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Sandy Spitzer and Christine Phelps-Gregory

For prospective teachers (PTs) to engage in lifelong systematic learning, they must be prepared to analyze teaching on the basis of its effects on student learning. We present the results of an intervention study aimed at developing PTs' ability to analyze a classroom video sample. The intervention used an online discussion board activity structured along three research-based dimensions, which allowed PTs to build their analysis skills outside of class time. Evidence for the effectiveness of this intervention includes findings that PTs engaged deeply with their peers' ideas, with many changing their mind about the lesson's success, and that PTs' final reflections showed increased attention to the mathematics of the learning goal. However, after the intervention, many PTs continued to take nonmathematical evidence as indicators of student learning. Implications illuminate key design features of interventions as well as the affordances and challenges of using online interactions for improving PTs' lesson analysis skills.

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Christine Phelps-Gregory and Sandy M. Spitzer

One goal in teacher education is to prepare prospective teachers (PTs) for a career of systematic re_ ection and learning from their own teaching. One important skill involved in systematic re_ ection, which has received little research attention, is linking teaching actions with their outcomes on student learning; such links have been termed hypotheses. We developed an assessment task to investigate PTs' ability to create such hypotheses, prior to instruction. PTs (N = 16) each read a mathematics lesson transcript and then responded to four question prompts. The four prompts were designed to vary along research-based criteria to examine whether different contexts in_ uenced PTs' enactment of their hypothesizing skills. Results suggest that the assessment did capture PTs' hypothesizing ability and that there is room for teacher educators to help PTs develop better hypothesis skills. Additional analysis of the assessment task showed that the type of question prompt used had only minimal effect on PTs' responses.

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Frank Smith

Slightly over ten years ago Wilbur H. Dutton presented the results of a study of the attitudes of prospective teachers toward arithmetic as determined by an objective evaluation instrument. This present study compares the attitudes toward arithmetic of prospective teachers today with those reported by Dutton.

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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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Kathryn B. Chval

Most of the prospective teachers who enter my methods courses assume that teaching mathematics to elementary students will be easy. For example, Jenny wrote, “I thought, ‘I can teach math. How can it be so hard? It's elementary math!’ But I have been proven wrong.” Based on comments such as Jenny's, I realized the importance of giving prospective teachers opportunities to understand that effectively teaching mathematics to elementary students is complex and challenging. I recognized that field experience in my mathematics methods courses had to make the complexities of teaching more visible for prospective teachers. In other words, prospective teachers must study teaching practices. Such study would not only require viewing, analyzing, and discussing practices but also include the opportunity for prospective teachers to practice and analyze their own teaching.

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Lynette DeAun Guzmán

In this conceptual piece, I explore complex and contradictory conversations during an idea mapping task in which prospective elementary teachers interrogated dominant discourses within mathematics education, such as “mathematics is everywhere” and “being a math person.” I argue that this exercise of engaging with contradictions provided prospective teachers with opportunities to tease out nuances for reconstructing ideas that generate new perspectives for teaching and learning mathematics. Sharing my experience with the idea mapping task as a case study, I offer an alternative role for mathematics teacher educators to consider-as facilitators who create spaces for prospective teachers to interrogate complex and contradictory conversations within mathematics education.

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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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Theresa J. Grant, Jane-Jane Lo and Judith Flowers

This article discusses the challenges and opportunities that arose in attempting to support prospective elementary teachers in developing mathematical justifications in the context of wholenumber computation. Justification for whole-number computation is important for three reasons. First, this is the introductory topic in the first of three mathematics courses for prospective elementary teachers. Second, the number and operations strand is a major focus in elementary school. Third, in our experience as teacher educators, prospective elementary teachers have a difficult time considering how and why to teach whole-number computation in a conceptual manner. If prospective teachers' reasoning and justifications can be shaped in this area of mathematics, sense making and mathematical justification in other areas of mathematics can be shaped as well (Simon and Blume 1996).

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Kathryn B. Chval, John K. Lannin, Fran Arbaugh and Angela D. Bowzer

Educators who can elicit preservice teachers' beliefs about teaching mathematics can effectively challenge and change unrealistic expectations.

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W. Robert Houston

During the past decade, major changes have occurred in mathematics in the elementary school, both in content and approach. Indeed, “modern mathematics” has become a household term epitomizing the radical changes in today's curriculum which separate the younger generation from their parents. Parents complain they can no longer understand, much less help their children with mathematics. Innovations in experimental programs, technical advancements, and new insights into human learning promise even greater changes in the future.