In the following I present an account and an evaluation of the tutorial observation and the interviews with the participants.

I first present a factual and then an evaluative account of tutorial observation.

The body of participating tutors and students was formed in the first two weeks of Michaelmas Term 1993. Observation started on the 3rd week and lasted for the remaining six weeks of the term. In Hilary Term observation lasted eight weeks. As mentioned in Part I, missing the first two weeks of Michaelmas Term was inevitable. Given that

it is fair to assume that the tutorials of Weeks 1 and 2 were, respectively, a general introductory session and a tutorial on the first weekly problem sheets. For the sake of continuity I informally discussed the contents of the first two tutorials with the tutors. From a methodological point of view my presence in the sessions from the very beginning might have created tensions, since the tutorial on Week 1 is the novice's first entering of this learning context. From the point of view of the thematic focus of the study (the novice's encounter with mathematical abstraction), however, the experience might have been valuable even though, according to most tutors, students do not participate extensively in the first sessions. In sum I think that giving the students the opportunity to experience tutorials for a couple of sessions and then asking them to decide whether they wish to participate in the study was fair and possibly made my suggestion to attend the sessions more easily accepted.

In Michaelmas Term observation of tutorials was 10 hours per week for individual or pair sessions plus 3 hours of group sessions in two of the four colleges. The allocation of time per college, hence per tutor, for Michaelmas Term is as follows:

I note that in the second college of the above table the pairing of the five students alternated so that by the end of term all of them had an equal number of pair and individual sessions. From the 22 students participating in Michaelmas Term, one left Oxford in Hilary Term for personal reasons and another one had to be excluded from the observation because of timetable clashes and because the observation and recording of material in one of the four colleges was drastically reduced in Hilary Term. Also one of the tutors, due to sabbatical leave, was replaced by two tutors of the same college and also changed the pairing of students. This implied that one of this college's Michaelmas student volunteers had to be replaced by another student from the same college. Therefore the picture for Hilary Term is as follows for the five tutors in the four colleges:

Group sessions were also observed in Hilary Term.

The mathematical content of the tutorials can be rather loosely defined as relevant to the content of the lectures and problem sheets of the Mathematical Institute for Michaelmas Term on

and for Hilary Term on

Occasionally other topics are addressed but the main bulk of discussions were on the above.

The tutorials were held in Oxford college rooms. In two of the four colleges the observer shared the same table with the tutor and the students. In the other two the observer was seated in a corner of the room having auditory and visual access in one of the colleges and only auditory to the other.

The individual and pair sessions were tape-recorded and the observer made notes during observation. The need to obtain a record of the sessions beyond fieldnotes was an aftermath of the Pilot Study, the data of which were of a strongly fragmentary and derivative character (see Chapter 3, Part Ib). Data originating in fieldnotes are instant reconstructions of the events and thus lack the accuracy of non-selective recording. Initially video-recording was considered as a viable option. At the very early stages of the negotiations with the tutors it became clear that video was too intimidating for most prospective participants and a too-visible interference in the intimate environment of college rooms where the tutorials were held. So, with the risk of missing non-audible communications, I eventually decided to proceed with suggesting audio recording to the participants.

Tape recording was more easily welcomed. It also provides a non-selective account of the events in the tutorials and allows the observer to keep a complementary record of the aspects of the tutorial that an audio tape cannot capture, such as writing on paper or on the blackboard and gestures. Depending on the situation, the observer's comments and observations might also fit in the fieldnotes. As mentioned above the intention was that through the available documents, the tape recordings and the fieldnotes, as comprehensive as possible an understanding of the learning occurrences in a tutorial would be achieved.

By the end of data collection my general impression was that the expectations set up during the Pilot Study, regarding the didactical content of tutorials, had been fulfilled. Tutorials are indeed a rich source of learning incidents and the primary stages of Data Analysis (see Chapter 5, Part I) indicated that the technical strategies employed provided a reliable record of these learning instances. However I also felt strongly that processing these data would be a more complex task than previously because of the diversity of teaching style, mathematical content and quantity of the data. Briefly I would describe the above impression as a realisation of the increased number of variables determining the nature of the data. As a sample of these new variables I note the shift from one tutor to four and the shift from the uniform content and structure of the tutorials observed in the Pilot Study to the nearly chaotic diversity of the mathematical content, tutoring styles and student personalities in the Main Study.

Another notable difference concerns the content of the fieldnotes: in the Pilot Study these fieldnotes were the data; in the Main Study they were a supplement consisting mostly of things not captured in the recording and some primary indicators of where analysis could possibly focus.

