Thematic Group 9 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

F. Arzarello 1
MATHEMATICAL OBJECTS AND PROOFS WITHIN TECHNOLOGICAL
ENVIRONMENTS: AN EMBODIED ANALYSIS
Ferdinando Arzarello
Dipartimento di Matematica, Universit di Torino
Domingo Paola
Liceo Scientifico A. Issel Finale Ligure
G.R.E.M.G. Dipartimento di Matematica Universit di Genova
Abstract. The paper faces an approach to Calculus in secondary schools within
technological environments. It illustrates a case study, where the concept of function
is approached in the 9-th grade using a sensor for motion connected to a calculator.
Pupils can move and see the Cartesian representation of their movement produced by
the sensor in real time. It analyses some excerpts of the discussion in the class after
that students have tried to reproduce with their movement the graphics drawn at the
blackboard by the teacher. The analysis uses a Vygotskian approach and the tools of
embodied cognition to interpret the situation: a theoretical model is sketched, which
stresses the embodied components. Some open problems are pointed out in the end.
1. Introduction. The paper analyses (a part of) an ongoing teaching experiment
conducted from the first years of secondary school (9
th
grade up), where Calculus is
approached early within different experience fields (Boero, 1995). Even if the
experiment concerns all the basic subjects in Calculus (limits, derivative, integrals, as
required by the Italian curriculum in the Liceo Scientifico), since our space is limited
we shall illustrate only the approach to the function concept within the experience
field of pupils motion and shall discuss some problems that one meets approaching
the theorems in Calculus. The experimental part of our study is an example of
research for innovation (Arzarello & Bartolini Bussi, 1998), in which action in the
classroom is both a means and a result of progressive knowledge of classroom
processes.
2. The theoretical framework. Using instruments is crucial in teaching-learning
activities because it can support and enhance learning abilities, putting forward the
different aspects after which a mathematical object can be looked at. For example, a
symbolic-graphic software can use both visualisation and mathematical symbols.
Since vast tracts of the brain are engaged in perception and construction of
imagery, [but] there are also huge areas of the cortex that are plastic and useable for
a variety of activities including the many processes involved in thinking
mathematically (Tall, 2000) such a representation can provide a powerful
environment for doing mathematics and, with suitable guidance, to gain conceptual
insight into mathematical ideas. In fact symbols can be used as cognitive pivots
between concepts for thinking about mathematics (Tall, ibid.). The dynamic of such
a conceptualisation can be described within a Vygotskian frame, since it represents a
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F. Arzarello 2
transition from the immediate intellectual processes to the operations mediated by
signs and illustrates the dialectic between everyday and scientific concepts.
To investigate the specificity of such a dialectic within our teaching experiment,
namely to describe the crucial cognitive aspects of pupils learning processes in
interaction with technological tools, we use three analysis tools: (i) the embodied
cognition approach by Lakoff & Nez (2000) (see also: Arzarello, 2000; Arzarello
et al. 2002; Ferrara, this CERME); (ii) the instrumental analysis by Rabardel (1995)
and others (Artigue, 2001; Lagrange, 2000; Verillon & Rabardel, 1995); (iii) the
definition of concept given by G.Vergnaud (1990)
1
, in particular the notion of
operating invariant.
The embodied approach reveals crucial means for describing pupils cognitive
evolution within technological environments and for designing suitable teaching
experiments. It shows a basic unity from the very perceptions, gestures, actions to the
most theoretical aspects within a systemic relationship. Specifically, in the
operational invariants of the mathematical concepts built up by students in our
experiment, are evident the traces of their actions. For two examples in this direction
and in different environments, see Arzarello (2000) and Bartolini Bussi et al. (1999).
Embodied cognition is also useful to analyse the dynamics of the social construction
of knowledge by pupils: specifically the metaphors, introduced by students in a group
or class discussion, or by the teacher when (s)he wants the students to concentrate on
a particular, possibly new, concept reveal powerful tools for sharing new ideas.
