions is often
at variance with the logic of mathematics. It is not always pure logic that gives us insight, nor is it
chance that makes us make mistakes ... We shall use the term concept image to describe the total
cognitive structure that is associated with the concept, which includes all the mental pictures and
associated properties and processes. It is built up over the years through experiences of all kinds,
changing as the individual meets new stimuli and matures. ... As the concept image develops it need
not be coherent at all times.The brain does not work that way. Sensory input excites certain neuronal
pathways and inhibits others. In this way different stimuli can activate different parts of the concept
image, developing them in a way which need not make a coherent whole. (Tall & Vinner 1981)
In this way it is possible for conflicting views to be held in the mind of a given individual
and to be evoked at different times without the individual being aware of the conflict until
they are evoked simultaneously.
The mature mathematician is not immune from internal conflicts, but he or she has been
able to link together large portions of knowledge into sequences of deductive argument. To
such a person it seems so much easier to categorize this knowledge in a logically structured
way. Thus a mature mathematician may consider it helpful to present material to students
in a way which highlights the logic of the subject. However, a student without the
experience of the teacher may find a formal approach initially difficult, a phenomenon
which may be viewed by the teacher as a lack of experience or intellect on the part of the
student. This is a comforting viewpoint to take, especially when the teacher is part of a
mathematical community who share the mathematical understanding. But it is not realistic
in the wider context of the needs of the students. What is essential – for them – is an approach
to mathematical knowledge that grows as they grow: a cognitive approach that takes
account of the development of their knowledge structure and thinking processes. To
become mature mathematicians at an advanced level, they must ultimately gain insight into
the ways of advanced mathematicians but, en route, they may find a stony path that will
require a fundamental transition in their thinking processes.
1.4 COGNITIVE DEVELOPMENT
There are many competing theories in psychology. Behaviourist theory, built on external
observation of stimulus and response, refuses to speculate about the internal workings of
the mind. It provides observable and repeatable evidence of the behaviour of animals,
including humans, under repeated stimuli, but it has limited application to mathematical
thinking beyond the mechanics of routine algorithms. Constructivist psychology, on the
other hand, attempts to discuss how mental ideas are created in the mind of each individual.
This may pose a dialectic problem for the mathematician with a Platonic ideal of
mathematics existing independently of the human mind, but it proves to give significant
insight into the creative processes of research mathematicians as well as the dfficulties
experienced by mathematics students.
The great Swiss psychologist Piaget saw the individual’s need to be in dynamic
equilibrium with his environment as an underlying theme in his work. This equilibrium
could be disturbed through the confrontation with new knowledge that conflicted with the
old, and so a transition period might occur in which the knowledge structure is reconstructed to give a more mature level of equilibrium.
8
DAVID TALL
Piaget saw the child grow into the adult through a series of stages of equilibrium, each
one richer than the one before. He identifed four main stages. The first is the sensori-motor
stage prior to the development of meaningful speech, followed by a pre-operational stage
when the young child realizes the permanence of objects, which continue to exist even if
they are temporarily out of sight. The child then goes through a transition into the period
of concrete operations where he or she can stably consider concepts which are linked to
physical objects, thence passing into a period of formal operations in the early teens when
the kind of hypothetical “if–then” becomes possible.
Piagetian stage theory has been extended to higher levels to encompass advanced
mathematical thinking. For instance, Ellerton (1985) suggested that Piaget’s cycle of
sensori-motor, pre-operational and concrete is the first level of a spiral cognitive development in which the formal stage is the beginning of another cycle of the same type at a higher
level of abstraction. Biggs & Collis (1982) suggested a repetition of formal operations at
successively higher levels, each repeating the learning cycle: unistructural, multistructural,
relational.
A difficulty of applying such theory to college mathematics teaching is that many –
probably most – college students are not able to perform at the abstract level of formal
operations, which Piaget reported occurring in children during their early teens. Ausubel
criticized the stage theory:
... because such a high percentage of American high school and college students fail to reach this
abstract level of cognitive logical operations.
(Ausubel et al 1968, p. 230)
Representative studies have indicated that only 15% of junior high school students ... 13.2% of
high school students ... and 22% of college students were at this level.
(ibid, p. 238)
The concrete/formal distinction has proved to be a useful starting point in developing local
hierarchies of difficulty in extensive studies such as Hart (1981) in the 11 to 16 age range,
and the development of early calculus concepts by Orton (1980). But a significant failure
of Piaget’s stage theory for the design of new teaching strategies is his own assertion that
the movement from one stage to another cannot be greatly accelerated by the affects of
teaching. Differences of cognitive demand have often been used in a negative sense to
describe students’ difficulties, but rarely to provide positive criteria for designing new
approaches to the subject. Papert (1980) asserted:
The Piaget of stage theory is essentially conservative, almost reactionary, in emphasizing what
children cannot do. I strive to uncover a more revolutionary Piaget,one who see pistemological ideas
might expand the known bounds of the human mind.
Advanced mathematics provides us with a useful metaphor which expands the vision of
stage theory to a theory more valuable in the development of advanced mathematical
thinking. Piaget used an analogy with group theory to underpin his sense of the dynamic
equilibrium of cognitive growth. He saw the identity element as representing the stable
state, and noted that stability could be maintained if any transformation from this state could
be reversed, thus suggesting a group structure in which every element has an inverse. But
the maintenance of a dynamic state of equilibrium has a more obvious mathematical
metaphor in dynamical systems and catastrophe theory. Here a system controlled by
continuously varying parameters can suddenly leap from one position of equilibrium to
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
9
another when the first becomes untenable. Depending on the history of the varying
parameters, the transition may be smooth, or it may be discontinuous. This analogy
suggests that stage theory may just be a linear trivialization of a far more complex system
of change, at least this may be so when the possible routes through a network of ideas
become more numerous, as happens in advanced mathematical thinking.
1.5 TRANSITION AND MENTAL RECONSTRUCTION
A far more valuable aspect of Piaget’s theory is the process of transition from one mental
state to another. During such a transition, unstable behaviour is possible, with the
experience of previous ideas conflicting with new elements. Piaget uses the terms
assimilation to describe the process by which the individual takes in new data and
accommodation the process by which the individual’s cognitive structure must be
modified. He sees assimilation and accommodation as complementary. During a transition
much accommodation is required. Skemp (1979) puts similar ideas in a different way by
distinguishing between the case where the learning process causes a simple expansion of
the individual’s cognitive structure and the case where there is cognitive conflict, requiring
a mental reconstruction. It is this process of reconstruction which provokes the difficulties
that occur during a transition phase.
Such transitions occur often in advanced mathematics as the individual struggles with
new knowledge structure. Conflict is aphenomenon well-known to the mathematical mind.
1.6 OBSTACLES
The most serious problem occurs when the new ideas are not satisfactorily accommodated.
In this case it may be possible for conflicting ideas to be present in an individual at one and
the same time:
New knowledge often contradicts the old, and effective learning requires strategies to deal with such
conflict. Sometimes the conflicting pieces of knowledge can be reconciled, sometimes one or the
other must be abandoned, and sometimes the two can both be “kept around” if safely maintained
in separate compartments.
(Papert, 1980, p. 121)
The thesis of Comu (1983) studies the conceptual development of the limit process from
school to university and underlines how the colloquial use of the term “limit” effects the
mathematical usage. He discusses the notion of an “obstacle”, introduced by Gaston
Bachelard (1938):
An obstacle is apiece of knowledge; it is part of the knowledge of the student. This knowledge was
at one time generally satisfactory in solving certain problems. It is precisely this satisfactory aspect
which has anchored the concept in the mind and made it an obstacle. The knowledge later proves
to be inadequate when faced with new problems and this inadequacy may not be obvious.
(Comu 1983, (original in French))
The obstacles found by Comu include the problems student face when they must calculate
limits using techniques more subtle than simple numerical and algebraic operations. He
discusses how the concept of infinity is introduced and is “surrounded in mystery”, yet the
10
DAVID TALL
new techniques “work” without the students understanding why. He demonstrates how
students’ experiences can lead to belief in the infinitely large and the infinitely small, with
“nought point nine recurring” being a number “just less than one” and the symbol ε
representing to many students a quantity that is smaller than any positive real number, but
not zero. There are implicit assumptions that the limiting process “goes on forever”, that
the limit “can never be attained”. (See chapter 10.)
Tall (1986a) suggests an explanation is given for these phenomena as the generic
extension principle:
If an individual works in a restricted context in which all the examples considered have a certain
property, then, in the absence of counter-examples, the mind assumes the known properties to be
implicit in other contexts.
For example, most convergent sequences described to beginning students are of a simple
kind given by a formula such as l/n, which tends to the limit( in this case zero), but the terms
never equal the limit. In the absence of any counter-examples students begin to believe that
this is always so. The rich experience of colloquial language supports this belief
(Schwarzenberger & Tall, 1978), with phrases like “gets close to” suggesting that the terms
of a sequence can never be coincident with the limit. Thus the implicit belief is slowly
formed that a sequence of terms converging to a limit gets closer and closer, but never
actually gets there.
Furthermore, if all the terms of a sequence have a certain property, it is natural to believe
that the limit has the same property. Thus the sequence 0.9, 0.99, ... has terms all less than
1, so the limit “nought point nine recurring” must also be less than one... This leads to the
mental image of a limiting object termed a generic limit in Tall (1986a). Ageneric limit need
not be a limit in the mathematical sense, but it is the concept of the limit that the individual
holds in his or her mind as a result of extrapolating the common properties of the terms of
the sequence.
This phenomenon happens not just with sequences of numbers, but sequences of
functions and other mathematical objects hat share a common property. Historically this
is enshrined in the “principle of continuity” of Leibniz:
In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning,
in which the final terminus may also be included. (Leibniz in a letter to Bayle, January 1687.)
It arises even earlier in the work of Nicholas of Cusa (1401–1464) who regarded the circle
as a polygon with an infinite number of sides, and inspired Kepler (1571–1630) to formulate
a metaphysical “bridge of continuity” in which normal and limiting forms of a figure are
characterized under a single definition. Thus Kepler (Opera Omnia II page 595) saw no
essential difference between a polygon and a circle, between an ellipse and a circle, between
the finite and the infinite, and between an infinitesimal area and a line.
The generic extension principle arises time and again in history. For example, Cauchy’s
assertion that the limit of continuous functions is continuous and Peacock’s “Principle of
Algebraic Permanence”, in which the properties of extended number systems, such as the
real and complex numbers, were based on the principle that the any algebraic law which
held in the smaller system also held in the extension. The latter held sway for some time
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
11
in the nineteenth century until Hamilton invented (discovered?) the quatemions, an
extension of the complex numbers whose multiplication fails to be commutative.
Obstacles arising from deeply held convictions about mathematics are rarely easy to
erase from the mind. We all carry with us a mental rag-bag of such beliefs, many of which
we suppress, but do not eliminate, when faced with the logic of mathematics. Often the only
trace of such an obstacle is through a sense of unease when there is a logical deduction that
does not “feel right”. We view this as an instance of cognitive conflict between inconsistent
portions of the individual’s concept image.
1.7 GENERALIZATION AND ABSTRACTION
A common difficulty observed in students learning advanced mathematics is their
complaint that the subject is “too abstract”. What is the cognitive reason for their difficulty?
The terms “generalization” and “abstraction” are used in mathematics both to denote
processes in which concepts are seen in a broader context and also the products of those
processes. For instance, we generalize the solution of linear equations in two and three
dimensions ton dimensions and we abstract from this context the notion of a vector space.
n
In doing so two very different mental objects are produced: the generalization and the
abstraction, a vector space Vovera field F. Practising mathematicians often regard a vector
space as both an abstraction and a generalization of aspects of two dimensional space and
it is therefore important to use the terms in a way which is consonant with their use in
mathematics. But the mathematics educator must look at the cognitive processes which are
involved, and here we see subtle differences between the two examples just given. The
2 to
3 , and so on, which
generalization n simply extends the chain ofideas from 1 to
is described by applying the usual arithmetic processes to each coordinate. The abstraction
Vis a very different mental object, which is defined by a list of axioms. Whilst the former
simply involves an extension of familiar processes, the latter requires a massive mental
reorganization.
As Dreyfus will discuss in greater detail in chapter 2, the process of defining the abstract
vector space must be followed by a sequence of theorems deducing the properties of a
vector space which follow from the axioms. Cognitively this is not just a deduction process
but a construction process in which the learner is building properties of the abstract object,
for example, that the axioms guarantee the “usual” properties of addition of vectors and
multiplication by scalars, that a linearly independent set of vectors will contain at most the
same number of vectors as a spanning set, that a space with a finite spanning set has a precise
“dimension” given in terms of an independent spanning set, or “basis”, and so on. In this
etc.) act
process of construction, existing examples of vector spaces (for example, 2,
both as supports and potential conflict factors. They support because they suggest
properties which are likely to hold, but they are potential sources of difficulty on the one
hand because the learner is constrained to prove something that may seem “self-evident”
from the examples and on the other because subtle properties common to all the examples
may be believed to be “generically true” for the abstract concept. During this period there
is a conflict between the properties of the examples which the learner knows, and the
properties of the new abstract concept which must be deduced from the definition. A period
of re-construction and consequent confusion is inevitable.
12
DAVID TALL
In Harel & Tall (to appear), we propose that a cognitive distinction be made between
different types of generalization in accordance with the cognitive activities involved. We
term an expansive generalization one which extends the student’s existing cognitive
structure without requiring changes in the current ideas. On the other hand, a generalization
which requires reconstruction of the existing cognitive structure we call a reconstructive
generalization. In this terminology we see that the general vector space n is, for most
students, an expansive generalization, whilst the abstract vector space is both an abstraction
and a reconstructive generalization.
We also note that it is possible for students in difficulties to operate in a third,
subsequently disastrous, way which simply involves remembering the new ideas as an
additional collection of information to be learned by rote and added to current knowledge
without any attempt at integration with the old ideas. This we call a disjunctive generalization. It is a generalization in the sense that the student may now be able to operate on a
broader range of examples, but it is likely to be of little lasting value to the student as it
simply adds to the number of disconnected pieces of information in the student’s mind
without improving the student’s grasp of the broader abstract implications.
The expansive generalization is a good teaching technique to adopt when it is necessary
to be able to deal with a wider class of applications without having to go through too much
stressful cognitive change. For instance, students who can carry out the process of solving
simultaneous linear equations in two variables are usually able to generalize (expansively)
to three, four, or more variables without difficult (though the calculations may soon
become tedious). Indeed, it is relatively straightforward to describe the process in general
terms whilst referring to a specific set of equations in, say, three variables, x, y, z. (“Subtract
suitable multiples of the first equation from the second and third to eliminate x, then
eliminate y from the resulting equations, solve for z and substitute back to find y, then x.”)
The process is easier Seen by enacting the solution than by describing it. Of course there
may be exceptions (for instance “what to do when the first equation does not contain x”),
but these may also be dealt with at the specific level. At risk of overusing an adjective we
have already used before, we will regard this type of expansive approach as generic, in the
sense that it describes the typical (general) procedure by referring to a specific case.
Such a generic approach is seen both an easy method of generalization because it applies
a well-known process in a broader context and also as a first step towards formal abstraction
as it does not involve a major cognitive reconstruction. Indeed, once the students have
reflected on the general process and seen it as a conscious act of widening the applicability
of a specific method, it may be viewed as a (relatively painless) form of abstraction which
we term a generic abstraction (Harel & Tall, to appear). This furnishes an approach which
is of particular value for students whose main interest is in applications rather than formal
mathematics. It may also provide a suitable transition phase for students passing on to the
formal abstraction, however, the latter will still require a cognitive re-organization, albeit
one which is better prepared.
Dubinsky encourages students to write programs in a computer language where many
of the constructs parallel the constructs of mathematical thinking: sets, sequences, ordered
pairs, relations, functions, and so on. By writing computer code which specifies the
procedure to carry out a function process, including an initial test to see if the input satisfies
conditions which define the domain of the function, the student is required to think through
the enactment of the function process. The act of programming is a generic process: it
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
13
carries out what may be seen as a more general construct in particular cases and gives rise
to a generic abstraction of the function concept. Given the theory just described, this
suggests a further stage is necessary to pass from the generic example of programming,
where the general is seen in the particular instances of functions programmed by the
student, to the formal abstraction which requires a new level of abstract construction from
the definition. Dubinsky formulates this transition within a Piagetian framework of
reflective abstraction, in which processes are encapsulated as objects, so that the function
process leads to the function as a mental object. This theory is further elaborated in chapters
7 and 15.
1.8 INTUITION AND RIGOUR
Mathematicians often regard the terms “intuition” and “rigour” as being mutually exclusive
by suggesting that an “intuitive” explanation is one that necessarily lacks rigour. There is
a grain of truth in this, for usually an intuition arrives whole in the mind and it may be
difficult to separate its components into a logical deductive order. But the opposition
between the two concepts is a false dichotomy as we shall soon see.
In a sense we have not one, but two brains. In attempting to assist patients who had
serious epileptic fits, Sperry and his colleagues took the drastic action of partial or total
severance of the corpus callosum that links the two hemispheres of the brain and found that
each could essentially operate independently, though carrying out totally dfferent functions:
Though predominantly mute and generally inferior in all performances involving language or
linguistic or mathematical reasoning, the minor hemisphere is nevertheless clearly the superior
cerebral member for certain types of tasks. If we remember that in the great majority of tests it is the
disconnected left hemisphere that is superior and dominant, we can review quickly now some of the
kinds of exceptional activities in which it is the minor hemisphere that excels. First, of course, as
one would predict, these are all non-linguistic non-mathematical functions, largely as they involve
the apprehension and processing of spatial patterns, relations and transformations. They seem to be
holistic and unitary rather than analytic and fragmentary, and orientational more than focal, and to
involve concrete perceptual insight rather than abstract, symbolic sequential reasoning.
(Sperry, 1974)
This evidence resonates strongly with the observation of the two different kinds of
mathematical mind suggest at the turn of the century by Poincaré. However, subsequent
research suggests that the brains of different individuals need not follow such a simplistic
division of functions. Gazzigna (1985) sees brain activity as a collection of different
modules functioning independently in parallel, with a control unit (usually in the left brain)
making decisions based on the information provided by the various modules. Thus it would
be incorrect to divide human activity simplistically into two different modes, just as it is
inappropriate to consider just two contrasting types of mathematical mind.In particular we
may envisage that the human mind immersed in logical thought may eventually develop
intuitions that are themselves logically based. Poincaré, speakmg of Hermite, said:
His eyes seem to shun contact with the world; it is not without, it is within he seeks the vision of truth.
... When one talked to M. Hermite, he never evoked a sensuous image, and yet you soon
perceived that the most abstract entities were for him like living beings. He did not see them, but
14
DAVID TALL
he perceived that they are not an artificial assemblage and that they have some principle of internal
unity.
(Poincaré, 1913, pp. 212, 220)
The conclusion is inescapable. Intuition is the product of the concept images of the
individual. The more educated the individual in logical thinking, the more likely the
individual’s concept imagery will resonate with a logical response. This is evident in the
growth of thinking of students, who pass from initial intuitions based on their pre-formal
mathematics, to more refined formal intuitions as their experience grows:
We then have many kinds of intuition; first the appeal to the senses and the imagination; next,
generalization by induction, copied, so to speak, from the procedures of the experimental sciences;
finally we have the intuition of pure num ber...
(Poincaré, 1913, p. 215.)
From a psychological viewpoint, Fischbein (1978) comes to similar conclusions, citing two
different types of intuition:
Primary intuitions refer to those cognitive beliefs which develop themselves in human beings, in
a natural way, before and independently of systematic instruction.
Secondary intuitions are those which are developed as a result of systematic intellectual training
... In the same meaning, Felix Klein (1898) used the term “refined intuition”: and F. Severi wrote
about “second degree intuition” (1951).
(Fischbein, 1978, p. 161)
Thus aspects of logic too can be honed to become more “intuitive” to the mathematical
mind. The development of this refined logical intuition should be one of the major aims of
more advanced mathematical education.
2. THE GROWTH OF MATHEMATICAL KNOWLEDGE
As we have seen, the nature of mathematical thinking is inextricably interconnected with
the cognitive processes that give rise to mathematical knowledge. We now focus on the full
cycle of mathematical thinking to see mathematical proof as the final stage of this
developmental process rather than just the formal framework of the completed knowledge
structure.
2.1 THE FULL RANGE OF ADVANCED MATHEMATICAL THINKING
Mathematical proof, according to Hadamard (1945), is but the last, “precising” phase of
mathematical thinking. Before a theorem can be conjectured, let alone proved, there is
much work to be done in conceiving of what ideas will be fruitful and what relationships
will be useful. Hadamard considers Poincaé’s description of his own personal research
activities and notes:
.. the very observations of Poincaré show us three kinds of inventive work essentially different if
considered from our standpoint, viz.,
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
a. fully conscious work
b. illumination preceded by incubation
c. the quite peculiar process of the sleepless night.
15
(Hadamard, 1945, p. 35)
Here Poincaré reports the necessity of working hard at a new problem, then relaxing to
allow the ideas to incubate in his subconscious, during which time he had a sleepless night
thinking vigorously about new ideas until suddenly, some time later, a sudden illumination
bursts into his consciousness with a solution. After a further time had elapsed, at his leisure,
he was able to analyse what had happened and build up a formal justification of his theory
in the final “precising” phase when the results of the illuminative break-through are
subjected to the cold analysis of the light of day, refining the assumptions so that the
deductions will stand analytic scrutiny.
What becomes apparent is that the initial phases of the creative cycle may rely in part
on logic and deduction, but they also need flexible mental activity to produce mental
resonances between previously unconnected concepts. According to Gazzigna’s model of
brain activity, they may occur as juxtapositions from different modules in the brain
processing simultaneously. Part of the success of this phase of mathematical thinking
seems to be due to working sufficiently hard on the problem to stimulate mental activity,
and then relaxing to allow the processing to carry on subconsciously.
2.2 BUILDING AND TESTING THEORIES: SYNTHESIS AND ANALYSIS
Poincaré was at pains to show the complementary roles of synthesis and analysis in
mathematical thinking. Synthesis begins with the conscious act of the initial phase to begin
to put ideas together, followed by a more intuitive activity, in which subconscious interplay
between concept images takes place, until a powerful resonance forces the newly linked
concepts to erupt once more into consciousness. Analysis, on the other hand, is a much more
cool and logical conscious activity which organizes the new ideas into logical form and
refines them to give precise statements and deductions.
Teaching of younger children emphasizes the synthesis of knowledge, starting from
simple concepts, building up from experience and examples to more general concepts. The
emphasis at this level is now changing to include more problem solving and open-ended
investigations. Teaching at university often emphasizes the other side of the coin: analysis
of knowledge, beginning with general abstractions and forming chains of deduction from
them which may be applied in a wide variety of specific contexts.
Working with much younger children, Dienes (1960) proposed a theory for building
concepts from concrete examples, yet Dienes & Jeeves (1965) formulates a far more
general deep-end principle in which “there is a preference for extrapolation by leaps and
interpolation, rather than always by step-by-step”. They respond to their own question
“When is it possible to generalize from a simple case to a more general case and when is
it better for them to particularize from a more complex case to the simple case?” with the
remark that “this is not likely to be answered by a simple positive or negative statement”.
They suggest that it is more a question of “the optimum degree of complexity required to
start with” – a response which is just as valid for teaching and learning at more advanced
levels. It is likely to require synthesis of knowledge to build up theories cognitively as well
as analysis of knowledge to give the total structure a logical coherence.
16
DAVID TALL
2.3 MATHEMATICAL PROOF
Viewed as a problem-solving activity, we see that proof is actually the final stage of
activity in which ideas are made precise. Yet so much of the teaching in university level
mathematics begins with proof. In his preface to The Psychology of Learning Mathematics,
Skemp succinctly refers to this as showing the students the product of mathematical
thought, instead of teaching them the process of mathematical thinking. The splendid tomes
of Bourbaki are a monument to the intellect of the mathematical mind, and may be used to
help the learner appreciate the formal structure of mathematics. But once again, Poincaré
has pertinent observations to make:
To understand the demonstration of a theorem, is that to examine successively each of the syllogisms
composing it and to ascertain its correctness, its conformity to the rules of the game? ... For some,
yes; when they have done this, they will say: I understand. For the majority, no. Almost all are much
more exacting they wish to know not merely whether all the syllogisms of a demonstrations are
correct, but why they link together in this order rather than another. In so far as to them they seem
engendered by caprice and not by an intelligence always conscious of the end to be attained, they
do not believe that they understand.
(Poincaré,1913, p.431)
Perhaps you think I use too many comparisons; yet pardon still another. You have doubtless seen
those delicate assemblages of silicious needles which form the skeleton of certain sponges. When
the organic matter has disappeared, there remains only a frail and elegant lace-work. True, nothing
is there except silica, but what is interesting is the form this silica has taken, and we could not
understand it if we did not know the living sponge which has given it precisely this form. Thus it
is that the old intuitiv enotions of our fathers, evenwhen we have abandoned them, still imprint their
(ibid, p. 219)
form upon the logical constructions we have put in their place.
Thus it is that so many mathematicians demand that a proof should not only be logical, but
that there should be some over-riding principle that explains why the proof works. The
proof of the four colour theorem, by exhaustion of all possible configurations using a
computer search (Appel & Haken, 1976) seems logical, yet many professional mathematicians, though keen to see the theorem proved once and for all, are nevertheless sceptical
that there may be some subtle flaw in the computer “proof”, because there seems to be no
rhyme or reason to illuminate why it works as it does.
Yet this principle is not always passed on to students. Sawyer (1987) reports how he tried
to teach theorems in functional analysis by referring back to theorems in real variables that
he expected his students to know, only to find that they had no recollection of them.
The reason for this was that in their university lectures they had been given formal lectures that had
not conveyed any intuitive meaning; they had passed their examinations by last-minute revision and
by rote.
He tells how he was shocked to learn of a lecturer who became stuck in the middle of a proof,
turned his back on the class to draw a picture to aid him, then erased it and carried on with
the formal proof without enlightening the class how he had used his intuition to rebuild it.
He observes:
... to teach calculus well is a very demandmg task. Three things have to be done: first to show by
a drawing that some result is extremely plausible; second, to give counter-examples, which indicate
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
17
the circumstances in which the conjecture would fail; third, to extract from these considerations a
formal proof of the result.
These remarks do not apply only to lectures and books for undergraduate. Felix Klein pointed
out that in papers for research journals the suppression of intuitive considerations was a common
(Sawyer, 1987)
and highly undesirable practice.
Many mathematicians have learned to present their best face in public, showing their ideas
in polished form and concealing the toil and false turnings that littered their growth. It is
therefore essential to pose the following question:
How it is possible to initiate students into the wider vision of the nature of
mathematical thinkng that includes the arduous growth of mathematical
thinking in a manner appropriate for a learner?
3. CURRICULUM DESIGN IN ADVANCED MATHEMATICAL LEARNING
3.1 SEQUENCING THE LEARNING EXPERIENCE
During the difficult transition from pre-formal mathematics to a more formal understanding of mathematical processes there is a genuine need to help students gain insight into what
is going on. A mathematician’s logic may here fail him (or her) in designing a teaching
schedule. A mathematician often takes a complex mathematical idea and “simplifies” it by
breaking it into smaller components ready to teach each component in a logical sequence.
From the expert’s viewpoint the components may be seen as parts of a whole. But the
student may see the pieces as they are presented, in isolation, like separate pieces of a jigsaw
puzzle for which no total picture is available. In fact the scenario may be worse. As the
student encounters each piece of the puzzle (s)he forms a personal concept image from the
particular context which may be at variance with the formal idea. Thus, not only is no picture
available for the puzzle, the pieces themselves may now have different shapes so that they
no longer fit.
For example, a mathematical analysis of the notion of the derivative f'(x) requires the
notion of the limit of (f(x+h)–f(x))/h as h tends to zero, so mathematically the derivative
must be preceded by the discussion of the notion of a limit. To make the process
mathematically easier the limit process is initially carried out with x fixed; only at a later
stage is x allowed to vary to give the notion of a function. Thus the sequence suggested by
a formal mathematical analysis is:
(1) notion of a limit,
(2) for fixed x, consider the limit of (f(x+h)–f(x))/h as h tends to zero,
(3) call the limit f'(x), then allow x to vary to give the derived function.
However, when the learner is at stage (1), the limit notion is mysterious because it seems
“plucked out of the air”, without any real reason. There are already cognitive obstacles here,
as observed by Comu (1983), and others. At stage (2) the limiting process introduces further
obstacles (Tall & Vinner, 1981) which will be discussed in more detail in chapter 10. Nor
18
DAVID TALL
is the passage from (2) to (3) as easy cognitively as it seems mathematically. Many students
see (2) as a purely symbolic activity, and do not see the derivative f'(x) as a function, with
a graph which is everywhere the gradient of the graph of f(x) (Tall, 1986).
The problem ofcurriculum development is therefore to present the student with contexts
in which cognitive growth is possible, leading ultimately to meaningful mathematical
thinking in which the formalism plays an appropriate part.
In analysis, for instance, one method which has proved successful might involve a more
flexible approach that complements numerical and algebraic approaches to the derivative
with a global, visual appreciation of the gradient of a graph generated on a computer.
In general it may be possible to use the complementary power of visualization to give
a global gestalt for a mathematical concept, to show its strengths and weaknesses, its
properties and non-properties, in a way that makes it a logical necessity to formulate the
theory clearly. Visual ideas without links to the sequential processes of computation and
proof are insights which lack mathematical fulfillment. On the other hand, logical
sequential processes without a vision of the total picture, are blinkered and limiting. It is
therefore a worthy goal to seek the fruitful interaction of these very different modes of
thought.
3.2 PROBLEM-SOLVING
For many undergraduates, problem-solving means learning the contents of a set of lecture
notes and applying this knowledge to specific problems clearly related to the material
taught. For research mathematicians, problem-solving is a more creative activity, which
includes the formulation of a likely conjecture, a sequence of activities testing, modifying
and refining until it is possible to produce a formal proof of a well-specified theorem.
Polya (1945) suggested four phases as a framework for problem-solving:
• understand the problem,
• devise a plan,
• carry out the plan,
• look back at the work.
This framework has formed the backbone of many subsequent attempts at formulating
problem-solving strategies, though Mason et al (1982) and Schoenfeld (1985) have seen
the need to make the actual heuristics much more explicit and more appropriate for the
learner. The idea of “devising a plan” is extremely daunting for the novice. More empathetic
is the version suggested by Mason, who proposes three phases:
• entry,
• attack,
• review.
The entry phase covers the first two stages of Polya whilst attack and review correspond
to Polya’s third and fourth stage. In the entry phase the potential problem-solver gets
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
19
acquainted with the problem-solving context – getting a sense of the problem by playing
with the ideas, perhaps through simple specializations, moving to a position which attempts
to specify clearly what is known and what is wanted, and considering carefully what can
be introduced (notation, procedures of solution, etc.) that might take the problem-solver
from what is known to what is wanted. Then a qualitative change occurs with a committed
attack on the problem using the ideas that have been introduced. This may be successful,
but it can more often lead to an impasse, a seeming dead-end from which the individual
should review what has been done and return to the entry phase to consider a new attack.
Once some kind of solution is achieved the mood changes yet again to one of sober review
– checking the results to make sure no error has been made, reviewing what has been done
to learn of strategies that may prove useful on other occasions and then being prepared to
extend the problem to new levels of sophistication, re-starting the entry cycle at a more
sophisticated level.
The author has had several years of experience teaching problem-solving within this
framework. It has proved possible to get undergraduates to develop original ways of solving
problems although the process requires longer initial periods for the students to reach a
point of insight than may be apparent when giving the information in a lecture. However,
the pay-off is in the way it can stimulate reflective thinking and develop an internal monitor
within the student’s mind to sense the progress and appropriate direction of the solution
process.
3.3 PROOF
Students starting out in advanced mathematics have great difficulty with proof before
they attain familiarity with the workings of the mathematical culture. In a questionnaire
investigating which proof of the irrationality of 2 was more clear, students preferred a
proof that showed that the square of any rational must have an even number of prime factors,
and therefore such a square could not be 2 because the prime 2 occurs an odd number of
times (namely once). They preferred this to the standard proof by contradiction and another
more general demonstration taken out of Hardy’s Pure Mathematics. This is despite the fact
that this “proof” is not a formal proof at all, but a discursive explanation with examples
demonstrating what form was taken by the square of a typical rational (Tall, 1979). Once
more we see that a generic proof: explaining the general concept by considering a typical
example, is an easier first step to understanding rather than the reconstructive leap to the
general formalism.
Of course it is essential in advanced mathematics to take the step from (generic)
explanation to formal proof. Some educators, such as Leron (1983ab, 1985ab), see their role
as making the structure of proof more meaningful to students. His method is, essentially,
to properly structure the proof, so that it is clear what is going on at any given time, and to
make the proof as direct as possible. Thus contradiction proofs are re-written so that they
are initially direct and constructive, with any contradiction being introduced as late as is
practicable in the proof.
Others see their duty as the wider role of introducing students to the full range of
mathematical thinking, including conjecture, positive verification through a convincing
argument or refutation through a counter-example. Thus the Grenoble school (Legrand et
al, 1984, 1988; Alibert, 1988) have introduced “scientific debate” into their courses in
20
DAVID TALL
which a full lecture audience is invited to group together to think up likely theorems in the
mathematical topic under consideration, and then to attempt to prove or disprove them. It
is important that the teacher does not comment on the truth or falsity of the conjectures in
the initial stages, so that the students are genuinely faced with the task of convincing their
peers of the truth of their arguments. (See chapter 13.)
3.4 DIFFERENCES BETWEEN ELEMENTARY
AND ADVANCED MATHEMATICAL THINKING
It is ironic that the National Curriculum in the UK and the NCTM standards in the USA for
school mathematics advocate a level of open ended problem solving which is rarely
specified in undergraduate courses at universities. The problem-solving procedures of
entry, attack and review can and are being performed by younger children in such
mathematical investigations. Thus many of the processes of advanced mathematical
thinking are already found at a more elementary level. However, Mason et al (1982)
describe the process of verification in Thinking Mathematically at three levels:
• convince yourself,
• convince a friend,
• convince an enemy.
Convincing oneself involves having an idea of why some statement might be true, but
convincing a friend requires that the arguments be organized in a more coherent way.
Convincing an enemy means that the argument must now be analysed and refined so that
it will stand the test of criticism. This is the closest that Thinking Mathematically gets to the
notion of proof. What is entirely absent is the notion of formal definitions and the logic of
formal deductions from those definitions.
It may be hypothesized that mathematical thinking at every level can include the phases
entry, attack and review, including a level of mathematical justification, but that elementary
mathematical thinking lacks the process of formal abstraction and does not include the final
“precising phase” in its most formal guise.
The move from elementary to advanced mathematical thinking involves a significant
transition: that from describing to defining, from convincing to proving in a logical manner
based on those definitions. This transition requires a cognitive reconstruction which is seen
during the university students’ initial struggle with formal abstractions as they tackle the
first year of university. It is the transition from the coherence of elementary mathematics
to the consequence of advanced mathematics, based on abstract entities which the
individual must construct through deductions from formal definitions.
4. LOOKING AHEAD
It is a truism that we can only think with the cognitive structure that we have available to
us. When we look at the psychology of advanced mathematical thinking, it is no wonder
that we each find it easier to use our own knowledge structure to formulate our own theories.
THE PSYCHOLOGY OF ADVANCED MATHEMATICAL THINKING
21
As a mathematician entering mathematics education it is no surprise that the author fist
attempted to use catastrophe theory to describe the discontinuities in learning (Tall, 1977).
Likewise those who begin mathematics education with a background of Piagetian theory
are likely to attempt to explain things in these terms, those with experience in computer
studies are more likely to use computer analogies, mathematicians are likely to attempt to
use mathematical constructs, and so on. In trying to formulate helpful ways of looking at
advanced mathematical thinking, it is important that we take a broad view and try to see the
illumination that various theories can bring, the useful differences that arise and the
common links that hold them together.
In the remainder of the fiist part of this book we consider the cognitive processes
involved in advanced mathematical thinking and the two complementary attributes of the
discipline: creativity in generating new ideas and the mathematician’s notion of proof in
convincing his peers of the truth of his assertions.
In the second part of the book we turn to cognitive theories that are proving of value in
analysing the difficulties that students face and providing insights into the learning process
that can be used in designing new ways of helping students construct mathematical ideas
for themselves. First the differences between concept definitions and students’ concept
images are considered, then the nature of the mental objects which mathematicians
construct: the conceptual entities that are the essence of advanced mathematics. This leads
to the theory of reflective abstraction in which processes are encapsulated as mental objects
which prove to be easier to manipulate at higher levels of abstraction.
In the third part of the text we review various advanced mathematical concepts from a
cognitive viewpoint, showing the cognitive obstacles that can occur during their development and reporting empirical evidence on the success of instructional sequences designed
from cognitive viewpoints. These involve the central ideas of function, limit, more
advanced concepts of analysis, infinity and proof. We then move on to look at the new
paradigm: the use of the computer and its cognitive effects in advanced mathematical
thinking.
Finally, in chapter 15, as editor of the book, I take the opportunity of reflecting on the
development of the theories of advanced mathematical thinking and its teaching and
learning over the last decade and highlight the important themes which recur, the questions
that have been posed and the partial answers that are beginning to show us the way ahead.
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I : THE NATURE OF
ADVANCED MATHEMATICAL THINKING
What is it that is so difficult about Advanced Mathematical Thinking? In Part I
of this book we have three chapters which consider the fundamental nature of
advanced mathematical thinking to lay the groundwork for the cognitive theory
and research reports to follow. First and foremost we acknowledge Advanced
Mathematical Thinking in terms of creative process rather than just proof and
deduction. In Chapter Two, Tommy Dreyfus recognizes that many of the
processes of advanced mathematical thinking are also found in elementary
mathematics. He considers the standard form of conveying information through
lectures and reports consequent student difficulties in coping with anything
which differs even marginally from what is taught. He focuses on the complexity
of advanced concepts, the need to represent them and abstract their essential
properties to control their complexity, and discusses the cognitive difficulties in
carrying out these processes. In Chapter Three, Gontran Ervynck considers the
enigmatic nature of the creative process in mathematics, which is the focus of the
research process yet plays so little part in student development. In Chapter Four,
Gila Hanna analyses the nature of mathematical proof from a philosophical and
pragmatic viewpoint to show that it is dependent on context and beliefs rather
than an immutable standard shared by all mathematicians.
In these three chapters we therefore take a critical look at advanced mathematical thinking as part of the living process of human thought rather than the
immutable final product of logical deduction.
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CHAPTER 2
ADVANCED MATHEMATICAL THINKING PROCESSES
TOMMY DREYFUS
Understanding, more than knowing or being skilled, has always been considered an
important goal by mathematics teachers. Understanding, as it happens, is a process
occurring in the student’s mind; it may be quick, an “Aha-Erlebnis”, a click of the mind;
more often, it is based upon a long sequence of learning activities during which a great
variety of mental processes occur and interact. Therefore, what it means to come to
understand a mathematical notion or concept is extremely difficult to analyze. Researchers
in psychology (e.g., Brown, Bransford, Ferrara, & Campione, 1983) have been asking
themselves what “understanding” means, in particular what are its components, what
mental processes may intervene and combine together to form that meta-process of
understanding. Researchers in mathematics education, in particular, have become conscious of the importance of the component processes for understanding advanced mathematics and their interactions.
Why would researchers be interested in the processes involved in learning advanced
mathematics? One reason is to gain basic theoretical knowledge about what is going on in
the student’s mind. There certainly is some intrinsic interest in this fundamental question.
But there are also very important applied aspects to this strand of research, and these
concern all teachers of advanced mathematics. The processes the teacher hopes to provoke
in the student do not happen by themselves nor, if they happen, are they necessarily
conscious on the students part. It is not sufficient, for example, to define and exemplify an
abstract concept such as vector space. Students must then construct the properties of such
a concept through deductions from the definition. They may involve being through
activities that promote abstraction on their part and it has to be brought to their attention that
this is what is being done, that this is the aim of the exercise. In this chapter, processes,
among them abstracting, are analyzed and discussed with the aim of making teachers of
advanced mathematics more conscious of what is going on during these processes.
Hopefully, this will help teachers introduce such action explicitly in their classrooms.
Recently some controlled trials with similar aims have been made, and they have met with
some success. A conscious process approach to abstracting has been described by Mason
(1989) and experiences with making student teachers reflect upon their mathematical
activity have been reported by Southwell (1988).
Reflection about one’s mathematical experience is of particular importance in the
solution of non-trivial problems (as opposed to standard exercises). And it is in this
connection that the importance of processes has first been realized by mathematics
educators (Schoenfeld, 1985). Reflection about one’s mathematical experience is an
important aspect of meta-cognizing, another meta-process. Such reflection is a characteristic of advanced mathematical thinking. We would not usually expect an elementary math
student to stop, after having solved a problem, and think or recount how he went about
solving this problem. We would, however, definitely like to see much more of this in our
advanced students and, in particular, in our high school teachers.
26
TOMMY DREYFUS
There is no sharp distinction between many of the processes of elementary and advanced
mathematical thinking, even though advanced mathematics is more focussed on the
abstractions of definition and deduction. Many of the processes to be considered in this
chapter are present already in children thinking about elementary mathematics concepts,
say number or place value. They are not exclusively used in advanced mathematics, nor,
indeed, are they exclusively used in mathematics. Abstractions are made in physics,
representations are used in psychology, analysis is used in economics and visualization in
art. Here, however, we will describe the processes as they are relevant for advanced
mathematical thinking, in particular focussing on those processes whose characteristics
make the mathematical thinking advanced.
It is possible to think about advanced mathematical topics in an elementary way (e.g.,
many standard exercises on rings or groups can be answered by just plugging in the right
numbers), and there is rather advanced thinking about elementary topics (look at some of
the problems in mathematics olympiads). One distinctive feature between advanced and
elementary thinking is complexity and how it is dealt with. Advanced concepts, such as
rings or Lie groups, are likely to be very complex. The distinction is in how this complexity
is managed. The powerful processes are those that allow one to do this, in particular
abstracting and representing. By means of abstracting and representing, one can move from
one level of detail to another and thus manage the complexity.
The processes to be discussed in this chapter are mathematical and psychological ones,
and in many cases they are both: in fact, the mathematical and the psychological aspects
of a process can rarely be separated. For example, when you build a graph of a function,
you are executing a mathematical process, following certain rules which can be stated in
mathematical language; at the same time, however, you are very likely generating a visual
mental image of that graph; in other words, you are visualizing the function in a way that
can later help you reason about the function. The mental and the mathematical images are
closely linked here. Neither can arise without the other, and they are in fact generated by
the very same process; they are, respectively the mathematical and the psychological
aspects of this process. A similar linkage between mathematics and psychology exists with
respect to the other processes of advanced mathematical thinking. In fact, it is precisely this
linkage which makes the processes interesting and relevant for understanding learning and
thinking in advanced mathematics.
1. ADVANCED MATHEMATICAL THINKING AS PROCESS
The typical mathematics course at, say, first year university level has a neatly defined
syllabus, which tells the instructor what material he is supposed to cover by the end of the
term. Whether this is a course in calculus, algebra, finite mathematics, numerical methods
or other, for the instructor the content to be taught is a well known, unassailable, accepted
segment of mathematical knowledge: although (s)he will probably think of several
possibilities to organize this material into a logically clean structure, each of these structures
will basically consist of a number of theorems, to be proved, and a number of applications
of these theorems to topics in mathematics and beyond. The instructor will probably
distribute these into as many class periods as are available and lecture during a considerable
part of these class periods, making extensive use of the strikingly convenient formalisms
ADVANCED MATHEMATICAL THINKING PROCESSES
27
of the specific domain of mathematics concerned. In so doing, a very important aspect of
the mathematics which is being taught is presented to the students, namely the finished and
polished product into which that well known, unassailable, fully accepted segment of
mathematics has grown.
Our instructor presumably knows very well that mathematics is not being created in final
and polished form, but through trial and error, through partially correct (and partially
wrong) statements, through intuitive formulations in which loose terms and imprecisions
have intentionally been introduced, through drawings that try to visually present parts of
the mathematical structures being thought about, through dynamic changes being made to
these drawings, etc., etc.. But the fact that our instructor knows about these other aspects
of mathematics, indeed, is very likely to experience them daily in his or her research work,
does not usually prevent him or her from almost exclusively teaching the one very
convenient and important aspect of mathematics which has been described above, namely
the polished formalism, which so often follows the sequence theorem–proof–application.
This manner of teaching has several advantages: for example, it allows for a well-planned
structure of the course, as well as for predictable progress through the material, and thus for
a fairly certain guarantee that most of the material in the syllabus can be covered.
Unfortunately, it also has at least one very serious disadvantage: it is inflexible in terms of
adaptability to the students. It may work rather well for students who major in mathematics
and who, from some exceptional teacher or on the basis of their own talent and investigative
nature, have already had the opportunity to acquire a mathematical attitude. But as is shown,
for instance, by the present calculus crisis, it does not work for the vast majority of students,
those majoring in science, engineering, medicine or the liberal arts and taking mathematics
as a required service subject.
What these students learn, and what they don’t learn, is illustrated very well by the results
of a recent study of students who had passed a traditional first quarter calculus course at
Tennessee Technological University (Selden, Mason & Selden, 1989). These were well
prepared students who had been taught in small groups by experienced teachers and
teaching assistants, obtaining at least a C grade. The students were presented with five
moderately difficult problems that could easily be solved with the techniques at the
students’ disposal. These problems were formulated in a manner that was somewhat nonstandard, for example:
Find at least one solution to the equation 4x 3 –x 4 =30 or explain why no such
solution exists.
The function f(x)=4x3–x4 has a maximum value of 27 and thus no solution to the given
equation exists. Although the students in the study were perfectly capable to carry out the
required function discussion, they could not answer the question as given. The situation was
similar with respect to the other four problems. Infact, the authors state that not one student
got an entire problem correct. Most couldn’t do anything.
This is not an exceptional finding, nor is it limited to average university freshman. Davis
(1988), in discussing a class of unquestionably excellent high school seniors, came to the
conclusion that when one looks carefully at how these “apparently successful” students
deal with mathematical problems, one finds that they hold many grotesque misconceptions
about mathematics and that the following reasons contribute to their success in passing
28
TOMMY DREYFUS
tests: Most mathematics instruction, from elementary school through college courses,
teaches what might be called rituals: "do this, then do this, then do this ..." and Teachers
... will typically accept the correctly-performed ritual as enough success for the time being.
In other words, what most students learn in their mathematics courses is, to carry out a
large number of standardized procedures, cast in precisely defied formalisms, for
obtaining answers to clearly delimited classes of exercise questions. They thus acquire the
capability to perform, albeit much slower, the kind of operation which a computer can
perform by means of a suitable program such as Mathematica. They end up with a
considerable amount of mathematical knowledge but without the working methodology of
the mathematician, that is they lack the know-how that allows them to use their knowledge
in a flexible manner to solve problems of a type unknown to them. They are examples of
the phenomenon described in the previous chapter: they have been taught the products of
the activity of scores of mathematicians in their final form, but they have not gained insight
into the processes that have led mathematicians to create these products.
While there is no need to form every student of science or engineering into a full-fledged
mathematician, most teachers of calculus would like their graduates to be able to answer
questions such as
What conditions are sufficient to ensure that the function f(x)=ax3+bx2+ cx+d is
increasing at x=0?
and to immediately start searching for a mistake when they sce a result that obviously has
the wrong sign such as in
Moreover, one would expect that they realize rather quickly that
for any continuous function g and that they have little dfficulty in determining which of
the statements (i) f'(a)= –f'(a), (ii) f'(–a)=f'(a), (iii) f'(–a)= –f'(a), (iv) f'(–a)= –f'(–a) are true
for any odd differentiable function f. Experience shows, however, that such tasks are
difficult for calculus students, and that a similar situation pertains to students of other
courses such as pre-calculus mathematics, linear algebra and differential equations.
The discrepancy between teacher expectations and student performance on such tasks
occurs because we, as teachers, often do not realize how much of our experience with
mathematical processes and concepts we use. Take, for example, the equality
ADVANCED MATHEMATICAL THINKING PROCESSES
29
here one needs to deal with the function g as an object that is operated on in two ways,
namely integrated from a to b to give a number as well as translated to the right and then
integrated over a translated interval: one possibility is to visualize the translations of the
function and the interval and to compare them. Although all of this may hardly take more
than a few seconds for the expert, his mental processing has probably included components
of representing (the function, maybe graphically), transforming (by translation), visualizing (the function, the translation, the area under the graph), checking (that the two
translations go in the same direction), and deducing (that the resulting numbers are equal).
Possibly processes of specializing (e.g., to positive functions only) and then generalizing
again have also been involved. Moreover, all of this was most certainly based on extensive
prior processing of functions and integrals which included repeated phases of generalizing,
abstracting and formalizing, that allow the expert to view functions and integrals as objects
but may not be available to the student. The message that I am trying to convey here is that
advanced mathematical thinking, as in the expert’s treatment of the equality of these two
integrals, is an extremely complex process, in which a large number of component
processes interact in intricate ways.
One place to look for ideas on how to find ways to improve students’ understandings
is the mind of the working mathematician. Not much has been written on how mathematicians actually work; certainly the deepest and most elaborate document on this topic is
Hadamard’s book (1945), which refers back extensively to Poincaré’s ideas and was the
source of inspiration for much of the discussion in the previous chapter. Hadamard
explicitly stresses the importance of informal reasoning, of thinking in the absence of
words, of visual imagery, of mental images (which cannot necessarily be expressed in
words, at least not when they first occur), and of playing around with ideas, for instance by
repeatedly trying to fit different elements together much like in a puzzle. This experimental
aspect of mathematics, as well as the visual means being used, have received added
importance in recent years due to the technological progress that has been made in computer
graphics. Investigations have become possible which are in principle qualitative and visual.
The prime example for this is research on non-linear systems,chaos and fractals. According
to Peitgen & Jürgens (1989), the fundamental mathematical developments in this area were
made possible only through computer-graphical experiments.
Hoffman (1989) has proposed a philosophy of mathematics education based on the
simple recognition that mathematics is a human activity, useful in the real world; on this
basis he requires that we should transmit to our students a picture of mathematics as a
science which incorporates observation, experiment and discovery.
Several projects have been carried out in recent years following basically this philosophy. For example, Breuer, Gal-Ezer & Zwas (1990) proposed to teach numerical methods
at calculus level in a computational laboratory and Ruthven (1989) has reported how the
use of graphic calculators has led students to use a graphic-trial approach and a numerictrial approach in parallel to an analytic-construction approach to pre-calculus problems.
Many (though not all) similar projects make extensive use of computers as experimental
30
TOMMY DREYFUS
tools. Computers can serve as heuristic tools for the mathematician and the mathematics
student in much the same way as a microscope serves the biologist: if the tool is directed
onto interesting phenomena and correctly focussed, it may show an unexpected picture,
often a visual one, of the phenomenon under study, and thus lead to new ideas, to the
recognition of heretofore unknown relationships. In the case of the researcher, these ideas
and relationships may be expected to be original; in the case of the student, they were very
likely known to many other people before but they are new to this particular individual.
By using computer learning environments many usually implicit relationships, for
instance between different representations for the same concept, may be made explicit. This
explicitness contributes to students’ recognition of such relationships and to the emergence
of related ideas, in brief to their formation of concepts. This process thus closely parallels
the recognition of relationships and the emergence of ideas in the research process.
Admittedly, there are clear differences between the research process and the learning
process; for instance, in the learning process the material to be learned is presented in a
manner judged by experts to be digestible; and the average learner should be expected to
be considerably less talented for mathematics than the average researcher; but the point here
is to stress the very important similarities between the learning process and the research
process; namely that in both cases the individual has to mentally manipulate, investigate
and find out about objects, about which his knowledge is very partial and fragmented. Thus,
just as the research process is extraordinarily complex, so is the corresponding learning
process. It contains the gist of what advanced mathematical thinking is all about. It is likely
to comprise, at any stage and in tight interaction, several of the processes mentioned in the
discussion of the translated integral, e.g. representing, visualizing, generalizing, as well as
others such as classifying, conjecturing, inducing, analyzing, synthesizing, abstracting or
formalizing. In other words, advanced mathematical thinking consists of a large array of
interacting component processes. It is important for the teacher of mathematics to be
conscious of these processes in order to comprehend some of the difficulties which their
students face.
2. PROCESSES INVOLVED IN REPRESENTATION
2.1 THE PROCESS OF REPRESENTING
Representations have a very important function in mathematics: If we want to talk about
the group of permutations of n objects, for instance, it will often be convenient to call it the
symmetric group of degree n and denote it by Sn. The notation Sn is a sign that refers to,
and thus represents, or symbolizes, the group in question; it is a symbolic representation of
the group. Such symbols are absolutely indispensable in modem mathematics, but there is
also some danger associated with them. As has been stressed by Olson& Campbell (in
press), symbols involve relations between signs and meanings; they serve to make a
person’s implicit knowledge – the meaning – explicit in terms of symbols. There must be
some meaning associated with a notion before a symbol for that notion can possibly be of
any use; in the educational discourse of teaching mathematics, this is too often overlooked
leading to the well known phenomenon of “symbol pushing”. The role of symbols is
ADVANCED MATHEMATICAL THINKING PROCESSES
31
discussed in greater detail in chapter 6 by Harel and Kaput.
Another meaning of representations is even more central for learning and thinking in
mathematics. When we talk or think about a group, an integral, an approximation, about
any mathematical object or process at all, each one of us relates to something we have in
mind – a mental representation of the object or process under consideration. Although most
mathematicians can be expected to come up with roughly equivalent definitions of, say, a
function, their respective mental representations of the notion may be vastly different. Have
you ever asked mathematicians working in different areas what comes to their mind when
they think about functions? When you also ask mathematics teachers and students, these
differences become not only more pronounced but also much more important. For example,
a student’s notion of a function may be limited to processes (of computation or mapping),
whereas the teacher teaching indefinite integrals may think of the function in the integral
as an object to be transformed. Such discrepancies easily lead to situations where students
are unable to understand their teachers.
To represent a concept, then, means to generate an instance, specimen, example, image
of it. But this short description is insufficient for us, because it does not specify whether the
generated instance is symbolic or mental, nor does it indicate what “generate” means in
terms of the processes by which mental representations come into existence and how they
develop. A symbolic representation is externally written or spoken, usually with the aim
of making communication about the concept easier. A mental representation, on the other
hand, refers to internal schemata or frames of reference which a person uses to interact with
the external world. It is what occurs in the mind when thinking of that particular part of the
external world and may differ from person to person. In case of the symmetric group, one
person’s mental representation may consist of nothing but the symbol Sn, another may think
of a set of colored cubes which are being permuted, a third may see “in the mind’s eye”
symbols like (1 3 5)(2 4 6 7) which may or may not have an associated meaning, and still
another person may conceive of the group by way of its irreducible representations.
Another example is that of vector space. When I thinkofavector space, I may “see” arrows
(before my mind’s eye), and I may be able to think in terms of these arrows when dealing
with bases, transformations etc. Others may evoke n-tuples of numbers or abstract symbols
which satisfy the axioms.
Visualization plays an essential role in the work of many eminent mathematicians. For
instance, Einstein wrote to Hadamard:
Words and language, written or oral, seem not to play any role in my thinking. The psychological
constructs which are the elements of thought are certain signs or pictures, more or less clear, which
can be reproduced and combined at liberty.
(Hadamard, 1945, p. 82)
Visualizing is one process by which mental representations can come into being. A more
general description of how mental representations of mathematical concepts may be
generated has been proposed by Kaput (1987b); according to his theory, the act of
generating a mental representation, relies on representation systems, i.e. concrete, external
artefacts, which can be materially realized. In the case of functions, graphs are one such
artefact, algebraic formulas are another, arrow diagrams and value tables still other ones.
Mental representations arc created in the mind on the basis of these concrete representation
systems. A person may thus create a single or several competing mental representations for
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the same mathematical concept. The topic of students’ mental images of various mathematical concepts has already been mentioned in the opening chapter of this volume and
will be discussed further by Vinner in chapter 5.
To be successful in mathematics, it is desirable to have rich mental representations of
concepts. A representation is rich if it contains many linked aspects of that concept. A
representation is poor if it has too few elements to allow for flexibility in problem solving.
Such inflexibility we often observe in our students: The slightest change in the structure of
a problem, or even in its formulation, may completely block them (see, e.g., the study by
Selden, Mason & Selden described in the previous section). Poor mental images of the
function concept, for instance, are typical among beginning college students, who think
only in terms of formulas when dealing with functions, even if they are able to recite a more
general set-theoretic definition (see Eisenberg, chapter 9).
Several competing mental representations of a concept may coexist in somebody’s
mind, and be taken advantage of different ones may be called up for considering different
mathematical situations. However, different mental representations may also enter into
conflict such as, for example, in a calculus student described by Schoenfeld, Smith &
Arcavi (in press), who simultaneously held four competing and conflicting interpretations
of the y-intercept of a straight line. In more favorable cases, several mental representations
for the same concept may complement each other and eventually may be integrated into
a single representation of that concept. This process of integration is related to abstraction
and is further discussed below. As a result of this process, one has available what is best
described as multiple-linked representations, a state that allows one to use several of them
simultaneously, and efficiently switch between them at appropriate moments as required
by the problem or situation one thinks about.
2.2 SWITCHING REPRESENTATIONS AND TRANSLATING
Although it is important to have many representations of a concept, their existence by itself
is not sufficient to allow flexible use of the concept in problem solving. One does not get
the support that is needed to successfully manage the information used in solving a problem,
unless the various representations are correctly and strongly linked. One needs the
possibility to switch from one representalion to another one, whenever the other one is more
efficient for the next step one wants to take. The process of switching representations is thus
closely associated with that of representing. Switching must always be carried out between
existing representations. In our context, it means going over from one representation of a
mathematical concept to another one. And again, functions are probably the clearest
example. A function is an abstract concept with which we usually work in one of several
representations, or preferably in several representations at once; often these include the
graphical and the algebraic representation. Teaching and learning this process of switching
is not easy because the structure is a very complex one; think, for example, of a
trigonometric function, which already has the properties of frequency, amplitude and
phase; now consider the algebraic formula for this function, its graph, and a table containing
the values of the function at special points such as extrema and zeros; moreover establish
the links between these three representations, e.g. clarify to yourself where some of the
points in the table of values appear on the graph or how the value of the phase determines
ADVANCED MATHEMATICAL THINKING PROCESSES
33
the position of the graph; this is already a lot of information to be dealt with, especially for
students who lack extensive experience; but all of this information may only be the starting
point for a problem such as the behavior of the trigonometric function under a shift or stretch
transformation, i.e. a change of frequency or phase. As a consequence, students very often
limit themselves to working in a single representation; for example, even when they are
required to draw a sketch, say before integrating an absolute value function, they often
ignore their own sketch and thus fail to solve the problem correctly (Mundy, 1984). One
possible approach is to systematically use several representations in teaching and to stress
the process of switching representations from the beginning. Computer environments have
been successfully used to achieve this in curricula for functions (Schwarz, Dreyfus &
Bruckheimer, 1990), calculus (Tall, 1986a, 1986b), and differential equations (Artigue,
1987). These will be further discussed in chapter 11 by Artigue and chapter 14 by Tall &
Dubinsky. The way in which multiple-linked representations may be treated is well
illustrated by the functions curriculum. This introductory curriculum is based on a
computer micro-world, which has been specifically designed to encourage switching
representations. Students are asked to solve open ended problems such as maximizing, to
a given accuracy, the volume of an open box constructed from a given sheet of cardboard
(before they learn any calculus!). To successfully solve this problem they have to use at least
two representations, and they need to transfer information obtained in one representation
in order to use it in another one. A large majority of students in the study have been found
to transfer information between representations and to successfully use the transferred
information for solving problems. It has been concluded that, for these students, functional
representations are symbols in the sense described above, namely signs with associated
meaning; moreover, that meaning was common to several representations of the function;
in other words, these students developed a satisfactory function concept incorporating three
representations, between which they were able to switch during problems solving processes. (Schwarz & Dreyfus, 1991).
A process which is closely connected to switching representations is translating. One
meaning of translating which is relevant for advanced mathematical thinking is going over
from one formulation of a mathematical statement or problem to another one. Applied
problems are a case in point. A second order linear differential equation with constant
coefficients may be presented as an oscillation problem, possibly with friction; its solution
may then be discussed in terms of permanent and transient states. From the present point
of view, this constitutes an additional representation, and one that introduces considerable
additional difficulties for the beginning student; at least my students are very apprehensive
about “applied problems” in examinations. This can be easily explained in the light of the
above discussion. Not only does the student need to understand the context of the applied
problems, e.g. an electric circuit, but more importantly, he needs to establish a close and
clear correspondence between quantities referred to in terms of electric circuits and
quantities referred to in terms of differential equations. This correspondence may be
obvious to the teacher, but for the student the construction of the appropriate mental
schemata is a difficult task which needs to be supported by explicit teacher action.
In this subsection, the distinction, made above, between mental and concrete and
symbolic representations has been blurred. To some extent, this is necessary when learning
processes are considered, because mental representations arise from concrete ones. The
section on representing will now be concluded with a process of concrete representing that
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has particular importance in applied mathematics and takes on ever greater weight in
university, and more recently also in school curricula, namely modelling.
2.3 MODELLING
Typically, the term modelling refers to finding a mathematical representation for a nonmathematical object or process. In this case, it means constructing a mathematical structure
or theory which incorporates essential features of the object, system or process to be
described. This structure or theory, the model, can then be used in order to study the
behavior of the object or process being modelled. For example, the Schrödinger equation
models the behavior of certain physical systems which obey the rules of quantum
mechanics; or a crystallographic group models the symmetry properties of a chemical
compound. A mathematical model thus has the status of a representation of a (physical)
situation; but for the person thinking about, say the symmetry properties of a silicate crystal,
this is not enough; that person also needs a mental representation of the silicate’s symmetry
group. This leads to an interesting connection between a mode land a mental representation.
The process of representing is, to some extent, analogous to the modelling process, but on
another level. In modelling the situation or system is physical and the model is mathematical; in representing the object to be represented is the mathematical structure, and the model
is a mental structure. Thus the mental representation is related to the mathematical model
as the mathematical model is related to the physical system. Each is a partial rendering of
the other. Each reflects some (but not all) properties of the other. And each enhances one’s
capacity to mentally manipulate the system under consideration.
3. PROCESSES INVOLVED IN ABSTRACTION
Many of the processes mentioned in this book occur at any level of mathematical thinking:
Certainly, even small children create mental representations of anything they think about,
and particularly of mathematical objects of thought, such as numbers or triangles. Starting
no later than in elementary school, children also work with these objects, especially
numbers, in different representations. Other processes, however, take on added importance
as students’ mathematical experience and abilities develop and as the mathematical
contents they deal with become more advanced; the most important among these advanced
processes is abstracting. If a student develops the ability to consciously make abstractions
from mathematical situations, he has achieved an advanced level of mathematical thinking.
Achieving this capability to abstract may well be the single most important goal of
advanced mathematical education.
Two processes, in addition to representing, form a prerequisite basis to abstracting:
generalizing and synthesizing; we briefly discuss these first.
ADVANCED MATHEMATICAL THINKING PROCESSES
35
3.1 GENERALIZING
To generalize is to derive or induce from particulars, to identify commonalities, to expand
domains of validity. A student may know from experience that a linear equation in one
variable has one solution, and that “most” systems of two (three) linear equations in two
(three) variables have a unique solution. (S)he may then generalize this to systems of n
linear equations in n variables. More importantly, with appropriate guidance, (s)he may be
led to examine the meaning of “most” for n=2 and n=3 in the above statement, formulate
it as an appropriate condition, and generalize also that condition to n>3. In this process, one
needs to make the transition from the particular cases of n=l, 2,3 to general n, one needs
to identify what thecondtions for n=2 and n=3 have in common, and to conjecture and then
establish that the domain of validity of the conclusion “there is a unique solution” can be
extended to general n.
The generalization in the previous example is important in that it establishes a result for
a large class of cases – all systems of n independent linear equations in n variables. It is,
however, limited to establishing analogies between the concrete cases of n=2 and n=3, and
extending them to the case of n equations in n variables, which may be less concrete but
presents no essentially new features. In particular, the general case does not require the
formation of any mathematical concepts that were not present for n=3. Other generalizations do include the need for such concept formation. An example is the transition from
convergence of a sequence of numbers to convergence of a sequence of functions, which
gives rise to the need for a topology on the space of functions. The cognitive requirements
in the process of generalization are thus increased considerably, and for the specific case
of convergence of functions the degree of difficulty of these requirements is well
documented by several decades of discussions between Cauchy, Fourier and Abel at the
beginning of the 19th century (Lakatos, 1978). Students must thus be expected to have a
hard time with such generalizations, and experience confirms this. It must be pointed out,
however, that even in this case, the generalization takes place with respect to given
(mathematical) objects, equations in the first case, numbers and functions in the second.
The presence of these objects is helpful to the student because it leaves him on (hopefully)
well known, solid ground while trying to grapple with the added generality of the situation.
3.2 SYNTHESIZING
To synthesize means to combine or compose parts in such a way that they form a whole,
an entity. This whole then often amounts to more than the sum of its parts. For example,
in linear algebra, students usually learn quite a number of isolated facts about orthogonalization of vectors, diagonalization of matrices, transformation of bases, solution of systems
of linear equations, etc. Later in the learning process, all these previously unrelated facts
hopefully merge into a single picture, within which they are all comprised and interrelated.
This process of merging into a single picture is a synthesis. Thurston (1990) has recognized
that, due in part to this possibility of synthesis, mathematics is tremendously compressible.
He has also noted, that while the insight that goes with this compression is one of the real
joys of mathematics, this process is irreversible; therefore, it is very hard for the
mathematician to put himself in the frame of mind of the student who has not yet achieved
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this synthesis, and to see not only how much detail is involved in learning even simple
concepts and operations but how much detailed work with these concepts and operations
is needed to be able to start synthesizing.
Classroom practice often does not put enough stress on this process of synthesis. While
the details are explained at length by the teacher and exercised by the student, few if any
activities are designed to lead the student to synthesize different aspects of a concept, and
even less different concepts within a domain or even different domains. Obviously, the
good teacher does his part of summarizing and this often includes some synthesis. He does
state, for the students’ benefit, what the connections, relationships, etc. are. But the fact that
it is done by the teacher rather than in a student activity conveys to the students that this is
what the mathematicians see, and is of no direct relevance to the problems the student has
to solve. These problems are standard exercises, which do not require synthesis. Consequence: I do not need it for the exam, why should I bother? As we saw earlier in the chapter,
non-standard questions, even if almost trivial, but requiring some amount of flexibility of
thinking and synthesis, are usually out of reach, at least for the average student (Selden,
Mason & Selden, 1989). Students, specifically high school students who do well in
mathematics, believe that solving a mathematics problem should typically take one minute,
and never more than ten; they also think that memorizing is extremely important for success
in mathematics and that there is little relationship between the different mathematics
courses (algebra, geomety, trigonometry) which they have taken (Schoenfeld, 1989).
Again, even if synthesis may be in the teacher’s mind, it is sorely lacking from the student’s.
3.3 ABSTRACTING
In the transition from the concrete vector space 3 to the notion of an abstract vector space,
the relationships between the vectors become the focus of attention, whereas their specific
realization as triples of numbers is dropped. In order to make this transition, one needs to
be able to conceive of the object “vector” purely in terms of its relationships to other similar
or different objects (vectors or scalars), and accept that the object itself is not further
specified by any intrinsic properties. Considering only these relationships, enables one to
draw conclusions from them which will be generally valid, independently of the specific
intrinsic properties of the vectors. In this manner, much of the power of mathematics derives
from abstraction.
The process of abstraction is thus intimately linked to generalization. One of the main
incentives for abstraction is the general nature of the results that can be obtained. Another
incentive is the achievement of synthesis. Groups show to the student that it is possible to
describe in a unifying manner a vast number of situations that have heretofore been
considered separately and independently of each other. But neither generalizing nor
synthesizing make the same heavy cognitive demands on students as abstraction. As we
saw in chapter 1, generalization usually involves an expansion of the individual’s
knowledge structure whilst abstraction is likely to involve a mental re-construction. In the
transition, say, from real to complex numbers, we achieve generalization by not insisting
any more on order but we continue working with objects that are represented explicitly
using numbers that we can add and multiply in a familiar way. Similarly, a student may well
learn about the connections between matrix calculus and planar or spatial symmetry
ADVANCED MATHEMATICAL THINKING PROCESSES
37
transformations such as principal axis transformations or crystallographic point groups
without forgoing their explicit concrete realization. Abstraction, however, requires giving
up exactly this explicitness: the student is required to focus on the relationships that exist
between numbers in order to be able to grasp what a field is, rather than on the numbers
themselves, and similarly for other notions such as function, group and vector space.
Abstraction thus contains the potential for both generalization and synthesis; vice versa,
it gets its purpose mainly from this potential of generalization and synthesis. The nature of
the mental process of abstracting is, however, very much different from that of generalizing
and from that of synthesizing. Abstracting is first and foremost a constructive process – the
building of mental structures from mathematical structures, i.e. from properties of and
relationships between mathematical objects. This process is dependent on the isolation of
appropriate properties and relationships. It requires the ability to shift attention from the
objects themselves to the structure of their properties and relationships. Such constructive
mental activity on the part of a student is heavily dependent on the student’s attention being
focussed on those structures which are to form part of the abstract concept, and drawn away
from those which are irrelevant in the intended context; the structure becomes important,
while irrelevant delails are being omitted thus reducing the complexity of the situation. The
role of mathematical and mental structure in abstraction has been examined by Thompson
(1985a) and Harel & Tall (in press). The cognitive aspects of focussing and shifting
attention during the process of abstraction have been investigated by Dörfler (1988) and
Mason (1989). Abstraction will be further discussed by Dubinsky in chapter 7 from a
Piagetian viewpoint. We here only raise a few points which may serve as background for
the coming chapters.
It is an open didactical problem, whether one should lead students to abstract from many
cases or from a single one. Schoenfeld, Smith & Arcavi give a very detailed description of
how one above average student constructed her understanding of y-intercept (of a straight
line). They observed her give four different, inconsistent interpretations of y-intercept
depending on the context she had to deal with; e.g., she interpreted y-intercept differently
according to whether the line was given by two points on the same or on different sides of
the y-axis. It took seven hours of work over several weeks for the student to decontextualize
the notion and achieve an abstract unified concept of y-intercept. Given the instability in
the student’s interpretations, it must be considered unlikely that a single example and an
explicit formal definition would have helped her avoid later misinterpretations. More
generally, having several examples, e.g. of concrete groups, will enable students to identify
commonalities; this is one way for the teacher to focus students’ attention on those
properties and relationships which are important for the intended abstraction. And this way
of focussing attention may work well, if the amount of information in the detailed
description of the internal structure of the examples is limited. If, however, the examples
are too complex, i.e. if they have many properties that are to be ignored in the process of
abstracting, it may be difficult to achieve such focussing. Therefore, it is sometimes better
to abstract from a single case, combined with a definition of the abstract concept. This single
case then needs to be chosen so that the intended properties and relationships take some
evidence, e.g. by being useful in an activity the students engage in. Students’ experience
with making abstractions is also likely tobe a factor here: Grade schoolers who should learn
about place value (a very abstract and difficult concept) are unlikely to have much use for
a definition; mathematics students who already have a good grasp of what vector spaces
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TOMMY DREYFUS
and groups are, may not need dozens of examples of rings before king able to digest a
definition. The question is thus one of finding the good measure; that this is not easy is well
known, for instance, from all those students in differential equations courses who lack an
abstract concept of a function as a mathematical object and therefore fail to understand that
a function rather than a number constitutes the solution to a differential equation.
There is a inherent difficulty in abstracting: How can we generate mental structures,
which are so often linked to visual images, if they should represent relationships that are
removed from the concrete objects which they were originally linked to? What is the role
of visualization in the process of abstraction? Again, there is no definite answer to this
question. Visual images are usually global and stress structural aspects. Therefore, if
appropriate visual images can be found, they are likely to be of great help to students
engaged in abstracting. The well known infinite row of domino stones as a model for
mathematical induction is a case in point. It incorporates exactly those features that are
common to all inductive processes and it does this in a manner that exhibits the structure
of these common features. For instance, if one stone falls, then so does the next one; this
is so at any place in the whole row of stones; therefore, if one falls, not only the next one
but all following ones fall, each one causing the next one to fall. Furthermore, and this is
central, it is obvious that the if-then-statement does not make the stones fall, but that in
addition to the if-then-statement being correct, one, not necessarily the first, stone needs to
fall. The picture of the domino stones contains the relevant elements of induction without
many extraneous features. It captures the structure of the entire process, globally, into a
single entity. Such a visual image undoubtedly helps students in building and strengthening
their mental representation of induction. It happens, however, that visual models appropriate for abstract mathematical concepts are non-existent, incomplete or even misleading,
and then care must be exercised. A detailed investigation of visually supported abstraction
has been reported by Kautschitsch (1988); he found that dynamic visual sequences were
strongly supporting abstraction because of their analogy to squences of actions.
4. RELATIONSHIPS BETWEEN REPRESENTING AND ABSTRACTING
(IN LEARNING PROCESSES)
Frequently, the concrete properties we would like to abstract from are linked to particular
representations of an abstract mathematical concept; functions are a case in point, but so
are vectors, vector spaces, groups, fields, C*-algebras, categories and functors. The
properties and relationships of the abstract concept are the representation-independent
ones.
Representing and abstracting are thus complementary processes in opposite directions:
On the one hand, a concept is often abstracted from several of its representations, on the
other hand representations are always representations of some more abstract concept.
When a single representation of a concept is used, attention may be focussed on this instead
of the abstract object. However, when several representations are being considered in
parallel, the relation to the underlying abstract concept becomes important. Often representations are needed to carry out some specific work with the concept; for instance, group
representations rather than abstract groups are used to carry out group theoretical calculations. This need for concrete representations in order to carry out some specific thought
ADVANCED MATHEMATICAL THINKING PROCESSES
39
process is not purely mathematical. There is a parallel cognitive need: The thinking of many
mathematicians and mathematics students is enhanced if they are able to place themselves
mentally in a particular representation, e.g. a visual one. It is even more enhanced, when
they are able to use several representations in parallel. Again, there is a complementarity,
this time between the mathematical and the cognitive aspects of representing mathematical
structures.
Both these complementarities, the one between abstracting and representing and the one
between mathematical and mental representations, may be and have been put to didactic
use in learning processes. Learning processes may then be seen as consisting of four stages:
• Using a single representation,
• Using more than one representation in parallel,
• Making links between parallel representations,
• Integrating representations and flexible switching between them.
In stage one, processes start from a concrete case, a single representation. But in learning
the function concept, for example, students usually meet several representations (graphical,
tabular, algebraic, arrow diagrams, ...). In the second stage, thus, several representations of
the same mathematical object are used in parallel. The difficult process of transition to the
abstract concept depends in an essential manner on the links between the representations
that are formed. The establishment of these links constitutes the third stage. Strong links
allow students to switch representations, which in turn makes them aware of the underlying
concept and is thus likely to positively influence abstraction. At the fourth stage, a process
of integration between the different representations is happening, a synthesis of the kind
that has been shown above to be partial to the process of abstraction: the links, the
relationships, the common properties remain to form the abstract concept, whereas the
representation specific aspects retract to the background. Once this process has been
completed, one has formed an abstract notion of a given concept, one somehow “owns” that
concept. When one then needs to solve a problem in which this concept occurs, one will
often need to go back to one (or several) of its concrete representations. The wonderful thing
about the abstract concept is that one is able to do precisely that, and to do it in a controlled
manner: One has control over the representations one wants to use.
The use of several representations to help students make the transition from a limited
concrete understanding of a certain topic to a more abstract and flexible understanding, has
been investigated by Kaput and his co-workers (1988). While dealing with ratio and
proportion, they used concrete, visual, computer-implemented representations whose
design was built on a cognitive basis. They called their approach a “concrete-to-abstract
software ramp”. The functions curriculum described in section 2.2 is another example: It
was systematically built towards the establishment of links between different function
representations and, as results from cognitive research have shown, this eventually lead
most of the students in the experiment to an abstract understanding of the function concept.
For some particularly gifted students, such carefully designed constructions of concepts
appear to be superfluous; the eight year old Terence Tao (Clements, 1983) is one such case:
in spite of his age, he was able to learn directly about abstract algebraic structures, and
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concrete representations tended to disturb him, if anything. When asked whether a given
structure was a ring, he immediately realized that he only needed to prove three things and
proceeded to prove these with maximal reliance on earlier proved structural results. A
similar situation may pertain to advanced mathematics students who have had the
opportunity to acquire considerable experience with abstraction; this experience is likely
to make some of the above stages superfluous and, for complex mathematical structures,
even a hindrance to abstraction; as has been pointed out above, abstraction from one, or even
from zero cases, may be easier for such students. But most students taking college and high
school mathematics courses do not belong to this category, and for them abstracting is
probably the most demanding of all advanced mathematical processes.
5. A WIDER VISTA OF ADVANCED MATHEMATICAL PROCESSES
The processes of representing and abstracting which have been discussed in some detail are
among the most important ones for advanced mathematical thinking; nevertheless, they are
only some among many processes which should and do occur as interacting links in chains
that may also include discovering, intuiting, checking, proving, defining and others.
Discovering or rather rediscovering relationships, for instance, is often considered
among the most effective ways for children to learn mathematics. To some extent, this
effectiveness may be attributed to the psychological aspects of the process of discovery: the
personal involvement, the intensity of the attention, the feeling of achievement and success.
