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The transition to formal thinking in mathematics

The transition to formal thinking in mathematics This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of ‘three worlds of mathematics’- the ‘conceptual-embodied’ world based on perception, action and thought experiment, the ‘proceptual-symbolic’ world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the ‘axiomatic-formal’ world of set-theoretic concept definitions and mathematical proof. Each ‘world’ has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics Education Research Journal Springer Journals

The transition to formal thinking in mathematics

Mathematics Education Research Journal , Volume 20 (2) – Jun 25, 2011

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References (31)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Mathematics Education Research Group of Australasia Inc.
Subject
Education; Mathematics Education
ISSN
1033-2170
eISSN
2211-050X
DOI
10.1007/BF03217474
Publisher site
See Article on Publisher Site

Abstract

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of ‘three worlds of mathematics’- the ‘conceptual-embodied’ world based on perception, action and thought experiment, the ‘proceptual-symbolic’ world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the ‘axiomatic-formal’ world of set-theoretic concept definitions and mathematical proof. Each ‘world’ has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds.

Journal

Mathematics Education Research JournalSpringer Journals

Published: Jun 25, 2011

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