Mathematics Education Research Journal

, Volume 25, Issue 2, pp 257–278 | Cite as

Primary teachers’ representations of division: assessing mathematical knowledge that has pedagogical potential

Original Article

Abstract

This article reports on a study that was conducted with 378 primary teachers from Catholic schools in Victoria who participated in the first year of a 2-year research and professional learning program in mathematics. One aim of the program was to enhance teacher knowledge in mathematics in its many forms. As part of the larger study, the teachers were assessed at the beginning and the end of school year (February and October, respectively) on their Mathematical Knowledge for Teaching (MKT), through the use of a questionnaire involving teachers’ responses to hypothetical teaching, planning, or assessment scenarios. We report here the results from one item that assessed teachers’ MKT in relation to representations of division. Results indicated that teachers were more familiar with partitive than quotitive division, and found connecting appropriate story problems with a given form of division difficult. Teachers’ relating their understanding of the forms of division to the context of division by a decimal number was also challenging. There were interesting variations in the data across primary grade levels, particularly in relation to change over time. Professional learning on these topics and other support within the project appeared to improve teachers’ MKT in this area.

Keywords

Teacher knowledge Mathematical knowledge for teaching Teacher professional learning Assessment Division Story problems 

References

  1. Anghileri, J. (1995). Language, arithmetic, and negotiations of meaning. For the Learning of Mathematics, 15(3), 10–14.Google Scholar
  2. Australian Curriculum, Assessment and Reporting Authority (ACARA). (2008). National assessment program literacy and numeracy (NAPLAN). Sydney, NSW: Ministerial Council for Education Early Childhood Development and Youth Affairs.Google Scholar
  3. Australian Curriculum, Assessment and Reporting Authority (ACARA). (2009). National assessment program literacy and numeracy (NAPLAN). Sydney, NSW: Ministerial Council for Education Early Childhood Development and Youth Affairs.Google Scholar
  4. Ball, D. (1990a). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466.CrossRefGoogle Scholar
  5. Ball, D. (1990b). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132–144.CrossRefGoogle Scholar
  6. Ball, D., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83–104). Westport, CT: Ablex.Google Scholar
  7. Ball, D. L., Hill, H. C., Bass, B. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14–46.Google Scholar
  8. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59, 389–407.CrossRefGoogle Scholar
  9. Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in verbal arithmetic problems: the effects of number size, problem structure and context. Educational Studies in Mathematics, 15, 129–147.CrossRefGoogle Scholar
  10. Bell, C. A., Wilson, S. M., Higgins, T., & McCoach, D. B. (2010). Measuring the effects of professional development on teacher knowledge: the case of developing mathematical ideas. Journal for Research in Mathematics Education, 41(5), 479–512.Google Scholar
  11. Boaler, J. (2003). Studying and capturing the complexity of practice: the case of the ‘dance of agency’. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 2003 psychology of mathematics education group North America (Vol. 1, pp. 3–16). Hawaii: University of Hawaii.Google Scholar
  12. 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 to easily? Journal for Research in Mathematics Education, 23(3), 194–222.CrossRefGoogle Scholar
  13. Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211–230.CrossRefGoogle Scholar
  14. Chick, H., Baker, M., Pham, T., & Cheng, H. (2006). Aspects of teachers’ pedagogical content knowledge for decimals. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 297–304). Prague: PME.Google Scholar
  15. Clarke, D. M. (2011). Do demonstration lessons work? (Professor’s page). Australian Primary Mathematics Classroom, 16(2), 12–13.Google Scholar
  16. Clarke, D. M., Roche A., Downton, A. (2009). Assessing teacher pedagogical content knowledge: Challenges and insights. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), In search of theories in mathematics education (Proceedings of the 33rd Conference of the International Group of Psychology of Mathematics Education, Vol. 1, p. 362). Thessaloniki, Greece: PME.Google Scholar
