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10 - Foundational Considerations

Does Primitive Number Sense Provide a Foothold for Learning Formal Mathematics?

from Subpart II.1 - Infancy: The Roots of Human Thinking

Published online by Cambridge University Press:  24 February 2022

By
Kristy vanMarle
Edited by
Olivier Houdé  and
Grégoire Borst
Olivier Houdé
Affiliation:
Université de Paris V
Grégoire Borst
Affiliation:
Université de Paris V
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Summary

Children with below average school entry mathematics knowledge are likely to remain behind their peers throughout schooling, and go on to face poor employment prospects and low wage-earning potential as adults. Previous research has identified key skills (number system knowledge) in the first-grade that predict math achievement ten years later. Recent studies have tried to identify early-emerging skills in infancy and during the preschool years that may provide a foundation upon which the formal skills they learn in school can be built. One candidate skill is the non-verbal ability to estimate and reason about numerosities that depends on an evolutionary ancient, and phylogenetically common, mechanism for representing quantities – the approximate number system (ANS). While there is evidence that ANS acuity (precision) predicts math achievement concurrently and prospectively, several studies have failed to find a link, or identified other quantitative skills (e.g., ordinal knowledge) or domain-general skills (e.g., intelligence, executive function, working memory) that are better predictors. Evidence for and against the claim that the ANS is a foundational skill that is critical for the development of formal mathematical knowledge is reviewed, followed by a synthesis of work by Geary and colleagues in which they examined the growth of many different domain-specific (quantitative) and domain-general skills longitudinally over four years. The findings suggest that the link between ANS acuity and math achievement is mediated by other skills, especially cardinal knowledge. The ANS still remains a significant factor, however, because our findings also suggest that it plays an important role in the acquisition of cardinal knowledge, and the conceptual insight children appear to have when they first come to understand counting and the natural numbers.

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The Cambridge Handbook of Cognitive Development , pp. 216 - 241
Publisher: Cambridge University Press
Print publication year: 2022

