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Overview 

Standard 10: Representation

Standard 1: Number and Operation

Mathematics instructional programs should foster the development of number and operation sense so that all students-

• understand numbers, ways of representing numbers, relationships among numbers, and number systems;
• understand the meaning of operations and how they relate to each other;
• use computational tools and strategies fluently and estimate appropriately.

Elaboration: Pre-K-12

Number, operation, and computation have a long and prominent history in the school mathematics curriculum. In addition, this area of mathematics, perhaps more than any other, is widely-recognized and valued beyond the school setting. Central to this standard is the goal of developing number sense-understanding what numbers mean, how they relate to one another, their relative size, how they can be thought about and represented in many ways, and the effect of operating with numbers. With guidance and meaningful experiences, over time students develop "good intuition about numbers and their relationships" (Howden 1989). Students with number sense naturally decompose numbers, develop and use benchmarks as referents, use the relationships among operations and their knowledge about the base-ten number system to solve problems, estimate a reasonable result for a problem, and have a disposition to make sense of numbers, problems, and results (Sowder 1992). Students with skills embedded in a sense of number and operations are confident users of mathematics.

Knowledge of basic number facts is essential to the development of number sense and is fundamental in enabling students to solve problems. Students must be able to recall basic facts easily. These facts include the single-digit addition combinations and the counterparts for subtraction, multiplication, and division. Basic fact understanding and skill can be developed through the exploration of thinking strategies such as "7+8 is the same as 7+7+1." Understandings and skills can also be developed through varied and systematic practice sessions at home and school. Most students should be able to recall addition and subtraction facts quickly by the end of grade 2 and recall multiplication and division facts with ease and facility by the end of grade 4.

Likewise, computational fluency- having and using efficient and accurate methods for computing-is essential in the development of number sense and to success in most areas of mathematics. In some cases, students will use mental strategies, such as thinking of "6 x 2.5" as "six twos plus six halves." In other cases, students use a combination of mental strategies and jottings on paper to produce quick and accurate results. In still other cases, students may use paper and pencil to perform student-generated or conventional computational algorithms, particularly when the numbers are large or complicated. The main point is that students must have methods that they can use efficiently and that produce correct answers. Using and manipulating materials and reflecting on and comparing strategies helps students develop an understanding of numbers, operations, and their properties and will lead to knowledge of basic facts as well as computational fluency.

All students should acquire strategies for estimating in computational situations and the inclination to judge the reasonableness of numerical data, including computed results. Ability and inclination to estimate depends on understanding numbers-their size, position in the number system, and equivalent forms-and on the effect of operating on those numbers (e.g., "What happens when a whole number is multiplied by a number less than 1?"). Estimation can be used to answer a question directly, such as "How much pizza should we order?" or used to evaluate the reasonableness of an answer resulting from paper and pencil or calculator computation. By the time they reach high school, students should understand measurement error and its effects on computations and should develop the ability to distinguish between estimation and approximation.

Calculators are accessible and reliable tools for computing. All students should use calculators at appropriate times as computational tools. The calculator should be considered a legitimate tool within the mathematics classroom for computing, particularly when many or cumbersome computations are necessary for solving problems. However, when the instructional focus is on developing student-generated or conventional computational algorithms, the calculator should be set aside to allow for this focus. Today, the calculator is a commonly used computational tool outside the classroom. The environment inside the classroom should reflect this reality.

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

Students' knowledge of number concepts and their properties should grow more sophisticated throughout their school experience. In grades pre-K-2, students learn to count, represent, and compare the size of numbers in a variety of ways, aided by physical representations that they can manipulate, such as counters and base-ten blocks. Students in grades pre-K-2 will encounter and should explore numbers larger than they can manipulate physically. In fact, they are often very interested in large numbers, particularly if embedded in contexts from their own experiences. For example, young students can count and keep an ongoing record of the number of coins collected in the school penny drive or the number of soda-can tabs collected in a conservation project. Students in the early grades explore and use part-part-whole relationships. In this way, "24" is seen as "two tens and four ones"; it is also "two sets of twelve." Viewing numbers in flexible ways provides a basis for understanding the way numbers are represented within the base-ten number system.

