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Anthony Fernandes

This article describes the 3rd cycle of an intervention in a mathematics content course that was designed to foster awareness among middle school mathematics preservice teachers (PSTs) of the challenges that English language learner (ELL) students face and the resources they draw on as they learn mathematics and communicate their thinking in English-only classrooms. Pairs of PSTs engaged 2 different ELL students in a videotaped task-based interview using 4 measurement tasks. Following each interview, the PSTs wrote a structured report guided by Mason's (2002) framework of noticing. The results of the intervention indicated that the PSTs went beyond awareness of ELLs' needs and challenges and also adopted strategies outlined in the literature that were aligned with best practices for teaching ELLs. The article also discusses the potential of the intervention and how it can be used by other mathematics educators.

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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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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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Erik D. Jacobson

This study (n = 1,044) used data from the Teacher Education and Development Study in Mathematics (TEDS-M) to examine the relationship between field experience focus (instruction- or exploration-focused), duration, and timing (early or not) and prospective elementary teachers' intertwined knowledge and beliefs about mathematics and mathematics learning. Early instruction-focused field experience (i.e., leading directly to classroom instruction) was positively related to the study outcomes in programs with such field experience of median or shorter duration. Moreover, the duration of instruction-focused field experience was positively related to study outcomes in programs without early instruction-focused field experience. By contrast, the duration of exploration-focused field experience (e.g., observation) was not related to the study outcomes. These findings suggest that field experience has important but largely overlooked relationships with prospective teachers' mathematical knowledge and beliefs. Implications for future research are discussed.

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Denise S. Mewborn

Four preservice elementary teachers were studied during a field-based mathematics methods course. The purpose of the study was to investigate the elements of mathematics teaching and learning the preservice teachers found problematic and how they resolved those problems. Data were collected in the form of individual interviews, group discussions, and individual journals. The preservice teachers exhibited concerns about the classroom context, pedagogy of mathematics, children's mathematical thinking, and, to a lesser extent, the mathematics content. The data indicate a relationship between the preservice teachers' locus of authority and the reflective quality of their thinking.

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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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Gwendolyn Lloyd

This article describes one student teacher's interactions with mathematics curriculum materials during her internship in a kindergarten classroom. Anne used curriculum materials from two distinct programs and taught lessons multiple times to different groups of children. Although she used each curriculum in distinct ways, her curriculum use was adaptive in both cases. Anne's specific ways of reading, evaluating, and adapting the curriculum materials contrast with previous results about beginning teachers' curriculum use. Several key factors appeared to contribute to Anne's particular ways of using the curriculum materials: features of her student-teaching placement, her personal resources and background, and characteristics of the materials. Directions for future research about student teachers' and other teachers' curriculum use are suggested in accord with these factors.

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Eva Thanheiser

Although preservice elementary school teachers (PSTs) lack the understanding of multidigit whole numbers necessary to teach in ways that empower students mathematically, little is known about their conceptions of multidigit whole numbers. The extensive research on children's understanding of multidigit whole numbers is used to explicate PSTs' conceptions of these numbers. A grounded theoretical approach leads to the development of a framework for PSTs' conceptions of multidigit whole numbers, and that framework facilitates description of their conceptions and their difficulties in the context of the standard algorithms. The framework also enables discussion of the PSTs' performance in other contexts.

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Wim Van Dooren, Lieven Verschaffel and Patrick Onghena

The study reported here investigated the arithmetical and algebraic problem-solving strategies and skills of preservice primary school and secondary school teachers in Flanders, Belgium, both at the beginning and at the end of their teacher training. The study then compared these aspects of the preservice teachers' own problem-solving behavior with the way in which they evaluated students' arithmetical and algebraic solutions to problems. Future secondary school teachers clearly preferred the use of algebra, both in their own solutions and in their evaluations of students' work, even when an arithmetical solution seemed more evident. Some future primary school teachers tended to apply exclusively arithmetical methods, leading to numerous failures on difficult word problems, whereas others were quite adaptive in their strategy choices. Taken as a whole, the evaluations of the preservice primary school teachers were more closely adapted to the nature of the task than those of their secondary school counterparts.

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Rose Mary Zbiek

This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated.