**Return to Proportional Reasoning 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.*

Ranking Proportions Task (e.g., Mazzocco & Devlin, 2008): Magnitude comparison for fractions and rational numbers where students rank order different representations of fractions including shaded circles, numerical fractions, and decimals.

Missing value problems (e.g., Lamon, 2007): Students are asked to determine which value is missing in a ratio equivalence context (e.g., how many sugars do you need to add to 6 lemons so your lemonade is the same sweetness as my lemonade with 2 sugars and 3 lemons?).

Agostino, A., Johnson, J., & Pascual-Leone, J. (2010). Executive functions underlying multiplicative reasoning: Problem type matters. *Journal of Experimental Child Psychology, 105*(4), 286-305.

Bailey, D. H., Siegler, R. S., & Geary, D. C. (2014). Early predictors of middle school fraction knowledge. *Developmental Science, 17*(5), 775-785.

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.

Carney, M. B., Smith, E., Hughes, G. R., Brendefur, J. L., & Crawford, A. (2016). Influence of proportional number relationships on item accessibility and students' strategies. *Mathematics Education Research Journal, 28*(4), 503-522.

Empson, S. B., & Turner, E. (2006). The emergence of multiplicative thinking in children's solutions to paper folding tasks. *Journal of Mathematical Behavior, 25*(1), 46-56.

Hansen, N., Jordan, N. C., Fernandez, E., Siegler, R. S., Fuchs, L., Gersten, R., & Micklos, D. (2015). General and math-specific predictors of sixth-graders' knowledge of fractions. *Cognitive Development, 35*, 34-49.

Hecht S., Close L., & Santisi, M. (2003). Sources of individual differences in fraction skills. *Journal of Experimental Child Psychology*, (86), 277-302.

Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. *Journal of Educational Psychology, 102*(4), 843-859.

Jeong, Y., Levine, S. C., & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities. *Journal of Cognition and Development, 8*(2), 237-256.

Kleemans, T., Segers, E., & Verhoeven, L. (2018). Role of linguistic skills in fifth-grade mathematics. *Journal of Experimental Child Psychology, 167*, 404-413.

Lamon, S. J., (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. Lester (Ed.), *Second handbook of research on teaching and learning mathematics, Vol. I* (pp. 629 - 667)*.* Reston, VA: National Council of Teachers of Mathematics.

Lortie-Forgues, H., Tian, J., & Siegler, R. S. (2015). Why is learning fraction and decimal arithmetic so difficult? *Developmental Review, 38*, 201-221.

Mazzocco, M. M. M., & Devlin, K. T. (2008). Parts and "holes": Gaps in rational number sense among children with vs. without mathematical learning disabilities. *Developmental Science, 11*(5), 681-691.

Mazzocco, M. M. M., Myers, G. F., Lewis, K. E., Hanich, L. B., & Murphy, M. M. (2013). Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement. *Journal of Experimental Child Psychology, 115*(2), 371-387.

Mohring, W., Newcombe, N. S., Levine, S. C., & Frick, A. (2016). Spatial proportional reasoning is associated with formal knowledge about fractions. *Journal of Cognition and Development, 17*(1), 67-84.

Namkung, J., Fuchs, L. S., & Koziol, N. (2018). Does initial learning about the meaning of fractions present similar challenges for students with and without adequate whole-number skill? *Learning and Individual Differences, 61*, 165-171.

National Research Council, & Mathematics Learning Study Committee. (2001). *Adding it up: Helping children learn mathematics*. National Academies Press.

Panaoura, A., Gagatsis, A., Deliyianni, E., & Elia, I. (2009). The structure of students' beliefs about the use of representations and their performance on the learning of fractions. *Educational Psychology, 29*(6), 713-728.

Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings In algebraic concepts in the curriculum K-12. Reston, VA: National Council of Teachers of Mathematics.

Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. *Trends in Cognitive Sciences, 17*(1), 13-19.

Thomas, N. (2004). The development of structure in the number system. *Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4,* 305-312.

Thompson, P. W., & Saldanha, L. A. (2003). Fractions and Multiplicative Reasoning. In J. Kilpatrick, G. Martin, & Schif (Eds.), *Research companion to the Principles and Standards for School Mathematics* (pp. 95-114).

Ye, A., Resnick, I., Hansen, N., Rodrigues, J., Rinne, L., & Jordan, N. C. (2016). Pathways to fraction learning: Numerical abilities mediate the relation between early cognitive competencies and later fraction knowledge. *Journal of Experimental Child Psychology, 152*, 242-263.

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