# Student Interpretations of Written Comments on Graded Proofs

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## Abstract

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.

### Keywords

Proof writing Proof grading Proof instruction Proof revision Student thinking## Supplementary material

40753_2017_59_MOESM1_ESM.pdf (1.9 mb)

### References

- Alcock, L. (2013).
*How to study as a mathematics major*. Oxford: Oxford University Press.Google Scholar - Alexander, D. S., & DeAlba, L. M. (1997). Groups for proofs: Collaborative learning in a mathematics reasoning course.
*Primus, 7*, 193–207.CrossRefGoogle Scholar - 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 - Chartrand, G., Polimeni, A. D., & Zhang, P. (2012).
*Mathematical proofs: A transition to advanced mathematics*(3rd ed.). Boston: Pearson.Google Scholar - Chierchia, G., & McConnell-Ginet, S. (2000).
*Meaning and grammar: An introduction to semantics*(2nd ed.). Cambridge: MIT Press.Google Scholar - Cupillari, A. (2013).
*The nuts and bolts of proof: An introduction to mathematical proofs*(4th ed.). Waltham: Academic Press.Google Scholar - De Villiers, M. (1990). The role and function of proof in mathematics.
*Pythagoras, 24*, 7–24 Retrieved from http://mzone.mweb.co.za/residents/profmd/proofa.pdf.Google Scholar - De Villiers, M. (1999).
*Rethinking proof with sketchpad*. Oakland: Key Curriculum Press.Google Scholar - 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 - Epp, S. S. (2003). The role of logic in teaching proof.
*American Mathematical Monthly, 110*, 886–899.CrossRefGoogle Scholar - Franklin, J., & Daoud, A. (2011).
*Proof in mathematics: An introduction*. Sydney: Kew Books.Google Scholar - Fukawa-Connelly, T. (2005). Thoughts on learning advanced mathematics.
*For the Learning of Mathematics, 25*, 33–35.Google Scholar - 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 - 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 - Halliday, M. A. K. (1978).
*Language as a social semiotic: The social interpretation of language and meaning*. Baltimore: University Park Press.Google Scholar - 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 - Hattie, J., & Timperley, H. (2007). The power of feedback.
*Review of Educational Research, 77*, 81–112.CrossRefGoogle Scholar - Herschensohn, J., & Young-Scholten, M. (Eds.). (2013).
*The Cambridge handbook for second language acquisition*. Cambridge: Cambridge University Press.Google Scholar - Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs.
*Journal for Research in Mathematics Education, 43*, 358–390.CrossRefGoogle Scholar - 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 - 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 - Lakatos, I. (1976).
*Proofs and refutations*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 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 - 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 - Lyster, R., & Ranta, L. (1997). Corrective feedback and learner uptake.
*Studies in Second Language Acquisition, 19*, 37–66.CrossRefGoogle Scholar - 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 http://sigmaa.maa.org/rume/RUME_XIV_Proceedings_Volume_2.pdf.Google Scholar - Moore, R. C. (1994). Making the transition to formal proof.
*Educational Studies in Mathematics, 27*, 249–266.CrossRefGoogle Scholar - 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 - Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions.
*For the Learning of Mathematics, 19*(1), 11–19.Google Scholar - Pimm, D. (1987).
*Speaking mathematically: Communication in mathematics classrooms*. New York: Routledge & K. Paul.Google Scholar - Pinker, S. (2009).
*Language learnability and language development, with new commentary by the author*(Vol. 7). Cambridge: Harvard University Press.Google Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*(1), 5–41.CrossRefGoogle Scholar - Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: A research review.
*Reading and Writing Quarterly, 23*, 139–159.CrossRefGoogle Scholar - 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 - 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 - Smith, D. D., Eggen, M., & St. Andre, R. (2014).
*A transition to advanced mathematics*(8th ed.) Boston: Cengage Learning.Google Scholar - 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 - 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 - 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 - Tedick, D. J., & de Gortari, B. (1998). Research on error correction and implications for classroom teaching.
*ACIE Newsletter, 1*(3), 1–6.Google Scholar - Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge.
*Educational Studies in Mathematics, 48*, 101–119.CrossRefGoogle Scholar - Weber, K. (2006). Investigating and teaching the processes used to construct proofs.
*Research in Collegiate Mathematics Education, 6*, 197–232.CrossRefGoogle Scholar - 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 - 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

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