Educational Studies in Mathematics

, Volume 79, Issue 1, pp 3–18 | Cite as

An assessment model for proof comprehension in undergraduate mathematics

  • Juan Pablo Mejia-Ramos
  • Evan Fuller
  • Keith Weber
  • Kathryn Rhoads
  • Aron Samkoff
Article

Abstract

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin’s (Educational Studies in Mathematics 67:59–76, 2008) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof’s high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory.

Keywords

Proof comprehension Proof reading Assessment Undergraduate mathematics education 

Notes

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DRL0643734.

References

  1. Alcock, L. (2009). e-Proofs: Students experience of online resources to aid understanding of mathematical proofs. In Proceedings of the 12 th Conference for Research in Undergraduate Mathematics Education. Available for download at: http://sigmaa.maa.org/rume/crume2009/proceedings.html. Last downloaded January 20, 2011.
  2. Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24, 125–134.CrossRefGoogle Scholar
  3. Alibert, D., & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 215–230). Dordrecht: Kluwer.Google Scholar
  4. Anderson, J. R., Boyle, C. F., & Yost, G. (1986). The geometry tutor. The Journal of Mathematical Behavior, 5, 5–19.Google Scholar
  5. Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42, 225–235.CrossRefGoogle Scholar
  6. Davis, P. J., & Hersh, R. (1981). The Mathematical Experience. Boston: Birkhäuser.Google Scholar
  7. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.Google Scholar
  8. Fuller, E., Mejia-Ramos, J. P., Weber, K., Samkoff, A., Rhoads, K., Doongaji, D., Lew, K. (2011). Comprehending Leron’s structured proofs. In Proceedings of the 14th Conference for Research in Undergraduate Mathematics Education. Google Scholar
  9. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.CrossRefGoogle Scholar
  10. Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM, 40, 345–353.CrossRefGoogle Scholar
  11. Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389–399.CrossRefGoogle Scholar
  12. Leron, U. (1983). Structuring mathematical proofs. The American Mathematical Monthly, 90(3), 174–184.CrossRefGoogle Scholar
  13. Malek, A., & Movshovitz-Hadar, N. (2011). The effect of using transparent pseudo proofs in linear algebra. Research in Mathematics Education, 13(1), 33–58.CrossRefGoogle Scholar
  14. Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24, 385–401.CrossRefGoogle Scholar
  15. Mejia-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education (Vol. 2, pp. 88–93). Taipei, Taiwan.Google Scholar
  16. Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319–325.CrossRefGoogle Scholar
  17. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5–41.Google Scholar
  18. Resnick, L. B., & Resnick, D. P. (1992). Assessing the thinking curriculum: New tools for educational reform. In B. R. Gifford & M. C. O'Connor (Eds.), Changing assessments: Alternative views of aptitude, achievement, and instruction (pp. 37–75). Boston: Kluwer.Google Scholar
  19. Rowland, T. (2001). Generic proofs in number theory. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 157–184). Westport: Ablex.Google Scholar
  20. Roy, S., Alcock, L., & Inglis, M. (2010). Undergraduates proof comprehension: A comparative study of three forms of proof presentation. In Proceedings of the 13th Conference for Research in Undergraduate Mathematics Education. Available for download from: http://sigmaa.maa.org/rume/crume2010/Abstracts2010.htm
  21. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.CrossRefGoogle Scholar
  22. Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29, 123–151.CrossRefGoogle Scholar
  23. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4–36.CrossRefGoogle Scholar
  24. Thurston, W. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar
  25. Weber, K. (2002). Beyond proving and explaining: Proofs that justify the use of definitions and axiomatic structures and proofs that illustrate technique. For the Learning of Mathematics, 22(3), 14–17.Google Scholar
  26. Weber, K. (2009). Mathematics majors’ evaluation of mathematical arguments and their conception of proof. In Proceedings of the 12 th Conference for Research in Undergraduate Mathematics Education. Available for download at: http://sigmaa.maa.org/rume/crume2009/proceedings.html. Last downloaded April 10, 2010.
  27. Weber, K. (2010). Proofs that provide insight. For the Learning of Mathematics, 30(1), 32–36.Google Scholar
  28. Weber, K. (2011). The pedagogical practice of mathematicians. International Journal of Mathematics Education in Science and Technology, in press.Google Scholar
  29. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34–38.Google Scholar
  30. Weber, K., Brophy, A., & Lin, K. (2008). How do undergraduates learn about advanced mathematical concepts by reading text? In Proceedings of the 11 th Conference on Research in Undergraduate Mathematics Education. San Diego.Google Scholar
  31. Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs. Educational Studies in Mathematics, 76, 329–344.CrossRefGoogle Scholar
  32. Yang, K.-L., & Lin, F.-L. (2008). A model of reading comprehension of geometry proof. Educational Studies in Mathematics, 67, 59–76.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Juan Pablo Mejia-Ramos
    • 1
  • Evan Fuller
    • 2
  • Keith Weber
    • 1
  • Kathryn Rhoads
    • 1
  • Aron Samkoff
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA
  2. 2.Montclair State UniversityMontclairUSA

Personalised recommendations