Preuve Proof Prueba

Web Newsletter
Septembre/Octobre 1997

 

31 08 1997

>1000

Nouveautés

News

Noticias

Keith J. (1997) Student-Teachers' Conceptions of Mathematical Proof. Mathematics Education Review 9, 23 - 32
Ibañes M. (1997) Alumnos de bachillerato interpretan une demostración y reconocen sus funciones. Uno 13, 95-101.

Johson-Laird P. N., Byrne R. M. J. (1997) Deduction. Psychology Press. USA: Washington D.C.

Textes "on-line" :

Boero P., Pedemonte B., Robotti E. (1997) Approaching theoretical knowledge through voices and echoes: a Vygotskian perspective. PME XXI (pp.81-88) Lahti, Finland. English Italiano

PME 21 research reports

Barnard T., Tall D. (1997) Cognitive units, connections and mathematical proofs. PME XXI (Vol.2 pp. 41-48). Lahti, Finland.
Brown L., Coles A. (1997) The story of Sarah: seing the general in the particular? PME XXI (Vol.2 pp. 113-120). Lahti, Finland.
Furinghetti F., Paola D. (1997) Shadows on proof. PME XXI (Vol.2 pp. 273-280). Lahti, Finland.
Godino J. D., Recio A. M. (1997) Meaning of proofs in mathematics education. PME XXI (Vol.2 pp. 313-320). Lahti, Finland.
Moc Ida Ah Chee (1997) A hierarchy of students' formulation of an explanation. PME XXI (Vol.3 pp. 248-255). Lahti, Finland.
Reid D. A. (1997) Constraints and opportunities in teaching proving. PME XXI (Vol.4 pp. 49-55). Lahti, Finland.
Potari D., Triadafillidis T. A. (1997) Studying children's argumentation by incorporating different representational media. PME XXI (Vol.4 pp. 230-237). Lahti, Finland.
Zack V. (1997) "You have to prove as wrong": Proof at the elementary school level. PME XXI (Vol.4 pp. 299-306). Lahti, Finland.

 


Research on Mathematical Proof

a PME 21 Research Forum
(Lahti, Finland, July 14-19)

Chairperson: Carolyn Maher


Maria-Alessandra Mariotti and Carolyn Maher in Lahti

The research workhop was organised around a common presentation of three Italian research projects conducted in Genoa, Modena and Pisa. The Italian team was composed of Maria Allessandra Mariotti, Mariolina Bartolini-Bussi, Paolo Boero and Rosella Garutti. Mariotti presented, during a first session, the theoretical framework and the main findings of the common approach of geometry theorems in contexts.
   The key issue of the research framework presented is the concept of cognitive unity "between the process of statement production and the construction of a proof"; in other words, the unity of the processes of conjecturing and proving.
   The reader can get the text associated to this presentation on this web site. Mariotti's talk was followed by a reaction by Michael de Villiers and by a reaction by Guershon Harel.
   The second session of the research forum began with the response of our Italian colleagues to the reactors, and followed by a stimulating discussion.

My own reaction to the Italian research project is to question the links between modeling and mathematics. I see the didactical sequences and ideas presented more clearly as providing pupil the opportunity of an introduction to a scientific enculturation, rather than an introduction to the meaning and the practice of proving in mathematics. In the situations presented, the pupil can expect feedback from a "real world" (a material milieu ) which could refute or confirm their conjectures, and so regulate the social interaction, whereas mathematics deal with "abstract entities" (intellectual constructs ) -- and make more complex the confrontation of pupils'conceptions.
   This suggest me some questions for further discussions:

What becomes the specificity of mathematics in such "realistic" contexts?
Would computers, which allow the reification of mathematical objects (virtual ontologies ), constitute specific media as compared to other material devices?
How could learners shift from semi-empiricism to apodictic reasoning?

Axiomatization has been more or less completely hidden or forgotten since its golden age, in the 70s. But is it possible to teach mathematical proof without addressing the question of axiomatization? If one shares the idea that mathematical conceptualization has to pass through the stage of experimentations and situated problem-solving, would making explicit the role of axiomatization -- and the problems it raises about the nature of mathematical objects -- help in leaving the world of pragmatic proofs to reach the world of intellectual proofs in which mathematics develop?

N. B.

 

The Italian presentation as well as de Villiers reaction are published in the
PME XXI proceedings.



