FOCUS ISSUE: CALCULATORS || Calculators and the Mathematics Curriculum

Calculators and the Mathematics CurriculumAuthor(s): James H. WiebeSource: The Arithmetic Teacher, Vol. 34, No. 6, FOCUS ISSUE: CALCULATORS (February 1987),pp. 57-60Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193099 .

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Calculators and the Mathematics Curriculum

By James H.Wiebei

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While the computer revolution has been making headlines, another much quieter revolution has taken place - in the way people in our society do arithmetic. With electronic calculators selling for less than $5, most people now use them to do such everyday computations as balancing a check- book or determining how large a re- fund is due them at income-tax time. This revolution will and should have

James Wiebe teaches mathematics methods and computer education courses at California State University, Los Angeles, CA 90032. He is interested in the use of technology in mathematics and other subjects.

more of an impact than computers on the types of things we teach in the elementary school mathematics classroom. The abundance of cheap electronic calculators and the presence of sophisticated cash registers in virtu- ally all retail outlets have nearly elim- inated the need for pencil-and-paper computations, both at home and in the workplace.

Despite this revolution, mathematics instruction in the elementary school classroom has changed little: the majority of class time is spent preparing children for the world of 1950, a time when calculators were expensive and cumbersome and the

types of computations and numbers they could work with were extremely limited. Although today most people reach for a calculator when they bal- ance their checkbooks or do other computations, the major emphasis of elementary school mathematics - as evidenced by textbook series, mini- mum competencies, and achievement tests - remains the mechanics of paper-and-pencil computations. It is time that we move to a curriculum that prepares students for a world where calculators are almost always available.

What types of mathematical skills are needed in a world where most arithmetical computations are done by machine? Of course, students still need to understand mathematical con- cepts: the meaning of multiplication, geometric principles, the associative property of addition, the meanings of various kinds of numbers, and so on. If anything, the need to know mathematics has increased as we prepare students to design, program, and use electronic devices.

Mental arithmetic and estimation of answers to computations done with pencil and paper or machines has always been important; except for a knowledge of the "basic facts," however, these skills have rarely been taught in our schools. Since the calculator is the dominant method of com- puting answers to arithmetical problems, the need for mental arithmetic has increased greatly- the possibility of making an error when entering

February 1987 57



numbers or operations into a calculator is rather large, and one needs to know if the result obtained is reasonable. Also, in many situations the arithmetic can be handled mentally. For example, when a bill is $3.95 and we give the cashier $5.00, we want to determine quickly if our change is correct. Mentally performing the standard pencil-and-paper alogrithm is not appropriate. It is better first to add $0.05 and then $1.00 to $3.95 to get $5.00. We need to teach children appropriate strategies for solving such problems.

The widespread use of calculators will not cause the complete demise of pencil and paper: calculators are not always available, and some problems are too complex to be handled mentally. Also, pencil-and-paper computations will help us develop appropriate mental-computational strategies. For example, in developing front-end mental addition and subtraction techniques, a teacher's initial instruction would include the student's writing down of the intermediate steps.

Given the fact that calculators are and will continue to be the dominant method of doing arithmetic in the home and workplace, changes in the following areas should be made in arithmetic instruction:

Mental Arithmetic Systematic instruction in mental arithmetic should be a standard part of the elementary school mathematics curriculum, receiving as much emphasis as pencil-and-paper computation. Al- though substantial research has al- ready been done and numerous devel- opmental activities have been devised in this area (e.g., Reys et al. 1984), more work is required. The best mental-computation strategies need to be identified, and appropriate instructional strategies and materials should be developed and incorporated into our mathematics textbooks and curriculum guides.

Since mental-computation strategies are based on basic facts and on such processes as halving, doubling, and moving decimals, the curriculum should continue to emphasize strongly the immediate recall of the

sums and products of single-digit numbers and the related subtraction and division facts. It should also be broadened to include instruction in fundamental operations required in mental arithmetic, such as multiplying and dividing by powers and multiples of ten, multiplying and dividing by two, three, four, and so on.

Numeration The meaning of numerals for whole numbers should receive much greater emphasis. Such manipulatives as numeration blocks and place-value charts should be used by every child in learning the meaning of multidigit numerals. Rounding strategies and number adjacencies (e.g., Is 259 closer to 240 or to 300?) should be emphasized, since important strategies for mental arithmetic include rounding numbers or changing numbers to adjacent numerals that are easy to work with mentally (e.g., in a problem such as 267 ■*• 9, changing 267 to 270 allows the quotient to be determined mentally).

