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Stephanie M. Butman

Research on students' learning has made it clear that learning happens through an interaction with others and through communication. In the classroom, the more students talk and discuss their ideas, the more they learn. However, within a one-hour period, it is hard to give everyone an equal opportunity to talk and share their ideas. Organizing students in groups distributes classroom talk more widely and equitably (Cohen and Lotan 1997).

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Amy F. Hillen and Tad Watanabe

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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Allison B. Hintz

Teachers can foster strategy sharing by attending to the cognitive demands that students experience while talking, listening, and making mistakes.

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A cartoon highlighting growth of a retirement fund is coupled with a full-page activity sheet.

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Gabriel T. Matney and Brooke N. Daugherty

Cans on a grocery store shelf and Hirst's Capric Acid Amide can illustrate dot arrays, thus helping students understand the distributive property, partial products, and the standard algorithm for multiplication.

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Lisa A. Brooks and Juli K. Dixon

A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.

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Wendy P. Ruchti and Cory A. Bennett

Solutions coupled with drawings can illustrate students' understandings or misunderstandings, particularly in the area of proportional reasoning.

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Pamela J. Dunston and Andrew M. Tyminski

Techniques for teaching mathematics terminology allow adolescents to expand their abstract reasoning ability and move beyond operations into problem solving.

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Gloriana González and Anna F. DeJarnette

Students develop ownership and increase their understanding of mathematics when they are allowed to discuss alternative perspectives.

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Tutita M. Casa

This instructional tool helps students engage in discussions that foster student reasoning, then settle on correct mathematics.