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IES Grant

Title: Improving Children’s Numerical Understanding
Center: NCER Year: 2008
Principal Investigator: Siegler, Robert Awardee: Carnegie Mellon University
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $1,184,676
Type: Development and Innovation Award Number: R305A080013

Purpose: The ability to estimate numerical magnitude is important, not only because of its role in mathematical thinking but also because it is a part of daily life. Estimation is also related to other aspects of mathematical ability, including arithmetic skill, conceptual understanding of computational procedures, and math achievement test scores. Preliminary evidence suggests that increasing young children's use of appropriate representations of numerical magnitudes can improve their estimation and, by extension, their math achievement. The research team has previously developed interventions for preschool and elementary school children to support the acquisition of linear representations of numerical magnitudes. The goals of the current project are to test whether: (a) the intervention increases ability to learn arithmetic; (b) the intervention can be effectively executed by Head Start personnel working with small groups of children; (c) lengthening the intervention increases learning; and (d) the intervention can be extended to what may be a manifestation of the same underlying problem: middle-school students' poor understanding of rational numbers (fractions, decimals, percentages).

Project Activities: In this project, board games are used to teach preschoolers from low-income backgrounds linear representations of numerical magnitude for numbers between 0 and 10 and second graders with learning disabilities, linear representations for numbers between 0 and 100. In an additional set of studies, the research team examines middle school and college students' understanding of fractional magnitudes.

Products: Products from this project include updated versions of a numerical board game for preschoolers, and published reports on students' abilities to estimate whole number and fractional magnitudes.

Structured Abstract

Purpose: This project applies cognitive science theories and methods to develop programs that improve numerical understanding among preschoolers, elementary and middle school students, and college students.

Setting: The research is being conducted in Head Start centers, public schools, and colleges in Pennsylvania and Washington, DC.

Population: There are 754 participants across the eight studies. Participants include preschoolers as well as second, fourth, sixth, and eighth graders, and college students; Head Start teacher aides are also participating. Participants are of mixed socioeconomic and ethnic backgrounds.

Intervention: In Studies 1–3 and 5, a board game, The Great Race, developed by the research team in a previous IES study, is used to teach preschoolers linear representations of numerical magnitude for numbers between 0 and 10. In Study 4, the commercial number board game, Chutes and Ladders, is used to teach second graders linear representations of numerical magnitude for numbers between 1 and 100. In three additional studies, fourth, sixth, and eighth graders and college students work with number lines to estimate magnitudes using whole numbers, common fractions, decimal fractions, and percentages.

Research Designs and Methods: The research team is conducting Studies 1–5 in order to further refine their numerical board game, The Great Race. In Studies 6–8, the research team is gaining background knowledge important to further development of age-appropriate materials. Studies 1–5 and 8 use an experimental design. Studies 6–7 are quasi-experimental. In Studies 1–4, preschoolers and second graders are randomly assigned to either an experimental condition in which they work individually with a Research Assistant to use the linear 0 to 10 Great Race board game or the 1 to 100 Chutes and Ladders number board game, or to a control condition. Alternative experimental conditions for the preschool classes vary the length of playing time, use circular board games, or board games in which numbers descend rather than ascend. In Study 5, preschoolers are randomly assigned to work in groups of three to play either The Great Race number board game or the color board game with their Head Start teacher's aide. In Study 6, 40 high-achieving university undergraduates are presented four estimation tasks in counterbalanced order with endpoints 0 to 1000, 0 to 1.000, and 0% to 100.0%. In Study 7, number line problems and procedures parallel to Study 6 are being used to examine the understanding of fractional magnitude among fourth, sixth, and eighth graders, and community college students. To test the hypothesis that a carefully chosen presentation order can improve students' estimation of fractional magnitude, sixth-grade students in Study 8 are randomly assigned to groups that receive fractions in different orders.

Control Condition: In Studies 1–3 and 5, preschoolers in the control conditions use a board game based on colors rather than on numbers, or engage in verbal counting tasks. In Study 4, second graders in the control condition practice counting from 1 to 100 and solving addition and subtraction problems with numbers 1 to 100 for the same amount of time. In Study 8, sixth graders in the control condition are given the same fraction families presented to the experimental condition, but in a different order. Studies 6 and 7 are quasi-experimental.

Key Measures: Almost all of the materials are pencil and paper or computerized estimation and arithmetic problems. In the studies involving preschoolers and second graders, dependent measures include counting, number line estimation, numerical magnitude comparison, numerical identification, and estimation of answers to two-digit addition problems. In Study 5, social interaction measures are also being collected and coded using videotaped recordings of the board game play. In the studies involving fourth, sixth, and eighth graders and college students, math achievement scores are being obtained.

