Why and how mathematicians read proofs: an exploratory study
- 751 Downloads
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.
KeywordsAdvanced mathematical thinking Argumentation Mathematical practice Proof Proof comprehension Proof reading Validation
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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.
- Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39, 431–459.Google Scholar