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References

Baddeley, A. D. (1996). Exploring the central executive. The Quarterly Journal of Experimental Psychology Section A, 49, 5–28.Google Scholar
Baddeley, A. D. (2002). Is working memory still working? European Psychologist, 7, 85–97.CrossRefGoogle Scholar
Baddeley, A. D., & Hitch, G. (1974). Working memory. In Bower, G. H. (ed.), The Psychology of Learning and Motivation: Advances in Research and Theory (Vol. 8, pp. 47–89). New York: Academic Press.Google Scholar
Blair, C., & Razza, R. P. (2007). Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Child Development, 78, 647–663.Google Scholar
Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607–618.CrossRefGoogle Scholar
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, 205–228.CrossRefGoogle ScholarPubMed
Bull, R., & Lee, K. (2014). Executive functioning and mathematics achievement. Child Development Perspectives, 8, 36–41.Google Scholar
Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children’s mathematics ability: inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293.Google Scholar
Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46, 3–18.Google Scholar
Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541.CrossRefGoogle ScholarPubMed
Bynner, J. (1997). Basic skills in adolescents’ occupational preparation. The Career Development Quarterly, 45, 305–321.Google Scholar
Carey, S. (2004). Bootstrapping and the origins of concepts. Daedalus, 133, 59–68.Google Scholar
Carey, S. (2009). The Origin of Concepts. New York: Oxford University Press.Google Scholar
Carey, S. (2011). Précis of the origin of concepts. Behavioral and Brain Sciences, 34, 113–124.CrossRefGoogle ScholarPubMed
Cheung, P., Rubenson, M., & Barner, D. (2017). To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count. Cognitive Psychology, 92, 22–36.Google Scholar
Chu, F. W., vanMarle, K., & Geary, D. C. (2013). Quantitative deficits of preschool children at risk for mathematical learning disability. Frontiers in Psychology, 4, 195.CrossRefGoogle ScholarPubMed
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.Google Scholar
Chu, F. W., vanMarle, K., & Geary, D. C. (2016). Predicting children’s reading and mathematics achievement from early quantitative knowledge and domain-general cognitive abilities. Frontiers in Psychology, 7, 1–14.CrossRefGoogle ScholarPubMed
Chu, F. W., vanMarle, K., Rouder, J., & Geary, D. C. (2018). Children’s early understanding of number predicts their later problem-solving sophistication in addition. Journal of Experimental Child Psychology, 169, 73–92.Google Scholar
Clark, C. A. C., Pritchard, V. E., & Woodward, L. J. (2010). Preschool executive functioning abilities predict early mathematics achievement. Developmental Psychology, 46, 1176–1191.CrossRefGoogle ScholarPubMed
Coubart, A., Izard, V., Spelke, E. S., Marie, J., & Streri, A. (2014). Dissociation between small and large numerosities in newborn infants. Developmental Science, 17, 11–22.CrossRefGoogle ScholarPubMed
Cowan, R., Donlan, C., Shepherd, D. L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011). Basic calculation proficiency and mathematics achievement in elementary school children. Journal of Educational Psychology, 103, 786–803.CrossRefGoogle Scholar
Cowan, R., & Powell, D. (2014). The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability. Journal of Educational Psychology, 106, 214–229.CrossRefGoogle ScholarPubMed
Davidson, K., Eng, K., & Barner, D. (2012). Does learning to count involve a semantic induction? Cognition, 123, 162–173.Google Scholar
de Hevia, M. D., Izard, V., Coubart, A., Spelke, E. S., & Streri, A. (2014). Representations of space, time, and number in neonates. Proceedings of the National Academy of Sciences, 111, 4809–4813.Google Scholar
De Smedt, B., & Gilmore, C. K. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108, 278–292.Google Scholar
De Smedt, B., Janssen, R., Bouwens, K., Verschaffel, L., Boets, B., & Ghesquière, P. (2009). Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade. Journal of Experimental Child Psychology, 103, 186–201.Google Scholar
De Visscher, A., & Noël, M.-P. (2014). Arithmetic facts storage deficit: The hypersensitivity-to-interference in memory hypothesis. Developmental Science, 17, 434–442.CrossRefGoogle ScholarPubMed
Deary, I. J., Strand, S., Smith, P., & Fernandes, C. (2007). Intelligence and educational achievement. Intelligence, 35, 13–21.Google Scholar
Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506.Google Scholar
