**ERIC Number:**EJ820231

**Record Type:**Journal

**Publication Date:**2008-Sep

**Pages:**20

**Abstractor:**As Provided

**Reference Count:**26

**ISBN:**N/A

**ISSN:**ISSN-1033-2170

The Transition to Formal Thinking in Mathematics

Tall, David

Mathematics Education Research Journal, v20 n2 p5-24 Sep 2008

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. (Contains 8 figures.)

Descriptors: Mathematical Logic, Mathematics Instruction, Mathematical Concepts, College Mathematics, Computation, Mathematics Education, Secondary School Mathematics, Cognitive Development, Cognitive Processes, Genetics, Concept Formation, Formal Operations, Learning Theories, Mathematics

Mathematics Education Research Group of Australasia. 67 Kokoda Avenue, Wahroonga 2076, Australia. Fax: +61-9688-5001; e-mail: vp.conferences@merga.net.au; Web Site: http://www.merga.net.au/publications/merj.php

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education; Postsecondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A