The Korea Society Of Educational Studies In Mathematics

Current Issue

Journal of Educational Research in Mathematics - Vol. 30

[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.1-13
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020

Proof in Mathematics Education, 1980-2020: An Overview
Gila Hanna*, ; Christine Knipping**
*Professor, University of Toronto, Canada
**Professor, Universität Bremen, Germany

Correspondence to : Email:,

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Please cite this article as: Hanna, G. & Knipping, C. Proof in mathematics education, 1980-2020: An Overview.


This paper looks at the evolution of ideas on the role of proof in mathematics education from 1980 to 2020, examining in particular the contributions of both theoretical and empirical research to the teaching of mathematical proof. In so doing, it describes some of the major epistemological themes that emerged in the last forty years, primarily in the philosophy of mathematics and in mathematical education, and informed both the mathematics curriculum and research in mathematics education. The paper also discusses selected research studies that shed light on the opportunities and limitations students face when they engage in proof. Finally, it describes briefly a number of developments in proof technology and the potential of automated theorem provers for enhancing the teaching of proof.

Keywords: Proof, Reasoning and proof, Epistemological perspectives, Empirical research, Proof assistants


We are grateful to the reviewers for their helpful comments. We wish to acknowledge the generous support of the Social Sciences and Humanities Research Council of Canada.

