Issues in Teaching Mathematical Problem Solvingin the Elementary Grades

Frank K. Lester, Jr.Mathematics Education Department

Indiana UniversityBloomington, Indiana 47405

Among the most important goals of school mathematics is to developin each student an ability to solve problems. To a certain extent, this goalhas long been recognized by secondary teachers and textbook authors. Inrecent years, more and more emphasis has been placed on problem solv-ing in the elementary mathematics curriculum as well. A cursory look atthe scope and sequence charts of the most popular textbook series, syl-labi, and recent mathematics education conference reports supports thisclaim. I wish to address some of the important issues related to teachingproblem solving in the elementary grades with particular emphasis on theintermediate grades (4-6). The paper is divided into two sections. Partone presents my point of view about teaching problem solving at the ele-mentary level. The second section considers some specific questionswhich are often raised by teachers who are considering ways to incor-porate problem solving into the mathematics program.

A Point of View About Problem Solving in the Elementary MathematicsCurriculum

The current concern about and interest in teaching elementary studentshow to solve problems raises a number of questions which should be con-sidered by all elementary teachers. Before raising these questions andtrying to answer them, it is appropriate that a definition of problem solv-ing be given in order to insure that you know what I mean when I refer tothe complex and often nebulous term "problem solving." For my pur-pose any mention of problem solving refers to the process of co-ordinat-ing previous experiences, knowledge, and intuition in an effort to deter-mine an outcome of a situation for which a procedure for determiningthe outcome is not known. Thus, finding a correct answer to thequestion, "What is 2346 divided by 9?", does not involve problem solv-ing for a student who understands and has skill with a long division algo-rithm. This is so because skill with a long division algorithm would pro-vide the student with ^procedure for finding a correct answer. However,for the student with only a vague understanding of division or with no fa-miliarity with any division algorithm, determining a correct answer tothis question may involve problem solving of a complex nature. Thisdefinition of problem solving is, then, a rather restricted one which ex-cludes the direct translation of words to mathematical equations or the

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simple application of a previously learned skill or algorithm. Nowconsider the questions alluded to at the beginning of this paragraph.

Question 7. Why is it so important/or children to become better prob-lem solvers?

A critical consideration in determining an answer to this question is thepace of change in our society. We are faced today with the prospect thatthe rate of change will accelerate so rapidly that every part of life will beaffected. Such rapid change makes it extremely difficult to prepare forthe future since there is no precise way to predict what scientific and tech-nological discoveries will be made. At the same time, there is no suffi-cient way to learn now everything that will need to be known in the fu-ture about a subject like mathematics. Today mathematical activity insuch areas as engineering, the physical sciences, the medical sciences, thesocial and behavioral sciences, and business are so extensive that the needfor a mathematically literate citizenry is highly important. The schoolmathematics curriculum will have to focus on ways to equip our studentswith an ability to learn things that no one yet knows. Such a change infocus implies a different role for mathematics teachers. Traditionally,mathematics teachers have concentrated their efforts on helping theirstudents acquire computational skills. The fact is, however, that we nolonger need the skill to perform very complicated calculations; machinescan do these much better (it should be pointed out that it is of course im-portant for students to gain computational proficiency�only the over-emphasis on this aspect of learning mathematics must be changed).There is an ever increasing demand for people who can analyze a prob-lem and devise a means of solving it. Thus, any mathematics curriculumof the future which does not give direct, serious attention to developingproblem solving ability in students will not be satisfactory.

Question 2. Can students be taught to be better problem solvers?

The answer is an unqualified "yes." What is less clear is the identifica-tion of the best ways to teach problem solving. A reason why the most ef-fective instructional approaches have not been determined is that thevery nature of problem solving is so complex it is difficult to knowexactly what causes changes in problem solving performances. For ex-ample, factors such as interest level of a problem, student motivation,readability of a problem statement, mathematical content involved, andlogical reasoning ability required are but a few of the many factors whichinfluence problem solving success.

Question 3. As a teacher I can’t wait until research finds answers/orall of the questions about problem solving before takingsome action in the classroom. Are there any guidelines Ican follow to help me plan my own problem solving expe-riences for my students?

