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Children’s number sequences as predictors of later mathematical development

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Abstract

Number sequences are defined in terms of children’s abilities to construct and transform units. Children operating with the initial number sequence (INS) construct units of 1. They construct other numbers as strings of 1s and can count on, by 1s, from one number to a subsequent number. A critical benchmark in children’s further numerical development is the construction of units of units, or composite units. This development corresponds to the tacitly nested number sequence (TNS) wherein numbers are nested within other numbers. We report on a large-scale quantitative study of children’s available number sequences and their relationships with later mathematical development, such as multiplicative reasoning and fractions knowledge. This study involved 5747 children from three cohorts surveyed at the beginning of second grade in 2013, 2014, and 2015. We document strong relationships between children’s early construction of an INS and a TNS and the likelihood of their later development of multiplicative reasoning, a measurement meaning of fractions, and general mathematics achievement, while controlling for rote computational skills. Implications for teacher instruction are discussed.

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Notes

  1. Initially, we used a logistic model that included interaction terms between INS and COMP and TNS and COMP: \( \mathrm{Logit}\kern0.2em (Y)=\ln \left(\frac{\pi }{1-\pi}\right)=\kern0.5em {b}_0+{b}_1\mathrm{INS}+{b}_2\mathrm{TNS}+{b}_3\mathrm{COMP}+{b}_4\left(\mathrm{INS}\times \mathrm{COMP}\right)+{b}_5\left(\mathrm{TNS}\times \mathrm{COMP}\right) \). No statistically significant interaction effects resulted from estimating these models. Therefore, the model used in the study only includes main effects.

  2. Two other pseudo-R2 indices are often reported as measures of goodness-of-fit for logistic models (Cox and Snell, and Nagelkerke); however, given their lack of clear interpretability and lack of predictive efficiency, we chose not to report them based on the recommendations of Osborne (Osborne 2015; also see Peng et al. 2002).

References

  • Behr, M., Harel, G., Post, T. A., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 296–333). New York: Macmillan Publishing Company.

    Google Scholar 

  • Blums, A., Belsky, J., Grimm, K., & Chen, Z. (2017). Building links between early socioeconomic status, cognitive ability, and math and science achievement. Journal of Cognition and Development, 18(1), 16–40.

    Article  Google Scholar 

  • Boulet, G. (1998). On the essence of multiplication. For the Learning of Mathematics, 18, 12–19.

    Google Scholar 

  • Boyce, S., & Norton, A. (2016). Co-construction of fractions schemes and units coordinating structures. The Journal of Mathematical Behavior, 41, 10–25.

    Article  Google Scholar 

  • Boyce, S., & Norton, A. (2019). Maddies’ units coordinating across contexts. Journal of Mathematical Behavior, 55. https://doi.org/10.1016/j.jmathb.2019.03.003.

  • Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293.

    Article  Google Scholar 

  • Clements, D. H. (1999). Subitizing: what is it? Why teach it? Teaching Children Mathematics, 5, 400–405.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Colorado Department of Education. (n.d.). Assessment. Retrieved August 7, 2019 from https://www.cde.state.co.us/communications/factsheetsandfaqs-assessment.

  • Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

    Google Scholar 

  • Davis-Kean, P. E. (2005). The influence of parent education and family income on child achievement: the indirect role of parental expectations and the home environment. Journal of Family Psychology, 19(2), 294–304.

    Article  Google Scholar 

  • Davydov, V. V. (1992). The psychological analysis of multiplication procedures. Focus on Learning Problems in Mathematics, 14(1), 3–67.

    Google Scholar 

  • De Smedt, B., Janssen, R., Bouwens, K., Verschaffel, L., Boets, B., & Ghesquière, P. (2009). Working memory and individual differences in mathematics achievement: a longitudinal study from first grade to second grade. Journal of Experimental Child Psychology, 103(2), 186–201.

    Article  Google Scholar 

  • Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: a revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26(1), 27–47.

    Article  Google Scholar 

  • Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: a critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28(1), 1–18.

    Article  Google Scholar 

  • Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. London: Sage Publishers.

    Google Scholar 

  • Harel, G., & Confrey, J. (Eds.). (1994). Development of multiplicative reasoning in the learning of mathematics, the, The. Suny Press.

  • Hosmer Jr., D. W., & Lemeshow, S. (2000). Applied logistic regression (2nd ed.). New York, NY: Wiley.

    Book  Google Scholar 

  • Hunt, J., & Tzur, R. (2017). Where is difference? Processes of mathematical remediation through a constructivist lens. The Journal of Mathematical Behavior, 48, 62–76.

