Discussion I of part II: Representing and meaning-making: The transformation of transformation

Abstract

This commentary simultaneously offers a broad and a narrow perspective. It is narrow in that it does not attempt to synthesize or digest the chapters in this section. It can be called broad as it attempts to sketch some salient features of the future development of and learning geometry, of the transforation of transformation. Four such strands of reasoning are discussed: the paramount significance of meaning-making, the role of artefacts as socially and culturally embedded, embodiment and enactment, and, finally, emotions, meaning-making, and triangulation. All chapters in this section circle around the question how (technical) artefacts can be thought of as mediating mathematical meaning, especially how mathematical meaning emerges in the interaction of a subject with a (technical) artefact embedded in an educational situation, said shortly. In all chapters, geometry is the kind of representation that is investigated. I will not go into the question of the special kind of representations that geometry is incorporating. I will restrict my commentary to a discussion of multiple forms of psychosocial and semiotic triangulation that are salient in the teaching and learning of geometry, in meaning-making, emotion, and development. By triangulation I mean those meaning-making relationships that include three instances: a subject-an object-another subject, or a subject-an artefact-an object, or simply three subjects. In my commentary, I will discuss the consequences of what has been called an embodied perspective on human activity and thinking following the seminal volume of Varela et al. (1991). They put this fundamentally different perspective on cognition as embodied action as follows:Reading this quotation again today, one is surprised to see that emotion is missing from this perspective-just as it is missing from the whole volume by Varela et al. It seemed easier then to think of cognition as having a bodily basis, leaving emotion unmentioned. It comes as no surprise that emotion is practically missing in a recent volume on the state of the art of mathematics education (Sriraman and English 2010), although today, emotion appears as the psychological function paradigmatic for a perspective of embodiment.In a sense and finally, my commentary is also meant to be a discussion and critical reflection of the main concept of this volume, transformation.