Learning by discovery, however, is time-consuming, and this is one reason why teachers,
especially teachers of more advanced mathematics, tend not to use it. But the central
question is whether learning by discovery in the early and middle stages of mathematical
education develops reasoning processes which make later learning so much more efficient
that there is, in the long run, no time loss. (Here, efficiency must be measured not only in
terms of topics covered but also in terms of depth of understanding.) More generally, to
what extent could more stress on processes and less on content enable our students to learn
abstract mathematics in speedy, independent, and understanding ways?
Intuiting, i.e., apprehending by intuition, by immediate direct cognition without
evidence of rational thought, has a central role in any sequence of processes that starts from
discovery; the role of intuition will be discussed further in chapter 12 by Tirosh. Let us here
only mention its close links to processes of visualizing. For instance, being interested in the
intrinsic description of curves in terms of curvature (g) and arc-length (s), I was recently
looking at ellipses; on the basis of the periodic increase and decrease of the curvature when
proceeding around the ellipse, my visual intuition told me that a good try could be a
trigonometric function of the form g=A+Bcos(ks) with suitable constants A, B, and k. By
the way, this intuition was soon proven wrong by more detailed analysis that included
checking particular cases.
Checking means taking actions to convince oneself that a result indeed does answer the
question that was asked, and does answer it correctly. One useful way of checking is to use
an inverse procedure, such as differentiating to check whether one has correctly found a
primitive function. All too often, checking is not seen by students as an essential part of
mathematical activity. Although checking could give them a lot of security, most students
appear not to be very interested in this security. This could and should be changed by
ADVANCED MATHEMATICAL THINKING PROCESSES
41
transferring more of the responsibility for learning processes from the teacher to the student,
in line with the independence mentioned in connection with discovering. Giving students
open-ended activities rather than one-minute exercises to work on, is one step in this
direction.
Discovering, intuiting and checking, however, are only the beginning of a sequence of
mathematical processes – the goal remains understanding of abstract relationships.
Students’ activity must therefore proceed from here to the more formal processes of
defining and proving, which will be analyzed respectively in chapter 5 by Vinner and
chapter 13 by Alibert & Thomas.
It will be seen in the following chapters that many features of these processes need to
be made very explicit to students, down into the smallest details. This does not mean that
students should be told about these details, but that student activities have to be designed
with these details in mind, in such a way that students realize them. These may, but need
not, be details of mathematical facts or relationships; more often, they are parts of the
processes. For example, with respect to switching representations, students must be made
conscious of their act of pulling information out of one function representation and using
this same information in another one (see Schwarz, Dreyfus & Bruckheimer, 1990, for
details on how this can be achieved). Similarly, students need to become conscious of the
interactions between processes such as representing and abstracting. The working mathematician is using many processes in short succession, if not simultaneously, and lets them
interact in efficient ways. Our goal should be to bring our students’ mathematical thinking
as close as possible to that of a working mathematician’s. Understanding advanced
mathematical processes and their interplay is a necessary prerequisite for achieving this
goal.
ACKNOWLEDGEMENT
This chapter has profited from numerous discussions with friends and colleagues, among
whom were Michèle Artigue, Bernard Comu, Ed Dubinsky, Janet Duffin, Jim Kaput, Uri
Leron, John Mason, Baruch Schwarz, Pat Thompson, and Shlomo Vinner. But more than
any others, Ted Eisenberg, David Tall, and Dina Tirosh have helped shape the chapter with
well-taken comments and suggestions based on their careful reading of earlier versions.
CHAPTER 3
MATHEMATICAL CREATIVITY
GONTRAN ERVYNCK
Creativity plays a vital role in the full cycle of advanced mathematical thinking. It
contributes in the first stages of development of a mathematical theory when possible
conjectures are framed as a result of the individual’s experience of the mathematical
context; it is also plays a part in the formulation of the final edifice of mathematics as a
deductive system with clearly defined axioms and formally constructed proofs. It is an
essential factor in research mathematics when new ideas are formulated in a manner
previously unknown to the mathematical community. Yet it is external to the theory of
mathematics. It is a human activity, a meta-process, which acts upon and generates new
mathematics. As such it is often viewed as a mysterious phenomenon. Most mathematicians seem to be not interested in the analysis of their own thinking procedures and do not
describe how they work or conceive their theories. Only a few (such as Poincaré,
Hadamard) explicitly attempt to describe ideas related to mathematical creativity. The best
known reference (at least to mathematicians) is probably Hadamard (1945), which has been
followed recently by Muir (1988).
The present chapter will not attempt to give an exhaustive description of the nature of
mathematical creativity and how it works. From a somewhat closer look at the aspects of
different kinds of mathematical activity as an heuristic procedure to register examples of
mathematical creativity, we derive some striking characteristics of the phenomenon and
frame a tentative definition. The reader is invited to activate his/her own imagination and
attentiveness and to rectify deficiencies in the text with their own personal observations.
1. THE STAGES OF DEVELOPMENT OF MATHEMATICAL CREATIVITY
Mathematical creativity does not, presumably cannot, occur in a vacuum. It needs a context
in which the individual is prepared by previous experiences for the significant step forward
in a new direction. Such preparation occurs through previous activities which form an
appropriate environment for creative development. We hypothesize that the context for
creativity is set by a preparatory stage in which mathematical procedures become
interiorized through action before they can be the objects of mathematical thought.
Preliminary to thismay be an initial stage where the procedures might be used without even
a full appreciation as to their theoretical status.
Stage 0: A preliminary technical stage
We hypothesize that genuine mathematical activity may be preceded by a preliminary
stage consisting of some kind of technical or practical application of mathematical rules and
procedures, without the user having any awareness of their theoretical foundation. We refer
here to the art of the craftsman who applies a set of mathematical procedures as a toolkit
42
MATHEMATICAL CREATIVITY
43
providing him with the necessary tools to manufacture his product. The justification for
these procedures is that it has been checked empirically that they work, in the sense that a
correctly applied rule always yields the desired result. An example of such a practical
procedure is the rule used in Ancient Mesopotamia and Egypt to stake out a right angle: they
used a rope with marks dividing it into three parts of length 3, 4 and 5. Forming the contour
of a triangle, they obtained a right angle between the sides of length 3 and 4. This preparatory
stage has become part of modem theories of mathematics learning, for instance, the “toolobject” dialectic of Douady (1986) which proposes that an idea should be introduced first
as a tool as part of a problem-solving activity, to become part of the individual’s experiential
cognitive structure before being reflected on as an object in its own right.
Stage I : Algorithmic activity
At this stage procedures are used to carry out mathematical operations, to calculate,
manipulate, solve. Algorithmic activity is essentially concerned with performing mathematical techniques. Examples of such techniques are: application of an algorithm,
working out formulae, factorizing a polynomial, calculating an integral, computational
activities involving computer programs such as in numerical methods for solving differential equations. A characteristic of such activities pertaining to this first stage is that they
need to be quite explicit. All intermediate steps have to be considered, at least implicitly;
if not, a serious error may occur and totally invalidate the result. In the case of a computer
algorithm, no steps, not even trivial ones, may be forgotten. There is no regeneration of
missing steps in an algorithm.
Such activities are an acceptable part of advanced mathematics because they may be
seen as part of an overall theory, created in accordance with the principles of the higher
activities to be described in stage 2. We hypothesize that algorithmic activity is an essential
part of the learning of mathematics because such processes must be interiorized and
become routinized before they can be reflected upon as manipulable mental objects in a
higher order theory. (See chapter 7 for a discussion of reflective abstraction, in which a
process is encapsulated as a concept). As with the tool-object dialectic, it is essential that
the tool become familiar in action before it becomes the focus of reflective activity.
Stage 2: The creative (conceptual, constructive) activity
It is at this second stage that true mathematical creativity is likely to occur and act as a
motive power in the development of a mathematical theory. A non-algorithmic decision is
taken in a manner which seems to signify a bifurcation of the underlying concept structure.
Mathematical creativity is the ability to perform such steps. The decisions that have to be
taken may be of a widely divergent nature and always involve a choice, such as a choice
of a certain concept to be defined (for instance, as in Hausdorff’s choice of the notion of
an open set, which proved to be of tremendous importance in major parts of mathematics)
or the decision to state and prove a theorem. The latter requires two distinct creative steps:
the choice of adequate hypotheses such that the resulting conclusion is of value within a
wider theory, and the actual deduction from the hypotheses to establish the proof of the
theorem. Note that the initial choice demands also a formulation of the possibilities from
which the choice is made. It is in such complex activities that mathematical creativity comes
to the fore.
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GONTRAN ERVYNCK
In order that mathematical creativity should be activated, there is no need to have a
formal theory at one’s disposal; the most active part of creativity acts at the intuitive level
in a spirit of regeneration and renovation. Davis & Hersh (1981) suggest that it comes
through a passage from the coarse (the intuitive) to the fine (the formalized).
What is essential in the individual is a state of mind prepared for mental activity that
relates previously unrelated concepts. It is often observed to occur after a period of intense
activity involving a heightened state of consciousness of the context and all the constituent
parts. And yet it is more likely to bear fruit when the mind is subsequently relaxed and able,
subconsciously, to relate the ideas in a manner which benefits from quiet, unforced,
contemplation.
High levels of creativity demand a subtlety of mental structure that is tuned to be able
to resonate with underlying patterns that may fail to come to light in less refined
circumstances. We illustrate this by looking at an example of a problem which can be solved
at different levels of mathematical sophistication. The three levels may be characterized in
a very typical manner, based on the status of the method used in the solution. This
classification proves to be parallel to the description of the stages of development of
creativity just described. A first (low) level relies heavily on the application of an algorithm;
the creativity involved requires only recognition of the overall positioning of the problem
in the whole of mathematics and the construction of the appropriate model (for instance,
a system of linear equations or a truth table). A second (higher) level abandons the
straightforward application of the algorithm, but is based on direct reasoning inside the
mathematical model. Some insight and intuition are needed to develop the right method of
solution. The environment (the model) is still borrowed from a general theory, but solving
the problem at hand is done by inferring directly from the given situation. A third level (the
highest) abandons the model completely, reasoning outside a formalized theory, constructing a solution ab ovo by an intelligent inspection of what is stated in the problem.
• Problem: A man was a child for one sixth, a young man for one twelfth and a
bachelor for one seventh of his life. His son was born five years after his marriage
and died four years earlier than his father. The lifetime of the son was half of the
lifetime of the father. How old was the father when he died?
• Solution at a low level of mathematical creativity
It is enough to realize that the problem is subject to strict conditions and can be
modelled in a system of algebraic equations. The abilities involved are the
introduction of the necessary unknowns and the formulation of the equations.
Let
x = the age of the father at his death;
y = the time the father has been married;
z = the age of the son at his death.
A careful translation of the problem gives the equations
MATHEMATICAL CREATIVITY
5+z+4=y
and a knowledge of the solution of linear equations gives the solution x=84.
• Solution at a higher level of mathematical creativity
An endeavour to formulate a concept image of the internal structure of the
problem may yield the awareness that the situation has a linear model. As the
lives of human beings occur in time, a time axis is all that is required to represent
all the events occurring in the problem. Moreover, all events refer to precise
moments which may easily be located on the time axis. A simple graphical
representation:
Figure 1: Visualizing the solution
gives (suitably interpreted) the following equation:
and the solution x=84.
• A third level solution
A much more sophisticated method for solving the problem is based on
intuition, experience and some plausible (in the sense of Polya) assumptions
embedded in the nature of the problem.
• A first hypothesis is the assumption that the age is generally expressed as a
positive integer in the range 0 to 100. A plausible starting point is to look for an
integer solution.
1
–1 , –17 occurring refer to
• A second step is the assumption that the fractions –6, 12
periods in the life of the father which are probably whole numbers as well.
• A decisive step is to realize that the denominators, 6, 12 and 7 have few
common multiples between 0 and 100, hence it may be valuable to compute the
lowest common multiple – indeed the only common multiple in this range –
which is 84. Verification confirms that 84 is the (only) solution.
45
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GONTRAN ERVYNCK
This higher level of mathematical activity involves a highly tuned experience with number
theory as well as an insight into the working methods of the problem-poser. It illustrates the
unusual pathways that might be taken by the creative mathematician in solving an old
problem in a new way. But to be able to analyse the role of creativity of new mathematics
in new contexts we must first consider the nature of advanced mathematics as a major goal
of mathematical creativity.
2. THE STRUCTURE OF A MATHEMATICAL THEORY
It is essential to have an overall view of the structure of mathematics as a mental construct
before concentrating on the nature of the creative processes that bring it into existence. We
see a formal theory of mathematics as a framework consisting of definitions of concepts
and relations between the defined concepts, the latter being of a very particular kind: the
relations emerge from the implementation of very strictly prescribed (deductive) rules. This
entails the necessity to determine (to define) the concepts is a very precise manner. The
concepts may be thought of as the nodes of a network and the relations are directed arrows
connecting the nodes. Moreover, the network has an additional feature: the connections are
ordered proceeding from the logically basic nodes towards the more complicated ones.
Mathematical creativity involves both the vision to build up parts of such a framework by
conjecture and argument and also to refine the structure into a mathematically deductive
framework.
We suggest that an act of creativity requires the realization of at least one of the following
objectives:
(i) to create a useful new concept, where ‘useful’ means favourable to the further
unfolding of the theory at hand;
(ii) to discover a formerly unnoticed relation between two (or more) nodes, with the
required ordering;
(iii) to construct a useful ordering: to organize a part of a theory such that its logical,
deductive order becomes more apparent.
The specification of a successful set of axioms for a previously unaxiomatized theory (as
with group theory) may be considered as an instance of mathematical creativity where all
three objectives have been realized.
3. A TENTATIVE DEFINITION OF MATHEMATICAL CREATIVITY
Examples of creativity in mathematics are: the ability to formulate a valuable definition
using concepts which assure the usefulness of the defined object in the subsequent theory;
or the formalization of a basic idea borrowed from the physical context which was initially
at the base of the mathematical problem. Hence, we look at mathematical creativity
essentially as the ability to create mathematical objects, together with the discovery of their
mutual relationships. This activity is considered here as different from, and even opposed
MATHEMATICAL CREATIVITY
47
to, algorithmic mathematical objects (the “first stage” mentioned earlier).
A tentative definition might be:
Mathematical creativity is the ability to solve problems and/or to develop
thinking in structures, taking account of the peculiar logico-deductive nature of
the discipline, and of the fitness of the generated concepts to integrate into the
core of what is important in mathematics.
4. THE INGREDIENTS OF MATHEMATICAL CREATIVITY
The working procedures of mathematical creativity are intimately linked with the stages
discussed in section 2. They are essentially a working-out of the impulses which steer the
creativity of the working mathematician and operate generally in the following order:
(1) study, yielding familiarity with the subject,
(2) intuition of the deep structure of the subject,
(3) imagination and inspiration,
(4) results, framed within a deductive (formal) structure.
It is the effort involved in studying and becoming familiar with the subject that sets in the
mind conceptual structures that contain the potential for mathematical creativity. Intuition
is the product of the action of these conceptual structures on newly acquired data. As we
saw in chapter 1, intuition can be honed and polished into a refined tool. The more refined
the mental structure, the more likely it is to produce refined intuitions. It is by reflection on
the deep structure of the subject that such intuitions may lead to the imagination and
inspiration which fomulate the required results, at first in an imperfect form, but then honed
by reflection into formal deductive order.
5. THE MOTIVE POWER OF MATHEMATICAL CREATIVITY
The power of mathematical creativity results from the interaction of a certain number of
elements which may be listed as follows (although there is no reason to believe that the list
is exhaustive):
• Understanding: the ability to regenerate the steps of the mathematical creativity
of the author(s) of a theorem, a part of a theory ... Mathematical creativity is
based on, and brings with it, a simultaneous deepening of understanding and
insight in a concept.
• Intuition: the formation of concept images which are sufficiently close to the
formal concept to allow the conception of plausible conjectures. Intuition
enables the mathematician to perform a fruitful selection as well (see below).
Other factors of equal importance, related to intuition and acting as driving
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GONTRAN ERVYNCK
forces in the process of mathematical creation are imagination, mathematical
phantasy and curiosity (Dieudonné, 1974).
• Insight: the driving force required to move towards a formulation of new
knowledge. This involves a refocussing of interest and a reorientation to
consolidate what is important, and even more, to envision will be important in
the future.
• Generalization: the ability to generalize is linked with insight because it
depends heavily on the ability to foresee what will be important in the future.
Generalization is a form of mathematical creativity, but sometimes only a weak
form: generalizing a theory is sometimes hard, sometimes straightforward.
Sometimes it may be an illusion: as any finite group has a representation as a
group of permutations, the generalization of the theory of permutation groups
of Galois and Jordan in the theory of finite groups is only a reformulation, though
undoubtedly a more handsome embodiment, of the former.
We see these four ingredients being parallel to the four topics mentioned in the previous
section. By understanding, we mean not just the instrumental understanding involved in
being able to carry out processes, but the relational understanding, in the sense of Skemp
(1976), which involves a meaningful grasp of the relationships between the concepts. Even
this is not enough, for it suggests a meaningful relationship between the concepts in the
context which they are currently known. Creativity demands an extension of this context
in a way that has not before been conceived. It therefore requires the individual to create
new ideas and to put old ideas together in a new way, It is not something which can be carried
out on demand.
“The philosopher’s stone can only be found when the search lies heavily on the searcher. Thou
seekest hard and findest it not. Seek not and thou willst find.”
It requires relaxation and incubation in the sense of Poincaré (see page 15). Given good
preparation and good fortune, the incubation may provoke intuitions that lead to the
fundamental insights that break through to give the creative leap.
The latter may be a generalization of previous knowledge, which means the extension
of current schemas to a broader context. In chapter 1 we saw that there were two
fundamentally different kinds of generalization: the expansive generalization, which
broadens the applicability of the theory without changing the nature of the cognitive
structure, and the reconstructive generalization which requires the knowledge structure to
be reorganized. Whilst the former may be relatively easy, even when it occurs creatively
for the first time, the latter involves a cognitive transition of great difficulty which requires
special personal qualities of character to succeed in the struggle.
MATHEMATICAL CREATIVITY
49
6. THE CHARACTERISTICS OF MATHEMATICAL CREATIVITY
In making the great leap of mathematical creation, we see certain characteristics coming
to the fore. Mathematical creativity is:
• Relational (in the sense of Skemp). It stimulates through interaction: it establishes a conceptual link between two or more concepts, such that a new idea
emerges which integrates different aspects form the initial concepts into a single
one. Interaction of ideas in the mind of the mathematician is perhaps the most
important driving force of mathematical creativity. Mathematical ideas and
concepts arise as mobile building blocks and combine (if the subject is not
mathematically totally insignificant) to form some new configuration. If the
configuration is favourable, it enters into the theory. This has already been
described by Poincaré.
A deeper view of the process entails the question:
is mathematical creativity acting just as mutations in biology?
A mathematical mutation occurs when a chain of ideas undergoes a restructuring, maybe
in one single place. Among all restructurings some are useless, others are useful. Some
survive, others are eliminated although they are entirely correct from a formal viewpoint.
An example of such a case is the theory of cubics, algebraic curves of degree three,
developed as a generalization of the theory of conics; this theory was developed in the
nineteenth century, but is seldom taught today.
We therefore also see that mathematical creativity is:
• Selective. This analogy with biology arises through the struggle for life amongst
mathematical concepts, with a natural selection and survival of the fittest. For
example, the several theories of integration established at the end of the
nineteenth and the beginning of the twentieth century to generalize the Riemann
integral entered into competition with each other and finally the Lesbegue
integral survived to dominate mathematical analysis (Van Dalen & Monna,
1972).
Selectivity gives rise to a related criterion:
• Fitness. This is a qualifying criterion for the value of definitions and theorems
and sets of axioms in mathematics. The well-known estimation by Stanislas
Ulam of the 200,000 yearly produced theorems makes it clear that a sieve seems
very necessary. In fact, the sieve exists, and in the first place does not consist of
the referees of the numerous journals, but acts spontaneously and unconsciously
in time, through the action of the struggle for mathematical life and the survival
of the fittest ideas.
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GONTRAN ERVYNCK
Finally, mathematical creativity must lead to new ways of handling the complexity of the
relationships between more complex concepts. It does this by encapsulating new structures
into single objects which are easier to manipulate mentally. It is therefore:
• Condensing. Mathematical creativity includes the ability to choose the appropriate wording and symbols for the representation of mathematical concepts.
The importance of symbolic representations in mathematics cannot be overestimated. Well-chosen symbols allow for a condensation of several aspects of
one concept into a single whole which is evoked every time the symbol occurs
in a text. In this manner the use of the symbol frees “memory space” in the mind
which becomes available for other, till then unknown or unclear concepts.
7. THE RESULTS OF MATHEMATICAL CREATIVITY
After the process of mathematical creativity, there are various qualities that the new ideas
must exhibit in order that they might be accepted and survive in the mathematical
community at large. MacLane (1986) suggests a number of criteria which are required so
that the new idea can be labelled “good mathematics”. It must be:
• Illuminating. This seems to be a necessary characteristic of mathematical
creativity. Good mathematics should be of help in understanding. A result that
obscures is not creative, or is creativity used in an inappropriate direction, for
example through indulging in long technical calculations. For the same reason
we say that mathematical creativity in the first stage (algorithmic activity) is
very low.
• Deep. Mathematical creativity is supposed to uncover hidden relationships. A
deep result is not necessarily difficult to prove, but it is usually wide in its
relevance and application.
• Responsive or fruitful. The successful product of creativity is based on former
results and often responds to current needs. If it is to survive, it also provides a
basis for future development so that it remains an essential part of living
mathematics.
• Original. There should be something unexpected in the results, something new
in the field, if it is just a rearrangement of known results, there will be strong
doubts concerning the creative aspect of the achievement.
In addition, there are subtle qualities of surprise, even humour, which cause a mathematical
result to appeal to a professional mathematician. The following example illustrates the
latter (though it is not put forward as an example of particularly deep mathematics). It is an
inference which occurs using well accepted methodology (use of axioms, logical deductions, and so on) and is analogous to the usual reasoning in mathematical papers, but the
result of the inference is strange and unexpected.
The problem (from Wille, 1984) runs as follows:
MATHEMATICAL CREATIVITY
51
In the teaching of geometry, how shall the teacher proceed in order to draw a
“general” triangle on the blackboard?
The problem is ill-posed as long as we have no agreement on what a general triangle is;
hence a clear description of the term “general” is required. Thus the first step in
mathematical theory is encountered: the formulation of clear definitions. As we want to
serve the purposes of teaching for mathematically inexperienced students, it is quite
acceptable to say that “general” means a triangle without any particular geometrical
property. Moreover, granted the students lack of mathematical experience, we claim that
it must be seen with the naked eye that the triangle is general. For example, a triangle with
angles 89°, 45°, 46° will be perceived by the students as a right-angled, equilateral triangle,
and so does not fit our requirements. The didactical principle underlying these concerns is
that mathematical concepts are understood and developed on the base of concept images
which are present in the learner’s mind. A correct concept image may be generated by a
selection of appropriate examples; hence, in the case of geometry, the selection of the
pictures to be drawn on the blackboard may yield a correct understanding (or not) of the
formal ideas.
The claim that the students recognize the generality of the triangle by naked eye requires
empirical investigations. Based on experiments, a model has been constructed that
describes the ability of the human eye to distinguish between angles of different sizes. It
appears that inability to recognize a second angle as different from a first given one is
normally distributed according to the difference between the two angles. Experimentally
the standard deviation is σ =5. 770 with a 99% certainty obtained by a difference interval
of size 2.6σ = 15°. We therefore adopt the statement that a triangle is “general” if it is
considered as such by 99% of the students.
This leads to the following axiom system which formalizes the condtions established
in the previous paragraphs:
Axiom I: The triangles not isosceles.
Axiom II: The triangle is not right-angled.
Axiom III: Two angles differing by less than 15° are considered as being equal.
With these axioms we may prove the following remarkable theorem:
Theorem There is, up to similarity (axiom III) precisely one general triangle with acute
angles, namely, 45°, 60°, 75°.
(we note that there are an infinite number with one obtuse angle.)
Proof Let the triangle have angles A, B, C where 90 > A > B > C.
As A differs from 90° by at least 15°, we have A = 90 — 15 — a = 75 — a where a ≥ 0.
Similarly, B differs from A by at least 15°, so
A = 75 – a – 15 – b = 60 – a – b
where b ≥ 0 and finally
C = 60 – a – b – 15 – c = 45 – a – b – c where c ≥ 0.
But the angles add up to 180°, so
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GONTRAN ERVYNCK
180 = 75 – a + 60 – a – b + 45 – a – b – c = 180 – 3a –2b – c
and the non-negative numbers a, b, c satisfy
3a + 2b + c = 0,
hence a = 0, b = 0, c = 0 and
A = 75, B = 60, C = 45,
as stated.
8. THE FALLIBILITY OF MATHEMATICAL CREATIVITY
A major characteristic of mathematical creativity which distinguishes it from the generally
accepted qualities of a mathematical theory is that it is sometimes fallible. It puts together
new ideas in a way which may prove to be insightful, or may equally lead to error. There
is no guarantee that theorems may be formulated correctly, or even that such theorems are
accompanied by correct proofs. Famous examples are early proofs of the Four Colour
Theorem, the numerous “proofs” of the fifth postulate of Euclid and the recent “proof” of
the Poincaré Conjecture which seemed plausible for some months before a flaw was
discovered.
Given the view of Lakatos (1976), mathematics does not proceed in a Vauban-like
manner, making step-by-step sure advances in a pre-determined direction, but like the
daring exploits of the cavalryman of the advance guard, the forays into new territory may
be flawed. Mathematical thinking, as opposed to the reflected organization of mathematical
thought, is a creative activity that brings with it the possibility of human error. Indeed the
very possibility of error is what makes the major advances such monuments of human
success.
9. CONSEQUENCES IN TEACHING
ADVANCED MATHEMATICAL THINKING
The fallibility of this vital stage in mathematical thinking is something that students may
find hard to accept. Their whole mathematical training is usually accompanied by the
provision of algorithms that provide certainty to solve a given class of problems, and with
it the (false) belief that, given sufficient time and study, there will be an algorithm that will
solve any given problem. When they study differential equations, they see the solution of
various types of equation: separable equations of first degree, those that can be solved using
an integrating factor, or a power series approach, the special case of simple harmonic
motion, then higher order differential equations with constant coefficients. It will come as
a surprise to such students that the subset of differential equations that can be solved is, in
a genuine cardinal number sense, an insignificant minority of all differential equations.
Students are so often given the impression that, in mathematics, all is logical, certain,
accurate, provable, amenable to clear explanation. Yet mathematical creativity is none of
these things. It offers a major difference between the actual working practices of research
mathematicians and the facets of the mathematician’s art that are selected to teach to the
next generation.
MATHEMATICAL CREATIVITY
53
We have seen that there are certain requirements of mathematical creativity which seem
to preclude its operation in all but the most gifted. In particular it requires a sophisticated
understanding of mathematics in a given context to make creative developments which
extend known theory. Clearly we should not expect students to (re-)invent what has taken
centuries of corporate mathematical activity to achieve. Yet if we do not encourage them
to participate in the generation of mathematical ideas as well as their routine reproduction,
we cannot begin to show them the full range of advanced mathematical thinking.
Such approaches are already beginning in elementary schools where children are being
asked to carry out extended mathematical investigations starting from a context which is,
to them, novel. For them, such an enterprise is creative. It provides an activity which is
complementary to traditional methods of learning mathematics without in any way
replacing them. It allows children to begin to explore, to conjecture and test, to formulate
and prove, in ways which give deeper meaning to mathematical processes.
In this way, at a time when the content and approach to elementary mathematics is
becoming more clearly prescribed in national curricula and national standards, there are
complementary moves to encourage younger children to play their own part in knowledge
generation, to make conjectures, to expect errors, to need to check, to convince, to prove.
In a society which is fast changing, such flexible thinking beyond the mere application of
algorithms is becoming not just desirable, but increasingly necessary. Creativity at only the
lowest level is no longer acceptable.
In the next chapter we will see that the apparently unimpeachable bastion of mathematical truth, the formal proof, is in practice context bound and dependent upon stylistic
conventions of the mathematical community. It is therefore somewhat more fallible than
mathematicians may care to admit. The wider appreciation of the full range of advanced
mathematical thinking, including knowledge generation and creative problem-solving
through conjecture, debate and proof is therefore an objective which is worthy of
consideration. In chapter 13 we will return to the question of conjecture and debate in the
creation of mathematical proof. Within this broader framework of advanced mathematical
thinking, we therefore see mathematical creativity, so totally neglected in current undergraduate mathematics courses, as a worthy focus of more attention in the teaching of
advanced mathematics in the future.
CHAPTER 4
MATHEMATICAL PROOF
GILA HANNA
The hallmark of the mathematics curriculum adopted in the sixties was an emphasis on
formal proof. Among the manifestations of this emphasis were an axiomatic presentation
of elementary algebra and increased attention to the precise formulation of mathematical
notions and to the structure of a deductive system.
Indeed mathematics itself had grown tremendously since the beginning of the twentieth
century. Entire new fields had come into being in the first half of the century: modem
mathematical statistics, the theory of games, queuing theory, graph theory, and techniques
such as linear programming, often included in the general category of operational research,
which had gained prominence through their successful application during World War II.
The growth of mathematics was accompanied by change of outlook on the part of
practising mathematicians. The work of the logicist, formalist, and intuitionist schools on
the foundations of mathematics had given an impetus to the concern for precision in
definition and for the careful use of language. Also, the axiomatic approach, which these
schools shared despite their many differences, had become a common denominator of most
mathematical endeavours. The new status of deductive rigor as a standard in mathematical
work was stated clearly by the prominent French mathematician Dieudonné (1971):
Hence the absolute necessity from now on for every mathematician concerned with intellectual
probity to present his reasonings in axiomatic form, i.e., in a form where propositions are limited
by virtue of rules of logic only, all intuitive “evidence” which may suggest expressions to the mind
being deliberately disregarded.
(p. 253)
Many important mathematicians saw the axiomatic method not only as the prescribed form
for each individual mathematical discipline, but also as a means of consolidating many
previously unconnected disciplines into a small number of “mathematical structures”. This
point of view was promulgated by the group of influential French mathematicians who
wrote under the name of Bourbaki. The Bourbaki group exerted a great deal of influence
internationally on mathematical research. The focus of the group, apart from its attention
to newer mathematical subject matter, was on what has been called the “Bourbaki
approach”: a formal, abstract, and rigorous approach, emphasizing precise definitions and
formal proof.
This chapter discusses the origins of this emphasis on formal proof and considers its
limitations as a focus for advanced mathematical thinking in light of those aspects of
mathematical practice which complement and go beyond formal proof.
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55
1. ORIGINS OF THE EMPHASIS ON FORMAL PROOF
The curriculum revolution of the sixties was predicated upon a number of beliefs, one of
which was that formal proof is the most important characteristic of modem mathematics.
This view was no doubt due in large part to the impressive work done during the first half
of the century in clarifying the very foundations of mathematics, work which had
demonstrated the enormous power of formal systems constructed step by step from a base
of definitions, axioms and rules of inference.
The brilliant mathematicians who were so united in a desire to lay a firm foundation for
mathematics were by no means united in their approach to this task. But though the schools
of thought which came to be known as logicism, formalism and intuitionism differed
greatly in their philosophical accounts of mathematics and even in their criteria for the
validity of a proof, they did share an emphasis on the importance of formal proof, and it is
this emphasis, rather than the differences among the schools, that has so greatly influenced
the mathematics curriculum.
The central assertion of logicism is that mathematics is part of logic. Accordingly, the
aim of the logicists was to produce the corpus of mathematics without introducing concepts
indefinable in logical terms or theorems which cannot be proved from the primitive
sentences of a logical calculus using its tightly-defined rules of proof. Thus formal proof
played a central role in the logicist agenda.
This was true of the formalist effort as well. In fact, the thesis of the formalist school was
precisely that mathematics is a science of formal systems: that it deals with the manipulation
of strings of symbols to which no meaning need be assigned. In the formalist view, the
validity of any mathematical proposition rests upon the ability to demonstrate its truth
through rigorous proof within an appropriate formal system.
The intuitionists, too, assigned importance to formal proof. Their differences with the
logicist and formalist schools centred upon the types of proof which should be admitted as
valid, with the intuitionists taking a more restrictive view. Intuitionism is the belief that
mathematics and mathematical language are two separate entities, mathematics being
essentially a languageless activity of the mind. Mathematical activity then consists of
“introspective constructions”, rather than axioms and theorems. But for the intuitionist the
assertion of a mathematical proposition was equivalent to the assertion that there is a
construction of a finite nature which produces the proposition – and such a construction had
to obey rules of rigour.
2. MORE RECENT VIEWS OF MATHEMATICS
In the last two decades several mathematicians and mathematics educators have challenged
the tenet that the most significant aspect of mathematics is reasoning by deduction,
culminating in formal proofs. In their view, there is much more to mathematics than formal
systems. This view recognizes the realities of mathematical practice. Mathematicians
admit that their proofs can have different degrees of formal validity– and still gain the same
degree of acceptance. Mathematicians agree, furthermore, that when a proof is valid by
virtue of its form only, without regard to its content, it is likely to add very little to an
understanding of its subject and ironically may not even be very convincing.
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In these more recent views, a proof is an argument needed to validate a statement, an
argument that may assume several different forms as long as it is convincing. Proof has been
described as a “debating forum” (Davis, 1986), as having “a certain openness and
flexibility” (Tymoczko, 1986), and as possibly depending for its validity on “correct or
reasonable social practice” (Kitcher, 1984).
In examining how proof in mathematics takes into account a social process and hence
goes beyond the concept of formal proof often reflected in mathematics teaching, ideas
advanced by Lakatos (1976), Kitcher (1984), Tymoczko (1986) and Davis (1986) will be
discussed.
Lakatos has expressed the point of view that mathematics is fallible by its very nature.
His account of mathematics is thus at odds with both logicism and formalism, undoubtedly
influenced by their failures. Though mathematics is not an empirical science, Lakatos
shows that its methods are very similar to those of the empirical sciences; he refers to
mathematics as quasi-empirical. Mathematics, in fact, grows through an incessant “improvement of guesses by speculation and criticism, by the logic of proof and refutation”
(Lakatos, 1976). Thus no proof is final, and indeed it is the essentially social process of
negotiation of meaning, rather than the application of formal criteria from the outset, which
leads to the improvement of a proof and its growing acceptance.
According to Kitcher (1984), to understand the development of mathematical knowledge one must focus on the development of mathematical practice: mathematical knowledge owes its growth to rational modifications to this practice. Mathematical practice has
five components:
(1) a language,
(2) a set of accepted statements,
(3) a set of accepted questions,
(4) a set of accepted reasonings,
and
(5) a set of metamathematical views.
The latter component includes standards for proof and definition, as well as claims about
the scope and structure of mathematics (p.163). Thus in his view it is not only the corpus
of mathematical results which develops, but also the very ways in which mathematics is
done.
Further, Kitcher does not accept the a priorist view that mathematical knowledge is
based on proof. He attacks the conception that “proposes to characterize the types that count
as proofs in structural terms” (p. 36). It is furthermore historically incorrect to assume that
change in mathematics has consisted only of the discovery of earlier mistakes and their
replacement by new, correct demonstrations. In his view mathematical knowledge is
always sensitive to peer challenges and is sustained, in part, by community approval of
assumptions and techniques.
Citing examples from the work of Euler, Cauchy, Weierstrass and Newton, Kitcher
concludes that mathematical proof is not always necessary to mathematical knowledge,
MATHEMATICAL PROOF
57
and that it may not even be rational to attempt to accumulate a series of certainties; a demand
for rigour may even be a hindrance to the growth of mathematics, because it impedes
problem solving. Indeed, within the set of accepted reasonings (mentioned above as a
component of mathematical practice), the most interesting are those which occupy an
intermediate position in this set: the unrigorous ones.
Tymoczko is a philosopher of mathematics who thinks that what mathematicians
actually do has a bearing on philosophical questions about mathematical knowledge.
According to him, the concept of the “ideal mathematician”, the totally rational agent who
needs only follow formal deductive procedures to generate eternal and infallible knowledge, is not one which is helpful in the philosophy of mathematics (Tymoczko, 1986).
Tymoczko’s account of mathematical knowledge centres upon the community. It views
not only mathematics teaching, but also the concept of proof and the practice of proving
theorems, as public activitics. Regarding the concept of proof, he agrees with Lakatos that
“proof ideas” are subject to criticism and even invite it. In his view
Mathematical proofs ... generally have a certain openness and flexibility. They can be paraphrased,
restated and filled out invarious ways, and to this extent they transcendany particular formal system.
We might say that an informal proof determines an open-ended class, or family, to use Wittgenstein’s
term, of more specific proofs.
(P. 49)
Tymoczko goes on to say that informal proofs are often convincing and can lead to new
discoveries. They are codified in terms of simple proofs ideas and become the property of
a network of mathematicians.
Davis (1986) states that a proof can play several different roles. It can serve as a
validation, it can lead to new discoveries, it can be a focus for debate, and it can help
eliminate errors. According to him, as to Tymoczko, the traditional philosophies of
logicism, formalism and intuitionism are “private theories” that describe an ideal mathematics. But mathematics, being a social activity, requires a public theory.
In the real world of mathematicians, Davis believes, a proof is never complete and
furthermore cannot be completed. Routine calculations will invariably be omitted. There
will always be an appeal to intuition, to pictures. There will be some metamathematical
objections, but such a part-proof will nevertheless be convincing because it is addressed to
people who share a mathematical subculture in which an incomplete argument is understood, appreciated and seen as adequate. A typical college lecture in advanced mathematics
will include formulations such as “it is easy to show”, “you can verify that”, “by an
elementary computation which I leave to you”, and so forth. It is considered perfectly
proper to transmit mathematics in this elliptical way.
Davis is quite explicit in his view of formal proof:
There is a view of proof or a view of mathematics which I disagree with and which I think is a myth,
which says that mathematics is potentially, totally formalizable, and therefore, one can say, in
advance, what a proof is, how it should work, etc.
(p. 336).
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3. FACTORS IN ACCEPTANCE OF A PROOF
Clearly the acceptance of a theorem by practising mathematicians is a social process which
is more a function of understanding and significance than of rigorous proof. Indeed, the
presence of any proof, rigorous or otherwise, is only one of several determining elements
in acceptance. This process is by no means capricious; the community judges by certain
criteria, as I will discuss. But the significance of a theorem for mathematics as a whole, and
an understanding of its underlying concepts, play a much greater role in creating this
acceptance than does the existence of a rigorous proof.
The development of mathematics and the comments of practising mathematicians
suggest that most mathematicians accept a new theorem when some combination of the
following factors is present:
• They understand the theorem, the concepts embodied in it, its logical antecedents, and its implications. There is nothing to suggest it is not true;
• The theorem is significant enough to have implications in one or more branches
of mathematics (and is thus important and useful enough to warrant detailed
study and analysis);
• The theorem is consistent with the body of accepted mathematical results;
• The author has an unimpeachable reputation as an expert in the subject matter
of the theorem;
• There is a convincing mathematical argument for it (rigorous or otherwise), of
a type they have encountered before.
If there is a rank order of criteria for admissibility, then these five criteria all rank higher
than rigorous proof.
Perhaps the situation is best discussed in terms borrowed from Maslow’s theory of social
motivation (Maslow, 1970). Understanding, significance, compatibility, reputation, and
convincing argument are “positive motivators” to acceptance: it is these factors which
focus the attention of practising mathematicians on a new theorem and move them to its
active acceptance, lifting it above the great body of equally valid but less attractive theorems
which confront them in the mathematical literature.
On the other hand, the structural validity of the mathematical argument for a new
theorem, that is, the actual or potential validity of its form as distinct from its content, is
merely a “hygiene factor”, a factor recognized as essential but taken for granted. There is
a presumption that any convincing proof appearing in a reputable journal is in fact valid in
terms of its form, or could be made so without violence to its content. The publication of
a rigorous proof would provide no additional positive motivation for active acceptance, and
in fact such a proof would not be examined at all in the absence of the motivating factors
enumerated above.
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59
4. THE SOCIAL PROCESS
The following discussion will establish the fact of a social process in acceptance, and the
central role played in that process by the factors of understanding, significance, compatibility, reputation, and convincing argument. The Russian logician Manin is among those who
have stressed the fact that the acceptance of a proof depends much more on a social process
than on some ideal objective criterion:
A proof becomes a proof after the social act of “accepting it as a proof ”. This is true of mathematics
(Manin, 1977, p. 48)
as it is of physics, linguistics, and biology.
Manin then goes on to explain that a new proof needs to be accepted and approved by other
mathematicians – who often decide to refine and improve it. The scrutiny to which
mathematicians subject a proof, he points out, is aimed more at weighing the plausibility
of the results than at verifying the deductive process. It is only when they are skeptical of
a result that mathematicians will put any great effort into discovering counter-examples.
Manin cites this as the reason why the truth of a theorem in the eyes of the mathematical
community becomes established indirectly, that is, not because the proof has been verified
as error-free, but because the results are compatible with other accepted results and the
arguments used in the proof are similar to ones used in other accepted proofs.
Of the estimated 200,000 theorems published yearly (Ulam, 1976), only a very few are
actively accepted by the mathematical community. It is the theorems judged significant that
have their proofs scrutinized, corrected, and refined, while the proofs of other theorems go
unexamined. Clearly an alleged proof of Fermat’s last theorem or the four-color theorem,
when submitted by reputable mathematicians, would attract meticulous review, while the
proof of a theorem of no apparent consequence is likely to be ignored, no matter how
original or sophisticated the proof might be in its own right.
Indeed, as Davis (1972) notes, most proofs in research papers are never checked. Many
of them are rife with errors, in fact. This is borne out by the many mistakes found in those
published proofs which have been checked, and is also supported by the contention of a
former editor of Mathematical Reviews that as many as half of the proofs published are
false, though the theorems they purport to prove are essentially true. When an error is
detected in the proof of a significant theorem, it is often the proof that is changed, of course,
while the theorem itself stands unquestioned.
The role of proof in the process of acceptance is similar to its role in discovery.
Mathematical ideas are discovered through an act of creation in which formal logic is not
directly involved. They are not derived or deduced, but developed by a process in which
their significance for the existing body of mathematics and their potential for future yield
are recognized by informal intuition. While a proof is considered a prerequisite for the
publication of a theorem, it need be neither rigorous nor complete. Indeed the surveyability
of a proof, the holistic conveyance of its ideas in a way that makes them intelligible and
convincing, is of much more importance than its formal adequacy (Hanna, 1983). Since
fully adequate, step-by-step proof is in most cases impracticable, and since surveyability
is lost when proofs become too long, proofs are conventionally elliptical and brief.
The conclusion therefore is that an orientation towards extreme formalism in proof is
not reflective of current mathematical practice or current philosophies of mathematics.
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There are, as has been shown above, good reasons for this. As Tymoczko has put it,
“Mathematicians, even ideal mathematicians, are able to do mathematics and to know
mathematics only by participating in a mathematical community.”
5. CAREFUL REASONING
Despite the secondary nature of proof, it is easy to see how misunderstandings about the
nature of mathematics arise. Mathematical results published for a mathematical audience
are invariably presented in the form of theorems and proofs. They retain this form,
reflectingas it does the nature of mathematics as a highly structured body of knowledge held
together by the concept of logical precedence, even though the proofs are not judged by
criteria of completeness or rigour. To a person only partially trained in mathematics, to
someone who is neither fully equipped to assess significance nor able to make the intuitive
judgments necessary in successfully surveying a proof, it might easily appear that the
manner of presentation – with its possible implication that full rigour is the ideal form – is
the core of mathematical practice. Thus competence in mathematics might readily be
misperceived as synonymous with the ability to create the form, a rigorous proof.
It is only one step further, then, to assume that learning mathematics must involve
training in the ability to create this form. To teach a beginning student is assumed to involve
teaching the formalities of proof. Paradoxically, such an emphasis omits the crucial
element. When a mathematician reads a proof, it is not the deductive scheme that
commands most attention. It is, in fact, the mathematical ideas, whose relationships are
illuminated by the proof in anew way, which appeal for understanding, and it is the intuitive
bridging of the gaps in logic that forms the essential component of that understanding.
When a mathematician evaluates an idea, it is significance that is sought, the purpose of the
idea and its implications, not the formal adequacy of the logic in which it is couched.
It would therefore appear that what needs to be conveyed to students is the importance
of careful reasoning and of building arguments that can be scrutinized and revised. While
these skills may involve a degree of formalization, the emphasis must be clearly placed on
the clarity of the ideas.
6. TEACHING
That reasoning is a pedagogical issue at all bespeaks a conviction that the learning of
mathematics is a dynamic rather than static process, in which students progress towards
deeper level of insight and skill. Thus a teaching activity that includes formal or informal
reasoning can be judged to be of value only to the degree that it promotes greater
understanding.
The starting point for understanding is the naive mathematical idea rooted in everyday
experience. To provide a basis for further progress, this naive idea must be developed and
made explicit. This requires a degree of formalism. A language must be created: symbols
defined, rules of manipulation specified, the scope of mathematical operations delineated.
Greater precision must be taught, so that the essential can be separated from the nonessential and greater generality achieved.
MATHEMATICAL PROOF
61
But this has its price. Distanced from the original intuitive context, the student may lose
sight of reality and become a symbol pusher. Experienced mathematicians have learned to
handle this danger by acquiring the ability to make mental shifts in moving among levels
of generality and formalism, and by building on specific examples, drawing only upon
those characteristics pertinent to the more general situation under study. They are able to
exploit symbolism and algorithms to work automatically and efficiently, while retaining
the ability to intervene in their own work to monitor its accuracy and effectiveness.
What are the issues to be kept in mind in teaching mathematics, then, and in particular
in developing the power of reasoning?
1. Formalism should not be seen as a side issue, but as an important tool for
clarification, validation and understanding. When a need for justification is felt,
and when this need can be met with an appropriate degree of rigour, learning will
be greatly enhanced.
2. It is not enough to provide mathematical experiences. It is the reflection on one’s
experiences which leads to growth. As long as students see mathematics as a
black box for the instantaneous production of “answers”, they will not develop
the patience necessary to cope with the many and erratic paths their minds will
take in trying to grasp what mathematics is about. One goal of pedagogy should
be to help pupils maintain the level of concentration required to negotiate a line
of reasoning.
3. Ironically for a discipline touted as precise, the student of mathematics has to
develop a tolerance for ambiguity. Pedantry can be the enemy of insight.
Sometimes an explanation is better given pictorially, loosely, by example or by
analogy. Sometimes distinctions are better left blurred (e.g., the various roles of
the minus sign and the use of “f(x)” as both the function and the value of the
function at x). Sometimes the role of a symbol in the discussion should be
allowed to vary (e.g., the parameter which is sometimes held constant, sometimes allowed to vary).
4. At the same time, when there is a danger that genuine confusion might develop,
the student must learn to become conscious of looseness and to apply the
necessary amount of rigour. It is this judgemental aspect of reasoning, so
essential in mathematics education, that must be communicated to students.
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II : COGNITIVE THEORY
OF
ADVANCED MATHEMATICAL THINKING
In this part of the book we begin to develop theories of cognitive development of
particular value in advanced mathematical thinking. In Chapter One we singled
out the importance of abstract definition and deduction at this level. In Chapter
Two we saw how the processes of representation and abstraction play a crucial
role. In Chapter Five Shlomo Vinner considers the diffrerences between the
abstract definition of a concept as given in a mathematical theory and the concept
image as conceived in the mind of an individual. The research in the last decade
clearly shows a wide gulf between desirable theory and actual practice. In
Chapter Six, Guershon Harel and James Kaput consider the ways in which
mathematical processes are encapsulated as conceptual entities and symbolized
by notations in ways which may be more or less appropriate in different contexts.
They too see that the formal definitions often play only a subsidiary role in
mathematical thinking and continue the discussion of Dreyfus from Chapter Two
by moving from mathematical processes to mental objects that can be manipulated. They reflect on the use of symbols in this thinking process and the manner
in which the representation may be appropriately elaborated to enhance its
meaning.
The encapsulation of a process as a mental object is subjected to deep analysis
in Chapter Seven. Here Ed Dubinsky takes an in-depth look at the process of
reflective abstraction, as originally conceived by Piaget for younger children,
and extends Piaget’s theories to advanced mathematics. He has a different
emphasis from other authors in that he sees encapsulation of processes as objects
as the main driving force in mathematical thinking and does not accept the
prominent role given to visualization proposed in several other chapters. This
difference exemplifies the divergence between two different kinds of mind cited
from the observations of Poincaré in Chapter One. It is a fitting point to end the
first half of our book.
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CHAPTER 5
THE ROLE OF DEFINITIONS
IN THE TEACHING AND LEARNING OF MATHEMATICS
SHLOMO VINNER
1. DEFINITIONS IN MATHEMATICS
AND COMMON ASSUMPTIONS ABOUT PEDAGOGY
Definition creates a serious problem in mathematics learning. It represents, perhaps, more
than anything else the conflict between the structure of mathematics, as conceived by
professional mathematicians, and the cognitive processes of concept acquisition. It seems
that no-one in the mathematical community disagrees with the claim that mathematics is
a deductive theory and as such, it starts with primary notions and axioms. By means of the
primary notions all other notions are defined. All the theorems, which are not axioms, are
proved from the axioms by means of certain rules of inference. This might be a too short
and oversimplified description, but essentially, it represents the common view of mathematicians about mathematics. It does not necessarily reflect the process by means of which
mathematics is created, but it tends to be the way mathematics is presented in higher
mathematics text books and mathematical periodicals. Of course, it is not possible to start
with primary notions and axioms in every situation. Typically, one starts with well known
notions and well known theorems and proceeds by defining new notions and by proving
new theorems. This might have certain consequences for the way mathematics is taught,
even before one starts to think about the appropriate pedagogy. Thus, mathematics teachers
might form in their classes a sequence of definitions, theorems and proofs as a skeleton for
their course. Following these consequences may be pedagogically wrong since the teaching
should take into account the common psychological processes of concept acquisition and
logical reasoning.
Let us describe some of the possible consequences which can be derived from
considering the role of definition in mathematics. We claim that the presentation and the
organization of mathematics in many text books and classrooms are partly based on the
following assumptions:
1. Concepts are mainly acquired by means of their definitions.
2. Students will use definitions to solve problems and prove theorems when
necessary from a mathematical point of view.
3. Definitions should be minimal. (By this we mean that definitions should not
contain parts which can be mathematically inferred from other parts of the
definitions. For instance, if one decides to define a rectangle in Euclidean
geometry by means of its angles it is preferable to define it as a quadrilateral with
3 right angles and not as a quadrilateral having 4 right angles. This is because
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in Euclidean geometry, if a quadrilateral has 3 right angles one can prove that
its fourth angle is also a right angle.)
4. It is desirable that definitions will be elegant. For instance, some mathematicians
think that the definition of the absolute value as |x|= (x 2) is more elegant than
its definition as
Also, some mathematicians believe that the definition of a prime number (in the
domain of whole numbers) as a number having exactly two different divisors
is more elegant than its definition as a number greater than 1 divisible only by
1 and itself.
5. Definitions are arbitrary. Definitions are “man made”. Defining in mathematics
is giving a name. (For instance, when defining a trapezoid, one can define it as
a quadrilateral having at least one pair of opposite sides which are parallel. On
the other hand, he or she can define it, if they wish, as a quadrilateral having
exactly one pair of opposite sides which are parallel. If you choose the first
definition, a parallelogram is also a trapezoid. If you choose the second one, it
is not. Now, if the idea that definitions are arbitrary is well understood the above
fact will not cause a confusion, otherwise it might cause a great deal.)
The above five assumptions do not necessarily reflect all the aspects of definitions in higher
mathematics. As claimed above, these assumptions are reflected very often in the pedagogy
of teaching mathematics. A quick look at the majority of high school and college text books,
and these demonstrate some concern to pedagogy, will show that definitions have major
role in the presentation of course materials. Take for instance, the notion of absolute value
of a number. Its best characterization is that it is the number without its sign or signs, This
is quite clear to the students and this is what most of them tell you when you ask them about
the absolute value. You can hardly find a text book which mentions it. Another possibility
to characterize the absolute value of a number is to say that it is the distance of the number
from zero on the number line. This is also quite clear to the students but perhaps less clear
than the former characterization. You can find some text books and teachers who use it, but
still the majority of teachers and text books will avoid it. So the majority of teachers and
textbooks will use one of the definitions mentioned above. However, some teachers know
that these definitions are quite unclear and confusing for most of the students. By
advocating that it is possible not to use these formal definitions we do not ignore the need,
at a later stage, to know that
THE ROLE OF DEFINITIONS IN TEACHING AND LEARNING
67
The student should use it when solving algebraic equations and inequalities with absolute
value. However, the above formula can be given and explained to the student at a later stage
as a claim about the absolute value and not as its formal definition.
The point that we would like to make by discussing the example of the absolute value
is the following: when coming to decide about the pedagogy of teaching mathematics one
has to take into account not only the question how students are expected to acquire the
mathematical concepts but also, and perhaps mainly, how students really acquire these
concepts.
2. THE COGNITIVE SITUATION
“Against definition” is a title of a paper by Fodor et al (1980). The paper discusses the way
that “the notion of definition has served to connect several aspects of classical theory of
language with one another and with widely credited accounts of concept acquisition”. The
authors argue with some of the widely accepted views in cognitive psychology, especially
with the following three:
1. “The definition of a word determines its extension” (p. 266).
2. “To understand a word is to recover its definition” (p. 274).
3. “Definitions express the decomposition of concepts into their elements” (p. 276).
Fodor et al claim that these views have no psychological ground. They bring some
experimental evidence which disconfirms these views. Especially, according to Fodor et
al, the following claim is refuted: “Understanding a sentence token involves recovering (i.e.
displaying in working memory) the definition of such lexical items as the sentence
contains”. Thus, when understanding a sentence token, or when trying to understand it,
people usually do not consult the definitions of the terms which occur in the sentence. Fodor
et al deal with sentences taken from everyday life contexts. A careful examination of their
claims, even without considering their experimental evidence, might lead to the conclusion
that these claims are not only extremely reasonable but that they are even trivial. This is
mainly because many words in everyday language do not have definitions (although they
are “defined” somehow in dictionaries). Think of “car”, “table”, “house”, “green”, “nice”,
etc., and you realize immediately that when understanding, for instance, the sentence “my
nice green car is parked in front of my house” you do not consult definitions. There is still
the question what you do consult when understanding this sentence and we are not sure
whether Fodor et al, give a clear answer to this question. With this particular sentence you
will not consult definitions because there are no definitions for the words involved. On the
other hand, contrary to everyday life contexts, there are the “technical contexts”. In these
contexts meaning is assigned to a term by a stipulation. Terms are defined as in
mathematics. Hence, if you are in a “technical context” you should consult definitions,
otherwise mistakes might occur. Of course, there is no need to consult definitions (which
do not exist) when trying to understand the sentence “among all the cars at the parking lot
my green car is the nicest”. However, it is necessary to consult definitions when trying to
understand the sentence: “among all rectangles with the same perimeter the square is the
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one which has the maximal area”. Note that in everyday life contexts, a square is not
considered as a rectangle by most of the people, whereas in all mathematical contexts a
square is a rectangle.
When Fodor et al spoke “against definitions” they meant it in non-technical contexts.
They wanted to refute a certain linguistic theory about the role of definitions in nontechnical thought processes. However, in technical contexts, contrary to non-technical
ones, the question is not how people behave but how they should behave. In technical
contexts people are supposed to consult definitions of the technical terms involved. On the
other hand, knowing the enormous impact that everyday life has on any situation, it will be
reasonable to predict that definitions will be ignored by many people also in technical
contexts. This really happens as we will show in the following. So, what do people consult
when dealing with technical terms in technical situations? We will try to answer this
question in the next section
3. CONCEPT IMAGE
A concept name when seen or when heard is a stimulus to our memory. Something is
evoked by the concept name in our memory. Usually, it is not the concept definition, even
in the case the concept does have a definition. It is what we call “concept image” (Tall &
Vinner, 1981; Vinner, 1983) and others (Davis, 1984) call it “concept frame”.
The concept image is something non-verbal associated in our mind with the concept
name. It can be a visual representation of the concept in case the concept has visual
representations; it also can be a collection of impressions or experiences. The visual
representations, the mental pictures, the impressions and the experiences associated with
the concept name can be translated into verbal forms. But it is important to remember that
these verbal forms were not the first thing evoked in our memory. They came into being
only at a later stage. For instance, when hearing the word “table”, a picture of a certain table
can be evoked in your mind. Experiences of sitting at a table, eating at a table, etc., can be
evoked as well. You can recall that many tables are made of wood, most of them have four
legs; usually you do not lie on a table, you can sit on a table but this can be regarded by some
people as an impolite behavior. When you hear the word “function”, on the other hand, you
might recall the expression “y = f(x )”, you might visualize a graph of a function, you might
think of specific functions like y = x2 or y = sinx, y = lnx, etc. From what we have said, it
is clear that it is only possible to speak of a concept image in relation to a specific individual.
In addition, the same individual might react differently to a certain term (concept name) in
different situations. In Tall & Vinner (1981) the term “evoked concept image” is introduced
to describe the part of the memory evoked in a given context. This is not necessarily all that
a certain individual knows about a certain notion. In general, although we may not always
use the term “evoked concept image” in what follows, the reader should always keep this
in mind.
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4. CONCEPT FORMATION
We assume that to acquire a concept means to form a concept image for it. To know by heart
a concept definition does not guarantee understanding of the concept. To understand, so we
believe, means to have a concept image. Certain meaning should be associated with the
words. To know, for instance, that the power set of a given set is the set of all subsets of that
given set, does not mean anything unless one can construct some power sets of given sets.
Hence, the image of the power set concept might include some memories of the
construction of some power sets.
Most concepts in everyday life, like house, orange, cat, etc., are acquired without any
involvement of definitions. On the other hand, some concepts, even everyday life concepts,
might be introduced by definitions. The word “forest” might be introduced to a child by
saying “many, many trees together” (the Merriam Webster dictionary definition “a large
thick growth of trees and underbrush” is, of course, a useless definition for a little child).
Definitions like this help to form a concept image. But the moment the image is formed,
the definition becomes dispensable. It will remain inactive or even be forgotten when
handling statements about the concept in consideration. Thus, the “scaffolding metaphor”
can be suggested for the role of definition, in concept formation: the moment a construction
of a building is finished, the scaffolding is taken away.
5. TECHNICAL CONTEXTS
In technical contexts, definitions might have extremely important roles. Not only that they
help forming the concept image but they very often have a crucial role in cognitive tasks.
They have the potential of saving you from many traps which are set by the concept image.
For instance, if you are asked to find a maximal value of a function in a closed interval and
you recall a graph that corresponds to a local maximum and you try to differentiate the given
function and to find the zeros of the derivative, then the explicit definition of a maximal
value in a closed interval might help you to consider other possibilities different from local
maximums. Sometimes, this can prevent mistakes. Not consulting the definition in the
above case, might cause a fixation on the differentiating technique associated with the
maximal value concept in the mind of many students. The differentiating technique leads
to the desirable results in many cases but not in all.
Thus, technical contexts impose on students some thought habits which are totally
different from those typical to everyday life contexts. One can predict that, at least in the
beginning of the learning process, the thought habits of everyday life will take over the
thought habits imposed by the technical contexts.
6. CONCEPT IMAGE AND CONCEPT DEFINITION –
DESIRABLE THEORY AND PRACTICE
In order to present our ideas by means of diagrams (as in Vinner, 1983), assume the
existence of two different “cells” in our cognitive structure (to avoid confusion, we do not
mean biological cells). One cell is for the definition(s) of the concept and the second one
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is for the concept image. One cell or even both of them might be void. (The concept image
cell is considered to be empty as long as some meaning is not associated with the concept
name. This can happen in many situations where the concept definition is memorized in a
meaningless way.) There might be some interaction between the two cells although they
can be formed independently. A student might have a concept image of the notion of
coordinate systems as a result of seeing many graphs in various situations. According to this
concept image, the two axes of a coordinate system are perpendicular to each other. Later
on, the student’s mathematics teacher might define a coordinate system as any two
intersecting straight lines. As a result of this, three scenarios might occur:
(I) The concept image may be changed to include also coordinate systems whose
axes do not form a right angle. (This is satisfactory reconstruction or accommodation.)
(II) The concept image may remain as it is. The definition cell will contain the
teacher’s definition for a while but this definition will be forgotten or distorted
after a short time, and when the student will be asked to define a coordinate
system he or she will talk about axes forming a right angle. (In this case the
formal definition has not been assimilated.)
(III) Both cells will remain as they are. The moment the student is asked to define
a coordinate system he will repeat his or her teacher’s definition, but in all other
situations he or she will think of coordinate system as a configuration of two
perpendicular axes.
A similar process might occur when a concept is first introduced by means of a definition.
Here, the concept image cell is empty in the beginning. After several examples and
explanations, it is gradually filled. However, it does not necessarily reflect all the aspects
of the concept definition. Similar scenarios to (I)–(III) above might occur. This is shown
in Figure 2.
Figure 2 : Interplay between concept image and concept definition
Another illustration of (II) above is the following:
There are many students who are ready to swear that the definition of a limit of a
sequence is a number to which the elements of a given sequence get closer and closer but
never reach it. Thus the sequence whose nth element is given by an =(–1)2n does not have
a limit (see also §7).
A further illustration of (II) above is the following:
Some students, after studying the modem concept of function, will say that a function
is any correspondence between two sets which assigns to every element of the first set
THE ROLE OF DEFINITIONS IN TEACHING AND LEARNING
71
exactly one element of the second set. On the other hand they will not admit that the
correspondence which assigns to every non-zero number its square and which assigns –1
to zero is a function (see also §7).
Figure 2 refers to the long term processes of concept formation. It seems to us that many
teachers at the secondary and the collegiate levels expect a one way process for the concept
formation as shown in figure 3, namely, they expect that the concept image will be formed
by means of the concept definition and will be completely controlled by it.
Figure 3 :The cognitive growth of a formal concept
In addition to the process of the concept formation there are also the processes of
problem solving or task performance. When a cognitive task is posed to a student the
concept image and the concept definition cells are supposed to be activated. Again, it seems
to us that many teachers at the secondary and the collegiate level expect that the intellectual
processes involved with the performance of a given intellectual task should be schematically expressed by one of the three following figures (the figures represent only the aspect
of concept image and concept definition involved in the process). The arrows in the figures
represent different ways in which a cognitive system might function.
Figure 4 : Interplay between definition and image
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Figure 5 : Purely formal deduction
Figure 6 : Deduction following intuitive thought
The common feature of all the processes illustrated in Figures 4–6 is the following: no
matter how your association system reacts when a problem is posed to you in a technical
context, you are not supposed to formulate your solution before consulting the concept
definition. This is, of course, the desirable process. Unfortunately, the practice is different.
It is hard to train a cognitive system to act against its nature and to force it to consult
definitions either when forming a concept image or when working on a cognitive task.
Hence, a more appropriate model, for the processes which really occur in practice, is the
following:
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73
Figure 7 : Intuitive response
Here, the concept definition cell, even if non-void, is not consulted during the problem
solving process. The everyday life thought habits take over and the respondent is unaware
of the need to consult the formal definition. Needless to say, that in most of the cases, the
reference to the concept image cell will be quite successful. This fact does not encourage
people to refer to the concept definition cell. Only non-routine problems, in which
incomplete concept images might be misleading, can encourage people to refer to the
concept image. Such problems are rare and when given to students considered as unfair.
Thus, there is no apparent force which can change the common thought habits which are,
in principle, inappropriate for technical contexts.
Before closing this section we would like to remind the reader about the “evoked concept
image” mentioned earlier in §3. In a specific cognitive task we deal only with one’s evoked
concept image. We do not claim that under different circumstances the same image will be
evoked again. Thus, in our discussion, we do not evaluate somebody’s cognitive system.
Our analysis relates only to the part of the cognitive system which was activated when
working on a given cognitive task.
7. THREE ILLUSTRATIONS OF COMMON CONCEPT IMAGES
In this section we will bring some experimental evidence to support our claim that the
majority of the students do not use definitions when working on cognitive tasks in technical
contexts. To be more specific, our claim is that the common high school and college courses
do not develop in the science students, not majoring in mathematics, the thought habits
needed for technical contexts. The students continue to use everyday life thought habits also
in technical contexts. (Luckily enough for the students, this does not prevent them from
passing exams.)
The concepts that we are going to discuss are the concept of function, the concept of
tangent and the concept of limit of a sequence. Since a more detailed report about these can
be found elsewhere (Davis & Vinner, 1986; Tall & Vinner, 1981; Vinner 1982, 1983) we
will present here only the main aspects of the findings and the method of getting them. A
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natural method to learn about somebody’s concept definition is by a direct question (what
is a function? what is a tangent? and so on). This is because definitions are verbal and
explicit. On the other hand, in order to learn about somebody’s concept image usually
indirect questions should be posed, as the concept image might be non-verbal and implicit.
Thus the main task of the researcher is to invent questions that have the potential to expose
the respondent’s concept image. We will bring some of them. The following questionnaire
was given to 147 students who studied mathematics at a high level in grades 10 and 11. In
the first three questions the students were asked to choose between “yes” or “no” and to
explain their answers.
1. Is there a function in which each number different from 0 corresponds to its
square and 0 corresponds –l?
2. Is there a function in which each positive number corresponds 1, each negative
number corresponds to –1, and 0 corresponds to 0?
3. Is there a function the graph of which is the following?
Figure 8 : Does this graph arise from a function?
4. In your opinion what is a function?
The concept of function was taught to all the students according to the modem approach,
namely, a function is a correspondence between two sets which assigns to each element in
the first set exactly one element in the second set. In spite of that, only 57% of the students
gave this definition or something which is partly equivalent to it as an answer to question
4. (Note that we are dealing with good students. Thus, the figure 57% which can be
considered as a great achievement in other circumstances is not so in this situation.) 14%
of the students said that a function is a rule of correspondence and eliminated the possibility
of an arbitrary correspondence. Rules cannot be arbitrary. They have to have a logical or
mathematical ground. An additional 14% claimed that a function is an algebraic term, a
formula, an equation or an arithmetical manipulation. The rest gave no answer or no
satisfactory answer. When it came to concept images it turned out that at certain situations
THE ROLE OF DEFINITIONS IN TEACHING AND LEARNING
75
(questions 1 and 2) between one third to two thirds of the students think that a function
should be given by one rule or, if two rules were given their domains should be half lines
or intervals. A rule for a single point (like in question 1) is not permitted. Some students
believe that correspondences which are not given by an algebraic rule are not functions
unless the mathematical community has declared them as functions by giving them a name
or a special notation. (This was reflected in answers to question 2). Other students (about
2/5) believe that a graph of a function should be regular, persistent, reasonably increasing
etc. (This was reflected in answers to question 3.) Thus, many students who defined
“function” correctly were not using their definition when replying to questions 1–3. In fact,
only one third of the students who gave the correct definition of function also answered
questions 1–3 correctly. No student with an incorrect definition answered questions 1–3
correctly.
Consider now the concept of tangent. It is usually introduced to mathematics students
at a geometry course in the context of the circle. The definition of a tangent to a circle is an
easy one and its visual representations is:
Figure 9 : A tangent to a circle
This picture can serve as a means to construct an image for the tangent concept in
additional cases like:
Figure 10 : A mental image for a tangent
Students who take a calculus course usually get a formal or a semi-formal definition of
the tangent to a graph of a differentiable function. However, their concept image, built up
from experiences involving pictures like figures 9 and 10 may contain coercive elements
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which insist that a tangent may only meet the curve at one point and may not cross the curve
at that point. As we shall see, such a concept image may lead the students to respond by
drawing a line that is not a tangent at the required point, yet has the generic properties of
the concept image. As in chapter 1, Tall (1987) termed such a concept a generic tangent.
The following questionnaire was given to 278 first year college students in calculus
courses designed for science students (not majoring in mathematics).
Here are three curves. On each one of them a point P is denoted. Next to each
one of them there are three statements. Circle the statement which seems true
to you and follow the instruction in the parentheses.
A. Through P it is possible to draw exactly one tangent to the curve (draw it).
B. Through P it is possible to draw more than one tangent (specify how many, one,
two, three, infinitely many. Draw all of them in case their number is finite and
some of them in case it is infinite).
C. It is impossible to draw through P a tangent to the curve.
Figure 11: Which graphs have tangent(s) at P ?
4. What is the definition of the tangent as you remember it from this course or from
previous courses. If you do not remember the definition of the tangent try to
define it yourself.
and
The curves in 1,2,3 are the graphs of y = x3, y =
but this was not given to the students. The tangent was defined in the above courses either
as a limit of secants or as a line having a common point with the function graph whose slope
is the derivative at this particular point. However, only 41 % of the students gave one of the
course definitions as an answer to question 4.35% gave descriptions that suit the case of
the tangent to a circle. They claim that a tangent touches the curve but does not intersect
it, or that it meets the curve but does not cut it or that it has a common point with the curve
but it is on one side of the curve. The rest gave no definition or meaningless definitions. The
students concept images were expressed in the answers to questions 1, 2, 3 and are given
in the following tables.
THE ROLE OF DEFINITIONS IN TEACHING AND LEARNING
Table I : Distribution of student drawings to question 1 ( N=278)
Table II : Distribution of student drawings to question 2 (N=278)
Table III : Distribution of student drawings to question 3 (N=278)
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Some of the drawings are especially interesting. For instance, in 1B, 2B and 3B the
students try to force the graph in order to meet the image formed by the tangent to the circle.
1B and 3B seem to be classic ‘generic tangents’ generated by their concept image, 2D is
a generalization in which the ‘tangent’ is balanced on the cusp. In 1C, 2D (the bottom
drawing) and 3C there is another phenomenon. It may be that the old concept image (a
tangent to a circle) and the new concept image (constructed by the course definition) act
at the student’s mind simultaneously. It is a well known phenomenon in science learning,
where, very often, old schemas are found together with new schemas in students’ thinking.
Tall (1987) found some students responding with a dynamic image of a tangent, for
example, intimating that the picture in figure 3B is such that the tangent “begins to turn”
at the point in question, and so the tangent is drawn at a tilt to represent this tendency, even
though the student concerned might sense that the turning does not actually begin until after
the point concerned.
In 2C and 3D the students even invent the case of infinitely many tangents, on one hand,
in order to meet the old image formed by the circle and on the other hand, realizing that there
is no reason to prefer one “tangent”, drawn according to the old image, to other, infinitely
many, “tangents”. Contrary to these students, there are the students in 2D (the top drawing)
and 3B who, perhaps, prefer a kind of symmetry and thus stay with only one tangent, or
perhaps, take as a starting point that there should be only one tangent and therefore conclude
that it should be the one that has symmetry.
Finally, we will say few words about the notion of a limit of a sequence. Although our
findings here come from a very small sample (N=15), they are more than typical because
of the following reasons:
(1) The respondents are mathematically gifted students at a university high school.
(2) An “appropriate pedagogy” was used to teach them the notion of limit (with the
teacher being aware of the necessity of bringing typical and non-typical
examples of sequences which tend or do not tend to a limit. This is, of course,
in addition to the formal definition. For more details see Davis & Vinner, 1986).
The concept was taught to the students at the end of their eleventh grade. Immediately after
the summer vacation, on the first day of class, the following was given to the students by
their teacher as a written test:
I need to know how much you remember about the concept of a limit of a
sequence. Please, write a few paragraphs to show me what you remember. I
suggested you may want to include:
1) A description of a sequence in intuitive or informal terms.
2) A precise formal definition.
Out of the 15 students only one gave an answer that can be considered as indication of
reasonably deep understanding of the concept. This was:
The limit of a sequence is the number from which all the terms in the sequence,
after a certain point, vary only by a little number ε.
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(This answer misses the most important element of the formal definition, namely, a
statement that the above is true for every ε>0. Thus, this answer is treated quite literally.
If only tough measures would have been taken then the result would be that not a single
student showed a deep understanding of the formal definition. The ability to construct a
formal definition is for us a possible indication of deep understanding. Of course, it is not
sufficient, since the reconstruction of a formal definition can be obtained by rote
memorization).
In the other 14 students some typical misconceptions were found which influenced the
formal definitions that the students were asked to give. In our terminology, the concept
definition was reconstructed by referring to the concept image. Since the concept image
was incorrect this resulted in an incorrect formal definition. The main misconceptions were:
(1) A sequence “must not reach its limit” (thus, the sequence: 1, 1,1,... would be said
not to converge to a limit),
(2) The sequence should be either monotonically increasing or monotonically
decreasing (thus, for instance the sequence whose nth element is given by
an = 1+(–1 )n/n does not tend to a limit),
(3) The limit is the ‘‘last’’ term of the sequence. You arrived to the limit after “going
through” infinitely many elements.
In the three central concepts discussed above there is a conflict between the formal
definition and the concept typical examples which might cause an incorrect concept image.
The findings show that, in spite of the emphasis which was given to the definition of the
concepts, many students did not use them when working on tasks in which formal
definitions should have been used. This can lead to two opposite conclusions:
(1) Giving up the goal of changing the students’ thought habits from the everyday
mode to the technical mode.
(2) Trying to change the students’ thought habits by an appropriate treatment
(perhaps as an independent topic which might lead to more awareness. The
integration of this topic in the common courses does not attract enough attention
that can lead to the desired results). More about this dilemma in the next section.
8. SOME IMPLICATIONS FOR TEACHING
We would like to recommend here two didactical rules which are relevant to the problem
raised in this paper.
(1) To avoid unnecessary cognitive conflicts with students,
(2) To initiate cognitive conflicts with students when these conflicts are necessary
in order to enhance the students to a higher intellectual stage. (This should be
done only when the chance of reaching a higher intellectual stage is reasonably
high.)
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We claim earlier that one of the gods of teaching mathematics should be changing the
thought habits from the everyday life mode to the technical mode. This cannot be done in
a short period and cannot be successful with everybody. Our belief is that mathematical
concepts, if their nature allows it, should be acquired in the everyday life mode of concept
formation and not in the technical mode. One should start with various examples and nonexamples by means of which the concept image will be formed.
This does not mean that the formal definition should not be introduced to the student.
However, the teacher or the text book writer should be aware of the effect that such
introduction can have on the student’s thinking. (If the concept is not too complicated the
teacher can even ask the students to suggest their definition for the concept.) If our students
are candidates for advanced mathematics then, no doubt, they should be trained to use the
definition as an ultimate criterion in various mathematical tasks. But in order to achieve this
goal, one should do more than introducing the definition. One should point at the conflicts
between the concept image and the formal definition and deeply discuss the weird examples
(like the tangent to the graph of y = x 3 at (0,0) or the limit of the sequence whose nth element
is (–1)2 n, n=1,2,3 ..., etc.). If, on the other hand, our students are not candidates for advanced
mathematics, then it is better to avoid the conflicts. There is no harm if the students
memorize the formal definition and repeat it in various occasions. The teacher and the text
book writer, on the other hand, may even feel that they have completed their task by
introducing the formal definition. But they should have no illusions about the cognitive
power that this definition has on the student’s mathematical thinking.
Thus the role of definition in a given mathematics course should be determined
according to the desired educational goals supposed to be achieved with the given students.
If the students are candidates for advanced mathematics then, not only that definitions
should be given and discussed, the students should be trained to use them as an ultimate
criterion in mathematical tasks. This goal can be achieved only if the students are given
tasks that cannot be solved correctly by referring only to the concept image. As long as
referring to the concept image will result in a correct solution, the student will keep referring
to the concept image since this strategy is simple and natural. Only a failure may convince
the student that he or she has to use the concept definition as an ultimate criterion for
behavior. Thus we do believe that changing students’ thought habits from the every day
mode to the technical mode is an important goal for teaching mathematics. Contrary to
Fodor et al (1980) we campaign for definition and not against definition but we claim that
this aspect of definition cannot be achieved with all students. There might be various
opinions about the percentage of students who are capable of this aspect and there is also
the practical question how to decide whether a certain student can change his thought habits
from the everyday life mode to the technical mode. We do not have answers to these
questions yet. Therefore, the decisions about the goals of teaching definitions should be left
to the intelligent and sensitive mathematics teacher.
The role of definition in mathematical thinking is somehow neglected in official
contexts (text books, documents about goals of teaching mathematics, etc.). We are not sure
whether this is because it is taken for granted or because it is overlooked. It is obligatory
to remember that there are some contexts in which referring to the formal definition is
critical for a correct performance on given tasks (among them there are the identification
of examples and non-examples of a given concept, problem solving and mathematical
proofs). On the other hand we want to be realistic about the chance of achieving the above
THE ROLE OF DEFINITIONS IN TEACHING AND LEARNING
81
goals. We do not believe in “mathematics for all”. We do believe in some mathematics for
some students. And even this can be achieved only by appropriate pedagogy under
appropriate conditions for learning.
CHAPTER 6
THE ROLE OF CONCEPTUAL ENTITIES
AND THEIR SYMBOLS
IN BUILDING ADVANCED MATHEMATICAL CONCEPTS
GUERSHON HAREL & JAMES KAPUT
Mathematical thinking is carried out using mental objects. For example, suppose one asks
if a vector space Vand its double dual V** are isomorphic. At one level, one is asking about
the “objects” Vand V** and, to begin describing an isomorphism, one may go on to describe
a correspondence between respective vectors in the two spaces, which again, are treated
mentally as objects, although they might be n -tuples or matrices, for example. Similarly,
one may need to define a mapping between two function spaces, where the elements of the
domain and range of the mapping must be treated cognitively as objects, as opposed to the
mapping itself, which may be treated as a process, with inputs and outputs. In yet another
instance, one may need to reinterpret a universal construction in the sense of MacLane
(1971) as an adjoint functor pair, where the existence of a unique mapping with a certain
property in fact defines a natural transformation between functors – so the mapping must
play the role of an object on which the natural transformation acts. Such experiences are
quite common in mathematics at all levels, but they feature widely throughout advanced
mathematical thinking. The aim of this chapter is to begin to discuss them and their roles
in helping us to build ever more complex mathematical concepts.
