STANDARD 10 - ESTIMATION
Estimation is a process that is used constantly by mathematically capable adults, and that can be mastered easily by children. It involves an educated guess about a quantity or a measure, or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that decision is through estimation. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact answer.
Meaning and Importance
As used in this standard, estimation is the process of determining approximate values in a variety of situations. Estimation strategies are used universally throughout daily life, but an examination of the mathematics curriculum of the past leads to the view that the strength of mathematics lies in its exactness, in the ability to determine the "right" answer. The growing use of calculators in the classroom requires greater emphasis on determining whether the answer given by a calculator or paper-and-pencil method is reasonable, a process that requires estimation ability, but efforts in support of this goal have been minimal compared to the time devoted to getting that one right answer. As a result, students have developed the notion that exactness is always preferred to estimation and their potential development of intuition may have been hindered with unnecessary calculations and detail.
People who use mathematics in their lives and careers find estimation to be preferable to the use of exact numbers in many circumstances. Frequently, it is either impossible to obtain exact answers or too expensive to do so. An air conditioning salesperson preparing a bid would be wasting time and money by measuring rooms exactly. Astronomers attempting to determine movements of celestial objects cannot obtain precise measurements. Many people use approximations because it is easier than using exact numbers. Shoppers, for example, use approximations to determine whether they have sufficient funds to purchase items. Travelers use rough estimates of time, distance, and cost when planning trips. Commonly reported data often use levels of precision which have been accepted as appropriate, even though they may not be considered "exact." Astronomers usually report information to two significant digits, and batting averages for baseball players are always reported as three-place decimals.
K-12 Development and Emphases
Part of being functionally numerate requires expertise in using estimation with computation. Such facility demands a strong sense of number as well as a mastery of the basic facts, an understanding of the properties of the operations as well as their appropriate uses, and the ability to compute mentally. As these skills and understandings are developed throughout the mathematics curriculum, students should have frequent opportunities to develop methods for obtaining estimates, and to recognize that estimation is useful. Estimation can help determine the correct answer from a set of possible answers, and establish the reasonableness of answers. Ideally, students should have an idea of the approximate size of an answer; then, if they recognize that the result they have obtained is incorrect, they can immediately rework the problem. This ability becomes increasingly important as students use calculators more and more.
Instruction in estimation has traditionally focused on the use of rounding. There are times when rounding is an appropriate process for finding an estimate, but this standard emphasizes that it is only one of a variety of processes. Computational estimation strategies are a new and important component in the curriculum. Clustering, front-end digits, compatible numbers, and other strategies are all helpful to the skillful user of mathematics, and can all be mastered by young students. Selection of the appropriate strategy to use depends on the setting, and the numbers and operations involved. Students should realize that there is not necessarily a "right" answer when estimating; different techniques may yield different estimates, and that is quite acceptable.
The foregoing discussion describes a new emphasis on the use of estimation in computational settings, but students should also be thoroughly comfortable with the use of estimation in measurement. Students should develop the ability to estimate measures such as length, area, volume, and angle size visually as well as through the use of personal referents, such as the width of a finger or the length of a pace. Measurement is rich with opportunities to develop an understanding that estimates are often used to determine approximate values which are then used in computations, and that results so obtained are not exact but fall within a range of tolerance.
Estimation should be emphasized in many other areas of the mathematics curriculum in addition to the obvious uses in numerical operations and measurement. Within statistics, for example, it is often useful to estimate measures of central tendency for a set of data; estimating probabilities can help a student determine when a particular course of action would be advisable; problem situations related to algebraic concepts provide opportunities to estimate rates such as slopes of lines and average speed; and working with sequences in algebra and increasing the number of sides of a regular polygon in geometry yield opportunities to estimate limits.
In summary, estimation is a combination of content and process. Students' abilities to use estimation appropriately in their daily lives develops as they have regular opportunities to explore and construct estimation strategies and as they acquire an appreciation of its usefulness in the solution of problems.
Note: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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