 
 
Principal Investigator: Jon Star
CoPrincipal Investigator: Bethany RittleJohnson, Vanderbilt University
Title: Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics
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 flexible knowledge that would allow them to apply what they have learned to new situations and novel problems. Flexible problem solving requires that students combine their knowledge of how to perform specific problemsolving tasks (procedural knowledge) with their understanding of general mathematical principles (conceptual knowledge).
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, and students are asked to compare them. This instructional approach should foster students’ awareness of critical features of the procedures and help them abstract their 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 the use of contrasting examples improves 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 seventhgrade students are participating in this research. The majority of the children participating are white; approximately 10 percent are African American. Approximately onequarter of the participating students qualify for free or reducedprice 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 workedout 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 wholeclass discussions. In Study 5, the classroom intervention will be scaled up to more diverse classrooms in public schools as first steps toward assessing the generalizability of this teaching approach. Studies 1, 3, and 5 will be on linear equation solving with seventhgrade students, and Studies 2 and 4 will be on mental math and computational estimation with fifthgrade 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 solutions.
Key Measures: Students are completing experimenterdeveloped 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.


