cK¢, a model to understand learners' understanding -- Discussing the case of functions

This text develops the invited talk I presented to the international meeting on learning and teaching calculus to be held in Mexico in September 2015. It addresses the problem of understanding and modelling students' conceptions taking as a theme the case of function. To set the problématique, the introduction reports the Arsac study of the development of the Cauchy's conception of uniform convergence. Then the issue of understanding students' understanding is discussed, and a framework is proposed: the model cK¢. Then conceptions of function across history and from a learning perspective are described with the tools provided by the model with a special emphasis on controls illustrating the key role they play