Journal of Mathematics Teacher Education

, Volume 22, Issue 6, pp 545–574 | Cite as

Primary school teachers implementing structured mathematics interventions to promote their mathematics knowledge for teaching proportional reasoning

  • Annette HiltonEmail author
  • Geoff Hilton


Proportional reasoning is the ability to use multiplicative thinking and make multiple comparisons. It is known to be challenging for many students and at the same time, many teachers require support to develop sufficient subject matter knowledge and pedagogical content knowledge to teach the diverse concepts that underpin proportional reasoning. The data reported in this paper are drawn from the first year of a broader study aiming to promote the teaching and learning of elements of proportional reasoning across the curriculum by engaging primary school teachers in ongoing professional development that includes the implementation of a series of mathematics interventions, each of which included a research component. This paper focuses on the impact of implementing the interventions on the teachers’ mathematical knowledge for teaching. Three structured interventions were implemented by eight teachers (Years 3, 4, 5) during each of three school terms. Data collected showed that engaging in this scaffolded type of practitioner research, the structured nature of the interventions, and reflection on the outcomes of each intervention promoted teachers’ subject matter knowledge and pedagogical content knowledge.


Teacher professional development Primary school mathematics teacher education Mathematical knowledge for teaching Pedagogical content knowledge 



This work was not funded by a grant. It was conducted as part of a university–school collaboration.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Informed consent

Informed consent was obtained from all participants prior to the study.


