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Reinforcing Mathematical Concepts and Developing Mathematical Practices Through Combinatorial Activity

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Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

As a branch of discrete mathematics, combinatorics is an area of mathematics that offers students chances to engage with accessible yet complex mathematical ideas and to develop important mathematical practices. In this chapter, we focus on a combinatorial task involving counting passwords, and we provide examples of affordances that undergraduate students gained by engaging with the task. We highlight two kinds of affordances—those that strengthened understanding about fundamental combinatorial ideas, and those that fostered meaningful mathematical practices. We hope that these examples of rich and sophisticated student work will contribute to an overall goal of elevating the status of combinatorics specifically, and discrete mathematics more broadly, in the K–16 curriculum. We conclude with a handful of pedagogical implications.

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Notes

  1. 1.

    Again, we recognize that these numbers are rows in Pascal’s triangle, but pursuing the relationship with Pascal’s triangle is not our goal in this set of tasks.

  2. 2.

    Note, in organizing the results, in each example we pair an affordance related to combinatorial content with one related to mathematical practice. We do so to be efficient in our presentation of three student examples, but we do not claim that the affordances must be paired in this way. Indeed, in investigating some combinatorial idea (like the multiplication principle) students may engage in a variety of practices.

  3. 3.

    All names are pseudonyms.

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Acknowledgements

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1419973. The authors wish to thank Sarah Erickson for her input on early drafts of the manuscript.

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Authors and Affiliations

  1. Oregon State University, Corvallis, OR, USA

    Elise Lockwood & Zackery Reed

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  1. Elise Lockwood
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  2. Zackery Reed
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Correspondence to Elise Lockwood .

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Editors and Affiliations

  1. Mathematics Department, Grand View University, Des Moines, Iowa, USA

    Eric W. Hart

  2. Department of Mathematics and Statistics, Georgetown University, Washington, District of Columbia, USA

    James Sandefur

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Lockwood, E., Reed, Z. (2018). Reinforcing Mathematical Concepts and Developing Mathematical Practices Through Combinatorial Activity. In: Hart, E., Sandefur, J. (eds) Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70308-4_7

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  • DOI: https://doi.org/10.1007/978-3-319-70308-4_7

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-70308-4

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