By grade 2, students can make a transition from viewing a "ten" as simply the accumulation of "ten ones" to seeing it both as "ten ones" and as "one ten." This recognition is a first step toward understanding the structure of the base-ten number system (Cobb and Wheatley 1998). In the early grades, students can also encounter and learn about common fractions (e.g., 1/2, 3/4) prompted by meaningful contexts and building on existing language. For example, most young students already use the word half in their vocabulary outside of school.

Students in grades 3-5 continue to develop and extend their ideas and range of strategies for thinking about and using whole numbers. The knowledge that "3408" is "three thousands, four hundreds, and eight ones" is a foundation for students' understandings of how "3408" is related to "4408," to "3308," and to "3500." Such understandings are part of the ongoing development of number sense and contribute to generating and using strategies for computing.

At grades 3-5, students will study and represent fractions and decimals as numbers, with emphasis on how they are related to whole numbers and to one another. Understanding how a fraction or decimal part of a quantity is related to the unit is a key idea in these grades. Instructional emphasis at this grade band should be on the development of rational number concepts rather than on rational number computational strategies. Useful experiences for students in grades 3-5 include creating physical representations of fractions and decimals, comparing fractions to familiar benchmarks such as 1/2, representing fractions and decimals on a number line, and generating equivalent representations of fractions and decimals. With these understandings, students should be able to estimate fraction sums. For instance, "1/2 + 3/8" must be less than 1 because each addend is less than or equal to 1/2.

As students in grades 3-5 study numbers, they also learn about classes of numbers and their characteristics, such as which numbers are odd, even, prime, composite, or square. Recognizing such characteristics lays a foundation for checking divisibility rules, finding prime factors, or understanding functional relationships.

In grades 6-8, students expand their work with fractions, decimals, and percents, so that they are able to move flexibly among equivalents and apply a range of strategies to order and compare rational numbers. The shift in thinking of fractions as "parts of" specific units to understanding fractions as numbers is completed in the middle grades. Students' knowledge about, and use of, decimals in the base-ten system also extends in these grades. In addition to estimating with rational numbers , students in grades 6-8 also develop computational strategies for fractions and decimals.

The understanding of very large numbers and what they represent continues to develop through the middle grades. Students use tools such as calculators and spreadsheets to organize and analyze numbers in applied contexts, and they learn to represent very large and very small numbers in scientific notation. By extending from whole numbers to integers, middle-grade students' intuitions about order and magnitude expand. They also encounter irrational numbers, such as sqrt(2) and ?, in the course of studying the Pythagorean Theorem and circumference.

Other areas of the curriculum are more prominent than number in grades 9-12; nevertheless, students continue to deepen their understanding of properties of numbers with which they are already familiar and see number systems from a more global perspective. Scientific notation and matrix representation are possible. Complex numbers are also added to students' repertoire, and students learn that not all properties of real numbers are preserved when the system is enlarged.

Understand the meaning of operations and how they relate to each other

As they develop computational fluency, students must understand the meaning of arithmetic operations. This includes deciding which operations should be used for a particular problem, how the same operation can be applied to problem situations that appear to be quite different from one another, how operations relate to one another, and what kind of result to expect.

During the primary grades, students encounter a variety of meanings for addition and subtraction. Subtraction can be viewed through a "take away" interpretation or as a comparison of two sets. Missing-addend situations highlight the relationship between addition and subtraction. Multiplication and division can begin to have meaning for students in grades pre-K-2 as they solve problems that arise within their environment. Such problems might include these: How many slices of bread are needed to make four sandwiches? How can this bag of raisins be shared fairly among four people? Although the major instructional emphasis in pre-K-2 is on addition and subtraction, students who naturally encounter and are curious about situations involving other operations, such as multiplication and division, should be encouraged.

