Early Numeracy: The Transition from Infancy to Early Childhood
Kelly S. Mix, PhD
Michigan State University, USA
Number concepts emerge before formal schooling. Preschool children exhibit verbal skills, such as counting, and basic concepts of equivalence, ordinality, and quantitative transformation. Although researchers agree that these abilities exist in early childhood, they continue to debate when, and by what mechanisms, these abilities emerge. In other words, what are the developmental origins of early numeracy?
Research on numeracy has traditionally focused on verbal counting. However, the notion that numeracy might emerge in infancy and toddlerhood shifted the focus toward nonverbal abilities. This shift expanded the range of behaviours included in early numeracy, a change that has direct implications for early childhood education and assessment. This shift also raised questions about the developmental origins of math disabilities and gaps in mathematical achievement, such as those associated with difference socioeconomic groups.
Current developmental accounts differ in the weight given to nonverbal versus verbal representations.
Some argue the core conceptual structure for number is inborn and takes the form of a nonverbal representation that is similar to verbal counting.1,2,3 On this view, a major developmental achievement is mapping verbal number words onto their nonverbal referents.
Others claim innate processes contribute to numerical development but do not constitute a complete conceptual system for number.4,5 These accounts incorporate both preverbal counting and a second representational format based on object tracking. They characterize verbal counting as a conceptual catalyst that permits integration of the two nonverbal representations,5 thereby transcending their inherent limitations and achieving a true concept of number.4
Yet other accounts incorporate object-based representations but claim that these representations develop during early childhood.6 On this view, object representations of number are not precise, even for small sets. Instead, they are thought to approximate number with increasing accuracy due to (1) age-related increases in working memory capacity, and (2) interactions between partial knowledge of the number words and recognition of small numerosities in specific contexts.6,7,8
Some argue that number concepts are extracted from the counting system itself, without support from nonverbal representations. Studies have shown that children do not understand counting principles until they have mastered counting procedures.9,10 It has also been argued that children cannot connect labels for small sets to the conventional counting system because they cannot pick out the natural number sequence from other sequences.11
Because research has focused on the emergence of verbal numeracy within a nonverbal conceptual base, existing experiments include a mixture of verbal and nonverbal methods. In the verbal realm, investigators measure various subcomponents of counting (e.g., asking children to recite the count list, count a set of objects, or name a set’s cardinality). In the nonverbal realm, investigators use object-based tasks that do not require verbal counting. With very young children and infants, looking time procedures (e.g., habituation) and reaching tasks are common.
Key Research Questions
A major aim has been to describe the numerical sensitivity of infants and very young children. Researchers want to know how much children understand about number before acquiring conventional skills. The specific profile of nonverbal strengths and weaknesses is sometimes used to argue for a particular developmental account. Another major research goal is describing the emergence of verbal numeracy in great detail. In this research, the potential interactions between verbal and nonverbal numeracy is carefully considered.
Recent Research Results
Numerical sensitivity in infants
Early habituation research indicated infants could discriminate between small sets of objects. For example, when babies were shown a series of object sets equated for number (e.g., two), but varying in color, shape, and position, their looking time gradually decreased. When a new number of objects was shown (e.g., three), looking times increased, suggesting that infants detected the change in number.12,13 Similar experiments have suggested infants can discriminate large sets of items in both visual and auditory displays,14,15 perform simple calculations over objects,3 and detect numerical relations across modalities.16,17
Nonverbal measures in early childhood
Children perform object-based number tasks much earlier than they demonstrate similar understandings in verbal tasks. For example, preschoolers solve simple object-based addition and subtraction problems (e.g., 2 + 2) years before they can solve analogous verbal problems.6,8,18 Similarly, children judge ordinality and equivalence in forced choice tasks much earlier than they can compare the same sets verbally, via counting.6,19,20,21,22,23,24 Competence on nonverbal measures emerges between 2½ and 3 years of age.
