grade placement and meaningful learning

Grade Placement and Meaningful Learning

Percy KrichSchool of Education, Long Beach State College, Long Beach, California

Teaching arithmetic concepts sequentially has been the acceptedpractice for many years. Addition, subtraction, multiplication andthen division have been the basic concepts and they are usuallytaught in that order. During the last several decades teachers havebeen injecting meanings into the learning experience by demon-strating the interrelationship of the arithmetic processes. The appar-ent difficulty children had in learning particular concepts and proces-ses (as judged by test results) determined the grades where the con-cepts should be taught. Currently, efforts are being made to introduceconcepts at earlier places in the child’s school life than was previouslyattempted.Although many schools are scheduling the learning of basic arith-

metic concepts in earlier grades than previously, there has been littleshift in the upper elementary grades. Little consideration has beengiven to presenting concepts involving multiplication or division ofdecimals or fractions to children in grades below sixth. Multiplicationand division of fractions or decimals is still reserved for the sixthgrade on the basis that it would be too difficult for younger childrento learn and understand such material.

Periodically, experiments test children^ ability to learn materialgenerally considered too difficult for them. This report discusses anexperiment that was concerned with the concept of dividing a fractionby a fraction.

PROBLEM

The purpose of this study was to determine if the process of divid-ing fraction by a fraction, which is usually taught during the latterhalf of the sixth grade, can be taught effectively at the beginning ofthe sixth grade.

Consensus of experiments involving meaningful versus mechanical(rote) teaching methods concluded that, generally, meaningful teach-ing produces more effective results than rote. A program whereby theprocess of division of a fraction by a fraction could be taught mean-ingfully, was developed by the author to use experimentally in orderto determine the optimum grade for teaching division of a fraction bya fraction.

MEANINGFUL TEACHINGMeaningful teaching is that which provides the rationale for the

concepts taught, permitting and offering the opportunity for the

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132 School Science and Mathematics

learner to understand why processes work and what symbols repre-sent.

MECHANICAL TEACHINGMechanical teaching refers to the method which develops an under-

standing, in the child, of the concepts and rationale of the process andthe relationship of the processes to each other. Mechanical teachingrequires rote learning since it expects the learner to memorize rules forarithmetic processes and provides drill as the only experience forlearning to use the rule.

HYPOTHESISThis experiment set out to determine if children who were taught

the division of a fraction by a fraction process meaningfully couldlearn the process as well as children who were older but were taughtthe process by memorizing and applying the rule used for the solution.

Previous studies have indicated that when children are taughtmeaningfully they learn better; therefore it was assumed in thisstudy that the B6 (lower half of the sixth year) children might learnat least as well as A6 grade children.

PROCEDUREChildren for this study were selected from one B6 grade class and

one A6 grade class from each of three schools in the Los Angeles CitySchool District. The B6 group was designated as the experimentalgroup, while the A6 group was designated the control group. Thegroups were then matched by IQ scores and three IQ ranges wereestablished. These are presented in Table I.

TABLE I. MEANS op IQ SCORES WITHIN IQ GROUPS

IQ n Experimental SD Control SD

Low20100.54.499.73.99Mid.26110.35.1110.82.69High26125.78.7121.44.5

A self-tutoring program was designed for the experimental group.In this program mathematical meanings involved in division of frac-tions was presented. Division synbols were described and their vari-ous usages explained. Symbols representing quantities and portionsof numbers were explained with the use of concrete instructional ma-terials. The rationale of why the process of multiplication contributesto the solution of a division problem in fractions was developed.The control group was taught the rules and procedures involved in

Grade Placement and Learning 133

division of fractions by regular classroom teachers. Particularly, theywere taught to learn the rule "invert the divisor and multiple." Theteachers used the method recommended by assigned arithmetic text.Fraction kits were used to demonstrate division by fractions. Next,examples in abstract form were presented and the computation pro-cedures explained. The sequence of explanation was to, 1) changedivision sign to a multiplication sign, 2) change fraction divisor to aninverted position, and 3) multiply inverted fraction times the ori-ginal dividend. This procedure was later (second day) simplified andthe procedure "invert and multiply" was emphasized. Each day’slesson concluded with computation drill on the material covered.Division of a fraction by a fraction was taught on the fifth day ofworking with fraction divisors. At this time the full rule, "To dividea fraction by another fraction, invert the divisor and multiply," wastaught as the tool to be used to compute and solve examples of divid-ing fractions by fractions. Drill work was assigned to practice apply-ing the rule. Since the text indicated five teaching days should be usedfor teaching division of a fraction by a fraction only five days wereused by the experimental group.A pre-test in two parts was prepared. Part I included thirty-five