Focusing on the meaning of multiplication and division, particularly with whole numbers, becomes central in grades 3-5. By creating representations of multiplication and division situations, either with diagrams or concrete objects, students gain a sense of the relationships among addition, subtraction, multiplication, and division. By inventing strategies for computing as well as by using conventional computational strategies, students use and recognize such properties as associativity, commutativity, and distributivity and recognize 0 and 1 as identities. The development and comparison of computational strategies provides a chance to focus on the nature of algorithms. For example, many multiplication algorithms illustrate distributivity, and division strategies sometimes involve an iterative process.

Understanding operations with rational numbers is emphasized in grades 6-8. Students at this level also represent and develop operations on integers. Students' intuitions about operations need to be revised with movement into an expanded number system (Graeber and Campbell 1993). For example, the multiplication of a positive number by a fraction smaller than 1 produces a result smaller than the number, counter to students' prior experience that multiplication always results in a bigger number. And, working with integers, subtraction no longer "makes smaller". Students also should focus on the inverse relationship between multiplication and division and the relationship between a fraction and its reciprocal. Proportional reasoning is fundamental for operating with fractions, decimals, rates, and ratios in grades 6-8. Prior to these grades, most of the comparisons students have done are additive, such as "how much taller?" or "how many more?" In the middle grades, students should become proficient in creating ratios to make comparisons in situations that involve pairs of numbers, as in the problem, "If three packages of cocoa make fifteen cups of hot chocolate, how many packages are needed to make sixty cups?"

In grades 9-12, students continue to study operations and establish connections between the operations and other topics. Addition of complex numbers is equivalent to addition of vectors, whereas multiplication by complex numbers has the geometrical interpretation of a rotation combined with stretching (or shrinking). Functional operations-such as finding nth roots, absolute values, and raising to powers-build on earlier work and familiarity with number. Properties of operations, such as closure, are part of understanding algebraic systems.

Use computational tools and strategies fluently and estimate appropriately

A variety of efficient computational tools exist and are used regularly by adults, including mental computation, paper and pencil strategies, estimation, and calculators. Students need experiences that help them choose among tools appropriately. The particular context, question, and numbers involved should be considered when choosing a method. Do the numbers allow a mental strategy? Does the context call for an estimate? Does the problem require repeated and tedious computations? Students should evaluate problem situations to determine whether an estimate or an exact answer is needed and be able to give a rationale for their decision. Estimation strategies and exact computation strategies should be used in tandem as students solve problems.

From pre-K to grade 8, students are expected to develop computational strategies based on their knowledge about numbers and operations and to explain and justify their procedures. By describing the algorithms they use to the teacher and their peers, students see that multiple solution procedures are possible and that some are more efficient than others. Students should become computationally fluent-that is, possess efficient and accurate methods for solving computational problems. Developing fluency requires a balance and connection between conceptual understanding and procedural computation and relies on quick access to basic number facts. On the one hand, computational strategies practiced and applied without a conceptual base are often forgotten or remembered incorrectly (Kamii 1998). On the other hand, understanding without fluency can inhibit the problem-solving process.

As children in grades pre-K-2 develop understanding of whole numbers and the operations of addition and subtraction, instructional attention should focus on developing strategies for computing with numbers of varying size, such as single-digit and multidigit whole numbers. Student-generated strategies should be shared and discussed. By the end of grade 2, students should be able to recall basic addition and subtraction facts, should be fluent in adding two-digit numbers, and should have methods for subtracting two-digit numbers.