  1. Ahl, V. A., Moore, C. F., & Dixon, J. A. (1992). Development of intuitive and numerical proportional reasoning. Cognitive Development,7, 81–108.Google Scholar
  2. Akatugba, A. H., & Wallace, J. (2009). An integrative perspective on students’ proportional reasoning in high school physics in a West African context. International Journal of Science Education,31(11), 1473–1493.Google Scholar
  3. Ammon, P., & Kroll, L. R. (2002). Learning and development in constructivist teacher education. In J. Rainer (Ed.), Reframing teacher education: Dimensions of a constructivist approach (pp. 193–240). Dubuque, IA: Kendall/Hunt.Google Scholar
  4. Aslan-Tutak, F., & Adams, T. L. (2015). A study of geometry content knowledge of elementary preservice teachers. International Electronic Journal of Elementary Education,7(3), 310–318.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.Google Scholar
  6. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York: Macmillan.Google Scholar
  7. Ben-Chaim, D., Keret, Y., & Ilany, B.-S. (2007). Designing and implementing authentic investigative proportional reasoning tasks: The impact on pre-service mathematics teachers’ content and pedagogical knowledge and attitudes. Journal of Mathematics Teacher Education,10, 333–340.Google Scholar
  8. Boyer, T. W., & Levine, S. C. (2012). Child proportional scaling: Is 1/3 = 2/6 = 3/9 = 4/12? Journal of Experimental Child Psychology,111, 516–533.Google Scholar
  9. Campbell, A., & Groundwater-Smith, S. (Eds.). (2010). Action research in education. Historical perspectives in action research in schools: From curriculum development to enhancing teacher professional learning (Vol. 1). London: Sage.Google Scholar
  10. Campbell, A., & McNamara, O. (2009). Mapping the field of practitioner research, inquiry and professional learning in educational contexts: A review. In A. Campbell & S. Groundwater-Smith (Eds.), Connecting inquiry and professional learning in education: International perspectives and practical solutions (pp. 10–26). Abingdon, Oxon: Routledge.Google Scholar
  11. Carrillo, J., Climent, N., Contreras, L. C., & Muñoz-Catalán, M. C. (2013). Determining specialized knowledge for mathematics teaching. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.) Proceedings of the VIII congress of the European society for research in mathematics education (CERME 8) (pp. 2985–2994). Antalya, Turkey: Middle East Technical University, Ankara.Google Scholar
  12. Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education (6th ed.). London: Routledge Falmer.Google Scholar
  13. Darling-Hammond, L., & Richardson, N. (2009). Teacher learning: What matters? Educational Leadership,66(5), 46–53.Google Scholar
  14. Day, C., & Sachs, J. (2004). International handbook on the continuing professional development of teachers. Maidenhead: Open University Press.Google Scholar
  15. Ellis, N. L. (2012). Teachers’ experiences as practitioner researchers in secondary schools: A comparative study of Singapore and NSW (Doctoral dissertation). The University of Sydney, Australia. Retrieved from
  16. English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  17. Fielding-Wells, J., Dole, S., & Makar, K. (2014). Inquiry pedagogy to promote emerging proportional reasoning in primary students. Mathematics Education Research Journal,26, 47–77.Google Scholar
  18. Flores, E., Escudero, D. I., & Carrillo, J. (2013). A theoretical review of specialized content knowledge. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.) Proceedings of the VIII congress of the European society for research in mathematics education (CERME 8) (pp. 3055–3064). Antalya, Turkey: Middle East Technical University, Ankara.Google Scholar
  19. Gee, J. P. (2005). An introduction to discourse analysis: Theory and method (2nd ed.). New York: Routledge.Google Scholar
  20. Goldsmith, L. T., Doerr, H. M., & Lewis, C. C. (2014). Mathematics teachers’ learning: A conceptual framework and synthesis of research. Journal of Mathematics Teacher Education,17, 5–36.Google Scholar
  21. Heikkinen, H. L. T., de Jong, F. P. C. M., & Vanderlinde, R. (2016). What is (good) practitioner research? Vocations and Learning,9, 1–19.Google Scholar
  22. Hiebert, J. S., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Reston: National Council of Teachers of Mathematics.Google Scholar
  23. Hill, H., & Ball, D. L. (2009). The curious and crucial case of mathematics knowledge for teaching. Kappan,91(2), 68–71.Google Scholar
  24. Hill, H., Blunk, M., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., et al. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction,26(4), 430–511.Google Scholar
  25. Hilton, A., & Hilton, G. (2016). Proportional reasoning: An essential component of scientific understanding. Teaching Science, 62(4), 31–41.Google Scholar
  26. Hilton, A., & Hilton, G. (2017). The impact of conducing practitioner research projects on teachers’ professional growth. Australian Journal of Teacher Education, 42(8), 77–94.Google Scholar
  27. Hilton, A., & Hilton, G. (2018). A string number line lesson sequence to promote students’ relative thinking and understanding of scale, key elements of proportional reasoning. Australian Primary Mathematics Classroom, 23(1), 13–18.Google Scholar
  28. Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle years students’ proportional reasoning. Mathematics Education Research Journal, 25, 523–545.Google Scholar
  29. Hilton, A., Hilton, G., Dole, S., & Goos, M. (2015). Using photographic images to enhance conceptual development in situations of proportion. Australian Primary Mathematics Classroom, 20(1), 3–9.Google Scholar
  30. Hilton, A., Hilton, G., Dole, S., & Goos, M. (2016). Promoting students’ proportional reasoning skills through an ongoing professional development programme for teachers. Educational Studies in Mathematics, 92, 193–219.Google Scholar
