Estimation and Mental Computation It's "About" TimeAuthor(s): Barbara J. ReysSource: The Arithmetic Teacher, Vol. 34, No. 1 (September 1986), pp. 22-23Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194192 .

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Estimation and Mental Computation: It's "About" Time

By Barbara J. Reys

Mental computation and estimation are topics we keep hearing about. Although mental computation is not a stranger to the history of mathematics education, estimation is a relative newcomer to the curriculum. The his- tory of mental computation dates back to when arithmetic was first taught. Historically, it was empha- sized because of its social utility - shopkeepers needed to "cipher" to- tals of grocery bills quickly. The cur- rent emphasis on both mental compu- tation and estimation has a different origin. As teachers, we want students to take advantage of whatever compu- tational tool is most appropriate for the situation at hand. Yet unless stu- dents have developed the skills both to compute mentally and to estimate and the awareness to take advantage of whichever is appropriate, we will not see it happen.

This year a special department de- voted to both these topics will appear in each issue (see p. 24). We'll explore various strategies and methods to teach each topic and provide some sample activities. But first, let's lay the groundwork: What are estimation and mental computation? What do we know about each process? What can we do to help students develop these skills?

What Are Mental Computation and Estimation? Although these two topics are often

Barbara J. Reys teaches at the University of Missouri, Columbia, MO 65211. She has an interest in mathematics education at all levels.

mentioned in the same breath, they have some significant differences. For one thing, mental computation pro- duces an exact answer, whereas many different but reasonable estimates can exist for a given problem. Therefore, every arithmetic problem can be esti- mated, but only a subset of problems lie within the realm of most students' ability to compute mentally.

Mental Computation The process of producing an exact answer to a computational problem without any external computational aid

Estimation The process of producing an answer that is sufficiently close to allow deci- sions to be made

What We Know The body of available research in both areas is growing rapidly. The follow- ing is a brief summary of what this research tells us: • More than 80 percent of all real-

world uses of mathematics by adults involve computations done mentally.

• Students have not performed well on assessments of estimation. In fact, NAEP reports that students do worse when estimating the results of computational exercises than when finding the exact answers to the same exercises. (See Lindquist, Mary Montomery, Thomas P. Car- penter, Edward A. Silver, and Westina Matthews. 'The Third Na- tional Mathematics Assessment: Results and Implications for Ele-

22 Arithmetic Teacher

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• Poor concept development of frac- tions, decimals, and percents inhib- its application of estimation and mental computation to problems containing these numbers.

• Good estimators have exceptional mental-computational skills, but the converse is not true. That is, people can be good at mental computation but not necessarily be good estimators.

• Students good at mental computa- tion use a variety of different think- ing strategies. Similarly, students good at estimation use many dif- ferent strategies.

• Systematic attention to mental com- putation and estimation can pro- duce significant changes in stu- dents' performance as well as in their thinking processes. Although distributed practice is important, it must be accompanied by teaching designed to stimulate thinking and discussion of effective strategies.

• Changes in thinking processes take time. Progress can and should be made each year, but for it to occur, instructional time must be devoted to both mental computation and es- timation throughout the year in ev- ery grade.

What Can You Do? Throughout the year we shall high- light strategies and suggest activities

September 1986

for you to try with your students. Here are some general suggestions:

• Make a commitment to devote time each week to these topics. Plan to teach strategies and offer opportuni- ties for practice on a regular basis. Take advantage of oppportunities to highlight mental computation and estimation wherever appropriate as you proceed through the curricu- lum. For example, in the primary grades include "extended" basic facts, such as 60 + 70 or 600 + 700. Instead of limiting practice of basic multiplication facts to 6 x 7, prac- tice 6 x 70, 60 x 70, or even .6 x 700 in the upper grades. This ap- proach not only allows you to vary basic-fact practice sessions but sug- gests many interesting patterns and helps develop number sense.

• Make a list of mental-computation strategies you use and decide which are most appropriate for your stu- dents. Do the same with estimation techniques. If your lists are short, examine some of the excellent re- sources available, such as the 1986 NCTM Yearbook, Mental Compu- tation and Estimation, and the sub- sequent issues of the Arithmetic Teacher throughout this volume.

• Encourage discussion and sharing of strategies. For example, how would you estimate the cost of one greeting card if a box of twenty-four cards costs $8.69? Did you think 8.00 -s- 20? 8.80 -î- 22? 8.00 + 25? 9.00 -s- 25? perhaps 10.00 - 25?

Each of these pairs of "compatible numbers" changes the original com- putation into a much easier and more mentally manageable prob- lem. As students become aware that they can change the problem to "easier numbers" to allow them to compute mentally an approximate value, they gain a greater apprecia- tion for the power of estimation. Once students realize that many dif- ferent approaches to a problem are possible, they become more open to alternative strategies that may be offered by other class members. Ul- timately they will need to decide which strategy to use, but they must first be comfortable with the notion that many different strategies not only exist but are often used by their peers.

• Develop an evaluation plan that re- flects your instructional commit- ment. Regular tests on the mental- computation and estimation skills that have been emphasized will re- mind students that you are serious about these skills. They will also serve as clear documentation of their progress.

• Recognize that instructional atten- tion to mental computation and es- timation is really directed toward the development of higher-order thinking skills. The acquisition of multiple strategies and the growth of number sense will contribute not only to improved problem solving but to a student's broader under- standing of mathematics, m

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