Abstract
In this chapter, we draw on evolutionary developmental psychology theory and Dehaene and colleagues’ triple-code model to describe quantity representation, which is the basis for a set of numerical abilities selected during evolution, including numerosity, which involves quickly determining the quantity of a set without counting, and ordinality, which involves recognizing that one set contains more than another without counting. We present research using innovative behavioral and cognitive neuroscience methods indicating that sensitivity to magnitude is present at birth and increases in precision into adulthood, including work investigating two quantity representation systems: the Parallel Individuation (PI) system that allows humans to precisely track a small number of individual objects through space and time; and the Approximate Number system, or number sense, that allows humans to approximate the numerosities of sets of items without using symbols. Research establishing a relationship between quantity representation and mathematics achievement during childhood and adolescence is also described. We present results from a functional Magnetic Resonance Imaging (fMRI) study demonstrating that brain activation in the inferior occipital gyrus, lingual gyrus, and bilateral intraparietal sulcus (IPS) during magnitude comparison is positively related to adolescents’ mathematics achievement, whereas deactivation of the Default Mode Network (DMN) during magnitude comparison is negatively related to adolescents’ mathematics achievement, indicating that abstract quantity representation may be foundational for the development of calculation skills.
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References
Antell, S. E., & Keating, D. P. (1983). Perception of numerical invariance in neonates. Child Development, 54(3), 695–701. https://doi.org/10.2307/1130057
Barth, H., Kanwisher, N., & Spelke, E. (2003). The construction of large number representations in adults. Cognition, 86(3), 201–221. https://doi.org/10.1016/S0010-0277(02)00178-6
Bonny, J. W., & Lourenco, S. F. (2013). The approximate number system and its relation to early math achievement: Evidence from the preschool years. Journal of Experimental Child Psychology, 114(3), 375–388. https://doi.org/10.1016/j.jecp.2012.09.015
Byars, A. W., Holland, S. K., Strawsburg, R. H., Bommer, W., Dunn, R. S., Schmithorst, V. J., & Plante, E. (2002). Practical aspects of conducting large-scale functional magnetic resonance imaging studies in children. Journal of Child Neurology, 17(12), 885–890. https://doi.org/10.1177/08830738020170122201
Cantlon, J. F., Brannon, E. M., Carter, E. J., & Pelphrey, K. A. (2006). Functional imaging of numerical processing in adults and 4-y-old children. PLoS Biology, 4(5), e125. https://doi.org/10.1371/journal.pbio.0040125
Cantrell, L., & Smith, L. B. (2013). Open questions and a proposal: A critical review of the evidence on infant numerical abilities. Cognition, 128(3), 331–352. https://doi.org/10.1016/j.cognition.2013.04.008
Carey, S. (2009). The origin of concepts. New York, NY: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780195367638.001.0001
Chen, Q., & Li, J. (2014). Association between individual differences in nonsymbolic number acuity and math performance: A meta-analysis. Acta Psychologica, 148, 163–172.
Chu, F. W., vanMarle, K., & Geary, D. C. (2015). Early numerical foundations of young children’s mathematical development. Journal of Experimental Child Psychology, 132, 205–212. https://doi.org/10.1016/j.jecp.2015.01.006
Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1), 87–185. https://doi.org/10.1017/S0140525X01003922
Davis, N., Cannistraci, C. J., Rogers, B. P., Gatenby, J. C., Fuchs, L. S., Anderson, A. W., & Gore, J. C. (2009). Aberrant functional activation in school age children at-risk for mathematical disability: A functional imaging study of simple arithmetic skill. Neuropsychologia, 47(12), 2470–2479. https://doi.org/10.1016/j.neuropsychologia.2009.04.024
De Smedt, B., Noël, M. P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2, 48–55. https://doi.org/10.1016/j.tine.2013.06.001
Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1–2), 1–42. https://doi.org/10.1016/0010-0277(92)90049-N
Dehaene, S. (2011). The number sense: How the mind creates mathematics (Rev. ed.). New York, NY: Oxford University Press.
Dehaene, S., & Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120.
