Journal of Mathematics Teacher Education

, Volume 20, Issue 1, pp 5–29 | Cite as

On mathematical understanding: perspectives of experienced Chinese mathematics teachers

Article

Abstract

Researchers have long debated the meaning of mathematical understanding and ways to achieve mathematical understanding. This study investigated experienced Chinese mathematics teachers’ views about mathematical understanding. It was found that these mathematics teachers embrace the view that understanding is a web of connections, which is a result of continuous connection making. However, in contrast to the popular view which separates understanding into conceptual and procedural, Chinese teachers prefer to view understanding in terms of concepts and procedures. They place more stress on the process of concept development, which is viewed as a source of students’ failures in transfer. To achieve mathematical understanding, the Chinese teachers emphasize strategies such as reinventing a concept, verbalizing a concept, and using examples and comparisons for analogical reasoning. These findings draw on the perspective of classroom practitioners to inform the long-debated issue of the meaning of mathematical understanding and ways to achieve mathematical understanding.

Keywords

Mathematical understanding Achieving mathematical understanding Experienced teachers Chinese teachers Perspective of classroom practitioners 

References

  1. Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review. http://actr.psy.cmu.edu/wordpress/wp-content/uploads/2012/12/146Applic.MisApp.pdf.
  2. Baroody, J. (1987). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York, NY: Teachers College Press.Google Scholar
  3. Baroody, A. J. (1999). Children’s relational knowledge of addition and subtraction. Cognition and Instruction, 17, 137–175.CrossRefGoogle Scholar
  4. Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115–131.Google Scholar
  5. Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotna (Ed.), Proceedings of the 30th conference of the international group for the psychology of mathematics education. Prague, Czech Republic: PME.Google Scholar
  6. Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.CrossRefGoogle Scholar
  7. Borgen, K. L., & Manu, S. S. (2002). What do students really understand? Journal of Mathematical Behavior, 21, 151–165.CrossRefGoogle Scholar
  8. Brownell, W. A. (1935). Psychological considerations in the learning and the teaching of arithmetic. In W. D. Reeve (Ed.), The teaching of arithmetic (Tenth Yearbook of the National Council of Teachers of Mathematics) (pp. 1–31). New York: Columbia University.Google Scholar
  9. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: Harvard University Press.Google Scholar
  10. Cai, J. (2000). Mathematical thinking involved in U.S. and Chinese students' solving process-constrained and process-open problems. Mathematical Thinking and Learning: An International Journal, 2, 309–340. CrossRefGoogle Scholar
  11. 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.Google Scholar
  12. 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
  13. Cai, J., Ding, M., & Wang, T. (2014). How do exemplary Chinese and U.S. mathematics teachers view instructional coherence? Educational Studies in Mathematics, 85, 265–280.CrossRefGoogle Scholar
  14. Cai, J., & Knuth, E. (2011). Early algebraization: A global dialogue from multiple perspectives. New York: Springer.CrossRefGoogle Scholar
  15. Chi, M. T. H., de Leeuw, N., Chiu, M., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439–477.Google Scholar
  16. Chi, T. H., & VanLehn, K. A. (2012). Seeing deep structure from the interactions of surface features. Educational Psychologist, 47(3), 177–188.CrossRefGoogle Scholar
  17. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.CrossRefGoogle Scholar
  18. Darling-Hammond, L. (1994). Performance-based assessment and educational equity. Harvard Education Review, 64(1), 5–30.CrossRefGoogle Scholar
  19. Davis, R. B. (1992). Understanding of “Understanding”. Journal of Mathematical Behavior, 11(3), 225–242.Google Scholar
  20. Ding, M., & Li, X. (2010). A comparative analysis of the distributive property in the US and Chinese elementary mathematics textbooks. Cognition and Instruction, 28, 146–180.CrossRefGoogle Scholar
  21. Ding, M., Li, X., Piccolo, D., & Kulm, G. (2007). Teacher interventions in cooperative-learning mathematics classes. Journal of Educational Research, 100, 162–175.CrossRefGoogle Scholar
  22. Ding, M., Li, Y., Li., X, & Gu, J. (2013). Knowing and understanding instructional mathematics content through intensive studies of textbooks. In Y. Li, & R. Huang (Eds.), How Chinese teach mathematics and improve teaching (pp. 66–82). New York: Routledge.Google Scholar
  23. Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. The Journal of The Learning Sciences, 16, 221–262.CrossRefGoogle Scholar
  24. Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer Academic Publishers.Google Scholar
  25. Gay, L. R., & Airasian, P. (2000). Educational research: Competencies for analysis and application (6th ed.). Upper Saddle River: Merrill.Google Scholar
  26. Gravemeijer, K. P. E. (1994). Developing realistic mathematics education. Utrecht: CD Beta Press.Google Scholar
  27. Greeno, J. G. (1978). Understanding and procedural knowledge in mathematics instruction. Educational Psychologist, 12, 262–283.CrossRefGoogle Scholar
  28. Greeno, J., & Riley, M. (1987). Processes and development of understanding. In R. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 289–313). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  29. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 309–347). Singapore: World Scientific.CrossRefGoogle Scholar
