1. From the laboratory to the classroom: Creating and implementing a research-based curriculum around the use of comparison Courtney Pollack, Harvard University Dr. Jon R. Star, Harvard University
I. Abstract
This poster presents a research program that seeks to improve educational practice and student learning in mathematics by developing, implementing, and testing curriculum materials built on findings from cognitive science. We convey the process of conducting experimental classroom studies built on lab-based cognitive science research and the subsequent design and implementation of a supplemental first-year algebra curriculum based on comparison.
II. Learning & comparison: Cognitive science research
There is a great deal of cognitive research showing the benefits of comparison for learning in young children (e.g., Loewenstein & Gentner, 2001; Oakes & Ribar, 2005) and adults (e.g., Gentner, Loewnstein, & Thompson, 2003; Namy & Gentner, 2002). Yet, little of this type of research has been done in classrooms. Building on these findings, we engaged in small-scale experimental classroom studies to explore the benefits of comparison for students’ learning of mathematics, focusing on equation solving.
IV. Curriculum Design and Development
During , we worked with a small group of expert teachers to transform our experimental materials (see Figure 1) into a supplementary Algebra I curriculum that embodied the principles derived from previous experimental research (see Figure 2).
Side-by-side comparison
Labeled solution steps
Prompts to identify similarities & differences
Principles of comparison
III. Experimental classroom research
Our experimental research showed positive effects of comparison on student learning in controlled settings.
Two-year randomized controlled trial
Pilot testing
Curriculum design
Experimental classroom studies
Cognitive science research
Figure 1. Experimental comparison materials from Rittle-Johnson & Star (2007)
“Infuse” comparison into first-year algebra classes
Facilitate comparison of and reflection on multiple strategies
Goals
About 80 “worked example pairs”
Characters: Alex and Morgan
Four comparison types
Which is better?
Why does it work?
How do they differ?
Which is correct?
Discussion phases
Understand, compare, make connections
Help teachers facilitate comparison conversations
Supplementary materials
Usability
Beyond linear equation solving to include Algebra I content
New comparison type: Correct versus incorrect
Scope
7th grade students
Worked examples side-by-side
Compare side-by-side or reflect sequentially
Comparison condition: Greater procedural knowledge and flexibility
Rittle-Johnson & Star (2007)
7th and 8th grade students
What you compare matters
Solution methods: largest gains in conceptual knowledge and flexibility
Problem types: Support both to a lesser extent
Rittle-Johnson & Star (2009)
How does comparison affect student learning in real classroom environments?
V. Pilot testing
During the school year, we worked with 12 middle and high school teachers to test our materials in classrooms.
VI. Randomized controlled trials
Our revised materials are currently being tested using about 80 first-year algebra classrooms across Massachusetts.
One-week summer PD
Comparison activities
Create own worked example pairs
Model teaching with comparison
Tasks
Use materials 1-2 x per week
Videotape 2 x per month
Submit log after material use (e.g., time spent, student learning, teacher satisfaction)
Student assessments
Feedback
Ongoing feedback throughout the year
Student and teacher end-of-year interviews
Pilot year
Figure 2. Worked example from comparison curriculum
VII. Conclusion
Atkinson, Derry, Renkl, and Wortham (2000) acknowledge the gap between mathematics research in controlled laboratory settings and the relevance of its findings to classroom settings. They note that lab findings cannot alone improve classroom practice, but that “controlled experimental research grounded in cognitive science has substantially improved educational practice,” (p. 184). We hope to illustrate one way that building on experimental research can improve educational practice and student learning, by extending the benefits of learning through comparison to authentic classroom settings.
References
Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2),
Gentner, D., Loewnstein, J., & Thompson, L. (2003). Learning and transfer: A general role of analogical encoding. Journal of Educational Psychology, 95(2),
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.
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),
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),
Figure 3. Take-away page excerpt from the revised curriculum
We are currently finishing data collection of videos, logs, and student assessments for the first year of the RCT. We will continue to collect data during the school year.
Based on our pilot year feedback, we expanded our curriculum to include 150 worked example pairs. We also added a second page for each (see Figure 3).