Data Analytic Strategy: Standard regression analysis, analysis of variance, MANOVA's, correlations, and t-test techniques are used to compare student outcomes across experimental conditions.

Related IES Projects: Using Cognitive Analyses to Improve Children's Math and Science Learning (R305H020060) and Improving Children's Pure Numerical Estimation (R305H050035)


Book chapter

Geary, D.C., Berch, D.B., Boykin, A.W., Embretson, S., Reyna, V., and Siegler, R.S. (2011). Learning Mathematics: Findings From the National (United States) Mathematics Advisory Panel. In N. Canto (Ed.), Issues and Proposals in Mathematics Education (pp. 175–221). Lisbon, Portugal: Gulbenkian.

Opfer, J.E., and Siegler, R.S. (2012). Development of Quantitative Thinking. In K.J. Holyoak, and R.G. Morrison (Eds.), Oxford Handbook of Thinking and Reasoning (pp. 585–605). Cambridge, UK: Oxford University Press.

Ramani, G.B., and Siegler, R.S. (2015). How Informal Learning Activities can Promote Children's Numerical Knowledge. In R. Kadosh, and A. Dowker (Eds.), The Oxford Handbook of Numerical Cognition (Advance online publication).

Siegler, R.S. (2010). Robbie Case: A Modern Classic. Preface for Developmental Interplay Between Mind, Brain, and Education. In M. Ferrari, and L. Vuletic (Eds.), Essays in Honor of Robbie Case (pp. 1–6). New York: Springer Press.

Siegler, R.S. (2012). From Theory to Application and Back: Following in the Giant Footsteps of David Klahr. In J. Shrager, and S. Carver (Eds.), The Journey From Child to Scientist: Integrating Cognitive Development and the Education Sciences (pp. 17–36). Washington, DC: American Psychological Association.

Siegler, R.S., Fazio, L.K., and Pyke, A. (2011). There is Nothing so Practical as a Good Theory. In J.P. Mestre, and B.H. Ross (Eds.), The Psychology of Learning and Motivation, Volume 55: Cognition in Education (pp. 171–197). San Diego: Elsevier Academic Press.

Journal article, monograph, or newsletter

Bailey, D.H., Zhou, X., Zhang, Y., Cui, J., Fuchs, L.S., Jordan, N.C., Gerstene, R., and Siegler, R.S. (2015). Development of Fraction Concepts and Procedures in US and Chinese Children. Journal of Experimental Child Psychology, 129: 68–83.

Fazio, L.K., and Siegler, R.S. (2013). Microgenetic Learning Analysis: A Distinction Without a Difference: Commentary on Parnafes and DiSessa. Human Development, 56(1): 52–58.

Opfer, J.E., Siegler, R.S., and Young, C.J. (2011). The Powers of Noise-Fitting: Reply to Barth and Paladino. Developmental Science, 14(5): 1194–1204.

Ramani, G.B., and Siegler, R.S. (2011). Reducing the Gap in Numerical Knowledge Between Low- and Middle-Income Preschoolers. Journal of Applied Developmental Psychology, 32(3): 146–159.

Ramani, G.B., Siegler, R.S., and Hitti, A. (2012). Taking it to the Classroom: Number Board Games as a Small Group Learning Activity. Journal of Educational Psychology, 104(3): 661–672.

Schneider, M., and Siegler, R.S. (2010). Representations of the Magnitudes of Fractions. Journal of Experimental Psychology: Human Perception and Performance, 36(5): 1227–1238.

Siegler, R.S. (2010). Playing Numerical Board Games Improves Number Sense in Children From Low-Income Backgrounds. Understanding Number Development and Number Difficulties, No. 7, British Journal of Educational Psychology, Monograph Series II: Psychological Aspects of Education—Current Trends: 15–29.

Siegler, R.S., and Thompson, C.A. (2014). Numerical Landmarks Are Useful – Except When They're Not. Journal of Experimental Child Psychology, 120: 39–58.

Siegler, R.S., Duncan, G.J., Davis-Kean, P.E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M.I., and Chen, M. (2012). Early Predictors of High School Mathematics Achievement. Psychological Science, 23(7): 691–697.

Siegler, R.S., Fazio, L.K., Bailey, D.H., and Zhou, X. (2013). Fractions: The New Frontier for Theories of Numerical Development. Trends in Cognitive Science, 17(1): 13–19.

Siegler, R.S., Thompson, C.A., and Schneider, M. (2011). An Integrated Theory of Whole Number and Fractions Development. Cognitive Psychology, 62(4): 273–296.

Thompson, C.A., and Siegler, R.S. (2010). Linear Numerical-Magnitude Representations Aid Children's Memory for Numbers. Psychological Science, 21(9): 1274–1281.