In grades 3-5, both student-generated and conventional computational strategies for adding, subtracting, multiplying, and dividing whole numbers are studied, applied to larger numbers, and practiced for fluency. Based on his view of relevant research, Gravemeijer (1998) notes:

The underlying idea is that the students develop mathematical concepts, notations, and procedures as organizing tools when solving problems. In such a process, informal algorithms may come to the fore as forms of well-organized routines for solving certain types of problems. With guidance of the teacher, these informal algorithms can be developed into conventional algorithms. However, one may also opt for fostering the informal algorithms, which are valuable as end goals as well. (p. 4)

Students also develop and begin to apply computational strategies to decimals. In these grades, students develop recall of basic multiplication and division facts. Rational number concepts are a major instructional goal at this level, and these concepts lead to informal methods for calculating with fractions. For example, in grade 5, a problem such as "1/4 + 1/2" should be solved mentally with ease, because students should have clear geometric images of 1/2 and 1/4 or be able to use decomposition strategies, such as "1/4 + 1/2 = 1/4 + (1/4 + 1/4)."

Fraction and decimal computation is an instructional focus in grades 6-8. Strategies for computing with fractions should build on conceptual knowledge developed in earlier grades. Students should come to grades 6-8 able to compute with commonly encountered fractions based on mental and concrete representations. In grades 6-8, students should develop more general computational strategies that can be applied to a full range of fraction situations. They also should extend the strategies for computing with whole numbers to decimal numbers. In grades 6-8, students are expected to become fluent in computing with rational numbers. As they develop understanding of the meaning and representation of integers, they should also develop ways of computing with integers.

In grades 9-12, students should analyze and compare algorithms as part of studying the role of algorithms in mathematics. By comparing algorithms, they consider which are easy to explain, easy to use, and are most efficient. They should be able to read a flow chart and decide if it describes a correct method for determining whether a number is divisible by 3. Students analyze why and how algorithms are constructed. At this level, students can study common algorithms for computing with whole and rational numbers as well as unfamiliar algorithms that they encounter for the first time in high school, including, for example, algorithms for finding real roots or finite differences of sequences.

Standard 1: Number and Operation

Standard 10: Representation

Mathematics instructional programs should emphasize mathematical representations to foster understanding of mathematics so that all students-

• create and use representations to organize, record, and communicate mathematical ideas;
• develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately;
• use representations to model and interpret physical, social, and mathematical phenomena.

Elaboration: Pre-K-12

The ways in which mathematical ideas are represented has a fundamentally important role in shaping the ways people can understand and use those ideas. To make the point simply, consider the following arithmetic questions:

What is the product of the two numbers represented by the Roman numerals MCMLXXXVII and MDCCXLIII? What is the quotient when the first is divided by the second?

In the given representation, multiplication is exceedingly difficult and division is seemingly impossible. Yet, when the representations are converted to standard base-ten notation, it a straightforward process to find the product and quotient, respectively, of the numbers 1987 and 1743.

This example illustrates a general point that mathematical symbolism and representation is one of the most significant achievements of humankind. Mathematical representations have been polished and refined over centuries. When students gain access to mathematical representations and the ideas that are represented, they have a set of tools that provides a significant expansion of their capacity to think mathematically.

Representation is a multifaceted concept, and there are subtle and complex issues of language associated with it. The term "representation" refers both to process and to product-in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself. Moreover, the term applies both to externally observable processes and products and to those that occur "internally" in the minds of people doing mathematics. All these meanings of representation are important to consider in school mathematics.

Some forms of representation-such as diagrams, graphical displays and symbolic expressions-have long been part of school mathematics. Unfortunately, these representations and others have often been taught and learned as if they were ends in themselves. This approach limits the power and utility of representations as tools for learning and doing mathematics. Representations should be treated as critical elements in supporting student understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understanding to one's self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations via modeling. In addition to these long-standing reasons for including representation as an important process of school mathematics, new forms of representation associated with electronic technology create a need for even greater instructional attention.

