Abstract
Students in the elementary grades often experience difficulty setting up and solving word problems. Using an equation to represent the structure of the problem serves as an effective tool for solving word problems, but students may require specific pre-algebraic reasoning instruction about the equal sign as a relational symbol to set up and solve such equations successfully. We identified students with mathematics difficulty (n = 138) from a sample of 916 third-grade students. We randomly assigned students to a word-problem intervention with a pre-algebraic reasoning component, a word-problem intervention without pre-algebraic reasoning, or the business-as-usual. Students in the 2 active intervention conditions participated in 45 individual sessions and learned about 3 additive word-problem schemas. Students who received word-problem intervention with a pre-algebraic reasoning component demonstrated improved nonstandard equation solving, equal sign understanding, and word-problem solving compared to students in the other two conditions.
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Boonen, A. J. H., de Koning, B. B., Jolles, J., & van der Schoot, M. (2016). Word problem solving in contemporary math education: A plea for reading comprehension skills training. Frontiers in Psychology,7(191), 1–10.
Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Science,14, 534–541.
Capraro, R. M., Capraro, M. M., Ding, M., & Li, X. (2007). Thirty years of research: Interpretations of the equal sign in China and the USA. Psychological Reports,101, 784–786.
Clarke, B., Doabler, C. T., Smolkowski, K., Turtura, J., Kosty, D., Kurtz-Nelson, E., et al. (2019). Exploring the relationship between initial mathematics skill and a kindergarten mathematics intervention. Exceptional Children,85, 129–146.
Cohen, J. (1992). A power primer. Psychological Bulletin,112, 155–159.
Cook, S. C., Collins, L. W., Morin, L. L., & Riccomini, P. J. (2019). Schema-based instruction for mathematical word problem solving: An evidence-based review for students with learning disabilities. Learning Disability Quarterly.
Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology,79, 347–362.
De Smedt, B., & Gilmore, C. K. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology,108, 278–292.
Depaepe, F., De Corte, E., & Verschaffel, L. (2010). Teachers’ approaches towards word problem solving: Elaborating or restricting the problem context. Teaching and Teacher Education,26, 152–160.
Doabler, C. T., Clarke, B., Stoolmiller, M., Kosty, D. B., Fien, H., Smolkowski, K., et al. (2017). Explicit instructional interactions: Exploring the black box of a Tier 2 mathematics intervention. Remedial and Special Education,38, 98–110.
Driver, M. K., & Powell, S. R. (2015). Symbolic and nonsymbolic equivalence tasks: The influence of symbols on students with mathematics difficulty. Learning Disabilities Research and Practice,30, 127–134.
Flores, M. M., Hinton, V. M., & Burton, M. E. (2016). Teaching problem solving to students receiving tiered interventions using the concrete-representational-abstract sequence and schema-based instruction. Preventing School Failure,60, 345–355.
Fuchs, L. S., Powell, S. R., Cirino, P. T., Schumacher, R. F., Marrin, S., Hamlett, C. L., et al. (2014). Does calculation or word-problem instruction provide a stronger route to pre-algebraic knowledge? Journal of Educational Psychology,106, 990–1006.
Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., et al. (2010). The effects of strategic counting instruction, with and without deliberate practice, on number combinations skill among students with mathematics difficulties. Learning and Individual Differences,20, 89–100.
Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., et al. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology,105, 683–700.
Fuchs, L. S., Seethaler, P. M., Powell, S. R., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2008). Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties. Exceptional Children,74, 155–173.
Fuchs, L. S., Sterba, S. K., Fuchs, D., & Malone, A. S. (2016). Does evidence-based fractions intervention address the needs of very low-performing students? Journal of Research on Educational Effectiveness,9, 662–677.
García, A. I., Jiménez, J. E., & Hess, S. (2006). Solving arithmetic word problems: An analysis of classification as a function of difficulty in children with and without arithmetic LD. Journal of Learning Disabilities,39, 270–281.
Gersten, R. (2016). What we are learning about mathematics interventions and conducting research on mathematics interventions. Journal of Research on Educational Effectiveness,9, 684–688.
Gilmore, C. K. (2006). Investigating children’s understanding of inversion using the missing number paradigm. Cognitive Development,21, 301–316.
Hiebert, J. (1982). The position of the unknown set and children’s solutions of verbal arithmetic problems. Journal for Research in Mathematics Education,13, 341–349.
Hinton, P. R., Brownlow, C., McMurray, I., & Cozens, B. (2004). SPSS explained. New York: Routledge.
