pp 1–12 | Cite as

Type of mathematical proof: personal preference or adaptive teaching behavior?

  • Esther BrunnerEmail author
  • Kurt Reusser
Original Article


In our study, 32 German and Swiss 8th/9th-grade classes of lower-secondary school worked with their teacher on the same proving problem. The sample belongs to the Swiss-German study “Quality of Instruction, Learning Behavior and Mathematical Understanding”. Our data analyses relate to the teachers’ approaches to generating a specific form of evidence with their classes when dealing with a particular elementary number-theory problem. We address the question of how the different strategies can be characterized as manifestations of a certain approach to proving and try to clarify in which way the observed approach can be interpreted as adaptive teaching behavior. For this purpose, we searched for possible correlations between three main strategies or types of generating a specific form of evidence (experimental, operative, formal-deductive approach) on the one hand and (a) the teachers’ beliefs and personal characteristics and (b) the students’ prior knowledge of algebra and mathematics in general on the other hand. As our analyses show, three main approaches to proving occurred but not in equal proportions: there is a predominance of the approach that entails the highest extent of formalization and abstraction. Nevertheless, an operative way of proving is widespread too. On the whole, the findings indicate that one particular approach to proving can be interpreted as a personal preference of a specific group of teachers and, at the same time, with respect to the students’ mathematical skills as a manifestation of adaptive teaching behavior.


Mathematical proving Mathematics education Lower-secondary level Teacher beliefs Generation of evidence 

Mathematics Subject Classification

03F03 97B20 97B50 97C70 97D40 



We thank the Swiss National Science Foundation (SNSF) for supporting the project.


