Abstract
This thesis falls within the field of didactics of university mathematics and aims at studying the didactic transposition of the concept of ideal. This mathematical concept is taught in abstract algebra at the Bachelor's level (the Licence in France). Its teaching coincides with the entry into structuralist algebra for students, which is accompanied by significant difficulties in learning. This study, carried out between France and Switzerland in institutions at the Bachelor level, aims to study the teaching of this concept and, through it, the way in which the entry into structural algebra is handled by the institutions and their teachers. To do this, we first conducted an epistemological analysis highlighting the role of this concept in the construction of structuralist algebra, in relation to its formalizing, unifying, generalizing and simplifying character. We then chose to undertake case studies covering the different institutions that the student goes through, in his learning of the concept of ideal. Our corpus of data is thus made up of pedagogical material and interviews with nine professors: two teachers in preparatory classes for the grandes écoles in France, two in the second year of a Bachelor's degree in France, three in the third year of a Bachelor's degree in France, and two in the first and second year of a Bachelor's degree in Switzerland. To conduct our analyses, we place ourselves within the framework of the anthropological theory of didactics. The analysis of the interviews allows us to characterize the personal relationships of the teachers with regard to the different institutional relationships to which they are subject and the epistemological characteristics of the concept. These analyses also allow us to obtain descriptions of the processes of didactic transposition (external and internal) as well as the constraints that weigh on them, classified according to the different levels of the didactic co-determination scale. The effect of these constraints on the development of mathematical and didactic organizations is studied through praxeological analyses mobilizing the notion of structuralist praxeology introduced by Hausberger (2018), together with other tools (object-structure dialectic, structuralist levels and type 1 and 2 transitions). The latter lead to the construction of praxeological reference models per teacher and per institution. These models allow us to compare the choices of didactic transposition in relation to the problem of entry into structuralist algebra, the characterization of the ecology of mathematical organizations which we will try to relate to the constraints identified, and finally the study of transitions between different institutions.