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Instructional programs from prekindergarten through grade 12 should enable all students to--
  • create and use representations to organize, record, and communicate mathematical ideas;
  • select, apply, and translate among mathematical representations to solve problems;
  • use representations to model and interpret physical, social, and mathematical phenomena.

Representations are necessary to students' understanding of mathematical concepts and relationships. Representations allow students to communicate mathematical approaches, arguments, and understanding to themselves and to others. They allow students to recognize connections among related concepts and apply mathematics to realistic problems.

To become deeply knowledgeable about fractions, for example, students need a variety of representations that support their understanding. They need to understand various interpretations of fractions, such as ratio, indicated division, or fraction of a number. They need to understand other common representations for fractions, such as points on a number line.

Some forms of representation--such as diagrams, graphical displays, and symbolic expressions--have long been part of school mathematics. Unfortunately, these representations and others have often been taught and learned as if they were ends in themselves. This approach limits the power and utility of representations as tools for learning and doing mathematics.

It is important to encourage students to represent their mathematical ideas in ways that make sense to them, even if those representations are not conventional. At the same time, students should learn conventional forms of representation in ways that facilitate their learning of mathematics and their communication with others about mathematical ideas. The integration of technology into mathematics instruction further increases the need for students to be comfortable with new mathematical representations.

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