Defining Representation
 The Representation Standard | Connections | Structure | Additional Points | Summary | Your Journal
"Electronic technologies ... furnish visual images of mathematical ideas, they facilitate organizing and analyzing data, and they compute efficiently and accurately. They can support investigation by students in every area of mathematics."

## (NCTM, 2000, p. 24)

The Role of the Teacher

Students need experiences that go beyond textbook examples and skill-building practice exercises with a particular representation (for instance, graphing quadratic equations). Teachers must provide them with problems that prompt good thinking about representations in a variety of situations. This helps them learn both how and when to apply a particular representation. It will also help students build a repertoire of tools that will be of use in problem solving and in later study, when representations of increasing sophistication and abstraction are encountered.

Working this way serves a key goal of this standard: to prepare students to work with representations in a wide variety of contexts. Examples include interpreting scientific or business data, working on personal finance questions, or evaluating news reports. In these contexts and others, students must be able to decide which type of representation will be of most use in interpreting and solving a particular problem. To do this, students need to be asked to look at a situation and critique the benefits and drawbacks of representing it in various ways, in light of specific goals. These ways include, but are not limited to, graphs, equations, and verbal descriptions. And whether they are evaluating others' representations or creating their own, students must be able to communicate about them and recognize connections.

So how can teachers create a classroom where this kind of development is encouraged? One way is to always look for problems and larger tasks that are connected to significant mathematical ideas and are conducive to the use of a variety of representations. Sometimes such tasks are highlighted in the curricular material, sometimes they are buried within a problem set, and sometimes they need to be collected from other teachers, from workshops, or other resources. The classroom itself may be a resource: Students working on graphing calculator may get an unexpected result, and this may be the springboard to an investigation of this representation and the underlying mathematics.

The teacher's skill at posing questions, scaffolding learning, and forming and coaching problem-solving student groups also comes into play. Students must encounter a certain amount of struggle and then resolve the issue in order to expand their capacity to work with a variety of representations. Through questioning and small-group discussions, students may be guided as they develop understanding of and ways to remember the features of a particular type of representation. For example, students may be asked to work in a small group to find similarities and differences between four equations that are related to a carefully chosen set of ellipses. A group may be asked to create and discuss graphs and their related equations and then to write down their observations about how any two equations, such as an equation symmetric to the x-axis and one that is not symmetric, can be transformed back and forth.

The Role of Technology

"Computers and calculators change what students can do with conventional representations and expand the set of representations with which they can work. For example, students can flip, invert, stretch, and zoom in on graphs using graphing utilities or dynamic geometry software" (NCTM, 2000, p. 68).

Computer-based technology offers students many opportunities to explore representations. A graphing calculator, for instance, makes it possible to examine the effects of changing the values of coefficients in a family of equations. Spreadsheets and computer algebra systems facilitate analysis of large quantities of real data or the solution of systems of equations. Dynamic geometry software gives students the chance to explore geometric properties by rotating and changing perspectives of figures; such software also provides greater access to concepts involving geometric solids. All of these tools have the potential to extend the range of topics and examples available to the high school mathematics learner. In all cases, technology should be used to enhance, not replace, basic skills and understandings.

Students benefit from explicit instruction in the different representational conventions used in such technological tools as well as in the connections to standard non-technological forms. For example, in the Observe section we saw students solve a probability problem using both tree diagrams and matrices. A basic calculator can easily be used to multiply the probabilities given on the tree diagram. But greater familiarity with a more advanced calculator or computer program is needed in order to correctly input the matrix information and to multiply the matrices in an efficient order. Such skill can be gained through exploration that is guided by the teacher while solving a meaningful problem. Attention should be paid to the students' understanding of what the technological tool is doing, what the resulting representation means, how to speak about it using standard vocabulary, and how to use the representation in problem solving.

The resources provided by the World Wide Web, including the interactive activities in this course, the Annenberg Learning Math online course, and "e-examples" from the NCTM Web site, use technology to represent mathematical ideas in an accessible, hands-on way. These activities should be considered for use in your classroom.

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