No complex field of endeavor can be defined neatly in a few sentences or paragraphs. Not science or art or technology. Not mathematics. But the outsider can gradually develop a rich sense of the nature of any one of them by examining it from various perspectives and by doing some of the things that the insiders do. These same two kinds of experience—learning about a domain and practicing it—also lead to the development of skills that have wide utility in adult life, as well as to understanding.

As the science of patterns and relationships, mathematics shares many of the features of other sciences, as already described in Chapter 1: The Nature of Science. Particularly relevant similarities are the belief in underlying order, the ideals of honesty and openness in reporting research, the importance of criticism by colleagues in judging the value of new work, and the essential role played by imagination. Mathematics is also like science and technology in that it incorporates both finding answers to fundamental questions and solving practical problems.

The richness and ubiquity of mathematics underlie its treatment by Project 2061. One component of the project's approach is evident in both Science for All Americans and Benchmarks for Science Literacy. In both documents, Chapter 2 outlines what students should know about mathematics as a unique endeavor; Chapter 9 recommends what mathematical ideas students should acquire; Chapter 11 presents some powerful mathematical concepts, such as scale and models, that are widely useful as analytical tools; and Chapter 12 lists the mathematical skills students need, along with scientific and technological skills, to deal effectively with practical affairs in everyday life. Of course, the curriculum would not separate knowing and doing in that way.

Indeed, it is in curriculum design that the other component of the Project 2061 approach to mathematics will be found. As will be described fully in a forthcoming report, Designs for Science Literacy, nearly all of the building blocks of a Project 2061 curriculum will include mathematics. Students will encounter mathematics at every grade level, and learn mathematics in many different subject-matter and real-world contexts (some explicitly labeled as mathematics, some not).

The universe is made up of galaxies, mountains, creatures, vehicles, and all manner of other things, each seemingly unique. Moreover, it is a chaotic affair in which those things intrude on one another in all sorts of ways, often violently but sometimes with great subtlety. But thanks to mathematics, people are able to think about the world of objects and happenings and to communicate those thoughts in ways that reveal unity and order.

The numbers, lines, angles, shapes, dimensions, averages, probabilities, ratios, operations, cycles, correlations, etc., that make up the world of mathematics enable people to make sense of a universe that otherwise might seem to be hopelessly complicated. Mathematical patterns and relationships have been developed and refined over the centuries, and the process is as vigorous and productive now as at any time in history. Perhaps that is because today mathematics is used in more fields of endeavor than ever before and has also become more essential in everyday life.

For purposes of general scientific literacy, it is important for students (1) to understand in what sense mathematics is the study of patterns and relationships, (2) to become familiar with some of those patterns and relationships, and (3) to learn to use them in daily life. The latter two of these general goals should be sought in parallel rather than sequentially. For the most part, learning mathematics in the abstract before seeking to use it has not proven to be effective. Thus, the curriculum should arrange instruction so that students encounter any given mathematical pattern or relationship in many different contexts before, during, and after its introduction in mathematics itself.

From time to time thereafter, in pursuit of the goal of understanding the nature of mathematics, students should have an opportunity in mathematics to reflect on the nature of patterns and relationships in a purely abstract way. Individual or class portfolios of examples of patterns and relationships collected over time could be used as the raw material for reflecting on how mathematics defines a pattern or relationship so that it transcends and is more powerful than individual instances of it.

The individual ideas created and used by mathematicians are traditionally aggregated, often for pedagogical reasons as well as for strictly conceptual ones, into families such as arithmetic, geometry, algebra, trigonometry, statistics, and calculus. Mathematicians look for patterns and relationships that link different ideas (themselves patterns and relationships) within such families and in between separate ones. Few achievements in mathematics are as satisfying as showing that what were previously thought of as two separate parts of mathematics are parallel or different examples of a single, more abstract formulation. If at all possible, all students should have the experience of discovering for themselves that an idea can be represented in different but analogous ways.

One line of research on how people learn emphasizes the helpfulness of making multiple representations of the same idea and translating from one to another. When a student can begin to represent a relationship in tables and in graphs and in symbols and in words, one can be confident that the student has really grasped its meaning. And, as the theory goes, the way students learn to make those representations and translations is to see them and practice them in contexts in which they care about what the answer is. Students engaged in this kind of activity will eventually get the idea of connectedness in mathematics—although they may need occasionally to look back on their own work and recognize the many connections they have made.

In the first few grades, children think in very specific and concrete terms. They are little interested in grand categories such as mathematics, science, and technology, but they usually respond positively to the challenge of learning numbers and how to manipulate them, identifying shapes and simple patterns, collecting and describing collections, and building things. At some point, of course, they need to see that certain kinds of ideas and activities are mathematical, certain kinds scientific, and others technological. But when that labeling happens is less important than that, from the start, children study numbers and shapes and simple operations on them and do so in as many different contexts as possible.