The analysis by Rabardel explores the interactions among students, mathematical
concepts and technologies at school. It considers the way in which the technological
tools act on the mathematical concepts and the way by which such concepts can
model the didactic transposition (instrumentation process). It pinpoints that an
artefact can become an instrument for a student through the appropriation of its
schemes of use (instrumental genesis). The operating invariants in conceptualisation
have traces of their actions as schemes of use of the artefacts: gestures, metaphors of
their activity with them are mirrored in their conceptual elaboration.
We think it is possible to integrate the instrumental approach with the new results by
cognitive science, in particular embodied cognition. These two approaches help us to
analyse the students activities from a new point of view. In fact, if the instrumental
approach can give us the tools to analyse the use of technologies by students, in terms
of schemes of use, it is not sufficient to support the interpretation of their mental
activities, especially for the conceptualisation. On the other hand, cognitive sciences
are perfectly aimed to study pupils mental activities; however, their approach to
conceptualisation processes in mathematics focuses some fundamental aspects but

1
According to Vergnaud, a concept consists of:
- a reference (lensemble des situations qui donnent du sens au concept);
- operating invariants (invariants opratoires: they allow the subject to rule the relationship between the reality and
the practical and theoretical knowledge about that);
- external representations (language, gestures, symbols,).
Thematic Group 9 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
F. Arzarello 3
does not give reason of all the theoretical and symbolic features of the mathematical
thought. Hence we find it useful to embed our analysis within the framework of
Vergnauds definition of concept.
In short, our studies are aimed to an integrate analysis of students cognitive
processes while doing activities in which they use artefacts (a sensor and a calculator)
and transform them into tools, through appropriate schemes of use, hence building
the concept of function because of their active interaction with such signs as
Cartesian graphs and numerical tables. In this paper we will concentrate mainly on
the first aspect (embodied approach).
3. The teaching experiment. Our teaching experiments main goal consists in
introducing some basic concepts of Calculus starting with young pupils (14-15 y.);
the experiment is pursued within different fields of experience, some of which use the
support of suitable technological environments. As a research for innovation in the
class, it has also some research goals, the most important of which is analysing such
mediation phenomena through the lens of embodied cognition. Namely we wish to
see how the cognitive tools put forward by Lakoff & Nez can allow to understand
the learning processes of pupils who interact with a technological artefact (e.g. a
CBR), used to support their building of mathematical concepts. We limit to give an
example of a 9-th grade class, where we have designed a dynamic approach to the
function concept based on the following points: a) the newtonian idea of quantities
which change in time, focusing on the first and second variations of the dependent
variable; b) functions as modelling tools of concrete situations; c) using graphic-
symbolic calculators and movement sensors as mediators in the teaching-learning of
the function concept; d) using different social interactions in the class (working and
discussing freely in small groups; general discussion orchestrated by the teacher in
the whole class; see Bartolini Bussi, 1996); e) taking into consideration the affective
and emotional aspects of learning mathematics: a friendly environment is created,
where evaluation is put forward in not a stressing fashion.
We illustrate pupils learning processes analysing some protocols of the teaching
experiment through the theoretical tools sketched in Section 2; a particular attention
is given to pupils gestures, to exploit their embodied approach to concepts. We
analyze an activity carried out a couple of months after the beginning of the teaching
experiment: a pupil in each students group (3 persons each) must move w.r.t. the
sensor so that the calculator reproduces a graphic equal as much as possible to the
one drawn at the blackboard by the teacher. The pupils observe their mates
movement and comment her/his possible mistakes, pointing out the reasons why
there are differences between the graphic at the blackboard and that produced by the
calculator. We have many recorded videos, but for reasons of space here we limit
ourselves to comment some protocols of a group, where a student (St1) tries to
reproduce through his movement the graphic sketched by the teacher at the
blackboard (fig. 1) and his mates discuss what he has done. While running, St1 looks
Thematic Group 9 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
F. Arzarello 4
at the blackboard and at the screen to co-ordinate suitably his movements. Moreover
a mate (St2) comments St1s movement with expressive gestures of his hands.
