In Chapter 1, particularly Part III, focus was placed on these developments within PME-AMT which are about the students' difficulties with advanced mathematical concepts and reasoning. Here I specify in more detail the nature of the learning phenomena that this study aims to investigate, the environment of the investigation and in the subsequent parts I describe the methodological tools that were employed in this investigation.

To achieve its purpose this study has adopted a strongly phenomenological approach. Here the phenomena, that is the things apprehended by the senses, on which typifications will be assigned are the novice's verbal expressions of their thought processes during tutorials and interviews.

Phenomenology (Burrell & Morgan 1979), that is the study of direct experience at face value, as a theoretical standpoint believes in the importance of subjective consciousness that is active and meaning bestowing. To comprehend the structures of consciousness Husserl encourages the act of transcendence, an Epoché during which the observer is freed from all preconceptions about the observed phenomena by questioning their taken for granted features. In this study, this transcendence is achieved by the constant clarification of the assumptions made and by adopting a very open series of data collection techniques such as unsystematic observation and loosely structured interviewing. In this sense Chapter 1 aims at declaring and clarifying the principles underlying the observation of cognitive phenomena and the techniques described in Part II of this Chapter are chosen because they allow the effortless emergence of the features of these phenomena.

Schutz (Burrell & Morgan 1979) suggests that, once this transcendence has been achieved, then meaning can be assigned to the observed phenomena retrospectively. That is, once observation of the phenomena (apprehension by the senses) is completed in the transcendental manner outlined above, then a meaning-bestowing process can begin through which the observer makes sense of the phenomena. In this study this means that theorising begins, once cognition has been observed in its full naturality and complexity and once the principles of observation and interpretation have been declared. This position is reflected in the theory emerging procedures that determine the data analysis.

An idea that is particularly significant in determining the methodological approaches of this study comes from the linguistic, ethnomethodological tradition (Garfinkel 1968). What is under investigation here is the novice's thinking processes from the perspective of the assumptions made, the tacit meaning attributed to external stimuli, the conventions utilised and the practices the novice adopts as these are reflected mainly in verbal and secondarily in written expressions. It is assumed here that the learners' indexical expressions in their interaction with peers, their tutor or the interviewer convey much more than it is actually said. Indexical expressions here is used as a term for the designations assigned to a stimulus so that this is located, labelled and interpreted. As seen in Chapter 1, these expressions are interesting and worth exploring because, in an interactionist frame of thought (Mead 1934) the learners continuously act — and in an educational study it is important to stress that they also interact — on the basis of their psychology as formed partly by these attributed meanings. So access to the learners' thinking, and consequently a consideration of the didactical implications of such a study, can only be possible if the learners' expression of their thinking is understood.

The typifications over the learner's expressions in this investigation will be data-grounded generated theory in the sense Glaser and Strauss (1967) use the term (see Part III).

I note that the use of the term 'phenomena' in this study is not to be confused with its phenomenographic use. Unlike Piaget and the constructivists, phenomenographic research, for instance as advocated by Marton (1988), is underlain by the sharp distinction between the knower and the objects of knowing. Phenomenographic cognitive phenomena then, in Marton's terms, are the observable relations between knowledge and the knower. As a result cognition can be described in a limited and finite number of ways: students' conceptions can be presented as an ordered, hierarchical structure (outcome space) and are not psychological, that is not context-specific and generalisable. The epistemological, psychological and sociocultural characteristics of this study are in opposition to this description of mathematical cognition. Below I outline briefly how cognition is perceived in this study through the words of Jacques Lacan.

Brown, Hardy & Wilson (1993) in transferring some of Lacan's ideas to inquiries about learning seem to resist the finitist ideas of phenomenography according to which phenomena are describable in a finite number of categories. These categories are, for the phenomenographist, descriptions of the learner's images of a mathematical idea which act

(Brown et al. 1993)

Expressing then their belief that however elaborate a description of the learning phenomena is, the sum is always greater than its parts, they claim that what is lacking is 'something that cannot be expressed in language or symbols or mathematical images'. This is 'the elusive reality of mathematics'. They then quote Zizek who interprets Lacan and his motif of symbolisation...

(ibid.)

In Chapter 1 mathematical activity was acknowledged as both personal and social. Rephrasing appropriately for this study Brown et al's quoting of Gattegno's 'only awareness is educable' to 'only awareness is communicable', I agree with them that 'the move [from the personal] into the social is always rooted in an attempt to label the personal'. But then, the same authors wonder, 'how do I articulate my awareness, my shifting from conscious to unconscious, into something usable' in communication?

