If mathematics is the "science of patterns" (Steen 1988), representations are the means by which those patterns are recorded and analyzed. 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. Mathematical modeling requires representations, as illustrated in the "drug dosage" problem and in the "pipe offset" problem. 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. By using various representations for the "counting rectangles" problem in the "Problem Solving" section, students could find different solutions and compare them. The use of algebraic symbolism to explain a striking graphical phenomenon is central to the "string traversing tiles" task in the "Communication" section. Representations facilitate reasoning and are the tools of proof: they are used to examine statistical relationships and to establish the validity of a builder's shortcut. They are at the core of communication and support the development of understanding in Marta's and Nancy's work on the "string traversing tiles" problem. Although at one level the story of Mr. Robinson's class is about connections, at another level it is about representation: one group of students places coordinates that "make things eeeasy," the class gains insights from dynamic representations of geometric objects, and the students produce proofs in coordinate and Euclidean geometry. A major lesson of that story is that different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses. »

What should representation look like in grades 9 through 12?

In grades 9–12, students' knowledge and use of representations should expand in scope and complexity. As they study new content, for example, students will encounter many new representations for mathematical concepts. They will need to be able to convert flexibly among these representations. Much of the power of mathematics comes from being able to view and operate on objects from different perspectives.

In elementary school, students most often use representations to reason about objects and actions they can perceive directly. In the middle grades, students increasingly create and use mathematical representations for objects that are not perceived directly, such as rational numbers or rates. By high school, students are working with such increasingly abstract entities as functions, matrices, and equations. Using various representations of these objects, students should be able to recognize common mathematical structures across different contexts. For example, the sum of the first n odd natural numbers, the areas of square gardens, and the distance traveled by a vehicle that starts at rest and accelerates at a constant rate can be represented by functions of the form f(x) = ax². The fact that these situations can be represented by the same class of functions implies that they are alike in some fundamental mathematical way. Students are ready in high school to see similarity in the underlying structure of mathematical objects that appear contextually different but whose representations look quite similar.

High school students should be able to create and interpret models of more-complex phenomena, drawn from a wider range of contexts, by identifying essential features of a situation and by finding representations that capture mathematical relationships among those features. They should recognize, for example, that phenomena with periodic features often are best modeled by trigonometric functions and that population growth tends to be exponential, or logistic. They will learn » to describe some real-world phenomena with iterative and recursive representations.

Consider the graph of the concentration of CO₂ in the atmosphere as a function of time and latitude during the period from 1986 through 1991 (see fig. 7.39) (Sarmiento 1993). Teachers might use an example such as this to help students understand and interpret several aspects of representation. Students could discuss the trends in the change in concentration of CO₂ as a function of time as well as latitude. Doing so would draw on their knowledge about classes of functions and their ability to interpret three-dimensional graphs. They should be able to see a roughly linear increase across time, coupled with a sinusoidal fluctuation with the seasons. Focusing on the change in the character of the graph as a function of latitude, students should note that the amplitude of the sinusoidal function lessens from north to south. Students can test whether the trends they observe in the graph correspond to recent theoretical work on CO₂ concentration in the atmosphere. For example, the author of the article attributes the sinusoidal fluctuation to seasonal variations in the amount of photosynthesis taking place in the terrestrial biosphere. Students could discuss the differences in amplitude across seasons in the Northern and Southern Hemispheres.

What should be the teacher's role in developing representation in grades 9 through 12?

An important part of learning mathematics is learning to use the language, conventions, and representations of mathematics. Teachers should introduce students to conventional mathematical representations » and help them use those representations effectively by building on the students' personal and idiosyncratic representations when necessary. It is important for teachers to highlight ways in which different representations of the same objects can convey different information and to emphasize the importance of selecting representations suited to the particular mathematical tasks at hand (Yerushalmy and Schwartz 1993; Moschkovich, Schoenfeld, and Arcavi 1993). For example, tables of values are often useful for quick reference, but they provide little information about the nature of the function represented. Consider the table in the "Algebra" section in this chapter that gives the number of minutes of daylight in Chicago every other day for the year 2000. The values in the table suggest that the function is initially increasing and then becomes decreasing. Knowledge of the context of a graph of those values suggests that the behavior is actually periodic. Similarly, algebraic and graphical representations of functions may provide different information. Some global properties of functions, such as asymptotic behavior or the rate of growth of a function, are often most readily apparent from graphs. But information about specific aspects of a function—the exact value of f() or exact values of x where f(x)has a maximum or a minimum—may best be determined using an algebraic representation of the function. Suppose g(x) is given by the equation g(x) = f(x) + 1, for all x. The analytic definitions of f(x) and g(x) may offer the most-effective ways of computing specific values of f(x) and g(x), but graphing the function reveals that the "shape" of g(x) is precisely the same as that of f(x)—that the graph of g(x) is obtained by translating the graph of f(x) one unit upward.

As in all instruction, what matters is what the student sees, hears, and understands. Often, students interpret what teachers may consider wonderfully lucid presentations in ways that are very different from those their teachers intended (Confrey 1990; Smith, diSessa, and Roschelle 1993). Or they may invent representations of content that are idiosyncratic and have personal meaning but do not look at all like conventional mathematical representations (Confrey, 1991; Hall et al. 1989). Part of the teacher's role is to help students connect their personal images to more-conventional representations. One very useful window into students' thinking is student-generated representations. To illustrate this point, consider the following problem (adapted from Hughes-Hallett et al. [1994, p. 6]) that might be presented to a tenth-grade class:

A flight from SeaTac Airport near Seattle, Washington, to LAX Airport in Los Angeles has to circle LAX several times before being allowed to land. Plot a graph of the distance of the plane from Seattle against time from the moment of takeoff until landing.

Students could work individually or in pairs to produce distance-versus-time graphs for this problem, and teachers could ask them to present and defend those graphs to their classmates. Graphs produced by this class, or perhaps by students in other classes, could be handed out for careful critique and comment. When they perform critiques, students get a considerable amount of practice in communicating mathematics as well as in constructing and improving on representations, and the teacher gets information that can be helpful in assessment. One representation of the flight that a student might produce is shown in figure 7.40. »

Fig. 7.40. A representation that a student might produce of an airplane's distance from its take-off point against the time from takeoff to landing

This representation indicates a number of interesting and not uncommon misunderstandings, in which literal features of the story (the plane flying at constant height or circling around the airport) are converted inappropriately into features of the graph (Dugdale 1993; Leinhardt, Zaslavsky, and Stein 1990). Representations of this type can provoke interesting classroom conversations, revealing what the students really understand about graphing. This revelation puts the teacher in a better position to move the class toward a more nearly accurate representation, as sketched in figure 7.41.