Modelling mathematics for educational research and practice: a comparison of two theoretical approaches

The production and use of explicit models of mathematics are important elements of research in Didactics of Mathematics. In this paper, we present and compare two different European approaches to didactical modelling. The first, centred around the notion of basic idea (Grundvorstellung), arose in German-speaking countries as a development within Subject Matter Didactics (Stoffdidaktik). The second is the notion of praxeology from the Anthropological Theory of the Didactic, which originated in France. After presenting the genesis and interpretation of these notions as theoretical modelling tools, we provide a detailed case study of actual models produced with these tools, concerning the notion of limits and derivatives. Furthermore, we identify similarities and differences of both models in how they can support didactical research and practice, for instance, by identifying important phenomena or problems. We conclude that the two approaches offer important, different and, to some extent, complementary positions and perspectives.