Encouraging students to justify earlier as they attempt to solve an open-ended task can lead to greater understanding and engagement.

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## Patterns on a Calendar

### big solutions to little problems

### Jo Ann Cady and Pamela J. Wells

Solutions to a previous Solve It problem are discussed, and the procedures used with problem solving are explored.

## Mathematical Explorations: A New Twist on Collaborative Learning

### classroom-ready activities

### 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).

## Mathematical Explorations: Find the Distance: No Formula Necessary

### classroom-ready activities

### Ryota Matsuura and Yu Yan Xu

This activity involves finding the distance between two points in a coordinate plane and emphasizes reasoning from repeated calculations, which is one of the mathematical practices specified by the Common Core State Standards for Mathematics.

## Math for Real: Shaking Things Up with the Richter Scale

### “when will I ever use this?”

### Fred Dillon and Kevin Dykema

This problem ties into the real-life measurement found in the Richter scale.

### D. Bruce Jackson

Given two slices of bread—a problem and the answer—students fill in the fixings: their own mathematics reasoning.

### Jane M. Wilburne and Ashley Kulbacki

A sixth-grade teacher's word task uncovers higher-level thinking and engages her students in the Standards for Mathematical Practice.

### 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.

### Hyewon Chang and Barbara J. Reys

Using Clairaut's historic-dynamic approach and dynamic geometry tools in middle school can develop students' conceptual understanding before they encounter formal proof in geometry.

## Redeem Reasoning

### readers speak out

### Kuo-Liang Chang

This opinion piece discusses how simplicity, ease, and efficiency—in the guise of shortcuts, tips, and packed procedures—kill mathematical reasoning.