Multiple representations as sites for teacher reflection about mathematics learning

  • Amy E. RykenEmail author


This documentary account situates teacher educator, prospective teacher, and elementary students’ mathematical thinking in relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning skill—creating and analyzing representations. The author examines responses of prospective teachers to a visual representation task and, in turn, their examination of school children’s responses to mathematical tasks. The analysis revealed the initial tendency of prospective teachers to create pictorial representations and highlights the importance of looking beyond the pictures created to how prospective teachers use mathematical models. In addition, the challenges prospective teachers face in moving beyond a ruled-based conception of mathematics and a right/wrong framework for assessing student work are documented. Findings suggest that analyzing representations helps prospective teachers (and teacher educators) rethink their teaching practices by engaging with a culture of teaching focused on reading for multiple meanings and posing questions about student thinking and curriculum materials.


Assessment Case studies Diagrams Elementary education Inter-relating theory and practice Mathematics Problem solving Professional development Reflection Representations Teacher learning Teaching knowledge Prospective teachers Teacher educators 



I am grateful to my students for their willingness to engage in mathematical and scientific problem-solving. Peter Haslam and Kathy Heimann graciously gave permission to share excerpts of their master’s projects. Holly A. Senn, my supportive partner, and an artist, who shares my obsession for thinking in pictorial terms provided thoughtful feedback on evolving drafts of this article. My friend and colleague Joseph Flessa helped me see the implicit assumptions in my work. Three anonymous reviewers asked supportive questions and provided useful resources to enhance my theoretical analysis.


