A look at problem solving in elementary school mathematicsAuthor(s): KATHRYN V. HERLIHYSource: The Arithmetic Teacher, Vol. 11, No. 5 (May 1964), pp. 308-311Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41184966 .

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A look at problem solving in elementary school mathematics

KATHRYN V. HERLIHY Hayward Unified School District, Hayward, California Mrs. Herlihy is a third-grade teacher at the H Merest School in Hayward.

JL he development of problem-solving ability may well be the main objective of education. One can think of all knowledge as an integrated unit, like a great ball with its components interlaced, bearing relationships waiting to be discovered. How a child goes about discovering these relationships and how effectively he ap- plies his discoveries in unfamiliar situa- tions will determine his problem-solving ability. Therefore, a vital role of the teacher of mathematics is helping the child explore life situations which contain mathematical relationships. Dutton and Adams, referring to problem solving, propose: "It may be that the next area of emphasis in the teaching of arithmetic will be in this field, which would be a logical consequence of the emphasis on drill, incidental learning, and the meaning theory.7'1

The teacher whose aim is to increase the chilďs competence in independent thinking will guide him toward learning the skills, concepts, and processes neces- sary for solving problems. First, let us consider what constitutes a problem and what are its relationships to learning. Al- though human behavior is largely deter- mined by custom and habit, there is an area in which deliberate choice-making takes place. It is this area of the unfa- miliar situation, incompletely sensed, im- properly or inadequately grasped, where- in the "problem" lies. Uncertainty is not the only characteristic of the situation we

1 Wilbur H. Dutton, and L. J. Adams, Arithmetic for Teach- ers (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1961), p. 43.

shall call a "problem." The unknown must be accepted as a question he desires to resolve before there is a problem for an individual. If the pupil produces an ap- propriate response from habit, or if he sees no relationships in the conditions, there is no problem for him.

If self-involvement characterizes a child's acceptance of a problem, it follows that problems are not limited to "word exercises" in texts, but could include ab- stract arithmetic sentences. Thus, 5X4 = ? could represent a question in- volving doubt and engage the pupil's de- sire to resolve it. This abstract statement could be as truly a problem as "The five boys at this table will need four sheets of paper each for their spelling booklets. How many sheets of paper will be needed in all?" It is interesting that studies to assess the value of the form and nature of problems reveal significant but incon- clusive and sometimes contradictory re- sults. Evaluation of the findings is com- plicated by such factors as the child's ex- perience, intelligence, verbal and number abilities, motivation, and goals. Experi- ments to determine the nature of prob- lems, as contrasted to the processes re- quired to solve them, have shown that there is little evidence that problem- solving success is enhanced by greater concreteness. The degree of familiarity in the situation does, however, appear to be a factor in problem-solving success.

Although there is an indication of ad- vantage in using the familiar context, there may be more of a predisposition to

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view the problem in a given way, pro- ducing a "set" which could prevent in- novational thinking. We can think of the process of problem solving as moving from a given situation to a desired situation (the child's goal). Attempting to resolve the difference between where he is and where he wants to go produces psycho- logical tension which evokes, directs, and sustains his behavior to reduce this ten- sion. A consciously held and clearly de- fined goal helps the child select and or- ganize behavior so that he may feel the satisfaction of attaining the goal and eliminating the tension.2 In the process of moving toward his goal, the child en- counters the conditions of the problem which block the process. Habitual re- sponses are not sufficient for removing the block and the child must deliberate; choose among alternatives; and advance, test, and verify provisional hypotheses. An understanding of the psychology of the process of problem solving should guide the teacher in the selection of appropriate examples employing various approaches.

There are some emotional conflicts which might prevent effective learning, and treatment of these difficulties may be a powerful way to improve a child's problem-solving difficulty. Fear of failure, disapproval, or ridicule has a paralyzing effect on a child's innovational develop- ment. Feelings of inadequacy result in rigidity of approach. Even though he has not achieved success, the child continues to try the same old process because "he is afraid of the naked feeling that he will have if he drops it and stands for a mo- ment without a hypothesis."3

Closely related to rigidity is the tend- ency to dismiss a problem as insoluble. Lack of confidence in the fidelity of arith- metic may cause the child to "give up"

2 Kenneth B. Henderson and Robert E. Pingry, "Problem- solving in Mathematics," The Learning of Mathematics, Twenty-first Yearbook (Washington, D.C.: The National Council of Teachers of Mathematics, 1955). p. 233

1 Lee J. Cronbach, "The Meanings of Problems," Univer- sity of Chicago Supplementary Educational Monographs, No. 66, October, 1948, p. 32.

mentally, even though he continues to give the appearance of exploring addi- tional possibilities. The teacher should try to determine the point at which the pupil's thinking, for instance in terms of number, is characterized by a sense of security. From this point the child may be guided into new experiences using this knowledge.

When the pupil understands the con- cepts and principles involved, the teacher can promote problem-solving ability by guiding him to determine what is given, devise a plan, execute it, and check the solution. These four aspects are essential to successful problem solving; however, they are a gross oversimplification and cannot be identified within precise limits, nor necessarily in that order. G. Polya has suggested that the teacher proceed by asking a series of questions which will lead the pupil through these steps to a satisfaction of his goal.4 Since the child first observes and then imitates, the teacher can ask himself the questions so that the pupil may discover the right use of these questions. This discovery will be more important to the pupil than the knowledge of any particular mathematical fact. "What do I want to find? What is given? How can I use this information? Is there anything else I can use?" The teacher should begin with questions that are simple, short, natural, and of a gen- eral nature so that the child may see the properties and relationships in the ex- ample. If he sees these, he can use them as a reference in solving future problems. If there is no response, the questions may gradually become more specific until there is a level at which the child feels confident. The problem situation should be presented in such a way that the child may discover a pattern which can be dem- onstrated as consistent. After some ex- perience with similar problems, the pupil may see the underlying general ideas and a symbolism for the emerging pattern

< G. Polya, How to Solve It (Garden City, New York: Doubleday & Company, Inc., 1957).

