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DISTRIBUTIVE PARTITIONING OPERATION IN MATHEMATICAL SITUATIONS INVOLVING FRACTIONAL QUANTITIES

  • Soo Jin Lee
  • Jaehong ShinEmail author
Article

Abstract

In the present study, we describe a participating student’s (Carol’s) distributive partitioning scheme and operations along with Steffe’s and his colleagues’ studies about children’s constructions of fraction knowledge as a particular model of mathematical learning. Analysis of Carol’s mathematical behaviors indicates that an operationally common mathematical behavior (distributive partitioning operation) was revealed in various mathematical problem situations such as fraction multiplication, fraction division, and multiplicative transformation between fractional quantities. It both provides a rationale for why becoming versed in one mathematical subject could facilitate working with another mathematical subject and also implies the necessity of describing and defining students’ mathematical behaviors from an operational view of knowledge, which might lead to building foundations of a substantial cognitive map for students’ mathematical development.

Key words

distributive partitioning scheme fraction multiplication and division multiplicative transformation scheme theory 

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References

  1. Behr, M. J., Wachsmuth, I., Post, T. R. & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323–341.CrossRefGoogle Scholar
  2. Biddlecomb, B. D. (1994). Theory-based development of computer microworlds. Journal of Research in Childhood Education, 8(2), 87–98.CrossRefGoogle Scholar
  3. Biddlecomb, B. D. & Olive, J. (2000). JavaBars [Computer software]. Retrieved July 7, 2005 from http://jwilson.coe.uga.edu/olive/welcome.html
  4. Confrey, J. & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 231–266). Mahwah, NJ: Erlbaum.Google Scholar
  5. Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432.CrossRefGoogle Scholar
  6. 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), 1–18.CrossRefGoogle Scholar
  7. Hall, R. (2000). Videorecordings as theory. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 647–664). Mahwah, NJ: Erlbaum.Google Scholar
  8. Hunting, R. P., Davis, G. & Pearn, C. (1996). Engaging whole-number knowledge for rational-number learning using a computer-based tool. Journal for Research in Mathematics Education, 27(3), 354–379.CrossRefGoogle Scholar
  9. Kieren, T. E. (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. In R. Leinhardt, R. Putnam & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323–371). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  10. Lamon, S. J. (1999). Teaching fractions and ratios for understanding. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  11. Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. II, pp. 629–667). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  12. Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16–33.CrossRefGoogle Scholar
  13. Mack, N. K. (2000). Long-term effects of building on informal knowledge in a complex content domain: The case of multiplication of fractions. Journal of Mathematical Behavior, 19(3), 307–332.CrossRefGoogle Scholar
  14. 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.CrossRefGoogle Scholar
  15. Norton, A. & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583.CrossRefGoogle Scholar
  16. Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical thinking and learning, 1(4), 279–314.CrossRefGoogle Scholar
  17. Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an extrapolation to rational number of arithmetic. The Mathematics Educator, 11(1), 1–9.Google Scholar
  18. Olive, J. & Steffe, L. P. (2002). Schemes, schemas and director systems—an integration of Piagetian scheme theory with Skemp’s model of intelligent learning. In D. Tall & M. Thomas (Eds.), Intelligence, learning and understanding in mathematics. Flaxton, Australia: Post Press.Google Scholar
  19. Piaget, J. (1954). The child’s construction of reality. London: Routledge & Kegan Paul.CrossRefGoogle Scholar
  20. Pitkethly, A. & Hunting, R. P. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30(1), 5–38.CrossRefGoogle Scholar
  21. Shin, J. & Lee, S. J. (2010). Eighth graders’ use of distributive reasoning for multiplicative transformation of unit fractions. Paper presented at the 32st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH.Google Scholar
  22. Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307.CrossRefGoogle Scholar
  23. Steffe, L. P. (2003). Fractional commensurate, composition, and adding scheme Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22(3), 237–295.CrossRefGoogle Scholar
  24. Steffe, L. P. (2008). Radical constructivism and “school mathematics”. Rotterdam, the Netherlands: Sense.Google Scholar
  25. Steffe, L. P. (2010). The partitive and the part–whole schemes. In L. P. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 75–122). New York: Springer.CrossRefGoogle Scholar
  26. Steffe, L. P., Cobb, P. & von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York: Springer.CrossRefGoogle Scholar
  27. Steffe, L. P. & Olive, J. (1990). Children’s construction of the rational numbers of arithmetic. Athens, GA: National Science Foundation, University of Georgia.Google Scholar
  28. Steffe, L. P. & Olive, J. (2010). Children’s fractional knowledge. New York: Springer.CrossRefGoogle Scholar
  29. Steffe, L. P. & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Erlbaum.Google Scholar
  30. Steffe, L. P. & Tzur, R. (1994). Interaction and children’s mathematics. Journal of Research in Childhood Education, 8(2), 99–116.CrossRefGoogle Scholar
  31. Steffe, L. P. & Ulrich, C. (2010). Equipartitioning operations for connected numbers: Their use and interiorization. In L. P. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 225–275). New York: Springer.CrossRefGoogle Scholar
  32. Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht, the Netherlands: Kluwer.CrossRefGoogle Scholar
  33. Thompson, P. W. & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  34. Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390–416.CrossRefGoogle Scholar
  35. von Glasersfeld, E. (1982). An interpretation of Piaget’s constructivism. Revue Internationale de Philosophie, 36(4), 612–635.Google Scholar
  36. von Glasersfeld, E. (1984). An introduction to radical constructivism. In P. Watzlawick (Ed.), The invented reality (pp. 17–40). New York: Norton.Google Scholar
  37. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. New York: Routledge Falmer.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2013

Authors and Affiliations

  1. 1.Korea National University of EducationCheongwon-gunSouth Korea

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