If a basic strategy for learning about the nature of mathematics is to reflect on things already learned in various contexts, one possibility would be to have a list in the classroom, headed "What We Do In Mathematics," to which new items could be added month by month. From time to time, items could be grouped, or shown to be subsets of others, or shown to be similar to those in other lists called "What We Do In Science" and "What We Do In Language." If subject differences are to be de-emphasized, an alternative would be to use "Numbers We Have Used," "Shapes we Have Used (or Made)," "Observations We Have Made," and so on.

The mathematics list might, for example, first include count, measure, estimate, and see shapes in things, then expand to include add, subtract, etc. Later, students could group together add, subtract, etc. under do operations on numbers, and group make graphs, spread out data, and compare two groups of data under analyze data. Measure would have to be included in the math and science lists, as would most of the data items, and would demonstrate linkages. Items such as find patterns, describe relationships, and give reasons would appear in the language list as well as the other lists. In this way, students would build their own inventory of mathematics and have a history of what they were learning, and newcomers could get an idea of what ideas (and language) would be expected of them.

In earlier grades, students studied mathematical patterns and relationships in mathematics and, hopefully, in other classes as well. By now, they should have had considerable experience in making data tables, graphs, and geometric sketches and using them, along with symbols and clear English, to describe a wide variety of patterns and relationships. Thus students are now ready to concentrate more intensely than before on the creative aspects of mathematical problem-solving and to begin to develop a sense of how mathematicians go about their work.

Students start by beginning to reflect on what they do in mathematics. Groups of students should independently propose solutions to problems and compare their solutions with one another, defending and discussing differences. Groups should be encouraged to invent some of their own methods for making computations. One result could be recognition that more than one way works. But different groups might develop strong feelings about which method is best—thereby experiencing some of the heat that disagreements about mathematical abstractions have produced historically. Investigations of data sets should enable different groups of students to find different, perhaps even contradictory, relationships in them. Students can also begin to invent their own problems and see how they differ from those that other students find interesting.

To many students, the most "elegant" mathematics might seem to be the most complicated. Repeated reinforcement is necessary to establish that the simplest way of representing and connecting ideas is often what mathematicians value most. But a simple mathematical connection may have been found by very messy and prolonged study, which may include jumping back and forth from one part of the problem to another and sometimes running into dead ends.

In addition to reflecting on personal problem-solving experience, case studies of how advances in mathematics have been made can be used to bring out some of the main features of how mathematics works and the kinds of patterns and relationships that have resulted from mathematical investigation. Students may occasionally discover mathematics for themselves, and although such discoveries are unlikely to be novel, a great deal should be made of them to encourage what may turn out to be a talent for mathematics.

Getting a grasp on the nature of today's mathematics, which is still engaged in eliciting new patterns and relationships, is likely to be a challenge for nearly all students. Modern theoretical mathematics may have to be suggested by the kind of practical problems it helps to solve—the coloring of maps, the optimizing of air routes, the recovery of detail from blurry images. If students believe that abstractions relevant to one practical situation are likely to be relevant to others as well, making the pedagogical transition from applied to abstract may not undermine the notion that the mathematician's interest is theoretical.

In the earliest grades, students make observations, collect and sort things, use tools, and build things. They are, for their level of development, doing science and using technology. In school practice, science and technology should contribute to understanding the value of mathematics, and mathematics should help in doing science and engineering. The usefulness of mathematics in science and technology will be clear to students if they experience it often."

No benchmarks for this level.

The interaction should become more frequent and more sophisticated as students progress through the upper elementary and middle grades. Graphing, making tables, and making scale drawings should become commonplace in student inquiry and design projects, as should the use of geometric and mathematical concepts such as perpendicular, perimeter, volume, powers, roots, and negative numbers. Problems that are used to challenge students may take the form of contests and games, but at least some of the problems should stem directly from the science and technology being studied."

No benchmarks for this level.

Science and technology are rich and especially important contexts in which to learn the value of mathematics and to develop mathematical problem-solving skills. But they are not the only ones. Art, music, social studies, history, physical education and sports, driver education, home economics, and other school subjects are appropriate places to learn, as well as use, mathematics.

Students in this age range should be exposed to historical examples of how mathematics has contributed to the advancement of science and technology—and vice versa. The instances are so numerous that there is no trouble in finding some that are related to whatever mathematics is being studied. At some point, special attention should be paid to the use of mathematical models in both science and technology. Also, the curriculum needs to provide opportunities for students to examine explicitly the relationship of mathematics to science and technology.

