The Relation Between Spatial Reasoning and Mathematical Achievement in Children with Mathematical Learning Difficulties

  • Ilyse ResnickEmail author
  • Nora S. Newcombe
  • Nancy C. Jordan


There is evidence that spatial reasoning and mathematics achievement are related in typically developing students (e.g., Mix, K. S., & Cheng, Y. L., Advances in Child Development and Behavior, 42, 197–243, 2012). This chapter expands discussion to consider spatial reasoning in children with mathematics learning difficulties (MD). We begin by showing that spatial reasoning and mathematics achievement are multidimensional constructs comprised of dissociable, yet interconnected, skills. In this context, we consider how selected spatial and mathematical tasks intersect and how these connections may differ for students with MD. Spatial reasoning supports understanding numerical magnitude, which is commonly recognized as a core deficit in children with MD. Studies suggests, however, that children with and without MD may have similar spatial skills, with the exception of spatial working memory, where typically developing children have an advantage. These findings raise the possibility that spatial reasoning might be an untapped reservoir of strength for some children with MD. Visual number line activities may be particularly effective for connecting spatial and numerical skills.


Spatial reasoning Mathematics achievement Learning disability Fractions Number line 



This work was supported by the National Science Foundation Grant SBE-1041707 which supports the NSF funded Spatial Intelligence and Learning Center.


  1. Alloway, T. P. (2007). Working memory, reading and mathematical skills in children with developmental coordination disorder. Journal of Experimental Child Psychology, 96, 20–36. CrossRefGoogle Scholar
  2. Andersson, U., & Ostergren, R. (2013). Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities. Learning and Individual Differences, 22(6), 701–714. CrossRefGoogle Scholar
  3. Ansari, D., Donlan, C., Thomas, M. S. C., Ewing, S. A., Peen, T., & Karmiloff-Smith, A. (2003). What makes counting count: Verbal and visuo-spatial contributions to typical and atypical number development. Journal of Experimental Child Psychology, 85, 50–62. CrossRefGoogle Scholar
  4. Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447–455. CrossRefGoogle Scholar
  5. Burnett, S. A., Lane, D. M., & Dratt, L. M. (1979). Spatial visualization and sex differences in quantitative ability. Intelligence, 3, 345–354.CrossRefGoogle Scholar
  6. Butterworth, B. (1999). Perspectives: Neuroscience – A head for figures. Science, 284(5416), 928–929. CrossRefGoogle Scholar
  7. Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3–18. CrossRefGoogle Scholar
  8. Butterworth, B., & Reigosa-Crespo, V. (2007). Information processing deficits in dyscalculia. In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins or mathematical learning difficulties and disabilities. Baltimore, MD: Paul H. Brookes.Google Scholar
  9. Casey, B. M., Dearing, E., Vasilyeva, M., Ganley, C. M., & Tine, M. (2011). Spatial and numerical predictors of measurement performance: The moderating effects of community income and gender. Journal of Educational Psychology, 103(2), 296–311. CrossRefGoogle Scholar
  10. Caviola, S., Mammarella, I. C., Lucangeli, D., & Cornoldi, C. (2014). Working memory and domain-specific precursors predicting success in learning written subtraction problems. Learning and Individual Differences, 36, 92–100. CrossRefGoogle Scholar
  11. Cheng, Y. L., & Mix, K. S. (2014). Spatial training improves children’s mathematics ability. Journal of Cognition and Development, 15(1), 2–11. CrossRefGoogle Scholar
  12. Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248.Google Scholar
  13. Cunnington, M., Kantrowitz, A., Harnett, S., & Hill-Ries, A. (2014). Cultivating common ground: Integrating standards-based visual arts, math and literacy in high-poverty urban classrooms. Journal for Learning through the Arts: A Research Journal on Arts Integration in Schools and Communities, 10(1).Google Scholar
  14. de Hevia, M. D., & Spelke, E. S. (2010). Number-space mapping in human infants. Psychological Science, 21, 653–660. CrossRefGoogle Scholar
  15. Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371–396.CrossRefGoogle Scholar
