A Comparison of a Structured-Equation Approachto Problem Solving With a Traditional Approach

Harold H. Lerch*Southern Illinois University^ CarbondalQ^ Illinois

Helen HamiltonfHerrin Elementary Schools, Herrin, Illinois

THE PROBLEM

Developing pupils^ ability to solve word problems continues to be amajor general objective of elementary mathematics programs. Sincepupils in out-of-school situations will rarely need to compute arithme-tic examples that are not related to problem solving situations, theimportance of developing problem solving abilities can hardly beover-emphasized. Elementary teachers report that pupils are oftenable to perform at high levels of accuracy on example type assign-ments, but are unable to do so when similar situations are stated inproblem solving situations. Results of standardized measuring in-struments confirm these reports.Authors of texts dealing with methods of teaching arithmetic tend

to agree that pupil difficulties in problem solving stem from inabilitiesto see and to understand the relationships involved in the problemsolving situation and from inabilities to perform the necessary com-putations. Several have tried to isolate factors of difficulty. Marks,Purdy, and Kinney (8: 311-312) have put pupil difficulties into fourcategories: (a) vocabulary difficulties; (b) failure to see the relation-ships between the different elements of a situation; (c) improper useof concepts and skills to interpret a problem; and, (d) incorrect com-putation. Banks (1:367) suggests three sources of trouble: (a) lack ofcommand of the fundamental processes; (b) reading difficulties; and,(c) use of poor techniques.Techniques for improving the problem solving abilities of pupils

suggested by authors of methods textbooks and most of the reportedresearch indicate a general dissatisfaction with the traditional ap-proach to problem solving. In this approach the pupil is to be taughtto see the relationship between the known facts in a problem situationand what he wants to find. He is encouraged to ask himself such ques-tions as: "What is given?" "What question is asked?" "What processshould be used?" Studies which have compared a traditional oranalytical approach to problem solving with some experimentalapproach generally indicate favorable gains by the experimentalgroup or no significant differences between the groups.

* Dr. Lerch is Associate Professor of Education and Mathematics at Southern Illinois University.

+ Miss Hamilton is fifth grade teacher at Northside School in Herrin, Illinois.

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242 School Science and Mathematics

In a study in which the experimental group utilized vocabularyexercises, discussion of the total problem situation, diagrams, estimat-ing and computing, and the control group followed the traditionalprocedure, Faulk and Landry (4) found the experimental procedureslightly more effective. In a very similar study, Monroe and Engle-hart (9) found no significant differences, dark and Vincent (3) con-cluded that pupils taught by an approach in which pupils began withthe expected answer and proceeded to graphically determine thecomputational method showed more improvement in problem solvingthan those studying a traditional approach. Greene (5) reports thatpupils who drill on the selection of processes necessary to solve prob-lems gain significantly in problem solving ability over pupils who donot drill on the selection of process. Thiele (10) found that a methodin which pupils compared their solutions with a model was more effec-tive in teaching problem solving than a method emphasizing cuewords and the traditional method. Washborn (11) found no significantdifferences in problem solving achievement between a group whichdrilled on new processes before using them in problem situations and agroup which was introduced to new processes with problems. A studyby Hanna (6) indicates that pupils who pictured their work graphi-cally and pupils who used individual approaches were significantlybetter at problem solving than those wdio used a traditional approach.However, Leno (7) found no significant differences in gains in problemsolving between pupils who used individual approaches and pupilswho followed a standard method. Burch (2) reports that pupilsscored higher on a test of problem solving when they were not re-quired to follow certain traditional steps than they did when theywere required to follow a step-by-step procedure. Methods of im-proving pupils’ problem solving abilities suggested by authors and byresearch, although quite varied, generally do not differ greatly fromthe traditional approach and are often pointed at certain types ofproblems.

Pupil difficulties in problem solving as cited by authorities can beplaced into two broad categories: (1) The inability to program or todetermine the procedure to be followed in solving the problems; and,(2) The inability to process or to perform the computations necessaryto solve the problem. It would seem that a procedure which is directedtoward improving pupils^ abilities in the first category and whichwould be useful in all types of problem situations would be moreeffective in helping pupils learn to solve word problems.One of the characteristics of modern programs in elementary

mathematics is the early introduction, development, and use of thenumber sentence or equation. One of the suggested purposes for

Problem Solving 243

emphasizing the study and use of number sentences is the possibilitythat the use of equations will aid in developing pupils’ abilities tosolve problems. The assumptions underlying this hypothesis are:(1) That in order to structure the number sentence pertaining to aproblem, pupils must see the problem in its entirety or the wholequantitative situation; and, (2) That writing the problem in a sym-bolic form or equation will help the pupil in programming or deter-mining the computations to be performed.

