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Developing a coherent approach to multiplication and measurement

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Abstract

We examine opportunities and challenges of applying a single, explicit definition of multiplication when modeling situations across an important swathe of school mathematics. In so doing, we review two interrelated conversations within multiplication research. The first has to do with identifying and classifying situations that can be modeled by multiplication, and the second has to do with identifying what is consistently characteristic of the operation when considering nonnegative real numbers. We review seminal lines of research––including those of Vergnaud and Davydov––and highlight ways that these lines do not provide a thoroughly unified view of multiplication. Then we offer our own approach based in measurement. To underscore consequences of the approach we outline, we use rectangular area and division to illustrate that, as a field, we may need to adjust how we think about connections between multiplication equations and at least some problem situations. We close with a set of questions about unified approaches to topics related to multiplication.

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Notes

  1. The distinction between sense making and making sense is reminiscent of Vygotsky’s (1986) discussion of spontaneous and scientific concepts.

  2. This definition of is a refinement of the one that appeared in Beckmann and Izsák (2015).

  3. Our definition for fractions refines that stated in the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 24).

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Acknowledgement

We thank Andy Norton, Jack Smith, and Bill McCallum for conversations that influenced our thinking about multiplication. This research was supported by the National Science Foundation under Grant No. DRL-1420307. The opinions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Education, Tufts University, 203 Paige Hall, Medford, MA, 02155, USA

    Andrew Izsák

  2. Department of Mathematics, University of Georgia, Boyd Graduate Studies Building, 200 D. W. Brooks Drive, Athens, GA, 30602, USA

    Sybilla Beckmann

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  1. Andrew Izsák
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Correspondence to Andrew Izsák.

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Izsák, A., Beckmann, S. Developing a coherent approach to multiplication and measurement. Educ Stud Math 101, 83–103 (2019). https://doi.org/10.1007/s10649-019-09885-8

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