A DVD Math
A DVD Math
A DVD Math
Patrick W. Thompson
San Diego State University
Center for Research in Mathematics and Science Education
education based on artifacts from the educational research community, she might conclude that, as
All joking aside, it is fair to say that the preponderance of mathematics education research
has been on elementary mathematics, and this research is often carried out with such a tight focus
that it is hard to imagine by what mechanisms one level of concept development can possibly be
transformed into more sophisticated levels. There might be several reasons for this, but I suspect a
primary one is that much of mathematics education research has not been conducted within a larger
perspective of mathematical thinking—a perspective that keeps firmly in mind that children, over
time and when taught appropriately, often do learn sophisticated and advanced mathematics. For
example, models of children’s competent additive reasoning will tend to be quite different when the
researcher focuses only on their ability to solve simple addition and subtraction problems than
when she keeps clearly in mind that the knowledge structures presently being imputed to children
must provide a foundation for their conceptualization of integers (Steffe, Cobb, & von Glasersfeld,
1988; Thompson, in press; Thompson & Dreyfus, 1988; Vergnaud, 1982). Similarly, models of
children’s multiplicative reasoning will tend to be quite different when the researcher focuses only
on their ability to solve simple multiplication and division problems than when she keeps clearly in
mind that the knowledge structures presently being imputed to children might provide a foundation
for children’s conceptualization of multiplicative variation—viz., direct and inverse variation, linear
function, exponential function, and concomitant rates of change (Harel & Confrey, in press).
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Until recently, mathematics education researchers had to rely mainly on their own
concepts within a larger curricular and conceptual perspective. The bits of research on advanced
mathematical reasoning and understanding were scattered and not widely accessible. On the other
hand, while mathematics education researchers have focused largely on the development of
elementary concepts, university mathematicians have seen little relevance of this research to their
practice or to their teaching. With the publication of Advanced Mathematical Thinking this situation
is changed considerably. This book can help mathematics educators place their research within a
broader perspective of the development of mathematical reasoning, and it can help mathematicians
see the deep connections between mathematics education research and their own teaching.
Advanced Mathematical Thinking is the product of five years of collaboration among the
authors, under the auspices of the Advanced Mathematical Thinking Working Group of the
International Group for the Psychology of Mathematics (PME). The authors met each year as part
of the PME’s annual meetings and communicated among themselves between meetings.
David Tall introduces the book with an overview and closes the book with reflections on its
contents. Tall’s introduction, besides giving an overview, sets the tone for the book with a
masterful weaving of current cognitive, pedagogical, and curricular issues with a historical account
of tensions within the community between mathematical practice and teaching. I especially enjoyed
his quotes from famous mathematicians, such as Poincaré, which served as a reminder that
thoughtful reflection on learning and teaching are not unique to mathematics education. In his
The book’s structure is nice. The first part portrays that which needs
at the same time making it evident that it can be a natural outgrowth of issues entailed in elementary
mathematics. The second part provides a more technical focus on the elements of advanced
mathematical reasoning, making specific what was portrayed more generally in the first part. The
third part focuses on the learning and teaching of important conceptual domains.
The chapters on the nature of advanced mathematical thinking are by Tommy Dreyfus,
Gontran Ervynck, and Gila Hanna. Dreyfus’ ostensible task is to explicate the dialectic between
representing and translating as advanced mathematical processes. But he does much more. He also
illustrates that this dialectic is present when students reason competently in elementary
mathematics, and argues convincingly that teachers must cultivate this dialectic early on and
consistently for students to progress to advanced levels. Dreyfus also does an excellent job of
making it evident that many university students often “succeed” in advanced courses without these
understanding. Dreyfus’ observations are completely consistent with an emerging research trend
that shows correct performance cannot be taken as an indicator of understanding (Seldon, Mason,
& Seldon, 1989). While Dreyfus focuses on reasoning processes, Ervynck attempts to elucidate
the leaps of insight that we swear happen but have little control over—either personally or
creativity than does Ervynck’s, Ervynck assumes the enormous task of exploring how we might
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shape instruction to engender students’ creativity, a task that was completely aside Hadamard’s
objective. The section closes with Hanna’s brief discussion of the necessity to consider social
historically.
