In grades 5-8, the mathematics curriculum should develop the concepts underlying computation and estimation in various contexts so that students can--

• compute with whole numbers, fractions, decimals, integers, and rational numbers;
• develop, analyze, and explain procedures for computation and techniques for estimation;
• develop, analyze, and explain methods for solving proportions;
• select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods;
• use computation, estimation, and proportions to solve problems;
• use estimation to check the reasonableness of results.

Focus

Although computation is vital in this information age, technology has drastically changed the methods by which we compute. Whereas inexpensive calculators execute routine computations accurately and quickly and computers execute more complex computations with ease, many current mathematics programs focus on traditional paper-and-pencil algorithms. This standard prepares students to select and use appropriate mental, paper-and-pencil, calculator, and computer methods.

It is no longer necessary or useful to devote large portions of instructional time to performing routine computations by hand. Other mathematical experiences for middle school students deserve far more emphasis. The facility in computation that calculators and computers offer should imbue the curriculum with an expanded potential for interesting problem-solving experiences. Students need more experience in developing procedures and evaluating their work and in interpreting the results of computations done by machines.

It is beneficial to view computation in relation to other topics in the mathematics curriculum. Computation, estimation, or methods for solving proportions should not be considered or taught as ends in themselves. In grades 5-8, computation and estimation should be integrated with the study of the concepts underlying fractions, decimals, integers, and rational numbers, as well as with the continuing study of whole number concepts. Similarly, methods for solving proportions should be viewed as one aspect of students' growing but informal understanding of ratio and proportions. These computational procedures should be developed in context so that the learner perceives them as tools for solving problems.

Discussion

This standard extends the K-4 standard on computation with whole numbers to include new number systems. Broadly, its premise is that computation should support meaningful experiences in geometry, probability, measurement, and other areas of mathematics. Estimation should be used to solve problems for which exact answers are inappropriate and to check computation results. This more general notion of computation is an important part of mathematical communication. In the middle grades, computation requires that students be able to select a symbol system appropriate to a particular context and represent an idea, a solution to a problem, or a particular situation with a symbolic procedure.

The greatest revisions to be made in the teaching of computation include the following:

• Fostering a solid understanding of, and proficiency with, simple calculations
• Abandoning the teaching of tedious calculations using paper-and-pencil algorithms in favor of exploring more mathematics
• Fostering the use of a wide variety of computation and estimation techniques--ranging from quick mental calculation to those using computers--suited to different mathematical settings
• Developing the skills necessary to use appropriate technology and then translating computed results to the problem setting
• Providing students with ways to check the reasonableness of computations (number and algorithmic sense, estimation skills)

The ability to compute 0.17 x 45 correctly is not as interesting for its own sake as it is for estimating the number of times a certain result will occur on a spinner in a game or in determining a discount when buying a new tennis racquet. Students should know when it is appropriate to multiply 0.17 x 45 in problem-solving situations and how to multiply 0.17 x 45 with a calculator.

Despite these fundamental revisions, certain aspects of computation continue to be important. A knowledge of basic facts and procedures is critical in mental arithmetic and estimation. Knowing that 8 x 7 = 56 is a basis for finding 8 x 700 mentally, multiplying (+8) x (-7), estimating 824 x 689, and estimating 8.24 x 6.89.

Valuable class time should not be devoted to developing students' proficiency in calculating 824 x 689 or 8.24 x 6.89 with paper and pencil, since these exercises can be done more readily with a calculator.

As they begin to understand the meaning of operations and develop a concrete basis for validating symbolic processes and situations, students should design their own algorithms and discuss, compare, and evaluate them with their peers and teacher. Students should analyze the way the various algorithms work and how they relate to the meaning of the operation and to the numbers involved.

Similar experiences can stimulate mental arithmetic, a particularly good topic for enhancing students' understanding of numbers and their sense of power in mathematics. In the following example, a teacher asks students to add in their heads the number of cans collected this week (157) to the number already collected (1950) for sale to a recycling depot.

Tony:          2007.



Julie:         2107.



Ms. Clark:     Tony, how did you do it?



Tony:          Well, I added 50 to 1950 and got 2000. Then I put on .....



                Oh! Oh! I forgot the hundred.



Ms. Clark:    Interesting. Julie?



Julie:         I got 21 hundreds: 1 + 19 and 1 more from 50 + 50.



                Then I added 7.



Ms. Clark:     O.K., Julie, so we'll have 2107 cans. How many boxes will



                we need to carry them out to the truck?

