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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.

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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.

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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.

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Peter Wiles, Travis Lemon, and Alessandra King

Students move from slides, flips, and turns into reasoning about the characteristics of rigid transformations.

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Sarah Ferguson

Explore the creation of a unique problem-based learning (PBL) experience.

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Edited by Brian Carvalho

You may have heard that if you are outside and see a flash of lightning, you can estimate the distance between you and the lightning strike fairly well by counting the number of seconds that pass between the lightning flash and the clap of thunder. The rule of thumb is that for every 5 seconds that pass before you hear the thunder, the lightning strike is 1 mile away.

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Joseph Muller and Ksenija Simic-Muller

What happens with cat populations when they are not controlled? Consider the case of Aoshima Island in Japan. Aoshima Island is called a cat island: Its cat population is 130 and growing; its human population is 13. The cats live in colonies and are fed and cared for by people who live on the islands.