# Why and how mathematicians read proofs: an exploratory study

- 792 Downloads
- 29 Citations

## Abstract

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, *Journal for Research in Mathematics Education* 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.

## Keywords

Advanced mathematical thinking Argumentation Mathematical practice Proof Proof comprehension Proof reading Validation## Notes

### Acknowledgments

The research in this paper was supported by a grant from the National Science Foundation (NSF #DRL0643734). The views expressed here are not necessarily those of the National Science Foundation. We would like to thank Sean Larsen for helpful insights on developing our theoretical model, as well as the anonymous reviewers for their comments on earlier drafts of this manuscript.

## References

- Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and evaluating warrants.
*Journal of Mathematical Behavior, 24*(2), 125–134.CrossRefGoogle 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*. Rotterdam: Sense Publishers.Google Scholar - Balacheff, N. (2002). The researcher epistemology: A deadlock from educational research on proof. In F. L. Lin (Ed.),
*2002 International conference on mathematics education- understanding proving and proving to understand*. Taipei: NCS and NUST.Google Scholar - Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education.
*International Newsletter on the Teaching and Learning of Mathematical Proof.*Retrieved August 2, 2010, from http://www.lettredelapreuve.it/OldPreuve/Newsletter/990708Theme/990708ThemeUK.html - Davis, P. (1972). Fidelity in mathematical discourse: Is one and one really two?
*American Mathematical Monthly, 79*, 252–263.CrossRefGoogle Scholar - De Milo, R., Liptus, A., & Perlis, A. (1979). Social processes and proofs of theorems and programs.
*Communications of the ACM, 22*, 271–280.CrossRefGoogle Scholar - de Villiers, M. D. (1990). The role and function of proof in mathematics.
*Pythagoras, 24*, 17–24.Google Scholar - Duval, R. (2007). Cognitive functioning and the understanding of the mathematical processes of proof. In P. Boero (Ed.),
*Theorems in schools: From history, epistemology, and cognition to classroom practice*. Rotterdam: Sense Publishers.Google Scholar - Fischbein, E. (1982). Intuition and proof.
*For the Learning of Mathematics, 3*(2), 9–18.Google Scholar - Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.),
*Advanced mathematical thinking*. Dordrecht: Kluwer.Google Scholar - Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge.
*ZDM, 40*, 345–353.CrossRefGoogle Scholar - Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.),
*Learning and teaching number theory*. New Jersey: Ablex Publishing Company.Google Scholar - Harel, G., & Sowder, L. (1998). Students proof schemes.
*Research in Collegiate Mathematics Education, 3*, 234–282.Google Scholar - Harel, G., & Sowder, L. (2007). Towards a comprehensive perspective on proof. In F. Lester (Ed.),
*Second handbook of research on mathematical teaching and learning*. Washington, DC: NCTM.Google Scholar - Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra.
*Journal for Research in Mathematics Education, 31*, 396–428.CrossRefGoogle 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 - Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification.
*Educational Studies in Mathematics, 66*, 3–21.CrossRefGoogle Scholar - Jackson, A. (2006). Conjectures no more? Consensus forming on the proof of the Poincare and geometrization conjectures.
*Notices of the American Mathematical Society, 53*, 897–901.Google Scholar - Konoir, J. (1993). Research into the construction of mathematical texts.
*Educational Studies in Mathematics, 24*, 251–256.CrossRefGoogle Scholar - Kreisel, G. (1985). Mathematical logic: Tool and object lesson for science.
*Synthese, 62*, 139–151.CrossRefGoogle Scholar - Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching.
*American Education Research Journal, 27*, 29–63.Google Scholar - Maher, C., Muter, E., & Kiczek, G. (2007). The development of proof making by students. In P. Boero (Ed.),
*Theorems in schools*. Rotterdam: Sense Publishers.Google Scholar - Manin, Y. (1977).
*A course in mathematical logic*. New York: Springer Verlag.Google Scholar - Martin, W. G., & Harel, G. (1989). Proof frames of pre-service elementary teachers.
*Journal for Research in Mathematics Education, 20*(1), 41–51.CrossRefGoogle Scholar - Mejia-Ramos, J.P. (2008).
*The construction and evaluation of arguments in undergraduate mathematics: A theoretical and a longitudinal multiple-case study*. Unpublished Ph.D. dissertation. University of Warwick, U.K.Google Scholar - Otte, M. (1994). Mathematical knowledge and the problem of proof.
*Educational Studies in Mathematics, 26*, 299–321.CrossRefGoogle Scholar - RAND Mathematics Study Panel. (2003).
*Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education*. Santa Monica: RAND corporation.Google Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*, 5–41.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 - Stylianides, A. & Stylianides, G. (2008). Enhancing undergraduates students’ understanding of proof. In
*Proceedings of the 11th Conference for Research in Undergraduate Mathematics Education*. Last downloaded March 18, 2009: http://www.rume.org/crume2008/Stylianides_LONG(21).pdf. - Thurston, W. P. (1994). On proof and progress in mathematics.
*Bulletin of the American Mathematical Society, 30*, 161–177.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