Improving Instruction in Elementary Mathematics with Calculators

633

Improving Instruction in ElementaryMathematics with Calculators

Evelyn M. VanDevenderDale R. Rice

Few people can dispute the tre-mendous impact technology is hav-ing on the American way of life andon education. The invention of themicroprocessor or "chip" has madepossible the creation of sophisti-cated, low-cost hardware capable ofperforming advanced mathematicalfunctions. The hand-held calcu-lator, for example, available tenyears ago for more than one hun-dred dollars ($100) is now availablefor less than ten dollars ($10), wellwithin the reach of most familybudgets. The microprocessor hasalso made possible a new line of"toys." Products such as Lit’ Pro-fessor, Genius, Speak and Mathhave provided an opportunity forchildren to have instruction inmathematics two or three years be-fore entering school. Most studentsentering school, therefore, have hadsome experience with mathematicsand the new technology. The chal-

lenge for educators is to learn how to use this technology to improvemathematics instruction.

School Science and MathematicsVolume 84 (8) December 1984

634 Improving Instruction mth Calculators

Many teachers, unfortunately, fear the use of the new technology,such as calculators, for instruction in their elementary mathematics class-room. One reason given is that students would not learn the basic con-cepts necessary to do addition, subtraction, multiplication and division.Another common response is that the use of calculators will make stu-dents lazy when doing mathematics. Studies, however, have indicatedthat the calculator can be used to improve mathematical skills (Fuys &Tischler, 1979, Leechford & Rice, 1982).

<(. . . the calculator can be a valuable instructional tool inthe process of teaching for mastery learning/’

According to Bartalo (1983) the calculator frees students from theworry of computation long enough for them to concentrate on funda-mental mathematical operations and to learn basic fact understandingsalong with developing problem-solving skills. Adams (1977) cites threeroles of the calculator for teaching mathematics: (1) Students can use thecalculator to check their work, which motivates some students; (2) thecalculator can be used in problem solving to encourage students to directtheir full attention to the process; and (3) some teachers are finding waysto team up with calculators to improve instruction. Thus, the calculatorcan be a valuable instructional tool in the process of teaching for masterylearning. Another prime use of the calculator identified by Kennedy(1980) is engaging students in learning activities and investigations thatare both instructive and interesting. Several activities suggested includechecking written work, practicing basic facts, gaining insights into opera-tions, and extending mathematical experiences through games, makingestimates or investigating magic squares.

Beardslee (1978) found that many students were delighted to workwith a calculator. Beardslee identified the following list of instructionaluses for the calculator: developing skills in reading numbers and count-ing, practicing basic facts, playing calculator games, building concepts,solving problems, investigating patterns, and using decimal exercises.Bell (1980) supports Beardslee’s conclusions and suggests that calculatorsalso be used to explore number concepts and operations to check compu-tations, practice mental skills, learn facts and algorithms, and exploreand discover principles.


Improving Instruction with Calculators 635

Calculators can be used to expand students’ mathematical understand-ings to lay the foundation for their functioning effectively in the techno-logical age of the future. Teachers, therefore, need to appreciate the cal-culator’s potential as an instructional aid and be trained to develop pro-cedures for providing students access to its teaching power. Calculatorscan give students the experiences of communicating with a machine andusing a set of procedures that a machine can follow. Fuys and Tischler(1979) challenge teachers "to use calculators to expand children’s mathe-matical horizons" (p. 570).

Instructional Uses

The calculator offers an excellent opportunity to develop skills in readingnumbers. For example, in a small mathematics group or in pairs (whereone student has already acquired the skill), have students practice read-ing various numbers on the display (See Figure 1).

Read: 9"Nine" (one-digit)

28"Twenty-eight" (two-digits)

102"One hundred two" (three-digits)

Figure 1: Reading numbers.

A teacher can also devise a game between two students or two teams.The object is to see which team/students can correctly identify variousnumbers starting at one-digit and moving on up to 8- or 12-digit num-bers.The constant feature enables students to count with the calculator.

