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IES Grant

Title: Improving Children's Pure Numerical Estimation
Center: NCER Year: 2005
Principal Investigator: Siegler, Robert Awardee: Carnegie Mellon University
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $851,346
Type: Development and Innovation Award Number: R305H050035
Description:

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 children's lives in and out of school. Estimation is also related to other aspects of mathematical ability, including arithmetic skill, conceptual understanding of computational procedures, and math achievement test scores. Although research on estimation is limited, we do know that children's skill at estimation is poor. Prior research conducted by this research team has demonstrated that a major reason for children's poor skill is that they rely on logarithmic rather than linear representations of numerical magnitudes. In other words, children's abilities to discriminate between the magnitudes of numbers decreases sharply as these magnitudes increase. Preliminary evidence suggests that increasing young children's use of appropriate representations of numerical magnitudes would improve their estimation and, by extension, their math achievement. The researchers have developed two interventions-one for preschool children and one for young elementary school children to support the acquisition of linear representations of numerical magnitudes. The purpose of this research project is to test whether learning produced by these interventions lasts over time, generalizes to other estimation tasks, enhances arithmetic learning, and can be scaled up to the classroom level. At the conclusion of this project, the research team will have validated both instructional interventions for use in one-on-one settings and in whole group instruction.

Structured Abstract

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 children's lives in and out of school. Estimation is also related to other aspects of mathematical ability, including arithmetic skill, conceptual understanding of computational procedures, and math achievement test scores. Although research on estimation is limited, we do know that children's skill at estimation is poor. Prior research conducted by this research team has demonstrated that a major reason for children's poor skill is that they rely on logarithmic rather than linear representations of numerical magnitudes. In other words, children's abilities to discriminate between the magnitudes of numbers decreases sharply as these magnitudes increase. Preliminary evidence suggests that increasing young children's use of appropriate representations of numerical magnitudes would improve their estimation and, by extension, their math achievement. The researchers have developed two interventions-one for preschool children and one for young elementary school children to support the acquisition of linear representations of numerical magnitudes. The purpose of this research project is to test whether learning produced by these interventions lasts over time, generalizes to other estimation tasks, enhances arithmetic learning, and can be scaled up to the classroom level. At the conclusion of this project, the research team will have validated both instructional interventions for use in one-on-one settings and in whole group instruction.

Setting: Participating children attend Head Start and childcare centers and public and Catholic elementary schools in the Pittsburgh area.

Population: The participants are 440 children. These include 240 4-year-olds from low-income, predominately African American, Head Start, and childcare centers and 200 white and African American second- and third-graders from low- and middle-income backgrounds.

Intervention: A board game The Great Race is used to teach children linear representations of numerical magnitude for numbers between 0-10. A second technique, The 150 Procedure, asks students to generate estimations at an optimal point on the number line and provides feedback to students. The 150 Procedure is used to help children improve their estimation skills on number lines from 0-1,000.

Research Design and Methods: At least seven different experiments are being carried out to test the effect of this instructional intervention on estimation skill. In the experiments with the second-graders, the researchers are examining instructional techniques for improving children's estimation skills within a 1-1,000 range. The researchers also are examining the influence of The 150 Procedure when implemented at the classroom level. In the school-based studies, classrooms are randomly assigned to condition.

Control Condition: The game board used in the control conditions for the evaluation of the effect of The Great Race is identical to The Great Race game board except that it does not include the numbers 1 to 10 ordered from left to right printed on the 10 squares on the game board. Children use the color of the squares to play the game in the control condition. In the experimental evaluation of The 150 Procedure, experimental and control children receive the same problems but control children do not receive feedback on the accuracy of their answers.

Key Measures: Almost all of the materials are pencil and paper or computerized estimation and arithmetic problems. In the elementary school studies, math achievement scores are being obtained.

Data Analytic Strategy: Analysis of variance and t-test techniques are used to compare student outcomes across experimental conditions.

Project Website: http://www.psy.cmu.edu/~siegler/publications-all.html

Related IES Projects: Using Cognitive Analyses to Improve Children's Math and Science Learning (R305H020060) and Improving Children's Numerical Understanding (R305A080013)

Publications

Book chapter

Lin, X., Siegler, R.S., and Sullivan, F.R. (2010). Students' Goals Influence Their Learning. In D.D. Preiss, and R.J. Sternberg (Eds.), Innovations in Educational Psychology: Perspectives on Learning, Teaching, and Human Development (pp. 79–104). New York: Springer.

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. (2006). Microgenetic Analyses of Learning. In D. Kuhn, and R.S. Siegler (Eds.), Handbook of Child Psychology: Volume 2: Cognition, Perception, and Language (6th ed., pp. 464–510). Hoboken, NJ: Wiley.

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., and Svetina, M. (2008). Relations Between Short-Term and Long-Term Changes in Children's Thinking. In S. Vosniadou (Ed.), International Handbook of Research on Conceptual Change (pp. 102–123). New York: Routledge/Taylor & Francis Group.

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.

Booth, J.L., and Siegler, R.S. (2006). Developmental and Individual Differences in Pure Numerical Estimation. Developmental Psychology, 42(1): 189–201.

Booth, J.L., and Siegler, R.S. (2008). Numerical Magnitude Representations Influence Arithmetic Learning. Child Development, 79(4): 1016–1031.

Laski, E.V., and Siegler, R.S. (2007). Is 27 a Big Number? Correlational and Causal Connections Among Numerical Categorization, Number Line Estimation, and Numerical Magnitude Comparison. Child Development, 76(6): 1723–1743.

Opfer, J.E., and Siegler, R.S. (2007). Representational Change and Children's Numerical Estimation. Cognitive Psychology, 55(3): 169–195.

Ramani, G.B., and Siegler, R.S. (2008). Promoting Broad and Stable Improvements in Low-Income Children's Numerical Knowledge Through Playing Number Board Games. Child Development, 79(2): 375–394.

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. (2007). Cognitive Variability. Developmental Science, 10(1): 104–109.

Siegler, R.S. (2009). Improving the Numerical Understanding of Children From Low-Income Families. Child Development Perspectives, 3(2): 118–124.

Siegler, R.S., and Chen, Z. (2008). Differentiation and Integration: Guiding Principles for Analyzing Cognitive Change. Developmental Science, 11(4): 433–448.

Siegler, R.S., and Mu, Y. (2008). Chinese Children Excel on Novel Mathematics Problems Even Before Elementary School. Psychological Science, 19(8): 759–763.

Siegler, R.S., and Ramani, G.B. (2006). Early Development of Estimation Skills. APS Observer, 19(5): 34–44.

Siegler, R.S., and Ramani, G.B. (2008). Playing Linear Numerical Board Games Promotes Low-Income Children's Numerical Development. Developmental Science, 11(5): 655–661.

Siegler, R.S., and Ramani, G.B. (2009). Playing Linear Number Board Games, but not Circular Ones, Improves Low-Income Preschoolers' Numerical Understanding. Journal of Educational Psychology, 101(3): 545–560.

Siegler, R.S., and Svetina, M. (2006). What Leads Children to Adopt New Strategies?: A Microgenetic/Cross-Sectional Study of Class Inclusion. Child Development, 77(4): 997–1015.

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., Thompson, C.A., and Opfer, J. E. (2009). The Logarithmic-to-Linear Shift: One Learning Sequence, Many Tasks, Many Time Scales. Mind, Brain, and Education, 3(3): 143–150.

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. 

Torbeyns, J., Schneider, M., Xin, Z., and 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.


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