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IES Grant

Title: Improving Children's Understanding of Mathematical Equivalence
Center: NCER Year: 2011
Principal Investigator: McNeil, Nicole Awardee: University of Notre Dame
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $565,456
Type: Development and Innovation Award Number: R305A110198

Purpose: Mathematical equivalence is widely regarded as one of the most important concepts for developing children's algebraic thinking. Although decades of research have shown that children struggle to understand this fundamental concept, researchers have yet to develop an intervention that produces mastery-level understanding. The researchers in this project will build on the success of their previous IES development award ( to develop and pilot test a comprehensive intervention that helps elementary school children achieve a mastery-level understanding of mathematical equivalence. The purpose of this project is to expand and integrate four successful component interventions into a comprehensive intervention that helps children reach mastery-level understanding of mathematical equivalence.

Project Activities: This development project involves both an iterative development phase, and a classroom-based pilot study. The four components will be integrated and optimized into a student workbook and classroom activities package following an iterative development process, which will include one-on-one design experiments. When the team has completed the development phase, they will carry out a classroom-based pilot study in which the promise of the comprehensive intervention will be tested.

Products: The products of these design experiments include student workbooks, activities, and a teacher manual. Scholarly reports of the findings will also be produced.

Structured Abstract

Setting: The study will take place in South Bend, Indiana.

Population: Approximately 200 second grade children from private and public schools will participate across two design experiments and a pilot study. Based on the characteristics of the targeted communities and schools, it is expected that the ethnic/racial composition will be approximately 59% White, 18% Hispanic or Latino, 15% African American, 5% multiethnic, and 3% Asian. Approximately 46% will be children from low-socioeconomic backgrounds.

Intervention: The intervention will be designed to take place over the course of 48 15-minute sessions. It will include four components supported by cognitive science research: (a) non-traditional arithmetic practice, such as problems where students are asked to solve equations where the placement of the equal sign is before the addends as opposed to after them, and where "=" is replaced with the words "is equal to"; (b) lessons that introduce the equal sign outside of arithmetic contexts; (c) "concreteness fading" exercises that strengthen the mappings between real-world relational contexts and the corresponding mathematics symbols; and (d) activities that require children to compare and explain different problem formats and problem-solving strategies.

Research Design and Methods: The research plan involves an iterative development process (Years 1 and 2) and a classroom-based pilot study (Year 3). During the first two years, the research team will focus on refining lessons that introduce the equals sign outside of an arithmetic context. The series of lessons will begin with examples of equivalence outside of arithmetic, then introduce concreteness fading in the middle sessions, and conclude with activities that require comparison and explanation of different problems and strategies. The lessons will use nontraditional arithmetic practice throughout all phases of the intervention. The researchers will make workbooks that use different combinations of nontraditional practice forms and will test the differences in student performance subsequent to using the different workbooks. Throughout the first two years, they will revise the workbooks and teaching materials for the intervention. After the iterative development process, they will perform a classroom-based pilot study to demonstrate the promise of the comprehensive intervention. Pretest and posttest measures will be taken, and results will be compared to benchmarks and the performance of students who received nontraditional instruction but not the full series of nontraditional practice forms developed in the first phase of the study.

Control Condition: The researchers will use benchmarks based on data collected in their previous studies. In addition to these benchmarks, they will include a comparison group in the pilot study in which children receive the nontraditional arithmetic component intervention that was found to be effective in our previous IES development grant instead of the comprehensive intervention for all 48 sessions.

Key Measures: All students will complete both standardized tests (e.g., the Ohio Test of Basic Skills) and experimenter-developed measures of understanding of mathematical equivalence and computational fluency. In addition to the student measures, teachers will complete measures that allow us to assess feasibility and usability of the teaching materials.

Data Analytic Strategy: The researchers will run both benchmark comparisons and statistical analyses using repeated-measures ANOVA to examine gains in understanding of mathematical equivalence and computational fluency from pretest to posttest in both groups.

Products and Publications

Book chapter

McNeil, N.M., Byrd, C.E., Fuhs, M.W., and O'Rear, C. (in press). Understanding and Alleviating Children's Difficulties With Mathematical Equivalence. In D.C. Geary, D.B. Berch, and K. Mann Koepke (Eds.), Mathematical Cognition and Learning, Volume 3. San Diego: Elsevier Academic Press.

Journal article, monograph, or newsletter

Fyfe, E.R., McNeil, N.M., Son, J.Y., and Goldstone, R.L. (2014). Concreteness Fading in Mathematics and Science Instruction: A Systematic Review. Educational Psychology Review, 26(1): 9–25.

McNeil, N.M. (2014). A Change–Resistance Account of Children's Difficulties Understanding Mathematical Equivalence. Child Development Perspectives, 8(1): 42–47.