The proving process within a dynamic geometry environment

Federica Olivero

PhD Thesis, Graduate School of Education, University of Bristol, 2003

Proof is a crucial aspect of mathematics education and has increasingly been recognised as such within the UK (Royal Society -Joint Mathematics Council working group on the Teaching and learning geometry 11-19) and more globally (e.g. NCTM Standard 2000, Robutti, 2001). Proof is also a crucial activity within mathematical practices (e.g Barbin, 1988; Balacheff, 1999; Rav, 1999). However, research has indicated the many difficulties that students have when approaching proof and proving in the classroom (e.g Arsac, 1992; Duval, 1996; Hanna, 1996; Bartolini Bussi & Mariotti, 1998). The main difficulty with respect to proof and proving that emerges from current research is the gap between empirical and theoretical elements involved with these activities. However, studies are beginning to show ways in which new technologies can be used as tools to support the proving process. In particular, a strand of research has investigated specifically the impact of dynamic geometry software with respect to the teaching and learning of proving in geometry (e.g. Goldenberg, 1995; Arzarello, Gallino, Micheletti, Olivero, Paola & Robutti, 1998; Laborde, 1998; Jones, 2000; Mariotti, 2000; Healy & Hoyles, 2001).

My doctoral study addressed the problem of how a dynamic geometry software (Cabri-Géomètre) may support students in managing the relationship between the empirical (spatio-graphical) field and the theoretical field, i.e. in their approach to proving. In particular, the main aims of the research were the following:

- To investigate the development of the proving process (i.e. the construction of conjectures and proofs) within a dynamic geometry environment.

- To investigate the interactions taking place in the proving process between the students and between the students and the tools used.

Observations of pairs of students were carried out within a number of classrooms in Italian and English schools. The methods of data collection used were video-recording and collection of students' materials. The data available for the analysis were transcripts from the video tapes, the Cabri files and the students' worksheets. The analysis of data included in-depth analysis of selected cases studies. An inductive process of analysis was carried out, starting from some theoretical assumptions, but being open to 'read' the data in order to identify the categories of analysis, with the purpose of developing an analytical and explanatory framework for the development of the proving process in dynamic geometry environments.

The research findings suggest that proving within a dynamic geometry environment develops as a focusing process, in which the shifts between ascending processes, from the empirical to the theoretical field, and descending processes, from the theoretical field to the empirical field, appear to be key elements for the construction of conjectures and proofs. Tools for focusing, used by students in the process, were identified (e.g. dragging in Cabri, the use of construction elements, the use of paper sketches). The findings further suggest that there is not a hierarchy from empirical to deductive/theoretical proof, but the key element of the proving process is a successful management of empirical and theoretical elements. Moving towards the theory does not mean abandoning the empirical field, but looking at it from another point of view. The analysis also shows that Cabri works as a shared workspace, i.e. as a space which supports the interaction between students' internal contexts and the construction of shared knowledge towards the production of conjectures and proofs.

These results will have an impact on teachers' practices, providing suggestions on how to use dynamic geometry as a support for proving activities in the classroom. The explanatory framework developed suggests a number of tools for focusing which students use when they are proving within a dynamic geometry environment. It is important that teachers become aware of these tools and that they are made explicit to students. Students could then transform these tools into appropriate instruments for supporting the proving process.