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., … Sexton, H. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446.Google Scholar
Eason, S. H., & Levine, S. C. (2017). Math learning begins at home. Zero to Three, 37, 35–44.Google Scholar
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, 757–769.Google Scholar
Espy, K. A., McDiarmid, M. M., Cwik, M. F., Stalets, M. M., Hamby, A., & Senn, T. E. (2004). The contribution of executive functions to emergent mathematic skills in preschool children. Developmental Neuropsychology, 26, 465–486.CrossRefGoogle ScholarPubMed
Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568–584.Google Scholar
Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants’ choice of more: Object-files versus analog magnitudes. Psychological Science, 13, 150–156.Google Scholar
Feigenson, L., Libertus, M. E., & Halberda, J. (2013). Links between the intuitive sense of number and formal mathematics ability. Child Development Perspectives, 7, 74–79.CrossRefGoogle ScholarPubMed
Friso-van den Bos, I., Van Der Ven, S. H., Kroesbergen, E. H., & Van Luit, J. E. (2013). Working memory and mathematics in primary school children: A meta-analysis. Educational Research Review, 10, 29–44.Google Scholar
Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., Seethaler, P. M., … Schatschneider, C. (2010). Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities? Developmental Psychology, 46, 1731–1746.CrossRefGoogle ScholarPubMed
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, 136–148.Google Scholar
Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43–74.Google Scholar
Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4, 59–65.CrossRefGoogle ScholarPubMed
Gallistel, C. R., & Gelman, R. (2005). Mathematical cognition. In Holyoak, K., & Morrison, R. (eds.), The Cambridge Handbook of Thinking and Reasoning (pp. 559–588). New York: Cambridge University Press.Google Scholar
Gathercole, S. E., & Pickering, S. J. (2000). Working memory deficits in children with low achievements in the national curriculum at 7 years of age. British Journal of Educational Psychology, 70, 177–194.Google Scholar
Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.Google Scholar
Geary, D. C. (2005). The Origin of Mind: Evolution of Brain, Cognition, and General Intelligence. Washington, DC: American Psychological Association.CrossRefGoogle Scholar
Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47, 1539.CrossRefGoogle ScholarPubMed
Geary, D. C., Bailey, D. H., & Hoard, M. K. (2009). Predicting mathematical achievement and mathematical learning disability with a simple screening tool: The number sets test. Journal of Psychoeducational Assessment, 27, 265–279.Google Scholar
Geary, D. C., Brown, S. C., & Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787–797.Google Scholar
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, 1343–1359.CrossRefGoogle ScholarPubMed
Geary, D. C., Hoard, M. K., & Nugent, L. (2012b). Independent contributions of the central executive, intelligence, and in-class attentive behavior to developmental change in the strategies used to solve addition problems. Journal of Experimental Child Psychology, 113, 49–65.Google Scholar
Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS ONE, 8, e54651.CrossRefGoogle ScholarPubMed
Geary, D. C., & Moore, A. M. (2016). Cognitive and brain systems underlying early mathematical development. Progress in Brain Research, 227, 75–103.Google Scholar
Geary, D. C., Nicholas, A., Li, Y., & Sun, J. (2017). Developmental change in the influence of domain-general abilities and domain-specific knowledge on mathematics achievement: An eight-year longitudinal study. Journal of educational Psychology, 109, 680–693.Google Scholar
Geary, D. C., & vanMarle, K. (2016). Young children’s core symbolic and nonsymbolic quantitative knowledge in the prediction of later mathematics achievement. Developmental Psychology, 52, 2130–2144.CrossRefGoogle ScholarPubMed
Geary, D. C., & vanMarle, K. (2018). Growth of symbolic number knowledge accelerates after children understand cardinality. Cognition, 177, 69–78.Google Scholar
Geary, D. C., vanMarle, K., Chu, F. W., Rouder, J., Hoard, M. K., & Nugent, L. (2018). Early conceptual understanding of cardinality predicts superior school-entry number-system knowledge. Psychological Science, 29, 191–205.Google Scholar
Gelman, R. (1972). Logical capacity of very young children: Number invariance rules. Child Development, 43, 75–90.Google Scholar
Gelman, R. (1993). A rational-constructivist account of early learning about numbers and objects. Learning and Motivation, 30, 61–96.Google Scholar
