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.

## For ELLs: Vocabulary beyond the Definitions

### Nancy S. Roberts and Mary P. Truxaw

A classroom teacher discusses ambiguities in mathematics vocabulary and strategies for ELL students in building understanding.

## Counterexamples as Starting Points for Reasoning and Sense Making

### David A. Yopp

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

## Connecting Research to Teaching: Classroom Collaboration: Moving beyond Helping

### Corey Webel

Do you use group work in your mathematics class? What does it look like? What do you expect your students to do when they work together? Have you ever wondered what your students think they are supposed to do?

## From the Editorial Panel: Fostering Flexible Mathematical Thinking

Understanding and making connections is at the heart of flexible mathematical thinking. Flexible thinking is generally observed in students' ability and willingness to work with multiple representations, but it can also be seen in their facility in adapting problem-solving strategies when faced with novel situations, reversing thought processes, viewing notation as a process as well as an object, interpreting others' ideas, and posing problems. Although teachers may characterize flexible mathematical thinking differently according to context, we can probably all agree that providing students with opportunities to engage in flexible mathematical thinking is vital for fostering the kind of mathematical understanding that we want them to have

## Delving Deeper: Approximating π Using Similar Triangles

### Ryota Matsuura

This article presents a method for approximating π using similar triangles that was inspired by the author's work with middle school teachers. The method relies on a repeated application of a geometric construction that allows us to inscribe regular polygons inside a unit circle with arbitrarily large number of sides.

## Sound Off!: The Mathematics Teacher's Bookshelf

### Vena M. Long

Books are an excellent source of information and enjoyment and should have a role in every mathematics teacher's professional development plan. This article encourages everyone to have a bookshelf, literal or virtual, and lists several of the author's favorites, noting what each has to offer the reader.

## Sketching Up the Digital Duck

### Kathryn G. Shafer, Gina Severt, and Zachary A. Olson

Two preservice teachers describe how using Google SketchUp, Terrapin Logo, and The Geometer's Sketchpad fosters a deeper understanding of measurement concepts.

## Connecting Research to Teaching: Making Student Thinking Public

### Shari L. Stockero and Laura R. Van Zoest

Teachers often have students publicly share their mathematical thinking as part of classroom instruction. Before reading further, we invite you to stop and think about this practice by writing down your responses to the following two questions: