Measures and References: Algebraic Thinking

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Numerous measures exist to gain a full picture of a student's learning strengths and challenges. Following are examples of measures used to assess this Learner Factor. These measures should be administered and interpreted by experienced professionals.

Early algebra assessment (Blanton et al., 2015): Written assessment to measure students' understanding of core algebraic concepts and practices


Andersson, U. (2008). Working memory as a predictor of written arithmetical skills in children: The importance of central executive functions. The British Journal of Educational Psychology, 78(Pt 2), 181-203.

Best, J. R., Miller, P. H., & Naglieri, J. A. (2011). Relations between executive function and academic achievement from ages 5 to 17 in a large, representative national sample. Learning and Individual Differences, 21(4), 327-336.

Blanton, M., Stephens, A., Knuth, E., Gardiner, A., Isler, I., & Kim, J. (2015). The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39-87.

Blanton, M., Brizuela, B., Stephens, A., Knuth, E., Isler, I., Gardiner, A., … Stylianou, D. (2018). Implementing a framework for early algebra. In Teaching and Learning Algebraic Thinking with 5-12 Year Olds. ICME-13 Monographs. (pp. 27-49). Springer, Cham.

Booker, G. (2009). Algebraic thinking: Generalising number and geometry to express patterns and properties succinctly. Griffith University, Brisbane.

Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. Journal of Mathematical Behavior, 32(3), 613-632.

Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization: A global dialogue from multiple perspectives. Heidelberg, Germany: Springer.

Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27(2-3), 147-164.

Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester, Jr. (Ed.), _Second handbook of research on mathematics teaching and learning _(Vol. 2, pp. 669-705). Charlotte, NC: Information Age Publishing.

Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. Algebra in the early grades, 235-272.

Empson S.B., Levi L., Carpenter T.P. (2011) The algebraic nature of fractions: Developing relational thinking in elementary school. In: Cai J., Knuth E. (Eds) Early Algebraization. Advances in Mathematics Education. Springer: Berlin, Heidelberg

Geary, D. C., Hoard, M. K., Nugent, L., & Rouder, J. N. (2016). Individual differences in algebraic cognition: relation to the approximate number and semantic memory systems. Journal of Experimental Child Psychology, 15(4), 942-953.

Irwin, K. C., & Britt, M. S. (2005). The algebraic nature of students' numerical manipulation in the New Zealand Numeracy Project. Educational Studies in Mathematics, 58(2), 169-188.

Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York, NY: Lawrence Erlbaum Associates.

Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. Lester (Ed.), _Second Handbook of Research on Mathematics Teaching and Learning _(pp. 707-762). Reston, VA: NCTM.

Lee, K., Ng, E. L., & Ng, S. F. (2009). The contributions of working memory and executive functioning to problem representation and solution generation in algebraic word problems. Journal of Educational Psychology, 101(2), 373-387.

Lee, K., Ng, S. F., Ng, E. L., & Lim, Z. Y. (2004). Working memory and literacy as predictors of performance on algebraic word problems. Journal of Experimental Child Psychology, 89(2), 140-158.

MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. _Journal for Research in Mathematics Education. 30, _449-467.

Mix, K. S., Levine, S. C., Young, C., & Hambrick, D. Z. (2016). Separate but correlated: The latent structure of space and mathematics across development. Journal of Experimental Psychology: General, 145(9), 1206-1227.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core standards mathematics. Washington, DC: Authors.

Stacey, K., & MacGregor, M. (1997). Building foundations for algebra. Mathematics teaching in the middle school, 2(4), 252-60.

Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565-579.

Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for Research in Mathematics Education. Reston, VA: National Council of Teachers of Mathematics.