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Vol. 6 No. 3 (2003): Noviembre

MODELS AND THEORIES OF THE MATHEMATICAL UNDERSTANDING: COMPARING PIRIE AND KIEREN’S MODEL OF THE GROWTH OF MATHEMATICAL UNDERSTANDING AND APOS THEORY

Submitted
December 23, 2024
Published
2003-11-30

Abstract

The search for a meaningful cognitive description of understanding has ensued for the past half a century.  Within the past three decades, new and integrative perspectives have grown out of Richard Skemp's distinctions between instrumental and relational understanding. The growth of these perspectives, up until 1987, was documented by Tom Schroeder in his PME synthesis of the work on understanding resulting from Richard Skemp's instrumental/relational contrasts.  Since 1987, the work on understanding has progressed and this paper examines the new, more recent theoretical frameworks of understanding which have arisen from these roots. This paper focuses on two theoretical frameworks, Pirie and Kieren's model of the growth of mathematical understanding and Dubinsky's APOS theory, and discusses other contemporary theoretical frameworks such as the work by Cornu and Sierpinska on cognitive or epistemological obstacles, the investigations into concept definition and concept image by Vinner and Tall, Kaput's explorations of multiple representations, and Sfard's distinctions between operational and structural conceptions. Besides explicating the definitions of understanding proposed by these two frameworks, the discussion addresses their elements and constructs as well as their linkages to historical and recent characterizations of understanding. The paper then argues why Pirie and Kieren's model and APOS theory satisfy the Schoenfeld (2000) criteria for classification as a theory and finally concludes with discussions of a variety of interconnections between these two theories as well as the elements which make them distinct from each other such as their origins, organizations, relationships to other frameworks, and implications of the two theories for both assessment and pedagogical practices.

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