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Dissonances in theories of number understanding

Published online by Cambridge University Press:  11 December 2008

Lance J. Rips ,
Amber Bloomfield  and
Jennifer Asmuth
Lance J. Rips
Affiliation:
Psychology Department, Northwestern University, Evanston, IL 60208; rips@northwestern.eduasmuth@northwestern.eduhttp://mental.psych.northwestern.edu
Amber Bloomfield
Affiliation:
Psychology Department, DePaul University, Chicago, IL 60614. abloomfi@depaul.edu
Jennifer Asmuth
Affiliation:
Psychology Department, Northwestern University, Evanston, IL 60208; rips@northwestern.eduasmuth@northwestern.eduhttp://mental.psych.northwestern.edu
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Abstract

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.

Type
Authors' Response
Information
Behavioral and Brain Sciences , Volume 31 , Issue 6 , December 2008 , pp. 671 - 687
Copyright
Copyright © Cambridge University Press 2008

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References

Braine, M. D. S. & O'Brien, D. P. (1998) Mental logic. Erlbaum.Google Scholar
Carey, S. (2004) Bootstrapping and the origins of concepts. Daedalus Winter issue, pp. 59–68.CrossRefGoogle Scholar
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.Google Scholar
Dedekind, R. (1888/1963) The nature and meaning of numbers. Dover. (Original work published 1888).Google Scholar
De Millo, R., Lipton, R. & Perlis, A. (1979) Social processes and proofs of theorems and programs. Communications of the ACM 22:271–80.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Gelman, R. & Butterworth, B. (2005) Number and language: How are they related? Trends in Cognitive Sciences 9:6–10.CrossRefGoogle ScholarPubMed
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).Google Scholar
Giaquinto, M. (2001) Knowing numbers. Journal of Philosophy 98:5–18.Google Scholar
Ginsburg, H. P. & Baroody, A. J. (2004) Test of early mathematics ability. Pro-Ed.Google Scholar
Heck, R. G. Jr. (2000) Cardinality counting and equinumerosity. Notre Dame Journal of Formal Logic 41(3):187–209.Google Scholar
Inhelder, B. & Piaget, J. (1964) The early growth of logic in the child. Harper & Row.Google Scholar
Johnson-Laird, P. N. (1983) Mental models: Towards a cognitive science of language, inference, and consciousness. Harvard University Press/Cambridge University Press.Google Scholar
Kitcher, P. (1983) The nature of mathematical knowledge. Oxford University Press.Google Scholar
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 42:59–69.CrossRefGoogle ScholarPubMed
Kripke, S. A. (1982) Wittgenstein on rules and private language. Harvard University Press.Google Scholar
Lakatos, I. (1976) Proofs and refutations. Cambridge.CrossRefGoogle Scholar
Lakoff, G. & Núñez, R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.Google Scholar
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.Google Scholar
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).Google Scholar
Le Corre, M. & Carey, S. (2007) One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition 105:395–438.CrossRefGoogle ScholarPubMed
Maddy, P. (2007) Second philosophy. Oxford University Press.Google Scholar
Osherson, D. N. (1974) Logical abilities in children. Erlbaum.Google Scholar
Rips, L. J. & Asmuth, J. (2007) Mathematical induction and induction in mathematics. In: Induction, ed. Feeney, A. & Heit, E.. Cambridge University Press.Google Scholar
Rips, L. J., Asmuth, J. & Bloomfield, A. (2006) Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition 101:B51–B60.CrossRefGoogle Scholar
Rips, L. J., Asmuth, J. & Bloomfield, A. (2008) Do children learn the integers by induction? Cognition 106:940–51.Google Scholar
Spelke, E. S. (2000) Core knowledge. American Psychologist 55:1233–43.Google Scholar
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.Google Scholar
Wilcox, L. V (1990) Prealgebra for problem solvers. Brooks/Cole.Google Scholar
Wilder, R. (1998) The cultural basis of mathematics. In: New directions in the philosophy of mathematics, ed. Tymoczko, T.. Princeton University Press.Google Scholar
Wittgenstein, L. (1958) Philosophical investigations. Macmillan.Google Scholar
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: http://plato.stanford.edu/archives/sum2008/entries/frege-logic/.Google Scholar