This example illustrates many points. First, computation arises from the need to do or know, not simply for its own sake. Second, students can and do make many interesting algorithms. Third, erroneous results offer opportunities for learning when students can describe what they are doing and can check their answers. The teacher can follow this exercise with a discussion of the role of the numeration system in the students' mental computations and extend the lesson to computation itself: "Would the same methods work for computing the sum of three items in a store: $19.50, $1.57, and $22.00?"

The mastery of a small number of basic facts with common fractions (e.g., 1/4 + 1/4 = 1/2; 3/4 + 1/2 = 1 1/4; and 1/2 x 1/2 = 1/4) and with decimals (e.g., 0.1 + 0.1 = 0.2 and 0.1 x 0.1 = 0.01) contributes to students' readiness to learn estimation and for concept development and problem solving. This proficiency in the addition, subtraction, and multiplication of fractions and mixed numbers should be limited to those with simple denominators that can be visualized concretely or pictorially and are apt to occur in real-world settings; such computation promotes conceptual understanding of the operations. This is not to suggest, however, that valuable instruction time should be devoted to exercises like 17/24 + 5/18 or 5 3/4 x 4 1/4, which are much harder to visualize and unlikely to occur in real-life situations. Division of fractions should be approached conceptually. An understanding of what happens when one divides by a fractional number (less than or greater than 1) is essential.

Similarly, students should learn to compute decimal products like 0.3 x 0.6, especially as a means of locating the decimal point. Although such problems train students to estimate more difficult computations, valuable instructional time should not be devoted to calculating products such as 0.31 x 0.588 with paper and pencil.

Instruction should stress informal but effective methods for solving proportions, including ways to identify integer multiples in ratios and processes in which changing one of the ratios to an equivalent unit ratio is an intermediate step. Cross-multiplication methods should be deferred until students understand these methods in algebraic terms.

Basic facts, processes, and translations within and among common and decimal fractions, percents, proportions, and integers are important to students' understanding of computation. Performing two-digit computations with whole numbers or decimals aids students in understanding connections between computation and numeration. Even though students can explore paper-and-pencil computations with numbers of any size and with various systems, they should not be expected to become proficient with paper-and-pencil computations with several digits. A curriculum that incorporated this standard would not include paper-and-pencil practice for proficiency with tedious computations, such as those with three-digit multipliers or divisors, or with operations on fractions beyond the extent suggested previously. Students should possess adequate mental arithmetic skills so that they are not dependent on calculators to do simple computations and are able to detect unreasonable answers when using calculators to solve harder computations. But this standard concentrates to a far greater degree on teaching students to use computations in context, to frame and execute computations using different methods, and to estimate.

Appropriate activities can build in students the kind of computational sense, capability, and confidence intended by this standard. Here is one example:

A set of cards is prepared, each one bearing the price of an object and a particular discount in percentages (e.g., $10.95, 15%). Each of the two players has a calculator. One player turns over a card to reveal a price and a discount. Then both players estimate the final, discounted price. They use the calculators to find the discounted price, and the player who comes closest to the actual discounted price earns one point. A game played to ten points takes ten minutes or less.

Choosing an appropriate method for performing a computation is more important than it has been in the past. For example, consumers might use mental arithmetic to estimate the amount of a tip in a restaurant, use a calculator to fill out a tax form, or use a computer to determine how long it will take to pay off a credit-card balance. Students need to learn how to do each method of computation and to choose which one to use in a given situation. Students should translate results from various computational devices into solutions that fit particular problems and settings. For example, consider the following situation:

A tray can hold 12 salads. How many trays are required for 244 persons?

If a student's calculator shows the answer 20.333333, he or she must be able to interpret this as 20 full trays plus 1 partially filled tray (or 21 trays).

Similar interpretations are required with computer software. If students use a spreadsheet to represent pay rates for newspaper carriers, they need to learn how to relate the information on the screen or printout to decisions they are trying to make.

Estimation is a powerful mathematical idea to be used both in solving problems and in checking the reasonableness of results. When a student wants to know about how long it will take to earn enough baby-sitting money to buy a new bicycle, he or she can estimate the answer. The use of basic facts as well as various techniques for estimation (e.g., rounding and comparison) are used in the following examples to check the reasonableness of results:

Sarah's calculator showed the answer 676.8 after she multiplied 9.4 x 7.2 (she forgot to enter the decimal point in 7.2). Her estimate suggested the answer should be about 9 x 7, or about 63.

Hector got 4/6 when he added 3/4 + 1/2 (he confused the rules for adding fractions with those for multiplying fractions and added the numerators and then the denominators). His estimate suggested the answer should be greater than l/2 + 1/2, or larger than 1.

It is the intent of this standard that computation be viewed not as a goal in itself but as a multifaceted tool for knowing and doing.