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
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
Edwards, L. A., Wagner, J. B., Simon, C. E., & Hyde, D. C. (2016). Functional brain organization for number processing in pre-verbal infants. Developmental Science, 19(5), 757–769. https://doi.org/10.1111/desc.12333
Emerson, R. W., & Cantlon, J. F. (2012a). Early math achievement and functional connectivity in the fronto-parietal network. Developmental Cognitive Neuroscience, 2(Suppl. 1), S139–S151. https://doi.org/10.1016/j.dcn.2011.11.003
Emerson, R. W., & Cantlon, J. F. (2012b). ‘Early math achievement and functional connectivity in the fronto-parietal network’: Erratum. Developmental Cognitive Neuroscience, 2(2), 291.
Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. https://doi.org/10.1016/j.jecp.2014.01.013
Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6(5), 568–584. https://doi.org/10.1111/1467-7687.00313
Feigenson, L., & Carey, S. (2005). On the limits of infants’ quantification of small object arrays. Cognition, 97(3), 295–313. https://doi.org/10.1016/j.cognition.2004.09.010
Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants’ choice of more: Object files versus analog magnitudes. Psychological Science, 13(2), 150–156. https://doi.org/10.1111/1467-9280.00427
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://doi.org/10.1016/j.tics.2004.05.002
Féron, J., Gentaz, E., & Streri, A. (2006). Evidence of amodal representation of small numbers across visuo-tactile modalities in 5-month-old infants. Cognitive Development, 21(2), 81–92. https://doi.org/10.1016/j.cogdev.2006.01.005
Fuhs, M. W., & McNeil, N. M. (2013). ANS acuity and mathematics ability in preschoolers from low-income homes: Contributions of inhibitory control. Developmental Science, 16(1), 136–148. https://doi.org/10.1111/desc.12013
Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59–65. https://doi.org/10.1016/S1364-6613(99)01424-2
Gallistel, C. R., & Gelman, R. (2005). Mathematical cognition. In K. J. Holyoak & R. G. Morrison (Eds.), The Cambridge handbook of thinking and reasoning (pp. 559–588). New York, NY: Cambridge University Press.
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
Geary, D. C. (2005). The origin of mind: Evolution of brain, cognition, and general intelligence. Washington, DC: American Psychological Association. https://doi.org/10.1037/10871-000
Gilmore, C. C., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N., … Inglis, M. (2013). Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PLoS One, 8, e67374. https://doi.org/10.1371/journal.pone.0067374
Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2010). Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition, 115(3), 394–406. https://doi.org/10.1016/j.cognition.2010.02.002
Ginsburg, H., & Baroody, A. (2003). TEMA-3 examiners manual (3rd ed.). Austin, TX: PRO-ED.
Haist, F., Wazny, J. H., Toomarian, E., & Adamo, M. (2015). Development of brain systems for nonsymbolic numerosity and the relationship to formal math academic achievement. Human Brain Mapping, 36(2), 804–826. https://doi.org/10.1002/hbm.22666
Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the ‘number sense’: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44(5), 1457–1465. https://doi.org/10.1037/a0012682
Halberda, J., Ly, R., Wilmer, J. B., Naiman, D. Q., & Germine, L. (2012). Number sense across the lifespan as revealed by a massive Internet-based sample. Proceedings of the National Academy of Sciences of the United States of America, 109(28), 11116–11120. https://doi.org/10.1073/pnas.1200196109
Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455(7213), 665+. Retrieved from http://go.galegroup.com.proxy.libraries.uc.edu/ps/i.do?p=EAIM&sw=w&u=ucinc_main&v=2.1&it=r&id=GALE%7CA188899858&sid=summon&asid=21a895d10d909bd02722a4786fe91bee
Halberda, J., & Odic, D. (2015). The precision and internal confidence of our approximate number thoughts. In D. C. Geary, D. B. Berch, & K. M. Koepke (Eds.), Evolutionary origins and early development of number processing (pp. 305–333). San Diego, CA: Elsevier Academic Press. https://doi.org/10.1016/B978-0-12-420133-0.00012-0