Gelman, R., & Gallistel, C. R. (1978). The Child’s Understanding of Number. Cambridge, MA: Harvard University Press.Google Scholar
Gelman, R., & Greeno, J. G. (1989). On the nature of competence: Principles for understanding in a domain. In Resnick, L. B. (ed.), Knowing and Learning: Issues for a Cognitive Science of Instruction (pp. 125–186). Hillsdale, NJ: Erlbaum.Google Scholar
Gelman, R., & Meck, E. (1983). Preschoolers’ counting: Principles before skill. Cognition, 13, 343–359.Google Scholar
Gilmore, C. K., 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.Google Scholar
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, 394–406.Google Scholar
Ginsburg, H. P., & Baroody, A. J. (2003). Test of Early Mathematical Ability (3rd ed.). Austin, TX: Pro-Ed.Google Scholar
Gunderson, E. A., Spaepen, E., & Levine, S. C. (2015). Approximate number word knowledge before the cardinal principle. Journal of Experimental Child Psychology, 130, 35–55.Google Scholar
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, 1457.CrossRefGoogle ScholarPubMed
Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665.Google Scholar
Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1992). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284–296.CrossRefGoogle Scholar
Hyde, D. C. (2011). Two systems of non-symbolic numerical cognition. Frontiers in Human Neuroscience, 5, 1–8.Google Scholar
Iuculano, T., Tang, J., Hall, C. W., & Butterworth, B. (2008). Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science, 11, 669–680.Google Scholar
Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy Sciences (USA), 106, 10382–10385.Google Scholar
Kahneman, D., Treisman, A., & Gibbs, B. J. (1992). The reviewing of object files: Object specific integration of information. Cognitive Psychology, 24, 174–219.Google Scholar
Klibanoff, R. S., Levine, S. C., Huttenlocher, J., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children’s mathematical knowledge: The effect of teacher “math talk.” Developmental Psychology, 42, 59–69.Google Scholar
Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.Google Scholar
Lee, M. D., & Sarnecka, B. W. (2011). Number-knower levels in young children: Insights from Bayesian modeling. Cognition, 120, 391–402.Google Scholar
Leslie, A. M., Gallistel, C. R., & Gelman, R. (2007). Where integers come from. In Carruthers, P., Laurence, S., & Stich, S. (eds.), The Innate Mind, Vol. 3: Foundations and the Future (pp. 109–138). New York: Oxford University Press.Google Scholar
Leslie, A. M., Gelman, R., & Gallistel, C. R. (2008). The generative basis of natural number concepts. Trends in Cognitive Sciences, 12, 213–218.Google Scholar
Libertus, M. E. (2015). The role of intuitive approximation skills for school math abilities. Mind, Brain, and Education, 9, 112–120.CrossRefGoogle Scholar
Libertus, M. E. (2019). Understanding the link between the approximate number system and math abilities. In Geary, D. C., Berch, D. B., & Mann-Koepke, K. (eds.), Mathematical Cognition and Learning: Cognitive Foundations for Improving Mathematical Learning, (Vol. 5, pp. 91–106). New York: Elsevier.Google Scholar
Libertus, M. E., & Brannon, E. M. (2010). Stable individual differences in number discrimination in infancy. Developmental Science, 13, 900–906.Google Scholar
Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14, 1292–1300.Google Scholar
Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense: Large number discrimination in human infants. Psychological Science, 14, 396–401.Google Scholar
Lyons, I. M., & Beilock, S. L. (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121, 256–261.Google Scholar
Lyons, I. M., & Beilock, S. L. (2013). Ordinality and the nature of symbolic numbers. Journal of Neuroscience, 33, 17052–17061.CrossRefGoogle ScholarPubMed
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, 714–726.Google Scholar
Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Development, 82, 1224–1237.Google Scholar
Moore, A. M., vanMarle, K., & Geary, D. C. (2016). Kindergartners’ fluent processing of symbolic numerical magnitude is predicted by their cardinal knowledge and implicit understanding of arithmetic 2 years earlier. Journal of Experimental Child Psychology, 150, 31–47.Google Scholar
Nosworthy, N., Bugden, S., Archibald, L., Evans, B., & Ansari, D. (2013). A two-minute paper-and-pencil test of symbolic and nonsymbolic numerical magnitude processing explains variability in primary school children’s arithmetic competence. PLoS ONE, 8, e67918.Google Scholar
Parsons, S., & Bynner, J. (1997). Numeracy and employment. Education and Training, 39, 43–51.Google Scholar
