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Developing Fractions as Multiplicative Relations: A Model of Cognitive Reorganization

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Constructing Number

Part of the book series: Research in Mathematics Education ((RME))

Abstract

In this chapter, I propose a stance on learning fractions as multiplicative relations through reorganizing knowledge of whole numbers as a viable alternative to the Natural Number Bias (NNB) stance. Such an alternative, rooted in the constructivist theory of knowing and learning, provides a way forward in thinking about and carrying out teaching-learning of fractions, while eschewing a deficit view that seems to underlie the ongoing impasse in this area. I begin with a brief presentation of key aspects of NNB. Then, I discuss key components of the alternative framework, called reflection on activity-effect relationship, which articulates the cognitive process of reorganizing one’s anticipations as two types of reflection that give rise to two stages in constructing fractions as numbers. Capitalizing on this framework, I then delineate cognitive progressions of nine fractional schemes, the first five drawing on operations of iterating units and the last four on recursive partitioning operations. To illustrate the benefits of the alternative, conceptually driven stance, I link it to findings from a recent brain study, which includes significant gains for adult participants and provides a glance (fMRI) into circuitry recruited to process whole number and fraction comparisons.

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Notes

  1. 1.

    The first and second activities differ, as we often see children selecting 3 to complement 7 into 10.

  2. 2.

    Their work did not include E for effect; in my analysis, such a reorganization will be signified as the change from [G0➔[A0➔E0]] to [G1➔[A0➔E1]].

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  1. School of Education and Human Development, University of Colorado Denver, Denver, CO, USA

    Ron Tzur

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  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Anderson Norton

  2. Department of Psychology, University of Wisconsin–Madison, Madison, WI, USA

    Martha W. Alibali

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Tzur, R. (2019). Developing Fractions as Multiplicative Relations: A Model of Cognitive Reorganization. In: Norton, A., Alibali, M.W. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-00491-0_8

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