Educational Studies in Mathematics

, Volume 95, Issue 2, pp 123–142 | Cite as

Values and norms of proof for mathematicians and students



In this theoretical paper, we present a framework for conceptualizing proof in terms of mathematical values, as well as the norms that uphold those values. In particular, proofs adhere to the values of establishing a priori truth, employing decontextualized reasoning, increasing mathematical understanding, and maintaining consistent standards for acceptable reasoning across domains. We further argue that students’ acceptance of these values may be integral to their apprenticeship into proving practice; students who do not perceive or accept these values will likely have difficulty adhering to the norms that uphold them and hence will find proof confusing and problematic. We discuss the implications of mathematical values and norms with respect to proof for investigating mathematical practice, conducting research in mathematics education, and teaching proof in mathematics classrooms.


Proof Values Norms Mathematical culture 


  1. Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. In F. Hitt, D. Holton, & P. Thompson (Eds.), Issues in mathematics education: Vol. 16. Research in collegiate mathematics education VII (pp. 63–91). Providence, RI: American Mathematical Society.Google Scholar
  2. Alcock, L., & Simpson, A. (2002). Definitions: Dealing with categories mathematically. For the Learning of Mathematics, 22(2), 28–34.Google Scholar
  3. Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24(2), 125–134.CrossRefGoogle Scholar
  4. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London, UK: Hodder & Stoughton.Google Scholar
  5. Ball, D. L., & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.), Yearbook of the national society for the study of education, constructivism in education (pp. 193–224). Chicago, IL: University of Chicago Press.Google Scholar
  6. Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351–382.Google Scholar
  7. Bieda, K. N., & Lepak, J. (2014). Are you convinced? Middle-grade students’ evaluations of mathematical arguments. School Science and Mathematics, 114(4), 166–177.CrossRefGoogle Scholar
  8. Borwein, J. (2008). Implications of experimental mathematics for the philosophy of mathematics. In B. Gold & R. Simons (Eds.), Proof & other dilemmas: Mathematics and philosophy (pp. 33–60). Washington, DC: Mathematics Association of America.Google Scholar
  9. Brown, S. A. (2014). On skepticism and its role in the development of proof in the classroom. Educational Studies in Mathematics, 86(3), 311–335.CrossRefGoogle Scholar
  10. Buldt, B., Löwe, B., & Müller, T. (2008). Towards a new epistemology of mathematics. Erkenntnis, 68(3), 309–329.CrossRefGoogle Scholar
  11. Burton, L., & Morgan, C. (2000). Mathematicians writing. Journal for Research in Mathematics Education, 31(4), 429–453.CrossRefGoogle Scholar
  12. Carlson, M. P., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58(1), 45–75.CrossRefGoogle Scholar
  13. Chinn, C. A., Buckland, L., & Samarapungavan, A. (2011). Expanding the dimensions of epistemic cognition: Arguments from philosophy and psychology. Educational Psychologist, 46(3), 141–167.CrossRefGoogle Scholar
  14. Cirillo, M., Kosko, K., Newton, J., Staples, M., & Weber, K. (2015). Conceptions and consequences of what we call argumentation, justification, and proof. In T. Bartell, K. Bieda, R. T. Putnam, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the Psychology of Mathematics Education (pp. 1343–1351). East Lansing, MI: Michigan State University.Google Scholar
  15. Common Core State Standards for Mathematics. (2012). Standards for mathematical practice. Retrieved from
  16. Davis, P., & Hersh, R. (1981). The mathematical experience. Boston: Houghton-Mifflin.Google Scholar
  17. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24(1), 17–24.Google Scholar
  18. De Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. International Journal of Mathematical Education in Science and Technology, 35(5), 703–724.CrossRefGoogle Scholar
  19. DeBellis, V. A., & Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131–147.CrossRefGoogle Scholar
  20. Department of Education (2013). Mathematics: Programmes of study: Key Stages 1–2 (National Curriculum in England). Retrieved from
  21. Devlin, K. (2003). The shame of it. Devlin’s Angle. Retrieved from
  22. Douek, N. (2009). Approaching proof in school: From guided conjecturing and proving to a story of proof construction. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), ICMI study 19: Proof and proving in mathematics education (Vol. 1, pp. 148–153). Taipei, Taiwan: National Taiwan Normal University.Google Scholar
  23. Dowling, P. (1998). The sociology of mathematics education: Mathematical myths/pedagogic texts. London: Falmer Press.Google Scholar
