Gérard Vergnaud

The Theory of Conceptual Fields

The theory of conceptual fields is a developmental theory. It has two aims: (1) to describe and analyse the progressive complexity, on a long- and medium-term basis, of the mathematical competences that students develop inside and outside school, and (2) to establish better connections between the operational form of knowledge, which consists in action in the physical and social world, and the predicative form of knowledge, which consists in the linguistic and symbolic expressions of this knowledge. As it deals with the progressive complexity of knowledge, the conceptual field framework is also useful to help teachers organize didactic situations and interventions, depending on both the epistemology of mathematics and a better understanding of the conceptualizing process of students.

Towards a Cognitive Theory of Practice

As I view it, the topic of the relations between practice and theory in mathematics education conveys two main problems. The first problem concerns the relationship between mathematics as a practical activity, and mathematics as a theoretical body of knowledge. The second problem concerns mathematics education, and the relationship between the teacher’s activity in the classroom and research in mathematics education and psychology. I think that, for both problems, we need a theory of practice and of articulation between theory and practice. This theory must be quite general since it does not concern mathematics or mathematics education alone, but all human activities. To do this, we also need some practice of theory, or theorizing. And we need examples, since theory without examples is usually incomprehensible.

Students' Conceptions in Physics and Mathematics: Biases and Helps

Abstract The thought modes that children and adolescents develop to solve problems encountered in everyday living are analyzed from two points of view: (1) sources of systematic error in solving mathematics and physics problems, and (2) the possible factors of knowledge development in these fields. Taking into account the gaps in thought modes necessarily involved in gaining access to scientific knowledge, and considering the possible links between the initial conceptions of individuals and the conceptions of experts, the authors discuss some potential means for providing cognitive guidance to foster the development of knowledge.

Conceptual Fields, Problem Solving and Intelligent Computer Tools

The goal of this paper is to present an overview of the theory of conceptual fields, illustrated by the examples of additive structures and multiplicative structures. The main difficulties encountered by students in learning mathematics and other scientific topics are conceptual. These difficulties cannot be analysed if student’s conceptions are investigated independently of the schemes that organize their behavior in problem solving situations, and if a concept or a class of situations is examined in isolation. A conceptual field can be defined as both a set of classes of situations, and as a set of interconnected concepts. This paper presents a theoretical analysis of the relationship between explicit knowledge and the implicit operational invariants that underlie schemes. Examples of theorems-in-action and of the relationships between concepts and symbolic representations are presented, as well as ways of using the framework of conceptual fields to devise intelligent computer tools.

On the Learning of geometric concepts using Dynamic Geometry Software

Many results on computer mediated geometry learning conclude about different heuristic approaches to problem solving with Dynamic Geometry Software (DGS). However, little is described concerning conceptualization process. We used a theoretical framework built upon constructivist foundations for analyzing mediated learning of specific geometrical concepts. Our point is illustrated in a case study in which we analyzed studentsi interaction with a DGS. Our results points to a clear mapping of potential conceptualization of geometry in software using.

Long term and short term in mathematics learning

“Long term” refers necessarily to a developmental perspective: between the first competencies acquired by young children, at the age of five or six, and 1 Universidade de Paris 8, Franca, CNRS (Centre Nationale des Recherches Scientifiques). VERGNAUD, G. O longo e o curto prazo na aprendizagem da matematica Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 15-27, 2011. Editora UFPR 16 those acquired by fifteen year-old adolescents (only part of them), there are several steps and processes: continuities and discontinuities can be observed along this development. “Short term” concerns situations that are likely to be offered to students, for them to learn, at some crucial moment of this development. It is therefore concerned by the discovery, even partially, of new properties by students, owing to the teacher’s help in conducting and facilitating the acquisitions course. The additive structures, for example, provide many different situations in which a particular reasoning and the choice of an addition or subtraction operation are clearly more delicate than in other situations: a case that requires a specific scenario, the assistance from the adult and, eventually, an original symbolic representation. The same kind of phenomena takes place with multiplicative structures, geometry, algebra, and other domains, far from mathematics, like morals and text comprehension. The present article limits this discussion to the case of mathematics learning.