The chapters on cognitive theory are by Shlomo Vinner, Guershon Harel and Jim Kaput,
and Ed Dubinsky. Vinner expands his and David Tall’s distinction between concept definitions and
concept images (Tall & Vinner, 1981; Vinner & Dreyfus, 1989) to provide insight into college
students’ understandings of important ideas like function and derivative. It is important to note that
the distinction between concept definition and concept image does not explicate competent
understanding of any specific concept. Rather its importance is that it enables us to make
constructive sense of students’ frequent inability to reason coherently from the basis of a
technically-defined vocabulary and gives hints as to pedagogical and curricular directions that
might support students’ development of these abilities. Harel and Kaput’s chapter, on conceptual
mathematical objects (Dubinsky & Lewin, 1986; Greeno, 1983; Harel, 1989; Sfard, 1991; Sfard &
Linchevski, in press; Thompson, 1985). Their chapter is a welcome synthesis of this line of theory
development, presented in the format of case studies of specific concepts. It also extends prior
work on mathematical objects by explicating the important role of symbolization in the entification
process. While Cajori’s (1929) analysis of mathematical notations is crucial to understanding the
direction. They highlight the importance of a dialectic among students’ creating mathematical
objects, their use of notation to express their reasoning, and notational characteristics that can
facilitate or obstruct students’ achievements in both regards. For example, the derivative of a
df(x)
function f might be expressed as dx or Dx(f), but the two often do not coincide in terms of what
mathematicians have in mind when using them. Attending to students’ internalization of notations
as a vehicle for expression is quite different from the community’s adoption of customary and
conventional notational systems (Thompson, 1992). Ed Dubinsky closes this section with an
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excellent discussion of Piaget’s notion of reflective abstraction. His chapter, together with von
Glasersfeld (1991), should constitute basic reading for anyone wishing to gain an understanding of
The last section, on teaching and learning advanced mathematical thinking, is where many
of the ideas brought out in the first two sections get applied to the undergraduate curriculum. Aline
Robert and Rolph Schwarzenberger open the section with a general overview of research issues in
Cornu discusses research on concepts of limit, Michèle Artigue discusses research on students’
learning of functional analysis (including differential equations), Dina Tirosh discusses students’
concepts of infinity and transfinite cardinal numbers, and Daniel Alibert and Michael Thomas
discuss research on proof. Eisenberg, Cornu, and Artigue each draw heavily on the
Tall/Vinner/Dreyfus notion of concept image to gain insight into students’ difficulties within their
various areas of focus. Alibert and Thomas continue Hana’s earlier discussion of social
dimensions of proof, but also address cognitive obstacles to students’ understanding of proof by
relating Uri Leron’s very interesting approach to the structuring of proof presentation (Leron,
1983; Leron, 1985). They close with a discussion of Grenoble’s creative experiment in teaching
debate” they mean that students generate mathematical propositions and then debate their validity;
demonstration is sufficiently convincing. It is interesting that false statements, along with their
The final chapter (aside from Tall’s reflections), by Ed Dubinsky and David Tall, is about
advanced mathematical thinking and computers. On the one hand, it is largely about Tall’s Graphic
Calculus and Dubinsky’s ISETL programming language. On the other hand, it is about a
fundamental re-thinking of the calculus (including differential equations) and students’ engagement
with the concept of function. I suspect that anyone whose interest is piqued by this chapter will
need to go to original sources before they can appreciate the power of these uses of computers.
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education community regain the sense that mathematics is a deep and abstract intellectual
achievement. Lest I be misunderstood, I should also say that I do not support the view that school
mathematics should focus on preparing students for college. But at the same time, we must
acknowledge that school mathematics must provide an adequate foundation for advancement, and
without a vision for what they might grow into, it is highly unlikely that students’ intellectual
more than its revelatory value. The book’s authors are all mathematicians, but at the same time
most have conducted research in school mathematics education. This shows itself in the book’s
continuing emphasis that advanced mathematical thinking does not begin after high school. As far
as professional mathematics education can influence it, it must begin in first grade.
My final comment is that, if the book has a shortcoming, it is that it is the first of its kind
and has little to build on. Tall remarked that the preponderance of cited literature is published
within the previous ten years. As such, it is understandable that the content has more to do with the
authors’ emerging reconceptions of mathematics and less to do with actual thinking. David Tall
shortcoming, for research on thinking is highly influenced by the kind of thinking researchers
seek, and this book has the potential of changing our image of what to seek.
Finally, I give two quotations from David Tall’s reflections that he offered to university
We cheated our students because we did not tell the truth about
the way mathematics works, possibly because we sought the Holy
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References
Dubinsky, E., & Lewin, P. (1986). Reflective abstraction and mathematics education: The genetic
decomposition of induction and compactness. Journal of Mathematical Behavior, 5(1), 55-
92.
Greeno, J. (1983). Conceptual entities. In D. Gentner & A. L. Stevens (Eds.), Mental Models
(pp. 227-252). Hillsdale, NJ: Erlbaum.
Hadamard, J. (1954). The psychology of invention in the mathematical field. New York:
Dover.
Harel, G. (1989). Learning and teaching linear algebra: Difficulties and an alternative approach to
visualizing concepts and processes. Focus on Learning Problems in Mathematics, 11, 139-
148.
Harel, G., & Confrey, J. (Eds.). (in press). The development of multiplicative reasoning in the
learning of mathematics. Albany, NY: SUNY Press.
Leron, U. (1985). A direct approach to indirect proof. Educational Studies in Mathematics, 16,
321-325.
Seldon, J., Mason, A., & Seldon, A. (1989). Can average calculus students solve non-routine
problems? Journal of Mathematical Behavior, 8(1), 45-50.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflectons on processes and
objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-
36.
Sfard, A., & Linchevski, L. (in press). The gains and the pitfalls of reification: The case of
algebra. Educational Studies in Mathematics.
Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetic meanings and
strategies. New York: Springer-Verlag.
Tall, D., & Vinner, S. (1981). Concept images and concept definitions in mathematics with
particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-
169.
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Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in
Mathematics Education, 19, 115-133.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for
Research in Mathematics Education, 20, 356-366.