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Calculation

  • Rhonda Douglas Brown
  • Vincent J. Schmithorst
  • Lori Kroeger
Chapter

Abstract

In this chapter, we present neuroscience research that addresses the development of the more complex skill of calculation from childhood into adulthood. Cognitive processes related to mathematics achievement are described including the quantity, verbal, and visual systems of Dehaene and colleagues’ triple-code model and domain-specific and domain-general processes. We present results from our research using functional Magnetic Resonance Imaging (fMRI) to examine relationships between neural correlates of calculation and mathematics achievement. Activation in critical brain regions and deactivation of the Default Mode Network (DMN) for a variety of tasks, including exact and approximate calculation and error detection, are illustrated. We also discuss our research using exploratory group Independent Component Analysis (ICA) to reveal separate components of functional activation in bilateral inferior parietal, left perisylvian, and ventral occipitotemporal areas during the mental addition and subtraction of fractions. Taken together, our work provides support for the triple-code model for a variety of tasks. Furthermore, it indicates that domain-specific neuroarchitecture for quantity processing and domain-general processes related to the DMN may act in coordination to perform calculation.

Keywords

Mathematics achievement Triple-code model Default Mode Network (DMN) Exact calculation Addition Multiplication Working memory Approximate calculation Error detection Fractions 

References

  1. Ansari, D., & Dhital, B. (2006). Age-related changes in the activation of the intraparietal sulcus during nonsymbolic magnitude processing: An event-related functional magnetic resonance imaging study. Journal of Cognitive Neuroscience, 18(11), 1820–1828.  https://doi.org/10.1162/jocn.2006.18.11.1820 CrossRefPubMedGoogle Scholar
  2. Ansari, D., Garcia, N., Lucas, E., Hamon, K., & Dhital, B. (2005). Neural correlates of symbolic number processing in children and adults. Neuroreport, 16(16), 1769–1773.  https://doi.org/10.1097/01.wnr.0000183905.23396.f1 CrossRefPubMedGoogle Scholar
  3. Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: A test of three verification models. Memory & Cognition, 9(2), 185–196.  https://doi.org/10.3758/BF03202334 CrossRefGoogle Scholar
  4. Beran, M. J., & Beran, M. M. (2004). Chimpanzees remember the results of one-by-one addition of food items to sets over extended time periods. Psychological Science, 15(2), 94–99.  https://doi.org/10.1111/j.0963-7214.2004.01502004.x CrossRefPubMedGoogle Scholar
  5. Boysen, S. T., & Berntson, G. G. (1989). Numerical competence in a chimpanzee (Pan troglodytes). Journal of Comparative Psychology, 103(1), 23–31.  https://doi.org/10.1037/0735-7036.103.1.23 CrossRefPubMedGoogle Scholar
  6. Bull, R., Espy, K. A., & Wiebe, S. A. (2008). Short-term memory, working memory, and executive functioning in preschoolers: Longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33(3), 205–228.  https://doi.org/10.1080/87565640801982312 CrossRefPubMedPubMedCentralGoogle Scholar
  7. Butterworth, B., & Reigosa, 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 of mathematical learning difficulties and disabilities (pp. 65–81). Baltimore, MD: Paul H. Brookes Publishing.Google Scholar
  8. Cantlon, J. F., Libertus, M. E., Pinel, P., Dehaene, S., Brannon, E. M., & Pelphrey, K. A. (2009). The neural development of an abstract concept of number. Journal of Cognitive Neuroscience, 21(11), 2217–2229.  https://doi.org/10.1162/jocn.2008.21159 CrossRefPubMedPubMedCentralGoogle Scholar
  9. Cantlon, J. F., Merritt, D. J., & Brannon, E. M. (2016). Monkeys display classic signatures of human symbolic arithmetic. Animal Cognition, 19(2), 405–415.  https://doi.org/10.1007/s10071-015-0942-5 CrossRefPubMedGoogle Scholar
  10. Chochon, F., Cohen, L., van de Moortele, P. F., & Dehaene, S. (1999). Differential contributions of the left and right inferior parietal lobules to number processing. Journal of Cognitive Neuroscience, 11(6), 617–630.  https://doi.org/10.1162/089892999563689 CrossRefPubMedGoogle Scholar
