Establishing Appropriate Conditions: Students Learning to Apply a Theorem

  • Giovanna Scataglini-Belghitar
  • John MasonEmail author


During a sequence of tutorials conducted by the first author, it became evident that students were not seeing how to apply the theorem concerning a continuous function on a closed and bounded interval attaining its extreme values in situations in which it is necessary first to construct the closed and bounded interval by reasoning about properties concerning asymptotes. The paper describes 1 tutorial with 10 students and a follow-up revision session with 2 of the students the same day concerning the use of this theorem in which the tutor used the pedagogic strategy of learner-generated examples. The students’ responses and the tutor’s observations are analysed using constructs of example spaces, structure of attention and dimensions of possible variation.


appropriate conditions attention coming-to-mind example construction examples 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abdul-Rahman, S. (2005). Learning with examples and students’ understanding of integration. In A. Rogerson (Ed.), Proceedings of the Eighth International Conference of Mathematics Education into the 21st Century Project on “Reform, Revolution and Paradigm Shifts in Mathematics Education”. Johor Bahru, Malaysia, November 25–December 1, 2005.Google Scholar
  2. Bauersfeld, H. (1988). Interaction, construction, and knowledge–Alternative perspectives for mathematics education. In D. A. Grouws & T. J. Cooney (Eds.), Perspectives on research on effective mathematics teaching: Research agenda for mathematics education (Vol. 1, pp. 27–46). Reston: NCTM/Lawrence Erlbaum.Google Scholar
  3. Benham, C. (1704). Benham’s book of quotations, proverbs and household words. London: Ward, Lock.Google Scholar
  4. Berger, M. (2006). Making mathematical meaning: From preconcepts to pseudoconcepts to concepts. Pythagoras, 63, 14–21.Google Scholar
  5. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. Steiner (Ed.), Theory of mathematics education, Paper 54 (pp. 110–119). Bielefeld: Institut fur Didaktik der Mathematik der Universitat.Google Scholar
  6. Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactiques des mathématiques, 1970–1990, N. Balacheff, M. Cooper, R. Sutherland, V. Warfield (Trans.), Dordrecht, Netherlands: Kluwer.Google Scholar
  7. Burke, E. (1796). Thoughts on the prospect of a regicide peace. London: Owen.Google Scholar
  8. Burkhardt, H. (1981). The real world of mathematics. Glasgow: Blackie.Google Scholar
  9. Chi, M. & Bassok, M. (1989). Learning from examples via self-explanation. In L. Resnick (Ed.), Knowing, learning and instruction: Essays in honour of Robert Glaser. Hillsdale: Erlbaum.Google Scholar
  10. Chi, M., Bassok, M., Lewis, P., Reiman, P. & Glasser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.CrossRefGoogle Scholar
  11. Courant, R. (1981). Reminiscences from Hilbert’s Göttingen. Math Intelligencer, 3(4), 154–164.CrossRefGoogle Scholar
  12. Dahlberg, R. P. & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33, 283–299.CrossRefGoogle Scholar
  13. Davis, B. (1996). Teaching mathematics: Towards a sound alternative. New York: Ablex.Google Scholar
  14. Dreyfus, T. & Eisenberg, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics. MAA Notes No. 19 (pp. 25–37). Washington: Mathematical Association of America.Google Scholar
  15. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.CrossRefGoogle Scholar
  16. Floyd, A., Burton, L., James, N. & Mason, J. (1981). EM235: Developing mathematical thinking. Milton Keynes: Open University.Google Scholar
  17. Gallagher, W. (2009). Rapt: Attention and the focused life. New York: Penguin.Google Scholar
  18. Halmos, P. (1983). Selecta: Expository writing. In D. Sarasen & L. Gillman (Eds.), New York: Springer.Google Scholar
  19. Halmos, P. (1994). What is teaching? The American Mathematical Monthly, 101(9), 848–854.CrossRefGoogle Scholar
  20. Holt, J. (1964). How children fail. Harmondsworth: Penguin.Google Scholar
  21. James, W. (1890, reprinted 1950). Principles of psychology, vol 1. New York: Dover.Google Scholar
  22. Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  23. Malara, N., & Navarra, G. (2003). ArAl: A project for an early approach to algebraic thinking. Retrieved January 2007 from
  24. Marton, F. & Booth, S. (1997). Learning and awareness. Mahwah: Lawrence Erlbaum.Google Scholar
  25. Mason, J. (2001). Teaching for flexibility in mathematics: Being aware of the structures of attention and intention. Quaestiones Mathematicae, 24(Suppl 1), 1–15.Google Scholar
  26. Mason, J. (2002). Generalisation and algebra: Exploiting children’s powers. In L. Haggerty (Ed.), Aspects of teaching secondary mathematics: Perspectives on practice (pp. 105–120). London: Routledge Falmer.Google Scholar
