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.

Contrasting Cases Algebra Assessment (e.g., Star et al., 2015): A researcher-designed measure of Algebraic Thinking that focuses on conceptual knowledge, procedural knowledge, and flexibility in problem solving.


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.

Blatto-Vallee, G., Kelly, R. R., Gaustad, M. G., Porter, J., & Fonzi, J. (2007). Visual-spatial representation in mathematical problem solving by deaf and hearing students. The Journal of Deaf Studies and Deaf Education, 12(4), 432-448.

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

Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118(1), 110-118.

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.

Dindyal, J. (2004). Algebraic thinking in geometry at high school level: Students' use of variables and unknowns. In 27th Annual Conference of the Mathematics Education Research Group of Australasia Incorporated (MERGA 2004) on "Mathematics Education for the Third Millennium, Towards 2010", (pp. 183-190).

Fernandes, S. H. A. A., & Healy, L. (2014). Algebraic expressions of deaf students: Connecting visuo-gestural and dynamic digital representations. Proceedings of PME 38 and PME-NA, 36(3), 49-56.

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.

Kyttala, M., & Lehto, J. E. (2008). Some factors underlying mathematical performance: The role of visuospatial working memory and non-verbal intelligence. European Journal of Psychology Education, 23(1), 77-94.

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.

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

Nichols, J. D., & White, J. (2001). Impact of peer networks on achievement of high school algebra students. The Journal of Educational Research, 94(5), 267-273.

Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41-54.

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.

Susac, A., Bubic, A., Vrbanc, A., & Planinic, M. (2014). Development of abstract mathematical reasoning: The case of algebra. Frontiers in Human Neuroscience, 8(679), 1-10.

Trezise, K., & Reeve, R. A. (2014). Working memory, worry, and algebraic ability. Journal of Experimental Child Psychology, 121(1), 120-136.