As explained in Part I data collection was completed in March '94 and my Transfer to DPhil status took place in April '94. By then Data Analysis was at the state of having constructed Scripts for the tutorials. During May and June and from September to mid-December '94 interviews and tutorials were transcribed, the former fully and the latter selectively. In the following I account for the analytical procedures during and after transcribing.

During data collection the interview material appeared to me as less intellectually intimidating and hence more manageable than the tutorial material. The interviews

On the grounds of criteria a, b and c, I decided to commence analysis from the interviews despite my initial intention to use them as supportive material to the tutorial material.

Full transcribing of the interview recordings was based upon

As expected, criteria a and b contributed to an almost immediate ordering of the interview data. The expectation however expressed in criterion c was not fully satisfied as I explain in the following.

In May '94 the interviews were fully transcribed. In the subsequent months seven texts, related to

were constructed and two papers (on Accumulation Points (Nardi 1995), on Spanning Sets (Nardi, to appear)) have come out of further analysis. These texts are compilations of the students' conceptions as expressed in their definitions, examples and images of the concepts. In the interviews the students also discussed what they find elusive about some of these concepts. The interview material is used as supportive reinforcement of some findings from the tutorial material.

The interviews did indeed illustrate and elaborate upon the students' difficulties with the mathematical areas I had touched upon; they even incidentally illuminated some other areas. However the interview data were strongly topically-centred and this was in a slight opposition to the evolving psychological directions of the study. This is a study of some general cognitive phenomena that characterise the novice's experience of advanced mathematics and the interviews explored the students' perceptions of a number of specific mathematical concepts. Taken out of the context of cognition-in-action — as this action was observed in the tutorials — the students' discourse on these mathematical concepts can be seen as a supplement of the rich evidence of the tutorials. Therefore the interview material did not provide the foci of further analysis, as expected in criterion c, but potentially helpful evidence for the findings from the tutorial material.

In the following I present an account of the construction of Selective Transcripts, the Episode Extracting and the subsequent stages of analysis that the extracted Episodes were submitted.

Selective transcribing of the tutorial recordings followed the transcribing of the interviews. Transcribing was based upon

Immediately after the concerted study of the above, an extended and iterated listening of the tape led to the construction of Selective Transcripts. These protocols constitute the central bulk of data used in the subsequent stages of data analysis. In the following I explain the process of construction as well as the criteria on which it was based.

Unlike the interviews, which were fully transcribed, the tutorial material which was strongly characterised by the complexity of phenomena unfolding in a very natural learning environment, had to be submitted to an ordering and filtering process. The criteria of the selective transcribing process are crucial and I have tried to maintain them consistently throughout the construction of the Selective Transcripts. So, I excluded from transcribing but for the sake of continuity summarised:

As a result, the final Selective Transcripts have the benefit of continuity and are more focused than the Scripts which were non emphatic telegraphic texts. To allow quick reference to the audio recordings Selective Transcripts are numbered according to the recording equipment's counter.

So far the criteria that I have described as the basis for the selection of the material to be transcribed are largely structural, that is they are criteria relating to the preferred structure of the learning interactions that were transcribed in detail. From these criteria it is evident that my intention was to filter out of the complex tutorial material the instances of higher student participation; hence the parts of the tutorial recordings that are fully transcribed are mostly the dialogues between tutor and students and student monologues. In the following I explain how the selection of the material to be fully transcribed related to my reflection on the focus of the study.

Before addressing further the issue of selecting material for transcription that relates to the objectives of the study, I digress in order to refer briefly to the technical difficulties of transcribing. Having very little experience in transcribing, not knowing with precision which of the material would eventually turn out to be useful, and having pressing time constraints, I decided that a minimum of one (towards the end of the transcribing period: two) tape every day should be dealt with — so that by the end of 1994 the material would have been fully covered. This implied that where there were substantial difficulties (inaudibility, restricted understanding of the mathematical discussion) transcribing was left for later and these parts of the tutorial were summarised. Still in the cases where transcribing of some significant parts of the recording was postponed for the future, I concentrated on pointing out the necessity for further listening so that in subsequent readings of the Selective Transcripts these parts were revisited with extra care.

Considering both the material that was excluded from transcription and the material that was included, it is evident that the parts of the tutorials that mostly attracted my attention were instances in which the students participated in a protagonistic way (hence the preference for the students' monologues or dialogues with peers or tutor). Progressively it was becoming clear that, as far as the structure of the sought-for incidents was concerned, I was seeking learning events with a beginning, a middle and an end, full stories on learning (I note however that at that stage any evidence of the students' cognition, however fragmentary and incomplete, was also attracting my attention). True to the phenomenological agenda of the study, the aspiration here was to extract information about the problematic aspects of the learners' cognition from their expressions of this cognition. As far as this material is concerned I searched for these expressions in the students' interaction with their tutors and their peers during the tutorials. At this stage I realised that the objective recordings, juxtaposed with the less objective (but still avoiding decisive selection) Scripts and fieldnotes, were transforming into subjective, researcher-processed data. In other words the construction of Selective Transcripts was the first genuine attempt to impose my loose but substantial theoretical framework on the data. This personal imposition on the material increased through the various stages of Data Analysis and it is the researcher's task to guarantee that this process preserves and highlights the essence of the phenomena to be studied.

The subsequent stage of analysis was to determine which parts of the transcribed material would become the object of detailed analysis. The analysis would be informed by the theoretical tools developed in the field of Advanced Mathematical Thinking (elaborated upon in Chapter 1) but before reaching the stage of being able to use these tools, the material which would be submitted to this analysis had to be further ordered and purified. I quote from the Transfer Paper my first declaration of analytical intentions as tentatively made at that very early stage:

The above intentions are heavily influenced by the then-recent experience of the Pilot Study and are also largely mathematically-centred in the proposed restructuring of the data. The concern for a global psychodidactical approach to the novice's cognition is also evident. In the following section I explain how the heavily mathematically-centred approach to the structuring of the data was reconciled with the global cognitive approach of the study.