  1. Alsup, J. (2006). Teacher identity discourses: Negotiating personal and professional spaces. New Jersey: Lawrence Erlbaum Associates, Inc.Google Scholar
  2. Ameis, J. A. (2002). Stories invite children to solve mathematical problems. Teaching Children Mathematics, 8(5), 260–264.Google Scholar
  3. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241. doi: 10.1023/A:1024312321077.CrossRefGoogle Scholar
  4. Ball, D. (1997). From the general to the particular: Knowing our own students as learners of mathematics. The Mathematics Teacher, 90, 732–737.Google Scholar
  5. Ball, D. L. (1990). The mathematical understanding that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466. doi: 10.1086/461626.CrossRefGoogle Scholar
  6. Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. American Educator, 14–22, 43–46.Google Scholar
  7. Battista, M. T., & Clements, D. H. (1991). Using spatial imagery in geometric reasoning. The Arithmetic Teacher, 39(3), 18–21.Google Scholar
  8. Brown, C. A., & Borko, H. (1992). Becoming a mathematics teacher. In D. A. Grouws (Ed.), Handbook on research on mathematics teaching and learning (pp. 209–239). New York: MacMillan Publishing Company.Google Scholar
  9. Brown, M. (1961). Once a mouse. New York: Charles Scribner’s Sons.Google Scholar
  10. Costello, C. Y. (2005). Professional identity crisis: Race, class, gender, and success at professional schools. Nashville, TN: Vanderbilt University Press.Google Scholar
  11. Crespo, S. M., & Kyriakides, A. O. (2007). To draw or not to draw: Exploring children’s drawings for solving mathematics problems. Teaching Children Mathematics, 14(2), 118–125.Google Scholar
  12. Davis, E. A., & Krajcki, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. doi: 10.3102/0013189X034003003.CrossRefGoogle Scholar
  13. Edens, K., & Potter, E. (2007). The relationship of drawing and mathematical problem solving: Draw for math tasks. Studies in Art Education, 48(3), 282–298.Google Scholar
  14. Eisner, E. (2004). Preparing for today and tomorrow. Educational Leadership, 61(4), 6–10.Google Scholar
  15. Elliott, P. C. (2005). Algebra in the pre-K-2 curriculum? Teaching Children Mathematics, 12(2), 100–104.Google Scholar
  16. Feiman-Nemser, S., & Buchman, M. (1985). Pitfalls of experience in teacher preparation. Teachers College Record, 87(1), 53–65.Google Scholar
  17. Grossman, P. (2005). Research on pedagogical approaches in teacher education. In M. Cochran-Smith & K. M. Zeichner (Eds.), Studying teacher education: The report of the AERA panel on research and teacher education (pp. 425–476). Washington, DC: American Educational Research Association.Google Scholar
  18. Hamilton, M. L. (Ed.). (1998). Reconceptualizing teaching practice: Self-study in teacher education. London: Falmer Press.Google Scholar
  19. Haslam, P. (2004). “All sides equal”: Looking for meaning in a written math response. Unpublished master’s thesis, University of Puget Sound, Tacoma, WA.Google Scholar
  20. Hawking, S. (1996). A brief history of time (10th ed.). New York: Bantam.Google Scholar
  21. Hedges, M., Huinker, D., & Steinmeyer, M. (2005). Unpacking division to build teachers’ mathematical knowledge. Teaching Children Mathematics, 11(9), 478–483.Google Scholar
  22. Heimann, K. E. (2003). “It is haerd to exsplan”: Creating a system of multiple assessment and student feedback in the classroom. Unpublished master’s thesis, University of Puget Sound, Tacoma, WA.Google Scholar
  23. Herbel-Eisenmann, B., & Phillips, E. D. (2008). Analyzing students’ work: A context for connecting and extending algebraic knowledge for teaching. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 295–311). Reston, VA: The National Council of Teachers of Mathematics, Inc.Google Scholar
  24. Lee, H. (2005). Understanding and assessing preservice teachers’ reflective thinking. Teaching and Teacher Education, 21(6), 699–715. doi: 10.1016/j.tate.2005.05.007.CrossRefGoogle Scholar
  25. Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future (pp. 429–459). Rotterdam: Sense Publishers.Google Scholar
  26. McGinn, M., & Roth, W. M. (1999). Preparing students for competent scientific practice: Implications of recent research in science and technology studies. Educational Researcher, 28, 14–24.Google Scholar
  27. Mewborn, D. S. (1999). Reflective thinking among preservice elementary mathematics teachers. Journal for Research in Mathematics Education, 30(3), 316–341. doi: 10.2307/749838.CrossRefGoogle Scholar
  28. Miles, M. B., & Huberman, M. A. (1994). An expanded sourcebook: Qualitative data analysis (2nd ed.). Thousand Oaks, CA: SAGE Publications.Google Scholar
  29. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.Google Scholar
  30. Office of Superintendent of Public Instruction. (1997). Mathematics essential academic learning requirements. Retrieved March 11, 2006 from
  31. Ogawa, R. T., Crain, R., Loomis, M., & Ball, T. (2008). CHAT-IT: Toward conceptualizing learning in the context of formal organizations. Educational Researcher, 37(2), 83–95. doi: 10.3102/0013189X08316207.CrossRefGoogle Scholar
  32. Otero, V. K. (2006). Moving beyond the “get it or don’t” conception of formative assessment. Journal of Teacher Education, 57(3), 247–255. doi: 10.1177/0022487105285963.CrossRefGoogle Scholar
  33. Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future (pp. 205–235). Rotterdam: Sense Publishers.Google Scholar
  34. Ryken, A. E. (2004). A spider and a fly in a web: Seeing myself in the details of praxis. Reflective Practice, 5(1), 111–123. doi: 10.1080/1462394032000169983.CrossRefGoogle Scholar
  35. Schereer, P., & Steinbring, H. (2006). Noticing children’s learning processes—teachers jointly reflect on their own classroom interaction for improving mathematics teaching. Journal of Mathematics Teacher Education, 9, 157–185.Google Scholar
  36. Schifter, D., Bastable, V., Russel, S. J., Seyferth, L., & Riddle, M. (2008). Algebra in the grades K-5 classroom: Learning opportunities for students and teachers. In C. Greens & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 263–277). Reston, VA: The National Council of Teachers of Mathematics, Inc.Google Scholar
  37. Schoenfeld, A. H. (2003). Math wars. Retrieved March 15, 2006, from University of California, Berkeley, Graduate School of Education Web site
  38. Schön, D. A. (1983). The reflective practitioner: How professionals think in action. USA: Basic Books.Google Scholar
  39. Segall, A. (2004). Revisiting pedagogical content knowledge: The pedagogy of content/the content of pedagogy. Teaching and Teacher Education, 20(5), 489–504. doi: 10.1016/j.tate.2004.04.006.CrossRefGoogle Scholar
  40. Sierpinska, A., & Lerman, S. (1996). Epistemologies of mathematics and of mathematics education. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatric, & C. Laborde (Eds.), International handbook of mathematics education (pp. 827–876). Dordrecht: Kluwer Academic Publishers.Google Scholar
  41. Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10, 145–166. doi: 10.1007/s10857-007-9034-z.CrossRefGoogle Scholar
  42. Ticha, M., & Hospesova, A. (2006). Qualified pedagogical reflection as a way to improve mathematics education. Journal of Mathematics Teacher Education, 9, 129–156. doi: 10.1007/s10857-006-6893-7.CrossRefGoogle Scholar
  43. Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston, MA.: Pearson Education, Inc.Google Scholar
  44. Ward, J. R., & McCotter, S. S. (2004). Reflection as a visible outcome for preservice teachers. Teaching and Teacher Education, 20, 243–257. doi: 10.1016/j.tate.2004.02.004.CrossRefGoogle Scholar
  45. Wheatley, G. H. (1991). Enhancing mathematics learning through imagery. The Arithmetic Teacher, 39(1), 34–36.Google Scholar
  46. Young, E., & Marroquin, C. L. (2006). Posing problems from children’s literature. Teaching Children Mathematics, 12(7), 362–366.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.University of Puget SoundTacomaUSA

Personalised recommendations