Une comparaison de l’enseignement de la géométrie dans le secondaire en France et en Italie

Résumé

L’enseignement de la géométrie est encore aujourd’hui au cœur d’un débat international. Cet article propose une modeste contribution à ce débat à travers quelques résultats d’une recherche comparative sur l’enseignement actuel de la géométrie dans les systèmes scolaires français et italien (Celi, 2002). La comparaison s’appuie sur les programmes officiels et les manuels scolaires mais aussi sur les réponses d’élèves français et italiens de seize ans à des problèmes portant sur des contenus d’enseignement communs. Comment les auteurs de manuels interprètent-ils les programmes? Dans quelle mesure les résultats des productions écrites recueillies peuvent-ils être mis en relation avec l’enseignement reçu tel qu’il apparaît à travers les manuels scolaires? Les difficultés communes aux élèves des deux pays ouvrent des pistes pour des réflexions d’un point de vue cognitif et orientent vers un questionnement plus précis sur les pratiques professionnelles.

Executive Summary

What objectives can be attributed to geometry instruction currently? Why is it appropriate and/or necessary to teach geometry? In 1995, the International Commission on Mathematics Instruction began a discussion about educational perspectives in the 21st century in order to answer these questions.* The research presented in this article is intended to be a contribution to a discussion initiated well before 1995 and whose culmination is unlikely to occur in the near future. Instruction in this discipline had already been called into question in the 1960s (Colloque de Royaumont, 1959; Congrès de Dubrovnik, 1960; see Vita, 1986), and since then, France and Italy have followed different approaches. Since the beginning of the 1970s, French geometry instruction has been based on transformational geometry. In Italian geometry instruction, geometric transformations are featured alongside the more traditional objects of Euclidian geometry.† It was in wondering about the consequences of these contrasting choices that I determined the problematics and hypotheses of this research, which are found in the first section of the article.

The second section describes the principal methodological elements developed to carry out this research and the primary theoretical tools these draw on; in particular, the anthropological approach to the didactic (Chevallard, 1989, 1992) and the categories of proof developed by Balacheff(1987).

Official texts and textbooks are a cultural reference useful for bringing out certain aspects characteristic of each institution, and it was essential to this research to draw on these official sources in order to compare the two systems of instruction. The textbook—considered as the approximate image of possible instructional practices—was, in particular, an object of study, but it also served as a tool for the research as a whole, making it possible to determine the main elements of the significant points common to instruction in plane geometry in the two institutional contexts. I drew on these objects when designing problems to be given to sixteen-year-old French and Italian students.

Some textbooks among the most widely used in the two countries were analysed in greater depth. The third section describes the results of this analysis and the written work collected. The process of determining the principal categories for analysing the students’ responses provided an opportunity to reconsider the theoretical frameworks.

The article concludes with some general reflections on current geometry instruction in the two countries. Views are put forward regarding an analysis of geometry instruction practices—the role of the instructor as a mediator between the knowledge to be taught and the students being essential for better understanding the operation of the two educational systems studied here.