Mathematics Learning Development: the Role of Long-Term Retrieval



This study assessed the relation between long-term memory retrieval and mathematics calculation and mathematics problem solving achievement among elementary, middle, and high school students in nationally representative sample of US students, when controlling for fluid and crystallized intelligence, short-term memory, and processing speed. As hypothesized, structural equation modeling comparing elementary school students and middle and high school students revealed that long-term retrieval skills became a better predictor of both mathematics calculation and mathematics problem solving as age and grade increased. Future research should focus on the effectiveness of interventions to improve long-term retrieval skills in general, and arithmetic facts retrieval and problem solving procedures in particular, at all grades, including high school.


Calculation High school students Long-term retrieval Mathematics achievement Problem solving 



The authors would like to thank Kathryn Nakagawa, Ph.D. and Kevin McGrew, Ph.D. for their valuable contributions.


  1. Baddeley, A. D. & Hitch, G. J. (1974). Working memory. In G. H. Bower (Ed.), The psychology of learning and motivation: Advances in research and theory (Vol. 8, pp. 47–90). New York, NY: Academic.Google Scholar
  2. Calderón-Tena, C. O. (2015). Mathematical development: The role of broad cognitive processes. Manuscript submitted for publication.Google Scholar
  3. Carroll, J. B. (1993). Human cognitive abilities. Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
  4. Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552.CrossRefGoogle Scholar
  5. Gelman, R. & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.Google Scholar
  6. Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009–4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from
  7. Hitch, G. J. (1978). The role of short-term working memory in mental arithmetic. Cognitive Psychology, 10, 302–323.CrossRefGoogle Scholar
  8. Horn, J. L. & Cattell, R. B. (1966). Refinement and test of the theory of fluid and crystallized general intelligence. Journal of Educational Psychology, 57, 253–270.CrossRefGoogle Scholar
  9. Hu, L. & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55.CrossRefGoogle Scholar
  10. Kearns, D. M. & Fuchs, D. (2013). Does cognitively focused instruction improve the academic performance of low-achieving students? Exceptional Children, 79, 263–290.Google Scholar
  11. Kline, R. B. (1998). Principles and practice of structural equation modeling. New York, NY: Guilford Press.Google Scholar
  12. McGrew, K. S. & Wendling, B. J. (2010). Cattell-Horn-Carroll cognitive-achievement relations: What we have learned from the past 20 years of research. Psychology in the Schools, 47(7), 651–675.Google Scholar
  13. Muthén, B. O. & Muthén, L. K. (2014). Mplus (Version 7.2) [Computer software]. Los Angeles, CA: Muthén & Muthén.Google Scholar
  14. Newton, J. H. & McGrew, K. S. (2010). Introduction to the special issue: Current research in Cattell-Horn-Carroll-based assessment. Psychology in the Schools, 47(7), 621–634.Google Scholar
  15. Pauly, H., Linkersdörfer, J., Lindberg, S., Woerner, W., Hasselhorn, M. & Lonnemann, J. (2011). Domain-specific rapid automatized naming deficits in children at risk for learning disabilities. Journal of Neurolinguistics, 24, 602–610.CrossRefGoogle Scholar
  16. Schmiedek, F., Lövdén, M. & Lindenberger, U. (2014). Younger adults show long-term effects of cognitive training on broad cognitive abilities over 2 years. Developmental Psychology 50(9),2304-2310. doi: 10.1037/a0037388.
  17. Taub, G. E., Floyd, R. G., Keith, T. Z. & McGrew, K. S. (2008). Effects of general and broad cognitive abilities on mathematics achievement. School Psychology Quarterly, 23(2), 187–198.CrossRefGoogle Scholar
  18. Van der Sluis, S., De Jong, P. F. & Van der Leij, A. (2004). Inhibition and shifting in children with learning deficits in arithmetic and reading. Journal of Experimental Child Psychology, 87(3), 239–266.CrossRefGoogle Scholar
  19. Wechsler, D. (2014). Wechsler Intelligence Scale for Children-Fifth Edition. San Antonio, TX: Pearson.Google Scholar
  20. Weston, R. & Gore, P. A. (2006). A brief guide to structural equation modeling. The Counseling Psychologist, 34, 719–751.CrossRefGoogle Scholar
  21. Willburger, E., Fussenegger, B., Moll, K., Wood, G. & Landerl, K. (2008). Naming speed in dyslexia and dyscalculia. Learning and Individual Differences, 18, 224–236.CrossRefGoogle Scholar
  22. Woodcock, R. W., McGrew, K. S. & Mather, N. (2007). Woodcock-Johnson III Normative Update (NU) Complete. Itasca, IL: Riverside Publishing.Google Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  1. 1.Department of PsychologyCalifornia State University, FresnoFresnoUSA
  2. 2.Mary Lou Fulton Teachers CollegeArizona State UniversityTempeUSA

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