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MARIA ALESSANDRA MARIOTTI and EFRAIM FISCHBEIN DEFINING IN CLASSROOM ACTIVITIES La mathematique est l’art de donner le mˆ eme nome ` a des choses diff´ erentes (Poincar´ e, 1909) ABSTRACT. This paper discusses some aspects concerning the defining process in geo- metrical context, in the reference frame of the theory of ‘figural concepts’. The discussion will consider two different, but not antithetical, points of view. On the one hand, the problem of definitions will be considered in the general context of geometrical reasoning; on the other hand, the problem of definition will be considered an educational problem and con- sequently, analysed in the context of school activities. An introductory discussion focuses on definitions from the point of view of both Mathematics and education. The core of the paper concerns the analysis of some examples taken from a teaching experiment at the 6th grade level. The interaction between figural and conceptual aspects of geometrical reas- oning emerges from the dynamic of collective discussions: the contributions of different voices in the discussion allows conflicts to appear and draw toward a harmony between figural and conceptual components. A basic role is played by the intervention of the teacher in guiding the discussion and mediating the defining process. 1. INTRODUCTION In mathematics, as a theoretical knowledge, a basic role is played by the processes of definition and validation. First of all the objects which we are dealing with, must be stated and clearly defined. Then properties about certain objects may be considered as true only if they are derived from other true statements via arguments on which there is an agreement by the ‘scientific’ community. Defining is a basic component of geometrical knowledge, and learning to define is a basic problem of mathematical eduction. This paper aims to discuss this problem. The discussion will consider two different, but not antithetical, points of view: on the one hand the problem of definitions will be considered in the general context of geometrical reasoning. On the other hand the problem of Educational Studies in Mathematics 34: 219–248, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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MARIA ALESSANDRA MARIOTTI and EFRAIM FISCHBEIN

DEFINING IN CLASSROOM ACTIVITIES

La mathematique est l’art de donner le memenomea des choses differentes

(Poincare, 1909)

ABSTRACT. This paper discusses some aspects concerning the defining process in geo-metrical context, in the reference frame of the theory of ‘figural concepts’. The discussionwill consider two different, but not antithetical, points of view. On the one hand, the problemof definitions will be considered in the general context of geometrical reasoning; on theother hand, the problem of definition will be considered an educational problem and con-sequently, analysed in the context of school activities. An introductory discussion focuseson definitions from the point of view of both Mathematics and education. The core of thepaper concerns the analysis of some examples taken from a teaching experiment at the 6thgrade level. The interaction between figural and conceptual aspects of geometrical reas-oning emerges from the dynamic of collective discussions: the contributions of differentvoices in the discussion allows conflicts to appear and draw toward a harmony betweenfigural and conceptual components. A basic role is played by the intervention of the teacherin guiding the discussion and mediating the defining process.

1. INTRODUCTION

In mathematics, as a theoretical knowledge, a basic role is played by theprocesses of definition and validation. First of all the objects which weare dealing with, must be stated and clearly defined. Then properties aboutcertain objects may be considered as true only if they are derived fromother true statements via arguments on which there is an agreement by the‘scientific’ community.

Defining is a basic component of geometrical knowledge, and learningto define is a basic problem of mathematical eduction. This paper aims todiscuss this problem.

The discussion will consider two different, but not antithetical, points ofview: on the one hand the problem of definitions will be considered in thegeneral context of geometrical reasoning. On the other hand the problem of

Educational Studies in Mathematics34: 219–248, 1997.c 1997Kluwer Academic Publishers. Printed in the Netherlands.

GR: 201007275, Pipsnr.: 136017 HUMNKAPeduc705.tex; 29/05/1998; 11:34; v.7; p.1

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definition will be considered as an educational problem and consequently,analysed in the context of school activities.

As far as geometrical reasoning is concerned, the following discussion issituated in the reference frame of the theory of figural concepts (Fischbein1963, 1993). According to this theory geometrical concepts have a doublenature characterised by two aspects, the figural and the conceptual. Thefigural aspect concerns the fact that geometrical concepts refer to space,while the conceptual aspect refers to the abstract and theoretical nature thatgeometrical concepts share with all the other concepts. According to thisinterpretation, geometrical reasoning can be characterised by a dialecticinteraction between these two aspects. Despite the fact that, in principle,the two aspects must interact harmoniously, in reality the harmony is oftenbroken and a temporary autonomy of each of the two aspects appears.Conflicts and difficulties, commonly observed in the school practice, canbe interpreted according to this theory. The investigation carried out in thepast years (Mariotti, 1991, 1992, 1993), had focused on the analysis andthe description of the process of interaction between the figural and theconceptual aspects in geometrical reasoning.

As far as the educational point of view is concerned, the results of theprevious investigations led to the following main hypothesis. The processof interaction between the two aspects of geometrical reasoning and, con-sequently, the harmonisation of them is not a ‘spontaneous’ achievement.On the contrary, it depends on teaching interventions. Within this generalframework the following discussion will focus on particular aspects relatedto the defining process.

2. DEFINING AS A MATHEMATICAL ACTIVITY

As far as the mathematical activity is concerned different perspectivesare possible. Mathematics is a theoretical system, in which definitionsplay a crucial role. Through definitions, the ‘objects’ of the theory areintroduced: definitions express the properties which characterize them andrelate them within a net of stated relations; new properties of the definedobjects and new relations between them and the objects of the theory canbe further established through the process of deduction. But the theoreticalsystematisation is only a final stage of a long productive process in whichdefinitions result from a negotiation between logical rigor and creativity.

In mathematics two main types of definition are possible: (a) the intro-duction of the basic objects of the theory, and (b) the introduction of a newelement (a constant or a predicate) within the theory itself. The followingdiscussion will follow this distinction.

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2.1. The introduction of the basic objects

Implicit definitions stated through the axioms characterize the objectswhich the theory is dealing with. For instance, consider the concept of‘group’ and its definition through its axioms, i.e. a set is a ‘group’ if andonly if its elements satisfy the specific properties described by the axioms.According to this formal approach, the autonomy of a theory from anyexternal reference is definitely stated.

Let us consider Geometry, the independence of a theory in respect to itsinterpretation in terms of reality (concrete objects and actions) reminds oneof the famous sentence attributed to Hilbert: “Instead of treating ‘points’,‘straight lines’ and ‘planes’, one must always be able to discuss ‘tables’,‘chairs’ and ‘beer-mugs”. Certainly, this sentence had and still has a fun-damental metatheoretical meaning, but it is quite hard to forget the factthat geometry has intuitive roots in experience.

Actually, the definitions of the basic geometrical figures are not mereconventions in the field of pure arbitrary facts; elementary geometry andgeometrical concepts are deeply rooted in common experience. In thewords of Poincare:

Thus we may conclude that the principles of geometry are only conventions; but, theseconventions are not arbitrary, and if we were carried in another world (which I call non-Euclidean and which I try to imagine) we could be obliged to change them.

(Poincare, 1968/1902: p. 26)

There is a privileged link between geometry and reality. Nonetheless,geometry is not an empirical science: it is not possible and it would be amistake to try to reduce geometry to empirical knowledge.

In fact, if experimental geometry means that the student makes experiments, then a greatpart of his mathematical activity should be experimental, as is the activity of the creativemathematician. If it should remind us of experimental physics, it is wholly mistaken.