The idea of conceptual entities formation was suggested by Piaget (1977) in his
distinction between form and content. Recently, several researchers have recognized its
value in the learning of mathematics. It has been called encapsulation (Ayers, Davis,
Dubinsky & Lewin, 1988), reification (Sfard, 1989), integration operation (Steffe &
Cobb, 1988), for example, this process is an instance of reflective abstraction (Beth &
Piaget, 1966), in which “a physical or mental action is reconstructed and reorganized on a
higher plane of thought and so comes to be understood be the knower” (p. 247). Greeno
(1983) defines a conceptual entity as a cognitive object for which the mental system has
procedures that can take that object as an argument, as an input. He distinguishes cognitive
objects from attributes, operations and relations, which attach to or act on objects. Further,
he suggests that to qualify as objects, they must be permanently available in the individual’s
mental representation (p. 277).
The construction of function as a conceptual entity is an example of the entitication
process (Thompson, 1985a; Harel, 1985; Ayers et al, 1988). One level of understanding
the concept of function is tothink of a function as a process associating elements in a domain
with elements in a range. This level of understanding may be sufficient to deal with certain
situations, such as interpreting graphs of functions point-wise or solving for x in an equation
of the form f(x)=b, but it would not be sufficient to deal meaningfully with situations which
involve certain operators on functions, such as the integral and differential operators, as we
will see later in this chapter. For the latter situations, the three components of function – the
rule, the domain, and the range – must be encapsulated into a single conceptual entity so
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83
that these operators can be considered as procedures that take functions as arguments.
Incidentally, a formal definition of a function as a single set of ordered pairs, a mathematical
entity, does not appear to play a role in these situations – when would one conceive of a
function as a set of ordered pairs in the context of applying a differential operator to that
function? In this way the concept image evoked in a given context may be different from
the formal definition, and may even at times be in conflict with that definition, as discussed
in the previous chapter.
The construction of conceptual entities embodies the “vertical” growth of mathematical
knowledge (in the sense of Kaput, 1987). For example, at lower levels, the act of counting
leads to (whole) numbers as objects, taking part-of leads to fraction numbers, functions as
rules for transforming objects become themselves objects that can then be further operated
upon, for instance they may be differentiated or integrated. This complements the kind of
“horizontal” growth associated with the translation of mathematical ideas across representation systems and between non-mathematical situations and their mathematical models.
In the next section of this chapter we lay out some of the circumstances under which
conceptual entities are created and used and what their cognitive function might be, often
by pointing to consequences in students’ reasoning processes where they have not yet been
mentally constructed. In the following part we will shift attention to the complex roles of
notation systems in building and using conceptual entities. We regard this chapter as a foray
into relatively unexplored territory, and do not make claims of completeness or of empirical
substantiation for the framework being suggested.
1. THREE ROLES OF CONCEPTUAL ENTITIES
We will discuss the concepts of function, operator, vector space, and limit in terms of the
role that conceptual entities have for:
1. Alleviating working memory or processing load when concepts involve multiple constituent elements.
2. Facilitating comprehension of complex concepts: the cases of “uniform”
operators, “point-wise” operators, and “object-valued” operators.
3. Assisting with the focus of attention on appropriate structure in problem solving.
Greeno (1983) suggested a number of functions of representational knowledge involving
conceptual entities: forming analogies between domains, reasoning with general methods,
providing computational efficiency, and facilitating planning. He offered empirical
findings that are consistent with his suggestions; these findings deal with elementary
mathematics – geometry proofs and multi-digit subtraction – as well as physics, puzzle
problems, and binomial probability. He also suggests that instructional activities with
concrete manipulatives can lead to an acquisition of representational knowledge that
includes conceptual entities. Other researchers suggest different types of instructional
activities for the construction of conceptual entities. For example, Ayers et al, (1988)
demonstrate how computer activities in learning mathematical induction and composition
of functions can facilitate the construction of these concepts as entities (see the next chapter).
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1.1 WORKING-MEMORY LOAD
One psychological justification for forming conceptual entities lies in their role in
consolidating or chunking knowledge to compensate for the mind’s limited processing
capacity, especially with respect to working memory. To avoid loss of information during
working memory processes, large units of information must be chunked into single units,
or conceptual entities. Thus, thinking of a function as a process would require more
working-memory space than if it is encoded as a single object. As a result, complex
concepts that involve two or more functions would be more difficult to retrieve, process,
or store if the concept of function is viewed as a process. This is true for many concepts in
advanced mathematics. Imagine, for example, the working-memory strain in dealing with
the concept of the double dual space of a space of nx n matrices if none or only a few of the
concepts, matrix, vector space, functional, and field are conceived as consolidated entities.
1.2a COMPREHENSION: THE CASE OF “UNIFORM”
AND “POINT-WISE” OPERATORS
Despite the heavy working-memory load involved in understanding the dual space of an
nxn matrix space without most of its subconcepts being entities, it is still possible to make
sense out of it, at least momentarily. In some situations, however, the justification for the
formation of conceptual entities is more than just a matter of cognitive strain that results
from a memory load. In such situations comprehension requires that certain concepts act
mentally as objects due to an intrinsic characteristic of the construct involved. Examples
of such situations include those which involve the integral or differential operators. These
types of “uniform” operators cannot be understood unless the concept of function is
conceived as a total entity. We distinguish these from other types of operators on functions
which could be termed “point-wise” operators, and for which there is no need to conceive
functions as objects, but only as processes acting on individual elements of their domains.
For example, sum and composition can be treated as point-wise operators; this position is
different from Ayers et al ’s (1988) position who argue that composition of functions
requires the encapsulation of function as an entity. Further research is needed to examine
the two arguments. The cognitive process of understanding these operators involves the
conception of a function as a process acting on individual elements of the domain. In
constructing the composition of two functions f and g, say f°g, one must first perform the
process g on an arbitrary element x of the domain, generating a result g(x), and then perform
the process of f on that result to obtain f(g(x)), all conceivable as acting on individual
elements of the domain. These two separate operations are coordinated to produced a new
process. Similarly, in constructing f+g, for every input x, the outputs, f(x) and g(x), are
produced to construct the sum, f(x)+g(x). This sum can even be illustrated graphically by
using a sample set of directed line segments for the distances between the horizontal axis
and the graphs off and g, respectively. Then the graph of f+g is the graph whose distance
from the horizontal axis is given by the vector sum of the directed line segments. Clearly,
the sum f+g can be illustrated point-wise.
The limit of a one variable function is another case which may be regarded as a pointwise operator. To understand this complex concept, many clusters of knowledge about
different concepts in mathematics are required whose rich conceptual content is reflected
CONCEPTUAL ENTITIES AND SYMBOLS
85
in the complexity of its historical development. We will not attempt to analyze this
knowledge here; however, the process-conception of function is sufficient (and necessary)
to understand the limit concept. This is so because lim f(x) = L may be viewed in terms
x→a
of the point-wise dependency between the behavior of the numbers “near” a , x ’s, or inputs
off, and the behavior of their outputs, f(x)’s, “near” L.
By contrast, “uniform” operators arise when the point-by-point process is inapplicable.
For example, to understand the meaning of:
as a function of t, it is necessary to think of I(t) as an operator that acts on the process x →
f(x) as a whole to produce a new process:
It is the awareness of acting on a process as a whole, as a totality – not point-by-point – that
constitutes the conception of that process as an object.
Mathematically unsophisticated students attempt to interpret “uniform” operators as
“point-wise” operators apparently because they cannot conceive of a function as an object.
Consider the derivative operator. Our experience in the classroom suggests that many
students understand that f'(x) means: for the input x there is the output f(x), and for that
output we get the derivative f'(x). Faced with the question,
find the derivative of the function f(x) =
a common response is:
The student is no longer treating differentiation as a limit process, but as an algorithm to
be applied to the formula at each point (or to the two separate formulas in the expression).
To be able to handle this problem, the student needs to be able to consider the values of the
function near x and renegotiate the limit process. In Greeno’s terms, the function f must act
as an argument for the (cognitive) differentiation operator, which it cannot do unless the
function is conceived as a conceptual entity.
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GUERSHON HAREL & JAMES KAPUT
1.2b COMPREHENSION: THE CASE OF OBJECT-VALUED OPERATORS
As the notion of function develops, it can have different objects as inputs and outputs, in
particular, it can output another function. For instance, the real-valued function f(x,y) is
usually thought of as a process mapping points on the plane, (x, y), into points on the real
line, f(x,y); thus, students who possess the process-conception of function would likely
have no difficulty dealing with this interpretation. A more subtle interpretation can view
f(x,y) as a process which associates points on the real line, x, with functions, fx(y) where the
latter assigns the value f(x, y) to y. In this interpretation f is regarded as a function with input
x and output the function fx . We believe that, cognitively, thinking of a function as an output
is not different from thinking of it as an input, in the sense that in both cases a function must
be treated as a variable, as a conceptual entity. In this respect, this interpretation of f( x,y),
like the “uniform” operator, demands that the concept of function will be treated as an
object. However, the cognitive demands of such a viewpoint are often great.
This analysis, which has yet to be empirically substantiated, is supported by our informal
observations while teaching undergraduate mathematics classes the concepts of double
limit, lim(x, y) → (a,b) f( x , y ), and the iterated limit, lim lim f(x, y). As some textbook
x→a y→b
authors have indicated (e.g.,Munroe, 1965, p. 108), we observed that while computationally
the iterated limit is easier than the double limit, conceptually the iterated limit involves a
more sophisticated idea, which causes difficulty for students in particular circumstances.
In stating and proving certain theorems on iterated limits (e.g., theorems concerning
conditions on equality between this limit and the double limit), one needs to regard
lim lim f(x,y) as a composition of the following three mappings (see figure 12):
x→a y→b
Figure 12 : lim lim f(x,y ) as a composition of three mappings
x→a y→b
1. M: x → fx(y), whose domain is a set of real numbers and whose range is a set
of functions;
2. lim : fx (y) → f(x), whose domain and range are sets of functions;
x→a
3. lim : f(x)→c, whose domain is a space of functions and range is a set of
x→a
numbers.
CONCEPTUAL ENTITIES AND SYMBOLS
87
Students responses and questions indicate difficulty in dealing with aspects concerning the
operator M, which, as indicated earlier, requires the object-conception of function. While
the operator M must be understood as an object-valued operator, the other two operators,
lim and x→a
lim can be viewed in two ways, which determine different levels of
y→b
understanding the concept of iterated limit. In one way y→b
lim and x→a
lim are uniform
operators acting on objects which happen to be functions. This level of understanding,
although desirable, is not achieved by the average student, who usually views these limits,
and the concept of limit in general, in a less sophisticated way as point-wise operators.
Besides the iterated limit, the undergraduate mathematics curriculum is replete with
situations involving object-valued operators, for example those which concern parametric
functions, such as f(x)=ax+b, f(x)=sin(ax), f(x)=logsax, etc., or parametric equations
involving such functions. In these situations the correspondence between the parameters
and the function, or the equation, constitutes an object-valued operator. The difficulties
involved in understanding object-valued operators was investigated by Harel (1985) in the
context of linear algebra (taught to advanced high-school students in Israel). It was found
that students usually had difficulty dealing with such a correspondence, unless they were
able to tag the outputs of the correspondence with familiar geometric figures, such as lines
or planes (e.g., t → (a, b)+t(c, d) or (t1, t2) → (a,b) + t1(c, d) + t2(e,f)). These geometric
figures, which were manipulable objects for the students, apparently helped the students
to construct such a correspondence as an object-valued operator.
Another common example involves the construction in abstract algebra of the quotient
object associated with a “normal” sub-object, e.g., in the case of groups. The cosets must
be conceived as objects if they are to participate as elements of a group. However, the
existence of a “representative element” for a coset, where the operation defined on cosets
can be given in terms of an operation on their representatives, makes it possible to deal
successfully with many aspects of the quotient group on a symbol manipulation level
without treating the subsets of the group as objects, or even as subsets. Students’ inadequate
conceptions are revealed when one asks them to attempt to create a group using a nonnormal subgroup’s cosets – they often cannot understand why the subsets “fall apart” when
they attempt to multiply them together as sets, or by using representatives.
Finally, data reported by Kaput (in press) can further support the cognitive distinctions
among the different types of operators made above. Secondary level students were asked
to determine an algebraic rule that fits a student-controllable set of numerical domain-data
(they pick the x’s and the computer provides the f(x)’s). Examination of their behavior
revealed a clear and stable decomposition of the group of students (in a sample of over 40
high school students) into two sets, one of whom consistently used a point-by-point patternmatching process, mediated by natural language formulations of their proposed “rules,”
while the other searched for and applied a parametrically mediated formulation of their
proposed rules. The latter, for example, would look for constant change in the dependent
variable, identify this as the “m” in y=mx+b, and proceed from there. For them the process
was a search for parameters that indexed functions as objects. In effect, they were dealing
with a space of functions (albeit a limited one), whereas the other group of students
conceptualized the task as a point-wise attempt to build a function whose point-wise
behavior matched the rule that they had formulated using natural language.
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1.3 CONCEPTUAL ENTITIES AS AIDS TO FOCUS
The third role of conceptual entities we have identified involves facilitating focus on those
aspects of a problem representation that are most relevant to the solution of a problem. In
a one-on-one interview with an experimental group of Israeli high-school students
regarding the concept of vector space (after several instructional sessions in which this
concept was gradually abstracted from two and three dimensional representations; see
Harel, 1989a, 1989b), the first author asked the following question:
Let V be a subspace of a vector space U, and let β be a vector in U but not in V.
Is the set V+β={v +β| v is a vector in V} a vector space?
There were clearly two groups of students: those who answered this question by checking
the whole list of the vector-space axioms, and those whose answer was something like, “you
moved the whole thing, it doesn’t have the zero vector any more”, or “the new thing, V+β,
is not closed under addition”. Clearly, the latter group of students viewed V as a total entity,
a “thing,” and thus they were able to view +β as a shift operator which takes V as an
argument, an input. This enabled them to focus on those vector space properties that are
most relevant to the solution of the given problem, namely, the zero property or one of the
closure properties. The other group of students, on the other hand, relied on the formal
definition of vector space by checking whether the individual axioms apply. That V+β is
a subset of the vector space U, which guarantees the existence of most of the axioms, was
not visible to these students. Moreover, many of these students failed to check some of the
axioms, including those essential to the solution of the problem (e.g., the existence of zero).
2. ROLES OF MATHEMATICAL NOTATIONS
The power of mathematics associated with the roles of conceptual entities is closely related
to the roles of mathematical symbolism. Using mathematical notations, complex ideas or
mental processes can be chunked and thus represented by physical notations which, in turn,
can be reflected on or manipulated to generate new ideas. In this section we will discuss
three aspects of the interaction between formation of conceptual entities and mathematical
notation:
1. The role of mathematical notation in forming conceptual entities.
2. Different types of mathematical notations, elaborated and tacit notations, and
the manner in which they represent conceptual structure.
3. Notations as substitutes for concepts.
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2.1 NOTATION AND FORMATION OF COGNITIVE ENTITIES
Greeno (1983) stated two conditions that help distinguish entities from other mental events.
One is its continual presence in a mental representation; the other (mentioned earlier) is its
ability to act as an argument in another mental procedure or argument. By providing
continual perceptual experience, material notations help provide the basis for continuing
conceptual presence. This role is based simply on notations as names – the notation serves
to name an item in our conceptual world. We might term this the “nominal” role. Note that
the parts of the syntax of a notation system associated with identification and discrimination
of notational objects plays an important role here. Having an explicit name for a mental
event helps objectify it through a kind of transference of object permanence – from the
permanence of the physical notational name (which produces perceptual experience on a
more or less continuous basis) to a cognitive permanence. Of course, the perceptual item
must somehow come to be integrated with the conceptual one. Other wise, all one mightend
up with is, say, an easily reproducible mental experience of a mark or character string, with
no other mental activity or structure beyond that primitive experience – which is the
experience of altogether too many students.
The nominal role of symbols is frequently played out using conventions that help
distinguish the status or differing roles of objects in complex situations – convention-based
variations in the names of objects help distinguish the classes to which they belong.
Suppose a concept involves a process which takes entities of a different order as inputs and
outputs, e.g. differentiation operating on functions. Then there is a need to distinguish
between the higher level process and its lower level inputs and outputs, a need which is
typically satisfied by using systematically different symbols for the items at each level.
Then the conceptual activity of keeping the things distinguished is off-loaded onto the
notation system. For example, many higher level mathematical activities involve defining
functions between sets of functions – as between a vector space and its double dual. Another
typical example occurs in topology, when one defines various compactifications, e.g., the
Stone-Czech compactification of a regular Hausdorff space based on sets of continuous
functions on the unit interval. In all such cases, one finds that, typically, different classes
of characters are used to distinguish the different levels of functions - say, one Greek and
the other contemporary English-based.
Systematic variation in names also is employed through the use of different classes of
symbols to distinguish when an object is being treated in two different ways, where it has
essentially two different identities. Consider the conventions used to distinguish the
identity of a real number x from its identity as a member of the field of complex numbers,
where it may be denoted by x+0i. Similar distinctions are made whenever a canonical
embedding is being employed, not merely in the case of algebraic closures, because it is a
characteristic of “canonicalness” that the substructure is maintained within the larger
structure. A related case involves the distinction between a constant function and its value.
In all these cases, object identity is identified and maintained notationally.
Relative to Greeno’s second condition for cognitive entities acting as arguments in other
procedures, the syntax of a notation system specifically structures the place of the material
notational objects in a coherently organized physical system. Such a system is designed to
support a given type of thinking. For example, the character string notation for functions
supports highly sophisticated manipulations, which in turn, are used to facilitate a wide
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GUERSHON HAREL & JAMES KAPUT
variety of mental operations on the conceptual objects that those character strings denote.
Thus, the act of factoring the character-string representation of a polynomial function to
help identify its roots may be based on some syntactic rule (e.g., applied to the difference
of cubics), which obviates the need to justify all the steps of the process. The strength of
anotation system maybe measured by whether, and to what degree of fidelity, syntactically
guided actions on its objects reflect and/or subsume important mental operations.
We conclude this section with two specific examples to illustrate the variety of ways
notations either help encapsulate mathematical concepts as entities or supplant conceptual
entities in reasoning processes. Goldin (1982) discusses the impacts of languages or
notations on the different stages of the problem solving process, citing his own data as well
as the well-known problem-isomorph work by Simon and colleagues. The following
discussion can be thought of as somewhat preliminary to the issues discussed by these
researchers in the sense that we are dealing with the concept-notation relationship at a more
primitive level.
Example 1: Consider the use of graphical notation, the slope of straight lines, to
facilitate the order comparison between ratios described as linear functions
between sets of objects, measures, or even numbers. To compare two such on
the basis of a table of data (a sequence of ordered pairs) or even on the basis of
a pair of fractions is not as easy as comparing the slopes of their associated
straight lines in acoordinate plane. In this case one need only attend to two things
(2 lines) as distinguished by their most salient attribute, their slope. Each single
line embodies an infinite set of equivalent pairs of ratio values. This seems to
be an instance of a one-for-many substitution of a single notational object for a
set of mental objects, although from another perspective it amounts to an
integration of detailed features into a single object.
Example 2: Recall the study mentioned in § 1.2 where students were determining
functions from numerical data. There were two types of students: One type of
students were essentially “pre-algebraic” in their thinking, and treated every
potential rule that they inferred from the numerical data in a table (which they
generated) as a natural language-based rule. That is, they thought of 2x+1 as
doubling and adding one, in terms of a natural language interpretation, rather
than in terms of parameters m and b in mx+b. Thus they did not see growth in
the numerical data in the same way as those who were looking for values of these
parameters. Basically, the latter were looking for growth rates, which they
interpreted as the first parameter’s value, etc. For them, a linear function was
experienced as a “thing”, a conceptual entity, whose identity is determined by
the two parameters. The other students were looking for a way to translate from
their natural language-based encoding of an unencapsulated process to algebra.
They quite often succeeded – as long as the parameters involved were positive
whole numbers. For negatives, they fell apart, because they were not able to get
easy natural language encodings of what for them was a process rather than a
thing (Kaput, in press). An open question is what is the relation between the
conceptualentity and the parameter notation? Which came first? Or did they coevolve? In any case, this example seems to offer an instance of the functional
power of the nominal use of symbols – as do most systematic uses of parameters.
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91
2.2 REFLECTING STRUCTURE IN ELABORATED NOTATIONS
The inventors of mathematical notations created them to express the contents of their own
minds, both to themselves, to aid their own thinking, and to others, to aid in the
communication of their conceptions. As Leibniz, that great master of notation-invention
put it,
In signs one observes an advantage in discovery which is greatest when they express theexactnature
of a thing briefly, and, as it were, picture it; then indeed the labor of thought is wonderfully
(Quoted in Cajori, 1929, p. 184)
diminished.
Extending his remark, we might add that the structure of the conceptions is, in some way,
being reflected in the structure of the notations, especially in their syntax. Or, put more
constructively, the experience of perceiving the notations shares important features with
the experience of the conception apart from any perceptual act. Extending this observation
further, we suggest that it is even more important that actions on notational objects in some
regular way reflect mental actions on the conceptions. (We again hasten to add, however,
that we are not suggesting any kind of simple relationship between notation and conception!)
But mathematical symbols differ in the extent to which they include features that reflect
the structure of the mathematical objects, relations or operations that they stand for. Some
are more elaborated than others (Harel, 1987). For example, the place-valued symbol 324
expresses a specific structure of thequantity it represents: three hundreds, two tens, and four
ones. Of course, this number written in expanded notation is even more elaborated.
Similarly, the more abstract symbols, (x, y) for an ordered pair of numbers, f(x) = 3x 2 for
a specific real-valued function, AB for a line segment whose endpoints are A and B, and
for an mxn matrix, areall relatively elaborated symbols, because they encode the structures
or relationships among components of their referents.
On the other hand, for example, the concept “matrix of the linear transformation T
relative to the pair of ordered bases µ and v” can be symbolized by the significantly less
elaborated symbol [T]µ,v . A more elaborated symbol for this concept could be [T]µ→v ,
which indicates that the matrix representation of a transformation T depends on the
relationship between the bases in its range and in its domain. An even further elaborated
symbol for this concept is:
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GUERSHON HAREL & JAMES KAPUT
[[T(µ 1)] v : [T(µ 2)] v : . . . : [T(µ n)]v ] (usedbyAnton, 1981),
which encodes many of the variables included and its referent. In contrast, the symbol [tij]
(used by Nering, 1970, to represent the same concept) is far less elaborated, whilst the bare
symbol T is a non-elaborated, or tacit, symbol. Tacit symbols provide essentially an
indexical function – they name things, without denoting aspects of the structure of what is
named.
One category of tacit symbols consists of those which, during a discussion or proof, are
used to represent variables. For example, the statement, “let βbe an ordered pair ...”, typifies
a context in which such a tacit symbol is used; here, the symbol β does not encode the
structure of its referent – an ordered sequence of two objects – but it, together with the
surrounding phrases, does name the set over which the variable varies.
The extent to which a notation is elaborated is determined by the extent to which it ties
to prior mathematical knowledge, which is very much a cognitive matter. Indeed, what is
elaborated for one person may appear very bare and tacit for another. Nonetheless, the act
of connecting a bare notation to an elaborated one is a translation act, which, depending on
circumstances, may operate in either direction. The notation’s perceived connection with
prior knowledge takes the form of perceived features that reflect features of the prior
knowledge. For example, two different symbols are usually used to represent the composition of two functions f and g: f(g(x)) and (f°g)(x). The symbol f(g(x)) expresses the process
in which the two functions are composed: the input x in the function-machine g produces
the output g(x), where g(x) now acts as an input in the function-machine f to produce the
output f(g(x)). (Note the strong use of temporality here.) Thus the symbol f(g(x)) is
amenable to the thinking of a function as a process, but depends on the prior knowledge of
input-output relations expressed using the standard f(x) notation. The symbol (f°g)(x), on
the other hand, describes an operation between two functions – f and g – which produces
a third one – (f°g)(x). This symbol describes f and gas inputs in the [meta] function-machine
°, and thus to understand its meaning functions must be viewed as conceptual entities. In
this example, the prior knowledge is that of operating on inputs to functions, and the
notation feature is reflected in a parallelism of structure, except that the first function in the
composition acts as the input.
The pedagogical importance of this example is that some mathematical symbols cannot
be understood via the symbol f(g(x)); for example, the ‘‘uniform” operator:
Students have trouble thinking of the integral as a function of x – which is revealed when
they are asked to treat it like a function. Our notation I(x) for it is itself intended to help with
this – it assists entification by treating it notationally as a function, elaborating it in such a
way that the functional dependence on the variable x is highlighted.
The distinction between elaborated symbols and tacit symbols has important consequences for learnability and usability. In Harel (1987) it was hypothesized that an
elaborated symbol would be better understood and remembered if it expresses the main and
salient variables in its referent. Here, we additionally hypothesize that a tacit symbol can
CONCEPTUAL ENTITIES AND SYMBOLS
93
be more meaningfully used when its referent is encapsulated into a conceptual entity. That
is, in developing a symbol for a concept one must try to match the degree of elaboration of
the symbol with the degree of elaboration of the user’s concept, which in turn must match
the user’s needs for the task at hand. After all, in some cases it is important to suppress detail,
and in others the detailed structure plays a role in what one is trying to do. It seems, then,
that one’s control of the amount of structure explicitly represented in the symbolism is a
major factor in mathematical thinking, because one can adjust the “focus of one’s mental
microscope” by adjusting the notation. This we believe to be an important facilitating factor
that notations offer us.
3. SUMMARY
We hope to have introduced some useful ways of thinking about some important aspects
of the learning of mathematics that highlight the role of conceptual entities and their
relationships with mathematical notations. We regard this chapter as but a beginning into
an area of research that others may find productive to pursue in the future.
In § 1 we laid out some of the circumstances under which conceptual entities are created
and used and what their cognitive function might be, often by pointing to consequences in
students’ reasoning processes where they have not yet been mentally constructed. We
observed three cognitive functions:
• Alleviating working memory or processing load when concepts involve multiple constituent elements, facilitating comprehension of complex concepts,
• the cases of “uniform” operators, “point-wise” operators, and “object-valued
operators”,
• assisting with the focus of attention on appropriate structure in problem solving.
These functions, undoubtedly, play an important role in mathematical thinking and in
fostering the vertical growth of mathematical ideas, at all levels.
In §2 we analyzed the key role that notations play in the entification process by helping
substitute names for complex conceptual structures and/or operations. We have discussed
three aspects of the interaction between formation of conceptual entities and mathematical
notation:
• the role of mathematical notation in forming conceptual entities,
• different types of mathematical notations – elaborated and tacit notations, and
the manner in which they represent conceptual structure,
• notations as substitutes for concepts.
Just as notations can help the formation and application of mental entities, notations can act
as substitutes for conceptual entities, supplanting the need for them. It is here where both
the great power and the great danger in using mathematical notation systems become
particularly and unavoidably evident. Accompanying the great power of notations as aids
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GUERSHON HAREL & JAMES KAPUT
to mathematical thought based on their identity-management role and their structuresubstitution role is the great danger that the notations do not refer to any mental content
beyond the experienced physical structure of the notations themselves, e.g., as when one
deals with an algebraic statement as a character string. This seems to be the case with
altogether too many students. While the inventors of notations created them to express and
perhaps elaborate their own pre-existing conceptions, in schools we often begin in reverse
order, concentrating on manipulation of notations, e.g., the techniques of differentiation
and integration in calculus, before providing sufficient experience that would enable the
building of mental referents for those notations (Davis, 1986). Students should be given
opportunities to build their own notational expressions of their ideas, which can then be
guided in the direction of the standard ones. In this way, one builds both notations and
conceptions simultaneously, rather than building one or the other first and then attempting
to connect the two.
CHAPTER 7
REFLECTIVE ABSTRACTION
IN ADVANCED MATHEMATICAL THINKING
ED DUBINSKY
Our purpose in this chapter is to propose that the concept of reflective abstraction can be
a powerful tool in the study of advanced mathematical thinking, that it can provide a
theoretical basis that supports and contributes to our understanding of what this thinking
is and how we can help students develop the ability to engage in it. To make such a case
completely, it would be necessary to do at least several things:
• explain exactly what we mean by reflective abstraction;
• show how it can be used to describe the epistemology of various mathematics
concepts;
• indicate how it can suggest explanations of some of the difficulties that students
have with many of these concepts;
and
• establish that it can influence the design of instruction in ways that result in a
significant improvement in th extent to which students appear to acquire these
concepts.
We are certainly not ready to do an exhaustive job on all four of these tasks. Indeed, our main
concern here is to make some progress with the first two. There will be a few examples of
the third, and we will make reference to other papers in which we have made a start on the
fourth especially involving the use of computer activities to help students make mental
constructions, with results that are encouraging.
Reflective abstraction is a concept introduced by Piaget to describe the construction of
logico-mathematical structures by an individual during the course of cognitive development. Two important observations that Piaget made are first that reflective abstraction has
no absolute beginning but is present at the very earliest ages in the coordination of sensorimotor structures (Beth & Piaget, 1966, pp. 203–208)l and second, that it continues on up
through higher mathematics to the extent that the entire history of the development of
mathematics from antiquity to the present day may be considered as an example of the
process of reflective abstraction (Piaget, 1985, pp. 149–150).
In the majority of his own work, however, Piaget concentrated on the development of
mathematical knowledge at the early ages, rarely going beyond adolescence. What we feel
is exciting is that, as he suggested, this same approach can be extended to more advanced
topics going into undergraduate mathematics and beyond. It seems that it is possible not
1 Piaget repeated many of his comments on reflective abstraction in several places, but was
quite consistent on this topic. Hence, the references we give should be taken as representative.
95
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ED DUBINSKY
only to discuss and conjecture, but to provide evidence suggesting, that concepts such as
mathematical induction, propositional and predicate calculus, functions as processes and
objects, linear independence, topological spaces, duality of vector spaces, duality of
topological vector spaces, and even category theory can be analyzed in terms of extensions
of the same notions that Piaget used to describe children’s construction of concepts such
as arithmetic, proportion, and simple measurement.
This is a strong claim embodied in the phrase “can be analyzed” and, before going
further, it is necessary to explain what sort of analysis we mean. The goal of our study of
reflective abstraction is a general framework which can be used, in principle, to describe
any mathematical concept together with its acquisition. We refer to this as a general theory
of mathematical knowledge and its acquisition. This is the first ingredient of the analysis,
but it does not, by itself, lead to any particular description. In addition, the investigator needs
to make use of her or his understanding of the mathematics. Together these two are enough
to obtain a description of any concept but the result would be far too ex post facto to expect
it to have any relation to how students actually might go about constructing the concept. A
third and essential ingredient in the study of any concept is a long drawn-out, time
consuming effort of observation of students as they try to construct mathematical concepts
in order to make sense out of situations in which they find themselves (presumably, but not
necessarily, as the result of activities of a teacher). The analysis then consists of a synthesis
of these three ingredients brought to bear on the question of how a particular topic in
mathematics may be learned. The starting point of our general theory is Piaget’s notion of
reflective abstraction. Unfortunately, this is not a simple idea clearly explained in one place,
but rather something that Piaget appeared to work with over a long period of time after he
completed his empirical studies of children in development. It is important, however, that
we begin with a solid understanding of what he meant by it before trying to extend it to a
wider class of mathematical topics. Therefore we begin this chapter with a section that gives
a brief summary of this concept as Piaget elaborated it in a number of books and papers,
mostly written in the last 15 years of his life. We will emphasize the construction aspects
of reflective abstraction because these are the most important for the development of
mathematical thought during adolescence and beyond.
In the second section we will show how Piaget’s ideas can be extended and reorganized
to form ageneral theory of mathematical knowledge and its acquisition which is applicable
to those mathematical ideas that begin to appear at the post-secondary level and continue
to be constructed in the course of mathematical and other scientific research. It is here, in
§ 2 that we relate various aspects of the general theory to specific topics in advanced
mathematical thinking and give several examples of how reflective abstraction can suggest
explanations of student difficulties.
Our analysis of a particular mathematical concept leads to what we call a genetic
decomposition of the concept which is a description, in terms of our theory, and based on
empirical data, of the mathematics involved and how a subject might make the constructions that would lead to an understanding of it (which, in our theory, are not very different).
It is important to note that we do not suggest that a concept has a unique genetic
decomposition or that this is the way every subject will learn it. We only claim that
observations of learning in progress form an important source for our genetic decompositions and we offer them as a guide for one possible way of designing instruction. In § 3 we
present genetic decompositions for three concepts: mathematical induction, predicate
REFLECTIVE ABSTRACTION
97
calculus, and function, insofar as we have constructed them. The references given in § 3
contain more information about examples of instructional treatments based on these
genetic decompositions, using computer experiences, and about the generally encouraging
results of implementing these treatments.
Finally, in § 4 we discuss some of the educational implications of our theory of
knowledge and learning and give an overview of how we go about designing an
instructional treatment based on it. We feel that the material in this section is very much akin
to the ideas in Thompson (1985a).
1. PIAGET’S NOTION OF REFLECTIVE ABSTRACTION
1.1 THE IMPORTANCE OF REFLECTIVE ABSTRACTION
Piaget distinguished three major kinds of abstraction. Empirical abstraction derives
knowledge from the properties of objects (Beth &Piaget, 1966, pp. 188–189). We interpret
this to mean that it has to do with experiences that appear to the subject to be external. The
knowledge of these properties is, however, internal and is the result of constructions made
internally by the subject. According to Piaget, this kind of abstraction leads to the extraction
of common properties of objects and extensional generalizations, that is, the passage from
“some” to “all”, from the specific to the general (Piaget & Garcia, 1983, p. 299). We might
think, for example of the color of an object, or its weight. These properties might be
considered to reside entirely in the object but one can only have knowledge of them by doing
something (looking at the object in a certain light, hefting it) and different individuals under
different conditions might come to different conclusions about these properties.
Pseudo-empirical abstraction is intermediate between empirical and reflective abstraction and teases out properties that the actions of the subject have introduced into objects
(Piaget, 1985, pp. 18–19). Consider, for example the observation of a 1-1 correspondence
between two sets of objects which the subject has placed in alignment (ibid, p. 39).
Knowledge of this situation may be considered empirical because it has to do with the
objects, but it is their configuration in space and relationships to which this leads that are
of concern and these are due to the actions of the subject. Again, of course, understanding
that there is a 1-1 relation between these two sets is the result of internal constructions made
by the subject.
Finally, reflective abstraction is drawn from what Piaget (1980, pp. 89–97) called the
general coordinations of actions and, as such, its source is the subject and it is completely
internal. We will see many instances of reflective abstraction, but a very early example we
can mention now is seriation, in which the child performs several individual actions of
forming pairs, triples, etc., and then interiorizes and coordinates the actions to form a total
ordering (Piaget, 1972, pp. 37–38). This kind of abstraction leads to a very different sort
of generalization which is constructive and results in “new syntheses in midst of which
particular laws acquire new meaning” (Piaget & Garcia, 1983, p. 299). An example of this
is the concept of euclidean ring which is certainly an abstraction and generalization. It might
be considered, however, to derive from the properties of a single example – the integers.
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We can see, therefore, that these different kinds of abstraction are not completely
independent. The actions that lead to pseudo-empirical and reflective abstraction are
performed on objects whose properties the subject only comes to know through empirical
abstraction. On the other hand, empirical abstraction is only made possible through
assimilation schemas which were constructed by reflective abstraction (Piaget, 1985, pp.
18–19). Consider, for example a physics experiment which may have the purpose of
making an empirical abstraction to obtain factual data about a certain object. The
experiment presupposes, however, an enormous range of logico-mathematical preliminaries – in deciding how to pose the question, in the construction of apparatus for “indirect
observations” (e.g., triangulation to obtain distances between stars), in the use of particular
forms of measurement, and finally, in setting out the results in logico-mathematical
language. All of these are concepts that must have been constructed using reflective
abstraction. (Piaget, 1980, p. 91). This mutual interdependence can be roughly summarized
as follows. Empirical and pseudo-empirical abstraction draws knowledge from objects by
performing (or imagining) actions on them. Reflective abstraction interiorizes and coordinates these actions to form new actions and, ultimately new objects (which may no longer
be physical but rather mathematical such as a function or a group). Empirical abstraction
then extracts data from these new objects through mental actions on them, and so on. This
feedback system will be reflected in our extension of these ideas in the next section.
In empirical abstraction the subject observes a number of objects and abstracts a
common property. Pseudo-empirical abstraction proceeds in the same way, after actions
have been performed on the object. Reflective abstraction, however, is much more
complicated. This is not surprising since, according to Piaget, “The development of
cognitive structures is due to reflective abstraction ...” (Piaget, 1985, p. 143). Before going
into the nature of this fundamental process, therefore, we should say a few things about its
importance, in Piaget’s view, to cognitive thought in general and mathematics in particular.
In two books Piaget (1976, 1978) interpreted the results of many experiments with
children in terms of reflective abstraction. But its role is not restricted to the intellectual
development of children. From Piaget’s psychological viewpoint, new mathematical
constructions proceed by reflective abstraction (Beth & Piaget, 1966, p. 205). Indeed, he
considered it to be the method by which all logico-mathematical structures are derived
(Piaget, 1971, p. 342); and that “it alone supports and animates the immense edifice of
logico-mathematical construction” (Piaget, 1980, p. 92).
In support of his position on the role of reflective abstraction in advanced mathematical
thinking, Piaget tried to explain a number of major mathematical concepts in terms of the
constructions that result from this psychological process. These included the idea of
Gödel’s incompleteness theorem (Beth & Piaget, 1966, p. 275), the abstract concept of
groups (1980, p. 19), Bourbaki’s attempts to encompass all of mathematics within three
“mother structures” (1970a, p. 24), the general theory of categories (Piaget 1970b, p. 28),
the impossibility of constructing the set of all sets (1970b, pp. 70–71), and the mathematical
concept of function (Piaget et al, 1977, p. 168). More generally, Piaget considered that it
is reflective abstraction in its most advanced form that leads to the kind of mathematical
thinking by which form or process is separated from content and that processes themselves
are converted, in the mind of the mathematician, to objects of content (Piaget, 1972, pp. 63–
64 and pp. 70–71).
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Returning to the ideas of Piaget, it is important to emphasize that there is no suggestion
here that all (or any) of the advanced mathematics described above is actually done by any
kind of direct application (conscious or otherwise) of reflective abstraction. This was not
Piaget’s purpose in trying to analyze that aspect of thinking. The point, rather, is that when
properly understood, reflective abstraction appears as a description of the mechanism of the
development of intellectual thought. It is important for Piaget’s theory that this same
process that describes advanced mathematical thinking appears in cognitive development
throughout life from the child’s very first coordinations that lead to concepts such as
number, measurement, multiplication, and proportion (Piaget, 1972, pp. 70–71). An
important ingredient of Piaget’s general theory (on which he worked for 60 years) that
relates biological evolution to the development of intelligence is the idea that reflective
abstraction is one isolated case of certain very general processes that are found throughout
living creation (Piaget, 1971, p. 331).
1.2 THE NATURE OF REFLECTIVE ABSTRACTION
As we have seen, reflective abstraction differs from empirical abstraction in that it deals
with action as opposed to objects and it differs from pseudo-empirical abstraction in that
it is concerned, not so much with the actions themselves, but with the interrelationships
among actions, which Piaget (1976, p. 300) called “general coordinations”.
According to Piaget, the first part of reflective abstraction consists of drawing properties
from mental or physical actions at a particular level of thought (Beth & Piaget, 1966, pp.
188–189). This involves, amongst other things, cognizance or consciousness of the actions
(1971, p. 320). It can also include the act of separating a form from its content (1972, pp.
63–64). Whatever is thus “abstracted” is projected onto a higher plane of thought (1985,
pp. 29–31) where other actions are present as well as more powerful modes of thought.
It is at this point that the real power of reflective abstraction comes in for, as Piaget
observes, one must do more than dissociate properties from those which will be ignored or
separate a form from its content (1975a, p. 206). There is “a process which will become
increasingly evident over time: the construction of new combinations by a conjunction of
abstractions” (Piaget, 1972, p. 23).
Piaget seemed to feel that this construction aspect of reflective abstraction is more
important than the abstraction (or extraction) aspect (ibid, p. 20). Not only did he assert, as
we observed earlier, that construction of this kind is the essence of mathematical
development, and that combining formal structures is a natural extension of the development of thought (ibid, p. 64), but he also used his analysis of this process to deal with the
philosophical question of the nature of mathematical thought (Beth & Piaget, 1966).
Certainly for our purposes, the construction aspect of reflective abstraction will play the
major role.
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1.3 EXAMPLES OF REFLECTIVE ABSTRACTION
IN CHILDREN’S THINKING
We begin with some of Piaget’s examples of reflective abstraction in logico-mathematical
thinking at the earlier ages. This is important because of his insistence on the continuity of
development as part of his search for a single process or set of processes that related to
biological development as well as intellectual development (Piaget, 1971, p. 331). Our
suggestion in this chapter is that the specific construction processes that can be used to build
sophisticated mathematical structures can be found, already, in the thinking of young
children.
commutativity of addition. The discovery that the number of objects in a
collection is independent of the order in which the objects are placed requires
first that the child count the objects, reorder them, count them again, reorder and
count, etc. Each of these actions are interiorized and represented internally in
some manner so that the child can reflect on them, compare them, and realize
that they all give the same result Piaget, 1970a, pp. 16–17).
number. According to Piaget (1941), the concept of number is constructed by
coordinating the two schemas of classification (construction of a set in which
the elements are units, indistinguishable from each other) and seriation (which,
as we observed earlier, is itself a coordination of the various actions of pairing,
tripling, etc.).
trajectory. The traversal of a path is understood as a coordination of successive
displacements to form a continuous whole (Piaget, 1980, p. 90).
see-saw . The balancing of objects on two sides of a see-saw by a combination
of actions on both sides involves more than just keeping two things in mind at
the same time. Because he observed a considerable delay between the time that
a child could create the balance and the time that the child appeared to
understand how he or she had done it, Piaget saw this as a coordination of two
actions into a single system (Piaget, 1978, p. 96).
multiplication. Both psychologically and mathematically, multiplication is the
addition of additions. It is, however, objects that are added in the sense that
addition is an operation applied to something. In order, therefore, to multiply,
it is necessary first to encapsulate the (mental) action of addition into an object
(or set of objects) to which addition can be applied (Piaget, 1985, p. 31).
fluid levels. In an experiment asking children to predict the level to which a
known amount of fluid would rise in a vessel with sloping sides and markings
at equal height divisions (Piaget et al, 1977, chapter 7). Piaget pointed out that
this situation is a case of “variation of variations”. That is, the differential in two
vertical markings is a variation, but the amount of change also varies because
of the sloping sides. Hence, the first variation must become an object to which
an action is applied (sloping sides) resulting in a higher order variation.
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1.4 VARIOUS KINDS OF CONSTRUCTION
IN REFLECTIVE ABSTRACTION
In considering the aboveexamples of reflective abstraction as methods of construction, we
can isolate four different kinds which will be important for advanced mathematical
thinking. We add a fifth which Piaget considers at length, but was not, for him, part of
reflective abstraction.
• With the appearance of the ability to use symbols, language, pictures, and
mental images, the child performs reflective abstractions to represent (piaget,
1970a, p. 64), that is, to construct internal processes as a way of making sense
out of perceived phenomena. Piaget called this interiorization (1980, p. 90) and
referred to it as “translating a succession of material actions into a system of
interiorized operations” (Beth & Piaget, 1966, p. 206). The commutativity of
addition described above is one example of this. (See also Thompson, 1985a,
p. 197.)
• Several of our examples such as trajectory and see-saw involve the composition
or coordination of two or more processes to construct a new one. This is to be
distinguished from Piaget’s phrase, “general coordinations of actions” which
refers to all ways of using one or more actions to construct new actions or
objects.
• Multiplication, proportion and variation of variation exemplify the construction
which is perhaps the most important (for mathematics) and most difficult (for
students). This is encapsulation or conversion of a (dynamic) process into a
(static) object. As Piaget put it (1985, p. 49), “ ... actions or operations become
thematized objects of thought or assimilation”. He considered that “The whole
of mathematics may therefore be thought of in terms of the construction of
structures, ... mathematical entities move from one level to another; an operation
on such ‘entities’ becomes in its turn an object of the theory, and this process is
repeated until we reach structures that are alternately structuring or being
structured by ‘stronger’ structures” (Piaget, 1972, p. 70). From a philosophical
point of view, Piaget was applying the idea of encapsulation to the relativity
between form and content when he referred to “...building new forms that bear
on previous forms and include them as contents” and “reflective abstractions
that draw from more elementary forms the elements used to construct new
forms” (Piaget, 1985, p. 140).
• When a subject learns to apply an existing schema to a wider collection of
phenomena, then we say that the schema has been generalized. This can occur
because the subject becomes aware of the wider applicability of the schema. It
can also happen when a process is encapsulated to an object as, for example, the
ratio of two quantities, or addition, so that an existing schema such as equality
or addition can then be applied to it to obtain, respectively, proportion or
multiplication. The schema remains the same except that it now has a wider
applicability. The object changes for the subject in that he or she now under-
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stands that it can be assimilated by the extended schema. Piaget referred to all
of this as a reproductive or generalizing assimilation (1972, p. 23), and he called
the generalization extensional Piaget & Garcia, 1983, p. 299).
Once a process exists internally, it is possible for the subject to think of it in
reverse, not necessarily in the sense of undoing it, but as a means of constructing
a new process which consists of reversing the original process. Piaget did not
discuss this in the context of reflective abstraction, but rather in terms of the
INRC group. We include it as an additional form of construction.
2. A THEORY OF THE DEVELOPMENT OF CONCEPTS
IN ADVANCED MATHEMATICAL THINKING
2.1 OBJECTS, PROCESSES, AND SCHEMAS
Although, as we have pointed out, Piaget believed that reflective abstraction was as
important for higher mathematics as it was for children’s logical thinking, his research was
mainly concerned with the latter. In order to try to develop the notion of reflective
abstraction for advanced mathematical thinking, we will isolate what seem to be the
essential features of reflective abstraction, reflect on their role in higher mathematics, and
reorganize or reconstruct them to form a coherent theory of mathematical knowledge and
its construction.
For us, reflective abstraction will be the construction of mental objects and of mental
actions on these objects. In order to elaborate our theory and relate it to specific concepts
in mathematics, we will use the notion of schema. A schema is a more or less coherent
collection of objects and processes. A subject’s tendency to invoke a schema in order to
understand, deal with, organize, or make sense out of a perceived problem situation is her
or his knowledge of an individual concept in mathematics. Thus an individual will have a
vast array of schemas. There will be schemas for situations involving number, arithmetic,
set formation, function, proposition, quantification, proof by induction, and so on throughout all of the subject’s mathematical knowledge. Obviously, these schemas must be
interrelated in a large, complex organization. For example, there will be a proof schema,
which can include a schema for proof by induction. This latter in turn could include a
schema for proposition valued functions of the positive integers (seep. 112). Hence there
would be a relation with the schemas for number, for function, and for proposition. On the
other hand, there is a sense in which a proof is an action applied to a proposition, so that the
proof schema might be one of the processes in the proposition schema.
We will also sometimes use the term process or mental process instead of mental action
when we are emphasizing its internal (to the subject) nature. Finally the term object will
refer to amental or physical object (avoiding any discussion of the nature of the distinction).
One of our goals in elaborating the general theory is to isolate small portions of this
complex structure and give explicit descriptions of possible relations between schemas.
When this is done for a particular concept, we call it a genetic decomposition of the concept.
We should also point out that although we only give, for each concept, a single genetic
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decomposition, we are not claiming that this is the genetic decomposition, valid for all
students. Rather it represents one reasonable way that students might use to construct a
concept.
It is not easy to separate a description of mathematical knowledge from its construction.
As Piaget put it, “ ... the problem of knowledge, the so-called epistemological problem,
cannot be considered separately from the problem of the development of intelligence”
(Piaget, 1975a, p. 166). It is not possible to observe directly any of a subject’s schemas or
their objects and processes. We can only infer them from our observations of individuals
who may or may not bring them to bear on problems – situations in which the subject is
seeking a solution or trying to understand a phenomenon. But these very acts of recognizing
and solving problems, of asking new questions and creating new problems are the means
(in our opinion, essentially the only means) by which a subject constructs new mathematical
knowledge.
This is where reflective abstraction comes in. Thus, although we might say that
mathematical knowledge consists of a collection of schemas, we have little to say about
how that knowledge exists inside a person. It does not seem to reside in memory or in a
physiological configuration. All we can say is that a subject will have a propensity for
responding to certain kinds of problems in a relatively (but far from totally) consistent way
which we can (as far as our theory has been developed) describe in terms of schemas. When
the subject is successful, we say that the problem has been assimilated by the schema. When
the subject is not successful then, in favorable conditions, her or his existing schemas may
be accommodated to handle the new phenomenon. This is the constructive aspect of
reflective abstraction to which we referred as forming the main object of our concern.
In this sense, the study of reflective abstraction is complementary to investigation of
notions such as epistemological obstacles as studied by Cornu (1983), and Sierpinska
(1985a, 1985b) or the conflict between concept image and concept definition as investigated by Schwarzenberger & Tall, 1978; Tall & Vinner, 1981; Dreyfus & Vinner, 1982;
Vinner, 1983;Tall, 1986a; Vinner & Dreyfus, 1989). One can think of reflective abstraction
as trying to tell us what needs to happen whereas the other notions attempt to explain why
it does not. It is possible that our idea of using computer experiences (Ayers et al , 1988;
Dubinsky, 1990a, 1990b) to help students make reflective abstractions can be a way of
dealing with these obstacles and conflicts. But these are matters for other investigators and
other papers.
2.2 CONSTRUCTIONS IN ADVANCED MATHEMATICAL CONCEPTS
In the previous section, we isolated five kinds of construction that Piaget found in the
development of children’s logical thinking: interiorization, coordination, encapsulation,
generalization, and reversal. We will reconsider each of them in the context of advanced
mathematical thinking to describe how new objects, processes and schemas can be
constructed out of existing ones.
Some of the following examples will apply to a single one of the five kinds of
construction and others will apply to a combination of two or more of them. Some of the
statements we make are based on observations of students and others are only suppositions,
derived as a preliminary to observations, from the general theory and our knowledge of the
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mathematics.
As we make these statements about constructions that we have seen students appear to
make or that our investigations suggest they need to make, or as we conjecture that certain
concepts could be constructed in these ways, the reader should be aware that we are not
suggesting that it is automatic, natural, or easy for students to take these steps. An important
aspect of the whole problem of education that we do not consider in this paper is to explain
why students do or do not make these particular constructions and what can be done to help
them. This is an important issue for research in mathematics education.
An important part of understanding a function that we have observed is to construct a
process (Dubinsky et al, 1989). For individual examples this means that the subject
responds to a situation in which a function may appear (via formula, as an algorithm, or
represented by data) and for which there is a process by which the value of the function, for
a particular value in the domain is obtained. Given such a situation, the subject may respond
by constructing, in her or his mind, a mental process relating to the function’s process. This
is a prime example of interiorization.
An example of the same kind of mental activity in a completely different mathematical
situation could arise in understanding proofs. When the mathematician exclaims (as which
of us has not?) that “I can understand each step of the proof, but I don’t see the whole
picture”, it could be the case that he or she is expressing the necessity of interiorizing a whole
collection of processes and coordinating them to obtain a single process. The interiorization
of the total process can be, in our opinion, the final step in “making a proof your own”.
Interiorization may not always be difficult. Most students seem to have little trouble with
constructing a mental process for multiplying a matrix and a vector, or two matrices. This
could be because there is a straightforward “hand-waving” action, used by most teachers,
that is a physical representation of the multiplication and could form an intermediary
between the external action and its interiorization. It seems that mathematics becomes
difficult for students when it concerns topics for which there do not exist simple physical
or visual representations. One way in which the use of computers can be helpful is to provide
concrete representations for many important mathematical objects and processes (see
Chapter 14).
Turning now to coordination, one of the most important examples that we have seen
occurs in the formation of the composition of two functions. Based on our research (Ayers
et al, 1987; Dubinsky et al, 1989), we would like to propose the following psychological
description. Composition is a binary operation which means that it acts on two objects to
form a third. Thus, it is necessary to begin with two functions, considered as objects. The
subject must “unpack” these objects, reflect on the corresponding processes, and interiorize
them. Then the two processes can be coordinated to form a new process that can then be
encapsulated into an object which is the function that results from the composition. This
is much more complicated than simple substitution and perhaps explains why students have
so much difficulty with ideas like the chain rule for differentiation, in which it would be
necessary to coordinate this view of composition with the notion of derivative. It could also
explain those results of Ayers et al (1988) in which students seem to improve their
understanding of composition as a result of performing computer tasks designed to foster
these mental operations.
A whole class of examples of that could be described as coordination of schemas in
advanced mathematics is given by the “mixed” structures: topological vector spaces,
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differentiable manifolds, homotopy groups, etc.
It is often possible to observe students having difficulty with determining the cardinality
of sets such as
{4, {–3, 2, –1/7}, {{17, 5}}}.
Very often, even undergraduates will think that this set has 6 elements (rather than 3). We
suggest that the difficulty is that the students have not encapsulated the sets {–3, 2, –1/7},
{17, 5} into objects so as to understand the nested structure of the given set.
The indefinite integral forms an important example that can be interpreted as encapsulation together with interiorization. Estimating the area under a curve with sums and passing
to a limit is, of course, a process. Students who seem to understand this often have difficulty
with the next step of varying, say, the upper limit of the integral to obtain a function. What
is lacking, we suggest, is the encapsulation of the entire area process into an object which
could then vary as one of its parameters vary. This would then form a “higher-level” process
which specifies the function given by the indefinite integral. The complexity of this total
process might then explain why students have such difficulty with not only the Fundamental Theorem of Calculus, but such powerful definitions as
A rather pervasive example that can be interpreted as encapsulation in mathematics is
duality. The dual of a vector space, for example, is obtained by considering all of the linear
transformations from the space to its scalar field as objects, collecting them in a set, and
introducing a natural algebraic structure on this set. It seems to us that this is an act of
encapsulation that is essential in this branch of mathematics.
The simplest and most familiar form of reflective abstraction is generalization.
According to our investigations, we can say that a subject’s function schema, in which
functions transform numbers, is generalized to include functions which transform other
kinds of objects (once they have been encapsulated) such as vectors, sets, propositions, or
other functions. Similarly it would seem that the schema of factorization of positive integers
can be generalized in this way to factoring polynomials, and then to an arbitrary euclidean
ring. Vectors of dimension two and three can be generalized to include higher, and even
infinite, dimensional vectors. All of these and a host of other examples in mathematics seem
to involve the application of an existing schema, essentially unchanged, to new objects
(which are often the result of encapsulation).
Finally there is reversal of a process. We can mention a number of familiar activities in
mathematics that appear to involve the reversal of a process: subtraction and division,
solving an equation, inverting a function, proving an inequality (in which one often starts
with the conclusion, manipulates until something known to be true is obtained, and then
sees if the argument can be reversed), and the mysterious choice of expressions such as
in proving limit theorems.
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2.3 THE ORGANIZATION OF SCHEMAS
In the previous section, we suggested how the construction of various concepts in advanced
mathematics could be described in terms of the five forms of construction in reflective
abstraction: interiorization, coordination, encapsulation, generalization, and reversal. We
offer the conjecture that the construction of all mathematical concepts can be described in
these terms. It may be that additional forms of reflective abstraction will have to be added
as additional concepts are considered, but we suggest that the five given here tell something
like the full story.
Of those concepts (mathematical induction and predicate calculus) for which we have
made a more or less complete genetic decomposition (Dubinsky, 1986; Dubinsky,
Elterman & Gong, 1988), our analysis has been greatly influenced by data obtained from
observations (interviews, written tasks, computer work, etc.) of students while they are
trying to understand the concept in question. The genetic decomposition is then derived
from a synthesis of these empirical results, our general theory, and our mathematical
knowledge of the concept in question. This is why it takes a long time and has only been
done extensively for two concepts. Work on other concepts (e.g., function, limit) is
proceeding slowly and, we hope, deliberately.
The following description of the organization of a schema is just a summary of what we
have seen in the concepts investigated thus far and, therefore, is somewhat tentative. We
give it here in general terms and then, in the next section, see how it looks in the context
of mathematical induction and predicate calculus. In addition, with more anticipation than
certainty, we will suggest how it might look for the concept of function, after considerably
more data has been gathered.
The structure of a schema is displayed in figure 13.
As we have already indicated, one should not think of a schema statically, but rather as
a dynamic activity (or propensity for such activity) by the subject. Moreover, the existence
of a schema is inseparable from its continuous construction and reconstruction. Thus, in
describing the system in Figure 1 we will try to do several things simultaneously: describe
what is there, describe what happens, describe how things are constructed, and refer to some
of the examples we have discussed previously. An additional complication is that, as
indicated in the picture, a schema is not a linear list of items but rather a circular feedback
system. Our description, necessarily linear, must break in at some point. In any case, the
following discussion is an alternative way of organizing the five kinds of construction
analyzed in the previous two sections. Here we also include the results of the constructions
(objects and processes).
We begin with objects. These encompass the full range of mathematical objects:
numbers, variables, functions, topological spaces, topologies, groups, vectors, vector
spaces, etc., each of which must be constructed by an individual at some point in her or his
mathematical development.
At any point in time there are a number of actions that a subject can use for calculating
with these objects. These actions go far beyond numerical calculation resulting in
numerical answers. The computation of the homotopy group of a topological space is a
calculation. So is the determination of the (topological) dual of a (locally convex
topological) vector space. We will return to this example a few paragraphs below when we
discuss coordination.
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Figure 13: Schemas and their construction
It is possible for a subject to work with actions in ways other than just applying them to
objects. First, an action must be interiorized. As we have said, this means that some internal
construction is made relating to theaction. An interiorizedaction is a process. Interiorization
permits one to be conscious of an action, to reflect on it and to combine it with other actions.
For example, the computation of the dual of a particular vector space is an action on that
object. The idea, independent of any particular vector space, that it may have a dual and it
can often be computed, is the process that results from interiorizing this action.
Interiorizing actions is one way of constructing processes. Another way is to work with
existing processes to form new ones. This can be done, for example, by reversal. A calculus
student may have interiorized the action of taking the derivative of a function and may be
able to do this successfully with a large number of examples, using various techniques that
are often taught and occasionally learned in calculus courses. If the process is interiorized,
the student might be able to reverse it to solve problems in which a function is given and
it is desired to find a function whose derivative is the original function. This is antidifferentiation or integration, and it too, is fist an action and then must be interiorized to
become a process. Encapsulating both the differentiation and integration processes – at
least to the point of having them as objects of reflection – would seem to be an essential
prerequisite for understanding the fundamental theorem of calculus.
Another way of making new processes out of old ones is to compose or coordinate two
or more processes. For example, let us return to the dual of an infinite dimensional vector
space and imagine (this is purely conjectural) how a subject might think about it. A subject
may have a schema (discussed in the previous section) for constructing the dual of a finite
dimensional vector space. If an infinite dimensional vector spac ecomes along, then it seems
that exactly the same schema can be used to construct its dual, as well. We would say that
the new phenomenon (infinite dimensional vector space) has been assimilated to this
schema, As mathematical experience goes further, however, this result would not be very
satisfactory, and it is particularly convenient to make use of topological structures. If there
is, in the subject’s schema, a process for equipping a set with a topology, then this could
be coordinated with the vector space schema to obtain a topological vector space. Now
within a schema for topological space there should be a schema for the concept of
continuous function and within a vector space schema there should be a notion of linear
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function. Coordinating continuity and linearity, one can obtain the idea of a continuous
linear function. This coordination would permit the subject to extend and reorganize the
process for constructing the dual of a vector space to apply to the set of those functions from
the original set to the scalar field which are continuous as well as linear, thereby obtaining
the topological dual. In such a situation we would say that the schema for duals has been
accommodated to the new phenomena (involving topologies) and experiences which made
the old schema less than satisfactory.
In addition to using processes to construct new processes, it is also possible to reflect on
a process and convert it into an object. Anytime a set of functions is considered, it seems
necessary to think of the functions in question as objects. Initially, functions are processes
and so the subject must have performed an encapsulation in order to consider them as
objects. It is important, for example in composition of functions, for the subject to alternate
between thinking about the same mathematical entity as a process and as an object. (cf. p.
104.)
A more advanced, and yet more fundamental example where encapsulation may occur
is in the concept of a topology. Initially, there is the notion of nearness or convergence,
which is a process. One of the accomplishments of twentieth century mathematics is to
capture this idea with the device of a collection of subsets (so-called “open sets”) which
must satisfy certain condtions but is otherwise arbitrary. The interaction (really another
form of coordination) between, on the one hand, a collection of sets which may be taken
as arbitrary in order to investigate general topological properties related to but not identical
with notions of “nearness”, and on the other hand, a very specific choice of this collection
so as to apply those properties to important concrete situations, say in analysis, and the use
of the resulting observations to stimulate the development of further general properties, and
so on, has led to a great deal of important new mathematics of both abstract and concrete
natures. A key step in this progress may be described as the encapsulation of the process
“nearness” to the object “topology”.
We conclude this section with a recapitulation of our description of the construction of
schemas in the context of the example, already mentioned on several occasions above, of
the (topological) dual of a (topological) vector space. This suggestion of a genetic
decomposition for the concept of dual is totally speculative in the sense that it depends
entirely on our theory and our understanding of the relevant mathematics. We have
gathered no data (other than introspection on our own experience) to support our
suggestions. On the other hand, it may be interesting for those with a background in
mathematics to see that our theory at least appears to be reasonably compatible with a topic
from the arena of mathematical research. It is an important point that the same ideas that
described the thinking of young children and adolescents can be used to talk about higher
mathematics.
In the beginning, there are vectors, which are the objects, and actions on vectors
including addition, scalar multiplication and the gathering together of vectors in a set with
these operations, to form a vector space. This is a schema that we assume the subject
possesses. We also assume that the subject has a schema for functions that transform
numbers into other numbers.
The first step, according to our conjecture, is to generalize the function schema to include
as a function any process that transforms vectors into scalars. This could then be
coordinated with the addition of vectors and their multiplication by scalars to restrict the
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functions to processes that transform vectors into scalars, but preserve the algebraic
operations of addition and scalar multiplication.
We would then say that these functions are encapsulated into objects called linear
functionals and collected together in a single set. At this point we would like to suggest that,
although the assigning of a name like linear functional to a process is closely connected with
its encapsulation into an object, it is the encapsulation that is fundamental and gives
“meaning” to the name. To name processes without encapsulating them is the essence of
jargon. When there is a complaint that a particular discourse has too much terminology and
not enough meaning, we feel that the real difficulty is that labels are being assigned without
an opportunity for encapsulating that which is being labeled.
In any case, the set of linear functionals can be assimilated to the vector space schema
(which may have to be accommodated to this purpose – that is, it may be necessary to project
and reconstruct this schema on the higher level of a vector space whose elements are linear
functionals) by defining addition and scalar multiplication of these functionals. This can
be done very naturally, interpreting the functions as processes and using “point-wise
operations”. In this way, the set of linear functionals becomes a vector space, called the
algebraic dual.
Now comes a major interiorization. What we have been describing is an action applied
to a vector space E that constructs its algebraic dual E*. When this has been interiorized,
one has constructed the beginning of duality theory. One can reverse the process to look
for a “pre-dual”, that is, given a vector space F, can one find a vector space E whose
algebraic dual is F? (The answer is yes if E is “finite-dimensional”, but otherwise it may
or may not be possible.) Or one can perform the process twice. When two instantiations are
coordinated, one obtains the bidual E**.The concept of reflexivity (fairly simple in the case
of the algebraic dual) has to do with whether E=E**.
Next, as we mentioned above, topology and algebra can be coordinated to obtain the
concept of topological vector space and the schema for dual can be projected onto this
higher plane and reconstructed by introducing considerations of continuity, to obtain the
topological dual E' of a topological vector space E.
Again the action of constructing the topological dual can be interiorized into a process
and the concepts of pre-dual and reflexivity (much more interesting in the topological case)
can be reconstructed and their properties investigated. Even more interesting, the content
of forming the topological dual can be removed from the form of this process (by reflecting
on it) and this would give rise to the idea of dual pairs 〈 E,F 〉 in which algebra and topology
are mixed in free and varying combinations to obtain the modern theory of dual systems
in linear topological spaces.
3. GENETIC DECOMPOSITIONS OF THREE SCHEMAS
We will consider three schemas in some detail: mathematical induction, predicate calculus,
and function. Our goal is to show how the general theory elaborated in the previous section
can be used in possible descriptions of the nature and construction of these specific
schemas. Thus in each case we will point out the relevant objects and processes as well as
the instances of reflective abstractions that seem to us can be used in constructing them.
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The details that we are about to present come from our three sources. First, there is the
psychological data that we have gathered through observations of students in the midst of
trying to learn these concepts. These experiments are described in full detail in Dubinsky
(1986, in press a,b), Dubinsky et al (1986, 1989, in press). This data, along with the ideas
of Piaget formed the basis for the derivation of our theory, which is the second source of
the genetic decompositions. That is, for each phenomenon that was observed, we tried to
use our theory to describe it, adjusting the theory when necessary. (As the necessity for
adjustment occurs less often, our confidence in the theory increases.) The third source of
the descriptions is our mathematical understanding of the concepts in question. It seems
important that a genetic decomposition should make sense from a mathematical point of
view, although it might not be exactly how the mathematician might have analyzed the
subject in thinking about how to teach it.
These three sources actually only apply in full to the first two examples: mathematical
induction and predicate calculus. Because our data, and the analysis that leads to our
conclusions, already appears in the above references, we do not repeat it here. In the case
of function, we have begun to gather data, but our studies were not yet complete at the time
of writing and so we make some mention of it, although very limited. Thus the genetic
decomposition of function given here is based mainly on the theory and our mathematical
understanding of function. As such, it must be taken as speculation that may form a bridge
for future work. As we obtain and analyze data on students’ learning the concept of function,
it will be interesting to see how close the genetic decomposition postulated here comes to
what is derived when the genetic data are taken into account. In a sense, this can provide
an indication of the predictive value of our theory as it has been developed so far.
3.1 MATHEMATICAL INDUCTION
The aspect of induction that we are interested in has to do with a subject’s understanding
of the induction process, why it “works” to establish something and how to construct an
induction proof. Ultimately, this has to be coordinated with a notion of infinity but it may
be that understanding the induction process is a precursor to constructing a notion of
infinity. It would be an interesting investigation to apply, to the concept of infinity, our
method of helping students learn induction (Dubinsky, 1986, in press).
In the first instance, mathematical induction is a process in that one interiorizes the
actions of moving along (as “n increases”) from one proposition to the next and, after an
initial independent determination, establish the truth of a statement by applying a tool (truth
of an implication) that was previously constructed.
Mathematical induction is also an object in the subject’s general schema for proofs. This
means that the induction process must have been encapsulated in order that the subject can
reflect on it, along with other methods, when confronted with a theorem to prove, so as to
select induction as the method for a particular problem.
The method itself is constructed by working with two major schemas: function and
logic. The developments of these two schemas are intertwined through various coordinations. We can illustrate the process with a chart as shown in figure 14.
We start with the assumption that the subject possesses a function schema and a logic
schema that are already developed to the point where, for example, the function schema
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Figure 14: Genetic Decomposition of Mathematical Induction
includes the ability to construct a process relating to a particular transformation of numbers
(see § 3.3), and the logic schema can construct statements in the first order propositional
calculus (see § 3.2). In particular we assume that the function schema includes the process
of evaluation of a function for a given value in its domain and that the logic schema includes
a process for logical necessity, that is, in certain situations, the subject will understand that
if A is true then of necessity B will be true. Of course we are not asserting that the subject
will necessarily be aware of these schemas in this terminology. What we mean, for example,
is that the subject will be able to think in terms of plugging a value of a positive integer into
a statement and asking if the result is a true statement. This is a function and we can infer
from a subject’s actions that it may exist in her or his mind as aschema – but we would hardly
require young subjects to be aware of it as such in order to understand induction.
The formation of first order propositions is a process in the logic schema which can come
from interiorizing actions (conjunctions, disjunctions, implications, negations) on declarative statements (objects). The subject can perform a reflective abstraction on this process
to obtain new objects which are the propositions of the first order propositional calculus,
on which the same actions can be performed. Consider for example, a simple proposition
such as,
where P, Q and R are simple declarations. The formation of the disjunction P ∨ Q can be
described as an action on the statements P,Q. It is not just the action of putting these symbols
in this expression. The subject must also construct a mental image involving the two
statements and the determination of the truth or falsity of the disjunction in various
situations. If nothing further is done after this action is interiorized, then it will not be
possible to combine this with R to get the full proposition. First, the disjunction process must
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be encapsulated to form a new object (P∨Q) which is a statement that can be conjoined with
another statement, such as R. Note how the use of parentheses in mathematical notation
corresponds to encapsulation.
Iterating this procedure, the subject enriches her or his logic schema to obtain a host of
new objects consisting of first order propositions of arbitrary complexity. Next the function
schema comes in. We are assuming that this schema can be used by the subject to construct
processes that transform numbers (for example an integer) into other numbers. It must be
generalized to permit the subject to construct processes that transform positive integers into
propositions, to obtain what we shall call a proposition valued function of the positive
integers. Consider for example, a statement such as,
Given a number of dollars, it is possible to represent it with $3 chips and $5 chips.
For such a statement, the subject must construct a process whereby, for each positive integer
n, a proposition is constructed which is the same statement, but with “a number of dollars”
replaced by that value of n. This is the proposition valued function. In order to evaluate it,
the subject must construct another process whereby, given n , a search is made and it is
determined whether it is possible to find non-negative integers kj such that
n = 3j + 5k.
It is useful for the subject to discover that the value of this function is true for n = 3, 5, false
for n = 1, 2, 4, 6, 7 and then appears to be true for all higher values.
It is only at this point that the subject can realize that the problem of “proving” the
statement consists of determining that the value of the function is true for all values of
n ≥ 8. For this, the proof schema can be invoked. If it contains the schema for induction,
it can be used, if not, further (re-)construction must take place. In describing this
construction, we reiterate that, in the context of this theory, it is never clear (nor can it be)
whether one is talking about a schema that is present or one that is being (re-)constructed.
Before going on with the description, there is a side issue that should be considered.
Whether the subject is able to construct a proposition valued function of the positive
integers to deal with a particular statement depends not only on the existence of the schemas
we are talking about, but also may require additional knowledge about the particular
situation – so-called “domain knowledge”. Thus, although the above example of chips is
probably well within the domain knowledge of most students who find themselves trying
to learn induction, others may not be. We have found, for example, that the following
statement provides difficulty for university undergraduates.
An integer consisting of 3n identical digits is divisible by 3n.
The trouble could lie in understanding the relationship between the value of an integer and
its representation with digits. It is a sort of “grown-up” version of the difficulty with the
concept of place value and it suggests that many students have not really constructed this
concept – at least in a sufficiently powerful form.
Returning now to the construction of proof by induction, the next development provides
an example of a cognitive step which our research has pointed out as providing a serious
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difficulty, whereas if one takes only the mathematical point of view, there is not even a step
that needs to be taken. This is the case even though it relates to an overt difficulty
encountered by everyone who has tried to teach mathematical induction.
We are referring to the notion that the essential point in an induction proof is that one
does not prove the original statement directly, but rather the implication between two
statements derived from it. This is the major difficulty for students. It requires a cognitive
step which is not necessary as a mathematical step. To explain, let us denote by P the
proposition valued function to be proved. Now P(n) can be any proposition, in particular,
it can be an implication. Therefore, if we define the proposition valued function Q by
Q(n) = (P(n) ⇒ P(n+1))
then, from a mathematical point of view there is nothing new in Q, that is, once one
understands P then, as a special case, one understands Q. We have observed, however, that
with students, this is not the case from the cognitive point of view. In the first place,
implications are the most difficult propositions for students and they are generally the last
to be encapsulated. Furthermore, there is a difference between constructing P from a given
statement and constructing Q from P. This is the step which must be taken. If there is some
subtlety here, then it might help explain the difficulty that students have at precisely this
point.
To summarize, this step appears to require the encapsulation of the process of
implication so that an implication is an object and can be in the range of a function, the
generalization of the function schema to include implication valued functions, and the
interiorization of the process of going from a proposition valued function of the positive
integers to its corresponding implication valued function.
The next step is to add to the logic schema a process which we shall call modus ponens.
This process is an interiorization of an action applied to implications (assuming as above
that they have been encapsulated into objects). The action consists of beginning at the
hypothesis, determining that it is true, and then “crossing the bridge” to the conclusion and
asserting its truth.
Finally, there is a coordination of the function schema, as it applies to an implication
valued function Q (obtained from a proposition valued function P) and the logic schema
as it applies to the process modus ponens which has just been constructed. Included in the
function schema is the process of evaluation, that is, sampling values n of the domain
(positive integers in this case) and computing the value of the function, Q(n), that is, P(n)
⇒ P(n+1). Suppose that it has been established that Q has the constant value true. The first
step in this new process which must be constructed is to evaluate P at 1 and to determine
that P(1) is true (or, more generally, to find a value n0 such that P(n0) is true). Next, the
function Q is evaluated at 1 to obtain P(1) ⇒ P(2). Applying modus ponens and the fact
just established) that P(1) is true yields the assertion P(2). The evaluation process is again
applied to Q but this time with n =2 to obtain P(2) ⇒ P(3). Modus ponens again gives the
assertionP(3). This is repeated ad infinitum, alternating the processes of modus ponens and
evaluation. Thus we have a rather complex coordination of two processes that we believe
leads to an infinite process.
This infinite process is encapsulated and added to the proof schema as a new object,
proof by induction.
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3.2 PREDICATE CALCULUS
The predicate calculus schema appears to be obtained through are construction of a schema
resulting from coordinating a schema for first order propositional calculus with a function
schema. The construction is illustrated in Figure 15. According to this analysis, the objects
in the propositional calculus schema are the propositions. The most important process is
the determination of the truth or falsity of a proposition. Other processes include the
formation of new propositions by the standard logical operations such as conjunction,
disjunction, implication and negation. They also include the process of expressing an
English statement in the formal language of symbolic logic and translating from that syntax
back to English. Then of course there are all the usual tasks that students are asked to
perform such as manipulation of the formulas, construction of truth tables, determination
of the validity of arguments and so on. Finally, we can mention the process of reasoning
about a statement, for example, to know if the truth or falsity of the statement
(P ⇒ Q) ∨ (not (Q ∧ R))
is determined once you know that P ⇒ R is false.
Amongst the various manipulations of logical expressions, one in particular will be
important in the sequel. That is the process of applying the conjunction operation (“and”
or ∧) to a set of propositions as in
(x1>b1 )∧ (x 2> b2 )∧ . . .∧ (xn>bn).
There is a similar process for disjunction (“or” or ∨). This is a manipulation of symbols, but
there is an underlying process connected with the truth value of the resulting proposition.
In a sense, the objects in the first order propositional calculus are constants. In an
expression such as (P ⇒Q) ∨ (not (Q ∧ R)) the quantities P, Q and R are constants whose
value maybe unknown, but fixed. The subject’s thinking about such matters can be elevated
to a higher plane when the propositional calculus schema is coordinated with the function
schema (appropriately reconstructed on this higher plane) to consider such an expression
as determining a function — in this case of the three variables, P, Q and R. This is the
beginning of the predicate calculus schema. Of course, a part of this coordination and
reconstruction was discussed already in the previous section for the special case of
proposition valued function of the positive integers.
As before, an important new process that can be constructed is the iteration (in the
subject’s mind) through the domain of a proposition valued function, checking the truth or
falsity of the resulting proposition for each value of the variable. Consider, for example a
statement such as
Given a car in the parking lot, if the tire fits the car, then the car is red.
Here, tire may be considered to be a constant, but car should be thought of as a variable
whose domain is the set of cars in the parking lot. There is an obvious action of walking
through the parking lot, checking each car to see if the tire fits and, if it does, seeing if the
car is red. When such a statement appears in a mathematical context, as in
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Figure 15: Genetic Decomposition of Predicate Calculus
Given x ∈ domino(F), if |x–x0| ≤ δ , then |F(x)–F(x0)|≤ε
then the mental process seems to consist in looking at each x ∈ domain(F) to see if
|x–x0| ≤ δ and, if so, seeing if |F(x)–F(x 0)|≤ ε.
This iteration process must now be coordinated with one of the two processes we
mentioned earlier: applying conjunction or disjunction to a set of propositions. The
resulting process can be encapsulated which leads to a single existential or universal
quantification as in
For all cars in the parking lot, if the tire fits the car, then the car is red.
∀ x ∈ domain(F), |x–x0| ≤ δ ⇒ |F(x)–F(x0)|≤ ε
We call this a single-level quantification.
The single-level quantification creates new objects which are again propositions so that
all of the previous processes of logical operations, negation and reasoning about statements
are reconstructed on this higher plane. Particularly important for understanding many
mathematics topics is the interiorization of a statement given as a quantification. The
subject seems to need a strong mental image of the iteration and quantification process that
we have described in order to relate the statement to the mathematical situation that is being
considered.
Next comes two-level quantifications in which two (usually different type) quantifiers
are applied in succession to a proposition valued function of two variables. For example,
the statements we have considered may be extended to obtain,
There is a tire in the library such that for all cars in the parking lot, if the tire fits
the car, then the car is red.
or
∃δ > 0 ∋∀ x∈ domain(F), | x – x0| ≤ δ ⇒ |F(x)–F(x 0)| ≤ ε.
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The process which we just described for constructing single-level quantifications ended
with an encapsulation so that the result becomes a proposition which is a mental object.