Rigueur et formalisation

Table ronde de la IX° Ecole d'été de
Didactique des Mathématiques

Houlgate, 19-27 août 1997

Dans le cadre du thème "Analyse didactique et épistémologique de contenus mathématiques au lycée et à l'université", organisé sous la responsabilité scientifique de Marc Rogalski et coordonné par Jean-Luc Dorier, cette table ronde a réuni : Catherine Goldstein, Gilbert Arsac, Yves Chevallard, Daniel Perrin et Marc Rogalski.

Présentation: La table ronde sera introduite par une intervention de quelques minutes de chacun des participants et sera suivie d'un débat avec la salle. Ces interventions ont été préparées à la suite d'un échange épistolaire organisé autour des pôles suivants:

  • Les rôles de la rigueur et de la formalisation dans les mathématiques: histoire, épistémologie, pratique des mathématiciens, construction de théories et résolution de problèmes, rôle des formalismes dans les diverses représentations sémiotiques utilisées en mathématiques, diversité de nature de différents concepts mathématiques, etc.
  • Rigueur et formalisation dans l'apprentissage des mathématiques: didactique, programmes et commentaires, manuels, place, présence, flou ou absence de définitions, énoncés, pratique des maîtres, contrats didactiques, rapports avec l'accès ou non des élèves à la conceptualisation, etc.
  • D'autres aspects de la rigueur et de la formalisation: l'accès à une véritable culture mathématique, les rapports avec d'autres disciplines et leurs modes de pensée, avec la formation de l'esprit critique (y en a-t'il un ?), etc.

La question sous-jacente à cette table ronde est: y a-t'il un écart trop grandissant entre ces trois pôles dans l'institution scolaire ? cet écarct n'est-il qu'une illusion idéologique, voire un argument élitiste ? un regret nostalgique d'un passé idéalisé ? si cet écart est réel, comment fonctionne-t-il didactiquement ? qu'implique-t-il pour l'apprentissage des mathématiques ? pour la formation du citoyen cultivé ? a-t-il un rapport avec des tendances lourdes de l'évolution de la société ? peut-on stopper cet écart ? comment , etc.

 

CLASSIC FALLACIES

http://www.math.toronto.edu/mathnet/falseProofs/fallacies.html

 

"Startling discoveries" from the Univ. of Toronto Mathematics Network, including "conclusive proof that 1 is equal to 2, that every person in Canada is the same age, that a ladder will fall infinitely fast if you pull on it, and many other results that threaten the very fabric of common sense."
Each of the 'proofs' is presented in steps: when you think you know where a fallacy lies, select that step and you will be told whether or not you are right, with an explanation of why the step is or is not valid.
See how many tries it takes you to identify the fallacy!

This is a news from the
Math Forum Newsletter

Une thèse

Le 29 septembre 1997,
Vanda Luengo soutiendra à Grenoble une thèse de doctorat sous le titre :

Cabri-Euclide
Un micromonde de preuve intégrant la réfutation.

Principes didactiques et informatiques
Réalisation

Laboratoire Leibniz - IMAG
Equipe Environnements Informatiques d'Apprentissage Humain.

46 Avenue Félix Viallet
38000 GRENOBLE
FRANCE

Current call for papers

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

See also Proof Newsletter January/February 1997

Reminder
ICME8 Topic Group on Proof

The 300 pages proceedings of the Topic Group on Proof at the 8th International Congress on Mathematical Education (ICME 8) is available at $19 (USD) which includes postage (surface mail only) and can be ordered from:

Prof Michael de Villiers
Mathematics Education
University of Durban-Westville
4000 Durban, South Africa

Please make out all cheques to: AMESA. (Only pre-paid orders will be processed).

For more information see the July/August Newsletter

Services

La bibliographie
Outil de recherche
Cours en ligne
Questions et réponses

A propos du site

Services

The bibliography
Search tool
Online course
Questions and answers

About the site

Servicios

La bibliografia
Herramienta de busqueda
Curso electronico
Preguntas y repuestas

con respecto a este servidor

Adresser suggestions et remarques à...
Send remarks and suggestions to...
Enviar comentarios y sugerencias a ...

Nicolas Balacheff

Lettres précédentes

Past Newsletters

Noticias anteriores

[97 07/08] [97 05/06] [97 03/04] [97 01/02]