  31. Hoover, M., Mosvold, R., Ball, D. L., & Lai, Y. (2016). Making progress on mathematical knowledge for teaching. The Mathematics Enthusiast,13(1 & 2), 3–34.Google Scholar
  32. Howe, C., Nunes, T., & Bryant, P. (2011). Rational number and proportional reasoning: Using intensive quantities to promote achievement in mathematics and science. International Journal of Science and Mathematics Education,9, 391–417.Google Scholar
  33. Hurrell, D. (2013). Effectiveness of teacher professional learning: Enhancing the teaching of fractions in primary schools (Doctoral dissertation). Edith Cowan University, Australia. Retrieved from
  34. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. New York: Basic Books.Google Scholar
  35. Jeong, Y., Levine, S., & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities. Journal of Cognition and Development,8, 237–256.Google Scholar
  36. Kastberg, S. E., D’Ambrosio, B., & Lynch-Davis, K. (2012). Understanding proportional reasoning for teaching. Australian Mathematics Teacher,68(3), 32–40.Google Scholar
  37. Koellner, K., Jacobs, J., Borko, H., Schneider, C., Pittman, M. E., Eiteljorg, E., et al. (2007). The problem-solving cycle: A model to support the development of teachers’ professional knowledge. Mathematical Thinking and Learning,9(3), 273–303.Google Scholar
  38. Kroll, L. R. (2005). Making inquiry a Habit of Mind: Learning to use inquiry to understand and improve practice. Studying Teacher Education,1(2), 179–193.Google Scholar
  39. Lamon, S. J. (1993). Ratio and proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education,24(1), 41–61.Google Scholar
  40. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). Charlotte, NC: Information Age Publishing.Google Scholar
  41. Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). New York: Routledge.Google Scholar
  42. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics.Google Scholar
  43. Lo, J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader. Journal for Research in Mathematics Education,28(2), 216–236.Google Scholar
  44. Lobato, J., Orrill, C. H., Druken, B., & Jacobson, E. (2011). Middle school teachers’ knowledge of proportional reasoning for teaching. In Paper presented in the Symposium Extending, expanding, and applying the construct of mathematical knowledge for teaching, at the Annual meeting of the American Educational Research Association, New Orleans, USA.Google Scholar
  45. Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010). Designing professional development for teachers of science and mathematics (3rd ed.). Thousand Oaks: Corwin.Google Scholar
  46. Loughran, J., Berry, A., & Mulhall, P. (2012). Understanding and developing science teachers’ Pedagogical content knowledge (2nd ed.). Rotterdam: Sense Publishers.Google Scholar
  47. Meijer, M.-J., Geijsel, F., Kuijpers, M., Boei, F., & Vrieling, E. (2016). Exploring teachers’ inquiry-based attitude. Teaching in Higher Education,21(1), 64–78.Google Scholar
  48. Park, S., & Oliver, J. S. (2008). Revisiting the conceptualisation of Pedagogical Content Knowledge (PCK): PCK as a conceptual tool to understand teachers as professionals. Research in Science Education,38, 261–284.Google Scholar
  49. Patton, M. Q. (2002). Qualitative research & evaluation methods (3rd ed.). Thousand Oaks: Sage Publications.Google Scholar
  50. Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. New York: Norton [Original work published 1951].Google Scholar
  51. Saldaña, J. (2013). The coding manual for qualitative researchers (2nd ed.). Thousand Oaks, CA: Sage Publications.Google Scholar
  52. Scheerens, J. (Ed.) (2010). Teachers’ professional development: Europe in international comparison. Retrieved from
  53. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher,15(2), 4–14.Google Scholar
  54. Siebert, D. (2002). Connecting informal thinking and algorithms: The case of division of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 247–256). Reston: National Council of Teachers of Mathematics.Google Scholar
  55. Smestad, B. (2015). When HPM meets MKT—Exploring the place of history of mathematics in the mathematical knowledge for teaching. In E. Babin, U. T. Jankvist, & T. H. Kjeldsen (Eds.) History and epistemology in mathematics: Proceedings of the seventh European Summer University ESU7 (pp. 539–549). Copenhagen, Denmark: Danish School of Education, Aarhus University.Google Scholar
  56. Sowder, J. (2007). The mathematical education and development of teachers. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 157–223). Reston: National Council of Mathematics Teachers.Google Scholar
  57. Sowder, J., Armstrong, B., Lamon, S. J., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education,1, 127–155.Google Scholar
  58. Taber, S. B. (2002). Go ask Alice about multiplication of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 61–71). Reston: National Council of Teachers of Mathematics.Google Scholar
  59. Thomas, E. (2007). Thoughtful planning fosters learning transfer. Adult Learning,18(3–4), 4–8.Google Scholar
  60. Thompson, A. G., & Thompson, P. W. (1996). Talking about rates conceptually, Part II: Mathematical knowledge for teaching. Journal for Research in Mathematics Education,27, 2–24.Google Scholar
  61. van den Kieboom, L. A. (2013). Examining the mathematical knowledge for teaching involved in pre-service teachers’ reflections. Teaching and Teacher Education,35, 146–156.Google Scholar
  62. Veal, W. R., & MaKinster J. G. (1999). Pedagogical content knowledge taxonomies. Electronic Journal of Science Education 3. Retrieved from
  63. Willegems, V., Consuegra, E., Struyven, K., & Engels, N. (2017). Teachers and pre-service teachers as partners in collaborative teacher research: A systematic literature review. Teaching and Teacher Education,64, 230–245.Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of EducationUniversity of Technology SydneyBroadwayAustralia
  2. 2.School of EducationThe University of QueenslandSt LuciaAustralia

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