At the very beginning of my research I knew in the broadest terms that I wanted to study an investigative approach to mathematics teaching. 1 It was obvious that the subjects of my research would be teachers of mathematics and their students, and that the research would take place in mathematics classrooms and their relevant surroundings. I wished to explore what such an approach might look like in practice and to attempt to characterize it from practical manifestations in classrooms, rather than from my own preconceived notions. I did not intend to study my own teaching as I was no longer a classroom teacher. I therefore wished to study other teachers, but it was not clear who they should be or what form the study should take.
Sharrock and Anderson (1982) write,
…we should bear in mind that whilst the question of what we are to look at is by no means a trivial one, it is a little less important than the question of how we are to look at whatever we do look at. (Sharrock and Anderson, 1982)
I recognize now both the potential and the insecurity of this stage of beginning research. I had been, very recently, a classroom teacher, and had no confidence in myself as a researcher. Indeed I was not sure what ‘research’ actually meant, and I went around for some time asking experienced researchers in mathematics education ‘what is research?’ The replies which I received were not very helpful, often assuming that I was after some rather abstruse definition, and simply not understanding the naivety of my question. It was hard to see from my reading of other educational studies, how the methodologies employed might be adapted to what I wanted to find out. And here was perhaps the root of the problem, what did I want to find out?
Thus I entered into a voyage of discovery, in which the first tentative steps I took were exploratory both in terms of my research goals, and of the research methods which would be appropriate. I was reassured by statements such of that of Sharrock and Anderson above, and of Edwards and Furlong (1985) who wrote,