verbal problem-solving questions with multiple choice answers. Thistest was submitted to an A6 class other than any of the ones used inthe study and the results were used to test the reliability of the instru-ment. Since the test achieved an r of .88 it was considered reliable.Part II comprised thirty computational problems dealing with addii-tion, multiplication and division of fractions problems, from simpleto complex. These were similar to computational problems used instandard achievement tests and in the textbook. Only four of theproblems utilized addition or multiplication symbols.The post-test was also in two parts but Part II was changed. In the

post-test Part II was made more difficult. The addition and multipli-cation examples were eliminated and replaced by division of frac-tion problems. Fourteen examples on the post-test Part II were thesame as examples on the pre-test Part II but randomly placed indifferent positions. The rest of the examples tested more difficultconcepts than were actually taught in the experimental program.The pre-test was administered to both groups early in the semester

to insure that the control group had not been taught to divide a frac-tion by a fraction. After the pre-test the experimental group beganthe program which was presented in self-tutoring booklets the pres-ent author had provided for them. Near the end of the semester whenit was certain that the division of a fraction by a fraction concept hadbeen taught to the control group the post-test was administered toboth groups. Differences in the results on these tests between groups


were measured. The significance of the differences of the means wastested with the test. Means of scores of the total population of bothgroups was compared and then between sub-groups within the popu-lation as determined by the IQ divisions.

ANALYSIS OF RESULTS

Table II shows the mean differences between experimental andcontrol groups on sub-text I (Part I).

TABLE II. SUB-TEST I, (PART I) MEANS or SCORES ON VERBALPROBLEM-SOLVING TEST, PRE TO POST-TEST

n Pre Post t P

Experimental 72 5.0 7.56 1.97 .06Control 72 5.2 9.77 2.87 .01

Study of the results showed that the experimental group hadachieved some learning on Part I but had failed to reach significance.The learning achieved by the control group was significant to the .01level.

Results for sub-test II (Part II) is presented in Table III below.

TABLE III. PART II MEANS OF SCORES ON COMPUTATIONALTEST. TOTAL POPULATION

nPrePost t P

Experimental 72 1.35Control 72 1.1

4.93 2.42 .0225.5 7.8 .001

There is no doubt that the control group achieved results on PartII that were far superior to the results of the experimental group al-though the experimental group did show significant learning to the.02 level.

Part I of the tests was so designed as to require some reasoningability (verbal problem solving) in order to choose the correct answer,from four choices, for each question. Part II, when used immediatelyafter teaching the rule "invert the divisor and multiply/’ could beanswered by anyone who had memorized the rule and could apply itin solving division of fractions problems.Although the control group had a strong immediate recall of the

"invert the divisor and multiply" rule they failed to show an under-standing of the process involved, that was comparable to their show-ing on the computation work.

It was different with the experimental group whose mean score


failed to reach significance on Part I. The means of their scores onPart II showed better results reaching significance at the .02 level. Ina relative sense the experimental group was stronger in their applica-tion of the rule than in the understanding of the process; yet, in theteaching program used by the experimental group they were not in-structed to learn or apply a specific rule. The burden was placed onthem to apply their understanding to infer a working rule withoutspecific directions on how to do so.More detailed findings were derived by measuring differences in

means between experimental and control groups by IQ sub-groups.Table IV presents the results on Part I of the tests.

