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Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds pp 237–260Cite as

Generalizing Fractional Structures: A Critical Precursor to Algebraic Thinking

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Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Our research focuses on how students find an unknown whole, when given a known fractional part of the whole, and its equivalent quantity. This chapter will show how Year 5 and Year 6 students, who have yet to meet formal algebraic notation, create algebraic meaning and syntax through their solutions of these fraction problems. Some students rely on diagrammatic representations using different mixes of multiplicative and additive strategies. Other students use fully multiplicative approaches to find the whole. Some students’ solutions show how they use “best available” symbols to move beyond arithmetic calculation and show evidence of algebraic thinking, especially when students are able to treat particular numerical and fractional values as quasi-variables. This chapter sets out to identify those precursors of algebraic thinking that allow students to move beyond particular fraction values to generalize their solutions.

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References

  • Australian Curriculum, Assessment and Reporting Authority [ACARA] (2016). Australian Curriculum: Mathematics. V8.3 Australian Curriculum, Assessment and Reporting Authority. http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1 Accessed: 20 February 2017.

  • Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–443.

    Google Scholar 

  • Empson, S., Levi, L., & Carpenter, T. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 409–428). Heidelberg: Springer.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2001). Fostering an understanding of algebraic generalisation through numerical expressions: The role of quasi-variables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The future of the teaching and learning of algebra (pp. 258–264). Melbourne: University of Melbourne.

    Google Scholar 

  • Hackenberg, A. & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46 (2), 196–243.

    Google Scholar 

  • Jacobs, V., Franke, M., Carpenter, T., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258–288.

    Google Scholar 

  • Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.

    Google Scholar 

  • Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lee, M. Y. (2012). Fractional knowledge and equation writing: The cases of Peter and Willa. Paper presented at the 12th International Congress on Mathematical Education 8–15 July 2012, Seoul, Korea. Retrieved 23rd May 2017 from https://www.researchgate.net/publication/299560857_FRACTIONAL_KNOWLEDGE_AND_EQUATION_WRITING_THE_CASES_OF_PETER_AND_WILLA.

  • Lee, M. Y., & Hackenberg, A. (2014). Relationships between fractional knowledge and algebraic reasoning: The case of Willa. International Journal of Science and Mathematics Education, 12(4), 975–1000.

    Google Scholar 

  • Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Mason, J. (2017). Exchanging, trading and substituting. Retrieved 23rd May 2017 from https://educationblog.oup.com/secondary/maths/exchanging-trading-and-substituting.

  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10–32.

    Google Scholar 

  • National Mathematics Advisory Panel [NMAP] (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education. http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf Accessed: 20 February 2017.

  • Pearn, C., & Stephens, M. (2015). Strategies for solving fraction tasks and their link to algebraic thinking. 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. 493–500). Sunshine Coast, AU: MERGA.

    Google Scholar 

  • Pearn, C., & Stephens, M. (2016). Competence with fractions in fifth or sixth grade as a unique predictor of algebraic thinking? In B. White, M. Chinnappan, & S. Trenholm (Eds.), Opening up mathematics education research. Proceedings of the 39th Annual Conference of MERGA (pp. 519–526). Adelaide, AU: MERGA.

    Google Scholar 

  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M., & Chen, M. (2012). Early predictors of high school mathematics achievement. http://pss.sagepub.com/content/early/2012/06/13/0956797612440101 Accessed: 20 February 2017.

  • Stephens, M., & Ribeiro, A. (2012). Working towards algebra: The importance of relational thinking. Revista Latinoamericano de Investigacion en Matematica Educativa, 15(3), 373–402.

    Google Scholar 

  • Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10–17.

    Google Scholar 

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Authors and Affiliations

  1. Melbourne Graduate School of Education, The University of Melbourne, Melbourne, VIC, 3010, Australia

    Catherine Pearn & Max Stephens

Authors
  1. Catherine Pearn
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  2. Max Stephens
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Correspondence to Catherine Pearn .

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Editors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada

    Carolyn Kieran

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Pearn, C., Stephens, M. (2018). Generalizing Fractional Structures: A Critical Precursor to Algebraic Thinking. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_10

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  • DOI: https://doi.org/10.1007/978-3-319-68351-5_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68350-8

  • Online ISBN: 978-3-319-68351-5

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