Calculators and Problem-Solving Instruction: They Were Made for Each Other

Calculators and Problem-Solving Instruction: They Were Made for EachOtherAuthor(s): Donald B. BartaloSource: The Arithmetic Teacher, Vol. 30, No. 5 (January 1983), pp. 18-21Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192160 .

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Calculators and Problem-Solving Instruction:

They Were Made for Each Other By Donald B. Bartalo

One of the major reasons for teaching mathematics in school is to help all students learn how to solve common, everyday problems - those practical situations that all of us face as citizens and consumers. Elementary school teachers know the importance of teaching their students how to think through problems instead of guessing at possible solutions. Because of these two factors, techniques for im- proving children's problem-solving skills deserve special attention.

Several years ago, I began to won- der if youngsters in the Dansville Ele- mentary School could improve their problem-solving skills by using hand- held calculators. My thinking was that by having a calculator to use, children could concentrate on the solution process and not just on the required computation. I also thought they could do more problems this way.

In order to test my theory, since 1979 I have taught nearly 300 mathematics lessons to more than 600 students in grades 3-6. My purpose for this project was to develop instructional techniques that effectively improve traditional problem-solving skills. On the basis of this experience, I have come to the following conclusions about problem-solving instruction: • Elementary school students enjoy

doing problems, but have trouble thinking them out.

Donald Bartalo is the principal at the Ellis B. Hyde Elementary School in Dansville, New York. He developed and implemented procedures for teaching problem-solving skills to elementary school children using hand-held calculators.

• More time needs to be spent helping students learn the ' 'basics of problem-solving," especially how to an- alyze a problem.

• Students need to work on a variety of types of problem-solving situations.

• Use of inexpensive, hand-held calculators can definitely help elementary students to become better problem solvers.

• Students need to learn to work in teams and to make group decisions.

Getting the Calculators Only a few of the classrooms I worked in had hand-held calculators as a part of their regular supplies. About one week before the unit was scheduled to begin, the children asked their parents about bringing a calculator to school. For each class, it was necessary to have one calculator for every two students. Finding enough calculators was never a serious problem. Occasionally we would have to borrow a few from other staff members.

As a security measure, the calculators were always labeled with the students' names, placed in a bag or box, and locked in the office safe. In every class one student was given the responsibility for taking care of the calculators. Throughout the three-year period we never lost a calculator. By having the students bring their own calculators to school, we were able to not only work with a variety of instru- ments, but also stimulate parent inter- est.

Grouping and Instruction Learning groups - small numbers of children working together in class - are highly recommended for improv- ing problem-solving skills. Young people must learn how to work together and to make decisions coopera- tively. Time on task is increased tre- mendously by the interaction and discussion among the students. A more practical reason for teaming, however, is that it is only necessary to have fourteen calculators for a class of twenty-eight students.

Team members are encouraged to help each other as much as possible and to work together to find correct solutions. The model that I use for this kind of teaching relies heavily on the students' experiences with organized clubs and sports like Little Leagues and Scouting. In most cases, the groups or teams are established by the regular teachers and have from 2- 4 students. The conversations between team members turned out to be one of the most significant aspects of this project.

A Five-Step Approach In the second lesson (after the pretest) every one in the class was asked to memorize five steps to becoming a better problem solver:

1. Get the correct information. 2. Ask yourself: What are we sup-

posed to find out? 3. Make plans to solve it. (First, we

will . . . , then . . . .)

18 Arithmetic Teacher



4. Solve the problem. (Calculators on!)

5. Check to see if your answer makes sense.

In each class, one student was asked to make a poster containing these five points; the poster was then displayed in the room throughout the duration of the unit.

To help the students understand the importance of solving problems in an organized fashion, these five steps were first used to deal with situations that had nothing to do with arithmetic. While they are growing up, children learn to deal with everything from bullies to getting Mom and Dad to say yes about going to the movies. As teachers, we must remember that the process of solving problems is not new to our students. The key is to help children think about solving arithmetic problems the same way they think about solving other types of problems. When arithmetic problems have some real meaning for students, we increase the chances that the students will be able to apply skills learned from their past experiences.

