Meta Level in the Teaching of Unifying and Generalizing Concepts in Mathematics

Certain concepts in mathematics were not invented only to solve new problems; their aim was mainly to find general methods to solve different problems with the same tools. Such concepts, as those of the axiomatic theory of vector spaces or groups or the modern definition of limit, will be called in this paper "unifying and generalizing concepts". I will point out some epistemological specificities of these concepts and subsequently analyze their influence on teaching. I will explain the reasons which led me to the conclusion that it is necessary to introduce some "meta" aspects into the teaching of unifying and generalizing concepts, and I will present the theoretical framework I adopted for my purpose, in relation to other theoretical approaches. I will then present and analyze one example, from which I will draw conclusions about theoretical questions of evaluation in a long term experiment which includes a meta dimension for the teaching of unifying and generalizing concepts in mathematics.