The STAAR Project: Supporting the Transition from Arithmetic to Algebraic Reasoning
Preparing Teachers to Foster Algebraic Thinking: A STAAR Professional Development Program
University of Colorado—Boulder
Project Team: Hilda Borko, Jeff Frykholm, Jennifer Jacobs, Karen Clark, Eric Eiteljorg, Mary Pittman, Mary Nelson, Craig Schneider, Kimberly Bunning
A grant funded by the Interagency Education Research Initiative (IERI)
Overview
The Professional Development Program
The STAAR research team at the University of Colorado—Boulder completed its second year of a professional development program intended to help teachers foster algebraic thinking in their classrooms.
The conceptual framework for the program is grounded in a situative perspective on teacher learning and consists of two primary threads: (1) the central role of community in fostering teacher learning, and (2) enhancing teachers' mathematical and pedagogical knowledge. The program has two major components: a summer institute and ongoing monthly professional development workshops. As an extension of the monthly workshops, the professional development/research team makes regular visits to the classrooms of participating teachers to videotape lessons that are later used by both the teacher-participants and the researchers as seminal pieces of the professional development effort.
The Professional Development Research Component
We conducted an extensive program of research in parallel with the summer institute and subsequent professional development activities. All professional development workshops (including the summer institute) were videotaped for later analysis. In addition, we regularly videotaped classroom lessons taught by participating teachers, interviewed the professional development facilitators and teachers, and collected artifacts of practice from professional development workshops and classroom lessons.
Conceptual Framework
In this project a situative perspective on teacher learning is foundational for both designing and studying our professional development activities. The situative perspective provides a framework for connecting two constructs that are central to the professional development program and research: learning communities and teachers' knowledge.
Our professional development program and research initiatives are firmly rooted in two assertions that are central to the situative perspective: (1) teacher learning about content and about pedagogy is situated; that is, how a teacher learns a particular set of knowledge and skills, and the situations in which a teacher learns, are fundamental parts of what is learned (Greeno et al., 1996); (2) teacher learning can be helpfully understood as "a process of increasing participation in the practice of teaching, and through this participation, a process of becoming knowledgeable in and about teaching" (Adler, 2000, p. 37).
- "Learning should be viewed as both a process of active individual construction and a process of enculturation into the … practices of wider society" (Cobb, 1994, p. 13).
- To understand teacher learning, we must study it within multiple contexts and consider teachers as individual learners as well as the social contexts within which they participate in their own professional growth and development (Putnam & Borko, 2000).
The Central Role of Community in Teacher Learning
Our professional development program is grounded in the principle that participation in community is an important component of meaningful learning. Thus, efforts to create and maintain a professional learning community characterized by trust and respect, as well as norms for critical dialogue about teaching, have been central to our work.
- "Only when teachers continually find themselves in discussions about learners, pedagogy, mathematics and reform - when they, out of habit, develop a critical consciousness about teaching - only then will they be able to interrupt the traditional expositional model that has been perpetuated for decades in mathematics classrooms" (Frykholm, 1998; p. 320).
- "Teacher development consists of teachers developing a deeper knowledge of children's mathematical thinking, but in the context of the 'community'" (Cooney, 1994; p. 613).
Teachers' Mathematical and Pedagogical Knowledge
The NCTM Principles and Standards document suggests that "teachers must know and understand deeply the mathematics they are teaching and be able to draw on that knowledge with flexibility in their teaching tasks" (NCTM, 2000, p. 17). Over 25 years of research have indicated that teachers do not typically possess this rich and connected knowledge of mathematics (Mewborn, 2003). This may be particularly true within the domain of algebra with its many related and layered constructs.
- "[Many] teachers possess weak knowledge and narrow views of mathematics and mathematics pedagogy that include conceptions of mathematics as a closed set of procedures, teaching as telling, and learning as the accumulation of information" (Lloyd & Frykholm, 2000; p. 576).