Note that the effect of a quantification is to eliminate a variable. If the original proposition
valued function had two variables, then the resulting object actually depends on the value
of the other variable and the schema for single-level quantifications can again be applied
to this proposition valued function. For example, in the case of the tires and cars, the
universal quantification over cars results in a proposition valued function of the single
variable, tire. This function can then be subjected to an existential quantification to obtain
a single, constant proposition. Thus, when analyzing a statement which requires a two-level
quantification over two variables, the subject can begin by parsing it into two quantifications. There is an inner quantification over one of the variables in a proposition valued
function of two variables. There is also an outer quantification over the other variable. What
we have described is a coordination of these two quantifications to obtain a process which
will be a two-level quantification. In order to proceed to higher-level quantifications this
new process is again encapsulated to obtain a new object. Once it is encapsulated, it can then
be subjected to the same processes (thereby generalized) as were the single level
quantifications.
Given a statement which is a three-level quantification, such as the definition of
continuity of F at x0,
∀ε > 0,∃δ > 0 ∋ ∀x ∈ domain(F), |x–x0| ≤ δ ⇒ |F(x)–F(x0)|≤ ε.
the subject can group the two inner quantifications and apply the two-level schema to again
obtain a proposition which depends on the outermost variable (in this case ε). This
proposition valued function is then quantified as before to obtain a single proposition. The
entire procedure can now be repeated indefinitely to obtain quantifications of any level. At
each level, the same processes of logical operations, negation, reasoning, etc. are reconstructed.
3.3 FUNCTION
As we indicated earlier, the thoughts about the function concept given here are based mainly
on the general theory and our understanding of this concept from the mathematical point
of view. Our purpose for including it and giving some examples of preliminary data is to
illustrate the explanatory power of our theory and to set guideposts for subsequent empirical
work. In the past decade, the function concept has been investigated by a number of authors
in ways that are quite different from the approach described here (see especially Dreyfus
& Eisenberg, 1983,1984; Dreyfus & Vinner, 1982; Vinner & Dreyfus, 1989). For a fuller
discussion of research on learning the concept of function, see chapter 9.
For most students, and indeed for many scientists, the idea of function is completely
contained in the “formula”. If you ask students for an example of a function, you will often
get an algebraic expression such as x2+3 with no mention of any kind of transformation.
Just as with the concept of variable in which the student insists that x “stands for” a single
number (which may not be known), the concept of function as formula has a very static
flavor.
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There are a number of ways in which such a function schema is inadequate. For one
thing, the objects are restricted to those functions which can be conveniently expressed with
a formula. This may suffice for elementary mathematics but it will not do for advanced
mathematical thinking. When a function is the same as a formula, the action of evaluation
on this object consists of plugging in numbers for letters and composition of two functions
is restricted to substitution of a formula for each occurrence of a letter. The notions of
domain and range have no place in this schema and graphs, while manageable in themselves
(because of their concrete and visual nature), have no connection with functions for the
student with a function-as-formula schema. When the graph does not display a clear picture
(as is the case with the characteristic function of the irrationals), then the student is unable
to think about it.
A more powerful schema for functions will involve interiorization of actions. When a
subject perceives a situation that can be dealt with in terms of a function, then we suggest
that he or she can view the situation as an action on objects that transforms them into other
objects. This action is interiorized. Thus, an important part of what it means to know a
function is to construct a certain kind of process that can be used to make sense of a certain
kind of phenomenon. Some may refer to this as a mental representation of the function, but
we prefer to avoid such terminology because of its tendency towards the misleading
suggestion that the internal process is a copy of some “external reality”. The imprtant point
is that when a function is known as an interiorized process, then this knowledge has a dynamic
flavor which affects the nature of the subject’s interaction with the function situation.
Evaluation becomes the action of taking a particular value (in the domain of the
function) and performing the process on it to obtain a new value (in the range of the
function). It may then be possible for the subject to coordinate a function’s process and its
graph. That is, there is the understanding that the height of the graph of a function f at a point
x on the horizontal axis is precisely the value f(x). The subject can then relate to the full
power of graphing which is the relationship between the physical shape of the graph and
the behavior of the function.
Several important ideas in mathematics can be described as doing some of the things we
have discussed with the process of a function. For example, the coordination of two
processes and the composition of the functions (see Ayers et al, 1988). A function’s process
can be reversed, thereby obtaining the inverse function. It is by reflecting on the totality of
a function’s process that one makes sense of the notion of a function being onto. Reflection
on the function’s process and the reversal of that process seem to be involved in the idea
of a function being one-to-one.
We have done some preliminary empirical work relative to the points in the preceding
paragraph. We find, for example, that students seem to have more difficulty with the
concept of one-to-one than with onto. We suggest that the presence of the reversal in oneto-one explains this observation. Similarly on several occasions we have given subjects the
following kinds of problems relative to three specific functions, F, G, H. (See Ayers et al,
1988 for details.)
1. Given F, G find H such that H=F°G.
2. Given G, H find F such that H=F°G.
3. Given F, H find G such that H=F°G.
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Of course the first is much easier than the other two, and we find invariably that the third
is harder than the second. We can suggest an explanation derived from our theory. The first
kind of problem seems to require only the coordination of two processes that, presumably,
have been interiorized by the subject. The second, however may require that the following
be done for each x in the domain of H.
2a. Determine what H does to x obtaining H(x).
2b. Determine what G does to x obtaining G(x).
2c. Construct a process that will always transform G(x) to H(x).
The thud kind of problem may be solved by doing the following for each x in the domain
of H.
3a. Determine what H does to x obtaining H(x) .
3b. Determine value(s) y having the property that the process of F will transform
y to H(x) .
3c. Construct a process that will transform any x to such a y.
Comparing 2b with 3b (the only point of significantdifference), we can see that 2b is a direct
application of the process of G whereas 3b requires a reversal of the process of F.
It is perhaps interesting to note that this difference in difficulty (between 2 and 3), which
is observed empirically and explained epistemologically, is completely absent from a
purely mathematical analysis of the two problems. They are, from a mathematical point of
view, the calculation of H°G-1 and F-l°G, respectively, which appear to be problems of
identical difficulty. This seems to be another important example in which the psychological
and mathematical natures of a problem are not the same ( cf. p. 113).
Another situation in which relative difficulty can be explained by the requirement of
reversing a process occurs in the development of children’s ability in arithmetic. According
to Riley, Greeno & Heller (1983, p. 157), “Problems represented by sentences where the
unknown is either the first (? + a = b) or second (a + ? = c) number are more difficult than
problems represented by equations where theresult is theunknown ( a + b = ?).”Obviously
the first two problem types involve a reversal of the process which, in the thud type, can
be applied directly.
A number of important mathematical activities may require that the function schema be
reconstructed at yet a higher level where a function is not only an interiorized process, but
as a result of encapsulation, this process can be treated as an object by the subject. One
representation that could help with this is the set of ordered pairs (with the “uniqueness to
the right” condition) and another is the graph. We refer to chapter 9 for a discussion of some
of the difficulties in this connection. Inorder for a function to be the result of a mathematical
activity (such as solving a differential equation or setting up an indefinite integral) it must
be an object. Similarly, it seems to us that the elements of a set must be (epistemological)
objects and thus, all of functional analysis with its sets and even structured spaces of
functions depends on the object nature of a function.
REFLECTIVE ABSTRACTION
119
At the same time, and this may be a further reconstruction of the function schema, it
seems necessary in many situations that the subject think of a function simultaneously (or
at least in rapid succession) as both a process and an object. Consider, for example, the
various binary operations on functions such as point-wise addition, point-wise multiplication or composition. In reflecting on the addition of two functions, the subject must see this
as a binary operation which takes two objects and transforms them in a new, third object.
To actually do this, however, it would seem that the original two objects must be unpacked
or “decapsulated” back into processes, these two processes coordinated (by means of
“point-wise addition”) and the resulting process encapsulated into an object which is the
new function that appears as the result of the operation of addition. The same kind of
description can be used, as we have indicated above (see page 104), for composition of
functions.
As a final example, consider how complex, in these terms, is the following mathematically straightforward statement.
In the semigroup hom( G, °) of endomorphisms of a group G under the operation
of composition, the subset of those endomorphisms which are isomorphisms
form a group.
From our point of view it seems that to understand this statement (and check that it is true)
the subject must think of functions as objects since they form a set, and later a subset, and
then understand composition as we have described it to get a firm grasp on hom(G, °). Now,
in dealing with the group axioms, the cognitive interpretation of function goes back and
forth between process and object. The two interpretations must be coordinated in order for
the subject to grasp the somewhat subtle idea that the group identity is the identity function
and the group inverse of a function is its function theoretic inverse — and this connection
is not exactly an accident.
4. IMPLICATIONS FOR EDUCATION
We conclude this chapter with some comments on teaching mathematics in light of the
theory we have expounded. Our theory does not have anything to say about the affective
aspects of the teaching/learning situation. In particular, we have ignored Piaget’s notion of
equilibration (1985) which for him was the driving force behind the (re-)construction of
schemas. We have also omitted consideration of various issues such as discovery versus
guided learning, and large classes versus individual instruction versus small-group
problem solving. The main implication for education that our theory has, as far as we have
taken it, is that, whatever happens, in or out of the classroom, the main concern should be
with the students’ construction of schemas for understanding concepts. Instruction should
be dedicated to inducing students to make these constructions and helping them along in
the process.
We can offer one general conjecture about motivation. Whatever is the mechanism (le
source according to Piaget & Garcia, 1983) that moves students to make cognitive
constructions, tol earn, it seems to us to be a very natural human drive, on a par with the drive
for food or sex. We admit that this suggestion is inconsistent with the experience of most
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ED DUBINSKY
mathematics teachers, especially at the post secondary level, where students, other than
those with obvious talent for mathematics, do not seem to be interested at all. Our conjecture
is that this is due to the overall approach in the traditional classroom, where the goal, as
presented and defended by the teacher, is for the student to develop skills in computational
procedures, to display on examinations, and to “get a good grade”. For reasons which we
will elaborate below, the student cannot learn these procedures through understanding,
whereas he or she is presented by the teacher with a conflict-free way out — imitate and
memorize. Unsurprisingly, most students accept the offer and take this route. But imitation
and memorization do not lead to cognitive constructions and the result is that the students’
desire to learn through growth is suppressed. He or she is “turned off mathematics”.
Our experience has been that when a student is presented with concepts that he or she
is capable of understanding, when the constructions are possible for the student, and if this
capability is apparent to the student, then a natural drive to learn, to understand, to construct
is released and the level of effort and concentration on mathematical ideas leaves little to
be desired. This happens even in the presence of difficulty, when the student is confronted
with mathematical problems that her or his existing schemas cannot handle. As long as there
is something for the student to think about, as long as he or she perceives that cognitive
activity is leading to some sort of growth that could, eventually, lead to a solution of the
problem, then there is little difficulty in maintaining the students’ interest.
We will present, therefore, some examples of how traditional teaching methods do not
relate to conceptual understanding as the theory presented here explains it and close with
a few brief words about what directions an alternative approach might take.
4.1 INADEQUACY OF TRADITIONAL TEACHING PRACTICES
If we are correct in our hypotheses that learning involves applying reflective abstraction to
existing schemas in order to construct new schemas for understanding concepts, then it is
a trivial but critical observation that a schema can not be constructed in the absence of
prerequisite existing schemas. Traditional teaching often ignores this. Consider, for
example, a lecture on induction which begins, “Today, we are going to learn how to make
proofs by induction”. This statement assumes that the listener has a “proof schema”, that
is, he or she is conscious of various methods of proof which could be applied in a given
situation and is therefore capable of adding a new one. For any students in the class who
do not possess such a schema, the statement is not very meaningful. It gets worse when
actual problems, theorems to be proved by induction, are introduced. If a student’s function
schema does not include functions that deal with transforming integers into propositions,
then the very statement of a problem can be meaningless. Many students are probably
somewhat bemused when, later, the teacher is roaring the admonition, “You don’t prove
the statement for every n, you prove the implication from n to n+ 1!” If proof is meaningful
at all, it means that you prove something. For students who have not encapsulated the
process of implication and for whom proposition valued functions of the positive integers
are not objects, there may be no “somethings” in that admonition. If such prerequisites are
not dealt with, then it is no wonder that the student gives up on trying to understand (he or
she does not have the right tools) and, because success on examinations is both essential
and possible, looks for something to imitate.
REFLECTIVE ABSTRACTION
121
Another kind of difficulty arises with the predicate calculus. For many teachers,
understanding the meaning of a statement such as,
For every function f in A there is another function g in A such that f(g(x))=x for
all x
is essentially a language problem, not very different from understanding statements such as,
Every student in the class has a counselor who will be available to give advice
every Monday at 9 am.
But there is much more than language present — in both statements. For the first, according
to our theory, the student must have constructed (in her or his mind) a set of functions,
interiorized a process of iterating through this set picking an object, iterating again to pick
another object, and converting the two objects back to their function processes so that it is
possible to iterate once again, this time through the domain of the functions, testing the
equality. Only after these constructions are made can the problem be treated linguistically.
From our point of view, it is the constructions that provide the essential difficulties, the
language aspect being fairly trivial. Similar comments can be made about the second
statement which most students have little difficulty in understanding. This is because each
construction required to understand the second statement is made naturally, in the course
of normal student life and every day experience.
This point about languages, if generalized, suggests to us that the traditional lecture
itself, depending largely on linguistic transmission, is not very useful in helping students
acquire concepts in mathematics. Mental objects and processes, although they may well
exist in the mind of the teacher, cannot be transmitted verbally, or even with pictures, to
listeners. It is necessary that the listener engage in active construction.
Another difficulty, related to the problems of imitation, memorization, and verbal
transmission arises with examples. It is an article of faith with most mathematics instructors
that “lots of examples” must be an integral part of any instructional treatment. It is certainly
the case that involvement wih examples, whether it be doing exercises or thinking about
illustrations and demonstrations, serves to reinforce the concepts that are present in the
mind of the subject. We suggest, however, that working with examples may not help very
much with the construction of concepts. Indeed, we agree with Tall (1986) and it is a major
aspect of our theory that understanding mathematical ideas come from sources other than
looking at many examples and “abstracting their common features”, which is what happens
if there is only empirical abstraction. Something more is needed and we suggest that it is
precisely the construction aspects of reflective abstraction that we have discussed. It is not
clear that more than a very few examples are necessary to construct a concept: in some cases
(such as the integers in the initial construction of the concept of a ring) a single example
might suffice to induce appropriate reflective abstraction. As we have said, we cannot in
this chapter give full consideration to the question of how to induce conceptual learning,
but one might well reflect on the contrast between the repetitive examples that seem to be
required by conventional wisdom and the single, representative example which so often
seems to be in the mind of the mathematician who understands a particular concept. Tall
(1986) has referred to this as the generic example and it is a promising notion well worth
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ED DUBINSKY
further investigation.
We would go farther in our critical view of repetitive examples and suggest that practice
can even be harmful. Yes, the effect of practice will be to reinforce structures that are
present. But we would raise the question, what structures are these? Are they part of a
student’s concept image which conflicts with the concept definition (see Tall, 1977)?
Consider what happens when a teacher is explaining, with reference to conceptual
understanding, how to solve a certain kind of problem. As we have indicated, the student
may not be able to understand the concept behind the method. A general investigation of
what drives cognitive development may reveal that whenever a subject is subjected to
phenomena, some sort of construction takes place. To say that the student does not
understand could mean that the student has not and does not construct an appropriate
schema for the concept being explained. But if it is the case that something is constructed,
then it would have to be an inappropriate schema. This result is not inconsistent with what
teachers seem to observe in their students after making explanations. What, then, will be
the effect of following the explanation with “lots of examples”. The inescapable conclusion
is that the incorrect interpretations will be reinforced, and teachers will pay a heavy price
later on in efforts to correct students’ misunderstanding. This may well be a source of
epistemological obstacles (Cornu, 1983).
This argument is not sophistry. It is offered as an explanation of a phenomenon in
education that seems to be generally recognized, but not very well understood. It seems that
Van Lehn (1980) was referring to it quite specifically when he wrote, in a study of the
procedural “bugs” observed in students doing subtraction, “When a student has just
invented a bug, practice may solidify the bug in memory, thus making remediation more
difficult” (p. 47). It is possible that this effect also explains the near impossibility of
disavowing undergraduates of various misconceptions observed by Tall (1986), Cornu
(1983) and others concerning the concept of limits as well as the persistence, in the face of
a variety of instructional treatments, of reversal errors in algebra (Clement et al , 1981).
It maybe argued that these difficulties can be avoided by giving both examples and nonexamples with the examples graded so as to display various features gradually. This could
be reasonable, but there are dangers. The decomposition should be based on more than the
curriculum developer’s understanding of the mathematics. Also, there is no certainty that
the student will see the examples in the same way that the instructor did. Finally, this really
avoids the issue which is that in order to construct a mathematical idea it is necessary to be
mentally active. The really important issues in mathematics education have to do with the
nature of this activity and what can be done to foster it.
We do not conclude from this discussion that practice with examples should be
eliminated. In addition to reinforcing concepts, they may be important for students to
become facile with calculations, to develop a “feeling” that something is wrong, or that it
all “hangs together properly”. Indeed, it is pure speculation but it may be that practice with
a process will tend to induce the subject to encapsulate it. It could be that this is the essential
point in the relationship between procedural knowledge and conceptual knowledge
(Hiebert, 1986). We do not, therefore reject examples and practice. We only caution the
instructor to pay attention to what concepts the students have and what exactly is being
reinforced when they are set to do “all the even numbered exercises”. It is also important
to be aware of the types of mistakes that a student makes, how he or she tries to justify an
answer (whether it is “correct” or not) or just explain how it was obtained.
REFLECTIVE ABSTRACTION
123
4.2 WHAT CAN BE DONE
At this point we must conclude, not, unfortunately, with a prescription for putting things
right, but with a brief indication of a research and development program that we are engaged
in with the hope of constructing a viable alternative to traditional practice for helping
students develop advanced mathematical thinking. There are important connections
between what is written here and the ideas found in (Thompson, 1985a; Dreyfus &
Thompson, 1985).
Our instructional approach to fostering conceptual thinking in mathematics has four
steps.
• Observe students in the process of learning a particular topic or set of topics to
see their developing conceptual structures, that is, their concept images.
• Analyze the data and, using these observations, along with the theory we have
elaborated in this paper and the designer’s understanding of the mathematics
involved, develop a genetic decomposition for each topic of concern that
represents one possible way in which a subject might construct the concept.
• Design instruction that attempts to move the student along the cognitive steps
in the genetic decomposition; develop activities and create situations that will
induce students to make the specific reflective abstractions that are called for.
• Repeat the process, revising the genetic decomposition and the instructional
treatment, and continue as long as possible or until stabilization occurs (if it does).
To this general description we can add the fact that, in designing instruction, we have found
activities with computers to be a major source of student experiences that are very helpful
in fostering reflective abstractions. For example, it seems that if a student implements a
process on a computer, using software that does not introduce programming distractions
(such as complex syntax, constructs that do not relate to mathematical ideas, etc.), then the
student will, as a result of the work with computers, tend to interiorize the process. If that
same process, once implemented, can be treated on the computer as an object on which
operations can be performed, then the student is likely to encapsulate the process. It turns
out to be possible to create such opportunities for computer experiences relative to
reflective abstractions necessary to construct a wide variety of concepts in mathematics, but
that is a topic for another chapter.
We have used this approach to design instruction, with extensive involvement of
computers, to help students learn mathematical induction, predicate calculus and many
other topics in discrete mathematics. Present efforts are directed towards applying the
method to functions and to calculus.
ACKNOWLEDGEMENTS
We would like to thank P. Davidson, T. Dreyfus, H. Sinclaire, L. Steffe, and D. Tall who
read and commented on a draft of this chapter.
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III : RESEARCH INTO THE TEACHING
AND LEARNING OF
ADVANCED MATHEMATICALTHINKING
The final part of the book is devoted to a wide review of the literature of research
in the teaching and learning of advanced mathematics. Much of it comes from the
last decade and the task is only just begun. In Chapter Eight, A line Robert and
Rolph Schwarzenberger pause to consider the role of Advanced Mathematical
Thinking by looking at the transition from school to university to see if there is a
noticeable change. Although there is an increase in complexity and in the need
for formal definitions and proof, they find that the intellectual viewpoints already
developed by students are often carried over to advanced mathematics with
serious consequences in lack of success. In particular the training in elementary
mathematics to expect an algorithm to carry out the solution of a problem leads
students to seek similar success in contexts where this is no longer appropriate.
In Chapter Nine, Theodore Eisenberg considers the function concept, which is
given as a formula or a graph in elementary school, and shows that this proves
to be resistant to traditional methods of teaching via formal definitions. Bernard
Cornu reveals a crucial example of the discontinuity between elementary and
advanced mathematics when he investigates the concept of limit in Chapter Ten.
In elementary mathematics there are algorithms for arithmetic, for calculations
in trigonometry, for solving equations, but in advanced mathematics a limit
usually needs to be calculated by indirect methods which are quite diferent from
the student’s previous experience. In particular, the limit, as a process of getting
closer, may be encapsulated in terms of an “arbitrarily small quantity” rather
than conceived in terms of the definition, leading to serious conflict between
concept definition and concept image. The same story continues with Michèle
Artigue’s review of research into the teaching of analysis in Chapter Eleven and
with Dina Tirosh’s consideration of the concept of infinity in Chapter Twelve.
Here we find explicit conflicts between the concept of infinity in the limiting
process and the concept of infinity met in set theory. Often they are kept mentally
in different compartments, but when intuitionsfrom one area are brought to mind
in an inappropriate context, then conflict isinevitable. At this stage an experiment
is reported in detail which is designed to encourage students to reflect on the
126
nature of their beliefs and to reconstruct their knowledge.
Chapter Thirteen focuses on the way in which students build the process of proof.
Daniel Alibert and Michael Thomas review both the student’s success and
difficulties with proofs presented to them through traditional exposition and also
look to the possibilities of students engaging in the process of conjecture and
debate appropriate for the creation of new advanced mathematical ideas.
Finally, in Chapter Fourteen, we close this review by looking topresent research
andfuture use of the computer in advancedmathematics. Ed Dubinsky and David
Tall return to the question of the increasing use of computersand the way in which
the new technology may be changing the nature of the subject.
CHAPTER 8
RESEARCH IN TEACHING AND LEARNING MATHEMATICS
AT AN ADVANCED LEVEL1
ALINE ROBERT & ROLPH SCHWARZENBERGER
The seven preceding chapters of this book have examined various aspects of “advanced
mathematical thinking”; the six chapters which follow report research into the teaching and
learning of specific topics. At this point of transition it is therefore apposite to look back
to see in what sense the previous chapters have specified aspects of advanced mathematical
thinking which are distinct from mathematical thinking at a more elementary level. In
particular, in the first part of this chapter we will address the question:
• To what extent are there aspects of advanced mathematical thinking which are
specific to the learning of advanced mathematics at college and university?
We will also take the opportunity to look forward to the specific research to come. We will
find that the remaining chapters of the book address themselves mainly to specific topics
in advanced mathematics, to study the concept images which students develop and the
consequent difficulties which they face in their encounters with the subject. We will
therefore spend the second part of this chapter casting the net wider to look at other research
in advanced mathematical thinking, in particular to ask:
• What research has been done in the development of advanced mathematical
thinking which goes beyond the specifics of the acquisition of individual
concepts and the associated learning difficulties?
We shall find that the work to come in the book overlaps with the ideas discussed in earlier
chapters. For in studying the learning difficulties of individual students we are gaining some
insights into those who are becoming part of the community of advanced mathematical
thinkers. Certainly we will find differences between those who study mathematics as a
means to an end and those who study mathematics as an end in itself. But since mathematics
may be defined as “what mathematicians do”, to observe and reflect upon the activities of
advanced mathematical thinkers is in principle the only possible way to define advanced
mathematical thinking. And to study the difficulties of students will focus on central
epistemological problems in the growth of mathematical thought. Conversely the aspects
of advanced mathematical thinking discussed in the preceding chapters need to be used in
the analysis of particular learning difficulties, if only for the formulation of hypotheses
which may be tested empirically.
In this chapter, with an eye on earlier chapters and those to come, we therefore consider
the teaching and learning of mathematics at the post-secondary level, that is, in most
countries, students aged 18 and above taking specialist mathematics courses at colleges or
universities. We concentrate in turn on the two questions formulated above.
1 Thanks are extended to Ed Dubinsky for his initial translation of the draft of this chapter into Ehglish.
127
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ALINE ROBERT & ROLPH SCHWARZENBERGER
1. DO THERE EXIST FEATURES SPECIFIC TO THE LEARNING
OF ADVANCED MATHEMATICS?
In an attempt to answer this question we will take a broader view than just the psychological
factors which have been considered in the first part of the book to attempt to identify
differences between elementary mathematics in compulsory education and specialist
mathematics at colleges and university.
1.1 SOCIAL FACTORS
A sociological viewpoint focuses first upon the characteristics of the group of students
taking specialist mathematics courses. At first sight it appears that there is a major
discontinuity from secondary school education: the students are no longer attending
compulsory mathematics courses but are continuing voluntarily after a selection process.
One might think that, at least above a certain level, the students might already regard
themselves as professional mathematicians or alternatively that some learning difficulties
might have been removed by enhanced self-motivation and greater willingness to work.
However, a closer look suggests greater continuity than discontinuity. The students are
young adults but are not yet financially independent; they usually have to study a mixture
of subjects among which mathematics may not be a first priority and do not usually display
an attitude to work much different from that at secondary school. In some cases mathematics is a compulsory pre-condition for studying another subject which is the student’s main
interest. Teaching methods rarely treat the student as an expert but continue to lay stress on
the greater and more accurate knowledge of the professional mathematician. One cannot
compare “experts” and students. From a sociological viewpoint, a class of mathematics
students at college or university does not look much different from a class in secondary
school.
We conclude that it is necessary to look elsewhere for features specific to advanced
mathematical thinking.
1.2 MATHEMATICAL CONTENT
From a mathematical viewpoint we see an immediate change in the nature of mathematics
being taught. In particular there are more new concepts to teach in less time. Furthermore,
from a certain stage onwards there is a greater concentration on a small number of fairly
similar mathematical topics with each benefiting from the student’s experience in the
others, in a manner qualitatively different from anything experienced in earlier years. On
the other hand the student is faced by a wider range of possible problems arising in a variety
of different contexts which cannot all be discussed in full detail.
This has important consequences. The change in the ratio of quantity of knowledge to
available acquisition time means that it is no longer possible for the student to learn all new
concepts in class time alone; significant individual activity outside the mathematics class
is now an absolute necessity. The concepts themselves are also radically different from the
student’s previous experience; they often involve not merely a generalization but also an
abstraction and a formalization, as outlined by Tall in chapter 1 and Dreyfus in chapter 2.
RESEARCH INTO TEACHING AND LEARNING
129
(See also Robinet, 1984.) The student is required to absorb formalized concepts very
quickly, which historically evolved more slowly from a mass of special solutions to special
problems by many mathematicians. At the same time the student is expected to adopt new,
and often strange, standards of rigourous proof (discussed further in chapter 13).
This formalization involves the abstraction of specific properties which apply now not
only to the objects from which they were abstracted but also to any objects which obey the
properties. This involves the construction of a new mental object which is different from,
and therefore may conflict with, the old objects. It causes the long period of confusion which
first year university students meet and is a significant barrier to formal advanced
mathematical thinking. It gives rise to a fundamental discontinuity in the difficult transition
from elementary to advanced mathematics.
Examples are:
• the concept of convergence of sequences and series (completely formalized for
the first time by Cauchy in 1827, although the practicalities were already well
known); this represents a major generalization and unification.
• The concept of vector space which appeared as a formalization in the nineteenth
century although specific properties had already been used and understood by
physicists in special cases.
• The concept of group which was formalized as a useful unification after
approximately fifty years of working with special cases of groups of permutations and groups of transformations arising from analysis and geometry.
In each case note that the problems from which these concepts arose in an essential manner
are not accessible to students who are beginning to study (and expected to understand) the
concepts today: for example, from problems like Céaro’s lemma to the definition of
convergence via ε−δ methods, or from function spaces to the concept of an infinite
dimensional vector space, or from solutions of differential equations to the concept of a
group of substitutions.
This leads to an apparent feature that distinguishes advanced mathematical teaching
from teaching in elementary mathematics: at this level the teacher is usually obliged to
present the notions in a lecture course before getting the students to work with them: there
seems to be no question of allowing, or making, the students (re)discover certain aspects
of the notions before they are formalized. For example, there does not seem to be a way to
make the concept of convergence (in a rigourous sense) accessible to students in which the
ε-δ definition is likely to be constructed spontaneously. A lecture course at this level seems
to have a specific purpose, quite different from the way in which mathematics developed
historically.
Nevertheless we must acknowledge that, far from being a feature specific to advanced
mathematical thinking, this may be a result of the unimaginative teaching methods
currently adopted in colleges and universities. In the past, similar remarks could have been
made of most mathematical instruction at primary or secondary school level. But today,
with the advent of more use of concrete materials and computer simulations, and greater
emphasis on investigative work in many countries, it is less common to find new concepts
imposed prior to the students’ ability to solve problems and to construct the concepts for
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ALINE ROBERT & ROLPH SCHWARZENBERGER
themselves. One may hope therefore that improvements in teaching methods at advanced
level may ultimately remove this apparent feature specific to advanced mathematical
teaching and lead to a fuller appreciation of the full cycle of advanced mathematical
thinking processes mentioned in earlier chapters. In view of the cognitive conflicts
involved, such improvements must pay particular attention to the treatment of definitions
and to agreement on the acceptability of given levels of proof.
1.3 ASSESSMENT OF STUDENTS’ WORK
The teacher of advanced mathematics must also double as an assessor. Therefore much of
what is taught must also feature in the assessment process, indeed may be required by the
assessment process. Such evaluation usually requires the student to be able to correctly state
definitions and reproduce correct proofs as well as apply the theory in related problems.
Research into concept definitions (chapter 5) underlines that the correct statement of
definitions in examinations is liable to degenerate into learning by rote and that many
students are often unable to relate directly to the form of the definition. Corresponding
research into students’ ability to follow or produce proofs (to be reported in chapter 13)
confirms that students find proof difficult, with proofs by induction and proofs by
contradiction presenting particular difficulty. The only generality on which there is any
unanimity seems to be that the required proofs often appear to the student more as a diffcult
exercise in the use of stylistic set-pieces rather than as an exercise in convincing somebody
else of an uncertain result.
It is therefore necessary to question the fundamental premises behind the teaching of
advanced mathematics. Is the purpose of learning such a vast quantity of abstract concepts
part of a wider scientific, critical and even creative form of advanced mathematical
thinking, or is it merely to be able to reproduce learned materials and mechanical skills?
Does the assessment process actually reflect the learning that is desired?
It is difficult at this level of complexity, so eloquently formulated in earlier chapters, to
decompose the student’s activities into elementary tasks. The various aspects of their work
are so diverse, yet so interrelated and interdependent, that they will necessarily depend upon
components derived from different points in the course which may be widely separated in
time. It follows therefore that any short-term assessment of learning is likely to be unreliable
and may even be impossible. We conclude that the cumulative character of mathematical
knowledge implies that assessment must be long-term. Only in the long run might it be
possible to establish accurately any links of cause and effect between instruction and
acquisition of knowledge and understanding.
In this respect there seems little distinction between the problems of assessing advanced
mathematics and elementary mathematics in primary or secondary school. The experience
of short-term assessment in the U.S.A., and the inadequacy of current plans to test pupils
at ages 7+, 11+, 14+ and 16+ in England and Wales, suggest that the necessity of long-term
assessment actually applies to mathematical thinking at all levels.
RESEARCH INTO TEACHING AND LEARNING
131
1.4 PSYCHOLOGICAL AND COGNITIVE CHARACTERISTICS OF STUDENTS
A psychological viewpoint is more likely to focus on older student’s enhanced capacity to
reflect on their own activity, as advocated in chapter 2 by Dreyfus and chapter 7 by
Dubinsky. According to Skemp (1979) this is a feature of all intelligent behaviour which
we surely may assume is common to all of those who advance to college or university,
particularly to study mathematics. We could therefore hypothesize that advanced mathematical thinking includes the ability to distinguish between mathematical knowledge and
meta-mathematical knowledge (e.g. of the correctness, relevance, or elegance of a piece of
mathematics); in addition, that students at this level should carry a substantial quantity of
mathematical knowledge, experience of mathematical strategies, and working methods,
and be able to communicate them at least at some minimal level. However, an investigation
of pupils in the upper section (age 18) of French lycées (Bautier & Robert, 1987) showed
that the existence of such abilities was by no means uniform and depended crucially upon
the current mathematics teacher of the pupil concerned. No doubt studies of sixth form
pupils (age 17–18) in England would yield similar results. We conclude from this that these
factors may be relatively easy to change but, to the extent that they help to determine the
work habits of students at college and university, they must be explicitly taught.
During the years of secondary schooling students seem to develop preferred methods
of approach to solving problems which are relatively stable yet may prove too narrow for
a wider range of applications. For example, in geometry some students never use vectors
and always employ analytic methods, in analysis some students make systematic use of the
graphical approach to solve problems, some rely on formal symbolism and others on
numerical methods. Some students systematically attempt to use algorithms to solve
problems even when they are inappropriate for the problem in question. A study of students
beginning real analysis in the first year of university (Robert & Boschet, 1984) revealed a
range of systematic behaviour which was not always correct and showed that good students
were often characterized by their versatility in being able to change their mode of approach
to suit the particular task. Even in countries where entry to university is restricted to a low
percentage of the population, it should be noted that the level of attainment may vary
significantly and this difference in ability adds to the divergence in student performance.
In England and Wales this is indicated clearly by the wide variation in A-level grades (taken
at 18+) attained by students taking mathematics courses. In France the same pre-test on
mathematical analysis for students entering university (at age 18+) resulted in a 20%
difference between the mean score of students registered at the university and those
registered in a class preparing for the entrance competition of the grandes écoles for
engineering (Robert, 1984).
Later chapters look at specific research into topics in advanced mathematics. This will
reveal a wide range of difficulty which confirms the loss of meaning encountered in the
work of students at university level. It is as if the great complexity encountered by students
causes them to lose all means of control over the material.
To summarize, although the students in question may have full capacity for metamathematical reflection, they are often hampered by having too narrow a perspective using
particular methods of approach or through having conceptual difficulties with the subject.
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1.5 HYPOTHESES ON STUDENT ACQUISITION OF KNOWLEDGE
IN ADVANCED MATHEMATICS
Although advanced mathematical thinking in research mathematicians may be characterized by the full cycle of activity from initial intuitions and conjectures through abstractions
to definition and proof, it must be clear from the foregoing discussion that little of these
creative activities are found in most advanced mathematical students. It is a reasonable
hypothesis to suppose that the various cognitive mechanisms which govern individual
learning are not qualitatively different for students from those which apply to younger
children. Following Piaget, it is therefore important to stress the central role of individual
action at this level, especially through the active solution of problems and the idea that a
driving force for the construction of knowledge comes from the process of disequilibrium
and re-equilibrium as outlined in chapter 1. Also fundamental is Piaget’s notion of
reflective abstraction (chapter 7) in the meta-theory of mathematical learning and Vygotsky’s
hypotheses on the role of communication, with the teacher or with other students, in the
formation of personal concepts. From a social psychological viewpoint we also underline
the importance of socio-cognitive conflicts, of different conceptions about mathematics,
and methods of working in mathematics, as envisaged and expressed by pupils and
teachers.
The psychological and cognitive characteristics described in §1.4 above lead to a
particular interest in the existence of preferred methods for different students. Our
hypothesis is that there is a better prospect of successful mathematical learning for the
student with knowledge (however imperfect) in many contexts than for a student having
greater knowledge in a single context. This preference for versatile learning is based upon
repeated verification of the hypothesis with first year university students studying the
beginnings of real analysis (Robert & Boschet, 1984). It was found that the weakest
performances in the course of the year were by students having initial knowledge in very
few contexts (usually numerical) whereas the more successful students also had initial
knowledge in the graphical or even the symbolic context. Such results are easily verified
but must be viewed with caution – there is of course a danger that they say no more than
that those who succeed on a university course are those who knew the material already. The
crucial difference appears to be not the mere existence of prior knowledge but the difference
between two very different ways of thinking: the reductive effect of systematically
functioning through a preferred context as against the liberating effect of bringing to bear
changes of context in problems.
A final hypothesis concerns the possibility of explicitly involving students in their own
learning. This might be by helping students to participate consciously in their own learning,
that is by helping them to learn how to learn or, on the other hand, by the adoption of a
specific and explicit didactic contract by the whole class. It takes account of the reflective
capacity of older pupils and is particularly relevant where not all the desired concepts can
be approached via meaningful problems.
One can cite here similar ideas in the work of Schoenfeld (1985) who is concerned with
problem-solving, and not with the mere acquisition of knowledge. For him, performance
in “problem-solving” requires not only knowledge (“resources”) but also the use of
heuristics much more precise and detailed than that of Polya (1945). This use is governed
by a conscious control on what one is in the process of doing and requires compatible
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133
conceptions of mathematics. To achieve this, Schoenfeld engages in explicit teaching of
heuristics and explains the rules of control (that is, in our terminology, gives instructions
of a meta-mathematical kind).
1.6 CONCLUSION
The search for single features which are specific to the learning of advanced mathematics
proves to be inconclusive. Many proposed features are seen, on closer examination, to
display strong continuity with the learning of mathematics at younger ages. Nevertheless,
it seems that, when all these features are taken together, there is a quantitative change: more
concepts, less time, the need for greater powers of reflection, greater abstraction, fewer
meaningful problems, more emphasis on proof, greater need for versatile learning, greater
need for personal control over learning. The confusion caused by new definitions coincides
with the need for more abstract deductive thought. Taken together these quantitative
changes engender a qualitative change which characterizes the transition to advanced
mathematical thinking.
2. RESEARCH ON LEARNING MATHEMATICS AT THE ADVANCED LEVEL
We now turn our viewpoint from looking back at the nature of advanced mathematical
thinking and the characteristics of thecognitive nature of the processes to explicit empirical
research that has been performed in recent years. For the purpose of this chapter it is useful
to distinguish three broad types of research:
• First one can distinguish research centred upon student’s acquisition of specific
concepts. In general the object here is essentially to diagnose those difficulties
of students which relate specifically to the mathematical structure of the
concepts in question. This involves working “inside” the student, for example,
by attempting to define those cognitive processes which are assumed to be
involved in the student’s acquisition of the concept. Alternatively, the student’s
failure to acquire the concept may be studied by analysing possible conceptual
obstacles in the way.
• Secondly, there is research which is centred upon the organization of mathematical content. Such research may focus upon the sequence of problems and
courses offered to the students, on the advantages or disadvantages of particular
programmes (e.g. in France for engineers), on methods of making use of the
student’s own resources to enable them to discover concepts for themselves
prior to explanation by the teacher, and so on. Here one tries to adapt to advanced
mathematical concepts the work by Piaget on action and by Vygotsky on
communication, but again with an analysis appropriate to advanced mathematical learning (e.g. Brousseau, 1988; Douady, 1984).
• Thirdly, there is research which concentrates upon the external conditions
under which teaching and learning take place: can one discover and recreate
environments that are productive of advanced mathematical thought? For
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example, such an environment might be created by a meta-mathematical style
of teaching, or by appropriate explicit contracts between teacher and pupils, or
by some combination of these.
2.1 RESEARCH INTO STUDENTS’ ACQUISITION OF SPECIFIC CONCEPTS
This is the area that has attracted most research into the development of advanced
mathematical thinking. At the beginning of the 198Os, simultaneous research by Cornu in
France on the learning of limits (inspired by the work of Brousseau and the earlier ideas of
Bachelard on epistemological obstacles), by Vinner in Israel on concept images in the
learning of geometry and Tall on cognitive conflict in the learning of limits and continuity,
all focussed attention on the mental images conjured up by students which conflicted with
accepted mathematical definitions. This research will feature in chapter 10 in the writing
of Bernard Cornu. His idea of “spontaneous conceptions” (Cornu, 1981) and Tall &
Vinner’s joint work on “concept image” were the beginning of a whole sequence of pieces
of research on cognitive conflict in advanced mathematical concepts. At the same time
Robert (1982a, 1982b) was investigated student conceptions of sequences, distinguishing
between responses which were dynamic (with a sense of motion towards the limit) and
static (being “close” to the limit, or using a formal ε–δ definition). This contrast between
dynamic and static, which also featured in the work of Schwarzenberger & Tall (1978), Tall
& Vinner (1981), Cornu (1982) was an early characterization of the process-concept
duality which appeared quite separately in the work on conceptual entities (chapter 6) and
reflective abstraction (chapter 7). The process-concept duality is at the heart of the
difficulties with the function concept in all its complexity, which will feature in the next
chapter. The conflicts in the limit concept extend into the topics of more advanced analysis
to be discussed in chapter 11 and conflict, in a different way, with the concept of cardinal
infinity, to be discussed in chapter 12. Only in chapter 13 does the flavour of the research
change from conceptual difficulties with individual concepts to the procedural difficulties
with mathematical proof.
We shall leave the discussion of these conceptual difficulties to their proper context in
later chapters, here we will dwell in more detail on the second and third broad types of
research mentioned above.
2.2 RESEARCH INTO THE ORGANIZATION OF MATHEMATICAL CONTENT
AT AN ADVANCED LEVEL
We begin with an example from France of research which falls within the second broad
type. Various researchers, in particular Douady (1986), have examined the efficacy, for the
development of student’s problem-solving abilities, of organizing mathematical content
according to the following prescription:
• give a new concept which it is hoped that students will learn, design problems
which are undertaken before the formal study of the concept and which contain
the possibility of using the concept in their solution.
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135
If possible, the concept should arise in at least two contexts and in one of these the student
should have sufficient knowledge or, if the need should arise, insight to make good progress
on the problem.
In research which follows this prescription, based on “old” knowledge, the instruction
follows in clearly defined stages:
• explanation of the role of the concept in problems,
• institutionalization through the taught course,
• familiarization by means of further problems for reinforcement,
and
• transfer to contexts where the concept is not explicitly apparent.
To summarize this cyclic programme, due to Douady, one may say that the knowledge of
the pupils begins with old tools available for new objects (explanation), which are
successively brought into play (institutionalization) and become fully available (familiarization) to once again be new tools for yet newer objects (transfer). To be effective this
programme must be realized for a sufficiently large number of concepts and especially for
those concepts which cause persistent errors. Thus plans for instruction according to this
prescription must be based upon research of the first broad type into the mathematical
structure of content and persistent difficulties of students.
A good example of such work is that of Artigue (1987) on teaching the qualitative theory
of differential equations which is to be discussed in greater detail in chapter 11. She analyses
the relationship between the qualitative approach and the algebraic approach and, using the
above prescription, arrives at a programme of teaching in four phases:
Phase 1. Introduction to the qualitative approach and discussion of elementary
tools (isoclines, translations, invariance under symmetry and direction of
variation of solutions) with the help of simple examples of curve tracing
(explanation).
Phase 2. Exploitation of the new concepts through examples which require
matching of pictures of solution curves to the corresponding differential
equations (institutionalization).
Phase 3. Comparisons between the methods available for solving problems
using the qualitative or the algebraic approach by particular reference to the
differential equation
(familiarization).
Phase 4. Concepts and fundamental theorems of the qualitative theory of
differential equations, with proofs based upon pictures of solution curves,
dealing with barriers, trapping regions, funnels, attractors and so on ( transfer).
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This teaching programme was given to three groups each containing about 30 students.
They were then tested by an examination which included both the algebraic solution of a
linear differential equation and the qualitative study of the non-linear equation:
We note however that it is practically impossible to obtain valid comparisons of the
effectiveness of such a teaching programme and a more traditional course; the assessment
items appropriate in the two cases will inevitably differ. Nor is the above prescription
adaptable to every mathematical concept in higher education. First, there are some concepts
which do not admit a “good” initial problem which can precede the formal study of the
concept. Secondly, it may be difficult to find suitable problems from a new domain where
the concept is not immediately apparent.
In like manner, Tall (1990) seeks starting points in learning sequences which are
meaningful to students at their current state of development yet contain the potential to grow
into fully fledged mathematical concepts. His archetypal example is that of “local
straightness” which students can visualize in a simple intuitive way, giving meaning to the
derivative as the gradient of a locally straight curve, yet containing the seeds of the
definition of a differentiable manifold. He calls such a concept a “cognitive root” for a
theory, as it is intended to take root in the current cognitive understanding of the student yet
grow into a fully fledged formal mathematical notion.
Such cognitive roots also prove hard to find. Indeed, a good cognitive root may often
prove to be a process that needs encapsulating as a concept, such examples might include
the process of counting as the cognitive root of whole number arithmetic, an input-output
process as a cognitive root for function, or the idea of an input-output machine, whose
output accuracy can be controlled by controlling the input accuracy, as a cognitive root for
a continuous function. All of these examples later require an encapsulation of the process
as a meaningful object and then an abstraction of the fundamental properties to give a formal
definition requiring the construction of the corresponding abstract object. Such routes from
informal sources still need to pass through difficult encapsulation and reconstruction
phases before being brought to formal fruition.
2.3 RESEARCH ON THE EXTERNAL ENVIRONMENT
FOR ADVANCED MATHEMATICAL THINKING
Most research which falls in this third broad type has been concerned with teaching which
encourages meta-mathematical reflection. There are, a priori, many ways of achieving this.
One method is to teach directly in the manner in which we think that one learns by
explaining the importance to discover and learn simultaneously, by making clear the
effectiveness of general methods, and by clarifying the basis of the new knowledge. This
is exemplified by the Grenoble experiment, described by Alibert in chapter 13, in which
scientific debates are held between students in a lecture room. It is designed to encourage
the creative, synthetic side of mathematics through conjecture, discussion and argument to
give the full cycle of mathematical activity culminating in formal proof that was suggested
RESEARCH INTO TEACHING AND LEARNING
137
as the true nature of advanced mathematical thinking in chapter 1. A basic difficulty of this
method is that not all students do in fact learn in the same way, so that some way must be
found of enabling students to make their own choices and decisions. It is important to define
clearly the contract between teacher and students in the first few sessions; this may then be
justified by the dynamics of learning which it can generate, and used subsequently as a
procedural framework.
The objectives and hypotheses behind such a contract are articulated by Legrand et al
(1988). The objective is to permit the majority of students to understand the meaning of the
algorithms that they are using and to achieve positive ownership of the mathematical
concepts which arise. For this theresearchers recommend a contract based on the following
conditions :
• suppression of indications of truth, especially by the teacher,
• restitution of an official, and approved, status for uncertainty,
• creation of a climate that encourages discussion and debate between students,
• direct engagement in mathematical material by students,
• devolution by the teacher of authority and competence to the students collectively,
• use of complex problems to prevent premature or simplistic algorithms which
inhibit concept formation.
Further details will be given in chapter 13.
We note that a very early example of such an approach, in this case at graduate level,
was the Texas seminar in point-set topology of R. L. Moore in the 1940s. In this activity
Moore also encouraged the students to formulate and prove their own theorems without
having formal lectures on the topic. For a personal description, and other remarks on
advanced mathematical thinking, see Halmos (1985).
A second way to encourage meta-mathematical reflection is to teach generalized
method (Robert et al, 1987). By this is meant a set of procedures applicable to a collection
of similar problems; thus the use of the method implies a certain class of problems to solve
and the fixing of the available tools, techniques and methods of attack. It may also imply
the development of a certain number of general ideas such as the usefulness of changing
the context, the strategy, or even the formulation of the problem. It may include more
specifically mathematical points such as the idea of considering the parameters which arise
in a problem as variables, or the idea of looking for invariants which characterize the
problem that is being considered. It may include the heuristics of Polya or specific
techniques of different conceptual fields (geometry, analysis, ...). The work of Schoenfeld
already cited illustrates this point of view, clearly showing the difference between explicit
instruction in problem-solving techniques and the more common, but much less effective,
implicit instruction based on unspoken imitation. Schoenfeld also demonstrates that it is
absolutely necessary to clarify the heuristics of Polya by developing more examples of their
use in different contexts and he insists on the necessity of the student’s control of their own
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problem-solving. A similar approach is advocated by Mason et al (1982) where the focus
is purely on the meta-mathematics of formulating, refining, attacking, reviewing problems
and their solutions, using general techniques such as specializing to simple cases, or
generalizing through systematic specialization seeking patterns. It seeks to give students
confidence through negotiating difficulties when they come up against a seemingly
impenetrable barrier and the sense to check insights that come suddenly yet may be flawed.
Thirdly, one can suggest instruction based upon the activities of mathematicians
themselves, for example through the study of historical mathematical texts. A difficulty
here is the barrier of notation and language as well as the extreme difficulty of many
concepts when formulated in their original contexts.
We note that each of these three ways of encouraging meta-mathematical reflection may
be suitable in one mathematical context but not in another. None is a prescription for
automatic success in any area of advanced mathematics. Moreover, there is no guarantee
of successful transfer of such meta-mathematical knowledge from one context to another;
no more, in any case, than of successful transfer of mathematical knowledge.
To encourage the transfer of meta-mathematical knowledge it is clearly necessary to
create opportunities for such knowledge to be used. It is therefore essential to introduce
situations which complete a round trip between meta-mathematical instruction and
mathematical experience. For example, Robert & Tenaud (1989) have conducted research
on the learning of geometry in the final year (age 18) of the French lycée based upon the
following scenario: the students receive explicit instruction in the methods of geometry
and, simultaneously, work in small groups on geometric exercises, set without hints, for
which the solution is facilitated by bringing these methods to bear. Our hypothesis is that
this scenario allows the students to improve their ability to get started on a problem, that
it permits the teacher to then usefully continue the instruction in methods, and that all this
can serve to accelerate the students’ learning of geometry. A similar philosophy, though
less carefully structured, lies behind the current research into explicit instruction in
investigative methods for secondary school children aged 11-16 in England and Wales.
An additional feature of the research of Robert & Tenaud is that it provides case studies
for the use of groupwork to encourage meta-mathematical reflection. They place three to
four students in each group, and find that the recent instruction in methods leads to group
discussion about what method to choose to get started on the problem. Not only do the
students become more aware of the effectiveness of the methodological approach but they
become more receptive to further instruction in methods (again, always returning to further
groupwork on problems), and they appear to become better able to solve more difficult
problems. Thus it is the combination of meta-mathematical instruction, return to appropriate problems, and the use of groupwork which together seem to accelerate the students’
learning.
Despite all these positive indications, many questions on the encouragement of metamathematical reflection remain unanswered. What is the optimal mix of meta-mathematical instruction and ordinary mathematical instruction? What method of meta-mathematical
instruction should be used, and how does this depend upon the particular area of
mathematics? Where more traditional teaching has led to particular gaps in understanding,
or to misconceptions, how can the use of meta-mathematical instruction avoid the same
failure? Again, does the answer to this question depend upon the previous experience of the
students?
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139
On the one hand there is a need for much more experimental research to answer all these
questions. On the other hand, such research can do no more than provide illuminating case
studies, and it might be argued that there are unlikely to be general answers to such
questions. Background of the students and the social context of their study are important
variables which will vitally affect any results. The setting up of research which requires
non-traditional organization of teaching may in any case provoke objections both from the
institution, which in many countries works under rigid constraints, and from the students
and teachers who have deeply held beliefs about the teaching contract between them. We
have seen that such views, at least on the part of students, are changeable. Teachers also can
change their views, but may sometimes use alleged rigidity of institutional structures as an
excuse not to do so. Added to these difficulties is the problem of objective evaluation and
comparison. At the level of the individual student we have already mentioned the need for
measures of long-term as well as short-term progress; at the level of different instructional
treatments it is necessary, for any valid comparison, to find tasks that are symmetrical with
respectto, or “equidistant” from, the teaching and this may not be possible. We cannot count
on a clear and convincing general result which will influence the views of unconvinced
teachers. We can, however, hope for case studies which follow students’ long-term
development provided that they are not dispersed too widely. All these caveats explain,
perhaps, the still very tentative and hypothetical character of this type of research.
3. CONCLUSION
How can we bring together the discussion of what features, if any, are specific to the
learning of advanced mathematics in the first part of this chapter, with the examples of
research on the learning of advanced mathematics in the second part?
• One still finds over and over again a certain number of difficulties, mentioned
in the first part of this chapter, that are related to the complexity of the contents
of advanced mathematics: abstraction and formalization being particular stumbling blocks. All research in mathematical education shows that there seems to
be no easy way of avoiding the difficulty of abstraction discussed in earlier
chapters by Tall, Dreyfus and Dubinsky. On the contrary it seems essential to
develop new ways of approaching it.
• Putting aside the research directed at particular complexity of mathematical
content, examples of which are described in subsequent chapters, the remaining
research (of the second and third broad types described in the second part of this
chapter) does have a common feature: the attempt to change the scientific
environment of the students to give them a new and more authentic relationship
to knowledge that is more akin to that of experts (i.e. researchers and practitioners) than to that of school pupils. It is here perhaps that we might find a genuine
application of advanced mathematical thinking: having available in full the
resources of the scientific spirit to control, create, and systematically introduce
methods of learning and even, perhaps, of effective research.
CHAPTER 9
FUNCTIONS AND ASSOCIATED LEARNING DIFFICULTIES
THEODORE EISENBERG
1. HISTORICAL BACKGROUND
The function concept has become one of the fundamental ideas of modem mathematics,
permeating virtually all the areas of the subject. Yet, despite being a powerful foundation
for the final edifice of mathematics organized in a formal Bourbaki style, it proves to be one
of the most difficult concepts to master in the learning sequence of school mathematics. In
part this is due to the layers of complexity and the numerous sub-notions associated with
the concept, for even at the most elementary level functions can beapproached in a variety
of contexts, and depending upon the approach taken, various difficulties surface from the
outset. It has been suggested that the problem of mastering this concept, or any mathematical concept, is simply a task-sequencing problem: provide the student with good exposition
and appropriately structured exercises to reveal various aspects of the notion, and students
will understand, internalize and master the notion. But this is has proved to be theoretical
fantasy. For at this point in time, after millions of dollars have been spent on research in an
effort to understand how we acquire mathematical concepts, we still do not know how
people learn. It is true that learning takes place, but the precise mechanism by which it
occurs is unknown. Indeed, we do not even know how to structure activities to teach
children how to order the natural numbers (Sinclaire, 1987). Children eventually learn to
do this, and we can roughly identify the stages through which this learning passes. But how
it occurs, and identifying and sequencing the essential experiences children must have to
acquire this skill, remains a mystery. It is the same with acquiring a deep understanding of
functions. Initially this was approached as a problem of task-sequencing in order to give
children the foundations for the logic of mathematics, but in thls regard, with hindsight, we
can see that such an effort was a total failure.
In the 1960s new mathematics movement a succession of influential committees
proposed that the function concept be used as a unifying factor in school mathematics.
By grade 6 the word function ... (and the ideas behind it) should be established firmly ... .
(Cambridge Conference, 1963, p. 98)
Concepts like . . . function . . . can be introduced in rudimentary form to very young children, and
repeatedly applied until a sophisticated comprehension is built up. We believe that these concepts
belong in the curriculum not because they are modern, but because they are useful in organizing the
material we want to present.
(Cambridge Conference, 1967, p. 10)
A formalist approach to functions soon found its way into the classroom based on the
encouragement of influential professors, including Adler (1966), Beberman (1956), Begle
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141
(1968) and Fehr (1966). The warnings of others, including Kline (1958), MacLane (1965),
Wilder (1967), and Buck (1970), went initially unheeded until it began to be seen that this
supposed logical approach to the curriculum did not work.
At the definition level the function concept can be introduced in a variety of contexts,
through arrow diagrams, tables, algebraic description, as a black input-output box, as
ordered pairs, etcetera. Of all of these approaches the pedagogically weakest and nonintuitive one seems to be the approach using ordered pairs. Here, a function is defined as
a certain sort of set; one which is made up of ordered pairs in which no two ordered pairs
have the same first element and different second elements. This seemingly innocent
definition proved to conjure up all kinds of logistic and epistemological problems, which
incredibly, were often addressed explicitly in some school curricula. For example, logically
it might be considered necessary to explain what is meant by an ordered pair and the phrases
first element and second element? Norbert Wiener found a way around this problem by
defining the ordered pair (a, b) to be { { {a}, ø} , { {b} } }, and variations on this notion of a
function were taught at literally all levels of the curriculum, from high school (Kline,
Oesterle & Willson 1959) to graduate school (Cohen & Ehrlich, 1963).Commenting on this
approach to functions Buck stated:
Experience seems to show that the ‘a function is a class of ordered pairs’ approach is one which
imposes severe limitations upon the student and provides a poor preparation for any further work
(1970, p. 255)
with functions, either in school or later.
Buck was being kind. Others made sharper criticism of this formalistic approach to school
mathematics (Hammersley, 1968), and of the perceived philosophical error of building the
new mathematics movement upon a Bourbaki-type foundation prominent in higher
mathematics at that time, Thom (1971) and Kline (1970).
Many teachers soon became aware of the limitations of the formalistic approach and
used more than one of the aforementioned settings for introducing the concept. This was
done with the hope that the notion would be well understood in one context and that this
deep understanding would be transferred to the other context settings. It did not happen.
Students with a clear idea of function notions in one setting had no idea how they applied
in other settings. For example extending the concept of a zero of a function in one variable
to functions of two or more variables was beyond most high school students. If transfer of
learning occurs between contexts research has shown that the situations have to be very
similar, and often the transfer has to be specifically pointed out (Carter, 1970). This has been
known to researchers for a long time, and was also familiar to many teachers. Indeed, the
inability of students to make “obvious” connections led Sweller (1990) toquestion whether
or not transfer of learning exists in the real world as something more than a theoretical
construct. Although some recent studies show that transfer of learning can in general be
obtained (Brown & Kane, 1988; Case & Sandleson, 1988; Lehman, Lempert & Nisbett,
1988), it seems that in the domain of school mathematics, situations in which transfers do
occur are only epsilons apart from one another; the medium level jumps and quantum leaps
which might be hoped for seem to escape all but the most able students.
This phenomenon necessarily led to a multiple embodiment approach – introducing the
function concept in a variety of settings with the hope of effecting transfer of learning. Some
textbooks developed all new function notions in one base setting (such as a graph, ordered
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pairs, or table) and then applied them to other settings. Other texts used a smorgasbord
approach, letting learners sift-out and build their own concept images and modus operandi
for internalizing the concepts. (See Harel, 1988.) But in spite of all of these efforts, the
function concept was and remains difficult for students to learn (Vinner & Dreyfus, 1989).
2. DEFICIENCIES IN LEARNING THEORIES
A major obstacle in discussing the learning problems associated with functions in
particular, and mathematics in general, is that there is no generally accepted theoretical
framework as a basis for discussion. Functions are found everywhere in mathematics; the
binary operations of ordinary addition and multiplication can be thought of as mappings
from X into , the mechanics of solving the standard inequalities problems encountered
in algebra and trigonometry can be thought of in a function format, as can many of the
standard problems in differential and integral calculus. All techniques, which form a major
component of school and university mathematics courses, can be discussed from a
functional approach.
Many attempts have been made to build theoretical models for such discussions.But two
general types of problems emerge with these models. One is that they are episodic in nature,
trying to explain why certain errors occur. But the errors are context dependent – and the
explanations necessarily differ from context to context. The second general problem is that
when more global schemas are developed, they seem to be too general to be of any use to
describe and prescribe remedies to overcome the learning difficulties of specific situations.
In short, models of learning are either too specific or too general. Let us look at three models
currently in vogue which try to describe how mathematical concepts are learned which have
been applied to the learning of functions.
Gagné (1970) developed a theory of learning mathematics which is based within a
behavioristic framework. He claims that if learning occurs as a result of instruction, then
one can do something after instruction that he could not do before instruction. Learning
implies a change in behavior; changes are observable and therefore measurable. Hence, for
Gagné, evidence of having learned topic X is being able to perform a specific set of tasks
which are related to X. He decomposes topic X into sub-topics and each sub-topic into its
pre-requisites. For each sub-topic there is a set of tasks which can be used as evidence that
one has mastered that particular subtopic. A tree of pre-requisites unfolds, and continues
to unfold until it can be assumed that all tasks for a particular level are within the knowledge
base of the learner. That is to say, Gagné builds a hierarchy of tasks through which one must
progress to master a particular topic. The problem here is that such hierarchies rarely
coincide with real learning. It may be possible to have all the prescribed sub-topics yet not
be able to learn the next stage; the subdivision may be such that the low-level detail may
obscure the whole picture. The problem with this theory is that it is task-oriented and says
little about the learning characteristics of the student.
Schoenfeld, Smith, & Arcavi (1989), on the other hand, have developed a modus
operandi for analysing how mathematical understanding evolves. Their chapter in Glaser’s
text, Advances In Instructional Psychology “. . . focuses on the changes in the mathematical
understandings of one student as she explored.. . the graphs of simple algebraic functions
in the Cartesian plane.. .”. It provides a fine-grained characterization of the structure of the
FUNCTIONS AND ASSOCIATED LEARNING DIFFICULTIES
143
student’s subject matter understanding, and a description of the nature of the change in her
knowledge structures as a result of her interactions with the learning environment (page 1).
The essence of their approach is that acquired knowledge must pass through four levels of
understanding.
Level 1 is concerned with a macro-organization of knowledge at a schema level;
for example understanding that in the equation of the line L: y=mx+b, the mand
b represent the slope and the y-intercept.
Level 2 deals with compiled knowledge, macro-entities and entailments; for
example if m>0 the line rises, if |m| is large, the line is steep, the point (0,b) is
the y-intercept of the line, etcetera.
Level 3 relates to the fine grained superstructure supporting the knowledge, such
as realizing that the slope (y 2–y 1)/(x 2–x 1) of a line through two given points can
be thought of as two directed line segments and that when x=0 in y=mx +b one
gets the y-intercept of the line.
Level 4 is when the limited applications context out, and individuals construct the
conceptual atoms that are seen at level 3.
In acquiring a deep understanding of any topic one must necessarily pass through these
levels.
Dubinsky’s theory of learning adapts aspects of Piagetian theory to the acquisition of
mathematical concepts (1988 and Chapter 7 this text). Each individual constructs his own
mathematical knowledge through the process of reflective abstraction. The theory is
concerned with the way in which processes are interiorized to become routinized,
encapsulated to be considered as concepts, coordinated (by following one procedure by
another), reversed (to be performed in the reverse direction) and generalized by being
placed in a broader context. The meanings of these notions are given more fully in Chapter
7. Dubinsky believes that an individual’s mathematical knowledge is concerned with their
tendency to respond to a perceived problem situation by (re-)constructing (new-) schema
in an effort to deal with the situation. Learning is episodic. He analyses such episodes and
puts a partial ordering on the subject matter to produce what he terms a genetic decomposition, then investigates how the genetic decomposition meshes with the schemata of the
student. (In addition to the examples given in Chapter 7, see his paper with Lewin on the
genetic decomposition of induction and compactness, 1986).
There are obvious similarities between these theories. Each has aspects of decomposing
the subject matter into a learning sequence, in which Schoenfeld et al and Dubinsky try to
determine a closeness of fit between the content decomposition and the changing schemata
in the student’s mind. In particular, Dubinsky’s formulation proves to be specifically
apposite for the function concept because the function is both a process (input - output) and
a mathematical object that needs to be treated as a conceptual entity(as described in chapter
6). Therefore the technique of encapsulation of the function process into a mathematical
object seems to be a perfect match for the development of the function concept. Indeed, by
getting the students to concentrate on the process of programming functions in an
appropriate language which allows the functions (or rather their symbols) to be used as
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objects, Dubinsky has shown some success in the encapsulation process.
But herein comes the major problem which Dubinsky (1988) explicates:
It is not possible to observe directly any of a subject’s schemas or their objects and processes.
(p.7)
If this is the case, then it makes it very difficult to map out how one learns in any terms but
general ones. States of knowledge can be observed, but the actual movement from one state
to another cannot. As Sinclaire (1987) stated, learning occurs, but when observing its
growth, one must be content with seeing it in progressively higher but static states. Because
of its complexity it is impossible to observe a continuous development. And it is only
possible to infer the cognitive structure of students’ conceptualizations through the concept
images they evoke in written and spoken communication. So how can one be absolutely
sure that the student has an abstract function concept, related to the function definition, as
opposed to a generic function concept that can handle all the tasks in a given context, such
as the programming of specific functions?
A major problem with developing Gagnéan type hierarchies and “fine structures” of
genetic decompositions is that they can get very complex. The lists of prerequisites which
must be mastered for even the simplest of the tasks get unwieldy; the topic gets over
atomized – and the whole seems to be much more than the sum of its parts. In a Gagnéan
hierarchy the integration of the component steps may not be made and, in many cases, seem
to result in the accumulation of isolated skills. So to this point in time, although there are
bench marks against which one’s knowledge of functions can be measured, the content
matter delineates the particular states of one’s knowledge. Although a lexicon seems to be
developing à la Schoenfeld et al and Dubinsky to discuss learning in more general terms,
attention is mainly drawn to episodc learning, and to catalog the troublesome areas students
have with the function concept. We rarely know why these problems occur, nor do we really
know how to guarantee a cure. With this in mind, we now focus on some of the fundamental
and documented problems in understanding basic notions of functions.
3. VARIABLES
The role of a variable is imperfectly understood. Although it is the building block for all
abstractions in mathematics, its meaning escapes many students. Wagner, Rachlin &
Jensen (1984) worked with small but carefully chosen groups of ninth grade students in
Athens, Georgia and Calgary, Canada. Collectively, the students represented the whole
spectrum of ability levels. The purpose of their study was to investigate learning difficulties
in elementary algebra. One of the tasks given was to solve an equation for a particular
variable: the investigators changed the name of the variable and the students were asked
to solve the “new” equation for the “new” variable. With the solution to the original problem
in front of them, one-third of the students resolved the problem from the start. Wagner
(1981) found similar results with even older high school students, average age 16+. The
students seemed to react to such exercises in one of two ways. Either they would accept the
change of variable name and state that the letter makes no differenceas long as the numbers
stayed the same, or they would regard the change of variable name as producing a
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completely new problem. Transfer of learning was not there. Wagner related this lack of
understanding to Piaget’s theory of conservation of variable and she tried to categorize
several of the misconceptions. But why didn’t transfer occur, especially with such a basic
idea? On a large scale, even if 25% or even 15% react as did the older students in Wagner’s
study, the situation is alarming. Who is at fault? The teacher? The students? The authors?
The material? The concept of variable is surely not that difficult to understand – or is it?
Arcavi & Schoenfeld (1987) developed a unit to sensitize teachers to how elusive the
concept of variable can be and why their students often have trouble with it. Part of their
unit demonstrates the variable meaning of variable. The notion of concept images and
concept definitions have been discussed in chapter five and also in several papers (Tall &
Vinner, 1981, Vinner, 1983), but the main idea therein is that students develop mental
pictures of concepts and definition by circumscription. Exemplars and non exemplars forge
the concept, which emerges intobeing inclusive and exclusive. But “the set of mathematical
objects considered by the student to be examples of the concept is not necessarily the same
as the set of mathematical objects determined by the definition” (Vinner & Dreyfus, 1989).
Arcavi and Schoenfeld’s work alerts teachers to the hidden problems of learning the
concept of variable itself. It is not as simple as writing a definition on the board. Wagner’s
research drives this point home; but teachers, who themselves have internalized the variable
concept, seem to pay little attention to it. As Freudenthal (l983; p. 469) states:
I have observed, not only with other people but also with myself ..., that sources of insight can be
clogged by automatism. One finally masters an activity so perfectly that the question of how and
why [students don’t understand them] is not asked anymore, cannot be asked anymore and is not
even understood anymore as a meaningful and relevant question.
4. FUNCTIONS, GRAPHS AND VISUALIZATION
Although most students can graph simple functions, they often treat the graph of a function
as something external to the function itself and not really part of its essence (Vinner &
Dreyfus, 1989). Moreover, they may incorrectly relate to data in graphs of functions they
themselves drew. The following problem was on a recent matriculation exam in Israel for
students in the least demanding mathematics track.
A circle of radius 8.5 cm is circumscribed about a triangle with angles of 100°,34°, and 46°. Find
the radius of the inscribed circle.
It is almost impossible to start this problem without first drawing a sketch. But thousands
of the students who took this exam drew the sketch incorrectly, and never referred to it
again. Students drew the diagram without taking into account the givens of the problem,
which force the triangle to sit in a half-circle. But the odd part in all of this is that it really
doesn’t matter; drawing the diagram correctly seems to complicate the solution process!
This problem was also given to a group of ten high school mathematics teachers. Eight
of them drew the problem incorrectly, and only half of them solved the problem within the
15 minute time span allotted. In another problem, more than 300 students were asked to find
the equations of the tangent lines to the circle x 2+ y 2 = 10 which pass through the point (5,5).
Eighty percent of them, including their instructors, did not draw a sketch for the problem.
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Similar observations were obtained when students were asked to find values for a and b
such that the line 2 x + 3y = a is tangent to f( x )= bx 2 at x =3, and when students were asked to
solve the inequality x–3≥ (2 x +9), both were approached analytically – without utilizing the
visual interpretation of the givens (Selden, Mason & Selden, 1989).
It seems natural to view many aspects of general mathematics, and functions in
particular, in a graphical way. But students simply do not have this concept image of a
function. They seem tied to processing information and solving exercises analytically, not
visually. This is not a new problem. Historically, the “visual concept image” for a function
versus an “analytic characterization” was debated for hundreds of years, with the visual
image eventually losing out (Kleiner, 1988).
In the wake of this development, the geometric conception of function is gradually abandoned. A
new tug of war soon ensues (and is, in one form or another, still with us today) between this novel
“logical” (“abstract”, “synthetic”, “postulational”) conception of function and the old “algebraic”
(“concrete”, “analytic”, “constructive”) conception.
The tug of war between thesecharacterizations actually became a three way battle, with the
“logical” description entering into the picture too. Today, this conception problem seems
to have been settled in favor of the analytic description, at least mathematically speaking.
But something much deeper seems to have happened. The nature of mathematics itself
seems to have been determined in the process. Consider Hilbert’s comment (quoted in
Hadamard, 1945):
I have given a simplified proof of part (a) of Jordan’s theorem. Of course, my proof is completely
arithmetizable, (otherwise it would be considered non existent but, investigating it, I never ceased
thinking of the diagram (only thinking of a very twisted curve), and so do I still when remembering
it. I cannot even say that I explicitly verified (or can verify) every link of the argumentas to its being
arithmetizable (in other words, the arithmetized argument does not generally appear in my full
consciousness). However, that each link can be arithmetized is unquestionable as well as for me as
for any mathematician ... I can give it instantly in its arithmetized form, which proves that
arithmetized form is present in my fringe-consciousness.
Hence, although he had an intuitive and visual understanding of the theorem, he did not
consider it mathematics unless it was arithmetizable. This point of view has dominated the
20th century mathematics (Davis & Hersh (1986), it is perpetuated in the classroom and
it is the view most students have of mathematics in general and functions in particular.
Mathematics is analytic in nature – that is the nature of the beast.
Clements (1984) has shown that although the mathematically talented can think
visually, they have a strong tendency not to do so. Krutetskii (1976) went even further. He
claimed that the ability to visualize is not a pre-requisite for having mathematical talent.
This certainly provides evidence for the Hilbertian view that mathematics should be
formalistic, even though as in Hilbert’s proof of Jordan’s theorem, the formalism emerges
from an intuitive-visual base. This visual base is often down-played in the classroom, With
the emphasis being placed on the analytic formalization of it. For many students this is
wrong; they can “see” the mathematics, they just cannot formalize it. They need to be taught
how to analytically describe their visualizations (Vinner, 1983).
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Calculus is a case in point. Tall (1978) and Mundy (1985) have shown that students
simply ignore the inconvenient. Ninety percent of first year calculus students simply
dropped the absolute value sign in integrating
|x| dx. Dreyfus & Eisenberg (1987)
have observed that students seldom relate to the graphs they themselves draw. Students
sketched a correct graph to evaluate
|x2+5|x|+6| d x but they simply ignored the
sketch when evaluating the integral. Schoenfeld ( 1985) has made similar observations. The
list could go on, but the point is that students do not see elementary functions of a single
variable as being inherently tied to a graph. What is worse, neither do their instructors.
Dreyfus & Eisenberg (1986) gave the following problem to professors of mathematics,
sin(x) [cos(x)+3x2 – xsin(x)] dx,
and asked them how they would go about solving the problem. All started by saying that
they would try integration by parts or substitution, but not one saw initially that the function
is odd, and because of the limits of integration, had to be zero. These professors of
mathematics did not see the function as a graph because they failed to spot the oddness of
the function which trivializes the solution. Other problems, in which both visual and
analytic solution methods were equally likely, were presented to the professors. Their
overall tendency was to approach the problems and process the information therein
analytically. Thinking visually was foreign to them, so how can their students be expected
to develop their visualization skills?
The work of Janvier (1978), Karplus (1979) and Ponte (1984) also point to the fact that
students simply do not understand some of the graphs that they themselves drew. Indeed,
these studies show that many students cannot interpret graphs. That is, they do not
understand the relationship the graph describes between the independent and dependent
variables. As an example to illustrate this dichotomy between the graph of a function and
the function itself, consider the case of a group of 40 post calculus students asked to find
the inverse of a graph which was given in both algebraic and graphical form. Ninety percent
of them were able to do this for the algebraic case with 55% being able to justify why their
procedure worked. Thirty percent knew the “reflecting through the line y=x” technique for
the geometric case, but not one could justify the procedure. (And not one knew the
technique of flipping the paper and rotating it 90°). The students obtained correct results
without understanding why they worked. This conceptof the inverse function wasdivorced
from a visual interpretation. Hadar, Zaslavsky & Inbar (1987) have categorized errors like
these that students make with functions. Thomas (1969) has tried to identify the ages and
stages of development students must be at in order to learn specific function topics. But such
categorizations and identifications are only first steps and no one to date, seems to have
progressed further. Why these errors and misunderstandings continue to occur with
students studying from the newest of curricula, remains a topic for further research.
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It is true that a certain type of elementary mathematics cannot be done without a fair
amount of visualization skills, and that this precedes most work with functions. Again we
look at calculus. Here, for example, students must be able to visualize the common volume
of two right cylinders of equal radii which intersect at right angles. They must also be able
to visualize an object if it is known that it has a circular base and that every plane passing
through it, which is perpendicular to a fixed diameter, generates an equilateral triangle.
Exercises such as these form the heart of calculus. But the analytic description of the
volume, i.e. the function to be integrated, is the key. Tall (1986a), Blackett (1987), Rival
(1987), and Thomas (1988) have shown that students can develop a deep understanding of
higher level function concepts by developing them visually. Moreover, they have shown
that when the emphasis is placed on the visual development, there is higher retention of
them than if they were developed in an analytic framework. But is mathematics the
visualization of such situations or is it the symbolization of them? It is with the symbolization of situations that students seem to have trouble not, when applicable, with their visual
interpretation. (Leinhardt, Zaslavsky & Stein, 1990).
As evidenced at a debate at a meeting of the Working Group on Advanced Mathematical
Thinking, there are two schools of thought among researchers as to whether or not school
and university mathematics should emphasize the visual aspects surrounding elementary
functions (see, for instance, Dubinsky, 1989; Dreyfus 1991). The majority seemed to be in
favor of a pro-visualization stance, but there were eloquent pleas for the symbolic view.
Those in favor of visualization considered that functions should be thought of graphically
wherever possible and when applicable, operationson them should be thought of in this way
too.Forexample,thesimple transformations f(x)±k, ±kf(x), f(x±k),f(±kx ), If(x)|,f(|x|),f2(x),
1/f(x), etcetera, should be viewed graphically and that this method of viewing functions can
be instilled in students through carefully chosen sequences of exercises. The evidence
presented in this chapter suggests that progress will be made only through a major shift in
how we approach school and university level mathematics. The Hilbert-Bourbakian view
of mathematics has produced generations of semi-literates, in part because the pictures
which motivate the proofs and which are behind the big ideas are seldom emphasized in
the classroom.
It is likely that the situation can be improved by emphasizing the role played by the visual
representation of a function. Ben-Chaim (1985) has shown that visualization skills can be
learned and should be emphasized. Cipra’s supplemental text for calculus (1983) which
builds intuition and visualization skills is a good example of such an approach. It is my belief
that the present tendency not to teach in a visual way needs to be reversed, unless of course
mathematicians are satisfied with the semi-literate and mathematical phobics produced by
the present methods (Paulos, 1988).
5. ABSTRACTION, NOTATION, AND ANXIETY
Functions and their associated sub-concepts have degrees of abstraction. Open any higher
level mathematical journal or textbook at random and a page of symbols and formulae will
generally appear. Students often scan the page to see if the symbols and formulae are
familiar; But far too often they are foreign to them and it soon becomes apparent that a
tremendous amount of work will have to be done to understand what is written. Davis &
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Hersh (1986, p. 269) label this: the loss of meaning through the intellectual process of
mathematical abstraction and, as we have seen in chapter 8, it is a widespread problem
amongst students. But abstraction is what mathematics is all about, at least according to
Hilbert, the Bourbakists and most university professors. And this abstraction, along with
the pace with which it comes in the university classroom, is often the downfall of many
students. Obviously there are levels of abstraction. But too often this is not realized in the
classroom for the topics have been internalized by the instructor; they have mastered the
topic and they expect their students to do it too, in record time. Freudenthal’s comments on
automaticity (op.cit.) are certainly apposite. But this feeling that instructors of mathematics
can absorb a page of mathematics as though it were an article in a newspaper is often
conveyed to the student. And in many cases, students are discouraged before extending an
effort. In the literature this is called math phobia or anxiety and surely enough, it is a cause
of student failure. That is, learning fails for affective (emotional) reasons, not cognitive
ones. But there is irony in all of this, for math anxiety touches every individual – even
instructors of mathematics.
Hard as it is for many students to believe, even mathematicians have entire domains of
mathematics with which they do not feel comfortable. For some this may be working in 3space, for others it may be Galois, probability, or ergodic theory; but every individual has
“grey areas” which they often try to avoid. It is only within recent years that mathematicians
have recognized this obstacle to learning. The “humanistic mathematics movement” led by
Alvin White is devoted to sensitizing instructors to this learning difficulty (White, 1988).
