U.S. and Chinese Teachers' Conceptions and Constructions of Representations: A Case of Teaching Ratio Concept

  • Jinfa CaiEmail author
  • Tao Wang


This study examines U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach the concept of ratio. All Chinese lesson plans are very similar with details in teaching contents and procedure. The U.S lesson plans are extremely varied although they all adopted the ‘outline and worksheet’ format. Both the Chinese and the U.S. teachers relied on concrete representations in introducing the concept of ratio, but they have very different thinking in selecting and presenting the concrete representations, as well as in the functions of the representations. The U.S. teachers are much more likely than Chinese teachers to predict drawing and guess-and-check strategies. Chinese teachers are much more likely than U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score than U.S. teachers. This study contributes to our understanding about U.S. and Chinese teachers' beliefs about mathematics teaching and learning.

Key Words

classroom instruction cross-national perspective mathematical problem solving pedagogical representation ratio concept solution representation 


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  1. Ball, D.L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T.P. Carpenter, E. Fennema & T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 328–375). Hillsdale, NJ: Erlbaum.Google Scholar
  2. Becker, J.P. Sawada, T. & Shimizu, Y. (1999). Some findings of the U.S.–Japan cross-cultural research on students' problem-solving behaviors, In G. Kaiser, E. Luna & I. Huntley (Eds.), International comparisons in mathematics education (pp. 121–139). London: Falmer Press.Google Scholar
  3. Bishop, A. (1992). International perspectives on research in mathematics education. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 710–723). New York: Macmillan Publishing Company.Google Scholar
  4. Bradburn, M.B. & Gilford, D.M. (1990). A framework and principles for international comparative studies in education. Washington, DC: National Academic Press.Google Scholar
  5. Brophy, J.E. & Good, T.L. (1986). Teacher behavior and student achievement. In M.C. Wittrock (Ed.), Handbook of research on teaching (3rd Ed.) (pp. 328–375). New York: Macmillan.Google Scholar
  6. Bruner, J. (1996). The culture of education. Cambridge, MA: Harvard University Press.Google Scholar
  7. Cai, J. (1995). A cognitive analysis of U.S. and Chinese students' mathematical performance on tasks involving computation, simple problem solving, and complex problem solving. (Journal for Research in Mathematics Education monograph series 7), Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  8. Cai, J. (2000). Mathematical thinking involved in U.S. and Chinese students' solving process-constrained and process-open problems. Mathematical Thinking and Learning, 2, 309–340.CrossRefGoogle Scholar
  9. Cai, J. (2001). Improving mathematics learning: Lessons from cross-national studies of U.S. and Chinese students. Phi Delta Kappan, 82(5), 400–405.Google Scholar
  10. Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers' beliefs. Journal of Mathematical Behavior, 23, 135–167.CrossRefGoogle Scholar
  11. Cai, J. (2005). U.S. and Chinese teachers' knowing, evaluating, and constructing representations in mathematics instruction. Mathematical thinking and learning: An International Journal, 7, 135–169.CrossRefGoogle Scholar
  12. Cai, J. & Hwang, S. (2002). U.S. and Chinese students' generalized and generative thinking in mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401–421.CrossRefGoogle Scholar
  13. Cai, J. & Sun, W. (2002). Developing students' proportional reasoning: A Chinese perspective. In B.H. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios and proportions (pp. 195–205). National Council of Teachers of Mathematics 2002 Yearbook. Reston, VA: NCTM.Google Scholar
  14. Carpenter, T.P. Franke, M.L. Jacobs, V.R. Fennema, E. & Empson, S.B. (1998). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20.CrossRefGoogle Scholar
  15. Cifarelli, V.V. (1998). The development of mental representations as a problem solving activity. Journal of Mathematical Behavior, 17(2), 239–264.CrossRefGoogle Scholar
  16. Clark, C.M. & Peterson, P.L. (1986). Teachers' thought process. In M.C. Wittrock (Ed.), Handbook of research on teaching (pp. 255–327). New York, MaCmillan Publishing Company.Google Scholar
  17. Cobb, P. Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.CrossRefGoogle Scholar
  18. Cooney, T.J. Shealy, B.E. & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306–333.CrossRefGoogle Scholar
  19. Doyle, W. (1983). Academic work. Review of Educational Research, 53, 159–199.Google Scholar
  20. Doyle, W. (1988). Work in mathematics classes: The context of students' thinking during instruction. Educational Psychologist, 23, 167–180.CrossRefGoogle Scholar
  21. 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). Hillsdale, NJ: Erlbaum.Google Scholar
  22. Dufour-Janvier, B. Bednarz, N. & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning mathematical problem solving (pp. 109–122). Hillsdale, NJ: Erlbaum.Google Scholar
  23. English, L.D. (1997). Mathematical reasoning: Analogies, metaphors and images. Mahwah, NJ: Erlbaum.Google Scholar
