volume by the factor n³, they can understand many situations in which objects of all shapes are proportionally expanded or shrunk. (They can understand, for example, why a 16-ounce cup that has the same shape as an 8-ounce cup is much less than twice as tall.)

Knowledge learned with understanding provides a foundation for remembering or reconstructing mathematical facts and methods, for solving new and unfamiliar problems, and for generating new knowledge. For example, students who thoroughly understand whole number operations can extend these concepts and procedures to operations involving decimals.

Understanding also helps students to avoid critical errors in problem solving—especially problems of magnitude. Any student with good number sense who multiplies 9.83 and 7.65 and gets 7,519.95 for an answer should immediately see that something is wrong. The answer can’t be more than 10 times 8 or 80, as one number is less than 10 and the other is less than 8. This reasoning should suggest to the student that the decimal point has been misplaced.

(2) Computing: Carrying out mathematical procedures, as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.

Computing includes being fluent with procedures for adding, subtracting, multiplying, and dividing mentally or with paper and pencil, and knowing when and how to use these procedures appropriately. Although the word computing implies an arithmetic procedure, in this document it also refers to being fluent with procedures from other branches of mathematics, such as measurement (measuring lengths), algebra (solving equations), geometry (constructing similar figures), and statistics (graphing data). Being fluent means having the skill to perform the procedure efficiently, accurately, and flexibly.

Students need to compute basic number combinations (6+7, 17−9, 8×4, and so on) rapidly and accurately. They also need to become accurate and efficient with algorithms—step-by-step procedures for adding, subtracting, multiplying, and dividing multi-digit whole numbers, fractions, and decimals, and for doing other computations. For example, all students should have an algorithm for multiplying 64 and 37 that they understand, that is reasonably efficient and general enough to be used with other two-digit numbers, and that can be extended to use with larger numbers.

The use of calculators need not threaten the development of students’ computational skills. On the contrary, calculators can enhance both understanding