sary nor efficient. Students should be able to use a variety of mental strategies to multiply by 10, 20, or 300 (or any power of 10 or multiple of 10). Also, students should be able to perform such operations as finding the sum of 199 and 67 or the product of 4 and 26 by using quick mental strategies rather than relying on paper and pencil. Further, situations vary in their need for exact answers. Sometimes an estimate is good enough, as in calculating a tip on a bill at a restaurant. Sometimes using a calculator or computer is more appropriate than using paper and pencil, as in completing a complicated tax form. Hence, students need facility with a variety of computational tools, and they need to know how to select the appropriate tool for a given situation.

Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy.12 As we noted earlier, the two are interwoven. Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding, and using procedures can help strengthen and develop that understanding. For example, it is difficult for students to understand multidigit calculations if they have not attained some reasonable level of skill in single-digit calculations. On the other hand, once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.13 In an experimental study, fifth-grade students who first received instruction on procedures for calculating area and perimeter followed by instruction on understanding those procedures did not perform as well as students who received instruction focused only on understanding.14

Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems. The attention they devote to working out results they should recall or compute easily prevents them from seeing important relationships. Students need well-timed practice of the skills they are learning so that they are not handicapped in developing the other strands of proficiency.

When students practice procedures they do not understand, there is a danger they will practice incorrect procedures, thereby making it more difficult to learn correct ones. For example, on one standardized test, the grade 2 national norms for two-digit subtraction problems requiring borrowing, such as 62–48=?, are 38% correct. Many children subtract the smaller from the larger digit in each column to get 26 as the difference between 62 and 48 (see Box 4–2). If students learn to subtract with understanding, they rarely make

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