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Early Awareness of Mathematical Pattern and Structure

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Reconceptualizing Early Mathematics Learning

Part of the book series: Advances in Mathematics Education ((AME))

Abstract

This chapter provides an overview of the Australian Pattern and Structure Project, which aims to provide new insights into how young students can abstract and generalize mathematical ideas much earlier, and in more complex ways, than previously considered. A suite of studies with 4- to 8-year old students has shown that an awareness of mathematical pattern and structure is both critical and salient to mathematical development among young students. We provide a rationale for the construct, Awareness of Mathematical Pattern and Structure (AMPS), which our studies have shown generalizes across early mathematical concepts, can be reliably measured, and is correlated with mathematical understanding. A study of Grade 1 students and follow up case studies enabled us to reliably classify structural development in terms of a five structural levels. Using a Pattern and Structure Assessment (PASA) interview involving 39 tasks, students identified, visualized, represented, or replicated elements of pattern and structure. Students with high AMPS are likely to have a better understanding of Big Ideas in mathematics than those with low AMPS. They are likely to look for, remember and apply spatial and numerical generalizations and in particular are likely to grasp the multiplicative relationships that underlie the majority of the concepts in the elementary mathematics curriculum.

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References

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

    Google Scholar 

  • Battista, M. C., & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27, 258–292.

    Article  Google Scholar 

  • Battista, M. C., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29, 503–532.

    Article  Google Scholar 

  • Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36, 412–446.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37, 87–115.

    Google Scholar 

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

    Article  Google Scholar 

  • Clarke, B., Clarke, D., & Cheeseman, J. (2006). The mathematical knowledge and understanding young students bring to school. Mathematics Education Research Journal, 18(1), 78–103.

    Article  Google Scholar 

  • Clements, D., & Sarama, J. (2009). Learning and teaching early maths: the learning trajectories approach. New York: Routledge.

    Google Scholar 

  • Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young students in an intervention based on learning trajectories. Journal for Research in Mathematics Education, 42, 167–198.

    Google Scholar 

  • Cobb, P., Gravemeijer, K., Yackel, E., McClain, K., & Whitenack, J. (1997). Mathematizing and symbolizing: the emergence of chains of signification in one first-grade classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition: social, semiotic, and psychological perspectives (pp. 151–233). Mahwah: Erlbaum.

    Google Scholar 

  • English, L. D. (1993). Children’s strategies for solving two-and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24, 255–273.

    Article  Google Scholar 

  • English, L. D. (1999). Assessing for structural understanding in students’ combinatorial problem solving. Focus on Learning Problems in Mathematics, 21(4), 63–82.

    Google Scholar 

  • English, L. D. (2004). Promoting the development of young students’ mathematical and analogical reasoning. In L. D. English (Ed.), Mathematical and analogical reasoning of young learners. Mahwah: Erlbaum.

    Google Scholar 

  • English, L. D. (2012). Data modelling with first-grade students. Educational Studies in Mathematics Education, 81, 15–30.

    Article  Google Scholar 

  • Gray, E., Pitta, D., & Tall, D. (2000). Objects, actions, and images: a perspective on early number development. The Journal of Mathematical Behavior, 18, 401–413.

    Article  Google Scholar 

  • Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98–112.

    Article  Google Scholar 

  • Hunting, R. (2003). Part-whole number knowledge in preschool students. The Journal of Mathematical Behavior, 22, 217–235.

    Article  Google Scholar 

  • Lamon, S. (1996). The development of unitizing: its role in students’ partitioning strategies. Journal for Research in Mathematics Education, 27, 170–193.

    Article  Google Scholar 

  • Lehrer, R., & Schauble, L. (2005). Developing modeling and argument in elementary grades. In T. Romberg, T. Carpenter, & F. Dremock (Eds.), Understanding mathematics and science matters (pp. 29–53). Mahwah: Erlbaum.

    Google Scholar 

  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating structure for all. Mathematics Education Research Journal, 2(2), 10–32.

    Article  Google Scholar 

  • Mulligan, J. T. (2011). Towards understanding the origins of children’s difficulties in mathematics learning. Australian Journal of Learning Difficulties, 16(1), 19–39.

    Article  Google Scholar 

  • Mulligan, J. T., & Mitchelmore, M. C. (1997). Young students’ intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28, 309–331.

    Article  Google Scholar 

  • Mulligan, J. T., & Mitchelmore, M. C. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.

    Article  Google Scholar 

  • Mulligan, J. T., & Vergnaud, G. (2006). Research on children’s early mathematical development: towards integrated perspectives. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present and future (pp. 261–276). London: Sense Publishers.

