Transcript 72x48 Poster Template

From the laboratory to the classroom: Designing a researchbased curriculum around the use of comparison
Courtney Pollack, Harvard University
Abstract
Comparison Curriculum
Dr. Jon R. Star, Harvard University
Comparison Curriculum
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.
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 1. Worked example pair focused on strategy choice.
Figure 3. Worked example pair focused on understanding
mathematical concept(s) by examining how two problems
differ.
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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.
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
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.
Development: Moving from the Lab to the Classroom, cont.
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.
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.