Preuve Proof Prueba

Web Newsletter
Janvier/Février 1998


1997 : visites






Galindo E., Birgisson G., Cenet J.-M., Krumpe N., Lutz M. (1997). The development of students' notions of proof in high school classes using dynamic geometry software. PME NA XIX (pp.207-214).
Walen S., Anderson D. (1997). Pre-service teachers' validations of mathematical solutions. PME NA XIX (pp. 502-503).
Knuth E., Elliott R. (1997). Preservice secondary mathematics teachers' interpretations of mathematical proof. PME NA XIX (pp. 545-551).
Heid M. K., Blume G., Flanagan K., Kerr K., Marshall J., Iseri L.(1997). Conjecturing and representational style in CAS-assisted mathematical problem solving. PME NA XIX (pp. 585-592).
Reid D. (1997). Jill's use of deductive reasoning: A case study from grade 10. PME NA XIX (p. 667).


The Theme of the Letter

Proof and perception
an original contribution by Michael Otte


The goal of the Theme of the Letter is the stimulation of exchanges on current questions about the learning and the teaching of mathematical proof. I have invited Michael Otte to offer a contribution related to the theme visualization which was initiated in the May/June Newsletter 1997.


Our humanistic and philosophical culture is completely permeated and entrenched with language. Language controls reasoning and thought and even emotion. And the separation of knowledge from speech or language is an extraordinarily difficult accomplishment, which each literate society must struggle over a prolonged span of time with a very mixed outcome in general. In mathematics as well as in the philosophy of mathematics the dominance of language has had a widespread influence up to now. Even intuitionism in general has abandoned Kant's a priori intuition of space, adhering the more resolutely to the apriority of time as an inner sense (see for instance Brouwer's inaugural address of 1912). Or, to mention just one example from the philosophy of science and mathematics, the analysis-synthesis distinction has been obscured largely by the fact that the problem of synonymy and of the indeterminacy of translation was considered more fundamental than the problem of perspectivity and theory-ladeness of empirical observation.

Mathematics, against that, has since Greek antiquity been a science of the eye and of form, or a visual art. Still since the Renaissance views began to become mixed. On the one hand the Great Book of Nature is written, as Galilei stressed, in mathematical language, in triangles and in other geometrical figures. On the other hand, it was widely believed that if you want "to write for people who are interested but not learned, and make this subject [geometry; M.O.] accessible to the common people and easily understood by anyone who studies it from your book", you must "employ the terminology and style of calculation of Arithmetic, as I did in my Geometry", Descartes writes in 1639 to Desargues. Nevertheless Descartes did believe that mathematical truth is constituted by intuition or perception. But it was Kant above all others, who had emphasized that mathematical "judgments are always visual, viz., intuitive" and who combined this view with a constructive epistemology. The fundamental question of mathematical epistemology therefore is, how activity (conceptualization, construction and deduction) and perception interact.

Let us consider some examples:

Elsewhere (cf. Otte 1994, chap 9, 252ff), I have shown that the function of the logical approach and of the concepts of mathematics and of the natural sciences consists in transforming a dynamic unclearness and chaotic motion of temporal processes and activities into surveyable images or into a form. Mathematics - a Human Endeavor by H.R. Jacobs provides a very simple but pertinent example:
   Somebody wants to cut up a cube that measures 3 meters on an edge into twenty-seven 1-meter cubes. Six cuts will certainly suffice. "Is it possible to do the job with less than six cuts if the pieces are rearranged between each cut? ... This seems like a very difficult problem, since the number of pieces increases with each cut and there are so many ways of rearranging them" (Jacobs loc.cit.).
   Now every cube except one will have at least one face that was originally part of the surface of the big cube. The one exception is the cube at the center, which has every one of its faces formed by cuts. Since this cube has six faces, six cuts are necessary to form it. Thus the problem is solved by forming the appropriate concepts ("cube at the center") or by finding the appropriate perspective and from there then reason deductively, rather than performing a lot of simulations or inductive trials.

This idea of seeking for a being in which the theoretical concept can be anchored has come down on us from Parmenides (ca. 500 BC). To Parmenides, modernity has added the element of construction or of activity which is the very factor which permits the perceptual element in mathematics and the natural sciences to attain its full effect because what we perceives is not the world in itself but rather our own constructions. Instead of looking outward into nature, merely receiving it, we are conducting experiments. Instead of merely analyzing the premises of a mathematical theorem to be proved, the mathematician constructs a diagram and a concept which help him on.

