What research methods are appropriate to studying the nature of an investigative approach to mathematics teaching? How does one decide? What issues does this raise? What consequences are there for the research process? In this chapter I shall address these questions from a methodological perspective for the research project as a whole. Although it may be possible to discuss the methods used in a research study in isolation from other aspects of the research, I have found it unrealistic to try to do this. I have found research methodology here an integral part of the research itself, closely bound up in the theory and practice of the research enquiry. Therefore this chapter will draw on theoretical aspects of the research process as well as methodological issues, and subsequent chapters which focus on classroom observations will continue the methodological discussion.
The theory-practice relationship in this research has been dialectical, maintaining a tension for the researcher while stimulating the research. I have sought to characterize an investigative approach through a study of its practical manifestations in classrooms! 1 On the one hand it is possible to look at classroom manifestations of aspects of theory. For example, an investigative approach might be thought to involve aspects of enquiry, so a researcher could look out for classroom events which could be seen to show some form of enquiry and describe those events. On the other hand, the researcher could observe and describe occurrences in a classroom and analyse these to distil essences of the approach, from which for example, something like enquiry might arise. This is theory generation. Throughout the research, it was impossible to avoid flipping between these perspectives, and tension arose in trying to rationalize the result. The methodology of the research is central to this dialectical relationship. The researcher has to act, so decisions have to be taken regarding the form of such acts and their consequences for the direction of the research.
In Chapter 3, I focused on the early thinking which led to decisions about appropriate methods. The purpose of the present chapter is to discuss these methods alongside research issues which either influenced their use or resulted from their use. Many of these methods are well documented in research literature and I shall try to show how and why my use accords with, or diverges from, what