Each "=" increases the display number on most calculators. Studentscan work in pairs and orally read the counting numbers as they appear onthe display. This activity not only teaches calculator usage but providespractice in counting and skip counting by twos, fives, and tens (See Fig-ure 2).

Calculators can also be used to tutor students in practicing basic num-ber facts. This activity can be done individually. The students should pre-dict the answers before pushing (< = ". Then they can check their answersand receive immediate feedback. Furthermore, many interesting basicfact games and activities can reinforce instruction by using a calculator.For example, two players take turns pushing a digit from (1) to (9) andthe (+) key. The first player to display 21 or more loses the game.


636Improving Instruction with Calculators

+ 1 . - = 3

Skip count by twos, fives, and tens.

+ 2 = = = = 8

+ 5 ==--== 20

+ 10 - ===== 50

Figure 2: Counting.

A calculator is an excellent tool to introduce students to multiplicationand division. Multiplication can be introduced as repeated addition, anddivision as repeated subtraction. After students have been introduced tothe concepts of repeated addition for multiplication and repeated sub-traction for division, they can be given practice sheets to further developthis concept (See Figure 3).

Multiplication: 24 \ 10 (push �+ 24, then * =" 10 times

Division: 32 : I -- (32 -- 4, then "=’’ Count the number of subtractions)

Answer: ^ ~- - ^ --===--

Figure 3:Multiplication concepts.

3 1 45301515

23 ^483^

10 First partial quotient

5 Second partial quotient

15 Answer

5 ^ 205

27 r 6308’First partial quotientSecond partial quotient

�Answer

First partial quotient

Second partial quotient

Answer

First partial quotientSecond partial quotientThird partial quotient

--. -Answer

Figure 4: Division concepts.


Improving Instruction "with Calculators 637

Long division is one of the most difficult concepts presented in the ele-mentary classroom. With the calculator, students can see each step usedto get the partial quotients. The student must first estimate the range ofthe quotient and then divide by ones, tens, hundreds, and so on. Someexamples the teacher can use for long division concept development areshown in Figure 4.A chain of calculations containing different operations is easily com-

puted with most calculators (not all calculators perform the followingorder of operations). Students should use the rules for order of opera-tion. Do the multiplication and division first. Record the answers, thendo the addition and subtraction (See Figure 5).

Step 1: 4 + (28 - 2) - 8 x 2

Step 2: 4+14-16

Figure 5: Combined operations.

Calculators speed up the computations, thus allowing students moretime to tackle complex problems. The problems found in Figure 6 andothers can be placed on flash cards with relay teams taking turns per-forming the computations on the calculator. Each team scores one pointfor a correct computation. The team with the highest score wins.

21 + (60 x 17) = 165 + (168 x 337) =

586 + 33 + (483 - 7 ) = 405 - (31 x 2.2 ) + 8.7 =

(4 x 4) + (18 - 2) 462 + (328 - 4) - (154 x 3) - (42 x 2) =

68-61-2

Figure 6: Speed compulations.

One of the most important functions of using a calculator in the ele-mentary classroom is that it aids in developing students’ problem solvingskills in students. Some realistic problems students could solve includebalancing a checkbook, computing the cost of a party or trip, averagingbasketball scores, buying a car, or figuring interest paid and earned.These problems can be placed on activity cards in the classroom mathe-matics center. Since there is no written record of errors, students are freeto explore without fear of exposed failure (See Figure 7).



A classroom party might cost:

$10.50 for a cake

.99 per 2 liter soft drink (8 liters are needed)2.52 paper cups and plates

1.38 napkins

1.87 plastic forks

1.94 mixed nuts

.83 mints

How much will the party cost?

If the cost is divided equally among the 23 studentsin the class, how much should each student bring tocover the cost of the party?

Don’t forget sales tax.

Figure 7:Problem solving.

Problem

Problem

7612-4269

40UO Estimate-

Problem

6.817.2.2

+ .99

Round

Round

8000-4000

Round

7.17.2.1.

