title: Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality.
abstract: Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children's mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children's understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32-36 months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.
authors.affiliation: Laboratoire Psychologie de la Perception, Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France; CNRS UMR 8158, 45 rue des Saints-Pères, 75006 Paris, France; Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138, USA. Electronic address: veronique.izard@polytechnique.org.
authors.forename: Véronique
authors.initials: V
authors.lastname: Izard
authors.affiliation: Laboratoire Psychologie de la Perception, Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France; CNRS UMR 8158, 45 rue des Saints-Pères, 75006 Paris, France.
authors.forename: Arlette
authors.initials: A
authors.lastname: Streri
authors.affiliation: Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138, USA.
authors.forename: Elizabeth S
authors.initials: ES
authors.lastname: Spelke
mesh_terms: D002657:Child Development; D002675:Child, Preschool; D003210:Concept Formation; D005260:Female; D006801:Humans; D007858:Learning; D008433:Mathematics
keywords: Cognitive development; Foundations of the integers; Mathematical cognition; Number concepts; Preschool children