Skip to main content
SpringerLink
Log in

The development and assessment of counting-based cardinal number concepts

Educational Studies in Mathematics volume 111, pages 185–205 (2022)Cite this article

Abstract

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Availability of data and material

The data and materials are available upon request to an author.

Code availability

Not applicable.

Notes

  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.

References

  • Barner, D., & Bachrach, A. (2010). Inference and exact numerical representation in early language development. Cognitive Psychology, 60, 40–62. https://doi.org/10.1016/j.cogpsych.2009.06.002

    Article  Google Scholar 

  • Baroody, A. J., Clements, D. H., & Sarama, J. (2021). Lessons learned from 10 experiments that tested the efficacy and assumptions of hypothetical learning trajectories Education Science, 12. https://doi.org/10.3390/educsci12030195

  • Baroody, A. J., Lai, M. L., & Mix, K. S. (2006). The development of young children’s number and operation sense and its implications for early childhood education. In B. Spodek & O. Saracho (Eds.), Handbook of research on the education of young children (pp. 187–221). Erlbaum.

  • Baroody, A. J., & Purpura, D. J. (2017). Early number and operations: Whole numbers. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 308–354). National Council of Teachers of Mathematics.

    Google Scholar 

  • Bakeman, R., & Quera, V. (2011). Sequential analysis and observational methods for the behavioral sciences. Cambridge University Press. https://doi.org/10.1017/CBO9781139017343

    Article  Google Scholar 

  • Beckwith, M., & Restle, F. (1966). Process of enumeration. Psychological Review, 73(5), 437–444. https://doi.org/10.1037/h0023650

    Article  Google Scholar 

  • Benoit, L., Lehalle, H., & Jouen, F. (2004). Do young children acquire number words through subitizing or counting? Cognitive Development, 19, 291–307. https://doi.org/10.1016/j.cogdev.2004.03.005

  • Carey, S. (2004). Bootstrapping & the origin of concepts. Daedalus, 133, 59–68. https://doi.org/10.1162/001152604772746701

    Article  Google Scholar 

  • Carey, S., & Barner, D. (2019). Ontogenetic origins of human integer representations. Trends in Cognitive Sciences, 23(10), 823–835. https://doi.org/10.1016/j.tics.2019.07.004

  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Routledge. ISBN 1–134–74270–3.

  • Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159.

    Article  Google Scholar 

  • Condry, K. F., & Spelke, E. S. (2008). The development of language and abstract concepts: The case of natural number. Journal of Experimental Psychology: General, 137, 22–38. https://doi.org/10.1037/0096-3445.137.1.22

    Article  Google Scholar 

  • Darlington, R. B. (1974). Radicals and squares and other statistical procedures for the behavioral sciences. Logan Hill Press.

  • Dixon, J. A., & Moore, C. F. (2000). The logic of interpreting evidence of developmental ordering: Strong inference and categorical measures. Developmental Psychology, 36, 826–834. https://doi.org/10.1037/0012-1649.36.6.826

    Article  Google Scholar 

  • Fischer, J. P. (1992). Subitizing: The discontinuity after three. In J. Bideaud, C. Meljac, & J. P. Fischer (Eds.), Pathways to number (pp. 191–208). Erlbaum.

  • Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. http://ies.ed.gov/ncee/wwc/practiceguide.aspx?sid=18.

  • Frye, D., Braisby, N., Lowe, J., Maroudas, C., & Nicholls, J. (1989). Children’s understanding of counting and cardinality. Child Development, 60, 1158–1171. https://doi.org/10.2307/1130790

    Article  Google Scholar 

  • Fuson, K. C. (1988). Children’s counting and concepts of number. Springer-Verlag.

    Book  Google Scholar 

  • Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59–65. https://doi.org/10.1016/s1364-6613(99)01424-2

  • Gelman, R. (1993). A rational-constructivist account of early learning about numbers and objects. Advances in research theory. In D. L. Medin (Ed.), The Psychology of Learning and Motivation (pp. 61–96). Academic Press.

  • Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525. https://doi.org/10.2307/1418556

  • Klahr, D., & Wallace, J. G. (1976). Cognitive development: An information-processing view. Erlbaum.

  • Krajcsi, A. (2021). Follow-up questions influence the measured number knowledge in the give-a-number task. Cognitive Development, 57, 100968. https://doi.org/10.1016/j.cogdev.2020.100968

    Article  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. https://doi.org/10.1016/j.cognition.2006.10.005

  • Le Corre, M. & Carey, S. (2008). Why the verbal counting principles are constructed out of representations of small sets of individuals: A reply to Gallistel. Cognition, 107, 650–662. https://doi.org/10.1016/j.cognition.2007.09.008

  • Le Corre, M., Van de Walle, G. A., Brannon, E., & Carey, S. (2006). Revisiting the performance/competence debate in the acquisition of counting as a representation of the positive integers. Cognitive Psychology, 52, 130–169. https://doi.org/10.1016/j.cogpsych.2005.07.002

    Article  Google Scholar 

  • Lipsey, M. W., Puzio, K., Yun, C., Hebert, M. A., Steinka-Fry, K., Cole, M. W., Roberts, M., Anthony, K. S., & Busick, M. D. (2012). Translating the statistical representation of the effects of education interventions into more readily interpretable forms. IES National Center for Special Education Research, Institute of Education Sciences.

