Advertisement

ZDM

, Volume 51, Issue 5, pp 717–730 | Cite as

Acceptance criteria for validating mathematical proofs used by school students, university students, and mathematicians in the context of teaching

  • Daniel SommerhoffEmail author
  • Stefan Ufer
Original Article

Abstract

Although there is no generally accepted list of criteria for the acceptance of proofs in mathematical practice, judging the acceptability of purported proofs is an essential aspect of handling proofs in daily mathematical work. For this reason, school students, university students, and mathematicians need to hold certain acceptance criteria for proofs, which are closely tied to their epistemology, notion of proof, and concept of evidence. However, acceptance criteria have so far received little academic attention and results have often been related to mathematicians’ acceptance criteria in a research context. Still, as the role of proof changes during mathematical education and differs from research, it can be assumed that acceptance criteria differ substantially between educational levels and different communities. Thus, we analyzed and compared acceptance criteria by school students, university students, and mathematicians in the context of teaching. The results obtained reveal substantial cross-sectional differences in the frequency and choice of acceptance criteria, as well as major differences from those acceptance criteria reported by prior studies in research contexts. Structure-oriented criteria, regarding the logical structure of a proof or individual inferences, are most often referred to by all groups. In comparison, meaning-oriented criteria, such as understanding, play a minor role. Finally, social criteria were not mentioned in the context of teaching. From an educational perspective, the results obtained underline university students’ need for support in implementing acceptance criteria and suggest that a more explicit discussion of the functions of proof, their evidential value, and the criteria for their acceptance may be beneficial.

Keywords

Mathematical proof Acceptance criteria Enculturation Argumentation Proof validation Socio-mathematical norms 

