Elham Kazemi

Promoting Conceptual Thinking in Four Upper-Elementary Mathematics Classrooms.

Informed by theory and research in inquiry-based mathematics, this study examined how classroom practices create a press for conceptual learning. Using videotapes of a lesson on the addition of fractions in 4 primarily low-income classrooms from 3 schools, we analyzed conversations that create a high or lower press for conceptual thinking. We use examples of interactions from these fourth- and fifth-grade lessons to propose that a high press for conceptual thinking is characterized by the following sociomathematical norms: (a) an explanation consists of a mathematical argument, not simply a procedural description; (b) mathematical thinking involves understanding relations among multiple strategies; (c) errors provide opportunities to reconceptualize a problem, explore contradictions in solutions, and pursue alternative strategies; and (d) collaborative work involves individual accountability and reaching consensus through mathematical argumentation.

Keeping It Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching

We analyze a particular pedagogy for learning to interact productively with students and subject matter, which we call “rehearsal.” Our goal is to specify a way in which teacher educators (TEs) and novice teachers (NTs) can interact around teaching that is both embedded in practice and amenable to analysis. We address two main research questions: (a) What do TEs and NTs do together during the kind of rehearsals we have developed to prepare novices for the complex, interactive work of teaching? and (b) Where, in what they do, are there opportunities for NTs to learn to enact the principles, practices, and knowledge entailed in ambitious teaching? We detail what happens in rehearsals using quantitative and qualitative methods. We begin with the results of our quantitative analyses to characterize how typical rehearsals were structured and what was worked on. We then show how NTs and TEs worked together to enable novices to study principled practice through qualitative analyses of a particularly salient aspect of ambitious teaching, namely, eliciting and responding to students’ performance.

Core Practices and Pedagogies of Teacher Education A Call for a Common Language and Collective Activity

Currently, the field of teacher education is undergoing a major shift—a turn away from a predominant focus on specifying the necessary knowledge for teaching toward specifying teaching practices that entail knowledge and doing. In this article, the authors suggest that current work on K-12 core teaching practices has the potential to shift teacher education toward the practice of teaching. However, the authors argue that to realize this vision we must reimagine not only the curriculum for learning to teach but also the pedagogy of teacher education. We present one example of what we mean by reimagined teacher education pedagogy by offering a framework through which to conceptualize the preparation of teachers organized around core practices. From our perspectives, this framework could be the backbone of a larger research and development agenda aimed at engaging teachers and teacher educators in systematic knowledge generation regarding ambitious teaching and teacher education pedagogy. We conclude with an invitation to the field to join with us in imagining approaches to generating and aggregating knowledge about teaching and the pedagogy of teacher education that will move not only our individual practice but also our collective practice forward.

Using Designed Instructional Activities to Enable Novices to Manage Ambitious Mathematics Teaching

If teacher education is to prepare novices to engage successfully in the complex work of ambitious instruction, it must somehow prepare them to teach within the continuity of the challenging moment-by-moment interactions with students and content over time. With Leinhardt, we would argue that teaching novices to do routines that structure teacher–student–content relationships over time to accomplish ambitious goals could both maintain and reduce the complexity of what they need to learn to do to carry out this work successfully. These routines would embody the regular “participation structures” that specify what teachers and students do with one another and with the mathematical content. But teaching routines are not practiced by ambitious teachers in a vacuum and they cannot be learned by novices in a vacuum. In Lampert’s classroom, the use of exchange routines occurred inside of instructional activities with particular mathematical learning goals like successive approximation of the quotient in a long division problem, charting and graphing functions, and drawing arrays to represent multi-digit multiplications. To imagine how instructional activities using exchange routines could be designed as tools for mathematics teacher education, we have drawn on two models from outside of mathematics education. One is a teacher education program for language teachers in Rome and the other is a program that prepares elementary school teachers at the University of Chicago. Both programs use instructional activities built around routines as the focus of a practice-oriented approach to teacher preparation.

