Suggestions for Teaching Problem Solving�A Baker’s Dozen

Stephen Krulik

andJesseA. Rudnick

Mathematics EducationTemple University

Philadelphia Pennsylvania 19122

The current re-emphasis on problem solving has led to a rash of jour-nal articles and presentations at many meetings. Indeed, the entire March1978 issue of School, Science and Mathematics and the November 1977issue of The Arithmetic Teacher were devoted to this area of study.We consider problem solving to be of primary importance in today’s

school mathematics curriculum. Without problem solving capabilities,our students will be disadvantaged regardless of the amount of basicskills they possess. And yet, most of the articles are attempting to devel-op and put forth heuristic models for problem solving. Are these modelsreally what classroom teachers need? Or, do they need concrete sugges-tions for classroom implementation of heuristical systems? We firmly be-lieve that the problem solving model in itself is not nearly as important ashow the classroom teacher approaches problem solving in the classroom.

In this article, we will present some thoughts on problem solving andsome specific suggestions for achieving the important goal of having allof our students become better problem solvers.As we see problem solving, several common threads exist:(1) Problem solving should not be considered as a vehicle for practic-

ing writing equations and/or other mathematical concepts, but should bepresented as a unique concept, and then integrated throughout the entiremathematics program. Problem solving should be regarded as the basicskill of mathematics, and should be continually emphasized. Many of theskills and concepts that we expect our students to master are basic toproblem solving.

(2) The sections of most textbooks labeled "Problem Solving" do notdeal with problem solving at all!! Rather, the "word problems" (such asmotion, mixture, age, interest, etc.) as presented in these textbooks arenot really problems in the problem solving sense. Rather, these are rein-forcement exercises or "routine problems" and should be taught as such.In practice, students solve these by following a previously learned al-gorithm or model; this is not problem solving�this is solving problems.

(3) The way to make students into better problem solvers is to exposethem to more and more problem solving. Teachers should build files ofnon-routine problems, and then spend time at all grade levels involvingstudents in the problem solving process.

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(4) The problem solving process suggests that a set of heuristics be de-veloped jointly by the teacher and students. Whether these be Polya’sfour-step heuristics, or some other set of five, seven, eight, or even moresteps is not important. What is important, is that the student develops anorganized set of "questions" to ask himself, and that he constantly referto them when he is confronted by a problem situation. Indeed, the ques-tion of who is a good problem solver might be answered by saying thatthe best problem solver is the person who holds the best conversationwith himself, and asks himself the right questions as he attacks a prob-lem.We are not proposing a specific heuristic model. The references at the

end of this article \\\\\ direct the reader to several such models. Any oneof these could serve as a guide. Regardless of which you decide to devel-op, here are some suggestions to assist you in helping students to becomebetter problem solvers.

1. Create an atmosphere of success in problem solving. The old adage,"Nothing succeeds like success" holds true in the classroom. Studentsare easily frustrated; indeed, beginning with problems that are too diffi-cult will usually "turn the students off." This makes problem solvingmore difficult than ever. Insure success. Choose your initial problemsw^ith a great deal of care.

2. Encourage your students to solve problems. We have already statedthat, in order to become good problem solvers, students must be con-stantly exposed to problem solving. Therefore, find problems that are ofinterest to students. Listen to them as they talk; they will often tell youthe things in which they are interested. Problems from TV and sports al-ways generate enthusiasm.

EXAMPLE: The weight of a $1 bill is 1 gram. A basketball player’s salary is $1 million,to be paid in $1 bills. Could the player carry his salary in a suitcase? Suppose he agreedto accept $20 bills?

Consider that the example concerns itself not only with weight, but withvolume as \\’e\\. Yet, no dimensions have been given. The weight of $1

million is one metric ton; this should cause some lively conversation inclass by itself.

3. Teach students how to read the problem. Students often overlookessential ideas and implications within the context of a problem. Sincemost problems are presented in written form, proper reading habits areessential. Have students underline or circle key wwds in the problem.Give one student the problem on a single slip of paper. Have him readthe problem, and tell it in his own words to the class. Students \\\\\ revealwhether or iiot they are discovering the parts of the problem, facts thatare really important to its solution, or whether they have missed the pointentirely. Another technique is to show the problem on the overhead pro-

Teaching Problem Solving 39

jector for about one minute. Then, turn the projector off, and allow stu-dents to tell the rest of the class what the problem said.

4. Put your students into the problem. Let them be a part of the ac-tion. Involve them in the "story".

EXAMPLE: How many times can you dribble a basketball in one minute? In fiveminutes?. . .

EXAMPLE: How fast can you roll a marble along the chalkboard tray in your class-room?

5. Require your students to create their own problems. Having stu-dents create their own problems will heighten their interest in solvingproblems. These may be variations on other problems that they haveseen, or may be original creations. Remember that not all of their prob-lems need have a solution; it is the problem solving process that is im-portant.

EXAMPLE: How much does a 100 pound lion eat in one week?

Creating problems with excess, insufficient, and just enough informationis an interesting and worthwhile task for your students.

6. Have your students work together in pairs or in small groups. Theinteraction that is provided by cooperating students helps them to modi-fy each other’s thinking, defend their position, and express theirthoughts in a clearer manner, by using precise mathematical language.

