Abstract

1. From the laboratory to the classroom: Designing a research-based curriculum around the use of comparison Courtney Pollack, Harvard University Dr. Jon R. Star, Harvard University Abstract Comparison Curriculum Comparison Curriculum Development: Moving from the Lab to the Classroom, cont. This poster shares our experience designing and implementing an Algebra I curriculum based on research findings about how students learn using comparison. Our goal was to bridge the gap between experimental lab-based research and classroom practice by transforming a body of research emerging from the psychological literature into usable, palatable, and effective materials for the classroom. We discuss the questions we encountered regarding the design and pilot implementation of the curriculum materials and our resulting decisions. We offer our story in an effort to assist future researchers and curriculum developers who seek to bridge the research-practice gap. The comparison problems used in our prior research only covered linear equation solving and were restricted to a small set of linear equation types, so we knew the curriculum would require greater coverage of topics in Algebra I. We considered questions about how many worked examples to include, and which mathematical topics would be amenable to learning via comparison. To address some of these issues, we examined the scope of current Algebra I curricula to create a set of topics and sub-topics that we felt were conducive to learning through comparison. We also considered what types of worked example pairs to include. Based on our prior research, we included worked example pairs for comparing solution methods and problem types. We included a new comparison type that we thought would be useful, though it represented a departure from our research findings. In this new problem type, one method is correct, while the other method contains an error and resulting incorrect answer. We present an example of each problem type in Figures 1-4. Finally, we considered how to distribute the worked example pairs across a typical Algebra I curriculum. We found that some topics lent themselves more or less well to a specific worked example pair type. We created multiple worked example pairs for other topics that were more favorable to more than one worked example pair type. Usability of Materials We never intended for classroom teachers to implement the worked example pairs from our research materials as-is. Our experimental materials were designed for use with student pairs and with minimal teacher or whole class time, which would likely not be suitable for most classrooms. To accommodate the need for flexibility in instructional formats, the curriculum materials were given to teachers in both print and electronic form. We also made the materials more engaging by designing two characters, Alex and Morgan, who each correspond to a solution method. To facilitate discussion, each worked example pair has accompanying questions, intended to guide a three-part discussion. Research on Comparison There is a great deal of cognitive research showing the benefits of comparison for learning (e.g., Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005). However, little of this type of research has been done in classrooms. Comparison has also played a fundamental role in mathematics education reform. The National Council of Teachers of Mathematics Standards underscores the sharing and comparing of solution methods (1989, 2000). Comparing, sharing, and discussing solution strategies have been central to the principles of reform pedagogy (Silver, Ghousseini, Gosen, Charalambous, & Strawhun, 2005). Recently, building on existing laboratory studies, we have been engaged in small-scale experimental classroom studies to explore the benefits of comparison for students’ learning of mathematics, focusing on equation solving. Rittle-Johnson and Star (2007) showed initial empirical evidence for the efficacy of comparison for linear equation solving. In this study, seventh grade students compared either a pair of worked examples presented side-by-side on the same page or reflected on a pair of worked examples presented sequentially. In each problem pair, two different solution methods were presented, one conventional method and either a shortcut method or less efficient method. Results of this study showed that students in the comparison condition showed greater procedural knowledge and flexibility than students in the sequential condition. Rittle-Johnson and Star (2009) lent further support to these results, showing that the use of comparison can increase procedural knowledge flexibility and support conceptual knowledge for linear equation solving. Additionally, this study extended the findings in Rittle-Johnson and Star (2007), by showing that comparison of two different solution methods or two different problem types supported students’ procedural flexibility and conceptual knowledge. Taken together, these studies formed the foundation for the beginning of the creation of our research-based curriculum. The central focus of this research foundation is a “worked example pair,” a one page, side-by-side presentation of two problems that differ either by problem type or solution method. The worked example pairs serve as a medium to facilitate students’ comparison of and reflection on multiple strategies. Figure 3. Worked example pair focused on understanding mathematical concept(s) by examining how two problems differ. Figure 1. Worked example pair focused on strategy choice. Future Directions Even when basing the development of our comparison curriculum on the principles we gained from prior research, we needed to make decisions for which research did not exist. We believe we have maintained the specific research findings that we sought to instantiate. The creation of our comparison curriculum represents the first step in an iterative development process. As of August 2010, the refined version of our curriculum is in use; the curriculum is being tested for efficacy as well. As we look ahead to further iterations, we will continue to consider questions regarding the nature of research-based curricula and curriculum development more generally. References Loewenstein, J., & Gentner, D. (2001). Spatial mapping in preschoolers: Close comparisons facilitate far mappings. Journal of Cognition and Development, 2, 189–219. Namy, L. L., & Gentner, D. (2002). Making a silk purse out of two sow’s ears: Young children’s use of comparison in category learning. Journal of Experimental Psychology: General, 131, 5-15. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Oakes, L. M., & Ribar, R. J. (2005). A comparison of infants’ categorization in paired and successive presentation familiarization tasks. Infancy, 7, 85–98. Rittle-Johnson, B. & 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. & 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. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhum, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior,24, 287-301. Development: Moving from the Lab to the Classroom When beginning development of the comparison curriculum program, we anticipated that the curriculum would need to be altered considerably in terms of the density and usability of the materials that had been created for our research studies. Density of Materials In moving from the laboratory to the classroom, the goal was to create supplemental materials that ‘infused’ comparison into the classroom and could be used in conjunction with existing curricula. Figure 2. Worked example pair focused on understanding why a strategy works. Figure 4. Worked example pair focused on identifying and explaining which strategy is correct.