In this part I present my search for participants as it took place in the first two weeks of Michaelmas Term 1993. The way the volunteering tutors and students were found is relevant to their profile. The findings of this study are derived from observations made during tutorials given by a small number of selected tutors to a small number of selected students in Oxford. This selection was based, not on the principles of statistical representativity, but on willingness to participate; also on timetable concerns such as clashes between tutorials.

Willingness to participate in an educational study possibly reflects some features in the psychological profile of the volunteering tutors and students which I, as the person in charge of the final selection of participants, hoped for and pursued. Evidently friendly and conversational tutors and open, expressive students who would not hesitate about vocalising their mathematical difficulties were sought for. The quest for these features was not explicit in the search campaign but the procedures that led to the formation of the body of participants constituted a reconciliation process between what was sought for and what was available.

At first a number of tutors from a variety of Oxford Colleges was approached. The experience of the Pilot Study had produced an estimate of the number of students that could fit reasonably in a week's schedule of observation (approximately twenty) so, given that most colleges had between 4 and 10 first year mathematics undergraduates that year, that implied that I should aim at an estimated number of three to five colleges. Negotiations with college tutors led to six positive responses, and three negative ones (the tutor was too busy to dedicate time to helping the researcher contacting and convincing the students; the tutor was concerned about the interference of the researcher's presence with the undergraduates' learning during the tutorials but was willing to allow one interview per student; the tutor thought students were too shy on the first term of their studies but was willing to participate in the study in Hilary Term). In the cases of negative responses the Pilot Study tutor's reassuring recommendations — at my request the Pilot Study tutor had informally discussed her experience with some of the potential participating tutors — had not succeeded in battling out these tutors' concerns. Moreover one of the positive responses had to be put aside because it turned out that the applied mathematical content of this tutor's tutorials did not coincide with the pure mathematical content of the others.

Subsequently the students (that were given tutorials by the tutors who had responded positively) were contacted. The introductory meetings were arranged by the tutors and were held in the college rooms where the tutorials were given. A short presentation of the researcher — on her work in general and on the nature of the requested participation in particular — was followed by questions from the students. The meetings ended with the researcher offering reassurances on the unintrusive, non-assessing character of her presence, on confidentiality and on the participants' right to withdraw from the study at any moment. It is worth noting that explanations about the nature of the investigation were not extensively given and particularly geared towards preventing the possibility that the study was perceived by the students as an assessment of their mathematical ability. It was clearly stated that this is an investigation of the novice mathematician's understanding and in particular of the problematic aspects of this understanding. Students then were given some time to decide and were asked to communicate their decisions to the tutor who would in turn inform the researcher.

By the end of the week an overwhelming 24 positive responses and 4 negative (these students were concerned about having the privacy of their individual tutorials disturbed but voted for the presence of the researcher at the group tutorial) accelerated the completion of the search for participants which had lasted the first two weeks of Michaelmas Term. Twenty one students were selected from four colleges. Final selection was based mostly on timetable concerns; also on maintaining a male-female balance among tutors and students.

The search for participants lasted the first two weeks of Michaelmas Term which is a crucial time, given that the purpose of this investigation is to capture aspects of the novel experience of advanced mathematics. Sparing these first few days of the novices' experiences was inevitable since attempts to contact tutors and students before the beginning of term had repeatedly failed. Moreover even if contact had been possible, the likelihood of tutors accepting collaboration before meeting their students, or of students accepting the researcher's presence in a teaching session they had never experienced before is small.

The background of the students participating in this study is diverse but their A-level experience is taken as a homogenising factor. Also despite the absence of any sign of linguistic difficulty, it must be noted that one of the students was Kenyan, one was of Turkish origin and another of Swiss-French. In the presentation of the findings, I shall refer to some special cases where significant differences in the students' background have been identified.

In retrospect it can be said that the overall majority of students who responded positively were friendly and open even though in some cases I felt that their positive response may have been the result of their reluctance to come to conflict with their tutor's willingness to participate in the study. However, no explicit evidence of that is available. As for the tutors, the willingness with which they responded positively to the invitation, constitutes evidence of a genuine interest in aiding a didactical investigation of tutorials. The role played by personality factors will become more evident throughout the study but also in the subsequent parts of this chapter.

Having finalised the body of participants, I then realised that, given my limited knowledge of the course structure and protocol, I needed to explore the learning context of the investigation, that is the tutorials. My mathematical background guaranteed a relatively immediate access to the mathematical content of the sessions. Gathering the relevant information regarding the course in general was the next step before moving on to the actual observation. Before and during observation the following documents constituted the body of information that helped me familiarise with the course and in particular with the essentially private learning context that tutorials are:

I note here that the latter, with its chapters on University Study, University Mathematics, The Formulation of Mathematical Statements and on Proofs, has the enormous ambition of initiating the novice into the rigorous mathematical thinking that is to succeed school mathematical thinking. Expectations regarding its influence were high but informal conversations with tutors led to a more sceptical approach: its thorough written style was deemed to be difficult by novices. Therefore the document had been practically marginalised. The few scattered indirect references to the Guide made by the students during the six months of observation will be mentioned when the novice's difficulties with the shift to a more rigorous way of thinking mathematically is presented in the subsequent chapters.

The documents mentioned above were circulated in the Mathematical Institute. There is however a bulk of documents circulated at college level and these will be looked at separately when necessary. These include documents whose content ranges from:

to

There was no intention to include in the above a study of the students' written work. Occasionally however I was allowed by the students to survey briefly their weekly drafts.