Such mental-computation strategies as front-end (i.e., focusing only on the first or second digits in a number) and left-to-right operations rely more heavily on a knowledge of place value than do the corresponding pencil-and- paper algorithms. For example, when we mentally add 3792, 4627, and 348, we cannot simply concentrate on the first digit or two in each of the addends - we must realize that the place value of the 3 in 348 corresponds to the place value of the second digit in the other two addends. Then, when we obtain 8 or 9 after adding the front digits and adjusting for the carry from the hundreds place, we must associate a place value with the sum. By con- trast, when we add these three values on paper, we mechanically align the digits from right to left and do not worry about the place value of the significant digits. Also, one of the most important aspects of determining if a displayed number is the correct answer is knowing whether the most significant digits of the answer are in the correct place (e.g., should the answer be 2300 or 23 000?). One of the easiest mistakes to make when

entering values into the calculator is to enter too many or too few zeros in numbers like 5000.

In both mental and pencil-and- paper computation, regrouping is extremely important. Regrouping out- side the context of operations (e.g., 723 means 7 hundreds, 2 tens, and 3 ones and also 7 hundreds, 1 ten, and 13 ones) should be emphasized. Ma- nipulative materials like numeration blocks or bundled sticks should be used to develop these meanings.

Decimals and Fractions Electronic computational devices do not handle fractional numerals well at all, and thus, the representation of rational numbers as decimals has be- come extremely important. Pupils will encounter decimals almost as soon as they start using calculators and computers. The present practice of delay- ing instruction in decimals until after students have mastered operations on fractions should be abandoned. In- stead, pupils should briefly be intro- duced to the meaning of fractions through such manipulatives as Frac- tion Pieces and Fraction Bars. They should also be taught to read and write numerals that represent simple fractions in the first and second grades. Decimal numeration should be taught starting in the second or the third grade as an extension of whole- number numeration and activities with fractions.

In the United States, where the metric system of measurement is not yet in common use, one can, perhaps, justify minimal instruction in operations on fractions. In other countries, such as Canada, however, real-life problems requiring operations on fractions rarely occur.

Original instruction in measurement in the United States and elsewhere should be with intuitive units (e.g., the pace, hand span) and the metric system. In the United States, instruction in the customary system of measurement and operations on fractions should not begin until after children have thoroughly mastered decimal numeration and the metric system - probably not before the sixth or seventh grade. A decision that educators

58 Arithmetic Teacher



Fig. 1 Calculator solution to problem containing fractions

Problem: What is the area of a floor that measures 141/2 feet by 201/4 feet?

Solution:

i -r 2 + 14 = [m+] ipn* + 2o[Z]H0[jl] To use this method, students need to understand the meaning of improper fractions (e.g., 141/2 means 14 + Y2), the commutative property, and so on. To solve this problem from left to right, a calculator with parentheses or multiple memories is necessary. The solution shown, however, requires only a four-function calculator with one memory.

Fig. 2 Subtractive algorithm

/Or<?j/' I Step 1: Estimate the quotient. Write the estimate loj Of O I to the right of the vertical line.

36tO¡2o - -. I Step 2: Multiply the estimate by the divisor and t50 / subtract it from the dividend. 3&Ò PO I Step 3: Estimate the quotient of the divisor and Ç 0 I the remaining value. Repeat steps 2 and 7B I // 3 until the remaining value is less than

y g I the divisor.

/ P I I Step 4: Add the values to the right of the vertical y, ̂ - line. The answer is the quotient.

Fig. 3 Left-to-right addition algorithm for two three-digit numbers

375 Step 1: Check the place to the right of the place where + 283 addition will occur- if the sum > = 9 and the

sum in the second place to the right > 9, add 1 to where the addition will occur.

375 + 283 Step 2: Add the digits in the desired place, including the 1 ,

if any. 375

+ 283 Step 3: RePeat these stePs for the next Place t0 the ri9ht« and so on.

in the United States need to face in the future is whether we should at- tempt to teach operations on fractions at all.

Perhaps operations on fractions should originally be taught with decimals - when operating on fractions, first convert the operands to decimals, then operate on the resulting decimals. This approach would allow calculators to be used for operations on fractions. Further instruction about fractions could be given in high school algebra classes, where such opera-

tions are used for simplifying alge- braic equations. Such an approach would not only simplify the curriculum but would also create room for more important parts of the mathematics curriculum and allow calculators to remain integrated with instruction. Furthermore, children would need to learn only four basic algorithms - addition, subtraction, multiplication, and division - that would be extended for use with decimals and fractions. If this approach is adopted, another addition to the "facts" we

ask children to memorize would be the decimal equivalents of the more frequently occurring fractions.

