# Browse

### Rachel B. Snider

Examples are an essential part of mathematics teaching and learning, used on a daily basis to teach and practice content. Yet, selecting good examples for teaching is complex and challenging. This article presents ideas to consider when selecting examples, drawn from a research study with algebra 2 teachers.

## Informing Practice: A Hybrid Perspective on Functions

### research matters for teachers

### Eric Weber

Formal notions of function, which appear in middle school, are discussed in light of how teachers might complement the input-output notion with a covariation perspective.

### 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.

### Allison B. Hintz

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

### Jennifer Suh and Padmanabhan Seshaiyer

Skills that students will need in the twenty-first century, such as financial literacy, are explored in this classroom-centered research article.

### Lynn M. McGarvey

A child's decision-making photo activity about pattern identification presents implications for teaching and learning patterns in the early years.

## Connecting Research to Teaching: Why Did You Do That?

### Reasoning in Algebra Classrooms

### Daniel Chazan and Dara Sandow

Secondary school mathematics teachers are often exhorted to incorporate reasoning into all mathematics courses. However, many feel that a focus on reasoning is easier to develop in geometry than in other courses. This article explores ways in which reasoning might naturally arise when solving equations in algebra courses.

### Tad Watanabe

**“A mile wide** and an inch deep” is an oftenrepeated criticism of U.S. mathematics curriculum. In 2006, NCTM published *Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence* to suggest important areas of emphasis for instruction. Many states produced new standards that were informed by the book. However, Charles (2008/2009) argues that we must address not only the mile-wide issue, by reducing the number of skill-focused standards, but also the inch-deep issue, by making essential understanding more explicit. Charles suggests that many useful resources are available to deal with the latter.