The role of geometry in reasoning and teaching

Ever since the very beginning of ancient philosophy, from Pythagoras to Plato, we know that the world is made up of numbers and figures. Greek mathematicians used drawings as a natural tool for proofing their theories, as the works of Archimedes and Euclid clearly show. Actually, the ruler-and-compass constructions are the most ancient examples of a perfectly well designed formal language, whose power is equivalent to up to second-degree equations. Drawings often provide wordless proofs that everybody can easily see: for instance, Pythagoras’ theorem and the statement that the sum of odd numbers in increasing order is a perfect square can be proved through a self-explaining drawing. The invention of symbolic algebra in the early seventeenth century, led mathematicians to a more abstract approach to mathematics. These tools are indeed very powerful, and they often need only a calculating capability instead of a deep understanding of the problems. However, especially in the nineteenth century, an analytical approach seemed to be safer than a geometrical one, and the drawing as a means was excluded from most books of mathematics, which had a negative impact on learning. On the other hand, functional analysis introduced a geometrical language in order to be able to describe many abstract concepts. Nowadays students have a very poor geometrical insight, the main fault for which lies in the scholastic institutions. Most of them cannot comprehend the large amount of information that a drawing contains, in spite of the existence of numerous geometrical softwares designed to construct and dynamically modify figures in order to verify guesses about their properties (to be eventually proved by a formal demonstration, of course). This naturally affects all branches of knowledge, as mathematics is ubiquitous. What can we do in order to improve their skills?