Student Interpretations of Written Comments on Graded Proofs

  • Martha Byrne
  • Sarah Hanusch
  • Robert C. Moore
  • Tim Fukawa-Connelly


Instructors often write feedback on students’ proofs even if there is no expectation for the students to revise and resubmit the work. It is not known, however, what students do with that feedback or if they understand the professor’s intentions. To this end, we asked eight advanced mathematics undergraduates to respond to professor comments on four written proofs by interpreting and implementing the comments. We analyzed the student’s responses using the categories of corrective feedback for language acquisition, viewing the language of mathematical proof as a register of academic English.


Proof writing Proof grading Proof instruction Proof revision Student thinking 

Supplementary material

40753_2017_59_MOESM1_ESM.pdf (1.9 mb)
Appendix Proofs A, B, C, and D were presented to the participants during the interviews. For each proof, first the interviewer presented to the participant a version of the proof without the professor’s comments, then later presented a version of the proof with the professor’s comments. The participants did not see the numbering of the comments. (PDF 1940 kb)


  1. Alcock, L. (2013). How to study as a mathematics major. Oxford: Oxford University Press.Google Scholar
  2. Alexander, D. S., & DeAlba, L. M. (1997). Groups for proofs: Collaborative learning in a mathematics reasoning course. Primus, 7, 193–207.CrossRefGoogle Scholar
  3. Bean, J. C. (2011). Engaging ideas: The professor's guide to integrating writing, critical thinking, and active learning in the classroom (2nd ed.). Hoboken: John Wiley & Sons.Google Scholar
  4. Chartrand, G., Polimeni, A. D., & Zhang, P. (2012). Mathematical proofs: A transition to advanced mathematics (3rd ed.). Boston: Pearson.Google Scholar
  5. Chierchia, G., & McConnell-Ginet, S. (2000). Meaning and grammar: An introduction to semantics (2nd ed.). Cambridge: MIT Press.Google Scholar
  6. Cupillari, A. (2013). The nuts and bolts of proof: An introduction to mathematical proofs (4th ed.). Waltham: Academic Press.Google Scholar
  7. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 7–24 Retrieved from Scholar
  8. De Villiers, M. (1999). Rethinking proof with sketchpad. Oakland: Key Curriculum Press.Google Scholar
  9. Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. H. Schoenfeld, J. Kaput, C. Kessel, & M. Keynes (Eds.), Research in collegiate mathematics IV (pp. 239–289). Providence: American Mathematical Society.Google Scholar
  10. Epp, S. S. (2003). The role of logic in teaching proof. American Mathematical Monthly, 110, 886–899.CrossRefGoogle Scholar
  11. Franklin, J., & Daoud, A. (2011). Proof in mathematics: An introduction. Sydney: Kew Books.Google Scholar
  12. Fukawa-Connelly, T. (2005). Thoughts on learning advanced mathematics. For the Learning of Mathematics, 25, 33–35.Google Scholar
  13. Fukawa-Connelly, T. (2016). Responsibility for proving and defining in abstract algebra class. International Journal of Mathematical Education in Science and Technology, 5, 1–17. doi: 10.1080/0020739X.2015.1114159.CrossRefGoogle Scholar
  14. Gass, S. M. (2003). Input and interaction. In C. J. Doughty & M. H. Long (Eds.), The handbook of second language acquisition (Vol. 27). Malden: Blackwell.Google Scholar
  15. Halliday, M. A. K. (1978). Language as a social semiotic: The social interpretation of language and meaning. Baltimore: University Park Press.Google Scholar
  16. Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Charlotte: Information Age.Google Scholar
  17. Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77, 81–112.CrossRefGoogle Scholar
  18. Herschensohn, J., & Young-Scholten, M. (Eds.). (2013). The Cambridge handbook for second language acquisition. Cambridge: Cambridge University Press.Google Scholar
  19. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43, 358–390.CrossRefGoogle Scholar
  20. Ko, Y., & Knuth, E. J. (2013). Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors. Journal of Mathematical Behavior, 32, 20–35.CrossRefGoogle Scholar
  21. Lai, Y., Weber, K., & Mejía-Ramos, J. P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof. Cognition and Instruction, 30, 146–169.CrossRefGoogle Scholar