Although there are many facets to mathematical anxiety, notational complexities are
often obstacles in preventing understanding of function concepts. Again we meet the
problem that it is not the mathematics, but the representation of the mathematics. Indeed,
Pimm (1984) posits that a positive correlation will be found between learning in texts which
minimize the notation and personalize the presentation. His arguments are persuasive, but
heretofore untested. Notational difficulties sneak in everywhere in elementary mathematics, but we will chronicle only a few of the pitfalls associated with initial notions on
functions.
1. The f(x) notation itself is confusing because f(x) is usually read to mean x under the
function f goes to.. . In the early 70’s a movement led by Howard Fehr (1974) used the
standard algebraists notation of x
but the movement never caught on. Nevertheless, this
notation captures the dynamic aspects of a function which should initially be emphasized.
Perhaps then students will have less trouble with understanding composite functions of the
form x g(x), g(x) f(g(x)). Moreover, this notation should further help students
understand the meaning of the argument. E.g., seventy percent of beginning calculus
students at Ben-Gurion University could not solve the following problem:
If 2 and 4 are the values of x for which f (x )=0, what are the values of x for which f(4x )=0?
This same problem was rephrased:
Only the values of 2 and 4 go to zero under the function f. What values multiplied by four will go
to zero under the function f ?
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Seventy percent of the students could answer the problem in this form, and they seemed to
have a basic understanding of what they were doing. This is a common phenomenon even
in simple word problems: that the phrasing of the question greatly affects the students
ability to answer it.
Students often do not realize that functions transform every point in the domain to a new
position. Until this is understood, problems such as finding the zeros of f(kx) given the zeros
of f(x) cannot be fully understood.
2. Realizing that
is of the form ∫ du/u is already evidence of looking at functions
in a more general way, although even this may only involve manipulation of symbolic
x
formulae.But defining a functionintermsofan integral suchasin1n(x)= ∫1 dt/t is beyond
most students in elementary courses – a common difficulty in the first stages of advanced
mathematics where the idea of definition, rather than description, is so new. Likewise,
visualizing functions in parametric form, also proves to be intolerably difficult, especially
when the representation moves from two dimensions to three.
Functions can be thought of as representing forms, and students need a wide set of
experiences in looking at functions in this way. Moreover, this inability to work with
concepts like functions, as though they are objects is a major obstacle for many students,
many of whom have demonstrated an ability to work with lower level abstractions in the
past.
As Harel and Kaput have already asserted in chapter 6, the isomorphism between a
vector space and the dual of its dual in linear algebra provides fertile ground to illustrate this
learning difficulty. A linear transformation from a vector space Vto the vector space 1 is
called a linear functional. The set of all linear functionals from V to 1 forms a vector space
called the dual of V; this vector space is usually denoted as V*. Here, the vectors in V* are
functions, which are combined in the usual way. But the dual of V*, is also a vector space,
V**, which is isomorphic to V. Students have a very hard time in understanding the role of
the vectors in V **. Certainly they are functions. But functions do things to objects in a
domain. What do these functions do to the functions in V* ? The functions in V** must send
the functions in V* somewhere, but to where ? As has been discussed in chapter 6 in terms
of conceptual entities and chapter 7 in terms of encapsulation, it is very difficult for students
to grasp that the functions are acting as operators as well as objects. This dual role is difficult
for students to understand because of the layering of the abstractions. And higher level
mathematics is full of such situations.
The problem solving research of Major & Clark (1963) has shown that when experts are
presented with a problem that they are seeing for the first time and which is not even close
to their domain of expertise, they approach a solution in the same way as novices. That is,
there is a playing around period, and generally an incubation period before realizing a
solution. For example, consider the following two mathematical problems:
1. For each real number x, let f(x) be the minimum of the numbers 4x+1, x+2, and –2x+4. What is
the maximum value of f(x)?
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2. There is a kingdom where if a person drinks poison he will die. The only way to counteract the
poison is to drink a stronger one. Then the reaction stops. The king decides that he must have the
strongest poison available in his possession. So, he sets up a contest between his court adviser and
his wizard. Each must find the strongest poison in the kingdom and give it to him. But to be sure that
he will get the strongest poison, he will force each to first drink the others’ poison. Then they will
drink their own. One of them will die, but the king will certainly then have the strongest poison in
his possession.
The court adviser knows that he can never out smart the wizard, and bemoans the fact that in a
few days time he will die. But soon, he and his wife think of a plan to outsmart the wizard.
The day of the contest arrives. The wizard drinks the adviser’s poison, and vice-versa. Then they
drink their own and the wizard dies.
What was the adviser’s trick?
These problems were given to novice and experts in mathematics. Both groups were
observed trying to set up functional relationships to describe the various situations in
problem two and to substitute values into problem one. But the two groups essentially
attacked the problems in the same way. Polya’s (1954) and Schoenfeld’s (1985) map of
problem-solving tactics was followed almost to the letter, including the fact that not one of
the interviewees wanted to spend more than two minutes on the problem. This, Schoenfeld
claims is characteristic of most problems in the curriculum, from grade school through the
university. Interestingly, most of the interviewees felt the second problem wasn’t mathematics, in spite of their symbolic-logistic solution paths. There is relevance in these
problems for our discussion on functions.
In order to solve the above problems a reversal tactic must be used. This is exactly the
same sort of skill which must be used to find g( x ), given f(g( x )) and f( x ), or f( x ), given f(g( x ))
and g(x). Moreover such reversals are necessary when determining, for example, the
function under consideration if it is known that the integral
represents the volume generated when a curve is revolved around a certain line and you
must determine the volume generated when this curve is revolved around the y-axis. Such
problems are very difficult for students, even though they often have all the skills to work
out the solution. They need time to internalize the skills involved in working with functions
in this way. Freudenthal’s admonitions on automaticity must always be in our minds;
concepts and topics arenot self-evident to students, even after our lucid explanations of them.
6. REPRESENTATIONAL DIFFICULTIES
Birkhoff (1956) developed a metric for aesthetics which showed that a mathematical
object’s appeal, be it a function, axiomatic system or whatever, is inversely proportional
to its complexity. How one measures this is another matter. But it seems obvious that the
more symbols, signs, etcetera, the more complex – and everyone, not just students, has
trouble with complexity.
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is a function which gives, for positive integers x the largest prime factor of x (Boas, 1960).
Notation often causes problems with students, even though, like many of the problems
discussed above, the basic underlying ideas which they represent are simple. Anxiety, the
nature of mathematics and the loss of meaning through the “intellectual” process of over
ambitious use of symbolism and abstraction have been discussed above. Good mathematics is not necessarily complex mathematics, and complex mathematics is not necessarily
good mathematics. But, given the choice, it seems obvious that one would opt for the less
complex and the more intuitive. Unfortunately, this has not been the option chosen in the
past.
7. SUMMARY
In this chapter we have surveyed some of the learning difficulties students have with
function concepts, and why they occur from historical and psychological points of view.
Perceived helplessness and anxiety seem to hinder learning, and literature has been cited
which studies these psychological barriers. But the major theme of this chapter is that
functions and their associated notions are not conceived visually, and that this non-visual
approach hinders one’s development of having a sense for functions. Students seem to think
of function concepts in only a symbolic representational mode. Indeed, not only are
functions thought of this way, but this seems symptomatic of the fact that the majority of
topics constituting mathematics are considered non-visual. It is the conclusion of the author
that this unwillingness to stress the visual aspects of mathematics in general, and of
functions in particular, is a serious impediment to students’ learning.
CHAPTER 10
LIMITS1
BERNARD CORNU
The mathematical concept of a limit is a particularly difficult notion, typical of the kind of
thought required in advanced mathematics. It holds a central position which permeates the
whole of mathematical analysis – as a foundation of the theory of approximation, of
continuity, and of differential and integral calculus.
One of the greatest difficulties in teaching and learning the limit concept lies not only
in its richness and complexity, but also in the extent to which the cognitive aspects cannot
be generated purely from the mathematical definition. The distinction between the
definition and the concept itself (discussed in detail by Vinner in chapter 5) is didactically
very important. Remembering the definition of a limit is one thing, acquiring the
fundamental conception is another. One facet is the idea of approximation, usually first
encountered through a dynamic notion of limit, and the way in which the concept of limit
is put to work to resolve real problems which rely not on the definition but on many diverse
properties of the intuitive concept. Starting from such a point of view students often believe
that they “understand” the definition of a limit without truly acquiring all the implications
of the formal concept. Students are often able to complete many of the exercises they are
asked to perform without having to understand the formalism of the definition at all.
Meanwhile, the quantifiers “for all”, “there exists”, which occur in epsilon-delta definitions, have their own meanings in everyday language subtly different from those encountered in the definition of the limit concept. From such beginning arise conceptual obstacles
which may cause serious difficulties.
In teaching mathematics, certain aspects of the limit concept are given greater emphasis
which are revealed by a review of the curriculum, the textbooks, exercises and examinations. In the first half of the twentieth century, French mathematics texts used be notion of
limit in an intuitive manner without a formal definition to introduce the definition of the
derivative. Later in the same text a definition would be given which is more in the manner
of an “explanation” in a note at the foot of the page. The official French programme of
school mathematics first cited the term limit with respect to the derivative as long ago as
1947. In 1966 the notion was properly introduced into the programme. Books generally
devoted a chapter to the general limit concept including a formal definition, a statement of
its uniqueness, and theorems about arithmetic operations applied to limits. The exercises,
however, did not concentrate on the limit concept, but on inequalities, the notion of absolute
value, the idea of a sufficient condition and, above all, on operations: the limit of a sum,
of a product, and so on. These exercises are far more related to algebra and the routines of
formal differentiation and integration than to analysis. They take on such an overwhelming
importance that one textbook cited thirty one different theorems on operations on limits!
1 The author wishes to thank Rebecca Tall for her initial translation of the draft of this chapter
into English, and the editor for help and assistance in completing the task.
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Given such a bias in emphasis it is therefore little wonder that students pick up implicit
beliefs about the way in which they are expected to operate.
Different investigations which have been carried out show only too clearly that the
majority of students do not master the idea of a limit, even at a more advanced stage of their
studies. This does not prevent them from working out exercises, solving problems and
succeeding in their examinations!
In this chapter we will study some didactic aspects of the idea of limits: concepts linked
to this notion, various obstacles which stand in the way of students learning the limit
concept, and discuss various strategies for teaching the limit concept.
1. SPONTANEOUS CONCEPTIONS AND MENTAL MODELS
For most mathematical concepts, teaching does not begin on virgin territory. In the case of
limits, before any teaching on this subject the student already has a certain number of ideas,
intuitions, images, knowledge, which come from daily experience, such as the colloquial
meaning of the terms being used. We describe these conceptions of an idea, which occur
prior to formal teaching, as spontaneous conceptions (Cornu 1981,1983). When a student
participates in a mathematics lesson, these ideas do not disappear – contrary to what may
be imagined by most teachers. These spontaneous ideas mix with newly acquired
knowledge, modified and adapted to form the students personal conceptions. We know that
in order to resolve a problem, we do not in general call uniquely on adequate scientific
theory, but on natural or spontaneous reasoning, which is founded on these spontaneous
ideas. This phenomenon is well-known in the empirical and theoretical development of
scientific concepts since Bachelard in the nineteen-thirties, but it is only in the last decade
that it has been fully realized that exactly the same forces operate in the apparent logic of
mathematics.
In the case of the limit concept, we observe that the words ‘tends to’ and ‘limit’ have a
significance for the students before any lessons begin (Schwarzenberger & Tall, 1978), and
that students continue to rely on these meanings after they have been given a formal
definition. Investigations have revealed many different meanings for the expression ‘tends
towards’:
• to approach (eventually staying away from it)
• to approach ... without reaching it
• to approach ... just reaching it
• to resemble (without any variation, such as “this blue tends towards violet”)
The word limit itself can have may different meanings to different individuals at different
times. Most often it is considered as an ‘impassible limit’, but it can also be:
• an impassible limit which is reachable,
• an impassible limit which is impossible to reach,
• a point which one approaches, without reaching it,
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• a point which one approaches and reaches,
• a higher (or lower) limit,
• a maximum or minimum,
• aninterval,
• that which comes ‘immediately after’ what can be attained,
• a constraint, a ban, a rule,
•
the end, the finish.
(Cornu, 1983)
From one student to another the meaning given to words varies; for one student it may have
many meanings, according to the situations. Spontaneous ideas live on a long time;
investigations show that they may remain with students at a much more advanced stage of
learning. In the face of a variety of spontaneous notions and the student’s growing
awareness of the formalisms it easily happens that contradictory ideas may be held
simultaneously in the mind of an individual, leading to a global “concept image” which
contains potential conflicting factors in the sense of Tall & Vinner (1981), as discussed in
chapter 5.
Aline Robert (1982a,b) has studied different models which students may hold of the
notion of the limit of a sequence. Despite the fact that students have been given a formal
definition of a sequence, when asked to describe the notion of a sequence, like as not they
would be liable to evoke conceptions relating to various aspects of their previous
experience. Some students suggested primitive, rudimentary models, reminiscent of those
which might be evoked spontaneously, such as:
• stationary: “The final terms always have the same value”,
• barrier: “The values cannot pass l ”.
In addition there were more models which arose more from the formal teaching:
• Monotonic and dynamic-monotonic :
“a convergent sequence is an increasing sequence bounded above (or decreasing bounded below)”;
“a convergent sequence is an increasing (or decreasing) sequence which
approaches a limit”.
• Dynamic :
“un tends to l ”;
“un approaches l ”;
“the distance from un to l becomes small”;
“the values approach a number more and more closely”.
• Static :
“The un are in an interval near l ”;
“the un are grouped round l”;
“The elements of the sequence end up by being found in a neighbourhood of l ”.
• Mixed : a mixture of those above.
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Once more she found these models influencing the manner in which students at university
solved problems. There is clearly no single notion of limit in the minds of students. It is
evident that the students have a variety of concept images.
Moreover, it is also evident that the initial teaching tends to emphasize the process of
approaching a limit, rather than the concept of the limit itself. The concept imagery
associated with this process, as exemplified above, contains many factors which conflict
with the formal definition (“approaches but cannot reach”, “cannot pass”, “tends to”, etc).
Thus it is that students develop images of limits and infinity which relate to misconceptions
concerning the process of “getting close” or “growing large” or “going on forever”.
In an ethnographic study of the conceptions of students concerning limits and infinity,
Sierpinska (1987) analysed the concept images of 31 sixteen year-old pre-calculus
mathematics and physics students. She then classified them into groups which she labelled
with a single name for each group:
Michael and Christopher are unconscious infinitists (at least at the beginning): they say “infinite”,
but think “very big”. ... For both of them the limits hould be the last value of the term ... for Michael
this last value is either plus infinity (a very big positive number) or minus infinity. ... It is not so for
Christopher who is more receptive to the dynamic changes of values of the terms. The last value is
not always tending to infinity, it may tend to some small and known number.
George is a conscious infinitist: Infiinity is about something metaphysical, difficult to grasp with
precise definitions. If mathematics is to be an exact science then one should avoid speaking about
infinity and speak about finite numbers only. In formulating general laws one can use letters
denoting concrete but arbitrary finite numbers. In describing the behaviour of sequences the most
important thing is to characterize the nth term by writing the general formula. For a given n one can
then compute the exact value of the term or one can give an approximation of this value.
Paul and Robert are kinetic infinitist s: the idea of infinity in them is connected with the idea of time.
... Paul is a potentialist: To think of some whole, a set or a sequence, one has to run in thought through
every element of it. It is impossible to think this way of an infinite number of elements. The
construction of an infinite set or sequence can never be completed. Infinity exists potentially only.
Robert is a potential actualist : it is possible [for him] to make a “jump to infinity” in thought: the
infinity can potentially be ultimately actualized. For both, Paul and Robert, the important thing is
to see how the terms of the sequence change, if there is a tendency to approach some fixed value.
For Paul, even if the terms of a sequence come closer and closer so as to differ less than any given
value they will never reach it. Robert thinks theoretically the terms will reach it in the infinity.
In this way she exhibits timeless conflicts about limits and infinity which have been with
us since time immemorial and which continue to hold in our students today.
Other limiting processes, such as the concept of continuity, differentiation, integration,
and so on, whilst on the surface being very different, in cognitive terms exhibit similar
difficulties. For instance, continuity suffers from having a spontaneous conception that is
evoked through the use of everyday language in phrases such as “it rained continuously all
day” (meaning there was no break in the rainfall) or “the railway line is continuously
welded” (meaning that there are no gaps in the rail). This viewpoint is often reinforced by
teacher’s attempts to give a simple insight into the notion of continuity by speaking of the
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graph “being in one piece” or “drawn without taking the pencil off the paper”, thereby
confusing the mathematical notions of continuity and connectedness.
A questionnaire administered to first year university mathematics students (Tall &
Vinner 1981) included the a question to investigate the students’ concept images of
continuity (figure 16).
Mathematically f1, f2 and f3 are continuous, whilst f4 and f5 are not. But the students’
concept images suggest otherwise (Table IV – “correct” responses in bold print).
Although all theresponses to f1 are “correct”, the majority are “right answers for wrong
reasons”, such as the idea that f1 is continuous “because it is given by only one formula”.
The function f2 often causes dispute even amongst seasoned mathematicians. It is
continuous according to the ε− δ definition on the domain {x∈ | x≠ 0}. But the students’
Which of the following
functions are continuous?
If possible give reasons
for your answer.
Figure 16 : the concept image of continuity
Table IV : Responses to figure 16
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concept images suggest:
It is continuous because:
“the function is given by a single formula”.
It is not continuous because
“the graph is not in one piece”,
“the function is not defined at the origin”,
“the function gets infinite at the origin”.
In the initial stages of learning, we therefore see spontaneous conceptions arising which are
often in conflict with the formal definition.
2. COGNITIVE OBSTACLES
The notion of a cognitive obstacle is interesting to study to help identify difficulties
encountered by students in the learning process, and to determine more appropriate
strategies for teaching. It is possible to distinguish several different types of obstacles:
genetic and psychological obstacles which occur as a result of the personal development
of the student, diductical obstacles which occur because of the nature of the teaching and
the teacher, and epistemological obstacles which occur because of the nature of the
mathematical concepts themselves. In planning to teach a mathematical concept it is of the
utmost importance to determine the possible obstacles, particularly the endemic epistemological obstacles.
The term was introduced by Gaston Bachelard (1938):
“We must pose the problem of scientific knowledge in terms of obstacles. It is not just a question
of considering external obstacles, like the complexity and the transience of scientific phenomena,
nor to lament the feebleness of the human senses and spirit. It is in the act of gaining knowledge itself,
to know, intimately, what appears, as an inevitableresult of functional necessity, to retard the speed
of learning and cause cognitive difficulties. It is here that we may find the causes of stagnation and
even of regression, that we may perceive the reasons for the inertia, which we call epistemological
obstacles.”
He goes on to say:
“We encounter new knowledge which contradicts previous knowledge, and in doing so must destroy
ill-formed previous ideas.”
He indicated that epistemological obstacles occur both in the historical development of
scientific thought and in educational practice. For him, epistemological obstacles have two
essential characteristics:
• they are unavoidable and essential constituents of knowledge to be acquired,
• They are found, at least in part, in the historical development of the concept.
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Many authors have become interested in epistemological obstacles. Guy Brousseau defines
an epistemological obstacle as knowledge which functions well in a certain domain of
activity and therefore becomes well-established, but then fails to work satisfactorily in
another context where it malfunctions and leads to contradictions. It therefore becomes
necessary to destroy the original insufficient, malformed knowledge, to replace it with new
concept which operates satisfactorily in the new domain. The rejection and clarifying of
such an obstacle is an essential part of the knowledge itself; the transformation cannot be
performed without destabilizing the original ideas by placing them in a new context where
they are clearly seen to fail. This therefore requires a great effort of cognitive reconstruction.
3. EPISTEMOLOGICAL OBSTACLES IN HISTORICAL DEVELOPMENT
It is useful to study the history of the concept to locate periods of slow development and
the difficulties which arose which may indicate the presence of epistemological obstacles.
In the case of the history of the limit concept, we see that this notion was introduced to
resolve three principal types of difficulty:
• geometric problems (area calculations, consideration of the nature of geometric
lengths, “exhaustion”),
• the problem of the sum and rate of convergence of a series,
• the problems of differentiation, (which come from the relationship between two
quantities which simultaneously tend to zero).
There are four major epistemological obstacles in the history of the limit concept:
1) The failure to link geometry with numbers.
When the Greeks became interested in mathematics about 400-300 BC, we must ask why
it happened that the limit concept was not clarified at the time. The problem of calculating
the area of a circle, for example, supplied an opportunity to develop the tools very similar
to the limit concept. Hippocrates of Chios (430 BC) wanted to prove that the ratio between
the area of two circles is equal to the ratio of the squares of their diameters. He inscribed
regular polygons within the circles and, by indefinitely increasing the number of sides, he
approached the areas of the two circles. At each step the ratio of the areas of the inscribed
polygons is equal to the ratio of the squares of the diameters, and it followed that, “in the
limit”, it would be true also for the areas of the circles.
This passage towards the limit, very sparingly explained, would be defined a year later,
in terms of the method of exhaustion, credited to Eudoxus of Cnidos (408-255 BC). The
method is based on the principle of Eudoxus (Euclid’s Elements, book 10, proposition 1)
that “given two unequal lengths, if from the first is taken apart larger than its half, then from
the remainder a part more than half what remains, and the process is repeated, then there
will come a time when what remains will be less than the second length”. In other words,
by successive halving we can attain a size as small as we wish. From this the principle of
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exhaustion follows which allows us to state that for any ε>0 there exists a regular polygon
inscribed within a circle whose area differs from that of the circle by less than ε. If the ratio
of areas of two circles is A1/A 2 and that of the squares of the radii is r12/ r22 , then we have
one of three possible cases:
We eliminate the first two by the principle of exhaustion, and hence deduce the truth of the
desired equality.
However, despite the fact that the exhaustion method seems extremely close to the
notion of limit, we cannot affirm that the Greeks possessed the modem limit concept. The
method of exhaustion is in essence a geometrical method which allows the proof of results
without having to deal with the problem of infinity. It is applied to geometrical magnitudes,
not to numbers. Each case is dealt with on an individual basis using a specific argument
tailored to the geometrical context. There is no transfer from geometrical figures to a purely
numerical interpretation, so the unifying concept of limit of numbers is absent. The
geometrical interpretation, and its success in resolving pertinent problems, is therefore seen
to cause an obstacle which prevents the passage to the notion of a numerical limit.
2) The notion of the infinitely large and infinitely small.
Throughout the history of the notion of limit we meet the supposition of the existence of
infinitesimally small quantities. Is it possible to have quantities which are so small as to be
almost zero, and yet not having a specific ‘assignable’ size? What happens at the instant
when one of these quantities becomes zero? Such philosophical problems have occupied
the attention of numerous mathematicians who, like Newton, spoke of the “soul of departed
quantities” at the time that they disappear to enable him to calculate their “ultimate ratio”.
Euler freely used the notion of the infinitely small as a quantity that can, where appropriate,
be considered equal to zero. D’Alembert opposed the use of infinitely small quantities and
sought to remove them from the differential calculus. He reasoned that a quantity is either
something or nothing. If it is something it cannot be made zero and if it is nothing it is already
zero. Thus the supposition that there is an intermediate state between the two he described
as a wild dream.
Cauchy also used the language of the infinitely small. In his Cours d’analyse de l’Ecole
Polytechnique of 1821, he defined a continuous function in these terms:
The function f( x) is continuous within given limits if between these limits an infinitely small
increment i in the variable x produces always an infinitely small increment, f( x+i)–f(x), in the
(as translated in Boyer, 1939, p. 277)
function itself.
He explained the idea of an infinitesimal as follows:
One says that a variable quantity becomes infinitely small when its numerical value decreases
indefinitely in such a way as to converge to the limit zero. (quoted from Boyer, 1939, p. 273).
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For Cauchy an infinitesimal is simply a variable which tends to zero.
The idea of an ‘intermediate state’ between that which is nothing and that which is not is
frequently found in modem students. They often view the symbol ε as representing a
number which is not zero yet is smaller than any positive real number. In the same way
individuals may believe that 0.9 99... is the ‘last number before 1’ yet is not equal to one.
There is a corresponding belief in the existence of an integer bigger than all the others, yet
which is not infinite.
3) The metaphysical aspect of the notion of limit.
The notion of limit is difficult to introduce in mathematics because it seems to have more
to do with metaphysics or philosophy. Mathematicians are often reticent in speaking of
such concepts, from the time of the Greeks through to D’Alembert who wrote “One can
quite easily do without the rest of all this metaphysics of the infinite in the differential
calculus”. Lagrange expressed a similar horror of the metaphysical aspects. Although in his
early career he believed he could make the use of infinitesimals rigourous, he later
considered that the infinitesimals of Leibniz has no satisfactory metaphysical basis and
recast the foundations of the calculus using infinite series in purely algebraic terms.
However, this too proved elusive, for
When Lagrange endeavored to free the calculus of its metaphysical difficulties, by resorting to
common algebra, he avoided the whirlpool of Charybdis only to suffer wreck against the rocks of
Scylla. The algebra of his day, as handed down to him by L. Euler, was founded on a false view of
infinity. No rigorous theory of infinite series had been established.
(Cajori, 1980, p. 257)
In this way, whichever way mathematicians seemed to turn in the historical development
of the subject, they came against profound theoretical difficulties.
The metaphysical aspect of the notion of limit is one of the principal obstacles for today’s
students. In an interview one said, “It is not really mathematics”, because the initial stages
of calculus no longer rely purely on simple arithmetic and algebra. The students may have
difficulties handling the concept of infinity, “It isn’t rigourous, but it works”; “it doesn’t
exist”, “it is very abstract”, “the method is all right, provided you are content with an
approximate value”. This obstacle makes the comprehension of the limit concept extremely
difficult, particularly because a limit cannot be calculated directly using familiar methods
of algebra and arithmetic.
4) Is the limit attained or not?
This is a debate which has lasted throughout the history of the concept. For example, Robins
(1697-1751) estimated that the limit can never be attained, just as regular polygons
inscribed in a circle can never be equal to the circle. He asserted “We give the name ultimate
magnitude to the limit which a variable quantity can approach as near as we would like, but
to which it cannot be absolutely equal”. On the other hand, Jurin (1685-1750), said that the
“ultimate ratio between two quantities is the ratio reached at the instant when the quantities
cancel out”, “it is not a question whether the increment is zero, but that it is disappearing,
or on the point of vanishing”, “there is a last ratio of increments which vanish”, “an
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increment born is an increment which starts to exist from nothing, or which begins to be
generated, but which has yet to attain a magnitude that may be assigned to such a small
quantity”. D’ Alembert insisted that a quantity should never become equal to its limit: “To
speak properly, the limit never coincides, or never becomes equal to the quantity of which
it is the limit, but is always approaching and can differ by as small a quantity as one desires”.
This debate is still alive in our students. In adiscussion one asked, “When n tends to zero,
isn’t n equal to zero?” The following dialogue between students clearly illustrates the
epistemological obstacle:
– the more n grows, the more 1/n approaches zero.
– as much as one would like?
– no, because one day they will meet.
There are certainly many other obstacles to the notion of limit other than these four. The
mistakes which students make are valuable indications for locating obstacles. The
construction of pedagogical strategies for teaching students must then take such obstacles
into account. It is not a question of avoiding them but, on the contrary, to lead the student
to meet them and to overcome them, seeing the obstacles as constituent parts of the revised
mathematical concepts which are to be acquired.
4. EPISTEMOLOGICAL OBSTACLES IN MODERN MATHEMATICS
It is interesting for mathematicians to look back at history and note the struggles that gave
birth to modem ideas, leading to the logical state of the art today. However the twentieth
century quest for certainty based on a secure axiomatic foundation begun by Hilbert
foundered on Gödel’s incompleteness theorem, and so uncertainty remains. In chapter 4,
Hanna has shown that acceptance of proof remains strongly dependent on peer approval.
The introduction of Weierstrassian analysis, depending only on logical definitions of
number concepts failed to eliminate the infinitesimal concepts that were an essential part
of earlier mathematical culture. Although we may formulate definitions of limits and
continuity in epsilon-delta terms, we still have occasion to use dynamic language of
“variables tending to zero” in a manner analogous to that of Cauchy, with the resultant
mental imagery linked to the “arbitrarily small”.
Cognitively this phenomenon is to be expected. The idea of an “arbitrarily small”
number is but the object produced by the encapsulation of the process of getting small in
terms of Dubinsky’s theory of encapsulation (chapter 7). As TaIl hypothesizes (chapter 1),
the formation of a mental concept of an “arbitrarily small number” is a generic limit concept
where the encapsulated object is believed to have the properties of the objects in the process.
Thus the generic limit of a set of numbers which tend to zero is an arbitrarily small, yet nonzero, number. The concept is a natural consequence of the way in which the mind is
hypothesized to work.
Hence, despite the attempts atbanning infinitesimals from modem analysis, it continues
to live in the minds and communications of professional mathematicians, even if it was
eliminated from formal proofs. The return of the logically based infinitesimal in the work
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of Robinson (1966) re-opened the debate, which continues to be hotly contested. Although
Robinson thought that his neat logical solution would solve the three hundred year conflict,
this proved not to be so. For Robinson’s construction of a hyperreal system containing real
numbers and infinitesimals depends on a version of the axiom of choice and is therefore
non-constructive. This is becoming more a bone of contention as the arrival of computers
begins to focus mathematicians on the pragmatic need to provide finite algorithms for
construction of concepts. For instance, the intermediate value theorem is seen to be
constructive but the existence of a maximum value of a continuous function is not. The
former asserts the existence of a zero of a continuous function between two places where
the function has opposite signs and can be programmed on a computer by a simple bisection
argument, but the latter depends essentially on a non-constructive proof by contradiction.
In this way we see a recurrence of the problem of Lagrange as he attempted to remove
the metaphysical ambiguity from the calculus: just as difficulty seems to be resolved,
another seems to appear to take its place. This is typical of the complexity of the ideas in
analysis and of the fundamental limit concept.
That the limit concept is essentially difficult may be seen in the way that it is defined in
terms of an unencapsulated process: “give me an ε>0, and I will find an N such that . . .”
rather than as a concept , in the form “there exists a function N( ε ) such that.. .” This means
that the proof of the first theorem on the algebra of limits (that the sum, product etc of the
limit of two sequences is the sum, product etc of the limits) is framed in process terms as
the coordination of two processes, rather than as the combination of two concepts. Were
the latter to be the case, then the proof would follow a similar format, but it would have the
advantage that it could be programmed on a computer in such a way that the proof of
continuity is merely the operation of a computer algorithm. Yet this unencapsulated
pinnacle of difficulty occurs at the very beginning of a course on limits presented to a naïve
student. No wonder they find it hard!
5. THE DIDACTICAL TRANSMISSION OF EPISTEMOLOGICAL OBSTACLES
Given the complexities of modem mathematics and the cultural colourations in meaning,
it is no surprise that such complex interactions affect students in their learning. In their
human interactions they are very sensitive to tone of voice and implicit meanings and such
ideas are conveyed to them by their teachers. Although such meanings may be avoided in
written texts, they can passed on inadvertently from generation to generation as the teacher
tries to “simplify” the complexities to “help” the students. When Orton ( 1980) investigated
the limit concept in terms of a “staircase with treads”, he showed a student the picture in
figure 17 and asked
(a) If this procedure is repeated indefinitely, what is the final result?
(b) How many times will extra steps have to be placed before this “final result” is reached?
(c) What is the area of the final shape in terms of “ a”, i.e. what is the area below the “final staircase”?
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Figure 17 : A limiting staircase
If a student gave a formula in response to (c) he asked:
Can you use this formula to obtain the ‘final term’ or limit of the sequence ?,
His justification for using this terminology was that:
The expression “final term” was again used in an attempt to help the students understand the
meaning of limits.
However, in the light of what has been said in this book about generic limit concepts, it is
evident that a phrase such as “the final staircase”, far from helping the students with the
formalities, is likely to create a generic limit concept in which the student imagines a
staircase with an “infinite number of steps”. This is precisely the response that it evoked.
In such ways, despite all our attempts to help students through the complexities, our
attempts to “simplify” can lead directly to the cognitive obstacles which we have described
earlier.
Such obstacles are almost certainly essential parts of the learning process. Davis &
Vinner (1986) suggest that there are seemingly unavoidable stages in which misconceptions are bound to occur, in line with our assertion that such obstacles require a cognitive
re-construction which are bound to involve a period of conflict and confusion. They too
highlight the misconceptions that arise from the use of language evoking inappropriate
images in spontaneous conceptions. Even though they attempted to teach a course in which
the word “limit” was not used in the initial stages, they concluded that “avoiding appeals
to such pre-mathematical mental representation fragments may very well be futile”. They
observed that another problem arises from the sheer complexity of the new ideas which
cannot appear “instantaneously in complete and mature form” and so “some parts of the
idea will get adequate representations before other parts”. They give evidence, substantiating the discussion of Robert and Schwarzenberger in the previous chapter, that specific
examples dominate the learning, so that when monotonic sequences feature heavily in the
student’s earlier experiences, it is not surprising that they dominate the students’ concept
images.
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6. TOWARDS PEDAGOGICAL STRATEGIES
The diversity of conceptions, the richness and complexity of notions, and the cognitive
obstacles makes the teaching of the limit concept extremely difficult. Numerous attempts
have been made and the problem remains unresolved! On considering these attempts, it is
possible to focus on certain fundamental points and to pose essential questions.
In the first place, far too many teachers seem to consider that it is sufficient to present
a clear exposition of the limit concept to enable the students to understand. It is far more
important that the students are made aware of the complexity of the notion and to reflect
on their own ideas and epistemological obstacles. Research so far shows clearly that the
students own conceptions are very varied, that they make fundamental mistakes and that
they do not necessarily overcome epistemological obstacles. It is necessary for teacher
education to take place to help teachers become aware of the problems involved. It is
equally important for students to become explicitly aware of the essential difficulties.
Experiments have been carried out in which, before starting a session on the notion of limit,
the students were given appropriate activities to help them become aware of their own
spontaneous ideas, images, intuitions, experiences which they possessed before and which
would necessarily be brought into play during the learning process. In particular, they were
made aware of the different meanings of the words which they were going to use. This
proved to be a valuable technique and enabled them to build on their own knowledge and
understanding (Cornu, 1983; Robert, 1982a).
A further problem is that of the context in which the learning takes place. An effective
apprenticeship needs to take place in a problem-solving context. The notion of limit has to
be used to solve specific problems. It is therefore necessary to present situations in which
the student can see that the limit is a useful tool, in which the limit is seen as part of the answer
to questions which the student may have asked for him (or her)self. This is often lacking
in contemporary teaching. A definition of the notion of limit is given, followed by a
sequence of problems and exercises, usually based solely on handling the algebra of the
limit concept the limit of a sum, of a product, of the composition of two functions or of two
series . . . We have already seen just how difficult the unencapsulated logical form of the
limit definition is to handle for experts, let alone beginners.
It is important to consider the order in which the limit concepts are presented. Not only
is there the question of designing a logical mathematical order of concepts, but also the
cognitive appropriateness of the curriculum sequence and of the problems to be solved. It
is now well-established that in the transition to advanced mathematical thinking a purely
logical sequence of topics, in which the mathematical concepts are introduced through
definitions and logical deductions, is likely to be insufficient.
Some alternative methods of approach will be discussed in later chapters. Tall (1986),
for example, decided that the research evidence on students’ difficulties with the limit
concept made it totally inappropriate to approach it either through the formal definition or
even (in the case of the derivative) through a geometric experience of a secant ” tending to
a tangent”. He hypothesized that the limit process should be used implicitly in the calculus
as part of a magnification process to “see” the gradient of a curve and gain experience of
the concept in action before the concept becomes the focus of formal discussion. In this
sense he is following a similar path to that of Douady (1986) in her “tool-object” dialectic
in which the concept is first used implicitly and informally as a tool to gain appropriate
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cognitive experiences before it becomes the explicit focus of attention as a mathematical
object. (Douady’s ideas are also discussed in §2.1 of chapter 8 and Tall’s graphic approach
to limits in the calculus will be discussed in the next chapter).
Dubinsky (chapter 7) has formulated the notion of a “genetic decomposition” of
mathematical concepts, that is to say a collection of reflective abstractions which are
approached in a certain order to provoke the learning of the concept envisaged. Thus if we
want to introduce the concept of the limit of a series we must first try to describe the concept
image and the sequence of mental constructions which are necessary for the student to
make. As students with different experiences and different cognitive structures are unlikely
to require exactly the same sequence, one may hypothesize that it is essential that the
students actively and consciously participate in the reflective abstraction process to
reconstruct their knowledge structure and build the limit concept.
As we shall see in chapter 14, the computer may very well play a significant role in
providing an environment where the student may gain appropriate experiences to construct
the limit concept. However, such approaches are very likely to contain their own peculiar
epistemological obstacles (Tall & Winkelmann 1988) and it is necessary to reflect deeply
on student experiences in the new environment to see precisely what is learnt and in what
form the knowledge is held in the mind. The interaction with the computer may involve
programming, for the individual to construct computer processes which, through reflection, may permit the acquisition and mastery of the correspondingmathematical constructs.
It may involve pre-prepared software to enable the student to experience carefully selected
environments which model the idea of a limit. It is equally possible to imagine a kind of
computer “toolbox” for the learning of the limit concept: a computer environment which
will permit the students to manipulate objects and to construct knowledge: to recognize and
construct sequences, to operate on them, constructing new sequences, transforming and
manipulating them, studying their behaviour and the nature of their convergence.
Various other approaches are possible. In a context such as that of studying limits it is
vital that the computer software is designed within a teaching strategy based on the careful
analysis of the concept due to be acquired. Spontaneous conceptions, concept images,
obstacles, reflective abstraction, and genetic decomposition, are all conceptual tools
designed to assist in the design of such pedagogical strategies.
CHAPTER 11
ANALYSIS
MICHÈLE ARTIGUE1
The conceptual field of analysis is vast. At the elementary level it is structured around the
notions of:
• real number,
• function,
• limits of numerical sequences and functions,
• continuity,
• the derivative and integral of functions of one real variable.
At more advanced levels these extend to analysis of several variables, complex analysis,
functional analysis, measure theory and so on.
The two preceding chapters have summarized empirical research and cognitive theory
relating to the first four of these headings. This chapter concentrates on the smaller body
of work focussing on the fundamental notions of differential and integral calculus.
In the initial section we will briefly review major points in the historical evolution of the
concepts and the ways that they have been taught. In the second we will use empirical
research and theoretical interpretations to draw up a catalogue of the mental conceptions
constructed by students engaged in traditional education. Finally, in the third, we will
present some instructional treatments taking this research into account and designed to
improve student understanding.
1 Thanks are due to Ed Dubinsky for his initial translation of the fist draft of this chapter and
to the editor for his assistance with the final version.
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1. HISTORICAL BACKGROUND
1.1 SOME CONCEPTS EMERGED EARLY
BUT WERE ESTABLISHED LATE
It is well known that the fundamental notions of differential and integral calculus appeared
on the mathematical scene very early, but their development was very slow.
From the time of antiquity, calculations of length, ara and volume, based on the method
of exhaustion, opened the way to integral calculus. In the seventeenth century, the problems
of tangent, maxima and minima, linked in particular to the study of celestial mechanics and
ballistics in their turn opened the way to differential calculus. This set the scene for the
independent development of the calculus by Newton and Leibniz, culminating in the
reciprocity of the operations of integration and differentiation. The first text-book: Analysis
of the infinitely small for understanding curved lines was published by the Marquis de
l’Hospital (l696).
It was not until the beginning of the nineteenth century that Cauchy developed a firmer
theoretical basis for the calculus using the notion of limit, and integration was developed
using continuous functions. In the remainder of the nineteenth century the arithmetization
of analysis was carried out, through formal definitions of the real line by Dedekind cuts or
Cauchy sequences, and formal definitions of limits and continuity using ε−δ methods in
a purely arithmetic form by Weierstrass. This led Boyer (1939) to claim:
the unequivocal symbolism of Weierstrass may be regarded as effectively banishing from the
calculus the persistent notion of infinitesimal.
Meanwhile, it was not until 1893 that Stolz introduced the notion of differentiability for
functions of several real variables and only in 1911 that the development of functional
analysis led to Fréchet introducing the differential in its modem interpretation in terms of
linear tangent maps.
The latest twist in the story occurred in the 1960s, when Robinson formulated a rigorous
theory of non-standard analysis, reintroducing infinitesimals on a logical basis after a
century of rejection.
1.2 SOME CONCEPTS CAUSE BOTH ENTHUSIASM
AND VIRULENT CRITICISM
It is well known that from its birth, infinitesimal calculus has excited passions. On the
one hand there is the enthusiasm of those who, like the Marquis de l’Hospital, are astonished
by the possibilities opened up by the algebraisation of calculus:
The extent of calculus is immense: it is as easy for mechanical curves as for geometrical ones; it is
indifferent to radical signs and even makes use of them; it extends to as many variables as one would
wish; comparisons of all kinds of infnitesimals are equally easy. Furthermore, an infinity of
surprising discoveries has come out of it.
(preface to de l’Hospital 1696)
ANALYSIS
169
On the other there is virulent criticism from those for whom infinitesimals are beings
without roles, carriers of paradoxes, the manipulations of which are based on dubious
practices. Thus Berkeley fiercely criticizes the arguments of Newton:
This reasoning seems neither correct nor honest. For when one says that increments are no longer
anything or that there are no more increments, the preceding supposition to the effect that increments
were something or that there were increments is destroyed, yet a consequence of the supposition is
retained ... This is a false reasoning.
Likewise, D’Alembert wrote (in his article “Differentiel” in the Encyclopedie Methodique):
It is not at all a question of how one speaks ordinarily of infinitesimal quantities in differential
calculus: it is just a question of limits of finite quantities. Thus the metaphysics of infinity and some
infinitesimal quantities being larger or smaller than others, is totally useless in differential calculus.
One only uses the term infinitesimal to abbreviate expressions. We would not say therefore, as do
many geometers, that a quantity is infinitely small neither before it vanishes nor after it is vanished,
but in the very instant at which it vanishes: for wouldn’t that mean a very false definition, a hundred
times more obscure than that which one wishes to define?
A little later Lagrange (1797) would attempt to liberate analysis simultaneously from both
limits and infinitesimals, judging each of the two approaches to be as subject to criticism
as the other.
1.3 THE DIFFERENTIAL/DERIVATIVE CONFLICT
AND ITS EDUCATIONAL REPERCUSSIONS
At the heart of these differences of opinion is the differential/derivative conflict, originally
a debate between the English school using Newton’s fluxions and the continental school
using the differential of Leibniz. On the continent during the eighteenth century, the
differential of Leibniz was one of the essential motors of the development of differential
and integral calculus.
But the structuring of differential calculus around the notion of limit led to the
progressive decline of infinitesimals and in their wake differentials were supplanted by
partial derivatives (defined in terms of the limit concept). The differential survived in
analysis – reduced to the role of a formal expression invariant with respect to a change of
variables and therefore a useful tool for calculation and memorization. It survived also in
applications, especially in physics, with a role approaching its original status under Leibniz
– as an infinitesimal increase – a useful tool for substitution in equations, but suspected of
being based on less rigorous practices.
The article “Differential Calculus”, in the French version of the German Encyclopedia
of Mathematical Sciences, gave a very good account of the situation at the end of the 19th
century. The differential was defined only at the end, in reference to work of Cauchy, as
the product of the derivative by an arbitrary increase of the variable. The author accompanied his definition with the following comment:
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In fact the Leibnizian notation is theoretically superfluous. Practically, it has a great importance...
In applications, usage is facilitated if one agrees, once and for all that one understands by the symbol
dy not the differential f ’(x) dx but a very small increase in the function y =f(x) ... A differential formula,
that is, arelation between x ,y, d x and dy, coming before expressing for example, the law of a physical
phenomenon, no longer has, it is true, any precise meaning. It is, however, easy to establish and there
is no inconvenience in using it, for in order for it to acquire a precise meaning, it suffices to divide
both sides by d x and pass to the limit as d x tends toward 0...
(Voss, 1899, translated from the French edition p. 279)
It is only in the course of the twentieth century that the differential reappeared, in the
development of functional analysis, as the key notion of local approximation, but this time
with a subtly different role: that of a tangent linear functional.
These debates and conflicts, and the atmosphere of scientific uncertainty that they
engendered, reverberated throughout education for an even longer time.
Differential and integral calculus was introduced into secondary education at the
beginning of the century in many countries and in 1913 the Commission Internationale pour
l’Enseignement des Mathématiques (CIEM)2 organized a study of the subject, published
by the journal “Enseignement Mathématique” the following year. The general reporter
wrote:
Scientific literature itself has not made a clear resolution of the diverse definitions of differential.
(Beke, 1914, p. 272)
The atmosphere of uncertainty still persisted in 1930:
This question (differentials) is one of the points in mathematics where those who search for precision
do not always find, even with the best authors, the desired clarity. Prudent recommendations as to
the mode of using differentials of higher order which generally accompany the exposition, doubtless
permit correct calculation with differentials: but the goal of education is more than that.
(Delens, 1930, p. 333)
To this is added the problem of developing the most rigorous possible instruction, without
metaphysics and therefore without infinitesimals, and to eliminate errors by beginners
based on formal and automatic manipulation of differentials influenced by their status
during this period. In an article in the same journal, on definitions in mathematics, Poincaré wrote:
As soon as one passes to derivatives of the second order, one swims in absurdity: let z be a function
of a variable y which is itself a function of x; I write:
In this formula1 write d2z twice, and the symbol has two different meanings ...
2 The CIEM later became ICMI (International Commission for Mathematical Instruction).
ANALYSIS
171
The difficulty is aggravated if one has several independent variables. I write:
Here again we have three occurrences of the symbol d z with three different meanings ...
How can we avoid these traps? Beginners will not be able to do so.
(Poincaré, 1899, p. 107)
The author cites a delightful story, where a student came to the equation of propagation of
sound, and after having simplified dt2, extracted the square roots of differential elements
by simply suppressing the indices “2”:
Beke concluded the paragraph of his CIEM report dedicated to differentials in these terms:
We are, I believe,unanimous in desiring that the metaphysical haze of infiintesimals should not enter
into secondary education. I am of the opinion that the wisest method is to not introduce differentials
at all in secondary education. This view is justified by the actual tendency to eliminate differentials
in the whole of mathematical science. How much more necessary would it appear to reject the
teaching of all notions which give rise to so much misunderstanding.
Thus in secondary education, teaching differential calculus was based on the notion of
derivative, defined as a limit of a quotient and associated with the geometric picture of the
tangent as a limiting position of secants. In higher elementary education, preference was
given to derivatives and partial derivatives, whilst differentials were defined in terms of
these and limited to first order.
Even so, the arguments between supporters and opponents of differentials continued to
be virulent (see, for example, Laurent, 1899). In the Mathematical Gazette, controversy
raged for several years following the appearance of an article of E.G. Phillips (1931)
regretting that students might come to the university without having heard a mention of
differentials. He proposed to introduce them from the beginning of “Elementary Calculus”
using the modern definition of differentiability arising from the increase of the function
proportional to the increase of the variable. There followed a debate on this question at the
annual meeting of the Association in 1934. Subsequently the derivative was introduced as:
dy
— now has the status of a single
where δy=f( x +δx)–f(x)forasmallvalueof δx. The symbol dx
indivisible symbol where dy and dx are given no individual meaning:
dy/dx must, at least for some considerable time, be regarded as an inseparable whole, just as δx is.
It does not in any simple or straightforward way mean anything like ‘dy divided by dx’ and a
statement such as ‘dy/dx dx/dt = d y/dt by cancelling d x’ is just so much gibberish.
(SMP Advanced Mathematics Book I, 1967, p. 221)
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1.4 THE NON-STANDARD ANALYSIS REVIVAL
AND ITS WEAK IMPACT ON EDUCATION
The publication in 1966 of Robinson’s book Non-Standard Analysis constituted, in some
sense, a rehabilitation of infinitesimals which had fallen into disrepute since the
arithmetization of analysis. By using a logical construction based on ultrafilters, he
proposed a rigorous foundation for the approach to differential and integral calculus using
the infinitely small and the infinitely large. It was met with suspicion, even hostility, by
many mathematicians who saw it only as a useless reintroduction of discredited, even
dangerous, archaic tools whose rejection had done nothing to hinder the development of
mathematics for more than a century. Nevertheless, despite the obscurity of this first work,
non-standard analysis developed rapidly both in mathematical research and in research into
logical foundations. In the latter case a major goal was that of simplifying the initial
construction of Robinson or to propose axiomatic approaches (for example, Nelson, 1977).
The attempts at simplification were often conducted with the aim of constructing an
elementary way of teaching non-standard analysis. This was the case with the work of
Keisler (1971,1976) and Henle & Kleinberg (1979). The first work of Keisler served as a
reference text for a teaching experiment in the first year of university in the Chicago area
during 1973–74. Sullivan (1976) used two questionnaires to evaluate the effects of the
course: one designed for teachers, the other for students. The eleven teachers involved gave
a very positive opinion of the experience. The student questionnaire revealed no significant
difference in technical performance between standardists and nonstandardists, but showed
that those following the non-standard course were beater able to interpret the mathematical
formalism of calculus and to make sense of it.
The appearance of the second book by Keisler (1976) led to a virulent criticism by
Bishop (1977) in the Bulletin of the American Mathematical Society, accusing Keisler of
seeking the goal of modem mathematicians: to convince students that mathematics is only
“an esoteric and meaningless exercise in technique”, detached from any reality. These
criticisms were in opposition to the declarations of the partisans of non-standard analysis
who affirmed with great passion its simplicity and intuitive character. For example Henle
and Kleinberg wrote in the preface to their work:
Thus we were led to the ε −δ approach to calculus, an approach that, although totally precise and
rigorous, was a disaster for students to learn and teachers to teach... A most natural place for
Robinson’s insight is a next (and possibly final) point in the evolution of the teaching of calculus.
We can now develop calculus using infinitesimals and enjoy all of their simplicity and intuitive
power, yet at the same time work in a mathematically precise and rigorous atmosphere.
We shall return to this question at the end of this chapter. However, it is necessary to
emphasize the weak impact of non-standard analysis on contemporary education. The
small number of reported instances of this approach are often accompanied with passionate
advocacy, but this rarely rises above the level of personal conviction.
ANALYSIS
173
1.5 CURRENT EDUCATIONAL TRENDS
As the majority of research cited in the remainder of this chapter is from France and
England, we restrict ourselves to very brief descriptions of recent educational trends in
these two countries.
Calculus/Analysis Teaching in France:
After the reform of 1902, the derivative was introduced in secondary education to students
aged 16–17. In 1971 the classical definition, interms of the limit of a quotient of differences,
gave way to a definition in terms of affine approximation, the derivative appearing as a byproduct of the approximation – the coefficient of the linear part, as evidenced by the
following extracts from the curriculum:
“Linear function tangent at a point to a given function; derivative at this point...”
(1971)
“Expansion limited to order 1; derived number, dynamic interpretations (velocity) and geometric
interpretations (tangent) ...”
(1982)
“Approximation by an affine function in a neighborhood of 0, functions which associate to a given
h: (1+h)2, (1+h)3, 1/(1+h), √( 1+h). When, for a neighborhood of h=0, f(a+h) can be written in the
form f(a+h)=f(a)+Ah +hε(h) with lim ε(h)=0 when h tends to 0, one says that the function f has A
(1985)
for its derived value at a...”
Correspondingly, the tangent is presented as the straight line of best local approximation
to the curve associated with the function. But, since the reform of 1982, the ε∠δ
formalization of the limit has been omitted.
Integration is now introduced in the last year of secondary education (age 17–18),
traditionally defined in terms of the primitive, whose existence is assumed for a function
continuous on an interval. The calculation of primitives is immediately applied to the
calculation of areas (area of a domain in the plane defined in a rectangular coordinate system
by the relations a≤x≤b and 0≤y≤f(x), f being a continuous positive function): it is specified
in the syllabus that the difficulties involved in the notion of area will not be introduced.
The reform of 1972 also introduces a more ambitious program in the domain of integral
calculus, with the definition of Riemann sums for a numerical function of a real variable
on a bounded interval: the theorems on the integrability of continuous or piecewise
monotone functions are admitted.
The reform of 1982 sees the return of the integral as primitive and as the area under a
positive function, and introduces examples of approximating the value of an integral by
various numerical methods.
Differential equations in the syllabus are only concerned with algebraic solutions in the
most simple cases. The latest programs only mention linear differential equations with
constant coefficients of the first and second order without a second term.
At university, in mathematics or physics, formal instruction in analysis constitutes the
major part of the first two years. In differential calculus, the derivatives, partial derivatives
and Jacobian matrices occupy centre stage. The notion of differential is introduced at the
beginning of the study of functions of several variables. For the last twenty years this has
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been in terms of its modem role in the tangent linear map. Integration is concerned with the
classical development of the Riemann integral. Differential equations are taught, but
essentially only with the goal of obtaining algebraic solutions.
Calculus/Analysis Teaching in England:
In England prior to the Education Reform Act of 1989 there were no national directives on
the curriculum and this act only concerns itself with education to the age of 16. The
syllabuses at “advanced level” in school (aged 16–18) are determined by external
examinations which are offered by a variety of competing examination boards who decide
their own content subject to the agreement of advisors from the teaching profession and the
universities, subject to an agreed “common core”. The pure mathematics content of the
various syllabuses is based on the algorithms for differentiation, integration and simple
ideas about differential equations. The concepts are explained in a dynamic way (“as x tends
to a” or “x→a”) and the course is mainly concerned with the methods and applications of
differentiation and integration; only a very few may see the ε−δ definitions at a later stage.
The methods of the calculus may be applied in other areas such as mechanics. At university,
mathematics students study the logical foundations of analysis using ε −δ definitions (or
equivalent topological formulations), whilst other students study calculus methods or
analytic theory to a level appropriate for their main subject. Formal analysis is known to
trouble all but the most able mathematics students and in some universities there is a trend
to reduce the formalities of the subject and concentrate more on methods and applications.
2. STUDENT CONCEPTIONS
One can associate some a priori concepts with the notions of derivative, integral, tangent
and tangent plane. For example, one can conceive of the tangent to a curve at a point A as:
• a line passing through A but not crossing the curve in a neighbourhood of A (the
point of view used notably by Appollonius to determine the tangents to the
conics and not requiring a differential approach),
• a line having a double intersection with the curve at A (a point of view present
in the works of Euler and Cramer for example, then later systematized in the
context of algebraic geometry),
• a line passing through two points infinitely close to A on the curve (the point of
view of Fermat, Leibniz, ...) or the line which the curve becomes when one
magnifies it in a neighborhood of A,
• the limit of the secants ( AM ) as the point M tends toward A along the curve, as
in figure 18 (the viewpoint of D’Alembert for example, traditional in education),
• the best linear approximation or the only linear approximation of the first order
to the curve in a neighborhood of A (leading to the more sophisticated idea of
the tangent linear map),
ANALYSIS
175
Figure 18 : The dynamic movement of a secant to a tangential position
• the line passing through A whose slope is given by the derivative at A of the
function associated with the curve (where the derivative is assumed to exist).
Similarly, one can see in the derivative at x=a of the function f as:
• the limit of the ratio (f(x +h)–f(x))/h when h tends toward 0,
• the first order coefficient of the expansion limited to order 1 of the function at
a (as in the contemporary French programme),
• the coefficient of the first order term in the full series expansion of f around a
(point of view of Lagrange),
• the coefficient characterizing the linear map, tangent to fat a,
• the slope of the tangent at a,
• the number or the function obtained by applying the usual rules of differentiation, knowing the derivatives of the elementary functions
or again,
• the slope of a highly magnified portion of the graph itself (for a “locally straight”
graph – the view point advocated by Tall (1986a) and now adopted by the British
School Mathematics Project in its new curriculum).
Similarly the integral has several different conceptions: the inverse operation of differentiation, a process for obtaining lengths, areas, volumes, a continuous linear form on a space
of functions, or more generally a process of measure.
One can imagine that these points of view can coexist in mathematicians, some of them
being preferred because of the mathematical context or because of individual preference.
One or another situation can lead to their being called forth and put into effect. What
happens for students? In particular, which viewpoints are preferred and which are difficult
to put into effect? Are there subtle transitions between different levels of functioning, are
there conflicts, obstacles? What role does education play in all this?
Whilst the research discussed in the next section was not designed to answer these
specific questions, it furnishes a good indication of the conceptions of students and the
manner in which they develop.
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2.1 A CROSS-SECTIONAL STUDY OF THE UNDERSTANDING OF
ELEMENTARY CALCULUS IN ADOLESCENTS AND YOUNG ADULTS
The study conducted by Orton in his thesis (1980) is experimentally based on individual
interviews conducted with 110 students aged 16 to 22 (60 in their last year of secondary
school – the English “sixth form” – and 50 in college) all having chosen to study
mathematics and having taken at least one course in calculus. The different tasks proposed
in the interview were regrouped by items in the analysis, each concerning just one aspect
of elementary calculus, and the responses were evaluated for each item on a scale of from
0 to 4. Tables V and VI, below – extracted from Orton (1983a, 1983b) – give a global idea
of the levels of success corresponding to each item for the two populations involved.
Following the work of Donaldson (1963), Orton classified the errors into three
categories: “structural errors”, “executive errors”, and “arbitrary errors”.
Structural errors were those which arose from some failure to appreciate the relationships involved
in the problem or to grasp some principle essential to solution. Arbitrary errors were said to be those
in which the subject behaved arbitrarily and failed to take account of the constraints laid down in
what was given. Executive errors were those which involved failure to carry out manipulations,
though the principles involved may have been understood.
Without going into further details, it seems important to note that the research showed:
• a reasonable mastery of algorithmic algebra in terms of calculation of derivatives and primitives, at least for the simple functions, as indicated by the degree
of success in tables V and VI.
• significant difficulty in conceptualizing the limit processes underlying the
notions of derivative and integral: For instance, when questioned what happens
in figure 19 to the secants PQ on a sketched curve as the point Qn tends towards
P on the circle, 43 students seemed incapable, even when strongly prompted, to
Mean Scores (out of 4)
School
College
Infinite geometric sequences
2.88
2.56
Limits of geometric sequences
2.92
2.78
3.32
3.68
Substitution and increases from equations
2.22
2.02
Rate of change from straight line graph
Rate, average rate, instantaneous rate
0.88
1.18
2.22
2.02
Average rate of change from curve
Carrying out differentiation
3.62
3.50
1.88
1.14
Differentiation as a limit
1.52
1.40
Use of d-symbolism
Significance of rate of change from differentiation
3.43
3.62
Gradient of tangent to curve by differentiation
3.63
3.76
2.30
2.54
Stationary points on a graph
Description of task
Table V: Performance on calculus tasks
ANALYSIS
177
Mean Scores (out of 4)
School
College
3.28
3.06
Limits of sequences of numbers
2.82
2.90
Limits from general terms
Heights of rectangles under graphs
2.68
3.42
2.40
3.12
Use of previous heights in a new situation
Calculation of areas of rectangles
3.03
3.62
Simplification of sum of areas of rectangles
2.43
3.52
Sequence of approximations to area under graph
2.18
3.22
0.78
1 .00
Limit of sequence equals area under graph
Limit from sequence of fractions, from general term
1.67
2.48
Carrying out integration
2.98
3.40
1.10
0.60
Integral of sum equals sum of integrals
Complications in area calculations
2.55
2.78
Volume of revolution
0.95
0.88
Description of task
Table VI
see that the process led to the tangent to the curve?
There appeared to be a considerable confusion in that the secant was ignored by
many students, they appeared only to focus their attention on the chord PQ,
despite the fact that the diagram and explanation were intended to try to insure
that this did not happen ... Typical unsatisfactory responses included : “the line
gets shorter”, “it becomes a point”, “the area gets smaller” ...
These responses are entirely consistent with the nature of the geometric obstacle studied
separately by Comu and Sierpin´ska (see Chapter 10).
Similarly, although students showed certain competencies in calculating limits of
sequences given explicitly in terms of numbers or simple functional expressions, only 10
were capable of expressing that the exact area under part of a parabola could be obtained
as the limit of the sums of approximating rectangular strips. Orton concluded:
Figure 19 : Secants “tending to” a tangent
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Students were able to obtain a limit from a sequence when the sequence was directly requested but
were not able to appreciate when a limit would solve a problem.
A similar difficulty arises evaluating areas bounded by curves in slightly more general
circumstances (the presence of negative values, discontinuities, or curves associated with
functions x=f(y) for example):
Many students appeared to know what to do, but when questioned about their method, didn’t really
know why they were doing it.
Further difficulties included:
• the difficulty of using relevant graphical representations. Students could
usually calculate derivatives of polynomials correctly and were equally successful with a task in the form:
Find the gradient of the tangent to the curve y=x3 –3x2+4 when x = 3.
But having to evaluate these same rates of growth from the graphs for functions
of similar complexity, a non-negligible proportion made errors, confusing
average and instantaneous rate of growth or simply giving the value of the
function at the point in question. In a graphical context the expression of the
derivative as a limit was poorly understood: 96 students, after having found the
expression for average rate of growth of the function f(x)=3x 2+ 1, between a and
a+ h, could not see how to obtain the rate of growth at 2.5 or at a general x.
• the minimal meaning ascribed to the symbols used.
For instance, when asked to explain the meaning of dx, dy , dy/dx , 71 gave
incorrect responses for the rate of growth:
rate of change of y
” “rate of change at a point”, “small increase in the rate
rate of change of x ’
of change”
and 25 interpreted dx as the limit of δx when δx tends toward 0.
This strength in the algorithms of algebra as opposed to weakness in graphs and geometry
is also found by other authors, some already cited in the preceding chapter, for example,
Tall, 1977, Artigue & Szwed, 1983. The latter presents an account of the responses of 89
first year university mathematics students of mathematics to the question in figure 20.
The first question yielded 28 correct responses (with some minor errors in calculation),
29 incorrect responses, and 31 incomplete responses, the errors being principally based on
confusion between continuity and differentiability or between differentiability and existence of derivatives to the left and to the right.
ANALYSIS
179
Figure 20 : A conceptual task on differentiability and integration
For the graph of f':
• 67 students (out of 89) gave a correct graph on the portion ]-4,2[ and 5 others
had an incorrectly positioned horizontal segment on this portion,
• 63 considered the sign of the derivative on the interval ]2,5[, but only 18 students
gave an acceptable form for the graph and, when the values attributed to f '(3) and
f' (5) are taken into account, only 10 pupils gave a satisfactory graph.
The graph of the function g was only attempted by 35 students. The curves produced were
extremely diverse and seemed to have only one property in common: the graph is a line
segmenton [–2,2]. Only 14 gave the correct slope and at least 14 graphs were discontinuous.
Of the 35 graphs produced, only 13 considered the direction of variation in g, and only 3
could be considered acceptable solutions.
Analysis of the transcript and the errors committed shows that many students did not
work directly with the graphs but sought to obtain algebraic expressions for f on each
interval in order to differentiate or integrate them, in the latter case making numerous errors.
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When the results produced were inconsistent with the graph, there seemed little awareness
of conflict, their confidence being placed more in the calculation than the picture.
A simplified version of the derivative part of this test has been used for several years in
university entrance examinations for potential mathematics students. The results obtained
are in the same spirit: significant confusion over continuity and differentiability of the
graphs, with correct responses only for linear parts of functions (cf. Robert, 1983; Authier,
1986).
2.2 A STUDY OF STUDENT CONCEPTIONS OF THE DIFFERENTIAL,
AND OF THE PROCESSES OF DIFFERENTIATION AND INTEGRATION
To investigate students conceptions in the related disciplines of mathematics and physics,
two teams from mathematics education and one from physics education collaborated in a
research study of the effects of current educational practices in the first two years of
university (cf. Alibert et al, 1987; Artigue & Viennot, 1987; Artigue et al, 1989). The
researchers conducted their work in three directions:
• analysis of the historical evolution of the concepts and how they were taught,
• analysis of student conceptions and, to a lesser extent, of the teachers, through
approximately 10 questionnaires from a mathematical or physical viewpoint,
together with individual interviews,
• experimentation with, and evaluation of, sequences of instruction.
The analysis of historical evolution suggested three directions for analysis of students’
conceptions:
• the meaning and usefulness of differentials and differential procedures,
• approximation and rigour in reasoning,
• the role of differential elements.
We consider the role of each of these in turn.
2.2.1 THE MEANING AND USEFULNESS OF DIFFERENTIALS
AND DIFFERENTIAL PROCEDURES
Mathematical questionnaires, completed by 85 third year university students, revealed an
important difference between the declarative level (how the students described the
concepts) and the procedural level (how they carried them out). At the declarative level, the
tangent linear approximation differential dominated, conforming to the definition in the
course. At the procedural level, the differential tended to lose its functional role and the
status of approximation disappeared, to be replaced by algebraic algorithms using partial
derivatives and Jacobian matrices.
ANALYSIS
181
A typical manifestation of this was revealed in the responses to the following two
questions:
“If you had to explain what a differential is to a first year student:
1 a) What definition would you give?
1 b) What notations would you introduce?
1c) What examples would you use?
1 d) What important points would you stress?”
and
Is the function
defined by f(x ,y ) =
at the point (0,0)? Justify your answer.
differentiable
The responses for 1a) were dominated by the notion of tangent linear map; in the remainder
of the question:
• 33 mentioned the relation between the differential and related notions
(continuity, differentiability, existence of partial derivatives),
• 15 mentioned the idea of local approximation, the functional and linear aspect,
• only 11 mentioned the algorithmic procedures of calculation.
In the second case, a small minority (13%) recognized that the given function is already in
the form of a linear expansion of order one and almost all, in spite of the complication caused
by the vanishing of the square root at the point in question, went directly to the calculation
of partial derivatives. Even among those who recognized the linear part, very few
succeeded in proving that the remainder is of higher order.
The percentages of responses to other questions confirm these impressions: 86%
responded to a complicated calculation of second partials of a composite function at the end
of the questionnaire; less than 10% responded to questions requesting a justification of
classical approximations for calculating a volume by cutting it into slices. Some students
even complained that they thought the latter was off the syllabus.
The questionnaires formulated from a physical viewpoint show clearly that first and
second year students do not understand the differential procedures. For example, they were
provided with the beginning of a classical calculation of atmospheric pressure leading to
the differential expression dp = –ρg dz. (figure 21).
The responses, summarized in figure 21, show:
• the strong conviction of students that the atmosphere needed to be cut into
infinitesimal slices (90%),
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Figure 21
but
• their justifications were based mainly on the mistaken view that p is a function
of z, rather than the fact that ρg is not constant (it is a function of z).
With water the factor p g is constant and cutting into slices is no longer necessary, but
students who give a wrong reason for “air” do not see this, still suggesting that it is necessary
to cut into slices.
2.2.2 APPROXIMATION AND RIGOUR IN REASONING
In general requests on the mathematics questionnaires for justification of approximations
were poorly answered and were considered “off the syllabus” unless they could be solved
by directly quoting classical theorems. In specific examples, the approximations suggested
by the students were incorrect more than half of the time and the great majority of errors
were concerned with the remainders, as if the fact of writing an ε arbitrarily at the end of
a formula is sufficient to make it rigorous.
In physics, the problem of rigour is handled differently, with phrases such as “provided
that dz is sufficiently small” being used instead of ε−δ methods.
These conceptions remained persistent, even with more advanced students. The
problem in figure 22 was presented to 50 third year mathematics students, 22 fourth year
physics students and 13 fourth year physics students in a very selective group preparing for
teaching.
ANALYSIS
183
Figure 22
Only a small proportion of those questioned (less than 25%) seemed capable of giving
an acceptable response. For instance, a future teacher wrote:
“Maybe it is just by chance that it works (computation of the volume). Indeed, the relationship
dV =S (z ) dz is not true. It could be the reason why the other computation about the area of the sphere
does not work.”
The results obtained here are completely compatible with those obtained in another
mathematical questionnaire given to 35 students of the third year (cf. figure 23).
The majority of students responding considered the different approximations to be valid
because the given slices tend towards a spherical slice. One has the impression that, for
them, this geometric convergence guarantees the convergence of all quantities associated
with the figure, even though this is false.
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Figure 23 : Elemental slices
2.2.3 THE ROLE OF DIFFERENTIAL ELEMENTS
The responses to the questionnaire show that the functional role of differentials is hardly
present in mathematics students at the declarative level less than a third of the differentials
given in response to various questions were in the form of a function. Moreover, the
geometric images associated with the concept were weak and restricted to one dimension.
For instance, this is manifested in the responses to the following questions:
“The map f:
is defined by:
f(x,y)=exp(y2+ x)*sin(xy)
Find its differential at the point (1,0) and give a geometrical interpretation.”
Of the 85 third year students questioned, only 31 dealt with the geometric interpretation and
only 8 gave a correct interpretation in terms of the tangent plane. In particular, many spoke
of the tangent to the curve as if they were still in the one dimensional case.
In physics, the results obtained show that the role of differential elements oscillates
between two poles:
• At one extremity, the differential elements have a purely formal role of
indicating the variable of integrationanditwasbetter to avoid thinking too much
about what they could mean when manipulating them:
“To integrate, it is essential not to think about what d l represents, but to proceed
mechanically, otherwise we are done for” (student of the first year) – “d x is not
real” – “immaterial” –“the length is fictitious” – “in fact, it does not matter at
all, when integrating, dl becomes a variable of integration”.
• At the other end, the differential elements have a very strong material existence
which can exclude all other meaning:
“dl is a small length”, “a little bit of wire”,
ANALYSIS
185
Figure 24 : The role of differentid elements
Between these two poles, there are a wide range of views such as,
“dz is the limit of z when →0”, “dl is an infinitely small element”, “one
cannot find anything smaller”, “it means very simple” (which apparently means
that differentials are often used to simplify the situation).
Often, to solve a problem, it is necessary to give a functional meaning to the differential
elements. The authors consider that strong conceptions, either purely formal or purely
material, can make this process more difficult. Thus, faced with the problem of calculating
the induction created by a straight electric wire through which a current is passed (figure
24), the only students who did not make the classic error of mechanically transforming the
differential element dl into a function of r,θ as dl = r dθ, were exactly those who were also
capable of giving a functional role to differential elements.
From the various results obtained, it Seems that two differentials exists simultaneously
for the students, potentially in conflict with the definitions, but governed by certain
tendencies:
• the algebraic algorithms overwhelm the meaning and process of linear approximation,
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• questions of rigour are reduced to formalism,
• the functional mode of thought is weak.
It should be emphasized that the existing gap between teaching in the two disciplines
usually prevents the potential conflicts from being realized.
2.3 THE ROLE OF EDUCATION
The empirical results discussed so far show a broad coherence over a wide range of
students: in every case there is a dominance of the algebraic mode of procedure contrasting
with a frailty in geometric and graphic modes, and a lack of meaning for limits and/or
approximation. Questions of rigour or justification linked to the treatment of approximations are conceived as secondary.
But how are these observations influenced by the educational process? Alibert et al
(1987) reinterpret the results obtained in the context of their current instructional procedures in analysis. Two contexts appear: algorithmic procedures and the conceptual
viewpoint involving questions of meaning and legitimacy of the theory. Each discipline
tries to manage the relations between these two contexts, in a manner appropriate to the
subject, whilst seeking an optimal balance between rigour and operational practice. The
conceptions developed by students are essentially a reflection of the very unsatisfactory
equilibria found by education.
In mathematics the means of justification is classically that of proof. However, from the
start, education distorts real difficulties concerning limits, functions, basic tools of
approximation (such as inequalities, absolute values, reasoning with sufficient conditions,
&c), and the understanding and manipulation of quantified statements. Instead it conceals
them all by using powerful algebraic algorithms (calculation of derivatives, partial
derivatives, Jacobian matrices, primitives) and potent theorems which reduce theoretical
considerations to algebraic techniques (such as theorems involving the sum, product and
composition of C1 functions). Regrettably this premature algebraic algorithmization, on
the one hand gives too privileged a role to the algebraic setting, and on the other tends to
drain the differential and integral procedures of their real meaning.
In physics, not being constrained to requiring proofs, it is possible to take refuge in the
conviction that it works, even if one does not know why, once again denying students a
satisfactory scientific experience.
3. RESEARCH IN DIDACTIC ENGINEERING
The work discussed in this section complements the research described in the preceding one
by using acquired knowledge about the learning process, and the effects of the usual
pedagogy, to develop and test new methods of teaching and learning. The French term
ingénierie didactique for this activity translates literally as “didactic engineering”.
ANALYSIS
187
3.1 “GRAPHIC CALCULUS”
“Graphic Calculus” is an approach to the calculus developed by Tall (1986a, 1986c, 1990)
which acknowledges the known conceptual obstacles in the limit concept and proposes
instead a new learning sequence built on the visualization of the “local straightness” of
graphs. It uses the graphic and dynamic possibilities of the computer to give a cognitive base
for the notions of derivative and integral in secondary education which can lead to later
formalizations in either standard or non-standard analysis.
In order to achieve this objective, the student is furnished with a computer environment
(or microworld) designed to encourage the exploration of examples of specific mathematical processes and concepts. This requires a special kind of software:
An environment that provides the user the facilities of manipulating examples (and, where possible,
non-examples) of aconcept, I term a generic organizer. The word “generic” means that the learner’s
attention is directed to certain aspects of the examples which embody the more abstract concept.
Thus the equality 3+2=2+3 may be seen as a specific example of the commutative property of
addition. The generic example is seen as a representative of the whole class of examples which
embody the general property.
Tall recognizes that ageneric organizer does not guarantee its use by the student as a tool
of abstraction and suggests the need for an “organizing agent”:
“guidance from a teacher, a textbook or appropriate computer material.”
For this reason the instructional treatment is based on three phases:
• a first phase of familiarization and negotiation of meaning. It is conducted in the
form of a dialogue between the teacher and the students, a dialogue designated
as the “enhanced Socratic mode”, the term “enhanced” referring to the aid
provided by the computer to communication:
“The mathematics is no longer just in the head of the teacher, or statically
recorded in a book. It has an external representation on the computer as a
dynamic process.”
• a second phase of autonomous work by students with the generic organizer,
and finally,
• a last phase of discussion and evaluation looking towards establishing the point
and making sure that the concept images constructed by the pupils are compatible with those of the community of mathematicians.
The generic organizers elaborated by the author are specifically adapted for introducing the
ideas of derivatives, integrals and differential equations.
For instance, the software “Gradient” contains various modules permitting the graphing
of functions defined by a formula, magnifying a graph around a point, superimposing onto
the graph y=f(x) the line passing through the points ( x,f(x)) and (x+c,f(x+c)) (for fixed c),
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f (x)= sin x
Figure 25 : a locally straight curve (magnified a little)
and dynamically seeing the evolution of this line and the curve representing its slope, as x
varies. It also allows the superimposition of a graph, to test conjectures made about the
derivative of this or that function (monomials or trigonometric functions for example).
Finally, as an archetypal non-example, it contains a module which shows the construction
by steps of a function continuous and non-differentiable at every point: “the blancmange
function”.
As conceived by the author, these generic organizers are designed to permit the pupil
at the pre-calculus level to develop a concept image of the notion of derivative based on:
• The conception of a function differentiable at a point as a function which, by
magnification around the point, eventually becomes like a straight line.
• A global image of a derived function associated with the notion of “practical
tangent”: a straight line passing through two points very close to each other on
the curve.
Corresponding generic organizers have also been designed for integration and differential
equations. These are discussed in greater detail in chapter 14.
Experimental studies using generic organizers described above were done in several
classes of secondary education with other classes as controls. The results confirm on the
one hand the difficulties already cited in § 2 of conceptualization of the notion of a limit, both
in terms of tangent as a limit of secants and of the derivative as a limit of slopes of secants.
The results of the post-test differed little from those of the pre-test, in the experimental
groups as well as the control groups. Tall concluded:
ANALYSIS
189
f(x) = b1(x)
Figure 26 : a highly wrinkled function that is nowhere locally straight
‘“The low level of responses indicate the high cognitive demand of this general concept, reinforcing
the opinion that, although the notion of a limit is the natural foundation of a mathematical
development of the calculus, it is not a natural starting point for a cognitive development”.
(Tall, 1986d)
As a result of the instruction, a significant number of the experimental pupils tended to
conceive of the tangent as a straight line passing through two very close points on the curve.
But they were far better at recognizing and drawing derivatives, even attaining performances comparable to those of university students. For example, 67% of the experimental
students recognized and justified that graph (2) in figure 27 (overleaf) is the one whose
derivative is graph (1), (with 68% for university students), while only 8% in the control
group were successful.
Tall concludes in these terms:
“Once more empirical research has demonstrated a process of didactic inversion that gives an
attractive cognitive approach. In this case, the cognitive approach, in the shape of the practical
tangent, proves to be surprisingly good mathematics.”
(Tall, 1986d)
Other approaches using the computer to reorganize the syllabus and to introduce the
calculus without using the limit as a prerequisite have shown success. D’Halluin & Poisson
(1988), for instance, pursued research on “a strategy for teaching mathematics: the
mathematization of situations integrating the computer as a tool and as a mode of thought”
as continuing education for adults and school drop-outs. A function has three formal objects
associated in interaction : Picture, Graph, Formula (the triple “PGF”) on which the
computer permits global operations. The introduction of the differentiation-integration
concept takes place first at the numerical and graphical level, exploiting the computer,
starting from a physical situation: the construction of a road. The problems of pitch and of
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Figure 27 : Recognizing an anti-derivative
digging and filling give a meaning to the derivative corresponding to that of the practical
tangent of Tall and to the integral in terms of measure. From the beginning they emphasize
reciprocity of the operations of differentiation and integration through calculation of
difference tables for the slope, tables of sums for the areas, global visualization of the slope
and area curves. They next ask the students to study situations of speed (motion approach)
and of distribution of salaries (statistical approach to the integral). The algebraic
operationalization comes later, building on simple calculations of slopes and areas, using
previously developed tools.
