U.S. and Chinese teachers' conceptions and constructions of representations: A case of teaching ratio concept

International Journal of Science and Mathematics Education

Volumn 4, Issue 1, 2006, Pages 145-186

  Cai, Jinfa ^a   Wang, Tao ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b UNIVERSITY OF TULSA   (United States)

Author keywords

Classroom instruction; Cross national perspective; Mathematical problem solving; Pedagogical representation; Ratio concept; Solution representation

Indexed keywords

EID: 33745596114     PISSN: 15710068     EISSN: 15731774     Source Type: Journal    
DOI: 10.1007/s10763-005-9006-7     Document Type: Article

References (52)

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.
2. Decker, 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.
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.
4. Bradburn, M.B. & Gilford, D.M. (1990). A framework and principles for international comparative studies in education. Washington, DC: National Academic Press.
5. Brophy, J.E. & Good, T.L. (1986). Teacher behavior and student achievement. In M.C. Wittrock (Ed.), Handbook of research on teaching, 3rd edn. (pp. 328-375). New York: Macmillan.
6. Bruner, J. (1996). The culture of education. Cambridge, MA: Harvard University Press.
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.
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.
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.
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.
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.
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.
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.
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.
15. Cifarelli, V.V. (1998). The development of mental representations as a problem solving activity. Journal of Mathematical Behavior, 17(2), 239-264.
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.
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.
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.
19. Doyle, W. (1983). Academic work. Review of Educational Research, 53, 159-199.
20. Doyle, W. (1988). Work in mathematics classes: The context of students' thinking during instruction. Educational Psychologist, 23, 167-180.
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.
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.
23. English, L.D. (Ed.) (1997). Mathematical reasoning: Analogies, metaphors and images, Mahwah, NJ: Erlbaum.
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.
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.
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.
27. Goldin, G.A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137-165.
28. Good, T., Grouws, D. & Ebmeier, M. (1983). Active mathematics teaching. New York: Longman.
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.
30. Harmon, M., Smith, T.A., Martin, M.O. et al. (1997). Performance assessment in lEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, MA, Boston College: TIMSS International Study Center.
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: Macmilian.
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.
33. Janvier, C. (1987). Problems of representation in the teaching and learning of mathematical problem solving. Hillsdale, NJ: Erlbaum.
34. Krutetskii, V.A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.
35. Leinhardt, G. (1993). On teaching. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 4, pp. 1-54). Hillsdale, NJ: Erlbaum.
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.
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.
38. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers ' understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Erlbaum.
39. National Council of Teachers of Mathematics (1991). Professional standards for school mathematics. Reston, VA: The Author.
40. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: The Author.
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.
42. Perkins, D.N. & Unger, C. (1994). A new look in representations for mathematics and science learning. Instructional Science, 2, 1-37.
43. 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.
44. Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge.
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.
46. Putnam, R.T., Lamport, 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.
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, 25(1), 35-54.
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.
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.
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.
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.
52. Wang, T. (2003). Culture and mathematics classroom discourse: A comparative perspective. Unpublished qualifying paper, Cambridge, MA: Harvard University.