  17. Clarke, D. M., Downton, A., Clarkson, P., Roche, A., Sexton, M., Hamilton, L., Brown, J., Horne, M., McDonough, A., Wilkie, K., & Wright, V. (2012). Contemporary teaching and learning of mathematics: report to CEO Melbourne 2011. Melbourne: Mathematics Teaching and Learning Research Centre, Australian Catholic University.Google Scholar
  18. Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415.CrossRefGoogle Scholar
  19. Downton, A. (2009). It seems to matter not whether it is partitive or quotitive division when solving one step division problems. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides (proceedings of the 32nd annual conference of the mathematics education research group of Australasia (pp. 161–168). Palmerston North, NZ: MERGA.Google Scholar
  20. Ellerton, N. F. (1988). Exploring children’s perceptions of mathematics through letters written by children. In A. Borbas (Ed.), Proceedings of the 12th international conference of the psychology of mathematics education (Vol. 1, pp. 280–287). Veszprem, Hungary: International Group for the Psychology of Mathematics.Google Scholar
  21. Fennema, E., & Franke, M. (1992). Teachers’ knowledge and its impact. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147–164). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  22. 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
  23. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17.CrossRefGoogle Scholar
  24. Graeber, A. O., Tirosh, D., & Glover, R. (1989). Preservice teachers’ misconceptions in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 20(1), 95–102.CrossRefGoogle Scholar
  25. Greer, B. (1997). Modeling reality in the mathematics classrooms: the case of word problems. Learning and Instruction, 7(4), 293–307.CrossRefGoogle Scholar
  26. Hargreaves, A. (1994). Changing teachers, changing times: teachers’ work and culture in the postmodern age. London: Cassell.Google Scholar
  27. Hattie, J. A. (2002). What are the attributes of excellent teachers? In Teachers make a difference: what is the research evidence (pp. 3–26). Wellington: New Zealand Council for Educational Research.Google Scholar
  28. Hill, H. C. (2010). The nature and predictors of elementary teachers’ mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41(5), 513–545.Google Scholar
  29. Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11–30.CrossRefGoogle Scholar
  30. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.CrossRefGoogle Scholar
  31. Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: what knowledge matters and what evidence counts. In K. F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111–155). Reston, VA: NCTM.Google Scholar
  32. Hill, H., Ball, D., & Schilling, S. (2008). Unpacking pedagogical content knowledge: conceptualising and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.Google Scholar
  33. Li, Y., & Huang, R. (2008). Chinese elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division. ZDM Mathematics Education, 40, 845–859.CrossRefGoogle Scholar
  34. Li, Y., & Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: the case of fraction division. ZDM Mathematics Education, 40, 833–843.CrossRefGoogle Scholar
  35. Li, Y., Ma, Y., & Pang, J. (2008). Mathematical preparation of prospective elementary teachers. In P. Sullivan & T. Wood (Eds.), International handbook of mathematics teacher education (Knowledge and beliefs in mathematics teaching and teaching development, Vol. 1, pp. 37–62). Rotterdam: Sense.Google Scholar
  36. Ma, L. (1999). Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.Google Scholar
  37. Maier, E. (1991). Folk mathematics. In M. Harris (Ed.), School, mathematics and work (pp. 62–66). London: Palmer.Google Scholar
  38. McIntosh, A., Reys, B. J., Reys, R. E., Bana, J., & Farrell, B. (1997). Number sense in school mathematics: student performance in four countries. Perth WA: Mathematics, Science & Technology Education Centre.Google Scholar
  39. Mousley, J. (2000). Understanding multiplication concepts. Australian Primary Mathematics Classroom, 5(3), 26–28.Google Scholar
  40. Mulligan, J. (1992). Children’s solutions to multiplication and division word problems: a longitudinal study. Mathematics Education Research Journal, 4(1), 24–41.CrossRefGoogle Scholar