The Instructional Lessons Each lesson consisted of structured procedures that were repeated every day throughout the unit. The time allotted for each step was gradually reduced as the students got better at making decisions. The children were constantly reminded that part of their responsibility as a group member was to help their teammate(s) understand the problem.

First day All students took a pretest made up of 6-8 questions. The test questions were given orally and written on the blackboard. Three minutes were allowed for completing each problem. The problems varied in difficulty, depending on the grade level and the abilities within the class. Students were not allowed to use their calculators during this initial testing situation.

Second day Time was spent identifying the important steps to becoming a better problem solver. Children were asked to relate personal experiences involving problem-solving situations. The remainder of the lesson was devoted to role playing the procedures and steps used to solve each problem. The emphasis was on making sure all children understood what was expected of them as a member of a learning group.

Third-ninth days Each day students tackled three to five practice problems. The learning groups worked in the following fashion: • A real life problem situation was

given orally only once; it was never repeated. (By not repeating the problem, we forced students to lis- ten carefully, an important feature of problem solving.)

• One team member recorded the information on paper.

• Each member of the team stated the purpose(s) of the problem.

• Team members developed a plan. "Making plans" included drawing pictures, writing notes, and charting the steps to follow, 1-2-3.

• Learning teams were given 2-3 minutes to solve the practice problem using their calculator.

• Each team had to agree on one answer and circle it.

• The correct solution was given orally and written on the board.

• One successful team was asked to "think aloud." This meant they ex- plained to everyone what they did to find the correct solution.

Tenth day All students were given a posttest, but this time they were allowed to use a calculator.

As you can see, this was a very structured process with timed tasks completed by all of the learning groups. Usually a learning goal was set for the class, such as 9 out of 14 teams obtaining the correct solution to earn an "A." Most of the problem

situations involved more than one op- eration and were typically more difficult than those normally found in the regular curriculum.

Before the reader jumps to any neg- ative conclusions about the teaching methods just described, a few words of explanation are in order. First, for purposes of this particular project only, time was an important factor because each learning team was required to concentrate on the thinking skills in a sequential manner. There- fore, students worked together in a directed fashion and were always told how much time they would have to complete a specific task. Later, as students gained experience with this type of process, rigid time limitations were removed and students worked at their own pace.

Second, the reason for ten-day units was to conduct the action re- search necessary for evaluating the teaching procedures. This unit approach would certainly not need to be repeated by other instructors; problem solving is normally taught throughout the school year. Both the ten-day unit approach and the time restrictions were designed to help young students remain on task.

Selecting the Problems Choosing good problems for instruction and practice requires thorough planning. Naturally you need to con- sider the grade level and the general abilities of the students, but there is more. Whenever possible, students in each class should be named as the participants in the problem situations. The main point here is that the kinds of problems presented do make a great deal of difference to the students. Most examples found in cur- rent textbooks are not very motivat- ing; avoid as much as possible using word problems that have not been modified to meet local situations.

Teachers need to hunt for problems in new and creative ways. No source should be overlooked. Elementary school students need to begin working with all types of problems. Some graded, sample problems, which illus- trate the points just mentioned, are included here.

January 1983 19



1. Mrs. Ullyette asked Laura to go to the school store to buy her 4 notebooks. Notebooks cost 650 apiece, and Mrs. Ullyette gave Laura $5.00 to spend. You have 2 problems to solve: First, did Laura have enough money to buy the 4 notebooks?, how much change would she bring back? (Grades 3-4)

2. Tom was asked to handle a problem at the Outdoor Store. Seven hun- dred fishing hooks had been separated from their proper containers. Each package of hooks was supposed to have exactly 24 hooks. Your problem: How many "full" packages of hooks was Tom able to make? (Grades 4-5)

3. Radio station WDNY was con- ducting a "call-in" show to raise money for a very sick junior high school student. The average caller pledged $3.00. If 78 people called the station, how much money did WDNY raise? (Grades 4-5)