Here is where symbolism and language come in as illustrated in the Lacanian strong metaphor below. I note that Lacan (1977) believed in the interpretation of the subconscious as a language:

(ibid.)

In this study the observed phenomena, the novices' verbal expressions of their conscious thinking are seen from the point of view of a departure from the primitive language of pre-university mathematics to the 'language of the tribe', the formalism of advanced mathematical thinking. Within the observed learning context, this departure is a constructive process which is far from smooth and unproblematic. The separation from the Mother, in other words, and the entrance in the world of the Father is a process of initiation into the culture of the tribe through the communication of language, a process of enculturation. The signs of this process are the students' expressions of 'retroactive naming' of the world they are trying to understand, give structure to and take position in.

Sexton (1988), who studied problem solving thought processes through the students' expressions of their thinking, notes about the reliability of these expressions:

(Sexton 1988)

This 'elusive reality' of mathematical cognition, this approximation of thought processes achieved through linguistic expression, will be discussed further in Part Ib, the section on the links of the study with Cognitive Psychology.

As far as research techniques are concerned, this study espouses some of the approaches used in Cognitive Psychology. In 1981 Greer was somewhat concerned with the absence of strong links between Cognitive Psychology and Mathematics Education since he thought that work was done in parallel in the two fields where interaction and exchange of ideas would be mutually helpful. This was 15 years ago but his outline of the reasons why a Cognitive Psyhology approach might apply to studies of mathematical thinking still holds.

In the same vein Vergnaud (1990) notes that 'cognitive and developmental psychology are certainly essential in that they really question what a concept is; what an operational behaviour is; how they develop; what part is played by action, perception, and language in concept formation; and what part is played by social interaction.'

Greer goes on to stress that cognitive psychologists, unlike behaviourists, are essentially interested in the mental processes that intervene between stimuli and responses. 'Theories', he contends, 'about these mental processes have to be developed on the basis of indirect inferences from observed behaviour (which may include what the subject says about their own thinking)'. One of the methods he illustrates, evolved for this purpose, is the use of verbal protocols. In his discussion he raises the issue of the validity of an account given by a cognising subject of their own thought processes. He continues: '...asking subjects to describe their own thought processes... is likely to yield much more insight, though the findings must be treated with circumspection because of subjectivity in interpreting them, ...and because of the lack of control and standardisation. There is also the unavoidable problem that the act of verbalising cognitive processes itself affects those processes, and there is the whole question of validity, i.e. whether subjects can report accurately on their own thinking ... '. He then gives examples of sceptics, like Evans, and of keen users of the approach, like Dominowski who studied concept learning.

Despite the traces of strong positivism, which is not compatible with the perspective of this study, I have used Greer's words to highlight some of the methodological constraints underlying an analysis of the novice mathematician's expressions of their thinking processes. While I am at more ease than Greer in recognising the situational character of these verbal expressions and the context-embeddedness of the interpretation, I am also aware of the risk that cognising subjects cannot necessarily report accurately on their thought processes and also that, while reporting them, these processes may be altered.

As illustrated in the next sections of this Chapter and in Chapters 3, 4 and 5, the methodology of this study is designed with the intention to capture evidence of the students' thought processes — while the students are engaged in cognitive activities — in its least artificial form: naturalistic observation and semi-structured interviewing are the tools of data collection and I note that the students are rarely asked by the tutors directly in the tutorials to explain their thought processes. They are mostly asked to explain their actions. In the interviews I always asked them to talk in detail about a number of mathematical concepts or theorems. Therefore the study is designed to avoid the artificiality and the risk of distortion that questions the validity of the cognising subjects' accounts of their own thought processes.

The approach taken in this study is a combination of elements from the phenomenological and cognitive psychology approaches outlined above: this is an exploration of the novice mathematician's thought processes as reflected in the novice's indexical expressions and in their accounts of their own thought processes. It is because, as Balacheff points out in (1990a), 'borrowing the theoretical framework' from psychological theories on thinking is not a way to cope sufficiently with issues of knowing in a teaching/learning situation, that a multi-disciplinary approach is required.

In the above what seems to be implicitly assumed is the accessibility of cognitive structures. This however should be done with modesty. As Balacheff poignantly stresses

(Balacheff 1990a)

On a more methodologically specific note he suggests that

(ibid.)

