Young Natural-Number Arithmeticians

When preschoolers count to check their arithmetic predictions, their counts are better than when they simply count a set of items on count-only tasks. This is so even for 2 1/2and 3-year-olds dealing with small values. Such results lend support to the view that learning about verbal counting benefits from a nonverbal count-arithmetic system and challenge theories that place understanding of verbal counting at 4 1/2 or 5 years. That preschoolers readily engage in predicting-and-checking number tasks has implications for educational programs. KEYWORDS—informal cognition; preschool arithmetic; early counting; cognitive development; abstraction; preschool children The idea that preschoolers (2 to about 5 years) are budding arithmeticians with respect to the natural (i.e., counting) numbers will strike many as odd. After all, mathematical ideas are not ‘‘out there’’ for inspection. They are abstract ideas, the kind that many assume preschoolers lack. I differ from several theorists who share the view that children younger than 4 or 5 do not understand the cardinal counting principle—the last word in a count represents the cardinal numerosity of the collection—let alone its relationship to addition and subtraction (e.g., Carey, 2004; Fuson, 1988; Mix, Huttenlocher, & Levine, 2002; Piaget, 1952). My viewpoint gains support from arithmetic counting tasks. Young children’s success on such tasks is consistent with the theme that the meaning of the counting procedure is tied to its relation to principles of natural-number arithmetic, as opposed to the idea (expressed by other theorists) that addition and subtraction is an induction that follows after an understanding of counting is achieved. Given the understanding of counting, the children can notice that the repeated placement of one item into a collection increases its value. For some theorists, children’s understanding of the count numbers requires a conceptual change. For example, Carey argues that the first few count words (one, two, three, etc.) are initially treated like the quantities one, some, more, a lot, all, and so on. When children switch to treating count terms above four and five as unique representations of successively larger cardinal values, they can move on to the idea that each successive count word in the list represents one more than the previous word. Mix et al. (2002) propose a gradual march from associations of number words to small collections of homogeneous items to associations of number words to larger collections and then to collections of heterogeneous items, and so on. Finally, the child achieves the abstract concept of natural number. Learning about the relation of counting to addition and subtraction follows. Fuson (1988) offers a closely related view. Piaget (1952) requires a qualitative shift from preoperational mental structures to concrete-operational ones, the latter supporting the use of oneto-one correspondences to assess equivalence or nonequivalence of sets, no matter what the set sizes are. The list of abstraction abilities demonstrated by young children is ever growing, and includes the abilities to assign pairs of look-alike photos, such as a statue and an animal, to their different ontological categories; to reason about unseen causal conditions; to answer questions about the insides of novel items; to attribute mental states to people; and so on (see Gelman & Lucariello, 2002). Still, all this is about people, objects, and events that occur in the natural world, not about arithmetic, which is really abstract. Or is it? My inclusion of ideas about the natural numbers and their relation to addition and subtraction was motivated by the ease with which a sample of 5-year-olds who failed a battery of conservation pretests, including those for number, length, and liquid and quantity amounts, responded to training. The conservation tasks assess whether a child understands that a given quantity does not change in amount when it is rearranged or moved, as when one of 2 rows of N items is spread out to look longer than the other, one of 2 sticks is moved to extend beyond the other, one ball of clay is pounded into a flat disk, or one glass of water is poured into a thin, tall beaker (see below for further details). Children younger than 6 years of age typically failed to conserve on any of these tasks. Yet they had no trouble choosing two of three rows of objects that had the same or Address correspondence to Rochel Gelman, Rutgers Center for Cognitive Science, 152 Frelinghuysen Road, Piscataway, New Brunswick, NJ 08854-8020; e-mail: [email protected]. CURRENT DIRECTIONS IN PSYCHOLOGICAL SCIENCE Volume 15—Number 4 193 Copyright r 2006 Association for Psychological Science different number. Similarly, they did very well on length problems for which they had to choose two of three sticks that were either the same or different lengths as the experimenter continually altered their configuration (Gelman, 1969). The children also achieved high levels of success on posttests of conservation of number and length, as well as of liquid and mass amount. It seemed most unlikely that training instilled a new set of mental structures that would support belief in equivalence despite the contradiction of perceptual data. More plausibly, the children were ready to conserve number across irrelevant transformations as well as conserve the pretested other quantities. My search was on for ways to show early number knowledge and other abstract abilities. As supporting data came in, I started to move toward a domain-specific view of cognitive development, with the assumption that the presence of a small set of implicit mental structures support learning on the fly. The idea that the mind initially has a small set of skeletal structures, each with their own principles, contrasts with the initial-blank-slate view held by associationists and Piaget. Despite their skeletal format, they are mental structures, and we know from cognitive psychology that when one already has a domain-specific mental structure, it is rather easy to learn more about content that fits that structure. The mind will use these different structures to actively engage and assimilate environmental data that share their internal relations. How does this bear on the acquisition of verbal knowledge of counting and arithmetic? It is known that a nonverbal numericalreasoning system that supports counting, addition, and subtraction is available to young children, infants, and animals (Barth, La Mont, Lipton, & Spelke, 2005; Cordes & Gelman, 2005). This supports Gelman and Gallistel’s (1986) proposal that implicit principles of nonverbal enumeration facilitate identifying and learning about both the words for counting and rules for how to use them. The nonverbal principles—one and only one unique tag for each to-be-counted item, stable ordering of the tags, and use of the last tag to represent the cardinal value—correspond to the use rules of verbal counting. So, given a counting list that maps to nonverbal counting principles, the stage is set for the child’s nonverbal structure to identify and start to assimilate the count list to a relevant structure. Since verbal counts and numerical estimates do not stand alone—that is, they gain their meaning from their relation to a mental structure of arithmetic reasoning that includes principles of addition, subtraction, and ordering—assessment of understanding of the language of counting might best be tapped in arithmetic counting tasks as opposed to tasks that only assess counting ability.