As students become mathematically sophisticated, they develop an increasingly large repertoire of mathematical representations and the knowledge of how to use them productively. This knowledge includes choosing specific representations in order to gain particular insights or achieve particular ends. The importance of representations can be seen in every section of this chapter. If large or small numbers are expressed in scientific notation, their magnitudes are easier to compare and they can more readily be used in computations.
Representation is pervasive in algebra.
Graphs convey particular kinds of information visually, whereas symbolic expressions may be easier to manipulate, analyze, and transform. The use of matrices to represent transformations in the plane illustrates how geometric operations can be represented visually yet also be amenable to symbolic representation and manipulation in a way that helps students understand them. The various methods for representing data sets further demonstrate the centrality of this topic. A wide variety of representations can be seen in the examples in this chapter. A major lesson of that story is that different representations support different ways of thinking about and manipulating mathematical objects.