Statement of Permission to Copy In presenting this professional paper in partial fulfillment of the requirements for an advanced degree at Montana State University, I agree that the Library shall make it freely available for inspection. I further agree that permission for extensive copying of this profes¬ sional paper for scholarly purposes may be granted by my major profes¬ sor, or, in his absence, by the Director of Libraries. It is under¬ stood that any copying or publication of this professional paper for financial gain shall not be allowed without my written permission. THE EFFECTS OF PRACTICE AND USAGE ON BASIC ARITHMETIC SKILLS by WILLIAM DOUGLAS TWILLING A professional paper submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION with concentration in Secondary Education Approved: Chairman, Examining Committee Graduate Dean MONTANA STATE UNIVERSITY Bozeman, Montana August, 1972 iii ACKNOWLEDGMENT The writer would like to express his sincere appreciation to Mr. G. V. Erickson, Head of the Secondary Education Department at Montana State University, for his helpful recommendations on content which were most helpful in developing this paper. A special note of thanks goes to Mr. Gerry Lenander, Principal of Lincoln Junior High School, in Billings, Montana, who gave permis¬ sion for this study to be carried out. Finally, a tribute is owed to my wife for her support and inspiration. TABLE OF CONTENTS Page LIST OF TABLES vi ABSTRACT vii Chapter 1. INTRODUCTION 1 Statement of the Problem 1 Purpose of the Study 2 General Questions to be Answered 3 General Procedures 3 Limitations 5 Definition of Terms 6 Summary 7 2. REVIEW OF THE LITERATURE 8 Introduction 8 Ideas and Thoughts on Practice and Usage 8 .Studies Done on the Effects of Practice and Usage. ... 11 Summary 15 3. PROCEDURES 17 Introduction 17 Subjects of the Study. 17 Experimental Treatment 18 Methods of Collecting Data 19 V Chapter Page Methods of Organizing Data 20 Statistical Hypotheses 21 Analysis of Data 21 Summary of the Procedures 22 4. ANALYSIS OF DATA 24 Summary 27 5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 28 Summary 28 Conclusions 28 Recommendations 29 APPENDICES 30 A. Diagnostic Test on Fundamentals 31 B. Instructional Worksheets Keyed to Diagnostic Test 33 C. Experimental Worksheets 37 LITERATURE CITED 44 vi LIST OF TABLES Table Page 1. Raw Score Data Compiled from the Study 25 2. Mean Differences of Subgroups 27 vii ABSTRACT The purpose of this study was twofold: (1) to develop a pro¬ gram for improving and maintaining arithmetic skills, and (2) to develop a method of testing that program to determine if it had any effect upon those exposed to it. The first half of this twofold purpose was carried out by developing a series of worksheets which covered the basic areas of arithmetic computation. These worksheets were designed so as to require a minimal amount of time to complete. The purpose of the work¬ sheets was to give continued exposure to the basic arithmetic skills without spending a gvoat deal of time. The second half of the purpose of this study was to create an experimental situation which would determine the effect the worksheets had on the ::objects’ ability to perform basic arithmetic skills. This was done by using three first year algebra classes as the subjects of the study. One class was designated the experimental group and the other two the control group. After a week of instruction in basic arithmetic skills was given to both groups, the experimental group then received twenty-four experimental worksheets over an eight week period. At the conclusion of the eighth week, both the control group and experi¬ mental group were tested. The data collected was then compared on the basis of matched- pair samples. The comparison was carried out by means of a ’t’ test analysis at thecC=.05 level of confidence in order to determine if there was any significant difference between the control and experimental groups at the conclusion of the experiment. It was found that a signi¬ ficant difference did occur in favor of the experimental group. There¬ fore, the conclusion drawn was that practice and usage, as applied in this study, did indeed have a positive effect in improving and main¬ taining arithmetic skills. Chapter 1 INTRODUCTION Today’s world seems to be one filled with all kinds of modern conveniences that make life more pleasant for the average man. One of the more popular of these conveniences is the small calculator which carries out the basic operations of arithmetic with great speed and accuracy. To operate such a calculator requires only that the operator know when to add, subtract, multiply, or divide, and not - necessarily how to perform these operations. However, it seems logical to think, at least in the near future, that, most people will still find the need to possess a working knowledge of the arithmetic skills, if only to balance their checkbook. The question that seems to arise from this last statement is do today's mathematics' programs provide for the development of mathematical concepts around which most "Modern Math" programs seem to be centered. Statement of the Probl^■:‘i!,. As a mathematics teacher, one of the most common complaints this researcher hears from fellow educators and others is that students today seem to be weak in the area of arithmetic skills. Of course, blame in many cases is placed on the failure of the "Modern Math" program to develop fundamental skills of arithmetic computation. However, it is the feeling of this researcher that such blame is misplaced. The 2 problem lies not so much in the developmental process but rather in the lack of most programs to provide for practice and usage of the skills once they have been developed. Therefore, the problem this study addressed itself to was to determine the effect practice and usage had on the maintenance and improvement of basic arithmetic skills in a first year algebra class. Purpose of the Study During the past few years, this researcher has been looking for a program which would help develop and maintain the arithmetic skills of students. While some programs were found, most required more class time than this researcher felt could be justified. With this in mind, a twofold problem was set forth which became the purpose of this study. The first was to develop a program for the purpose of developing and maintaining arithmetic skills that was so simple that very little class time or student study time would be sacrificed. This was accomplished by developing a set of worksheets which gave continued exposure to the four fundamental operations of addition, subtraction, multiplication, and division of whole numbers, mixed numbers, fractions, and decimal fractions. Also included were problems dealing with per cent. Each worksheet was designed to cover a certain area of these skills and at the same time not to require more than three or four minutes time for the average student. 3 The second part of the purpose of this study was to develop a method of testing the new program to see if the worksheets did create any significant difference in the students’ fundamental arithmetic skills. This was done by creating a control group which was not given the worksheets and an experimental group which was given the worksheets. Then, by using the numerical aptitude scores of the Differential Apti¬ tude Test, matched pairs were created so that upon completion of the experiment the researcher could administer a standardized test and compare the results using the theoretical model of the distribution of ’t* to determine if any significant difference between the experimental and the control group was present. General Questions to be Answered At the conclusion of this study, it was hoped that the results would answer two basic questions: (1) Did the worksheets designed by this researcher bring about any significant improvement in the experi¬ mental group as opposed to the control group?, and (2) If any signifi¬ cant change was noted, did it appear throughout the experimental group or did the change manifest itself within some subgroup of the experi¬ mental group? General Procedures This nine week study focused around three first year algebra classes which were assigned to this researcher at Lincoln Junior High 4 School in Billings, Montana. At the outset of the study, one of the three classes was selected to serve as the experimental group, while the other two classes combined served as the control group. The first week of the study served as an instructional period. During this week, the students were given a diagnostic test (see Appen¬ dix A, page 31) which was designed to help them determine their weak¬ nesses in performing the basic fundamental operations of addition, subtraction, multiplication, and division as they are carried out with whole numbers, mixed numbers, fractions, and decimal fractions, as well as some problems involving per cent. Using the results of this test, students were then able to obtain instructional worksheets (see Appen¬ dix B, page 33) which corresponded to those problems which they missed on the test. Following this, this researcher spent ten or fifteen minutes during each of the next four class periods giving instructions in each of the skill areas. The remainder of the period was reserved for the student to complete his instructional worksheets, check his own work, and get individual help if he felt he needed it. ' At the conclusion of the first week of the study, no further instruction was initiated by this researcher, except as it arose in the regular subject matter in a first year algebra class. However, all student initiated questions were answered upon request. During the next eight weeks of the study, the experimental group began receiving the experimental worksheets (see Appendix C, page 37) on Monday, Wednesday, 5 and Friday. Monday’s worksheet was due on Wednesday, Wednesday’s was due on Friday, and Friday’s was due the following Monday. Each weeks’ worksheets were designed to cover all the areas of concern to this study. On Monday, Wednesday, and Friday, the first few seconds of the period were taken to read the answers to