Hyde, D. C. (2011). Two systems of non-symbolic numerical cognition. Frontiers in Human Neuroscience, 5, 150. https://doi.org/10.3389/fnhum.2011.00150
Hyde, D. C., Boas, D. A., Blair, C., & Carey, S. (2010). Near-infrared spectroscopy shows right parietal specialization for number in pre-verbal infants. Neuroimage, 53(2), 647–652. https://doi.org/10.1016/j.neuroimage.2010.06.030
Hyde, D. C., & Mou, Y. (2016). Neural and behavioral signatures of core numerical abilities and early symbolic number development. In D. B. Berch, D. C. Geary, & K. Mann Koepke (Eds.), Development of mathematical cognition: Neural substrates and genetic influences (pp. 51–77). San Diego, CA: Elsevier Academic Press. https://doi.org/10.1016/B978-0-12-801871-2.00003-4
Hyde, D. C., & Spelke, E. S. (2011). Neural signatures of number processing in human infants: Evidence for two core systems underlying numerical cognition. Developmental Science, 14(2), 360–371. https://doi.org/10.1111/j.1467-7687.2010.00987.x
Izard, V., Dehaene-Lambertz, G., & Dehaene, S. (2008). Distinct cerebral pathways for object identity and number in human infants. PLoS Biology, 6(2), e11. https://doi.org/10.1371/journal.pbio.0060011
Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences of the United States of America, 106(25), 10382–10385. https://doi.org/10.1073/pnas.0812142106
Kobayashi, T., Hiraki, K., & Hasegawa, T. (2005). Auditory-visual intermodal matching of small numerosities in 6-month-old infants. Developmental Science, 8(5), 409–419. https://doi.org/10.1111/j.1467-7687.2005.00429.x
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
Lemer, C., Dehaene, S., Spelke, E., & Cohen, L. (2003). Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 41(14), 1942–1958. https://doi.org/10.1016/S0028-3932(03)00123-4
Libertus, M. E., & Brannon, E. M. (2010). Stable individual differences in number discrimination in infancy. Developmental Science, 13(6), 900–906. https://doi.org/10.1111/j.1467-7687.2009.00948.x
Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14(6), 1292–1300. https://doi.org/10.1111/j.1467-7687.2011.01080.x
Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense: Large-number discrimination in human infants. Psychological Science, 14(5), 396–401. https://doi.org/10.1111/1467-9280.01453
Lyons, I. M., Ansari, D., & Beilock, S. L. (2012). Symbolic estrangement: Evidence against a strong association between numerical symbols and the quantities they represent. Journal of Experimental Psychology: General, 141(4), 635–641. https://doi.org/10.1037/a0027248
Lyons, I. M., Price, G. R., Vaessen, A., Blomert, L., & Ansari, D. (2014). Numerical predictors of arithmetic success in grades 1–6. Developmental Science, 17(5), 714–726. https://doi.org/10.1111/desc.12152
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
Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9), e23749. https://doi.org/10.1371/journal.pone.0023749
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
Mix, K. S., Levine, S. C., & Huttenlocher, J. (1997). Numerical abstraction in infants: Another look. Developmental Psychology, 33(3), 423–428. https://doi.org/10.1037/0012-1649.33.3.423
Moore, D., Benenson, J., Reznick, J. S., Peterson, M., & Kagan, J. (1987). Effect of auditory numerical information on infants’ looking behavior: Contradictory evidence. Developmental Psychology, 23(5), 665–670. https://doi.org/10.1037/0012-1649.23.5.665
Mou, Y., & vanMarle, K. (2014). Two core systems of numerical representation in infants. Developmental Review, 34(1), 1–25. https://doi.org/10.1016/j.dr.2013.11.001
Park, J., Li, R., & Brannon, E. M. (2014). Neural connectivity patterns underlying symbolic number processing indicate mathematical achievement in children. Developmental Science, 17(2), 187–202. https://doi.org/10.1111/desc.12114
Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., … Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116(1), 33–41. https://doi.org/10.1016/j.cognition.2010.03.012
Revkin, S. K., Piazza, M., Izard, V., Cohen, L., & Dehaene, S. (2008). Does subitizing reflect numerical estimation? Psychological Science, 19(6), 607–614. https://doi.org/10.1111/j.1467-9280.2008.02130.x
