Images of Expertise in Mathematics Teaching

  • Rosemary S. Russ
  • Bruce Sherin
  • Miriam Gamoran Sherin
Chapter

Abstract

In this chapter we present a brief portrait of how researchers engaged in the study of mathematics teaching have understood teaching expertise, a portrait that is attentive to the diversity that has existed and continues to exist in the field. To do so we first adopt a historical perspective and attempt to capture some of the trends in how teaching expertise has been conceptualized, with an emphasis on how these trends were driven by broader changes in educational research. In particular, we trace the study of mathematics teaching through the traditions of process-product research, cognitive research, subject-specific cognitive research, situated cognition research, and design research. We then provide some sense for the diversity of perspectives and approaches to mathematics teaching that are currently prominent by presenting four images of mathematics teaching practice. We describe how researchers have tacitly conceived of mathematics teachers as either diagnosticians of students’ thinking, conductors of classroom discourse, architects of curriculum, or river guides who are flexible in the moments of teaching. An awareness of these images of expertise will help the field both recognize and situate new images, allowing us to use them in productive ways to further understand the work of mathematics teaching.

Keywords

Mathematics Teaching Expertise 

Notes

Acknowledgement

This research is supported by the National Science Foundation under Grant No. REC-0133900 and by a grant from the Martinson Family Foundation. The opinions expressed are those of the authors and do not necessarily reflect the views of the supporting agencies.

