Through the middle grades, children's mathematical representations usually are about objects and actions from their direct experience. Primary school students might use objects to represent the number of wheels on four bicycles or the number of fireflies in a story. In the middle grades, students can begin to create and use mathematical representations for more-abstract objects, such as rational numbers, rates, or linear relationships. High school students should use conventional representations as a primary means for expressing and understanding more-abstract mathematical concepts. Through their representations, they should be ready to see a common structure in mathematical phenomena that come from very different contexts.
Representations can help students organize their thinking.
Students' use of representations can help make mathematical ideas more concrete and available for reflection. In the lower grades, for example, children can use representations to provide a record for their teachers and their peers of their efforts to understand mathematics. In the middle grades, they should use representations more to solve problems or to portray, clarify, or extend a mathematical idea. They might, for example, focus on collecting a large amount of data in a weather experiment over an extended time and use a spreadsheet and related graphs to organize and represent the data. Computers and calculators change what students can do with conventional representations and expand the set of representations with which they can work.