(Freudenthal, 1973)

Geometry maintains its autonomy as a theoretical domain, but at thesame time, it depends on reality as a model of some of its aspects. Thatis, geometrical concepts belong to a theoretical system, but at the sametime, are not completely free. They must refer to a meaning which has itsown root in reality and over the centuries, has been well settled. Enriquesclearly expressed the complexity of the nature and the origin of geometricalconcepts, which should have been taken into account for the setting up ofan axiomatic system.

The concept of space originates in the external facts, in their representation given by thesenses to the mind. Geometry studies this concept, already formed in the mind of a geometer,

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without rising the (psychological but not mathematical) problem of its genesis. Thus, therelationships among the elements (points, lines, planes, etc.) which constitute the conceptof space: these relationships are called spatial properties or geometrical properties.

[...]Mathematicians study these properties in two ways:1. Using their (psychological) intuitions on spatial concepts,2. Deducing through logical thinking new properties from those already given by intuition(the new properties obtained are considered proved).

(Enriques, 1920: pp. 1–2)

The long tradition of geometrical concepts makes it difficult for us tobe aware of the complexity of this process. On the one hand, the intuitivemeaning of geometrical concepts is rooted in experience, but the theoreticalaspect of such concepts requires their complete autonomy from empiricaldata. On the other hand, the fact that elementary mathematics presentsitself as a ready made corpus of knowledge hides the long process whichoriginated it and leads one to emphasise the spontaneous links with exper-ience. We are so used to certain concepts, such as that of line, circle, orperpendicularity, that we could not think differently; they seem so obviousthat they appear the only possible ideas coming from our experience andperfectly suitable to explain it. Actually, the distance between geometryand spontaneous conceptualisation of physical experience is much greaterthan it is generally supposed and it is often underestimated. In this per-spective the theory of figural concept clearly focus on the complexity of theproblem from the cognitive point of view, while from the didactic point ofview interesting analyses and proposals can be found (Berthelot & Salin,1992; Laborde, 1992).

2.2. The introduction of a new element

Let us take the point of view of formal logic. When a new constant or a newpredicate is introduced within a theory, the introduction of the new element,the new name, is allowed by a theorem which states the existence of such anelement of the theory characterised by specific properties. That means thatthe new element must respect a ‘criterion of elimination’ (Rogers, 1978,p. 96), which justify the introduction of the new symbol only as a reversibleprocess of abbreviation; through the substitution of thedefiniendumwiththedefiniensthe previous status of the system is restored. In other terms, inthe new theory, it is not possible to prove anything which was not alreadypossible to prove in the old one. From the formal point of view, a definitiondoes not enlarge the power of the theory. A definition is rather a correctdefinition just because it can be eliminated. Again the formal approachdoes not grasp the very process of defining.

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From the Mathematical point of view, i.e. considering Mathematics asproduct of human thinking definitions of new elements have a constructive(creative) role. In fact, a new definition introduces a new concept which,although related to all the others, did not ‘exist’ before. An interestingexample is provided by the discussion of Waterhouse (1972–73) on the‘history of the regular polyhedra’.The study of the regular solids started with a sort of ‘prehistory’; in thatstage they were investigated as individual objects, without recognising theunifying idea of regularity connecting them.

‘To discuss the solids one by one misses the point; we must study non just their individualhistory but above all their joint history. The real history of the regular solids thereforebegins at the point when men realised there was such a subject. The discovery of this orthat particular body was secondary. The crucial discovery was thevery concept of a regularsolid (stress is mine).’

(Waterhouse, 1972–1973: p. 214)

Because of the long acquaintance with regular solids it is nearly impos-sible to consider as a real difficulty the recognition of this regularity asa common characteristic of these objects; nonetheless, the mere fact oflooking at them was not, and still it may be not sufficient to regard themas ‘regular solids’.

We have the mathematical concept of a regular solid only because some mathematicianinvented it. (ibid., p. 214).

What required discovery was not so much the object itself as its significance. (ibid., p. 216).

A full discussion about the origin of the very idea of regular polyhedronis not our objective, but it is interesting to stress the fact that such ameaningful idea as that of regularity does not arise in a vacuum. There area lot of common properties that can be stated for a collection of solids, butany choice can arise only in relation to a specific goal. The characterisationof idealised objects is the product of a process of abstraction, but actuallydifferent problem situations may determine different interests and originatedifferent process of abstraction, leading to different possible definitions.

In conclusion, not only a definition must be correct from the theor-etical point of view, but it must also be productive by the way it opensnew problems or new perspectives to think and solve old problems: adefinition is to be considered a ‘good’ definition as far as the new objectstarts to live by itself and may become the subject of a new theory. In thissense, the Lakatos’ description of “the logic of mathematical discovery”(1979/76) offers a interesting model of the production of mathematicalknowledge, in which the author discusses the complex dynamic between

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creativity in terms of solution of problems and rigour in terms of form-al constraints. Lakatos’ discussion considers both the synchronic and thediachronic dimension, highlighting the negotiation process which emergesboth along the historical evolution and within the community of contem-porary mathematicians.

2.3. Systemic organisation of concepts

Besides the origin of a concept it is interesting to consider the consequencesof the new perspective that it opens. Once a definition is stated, the meaningand the nature of the object are determined, so that specific dynamics ofthinking are mobilised; for instance in geometry, the definition of a figuralconcept determines the dynamic of its figural and conceptual aspects.

Consider the previous example. Following the analysis of Waterhouse,a trace of a difference between empirical and theoretical concepts can berecognized in the history of regular solids. It is possible to observe thepresence of two stages testified by the use of different terms referringto the same solid. At a first stage, the cube and the pyramid were thenames for the corresponding solids, while the dodecahedron was usuallyreferred to as the ‘sphere of the twelve pentagons’ (ibid. p. 216). At asecond stage, the change of perspective corresponding to the introductionof the idea of regular solids, determines the introduction of the new termshexahedron, tetrahedron, dodecahedron, octahedron, icosahedron (and thelast two solids were never named differently): these terms are not onlytechnical terms but also ‘systematic’ terms. They witness the appearanceof a new systematic way of analysis, which generated something new,i.e. the category of regular solids, but at the same time, reconsidered oldelements, i.e. the cube or the ‘sphere of the twelve pentagons’, from a newperspective.

In conclusion, from the psychological point of view, defining in thegeometrical field seems to present a great complexity due both to generalcharacteristics of the defining process and specific characteristics of thegeometrical concepts. The theory of figural concepts, which ‘[...] constituteonly the limit of a process of fusion and integration between the logical andthe figural facets’ (Fischbein, 1993: p. 150), provides an adequate approachto the problem.

When spontaneous concepts are concerned, we have no clearly definedclassifications, they are often incomplete and characterised by instability.In particular, when geometry is concerned ‘spontaneous’ classificationsare achieved according to similarities and differences which are figurallypertinent. For instance, from the figural point of view squares and non-square rectangles look so different that they impose the need of being

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distinguished at least as much as triangles and quadrilaterals; on the con-trary, although rectangular prisms and rectangles cannot be consideredmathematically ‘similar’, from a certain point of view, they can be said ‘tohave the same shape’ (Freudenthal, 1983: p. 228).

