Tracking Change in Primary Teachers’ Understanding of Mathematical Reasoning Through Demonstration Lessons
Keywords:
teacher professional learning, demonstration lessons, mathematical reasoning, post lesson discussion, inquiry communityAbstract
This paper reports on the impact of a professional learning program on participating teachers’ knowledge of mathematical reasoning. A total of 26 teachers participated in this study from four schools in Victoria, Australia and one school in British Columbia, Canada. The participants observed two demonstration lessons prepared and taught by the research team, attended pre- and post-demonstration lesson group discussions and taught each lesson in their classroom. Interviews with participating teachers before beginning the program; after the first demonstration lesson; and after trialling the lessons provided data for analysis. The Primary Teachers' Perceptions of Mathematical Reasoning Framework previously established by the research team was used to track the shifts in teachers’ perceptions and understanding of mathematical reasoning across the program. We theorise that intentional foci on salient aspects of reasoning demonstration lessons, highly collaborative reflections, and teacher enactment of the demonstrated lessons have the potential to develop teachers’ perceptions and understanding of reasoning.
References
Anderson, J. (2007). Inquiry into Effective Strategies for Teacher Professional Learning. Melbourne: Parliament of Victoria Education and Training Committee.
Australian Curriculum Assessment and Reporting Authority (2012). Australian Curriculum: Mathematics. Retrieved 24 June, 2014, from http://www.australiancurriculum.edu.au/mathematics/content-structure
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.
Boero, P. & Dapueto, C. (2007). Problem solving in mathematics education in Italy: Dreams and reality. ZDM, 39(5–6), 383–393.
Booth, S. (1997). On phenomenography, learning and teaching. Higher Education Research and Development, 16(2), 135–158.
Bowden, J. A., & Green, P. (Eds.). (2005). Doing develomental phenomenography. Melbourne, Australia: RMIT University Press.
Bowden, J. A., & Marton, F. (2004). The university of learning. London: Routledge.
Brodie, K. (Ed.). (2010). Teaching mathematical reasoning in secondary school classrooms. New York Dordrecht Heidelberg London: Springer.
Bruce, C. D., Ross, J., Flynn, T. & McPherson, R. (2009). Lesson study and demonstration classrooms: Examining the effects of two models of teacher professional development. Retrieved 20 November 2014 from http://www.tmerc.ca/digitalpapers/samples/WholeResearchStory.pdf
Choy, B. H. (2013). Productive Mathematical Noticing: What it is and why it matters. Paper presented at the 36th annual conference of the Mathematics Education Research Group of Australasia (MERGA 36), 7-11 July 2013, Melbourne, Australia.
Clarke, D. M. (2011). Do demonstration lessons work? Australian Primary Mathematics Classroom, 16(2), 12-13.
Clarke, D. M., Clarke, D. J., & Sullivan, P. (2012). Reasoning in the Australian Curriculum: Understanding its meaning and using the relevant language. Australian Primary Mathematics Classroom, 17(3), 28-32.
Clarke, D. M., Roche, A, Wilkie, K., Wright, V., Brown, J., Downton, A., . . . Worrall, C. (2013). Demonstration lessons in mathematics education: teachers’ observation foci and intended changes in practice. Mathematics Education Research Journal, 25(2), 207-230.
Darling-Hammond, L., Wei, R. C., Andree, A., Richardson, N., & Orphanos, S. (2009). Professional learning in the learning profession: A status report on teacher development in the United States and abroad. National Staff Development Council & The School Redesign Network: Stanford University.