Jitendra, A. K., Griffin, C. C., Haria, P., Leh, J., Adams, A., & Kaduvettoor, A. (2007). A comparison of single and multiple strategy instruction on third-grade students’ mathematical problem solving. Journal of Educational Psychology,99, 115–127.
Jitendra, A. K., Petersen-Brown, S., Lein, A. E., Zaslofsky, A. F., Kunkel, A. K., Jung, P. G., et al. (2015). Teaching mathematical word problem solving: The quality of evidence for strategy instruction priming the problem structure. Journal of Learning Disabilities,48, 51–72.
Jordan, N. C., & Hanich, L. B. (2000). Mathematical thinking in second-grade children with different forms of LD. Journal of Learning Disabilities,33, 567–578.
Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator,8, 139–151.
Kingsdorf, S., & Krawec, J. (2014). Error analysis of mathematical word problem solving across students with and without learning disabilities. Learning Disabilities Research and Practice,29, 66–74.
Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review,92, 109–129.
Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences,13, 129–164.
Koponen, T., Aro, M., Poikkeus, A.-M., Niemi, P., Lerkkanen, M.-K., Ahonen, T., et al. (2018). Comorbid fluency difficulties in reading and math: Longitudinal stability across early grades. Exceptional Children,84, 298–311.
Krawitz, J., Schukajlow, S., & Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing information: What are important barriers? Educational Psychology,38(10), 1221–1238.
Lai, Y., Zhu, X., Chen, Y., & Li, Y. (2015). Effects of mathematics anxiety and mathematical metacognition on word problem solving in children with and without mathematical learning difficulties. PLoS One,10(6), e0130570.
MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology,58, 593–614.
Matthews, P., & Rittle-Johnson, B. (2009). In pursuit of knowledge: Comparing self-explanations, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology,104, 1–21.
McNeil, N. M. (2008). Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence. Child Development,79, 1524–1537.
McNeil, N. M., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Brletic-Shipley, H. (2011). Benefits of practicing 4 = 2 + 2: Nontraditional problem formats facilitate children’s understanding of mathematical equivalence. Child Development,82, 1620–1633.
Molina, M., Castro, E., & Castro, E. (2009). Elementary students’ understanding of the equal sign in number sentences. Electronic Journal of Research in Educational Psychology,17, 341–368.
Mononen, R., Aunio, P., Koponen, T., & Aro, M. (2014). A review of early numeracy interventions for children at risk in mathematics. International Journal of Early Childhood Special Education,6, 25–54.
Morgan, P. L., Farkas, G., & Wu, Q. (2009). Five-year growth trajectories of kindergarten children with learning difficulties in mathematics. Journal of Learning Disabilities,42, 306–321.
Nelson, G., & Powell, S. R. (2018). A systematic review of longitudinal studies of mathematics difficulty. Journal of Learning Disabilities,51, 523–539.
Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education,40, 282–313.
O’Shea, A., Booth, J. L., Barbieri, C., McGinn, K. M., Young, L. K., & Oyer, M. H. (2016). Algebra performance and motivation differences for students with learning disabilities and students of varying achievement levels. Contemporary Educational Psychology,50, 80–96.
Panayides, P. (2014). Misinterpreting the meaning of the equal sign: A study of pupils in the final grades of primary school in Cyprus. Global Journal of Interdisciplinary Social Sciences,3(5), 16–26.
Peake, C., Jiménez, J. E., Rodríguez, C., Bisschop, E., & Villarroel, R. (2015). Syntactic awareness and arithmetic word problem solving in children with and without learning disabilities. Journal of Learning Disabilities,48, 593–601.
Peltier, C., & Vannest, K. J. (2017). A meta-analysis of schema instruction on the problem-solving performance of elementary school students. Review of Educational Research,87, 899–920.
Pillay, H., Wilss, L., & Boulton-Lewis, G. (1998). Sequential development of algebra knowledge: A cognitive analysis. Mathematics Education Research Journal,10, 87–102.
Powell, S. R. (2007). Open Equations. Available from S. R. Powell, 1912 Speedway, Austin, TX 78712.
Powell, S. R. (2012). Equations and the equal sign in elementary mathematics textbooks. The Elementary School Journal,112, 627–648.
Powell, S. R. (2015). Equivalence Problems. Available from S. R. Powell, 1912, Speedway, Austin, TX 78712.
