# Seeing structure & generalizing (MP7, MP8)

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Helping All Students Learn Algebra

An eighth-grade algebra teacher talks about how he builds skills necessary for algebra concurrently with the teaching of algebra standards. He describes his method and shares experiences with linear equations that reflect a coherent approach to learning and skill development. The discussed tasks are available.

*Source:* Doing What Works

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A Coherent Algebra Framework

Hung-Hsi Wu, a professor of mathematics at UC Berkeley, discusses the importance of seeing algebra as a whole rather than a collection of isolated facts and shortcuts to be memorized. He goes on to state the topics that would make up a coherent algebra framework as supported by the National Math Advisory Panel.

**Featuring: **Hung-Hsi Wu, University of California at Berkeley

*Source:* Doing What Works

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What Teachers Need to Know About Teaching Fractions

Francis “Skip” Fennell describes the knowledge teachers need to teach fractions effectively, paying special attention to multiple strategies for representing problems involving fractions. He discusses the convergence of curriculum and explores why students and teachers have difficulty with certain fraction concepts.

**FEATURING: **Francis Fennell, McDaniel College

*SOURCE:* Doing What Works

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The Concepts Behind Operations

David C. Geary describes how to build students’ understanding of the concepts underlying operations with fractions, common misconceptions that children have, and what can be learned from their errors.

**FEATURING: **David C. Geary, University of Missouri

*SOURCE: *Doing What Works

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Making Sense of Computational Procedures

Watch this multimedia overview to learn about the importance of focusing on conceptual understanding and procedural fluency with fractions operations and how they connect. The overview describes recommended instructional practices for developing understanding of computational procedures and ways to address typical student misconceptions.

*SOURCE: *Doing What Works

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Developing Proportional Reasoning

This multimedia overview offers an introduction to students’ development of proportional thinking and its relationship to cross-multiplication. It describes recommended instructional strategies for solving problems related to ratio, rate, and proportion, including buildup and unit ratio strategies, and illustrates examples of real-world context for such problems. This recommended practice contains three tools to support the teaching and learning of ratio, rate, and proportion.

*SOURCE:* Doing What Works

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Building on Intuitive Understanding

Thomas P. Carpenter discusses the importance of building children’s conceptual understanding of fractions based on intuitive ideas about sharing and apportioning objects. He demonstrates examples of children’s responses to challenging problems that help them expand and articulate their thinking.

**FEATURING:** Thomas P. Carpenter, University of Wisconsin - Madison

*SOURCE:* Doing What Works

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A Learning Trajectory for Fractions

Mathematics coaches describe what they’ve observed about how children deepen their understanding of fractions. They unpack the skills required to perform fraction operations using a technique called “tip of the iceberg.”

**FEATURING:** Renee Sherry, Ken Jensen, and Kim Pippenger, Math Coaches, Tollgate Elementary School (CO)

*SOURCE: *Doing What Works