Advanced Mathematical Thinking

Mathematics Education LibraryAdvanced Mathematical Thinking by David Tall (Mathematics Education Library) pdf

Advanced Mathematical Thinking has played a central role in the development of human civilization for over two millennia. Yet in all that time the serious study of the nature of advanced mathematical thinking – what it is, how it functions in the minds of expert mathematicians, how il can be encouraged and improved in the developing minds of students – has been limited to the reflections of a few significant individuals scattered throughout the history of mathematics.

In the twentieth centuiy the theory of mathematical education during the compulsory yea is of schooling to age 16 has developed its own body of empirical research, theory and practice. But the extensions of such theories to more advanced levels have ally occurred in the last few years.

In 1976 The International Group for the Psychology of Mathematics (known as PME) was formed and has met annually’ at different venues round the work to share research ideas. In 1985 a Working Group of PME was famed to focus ai Advanced Mathematical Thinking with a major aim ofproducing this volume.

The text begins with an introductory chapter on the psychology of advanced mathematical thinking, with the remaining chapters grouped under three headings:

– the nature of advanced mathematical thinking,
– cognitive theory,
– reviews of the progress of cognitive research into different areas of advanced mathematics.

– reviews of the progress of cognitive research into different areas of advanced mathematics.

It is writt en in a style intended both for mathematicians and for mathematics educators, to encourage an interest in the cognitive difficulties experienced by students of the former and to extend the psychological theories of the latter through to later stages of development. We are cognizant of the fact that it is essential to understand the nature of the thinking erf mathematical experts to see the full spectrum of mathematical growth.

We therefore begin with an introductory chapter on the psychology of advanced mathematical thinking. This is followed by three chapters which focus ern the nature of advanced mathematical thinking: a study of tire mental processes involved, tire essential qualities of mathematical creativity and the mathematician’s view of proof.

The processes prove to be subtle and complex and, sadly, few of the more advanced processes are made available to the average student in an advanced mathematical course. Creativity is concerned with how the subtle ideas of research are built in the mind Proof is how they are ordered in alogical development both to verify the nature of the relationships and also to present them f approval to the mathematical community.

However, there is a huge gulf between the way in which ideas are built cognitively and the way in which they are arranged and presented in a deductive order. This warns us that s imply presenting a mathematical theory as a sequence of definitions, theorems and proofs (as happens in a typical university course) may show the logical structure of the mathematics, but it fails to allow fa- the psychological growth of the developing human mind

We begin the part of the book on cognitive theory by considering the way in which formal mathematical definitions are conceived by students and how this can be at variance with the formal theory. As a result of mentally manipulating a (mathematical) concept an individual develops an idiosyncratic personal concept image which is the product of experience and mental activity. Empirical research shows how this can give rise to subtle conflicts that can cause cognitive obstacles in the mind of the developing student and act as a barrier to attaining the formal ideas in the theory.

The next chapter looks at the mental objects that arc the material of mathematical thought – the conceptual entities that are manipulated in the mind during advanced mathematical thinking, and how’ these entities are represented by different kinds of symbolism. The final chapter in this part considers how these conceptual entities are fanned – through the process of reflective abstraction.

All advanced mathematical concepts are “abstract”. This chapter postulates a theory’ of how these concepts start as processes which are encapsulated as mental objects that arc then available for higher level abstract thought. Such a theory’ can give insight into how mathematicians develop advanced mathematical ideas, yet may fail to pass these thinking processes on to students, and what might be done to improve the situation.

The remainder of the bode is concerned with overviews of empirical research and theory in various specific topics. First the question of the nature of advanced mathematical thinking is addressed and how (if at all) it differs from more elementary thinking occurring in younger children. Then there follow chapters on functions, limits, analysis, infinity, proof, and the growing use of the computer in advanced mathematics.

Each one of these reveals a wide variety’ ofobstacles in students’ mental imagery and often extremely limited conceptions of formal concepts which are the unforseen consequences of the manner in which the subject is presented to the student. A variety of more cognitively appropriate approaches are postulated, some with empirical evidence of success. These include:

♦ the parincipal of the student in the process erf mathematical thinking through an active process of “scientific debate”, rather than passive receipt of preorganized theory,

♦ the direct confrontation of the student with conflict which occurs in developing new theoretical constructs, to help them reflect on the problem and build a new, more coherent, cognitive structure.

♦ the building up of appropriate intuitive foundations fa- the advanced mathematical concepts, through an approach which balances cognitive growth and an appreciation of logical development.

♦ the use of programming to cause the student to think through mathematical processes in a way which can be encapsulated by reflective abstraction.

The cognitive theory of advanced mathematical thinking is developing apace. This study is the first step hi making the broad sweep of current ideas hi the advanced mathematical education community available to a wider readership.

Language: English
Format: djvu
Pages: 310