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IES Grant

Title: Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics
Center: NCER Year: 2005
Principal Investigator: Star, Jon Awardee: Harvard University
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $1,014,175
Goal: Development and Innovation Award Number: R305H050005

Previous Grant Number: R305H050179
Previous Institution: Michigan State University

Co-Principal Investigator: Bethany Rittle-Johnson (Vanderbilt University)

This research team argues that there are times when students have the computational skills to solve specific mathematics problems, but do not understand the underlying mathematical principles or reasons why a specific set of computations results in the correct answer. The investigators hypothesize that structuring lessons so that contrasting examples are used to highlight specific principles will lead to better understanding and better integration of computational skills and conceptual knowledge. The research team will test whether or not the use of contrasting examples will improve students' ability to apply what they have learned and adapt existing procedures to solve novel math problems.

Structured Abstract

Purpose: Both international and national mathematics assessments indicate that, while U.S. students may learn to execute mathematical procedures, they often fail to gain the kind of robust, flexible knowledge that would allow them to apply what they have learned to new situations and novel problems. Flexible problem solving requires that students integrate the procedural knowledge that they have gained of specific actions for solving problems with conceptual knowledge of general mathematical principles. The purpose of this research project is to develop and evaluate an instructional approach that uses contrasting examples in order to foster flexible mathematical problem solving. Contrasting examples of solution procedures are presented during discussions in mathematics classrooms. Students are asked to compare these contrasting examples. This instructional approach should foster students' awareness of critical features of the procedures and to abstract their common underlying structure.

In this project, the use of contrasting examples is examined both in the context of algebra problem solving and computational estimation. At the conclusion of this project, the research team will have tested whether or not the use of contrasting examples will improve students' ability to apply what they have learned and adapt existing procedures to solve novel math problems.

Setting: The research studies take place in a private school in an urban center in the southern United States, and in public suburban and rural middle schools in both the southern United States, and in the Midwest.

Population: A total of approximately 825 fifth- and seventh-grade students are participating in this research. The majority of the children participating are Caucasian; approximately 10 percent are African-American. Approximately one-quarter of the participating children qualify for free and reduced lunch.

Intervention: Materials are being developed to support both students and teachers in the use of contrasting examples. In each experiment, students are presented with worked-out examples of mathematics problems and are asked to answer questions about the examples. Students using contrasting examples are shown a pair of worked examples illustrating different solutions to the same problem and are asked to compare and contrast the solution procedures.

Research Design and Methods: The researchers are comparing learning from contrasting examples to learning from sequentially presented examples (a more common educational approach) in five studies. In Studies 1 and 2, pairs of students are randomly assigned to condition, and the manipulation occurs while each pair studies worked examples and solves practice problems in their mathematics classrooms.

In Studies 3 and 4, classrooms are randomly assigned to condition, and the manipulation occurs both in partner activities and in whole-class discussions. In Study 5, the classroom intervention will be scaled up to more diverse classrooms in public schools as first steps towards assessing the generalizability of this teaching approach. Studies 1, 3, and 5 will be on linear equation solving with seventh-grade students, and Studies 2 and 4 will be on mental math and computational estimation with fifth-grade students.

Control Condition: Students in the control condition are presented the same worked examples as the treatment students, but are shown each worked example separately and are asked to think about the individual solutions.

Key Measures: Students are completing experimenter-developed tests that measure their ability to perform the linear equation solving or computational estimation that they are currently being taught.

Data Analytic Strategy: For the studies where pairs of students are assigned to condition, MANOVA techniques are used to compare performance of students in the two conditions. For studies where classrooms are assigned to condition, hierarchical linear modeling techniques are used.


Book chapter

Rittle-Johnson, B., and Star, J.R. (2011). The Power of Comparison in Learning and Instruction: Learning Outcomes Supported by Different Types of Comparisons. In J.P. Mestre, and B.H. Ross (Eds.), The Psychology of Learning and Motivation, Volume 55 (pp. 199–226). San Diego: Elsevier.

Journal article, monograph, or newsletter

Durkin, K., and Rittle-Johnson, B. (2012). The Effectiveness of Using Incorrect Examples to Support Learning About Decimal Magnitude. Learning and Instruction, 22(3): 206–214.

Rittle-Johnson, B., and Star, J.R. (2007). Does Comparing Solution Methods Facilitate Conceptual and Procedural Knowledge? An Experimental Study on Learning to Solve Equations. Journal of Educational Psychology, 99(3): 561–574.

Rittle-Johnson, B., and Star, J.R. (2009). Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving. Journal of Educational Psychology, 101(3): 529–544.

Rittle-Johnson, B., Star, J.R., and Durkin, K. (2009). The Importance of Prior Knowledge When Comparing Examples: Influences on Conceptual and Procedural Knowledge of Equation Solving. Journal of Educational Psychology, 3(4): 836–852.

Rittle-Johnson, B., Star, J.R., and Durkin, K. (2012). Developing Procedural Flexibility: Are Novices Prepared to Learn From Comparing Procedures?. British Journal of Educational Psychology, 82(3): 436–455.

Star, J.R., and Rittle-Johnson, B. (2008). Flexibility in Problem Solving: The Case of Equation Solving. Learning and Instruction, 18(6): 565–579.

Star, J.R., and Rittle-Johnson, B. (2009). It Pays to Compare: An Experimental Study on Computational Estimation. Journal of Experimental Child Psychology, 102(4): 408–426.

Star, J.R., and Rittle-Johnson, B. (2009). Making Algebra Work: Instructional Strategies That Deepen Student Understanding, Within and Between Algebraic Representations. ERS Spectrum, 27(2): 11–18.

Star, J.R., Kenyon, M., Joiner, R., and Rittle-Johnson, B. (2010). Comparison Helps Students Learn to be Better Estimators. Teaching Children Mathematics, 16(9): 557–563.

Star, J.R., Kenyon, M., Joiner, R., and Rittle-Johnson, B. (in press). Comparison Helps Students Learn to Solve Equations. Mathematics Teacher. Star, J.R., Rittle-Johnson, B., Lynch, K., and Perova, N. (2009). The Role of Prior Knowledge and Comparison in the Development of Strategy Flexibility: The Case of Computational Estimation. ZDM—The International Journal on Mathematics Education, 41(5): 569–579.

Nongovernment report, issue brief, or practice guide

Star, J.R. (2008). It Pays to Compare! Using Comparison to Help Build Students' Flexibility in Mathematics. Washington, DC: The Center for Comprehensive School Reform and Improvement.