©Leiter 1996, Le Monde 1996

Let us further consider the following related problem.
   27 cubes are given to 4-5 year old children. The cubes are coloured in a variety of ways having red, blue, green, yellow etc. faces in various combinations. The children are asked to construct a big cube from these 27 little cubes but in such a way that it is blue everywhere outside. The childrem at first eagerly pick up those little cubes that show predominantly the desired colour "blue" and begin the construction immediately. But they soon end up in frustration by discovering that there are not sufficient "blue" cubes.
   At this moment they are asked to analyze how many blue faces a little cube needs, depending on its location within the overall construction. The cubes at the vertices of the big cube obviously must have 3 blue faces etc. ... After having in this manner formed the "concepts": vertex-cube, edge-cube, face-cube, inner-cube they encounter no difficulties in completing the task rapidly.

A third example: Intuition is forceful. On the other hand an absolute insight or intuition does not exist. This is very often misunderstood.
   For example, the well-known Gestalt psychologist Max Wertheimer (1880-1943) comments on the presentation and solution of Zeno's paradoxes by means of a geometric series that is current in present day mathematics. Rather, he comments on the current proof of the convergence of that series, which is accomplished by multiplying the series by a and subtracting afterwards. Set S = 1 + a = a2 + ... Then S - aS = 1 or S = 1/(1 - a).

"It is correctly derived, proved, and elegant in its brevity. A way to get real insight into the matter, sensibly to derive the formula is not nearly so easy; it involves difficult steps and many more. While compelled to agree to the correctness of the above proceeding, there are many who feel dissatisfied, tricked. The multiplication of (1 + a + a2 + a3 + ...) by a together with the subtraction of one series from the other, gives the result; it does not give understanding of how the continuing series approaches this value in ist growth."

Mathematics does not operate with objects but with thoughts or ideas and makes the latter perceivable or observable as forms or diagrams. Diagrams are models in the strict logical sense. This is what intuition is essentially about: seeing the essence of a thought or object as a form or object itself.

This definition, however, reappears also in the determination of formal logic and is summed up in the so-called immediacy claim for formal systems. Only the text, the sign, the formal can in the last instance, be given in a pure state and thus does not entail any further problems of meaning and justification. On this basis Hilbert had called formal logic a material logic. "The problem of logic is a very direct one: how can a proposition tell something about itself?" The starting point of this problem is the assumption that every proposition immediately implies itself. If I say "p is true" this means "p" is true. Nothing is added to the proposition "p" by the words "is true". The difference between intuition and logic must thus be sought in other circumstances. But what is the resulting difference? Where do we find it? Upon closer consideration it seems as if the intuitive were nothing but a compressed and unresolved logic. So the difference is a matter of time, and thus a matter of the finite or infinite character of the cognitive subject. This exactly was Descartes' view.

In his Rules for the Direction of the Mind , Descartes wrote:

"We have now indicated the two operations of our understanding, intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge. ... Truly we shall learn how to employ our mental intuition from comparing it with the way in which we employ our eyes. For he who attempts to view a multitude of objects with one and the same glance, sees none of them distinctly; and similarly the man who is wont to attend to many things at the same time by means of a single act of thought is confused in mind. ... It is a common failing of mortals to deem the more difficult the fairer; and they often think that they have learned nothing when they see a very clear and simple cause for a fact, while at the same time they are lost in admiration of certain sublime and profound philosophical explanations, even though these for the most part are based upon foundations which no one has adequately surveyed - a mental disorder which prizes the darkness higher that the light" (Rule IX).

To be able to perceive or intuit something we thus have first of all to invest into its analysis and into constructive synthesis, in order to end up with something which we can perceive clearely.

Reaction? Remarks?

Reactions to Otte's contribution will be published in the March/April Newsletter
© Michael Otte 1998


Trimestre Intensiu en Educaciò Matematica

Tomy Dreyfus
The role and nature of proof in highschool

24 de febrer, 18:30, als locals del Centre de Recerca Matemàtica ubicat a la Facultat de Ciencies de la Universitat Autònoma (Bellatera).

el TIEM Web



Vanda Luengo a soutenu le lundi 29 septembre 1997 une thèse pour le doctorat de didactique des mathématiques de l'Université Joseph Fourier, sous le titre :

Un micromonde de preuve intégrant la réfutation.

Cette thèse a été préparée au sein du projet Cabri-géomètre et de l'équipe EIAH du laboratoire Leibniz sous la direction de Nicolas Balacheff.

