The Training of Mathematics Teachers: A Higher Standpoint

Taking a cue from the recent editions of Klein's three volumes on "Elementary Mathematics from a higher standpoint", we examine university courses aimed at teacher training. We discuss the well-known double discontinuity underlined by Klein and some opportunities to overcome it. In this framework, we compare the intensive teaching of advanced courses with the extensive teaching of elementary mathematics courses. Throughout the paper, we mostly refer to the Italian situation, but almost all the considerations can be adapted to different contexts.