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.

Project Activities: 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

Products and Publications

ERIC Citations: Find available citations in ERIC for this award here.

Center Website: http://www.udel.edu/soe/fractions

Book chapter

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., Fuchs, L., Jordan, N., Gersten, R., and 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: Routledge.

Siegler, R.S. (2013). Cognitive Development in Childhood. In E. Diener, and R. Biswas-Diener (Eds.), Noba Textbook Series: Psychology. Champaign, IL: DEF Publishers.

Journal article, monograph, or newsletter

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

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

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

Fazio, L., and Siegler, R.S. (2012). Teaching Fractions. Educational Practice Series, Volume 22 : 1–25. Full text

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

Fazio, L.K., DeWolf, M., and Siegler, R.S. (2016). Strategy Use and Strategy Choice in Fraction Magnitude Comparison. Journal of Experimental Psychology: Human Perception and Performance, 42(1): 1–16. doi:10.1037/xlm0000153

Fazio, L.K., Kennedy, C.A., and Siegler, R.S. (2016). Improving Children's Knowledge of Fraction Magnitudes. PLOS ONE. doi:10.1371/journal.pone.0165243 Full text

Fuchs, L.S., Fuchs, D., Compton, D.L., Wehby, J., Schumacher, R.F., Gersten, R., and 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(2): 134–157. Full text

Fuchs, L.S., Malone, A., Schumacher, R.F., Namkung, J., Hamlett, C.L., Jordan, N.C., Siegler, R.S., Gersten, R., and Changas, P. (2016). Supported Self-Explaining During Fraction Intervention. Journal of Educational Psychology, 108(4): 493–508. doi:10.1037/edu0000073

Fuchs, L.S., Schumacher, R.F., Long, J., Namkung, J., Hamlett, C.L., Cirino, P.T., Jordan, N.C., Siegler, R.S., Gersten, R., and Changas, P. (2013). Improving At-Risk Learners' Understanding of Fractions. Journal of Educational Psychology, 105(3): 683–700. doi:10.1037/a0032446 Full text

Fuchs, L.S., Schumacher, R.F., Long, J., Namkung, J., Malone, A., Hamlett, C.L., Jordan, N.C., Gersten, R., Siegler, R.S., and Changas, P. (2014). Does Working Memory Moderate the Effects of Fraction Intervention? An Aptitude-Treatment Interaction. Journal of Educational Psychology, 106(2): 499–514. doi:10.1037/a0034341

Fuchs, L.S., Schumacher, R.F., Long, J., Namkung, J., Malone, A., Wang, A., Hamlett, C.L., Jordan, N.C., Siegler, R.S., and Changas, P. (2016). Effects of Intervention to Improve At-Risk Fourth Graders' Understanding, Calculations, and Word Problems With Fractions. Elementary School Journal, 116(4): 625–651. doi:10.1086/686303

Fuchs, L.S., Sterba, S.K., Fuchs D., and Malone, A. (2016). Does Evidence-Based Fractions Intervention Address the Needs of Very Low-Performing Students?. Journal of Research on Educational Effectiveness, 9(4): 662–677.

Hansen, N., Jordan, N.C., Fenandez, E., Siegler, R.S., Fuchs, L., Gersten, R., and 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?

Jordan, N.C., Hansen, N., Fuchs, L.S., Siegler, R.S., Gersten, R., and Micklos, D. (2013). Developmental Predictors of Fraction Concepts and Fraction Procedures. Journal of Experimental Child Psychology, 116(1): 45–58. doi:10.1016/j.jecp.2013.02.001

Kurz, A., Elliott, S.N., Kettler, R.J., and Yel, N. (2014). Assessing Students' Opportunity to Learn the Intended Curriculum Using an Online Teacher Log: Initial Validity Evidence. Educational Assessment, 19(3): 159–184. doi:10.1080/10627197.2014.934606

Lortie-Forgues, H., and Siegler, R. S. (2017). Conceptual Knowledge of Decimal Arithmetic. Journal of Educational Psychology, 109(3): 374–386. doi:10.1037/edu0000148