Learning to Teach Mathematics: Focus on Student Thinking

Megan Loef Franke is associate professor of education at the University of California, Los Angeles; Elham Kazemi is assistant professor of education at the University of Washington, Seattle. T FIELD OF MATHEMATICS EDUCATION has made great strides in developing theories and research-based evidence about how to teach elementary school mathematics in a way that develops students’ mathematical understanding. Much of this progress has grown out of research projects that engage teachers in learning to teach mathematics. These projects have not only shown what is possible for teachers and students but have also provided insight into how to support teachers in their own learning. We use one particular research and development project, Cognitively Guided Instruction (CGI), as an illustration of how theory and research inform the teaching and learning of mathematics. The key to CGI has been an explicit, consistent focus on the development of children’s mathematical thinking (Carpenter, Fennema, & Franke, 1996). This focus serves as the guide for understanding CGI’s contributions as well as its evolution. Initiated over 15 years ago by Carpenter, Fennema, and Peterson, CGI sought to bring together research on the development of children’s mathematical thinking and research on teaching (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). This research project engaged first-grade teachers with the research-based knowledge about the development of children’s mathematical thinking. We studied the teachers’ use of this knowledge within their classroom practice and examined the changes in teachers’ beliefs and knowledge. This work created a rich context for our own learning as well as for the teachers. We learned a great deal about teachers’ use of children’s thinking, teacher learning, and professional development. Since the initial project, our understandings and our work with teachers and students have continued to evolve. We now work with K-5 teachers across a number of different content areas (Franke & Kazemi, in press; Kazemi, 1999). Our learning reflects much of the learning occurring in the field. In this article, we divide our learning about CGI into two sections. These sections represent an ongoing shift in our thinking from a consistent cognitive paradigm to a more situated paradigm. These different notions of learning have influenced our views of student and teacher learning and how to maximize that learning. We tell this story by describing our initial CGI work and what we learned about students and teachers. We then elaborate by describing and characterizing our current work and how that has influenced our learning about creating a focus on students’ mathematical thinking within professional development.

Conceptualizing the Work of Leading Mathematical Tasks in Professional Development

Filling the knowledge gap in the limited research on professional development leaders is an urgent issue if teacher learning is to be improved. This research and development project is studying how leaders learn to cultivate mathematically rich professional development environments. The authors adapted two frameworks from classroom-based research—sociomathematical norms and practices for orchestrating productive discussion—to support leaders’ understanding of facilitation of mathematics professional development. In this article, the authors describe the use of these frameworks in their work and argue for a third framework—the mathematical knowledge for teaching. Based on the analysis of their work, they believe that mathematics professional development leaders need to cultivate particular sociomathematical norms for teacher explanation and employ practices for orchestrating discussions to achieve the purposeful development of teachers’ specialized knowledge of mathematics for teaching.

Getting Inside Rehearsals Insights From Teacher Educators to Support Work on Complex Practice

In recent years, work in practice-based teacher education has focused on identifying and elaborating how teacher educators (TEs) use pedagogies of enactment to learn in and from practice. However, research on these pedagogies is still in its early development. Building on prior analyses, this article elaborates a particular pedagogy of enactment, rehearsal, developed through a collaboration of elementary mathematics TEs across three institutions. Rehearsals are embedded within learning cycles that provide repeated opportunities for novice teachers (NTs) to investigate, reflect on, and enact teaching through coached feedback. This article shares a set of insights gained from 5 years of developing, studying, and learning how to support NTs’ enactment in rehearsal. The insights we share in this article contribute to building a knowledge base for pedagogies of teacher education.

Practice Makes Practice: Learning to Teach in Teacher Education

In this article, we argue that teaching is and should be a central element to learning to teach, particularly as teacher education once again turns toward practice. From this perspective, we must elaborate how such a shift addresses the need to bridge the gap between knowledge for teaching and knowledge from teaching, between theory and practice, and among university courses and fieldwork. If the intent of such a shift is to fundamentally change the preparation of teachers, we argue that it requires teacher education programs to do more than increase the amount of time candidates spend in clinical field placements. It requires, we argue, that teacher educators engage in simultaneous innovation in three related, but distinct aspects of program design and implementation: organizational structures and policies, content and curriculum, and teacher education pedagogy. Without such dynamic engagement, the practice-turn will go the way of many past reforms in teacher education—it will be symbolic but not significant or meaningful.

Making Sense of “The Boy Who Died”: Tales of a Struggling Successful Writer

This article presents a case study of a fourth grade boys experiences in writing, preceding and following a story he wrote about a boy whose struggles in writing led directly to his death. We explore how Maxs writing experiences related to his identity, specifically his sense of himself as a writer, his struggle to communicate his ideas, and his discomfort with expressing private thoughts and emotions in print. Drawing on a range of qualitative data, we examine Maxs experiences with writing workshop, journal writing, responding to literature, and a state writing assessment. Maxs story argues for the importance of considering issues of identity in the writing classroom to help students build on the successes that often hide behind the surface struggles of their writing.

Exploring test performance in mathematics: the questions children’s answers raise

Abstract This article investigates children’s mathematical performance on test items, specifically multiple-choice questions. Using interviews with 90 fourth-graders, it reveals why particular kinds of items are more or less difficult for students. By using multiple-choice questions and juxtaposing them with similar open-ended problems, the findings underscore the costs of not attending to children’s thinking in designing and interpreting problems. The data from this study suggest that when answering multiple-choice questions, students’ attention is drawn to the choices themselves. They do not necessarily think through the problem first and thus make their choices based on (often incorrect) generalizations they have made about problem-solving. Whether students answered a multiple-choice question or a similar open-ended problem first impacted both their performance and their reasoning. Moreover, children draw on their life experiences when the context of the problem is salient, thus ignoring important parameters of the stated problem. Implications for investigating children’s thinking, instruction, and test design are discussed.