The “metamorphosis” can be explained quickly as follows: it is the replacement in teaching of geometrical proofs using inspection of classical figures as segments, triangles, conics, with geometrical transformations and also algebraic calculations, by the exclusive use of calculations in linear algebra with matrices. The visual aspect in the geometric sense disappears in favor of algebraic calculation alone. The back and forth between figures and calculations are forgotten but the capacity of calculations increases. My purpose is only to provide some clarifications about the sources and conditions of such a metamorphosis in France. When and how the organization of changes of geometrical teaching had been thought? Two precautions are necessary: a distinction must be made between teaching in secondary level (students aged 14 to 18) and in tertiary level (students at the university); and this must be examined especially before the “Réforme des mathématiques modernes” [new maths]. I show that, in fact, events started almost ten years before the meeting at Royaumont in 1959, with Gustave Choquet as one of the main actors. He tried to introduce a modern teaching, with linear structures (the “royal road”), but at the same time he wanted to preserve the geometrical intuition given by experience with real objects, with several “intuitive” axiomatics. At the end, the Reform did not take this moderate direction, and it was Jean Dieudonné’s purely linear approach that was privileged, up to excess of formalizations and calculations. Hence the metamorphosis arrived, when the view of the figures was replaced by the view of equations, and worse by formalization. As a testimony I explain how under these circumstances I learned geometry in the 1960s and taught geometry at the early 1970s at the university of Paris 7. Keywords: reform of modern mathematics, geometry, teaching, France, APMEP, CIEAEM, Gustave Choquet, Jean Dieudonné, Lucienne Félix, Caleb Gattegno, Morris Kline, Georges Papy, Jean Piaget, André Revuz, Gilbert Walusinski
The Prehistory of Mathematical Structuralism