Project

Leveraging Comparison and Explanation of Multiple Strategies (CEMS) to Improve Algebra Learning

Goal: This NSF-sponsored grant refines our hypothesis that productive learning of algebra is supported by reflection on multiple solution strategies through comparison and explanation of the reasons behind the strategies (Comparison and Explanation of Multiple Strategies: CEMS). Existing theories of algebra learning focus on building conceptual knowledge and place less emphasis on how students gain expertise with symbolic strategies. Working with symbolic strategies is essential in algebra learning. Students need to develop procedural flexibility - knowing multiple strategies for solving a problem and selecting the most appropriate strategy for a given problem - and understand the conceptual rationale behind commonly used strategies. Knowledge of strategies (procedural knowledge) supports gains in both procedural flexibility and conceptual knowledge of algebra (Schneider, Star & Rittle-Johnson, 2011). In small-scale studies, redesigning lessons on equation solving to integrate a CEMS approach supported greater procedural knowledge, flexibility and/or conceptual knowledge than completing the lessons without a CEMS approach (Rittle-Johnson & Star, 2007, 2009; Rittle-Johnson, Star, & Durkin, 2009, 2012; Star & Rittle-Johnson, 2009). A preliminary set of supplemental materials to support a CEMS approach across the Algebra I curriculum has been developed, with evidence that classroom teachers can implement the materials with good fidelity (Star, Pollack, et al., 2015).

Across three years, we will work with teachers to integrate a CEMS approach into their teaching of four Algebra I units. In Year 1, we will work with a small number of teachers to refine our existing CEMS materials, to integrate the materials into their curriculum, and to validate outcome measures that assess multiple types of knowledge (e.g., procedural flexibility, conceptual knowledge, and procedural knowledge). In Year 2, we will evaluate the effects of teachers using our materials versus a “business as usual” control for each of the four units. In Year 3, we will study the effects of the CEMS approach versus business as usual with a larger group of teachers; we will also study the quality of implementation and impact on student outcomes after treatment teachers have gained some proficiency with the CEMS approach. Using both quantitative and qualitative analyses, we will evaluate the hypotheses that: a) Classroom teachers can successfully and consistently integrate a CEMS approach in their algebra instruction, b) Students’ procedural flexibility, procedural knowledge, and conceptual knowledge for a variety of algebra topics can be reliably assessed and each type of knowledge is positively related and predictive of one another over time, and c) Integrating a CEMS approach supports better procedural flexibility, conceptual knowledge, and procedural knowledge for a variety of algebra topics (units) than business as usual instruction.

Updates
0 new
7
Recommendations
0 new
1
Followers
0 new
25
Reads
0 new
415

Project log

Abbey M. Loehr
added an update
Loehr, A. M., Durkin, K., Rittle-Johnson, B., Star, J. R. (2019, April). Impact of comparison and explanation of multiple strategies on learning and flexibility in algebra. Paper presented at the American Educational Research Association (AERA), Toronto, Canada.
 
Bethany Rittle-Johnson
added an update
Loehr, A. M., Rittle-Johnson, B., Star, J. R., & Desharnais, C. (2018, April). Developing a more comprehensive measure of formal algebra knowledge. Poster presented at the American Educational Research Association (AERA), New York City, NY.
 
Bethany Rittle-Johnson
added an update
Durkin, K., Loehr, A. M., Rittle-Johnson, B., Star, J. (2018, April). Effects of encouraging comparison and explanation of multiple strategies on instructional practices in algebra classrooms. Roundtable presentation at the American Educational Research Association (AERA), New York City, NY
 
Bethany Rittle-Johnson
added an update
Poster on our new procedural flexibility measure for Algebra I content
Zhang, Y., Fine, S., Loehr, A., Star, J., & Rittle-Johnson, B. (2018, May). Procedural Flexibility for Algebra: Assessment Development. Poster presentation at the 8th East Asia Regional Conference on Mathematics Education. Taipei, Taiwan.
 
Bethany Rittle-Johnson
added an update
Upcoming presentation summarizes past findings and outlines design of new materials and study.
Rittle-Johnson, B. Star, J., Durkin, K. & Loehr, A. (2018, May). Comparing solution strategies to promote algebra learning and flexibility. Paper to be presented at the 8th East Asia Regional Conference on Mathematics Education. Taipei, Taiwan.
 
