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The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework

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Abstract

Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults’ use of shortcut procedures that make use of the inverse relation on two kinds of problems: inversion problems (e.g., \( {9} \times {24} \div {24} \)) and associativity problems (e.g., \( {9} \times {24} \div {8} \)). Both can be solved more easily if the division of the second and third numbers is performed before the multiplication of the first and second numbers. The findings we reviewed suggest that understanding and use of the inverse relation between multiplication and division develops relatively slowly and is difficult for both children and adults to implement in shortcut procedures if they are not flexible problem solvers. We use the findings to expand an existing model, highlight some similarities and differences in solvers’ use of conceptual knowledge across operations, and discuss educational implications of the findings.

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Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada through Discovery Grants to K. M. Robinson, and to J. LeFevre.

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Authors and Affiliations

  1. Department of Psychology, University of Regina, 3737 Wascana Parkway, Regina, Saskatchewan, S4S 0A2, Canada

    Katherine M. Robinson

  2. Department of Psychology and Institute of Cognitive Science, Carleton University, Ottawa, ON, Canada

    Jo-Anne LeFevre

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  1. Katherine M. Robinson
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  2. Jo-Anne LeFevre
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Correspondence to Katherine M. Robinson.

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Robinson, K.M., LeFevre, JA. The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework. Educ Stud Math 79, 409–428 (2012). https://doi.org/10.1007/s10649-011-9330-5

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