Understanding Students’ Mathematical Thinking for Effective Teaching: A Comparison between Expert and Nonexpert Chinese Elementary Mathematics Teachers
Yan Zhu 1,  
Wenhui Yu 2,  
Jinfa Cai 3  
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East China Normal University, CHINA
Education Development and Research Center, CHINA
University of Delaware, U.S.A.
Online publish date: 2017-10-05
Publish date: 2017-11-02
EURASIA J. Math., Sci Tech. Ed 2018;14(1):213–224
It is widely believed that teachers’ knowledge of students’ thinking has a significant impact on teachers’ teaching and students’ learning. However, there is far less research on how teachers acquire their knowledge of students’ thinking before, during, and after lessons. This study is designed to compare the differences between expert and nonexpert mathematics teachers on their behaviors and perceptions related to understanding students’ mathematical thinking. Based on 554 Chinese elementary mathematics teachers’ responses to a survey, the study found that teachers took actions to understand students’ thinking more often when students were learning new topics or encountering difficulties, and they were more likely to do so before lessons than during or after lessons. The comparison revealed that significantly more expert elementary mathematics teachers attempted to understand students’ thinking from a variety of perspectives before making the necessary adjustments to their predetermined teaching plans than did nonexpert teachers. Significantly more expert teachers also relied on their own teaching experiences to understand students’ thinking. In contrast, significantly more nonexpert teachers claimed that they did not rely on prior teaching experiences because they “did not know how to.”
Jinfa Cai   
University of Delaware, United States
1. An, S., Kulm, G., Wu, Z., Ma, F., & Wang, L. (2006). The impact of cultural differences on middle school mathematics teachers’ beliefs in the U.S and China. In F. K.-S. Leung, K.-D. Graf, & F. J. Lopez-Real (Eds.), Mathematics education in different cultural traditions – A comparative study (pp. 449464). New York, NY: Springer.
2. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 371397.
3. Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire to hear children. In B. J. Biddle, T. L. Good, & I. F. Goodson (Eds.), International handbook of teachers and teaching (Vol. 2, pp. 769818). Dordrecht, the Netherlands: Kluwer.
4. Beswick, K. (2008). Improving middle school students’ proportional reasoning. In J. Vicent, R. Pierce, & J. Dowsey (Eds.), Connected maths: Proceedings of the 45th annual conference of the Mathematical Association of Victoria (pp. 2539). Brunswick, Australia: Mathematical Association of Victoria.
5. Bishop, A. J. (1976). Decision-making, the intervening variable. Educational Studies in Mathematics, 7(1/2), 4147.
6. Borko, H., Roberts, S. A., & Shavelson, R. (2008). Teachers’ decision making: From Alan J. Bishop to today. In P. Clarkson & N. Presmeg (Eds.), Critical issues in mathematics education: Major contributions of Alan Bishop (pp. 3770). New York, NY: Springer.
7. 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.
8. 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.
9. Cai, J. (2005). U.S. and Chinese teachers’ knowing, evaluating, and constructing representations in mathematics instruction. Mathematical Thinking and Learning, 7(2), 135169.
10. Cai, J., & Ding, M. (2017). On mathematical understanding: Perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, 20(1), 529.
11. Cai, J., Ding, M., & Wang, T. (2014). How do exemplary Chinese and U.S. mathematics teachers view instructional coherence? Educational Studies in Mathematics, 85(2), 265–280.
12. Cai, J., Kaiser, G., Perry, R., & Wong, N-Y. (2009) (Eds.). Effective mathematics teaching from teachers’ perspectives: National and international studies. Rotterdam, the Netherlands: Sense.
13. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499531.
14. Carter, I. S. W., & Amador, J. M. (2015). Lexical and indexical conventional components that mediate professional noticing during lesson study. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 13391361.
15. Clarkson, P., & Presmeg, N. C. (2008). Critical issues in mathematics education: Major contributions of Alan Bishop. Dordrecht, the Netherlands: Springer.
16. Cohen, J. (1988). Statistical power and analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
17. Confrey, J. (1990). What constructivism implies for teaching. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning mathematics (pp. 107–122). Reston, VA: National Council of Teachers of Mathematics.
18. Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ interpretations of students’ mathematical work. Journal of Mathematics Teacher Education, 3, 155181.
19. Darling-Hammond, L. (1994). Performance-based assessment and educational equity. Harvard Education Review, 64(1), 530.
20. Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York, NY: Garland.
21. Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28, 355376.
22. Gardner, H. (1999). The disciplined mind: What all students should understand. New York, NY: Simon & Schuster.
23. Henderson, B. (2003). Teacher thinking about students’ thinking. MountainRise: The International Journal for the Scholarship of Teaching and Learning, 1(1). Retrieved from
24. Huang, C. (1997). The analects of Confucius (Lun Yu): A literal translation with an introduction and notes. New York, NY: Oxford University Press.
25. Isaacs, T., Creese, B., & Gonzalez, A. (2015). Aligned instructional systems: Shanghai. Center for International Education Benchmarking (CIEB). Retrieved from
26. Jacobs, V. R., Lamb, L. L., Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41, 169202.
27. Kamii, C. (1989). Young children continue to reinvent arithmetic: Second grade. New York, NY: Teachers College Press.
28. Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing (pp. 3550). New York, NY: Routledge.
29. Moulthrop, D., Calegari, N. C., & Eggers, D. (2006). Teachers have it easy: The big sacrifices and small salaries of America’s teachers. New York, NY: Perseus Books.
30. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
31. Perry, R., MacDonald, A., & Gervasoni, A. (Eds.). (2015). Mathematics and transition to school: International perspectives. Singapore: Springer.
32. Star, J., & Strickland, S. (2008). Learning to observe: Using video to improve perspective teachers’ ability to notice. Journal of Mathematics Teacher Education, 11, 107125.
33. Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolising, modeling, and tool use in mathematics education (pp. 191–212). Dordrecht, the Netherlands: Kluwer.
34. Yu, W., & Cai, J. (2017). Survey of senior-ranked teachers and non-senior-ranked elementary mathematics teachers’ knowledge of students [in Chinese]. Journal of Mathematics Education, 26(2), 613.