PROGRESS REPORT OF AN ONGOING PROJECT ON TEACHING NON-EUCLIDEAN GEOMETRIES AT HIGH SCHOOL
This paper deals with the progress of an ongoing project, first reported in , on teaching non-Euclidean geometry at high school. The main goal of the research work is:
(1) to show whether non-Euclidean geometries can be a tool to allow students to consolidate the knowledge of Euclidean geometry by developing it in a critical way;
(2) to promote the understanding of the modern axiomatic method in geometry; and
(3) to give students a new perspective on mathematics so that they can see it as a creative activity and a widely discussed topic with a historical background. The idea itself is not novel, but one of the main issues related to the state of the art is the lack of experimental studies with students. Specifically, up to the authors’ knowledge, there is only one study that has conducted an extensive experimentation with high-school students, showing data that seem to support the hypothesis (3): teaching non-Euclidean geometries lets students change their view of mathematics . Experimental studies conducted with high-school students related to the first two points seem to be absent. For this reason, our aim is to gather further experimental evidence of the potential benefits of teaching non-Euclidean geometries at high school.
The authors have carried out a pilot study based on class activities on non-Euclidean geometries that involved four sets of Italian high-school students (10-18-20-29 students). The pilot study was aimed to understand how to improve the class activities and, most importantly, how to better design useful questionnaires to collect data for the final experimental study. To gather data during the pilot study, the authors used, before and after the class activities, the Van Hiele geometry test as formulated by Usiskin  (translated in Italian) and questionnaires previously designed by the authors.
The pilot study made us aware of some students’ misconceptions and difficulties. The paper reports on these findings and on how they led us to change some of the class activities proposed to the students and to elaborate the questionnaires in a different way. Moreover, the paper presents data that seem to support the hypothesis (3) as in .
 S. Benvenuti, A. Cardinali, "The mental telescope: understanding the geometry of Euclid by learning the non-Euclidean geometry", INTED2018 Proceedings.
 P. Schiano, "Convinzioni e cambi di convinzioni degli studenti sugli errori e sullo sviluppo della conoscenza in matematica (studenti di età 14-18)", PhD thesis, Consortium between the University of Bologna, the University of Catania, the University of Pavia, the University “Federico II” of Napoli, the University of Bratislava, the University of Nitra, the University of Alicante, the University of Palermo, the University of Cipro, C.I.R.E “Centro interdipartimentale ricerche educative” Palermo.
 Z. Usiskin, "Van Hiele Levels and Achievement in Secondary School Geometry", University of Chicago, 1982.