  11. Cohen, L., & Dehaene, S. (1991). Neglect dyslexia for numbers? A case report. Cognitive Neuropsychology, 8(1), 39–58.  https://doi.org/10.1080/02643299108253366 CrossRefGoogle Scholar
  12. Dahmen, W., Hartje, W., Büssing, A., & Sturm, W. (1982). Disorders of calculation in aphasic patients—spatial and verbal components. Neuropsychologia, 20(2), 145–153.  https://doi.org/10.1016/0028-3932(82)90004-5 CrossRefPubMedGoogle Scholar
  13. Davis, N., Cannistraci, C. J., Rogers, B. P., Gatenby, J. C., Fuchs, L. S., Anderson, A. W., & Gore, J. C. (2009). The neural correlates of calculation ability in children: An fMRI study. Magnetic Resonance Imaging, 27(9), 1187–1197.  https://doi.org/10.1016/j.mri.2009.05.010 CrossRefPubMedPubMedCentralGoogle Scholar
  14. De Pisapia, N., Slomski, J. A., & Braver, T. S. (2007). Functional specializations in lateral prefrontal cortex associated with the integration and segregation of information in working memory. Cerebral Cortex, 17(5), 993–1006.  https://doi.org/10.1093/cercor/bhl010 CrossRefPubMedGoogle Scholar
  15. Dehaene, S. (1989). The psychophysics of numerical comparison: A reexamination of apparently incompatible data. Perception & Psychophysics, 45(6), 557–566.  https://doi.org/10.3758/BF03208063 CrossRefGoogle Scholar
  16. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1–2), 1–42.  https://doi.org/10.1016/0010-0277(92)90049-N CrossRefPubMedPubMedCentralGoogle Scholar
  17. Dehaene, S. (2011). The number sense: How the mind creates mathematics (Rev. ed.). New York, NY: Oxford University Press.Google Scholar
  18. Dehaene, S., & Cohen, L. (1991). Two mental calculation systems: A case study of severe acalculia with preserved approximation. Neuropsychologia, 29(11), 1045–1074.  https://doi.org/10.1016/0028-3932(91)90076-K CrossRefPubMedGoogle Scholar
  19. Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120.Google Scholar
  20. Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33(2), 219–250.  https://doi.org/10.1016/S0010-9452(08)70002-9 CrossRefPubMedPubMedCentralGoogle Scholar
  21. Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16(3), 626–641.  https://doi.org/10.1037/0096-1523.16.3.626 CrossRefPubMedGoogle Scholar
  22. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3–6), 487–506.  https://doi.org/10.1080/02643290244000239 CrossRefPubMedPubMedCentralGoogle Scholar
  23. Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284(5416), 970–974.  https://doi.org/10.1126/science.284.5416.970 CrossRefPubMedGoogle Scholar
  24. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2), 43–74.  https://doi.org/10.1016/0010-0277(92)90050-R CrossRefPubMedGoogle Scholar
  25. Geary, D. C. (1995). Reflections of evolution and culture in children's cognition: Implications for mathematical development and instruction. American Psychologist, 50(1), 24–37.  https://doi.org/10.1037/0003-066X.50.1.24 CrossRefPubMedPubMedCentralGoogle Scholar
  26. Geary, D. C. (2007). An evolutionary perspective on learning disability in mathematics. Developmental Neuropsychology, 32(1), 471–519.  https://doi.org/10.1080/87565640701360924 CrossRefPubMedPubMedCentralGoogle Scholar
  27. Geary, D. C. (2010). Mathematical disabilities: Reflections on cognitive, neuropsychological, and genetic components. Learning and Individual Differences, 20(2), 130–133.  https://doi.org/10.1016/j.lindif.2009.10.008 CrossRefPubMedPubMedCentralGoogle Scholar
  28. Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552.  https://doi.org/10.1037/a0025510 CrossRefPubMedPubMedCentralGoogle Scholar
  29. Geary, D. C., Hoard, M. K., Byrd-Craven, J., & DeSoto, M. C. (2004). Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88(2), 121–151.  https://doi.org/10.1016/j.jecp.2004.03.002 CrossRefPubMedGoogle Scholar
  30. Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study. Journal of Educational Psychology, 104(1), 206–223.  https://doi.org/10.1037/a0025398 CrossRefPubMedPubMedCentralGoogle Scholar
