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Matt B. Roscoe

Symmetric dot patterns are a particularly powerful object for investigation, providing opportunities for foundational learning across PK–5. We found that second-grade students naturally used repeated addends to count symmetric dot patterns created using the new software TileFarm.

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Allison B. Hintz

Teachers can foster strategy sharing by attending to the cognitive demands that students experience while talking, listening, and making mistakes.

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Jon R. Star, Martina Kenyon, Rebecca M. Joiner, and Bethany Rittle-Johnson

The ability to estimate is not only a valuable math skill but also an essential life skill. Many adults use estimation daily: when tipping a waitress, determining the cost of a sale item, or converting units. Within mathematics, the ability to estimate is linked to deep understanding of place value, mathematical operations, and general number sense (Beishuizen, van Putten, and van Mulken 1997) and allows students to check the reasonableness of their answers to mathematics problems in a variety of contexts.

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Tad Watanabe

“A mile wide and an inch deep” is an oftenrepeated criticism of U.S. mathematics curriculum. In 2006, NCTM published Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence to suggest important areas of emphasis for instruction. Many states produced new standards that were informed by the book. However, Charles (2008/2009) argues that we must address not only the mile-wide issue, by reducing the number of skill-focused standards, but also the inch-deep issue, by making essential understanding more explicit. Charles suggests that many useful resources are available to deal with the latter.