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Journal of Mathematics Teacher Education

, Volume 22, Issue 2, pp 129–151 | Cite as

Teachers’ conceptions of prior knowledge and the potential of a task in teaching practice

  • Hyung Sook LeeEmail author
  • Jacqueline Coomes
  • Jaehoon Yim
Article

Abstract

In this study, we specify teacher’s knowledge and beliefs that influence their enactment of mathematics tasks. The use of tasks in lessons requires teachers to consider students’ mathematical thinking and to know how to effectively implement tasks. Among the many aspects of teachers’ understanding of students and ability to implement tasks for learning, this study focuses on teachers’ knowledge and beliefs of students’ prior knowledge and the potential to develop new knowledge while solving mathematical tasks; we exemplify these knowledge and beliefs through the practices of three Algebra I teachers. Three distinct conceptions of prior knowledge and two conceptions of the potential of a task emerged. From combinations of these conceptions, we defined three types of teaching. Two types of teaching reduced potential new knowledge to the prior context, whereas one type of teaching promoted prior knowledge to develop new knowledge. In particular, the type of teaching we call “reviewing” explains why teachers repeat the way they taught a concept the first time. Our study suggests that it is important for teachers to conceive students’ prior knowledge as the knowledge to be developed and the task as having potential for developing new knowledge in order to teach for coherence.

Keywords

Teacher knowledge Teacher conception Prior knowledge Mathematical potential of tasks Teaching practice Mathematical knowledge used in teaching 

Notes

Acknowledgements

This article is based on a professional development project funded by a Grant authorized under Title II Part A Subpart 3 and Title II Part B of the Elementary and Secondary Education Act, administered at the federal level by the US Department of Education and at the state level by the Washington Student Achievement Council and Office of Superintendent of Public Instruction, respectively. These federal and state agencies are not responsible for statements made in the paper