May 1964 309

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may seem almost a necessity.5 By vary- ing the data, the teacher can enlarge the pupil's understanding. She might ask, "Will the product be larger or smaller if the factors are increased?" or "Suppose Billy was 10 years old instead of 5."

To understand a written problem, the child must be able to read not only ac- curately, but comprehensively. He must understand what is given and what is re- quired. This understanding includes skills and meanings which are developed before the pupil encounters the problem. A child may consider, "John paid 2 cents for candy and 3 cents for gum. How much did he spend?" only after he knows the meaning of 2 cents and 3 cents and under- stands the arithmetic statement 2 plus 3. It is true that material is not enough to make a coat, but a coat cannot be made without it. Before he can formulate his procedure, the child must learn to deter- mine the materials available for solving the problem. These may be present in the situation or available through past ex- perience or knowledge.6

To approach the main achievement in problem solving, devising a plan, the pupil must not only understand meanings and identify available materials, but also discern the relationships between them. Here we come to the point of nonhabitual, choice-making behavior. The teacher may guide at this difficult and critical phase by asking questions which will help the child recall a similar problem which is related closely enough that the child may use it to discover a workable pattern. The question may be restated, varied, and modified. To conceive an idea which will lead to the solution, the pupil will need formerly acquired knowledge, good mental habits, concentration upon his purpose, and perseverance. It will be helpful for the pupil to learn to look for the conditions in the problem rather than

8 George Baird, "Children Discover Own Math," Insight (Chicago: Science Research Associates, Inc.). II. 1 (1962).

8 Frederick J. McDonald, Educational Psychology (San Francisco: Wadsworth Publishing Company, Inc., 1959).

the "answer." In learning that 2 plus 3 equals 5, pupils may be encouraged to make up their own problems by telling a story, writing it, drawing illustrations, or demonstrating it with manipulative ma- terials.

Once the plan is devised, the teacher needs patience while the pupil proceeds with computational skills. If the pupil conceived the plan rather than received it, he will not lose the idea easily, but he must check each step, not only for arith- metic accuracy, but also with relation- ship to the framework of his plan. When the pupil has arrived at a result, the process of checking the steps and solution is especially valuable to develop insight into the solution. Can the result be de- rived differently? Can either the result or method be used for some other problem? Exploring these possibilities will increase the pupiPs confidence in the skills and facts he has used to solve the problem.

Perhaps one of the most significant ways of looking at problem solving in the elementary school is persistent recon- sidering and reexamining of the material presented. Although each subsequent presentation is related to prior learning, it expands in increasingly broader applica- tions and deeper insights. For this reason, it is vital for the teacher to devote early attention to fostering sound problem- solving approaches and techniques which will neither stifle innovational thinking nor have to be unlearned in a later phase of education.

Developing problem-solving ability in mathematics is broader than a compari- son of conventional exercises with some imaginative form of statement. It is a rich opportunity to prepare children for effective living in our quantitative so- ciety.

Bibliography

Baird, George. "Children Discover Own Math," Insight (Chicago: Science Re- search Associates, Inc.), II, 1 (1962).

Cronbach, Lee J. "The Meanings of

310 The Arithmetic Teacher

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Problems/ ' University of Chicago Sup- plementary Educational Monographs, No. 66, October, 1948.

Dutton, Wilbur H., and Adams, J. J. Arithmetic for Teachers (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1961).

Henderson, Kenneth В., and Pingry, Robert E. "Problem-solving in Mathe- matics, " The Learning of Mathematics, Twenty-first Yearbook (Washington, D. C: The National Council of Teachers of Mathematics, 19*55).

McDonald, Frederick J. Educational Psychology (San Francisco: Wadsworth Publishing Co., Inc., 1959).

Polya, George. How to Solve It (Garden City: Doubleday & Company, Inc., 1957).

Wheat, Harry G., "The Nature and Sequences of Learning Activities in Arithmetic," The Fiftieth Yearbook, the National Society for the Study of Education (Chicago: University of Chicago Press, 1951).

Request for information One phase of a U.S. Office of Education study presently under way at the Ohio State University involves abstracting re- ports of comparative experiments in arithmetic.

The abstracts are being compiled to facilitate a study of factors which are common to several experiments - for in- stance, an experiment in which the same teacher taught both a control and an ex- perimental section, or one in which a visit-

ing college person taught one class while the regular teacher taught the other class.

If you know of any comparative ex- periments in arithmetic, either published or unpublished, successful or disappoint- ing, please send the name of the experi- menter and information on how to locate the report to Lewis H. Coon, The Haw- thorne Effect Project, 202 Townshend Hall, 1885 Neil Avenue, Columbus, Ohio. 43210

As we read (Continued from page 289)

these people from building the walls against learning. Of course, something of an inventory listing the strengths and weaknesses of the students, as suggested by Redbirďs preparation for individual- ized instruction, would help to insure sympathetic treatment for these children.

Young's article treats a much dif- ferent topic. At least part of her ac- count of the growth of number ideas is the birthright of every intelligent student. Her bibliography will be useful to students and teachers alike in exploring the history of arithmetic.

May 1964 311

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