Clearly observing learners while in the process of knowing in the 'informed' way Balacheff illustrates is a potentially illuminating approach as far as this knowing is concerned. Similarly Dubinsky and Lewin, referring to the act of thought, note that,

(Dubinsky & Lewin 1986)

In this study it has been assumed that access to cognitive structures can be relatively achieved by means of an extensive and close observation of the learner in action. As the last quotation points out, encouraging metacognition on the part of the observer and of the learner enhances the chances of success. In this study opportunity to encourage metacognition on the part of the learner was given during interviewing. However the main bulk of data for this study has its origins in a learning environment on which interventions were neither possible nor intended. In the following I present the Oxford tutorial as the selected natural learning environment in which the cognitive phenomena elaborated upon in this study were explored.

Theory building is, as Davis says (1989), the 'trademark of science'. However, as illustrated in the introduction to this chapter, PME-AMT is at a pre-paradigm phase. Therefore unlike in mature disciplines, such as the natural sciences, where theory generating tools are sharp and accurate enough to yield theory and subsequently engage in a process of verification, in disciplines at a pre-paradigm phase a hypothesis testing approach is not feasible due to the absence of testable hypotheses. Hence the theory generated from this study is data-grounded. Data has been obtained by turning to potentially rich sources of evidence: in other words it is suggested that in an inquiry that aims at the study of the novice mathematician's problematic encounter with mathematical abstraction, a learning context within which this encounter takes place must be identified and subsequently provide the raw evidence of the searched-for phenomena. It is assumed that this approach that is at the same time conscious of preconceptions and informed by previous developments can be an efficient one. In Part Ia this approach was described in more detail.

For the reasons outlined above one of the primary concerns of this study was to identify contexts where the novice mathematician's learning takes place. Initial considerations included the prospects of studying mathematicians' written accounts of their own thinking, triggered by Jacques Hadamard's work (1954). Given that

the search for a learning context continued elsewhere.

Next, the study of the novices' written work was considered. The inadequacies of this approach are also clear: written work is a form of monologue on the part of the learner and can only give an in-depth account of the learner's first unnegotiated (with peers, with tutors) response to a mathematical idea. To achieve access to the learner's evolving train of thought, one must trigger its expression, its visible manifestation. Given the prerequisite for the naturality of this process, as explained earlier in this Part, it was deemed that the first year of mathematical studies at university level, provides a conveniently organised context. In particular the individual or pair tutorials, given to students on a weekly basis constitute a habitat in which it is most likely that the students are offered a forum for expressing themselves mathematically.

In Chapters 3, 4 and 5 a detailed account is given of why a tutorial is a rich source of raw data that satisfies the purposes of this study, but here I outline briefly its basic features that are essential in comprehending the process of selecting and using observation and interviewing which will be elaborated upon in Part II: a tutorial, as semi-officially defined by the mathematics tutors participating in this study, is usually a 30-60 minute session the main focus of which is resolving the learner's queries as well as a series of other activities, such as presentation of new extra-curricular material (new definitions, theorems and proofs that were either omitted or not quite emphasised in the lectures or the problem sheets) and mainly exposition on the solutions for the questions in the problem sheets. Given that this material is the same for all first year mathematics undergraduates, one can thus guarantee the relatively uniform basis of the observation sessions. Depending on the degree of permissiveness on the part of the tutor — as to replacing the traditional monologue with a more conversational teaching style — and of openness on the part of the students — as to how exposed they allow themselves to be with regard to their mathematical understanding — observing tutorials can be variably productive of incidents on mathematical thinking. In this sense tutorials can be ideal for the purposes of this inquiry.

In sum this is a piece of qualitative research. As such, in order to justify its theorisation process, the definition of scientific validity needs to be widened so that vertical studies (ones that probe into the depths of the explored phenomena with regard to a small number of participants) share the traditional prestige of horizontal ones (mostly statistically valid studies that explore a large number of variables over large samples of participants). Sierpinska et al (1993), Abbott-Chapman (1993) and the authors of Research Issues in Undergraduate Mathematics Learning (Kaput & Dubinsky 1994), in their texts on what is research in mathematics education and what are its results, also make the point: in mathematics education the debate between qualitative and quantitative research is an unnecessary and false dichotomy. Each methodology is underlain by a number of theoretical assumptions and governed by its own technical rules: as long as researchers make these rules and assumptions competently overt and explicit, and as long as they accept that the validity of their findings is also subject to/ yielded from these rules, the two traditions can develop in parallel. Ideally a merge of quantitative and qualitative approaches is achieved in cases where the hypotheses tested in a quantitative piece of research have been generated via a qualitative approach.