A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

REFERENCES

• Borasi, R. (1991). Learning mathematics through inquiry. Portsmouth, NH: Heinemann.
   Google Scholar
• Coles, A., & Brown, L. (2001). Needing to use algebra. Research in Mathematics Education, 3(1), 23–36. doi:10.1080/14794800008520082.
   Google Scholar
• Conference Board of the Mathematical Sciences. (2001). Issues in Mathematics Education: Vol. 11. The mathematical education of teachers. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.
   Google Scholar
• Conference Board of the Mathematical Sciences. (2012). Issues in Mathematics Education: Vol. 17. The mathematical education of teachers II. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.
   Google Scholar
• Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For the Learning of Mathematics, 15(2), 2–9.
   Google Scholar
• Edwards, B. (1997). Undergraduate mathematics majors’ understanding and use of formal definitions in real analysis (Unpublished doctoral dissertation). The Pennsylvania State University, University Park, PA.
   Google Scholar
• Edwards, B., & Ward, M. (2004). Surprises from mathematics education research: Student (mis)use of mathematical definitions. The American Mathematical Monthly, 11, 411–424. doi:10.2307/4145268.
   Google Scholar
• Edwards, B., & Ward, M. (2008). The role of mathematical definitions in mathematics and in undergraduate mathematics courses. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 223–232). Washington, DC: Mathematical Association of America.
   Google Scholar
• Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: D. Reidel.
   Google Scholar
• Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Mahwah, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Leikin, R., & Winicki-Landman, G. (2000). On equivalent and non-equivalent definitions: Part 2. For the Learning of Mathematics, 20(2), 24–29.
   Google Scholar
• Leikin, R., & Winicki-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
• Levenson, E. (2012). Teachers’ knowledge of the nature of definitions: The case of the zero exponent. Journal of Mathematical Behavior, 31, 209–219. doi:10.1016/j.jmathb.2011.12.006.
   Google Scholar
• Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 219–248. Retrieved from http://link.springer.com/journal/10649
   Google Scholar
• Pimm, D. (1993). Just a matter of definition [Review of the book Learning mathematics through inquiry]. Educational Studies in Mathematics, 25, 261–277. doi:10.1007/BF01273865.
   Google Scholar
• Poincaré, H. (1914). Science and method. New York: Thomas Nelson and Sons.
   Google Scholar
• Reid, D. A. (1996). Enactivism as a methodology. In L. Puig & A. Gutierrez (Eds.), Proceedings of the Twentieth Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 203–209). Valencia, Spain: PME.
   Google Scholar
• Sierpinska, A. (2004). Research in mathematics education through a keyhole: Task problematization. For the Learning of Mathematics, 24(2), 7–15.
   Google Scholar
• Sowder, L. (1980). Concept and principle learning. In R. Shumway (Ed.), Research in mathematics education (pp. 244–285). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Stake, R. E. (2005). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (pp. 443–466). Thousand Oaks, CA: SAGE Publications.
   Google Scholar
• Sumara, D. J., & Davis, B. (1997). Enactivist theory and community learning: Toward a complexified understanding of action research. Educational Action Research, 5, 403–422. doi:10.1080/09650799700200037.
   Google Scholar
• 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. doi:10.1007/BF00305619.
   Google Scholar
• van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case for periodicity. Journal of Mathematical Behavior, 22, 91–106. doi:10.1016/S0732-3123(03)00006-3.
   Google Scholar
• Varela, F., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and the human experience. Cambridge, MA: MIT Press.
   Google Scholar
• Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer.
   Google Scholar
• Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Erlbaum.
   Google Scholar
• Wolcott, H. F. (1994). Transforming qualitative data: Description, analysis, and interpretation. Thousand Oaks, CA: SAGE Publications.
   Google Scholar
• Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. Journal of Mathematical Behavior, 29, 57–75. doi:10.1016/j.jmathb.2010.01.001.
   Google Scholar
• Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36, 317–346. Retrieved from http://www.nctm.org/publications/jrme.aspx
   Google Scholar
• Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69, 131–148. doi:10.1007/s10649-008-9131-7.
   Google Scholar
• Zbiek, R. M., Heid, M. K., & Blume, G. W. (2012, July). Seeing mathematics through processes and actions: Investigating teachers’ mathematical knowledge and secondary school classroom opportunities for students. Paper presented at the 12th International Congress on Mathematical Education, Seoul, Republic of Korea.
   Google Scholar