Introduction
Mathematics, both as a way of knowing (Bishop 1991) — the activity of doing mathematics — and as a body of knowledge — the outcome of this activity — grows in a physical, a sociocultural and a self-referential context. By the first I mean the physical world which mathematics takes into account and explains; by the second I refer to the particular social and cultural habitat in which mathematical activity is embedded; and by the third I refer to the mathematical community of all individuals engaged in the creation and dissemination of mathematics. These three dimensions reflect, interdependently and not exclusively, the complex environment in which mathematics develops.
The learning of mathematics takes place also in this tripartite context. If a global understanding of the process of learning mathematics is sought, one ought to add to this socio-cultural context the individual nature of learning. The dialectic consideration of both contextual (environmental) and mental (psychological) aspects is then likely to illuminate the process of mathematical learning.
Education is an institutionalised form of learning. Hence it serves as the milieu within which learning takes place and in which learning can therefore be studied. In the dialectics suggested above, Education, as an institution, is an inextricable part of the learning environment and therefore, in an inquiry regarding learning, contextual considerations are important.
One implication from the above can be that no study of the learner's thinking processes can be undertaken that is void of the impact of epistemological, psychological and educational theoretical assumptions. In this chapter I thus present a declaration of the principles underlying the study which is located within the area of research on the Psychology of Mathematics Education recently known as Advanced Mathematical Thinking (PME-AMT; also a Working Group of the International Group for the Psychology of Mathematics Education since 1985).
PME-AMT is at a Kuhnian pre-paradigm stage (Kuhn 1962) or, in R.B. Davis' words (1989), is a not 'data-poor' but a 'metaphor-poor' field. The meaning of this will be elaborated in Chapter 2 but briefly it means that PME-AMT is still in search of unified theoretical frameworks, of explanatory systems within which its researchers can work equivocally and unambiguously. I quote from The Structure of Scientific Revolutions in order to illustrate what makes the declaration of principles in this Chapter necessary:
When the individual scientist can take a paradigm for granted, he need no longer, ..., attempt to build his field anew, starting from first principles and justifying the use of each concept introduced...The creative scientist can begin his research where it leaves off and thus concentrate exclusively upon the subtlest and more esoteric aspects of the...phenomena that concern his group.
(Kuhn 1962, p.19)
Exactly because a researcher in PME-AMT can currently 'take no paradigm for granted', in what follows, an account is given of the Theoretical Background of the study, i.e. a declaration of its underlying philosophical (Part I) and psychological (Part II) principles. Subsequently the study is embedded within current PME-AMT developments (Part III).