  27. Mason, J. (2003). On the structure of attention in the learning of mathematics. Australian Mathematics Teacher, 59(4), 17–25.Google Scholar
  28. Mason, J. & Davis, J. (1989). The inner teacher, the didactic tension, and shifts of attention. In G. Vergnaud, J. Rogalski, & M. Artigue, (Eds.). Proceedings of PME XIII, Paris, vol 2, pp. 274–281.Google Scholar
  29. Mason, J. Drury, H. & Bills, E. (2007). Explorations in the zone of proximal awareness. In J. Watson & K. Beswick (Eds.) Mathematics: Essential research, essential practice: Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia. Adelaide, SA, Australia: MERGA, vol. 1, pp. 42–58.Google Scholar
  30. Mason, J. Burton L. & Stacey K. (1982/2010). Thinking Mathematically, London, UK: Addison Wesley.Google Scholar
  31. Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education. London: Routledge Falmer.Google Scholar
  32. Mason, J. & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks (2nd ed.). St. Albans: Tarquin.Google Scholar
  33. Mason, J. & Pimm, D. (1984). Generic Examples: Seeing the General in the Particular. Educational Studies in Mathematics, 15(3), 277–290.CrossRefGoogle Scholar
  34. Mason, J. & Watson, A. (2001). Stimulating Students to Construct Boundary Examples. Quaestiones Mathematicae, 24(Suppl 1), 123–132.Google Scholar
  35. Nardi E. (1998). Mapping Out the First-year Mathematics Undergraduate’s Difficulties with Abstraction in Advanced Mathematical Thinking: Didactical observations related to the teaching of advanced mathematics from a cross-topical study, in Proceedings of the International Conference on the Teaching of Mathematics, Pythagorion, Greece. p224–226. New York, NY, USA: Wiley.Google Scholar
  36. Oxford (private communication). Internal mock exam set for mathematics undergraduates at Oxford University.Google Scholar
  37. Pirie, S. & Kieren, T. (1989). A Recursive Theory of Mathematical Understanding. For the Learning of Mathematics, 9(4), 7–11.Google Scholar
  38. Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190.CrossRefGoogle Scholar
  39. Pólya, G. (1962). Mathematical Discovery: on understanding, learning, and teaching problem solving. New York: Wiley.Google Scholar
  40. Pólya, G. (1965). Let Us Teach Guessing, (film). Washington: Mathematical Association of America.Google Scholar
  41. Popper, K. (1965). Conjectures and Refutations: the growth of scientific knowledge. London: Routledge & Kegan Paul.Google Scholar
  42. Schoenfeld, A. (1985). Mathematical Problem Solving. New York: Academic.Google Scholar
  43. Schön, D. (1983). The Reflective Practitioner: how professionals think in action. London: Temple Smith.Google Scholar
  44. Sierpinska, A. (1994). Understanding in Mathematics. London: Falmer Press.Google Scholar
  45. Sweller, J. & Cooper, G. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 58–89.CrossRefGoogle Scholar
  46. Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  47. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Chicago: University of Chicago.Google Scholar
  48. van der Veer, R. & Valsiner, J. (1991). Understanding Vygotsky. London: Blackwell.Google Scholar
  49. Van Hiele, P. (1986). Structure and Insight: a theory of mathematics education. Developmental Psychology Series. London: Academic.Google Scholar
  50. Watson, A. & Mason, J. (2002). Student-Generated Examples in the Learning of Mathematics, Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249.Google Scholar
  51. Watson, A. & Mason, J. (2005). Mathematics as a Constructive Activity: students generating examples. Mahwah: Erlbaum.Google Scholar
  52. Whitehead, A. (1911). (reset 1948). An Introduction to Mathematics. Oxford University Press, London.Google Scholar
  53. Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul.Google Scholar
  54. Zaslavsky, O. & Lavie, O. (2005). Teachers’ Use of Instructional Examples. Paper presented at the 15th ICMI study conference: The Professional Education and Development of Teachers of Mathematics. Águas de Lindóia, Brazil.Google Scholar
  55. Zaslavsky, O. & Peled, I. (1996). Inhibiting Factors in Generating Examples by Mathematics Teachers and Student-Teachers: The Case of Binary Operation. Journal for Research in Mathematics Education, 27(1), 67–78.CrossRefGoogle Scholar
  56. Zazkis, R. & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27(2), 15–21.Google Scholar

Copyright information

© National Science Council, Taiwan 2011

Authors and Affiliations

  1. 1.Balliol College, University of OxfordOxfordUK
  2. 2.OxfordUK
  3. 3.Department of EducationUniversity of OxfordOxfordUK
  4. 4.Open University (emeritus)Milton KeynesUK

Personalised recommendations