IES PRACTICE GUIDE WHAT WORKS CLEARINGHOUSE Developing Effective Fractions Instruction for Kindergarten Through 8th Grade NCEE 2010-4039 U.S. DEPARTMENT OF EDUCATION Review of Recommendations The Institute of Education Sciences (IES) publishes practice guides in education to bring the best available evidence and expertise to bear on current challenges in education. Authors of practice guides combine their expertise with the findings of rigorous research, when available, to develop specific recommendations for addressing these challenges. The authors rate the strength of the research evidence supporting each of their recommendations. See Appendix A for a full description of practice guides. The goal of this practice guide is to offer educators specific evidence-based recommendations that address the challenge of improving students’ understanding of fraction concepts in kindergarten through 8th grade. The guide provides practical, clear information on critical topics related to the teaching of fractions and is based on the best available evidence as judged by the authors. Practice guides published by IES are offered on our website at whatworks.ed.gov/publications/ practiceguides. Practice guides published to date are shown in the following table. Relevant for Relevant for Relevant for All Grade Elementary Secondary Practice Guides Published Levels School School Encouraging Girls in Math and Science (September 2007) Organizing Instruction and Study to Improve Student Learning (September 2007) Turning Around Chronically Low-Performing Schools (May 2008) Using Student Achievement Data to Support Instructional Decision Making (September 2009) Assisting Students Struggling with Reading: Response to Intervention (RtI) and Multi-Tier Intervention in the Primary Grades (February 2009) Effective Literacy and English Language Instruction for English Learners in the Elementary Grades (December 2007) Improving Reading Comprehension in Kindergarten Through 3rd Grade (September 2010) Reducing Behavior Problems in the Elementary School Classroom (September 2008) Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools (April 2009) Developing Effective Fractions Instruction for Kindergarten Through 8th Grade (September 2010) Improving Adolescent Literacy: Effective Classroom and Intervention Practices (August 2008) Structuring Out-of-School Time to Improve Academic Achievement (July 2009) Dropout Prevention (August 2008) Helping Students Navigate the Path to College: What High Schools Can Do (September 2009) IES Practice Guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade September 2010 Panel Robert Siegler (Chair) CARNEGIE MELLON UN I V E R S I T Y Thomas Carpenter UN I V E R S I T Y OF WISCONSIN –MA D I S O N Francis (Skip) Fennell MCDANIEL COLLEGE , WESTMINSTER , MD David Geary UN I V E R S I T Y OF MI S S O U R I AT CO L U M B I A James Lewis UN I V E R S I T Y O F NEBRASKA –lI N C O L N Yukari Okamoto UN I V E R S I T Y OF CALIFORNIA –sA N T A BA R B A R A Laurie Thompson EL E M E N T A R Y TE A C H E R Jonathan Wray HOWARD COUNTY (MD) PU B L I C SC H O O L S Staff Jeffrey Max Moira McCullough Andrew Gothro Sarah Prenovitz MATHEMATICA PO L I C Y RE S E A R C H Project Officer Susan Sanchez IN S T I T U T E O F ED U C A T I O N SC I E N C E S NCEE 2010-4039 U.S. DEPARTMENT OF EDUCATION ( i ) Review of Recommendations This report was prepared for the National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences under Contract ED-07-CO-0062 by the What Works Clearinghouse, which is operated by Mathematica Policy Research. Disclaimer The opinions and positions expressed in this practice guide are those of the authors and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education. This practice guide should be reviewed and applied according to the specific needs of the educators and education agency using it, and with full realization that it rep­ resents the judgments of the review panel regarding what constitutes sensible practice, based on the research that was available at the time of publication. This practice guide should be used as a tool to assist in decisionmaking rather than as a “cookbook.” Any references within the document to specific education products are illustrative and do not imply endorsement of these products to the exclusion of other products that are not referenced. U.S. Department of Education Arne Duncan Secretary Institute of Education Sciences John Q. Easton Director National Center for Education Evaluation and Regional Assistance Rebecca Maynard Commissioner September 2010 This report is in the public domain. Although permission to reprint this publication is not necessary, the citation should be: Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from whatworks.ed.gov/ publications/practiceguides. What Works Clearinghouse Practice Guide citations begin with the panel chair, followed by the names of the panelists listed in alphabetical order. This report is available on the IES Web site at http://ies.ed.gov/ncee and whatworks.ed.gov/ publications/practiceguides. Alternate Formats On request, this publication can be made available in alternate formats, such as Braille, large print, or computer diskette. For more information, contact the Alternate Format Center at 202–260–0852 or 202-260-0818. ( ii ) Table of Contents Developing Effective Fractions Instruction for Kindergarten Through 8th Grade Table of Contents Review of Recommendations ......................... 1 Acknowledgments . 2 Institute of Education Sciences Levels of Evidence for Practice Guides . 3 Introduction to the Developing Effective Fractions Instruction for Kindergarten Through 8th Grade Practice Guide . 6 Recommendation 1. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts . 12 Recommendation 2. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward . 19 Recommendation 3. Help students understand why procedures for computations with fractions make sense . 26 Recommendation 4. Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems . 35 Recommendation 5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them ........................... 42 Glossary .................................. 47 Appendix A. Postscript from the Institute of Education Sciences . 49 Appendix B. About the Authors . 51 Appendix C. Disclosure of Potential Conflicts of Interest . 54 Appendix D. Rationale for Evidence Ratings . 55 Appendix E. Evidence Heuristic . 68 Endnotes ................................. 70 References ................................. 76 Index of Key Mathematical Concepts ..................... 84 ( iii ) Table of Contents continued List of Tables Table 1. Institute of Education Sciences levels of evidence for practice guides . 4 Table 2. Recommendations and corresponding levels of evidence . 11 Table D.1. Studies of interventions that used number lines to improve understanding of whole number magnitude that met WWC standards (with or without reservations) . 58 Table D.2. Studies of interventions that developed conceptual understanding of fraction computation that met WWC standards (with or without reservations) . 61 Table E.1. Evidence heuristic . 68 List of Figures Figure 1. Sharing a set of objects evenly among recipients . 14 Figure 2. Partitioning both multiple and single objects . 15 Figure 3. Student work for sharing four pizzas among eight children . 16 Figure 4. Finding equivalent fractions on a number line . 23 Figure 5. Using fraction strips to demonstrate equivalent fractions . 24 Figure 6. Fraction circles for addition and subtraction . 28 Figure 7. Redefining the unit when multiplying fractions . 29 Figure 8. Using ribbons to model division with fractions . 30 Figure 9. Ratio table for a proportion problem . 39 Figure 10. Ratio table for exploring proportional relations . 40 List of Examples Example 1. Measurement activities with fraction strips . 21 Example 2. Introducing fractions on a number line . 22 Example 3. Strategies for estimating with fractions . 31 Example 4. Problems encouraging specific strategies . 38 Example 5. Why cross-multiplication works . 39 ( iv ) Review of Recommendations Recommendation 1. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. • Use equal-sharing activities to introduce the concept of fractions. Use sharing activities that involve dividing sets of objects as well as single whole objects. • Extend equal-sharing activities to develop students’ understanding of ordering and equivalence of fractions. • Build on students’ informal understanding to develop more advanced