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## Using Covariation Reasoning to Support Mathematical Modeling

Table representations of functions allow students to compare rows as well as values in the same row.

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## “Tricks” Work!

The use of mnemonic devices, or “tricks,” in the mathematics classroom has been criticized by some authors. However, when used in the proper context, such “tricks” can be extraordinarily helpful in motivating students and helping them remember procedures while understanding concepts and mastering appropriate mathematical vocabulary.

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## Mysterious Subtraction

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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## Strengthening Discussions

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

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## Mathematical Micro-Identities: Moment-to-Moment Positioning and Learning in a Fourth-Grade Classroom

Identity is an important tool for understanding students' participation in mathematics lessons. Researchers usually examine identity at a macro-scale: across typical classroom activity and in students' self-reports. However, learning occurs on a micro-scale: in moments during a lesson. To capture identity in these moments, I used positioning theory to develop a framework of micro-identity and then to examine the identities and learning of 1 fourth-grade student during 1 mathematics lesson. This study demonstrates how mathematical identities can shift in dramatic ways in response to minor changes in context so that a student might be, in one moment, engaged in an identity that undermines learning and then later engaged in an academically productive identity. These shifting micro-identities have important implications for mathematical learning, classroom contexts, and macro-identities.

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## Cartoon Corner: Investing in Whose Retirement?

A cartoon highlighting growth of a retirement fund is coupled with a full-page activity sheet.

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## Open-ended Questions and the Process Standards

Educating students—for life, not for tests—implies incorporating open-ended questions in your teaching to develop higher-order thinking.

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## Seeing Spots and Developing Multiplicative Sense Making

Cans on a grocery store shelf and Hirst's Capric Acid Amide can illustrate dot arrays, thus helping students understand the distributive property, partial products, and the standard algorithm for multiplication.

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## Spaghetti Sine Curves: Virtual Environments for Reasoning and Sense Making

The Spaghetti Sine Curves activity, which uses GeoGebra applets to enhance student learning, illustrates how technology supports effective use of physical materials.

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## Changing the Rules to Increase Discourse

A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.