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A metacognitive intervention for teaching fractions to students with or at-risk for learning disabilities in mathematics

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Abstract

Assessment data from the United States and international reports of student achievement indicate that upper elementary students are failing to meet basic levels of proficiency in fractions and writing, and that this is particularly prevalent with students with or at-risk for learning disabilities in mathematics. Proficiency with fractions has been identified as foundational for learning higher-level mathematics but remains one of the most difficult skills for students to learn. In addition, students’ difficulty with fractions is exacerbated because of increased chances of comorbidity with language learning problems, particularly difficulties constructing arguments and communicating using writing. We describe FACT + R2C2, a language-based, metacognitive instructional intervention that was designed using the Self-Regulated Strategy Development model (SRSD) for teaching foundational concepts of fractions. The results from two studies in which the intervention was administered to upper elementary students who exhibit mathematics difficulties indicated selected increases in students’ computational accuracy, quality of mathematical reasoning, number of rhetorical elements, and total words. With evidence of improved performance in these areas, FACT + R2C2 holds promise for helping these students become proficient self-regulated learners.

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References

  • Agrawal, J., & Morin, L. L. (2016). Evidence-based practices: Applications of concrete representational abstract framework across math concepts for students with mathematics disabilities. Learning Disabilities Research & Practice, 31, 34–44.

    Article  Google Scholar 

  • Aunio, P., & Tapola, A. (2015). Children with low performance in mathematics and interventions. In P. Aunio, R. Mononen, & A. Laine, Mathematical learning difficulties—snapshots of current European research. LUMAT, 5, 664–667.

  • Ball, D. L., & Forzani, F. M. (2011). Building a common core for learning to teach: And connecting professional learning to practice. American Educator, 35, 17.

    Google Scholar 

  • Bangert-Drowns, R. L., Hurley, M. M., & Wilkinson, B. (2004). The effects of school-based writing-to-learn interventions on academic achievement: A meta-analysis. Review of Educational Research, 74, 29–58.

    Article  Google Scholar 

  • Bryant, B. R., & Bryant, D. P. (2008). Mathematics and learning disabilities. Learning Disability Quarterly, 31, 3–11.

    Article  Google Scholar 

  • Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105, 380–400.

    Article  Google Scholar 

  • Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. The Journal of Special Education, 26, 1–19.

    Article  Google Scholar 

  • Cassel, J., & Reid, R. (1996). Use of a self-regulated strategy intervention to improve word problem-solving skills of students with mild disabilities. Journal of Behavioral Education, 6, 153–172.

    Article  Google Scholar 

  • Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understanding of fractions. Educational Studies in Mathematics, 64, 293–316.

    Article  Google Scholar 

  • Cuenca-Carlino, Y., Freeman-Green, S., Stephensen, G. W., & Hauth, C. (2916). Self-regulated strategy development for teaching multi-step equations to middle school students struggling in math. The Journal of Special Education, 50, 75–85.

    Article  Google Scholar 

  • De Corte, E., Versaffel, L., & Eynde, P. O. (2000). Self-regulation: A characteristic and a goal of mathematics education. In M. Boekaerts, P. R. Pintrich & M. Ziedner (Eds.), Handbook of self-regulation (pp. 687–726). San Diego: Academic Press.

    Chapter  Google Scholar 

  • De La Paz, S. (2005). Effects of historical reasoning instruction and writing strategy mastery in culturally and academically diverse middle school classrooms. Journal of Educational Psychology, 97, 139–156.

    Article  Google Scholar 

  • Desimone, L. M. (2009). Improving impact studies of teachers’ professional development: Toward better conceptualizations and measure. Educational Researcher, 38, 181–199.

    Article  Google Scholar 

  • Desoete, A. (2015). Language and math. In P. Aunio, R. Mononen, & A. Laine, Mathematical learning difficulties—snapshots of current European research. LUMAT, 5, 647–674.

  • Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht: Kluwer Academic Press.

    Google Scholar 

  • Faulkner, V. N., & Cain, C. R. (2013). Improving the mathematical content knowledge of general and special educators: Evaluating a professional development module that focuses on number sense. Teacher Education and Special Education, 36, 115–131.

    Article  Google Scholar 

  • Fritjters, S., Van Dam, G. I., & Rijlaarsdam, G. (2006). Effects of dialogic learning on value-loaded critical thinking. Learning and Instruction, 18, 66–82.

    Article  Google Scholar 

  • Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C., Cirino, P. T., Jordan, N. C., Siegler, R., Gersten, R., & Changas, P. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105, 683–700.

    Article  Google Scholar 

  • Fuson, K. C. (2013). Math Expressions. Boston: Houghton Mifflin Harcourt.

    Google Scholar 

  • Gafurov, B. S., & Levin, J. R. (2018). ExPRT (Excel Package of Randomization Tests): Statistical Analyses of Single-Case Intervention Data; current Version 3.1 (June 2018) is retrievable from the ExPRT website at http://ex-prt.weebly.com.

