Abstract : Since the beginning of the counter-reform of modern mathematics, the teaching of the notion of function in the beginning of upper secondary school in France has been subject to profound modifications. One of the most important transformations deals with the progressive reinforcement of various modes of representation of functions. In this sense, there has been a decline of the algebraic register along with a larger use of the graphic register and there is a strong demand of the institution for the use in new conditions of the tables of values and the table of variations. It is our research hypothesis, following Duval's work, that the multiplicity of possible semiotic registers brings the concept and its properties to life.
In our work, we first make an institutional analysis of the notion of function in an ecological perspective, in order to reveal the different systems of constraints and conditions that deals with evolutions of this piece of knowledge within the process of internal didactical transposition. This work is made throughout all of the counter-reform of modern mathematics, since the 80s. Then, we establish the state of art of the present teaching in order to see how textbooks and teachers (through a questionnaire) deal with the novelties of the curriculum. We also give an account of students' abilities, through results to a written test.
Based on this first analyses, we determine some characteristics of the “praxeological organisation” (according to Chevallard's terminology) included in the last reform of 2000 about the notion of function, which have difficulties to live in the actual teaching. At last, we propose a didactical enginery aiming at making these aspects of the curriculum possible in the classroom and we evaluate their viability with an experimentation.