In the following I present a concise theoretical profile of minimally participant, unsystematic observation and semi-structured interviewing, the methodological techniques employed in this study.

The rationale behind the selection of minimally participant, unsystematic observation and semi-structured interviewing as the main methodological techniques of this study, lies in the conceptualisation of the study as a data-grounded theory generating project. The theoretical origins of this concept lie in the work of Glaser and Strauss (1967; Strauss 1990) whose ideas I present below, and in Part III, as they have been filtered through the data collection and analysis processes.

In brief (see Part III for more details) Data Grounded Theory is a cumulative plan for progressive building up from facts: from data to substantive theory and then to more conceptual forms and formal theory. First however it is necessary to specify the features and qualities of the data that constitute the body of substantial evidence on which the theory generation process is grounded. To do so I outline below how Data Collection for this study is reconciled with the principles of theoretical sampling in the Glaser and Strauss theory.

Theoretical sampling — or naturalistic sampling as Ball calls it in (Hammersley 1993) — is the process of data collection for generating theory whereby the analyst jointly collects, codes and analyses data, decides which data to collect and where to find them in order to develop emerging theory. Initial decisions are not based on a firmly preconceived theoretical framework. A partial framework of local concepts is the basis and the analyst is open and sensitive to influences from a variety of sources. This sensitivity is valuable since it secures data collection from the exertion of power from one specific preconceived theoretical frame.

Data collection in this study did not take place within a firm theoretical frame. The focus of the study is conceived as a general address of an issue that seems to be significant in the field — the novice mathematician's problematic encounter with mathematical abstraction. The local concepts that are enrolled into the study come from the research literature on difficulties that the novice has in particular mathematical topics and in mathematical reasoning (see Chapter 1 Part III). The psychological background of the study is also diverse and locally valid (see Chapter 1 Part II). In other words this remarkable lack of unified theory in the field, in a way, forces an open-mindedness on the analyst who addresses the relevant issues. This open-mindedness is liberating and excruciating at the same time. Why this is so will become manifest in the course of the theory emerging process.

Previously in this chapter the nature of the sought-for evidence has been outlined (the indexical expressions of the novice's difficulties in the encounter with mathematical abstraction) as well as the context within which the inquiry takes place (tutorials). Some characteristics of the Main Study (such as the number of participants) were determined on the basis of the Pilot Study's experience. In the Pilot, as explained in Chapter 3, it became evident that the relatively uniform mathematical content of the tutorials as well as their regularity (once a week) provided a fairly uniform learning environment for observing a group of students in cognitive action. To use Glaser and Strauss terminology, this uniformity makes theoretical prediction possible.

The nature of the sought-for evidence and the context of inquiry coupled with the openness imposed by the theoretical espousals of the study led to determining the intended technical features of the observation and interviewing techniques. In subsequent chapters the implementation of these intentions is discussed.

Here it is necessary to stress that the observation techniques used in the two stages of the study Pilot and Main: details follow in chapters 3-5) were substantially different. In the latter the use of audio recording guaranteed an accurate account of the events and thus gave me, as the observer, the latitude to make less descriptive and potentially more insightful observations. These observations operated as pivots for the primary stages of data analysis. The differences between the two phases of observation are elaborated in Chapters 3 and 4. Here I outline the intended use of observation in the Main Study.

The use of the term 'unsystematic observation' contains a potential misunderstanding (Medley, Mitzel & Gage 1963): entering a very rich learning environment, such as a mathematics tutorial, with the intentions to 'observe unsystematically' does not imply that the observer has not clarified, at least generally, the dimensions of the observed events that are of particular interest to the inquiry (Cohen & Manion 1989). The existence of a primary focus is crucial (Van Dalen 1966); the degree of specificity may vary (Anderson 1995). Here the focus was the psychological aspects of the novice's learning as expressed in the context of the tutorial and the degree of specificity was low. The former made the inquiry relatively tight; the latter theoretically allows the emergence of some cognitive phenomena with regard to the novice's transition to abstract mathematical thinking that under a tighter focus might have remained unnoticed. By the same token (Anderson 1995) the interviews, carried out during the Main Study, were chosen to be semi-structured and broadly based on some critical observations on the tutorials. Details on their content follow in Chapter 4. The scope of mathematical topics these interviews cover as well as the openness of the questions regarding these mathematical topics are again indications of the low specificity of focus selected for the data collection of this study.

