### Developing Pre-service Teachers’ Knowledge for Teaching in the Early Years: Selecting and Sequencing

#### Abstract

Developing pre-service teachers’ (PSTs’) knowledge of pedagogical practices can be particularly challenging within university classrooms. Teacher educators are well placed to provide PSTs with theoretical perspectives on pedagogical practices; what is particularly challenging, however, is linking theory with practice and developing PSTs’ breadth and depth of knowledge in mathematical concepts. In this article, we describe the experiences undertaken by two cohorts of PSTs during their tutorials designed to assist them to notice and discuss Year 2 students’ responses to an array task. Open-coding was used to analyse PSTs’ selection and sequencing of five different work samples. Findings indicated that while the authentic work samples assisted the PSTs to make connections with the students’ mathematical understandings, the lesson also provided an insight into the PSTs’ own foundation knowledge of mathematical understandings relating to children’s development of multiplicative thinking.

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Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20, 367-385.

Anthony, G., Cooke, A., & Muir, T. (2016). Challenges, reforms, and learning in initial teacher education. In K. Makar, S. Dole, J. Visnovska, M. Goos, A. Bennison, & K. Fry (Eds.), Research in mathematics education in Australasia 2012-2015. Singapore: Springer.

Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective teachers of numeracy. London: King’s College.

Australian Curriculum, Assessment and Reporting Authority (ACARA). (2016). Australian curriculum: Mathematics. http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Australian Institute for Teaching and School Leadership (AITSL). (2011). Accreditation of initial teacher education programs in Australia: Standards and procedures April 2011., 1-22. Retrieved from http://www.aitsl.edu.au/verve/_resources/Accreditation_of_initial_teacher_education.pdf

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449-499.

Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematics knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 3-14). Edmonton: AB:CMESG/GCEDM.

Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners' mathematical futures. Paper presented at the 43rd Jahrestagung für Didaktik der Mathematik, Oldenburg, Germany.

Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503-532.

Beswick, K. (2006). Changes in preservice teachers’ attitudes and beliefs: The net impact of two mathematics education units and intervening experiences. School Science and Mathematics, 106(1), 36-47.

Beswick, K., & Goos, M. (2012). Measuring pre-service primary teachers’ knowledge for teaching mathematics. Mathematics Teacher Education and Development, 14(2), 70-90.

Beswick, K., & Muir, T. (2013). Making connections: Lessons on the use of video in pre-service teacher education. Mathematics Teacher Education and Development, 15(2), 27-51.

Clarke, B., Grevholm, B., & Millman, R. (2009). Tasks in primary mathematics teacher education. New York: Springer

Cohrssen, C. & Tayler, C. (2016). Early childhood mathematics: A pilot study in preservice teacher education. Journal of Early Childhood Education, 37(1), 25-40.

Dole, S. (2008). Ratio tables to promote proportional reasoning in the primary classroom. Australian Primary Mathematics Classroom, 12(2), 19-22.

Downton, A., & Sullivan. P. (2013). Fostering the transition from additive to multiplicative thinking. In A. M. Lindmeier, & A. Heinze (Eds.), Mathematics learning across the life span (Proceedings of the 37th Conference of the International Group of Psychology of Mathematics Education, Vol 2, pp. 241-248). Keil, Germany: PME.

Flick, U. (2009). An Introduction to Qualitative Research (4th ed.). London: Sage Publications.

Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276-295). NY: Macmillan.

Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re‐imagining teacher education. Teachers and Teaching: theory and practice, 15(2), 273-289.

Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of mathematics teacher education, 9(2), 187-211.

Klein, M., Ralya, T., Pollak, B., Obenza, R., & Harbour, M. G. (2012). A practitioner’s handbook for real-time analysis: guide to rate monotonic analysis for real-time systems. Springer Science & Business Media.

Ma, L. (1999). Knowing and teaching elementary mathematics. Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc.

Maher, N., & Muir, T. (2013). “I know you have to put down a zero but I don’t know why”: Exploring the link between teachers’ content knowledge and pedagogical content knowledge. Mathematics Teacher Education and Development, 15(1), 72-87.

Meikle, E. M. (2016). Selecting and Sequencing Students' Solution Strategies: Reflect and Discuss. Teaching Children Mathematics, 23(4), 226-234.

Mulligan, J., & Mitchelmore, M. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28(3), 309-330.

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing Primary Mathematics Teaching: Reflecting on practice with the Knowledge Quartet. London: Sage.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Siemon, D., Bleckly, J. & Neal, D. (2012). Working with the big ideas in number and the Australian Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen & D. Siemon, (Eds.). (2012). Engaging the Australian National Curriculum: Mathematics: Perspectives from the Field. Online Publication: Mathematics Education Research Group of Australasia pp. 19- 45.

Smith, S., & Stein, M. (2011). 5 Practices for Orchestrating Productive Mathematical Discussions. Reston, VA: National Council of Teachers of Mathematics.

Sullivan, P. (2016). Exploring the potential of using challenging mathematical tasks: A resource to support teachers in Years 3-6 semester 1, 2016. Monash University. Melbourne.

Sullivan, P., Aulert, A., Lehmann, A., Hislop, B., Shepherd, O., & Stubbs, A. (2013). Classroom culture, challenging mathematical tasks and student persistence. In V. Steinle, L. Ball & C. Bardini (Eds.), Mathematics education: Yesterday, today and tomorrow (Proceedings of the 36th Annual Conference of the Mathematics Education Research of Australasia, pp. 618-625). Melbourne, VIC: MERGA.

Sullivan, P., Cheeseman, J., Michels, D., Mornane, A., Clarke, D., Roche, A., & Middleton, J. (2011). Challenging mathematics tasks: What they are and how to use them. Proceedings of the 48th Annual Conference of the Matheamtical Association of Victoria. In L. Bragg (Ed.), Challenging mathematics tasks: What they are and how to use them. Proceedings of the 48th Annual Conference of the Mathematica Association of Victoria. (pp. 33-46). Melbourne: The Mathematical Association of Victoria.

Sullivan, P., Clarke, D., Cheeseman, J., Mulligan, J. (2001). Moving Beyond Physical Models in Learning Multiplicative Reasoning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol.4, pp. 233-240). Utrecht –The Netherlands: PME.

Sullivan, P., Holmes, M., Ingram, N., Linsell, C., Livy, S., & McCormack, M. (2016). The intent and processes of a professional learning initiative seeking to foster discussion around innovative approaches to teaching. Proceedings of the 39th annual conference of the Mathematics Education Research Group of Australasia. Adelaide: MERGA.

Sullivan, P., Walker, N., Borcek, C. & Rennie, M. (2015). Exploring a structure for mathematics lessons that foster problem solving and reasoning. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia, pp. 41–56). Sunshine Coast: MERGA.

Teacher Education Ministerial Advisory Group (TEMAG). (2015). Action now: Classroom ready teachers. Retrieved from http://docs.education.gov.au/system/files/doc/other/action_now_classroom_ready_teachers_print.pdf.

Thames, M. H., & Ball, D. L. (2010). What math knowledge does teaching require? Teaching Children Mathematics, 17(4), 220-229.

Turner, F. & Rowland, T. (2011). The Knowledge Quartet as an organising framework for developing and deepening teachers’ mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.195-212). New York: Springer.

Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp.141-161). Hillsdale, NJ: Lawrence Erlbaum.

Yin, R.K. (2009). Case study research: Design and methods (4th ed.). Thousand Oaks, CA: Sage Publications.

Young-Loveridge, J. (2005). Fostering multiplicative thinking using array-based materials. The Australian Mathematics Teacher, 61(3), 34-39.

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