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.