History of teaching geometry

Since the beginnings of a transmission of geometric knowledge two different aspects of geometry are present: the abstract “speculative”, represented by Plato and by the Elements of Euclid, and the practical, represented by the work of Heron and by the applications of geometry. Besides the original rationale, the two aspects will develop and assume different meanings in the teaching of geometry in the essential tension between a deductive/rational science and a pseudo-practical/intuitive one. The stressing of the second aspect in the 19th and 20th century, in a period of major attention to the child’s mind, will lead to experimental geometry and to methods that allow the shifting of the subject to the lower grades. Particularly after the work of Descartes, another tension will develop in the teaching of geometry and particularly in the solution of problems, between the pure methods of reasoning and methods that make use of arithmetic, algebra or analysis. These attempts to find a new language for school geometry will have their apex in the substitution of geometry with linear algebra in the 1960s.