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Counting and Basic Numerical Skills

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Abstract

The following chapter outlines a typical developmental trajectory of children’s early number knowledge and counting skills. Using a series of anecdotal demonstrations of a young child’s emergent knowledge as a guide, the chapter first outlines the conceptual and procedural building blocks for counting and basic numerical skills, proceeds to an extended discussion of major conceptual achievements in counting, and concludes with a review of our emerging understanding on how to best support and facilitate the development of these skills. Throughout each of these sections, seminal studies are discussed to more clearly demonstrate the role of children’s intuitive number sense in the construction of natural-number concepts, specific challenges that children confront as they acquire the verbal count list (including several conceptual and linguistic obstacles that are often overlooked in early childhood curricula and assessments), and the effectiveness of low-cost, practical interventions that can be adopted by educators and parents to support and facilitate development.

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Notes

  1. 1.

    The term “subitize” also enjoys many definitions across early childhood curricula and assessment. The present chapter, however, will adopt and adhere to the definition provided above.

  2. 2.

    Note that approximate quantifiers such as “more” and “a lot” can take a wide range of referents, with few constraints, while number words refer only to collections of discrete individuals.

  3. 3.

    Children in the other conditions received either a set of ethnically diverse dolls, dress-up clothes, or wooden blocks.

References

  • Arnold, D. H., Fisher, P. H., Doctoroff, G. L., & Dobbs, J. (2002). Accelerating math development in head start classrooms. Journal of Educational Psychology, 94, 762–770.

    Article  Google Scholar 

  • Aunio, P., & Niemivirta, M. (2010). Predicting children’s mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20, 427–435.

    Article  Google Scholar 

  • Barner, D., Thalwitz, D., Wood, J., & Carey, S. (2007). On the relation between the acquisition of singular-plural morpho-syntax and the conceptual distinction between one and more than one. Developmental Science, 10, 365–373.

    Article  Google Scholar 

  • Bartelet, D., Vaessen, A., Blomert, L., & Ansari, D. (2014). What basic number processing measures in kindergarten explain unique variability in first-grade arithmetic proficiency? Journal of Experimental Child Psychology, 117, 12–28.

    Article  Google Scholar 

  • Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement in school. Science, 350, 196–198.

    Article  Google Scholar 

  • Bloom, P., & Wynn, K. (1997). Linguistic cues in the acquisition of number words. Journal of Child Language, 24, 511–533.

    Article  Google Scholar 

  • Carey, S. (2010). The origin of concepts. New York: Oxford University Press.

    Google Scholar 

  • Carey, S., Shusterman, A., Haward, P., & Distefano, R. (2017). Do analog number representations underlie the meanings of young children’s verbal numbers? Cognition, 168, 243–255.

    Article  Google Scholar 

  • Chu, F., van Marle, K., & Geary, D. (2015). Early numerical foundations of young children’s mathematical development. Journal of Experimental Child Psychology, 132, 205–212.

    Article  Google Scholar 

  • Clearfield, M. W., & Mix, K. S. (1999). Number versus contour length in infants’ discrimination of small visual sets. Psychological Science, 10, 408–411.

    Article  Google Scholar 

  • Cordes, S., Gelman, R., Gallistel, C. R., & Whalen, J. (2001). Variability signatures distinguish verbal from nonverbal counting for both large and small numbers. Psychonomic Bulletin & Review, 8, 698–707.

    Article  Google Scholar 

  • Dale, P. S., & Fenson, L. (1996). Lexical development norms for young children. Behavior Research Methods, Instruments, & Computers, 28, 125–127.

    Article  Google Scholar 

  • Dantzig, T. (1967). Number: The language of science. New York: Free Press.

    Google Scholar 

  • Davidson, K., Eng, K., & Barner, D. (2012). Does learning to count involve a semantic induction? Cognition, 123, 162–173.

    Article  Google Scholar 

  • Dehaene, S. (2011). The number sense: How the mind creates mathematics. Oxford, England: Oxford University Press.

    Google Scholar 

  • Dowker, A. (2008). Individual differences in numerical abilities in preschoolers. Developmental Science, 11, 650–654.

    Article  Google Scholar 

  • Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., et al. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428.

    Article  Google Scholar 

  • Feigenson, L., & Carey, S. (2003). Tracking individuals via object files: Evidence from infants’ manual search. Developmental Science, 6, 568–584.

    Article  Google Scholar 

  • Feigenson, L., Dehaene, S., & Spelke, E. S. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314.

    Article  Google Scholar 

  • Frank, M. C., Everett, D. L., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108, 819–824.

    Article  Google Scholar 

  • Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer Science & Business Media.

    Book  Google Scholar 

  • Gelman, R., & Gallistel, C. (1978). Young Children’s understanding of numbers. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33, 18–28.