  31. González, E. G., & Kolers, P. A. (1982). Mental manipulation of arithmetic symbols. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8(4), 308–319.  https://doi.org/10.1037/0278-7393.8.4.308 CrossRefGoogle Scholar
  32. Grabner, R. H., Ischebeck, A., Reishofer, G., Koschutnig, K., Delazer, M., Ebner, F., & Neuper, C. (2009). Fact learning in complex arithmetic and figural-spatial tasks: The role of the angular gyrus and its relation to mathematical competence. Human Brain Mapping, 30, 2936–2952.CrossRefGoogle Scholar
  33. Gruber, O., Indefrey, P., Steinmetz, H., & Kleinschmidt, A. (2001). Dissociating neural correlates of cognitive components in mental calculation. Cerebral Cortex, 11(4), 350–359.  https://doi.org/10.1093/cercor/11.4.350 CrossRefPubMedGoogle Scholar
  34. Hauser, M. D. (2000). Homologies for numerical memory span? Trends in Cognitive Sciences, 4(4), 127–128.  https://doi.org/10.1016/S1364-6613(00)01473-X CrossRefPubMedGoogle Scholar
  35. Ischebeck, A., Schocke, M., & Delazer, M. (2009). The processing and representation of fractions within the brain: An fMRI investigation. Neuroimage, 47(1), 403–413.  https://doi.org/10.1016/j.neuroimage.2009.03.041 CrossRefPubMedGoogle Scholar
  36. Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45(3), 850–867.  https://doi.org/10.1037/a0014939 CrossRefPubMedPubMedCentralGoogle Scholar
  37. Kroeger, L. (2012). Neural correlates of error detection in math facts (Order No. 3554345). Available from ProQuest Dissertations & Theses Global (1315765851). Retrieved from https://search-proquest-com.proxy.libraries.uc.edu/docview/1315765851?accountid=2909
  38. Kucian, K., von Aster, M., Loenneker, T., Dietrich, T., & Martin, E. (2008). Development of neural networks for exact and approximate calculation: A FMRI study. Developmental Neuropsychology, 33(4), 447–473.  https://doi.org/10.1080/87565640802101474 CrossRefPubMedPubMedCentralGoogle Scholar
  39. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8-9-year-old students. Cognition, 93(2), 99–125.  https://doi.org/10.1016/j.cognition.2003.11.004 CrossRefPubMedGoogle Scholar
  40. Locuniak, M. N., & Jordan, N. C. (2008). Using kindergarten number sense to predict calculation fluency in second grade. Journal of Learning Disabilities, 41(5), 451–459.  https://doi.org/10.1177/0022219408321126 CrossRefPubMedPubMedCentralGoogle Scholar
  41. Mason, M. F., Norton, M. I., Van Horn, J. D., Wegner, D. M., Grafton, S. T., & Macrae, C. N. (2007). Wandering minds: The default network and stimulus-independent thought. Science, 315(5810), 393–395.  https://doi.org/10.1126/science.1131295 CrossRefPubMedPubMedCentralGoogle Scholar
  42. Mazzocco, M. M., & Kover, S. T. (2007). A longitudinal assessment of executive function skills and their association with math performance. Child Neuropsychology, 13(1), 18–45.  https://doi.org/10.1080/09297040600611346 CrossRefPubMedPubMedCentralGoogle Scholar
  43. Mazzocco, M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disabilities Research & Practice, 20(3), 142–155.  https://doi.org/10.1111/j.1540-5826.2005.00129.x CrossRefGoogle Scholar
  44. McKeown, M. J., Makeig, S., Brown, G. G., Jung, T., Kindermann, S. S., Bell, A. J., & Sejnowski, T. J. (1998). Analysis of fMRI data by blind separation into independent spatial components. Human Brain Mapping, 6(3), 160–188.  https://doi.org/10.1002/(SICI)1097-0193(1998)6:3<160::AID-HBM5>3.0.CO;2-1 CrossRefPubMedGoogle Scholar
  45. McKiernan, K. A., D’Angelo, B. R., Kaufman, J. N., & Binder, J. R. (2006). Interrupting the “stream of consciousness”: An fMRI investigation. Neuroimage, 29(4), 1185–1191.  https://doi.org/10.1016/j.neuroimage.2005.09.030 CrossRefPubMedPubMedCentralGoogle Scholar
  46. Naccache, L., & Dehaene, S. (2001). The priming method: Imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. Cerebral Cortex, 11(10), 966–974.  https://doi.org/10.1093/cercor/11.10.966 CrossRefPubMedGoogle Scholar
  47. Passolunghi, M. C., Vercelloni, B., & Schadee, H. (2007). The precursors of mathematics learning: Working memory, phonological ability and numerical competence. Cognitive Development, 22(2), 165–184.  https://doi.org/10.1016/j.cogdev.2006.09.001 CrossRefGoogle Scholar