Produire et lire des textes de démonstration

Rennes 23 et 24 Janvier 1998

Colloque organisé par le Laboratoire de Didactique des Mathématiques de l'Université de Rennes I



Vendredi 23 Janvier

9h30 : Présentation des journées

Les textes de démonstration

9h40 : Une approche linguistique des textes de raisonnement, Conférence d'Isabelle Beck
10h50 : La diversité des textes de démonstration ; Conférence de Jean Houdebine

Les textes des enseignants

14h : Ateliers

1 - Des propositions d'enseignants : Dominique Hilt et Marie Annick Juhel
2 - Les manuels et la démonstration : Marie Agnès Egret
3 - A partir de quelques textes historiques : Jean Paul Guichard
4 - Ecrire pour apprendre et apprendre à écrire. Critères d'un langage pour décrire, démonter, démontrer : Françoise Van Dieren et Luc Lismont

15h45 : Etudes de textes historiques ; Evelyne Barbin

Les textes des logiciels

17 h : La démonstration dans les EIAO de géométrie ; conférence de Dominique Py

18h : Ateliers

5 - Premier pas ; André Simon
6 - Menthoniez ; Dominique Py
7 - Une analyse de messages d'un logiciel d'apprentissage : DEFI ; Bahia El Gass et Italo Giorgiutti
8 - Cabri-Euclide ; Vanda Luengo

Samedi 24 janvier

Les textes des élèves

8h30 : Généalogie cognitive des textes ; Conférence de Raymond Duval
10h : Ateliers

9 - Difficultés d'élèves de 3ème dans un problème de démonstration en géométrie ; Hanène Abrougui-Hattab
10 - Productions d'élèves de quatrième ; Nicole Bellard et Martine Lewillion
11 - Analyse de copies d'élèves ; Jean Houdebine
12 - Narrations de recherche point d'appui pour la démonstration ; F. C. Combes et F. Bonafé
13 - Etudes de cas ; Italo Giorgiutti

En guise de conclusion

14h : Analyse d'un texte de démonstration dans des cadres théoriques différents : Evelyne Barbin, Raymond Duval, Jean Houdebine, Colette Laborde
15h15 : Débat
16h30 : Clôture


Laboratoire de Didactique des Mathématiques
Université de Rennes I - Campus de Beaulieu
Avenue du General Leclerc
35042 Rennes Cedex

Tel : 0299286003, Fax : 0299281638

Université de Paris VII

Séminaire de didactique des mathématiques 1997/98

Le mercredi 21 janvier 1998, de 14h a 16h, Tour 46-0, salle 408, campus Jussieu à Paris


Conditionnels, necessites et contingence dans la classe de mathematiques : aspects theoriques et illustration.


Université Joseph Fourier, Grenoble
Laboratoire Leibniz

Seminaire DidaTech

Mercredi 4 février 1998, de 14h a 16h bâtiment C,SALLE 310, 46 avenue Felix Viallet, Grenoble

Equipe EIAH, Laboratoire Leibniz

La démonstration dans l'enseignement tunisien : étude d'exigences d'enseignants et de difficultés d'élèves.


Résumé : Beaucoup d'élèves éprouvent, en quatrième et au delà, des difficultés pour comprendre ce qu'est une démonstration ou pour rédiger une démonstration. Les enseignants, de leur côté, restent, pour la plupart, persuadés qu'il est très difficile de faire progresser, de manière sensible, un élève "peu doué". Même s'ils réussissent à transmettre les connaissances indiquées dans le programme, il leur est difficile de développer les capacités à démontrer. Cela constitue l'un des problèmes les plus connus contre lequel bute l'enseignement des mathématiques. Ce constat est l'une des raisons essentielles qui ont motivé notre intérêt pour l'étude de cette notion. En effet, pour essayer de comprendre cette résistance, il nous a paru nécessaire de connaître, de façon précise, les phénomènes d'enseignement de la démonstration en jeu, en Tunisie.
   Nous avons décidé de nous pencher sur le problème en nous intéressant aux deux pôles du système didactique: l'enseignant et l'élève. L'étude menée auprès des enseignants vise à identifier leurs exigences à propos de l'élaboration d'une démonstration et ceci à partir de l'évaluation qu'ils font de copies d'élèves. Quant à l'expérience menée auprès des élèves, elle vise l'étude des interprétations qu'ils donnent au passage d'une hypothèse à une conclusion, dans un pas de la démonstration.
   Dans l'exposé que nous ferons, nous présenterons la problématique de notre recherche, en précisant les questions que nous nous sommes posées et nos hypothèses de départ. Ensuite, nous décrirons la méthodologie mise en place afin de répondre à ces questions. Enfin, nous présenterons quelques résultats issus des expérimentations qui se sont déroulées auprès des enseignants et des élèves.

Responsable du séminaire : Madeleine Eberhard

Current call for papers

NCTM 1999 Yearbook on Mathematical reasoning.
Information and guidelines:  under "Educational Materials / 1999 Yearbook"

See also Proof Newsletter January/February 1997


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