Since the beginning of the transmission of geometric knowledge, two aspects of geometry have been present: the abstract “speculative” and the practical. These two aspects correspond to an essential dialectic in geometry teaching, between a deductive/rational science and a practical/intuitive one. In the nineteenth and twentieth centuries, the stress on the second aspect led to experimental geometry. After the work of Descartes, another tension concerned the solution of problems: between “pure” methods and methods coming from algebra or analysis. Attempts to find a new language for school geometry reached their apex when geometry was substituted by linear algebra in the 1960s.