ABSTRACT
The theory of praxeologies, which includes praxeologies to be taught and praxeologies for teaching, was developed as an answer to the following question: where do personal and institutional relations to objects come from? This chapter explains the risk of mixing both aspects because the frontier between praxeologies to be taught and praxeologies for teaching seems to be of importance to explain the formation of personal and institutional relations. It provides an example of a class studying right-angled triangles and highlights that the diffraction results from the integration of some ingredients of the didactic organisation into the mathematical organisation. The chapter examines how to integrate a didactic element, the dialectic of media and milieus, in a mathematical organisation. To continue the study of the integration of didactic elements into the praxeologies to be taught and its effects, the chapter considers the case of the dialectic of media and milieus.
