teaching problem solving to preservice teachers

Teaching Problem Solving to Preservice TeachersAuthor(s): Stephen Krulik and Jesse A. RudnickSource: The Arithmetic Teacher, Vol. 29, No. 6 (February 1982), pp. 42-45Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192013 .

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Teaching Problem Solving to Preservice Teachers

By Stephen Krulik and Jesse A. Rudnick

Problem solving! Problem solving! Problem solving! Wherever mathematics teachers turn today, they are confronted by these two words, whether it be when reading NCTM' s An Agenda for Action, current educa- tional sections of their newspapers, professional journals, or when attend- ing a mathematics education conference. In fact, at the annual NCTM meeting in St. Louis approximately 20 percent of the sections had problem solving in either the description or the title of the talk.

We are pleased that this vital topic is receiving the attention it deserves. This special edition of the Arithmetic Teacher, for example, illustrates the Council's commitment to emphasis on problem solving in the mathemathics curriculum of the 1980s.

If we are to develop the problem- solving capabilities of children, it is necessary to prepare their teachers in this crucial art and skill. The preservice teacher preparation program is a good starting point. The preservice teacher is young, full of enthusiasm, and has not as yet adopted the errone- ous belief that teaching specific algo- rithms to solve specific problems is teaching problem solving.

Stephen Krulik and Jesse Rudnick are profes- sors at Temple University in Philadelphia, where they conduct courses in mathematics education and microcomputers at both the un- dergraduate and graduate levels. Krulik was editor of the 1980 NCTM yearbook on problem solving and Rudnick was a member of the editorial panel. Rudnick is also a former NCTM director.

If a teacher is to be an effective guide for the learning of problem- solving skills, then he or she must first become a problem solver. This is not necessarily the case in other profes- sions. For example, coaches instruct their athletes in the performance of tasks that they themselves may not be able to perform. Architects, although they design a building, are not capable of constructing it. It is impossible, however, to teach problem solving if the teachers themselves are not ade- quate problem solvers.

Fundamental to a preservice program for teachers of mathematics is the clarification of the meaning of the terms problem and problem solving. Many teachers use the words question, exercise, and problem inter- changeably, but there are differences. As an example, let's look at the following:

What is the product of 7 and 6? or What is 7 x 6?

Is this a question? An exercise? A problem? Or is it all three? It depends! If we ask this of our future mathematics teachers, we expect an immediate response. The answer, 42, should be automatic - a recall based on many years of experience with multiplication. For this audience, What is 7 x 6? is a question - it involves only recall.

On the other hand, if we ask this of an elementary school student who has been studying the basic multiplication facts, then the purpose of our inquiry is quite different. Here, we are providing drill and practice for that youngster. Thus "What is 7 x 6?" is now an exercise - it involves practice.

Finally, we ask this of a first or second grader who has not as yet learned the meaning of multiplication. Now the meaning of product must be discussed and the youngster led to understand that 7x6 means taking 7 groups of 6 objects and finding the total number of objects. To this students we have presented a problem - something that requires thought.

Thus a question requires recall, an exercise provides drill and practice, and a problem requires careful thought and synthesis of knowledge. What is a problem at one time to one person may be an exercise or even a question at some later stage of that person's mathematical development. Furthermore, what is a problem for one person can easily be an exercise or a question for someone else.

A preservice program should also make it quite clear to the students that problem solving is a process. As a process, problem solving is the means by which an individual uses previously acquired knowledge and under- standing to satisfy the demands of an unfamiliar situation. The student must synthesize what he or she has learned and apply it to the new and different situation. We have previously stated that what is a question for one person can easily be an exercise or even a problem for someone else, or to the same person at another time. Yet a creative teacher can take any question, any exercise, or any routine word problem and use it as a basis for a worthwhile problem-solving experience. It all depends on the way in which the teacher presents the materi- al.

42 Arithmetic Teacher



We do feel it is important that teachers accept the notion that the goal of teaching problem solving is to have students successfully develop a process; that although the answer is important, it is the process that is at the heart of problem solving. In the real world the final answer is the important thing, but our responsibility as teachers is to help students develop a process for finding the answer.

Becoming a Problem Solver A preservice program should take future teachers through the same kinds of activities and experiences that they will require of their own students a few years hence. Like any other skill, problem solving requires lots and lots of practice.

Figure 1 shows a flowchart of the problem-solving process. The boldly outlined command boxes indicate the procedures that are basic to the problem-solving process. In our class- rooms this set of rules occupies a central position. Reference is con- stantly made to it in our attempt to help the preservice teachers become good problem solvers. The same flowchart will later serve as the basis for their own teaching.

Select a strategy For most preservice teachers, the most difficult and important part of the problem-solving approach is to "select a strategy." A wide variety of problem-solving strategies must be identified and illustrated by the use of many problem situations. Some problems and solution strategies just natu- rally fit together. Indeed, more than one strategy may be necessary to solve a particular problem. Preservice teachers should be encouraged to find multiple solution strategies (fig. 2). It is important to continually emphasize that the goal of problem solving lies in the solution process y not necessarily in the answer itself.

