PART I Data Processing During Data Collection
In the following I give an account of the initial stages of ordering the collected material as well as of the first attempt to explore its contents which led to the construction of the Scripts and to the first steps towards the tightening of the research focus.
Ia. Initial ordering of the Data
As explained in Chapter 4, data collection, from tutorial observation and interviews, took place during the first two terms of the participants' first year in Oxford. During the 14 weeks of data collection — interviews were taken on the 6th and on the 14th week of observation — it was imperative that a neat record of the procedures was kept. The vast amount of collected information (recordings, relevant documentation, fieldnotes, personal experience of immersing into the field) as well as the short time during which this information was collected made it necessary to order the data in a way that would facilitate the re-evoking of the research experience in the subsequent months and would allow quick access to the material.
At a practical level this implied that
• all recordings were immediately labelled (date, time, college, participants/ interviewees),
• all documents were also labelled with dates and in most cases with a brief note on the document's relevance and reason to keep it in file and,
• all recordings were matched with problem sheets and fieldnotes; interview recordings were matched with the interviewees' writings while interviewed.
At a more reflective level and in an attempt to capture and preserve the atmosphere of the data collection period, as well as register the off-the-record input to the study that sprang from the informal contact with the participants, a Researcher's Log was kept from the days of the search for volunteers up to and including the earlier stages of data processing. Unfortunately my consistency in keeping the Log flagged as data processing carried on, possibly because most of the notes and observations I deemed important became part of the data analysis. As a paradoxical result, while reporting the procedures of Data Collection and Data Analysis in Chapters 4 and 5, I found it increasingly difficult to reconstruct an account for the latter despite its being considerably more recent. This is the major reason why the account of the procedures of Data Analysis consists mainly of the final techniques used in the study and not of the various attempts at other techniques that were abandoned in the course of analysis.
The Log contains a large amount of information that turned out to be variably useful in the course of the study. As far as tutorials are concerned the log was used as a register of immediate off-the-record (namely off the recordings and off the fieldnotes) observations made during informal conversation with the participants from the time when volunteers to participate in the study were sought, until the completion of the data collection. So for instance the final selection of participants, tutors and students, was largely based on the impressions registered in the Log (for example with regard to the ambience of my first meeting with the tutor or of the introductory meetings with the students); similarly the selection of interviewees for the first interviews — only half of the students were interviewed on the 6th week — was largely based on it (for example I chose to interview approximately half of the students in each college and I based my choice on the expectation, formed during the first five weeks of observation, that some of the students would be more open).
Moreover, since during interviewing I made no notes but observed and listened/ responded carefully to the interviewees, the Log served as a memoir of my off-the-record impressions of the interviews with regard to:
• the interviewees' tone of voice and body language,
• their amicability, openness, fatigue/willingness to participate and elaborate,
• the interview environment: for instance noise, privacy.
• self evaluation: formative comments that allowed me to improve the quality of subsequent interviews (for instance the need to allow more time for the interviewees to respond; the need to be more flexible and not insist on covering all the issues by all means and at the expense of the quality of the coverage).
The part of the Log devoted to the observations made during the tutorials mostly contains the students' evaluative comments of their day-to-day experience of the lectures, the tutoring, the problem sheets, the textbooks, their peers. Time constraints and difficulty of lectures and problem sheets dominate these comments. Some are the students' queries about the study and the goals of my observation and interviewing. As for the tutors, in their comments registered in the Log, they address general issues of the difficulty of some courses and their expectations of the students. Occasionally they inquire about my findings or explain some of the approaches they took during teaching. In the presentation of the data, Chapters 6 to 10, references to the Log are made whenever it is deemed necessary or illuminating.
In conclusion it seems that good management of the collected material is crucial for
• practical reasons, such as not wasting time on looking for items like tapes or documents, and,
• for reasons directly related to the quality of the research, such as the degree to which the ordering of the material allows easier references and making of connections.
Apart from keeping a reasonably neat and comprehensive record of the collected material, during Data Collection, the construction of Scripts took place.
Ib. The construction of Scripts
A Script is a summary of the tutorial events that highlights the structure of the tutorial. It constitutes a primary compartmentalisation of the tutorial to its component incidents. It contains no transcribed parts but indicates where transcribing might eventually prove necessary. It also contains my reflective comments made in the freshness of the moment. Especially because of the value of the latter there was a general attempt towards constructing the Scripts as close to the actual time of the tutorial as possible. So it was intended that the Scripts were constructed within the same day, or at least within the first couple of days, of the time when the recording had been made. The text of the Scripts was based upon
• one non-stop listening to the tape
• fieldnotes
• general consulting of problem sheets and other documents.
The aim of employing the freshness of the moment to provide a concise and accurate account of the events in the tutorial was achieved with variable degrees of success during the two terms of observation. Success was considerably higher during the second term of observation since more convenient time allocation of the tutorial recordings during the day allowed a day-by-day construction of the Scripts . Accumulation of research experience also contributed to a more efficient exploitation of time.
