Grant Closed

Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics

NCER
Program: Education Research Grants
Program topic(s): Cognition and Student Learning
Award amount: $1,014,175
Principal investigator: Jon Star
Awardee:
Harvard University
Year: 2005
Project type:
Development and Innovation
Award number: R305H050005

Purpose

In this project, researchers proposed to develop and evaluate an instructional approach that uses contrasting examples in order to foster flexible mathematical problem solving. 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 instructional approach asks students to compare solution procedures in the context both of algebra problem solving and computational estimation that are presented during discussions in mathematics classrooms. At the conclusion of this project, the research team aimed to have developed an approach to integrate contrasting examples into instruction and to have tested whether or not it improves students' ability to apply what they have learned and adapt existing procedures to solve novel math problems.

Structured Abstract

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.

Sample

A total of approximately 825 grade 5 and 7 students are participating in this research. The majority of the children participating are White, and approximately 10 percent are African American. Approximately 25 percent of the participating children qualify for free and reduced-price 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.

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, multivariate analysis of variance (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.

People and institutions involved

IES program contact(s)

Elizabeth Albro

Elizabeth Albro

Commissioner of Education Research
NCER

Products and publications

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

Select Publications:

Book chapters

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 articles

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.

Additional project information

Previous award details:

Previous award number:
R305H050179
Previous awardee:
Michigan State University

Supplemental information

Co-Principal Investigator(s): Rittle-Johnson, Bethany

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.

Questions about this project?

To answer additional questions about this project or provide feedback, please contact the program officer.

 

Questions about this project?

To answer additional questions about this project or provide feedback, please contact the program officer.

 

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