St2: That is first slow [he moves his right hand horizontally towards right], then fast
[he hands up his right hand very fast], then down fast [he hands down his hand fast
towards link], then slows down [he moves his hand towards link describing a concave
descending curve in the air], down his hand fast towards link], then slows down [he
moves his hand towards link describing a concave descending curve in the air], then
fast again [again his hand up to the right]then it stops [he moves his hand towards
right horizontally]. St2s gestures show clearly that he has
understood both the movement and the graphic. His hand
gestures incorporate in a compressed way (Tall, 2002) the
features of the time law. His hand gestures incorporate in a
compressed way (Tall, 2002) the features of the time law.
In fact when the speed is increasing, his hand moves faster,
and when the speeds decreases his hand moves slower. In a
Cartesian graphic the information concerning the function
variation and its derivative is coded in a unique sign Fig. 1
(i. e. the graphic) and as such it is not accessible to all. The movement of St2s hand
has two aspects: the first (namely the trajectory of his hand) expresses how the
function varies (the time law form); the second (his hand speed) incorporates the
velocity of the moving body. This double embodiment of information is not a coding
into an unknown language; it is a natural representation of the movement. In fact,
St2s gestures are more direct representations than the blackboard graph (i.e. a
Cartesian plane with different quantities on the two axes): they are a mediating tool
for grasping the situation in a more feasible way (no transcoding is needed, apart the
embodied one). In a certain sense, St2s intervention represents an intermediate level
between the external movement and the time law, (i.e. through the Cartesian graph),
which is useful to start a comprehension process of the scientific features of the
motion. It represents a stage towards the interiorisation of this scientific meaning for
St2, but it also creates a possible space of communication for the class, which was not
evident before. Eventually this allows pupils evolution towards the scientific
meaning of the sign introduced by the teacher at the blackboard. In factSt2s words
and gestures are taken again by other students in the class: most of them use the same
type of gestures than St2 while discussing the problem.
Another important issue to point out consists in the role of the teacher. His
interventions are crucial in order to make pupils conscious of the scientific concepts
they are learning. Their experience lives still in everyday concepts, but their gestures
incorporate the scientific aspects; they are typically in a zone of proximal
development (in the sense of Vygotsky): the teacher supports them linguistically to
transcode their conceptualisation into the scientific language. In the discussion, the
scientific words suggested by the teacher give a name to the gestural description used
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F. Arzarello 5
by pupils to describe the situation: they repeat the words and make the same gestures.
In this blending of representations they conceptualise in a conscious, intentional and
willing way, namely they conceptualise a scientific idea according to Vygotsky.
However the blending of gestures and words they use show that their
conceptualisation embody the traces of their actions.
Two more observations seem important. First, in the usual teaching activities for
Calculus, functions, (first and second) derivatives are introduced with split meanings,
generally in different moments of the curriculum. This inhibits important links
between them and also any possible meaning compression (Tall, 2002). Using
sensors allows to approach all the aspects within the same experience field. This can
support the emerging of a cognitive unit, in the sense of Tall (2000).
Second, the social interaction seen in the class is a typical vygotskian situation where
there are signs (the two diagrams: at the blackboard and on the screen), actions (St1s
movement, St2s gestures) and their interpretation: the pupils who mimic St2 enter in
a common interpretative process which allows them to grasp the signs meaning.
Through that, the meaning is interiorised and gestures become a
communication/interpretation tool for successive similar situations. In particular such
a tool does not block the lowest students, who would encounter difficulties using a
more formal language.
