Discovering Neglected Synthetic Geometry on Social Networks: Learning Maths as in the Historical Italian Academies

In this paper, we present the work of an ongoing project, devoted to 13–15 years old students in southern Italy, aiming to improve their mathematical literacy. Starting point of our research was the way to spread the culture in the sixteenth century Academies. There were about 800 Academies in Italy in 1525-1700 years: they were fundamental for the development of intellectual networks and the dissemination of ideas in Europe. In these Academies, interdisciplinary paths were developed, including literature, arts, natural sciences, medicine and mathematics. In Mathematics, there was a fervent discussion about “new geometry”, i.e. calculus vs “old” synthetic geometry. Because of calculus success, some very interesting results about synthetic geometry were neglected. Aim of present work is planning and experimenting an interdisciplinary learning unit, focused on some hidden historical theorems of synthetic geometry developed for 1500-1800 years about “Cevian, Orthic and Pedal Triangles”, by simulating “old Italian Academies” in a modern key, also by using ITC. These theorems are quite interesting from a Maths and Physics point of view, too. We studied, in particular, Ceva's theorem and Fagnano's problem. Both Ceva and Fagnano theorems may be generalized, under some conditions, to orthic quadrilaterals of a convex quadrilateral and Fagnano's problem may be extended to “polygonal billiard” physics, too. In order to re-discover these hidden theorems, blended learning and flipped lessons methodologies have been also used. The planned learning unit has been experimented with about thirty students, attending the second year of a Secondary School in South of Italy. Italian Academies have been simulated in two ways: virtual and blended academies were founded, respecting main characteristics of ancient Academies. English has been chosen as universal language, instead of Latin used in the past. Nevertheless, some historical documents (e.g. Ceva theorem) have been studied in Latin and in archaic Italian, to critically analyze original fonts. Information Technologies can provide significant advantages in this learning – teaching approach: they have been used to share and communicate main results, to create “virtual Academies” and to realize multimedia materials (a final video, summarizing all the activities), too, which are “modern fonts” to be archived on cloud software’s. The experimented educational path shows that, although students generally consider Geometry as a difficult branch, they may excite if they study it in an innovative way and in an historical environment.