  24. Durand-Guerrier, V. (2008). Truth versus validity in mathematical proof. ZDM, 40(3), 373–384.CrossRefGoogle Scholar
  25. Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2012). Argumentation and proof in the mathematics classroom. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 349–367). Dordrecht, The Netherlands: Springer.Google Scholar
  26. Edwards, B., & Ward, M. (2008). The role of mathematical definitions in mathematics and in undergraduate mathematics courses. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education MAA notes #73 (pp. 223–232). Washington, DC: Mathematics Association of America.CrossRefGoogle Scholar
  27. Epp, S. S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110(10), 886–899.CrossRefGoogle Scholar
  28. Ernest, P. (1991). The philosophy of mathematics education. London: Routledge.Google Scholar
  29. Fallis, D. (2002). What do mathematicians want? Probabilistic proofs and the epistemic goals of mathematicians. Logique et Analyse, 45(179–180), 373–388.Google Scholar
  30. Grcar, J. F. (2013). Errors and corrections in mathematics literature. Notices of the American Mathematical Society, 60(4), 418–425.CrossRefGoogle Scholar
  31. Hales, T. (2003). The Flyspeck Project Fact Sheet. Retrieved from
  32. Halmos, P. R. (1970). How to write mathematics. L’Enseignement Mathématique, 16(2), 123–152.Google Scholar
  33. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.CrossRefGoogle Scholar
  34. Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42–49.Google Scholar
  35. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Issues in mathematics education: Vol. 7. Research in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.Google Scholar
  36. Harel, G., & Sowder, L. (2009). College instructors’ views of students vis-à-vis proof. In M. Blanton, D. Stylianou, & E. Knuth (Eds.), Teaching and learning proof across the grades (pp. 275–289). New York: Routledge.Google Scholar
  37. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.CrossRefGoogle Scholar
  38. Herbst, P., & Balacheff, N. (2009). Proving and knowing in public: The nature of proof in a classroom. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades (pp. 40–64). New York, NY: Routledge.Google Scholar
  39. Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73–122.CrossRefGoogle Scholar
  40. Herbst, P., Nachlieli, T., & Chazan, D. (2011). Studying the practical rationality of mathematics teaching: What goes into “installing” a theorem in geometry? Cognition and Instruction, 29(2), 218–255.CrossRefGoogle Scholar
  41. Holland, D., Lachicotte, W. J., Skinner, D., & Cain, C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press.Google Scholar
  42. Inglis, M., & Aberdein, A. (2016). Diversity in proof appraisals. In B. Larvor (Ed.), Mathematics cultures (pp. 163–179). Dordrecht: Springer.Google Scholar
  43. Knuth, D., Larrabee, T., & Roberts, P. (1989). Mathematical writing. Cambridge, UK: Cambridge University Press.Google Scholar
  44. Ko, Y. Y., & Knuth, E. (2009). Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions. The Journal of Mathematical Behavior, 28(1), 68–77.CrossRefGoogle Scholar
  45. Konoir, J. (1993). Research into the construction of mathematical texts. Educational Studies in Mathematics, 24(3), 251–256.CrossRefGoogle Scholar
  46. Kuhn, T. S. (1962). The structure of scientific revolutions. Chicago, IL, USA: University of Chicago Press.Google Scholar
  47. Lai, Y., & Weber, K. (2014). Factors mathematicians profess to consider when presenting pedagogical proofs. Educational Studies in Mathematics, 85(1), 93–108.CrossRefGoogle Scholar
  48. Lai, Y., Weber, K., & Mejía-Ramos, J. P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof. Cognition and Instruction, 30(2), 146–169.CrossRefGoogle Scholar
  49. Laudan, L. (1984). Science and values. Cambridge, UK: Cambridge University Press.Google Scholar
  50. Lew, K. (2016). Conventions of the language of mathematical proof writing at the undergraduate level. (Doctoral dissertation). Retrieved from ProQuest Dissertations and Theses Database.Google Scholar
  51. Livingston, E. (2006). The context of proving. Social Studies of Science, 36(1), 39–68.CrossRefGoogle Scholar
  52. Maddy, P. (1997). Naturalism in mathematics. Oxford: Clarendon.Google Scholar
  53. Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24(3), 385–401.CrossRefGoogle Scholar
  54. Mamona-Downs, J., & Downs, M. (2010). Necessary realignments from mental argumentation to proof presentation. In V. Durrand-Gurrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2337–2345). Lyon: Institute National de Recherche Pedagogique.Google Scholar
  55. Manin, Y. I. (1998). Truth, rigour, and common sense. In H. G. Dales & G. Oliveri (Eds.), Truth in mathematics (pp. 147–159). New York, NY: Oxford University Press.Google Scholar