ANALYSIS
191
3.2 TEACHING INTEGRATION THROUGH SCIENTIFIC DEBATE
In recent years, a form of scientific debate has been introduced at the University of Grenoble
(Legrand et al , 1986). Full details of the methodology will be discussed in the chapter 13
on proof in advanced mathematical thinking. Here we will concentrate on the outcomes of
this approach in the learning of concepts in analysis. It occurred in a context where students
were encouraged to conjecture and debate ideas in groups within a large class, where
arguments were proposed and addressed to other students rather than the teacher.
The concept of integral was studied in great depth during this research. The associated
didactic engineering was presented to first year university students who had already studied
the secondary curriculum described earlier, including calculation of simple primitives and
the conception of the integral both as the inverse operation of differentiation and the area
under a curve. The new curriculum was designed to enrich the conceptions of students by
giving a meaning to the notion of integral procedure.
It began with the following problem (Alibert et al , 1987b):
figure 28 : A problem to initiate debate In analysis
The researchers hypothesize that, appropriately managed, scientific debate can help to
solve this problem. One method is through visualizing the bar as being made up of tiny
slices, calculating the force, then refining and passing to the limit through integration.
However, the vast majority of students suggest a solution by conceiving the mass of the rod
concentrated at the centre of gravity, which proves erroneous. In the course of the
experiments, students were always found who proposed testing the validity of such a
calculation by cutting the bar in two and applying the principle of centre of gravity to each
half of the bar, which gives a different answer, so the principle is seen to be in conflict with
itself. If the idea of concentrating the mass at a point is to be retained, new methods are
suggested by placing the point mass at one end, or the other, to obtain inequalities.
Repeating such process on each half of the rod, then on each quarter, and so on, leads to a
conviction that one is going to be able to obtain a value to any desired precision, prior to
the passage to the limit.
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This first problem was followed by others aimed at extracting the integral procedure
from other situations (averages, problems using time as a variable, space variables in two
or three dimensions). Only after these problems is the study of mathematical properties of
the integral operation begun, largely based on the work developed by the students.
The results of the first two experiments conducted in first year at Grenoble University
with 105 students in 12 two hour sessions and 101 students in 14 two hour sessions are
reported in Legrand et al, (1986). The effects of scientific debate within the overall
instruction revealed improved understanding of integration in the final exam. For instance,
the problem in figure 29 was given in 1986.
Figure 29
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193
89 students (84%) passed this test and only 20% tried to solve the problem by looking
for primitives. This led the authors to conclude:
“The results show that a majority of students acquired a satisfactory level of understanding of the
integral concept introduced by the method of scientific debate and that they understand it in
sufficient depth as to know how to explore their knowledge, even when the usual algorithms are not
applicable.”
(Alibert et al , 1987b)
3.3 DIDACTIC ENGINEERING IN TEACHING DIFFERENTIAL EQUATIONS
Traditionally differential equations have been taught as a catalogue of recipes for algebraic
solution in the classical integrable cases. Recently software has been developed for the
effective teaching of differential equations by a number of authors (e.g. Tall, 1986b; Tall
et al, 1990; Koçak, 1986; Hubbard & West, 1990). The research reported here, conducted
by Artigue (l987), is aimed at studying the viability of a teahing approach which tries from
the beginning to coordinate the algebraic, numerical, and graphic approaches with the
solution of associated differential equations. The author interpreted present-day instruction
as a position of stable equilibrium of a system subject to a set of constraints (epistemological, cognitive, didactic conventions, the available mental representations of the students
and teachers) and seeks to modify some of these to permit the system to come to another
stable equilibrium that is more satisfying from the point of view of the epistemology of the
field. Computer software constitutes the principal lever employed to modify the space of
constraints, facilitating access to numerical and graphical representations. In the new
curriculum it is used in both interactive and ready prepared form (using supplied computergenerated graphs) to take account constraints of time and material, and to optimize
management.
For example, in the graphical context, it is first used in prepared form to give meaning
to the qualitative solution: drawing curves compatible with a field of tangent directions. It
is then used to introduce elementary qualitative tools: isoclinic lines, in particular, the
isoclinic line 0 as being essential to determine the direction of variations of the solutions
that act as barriers or separatrices between different types of solutions, to identify stability
by simple geometric transformations (such as symmetry, translation). Such an approach
proves to be motivating in significantly improving students’ abilities to associate pictures
of solutions with algebraic equations in a way which is far less complex than having to make
drawings by hand (figure 30).
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Figure 30 : Relating graphical and symbolic representations
The computer is also used in an interactive manner. For instance, when studying
differential equations depending on a parameter; the software allows experimental
determination of the different types of pictures of possible phase portraits, then students
have to justfy the most likely graphs obtained by using algebraic and/or graphical
arguments, to address a list of unsolved problems and to formulate, if possible, appropriate
conjectures.
Although only a short time was available for the instruction, research revealed very
positive results. The students showed themselves capable of giving meaning to the
qualitative approach, to describe and draw solutions without algebraic integration in simple
cases such as the equation:
and to coordinate the algebraic and graphical contexts.
ANALYSIS
195
3.4 SUMMARY
It is important to emphasize once more the convergence of the research. In the work on
didactic engineering there are themes which relate the different approaches:
• focus on construction and the control of meaning,
• search for a better equilibrium between the different representations for the
concepts, in particular, concern with a better use of a graphical context,
• concern to use the possibilities offered by the computer to rethink the content
of education in terms of epistemological adequacy of the domains considered
and the cognitive capacity of students.
As for the precise contents envisaged by the research, even where the works concern
different levels, one finds again common preoccupations:
• concern with developing a functional approach,
• concern to focus the notion of derivative on the existence of a good approximation of the first order, the computer allowing exact visualization of this property
by magnification of the graph, even before the notion of limit is mastered.
However, the research is largely focussed on the intuitive beginnings of the subject and it
is natural to ask the question: can the development of strong conceptions of this type
subsequently form an obstacle to the construction of more formal concepts such as the
measure interpretation of the integral procedure? On this point there is need for more
empirical research.
The differences between these various experiments are more concerned with the
proposed management of instruction. In all didactic engineering there is a place for an
experimental approach to mathematics and organization of the construction of knowledge
around the activity of the pupils. But the researchers at Grenoble also have the conviction
that in order to allow the students to establish a correct epistemology of mathematical
knowledge, it is necessary to change their relationship with mathematics created during
their schooling, which is based on the predominance of an algorithmic approach and on a
vision of proof as a simple contractual agreement rather than a means of convincing or of
lifting uncertainty. In association with this, there is a conviction that this change can only
be made by a break with the traditional learning contract through the notion of scientific
debate. Such a conviction is developing in a number of different countries (cf. for example
Schoenfeld, 1985 or Robert in chapter 8) and researchers seek to exploit the capacities of
pupils at the end of secondary education to help them reflect explicitly on the mathematical
processes.
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4. CONCLUSION AND FUTURE PERSPECTIVES IN EDUCATION
The results obtained in the various research projects described in this chapter, present a
strong coherence not only between themselves but also with those of the preceding chapter
on pre-calculus concepts. Learning the beginnings of analysis presents certain difficulties
which appear to be due to different factors, in particular:
• the highly sophisticated level of structure of the objects in the foundations of this
conceptual field, such as sequences and functions,
• the existence of various obstacles, including those evident in the historical
development, those due to the conflicting everyday meaning of some of the
terms, obstacles due to the all-pervading problem of infinity, and to pupils’
conceptualizations of the reals more consonant with the non-standard theory
than the standard formalization,
• difficulties posed in learning specific techniques of the field: such as use of
upper and lower bounds, use of thecompleteness axiom, reasoning by sufficient
condition which forces the acceptance of loss of information,
• finally the difficulties due to formalization in this field: first because it
introduces structural definitions which may conflict in the students mind with
more intuitive spontaneous conceptions, and secondly because it bases proofs
on complex propositions involving quantifications which operate in a direction
seemingly contrary to the dynamic flow of intuitive thought.
These difficulties are far from being resolved by students in secondary education and are
reflected in learning calculus which constitutes the most important part of education in
mathematics during the first two years of the university. The research suggests that, faced
with these difficulties, the usual instruction takes refuge in an intensive “algebraisation” of
analysis: manipulating formulae rather than functions, emphasizing the calculation of
derivatives rather than the theory of linear approximations, calculating primitives in
integration rather than delving into the meaning of the integration procedure and learning
recipes for solving differential equations without developing a general numerical or
graphical approach to the solution. Moreover, one tries to resolve difficulties due to
formalization by first giving definitions then quickly proving or quoting powerful theorems
which permit the learner to move on from the subtle theory to return to algebraic algorithms.
The results of the research cited bring out clearly the perverse effects of this avoidance:
in avoiding difficulties of formalization and techniques of approximation, a real chasm is
created between concept definition and concept image; in excessively emphasizing the
algebraic approach through facile algorithmization, the possibilities of changing points of
view, essential to the real practice of the mathematician, are reduced. Furthermore it
excludes the possibility of taking advantage of cognitive diversities which can exist among
students. By avoiding the problems of legitimization one leads students to see proofs as a
simple matter of didactic contract – something to be done to fulfil the requirements of the
course. Certainly the students come to obtain a reasonable level of success in a certain
number of algorithmic tasks, but it must be emphasized that there is not, in the context of
ANALYSIS
197
this instruction, a real introduction to analysis: the conceptions developed by the students
are poor and the subtle techniques in the field are not adopted.
However, the different experiences of didactic engineering presented here prevent
fatalism. They tend to show in particular that the cognitive capacities of students, properly
exploited, could warrant a more satisfying equilibrium between conceptualization and
algorithmization as well as between the different contexts in which the concepts arise. And,
as Dubinsky and Tall will discuss in chapter 14, use of appropriate computer languages can
help the students handle quantifiers and construct a meaningful concept image of rigorous
proof. Thus we see that the computer offers a number of didactic advantages:
• it provides possibilities for dynamic visualization to make the geometric and
graphical contexts much more accessible and, properly exploited, it can help to
bring out the necessary relations between algebraic and geometric representations,
• if the graphical context becomes more familiar, the unity of the graphic
representation of the functional object which it furnishes can help to establish
the concept image of the foundational concepts by enriching the stock of mental
images,
• through experimental activities with the interactive simulations, students may
be initiated into mathematics as a constructive scicntific activity.
• appropriate computer languages can help with the problems of formalization,
the constructive use of quantifiers and the development of rigour (as we shall see
in chapter 14).
However, the computeris not a tool that will miraculously solve all the problems of teaching
analysis. It can without doubt help overcome certain difficulties, but the different studies
reported in this chapter show clearly that it will only be effective within a coherent teaching/
learning context. Elaboration, experimentation and evaluation of such an approach is costly
work. Moreover, even if it can help solve certain problems, the introduction of the computer
tool into education cannot fail to create, in its turn, new problems in classroom management
(availability of computers, coordination of computer use with other supports for learning,
and so on) and even cognitive problems which, in their turn, should be the subject of further
research.
We mentioned non-standard analysis in the first part of the chapter and at this point we
return for further consideration.
The results of the research presented in thepreceding chapter show that students’ mental
representations of the reals seems closer to non-standard representations then to standard
representations. Some of them are not in perfect agreement with classical non-standard
representations: for example when students said that 0.999 ... is the last number before 1,
they adopt an atomistic point of view incompatible with the axioms of non-standard
analysis, but one could imagine that it would be easier to move these conceptions towards
coherent non-standard conceptions, which might then lead back into standard analysis.
The results of the research show equally the importance of the void that separates
concept definition and concept image for school and university students and the operational
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deficiency of given definitions. Non-standard definitions are closer to the descriptions of
differential and integral problems in physics than standard analysis. They also have fewer
quantifiers and do not require the reversing of direction of the standard ε – d or ε – N
formulations: for example, a sequence un is convergent to a limit 1 if and only if for every
infinitely large N, uN –1 is an infinitesimal. Perhaps the definitions are more useable by
students and the chasm between concept image and concept definition may be diminished
by permitting a more gradual initiation to formalization.
The few pieces of research to date suggest that the logical baggage for current simplified
introductions does not constitute a severe obstacle for students (cf. for example, Tall,
1980b, Artigue et al, 1985). But at the present time we cannot say what difficulties will be
introduced by a non-standard approach to analysis, no more than we can say how standard
and non-standard concepts might be coordinated in the minds of the students, nor what
problems this coordination could pose. At the very least, however, the non-standard
approach seems an interesting road to pursue in future research, if institutional conditions
permit.
CHAPTER 12
THE ROLE OF STUDENTS’ INTUITIONS OF INFINITY
IN TEACHING THE CANTORIAN THEORY
DINA TIROSH
Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have
one line segment longer than another, each containing an infinite number of points, we are forced
to admit that, within one and the same class, we may have something greater than infinity, because
the infinity of points in the long line segment is greater than the infinity of points in the short line
segment. This assigning to an infinite quantity a value greater than infinity is quite beyond my
comprehension.
(Galileo, 1638)
Infinity is undeniably one of the central concepts in philosophy, science and mathematics.
In this chapter we review the nature of this concept and find that in different contexts the
term infinity means different things, it might be potential infinity (representing a process
that is finite and yet could go on for as long as is desired), or actual infinity in the sense of
the cardinal infinity of Cantor, or ordinal infinity, also in the sense of Cantor, but this time
representing correspondences between ordered sets, or non-standard infinity which arises
in the study of non-standard analysis, and, unlike the others, admits all the operations of
arithmetic, including division to give infinitesimals. It is clear that with this wide variety
of technical meanings which often have quite different, even conflicting, properties, the
possible intuitive meanings that arise in various contexts are likely also to be varied and in
conflict. Indeed this is a common fact to be found throughout the research on the cognitive
nature of the concept images associated with infinity. They are usually transient, unstable
and conflicting. In this chapter we will first review the different perceptions of infinity in
which we shall see that experiences of everyday life give little preparation for the nature
of the cardinal infinity encountered in set theory.
Given the conflict between previous experience and the formal theory, this is therefore
an ideal opportunity to test the theories enunciated in previous chapters in which students
are confronted with the cognitive obstacles and encouraged to reflect on them in an effort
to re-construct their knowledge to come to a new and richer cognitive equilibrium. The
second part of the chapter lays the groundwork by reporting a sustained investigation into
the intuitive criteria that students use to determine whether two infinite sets have the same
number of elements. The final part of the chapter considers a research study to help students
develop a formal knowledge of the Cantorian set theory supported by an adequate intuitive
background. It offers a possible model for the way in which students may be helped to come
to terms with the kind of abstract thinking that causes such difficulty in the transition to
advanced mathematical thinking. In this case the students are in the later years of secondary
school and are therefore at a suitable stage of development to test out value of reflection on
cognitive obstacles to assist transition to the more abstract forms of thinking at higher
levels.
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1. THEORETICAL CONCEPTIONS OF INFINITY
In the early history of mathematical development the two competing ideas of infinity
were potential infinity in which a mathematical process can be carried out for as long as
required to approach a desired objective, and actual infinity in which one contemplates the
totality of infinity, through, for example, conceiving the totality of all natural numbers at
one time.
Ever since Aristotle, philosophers and mathematicians have invariably rejected the
concept of actual infinity. Aristotle himself argued that
“The infinite is potential, never actual”
(Aristotle, Physics, Book 3, Ch. 7).
Similarly, in 1831 Gauss stated
“I protest above all the use of an infinite quantity as a completed one, which in mathematics is never
allowed. The infinite is only a façon de par ler in which one properly speaks of limits”.
(Gauss, in Dauben, 1983).
Rejection of the notion of actual infinity can be found even at the beginning of the twentieth
century: Poincaré, in an essay on “The logic of infinity”, noted that
There is no actual infinity, and when we speak of an infinite collection, we understand a collection
(Poincaré, 1963/1913, p. 47).
to which we can add new elements unceasingly.
Why have mathematicians argued so consistently against actual infinity? A main source of
their opposition to the idea is that it has given rise to numerous paradoxes and difficulties
in mathematics. Even those mathematicians who essentially accepted the existence of
actual infinity, such as Galileo, Bolzano, Dedekind, Hahn, Hilbert and Russell, were aware
of the difficulties involved. Galileo, for example, pointed out that if the number of natural
numbers is not only potentially but actually infinite, then there are as many perfect squares
as there are natural numbers, since for every natural number there is a perfect square and
every perfect square has a square root. He further noted that it is also possible to determine,
on the basis of the “part-whole” principle, that there are more natural numbers than square
numbers. Galileo concluded that infinite quantities are incomparable. He was one of the
first to mention that
Difficulties arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those
properties which we give to the finite and limited.
(Galileo, 1954/1638, p. 31).
Cantor made a significant and surprising breakthrough in creating a theory of actual
infinity. He defined not one, but two, distinct kinds of infinite numbers: transfinite ordinal
numbers, which are denoted by ω, ω+1, ω+2,... 20, etc.; and transfinite cardinal numbers
whicharedenotedby ℵ0, ℵ1, ℵ2, etc. The transfinite ordinal numbers, which were the first
to be introduced, are an extension of the notion of ordinal numbers to the infinite case. They
were defined only for ordered sets.Two ordered sets are considered to have the same ordinal
number if they can be put into a 1-1 correspondence with one another in such a manner as
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
201
to maintain the order relation between corresponding elements. The transfinite cardinal
numbers are an extension of the notion of counting. Two sets are considered to have the
same cardinal number if they can be put into a 1-1 correspondence with each other. These
two notions of transfinite numbers are clearly distinct from one another. In fact, Cantor
showed that it is possible to construct an infinite number of infinite sets having different
ordinal numbers but the same cardinal number.
Cantor’s treatment of infinite sets leads to unexpected conclusions, such as: there are as
many odd numbers as natural numbers; the number of even numbers is equal to the number
of rational numbers; and the number of points in a line segment is greater than the number
of natural numbers. These properties of infinite numbers are so startling that not a few of
Cantor’s contemporary mathematicians and philosophers were reluctant to accept the new
doctrine. Even Cantor himself admitted that certain conclusions deriving from it appeared
to be counter-intuitive.
2. STUDENTS’ CONCEPTIONS OF INFINITY
The Cantorian set theory is the most commonly used theory of infinity today. Yet, recent
psycho-didactical studies have shown that students face great difficulties in acquiring
various properties of cardinal infinity that give the impression of being impossible or even
self-contradictory (Fischbein, Tirosh, & Hess, 1979; Fischbein, Tirosh & Melamed, 1981;
Duval, 1983; Borasi, 1984; Borasi, 1985; Tirosh, 1985; Martin &Wheeler, 1987; Wheeler
& Martin, 1988; Tall, in press). It has been found that:
1. There are profound contradictions between the concept of actual infinity and our
intellectual schemes, which are naturally adapted to finite objects and finite
events. Consequently, some of the properties of cardinal infinity, such as the fact
that ℵ0 +1=ℵ0 and 2ℵ0=ℵ0 are very difficult for many of us to swallow
(Fischbein, Tirosh, & Hess, 1979; Fischbein, Tirosh & Melamed, 1981; Tall,
1980c, 1981, in press; Duval, 1983).
2. Intuitions of actual infinity are very resistant to the effects of age and of schoolbased instruction (Fischbein, Tirosh & Hess, 1979; Martin & Wheeler, 1987;
Wheeler & Martin, 1988). This means that what we consider as self-evident
concerning the magnitude of infinite sets remains largely unchanged from the
age of 12 on, and these intuitions are unaffected by regular mathematical
training which strengthens the logical schemes which are genuinely finitist.
3. Intuitions of actual infinity are very sensitive to the conceptual and figural
context of the problem posed (Fischbein, Tirosh & Hess, 1979; Martin &
Wheeler, 1987).
4. Students possess different ideas of infinity which largely influence their ability
to cope with problems that deal with actual infinity ( Sierpin`ska , 1987,1989).
These ideas are usually based on the notion of potential infinity (Fischbein,
Tirosh & Hess, 1979).
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DINA TIROSH
5. The experiences that children encounter with actual infinity rarely relate to the
notion of transfinite cardinal numbers. But they do have increasing experiences
in school of quantities which grow large or small (Tall, 1980c, 1981).
Based on this observation, Tall formulated another notion of actual infinity (1981), infinite
measuring numbers, which generalize the notion of measuring from real numbers to a
larger number system. This corresponds in formal mathematics to an extension of the field
of real numbers, such as the hyperreal number system of non-standard analysis. In this
notion of infinity a line segment twice as long as another line segment contains twice as
many infinitesimally small points as the other, and a line contains more points than a line
segment. Tall argues that experiences of infinity that children encounter are more related
to the notion of infinite measuring number and are closer to the modem theory of nonstandard analysis than to cardinal number theory. For instance, in Tall, 1980d, he asked
students to compute various limits, including the limits of
as n tends to infinity. A student who wrote
was shown that a similar argument would give
but she replied firmly “no it wouldn’t, because in this case the denominator is a bigger
infinity, and the result would be zero”. In his case Tall claims that her intuition is based
more on extending experiences of comparative size that on potential infinity, and therefore
is more akin to measuring infinity.
Fischbein et al (1979) cite an example where
is stated to be
1
s=2–—
∞ , “because there is no end to the sum of segments”.
Here the potential infinity of the limiting process leads to a limit concept where the student
divides by an infinitely large number to get an infinitely small one. This too is a closer fit
with non-standard analysis than with cardinal numbers where infinities cannot be divided.
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
203
Following such cases Tall (to appear) suggests:
Most experiences with limits relate to things getting large, or small, or close to one another. All of
these extrapolate experience from arithmetic rather than comparisons between sets and are more
likely to evoke measuring infinity, rather that cardinal infinity. It follows that the ideas of limits and
infinity , which are often considered together, relate to two different and conflicting paradigms.
These findings clearly indicate that our primary intuitions are not adapted to the notion of
cardinal infinity. Thus, it would seem to require a considerable effort to develop appropriate
“secondary intuitions” (i.e., intuitions which are acquired through educational intervention) of the notion of cardinal infinity. Such secondary intuitions are in conflict with some
of our deeply held convictions, such as that the whole can not be equivalent to any of its
parts and that there is only one level of infinity.
In the next section we consider the intuitive criteria adopted by students when
determining whether two infinite sets have the same number of elements. In the section
which follows we describe a research study in which students are assisted in constructing
anadequateintuitivebackground for Cantorian set theory to lay the foundations for a formal
knowledge of cardinal infinity.
2.1 STUDENTS’ INTUITIVE CRITERIA
FOR COMPARING INFINITE QUANTITIES
Only a few studies have investigated students’ intuitions concerning the comparison of
infinite quantities (Duval, 1983; Fischbein, Tirosh & Hess, 1979; Martin & Wheeler, 1987;
Sierpin`ska, 1989). In one of these studies, which is described fully in Tirosh (1985), 1381
students in the age range 11-17 years (grades 6-11) were given 32 mathematical problems
that called for a comparison of infinite quantities. In each of these problems two infinite sets,
with which the students were relatively familiar, were given. The students were asked to
determine whether the two sets were equivalent and to justify their answers. A sample of
the problems and the distribution of the students’ answers appears in Table VII.
In line with the results of the above mentioned studies, it was found that students’
responses to the problems included in Table 1, as well as to other problems which are not
Same Different No
The Sets compared
A vs B
Cardinal Cardinal Answer
85*
15
0
The positive even numbers The positive odd numbers
All the points in a line
The natural numbers
80
19*
1
All points in a line segment The natural numbers
56
42*
2
The natural numbers
The positive even numbers
48*
51
1
The natural numbers
The rational numbers
46*
51
3
All points in a line segment All points in a line
40*
59
1
All points in a line segment All points in a square
39*
59
2
* Correct answers
Table VII: Solutions to Problems dealing with Equivalent Sets (%)
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DINA TIROSH
presented here, were relatively stable across the age groups. Moreover, students of various
ages used the same intuitive criteria to compare two infinite sets.
The main argument used by the students to justify their claim that two sets have the same
number of elements was: “All infinite sets have the same number of elements”. For
instance, 80% of the students claimed that there is an infinite number of natural numbers
and also an infinite number of points in a line, and therefore there are as many natural
numbers as points in a line.
The claim that two infinite sets were not equivalent was justified by one of three
arguments:
(a) “A proper subset of a given set contains fewer elements than the set itself.” For
example, 51% of the students claimed that there are fewer positive even
numbers than natural numbers, since the former is a proper subset of the latter;
(b) “A bounded set contains fewer elements than an unbounded set.” For instance,
12% of the students used this argument to justify their claim that the number of
the points in a square is greater than that in a line segment;
(c) “A linear set contains more elements than a two dimensional set.” This argument
was used by 38% of the students to justify the claim that there are more points
in a square than in a line segment.
The following observations were made with respect to the students’ responses.
(a) A very small percentage of the students (less than 1 %) intuitively employed the
notion of 1-1 correspondence, which is the rigorous criterion used in the
Cantorian theory, to compare the cardinality of infinite quantities.
(b) The students tended to think that all infinite sets have the same number of
elements. This belief stems from their intuitive understanding of infinity as
identical to inexhaustibility.
(c) Many children and adolescents incorrectly assumed that all methods suitable for
comparing finite sets are adequate for infinite sets as well. Thus, most of the
students who argued that two infinite sets were not equivalent based their claims
on the assumption that the maxim “the whole is greater than each of its parts,”
which is adequate for comparing finite sets, holds for infinite sets as well. This
maxim was well-rooted in the students’ minds and they expressed a high degree
of confidence in it.
(d) The intuitive criteria that the students used to compare infinite quantities were
inconsistent with each other and led to conflicting responses and to contradictions of which most of them were unaware. In fact, all but 16% of the students
treated each problem separately, and were greatly influenced by the figural
context of the problem itself. They justified some of their answers by arguing
that all infinite sets had the same number of elements, while justifying other
answers with the claim that one infinite set had fewer elements than the other.
Only about 8% of them mentioned that they realized the inconsistencies in their
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
205
responses. One of the students wrote: “All these problems dealt with comparison of infinite sets. When answering some of the problems, I instinctively felt
that all infinite sets have the same number of elements, because they all have an
infinite number of elements. However, when answering other problems, such
as the problem which dealt with a square and a plane, I felt that there are more
elements in the plane, since the plane contains the square. But it seems
impossible that one infinite set is greater than another infinite set. I realized, after
answering these questions, that there are inconsistencies in my answers, and I
would like to know if all infinite sets have the same number of elements”.
The 16% of the students who were consistent in their answers concerning the
comparison of infinite quantities justified their answers to each of the mathematical problems by claiming that all infinite sets have the same number of
elements.
(e) There are conflicts between the intuitive criteria that the students used to
compare infinite quantities and the formal definitions and theorems of set
theory, i.e., between the intuitively accepted statement, “the cardinality of a
proper subset is smaller than that of the entire set”, and the formal statement,
“every infinite set has a proper subset with the same cardinality”.
The contradictory and persistent nature of the students’ intuitive beliefs in regard to the
comparison of infinite sets, as well as the conflicts between these beliefs and the theorems
of the Cantorian set theory, is a real challenge for those attempting to teach this theory. In
fact, there is evidence, both in the science and mathematics education literature, that
contradictory intuitions may be a main obstacle to acquiring formal knowledge (Fischbein
& Gazit, 1984; Stavy, Eisen & Yakobi, 1987). Moreover, inadequate intuitive beliefs often
continue to affect student’s choices of solutions to problems even after formal instruction
of the relevant theories (Clement, 1983; McCloskey, 1983). Therefore, instruction of the
Cantorian theory of transfinite numbers must take into account the intuitive biases of the
learners. It should attempt not only to help learners acquire the definitions and theorems of
set theory, but also to assist them in developing efficient secondary intuitions about actual
infinity.
3. FIRST STEPS TOWARDS IMPROVING STUDENTS’ INTUITIVE
UNDERSTANDING OF ACTUAL INFINITY
In what follows we shall fist present a learning unit of the Cantorian set theory, which is
called: “Finite and Infinite Sets”. We shall then describe a study which was aimed at
assessing the effects of the unit on students’ formal and intuitive understanding of infinity.
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3.1 THE “FINITE AND INFINITE SETS” LEARNING UNIT
This unit consists of 20 lessons and is subdivided into four sections:
1. Basic Notions of Set Theory,
2. Equivalence of Infinite Sets,
3. Enumerable Sets,
4. Non-enumerable, Linear Sets.
A special attempt was made, throughout the unit, to interact with the students’ intuitive
background in regard to infinity and to change their altitude towards their primary intuitive
reactions. The following strategies were used to help the students overcome the inner
contradictions in their intuitive understanding of actual infinity.
3.2 RAISING STUDENTS’ AWARENESS OF THE INCONSISTENCIES
IN THEIR OWN THINKING
In our opinion, in order to forestall the use of intuitive methods that are inadequate for
comparing infinite quantities in terms of the Cantorian theory, students should realize that
these intuitive criteria lead to contradictory answers. Only after they recognize these
contradictions, can we proceed to raising their awareness of the need to use rigorous criteria
for comparing infinite sets.
Several methods were employed to raise the students’ awareness of the inconsistencies
in their own thinking about infinity. The most prominent one is the conflict teaching
approach based upon Piaget’s notion of cognitive conflict Paiget, 1975). It is aimed at
involving students in discussion of and reflection on the inconsistencies in their thinking.
Awareness of inconsistencies is expected to lead to a state of inner disequilibrium which
can be used to help students resolve the apparent conflicts in their thinking, create new
modified concepts and lead to a new equilibrium.
The following example of an activity which makes use of the cognitive teaching
approach comes from the section on “Equivalence of Infini te Sets”. In this activity the class
is divided into teams of four. Each student is asked to answer the following question:
Two sets are given:
M = {4, 8, 12,16, 20, ...}, N = {2, 4, 6, 8, 10, ...}.
Is the number of elements in set M equal to the number of elements in set N ?
Explain your answer.
Each team is then instructed to discuss its answers and to come to mutual agreement about
the correct response. It is likely that in each team some students may claim that “both sets
have the same number of elements because they are both infinite”, whereas others may
argue that “the set M is smaller because it is a proper subset of the set N”. Consequently,
the participating students will note that both answers seem reasonable and that they are
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
207
unable to decide which of them is correct.
Then the class is encouraged to discuss the consequences of relying only on intuitive
criteria. It is concluded that the 1-1 correspondence may be used to compare the number
of elements in both finite and infinite sets.
3.3 DISCUSSING THE ORIGINS OF STUDENTS’ INTUITIONS
ABOUT INFINITY
It is widely accepted that students’ understanding of the sources of their intuitive beliefs is
essential to enable them to develop their ability to monitor and control the effects of primary
intuitions on their thinking processes. Therefore, the learning unit includes explanations
about the sources of relevant intuitive beliefs. For instance, the unit explains that our mental
schemes, built as they are on our real life experiences, are naturally adapted to finite sets.
We tend to apply these schemes to infinite sets and to accept intuitively generalizations such
as: “A proper subset of an infinite set contains fewer elements than the entire set”. The
tendency to relate properties of finite sets to infinite sets is one of the main sources of the
inadequacy of intuitive beliefs with respect to the Cantorian set theory.
Students are also asked questions such as: “What is infinity?” or “How would you
explain the idea of infinity to a friend of yours?” Such questions are aimed at eliciting
spontaneous responses reflecting the idea that infinity is identical to inexhaustibility, or, the
intuitive interpretation of infinity as pure potentiality. It is then explained that this
interpretation of infinity is the source of the intuitive belief that two infinite sets are always
equivalent, which is inadequate in respect to the notion of transfinite numbers.
3.4 PROGRESSING FROM FINITE TO INFINITE SETS
It is largely acknowledged that infinity can be viewed as an extrapolation of our finite
experiences (Tall, 1981; Rucker, 1982; Dauben, 1983). In particular, infinite sets may be
seen as an extrapolation of finite sets. Therefore, throughout the unit an effort is made to
refer first to finite sets, with which the students are already familiar, and then to deal with
infinite sets, discussing the similarities and differences between them.
For instance, the students are asked to solve several problems that deal with a
comparison of the number of elements in finite sets. They are instructed to use various
methods, such as counting, 1-1 correspondence and the part-whole principle, in order to
solve these problems. This activity evokes a number of questions such as:
• What is the basis of each of these methods?
• Can the same methods be used to compare the number of elements in two given
infinite sets?
• Is there one general criterion that would enable a comparison of any two sets,
finite or infinite?
• What might have led Cantor to choose 1-1 correspondence as the criterion for
comparing infinite quantities?
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DINA TIROSH
• What are the similarities and differences between the comparison of finite sets
and that of infinite sets?
A discussion of these issues may lead to an examination of the adequacy of each of the
intuitive methods used by the students to compare infinite sets with reference to the
cantorian set theory.
3.5 STRESSING THAT IT IS LEGITIMATE TO WONDER
ABOUT INFINITY
Some of the comments of mathematicians on the puzzling aspects of infinity are quoted in
the unit in order to give the students the feeling that it is legitimate to find these aspects
perplexing. For instance, we quoted Hahn’s comments about the theorem:
An infinite set is equivalent to at least one of its proper subsets. If we look for examples of
enumerable infinite sets we arrive immediately at highly surprising results. The set of all the positive
even numbers is an enumerable infnite set and has the same cardinal number as the set of all the
natural numbers, though we would be inclined to think that there are fewer even numbers than
natural numbers.
Later, he suggests his own intuitive explanation of this discrepancy by drawing an analogy
between the intuitive definition of cardinal numbers and the discovery of a new species of
animals:
This species must be different in some way from the known ones, otherwise it would not be a new
species.
(Hahn, 1956, p. 1604)
Comments of other mathematicians, such as Uilbrt (1964), Russell (1956), Frankel (l953)
and Cantor himself, which are included in the unit, illustrate that these mathematicians were
aware that a major breakthrough was needed in the concepts of number, comparison and
infinity in order to make the transition from finite to infinite numbers.
3.6 EMPHASIZING THE RELATIVITY OF MATHEMATICS
Yet another strategy is to describe several alternative concepts of infinte numbers
developed by different mathematicians such as Bolzano (1950), Cantor (1955), Robinson
(1966) and Tall (1980c). Emphasis is placed on analyzing the possible reasons that led each
of them to devise his own way of perceiving infinity. Several extensions of the concept of
natural numbers are introduced. It is hoped that this exposure will help the students gain a
clearer realization of the relativity of mathematics.
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209
3.7 STRENGTHENING STUDENTS’ CONFIDENCE
IN THE NEW DEFINITIONS
Students are provided with opportunities to perform mental activities developed with an
eye to strengthening their confidence in the new definitions and theorems they have just
learned. For example, when dealing with the equivalence between the set of the natural
numbers and the set of positive even numbers, students are asked to write down 10 pairs
of elements with each containing one element from the set of natural numbers and one from
the set of positive even numbers. In this way an attempt is made to offset their tendency to
treat each of the sets separately. Rather, they are directed to look at both sets simultaneously
by referring to the corresponding pairs. The next step is to guide them to discovering for
themselves the formula, f(x) =2x, which describes a 1-1 correspondence between these sets.
In the discussion that follows, emphasis is put on the need to use formal methods to
determine whether two infinite sets are equivalent, rather than relying on intuitions alone.
4. CHANGES IN STUDENTS’ UNDERSTANDING OF ACTUAL INFINITY
In order to examine the impact of the “Finite and Infinite Sets” learning unit on high-school
students’ understanding of actual infinity, the unit was taught to students aged 15-16 in four
tenth-grade classes. Although our main aim was to assess its effects on the students’
intuitive understanding of actual infinity, such an evaluation would be meaningless without
an assessment of the extent to which the students also acquired the definitions and the
theorems which they were taught. Thus, two questionnaires were employed to examine the
effects of the instruction on both the students’ formal and their intuitive understanding of
infinity.
Questionnaire A was designed to check the extent to which students were able to use the
concepts and procedures they were taught. It contained 10 mathematical problems which
dealt with the definition of equivalent sets, equivalent and non-equivalent sets, and the
equivalence between a set and one of its proper subsets. It was administered to the students
twice, the first time immediately after instruction and again two months later.
Questionnaire B was designed to assess the effects of instruction on students’ intuitions
of infinity. It too was administered to the students twice, the first time before instruction and
again two months after instruction. This questionnaire contained 14 mathematical problems. In each, two infinite sets were given. The students were asked to determine whether
the two sets were equivalent and to justify their claims. At least one of the sets in each
problem was two dimensional (i.e., the set ofpoints in a square or the set of points in a plane).
Two dimensional sets were not introduced in class during instruction and thus all these
problems presented situations that were not dealt with in the learning unit. A brief
description of these 14 problems appears in Tables 2 and 3.
The data obtained from Questionnaire A show that 86% of the students acquired the
basic concepts, definitions, theorems and strategies necessary to establish the cardinality
of and equivalence between infinite sets. These students gave correct solutions to problems
that dealt with equivalent and non-equivalent sets, used only formal strategies to justify
their claims, and agreed with theorems that contradict maxims which they had regarded as
self-evident prior to instruction. The substantial gains made by the students during the
instruction phase were maintained over the two-month period between the administration
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of the two post-tests.
Only 14% of the students used primary intuitive arguments to justify their responses to
at least one of the problems in Questionnaire A. Most of these students claimed that an
infinite set can not be equivalent to any of its proper subsets and used the part-whole
principle to justify this inadequate claim. The intuitive claim that all infinite sets have the
same number of elements was rarely used. It is noteworthy that some students reported that
they felt uncomfortable with their answers although they knew they were correct. For
instance, 8% mentioned that: “It is odd that the set of natural numbers is equivalent to the
set of square numbers, which is its proper subset”. Similarly, 14% claimed that: “Although
the line segment [0,2] is longer than the line segment [0,1], these two sets are equivalent”.
Several comments about the extent to which students acquired the concepts they were
taught seem appropriate here.
1. After instruction, about 10% of the students incorrectly argued that “a finite set
might be equivalent to one of its proper subsets”. These students apparently
over-generalized the theorem “an infinite set is equivalent to at least one of its
proper subsets”. A possible source of this misapprehension is our natural
tendency to generalize theorems in order to use them in a variety of situations.
This difficulty could probably be overcome by a greater emphasis of the
differences between finite and infinite sets with respect to the part-whole
principle. The idea that properties that hold true for infinite sets may not
necessary hold true for finite sets should be discussed.
2. Students’ performance on problems involving equivalent sets was better than
their performance with non-equivalent sets; the percentage of adequate responses (correct claims and full justification) to problems dealing with equivalent sets ranged from 88% to 97%, whereas the percentage of correct responses
to problems dealing with non-equivalent sets ranged from 74% to 80%. One
possible explanation for this difference is the fact that the students spent more
time and had more practice with enumerable sets. Another possible explanation
is that claims of equivalence are verified by direct methods of proof while claims
of non-equivalence are verified by the indirect method of proof, which is less
familiar to and more problematic for high school students (Roberti, 1987). It is
reasonable to assume that explanations about the indirect method of proof, as
well as more practice with non-enumerable sets, could improve students’
performance on these problems.
In general, our findings indicate that the learning unit on infinite sets may be introduced
without particular difficulty starting from the tenth grade. Although some problems were
identified, we assume that an improved version of the learning unit would be able to
overcome them.
The situation in regard to the effects of instruction on students’ intuitions of infinity is
far more complex. Tables VIII and IX present students’ responses to Questionnaire B.
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
211
* Correct answers
Table VIII : Solutions to Problems Dealing with Equivalent Sets (%)
* Correct answers
Table IX : Solutions to Problems Dealing with Non-equivalent Sets (%)
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DINA TIROSH
The above data indicate that the percentage of adequate responses increased after
instruction. The lowest percentage of adequate responses after instruction, were yielded by
problems that dealt with the comparison of a linear set and a two dimensional unbounded
set (problems 3, 5, 11 and 13). All students who gave inadequate answers to these problems
argued that the cardinality of a two dimensional unbounded sets (i.e., the set of all points
in a plane or the set of all rational points in a plane) was higher than c, the cardinal number
of the set of reals.
The changes in students’ responses to problems dealing with a comparison of infinite
quantities are also reflected in the criteria they used to determine their solutions. After
instruction, the majority of the students made an attempt to implement the formal strategies
and theorems that they had learned. For example, on problem 9, 82% of the students
claimed that: “We have learned that the cardinality of natural numbers is ℵ0 . The
cardinality of the set of points in a plane is at least c, since it is at least the same as the
cardinality of the set of points in a line. Thus the cardinality of the set of points in a plane
is greater than that of the set of natural numbers”.
Another strategy used by the students to justify their solutions was that of indicating an
analogy between the given problem and one they had discussed in class. For example, in
the case of problem 7, 19% of the students claimed that “the problem of comparing the
number of points in a square with the number of points in a larger square is similar to that
of comparing the number of points in a line segment with the number of points in a larger
line segment. The set of points in a line segment is equivalent to the set of points in a larger
line segment; thus, the set of points in a square is equivalent to the set of points in a larger
square.” Only 13% of the students used their former intuitive arguments to justify their
solutions to at one or more of the problems in Questionnaire B.
The students’ responses to the various problems showed that as a result of instruction
the vast majority of them realized that the intuitive criteria they had used when comparing
infinite quantities were inadequate in respect to the Cantorian set theory. The students
became aware of the need for formal mathematical proof asopposed to intuitive evaluation.
Some of their responses clearly illustrate that they became critical about their natural
intuitive attitudes. For instance, in solving problem 7, one of the students wrote: “According
to my natural reasoning the two sets are equivalent. However, I realize that sometimes my
reasoning is misleading. Therefore, I cannot depend only on my natural reasoning and I will
try to prove, formally, that the two sets are equivalent.” Another student commented, with
reference to the same problem, that: “This case is similar to that of the line segment [0,1]
and the line segment [0,2]. Therefore, it seems that these sets are equivalent. However, I
am not sure since unexpected things happen with infinite sets.”
Can one conclude, based on these findings, that as a result of instruction the students
developed modified intuitions towards the comparison of infinite sets? Our data do not
allow us to draw such a conclusion with confidence. They do not provide evidence to
support the claim that ideas that were originally regarded as preposterous by the students,
such as the possibility of equivalence between an infinite set and a proper subset of it, now
became self-explanatory concept for them. The changes in students’ reactions to nonstandard situations may indeed be due to modified intuitions, but it is also possible that these
changes are a result of the newly acquired formal knowledge of the Cantorian set theory
together with an increased awareness of the need to control intuitively based reactions. The
fact that some students reported that they felt uncomfortable with their answers although
INTUITIONS OF INFINITY AND THE CANTORIAN THEORY
213
they knew they were correct may be viewed as supporting the claim that as a result of
instruction, the students realized that when comparing infinite sets they had to consciously
monitor their reactions and not base them on their intuitions, but that their intuitions per se
were not modified.
Other data which may be viewed as providing support for this claim derive from the
students’ responses tomathematical problems involving two dimensional, unbounded sets.
As mentioned above, a substantial number of them claimed that a two dimensional,
unbounded set contains more elements than a linear set. This may indicate that when
dealing with mathematical problems with which they were unfamiliar, the students were
still, implicitly, influenced by their primary intuitions. However, it is equally reasonable to
argue that students were reluctant to accept the possibility of equivalence between a
bounded linear set and a two dimensional, unbounded set because during instruction they
had no experience of equivalent sets that differ in dimension. If so, the learning unit may
have effectively changed the students’ intuitions concerning the equivalence of bounded
and unbounded sets, but the change in this specific intuition may not necessarily lead to a
change in their intuitions in respect to the equivalence of sets that differ in dimension. It is
certainly feasible that eradicating one inadequate intuition does not necessarily eliminate
another.
Thus, it seems safe to conclude that the instruction influenced the explicit decisions of
the vast majority of the students concerning the comparison of infinite sets. They learned
to give conceptually controlled answers rather than spontaneous intuitive ones. A substantial portion of the students developed an “alarm technique” for problems dealing with a
comparison of infinite sets. This technique was an upshot of their realization that it is risky
to support a mathematical statement by intuitive evaluations alone. These students
mentioned that the intuitive criteria they had previously used to compare infinite quantities
had led them to contradictions and inadequate solutions. It became apparent to them that
their solutions to problems dealing with infinitesets should be based on the formal theorems
and strategies that they were taught.
This study shows that the comparison of infinite sets, which illustrates some of the more
perplexing aspects of mathematics, can be used to enable students
(a) to accept conclusions which at first appear paradoxical,
(b) to recognize the coercive nature of intuitive thinking,
(c) to understand the need to control their primary intuitions,
(d) to refrain from responding intuitively,
and
(e) to use explicit theorems in order to determine their solutions to the problems.
It is hoped that such instruction would raise students’ awareness of the role of intuition in
their thinking and affect their attitude towards intuitive responses not only in respect to
infinity but also with reference to other mathematical and scientific processes.
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DINA TIROSH
5. FINAL COMMENTS
1. Developing a learning unit that takes into account the intuitive backgroundof the learners
requires a profound knowledge of the nature of students’ intuitions towards the specific
mathematical theory. The identification of relevant, inadequate intuitions held by the
students is extremely important. This study has shown that in the case of comparing infinite
sets, many of the students’ primary intuitions were similar to those experienced by
mathematicians in the history of the development of the concept. Such palpable parallelism
between phylogeny (historical development of the species) and ontogeny (development of
the individual), reveals the former as a potential source for identifying students’ intuitions.
2. We have already seen that some of the students justified their responses to certain
problems by pointing out the analogy between the given problem and one discussed in class.
Analogies were not discussed during instruction, yet students used them spontaneously and
correctly. Recent studies have shown that analogical reasoning can be useful for helping
students achieve conceptual changes (Strauss & Perlmutter, 1986; Clement, 1987; Stavy,
in press). The potential of analogies to help students recognize the dissonance in their
thinking about infinity and the consequences of including them in instruction examples
which illustrate possible correct and incorrect uses of analogies should be explored.
3. In a paper on “Mathematics and the Metaphysicians”, Bertrand Russell argues that:
On the subject of infinity it is impossible to avoid conclusions which at first sight appear paradoxical,
and this is the reason why so many philosophers have supposed that there were inherent
contradictions in the infinite. But a little practice enables one to grasp the true principles of Cantor’s
doctrine, and to acquire new and better instincts as to the true and the false. The oddities then become
no odder than the people at the antipodes, who used to be thought impossible because they would
(Russell, 1956, p. 1578).
find it so inconvenient to stand on their heads.
Russell here distinguished between gaining a conceptual knowledge of infinity and
acquiring new intuitions. He also claimed that a little practice would enable the learners to
acquire both. Our study has shown that the process of acquiring new instincts is not quite
that simple. Further, our data indicate that the road from a conceptual grasp of the principles
of set theory to acquisition of better intuitions in respect to the comparison of infinite sets
may be a long one.
It seems that the various strategies that were used in the learning unit “Finite and Infinite
Sets” did indeed enable the students to progress towards acquiring intuitions which are
consistent with the theory they learned. However, we lack the means to evaluate these
effects systematically. In order to proceed in devising instructional strategies that take into
account the intuitive background of the learners we need to develop means to measure
“degrees of intuitiveness”. A preliminary attempt to measure the intuitive acceptance of a
mathematical statement is described in Fischbein, Tirosh & Melamed (1981). Yet, if we
believe that the intuitive attitudes of the learner have a crucial effect on his or her concepts
and capacity to understand mathematical theories, and if we further believe that intuitions
can be modified, we need to devote much greater efforts to devising means that will enable
us to base our assessments of the effectiveness of various methods of instruction in
modifying students’ intuitions on systematic evaluation.
CHAPTER 13
RESEARCH ON MATHEMATICAL PROOF
DANIEL ALIBERT & MICHAEL THOMAS
1. INTRODUCTION
The formulation of conjectures and the development of proofs are two fundamental aspects
of a professional mathematician’s work. They have a dual character. Firstly there is the
personal, intimate side, which aims at clarifying the position the researcher has reached in
his/her own understanding, through the statement of explicit hypotheses. Secondly there
is the collective side, where a conjecture is proposed for the reflection of other mathematicians, sharing ideas, as yet unsure. In this context a proof is a means of convincing oneself
whilst trying to convince others.
These two facets of advanced mathematical thinking are generally absent in undergraduate mathematics at university, where the subject matter is presented as a finished
theory, where “all is calm ... and certain” and proofs are developed along traditional ‘linear’,
deductive lines.
The epistemology (the understanding of the structure of knowledge) generated by such
teaching practices is thus diametrically opposed to the reality of the mathematical
community.
A study of textbooks for students at this level appears to confirm that the semantic
characteristics of the mathematics – the control of meaning – is not a primary aim. Instead
emphasis is placed on the syntactic aspects in carrying out and using the results of
algorithms.
This apparent conflict between the practice of mathematicians on the one hand, and their
teaching methods on the other, creates problems for students. They exhibit a lack of concern
for meaning, a lack of appreciation of proof as a functional tool and an inadequate
epistemology.
It may well be that the students’ view of whether proof is a necessary mathematical
activity, their understanding of the need for rigour, and their preference for one type of proof
over another, are concerns which have been neglected by some mathematics educators in
favour of a perceived need to preserve the precision and the beauty of mathematics. A
consideration of the students’ view may be especially important during the transition phase
when they are first exposed to the rigour of formal proof as it often occurs in a first year
university mathematics course.
Researchers in this area of mathematics education have demonstrated that there is an
important difference between communicating sufficient understanding of a proof to
convince students of a result, and a formal, rigorous proof that it is true. Balacheff (1982,
1988) has described various levels on which proof may exist, and the importance of
distinguishing between convincing arguments and rigorous proof. The latter may well be
a suitable instrument to be used in the kind of formal text that mathematicians write in books
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DANIEL ALIBERT & MICHAEL THOMAS
or research articles, but may not be suitable when initially passing on acquired knowledge
to students. One difficulty associated with achieving a proof which is both meaningful and
formally acceptable to students is:
How do we include the main ideas through which we understand why the result
is true at the same time as the necessary details to make it rigorous?
We shall discuss this and other aspects relating to how understanding of proofs may be
better communicated to students. We shall also pay particular attention to studies
emphasizing the nature of proof as an activity with a social character, a way of communicating the truth of a mathematical statement to other people, helping them to understand
why it is true.
A major area of difficulty linked with this social character of proof which we shall
consider here is:
How can we manage to make students see proof as a necessary step in the
scientific process, alongside activities such as research, the formulation of
conjectures etc. and not just as a formal necessity required by the teacher, or as
an answer given by the teacher in response to a question which the student may
not have asked?
These two problems, respectively, have been the subject of research by Leron in the
Department of Science Education of Haifa University, Israel and Alibert, Grenier, Legrand
and Richard in the Research Group in the Didactics of Mathematics at the University of
Grenoble, France. Leron (1983a, 1985a) has proposed a method of structuring proofs to
improve the way students understand them, while Alibert et al (1986, 1987, 1988abc,
1991), Grenier et al (1984, 1985) and Legrand & Richard (1984) have designed a new
teaching method involving scientific debates in order to encourage students to see the
necessity for proof as a mathematical activity.
2. STUDENTS’ UNDERSTANDING OF PROOFS
First we shall turn our attention to the students’ perspective of proof in a mathematics
course. What are the characteristics of the proofs which they prefer and claim to understand
better, and how good is their understanding?
Several researchers, including Fischbein (1982), Movshovitz-Hadar (1988) and Tall
(1979) have investigated aspects of the teaching of proofs which may be appropriate for
presenting material in a potentially meaningful manner for the learner. There are proofs,
for example where the inner parts of the proof are not trivial, where structured proofs and
linear (or formal deductive) proofs display similar pedagogical problems.
Tall (1979) is concerned with the students’ first acquaintance with proof at university
and investigates which of several types of proof they find more understandable. Following
Steiner (1976), he suggests the concept of a generic proof as a potential way round such
problems. Such a proof works at the example level but is generic in that the examples chosen
are typical of the whole class of examples and hence the proof is generalizable. This may
RESEARCH ON MATHEMATICAL PROOF
217
be contrasted with general proof which works at a more general level but consequently
requires a higher level of abstraction. Whilst there may be no replacement for the formal
proof from the purely logical point of view, the generic proof may sometimes be preferable
if it results in improved understanding on the part of the students.
Discussing the proof of the irrationality of 2,Tall describes a study in which 33 first
year university students were presented with three proofs of the result: one general, one
generic and the ‘standard’ proof by contradiction. In a second questionnaire, 37 students
responded to a generic and a contradiction proof for the irrationality of
.
The generic proof as used here was :
We will show that if we start with any rational p/q and square it, then the result p 2/q 2 cannot be
5/8.
On squaring any integer n, the number of times that any prime factor appears in the factorization of
n is doubled in the prime factorization of n 2, so each prime factor occurs an even number of times
in n2. (For instance, if n = 12 = 22x3, then 122=24x32.)
In the fractionp 2/q2, factorize p2 and q2 into primes and cancel common factors wherepossible. Each
factor will either cancel exactly or we are left with an even number of appearances of that factor in
the numerator or denominator of the fraction. The fraction p2/q2 can never be simplified to 5/8 for
the latter is 5/23, which has an odd number of 5’s in the numerator (and an odd number of 2’s in the
denominator).
The results showed that the generic proof for the irrationality of
was significantly
preferred to the proof by contradiction, both in terms of understanding and lack of
confusion. Furthermore there was a highly significant preference for both the generic proof
and the proof by contradiction over the general proof of the irrationality of 2.
Dreyfus & Eisenberg (1986) gave five proofs of the irrationality of 2 to mathematicians
who were asked to rank them according to elegance. It is of interest that the experts’
personal preferences were for the proofs which were older and more elementary, including
a proof along the lines of that described above for
Dreyfus and Eisenberg conclude
that clarity and simplicity of argument arc two principal factors which should guide one
when trying to nurture mathematical appreciation. The use of generic examples in proofs
may be a way to promote such arguments.
Movshovitz-Hadar (1988) also recommends a “generic-example assisted” type of
proof. Applying this method of proof to the theorem:
For any nxn matrix, n a positive integer, such that the rows form arithmetic progressions with the
same common difference d, then the sum of any n elements, no two of which are in the same row
or column, is invariant.
She uses an 8x8 matrix as an example:
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DANIEL ALBERT & MICHAEL THOMAS
small enough to serve as a concrete example, yet large enough to be considered a non-specific
representative of the general case. The proof for the 8x8 case ... is kind of “transparent”, one can see
the general proof through it because nothing specific to the 8x8 case enters the proof.
(Movshovitz-Hadar, 1988 p. 19)
It would seem that the explanatory power of such a proof may supplant the generality of
general proofs for the student, resulting in more meaningful understanding. Mathematical
insight in proof may be more important than precision in these circumstances.
Research by Vinner (1988) shows that when taking this generic approach to proofs, one
should be aware that students may be resistant to accepting the proof due to cognitive
obstacles arising from their pre-disposition to linear formalism. He gave students two
proofs of the mean value theorem which states:
If a function f is differentiable between a and b, and continuous at a and b, then there is a point
between a and b such that f( ) =
The first proof was the standard algebraic proof, applying Rolle’s Theorem to
The second was a visual proof involving moving the chord AB as shown below, parallel to
itself until it becomes a tangent.
Figure 31 : The mean value theorem
Of 74 students, 29 found the visual proof more convincing, 28 the algebraic proof and
17 considered them of equal value. Those preferring the algebraic method tended to remark
that there was something wrong or ‘illegal’ in the visual approach, and Vinner considered
that students develop an algebraic bias through environmental effects to do with ‘habit’ and
RESEARCH ON MATHEMATICAL PROOF
219
‘convenience’ rather than cognitive necessity. This feeling of needing a formal deductive
proof may emanate from a lack of confidence in any other approach rather than an affinity
for the aesthetics of algebra.
Research by Fischbein has uncovered another aspect to the understanding of proof by
students. He has found that they may understand the theorem statement itself, they may
even, through the use of a structured proof, or otherwise, grasp the structure of the proof,
and yet still they may faiI to appreciate the universal validity of the statement as guaranteed
and imposed by the validity of the proof. This conclusion was reached following a research
project in which about four hundred high school pupils with advanced training in
mathematics were presented with a correct proof of the theorem:
n3– n is divisible by 6 for every integer n.
The students were then given various questions about the validity of the theorem. Whilst
81 % checked the proof and claimed it to be correct in every detail, 68.5% agreed with the
theorem and 60% considered the generality of the theorem guaranteed by the proof, only
41% of the students accepted all three of these. Further only 24.5% accepted the correctness
of the proof and at the same time answered that additional checks are not necessary, and
only 14.5% were completely consistent in their answers.
To a mathematician the proof of a theorem is
the absolute guarantee of the universal validity of the theorem. He believes in that validity.
(Fischbein, 1988, p. 17)
The question is how does one convey, in a proof or otherwise, the information necessary
for the individual student to synthesize cognitively the formal understanding of the truth
of the result and an acceptability of its universal validity? This pedagogical necessity in
mathematical proof exposition may be one which still needs to be addressed.
3. THE STRUCTURAL METHOD OF PROOF EXPOSITION
Mathematical proofs have long tended to have a format which requires them to be read in
a strictly serial/sequential manner, with sub-proofs, or lemmas, which are themselves also
strongly sequential. Such a style of proof makes the acquiring of a global over-view
something which requires sufficient mathematical sophistication to understand the details
of the sequence well enough to be able to relate them to the overall theme as one progresses
through the proof. Such an ability to switch as and when necessary from a sequential view
of the mathematics to a global one and vice-versa is a characteristic of one who has been
described as a versatile learner (Tall & Thomas, 1989). In order to promote versatility in
students, Tall & Thomas (1990) have highlighted the importance and value of actively
encouraging a global view of the mathematics, indeed promoting it in one’s teaching in
addition to the more familiar serialist presentations of mathematics. Using the benefits
provided by the computer paradigm they have obtained some evidence that a versatile
learner, who is able to switch between a global and a sequential view of the mathematics,
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DANIEL ALIBERT & MICHAEL THOMAS
is more likely to be successful in the early learning of algebra as generalized arithmetic
(with 12–14 year old pupils) and in the initial stages of the calculus, and have placed this
improved ability in a theoretical context applicable to other areas of mathematics.
Leron (1983a, 1985a) has attempted to fuse formal and informal methods of presentation into a proof which is rigorous and yet explanatory. The two informal practices
(heuristics) he has built into the formalism are:
• the prefacing of a long, complex proof with a short, intuitive overview,
• a method of constructing a mathematical object, a solution-object, to satisfy a
system of constraints by using the given constraints to search for the solutionobject and then using its form to define it.
This kind of proof he calls a structural proof.
Here the primary aim is not merely to convince, but to help the listener or reader gain
a real understanding of the ideas behind the proof and its connections with other
mathematical results. In comparison, the usual ‘linear code’ type of proof not only often
fails to elucidate the main ideas, but may even obscure them. This type of proof may well
be suited to ensuring the validity of a proof, but it is unsuitable for the role of mathematical
communication. It has even happened, in some extreme cases, that the author recognizes
that his/her own proof fails to give any real insight into the understanding of the
mathematics. For instance Deligne, one of the most famous contemporary mathematicians
(and winner of a Fields Medal), wrote after a very formal proof about derived functors and
categories,
“I would be grateful if anyone who has understood this demonstration would explain it to me”.
(Deligne, page 584)
Clearly proving and explaining seem to be two different kinds of mathematical activities.
The linear formalism of traditional proof may be described as the minimal code
necessary for the transmitting of the mathematical knowledge. It appears, however, that in
several important respects, it is a sub-minimal code, resulting in an irretrievable loss of
information vital for understanding.
Whilst most of the work in mathematics education rightly seeks to improve the learning
and communication of mathematics by supplementing the formalism, it is also important
to look at the formalism itself and consider how it too might be improved, leading to better
communication and understanding. It is certainly to be hoped that students of mathematics
are actively engaged in discovering and constructing as much of the mathematics they learn
as possible, but it is also necessary to find better ways of communicating the products of
such mathematical activity to others.
The fundamental concept underlying the structural method of presenting proof is to
arrange the proof in levels, proceeding from the top down. Each level consists of short
autonomous modules, each embodying one main idea of the proof. This type of structure
is already recognized and well-known in computer science as a method of structuring
complex computer programs, where it is called top-down programming.
RESEARCH ON MATHEMATICAL PROOF
221
It should be noted that the term ‘main idea’ which we here refer to is used more in the
abstract sense of that idea which enables one to gain an overview of a sub-section of the
proof rather than indicating that which is mathematically among the most important ideas
in the proof.
It is useful to analyze some examples of this type of structural proof (see below) in order
to ascertain the difference in the treatment given by the two methods and to see specifically
what is meant by the ‘main ideas’ of a proof. A main idea is often contained by the
construction of a new, intermediate object called the pivot (so named because the rest of the
proof hinges upon it), designed to mediate between the hypotheses and the conclusion. The
pivot occupies a central position in the proof (or sub-proof), and so it offers a vantage point
from which one may view the global architecture of the proof. In the linear approach, the
pivot is often poorly treated and its potential for improving understanding wasted. Rather
the proof begins to resemble the pulling of a rabbit from the hat, since the pivotal concept
may be introduced near the beginning of the proof, possibly by simply a bare statement of
its definition.
In the first level of the proof one tool-pivot is identified (e.g. set, relation, function etc ...),
the existence of which is essential for the proof to be developed. Since this is to be a tool,
it is then given some properties which are also to be used in the proof, although the actual
existence of the tool-pivot is not proved at this stage of the proof, but at a later, deeper one.
In the second level of the proof the tool-pivot becomes an object-pivot which is to be
constructed, subject to certain constraints. An heuristic discussion follows concerning the
possibility of achieving the construction under the given constraints. This construction
itself may, if necessary, be further divided up and treated on several levels. Proceeding in
this way avoids the view that the pivot is just a construction which in the eyes of the student
is ‘an extremely clever answer to a question which was never asked’. These concepts are
probably best understood by considering some examples of structural proofs and their
‘linear’ equivalents. We shall consider here two of the examples given by Leron (1983a).
Theorem : There exist infinitely many triadic primes
(i.e. numbers of the form 4k + 3, for integral k)
3.1 A PROOF IN LINEAR STYLE
Consider the product of two monadic numbers :
(4k + 1)(4m + 1) = 4k.4m + 4k+ 4m + 1 = 4(4km + k +m) + 1
which is again monadic. Similarly, the product of any number of monadic primes is monadic.
Now assume the theorem is false, so there are only finitely many triadic primes, say p1, p2, . . . , pn.
Define
M = 4 p1 p2 p 3 ... pn – 1.
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If pi | M then pi | 1Since pi | 4 p 1 p 2 p 3 . . . pn. Since this is impossible, we conclude that no pi divides
M. Also 2 does not divide M as M is odd. Thus all M’s prime factors are monadic, henceM itself
must be monadic. But
M = 4 p1 p2 p3 ... Pn – 1 = 4( p 1 p 2 p 3 ... pn – 1) +3
is clearly triadic – a contradiction. Thus the theorem is proved.
We note that in the above proof the general plan is never revealed, neither is the purpose
of the various steps taken. In the absence of any explanation as to why certain steps are taken
(such as considering the product of two monadic numbers) the student may be reduced to
merely checking the validity of the deduction at each step.
3.2 A PROOF IN STRUCTURAL STYLE
Level 1 – suppose the theorem is false and let p1 , p 2. p3, ... , pn be all the triadic primes. We construct
(in level 2) a number M [the pivot] having the following two properties :
(a) M as well as all its factors is different from p1 , p2 , p3 , ... , pn ,
(b ) M has a triadic prime factor.
These two properties clearly produce a contradiction, as we get a triadic prime which is not one of
pl, p2, p3 , ... , pn . Thus the theorem is proved.
In the Elevator – (a metaphor for the process of descending in levels)
How shall we approach the definition of M? In the light of Euclid’s classical proof, it is natural to
try M = 4p1 p2p3 . . . pn + 1. This indeed meets requirement (a) but not (b). In fact, since for all we
know M itself may turn out to be prime, it must be triadic to meet (b).
Thus a natural second guess is M =4p1 p2 p3 ... pn + 3.However, this has another “bug”– since one
of the pi’s is 3, M is divisible by 3, in violation of (a). But this bug, once discovered, is easy to fix
– simply eliminate 3 from the product in the bugged definition.
Level2 –LetM=4p2p3 . . . p n +3 (we assumep1= 3). We show thatM satisfies the two requirements
from Level 1.
Requirement (a) means that no p should divide M. Indeed, p 2, p 3, . . . , pn do not divide M as they
leave a remainder of 3; and 3 does not divide M as it does not divide 4 p2 p 3 . . . pn.
As for requirement (b), suppose on the contrary that all of M’s prime factors were monadic. Then
M, as a product of monadic numbers, would itself be monadic (Lemma, Level 3 ) – a contradiction.
Thus (a) and (b) are satisfied.
RESEARCH ON MATHEMATICAL. PROOF
223
Level 3 – Lemma. A product of monadic numbers is again a monadic number. (The proof is given
above.)
We note here that the top level 1 gives aglobal view of the proof. The ‘elevator’ affords the
opportunity for informal discussion within the proof, including part of the process of
finding a proof. The necessary lemma is only introduced when it is needed, avoiding its
introduction at the start of the proof with no mention as to its later use.
As a second example, Leron considers the proof of the theorem:
A standard text-book proof starts with an ε>0 and shows by a number of intermediate steps
how a δ>0 can be found so that, when l|x–a|<δ we have If(x)g(x)-LM |<ε. At level 1, Leron
starts with ε>0 and assumes that the pivot, δ>0, can be found at level 2, thus proving the
theorem subject to the level 2 construction. He then sets out to find such a δ, by looking at
what he is trying to achieve, using the inequality
to break this down to level 3 problem seeking to bound the terms in the second line.
After a process of trial and improvement to achieve this, he is able to take the ‘elevator’
back through the levels to complete the proof.
The format for structural proofs suggested by Leron and seen in these examples is:
1. Introduce the pivot as a system of constraints, i.e. define it implicitly by postulating its
properties.
2. Without actually solving the system, use the pivot as introduced in step 1 to derive the
conclusion of the theorem.
3. Discuss heuristically the solution of the system to find how the pivot might be
constructed.
4. (Recursionstep). Solve the system, repeating steps 1–4 if necessary. That is construct (or
prove the existence of) the pivot then prove that it satisfies the postulated properties. If
some of the sub-proofs are themselves complicated, introduce sub-pivots and repeat the
four step procedure.
(Leron, 1985a, p. 12)
Though such proofs are clearly longer than standard linear type proofs, the claim for them
is that loss of economy is more than balanced by a gain in understanding and learning, and
that it would be better to leave out some of the low-level details from these structural proofs,
if brevity is required, than to exclude the high-level ideas and connections left out by linear,
deductive expositions.
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It may be argued that the knowledge of the structure of the proof, as illustrated above, is best
left for the students themselves to discover, however experience shows that this kind of task
is beyond the capability of most undergraduates with a standard mathematical training.
They are simply unable to decode the proof and are reduced to meaningless manipulation
of the formal code itself, with no awareness of the ideas and concepts it represents. One of
the goals of the structural method is to train students to structure linear proofs. Using the
structural method of proof also leads to the possibility of several new structure related
activities which encourage the learner to reflect on the process of theorem proving itself.
For example they may
• complete the lower levels of a proof, given the higher levels.
• take a standard proof and examine its structure (not an easy exercise but one
which may result in deeper understanding of the proof).
• examine the depth of similarity of two theorems which exhibit some similarity
in terms of the levels of the theorems.
The major difference between the approach outlined above and the traditional linear proof
style is that the students are given a means of understanding the choices that, generally, the
teacher presents without any indication that there had actually been a choice involved.
Previously some questions about the choices made may have arisen in the minds of some
alert students (although no answers would generally be provided during the presentation
of the proof) but for many of them understanding a proof is synonymous with merely
checking in a sequential manner the validity of the deduction at each step, much as a
computer might execute a program. Unable to construct any personal meaning for proofs
like these, even simple ones, many of them must feel either cheated or stupid, and certainly
they are not in a good position to further develop their scientific capability. In contrast, the
understanding gained by the students from a structural type of proof may lead to real
scientific autonomy on their part.
4. CONJECTURES AND PROOFS – THE SCIENTIFIC DEBATE
IN A MATHEMATICAL COURSE
A further step in the direction of scientific autonomy is taken by the Grenoble school
(Alibert et al, 1986), attempting to enable students to see proof as a necessary part of the
scientific process of advancing knowledge, rather than just a formal exercise to be done for
the teacher. An experimental teaching method, set in a theoretical framework, was devised
and applied to the teaching of mathematics in the first year of university. The theoretical
framework was based on the following general cognitive and didactic hypotheses:
1. Constructivism - the theory of knowledge acquisition in which students construct their
own knowledge through interactions, conflicts and re-equilibrations involving mathematical knowledge, other students and problems. The interactions are managed by the
teacher who makes the fundamental choices.
(Brousseau, 1986)
2. Knowledge is made firmer when it has been constituted and applied in more than one
RESEARCH ON MATHEMATICAL PROOF
appropriate conceptual setting.
225
(Douady, 1986)
3. The role of contradictions – how they may be made sharper and more explicit and how
they may be resolved.
(Balacheff, 1982)
4. The importance of the role of a group of students in the construction of individual
meaning (Bishop, 1985; Balacheff & Laborde, 1985)
5. The influence of meta-mathematical factors such as the systems of representations and
how these may be worked on explicitly in order to emphasize teaching points.
6. The construction of a learner’s epistemology which includes the set of problems,
situations, etc, which in the student’s mind give meaning to the concept through their
association with the introduction and progressive constitution of the concept.
The experiment took place at the University of Grenoble with sections of about one hundred
students (95 of the first year and 130 of the second), all of whom were in the first year of
a programme called DEUG A (First year university students all taking courses in
mathematics, physics and chemistry). The lessons were given to the whole of a section in
an ordinary lecture theatre and the experiment lasted the complete year with the group.
4.1 GENERATING SCIENTIFIC DEBATE
The classroom teaching was built around several new customs. Uncertainty in the learning
place is important and room should be left for it. In mathematical knowledge this
uncertainty is institutionalized in the notion of conjecture, and in this study the validation,
and even the production of these, was devolved to the community of students. They were
required to produce and validate conjectures relevant to their mathematics curriculum.
Underpinning this custom was the principle that the functional nature of proof only arises
in situations where students meet the uncertainty of mathematical propositions.
This worked in the classroom as follows:
• First step: the teacher initiates and organizes the production of scientific
statements by the students. These are written on the blackboard without any
immediate evaluation of their validity.
• Second step: the statements are put to the students for consideration and
discussion. They come to a decision about their validity by taking a vote, with
each opinion supported in some way, e.g. by scientific argument, by proof, by
refutation, by counter-example, etc.
• Third Step: the statements which can be validated by a full demonstration
become theorems, whilst those which are established as incorrect are preserved
as “false-statements”, with a corresponding counter-example.
The demonstrations are produced through interaction between the students and, when
necessary, the teacher, after the students have been confronted with the particular problem
during a debate. In this form of ‘scientific debate’ the arguments forming the proof are not
addressed by the students to the teacher, but rather to the other students. We have to
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distinguish here between ‘proofs to convince’ someone (such as another student) of
something that is not already a part of his institutionalized knowledge and ‘proofs to show’,
where the aim is to show someone (such as the teacher) that one has reached some
knowledge that he/she already possesses. One of the main hypotheses of research is that the
activity involved in the first process is fundamentally different from that involved in the
second, and is able to produce a deepening of knowledge and its meaning. The theoretical
description of such a teaching system, used since 1984, is based on this hypothesis and is
referred to as a ‘codidactic situation’ (Alibert, 1991). This isa situation in which the student
seeks to convince both himself/ herself, and others at the same time, of the truth of a
conjecture formulated in response to a problem which the whole group is trying to solve.
The students are all aware that the conjecture is not necessarily true and in particular that
it is not yet an established part of institutionalized knowledge.
4.2 AN EXAMPLE OF A SCIENTIFIC DEBATE
A scientific debate of the type described above starts with a statement such as in the
following actual example :
If I is an interval on the reals, a is a fixed element of I, and x an elementof I, then we set, for f integrable
over I,
The teacher then asks the question :
“Can you make some conjectures of the form : if f ... then F ... ?”
In response to this, in the example given by Alibert (1988a), about 20 statements of varying
complexity were produced by the students. One of the sessions started with an examination
of the first of these.
“If f is increasing then F is increasing too” (which happens to be false).
The variation of F is one of the items from the curriculum that the students had to learn.
During this session the following steps of the proof construction were observed:
(a) Counter-example : one student produced an example of a function f contradicting the statement. The class then concluded that the statement was false.
(b) Statement Modification : another student proposed that “If f is monotonic then F
is monotonic too.” (This, of course, is also false.)
(c) Counter-example : the same function as used before under (a), but defined on
a different interval, contradicted the modified statement too.
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227
(d) Observation : by considering the counter-example it seemed that if f ≥ 0 then F
is increasing.
(e) A New Conjecture : A student now proposed that “If f ≥ 0 then F is increasing.”
(The majority of the students thought that this statement was false though it is
of course true.)
(f) An Argument : the student produced the explanation:
F(x') – F(x )=
f(t ) dt ≥ 0 if f ≥ 0 and x' > x .
Interestingly, many of the students did not believe that this was always true and this stage
of the debate revealed to the teacher that some of the results and definitions, previously
discussed and settled, had been misunderstood by many of the students. In particular they
had not fully understood the convention that
f(t) d t is the Riemann integral on the segment [x, x' ] if x'>x,
and minus the integral if x'<x.
Analysing this phenomenon we may say that this was a reappearance of old, stable
knowledge about the integral learned in previous years.
(g) Validation : the students reached a validation of the argument of (f) with:
f(t ) dt ≥ (x’– x)inf(x), in this case.
This debate took up most of a two-hour class.
The above example illustrates that the propositions debated during the sessions were far
from trivial. They alsoallow students to tackle real problems involving important concepts.
Even though many of the statements considered are false they are still very important
because, firstly, they discover what students really think about a concept at that precise point
in the course and, secondly, the debate about their proposals enables students to be
convinced of any false ideas or deep misunderstandings of the concepts which they may
hold.
This experience teaches them that proof is really a tool which may be used to improve
ideas and separate false intuition, however natural it may appear, from true mathematical
statements; to communicate and hence validate or refute mathematical ideas.
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4.3 THE ORGANIZATION OF PROOF DEBATES
The organization of the type of debate outlined above involves the use of some precise
techniques, such as:
Initialization time–during the first lesson of the year the teacher outlines the way
that the course will proceed, partly through explanation and partly through
illustrating the process using a debate on a simple mathematics problem. The
teacher initiates this by asking a question and a debate about the validity of the
answers proceeds, without specific rules.
Reinforcement time – There should be two or three of such lessons at the start
of the course, with rules for the debates progressively introduced. Some are
simple ones – for example speak loudly; speak to one another; listen to the other
students – whilst others are more subtle.
The teacher’s position–The teacher has a precise role to play right from the first
lesson. If he/she faces aquestion which should be considered by all the students
then he/she should ask for a conjecture to be produced. The conjectures
produced in answer to the problem set should be written on the blackboard
without comment. After allowing two or three minutes for reflection the teacher
asks for a vote. The students are asked to vote true, false, can’t decide or refuses
to decide on each conjecture. Then each opinion has to be supported by
mathematical arguments.
In this way the students as a group learn that the formulation of conjectures is
a useful and necessary activity and that to make mistakes is a normal stage in the
learning process.
The rules – It has been observed that, at this stage in the course, the teacher is
frequently called on to close a debate by expressing his/her opinion on the
question in hand because the students have been unable to agree. This inability
to convince one another needs addressing and a special lesson called the circuit
is introduced. The aim of this lesson is to give the students the means to refute
statements. Students are told that: “In mathematics a statement is true if and only
if it has no counter-example.” This lesson uses some very simple situation to
produce conjectures, to refute them and so gradually build up the rules used in
mathematics. It is very important here that the mathematical context does not
hide the logical problems.
After this session the counter-example should become a very powerful tool for students to
use to refute a statement or, more generally, to understand a particular proof. For instance:
To express in mathematical form that a function F does not have an ‘infinite
positive limit’ as the variable becomes infinite is not a simple exercise for
students at this level. They do know, however, how to express the idea that F has
an ‘infinite limit’ as x becomes infinite (positive), namely :
For every real A there exists B such that if x>B then F(x)>A.
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Some have also learned that to negate such a statement one ‘replaces for every by there
exists and vice-versa’, but this becomes merely a meaningless manipulation of formal code,
as we have discussed above. The understanding of this manipulation becomes clear if it is
presented as the formulation of the existence of a counter-example for the previous
statement.
4.4 EVALUATING THE ROLE OF DEBATE
As a result of this teaching method, concerning the role of proof in mathematics, and the
formation of scientific truth, it has been observed that erroneous ideas are no longer
considered by the students as faults but as a normal scientific event, and a productive one
at that. The students in the study described above were given a questionnaire about the
comparison of teaching methods. Many of the students replied that they preferred the
course incorporating debates and emphasized that they allowed them to understand the
problem which the new mathematical knowledge was aimed at solving, and also what
errors may be made too. Some of the comments about the scientific debates from the
questionnaires were :
“It compels us to reflect more on the question. One often listens to a clear lecture without reflecting
deeply.”
“A concept introduced through some conjecture makes the problem that the concept poses much
clearer than in a lecture.”
“It allows us to have several views, to eliminate some intuitive ideas that are wrong.”
Many students also stated how difficult they found the study of a conjecture to be, involving
as it does the stating of the problem, the construction of the proof and the formulation of
ideas when one is uncertain about the truth of a proposition.
“For me the hardest thing is to find counter-examples when I think that a conjecture is false.”
“I often have difficulty forming an opinion, and following it up with a proof.”
Certainly the students felt involved and interested and, without excluding the value of
‘traditional’ lectures, found the debates very useful.