  16. Dehaene, S., & Cohen, L. (1994). Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosic patients. Journal of Experimental Psychology: Human Perception and Performance, 20(5), 958–975. CrossRefGoogle Scholar
  17. Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science (New York, N.Y.), 320(5880), 1217–1220. CrossRefGoogle Scholar
  18. Delgado, A. R., & Prieto, G. (2004). Cognitive mediators and sex-related differences in mathematics. Intelligence, 32, 25–32. CrossRefGoogle Scholar
  19. Friso-van den Bos, I., Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. P. M. (2014). Explaining variability: Numerical representations in 4- to 8-year-old children. Journal of Cognition and Development, 15(2), 325–344.CrossRefGoogle Scholar
  20. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide (NCEE 2014–4005). Washington, DC: U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance Retrieved from Google Scholar
  21. Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., Seethaler, P. M., et al. (2010). Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities? Developmental Psychology, 46(6), 1731–1746. CrossRefGoogle Scholar
  22. 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(3), 683–700. CrossRefGoogle Scholar
  23. Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J., Malone, A., et al. (2014). Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. Journal of Educational Psychology, 106(2), 499–514. CrossRefGoogle Scholar
  24. Geary, D. C. (2000). From infancy to adulthood: The development of numerical abilities. European Child & Adolescent Psychiatry, 9(Suppl 2), S11. CrossRefGoogle Scholar
  25. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37(1), 4–15. CrossRefGoogle Scholar
  26. Geary, D. C., & Burlingham-Dubree, M. (1989). External validation of the strategy choice model for addition. Journal of Experimental Child Psychology, 47, 175–192. CrossRefGoogle Scholar
  27. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78(4), 1343–1359. CrossRefGoogle Scholar
  28. Geary, D. C., Hoard, M. K., Nugent, L., & Byrd-Craven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33, 300–317. CrossRefGoogle Scholar
  29. Grissmer, D. W., Mashburn, A. J., Cottone, E., Chen, W. B., Brock, L. L., Murrah, W. M., et al. (2013). Play-based after-school curriculum improves measures of executive function, visuospatial and math skills and classroom behavior for high risk K-1 children. Paper at Society for Research in Child Development, Seattle, WA.Google Scholar
  30. Gunderson, E. A., Ramirez, G., Beilock, S. L., & Levine, S. C. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line. Developmental Psychology, 48(5), 1229–1241. CrossRefGoogle Scholar
  31. Hamdan, N., & Gunderson, E. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587–596. CrossRefGoogle Scholar
  32. Hammill, D. D., Pearson, N. A., & Voress, J. K. (1993). Developmental test of visual perception (2nd ed.). Austin, TX: Pro Ed.Google Scholar
  33. Hawes, Z., Moss, J., Caswell, B., Naqvi, S., & MacKinnon, S. (2017). Enhancing children’s spatial and numerical skills through a dynamic spatial approach to early geometry instruction: Effects of a seven-month intervention. Cognition and Instruction, 35(3), 236–264. CrossRefGoogle Scholar
  34. Hawes, Z., Moss, J., Caswell, B., & Poliszczuk, D. (2015). Effects of mental rotation training on childrens spatial and mathematics performance: A randomized controlled study. Trends in Neuroscience and Education, 4, 60–68. CrossRefGoogle Scholar
  35. Hegarty, M., & Waller, D. (2004). A dissociation between mental rotation and perspective-taking spatial abilities. Intelligence, 32, 175–191. CrossRefGoogle Scholar
  36. Hitch, G. J., & McAuley, E. (1991). Working memory in children with specific arithmetical learning difficulties. British Journal of Psychology, 82(3), 375–386. CrossRefGoogle Scholar
  37. Holmes, J., Adams, J. W., & Hamilton, C. J. (2008). The relationship between visuospatial sketchpad capacity and children’s mathematical skills. European Journal of Cognitive Psychology, 20(2), 272–289. CrossRefGoogle Scholar
  38. Jordan, N. C., Fuchs, L. S., & Dyson, N. (2015). Early interventions and mathematical cognition. In R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook on numerical cognition (pp. 1079–1097). Oxford, UK: Oxford University Press.Google Scholar
  39. Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116, 45–58. CrossRefGoogle Scholar
