Developing Students’ Algebraic Thinking in Earlier Grades: Lessons from China and Singapore

  • Jinfa CaiEmail author
  • Swee Fong Ng
  • John C. Moyer
Part of the Advances in Mathematics Education book series (AME)


In this chapter, we discuss how algebraic concepts and representations are developed and introduced in the Chinese and Singaporean elementary curricula. We particularly focus on the lessons to be learned from the Chinese and Singaporean practice of fostering Early algebra learning, such as the one- problem-multiple-solutions approach in China and pictorial equations approach in Singapore. Using the lessons learned from Chinese and Singaporean curricula, we discuss four issues related to the development of algebraic thinking in earlier grades: (1) To what extent should we expect students in early grades to think algebraically? (2) What level of formalism should we expect of students in the early grades? (3) How can we help students make a smooth transition from arithmetic to algebraic thinking? and (4) Are authentic applications necessary for students in early grades?


Word Problem Chinese Student Algebraic Approach Early Grade Algebraic Thinking 
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  1. AAAS (2000). Algebra for all: Not with today’s textbooks. See
  2. Bell, A. (1996). Problem-solving approaches to algebra: Two aspects. In N. Bernardz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra. Perspectives to Research and Teaching (pp. 167–187). Dordrecht, The Netherlands: Kluwer Academic Publishers. Google Scholar
  3. Cai, J. (2003). Singaporean students’ mathematical thinking in problem solving and problem posing: An exploratory study. International Journal of Mathematical Education in Science and Technology, 34(5), 719–737. CrossRefGoogle Scholar
  4. Cai, J. (2004a). Introduction to the special issue on developing algebraic thinking in the earlier grades from an international perspective. The Mathematics Educator (Singapore), 8(1), 1–5. Google Scholar
  5. Cai, J. (2004b). Developing algebraic thinking in the earlier grades: A case study of the Chinese mathematics curriculum. The Mathematics Educator (Singapore), 8(1), 107–130. Google Scholar
  6. Cai, J. (2004c). 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
  7. Cai, J., & Hwang, S. (2002). Generalized and generative thinking in U.S. and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401–421. CrossRefGoogle Scholar
  8. Cai, J., & Knuth, E. (Eds.) (2005). Developing algebraic thinking: Multiple perspectives. Zentralblatt fuer Didaktik der Mathematik (International Journal on Mathematics Education), 37(1). Special Issue. Google Scholar
  9. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann. Google Scholar
  10. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (Vol. II, pp. 669–705). Charlotte, NC: Information Age Publishing. Google Scholar
  11. Curriculum Planning & Development Division (1999). Mathematics Syllabus Primary. Singapore: Ministry of Education. Google Scholar
  12. Curriculum Planning & Development Division (2000). Mathematics Syllabus Primary. Singapore: Ministry of Education. Google Scholar
  13. Division of Elementary Mathematics (1999). Mathematics: Elementary School Textbook (number 1). Beijing, China: People’s Education Press. Google Scholar
  14. Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers Grades 6–10. Portsmouth, NH:: Heinemann. Google Scholar
  15. Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.), Mathematics Classrooms that Promote Understanding (pp. 133–155). Mahwah, NJ: Erlbaum. Google Scholar
  16. Kho, T. H. (1987). Mathematical models for solving arithmetic problems. In Proceedings of Fourth Southeast Asian Conference on Mathematical Education (ICMI-SEAMS). Mathematical Education in the 1990’s (pp. 345–351). Singapore: Institute of Education. Google Scholar
  17. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390–419). New York: Macmillan. Google Scholar
  18. Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator (Singapore), 8(1), 139–151. Google Scholar
  19. Mathematical Sciences Education Board (1998). The Nature and Role of Algebra in the K-14 Curriculum: Proceedings of a National Symposium. Washington, DC: National Research Council. Google Scholar
  20. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author. Google Scholar
  21. Ng, S. F. (2004). Developing algebraic thinking: A case study of the Singaporean primary school curriculum. The Mathematics Educator (Singapore), 8(1), 39–59. Google Scholar
  22. Ng, S. F., & Lee, K. (2005). How primary five pupils use the model method to solve word problems. The Mathematics Educator, 9(1), 60–83. Google Scholar
  23. Post, T. R., Behr, M. J., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings. In A. Coxford & A. Shulte (Eds.), The Ideas of Algebra, K-12, 1988 Yearbook (pp. 78–90). Reston, VA: NCTM. Google Scholar
  24. Schifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K-6. In L. V. Stiff & F. R. Curcio (Eds.), Developing Mathematical Reasoning in Grades K-12, 1999 NCTM Yearbook (pp. 62–81). Reston, VA: NCTM. Google Scholar
  25. Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1996). Characterizing Pedagogical Flow: An Investigation of Mathematics and Science Teaching in Six Countries. The Netherlands: Kluwer Academic Publishers. Google Scholar
  26. Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov. The Mathematics Educator (Singapore), 8(1), 60–87. Google Scholar
  27. Senk, S. L., & Thompson, D. R. (Eds.) (2003). Standards-Based School Mathematics Curricula: What Are They? What Do Students Learn? Mahwah, N.J: Lawrence Erlbaum Associates. Google Scholar
  28. Silver, E. A., & Kenney, P. A. (2001). Results from the Sixth Mathematics Assessment of the National Assessment of Educational Progress. Reston, VA: NCTM. Google Scholar
  29. Stacey, K., Chick, H., & Kendal, M. (2004). The Future of the Teaching and Learning of Algebra: The 12th ICMI Study. The Netherlands: Kluwer Academic Publishers. CrossRefGoogle Scholar
  30. Usiskin, Z. (1995). Why is algebra important to learn. In B. Moses (Ed.), Algebraic Thinking in Grades K-12: Readings from NCTM’s School-Based Journals and Other Publications (pp. 16–21). Reston, VA: NCTM. Google Scholar
  31. van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33, 319–351. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.National Institute of EducationNanyang Technological UniversitySingaporeSingapore
  3. 3.Department of Mathematics, Statistics and Computer ScienceMarquette UniversityMilwaukeeUSA

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