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The IERI Research Community :: Projects

Collaborative Research I: Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning

Sharon J. Derry

Martha W. Alibali, Eric Knuth, Mitchell Nathan


Background: Algebra functions as a "gatekeeper" to advanced mathematics as well as to future educational and employment opportunities. While educational reform shows progress in many areas of mathematics, algebra instruction and curricula have remained relatively unchanged. Our joint project (see also Collaborative Research II and Collaborative Research III) is significant because little research has focused on the development of algebraic reasoning in the middle grades or on middle school teachers' knowledge of algebra as pertaining to the integration of algebraic ideas into the middle school curriculum or to the development of students' algebraic reasoning.

Purpose: Our education goal was to perfect the design of a year-long graduate course that will continue to be offered through the University of Wisconsin-Madison for middle school mathematics and special education teachers. The course "Understanding and Cultivating Students' Algebraic Thinking" is designed to help teachers improve their ability to assist students in making the transition from arithmetic to algebraic reasoning, a shift that is difficult for many students and can hinder their ability to succeed with more advanced mathematics.

Our research goal was to document changes in teacher cognition and classroom practice and to examine the effectiveness of contrasting cases instructional activities used in the course. We are using these accounts to extend knowledge and cognitive theory about teacher learning and to inform the design of our course as we enter a "scaling-up" phase of our work. We hope to offer the course on a larger scale throughout Wisconsin as an innovative, theory-based online course.

Intervention: The courses evolved around a series of designed instructional activities nicknamed "SAMS" (Sample Algebraic Modules), the structure of which was derived from cognitive research on contrasting cases, transfer, and learning through analogy. SAMs aided teachers in abstracting common patterns (generalizing) across many cases of student and teacher mathematics work. The cases teachers examined were mostly examples of problem solving from mathematics teaching practice.

Setting: Middle school mathematics and special education teachers from 5 school districts in Wisconsin participated in an intensive summer workshop held on the campus of the University of Wisconsin-Madison, followed by a series of monthly meetings held in different locations throughout the academic year. They also participated in online activities. The "design experiment" started in August 2004 and continued through May 2006. We are still analyzing data, as well as planning for the next course iteration.

Research Design: We employed videography and other ethnographic methods (classroom observation, videotaping and focus groups) to study teachers' classroom interaction as well as quasi-experimental methods using pre- and post-tests to evaluate the summer workshop and end-of-course posttests. We used multiple instruments, including a video-assessment that we developed and validated with correlational methods. Methods that apply include: Correlational, descriptive, interview, narrative synthesis, quasi-experimental, videography.

We mailed recruitment letters describing our program to Dane County, WI schools, offering graduate credit and a small stipend for participation, accepting all teachers who applied. Teachers came from the same school district in year one but in year two came from five school districts. In year one twelve regular education and special education teachers completed the year-long course: five 6th grade teachers, three 7th grade teachers and four 8th grade teachers. In year two ten regular education and special education teachers completed the course: five 6th grade teachers, two 7th grade and three 8th grade teachers. The range of years teaching was one to 28 years. Only three teachers had certificates or formal training in mathematics.

In addition to ethnographic data collection, two formal assessments were used: One test examined teachers' content and pedagogical content knowledge and ability to reflect on their solutions [adapted from another project (Seago, Mumme, & Branca, 2004)]. The other "Analysis of Student Work" (ASW) assessment, which we developed and validated, examined teachers' ability to analyze video of student work (Hackbarth, Derry, & Wilsman, 2006). The rubric for scoring the ASW produces four sub-scales measuring different facets of a teacher's analytic capability: 1) Teacher's analysis of the problem representation; 2) Teacher's inferences about students' ability and understanding of math concepts and algebraic thinking; 3) Teacher's inferences about students' mathematical development trajectory as a basis for pedagogical decision making; and 4) Teachers' spontaneous "reflection-in-action" (Sch n, 1983) on their analyses. We analyzed assessment gains in teacher performance using ANOVA, used principal components analysis in our instrument-validation study, and we are now analyzing video data using mixed qualitative methods, including interaction analysis (Jordan & Henderson, 1995).

Findings: Three subscales of the ASW instrument appear to measure aspects of what Shulman (1987) calls "pedagogical content knowledge" (PCK). The third subscale measures what Sch n (1983) terms "reflection-in-action." In the first year the teachers improved significantly on the three PCK scales of the ASW but not on reflection. They did not improve on the algebra content test. The second-year teachers, who entered with higher scores on the ASW and the content test, exhibited a different pattern of improvement based on our preliminary analyses: those who scored less than 90 percent on the content pretest improved their performance significantly on the algebra content test. The only improvement on the ASW was related to reflection. This pattern of results raises many questions that we are now investigating as we complete analysis of our data and plan our next course offering.

References Cited
Hackbarth, A. J., Derry, S. J., & Wilsman, M. J. (2006). Measuring teachers' algebraic reasoning: Development and preliminary validation of a video assessment task. In Proceedings of the Seventh International Conference of the Learning Sciences. Bloomington, IN (forthcoming).

Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and practice. The Journal of the Learning Sciences, 4(1), 30-103.

Schon, D. A. (1983). The reflective practitioner: How professionals think in action. New York: Basic Books.

Seago, N., Mumme, J., & Branca, N. (2004). Learning and teaching linear functions: Video cases for mathematics teacher professional development, 6-10. Portsmouth, NH: Heinemann.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.

Derry, S. J. (in press). Video research in classroom and teacher learning (Standardize that!). In R. Goldman, R. Pea, B. Barron, S. J. Derry (Eds). Video research in the learning sciences.

Derry, S.J., Wilsman, M.J., & Hackbarth, A.J. (2006) Using Contrasting Case Activities to Deepen Teacher Understanding of Algebraic Thinking, Student Learning and Teaching. Mathematics Teaching and Learning Journal. (forthcoming)

Hackbarth, A. J., Derry, S. J., & Wilsman, M. J. (2006). Measuring teachers' algebraic reasoning: Development and preliminary validation of a video assessment task. International Conference of Learning Sciences. Bloomington, IN.

Wilsman, M.J., Hackbarth, A.J. & Derry (2006). Using Artifacts of Classroom Practice to Build Teachers' Expertise in Distinguishing Conceptual and Computational Understandings. Paper presented at the annual meeting of the National Council of Teachers of Mathematics. St. Louis, MO.

Hackbarth, A.J. & Wilsman, M.J. (2006). The Hat Problem. Annual meeting of the Wisconsin Mathematics Council. Green Lake, WI.

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