Undergraduate Students’ Combinatorial Proof of Binomial Identities

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  • 1 Oregon State University
  • | 2 Embry-Riddle Aeronautical University

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

Footnotes

This article is based in part on work supported by the National Science Foundation under Grant No. DRL-1419973. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors also wish to thank Keith Weber and the anonymous reviewers for helpful comments and insights on previous versions of the article.

This article was accepted under the editorship of Jinfa Cai.

Contributor Notes

Elise Lockwood, Department of Mathematics, Oregon State University, 064 Kidder Hall, Corvallis, OR 97330; Elise.Lockwood@oregonstate.edu

Zackery Reed, Department of STEM Education, Embry-Riddle Aeronautical University, 1 Aerospace Boulevard, Daytona Beach, FL 32114; reedz@erau.edu

Sarah Erickson, Department of Mathematics, Oregon State University, 368 Kidder Hall, Corvallis, OR 97330; ericksos@oregonstate.edu

Journal for Research in Mathematics Education
  • Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181199. https://doi.org/10.1023/a:1002954428327

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Benjamin, A. T., & Quinn, J. J. (2003). Proofs that really count: The art of combinatorial proof. Mathematical Association of America. https://doi.org/10.5948/9781614442080

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Czocher, J. A., & Weber, K. (2020). Proof as a cluster category. Journal for Research in Mathematics Education, 51(1), 5074. https://doi.org/10.5951/jresematheduc.2019.0007

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dawkins, P. C., & Roh, K. H. (2016). Promoting metalinguistic and metamathematical reasoning in proof-oriented mathematics courses: A method and a framework. International Journal of Research in Undergraduate Mathematics Education, 2(2), 197222. https://doi.org/10.1007/s40753-016-0027-0

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Engelke, N., & CadwalladerOlsker, T. (2010, February 25–28). Counting two ways: The art of combinatorial proof. Online proceedings for the thirteenth SIGMAA on Research in Undergraduate Mathematics Education Conference. http://sigmaa.maa.org/rume/crume2010/Archive/Engelke.pdf

    • Search Google Scholar
    • Export Citation
  • Engelke Infante, N., & CadwalladerOlsker, T. (2011). Student difficulties in the production of combinatorial proofs. In J. Hannah, M. Thomas, & L. Sheryn (Eds.), Proceedings of Volcanic Delta 2011 (pp. 93101). The University of Canterbury and The University of Auckland.

    • Search Google Scholar
    • Export Citation
  • Erickson, S. A., & Lockwood, E. (2021a). Investigating combinatorial provers’ reasoning about multiplication. International Journal of Research in Undergraduate Mathematics Education, 7(1), 77106. https://doi.org/10.1007/s40753-020-00123-8

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Erickson, S. A., & Lockwood, E. (2021b). Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof. The Journal of Mathematical Behavior, 62, Article 100868. https://doi.org/10.1016/j.jmathb.2021.100868

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 613. https://doi.org/10.1007/BF01809605

  • Harel, G. (2007). The DNR system as a conceptual framework for curriculum development and instruction. In R. A. Lesh, E. Hamilton, & J. J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 263280). Erlbaum. https://doi.org/10.4324/9781003064527-16

    • Search Google Scholar
    • Export Citation
  • Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In B. Gold & R. A. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy (pp. 265290). Mathematical Association of America.

    • Search Google Scholar
    • Export Citation
  • Harel, G., Fuller, E., & Rabin, J. M. (2008). Attention to meaning by algebra teachers. The Journal of Mathematical Behavior, 27(2), 116127. https://doi.org/10.1016/j.jmathb.2008.08.002

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389399. https://doi.org/10.1007/bf01273372

  • Lockwood, E. (2013). A model of students’ combinatorial thinking. The Journal of Mathematical Behavior, 32(2), 251265. https://doi.org/10.1016/j.jmathb.2013.02.008

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lockwood, E. (2014). A set-oriented perspective on solving counting problems. For the Learning of Mathematics, 34(2), 3136. https://flm-journal.org/Articles/E90F6EA084EACFD943E77656A3133.pdf

    • Search Google Scholar
    • Export Citation
  • Lockwood, E., & Caughman, J. S., IV. (2016). Set partitions and the multiplication principle. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26(2), 143157. https://doi.org/10.1080/10511970.2015.1072118

    • Search Google Scholar
    • Export Citation
  • Lockwood, E., Caughman, J. S., & Weber, K. (2020). An essay on proof, conviction, and explanation: Multiple representation systems in combinatorics. Educational Studies in Mathematics, 103(2), 173189. https://doi.org/10.1007/s10649-020-09933-8

