Educational Studies in Mathematics

, Volume 77, Issue 1, pp 1–14 | Cite as

Does generating examples aid proof production?

  • Paola IannoneEmail author
  • Matthew Inglis
  • Juan Pablo Mejía-Ramos
  • Adrian Simpson
  • Keith Weber


Many mathematics education researchers have suggested that asking learners to generate examples of mathematical concepts is an effective way of learning about novel concepts. To date, however, this suggestion has limited empirical support. We asked undergraduate students to study a novel concept by either tackling example generation tasks or reading worked solutions to these tasks. Contrary to suggestions in the literature, we found no advantage for the example generation group on subsequent proof production tasks. From a second study, we found that undergraduate students overwhelmingly adopt a trial and error approach to example generation and suggest that different example generation strategies may result in different learning gains. We conclude by arguing that the teaching strategy of example generation is not yet understood well enough to be a viable pedagogical recommendation.


Examples Example generation Fine function Proof Undergraduate 



This research was partially supported by a grant from the Maths, Stats & OR Network of the Higher Education Academy and a Royal Society Worshipful Company of Actuaries Research Fellowship.


  1. Antonini, S. (2006). Graduate students’ processes in generating examples of mathematical objects. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.), Proceedings of the 30th International Conference on the Psychology of Mathematics Education (Vol. 2, pp. 57–64). Prague, Czech Republic: IGPME.Google Scholar
  2. Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33, 283–299.CrossRefGoogle Scholar
  3. Ginsburg, H. (1981). The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(1), 4–11.Google Scholar
  4. Goldenberg, P., & Mason J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69, 183–194.CrossRefGoogle Scholar
  5. Housman, D., & Porter, M. (2003). Proof schemes and learning strategies of above-average mathematics students. Educational Studies in Mathematics, 53, 139–158.CrossRefGoogle Scholar
  6. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Mason, J. (2002). Mathematics teaching practice: A guide for university and college lecturers. Chichester, UK: Horwood.Google Scholar
  8. Meehan, M. (2007). Student generated examples and the transition to advanced mathematical thinking. In D. Pitta-Pantazi & G. Phillipou (Eds.), Proceedings of the fifth congress of the European Society for Research in Mathematics Education (pp. 2349–2358). Larnaca, Cyprus: ERME.Google Scholar
  9. Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266.CrossRefGoogle Scholar
  10. Watson, A., & Mason, J. (2001). Getting students to create boundary examples. MSOR Connections, 1, 9–11.Google Scholar
  11. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  12. Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts. Educational Studies in Mathematics, 69, 97–109.CrossRefGoogle Scholar
  13. Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and construct counterexamples in advanced mathematics. Journal of Mathematical Behavior, 28, 200–208.CrossRefGoogle Scholar
  14. Weber, K., Porter, M., & Housman, D. (2008). Worked examples and conceptual example usage in understanding mathematical concepts and proofs. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 245–252). Washington, DC: MAA.Google Scholar
  15. Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers’ professional development. For the Learning of Mathematics, 15(3), 15–20.Google Scholar
  16. Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69, 131–148.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Paola Iannone
    • 1
    Email author
  • Matthew Inglis
    • 2
  • Juan Pablo Mejía-Ramos
    • 3
  • Adrian Simpson
    • 4
  • Keith Weber
    • 3
  1. 1.School of Education and Lifelong LearningUniversity of East AngliaNorwichUK
  2. 2.Mathematics Education CentreLoughborough UniversityLoughboroughUK
  3. 3.Graduate School of EducationRutgers UniversityNew BrunswickUSA
  4. 4.School of EducationDurham UniversityDurham DH1 3DFUK

Personalised recommendations