Programmed instruction in elementary arithmetic

Programmed instruction in elementary arithmeticAuthor(s): GLEN E. FINCHER and H. T. FILLMERSource: The Arithmetic Teacher, Vol. 12, No. 1 (JANUARY 1965), pp. 19-23Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185074 .

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Programmed instruction in elementary arithmetic

GLEN E. FINCHER Kent State University, Kent, Ohio H. T. FILLMER Emory University, Atlanta, Georgia Professor Fincher is a member of the department of education at Kent State University. Professor Fillmer is a member of the teacher education department at Emory University.

1 here is a current interest in programmed learning in the areas of psychology and education. In the near future we can ex- pect to see numerous programmed textbooks prepared for use in the schools. Extensive research in the classroom use of such materials is necessary to insure that these materials are effective and that proper teaching methods and procedures are employed in the use of the materials. At present there is a dearth of research evidence evaluating this new approach to the teaching-learning process. Before programmed instruction can arrive at the po- tential many believe possible for it, there will need to be much more research conducted in public school classrooms. In recognition of this need, the data collected in this study were concerned with the effectiveness of programmed textbooks in the elementary school.1

Purpose It was the purpose of this study to com-

pare the achievements of a group of fifth- grade pupils using programmed materials on the addition and subtraction of fractions with a like group of pupils using a conventional classroom approach to the study of these topics. More specifically, this study tested the following null hypotheses:

There are no differences in either performance or retention of the arithmetic

1 Glen E. Fincher, "The Construction and Experimental Application of a Programed Course on the Addition and Subtraction of Fractions for Grade Five" (Ph.D. dissertation, Ohio University, 1963).

skills of addition and subtraction of fractions (1) between a group of fifth-grade subjects using programmed materials and a like group using a conventional classroom approach; (2) among the high, average, and low intelligence subgroups within each group; (3) among the high, average, and low achievers on an arithmetic skills test within each group, and (4) as a result of interaction of treatments by IQ levels and of treatments by grade levels of arithmetic skills.

Method of procedure Test construction The investigator first compiled the fol-

lowing list of twenty-eight behavioral objectives for the addition and subtraction of fractions.

Behavioral objectives for the addition and subtraction of fractions

1 Recognizing fractions 2 Learning terminology associated with addi-

tion and subtraction of fractions 3 Finding common denominators 4 Changing fractions to lowest terms 5 Adding like fractions; answer in lowest terms 6 Adding like fractions; answer must be

changed to lowest terms 7 Adding like fractions; the answer must be

changed to a mixed number, but not to lowest terms

8 Adding like fractions; the answer must be changed to a mixed number and changed to lowest terms

9 Adding a whole number and a fraction 10 Adding a whole number and a mixed number 11 Adding unlike fractions; the answer in low-

est terms 12 Adding unlike fractions; the answer must be

changed to lowest terms

January I960 19



13 Adding unlike fractions; the answer must be changed to a mixed number, but not to lowest terms

14 Adding unlike fractions; the answer must be changed to a mixed number and changed to lowest terms

15 Adding mixed numbers with unlike denominators; the answer in lowest terms

16 Adding mixed numbers with unlike denominators; the fraction in the answer must be changed to a mixed number, but not changed to lowest terms

17 Adding mixed numbers with unlike denominators; the fraction in the answer must be changed to a mixed number and changed to lowest terms

18 Adding three mixed numbers; the answer in lowest terms

19 Adding three mixed numbers; the fraction in the answer must be changed to a mixed number, but not to lowest terms

20 Subtracting like fractions; answer in lowest terms

21 Subtracting like fractions; answer must be changed to lowest terms

22 Subtracting unlike fractions; answer in lowest terms

23 Subtracting unlike fractions; answer must be changed to lowest terms

24 Subtracting with borrowing, no fraction in the minuend

25 Subtracting mixed numbers that have fractions with like denominators; the answer in lowest terms

26 Subtracting mixed numbers that have fractions with like denominators; the answer must be changed to lowest terms

27 Subtracting mixed numbers with unlike fractions; the answer in lowest terms

28 Subtracting mixed numbers with unlike fractions; the answer must be changed to lowest terms

Since standardized tests on these number operations were unavailable, two parallel forms of a test were constructed by the investigator who used the twenty-eight objectives as a guide. Each test consisted of fifty items. A test validation with over one hundred subjects revealed a coefficient of reliability between tests of .93 and an internal reliability of .95 for each separate test.

Construction of programmed textbook For each of the twenty-eight objectives

a series of frames was constructed to lead the student to perform the objective. These twenty-eight units were arranged in a logical sequence from the simple to the

complex. Additional frames were then constructed to effect a smooth transition from one unit to another. Finally, the programmed units were arranged in book form according to principles of linear pro- gramming explicated in the literature.

