### Number and Operations

 Instructional programs from prekindergarten through grade 12 should enable all students to— understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; compute fluently and make reasonable estimate

The Number and Operations Standard describes deep and fundamental understanding of, and proficiency with, counting, numbers, and arithmetic, as well as an understanding of number systems and their structures. The concepts and algorithms of elementary arithmetic are part of number and operations, as are the properties and characteristics of the classes of numbers that form the beginnings of number theory. Central to this Standard is the development of number sense—the ability to decompose numbers naturally, use particular numbers like 100 or 1/2 as referents, use the relationships among arithmetic operations to solve problems, understand the base-ten number system, estimate, make sense of numbers, and recognize the relative and absolute magnitude of numbers (Sowder 1992).

Historically, number has been the cornerstone of the entire mathematics curriculum internationally as well as in the United States and Canada (Reys and Nohda 1994). All the mathematics proposed for prekindergarten through grade 12 is strongly grounded in number. The principles that govern equation solving in algebra are the same as the structural properties of systems of numbers. In geometry and measurement, attributes are described with numbers. The entire area of data analysis involves making sense of numbers. Through problem solving, students can explore and solidify their understandings of number. Young children's earliest mathematical reasoning is likely to be about number situations, and their first mathematical representations will probably be of numbers. Research has shown that learning about number and operations is a complex process for children (e.g., Fuson [1992]).

In these Standards, understanding number and operations, developing number sense, and gaining fluency in arithmetic computation form the core of mathematics education for the elementary grades. As they progress from prekindergarten through grade 12, students should attain a rich understanding of numbers—what they are; how they are represented with objects, numerals, or on number lines; how they are related to one another; how numbers are embedded in systems that have structures and properties; and how to use numbers and operations to solve problems.

Knowing basic number combinations—the single-digit addition and multiplication pairs and their counterparts for subtraction and division—is essential. Equally essential is computational fluency—having and using efficient and accurate methods for computing. Fluency might be manifested in using a combination of mental strategies and jottings on paper or using an algorithm with paper and pencil, particularly when the numbers are large, to produce accurate results quickly. Regardless of the particular method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general. Students also need to be able to estimate and judge the reasonableness of results. Computational fluency should develop in tandem with understanding of the role and meaning of arithmetic operations in number systems (Hiebert et al., 1997; Thornton 1990).

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Calculators should be available at appropriate times as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when teachers are working with students » on developing computational algorithms, the calculator should be set aside to allow this focus. Today, the calculator is a commonly used computational tool outside the classroom, and the environment inside the classroom should reflect this reality.

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

Understanding of number develops in prekindergarten through grade 2 as children count and learn to recognize "how many" in sets of objects. A key idea is that a number can be decomposed and thought about in many ways. For instance, 24 is 2 tens and 4 ones and also 2 sets of twelve. Making a transition from viewing "ten" as simply the accumulation of 10 ones to seeing it both as 10 ones and as 1 ten is an important first step for students toward understanding the structure of the base-ten number system (Cobb and Wheatley 1988). Throughout the elementary grades, students can learn about classes of numbers and their characteristics, such as which numbers are odd, even, prime, composite, or square.

Beyond understanding whole numbers, young children can be encouraged to understand and represent commonly used fractions in context, such as 1/2 of a cookie or 1/8 of a pizza, and to see fractions as part of a unit whole or of a collection. Teachers should help students develop an understanding of fractions as division of numbers. And in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers. Students' knowledge about, and use of, decimals in the base-ten system should be very secure before high school. With a solid understanding of number, high school students can use variables that represent numbers to do meaningful symbolic manipulation.

Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. By the middle grades, students should understand that numbers can be represented in various ways, so that they see that 1/4, 25%, and 0.25 are all different names for the same number. Students' understanding and ability to reason will grow as they represent fractions and decimals with physical materials and on number lines and as they learn to generate equivalent representations of fractions and decimals.

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As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers. In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2. In grades 6–8, it is important for students to be able to move flexibly among equivalent fractions, decimals, and percents and to order and compare rational numbers using a range of strategies. By extending from whole numbers to integers, middle-grades students' intuitions about order and magnitude will be more reliable, and they have a glimpse into the way that systems of numbers work. High school students can use variables and functions to represent relationships among sets of numbers, and to look at properties of classes of numbers. »

Although other curricular areas are emphasized more than number in grades 9–12, in these grades students should see number systems from a more global perspective. They should learn about differences among number systems and about what properties are preserved and lost in moving from one system to another.

