**Return to Algebraic Thinking factor page.**

*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.

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