4. Gary did some babysitting over last vacation. On Monday he sat from 3:30-5:00 p.m., Wednesday from 4:00-7:00 p.m., Friday from 7:30-9:00 p.m. and Saturday from 10:00 a.m.- 3:00 p.m. If he makes $1.25 per hour, how much money did Gary make babysitting last week? (Grade 6)

5. The Dansville Electronic Shop is having a 2-for-l sale. Laura buys a package of cassette tapes for $2.95, 4 calculator batteries for $6.95, and a 3.5 m extension cable for $7.39. You have three problems: How much did Laura spend?, What did she get?, and How much did she save? (Grade 6)

6. The Dansville Fast Food Shoppe is helping to raise money for a handi- capped students camp. They have promised to donate 250 for every Huge Hamburger and 100 for every soft drink sold during a one-week period. At the end of the week they had sold 325 Huge Hamburgers and 1500 drinks. How much money did the Fast Food Shoppe raise for the camp? (Grades 4-5)

Lately I have been using the daily newspaper as a source of problem situations. It seems every section of the paper contains at least one or two exciting challenges for young students. A story about the Рас-Man

computer champion who scored 6 950 000 points in 8 1/2 hours led to a stimulating problem for fifth graders. They were asked to find out how many points the champion scored every minute. The sports page is loaded with potential problem-solving situations.

Although my study dealt mainly with problems that required thinking procedures related to the four basic operations, this approach was only a beginning. Future problem-solving instruction would most certainly in- clude nonroutine or process-oriented problems. More open-ended situations involving the concepts of pre- dicting, comparing, analyzing, and programming also would certainly be an important part of this kind of instruction.

The solution to any problem is based on an understanding of the problem to be solved. Traditional teaching techniques frequently fall short of this goal and unwittingly develop young people who are easily frustrated when they cannot "get the right answer" immediately. Educa- tors are hoping to turn out students who can use computers to find solutions to problems. The question is, Who's going to teach them how to write precise plans, to brainstorm with others, to look for patterns, to sequence specific operations, and to understand logical order? Our hope is that these skills will be taught as part of the basics.

Test Results and Evaluation As previously mentioned, all students were given a pretest on the first day of class. They worked on these tests independently and were not allowed to use calculators. The pretest consisted of 6 problems for grades 3-4 and 8 problems for grades 5-6. The pretest took 20-30 minutes to admin- ister.

The questions on the pretest were similar to the sample problems listed earlier. In most instances, the pretest items were more difficult than the problems the students had been doing in their regular class. There was al-

ways at least one example for each of the four basic arithmetic operations. The posttest was given on the last day of class. Students worked on these tests by themselves but were allowed to use a calculator. All other variables remained constant with the initial quiz.

As the case with the time limitations and ten-day units, pretesting and posttesting were conducted for purposes of this project only. Although testing is certainly a part of effective instruction, pretesting and posttesting are not necessary for teaching problem solving as described in this article. Both the teachers and I, however, were more than pleased with the results of this approach.

Suggestions for Teaching In order to help young people develop the thinking skills necessary to solve common, everyday problems, we must get them to stop playing a game I call, "add-subtract-multiply-or di- vide." This is a game many youngsters learn to play early in their school experience and perfect by fifth and sixth grade. When they are faced with a traditional word problem from a textbook or workbook, and depending on the grade level, students frequently attempt to solve problems by guessing at the required computation. Al- though this game can frequently lead to getting the right answer, it does little to encourage thinking.

The following suggestions may help: 1. Make a distinction between com-

putation skills and thinking skills. 2. Try to devote at least one- third of

your arithmetic teaching time to problem solving. (Actually, this percentage is rather low for fifth and sixth graders.)

3. Encourage children to practice solving problems together and to help each other understand how to do the problem. (Learning re- search clearly indicates that spac- ing out, repetition, and practice promotes more permanent learning.)

20 Arithmetic Teacher



4. Don't hesitate to ask children to think aloud and explain their plan for solving a problem.

5. Help children to relate solving mathematics problems to solving social problems.

6. Don't let problem solving become a mechanical process done accord- ing to rules.

7. Encourage students to write their own problems and to use examples from their immediate experiences.

Final Thoughts The major advantage of using calculators to help students practice problem solving is that it frees them from having to worry about computational operations. This is not meant to imply that elementary school children should not learn how to compute by

hand or study mathematics facts. They definitely have to understand the fundamental operations and know the basic facts. But students must also be taught how to apply those basic skills in practical situations.

The structured lessons described in this article were designed to help elementary school students focus on the process involved in solving common problems. This project was aimed at making students think and talk about what they were doing. Students were also encouraged to help their class- mates understand by thinking aloud. Today's generation of children are being raised in a world where technology is changing at an unbelievable rate. The challenge for educators is to learn how to use this technology to help our students learn better. For my money, calculators are a definite part of that technology, w

Statement of ownership, management and circulation (Re- quired by 39 U.S.C. 3685). 1. Title of publication, Arithme- tic Teacher. A. Publication Number 0004 136X. 2. Date of filing, 1 October 1982. 3. Frequency of issue. Monthly - September through May. A. No. of issues published annu- ally, nine. B. Annual subscription price, $30.00. 4. Loca- tion of known office of publication, 1906 Association Drive, Reston, Virginia 22091 . 5. Location of the headquar- ters or general business offices of the publishers, same as #4. 6. Names and complete addresses of publisher, editor, and managing editor. Publisher, National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091. Editor, none. Managing Editor, Jane M. Hill, 1906 Association Drive, Reston, Virginia 22091. 7. Owner, National Council of Teachers of Mathematics, 1906 Asso- ciation Drive, Reston, VA 22091. 8. Known bondholders, mortgagees, and other security holders owning or holding I percent or more of total amount of bonds, mortgages or other securities, none. 9. The purpose, function, and nonprofit status of this organization and exempt status for Federal income tax purposes have not changed during preceding 12 months. 10. Extent and nature of circulation. Average no. copies each issue during preceding 12 months. A. Total no. copies printed, 36 457. Bl. Paid circulation, sales through dealers and carriers, street vendors and counter sales, none. B2. Paid circulation, mail subscriptions, 31 717. С Total paid circulation, 31 717. D. Free distribution by mail, carrier or other means, samples, complimentary, and other free copies, 63. E. Total distribution, 31 780. Fl. Copies not distributed, office use, left over, unaccounted, spoiled after printing, 4 677. F2. Copies not distributed, returns from news agents, none. G. Total, 36 457. Actual no. copies of single issue published nearest to filing date. A. Total no. copies printed, 33 825. Bl. Paid circulation, sales through dealers and carriers, street vendors and counter sales, none. B2. Paid circulation, mail subscriptions, 30 094. C. Total paid circulation, 30 094. D. Free distribution by mail, carrier or other means, samples, complimentary, and other free copies, 63. E. Total distribution, 30 157. Fl. Copies not distributed, office use, left over, unaccounted, spoiled after printing, 3 668. F2. Copies not distributed, returns from news agents, none. G. Total, 33 825. 11.1 certify that the statements made by me above are correct and complete. James D. Gates, Business Man- ager.

TAC-TIC

The game of reverse tic-tac-toe (known to some as toe-tac-tic) has the same rules as the standard game with one exception. The first player with three markers in a row loses. Can the player with the first move avoid being beaten?

From The Best of Problematical Recreations, Volume I, Litton Indus- tries, Inc., 1964. (Available also in Dunn, Angela (ed.), Mathematical Bafflers, New York: Dover Publications, Inc.)

Answer to "Tac-Tic" aiBnbs eijsoddo

Ацеэщэше1р ец' ди'це' Aq sdAOtu sjueuoddo ещ ¿о цэвэ sjeiunoo иэщ pue ejenbs ю'иээ эщ se>|gi J8ÁB|d isjjj эщ *Ä|jsee Лдед

AT-1-83

January 1983 21

Problem tolving

Challenge: Fof-qble Xudent}



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Calculators and Problem-Solving Instruction: They Were Made for Each Other