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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Students' Understanding of Counterexamples
Rebecca Vinsonhaler and Alison G. Lynch
Avoiding the Ineffective Keyword Strategy
Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty
Try these meaningful alternative approaches to helping students make sense of word problems.
Capturing Mathematical Curiosity with Notice and Wonder
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.
Decomposing Rectangles
Annie Perkins and Christy Pettis
Students are given a problem to break down rectangles.
Designing for Voice and Agency
Laurie Speranzo and Erik Tillema
Specific teacher moves and lesson planning can facilitate student empowerment in the middle school classroom.
Emphasizing the M in STEAM activities
Emily Dardis and Megan H. Wickstrom
Modifications to a first- and second-grade STEAM activity, Elephant Toothpaste, highlight ways to emphasize mathematical thinking by running multiple experiments, posing mathematical questions, and having students make both qualitative and quantitative observations. Contributors to the iSTEM department share ideas and activities that stimulate student interest in the integrated fields of science, technology, engineering, and mathematics (STEM) in K–grade 5 classrooms.
Investigating a Super-Bear
Clayton M. Edwards, Rebecca R. Robichaux-Davis, and Brian E. Townsend
Three inquiry-based tasks highlight the planning, classroom discourse, positive results, and growth in one class's journey.
Letters to the editor
License to Do Math with a Full Tank
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.
