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Take It Away or Walk the Other Way? Finding Positive Solutions for Integer Subtraction

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Exploring the Integer Addition and Subtraction Landscape

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

Abstract

Practicing teachers as well as researchers, mathematicians, and teacher educators have offered opinions and theoretical critiques of the multiple models used to teach integer arithmetic. Few studies, however, have investigated what students learn with models or empirically compared affordances and constraints of integer models. This led me to investigate how 160 fifth- and sixth-grade students who were learning integer arithmetic for the first time could benefit from a particular model. Each integer model encouraged students to conceive of numbers using distinct conceptual metaphors and move in certain ways to represent integer subtraction. Thus, I used embodied cognition to illuminate ways a manipulative-based cancellation model (chip model) and a physically enacted number line model (walk-it-off model) differentially impacted students’ subtraction knowledge. Integer subtraction, particularly the idea that subtracting a negative number could create a positive solution is especially difficult for students regardless of age, so assessment of this construct deserved a special focus in the test design of the larger study. This chapter reports students’ accuracy and reasoning on this difficult subtraction type 5 weeks after instruction with their assigned model. Findings for practice suggest the walk-it-off model was more effective as the first model students used and more research is needed.

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Notes

  1. 1.

    I use the verb forms as opposed to the noun forms of the metaphors to better reflect the patterns of interacting with the world as part of an ongoing dynamic system, in other words “enactive metaphors” (Gallagher, & Lindgren, 2015; Nurnberger-Haag, 2014; Smith, 2005).

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Acknowledgments

This work was supported in part by Kent State University start-up funds and Michigan State University dissertation expense funds. Special thanks to Katherine Bryk in the School of Teaching, Learning, and Curriculum Studies, Kent State University, for her assistance with the figures.

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Authors and Affiliations

  1. School of Teaching, Learning & Curriculum Studies, Kent State University, Kent, OH, USA

    Julie Nurnberger-Haag

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  1. Julie Nurnberger-Haag
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Correspondence to Julie Nurnberger-Haag .

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Editors and Affiliations

  1. Department of Curriculum and Instruction, Purdue University, West Lafayette, Indiana, USA

    Laura Bofferding

  2. School of Education, George Fox University, Newberg, Oregon, USA

    Nicole M. Wessman-Enzinger

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Nurnberger-Haag, J. (2018). Take It Away or Walk the Other Way? Finding Positive Solutions for Integer Subtraction. In: Bofferding, L., Wessman-Enzinger, N. (eds) Exploring the Integer Addition and Subtraction Landscape. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-90692-8_5

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