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The IERI Research Community :: Projects

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

PRINCIPAL INVESTIGATOR:
Hilda Borko

CO-INVESTIGATORS:
Karen Koellner Clark, Jeffrey A. Frykholm

CATEGORIES:
Math

PROJECT OVERVIEW:
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 I and Collaborative Research II) 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 for teaching and its relationship to their instructional practice and student learning.

Purpose: The STAAR research team at the University of Colorado at Boulder has designed a professional development program intended to help teachers promote algebraic thinking in their classrooms. The conceptual framework for the program is grounded in a situative perspective which suggests that teachers' own classrooms are powerful contexts for their learning. Another central tenet of the situative perspective is that building a strong professional community is essential. In keeping with this framework, our program relies heavily on artifacts of classroom practice, including videotaped records of the participating teachers mathematics lessons. In addition, we focus on creating and maintaining an environment in which teachers are comfortable working together to expand their mathematical knowledge and critically examine their instructional practices.

Intervention: At the core of our program is the Problem-Solving Cycle (PSC), a series of three interconnected professional development workshops, which provides teachers with the opportunity to share a common mathematical and pedagogical experience. During Workshop 1 of the Problem-Solving Cycle, teachers collaboratively solve a rich mathematical task and develop plans for teaching it to their own students. The goals of this workshop are to help teachers develop content knowledge and planning skills. After the workshop, all participants teach the problem and their lessons are videotaped. Workshops 2 and 3 focus on the teachers experiences teaching the problem and rely heavily on selected clips from the videotaped lessons. The goals of these workshops are to help teachers learn more about the mathematical concepts and skills entailed in the problem, and to reflect on their instructional strategies (primarily in Workshop 2) and student thinking (primarily in Workshop 3).

Setting: The STAAR professional development program began in the summer of 2003 and includes 16 middle school mathematics teachers representing six schools in three school districts within Colorado.

Research Design: The 16 teachers began by attending a two-week algebra course. Over the next two academic years we held monthly, full-day professional development workshops with a subset of these teachers. During the 2003 2004 academic year, 8 teachers (with classroom experience ranging from 1 to 27 years) attended the workshops. In the 2004 2005 year, 7 teachers continued working with us and 3 new teachers joined the program. Each new teacher was a colleague of one of the current participants.

We conducted three iterations of the Problem-Solving Cycle. All three iterations included the key components of the model; however, they were built around different mathematical tasks and focused on different aspects of the teacher's role and students' mathematical reasoning. The STAAR research team aimed to select tasks that would support the development of algebraic reasoning for both the participating teachers and their middle school students.

In parallel with the two-year professional development program, we utilized a design experiment approach to study and refine the Problem-Solving Cycle model. We collected and analyzed a large amount of data on the processes involved in developing and enacting the model, and examined the impact of the professional development experience on participating teachers' professional knowledge and instructional practices.

The entire professional development program was videotaped with multiple cameras to capture small-group and whole-group conversations. In addition, we kept detailed notes on our planning processes, regularly interviewed the professional development facilitators, and collected all written materials from each workshop. We also videotaped mathematics lessons taught by the participating teachers throughout each school year, using two cameras (one focused on the teacher and another on a selected small group of students), interviewed the teachers after each classroom observation (and several other times throughout the program), and collected artifacts of practice from their classroom lessons.

Findings: Although we are in the midst of ongoing analysis and reporting, thus far we have written about (1) the nature of our professional development program, including a detailed description of the conceptual framework and Problem-Solving Cycle model, (2) the impact of the program on teachers' content knowledge, pedagogical content knowledge, and instructional practices, and (3) strategies for building discourse communities and using video as an artifact of practice in mathematics professional development. In addition, we are preparing a facilitators' guide intended for use by individuals interested in understanding and carrying out the PSC with teachers. This guide offers facilitators a description of, and rationale for, the types of activities involved in the PSC as well as examples from the STAAR project.

We have preliminary indications that our professional development program has had a positive impact on the participants. For example, we examined teachers' algebraic content knowledge through an assessment administered at three time points: prior to the summer algebra course, immediately after the course, and at the end of the second year of professional development workshops. We analyzed differences in the number of correct answers and the number of solution strategies employed by each teacher, and found significant increases in both. These results suggest the teachers gained algebraic content knowledge and retained these gains over several years.

Moreover, from interviews, observations of teachers classroom practices, and case studies of participating teachers, we are finding evidence of increases in teachers' pedagogical content knowledge. Our analyses also suggest connections between teachers' experiences in the professional development program and changes in their instructional practices. For example, we have observed teachers more frequently incorporating group work on open-ended tasks and encouraging students' sharing of mathematical explanations and justifications.

Methodology key words: descriptive, interview, longitudinal, observation, videography, design experiment

PROJECT PUBLICATIONS:
Borko, H. (2004). Professional development and teacher learning: Mapping the terrain. Educational Researcher, 33(8), 3-15.

Borko, H. (2006, April). The problem-solving cycle: An approach to mathematics professional development. In The problem-solving cycle: An approach to mathematics professional development. Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco.

Borko, H., Frykholm, J. A., Pittman, M., Eiteljorg, E., Nelson, M., Jacobs, J., Clark, K. K., & Schneider, C. (2005). Preparing teachers to foster algebraic thinking. Zentralblatt f r Didaktik der Mathematik: International Reviews on Mathematical Education, 37(1), 43-52.

Bunning, K., & Schneider, C. (2006, April). Being facilitators of the STAAR problem-solving cycle professional development. In The problem-solving cycle: An approach to mathematics professional development. Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco.

Clark, K. K., & Borko, H. (2004). Establishing a professional learning community among middle school mathematics teachers. In M. J. Hoines & A. Fuglestad (Eds.), Proceedings of the Twenty-eighth Conference of the International Group for the Psychology of Mathematical Education (Vol. 2, pp. 223-230). Bergen, Norway: Bergen University College.

Clark, K. K., & Jacobs, J. (2005). Using video to support teacher learning: Theory and practice response. AMTE Connections, 14(3), 9-11.
http://www.amte.net/newsletter/june05.pdf

Clark, K. K., & Jacobs, J. (2006, April). The role of individual teachers goals in the STAAR professional development. In The problem-solving cycle: An approach to mathematics professional development. Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco.

Clark, K. K., Jacobs, J. Pittman, M., & Borko, H. (2005). Strategies for building mathematical communication in the middle school classroom: Modeled in professional development, implemented in the classroom. Current Issues in Middle Level Education, 11(2), 1-12.
http://www.kennesaw.edu/education/mge/napomle/cimle/fall2005
/clark_fa05.pdf


Eiteljorg, E. & Borko, H. (2006, April). Uncovering student reasoning: The transformation of Ken s questioning. In The problem-solving cycle: An approach to mathematics professional development. Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco.

Pittman, M. A., Nelson, M., & Frykholm, J. (2006, April). The development of teachers knowledge of algebra for teaching. In The problem-solving cycle: An approach to mathematics professional development. Symposium conducted at the annual meeting of the American Educational Research Association, San Francisco.

ON THE WEB:
You can learn more about this project by visiting http://algebra.colorado.edu/.