We consider the following hypothesis: whereas, the institutions in charge of the teaching of mathematics for future professional mathematicians (IPM, in short) share a "didactic paradigm" (strongly based on logical arguments), there does not seem to be such a thing in secondary mathematics education. In this work, we analyse the notion of didactic paradigm and its possible uses in research, we sketch a description of the didactic paradigm shared in IPM, we indicate the problems that it would have in being adopted in secondary education, and we consider the failure of consecutive reforms in the teaching of secondary mathematics as an evidence of the difficulties in finding a suitable didactic paradigm at that level. Part of our study uses the "paradigm of mathematical modelling" as a reference paradigm.