    Article  Google Scholar 

  • Ginsburg, H., & Baroody, A. (2003). Test of early math ability (3rd ed.). Austin, TX: Pro-Ed.

    Google Scholar 

  • Göbel, S. M., Watson, S. E., Lervåg, A., & Hulme, C. (2014). Children’s arithmetic development: It is number knowledge, not the approximate number sense, that counts. Psychological Science, 1–10.

    Google Scholar 

  • Goldman, M. C., Negen, J., & Sarnecka, B. W. (2014). Are bilingual children better at ignoring perceptually misleading information? A novel test. Developmental Science, 17, 956–964.

    Article  Google Scholar 

  • Greenes, C., Ginsburg, H. P., & Balfanz, R. (2004). Big math for little kids. Early Childhood Research Quarterly, 19, 159–166.

    Article  Google Scholar 

  • Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14, 1021–1032.

    Article  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, 1457–1465.

    Article  Google Scholar 

  • Halberda, J., Ly, R., Wilmer, J., Naiman, D., & Germine, L. (2012). Number sense across the lifespan as revealed by massive internet-based sample. Proceedings of the National Academy of Sciences, 109, 11116–11120.

    Article  Google Scholar 

  • Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665–668.

    Article  Google Scholar 

  • Hespos, S. J., Ferry, A., & Rips, L. (2009). Five-month-old infants have different expectations for solids and substances. Psychological Science, 20, 603–611.

    Article  Google Scholar 

  • Huang, Y. T., Spelke, E., & Snedeker, J. (2010). When is four far more than three? Children’s generalization of newly acquired number words. Psychological Science, 21, 600–606.

    Article  Google Scholar 

  • Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45, 850.

    Article  Google Scholar 

  • Jordan, N. C., & Levine, S. C. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15, 60–68.

    Article  Google Scholar 

  • Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. The American Journal of Psychology, 62, 498–525.

    Article  Google Scholar 

  • Kritzer, K. L. (2009). Barely started and already left behind: A descriptive analysis of the mathematics ability demonstrated by young deaf children. Journal of Deaf Studies and Deaf Education, 1–13.

    Google Scholar 

  • Le Corre, M., Van de Walle, G., Brannon, E. M., & Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52, 130–169.

    Article  Google Scholar 

  • Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46, 1309–1319.

    Article  Google Scholar 

  • Li, P., Le Corre, M., Shui, R., Jia, G., & Carey, S. (2003). Effects of plural syntax on number word learning: A cross-linguistic study. In Proceedings of the 28th Boston University Conference on Language Development. Somerville, MA: Cascadilla Press.

    Google Scholar 

  • Libertus, M., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14, 1292–1300.

    Article  Google Scholar 

  • Libertus, M., Odic, D., & Halberda, J. (2012). Intuitive sense of number correlates with math scores on college-entrance examination. Acta Psychologica, 141, 373–379.

    Article  Google Scholar 

  • Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense: Large number discrimination in human infants. Psychological Science, 15, 396–401.

    Article  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.

    Article  Google Scholar 

  • Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111, 1–22.

    Article  Google Scholar 

  • Mix, K., Huttenlocher, J., & Levine, S. (2002). Quantitative development in infancy and early childhood. New York: Oxford University Press.

    Book  Google Scholar 

  • Mix, K., Sandhofer, C., Moore, J., & Russell, C. (2012). Acquisition of the cardinal word principle: The role of input. Early Childhood Research Quarterly, 27, 274–283.

    Article  Google Scholar 

  • Muldoon, K., Lewis, C., & Freeman, N. (2009). Why set-comparison is vital in early number learning. Trends in Cognitive Sciences, 13, 203–208.

    Article  Google Scholar 

  • Odic, D., Libertus, M., Feigenson, L., & Halberda, J. (2013). Developmental change in the acuity of approximate number and area representations. Developmental Psychology, 49, 1103.

    Article  Google Scholar 

  • Paliwal, V., & Baroody, A. (2017). How best to teach the cardinality principle? Paper presented at the American Education Research Association, San Antonio, TX.

    Google Scholar 

  • Piazza, M., Pica, P., Izard, V., Spelke, E., & Dehaene, S. (2013). Education enhances the acuity of the nonverbal approximate number system. Psychological Science, 24, 1037–1043.

    Article  Google Scholar 

  • Sarnecka, B., Goldman, M., & Slusser, E. (2014). How counting forms the basis for children’s first representations of the natural numbers. In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 291–309). Oxford, UK: Oxford University Press.

    Google Scholar 

  • Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662–674.

    Article  Google Scholar 

  • Sarnecka, B. W., & Gelman, S. A. (2004). Six does not just mean a lot: Preschoolers see number words as specific. Cognition, 92, 329–335.

    Article  Google Scholar 

  • Sarnecka, B. W., Kamenskaya, V., Yamana, Y., Ogura, T., & Yudovina, Y. (2007). From grammatical number to exact numbers: Early meanings of ‘one’, ‘two’, and ‘three’ in English, Russian, and Japanese. Cognitive Psychology, 55, 136–168.

    Article  Google Scholar 

  • Sarnecka, B. W., & Lee, M. D. (2009). Levels of number knowledge in early childhood. Journal of Experimental Child Psychology, 103, 325–337.

    Article  Google Scholar 

  • Sarnecka, B. W., & Wright, C. E. (2013). The exact-numbers idea: Children’s understanding of cardinality and equinumerosity. Cognitive Science, 37, 1493–1506.

    Article  Google Scholar 

  • Shusterman, A., Slusser, E., Halberda, J., & Odic, D. (2016). Acquisition of the cardinal principle coincides with improvement in approximate number system acuity in preschoolers. PLoS One, 11, 1–22.

    Article  Google Scholar 

  • Siegler, R. S., & Ramani, G. B. (2008). Playing linear numerical board games promotes low-income children’s numerical development. Developmental Science, 11, 655–661.

    Article  Google Scholar 

  • Slusser, E. (2010). The development of number concepts: Discrete quantification and numerosity. (Doctoral dissertation). University of California, Irvine, CA.

    Google Scholar 

  • Slusser, E., Chase, E., Berkowitz, T., George, E., Swee, M., Cho, D., Barth, H., et al. (2013). The power of play: Promoting preschoolers’ social and numerical development through independent play with toys. Poster presented at the Biennial Meeting for the Society for Research in Child Development, Seattle, WA.

    Google Scholar 

  • Slusser, E., Ditta, A., & Sarnecka, B. (2013). Connecting numbers to discrete quantification: A step in the child’s construction of integer concepts. Cognition, 129, 31–41.

    Article  Google Scholar 

  • Ribner, A., Shusterman, A., & Slusser, E. (2015). Preschool indicators of primary school math ability. Poster presented at the Biennial Meeting of the Society for Research in Child Development, Philadelphia, PA.

    Google Scholar 

  • Slusser, E., & Sarnecka, B. (2011). A picture of eight turtles: The child’s understanding of cardinality and numerosity. Journal of Experimental Child Psychology, 110, 38–51.

    Article  Google Scholar 

  • Slusser, E., Stoop, T., Lo, A., & Shusterman, A. (2017). Children’s use of newly acquired number words in novel contexts. Poster presented at the Biennial Meeting for the Society for Research in Child Development, Austin, TX.

    Google Scholar 

  • Spaepen, E., Coppola, M., Spelke, E., Carey, S., & Goldin-Meadow, S. (2011). Number without a language model. Proceedings of the National Academy of Sciences, 108, 3163–3168.

    Article  Google Scholar 

  • Starkey, P., & Cooper, R. G. (1980). Perception of numbers by human infants. Science, 28, 1033–1035.

    Article  Google Scholar 

  • Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19, 99–120.

    Article  Google Scholar 

  • Tosto, M. G., Petrill, S. A., Halberda, J., Trzaskowski, M., Tikhomirova, T. N., Bogdanova, O. Y., et al. (2014). Why do we differ in number sense? Evidence from a genetically sensitive investigation. Intelligence, 43, 35–46.

    Article  Google Scholar 

  • Trick, L., & Pylyshyn, Z. (1993). Why are small and large numbers enumerated differently? A limited-capacity preattentive stage in vision. Psychological Review, 101, 80–102.

    Article  Google Scholar 

  • van Marle, 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.

    Article  Google Scholar 

  • Wagner, J. B., & Johnson, S. C. (2011). An association between understanding cardinality and analog magnitude representations in preschoolers. Cognition, 119, 10–22.

    Article  Google Scholar 

  • Wang, H., Odic, D., Halberda, J., & Feigenson, L. (2016). Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology, 147, 82–99.

    Article  Google Scholar 

  • Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193.

    Article  Google Scholar 

  • Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251.

    Article  Google Scholar 

  • Wynn, K. (1996). Infants’ individuation and enumeration of actions. Psychological Science, 7, 165–169.

    Article  Google Scholar 

  • Xu, F. (2003). Numerosity discrimination in infants: Evidence for two systems of representations. Cognition, 89, B15–B25.

    Article  Google Scholar 

  • Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month old infants. Cognition, 74, B1–B11.

    Article  Google Scholar 

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Slusser, E. (2019). Counting and Basic Numerical Skills. In: Fritz, A., Haase, V.G., Räsänen, P. (eds) International Handbook of Mathematical Learning Difficulties. Springer, Cham. https://doi.org/10.1007/978-3-319-97148-3_31

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