4. Open problems: a dramatic gap? The example illustrates emblematically that a
technological artefact can support effectively students in building meaningfully
various advanced mathematical concepts of Calculus, e.g. functions, derivatives,
integrals (see the proposal of O. Robutti at this same CERME group). In fact, a
suitable coaching of the situation by the teacher makes accessible to the students the
cognitive roots (Tall, 2002) of delicate concepts in Calculus in an embodied way; this
is the very basis which can support the transition from the perceptual side to the
theoretical one. However things change when one passes from the mathematical
objects to the proofs even of the simplest theorems in Calculus. In this case, it is
necessary to take into account the conceptual discontinuities which feature reasoning
with such concepts. We shall discuss sketchily this issue, which points out some
crucial didactical points, which require new investigations to know how an embodied
approach can be useful to overcome the new discontinuities.
In other papers (Arzarello, 2000; Olivero et al., 2001; Arzarello et al., 2002) one of
the authors has analysed the dynamic geometric software as a support for the learning
of proofs in Geometry. The studies have pointed out the cognitive continuity (Garuti
et al, 1996) which features the transition from the exploring, conjecturing, arguing
phase to the proving one. That is in learning environments which can support them in
their observations and explorations, pupils systematically build up the proofs of their
conjectures using facts, ideas, words and sentences already produced during the
exploration and conjecturing phase. It is not always a one to one translation, since
some crucial restructuring processes may happen (see Arzarello, 2000), but the main
ingredients remain the same. Such a is opposite to the epistemological gap which is
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F. Arzarello 6
observed between the informal and the formal phases of proving (Arzarello et al.,
1998).
We distinguish between a conceptual continuity, when the ingredients remain the
same within a reference theory, but the way they are arranged together may change
and a structural continuity, when also the structure after which the ingredients of the
proof are put together does not change essentially. Such a terminology comes from E.
Pedemonte (2002), who defines reference system continuity the former and structural
continuity the latter. While in some geometric situations the structural continuity is
preserved, it is seems not always so in the case of more advanced arguments, e.g. in
Calculus,: in fact, the necessity of restructuring is an index of the difficulty of the
task. Context relevance is not surprising: the conjecturing and proving activities are
linked with the meaning of the mathematical objects they involve as well as with the
mutual relationships among them. Hence the existence and the nature of a possible
cognitive continuity depends strongly on the mathematical context within which the
proof is developed. Let us make a concrete example, comparing Geometry with
Calculus. In Calculus things as y=log(x), y=xsin(1/x), the tangent to y = x
3
, a
continuous function... are not so immediate as a triangle in Geometry; they exist as an
elaborated product of mathematics (generally incorporated by formulas). Links with
perception are problematic (e.g. through a picture of the continuum or through orders
of magnitude): a theoretical frame is necessary to guarantee some ideal perception
through the minds eye. Proofs are made of words and calculations, only apparently
similar to the algebraic ones: in fact logical complexity increases, with the use of
formulas; besides, calculations concern inequalities, nested within such
alternated quantifications and so their status changes deeply from the algebraic one.
Last but non least, the reference to drawings (figures) as generic objects is very
problematic in technological environments. While with a dynamic geometry software
one can give an idea of a generic triangle, the notion of a generic continuous function
is beyond the capabilities of nowadays technology, as far as we know. Even in the
case of arguments and proofs there are crucial differences between elementary
Geometry and Calculus: the latter has more obstacles and traps for the learner. To be
concrete, let us consider two different proving strategies, which are a possible cause
of breaking within the structural continuity between arguing and proving phases: the
first consists in arguments and proofs which require the limit definition in analysis;
the second is the use of abductive
2
strategies in geometry during the exploring and
conjecturing phase as a prelude to the proving one (the latter have been investigated
in Arzarello et al., 1998 and 2000; Arzarello, 2000). Both strategies need a sort of
mental somersault; however this has different features: while abductive one does not
break the cognitive unity (Arzarello et al., 1998), the cognitive continuity of

2
The following example (Peirce, 1960, p.372) is illuminating about abduction. Suppose I know that a certain bag is
plenty of white beans. Consider the sentences: A) these beans are white; B) the beans of that bag are white; C) these
beans are from that bag. A deduction is a concatenation of the form: B and C, hence A; an abduction is: A and B, hence
C (Peirce called hypothesis the abduction). An induction is: A and C, hence B. For more details, see Magnani (2001)
and Arzarello et al. (2000).
