# Why and how mathematicians read proofs: further evidence from a survey study

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## Abstract

In a recent paper (Weber & Mejia-Ramos, *Educational Studies in Mathematics*, 76, 329–344, 2011), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians (*N* = 118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians.

### Keywords

Mathematical practice Proof Proof reading### References

- Afflerbach, P., & Johnston, P. (1984). On the use of verbal reports in reading research.
*Journal of Literacy Research, 16*(4), 307–322.CrossRefGoogle Scholar - Auslander, J. (2008). On the roles of proof in mathematics. In B. Gold and R. A. Simons (Eds.),
*Proof & other dilemmas*:*Mathematics and philosophy*(pp. 61–77). Mathematical Association of America.Google Scholar - Arzarello, F. (2007). The proof in the 20th century: From Hilbert to automatic theorem proving. In P. Boero (Ed.),
*Theorems in schools: From history, epistemology, and cognition to classroom practice*(pp. 43–63). Rotterdam: Sense Publishers.Google Scholar - Burton, L. (2002). Recognising commonalities and reconciling differences in mathematics education.
*Educational Studies in Mathematics, 50*(2), 157–175.CrossRefGoogle Scholar - Burton, L. (2004).
*Mathematicians as enquirers: Learning about learning mathematics*. Dordrecht: Kluwer.CrossRefGoogle Scholar - Burton, L. (2009). The culture of mathematics and the mathematical culture. In O. Skovsmose, P. Valero, & O. R. Christensen (Eds.),
*University science and mathematics education in transition*(pp. 157–173). New York: Springer.Google Scholar - Davis, P. (1972). Fidelity in mathematical discourse: Is one and one really two?
*American Mathematical Monthly, 79*, 252–263.CrossRefGoogle Scholar - Ericsson, K. A., & Simon, H. S. (1980). Verbal reports as data.
*Psychological Review, 87*(3), 215–251.CrossRefGoogle Scholar - Fallis, D. (2003). Intentional gaps in mathematical proofs.
*Synthese, 134*, 45–69.CrossRefGoogle Scholar - Geist, C., Löwe, B., & Van Kerkhove, B. (2010). Peer review and knowledge by testimony in mathematics. In B. Löwe & T. Müller (Eds.),
*Philosophy of mathematics: Sociological aspects and mathematical practice*(pp. 155–178). London: College Publications.Google Scholar - Gosling, S. D., Vazire, S., Srivastava, S., & John, O. P. (2004). Should we trust web-based studies? A comparative analysis of six preconceptions about internet questionnaires.
*American Psychologist, 59*, 93–104.Google Scholar - Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.),
*Advanced mathematical thinking*. Dordrecht: Kluwer.Google Scholar - Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa).
*American Mathematical Monthly, 105*, 497–507.CrossRefGoogle Scholar - Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. Lester (Ed.),
*Second handbook of research on mathematics education*(pp. 805–842). Charlotte, NC: Information Age Publishing.Google Scholar - Heinze, A. (2010). Mathematicians’ individual criteria for accepting theorems as proofs: An empirical approach. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.),
*Explanation and proof in mathematics: Philosophical and educational perspectives*(pp. 101–111). New York: Springer.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 - Inglis, M., & Alcock, L. (2013). Skimming: A response to Weber and Mejia-Ramos.
*Journal for Research in Mathematics Education, 44*, 471–474.Google Scholar - Inglis, M., & Mejia-Ramos, J. P. (2009). The effect of authority on the persuasiveness of mathematical arguments.
*Cognition and Instruction, 27*, 25–50.CrossRefGoogle Scholar - Johnson-Laird, P. N., & Savary, F. (1999). Illusory inferences: A novel class of erroneous deductions.
*Cognition, 71*, 191–229.CrossRefGoogle Scholar - Krantz, J. H., & Dalal, R. (2000). Validity of web-based psychological research. In M. H. Birnbaum (Ed.),
*Psychological experiments on the Internet*(pp. 35–60). San Diego: Academic Press.CrossRefGoogle Scholar - Mejia-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research, in F.-L. Lin, F.-J. Hsieh, G. Hanna, and 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 - Moschkovich, J. N. (2002). An introduction to examining everyday and academic mathematical practices. In J. N. Moschkovich & M. E. Brenner (Eds.),
*Everyday and Academic Mathematics in the Classroom, Journal for Research in Mathematics Education Monograph (Vol. 11, pp.1-11)*. Reston, VA: NCTM.Google Scholar - Nathanson, M. (2008). Desperately seeking mathematical truth.
*Notices of the American Mathematical Society, 55*(7), 773.Google Scholar - Nisbett, R. E., & Wilson, T. D. (1977). Telling more than we can know: Verbal reports on mental processes.
*Psychological Review, 84*, 231–259.CrossRefGoogle Scholar - Porteous, K. (1986).
*Children’s appreciation of the significance of proof*(In*Proceedings of the tenth international conference for the psychology of mathematics education*(pp. 392–397)). London: England.Google Scholar - RAND Mathematics Study Panel. (2003).
*Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education*. Santa Monica, CA: RAND Corporation.Google Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*, 5–41.CrossRefGoogle Scholar - Reips, U. D. (2000). The web experiment method: Advantages, disadvantages, and solutions. In M. H. Birnbaum (Ed.),
*Psychological experiments on the Internet*(pp. 89–117). San Diego: Academic Press.CrossRefGoogle Scholar - 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, CT: Ablex Publishing.Google Scholar - Selden, A., & Selden, J. (2003). Validations of proofs written as texts: Can undergraduates tell whether an argument proves a theorem?
*Journal for Research in Mathematics Education, 36*(1), 4–36.CrossRefGoogle Scholar - Shanahan, C., Shanahan, T., & Misischia, C. (2011). Analysis of expert readers in three disciplines: History, mathematics, and chemistry.
*Journal of Literacy Research, 43*, 393–429.CrossRefGoogle Scholar - Weber, K. (2008). How mathematicians determine if an argument is a valid proof.
*Journal for Research in Mathematics Education, 39*, 431–459.Google Scholar - Weber, K. (2009). Mathematics majors’ evaluation of mathematical arguments and their conception of proof. In
*Proceedings of the 12th Conference for Research in Undergraduate Mathematics Education*. Available for download from: http://sigmaa.maa.org/rume/crume2009/proceedings.html - Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity, and proof.
*Mathematical Thinking and Learning, 12*, 306–336.CrossRefGoogle Scholar - Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study.
*Educational Studies in Mathematics, 76*, 329–344.CrossRefGoogle Scholar - Weber, K., & Mejia-Ramos, J. P. (2013a). On mathematicians’ proof skimming: A reply to Inglis and Alcock.
*Journal for Research in Mathematics Education, 44*, 464–471.CrossRefGoogle Scholar - Weber, K., & Mejia-Ramos, J. P. (2013b). The influence of sources in the reading of mathematical text: A reply to Shanahan, Shanahan, and Misischia.
*Journal of Literacy Research, 45*(1), 87–96.CrossRefGoogle Scholar