the worksheet due that day; then they were collected. No class time was used at this time to ans¬ wer questions. However, following the daily lesson if some time re¬ mained in the class period, questions would be answered individually while the remainder of the class worked on their algebra assignment. Upon completion of the eight week period and twenty-four work¬ sheets, both the experimental group and the control group wore given the Survey of Mathematics Achievement Test, and the results were recorded. The results of the experimental group were then compared to the control group on the basis of matched pairs and the use of a ’t' test to determine whether there was any significant difference at the e£ = .05 level. The matched pairs were formed by matching each inember of the experimental group with a member of the control group based on the numerical aptitude score of the Differential Aptitude Test. Limitations This experimental situation was limited to three first year algebra classes at Lincoln Junior High School in Billings, Montana. Since this study was based around matched pairs which were determined by the numerical aptitude score of the Differential Aptitude Test, the 6 results of this study were subject to the reliability and validity of the D.T.A. In addition, the results of this study were also limited to the reliability and validity of the Survey of Mathematical Achieve¬ ment Test, which was used to measure the difference between the control and experimental groups. Definition of Terms The language of this study is composed of terms commonly used in education. The following terms are listed so that the reader will be able to interpret them in the same manner as the researcher. Arithmetic skills. Those tools which are needed to perform the operations of addition, subtraction, multiplication, division, and work with percentages. Computational ability. One’s ability to perform arithmetic skills. Control group. The one of two or more groups that is not sub¬ jected to the experimental factor introduced into the treatment of the experimental group (Good, 1:191). D.A.T. The abbreviation for Differential Aptitude Test. Experimental group. The one of two groups that is subjected to the experimental factor; the effect of which it is the purpose of the 7 experiment to discover (Good, 1:191). Numerical aptitude. The numerical aptitude score of the D.A.T. S.M.A. The abbreviation of the Survey of Mathematical Achieve¬ ment Test. Summary The purpose of this study was to see whether the method devel¬ oped by the researcher had any effect on the ability of his students to perform arithmetic skills. This study was conducted at Lincoln Jim!' “ High School in Billings, Montana, and involved three first year algebra classes. One of the classes was designated as the experimental group and was sub¬ jected to the experimental factor. The other two were designated the control group and were not subjected to the experimental factor. The experiment covered a nine week period, and at the conclusion of that period both experimental and control groups were tested. Using these test results, the groups were compared on the basis of matched pairs. A ,tl test was used to determine if there was any significant difference. Chapter 2 REVIEW OF THE LITERATURE Introduction The scope of this professional study was twofold: (1) the development of a program which provided for practice and usage of basic arithmetic skills in a first year algebra class, and (2) the organiza¬ tion of an experimental situation which was used to evaluate the said program. The following review of available literature served as a guide to that end and was organized in two parts. The first part of the review deals with past and present ideas and thoughts of educators on practice and usage as a method of developing and maintaining basic skills. The second portion of this review examines other studies done on the subject of practice and usage as a method of improving skills. Ideas and Thoughts on Practice and Usage At the outset of this section of the review of literature, this researcher would like to point out to the reader that many authors use the term "drill" when referring to what this researcher calls practice or usage. It is the intent of this study to use the terms interchange¬ ably. In the past seventy years, the concept of practice or drill has undergone some significant changes in the minds of educators. Ben A. Sueltz (7:192) points out in an article written in 1953 that twenty-five 9 years ago drill was a very common method of learning. However, during the ten-year period from 1935 to 1945, drill had been carried to such extremes that it became frowned upon by educators as a method of learn¬ ing. More recently, drill has again become part of the learning process but in a new light; not drill for the sake of drill, but rather as a complement to the learning process so that what is learned can become functional for the individual. In the process of reviewing the literature, this researcher has found that most educators writing on the subject of practice or drill