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Seeking Big Ideas in Algebra

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– 1 – Editorial Manager for Journal of Mathematics Teacher Education Manuscript Number: JMTE322R1 Title: Seeking Big Ideas in Algebra: The Evolution of a Task Article Type: Special Issue Manuscript Section/Category: Short Research Paper Key Words: teacher change, school algebra, big ideas, active learning Corresponding Author: Eric Hsu Corresponding Author’ Institution: San Francisco State University First Author: Eric Hsu, Ph.D. Order of Authors: Eric Hsu, Judy Kysh, Katherine Ramage, and Diane Resek Corresponding Author’s Address: Eric Hsu, Mathematics Department, SFSU, 1600 Holloway Avenue, San Francisco, CA 94132 Corresponding Author’s Email: [email protected] Manuscript Region of Origin: USA Abstract: In this paper we describe the evolution of a task that we used with two cohorts of participants in a professional development program called Revitalizing Algebra (REAL). We first discuss our goals and describe the participants, and then we describe the construction the task followed by teacher responses. We reflect on the different iterations of the tasks and their impact on the teachers’ thinking and practice. We describe how some teachers, influenced by this task, were able to consider changes in their schools and how the school and department cultures contributed to such change.– 2 – Seeking Big Ideas in Algebra: The Evolution of a Task Eric Hsu, Judy Kysh, Katherine Ramage, and Diane Resek Abstract In this paper we describe the evolution of a task that we used with two cohorts of participants in a professional development program called Revitalizing Algebra (REAL). We first discuss our goals and describe the participants, and then we describe the construction the task followed by teacher responses. We reflect on the different iterations of the tasks and their impact on the teachers’ thinking and practice. We describe how some teachers, influenced by this task, were able to consider changes in their schools and comment on some the conditions for successful change. The Teachers of Revitalizing Algebra REAL (Revitalizing Algebra) is a National Science Foundation Mathematics and Science Partnership between San Francisco State University (SFSU) and five local Northern California school districts. The goal of the partnership is to improve the performance of all students (particularly minority students) in algebra, both in K-12 schools and in college. With the participating mathematics departments, we selected teacher leaders on the basis of their interest and their leadership potential. We did two years of work with two cohorts, each of 27 participants. Each cohort consisted of 9 secondary algebra teachers, 9 graduate student instructors for remedial algebra at SFSU, and 9 mathematics majors aspiring to be teachers. In addition to participating in the REAL class, each undergraduate mathematics major worked ten hours per week in an algebra class of one of the participating secondary teachers. The first year included an intensive two-semester weekly class and a three-week summer session. In this paper we will focus on the secondary teachers. The second year, secondary teacher leaders worked with teams of teachers in their own departments at their schools using common daily preparation time (paid for by the REAL project) to help each other examine the day-to-day effectiveness of their teaching. They also began to wrestle with longer-term pedagogical issues as a basis for permanent growth in their department teaching culture. We had introduced the leaders to the ideas of lesson study, along with some examples (e.g. Fernandez and Yoshida, 2004; Lewis, 2005), but we let teams find a structure that suited their local conditions. Basic Principles of Revitalizing Algebra Our plan for REAL takes into consideration what research and our own experience tell us about the qualities of effective professional development. We were aware of the ineffectiveness of “top-down” educational movements. Influential professional development needs to be highly adaptable, long-term, and relevant to the local conditions of the teachers, yet one cannot encourage growth without some underlying basic principles. We now discuss some basic principles, making frequent references and connections to the Taken As Shared document (Watson and Mason, this– 3 – volume) in the interest of brevity. First, we based our work on considerable research about the complexity of learning. Watson and Mason (this volume) describe theories of learning based on the work on Vygotsky, Piaget and others in which learners are challenged with actively reconciling new ideas and old ideas, intuitive understandings with conventional understandings. In brief, our first principle is: 1. Learners understand formal mathematical ideas by connecting them to their own intuitions and constructed understandings. This view could be seen as Piagetian, requiring individual construction of meaning, or Vygotskian, requiring dialogue to relate spontaneous and scientific conceptualizations. However, we are more interested in the bridge provided in the work of Skemp (1987), particularly his notion of instrumental versus relational understanding, where the teachers on the course may have a view of mathematics as instrumental because that view was adequate in their previous experience. Second, Watson and Mason note the importance of milieu, which refers both to mathematical and social environment. In American public school settings, the student body is usually highly diverse in race, gender and culture, so this environment is necessarily influenced by such diversity. Our second principle is 2. Assumptions about race, gender and culture affect the dynamics of a classroom, and must be consciously considered by teachers. For our chosen content focus, first-year algebra, race, gender and culture are essential parts of the milieu to be wrestled with. The widespread American system of tracking has resulted in a large proportion of non-white, poor students being segregated into mostly non-white, low-track, first-year algebra classes. These classes can offer an impoverished algebraic experience which we needed to address through REAL. We acknowledged the power of racial, gender and cultural discrimination in education and took as part of our charge the important and delicate task of addressing it. We were influenced here by the work of Weissglass (1997) and Delpit (1988) among others. We worked to create


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