7. Encourage the use of freehand drawings. Drawings do help stu-dents in problem solving. However, beginning problem solvers have atendency to want to measure to find an answer, rather than to think theproblem through or to demonstrate it mathematically. The use of free-hand drawings will teach them not to rely on the drawing for measure-ment. This does not mean, by the way, that you should allow sloppy,inaccurate sketches. Students should draw neat, carefully labeled, dia-grams.

8. Suggest alternatives when the present approach has apparentlyyielded all possible information. The mind-set (preconceived idea) thatstudents have often leads them to a dead end. This is not unusual, nor isit unexpected. This mind-set must be changed, and another approachundertaken once the student has exhausted all possible information andhas still not found a solution. Some teachers err by directing studentsthrough the most efficient path to the solution, rather than allowing fur-ther exploration to take place in a new direction.

EXAMPLE: In the figure shown, AB and CD are perpendicular diameters that inter-sect at point 0. Any point, F, is selected on arc AC. Perpendiculars FG and FH aredrawn to OA and OC respectively. If the radius of the original circle was 8, find thelength of segment GH.

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b

In trying to solve this problem, most students have a mind-set which sug-gests the use of the Pythagorean Theorem in triangle GHO. Once theyhave exhausted all the information in the problem, however, the teachershould suggest that the students examine tlie entire figure FGOH, a rec-tangle.

9. Raise creative, constructive questions. This provides some studentguidance, but at the same time, allows them a wide range of responses.Give them sufficient time to think before they answer. Research indicatesthat teachers do not allow sufficient time for children to think about thequestion being asked of them. The average elapsed time between thequestion and the expected student response is about 3 seconds or less.Problem solving is a complex process; it requires time for reflection.Don’t rush your students!

10. Emphasize creativity of thought and imagination. In a positiveatmosphere, students can be as free-thinking as they wish. Encourage ex-perimentation, trial and error, intuition, guessing and hunches. Youshould never penalize "way out" answers if they show some thought onthe students’ part. It is the process that is important.

EXAMPLE: A student’s response to the question," How can you divide 25 pieces ofcandy among three friends?" was, "I’ll take 23 pieces, and give cacti of my two friendsone."

Although this is not the answer that the textbook \vas seeking, it didshow that the student’s interpretation of the problem was a bit differentfrom that of the author, but still indicates appropriate thought processes.

11. Encourage your students to use a calculator. Over 100 millionhand-held minicalculators have been sold in the United States. The ma-jority of your students probably own one. This tool will enable your stu-dents to discuss concepts, long before they can do the mathematical skillsa problem calls for. Every student can now add, subtract, multiply anddivide. They can work problems that are interesting, significant, andmathematically important, even if tlie skills are beyond the capacity ofthe student.

Teaching Problem Solving 41

12. Use strategy games in your class. The processes used in analyzingstrategy games and developing winning strategies is quite similar to thestrategy used in problem solving. Games have a strong appeal for allstudents; they have been exposed to strategy gaming all of their lives.Have your students play and analyze games such as NIM, HEX, check-ers, chess, etc.

13. Don’t try to teach ne\v mathematics while teaching problem solv-ing. The development of problem solving skills, not the mathematics in-volved in the problem, is the paramount reason for discussing problems.Many of the textbook exercises labeled "verbal problems" are intendedto reinforce some new mathematical skills�this can often ^mask" theproblem solving process as the primary skill. Try to keep the mathema-tics level at least one or two year’s behind the students’ abilities. In manycases, (such as problems involving logical thought), the mathematics in-volved is already a relatively minor part of the problem.What we have tried to do in this article, is to show the teacher a variety

of techniques for helping students to become problem solvers. Basic toall of these, however, is the recognition that the problem solving processis a skill, and like any skill, it can be taught. However, it must not be con-fined to a single brief time period during any one month of a singlesemester, but must be carefully integrated throughout the entire mathe-matics program. Imagine the difficulties that teachers and studentswould face if they tried to master all of the concepts and skills associatedwith fractions in a single one or two day unit of work, and then never hadto work with fractions again! The same is true of problem solving. As thebasic skill which underlies all of mathematics, it must be emphasizedthroughout all of mathematics. Only then can our students become adeptat problem solving, not merely solving problems.

REFERENCES

1. Arithmetic Teacher, National Council of Teachers of Mathematics, Reston Virginia.November, 1977.

2. KINSELLA, JOHN. "Problem Solving." The Teaching ofSecondary School Mathematics,Thirty-third Yearbook of the National Council of Teachers of Mathematics, Reston,Virginia, 1970.

3. KRULIK, STEPHEN. "Problems, Problem Solving and Strategy Games," The Mathema-tics Teacher, November 1977. Pp. 649-654.

4. KRULIK, STEPHEN and RUDNICK, JESSE A. Problem Solving: A Handbook for Teachers.AIlynand Bacon, Boston, Massachusetts, 1980.

5. POLYA, GEORGE. How To Solve It. Princeton University Press, Princeton, N.J. 1945.6. SCHOOL, SCIENCE AND MATHEMATICS, School Science and Mathematics Association,

Inc. Indiana, Pennsylvania, March 1978.