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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
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.
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(Lahti, Finland, July 14-19) |
 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.
PME XXI proceedings.
Rigueur et formalisation
Didactique des Mathématiques
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.
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"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." Math Forum Newsletter |
Le 29 septembre 1997, Un micromonde de preuve intégrant la réfutation. Réalisation Laboratoire Leibniz - IMAG 38000 GRENOBLE FRANCE |
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NCTM 1999 Yearbook on
Mathematical reasoning. |
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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: Mathematics Education University of Durban-Westville 4000 Durban, South Africa For more information see the July/August Newsletter |
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