Proofs as Bearers of Mathematical Knowledge



Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999; 7: 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.


Quadratic Equation Mathematical Knowledge Mathematics Curriculum Isosceles Triangle Central Thesis 



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.

A previous version appeared in ZDM, The International Journal on Mathematics Education 2008;40(3):345–353. It is reproduced by permission from Springer.


  1. Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.CrossRefGoogle Scholar
  2. Balacheff, N. (2004). The researcher epistemology: a deadlock from educational research on proof. Retrieved April 2007 from
  3. Barbeau, E. (1988). Which method is best? Mathematics Teacher, 81, 87–90.Google Scholar
  4. Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23–40.CrossRefGoogle Scholar
  5. Bressoud, D. M. (1999). Proofs and confirmations: The story of the alternating sign matrix conjecture. Cambridge: Cambridge University Press.Google Scholar
  6. Bressoud, D., & Propp, J. (1999). How the alternating sign matrix conjecture was solved. Notices of the AMS, 46(6), 637–646.Google Scholar
  7. Corfield, D. (2003). Towards a Philosophy of Real Mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  8. Dawson, J. W. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 14, 269–286.CrossRefGoogle Scholar
  9. DeVilliers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 7–24.Google Scholar
  10. Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–18. 24.Google Scholar
  11. Fischbein, E. (1999). Intuition and schemata in mathematical reasoning. Educational Studies in Mathematics, 38(1–3), 11–50.CrossRefGoogle Scholar
  12. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.CrossRefGoogle Scholar
  13. Hanna, G. (1997). The ongoing value of proof in mathematics education. Journal für Mathematik Didaktik, 97(2/3), 171–185.Google Scholar
  14. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23. Special issue on “Proof in Dynamic Geometry Environments”.CrossRefGoogle Scholar
  15. Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7–16.Google Scholar
  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.CrossRefGoogle Scholar
  17. Jahnke, H. N. (2007). Proofs and hypotheses. ZDM. The International Journal on Mathematics Education, 39(1–2), 79–86.CrossRefGoogle Scholar
  18. Jones, K., Gutiérrez, A., & Mariotti, M. A. (Eds.) (2000). Proof in dynamic geometry environments. Educational Studies in Mathematics, 44(1–2); Special issue.Google Scholar
  19. Lucast, E. (2003). Proof as method: A new case for proof in mathematics curricula. Unpublished masters thesis, Carnegie Mellon University, Pittsburgh, PA.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. 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/Taipei: Sense Publishers.Google Scholar
  22. 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.Google Scholar
  23. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.CrossRefGoogle Scholar
  24. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.Google Scholar
  25. Reiss, K., & Renkl, A. (2002). Learning to prove: The idea of heuristic examples. Zentralblatt für Didaktikt der Mathematik, 34, 29–35.CrossRefGoogle Scholar
  26. Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123–151.CrossRefGoogle Scholar
  27. 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.CrossRefGoogle Scholar
  28. Tall, D. (1998). The cognitive development of proof: Is mathematical proof for all or for some. Paper presented at the UCSMP Conference, Chicago.Google Scholar
  29. 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, VA: National Council of Teachers of Mathematics.Google Scholar

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Authors and Affiliations

  1. 1.Department of Curriculum, Teaching and LearningOntario Institute for Studies in Education of the University of TorontoTorontoCanada

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