As a result of my observation and analysis, summarized in the case studies of the last three chapters, I am now able to discuss an investigative approach to mathematics teaching from a more knowledgeable position than when I began my study. My theoretical position, initially, was based on intuition, experience, and what others had written. The research has substantiated this position and advanced it. It has provided a practical perspective and highlighted issues and tensions. I saw some very good teaching by some very good teachers, from which it has been possible to synthesize characteristics. This teaching was by no means unproblematic—I found evidence of inherent tensions. In this chapter I shall focus on these characteristics and tensions, and, in Delamont and Hamilton’s words (1984), attempt to ‘clarify relationships, pinpoint critical processes and identify common phenomena’ leading to the formulation of ‘abstracted summaries and general concepts’. 1
From the classrooms I studied, I shall now highlight common themes and issues and attempt to distil characteristics which seem pervasive and in some sense germane to an investigative approach. This is not to say that investigative teaching, or teaching arising from a constructivist philosophy, will invariably have these characteristics, but rather that these have seemed to be significant in this study where I recognize that my sample of teachers is both small and selective.
I shall begin by setting the common scene as I saw it. In all cases throughout the three phases, the teachers worked with classes of twenty-five to thirty-two students. They set tasks which involved mathematics and on which students worked. These tasks and the way in which they were set varied considerably both from teacher to teacher, and from lesson to lesson for any one teacher, depending on the particular objectives declared and undeclared for any lesson. However, there were common features.