Le raisonnement en arithmétique : de l’analyse épistémologique à l’analyse didactique

Résumé

À l’ articulation entre analyses épistémologique et didactique, la recherche dont est issu cet article vise à identifier les potentialités de l’arithmétique pour l’apprentissage du raisonnement mathématique et à étudier l’écologie de celles-ci à la fin du cursus secondaire français où ce champ a été réintroduit depuis peu. Nous introduisons d’abord un outil d’analyse provenant du travail épistémologique, à l’aide d’une démonstration historique qui offre l’avantage de bien mettre en évidence la distinction, dans le raisonnement en arithmétique, entre deux dimensions que nous appelons ≪ organisatrice ≫ et ≪ opératoire ≫. Nous étudions ensuite la pertinence didactique de cet outil en l’exploitant pour l’analyse de raisonnements d’élèves confrontés à la résolution de problèmes arithmétiques dans deux contextes distincts.

Abstract

In French secondary education, the place of arithmetic—the arena of numbers par excellence—has varied greatly, both qualitatively and quantitatively, over the history of the curriculum. After spending years in mothballs, it reappeared in 1998 in the curriculum for the terminale (18 years-old, Grade 12) science course and in the enseignement de spécialité (enriched education with three possible concentrations) and has, since then, also figured in the courses of the troisième (15 years-old, Grade 9) and the seconde (16 years-old, Grade 10). In part, its reintroduction was a reflection of the concept that arithmetic could foster work on mathematical reasoning. Curricular developments of this kind inevitably bring into play questions of teaching methods. To be specific, Does the type of arithmetic covered in the current terminale curriculum genuinely encourage this type of work, and, if so, what are its specific characteristics? For more than 20 years, a significant amount of education research has takenup questions related to mathematical reasoning and proof—in other words, mathematical rationality. On the whole, the present study is positioned within this body of research, but at the same time, it adopts as its point of approach the examination of a specific field—namely, arithmetic conceived of as number theory. In contrast to what has occurred in the case of geometry, for obvious cultural reasons, the potentialities offered by this field of learning and the teaching of mathematical rationality seem, in my view, to have been less systematically explored, particularly at the relatively advanced level analyzed here. The answers to the questions posed above are thus far from being self-evident.

This is why, in this study, the choice has been made to intertwine analyses bearing on epistemology and on teaching methods. The purpose of the epistemological analysis is to provide a basis for studying the characteristics of modes of reasoning that bring into play the arithmetical concepts at stake at this level.¹The present analysis then goes on to discuss such concepts in terms of a reasoning in arithmetic, so as to fully distinguish it from the arithmetical type of reasoning identified in research on algebra and especially on the arithmetic-algebra transition (Schmidt, 2002). In cases of reasoning in arithmetic, algebraic symbolism functions as a tool that has presumably been mastered with sufficient proficiency by terminale science students. The epistemological analysis is based on the study of historical and present-day arithmetical proofs. Thus, the first part of this article offers an historical demonstration as a means of introducing a tool generated by this same demonstration—that is, on the one hand, a process of differentiation between two, so-called organizing and operative dimensions of reasoning and, on the other, the identification of both components’ characteristics. This differentiation and these characteristics served, to evaluate, from the outset, the potentialities of this field for the teaching and learning of mathematical reasoning. The objective of the methods-related analysis was to study the ecology of the potentialities revealed by the epistemological analysis in the curricular context considered here. This particular analysis was conducted along various main threads of inquiry. However, the present article only takes up an analysis of the reasoning of students confronted with arithmetical problem solving. As mathematical reasoning depends not only on the field involved but also on the context within which it is produced, two types of corpora were analyzed: (1) a student test paper from a baccalauréat (secondary school graduation certificate) training examination, and (2) the process adopted by a group of students in a classroom situation to produce an arithmetical proof. These two examples serve to demonstrate the relevance, in terms of teaching methods, of the analytical tool resulting from the epistemological research.