Thematic Group 9 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
F. Arzarello 7
processes seems not always preserved with limits, because of cognitive and
epistemological reasons. Let us see better why.
In the limit definition the somersault consists: (i) in the consideration of the x y
subordinate variations (x variations are seen as causes of y variations) during the
conjecturing or in the early proof construction phases; (ii) in the deductive enchaining
phase the y-universal quantification drives the x-existential/universal quantification.
A shift from x f(x) direct reasoning to inverse, -reasoning is needed. Hence,
we have a double inversion: from x y to y x and from - to -quantifiers.
In abduction within geometric explorations the sommersault consists in a switching:
(i) in the modality of control of the subject with respect to the geometrical figures:
ascending vs/ descending
3
; (ii) in the way the subject sees the mathematical objects,
with respect to what is considered as given and what is supposed to be found: such a
relationship usually change many times during the exploration (Arzarello et al., 1998
and 2002). In Calculus the somersaults involve also the genesis of some basic
structured mathematical objects necessary in that field: the natural (metaphorical and
possibly perceptual) ways after which such concepts are built up require deep
somersaults to get a rigorous mathematical definition. Since the cognitive genesis of
mathematical objects unlikely can avoid the natural metaphorical way, this poses
serious didactical questions from the very beginning of the teaching of Calculus. The
problem is hard since at the moment it seems difficult finding genuine cognitive roots
or other natural cognitive entities which can support a cognitive continuity while
proving the theorems in Calculus. Neither suitable mediating tools (e.g. software)
seem to exist that can support the apprenticeship to the proof in Calculus through its
structural continuity breaks, pointed out above. As we have already told, somersaults
present a double inversion (from x y to y x and from - to -formulas), which
seems very hard to overcome. In particular, we have not found any evident natural
(e.g. embodied) example of such somersaults: do they exist? At the moment the
answer seems negative (
4
) and two ways remain available for approaching proofs in
Calculus.
(i) A first solution consists in developing a suitable didactical engineering, namely
designing learning environments and situations centred on such conceptual
discontinuities. For example, it is possible to coach an evolution in the class from the
natural, embodied roots of the mathematical objects of Calculus (like in our example
or in Tall 2002) to the culture of theorems described by Boero. This can be achieved
through a cognitive apprenticeship (Arzarello et al., 1993), where pupils are nurtured
to the somersaults (i.e. to the structural continuity breaks). The role of the teacher is
crucial in such an apprenticeship, insofar (s)he is responsible for the transition to the

3
Ascending processes are from the drawings to the subject, who explores the situation, looking for regularities,
invariants, etc. with an open mind; descending processes are from the subject to drawings, in order to validate or refute
conjectures, to check properties, etc., which the subject has already in her/his mind (Arzarello & al., 2002).
4
The best example of a natural somersault that I know is the following (Lolli, 1992): For each criminal, it will arrive
the moment, when he will be obliged to pay for his faults
Thematic Group 9 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III
F. Arzarello 8
socially shared level of mathematics which incorporates the somersaults in its historic
evolution. During the exposition we shall present an example.
(ii) Another solution consists in changing technically the classic approach to Calculus
and avoiding systematically the conceptual somersaults. For example, there are
teaching experiments based on non-standard analysis (see the survey in Maschietto,
2002), or even books where a big ingenuity has been used to build up a Calculus
which avoids Weierstrass traps.
A major issue consists in finding suitable cognitive roots for the new gaps; our first
results suggest that an embodied approach can be the right way also in this case.
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