Basic Geometric Concepts in the Thinking of In-Service and Pre-Service Mathematics Teachers

This paper discusses a model of a mathematics teacher professional development implemented in Italy and Hungary with in-service and pre-service mathematics teachers. The model focuses on comparative geometry, and it develops with the use of an artifact: the Lenart spheres. The teacher training model is the result of several years of experience of the two authors both as regards the activities in the classroom with the Lenart spheres and as regards the training of teachers in this field. The proposed teachers' professional development, in addition to providing ideas for activities to be implemented in the classroom, has the objective of proposing reflective activities from a community of inquiry perspective; during the activities, mediated by the artifact, both the Pedagogical Content Knowledge and the Mathematical Content Knowledge are taken into consideration (Ball et al., 2008). The model has been implemented in Italy in more than 15 training courses taught in the last 5 years, both with primary school teachers and with secondary school teachers. In Hungary, the model is at the basis of elective courses under the title 'Ball Geometry' at ELTE University, Budapest, for decades. These courses have been aimed at prospective preschool and elementary school teachers at the Faculty of Primary and Preschool Education, as well as future secondary teachers at the Faculty of Natural Sciences. The subject of the teachers' professional development paths corresponds to the comparative geometry between the plane and the sphere. After the presentation of the model, some examples of activities implemented in Hungary during the pandemic period will be illustrated and commented from a didactic point of view, which will serve to exemplify the path described. The described path was carried out remotely in online mode through synchronous and asynchronous activities. The distance obviously changed the way we interacted with the artifact, but it did not prevent the achievement of the courses' objectives.