Using the computer as an environment for building and testing mathematical concepts: A Tribute to Richard Skemp

In this paper I shall show how the computer may be used to provide an environment for building and testing mathematical concepts as part of a long-term learning schema following the theory of Skemp, (1962, 1971, 1976, 1979). I do this in tribute to the inspiration I have received from working with Richard Skemp and reading his publications over the major part of my time in mathematics education. History may record that it was my fortune to be his last Ph.D. student before his retirement as Professor of Educational Theory at Warwick University. It is an honour that I cherish, and a responsibility that will be hard to maintain at a fraction of the standards that he has set in scholarship and insight. Background Fifteen years ago I was to give my first course of lectures on “The development of mathematical concepts” to undergraduates who were mathematics majors with excellent records in mathematics but no experience in studying how people learn. The course was based on the difference between the formal, the historical, and the cognitive development of mathematical concepts. The formal mathematics was familiar to students, the historical and cultural side could be taken from a number of sensible mathematical histories, but the cognitive development caused some difficulties. In the early seventies there were five major thinkers in the psychology of education who gave highly relevant insights into mathematics education : Piaget, Dienes, Bruner, Ausubel and Skemp. For my undergraduates, with little knowledge of children, Piaget proved difficult to penetrate. Although we could talk about assimilation and accommodation of concepts in a meaningful way, the notion of the stages: sensori-motor, concreteoperational and formal-operational, was purely a theoretical construct. To these students a “concrete operational child” was one who could conserve number, quantity, volume, etc, and could argue logically, provided there were concrete materials available to represent the concepts physically. The students had an enjoyable time playing with Dienes’ logic blocks and algebraic experience materials for factoring quadratics. Yet the precise nature of how this concrete