Abstract
What could two people stand to gain from sharing a taxi ride? We aimed to explore the extent to which this challenging yet accessible financial context might stimulate students’ mathematical exploration of multiplicative thinking and proportional reasoning. Through teaching experiment methodology, data were collected from 37 Year 5 and 6 students (10–12 years of age) in suburban Melbourne. The findings reveal that the majority of the students had some intuitive understanding of how to solve a financial problem that involved rate, and at least half of them used either multiplicative thinking or proportional reasoning. While the study reported is small and cannot claim to be representative, the findings confirm that well-designed financial problems have the potential to unveil sophisticated mathematical understandings among primary school students. This research demonstrates what young adolescents can do prior to formal exposure to ratio and proportion as part of the curriculum.
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Notes
The EPMC project was funded through an Australian Research Council Discovery Project (DP110101027).
References
Askew, M. (2018). Multiplicative reasoning: teaching primary pupils in ways that focus on functional relations. The Curriculum Journal, 29, 1–18. https://doi.org/10.1080/09585176.2018.1433545.
Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2017). F-10 curriculum: mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics. Accessed 12 Jan 2018
Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2018). F-10 curriculum: general capabilities. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/. Accessed 12 Jan 2018
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.
Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 243–307). Dordrecht: D. Reidel.
Dole, S., Wright, T., Clarke, D., & Hilton, G. (2012). The MC SAM Project: an integration of science and mathematics explorations. Retrieved from http://www.proportionalreasoning.com/uploads/1/1/9/7/11976360/mcsamproportionalreasoning.pdf
Downton, A., & Sullivan, P. (2017). Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections. Educational Studies in Mathematics, 95(3), 303-328. https://doi.org/10.1007/s10649-017-9751-x
English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In L. English & B. Sriraman (Eds.), Theories of mathematics education: seeking new frontiers (Advances in Mathematics Education series) (pp. 263–290). Heidelberg: Springer Science.
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel.
Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic Publishers.
Galbraith, P., Stillman, G., & Brown, J. P. (2006). Identifying key transition activities for enhanced engagement in mathematical modelling. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces (Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia) (pp. 237–245). Canberra: MERGA.
Goswami, U., & Bryant, P. (2007). Children’s cognitive development and learning (primary review research survey 2/1A). Cambridge: University of Cambridge Faculty of Education Retrieved from www.primaryreview.org.uk/Publications/Interimreports.html.
Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: a calculus course as an example. Educational Studies in Mathematics, 39, 111–129.
Greer, B. (1988). Non-conservation of multiplication and division: analysis of a symptom. Journal of Mathematical Behaviour, 7(3), 281–298.
Hilton, A., Hilton, G., Dole, S., Goos, M., & O’Brien. (2012). Evaluating middle years students’ proportional reasoning. In J. Dindyal, L. P. Cheng, & S. F. Ng (Eds.), Mathematics education: expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia) (pp. 330–337). Singapore: MERGA.
Hilton, A., Hilton, G., Dole, S., & Goos, M. (2016). Promoting students’ proportional reasoning skills through an ongoing professional development program for teachers. Educational Studies in Mathematics, 92(2), 193–219.
Lamon, S. J. (1993). Ratio and proportion: connecting content and children’s thinking. Journal for Research in Mathematics Education, 24(1), 41–61.
Lappan, G., Fey, T., Fitzgerald, W. M., Friel, S., & Phillips, E. D. (2006). Connected mathematics 2: implementing and teaching guide. Boston: Pearson, Prentice Hall.
Lesh, R., & Harel, G. (2003). Problem solving, modelling and local conceptual development. Mathematical Thinking and Learning, 3(2 & 3), 157–189.
Middleton, J. A. (1995). A Study of Intrinsic Motivation in the Mathematics Classroom: A Personal Constructs Approach. Journal for Research in Mathematics Education, 26(3), 254-279.
Organisation for Economic Cooperation and Development. (2013). PISA 2012 assessment and analytical framework: mathematics, reading, science, problem solving and financial literacy. OECD Publishing. https://doi.org/10.1787/9789264190511-en.
Organisation for Economic Co-operation and Development (OECD). (2017a). PISA 2015 assessment and analytical framework: science, reading, mathematic, financial literacy and collaborative problem solving. Revised edition. Paris: OECD. https://doi.org/10.1787/9789264281820-en.
