electronic calculators in the classroom

Electronic calculators in the classroomAuthor(s): LOWELL STULTZSource: The Arithmetic Teacher, Vol. 22, No. 2 (FEBRUARY 1975), pp. 135-138Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41188717 .

Accessed: 12/06/2014 18:58

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 62.122.78.64 on Thu, 12 Jun 2014 18:58:41 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/41188717?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Electronic calculators in the classroom

LOWELL STULTZ

Having spent six years teaching junior and senior high school mathematics and six years as a college instructor, Lowell

Stultz has most recently participated in an experimental

tutoring program using electronic calculators in fifth-

grade mathematics classes.

/'nyone who has noticed recent news- paper advertisements or attended book and equipment exhibits at teachers' conferences is aware of the exploding market in electronic calculators. The microcircuit technology spin-off from the space industry has given the consumer a handy mathematical tool that will soon become commonplace in the home, office, shop, and classroom, but it is too early to predict the full extent to which the availability of electronic calculators will affect the teaching of mathematics. This article presents some of the uses of calculators in the elementary classroom as well as some of ,the changes that may become necessary in the subject content itself.

All an educator need do to see some of these new electronic marvels is consult the yellow pages of a city telephone directory. A few calls will result in many salesmen coming with models of calculators for any budget and almost any calculating purpose.

At present, there are several types of calculators ranging in price from $50 to $4,000. Optional features- the most notable being the choice of either electronic display or

Note: The examples in this article are based on the author's use of a pocket calculator from Sears Roe- buck Co., which is identical to one made by Bowmar.

paper printout - are available for each type. The type most likely to be used in ele-

mentary schools is the small hand-sized simple calculator ($50 to $150) that per- forms the four basic operations: +, -, x, and -5-. Recommended features on these are keys for both "clear" and "clear entry," a "constant" switch for multiplication and division by a constant, and full "floating point" display. Many models of this size also come with rechargeable batteries. Other, more expensive models have a few more features for more complex calculation and would generally be more suited for high school and college.

Most of the calculators suitable for elementary school mathematics have the following four operation keys: ОФОО To solve the problem

7 - 5 = ?

the buttons О@®ф are pushed. If we use a rectangle for the displayed numbers, the calculation goes like this:

It may be desirable to call attention to the fact that the + and - symbols on the Q key and the £ key have a dual role - as the signs of numbers and also as the operations of addition and subtraction.

February 1975 135



The multiplication of signed numbers for the problem

3x2 = ? can be written

3 x 2+ = ?

and the calculation on the calculator is done in the following order:

Division is identical to the multiplication process except that the Q key is used in- stead of the© key.

Uses in the elementary classroom

Teachers are just beginning to imagine the uses that these devices might have in the classroom. As years go by, we shall see many innovative projects and experiments in which students use calculators. The following list of iďeas is a modest attempt to stir the imagination by showing how a calculator may be used as an instructional tool in the mathematics classroom. We shall assume that at least one calculator is available for every two students. Any higher ratio of students to calculator has been found to lead to frustration, confusion, and a dislike for the machine.

1. Preschool, kindergarten, and first- grade students enjoy using the calculator to count. First graders can make up their own . addition and subtraction problems with one-digit numbers. Allow the students to work their own problems on paper first and then to check them on the calculator. You can even explain negative differences as "in the hole" numbers, or "owed" numbers.

2. For any grade, the possibility of a student's making up his own problems presents a motivating experience never practical before. The teacher will need to limit the size and order of the numbers and spe- cify the operation. Some students at first will tend to make up oversimplified problems, but this becomes boring after a while. Others will experiment with numbers that are too large or with more difficult chains

of operations. Nevertheless, this whets their appetite for more advanced mathematics.

If students are required to do several problems on paper before checking them on the calculator, they will be challenged to work for more speed and accuracy. They often cannot wait to see how well they do with their inventions. Students who get their own problem wrong feel that the right answer on the calculator is something like a "slap on the hand."

3. At any grade level, the calculator may be used not only to check answers but also to debug the problem. In a multiplication problem, a student can check partial products. In addition problems, he can get partial sums as he adds a list of numbers. In division, he can check the products and the subtractions. In larger division problems in later grades, he can use the calculator to help guess partial quotients. Allow the student to do long division using only the ф key. He can then use the ф to check the answer when finished.