On the contrary, in the case of mathematical concepts – because oftheir theoretical nature – clear and unambiguous rules are stated accordingto which a certain figure belongs to a category. Geometrical definitionsrespond to specific theoretical demands and to a specific theoretical func-tionality (De Villier, 1994); they yield classifications of geometrical objectsaccording to well defined (explicitly stated) specific geometrical proper-ties, which could be said to be conceptually pertinent. For instance, theclassification of triangles can be carried out in terms of angles’ congruenceor of sides’ congruence. In the former case, we have the distinction amongright angle, acute and obtuse triangles; in the latter case we have the dis-tinction among equilateral, isosceles and scalene. Similarly, quadrilateralscan be classified according to the number of axes of symmetry. The ‘con-gruence’ of angles or of sides and the ‘axes of symmetry’ are geometricalconcepts according to which geometrical objects can be defined. Each ofthese classifications corresponds to a different point of view. But, as soonas a concept is decontextualized (Chevallard, 1985), the pertinence of theseproperties cannot be immediately clear, no trace of their origin is left; thusone is compelled to accept them and be convinced of their obviousness.

In fact, theoretically, a definition relates the new object to all the others,in such a way that a chain (system) of definitions is built up; this system isan organic and coherent whole. The gap between a spontaneous definingprocess and a mathematical defining process concerns both the origin ofthe concepts and their organisation within a theoretical system.

As discussed above, it is just this structure of stated relationships whichcharacterises a ‘theory’ and differentiates between theoretical and spon-taneous concepts.

But the absence of a system is the cardinal psychological difference distinguishing spontan-eous from scientific concepts... all the peculiarities of child thought stem from the absenceof a system in the child’s spontaneous concepts – a consequence of undeveloped relationsof generality.

(Vygotsky, 1962, p. 116)

3. DEFINING AS AN EDUCATIONAL PROBLEM

Defining holds an important position among school mathematical activit-ies: pupils must assimilate (Leont’ ev, 1964/76) intellectual objects belong-ing to a culturally relevant theory. As far as geometry is concerned, the

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educational task consists of introducing pupils to its objects and its prob-lems, emphasizing in particular their theoretical nature. In the specific caseof geometry, this means to achieve a good, active harmony between thetwo components, the figural and the conceptual.

In a previous investigation (Vinner, 1976) the problem of definition wasconsidered from the point of view of the common experience of the pupils,as the author calls it, from the point of view of an implicit, preliminaryapproach (i.p.a.). Despite the fact that ‘methodology of mathematics is nota part of regular curriculum’, the author claims that the student has someideas of his own about what a definition is... (ibid. p. 414). His conjecture isthat the naive student’s i.p.a. towards mathematical definitions is the sameas the plain man’s approach to lexical definitions. Conflicts between theempirical (lexical) approach and the theoretical approach can represent areal obstacle for the students’ understanding. That is the reason why theproblem of introducing the pupils to the mathematical process of definingconstitutes a crucial point in mathematics education which needs to befaced directly.

The process of defining must be considered from the point of view of itscomplexity as both a component of geometrical reasoning and a specificprocess of mathematical activity. It is this last aspect that often escapesthe domain of school practice: pupils never participate in the real game ofdefinitions; meaningful aspects are proposed to pupils without giving themany possibility to grasp the reason of their significance.

3.1. Producing Definitions in the Classroom by Collective Discussion

The conceptualisation discussion, as outlined by Bartolini Bussi (1991,p. 10), has the purpose to achieve a ‘definition’. The difficulties arisingin entering the theoretical world make the role of the teacher becomerelevant; the teacher has the difficult task of mediating between cultureand pupils, between mathematics, as a product of human activities, andpupil’s learning. We are convinced that the teacher has the main role of‘guiding the pupils’ work’ (ibid.), introducing them to mathematics andmaking them aware of mathematical activities.

As it will be shown in the following, when geometrical concepts areconcerned, defining simultaneously requires the intervention of the figurallevel and the conceptual level. In other words, the task requires the doublemovement between the figural and the conceptual level. This is the core ofthe interaction between the two aspects. Thus, a dialectic process may beexpected; the main steps may be summarized as follows:

� observing� identifying the main characteristics

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� stating properties according to these characteristics� returning to observation, checking the definition with regard to figural

differences, and so on.

As a matter of fact, no one step can be considered either purely figuralor purely conceptual. From the start, observation is performed from theperspective of a particular conceptualisation, yet any definition has itsorigin in the figural features of objects and their transformations. At thesame time, the process of elaborating a definition consists of a doubleprocess from the particular to the general and vice versa, from the generalto the particular.

As far as the dialectic of figural concepts is concerned, there seem tobe two relevant moments in the collective discussion:

� In the course of the argument pupils refer to both figural and concep-tual aspects. When they are confronted with a disagreement, the twoaspects interact and the two aspects can finally harmonise.

� In the attempt to convince classmates that their definitions are correct,pupils are compelled to make their thinking explicit, and to get aconceptual control of the situation.

Interactive conditions enable discourse to be shared and potentially produce a higher inter-individual functioning.

(Pontecorvo, 1989)

In the following, we will present and discuss some reports of schoolactivities. The analysis will be carried out from two complementary pointsof view, at first focusing on the description of the evolution of the interactionbetween the figural and the conceptual components, then focusing on thespecific intervention of the teacher affecting this evolution.

4. THE EXPERIMENT

The examples we consider in this paper come from a teaching experi-ment set up with the general aim of studying the dialectic process betweenthe figural and the conceptual aspects of geometrical reasoning (Mariotti,1996). Three different Italian 6th grade classes were involved during threesuccessive school years, always with the same teacher. The teacher col-laborated with the authors in planning the activities and in collecting data(in particular, in recording the classroom discussions). The experimentaldesign was included within the regular curriculum and foresaw a sequenceof activities centred on the problem of unfolding/folding a polyhedron; theteaching experiment developed along a period of approximately one month.

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For all of these classes, these activities were the first contact with geometryat this school level, but these pupils have already studied geometry in theprimary school; in particular, pupils know the main solids. The teacher hasalready experienced collective discussions with her classes, with the gen-eral objective of establishing a specific habit in the class and a particularattitude regarding the act of discussing, one may say a ‘culture of discus-sion’. This represents something new for these pupils, who have had a verytraditional school experience, both in the sense of the organisation of theclass (frontal lessons, ‘question-answer’ interaction with the teacher) andin the attitude towards the subject, in particular in what concerns ‘errors’.

In the following, the analysis concerns the transcript of some collectivediscussions which took place in two experimental classes. The discus-sion aims to highlight the dynamics of conceptualisation and definitionand is related to two different tasks: a classification task and the unfold-ing/folding task; differences emerge, related to the nature of the task andthe intervention of the teacher.

4.1. The classification task

The first examples concerns a classification task: some cardboard modelsof solids are presented and pupils are asked to name and characterise them.Pupils have already met the main geometrical solids in the primary school,they knew the names and for some of them they even knew the formula forthe volume. But, according to the tradition no explicit stimulus was givenjustifying the mathematical choice of the standard structural classification;thus it was reasonable to expect a discrepancy between spontaneous andtheoretical conceptualisation. The first discussion aimed at negotiating afirst characterisation of prisms, but the main objective was that of focusingon the very process of definition, stressing the necessity of stating andrespecting a characterisation.

4.1.1. The figural and the conceptual component in a classification taskTwo episodes will be analysed, the first concerning the parallelepipeds andthe seconds the more general class of prisms. At first, a characterisation ofa parallelepiped was achieved in terms of its rectangular faces which wasadequate for the description of the parallelepiped, but was inadequate fordefining the more general class of prisms (including the parallelepiped).According to the theory of figural concepts, a new harmonisation is requiredin order to achieve the geometrical concept of prism and a reorganisationis required to conceive a parallelepiped as a particular prism.

