Appropriation des proprietes ordinales du nombre par l'eleve du cours preparatoire

The aim of this research is the clarification of the process whereby the learner, in the first year of compulsory education (Cours Préparatoire) constructs and internalizes the concept of natural number. The research is based on a close and rigorous analysis of learners' behaviour when confronted with problem situations designed according to a priori established priorities. This analysis is based on one-to-one interviews, a method which enables the dynamics of the processes involved for each learner, and thus the evolution of his cognitive system with respect to the task in hand, to be apprehended.In this paper, in a problem situation called The Race, we examine the functioning of ordinal properties of natural numbers, the acquisition of which plays an important role, both at the level of the construction of the concept of number and at the operational level. We show, in particular, that while the majority of learners at the end of the Cours Préparatoire are capable of supplying the preceeding and following numbers in the numerical field [1, 30], and that while they can count starting from a number other than one within this field, they nevertheless experience many difficulties with respect to the functioning of properties, such as, for example, cardinal-ordinal relations, or the property of rank invariance.Furthermore, even when one of the properties under consideration functions in a given situation, this, in itself, does not mean that the learner has acquired it. The learner may, at a later date, no longer be able to use it, or, alternatively, when the situation is slightly modified he may overgeneralize, as the functioning of these properties depends both on the learner's current mathematical competence and on the tasks which confront him.