Early Algebraization pp 259-276 | Cite as

# Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable

## 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.

## Keywords

Middle School Grade Level Large Problem Grade Student Middle School Student## 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. Google Scholar - Bednarz, N., Kieran, C., & Lee, L. (1996).
*Approaches to Algebra: Perspectives for Research and Teaching*. Dordrecht: Kluwer Academic. Google Scholar - 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. Google Scholar - Carpenter, T., Franke, M., & Levi, L. (2003).
*Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School*. Portsmouth, NH: Heinemann. Google Scholar - 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). Google Scholar - 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. Google Scholar - 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. Google Scholar - 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. Google Scholar - Kaput, J., Carraher, D., & Blanton, M. (2008).
*Algebra in the Early Grades*. Mahwah, NJ: Erlbaum Associates. Google Scholar - Kieran, C. (1981). Concepts associated with the equality symbol.
*Educational Studies in Mathematics*,*12*(3), 317–326. CrossRefGoogle Scholar - 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 - 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. Google Scholar - Küchemann, D. (1978). Children’s understanding of numerical variables.
*Mathematics in School*,*7*(4), 23–26. Google Scholar - Lacampagne, C., Blair, W., & Kaput, J. (1995).
*The Algebra Initiative Colloquium*. Washington, DC: U.S. Department of Education & Office of Educational Research and Improvement. Google Scholar - 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. Google Scholar - MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15.
*Educational Studies in Mathematics*,*33*, 1–19. CrossRefGoogle Scholar - 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). Google Scholar - 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. CrossRefGoogle Scholar - 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). Google Scholar - National Council of Teachers of Mathematics (1997).
*A framework for constructing a vision of algebra: A discussion document*. Unpublished manuscript. Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and Standards for School Mathematics*. Reston, VA: NCTM. Google Scholar - National Research Council (1998).
*The Nature and Role of Algebra in the K-14 Curriculum*. Washington, DC: National Academy Press. Google Scholar - Philipp, R. (1992). The many uses of algebraic variables.
*Mathematics Teacher*,*85*, 557–561. Google Scholar - RAND Mathematics Study Panel (2003).
*Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education*. Santa Monica, CA: RAND. Google Scholar - 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. CrossRefGoogle Scholar - Steinberg, R., Sleeman, D., & Ktorza, D. (1990). Algebra students’ knowledge of equivalence of equations.
*Journal for Research in Mathematics Education*,*22*(2), 112–121. CrossRefGoogle Scholar - 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. Google Scholar