Representations

Representations are used daily in math instruction, yet students and teachers sometimes hardly realize that those representations are indeed...representations. That is, they are symbols used in place of a complex process as well as a function. For example, the + sign, we are told, tells us that 2 + 2 equals 4. In reality, that symbol represents a complex process - a mathematical concept.

Representations are not in and of themselves the process, but they are fundamentally important in learning and understanding mathematics. If an addition sign would not be the established representation for adding, we would inevitably create our own symbol at some point in life to describe a process in short hand. When we as teachers hand the children those representations, they quickly embrace the symbol and its function before grasping the process. Teachers should encourage students to create their own representations - perhaps even before the established mathematical symbol is presented them. This would help students think about the process and approach a meaningful symbol to represent that process, and would probably help the students to remember math better.
In addition to letting the students devise their own representations, teachers should encourage the students to share their representations with other students to increase flexibility in different types of representations for the same concepts. The article suggests that students tend to miss the connection between a circle divided into three equal parts, a division bar, and a fraction bar. If they are presented with a rigid, one-symbol approach to math, they will not be able to make appropriate connections between mathematical concepts.
Finally, once students are able to create representations and translate them to a variety of contexts, they should be able to use them as models for real-life situations. A model, as pointed out in the article, is an ideal picture of an object or a phenomenon. They give people something to work towards - a desired end. Many occupations use models, many of them containing mathematical representations, to engineer their product or goal. It is clear, then, that representations should not be used as an object, but as building blocks to deeper, more creative thinkers.

NCTM (2000). Representations. Principles and Standards for School Mathematics (pp. 76-70). Reston, VA: NCTM.