Critique of methodology and methods

The research paradigm, approach and method and the theoretical framework for interpretation in this research are mutually enhancing. The preceding discussion has illustrated the close match between acknowledgement of subjectivity in a qualitative, interpretative approach and the theoretical stances of socioculturalism and social

constructivism (as explored by Ernest, 1998 and Wenger, 1998, 2010). This relationship sits also between the analysis of interview data and recognition of the communities of practice in the dialogue between teachers and teacher researchers. Interpretativist knowledge perhaps inevitably reflects the values of the enquirer, even as it seeks to reconstruct others' sense of meaning and supporting beliefs. The interpretative framework exemplified within

communities of practice is complemented by the use of social constructivist theory in developing an honest and open analysis of data in ways that are respectful of the rights of interviewees. It might have been useful to draw on more than one theoretical framework although this may have further confused interpretation. The choice of communities of practice was used deliberately in order that there was the potential for seeing the extent and the ways in which teachers' beliefs are influenced by internal and external factors, conscious and unconscious dispositions, as drawn out by Wenger (1998, 2010). It might have been interesting to consider one of the other of the seven theorists.

It is important to recognise that the interpretation blends my own and the teachers' histories in ways that are not always clearly divisible. Furthermore, it was undoubtedly the case that I came to the research with my particular view of the nature of excellence in primary

mathematics teaching but emerged from it with a far more nuanced interpretation of what that might be. Researchers must give particular consideration to their position in relation to their enquiry as whatever the situation, their role, identity and relation to it will be of

profound importance to the way they conduct their enquiry and its results. I was investigating a field within which I have worked for well over a decade. As Ely (1991) reflected, researching the familiar may seem advantageous but it may also mean that I may have overlooked

important aspects of the situation, and so needed to ensure I maintained my critical gaze. It is important to acknowledge my pre-existing beliefs about the field of excellence in mathematics teaching and that these will have been brought.to bear in what was perceived in the data and in defining what was important in reporting the data. Thus the conclusions rest on the researcher's interpretation of the experiences, attitudes and beliefs of the teachers who were the respondents at the time and place during which the research took place.

In interpreting the views of individuals, it is incumbent on the researcher to represent those individuals as truthfully and honestly as possible (Kvale & Brinkmann, 2009). The threat to that came in the form of attempts at generalisation across the group. I may well have

unconsciously looked for commonalties and perhaps assumed group agreement, when the teachers were very much individuals. Their individuality was rooted in their past histories and their present teaching context and this was different for each one. Throughout the analysis there was movement from the individual to the group and back to the individual and this is illustrative of the difficulties involved in attempting to deconstruct practice as something logical (Grenfell & James, 1998). Therefore the search for commonality was perhaps more a shortcoming in interpretation and my use of the method, than a failing attributable to the method. That said, there was evidence of some patterns in the teachers' responses despite their widely differing modes of expression.

The untying of the complexities of teachers' understanding of their practice was a very important outcome of the research to address the research question with depth and breadth. The data were interviews in which teachers described their beliefs and practice, rather than observation of actual practice and as such there was no data against which actual practice could be tested against the intents and aspirations expressed. Additionally, this research has

focussed on teachers rather than children (or parents, governors or policy makers). It was clear that there were some findings specific to the children the teachers had encountered and hence it cannot be assumed to be true for other situations and could not be tested against the views of the children.

Nevertheless, potential breadth in analysis was considerable, made clear by the identification of themes and dichotomies each of which is possibly wide-ranging and each o f which could have made a single research focus. However, it is perhaps the nature of interpretative enquiry that attempts to define the research field might end in findings that cover more ground than was originally intended; the researcher cannot know either what the data analysis might generate or the ways in which the views and beliefs of the participants might be manifest (Greene, 2010). Social and educational interactions can only be understood within their context (Denzin & Lincoln, 2013b) and choices were made about the foci for analysis. These were governed principally and appropriately by the frame of communities of practice and situated cognition, which of itself is a broad construct open to many different definitions and interpretations by researchers.

It could have been useful to consider one particular view of excellence and use this as the point of reference in analysis of practice. For example, it might have made sense to draw solely on the theme of knowledge in interpreting teacher's practice because this aspect arises a great deal in the literature. Within this, for example, there might have been room for greater attention to and analysis of subject knowledge against the full range of ways in which this portrayed itself during the interviews. Another possibility might have been to focus on the extent to which teachers are affected by years of curriculum control (noted by Moore et al, 2002), relating to the interplay of successive changes in policy with teacher identity and the subsequent impact on their sense of agency to affect change. Alternatively, in conceptualising the challenges for teachers in achieving excellence in mathematics teaching, this research went some way to joining differing views from government policies and research about how

children learn mathematics and what is a desirable outcome of teaching. Analysis of this particular focus could have further explored possible links.