Create and use representations to organize, record, and communicate mathematical ideas

As the example of Roman numerals versus Arabic base-ten representation indicates, the forms of representation that are used will shape one's capacity to understand and do mathematics. The fact that it took centuries to develop the base-ten representation highlights another point. That is, many of the representations we now take for granted-such as numbers expressed in base-ten or binary form, fractions, equations, graphs, and spreadsheet displays-are the result of a process of cultural refinement that took place over many years. These tools are extremely powerful for solving problems. Equally important, a shared mathematical language is fundamentally important for purposes of communication. Written representations are central to reading and understanding mathematical text written by others. They also can be very useful in sharing with others one's own understandings about and approaches to mathematics. For these reasons, it is important that students learn conventional forms of representation in ways that facilitate their learning of mathematics and their communication with others about mathematical ideas.

Mathematical representations do not exist in isolation. They are usually developed and applied to various classes of situations and are connected to particular bodies of mathematical concepts. Part of understanding a representation involves having links to related mathematics, knowing the circumstances to which it applies, and knowing how that representation can be used in the service of particular goals. Given a particular problem, one might choose a representation that would help clarify and solve that problem. A different problem dealing with the same phenomenon might be more profitably approached with a different representation. Sometimes the treatment of conventional representational forms in school leads students to think of them as arbitrary and disconnected both from settings of use and from students' other knowledge and experience. Conventional representations should be viewed as powerful tools for organizing and communicating mathematical thinking about mathematics. It is useful to consider not only how conventional representations might be better taught to students, but also to consider the role that other, less conventional, representations might play in the mathematics classroom.

The fact that representations are such effective tools may obscure how difficult it was to develop them and, more importantly how much work it takes to understand them. Teachers in the elementary grades are aware of how difficult base-ten notation is for young children, and the curriculum allows room for a large amount of give-and-take between students' emerging understanding of the counting numbers and the structure of base-ten representation. But as students move through the curriculum, the focus is increasingly on presenting the mathematics itself, perhaps under the assumption that students who are old enough to think in formal terms do not, like their younger counterparts, need to negotiate between their naive conceptions and the mathematical formalisms. Research indicates, however, that students at all levels need to work at developing their understandings of the complex ideas captured in conventional representations (Greeno and Hall 1987). A representation as seemingly clear as the variable "x" can be difficult for students to comprehend.

The idiosyncratic representations constructed by students as they solve problems and investigate mathematical ideas can play an important role in supporting the development of students' understanding in mathematical learning and performance. Although personally constructed representations often lack the precision and generality of conventional representations, they appear to be valuable to students. In particular, they support the understanding and solution of problems; they provide meaningful ways to record a solution method and to describe the method to others; and they provide an experiential base from which students can develop an appreciation of the nature and power of other representations, including conventional ones. Representations constructed by students also give teachers opportunities to gain valuable insights into students' ways of interpreting and thinking about mathematics. Using this information, teachers can build bridges from students' personal representations to more conventional ones, when appropriate. In summary, it is important that students have opportunities not only to learn conventional forms of representation, but also to construct, refine, and use their own representations as tools to support learning and doing mathematics.

Computers and calculators change what students can do with conventional representations and expand the set of representations with which students can work. For example, students can flip, invert, stretch, and zoom in on graphs using graphing utilities or dynamic geometry software; they can use computer algebra software to manipulate expressions; and they can investigate complex data sets using spreadsheets. As students learn to use these new, versatile tools to enhance their methods for organizing and recording their ideas, increase their understanding of mathematical concepts and relationships, and facilitate communication of their thinking, they also can consider ways in which some representations used in electronic technology differ from conventional representations. For example, scientific calculators use a representational form for a number in scientific notation that differs from the form typically found in a textbook. A symbolic manipulator often changes the order or form of an algebraic expression. These technological tools offer students and teachers many new issues related to mathematical representation.

Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately

Different representations often illuminate different aspects of a complex concept or relationship. For example, students usually learn to represent fractions as sectors of a circle or as pieces of a rectangle or some other figure, sometimes using physical displays of pattern blocks or fraction bars. This form of representation is useful in conveying the part-whole interpretation of fractions. It supports the development of an understanding of fraction equivalence and it can help students visualize the meaning of the addition of fractions, especially when the fractions have the same denominator and when their sum is less than one. Yet this form of representation does not convey other interpretations of fraction, such as ratio, indicated division, or fraction as number. Nor does it support directly an understanding of the addition of fractions whose denominators are different or whose sum is greater than one. Similarly, other common representations for fractions, such as points on a number line or ratios of discrete elements in a set, convey some but not all aspects of the complex fraction concept. Thus, in order to become deeply knowledgeable about fractions-and many other concepts in school mathematics-students will need a variety of representations that support their understanding.

One of the powerful aspects of mathematics is its use of abstraction-the fact that symbolization strips away some features of a problem that are not necessary for analysis and that the "naked symbols" can be operated on easily. Moreover, recognizing that a particular object has a particular representation may provide a large amount of information about the object itself. In many ways, this fact lies behind the power of mathematical applications and modeling. For example, a problem may be stated in terms of the real world context that generated it:

From a ship on the sea at night, the captain can see three lighthouses and can measure the angles between them. If the captain knows the position of the lighthouses from a map, can the captain determine the position of the ship?

When one translates the problem into a mathematical representation, the ship and the lighthouses become points in the plane, and the problem can be solved without knowing that the problem is about ships. Many other problems from different contexts may have similar representations. As soon as the problem is represented in some form, the classic solution methods for that mathematical form may solve the problem. For instance, in mathematical modeling problems, if the phenomenon being modeled turns out to be a quadratic function, students should know that there is one critical value that can be determined analytically, that the graph is symmetric, and so forth.

Common underlying representations can also be useful in helping students build connections as they learn mathematics. Students who can represent the product (a + b)(c + d) as a rectangle whose area is composed of smaller rectangles with areas of ac, ad, bc, and bd have a way to link their study of the multiplication of algebraic expressions (e.g., (x + h)² = x² + 2xh + h²) to their previous study of multiplication of two-digit numbers (e.g., 15 x 27 = (10 x 20) + (10 x 7) + (5 x 20) + (5 x 7)). In this way, representations enable students to make connections across topics in the mathematics curriculum that would otherwise seem disconnected.

Technological tools now offer opportunities for students to have more and different experiences with the use of multiple representations. For example, several software packages allow students to view different representations of a function simultaneously. For example, a function can be displayed in tabular, graphical, and equation form. Such software can allow students to examine how certain changes in one representation, such as varying a parameter in the equation ax² + bx + c = 0, simultaneously affect the other representations. Similarly, dynamic geometry software allows students to consider geometrical, computational, and symbolic representations simultaneously.

During grades pre-K-12 students should encounter multiple representations for many mathematical concepts and relationships. As they expand their representational repertoire, it is also important for students to reflect on their use of these representations to develop an understanding of the relative strengths and weaknesses of various representations for different purposes. For example, students learn different representational forms for displaying statistical data, and it is important that their learning include opportunities to consider the kinds of data and questions for which it might be more appropriate to use a circle graph rather than a line graph, or a box-and-whiskers plot instead of a histogram.

Use representations to model and interpret physical, social, and mathematical phenomena

The term "model" has many different meanings. So, it is not surprising that the word is used in many different ways in discussions about mathematics education. For example, "model" is used to refer to physical materials with which students work in school, as in manipulative models. The term is also used in ways that suggest exemplification or simulation, as in when a teacher "models" the problem-solving process for her students. Yet another usage treats the term as if it were roughly synonymous with representation. Amidst this array of uses and meanings is another that is particular to mathematics, and it is the major focus of this part of the discussion about the role of representation in school mathematics. The term "mathematical model" in this context means a mathematical representation of the elements and relationships within an idealized version of a complex phenomenon. Mathematical models can be used to clarify understandings of the phenomenon and to solve problems. What it means "to model" in this sense includes not only representation, but also acting upon the representation and interpreting the meanings of one's actions within the mathematical model and with respect to the phenomenon being modeled.

