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Amber G. Candela, Melissa D. Boston, and Juli K. Dixon

We discuss how discourse actions can provide students greater access to high quality mathematics. We define discourse actions as what teachers or students say or do to elicit student contributions about a mathematical idea and generate ongoing discussion around student contributions. We provide rubrics and checklists for readers to use.

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Melissa D. Boston and Margaret S. Smith

Mathematics teachers' selection and implementation of instructional tasks were analyzed before, during, and after their participation in a professional development initiative that focused on selecting and enacting cognitively challenging mathematical tasks. Data collected from 18 secondary mathematics teacher participants included tasks and student work from teachers' classrooms, lesson observations, and interviews. Ten secondary mathematics teachers who did not participate in the professional development initiative served as the contrast group and participated in 1 lesson observation each. Analysis of the data indicated that, following their participation in the professional development initiative, project teachers more frequently selected highlevel tasks as the main instructional tasks in their classrooms and had improved the maintenance of high-level cognitive demands. Significant differences existed between project teachers and the contrast group in task selection and implementation. These differences were not influenced by the use of Standards-based or conventional curricula in project teachers' classrooms.

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Melissa D. Boston, Margaret S. Smith, and Amy F. Hillen

Middle-grades students' understanding of proportional relationships should be fostered through problem solving and reasoning (NCTM 2000). Toward this end, instruction in proportionality should expose students to a variety of strategies and allow students to gain experience modeling proportional situations (Langrall and Swafford 2000). Students should be given ample opportunities to develop intuitive strategies based on factor- of-change (“how many times as many”) relationships (Cramer and Post 1993). Research has shown that middle-grades students are more successful at method is appropriate to use” (NCTM 2000, p. 221). We begin our discussion by focusing on the events that unfold in Marie Hanson's sixth-grade classroom during a lesson on understanding ratios and proportions (Smith et al. forthcoming), and use this lesson as a context for considering how factor-of-change relationships might be used to assist students in understanding why cross multiplication works.