PURPOSE OF THE STUDYThe general purpose of this study was to compare the growth in

problem solving abilities of fifth grade pupils who studied a struc-tured-equation approach to problem solving with the growth inproblem solving abilities of fifth grade pupils who studied a tradi-tional approach to problem solving. Hypotheses for this study may bestated as:

1. Fifth grade pupils who have been instructed in the use of astructured equation approach to problem solving will be betterable to program problem solving situations than fifth gradepupils who studied a traditional approach to problem solving.

2. Fifth grade pupils who have been instructed in the use of astructured equation approach to problem solving will grow morein complete problem solving ability than fifth grade pupils whostudied a traditional approach to problem solving.

PROCEDUREBoth the experimental and control groups were fifth grade classes in

Herrin, Illinois. The experimental class was composed of 28 pupils,13 boys and 15 girls who ranged in age from ten to twelve years. Thecontrol class was composed of 17 pupils, 9 boys and 8 girls who rangedin age from ten to thirteen years. As illustrated by Table I, bothgroups varied widely in arithmetic ability, with the control groupaveraging somewhat higher.

TABLE 1. STUDENT GRADE PLACEMENT ON THE ARITHMETIC SECTIONOF THE IOWA TESTS OF BASIC SKILLS

Experimental Group

Control Group

Arithmetic Section

Range G. P.

3.6-7.2

4.2-6.9

Mean G. P.

5.3

5.7

Problem Solving Section

Range G. P.

3.3-7.8

4.8-7.8

Mean G. P.

5.2

5.9

244 School Science and Mathematics

Both classes were using the same textbook series, and no previouswork had been introduced in using equations to solve problems. How-ever, some work in both classes had been done with unknowns in theform 6XiV=54. At the beginning of the five month experimentalperiod, pupils in the experimental group reviewed and were instructedon the different ways of symbolically indicating arithmetical opera-tions to be performed. During the course of the experiment only thoseproblems in the textbook were used. Thus, the number of problemsundertaken by the experimental group was similar to the numberundertaken by the control group. When the experimental group en-countered word problems, emphasis was placed upon seeing the totalquantitative situation and writing a number sentence which describedthat situation. At the beginning of the experimental period, writingthe number sentences for problems was a group or class effort. Asabilities developed, individuals were encouraged to structure theirown equations. Emphasis was placed upon the idea that for mostproblem situations there is more than one correct way to structure anequation. Problem situations involving all operations with wholenumbers and all except division with fractions were utilized. Manyproblems required more than one operational step. Using ratios tosolve certain problems was a new procedure for the experimentalgroup.Both the experimental and control groups were taught by their

regular classroom teachers. One of the writers of this report taughtthe experimental group. During the experimental period, the controlgroup continued to receive instruction in a more traditional approachto problem solving as advocated by the textbook series. When prob-lems became a part of the class work, pupils in the control class wereasked to solve the problems by using an analytic approach, askingthemselves: "What is asked?" "What is given?" "What processshould be used?" Other phases of the arithmetic program were essen-tially the same for both groups because the material covered was thatpresented by the textbook.At the beginning of the experimental period, a pre-test of problem

solving ability was administered to both groups. A second form of thetest was given after the experimental period. These instruments weredeveloped by collecting word problems at each of the grade levelsthree through seven from the texts of three recent arithmetic series.Of the twenty-six problems selected for each test form, three werefrom third grade, five were from fourth grade, ten were from fifthgrade, five were from sixth grade, and three were from seventh gradetexts. Problems on the test were arranged in an easy to more difficultorder, and each item on one form of the test corresponded in difficulty

Problem Solving 245

level and process involved to the same-numbered item on the otherform.

Pupils were scored on their ability to program the problems, and ontheir ability to process the problems. Individual growth scores onprogramming and processing were computed by comparing scores onthe pre- and post-tests. Mean growth scores for the two groups werethen determined. Variances of the growth scores of the two groupswere compared with the F-test in order to determine whether or notthe variances could be presumed equal and so that appropriate /-tests

could be used in the comparisons of means. In all comparisons, the.05 level of significance was used.

