David Tall - Advanced Mathematical Thinking (Mathematics Education Library)

David Tall - Advanced Mathematical Thinking (Mathematics Education Library)
Kluwer | ISBN 0792328124 | May 1994 | 316 Page | djvu | 2,6 M
This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Its three main parts focus on the nature of advanced mathematical thinking, the theory of its cognitive development, and reviews of cognitive research. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. The reviews of recent research concentrate on cognitive development and conceptual difficulties with the notions of functions, limits, infinity, analysis, proof, and the use of the computer. They provide a wide overview and an introduction to current thinking which is highly appropriate for the college professor in mathematics or the general mathematics educator.

David Tall himself tells us:

My ideas have developed over my life, starting from my early textbooks for undergraduates, the edited book on Advanced Mathematical Thinking and the development of a theoretical framework of three distinct worlds of mathematics, embodied, (proceptual) symbolic, and formal. (eg Tall 2004, 2005).

My initial interest began as a mathematician in the early 70s, when I began to write textbooks on undergraduate mathematics with Ian Stewart, including Foundations of Mathematics, Complex Analysis, Algebraic Number Theory and Fermat's Last Theorem. There followed my own research in the late 70s with studies of students moving in transition from calculus to analysis, concepts of limits and infinity, and the concept images constructed by students to give meaning to formal mathematical concepts. Such images may or (more usually) may not be consonant with the formal theories. Research developed by Marcia Pinto in her PhD thesis under my supervision revealed that some individuals have a formal approach which builds directly from the definitions while others have natural approaches that build on their own imagery. To succeed, natural thinkers must reorganise their natural ideas into a formal sequence, conquering the difficulties of dealing with the multiply quantified statements. Natural thinkers may fall short of reconstructing their ideas to build the formalism, being guided implicitly or explicitly from their imagery. Formal thinkers build using formal deduction and must overcome difficulties such as the complexity of the quantifiers. They too may fail to cope with the complexity.

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