Development of verbal counting
Verbal counting encompasses three major subskills. First, children must learn the sequence of count words. The first 10 count words are usually memorized by age 3 years.25,26 Children learn to generate numbers using the decade structure (teens, twenties, etc.) around 6 years of age. Second, young counters must coordinate words and objects, such that each item in a set is tagged once and only once. Children make many errors as they discover and master tagging procedures, peaking in frequency between 36 and 42 months of age.25 Third, children learn that the last word in a count represents its cardinal value (e.g., when you count, “1-2-3,” you have three things). Interestingly, children gain this insight before mastering verbal counting procedures, which suggests they access the cardinal word principle via experiences with small sets.4,25,26,27,28,29 In fact, small set sizes (i.e., 1-3) may provide the only context for discovering the cardinal word principle because these can be both counted and labeled without counting.4,26,27,28,29,30,31,32,33
A persistent problem has been reconciling infants’ apparent precocity for number with the struggles exhibited by preschool children on similar tasks. For example, if infants can represent and compare large object sets as some have claimed,15 why can’t preschool children match large sets until after they have learned to count?34,35 Such discrepancies have fueled vigorous debate about the meaning of the infant work, and articulating these literatures remains a significant challenge. For example, researchers have only begun to ask whether infant sensitivity to quantity is connected to preschool numeracy and, likewise, whether preschool numeracy is connected to subsequent achievement in school mathematics.36
Another unexplored question is how children coordinate discrete and continuous quantity. Infants’ perception of continuous amount is well established. Some have argued use of continuous amount actually explains infants’ performance on numerical tasks.37,38 Regardless of whether infants process continuous quantity, discrete number or both, research is needed to determine what causes them to shift attention from one type of quantification to the other, as well as the developmental changes that occur as children learn how continuous and discrete quantity are related (e.g., size does not affect counting, unless you are counting measurement units).
Finally, much remains to be learned about the interactions between nonverbal quantification and verbal counting. Some contend that whatever preverbal infants can do or understand is necessarily innate because it emerges without verbal input.4 However, others have argued that even infants who do not speak about number themselves have nonetheless been exposed to number language, and thus it is not clear that infant competencies are either nonverbal or innate.39 A related issue is how children acquire the meanings of the number words and the extent to which this relies on a nonverbal foundation. Current research is also exploring whether acquisition of the plural mediates these interactions.40
Evidence of numerical competence in infants has raised interesting questions about the origins of numeracy and the conceptual resources young children use to acquire verbal counting. However, further research is needed to reveal what this infant competence entails and precisely how it connects to subsequent nonverbal and verbal development.
- Dehaene S. The number sense: How the mind creates mathematics. Oxford, England: Oxford University Press; 1997.
- Gallistel CR, Gelman R. Preverbal and verbal counting and computation Cognition 1992;44: 43-74.
- Wynn K. Origins of numerical knowledge. Mathematical cognition 1995;1:35-60.
- Carey S. Whorf versus continuity theorists: Bringing data to bear on the debate. In: Bowerman M, Levinson SC, eds. Language acquisition and conceptual development. New York, NY: Cambridge University Press: 2001;185-214.
- Spelke E. What makes us smart? Core knowledge and natural language. In: Gentner D, Goldin-Meadow S, eds. Language in mind. Cambridge, MA: MIT Press; 2003.
- Huttenlocher J, Jordan N, Levine SC. A mental model for early arithmetic. Journal of Experimental Psychology: General 1994;123:284-296.
- Mix KS, Sandhofer CM., Baroody A. Number words and number concepts: The interplay of verbal and nonverbal processes in early quantitative development. In: Kail RV, ed. Advances in Child Development and Behavior. New York, NY: Academic Press; 2005: 305-345.
- Rasmussen C, Bisanz J. Representation and working memory in early arithmetic. Journal of Experimental Child Psychology 2005; 91:137-157.
- Briars DJ, Siegler RS. A featural analysis of preschoolers’ counting knowledge. Developmental Psychology 1984;20:607-618.
- Frye D, Braisby N, Lowe J, Maroudas C, Nicholls J. Young children’s understanding of counting and cardinality. Child Development 1989;60:1158-1171.
- Rips LJ, Asmuth J, Bloomfield A. Giving the boot to the bootstrap: How not to learn natural numbers. Cognition 2006;101:B51-B60.
- Antell S, Keating DP. Perception of numerical invariance in neonates. Child Development 1983;54:695-701.
- Strauss MS, Curtis LE. Infant perception of numerosity. Child Development 1981;52:1146-1152.
