Didactical paradoxes observed in the management of Euclid's theorems

Abstract

Geometry teaching and learning processes have allowed us to pose several problems that have been intensively studied by the Mathematics Education research community in recent decades, focusing on aspects such as software use, student knowledge, mathematical complexity, teacher knowledge etc. However, today, there is still interest in understanding the difficulties associated with the teachers' practices, in particular, regarding the management of the classes they carry out, since this directly impacts geometry learning by the students. In this sense, we present a study that consists on the observation of the management of two teachers who have faced the teaching of Euclid's Theorems in their second-year courses in Middle Education in Chile. To this end, direct observation of the classes was carried out and both teachers were subsequently interviewed. For the analysis of pedagogical management, we used Guy Brousseau's Theory of Didactical Situations (TSD), specifically focusing on the notions of didactical contract, didactical milieu, and the processes of devolution and institutionalization. The methodology used in the study is qualitative, descriptive, and interpretive, where extracts of episodes from a class are retrieved. The analysis reveals numerous didactical paradoxes that do not favor the learning of the students in relation to the foreseen objectives, nor to the mathematical objects treated in class. Among the paradoxes, phenomena such as the topaze and resonance effects are observed, detected in the teacher-student interaction. The need to analyze in depth the professional practices of teachers is evident, since this work revealed the rupture between the institutional discourse and the pedagogical management in practice.

Keywords:
Didactical contract; Didactical environment; Return process; Institutionalization process; Topaze effect; Resonance phenomenon