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

### Catherine Tabor

A programming activity helps students give meaning to the abstract concept of slope.

### Barbara M. Kinach

The meaningful use of symbols links context and generality.

### Rachel Levy

The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.

### Jason Lee O'Roark

After teaching high school mathematics in Maryland for three years, I began teaching sixth-grade mathematics in one of the best school districts in Pennsylvania (according to state test scores) and have been teaching there for the past six years. My high school teaching background led me to differentiate differently from my colleagues. I share my observations of the result of the differences in methodology and my conclusions from those observations, and I offer a plan to implement changes in the way that mathematics is taught.

### Emily Sliman

Chalk Talk and Claim-Support-Question are two routines for developing students' ability to use multiple representations and encouraging classroom discussion.

### Lingguo Bu

The relationship between a midpoint and an average showcases the interplay between procedural knowledge and conceptual knowledge in learning mathematics for teaching.

### Michael Dempsey

When understood and applied appropriately, mathematics is both beautiful and powerful. As a result, students are sometimes tempted to extend that power beyond appropriate limits. In teaching statistics at both the high school and college level, I have found that one of students' biggest struggles is applying their understanding of probability to make appropriate inferences.

### Neil C. Schwertman and Kate Thomas

Analysis of a rare event such as a *blue moon* (defined today as the second full moon in a month) can provide an interesting exercise that develops quantitative reasoning skills. Research by Whitacre and Nickerson (2006), Grouws and Cebulla (2000), Hill and Ball (2004), and Greeno and Hall (1997), for example, shows that active learning such as classroom activities for individuals or small groups is an excellent tool for stimulating students.

### Jeffrey J. Wanko, Michael Todd Edwards, and Steve Phelps

The Measure-Trace-Algebratize (MTA) approach allows students to uncover algebraic relationships within familiar geometric objects.