    Article  Google Scholar 

  • Munter, C. (2010). Evaluating math recovery: the impact of implementation fidelity on student outcomes. Unpublished doctoral dissertation, Vanderbilt University.

  • Norton, A., & Boyce, S. J. (2015). Provoking the construction of a structure for coordinating n + 1 levels of units. Journal of Mathematical Behavior, 40, 211–232.

    Article  Google Scholar 

  • Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583.

    Article  Google Scholar 

  • Olive, J. (1999). From fractions to rational numbers of arithmetic: a reorganization hypothesis. Mathematical Thinking and Learning, 1(4), 279–314.

    Article  Google Scholar 

  • Olive, J. (2001). Children’s number sequences: an explanation of Steffe’s construct and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11(1), 4–9.

    Google Scholar 

  • Oliver, J., May, W. L., & Bell, M. L. (2017). Relative effect sizes for measures of risk. Communications in statistics: Theory and methods, 46(14), 6774–6781.

    Article  Google Scholar 

  • Osborne, J. W. (2015). Best practices in logistic regression. Thousand Oaks, CA: Sage Publications, Inc..

    Book  Google Scholar 

  • Peng, C. J., Lee, K. L., & Ingersoll, G. M. (2002). An introduction to logistic regression analysis and reporting. Journal of Educational Research, 96(1), 3–14.

    Article  Google Scholar 

  • Piaget, J., & Szeminska, A. (1952). The child’s conception of number. London: Routledge and Kegan Paul.

    Google Scholar 

  • Popham, W. J. (2001). Teaching to the test? Educational Leadership, 58(6), 16–20.

    Google Scholar 

  • SAS Institute, Inc. (2012). SAS for windows [Version 9.4]. Cary, NC: SAS Institute, Inc..

    Google Scholar 

  • Steffe, L. P. (1991). The constructivist teaching experiment: illustrations and implications. In E. von Glasersfeld (Ed.), Radical Constructivism in Mathematics Education (pp. 177–194). Netherlands: Springer.

    Chapter  Google Scholar 

  • Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309.

    Article  Google Scholar 

  • Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). Albany, NY: SUNY Press.

    Google Scholar 

  • Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307.

    Article  Google Scholar 

  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York: Springer.

    Book  Google Scholar 

  • Stein, M., Kinder, D., Silbert, J., & Carnine, D. W. (2005). Designing effective mathematics instruction: a direct instruction approach. Pearson.

  • Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5–13.

    Article  Google Scholar 

  • Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (part 1). For the Learning of Mathematics, 35(3), 2–7.

    Google Scholar 

  • Ulrich, C. (2016a). Stages in constructing and coordinating units additively and multiplicatively (part 2). For the Learning of Mathematics, 36(1), 34–39.

    Google Scholar 

  • Ulrich, C. (2016b). The tacitly nested number sequence in sixth grade: the case of Adam. Journal of Mathematics Behavior, 43, 1–19.

    Article  Google Scholar 

  • Ulrich, C., & Wilkins, J. L. M. (2017). Using written work to investigate stages in sixth-grade students’ construction and coordination of units. International Journal of STEM Education, 4, 23.

    Article  Google Scholar 

  • Wilkins, J. L. M., & Norton, A. (2011). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583.

    Google Scholar 

  • Wright, R. J., Ellemor-Collins, D., & Tabor, P. D. (2012). Developing number knowledge: assessment, teaching and intervention with 7–11 year olds. London: SAGE Publications Ltd.

    Book  Google Scholar 

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Authors and Affiliations

  1. Virginia Tech, War Memorial Hall, Rm 326, 370 Drillfield Drive, Blacksburg, VA, 24061, USA

    Jesse L. M. Wilkins

  2. Boulder Valley School District, 6500 Arapahoe, Boulder, CO, 80303, USA

    David Woodward

  3. Virginia Tech, 434 McBryde Hall, 225 Stanger Street, Blacksburg, VA, 24061, USA

    Anderson Norton

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  1. Jesse L. M. Wilkins
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  2. David Woodward
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  3. Anderson Norton
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Correspondence to Jesse L. M. Wilkins.

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Research methods and procedures for this study were conducted in accordance with human subjects guidelines and approved by the first author’s Institutional Review Board for research involving human subjects.

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Wilkins, J.L.M., Woodward, D. & Norton, A. Children’s number sequences as predictors of later mathematical development. Math Ed Res J 33, 513–540 (2021). https://doi.org/10.1007/s13394-020-00317-y

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  • DOI: https://doi.org/10.1007/s13394-020-00317-y

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