The Role of Tools in Conjecturing, Arguing and Proving in School Mathematics: The Case of Dynamic Geometry

Catia Mogetta

PhD thesis, University of Bristol - Graduate School of Education

Proving lies at the heart of mathematics and has increasingly been recognised as a crucial activity in school mathematics over the past few years. The processes of conjecturing, arguing and proving involve the interplay of experiential and theoretical knowledge in different activities, ranging from concrete, empirical actions to deductive reasoning.
Cognitive, historical and epistemological studies have highlighted potential and actual difficulties met by students when conjecturing, arguing and proving. The relatively recent introduction of Dynamic Geometry has changed the setting of the proving activity in school mathematics, providing a space in which the interaction between visualisation, exploration and heuristic strategies reduces the gap existing between the theoretical and the empirical plane.
Starting from the sociocultural position according to which the actions students perform together with the knowledge they construct are shaped by the tools used, the study provides both a description and an explanation of how tools included in the activity setting shape and transform students' production of arguments and possibly favour the construction of links between the empirical and the theoretical plane.
Drawing on a mediated activity framework, the notion of toolkit has been introduced as the main analytical tool, and developed throughout the study to account for both the heterogeneity of the tools available to students and the mutual relationships linking them within a structure.
The qualitative analysis carried out in this study highlights the mediatory role of tools in the proving activity, especially in the elaboration of conjectures and of arguments to justify or refute them. Some particular tools are indicated, with respect to the Dynamic Geometry setting, but the findings extend beyond the particular tools to explain their functions within the proving process.
The main findings of the study concern the identification of proving as a tool-mediated activity depending on cultural, social and contextual features of the setting.
The management of heterogeneity emerges as a crucial and transversal issue: the interplay of heterogeneous tools and the changes of perspective favoured by a dynamic exploration of the problem have been identified as crucial elements for an effective construction of links between the empirical and the theoretical plane.