Advertisement

Comparing US and Japanese Elementary School Teachers' Facility for Linking Rational Number Representations

  • Bryan James MoseleyEmail author
  • Yukari Okamoto
  • Junichi Ishida
Article

Abstract

Using cognitive ethnography as a guiding framework, we investigated US and Japanese fourth-grade teachers' domain knowledge of key fraction representations in individual interviews. The framework focused on revealing cultural trends in participants' organization of knowledge and their interpretations of that organization. Our analyses of the interviews, which included a representation sorting task, indicated three major differences that defined US and Japanese teachers' approaches to rational number representation: (1) Japanese teachers interpreted all rational number representations as conveying primarily mathematical information, whereas US teachers interpreted only some representations as conveying primarily mathematical information; (2) the US teachers focused more intently on part-whole relations than Japanese in their interpretations; and (3) Japanese teachers more easily linked rational number representations to more advanced upcoming content in the curriculum. A review of US textbooks used by the teachers reflected their consistency with US teachers' interpretations of the representations. These findings imply that strong cultural differences underlay the approaches that teachers in both nations take to rational number representation and that these differences may help explain established cross-national differences in student reasoning.

Key words

cognitive cross-national research ethnography rational number understanding teacher knowledge 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). NY: Macmillan.Google Scholar
  2. Berg, B.L. (2004). Qualitative research methods for the social sciences. Boston, MA: Pearson.Google Scholar
  3. Brenner, M.E., Mayer, R.E., Moseley, B., Brar, T., Duran, R., Smith-Reed, B., & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 4(34), 663–689.CrossRefGoogle Scholar
  4. Brenner, M.E., Herman, S., Ho, H.-Z., & Zimmer, J.M. (1999). Cross-national comparison of representational competence. Journal for Research in Mathematics Education, 30, 541–557.CrossRefGoogle Scholar
  5. Carraher, D. (1996). Learning about fractions. In L.P. Steffe, P. Nesher, P. Cobb, G.A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 241–266). Mahwah, NJ: Erlbaum.Google Scholar
  6. Confrey, J. & Scarano, G.H. (1995). Splitting reexamined: Results from a three year longitudinal study of children in grades three to five. Paper presented at the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH, October.Google Scholar
  7. D'Andrade, R.G. (1995). The development of cognitive anthropology. Cambridge: Cambridge University Press.Google Scholar
  8. Davis, R.B. & Maher, C.A. (1993). What are the issues? In R.B. Davis & C.A. Maher (Eds.), Schools, mathematics and the world of reality (pp. 9–34). Boston, MA: Allyn & Bacon.Google Scholar
  9. Dreyfus, T. & Eisenberg, T. (1996). On different facets of mathematical thinking. In R.J. Sternberg & T. Ben-zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah, NJ: Erlbaum.Google Scholar
  10. Harcourt Brace & Company (1998). Math advantage. New York: Harcourt School.Google Scholar
  11. Kerslake, D. (1991). The language of fractions. In K. Durkin & S. Beatrice (Eds.), Language in mathematical education: Research and practice (pp. 85–94). Bristol, PA: Open University Press.Google Scholar
  12. Kieren, T.E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R.A. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101–144). Columbus, OH: ERIC/SMEAC.Google Scholar
  13. Kieren, T.E. (1980). The rational number construct: Its elements and mechanisms. In T.E. Kieren (Ed.), Recent research on number learning (pp. 32–55). Columbus, OH: ERIC/SMEAC.Google Scholar
  14. Kieren, T.E. (1993). Rational and fractional number: from quotient fields to recursive understanding. In T.P. Carpenter, E. Fennema & T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 261–288). Hillsdale, NJ: Erlbaum.Google Scholar
  15. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representation in mathematics learning and problem solving. In C. Javier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hilsdale, NJ: Erlbaum.Google Scholar
  16. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Erlbaum.Google Scholar
  17. Mack, N.K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16–32.CrossRefGoogle Scholar
  18. Mack, N.K. (1993). Learning rational numbers with understanding: The case of informal knowledge. In T.P. Carpenter, E. Fennema & T.A. Romberg (Eds.), Rational numbers: an integration of research (pp. 85–106). Hillsdale, NJ: Erlbaum.Google Scholar
  19. Mack, N.K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26, 422–441.CrossRefGoogle Scholar
  20. Marshall, S.P. (1993). Assessment of rational number understanding: A schema-based approach. In T.P. Carpenter, E. Fennema & T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 261–288). Hillsdale, NJ: Erlbaum.Google Scholar
  21. Mayer, R.E., Sims V., & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32, 443–460.CrossRefGoogle Scholar
  22. Moseley, B. (2005). Students' early mathematical representation knowledge: The effects of emphasizing single or multiple perspectives of the rational number domain in problem solving. Educational Studies in Mathematics, 60, 37–69.CrossRefGoogle Scholar
  23. Mullis, I.V.S., Martin, M.O., Beaton, A.E., Gonzalez, E.J., Kelley, D.L., & Smith, T.A. (1997). Mathematics achievement in the primary school years: IEA's Third International Mathematics and Science Study (TIMSS). Boston, MA: Boston College.Google Scholar
  24. Nunes, T. & Bryant, P.E. (1996). Children doing mathematics. Oxford: Blackwell.Google Scholar
  25. Ohlsson, S. (1987). Sense and reference in the design of iterative illustrations for rational numbers. In R.W. Lawler & M. Yazdani (Eds.), Artificial Intelligence and Education (pp. 307–344). Norwood, NJ: Ablex.Google Scholar
  26. Quilici, J.L. & Mayer, R.E. (2002). Teaching students to recognize structural similarities between statistics word problems. Applied Cognitive Psychology, 16, 325–342.CrossRefGoogle Scholar
  27. Rahim, M.H. & Kieren, T E. (1988). A preliminary report on the reliability and factorial validity of the rational number thinking test in the Republic of Trinidad and Tobago. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Dekalb, IL.Google Scholar
  28. Schoenfeld, A.H. & Herrman, D.J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 484–494.CrossRefGoogle Scholar
  29. Spradley, J.P. (1979). The ethnographic interview. New York: Holt, Renehart & Wilson.Google Scholar
  30. Stevenson, H. (1985). An analysis of Japanese and American textbooks in mathematics. Office of Educational Research and Improvement (ED), Washington DC, Clearing house no. SO017450.Google Scholar
  31. Stigler, J.W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. NY: Free Press.Google Scholar
  32. Stoddart, K. (1986). The presentation of everyday life. Urban Life, 15(1), 103–121.Google Scholar
  33. Yoshida, H. (1991). Open image in new window [Children’s understanding of number]. Tokyo, Japan: Shinyo-sha.Google Scholar

Copyright information

© National Science Council, Taiwan 2006

Authors and Affiliations

  • Bryan James Moseley
    • 1
    Email author
  • Yukari Okamoto
    • 2
  • Junichi Ishida
    • 3
  1. 1.College of Education, Department of Educational & Psychological StudiesFlorida International UniversityMiamiUSA
  2. 2.UC Santa BarbaraSanta BarbaraUSA
  3. 3.Yokohama National UniversityYokohamaJapan

Personalised recommendations