Skip to main content

Advertisement

Log in

The Effects of Concrete-Representational-Abstract Sequence Instruction on Fractions for Chinese Elementary Students with Mathematics Learning Disabilities

  • Published:
International Journal of Science and Mathematics Education Aims and scope Submit manuscript

Abstract

Fractions are an important mathematical concept for elementary students. However, students with mathematics learning disabilities (MLD) struggle with learning fractions. Research has demonstrated that concrete-representational-abstract (CRA) instruction is an effective practice for students with learning and developmental disabilities. Here we investigated whether CRA instruction can support struggling learners’ understanding of fractions in China. Using a multiple probe across-students single-case design, we applied CRA instructions to 4 fourth-grade students with MLD. Results showed that CRA instruction improved students’ fraction proficiency. Limitations of the study, as well as implications on CRA instruction for elementary students with MLD in practice, are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Phase II was developed before the study was undertaken.

References

  • Bailey, D. H., Zhou, X., Zhang, Y., Cui, J., Fuchs, L. S., Jordan, N. C., Gersten, R., & Siegler, R. S. (2015). Development of fraction concepts and procedures in U.S. and Chinese children. Journal of Experimental Child Psychology, 129, 68–83.

  • Booth, J. L., & Newton, K. J. (2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37(4), 247–253.

    Article  Google Scholar 

  • Bouck, E. C., & Park, J. (2018). A systematic review of the literature on mathematics manipulatives to support students with disabilities. Education and Treatment of Children, 41(1), 65–106.

    Article  Google Scholar 

  • Bouck, E. C., Satsangi, R., & Park, J. (2018). The concrete–representational–abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedial and Special Education, 39(4), 211–228.

    Article  Google Scholar 

  • Braithwaite, D. W., Tian, J., & Siegler, R. S. (2018). Do children understand fraction addition? Developmental Science, 21(4), e12601.

    Article  Google Scholar 

  • Bryant, B. R., & Bryant, D. P. (2008). Introduction to the special series: Mathematics and learning disabilities. SAGE Publications Sage.

    Google Scholar 

  • Bryant, B. R., Bryant, D. P., Kethley, C., Kim, S. A., Pool, C., & Seo, Y.-J. (2008). Preventing mathematics difficulties in the primary grades: The critical features of instruction in textbooks as part of the equation. Learning Disability Quarterly, 31(1), 21–35.

    Article  Google Scholar 

  • Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99–111.

    Article  Google Scholar 

  • Butterworth, B., & Kovas, Y. (2013). Understanding neurocognitive developmental disorders can improve education for all. Science, 340(6130), 300–305.

    Article  Google Scholar 

  • Chan, W.-H., Leu, Y.-C., & Chen, C.-M. (2007). Exploring group-wise conceptual deficiencies of fractions for fifth and sixth graders in Taiwan. The Journal of Experimental Education, 76(1), 26–57.

    Article  Google Scholar 

  • Desoete, A., Ceulemans, A., De Weerdt, F., & Pieters, S. (2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82(1), 64–81.

    Article  Google Scholar 

  • Ferreira, F. D. O., Wood, G., Pinheiro-Chagas, P., Lonnemann, J., Krinzinger, H., Willmes, K., & Haase, V. G. (2012). Explaining school mathematics performance from symbolic and nonsymbolic magnitude processing: Similarities and differences between typical and low-achieving children. Psychology & Neuroscience, 5(1), 037–046.

    Article  Google Scholar 

  • Flores, M. M., & Milton, J. H. (2020). Teaching the partial products algorithm using the concrete-representational-abstract sequence. Exceptionality, 28(2), 142–160.

    Article  Google Scholar 

  • Flores, M. M., Hinton, V., & Strozier, S. D. (2014). Teaching subtraction and multiplication with regrouping using the concrete-representational-abstract sequence and strategic instruction model. Learning Disabilities Research & Practice, 29(2), 75–88.

    Article  Google Scholar 

  • 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: Alternative Education for Children and Youth, 60(4), 345–355.

    Article  Google Scholar 

  • Fuchs, L. S., Fuchs, D., & Prentice, K. (2004). Responsiveness to mathematical problem-solving instruction: Comparing students at risk of mathematics disability with and without risk of reading disability. Journal of Learning Disabilities, 37(4), 293–306.

    Article  Google Scholar 

  • Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37(1), 4–15.

    Article  Google Scholar 

  • Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental and Behavioral Pediatrics: JDBP, 32(3), 250–263.

    Article  Google Scholar 

  • Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7(2), 155–170.

    Article  Google Scholar 

  • Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304.

    Article  Google Scholar 

  • Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202–1242.

    Article  Google Scholar 

  • Ginsburg, H. P. (1997). Mathematics learning disabilities: A view from developmental psychology. Journal of Learning Disabilities, 30(1), 20–33.

    Article  Google Scholar 

  • Gonzales, P. A. (2001). Pursuing excellence: Comparisons of international eighth-grade mathematics and science achievement from a US perspective, 1995 and 1999: Initial findings from the Third International Mathematics and Science Study-Repeat. DIANE Publishing.

  • Handel, M. J. (2016). What do people do at work? Journal for Labour Market Research, 49(2), 177–197.

    Article  Google Scholar 

  • Harris, C. A., Miller, S. P., & Mercer, C. D. (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research and Practice, 10(3), 180–195.

    Google Scholar 

  • Hinton, V. M., & Flores, M. M. (2019). The effects of the concrete-representational-abstract sequence for students at risk for mathematics failure. Journal of Behavioral Education, 28(4), 493–516.