today generally tend to agree that it is an acceptable method of achiev¬ ing some of the objectives of a mathematics program. The authors of the text, The Teaching of Secondary Mathematics, state the general feeling in this statement: Drill must be recognized as an essential means of attaining some of the desired controls, just as a strong emphasis on con¬ cepts and meanings and patterns and relationships must be recog¬ nized as being essential for understanding. Both are necessary; neither alone is sufficient. Many of the operations of mathema¬ tics need to be performed not only correctly, but with consider¬ able facility and speed, if they are to be most useful. Some of them need to be actually automatized. Facility in such operations can be attained only through systematic and repeated practice in using them; i.e., through drill (Banks, 1:130). The question that arises now is when does a teacher employ practice or drill in the development of a specific skill? Max A. Sobel (8:292-95) in an article entitled "Skills" offers some specific sugges¬ tions for skill development. The most timely of these suggestions is that before development of the skill must come the understanding of the 10 skill. For example, in teaching addition of fractions, the teacher must first instill in the learner the need for a common denominator. Next, the teacher must show the student a procedure for finding a common denominator and why that procedure works. W. G. Quast (6:628), a pro¬ fessor at Slippery Rock Bible College whose major responsibility is teaching content and method courses in elementary school mathematics, re-emphasizes this in his article, "On Computation and Drill". Mr. Quast points out that drill, in itself, is not a good learning device. He says to tell Johnny to do thirty addition problems because he is having trouble in addition is ridiculous. All this does is reinforce in Johnny’s mind his lack of ability to perform the operations of addi¬ tion. The answer then to the question of when should a teacher employ practice and drill in skill development seems to be only after under¬ standing. Another area to which practice or usage would seem to have a direct effect would be in the maintenance of some skill. Surely a track coach would not expect his pole vaulter to maintain or improve his jump¬ ing height without practice. Does it seem logical then for a mathematics teacher to expect his students to maintain their basic skills without practice? The authors of The Teaching of Secondary Mathematics point out that if a skill is not used it soon becomes vague and fuzzy, and the procedures can become uncertain. Here it would seem that carefully planned, cumulative drill could play a significant part in the maintenance 11 of a skill (Banks, 1:132). Before leaving this section of the review of literature, it would be worthwhile to look at one more of Max A. Sobel's (8:295) sug¬ gestions for the development of skills. Mr. Sobel warns that routine drill that tends to be mechanical must be avoided. He states, "To be effective, drill must be varied. The structure of the drill situation should force the student to think about each problem." Studies Done on the Effects of Practice and Usage This review of literature now turns its attention to three studies, all of which have been done in the last ten years. All three of these studies deal with some aspect of the effect practice and usage have on a student’s aMlity to perform basic arithmetic operations. In 1963, G. H. Miller (5:117), a professor of mathematics at Edinberg State College in Pennsylvania, carried out a study which he entitled "Theory or Practice in Arithmetic—Which Shall it Be?". The hypothesis of the study was whether or not theory in mathematics elimi¬ nates the need for a great amount of computational drill? He drew his sample from 137 students in the class of entering college freshmen who did not meet the criterion of the fiftieth percentile of the ninth grade norms on the Madden-Peak Arithmetic Test. The 137 students were then divided into two groups. One of these was called the modified tradi¬ tional group and the other the modern group. The modified traditional 12 group was subjected to lecture class five days a week followed by a workshop three days a week. The textbook used by this group contained most of the modern concepts, but Mr. Miller purposely omitted the major topics of set theory and logic to allow more time for practice and drill. The second group of students, the modern group, were lectured to five days a week and were provided with workshops for extra help. Unlike the traditional group, the modern group covered all the material in the text and was not provided with the extra practice and drill time (Miller, 5:117). At the completion of seven weeks, the modern group and the modi¬ fied traditional group were compared, and Mr. Miller found a most signi¬ ficant difference in favor of the modified traditional group; and, therefore, concluded that arithmetic skills are greatly improved by practice (Miller, 5:119). In 1965, another study was carried out which