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I agree wholeheartedly that classroom teachers can’t wait for researchto provide definitive answers before taking some action. Steps should betaken now to help students become better problem solvers. We have anobligation to do our best to meet the needs of our students in this area.Here are some tips that have proven to be useful to me in planning andimplementing problem solving instruction.

Tip 1: Students may have difficulty with a story problem because they can’t read orunderstand it. If this happens, read it with them and then have them restate theproblem in their own words.

Tip 2: Students often interpret the meaning of problems differently from the intendedmeaning. Be sure to find out what they are thinking about after they read orhear a problem.

Tip 3: Pictures and diagrams which accompany a problem are often helpful, but theyare sometimes a source of confusion. Be sure to have students talk about thepicture or diagram which is associated with a problem.

Tip 4: Try to get the students to estimate about what the answer might be before try-ing to find an "exact" solution.

Tip 5: Have students act out the situation posed by a problem. This is an especiallyuseful technique with primary level children.

Tip 6: Sometimes students seem to understand a problem but have difficulty in get-ting started on a solution. When this happens it is often helpful to simplify theproblem by using smaller numbers or, if the problem has more than one part,do the problem one part at a time.

Tip 7: Encourage students to organize their work by making tables, keeping a recordof their work, looking for patterns, and similar techniques.

Tip 8: Students should be encouraged to "play their hunches." Guess and test strate-gies are often quite helpful.

Tip 9: A problem of the week (or day, perhaps) format using problems with a highlevel of interest for the children is a sensible approach.

Tip 10: Most importantly, students cannot develop any skill in problem solving unlessthey become actively engaged in solving problems. One good way to get themactively involved in problem solving is to pose problems which have relevanceto their lives. Problems related to interest rates, satellite launchings, and thelike are real-world situations but often are of little interest to students becausesuch situations are so far removed from their own personal experiences. En-courage students to being some of their everyday problems into the classroom.You and they may be surprised at how useful mathematics can be in solvingmany of these types of problems.

These ten tips are based on my own experience and the thoughtful sug-gestions of one of mathematics education’s most prominent educators,William A. Brownell. I urge anyone who is seriously interested in learn-ing more about problem solving to read BrownelPs ideas on this topic(Brownell, 1942). It is particularly interesting to note that Brownell’sarticle was written 39 years ago.

Specific Questions Related to Making Problem Solving a Real Part of theMathematics Program

7. (a) What types of problems should be used?(b) What is the relative emphasis that should be given to each type of

problem?

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Professor John LeBlanc, has written an excellent article about teach-ing problem solving (LeBlanc, 1977). In his article, he distinguishes be-tween two types of problems, typical textbook problems and "process"problems. Both types of problems should be considered. As LeBlanc says"the basic purpose of the typical textbook problem is to reinforce chil-dren’s understanding of a concept or to use a skill learned earlier by pre-senting a ’real-world* situation that embodies that concept or skill (p.17)." On the other hand, a process problem "lends itself to exemplifyingthe procedures inherent in problem solving (p. 18)". Process problemsare richer in their potential for developing general strategies which can beused to solve a variety of other types of problems. Also, they usually re-quire more time. Consequently, fewer process problems can be solved ina given amount of time than standard textbook problems. Both types areimportant depending on the purpose. When problem solving processesare the focus, process problems should be used; otherwise the other typecan be employed.

2. How much time should students spend solving problems each week?Each day?

Before answering this question I must make some assumptions:

a) "Time spent solving problems" means teacher-specified time devoted exclusively toproblem solving.

b) "Problems" are process-oriented problems (i.e., problems which require the use ofvarious thought processes in addition to, or instead of, simply using an algorithm ortranslating words to an equation).

c) Students are intermediate grade children (4-6).

In the beginning of the school year I think 30-45 minutes twice perweek is ample time. As the year progresses and students gain confidencein their ability perhaps more time will be appropriate. In any event, anhour per week is the absolute minimum (there is no maximum in myview). In addition, students should be encouraged to make use of un-structured time to work on problems. If the problems are interestingenough not much encouragement will be needed. It is not necessary tohave students work at solving problems every day, although it is desira-ble.

