Étude didactique et épistémologique des obstacles qui émergent chez les élèves au cours des activités de développement et de réduction des polynômes au collège : proposition d’un dispositif pour leur franchissement.

Abstract

Algebraic calculation is viewed from two points of view by researchers. On the one hand, it is considered an extension of arithmetic, on the other hand it is constructed in a break with arithmetic. This is the reason why, and in accordance with these points of view, we observe many difficulties in learning literal calculation among students in 4th and 3rd grade whose age varies from 12 to 16 years. The thesis we support is that the identification of epistemological and didactic obstacles followed by work based on these obstacles, and taking into account the semantic and syntactic aspects of the statements, can lead students to overcome them. For this, based on a test to which we submitted the students and an interview with them, we identified some obstacles emanating from the operational invariants identified, within the framework of the theory of conceptual fields. We also called upon the anthropological theory of didactics from which we carried out an analysis of textbooks and classroom practices, from the point of view of praxeology and didactic transposition. The results of the analyses showed, for example, that the reductive representation of equality as an announcement of a result in numerical calculation and of the "+" sign placed between two numbers as a sign of an operation leads students to reduce the result to a number by adding monomials or by assembling dissimilar monomials. Similarly, the fact that students associate the removal of parentheses with the application of distributive property leads them to distribute a monomial under the terms of a sum delimited by parentheses, in an algebraic sum. The results of the experiment with the proposed crossing device show that learning difficulties are less frequent among students in the experimental group than among those in the control group.