Notes

References

  1. Aberdein, A. (2009). Mathematics and argumentation. Foundations of Science, 14(1), 1–8.Google Scholar
  2. Alama, J., & Kahle, R. (2013). Checking proofs. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp. 147–170). Dordrecht: Springer.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.Google Scholar
  4. Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(2), 81–106.Google Scholar
  5. Balacheff, N. (2008). The role of the researcher’s epistemology in mathematics education: An essay on the case of proof. ZDM - The International Journal on Mathematics Education, 40(3), 501–512.Google Scholar
  6. Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. Dordrecht: Springer.Google Scholar
  7. Bieda, K., Holden, C., & Knuth, E. (2006). Does proof prove? Students’ emerging beliefs about generality and proof in middle school. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 2, pp. 395–402). Mérida: Universidad Pedagógica Nacional.Google Scholar
  8. Bieda, K., & Lepak, J. (2014). Are you convinced? Middle-grade students’ evaluations of mathematical arguments. School Science and Mathematics, 114(4), 166–177.Google Scholar
  9. CCSSI. (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.Google Scholar
  10. Chinnappan, M., & Lawson, M. J. (1996). The effects of training in the use of executive strategies in geometry problem solving. Learning and Instruction, 6(1), 1–17.Google Scholar
  11. Clark, M., & Lovric, M. (2009). Understanding secondary-tertiary transition in mathematics. International Journal of Mathematical Education in Science and Technology, 40(6), 755–776.Google Scholar
  12. Corriveau, C., & Bednarz, N. (2017). The secondary-tertiary transition viewed as a change in mathematical cultures: An exploration concerning symbolism and its use. Educational Studies in Mathematics, 95(1), 1–19.Google Scholar
  13. Dawkins, P. C., & Weber, K. (2017). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics, 95(2), 123–142.Google Scholar
  14. de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.Google Scholar
  15. Devlin, K. (2003). When is a proof? Retrieved from https://www.maa.org/external_archive/devlin/devlin_06_03.html. Accessed 2 Mar 2018.
  16. Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20–23.Google Scholar
  17. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23.Google Scholar
  18. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. Bishop, M. A. K. Clements, C. Keitel-Kreidt, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 877–908). Dordrecht: Springer.Google Scholar
  19. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, E. Dubinsky & T. Dick (Eds.), Research in collegiate mathematics education. III (pp. 234–283). Providence: AMS.Google Scholar
  20. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 21(4), 396–428.Google Scholar
  21. Heinze, A. (2010). Mathematicians’ individual criteria for accepting theorems and proofs: An empirical approach. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 101–111). Boston: Springer.Google Scholar
  22. Heinze, A., & Reiss, K. (2003). Reasoning and proof: Methodological knowledge as a component of proof competence. In M. A. Mariotti (Ed.), Proceedings of the third conference of the European society for research in mathematics education (Vol. 4, pp. 1–10). Bellaria, Italy.Google Scholar
  23. Heinze, A., & Reiss, K. (2009). Developing argumentation and proof competencies in the mathematics classroom. In D. A. Stylianou, M. L. Blanton & E. J. Knuth (Eds.), Teaching and learning of proof across the grades: A K-16 perspective (pp. 191–203). New York: Routledge.Google Scholar
  24. Hemmi, K. (2006). Approaching proof in a community of mathematical practice. Stockholm: Department of Mathematics, Stockholm University.Google Scholar
  25. Hilbert, D. (1931). Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen, 104(1), 485–494.Google Scholar
  26. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390.Google Scholar
  27. Inglis, M., Mejía-Ramos, J. P., Weber, K., & Alcock, L. (2013). On mathematicians’ different standards when evaluating elementary proofs. Topics in Cognitive Science, 5(2), 270–282.Google Scholar
  28. Lakatos, I. (1963). Proofs and refutations (I). The British Journal for the Philosophy of Science, 14(53), 1–25.Google Scholar
  29. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.Google Scholar
  30. Manin, Y. (2010). A course in mathematical logic for mathematicians. New York: Springer.Google Scholar
  31. Mariotti, M. A. (2006). Proof and proving in mathematics education. In Á Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 173–204). Rotterdam: Sense Publishers.Google Scholar
  32. Mariotti, M. A., & Balacheff, N. (2008). Introduction to the special issue on didactical and epistemological perspectives on mathematical proof. ZDM - The International Journal on Mathematics Education, 40(3), 341–344.Google Scholar
  33. Mayring, P. (2014). Qualitative content analysis: Theoretical foundation, basic procedures and software solution. Klagenfurt. http://nbn-resolving.de/urn:nbn:de:0168-ssoar-395173. Accessed 13 Apr 2018.
  34. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266.Google Scholar
  35. Perrenet, J., & Taconis, R. (2009). Mathematical enculturation from the students’ perspective: Shifts in problem-solving beliefs and behaviour during the bachelor programme. Educational Studies in Mathematics, 71(2), 181–198.Google Scholar
  36. Reid, D. A., & Knipping, C. (2010). Proof in mathematics education. Research, learning and teaching. Rotterdam: Sense Publishers.Google Scholar
  37. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), The handbook for research on mathematics teaching and learning (pp. 334–370). New York: MacMillan.Google Scholar
  38. Selden, A. (2011). Transitions and proof and proving at tertiary level. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 391–420). Dordrecht: Springer.Google Scholar
  39. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4–36.Google Scholar
  40. Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.Google Scholar
  41. Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica, 23(3), 295–310.Google Scholar
  42. Ufer, S., Heinze, A., Kuntze, S., & Rudolph-Albert, F. (2009). Beweisen und Begründen im Mathematikunterricht: Die Rolle von Methodenwissen für das Beweisen in der Geometrie. Journal für Mathematik-Didaktik, 30(1), 30–54.Google Scholar
  43. Weber, K. (2003). Students’ difficulties with proof. In A. Selden & J. Selden (Eds.), In The Mathematical Association of America online: Research sampler (Vol. 8). https://www.maa.org/programs/faculty-anddepartments/curriculum-department-guidelines-recommendations/teaching-and-learning/research-sampler-8-students-difficulties-with-proof.
  44. Weber, K. (2008). How mathematicians determine If an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–459.Google Scholar
  45. Weber, K., & Mejía-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76(3), 329–344.Google Scholar
  46. Weber, K., & Mejia-Ramos, J. P. (2015). On relative and absolute conviction in mathematics. For the Learning of Mathematics, 35(2), 15–21.Google Scholar
  47. Weber, K., Mejía-Ramos, J. P., Inglis, M., & Alcock, L. (2013). On mathematicians’ proof skimming: A reply to Inglis and Alcock / Skimming: A response to Weber and Mejía-Ramos. Journal for Research in Mathematics Education, 44(2), 464–475.Google Scholar
  48. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.Google Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Chair of Mathematics EducationLudwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations