The Role of Geometry in Reasoning and Teaching

Claudio Citrini ()
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Claudio Citrini: Politecnico di Milano, Department of Mathematics

A chapter in The Visual Language of Technique, 2015, pp 77-91 from Springer

Abstract: Abstract 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 enabling 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 a great variety of geometrical software packages 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 the branches of knowledge, as mathematics is ubiquitous. What can we do in order to improve their skills?

Keywords: Golden Section; Space Filling Curve; Platonic Solid; Cremonian Transformation; Polytechnic School (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-05326-4_8

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