  56. Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
  57. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266.CrossRefGoogle Scholar
  58. National Council of Teacher of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  59. Paseau, A. (2011). Mathematical instrumentalism, Gödel’s theorem, and inductive evidence. Studies in History and Philosophy of Science Part A, 42(1), 140–149.CrossRefGoogle Scholar
  60. Paseau, A. (2015). Knowledge in mathematics without proof. British Journal for the Philosophy of Science, 66(4), 775–799.CrossRefGoogle Scholar
  61. Quine, W. V. O. (1951). Two dogmas of empiricism. Philosophical Review, 60(1), 20–43.CrossRefGoogle Scholar
  62. Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.CrossRefGoogle Scholar
  63. Saxe, K., & Braddy, L. (2015). A common vision for undergraduate mathematical sciences programs in 2025. Washington, DC: Mathematical Association of America.Google Scholar
  64. Segal, J. (1999). Learning about mathematical proof: Conviction and validity. The Journal of Mathematical Behavior, 18(2), 191–210.CrossRefGoogle Scholar
  65. Selden, A., & Selden, J. (2003). Validation of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4–36.CrossRefGoogle Scholar
  66. Selden, A., & Selden, J. (2013). The genre of proof. In M. Fried & T. Dreyfus (Eds.), Mathematicians and mathematics educators: Searching for common ground (pp. 237–260). New York, NY: Springer.Google Scholar
  67. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.Google Scholar
  68. Solomon, Y. (2006). Deficit or difference? The role of students’ epistemologies of mathematics in their interactions with proof. Educational Studies in Mathematics, 61(3), 373–393.CrossRefGoogle Scholar
  69. Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447–462.CrossRefGoogle Scholar
  70. Steiner, M. (1978). Mathematical explanations. Philosophical Studies, 34(2), 135–151.CrossRefGoogle Scholar
  71. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.Google Scholar
  72. Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutierez, G. Leder, & P. Boero (Eds.), 2nd handbook on the psychology of mathematics education (pp. 315–351). Rotterdam: Sense Publishers.Google Scholar
  73. Stylianides, G., & Stylianides, A. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314–352.Google Scholar
  74. Stylianides, G., Stylianides, A. & Weber, K. (in press). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  75. Swart, E. R. (1980). The philosophical implications of the four-color problem. American Mathematical Monthly, 87(9), 697–707.CrossRefGoogle Scholar
  76. Tannen, D. (1984). Conversational style: Analyzing talk among friends. New York, NY: Oxford University Press.Google Scholar
  77. Tao, T. (2007). What is good mathematics? Bulletin of the American Mathematical Society, 44(4), 623–634.CrossRefGoogle Scholar
  78. Taylor, C. (1993). To follow a rule…. In C. Calhoun, E. LiPuma, & M. Postone (Eds.), Bourdieu: Critical perspectives (pp. 45–60). Chicago, IL: University of Chicago Press.Google Scholar
  79. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar
  80. Tymoczko, T. (1979). The four-color problem and its philosophical significance. The Journal of Philosophy, 76(2), 57–83.CrossRefGoogle Scholar
  81. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.CrossRefGoogle Scholar
  82. Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12(4), 306–336.CrossRefGoogle Scholar
  83. Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. International Journal of Mathematical Education in Science and Technology, 43(4), 463–482.CrossRefGoogle Scholar
  84. Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.CrossRefGoogle Scholar
  85. Weber, K., Inglis, M., & Mejia-Ramos, J. P. (2014). How mathematicians obtain conviction: Implications for mathematics instruction and research on epistemic cognition. Educational Psychologist, 49, 36–58.CrossRefGoogle Scholar
  86. Weber, K., & Mejia-Ramos, J. P. (2013). 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
  87. Weber, K., & Mejia-Ramos, J. P. (2015). The contextual nature of conviction in mathematics. For the Learning of Mathematics, 35(2), 9–14.Google Scholar
  88. Worrall, J. (1988). The value of a fixed methodology. The British Journal for the Philosophy of Science, 39(2), 263–275.CrossRefGoogle Scholar
  89. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.CrossRefGoogle Scholar
  90. Zazkis, D., Weber, K., & Mejia-Ramos, P. (2016). Bridging the gap between visual arguments and verbal-symbolic proofs in calculus. Educational Studies in Mathematics, 93(2), 155–173.CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA
  2. 2.Graduate School of EducationRutgers UniversityNew BrunswickUSA

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