Professional Development Program Design
Summer Algebra Institute (July 2003)
Institute Goals
- Build teachers' knowledge of algebra for teaching (KAT)
- Build teachers' pedagogical content knowledge (PCK)
- Demonstrate strategies for teaching algebra
- Focus on students' algebraic reasoning (STAAR Tier 1)
- Build a professional community
- Create a safe environment for mathematical explorations
- Distribute the social and intellectual work
Institute Design
- Sixteen participants (primarily middle school teachers)
- Three-credit university course, 60 contact hours
Institute Teaching Approach
- Use contextually based mathematics problems similar to middle school tasks
- Incorporate small group problem solving
- Have teachers share solution strategies in progressive sophistication
School Year Professional Development Workshops
(AY 2003-2004; AY 2004-2005)
Workshop Goals
- Group goals: build algebraic and pedagogical content knowledge
- Selecting and teaching rich mathematical tasks
- Increasing mathematical discourse in the classroom
- Developing and building on students' mathematical thinking
- Individual goals: teachers identify and work toward personal goals
Workshop Design
- Seven workshops per year (3 fall, 4 spring), 5 hours each
- Eight (year 1) and ten (year 2) middle school teachers
- Multiple iterations of the Problem-Solving Cycle
The Problem-Solving Cycle
- Workshop 1: Solve a rich algebraic mathematics problem and develop individual lesson plans around the problem
- Videotape teachers' implementation of the problem
- Workshops 2 & 3: Debrief teaching episodes using video
- Focus on the teachers' role (Workshop 2)
- Focus on students' mathematical thinking (Workshop 3)
The Problem-Solving Cycle (PSC) is the centerpiece of our professional development program within the STAAR Project. The PSC consists of a series of three workshops designed to help prepare middle school teachers to support their students' transition from arithmetic to algebraic reasoning.
The Problem-Solving Cycle focuses on a rich mathematical task that incorporates important concepts central to the middle school algebra curriculum. During the first workshop, teachers collaboratively solve the problem and develop plans for teaching it to their own students. Subsequent workshops focus on teachers' experiences using the problem in their classrooms. The participants consider more about the mathematical concepts and skills entailed in the problem, their own role in teaching it, and the student thinking that the problem generated. In all three workshops, we emphasize building a strong professional learning community and using artifacts of practice to situate teachers' learning opportunities in the context of their work.
The Problem-Solving Cycle model of professional development
Research Design and Methods
Research Goals
Two broad goals set the direction for our research agenda.
1. Examine the effectiveness of the professional development program. To address this goal, we designed a set of studies to investigate aspects of the program such as:
- Development of community and norms of discourse among the cohort of teachers.
- Knowledge of algebra for teaching and pedagogical content knowledge of teachers in the program.
- Degree to which teachers are incorporating ideas, experiences, and knowledge gained in the professional development setting into their own teaching practices.
2. Explore the degree to which the Problem-Solving Cycle may be adopted and adapted in different contexts. To begin to address this goal, we have enacted the PSC with different algebraic content, an expanding group of teachers, and multiple facilitators.
Data Collection
- Over 400 hours of video have been collected and catalogued
- Summer algebra institute
- Professional development workshops
- Classroom instruction of participating teachers
- Interviews of participants
- Ongoing interviews of teachers in the program
- Ongoing interviews of professional development facilitators
- Artifacts of the professional development
- Teachers' reflections and autobiographical statements
- Pre- and post-tests of teachers' knowledge of algebra for teaching
- Teachers' mathematical work
- Artifacts of practice (classroom artifacts)
- Teachers' lesson plans and instructional materials
- Samples of student work
Data Analysis
We are using a variety of methods to organize, categorize and analyze data, including:
- Summer Institute daily catalogue of activities
- Professional development workshop descriptions and catalogue of activities
- Transcriptions of interviews
- Pre- and post-test content knowledge assessments
- Analytic summaries of each participant's classroom instruction
- Quantitative coding and analyses of professional development activities, classroom observations, and interviews
Initial Research Findings
Although we are in the midst of ongoing analysis and reporting findings, preliminary indications of the impact of this program are positive. Results that have been shared at national and international conferences or in papers include:
1. Impact of the professional development program
Increases in teachers' knowledge of algebra for teaching
One way we examined teachers' knowledge of algebra for teaching was through a content assessment, administered at three points throughout the professional development program: prior to the Summer Institute, immediately after the Summer Institute, and at the end of the second year of professional development workshops. Problems involved pattern recognition, representational fluency, and writing and solving one and two variable equations. 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. Interviews, written reflections, and analyses of conversations during the professional development workshops provide further evidence of advances in teachers' knowledge of algebra.for teaching.
Increases in teachers' pedagogical content knowledge and changes in instructional practices
Based on 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 observed teachers more frequently incorporating group work on open-ended tasks and encouraging students' sharing of mathematical explanations and justifications.