  1. Aberdein, A. (2009). Mathematics and argumentation. Foundation of Science, 14(1), 1–8. Scholar
  2. Aebli, H. (1981). Denken. Das Ordnen des Tuns. Stuttgart: Klett-Cotta.Google Scholar
  3. Ainsworth, S. (1999). The functions of multiple representations. Computers and Education, 33(2/3), 131–152.Google Scholar
  4. Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25–61.Google Scholar
  5. Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2007). The transition to formal proof in geometry. In P. Boero (Ed.), Theorems in school. From history, epistemology and cognition to classroom practice (pp. 305–323). Rotterdam: Sense.Google Scholar
  6. Balacheff, N. (1988). Etude des processus de preuve chez des élèves de collège. Grenoble: Université Joseph Fournier.Google Scholar
  7. Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 115–135). New York: Springer.Google Scholar
  8. Baroody, A. J., & Dowker (Eds.). (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah: Erlbaum.Google Scholar
  9. Bieg, M., Goetz, T., Wolter, I., & Hall, N. C. (2015). Gender stereotype endorsement differentially predicts girls’ and boys’ trait-state discrepancy in math anxiety. Frontiers in Psychology, 17(6), 1404. Scholar
  10. Blömeke, S., & Kaiser, G. (2014). Theoretical framework, study design and main results of TEDS-M. In S. Blömeke, F.-J. Hsieh, G. Kaiser & W. H. Schmidt (Eds.), International perspectives on teacher knowledge, beliefs and opportunities to learn. TEDS-M results (pp. 19–48). Heidelberg: Springer.Google Scholar
  11. Brunner, E. (2013). Innermathematisches Beweisen und Argumentieren in der Sekundarstufe I. Münster: Waxmann.Google Scholar
  12. Brunner, E., Jullier, R., & Lampart, J. (2019). Aufgabenangebot zum mathematischen Begründen in je zwei aktuellen Mathematikbüchern für die fünfte bzw. achte Klasse (in prep.).Google Scholar
  13. Common Core State Standards Initiative. (2012). Mathematics standards. Accessed 10 March 2018.
  14. Corno, L., & Snow, R. E. (1986). Adapting teaching to individual differences among learners. In M. C. Wittrock (Ed.), Handbook of research on teaching (pp. 605–629). New York: Macmillan.Google Scholar
  15. de Villiers, M. (1990). The role and the function of proof in mathematics. Pythagoras, 24, 17–24.Google Scholar
  16. de Villiers, M. (2010). Experimentation and proof in mathematics. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 205–221). Dodrecht: Springer.Google Scholar
  17. D-EDK. (2014). Lehrplan 21. Mathematik. Bern: Projekt Lehrplan 21.Google Scholar
  18. Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving in the classroom. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI study (pp. 191–213). Dodrecht: Springer.Google Scholar
  19. Drüke-Noe, C. (2014). Aufgabenkultur in Klassenarbeiten im Fach Mathematik. Empirische Untersuchungen in neunten und zehnten Klassen. Wiesbaden: Springer.Google Scholar
  20. Duncker, K. (1935). Zur Psychologie des produktiven Denkens. Berlin: Springer.Google Scholar
  21. Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (Ed.), Theorems in school (pp. 137–161). Rotterdam: Sense.Google Scholar
  22. Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–18.Google Scholar
  23. Furinghetti, F., & Morselli, F. (2011). Beliefs and beyond: Hows and whys in the teaching of proof. ZDM, 43(4), 587–599.Google Scholar
  24. Hammer, S., Reiss, K., Lehner, M. C., Heine, J.-H., Sälzer, C., & Heinze, A. (2016). Mathematische Kompetenz in PISA 2015: Ergebnisse, Veränderungen und Perspektiven. In K. Reiss, C. Sälzer, A. Schiepe-Tiska, E. Klieme, & O. Köller (Eds.), PISA 2015. Eine Studie zwischen Kontinuität und Innovation (pp. 219–247). Münster: Waxmann.Google Scholar
  25. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.Google Scholar
  26. Hanna, G., & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM, 40(3), 345–353.Google Scholar
  27. Hanna, G., & Barbeau, E. (2010). Proofs as bearers of mathematical knowledge. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 85–100). Dodrecht: Springer.Google Scholar
  28. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 877–908). Dodrecht: Kluwer.Google Scholar
  29. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education, 7, 234–283.Google Scholar
  30. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.Google Scholar
  31. Heintz, B. (2000). Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin. Vienna: Springer.Google Scholar
  32. Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM, 41(5), 535–540.Google Scholar
  33. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense. Teaching and learning mathematics with understanding. Portsmouth: Heinemann.Google Scholar
  34. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., & Jacobs, J. (2003). Teaching mathematics in seven countries. Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, Institute of Education Sciences.Google Scholar
  35. Hugener, I., Pauli, C., Reusser, K., Lipowsky, F., Rakoczy, K., & Klieme, E. (2009). Teaching patterns and learning quality in Swiss and German mathematics lessons. Learning and Instruction, 19(1), 66–78.Google Scholar
  36. Jahnke, H. N. (2008). Theorems that admit exceptions, including a remark on Toulmin. ZDM, 40(3), 363–371.Google Scholar
  37. Jahnke, H. N., & Ufer, S. (2015). Argumentieren und Beweisen. In R. Bruder, L. Hefendehl-Hebeker, B. Schmidt-Thieme & H.-G. Weigand (Eds.), Handbuch der Mathematikdidaktik (pp. 331–355). Heidelberg: Springer Spektrum.Google Scholar
  38. Klieme, E., Pauli, C., & Reusser, K. (Eds.). (2006). Dokumentation der Erhebungs- und Auswertungsinstrumente zur schweizerisch-deutschen Videostudie “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis”. Videoanalysen. Frankfurt: DIPF.Google Scholar
  39. Klieme, E., Pauli, C., & Reusser, K. (2009). The Pythagoras Study. In T. Janik & T. Seidel (Eds.), The power of video studies in investigating teaching and learning in the classroom (pp. 137–160). Münster: Waxmann.Google Scholar
