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Mathematics Matters in Education pp 117–148Cite as

How Future Teachers Reasoned with Variable Parts and Strip Diagrams to Develop Equations for Proportional Relationships and Lines

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Part of the book series: Advances in STEM Education ((ASTEME))

Abstract

Findings are presented from an analysis of how six future middle-grade teachers reasoned with strip diagrams and a variable parts perspective on proportional relationships to develop and explain equations in two variables. One equation was for two quantities varying together and one was for a line through the origin in a coordinate plane. Both equations involved a constant of proportionality that was not a whole number. The future teachers’ arguments were mathematically valid and relied on reasoning quantitatively about strip diagrams. The arguments also treated variables as quantities but rarely described the variables as numbers of units. Some arguments combined an interpretation of fractions as multiples (or iterates) of unit fractions with an interpretation of multiplication as a whole number of equal groups. In contrast, most arguments involving a fractional multiplier interpreted multiplication as “of.” All the points of tension that the future teachers encountered while developing their equations concerned referent units for quantities. The points of tension were resolved by focusing on referent units or on equality. Based on the data, extensions to current theories on reasoning quantitatively and with variables are proposed.

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Acknowledgment

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 NSF.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Sybilla Beckmann

  2. Department of Mathematics and Science Education, University of Georgia, Athens, GA, 30602, USA

    Torrey K. Kulow

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  1. Sybilla Beckmann
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  2. Torrey K. Kulow
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Correspondence to Sybilla Beckmann .

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Editors and Affiliations

  1. Department of Teaching, Learning and Culture, Texas A&M University, College Station, Texas, USA

    Yeping Li

  2. Department of Mathematics, University of Nebraska – Lincoln, Lincoln, Nebraska, USA

    W. James Lewis

  3. Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA

    James J. Madden

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Beckmann, S., Kulow, T.K. (2018). How Future Teachers Reasoned with Variable Parts and Strip Diagrams to Develop Equations for Proportional Relationships and Lines. In: Li, Y., Lewis, W., Madden, J. (eds) Mathematics Matters in Education. Advances in STEM Education. Springer, Cham. https://doi.org/10.1007/978-3-319-61434-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-61434-2_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-61433-5

  • Online ISBN: 978-3-319-61434-2

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