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Nicholas H. Wasserman and William McGuffey

year following participation in the real analysis course. As a reflection on opportunities to learn from mathematical coursework, this article is particularly focused on teachers’ attributions for their actions during particular teaching moments

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Kara Jackson, Anne Garrison, Jonee Wilson, Lynsey Gibbons, and Emily Shahan

This article specifies how the setup, or introduction, of cognitively demanding tasks is a crucial phase of middle-grades mathematics instruction. We report on an empirical study of 165 middle-grades mathematics teachers' instruction that focused on how they introduced tasks and the relationship between how they introduced tasks and the nature of students' opportunities to learn mathematics in the concluding whole-class discussion. Findings suggest that in lessons in which (a) the setup supported students to develop common language to describe contextual features and mathematical relationships specific to the task and (b) the cognitive demand of the task was maintained in the setup, concluding whole-class discussions were characterized by higher quality opportunities to learn.

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Maryl Gearhart, Geoffrey B. Saxe, Michael Seltzer, Jonah Schlackman, Cynthia Carter Ching, Na'ilah Nasir, Randy Fall, Tom Bennett, Steven Rhine, and Tine F. Sloan

In this study we addressed 2 questions: (a) How can we document opportunities to learn aligned with the NCTM Standards? (b) How can we support elementary teachers' efforts to provide such opportunities? We conducted a study of the effect of curriculum (problem solving vs. skills) and professional development (subject-matter focused vs. collegial support) on practices and learning. From analyses of videotapes and field notes, we created 3 scales for estimating students' opportunities to learn. Analyses of fractions instruction in 21 elementary classrooms provided evidence of the technical quality of the indicators and indicated that support for teachers' knowledge may be required for a problem-solving curriculum to be beneficial.

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Tracking, that is, assigning students to homogeneous groups for educational purposes, is a topic of considerable study and heated debate for the educational research community (Dossey 2000; Edwards 1999; Harlen 2000; Loveless 1998). In his metanalysis of research on tracking, Kulik (1992) identified more than 700 studies. Rarely, however, has this research been informed by cross-national comparisons. This NSF-funded study is providing a cross-national perspective on tracking practices in mathematics as they relate to opportunity to learn and achievement. The study is using data from the Third International Mathematics and Science Study (TIMSS 1995 in Beaton et al. 1996) and TIMSSRepeat (TIMSS 1999, in Mullis et al. 2000).

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William Tate

The importance of mathematics communication that builds on the lives and experiences of African American students in urban schools, thereby creating additional opportunities to leant and explore mathematics, is the focus of this article.

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James Hiebert, Dawn Berk, Emily Miller, Heather Gallivan, and Erin Meikle

We investigated whether the mathematics studied in 2 content courses of an elementary teacher preparation program was retained and used by graduates when completing tasks measuring knowledge for teaching mathematics. Using a longitudinal design, we followed 2 cohorts of prospective teachers for 3 to 4 years after graduation. We assessed participants' knowledge by asking them to identify mathematics concepts underlying standard procedures, generate multiple solution strategies, and evaluate students' mathematical work. We administered parallel tasks for 3 mathematics topics studied in the program and one mathematics topic not studied in the program. When significant differences were found, participants always performed better on mathematics topics developed in the program than on the topic not addressed in the program. We discuss implications of these findings for mathematics teacher preparation.

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Rose Mary Zbiek and Jeanne Shimizu

Similarities and differences among multiple solutions to two different problems (How much skin covers the human body? and How many melon balls in a melon?) and how thinking about the differences among problems and their solutions can be used to extend students' understanding of mathematical concepts and skills. The article discusses the teachers' questioning skills as they focused on the conceptual and procedural aspects of the alternative solutions. The authors noted that when the solutions differ conceptually, the conceptual diversity of solutions helps students refine their understanding of definitions and nuances of underlying concepts.

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Denisse R. Thompson, Sharon L. Senk, and Gwendolyn J. Johnson

This article addresses the nature and extent of reasoning and proof in the written (i.e., intended) curriculum of 20 contemporary high school mathematics textbooks. Both the narrative and exercise sets in lessons dealing with the topics of exponents, logarithms, and polynomials were examined. The extent of proof-related reasoning varied by topic and textbook. Overall, about 50% of the identified properties in the 3 topic areas were justified, with about 30% of the addressed properties justified with a general argument and about 20% justified with an argument about a specific case. However, less than 6% of the exercises in the homework sets involved proof-related reasoning, with developing an argument and investigating a conjecture as the most frequently occurring types of proof-related reasoning.

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Karisma Morton and Catherine Riegle-Crumb

Using data from a large urban district, this study investigated whether racial inequality in access to eighth-grade algebra is a reproduction of differences in prior opportunities to learn (as evidenced by grades, test scores, and level of prior mathematics course) or whether patterns reflect an increase in inequality such that racial differences in access remain when controlling for academic background. We considered how this varies by the racial composition of the school; further, we examined differences in access between both Black and Hispanic students and their White peers as well as differences between Black and Hispanic students. The results point to patterns of reproduction of inequality in racially integrated schools, with some evidence of increasing inequality in predominantly Hispanic schools

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Richard Wilders and Lawrence VanOyen

Students will flip when they are given the opportunity to learn about art and transformations in math class.