was similar to G. H. Miller's, but it was done in a much different setting. This study was done by Elmer A. Koch Jr. (4:9), who is a fifth grade teacher in Calhoun Elementary School in Minneapolis. His study was entitled "Homework in Arithmetic", and it was built around the question, does homework increase arithmetic achievement. The study was done with three sixth grade classes in one school. These three classes were divided into three groups: (1) full group—which received long daily 13 homework assignments, (2) half group--which received a relatively short daily homework assignment, and (3) control group--which received no daily homework assignments. All three of these groups were taught the regular arithmetic curriculum by the regular classroom teachers. All of the groups used the same textbooks and covered the same material. Weekly conferences with the classroom teachers were held to determine the content of the homework assignments. Mr. Koch then produced and corrected the assign¬ ments himself (Koch, 4:10). While, the data obtained from this experiment was not sufficient to make the claim that homework will increase achievem nu in arithmetic, some of the data did lean in this direction. Based upon the results of the full group, it did appear that daily homev7ork assignments are a significant factor in raising the achievement level of learning in the area of arithmetic computation. The results of the half group were not significant in the area of computation, which the author points out might indicate that longer homework assignments are in order (Koch, 4:12). In the area of problem solving, neither the full group nor the half group showed a significant improvement. The author offers as an explanation the fact that problem solving involves the complex skill of reasoning, and this is something that cannot be developed by the prac¬ tice-type assignment (Koch, 4:12). 14 As a result of his study, Mr. Koch (4:13) draws a conclusion which seems to support what this review of literature earlier found to be the opinion of many educators. Koch concludes that the question is not "'What good is homework in arithmetic?' Rather, the question becomes, 'What is homework in arithmetic good for?'" In other words, not drill for the sake of drill, but meaningful drill which will help in the development of a skill. Roland F. Gray, a professor of education at the University of British Columbia (3:199), published a study entitled "An Experiment in Tccchi:-.; of Introductory Multiplication", in the March, 1965, issue of the Arithmetic Teacher. The h;>t.othesis of his study was that introduc¬ tory multiplication could be learned equally well, if not better, by understanding of the distributive property, as opposed to practice. Mr. Gray (3:199-200) performed his study with 480 third grade students. All the students received the same basic introduction to multiplication. That introduction was the method of repeated additions. Following this uniform introduction, the students were divided into two groups. In the first group, a great deal of time was allowed for prac¬ tice and memorization of the multiplication combinations, but no mention of the distributive property was made. In the second group, all of multiplication was explained in terms of the distributive property. As a result of this, the second group had less time for drill. Also, no attempt was made for the second group to memorize the multiplication 15 combinations. At the conclusion of his study, Mr. Gray (3:201-202) measured his results three ways. First, he used a paper and pencil test which was devised to determine the students' actual ability to perform the operation immediately following the experiment. This test revealed no significant difference between the two groups. At a later date, he employed a similar test to determine retention of the skill and found a significant difference which favored the second group. Finally, as a third method of measuring his results, Mv. Gray used the personal interview to determine if: (1) the subject could give a rational expla nation of the multiplication procedure or whether he gave a rote answer and (2) to see if there was any evidence of the ability to apply the distributive property. On both counts, Mr. Gray found that the second group performed at a higher level. Summary This review of literature was organized in a way which would supply this researcher and the reader some knowledge of the effects of practice and usage on the development of a skill. In order to do this, the review was divided into two parts. The first part dealt with past and present ideas and thoughts of educators on the subject and the second part with the results of some studies done on the subject. In review of the history of practice or drill, it seems that about the turn of the century this was a very acceptable method of 16 teaching and was used extensively. However, during the ten year period from 1935 to 1945, practice and drill had been carried to such extremes that they lost favor in the eyes of the educators. Foljowing this period, drill once again