THE MENTAL TELESCOPE: UNDERSTANDING THE GEOMETRY OF EUCLID BY LEARNING THE NON-EUCLIDEAN GEOMETRY
, A. Cardinali2
1Università di Camerino (ITALY)
2Scuola di Scienze e Tecnologie - Sezione di Matematica - Università di Camerino (ITALY)
We describe a research project about the teaching of non-Euclidean geometries. Our intent is to show that non-Euclidean geometries can be a tool to promote the understanding of the modern axiomatic method in mathematics, to stimulate students' aptitudes to logical thinking and to allow to consolidate the knowledge of Euclidean geometry by developing it in a critical way.
This thesis has been already object of debate by several mathematicians and, in the 90s, the PNI (Piano Nazionale Informatica, a plan experimented by many schools in Italy before the so-called Gelmini reform) suggested in his guidelines the teaching of Non-Euclidean geometries. In particular, they suggested the comparison of Euclidean and Non-Euclidean geometries to help the student to understand that there can be more than one truth if we start from different axioms. Also out of Italy, similar methods were proposed. However, as far as we know, none of these activities reported quantitative data or results regarding the efficacy of their methods.
With the Gelmini reform the guidelines of Piano Nazionale Informatica about Non-Euclidean geometry have been omitted, even if the new guidelines still require the comprehension of the axiomatic approach in its modern formalization and the understanding of the historical context of several scientific theories. Hence the introduction of some historical elements regarding the birth of non-Euclidean geometry can be of benefit in showing students how Mathematics develops from human histories, debates and it is not only the accumulation of concepts and theories. In fact, the rise of non-Euclidean geometries and the proof of their logical consistency, guaranteed by the models of Beltrami, Klein and Poincarè, represents a revolution in Mathematics (according to definition of revolution given by Joseph Dauben); radical innovation have occurred which have fundamentally altered mathematics, even if any earlier mathematics (i.e. Euclidean geometry) was not “irrevocably discarded”. Recalling the words of Imre Toth , the great novelty has been the establishment of a plurality of worlds: the universe of geometry is no longer a domain in which there is only one valid truth, but two opposing truths are equally valid, depending on the starting hypotheses. The problem of guaranteeing the consistency of a system of axioms chosen through an act of freedom and without basing their validity on some "evidence" becomes therefore of fundamental importance.
We believe that teaching non-euclidean geometries in high-schools at the moment is overlooked but, even more important, there has never been sufficient research on the efficacy of its teaching.
The research we intend to conduct will take into account the results obtained from other works already carried out both in Italy and abroad. In particular, it will be merged within the PLS (Progetto Lauree Scientifiche - Scientific Degrees) project of the University of Camerino and will start with the analysis of laboratories conducted since 2007 by Silvia Benvenuti, the project supervisor. A fundamental aspect will be the investigation aimed at establishing the effectiveness of this method.
 Toth, I. (1991) La geometria non euclidea come atto di creazione, Napoli, Università “Federico II”, video taken from Enciclopedia Multimediale delle Scienze Filosofiche – http://www.conoscenza.rai.it/site/it-IT/?ContentID=383&Guid=85321985043d44b5b46c9562ccc84d79#