Department of Education and Early Childhood Development. (2013). Professional Development. Retrieved 24 June, 2014, from http://www.education.vic.gov.au/school/teachers/profdev/pages/default.aspx
Department of Education UK. (2012). National curriculum for mathematics, Key stages 1 and 2: Draft. Retrieved 8 July, 2014, from http://media.education.gov.uk/assets/files/pdf/d/draft%20national%20
curriculum%20for%20mathematics%20key%20stages%201%202.pdf
Department of Education Western Australia. (2014). Child Protection Professional Learning Program. Retrieved 24 June 2014, 2014, from http://det.wa.edu.au/childprotection/detcms/inclusiveeducation/child-protection/public/training/child-protection.en?oid=Article-id-3160729&tab=Main
Francisco, J. M., & Maher, C. A. (2011). Teachers attending to students’ mathematical reasoning: lessons from an after-school research program. Journal of Mathematics Teacher Education, 14(1), 49-66. doi: 10.1007/s10857-010-9144-x
Franke, M.L., Webb, N.M., Chan, A.G., Ing, M., Freund, D., & Battey, D.(2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 380–392. http://dx.doi.org/10.1177/0022487109339906.
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 403–434.
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(1), 5-36. doi: 10.1007/s10857-013-9245-4
Grierson, A. L., & Gallagher, T. L. (2009). Seeing is believing: creating a catalyst for teacher change through a demonstration classroom professional development initiative. Professional Development in Education, 35(4), 567-584. doi: 10.1080/19415250902930726
Herbert, S. & Pierce, R. (2013). Gesture as data for a phenomenographic analysis of mathematical conceptions. International Journal of Educational Research, 60, 1 - 10.
Herbert, S. & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics. 81 (1), 85-101.
Herbert, S., Vale, C., Bragg, L., Loong, E., & Widjaja, W. (2015). A framework for primary teachers' perceptions of mathematical reasoning. International Journal of Educational Research 74, 26-37.
Huang, R., Su, H., & Xu, S.. (2014). Developing teachers’ and teaching researchers’ professional competence in mathematics through Chinese Lesson Study. ZDM, 46(2), 239-251. doi: 10.1007/s11858-013-0557-8
Hunter, J. & Back, J. (2011). Facilitating Sustainable Professional Development through Lesson Study. Mathematics Teacher Education & Development, 13(2), 94-114.
Jaworski, B. (2004). Grappling with complexity: Co-learning in inquiry communities in mathematics teaching development. Paper presented at the 28th Conference of the International Group for the Psychology of Mathematics Education.
Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development. In K. Krainer & T. Wood (Eds.), Participants in Mathematics Teacher Education (pp. 309-330): Sense Publishers.
Kazemi, E., & Franke, M.L. (2004). Teacher learning in mathematics: Using student work to promote collective inquiry. Journal of Mathematics Teacher Education, 7(3), 203–235.
Kilinc, A., & Aydin, A. (2013). Turkish student science teachers’ conceptions of sustainable development: A phenomenography. International Journal of Science Education, 35(5), 731-752.
Lewis, C.. (2002). Does lesson study have a future in the United States? Nagoya Journal of Education and Human Development, January(1), 1-23.
Marshall, J. C., & Horton, R. M. (2011). The Relationship of Teacher-Facilitated, Inquiry-Based Instruction to Student Higher-Order Thinking. School Science and Mathematics, 111(3), 93-101. doi: 10.1111/j.1949-8594.2010.00066.x
Marton, F., Runesson, U., & Tsui, A. B. M. . (2004). Classroom discourse and the space of learning. Mahwah, NJ: Lawrence Erlbaum.
Mason, J. (2002). Researching your own practice: the discipline of noticing. London: Routledge Falmer.
Ministry of Education Province of British Columbia. (2007). Introduction to Mathematics K-7: Integrated Resource Package. . Retrieved 4 November, 2014, from https://www.bced.gov.bc.ca/irp/pdfs/mathematics/2007mathk7.pdf
Ministry of Education Singapore. (2012). Primary mathematics teaching and learning syllabus. Retrieved 2 March, 2013, from http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-primary-2013.pdf
Murata, A. (2011). Introduction: Conceptual overview of lesson study. In L. C. Hart, A. S. Alston & A. Murata (Eds.) Lesson study research and practice in mathematics education (pp. 1-12). Springer Netherlands.