Estimate

Problem Round

f^’oblem Round

2187x 0.99 x Estimate

Problem Round

21 ^ 4284 Round ’

Estimate

Figure 8: Estimation.


Improving Instruction wth Calculators 639

Since the calculator does not keep a record of entries, it is difficult tobe sure that each number has been punched in correctly; therefore, thestudent should learn to estimate the reasonableness of answers. This abil-ity is a valuable mathematical achievement.The calculator gives the correct answer only if the correct keys are

pushed in the correct order. To estimate an answer, have students roundthe numbers and compute mentally.

Example: Problem Round34 3059 60

+ 102 + 100

190 EstimateThen students could estimate the answer for each of these problems in

FigureS.

Students then could use the calculator to find correct answers andcompare the calculator answers with estimates to check for the reason-ableness of their results.The three basic types of percent problems can be performed easily with

the calculator. This allows students time to concentrate on the process in-volved. In each case, students can set up the proportion, cross multiply,and divide (See Figure 9).

Exercises for individual players are presented in Figure 10.

The relationship between common fractions and decimal fractions iseasy to show using a calculator.

Example:

-^ + -1- = .25 + .204 5

Students can also change more difficult examples, such as:

8 11or �^~21 42

Many exercises can easily be adapted to using decimals on the calcu-lator (See Figure 11).



1.10% of 300 is what number?

2.26 is 20% of what number?

3.18 is what percent of 72?

Figure 9: Percent.

6% of $329.00 =

5% of $35.00 =

TM^ of $800.00

$14.00 is

$ 9.60 is

$ 1.75 is

30 =

12 =

9 =

32% of

9.0% of

5% of

% of 65

% of 15

% of 1.^7

Figure 10:Individual task cards on percents.

Stepi:^2^-4^ l^x^Step 2:0.2 ^ (2.125 - 4.25) + (1.5 x 0.75)

Step 3:0.3 + 0.5 + 1.125

Figure 11: Decimal calculations.

To provide reinforcement in working with decimals, one player can se-lect the correct answers of the examples in Figure 12. If more than threeare wrong, a few more examples might be provided.The calculator provides a chance for students to discover patterns that

would otherwise be too time consuming. Exercises in pattern investiga-tions are often stimulating as well as fun. The examples that follow arethree of the numerous patterns that can be investigated using a calcu-lator. Students should write the answer at each step to identify the pat-tern. One way to do this is to prepare a worksheet giving the exercisesand then leaving blanks for the answers (See Figure 13).


Improving Instruction "with Calculators 641

Examples_____________________Choicesa b c d

1..02 + 1.5 ^ 7.29 -=9.0210.018.81,7.26

2-dhiih -5 ’ilh1-97^1’762-163.ri-x -J, -.06.6.16.006

10 100

4. I- x $.84 = $1.68 $1.42 $ .42 $ .68

5. .62Y4 x $15.38 = $9.61 $9.30 $9.62 $9.68

Figure 12: Individual task cards on decimals.

14-3= _________ I2 =

1 + 3 + 5 = ______ II2 -1 + 3 + 5 + 7 = ______ 1112 =

1+3+5+7+9= ______ llll2 =

1 x 8 x 1 = ______

12 x 8 x 2 = ______

123 x 8 x 3 = ______

1234 x 8 x 4 = ______

Figure 13: Pattern investigation.

Probability investigations contribute to the fun side of the world ofnumbers. Students can work in groups of four or five to explore the fol-lowing probability situations. If you roll a dice, what are the chances ofrolling a six? What are the chances that you have won a prize if there areforty-four other applicants? What is the probability of drawing a markedcard from a hat containing nine cards, four of which are marked?


642 Improving Instruction with Calculators

Fun activities, puzzles, and games encourage students to practice skillsand to understand concepts. Students enjoy and learn from just playingwith the calculator. For example, two or more students can try the activ-ities in Figure 14 using their calculators.