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1� Episode. Toward the definition of a parallelepiped

Let us describe the situation in the classroom. Some cardboard models ofpolyhedra are on a desk: a cube, a rectangular prism, a hexagonal prismand a pyramid with a square base.

The teacher picks up the cube, shows it to the pupils and asks: ‘What isthis?’ Everybody agrees that it is a cube and they characterize the cube as‘a solid (three-dimensional object), delimited by six faces, each face beinga square’. Then, the teacher reminds them of the goal of the discussion:‘Let us try to describe these objects’ (pointing to the polyhedra standingon the desk). She picks up the rectangular prism, shows it to them andasks ‘What is this?’. The whole class answers: ‘A parallelepiped!’1 At thispoint the teacher goes back to the task and encourages the pupils to go oncharacterising the object.

Excerpt 1. (6th grade, 1st Collective Discussion, school year 1991–92)

60 Teacher: This is a parallelepiped, does everybody agree?

61 Riccardo: I want to say one thing: it is a parallelepiped with a rectangular base.

62 Teacher: It is a parallelepiped. You [she refers to Riccardo] say ‘with a rectan-gular base’. Why? How could the base be different?

63 Riccardo: Hexagonal, triangular.

64 Teacher: So, you say ...

65 Confusion, NO, NO.

66 Alex: No, a parallelepiped can only have a rectangular or a square base.

67 Teacher: Yes, Alex, you say that the parallelepiped can have only rectangularand square bases.

68 Alice: Yes.

69 Chiara: Yes, because parallelepiped means that it has the sides ... uhm ... the ...‘sides’ ... parallel, because it has ... parallel faces.

70 Matteo: Two by two parallel.

71 Teacher: Parallelepiped, because it has faces which are two by two parallel.

The cube is also a parallelepiped, isn’t it?

72 Matteo: Yes, parallelepiped, there is only the square and the ...[he can’t thinkof the word]

73 Priscilla: But, nearly all of them [she refers to the objects] are parallelepiped..

74 Matteo: I wanted to say the cube, not the square [actually he is going on withhis thoughts, and he is not following what his classmates are saying].

75 Chiara: Look here, these faces are parallel [she points at the parallelepiped].

76 Teacher: Oh, yes, O.K.. There is a difference, you wanted to say cube, not square[she is answering Matteo].

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77 Chiara: But this one [the prism with a trapezoid], doesn’t look like a paral-lelepiped, because it has only two bases which are oblique [she usesthe word ‘bases’ instead of the correct word ‘faces’].

[...]

108 Chiara: I want to say something about the definition of a parallelepiped,because, it’s a parallelepiped if it has parallel sides ...

[...]

110 Chiara: Because, in my opinion, if one says ‘a parallelepiped has sides paralleltwo by two’, this sounds a little bit generic, because, I mean, this figure(hexagonal prism) also has two by two parallel sides.

In this first approach, the discussion seems primarily influenced by thedifferences between the solids, and pupils describe them in terms of theshape of the faces. Thus, when Riccardo affirms that it must be explicitlysaid that ‘it is a parallelepiped with a rectangular base’ (61) and considersthe parallelepiped as a particular prism, which has a rectangle as the base,his classmates don’t agree. They can’t understand Riccardo. They make aclear distinction between prisms and parallelepipeds and according to thepicture in their mind, a parallelepiped is the solid that has faces having theshape of ‘rectangles’ (66).

Although a predominance of the figural component, not even in thiscase, can we speak of pure figural thinking. In fact, the characterisation ofa parallelepiped in terms of its rectangular faces requires the intervention ofthe conceptual component; this conceptualisation is adequate to describethe parallelepiped, as it appears, and to distinguish it from other solids, butit is inadequate to include the parallelepiped within the more general classof prisms.

Because of the need of settling the disagreement, the conceptual com-ponent is mobilised and suddenly, Chiara (69) shifts to a new point of view.She attempts to characterize the parallelepiped by means of the geometricalproperty of parallelism of faces. Matteo (70) follows Chiara’s conceptual-isation and makes it more precise, stating that ‘faces must be two by twoparallel’.

The interaction continues with the intervention of the teacher who sug-gests that they consider the cube a parallelepiped; in other words, shesuggests that they check the cube according to the property of the paral-lelepiped. What is their reaction? While Matteo stops and considers onlythe cube, Chiara has in mind the general idea of prism and goes on bychecking other objects, still following the property of parallelism of faces.She is interrupted by other interventions, actually there is a digression, but,as soon as she can, she comes back to the point. At first, Chiara explicitly

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refers to the definition (108) and repeats the characterising property ‘... it isa parallelepiped if it has parallel sites’; then she passes to check the figuralconstraints of the stated property on the hexagonal prism. The hexagon-al prism verifies the properties of parallelism required by the definition,although the hexagonal prism is not a parallelepiped (as is easily seen).Chiara considers the hexagonal prism as a counter-example in order toshow the inadequacy of the given definition of the parallelepiped, as shesays, this definition ‘sounds a little bit generic’ (110).

2� Episode. Differentiation between the concepts of prism andparallelepiped

The following episode shows the evolution of the interaction between thefigural and the conceptual components. Chiara attempts to clarify what dif-ferentiates a parallelepiped from a hexagonal prism (116); her statementis very confusing, but Alex helps her and in the dialogue (119–120–121) aprovisional ‘definition’ is negotiated.

Excerpt 2. (6th grade, 1st Collective Discussion, School year 1991–92)

116 Chiara: Because, I mean, this figure (parallelepiped) has sides which are twoby two parallel, but two are equal and two are not, so, in my opinion,a parallelepiped has sides which are parallel and two by two equal.

(The term ‘equal’ is used with the implicit meaning of ‘with the sameshape’, this is what Alex is going to make explicit.)

[...]

119 Alex: In my opinion, I mean, perhaps I understand what she wanted so say,that faces are reciprocally, two by two, parallel, but that, even if theyare smaller, they have the same shape, I mean.

120 Chiara: Two by two parallel, with the same shape.

121 Alex: I mean, that one [hexagonal prism] has the faces two by two [parallel],so, ... but here they are rectangular, there they are hexagonal; on theother hand, this one [parallelepiped] ... they [the faces] are two by two,but they all are rectangular.

In the dialogue Chiara plays the role of the ‘conceptualiser’, while Alexprovides the figural interpretation of Chiara’s conceptualisation (compare120 and 121). The verbal exchange between Chiara and Alex is the socialcounterpart of the mental interaction between the figural and the conceptualaspects, which is expected to take place at the individual level.

The conceptualisation of a prism is achieved but it is still insufficient. Itis still focused on the figural differences between the two solids observed(the parallelepiped and the hexagonal prism), so it fails to arrive at the

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necessary generality. The figural differences between a parallelepiped anda hexagonal prism lead to a classification which aims to separate thetwo classes of objects, whilst, in the standard mathematical classification,the class of the parallelepipeds is included in the more general class ofthe prisms. In order to get such structural conceptualisation, differencesbetween the particular objects should be overcome in favour of analogiesbetween them. In the following section this specific point will be discussed,showing how the intervention of the teacher can play a determinant role.

4.1.2. The intervention of the teacher in the process of generalisationLet us consider a new episode. In this case, the specific difficulty relatedto the process of generalisation appears dramatically; the pupils’ attitudeto differentiating according to figurally pertinent properties is particularlyevident. The awareness of this obstacle determined the teacher to inten-tionally cause the shift from differences to analogies, and through them toconceptual determinants.