  40. Jordan, N. C., Resnick, I., Rodrigues, J., Hansen, N., & Dyson, N. (2016). The Delaware longitudinal study of fraction learning: Implications for students with mathematics learning difficulties. Journal of Learning Disabilities, 50, 621. CrossRefGoogle Scholar
  41. Keeler, M. L., & Swanson, H. L. (2001). Does strategy knowledge influence working memory in children with mathematical disabilities? Journal of Learning Disabilities, 34, 418–434. CrossRefGoogle Scholar
  42. Kroesbergen, E. H., Van der Ven, S. H. G., Kolkman, M. E., Van Luit, J. E. H., & Leseman, P. P. M. (2009). Executieve functies en de ontwikkeling van (voorbereidende) rekenvaardigheid [Executive functions and the development of (preparatory) math skills]. Pedagogische Studiën, 86, 334–349.Google Scholar
  43. Lachance, J. A., & Mazzocco, M. M. M. (2006). A longitudinal analysis of sex differences in math and spatial skills in primary school age children. Learning and Individual Differences, 16, 195–216. CrossRefGoogle Scholar
  44. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8–9–year-old students. Cognition, 93, 99–125. CrossRefGoogle Scholar
  45. Lee, K. M., & Kang, S. Y. (2002). Arithmetic operation and working memory: Differential suppression in dual tasks. Cognition, 83(3), B63–B68. CrossRefGoogle Scholar
  46. Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. British Journal of Educational Psychology, 87, 170–186. CrossRefGoogle Scholar
  47. Lubinski, D., & Benbow, C. P. (1992). Gender differences in abilities and preferences among the gifted: Implications for the math/science pipeline. Current Directions in Psychological Science, 1(2), 61–66.CrossRefGoogle Scholar
  48. Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research and Practice, 20(3), 142–155. CrossRefGoogle Scholar
  49. McKenzie, B., Bull, R., & Gray, C. (2003). The effects of phonological and visual-spatial interference on children’s arithmetical performance. Educational and Child Psychology, 20, 93–108.Google Scholar
  50. McLean, J. F., & Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240–260.CrossRefGoogle Scholar
  51. Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: Developmental and educational implications. Advances in Child Development and Behavior, 42, 197–243. CrossRefGoogle Scholar
  52. Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., Hambrick, D. Z., & Ping, R. (2016). Separate but correlated: The latent structure of space and mathematics across development. Journal of Experimental Psychology: General, 145(9), 1206–1227. CrossRefGoogle Scholar
  53. Newcombe, N. S. (2018). Three kinds of spatial cognition. In Thompson-Schill, S. L. (Ed.), Stevens’ handbook of experimental psychology and cognitive neuroscience: Vol. 3. Language and thought (4th ed., pp. 521–552). Hoboken, NJ: John Wiley.
  54. Passolunghi, M. C., & Siegel, L. S. (2001). Short-term memory, working memory, and inhibitory control in children with difficulties in arithmetic problem solving. Journal of Experimental Child Psychology, 80, 44–57. CrossRefGoogle Scholar
  55. Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., et al. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116(1), 33–41. CrossRefGoogle Scholar
  56. Pinel, P., Dehaene, S., Rivière, D., & LeBihan, D. (2001). Modulation of parietal activation by semantic distance in a number comparison task. NeuroImage, 14(5), 1013–1026. CrossRefGoogle Scholar
  57. Pinel, P., Piazza, M., Le Bihan, D., & Dehaene, S. (2004). Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41, 983–993. CrossRefGoogle Scholar
  58. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. Child Development, 79(2), 375–394. CrossRefGoogle Scholar
  59. Rasmussen, C., & Bisanz, J. (2005). Representation and working memory in early arithmetic. Journal of Experimental Child Psychology, 91, 137–157.CrossRefGoogle Scholar
  60. Resnick, I., Jordan, N. C., Hansen, N., Rajan, V., Rodrigues, J., Siegler, R. S., & Fuchs, L. (2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52(5), 746–757. CrossRefGoogle Scholar
  61. Resnick, I., & Shipley, T. F. (2013). Breaking new ground in the mind: An initial study of mental brittle transformation and mental rigid rotation in science experts. Cognitive Processing, 14(2), 143–152. CrossRefGoogle Scholar
  62. Rittle-Johnson, B., Siegler, R., & Alibali, M. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362. CrossRefGoogle Scholar