was finding its way back into the educational process, but in a new light; not drill for the sake of drill, but only as a device to develop a skill so that skill could become functional. Today, there seems to be a general agreement that the use of drill should only follow understanding. The studies which were included in this review of literature seem to coincide with the feelings of the educators. For example, R. F. Gray's study emphasized understand:Mg, and the results revealed signifi¬ cant improvement in the areas of retention and understanding. This pre¬ sents a strong case for the idea that understanding must precede prac¬ tice. The studies done by Koch and Miller, on the other hand, showed that drill could be used effectively as a device for improving some skills. Perhaps if the feelings of the educators and the results of the research were combined, the conclusion would be that practice or drill could be effective in developing skills, but that they should not be employed until understanding of the skill is present. Chapter 3 PROCEDURES Introduction The purpose of this study was to determine the effects practice and usage had on the maintenance and improvement of basic arithmetic skills. It is the purpose of this chapter to explain the procedures followed in determining: (1) how those students involved in the study were chosen; (2) to explain the experimental variable to which the experimental group was exposed; (3) the devices and methods which were used in collecting data; (4) how the data was organized; (5) a statement of the statistical hypotheses; and (6) the procedures which were fol¬ lowed in analyzing the data. Subjects of the Study This study wa;^ focused around seventy-eight ninth grade algebra students who comprised three algebra classes assigned to this researcher at Lincoln Junior High School in Billings, Montana, during the 1971-1972 school year. One of these classes, which contained twenty-eight members, was selected to serve as the experimental group and was exposed to the experimental variable. Since no attempt was made to select these students at random, some method had to be employed to equate the control and experimental groups at the outset of the study. Prior to the outset of the study, all ninth grade students at Lincoln had taken the Differential Aptitude 18 Test. The numerical aptitude scores of this test were used to form matched pairs by matching members of the experimental group with members of the control group whose numerical aptitude scores were the same on the D.A.T. Experimental Treatment During the first week of this nine-week study, both the experi¬ mental and the control groups were subjected to a review situation in the four basic skills of addition, subtraction, multiplication, and division, as they are carried out with whole numbers, mixed numerals, fractions, and decimal fractions. Also included in this was the area of percentage problems. This review was carried out by administering a diagnostic test (see Appendix A, page 31) on the first day of the study. This test was designed to help the students locate the areas in which they were weak. The students then obtained instructional worksheets (see Appendix B, page 33), which were coordinated with the area in which they needed work. During the last four days of the week, this researcher spent the first ten to fifteen minutes of each period going over the various areas for groups of students who were having trouble with a specific area. The remainder of each period was set aside for class work which gave students a chance for individual attention. Upon completion of the first week, no further instruction was initiated by this researcher, except as it arose in a first year algebra class. However, all students' questions were answered upon their 19 request. At this point, the experimental group began receiving the experimental worksheets (see Appendix C, pages 37-43) on Monday, Wed¬ nesday, and Friday. Monday's worksheet was due on Wednesday, Wednes¬ day's worksheet was due on Friday, and Friday's worksheet was due the following Monday. On Monday, Wednesday, and Friday, the first few seconds of each class were taken to correct the worksheets, thus enabling the students to see which problems they missed so that they could seek help on their own time. Each weeks' worksheets covered all the areas of concern to this study and were designed so as to require a minimal amount of time by the students. At the conclusion of the eight weeks, or twenty-four worksheets, by the experimental group, the S.M.A. test was administered to both the experimental and the control groups. These results were then compared on the basis of matched pairs, which had previously been determined, by the numerical aptitude score of the D.A.T. to determine if any significant difference was present. Methods of Collecting Data The first data collected for this study was the numerical aptitude scores of the Differential Aptitude Test. These scores were obtained from the guidance department at Lincoln Junior High School and were used in the initial phase of the