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The Development of Cardinal Extension: From Counting to Exact Equality

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Title: The Development of Cardinal Extension: From Counting to Exact Equality
Language: English
Authors: Khuyen N. Le (ORCID 0009-0007-1051-838X), Rose M. Schneider, David Barner
Source: Developmental Psychology. 2025 61(6):1180-1195.
Availability: American Psychological Association. Journals Department, 750 First Street NE, Washington, DC 20002. Tel: 800-374-2721; Tel: 202-336-5510; Fax: 202-336-5502; e-mail: order@apa.org; Web site: http://www.apa.org
Peer Reviewed: Y
Page Count: 16
Publication Date: 2025
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 2000827
Document Type: Journal Articles
Reports - Research
Descriptors: Computation, Numbers, Accuracy, Preschool Children, Foreign Countries, Inferences
Geographic Terms: United States, Canada
DOI: 10.1037/dev0001922
ISSN: 0012-1649
1939-0599
Abstract: Numerate adults know that when two sets are equal, they should be labeled by the same number word. We explored the development of this principle--sometimes called "cardinalextension"--and how it relates to children's other numerical abilities. Experiment 1 revealed that 2- to 5-year-old children who could accurately count large sets often inferred that two equal sets should be labeled with the same number word, unlike children who could not accurately count large sets. However, not all counters made this inference, suggesting that learning to construct and label large sets may be a necessary but not sufficient step in learning how numbers represent exact quantities. Experiment 2 found that children who extended labels to equal sets were not actually sensitive to exact equality and that they often assigned two sets the same label when they were approximately equal, but differed by just one item (violating one-to-one correspondence). These results suggest a gradual, stagelike, process in which children learn to accurately count, learn to extend labels to perceptually similar sets, and then eventually restrict cardinal extension to sets that are exactly equal.
Abstractor: As Provided
Notes: https://osf.io/eswa4
Entry Date: 2026
Accession Number: EJ1502474
Database: ERIC
FullText Text:
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  Data: The Development of Cardinal Extension: From Counting to Exact Equality
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  Data: <searchLink fieldCode="AR" term="%22Khuyen+N%2E+Le%22">Khuyen N. Le</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0007-1051-838X">0009-0007-1051-838X</externalLink>)<br /><searchLink fieldCode="AR" term="%22Rose+M%2E+Schneider%22">Rose M. Schneider</searchLink><br /><searchLink fieldCode="AR" term="%22David+Barner%22">David Barner</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Developmental+Psychology%22"><i>Developmental Psychology</i></searchLink>. 2025 61(6):1180-1195.
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  Data: American Psychological Association. Journals Department, 750 First Street NE, Washington, DC 20002. Tel: 800-374-2721; Tel: 202-336-5510; Fax: 202-336-5502; e-mail: order@apa.org; Web site: http://www.apa.org
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  Data: National Science Foundation (NSF)
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  Data: Numerate adults know that when two sets are equal, they should be labeled by the same number word. We explored the development of this principle--sometimes called "cardinalextension"--and how it relates to children's other numerical abilities. Experiment 1 revealed that 2- to 5-year-old children who could accurately count large sets often inferred that two equal sets should be labeled with the same number word, unlike children who could not accurately count large sets. However, not all counters made this inference, suggesting that learning to construct and label large sets may be a necessary but not sufficient step in learning how numbers represent exact quantities. Experiment 2 found that children who extended labels to equal sets were not actually sensitive to exact equality and that they often assigned two sets the same label when they were approximately equal, but differed by just one item (violating one-to-one correspondence). These results suggest a gradual, stagelike, process in which children learn to accurately count, learn to extend labels to perceptually similar sets, and then eventually restrict cardinal extension to sets that are exactly equal.
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  Data: 2026
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      – Text: English
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        Type: general
      – SubjectFull: Numbers
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      – SubjectFull: Accuracy
        Type: general
      – SubjectFull: Preschool Children
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      – SubjectFull: Foreign Countries
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      – SubjectFull: Inferences
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      – SubjectFull: Canada
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      – TitleFull: The Development of Cardinal Extension: From Counting to Exact Equality
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