#### Understand meanings of operations and how they relate to one another

During the primary grades, students should encounter a variety of meanings for addition and subtraction of whole numbers. Researchers and teachers have learned about how children understand operations through their approaches to simple arithmetic problems like this:

Bob got 2 cookies. Now he has 5 cookies. How many cookies did Bob have in the beginning?

To solve this problem, young children might use addition and count on from 2, keeping track with their fingers, to get to 5. Or they might recognize this problem as a subtraction situation and use the fact that 5 – 2 = 3. Exploring thinking strategies like these or realizing that 7 + 8 is the same as 7 + 7 + 1 will help students see the meaning of the operations. Such explorations also help teachers learn what students are thinking. Multiplication and division can begin to have meaning for students in prekindergarten through grade 2 as they solve problems that arise in their environment, such as how to share a bag of raisins fairly among four people.

In grades 3–5, helping students develop meaning for whole-number multiplication and division should become a central focus. By creating and working with representations (such as diagrams or concrete objects) of multiplication and division situations, students can gain a sense of the relationships among the operations. Students should be able to decide whether to add, subtract, multiply, or divide for a particular problem. To do so, they must recognize that the same operation can be applied in problem situations that on the surface seem quite different from one another, know how operations relate to one another, and have an idea about what kind of result to expect.

In grades 6–8, operations with rational numbers should be emphasized. Students' intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g., 81/2) produces a result less than the whole number. This is counter to students' prior experience (with whole numbers) that multiplication always results in a greater number.

Working with proportions is a major focus proposed in these Standards for the middle grades. Students should become proficient in creating ratios to make comparisons in situations that involve pairs of numbers, as in the following problem:

If three packages of cocoa make fifteen cups of hot chocolate, how many packages are needed to make sixty cups?
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Students at this level also need to learn operations with integers. In grades 9–12, as students learn how to combine vectors and matrices arithmetically, they will experience other kinds of systems involving numbers in which new properties and patterns emerge. »

#### Compute fluently and make reasonable estimates

Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over-practiced without understanding are often forgotten or remembered incorrectly (Hiebert 1999; Kamii, Lewis, and Livingston 1993; Hiebert and Lindquist 1990). On the other hand, understanding without fluency can inhibit the problem-solving process (Thornton 1990). As children in prekindergarten through grade 2 develop an understanding of whole numbers and the operations of addition and subtraction, instructional attention should focus on strategies for computing with whole numbers so that students develop flexibility and computational fluency. Students will generate a range of interesting and useful strategies for solving computational problems, which should be shared and discussed. By the end of grade 2, students should know the basic addition and subtraction combinations, should be fluent in adding two-digit numbers, and should have methods for subtracting two-digit numbers. At the grades 3–5 level, as students develop the basic number combinations for multiplication and division, they should also develop reliable algorithms to solve arithmetic problems efficiently and accurately. These methods should be applied to larger numbers and practiced for fluency.

Researchers and experienced teachers alike have found that when children in the elementary grades are encouraged to develop, record, explain, and critique one another's strategies for solving computational problems, a number of important kinds of learning can occur (see, e.g., Hiebert [1999]; Kamii, Lewis, and Livingston [1993]; Hiebert et al. [1997]). The efficiency of various strategies can be discussed. So can their generalizability: Will this work for any numbers or only the two involved here? And experience suggests that in classes focused on the development and discussion of strategies, various "standard" algorithms either arise naturally or can be introduced by the teacher as appropriate. The point is that students must become fluent in arithmetic computation—they must have efficient and accurate methods that are supported by an understanding of numbers and operations. "Standard" algorithms for arithmetic computation are one means of achieving this fluency.

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The development of rational-number concepts is a major goal for grades 3–5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4). In these grades, methods for computing with decimals should be developed and applied, and by grades 6–8, students should become fluent in computing with rational numbers in fraction and decimal form. When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981). Most said it was close to 1, 19, or 21, all of which reflect common computational errors in adding fractions and suggest a lack of understanding of the operation being carried out. If students understand addition of fractions and have developed number sense, these errors should not occur. As they develop an understanding of the meaning and representation of integers, they should also develop methods for computing with » integers. In grades 9–12, students should compute fluently with real numbers and have some basic proficiency with vectors and matrices in solving problems, using technology as appropriate.

Part of being able to compute fluently means making smart choices about which tools to use and when. Students should have experiences that help them learn to choose among mental computation, paper-and-pencil strategies, estimation, and calculator use. The particular context, the question, and the numbers involved all play roles in those choices. 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, using their number sense to advantage, and be able to give a rationale for their decision.