Andy D. Jones

Abstract

Although time may not have been sufficient for publishing many criticisms of including representation as a process standard in the recently released Principles and Standards for School Mathematics (Standards 2000), there undoubtedly will be some group that objects. At this time those arguments will most likely revolve around the ideas that representation is just another form of mathematical modeling and is too limited to be its own process standard. Or perhaps the argument against its inclusion will be that representation does not reach the level of importance that the four previous process standards did.

Whatever the criticisms of including it, there appears to be adequate research on cognition and the learning of mathematics that indicates a strong argument can be made for its inclusion. The argument in this paper for the inclusion of representation as the fifth process standard centers on three main points: 1) fluency in translation between different representations forms a basis for being able to form concepts and think mathematically; 2) the external representations that teachers use have an impact on the internal representations that students form; and 3) use of invented representations allow students to develop meaningful support for problem solving skills and more conventional representations.

The first point is simple: representations should not be taught as ends in themselves. Physical representations as products are not necessarily what we want to achieve with our students. It is the sense of connections between representations that we want students to gain. It is the interplay and connections between different representations that develop a mathematical concept.

Representational theory has several implications on teaching. External representations such as materials, instructional strategies, problems, and technology to which teachers expose students have an impact on how students develop their own representations of mathematical concepts. Whether it be an inappropriate use of a manipulative for a desired teaching objective or the use of a graphing calculator to explore patterns in different representational forms, educators must gauge what internal representations will result.

The final point made in this argument to include representation as a process standard is in regard to the need for learners to construct and invent representations as they learn. These invented representations lend themselves to create correct inferences and processes for solving problems sometimes more often than do conventional representations. In addition, these inventions must be valued in that they serve as supports for the subsequent understanding of more conventional representations which have been deemed worthy of societal use.

These points address basic issues in mathematics and also discount some of the anticipated criticisms of including representation as a major process standard. Representations influence what mathematics is learned, how it is learned, and the process of learning. On the surface, it appears that representation is very important to mathematics education as it enters the 21st century and should play a prominent role in NCTM's vision of mathematics education.

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