In this paper, we consider the design of capstone courses to prepare university students for secondary school teaching professions in initial training. Our aim is to prepare prospective teachers to develop an approach for doing mathematics which they then impart to their pupils. This posture involves systematically questioning the mathematical work carried out, developing and clarifying general methods, and identifying links between the various aspects of the knowledge involved. This is not a research article as such, but presents a set of reflections and proposals in mathematics didactics. After summarising previous works on Klein's second discontinuity, we specify our aim in terms of the type of mathematical work to be conducted with students and their future pupils before summarizing our theoretical background on learning, practice and training. We then present various axes based on the hypotheses we have adopted regarding our objectives, meaningful representations of mathematics, and discourses fostering subsequent reinvestment. We outline a capstone course dedicated to Arithmetic for prospective French teachers following these principles. The conclusion sets out a discussion on further perspectives.