study to establish equality between the control group and the experimental group on the basis of matched pairs. The other data used in this study was obtained from the scores of the Survey of Mathematics Achievement Test taken by both the 20 experimental and control groups upon completion of the experiment. The scoring of the Differential Aptitude Test was done by machine, while the scoring of the Survey of Mathematical Achievement Test was done by the researcher with the aid of an answer key supplied by the test manufac¬ turer. The reliability and validity of the two tests are reported below. Test Reliability Validity D.A.T. .90 .62 S.M.A. .90 Methods of Organizing Data Once the researcher had received the numerical aptitude scores of the D.A.T. and the matched pairs had been established, a six column table was constructed. The first column identified the matched pair, and the second column the numerical aptitude score received by the pair. The third and fourth columns were reserved for the scores of the S.M.A. Test taken at the conclusion of the study. The fifth and sixth columns were used to report the difference in the scores of the S.M.A. which were received by the experimental and control member of each pair and the square of that difference. This last data was necessary for calcu¬ lating the statistics of this study. 21 Statistical Hypotheses Verbally, the null hypothesis of this study was: there was no difference in the achievement of the experimental group and the control group in performing basic arithmetic skills. The corresponding alter¬ nate hypothesis was: the achievement of the experimental group in per¬ forming basic arithmetic skills was greater than that of the control group. Algebraically, the null hypothesis and the alternate hypothesis were: H : JX - A 0 E C Hr where At was the mean achievement of the experimental group and was the mean achievement of the control group. Analysis of Data The data which was obtained from the Survey of Mathematical Achievement Test administered at the conclusion of the study was analy¬ zed to determine if any significant difference existed between the experimental group and the control group. The theoretical model that was used to determine this significance was the distribution of 't'. The statistical analysis was based around the matched pairs of the.experimental group and control group, as was determined by the numerical aptitude score of the D.A.T. It was assumed that this matching 22 equated the paired individuals in such a way that the difference between their abilities to perform basic arithmetic skills was zero. Once the test scores of the Survey of Mathematical Achievement Test had been compiled at the completion of the study, the researcher calculated the difference between the scores made by the paired indivi¬ duals. Following this, the mean of the distribution of differences was calculated along with the variance and standard deviation. The researcher then solved for the ’t' statistic. The level of significance for this study was the cC=.05. From the table of 't', the researcher was then able to determine whether to retain or reject the null hypothesis. The test was regarded to be a one-tailed test;. Summary of the Procedures Three classes of students taking first year algebra at Lincoln Junior High School in Billings, Montana, were the subjects of an experi¬ mental study to determine the effects of practice and usage on basic arithmetic skills. In order to accomplish this, one class was considered the experimental group and the other two the control group. The experi¬ mental group was exposed to the experimental variable for a period of eight weeks. At the conclusion of this eight week period, a standardi¬ zed test was administered to both the experimental and the control groups. With all other variables being held constant, the results of the test scores received by the experimental and the control groups were 23 compared. A 1t' test was then used to determine if the null hypothesis was to be retained or rejected. The level of significance was oC=.05 for a one-tailed test. Chapter 4 ANALYSIS OF DATA The purpose of this study was to determine the effects that practice and usage have on the basic arithmetic skills. In order to accomplish this, an experimental situation was established which allowed the researcher to compare ninth grade algebra students who had been exposed to the variable factor of practice and usage of arithmetic skills over an eight week period to those students who had not been exposed to the variable factor. This comparison was done by carrying out a one-tailed lt’ test at the oC=.05 level of significance on the data collected. Table 1, page 25, shows the data which was compiled at the con¬ clusion of the study. Using the data reported in Table 1, the following statistics were computed: (1) the mean score of the experimental group on the S.M.A., symbolized X; (2) the mean score of the control group on the S.M.A., symbolized Y; (3) the mean of the distribution of dijferences of scores received by experimental group and the control