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Journal of Mathematics Teacher Education

, Volume 18, Issue 4, pp 327–351 | Cite as

Secondary teachers’ conception of various forms of complex numbers

  • Gulden KarakokEmail author
  • Hortensia Soto-Johnson
  • Stephenie Anderson Dyben
Article

Abstract

This study explores in-service high school mathematics teachers’ conception of various forms of complex numbers and ways in which they transition between different representations of these forms. One 90-min interview was conducted with three high school mathematics teachers after they completed three professional development sessions, each 4 h, on complex numbers. Results indicate that, in general, these teachers did not necessarily have a dual conception of complex numbers. However, they demonstrated varying conceptions with different forms of complex numbers. Teachers worked at an operational level with the exponential form of complex numbers, but there was no evidence to indicate that they had a structural conception of this form. On the other hand, two teachers were very comfortable with the Cartesian form and exhibited a process/object duality by translating between different representations of this form. These results indicate that high school teachers need more opportunities to help them develop a dual conception of each form (multiple duals), which in turn can result in developing a dual conception of complex numbers. An interesting phenomenon that we found was that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks. This particular observation may suggest that teachers’ teaching assignments coupled with appropriate professional development activities could facilitate their understanding of these concepts.

Keywords

Complex numbers Mathematical content knowledge Operational conception Process/object duality Representations Structural conception 

References

  1. Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problems of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433–456). New York: Macmillan.Google Scholar
  2. Council of Chief State School Officers & National Governors Association Center for Best Practices (2010). Common core state standards for mathematics. Common Core State Standards Initiative. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
  3. Conner, M. E., Rasmussen, C., Zandieh, M., & Smith, M. (2007). Mathematical knowledge for teaching: The case of complex numbers. In Proceedings for the tenth special interest group of the mathematical association of America on research in undergraduate mathematics education. San Diego, CA.Google Scholar
  4. Cuoco, A. A., & Curcio, R. F. (2001). The roles of representation in school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  5. Danenhower, P. (2000). Teaching and learning complex analysis at two-British Columbia universities. Dissertation Abstract International 62(09). (Publication Number 304667901). Retrieved March 5, 2011 from ProQuest Dissertations and Theses database.Google Scholar
  6. Dannenhower, P. (2006). Introductory complex analysis at two British Columbia universities: The first week- complex numbers. In F. Hitt, G. Harel, & A. Selden (Eds.), Research in collegiate mathematics education VI (pp. 139–169). Rhode Island: AMS.Google Scholar
  7. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In Advanced mathematical thinking (pp. 95–126). Netherlands: Springer.Google Scholar
  8. English, L. D. (1997). Mathematical reasoning: Analogies, metaphors and images. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  9. Erickson, F. (2006). Definition and analysis of data from videotape: Some research procedures and their rationales. In J. L. Green, G. Camilli, & P. B. Elmore (Eds.), Handbook of complementary methods in educational research (pp. 177–191). Washington, DC: American Educational Research Association.Google Scholar
  10. Goldin, G. A. (2002). Representation in mathematical learning and problem solving. In L. English (Ed.), Handbook of international research in mathematics education (pp. 197–218). Mahwah, NJ: Lawrence Erlbaum Associates Inc.Google Scholar
  11. Goldin, G. & Shteingold, N. (2001). System of representations and the development of mathematics concepts. In A. Cuoco & F. R. Curcio (Eds.), The roles of representations in school mathematics (pp. 1–23). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  12. Harel, G. (2007). The DNR system as a conceptual framework for curriculum development and instruction. In R. Lesh, J. Kaput, & E. Hamilton (Eds.), Foundations for the future in Mathematics education (pp. 263–280). Mahwah, NJ: Erlbaum.Google Scholar
  13. Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts. Educational Studies in Mathematics, 40(1), 71–90.CrossRefGoogle Scholar
  14. Izsak, A., & Sherin, M. G. (2003). Exploring the use of new representations as a resource for teacher learning. School Science and Mathematics, 103, 18–27.Google Scholar
  15. Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. Highsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  16. Kaput, J. (1987). Representation systems in mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of Mathematics (pp. 19–26). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  17. Kieran, C. 1992. The learning and teaching of school Algebra. In Grouws, D. (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York.Google Scholar
  18. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press.Google Scholar
  19. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  20. Maracci, M. (2008). Combining different theoretical perspectives for analyzing students’ difficulties in vector spaces theory. ZDM, 40(2), 265–276.CrossRefGoogle Scholar
  21. Nahin, P. J. (1998). An imaginary tale: The story of \(\sqrt { - 1}\). Princeton, NJ: Princeton University Press.Google Scholar
  22. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  23. Nemirovsky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal for the Learning Sciences, 21(2), 287–323.CrossRefGoogle Scholar
  24. Panaoura, A., Elia, I., Gagatsis, A., & Giatilis, G. (2006). Geometric and algebraic approaches in the concept of complex numbers. International Journal of Mathematical Education in Science and Technology, 37(6), 681–706.CrossRefGoogle Scholar
  25. Patton, M. Q. (2002). Qualitative research and evaluation methods, 3rd. ed. Thousand Oaks, CA: SAGE.Google Scholar
  26. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies, 22(1), 1–36.CrossRefGoogle Scholar
  27. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: the case of function. In Harel, G., & Dubinsky, E. (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes 25 (pp. 59–84). Washington: MA.Google Scholar
  28. Soto-Johnson, H., Oehrtman, M., & Rozner, S. (2011). Dynamic visualization of complex variables: The case of Ricardo. In Proceeding from the 14th annual conference on research in undergraduate mathematics education (pp. 488–503).Google Scholar
  29. Soto-Johnson, H., Oehrtman, M., Noblet, K., Roberson, L., & Rozner, S. (2012). Experts’ reification of complex variables concepts: The role of metaphor. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of the 15th annual conference on research in undergraduate mathematics education (pp. 443–447). Portland, Oregon.Google Scholar
  30. Sowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Lenhardt, R. Putnam, & R. A., Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 1–52). Mahwah, NJ: Erlbaum.Google Scholar
  31. Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education. IV. CBMS Issues in Mathematics Education, 103–127.Google Scholar
  32. Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. P. (2007). Research on technology in mathematics education: A perspective of constructs. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1169–1207). Charlotte, NC: Information Age.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Gulden Karakok
    • 1
    Email author
  • Hortensia Soto-Johnson
    • 1
  • Stephenie Anderson Dyben
    • 1
  1. 1.University of Northern ColoradoGreeleyUSA

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