Une situation qui problématise les passages discret-dense-continu

Abstract

This article presents an analysis of the discussion and written work produced during an experiment in which two students in teacher training were asked to solve a version of the fixed-point problem initially proposed by Pontille, Feurly-Reynaud and Tisseron. Our interest lies in how the problem can be broken down into five tasks (proposed by us), and specifically the effect of an intermediary phase involving the set of numbers that have a maximum of five decimal places. We are also interested in how the students reorient their mathematical work based on the hints and clues provided by the researchers. In our analysis, we discuss─as much from an epistemological perspective as from a cognitive perspective─the problem of how to conceptualize/coordinate the properties of density and completeness when it comes to extending the set of rational numbers to the set of real numbers. We are particularly interested in the property of density (intrinsic, with respect to order) in ordered sets D and Q and the role it can play in the learner's ability to separate the numerical-arithmetic structure of the theoretical “real axis” from its representation in the figural register, which, according to our analysis, is one of the key aspects in the conceptualization of R.