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Patricia F. Campbell, Masako Nishio, Toni M. Smith, Lawrence M. Clark, Darcy L. Conant, Amber H. Rust, Jill Neumayer DePiper, Toya Jones Frank, Matthew J. Griffin and Youyoung Choi

This study of early-career teachers identified a significant relationship between upper-elementary teachers' mathematical content knowledge and their students' mathematics achievement, after controlling for student- and teacher-level characteristics. Findings provide evidence of the relevance of teacher knowledge and perceptions for teacher preparation and professional development programs.

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Dina Tirosh

In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.

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Kimberly Hufferd-Ackles, Karen C. Fuson and Miriam Gamora Sherin

The transformation to reform mathematics teaching is a daunting task. It is often unclear to teachers what such a classroom would really look like, let alone how to get there. This article addresses this question: How does a teacher, along with her students, go about establishing the sort of classroom community that can enact reform mathematics practices? An intensive year-long case study of one teacher was undertaken in an urban elementary classroom with Latino children. Data analysis generated developmental trajectories for teacher and student learning that describe the building of a math-talk learning community—a community in which individuals assist one another's learning of mathematics by engaging in meaningful mathematical discourse. The developmental trajectories in the Math-Talk Learning Community framework are (a) questioning, (b) explaining mathematical thinking, (c) sources of mathematical ideas, and (d) responsibility for learning.

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Anne K. Morris, James Hiebert and Sandy M. Spitzer

The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an essential starting point for studying and improving one's teaching. Thirty K–8 preservice teachers completed 4 written tasks. Each task specified a learning goal and then asked the preservice teachers to complete a teaching activity with this goal in mind. For example, preservice teachers were asked to evaluate whether a student's responses to a series of mathematics problems showed understanding of decimal number addition. The results indicate that preservice teachers can identify mathematical subconcepts of learning goals in supportive contexts but do not spontaneously apply a strategy of unpacking learning goals to plan for, or evaluate, teaching and learning. Implications for preservice education are discussed.

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Douglas L. Corey, Blake E. Peterson, Benjamin Merrill Lewis and Jared Bukarau

Previous research gives evidence that Japanese mathematics teachers “may have a more detailed and widely shared theory about how to teach effectively” when compared to their U.S. counterparts (Jacobs & Morita, 2002). This study explores the conceptions and cultural scripts of a group of Japanese mathematics teachers by analyzing the conversations between cooperating teachers and student teachers. It describes 6 principles of high-quality instruction that arose in at least half the conversations we analyzed. Each of these principles is examined in detail. Finally, some advantages of having a strong, shared conception of high-quality instruction and focusing on widely applicable instructional principles are presented.

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Heather C. Hill, Deborah Loewenberg Ball and Steven G. Schilling

There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items. Our results suggest partial success in measuring this domain among practicing teachers but also identify key areas around which the field must achieve conceptual and empirical clarity. Although this is ongoing work, we believe that the lessons learned from our efforts shed light on teachers' knowledge in this domain and can inform future attempts to develop measures.

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Nancy Nesbitt Vacc and George W. Bright

In this research, we examined changes in preservice elementary school teachers' beliefs about teaching and learning mathematics and their abilities to provide mathematics instruction that was based on children's thinking. The 34 participants in this study were introduced to Cognitively Guided Instruction (CGI) as part of a mathematics methods course. Belief-scale scores indicated that significant changes in their beliefs and perceptions about mathematics instruction occurred across the 2-year sequence of professional course work and student teaching during their undergraduate program but that their use of knowledge of children's mathematical thinking during instructional planning and teaching was limited. Preservice teachers may acknowledge the tenets of CGI and yet be unable to use them in their teaching. The results raise several questions about factors that may influence success in planning instruction on the basis of children's thinking.

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Judith L. Fraivillig, Lauren A. Murphy and Karen C. Fuson

In this article we present and describe a pedagogical framework that supports children's development of conceptual understanding of mathematics. The framework for Advancing Children's Thinking (ACT) was synthesized from an in-depth analysis of observed and reported data from 1 skillful 1st-grade teacher using the Everyday Mathematics (EM) curriculum. The ACT framework comprises 3 components: Eliciting Children's Solution Methods, Supporting Children's Conceptual Understanding, and Extending Children's Mathematical Thinking. The framework guided a cross-teacher analysis over 5 additional EM 1st-grade teachers. This comparison indicated that teachers often supported children's mathematical thinking but less often elicited or extended children's thinking. The ACT framework can contribute to educational research, teacher education, and the design of mathematics curricula.

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P. Holt Wilson, Hollylynne Stohl Lee and Karen F. Hollebrands

This study investigated the processes used by prospective mathematics teachers as they examined middle-school students' work solving statistical problems using a computer software program. Students' work on the tasks was captured in a videocase used by prospective teachers enrolled in a mathematics education course focused on teaching secondary mathematics with technology. The researchers developed a model for characterizing prospective teachers' attention to students' work and actions and interpretations of students' mathematical thinking. The model facilitated the identification of four categories: describing, comparing, inferring, and restructuring. Ways in which the model may be used by other researchers and implications for the design of pedagogical tasks for prospective teachers are discussed.

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Laura B. Sample McMeeking, Rebecca Orsi and R. Brian Cobb

The effect of a 15- to 24-month in-service professional development (PD) program on state accountability mathematics test scores for middle school students was examined using a quasi-experimental design. Middle level mathematics teachers (n = 128) from 7 school districts and 64 middle schools volunteered for a PD sequence of content-oriented summer courses and pedagogy-oriented structured follow-up experiences during the subsequent academic year. Student effects of the PD program were measured using Colorado's state mathematics test results for 2 cohorts of students: 1 that received mathematics instruction from participant teachers in the year prior to the PD and another cohort that received mathematics instruction in the year following the PD. The odds of a student achieving a Proficient or Advanced score on the state test were then compared between cohorts. Results showed that students' odds of achieving a score of Proficient or better increased with teacher participation in the PD program.