The task-situation is the same as in the previous episode: some card-board models are on the desk and pupils are asked to name and character-ise the different solids. The discussion achieves the ‘definition’ of a cube,everybody agrees with (91); then, the teacher reminds them of the goal ofthe discussion and calls for a characterisation of the rectangular prism2.

Excerpt 3: (6th grade, 1st Collective Discussion, School year 1992–93)

91 Teacher: [...]. Well, a cube is a solid limited by six square faces, equal to eachother and connected. And this [she handles a rectangular prism], whatis it? how is it called?

92 Alex: Parallelepiped.

93 Teacher: Parallelepiped. And this figure, how is it called? [she presents a rect-angular prism, with a square base].

94 Chorus: It is a parallelepiped too.

95 Teacher: You remember them from the primary school, don’t you? These areparallelepipeds. Can you say why? What are their characteristics?

96 Gio: Parallelepipeds have ... are nearly the same as a cube, but they haveone face longer than the other.

97 Sol: I’d say that a parallelepiped has two by two parallel faces.

98 Teacher: A parallelepiped has two by two parallel faces. Will, a cube, has twoby two parallel faces.

99 Sol: Yes, it has.

100 Teacher: Then, the cube is parallelepiped too, isn’t it?

101 Sol: Yes, it is. Chorus: No. No, it is not

102 Sol: I mean a cube may be a parallelepiped too, but it is a little bit different.

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103 Luc: In my opinion, it is not a parallelepiped, because it has square sides.

104 Teacher: What do you mean ‘square sides’?

105 Luc: I mean the walls.

106 Teacher: You mean the faces.

107 Luc: All of the faces of a cube are equal. On the contrary in a parallelepipedthey are not.

108 Teacher: You say that the faces of a parallelepiped are different.

109 Mat: Thus, the faces are two by two parallel, but a cube is not a paral-lelepiped. A parallelepiped has, by definition, six sides, two by twoparallel, but not equal.

110 Teacher: Have you said ‘six sides’?

111 Mat: I mean, six faces.

112 Teacher: Well, parallelepipeds have six faces, two by two parallel. It is O.K., foreverybody?

113 Fab: I want to say that they are two by two equal [parallel], but it dependson the figure, because that [he refers to the rectangular prism with asquare base] has four equal faces and two different, but equal to eachother; and the other has two by two equal faces.

114 Teacher: Well, here we have a nice collection, this figure has four equal facesand two different, but equal to each other, here there are two by twoequal faces, here all of the faces are equal. Our problem is that offinding a good definition for the parallelepiped.

[...]

122 Teacher: If all of them are squares it is a cube.

Well, you say that faces can be rectangles or squares, two by two equaland two by two parallel.

123 Chorus: Yes. Yes.

124 Teacher: Well, a parallelepiped is a solid bordered by six faces, which are twoby two parallel, two by two equal and of a rectangular shape.

125 Fab: Or of a square shape.

The main difficulty faced by pupils is represented by the fact that itseems impossible to assimilate the differences of the two solids into onlyone definition. Pupils refuse to accept the argument of the teacher whoconsiders the cube a parallelepiped. Although they accept the ‘definition’of the parallelepiped given by Solange (97: ‘I’d say that a parallelepiped hastwo by two parallel faces’), they do not accept that a cube is a parallelepipedtoo.

It is interesting to remark that the intervention of the teacher allows theemergence of an obstacle which otherwise would remain implicit. Insteadof accepting the given definition and stopping the process, she focuses onthe possible conflict between having a certain characteristic and belonging

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to a certain class. The attention is drawn to the fact that the cube hasthe properties required by the definition and for that reason it has to beconsidered a parallelepiped. The teacher makes the deduction necessaryto pass from the general to the particular case: the cube has two by twoparallel faces, then it is a parallelepiped.

The pupils’ reactions to this argumentation is very interesting. WhileSolange accepts both the premise and the conclusion, the rest of the classagrees with the premise, but definitely refuses the conclusion: a cube cannotbe a parallelepiped. The difference is so evident that even Solange is forcedto admit that the cube is ‘different’ (102).

All the contributions to the discussion aim to support the need of differ-entiating the particular cases, refusing to assimilate them into one generalclass and the possible definitions, as they are proposed, sound like a collec-tion of cases respecting figural differences. As a consequence, there seemsto be a regression from the ‘definition’ of Solange (97) to the ‘definition’accepted subsequently (122).

It is interesting to note the particular form of the definitions proposed,in particular the presence of a collection of cases, in contrast with the veryprocess of classification. As clearly shown, the difficulty consists in recon-ciling the process of differentiating, rooted in observing objects (the figuralaspect) and the process of generalisation which effaces differences throughthe equivalence relationship, stated according to common properties (theconceptual aspect). Pupils accept ‘in principle’ the definition proposed bySolange, but submit it to several exceptions. As Fabio says

(113) ‘[...] they are two by two equal [parallel], but it depends on the figure,because this (he refers to rectangular prism with a square base) hasfour equal faces and two different, but equal to each other, while theother has two by two equal faces’.

The complete symbiosis between the conceptual aspect and figuralaspect, under the rigorous control of the conceptual requirements, is notyet achieved. While the act of defining should consist, in mathematics, ofa double process from the particular to the general and vice versa, onlyone direction is considered. From the particular cases some features areabstracted (a parallelepiped has two by two parallel faces); but the oppositedirection is neglected: possessing a certain property does not guarantee theclassification of the cube as a parallelepiped.

This fact witnesses the difficulty in accepting the logical constraints ofa definition, in which all the figural differences are absorbed. As a lastconfirmation of this fact, let us consider one of the formulations given inthe following for the prism.

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(236) Mattia: So, the top and the bottom are equal and parallel, for some of them theyare triangles, for some of them they are other figures, and the lateralfaces are rectangles.

4.1.3. Classifications and definitionsThe previous examples clearly show the difficulty of harmonising the theor-etical dimension and the spontaneous dimension. As expected, the pupils’first reaction to a general task of classification produced a description of theobjects which does not have the characteristics of a mathematical defini-tion. The intervention of the teacher aims to point out the logical constrainsof a mathematical definition.

Actually, the basic aim of this activity was that of a first approach tothe process of generalisation, using the fact that pupils already knew someof those solids – their names and some of their properties – the teacherfocused on this process. Actually, she did not validate any definition, shejust stressed the necessity of respecting a stated characterisation.

As discussed above, the basic ambiguity of its objective is the weaknessof this kind of task. There is no explicit goal in respect to which theclassification should be undertaken; in particular, there is no element whichcould direct towards the standard structural characterisation of prisms.

On the contrary, in the following example a different kind of task willbe considered; in this case, the concept to be defined functionally emergesfrom the solution of a problem.

4.2. Figural and conceptual aspects in a problem-solving situation

The main topic of our experimental units deals with the unfolding/foldingproblem. Pupils were presented with two different kinds of tasks:

� Drawing a net of a certain solid (the cardboard models, a pyramid, ahexagonal prism, ... standing on the desk).

� Given a collection of drawings3, recognising the correct nets andcorrecting the incorrect ones (see Figure 2).

Collective discussions followed each of the two activities which wereperformed in small groups.

The first approach to the unfolding/folding problem is a practical one;the teacher explains what a net of a certain solid is in terms of unfold-ing/folding processes and pupils are asked to draw or to recognise a net ofa solid as the result of the unfolding/folding operation. Pupils immediatelyunderstand the problem, but perform the operation at different levels ofabstraction4.

