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The development and assessment of counting-based cardinal number concepts


The give-n task is widely used in developmental psychology to indicate young children’s knowledge or use of the cardinality principle (CP): the last number word used in the counting process indicates the total number of items in a collection. Fuson (1988) distinguished between the CP, which she called the count-cardinal concept, and the cardinal-count concept, which she argued is a more advanced cardinality concept that underlies the counting-out process required by the give-n task with larger numbers. One aim of the present research was to evaluate Fuson’s disputed hypothesis that these two cardinality concepts are distinct and that the count-cardinal concept serves as a developmental prerequisite for constructing the cardinal-count concept. Consistent with Fuson’s hypothesis, the present study with twenty-four 3- and 4-year-olds revealed that success on a battery of tests assessing understanding of the count-cardinal concept was significantly and substantially better than that on the give-n task, which she presumed assessed the cardinal-count concept. Specifically, the results indicated that understanding the count-cardinal concept is a necessary condition for understanding the cardinal-count concept. The key methodological implication is that the widely used give-n task may significantly underestimate children’s understanding of the CP or count-cardinal concept. The results were also consistent with a second aim, which was to confirm that number constancy concepts develop after the count-cardinal concept but before the cardinal-count concept.

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  1. The cases in cells A, D, E, H, K, and O are developmentally ambiguous, because it is consistent with a CP-priority (Fuson’s 1988 hypothesis), reverse-priority, or synchrony (Sarnecka and Carey’s 2008 non-distinct) hypothesis.

  2. The child who appeared to construct an understanding of the count-cardinal concept but had no success on the give-n task (cell G in Table 3) is not included in this analysis. Evidence of the former was collected in session 2; evidence of the latter was in session 1. So, it is unclear whether the girl would also have constructed the cardinal-count concept and been successful on the give-n task in session 2.


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The preparation of this report was supported by the Institute of Education Science [grant number R305A150243] and the National Science Foundation [grant number 1621470] to the first author. The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the Institute of Education Science or the National Science Foundation.

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Dr. Arthur J. Baroody was the primary contributor to the study conception and design. Material preparation, data collection, and analysis were performed by Dr. Menglung Lai. The first author wrote the first draft of the manuscript, and the second author commented on previous versions of the manuscript. Both authors read and approved the final manuscript.

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Correspondence to Arthur J. Baroody.

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Baroody, A.J., Lai, M. The development and assessment of counting-based cardinal number concepts. Educ Stud Math 111, 185–205 (2022).

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