Figure 1 presents a problem in which the calculator could be used to work with fractions:

Pencil-and-Paper Computation Instructional time with pencil-and- paper computation should be limited to computations with two- or three- digit numbers and to meaningful and easy-to-remember algorithms that can, in a pinch, be used to solve problems of any size. For example, instead of teaching the highly complex long-division algorithm, we should consider teaching only the subtractive algorithm (fig. 2). This algorithm has fewer steps to remember and is appropriate for various levels of mastery - beginners and experts both use the same technique but at different levels of efficiency.

We should also consider the possibility of teaching paper-and-pencil algorithms that parallel the standard techniques for mental computation. For example, we may wish to teach left-to-right addition (fig. 3), subtraction, and multiplication rather than the right-to-left algorithms because the former are more relevant to mental computations.

Order of Operations and Parentheses When performing any computation requiring more than one operation on a calculator, students need to know how to enter the values and operations to obtain the correct answer. Instruction should start with the standard hierarchy of operations and with the use of parentheses for changing that sequence. Students should also be taught how to determine in what sequence their calculator finds answers having more than two operands (e.g., in 3 + 4 x 5 will the calculator multiply or add first?) and how to enter operations on several operands.

Integers Students will encounter negative

February 1987 59



Fig. 4 Calculator problem with realistic values

Fig. 5 Problem-solving and the calculator

16 396 people watched the state championship game in high school football. 853 students from the two schools playing each paid $3.50 to attend. 762 people under age 13 each paid $1.75 and 12 480 people over 13 each paid $5.25 to attend. The rest got in free. How much money was collected at the game?

The Smith family (two parents and two children) decided that they want to visit Washington, D.C., for four days during their next summer vacation. They live in Los Angeles and want to travel as cheaply as possible- should they take the train, fly, or drive their own car? If they travel by car, they will eat in restaurants and sleep in motels. If they travel by train, they will eat in the dining car and rent a sleeper. If they fly or take the train, they will rent a car when they get to Washington.

(Note: You can give the pupils an average daily amount for food, lodging, and other expenses, or ¿ you can ask them to decide what ^ kind of meals the family will eat, I the cost of these meals, and so [ on.) I

numbers almost as soon as they start working with calculators. We should therefore expose them to the meaning of negative numbers at an earlier age than is currently the practice. We should also teach students to employ negative numbers to solve certain types of problems on calculators. For example, the easiest way to solve $35.00 - 2 x $1.75 on a calculator is first to multiply $1.75 by 2, then subtract $35.00. If this value is needed in further computation, the negative dis- play can be changed to a positive one.

The third or fourth grade might be the appropriate time to develop the meaning of negative numbers. In- struction in operations on negative integers might be delayed until the sixth or seventh grade, as is now the practice.

Problem Solving For the most parí, the verbal and nonroutine problems in elementary school textbooks contain numbers that can easily be operated on with

pencil-and-paper algorithms. Instead, these problems should contain values that are at the same time realistic and within the capabilities of a typical four-function calculator (e.g., see fig. 4).

And, when children are learning to use problem-solving strategies, they should use calculators where appropriate and examine their results to see if they are reasonable. See figure 5.

Instruction in Calculator Use Not only should we teach mathematics content that is relevant to the age of technology, but we should teach students how to use that technology. Students should be taught all the functions of typical inexpensive calculators, including the memory keys, the square-root and percentage functions, and the sign-change key. They should be taught how to clear the calculator's memory and how to recover if an incorrect value is added to, or sub- tracted from, memory.

Calculators should be an integral part of most phases of mathematics instruction, including recording ma- nipulations with concrete materials, solving verbal problems, exploring mathematical principles and patterns, checking answers to mental and pencil-and-paper computations, and solving applied problems. Calculators should be as much a part of the elementary school mathematics classroom as the textbook and manipula- tive materials.

Bibliography Musser, G. L> "Let's Teach Mental Algorithms

for Addition and Subtraction.'* Arithmetic Teacher 29 (April 1982):40-42.

Reys, Robert E., Paul R. Trafton, Barbara B. Reys, and Judith J. Zawojewski. Developing Computational Estimation Materials for the Middle Grades. Final Report, March 1984. National Science Foundation Grant No. NSF-81 13601.

Usiskin, Zalman. "One Point of View: Arith- metic in a Calculator Age." Arithmetic Teacher 30 (May 1983):2.

Wiebe, James H. Active Mathematics for a Technological Age. Scottsdale, Ariz.: Gorsuch Scarisbrick Publishers, forthcom- ing. W

60 Arithmetic Teacher



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FOCUS ISSUE: CALCULATORS || Calculators and the Mathematics Curriculum