PART III Data Analysis Methodology. Data-Grounded Theory
In this last part of the chapter an account is given of the principles underlying the methodological techniques espoused in the data analysis of the study. Details on how these principles were put into practice are given in Chapter 5. Here the data analysis methodology is presented as a data-grounded theory emerging process in which a blend of techniques has been used from a diversity of methodological traditions: most notably an inductive approach to analysis of qualitative data and discourse analysis. A number of texts on the methods of analysis in qualitative research have been consulted, most notably (Adelman 1981; Bliss, Monk, & Ogborn 1983; Hitchcock & Hughes 1991; Lofland & Lofland 1995; Merriam 1988 and Miles & Huberman 1984), and have influenced the formation of techniques presented in Chapter 5.
According to Glaser and Strauss effective theory should enable explanation, advance theory in the field, be usable in practical applications and provide a stance towards future data and style of research. Glaser and Strauss use the term 'comparative analysis' for a method of generating theory through drawing patterns or making deductions that underlie a set of analytical units. Typical uses of comparison include: fact replication with comparative evidence and spotting the indicators of the conceptual category the event in question belongs to, drawing empirical generalisations and specifying concepts.
The process of generating theory is closely linked with its form of presentation. A difference between data-grounded generated theories and logicodeductive theories is that the former is generally more prone to a discussional form of presentation as opposed to the propositional form. It must be noted though that the propositional form allows easier and possibly riskier leaps to deductions. As shown in subsequent chapters, the presentation of the study has been designed in a way that allows the inductive approach of its theory generation to become visible (details in Chapter 5).
Further characteristics of the study that have determined its analytical strategies originate in Discourse Analysis and also the theory of Epistemological Obstacles.
Links to Discourse Analysis. This is a study which aims at gaining access to thought processes through the participants' verbalisations (earlier called verbal or indexical expressions). This has led to the adoption of certain techniques often attached to the methodological tradition of discourse analysis (Dijk 1985 and Coulthard 1985). As demonstrated in Chapter 5 where particular qualitative techniques employed in the study are presented in detail, the main directive of the processing of the raw material has been towards the extraction of data relevant to the aims of the study. Successive filtering has resulted in a compilation of episodes which are all relevant to the novices' advanced mathematical cognition and are underlain by the iterated reappearance of a certain number of cognitive phenomena. These phenomena, or in Vergnaud's words these conceptual fields (1990), are the categories that Glaser and Strauss describe as expected to emerge from this inductive process.
Links to the Theory of Epistemological Obstacles. Mathematical learning as explicated in Chapter 1 is perceived in this study to evolve as a process of constant confrontation of epistemological obstacles. As a result the Theory of Epistemological Obstacles as conceived by Bachelard and revived and refined by Brousseau 'directs the thought' of this piece of research. So, according to Sierpinska (1994) to whom these words belong, in that sense Epistemological Obstacles emerge as another dimension of the 'categories' mentioned above. She describes the somewhat elusive nature of a category:
It is possible that the most characteristic feature of a category is that it is hard to grasp with a definition, difficult to enclose within a rigid theory. A category does not belong to the world of theories; if it functions the way it does — by directing the thought — it is because it works somewhere between and above the vernacular and the research field. It is better described by the use that was made of it in research, what questions did it lead to, what explanations did it provide, what kind of discourse has developed around it.
(Sierpinska 1994, p.134)
The categories that have 'directed the thought' and emerged from this study fulfil Sierpinska's description: the study is
• driven by issues relating to advanced mathematical cognition,
• directed by a frame of mind that describes learning as a process of conflict and confrontation of difficulties, and,
• leading to the formation of conceptual categories regarding advanced mathematical cognition.
In Chapters 1 and 2 the conceptualisation of the study has been presented. In the subsequent chapters the Pilot and the Main Study are presented as its realisation.