  • Garet, M. S., Porter, A. C., Desimone, L., Birman, B. G., & Yoon, K. W. (2001). What makes professional development effective? Results for a national sample of teachers. American Educational Research Journal, 38, 915–945.

    Article  Google Scholar 

  • Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental & Behavioral Pediatrics, 32, 250–263.

    Article  Google Scholar 

  • Gillespie, A., & Graham, S. (2014). A meta-analysis of writing interventions for students with learning disabilities. Exceptional Children, 80, 454–473.

    Google Scholar 

  • Graham, S., McKeown, D., Kiuhara, S. A., & Harris, K. R. (2012). A meta-analysis of writing instruction for students in the elementary grades. Journal of Educational Psychology, 204, 879–896.

    Article  Google Scholar 

  • Hacker, D. J. (1998). Metacognition: Definitions and empirical foundations. In D. J. Hacker, J. Dunlosky, & A. C. Graesser (Eds.), Metacognition in educational theory and practice, (pp. 1–23). Mahwah: Erlbaum.

    Google Scholar 

  • Hacker, D. J. (2018). A metacognitive model of writing: An update from a developmental perspective. Educational Psychologist, 53, 220–237.

    Article  Google Scholar 

  • Harris, K. R., & Graham, S. (2009). Self-regulated strategy development in writing: Premises, evolution, and the future. Teaching and Learning Writing, 6, 113–135.

    Google Scholar 

  • Harris, K. R., Graham, S., & Adkins, M. (2015). Practice-based professional development and self-regulated strategy development for Tier 2, at-risk writers in second grade. Contemporary Educational Psychology, 40, 5–16.

    Article  Google Scholar 

  • Hwang, Y., & Levin, J. R. (2019). Application of a single-case intervention procedure to assess the replicability of a two-component instructional strategy. Contemporary Educational Psychology, 56, 161–170.

    Article  Google Scholar 

  • Individuals With Disabilities Education Act, 20 U.S.C. § 1400 (2004).

  • Jonsson, B., Norqvist, M., Liljekvist, Y., & Lithner, J. (2014). Learning mathematics through algorithmic and creative reasoning. The Journal of Mathematical Behavior, 36, 20–32.

    Article  Google Scholar 

  • Kiuhara, S. A., Gillespie Rouse, A., Dai, T., Witzel, B., Morphy, P., & Unker, B. (in press). Self-regulated strategy development: Constructing arguments to develop fraction knowledge. Journal of Educational Psychology.

  • Kiuhara, S. A., Levin, J. R., MacKay, M., & Tolbert, M. (2019). FACT + R2C2: Illustration of a multiple baseline design with an associated randomization test (Manuscript in preparation).

  • Klein, P. D. (1999). Reopening inquiry into cognitive processes in writing-to-learn. Educational Psychology Review, 11, 203–270.

    Article  Google Scholar 

  • Klein, P. D., Haug, K. N., & Arcon, N. (2017). The effects of rhetorical and content subgoals on writing and learning. The Journal of Experimental Education, 85, 291–308.

    Article  Google Scholar 

  • Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and metacognitive training. American Educational Research Journal, 40, 281–310.

    Article  Google Scholar 

  • Krowka, S. K., & Fuchs, L. S. (2017). Cognitive profiles associated with responsiveness to fraction intervention. Learning Disabilities Research & Practice, 32, 216–230.

    Google Scholar 

  • Levin, J. R. (1983). Pictorial strategies for school learning: Practical illustrations. In M. Pressley & J. R. Levin (Eds.), Cognitive strategy research: Educational applications (pp. 213–237). New York: Springer.

    Chapter  Google Scholar 

  • Levin, J. R. (1992). Single-case research design and analysis: Comments and concerns. In T. R. Kratochwill & J. R. Levin (Eds.), Single-case research design and analysis: New directions for psychology and education (p. 213224). Hillsdale: Erlbaum.

    Google Scholar 

  • Levin, J. R., Ferron, J. M., & Gafurov, B. S. (2018). Comparison of randomization-test procedures for single-case multiple-baseline designs. Developmental Neurorehabilitation, 21, 290–311.

    Article  Google Scholar 

  • Lewis, K. E., & Fisher, M. B. (2016). Taking stock of 40 years of research on mathematical learning disability: Methodological issues and future directions. Journal for Research in Mathematics Education, 47, 338–371.

    Article  Google Scholar 

  • Marascuilo, L. A., & Busk, P. L. (1988). Combining statistics for multiple-baseline AB and replicated ABAB designs across subjects. Behavioral Assessment, 10, 1–28.

    Google Scholar 

  • 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 student with mathematics learning disability (dyscalculia) versus low math achievement. Journal of Experimental Child Psychology, 115, 371–387.

    Article  Google Scholar 

  • Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms. American Educational Research Journal, 3, 365–395.

