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Together yet separate: Students’ associating amounts of change in quantities involved in rate of change

Abstract

This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students’ reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between associated amounts, students identified sections (e.g., an interval on a graph) in which they could make comparisons between amounts of change in quantities. Although such reasoning is useful for interpreting a Cartesian graph as a representation of covarying quantities, it does not foster attention to variation in the intensity of change in covarying quantities (e.g., a decreasing increase). Focusing on the kinds of relationships students make between amounts of change in covarying quantities might provide further insight into how students could develop a robust understanding of rate of change.

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Notes

  1. In using variationally, I intend to communicate an envisioning of variables as varying, not to distinguish between variation and covariation.

  2. This choice, however, does not preclude the possibility of a learning effect.

  3. By task, I mean a problem that has been purposefully designed for a particular audience (Sierpinska, 2004).

  4. Due to scheduling constraints, Jacob was interviewed twice during 1 week.

  5. For more detail regarding students’ work on other tasks, see Johnson (2010).

  6. See Johnson (2010) for a more comprehensive discussion.

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Acknowledgments

This research was completed to fulfill the dissertation requirement for a doctoral degree at The Pennsylvania State University under the advisement of Rose Mary Zbiek. Results reported in this paper are based on subsequent analysis of dissertation data. I am grateful to Evan McClintock for his thoughtful comments on prior versions of this paper and for the insights that resulted from our conversations related to this paper. This paper is supported in part by the National Science Foundation under Grant ESI-0426253 for the Mid-Atlantic Center for Mathematics Teaching and Learning (MAC-MTL). Any opinions, findings, or conclusions expressed in this document are my own and do not necessarily reflect the views of the National Science Foundation.

A previous version of this article appeared in:

Johnson, H. L. (2012). Reasoning about quantities involved in rate of change as varying simultaneously and independently. In R. Mayes & L. L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (Vol. 2, pp. 39-53). Laramie, WY: University of Wyoming College of Education.

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Johnson, H.L. Together yet separate: Students’ associating amounts of change in quantities involved in rate of change. Educ Stud Math 89, 89–110 (2015). https://doi.org/10.1007/s10649-014-9590-y

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Keywords

  • Quantitative reasoning
  • Rate of change
  • Quantity
  • Covariational reasoning
  • Cartesian graphs