Dynamic geometry, construction and proof : making meaning in the mathematics classroom
The overall aim of this study was to investigate mathematical meaning making in relation to the areas of construction and proof through the use of a dynamic geometry environment (Cabri II as available on the TI 92 calculator). The experimental work was carried out with 11-14 year old pupils in four schools in the North of England between 1996 and 1999. The research involved working with whole classes and a range of groups of varying sizes. The research methodologies adopted were drawn from various areas (an approach advocated as suitable for classroom research by Klafid, 1998). The researcher acted as both teacher and participant observer. The study was conducted over several cycles, with previous cycles of analysis and reference to the literature being used to inform subsequent stages. After a pilot phase when recording methods and technical approaches were clarified, there were four cycles of investigation. Data collection was by means of participant observation, with audio recording of dialogue. Screens generated by pupils were recorded in field notes. There was emphasis from the outset of the study to relate the findings to classroom practice. This led to a consideration as an ongoing part of the study, of ideas of classroom and group dynamics and how these could be combined with, and related to, the use of the technology. The study illuminated two key areas; the processes of immediate individual and group meaning making and wider aspects of social dynamics in the mathematics classroom. Socio-cultural analysis of classroom and group discourses identified progression from spontaneous to scientific concepts, illuminating the development of pupils' powers of intuition and sense of conviction. The dynamic geometry environment was used to investigate constructions stable under drag, illuminating the way in which the dynamic aspects afforded by the technology affect pupils' appreciation of the relationship between construction and proof. Various aspects of proof were highlighted and in particular the function of proof as explanation was seen to be an important aspect in the development of pupils' mathematical meaning making. Further analysis illuminated a distinction between the immediate individual sense making of pupils and the way this sense making is brought to social and consensual meaning making. At the wider classroom level the study identified issue of transparency the importance of the social use of argumentation to take forward the 'taken as shared' and the development of socio-mathematical norms and whole-class zones of proximal development. These aspects of individual and group meaning-making and whole class dynamics are advanced as ways of promoting local communities of mathematical practice as advocated by Winbourne and Watson( 1998).