Defining Representation
 Introduction | Defining Representation | Connecting Representations | The Teacher's Role | Your Journal
"Learning to record or represent thinking in an organized way, both in solving a problem and in sharing a solution, is an acquired skill for many students. Teachers can and should emphasize the importance of representing mathematical ideas in a variety of ways. Modeling this process as they work through a problem with the class is one way to stimulate students to use and analyze representations"

## (NCTM, 2000, p. 207)

Students are frequently advised to "Draw a picture," "Use models," or "Write a number sentence." Students further develop their own problem-solving skills and their understanding of operations and procedures by using a variety of representations. However, the power of representation is greatly enhanced when the right balance is achieved between independent exploration, reflection on the work of others, and experiences with new types of representation that build upon each other.

The teacher can model the use of a variety of representations as a class or in a group in order to solve problems. By observing students as they work on rich problems, the teacher also has an opportunity to select a number of students or groups to share their representation and thinking. Through careful questioning, the similarities and differences, as well as the general usefulness, of different representations can be discovered.

Certain conventional tools of representation, such as number lines, bar graphs, and coordinate graphs, have important design elements that must be explicitly brought to the attention of all students. Other tools are more open organizational devices and can be developed over time through problem solving and comparisons of methods. However, "many students need support in constructing pictures, graphs, tables, and other representations. If they have many opportunities for using, developing, comparing, and analyzing a variety of representations, students will become competent in selecting what they need for a particular problem" (NCTM, 2000, p. 208).

It is beneficial to work from different starting points, which the teacher can help foster. For example, students might be given the equation n = 6(x); their task is to make up a situation that fits this equation and then to model it with manipulative materials. Similarly, a table of values might be given, and the situation and labels for the first and second columns will then be decided on by a class or small group of students.

Here is what one student group came up with for the n = 6(x) equation:

Selecting tasks is one of a teacher's most critical responsibilities of the teacher. Rich mathematical tasks that are compatible with a variety of representations offer students opportunities to make decisions as to which form of representation to use, to see other representations used and discussed, to make connections between representations, and to practice the use of new forms.

Representations are truly powerful thinking tools for upper elementary students. Through work with a variety of manipulative materials, diagrams, graphs, and equations, students can develop lasting understanding of and skill with important mathematical topics while also building strong foundations for future learning. By looking for connections among representations, students are able to continue to see mathematics as a unified, sensible subject area. Representations also facilitate communication with others and serve as records of information. In the course of solving and discussing rich problems under a teacher's guidance, all students can expand their use of representations.

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