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.

# Browse

### Sherin Gamoran Miriam and James Lynn

This article explores three processes involved in attending to evidence of students' thinking, one of the Mathematics Teaching Practices in *Principles to Actions: Ensuring Mathematical Success for All*. These processes, explored during an activity on proportional relationships, are discussed in this article, another installment in the series.

### Sarah K. Bleiler-Baxter, Sister Cecilia Anne Wanner O.P., and Jeremy F. Strayer

Explore what it means to balance love for mathematics with love for students.

### 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**.

### Wayne Nirode

To introduce sinusoidal functions, I use an animation of a Ferris wheel rotating for 60 seconds, with one seat labeled *You* (see **fig. 1**). Students draw a graph of their height above ground as a function of time with appropriate units and scales on both axes. Next a volunteer shares his or her graph. I then ask someone to share a different graph. I choose one student with a curved graph (see **fig. 2a**) and another with a piece-wise linear (sawtooth) graph (see **fig. 2b**).

### Low Chee Soon

Use freedom of choice to promote students' mathematical flexibility.

### Karen D. Campe

There is a distinction between using technology as a tool for doing mathematical tasks and using it to develop conceptual understanding (Dick and Hollebrands 2011). In this article, the table feature of the TI-84 Plus graphing calculator is used in the second role, enabling students to participate in the reasoning and sense-making process. This article showcases four classroom activities that use tables as a dynamic tool for inquiry, applying numerical representations to algebraic, graphical, and geometric phenomena. Although these activities are presented using the TI-84 Plus CE graphing calculator, other calculator and computer platforms can be employed; see the Teacher Guide in **more4U** for details.