Abstract:

“Let no one ignorant of Geometry enter” was, according to tradition, the motto engraved above the entrance of Plato’s academy. This shows the relevance given to this branch of mathematics as a gateway to a higher level of knowledge. In contrast, Geometry is typically relegated to the end of textbooks and currently receives less attention than Arithmetic or Calculus in primary and secondary school, respectively.

To give geometry back to the position it deserves as a tool to understand the world around us, we propose a complete redesign and update of the traditional way of teaching geometry. The latter is based on the memorization of definitions and the mechanical application of formulas, following a cooking recipe. To change this deep-seated tendency, we should start by training future teachers of primary and secondary school how to promote a meaningful and active learning of geometric notions.

In a nutshell, some of the main lines of our proposal, which will be detailed throughout the article, consist in showing prospective teachers how to design:
• Geometric puzzles to wake up the spatial intuition of students. For early stages we suggest physical jigsaws made of simple materials like wood, string, and foam. Later, new technologies will be used to create enigmas whose philosophy works like videogames, with increasing levels of difficulty and gamification strategies.
• Constructive activities by exploiting materials like geoboards, Lego bricks, origami, tangrams, the stomachion created by Archimedes … After construction, one can use geoboards so that students deduce by themselves the typical formulas of areas.
• Geometric excursions: depending on the level, it can be oriented to shape discovery, measurements, live solution of a real problem using trigonometry, … But there is a common idea: geometry is everywhere and, accordingly, it should be taught beyond the classroom walls. One can also plan virtual mathematical visits, for instance, to the Alhambra and use Geogebra to analyse it geometrically.
• Art and geometry in motion: use tiles with a recognisable pattern (a local symbol) to explain and construct frieze patterns and mosaics, as an excuse to teach symmetries and classify geometric motions on the plane.
• Activities to train inductive reasoning (like counting how many diagonals has a polygon of n sides) or deductive arguments (introducing how Euclidean geometry was born).
• Exhibition of mathematical shapes: there is life beyond the area of the square, for instance, what are the geometric ideas behind the yin and yang symbol? We propose to design a gallery of curious shapes explaining their main features (as an open exhibition, blog, magazine, or short videos, depending on the level of the students).
• Photograph competitions to catch geometric similarities or fractal structures.

Our teaching strategy includes an update of the Van Hiele levels of geometric understanding. The idea is to use this theory as a roadmap to design activities for the students, but after a revision from the perspective of new technologies and methodologies. In short, we will present how to produce a lecture series of Didactics of Geometry so that in future schools of education, one can stamp the following modification of Plato’s leitmotiv “let no one ignorant of geometry exit this classroom”.

Keywords:

Didactics of geometry, active learning, contextual teaching, learning by doing.