Advertisement

Developing Algebraic Thinking in the Context of Arithmetic

  • Susan Jo RussellEmail author
  • Deborah Schifter
  • Virginia Bastable
Part of the Advances in Mathematics Education book series (AME)

Abstract

Using classroom episodes from grades 2–6, this chapter highlights four mathematical activities that underlie arithmetic and algebra and, therefore, provide a bridge between them. These are:
  • understanding the behavior of the operations,

  • generalizing and justifying,

  • extending the number system, and

  • using notation with meaning.

Analysis of each episode provides insight into how teachers recognize the opportunities to pursue this content in the context of arithmetic and how such study both strengthens students’ understanding of arithmetic operations and enables them to develop ideas foundational to the study of algebra.

Keywords

Number Line Number System Distributive Property Negative Number Mathematical Activity 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Behr, M. J., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13–15. Google Scholar
  2. Blanton, M. (2008). Algebra and the Elementary Classroom. Portsmouth, NH: Heinemann. Google Scholar
  3. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann. Google Scholar
  4. Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286–290. CrossRefGoogle Scholar
  5. Harel, G., & Sowder, L. (1998). Students’ proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on Collegiate Mathematics Education (Vol. III, pp. 234–283). Providence: AMS. Google Scholar
  6. Harel, G., & Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning. Reston, VA: National Council of Teachers of Mathematics. Google Scholar
  7. Kaput, J., & Sims-Knight, J. (1983). Errors in translations to algebraic equations: Roots and implications. Focus on Learning Problems in Mathematics, 5(3–4), 63–78. Google Scholar
  8. Kaput, J., Carraher, D., & Blanton, M. (Eds.) (2008). Algebra in the Early Grades. New York: Lawrence Erlbaum Associates. Google Scholar
  9. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326. CrossRefGoogle Scholar
  10. Kieran, E. J., Slaughter, Ml., Choppin, J., & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In D. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Noney (Eds.), Proceedings of the 24 th Meeting for PME-NA, Athens, GA (Vol. 4, pp. 1693–1700). Google Scholar
  11. Martin, G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 12(1), 41–51. CrossRefGoogle Scholar
  12. Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematics proof. Educational Studies in Mathematics, 48, 83–99. CrossRefGoogle Scholar
  13. Russell, S. J., & Vaisenstein, A. (2008). Computational fluency: working with a struggling student. Connect, 22(1), 8–12. Google Scholar
  14. Russell, S. J., Schifter, D., & Bastable, V. (2006). Is it 2 more or less? Algebra in the elementary classroom. Connect, 19(3), 1–3. http://www.terc.edu/newsroom/776.html. Google Scholar
  15. Russell, S. J., Economopoulos, K., Wittenberg, L., et al. (2008). Investigations in Number, Data, and Space (2nd ed.). Glenview, IL: Scott Foresman. Google Scholar
  16. Schifter, D. (2009). Representation-based proof in the elementary grades. In D. A. Stylianou, M. Blanton, & E. Kieran (Eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective. Oxford: Routledge—Taylor Francis and National Council of Teachers of Mathematics. Google Scholar
  17. Schifter, D., Bastable, V., & Russell, S. J. (1999). Developing Mathematical Ideas. Number and Operations, Part 1: Building a System of Tens Video. Parsippany, NJ: Dale Seymour Publications. Google Scholar
  18. Schifter, D. V., Bastable, V., & Russell, S. J. (2008a). Developing Mathematical Ideas. Number and Operations, Part 3: Reasoning Algebraically about Operations. Parsippany, NJ: Dale Seymour Publications. Google Scholar
  19. Schifter, D., Bastable, V., & Russell, S. J. (2008b). Developing Mathematical Ideas. Patterns, Functions, and Change. Parsippany, NJ: Dale Seymour Publications. Google Scholar
  20. Schifter, D., Monk, S., Russell, S. J., & Bastable, V. (2008c). Early algebra: What does understanding the laws of arithmetic mean in the elementary grades. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades (pp. 413–447). New York: Lawrence Erlbaum Associates. Google Scholar
  21. Schifter, D., Russell, S. J., & Bastable, V. (2009). Early algebra to reach the range of learners. Teaching Children Mathematics, 16(4), 230–237. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Susan Jo Russell
    • 1
    Email author
  • Deborah Schifter
    • 2
  • Virginia Bastable
    • 3
  1. 1.Education Research CollaborativeTERCCambridgeUSA
  2. 2.Education Development CenterNewtonUSA
  3. 3.SummerMath for TeachersMount Holyoke CollegeSouth HadleyUSA

Personalised recommendations