Bethany Rittle-Johnson
added a research item
Education policy should aim to promote instructional methods that are easy for teachers to implement and have demonstrable, positive impact on student learning. Our research on comparison and explanation of multiple strategies illustrates the promise of this approach. In several short-term experimental, classroom-based studies, comparing different strategies for solving the same problem was particularly effective for promoting student learning. Thus, we developed a supplemental Algebra 1 curriculum to foster comparison in combination with explanation of multiple strategies. In a randomized control trial, teachers used our materials as intended, but much less often than expected, and student learning was not greater in experimental classrooms. Yet greater use of our comparison materials was associated with greater student learning, suggesting the approach has promise when used sufficiently often. These studies provide some evidence that easy-to-implement reforms can change teacher practice and improve student learning.
Bethany Rittle-Johnson
added a research item
Comparison is a fundamental cognitive process that supports learning in a variety of domains. To leverage comparison in mathematics instruction, evidence-based guidelines are needed for how to use comparison effectively. In this chapter, we review our classroom-based research on using comparison to help students learn mathematics. In five short-term experimental, classroom-based studies, we evaluated two types of comparison for supporting the acquisition of mathematics knowledge and tested whether prior knowledge moderated their effectiveness. Comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods. We next developed a supplemental Algebra 1 curriculum to foster comparison and evaluated its effectiveness in a randomized-control trial. Teachers used our supplemental materials much less often than expected, and student learning was not greater in classrooms that had been assigned to use our materials. Students’ procedural knowledge was positively related to greater implementation of the intervention, suggesting the approach has promise when used sufficiently often. This study suggests that teachers may need additional support in deciding what to compare and when to use comparison.
Kelley Durkin
added a research item
Comparison is a fundamental cognitive process that can support learning in a variety of domains, including mathematics. The current paper aims to summarize empirical findings that support recommendations on using comparison of multiple strategies in mathematics classrooms. We report the results of our classroom-based research on using comparison of multiple strategies to help students learn mathematics, which includes short-term experimental research and a year-long randomized controlled trial using a researcher-designed supplemental Algebra I curriculum. Findings indicated that comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods, but that teachers may need additional support in deciding what to compare and when to use comparison. Drawing from this research, we offer instructional recommendations for the effective use of comparison of multiple strategies for improving mathematics learning, including (a) regular and frequent comparison of alternative strategies, particularly after students have developed some fluency with one initial strategy; (b) judicious selection of strategies and problems to compare; (c) carefully-designed visual presentation of the multiple strategies; and (d) use of small group and whole class discussions around the comparison of multiple strategies, focusing particularly on the similarities, differences, affordances, and constraints of the different approaches. We conclude with suggestions for future work on comparing multiple strategies, including the continuing need for the development of, and rigorous evaluation of, curriculum materials and specific instructional techniques that effectively promote comparison.
Bethany Rittle-Johnson
added a project goal
This NSF-sponsored grant refines our hypothesis that productive learning of algebra is supported by reflection on multiple solution strategies through comparison and explanation of the reasons behind the strategies (Comparison and Explanation of Multiple Strategies: CEMS). Existing theories of algebra learning focus on building conceptual knowledge and place less emphasis on how students gain expertise with symbolic strategies. Working with symbolic strategies is essential in algebra learning. Students need to develop procedural flexibility - knowing multiple strategies for solving a problem and selecting the most appropriate strategy for a given problem - and understand the conceptual rationale behind commonly used strategies. Knowledge of strategies (procedural knowledge) supports gains in both procedural flexibility and conceptual knowledge of algebra (Schneider, Star & Rittle-Johnson, 2011). In small-scale studies, redesigning lessons on equation solving to integrate a CEMS approach supported greater procedural knowledge, flexibility and/or conceptual knowledge than completing the lessons without a CEMS approach (Rittle-Johnson & Star, 2007, 2009; Rittle-Johnson, Star, & Durkin, 2009, 2012; Star & Rittle-Johnson, 2009). A preliminary set of supplemental materials to support a CEMS approach across the Algebra I curriculum has been developed, with evidence that classroom teachers can implement the materials with good fidelity (Star, Pollack, et al., 2015).
Across three years, we will work with teachers to integrate a CEMS approach into their teaching of four Algebra I units. In Year 1, we will work with a small number of teachers to refine our existing CEMS materials, to integrate the materials into their curriculum, and to validate outcome measures that assess multiple types of knowledge (e.g., procedural flexibility, conceptual knowledge, and procedural knowledge). In Year 2, we will evaluate the effects of teachers using our materials versus a “business as usual” control for each of the four units. In Year 3, we will study the effects of the CEMS approach versus business as usual with a larger group of teachers; we will also study the quality of implementation and impact on student outcomes after treatment teachers have gained some proficiency with the CEMS approach. Using both quantitative and qualitative analyses, we will evaluate the hypotheses that: a) Classroom teachers can successfully and consistently integrate a CEMS approach in their algebra instruction, b) Students’ procedural flexibility, procedural knowledge, and conceptual knowledge for a variety of algebra topics can be reliably assessed and each type of knowledge is positively related and predictive of one another over time, and c) Integrating a CEMS approach supports better procedural flexibility, conceptual knowledge, and procedural knowledge for a variety of algebra topics (units) than business as usual instruction.