    Google Scholar 

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

    Article  Google Scholar 

  • Mulligan, J. T., Mitchelmore, M. C., Outhred, L. N., & Russell, S. (1997). Second graders’ representations and conceptual understanding of number. In F. Biddulp & K. Carr (Eds.), People in mathematics education. Proceedings of the 20th annual conference of the Mathematics Education Research Group of Australasia, Rotorua, NZ (Vol. 1, pp. 361–368). Sydney: MERGA.

    Google Scholar 

  • Mulligan, J. T., Mitchelmore, M. C., & Prescott, A. (2005). Case studies of children’s development of structure in early mathematics: a two-year longitudinal study. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1–9). Melbourne: PME.

    Google Scholar 

  • National Council of Teachers of Mathematics (2010). Curriculum focal points from prekindergarten through grade 8: a quest for coherence. Reston: NCTM.

    Google Scholar 

  • Outhred, L., & Mitchelmore, M. C. (2000). Young students’ intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 31, 68–144.

    Article  Google Scholar 

  • Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42, 237–268.

    Google Scholar 

  • Perry, B., & Dockett, S. (2008). Young children’s access to powerful mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 75–108). New York: Routledge.

    Google Scholar 

  • Pitta-Pantazi, D., Gray, E., & Christou, C. (2004). Elementary school students’ mental representations of fractions. In M. Høines & A. Fugelstad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41–48). Bergen: Bergen University College

    Google Scholar 

  • Reynolds, A., & Wheatley, G. H. (1996). Elementary students’ construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 27, 564–582.

    Article  Google Scholar 

  • Steffe, L. (1994). Students’ multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–41). Albany: State University of New York Press.

    Google Scholar 

  • Thomas, N., & Mulligan, J. T. (1995). Dynamic imagery in students’ representations of number. Mathematics Education Research Journal, 7(1), 5–26.

    Article  Google Scholar 

  • Thomas, N., Mulligan, J. T., & Goldin, G. A. (2002). Students’ representations and cognitive structural development of the counting sequence 1–100. The Journal of Mathematical Behavior, 21, 117–133.

    Article  Google Scholar 

  • van den Heuvel-Panhuizen, M., & van den Boogaard, S. (2008). Picture books as an impetus for kindergartners’ mathematical thinking. Mathematical Thinking and Learning, 10, 341–373.

    Article  Google Scholar 

  • van Nes, F., & de Lange, J. (2007). Mathematics education and neurosciences: relating spatial structures to the development of spatial sense and number sense. The Montana Mathematics Enthusiast, 2, 210–229.

    Google Scholar 

  • Warren, E., & Cooper, T. J. (2006). Using repeating patterns to explore functional thinking. Australian Primary Mathematics Classroom, 11(1), 9–14.

    Google Scholar 

  • Warren, E., & Cooper, T. J. (2008). Generalising the pattern rule for visual growth patterns: actions that support 8 year olds’ thinking. Education Studies in Mathematics, 67, 171–185.

    Article  Google Scholar 

  • Wright, R. J. (1994). A study of the numerical development of 5-year-olds and 6-year-olds. Educational Studies in Mathematics, 26, 25–44.

    Article  Google Scholar 

  • Young-Loveridge, J. (2002). Early childhood numeracy: building an understanding of part-whole relationships. Australian Journal of Early Childhood, 27(4), 36–42.

    Google Scholar 

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Acknowledgements

The studies described in this chapter were supported by grants from the Australian Research Council (ARC) and Macquarie University. Any opinions, findings, and conclusions or recommendations expressed in this chapter are those of the authors and do not necessarily reflect the views of the ARC or Macquarie University. The authors express thanks to Dr Marina Papic, Dr Coral Kemp, Dr Kristy Goodwin, Dr Kate Highfield and Jennie Marston, Macquarie University; Dr Anne Prescott, University of Technology Sydney; and teachers, teachers’ aides, students, and school communities for their generous support.

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Authors and Affiliations

  1. Macquarie University, Sydney, Australia

    Joanne T. Mulligan & Michael C. Mitchelmore

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  1. Joanne T. Mulligan
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  2. Michael C. Mitchelmore
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Correspondence to Joanne T. Mulligan .

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Editors and Affiliations

  1. School of STM Education, Queensland University of Technology, Victoria Park Rd, Kelvin Grove, Brisbane, 4059, Queensland, Australia

    Lyn D. English

  2. Macquarie University, University Avenue, Sydney, 2113, New South Wales, Australia

    Joanne T. Mulligan

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Mulligan, J.T., Mitchelmore, M.C. (2013). Early Awareness of Mathematical Pattern and Structure. In: English, L., Mulligan, J. (eds) Reconceptualizing Early Mathematics Learning. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6440-8_3

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