**Abstract** : Geometry is a school subject, but also and primarily geometry is a mathematical domain. As mathematical educators we are interested in geometry from both perspectives, as well as we are interested in the relationship between them. First of all, geometry is a formalized theory, within the broader frame of mathematical theories. Actually, geometry pervades (permeates) a large part of mathematics, even some of its very recent developments, influencing and affecting both their origin and their evolutions. Geometry as a theory of space offers a good model to control basic intuition in different and various mathematical domains. A privileged relationship between elementary Geometry and physical reality must be recognized. According to the theory of figural concepts geometrical concepts are "A mixture of two independent, defined entities that is abstract ideas (concepts), on one hand, and sensory representations reflecting some concrete operations, on the other" (Fischbein,1993, pag. 140). The question arises about the congruence between spatial cognition and abstract mathematical space, i.e. Geometry. Complete congruence between the two systems is not always assured. Far from being natural the move from intuition to geometry presents great difficulties . Two objectives can be stated for geometrical education. - the necessity of developing a flexible interaction between images and concepts. - the development of complex conceptual schemes, controlling the meanings, the relationships and the properties of a geometrical figure.