2. Strategies for building mathematical discourse communities in professional development workshops and middle school classrooms
We have focused on the following four strategies that appear to be fundamental in creating a mathematical discourse community in both professional development and classroom settings:
- Posing rich tasks that promote discussion
- Establishing and maintaining a safe environment
- Asking teachers/students to explain and justify their thinking
- Encouraging teachers/students to actively process each other's ideas
We have reported on the ways in which we used these ideas to establish a productive professional learning community for middle school mathematics teachers. In addition, we examined how these strategies were modeled in the Summer Institute, and then carried out in a participating teacher's eighth-grade algebra lesson. We have also written about the features of rich problem solving tasks that promote algebraic generalization.
Project Direction
Our goals for the remainder of this project include:
- Continued data analyses
- Continued reporting and sharing of findings
- Conducting follow-up classroom observations and interviews with teachers
- Developing a facilitator's guide to help others implement the Problem-Solving Cycle
- Detailed descriptions of the workshops
- Selecting mathematical tasks
- Videotaping classrooms and choosing video clips
- Orchestrating small and whole group conversations
Publications and Works in Progress
In addition to these manuscripts, our research findings have been presented at several national and international conferences.
- 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 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 2-223 - 230). Bergen, Norway: Bergen University College.
- This paper focuses on how a professional learning community began to develop in the summer algebra institute that initiated our PD program. We examined marker activities that fostered the development of community across five dimensions.
- 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 für Didaktik der Mathematik: International Reviews on Mathematics Education, 37(1), 43-52.
- This paper provides an overview of the summer algebra institute. The paper describes our conceptual framework, outlines the structure of the professional development we provided our teachers, and shares initial findings and implications.
- Clark, K.K., & Jacobs, J. (2005). Using video to support teacher learning: Theory & Practice Response. AMTE Connections, 14(3), 9-11.
- This response describes how we used video of teachers' classrooms as a tool for helping teachers analyze, discuss, and change their practice.
- Clark, K.K., Jacobs, J., Pittman, M. E., & Borko, H. (2005). Strategies for building mathematical communication: Modeled in professional development, implemented in the classroom. Current Issues in Middle Level Education, 11(2), 1-12.
- We describe specific strategies for fostering mathematical communication that were modeled in the summer algebra institute and then implemented by one of the participating teachers in her middle school classroom.
- Borko, H., Jacobs, J., Schneider, C. & Eiteljorg, E. (under review). Professional development to support the transition from arithmetic to algebraic reasoning: The Problem-Solving Cycle.
- This article presents the conceptual framework for the Problem-Solving Cycle, details of its enactment, and initial findings regarding its impact on teachers' knowledge and instructional practices.
- Clark, K.K., Frykholm, J. & Pittman, M. (under review). Algebraic thinking and generalization: The importance of problem solving tasks.
- We outline three features of problem solving tasks that promote algebraic generalization and illustrate how one such task was successfully carried out in a participating teacher's classroom.
- Borko, H., Jacobs, J., Eiteljorg, E., & Pittman, M. E. (in progress). Video as a tool for teacher learning: Building a productive discourse community.
- This paper presents an analysis of teachers' conversations in our professional development workshops centered around watching videotapes of classroom teaching and describes how video fostered a productive discourse community.
- Clark, K. K., & Jacobs, J. (in progress). The role of individual teacher goals in the Problem-Solving Cycle.
- One central component of the Problem-Solving Cycle involves having teachers generate and pursue their own pedagogical goals. This paper examines the goals that were identified and pursued by two teachers, and analyzes how these goals influenced changes in their classroom practices.
- Eiteljorg, E., Pittman, M., Borko, H., Frykholm, J., & Nelson, M. (in progress). A program for teacher professional development: Developing participation structures to support student understanding.
- This paper develops a case study of one teacher and his evolving teaching strategies throughout the first year of our professional development program.
- Pittman, M., Nelson, M., & Frykholm, J. (in progress). Enhancing the content knowledge of teachers of algebra.
- This paper explores gains in participants' content knowledge of algebra. Pre- and posttests were administered before and after the summer algebra institute; significant gains were found along multiple dimensions of algebraic content.
- Koellner, K., Borko, H., Frykholm, J., Schneider, C., Eiteljorg, E., Bunning, K. & Pittman, M. The Problem-Solving Cycle: The Conceptual Framework and Design of a New Model of Mathematics Professional Development.
- This article focuses on the Problem-Solving Cycle, a new model of professional development designed to assist middle-school teachers to support their students' development of algebraic reasoning.