Introduction
'Collegiate mathematics education' is, in Selden and Selden's words (1993), 'at a pre-paradigm stage', that is exploratory and seeking the construction of theoretical frames. Only recently has the field started being recognised as an autonomous field: Batanero et al (1994) report the teaching of mathematics at the undergraduate level as one of the fields attached to which there is a developing number of projects. Its publications appear scattered in general mathematics education journals and conferences and, as noted by Kaput and Dubinsky in the introduction of (1994), the field is in a transition: from identification of the phenomena of learning to interpretation and action, from the locality of studying particular mathematical topics/concepts to the global confrontation of the problematique of advanced mathematical cognition.
In other words the field seems currently to be shifting from the improvisational amateurism of quick diagnosis-prescription to informed professionalism in its confrontation of the phenomena of learning. Becker and Pence (1994) also point out the transitional stage of the field and subsequently categorise undergraduate students' learning as an increasingly autonomous area of research. This field of research, within which this study is located, has been denoted in Chapter 1 as PME-AMT.
Thomas Kuhn (1962), speaking of the 'route to normal science', offers an account which, I think, summarises vividly the state of the art in PME-AMT. To him 'normal science' means 'research firmly based upon one or more past scientific achievements, achievements that some particular scientific community acknowledges for a time as supplying the foundation for its further practice'. These works 'define the legitimate problems and methods of a research field for succeeding generations of practitioners' and share two characteristics:
Their achievement was sufficiently unprecedented to attract an enduring group of adherents away from competing modes of scientific activity. Simultaneously, it was sufficiently open-ended to leave all sorts of problems for the redefined group of practitioners to resolve.
(Kuhn 1962, p.10)
He then calls the achievements that share these characteristics 'paradigms'. Paradigms relate closely to 'normal science' which he defines as 'some accepted examples of actual scientific practice' which 'include law, theory, application and instrumentation together' and 'provide models from which spring particular coherent traditions of scientific research'. Researchers then 'whose research is based on shared paradigms are committed to the same rules and standards for scientific practice. That commitment and the apparent consensus it produces are prerequisites for normal science, i.e., for the genesis and continuation of a particular research tradition'. In his further clarification of the concepts of normal science and paradigm, Kuhn notes that
there can be a sort of scientific research without paradigms, or without any so unequivocal and so binding...Acquisition of a paradigm and of the more esoteric type of research it permits is a sign of maturity in the developments of any given scientific field.
(ibid., p.11)
The usual developmental pattern for mature science is competition among paradigms followed by successive transitions via revolutions. This pluralism is indicative of how 'science develops before it acquires its first universally received paradigm'. These competing paradigms all possess 'components of real scientific theories, of theories that had been drawn in part from experiment and observation and that partially determined the choice and interpretation of additional problems undertaken in research'. Kuhn (and this work was first published in 1962) also notes:
...and it remains an open question what parts of social science have yet acquired such paradigms at all. History suggests that the road to a firm research consensus is extraordinarily arduous.
(ibid., p.15)
He then highlights some of the reasons for the difficulties in this road:
In the absence of a paradigm or some candidate for paradigm, all of the facts that could possibly pertain to the development of a given science are likely to seem equally relevant. As a result, early fact-gathering is a far more nearly random activity than the one that subsequent scientific development makes familiar. Furthermore, in the absence of a reason for seeking some particular form of more recondite information, early fact-gathering is usually restricted to the wealth of data that lie ready to hand.
(ibid., p.15)
Kuhn then is sceptical about the possibility that this 'sort of fact-collecting' may 'produce a morass'. He is worried about whether 'facts collected with so little guidance from pre-established theories speak with sufficient clarity to permit the emergence of a first paradigm'. He however recognises that this complex and chaotic practice is essential to the building of scientific foundations. As a result of this practice it is
no wonder, then, that in the early stages of development of any science different men confronting the same range of phenomena, but not usually the same particular phenomena, describe and interpret them in different ways. What is surprising, and perhaps also unique in its degree to the fields we call science, is that such initial divergences should ever largely disappear.
(ibid., p.17)
The conditions in which these divergences disappear, if they do, are provided by the emergence of a theory 'better than its competitors' which 'need not, and in fact never does, explain all the facts with which it can be confronted'. Then both fact collection and theory articulation become highly directed activities. 'Truth' as Frances Bacon acutely noted, 'emerges more readily from error than from confusion'.
The reason I have so extensively quoted Kuhn is that, again in his words, 'it is hard to find another criterion that so clearly proclaims a field a science'. Also I find his account in complete resonance with the methodological and thematic ambience within PME-AMT which I consider as a field that is in search of its 'normal science' or, as worded earlier, 'at a pre-paradigm phase'. The methodology employed in this study reflects Kuhn's description of the 'fact-collecting' practice of a pre-paradigmatic field: the methodological tools of this study are chosen in a way that aspires at the identification, exploration and interpretation of the phenomena of undergraduate mathematics learning. The study is also underlain by an intention to submit these tools to evaluative testing (in tune with Jacob (1987) who acknowledges research 'on adapting qualitative traditions to the study of naturally occurring cognitive behaviour in classrooms' as one of the 'most exciting areas' of future methodological research). So in this sense research within a pre-paradigmatic field is required to work on these two levels: the thematic and the methodological.
As a result, this study is a piece of qualitative research and, in particular, it is a phenomenological study of advanced mathematical cognition. The theory that is generated from this study is the outcome of a data-grounded theory emergence process. In the following the theoretical underpinnings of this declaration are clarified. So:
• in Part I an account is given of the phenomenological character of the study, the cognitive nature of the phenomena to be explored and the learning environment within which they are explored,
• in Part II a theoretical justification is given of the methodological data collection techniques of observation and interviewing, and,
• in Part III a theoretical justification is given of the methodological data analysis techniques of data grounded theory.