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Thus, the didactic problem consists of how to develop their compre-hension of the process leading from the practical process to an ideal ‘geo-metrical’ notion of net. That is, identifying general properties of a ‘net’and giving up all those aspects which are not relevant for the geometricalcontext.

From the geometrical point of view the following definition can beaccepted.

A net is a plane figure, a union of polygons, connected by one and onlyone side, and in such a way that:

� each face of the polyhedron corresponds to one and only one polygonof the net;

� it is possible to pair the sides of the net’s polygons in such a way thateach pair corresponds to one and only one edge of the polyhedron.5

Such a formal definition of a net is not part of our aims at this level, butwe want to achieve a definition consistent with this one, where a net is a2-D representation of a solid, preserving certain properties (the number andthe shapes of the faces, together with a rule of connection) and neglectingmany others.

At the beginning of the slow development, we will describe the genesisof the idea of net arising from the interaction between the two components,the figural and the conceptual; the first rooted in the concrete operationof unfolding/folding, the second defining the properties that must be pre-served and which characterize the geometrical notion of net. As expected,this evolution, integrating the empirical and the theoretical aspects, is notspontaneous and the intervention of the teacher becomes fundamental tonegotiate the mathematical approach.

4.2.1. The problem of the flapsThe following discussion follows the collective work on the unfoldingproblem. The utterances’ exchangeclearly shows the difficulty in overcom-ing practical constraints which have no correspondence in the geometricalmodel.

The teacher observed that in their drawings some of the groups repres-ented the ‘flaps’, i.e. the small pieces of cardboard (in Italian, ‘linguette’,see Figure 1), which are used in the concrete folding in order to glue togeth-er the corresponding faces. The ‘flaps’ are needed to obtain a well mademodel, thus they are important elements from the concrete point of view,but they have no meaning when the unfolding problem is considered in thegeometrical domain. Thus, before starting the discussion on the unfolding

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Figure 1. A net of a cube with flaps.

strategies, the teacher decides to propose the problem to the whole class;she aims to introduce the problem of the relationship between the concreteoperation and the ideal notion of unfolding/folding. Here is an excerptfrom this preliminary discussion.

Excerpt 5: (6th grade, 2nd Collective Discussion, school year 1992–93)

1 Teacher: [...] the first thing to discuss, because the groups behaved differently,is the problem of the ‘flaps’; some groups did not draw them, othersdid. So, this is the first thing to discuss, it is important to clarify thispoint.

2 Davide: In my opinion, the ‘flaps’ shouldn’t be there; if we open a solid, anyreal [in Italian, ‘in natura’] solid, we don’t find the ‘flaps’, only if wewant to construct a cardboard solid, should they be there.

3 Mattia: I agree

4 Cinzia: I agree with Davide.

5 Teacher: You agree with Davide, do you want to say it in your own words?

6 Cinzia: Yes, in a geometrical figure there shouldn’t be any ‘flaps’. On thecontrary, if we want to construct it, we need them to glue the sidestogether.

7 Sara G.: I agree too, because this is the first thing I thought. I mean, if it is adrawing they don’t need to be there, we only imagine the figure, but fora construction that we really perform, they must be there, otherwise itdoesn’t close.

8 Teacher: Otherwise it doesn’t close.

9 Solange: I think that the ‘flaps’ must be put in because the net is the solid figuredetached, practically, and if we didn’t put the ‘flaps’ and we imaginedreconstructing it, some sides would be missing [in Italian, ‘ci sarebberodelle mancanze di lati’]

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[...]

25 Mattia: I don’t agree, because when we think of a geometrical figure, forinstance a cube, I immediately think of a cube, I don’t think thatit doesn’t glue (fold), so for a geometrical figure they [the ‘flaps’]shouldn’t be there.

26 Teacher: You are saying, as Cinzia said, if I have to construct an object with theshape of a cube, then the ‘flaps’ are required; but, if I think of a cubeas a geometrical figure, that is an idea and not an object, then this cubehas no ‘flaps’.

27 Giovanni: I agree with Davide. In a geometrical figure there are no ‘flaps’, becausethey keep themselves together [in Italian, ‘si tengono attaccati da soli’].

28 Teacher: Because they keep themselves together.

29 Fabio: I say, I agree with Davide too, because the ‘flaps’ are needed to con-struct cardboard objects, but for instance, this [he points to the top ofthe desk] is a geometrical figure and it doesn’t have any ‘flaps’.

30 Solange: I agree that in a geometrical figure there shouldn’t be any ‘flaps’, butfor the net, they are required; it is like we opened a solid figure.

[... the discussion goes on alternating arguments against and in favourof the necessity of the ‘flaps’; geometrical figures don’t have the ‘flaps’,but cardboard boxes do.]

[...]

84 Mattia: Because, when we imagine, when we draw a solid figure on the black-board, in the drawing the ‘flaps’ are not required, but, even if we unfoldit, I don’t think that somebody thinks that there are ‘flaps’, because thesides are lying over it.

85 Teacher: Do you mean the faces?

86 Mattia: Yes, the faces are laying over it, there shouldn’t be any ‘flaps’.

[...]

100 Solange: O.K., well, if the geometrical figure is an idea, then it has no thickness;but before, I thought, because it was a solid, it had a thickness and avolume.

After the first interventions, it becomes clear that the ‘flaps’ are requiredonly if a cardboard box must be really constructed (Fabio (29)). But theidea of net is still ambiguous, it moves between the concrete unfoldingprocess and the abstract idea of net. It is interesting to note that the crucialproblem is that ofharmonising the ideal of geometrical figures and theconcreteness of the unfolding process, as Solange (30) points out. Howeverthe ambiguity between a 3-D geometrical figure and a concrete object isstill present. Pupils mistake the third dimension of the geometrical figure

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for the thickness which determines the concreteness of a solid (Fabio (29),Solange (100)).

From this beginning we understand how difficult it is to reach a geomet-rical point of view harmonising the abstractness with the reference to theconcrete operation of folding/unfolding. As a pupil says: (Federica (diss.3 ’94) (140)) ‘The net is a way to represent the figure by unfolding it [...][in Italian, ‘lo sviluppoe un modo di rappresentare la figura aprendola’]’

4.2.2. The Geometrical Concept of NetPupils have already solved the folding task and a collective discussion isintroduced where the different solutions can be compared. The discussionaims at reaching the geometrical idea of a net of a solid and formulating afirst reasonable ‘definition’ of it. That is, conceiving and characterising anet as a particular 2-D representation of the solid, preserving specific geo-metrical properties, i.e., the shape of the faces, the connections through thecorrespondence between the sides of the net and the edges of the solid. Atthe beginning, the pupils’ solutions mainly resort to imagining the unfold-ing process: the pupils describe their strategies through the metaphors of‘blooming flowers’ or that of ‘peeling bananas’. For instance, Paola says:

We had a square pyramid. We put it on the table, standing on the square base, and weimagined unfolding it, like peeling a banana... From the vertex we pull the faces down.

Eleonora says:

We had a prism with a trapeze as base. We did not imagine peeling it. We cut it, we cut thebases and in doing this, it came out a rectangle with two small wings attached.

The comparison of different strategies applied to different solids andthe analysis of the different results for the same solid are the first approachto generalising the unfolding problem into the geometrical concept ofnet. During the first discussion, the teacher introduces the problem ofcharacterising a net, that is,she explicitly proposes to look for and state a‘definition’ of a net of a solid.

After the second activity referring to the recognition of the correct netsof a rectangular prism, the teacher comes again to the problem of defininga net and puts the question directly: ‘This morning we want to compare ourideas about nets, in order to understand which are the characteristics of nets.Let us start with drawing 1 (Figure 2). ‘Is it a correct net?’. Successively,all drawings are discussed.

