François Conne

What Concepts are and How Concepts are Formed

With respect to the question of how mathematical concepts are formed by children, we can draw upon several theoretical positions, going from the most general point of view, relative to the universal subject, to the most specific point of view, relative to the didactic subject. In this chapter I will rely on the theory of conceptual fields, whose originator, G. Vergnaud, describes it thus: “The theory of conceptual fields is a cognitivist theory, which sets out to provide a coherent framework and some basic principles for the study of the development and learning of complex competencies, notably those which arise in the sciences and technologies” (Vergnaud, 1991, p. 135) I shall try first to show the necessity of that theory for the didactics of mathematics in relation to that of genetic epistemology; in the second section I shall elaborate on what the theory of conceptual fields contributes, with the notion of scheme-algorithm, to questions on the cognitive sources of observable errors in written calculations.