In grades K-4, the curriculum should include estimation so students can--

• explore estimation strategies;
• recognize when an estimate is appropriate;
• determine the reasonableness of results;
• apply estimation in working with quantities, measurement, computation, and problem solving.

Focus

Estimation presents students with another dimension of mathematics; terms such as about, near, closer to, between, and a little less than illustrate that mathematics involves more than exactness. Estimation interacts with number sense and spatial sense to help children develop insights into concepts and procedures, flexibility in working with numbers and measurements, and an awareness of reasonable results. Estimation skills and understanding enhance the abilities of children to deal with everyday quantitative situations.

From children's earliest experiences with mathematics, estimation needs to be an ongoing part of their study of numbers, computation, and measurement. It is important that children learn a variety of methods of estimating, such as the front-end strategy for computation and the chunking procedure for measurement. They also need to develop reasoning, judgment, and decision-making skills in using estimation.

Instruction should emphasize the development of an estimation mind-set. Children should come to know what is meant by an estimate, when it is appropriate to estimate, and how close an estimate is required in a given situation. If children are encouraged to estimate, they will accept estimation as a legitimate part of mathematics.

Discussion

When children enter school, they are accustomed to estimating. They know that they are almost six years old, that they are a little shorter than a brother or sister, that a carton of milk can fill more than three glasses, and when it is about noon. This experiential knowledge provides a foundation for further development in estimating quantities. Consider the following example: As a referent, children can be told that the set at the left in figure 5.1 has ten balls. Without counting, they can be asked to quickly classify the other sets as fewer than ten, about ten, or more than ten.

Fig. 5.1

Children should also estimate larger quantities, such as the number of seeds in a pumpkin, beans in a bag, or Valentine candies in a jar. For larger quantities, it is usually more appropriate to use a referent set having 50 or 100 items.

Several important considerations in estimating quantities should be remembered. When checking estimates, a teacher can reinforce place-value ideas by having the children place the estimated items in groups of ten and then in hundreds whenever possible. It is also important for the teacher and the children to identify a range for "good estimates." Further, it should be emphasized that estimates that happen to be exact are no better than other estimates within the identified range; the goal is an approximation, not the exact number. Finally, children should always check their initial estimates and then make additional ones so that they can use the feedback to refine their estimating skills.

A particularly good estimating activity involving measuring uses interlocking cubes. Using a stick of ten cubes as a referent, children estimate how many cubes long a work table is, for example. Then they make a "train" of cubes as long as the work table and break the train into sticks of ten to check their estimates. See figure 5.2.

Fig. 5.2

Another measurement-and-estimating activity illustrates the process of chunking. In the task in figure 5.3, children estimate the number of boxes necessary to fill the classroom. A child mentally lines up seven boxes along one edge of the floor and uses them as a unit, or a "chunk," to estimate the total number of boxes.

Fig. 5.3

Children also should be taught specific strategies to aid them in computational estimation. A child who needs to evaluate 243 + 479 might estimate by thinking, "200 and 400 is 600, 43 and 79 is over 100, so the sum is a little more than 700." This is "front-end estimation." Another way of estimating is this: "243 is just under 250, 479 is just under 500, so the sum is less than 750." This is a flexible use of "rounding" for estimation or selecting "nice" numbers that are easy to work with. It is useful to discuss various strategies and to help students develop their own strategies. For example, a student adept at mental computation could estimate 243 + 479 in this way: "24 + 48 (tens) is 72 (tens) so the sum is about 720." Continual emphasis on computational estimation helps children develop creative and flexible thought processes and fosters in them a sense of mathematical power.

Estimation is especially important when children use calculators. If they need to compute 4783 ÷ 13, for example, a quick estimate can be found by using "compatible numbers." In this case, 4783 is about 4800 and 13 is about 12, so 4783 ÷ 13 is about 4800 divided by 12. The dividing can be done mentally, since 48 and 12 are "compatible numbers" for division. Thus, 4783 ÷ 13 is about 400. This rough estimate provides children with enough information to decide whether the correct keys were pressed and whether the calculator result is reasonable. Such uses of estimation reduce the incidence of errors with calculators, decrease the mindless use of calculators for computation, and contribute to children's development of number and operation sense.

Children often find that estimation skills are useful in their daily lives. Many children know when it is appropriate to estimate and how close an estimate should be, as the following anecdote indicates. Three children huddled together in a shopping mall, discussing the purchase of some clothing. One held a newspaper advertisement, another a calculator. Two children picked items from the ad and the third entered the appropriate prices into the calculator. In considering the calculator result, one of the children reasoned, "The total cost can't be more than $50 because two shirts cost $14 each; that's less than $30, and the pants cost $17.99." Classroom instruction on estimation should help children develop a similar estimation mind-set so they can use good judgment and logical reasoning to make decisions in their daily lives.