Perspective for Analyzing Mathematics Teachers’ Reflection from Viewpoints of Mathematical Knowledge: Praxeological Analysis within the Anthropological Theory of the Didactic

Abstract

　　This study aims to propose a perspective for describing and analyzing mathematics teachers’ reflection focusing on mathematical knowledge.  For this purpose, the notion of praxeology which is used within the Anthropological Theory of the Didactic (ATD) is employed for modelling both teachers’ activity and their knowledge.  In this paper, we conduct following two research tasks: 1) Setting up the perspective from the theoretical implication of praxeology, especially of paradidactic praxeology; 2) Validating the perspective through a case study which analyzes a post-lesson discussion by preservice teachers.  For the first task, we focus on relationships between different kinds of praxeologies: mathematical praxeology, didactic praxeology, and paradidactic praxeology.  As a rule, didactic praxeology refers to knowledge and activities for elaborating mathematical praxeology, and, paradidactic praxeology refers to that of didactic praxeology.  Thus, the subjects of paradidactic praxeology can be described as didactic praxeology (or elements of that).  Similarly, mathematical praxeology can be described as the viewpoints of that paradidactic practice.  For the second task, we use the lesson plan and the transcription of the post-lesson discussion as empirical date.  As a result, we could get a figure which showed the structure of knowledge constructed in the post-lesson discussion (Fig. 2). Moreover, the analysis suggested that persistence in practical aspects was one of the characteristics of reflection by preservice teachers.  With reference to validity of the analytical perspective, the case study showed its usefulness for considering relationships between praxeologies and elements of that.  Particularly, the value of the analytical perspective consisted in availability for considering not only observed phenomenon but also non-realized phenomenon.