References

  1. Askew, M. (2008). Mathematical discipline knowledge requirements for prospective primary teachers, and the structure and teaching approaches of programs designed to develop that knowledge. In P. Sullivan & T. Wood (Eds.), Knowledge and beliefs in mathematics teaching and teaching development (pp. 13–35). Rotterdam: Sense publishers.Google Scholar
  2. Askew, M., & Brown, M. (1997). Effective teachers of numeracy in UK primary schools: Teachers’ beliefs, practices and pupils’ learning. Paper presented at the European conference on educational research (ECER 97), Johann Wolfgang Goethe Universitat, Frankfurt am Main.Google Scholar
  3. Askew, M., Brown, M., Rhodes, V., Johnson, D., and William, D. (1997). Effective teachers of numeracy. Final report. London: King’s College.Google Scholar
  4. Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (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). Washington, DC: American Educational Research Association.Google Scholar
  5. 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
  6. Bennett, N., & Carre, C. (Eds.). (1993). Learning to teach. London: Routledge.Google Scholar
  7. Bennett, N., & Desforges, C. (1988). Matching classroom tasks to students’ attainments. Elementary School Journal, 88(3), 221–234.CrossRefGoogle Scholar
  8. Bishop, J. P., Lamb, L. C., Philipp, R. A., Whitacre, I., Schappelle, B. P., & Lewis, M. L. (2014). Obstacles and affordances for integer reasoning from the analysis of children’s thinking and the history of mathematics. Journal for Research in Mathematics Education, 45(1), 19–61.CrossRefGoogle Scholar
  9. 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(3), 194–222.CrossRefGoogle Scholar
  10. Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds., Trans.). Dordrecht: Kluwer.Google Scholar
  11. Brown, M., Askew, M., Baker, D., Denvir, H., & Millett, A. (1998). Is the national numeracy strategy research-based? British Journal of Educational Studies, 46(4), 362–385.CrossRefGoogle Scholar
  12. Calderhead, J. (1996). Teachers: Beliefs and knowledge. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 709–725). New York: Simon & Schuster Macmillan.Google Scholar
  13. Carlson, M., & Oehrtman, M. (2005) Key aspects of knowing and learning the concept of function. Research sampler series, MAA notes online.Google Scholar
  14. Carpenter, T., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3–20.CrossRefGoogle Scholar
  15. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.Google Scholar
  16. Charalambous, C. Y., (2008). Mathematical knowledge for teaching and the unfolding of tasks in mathematics lessons: Integrating two lines of research. In O. Figuras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (EDS.), Proceedings of the 32nd annual conference of the international group for the psychology of mathematics education, (Vol. 2, pp. 281–288).Morelia: PME.Google Scholar
  17. Charmaz, K. (2002). Qualitative interviewing and grounded theory analysis. In J. Gubrium & J. A. Holstein (Eds.), Handbook of interview research (pp. 675–694). Thousand Oaks: Sage.Google Scholar
  18. Clark, C. M., & Peterson, P. L. (1986). Teachers’ thought processes. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 255–296). New York: Macmillan.Google Scholar
  19. Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12(3), 311–329.CrossRefGoogle Scholar
  20. Cohen, D. K., & Ball, D. L. (1990). Policy and practice: An overview. Educational Evaluation and Policy Analysis, 12(3), 233–239.CrossRefGoogle Scholar
  21. Common Core State Standards Initiative (CCSSI). 2010 Common core state standards for mathematics. Retrieved December 10, 2015, from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf.
  22. Desforges, C., & Cockburn, A. (1987). Understanding the mathematics teachers: A study of practice in first schools. London: The Palmer Press.Google Scholar
  23. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180.CrossRefGoogle Scholar
  24. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434.CrossRefGoogle Scholar
  25. Franke, M. L., Kazemi, E., & Battey, D. (2007). Mathematics teaching and classroom practice. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 225–256). Greenwich, CT: Information Age.Google Scholar
  26. Gibson, J. J. (1986). The ecological approach to visual perception. Hillsdale, NJ: Erlbaum.Google Scholar
  27. Heinz, K., Kinzel, M., Simon, M. A., & Tzur, R. (2000). Moving students through steps of mathematical knowing: An account of the practice of an elementary mathematics teacher in transition. Journal of Mathematical Review, 19, 83–107.Google Scholar
  28. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 424–549.CrossRefGoogle Scholar
  29. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.Google Scholar
  30. Hiebert, J., & Stigler, J. W. (2000). A proposal for improving classroom teaching: Lessons from the TIMSS video study. Elementary School Journal, 101, 3–20.CrossRefGoogle Scholar
  31. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008a). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.Google Scholar
  32. Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., et al. (2008b). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511.CrossRefGoogle Scholar
  33. Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.CrossRefGoogle Scholar
  34. Klug, J., Bruder, S., Kelava, A., Spiel, C., & Schmitz, B. (2013). Diagnostic competence of teachers: A process model that accounts for diagnosing learning behavior tested by means of a case scenario. Teaching and Teacher Education, 30, 38–46. doi: 10.1016/j.tate.2012.10.004.CrossRefGoogle Scholar
