Ancient problems and contemporary issues.

In primary and secondary Italian schools, teachers teach Geometry basically by following textbooks. Textbooks contain the Propositions of Book 1 of the Elements, a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid (c. 300 BC). The Propositions are normally presented at school without any element that makes students understand their practical usefulness and links with the environment and current problems. So, we involved the pupils in a problem-solving activity: starting from a current problem, students are guided to understand and use some Propositions and Common Notions from Book I of the Elements. The students approached the resolution of the problem using teaching tools (called manipulatives) used in a context of scientific theory, just as the Elements are de-finable. We think, in fact, that the reconstruction of the demonstration using these tools is incisive because it allows students to independently discover the solution strategy, with a deductive chain that is generally simpler than the original Euclidean one and therefore more suitable for a mind still in formation of pre-adolescent students. Appropriate reflections on the ‘invariants’ that emerge make it possible to transfer what is learned in specific cases to more general problems, allowing the student to generalize that follow a certain rationality even if not yet supported by typically mathematical formalism. These procedures differ very much from the proofs in textbooks which we consider not very useful, as they are applicable only in specific cases and do not allow us to transfer what we have learned to more general cases. In this paper we show our proposal, the conclusions on the experience and some issues that support our choices.