From the beginning, and again and again, pupils propose solutionsresorting to mentally imagining the folding. That is, the solution processesremain very close to the practical meaning and have the structure of a‘recipe’. The general characters of the concept of net are not grasped.

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Since the first case (Figure 2, drawing 1), the inadequacy of this kind ofsolution clearly appears; Marco, Silvia and Roberta imagined folding thenet, but they do not agree on the result.

Marco affirms that the net is correct ‘because it folds’. Silvia andRoberta affirm that it is incorrect ‘because it does not fold’.

The teacher intervenes, but instead of validating one of the two answers,she asks how one could solve the problem differently; she tries to suggestsolutions at a ‘theoretical’ level and asks for general criteria. Only one ofthe pupils provides a reason which does not simply resort to the foldingprocess. Francesca says (6): ‘It doesn’t fold because (see Figure 2, drawing1) this face (A) cannot match with this one (B), because it is too long’. Butthe fact is not true and the argument does not seem to convince the otherpupils. They become convinced only after Marco succeeds in describingthe reconstruction of the solid. Actually, mentally folding the drawing 1(Figure 2) is a very complex process: successive transformations must becombined – and Marco has to overcome this difficulty. He says:

‘I fold this up (A) and also the other one (B) which drags the other two [faces] with it’.

For the following drawing (Figure 2, drawing 3), Giorgia refers to thelengths of the sides and refuses it, other pupils resort to the folding processand affirm that it folds, but Giorgia keeps to the point and Valentina mustadmit that ‘the side is a little bit too short’. The same problem arises forthe following drawing (Figure 2, drawing 4). Now the sides are too long.

The discussion shows how the abstract idea of folding is slowly grasped.In a dialectic way, pupils pass through different ‘approximations’ of theperfect folding; at first, the drawings whose faces are ‘too short’ are rejec-ted, then are the nets whose faces are ‘too large’. It is surprising howdifficult it is for pupils to overcome the concrete experience. For instance,when one of the faces is larger than it should be, the net can be acceptedbecause anyway, it can fold (‘close’) into the solid (see the following pro-tocol).

Excerpt 6: (6th grade, 3rd Collective Discussion, school year 1993–94)The discussion regards the drawing 4 (Figure 2) and pupils do not agreeon its correctness.

31 Marco: When I fold it, these [faces (A)] stick out.

32 Giorgia: I think that it is correct, because these two rectangles close, they close,I did it.

33 Marco: Yes, they close, but they stick out.

34 Nicola: Before, I thought that it was correct [in Italian, ‘andasse bene’], butnow, I understand that they are too large [in Italian, ‘grandi’].

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Figure 2. Given a collection of drawings, recognizing the correct nets and correcting theincorrect ones. This is a reduced copy of the drawings, the original was much larger anddifferences in lengths more evident.

[...]

38 Fabio: We had to unfold a geometrical figure, that is you [he refers to theteacher] gave us a geometrical figure.

39 Teacher: Let’s say, there was a certain ambiguity, because I gave you a cardboardobject and it was evident that the object had ‘flaps’. It was easy to seethem, because it was not perfectly glued, but we have to decide if wemust unfold that object or we must unfold the geometrical figure whichthat object somehow represents.

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40 Fabio: I think that we shouldn’t have to include the ‘flaps’, because you gaveus a cardboard figure, but we had to imagine it as if it were not madeof cardboard, rather of another material which doesn’t require ‘flaps’.

When one deals with the concrete folding/unfolding processes, thereis a certain degree of approximation which is accepted. On the contrary,as far as the geometrical net is concerned, a theoretical unfolding (orfolding) occurs, which is purely ideal and only in an ideal context can it beconceived. Actually, the task was formulated in an ambiguous way, that isit could be interpreted in both a concrete and an ideal way.

However, to reach the ideal level seems to be more difficult than wasforeseen and even if the reference to the cardboard model is overcome, theideal of the geometrical figure is not completely grasped (see Fabio (40)).The difficulty in abandoning the object and the concrete operation of fold-ing is shown by the persistence of this kind of mistake; in the case of thedrawing 7, one can imagine folding it, but two of the faces overlap. Fromthe point of view of the concrete folding, this fact is acceptable. However,it is not acceptable from the point of view of the geometrical idea of net.

Excerpt 7: (6th grade, 3rd Collective Discussion, school year 1993–94)

53 Teacher: What about drawing 7?

54 Moreno: In my opinion it is not correct, because there is an extra side [inItalian, ‘un lato in piu’].

55 Lisa: An extra face!

56 Teacher: There is an extra face, do you agree?

57 Valentina C: Yes, I do, but it can be folded anyhow.

58 Giorgia: But you can’t put one over the other.

So, the relationship between the concrete and the ideal notion of netmust be negotiated, and general characteristics of a net must be statedexplicitly.

The discussion goes on as following.

Excerpt 8: (6th grade, 3rd Collective Discussion, school year 1993–94)

135 Valentina B: The net is a way to represent the solid and first of all one mustcheck the number of faces.

136 Michela: Yes, let’s see if there are all the faces.

137 Lisa: But we make them smaller [in Italian, ‘le rimpiccioliamo’].

138 Valentina C: Yes, but they must be of the same shape.

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139 Teacher: Well, the shape is the same, and then, you make them smaller. So,each face of the solid appears in the net with the same shape.

140 Federica: Yes, a net is a way to represent the solid figure, by unfolding it,so, I agree on what we said about the faces, but the faces must beconnected, they cannot be one here and the other there (scatteredhere and there).

141 Teacher: Sure, this is important. So, this is not the only way to represent thesolid ...6

It is not possible, and beyond the scope of this paper, to describe inextent the teaching experiment, but it is important to say that, along theway towards the idea of net, pupils faced the need of introducing anddefining the geometrical concepts of ‘edge’ and ‘vertex’ (Mariotti, 1996).

The excerpts previously discussed, clearly show the evolution from thesituation, centred on the practical problem of constructing the object, tothe situation centred on the geometrical idea of net. The basic aim is that ofdeveloping a good harmony between the figural and the conceptual aspect,towards the geometrical idea of net.

Such an evolution is directed by a number of interventions by the teach-er, who explicitly refers to the aim of defining or asks for solutions which donot simply resort to the folding process, but consider the specific propertiesof the net. Later on, the relationship between the two kinds of situations– the practical and the theoretical ones – is openly discussed, when Fabio(Excerpt 6, (38)) points to the specificity of ‘unfolding geometrical fig-ures’ and the teacher explains the ambiguity of the task (Excerpt 6, (39)).It is important to stress the fact that even in this case where the conceptfunctionally emerges from the problem situation, the intervention of theteacher is determinant: she explicitly ask pupils to formulate a charac-terisation of a net overcoming the reference to the practical operation offolding/unfolding, in so doing she directs pupils to overcome the conflictbetween the practical and the theoretical approach.

5. DISCUSSION

The previous examples highlighted the great complexity of the definingprocess in the classroom situation. The episodes concerning the classi-fication task and that concerning the unfolding/folding task focus on thebig gap between the spontaneous processes of conceptualisation and thetheoretical approach to definitions.

A classification task consists of stating an equivalence among similarbut figurally different objects, towards a generalisation. That means over-coming the particular case and consider this particular case as an instance

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of a general class. In other terms, the process of classification consists ofidentifying pertinent common properties, which determine a category.

