Using task-based interviews to generate hypotheses about mathematical practice: mathematics education research on mathematicians’ use of examples in proof-related activities

Abstract

Mathematics education researchers frequently use task-based interviews to gain insight into mathematicians’ practice. However, there are a number of factors that should prevent mathematics educators from extrapolating how individual mathematicians respond to researcher-generated tasks in laboratory conditions, to how mathematicians practice their craft in authentic settings. In this paper we critically analyze the rationality of using task-based interviews to investigate mathematical practice, focusing on how task-based interview studies have been used to inform our understanding of mathematicians’ use of examples in mathematical practice. We discuss four specific generalizations about mathematical practice drawn from these studies, and suggest other types of studies that can be used to corroborate or challenge those generalizations.

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

    An abundant number is a natural number n whose positive divisors add up to more than 2n. A perfect number is a natural number n whose positive divisors add up to exactly 2n. A deficient number is a natural number n whose positive divisors add up to less than 2n.

  2. 2.

    Of course, this methodology has its own weaknesses. For instance, participants may be unable to provide an accurate account of their actual mathematical behavior (Inglis & Alcock, 2012). Our position is that all methodologies have weaknesses, but that through their complementary use we can obtain a picture of mathematical practice that is more accurate and robust than the one we obtain when we restrict its study to any one of those methodologies.

  3. 3.

    Having surveyed 291 mathematicians about their use of examples when exploring a new mathematical conjecture, Lockwood et al. explained that they had decided to conduct task-based interviews in order to get access to “what mathematicians do in practice”. Lockwood et al. claimed that data from the survey and the interview study were “meant to be complementary, and the interviews served to elaborate and refine the initial results from the survey” (p. 169).

  4. 4.

    All survey data reported by Lockwood et al. (2016) comes from mathematicians’ response to a single prompt: “If you sometimes use examples when exploring a new mathematical conjecture, how do you choose the specific examples you select in order to test or explore the conjecture? What explicit strategies or example characteristics, if any, do you use or consider?” (p. 169). As the aim of Lockwood et al. (2016) was to make a qualitative contribution, Lockwood et al. did not report any quantitative analyses of these survey data. We believe such quantitative analyses would need to be scrutinized to determine how well these generalizations are supported.

References

  1. Alcock, L. J., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111–129.

    Article  Google Scholar 

  2. Balacheff, N. (1987). Processus de preuves et situations de validation. Educational Studies in Mathematics, 18, 147–176.

    Article  Google Scholar 

  3. Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.

    Article  Google Scholar 

  4. Clark, H. H. (1973). The language-as-fixed-effect fallacy: A critique of language statistics in psychological research. Journal of Verbal Learning and Verbal Behavior, 12, 335–359.

    Article  Google Scholar 

  5. Clement, J. (2000). Analysis of clinical interviews: Foundation and model viability. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). New Jersey: Lawrence Erlbaum.

    Google Scholar 

  6. Coffield, F., Moseley, D., Hall, E., & Ecclestone, K. (2004). Learning styles and pedagogy in post-16 learning: A systematic and critical review. London: Learning & Skills Research Centre.

    Google Scholar 

  7. Dawkins, P. C., & Karunakaran, S. (2016). Why research on proof-oriented mathematical behavior should attend to the role of particular mathematical content. The Journal of Mathematical Behavior, 44, 65–75.

    Article  Google Scholar 

  8. DeFranco, T. C. (1996). A perspective on mathematical problem-solving expertise based on the performances of male Ph.D. mathematicians. In J. Kaput, A. Schoenfeld, & E. Dubinsky (Eds.), Research in collegiate mathematics II (pp. 195–213). Providence: American Mathematical Association.

    Google Scholar 

  9. Goldin, G. (2000). A scientific perspective on structures, task-based interviews in mathematics education research. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). New Jersey: Lawrence Erlbaum.

    Google Scholar 

  10. Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory (pp. 185–212). Norwood: Ablex.

    Google Scholar 

  11. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390.

    Article  Google Scholar 

  12. Inglis, M., Mejía-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3–21.

    Article  Google Scholar 

  13. Kidron, I., & Dreyfus, T. (2014). Proof image. Educational Studies in Mathematics, 87, 297–321.

    Article  Google Scholar 

  14. Lester, F. K., & Kehle, P. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 501–517). Mahwah: Erlbaum.

    Google Scholar 

  15. Lockwood, E., Ellis, A. B., & Lynch, A. G. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. International Journal of Research in Undergraduate Mathematics Education, 2(2), 165–196.

    Article  Google Scholar 

  16. Lynch, A., & Lockwood, E. (2019). A comparison between mathematicians’ and students’ use of examples for conjecturing and proving. Journal of Mathematical Behavior, 53, 323–338.

    Article  Google Scholar 

  17. Maher, C. A., & Sigley, R. (2014). Task-based interviews in mathematics education. In S. Lernman (Ed.), Encyclopedia of mathematics education. Dordrecht: Springer.

    Google Scholar 

  18. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the generic in the particular. Educational Studies in Mathematics, 15, 277–289.

    Article  Google Scholar 

  19. Mejía-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. Educational Studies in Mathematics, 85(2), 161–173.

    Article  Google Scholar 

  20. Melhuish, K. (2018). Three conceptual replication studies in group theory. Journal for Research in Mathematics Education, 49(1), 9–38.

    Article  Google Scholar 

  21. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers.

    Google Scholar 

  22. Samkoff, A., Lai, Y., & Weber, K. (2012). On the different ways that mathematicians use diagrams in proof construction. Research in Mathematics Education, 14(1), 49–67.

    Article  Google Scholar 

  23. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press Inc.

    Google Scholar 

  24. Stylianou, D. A. (2002). Interaction of visualization and analysis—The negotiation of a visual representation in problem solving. Journal of Mathematical Behavior, 21(3), 303–317.

    Article  Google Scholar 

  25. Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Mathematical Thinking and Learning, 6(4), 353–387.

    Article  Google Scholar 

  26. Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  27. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.

    Article  Google Scholar 

  28. 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 

  29. Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.

    Article  Google Scholar 

  30. Weber, K., Dawkins, P., & Mejía-Ramos, J. P. (2020). The relationship between mathematical practice and mathematics pedagogy in mathematics education research.

  31. Weber, K., Inglis, M., & Mejía-Ramos, J. P. (2014). How mathematicians obtain conviction: Implications for mathematics instruction and research on epistemic cognition. Educational Psychologist, 49(1), 36–58.

    Article  Google Scholar 

  32. Wilkerson-Jerde, M. H., & Wilensky, U. (2011). How do mathematicians learn math? Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78, 21–43.

    Article  Google Scholar 

  33. Yin, R. K. (1994). Case study research: Design and methods (2nd ed.). Thousand Oaks: SAGE Publications.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Juan Pablo Mejía-Ramos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mejía-Ramos, J.P., Weber, K. Using task-based interviews to generate hypotheses about mathematical practice: mathematics education research on mathematicians’ use of examples in proof-related activities. ZDM Mathematics Education 52, 1099–1112 (2020). https://doi.org/10.1007/s11858-020-01170-w

Download citation

Keywords

  • Task-based interviews
  • Mathematical practice
  • Examples
  • Conjecturing
  • Proof