Introduction

Summary

Psychology and Mathematics

There has always been a reciprocal interest between psychology and mathematics despite the fact that these two domains are structurally so different. Psychologists, following the example of other empirical sciences, have tried to express psychological phenomena in terms of mathematical models, quantitative laws, and even axiomatic deductive systems (such as that of Clark Hull). On the other hand, mathematicians like Poincaré, Hadamard, and, more recently, Polya and Freudenthal became interested in the psychology of mathematical reasoning. Their accounts based on introspective attempts or didactical experiences are genuinely illuminating for the psychological aspects of mathematical reasoning. Piaget and his co-workers have played a fundamental role in developing the psychology of mathematical reasoning as a research domain. Their works on concepts like classes and relations, number, geometrical representations, and proportional and probabilistic reasoning have had an essential role in deepening our understanding of mathematical notions.

Nevertheless, mathematics education took little advantage of these developments. For a very long time, mathematical curricula and teaching methods were inspired mainly by new mathematical ideas like those promoted by the Bourbaki group or by the current experience of schoolteachers. Psychology and mathematics education as research domains remained relatively disjoint.

The split persisted primarily because of a lack of communication and consequently a lack of cooperation. A few basic mathematical concepts attracted the interest of psychologists, but most of the major concepts of mathematics remained outside their concern.