Following the construction of the Selective Transcripts and the reflection upon the nature of the sought-for learning incidents, came a critical scanning of the transcribed material during which the tutorials were broken into their constituent learning incidents. A crucial step of that time was the decision to concentrate on the extraction of complete learning episodes. Already during transcribing this preference had begun to show. Episode extracting was generally based on

The outcome of this selection was a list of 110 Episodes, a large number of which was double, triple, quadruple or fivefold incidents of the same piece of mathematics dealt with different students. The tabulated form of the list was arranged in three-columns:

So an entry like

Subsequently the tighter focusing of the criteria for the sought-for Episodes led to a further drastic reduction to their number. To illustrate how the tightening of the focus occurred I note that this first list of Episodes was compiled between January and May '95. In the meantime the drafting of the first chapters of the Thesis as well as processing of the interview material led to a reflection on the data that made further filtering of the Episodes necessary and more obvious. The main criterion that drove the formation of the first Episode List was a general pursuit of whole learning incidents initiated by a declared learning difficulty of the students, either announced by them or identified by the tutor. This criterion was quite general and as a result the extracted Episodes were of a large number and of mixed quality. So the next step was to decide how many of these Episodes were in fact giving evidence that allowed me to go beyond the identification of learning difficulties and access potential sources of the students' troubled cognition. For instance, in most of the extracted Episodes the cognitive conflicts that initiated the discussions appeared to be resolved by the end of the incident. Further exploration of the incidents could lead to discovering whether cognitive conflicts had actually been resolved or not.

The new filtering process was carried out after another listening to the recordings; also bearing in mind that the focus was now on Episodes that not only reveal learning difficulties, but appear as potentially permitting some access to the sources of the novice's cognitive conflict as well as possibly providing answers with regard to whether and how the conflict has been resolved. The 70 Episodes that were selected were tabulated in a similar three-column table to the one described earlier. The excluded ones were summarised and incorporated in the rest of the material that was beginning to take the form of auxiliary, supportive data. At the same time a few mathematical areas appeared as dominating the mathematical content of the material which until then was arranged chronologically and in accordance with the course and the Problem Sheet number: the foundations of analysis, calculus, topology, linear algebra and abstract algebra. The observations that determined the filtering also began to become more stable. A tendency to reflect on the incidents, while trying to filter further, into the conceptual, reasoning and notational/linguistic components of the students' difficulties, was forming (possibly reminiscent of the S.DIFF triad of the Pilot Study). Also I felt it was increasingly necessary that I am very comfortable with the mathematics of the Episodes. A pile of notes on the solutions of the problems began to accumulate. My reading of the literature was becoming increasingly triggered by the things I could see in the Episodes. Reading the Selective Transcripts and listening to the tapes was becoming increasingly loaded with expectations. Predictions of what the outcome of the Episode or the next line in the mouth of a particular tutor or a student would be were becoming gradually more successful; and I knew it was not simply because I had listened to the tapes again and again. It was because there was a clear emergence of patterned behaviours which was fermented by my constant familiarisation with the data.

While Episode extracting was approaching completion, I began to see that the analytical intuitions described in Part IIb.ii were in need of discipline and order. The 70 Episodes were physically extracted from the Selective Transcripts . They formed the main body of data, denominated Episodic Material and were filed chronologically. The rest of the material, the Non-Episodic Material was also filed chronologically, summarised/ tightened up further and kept for reference and as supporting material for the Episode Analysis. In the meantime further processing of the interview material had produced the two publications mentioned in Part IIa. I quote from (Nardi 1995) to illustrate the form that the processing of the 70 Episodes took in the subsequent months:

Therefore each Episode was treated as an analytical unit. A text for each Episode was prepared consisting of the five sections cited above. The Non-Episodic Material helped form an introduction to each text regarding the context of the Episode. The fieldnotes and relevant mathematical documents helped form Section (M). The Episodes were rearranged in terms of the five topical areas (the foundations of analysis, calculus, topology, linear algebra and abstract algebra) that had emerged as dominating the mathematical content of the Episodic Material. In the process the 70 episodes were further reduced to 50 on the simple basis of excluding a few more mathematical areas (for instance Permutations in Abstract Algebra is the topic of discussion for a considerable number of Episodes and have been kept separately as material for future analysis and reference). From June to mid-July, September, October, and mid-November to mid-December the filtering of Episodes, their allocation to one of the five thematic/topical units and the construction of the five-section Analytical Texts was completed. In the meantime the Analytical Texts had taken the following form:

In the construction of the Interpretive Accounts a variety of tools, developed within the field of Advanced Mathematical Thinking, are used. The links between the theoretical underpinnings of the analysis and the actual analysis as presented in the subsequent chapters will be made individually in the Interpretive Account of each Episode. The interpretation of the incidents is also largely reinforced by the Non-Episodic Material and the Interview Material.

In the above I have attempted to illustrate the long and painstaking process of how 120 hours of recording transformed into 1500 pages of transcripts and distilled into the 32 learning incidents that constitute the main body of evidence for the theoretical cases of the study. In brief the presentation of the Analytical Texts follows mostly the structure: fact - interpretation - conclusions. Conclusions for each Episode inform the construction of a Conclusion Part for each of the four thematic/topical units featured in Chapters 6 to 9 (the reduction of the number of Episodes used as evidence to 32 and of the thematic areas from 5 to 4 was dictated by the space limitations of the thesis). Chapter 10 constitutes the cross-topical theorising part of the study and brings together the findings reported in Chapters 6 to 9.