  41. Newton, K. J. (2008). An extensive analyses of preservice elementary teachers’ knowledge of fractions. American Education Research Journal, 45(4), 1080–1110.CrossRefGoogle Scholar
  42. Roche, A. (in press). Making the story problem fit the mathematics: Choosing, creating, and using story problems: Some helpful hints. Australian Primary Mathematics Classroom.Google Scholar
  43. Roche, A., & Clarke, D. (2009). Making sense of partitive and quotitive division: a snapshot of teachers’ pedagogical content knowledge. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides (proceedings of the 32nd annual conference of the mathematics education research group of Australasia (pp. 467–474). Palmerston North: MERGA.Google Scholar
  44. Roche, A., & Clarke, D. (2011). Some lessons learned from the experience of assessing teacher pedagogical content knowledge in mathematics. In J. Clark, B. Kissane, J. Mousley, T. Spencer, & S. Thornton (Eds.), Mathematics: traditions and [new] practices (Proceedings of the 23rd biennial conference of the Australian Association of Mathematics Teachers Inc. and the 34th annual conference of the Mathematics Education Research Group of Australasia Inc., Vol. 2, pp. 658–666). Adelaide: MERGA.Google Scholar
  45. Seago, N., & Goldsmith, L. (2006). Learning mathematics for teaching. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 5, pp. 73–80). Prague: PME.Google Scholar
  46. Senk, S., Tatto, M. T., Reckase, M., Rowley, G., Peck, R., & Bankov, K. (2012). Knowledge of future primary teachers for teaching mathematics: an international comparative study. ZDM—The International Journal of Mathematics Education, 44(3), 307–324.CrossRefGoogle Scholar
  47. Shulman, L. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  48. Shulman, L. (1987). Knowledge and teaching: foundations of the new reform. Harvard Educational Review, 57(1), 1–22.Google Scholar
  49. Silver, E. A., Mamona-Downs, J., Leung, S. S., & Kenny, P. A. (1996). Posing mathematical problems: an exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.CrossRefGoogle Scholar
  50. Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499–511.CrossRefGoogle Scholar
  51. Simon, M. A. (1993). Prospective elementary teachers’ knowledge of divison. Journal for Research in Mathematics Education, 24(3), 233–254.CrossRefGoogle Scholar
  52. Son, J., & Crespo, S. (2009). Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions. Journal of Mathematical Teacher Education, 12, 235–261.CrossRefGoogle Scholar
  53. Swan, M. (1983). The meaning and use of decimals: calculator based diagnostic tests and teaching materials. Nottingham UK: Shell Centre for Mathematical Education.Google Scholar
  54. Tschoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141–164.CrossRefGoogle Scholar
  55. Ward, J. (2009). Teacher knowledge of fractions: An assessment. Unpublished doctoral dissertation, University of Otago, Dunedin.Google Scholar
  56. Watson, J. M. (2001). Profiling teachers’ competence and confidence to teach particular mathematics topics: the case of chance and data. Journal of Mathematics Teacher Education, 4, 305–337.CrossRefGoogle Scholar
  57. Watson, J., Callingham, R., & Donne, J. (2008). Proportional reasoning: student knowledge and teachers’ pedagogical content knowledge. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions (proceedings of the 31st annual conference of the mathematics education research group of Australasia (Vol. 2, pp. 563–571). Adelaide: MERGA.Google Scholar
  58. White, P., & Anderson, J. (2011). Teachers’ use of national test data to focus numeracy instruction. In J. Clark, B. Kissane, J. Mousley, T. Spencer, & S. Thornton (Eds.), Mathematics: Traditions and [new] practices (Proceedings of the 23rd biennial conference of the Australian Association of Mathematics Teachers Inc. and the 34th annual conference of the Mathematics Education Research Group of Australasia Inc., Vol. 2, pp. 777–785). Adelaide: MERGA.Google Scholar
  59. Yeo, K. K. J. (2008). Teaching area and perimeter: mathematics-pedagogical-content-knowledge-in-action. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions (proceedings of the 31st annual conference of the mathematics education research group of Australasia, Vol. 2, pp. 621–628). Adelaide: MERGA.Google Scholar

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2012

Authors and Affiliations

  1. 1.Australian Catholic UniversityMelbourneAustralia

Personalised recommendations