Research in Mathematical Education (한국수학교육학회지시리즈D:수학교육연구)

Korean Society of Mathematical Education (한국수학교육학회)
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Investigating Children's Informal Knowledge and Strategies: The Case of Fraction Division

  • Received : 2019.09.03
  • Accepted : 2019.12.27
  • Published : 2019.12.31
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Abstract

This paper investigates what informal knowledge and strategies fifth-grade students brought to a classroom and how much they had potential to solve fraction division story problems. The findings show that most of the participants were engaged to understand the meaning of fraction division prior to their formal instruction at school. In order to solve the story problems, the informal knowledge related to fractions as well as division was actively utilized in student's strategies and justification. Students also used various informal strategies from mental calculation, direct modeling, to relational thinking. Formal instructions about fraction division at schools can be facilitated for sense-making of this complex fraction division conception by unpacking informal knowledge and thinking they might bring to the classrooms.

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References

  1. Baroody, A. J. & Coslick, R. T. (1998). Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum Associates.
  2. Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23, 194-222. https://doi.org/10.2307/749118
  3. Bulgar, S. (2003). Children's sense-making of division of fractions. The Journal of Mathematical Behavior, 22(3), 319-334. https://doi.org/10.1016/S0732-3123(03)00024-5
  4. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.
  5. Corbin, J., & Strauss, A. (2008). Basics of qualitative research (3rd ed.). Los Angeles, CA: Sage.
  6. Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction first-grade classroom. Cognition and Instruction, 17(3), 283-342. https://doi.org/10.1207/S1532690XCI1703_3
  7. Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children's thinking. Educational Studies in Mathematics, 63, 1-28. https://doi.org/10.1007/s10649-005-9000-6
  8. Empson, S. B., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Innovations in Cognitively Guided Instruction. Portsmouth, NH: Heinemann.
  9. Flores, A. (2002). Profound understanding of division of fractions. In B. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions, 2002 NCTM Yearbook, (pp. 247-256). Reston, VA: National Council of Teachers of Mathematics.
  10. Ginsburg, H. P. (1977). Children's arithmetic: The learning process. New York: Van Nostrand.
  11. Ginsburg, H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. Cambridge, MA: Cambridge University Press.
  12. Hatano, G. (1996). A conception of knowledge acquisition and its implications for mathematics education. In L. P. Steffe et al. (Eds.), Theories of mathematical learning (pp. 197-217). Mahwah, NJ: Lawrence Erlbaum Associates.
  13. Kent, A. B., Empson, S. B., and Nielsen, L. (2015). The richness of children's fraction strategies. Teaching children mathematics, 22(2), 84-90. https://doi.org/10.5951/teacchilmath.22.2.0084
  14. Leinhardt, G. (1988). Getting to know: Tracing students' mathematical knowledge from intuition to competence. Educational Psychologist, 23(2), 119-144. https://doi.org/10.1207/s15326985ep2302_4
  15. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
  16. Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 26, 422-441. https://doi.org/10.2307/749431
  17. Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295. https://doi.org/10.2307/749828
  18. Merriam, S. B. & Tisdell, E. J. (2016). Qualitative research: A guide to design and implementation. San Francisco, CA: JosseyBass.
  19. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: The Author.
  20. Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26(2/3), 165-190. https://doi.org/10.1007/BF01273662
  21. Pothier, Y. & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14(5), 307-3 https://doi.org/10.2307/748675
  22. Resnick, L. B. (1989). Developing mathematical knowledge. American Psychologist, 44, 162-169. https://doi.org/10.1037/0003-066X.44.2.162
  23. Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analyses of arithmetic for mathematics teaching (pp. 373-429). Hillsdale, NJ: Erlbaum.
  24. Saldana, J. (2009). The coding manual for qualitative researchers. London: Sage.
  25. Saxe, G. B. (1988). Candy selling and math learning. Educational Researcher, 17(6), 14-21. https://doi.org/10.2307/1175948
  26. Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. Journal of Educational Research, 95, 333-348. https://doi.org/10.1080/00220670209596608
  27. Siebert, D. (2002). Connecting informal thinking and algorithms: The case of division of fractions. In B. Litwiller and G. Bright (Eds.), Making sense of fractions, ratios, and proportions, 2002 NCTM Yearbook, (pp. 247-256). Reston, VA: National Council of Teachers of Mathematics.
  28. Sinicrope, R., Mick, H., & Kolb, J. (2002). Interpretations of fraction division. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions, 2002 NCTM Yearbook, (pp. 247-256). Reston, VA: National Council of Teachers of Mathematics.
  29. Steffe, L.P. (1994). Children's multiplying schemes, in G. Harel and J. Confrey (eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). Albany, NY: State University of New York Press.
  30. Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22(3), 237-295. https://doi.org/10.1016/S0732-3123(03)00022-1
  31. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht: Kluwer Academic Publishers.
  32. Streefland, L. (1993). Fractions: A realistic approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 289-325). Hillsdale, NJ: Erlbaum.
  33. Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston, MA: Allyn & Bacon.
  34. Von Glasersfeld E. (1987). The construction of knowledge: Contributions to conceptual semantics. Salinas CA: Intersystems Publications.
  35. Warrington, M. A. (1997). How children think about division with fractions. Mathematics Teaching in the Middle School, 2(6), 390-394. https://doi.org/10.5951/MTMS.2.6.0390
  36. Yin, R. K. (2014). Case study research: Design and methods. London: Sage.