Abstract
Félix Klein has pointed out a double discontinuity in the mathematical study path: in addition to the transition from secondary to higher education, a second discontinuity occurs at the end of the university when a student becomes a mathematics teacher in secondary school. The latter generally has difficulty perceiving the relationship between the mathematics taught in secondary school and the knowledge acquired at university.This thesis, which is part of the field of didactics of mathematics, aims at studying Klein's second discontinuity by particularizing it to an object of knowledge: the integral. This mathematical concept is taught in the final year of high school in France, and then at the university during the first three years. Several integration theories are taught in higher education and students generally do not perceive the links between these theories, as well as with the high school integral. In order to analyze the different didactic phenomena that result, we place our study in the institutional perspective of the Anthropological Theory of the Didactic (ATD).First, we give a review of the literature in mathematics education on Klein's second discontinuity as well as on the integral, the latter being limited to the secondary-tertiary transition. Then we present an epistemological study of the main integration theories that are encountered in teaching. We continue with a study of the teaching of the integral at the university in France, in particular the external didactic transposition in the sense of the ATD. We report on the debates that have taken place in the noosphere (the community of mathematicians) concerning the choice of theories to be taught in the bachelor's degree program and we compare them with those made by a panel of universities. The different syllabi of teaching units related to the integral are studied using the tools of the ecology of knowledge.Our study of the didactic transposition of the integral is then particularized to the case of the university of Montpellier. Interviews with teacher-researchers allow us to shed light on the reasons for the choice of theories taught. The praxeological study of the course documents in the second and third years of the Bachelor's degree allows us to model the students' knowledge (their praxeological equipment) in the form of dominant praxeological models (in the sense of ATD). The comparison with a dominant praxeological model for the high school integral then allows us to point out links and breaks in these sets of knowledge, in terms of the constitutive blocks of praxeologies mobilizing the concept of integral. In particular, we show that the intuitive theory of areas which grounds the high school integral is not formalized at the university, apart from the theory of measurement - of a much higher level of generality and abstraction.In fact, our epistemological investigation has allowed us to identify an intermediate theory, the Jordan measure, from which we have developed an engineering that aims at making visible the relationships between measure theory, the Riemann integral and the intuitive theory of areas. These relations are described in terms of blocks of praxeologies. This leads us to introduce the notion of Kleinian praxeologies, which aim at an integrative relation to the knowledge taught at the university and in the secondary school, and to specify the associated didactic organizations by adapting the moments of study of the ATD. Our engineering, built in the form of a study and research activity in the sense of the ATD, was experimented in the context of the Master MEEF (student teachers). The analyses of the students' work, carried out with the different ATD tools (Question-answer maps, Herbartian scheme and moments of study), confirm the potentialities of our device while highlighting obstacles to the development of Kleinian praxeologies.