MARIA ALESSANDRA MARIOTTI and EFRAIM FISCHBEIN
DEFINING IN CLASSROOM ACTIVITIES
La mathematique est lart de donner le memenome a` des choses differentes
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.
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 thescientific 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 Mathematics 34: 219248, 1997.c
1997 Kluwer Academic Publishers. Printed in the Netherlands.
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220 MARIA ALESSANDRA MARIOTTI AND EFRAIM FISCHBEIN
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 objectsImplicit definitions stated through the axioms characterize the objectswhich the theory is dealing with. For instance, consider the concept ofgroup 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 the
same 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. 12)The long tradition of geometrical concepts makes it difficult for us to
be 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 elementLet 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 the definiendum withthe definiens the 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 (197273) on thehistory 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 the very concept of a regularsolid (stress is mine).
(Waterhouse, 19721973: 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 conceptsBesides 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. t...