By the end of the second term an amount of approximately 120 hours of recordings had been gathered. This is less than the sum of hours of observation mainly because of the instant editing that took place in one of the colleges. The hours of observation in this college decreased from 4 in Michaelmas Term to 2 in Hilary Term, because I decided that the tutor's long monologues that contain exposition and technical advice are not particularly useful to the purposes of this study. During the two hours of observation in this college in Hilary Term, recording was paused on such occasions. Other losses are due to the absence of students but these were miniscule. Also there have been minor losses of material due to faults of the recording equipment. Audibility of the recordings varies but is generally good. From the beginning this has been a factor that merited special attention given that the study is in my second language.

The effect of the researcher's presence can be described here, on the basis of my judgement and the participants' comments, as minimally significant. In the beginning however the degree of self-consciousness, mostly on the part of the tutors, is slightly higher and so was the number of instances in which the tutor would break the tacit rule of the researcher's invisibility and address her during a session. A slight increase in these instances has been observed towards the end of Hilary Term where all participants and the researcher felt the reasons for a strict attachment to the rules of minimally participant observation were no longer required. My invisibility was imperative in the beginning of observation. This was the time that a relation of trust and openness had to be established. Towards the end of the sixth month the relation had been evidently well established: I had been accommodated and my presence in the tutorials raised no questions; it was deemed natural.

Some students admit that they had noticed that at some instances my note-taking appeared keener but that they could not see a pattern in this behaviour. When they were convinced of the non-threatening character of my presence, they stopped paying attention. Notably, after the first interview, most of them say that they had a more solid understanding of the purposes of my investigation. According to tutors and students, no visible change in their attitude or behaviour can be linked to my presence.

A few students refer to their tutors as 'a little more polite and patient' during recordings and some tutors thought their students were 'more attentive'. Even if these changes occurred extensively — and they did not — they would promote a better realisation of the purposes of data collection: a more polite and patient tutor and a more attentive student is likely to enhance the chances for a more mathematically and didactically creative, and revealing, session.

On my part, instant understanding of the occurrences on the tutorials during observation depended upon the strength of my mathematical background in the relevant topic and the clarity of expression on the part of the tutor and the students. For example the instances where the tutor was correcting the students' drafts during the tutorials were usually unhelpful because I generally had no access to the drafts. Also the tutor's and the students' comments on the drafts, despite their relevance, most of the times, to the difficulties the students had with the questions in the problem sheets, could not always be helpful because of their fragmentary and unclear referencing. For example in quite a few occasions the tutor and the students point at various parts of the students' drafts and refer to what they point at as 'this', 'it', 'here' or 'there'. Since seating arrangements did not allow me to have any visual access to the drafts, these references were hard to understand. I used to work on the Institute's problem sheets beforehand and this proved a good strategy for enhancing understanding in most cases.

As opposed to these fragmentary and unclear references to previous problems, complete presentations of questions from the problem sheets, or new questions/theorems, by the tutor or the student, were usually understandable and transparent. They thus constitute the material that, as it turned out in the analysis, produced data of the highest quality.

In the following I present first a factual and then an evaluative account of the interviews with the participating students.

During observation a few generally problematic areas of the mathematical content discussed in the tutorials were identified and thus became the focus of the loosely structured interviews conducted on the eighth week of each term. Eleven students were interviewed in Michaelmas and nineteen in Hilary Term. The scenarios of the two interviews are as follows:

Interviews were 25-30 minutes long and were tape recorded. A compilation of the students writing and drawing during the interviews has been kept. None of the students who were invited to an interview declined and all except one kept the appointment. In Michaelmas Term only a selection of students was interviewed. Selection was based on the impression that these students made to the researcher as to their willingness to vocalise their mathematical thinking. Audibility varies depending on the surroundings of the interview but is generally very good.

The interviews from Michaelmas Term are relatively flawed because there is an eagerness on my part to cover all the topics in the scenario. As a consequence the interviewees are sometimes not given enough time to reflect on a question. Moreover in some occasions valuable time is lost when I, carried away by the persistent request of the interviewees for correction, indulged in exposition. These occurrences are rare and offered a remarkable learning experience of things to be avoided in Hilary Term interviews. These are more disciplined and better prepared. Eventually, nevertheless, I felt that both sets of interviews have substantially achieved their aims. Given the lack of time for a thorough preparation (the interviews had to be done in term time because of most students' absence during vacation) the identification of problematic topics to be touched upon in the interviews was generally successful.