Organisation for Economic Co-operation and Development (OECD). (2017b). PISA 2015 results (Volume IV): Students’ financial literacy. Paris: PISA, OECD Publishing. https://doi.org/10.1787/9789264270282-en.
Parish, L. (2010). Facilitating the development of proportional reasoning through teaching ratio. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education (Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia) (pp. 469–476). Fremantle: MERGA.
Perrenet, J., & Zwaneveld, B. (2012). The many faces of the modelling cycle. Journal of Modelling and Application, 1(6), 3–21.
Salgado, F. A. (2016). Investigating the impact of contexts on students’ performance. In B. White, M. Chinnappan, & S. Trenholm (Eds.), Opening up mathematics education research (Proceedings of the 39th annual conference of the Mathematics Education Research Group of Australasia) (pp. 102–109). Adelaide: MERGA.
Sawatzki, C. (2014). Connecting social and mathematical thinking: The use of “real life” contexts. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), Curriculum in focus: Research guided practice. (Proceedings of the 37th Annual Conference of the Mathematics Education Research Group of Australasia) (pp. 557–564). Sydney: MERGA.
Sawatzki, C. (2017). Lessons in financial literacy task design: Authentic, imaginable, useful. Mathematics Education Research Journal, 29(1), 25-43.
Sawatzki, C., & Goos, M. (2018). Cost, price and profit: What influences students’ decisions about fundraising? Mathematics Education Research Journal, 30(1), 1-20.
Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment framework for multiplicative thinking. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Mathematics in the centre. Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 113–120). Prague: PME.
Singh, P. (2000). Understanding the concept of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 14(3), 271–292.
Smith, M. S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematical discussions. Reston: National Council of Teachers of Mathematics.
Stacey, K. (2015). The real world and the mathematical world. In K. Stacey & R. Turner (Eds.), Assessing mathematical literacy: the PISA experience (pp. 57–84). Cham: Springer International Publishing.
Steffe, L. P. (1994). Children’s multiplying schemes. In: Harel G. & Confrey J. (eds.) The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). New York: SUNY Press.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale: Erlbaum.
Stillman, G. (2000). Impact of prior knowledge of task context on approaches to applications tasks. The Journal of Mathematical Behavior, 19(3), 333-361.
Strauss, A., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Thousand Oaks, CA: Sage Publications, Inc.
Sullivan, P. (2011). Teaching mathematics: using research informed strategies. Australian Education Review. Melbourne: ACER Press.
Sullivan, P., Zevenbergen, R., & Mousley, J. (2003). The contexts of mathematics tasks and the context of the classroom: are we including all students? Mathematics Education Research Journal, 15(2), 107–121.
Sullivan, P., Clarke, B., Cheeseman, J., Mornane, A., Roche, A., Sawatzki, C., & Walker, K. (2014). Students’ willingness to engage with mathematical challenges: Implications for classroom pedagogies. In J. Anderson, M. Cavanagh, & A. Prescott (Eds.), Curriculum in focus: Research guided practice (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia) (pp. 597–604). Sydney: MERGA.
Sullivan, P. A., Askew, M., Cheeseman, J., Clarke, D. M. Mornane, A., Roche, A., & Walker, N., (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18(2), 123-140.
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. In B. White, M. Chinnappan, & S. Trenholm (Eds.), Opening up mathematics education research. (Proceedings of the 39th annual conference of the Mathematics Education Research Group of Australasia) (pp. 669–673). Adelaide: MERGA.
Van den Heuvel-Panhuizen, M. (2005). The role of contexts in assessment problems in mathematics. For the Learning of Mathematics, 2, 2–9.
Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication…and back: the development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381.
Verschaffel, L., de Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modelling of school arithmetic word problems. Learning and Instruction, 4, 273–294.
Zbiek, R. M., & Conner, A. (2006). Beyond motivation: exploring mathematical modeling as a context for deepening students’ understandings of curricular mathematics. Educational Studies in Mathematics, 63(1), 89–112. https://doi.org/10.1007/s10649-005-9002-4.
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Sawatzki, C., Downton, A. & Cheeseman, J. Stimulating proportional reasoning through questions of finance and fairness. Math Ed Res J 31, 465–484 (2019). https://doi.org/10.1007/s13394-019-00262-5
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DOI: https://doi.org/10.1007/s13394-019-00262-5