Many of the symbolization-and-analysis activities in which students engage are a simplified form of modeling in which situations are characterized symbolically and then information about the situation is derived analytically. Consider, for example, a typical optimization problem in which students are asked to determine which price for a magazine will yield the highest profits, given that each time the price of the magazine is increased, the number of people who will buy it decreases. To solve such a problem, a student might determine an equation expressing profit as a function of magazine cost and then use graphical or analytic techniques to determine the best price. Although classroom versions of such tasks are often dramatically simplified, the work captures some aspects of the modeling process.

There are activities in which models allow a view of a real-world phenomenon through an analytic structure imposed on it. One example is building a model of traffic flow. An example of a general question explored might be, "How long should a traffic light stay green to let a reasonable number of cars flow through the intersection?" Students can gather data about how long (on average) it takes the first car to go through, the second car, and so on. They can represent these data statistically, or they can construct analytic functions to work on the problem in the abstract, considering the wait time before a car starts moving, how long it takes a car to get up to regular traffic speed, and so on. This kind of analysis allows for a simulation of traffic flow, which can be used to make judgments about how long a light should be left green.

Technological tools now allow students to explore iterative models for situations that were once studied in much more advanced courses. For example, it is now possible for students in grades 9-12 to model predator-prey relations. The initial set-up might be that a particular habitat houses so many wolves and so many rabbits, which are the wolves' primary food source. When the wolves are well fed, they reproduce well (and more wolves eat more rabbits); when the wolves are starved, they die off. The rabbits multiply readily when wolves are scarce, but lose numbers rapidly when the wolf population is large. Modeling software that uses difference equations allows students to input initial conditions and the rules for change and then see what happens to the system dynamically.

The act of modeling is a complex enterprise. One oversimplified diagram representing the process is given in figure 3.8.

Figure 3.8. Aspects of the modeling process.

A "tour" of figure 3.8 from the bottom left up to the top left (representing the modeling process), across the top (representing mathematical analyses) and down to the bottom right (representing the interpretation of the findings) suggests the path by which conclusions are drawn about the situation being modeled. Each of the three examples described above works in the way suggested here. Note, however, that each stage of the process is potentially problematic. Are the right features selected for analysis? Does the mathematical model capture the right relationships among them? Are the manipulations within the model justified and appropriate? Is the interpretation meaningful? And does it make sense when mapped back to the original situation? All of these questions must be kept in mind when students are engaged in the modeling process.

In summary, representations are vehicles for mathematical thought. Individuals often build their own idiosyncratic representations as a way of understanding ideas and relations. Society offers the product of its collective wisdom in conventional mathematical representations, and these have great power for problem solving and communication. One of our major tasks in classrooms is to bring these two together and provide students meaningful access to those powerful societal tools.

Conclusion

The grade band sections (Chapters 4 through 7 in the print version) of this draft discuss in detail the ideas that have been introduced in this chapter. Although ten standards have been chosen in an effort to show how mathematical ideas as well as students’ learning should progress over the pre-K-12 span, these overviews are, at best, a sketch of the goals that teachers and students might hope to achieve. In the grade band chapters that follow, a more detailed image is conveyed.

Because the standards have been stated somewhat generally in order to extend across the grade-bands, and because there are four grade-band divisions, a considerable amount of detail and guidance for teachers is embedded in the bulleted subtopics that elaborate each of the content standards’ focus areas. For the content standards, the image is conveyed largely through description of what students should know and be able to do with the focus areas. For the process standards, elaboration is provided through questions. It is, of course, not possible to describe fully what would occur for all students at all grade levels, but the intention of the grade-band discussions is that teachers and others will be able to use these ideas as they collaboratively and deliberately structure their own instructional programs, within all of the constraints and contextual features they must address.