Representations include symbols, equations, words, pictures, tables, graphs, manipulative objects, and actions as well as mental, internal ways of thinking about a mathematical idea. Representations are powerful thinking tools. However, for many students, this power is not accessible unless they receive purposeful guidance in expanding their repertoire.

The act of representing a concept or relationship may result in the use of manipulative materials, the construction of graphs or diagrams, the writing of number sentences, or the presentation of a written or oral explanation. When using representations to solve a problem or make sense of a new concept, students are likely to go back and forth, using the representation to help clarify the problem and using the problem to extend their understanding of the representation.

"Representations do not 'show' the mathematics to the students. Rather, the students need to work with each representation extensively in many contexts as well as move between representations in order to understand how they can use a representation to model mathematical ideas and relationships." (NCTM, 2000, p. 208)

  Defining the Representation Standard