Empirical diversity to make mathematical

objects exist

Thierry DiasHaute Ecole pédagogique du canton de Vaud

Lausanne, Suisse

thierry.dias@hepl.ch

Thierry Dias – juillet 2014

Main epistemological references for this communication:

Petitot, J. (1987). Refaire le Timée : introduction à la philosophie mathématiques d'Albert Lautman. Revue d'histoire des sciences, 40(1), 79-115.

Petitot, J. (1991). Idéalités mathématiques et réalité objective. Approche transcendantale. Hommage à Jean Toussaint Dessanti, Trans Europ-Repress, Mauvezin.

Descaves, A. (2011). L'apprentissage du sens, certes ! Mais dans quel ses prendre le sens ? Actes du colloque de la COPIRELEM, Tours.

Conne, F. (1999). Faire des maths, faire faire des maths, regarder ce que ça donne, In Le Cognitif en didactique des mathématiques, G. Lemoyne et F. Conne eds, presses universitaires de l'UdeM, p.31-69

Thierry Dias – juillet 2014

1. About reality of mathematical objects

2. Situation, knowledge/knowing/experience

3. Environments and things to make

mathematical objects exist.

Thierry Dias – juillet 2014

1. About reality of mathematical objects

• Necessity of an epistemological position

• Dealing with the question of sense

• Sharing the real

• Distinguishing things and objects

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Four pillars of the epistemology of the

mathematics(Petitot) :

- The constitutions of the mathematical activity

- The status of knowledge and of symbolic legality

- The problem of the donation and of the reality of

objects and of mathematical structures

- The nature of the applicability of mathematics to real-

world experiences.

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Mathematical objects can be considered as

principles of coherence.

Mathematical objects are correlated to acts

operating on a perceptive given.

(Descaves, 2001) (our translation)

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subsomption*(catégorisation)

langage formel

Faits(et divers)

empiriques

Faits(et divers)

empiriques

CONCEPTSschématisation*

modélisation

corrélationcorrélation

MATHEMATIQUES

pas de dénotation

*subsomption au sens Kantien: rapporter la pluralité des données de l'intuition à l'ensemble des concepts purs de l'entendement

choseschoses

expériences sensibles

expériences sensibles

*schématisation au sens (relativement) Kantien: passage par des schèmes intermédiaires entre entendement et sensibilité

objetmathématique

spécifique

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Meaning and reality ?

Let us dare to change the point of view:It is not concrete situations that give meaning (i.e. coherence) to mathematical objects, but rather mathematics and their objects which determine the forms of reality. " We must not identify the meaning of mathematical objects with their use in concrete situations. " ( Descaves) (our translation)

Thierry Dias – juillet 2014

In particular the coherence of rules operating

on symbolic system gives meaning to

situations and to the possible pupils’

experiences they cause.

See the iceberg metaphors proposed by Drijvers.

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• To overtaken juxtaposition of experiences.

• What makes sense is not the experience but the

theory connected to the mathematical objects (their

syntax, their semantics).

• To avoid believing in or expecting the fortuitous

meeting of knowledge during experiences:

experiment / vs manipulate.

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Sharing real (Lelong, 2004) : necessity (however

not sufficient) of interactive social experiences.

A community of individuals (students for

example) collaborating in acts and words in

order to elaborate concepts through

categorisation.

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Distinction things/objects (Conne)

To teach consists in featuring objects that

pupils perceive as things with which they can

interact.

For various persons, the same thing will not

refer necessarily to the same object.

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conclusion of the epistemological position:

Reality can be associated with the notion of

veracity (Granger, 1999): objects, things, acts and

thoughts are, or, at least, can occur.

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2. Situation: knowledge/knowing/experience

Situation is a theoretical modelling of a level of

reality taking into account various types of

interactions that it is necessary to distinguish

according to the status of the interacting

subjects and objects.

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Didactic purpose: the meeting between objects and subjects is successful

The environment created for exchanges of signs and of

knowledge and for the experiences on things does not

guarantee that mathematical objects will be met.

It is the consistency of the situations (the richness of

their milieu) that allows or not the process of

conceptualisation.

.

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savoirssavoirs

connaissancesconnaissances

situationexpériences

environnement d'apprentissageenvironnement d'apprentissage

faits et phénomènes divers empirique

objetsobjets

choseschoses

signesreprésentations

point de vue enseignantpoint de vue enseignant

point de vue apprenantpoint de vue apprenant

Thierry Dias – juillet 2014

3. Experiences in order to make mathematical objects exist.

Or how to operationalize the epistemological principles…

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Various contexts of implementation of experimental situations:

- Pupils with specific needs

- Prospective teachers

- In-service teachers

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• We propose to pupils specific and adapted milieu

able to provoke experiences and personal and/or

collective creations,

• We add in these environments specific constraints

(sometimes during the resolution),

• We observe and interact with the proposals made by

the pupils.

Thierry Dias – juillet 2014

The choice of Space and Geometry

To become geometrical, the a priori sensible space (the

representative space) must be idealized. Although

empirically forced, this process of idealization is

empirically (and experimentally) undecidable. It is a

matter of formal and a priori faculty of intellectual

abstraction, which is autonomous with regard to

sensible space. (Petitot, 1987). (our translation)

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The choice of Space and Geometry:

Escaping the influence of the numerical

formalism.

Favouring the move from local to global

(facilitation of the process of generalization).

No preliminary pregnant formalism.

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example 1 :Building in big

Originality of the situation: its variables of size (3-D, length and area). Things: baguettes, connectors, thread.Objects: plan, angles, symmetries, limit. Experimental task: build, observe, anticipate, understand.

des fuzzy constructions

des fuzzy constructions

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The experiment consists in intervening,

anticipating, transforming and verifying , in

a chronology of acts which belongs to every

subject but which call out to one another

constantly.

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Show, indicate, experiment things which can be the modelling of ideal mathematical objects.

Build in big: vary the registers of representation

Thierry Dias – juillet 2014

To vary the registers of representations of

the geometrical objects and to make links

may reveal knowledge and capacities of

pupils and/or teachers learning

mathematics.

Thierry Dias – juillet 2014

Spatial and physical stake in the construction of the geometrical knowledge

Meeting geometrical objects via categorization and networking of the knowledge

réel partagé

réel partagé

Thierry Dias – juillet 2014

Build in big: to make different experiences, to express and change one’s point of view.

Thierry Dias – juillet 2014

exemple 2 :Pave Space with the Platonic solids

Originality of the situation: its cultural and aesthetic

anchoring

Things: envelopes, boxes

Objects: plan, angles, symmetries, limit

Experimental task: explore, put in relation

Thierry Dias – juillet 2014

thierry.dias@hepl.ch

http://perso.orange.fr/dias.thierry

thierry.dias@hepl.ch

http://perso.orange.fr/dias.thierry