Sasanguie, D., De Smedt, B., Defever, E., & Reynvoet, B. (2012). Association between basic numerical abilities and mathematics achievement. British Journal of Developmental Psychology, 30(2), 344–357. https://doi.org/10.1111/j.2044-835X.2011.02048.x
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
Sharon, T., & Wynn, K. (1998). Individuation of actions from continuous motion. Psychological Science, 9(5), 357–362. https://doi.org/10.1111/1467-9280.00068
Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. https://doi.org/10.1111/cdep.12077
Starkey, P., & Cooper, R. G. (1980). Perception of numbers by human infants. Science, 210(4473), 1033–1035. https://doi.org/10.1126/science.7434014
Starkey, P., Spelke, E. S., & Gelman, R. (1983). Detection of intermodal numerical correspondences by human infants. Science, 222(4620), 179–181. https://doi.org/10.1126/science.6623069
Starkey, P., Spelke, E. S., & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36(2), 97–127. https://doi.org/10.1016/0010-0277(90)90001-Z
Starr, A., & Brannon, E. M. (2015). Evolutionary and developmental continuities in numerical cognition. In D. C. Geary, D. B. Berch, & K. M. Koepke (Eds.), Evolutionary origins and early development of number processing (pp. 123–144). San Diego, CA: Elsevier Academic Press. https://doi.org/10.1016/B978-0-12-420133-0.00005-3
Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences of the United States of America, 110(45), 18116–18120. https://doi.org/10.1073/pnas.1302751110
van Loosbroek, E., & Smitsman, A. W. (1990). Visual perception of numerosity in infancy. Developmental Psychology, 26(6), 916–922. https://doi.org/10.1037/0012-1649.26.6.911.b
vanMarle, K. (2013). Infants use different mechanisms to make small and large number ordinal judgments. Journal of Experimental Child Psychology, 114(1), 102–110. https://doi.org/10.1016/j.jecp.2012.04.007
vanMarle, K., Chu, F., Li, Y., & Geary, D. C. (2014). Acuity of the approximate number system and preschoolers’ quantitative development. Developmental Science, 17(4), 492–505. Retrieved from http://dx.doi.org.proxy.libraries.uc.edu/10.1111/desc.12143
vanMarle, K., & Wynn, K. (2011). Tracking and quantifying objects and non-cohesive substances. Developmental Science, 14(3), 502–515. https://doi.org/10.1111/j.1467-7687.2010.00998.x
Wilcox, T., Haslup, J. A., & Boas, D. A. (2010). Dissociation of processing of featural and spatiotemporal information in the infant cortex. Neuroimage, 53(4), 1256–1263. https://doi.org/10.1016/j.neuroimage.2010.06.064
Wilcox, T., Stubbs, J., Hirshkowitz, A., & Boas, D. A. (2012). Functional activation of the infant cortex during object processing. Neuroimage, 62(3), 1833–1840. https://doi.org/10.1016/j.neuroimage.2012.05.039
Wood, J. N., & Spelke, E. S. (2005). Infants’ enumeration of actions: Numerical discrimination and its signature limits. Developmental Science, 8(2), 173–181. https://doi.org/10.1111/j.1467-7687.2005.00404.x
Woodcock, R. W., McGrew, K. S., & Mather, N. (2001). Woodcock-Johnson III tests of achievement. Itasca, IL: Riverside Publishing.
Wynn, K. (1996). Infants’ individuation and enumeration of actions. Psychological Science, 7(3), 164–169. https://doi.org/10.1111/j.1467-9280.1996.tb00350.x
Wynn, K., Bloom, P., & Chiang, W. (2002). Enumeration of collective entities by 5-month-old infants. Cognition, 83(3), B55–B62. https://doi.org/10.1016/S0010-0277(02)00008-2
Xu, F. (2003). Numerosity discrimination in infants: Evidence for two systems of representations. Cognition, 89(1), B15–B25. https://doi.org/10.1016/S0010-0277(03)00050-7
Xu, F., & Arriaga, R. I. (2007). Number discrimination in 10-month-old infants. British Journal of Developmental Psychology, 25(1), 103–108. https://doi.org/10.1348/026151005X90704
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11. https://doi.org/10.1016/S0010-0277(99)00066-9
Xu, F., Spelke, E. S., & Goddard, S. (2005). Number sense in human infants. Developmental Science, 8(1), 88–101. https://doi.org/10.1111/j.1467-7687.2005.00395.x
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Brown, R.D., Schmithorst, V.J. (2018). Quantity Representation. In: Neuroscience of Mathematical Cognitive Development. Springer, Cham. https://doi.org/10.1007/978-3-319-76409-2_3
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