Just what is it exactly that mathematicians do when they are doing mathematics? Most people have some sense, however inaccurate in detail, of what different occupations are because they encounter them personally or indirectly through books, movies, and television. They have little opportunity, however, to watch mathematicians at work or to have them explain what they do. Learning how to solve certain kinds of well-defined mathematics problems is important for students but does not automatically lead them to a broad understanding of how mathematical investigations are carried out.

Mathematics can be characterized as a cycle of investigation that is intended to lead to the development of valid mathematical ideas. That is the approach taken in Science for All Americans and in this section of Benchmarks. (Some of the same ground is covered in Chapter 11: Common Themes, in which the use of mathematical models is considered along with physical and conceptual models.)

It is essential to keep in mind that mathematical discovery is no more the result of some rigid set of steps than is discovery in science. It is true that mathematical investigations sooner or later involve certain processes, but the order is not fixed and the emphasis placed on each process varies greatly. Each of the three parts of the cycle—representation, manipulation, and validation—should be studied in its own right as part of what constitutes learning mathematics. Students should have the chance to use the entire cycle in carrying out their own mathematical investigations. The purpose of this experience is to produce not professional mathematicians but adults who are familiar with mathematical inquiry.

Each part of the cycle poses some learning difficulties. The process of representing something by a symbol or expression is taken by many students to refer only to "real things." "Let A stand for the area of the floor in this room" is easier for young students to grasp than "Let Y equal the area of any rectangle." First, students have to be convinced that substituting abstract symbols for actual quantities is worth the effort. Then they need to work their way toward the realization that using symbols to represent abstractions, and abstractions of abstractions, also pays off in solving problems. Perhaps this means bringing students to see that in the world of mathematics, numbers, shapes, operations, symbols, and symbols that summarize sets of symbols are as "real" as blocks, cows, and dollars.

As to manipulation, there are two conditions that may seem contradictory to students. One is that there is always a set of rules that must be strictly adhered to; the other is that the rules can be made up. That is where the rigor and game-playing spirit of mathematics meet. Imagine some quantities, assign them properties, select some operations, represent everything by symbols, set a problem, and then, following the rules of logic that have been adopted, move the symbols around to see what solutions emerge.

But how good are the solutions? It depends—and that is what students may have trouble understanding. They are used to working mathematical problems in which the procedures are predetermined and "correct" answers are expected. But in real mathematical investigations, a good solution is one that results in new mathematical discoveries or that leads to practical outcomes in science, medicine, engineering, business, or elsewhere. Thus validation in mathematics is a matter of judgment, not authority. And where a solution is less than satisfactory, it may have as much to do with the sense of what is good enough or with how the problem was formulated as with how it was carried out.

Concrete objects should be employed routinely to help children discover and explain symbolic relationships. Students should come to see that numbers and shapes can be used to describe many things in the world around them. Eventually they should come to realize that just as letters and words make up a language in reading and writing, numbers and shapes make up a language in mathematics.

The routine use of concrete objects continues to be essential to help students connect real things and events with their abstract representations. The ability to picture and do things in their heads will be enhanced by frequent reference to real-world applications. Students should be encouraged to describe all sorts of things mathematically—in terms of numbers, shapes, and operations.

Students should begin to assign letters as temporary names of objects—mathematical or not—for purposes of discussing these objects when no other name is known. Gradually the notion of a symbol standing in for a particular unknown can be extended to its standing for any of a collection of possible unknowns. Undoubtedly students will often have to return to concrete ideas as they learn new mathematics.

Students should examine the limitations of some mathematical models in describing and predicting events in the real world. (Disappointing results of mathematical modeling may be due to unpredictable variation in the real world, as well as to use of an inappropriate mathematical model.) Students should be encouraged to state their own criteria for what is a satisfactory result and to discuss their judgments in terms of their purposes.

The artificiality of problems should be minimized, so that there is not always a clear-cut right answer—and so that improvements and alternatives in the solution can be made through the mathematical cycle of trial, evaluation, and revision. A distinction should be drawn between mistakes (such as faulty multiplication) and reasonable choices that turn out to be unsuccessful (and can be reconsidered).

So that students do not get the idea that there is always one best mathematical model for any science or technology problem, opportunities should be provided in which more than one mathematical description seems equally appropriate. The mathematical cycle of reasoning can first be considered explicitly by having students go back over how they solved problems before—and thereafter by recalling that to their attention whenever they approach new problems. The image of some mathematics as a "game" played with arbitrary rules should include the idea that the play is chosen with the goal that the results will be interesting and widely applicable. The rules of the game shouldn't be mutually contradictory—at least not within any intended applications.