University mathematics programs cater, in various ways, to future teachers of mathematics. The gap between the university contents and methods on one hand, and on the other hand secondary school mathematics, is investigated in studies of how students experience the beginnings of university mathematics programs. Here we consider the gap as it appears at the other end of the students’ experience at university, as they try to relate their new mathematical knowledge to contents and methods in secondary school teaching. The main points of the paper are to introduce a theoretical model to analyse the changes in relation to mathematical knowledge which may be operated at that point, and to show its use on a significant example (the construction of exponents or, more generally, of the exponential function).