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:
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understanding the behavior of the operations,
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generalizing and justifying,
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extending the number system, and
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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.
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References
Behr, M. J., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13–15.
Blanton, M. (2008). Algebra and the Elementary Classroom. Portsmouth, NH: Heinemann.
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann.
Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286–290.
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.
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.
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.
Kaput, J., Carraher, D., & Blanton, M. (Eds.) (2008). Algebra in the Early Grades. New York: Lawrence Erlbaum Associates.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326.
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).
Martin, G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 12(1), 41–51.
Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of mathematics proof. Educational Studies in Mathematics, 48, 83–99.
Russell, S. J., & Vaisenstein, A. (2008). Computational fluency: working with a struggling student. Connect, 22(1), 8–12.
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.
Russell, S. J., Economopoulos, K., Wittenberg, L., et al. (2008). Investigations in Number, Data, and Space (2nd ed.). Glenview, IL: Scott Foresman.
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.
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.
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.
Schifter, D., Bastable, V., & Russell, S. J. (2008b). Developing Mathematical Ideas. Patterns, Functions, and Change. Parsippany, NJ: Dale Seymour Publications.
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.
Schifter, D., Russell, S. J., & Bastable, V. (2009). Early algebra to reach the range of learners. Teaching Children Mathematics, 16(4), 230–237.
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Russell, S.J., Schifter, D., Bastable, V. (2011). Developing Algebraic Thinking in the Context of Arithmetic. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_4
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