3. What grouping procedures should be used (eg., small groups, individ-uals) and for what purpose(s) should each be used?

A variety of grouping procedures should be employed. There is a placein problem solving instruction for individual, small group (3-4 students),and whole class work; it depends on the purposes the teacher has in mindand the nature of the problem. For example, some problems almost de-mand silent work by students at their desks while others may be solvedbest by sharing ideas with classmates. Also, the teacher may want to fos-

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ter reflective thinking in some of the more impulsive children therebymaking individual work appropriate. At other times the teacher maysense a need to discuss a problem with the entire class. All three of thesegroupings, individual, small group, and whole class, are appropriate un-der certain conditions. My experience suggests that students will profitmost from an organization which allows them to talk among themselveswhenever necessary and appropriate.

4. To what extent should the standard textbook story problems used berelated to the computational skills recently developed by the teacher?

Let me begin by pointing out that the fundamental concepts related toan operation (addition, subtraction, multiplication, and division) withwhole numbers, integers, fractions, or decimals should arise from real-world situations (i.e., situations which are real to the students). Thus, asfar as possible the teacher should motivate the need for and value of spe-cific computational skills through the use of real-world problems. A sixstep instructional process is involved (see diagram):

(1) Real-world situations to indicate the need for the computational skill;(2) introduction to the operation as a part of mathematics;(3) learn basic facts associated with the operation;(4) apply newly learned skills to solve problems;(5) learn algorithms for the operation;(6) apply algorithmic skills to solve problems.

Real-WorldSituations -^

MathematicalModel of theSituation

What all of this means is that the acquisition of computational skills isvalueless unless those skills can be used to solve problems. That is, itwould be ludicrous for students to know the "times tables throughtwelve" but not know when, or if, to multiply when multiplication mightbe useful to solve a problem. Textbook word problems are important fortwo reasons.

1) They provide students with necessary practice in determining what operation to useand when to use it.

2) They help teachers determine if their students have a firm and meaningful under-standing of the mathematical operations involved.

On the other hand, I believe standard textbook word problems do verylittle, if anything, to enhance the development of good problem solvingbehavior. Thus, it seems there is a need for two types of word problems,standard textbook problems and "process" problems.

5. What, if any, evaluation procedures should be used?

This is the most difficult question of all. I think standardized problemsolving tests (eg., Iowa Test of Basic Skills, Stanford Achievement Test)

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do measure certain aspects of problem solving ability but they are very,very limited in value. Such tests may tell you if a student is a good prob-lem solver but they don’t provide any insights into the nature of the proc-esses students use and they don’t give you any clue about a student’spotential. Teacher-designed evaluation is probably best. Teacher-de-signed evaluation can have both formal and informal aspects. Formally,a teacher can look for the acquisition of certain "tool skills." Tool-skillsare problem solving skills which help students organize their work, sortrelevant from irrelevant information, and so on. Examples of tool-skillsare making tables, drawing appropriate diagrams, and using estimation.Evaluation of the attainment of these skills is accomplished best by look-ing at how the student attempted to solve a problem, not just looking tosee if the answer is correct. Informally, the teacher should observe stu-dents as they work on problems. How students get started on a problemis often a good indicator of how well they understand the problem andwhether or not they have any skills or strategies at hand. Above all, don’tbe content to check right and wrong answers only. Oftentimes the bestproblem solvers make minor errors and poor problem solvers get luckywith a wild guess. If the development of sound problem solving processesis your goal, you must look for signs that they are being developed.

Closing Comments

Teaching children to be better problem solvers is not only the most im-portant role of the mathematics teachers, it is also the most challengingand exciting. In the preceding pages, I have tried to provide perspectiveregarding the nature of mathematical problem solving in the elementarygrades and to answer some of the questions which teachers often ask. Ihope some of these suggestions and ideas prove to be helpful to you inyour classrooms.

REFERENCES

BROWNELL, W. A. Problem Solving. In The Psychology of Learning. Forty-first Yearbookof the National Society for the Study of Education, part 2, pp. 415-443. Chicago: TheSociety, 1942.

LEBLANC, J. F. You can teach problem solving. The Arithmetic Teacher, vol. 25(2), No-vember, 1977, pp. 16-20.

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