Didactic transposition and the knowledge to be taught: towards an archeorganisation for concave/convex functions

This study is framed within the anthropological theory of the didactic. The first part is a comparative study of the mathematical organisation of concave/convex functions on the one hand at university, and on the other hand in secondary education. Data for this analysis are from a university textbook on calculus and a secondary school textbook. The analytic results show how the mathematics (to be) taught at secondary level consists of a largely demathematised logos block. The corresponding institutional fragmentation of knowledge reveals a missing praxeological organisation from which the organisations of knowledge we consider in this paper could be derived by transposition. Responding to this insight, in the second part of the paper, we have outlined what we call an archeorganisation for concave/convex functions, a framework that teacher education institutions could refer to when trying to satisfy the specific constraints they are subjected to (notably because of its publics), without for that matter isolating themselves epistemically from other academic institutions. Formulating an archeorganisation represents a meta-curricular study, with the objective of transforming the institutional conditions and constraints that guide mathematics educators and curriculum designers in their deliberations about the concept of concave/convex functions.