FINDINGS AND CONCLUSIONSMean scores for both groups in regard to abilities to program and

abilities to process problem solving situations are reported in Table II.

TABLE II. MEAN SCORES ON PROBLEM SOLVING ABILITIES

ExperimentalGroup

Control Group

Programming

Pre-test

14.36

15.47

Post-test

18.57

17.35

Growth

4.21

1.88

Processing

Pre-test

10.71

10.53

Post-test

14.21

14.47

Growth

3.50

3.94

Variances of growth scores in programming could be presumedequal (jF==1.14) and a /-test which assumes equal variances for un-equal sized groups was used to compare means. A /-value of 2.84 wasfound. Reference to a table of /-values indicates a significant differ-ence in mean growth scores in programming beyond the .05 level ofsignificance. Fifth grade pupils who studied a structured equationapproach to problem solving were better able to program problemsolving situations than fifth grade pupils who studied a traditionalapproach.The variances of growth scores in processing could also be presumed

equal (F= 1.18) and the same /-test was used to compare the growthmeans. Reference of the computed /-value of �.488 to the tableindicated no significant difference between mean growth scores inprocessing. Fifth grade pupils who studied a structured-equationapproach to problem solving did not grow more in processing abilitythan fifth grade pupils who studied a traditional approach to problemsolving.

If complete problem solving is viewed as the ability to arrive at

246 School Science and Mathematics

correct solutions to problem solving situations, we cannot concludethat the experimental group grew more in complete problem solvingability than the control group. However, we may conclude that theexperimental group did grow more in the primary aspect of problemsolving: determining the procedure to be followed in solving problems.Table II indicates that some pupils programmed problems correctlyand made mistakes in processing. Very few pupils programmed incor-rectly and processed correctly. Obviously, computational skills con-tinue to be of major importance in problem solving and teachersshould continue to stress the development of these skills. Althoughthis study involved fifth grade pupils, there are indications that astructured-equation approach to problem solving might be moreeffective if introduced at earlier grade levels. The method seems tohave sufficient value to merit incorporation into elementary mathe-matics programs.

REFERENCES1. BANKS, J. HOUSTON. Learning and Teaching Arithmetic. Boston: Allyn and

Bacon, Inc., 1959.2. BURCH, ROBERT L. "Formal Analysis as a Problem-Solving Procedure,"

Journal of Education, CXXXVI (November, 1953), 44-47, 64.3. CLARK, JOHN R. AND VINCENT, E. LEONA. "A Comparison of Two Methods

of Arithmetic Problem Analysis," The Mathematics Teacher, XVIII (April,1925), 226-233.

4. FAULK, CHARLES J. AND LANDRY, THOMAS R. "An Approach to Problem-Solving," The Arithmetic Teacher, VIII (April, 1961), 157-160.

5. GREENE, HARRY A. "Directed Drill in the Comprehension of Verbal Prob-lems in Arithmetic," Journal of Educational Research, XI (January, 1925),33-40.

6. HANNA, PAUL R. "Methods of Arithmetic Problem Solving," MathematicsTeacher, XXIII (November, 1930), 442-450.

7; LENO, RICHARD S. "Children’s Methods of Problem Solving in Arithmetic,"Dissertation Abstracts, XIX (April, 1959), 2549.

8. MARKS, JOHN L.; PURDY, C. RICHARD; AND KINNEY, LUCIEN B. TeachingArithmetic for Understanding. New York: McGraw-Hill Book Company, Inc.,1958.

9. MONROE, WALTER S. AND ENGLEHART, MAX D. "The Effectiveness ofSystematic Instruction in Reading Verbal Problems," Elementary SchoolJournal, XXXIII (January, 1933), 377-381.

10. THIELE, C. Louis. "Comparison of Three Instructional Methods in ProblemSolving," Research on the Foundations of American Education. Official Reportof 1939 meeting of A.E.R.A., 1939, 11-15.

11. WASHBURN, CARLETON W. "Comparison of Two Methods of Teaching Pupilsto Apply the Mechanics of Arithmetic to the Solution of Problems," Ele-mentary School Journal, XXVII (June, 1927), 758-767.

Plan now to take thefamily to French Lick

CASMT CONVENTION�1966