We introduce the Into Math Graph tool, which students use to graph how “into" mathematics they are over time. Using this tool can help teachers foster conversations with students and design experiences that focus on engagement from the student’s perspective.

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## Student Engagement with the “Into Math Graph" Tool

### Amanda K. Riske, Catherine E. Cullicott, Amanda Mohammad Mirzaei, Amanda Jansen, and James Middleton

## Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs

### 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.

## Fractions and Long Division Predict Success in High School Mathematics

This department publishes brief news articles, announcements and guest editorials on current mathematics education issues that stimulate the interest of TCM readers and cause them to think about an issue or consider a specific viewpoint about some aspect of mathematics education. This month in the Coaches' Corner, take a closer look at CCSS Standard 3 for Mathematical Practice, Explain and Justify. Coaches may want to demonstrate the integration of math and writing with Speak, Write, Reflect, Revise, a five-step approach for integrating problem solving and the writing process.

## Intellectual Engagement and Other Principles of Mathematics Instruction

### Blake E. Peterson, Douglas L. Corey, Benjamin M. Lewis, and Jared Bukarau

What can American teachers learn about high-quality mathematics instruction from the Japanese teacher education process?