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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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Laurie Speranzo and Erik Tillema

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

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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.

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Stephen Phelps

Edited by Anna F. DeJarnette

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

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P. Reneé Hill-Cunningham

Hundreds of species of animals around the world are losing their habitats and food supplies, are facing extinction, or have been hunted or otherwise negatively influenced by humans. Students learn about some of these animals and explore multiple solution strategies as they solve this month's problems. Math by the Month features collections of short activities focused on a monthly theme. These articles aim for an inquiry or problem-solving orientation that includes four activities each for grade bands K–2, 3–4, and 5–6.

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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.