  35. Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500–507.CrossRefGoogle Scholar
  36. Leikin, R., & Zazkis, R. (2010). Teachers’ opportunities to learn mathematics through teaching. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 3–21). New York: Springer.CrossRefGoogle Scholar
  37. Liljedahl, P., Rolka, K., & Rösken, B. (2007). Affecting affect: The re-education of preservice teachers’ beliefs about mathematics and mathematics learning and teaching. In M. Strutchens & W. Martin (Eds.) 69th NCTM Yearbook. NCTM.Google Scholar
  38. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  39. National Board for Professional Teaching Standards. (2001). NBPTS early childhood generalist standards (2nd ed.). Arlington, VA: National Board for Professional Teaching Standards.Google Scholar
  40. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.Google Scholar
  41. Peterson, P. L. (1990). Doing more in the same amount of time: Cathy Swift. Educational Evaluation and Policy Analysis, 12, 261–280.CrossRefGoogle Scholar
  42. Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Charlotte, NC: Information Age.Google Scholar
  43. Polly, D., McGee, J. R., Wang, C., Lambert, R. G., Pugalee, D. K., & Johnson, S. (2013). The association between teachers’ beliefs, enacted practices, and student learning in mathematics. The Mathematics Educator, 22(2), 11–30.Google Scholar
  44. Polly, D., Neale, H., & Pugalee, D. K. (2014). How does ongoing task-focused mathematics professional development influence elementary school teacher’s knowledge, beliefs and enacted pedagogies? Early Childhood Education Journal. doi: 10.1007/s10643-013-0585-6.CrossRefGoogle Scholar
  45. Schwartz, D. L., Sears, D., & Chang, J. (2007). Reconsidering prior knowledge. In M. C. Lovett & P. Shah (Eds.), Thinking with data (pp. 319–344). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  46. Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 24(3), 233–254.CrossRefGoogle Scholar
  47. Simon, M. A., & Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics, 22, 309–331.CrossRefGoogle Scholar
  48. Simon, M. A., & Schifter, D. (1993). Toward a constructivist perspective: The impact of a mathematics teacher inservice program on students. Educational Studies in Mathematics, 25, 331–340.CrossRefGoogle Scholar
  49. Simon, M. A., & Tzur, R. (1999). Explicating the teacher’s perspective from the researchers’ perspectives: Generating accounts of mathematics teachers’ practice. Journal for Research in Mathematics Education, 30(3), 252–264.CrossRefGoogle Scholar
  50. Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20, 267–307.CrossRefGoogle Scholar
  51. Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–370). Greenwich, CT: Information Age Publishing.Google Scholar
  52. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques (2nd ed.). Newbury Park, CA: Sage.Google Scholar
  53. Sullivan, P. (2008). Knowledge for teaching mathematics. In P. Sullivan & T. Wood (Eds.), Knowledge and beliefs in mathematics teaching and teaching development (Vol. 1, pp. 1–9). Dordrecht: Sense Publishers.Google Scholar
  54. Swan, M. (2006). Designing and using research instruments to describe the beliefs and practices of mathematics teachers. Research in Education, 75, 58–70.CrossRefGoogle Scholar
  55. Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: A design research study. Journal of Mathematics Teacher Education. doi: 10.1007/s10857-007-9038-8.CrossRefGoogle Scholar
  56. Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction toward a theory of teaching. Educational Researcher, 41(5), 147–156.CrossRefGoogle Scholar
  57. Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105–127.CrossRefGoogle Scholar
  58. Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  59. Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany, NY: SUNY Press.Google Scholar
  60. Thompson, P. W. (2015). Researching mathematical meanings for teaching. In L. English & D. Kirshner (Eds.), Third handbook of international research in mathematics education (pp. 435–461). London: Taylor and Francis.Google Scholar
  61. Tzur, R. (2008). Profound awareness of the learning paradox: A journey towards epistemologically regulated pedagogy in mathematics teaching and teacher education. In B. Jaworski & T. Wood (Eds.), The international handbook for mathematics teacher education: The mathematics teacher educator as a developing professional (Vol. 4, pp. 137–156). Rotterdam: Sense.Google Scholar
  62. Tzur, R. (2010). How and what might teachers learn through teaching mathematics: Contributions to closing an unspoken gap. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 49–67). New York: Springer.CrossRefGoogle Scholar
  63. Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumption about tasks in teacher education. Journal of Mathematics Teacher Education, 10(4), 205–215.CrossRefGoogle Scholar
  64. Watson, A., & Ohtani, M. (2015). Task design in mathematics education: An ICMI study 22. Berlin: Springer.CrossRefGoogle Scholar
  65. Wilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2014). Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. Journal of Mathematics Teacher Education, 17(2), 149–175.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Eastern Washington UniversityCheneyUSA
  2. 2.Gyeongin National University of EducationIncheonKorea

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