In the previous sessions, we studied the Communication, Problem Solving, and Reasoning and Proof Standards. In this session, we examine the Representation Standard. Representation includes the ways that students depict their mathematical thinking as well the process they use to put their thinking into that form. Representations can include a variety of written formats, oral explanations, models with manipulative materials, or even the mental process one uses to do mathematics. While representation is the focus of this session, it is helpful to keep in mind that it often works in conjunction with other process standards; for instance, representation can be key to effective problem solving and communication.

We want students to represent their mathematical thinking for two purposes: so that they better understand the mathematics, and so that they can share their ideas with others. Some forms of representation are diagrams, graphical displays, and symbolic expressions. However, before students are ready to use these conventional forms, they need opportunities to express their thinking using their own invented, non-conventional forms of representation. In this session, we look at both non-conventional and conventional forms that students in the middle grades might use to represent their mathematical thinking.

Learning Objectives

This session shows how to help students do the following:

• Create representations from given data
• Use representations to solve problems
• Use representations to model mathematical problems
• Translate among representations
• Use representations to communicate mathematical knowledge

NCTM Representation Standard

Instructional programs. . . 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

## Principles and Standards of School Mathematics (NCTM, 2000, p. 67). For more information on this process standard, see the NCTM Web site.

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