1. Each player picks any three-digit number. They take turns

trying to find the square root of the other’s number.

2. Again select any three-digit number and repeat the digits

making a six-digit number. One player, who knows the

secret, uses his magic and says the number is divisible by

13 (- 13=). No remainder. Next, the magic player says

the number is also divisible by 11 (-i- 11=). Again, no

remainder. Magic player also suggests that it will be divisible

by 7 (- 7 = ). SHAZAM! Back to the original number on

the display.

3. What will you get if you eat three ounces of spinach every

day for 306 days? Multiply 3 x 306 and turn your calculator

upside down. Tho number will resemble the answer "BIG"-

Figure 14: Fun activities.

The new technology promises to alter the way instruction in mathe-matics takes place in the elementary classroom. To use this new tech-nology to aid students in learning mathematical concepts is important forteachers. Research has shown that the use of the calculator in elementarymathematics instruction increases motivation in students while at thesame time improving mathematical skills. If used properly, the calculatorcan effectively increase studentsy learning.The calculator uses described in this article were designed to help ele-

mentary mathematics teachers "get their feet wet." Hopefully, thesesuggestions will lead to an extension of the ideas and concepts presentedinto all areas of the mathematics curriculum. Although the instructional


Improving Instruction with Calculators 643

activities described are for the calculator, computer application or adap-tations are possible in many instances. The calculator can be used effec-tively to extend our human capabilities. Teacher’s responsibilities are todevelop these techniques for use as an integral component of their in-structional arsenal in mathematics.

References

Adams, S., L. C. Ellis, and B. F. Beeson. Teaching Mathematics with Emphasis on theDiagnostic Approach. New York: Harper and Row, 1977.Bartalo, D. B. Calculators and Problem-Solving Instruction: They Were Made for EachOther. Arithmetic Teacher, 1983. 30(5), 18-21.Beardslee, E. C. Teaching Computational Skills with a Calculator. Reston, Virginia: Na-tional Council of Teachers of Mathematics. 226-241. 1976.Bell, F. H. Teaching Elementary School Mathematics. Dubuque, Iowa: William C. BrownCompany. 1980.Bestgen, B. J. Calculators�Taking the First Step. Arithmetic Teacher. 1981. 29(1), 34:37.Cruishank, D. E., D. L. Fitzgerald, and L. R. Jensen. Young Children Learning Mathe-matics. Boston: Allyn& Bacon, 1980.Fuys, D. J. and R. W. Tischler. Teaching Mathematics in the Elementary School. Bos-ton: Little, Brown & Company, 1979.Kennedy, L. M. Guiding Children to Mathematics Discovery. Belmont, California: Wads-worth Publishing Company, 1980.Lappan, G., E. Phillips, and M. J. Winter. Powers and Patterns: Problem Solving withCalculators. Arithmetic Teacher. 1982. 30 (2), 42-44.Leechford, S. and D. R. Rice. The Effect of a Calculator Based Curriculum on Sixth GradeStudents’ Achievement in Mathematics. School Science and Mathematics, 1982. 82 (7),576-580.Lerch, H. H. Teaching Elementary School Mathematics. Boston: Houghton-Mifflin,1981.O’Neil, D. R. and Jensen, R. Let’s do it: Let’s Use Calculators. Arithmetic Teacher, 1982.29 (6), 6-9.Wiebe, J. H. Using a Calculator to Develop Mathematical Understanding. ArithmeticTeacher. 1981. 29(3), 36-38.

Evelyn M. VanDevenderDale R. RiceCollege of EducationCollege of EducationUniversity of South AlabamaUniversity of South AlabamaMobile, Alabama 36688Mobile, Alabama 36688

EDUCATIONAL STANDARDS

Higher standards of educational achievement will be assured only with longrange planning in: development of human resources; diverse needs of future stu-dents; relevant goals in education for industrial growth; essential employmentskills; requirements for instruction of higher order skills.


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Improving Instruction in Elementary Mathematics with Calculators