The programmed textbook was com- posed of 500 frames distributed over 100 pages. The teaching technique employed an explanation of each topic, using worked examples when appropriate, and a gradual reduction in the number of cues from frame to frame until all cues were removed and the pupil was required to work examples without the aid of further cues. Thus, each frame became more difficult than the preceding frame, and the pupil was led in small steps from the simple to the complex, from the known to the un- known. Review frames, which required the subjects to work several review examples, were interspersed throughout the textbook. Pupils missing any of the review examples were directed to go back to the unit in the book which presented the explanation of that example and work through that unit again.

When pupils came to the end of a unit, they informed the teacher and were given a unit test which they were required to complete before proceeding to the next unit. The pupil was required to score 70 percent before continuing on to the next unit. If he did not complete the test satis- factorily, he was required to review the section of the program in which he was weak. Six supplementary units were constructed as enrichment material for ac- celerated students who completed the programmed textbook prior to the end of the experimental period.

Pilot study A preliminary study was conducted

with fifty pupils in two fourth-grade classes to test the effectiveness of the programmed textbook as a teaching device, and to revise ambiguous frames, correct inappropriate vocabulary, and eliminate poor transition from one frame to the

20 The Arithmetic Teacher



next. Fourth-grade pupils were selected because they had not yet studied addition and subtraction of fractions, and it was acknowledged that if fourth-grade pupils could learn from the programmed textbook, it would be appropriate even for slow fifth-grade pupils.

A two-way analysis of variance for cor- related samples from pretest to posttest revealed that the improvement in scores as the result of experimental treatment was significant at the .01 level. Errors in the textbooks were tabulated and ana- lyzed, and students were interviewed so that programmed textbooks could be re- vised to best advantage. All results of the preliminary study were favorable, so plans were drawn to continue the investigation.

Sample The sample selected for this study con-

sisted of 309 fifth-grade students attend- ing three of the thirteen elementary schools in Zanesville, Ohio. The ten fifth- grade classes in the three schools were as- signed at random to experimental and control treatments, and provided five replications of the investigation. The Henman- Nelson Tests of Mental Ability and the arithmetic section of the Iowa Test of Basic Skills were administered to all experimental and control students participating in the study. A ¿-test of the significance of the difference between independ- ent means was then applied to these data to determine if the samples were drawn from the same population with regard to IQ and performance in arithmetic skills. The data obtained were consistent with the hypothesis of no difference between the mental ability and arithmetic skills of the experimental and control classes, either in replication or in combination.

Classroom procedure Prior to the implementation of this

study a conference was held with the ten fifth-grade teachers and the supervising principals of the participating schools. During the conference the purposes of the

study and the methods of procedure were explained in detail. All expressed an interest in programmed instruction and a de- sire to participate in the study. Both the teachers and principals were properly cer- tificated for their positions under the re- quirements of Ohio school law.

After assignments were made to treatments, teachers of experimental classes were instructed to distribute the programmed textbooks to the pupils at the beginning of the arithmetic period and to collect them at the end of each period to insure that no homework was done. Pupils spent exactly forty-five minutes each school day working with the programmed books. Teachers were requested to super- vise students closely to minimize "cheat- ing" which could render the reinforcement principle of the programmed textbook less effective. Teachers were also instructed to give individual attention to the less capa- ble students of arithmetic, but not to use workbooks or conventional textbooks as a means of giving this help.

The content outline of the programmed textbook was shown to the teachers of control classes. The teachers were requested to teach this same content by whatever method and with whatever aids they believed to be most effective, with the one exception of programmed materials. The only restriction placed upon the control teachers was that they were to de- vote only forty-five minutes of class time to arithmetic. There was no restriction placed upon the frequency or amount of arithmetic homework which they might wish to assign. It was decided that the effectiveness of the study might be en- hanced if all participating classes were to study arithmetic during the same time each day. The time from 9 to 9:45 a.m. was selected since fatigue could become a factor later in the school day.

Sources of data

Pupils participating in this study were divided into three IQ levels - above 110, 95-110, and below 95. They also were di-

January 1965 21



vided into three grade levels of arithmetic skills - above 5.4, 4.5-5.4, and below 4.5. Two forms of a performance test in addition and subtraction of fractions were constructed for use in this investigation. Form A was administered to three replications and Form B to two replications prior to the formal study of the addition and subtraction of fractions. The alternate form was administered as a posttest at the com- pletion of the four-week period of study. The mean gain between the pretest and posttest was computed for each subject.

Scores on the pretest were used to com- pute a two-way analysis of variance to determine if there was a significant difference between the control and experimental group and between IQ levels in performance on thç pretest. The same analysis was used for the three levels of arithmetic achievement.

The gain scores, which were computed by subtracting each subject's pretest score from his posttest score, were used to com- pute a treatments by levels by replications analysis of variance. This analysis drew on the previously established IQ and arithmetic achievement levels and the five replications.

