In the previous sessions, we studied the Communication, Problem Solving, and Reasoning and Proof Standards. In this session, we will examine the Representation Standard. Representations include all the different ways that students depict their mathematical thinking as well as the processes they use to put their thinking into those forms. Representations can include written work, oral explanations, models with manipulative materials, and even the mental processes one uses to do mathematics.

Students need to represent their mathematical thinking for two purposes: so that they better understand the mathematics they are doing and to share their ideas with others. There are many conventional forms of representation that are developmentally appropriate for young children, for example, diagrams, graphical displays, and symbolic expressions. However, before students are ready to use these forms, they need opportunities to express their thinking using their own, invented, non-conventional forms of representation, for example, movement, demonstrations, manipulative materials, drawings, and diagrams. In this session, we will look at both non-conventional and conventional forms that young students might use to represent their mathematical thinking and how the Representations Standard is part of the total process of helping students make sense of mathematics.

Learning Objectives

This session shows you how to help students do the following:

• Create representations
• Use multiple representations to solve problems
• Use representations to model their thinking
• Translate among different representations of the same idea
• Use representations to communicate mathematical understanding

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). For more information on this Process Standard, see the NCTM Web site

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