group on the S.M.A., symbolized D; (4) the variance of the distribution of differ- 2 ences, symbolized S^; and (5) the standard deviation of the distribution of difference, symbolized S^. The results were as follows: X = 48.0 = 27.01 Y = 44.4 SD = 5.20 D = 3.68 25 Table 1 Raw Score Data Compiled from the Study Matched Pair Numerical Aptitude X Y D D2 1 30 51 41 10 100 2 29 53 48 5 25 3 28 58 58 0 0 4 28 54 53 1 1 5 27 55 53 2 4 6 27 45 42 3 9 7 25 48 45 3 9 8 25 59 43 16 256 9 24 54 48 6 36 10 24 48 42 ' 6 36 11 24 49 48 1 1 12 24 49 43 6 36 13 24 44 35 9 81 14 23 50 40 10 100 15 22 48 47 1 1 16 22 46 48 -2 4 17 21 48 39 9 81 18 20 37 43 -6 36 19 19 41 39 2 4 20 19 38 42 -4 16 21 17 38 40 -2 4 22 16 44 39 5 25 £X=1057 £Y=976 XD=81 £D2=865 X = score on S.M.A. by experimental group Y = score on S.M.A. by control group D = (X - Y) Using the formula t = \ 26 XJ1 the researcher solved N Zb2- (ZD)2 N - 1 for the 't’ statistics and found it to be 3.32. Upon examining the table of 111 at thecL=.05 level with 21 degrees of freedom, it was found that in order for the experimental factor to have had a significant effect a ft’ score greater than 1.721 would have to occur. Based on the fact that the *t1 score determined by the data of this study ex¬ ceeded 1.721, this researcher made the decision to reject the null hypo¬ thesis of this study in iavor of the alternate hypothesis which stated, "The achievement of the experimental group in performing basic arithme¬ tic skills was greater than that of the control group." This result gave an affirmative answer to the first general question to be answered by this study, namely, "Did the worksheets designed by the researcher bring about any significant improvement in the experimental group as opposed to the control group?" The second question to be answered by this study was, "If any significant difference was noted, did it appear throughout the experi¬ mental group or did the change manifest itself within some subgroup of the experimental group?" Upon finding a significant difference, the researcher prepared Table 2, page 27, which subdivides the matched pairs into five subgroups based on their D.A.T. numerical aptitude score. As can be seen in Table 2, the mean difference of each subgroup 27 Table 2 Mean Difference of Subgroups Numerical Aptitude N ID 2D/N 28 - 30 4 16 4 25 - 27 4 24 6 22 - 24 8 31 3.9 19 - 21 4 1 .25 16 - 18 2 3 1.5 was calculated. While all the differences were of a positive nature, the data in this table do'S point out that the greatest improvement occurred in the top three groups which included approximately 72 per cent of those involved in the study. Summary In summary, the analysis of data shows that the experimental variable did indeed cause a significant improvement in the experimental group over the control group. The data also gave evidence that the improvement was greater in the top 72 per cent of the experimental group. Chapter 5 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS Summary The purpose of this study was twofold: (1) to develop a simple program for maintaining or improving basic arithmetic skills which would demand very little class time or student study time, and (2) to develop a method for testing that program. The first of these purposes was accomplished by developing a set of worksheets which covered the basic areas of arithmetic skills. The second purpose was accomplished by giving these worksheets to one of three first year algebra classes over a period of eight weeks. At the conclusion of these eight weeks, all three classes were tested and the results were compared on the basis of matched pairs. A ’t' test analysis at the °^“.05 level of confidence was used to see if there was any significant difference between the group receiving the worksheets -nd those not receiving the worksheets. The results of the 't' test showed that the worksheets had created a significant difference in the students' ability to perform basic arith¬ metic skills. Conclusions The results of this study lead this researcher to the conclusion that practice and usage can have a positive effect on students' ability to perform basic arithmetic skills. Results also lead to the conclusion that the greater improvement will occur in higher ability groups, 29 although some improvement was noted throughout the entire group. Recommendations Upon completion of this study, many questions arose in the mind of this researcher about what effect changing some of the variable factors or even the group studied would have upon the results of this experiment. It was these questions that lead this researcher to make the following recommendations for further study in this area. It is recommended that: 1. A study be carried out using the same number of worksheets extended over longer periods of time. 2. A similar study be carried out with a larger group of stu¬ dents. 3. Similar studies be carried out using other groups of stu¬ dents, such as general mathematics rather than ninth grade algebra students. APPENDICES 31 APPENDIX A DIAGNOSTIC TEST ON FUNDAMENTALS Use the symbols =, ^>, or