Abstract
Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.
Balacheff, N. (2004). The researcher epistemology: a deadlock from educational research on proof. Retrieved April 2007 from http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahiers2004/Cahier2004.xhtml.
Barbeau, E. (1988). Which method is best? Mathematics Teacher, 81, 87–90.
Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23–40.
Bressoud, D.M. (1999). Proofs and confirmations: The story of the alternating sign matrix conjecture. Cambridge: Cambridge University Press.
Bressoud, D., & Propp, J. (1999). How the alternating sign matrix conjecture was solved. Notices of the AMS, 46(6), 637–646.
Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.
Dawson, J.W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 14, 269–286.
DeVilliers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 7–24.
Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–18, 24.
Fischbein, E. (1999). Intuition and schemata in mathematical reasoning. Educational Studies in Mathematics, 38(1–3), 11–50.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Hanna, G. (1997). The ongoing value of proof in mathematics education. Journal für Mathematik Didaktik, 2/3(97), 171–185.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, Special issue on Proof in Dynamic Geometry Environments, 44(1–2), 5–23.
Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7–16.
Inglis, M., Mejia-Ramos, J.P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3–21.
Jahnke, H.N. (2007). Proofs and hypotheses. ZDM The International Journal on Mathematics Education, 39(1–2), 79–86.
Jones, K., Gutiérrez, A., & Mariotti, M.A. (Eds.) (2000). Proof in dynamic geometry environments. Special issue. Educational Studies in Mathematics, 44(1–2).
Lucast, E. (2003). Proof as method: A new case for proof in mathematics curricula. Unpublished masters thesis. Pittsburgh, PA, USA: Carnegie Mellon University.
Maher, C.A., & Martino, A.M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.
Mariotti, A. (2006). Proof and proving in mathematics education. In A. Gutiérrez, & P. Boero, (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam: Sense Publishers.
Moreno-Armella, L., & Sriraman, B. (2005). Structural stability and dynamic geometry: Some ideas on situated proof. International Reviews on Mathematical Education, 37(3), 130–139.
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.
Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples. Zentralblatt für Didaktikt der Mathematik, 34, 29–35.
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123–151.
Sowder, L., & Harel, G. (2003). Case studies of mathematics majors’ proof understanding, production, and appreciation. Canadian Journal of Science Mathematics and Technology Education, 3(2), 251–267.
Tall, D. (1998). The cognitive development of proof: Is mathematical proof for all or for some. Paper presented at the UCSMP conference. Chicago, USA.
Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston: National Council of Teachers of Mathematics.
Acknowledgments
Preparation of this paper was supported in part by the Social Sciences and Humanities Research Council of Canada. We are grateful to Ella Kaye and Ysbrand DeBruyn for their assistance. We wish to thank the anonymous reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hanna, G., Barbeau, E. Proofs as bearers of mathematical knowledge. ZDM Mathematics Education 40, 345–353 (2008). https://doi.org/10.1007/s11858-008-0080-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-008-0080-5