Teaching Mathematics: Discovering vs Storytelling

Abstract

There exist at least two approaches to the teaching of every subject, including mathematics. The first one is as old as the world (or, at least, as the education). According to this approach, to teach something means to outline what you teach. This approach is appreciated by the great majority of teachers. Because the realization of this approach in most cases reduces to the telling of stories about what is taught, I call this approach by storytelling.
There exists also another approach, which goes from the famous statement by Aristotle: We learn something only when we do what we learn. From this statement, I think, was born the following idea by Herbert Spencer: What does it mean to teach? It means to encourage systematically the students to their own discoveries (as it was formulated by George Pya and Gabor Segin the epigraph to the preface of their book Problems and Theorems in Analysis and repeated in Polya, G., (1981)). This approach to teaching is natural to call discovering.
In this paper, I intend to look closely to these approaches (mainly, with respect to mathematics), to identify their advantages and/or disadvantages, and to consider the work of brain hemispheres within these approaches.