In the previous sessions, we have studied the Communication, Problem Solving, and Reasoning and Proof Standards. In this session, we will examine Representation.

Representation is central to the learning and use of mathematics. But what is representation? Although our quick answer might be something like, "A representation is a graph of an equation," in this session we will broaden the scope. As the NCTM points out: "The term representation refers both to process and to production; in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself. Moreover, the term applies to processes and products that are observable externally as well as to those that occur 'internally,' in the minds of people doing mathematics" (NCTM, p. 67).

We expand our understanding of mathematical relationships as we use representations; for instance, as we decide upon the most appropriate form to use while working on a task (for example, graphs, equations, verbal statements, or tables of values), and as we interpret these forms, using them to gain insights. It is important to keep in mind that "different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses" (NCTM, 2000, p. 360).

This session shows how to help students do the following:

• Identify connections among representations
• Use representations to model relationships and solution methods
• Use representations to communicate mathematical knowledge
• Choose and create representations from given data

Instructional programs. . . should enable all students to do the following:

• 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.

  Observe a math problem