    Article  Google Scholar 

  • Ho, C. S. H. (2010). Understanding reading disability in the Chinese language: From basic research to intervention. In M. H. Bond (Ed.), Oxford handbook of Chinese psychology (pp. 109–121). Oxford University Press.

  • Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74(3), 834–850.

    Article  Google Scholar 

  • Judge, S., & Watson, S. M. R. (2011). Longitudinal outcomes for mathematics achievement for students with learning disabilities. The Journal of Educational Research, 104(3), 147–157.

    Article  Google Scholar 

  • Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320(5875), 454–455.

    Article  Google Scholar 

  • Kratochwill, T. R., Hitchcock, J. H., Horner, R. H., Levin, J. R., Odom, S. L., Rindskopf, D. M., & Shadish, W. R. (2013). Single-case intervention research design standards. Remedial and Special Education, 34(1), 26–38.

    Article  Google Scholar 

  • Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.

    Google Scholar 

  • Linn, M. C., Lee, H. S., Tinker, R., Husic, F., & Chiu, J. L. (2006). Teaching and assessing knowledge integration in science. Science, 313(5790), 1049–1050.

    Article  Google Scholar 

  • Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15(1), 10–21.

    Article  Google Scholar 

  • Manches, A., & O’Malley, C. (2011). Tangibles for learning: A representational analysis of physical manipulation. Personal and Ubiquitous Computing, 16(4), 405–419.

    Article  Google Scholar 

  • Mancl, D. B., Miller, S. P., & Kennedy, M. (2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 27(4), 152–166.

    Article  Google Scholar 

  • Martin, M. O., Mullis, I. V. S., & Chrostowski, S. J. (2004). TIMSS 2003 Technical Report: Findings from IEA’s Trends in International Mathematics and Science Study at the Fourth and Eighth Grades. ERIC.

  • 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.

    Article  Google Scholar 

  • McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month-old infants. Psychological Science, 18(8), 740–745.

    Article  Google Scholar 

  • McNeil, N. M., & Jarvin, L. (2007). When theories don’t add up: Disentangling the manipulative debate. Theory into Practice, 46(4), 309–316.

    Article  Google Scholar 

  • Miller, S. P., Stringfellow, J. L., Kaffar, B. J., Ferreira, D., & Mancl, D. B. (2011). Developing computation competence among students who struggle with mathematics. Teaching Exceptional Children, 44(2), 38–46.

    Article  Google Scholar 

  • Murnane, R. J., Willett, J. B., & Levy, F. (1995). The growing importance of cognitive skills in wage determination. National Bureau of Economic Research.

    Book  Google Scholar 

  • National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U. S. Department of Education.

  • Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26(3), 400–417.

    Article  Google Scholar 

  • Richland, L. E. (2015). Linking gestures: Cross-cultural variation during instructional analogies. Cognition and Instruction, 33(4), 295–321.

    Article  Google Scholar 

  • Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316(5828), 1128–1129.

    Article  Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.

    Article  Google Scholar 

  • Salimi, M., Suhartono, S., Hidayah, R., & Fajari, L. E. W. (2020). Improving mathematics learning of geometry through the concrete-pictorial-abstract (CPA) approach: Collaborative action research. Journal of Physics: Conference Series, 1663, 012046.

  • Scruggs, T. E., & Mastropieri, M. A. (1998). Summarizing single-subject research: Issues and applications. Behavior Modification, 22, 221–242.

    Article  Google Scholar 

  • Scruggs, T. E., Mastropieri, M. A., & Casto, G. (1987). The quantitative synthesis of single-subject research: Methodology and validation. Remedial and Special Education, 8, 24–33.

    Article  Google Scholar 

  • Shin, M., & Bryant, D. P. (2015). A synthesis of mathematical and cognitive performances of students with mathematics learning disabilities. Journal of Learning Disabilities, 48(1), 96–112.

    Article  Google Scholar 

  • Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697.

  • Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14(3), 237–250.

    Article  Google Scholar 

  • Tian, J., & Siegler, R. S. (2017). Fractions learning in children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 614–620.

    Article  Google Scholar 

  • Vallentin, D., & Nieder, A. (2008). Behavioral and prefrontal representation of spatial proportions in the monkey. Current Biology, 18(18), 1420–1425.

    Article  Google Scholar 

  • Wolery, M., Busick, M., Reichow, B., & Barton, E. E. (2010). Comparison of overlap methods for quantitatively synthesizing single-subject data. The Journal of Special Education, 44, 18–28.

    Article  Google Scholar 

  • Yoshida, H., & Sawano, K. (2002). Overcoming cognitive obstacles in learning fractions: Equal-partitioning and equal-whole 1. Japanese Psychological Research, 44(4), 183–195.

    Article  Google Scholar 

  • Zhang, X., Räsänen, P., Koponen, T., Aunola, K., Lerkkanen, M. K., & Nurmi, J. E. (2020). Early cognitive precursors of children’s mathematics learning disability and persistent low achievement: A 5-year longitudinal study. Child Development, 91(1), 7–27.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ting Jiang.

Ethics declarations

Conflict of Interest

The authors declare no competing interests.

Additional information

We would like to thank the Ministry of education of Humanities and Social Science General Project (20YJA880083) for funding this project. We would also like to thank administration, teachers, students, and parents who participated in our study.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (DOCX 870 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, S., Yu, S., Xiao, J. et al. The Effects of Concrete-Representational-Abstract Sequence Instruction on Fractions for Chinese Elementary Students with Mathematics Learning Disabilities. Int J of Sci and Math Educ 20, 1481–1498 (2022). https://doi.org/10.1007/s10763-021-10215-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10763-021-10215-9

Keywords

Navigation