Four weeks after the posttest was completed, the alternate form of the posttest was administered to all students participating in this investigation to test the amount of knowledge students had re- tained on the addition and subtraction of fractions. Retention scores were computed by subtracting each pupil's posttest score from his recall test score. A treatments by levels by replications analysis of variance was then used to analyze retention scores on the basis of the previously established IQ and arithmetic achievement levels and the five replications.

The .05 level of significance was selected for the F-ratios and the ¿-tests.

Findings 1 The difference between the control and

experimental groups in performance on the pretest was not significant.

2 Subjects with higher IQ's in both the experimental and control groups made a mean score on the pretest which was significantly higher than subjects with lower IQ's.

3 Subjects with a higher grade level in arithmetic skills in both the experimental and control groups made a mean score on the pretest which was significantly higher than the mean score for subjects with a lower grade placement.

4 The effect of the treatments on the gain scores was significant at the .05 level in favor of the experimental group.

5 There was a significant difference among the mean gain scores of the different IQ levels of both treatments. Those with higher IQ's made greater mean gains than those with lower IQ's.

6 There was a significant difference among the mean gain scores of the different grade levels of arithmetic skills of both treatments. Those in the higher grade levels made greater mean gains.

7 There were no significant treatments by levels interaction effects on gain scores. This was true when the analysis was made by IQ levels and by grade levels of arithmetic skills.

8 The effect of the treatments on the retention scores was not significant at the .05 level.

9 There was a significant difference among the mean retention scores of the different IQ levels of both treatments. Subjects with higher IQ's showed lower mean losses than subjects with lower IQ's.

10 There was no significant difference among the mean retention scores of the different grade levels of arithmetic achievement of either treatment.

1 1 There was no significant treatments by levels interaction effect on retention scores. This was true when the analysis was made by IQ levels and by grade levels of arithmetic skills.

22 The Arithmetic Teacher



Conclusions

As a result of the findings of this investigation the following conclusions were drawn:

1 The programmed textbooks used in this study were found to be more effective than the conventional classroom approach to the study of addition and subtraction of fractions.

2 The programmed textbooks were found to be as effective as the conventional classroom approach in the retention of

knowledge acquired on the addition and subtraction of fractions.

3 In both experimental and control groups there was a direct relationship between IQ level and achievement. Subjects with higher IQ's made greater mean gains than subjects with lower IQ's. This same relationship existed between grade level of arithmetic skills and achievement. Subjects with a higher grade level of arithmetic skills made greater mean gains than those with a lower grade level of arithmetic skills.

In answer to your questions- Why do children have difficulty with verbal problems? (Continued from page 18)

In a rural community several of the children should know about how much a Hereford steer weighs. These children might also find it in- teresting to discuss how much weight a ton- truck can haul without being overloaded.

Interpretation of verbal problems may be made more difficult for children because we have distorted their quantitative vocabulary. When we say "take away" rather than subtract, we may lead children to think all subtraction is an operation for finding a remainder. We may say that one number "goes into" another, that we "reduce" fractions, that we "borrow" when we subtract, and that we "turn it upside down" when we invert a fraction. It is no more difficult for teachers to use precise, definitive terms and to use them correctly. Unlearning confusing vocabulary is wasteful and frustrating, especially when the correct words are simple and easily taught. Mathematicians, for example, prefer the word "terms" to minuend and subtrahend, and they recommend that children learn to say "factors" rather than multiplicand and multi- plier.

In our eagerness to make verbal problem interpretation a little easier, we have over- emphasized word and number clues. Expressions such as "in all," "have left," "in each," and "altogether" appear at the right places and the numerals are written in convenient sequences. One child expressed his problem-solving technique as follows, "When there are three or four

numbers and they are about the same size, I add 'em; when there are only two and they are about the same size, I subtract 'em; but when there are only two and they are different in size, then I'm stuck. I divide 'em, and if they come out even I let 'em go. If they don't, I multiply 'em." Teachers contribute to this kind of in- tellectual delinquency when they provide pupils with verbal problems which permit it.

The responsibility for teaching verbal problem solving begins early in the primary grades and continues through each grade level. Prob- lems must be thoughtfully written and must say exactly what the teacher means them to say. Quantitative judgment is essential to an under- standing of any verbal problem situation, and experiences with measures will contribute much to such judgment. A mature, accurate, and forceful mathematical vocabulary adds dignity and meaning to verbal problems. Word and number clues deliberately placed in problems deny children the opportunity to do construc- tive thinking about the problems. Class discus- sion, a search for missing information, the development and testing of hypotheses, and the exercise of quantitative judgment are the most important outcomes of problem solving. No approach to problem solving is sound unless it provides the opportunity for children to develop skill in each of these activities. - Clyde G. Corle, The Pennsylvania State University, Uni- versity Park, Pennsylvania.

January 1965 23



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Programmed instruction in elementary arithmetic