Abstract

A four-stage model of mathematical concept development is proposed in this article. The model is based on a distinction between a weak and a strong sense of reification in mathematics education. More specifically, the learner’s language is reconstructed according to a distinction between the reification of functions, sets, and other mathematical concepts in terms of “constants” and the reification of those concepts in terms of “bound variables.” The model is based on a detailed discussion of the traditional theory of reification, mainly due to Anna Sfard and her collaborators, as well as empirical data concerning the main difficulties experienced by students with the concepts of function and set. Furthermore, some remarkable episodes from the history of the concept of function in the twentieth century (including Schönfinkel’s approach to functions) are examined from the perspective of the model. An example of reification of procedures in programming languages is also considered.