The focus of each of the three phases was a product of my own thinking at that particular stage of the research, not all of which is documented in the research chapters. The purpose of the interlude is to highlight aspects of this thinking, particularly as it influenced the moves from Phase 1 to Phase 2.
I now see the move from Phase 1 to Phase 2 largely as one from theory-validation to theory-construction. It was also the time when my seeking for objectivity shaded into a recognition of intersubjectivity. In this interlude I shall elaborate these terms and refer briefly to a strong influence on my thinking, external to my research.
When I began this study, my concept of investigational work was articulated in terms of classroom practices which had evolved from my own teaching, my own mathematical studies, and from discussions with colleagues. These included the overt use of mathematical processes (such as specializing, generalizing, conjecturing and convincing—Mason, 1988a), the value of group work and of student discussion, the importance of trust, the dangers of teacher-lust, etc. (Jaworski, 1985b). I made no attempt to define an investigative approach. Instead, I talked in theoretical and practical terms about what it might involve. This was based on my experiences in working with mathematics teachers, in different parts of England, on aspects of mathematical discussion, practical work and investigational work as expressed in the Cockcroft Report (DES, 1982, par. 243). The purpose of this work was to compile a video-tape, commissioned by the (then) Department of Education and Science (Open University, 1985).
This perspective directly influenced my early work at Amberley. Thus, I remarked in my diary that children were ‘specializing, but not very systematically’ and were ‘quick to spot patterns’, wrote ‘Students now need to be encouraged to write down findings coherently’, and commented that the teacher said to a student, ‘If you notice something happening, write it down.’ In retrospect, what I was doing, implicitly, was seeing manifestations of an investigative approach as I had theorized it.
It was an important realization for me that any theory is a generalization, and that classroom manifestations of such theory always include nuances or particularities that the general theory cannot predict. Thus the theory might claim that