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

### Rebecca Vinsonhaler and Alison G. Lynch

This article focuses on students use and understanding of counterexamples and is part of a research project on the role of examples in proving. We share student interviews and offer suggestions for how teachers can support student reasoning and thinking and promote productive struggle by incorporating counterexamples into the classroom.

### Aaron M. Rumack and DeAnn Huinker

Capturing students' own observations before solving a problem propelled a culture of sense making by meeting needs typical of middle school learners.

### Lee Melvin M. Peralta

One of the many benefits of teaching mathematics is having the opportunity to encounter unexpected mathematical connections while planning lessons or exploring ideas with students and colleagues. Consider the two problems in **figure 1**.

### Laurie Speranzo and Erik Tillema

Specific teacher moves and lesson planning can facilitate student empowerment in the middle school classroom.

### Ron Lancaster

Since its inception, the Mathematical Lens column has provided teachers with resources to use with their students to make connections between mathematics and the world around us through the use of photographs. The editors and the dozens of teachers who submitted material for columns have taken all of us on a journey around the world to discover where mathematics lives. These columns have offered teachers a license to do mathematics everywhere and to travel far with their students with a full tank of resources.

### Stephen Phelps

### Edited by Anna F. DeJarnette

A monthly set of problems targets a variety of ability levels.

### Jo Boaler

Engage your learners through tasks proven to significantly promote reasoning and problem solving, which touch on many of the Mathematics Teaching Practices in *Principles to Actions: Ensuring Mathematical Success for All*. These tasks are discussed in this article, another installment in the series.