Lortie-Forgues, H., Tian, J., and Siegler, R.S. (2015). Why is Learning Fraction and Decimal Arithmetic so Difficult? Developmental Review, 38: 201–221. doi:10.1016/j.dr.2015.07.008

Malone, A.S., and Fuchs, L.S. (2014). Comparing the Contribution of Teacher Versus Tutor Ratings of Inattentive Behavior in Predicting Mathematics Achievement. Remedial and Special Education, 35(6): 378–386. doi:10.1177/0741932514539855

Namkung, J.M., and Fuchs, L.S. (2016). Cognitive Predictors of Calculations and Number Line Estimation With Whole Numbers and Fractions Among At-Risk Students. Journal of Educational Psychology, 108(2): 214–228. doi:10.1037/edu0000055

Resnick, I., Jordan, N.C., Hansen, N., Rajan, V., Rodrigues, J., Siegler, R.S., and Fuchs, L.S. (2016). Developmental Growth Trajectories in Understanding of Fraction Magnitude From Fourth through Sixth Grade. Developmental Psychology, 52(5): 746–757. doi:10.1037/dev0000102

Seethaler, P.M., Fuchs, L.S., Star, J.R., and 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

Siegler, R. S., and Braithwaite, D. W. (2017). Numerical Development. Annual Review of Psychology , 68: 187–213. Retrieved from https://eric.ed.gov/?id=ED565890. Full text

Siegler, R.S. (2016). Magnitude Knowledge: The Common Core of Numerical Development. Developmental Science, 19(3): 341–361. doi:10.1111/desc.12395 Full text

Siegler, R.S., and Lortie-Forgues, H. (2014). An Integrative Theory of Numerical Development. Child Development Perspectives, 8(3): 144–150. doi:10.1111/cdep.12077

Siegler, R.S., and Lortie-Forgues, H. (2015). Conceptual Knowledge of Fraction Arithmetic. Journal of Educational Psychology, 107(3): 909–918. doi:10.1037/edu0000025

Siegler, R.S., and Pyke, A.A. (2013). Developmental and Individual Differences in Understanding of Fractions. Developmental Psychology, 49(10): 1994–2004. doi:10.1037/a0031200

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. doi:10.1016/ j.jecp.2013.11.014

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. doi:10.1177/0956797612440101

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. doi:10.1016/j.tics.2012.11.004

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

Tian, J. and Siegler, R.S. (2016). Fractions Learning in Children with Mathematics Difficulties. Journal of Learning Disabilities: 1–7. doi:10.1177/0022219416662032 Full text

Vukovic, R.K., Fuchs, L.S., Geary, D.C., Jordan, N.C., Gersten, R., and Siegler, R.S. (2014). Sources of Individual Differences in Children's Understanding of Fractions. Child Development, 85(4): 1461–1476. doi:10.1111/cdev.12218

Wang, Y., and Siegler, R.S. (2013). Representations of and Translation Between Common Fractions and Decimal Fractions. Chinese Science Bulletin, 58(36): 4630–4640. doi:10.1007/s11434–013–6035–4

Watts, T.W., Duncan, G.J., Chen, M., Claessens, A., Davis-Kean, P.E., Duckworth, P., Engle, M., Siegler, R., and Susperreguy, M.I. (2015). The Role of Mediators in the Development of Longitudinal Mathematics Achievement Associations. Child Development, 86(6): 1892–1907. doi:10.1111/cdev.12416

Watts, T.W., Duncan, G.J., Siegler, R.S., and Davis-Kean, P.E. (2014). What's Past is Prologue: Relations Between Early Mathematics Knowledge and High School Achievement. Educational Researcher, 43(7): 352–360. doi:10.3102/0013189X14553660?

Nongovernment report, issue brief, or practice guide

Fazio, L., and Siegler, R. (2012). Teaching Fractions, Volume 22 of Educational Practices Series. Geneva: UNESCO's International Bureau of Education. Full text