International Electronic Journal of Mathematics Education

A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Lee MY. A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade. Int Elect J Math Ed. 2019;14(1), 109-122. https://doi.org/10.12973/iejme/3980
APA 6th edition
In-text citation: (Lee, 2019)
Reference: Lee, M. Y. (2019). A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade. International Electronic Journal of Mathematics Education, 14(1), 109-122. https://doi.org/10.12973/iejme/3980
Chicago
In-text citation: (Lee, 2019)
Reference: Lee, Mi Yeon. "A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade". International Electronic Journal of Mathematics Education 2019 14 no. 1 (2019): 109-122. https://doi.org/10.12973/iejme/3980
Harvard
In-text citation: (Lee, 2019)
Reference: Lee, M. Y. (2019). A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade. International Electronic Journal of Mathematics Education, 14(1), pp. 109-122. https://doi.org/10.12973/iejme/3980
MLA
In-text citation: (Lee, 2019)
Reference: Lee, Mi Yeon "A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade". International Electronic Journal of Mathematics Education, vol. 14, no. 1, 2019, pp. 109-122. https://doi.org/10.12973/iejme/3980
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Lee MY. A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade. Int Elect J Math Ed. 2019;14(1):109-22. https://doi.org/10.12973/iejme/3980

Abstract

As a part of the larger study, a case study of two seventh grade students, Peter and Willa, was conducted. To examine links between their fractional knowledge and algebraic reasoning, the students were interviewed twice, once for their fractional knowledge and once for their algebraic knowledge in writing linear equations that required explicit use of unknowns. Peter and Willa’s fractional knowledge influenced the linear equations they wrote to represent quantitative situations. In particular, the findings showed that a reversible iterative fraction scheme is important to understand reciprocal relationships between two quantities in writing a basic linear equation of the form ax=b. Also, considering fractions as multipliers on unknowns is important to write linear equations. Implications for the possibility of introducing linear equations in an earlier grade and how to support the introduction are suggested.

References

  • Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 165-184). New York: Lawrence Erlbaum.
  • Bodanskii, F. G. (1991). The formation of an algebraic method of problem-solving in primary schoolchildren. In L.P. Steffe (Ed.), Psychological abilities of primary school children in learning mathematics (pp.275-338). Reston, VA: National Council of Teachers of Mathematics.
  • Carpenter, T. P., Franke, M. L., & Levi, L. W. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann.
  • Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235-272). New York: Lawrence Erlbaum.
  • Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics instruction. Journal for Research in Mathematics Education, 37(2), 87-115.
  • Choy, B. H., Lee, M. Y., & Mizzi, A. (2015). Textbook signatures: an exploratory study of the notion of gradient in Germany, Singapore and South Korea. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, (pp. 201-208). Hobart, Australia: University of Tasmania.
  • Dougherty, B. (2008). Measure up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early graders (pp.389-412). New York: Lawrence Erlbaum.
  • Driscoll, M. J. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. NH: Heinemann.
  • Empson, S. B., Levi, L., & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai & E. J. Knuth (Eds.), Early algebraization (pp. 409-428). Berlin: Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_22
  • Hackenberg A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. Journal of Mathematical Behavior, 28, 1-18. https://doi.org/10.1016/j.jmathb.2009.04.004
  • Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 1-50. https://doi.org/10.1080/07370008.2010.511565
  • Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196-243.
  • Han, D. H. (2010). Debates on the new national elementary mathematics curriculum content. Journal of Elementary Mathematics Education in Korea, 14(3), 633-658.
  • Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York: Lawrence Erlbaum.
  • Kaput, J. J., Carraher, D. W., & Blanton, M. L. (Eds.). (2008). Algebra in the early grades. New York: Lawrence Erlbaum.
  • Lee, M. Y. (2017). Pre-service teachers’ flexibility with referent units in solving a fraction division problem. Educational Studies in Mathematics, 96(3), 327-348. https://doi.org/10.1007/s10649-017-9771-6
  • Lee, M. Y., & Hackenberg, A. J. (2014). Relationships between Fractional Knowledge and Algebraic Reasoning: The case of Willa. International Journal of Science and Mathematics Education, 12(4), 975-1000. https://doi.org/10.1007/s10763-013-9442-8
  • Ministry of Education, Science and Technology (2009). The 7th elementary school curriculum. Seoul, Korea: Future & Culture.
  • Mizzi, A., Lee, M. Y., & Choy, B. H. (2016). Textbook signatures: An exploratory study of the notion of fractions in Germany, Singapore, and South Korea. Paper presented at the 13th International Congress on Mathematical Education (ICME-13). Hamburg, Germany.
  • Norton, A. (2008). Josh’s operational conjectures: Abductions of a splitting operation and the construction of new fractional schemes. Journal for Research in Mathematics Education, 39(4), 401-430.
  • Seo, G. H., Yu, S. A., & Jeong, J. Y. (2003). Analysis of the 7th Korean national elementary school mathematics curriculum: Focusing on the continuity, sequence, and integration of mathematical content. The Study of Elementary Education in Korea, 16(2), 159-184.
  • Sfard, A. (1995). The development of algebra: confronting historical and psychological perspectives, Journal of Mathematical Behavior, 14, 15-39. https://doi.org/10.1016/0732-3123(95)90022-5
  • Smith, E. (2003). Stasis and change: Integrating patterns, functions, and algebra throughout the K-12 How Does Students’ Fractional 48 curriculum. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 136-150). Reston, VA: National Council of Teachers of Mathematics. https://doi.org/10.1002/jsc.645
  • Steffe, L. P. (1991). Operations that generate quantity. Learning and Individual Differences, 3(1), 61-82. https://doi.org/10.1016/1041-6080(91)90004-K
  • 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. 3-39). Albany, NY: State University of New York Press.
  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York: Springer. https://doi.org/10.1007/978-1-4419-0591-8
  • Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Hillsdale, NJ: Erlbaum.
  • Steffe, L.P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 43, 259–309. https://doi.org/10.1016/1041-6080(92)90005-Y
  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.
  • Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199-219). Albany, NY: State University of New York Press.
  • Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54, 63-75. https://doi.org/10.1023/B:EDUC.0000005237.72281.bf
  • Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning (Vol. 6). New York, NY: Routledge Falmer. https://doi.org/10.4324/9780203454220

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