  48. Pinel, P., Dehaene, S., Rivière, D., & Le Bihan, D. (2001). Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14(5), 1013–1026.  https://doi.org/10.1006/nimg.2001.0913 CrossRefPubMedGoogle Scholar
  49. Raghubar, K., Cirino, P., Barnes, M., Ewing-Cobbs, L., Fletcher, J., & Fuchs, L. (2009). Errors in multi-digit arithmetic and behavioral inattention in children with math difficulties. Journal of Learning Disabilities, 42, 356–371.CrossRefGoogle Scholar
  50. Raichle, M. E., & Snyder, A. Z. (2007). A default mode of brain function: A brief history of an evolving idea. Neuroimage, 37(4), 1083–1090.  https://doi.org/10.1016/j.neuroimage.2007.02.041 CrossRefPubMedGoogle Scholar
  51. Reigosa-Crespo, V., Valdés-Sosa, M., Butterworth, B., Estévez, N., Rodríguez, M., Santos, E., … Lage, A. (2012). Basic numerical capacities and prevalence of developmental dyscalculia: The Havana Survey. Developmental Psychology, 48(1), 123–135.  https://doi.org/10.1037/a0025356 CrossRefPubMedGoogle Scholar
  52. Restle, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83(2, Pt. 1), 274–278.  https://doi.org/10.1037/h0028573 CrossRefGoogle Scholar
  53. Rubinsten, O., & Henik, A. (2009). Developmental dyscalculia: Heterogeneity might not mean different mechanisms. Trends in Cognitive Sciences, 13(2), 92–99.  https://doi.org/10.1016/j.tics.2008.11.002 CrossRefPubMedGoogle Scholar
  54. Sato, J. R., Salum, G. A., Gadelha, A., Picon, F. A., Pan, P. M., Vieira, G., … Jackowski, A. P. (2014). Age effects on the default mode and control networks in typically developing children. Journal of Psychiatric Research, 58, 89–95.  https://doi.org/10.1016/j.jpsychires.2014.07.004 CrossRefPubMedPubMedCentralGoogle Scholar
  55. Schmithorst, V. J., & Brown, R. D. (2004). Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group independent component analysis of the mental addition and subtraction of fractions. Neuroimage, 22, 1414–1420. Retrieved from http://dx.doi.org.proxy.libraries.uc.edu/10.1016/j.neuroimage.2004.03.021 CrossRefGoogle Scholar
  56. Simon, O., Mangin, J.-F., Cohen, L., Le Bihan, D., & Dehaene, S. (2002). Topographical layout of hand, eye, calculation, and language-related areas in the human parietal lobe. Neuron, 33(3), 475–487.  https://doi.org/10.1016/S0896-6273(02)00575-5 CrossRefPubMedGoogle Scholar
  57. Stanescu-Cosson, R., Pinel, P., van De Moortele, P., Le Bihan, D., Cohen, L., & Dehaene, S. (2000). Understanding dissociations in dyscalculia. A brain imaging study of the impact of number size on the cerebral networks for exact and approximate calculation. Brain, 123(11), 2240–2255.  https://doi.org/10.1093/brain/123.11.2240 CrossRefPubMedGoogle Scholar
  58. Swanson, H. L., Jerman, O., & Zheng, X. (2008). Growth in working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 100(2), 343–379.  https://doi.org/10.1037/0022-0663.100.2.343 CrossRefGoogle Scholar
  59. Torbeyns, J., De Smedt, B., Peters, G., Ghesquière, P., & Verschaffel, L. (2011). Use of indirect addition in adults’ mental subtraction in the number domain up to 1,000. British Journal of Psychology, 102(3), 585–597.  https://doi.org/10.1111/j.2044-8295.2011.02019.x CrossRefPubMedGoogle Scholar
  60. van Eimeren, L., Grabner, R. H., Koschutnig, K., Reishofer, G., Ebner, F., & Ansari, D. (2010). Structure-function relationships underlying calculation: A combined diffusion tensor imaging and fMRI study. Neuroimage, 52(1), 358–363.  https://doi.org/10.1016/j.neuroimage.2010.04.001 CrossRefPubMedGoogle Scholar
  61. Weddell, R. A., & Davidoff, J. B. (1991). A dyscalculic patient with selectively impaired processing of the numbers 7, 9, and 0. Brain and Cognition, 17(2), 240–271.  https://doi.org/10.1016/0278-2626(91)90076-K CrossRefPubMedGoogle Scholar
  62. Woodcock, R. W., McGrew, K. S., & Mather, N. (2001). Woodcock-Johnson III tests of achievement. Itasca, IL: Riverside Publishing.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Rhonda Douglas Brown
    • 1
  • Vincent J. Schmithorst
  • Lori Kroeger
  1. 1.Developmental & Learning Sciences Research CenterSchool of Education, University of CincinnatiCincinnatiUSA

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