Journal of Mathematics Teacher Education

, Volume 5, Issue 2, pp 153–186 | Cite as

Understanding and Learning-to-explain by Representing Mathematics: Epistemological Dilemmas Facing Teacher Educators in the Secondary Mathematics ``Methods'' Course

  • Barbara M. Kinach


Building on the work of Ball and McDiarmid,this study provides an equivalent at thesecondary level to the work of Liping Ma at theelementary level in that it provides a betterunderstanding of the conceptual knowledge ofschool mathematics held by prospectivesecondary teachers, along with examples of thesorts of knowledge needed to teach forunderstanding within the domain of integersubtraction. Part of an eight-yearlongitudinal study of secondary teachercandidates' conceptions of instructionalexplanations, this analysis of interaction inthe author's methods course and its discussionof epistemological obstacles and changescombines subject-matter and interactionistperspectives. The author concludes thatsecondary teacher candidates can deepen theirrelational knowledge of secondary mathematicswithin a methods course by focusing oninstructional explanations.


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  1. Ball, D.L. (1988). The subject-matter preparation of prospective mathematics teachers: Challenging the myths. Issue paper 88-3. East Lansing, MI: National Center for Research on Teacher Education.Google Scholar
  2. Ball, D.L. (1988). Unlearning to teach mathematics. Issue paper 88-1. East Lansing, MI: National Center for Research on Teacher Education.Google Scholar
  3. Ball, D.L. (1990). Breaking with experience in learning to teach mathematics: The role of a pre-service methods course. For the Learning of Mathematics, 10(2), 10–16.Google Scholar
  4. Ball, D.L. (1992). The mathematical understandings that prospective teachers bring to teacher education. In J. Brophy (Ed.), Advances in research on teaching (Volume 2, 1–48). Greenwich, CT: JAI.Google Scholar
  5. Ball, D.L. & McDiarmid, G.W. (1990). The subject-matter preparation of teachers. In W. Robert Houston (Ed.), Handbook of research on teacher education: A project of the Association of Teacher Educators. New York: Macmillan.Google Scholar
  6. Brown, H.I. (1977). Perception, theory and commitment: The new philosophy of science. Chicago: University of Chicago Press.Google Scholar
  7. Dewey, J. (1916/1944). Democracy and education. New York: The Free Press.Google Scholar
  8. Donald, J.G. (1991). Knowledge and the university curriculum. In C.F. Conrad & J.G. Haworth (Eds.), Curriculum in transition: Perspectives on the undergraduate experience (295–307). Needham Heights, MA: Ginn Publishing.Google Scholar
  9. Glaser, B. & Strauss, A. (1967). The discovery of grounded theory. New York: Aldine.Google Scholar
  10. Hershkowitz, R., Baruch, B.S. & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222.CrossRefGoogle Scholar
  11. Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics.Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  12. Kinach, B.M. (1996). Logical trick or mathematical explanation? Re-negotiating the epistemological stumbling blocks of pre-service teachers in the secondary mathematics methods course. Proceedings of the eighteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Volume 2 (414–420). Columbus, OH: Ohio State University.Google Scholar
  13. Kinach, B.M. (2001). Assessing, challenging, and developing prospective teachers' pedagogical content knowledge and beliefs: A role for instructional explanations. In T. Ariav, A. Keinan & R. Zuzovsky (Eds.), The ongoing development of teacher education: Exchange of ideas. Tel Aviv, Israel: The Mofet Institute.Google Scholar
  14. Kinach, B.M. (2002). A cognitive strategy for developing prospective teachers' pedagogical content knowledge in the secondary mathematics methods course: Toward a model of effective practice. Teaching and Teacher Education, 18(1), 51–71.CrossRefGoogle Scholar
  15. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  16. Martin, J.R. (1970). Explaining, understanding and teaching. New York: McGraw Hill.Google Scholar
  17. McDiarmid, G.W. (1990). Challenging prospective teachers' beliefs during an early field experience: A quixotic undertaking? Journal of Teacher Education, 41(3), 12–20.CrossRefGoogle Scholar
  18. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for mathematics. Reston,VA: author.Google Scholar
  19. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston,VA: author.Google Scholar
  20. Pape, S.J. & Tchoshanov, M.A. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118–127.CrossRefGoogle Scholar
  21. Perkins, D.N. (1992). Smart schools: Better thinking and learning for every child.New York: Free Press.Google Scholar
  22. Perkins, D.N. & Simmons, R. (1988). Patterns of misunderstanding: An integrative model for science, math, and programming. Review of Educational Research, 58(3), 303–326.Google Scholar
  23. Pimm, D. (1995). Symbols and meanings in school mathematics. New York: University of Oxford Press.CrossRefGoogle Scholar
  24. Schwab, J.J. (1978). Education and the structure of the disciplines. In I. Westbury & N.J. Wilkof (Eds.), Science, curriculum and liberal education: Selected essays (229–272). Chicago: University of Chicago Press.Google Scholar
  25. Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–22.Google Scholar
  26. Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10(3), 24–41.Google Scholar
  27. Skemp, R.R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26a.Google Scholar
  28. Skemp, R.R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26(3), 9–15.Google Scholar
  29. Stodolsky, S.S. (1985). Telling math: Origins of math aversion and anxiety. Educational Psychologist, 20(3), 125–133.CrossRefGoogle Scholar
  30. Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education, 1, 157–189.CrossRefGoogle Scholar
  31. Tall, D. (1978). The dynamics of understanding mathematics. Mathematics Teaching, 84, 50–52.Google Scholar
  32. Thompson, A.G. (1984). The relationship of teachers' conceptions of mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105–127.CrossRefGoogle Scholar
  33. Thompson, A.G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. InDouglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (127–146). New York: Macmillan.Google Scholar
  34. Watanabe, T. & Kinach, B. (1997). Creating a community of inquiry in an undergraduate mathematics course for prospective middle grade teachers: Voices from MCTP. Paper presented at the American Educational Research Association, Chicago.Google Scholar
  35. Winicki-Landman, G. & Leikin, R. (2000). On equivalent and non-equivalent definitions: Part 1. For the Learning of Mathematics, 20(1), 17–21.Google Scholar
  36. Wiske, M.S. (Ed.) (1998). Teaching for understanding: Linking research with practice. San Francisco, CA: Jossey-Bass.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Barbara M. Kinach
    • 1
  1. 1.Department of EducationUniversity of MarylandBaltimoreUSA

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