Analyse des problèmes de géométrie et apprentissage de la preuve au secondaire

Résumé

Le présent article rend compte de l’élaboration d’une grille d’analyse des problèmes de géométrie, et de sa mise à l’épreuve par la classification des problèmes et exercices de géométrie synthétique dans une collection de manuels du secondaire parmi les plus utilisées au Québec. Le cadre conceptuel sur lequel s’appuie cette élaboration s’inspire principalement des travaux de Balacheff (1987), Barbin (1988), Brousseau (1998), Hanna (1995) et Rouche (1989), et débouche sur une typologie des preuves de géométrie. La classification des problèmes à partir de cette grille et l’analyse qui en découle nous a permis de conclure sur les aspects de l’apprentissage de la preuve que nous évaluons comme mal « gérés » dans la collection: transition non suffisamment graduelle du sensible au formel (peu de problèmes qui sollicitent une validation hybride, niveau de formalisation trop longtemps stationnaire), prépondérance des applications directes et des déductions locales sur les séquences déductives, intérêt et mode de présentation des résultats qui ne favorisent pas une « attitude de preuve ».

Executive Summary

In this research project, I have attempted to understand how the notion of proof develops during the secondary school student’s learning process. From this perspective, I have first examined official texts in order to identify not only the objectives of the Québec Ministry of Education (MEQ) yearly curriculum (1993–1996) dealing with the learning of proof but also how such curricula propose to accomplishing these objectives. According to these curricula, the learning of proof occurs primarily through the study of geometry, in a more general context ofproblem solving. The next step was to understand, on a second level, how this learning has been transferred into textbooks. In that connection, it was necessary to be able to account for learning progression, in terms of continuity-discontinuity. As the objectives of the curriculum emphasize problem solving, this phase of my analysis required a close analysis of geometry problems. With the objective of developing an analytical grid to be applied to synthetic geometry problems in relation to the type of proof they require, I have attempted to synthesize the reflections on proof contained in the work of Balacheff (1987), Barbin (1988), Brousseau (1998), Hanna (1995), and Rouche (1989), on the basis of what I call ‘schemas of bipolarization.’ Using the schémas suggested by these authors, I then built both a typology of proof and, on the basis of this typology, an analytical grid proper. Both the typology and the grid were developed from the perspective of sources of validation that students are capable of drawing on. The grid broke down all of the problems and exercises of synthetic geometry into seven categories: the direct application (nothing to be validated); the spontaneously seeing the general in one particular case and empirical induction (source of validation: the ‘tangible’ or the ‘perceptible’); mental experience and the empirico-deductive argument (dual sources of validation: reasoned argumentation based on the perceptible); and, finally, the local deduction and the deductive chain or linkage (source of validation: logico-deductive reasoning). I tested out this grid by classifying problems and exercises of synthetic geometric contained in a collection of textbooks among those most widely used in Québec for all the secondary school grades (Secondary I to V, students ages 12 to 17). Based on an analysis of the results, I developed a series of conclusions about aspects of the learning of proof that I view as being poorly handled in these textbooks, including an insufficiently gradual transition from the perceptible to the formal (very few problems that draw on a hybrid validation; over-long stationary formalization; a break during Secondary V); the predominance of direct applications and local deductions over deductive sequences; and a focus and mode of presentation of results that do not foster a ‘proof-oriented attitude.’ This analysis also gave me an opportunity to examine various possible interpretations of what the MEQ curriculum has termed îlot déductif (or ‘local axiomatic’) and the role that has been planned for the geometry of transformations.