Skip to main content
SpringerLink
Log in

An essay on proof, conviction, and explanation: multiple representation systems in combinatorics

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

There is a longstanding conversation in the mathematics education literature about proofs that explain versus proofs that only convince. In this essay, we offer a characterization of explanatory proofs with three goals in mind. We first propose a theory of explanatory proofs for mathematics education in terms of representation systems. Then, we illustrate these ideas in terms of combinatorial proofs, focusing on binomial identities. Finally, we leverage our theory to explain audience-dependent and audience-invariant aspects of explanatory proof. Throughout, we use the context of combinatorics to emphasize points and to offer examples of proofs that can be explanatory or only convincing, depending on how one understands the claim being made.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Providing such a definition was not necessarily their intent.

  2. Balacheff (2009) also defined explanation as a psychological construct, proof as a social construct, and both characterized in terms of representation systems. However, at least in terms of student-generated proofs, he viewed a proof as a socially accepted explanation (i.e., all proofs are explanations). Our characterization says a student-generated proof might not be explanatory if it was generated with non-natural inferences or in an RS that a student did not personally value.

  3. The theorem extends to other, non-integer values, but we focus on this version.

References

  • Balacheff, N. (2009). Bridging knowing and proving in mathematics: A didactical perspective. In Pulte, H., Hanna, G., & Jahnke, H. J. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. New York, NY: Springer.

  • Bartlo, J. R. (2013). Why ask why: An exploration of the role of proof in the mathematics classroom. (Unpublished doctoral dissertation). Portland State University. Portland, OR.

  • Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181–199.

    Article  Google Scholar 

  • Brown, S. A. (2014). On skepticism and its role in the development of proof in the classroom. Educational Studies in Mathematics, 86(3), 311–335.

    Article  Google Scholar 

  • De Villiers, M. (1999). The role and function of proof. In M. de Villiers (Ed.), Rethinking proof with the Geometer’s Sketchpad (pp. 3–10). Emeryville, CA: Key Curriculum Press.

  • Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulty of proof. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 345–352). Stellenbosch, South Africa.

  • Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.

    Article  Google Scholar 

  • Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory (pp. 185–212). Norwood, NJ: Ablex.

    Google Scholar 

  • Harel, G., & Sowder, L. (1998). Students' proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education (vol. III, pp. 234–283). Providence, RI: American Mathematical Society.

    Chapter  Google Scholar 

  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.

    Article  Google Scholar 

  • Lange, M. (2009). Why proofs by mathematical induction are generally not explanatory. Analysis, 69(2), 203–211.

    Article  Google Scholar 

  • Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32, 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008

    Article  Google Scholar 

  • Mejia-Ramos, J. P., & Inglis, M. (2017). ‘Explanatory’ talk in mathematics research papers. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20thConference for Research in Undergraduate Mathematics Education (pp. 373–382). San Diego, California.

  • Nelsen, R. (1993). Proofs without words: Exercises in visual thinking. Washington, DC: MAA.

    Google Scholar 

  • Nunokawa, K. (2009). Proof, mathematical problem-solving, and explanation in mathematical teaching. In H. Pulte, G. Hanna, & H. J. Jahnke (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives. New York, NY: Springer.

  • Paseau, A. (2010). Proofs of the compactness theorem. History and Philosophy of Logic, 31(1), 73–98.

    Article  Google Scholar 

  • Pulte, H., Hanna, G., & Jahnke, H. J. (Eds.). (2009). Explanation and proof in mathematics: Philosophical and educational perspectives. New York: Springer.

    Google Scholar 

  • Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319–325.

    Article  Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

    Article  Google Scholar 

  • Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.

    Article  Google Scholar 

  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

    Google Scholar 

  • Stylianides, G. J., Sandefur, J., & Watson, A. (2016). Conditions for proving by mathematical induction to be explanatory. Journal of Mathematical Behavior, 43, 20–34.

    Article  Google Scholar 

  • Weber, K. (2010a). Proofs that develop insight: Proofs that reconceive mathematical domains and proofs that introduce new methods. For the Learning of Mathematics, 30(1), 32–37.

    Google Scholar 

  • Weber, K. (2010b). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306–336.

    Article  Google Scholar 

  • Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.

    Article  Google Scholar 

  • Weber, K. & Alcock, L. (2009). Semantic and syntactic reasoning and proving in advanced mathematics classrooms. Invited chapter on research on proof at the undergraduate level for M. Blanton, D. Stylinaiou, and E. Knuth (Eds.), The teaching and learning of proof across the K-16 curriculum (pp. 323–338). New York, NY: Routledge.

  • Weber, K., & Mejia-Ramos, J. P. (2019). An empirical study on the admissibility of graphical inferences in mathematical proofs. In A. Aberdein & M. Inglis (Eds.), Advances in experimental philosophy of logic and mathematics (pp. 123–144). London, UK: Bloomsbury.

Download references

Acknowledgments

We thank the anonymous reviewers for helpful comments.

Author information

Authors and Affiliations

  1. Oregon State University, Corvallis, OR, USA

    Elise Lockwood

  2. Portland State University, Portland, OR, USA

    John S. Caughman

  3. Rutgers University, New Brunswick, NJ, USA

    Keith Weber

Authors
  1. Elise Lockwood
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John S. Caughman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Keith Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elise Lockwood.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lockwood, E., Caughman, J.S. & Weber, K. An essay on proof, conviction, and explanation: multiple representation systems in combinatorics. Educ Stud Math 103, 173–189 (2020). https://doi.org/10.1007/s10649-020-09933-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-020-09933-8

Keywords

Search

Navigation