Advertisement

Educational Studies in Mathematics

, Volume 69, Issue 2, pp 131–148 | Cite as

Exemplifying definitions: a case of a square

  • Rina Zazkis
  • Roza Leikin
Article

Abstract

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the categories of accessibility and correctness, richness, and generality. Results shed light on participants’ understanding of what a mathematical definition should entail and, moreover, contrast their pedagogical preferences with mathematical considerations.

Keywords

Examples Learner-generated examples Example spaces Definitions Necessary and Sufficient conditions Prospective teachers Secondary mathematics teachers 

References

  1. Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann Educational Books.Google Scholar
  2. Cooney, T. J., & Wilson, M. R. (1993). Teachers’ thinking about functions: Historical and research perspectives. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 131–158). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  3. Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283–299.CrossRefGoogle Scholar
  4. De Villiers, M. (1998). To teach definitions in geometry or to teach to define? In A. Olivier, & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 248–255). Stellenbosch: RSA.Google Scholar
  5. Edwards, B. (1997). An undergraduate student’s understanding and use of mathematical definitions in real analysis. In J. Dossey, J. Swafford, M. Parmantie, & A. Dossey (Eds.), Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 17–22). Colulmbus, OH: The ERIC Clearing house for Science, Mathematics, and Environment Education.Google Scholar
  6. Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research: Student (mis)use of mathematical definitions. The American Mathematical Monthly, 111(5), 411–424.CrossRefGoogle Scholar
  7. Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht: Kluwer.Google Scholar
  8. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  9. Khinchin, A. Y. (1968). The teaching of mathematics. London: The English Universities Press.Google Scholar
  10. Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66(3), 349–371.CrossRefGoogle Scholar
  11. Leikin, R., & Winicky-Landman, G. (2000). On equivalent and nonequivalent definitions II. For the Learning of Mathematics, 20(2), 24–29.Google Scholar
  12. Leikin, R., & Winicky-Landman, G. (2001). Defining as a vehicle for professional development of secondary school mathematics teachers. Mathematics Teacher Education and Development, 3, 62–73.Google Scholar
  13. Linchevsky, L., Vinner, S., & Karsenty, R. (1992). To be or not to be minimal? Student teachers’ views about definitions in geometry. In W. Geeslin, & K. Graham (Eds.), Proceedings of the 16th Conference of the International Group for the Mathematics Education (vol. 2, pp. 48–55). Durham, New Hampshire.Google Scholar
  14. Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34(3), 219–248.CrossRefGoogle Scholar
  15. Mason, J. (1998). Researching from the inside in mathematics education. In A. Sierpinska, & J. Kilpatrick (Eds.), Mathematics Education as a research domain: A search for identity (pp. 357–377). Dordrecht: Kluwer.Google Scholar
  16. Musser, G. L., Burger, W. F., & Peterson, B. E. (2006). Mathematics for elementary teachers: A contemporary approach. New York: Wiley.Google Scholar
  17. Pimm, D. (1993). Just a matter of definition. Educational Studies in Mathematics, 25, 261–277.CrossRefGoogle Scholar
  18. Poincare, H. (1909/1952). Science and method. New York, NY: Dover.Google Scholar
  19. Solow, D. (1984). Reading, writing and doing mathematical proofs. Book I. Palo Alto, CA: Dale Seymour.Google Scholar
  20. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics—With particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar
  21. Tirosh, D., & Even, R. (1997). To define or not to define: The case of (−8)1/3. Educational Studies in Mathematics, 33(3), 321–330.CrossRefGoogle Scholar
  22. van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22(1), 91–106.CrossRefGoogle Scholar
  23. Vasco, C. E. (2006). Cronotopía: Un “Programa de Bogotá” para lo que se suele llamar “Geometría”. In C. Ruiz, et al. (Eds.), Memorias: XVI Encuentro de Geometría y sus aplicaciones-IV Encuentro de aritmética (vol. 1, pp. 1–28). Bogotá: Universidad Pedagógica Nacional.Google Scholar
  24. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 65–81). Dordrecht: Kluwer.Google Scholar
  25. Vinner, S., Linchevski, L., & Karsenty, R. (1993). How much information should include a geometrical definition? Zentralblatt für Didaktik der Mathematik, 25, 164–170.Google Scholar
  26. Vygotsky, L. S. (1982). Mishlenie i rech (Thought and language). In L. S. Vygotsky (Ed.), Sobranie Sochinenii, t.2. Moscow: Pedagogika (in Russian).Google Scholar
  27. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  28. Weinstein, E. (2007a). Wolfram MathWorld: Polygon. Retrieved March 15, 2008 from http://www.mathworld.wolfram.com/Polygon.html.
  29. Weinstein, E. (2007b). Wolfram MathWorld: Square. Retrieved March 15, 2008 from http://www.mathworld.wolfram.com/Square.html.
  30. Winicky-Landman, G., & Leikin, R. (2000). On equivalent and nonequivalent definitions I. For the Learning of Mathematics, 20(1), 17–21.Google Scholar
  31. Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.CrossRefGoogle Scholar
  32. Zazkis, R., & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27, 11–17.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Faculty of EducationSimon Fraser UniversityBurnabyCanada
  2. 2.Faculty of EducationUniversity of HaifaHaifaIsrael

Personalised recommendations