  22. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  23. Leeman, J. (2007). Feedback in L2 learning: Responding to errors during practice. In R. M. DeKeyser (Ed.), Practice in a second language: Perspectives from applied linguistics and cognitive psychology (pp. 111–137). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  24. Lew, K., Fukawa-Connelly, T. P., Mejía-Ramos, J. P., & Weber, K. (2016). Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education, 47(2), 162–198.CrossRefGoogle Scholar
  25. Lyster, R., & Ranta, L. (1997). Corrective feedback and learner uptake. Studies in Second Language Acquisition, 19, 37–66.CrossRefGoogle Scholar
  26. Mills, M. (2011). Mathematicians’ pedagogical thoughts and practices in proof presentation. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 14th annual conference on research in undergraduate mathematics education (pp. 283–297). Oregon: Portland Retrieved from Scholar
  27. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266.CrossRefGoogle Scholar
  28. Moore, R. C. (2016). Mathematics professors’ evaluation of students’ proofs: A complex teaching practice. International Journal of Research in Undergraduate Mathematics Education, 2(2), 246–278. doi: 10.1007/s40753-016-0029-y.CrossRefGoogle Scholar
  29. Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19.Google Scholar
  30. Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. New York: Routledge & K. Paul.Google Scholar
  31. Pinker, S. (2009). Language learnability and language development, with new commentary by the author (Vol. 7). Cambridge: Harvard University Press.Google Scholar
  32. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.CrossRefGoogle Scholar
  33. Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading and Writing Quarterly, 23, 139–159.CrossRefGoogle Scholar
  34. 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, 4–36.CrossRefGoogle Scholar
  35. Selden, A., & Selden, J. (2008). Overcoming students’ difficulties in learning to understand and construct proofs. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 95–110). Washington: Mathematical Association of America.CrossRefGoogle Scholar
  36. Smith, D. D., Eggen, M., & St. Andre, R. (2014). A transition to advanced mathematics (8th ed.) Boston: Cengage Learning.Google Scholar
  37. Strickland, S., & Rand, B. (2016). Learning proofs via composition instruction techniques. In R. Schwell, A. Steurer, & J.F. Vasquez (Eds.), Beyond Lecture: Resources and pedagogical techniques for enhancing the teaching of proof-writing across the curriculum. Washington: Mathematical Association of America.Google Scholar
  38. Stylianides, G., Stylianides, A., & Weber, K. (in press). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), First compendium for research in mathematics education. National Council of Teachers of Mathematics: Reston.Google Scholar
  39. Swain, M. (1998). Focus on form through conscious reflection. In C. Doughty & J. Williams (Eds.), Focus on form in classroom second language acquisition (pp. 64–81). Cambridge: Cambridge University Press.Google Scholar
  40. Tedick, D. J., & de Gortari, B. (1998). Research on error correction and implications for classroom teaching. ACIE Newsletter, 1(3), 1–6.Google Scholar
  41. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.CrossRefGoogle Scholar
  42. Weber, K. (2006). Investigating and teaching the processes used to construct proofs. Research in Collegiate Mathematics Education, 6, 197–232.CrossRefGoogle Scholar
  43. Weinberg, A., Wiesner, E., & Fukawa-Connelly, T. (2014). Students’ sense-making frames in mathematics lectures. The Journal of Mathematical Behavior, 33, 168–179.CrossRefGoogle Scholar
  44. Zerr, J. M., & Zerr, R. J. (2011). Learning from their mistakes: Using students’ incorrect proofs as a pedagogical tool. Primus, 21, 530–544.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsEarlham CollegeRichmondUSA
  2. 2.Department of MathematicsTexas State UniversitySan MarcosUSA
  3. 3.Department of MathematicsAndrews UniversityBerrien SpringsUSA
  4. 4.College of EducationTemple UniversityPhiladelphiaUSA

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