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Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections

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Abstract

While the general planning advice offered to mathematics teachers seems to be to start with simple examples and build complexity progressively, the research reported in this article is a contribution to the body of literature that argues the reverse. That is, posing of appropriately complex tasks may actually prompt the use of more sophisticated strategies. Results are presented from a detailed study of young children working on tasks that prompt multiplicative thinking. It was found that the tasks involving more complex number triples prompted the use of more sophisticated multiplicative thinking.

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References

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

    Article  Google Scholar 

  • Anghileri, J., & Johnson, D. C. (1992). Arithmetic operations on whole numbers: Multiplication and division. In T. R. Post (Ed.), Teaching mathematics in grades K-8 (pp. 157–200). Boston: Allyn & Bacon.

    Google Scholar 

  • Anghileri, J., Beishuizen, M., & van Putten, K. (2002). From informal strategies to structured procedures: Mind the gap! Educational Studies in Mathematics 49(2), 149.

  • Australian Curriculum, Assessment and Reporting Authority (ACARA). (2010). NAPLAN: National Assessment Program – Literacy and Numeracy. Retrieved from: http://www.naplan.edu.au/.

  • Battista, M. C. (1999). Spatial structuring in geometric reasoning. Teaching Children Mathematics, 6(3), 171–177.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • City, E. A., Elmore, R. F., Fiarman, S. E., & Teitel, L. (2009). Instructional rounds in education. Boston: Harvard Educational Press.

    Google Scholar 

  • Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27(1), 41–51.

    Article  Google Scholar 

  • Clarke, D. M., Cheeseman, J., Gervasoni, A., Gronn, D., Horne, M., McDonough, A., … Rowley, G. (2002). Early numeracy research project: Final report. Melbourne: Australian Catholic University.

  • Clarke, D. M., & Clarke, B. A. (2004). Mathematics teaching in Grades K–2: Painting a picture of challenging, supportive, and effective classrooms. In R. N. Rubenstein & G. W. Bright (Eds.), Perspectives on the teaching of mathematics (66th Yearbook of the National Council of Teachers of Mathematics, pp. 67–81). Reston: NCTM.

  • Clarke, B., Clarke, D., & Horne, M. (2006). A longitudinal study of mental computation strategies. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the International Group for Psychology of Mathematics Education (Vol. 2, pp. 329–336). Prague: PME.

    Google Scholar 

  • Delpit, L. (1988). The silenced dialogue: Power and pedagogy in educating other people’s children. Harvard Educational Review, 58(3), 280–298.

    Article  Google Scholar 

  • Downton, A. (2010). Challenging multiplicative problems can elicit sophisticated strategies. 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. 169–176). Fremantle: MERGA.

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

    Google Scholar 

  • Dweck, C. S. (2000). Self-theories: Their role in motivation, personality, and development. Philadelphia: Psychology Press.

    Google Scholar 

  • Ell, F., Irwin, K., McNaughton, S. (2004). Two pathways to multiplicative thinking. In I. Put, R. Faragher, & M. McLean (Eds.), Mathematics education for the third millennium, towards 2010. (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, pp. 199–206). Townsville: MERGA.

  • Greer, B. (1988). Non-conservation of multiplication and division: Analysis of a symptom. Journal of Mathematical Behaviour, 7(3), 281–298.

    Google Scholar 

  • 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). New York: Macmillan.

    Google Scholar 

  • Heirdsfield, A. M., Cooper, T. J., Mulligan, J., & Irons, C. J. (1999). Children’s mental multiplication and division strategies. In O. Zaslavsky (Ed.), Proceedings of the 23rd conference of the Psychology of Mathematics Education Conference (Vol. 3, pp. 89–96). Hiafa: PME.

  • Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20(2), 147–158.

    Article  Google Scholar 

  • Lamon, S. J. (1993). Ratio and proportions: Connecting content and children’s thinking. Journal for Research in Mathematics Education, 24(1), 41–61.

    Article  Google Scholar 

  • Masters, G. (2009). A shared challenge: Improving literacy, numeracy and science learning in Queensland primary schools. Camberwell, Victoria: Australian Council for Education Research.

  • Middleton, J. A. (1999). Curricular influences on the motivational beliefs and practice of two middle school mathematics teachers: A follow-up study. Journal for Research in Mathematics Education, 30(3), 349–358.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Mulligan, J. T., Prescott, A., & Mitchelmore, M. C. (2006). Integrating concepts and processes in early mathematics: The Australian Pattern and Structure Mathematics Awareness Project (PASMAP). 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. 4, pp. 209–216). Prague: PME.

    Google Scholar 

  • Mulligan, J., & Watson, J. (1998). A developmental multimodal model for multiplication and division. Mathematics Education Research Journal, 10(2), 61–86.

    Article  Google Scholar 

  • Nesher, P. (1988). Multiplicative school word problems: Theoretical approaches and empirical findings. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 19–40). Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 61–86.

    Article  Google Scholar 

  • Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Sherin, B., & Fuson, K. (2005). Multiplication strategies and the appropriation of computational resources. Journal for Research in Mathematics Education, 36(4), 347–395.

    Google Scholar 

  • Siegler, R. S. (2000). The rebirth of children’s learning. Child Development, 71(1), 26–35.

    Article  Google Scholar 

  • Siemon, D., Breed, M., & Virgona, J. (2005). From additive to multiplicative thinking: The big challenge of the middle years. In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics: Celebrating achievement (Proceedings of the 42nd conference of the Mathematical Association of Victoria, pp. 278–286). Brunswick: MAV.

  • Singh, P. (2000). Understanding the concept of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 14(3), 271–292.

    Article  Google Scholar 

  • Singh, P. (2005). Multiplicative thinking in children’s learning at early grades. Paper presented at the ICMI East Asia Regional Conference on Mathematics Education, Shanghai, China.

  • Sophian, C., & Madrid, S. (2003). Young children’s reasoning about many-to-one correspondence. Child Development, 74(5), 1418–1432.

    Article  Google Scholar 

  • Steffe, L. P. (1988). Children’s construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 119–140). Hillsdale: Lawrence Erlbaum.

    Google Scholar 

  • Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). Albany: State University of New York Press.

    Google Scholar 

  • Steffe, L. P., & Cobb, P. (1994). Multiplicative and divisional schemes. Focus on Learning Problems in Mathematics, 16(1–2), 45–61.

    Google Scholar 

  • Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.

    Article  Google Scholar 

  • Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason and analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50–80.

    Article  Google Scholar 

  • Sullivan, P. (2011). Teaching mathematics: Using research informed strategies. Australian education review. Melbourne: ACER Press.

  • 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: PME.

  • Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York: Academic.

    Google Scholar 

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

    Google Scholar 

  • Victorian Curriculum and Assessment Authority. (2012). The AusVELS curriculum: Mathematics. Retrieved from http://ausvels.vcaa.vic.edu.au/Mathematics/Overview/Content-structure.

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Authors and Affiliations

  1. Faculty of Education, Monash University, Clayton, Victoria, Australia

    Ann Downton & Peter Sullivan

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  1. Ann Downton
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  2. Peter Sullivan
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Correspondence to Ann Downton.

Appendix

Appendix

Table 7 Challenge/ extra challenge multiplication tasks for each semantic structure
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Downton, A., Sullivan, P. Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections. Educ Stud Math 95, 303–328 (2017). https://doi.org/10.1007/s10649-017-9751-x

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