The challenge of structuralist thinking in abstract algebra: a transition problem

Abstract

This paper deals with the issue of transitions in the teaching and learning of abstract algebra throughout the Bachelor in France, focusing on the notion of ideal in ring theory which is seen as a gateway to structuralist thinking. In the framework of the Anthropological Theory of the Didactic, the notion of structuralist praxeologies, endowed with structuralist levels, is used to analyse a corpus of course documents from three upper-secondary teachers which cover the teaching of ideals through the Bachelor. Hausberger's three-phase model for structuralist transitions, which generalises Winsløw's model of the concrete to abstract transition in analysis, is applied to discuss the issue of transitions. Our research results cover a description of structuralist praxeologies developed throughout the Bachelor around the notion of ideal, a discussion of discontinuities and ruptures that are observed, and the identification of strategies developed by teachers to handle these transitions as they carry out the internal didactic transposition.