  30. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Mcmillan.Google Scholar
  31. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth: Heinemann.Google Scholar
  32. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  33. Huang, R., & Li, Y. (2009). Pursuing excellence in mathematics classroom instruction through exemplary lesson development in China: A case study. ZDM-International Journal of Mathematics Education, 41, 297–309.CrossRefGoogle Scholar
  34. Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.CrossRefGoogle Scholar
  35. Lester, F. A. (2003). Research and issues in teaching mathematics through problem solving. Reston: National Council of Teachers of Mathematics.Google Scholar
  36. Lewis, C. (1988). Why and how to learn why: Analysis-based generalization of procedures. Cognitive Science, 12, 211–256.CrossRefGoogle Scholar
  37. Li, X., Ding M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children's understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and in the United States. Cognition and Instruction, 26, 195–217.CrossRefGoogle Scholar
  38. Lobato, J. (2006). Alternative perspectives on the transfer of learning: History, issues, and challenges for future research. The Journal of The Learning Sciences, 15, 431–449.CrossRefGoogle Scholar
  39. Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36, 101–136.Google Scholar
  40. Ma, L. (1999). Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  41. Marton, F., Tse, L. K., & dall’Alba, G. (1996). Memorizing and understanding: The keys to the paradox? In D. A. Watkins & J. B. Biggs (Eds.), The Chinese learner: Cultural, psychological and contextual influences (pp. 69–83). Hong Kong: Comparative Education Research Centre.Google Scholar
  42. Mason, J. (1989). Mathematical abstraction seen as a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.Google Scholar
  43. Meel, D. E. (2003). Models and theories of mathematical understanding: Comparing Pirie and Kieren’s model of the growth of mathematical understanding and APOS theory. In A. Selden, E. Dubinsky, G. Harel & F. Hitt (Eds.) Research in collegiate mathematics education. V. Issues in mathematics education (pp. 132-181). Providence, RI: American Mathematical Society.Google Scholar
  44. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: Author.Google Scholar
  45. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
  46. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.CrossRefGoogle Scholar
  47. Nisbett, R. E., Peng, K., Choi, I., & Norenzayan, A. (2001). Culture and systems of thought: Holistic versus analytic cognition. Psychological Review, 108, 291–310.CrossRefGoogle Scholar
  48. Piaget, J. (1952). The child’s conception of number. London: Routledge and Kegan Paul.Google Scholar
  49. Pirie, S., & Kieren, T. (1992). Creating constructivist environments and constructing creative mathematics. Educational Studies in Mathematics, 23, 505–528.CrossRefGoogle Scholar
  50. Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.CrossRefGoogle Scholar
  51. Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47, 189–203.CrossRefGoogle Scholar
  52. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.CrossRefGoogle Scholar
  53. Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem solving. Cognition and Instruction, 23, 313–349.CrossRefGoogle Scholar
  54. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362.CrossRefGoogle Scholar
  55. Rittle-Johnson, B., & Star, J. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574.CrossRefGoogle Scholar
  56. Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359–371.CrossRefGoogle Scholar
  57. Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20–26.Google Scholar
  58. Star, J. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.Google Scholar
  59. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press.Google Scholar
  60. Sun, X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educational Studies in Mathematics, 76, 65–85.CrossRefGoogle Scholar
  61. Sweller, J. (2006). The worked example effect and human cognition. Learning and Instruction, 16, 165–169.CrossRefGoogle Scholar
  62. Torbeyns, J., De Smedt, B., Stassens, N., Ghesquière, P., & Verschaffel, L. (2009). Solving subtraction problems by means of indirect addition. Mathematical Thinking and Learning, 11, 79–91.CrossRefGoogle Scholar
  63. Trends in International Mathematics and Science Study (2003). Highlights from the Trends in International Mathematics and Science Study (TIMSS) 2003. Retrieved from http://nces.ed.gov/pubs2005/2005005.pdf.
  64. Trends in International Mathematics and Science Study (2007). Highlights from TIMSS 2007: Mathematics and Science Achievement of US Fourth and Eighth-Grade Students in an International Context. Retrieved from http://nces.ed.gov/pubs2009/2009001.pdf.
  65. Wearne, D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers. Journal for Research in Mathematics Education, 19(5), 371–384.CrossRefGoogle Scholar
  66. Weiss, I. R., Smith, S., & O’Kelley, S. K. (2009). The Presidential award for excellence in mathematics teaching: Setting the standard. In J. Cai, G. Kaiser, R. Perry, & N.-Y. Wong (Eds.), Effective mathematics teaching from teachers’ perspectives: National and international studies (pp. 281–304). Rotterdam: Sense Publisher.Google Scholar
  67. Wertheimer, M. (1959). Productive thinking (Enlarged ed.). New York: Harper and Row (Original work published 1945).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.University of DelawareNewarkUSA
  2. 2.Temple UniversityPhiladelphiaUSA

Personalised recommendations