  • Lobato, J., & Walters, C. D. (2017). A taxonomy of approaches to learning trajectories and progressions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 74–101). National Council of Teachers of Mathematics.

    Google Scholar 

  • Maloney, A. P., Confrey, J., & Nguyen, K. H. (Eds.). (2014). Learning over time: Learning trajectories in mathematics education. Information Age Publishing.

  • Mou, Y., Zhang, B., Piazza, M., & Hyde, D. C. (2021). Comparing set-to-number and number-to-set measures of cardinal number knowledge in preschool children using latent variable modeling. Early Childhood Research Quarterly, 54, 125–135. https://doi.org/10.1016/j.ecresq.2020.05.016

    Article  Google Scholar 

  • Nieder, A. (2017). Number faculty is rooted in our biological heritage. Trends in Cognitive Sciences, 21(6), 403–404.

  • Paliwal, V., & Baroody, A. J. (2020). Cardinality principle understanding: The role of focusing on the subitizing ability. ZDM-Mathematics Education, 52(4), 649–661. https://doi.org/10.1007/s11858-020-01150-0

    Article  Google Scholar 

  • Potter, M. C., & Levy, E. I. (1968). Spatial enumeration without counting. Child Development, 39(1), 265–272. https://doi.org/10.2307/1127377

    Article  Google Scholar 

  • Posid, T., & Cordes, S. (2018). How high can you count? Probing the limits of children’s counting. Developmental Psychology, 54(5), 875–889. https://doi.org/10.1037/dev0000469

    Article  Google Scholar 

  • Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Erlbaum.

  • Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.

  • Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662–674. https://doi.org/10.1016/j.cognition.2008.05.007

    Article  Google Scholar 

  • Sarnecka, B. W., & Gelman, S. A. (2004). Six does not just mean a lot: Preschoolers see number words as specific. Cognition, 92, 329–352. https://doi.org/10.1016/j.cognition.2003.10.001

    Article  Google Scholar 

  • Sarnecka, B. W., & Wright, C. E. (2013). The idea of an exact number: Children’s understanding of cardinality and equinumerosity. Cognitive Science, 37, 1493–1506. https://doi.org/10.1111/cogs.12043

    Article  Google Scholar 

  • Schaeffer, B., Eggleston, V. H., & Scott, J. L. (1974). Number development in young children. Cognitive Psychology, 6, 357–379. https://doi.org/10.1016/0010-0285(74)90017-6

    Article  Google Scholar 

  • Schneider, R. M., Sullivan, J., Guo, K., & Barner, D. (2021). What counts? Sources of knowledge in children’s acquisition of the successor function. Child Development, 92(4), e476–e492. https://doi.org/10.1111/cdev.13524

  • Sella, F., Slusser, E., Odic, D., & Krajcsi, A. (2021). The emergence of children’s natural number concepts: Current theoretical challenges. Child Development Perspectives. https://doi.org/10.1111/cdep.12428

    Article  Google Scholar 

  • Shavelson, R. J., & Karplus, A. (2012). Reflections on learning progressions. In A. C. Alonzo & A. W. Gotwals (Eds.), Learning progressions in science: Current challenges and future directions (pp. 13–26). Sense.

    Chapter  Google Scholar 

  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. https://doi.org/10.2307/749205

    Article  Google Scholar 

  • Von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50, 191–218.

  • Wagner, K., Chu, J., & Barner, D. (2019). Do children’s number words begin noisy? Developmental Science, 22(1), e12752.

  • Wilkinson, L., APA Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594–604. https://doi.org/10.1037/0003-066X.54.8.594

  • Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193. https://doi.org/10.1016/0010-0277(90)90003-3

  • Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. https://doi.org/10.1016/0010-0285(92)90008-P

Download references

Funding

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.

Author information

Authors and Affiliations

  1. College of Education, University of Illinois, Champaign, IL, 61853, USA

    Arthur J. Baroody

  2. Department of Early Childhood Education, National Chiayi University, Chiayi, Taiwan

    Menglung Lai

Authors
  1. Arthur J. Baroody
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Menglung Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Corresponding author

Correspondence to Arthur J. Baroody.

Ethics declarations

Ethics approval

In Taiwan, researchers need only to obtain parent consent.

Consent to participate

Informed parental consent was obtained before testing.

Consent for publication

Not applicable.

Conflict of interest

Not applicable.

Additional information

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (DOCX 21 kb)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baroody, A.J., Lai, M. The development and assessment of counting-based cardinal number concepts. Educ Stud Math 111, 185–205 (2022). https://doi.org/10.1007/s10649-022-10153-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-022-10153-5

Keywords

search

Navigation