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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## Mathematical Explorations: A New Twist on Collaborative Learning

### classroom-ready activities

### Stephanie M. Butman

### Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

### Katie L. Anderson

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes a set of lessons where sixth graders use virtual pattern blocks to develop proportional reasoning. Students' work with the virtual manipulatives reveals a variety of creative solutions and promotes active engagement. The author suggests that technology is most effective when coupled with worthwhile mathematical tasks and rich classroom discussions.

## Mathematical Explorations: Moving beyond Factor Trees

### classroom-ready activities

### Terry L. Kurz and Jorge Garcia

An alternative method for teaching prime decomposition explores using tools rather than factor trees.

### Aryn A. Siegel and Enrique Ortiz

A simple problem-solving exercise encourages teachers to “start small” to reveal how third graders understand multiple math concepts simultaneously.

## Informing Practice: Examples as Tools for Constructing Justifications

### research matters for teachers

### Kristen N. Bieda and Jerilynn Lepak

Research explores how to help students build from, instead of building with, examples when justifying mathematical ideas.

### Terri L. Kurz

After analyzing advertising claims regarding water shooters, students present their findings.

### Gina Kling

Applying known facts to derive unknown facts results in efficiency, flexibility, and an understanding of number combinations for young students.

## Math for Real: Packaging Golf Balls

### “when will I ever use this?”

### Randy Hall

A package of three golf balls provides the real-world scenario for this ratio and area activity.

### Ben C. Sloop and S. Megan Che

This investigation builds on students' understandings of fairness as they explore chance using a set of nontraditional dice with special properties.