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Susan B. Empson

This article presents an analysis of two low-performing students' experiences in a firstgrade classroom oriented toward teaching mathematics for understanding. Combining constructs from interactional sociolinguistics and developmental task analysis, I investigate the nature of these students' participation in classroom discourse about fractions. Pre- and post-instruction interviews documenting learning and analysis of classroom interactions suggest mechanisms of that learning. I propose that three main factors account for these two students' success: use of tasks that elicited the students' prior understanding, creation of a variety of participant frameworks (Goffman, 1981) in which the students were treated as mathematically competent, and frequency of opportunities for identity-enhancing interactions.

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Susan B. Empson

It is surprising to learn that first graders know a lot about fractions. That is what two first-grade teachers and the author discovered when they collaborated on a five-week fraction unit. This article describes the highlights of a case study of fractions in a first-grade class then presents some preliminary findings suggesting that third-, fourth-, and fifth-grade children can learn fractions in similar ways.

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Laura B. Kent, Susan B. Empson and Lynne Nielsen

Fifth graders explore approaches to solving a division-of-fractions problem introduced within the context of hot chocolate servings.

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Katherine Baker, Naomi A. Jessup, Victoria R. Jacobs, Susan B. Empson and Joan Case

Productive struggle is an essential part of mathematics instruction that promotes learning with deep understanding. A video scenario is used to provide a glimpse of productive struggle in action and to showcase its characteristics for both students and teachers. Suggestions for supporting productive struggle are provided.

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Erin E. Turner, Debra L. Junk and Susan B. Empson

Building understanding of multiplicative relationships is a key goal of mathematics instruction in the upper elementary and middle grades. Multiplicative thinking includes comparing numbers through many processes: multiplication and division (rather than addition and subtraction), ratio, proportions, stretching and shrinking, magnification, scaling, and splitting. Research has shown that multiplicative thinking develops slowly in children, over long periods of time (Clark and Kamii 1996; Vergnaud 1988). Initially, children tend to reason additively about multiplicative situations, and this additive thinking is often resistant to change (Hart 1984). Students need practice with tasks that help develop multiplicative thinking—in particular, tasks that help them recognize and reason about multiplicative relationships.

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Thomas P. Carpenter, Megan L. Franke, Victoria R. Jacobs, Elizabeth Fennema and Susan B. Empson

This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1—3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.

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Elizabeth Fennema, Thomas P. Carpenter, Megan L. Franke, Linda Levi, Victoria R. Jacobs and Susan B. Empson

This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.