The title of this book ‘Investigating Mathematics Teaching’ has been used deliberately in a double sense. It refers to the use of investigative processes in the classroom teaching of mathematics. It also refers to a research process which has explored the teaching of mathematics and its development. The research described here was conducted from a constructivist philosophy of knowledge and learning. I have hinted, earlier, that many educators now see constructivism, particularly radical constructivism, to be inadequate to underpin the complexity of mathematical learning in classrooms. In this chapter, I look briefly at some current views and criticisms and justify my own theoretical position in this research.
I explored with mathematics teachers in Phase 1 what might be involved in implementing an investigative approach to teaching mathematics in their classrooms. It seemed that this early thinking fitted well with a radical constructivist philosophy of knowledge and learning as elaborated by von Glasersfeld (1984). In Phase 2, I studied the teaching of two teachers, both experienced and successful practitioners, who could be seen to implement an investigative approach. My analysis of the data collected sought to characterize the teaching I observed, and a theoretical device, the teaching triad, emerged from this analysis. Simultaneously I was developing my own understanding of constructivism and its implications for teaching. In Phase 3, the constructive processes of both teacher and students became clearer as I sought to rationalize the teacher’s view of his teaching from apparently contradictory positions. The teacher-researcher relationship was of fundamental importance to the research and led to a theoretical model to describe reflective practice and the development of teaching. All observations and analyses were rooted in classroom interaction, and in the teachers’ conceptions of that interaction.
Throughout my observations of teaching, issues and tensions emerged. An investigative approach to teaching mathematics proved problematic for the teachers in all three phases. Statements from the teachers themselves suggested that the status and communication of knowledge were at the root of problems perceived. For