1. Alcock, L., & Inglis, M. (2009). Representation systems and undergraduate proof productions: A comment on Weber. Journal of Mathematical Behavior, 28(4), 209-211.
2. Alcock, L., & Weber, K. (2010). Referential and syntactic approaches to proving: Case studies from a transition-to-proof course. Research in Collegiate Mathematics Education, 7, 101-123.
3. Atiyah, M. et al. (1994). ‘Responses to “Theoretical Mathematics”: Towards a cultural synthesis of mathematics and theoretical physics, by A. Jaffe and F. Quinn’, Bulletin of the American Mathematical Society, 30(2), 178-207.
4. Avigad, J., & Harrison, J. (2014). Formally verified mathematics. Communications of the ACM 57(4), 66-75.
5. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London: Hodder and Stoughton.
6. Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof. In A. J. Bishop, E. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 175-192). Dordrecht, Netherlands: Kluwer.
7. Beberman, M. (1958). An emerging program of secondary school mathematics. Cambridge: Harvard University Press.
8. Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183-203.
9. Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387.
10. Cobb, P., & Steffe, L. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83-94.
11. Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 2(1), 41-53.
12. Davis, P. J., & Hersh, R. (1983). The mathematical experience. Harmondsworth: Penguin Books.
13. Davydov, V. V. (1995). (Translated by Stephen T. Kerr). The influence of L. S. Vygotsky on education theory, research, and practice. Educational Researcher, 24(3), 12–21.
14. De Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. The International Journal of Mathematical Education in Science and Technology, 35(5), 703-724.
15. Dreyfus, T., & Hadas, N. (1996). Proof as an answer to the question why. Zentralblatt für Didaktik der Mathematik (ZDM), 28(1), 1-5.
16. Dunfield, W., & Thurston, W. (2003). The virtual Haken conjecture: experiments and examples. Geom. Topol. 7, 399-441.
17. Durand-Guerrier, V., Boero, P., Douek, N., Epp, S.S., & Tanguay, D. (2012). Examining the role of logic in teaching proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 369-389). Dordrecht: Springer.
18. Fischbein, E. (1982). Intuition and proof. For the learning of mathematics 3(2), 8-24.
19. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Kluwer.
20. Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38(1-3), 11-50.
21. Gowers, W. T. (2007). Mathematics, memory, and mental arithmetic, In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 33-58). Oxford, UK: Oxford University Press.
22. Grabiner, J. V. (1986). Is mathematical truth time-dependent? In T. Tymoczko (Ed.), New directions in the philosophy of mathematics: An anthology (pp. 201-213). Boston: Birkhäuser.
23. Gutiérrez, A., & Boero, P. (Eds.), (2006). Handbook of research on the psychology of mathematics education: Past, present and future. Rotterdam/Taipei: Sense Publishers.
24. Hadas, N., Herschkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environment. Educational Studies in Mathematics, 44(1&2), 127-150.
25. Hanna, G. (1983). Rigorous proof in mathematics education. Toronto: OISE Press.
26. Hanna, G. (1989a). More than formal proof. For the Learning of Mathematics, 9(1), 20-25.
27. Hanna, G. (1989b). Proofs that prove and proofs that explain. In Proceedings of the Thirteenth International Conference on the Psychology of Mathematics Education, Vol. 2, 45-52. Paris.
28. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
29. Hanna, G. (2003). Journals of mathematics education, 1900-2000. In Coray, D. and Furringhetti, F. (Eds.), One Hundred Years of L’Enseignement Mathématique : Moments of Mathematics Education in the 20th Century (pp. 67-84). Geneva: L’Enseignement Mathématique.
30. Hanna, G., & de Villiers, M. (Eds.), (2012). Proof and proving in mathematics education, New York, NY: Springer.
31. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In Bishop, A. et al (Eds.), International handbook of mathematics education (pp. 877-905). Dordrecht: Kluwer Academic Publishers.
32. Hanna, G., & Mason, J. (2014). Key ideas and memorability in proof. For the Learning of Mathematics, 34(2), 12-16.
33. Hanna, G., & Winchester, I. (Eds.), (1990). Creativity, thought and mathematical proof. Toronto: Ontario Institute for Studies in Education.
34. Harel, G. (2008). What is mathematics? A pedagogical answer to a philosophical question. In R. B. Gold, & R. Simons (Eds.), Proof and other dilemmas: Mathematics and philosophy (pp. 265-290). Washington, DC: Mathematical Association of America.
35. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education, III (pp. 234-283). Washington DC: Mathematical Association of America.
36. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.
37. Hemmi, K. (2008). Students’ encounter with proof: the condition of transparency. ZDM-– The International Journal on Mathematics Education, 40(3), 413-426.
38. Hohenwarter, M. (2002). GeoGebra - Ein Software system für dynamische Geometrie und Algebra der Ebene. Master thesis, University of Salzburg.
39. Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7-16.
40. Jaffe, A. & Quinn, F. (1993). “Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1) 1-13.
41. Jahnke, H. N. (2005). A genetic approach to proof. In M. Bosch (Ed.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp. 428-437). Sant Feliu de Guíxols.
42. Jahnke, H. N. (2007). Proofs and hypotheses. ZDM – The International Journal on Mathematics Education, 39(1–2), 79-86.
43. Jahnke, H. N., & Wambach, R. (2013). Understanding what a proof is: a classroom-based approach. ZDM – The International Journal on Mathematics Education 45(3), 4690-482.
