ELEMENTARY GEOMETRY SPLIT INTO DIFFERENT GEOMETRICAL PARADIGMS

Since several years, we have developed a scientific investigation on the teaching of Geometry, especially for pre-service teachers. This research is based on a specific theoretical frame. We have chosen to give some elements of our frame and we hope to present more consistent examples of our empirical studies during the working group. For us elementary geometry appears to be split into three various paradigms: natural geometry (Geometry I), natural axiomatic geometry (Geometry II) and formalist axiomatic geometry (Geometry III). We confront our approach with the approach of van Hiele, which to date is not used in France. We obtain a synthesis that we currently study in teachers training. During their school career, students are faced with different mathematical worlds, at least with a numerical one and a geometrical one. In the numerical world, the objects (e.g. the numbers) are represented by “abstract signs ” (juxtaposition of digits), that do not evoke the quantity they refer to. In contrast to this, in the geometrical world representations of objects often remain spatial objets. And in fact, the way is very long from a real spatial object to the notion of “figural concept ” described by E