In the case of geometry, theoretical classifications often resort to struc-tural criteria which are not immediately clear, and certainly are far fromthose perceptual criteria to which we are used to refer in our spontaneousactivity of classification. The previous example clearly shows this phe-nomenon and its consequences, in terms of definitions. Thus, very oftentheoretical classifications conflict with spontaneous ones, and generalisingrequires to overcome differences corresponding to perceptual properties.A process of generalisation, as it occurs in a theorisation, states an equi-valence (similarities), overcoming differences; but, as Montaigne said: ‘Itis not similarity that identifies, rather differences’ (de Montaigne, 1970).

The process of generalisation requested by a theoretical definition con-flicts with the need of differentiating. Difficulties arise when theoreticalconstraints state the equivalence between ‘different’ things, requiring tocancel the variety once for all.

Such complexity concerning classification and definitions suggests areflection about the van Hiele theory and the hierarchy of levels that itintroduces. As commonly accepted (see Clements & Battista, 1992), thereare the following five levels:

1. The visual level: The student identifies geometric figures as visualgestalts.

2. The descriptive-analytic-level: The student characterises shapes accord-ing to their properties.

3. Abstract-relational: The student is able to use abstract definitions andconsider the hierarchical organisation of figures.

4. Formal deduction: The student understands the organisation of geo-metry as a formal, deductive, axiomatic theory.

5. The student becomes able to reason about geometrical entities withoutreferring to figural models resorting only to mathematical statements(axioms, definitions and theorems).

Although the van Hiele model is a useful tool for analysing the develop-ment of geometrical reasoning in the individual, this model does not graspthe complexity of the defining process as it comes out from the previousexamples. What the van Hiele theory tends to overlook is the complexdynamics of concepts and figures; for instance, even a subject who knowsthe definition of a certain category may fail in identifying correctly thecategory to which a certain figures belongs. A student may quote correctly,some definitory properties (critical attributes) and overlook others or relysometimes implicitly on unnecessary properties. Concepts are often, impli-citly or explicitly, distorted by gestalts. A student may accept the statement

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(Excerpt 3, (97), (99), (102)) that a parallelepiped has two by two parallelfaces and, at the same time, reject the idea that a cube is a parallelepiped.Why? Because (Excerpt 3, (107) in a parallelepiped the faces are not equal.

The basic hypothesis of the van Hiele theory is that figural models tendto disappear with the progress of geometrical reasoning, as an effect ofinstruction. In our opinion, what changes with age and instruction, are therelationships between figure and concept in geometrical reasoning. Ini-tially, the figural-gestalt constraints are dominant. Stepwise, the role offormal constraints becomes more and more important until the constitu-tion of, what we have called, figural concepts: complex figural-conceptualentities in which the synthesis tends to be perfect, and in which the formalconstraints become rigorously dominant. But this is only the final, idealstage. Along this progressive evolution, the dialectical dynamics betweenconcept and figures in geometrical reasoning never disappears and con-flictual relationships between conceptual and figural constraints may arise,mainly because spontaneous conceptualisation may not match geomet-rical definitions. Neverthelss figures continue to play an essential role insuggesting meanings, interpretations, relationships, solutions. The mostabstract mathematician resorts to figural models, to images, in his geo-metrical reasoning. This is what Hilbert himself, one of the great foundersof axiomatics, has clearly stated: ‘Who does not always use, along withthe double inequalitya > b > c, the picture of these points following oneanother on a straight line, as the geometrical picture of the idea ‘between’?’(cf. Reid, 1970: p. 79).

From the point of view of figural concepts, achieving and correctlyusing a definition requires that a good harmony between figural and con-ceptual aspects is realised, which takes into account the theoretical con-straints of formal definitions and openly admits the possible discrepancybetween empirical and geometrical concepts. The great difficulty is causedby the conflict which emerges, very often, between the need to differenti-ate, imposed by strong figural structures and the requirement to unify, togeneralise imposed by the geometrical conceptualisation. A certain image,a certain instance tends to become the paradigmatic model (the prototype)for the entire class. But the prototype introduces properties, perceptuallyrelevant, which do not conform to the general requirements of the definition(‘in a parallelepiped the faces are not equal’ according to the usual model).And thus, the conflict emerges. In empirical domains, one tends to adapt thedefinitions to the empirical data – and exceptions are admissible (Excerpt3, (113)). In mathematics, the definition dictates rigorously, the meaning,sometimes in disagreement with the figural-perceptual constraints.

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As a consequence, a didactic problem arises: how to strengthen theharmonisation between these contrasting aspects. According to our hypo-thesis, the contribution of different opinions (Bartolini Bussi, 1995) in thecollective discussion, allows the conflict to appear.

As far as classifications are concerned, the dynamic of the discussionbecomes essential in order to stimulate the dialectic between the generaland the particular. It makes the exceptions emerge, instead of remainingimplicit in a general tacit agreement. According to our findings, a cru-cial role is played by the teacher in the very process of defining: certainproperties are stated, then all the members of the class must have theseproperties, but also all the objects with those properties must belong to theclass. The teacher mediates the mental process requested to verify whetheran object belongs to the class: in our example, she provocatively selectsthe cube. She guides the discussion and mediates that part of the definingprocess (from the general to the particular) necessary in order to transforman indefinite description into a ‘definition’.

But often the intrinsic difficulty of defining consists of making senseof the choice of the properties according to which the classification isachieved; in the majority of the cases, the pertinence of a property for aclassification may remain hidden to the pupils.

Although difficult, it seems useful to find a problematic context withinwhich the significance of a definition arises. Because of the complexityand the relevance of the process, it seems important to offer pupils theopportunity of experiencing the dialectic interplay between empirical andtheoretical dimensions of a task. The example concerning the definitionof a net can be considered from this point of view. Nets have an evidentfunctionality in respect to the unfolding task, that students immediatelyrecognise. Thus it seems possible to negotiate a ‘definition’, both correctfrom the mathematical point of view and efficient with respect to thesolution of the given problem. In the negotiation process the delicate roleof the teacher within the dynamic of a collective discussion becomesfundamental. The teacher promotes a specific attitude of the pupils towardsthe solution of a problem, so that the ‘theoretical’ aspect may be explicitlyintroduced and pupils progressively achieve a geometrical point of view.

ACKNOWLEDGEMENTS

We thank Claudia Costa for her precious and competent collaboration torealise the teaching experiment.

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NOTES

1This is a more general term but pupils have known it since the primary school and useit with the restricted meaning of rectangular prism (in the transcript and in the followingwe have maintained the original term parallelepiped).

2Pupils use the term parallelepiped, see footnote 1.3The drawings’ size is reduced, this makes figures appear different from how they

appeared to pupils.4A detailed description of the different kinds of solution processes in agreement with

other studies (Potari and Spilotopoulou, 1992), can be found in Mariotti, (1996).5It is interesting to remark that in such a definition any link with the unfolding/folding

operation is cut.6The last intervention of the teacher refers to the subsequent development of geomet-

rical activities: that is the project of introducing the geometrical problem of perspective.Certainly, the idea of a net as a ‘representation’ of the solid, preserving specific geometricalproperties is very far from the concrete problem of constructing a cardboard box, but thisis a characteristic of geometry: rooted in the concrete experience, geometry is a theoreticaldomain where different things become very close.

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Maria Alessandra MariottiDepartment of MathematicsUniversity of PisaPisa, Italy

Efraim FischbeinTel Aviv UniversitySchool of EducationTel Aviv, Israel

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