  24. Gallimore, R. (1996). Classrooms are just another cultural activity. In D.L. Speece & B. K. Keogh (Eds.), Research on classroom ecologies: Implications for inclusion of children with learning disabilities (pp. 229–250). Mahwah, NJ: Erlbaum.Google Scholar
  25. Geary, D.C. Bow-Thomas, C.C. Liu, F., & Siegler, R.S. (1996). Development of arithmetic competencies in Chinese and American children: Influence of age, language, and schooling. Child Development, 67, 2022–2044.PubMedCrossRefGoogle Scholar
  26. Goldin, G.A. (1987). Cognitive representational systems for mathematical problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning mathematical problem solving (pp. 125–145). Hillsdale, NJ: Erlbaum.Google Scholar
  27. Goldin, G.A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.CrossRefGoogle Scholar
  28. Good, T. Grouws, D. & Ebmeier, M. (1983). Active mathematics teaching. New York: Longman.Google Scholar
  29. Greeno, J.G. (1987). Instructional representations based on research about understanding. In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 61–88). New York: Academic Press.Google Scholar
  30. Harmon, M. Smith, T. A. Martin, M.O. et al. (1997). Performance assessment in IEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, MA, Boston College: TIMSS International Study Center.Google Scholar
  31. Hiebert, J. & Carpenter, T. (1992). Learning and teaching with understanding. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.Google Scholar
  32. Hiebert, J. & Wearne, D. (1993). Instructional tasks, classroom discourse, and students' learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425.Google Scholar
  33. Janvier, C. (1987). Problems of representation in the teaching and learning of mathematical problem solving. Hillsdale, NJ: Erlbaum.Google Scholar
  34. Krutetskii, V.A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.Google Scholar
  35. Leinhardt, G. (1993). On teaching. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 4, pp. 1–54). Hillsdale, NJ: Erlbaum.Google Scholar
  36. Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook for research on teaching, 4th Edn. (pp. 333–357). Washington, DC: American Educational Research Association.Google Scholar
  37. Lo, J.J., Cai, J. & Watanabe, T. (2001). A comparative study of the selected textbooks from China, Japan, Taiwan, and the United States on the teaching of ratio and proportion concepts. In R. Speiser, C.A. Maher & C. N. Walter (Eds.), Proceedings of the Twenty-Third Annual Meeting of the North American Chapter of the International Group of the Psychology of Mathematics Education (pp. 509–520). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  38. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Erlbaum.Google Scholar
  39. National Council of Teachers of Mathematics (1991). Professional standards for school mathematics. Reston, VA: The Author.Google Scholar
  40. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: The Author.Google Scholar
  41. Paine, L. & Ma, L. (1993). Teachers working together: A dialogue on organizational and cultural perspectives of Chinese teachers. International Journal of Educational Research, 19, 675–697.CrossRefGoogle Scholar
  42. Perry, M. VanderStoep, S.W. & Yu, S.L. (1993). Asking questions in first-grade mathematics classes: Potential influences on mathematical thought. Journal of Educational Psychology, 85(1), 31–40.CrossRefGoogle Scholar
  43. Perkins, D.N. & Unger, C. (1994). A new look in representations for mathematics and science learning. Instructional Science, 2, 1–37.CrossRefGoogle Scholar
  44. Pimm, D. (1995). Symbols and Meanings in School Mathematics. London: Routledge.Google Scholar
  45. Presmeg, N.C. (1997). Generalization using imagery. In L.D. English (Ed.), Mathematical Reasoning: Analogies, Metaphors and Images (pp. 299–312). Mahwah, NJ: Erlbaum.Google Scholar
  46. Putnam, R.T. Lampert, M. & Peterson, P.L. (1990). Alternative perspectives on knowing mathematics in elementary schools. In C.B. Cazden (Ed.), Review of research in education (pp. 57–150). Washington, DC: American Educational Research Association.Google Scholar
  47. Silver, E.A. Leung, S.S. & Cai, J. (1995). Generating multiple solutions for a problem: A comparison of the responses of U.S. and Japanese students. Educational Studies in Mathematics, 28(1), 35–54.CrossRefGoogle Scholar
  48. Stein, M.K. & Lane, S. (1998). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform Mathematics Project. Educational Research and Evaluation, 2, 50–80.CrossRefGoogle Scholar
  49. Stigler, J.W. & Hiebert, J. (1999). The Teaching gap: Best ideas from the world's teachers for improving education in the classroom. The Free Press.Google Scholar
  50. Stigler, J.W. Fernandez, C. & Yoshida, M. (1996). Traditions of school mathematics in Japanese and American elementary schools. In L.P. Steffe, P. Nesher, P. Cobb, G.A. Goldin, & B. Greer (Eds.), Theories of mathematics learning (pp. 149–177). Hillsdale, NJ: Erlbaum.Google Scholar
  51. Thompson, A.G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). NY: Macmillan.Google Scholar
  52. Wang, T. (2003). Culture and mathematics classroom discourse: A comparative perspective. Unpublished qualifying paper, Cambridge, MA: Harvard University.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.University of DelawareDelawareUSA
  2. 2.University of TulsaTulsaUSA

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