Abstract
Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.
This research is supported in part by the National Science Foundation under grant No. REC-0115661. The opinions expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation, the Department of Education, or the National Institute of Child Health and Human Development. This chapter is a reprint of an article published in ZDM—International Reviews on Mathematical Education, 37(1), 68–76. DOI 10.1007/BF0255899.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades (pp. 165–184). Mahwah, NJ: Erlbaum Associates.
Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to Algebra: Perspectives for Research and Teaching. Dordrecht: Kluwer Academic.
Carpenter, T., & Levi, L. (1999). Developing conceptions of algebraic reasoning in the primary grades. Paper presented at the annual meeting of the American Educational Research Association, Montreal (Canada), April, 1999.
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School. Portsmouth, NH: Heinemann.
Carraher, D., Brizuela, B., & Schliemann, A. (2000). Bringing out the algebraic character of arithmetic: Instantiating variables in addition and subtraction. In T. Nakahara & M. Koyama (Eds.), Proceedings of the Twenty-Fourth International Conference for the Psychology of Mathematics Education, Hiroshima (Japan), July, 2000 (pp. 145–152).
Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 56–60.
Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 276–295). New York: Macmillan.
Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. Paper presented at the Algebra Symposium, Washington, DC, May, 1998.
Kaput, J., Carraher, D., & Blanton, M. (2008). Algebra in the Early Grades. Mahwah, NJ: Erlbaum Associates.
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.
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.
Knuth, E., Stephens, A., McNeil, N., & Alibali, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.
Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4), 23–26.
Lacampagne, C., Blair, W., & Kaput, J. (1995). The Algebra Initiative Colloquium. Washington, DC: U.S. Department of Education & Office of Educational Research and Improvement.
Ladson-Billings, G. (1998). It doesn’t add up: African American students’ mathematics achievement. In C. E. Malloy & L. Brader-Araje (Eds.), Challenges in the Mathematics Education of African American Children: Proceedings of the Benjamin Banneker Association Leadership Conference (pp. 7–14). Reston, VA: NCTM.
MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33, 1–19.
McNeil, N., Grandau, L., Stephens, A., Krill, D., Alibali, M. W., & Knuth, E. (2004). Middle-school students’ experience with the equal sign: Saxon Math does not equal Connected Mathematics. In D. McDougall (Ed.), Proceedings of the Twenty-Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Toronto, October, 2004 (pp. 271–275).
McNeil, N. M., & Alibali, M. W. (2005). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6, 285–306.
Mevarech, Z., & Yitschak, D. (1983). Students’ misconceptions of the equivalence relationship. In R. Hershkowitz (Ed.), Proceedings of the Seventh International Conference for the Psychology of Mathematics Education. Rehovot (Israel), July, 1983 (pp. 313–318).
National Council of Teachers of Mathematics (1997). A framework for constructing a vision of algebra: A discussion document. Unpublished manuscript.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
National Research Council (1998). The Nature and Role of Algebra in the K-14 Curriculum. Washington, DC: National Academy Press.
Philipp, R. (1992). The many uses of algebraic variables. Mathematics Teacher, 85, 557–561.
RAND Mathematics Study Panel (2003). Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education. Santa Monica, CA: RAND.
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and Procedural Knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189.
Steinberg, R., Sleeman, D., & Ktorza, D. (1990). Algebra students’ knowledge of equivalence of equations. Journal for Research in Mathematics Education, 22(2), 112–121.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford (Ed.), The Ideas of Algebra, K-12 (pp. 8–19). Reston, VA: NCTM.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Knuth, E.J., Alibali, M.W., McNeil, N.M., Weinberg, A., Stephens, A.C. (2011). Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable. 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_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-17735-4_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17734-7
Online ISBN: 978-3-642-17735-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)