National Council of Teachers of Mathematics. (2014). Standards Overview. Retrieved 8 July, 2014, from http://www.nctm.org/standards/content.aspx?id=26798
Olson, J. C., White, P. & Sparrow, L. (2011). Influence of lesson study on teachers’ mathematics pedagogy. In L. C. Hart, A. S. Alston & A. Murata (Eds.), Lesson study research and practice in mathematics education (pp. 39-57). Springer Netherlands.
Piaget, J. (1970). Piaget's theory. In P. Mussen (Ed.), Carmichael's manual of child psychology. New York: Wiley.
Ramsden, P. (1988). Studying learning: Improving teaching. In P. Ramsden (Ed.), Improving learning: New perspectives (pp. 13–31). London: Kogan Page.
Reid, D. A. (2002). Conjectures and Refutations in Grade 5 Mathematics. Journal for Research in Mathematics Education, 33(1), 5-29. doi: 10.2307/749867
Reid, D. A. (2003). Forms and uses of abduction. Paper presented at the Proceedings of the CERME 3 international conference.
Schoenfeld, A.H. (2011). Noticing matters. A lot. Now what? In M. G. Sherin, V. R. Jacobs & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 223-238). New York: Roultledge.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard educational review, 57(1), 1-23.
Small, M. (2011). One, two, infinity. Retrieved 24 June, 2014, from http://www.onetwoinfinity.ca/
Stacey, K. . (2003). The need to increase attention to mathematical reasoning. In H. Hollingsworth, J. Logan & B.
Mcrae (Eds.), Teaching Mathematics in Australia: Results from the TIMSS 1999 Video Study. Melbourne: ACER.
Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307-332. doi: 10.1007/s10857-008-9077-9
Takahashi, A. (2011). Response to Part I: Jumping into lesson study—Inservice mathematics teacher education. In L. C. Hart, A. S. Alston & A. Murata (Eds.), Lesson Study Research and Practice in Mathematics Education (pp. 79-82). Springer Netherlands.
Yang, Y. (2009). How a Chinese teacher improved classroom teaching in Teaching Research Group: a case study on Pythagoras theorem teaching in Shanghai. ZDM, 41(3), 279-296. doi: 10.1007/s11858-009-0171-y
Widjaja, W., Vale, C., Groves, S., & Doig, B. (2015). Teachers’ professional growth through engagement with lesson study. Journal of Mathematics Teacher Education, 1-27. doi: 10.1007/s10857-015-9341-8.
Authors' works cited in paper
Bragg, L. A., Vale, C., Herbert, S., Loong, E., Widjaja, W., Williams, G., & Mousley, J. (2013, January). Promoting awareness of reasoning in the primary mathematics classroom. In MAV 2013: Mathematics of the planet earth: Proceedings of the MAV 50th Annual Conference 2013 (pp. 23-30). Mathematical Association of Victoria.
Bragg, L. A., Loong, E. Y. K., Widjaja, W., Vale, C., & Herbert, S. (2015). Promoting reasoning through the magic V task. Australian Primary Mathematics Classroom, 20(2), 10.
Herbert, S., Vale, C., Bragg, L. A, Loong, E., & Widjaja, W. (2015). A framework for primary teachers' perceptions of mathematical reasoning. International Journal of Educational Research 74 (26–37).
Loong, E. Y-K., Vale, C., Bragg, L. A, & Herbert, S. (2013). Primary school teachers' perceptions of mathematical reasoning. Paper presented at the 36th annual conference of the Mathematics Education Research Group of Australasia ' Mathematics education: Yesterday, today and tomorrow', Melbourne.
Vale, C., Widjaja, W., Herbert, S., Bragg, L. A., & Loong, E. Y. K. (2016). Mapping Variation in Children’s Mathematical Reasoning: The Case of ‘What Else Belongs?’. International Journal of Science and Mathematics Education, 1-22.
Widjaja, W. (2014, January). Year 3/4 children's forms of justification. In MERGA 2014: Curriculum in focus: research guided practice: Proceedings of the Mathematics Education Research Group of Australasia 2014 annual conference (pp. 694-697). Mathematics Education Research Group of Australasia.