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Dissonances in theories of number understanding

  • Lance J. Rips (a1), Amber Bloomfield (a2) and Jennifer Asmuth (a1)


Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end states. A survey of these proposals, however, shows that they do not succeed in bridging the gap to knowledge of the integers. We suggest that a better theory depends on starting with primitives that are inherently structural and mathematical.



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Braine, M. D. S. & O'Brien, D. P. (1998) Mental logic. Erlbaum.
Carey, S. (2004) Bootstrapping and the origins of concepts. Daedalus Winter issue, pp. 5968.
Carey, S. & Sarnecka, B. W. (2006) The development of human conceptual representations: A case study. In: Processes of change in brain and cognitive development, ed. Munakata, Y. & Johnson, M. H.. Oxford University Press.
Dedekind, R. (1888/1963) The nature and meaning of numbers. Dover. (Original work published 1888).
De Millo, R., Lipton, R. & Perlis, A. (1979) Social processes and proofs of theorems and programs. Communications of the ACM 22:271–80.
Denton, K. & West, J. (2002) Children's reading and mathematics achievement in kindergarten and first grade. National Center for Education Statistics (NCES 2002–125), Washington, D.C. U.S. Department of Education.
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., Pagani, L. S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K. & Japel, C. (2007) School readiness and later achievement. Developmental Psychology 43:1428–46.
Ehrlich, S. B. (2007) The preschool achievement gap: Are variations in teacher input associated with differences in number knowledge? Dissertation Abstracts International: Section B: The Sciences and Engineering 68(2–B):1337.
Gelman, R. & Butterworth, B. (2005) Number and language: How are they related? Trends in Cognitive Sciences 9:610.
Gelman, R. & Gallistel, C. R. (1978) The child's understanding of number. Harvard University Press/MIT Press. (Second printing, 1985. Paperback issue with new preface, 1986).
Giaquinto, M. (2001) Knowing numbers. Journal of Philosophy 98:518.
Ginsburg, H. P. & Baroody, A. J. (2004) Test of early mathematics ability. Pro-Ed.
Heck, R. G. Jr. (2000) Cardinality counting and equinumerosity. Notre Dame Journal of Formal Logic 41(3):187209.
Inhelder, B. & Piaget, J. (1964) The early growth of logic in the child. Harper & Row.
Johnson-Laird, P. N. (1983) Mental models: Towards a cognitive science of language, inference, and consciousness. Harvard University Press/Cambridge University Press.
Kitcher, P. (1983) The nature of mathematical knowledge. Oxford University Press.
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 42:5969.
Kripke, S. A. (1982) Wittgenstein on rules and private language. Harvard University Press.
Lakatos, I. (1976) Proofs and refutations. Cambridge.
Lakoff, G. & Núñez, R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
Laurence, S. & Margolis, E. (2005) Number and natural language. In: The innate mind: Structure and content, ed. Carruthers, P., Laurence, S. & Stich, S.. Oxford University Press.
Le Corre, M. (2005) The construction of the positive integers: A case study of human cognition as a product of evolution and culture. Dissertation Abstracts International: Section B: The Sciences and Engineering 65(12–B).
Le Corre, M. & Carey, S. (2007) One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition 105:395438.
Maddy, P. (2007) Second philosophy. Oxford University Press.
Osherson, D. N. (1974) Logical abilities in children. Erlbaum.
Rips, L. J. & Asmuth, J. (2007) Mathematical induction and induction in mathematics. In: Induction, ed. Feeney, A. & Heit, E.. Cambridge University Press.
Rips, L. J., Asmuth, J. & Bloomfield, A. (2006) Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition 101:B51B60.
Rips, L. J., Asmuth, J. & Bloomfield, A. (2008) Do children learn the integers by induction? Cognition 106:940–51.
Spelke, E. S. (2000) Core knowledge. American Psychologist 55:1233–43.
Tappenden, J. (2005) Proof style and understanding in mathematics, I: Visualization, unification and axiom choice. In: Visualization, explanation and reasoning styles in mathematics, ed. Mancosu, P.. Springer.
Wilcox, L. V (1990) Prealgebra for problem solvers. Brooks/Cole.
Wilder, R. (1998) The cultural basis of mathematics. In: New directions in the philosophy of mathematics, ed. Tymoczko, T.. Princeton University Press.
Wittgenstein, L. (1958) Philosophical investigations. Macmillan.
Zalta, E. N. (2008) Frege's logic, theorem, and foundations for arithmetic. In: The Stanford encyclopedia of philosophy (Summer 2008 edition), ed. Zalta, E. N.. Available at:
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Behavioral and Brain Sciences
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  • EISSN: 1469-1825
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