Representation and activity : two concepts intrically tied together

Gerard Vergnaud

Behaviorists wanted to get rid of the concept of representation. Not only did they fail, but representation is to-day the most central concept of psychology. Concerning the development of mathematical knowledge in children, representation is not made only of numbers, figures, drawings, diagrams, tables, graphs or algebras, but also of interiorised forms of activity in situations.

Activity is more than behavior : behavior is only the visible part of activity. Therefore when analysing mathematical behavior, one must look into the representational activity underlying it. The concept of scheme is essential to cover this problem .

The most important part of our knowledge consists of competences, and they cannot be put into words easily. This is true for every domain of knowledge, including  mathematics ; and it is even more true for children, as they are unable  to express the knowledge  they use in action.

Facing situations, children can progressively grasp relational entities between quantities and magnitudes,  between positions, figures and movements… Part-part-whole relationships, state-transformation-state relationships, isomorphic properties in problems of proportion cannot be reduced to numerical structures ; nor can they be considered as linguistic or symbolic  entities only. They are concepts and theorems-in-action.

The implicit character of a large part of our knowledge does not mean that explicit knowlege  is not operational. But we cannot be satisfied with a theory that would consider mathematics only as an explicit body of knowledge.

Even when one is interested in the function of language  and symbols in the development of the mind, it is necessary to identify safely which properties of the signifier  represent which properties of the signified. We are aware to-day that words mean different things for different individuals,  especially for the teacher and each student individually. Vygotski explained 70 years ago that the « sense » given to words is different from their conventional « meaning » . Therefore there is a theoretical need to analyse activity and representation as composed also of invariants that may be different from the meaning of words. This problem can be solved only if we accept the idea that schemes involve operational invariants : concepts and theorems-in-action. It is our job to identify them, together with the other components of schemes, and representation.

Several examples will illustrate this unexpected view.