44. Jones, K. (2000). Providing a foundation for a deductive reasoning: students’ interpretation when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1&2), 55-85.
45. Jones, K., Gutiérrez, A., & Mariotti, M. A. (Eds.), (2000). Proof in dynamic geometry environments. Educational Studies in Mathematics, 44(1-3). [Guest editorial for a PME special issue]
46. Kline, M. (1973). Why Johnny can't add: The failure of the New Math. New York: St. Martin's Press. ISBN 0-394-71981-6.
47. Laborde, J.-M. (1990). Cabri-Geometry [Software]. Université de Grenoble, France.
48. Lakatos, I. (1983). Proofs and refutations, 4th reprint. Cambridge: Cambridge University Press.
49. Lehrer, T. (1965). That was the year that was. Live album.
50. Lerman, S. (2019). Theoretical aspects of doing research in mathematics education: An argument for coherence. In Kaiser G., Presmeg N. (Eds.), Compendium for early career researchers in mathematics education (pp. 309-324). ICME-13 Monographs. Cham: Springer.
51. Maher, C. A., & Martino, A. M. (1996). The development of the idea of a mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194-214.
52. Mariotti, M. A. (2002). Influence of technologies advances on students' math learning. In L. English (Ed.), Handbook of International Research in Mathematics (pp. 695-724). New Jersey: Lawrence Erlbaum Associates.
53. Mariotti, M. A., Durand-Guerrier, V., Stylianides, G. J. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K. Ruthven (Eds.), Developing research in mathematics education. Twenty years of communication, cooperation and collaboration in Europe (pp. 75–89). Oxon, UK: Routledge.
54. Mariotti, M. A., & Pedemonte, B. (2019). Intuition and proof in the solution of conjecturing problems. ZDM-– The International Journal on Mathematics Education, 51(5), 759-777.
55. Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1), 87-125.
56. National Council of Teachers of Mathematics. (1963). An analysis of new mathematics programs. Washington, DC: NCTM.
57. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. [cit. 20150912]. Available from:
58. National Council of Teachers of Mathematics. (1980). An Agenda for Action: Recommendations for school mathematics of the 1980s. Reston, VA: NCTM.
59. Polya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
60. Polya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem-solving. New York, NY: John Wiley & Sons.
61. Polya, G. (1985). How to solve it: A new aspect of mathematical method. Princeton and Oxford: Princeton University Press.
62. Radford, L. & Roth, WM. (2017). Alienation in mathematics education: a problem considered from neo-Vygotskian approaches. Educational Studies in Mathematics, 96(3), 367–380.
63. Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319-325.
64. School Mathematics Study Group (SMSG). (1962). Grades 7-12 texts and commentaries. Yale: Yale University Press.
65. Selden, A., & Selden, J. (2013). Proof and problem solving at university level. Montana Mathematics Enthusiast, 10(1-2), 303-334.
66. Senk, S. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448-456.
67. Stanic, G. M. A., & Kilpatrick, J. (1988). Historical perspectives on problem solving in mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), Research agenda for mathematics education: The teaching and assessing of mathematical problem solving (pp. 1-22). Reston, VA: National Council of Teachers of Mathematics.
68. Stewart, I. N., & Tall, D. O. (2014). Foundations of mathematics, Second Edition. Oxford: OUP. ISBN: 9780198531654
69. Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. C., Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315-351). Rotterdam: Sense Publishers.
70. Stylianides, G., Stylianides, A., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237-266). Reston, VA: National Council of Teachers of Mathematics.
71. Tall, D. (2013). How humans learn to think mathematically. New York: Cambridge University Press. ISBN: 9781139565202.
72. Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y.-H. (2012). The cognitive development of proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 13-49). New York, NY: Springer.
73. Thurston, W. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161-177.
74. Thurston, W. (1995). On Proof and Progress in Mathematics. For the Learning of Mathematics, 15(1), 29-37.
75. Vergnaud, G. (1987). About constructivism. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference.
76. Voevodsky, V. (2014). The origins and motivations of univalent foundations. The Institute Letter, 8-9.
77. Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge, MA: Harvard University Press.
78. Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2), 209-234.
79. Weber, K., & Mejia-Ramos, J.P. (2009). An alternative framework to evaluate proof productions: A reply to Alcock and Inglis. Journal of Mathematical Behavior, 28(4), 212-216.
80. Wiedijk, F. (2008). Formal proof – getting started. Notices of the American Mathematical Society, 55(11), 1408-1414.
81. Williams, J. (2016). Alienation in mathematics education: critique and development of neo-Vygotskian perspectives. Educational Studies in Mathematics, 92(1), 59-73.
82. Yackel, E., & Cobb, P. (1994). The development of young children’s understanding of mathematical argumentation. Paper presented at the annual meeting of the American Educational Research Association (AERA), New Orleans.
83. Yan, X. (2017). Key ideas in proof in undergraduate mathematics classrooms. (Unpublished doctoral dissertation.) University of Toronto, Toronto, Canada.
84. 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.