Part IV Methodological Implications: A Comment on Observation As A Means to Gain Access to Advanced Mathematical Cognition
Minimally participant observation, as described in Chapters 2 and 4 and used in this study, allows natural access to expressions of cognition made by learners in action. How deep or illuminating this access was, relied to a great extent on the participants: the tutors and the students. Below I have collected a few observations with regard to the participants' influence on the study:
• Highly leading discursive practices, such as closed questioning, seemed to impede my attempt to observe the students' thinking because, by channelling the students' responses, the expressions of their thinking come under the clinical control of the tutor and thus lose a great amount of their authenticity. So the sense of the students' difficulties I was making seemed to be derivative of the sense being made by the tutor: by that I mean that the tutor chose a particular direction of closed questioning on the basis of her interpretation regarding the student's difficulty. Some of the closed-questioning material however seems to provide insights on the students' cognition and parts of it have been presented in Chapters 6-9.
• The influence of the students on the quality of the observation seemed to be dependent on the degree of openness with which they participated in the tutorials. It turned out that the higher the degree of openness and participation of the student in a piece of data, the higher the probability that this piece of data would be filtered through the final stages of analysis: a selection of these filtered pieces of data, the Episodes, was presented in Chapters 6-9 and were deemed as data in which the students' discourse is rich and at moments transparent as to their difficulties. Certainly, and given the considerations of this study with regard to the accessibility of cognition explained in Chapter 2, this transparency is relative and bounded by the students own explanatory power.
There was a plethora of moments during observation that further elaboration on the students' expressions would have enhanced my potential for subsequent interpretation. Given the non-participant intentions of the study this intervention was not possible — it was more possible in the interviews. In Part V I explain briefly how the naturalistic observations made in this study can be extended to more focused research around the themes elaborated in the analysis.