In conclusion the aim of maintaining a neat record and keeping up with the data during Data Collection was generally achieved. What did not so evidently succeed was progressive focusing of the thematic aims of the research. In the following I give an account of the primary attempts at tightening the focus of the study.
Ic. A step of Progressive Focusing: refining the focus of the study during data collection
Due to the density of the data collection, there was scarcely any time to reflect on the collected material. The only record of reflection during data collection are the impressionistic, telegraphic comments in the Log, in the fieldnotes or partly in the Scripts . So striking impressions of the students' overall learning behaviour (for example 'their difficulty to express formally', or 'the intimidating role of new mathematical notation') or primary interpretations of their learning behaviour (for example 'the conflict between the school-mathematics approach and the university-mathematics approach') were registered but only in a general and haphazardous way.
However a first attempt for a global reflection on the contents of the collected material became necessary immediately after the completion of the data collection on the occasion of the Transfer to DPhil Status, the administrative process during which the Oxford University Department of Educational Studies assesses the progress of Probationary Research Students. Due to the requirements of the Transfer process, a first scanning through the fieldnotes and the Scripts led to a first, primary collection of themes that drove the study. I quote from the Transfer Paper in order to illustrate the status of the Research Focus at the time:
...
A few directions regarding the handling of the material, each of which approaches the material from a different angle, are listed below. These reflect themes that it is intended to address in the analysis and are by no means exhaustive.
COMMENT: The process of data analysis is expected to be cyclic. That is, as themes emerge in the course of the analytical process, the material has to be revisited in the light of the new themes. Ceasing this seemingly never-ending process depends on the degree to which the aims of the study will have been achieved.
The First-Principles/Theorem -Deduced Proofs Debate. In their encounter with mathematical problems novices are very little aware of what knowledge is taken for granted, what is the continuity between school knowledge and university knowledge, in brief what is the status of the knowledge they are expected to handle. In most occasions the above is crystallised in the question of what a novice is allowed to assume in a deductive procedure, for instance whether in this procedure all deductions must be based on First Principles or on already known Propositions. This question is of a foundational nature and the absence or insufficiency of an answer given to novices poses severe hindrances in their route towards mathematical abstraction.
The Different Levels of Mathematical Knowledge. The mathematics that a novice encounters varies in content from arithmetical and algebraic calculations applied to numbers or other elements such as matrices, vectors, functions, sets, sets of sets etc., to calculational techniques, proofs, methods of proof, items on the content and use of mathematical objects, and also a variety of metamathematical information. This diversity in content implies the necessity for a diversity-aware approach to teaching. Tutoring lacking in this awareness leads to the novice's problematic handling of mathematical abstraction.
The portrait of a problematic question. An issue to be addressed is what renders a question in a problem sheet problematic. Gathering the queries of the students on the problem sheet questions might illuminate that.
Patterns in the Tutor's Advice to the Novices. As mentioned before, the tutor usually propagates ways of mathematical thinking — that range from practical advice to fostering a whole philosophical perspective. Studying the influence of these pieces of advice on the formation of the novice's mathematical behaviour might provide insight into aspects of the cognitive processes studied here.
Mathematics as a Debate. A large part of the novice's mathematical behaviour is determined by the way mathematics has been presented to them. Given the flexibility of the tutorial — technically it is almost entirely up to the tutor to determine its content and structure — these styles of presentation vary. Some general patterns are the following: tutoring as a proof-and-refutation process (optimisation of solutions/proofs is one example; tutor as a facilitator between two students who fail to communicate is another); the abundant use of the Socratic Method . Especially on the latter, it is worth noting that the novice is a naive believer and that the traces of an inquisitive/critical spirit are practically extinguished by this method which is highly tutor-driven and allows no exploratory deviations. It is worth investigating how the tutoring process of this kind hinders the novice's encounter with mathematical abstraction.
The (Un)Naturality of Concepts. Introducing new concepts to the novices is a process that most novices characterise as unnatural, that is lacking in providing with sufficient justification for the existence and the utility of the introduced concepts. This unnaturality appears as a major obstacle to abstraction. A cross-topic investigation might illuminate further details.
Other themes of a lesser range have also been primarily identified.
NOTE: The above approaches are tentative. The formation of the subsequent levels of analysis whilst analysis is taking place is inherent in qualitative research.
I note that the themes cited in the Transfer paper addressed a wide variety of issues (epistemological, contextual, psychological) related to the learning difficulties of the novice mathematician's encounter with mathematical abstraction. As a step towards the progressive focusing of the study, the subsequent stages of the data analysis more emphatically concentrated on the psychological issues but remained intensively — even though more implicitly — informed by epistemological and contextual issues.