  63. Robinson, N., Abbott, R., Berninger, V. W., & Busse, J. (1996). The structure of abilities in math-precocious young children: Gender similarities and differences. Journal of Educational Psychology, 88, 341–352.CrossRefGoogle Scholar
  64. Rotzer, S., Kucian, K., Martin, E., von Aster, M., Klaver, P., & Loenneker T. (2008). Optimized voxel-based morphometry in children with developmental dyscalculia. NeuroImage 39, 417–422 CrossRefGoogle Scholar
  65. Rousselle, L., & Noel, M. P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102, 361–395. CrossRefGoogle Scholar
  66. Rykhlevskaia, E., Uddin, L. Q., Kondos, L., & Menon, V. (2009). Neuroanatomical correlates of developmental dyscalculia: Combined evidence from morphometry and tractography. Frontiers in Human Neuroscience, 3, 51. CrossRefGoogle Scholar
  67. Saxe, G. B., Shaughnessy, M. M., Shannon, A., Langer-Osuna, J. M., Chinn, R., & Gearhart, M. (2007). Learning about fractions as points on a number line. In W. G. Martin, M. E. Strutchens, & P. C. Elliott (Eds.), The learning of mathematics: Sixty-ninth yearbook (pp. 221–237). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  68. Shea, D. L., Lubinski, D., & Benbow, C. P. (2001). Importance of assessing spatial ability in intellectually talented young adolescents: A 20-year longitudinal study. Journal of Educational Psychology, 93, 604–614.CrossRefGoogle Scholar
  69. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444.CrossRefGoogle Scholar
  70. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. CrossRefGoogle Scholar
  71. Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. CrossRefGoogle Scholar
  72. Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237–243.CrossRefGoogle Scholar
  73. Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004. CrossRefGoogle Scholar
  74. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. CrossRefGoogle Scholar
  75. Stannard, L., Wolfgang, C. H., Jones, I., & Phelps, P. (2001). A longitudinal study of the predictive relations among construction play and mathematical achievement. Early Child Development and Care, 167, 115–125.CrossRefGoogle Scholar
  76. Stieff, M., & Uttal, D. (2015). How much can spatial training improve STEM achievement? Educational Psychology Review, 27(4), 607–615. CrossRefGoogle Scholar
  77. Swanson, H. L., & Jerman, O. (2006). Math disabilities: A selective meta-analysis of the literature. Review of Educational Research, 76, 249–274. CrossRefGoogle Scholar
  78. Szucs, D., Devine, A., Soltesz, F., Nobes, A., & Gabriel, F. (2013). Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment. Cortex, 49(10), 2674–2688. CrossRefGoogle Scholar
  79. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139, 352–402. CrossRefGoogle Scholar
  80. Van’t Noordende, J. E., & Kolkman, M. E. (2013). Getallenlijnschatten door kinderen met en zonder rekenproblemen: accuratesse, representaties en strategiegebruik [Number line estimation in children with and without math learning problems: Accuracy, representations, and strategy use]. Orthopedagogiek: Onderzoek en Praktijk, 52, 322–335.Google Scholar
  81. Verdine, B. N., Irwin, C. M., Golinkoff, R. M., & Hirsh-Pasek, K. (2014). Contributions of executive function and spatial skills to preschool mathematics achievement. Journal of Experimental Child Psychology, 126, 37–51. CrossRefGoogle Scholar
  82. Voyer, D., Voyer, S., & Bryden, M. P. (1995). Magnitude of sex differences in spatial abilities: A meta-analysis and consideration of critical variables. Psychological Bulletin, 117(2), 250–270.CrossRefGoogle Scholar
  83. Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101, 817–835. CrossRefGoogle Scholar
  84. Whyte, J. C., & Bull, R. (2008). Number games, magnitude representation, and basic number skills in preschoolers. Developmental Psychology, 44, 588–596. CrossRefGoogle Scholar
  85. Wolfgang, C. H., Stannard, L. L., & Jones, I. (2003). Advanced constructional play with LEGOs among preschoolers as a predictor of later school achievement in mathematics. Early Child Development and Care, 173, 467–475. CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Ilyse Resnick
    • 1
    Email author
  • Nora S. Newcombe
    • 2
  • Nancy C. Jordan
    • 3
  1. 1.Department of PsychologyPenn State University Lehigh ValleyCenter ValleyUSA
  2. 2.Department of PsychologyTemple UniversityPhiladelphiaUSA
  3. 3.School of EducationUniversity of DelawareNewarkUSA

Personalised recommendations