- Lipton JS, Spelke ES. Origins of number sense: Large number discrimination in human infants. Psychological Science 2003;14: 396-401.
- Xu F, Spelke ES. Large number discrimination in 6-month-old infants. Cognition 2000;74: B1-B11.
- Starkey P, Spelke ES, Gelman R. Numerical abstraction by human infants. Cognition 1990;36:97-127.
- Jordan KE, Suanda SH, Brannon EM. Intersensory redundancy accelerates preverbal numerical competence. Cognition 2008;108: 210-221.
- Levine SC, Jordan NC, Huttenlocher J. Development of calculation abilities in young children. Journal of Experimental Child Psychology 1992;53:72-103.
- Cantlon J, Fink R, Safford K, Brannon EM. Heterogeneity impairs numerical matching but not numerical ordering in preschool children. Developmental Science 2007;10:431-440.
- Mix KS. Preschoolers’ recognition of numerical equivalence: Sequential sets. Journal of Experimental Child Psychology 1999;74:309-322.
- Mix KS. Similarity and numerical equivalence: Appearances count. Cognitive Development 1999;14:269-297.
- Mix KS. The construction of number concepts. Cognitive Development 2002;17:1345-1363.
- Mix KS. Children’s equivalence judgments: Crossmapping effects. Cognitive Development 2008;23:191-203.
- Mix KS, Huttenlocher J, Levine SC. Do preschool children recognize auditory-visual numerical correspondences? Child Development 1996; 67:1592-1608.
- Fuson KC. Children’s counting and conceptions of number. New York, NY: Springer-Verlag; 1988.
- Bermejo V. Cardinality development and counting. Developmental Psychology 1996;32:263-268.
- Mix KS. How Spencer made number: First uses of the number words. Journal of Experimental Child Psychology 2009;102: 427-444.
- Wynn, K. Children's understanding of counting. Cognition 1990;36:155-193.
- Klahr D, Wallace JG. Cognitive development: An information processing approach. Hillsdale, NJ: Erlbaum; 1976.
- Mix KS, Sandhofer CM, Moore JA. How input helps and hinders acquisition of the cardinal word principle. Paper presented at: The biennial meeting of the Society for Research in Child Development. April 2-4, 2009. Denver, CO.
- Schaeffer B, Eggleston VH, Scott JL. Number development in young children. Cognitive Psychology 1974;6:357-379.
- Spelke ES, Tsivkin S. Initial knowledge and conceptual change: Space and Number. In: Bowerman M, Levinson SC, eds. Language acquisition and conceptual development., New York, NY: Cambridge University Press; 2001:70-97.
- Wagner S, Walters JA. A longitudinal analysis of early number concepts: From numbers to number. In: Forman G, ed. Action and thought. New York: Academic Press; 1982:137-161.
- LeCorre M, Carey S. One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition 2007;105: 395-438.
- Siegel LS. The sequence of development of certain number concepts in preschool children. Developmental Psychology 1971;5:357-361.
- Jordan NC, Kaplan D, Ramineni C, Locuniak MN. Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology 2009;45: 850-867.
- Clearfield MW, Mix KS. Number versus contour length in infants’ discrimination of small visual sets. Psychological Science 1999;10:408-411.
- Feigenson L, Carey S, Hauser M. The representations underlying infants’ choice of more: Object files versus analog magnitudes. Psychological Science 2002;13:150-156.
- Mix KS, Huttenlocher J, Levine SC. Multiple cues for quantification in infancy: Is number one of them? Psychological Bulletin 2002;128: 278-294.
- Barner D, Libenson A, Cheung P, Takasaki M. Cross-linguistic relations between quantifiers and numerals in language acquisition: Evidence from Japanese. Journal of Experimental Child Psychology 2009;103: 421-440.
How to cite this article:
Mix KS. Early Numeracy: The Transition from Infancy to Early Childhood. In: Tremblay RE, Boivin M, Peters RDeV, eds. Bisanz J, topic ed. Encyclopedia on Early Childhood Development [online]. http://www.child-encyclopedia.com/numeracy/according-experts/early-numeracy-transition-infancy-early-childhood. Published June 2010. Accessed September 20, 2018.