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Developing a Concept of Multiplication of Fractions: Building on Constructivist and Sociocultural Theory

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Constructing Number

Part of the book series: Research in Mathematics Education ((RME))

Abstract

Promoting an understanding of multiplication of fractions has proved difficult for mathematics educators. I discuss a research study aimed at developing a concept of multiplication that supports both multiplication of whole numbers and multiplication of fractions. The study demonstrates how domain-specific theories grounded in two major psychological theories contribute to the development of an empirically based approach to developing this concept. Specifically, the researchers used Learning Through Activity, grounded in constructivism, and aspects of the Elkonin-Davydov Curriculum, grounded in Russian activity theory (sociocultural theory).

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Notes

  1. 1.

    See Simon, Kara et al. (in press) for the additional reasons that guided reinvention is a core principle of LTA.

  2. 2.

    The MARN researchers modified JavaBars by creating an “iterate” button. The iterate button takes a bar and a counting number (we will call it “n”) as inputs and creates a new bar that is n times as long as the original bar. We are grateful to Frank Iannucci for programming this modification.

  3. 3.

    Whether the multiplicand is a fraction or a whole number was not a problem for Kylie and tends not to be a problem for students with a multiple-groups concept of multiplication. They can easily think about iterating a fractional quantity a whole number of times.

  4. 4.

    This convention was used in the Elkonin-Davydov curriculum. It was useful to us because the order of factors was consistent with the student activity (i.e., creating a bar to represent the multiplicand and using the multiplier to act on the multiplicand).

  5. 5.

    The analyses of data can be found in Simon, Placa et al. (in press).

  6. 6.

    See in press, Placa for a detailed account.

  7. 7.

    We have switched from JavaBars to Fraction Bars (Kaput Center for Research and Innovation in STEM Education, 2015). The latter is a more user-friendly application based on JavaBars

  8. 8.

    In this discussion, we focus only on multiplication involving two factors. The MAKE button could be used to produce a product of three or more factors.

References

  • Bereiter, C. (1985). Toward a solution of the learning paradox. Review of Educational Research, 55(2), 201–226.

    Article  Google Scholar 

  • Biddlecomb, B., & Olive, J. (2000). JavaBars [Computer software]. Retrieved 12/10 from http://math.coe.uga.edu/olive/welcome.html

  • Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 3–38). Charlotte, NC: Information Age.

    Google Scholar 

  • Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3–4), 175–190.

    Article  Google Scholar 

  • Davydov, V.V. (1966). Logical and psychological problems of elementary mathematics as an academic subject. From D. B. Elkonin & V. V. Davydov (eds.), Learning capacity and age level: Primary grades, (pp. 54–103). Moscow: Prosveshchenie.

    Google Scholar 

  • Davydov, V. V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula, Soviet Studies in Mathematics Education, Volume 2 (J. Teller, Trans.). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Davydov, V. V., Gorbov, S., Mukulina, T., Savelyeva, M., & Tabachnikova, N. (1999). Mathematics. Moscow: Moscow Press.

    Google Scholar 

  • Davydov, V. V., & Tsvetkovich, Z. H. (1991). On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics, 13, 13–64.

    Google Scholar 

  • 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 grades (pp. 389–412). New York: Routledge.

    Google Scholar 

  • Dougherty, B. (March 31, 2016). Personal communication.

    Google Scholar 

  • Elkonin, D. B. (1972). Toward the problem of stages in the mental development of the child. Soviet Psychology, 10, 225–251.

    Article  Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28(1), 1–18.

    Article  Google Scholar 

  • Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26, 95–143.

    Article  Google Scholar 

  • Ji Yeong, I., & Dougherty, B. (2013). Linking multiplication models to conceptual understanding in measurement approach. 11th Hawaii International Conference of Education. Honolulu, Hawaii.

    Google Scholar 

  • Kaput Center for Research and Innovation in STEM Education. (2015). Fraction Bars [software]. Retrieved from http://www.kaputcenter.umassd.edu/products/software/fractionbars

  • Kennedy, L., & Tipps, S. (1997). Guiding children’s learning of mathematics (8th ed.). Albany, NY: Wadsworth Publishing Company.

    Google Scholar 

  • Lantolf, J. P. (2003). Interpersonal communication and internalization in the second language classroom. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky’s educational theory in cultural context. Cambridge, MA: Cambridge University Press.

    Google Scholar 

  • Lewin, K. (1951). In D. Cartwright (Ed.), Field theory in social science; selected theoretical papers. New York: Harper & Row.

    Google Scholar 

  • Luria, A. R. (1979). The making of mind: A personal account of Soviet psychology. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Mack, N. K. (2000). Long-term effects of building on informal knowledge in a complex content domain. The case in of multiplication of fractions. The Journal of Mathematical Behavior, 19, 307–332.

    Article  Google Scholar 

  • Montangero, J., & Maurice-Naville, D. (1997). Piaget or the advance of knowledge. New York: Psychology Press.

    Google Scholar 

  • Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical Thinking and Learning, 1(4), 279–314.

    Article  Google Scholar 

  • Piaget, J. (1971). Biology and knowledge. Chicago, IL: University of Chicago Press.

    Google Scholar 

  • Piaget, J. (1980). Adaptation and intelligence: Organic selection and phenocopy. Chicago, IL: University of Chicago Press.

    Google Scholar 

  • Piaget, J. (2001). Studies in reflecting abstraction (R. L. Campbell, Trans.). Sussex: Psychology Press.

    Google Scholar 

  • Prediger, S. (2008). The relevance of didactic categories for analyzing obstacles in conceptual change: Revisiting the case of multiplication of fractions. Learning and Instruction, 18, 3–17.

    Article  Google Scholar 

  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.

    Article  Google Scholar 

  • Simon, M. A. (2009). Amidst multiple theories of learning in mathematics education. Journal for Research in Mathematics Education, 477–490.

    Google Scholar 

  • Simon, M. A. (in press). An emerging methodology for studying mathematics concept learning and instructional design. The Journal of Mathematical Behavior.

    Google Scholar 

  • Simon, M. A., Kara, M., Norton, A., & Placa, N. (in press). Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program. Journal of Mathematical Behavior

    Google Scholar 

  • Simon, M. A., Kara, M., Placa, N., & Avitzur, A. (in press). Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework. The Journal of Mathematical Behavior.

    Google Scholar 

  • Simon, M. A., Placa, N., & Avitzur, A. (2016). Two stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47, 63–93.

    Article  Google Scholar 

  • Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (in press). Promoting a concept of fraction-as-measure: A study of learning through activity. The Journal of Mathematical Behavior.

    Google Scholar 

  • Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer.

    Google Scholar 

  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes (Cole, M., John-Steiner, V., Scribner, S. & Souberman, E., Eds.). Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Webel, C., & DeLeeuw, W. W. (2016). Meaning for fraction multiplication: Thematic analysis of mathematical talk in three fifth grade classes. The Journal of Mathematical Behavior, 41, 123–140.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Teaching & Learning, New York University, New York, NY, USA

    Martin A. Simon

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  1. Martin A. Simon
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Correspondence to Martin A. Simon .

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Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Anderson Norton

  2. Department of Psychology, University of Wisconsin–Madison, Madison, WI, USA

    Martha W. Alibali

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Simon, M.A. (2019). Developing a Concept of Multiplication of Fractions: Building on Constructivist and Sociocultural Theory. In: Norton, A., Alibali, M.W. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-00491-0_9

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  • DOI: https://doi.org/10.1007/978-3-030-00491-0_9

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