Methods for data collection

The cases of tutors’ teaching in SGTs were multiple across the stages of the study. The structure of the study included four stages: Pilot study 1 (stage 1); Main study 1 (stage 2); Pilot study 2 (stage 3); and Main study 2 (stage 4). Details can be found in the following Table 3.1 and in Appendix B.

Table 3.1: Details for the structure of the study. # of SGT observations # of tutors observed Details for # of tutors observed Semester#/Year#6 Pilot study 1 (stage 1) 26 7 Multiple observations of 7 tutors’ teaching. Semester1/Year 1 Main study 1 (stage 2) 10 (1 of 10 SGTs lasted for 3 hours) 1 (female) The teaching of 1 of the 7 tutors in Pilot

study 1 was observed systematically. Semester2/Year1 Pilot study 2 (stage 3)

23 21 In order to select the sample for Main

study 2: • 2 of the 21 tutors

had also been observed in Pilot

study 1; and • 2 of the 21 tutors were observed twice

in Pilot study 2. Semester2/Year1 Main study 2 (stage 4) 22 2 (male) The teaching of 2 of the 21 tutors in Pilot

study 2 was observed systematically.

Semester1/Year2

Total 81 26 - 3 semesters

Following each of the vast majority of SGT observations were interviews with the tutors about their underlying considerations. The nature of the interviews was friendly but professional; so discussions with the tutors was a more suitable term than interviews with them.

The common aspect of all sampling methods used for my pilot and main studies was the theoretically driven sequential sampling (Miles & Huberman, 1994). In other words, the sampling was not wholly prespecified, but evolved conceptually and sequentially during fieldwork. The levels of sampling were sequential; on the one hand, the initial sampling drove me to observe other cases within the same stage of the study. For example, within Pilot study 1, I started to approach and observe tutors who cooperated with the Mathematics Education Unit (MEU) of the University thus being

familiar with mathematics education (one level of sampling), and gradually moved to also observe tutors not familiar with mathematics education (another level of sampling). (The MEU is introduced in Chapter 1: Section 1.1.) On the other hand, the sequence was between the stages of the study. For example, I selected to study in depth one participant’s teaching from Pilot study 1 (stage 1), and this was the participant for Main study 1 (stage 3). So, there was a sequence between sampling in pilot studies and sampling in main studies.

There was also a theoretical process within the sequential sampling; that is, sampling of one stage of the study learnt from sampling of the previous stage of the study. This learning was theoretical, since it was based on issues discerned in analysis of the tutors’ teaching in each stage of the study. For example, when I started to analyse my cases of Main study 2, I had brought the emergent characterisation of teaching into tools and strategies from the case of Main study 1. I thus used the characterisation across the cases of my main studies. This characterisation nevertheless was theory emerged from data, and not prespecified.

Miles and Huberman (1994) stressed that in case theory emerges from data, Glaser’s and Strauss’ (1967) term for theoretically driven sampling is theoretical sampling. I gradually made the terminology for my sampling more precise by specifying that my theoretically driven sampling is in Glaser’s and Strauss’ (1967) term a theoretical sampling.

The sample came from a specific British University; first, because of convenience (distance, networking), and second, because of the profile of the University. The elements of context, which contributed to the profile of the University, were:

• There was an average-sized group of tutors in modules offered by the Mathematics department at the time of data collection (40 tutors).

• Unlike most universities, that university had the structure of SGTs, in which the tutors were not Ph.D. students.

• Tutors came from different educational backgrounds and cultures; the academic staff thus was multicultural.

• There were tutors with doctorates in mathematics as well as tutors with doctorates in mathematics education, and both taught mathematics in SGTs.

Data is qualitative and consists of field notes during observations in SGTs, audio- recordings of SGTs, and audio-recordings of follow up discussions with tutors. Data was collected during the pilot and main studies over three academic semesters.

Ÿ Field notes: My field notes consisted of the words I captured from tutor and students in a dialogue form; tutor’s and students’ writings on the board; and their body language such as hand movements and facial expressions. I was specifically careful to capture in my notes as much wording as I possibly could from students, since they were usually not speaking loud enough so that I could hear them afterwards in the recordings. The students’ articulations in SGTs usually consisted of some words or phrases some of which were inaudible in recordings. During SGTs, I also wrote comments on the tutors’ actions in the right margin of my field notes. When the students were getting ready to leave the tutorials in the end of the sessions, I revised these comments to organise them for the follow up discussions with the tutors about their underlying considerations for their actions. The discussions in all my studies were unstructured, informal and friendly. I selected this type of interview so that I could go in depth into the conceptual areas I explored through my research questions. (A sample of field notes is in Appendix C.)

Ÿ Audio-recordings: I audio recorded and transcribed all SGTs from my main studies. My recorder was a small noiseless device of 15 x 5 x 1.4 cm. I also audio recorded those tutorials from my pilot studies for which I had permission from the tutor and the students. I then enhanced all transcripts with information coming from my field notes. (A sample of transcript is in Appendix C.) For the tutorials I did not have permission to record, the most usual reason tutors said was that it was intrusive for students. In all tutorials, however, when I had permission from tutors, I also had permission from students. I had the feeling, but no data to support it, that students were forgetting the existence of the recorder after the first couple of minutes in the tutorial. There were also some discussions with tutors about their SGTs in the pilot studies, where the tutors did not give permission for audio recording. In that case, my endeavour was to capture as much from the tutors’ responses as I possibly could in notes. All tutors, whose teaching I present in this thesis, nevertheless gave me permission to audio record their SGTs and our follow up discussions.