5. CONCLUSION
In this chapter we have looked briefly at some of the research into mathematical proof and
its presentation. We have considered different methods of presenting proofs in order to
improve students’ understanding, including generic proofs and structural proofs. We have
also looked at the environment within which these proofs are examined by students with
particular regard to the scientific debate as an alternative to traditional presentations.
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Amongst the conclusions which some researchers are reaching with regard to students’
perceptions of the importance of proof and their understanding of individual proofs are:
(a) There should be a distinct and important difference between the kind of proofs
produced by a mathematician researching new areas of mathematics in order to
convince others that they have indeed broken new ground and the proofs of these
results which will later be used to transmit the results to students of higher
mathematics. The latter proofs may need to have extra material included which
gives a global view of the proof and its structure if they are to be meaningful to
the average student and not just a linear sequence of symbolic reasoning whose
step-by-step validity is to be checked.
(b) The context in which students meet proofs in mathematics may greatly
influence their perception of the value of proof. By establishing an environment
in which students may see and experience first-hand what is necessary for them
to convince others, of the truth or falsehood of propositions, proof becomes an
instrument of personal value which they will be happier to use in future.
Whatever the cognitive benefits of these approaches to students’ intellectual appreciation
of specific proofs, and of proof as a mathematical tool, the problem remains of transmitting
the methodologies to other teachers so that they may investigate their effectiveness for
themselves. The reader might thus like to consider his/her role in this, and review
(i) students’ reactions to the proofs presented. Do they display both appreciation
for them and understanding of them?
(ii) how proofs which (s)he currently teaches could be restructured to present a
potentially more meaningful face to students.
(iii) how a topic/course which (s)he currently teaches could be developed along the
lines of a proof debate in the light of the methodology described above, to
promote greater appreciation of the necessity of proof in mathematics.
Such a self-appraisal of one’s current practice could be very valuable both from the point
of view of one’s own appreciation of the results and the understanding of one’s students.
Those readers not so closely involved in these aspects of mathematics education (and
others) may like to assess their own understanding in depth of a proof which they are
familiar by re-writing the proof in a structural form. Alternatively (and rather more testing)
it might be more interesting to attempt to produce structural proofs of one or more of the
following results from the usual formats given in standard texts:
1. Cantor-Bernstein Theorem – Let A and B be two sets, If there exists one-to-one
maps from A into B and from B into A, then there exists a one-to-one map from
A onto B.
2. A connected manifold is path-connected.
3. Sylow’s Third Theorem – The number of Sylow p-subgroups of a finite group
G is equal to 1 modulo p.
CHAPTER 14
ADVANCED MATHEMATICAL THINKING
AND THE COMPUTER
ED DUBINSKY AND DAVID TALL
1. INTRODUCTION
The computer can be used as a tool to complement advanced mathematical thinking in a
variety of ways. In research it has been used to provide data to suggest possible theorems,
to seek counter examples and to carry out onerous computations to prove theorems
involving only a finite number of algorithmic cases. In education it can be used for the same
objectives, and for one other major purpose: to help students conceptualize, and construct
for themselves, mathematics that has already been formulated by others.
There are already many computer tools available for general use. Symbolic manipulators have been used in research, but with less initial success in education. We hypothesize
that success using the computer in education is enhanced by using the computer for explicit
conceptual purposes and report empirical research which supports this hypothesis. New
software environments are being developed which enable the student to explore concepts
in a directed and meaningful way, and which suggest new approaches to mathematics more
appropriate for the learner.
Programming can be used to support both mathematical research and mathematics
teaching. But when it is simply added to the curriculum without very specific aims in mind
it has not always been successful. We will discuss the way in which a computer language,
designed so that the programming constructs mirror mathematical constructs, can assist
students to carry out mathematical processes and encapsulate them as mathematical
concepts.
2. THE COMPUTER IN MATHEMATICAL RESEARCH
Mathematical research passes through several distinct stages of development, from the
germ of an idea to the formalities of proof:
In Mathematics, as in the Natural Sciences, there are several stages involved in a discovery, and
formal proof is only the last. The earliest stage consists in the identification of significant facts, their
arrangement into meaningful patterns and the plausible extraction of some law or formula. Next is
the process of testing this proposed formula against new experimental facts, and only then does one
consider the question of proof.
(Atiyah, 1984)
Computers have proved useful in every stage of this development. In the initial exploration
phase computer generated data has led to surprising new intuitions and new theory. The
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famous example is that of Lorenz, studying the outcome of differential equations to predict
the weather, who wished to repeat a cycle of events to analyse it in greater detail. Instead
of starting from the beginning of a run, he took numbers occurring part way through a
previous run and found, to his amazement, that the subsequent pattern diverged enormously
from his previous data. He then realized that the output of the previous run had given
numbers only to three places: 0.506 instead of the internally stored number 0.506127. The
small variation in initial conditions had given a large variation in long term behaviour –
knowing initial conditions in a practical sense cannot be used to predict the eventual
outcome and chaos theory was born (Lorenz, 1963).
Since that time, sensibly programmed environments have proved increasingly valuable
to produce data to suggest possible conjectures. Recent developments in the theory of
iteration of functions, leading to the beautiful fractal pictures that have become well known
even to the general public, arose from research begun, but abandoned, in the earlier part of
this century because of the massive computations involved. It was only with the arrival of
the computer that the results of the computations could be represented graphically, leading
to surprising pictures and new hypotheses to be tested first by drawing, then by a search for
formal proof. Likewise, in the theory of dynamical systems, computer graphics have
exhibited phenomena that might not have otherwise come to light. Software for the
investigation of such phenomena is now generally available. For instance, figure 32 shows
a model of a possible orbit of a tiny satellite round two larger bodies, alternately oscillating
between revolving round one then moving into a position of superior gravitational pull of
the other and moving, for a time, to revolve round the other (Koçak, 1986). It is interesting
to note that this book features a significant number of research problems for which there
is a clear visual idea of possible solutions but for which no formal proof was available at
the time of publication. The theory of dynamical systems and chaos is a paradigmatic
example of a new branch of mathematics in which the complementary roles of computer
generated experiments to suggest theorems and formal mathematical proof to establish
them with logical precision go hand in hand.
Chaos has become not just a theory but also a method, not just a canon of beliefs but also a way of
doing science. Chaos has created its own technique of using computers, a technique that does not
require the vast speed of Crays and Cybers but instead favours modest terminals that allow flexible
interaction. To chaos researchers, mathematics has become an experimental science, with the
computer replacing laboratories full of test tubes and microscopes. Graphic images are the key. “It’s
masochism for a mathematician to do without pictures,” one chaos specialist would say. “How can
they see the relationship between that motion and this, how can the develop intuition?”.
(Gleick, 1987, pp. 38–39)
In the second stage of mathematical thinking, where conjectures have been made more
precise and serious attempts are being made to test them, computers may be used somethes
to generate appropriate examples or counter-examples. Nearly two centuries ago, after a
prodigious number of calculations, Euler formulated the conjecture that a sum of at least
n positive nth powers of integers are required to produce an nth power. So forbidding were
the calculations required to investigate this that it stood without proof or refutation until a
computer search in 1969 by Lander and Parkin produced the counter-example:
275 + 845 + 1105 + 1335 = 1445 .
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
233
Figure 32 : Chaotic movement of a satellite round two larger bodies
This case was fortunate, in that the discovery of a counter-example showed the conjecture
to be false. On the other side of the coin, the inability to find such a counter-example will
not show a conjecture to be true. Goldbach’s conjecture, that any even number greater than
two is a sum of two primes, remains unproven, even though computers have found an
appropriate decompositions into two primes for all even numbers up to a formidable size.
In 1916 Bieberbach conjectured that an analytic function
z +a2z2 + ... +an zn+ ...
which was 1–1 on the unit disc satisfied
|an| ≤ n.
Bieberbach proved the case n=2, but by the early 1980s, only the cases up to and including
n=6 had been proved, by a variety of different methods. Louis de Branges worked for seven
years and in 1984 developed a technique which proved the Bieberbach conjecture subject
to a condition that could be checked algorithmically. A colleague, Walter Gautschi, ran the
method on the Purdue university super-computer – one of only three in the United States
at the time – and verified the method as far as the 25th coefficient. The computer proved
a vital confirmation at a difficult time for de Branges who had previously twice published
erroneous proofs of theorems and found his latest and most complex deductions considered
suspect by the mathematical community. His proof was subsequently vindicated when the
final steps were confirmed by other means (Kolata, 1984).
In the final stage of mathematical thinking, when a formal proof is being sought, the
computer may prove decisive when the question can be reduced to a finite number of cases,
each which can be investigated algorithmically. The most famous example is the four
colour problem, which Appel & Haken (1976) reduced to a finite (but large) number of
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alternatives which were resolved by computer. Now the computer is being widely used in
combinatorial problems in group theory, algebraic geometry, and other areas with an
algorithmic content that can be progmmmed, leaving the computer to carry out the complex
calculations.
The proof of the four colour theorem raises a significant issue in advanced mathematical
thinking. For, although there is an apparently impeccable logic in the listing of the
possibilities and their checking by computer, the proof itself seems to shed no light as to
why the theorem is true. Some mathematicians are happy with the situation. For them the
process of proof is a mechanistic sequence of deductions from axioms and it is important
that, in the actual proof process itself, there are no intuitive leaps that are not subject to
logical scrutiny. The logic of the computer is for them an acid test.
However, others involved with mathematical research sense the need not only for the
security of logical deduction from a proof, but also some kind of insight as to how the
concepts fit with other known results. Without such insight there is always for them the
insecurity that some small logical error may be found which renders the argument
fallacious. Without some overall view of the pattern there may be a distinct lack of vision
as to the possible direction of future research. And, given the ever growing complexity of
computer software, there may be errors in the programming which, if the principles are not
fully understood, may lead to precisely the weak links that those requiring only a logical
approach may fear.
Thus there is value in using the computer to complement the human creative thinking
process both in providing environments for exploration into possible new theorems and
also to carry out algorithmic calculations to provide mathematical proof, but it is necessary
to acknowledge that such methods have weaknesses as well as strengths.
3. THE COMPUTER IN MATHEMATICS EDUCATION – GENERALITIES
All the various ways that computers are used in research are potentially available for
teaching and learning advanced mathematics. For example, students may learn to program
in order to tackle certain types of problem, or they may use general purpose software as an
environment to explore ideas. The main difference between the activities of undergraduate
students and mathematical research is that the former usually covers knowledge domains
which are known to the more experienced members of the mathematical community,
whereas research is attempting to break new ground. Of course, to the student the
mathematics is new, and here there may be strong analogies with research, but the far
greater portion of a student’s work is concerned with mathematics that is already part of an
organized knowledge system. This opens up a further possibility for the use of the computer
in mathematical education, through the development of computer software designed to
help the student conceptualize mathematical ideas.
Recent research into concept development shows consistently the complexity of an
individual’s mental imagery: students can give the “right” answers for the wrong reasons,
whilst “wrong” answers may have a rational origin. In particular, many researchers have
realized that student errors are often the product of misconceptions brought about using old
knowledge in anew context where it no longer holds good. This leads to the hypothesis that
learning may be improved by helping students construct knowledge in their own minds in
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
235
a context which is designed to aid, or even stimulate, that construction. One way of doing
this is through providing richly endowed computer software which embodies powerful
mathematical ideas so that the student can manipulate and reflect on them. Another is to
have the student program mathematical constructions in a computer language designed so
that the act of programming parallels the construction of the underlying mathematical
processes.
A computer can also give much-needed meaning to mathematical concepts that students
may feel are “not of the physical world but in the mind, or in some ideal world. It is
generally agreed that ideas are easier to understand when they are made more “concrete”
and less “abstract”. When an abstract idea is implemented or represented in a computer,
then it is concrete in the mind, at least in the sense that it exists (electro-magnetically, if not
physically). Not only can the computer construct be used to perform processes represented
by the abstract idea, but it can itself be manipulated, things can be done to it. This tends to
make it more concrete, especially for the person who constructed it. Indeed, it is in general
true that whenever a person constructs something on a computer, a corresponding
construction is made in the person’s mind. It is possible to orchestrate this correspondence
by providing programming tasks in an appropriate programming language designed so that
the resulting mental constructions are powerful ideas that enhance the student’s mathematical knowledge and understanding. Moreover, once the various constructions exist on the
computer, it is very useful to reflect on what they are (in terms of how the computer makes
them) and what processes they can engage in.
4. SYMBOLIC MANIPULATORS
The use of symbolic manipulators has powerful advocacy from several quarters. Lane et
al (1986) suggests ways in which symbolic systems can be used to discover mathematical
principles and Small et al (1986) reports the effect of using a computer algebra system in
college mathematics. In the latter case the activities often consist of encouraging students
to apply a technique already understood in simple cases to more complicated cases where
the symbolic manipulator can cope with the difficult symbolic manipulations.
However, in the initial stages of use of symbolic manipulaton in education, Hodgson
observed:
In spite of the fact that symbolic manipulation systems are now widely available, they seem to have
had little effect on the actual teaching of mathematics in the classroom. (Hodgson, 1987, p. 59)
He quoted a report of Char et al (1986) on the experiences of using the symbolic system
Maple in an undergraduate course in which students were given free access to the symbolic
manipulator to experiment on their own or to do voluntary symbolic problems which they
could elect to count for credit. He noted a “somewhat limited acceptance of Maple by the
students”:
While many explanations can be put forward for such a reaction (little free time, no immediate
payoff, weaknesses of the symbolic calculator for certain types of problems, absence of numerical
or graphical interface, lack of user-friendliness), it is clear that the crux of the problem concerns the
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full integration of the symbolic system to the course in such a way thatitdoesnot remainjust an extra
activity. This calls for arevision of the curriculum, identifying which topics should be emphasized,
de-emphasized or even eliminated, and for the development of appropriate instruction materials.
(ibid.)
Subsequent developments have seen Maple extended to include both numerical and
graphical facilities and improved radically in user-friendliness. Yet there is an underlying
reason why there may be a major problem with symbolic manipulators in mathematical
education which is more than a question of interface, available facilities, and the need for
integration in the curriculum. A symbol manipulator is a tool – a very powerful tool – but
any tool can only be used to its fullest capabilities by those who know how to use it. The
situation is parallel to the use of simple calculators: they do not teach a child how to add
(or divide), but they are useful tools for adding or dividing when one knows what arithmetic
is all about. Once one knows how to cope with small numbers, perhaps the calculator can
be used to investigate facts with much larger numbers. Likewise, symbolic manipulators
are likely to prove more useful – as they have proved useful in mathematical research – once
the student has progressed to the stage of knowing what the tool is being used for.
The later generation of symbolic manipulators, particularly Mathemtica, have made
a step in helping the user come to terms with the nature of the concepts by including wordprocessing facilities as well as symbol manipulation. This allows the development of
teaching material in the form of electronic notebooks, in which symbols present may be
manipulated or edited at will by the user. In this way it is possible to introduce the user to
new concepts in a cybernetic environment which responds to the users needs in manipulating the symbols which appear. It promises to be an exciting development which has been
met with more enthusiasm than the environment which requires the user to type in the
complete command in the idiosyncratic syntax of the particular manipulator. Here words
can tell the user the meaning of a command and the user may just select it and instruct the
computer to carry it out. However, our experience in all the earlier chapters tells us to
beware of the simple solution. It is likely to contain seeds for misconceptions and cognitive
conflict. In order that students can re-construct their knowledge faced with the radical new
concepts of advanced mathematics, they need to gain experience of how the ideas work and
actively reflect on the cognitive changes required to integrate this new knowledge into a
more appropriate mental structure. Two thousand years ago Euclid is reported to have told
Ptolemy that there is no Royal Road to Geometry, given the nature of the human animal,
even in collaboration with the computer, we should not be deluded into believing that the
computer will provide an entirely smooth path to mathematical knowledge.
Having a computer to perform the algorithms, even to show how those algorithms work
is one thing, being able to cope with these concepts meaningfully is another. Some symbolic
manipulators include facilities to allow the user to step through the manipulation, seeing
what is done at each stage. This can be very helpful to the student who is trying to learn how
to reproduce the algorithm, but knowing how to differentiate symbolically is very different
from knowing what the derivative means. Likewise, knowing routines for solving
differential equations symbolically by reversing this symbolic differentiation process is a
very different process from being able to visualize a solution or a family of solutions. What
may help to broaden the student’s understanding is to set the use of the symbolic
manipulator in an appropriate conceptual environment.
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5. CONCEPTUAL DEVELOPMENT USING A COMPUTER
Heid (1985,1988) spent the first twelve weeks of a fifteen week applied calculus course
studying fundamental concepts using graphic and symbol-manipulation software to
perform routine calculations whilst she focussed the students on the underlying concepts.
Only in the last three weeks did they practice any routine algorithms for differentiation and
integration. She found the learning of fundamental concepts was greatly improved in the
experimental class:
Students showed deep and broad understanding of course concepts and performed almost as well
on a final exam of routine skills as a group who had studied the skills for the entire fifteen weeks.
(Heid, 1985, p. 2)
In the classes the experimental students were encouraged to use a large variety of concept
representations and to reason with them, for instance using computer generated graphs and
tables of values to solve real world problems and make conclusions about applications:
One student, for example, located the sales level for maximum profit by finding the x-value for the
greatest vertical difference between the revenue and cost curves. Another formulated consumers’
surplus as the sum of the areas of rectangles without the typical first translation to a Riemann sum
formula. A third gave a new integration formula for the area between curves by conjuring up an
alternative geometric explanation and translating it directly into a statement about integrals.
Reasoning in non-algebraic modes of representation characterized concept development in experi(Heid, 1988, p. 10)
mental classes.
By encouraging the students to think for themselves and to construct their own ways of
handling the concepts, it became apparent that they had integrated the ideas into their own
knowledge structure:
... when the students realized that they had made misstatements about concepts ... on many of these
occasions, on their own initiative, the students in the experimental classes reconstructed facts ... by
returning to basic principles. When [they] spoke about limits, functions, derivatives and Riemann
sums, the wording was often clearly their own.
(ibid, p. 15, 16)
In contrast:
When the students in the comparison class verbalized that they had made erroneous statements about
concepts, there was no evidence of attempts to reason from basic principles. They often alluded to
having been taught the relevant material but being unable to recall what had been said in class.
(ibid, p. 16)
Thus Heid’s research shows clear evidence of the value of giving meaning to the basic
concepts, even before the students have had any extended practice with the algorithmic
techniques.
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6. THE COMPUTER AS AN ENVIRONMENT FOR EXPLORATION
OF FUNDAMENTAL IDEAS
In her research, Heid used existing software for graphs and symbolic manipulation to build
conceptual insights. This software is built on mathematical principles: to draw graphs, to
carry out mathematical processes, and so on. Another possibility is to design software
which uses a combination of mathematical and cognitive principles – building on what
students already know in a way which is consistent with their cognitive development.
Students meeting advanced mathematical concepts such as infinite processes, limits,
continuity and differentiability for the first time are known to have serious cognitive
difficulties (see chapters 10, 11). The mathematics educator, with a knowledge of both the
mathematics and the cognitive development, can play a fundamental role by identifying
powerful ideas in the theory that can be presented in a meaningful way to the students at
their current point in development, yet play a fundamental role throughout the theory. To
illustrate this we return to the cognitive approach to the calculus illustrated in chapter 11
and concentrate on the computer environment which it uses.
Graphic Calculus (Tall, 1986, Tall et al, 1990) was conceived as an example of software
designed to provide students with a cognitive approach to the calculus and differential
equations. Because of students’ known conceptual difficulties in understanding the limit
concept, it was decided to found the approach on the notion of local straightness. Here the
possibility of computer magnification of graphs allows the limiting process to be implicit
in the computer magnification, rather than explicit in the limit concept. Students therefore
begin the calculus by exploring the magnification of graphs of functions of one variable.
They can see that most of the familiar graphs (polynomials, trigonometric, exponential,
logarithmic and their combinations) are all locally straight, but some, such as f(x)=|sinx|
have points where left and right gradients differ. They can be guided to look at graphs such
as f(x)=x sin(1/x) (with f(0)=0) which oscillates so wildly that it never looks straight at the
origin, whilst f(x)=(x+|x|)sin( 1/x) looks straight to the left from the origin, but not to the
right. Other functions are available for exploration, including fractal functions that are so
wrinkled that they never look straight under magnification, giving students mental images
of differentiability and various ways in which non-differentiability may arise. Thus the
local straightness of differentiable functions, and non-straightness of non-differentiable
functions allows the student to gain a fundamental insight into the notion of differentiability
from the very beginning, instead of founding their understanding on simpler ideas
concerned only with polynomials.
Local straightness also links naturally to the ideas of differential equations (building
locally straight curves, knowing their gradient) and to the general study of differentiable
manifolds (locally flat substructures of higher dimensional spaces). The idea is also
enshrined in nonstandard analysis (e.g. Keisler, 1976) where it is proved that under an
infinite magnification (the standard part of) an infinitesimal portion of a graph is precisely
straight.
As discussed in chapter 11, a student with the mental ability to view a small part of a
displayed graph and to see its gradient, can then conceptualize the numerical gradient
f(x+h)–f( x) for variable x and fixed h. By investigating the numerical gradient in simple
h
cases using the computer, it is found that students can conjecture the formula for the
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
239
stabilized numerical gradient, which is the derivative, before they have the ability to derive
the formula algebraically from first principles (Tall, 1986).
Furthermore, the pictorial idea can lead to the notion of a differential. If a graph is locally
straight, then a small portion of the tangent at a given point (x,y) will closely approximate
dy
the curve. Denoting the components of the tangent vector by dx,dy then f ' (x)=
and
dx
visually, one can see that the point (x+dx, y+dy) is on the tangent, closely approximating
the curve when dx is small.
This leads naturally into the notion of a first order differential equation
where the gradient at any point (x,y) is given as F(x,y). The Solution Sketcher (Tall, 1989)
allows the user to specify a first order differential equation, then move a pointer round a
screen window representing the ( x,y) plane, drawing a small line-segment through (x,y)
with gradient F(x,y). By a simple key stroke the line-segment may be left as a permanent
mark, and successive segments may be placed end to end to construct an approximate
solution to the differential equation. Thus the student can gain a physical idea of what the
solution of a first order differential equation actually means.
It is a simple matter to show that a higher order differential equation
can be written as two simultaneous differential equations by the substitution v=dx/dt to get
This too has a (locally straight) solution in (t,x,v) space with tangent direction given by
(dt, dx, dv) = (dt, v dt, F(t,x,v) dt)
which is in the direction (1, v, F(t,x,v)). Thus the simple idea that a solution “follows the
gradient direction” is true not only for first order differential equations, but for higher order
(simultaneous) differential equations in a suitable solution space.
Hubbard & West (1985) developed a computer graphics approach to differential
equations. They found that, without computer graphics, students had difficulty appreciating the notions of existence and uniqueness of solutions. When so much of their work had
involved routine symbolic manipulation to produce an answer, many students found it
difficult to comprehend how a solution could exist if it could not be expressed as a familiar
formula. The computer graphics helped them to see existence as the ability to draw a
solution – a solution that existed visually even thoughthey wereunable to provide a formula
for it. This links closely to the formal theory – the solution exists, and is unique, provided
that the differential equation properly specifies a direction to follow at each point. Solutions
fail to exist where the differential equation fails to specify a unique direction.
The fact that there are symbolic differential equations which lack symbolic solutions shows
the need to incorporate numerical and graphic representations with symbolic methods. A
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ED DUBINSKY AND DAVID TALL
computer is able to process a vast amount of numerical data and to present it in graphical
form. Even where the symbolic methods are available, they may need geometric interpretation. Tall (1986c) quotes the following example from a national examination in the U.K.:
It is easily solved by separating the variables:
and may be integrated to give:
But what does this mean? Regarding (*) as specifying the direction of the tangent vector
(dx,dy), to the solution curve through any point (x,y) enables a “direction field of short line
segments to be drawn in the appropriate directions through an array of points in the plane
(figure 33).
It can be seen that some solutions are closed loops whilst others may be conceived as
functions in the form y=f(x). The symbolic solution is in this case of little value without a
graphical representation of its meaning, whilst the graphical interpretation alone lacks the
precision of the symbolism. It therefore needs the complementary power of visualization
and symbolic manipulation to give a deeper mathematical insight.
Figure 33 : Solutions to a first order differential equation
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
241
7. PROGRAMMING
In recent years moves have been made to introduce student programming into mathematics
courses. Initially this tended to be in the form of enhancing ready existing mathematics
courses by introducing computers and calculators to carry out numerical algorithms and
perhaps represent the results graphically. It has met with mixed success. With younger
children there is considerable evidence that if programming is simply attached to a course
without any thought about conceptual integration, then there is no reason to expect an
improvement in conceptualization of the course content (Menis et al, 1980; Cheshire,
1981). Some research projects have shown that when programming is introduced as an
extra into the traditional curriculum it may reduce the time spent on traditional skills,
causing a lower level of performance in them (Reding, 1981; Robitaille et al, 1977).
However, Thomas & Tall (1988) found that teaching algebraic concepts in a module
including programming at first gave the usual initial losses in traditional skills to balance
gains in conceptual understanding, but after a brief review of skills at a later date, this was
changed into a gain on both skills and concepts.
At university level, Simons (1986) reported on the use of hand-held computers to be
programmed in BASIC to supplement the traditional teaching of calculus in these terms:
... the introduction of a personal computer into acourse of this nature, whilst enhancing teaching and
presentation in many areas, raises profound problems.
(Simons, 1986, p. 552)
There were evident gains in the immediate usefulness of the work, but a substantial number
of staff, long experienced in mathematics teaching yet new to the computer and numerical
analysis, did not like the course. Simons suggests that the aversion displayed by some
members of staff lies in the feeling of uncertainty in applying a numerical method:
The traditional mathematician ... is clearly aware that for every numerical method a function exists
for which the method produces a wrong answer. ... The statement that nothing is believed until it is
proved is the starting point for teaching mathematics and introducing the computer forces the teacher
(ibid, p. 552)
away from this starting point.
A recurring observation is the difficulty experienced by teachers, both at university and in
school, to come to terms with the new technology. We are at present in the throes of a
paradigmatic upheaval and cultural forces operate to preserve what is known and
comfortable, and to resist new ideas until they are proven better beyond doubt.
On the other hand, there is also evidence that when programming is used for conceptual
purposes, such as solving problems where the programming parallels the underlying
mathematical processes, or using computer activities to foster specific mental constructions that can lead to mathematical understanding, then there is a much higher level of
success.
Several universities in the U.K. now include mathematical problem-solving through
programming – usually in structured BASIC – as an element of the undergraduate
mathematics course. The problem-solving often requires program construction to give
numerical or graphical data and experience shows that the students gain considerably from
the task.
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Various programming languages are becoming available which are almost certainly
more appropriate for mathematics than BASIC. Some are specifically designed to make
concepts in mathematics easy to program. For example, Mathematica, as well as providing
a symbolic manipulation system within a word-processing package that will draw graphs,
also gives a complete programming system that allows a powerful blend of functional and
structural programming constructs. Such developments within a multi-purpose computer
environment are likely to prove of increasing use in advanced mathematical thinking in the
future. It should be noted, however, that the principal aim of the programming system of
Mathematica is predominantly for doing mathematics, rather than learning mathematics.
It is therefore better designed for the expert than the novice.
A language specifically designed for mathematics learning is ISETL (Interactive SET
Language). Dubinsky and his colleagues have found that having students make certain
constructions in the ISETL can lead to their making parallel mathematical constructions in
their minds and thereby come to understand various mathematical concepts (Ayers et al,
1987; Dubinsky, 1986, 1990a, 1990b; Dubinsky et al, 1988; Dubinsky et al, 1989;
Dubinsky & Schwingendorf, 1990a, 1990b). The specific use of computers in this work is
driven by the theoretical analysis laid out in Chapter 7 and a brief description of the language
is given in an appendix to this chapter.
These experiences, both positive and negative, tell us that the issue in using programming to help students learn mathematical concepts is not whether it should be done, nor is
it the particular language that is used. The main consideration is how the instructional
treatment uses the language through the design of the programming tasks for the students.
Although the nature of the computer language is not the primary consideration, it is an
important one. The inconvenience of working with Fortran or Pascal syntax introduces
difficulties for students and teachers that have nothing to do with mathematical issues. The
same is true to a lesser extent of LISP, APL and PROLOG. BASIC is easier to use and is
adequate for numerical algorithms and representing numerical data in a graphical form, but
it is inappropriate for arithmetic with large integers, for symbol manipulation and for most
higher-level mathematical thinking. LISP is particularly powerful for symbol manipulation
and LOGO is almost as good (for the purposes of mathematics) with much less syntactic
overhead. APL makes working with vectors and matrices especially easy while PROLOG
is designed for programming systems of complex logical inferences. ISETL supports most
of the standard mathematical constructs with a syntax very close to mathematical notation.
It is the only one of these languages that treats functions as data.
Graphics have often only been added to languages at a later stage. The omission of
graphics from the first version of BASIC led to a proliferation of different dialects. Given
the acclaim for turtle graphics, it is a salutary experience to realize that these were almost
an afterthought in the initial specification of LOGO. ISETL was also originally designed
without graphics which were added later.
To ask which kind of programming language is most beneficial to help students learn
mathematics, one must first ask what it is one is trying to teach and how:
Is mathematics a bag of tricks that may be useful to later life? Is mathematics taught because it is
an important part of our culture, or because it helps young people to teach logically and abstractly?
These are questions for mathematics teachers. In the long run, computer software can be adjusted
to their requirements.
(Grogono, 1989)
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
243
Grogono shows how different kinds of languages may be used to model different kinds of
thinking processes. The question is equally applicable at more advanced levels of
mathematics. If its answer is that one wishes to encourage students to think mathematically
about mathematical concepts, then a computer language is required that supports these
requirements.
8. THE FUTURE
Thus we see the computer already proving a powerful tool in advanced mathematical
thinking, both in mathematical research and in mathematics education at the higher levels.
The empirical evidence shows that it proves more successful in the educational process
when it is used to enhance meaning, either through programming in a language embodying
the mathematical processes or through the use of computer environments for exploration
and construction of concepts.
Computers are likely to prove a profound influence over the next N years, where the
reader may care to estimate the value of N. It is possible, but it may not be meaningful, to
speculate on the changes that new technology will bring. Already the promise of parallel
processing may bring new possibilities, for instance in the simultaneous processing of
several different representations. Intelligent tutoring systems currently seem to promise
more than they deliver, but it is conceivable that new techniques may bring greater success.
Already we have video discs carrying large amounts of information for the user to explore
in new and unforseen ways.
However, it is our belief that mathematics is not a spectator sport, and that advanced
mathematical thinking will continue to blossom through the constructive actions of the
human mind, albeit complemented by the enormous processing power of the computer.
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ED DUBINSKY AND DAVID TALL
APPENDIX TO CHAPTER 14
ISETL : A COMPUTER LANGUAGE
FOR ADVANCED MATHEMATICAL THINKING
ISETL is a computer language which has been designed and used to foster mathematical
thinking at advanced levels. The language and its use will be indicated by giving some
examples of actual code along with indications of how this relates to some specifics of the
constructivist analysis given in Chapter 7. We will use terminology such as process, object,
interiorize, encapsulate, coordinate, and reverse which are explained fully in that chapter.
The interactive set language ISETL is designed to implement many mathematical
constructions in ordinary mathematical language. Sets can be listed in the usual way within
braces {}, either as a list of elements separated by commas, or as a set defined by a property.
Square brackets [ ] denote sequences, and the notation [a..b] for integers a , b denote all the
integers from a to b.
The following line entered into ISETL:
P := {x : x in [2..1000] | not exists y in [2..(x–1)] | x mod y = 0 }
assigns to P the set of numbers x between 2 and I000 which do not have a smaller factor
y – in other words P is the set of primes less than 1000.
In full generality a set in ISETL can be specified as:
{ expr : x,y, ... in S, u,v, ... in T, ... | condition | condition ...}
where expr is an expression, generally involving variables x, y. u, v, etc whose domains
are previously constructed sets S, T, ... and each condition is an expression whose value is
true or false. It is important for the student to think about how the computer might handle
this construction: by iterating the variables through their domains, and for each value to
evaluate the conditions and, if it is true, placing the expression in the set.
The assumption made by those who use this language in education is that by writing such
code the student will interiorize the process of forming this set.
A set is not only a process of formation, it is an object with its own existence; for instance,
it has a cardinality operator, it can be itself a member of a set, etc. One way to check that
someone has an understanding of the process is to ask her or him to calculate the number
of elements in a set such as
{1+2, {1..4}, “cat”, {1,2,3}, {{“house”, “dog”, 3}, 3}}
(in this case it is 5). ISETL does this with a single operation. Thus,
#({1+2, {1..4}, “cat”, {1,2,3}, {{“house”, “dog”, 3}, 3}});
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
245
returns the value 5.
Again there is an assumption that if you write code that applies operators, then you will
tend to think of that to which an operator applies as an object. In this way, it is considered
that students will come to encapsulate the process of set formation and think of the resulting
set as an object.
A function can be represented in ISETL as a dynamic process which transforms
elements in one set to elements in another. For instance:
F := func(k);
return %+[i**2 : i in [2,4..k]]
end;
defines F as a function of k and returns the sum (denoted by %+) of the squares of all even
numbers between 2 and k.
An important effect of writing procedures that express mathematical actions is that, in
the sense of Chapter 7, the students tend to interiorize these actions and construct mental
processes that contribute to their understanding the underlying concepts.
As we pointed out in Chapter 7, it is important to encapsulate functions that are
understood as processes and think of them as objects. The best way to achieve this is to
operate on functions and/or make new ones. This is possible in ISETL because a function
is treated as data. It is possible to form sets of functions, have functions as parameters to
other functions and also to have a func construct and return a function. Consider the
following example.
co := func(f,g);
return func(x);
return f(g(x));
end;
end;
co is an operation which will take two representations offunctions, say f1 and g1 and return
a representation of their composition. The composite function co(f1,f2) may also be written
using infix notation as (f1 .co f2). Assuming that f1 and f2 represent functions and the value
of expr is in the right set, the computer will accept
(f1 .co g1) (expr);
and return the value of
f1 (g1(expr)).
A powerful way to use this idea is to have students construct co and use it, preferably
to solve problems of interest to them. The student will tend to have a number of important
experiences as a result of constructing co. First, it is necessary to think of functions as
objects in order to imagine applying some process to two functions. Then these two objects
must be unpacked to reveal their processes which can be coordinated by linking them
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ED DUBINSKY AND DAVID TALL
sequentially. The resulting process is then converted back to an object by the three lines
beginning with return func(x);. This code, which has the effect of returning a representation of a function whose domain variable will be denoted by x, is very difficult for students
and having them struggle to construct it in order to solve a problem can have a profound
positive effect on their conceptualization of functions.
A second way of representing functions in ISETL, which corresponds to one way that
mathematicians think of functions is to list the ordered pairs, for instance
H := {[x,x**2] : x in P};
assigns to the variable H the set of ordered pairs [x,x**2], where x is a prime less than 1000
and x**2 denotes x2.
Within ISETL a set of ordered pair works like a function, so an expression such as
H(3);H(7);H(4);
will print on the screen the values 9, 49 and om, the last symbol being the sign that H (4)
is not defined because 4 is not in P.
In a sense, this reverses the mental excursion. If a function is constructed as a func which
is then operated on, one is influencing students to think about a function first as a process,
then as an object. A set of ordered pairs, on the other hand, is most likely to be considered
to be an object, especially if previous study of the language has treated sets in this way.
Having students write such code and then do evaluations tends to have them think first of
a function as an object, and then as a process. Clearly, students should experience both
excursions and see them as two aspects of the same notion. The fact that ISETL will treat
sets of ordered pairs and funcs in many similar ways (for example, co will work just as well
if its inputsare sets of ordered pairs rather than funcs, or even a mixture) helps students unify
their thoughts about the two points of view.
An example of the inputs to a function being a combination of functions and numbers
is the following func to calculate a Riemann sum for the function f from a to b using n equal
width strips whose height is the left endpoint of each subinterval:
Riem Left := func(f,a,b,n);
x := [a+((b–a)/n)*(i-1) : i in [1..n+1]];
return %+[f(x(i))*(x(i+1)–x(i)) : i in [1..n]];
end;
Students can also encapsulate the notion of integration as a function operating on other
functions by defining:
Int := func(f,a,n);
return func(x,a,n);
return Riem Left(f,a,x,n);
end;
end;
ADVANCED MATHEMATICAL THINKING AND THE COMPUTER
247
Here Int(f,a,n) represents a function off, a and n where Int(f,a,n)(x) gives the Riemann
sum for f from a to x using n equal steps.
ISETL is also ideal for other mathematical concepts and the benefits to learning can also
be delineated in terms of the general theory presented in Chapter 7. We mention briefly a
few additional things one can do in this language and how they relate to understanding
mathematical concepts.
For instance, it is helpful for students to write programs to construct the truth table for
a given expression. With the first order calculus there is again the dichotomy and synthesis
of thinking of a logical expression as a process and as an object. Thus, in an expression such
as
(P ∧ Q) ⇒ ((–Q) ∨(P ∨ R))
the expression (P ∧ Q) can represent, in the mind of the student, a process consisting of
putting together P and Q and evaluating the truth or falsity for various values of the
variables. But in order to combine (P ∧ Q) with the rest of the expression, it must be treated
as an object.
Once boolean expressions (having the value true or false) are considered as objects, they
can be collected as elements in a set. This is a critical step in the transition to the second order
predicate calculus in which quantification is involved. In order to interpret the logical
statement
∃ x ∈ S ∋P(x)
one has to imagine a set of propositions indexed by x. The existential operator is performed
by iterating x through the domain S, evaluating the proposition valued function P at x and,
if once the result is true, declaring success and going home. This is exactly what the
computer does when given the ISETL command
exists x in S I| P(x)
and thinking about the ISETL procedure helps the student think about the corresponding
mathematical process. Beginning with a function P of two variables and applying two
quantifiers (generally one existential and one universal) leads to a second order quantification. Writing the code helps the student to coordinate two instances of the quantification
process and make the appropriate mental construction.
Formal definitions of mathematical structures are straightforward to implement (for
finite sets) in ISETL. For instance, if G is a finite set with binary operation op, then the
following ISETL func tests whether it is a group:
grouper := func(G,op);
return (forall x,y in G | x .op y in G)
and (forall x,y,z in G | (x .op y) .op z) = (x .op (y .op z))
and (exists e in G | (forall x in G | x .op e = x))
and (forall x in G | (exists y in G | x .op y = e));
end;
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Notice how closely this code resembles the formal definition of a group. It also fosters
the psychological constructions necessary to understand the group axioms. There are
several instances of processes and objects here as well as coordination of two processes.
In addition, the axiom for inverses requires a reversal of the process which arose in the
axiom for identity.
It turns out that, whether or not the students succeed in writing such a func, once they
have it and understand it, they can write funcs to test for subgroups and even normal
subgroups. Then, it is very effective to have them construct the set of cosets, define the
appropriate binary operation and use grouper to decide whether it gives a group. This can
be carried at least up to the fundamental theorem of homomorphisms.
EPILOGUE
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CHAPTER 15
REFLECTIONS
DAVID TALL
The production of this book is a first stage in a journey which sixteen authors and a wider
group of co-workers in Advanced Mathematical Thinking have shared. It is pertinent, given
the nature of the thinking processes that we have unfolded, to reflect upon what we have
done with the spiral of conceptual development in mind. First one begins with a problem
which may not be well-defied. Then one uses what tools are available to attack the problem
as it progressively becomes clearer, with all the false starts and hard-won minor advances
that are inevitable ingredients of the struggle. And now there is a calm after the storm to
reflect, to see what gains have been made and what remains to be done.
It would be good to be able to look back on the definitive book on Advanced
Mathematical Thinking , with all the resolutions of all the problems that occur and a
coherent theory that explains what it is and how to help others achieve it. This task is not
yet complete, certainly not in a definition-theorem-application format that a mathematician
might require of a theory. What has been done is to set out on a journey, on which the reader
has been encouraged to participate, to consider the way in which advanced mathematical
thinking functions, to understand what makes some thinkers successful and to help others
on their journey to greater success. "The journey is the reward”. And at this time we can
look back on the pathways we have taken to see what problems have been well-formulated
and what solutions have appeared as we move on to the next stage of the journey.
For me, as editor, it has been a fascinating study to see the development of various parts
of a theory, to see consonances and dissonances, some of which have been resolved whilst
others remain suspended in the ether. At the beginning of the journey I saw through a glass
darkly. I have yet to see face to face.
But now there are clearer avenues to follow, beginning with a more focussed picture of
the nature of the advanced mathematical thinking and moving towards pertinent questions
and partial answers. First we must highlight the different ways in which individual
mathematicians may think successfully. In particular, the need for all of us, successful in
our various ways, to give space to others to help them use their own particular talents to build
up their mathematical thinking processes. Then there is the realization of the thorny nature
of the full path of mathematical thinking, so much more demanding and rewarding than the
undoubted aesthetic beauty of the final edifice of formal definition, theorem and proof.
It is clear that the formal presentation of material to students in university mathematics
courses – including mathematics majors, but even more for those who take mathematics
as a service subject – involves conceptual obstacles that make the pathway very difficult
for them to travel successfully. And the changes in technology, that render routine tasks less
needful of labour, suggest that the time for turning out students whose major achievement
is in reproducing algorithms in appropriate circumstances is fast passing and such an
approach needs to move to one which attempts to develop much more productive thinking.
It is therefore no longer viable, if indeed it ever was, to lay the burden of failure of our
students on their supposed stupidity, when now the reasons behind their difficulties may
251
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be seen to be in part to be due to the epistemological nature of mathematics and in part to
misconceptions by mathematicians of how students learn. We often teach certain skills
because we know that these will bring visible, albeit limited, success, but we now know,
somewhat furtively, that the acquiring of those skills may develop concept imagery that
contains the seeds of future conflict. We have evidence that a formal approach, which
appeals to the sophisticated expert may be cognitively totally inappropriate for the naïve
learner and demands new forms of teaching to pass through the transition from elementary
mathematics to a point where the economy and structure of modem mathematics is seen
as a meaningful goal.
It seems incredible that our list of references is largely dominated by papers written in
the last decade with only a few honourable exceptions before the early eighties. What has
emerged from a meeting of individuals over a five year period, to reflect on this newly
developing area of concern, is a clearer understanding of the full cycle of mathematical
thinking: the need to begin with conjectures and debate, the need to construct meaning, the
need to reflect on formal definitions to construct the abstract object whose properties are
those, and only those, which can be deduced from the definition. Advanced mathematics,
by its very nature, includes concepts which are subtly at variance with naïve experience.
Such ideas require an immense personal re-construction to build the cognitive apparatus
to handle them effectively. It involves a struggle which virtually every author in this book,
both severally and individually, sees in terms of a reflection on personal knowledge and a
direct confrontation with the inevitable conflicts which require resolution and reconstruction.
College professors see this conflict daily in individual students as they struggle to come
to terms with new ideas. In the past they have often tried to help by providing clearer
lectures, making the transitions as simple as possible, presenting the ideas in a way which
reduces the strain. This may even lead to the successful professor being lauded by his or
her students for the clarity of their exposition, but the acid test is what do the students learn?
And this needs to be assessed in a wider sense than just which algorithms closely related
to their course they can carry out, or which definitions and proofs can they correctly
reproduce.
Our cognitive studies have shown the manifold differences between the formal
definitions of concepts and the images we use in our minds to work with these concepts.
They show how the complexity of the subject demands a “chunking” of information in an
efficient way so that it can be easily handled, and this is linked to the appropriate use of
symbolism for a given context and the appropriate meaning which the individual links to
that symbolism.
We have seen a divergence between the visualizers and verbalizers amongst us, just as
there appears to be a time-honoured difference between the mental processes of the
mathematical giants of the past. In recent months, as I have interacted with the various
authors in an attempt to either come to an agreement or to hone our differences into explicit
focus, I have been privileged to gain some additional insights.
It is clear that mathematics without process to give results is of little value, in other
words, visualizing an idea without being able to bring it to fruition is virtually useless. I
emphasize this fact even though a major thrust of my work is in the use of visualization. On
the other hand, simply to be able to carry out procedures in a narrow way, without being
able to see the overall connections, is also grossly limiting. For me this has led to a belief
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253
in a versatile form of thinking which complements the procedural with the global overview.
However, we have evidence of mathematicians, such as Hermite, steeped in the logic of
their subject who develop a powerful intuition of the processes and their symbolism in such
away as to render visualization – for them – redundant. Wealsohaveevidenceof successful
students (such as the case of Terence Tao, Clements, 1984) who vastly prefer the power of
logical deduction. We therefore need to cater for different types of minds.
Recently, however, in a very different context, I was able to obtain an insight which may
prove helpful in this apparent dichotomy. Mathematics – according to the Oxford
Dictionary – is said to be “the Science of Space and Number”. In recent months I have been
reflecting on the fundamental differences between these two different forms of mathematics and the manner in which they develop cognitively. Space, through the study of
geometry, begins with gestalts – “that is a triangle”, “this is a straight line”, “that there is
a rectangle” and “this is a square”. The child learns to recognize these visual gestalts from
examples and non-examples. “Yes, that is a square, but don’t think it is not a rectangle,
because a square is a special kind of rectangle”. Through exploration and interaction with
others, the child learns to discriminate between these various gestalts and to isolate some
of their properties: “a rectangle has four right-angles and opposite pairs of sides equal”, “a
square has four right-angles and all four sides equal” and to begin to see relationships “an
isosceles triangle has two equal angles and two equal sides”. From here the relationships
begin to build into deductions “if a figure is a square, then it is a rectangle”, “if a triangle
has two equal sides, then it has two equal angles”, definitions begin to be isolated and,
finally, these can be formulated in an axiomatic way to give the framework for logical
deduction. Indeed, what I have just described in outline was formulated about the
development of geometry more clearly as a hierarchy over thirty years ago by V an Hiele
(1959).
Number on the other hand is a very different animal. It begins with imitation of the
number names recited in sequence, “one, two, three, ...”, perhaps imperfectly at first, “ ...
four, five, nine, seven, ...”, then with more confidence, until the routine of pointing at objects
and reciting the number names in proper sequence leads to the concept of counting. This
is an encapsulation. The process of counting leads to the concept of number. By various
further strategies of process, “counting all” of two sets (a coordination of two processes),
or “counting on” (combining the concept of number of the first set with the process of
counting the second) leads from the process of counting to the concept of “sum”. A vital
phenomenon occurs here in that the symbolism 4+3 represents to the user both the process
of counting and the product of that process, the sum. The rest of the “number” part of
mathematics proceeds, in the same way, by encapsulating processes as concepts, often
using the same symbolism for both process and concept. Thus the process of “repeated
addition”, “five threes” becomes the concept of “product”, “5 times 3” – both written as 5x3.
The process of “repeated multiplication” becomes the concept of “power” and so on. Of
course this prescription is exactly parallel to the discussion of Dubinsky on reflective
abstraction. It is a phenomenon known to Piaget and to many an observant teacher since
time began – except that there is an amazing simplicity about what is being done. In the
number side of mathematics the mathematician makes progress by being ambiguous about
notation. (S)he uses the same notation for process and product deliberately, so that (s)he
can powerfully use whichever is appropriate for a given task. To calculate means to use the
process, to manipulate is easier with a single object which involves using the product.
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DAVID TALL
As whole number generalizes to signed integer, the symbols +2 and –7 also have dual
roles as process and the product: “shift two units right on the number line”, “the integer plus
two”, “shift seven units left”, “the number minus two”. The same happens with fractions:
“3/4” is both “divide three by four” and the product of the process: “three-quarters”. It is
the same with trigonometry, where
sine = opposite/hypotenuse
is both an instruction to calculate and a symbolism for the result.
Algebra too exhibits this same dualism of notation where 2+3x means both the process
of adding two to the product of three and x and also the result of that process.
In chapter4 Hanna remarks on the irony that in a “discipline touted as precise, the student
must develop a tolerance for ambiguity”. Instead of being defensive about this state of
affairs, it is more appropriate to note that the successful mathematician is the individual who
sees the duality of this kind of notation as process and product and who uses the ambiguity
in a flexible way. Given the importance of a concept which is both process and product, I
find it somewhat amazing that it has no name. So I coined the portmanteau term “procept”
for a process which is symbolized by the same symbols as the product. It seems that the
whole of number and algebra is built on procepts, so a theory of procepts and their use in
mathematics has a vast potential domain of application.
Yet space and geometry are different. They seem to be built on gestalts whose properties
are only slowly teased out and put into coherent relationships, then definitions and
deductions.
There are therefore (at least) two different kinds of mathematics. One builds from
gestalts, through identification of properties and their coherence, on to definition and
deduction at advanced levels of mathematical thinking. The other continually encapsulates
processes as concepts, to build up arithmetic, then generalizes these ideas in algebra before
formalizing them as definitions and deductive theorems in the advanced mathematics of
abstract algebra.
If we look at the discussion of Vinner in chapter 5, we find his theories originally began
with geometry, and his examples include “car”, “table”, “house”, “green”, “nice”, etc. None
of these are procepts. However, if we look at the discussion of Dubinsky in chapter 7, we
find his examples include “commutativity of addition”, “number”, “trajectory” (as a
coordination of successive displacements), “see-saw” (as the balancing of two objects),
“multiplication”, “fluid levels” (as a ‘variation of variations’). All these are processes
which become encapsulated as concepts. As they stand, they are not all procepts within the
narrow meaning of the term just defined. However, they all involve manipulation of
quantities, or balancing of quantities, or variation of quantities, and this in turn involves
number, which brings us back to proceptual ideas where symbolism is used both to
represent a process of manipulation and the result of that process.
The fascinating thing is that, by the time we reach the level of formalism in advanced
mathematics, these two different strands move to a similar formulation: the definition of
concepts and the deduction of properties of those concepts.
I believe that the major catastrophe of the new mathematics movement was due to the
unproven assumption that “if only the concepts are properly defined, then everything will
be OK’. The need for clear definitions and deductions caused mathematicians to be covert
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255
about the power of their ambiguous use of procepts. This move served our students badly
because it failed to acknowledge the methods of the working mathematician. The power
in mathematics is not given through unique and precise meaning to symbolism –“afunction
is a set of ordered pairs such that.. .” – but through a duality which gains flexibility through
ambiguity 1 – a function is both a process (to be able to calculate) and a concept (which can
be manipulated). It is as simple as that. We cheated our students because we did not tell the
truth about the way mathematics works, possibly because we sought the Holy Grail of
mathematical precision, possibly because we rarely reflected on, and therefore never
realized, the true ways in which mathematicians operate.
The evidence which we are collecting with a wide range of ability of much younger
children is that the most able naturally use this flexibility (Gray, 1991). In arithmetic they
soon learn a few facts then, when they are faced with a new arithmetic problem, they are
often able to relate it to one they know and derive new facts from old. The more able
therefore have a built-in knowledge generator that develops new arithmetical knowledge
from old. Once they grasp this, they realize that they do not need to remember so much
because they can soon derive what they want to know. They have a flexible proceptual
knowledge in which number problems such as 4+5 can be decomposed as the process 4+5,
which might be seen as 4+4+1 and (if they know 4+4=8) can be reassembled as 8+1=9.
Thus the procept 4+5 is decomposed into process and parts of this are recomposed back to
derive the concept, or result, 4+5=9.
Meanwhile the lower ability children remember few facts and continue to use the
process of counting to add numbers together. If asked 8+4, they faithfully count on four to
get “nine, ten, eleven, twelve” but this is rarely remember as a known fact and, instead of
having a knowledge generator, they have an unencapsulated process which produces
answers which are not manipulable objects. Thus there grows a “proceptual divide”
between the more able, using proceptual flexibility, and the less able, locked in process.
The same proceptual divide occurs with algebra. The child who sees algebraic notation
only as process, is faced with a nightmare, for how can (s)he conceive of 2+3x as a process
when, without knowing x, it is a process which cannot be carried out. And if x is known,
why is it necessary to use algebra anyway? Only the child who can give meaning to the
symbolism as a conceptual entity can begin to manipulate more complex expressions
meaningfully in the sense of Harel and Kaput in chapter 6.
This same division between those who conceptualize process as product and those
locked in process occurs again at higher levels. The limit concept lim an is again a
n→∞
procept. The same notation represents both the process of tending to the limit, and also the
value of the limit. But this phenomenon is very different from procepts met in elementary
mathematics. There the process could be used to calculate the product. Now we have the
phenomenon that Cornu identified as an obstacle in chapter 10 understanding the
dynamics of the process does not lead directly to the calculation of the limit. Instead indirect
alternative methods of computation must be devised.
Just as with arithmetic, the theory of limits has a structure for devising new facts from
old. But in arithmetic the new facts are derived from old using the calculation processes of
1 I am grateful to my colleague Eddie Gray for this phrase, which comes the title of a joint
paper (Gray & Tall, 1991) based on his work with the number processes of younger children.
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arithmetic and the new facts have the same status as the old they can be calculated by the
processes of arithmetic in the same way. In the case of the theory of limits, the “known facts”
are one or two “elementary” deductions from the definition: that lim 1/n is zero, or that
n→∞
a constant function and the identity function are continuous. All three of these “elementary”
facts are derived from the definitions in singularly peculiar ways which can cause initial
confusion. The fact that 1/n tends to zero might be deduced from Archimedes’ axiom, or
perhaps by some heuristic appeal to the fact that :”I can make 1/ n smaller than ε by making
n bigger than the integer part of 1/ε plus one”, both of which are strange ways of asserting
1/n gets small as n gets large – the student knows that anyway! To establish the fact that a
constant function is continuous is just “tell me ε and I will tell you δ, in fact you can take
any δ>0 you like, say δ=1066”. It is a joke that few students have the experience to find
funny. The continuity of the identity function is equally enigmatic “OK, take δ=ε then,
when |x–x0|<δ, we have |f(x)–f(x0)|<ε, because x=f(x),can’t you see,you dummy?”Unlike
arithmetic, once these few “elementary facts” are deduced, few, if any, other such “facts”
are calculated directly. Instead the “algebra of limits” is proved, using the coordination of
the “unencapsulated definition of the limit” as reported in chapter 10, which is at, or beyond,
the zone of competence of most students. The result is that the derived facts are “proved”
(any polynomial is “continuous” by an induction argument combining sums and products
of constant functions and the identity) yet the actual definition is no longer used because
the calculations become horrendous.
Thus it is that the procepts in advanced mathematics work in a totally different and
completely enigmatic way compared with the procepts in elementary mathematics. It is no
wonder that, faced with this confusion, so many students end up conceiving the limit either
as an (unencapsulated) process or in terms of meaningless rote-learned symbol pushing.
Likewise the gestalt geometric concepts work differently in advanced mathematics too.
Instead of being “described” and having coherent relationships, they are “defined” and
other properties must be “deduced” from the definitions. Again, given the conflictbetween
the elementary ideas where the facts are known and the abstract ideas where they need to
be deduced confusion, as discussed by Vinner (chapter 5), is almost inevitable.
So what is the solution? First it should be noted that the chapters of this book nowhere
give methods that will produce guarantied success. There is no dispute that, for the most
able, a formal presentation maybe sufficient to show the structure of the subject which they
may appreciate and build into a deductive system. But for the vast majority of students, the
way ahead is stony and littered with cognitive obstacles which, if not addressed, will only
be isolated in the mind in such a way that they lie there ready to cause conflict in future times
– if they do not cause outright confusion already.
The evidence is that students of a wide range of abilities prosper when they can give
meaning to the ideas. This does not mean that they must always relate the concepts back
to some concrete foundation that has physical meaning. Just as the child who counts objects
successfully moves on at a very early age to mentally manipulate number symbols in
arithmetic, so successive layers of encapsulation of process into procept only need refer to
the level of the previous proceptual layer. In fact, once the encapsulation has occurred, the
use of the same symbolism causes the process and concept to coalesce into a single level.
Thus the so-called hierarchy of concepts, which is an obstacle to learning, becomes, to the
successful encapsulator, a single level in which process and concept are dually represented,
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with the complexity disguised by the simplicity of the symbolism.
The question to be addressed is: if this is the way of success for the more able, what
should we do with, or for, or to, the vast majority of our students? The evidence in this book
is, that to give them a sense of the full range of advanced mathematical thinking, it is
essential to help them reflect on the nature of the concepts and the need for mental
reconstruction in an overt and explicit way, and to give them opportunities in which they
can learn to conjecture and debate, so that they may participate in mathematical thinking,
not just learn to reproduce mathematical thought.
This is not going to be an easy task. What stands against it is, in many cases, fear. Fear
of professional mathematicians for the unknown when they leave their neatly planned
course structures of theorem-proof-application and give open-ended opportunities for
problem-solving. Fear of the increased time that this will take and that they will not “get
through the course”. Fear that “standards will drop” because students will not be able to
exhibit theability to carry out all the processes that need to be taught in an “honours degree”.
Fear that they dare not make any changes whilst other institutions maintain the traditional
standards.
In recent years the fast changes in society are causing all of the well-established truths
to be reassessed. In Britain through the Institute of Physics, university departments of
physics have mutually agreed to reduce the content of the three year physics course by one
third to give more room for understanding what is actually taught. In mathematics a step
in similar direction might not be out of place. It is not necessary to change the whole of the
approach in a single step. Given a modest reduction incontent, a new flexibility could allow,
say, a single course in problem-solving, of a general nature, to be introduced early in the
course, to encourage creativity in mathematical thinking, even though it introduced no new
content, but compensated in terms of reflection on higher processes. For ten years I have
run such a problem-solving course and I know the way it changes students’ perceptions of
themselves and builds up confidence through success in small things that steadily grow
more complex. They learn to talk to each other, to verbalize mathematics, to speak
coherently. They even learn to enjoy interchanging information and helping each other,
whereas before they had often believed that good students only do mathematics for
themselves, on their own.
Given a modest reduction in content, it might be possible to allow time for students to
explore their own conjectures in a specific subject area. In my own analysis lectures I
regularly set up a problem scenario and leave the students to work in groups to try to solve
the problem. “OK, so the intermediate value theorem seems obvious, but suppose you knew
f was continuous between a, b and that f(a) and f(b) had opposite signs – how would you
prove that it is zero in between?” Setting this as homework does not have the same effect
as encouraging students to talk together in class time, and the best way to do this is for the
instructor to make sure that there is a good topic for investigation and then leave the room.
Some of my best teaching occurs when I am somewhere else drinking coffee and getting
paid for it! A return to the classroom after an appropriate passage of time may find that the
students have not solved the problem, but they often have experiences on which a proof can
then be constructed through a mutual dialogue. In this way they learn to participate in the
construction of mathematical knowledge rather than just remembering and repeating it.
Viewing the third part of this book – the review of the literature – we see authors adopting
very different stances. Robert and Schwarzenberger highlight the difficulties of the
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DAVID TALL
transition from school to university, Eisenberg begins with a catalogue of failure in the
teaching of the function concept. Cornu is fascinated by the processes of knowledge
creation and the parallel between the epistemological obstacles in the past and the cognitive
obstacles of students. Artigue continues the study to further levels of mathematical analysis
and certain avenues of hope begin to appear. Tirosh reviews the cognitive conflicts inherent
in contemplating the infinite and gives a detailed report of a single experiment exemplifying
how students may be taught to reflect on their knowledge and actively participate in its
reconstruction. Alibert and Thomas look at the process of proof and show the difficulties
of the formalism and how it might be tackled through debate. Finally Dubinsky and I look
forward to the use of the computer and the way in which this may change the nature of
mathematics and provide an environment for learning. Despite the different tone of some
chapters, the message of hope for reflective reconstruction of knowledge is there in all of
them.
My recent thinking has led me to realize that the computer can be used in a very special
way in learning – to carry out the processes, so that the user can concentrate on the product.
This is the essence of a spread-sheet, a graph-plotter, a symbol manipulator, and so on. In
other words, the computer allows a change in the encapsulation from process to object.
Instead, of forcing the student first to learn and interiorize the process, the computer can
carry out the process and allow the user to focus on the properties of the product. In this way
there can be a shift of attention away from the process (in which the less able may become
trapped) and towards the mathematical objects, and their relationships at a higher level.
Instead of just learning the processes of solving differential equations, students may first
appreciate the existence and uniqueness of solutions, and construct them in a meaningful,
quasi-physical way, building an approximate solution curve by putting together short
straight-line segments of the appropriate gradient.
Thus the final plank in the new charter of advanced mathematical thinking in the
information age is what I have termed the principle of selective construction of knowledge,
in which the learner is allowed, even encouraged, to focus separately on the processes of
mathematics and the procepts produced by those processes. It is now possible to get a
computer to carry out the algorithms so that the student can concentrate on the properties
of the product. In this way the student can be encouraged to construct the properties and
relationships enjoyed by the product whilst suppressing consideration of the process which
is constructed internally by the computer. The student may at one time selectively
concentrate purely on the process and at another on the higher level relationships. Both
activities remain essential, for the process is needed to be able to do mathematics and the
higher level relationships are essential to fit it together in a meaningful way. The interesting
factor is that the focus on the process need not always precede the construction of the
properties of the product. The intuitive idea of existence and uniqueness of differential
equations can be investigated before formulating any symbolic solution. In this way the use
of the computer gives new teaching and learning strategies in advanced mathematics.
We therefore arrive at a possible new synthesis in teaching and learning advanced
mathematics which offers a more complete cycle of advanced mathematical thinking to
students, even those of more modest abilities. The active participation in thinking is
essential for the personal construction of meaningful concepts. Students need to be
challenged to face the cognitive reconstruction explicitly, through conjecture and debate,
through problem-solving, and they may be assisted in the acquisition of insights at higher
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259
levels by selectively sharing the construction with the computer. This does not remove the
need to pass on information in the theorem-proof-application mode, for this is the crowning
glory of advanced mathematics. But students need to be assisted through a transition to a
stage where they see the necessity and economy of such an approach. Therefore, step by
step, through professors being given a little space to experiment, initially as part of a
traditional curriculum, a new balance may be struck, between the shining edifice of
advanced mathematics that is the rightful pride of the mathematical community and the
fuller range of advanced mathematical thinking that gave rise to its construction.
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BIBLIOGRAPHY
Adler, C., (1966), ‘The Cambridge conference report: blueprint or fantasy?’, Mathematics Teacher ,
58, 210–17.
Alibert, D., (1987), ‘Situation Codidactique et Délocalisation du Savoir’, Exposé au Séminaire de
Didactique des Mathématiques et de l’Informatique, Grenoble.
Alibert, D., (1988a). ‘Towards New Customs in the Classroom’, For the Learning of Mathematics,
8 (2). 31–35.
Alibert, D., (1988b). ‘Codidactic System in the Course of Mathematics : How to Introduce it?’,
Proceedings of the PME 12, Veszprem.
Alibert, D., (1988c), ‘How to Set Problems Before Giving Answers: An Experimental Mathematics
Course at the University of Grenoble’, Proceedings of the Sixth International Congress of
Mathematics Education, Budapest.
Alibert, D., (1991). ‘Sur le Rôle du Groupe–classe pour obtenir, et résoudre une situation a–
didactique’, Recherches en Didactique des Mathématiques, 10 (2). in press.
Alibert, D., Grenier, D., Legrand, M., Richard, F., (1986). ‘Introductiondu Débat Scientifique dans
un Cours de Première Annie du Deug A à l’Université de Grenoble 1’. Rapport de l’ATP
Transitions dans le systeme educatif No. 122601 du MEN.
Alibert D., Legrand, M. & Richard, F., (1987a), ‘Le thème “différentielles”, un exemple de
coordination maths–physique dans la recherche’, Actes du Colloque du Sèvres, Mai 1987,
Editions La Pensée Sauvage, Grenoble, 7–45.
Alibert, D., Legrand, M. & Richard, F., (1987b), ‘Alteration of didactic contract in didactic
situations’, Proceedings of PME 11, Montréal, 379–386.
Arton, H., (1981), Elementary linear algebra, John Wiley & Sons, New York.
Appel, K. & Haken, W., (1976). ‘The solution of the four colour map problem’, Scientific American
(October), 108–121.
Arcavi, A. & Schoenfeld, A., (1987), On the meaning of variable, Technical Report, School of
Education, University of California, Berkeley.
Artigue, M., (1987), ‘Ingénierie didactique à propos d’équations differentielles’, Proceedings of
PME 11, Montréal, 236–242.
Artigue, M., Gautheron, V. & Isambard, E., (1985), ‘Analyse non-standard et enseignement’,
Cahier de Didactique N0.15, IREM Paris VII.
Artigue, M. & Szwed, T., (1983), Représentations graphiques, IREM Paris Sud.
Artigue, M. & Viennot, L., (1987), ‘Students’ conceptions and difficulties about differentials’,
Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics , Comell, III 1–7.
Atiyah, M.F., (1984), ‘Mathematics and the computer revolution’, Nuovo Civilità della Macchine
(Bologna), 2 (3), reprinted in A. G. Howson & J.-P. Kahane (Eds.), The Influence of Computers
and Informatics on Mathematics and its Teaching , Cambridge University Press, Cambridge.
Ausubel, D.P., Novak, J. D. & Hanesian, H., (1968), Educational Psychology, a Cognitive View,
(2nd edition), Holt, Rinehart & Winston, New York.
Authier, H., (1986), ‘Étude comparative de diverses productions d’etudiants de première année de
DEUG scientifique selon les séries de baccalauréat d’origine’, Cahier de Didactique No. 31,
IREM Paris VII.
Ayers, T., Davis, G., Dubinsky, E. & Lewin, P., (1988), ‘Computer experiences in learning
composition of functions’, Journal for Research in Mathematics Education , 19 (3), 243–259.
261
262
BIBLIOGRAPHY
Bachelard, G., (1938). (reprinted 1983). La formation de l’esprit scientifique, J. Vrin, Paris.
Balacheff, N., (1982). ‘Preuve et Demonstration’, Recherche en Didactique des Mathematiques, 3
(3), 261–304.
Balacheff, N., (1988), Une Étude des Processus de Preuve en Mathématique chez des Élèves de
Collége, Thèse Université Grenoble 1.
Balacheff, N. & Laborde, C., (1985), ‘Social Interactions for Experimental Studies of Pupils’
Conceptions : Its Relevance for Research in the Didactics of Mathematics’, First International
Conference on the Theory of Mathematics Education, Bielefeld.
Bautier, E. & Robert, A., (1987). ‘Apprendre des mathématiques et comment apprendre des
mathématique’, Cahier de didactique des mathématiques 41 , IREM, Paris VII.
Beberman, M., (1956). ‘The university of Illinois school mathematics program’, in J. Bidwell & R.
Clason (Eds.), Readings in the History of Mathematics Education , NCTM, Washington D.C.,
655–663.
Begle, E., (1968), ‘SMSG: The first decade’, Mathematics Teacher, 62, 239–45.
Beke, E., (1914), ‘Rapport génénral sur les résultats obtenus dans l’ introduction du calcul différentiel
et intégral dans les classes supérieures des établissements secondaires’, L’Enseignement
Mathématique, 16, 246–284.
Ben-Chaim, D., (1982), Spatial visualization: sex differences, grade level differences and the effect
of instruction on performance and attitudes, Ph.D. dissertation, Michigan State University.
Berkeley, G., (1951). ‘The Analyst’, Collected Works, vol. 4 (ed. Luce, A. A. & Jessop, T.E.),
Nelson, London.
Beth, E.W. & Piaget, J., (1966), Mathematical Epistemology and Psychology (W. Mays, trans.),
Reidel, Dordrecht (originally published 1965).
Biggs, J. & Collis, K., (1982), Evaluating the Quality of Learning: the SOLO Taxonomy , Academic
Press, New York.
Birkhoff, G.D., (1956), ‘Mathematics of aesthetics’, in J. R. Newman (Ed.), The World of
Mathematics Vol. 4 , (7th edition), Simon & Schuster, New York, 2185–2197.
Bishop, A.. (1985). ‘The Social Construction of Meaning – a Significant Development for
Mathematics Education?’, For the Learning ofMathematics, 5 (1), 24–28.
Bishop, E., (1977). review of Elementary Calculus by H. J. Keisler, Bulletin American Mathematical Society , 83, 205–208.
Bishop, E., (1967), Foundations of Constructive Analysis, McGraw-Hill, New York.
Blackett, N., (1987), Computer graphics and children’s understanding of linear and locally linear
graph, (M.Sc. Thesis), University of Warwick, U.K.
Boas, R., (1960). A Primer of Real Functions, Carus Mathematical Monographs, John Wiley and
Sons, New York.
Bolzano, B., (1950). Paradoxes of the Infinite , Routledge & Kegan Paul, London.
Borasi, R., (1984). ‘Some reflections on and criticisms of the principle of learning concepts by
abstraction’, For the Learning of Mathematics, 4, 14–18.
Borasi, R., (1985). ‘Errors in the enumeration of infinite sets’, Focus on Learning Problems in
Mathematics, 7, 77–49.
Boyer, C.B., (1939),The History of the Calculus and its ConceptualDevelopment, (page references
as in reprint, Dover, New York, 1959).
Breuer, S., Gal-Ezer, J. & Zwas, G., (1990), ‘Microcomputer Laboratories in Mathematics
Education’, Computers in Mathematics and its Applications, 19 (3). 13–34.
Brousseau, G., (1986). ‘Fondements et Méthodes de la Didactique des Mathématiques’, Recherches
en Didactique des Mathématiques, 7 (2), 33–115.
Brousseau, G., (1988), Fondements et méthodes de la didactique des mathématiques, Thesis,
University of Bordeaux.
BIBLIOGRAPHY
263
Brown, A. L., Bransford, J. D., Ferrara, R. A. & Campione, J. C., (1983), ‘Learning, Remembering
and Understanding’, in P. H. Mussen (Ed.), Handbook of Child Psychology , Fourth Edition, Vol.
III: Cognitive Development (volume eds. J. H. Flavell & E. M. Markman), Wiley, New York,
77–166.
Brown, A.L. & Kane, M. J., (1988), ‘Pre-school children can learn to transfer: Learning to learn and
learning from example’, Cognitive Psychology, 20, 493–523.
Buck, R. C., (1970), “‘Functions”in mathematics education’, in E. G. Begle (Ed.), 69th Yearbook
of the National Society for the Study of Education, University of Chicago Press, Chicago, 236–
259.
Cambridge Conference, (1963), Goals for School Mathematics, (Report of the Cambridge Conference on School Mathematics) HoughtonMifflin Co., Boston, for Educational Services, Inc., 10.
Cambridge Conference, (1967), Goals for Mathematics Education for Elementary School Teachers, (Cambridge Conference on Teacher Training), Houghton Mifflin Co., Boston, for Educational Development Center, Inc., 98.
Cajori, F., (1929), A history of mathematical notations, Vol. 2 Notations mainly in higher
mathematics, The Open Court Publishing Co, La Salle, Illinois.
Cajori, F., (1980), History of Mathematics, (Third edition, originally published 1893), Chelsea
Publishing Co., New York.
Cantor, G., (1955). Contributions to the Founding of the Theory of Transfinite Numbers , Dover,
New York.
Carter, H. C., (1970), ‘A study of one learner cognitive style and the ability to generalize behavioral
competences’, Proceedings of AERA, ERIC No. ED 040–758.
Case, R. & Sandleson, R., (1988), ‘A developmental approach to the identification and teaching of
central conceptual structures in middle school science and mathematics’, in J. Hiebert & M. Behr
(Eds.), Number Concepts and Operations in the Middle Grades , Erlbaum, Hillsdale NJ.
Char, B. W., Fee, G. J., Geddes, K. O., Gonnet, G. H., Marshman, B. J. & Ponzo, P., (1986),
‘Computer Algebra in the Mathematics Classroom’, Proc. 1986 Symposium on Symbolic and
Algebraic Computation, Assoc. Comput. Mach., 135–140.
Cheshire, F.D., (1981), The effect of learning computer programming skills on developing cognitive
abilities , Doctoral dissertation, Arizona State University, Dissertation Abstracts International ,
42, 654A, (University Microfilms No.81–17163).
Cipra, B., (1983), Misteaks: (Mistakes) A Calculus Supplement, Birkhauser, Boston.
Clement, J., Lochhead, J. & Monk, J. S., (1981), ‘Translation difficulties in learning mathematics’,
American Mathematical Monthly, 286–290.
Clement, J., (1983), ‘A conceptual model discussed by Galileo and used intuitively by physics
students’, in D. Gentner and A. L. Stevens (Eds.), Mental Models , Lawrence Erlbaum
Associates, London, 325–340.
Clement, J., (1986), ‘Misconceptions in graphing’, Proceedings of PME 9 , Noordwijkerhout, 369–
375.
Clement, J., (1987), ‘Overcoming students’ misconceptions in physics: The role of anchoring
intuitions and analogical validity’, in J. D.Novak (Ed.), Proceedings of the Second International
Seminar on Misconceptions and Educational Strategies Strategies in Science and Mathematics,
3, 84–97. Comell University, Ithaca NY.
Clements, M. A., (1984), ‘Terence Tao’, Educational Studies in Mathematics , 15 (21), 32–38.
Cohen, L. & Ehrlich, G., (1963), The Structure of the Real Number System, D. van Nostrand Co.
Inc, Princeton.
Comu, B., (1981), ‘Apprentissage de la notion de limite: modèles spontanés et modèles propres’,
Actes du Cinquième Colloque du Groupe Internationale PME, Grenoble, 322–326.
Cornu, B., (1983), Apprentissage de la notion de limite: conceptions et obstacles, Thèse de doctorat
de troisième cycle, L’Université Scientifique et Medicale de Grenoble.
264
BIBLIOGRAPHY
D’ Alembert, J. L., (1975), ‘Differentiel’, Encyclopedie Méthodique Mathematiques, 1, 520–560,
Panc Koucke Ed., Paris.
D’ Halluin, C. & Poisson, D., (1988), Une stratégie d’enseignement des mathématiques: la
mathémematisation de situations intégrant l’informatique comme outil et mode de pensée,
Thèse de Doctorat, Université de Lille.
Dauben, J., (1983), ‘George Cantor and the origins of transfinite set theory’, Scientific American,
248, 112–121.
Davis, P. J., (1986), ‘The nature of proof’, in M.Carss (Ed.) Proceedings of the fifth international
congress on mathematical education, Birkhauser, Boston.
Davis, P. J., (1972), ‘Fidelity in mathematical discourse: is one and one really two?’, American
Mathematical Monthly, 79, 252–263.
Davis, P. J. & Hersh, R., (1981), The Mathematical Experience, Birkhauser, Boston.
Davis, P. J. & Hersh, R., (1986), Descartes’ Dream. The World According to Mathematics,
Houghton Mifflin Co, Boston.
Davis, R. B., (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics,
Ablex, Norwood NJ.
Davis, R. B., (1986), ‘Algebra in elementary schools’, in M. Carss (Ed.), Proceedings of the Fifth
International Congress on Mathematical Education, Birkhauser, Boston.
Davis, R. B., (1988), ‘The Interplay of Algebra, Geometry, and Logic’, Journal of Mathematical
Behavior, 7, 9–28.
Davis, R. B. & Vinner, S., (1986), ‘The Notion of Limit: Some Seemingly Unavoidable Misconception Stages’, Journal of Mathematical Behaviour, 5 (3), 281–303.
Delens, P., (1930), ‘Notations différntielles’, L’ Enseignement Mathématique,30,333–337.
Deligne, P., (1977). ‘Seminaire de Géométrie Algébrique du Bois Marie (SGA4)ThériedesTopos
et Cohomologie Étale des Schémas’, Lecture Notes in Mathematics no. 305, vol. 3, SpringerVerlag, Berlin.
Dienes, Z. P., (1960), Building up Mathematics, Hutchinson, London.
Dienes, Z. P., (1968), Fractions: an operations approach , Hutchinson, London.
Dienes, Z. P. & Jeeves, J., (1965), Thinking in Structures, Hutchinson, London.
Dieudonne, J. A., (1971), ‘Modem axiomatic methods and the foundations of mathematics’, in F.
Le Lionnais (Ed.), Great currents of mathematical thought (Vol. l), Dover, New York.
Dieudonné, J. A., (1975). ‘L’abstraction et l‘intuition mathématique’, Actes du Colloque sur Les
Mathématiques et la Réalité, Centre Universitaire de Luxembourg, 1974 , published in Dialectica,
Revue internationale de philosophie de la connaissance, 29 (1), 39–54.
Donaldson, M., (1963), A study of children’s thinking, Tavistock Publications, London, 183–185.
Dörfler, W., (1988), ‘Die Genese mathematischer Objekte und Operationen aus Handlungen als
kognitive Konstruktion’, in Willibald Dörfler (Ed.), Kognitive Aspekte mathematischer
Begriffsentwicklung , Hölder-Pichler-Tempsky, Vienna, Austria, 55–125.
Douady, R., (1984). ‘L’ingénierie didactique, une instrument privilegie pour une prise en compte
de la complexité de la class’, Proceedings of PME 11, Montréal, III 222–228.
Douady, R., (1986), ‘Jeu de Cadres et Dialectique Outil-objet’, Recherches en Didactique des
Mathématiques, 7 (2), 5–32.
Dreyfus T., (1991), ‘On the status of visualization and visual reasoning in mathematics and
mathematics education’, Proceedings of PME 15, Assisi, 1,33–48.
Dreyfus, T. & Vinner, S., (1982), ‘Some aspects of the function concept in college students and
junior high school teachers’, Proceedings of PME 6, Antwerp, 12–17.
Dreyfus, T. & Eisenberg, T., (1983), ‘The function concept in College Students: Linearity,
Smoothness and Periodicity’, Focus on Learning Problems in Mathematics, 5 (3 & 4), 119–132.
Dreyfus, T. & Eisenberg, T., (1984), ‘Intuitions on functions’, Journal of Experimental Education,
52, 77–85.
BIBLIOGRAPHY
265
Dreyfus, T. & Eisenberg, T., (1986), ‘On the Aesthetics of Mathematical Thought’, For the
Learning of Mathematics, 6 (1), 2–10.
