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Representational disfluency in algebra: evidence from student gestures and speech

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Abstract

In this investigation, we analyzed US middle school students’ (grades 6–8) gestures and speech during interviews to understand students’ reasoning while interpreting quantitative patterns represented by Cartesian graphs. We studied students’ representational fluency, defined as their abilities to work within and translate among representations. While students translated across representations to address task demands, they also translated to a different representation when reaching an impasse, where the initial representation could not be used to answer a task. During these impasse events, which we call representational disfluencies, three categories of behavior were observed. Some students perceived the graph to be bounded by its physical and numerical limits, and these students were categorized as physically grounded. A second, related, disfluency was categorized as spatially grounded. Students who were classified as spatially grounded exhibited a bounded view of the graph that limited their ability to make far predictions until they physically altered the spatial configuration of the graph by rescaling or extending the axes. Finally, students who recovered from one or more of these disfluencies by translating the quantitative information to alternative but equivalent representations (i.e., exhibiting representational fluency), while retaining the connection back to the linear pattern as graphed, were categorized as interpretatively grounded. Understanding the causes and varieties of representational fluency and disfluency contributes directly to our understanding of mathematics knowledge, learning and adaptive forms of reasoning. These findings also provide implications for mathematics instruction and assessment.

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Notes

  1. Another possible interpretation of Sophia’s use of the word “it” in line 9 is that she is referring to the graph as a whole unit, and views either the graph or the points plotted as malleable and organic.

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Acknowledgments

This research was funded in part by a grant entitled “Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning,” awarded to the second author by the Interagency Educational Research Initiative, an alliance of the National Science Foundation, the US Department of Education’s Institute of Educational Sciences, and the National Institute of Child Health and Human Development within the National Institutes of Health. We extend our appreciation to Andrew Garfield for his technical knowledge and assistance.

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Authors and Affiliations

  1. Michigan State University, 312 Erickson Hall, East Lansing, MI, 48824, USA

    Kristen N. Bieda

  2. School of Education, University of Wisconsin-Madison, 1025 West Johnson St, Madison, WI, 53706, USA

    Mitchell J. Nathan

Authors
  1. Kristen N. Bieda
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  2. Mitchell J. Nathan
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Corresponding author

Correspondence to Kristen N. Bieda.

Appendices

Appendix A: Students’ pattern generalization strategy codes

See Table 4.

Table 4  
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Appendix B: Transcripts of student responses

Gesture events coincide with boldface text and are described in parenthetical statements.

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Bieda, K.N., Nathan, M.J. Representational disfluency in algebra: evidence from student gestures and speech. ZDM Mathematics Education 41, 637–650 (2009). https://doi.org/10.1007/s11858-009-0198-0

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