References

  1. Artzt, A. F., & Armour-Thomas, E. (2002). Becoming a reflective mathematics teacher: A guide for observations and self-assessment. Studies in mathematical thinking and learning. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  2. Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433–456). New York: Macmillan.Google Scholar
  3. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.CrossRefGoogle Scholar
  4. Berliner, D. C. (1994). Expertise: The wonder of exemplary performances. In J. M. Mangier & C. C. Block (Eds.), Creating powerful thinking in teachers and students: Diverse perspectives (pp. 161–186). Fort Worth, TX: Holt, Rinehart, & Winston.Google Scholar
  5. Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23, 194–222.CrossRefGoogle Scholar
  6. Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. Remillard, B. A. Herbel-Eisenmann, & G. A. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17–36). New York: Routledge.Google Scholar
  7. Brown, A. L., & Campione, J. C. (1996). Psychological theory and the design of innovative learning environments: On procedures, principles, and systems. In L. Schauble & R. Glaser (Eds.), Innovations in learning: New environments for education (pp. 289–325). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  8. Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of students: Problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 385–401.CrossRefGoogle Scholar
  9. Clark, C. M., & Yinger, R. J. (1979). Teachers’ thinking. In P. L. Peterson & H. J. Walberg (Eds.), Research on teaching: Concepts, findings, and implications (pp. 231–263). Berkeley, CA: McCutchan Publishing Corporation.Google Scholar
  10. Clark, C. M., & Yinger, R. J. (1987). Teacher planning. In J. Calderhead (Ed.), Exploring teachers’ thinking (pp. 84–103). London: Cassell.Google Scholar
  11. Dunkin, M. J., & Biddle, B. J. (1974). The study of teaching. New York: Holt, Reinhart and Winston.Google Scholar
  12. Edelson, D. C. (2002). Design research: What we learn when we engage in design. Journal of the Learning Sciences, 11(1), 105–122.CrossRefGoogle Scholar
  13. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24, 94–116.CrossRefGoogle Scholar
  14. Even, R., & Tirosh, D. (2002). Teacher knowledge and understanding of students’ mathematical learning. In L. English (Ed.), Handbook of international research in mathematics education (pp. 219–240). Mahwah, NJ: Erlbaum.Google Scholar
  15. Even, R., & Wallach, T. (2004). Between student observation and student assessment: A critical reflection. Canadian Journal of Science, Mathematics, and Technology Education, 4(4), 483–495.CrossRefGoogle Scholar
  16. Floden, R. E. (2001). Research on effects of teaching: A continuing model for research on teaching. In V. Richardson (Ed.), Handbook of research on teaching (Vol. 4, pp. 3–16). Washington, DC: American Educational Research Association.Google Scholar
  17. Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527–548.CrossRefGoogle Scholar
  18. Franke, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (2001). Capturing teachers’ generative change: A follow-up study of professional development in mathematics. American Educational Research Journal, 38(3), 653–689.CrossRefGoogle Scholar
  19. Glaser, R., & Chi, M. T. H. (1988). Overview. In M. T. H. Chi, R. Glaser, & M. Farr (Eds.), The nature of expertise (pp. xv–xxviii). Hillsdale, NJ: Erlbaum.Google Scholar
  20. Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606–634.CrossRefGoogle Scholar
  21. Hammer, D., & Schifter, D. (2001). Practices of inquiry in teaching and research. Cognition and Instruction, 19(4), 441–478.CrossRefGoogle Scholar
  22. Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance. The practitioner inquiry series. New York: Teachers College Press.Google Scholar
  23. Hufferd-Ackles, K., Fuson, K., & Sherin, M. G. (2004). Describing levels and components of a math-talk community. Journal for Research in Mathematics Education, 35(2), 81–116.CrossRefGoogle Scholar
  24. Jacobs, V., Lamb, L. C., & Philipp, R. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.Google Scholar
  25. Kazemi, E., & Franke, M. L. (2004). Teacher learning in mathematics: Using student work to promote collective inquiry. Journal of Mathematics Teacher Education, 7, 203–235.CrossRefGoogle Scholar
  26. Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven, CT: Yale University Press.Google Scholar
  27. Lappan, G. (1997a, October). Lessons from the Sputnik era in mathematics education. Paper presented at a National Academy of Sciences Symposium, Washington, DC. http://www.nas.edu/sputnik/lappan1.htm
  28. Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1997b). Comparing and scaling: Ratio, proportion, and percent – The connected mathematics projects. Palo Alto: Dale Seymour.Google Scholar
  29. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  30. Marks, R. (1989). Pedagogical content knowledge in elementary mathematics. Unpublished doctoral dissertation, Stanford University, Stanford, CA.Google Scholar
  31. Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267.CrossRefGoogle Scholar
  32. Mason, J. (2002). Researching your own practice: The discipline of noticing. London: Routledge Falmer.Google Scholar
  33. Morine-Dershimer, G. (1978–79). Planning in classroom reality: An in-depth look. Educational Research Quarterly, 3(4), 83–89.Google Scholar
  34. Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24–30.Google Scholar
  35. Nathan, M., & Knuth, E. (2003). The study of whole classroom mathematical discourse and teacher change. Cognition & Instruction, 21(2), 175–207.CrossRefGoogle Scholar
  36. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  37. Nicol, C., & Crespo, S. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355.CrossRefGoogle Scholar
  38. O’Connor, M. C. (2001). “Can any fraction be turned into a decimal?” A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143–185.CrossRefGoogle Scholar
  39. Peterson, P. L., & Clark, C. M. (1978). Teachers’ reports of their cognitive processes during teaching. American Educational Research Journal, 15(4), 555–565.Google Scholar
  40. Putnam, R. T. (1992). Teaching the “hows” of mathematics for everyday life: A case study of a fifth-grade teacher. Elementary School Journal, 93, 163–177.CrossRefGoogle Scholar
  41. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.CrossRefGoogle Scholar
  42. Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352–388.CrossRefGoogle Scholar
  43. Rex, L., Steadman, S., & Graciano, M. (2006). Researching the complexity of clasroom interaction. In J. Green, G. Camilli, & P. Elmore (Eds.), Complementary methods for research in education (pp. 727–771). Washington, DC: American Educational Research Association.Google Scholar
  44. Rowe, M. B. (1974). Wait time and rewards as instructional variables, their influence on language, logic, and fate control: Part 1 – Wait time. Journal of Research in Science Teaching, 11, 81–94.CrossRefGoogle Scholar
  45. Sawyer, R. K. (2004). Creative teaching: Collaborative discussion as disciplined improvisation. Educational Researcher, 33(2), 12–20.CrossRefGoogle Scholar
  46. Schoenfeld, A. H. (1998). Toward of theory of teaching-in-context. Issues in Education, 4(1), 1–94.CrossRefGoogle Scholar
  47. Sfard, A. (2007). When the rules of discourse change but nobody tells you: Making sense of mathematics from a commognitive standpoint. Journal of the Learning Sciences, 16(4), 565–613.Google Scholar
  48. Sherin, M. G. (2002). When teaching becomes learning. Cognition and Instruction, 20(2), 119–150.CrossRefGoogle Scholar
  49. Sherin, M. G. (2007). The development of teachers’ professional vision in video clubs. In R. Goldman, R. Pea, B. Barron, & S. Derry (Eds.), Video research in the learning sciences (pp. 383–395). Hillsdale, NJ: Erlbaum.Google Scholar
  50. Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform-based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467–500.CrossRefGoogle Scholar
  51. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.Google Scholar
  52. Silver, E. A., Ghousseini, H., Charalambous, C. Y., & Mills, V. (2009). Exploring the curriclum implementation plateau: An instructional perspective. In J. Remillard, B. A. Herbel-Eisenmann, G. A. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 245–265). New York: Routledge.Google Scholar
  53. Silver, E. A., & Smith, M. S. (1996). Building discourse communities in mathematics classrooms: A worthwhile but challenging journey. In P. C. Elliott (Ed.), 1996 Yearbook: Communication in mathematics, K–12 and beyond (pp. 20–28). Reston, VA: NCTM.Google Scholar
  54. Smith, M. S. (2000). Balancing old and new: An experienced middle school teacher’s learning in the context of mathematics instructional reform. Elementary School Journal, 100(4), 351–375.CrossRefGoogle Scholar
  55. Son, J., & Crespo, S. (2009). Prospective teachers’ reasoning and response to a students’ non-traditional strategy when dividing fractions. Journal of Mathematics Teacher Education, 12, 235–261.CrossRefGoogle Scholar
  56. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.CrossRefGoogle Scholar
  57. Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers? Journal of Mathematical Behavior, 27, 33–47.CrossRefGoogle Scholar
  58. Sykes, G. (1990). Organizing policy into practice: Reactions to the cases. Educational Evaluation and Policy Analysis, 12(3), 349–353.Google Scholar
  59. van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers’ “learning to notice” in the context of a video club. Teaching and Teacher Education, 24(2), 244–276.CrossRefGoogle Scholar
  60. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.Google Scholar
  61. Wallach, T., & Even, R. (2005). Hearing students: The complexity of understanding what they are saying, showing, and doing. Journal of Mathematics Teacher Education, 8(5), 393–417.CrossRefGoogle Scholar
  62. Williams, S., & Baxter, J. (1996). Dilemmas of discourse-oriented teaching in one middle school mathematics classroom. Elementary School Journal, 97, 21–38.CrossRefGoogle Scholar
  63. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–47.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Rosemary S. Russ
    • 1
  • Bruce Sherin
    • 1
  • Miriam Gamoran Sherin
    • 1
  1. 1.School of Education and Social Policy, Northwestern UniversityEvanstonUSA

Personalised recommendations