TABLE IV. MEANS OF SCORES ON VERBAL PROBLEM-SOLVINGTEST (PART I). IQ SUB-GROUPS

LowIQn Pre Post t P

Experimental 20 3.80 4.95 .59 �

Control 20 2.6 6.1 1.89 .08

Middle IQn Pre Post t P


High IQn Pre Post t P

Experimental 26 8.59 11.90 1.01 �-Control 26 7.8 12.5 1.4 �

Analysis of the results shown in Table IV indicates that with the in-crease in IQ learning was better. Results on Part I demonstrates somedifferences between control and experimental groups but not much.In the low and high IQ sub-groups neither the control nor the experi-mental group showed statistically significant learning although bothgroups learned. Only the middle IQ sub-group scored significantlybetter on the post-test. However, it should be noted that the controlgroup started at about the level of the post-test score of the experi-mental group.The most impressive result on Part I occurred in the high IQ sub-

group where both experimental and control groups achieved nearlyidentical scores on both pre and post-tests. Neither group showedstatistically significant improvement.A glance at Table V shows that Part II tells an entirely different

story than Part I of the tests. There is no doubt about the superior re-sults achieved by the control group. Every control sub-group madesignificant gains, from the .01 to the .005 level. The experimentalgroup improved with increased IQ to the .03 level of significance for


the high IQ sub-group.

TABLE V. MEANS or SCORES ON COMPUTATION TEST. IQ SUB-GROUPS

LowIQn Pre Post t P

Experimental 20 .88 2.38 1.34 �

Control 20 .95 22.8 3.9 .01

Middle IQn Pre Post t P

Experimental 26 .58 4.33 2.06 .05Control 26 .95 26.7 4.95 .005

High IQn Pre Post t P


It would appear that the A6 children had a strong immediate recallof the rules for dividing a fraction by fractions according to theirscores on Part II of the post-test. By comparison to Part II, theyscored poorly on Part I which was testing their understanding ofwhat was involved in the division of fractions process. This poorshowing was true with the total control group population when meas-ured as a unit, as well as with the IQ sub-groups when studiedseparately.The B6 group did not score as well on either Part of the tests as the

A6 group although they nearly matched the A6 group on Part I.

NULL HYPOTHESIS

There is no significant difference in ability to divide a fraction by afraction between two groups based on the maturity of the groupsdespite the teaching method used to teach the process.

Since the pre and post-tests used were in two parts the nullhypothesis was tested against each part separately. On Part I the nullhypothesis is rejected since the control group showed a significantdifference to the .01 level. On Part II the null hypothesis is rejectedwith great confidence. Although both groups showed significantdifferences between tests the control group scored significantly betteron the post-test than the experimental group.

CONCLUSIONSIt must be noted that the B6 or experimental group was taught the

division of a fraction by a fraction concept in five days with no back-ground in division or multiplication of fractions. Within the teachingprogram one lesson was devoted to review of addition and subtractionof fractions and one to teaching multiplication of fractions. The other


lessons taught division of a fraction by a fraction. Through the bal-ance of the semester the B6 group did not receive instruction in divi-sion of fractions. The A6 group, on the other hand, did have back-ground in addition, subtraction and multiplication of fractions formost of a semester prior to learning to divide a fraction by a fraction.Interpreted against the background described here the achievementby the experimental group on the Part I becomes impressive, par-ticularly since the difference between the pre and post-tests^ means ofscores approached statistical significance. (t= 1.97)The results on Part II are also impressive in view of the fact that

the differences on tests were statistically significant. This differenceoccurred even though the children were not taught a rule nor givenpractice to help memorize a rule for division of a fraction by a frac-tion.

Older children can learn and apply the rule "invert the divisor andmultiply" even if taught by rote. However, children who were taughtmeaningfully can infer the rule and apply it in solving division offractions problems.

Children who were taught rules by rote do not develop nor acquiremuch understanding of the rationale involved in the process. Theachievement of the B6 group indicates that if meanings were stressed,more efficient learning would result than if only teaching by rote wereemphasized.

It would appear that teaching meaningfully can produce better re-sults than teaching mechanically (presenting rules and requiringtheir memorization); although initial results in rote recall were goodthere was not enough understanding to guarantee retention.

SUMMARYA comparison of learning achievement between two different grade

levels using different methods of teaching showed significant results.Older children can learn effectively, even if taught by rote but theyfail to develop equivalent understanding. Younger children taughtmeaningfully inferred necessary rules of procedures and retainedthem over a period of two months which implies that they understoodit.

Additional experimental research is necessary to determine, as accu-rately as possible, the best grade levels or mental ages at which arith-metic processes can be taught meaningfully and therefore effectively.

grade placement and meaningful learning

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