Teaching Non-Euclidean Geometries in High-School: an experimental study.

In these years it is becoming quite clear that the development of a young student into an adult citizen requires a solid scientific background. Facing the challenges of a quickly changing world where political decisions are not only concerned with economics or ethics, but also with climate sciences, medicine, etc., requires a good education. Citizens are required to exert logical thinking and know the methods of science in order to adapt, to understand and to develop as persons. At the core of all these required skills sits mathematics, with its formal methods to develop knowledge. Learning the axiomatic method is fundamental to understand how hard sciences work, and helps in consolidating logical thinking, which will be useful for the entire life of a student. In my experience as a secondary school teacher, I have tried to understand how students perceive mathematics and what difficulties they encounter. One observation I often made was that the axiomatic study of geometry was a problematic topic for students, even for those with an interest for mathematics. For this reason, I decided to focus my PhD work on the teaching and learning of geometry, focusing explicitly on its axiomatic foundations, in order to concentrate on those aspects that foster the development of a logical thinking, of the ability of proving a thesis, etc., which are necessary, as said, for the growth of a modern citizen. Axiomatic geometry exposes the students to plural worlds, where the choice of a few base axioms heavily influences the properties of the objects that can be observed. The students, who are used to an intuitive study of geometry in the Euclidean plane, can benefit from the discovery of non-Euclidean geometries in several regards. First, they are shown that different non-Euclidean geometries exist, then they discover how these geometries can be developed by slightly modifying the axioms. Finally, they can be taught how these are useful to model real problems. The importance of teaching non-Euclidean geometries in high school has been debated for decades and several experiences have been conducted with students in the past. However, all these works were often of qualitative nature, the experimental protocols were poorly documented, and the statistical data was missing. The main objective of this thesis is to investigate, by means of quantitative experimental protocols, the viability and effects of teaching an introductory course on non-Euclidean geometries to high-school students. The experimental nature of this study required the classroom work to be concise and limited to a short number of seminars and workshops. Several experiences are described, involving several high school classes and a total of 154 students and 57 teachers. These have been used to evaluate and refine the teaching tools and topics that are covered in this thesis and are reported in detail for use in future experiences. Statistical methods and evaluation questionnaires are discussed to assess the effectiveness of the approach, which will prove necessary in larger-scale experiments. The outline of the thesis follows. In Chapter 1, a more elaborate motivation and introduction to the topics of the thesis are given. Chapter 2 reports a brief history of the development of Euclidean and non-Euclidean geometries and then discusses whether the birth and the development of non-Euclidean geometries constitute a revolution in mathematics. Chapter 3 discusses the teaching of geometry in secondary schools, with a specific interest in the Italian education system. Chapter 4 gives an informative introduction to the main aspects of the study and open questions; summarise and critiques the studies that have been conducted on the teaching of non-Euclidean geometries; and states the research questions that are investigated in the present thesis: - RQ1:What features of a short introductory course in non-Euclidean geometries are effective in engaging high-school students? - RQ2: To what extent do students gain a new perspective on the concept of axiomatic system? - RQ3: How well do students learn the taught concepts of non-Euclidean geometries? - RQ4: To what extent do students’ critical thinking and proof skills improve over the duration of the course? - RQ5: Do students’ beliefs about mathematics change over the duration of the course? Chapter 5 discusses all the details about the experimental phase. Specifically, this chapter describes and justifies the research methods and justifies the choice of adopting an essentially positivist paradigm using quantitative methods; discusses a preliminary experimentation conducted to investigate a suitable methodology, and – in more detail – a second experimentation. In addition, this chapter contains two sections dealing, respectively, with topics related to the experimentations: an experience with high-school teachers; and the description of the necessary adaptations for the distance learning imposed by Covid-19 restrictions. While Covid-19 restrictions impaired the possibility of a large-scale experiment, it allowed me to observe some peculiarities of the distance learning paradigm that must be accounted for when conducting geometry seminars online. Chapter 6 discusses the results of the data analysis, and provide an interpretation of the data shown and many conjectures for future developments. Chapter 7 concludes the doctoral dissertation.