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.

### Samuel Otten and Andrew Otten

Students make strategic choices–and justify them–to solve a system of two linear equations.

### S. Asli Özgün-Koca

Student interviews inform us about their use of technology in multiple representations of linear functions.

A set of problems of many types.

### Wayne Nirode

Using technology to solve triangle construction problems, students apply their knowledge of points of concurrency, coordinate geometry, and transformational geometry.

### Dung Tran and Barbara J. Dougherty

The choice and context of authentic problems—such as designing a staircase or a soda can—illustrate the modeling process in several stages.

A set of problems of many types.

### J. Matt Switzer

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

### G. Patrick Vennebush, Thomas G. Edwards, and S. Asli Özgün-Koca

Students analyze items from the media to answer mathematical questions related to the article. This month's clips involve finding a mathematical error in an advertisement as well as working with ratios and proportions.