To illustrate the application of more than one solution strategy, consider the following problem:

There are 16 football teams in the Continental Football League. To

Fig. 1 A Flow Chart of the Problem-Solving Process

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Adapted from Krulik, Stephen and Rudnick, Jesse Problem Solving: a Handbook for teachers Allyn and Bacon, Inc. Boston, Mass. 1980

Fig. 2

Strategies for Problem Solving

1 . Pattern Recognition 8. Representing data 2. Working backwards 8.1 Graph 3. Guess and test 8.2 Equation 4. Simulation or experimentation 8-3 Algebraic expression 5. Reduction JJ Jah^rt 6. Exhaustive listing 86 Diagram 7. Logical deduction

February 1982 43



Fig. 3

Starting with a Smaller Number of Cities

2 cities ' / 3 cities 1 line 'r 3 lines

^^■>V 6 lines *^ 10 lines

Fig. 4

Suggestions for Teaching Problem Solving

1 . Create an atmosphere of success. 2. Provide an ample supply of challenging problems for

problem-solving practice. 3. Help students to develop techniques to read problems

analytically. 4. Require students to create their own problems. 5. Have students work together in pairs or small groups. 6. Encourage students to attempt alternate strategies. 7. Raise creative, constructive questions when leading a

problem-solving discussion, as a model for students to emulate.

8. Require students to estimate their answers. 9. Use strategy games to develop the problem-solving

process. 1 0. Have students make a flowchart of their own

problem-solving procedures.

conduct their annual draft of play- ers, teams in each city must have direct telephone lines to each of the other cities. How many direct telephone lines must be installed by the telephone company to accomplish this?

Three possible approaches are presented here.

First strategy

Some students may wish to act out this problem. For instance, they may set up 16 desks and tie pieces of string from one desk to another, counting them as they go (experimentation).

Second strategy

Others may examine the problem starting with a smaller number of cities (reduction), keeping track of their data (organized listing), and looking for some pattern that appears (pattern recognition). Using this approach (fig. 3), we begin with two cities, then consider three cities, then four cities, and so on. The data collected can be recorded in a table (see table 1).

Table 1 Data on the Number of Direct Lines

Number of cities 12 3 4 5 ... 16 Number of lines 0 1 3 6 10 ... ?

Some preservice teachers will actu- ally continue the table for 6, 7, 8, . . . , 16 cities. Others may be able to work with the pattern in the second row, namely, an increase of one between the first two terms; an increase of two between the next two terms; an increase of three between the next two terms, and so on, until the se- quence is recognized or the general term is found.

n(n - 1) 2

Students with a more sophisticated mathematics background may recognize this as the method of finite differences.

Third strategy

Other preservice teachers may use logical deduction. Since each city is connected to every other city, there will be 16 times 15 connections; but since A to В is the same as В to A, there will be one-half as many connections needed, resulting in

(16) (15) 2

Thus, 120 telephone lines needed. Some people may recognize this

problem as another formulation of the well-known ''handshake" problem, and may know how to solve it right

away. For these people, this is not a "problem," but merely a "question."

Notice that this problem was solved in several ways with a variety of strategies. Experience shows that no one strategy is any more important or val- uable than any other. Pattern recognition, however, seems to be called upon in conjunction with many of the other strategies.

We have illustrated briefly a tech- nique that we use in our preservice program to help our students become problem solvers. Now let us look at some suggestions that will help these preservice teachers teach problem solving to their students.

Teaching Problem Solving We still believe that the best method for teaching problem solving is for the teacher to be a good model, but there are specific things that teachers should do in the classroom. Ten suggestions for teaching problem solving are listed in figure 4. Let's take a closer look at "helping students to develop techniques to read problems analytically."

Being able to read the problem analytically is crucial to successful problem solving. After all, if students can- not interpret what is being asked or what information is supplied, then

44 Arithmetic Teacher



they can hardly be expected to solve the problem. Many teachers assume that if students can read the words, they understand the context and can interpret the information. This is far from true.

What steps can classroom teachers take to help students in this vital area? Five are essential:

1. Have students restate the problem in their own words and discuss with the teacher and with the class what is taking place.

a. Can they "see" what is happen- ing? b. Have they discovered all of the "given"? с Have they been able to elimi- nate the extraneous data? d. Are they aware of the hidden data, such as different time zones, units of measure, and so on?

2. Discuss the meanings of words. a. Mathematical terms that might be completely unknown to some students. b. Words with multiple meanings. (Many words assume a meaning in a mathematical context that is different from everday usage; for instance, prime, count, root, vol- ume, chord, and so on.

3. Have the students provide appro- priate questions for stories that you present. 4. Have the students provide stories for questions that you present. 5. Provide problems for which the student must determine whether there is an excess of, deficiency in, or just enough information. In the first two

cases, they should determine what is excess or supply what is deficient.

With regard to number two in figure 4, providing an ample supply of challenging problems for problem solving, we suggest the following:

Get a supply of 6-by-9 file cards in a variety of colors, on which you will write the problems you collect. De- cide upon your own color coding system. (For example, white cards might contain the simplest problems, blue the next simplest, and so on, with green cards having the most difficult problems. When you find or create problems that you feel would be good vehicles for problem-solving experiences for your students, put them on these cards. On the backs of the cards you can list the strategy or strategies that the problem suggests.

Problems can be found in a multitude of places. The Mathematics Teacher and the Arithmetic Teacher are valu- able sources, as are other journals. The IDEAS section of the AT is another excellent source, as are confer- ences at which talks on problem solving are given. Finally, go "back to basics" - very far back! Examine old mathematics text books, some over 100 years old. Many of the problems in these books are excellent vehicles for problem solving.

Whatever the source of the problems, in all cases preservice teachers should have an opportunity to actual- ly solve these problems and to identi- fy the strategy or strategies they used in the process. This not only provides them with new problems for their own problem file, but also gives them addi-

tional practice in becoming problem solvers.

In this brief paper we have de- scribed what can be done to help preservice teachers become good classroom teachers of problem solving, w

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