Mise en œuvre d'une ingénierie didactique de développement dans le cadre d'un travail collaboratif chercheur/enseignant lors de la conceptualisation des objets de l'Analyse au début du cursus dans le supérieur

At the start of the higher education curriculum, the conceptualization of local approximation objects of a function requires the articulation of knowledge and skills from Functional Analysis and Topology. In the study of functions, a number of studies have established the existence of difficulties encountered by students, mainly as a result of the change of didactic contract during the transition from secondary to higher education. The construction of a teaching-learning project, as part of a collaborative effort with the class teacher, a priori helps students to overcome the main difficulties inherent in conceptualizing the local approximation objects of a function in the first year of preparatory classes. In the case of the design and implementation of didactic development engineering, analysis of the reasoning produced by students confronted with a situation with an adidactic dimension will enable us a priori to study the nature and origin of these difficulties. Our methodology for analyzing student work is based on a model of reasoning analysis within the framework of the theory of didactic situations mathematics. This model has played an essential role in the development of didactic engineering, in the identification of students' conceptions, forms and functions of reasoning. It also enabled us to identify epistemological, didactic and cultural obstacles to learning the concept of local approximation of a function. These obstacles result either from the paradigm shift that takes place during the transition from secondary to higher education, or from working within the paradigm of Infinitesimal Analysis during the appropriation of this mathematical concept.