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.
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References
Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth, NH: Heinemann Educational Books.
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.
Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283–299.
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.
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.
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.
Fischbein, E. (1987). Intuition in science and mathematics. An educational approach. Dordrecht: Kluwer.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
Khinchin, A. Y. (1968). The teaching of mathematics. London: The English Universities Press.
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.
Leikin, R., & Winicky-Landman, G. (2000). On equivalent and nonequivalent definitions II. For the Learning of Mathematics, 20(2), 24–29.
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.
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.
Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34(3), 219–248.
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.
Musser, G. L., Burger, W. F., & Peterson, B. E. (2006). Mathematics for elementary teachers: A contemporary approach. New York: Wiley.
Pimm, D. (1993). Just a matter of definition. Educational Studies in Mathematics, 25, 261–277.
Poincare, H. (1909/1952). Science and method. New York, NY: Dover.
Solow, D. (1984). Reading, writing and doing mathematical proofs. Book I. Palo Alto, CA: Dale Seymour.
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.
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.
van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. Journal of Mathematical Behavior, 22(1), 91–106.
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.
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.
Vinner, S., Linchevski, L., & Karsenty, R. (1993). How much information should include a geometrical definition? Zentralblatt für Didaktik der Mathematik, 25, 164–170.
Vygotsky, L. S. (1982). Mishlenie i rech (Thought and language). In L. S. Vygotsky (Ed.), Sobranie Sochinenii, t.2. Moscow: Pedagogika (in Russian).
Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum.
Weinstein, E. (2007a). Wolfram MathWorld: Polygon. Retrieved March 15, 2008 from http://www.mathworld.wolfram.com/Polygon.html.
Weinstein, E. (2007b). Wolfram MathWorld: Square. Retrieved March 15, 2008 from http://www.mathworld.wolfram.com/Square.html.
Winicky-Landman, G., & Leikin, R. (2000). On equivalent and nonequivalent definitions I. For the Learning of Mathematics, 20(1), 17–21.
Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.
Zazkis, R., & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27, 11–17.
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Zazkis, R., Leikin, R. Exemplifying definitions: a case of a square. Educ Stud Math 69, 131–148 (2008). https://doi.org/10.1007/s10649-008-9131-7
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DOI: https://doi.org/10.1007/s10649-008-9131-7