Despite all the intense and international efforts of research into the teaching-learning processes of mathematics, Euclid's famous dictum is still valid according to which there is no royal way to mathematics. A growing number of approaches has as its focus the nature of mathematics and investigates whether, by taking into account this nature, the teaching-learning processes might be improved. A common pattern of these approaches can be called to be a "genetic" one, i.e., to establish a relation between the historical evolution of mathematics and the learning of mathematics. Evidently, the respective approaches are based on certain epistemological views about the nature of mathematics. The lecture will analyse several of these approaches, in particular Otto Toeplitz's genetic method and the prominent French conception of epistemological obstacles, discussing their strengths, pitfalls, and weaknesses.

The lecture will then discuss how interactions between epistemology and history of mathematics and semiotics in mathematics can contribute to better qualify teachers to cope with the conceptual problems inherent to the nature of mathematics. Given that the introduction of new terminology and notations in mathematics used not only to explicate hitherto hidden assumptions, but also to facilitate the understanding of difficult concepts, there has always been a considerable importance of semiotics in the history of mathematics. An analysis of mathematics textbooks of various epochs and cultures and of the impact of symbolism will enhance to make these interactions operative.