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Knowledge construction and diverging thinking in elementary & advanced mathematics
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Gray, Edward Martin, Pinto, Marcia, Pitta, Demetra and Tall, David. (1999) Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, Vol.38 (No.13). pp. 111133. ISSN 00131954

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Official URL: http://dx.doi.org/10.1023/A:1003640204118
Abstract
This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children's arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success. This revised version was published online in September 2005 with corrections to the Cover Date.
Item Type:  Journal Article 

Subjects:  L Education > L Education (General) 
Divisions:  Faculty of Social Sciences > Institute of Education 
Library of Congress Subject Headings (LCSH):  Mathematics  Study and teaching, Cognitive learning 
Journal or Publication Title:  Educational Studies in Mathematics 
Publisher:  Springer Netherlands 
ISSN:  00131954 
Date:  March 1999 
Volume:  Vol.38 
Number:  No.13 
Page Range:  pp. 111133 
Identification Number:  10.1023/A:1003640204118 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
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URI:  http://wrap.warwick.ac.uk/id/eprint/470 
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