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1997
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 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: 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.
In his Rules for the Direction of the Mind , Descartes wrote:
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. © Michael Otte 1998 |
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 |
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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 HoudebineLes 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 Lismont15h45 : 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 : Ateliers9 - 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 GiorgiuttiEn 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
Université de Paris
VII Séminaire de didactique des mathématiques 1997/98Le mercredi 21 janvier 1998, de 14h a 16h, Tour 46-0, salle 408, campus Jussieu à Paris
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Université Joseph
Fourier, Grenoble Seminaire DidaTech Mercredi 4 février 1998, de 14h a 16h bâtiment C,SALLE 310, 46 avenue Felix Viallet, Grenoble Equipe EIAH, Laboratoire Leibniz 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. |
NCTM 1999 Yearbook on
Mathematical reasoning. |
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