Advertisement

Developing a coherent approach to multiplication and measurement

  • Andrew IzsákEmail author
  • Sybilla Beckmann
Article
  • 63 Downloads

Abstract

We examine opportunities and challenges of applying a single, explicit definition of multiplication when modeling situations across an important swathe of school mathematics. In so doing, we review two interrelated conversations within multiplication research. The first has to do with identifying and classifying situations that can be modeled by multiplication, and the second has to do with identifying what is consistently characteristic of the operation when considering nonnegative real numbers. We review seminal lines of research––including those of Vergnaud and Davydov––and highlight ways that these lines do not provide a thoroughly unified view of multiplication. Then we offer our own approach based in measurement. To underscore consequences of the approach we outline, we use rectangular area and division to illustrate that, as a field, we may need to adjust how we think about connections between multiplication equations and at least some problem situations. We close with a set of questions about unified approaches to topics related to multiplication.

Keywords

Multiplication Division Multiplicative conceptual field Coherence Mathematical definitions 

Notes

Acknowledgement

We thank Andy Norton, Jack Smith, and Bill McCallum for conversations that influenced our thinking about multiplication. This research was supported by the National Science Foundation under Grant No. DRL-1420307. The opinions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20(4), 367–385.CrossRefGoogle Scholar
  2. Australian Curriculum, Assessment, and Reporting Authority (ACARA). (2010). F-10 curriculum, mathematics, key ideas. Retrieved 8 Dec 2018 from www.australiancurriculum.edu.au.
  3. Beckmann, S. (2017). Mathematics for elementary teachers with activities (5th ed.). New York, NY: Pearson.Google Scholar
  4. Beckmann, S., & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17–38.CrossRefGoogle Scholar
  5. Beckmann, S., & Izsák, A. (2018a). Generating equations for proportional relationships using magnitude and substance conceptions. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (pp. 1215–1223). San Diego, California.Google Scholar
  6. Beckmann, S., & Izsák, A. (2018b). Two senses of unit words and implications for topics related to multiplication. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 205). Umeå, Sweden: PME.Google Scholar
  7. Boulet, G. (1998). On the essence of multiplication. For the Learning of Mathematics, 18(3), 12–19.Google Scholar
  8. Brown, L. (Ed) (1993). The New Shorter Oxford English Dictionary. Oxford England: Clarendon Press.Google Scholar
  9. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Washington, DC: Author.CrossRefGoogle Scholar
  10. Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–330). Albany: State University of New York Press.Google Scholar
  11. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.CrossRefGoogle Scholar
  12. Cuoco, A., & McCallum, W. (2018). Curricular coherence in mathematics. In Y. Li, W. J. Lewis, & J. J. Madden (Eds.), Mathematics matters in education: Essays in honor of Roger E. Howe (pp. 245–256). Cham: Springer.CrossRefGoogle Scholar
  13. Davydov, V. V. (1992). The psychological analysis of multiplication procedures. Focus on Learning Problems in Mathematics, 14(1), 3–67.Google Scholar
  14. Davydov, V. V., & TSvetkovich, Z. (1991). On the objective origin of the concept of fraction. Focus on Learning Problems in Mathematics, 13(1), 13–64.Google Scholar
  15. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17.CrossRefGoogle Scholar
  16. Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York: MacMillan.Google Scholar
  17. Greer, B. (1994). Extending the meaning of multiplication and division. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 61–85). Albany: State University of New York Press.Google Scholar
  18. Hart, K. (1981). Ratio and proportion. In K. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 88–101). London: John Murray.Google Scholar
  19. Izsák, A., Kulow, T., Beckmann, S., Stevenson, D., & Ölmez, B. (in press). Using a measurement meaning of multiplication to connect topics in a content course for future teachers. Mathematics Teacher Educator.Google Scholar
  20. Kaput, J., & West, M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235–287). Albany: State University of New York Press.Google Scholar
  21. Lobato, J., & Ellis, A. B. (2010). Developing essential understandings: ratios, proportions, and proportional reasoning Grades 6–8. In R. M. Zbiek (Series Ed.), Essential understandings. Reston: National Council of Teachers of Mathematics.Google Scholar
  22. McCallum, W. (2018). Making sense of mathematics and making mathematics make sense. In Y. Shimizu & R. Vithal (Eds.), ICMI Study 24 Pre-conference proceedings: School mathematics curriculum reforms: Challenges, changes and opportunities (pp. 1–8). Tsukuba, Japan: University of Tsukuba.Google Scholar
  23. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, D.C.: Author.Google Scholar
  24. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.Google Scholar
  25. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Blackwell Publishers.Google Scholar
  26. Schmidt, W., Wang, H., & McKnight, C. (2005). Curriculum coherence: An examination of U.S. mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37(5), 525–559.CrossRefGoogle Scholar
  27. Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Reston, VA; Hillsdale, NJ: National Council of Teachers of Mathematics; Lawrence Erlbaum.Google Scholar
  28. Schwartz, J. (2009). Formulating measures: Toward modeling in the K-12 STEM curriculum. In B. Brizuela & B. Gravel (Eds.), “Show me what you know”: Exploring student representations across STEM disciplines (pp. 250–267). New York: Teachers College Press, Columbia University.Google Scholar
  29. Sekretariat der Ständigen Konferenz der Kulturminister der Länder in der Bundesrepublik Deutschland. (2005). Bildungsstandards im Fach Mathematik für den Primabereich, Beschluss fom 15.10.2004. Germany: Luchterland, Wolters Kluwer. Retrieved 8 Dec 2018 from https://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/2004/2004_10_15-Bildungsstandards-Mathe-Primar.pdf.
  30. Simon, M., Kara, M., Norton, A., & Placa, N. (2018). Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program. The Journal of Mathematical Behavior, 52, 151–173.CrossRefGoogle Scholar
  31. Simon, M., & Placa, N. (2012). Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding. For the Learning of Mathematics, 32(2), 35–41.Google Scholar
  32. 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: State University of New York Press.Google Scholar
  33. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York: Springer.CrossRefGoogle Scholar
  34. Steffe, L. P. (1988). Children’s construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 119–140). Reston, VA: Lawrence Erlbaum Associates & National Council of Teachers of Mathematics.Google Scholar
  35. Thompson, P., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Shifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  36. Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics, 17(4), 401–412.CrossRefGoogle Scholar
  37. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York: Academic Press.Google Scholar
  38. Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in middle grades (pp. 141–161). Reston, VA; Hillsdale, NJ: National Council of Teachers of Mathematics; Erlbaum.Google Scholar
  39. Vergnaud, G. (1994). The multiplicative conceptual field: What and why? In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 51–39). Albany: State University of New York Press.Google Scholar
  40. Vygotsky, L. (1986). Thought and language. Cambridge, MA: MIT Press.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of EducationTufts UniversityMedfordUSA
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

Personalised recommendations