A term which relates fittingly the aims of this study with its data collection is Naturalistic Observation (Fraenkel & Wallen 1990). Previously in this chapter the need to explore advanced mathematical cognition in a 'natural learning environment' was explained. An assumption underlying the selection of Unsystematic Minimally-Participant Observation is that it provides an account of mathematical learning of an unprecedented, strong internal validity (Anderson 1995). In Chapter 4, details are given of how participation of the observer was kept to minimal levels; also an evaluation of the technique as used in this study is provided.

As far as the external validity of an account based on Unsystematic Minimally-Participant Observation is concerned ((Van Dalen 1966) and (Merriam 1988)), as explained below, the aspirations of a qualitative study is to provide an accurate and profound account of intensely context-embedded phenomena. The intricacy of the task lies in giving an account whose detail and acuteness simultaneously highlight the idiosyncrasy as well as the generality of the situation. This requires that the observer is well aware of the generalities established in other pieces of research and that, at the time of observation (as well as at the subsequent stages of analysis) the observer is able to draw appropriate abstractions without stripping the situation of its specificity. Observation provides material of immense diversity, density and complexity which the observer and analyst is required to engage in disentangling. As repeatedly stressed in this chapter this is where the strengths and the weaknesses of this methodology lie (Mercer & Walford 1991). The details on the use of Unsystematic Minimally-Participant Observation given in Chapter 4 explain how some weaknesses were restrained.

As explained in the presentation of the Main Study in subsequent chapters, the interviews with the participants play a supportive role in the analysis of the main bulk of material that was produced via observation. Their loose structure was based on a selection of some striking difficulties of the participants with certain mathematical topics identified during tutorial observation (details are given in Chapter 4). Further investigation of these difficulties via interviewing was decided on the grounds of the following characteristics of clinical interviewing as identified by Jean Piaget (Ginsburg 1981; Piaget & Inhelder 1963) : despite its artificial setting a clinical interview is designed in a way that allows the identification, exploration and evaluation of the interviewee's thought processes. The first two aims are relevant to this study:

Moreover a clinical interview aims at clarifying the ambiguity of verbal statements as well as checking out alternative explanations of the interviewee's conceptions, images and reasoning. Contingency and open-endedness are definitive characteristics of the clinical approach. In Chapter 4 details are given on how the clinical interviews made in this study secured the fulfilment of the above characteristics.

In this study there has been a substantial deviation from the Glaser and Strauss plan for data collection: the only forms of data processing that took place during data collection were making of critical/evaluative notes during observation and constructing descriptive summaries of the events in each tutorial on the same day, that is construction of Scripts. Therefore the decision when to stop sampling, which would normally, in the Glaser and Strauss plan, be indicated by theoretical saturation — that happens when no additional data develops any further the properties of the category — was informed by the experience of the Pilot study and external factors such as time constraints of the study and recommendations by more experienced colleagues in the field. What was continually corrected however was the note and script making process. Details on that are given in Chapter 4 but briefly I would describe these corrections as geared towards giving a less judgmental and gradually more focused account.

It must also be made clear that the sense in which the term sampling is used here is quite distinct from the statistical use of the term: the primary aim here is not to obtain accurate evidence on a wide distribution of people but to discover categories and properties. Working on a small sample of Oxford undergraduates, that have been chosen through procedures of mutual agreement and volunteering is a process of theoretical sampling. The aim of the selection of participants here is not statistical representativity but gaining access to individuals who are willing to share their mathematical thinking in a way that will help the emergence of rich theoretical categories regarding the novice's mathematical cognition. In this sense the sampling of this study has been systematic (Hammersley 1990).

Finally, in Part III I outline briefly the principles underlying the data-analysis of the study.