Teaching Algebraic Equations with Variation in Chinese Classroom

  • Jing Li
  • Aihui Peng
  • Naiqing SongEmail author
Part of the Advances in Mathematics Education book series (AME)


This chapter gives a detailed analysis of how teaching with variation is helpful for students’ learning of algebraic equations by using typical teaching episodes in grade seven in China. Also, it provides a demonstration showing how variation is used as an effective way of teaching through the discussion after the analysis.


Algebraic Equation Problem Variation Word Problem Mathematical Thinking Classroom Observation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Ausubel, D. P. (1968). Educational Psychology: A Cognitive View. New York: Holt, Rinehart & Winston. Google Scholar
  2. Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equal sign. Mathematics Teaching, 92, 13–15. Google Scholar
  3. Biggs, J. B., & Watkins, D. A. (1996). The Chinese learner in retrospect. In D. A. Watkins & J. G. Biggs (Eds.), The Chinese Learner: Cultural, Psychological, and Contextual Influences (pp. 269–285). Hong Kong/Melbourne: Comparative Education Research Centre, the University of Hong Kong/Australian Council for Education Research. Google Scholar
  4. Cai, J., & Nie, B. (2007). Problem solving in Chinese mathematics education: Research and practice. ZDM, 39, 459–475. CrossRefGoogle Scholar
  5. Cai, J. et al. (2005). The developments of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective. ZDM, 37(1), 5–15. CrossRefGoogle Scholar
  6. Chen, Z., & Song, N. (1996). Mathematics instruction experiment for improving efficiency in class. Journal of Mathematics Bulletin, 8, 21–23. In Chinese. Google Scholar
  7. Gu, L. (1994). Theory of Teaching Experiment: The Methodology and Teaching Principle of Qinpu. Beijing: Educational Science Press. In Chinese. Google Scholar
  8. Gu, M. (1999). The Grand Educational Dictionary. Shanghai: Shanghai Education Publishing House. Google Scholar
  9. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Wong, J. Cai, & S. Li (Eds.), How Chinese Learn Mathematics: Perspectives from Insiders. New Jersey: World Scientific. Google Scholar
  10. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78. CrossRefGoogle Scholar
  11. Huang, R., & Leung, F. K. S. (2004). Cracking the paradox of the Chinese learners Looking into the mathematics classrooms in Hong Kong and Shanghai. In L. Fan, N. Wong, J. Cai, & S. Li (Eds.), How Chinese Learn Mathematics: Perspectives from Insiders. New Jersey: World Scientific. Google Scholar
  12. Husen, T. (1967). International Study of Achievement in Mathematics: A Comparison of Twelve Countries, Vols. I, II. New York: Wiley. Google Scholar
  13. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. Reston, VA: National Council of Teachers of Mathematics. Google Scholar
  14. Lapointe, A. E., Mead, N. A., & Phillips, G. W. (1989). A World of Differences: An International Assessment of Mathematics and Science. Princeton, NJ: Educational Testing Service. Google Scholar
  15. Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30, 39–65. CrossRefGoogle Scholar
  16. Ma, F. (2002). Curriculum Standard Experimental Textbook for Compulsory Education: Mathematics (for the Second Semester of Grade Seven). Beijing: Beijing Normal University Press. Google Scholar
  17. Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton, A. B. M. Tsui, Chik, P. Y. Ko, M. L. Lo, & I. A. C. Mok (Eds.), Classroom Discourse and the Space of Learning (pp. 3–40). New Jersey: Lawrence Erlbaum. Google Scholar
  18. Mok, I. A. C., Cai, J., & Fung, A. F. (2008). Missing learning opportunities in classroom instruction: Evidence from an analysis of a well-structured lesson on comparing fractions. The Mathematics Educator, 11, 111–126. Google Scholar
  19. National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Google Scholar
  20. Park, K. (2006). Mathematics lessons in Korea: Teaching with systematic variation. Tsukuba Journal of Educational Study in Mathematics, 25, 151–167. Google Scholar
  21. Pirie, S., & Martin, L. (1997). The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations. Educational Studies in Mathematics, 34, 159–181. CrossRefGoogle Scholar
  22. Sadovsky, P., & Sessa, C. (2005). The didactic interaction with the procedures of peers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 59, 85–112. CrossRefGoogle Scholar
  23. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14, 15–39. CrossRefGoogle Scholar
  24. Stevenson, H. W., & Lee, S. (1990). Contexts of Achievement: A Study of American, Chinese, and Japanese Children. Chicago: University of Chicago Press. Google Scholar
  25. Sun, X. (2009). “Problem variations” in the Chinese text: Comparing the variation in problems in examples of the division of fractions in American and Chinese mathematics textbook. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education. Google Scholar
  26. Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341–359. CrossRefGoogle Scholar
  27. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111. CrossRefGoogle Scholar
  28. Zhang, D., & Song, N. (2004). Introduction to Mathematics Education. Beijing: Higher Education Press. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingChina
  2. 2.School of Mathematics and InformationLangfang Teachers’ CollegeLangfangChina
  3. 3.Institute of Higher EducationSouthwest UniversityChongqingChina

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