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lockwood, E., & Purdy, B. (2020). An unexpected outcome: Students’ focus on order in the multiplication principle. International Journal of Research in Undergraduate Mathematics Education, 6(2), 213244. https://doi.org/10.1007/s40753-019-00107-3

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lockwood, E., & Reed, Z. (2020). Defining and demonstrating an equivalence way of thinking in enumerative combinatorics. The Journal of Mathematical Behavior, 58, Article 100780. https://doi.org/10.1016/j.jmathb.2020.100780

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lockwood, E., Reed, Z., & Caughman, J. S. (2017). An analysis of statements of the multiplication principle in combinatorics, discrete, and finite mathematics textbooks. International Journal of Research in Undergraduate Mathematics Education, 3(3), 381416. https://doi.org/10.1007/s40753-016-0045-y

    • Crossref
    • Export Citation
  • Lockwood, E., Swinyard, C. A., & Caughman, J. S. (2015). Patterns, sets of outcomes, and combinatorial justification: Two students’ reinvention of counting formulas. International Journal of Research in Undergraduate Mathematics Education, 1(1), 2762. https://doi.org/10.1007/s40753-015-0001-2

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lockwood, E., Wasserman, N. H., & McGuffey, W. (2018). Classifying combinations: Investigating undergraduate students’ responses to different categories of combination problems. International Journal of Research in Undergraduate Mathematics Education, 4(2), 305322. https://doi.org/10.1007/s40753-018-0073-x

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2011). Combinatorics and reasoning: Representing, justifying and building isomorphisms. Springer. https://doi.org/10.1007/978-0-387-98132-1

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maher, C. A., & Speiser, R. (1997). How far can you go with block towers? The Journal of Mathematical Behavior, 16(2), 125132. https://doi.org/10.1016/s0732-3123(97)90021-3

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mamona-Downs, J., & Downs, M. (2004). Realization of techniques in problem solving: The construction of bijections for enumeration tasks. Educational Studies in Mathematics, 56(2–3), 235253. https://doi.org/10.1023/b:educ.0000040414.92348.08

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Martin, G. E. (2001). Counting: The art of enumerative combinatorics. Springer. https://doi.org/10.1007/978-1-4757-4878-9

  • Mazur, D. R. (2010). Combinatorics: A guided tour. Mathematical Association of America.

  • Moore, R. C. (2016). Mathematics professors’ evaluation of students’ proofs: A complex teaching practice. International Journal of Research in Undergraduate Mathematics Education, 2(2), 246278. https://doi.org/10.1007/s40753-016-0029-y

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Reed, Z., & Lockwood, E. (2021). Leveraging a categorization activity to facilitate productive generalizing activity and combinatorial reasoning. Cognition and Instruction. Advance online publication. https://doi.org/10.1080/07370008.2021.1887192

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rosen, K. H. (2007). Discrete mathematics and its applications: With combinatorics and graph theory (6th ed.). McGraw Hill.

  • Speiser, R. (2011). Block towers: From concrete objects to conceptual imagination. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and reasoning: Representing, justifying and building isomorphisms (pp. 7386). Springer. https://doi.org/10.1007/978-94-007-0615-6_7

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267306). Erlbaum.

    • Search Google Scholar
    • Export Citation
  • Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Sage.

    • Search Google Scholar
    • Export Citation
  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289321.

  • Stylianou, D. A., Blanton, M. L., & Rotou, O. (2015). Undergraduate students’ understanding of proof: Relationships between proof conceptions, beliefs, and classroom experiences with learning proof. International Journal of Research in Undergraduate Mathematics Education, 1(1), 91134. https://doi.org/10.1007/s40753-015-0003-0

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tarlow, L. D. (2011). Pizzas, towers, and binomials. In C. A. Maher, A. B. Powell, & E. B. Uptegrove (Eds.), Combinatorics and reasoning: Representing, justifying and building isomorphisms (pp. 121131). Springer. https://doi.org/10.1007/978-94-007-0615-6_11

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tucker, A. (2002). Applied combinatorics (4th ed.). Wiley.

  • Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2–3), 209234. https://doi.org/10.1023/b:educ.0000040410.57253.a1

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Weber, K., & Alcock, L. (2009). Proof in advanced mathematics classes: Semantic and syntactic reasoning in the representation system of proof. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K–16 perspective (pp. 323338). Routledge. https://doi.org/10.4324/9780203882009-19

    • Search Google Scholar
    • Export Citation

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