Powell, S. R., & Berry, K. A. (2015). Texas Word Problems. Available from S. R. Powell, 1912 Speedway, Austin, TX 78712.
Powell, S. R., Driver, M. K., & Julian, T. E. (2015). The effect of tutoring with nonstandard equations for students with mathematics difficulty. Journal of Learning Disabilities,48, 523–534.
Powell, S. R., & Fuchs, L. S. (2010). Contribution of equal-sign instruction beyond word-problem tutoring for third-grade students with mathematics difficulty. Journal of Educational Psychology,102, 381–394.
Powell, S. R., Kearns, D. M., & Driver, M. K. (2016). Exploring the connection between arithmetic and pre-algebraic reasoning at first and second grade. Journal of Educational Psychology,108, 943–959.
Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction,5, 49–101.
Shalev, R. S., Auerbach, J., Manor, O., & Gross-Tsur, V. (2000). Developmental dyscalculia: Prevalence and prognosis. European Child and Adolescent Psychiatry,9, 58–64.
Shalev, R. S., Manor, O., & Gross-Tsur, V. (2005). Developmental dyscalculia: A prospective six-year follow-up. Developmental Medicine and Child Neurology,47, 121–125.
Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology,101, 88–100.
Stevens, J. J., Schulte, A. C., Elliott, S. N., Nese, J. F. T., & Tindal, G. (2015). Growth and gaps in mathematics achievement of students with and without disabilities on a statewide achievement test. Journal of School Psychology,53, 45–62.
Swanson, H. L., Lussier, C. M., & Orosco, M. J. (2013). Cognitive strategies, working memory, and growth in word problem solving in children with math difficulties. Journal of Learning Disabilities,48, 339–358.
Szücs, D., & Goswami, U. (2013). Developmental dyscalculia: Fresh perspectives. Trends in Neuroscience and Education,2, 33–37.
Tolar, T. D., Fuchs, L., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2016). Cognitive profiles of mathematical problem solving learning disabilities for different definitions of disability. Journal of Learning Disabilities,49, 240–256.
Van Dooren, W., Verschaffel, L., Greer, B., & De Bock, D. (2006). Modeling for life: Developing adaptive expertise in mathematical modelling from an early age. In L. Verschaffel, F. Dochy, M. Boekaerts, & S. Vosniadou (Eds.), Instructional psychology: Past, present and future trends (pp. 91–112). Oxford: Elsevier.
van Lieshout, E. C. D. M., & Xenidou-Dervou, I. (2018). Pictorial representations of simple arithmetic problems are not always helpful: A cognitive load perspective. Educational Studies in Mathematics,98(1), 39–55.
Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Educational Studies in Mathematics,42, 211–213.
Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Charlotte: Information Age Publishing.
Vincent, J., Bardini, C., Pierce, R., & Pearn, C. (2015). Misuse of the equals sign: An entrenched practice from early primary years to tertiary mathematics. Australian Senior Mathematics Journal,29(2), 31–39.
Wang, A. Y., Fuchs, L. S., & Fuchs, D. (2016). Cognitive and linguistic predictors of mathematical word problems with and without irrelevant information. Learning and Individual Differences,52, 79–87.
Wei, X., Lenz, K. B., & Blackorby, J. (2013). Math growth trajectories of students with disabilities: Disability category, gender, racial, and socioeconomic status differences from ages 7 to 17. Remedial and Special Education,34, 154–165.
Xin, Y. P., & Zhang, D. (2009). Exploring a conceptual model-based approach to teaching situated word problems. The Journal of Educational Research,102, 427–441.
Zhang, D., & Xin, Y. P. (2012). A follow-up meta-analysis for word-problem-solving interventions for students with mathematics difficulties. The Journal of Educational Research,105, 303–318.
Zhang, D., Xin, Y. P., Harris, K., & Ding, Y. (2014). Improving multiplication strategic development in children with math difficulties. Learning Disability Quarterly,37, 15–30.
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This research was supported in part by Grant R324A150078 from the Institute of Education Sciences in the U.S. Department of Education to the University of Texas at Austin. The content is solely the responsibility of the authors and does not necessarily represent the official views of the U.S. Department of Education.
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Powell, S.R., Berry, K.A. & Barnes, M.A. The role of pre-algebraic reasoning within a word-problem intervention for third-grade students with mathematics difficulty. ZDM Mathematics Education 52, 151–163 (2020). https://doi.org/10.1007/s11858-019-01093-1
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DOI: https://doi.org/10.1007/s11858-019-01093-1