Peng, P., & Fuchs, D. (2016). A meta-analysis of working memory deficits in children with learning difficulties: Is there a difference between verbal domain and numerical domain? Journal of Learning Disabilities, 49, 3–20.Google Scholar
Piaget, J. (1952). The Child’s Concept of Number. London: Routledge & Kegan Paul.Google Scholar
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, 33–41.Google Scholar
Price, G. R., Holloway, I., Räsänen, P., Vesterinen, M., & Ansari, D. (2007). Impaired parietal magnitude processing in developmental dyscalculia. Current Biology, 17, R1042–R1043.Google Scholar
Pylyshyn, Z. W., & Storm, R. W. (1988). Tracking multiple independent targets: Evidence for a parallel tracking mechanism. Spatial Vision, 3, 179–197.Google Scholar
Ramani, G. B., Rowe, M. L., Eason, S. H., & Leech, K. A. (2015). Math talk during informal learning activities in Head Start families. Cognitive Development, 35, 15–33.Google Scholar
Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24, 1301–1308.CrossRefGoogle ScholarPubMed
Rivera-Batiz, F. (1992). Quantitative literacy and the likelihood of employment among young adults in the United States. Journal of Human Resources, 27, 313–328.Google Scholar
Rose, H., & Betts, J. R. (2004). The effect of high school courses on earnings. Review of Economics and Statistics, 86, 497–513.Google Scholar
Rousselle, L., & Noël, M.-P. (2007). Basic numerical skills in children with mathematical learning disabilities: A comparison of symbolic vs. non-symbolic number magnitude processing. Cognition, 102, 361–395.Google Scholar
Sasanguie, D., Defever, E., Maertens, B., & Reynvoet, B. (2014). The approximate number system is not predictive for symbolic number processing in kindergarteners. The Quarterly Journal of Experimental Psychology, 67, 271–280.Google Scholar
Scholl, B. J. (2001). Objects and attention: The state of the art. Cognition, 80, 1–46.Google Scholar
Shalev, R. S., Manor, O., & Gross-Tsur, V. (2005). Developmental dyscalculia: A prospective six-year follow-up. Developmental Medicine and Child Neurology, 47, 121–125.Google Scholar
Siegel, L. S., & Ryan, E. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled children. Child Development, 60, 973–980.Google Scholar
Siegler, R. S. (1988). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development, 59, 833–851.Google Scholar
Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444.Google Scholar
Soltész, F., Szűcs, D., & Szűcs, L. (2010). Relationships between magnitude representation, counting and memory in 4-to 7-year-old children: A developmental study. Behavioral and Brain Functions, 6, 13.Google Scholar
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, 110, 18116–18120.Google Scholar
Swanson, H. L., & Sachse-Lee, C. (2001). Mathematical problem solving and working memory in children with learning disabilities: Both executive and phonological processes are important. Journal of Experimental Child Psychology, 79, 294–321.Google Scholar
Vanbinst, K., & De Smedt, B. (2016). Individual differences in children’s mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions. Progress in Brain Research, 227, 105–130.Google Scholar
vanMarle, K. (2013). Infants use different mechanisms to make small and large number ordinal judgments. Journal of Experimental Child Psychology, 114, 102–110.Google Scholar
vanMarle, K., Chu, F. W., Li, Y., & Geary, D. C. (2014). Acuity of the approximate number system and preschoolers’ quantitative development. Developmental Science, 17, 492–505.Google Scholar
vanMarle, K., Chu, F. W., Mou, Y., Seok, J. H., Rouder, J., & Geary, D. C. (2018). Attaching meaning to the number words: Contributions of the object tracking and approximate number systems. Developmental Science, 21, e12495.Google Scholar
Walberg, H. J. (1984). Improving the productivity of America’s schools. Educational Leadership, 41, 19–27.Google Scholar
Wechsler, D. (2001). Wechsler Individual Achievement Test – Abbreviated II. San Antonio, TX: Psychological Corp.Google Scholar
Wechsler, D. (2002). Wechsler Preschool and Primary Scale of Intelligence (3rd ed.). San Antonio, TX: Psychological Corp.Google Scholar
Wood, J. N., & Spelke, E. S. (2005). Infants’ enumeration of actions: Numerical discrimination and its signature limits. Developmental Science, 8, 173–181.Google Scholar
Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193.Google Scholar
Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251.Google Scholar
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1–B11.Google Scholar

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