    Article  Google Scholar 

  • Montague, M., & Jitendra, A. K. (2012). Research-based mathematics instruction for students with learning disabilities. In H. Forgasz & F. Rivera (Eds.), Towards equity in mathematics Education: Advances in Mathematics Education (pp. 481–502). Berlin: Springer.

    Chapter  Google Scholar 

  • Namkung, J., & Fuchs, L. (2016). Cognitive predictors of calculations and number line estimation with whole numbers and fractions among at-risk students. Journal of Educational Psychology, 108, 214–228.

    Article  Google Scholar 

  • National Center for Education Statistics. Institute of education sciences national assessment of educational progress for mathematics. (2017). The Nation’s Report Card: Mathematics 2017. Washington: Institute of Education Sciences, U. S. Department of Education.

  • Neuman, Y., & Schwarz, B. B. (1998). Is self-explanation while solving problems helpful? The case of analogical problem solving. British Journal of Educational Psychology, 68, 15–24.

    Article  Google Scholar 

  • Nussbaum, E. M., & Kardash, C. M. (2005). The effects of goal instructions and text on the generation of counterarguments during writing. Journal of Educational Psychology, 97, 157–169.

    Article  Google Scholar 

  • Odom, S. L., Brantlinger, E., Gersten, R., Horner, R. H., Thompson, B., & Harris, K. R. (2005). Research in special education: Scientific methods and evidence-based practices. Council for Exceptional Children, 71, 137–148.

    Google Scholar 

  • Organization for Economic Cooperation and Development. (2003). Literacy skills for the world of tomorrow. Further results from PISA 2000. OECD Publishing Paris.

    Google Scholar 

  • Pintrich, P. R. (2002). The role of metacognitive knowledge in learning, teaching, and assessing. Theory Into Practice, 41, 219–225.

    Article  Google Scholar 

  • Prain, V., & Hand, B. (2016). Coming to know more through and from writing. Educational Researcher, 45, 430–434.

    Article  Google Scholar 

  • Pressley, M., & Harris, K. R. (2006). Cognitive strategies instruction: From basic research to classroom instruction. In P. A. Alexander & P. H. Winne (Eds.), Handbook of educational psychology (pp. 265–286). Mahwah: Lawrence Erlbaum Associates Publishers.

    Google Scholar 

  • Pugalee, D. K. (2002). Writing, mathematics, and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, 101, 236–245.

    Article  Google Scholar 

  • Räsänen, P. (2015). Longitudinal studies on dyscalculia. In P. Aunio, R. Mononen, & A. Laine, Mathematical learning difficulties—snapshots of current European research. LUMAT, 5, 651–655.

  • Rau, M. A., & Matthews, P. G. (2017). How to make ‘more’ better? Principles for effective use of multiple representations to enhance students’ learning about fractions. ZDM Mathematics Education, 49, 531–544.

    Article  Google Scholar 

  • Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education, 49, 599–611.

    Article  Google Scholar 

  • Schleppegrell, M. J. (2013). The role of metalanguage in supporting academic language development. Language Learning, 63, 153–170.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334–370). New York: MacMillan.

    Google Scholar 

  • Schunk, D. H., & Zimmerman, B. J. (1997). Social origins of self-regulatory competence. Educational Psychologist, 32, 195–208.

    Article  Google Scholar 

  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Classens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691–697.

    Article  Google Scholar 

  • Swanson, H. L., & Fung, W. (2016). Working memory components and problem-solving accuracy: Are there multiple pathways? Journal of Educational Psychology, 108, 1153–1177.

    Article  Google Scholar 

  • Tindal, G., & Alonzo, J. (2012). easyCBM. Boston: Houghton Mifflin Harcourt.

    Google Scholar 

  • Vukovic, R. K., & Lesaux, N. K. (2013). The language of mathematics: Investigating the ways language counts for children’s mathematical development. Journal of Experimental Child Psychology, 115, 227–244.

    Article  Google Scholar 

  • Wolff, P., Levin, J. R., & Longobardi, E. T. (1974). Activity and children’s learning. Child Development, 45, 221223.

    Article  Google Scholar 

  • Zimmerman, B. J. (2008). Investigating self-regulation and motivation: Historical background, methodological developments, and future prospects. American Educational Research Journal, 45, 166–183.

    Article  Google Scholar 

  • Zito, J. R., Adkins, M., Gavins, M., Harris, K. R., & Graham, S. (2007). Self-regulated strategy development: Relationship to the social-cognitive perspective and the development of self-regulation. Reading & Writing Quarterly, 23, 77–95.

    Article  Google Scholar 

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Correspondence to Douglas J. Hacker or Sharlene A. Kiuhara.

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Hacker, D.J., Kiuhara, S.A. & Levin, J.R. A metacognitive intervention for teaching fractions to students with or at-risk for learning disabilities in mathematics. ZDM Mathematics Education 51, 601–612 (2019). https://doi.org/10.1007/s11858-019-01040-0

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