4. The concept of place value is impor- tant in all the operations at all levels. The calculator can be used to check the ex- panded form of a number by adding digits with the appropriate number of zeros after them. For example:

5000 + 300 + 70 + 4 = 5374

In multiplication, the partial products will add up to the answer only if their place value is preserved. In the vertical form of the problem

354 X 26 2124 708 9204

it is easily seen by students that the 708 must be in the tens column. On the calculator, these partial products will add up to the same answer as when the ф key is used only if zeros are entered to preserve place value: 26 x 354 = (20 + 6) x 354

= (20 x 354) + (6 X 354) = 7080 + 2124

136 The Arithmetic Teacher



Note: One difficulty on all small calculators available at the writing of this article is a lack of the capacity to perform the dis- tributive principle for multiplication over addition. For instance, in

7 x 36 = 7 x (30 + 6) = (7 x 30) + (7 x 6), the first two expressions can be performed in a chain of calculations, but the third cannot. There is no way to "hold" one product while another product is obtained and then add them. (This is possible on more expensive machines that have the capability of permanent storage of a constant.) Students can, however, jot down their partial solutions to problems on paper and continue with the electronic calculations.

5. Calculators can serve the same purpose as flash cards with quick oral or written response and immediate reinforcement. Singly, or in a group, students can be asked to enter a number, an operation, and another number. Then after giving the answer, they depress the ф key and find out whether they were right.

6. The fact that calculators can display fractions only as decimals is a drawback in long decimals or when division remainders are to be expressed in fractional form. But this feature is also a great aid in introduc- ing decimal fractions in later grades. Very soon a student using a calculator for some division problems wonders about the numbers after the decimal point. Even in early grades, a student can be shown that

Il-ž-8=l,r3=l + -f=l-f= 1.375. о о

In the lower grades where a student needs to check a division problem that has a re- mainder, he can learn to divide the re- mainder by the divisor using the calculator to see if the decimal fraction is the same as when he uses the ф key in the original problem.

7. Most small calculators will do a "chain" of calculations. In the later grades, much' can be done with the order of operations on signed numbers using the calcula-

tor. If this were a major effort in the seventh and eighth grades, the student's readiness for algebra should be enhanced tremen- dously. For example, the problem

3 - 6 I 5 -- [4 + 2 (7)] | must be performed in mathematically correct order on the calculator if the student is to be consistent in what he thinks is correct on paper. The order of calculations would be

8. The constant key is one of the more useful features on these small instruments. This key automatically stores the first number in multiplication or the second number in a division problem so that it can be used over again in the next calculation. For instance, after the calculation-

ooeoH pushing фф results inHT pushing only ф would multiply the last displayed number by 7. Because it allows for repeated multiplication, this procedure can be used to demonstrate powers of a number, as follows:

When this process is reversed, something very interesting happens: if the ф key is pushed, and then the фкеу, we get next

Continuing, we get still more repeated divi- sions by 3,

which are approximately the decimal fractions for 1/32, 1/33, and 1/34. If the operation is reversed again by depressing first the ф key and then the ф key, we get

February 1975 137



which is not exactly the 3 we started with. With students, this fascinating little experiment leads naturally to the question Why? Now the teacher has a perfect opportunity to talk about number approximations, truncation errors, rounding numbers off, and errors due to repeated calculation with approximate numbers.

In the elementary classroom we have no obligation to go into the higher concepts of "degree of error," "precision," or "range of accuracy," but we shall have to pay the price for the use of electronic machines by discussing their limitations.

9. One of the more practical uses of a calculator in junior high school mathematics is in the evaluation of formulas. For- mulas for measurement are encountered in science classes and geometry lessons even though the student has not yet had algebra.

10. "Ah ha! A function machine!" This is what a fifth grader told me when we started working with a small calculator a few days after we had struggled with the concept of a function.

An electronic calculator is a perfect example of the "machine," or "rule," idea of a function. Most elementary textbooks use this concept anyway in pictures and diagrams. The student sees quickly that given a certain input, the calculator can display only one number - it has no choice. If it is to work properly, it has to display something, and the output will always be the same for the same input.

It is worthwhile at this point to ask the student to find something that the machine cannot do. Even the youngest students sooner or later discover that dividing by zero makes the machine display an error code, and in some calculators everything is unworkable until it is cleared. On larger machines with a 'J button, the square

138 The Arithmetic Teacher

root of a negative number results in an error indication. These experiences lead naturally to a discussion of the domain of the division and square-root functions.

The first uses of functions are usually the basic +, -, x, and + operations with whole numbers. Here again the calculator can be used to check a student's work and allow him to make up his own domain or range values. For example, define a function f(n) = 7 X n and ask the student to make up his own partial table like this:

n ßn) 1 3 0

49 12

14 56 0

11

Allow him to fill in the whole table by hand first and then have him check it with the calculator.

1 1 . The graphing of functions also becomes more exciting with the use of a calculator. After the usual discussion of co- ordinates and the graphing of ordered pairs of whole numbers, ask the student to put random large and small numbers in the calculator function and see if the ordered pair results in a point on the same graph.

With a little thought and ingenuity, the reader can come up with many more appli- cations of these little electronic marvels. The suggestions made in this article would not require a classroom to have calculators all day, every day. A set of a dozen calculators, delivered occasionally to the room like other audiovisual equipment, would be enough to make them available to the whole school.



electronic calculators in the classroom

Documents