Dreyfus, T. & Eisenberg, T., (1987), ‘On the deep structure of functions’, Proceedings of PME II,
Montréal, 190–96.
Dreyfus, T. & Thompson, P. W., (1985), ‘Microworlds and Van Hiele Levels’, Proceedings of PME
9, Noordwijkerhout, 1, 5–1 1.
Dubinsky, E., (1986), ‘Teaching mathematical induction I’, The Journal of Mathematical Behavior,
5, 305–317.
Dubinsky, E., (1989), ‘The case against visualization in school and university mathematics’,
position paper presented to the Advanced Mathematical Thinking Group at PME 13, Paris,
(Available from the author: Department of Mathematics, Purdue University, West Lafayette,
Indiana, USA).
Dubinsky, E., (1990a), ‘On learning quantification’, Journal of MathematicalBehavior, (inpress).
Dubinsky, E., (1990b), ‘Teaching mathematical induction II’, The Journal of Mathematical
Behavior, (in press), 285–304.
Dubinsky, E. & Lewin, P., (1986), ‘Reflective abstraction and mathematics education: the genetic
decomposition of induction and compactness’, The Journal of Mathematical Behavior , 5, 55–
92.
Dubinsky, E. & Schwingendorf, K. E., (1990a), ‘Calculus, concepts and computers – innovations
in learning calculus’, CRAFTY (ed. Tucker T.), Math. Assoc. Amer.
Dubinsky, E. & Schwingendorf, K. E., (1990b), ‘Constructing calculus concepts: cooperation in a
computer laboratory’, MAA Notes Series, (ed. Leinbach C.), Math. Assoc. Amer.
Dubinsky E., Elterman, F. & Gong, C., (1988), ‘The student’s construction of quantification’, For
the Learning of Mathematics, 8 (2), 44–51..
Dubinsky E., Hawks, J., & Nichols, D., (1989), ‘Development of the Process Conception of
Function in Pre–Service Teachers in a Discrete Mathematics Course’, Proceedings of thePME
13, Paris.
Dubinsky, E, & Lewin, P., (1986), ‘Reflective Abstraction and Mathematics Education: the genetic
decomposition of induction and compactness’, The Journal of Mathematical Behavior, 5, 55–
92.
Duval, R., (1983). ‘L‘Obstacle du dedoublement des objets mathématiques’, Educational Studies
in Mathematics, 14, 385–414.
Edwards, E. M., (1987). ‘An appreciation of Kronecker’, The Mathematical Intelligencer,9(1), 28–
35.
Ellerton, N. F., (1985), The Development of Abstract Reasoning – Results from a large scale
mathematics study in Australia and New Zealand.
Fehr, H., (1 966), A unified mathematics program for grades seven though twelve, The Mathematics
Teacher, 59, 463.
Fehr, H., (1974). ‘The secondary school mathematics curriculum improvement study: a unified
mathematics program’, Mathematics Teacher, 25–33.
Fischbein, E., (1982), ‘Intuition and Proof’, For the Learning of Mathematics, 3 (2), 9–18, 24.
Fischbein, E., (1978). ‘Intuition and mathematical education’, Osnabrücker Schriftenzür Mathematik
1, 148–176.
Fischbein, E. & Gazit, A., (1984), ‘Does the teaching of probability improve probabilistic
intuitions?’, Educational Studies in Mathematics, 15, 1–24.
Fischbein, E., Tirosh, D. & Hess, P., (1979). ‘The Intuition of Infinity’, Educational Studies in
Mathematics, 10, 3–40.
Fischbein, E., Tirosh, D. & Melamed, U., (1982), ‘Is it possible to measure the intuitive acceptance
of the mathematical statement?’, Educational Studies of Mathematics, 12, 491–512.
266
BIBLIOGRAPHY
Fodor, J. A., Garret, M. F., Walker, E. C. & Parley, C. H., (1980), ‘Against definition’, Cognition,
8, 263–267.
Frankel, A. A., (1953). Abstract Set Theory, North-Holland Pub., Amsterdam.
Fréhet, M., (1911), ‘Surlanotion de différentielle’, Comptes-Rendus de l’Académie des Sciences,
152 (13), 845–847, 1050–1051.
Freudenthal, H., (1983), The Didactical Phenomenology of Mathematics Structures, Reidel,
Dordrecht.
Gagné, R. M., (1970). The Conditions of Learning , 2nd ed., Holt, Rinehart and Winston, New York.
Galileo, G., (1954), The New Sciences, (H. Crew and A. Salvio, Trans.), Dover, New York
(previously published 1811).
Gazzanigna, M. S., (1985), The Social Brain: Discovering Networks of the Mind , Basic Books, New
York.
Gleick, J., (1987), Chaos: Making a New Science, Penguin, London.
Glennon, V. J., (1980), ‘Neuropsychology and the Instructional Psychology of Mathematics’, The
Seventh Annual Conference of the Research Council for Diagnostic and Prescriptive Mathematics, Vancouver, B.C.
Goldin, G., (1982), ‘Mathematical language and problem solving’, Visible language (Special issue
on mathematical language, R. Skemp, Ed.), 16, 221–238.
Gray, E. M. &Tall D. O., (1991). ‘Duality, Ambiguity and Flexibility in Successful Mathematical
Thinking’, Proceedings of PME 15, Assisi, 2, 72–79.
Gray E. M., (1991). ‘An Analysis of Diverging Approaches to Simple Arithmetic : Preference and
its Consequences’, Educational Studies in Mathematics, (in press).
Greeno, G. J., (1983), ‘Conceptual entities’, in D. Genter & A. L. Stevens (Eds.), Mental Models,
227–252.
Grenier, D., Legrand, M. & Richard, F., (1984). ‘L‘Introductiondu Débat Scientifique à l’Intérieur
du Cours pour Provoquer chez les Étudiants un Processus du Découverte et de Preuve’, Paper
presented at the Séminaire de Didactique, Grenoble.
Grenier, D., Legrand, M. & Richard, F., (1985). ‘Une Séquence d’Enseignement sur l’Intégrale en
DEUG A Première Année’, Cahier de Didactique des Mathématiques No. 22, IREM Paris VII.
Grogono, P., (1989), ‘Meaning and Process of Mathematics and Programming’, For the Learning
of Mathematics, 9, 14–19.
Hadamard, J., (1945), The Psychology of Invention in the Mathematical Field., Princeton University
Press, (page references are to the Dover edition, New York 1954).
Hadar-Moscovitz, N., Zaslavsky, O., & Inbar, S., (1987), ‘An empirical classification model for
errors in high school mathematics’, Journal for Research in Mathematics Education, 3–14.
Hahn, H., (1956), ‘Infinity’, in J. R. Newman (Ed.), The World of Mathematics, Vol. 3, Simon &
Schuster, New York, 1593–1611.
Halmos, P. R., (1985), I want to be a mathematician, Springer-Verlag, New York.
Hammersley, J. M., (1968). ‘On the enfeeblement of mathematical skills by “modern mathematics”
and by similar soft intellectual trash’, Bulletin of the Institute of Mathematics and lts Applications, (Oct., 1968). 66–85.
Hanna, G., (1983), Rigorous proof in mathematics education, OISE Press, Toronto.
Harel, G., (1985), Teaching linear algebra in high-school, unpublished doctoral dissertation, BenGurion University of the Negev, Beer-Sheva, Israel.
Harel, G., (1987), ‘Variation in linear algebra content presentations’, For the Learning of
Mahematics, 7, 29–31.
Harel, G., (1989a), ‘Learning and teaching linear algebra: Difficulties and an alternative approach
to visualizing concepts and processes’, Focus onLearning Problems in Mathematics, 11,139–
148.
BIBLIOGRAPHY
267
Harel, G., (1989b). ‘Applying the principle of multiple embodiments in teaching linear algebra:
aspects of familiarity and mode of representation’, School Science and Mathematics, (in press).
Harel, G. & Tall, D. O.. (to appear): ‘The General, The Abstract and the Generic in Advanced
Mathematical Thinking’, For the Learning of Mathematics.
Hart, K. M. (ed), (1981), Children’s Understanding of Mathematics 11–16, John Murray, London.
Heid, K., (1984), Resequencing Skills and Concepts in Applied Calculus through the Use of the
Computer as a Tool, Ph.D. Thesis, Pennsylvania State University.
Henle, J. M. & Kleinberg, E. M., (1979), Infinitesimal Calculus, MIT Press, Cambridge.
Hiebert, J., (1986), Conceptual and Procedural Knowledge, Erlbaum, Hillsdale NJ.
Hilbert, D., (1981), ‘Axiomatisches Denken’, Mathematische Annalen, 78, 405–415.
Hilbert, D., (1964),’ On the infinite’ in P. Benacerraf & H. Putman (Eds.), Philosophy of
Mathematics (34–151), Prentice-Hall, New Jersey (original work published 1923).
Hodgson, B. R., (1987). ‘Symbolic and Numerical Computation: the computer as a tool in
mathematics’, in D. C. Johnson & F. Lovis (Eds.), Informatics and the Teaching of Mathematics,
North-Holland, Amsterdam, 55–60.
Hoffman, K. M., (1989). The Science of Patterns: A Practical Philosophy of Mathematics
Education, Lecture to SIG/RME, AERA Annual Meeting, San Francisco, CA.
Hubbard, J. H. & West, B. H., (1985), ‘Computer Graphics Revolutionize the Teaching of
Differential Equations’, Supporting Papers for the ICMI Symposium, IREM, Université Louis
Pasteur, Strasbourg.
Hubbard, J. H. &West, B. H., (1989), Ordinary differential equations, Springer-Verlag, New York.
Janvier, C., (1978), The interpretation of complex cartesian graphs representing situations – studies
and teaching experiments, (Doctoral dissertation), University of Nottingham, England.
Kaput, J. J., (1982), ‘Differential effects of the symbol systems of arithmetic and geometry on the
interpretation of algebraic symbols’, paper presented at the meeting of the American Educational
Research Association, New York.
Kaput, J. J., (1987a), ‘Towards a theory of symbol use in mathematics’, in C. Janvier (Ed.) Problems
of representation in mathematics learning and problem solving, Erlbaum, Hillsdale NJ.
Kaput,
J.
PME11, Montréal, 1, 345–354.
Kaput, J. J., (1989). ‘Linking representations in the symbol system of algebra’, in C. Kieran & S.
Wagner (Eds.) A research agenda for the teaching and learning of algebra, NCTM, Reston, VA
and Erlbaum. Hillsdale NJ.
Kaput, J. J. (in press), ‘Notations and representations as mediators of constructive processes’, in E.
von Glasersfeld (Ed.), Constructivism in mathematics education, Reidel, Dordrecht.
Kaput, J. J. (in press), ‘Creating cybemetic and psychological ramps from the concrete to the
abstract: Examples from multiplicative structures’, to appear in J. Schwartz, M. West., M. S.
Wiske & D. Perkins (Eds.) Making sense of the future: Technology in mathematics and science
education, Harvard University Press, Cambridge, MA.
Kaput, J. J. (in preparation), ‘Patterns in students: Formalization of quantitative patterns’, to appear
in G. Harel & E. Dubinsky (Eds.), The Concept of Function: Aspects of epistemology and
pedagogy, MAA.
Kaput, J. J., West, M. M., Luke, C. & Pattison-Gordon, L., (1988), ‘Concrete representations for
ratio reasoning’, Proceedings of the Tenth Annual Meeting of the North American Chapter of
the International Group for the Psychology of Mathematics Education, DeKalb, IL: Northern
Illinois University, 93–99.
Karplus, R., (1979), ‘Continuous functions: students’ viewpoints’, European Journal of Science
Education, 1, 397–415.
Kautschitsch, H., (1988). ‘Bild-unterstützte Abstraktion und Verallgemeinerung’, in W. Dörfler
(Ed.),Kognitive Aspekte mathematischer Begriffsentwicklung, Hölder-Pichler-Tempsky,Vienna,
268
BIBLIOGRAPHY
Austria, 191–258.
Keisler, H. J., (1971), Elementary calculus: an approach using infinitesimals, Prindle, Weber &
Schmidt, Boston.
Keisler, H. J., (1976), Elementary Calculus, Prindle, Weber & Schmidt, Boston.
Kitcher, P., (1984), The nature of mathematical knowledge, Oxford University Press, New York.
Kline W., Oesterle R., & Willson, L., (1959), Foundations ofAdvanced Mathematics, American
Book Co., New York, 239.
Kline, M., (1958), ‘The ancients versus the modems: A new battle of the books’, Mathematics
Teacher, 51, 418–27.
Kline, M., (1970), ‘Logic versus pedagogy’, American Mathematics Monthly, 77, 264–282.
Kleiner, I., (1989), ‘Evolution of the function concept: A brief survey’, The College Mathematics
Journal, 20 (4), 282–300.
Koçak, H., (1986), Differential and difference equations through computer experiments, SpringerVerlag, New York.
Kolata, G., (1984). ‘Surprise Proof of an Old Conjecture’, Science, 225, 1006–1007.
Krutetskii, V. A., (1976), The Psychology of Mathematical Abilities in School Children, J. Teller
(transl), J. Kilpatrick & I. Wirszup (Eds.), University of Chicago Press, Chicago.
L’Hospital, Marquis de., (1696), Analyse des infinient petits pour l‘intelligence des lignes courbes,
Paris.
Lakatos, I. M., (1976), Proofs and refutations: The logic of mathematical discovery, Cambridge
University Press, Cambridge.
Lakatos, I. M., (1978), ‘Cauchy and the Continuum: The Significance of Non-Standard Analysis
for the History and Philosophy of Mathematics’, Mathematical Intelligencer, 1 (3), 151–161.
Lander, L. J. & Parkin, T. R., (1967), ‘A counter-example to Euler’s sum of powers conjecture’,
Math. Comp., 21, 101–103.
Lane, K. D., Ollongren, A. & Stoutmyer, D., (1986), ‘Computer Based Symbolic Mathematics for
Discovery’, The Influence of Computers and Informatics on Mathematics and its Teaching, (ed.
Howson A. G. & Kahane J.-P.), Cambridge University Press, Cambridge, 133–146.
Laurent, H., (1899). ‘Considérations sur l’enseignement des mathématiques dans les classes de
Spéciales en France’, L’Enseignement Mathématique, 1, 38–44.
Legrand, M., et al, (1988), ‘Le débat scientifique’, Actes du colloques franco-allemands de
Marseille, 53–66.
Legrand, M., et al, (1986), Introduction du débat scientifique dans un cours de première année de
DEUG A a l’Université de Grenoble I, Rapport de recherche, Editions IMAG, Grenoble.
Legrand, M. & Richard, F., (1984), ‘Mathématiques Expérimentales ou une Approche de la
Conjectureet de la Preuve par l’Étudiant’, Collque “IngenieriePédagogique dansl’Enseignement
Superieur”, Paris.
Lehman, D. R., Lempert, R. O. & Nisbett, R. E., (1988), ‘The effects of graduate training on
reasoning: formal discipline and thinking about everyday-life events’, American Psychologist,
43, 431–442.
Leinhardt, G., Zaslavsky, O. & Stein, M., (1990), “‘Functions, Graphs, and Graphing: Tasks,
Learning, and Teaching”, review of Educational Research’, 60 (1), 1–64.
Leron, U., (1983a), Structuring Mathematical Proofs, The American Mathematical Monthly, 90 (3),
174–184.
Leron, U., (1983b). ‘The Diagonal Method’, Mathematics Teacher, 76 (9), 674–676.
Leron, U., (1985a), ‘Heuristic Presentations: the Role of Structuring’, For the Learning of
Mathematics, 5 (3), 7–13.
Leron, U., (1985b). ‘ A Direct Approach to Indirect Proof’, Educational Studies in Mathematics, 16,
321–325.
BIBLIOGRAPHY
269
Lorenz E., (1963). ‘Deterministic Nonperiodic Flow’, Journal of the Atmospheric Sciences, 20,
448–464.
MacLane, S., (1971), Categories for the working mathematician, Springer-Verlag, New York.
MacLane, S., (1986), ‘Criteria for excellence in mathematics’, Bull. de la Soc. Math. de Belg. Série
A, 38, 301–302.
MacLane, S., (1965). Proceedings of the Preliminary Meeting on College Level Mathmatics
(Katada Report 1964), Japanese International Printer.
Major, R. & Clark, C., (1963), ‘Explorations in students controlled instruction’, in G. Fiesh & H.
Meier (Eds), National Society for Programmed Instruction.
Manin, Y. I., (1977), A course in mathematical logic, Springer-Verlag, New York.
Martin, G. & Wheeler, M. M., (1987), ‘Infinity concepts among pre-service elementary school
teachers’, Proceedings of PME11, Montréal, 3, 362–368.
Maslow, A. H., (1970), Motivation and personality, Harper & Row, New York.
Mason, J. with Burton, L. & Stacey, K., (1982), Thinking Mathematically, Addison-Wesley,
London.
Mason, J., (1989). ‘Mathematical Abstraction as the result of a delicate shift of attention’, For the
Learning of Mathematics, 9 (2), 2–8.
McCloskey, M., (1983), ‘Intuitive physics’, Scientific American, 248, 114–122.
Menis, Y., Snyder, M. & Ben-Kohav, E., (1980), ‘Improving achievement in algebra by means of
the computer’, Educational Technology, 20, 19–22.
Monaghan, J. D., (1986). Adolescents’ Understanding of Limits and Infinity, unpublished Ph.D.
thesis, Warwick University.
Movshovitz-Hadar, N., (1988), ‘Stimulating Presentation of Theorems Followed by Responsive
Proofs’, For the Learning of Mathematics, 8 (2), 12–19.
Muir, A,, (1988), ‘The Psychology of Mathematical Creativity’, Mathematical Intelligencer, 10 (1),
33–37.
Mundy, J., (1984). ‘Analysis of errors of first year calculus students’, in Theory, Research and
Practice in Mathematics Education, A. Bell, B. Low & J. Kilpatrick (Eds.), Proceedings of
ICME 5, Adelaide, Working group reports and collected papers, Shell Centre, Nottingham,
U.K., 170–172.
Munroe, M. E., (1965), Introductory Real Analysis, Addison-Wesley, Atlanta.
Nelson, E., (1977), ‘Internal set theory – a new approach to non-standard analysis’, Bulletin
Americal Mathematical Society, 83, 1165–1198.
Olson, D. R. & Campbell, R., (in press), ‘Representation and Misrepresentation: On the beginning
of symbolization in young children’, in D. Tirosh (Ed.), Implicit and Explicit Knowledge: An
Educational Approach, Ablex, Norwood NJ.
Orton, A., (1980), A cross-sectional study of the understanding of elementary calculus in
adolescents and young adults, unpublished Ph.D. Thesis, Leeds University, U.K.
Orton, A., (1983a), ‘Students’ understanding of integration’, Educational Studies in Mathematics,
14 (1). 1–18.
Orton, A.,, (1983b), ‘Students’ understanding of differentiation’, Educational Studies in Mathematics, 14 (3), 235–250.
Papert, S., (1980). Mindstorm, Basic Books, New York & Harvester Press, Brighton, U.K.
Paulos, J. A., (1988), Innumeracy: Mathematical Illiteracy and its Consequences, Hill & Wang,
New York.
Peitgen, H. & Jürgens, H., (1989). ‘Fraktale: Computerexperimente (ent)zaubem komplexe
Strukturen’, Mathematik-Unterricht, 5, 4–19.
Phillips, E. G., (1931). ‘The teaching of differential’, The Mathematical Gazette, 15, 401–403.
Piaget, J., (1952), The Child’s Conception of Number, Norton, New York (original published 1941).
270
BIBLIOGRAPHY
Piaget, J., (1970a), Genetic Epistemology (E. Duckworth, trans.), Columbia University Press, New
York.
Piaget, J., (1970b), Structuralism (C. Maschler. trans.), Basic Books, New York (original published
1968).
Piaget, J., (1971), Biology and Knowledge (B. Walsh, trans.), University of Chicago Press, Chicago
(original published 1967).
Piaget, J., (1972a), The principles of Genetic Epistemology (W. Mays trans.), Routledge & Kegan
Paul, London (original published 1970).
Piaget, J., (1972b), ‘Comments on Mathematical Education’, in A.J. Howson (Ed.), Developments
in Mathematical Education, Proceedings of the Second International Congress in Education,
Cambridge University Press, Cambridge.
Piaget, J., (1975a), ‘Piaget’s Theory’, in P. B. Neubauer (Ed.), The Process of Child Development,
Jason Aronson, New York, 164–212.
Piaget, J., (1975b). ‘Piaget’s theory’, in P. H. Mussen (Ed.), Carmichael’s Manual of Child
Psychology (Vol. l), John Wiley & Sons, New York.
Piaget, J., (1976), The Grasp of Consciousness (S. Wedgwood, trans.), Harvard University Press,
Cambridge MA (original published 1974).
Piaget, J., (1978), Success and Understanding (A. J. Pomerans, trans.), Harvard University Press,
Cambridge MA (original published 1974).
Piaget, J., (1980), Adaptation and Intelligence (S. Eames, trans.), University of Chicago Press,
Chicago (original published 1974).
Piaget, J., (1985), The Equilibration of Cognitive Structures (T. Brown and K. J. Thampy, trans.),
Harvard University Press, Cambridge MA (original published 1975).
Piaget, J. & Garcia, R., (1983), Psychogenèse et histoire des sciences, Flammarion, Paris.
Piaget, J., Grize, J.-B., Szeminska, A., & Bang, V., (1977), Epistemology and Psychology of
Functions (J. Castellanos & V. Anderson, trans.), Reidel, Dordrecht (original published 1968).
Piaget, J., Inhelder, B., & Szeminska, A., (1960). The Child’s Conception of Geometry (E. A.
Lunzer, trans.), Norton, New York.
Pimm, D., (1984), ‘The role of the individual in mathematics texts’, Proceedings of PME 8, Sydney,
469–77.
Poincaré, H., (1899), ‘La notation différentielle et l’enseignement’, L’Emeignement Mathématique,
1. 106–110.
Poincaré, H., (1913), The Foundations of Science (translated by Halsted G.B.), The Science Press,
New York (page references as in University Press of America edition, 1982).
Poincaré, H., (1963), Mathematics and Science:Last Essays, (J. Bolduc trans.), Dover, New York,
(original published 1913).
Polya, G., (1945), How to Solve It, Princeton University Press, Princeton.
Polya, G., (1954), Mathematicsand Plausible Reasoning, (2 volumes), Princeton University Press,
Princeton.
Polya, G., (1966), ‘On teaching problem solving’, in The Role of Axiomatics and Problem Solving
in Mathematics, Ginn & Co., Washington, D.C., 123–29.
Polya, G., (1980), Mathematical Discovery, (2 volumes), Academic Press, New York.
Ponte, J., (1984). Functional reasoning and the interpretation of Cartesian graphs, (Doctoral
dissertation), University of Georgia, Athens, Georgia
Reding, A. H., (1981), The effects of computer programming on problem solving abilities of fifth
grade students, Doctoral dissertation, University of Wyoming, Dissertation Abstracts International, 42, 3484A, (University Microfilms No. 82–01793).
Riley, M. S., Greeno, J. G. & Heller, J. L., (1983), ‘Development of children’s problem solving
ability in arithmetic’, in H. P. Ginsburg (Ed.), The development of mathematical thinking,
Academic Press, New York, 153–196.
BIBLIOGRAPHY
271
Rival, I., (1987), ‘Picture puzzling: mathematicians are rediscovering the power of pictorial
reasoning’, The Sciences, 41–46.
Robert, A., (1982a), L‘Acquisition de la notion de convergence des suites numériques dans
l’Enseignement Supérieur, Thése de Doctorat d’État, Paris VII.
Robert, A., (1982b). ‘L‘Acquisition de la notion de convergence des suites numériques dans
1’Enseignement Supérieur’, Recherches en Didactique des Mathématiques, 3 (3), 307–341.
Robert, A., (1984), ‘Connaisances des élèves sur les debuts de l’analyse sur à la fin des études
scientifiques secondaire françaises’, Cahier de didactique des mathématiques 18, IREM, Paris
VII.
Robert, A., (1985). ‘Rapports enseignement/apprentissage (débuts de l’analyse sur ) – Analyse
d’une section de DEUG A première année, Cahier de didactique des mathématiques 18–1,
IREM, Paris VII.
Robert, A. & Boschet, F., (1984). ‘L‘acquistion des débuts de l’analyse sur dans un section
ordinaire de DEUG première année’, Cahier de didactique des mathématiques 7, lREM, Paris
VII.
Robert, A. & Tenaud, I., (1988). ‘Une expérience d’enseignement de la géometrie en Terminal C’,
Recherches en Didactique des Mathématiques, 9 (1) 31–70.
Robert, A., Rogalski, J. & Samurçy, R., (1987). ‘Enseigner des méthodes’, Cahier de didactique
des mathématiques 38, IREM, Paris VII.
Roberti, J. V., (1987), ‘The indirect method’, Mathematics Teacher, 80, 41–43.
Robinet, J., (1984), Ingénierie didactique de I‘elementaire au superieur, University of Paris VII.
Robinson, A., (1966), Non-Standard analysis, North Holland, Amsterdam, Holland.
Robitaille, D. F, Sherrill, J. M. & Kaufman, D. M., (1977), ‘The effect of computer utilization on
the achievement and attitudes of grade nine mathematics students’, Journal for Research in
Mathematics Education, 8, 26–32.
Rucker, R., (1982), Infinity and the mind, Birkhauser Boston Inc., Cambridge, Ma.
Russell, B. R., (1956), ‘Mathematics and the Metaphysicians’, in J. R. Newman (Ed.), The World
of Mathematics (Vol. 3, 1576–1592), Simon & Schuster, New York.
Ruthven, K., (1990), ‘The influence of graphic calculator use on translation from graphic to
symbolic forms’, Educational Studies in Mathematics, 21, 5, 431–450.
Sawyer, W. W., (1987), ‘Intuitive Understanding of Mathematical Proof’, Bulletin of the I.MA., 23,
61–62.
Schoenfeld, A. H., (1983), ‘Beyond the Purely Cognitive : Belief Systems, Social Cognitions and
Metacognition as Driving Forces andIntellectual Performance’, Cognitive Science, 7, 329–363.
Schoenfeld, A. H., (1985), Mathematical Problem Solving, Academic Press, Orlando.
Schoenfeld, A. H., (1986), ‘On having and using geometric knowledge’, in J. Hiebert (Ed.),
Conceptual and Procedural Learning, Lawrence Erlbaum, Hillsdale NJ.
Schoenfeld, A. H., Smith, J. P. & Arcavi, A., (to appear), ‘Learning: the microgenetic analysis of
one students evolving understanding of a complex subject matter domain’, in R. Glaser (ed.),
Advances in Instructional Psychology (vo1.4), Erlbaum, Hillsdale NJ.
Schoenfield, A. H., (1989), ‘Explorations of Students’ Mathematical Beliefs and Behavior’,
Journal for Research in Mathematics Education, 20 (4), 338–335.
Schoenfeld, A. H. (in press), ‘On mathematics as sense-making: an informal attack on the
unfortunate divorce of formal and informal mathematics’, in D. Perkins, J. Segal & J. Voss
(Eds.), Informal Reasoning and Education, Lawrence Erlbaum, Hillsdale NJ.
School Mathematics Project, (1967), Advanced Mathematics BookI, C.U.P., Cambridge.
Schwarz, B ., (1989), The use of a microworld to improve ninth grade’ concept image of a function:
The Triple Representation Model curriculum, PhD thesis, Weizmann Institute of Science,
Rehovot, Israel.
272
BIBLIOGRAPHY
Schwarz, B., Dreyfus, T. & Bruckheimer, M., (1990), ‘A model of the function concept in a threefold representation’, Computers and Education, 14 (3), 249–262.
Schwarz,B. & Dreyfus,T., (1991), ‘Assessment of thought processes with mathematical software’,
Proceedings of PME 15, Assisi.
Schwarzenberger, R. L. E. & Tall, D. O., (1978), ‘Conflicts in the learning of real numbers and
limits’, Mathematics Teaching, 82, 44–49.
Selden, J., Mason, A. & Selden, A., (1989), ‘Can Average Calculus Students Solve Non-routine
Problems?’ Journal of Mathematical Behavior, 8 (2), 45–50.
Sfard, A., (1989), ‘Transition from operational to structural conception: The notion of function
revisited’ Proceedings ofPME 13, Paris, 3, 151–158.
Sierpin`ska, A., (1985a). ‘La notion d’obstacle epistémologique dans l’ensignement des
mathematiques’, Proceedings of the 37th CIEAEM Meeting, Leiden.
Sierpin`ska, A , .(1985b), ‘Obstacles epistémologiques relatifs à la notion de limite’, Recherches en
Didactique des Mathématiques, 6 (1), 5–67.
Sierpin`ska, A., (1987), ‘Humanities students and Epistemological Obstacles Related to Limits’,
Educational Studies in Mathematics, 18 (4), 371–87.
Sierpin`ska, A., (1989), ‘How and when attitudes towards mathematics and infinity become
constituted into obstacles in students?’, Proceedings of PME 13, Paris, 3, 166–173.
Simons, F. H., (1986). ‘A course in calculus using a personal computer’, Int. J. Math. Ed. Sci. &
Tech., 17 (5), 549–552.
Sinclaire, H., (1987). ‘Constructivism and the psychology of mathematics’, Plenary Paper,
Proceedings of PME 11, Montréal, 1, 28–41.
Skemp, R. R., (1971), The Psychology of Learning Mathematics, Penguin, London.
Skemp, R. R., (1979), Intelligence, Learning and Action, Wiley, London.
Small, D., Hosack, H. & Lane, K. D., (1986), ‘Computer Algebra Systems in Undergraduate
Instruction’, The College Mathematics Journal, 17 (5). 423–433.
Southwell, B., (1988), ‘Construction and Reconstruction: The Reflective Practice in Mathematics
Education’, Proceedings of PME 12, Veszprem, 584–592.
Stavy, R., Eisen, Y. & Yakobi, D., (1987), ‘How students ages 13–15 understand photosynthesis’,
International Journal of Science Education, 9, 105–1 15.
Stavy, R., (1991). ‘Using analogy to overcome misconceptions about the conservation of matter’,
Journal of Research in Science Education, 28, 305–313.
Steffe, L., (1988), Construction of arithmetical meanings and strategies, Springer-Verlag. New
York.
Steiner, M., (1976), Mathematical Explanation, Mimeographed Notes, Columbia University.
Stolz, O., (1893), Grundzüge der Differential und lntegral Rechnung, vol. 1, (ed. Teubner), Leipzig.
Stoutemyer, D. et al, (1983), MuMath, The Soft Warehouse, Honolulu, Hawaii.
Strauss, S. & Perlmutter, A., (1986), Teaching a property of the arithmetic average via analogies
coded in different symbol systems: A case study in educational–developmental psychology,
unpublished manuscript, Tel-Aviv University, Israel.
Sullivan, K. A., (1976), The teaching of elementary calculus using the non-standard approach,
American Mathematical Monthly, 370–375.
Sweller, J., (1990), ‘On the limited evidence for the effectiveness of teaching general problem
solving strategies’, Journal for Research in Mathematics Education, 21 (5), 411–415.
Tall, D. O.. (1977), ‘Conflicts and catastrophes in the learning of mathematics’, Mathematical
Education for Teaching, 2 (4), 2–18.
Tall, D. O., (1978), Mathematical thinking and the brain, Proceedings of PME 2, Osnabrück, 333–
344.
Tall, D. O., (1979), ‘Cognitive aspects of proof, with special reference to the irrationality of 2,’
Proceedings of PME 3, Warwick, 203–205.
BIBLIOGRAPHY
273
Tall, D. O., (1980a), ‘Looking at graphs through infinitesimal microscopes, windows and telescopes’, Mathematical Gazette, 64, 22–49.
Tall, D. O., (1980b), ‘Intuitive infinitesimals in the calculus’, Abstracts of short communications,
Fourth International Congress on Mathematical Education, Berkeley, page C5.
Tall, D. O., (1980c, ‘The notion of infinite measuring numbers and its relevance in the intuition of
infinity’, Educational Studies in Mathematics, 11, 271–284.
Tall D. O., (1980d), ‘Mathematical intuition, with special reference to limiting processes’,
Proceedings of PME 4, Berkeley, 170–176.
Tall, D. O., (1981), ‘Intuitions of infinity’, Mathematics in School, 10, 30–33.
Tall, D. O., (1986a). Building and Testing a Cognitive Approach to the Calculus using Computer
Graphics, Ph.D. Thesis, Mathematics Education Research Centre, University of Warwick.
Tall, D. O., (1986b), Graphic Calculus I, II, III, (BBC compatible software),Glentop Press, London.
Tall, D. O., (1986c), ‘Lies, Damn Lies and Differential Equations’, Mathematics Teaching, 114, 54–
57.
Tall, D. O., (1986d). ‘Using the computer to represent calculus concepts’, Actes de la 4 iéme École
d’Été de Didactique des Mathématiques et de l’informatique, Orléans, Rapport de recherche,
IMAG Grenoble, 238–264.
Tall D. O., (1987), ‘Constructing the concept image of a tangent’, Proceedings of PME11, Montréal,
3, 69–75.
Tall, D. O., (1989). Real Functions & Graphs: SMP 16–19, (for BBC compatible computers),
Rivendell Software, prior to publication by Cambridge University Press (1991).
Tall, D. O. (in press), ‘The transition to advanced mathematical thinking: functions, limits, infininity
and proof’, to appear in The National Council of Teachers of Mathematics Handbook on
Research in Mathematics Education, Reston, Virginia.
Tall, D. O., Blokland, P. & Kok, D., (1990): A Graphic Approach to the Calculus, (I.B.M.
compatible software), Sunburst, Pleasantville NY.
Tall, D. O. &Thomas,M.O. J., (1989): ‘VersatileLeaming andthecomputer’, FocusonLearning
Problems in Mathematics, 11 (2), 117–125.
Tall, D. O. & Thomas, M. O. J., (1991): ‘Encouraging versatile thinking in algebra using the
computer’, Educational Studies in Mathematics, 22 (2), 125–147.
Tall, D. O. & Vinner, S., (1981). ‘Concept image and concept definition in mathematics with
particular reference to limits and continuity’, Educational Studies in Mathematics, 12 (2) 151–
169.
Tall, D. O. & Winkelmann, B., (1988), ‘ Hidden algorithms in the drawing of discontinuous
functions’, Bulletin of theI.M.A., 24, 111–115.
Thom, R., (1971), ‘Modem mathematics: an educational and philosophical error?’, American
Scientist, 59, 695–699.
Thomas, H. L., (1969), Ananalysis of stages in the attainment of a concept of function, Dissertation
Abstracts, 30A, 4163.
Thomas, M. O. J. & Tall, D. O., (1988), ‘Longer Term Effects of the Use of the Computer in the
Teaching of Algebra’, Proceedings of PME 12, Veszprem, 601–608.
Thomas, M. O. J., (1988), A conceptual approach to the early learning of algebra using a computer,
(Doctoral dissertation), University of Warwick, U.K.
Thompson, P. W. & Dreyfus, T., (1988), ‘Integers and algebra: Parallels in operations of thought’,
Journal for Research in Mathematics Education, 19 (2), 115–133.
Thompson, P. W., (1985a), ‘Experience, problem solving and learnng mathematics: considerations
indeveloping mathematics curricula’, in E. Silver (Ed.), Teaching and Learning Mathemical
Problem Solving, Erlbaum, Hillsdale NJ, 189–236.
Thompson, P. W., (1985b), ‘Computers in research on mathematical problem solving’, In E. Silver
(Ed.) Teaching andLearning MathematicalProblem Solving, Erlbaum, Hillsdale NJ, 417–436.
274
BIBLIOGRAPHY
Thurston, W. P., (1990). ‘Mathematical Education’, Notices of the American Mathematical Society,
37 (7), 844–850.
Tirosh, D., (1985). The intuition of infinity and its relevance for mathematics education, unpublished doctoral dissertation, Tel Aviv University, Israel.
Tymoczko, T., (1986), ‘Making room for mathematicians in the philosophy of mathematics’, The
Mathematical Intelligencer, 8 (3), 44–50.
Ulam, S. M., (1976). Adventures of a mathematician, Scribner, New York.
VanDalen, D. & Monna, A. F., (1972), Sets and lntegration, An outline of the development, WoltersNoordhof, Groningen.
Van Hiele, P. M., (1959). ‘La pensée de l’enfant et la géometrie’, Bulletin de l’Association des
Professeurs Mathématiques de L’ Enseignement Public, 198.
Van Lehn, K., (1980), ‘Bugs are not enough: empirical studies of bugs, impasses, and repairs in
procedural skills’, The Journal of Mathematical Behavior, 3 (2), 3–71.
Vergnaud, G., (1982), Quelques orientations théoriques et méthodiques des recherches françaises
en didactique des mathématiques, Recherches en Didactique des Mathématiques, 2 (2).
Vinner, S., (1982), ‘Conflicts between definitions and intuitions: the case of the tangent’,
Proceedings of PME 6, Antwerp, 24–28.
Vinner, S., (1983), ‘Concept definition, concept image and the notion of function’, International
Journal of Mathematical Education in Science and Technology, 14, 239–305.
Vinner, S., (1988), ‘Visual Considerations in College Calculus – Students and Teachers’, Theory
of Mathematics Education Proceedings of the Third International Conference, Antwerp, 109–
116.
Vinner, S. & Dreyfus T., (1989). ‘Images and Definitions for the Concept of Function’, Journal for
Research in Mathematics Education, 20 (4), 356–366.
Voss, A., (1899, 1916). ‘Differential und Integalrechnung’, Encyclopädie der Mathematischen
Wissenschaften mit einschluss ihrer anwendungen, vol II 1.1,54–134, Leipzig, (ed. Teubner),
French translation by J. Molk: ‘Calcul Différentiel ’, L’ Ency clopédie des Sciences Mathématiques,
vol II–1, 242–297.
Wagner, S., (1981). ‘Conservation of equation and function under transformations of variable’,
Journal for Research in Mathematics Education, 12 (2), 107-118.
Wagner, S., RachlinS . & Jensen, R., (1984), Algebra Learning Project: Final Report, Dept. of Math
Education, University of Georgia, Athens.
Wheeler, M. M. & Martin, G., (1988), ‘Explicit knowledge of infinity’, Proceedings of the 10th
Annual Meeting of the North American Chapter of PME, Northern Illinois University, Dekalb,
Illinois, 312–318.
White, A., (1988). Proceedings of the Second Conference on Humanistic Mathematics; Newsletter
N0.2, (Available from author, Dept. of Mathematics, Harvey Mudd College, Claremont,
California 91711).
Wilder, R., (1967), ‘The role of axiomatics in mathematics’, American Mathematical Monthly, 74,
115–27.
Wille, F., (1984), Humor in der Mathematik, Vandenhoeek und Ruprecht, Göttingen, 84–87.
INDEX
Abel, N. H. 35
abstracting 36–38, 38, 41
relationships with representing 38–39
abstraction 11–12, 36,37, 97, 98, 132, 139,
144,148–151, 217
as concept 11
asprocess 11
empirical 97, 99, 121
generic 12, 13
of properties 129
processes in 34
pseudo-empirical 97, 99
reflective 13, 21, 95–124, 97, 98, 99,
103, 105, 106, 121, 134, 253
using a generic organizer 187
accommodation 9, 103
acquisition of knowledge 132, 133
acquisition of specific concepts 134
Adler, C. 140
advanced mathematical thinking
as a process 26
differences from elementary mathematical
thinking 20, 26, 127, 133
full cycle of 42, 132, 136, 252, 259
Psychology of 3–22
taught as a finished theory 215
advanced mathematical thinking
processes 25–41
affine approximation 173
algebra 220
learning difficulties 144
algebraic permanence, principle of 10
algorithm 5, 43, 61, 104, 137, 163, 193
algebraic differentiation 180
as a replacement for proof 186
premature algebraic use 186
procedures in analysis 186
to solve a problem 125, 131
algorithmisation 197
Alibert, D. 19, 41, 126, 136, 180, 191, 215–
230, 216, 224, 226, 258
analysis 167–198
arithmetization of 168
complex 167
constructive 5
epistemological 11 8
functional 167, 168, 170
mathematical 125, 153
non-standard 6, 172, 187, 196, 197, 197,
202
of several variables 167
real 131
Weierstrassian 162
analysis of knowledge 15
analytic thinking 147
Anton, H. 92
anxiety 148–151,152
APL 242
Appel, K. 16, 233
Appollonius of Perga 174
approximation in reasoning 182–183
arbitrarily small 162
Arcavi, A. 32, 37,1 42, 145
Archimedes’ axiom 256
Aristotle 200
arithmetization of analysis 168
arithmetization of mathematics 146
Artigue, M. 41, 125, 135, 167–198, 178,
180, 193, 198, 258
assessment 130
assimilation 9
generalizing 102
Athens, Georgia 144
Atiyah, M.F. 231
attack phase of problem-solving 18, 19, .20
attainment, variation in 131
Ausubel, D. P. 8
Authier, H. 180
axiom of choice 163
axiomatic method 54
Ayers, T. 82, 83, 103, 104, 117, 118, 242
Bachelard, G.
134, 154, 158
Balacheff, N. 215, 225
ballistics 168
barrier (limit) 155
barrier to advanced mathematical
275
276
INDEX
thinking 129
BASIC 241,242
Bautier, E. 131
Beberman, M. 140
Begle, E. 140
behaviourist psychology 7
Beke, E. 170,171
Ben-Chaim, D. 148
Ben-Gurion University 149
Berkeley, G. 169
Beth, E. W. 82, 95, 97, 99
Bieberbach’s conjecture 233
Biggs J. 8
biological development 100
Birkhoff, G. D. 151
Bishop, A. 225
Bishop, E. 5,172
Blackett, N. 148
blancmange function 188
Boas, R. 152
Bolzano, B. 200,208
Borasi, R. 201
Boschet, F. 131, 132
Bourbaki 16, 54, 98, 140, 141, 149
Boyer, C. B. 160, 168
de Branges, L. 233
Bransford, J. D. 25
Breuer, S. 29
Brousseau, G. 133, 134, 159, 224
Brown, A. 25
Brown, A.L. 141
Bruckheimer, M. 33, 41
Buck, R. C. 141
built-in knowledge generator 255
Bulletin of the American Mathematical
Society 172
Cajori, F. 91, 161
calculus 8, 16,27, 105, 107, 142, 147, 148,
153,160,161,163,165, 220
based on limits 169
history of 168–198
infinitesimal 168
introduction into secondary
education 170
its metaphysical difficulties 161
rigorously based on infinitesimals
supplemented by programming in
BASIC 241
172
using graphic and symbol manipulating
software 237
Calgary, Canada 144
Cambridge Conference 140
Campbell, R. 30
Campione, J. C. 25
Cantor, G. 4–5, 6, 200, 201, 208, 214
Cantor-Bernstein Theorem 230
Cantorian set theory 199, 205, 207, 208, 212
constructing intuitive background 203
cardinality of sets 105
Carter, H. C. 141
Case, R. 141
category theory 98
Cauchy, A. L. 10, 35, 56, 129, 160, 161,
162,168,169
Cauchy sequence 168
celestial mechanics 168
Césaro’s lemma 129
chaos theory 232
Char, B. W. 235
checking 40.41
Cheshire, F. D. 241
chunking complex ideas 88, 252
Cipra, B. 148
Clark, C. 150
Clement, J. 122, 205, 214
Clements, M. A. 39, 146, 253
Cobb, P. 82
codidactic situation 226
cognitive characteristics of students 131
cognitive conflict 134, 206, 236
cognitive development 3, 7–8
cognitive mechanisms in learning 132
cognitive obstacles 9–11, 21, 158–159, 164,
165, 199, 256
cognitive re-construction 9, 114, 136, 159,
164
cognitive root 136
cognitive theory 63
Cohen, L. 141
Collis,K. 8
Commission Internationale
pour l’Enseignement de
Mathématiques 170
pour l’Enseignement des
Mathématiques 171
commutativity of addition 100
comparison of infinite quantities 203
completeness axiom 6, 196
complex analysis 167
INDEX
complex number 5
complexity 151
encountered by students 131, 139
of analysis 163
of function concept 140
comprehension
of complex concepts 83
of object-valued operators 86–88
of point-wise operators 84
compression of ideas 35
computer 126
and the need for finite algorithms 163
as an experimental tool 29, 166, 189
as environment for exploration of
ideas 238–240
aversion displayed by teaching staff 241
didactic advantages in analysis 197
for conceptual development 237
for implementing processes 123
for linking representations 33
for programming 197, 241–48
for providing concrete
representations 104, 187
for visualizing differential
equations 193, 239
for visualizing graphic
representations 193, 232
in advanced mathematical thinking
231–248
in mathematics education 234–235
in mathematics research 231–234
to construct solution of a differential
equation 239
to perform algorithms 236
used in mathematical proof 233
computer algebra system 235
computer generated experiments 232
concept acquisition 65
concept defintion 6–7, 21, 70, 71, 72, 73,
103, 122, 125, 130, 145, 196, 197, 198
in teaching and learning 65–80
of a limit 156
operational deficiency 197
theory and practice 69
concept formation 69
long term processes 71
concept frame 68
concept image 6–7,14, 17, 21, 68 , 69, 70,
71, 72, 73, 76, 78, 83,103, 122,123,
125, 127, 134, 145, 166, 196, 197, 198
277
changing 70
construction compatible with formal
mathematics 187
evoked 68, 73, 83, 144
of a limit 155
in geometry 134
of a function 74
of a function as a graph 146
of a limit 155, 156
of a limit of a sequence 78,164
of a limit of a series 166
of a tangent 75–78, 174–175
of continuity 156–158, 157
of derivative 175, 188
of infinity 156, 199
of rigorous proof 197
theory and practice 69
three illustrations 73
weakness of geometric image of
differential 184
conceptual entities 21, 82, 82–93, 84, 93,
134, 143, 150, 255
and symbolism 88
as aids to focus 88
construction of 83
threeroles 83
conceptual obstacles 133,153, 251
conceptualisation 197
concrete operations 8
concrete representations 38
condensing power of creativity 50
conflict 129, 155
between actual infinity and finite
experiences 201, 205
between concept image and
definition 125, 158
between different student
conceptions 175
between different theoretical
paradigms 203
between differential and derivative
169–171
between infinity in limits and set
theory 125, 203
between limit as a process and its
definition 156
between mathematics and cognition 65
between previous experience and formal
theory 199, 205
between secondary intuitions and
primitive convictions 203
278
INDEX
between spontaneous conceptions and
definitions 158,196
between two conceptions of a differential
185
cognitive 134, 206, 236
concerning limits and infinity 156
in comprehending cardinal infinity 206
in learning continuity 134
in learning limits 134,164
lack of awareness of 180
with infinity 199,204
confusion in first year university 129
conjecture 132,136,191, 224–225, 227,
229, 252, 257, 258
constructivism 224
constructivist psychology 7
continuity 156–158,167
conceptual difficulties 178
continuous function
definition of Cauchy 160
convergence
of sequences and series 129
of series 159
via epsilon-delta methods 129
convincing 20,130
coordination
103,104,106,114,143
of actions 97, 99, 101
of function schema 113
of processes 101, 107, 113, 115, 119
of quantifications 11 6
of schemas 104
Cornu, B. 9, 17, 41,1 03, 122, 125, 134,
153–166, 154, 155, 165, 177, 255, 258
coset 87
counter-example 226
generated by computer 232
Cours d’analyse (Cauchy) 160
Cramer, G., definition of tangent 174
creative activities, absent in students 132
creativity 21, 42–53, 257
a tentative definition 46
characteristics of 49
fallibility 52
ingredients 47
motive power 47
results of 50
stages of development 42
curriculum design 17,165
cybernetic environment 236
D’Alembert, J. L. 160,161,162,169
definition of tangent 174
Dalen, D. See Van Dalen, D.
Dauben, J. 207
Davis,G. 82
Davis, P. J. 44, 56, 57, 59, 146, 148
Davis, R. B. 27, 68, 73, 78, 94, 164
debating forum 56
decapsulation 119
Dedekind cut 168
Dedekind, J. W. R. 200
deep-end principle 15
defining 20, 41
definition 132, 254. See also concept
definition
cognitive situation 67
formal 125
in technical contexts 69
some common assumptions 65
Delens, P. 170
Deligne, P. 220
derivative 85, 107, 167, 176
as a first order approximation 195
as a limit of slopes of secants 188
as affine approximation 173
as gradient of locally straight curve 136,
175
as limit of gradient of secants 165
concept images 175
dy/dx as an indivisible symbol 171
of the second order 170
partial 169
describing 20
development
biological 100
intellectual 100
of concepts 102-103
D’Halluin, C. 189
didactic contract 132, 137
didactic engineering 186,195,197
in teaching differential equations 193–
194
through scientific debate 191
Dienes, Z. P. 15
Dieudonné, J. A. 48.54
differences between elementary and advanced
maths 20, 26, 127, 133
differentiable manifold 136
differential 171
algorithmic calculation 18 1
analyst’s view 6
INDEX
279
and local approximation 181
discrete mathematics 123
and related notions 181
disequilibrium 132
as a component of the tangent
domino stones, in mathematical
vector 239
induction 38
difficulties with symbols and
Donaldson, M. 176
meaning 178
Dörfler,W. 37
in education 170
Douady, R. 43,133,134,135,165, 225
in physics as infinitesimal increase 169
double limit 86
in terms of linear tangent map 168, 173
Dreyfus, T. 11, 23, 25, 25–41, 33, 41, 63,
in terms of tangent linear
103,116,123,128,131,139,142,145,
approximation 180
147,148, 217
its survival in analysis 169
drinking coffee 257
ofLeibniz 169
duality theory 109
student explanations 181
Dubinsky, E. 12, 13, 37, 41, 63, 82, 95–124,
student lack of understanding 181, 184–
104,106,110,126,131,139,143,144,
186
148,162, 166, 197, 231–248, 242, 253,
visualized pictorially through local
254, 258
straightness 239
Duffin, J. 41
differential calculus 160
Duval, R. 201, 203
based on derivative as a limit 171
dynamical systems 232
differential equation 119, 129, 135, 238
algebraic solution 173
existence of solution 239, 258
Education Reform Act (U.K.) 174
higher order 239
Edwards, E. M. 5
qualitative theory of 135, 193
Ehrlich,G. 141
simultaneous 239
Einstein, A. 31
solved symbolically or visually 236
Eisen, Y. 205
to predict the weather 232
Eisenberg, T. 32, 41, 116, 125, 140–152,
with no symbolic solution 239
147, 217, 258
differential formula 170
elaborated notation 88, 91, 93. See also
differentiation 174
notation: elaborated
algorithms 174
elaborated symbol 91, 92
as an algorithm for formulas 85
electronic notebooks 236
differentiation operator 85
elementary mathematical thinking
difficulties 186
differences from advanced mathematical
due to formalization 196
thinking 20, 26, 127, 133
in the beginnings of analysis 196
elevator, in structural proof 222, 223
representational 151–152
Ellerton, N. F. 8
with actual infinity 200
Elterman, F. 106
with cardinal infinity 201
empirical abstraction 97,99, 121
with continuity 178
encapsulation 13, 21, 63, 82, 101, 103,105,
with differentiability 178
106,108,112,115,116,136,143,144,
with graphical representations 178
253, 258
with infinity 161
failure to encapsulate 105
with limits of sequences 178
of a function 108
with symbols 178
of addition 100
with unencapsulated limit concept 165
of getting small as an infinitesimal 162
discontinuity between elementary and
of implication process 113
advanced math 125
of the function process 143
discovering 40, 41
Encyclopedie Methodique 169
discovery 231
280
INDEX
endomorphism 119
enhanced Socratic mode 187
Enseignement Mathématique 170
entification 92, 93
entry phase of problem-solving 18, 20
environments for learning 133
epistemological analysis 11 8
epistemological obstacles 134, 258
and didactical transmission 163–164
important characteristics 158
in historical development 159–162
in history and education 158
in modem mathematics 162–163
in the limit concept 162, 165
using a computer 166
epistemology 225
epistomological problems 141
ergodic theory 149
error
arbitrary 176
executive 176
structural 176
Ervynck, G. 23, 42–53
Euclid 222, 236
euclidean ring 97
Euclid’s Elements 159
Eudoxus of Cnidos 159
Euler, L. 56, 160, 161, 232
definition of tangent 174
exhaustion 159–162,168
expansion of cognitive structure 9
explaining 220
explanation 135
fallibility of mathematics 56
familiarisation 135
Fehr, H. 141,149
Fermat, P. 59
definition of tangent 174
Ferrara, R. A. 25
field 84
Fields Medal 220
final term of a sequence 164
finite and infinite sets learning unit 206,209
first order propositions 111
Fischbein, E. 14, 201, 202, 203, 205, 214,
216, 219
fitness of creativity 49
fluid levels 100
fluxion 169
focus of attention 83, 93
Fodor, J. A. 67, 68,80
formal operations 8
formal proof 136
formalism 55, 56,61, 146
formalist school 54
formalization 129
gradual initiation 198
Fortran 242
foundations of mathematics 54
four colour theorem 234
Fourier, J. 35
Frankel, A. A. 208
Fréhet, M. 168
Freudenthal, H. 145,149,151
full cycle of advanced mathematical
thinking 42, 132, 136, 252, 259
function 68, 70, 73, 98, 106, 109, 125, 167
and its graph 147
as a black input-output box 141
as a conceptual entity 82, 85, 86, 92
as a formula 32, 74, 104, 116
as a graph 118
as aprocess 12, 31, 82, 108
as a rule 74
as a set of ordered pairs 118, 141
its pedagogical weakness 141
as a table of values 141
as an algebraic formula 141
as an algebraic term 74
as an algorithm 104
as an arithmetical manipulation 74
as an arrow diagram 141
as an equation 74
as an object 87, 108, 118
as data 104
as process and object 119, 143
as uniform operator 92
as unifying factor in school
mathematics 140
composite
as a concept 84
as a process 84.92
of two functions 104
through substitution 117
concept image 74
considered graphically 148
continuous 107, 160
properties of 163
continuous linear 108
INDEX
determined by numerical data 90
different representations 32
differentiable 188, 238
geometric conception 146
historical background 140
inverse as graph 147
its complexity 140
limited to one representation 33
linear 90, 107
logical versus algebraic 146
multiple embodiment 141
non-differentiable 188, 238
one-to-one 117
onto 117
parametric 87
proposition valued 112, 113, 114, 11 6,
120
schema 114,117
space 82
through interiorization of actions 117
visual versus analytic 146
function schema 108,114
functional 84
functional analysis 167,168,170
functions and learning difficulties 140–152
functor 82
fundamental theorem of calculus 105, 107
Gagné, R. M. 142,144
Gagnéan hierarchy 144
Gal-Ezer, J. 29
Galileo, G. 199, 200
Galois, E. 48
Galois theory 149
Garcia, R. 97,120
Gauss, C. F. 200
Gautschi, W. 233
Gazit, A. 205
Gazzigna, M. S. 13
general coordinations 97, 99, 101
general triangle 51
generalization 11–12, 36, 37, 97, 103, 105,
106, 143
asconcept 11
as process 11
constructive 97
disjunctive 12
expansive 12, 48
extensional 102
281
generic 12
in mathematical creativity 48
of a schema 101
of function schema 113
reconstructive 12, 48
generalizing 34, 35, 138
generic abstraction 13
generic example 122, 217
generic extension principle 10
generic limit 10, 162, 164
generic organizer 187-190
for differential equations 188
for gradient (derived function) 187
for integration 188
generic proof 19, 216, 229
generic tangent 76, 78
genetic decomposition 96, 102, 106, 110,
123,143,144,166
of dual vector space 108
of function 110, 116–120
of predicate calculus 114–1 16
of three schema 109–1 19
geometry 138
Gleick, J. 232
Gödel, K. 5,98, 162
Gödel’s incompleteness theorem 98, 162
Goldbach’s conjecture 233
Goldin, G. 90
Gong, C. 106
Graphic Calculus 187, 238
graphs, interpretation of 147
Gray, E. M. 255
Greek mathematics 159
Greeno, G. J. 82, 83, 85, 89, 118
Grenier, D. 216
Grenoble 19, 136, 191, 192, 195, 216, 224,
225
Grogono, P. 242
group 87,106
as formal structure 129
Hadamard, J. 3, 14, 14–15, 29, 31, 42, 146
Hadar-Moscovitz, N. 147. See also
Movshovitz-Hadar
Hahn, H. 200,208
Haifa University, Israel 216
Haken, W. 16,233
Halmos, P. R. 137
Hamilton, W. 11
282
INDEX
Hammersley, J. M. 141
Hanna, G. 23, 54–61, 59, 162, 254
Hardy, T. 19
Harel, G. 12, 31, 37, 63, 82, 82–93, 87, 91,
92,142, 255
Hart,K.M. 8
Hausdorff, F. 43, 89
Heid, K. 237, 238
Heller, J. L. 118
helplessness 152
Henle, J.M. 172
Hermite, C. 13, 253
Hersh, R. 44,146, 149
Hess, P. 201, 203
heuristics 132,137,220
hierarchy of concepts 256
Hilbert, D. 5, 146, 149, 162, 200, 208
Hippocrates of Chios 159
historical texts 138
history of analysis 168
Hodgson, B. R. 235
Hoffman, K. M. 29
homotopy group 106
horizontal growth of knowledge 83
Hubbard, J. H. 193, 239
humanistic mathematics movement 149
hyperreal numbers 163
illumination 50
Inbar, S. 147
inconsistencies
raising students’ awareness of 206–207
inconsistencies in comparing infinite
quantities 204
induction, mathematical 38
infinitely large and infinitely small 160–161
infinitely small 160,168
infinitesimal 6, 160, 161, 198, 199
as a carrier of paradoxes 169
as an abreviation of an expression 169
as ‘banned’ by Weierstrass 168
as defined by Cauchy 160
decline in face of the limit notion 169
in minds of mathematicians 162
in non-standard analysis 162, 168
metaphysical haze 171
of Leibniz 161
infinitist 156
infinity
110, 125, 156, 196, 199–214
actual 199, 200
accepted by Galileo etc 200
rejected by Aristotle 200
rejected by Poincaré 200
student experiences 202
student understanding 209–213
cardinal 199, 201, 203
comparing infinite quantities 199,
203, 203–205
comparison between two infinite
sets 199
measuring 202,203
non-standard 199
ordinal 199
potential 199, 200, 202
student conceptions 201-205
student difficulties 161
teaching the Cantorian theory 199
theoretical conceptions 200–201
INRC group 102
institutionalization 135
instrumental understanding 48
integral 85, 151, 167, 176
as a continuous linear form 175
as a function 92
as a process of measure 175, 190
as an inverse to differentiation 191
as area under a curve 191
as encapsulation and
interiorization 105
Riemann 227
student conceptions 175
integration 107, 147, 173, 174
algorithms 174
in terms of the primitive 173
in terms of the Riemann sum 173
integration operation 82
intellectual development 100
intelligent behaviour 131
interionzation 103, 104, 106, 113, 143
of a statement 115
of actions 100, 101, 107, 111, 113,
117
intermediate value theorem 163, 257
International Commission
for Mathematical Instruction 170
intuiting 40, 41
intuition 13–14, 40, 125, 132, 154
criteria for comparing infinite
quantities 203–205
developed with computer
graphics 232
INDEX
in mathematical creativity 47
in research using the computer 231
of actual infinity 201
of infinity 199–214
origins in student experience 207
via experiences of comparative
size 202
primary 14
effects on thinking processes 207
secondary 14, 203, 205
intuitionism 55
intuitionist school 54
irrationality of 2 217
ISETL 242, 244–248
isomorphism 119
between vector spaces 82
iterated limit 86
iteration 114
Jacobian matrix 173,180,186
Janvier, C. 147
jargon 109
Jeeves, J. 15
Jensen, R. 144
Jordan, M. 48
Jordan’s theorem
Jurgens, H. 29
Jurin, J. 161
146
Kane, M. J. 141
Kaput, J. J. 31, 39, 41, 63, 82–93, 87, 255
Karplus, R. 147
Kautschitsch, H. 38
Keisler, H. J. 172, 238
Kepler, J. 10
King 151
Kitcher, P. 56
Klein, F. 14,17
Kleinberg, E. M. 172
Kleiner, I. 146
Kline, M. 141
Kline,W. 141
Koçak, H. 193, 232
Kolata, G. 233
Kronecker, L. 4, 5
Krutetskii, V. A. 146
283
Laborde, C. 225
Lagrange, J. L. 161,163
analysis without limits or
infinitesimals 169
Lakatos, I. 35,56
Lander, L. J. 232
Lane, K.D. 235
Laurent, H. 171
learning difficulties in algebra 144
learning theories, and their
deficiencies 142–144
Legrand, M. 19, 137, 191, 216
Lehman, D. R. 141
Leibniz, G. W. 10, 91,161,168,169
definition of tangent 174
notation for calculus 170
Leinhardt, G. 148
Lempert, R. O. 141
Leron, U. 19, 41, 216, 220 ,221, 223
levels of understanding 143
Lewin, P. 82,143
L’Hospital, Marquis de 168
limit 106, 125,134, 154
absence in Greek mathematics 160
as a barrier 155
as a concept 156
as a formal definition 153
as a procept 255
as a process 156
as being impassible 154
as the basis for calculus 168
colloquial meaning 154
conceptual difficulty 178, 188
conflicts in 134, 176
defined as unencapsulated process 163
double 86
dynamic 155
generic 10, 162, 164
given by operations 153
in epsilon-delta terms 162
in integration 105
in the derivative concept 17
is it attained? 161–162
iterated 86
metaphysical aspect 161
mixed 155
monotonic 155
obstacles in history 159
of a function 84, 85, 167
of a sequence 73, 78, 167, 177
of a series 166
284
INDEX
of a staircase 163
of polygons as a circle 159
of secants 76
static 155
stationary 155
limits 153–166
linear function
as conceptual entity 90
linear functional 109, 150
linear transformation 150
LISP 242
local approximation 170
local straightness 136, 187, 238
logic schema 111,112, 113
logical thinking 103
logicism 55,56
logicist school 54
logico-mathematical thinking 100
Logo 242
Lorenz, E. 232
MacLane, S. 50, 82, 141
Major, R. 150
Manin, Y. 59
Maple 235, 236
Martin, G. 201,203
Maslow, A. H. 58
masochism 232
Mason, A. 27, 32, 146
Mason, J. 18, 20, 25, 37, 41,138
Mathematica 28, 236, 242
mathematical content 128
Mathematical Gazette 171
mathematical induction 102, 106, 109, 110–
113, 113, 120, 123, 139. See also proof:
by induction
as aprocess 110
encapsulated as an object 110
mathematical minds 4,63
mathematical phobics 148
mathematical practice 56
Mathematical Reviews 59
mathematical theory
structure of 46
matrix 82, 84, 91, 104
McCloskey, M. 205
mean value theorem 218
meaning of calculus concepts 237
measure theory 167
Melamed, U. 201, 214
Menis, Y. 241
mental reconstruction 9
meta-mathematical instruction 138
meta-mathematical knowledge 131,138
meta-mathematical reflection 136, 137, 138
meta-mathematics 138
metaphysical aspect of limit 161
metaphysics of infinity 169
method of exhaustion 168
metric for aesthetics 151
misconceptions 27, 236
about “getting close”, “growing large”,
etc 156
in learning about limits 164
of a function 74
of a limit of a sequence 79
of a tangent 76
of a variable 145
modelling 34
models of learning 142
modus ponens 113
monadic number 221
Moore, R. L. 137
Movshovitz-Hadar, N. 216, 217, 218
Muir, A. 42
multiple embodiment 141
(picture, graph, formula) 189
multiplication
as addition of additions 100
Mundy, J. 147
Munroe, M. E. 86
mutation, mathematical 49
Nelson, E. 172
Nering 92
Newton, I. 56, 160, 168, 169
Nicholas of Cusa 10
Nisbett, R. E. 141
non-standard analysis 6, 172, 187, 196, 197,
198, 202
its weak impact on education 172–198
Non-Standard Analysis (Robinson) 172
notation 88, 148–151, 152
as substitute for a concept 88, 93
elaborated 88, 91, 93
f(x) for a function 149
graphical 90
in forming conceptual entities 88, 89–
INDEX
91, 93
tacit 88, 93
to encapsulate entities 90
to name a concept 89
notation system 89
number 100,106
as concept 253
as process 253
cardinal 200, 202
cardinal number of set of reals
hyperreal 202
infinite measuring 202
monadic 221
ordinal 200
triadic 221
numerical analysis 241
numerical methods
used to solve problems 131
212
object 102,106
object-pivot in structural proof 221
object-valued operator 83, 87,93
obstacle 9–11,166,175
cognitive 11, 21, 165
conceptual 133, 153
didactical 158
epistemological 103, 134, 158, 162
didactically transmitted 163–164
important characteristics 158
inability to overcome 165
genetic 158
in learning calculus/analysis 196
to construction of formal concepts 195
Oesterle, R. 141
Olson, D. R. 30
operator
object-valued 83, 87, 93
point-wise 83, 85, 87, 93, 109
uniform 83, 86, 93
organization of mathematical content 133,
134–136
organizing agent 187
Orton, A. 8, 163, 176, 177
ownership of the mathematical concepts
Oxford Dictionary 253
Papert, S. 8, 9
parameter 90
137
285
parametric equation 87
parametric function 87
Parkin, T. R. 232
Pascal 242
Paulos, J. A. 148
Peacock, G. 10
pedantry 61
Peitgen, H. 29
Perlmutter, A. 214
Phillips, E. G. 171
Piaget J. 7, 8, 9, 63, 82, 95, 96, 102, 103,
110, 119, 120, 132, 133, 143, 206, 253
notion of reflective abstraction 97–102
theory of conservation of variable 145
pivot (in a structural proof) 221
Poincaré, H. 4, 13, 14, 15, 16, 29, 42, 48,
63, 170, 171, 200
point-wise dependence 85
point-wise operator 83, 85, 87,93, 109
poison 151
Poisson, D. 189
Polya, G. 18, 132, 137, 151
Ponte,J. 147
potential conflicting factors 155
potentialist 156
practical tangent 188,190
pre-operational stage 8
predicate calculus 106, 109, 114–116, 121,
123
predicate calculus schema 114
primary intuitions 14
principle of continuity 10
principle of selective construction of
knowledge 258
probability theory 149
problem-solving 18, 32, 93, 132, 137, 150,
165, 257
tactics 151
procept 254, 258
proceptual divide 255
proceptual knowledge 255
process 102
process of representing 30
process-concept duality 134, 255
processing load 83
programming 197 , 231, 241–248
as a generic process 12, 144
choice of language 242
in BASIC 241
in ISETL 242, 244–248
in Mathematica 242
286
INDEX
the function concept 143
to teach algebra 241
PROLOG 242
proof 126, 130, 132
acceptance of 58, 162
analytic formalization 146
and uncertainty 225
as a contractual agreement 195
as a mathematical activity 216
as a necessary mathematical activity 215
as a stylistic exercise 130
as mechanistic deduction from
axioms 234
avoidance by using algebraic
algorithms 186
by computer checking 233
by contradiction 5, 130, 163, 217
by induction 102, 113, 120, 130
careful reasoning 60
cognitive aspects 19
direct methods for equivalence of infinite
sets 210
exposition by the structural
method 2 19–221
formal 54, 57, 125, 136, 217, 231
its origins 55
its validity 55
general 217
generic 19, 216, 229
in linear style 215, 221–222
in structural style 222–224
indirect method for non-equivalence 210
interiorization of steps 104
mathematical 16, 21, 54–61
of Jordan’s Theorem 146
research on learning 215–230
schema 120
standards of rigour 56,129
structural 220, 229
student understanding 216–219
the social process 59
to convince 20, 226
to show 226
using a generic example 217
proof and refutation 56
proof debate
organisation 228–229
propositional calculus schema 114
proving 20, 41, 220
pseudo-empirical abstraction 97, 99
psychological barriers 152
psychological characteristics of students 131
Psychology
behaviourist 7, 142
constructivist 7
psychology of advanced mathematical
thinking 3–21
Psychology of Invention in the Mathematical
Field 3
Psychology of Learning Mathematics 16
Ptolemy 236
F’urdue University 233
quantification 115
existential 116
higher-level 116
single-level 115, 116
three-level 116
two-level 115, 116
universal 116
quantifier
115,153,197,198
quaternions 11
quotient object 87
Rachlin, S. 144
re-equilibrium 132
reconstruction 114,136
of cognitive structure 9, 159, 164
of function schema 119
of knowledge 236, 252
through reflection on conflict 199
Reding, A. H. 241
recursion 286
reflection 25, 61, 252, 257
meta-mathematical 131, 137, 138
on a function process 117
oninfinity 125
Reflections 251–258
reflective abstraction 95–124, 97, 98, 99,
103, 105, 106, 121, 123, 134, 166, 253
as construction 99,101
in children’s thinking 100
its nature 99
reification 82
relational understanding 48, 49
representation 30, 225
computer-implemented 39
concrete 31, 38, 39
mathematical 34
INDEX
mental 31, 32
switching 32
symbolic 31
visual 39
representational difficulties 151–152
representing 38,41
relationships with abstracting 38
research in teaching and learning 127–139
reversal 103, 105, 106, 143
of a process 102, 118
review phase of problem-solving 18, 19, 20
Richard, F. 216
Riemann, G. F. B. 4
Riemann integral 227, 237
rigour 13–14, 182–183, 197
Riley, M. S. 118
Rival, I. 148
Robert, A. 125, 127–139, 131, 132, 134,
137, 138, 155, 164, 165, 180, 195, 257
Roberti, J. V. 210
Robinet, J. 129
Robins, B. 161
Robinson, A. 163, 168, 172, 208
Robitaille, D. F. 241
Rolle’s theorem 218
Royal Road to Geometry 236
Rucker, R. 207
Russell, B. 200, 208, 214
Ruthven, K. 29
Sandleson, R. 141
Sawyer, W. W. 16–17
schema 102,106, 109, 143
organization of 106–110
Schoenfeld, A. H. 18,25, 32, 37, 132, 133,
137, 142, 143, 144, 145, 147,151,195
School Mathematics Project 171, 175
Schwarz, B. 33, 41
Schwarzenberger, R. L. E. 10, 103, 125,
127–139, 134, 154, 164, 257
Schwingendorf, K. 242
science students 73
scientific debate 136, 191–193, 195, 216,
224–225, 225
an example in analysis 224–227
evaluating its role 229
generating 225–226
secondary intuition 14, 203, 205
see-saw 100
Selden, A. 27, 32, 146
287
Selden, J. 27, 32, 146
selective nature of creativity 49
semigroup 119
sensori-motor stage 8
sequence 70
‘final term’ 164
formal definition 155
limitof 78–79
student conceptions 134
series 161, 166
set theory 98, 199, 203, 205, 207, 208, 212
Severi, F. 14
Sfard, A. 82
Sierpinska, A. 103,156,177, 201, 203
Simons,F.H. 241
Sinclaire, H. 140, 144
sixth form 131
Skemp, R. R. 3, 9, 16, 48, 49, 131
slope of straight lines 90
Small, D. 235
small group work 138
Smith, J. P. 32, 37, 142
smorgasbord 142
social factors 128
socio-cognitive conflict 132
Socratic mode, enhanced 187
Solution Sketcher 239
soul of departed quantities 160
Southwell, B. 25
specializing 138
Sperry, R. W. 13
spontaneous conceptions 154–158, 164, 166
stagetheory 8
staircase 163
stationary (limit) 155
Stavy, R. 205, 214
steffet, L. 82
Stein, M. 148
Steiner, M. 216
Stolz, O. 168
Stone-Czech compactification 89
Strauss, S. 214
structural proof 219–221, 229
elevator 222, 223
student conceptions
of derivative 175
of derivative, integral, tangent 174–175
of differential 180
of differentiation 180
of infinity 201–205
of integral 175
288
INDEX
of integration 180
student conceptions of sequences 134
study of historical texts 138
sub-object 87
subspace 88
Sullivan, K. A. 172
Sweller, J. 141
Sylow Theorem 230
symbols 82–93
and structure of mathematical objects 91
conventions 89
elaborated 92
nominal role 89, 92
tacit 92
symbol manipulator 231, 235–236, 236, 258
as a tool 236
symbol pushing 30, 61
symbolic representation 3 1
symbolism 61
used for process and concept 253
used to solve problems formally 131
synthesis 37
synthesis of knowledge 15
synthesizing 34, 35
Szwed, T. 178
tacit notation 88, 93
tacit symbol 92
Tall, D. O. 3–21, 6, 7, 10, 12, 17, 18 , 21, 33,
37, 41, 68, 73, 78, 103, 121, 122, 126,
128, 134, 136, 139, 145, 147, 148, 154,
155, 157, 162, 165, 166, 175, 178, 187,
188, 189, 190, 193, 197, 198, 201, 202,
203, 207, 208, 216, 217, 219, 231–248,
238, 239, 240, 241, 251–258
Tall, R. 153
tangent 73,168
as a close approximation to the
curve 239
as limit of secants 76,188
concept image 75–78, 174–175
generic 76,78
practical 188, 190
student drawings 77
tangent linear functional 170
task-sequencing 140
teacher, effect of 131
teaching
consequences of . . .
complex conceptions 165
concept images 79
creativity 52
practice of proof 60
reflective abstraction 119
in analysis 186
in proof debates 228
inadequacy of traditional practices 120
integration through scientific debate 191
of the limit concept 165
teaching calculus/analysis
in England 174
in France 173
Tenaud, I. 138
tends to, meanings of the word 154
Tennessee Technological University 27
Terence Tao 39, 253
theoretical fantasy 140
Thinking Mathematically 20
Thom, R. 141
Thomas, H. L. 147
Thomas, M. O. J. 41, 126, 148, 215–230,
219, 241, 258
Thompson, P. W. 37, 41, 82, 123
Thurston, W. P. 35
Tirosh, D. 40, 41, 125, 199, 201, 203, 214,
258
tool-object dialectic 165
tool-pivot in structural proof 221
topological dual 109
topological space 106
topological vector space 109
toplogy 107, 108
trajectory 100
transfer 135, 141
transfer of learning
its absence in algebra 145
transfinite numbers 200, 205
transition
from elementary to advanced
thinking 20, 129, 165, 199
from one mental state to another 9
from school to university 125
translating between representations 32, 33
triadic number 22 1
Tymoczko, T. 56, 57, 60
Ulam, S. M. 49.59
ultimate magnitude 161
INDEX
289
ultimate ratio 160, 161
understanding
in mathematical creativity 47
instrumental 48
intuitive 146
lack of 145
relational 48, 49
visual 146
uniform operator 83, 85, 86, 87, 93
White, A. 149
Wiener,N. 141
Wilder, R. 141
Wille, F. 50
Willson, L. 141
Winkelmann, B. 166
wizard 151
working memory 83
working memory load 84, 93
validation 227
Van Dalen, D. 49
Van Hiele, P. M. 253
Van Lehn, K. 122
variable 106
variables 144–145
vector space 11–12, 36, 82, 83, 84, 88, 105,
106,107,108
as formalized in 19th century 129
double dual 82, 84, 89, 150
dual 105, 106, 108, 150
of infinite dimensional space 107
infinite dimensional 129
interiorization of dual 109
topological 107
vector space schema 109
versatile learner 219
versatile learning 132
versatile thinkiig 253
versatility in solving problems 131
vertical growth of knowledge 83, 93
Viennot, L. 180
Vinner, S. 6, 7, 17, 32, 41, 63, 65–80, 68,
69, 73, 78, 103, 116, 134, 142, 145, 146,
155, 157, 164, 218, 254, 256
visual thinking 146,147
visualization 31, 63, 148, 152, 197, 252
in abstraction process 38
of area curve 190
of slope of a graph 190, 195
to complement symbolism 240
Voss, A. 170
Vygotsky, L. S. 132, 133
Yakobi,D.
Wagner, S. 144,145
Weierstrass, K. T. W. 4, 56, 168
West, B. H. 193, 239
Wheeler, M. M. 201, 203
205
Zaslavsky, 0. 147, 148
Zwas, G. 29
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Mathematics Education Library
Managing Editor: A.J. Bishop, Cambridge, U.K.
1. H. Freudenthal: Didactical Phenomenology of Mathematical Structures. 1983
ISBN 90-277-1535-1; Pb 90-277-2261-7
2. B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives on Mathematics Education. Papers submitted by Members of the Bacomet Group.
1986.
ISBN 90-277-1929-2; Pb 90-277-21 18-1
A.
Treffers:
Three
Dimensions.
A
Model
of Goal and Theory Description in
3.
Mathematics Instruction – The Wiskobas Project. 1987 ISBN 90-277-2 165-3
4. S. Mellin-Olsen: The Politics of Mathematics Education. 1987
ISBN 90-277-2350-8
5. E. Fischbein: Intuition in Science and Mathematics. An Educational Approach. 1987
ISBN 90-277-2506-3
6. A.J. Bishop: Mathematical Enculturation. A Cultural Perspective on
Mathematics Education. 1988
ISBN 90-277-2646-9; Pb (1991) 0-7923-1270-8
7. E. von Glasersfeld (ed.): Radical Constructivism in Mathematics Education.
1991
ISBN 0-7923-1257-0
8. L. Streefland: Fractions in Realistic Mathematics Education. A Paradigm of
Developmental Research. 1991
ISBN 0-7923-1282-1
9. H. Freudenthal: Revisiting Mathematics Education. China Lectures. 1991
ISBN 0-7923-1299-6
10. A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.): Mathematical
Knowledge: Its Growth Through Teaching. 1991
ISBN 0-7923-1344-5
11. D. Tall (ed.): Advanced Mathemutical Thinking. 1991
ISBN 0-7923-1456-5
12. R. Kapadia and M. Borovcnik (eds.): Chance Encounters: Probability in
Education. 1991
ISBN 0-7923-1474-3
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