  40. KMK. (2005). Bildungsstandards der Kultusministerkonferenz. Erläuterungen zur Konzeption und Entwicklung. Munich: Luchterhand.Google Scholar
  41. Knuth, E. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 33(5), 61–82.Google Scholar
  42. Kotelawa, U. (2009). A survey of teacher beliefs on proving. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education. ICMI Study 19 Conference Proceedings (Vol. 1, pp. 250–255). Taipei: The Department of Mathematics, National Taiwan Normal University.Google Scholar
  43. Lipowsky, F., Drollinger-Vetter, B., Hartig, J., & Klieme, E. (2006). Dokumentation der Erhebungs- und Auswertungsinstrumente zur schweizerisch-deutschen Videostudie “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis” (Vol. 14). Frankfurt: DIPF. Leistungstests (Materialien zur Bildungsforschung.Google Scholar
  44. Lorenz, J. H. (2017). Einige Anmerkungen zur Repräsentation von Wissen über Zahlen. JMD, 38(1), 125–139.Google Scholar
  45. Malek, A., & Movshovitz-Hadar, N. (2011). The effect of using transparent pseudo-proofs in linear algebra. RME, 13(1), 33–58.Google Scholar
  46. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.Google Scholar
  47. Muijs, D., & Reynolds, D. (2002). Teachers’ beliefs and behaviors: What really matters? Journal of Classroom Interaction, 37, 3–15.Google Scholar
  48. NCTM (Ed.). (2000). Principles and standards for school mathematics. Reston: NCTM.Google Scholar
  49. Newell, A., & Simon, H. (1972). Human problem solving. Englewood Cliffs: Prentice-Hall.Google Scholar
  50. OECD (Ed.). (2016). PISA 2015 results: Excellence and equity in education (Vol. 1). Paris: OECD.Google Scholar
  51. Pfeiffer, K. (2011). Features and purposes of mathematical proofs in the view of novice students: Observations from proof validation and evaluation performances. Galway: National University of Ireland.Google Scholar
  52. Rakoczy, K., Buff, A., & Lipowsky, F. (2005). Befragungsinstrumente. In E. Klieme, C. Pauli & K. Reusser (Eds.), Dokumentation der Erhebungs- und Auswertungsinstrumente zur schweizerisch-deutschen Videostudie “Unterrichtsqualität, Lernverhalten und mathematisches Verständnis” (Vol. 1). Frankfurt: DIPF.Google Scholar
  53. Reid, D. (2005). The meaning of proof in mathematics education. In M. Bosch (Ed.), European research in mathematics education IV. Proceedings of CERME 4, San Feliu de Guixols, Spain. Barcelona: University Ramon Llull.Google Scholar
  54. Reid, D. A., & Knipping, C. (2010). Proof in mathematics education. Research, learning and teaching. Rotterdam: Sense.Google Scholar
  55. Reusser, K. (1984). Problemlösen in wissenstheoretischer Sicht. Problematisches Wissen, Problemformulierung und Problemverständnis. Berne: University of Berne.Google Scholar
  56. Reusser, K. (1993). Tutoring systems and pedagogical theory: Representational tools for understanding, planning, and reflection in problem solving. In S. P. Lajoie & S. J. Derry (Eds.), Computers as cognitive tools (pp. 143–177). Hillsdale: Erlbaum.Google Scholar
  57. Reuterswärd, E., & Hemmi, K. (2011). Upper secondary school teachers’ views of proof and the relevance of proof in teaching mathematics. In M. Pytlak, T. Rowland & E. Swoboda (Eds.), Proceedings of the seventh congress of the European Society for Research in Mathematics Education (pp. 253–262). Rzeszów: University of Rzeszów.Google Scholar
  58. Rowland, T. (2002). Generic proofs in number theory. In S. R. Campbell & R. Zazkis (Eds.), Learning and teaching number theory (pp. 157–183). Westport: Ablex.Google Scholar
  59. Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.Google Scholar
  60. Spiro, R. J., & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matter. In D. Nix & R. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology (pp. 163–205). Hillsdale: Erlbaum.Google Scholar
  61. Staub, F. C., & Stern, E. (2002). The nature of teachers’ pedagogical content beliefs matters for students’ achievement gains: Quasi-experimental evidence from elementary mathematics. Journal of Educational Psychology, 94(2), 344–355.Google Scholar
  62. Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17, 213–226.Google Scholar
  63. Stylianides, G. L. (2008). An analytic framework of reasoning and proving. For the Learning of Mathematics, 28(1), 9–16.Google Scholar
  64. Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253–295.Google Scholar
  65. Vygotsky, L. S. (1978). Mind in society. Cambridge: Harvard University Press.Google Scholar
  66. Wagenschein, M. (1968). Verstehen lehren. Genetisch-Sokratisch-Exemplarisch. Weinheim: Beltz.Google Scholar
  67. Warner, L. B., Schorr, R. Y., & Davis, G. E. (2009). Flexible use of symbolic tools for problem solving, generalization, and explanation. ZDM, 41(5), 663–679.Google Scholar
  68. Wertheimer, M. (1964). Produktives Denken (2nd ed.). Frankfurt: Kramer.Google Scholar
  69. Wittmann, E. C., & Müller, N. G. (1988). Wann ist ein Beweis ein Beweis? In P. Bender (Ed.), Mathematikdidaktik—Theorie und Praxis. Festschrift für Heinrich Winter (pp. 237–258). Berlin: Cornelsen.Google Scholar
  70. Wittmann, E. Ch. (2009). Operative proof in elementary mathematics. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. De Villiers (Eds.), Proof and proving in mathematics education. ICMI Study 19 Conference Proceedings (pp. 251–256). Taipei: Department of Mathematics National Taiwan Normal University.Google Scholar
  71. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation and autonomy in mathematics. The Journal of Research in Mathematics Education, 27, 458–477.Google Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Thurgau University of Teacher EducationKreuzlingenSwitzerland
  2. 2.Institute of EducationUniversity of ZurichZurichSwitzerland

Personalised recommendations