Topic: Cognition and Mathematics Instruction
Purpose: The purpose of the National Center on Mathematics Instruction for Students with Mathematics Difficulties is to increase knowledge of how children acquire and fail to acquire an understanding of rational numbers (i.e., fractions) and how children with math difficulties can be taught to understand and operate fluently with rational numbers.
The new center will conduct exploratory research including both small-scale experimental and longitudinal studies to examine the cognitive processes, such as working memory and inhibition, that impede understanding and operating with fractions. Findings from these studies will be used to inform the design of an intervention package intended to improve fraction skills in students with math difficulties.
Established through a five-year, $9.9 million grant from the Institute of Education Sciences (IES) of the U.S. Department of Education, the center is staffed with nationally-recognized experts in math education and instruction, cognitive psychology, mathematics difficulties, and research evaluation and methodology.
The experimental and longitudinal studies will establish the conceptual and procedural knowledge that is critical for understanding and operating with fractions as well as identify the key cognitive processes, including working memory for numbers, attentional control, and inhibitory control, that underlie this understanding. Individual differences in the emergence and development of fractions skills will also be explored in order to identify early predictors of difficulty in understanding and operating with fractions. Findings from the exploratory studies will be used to inform the development of an intervention package.
An intervention component will be developed each year. At the completion of the project, the research team will have developed an intervention package intended to promote a solid foundation of conceptual understanding of fractions by addressing concepts and procedures in an integrated manner. Each intervention component will incorporate instructional principles considered effective for students with math difficulties across academic domains (e.g., explicit instruction, embed systematic cumulative practice) and designed to compensate for the cognitive deficits these students experience (e.g., minimize cognitive load, promote attentive behavior, incorporate metacognitive strategies). The research team will use an iterative process to develop each component as well as develop a rubric to measure fidelity. The intervention components will focus on division, multiplicative relations, and fractions and ratios within word problems. The intervention will be designed to be implemented in small groups. During the development process, the research team will collect data on (1) instructional effectiveness; (2) appropriate use of instructional vocabulary; (3) ease of implementation; (4) quality of student engagement; and (5) quality of student work.
Key Personnel: Nancy Jordan, Robert Siegler, Lynn Fuchs, Russell Gersten
Center Website: http://www.udel.edu/soe/fractions
IES Program Contact: Jacquelyn Buckley
Phone: (202) 219-2130
Publications from this project:
Seethaler, P. M., Fuchs, L. S., Star, J. R., & Bryant, J. R. (2011). The cognitive predictors of computational skill with whole versus rational numbers: An exploratory study. Learning and Individual Differences, 21 (5), 536–542. doi: 10.1016/j.lindif.2011.05.002.
Fazio, L., & Siegler, R.S. (2012). Teaching Fractions. Vol 22 of Educational Practice Series. Geneva: International Academy of Education—International Bureau of Education.
Opfer, J.E., & Siegler, R.S. (2012). Development of quantitative thinking. In K. Holyoak & R. Morrison (Eds.), Oxford Handbook of Thinking and Reasoning (pp. 585–605). Cambridge, UK: Oxford University Press.
Siegler, R.S., Duncan, G.J., Davis-Kean, P.E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M.I. & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23 (7), 691–697. doi: 10.1177/0956797612440101.
Fazio, L. K. & Siegler, R. S. (2013). Microgenetic learning analysis: A distinction without a difference. Commentary on Parnafes and diSessa. Human Development, 56, 52–58. doi: 10.1159/000345542
Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., Jordan, N. C., Siegler, R. S., Gersten, R., & Changas, P. (2013). Improving at-risk learners' understanding of fractions. Journal of Educational Psychology, 105, 683–700.
Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and fraction procedures. Journal of Experimental Child Psychology. doi:dx.doi.org/10.1016/j.jecp.2013.02.001
Siegler, R. S. (2013). Cognitive development in childhood. In E. Diener & R. Biswas-Diener (Eds.), Noba textbook series: Psychology. Champaign, IL: DEF Publishers.
Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Science, 17, 13–19.
Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994–2004. doi: 10.1037/a0031200
Wang, Y., & Siegler, R. S. (2013). Representations of and translation between common fractions and decimal fractions. Chinese Science Bulletin, 58, 4630–4640. doi: 10.1007/s11434–013–6035–4
Bailey, D. H., Siegler, R. S., & Geary, D. C. (2014). Early predictors of middle school fraction knowledge. Developmental Science, 17, 775–785. doi: 10.1111/desc.12155
Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. doi: 10.1016/j.jecp.2014.01.013
Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J., Malone, A., & Changas, P. (2014). Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. Journal of Educational Psychology, 106, 499–514.
Malone, A. S., & Fuchs, L. S. (2014). Comparing the contribution of teacher versus tutor ratings of inattentive behavior in predicting mathematics achievement. Remedial and Special Education, 35, 378–386.
Siegler, R. S. & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8, 144–150. doi: 10.1111/cdep.12077
Siegler, R. S., & Thompson, C. A. (2014). Numerical landmarks are useful – Except when they're not. Journal of Experimental Child Psychology, 120, 39–58. doi: 10.1016/ j.jecp.2013.11.014
Vukovic, R.K., Fuchs, L.S., Geary, D.C., Jordan, N.C., Gersten, R., & Siegler, R.S. (2014). Sources of individual differences in children's understanding of fractions. Child Development, 85, 1461–1476. doi:10.1111/cdev.12218
Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What's past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43, 352–360. doi: 10.3102/0013189X14553660
Bailey, D. H., Zhou, X., Zhang, Y., Cui, J., Fuchs, L. S., Jordan, N. C., Gersten, R., & Siegler, R. S. (2015). Development of fraction concepts and procedures in U.S. and Chinese children. Journal of Experimental Child Psychology, 129, 68–83. doi: 10.1016/j.jecp.2014.08.006
Fuchs, L.S., Fuchs, D., Compton, D.L., Wehby, J., Schumacher, R.F., Gersten, R., & Jordan, N.C. (2015). Inclusion versus specialized intervention for very low-performing students: What does access mean in an era of academic challenge? Exceptional Children, 81, 134–157. doi: 10.1177/0014402914551743
Hansen, N., Jordan, N. C., Fenandez, E., Siegler, R. S., Fuchs, L., Gersten, R., & Micklos, D. (2015). General and math-specific predictors of sixth-graders' knowledge of fractions. Cognitive Development, 35, 34–49. doi: 10.1016/j.cogdev.2015.02.001
Siegler, R., Fuchs, L., Jordan, N., Gersten, R., & Ochsendorf, R. (2015). The Center for Improving Learning of Fractions: A progress report. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 292–303). New York, NY: Routledge.
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. doi: 10.1016/j.learninstruc. 2014.03.002
Fuchs, L. S., Malone, A., Schumacher, R. F., Namkung, J., Hamlett, C. L., Jordan, N. C., Siegler, R. S., Gersten, R., & Changas, P. (in press). Supported self-explaining during fraction intervention. Journal of Educational Psychology.
Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Malone, A., Wang, A., Hamlett, C. L., Jordan, N. C., Siegler, R. S., & Changas, P. (in press). Effects of intervention to improve at-risk fourth graders' understanding, calculations, and word problems with fractions. Elementary School Journal.
Lortie-Forgues, H., Tian, J., & Siegler, R. S. (in press). Why is learning fraction and decimal arithmetic so difficult? Developmental Review.
Namkung, J. M., & Fuchs, L. S. (in press). Cognitive predictors of calculations and number line estimation with whole numbers and fractions among at-risk students. Journal of Educational Psychology.
Siegler, R. S., & Lortie-Forgues, H. (in press). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology.
Watts, T. W., Duncan, G. J., Chen, M., Claessens, A., Davis-Kean, P. E., Duckworth, P., Engle, M., Siegler, R., & Susperreguy, M. I. (in press). The role of mediators in the development of longitudinal mathematics achievement associations. Child Development.