i AN EXPERIMENTAL STUDY TO DETERMINE THE EFFECTIVENESS OF GROUP INSTRUCTION USE OF CERTAIN MANIPULATIVE MATERIALS IN CONTRIBUTING TO AN UNDERSTANDING OF DECIMAL CONCEPTS by George James Greenaway B.A.,- University of Manitoba, 1939 B.Ed., University of British Columbia, 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Education W-e accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1958 i l ABSTRACT AN EXPERIMENTAL STUDY TO DETERMINE THE EFFECTIVENESS OF GROUP INSTRUCTION USE OF CERTAIN MANIPULATIVE MATERIALS IN CONTRIBUTING TO AN UNDERSTANDING OF DECIMAL CONCEPTS The increasing emphasis on teaching arithmetic meaningfully Intensifies the search for materials of instruction which can effectively communicate arithmetical understandings to children. Though manipulative aids are widely approved as effective teaching media for achieving this purpose, most of the endorsements are subjective opinions rather than objective evaluations based on experimentation. This study is an attempt to determine the effectiveness of group instruction use of certain manipulative aids in teaching decimal fraction concepts to Grade VII pupils. The effectiveness was i determined by comparing the achievement of two unselected groups, randomly assigned, on a test of understanding of the processes involved in decimal fractions. The two groups were given teaching treatments identical except in so far as the materials of instruction were concerned. One group used manipulative aids; the other used static representations of these aids. These materials were intended to differ only with respects to the characteristic of manipulability. Since manipulability of concepts is the most essential property of manipulative aids, i t was i i i isolated as the experimental variable. Because the groups were randomly assigned, analysis of covariance was selected to control statistically the initial differ-ences between groups in the four variables considered likely to influence achievement on the criterion test: initial understanding of the processes involved in decimal fractions, computational ability in decimal fractions, mental ability, and reading ability. The data obtained from the investigation were analyzed and the following conclusions reached. 1. The pupils taught by means of group instruction with the manipulative aids used in this investigation did not acquire a sig-nificantly better understanding of decimal fractions than did the pupils taught with static representations of these aids. In other words, the manipulation of the concepts, performed by using the mani-pulative aids in group demonstrations, was not effective in contributing to the pupils1 understanding of these concepts. 2. A study of the correlations for both treatment groups between achievement on the criterion variable and achievement on each of the independent variables indicates that the manipulative aids proved to be neither more nor less effective than the static representations as media for conveying an understanding of decimal fractions to pupils of any particular ability in the areas represented by the independent variables. 3 . It must not be inferred that any generalization concerning the effectiveness of these specific materials of instruction, used iy; exclusively by the teacher for group demonstration purposes, would be applicable also to similar materials i f they were used in a teaching procedure in which the pupils themselves, participated individually in the manipulative activity. It must not be inferred that any generalization concerning the effectiveness of these specific materials of instruction, which were used in a.brief teaching assignment devoted exclusively to the rational-ization of processes, would be applicable also to the same materials i f they were used in a teaching assignment of longer duration, and/or a teaching assignment in which the emphasis on the WHY of the processes was taught concurrently with, or preceded, the emphasis on the HOW of the processes. 5. Independently of treatment groups, the achievement on the i n i t i a l test of understanding of the processes involved in decimal fractions was the variable most predictive of achievement on the final test of understanding. Computational ability in decimal fractions and mental ability each shared approximately one-half the predictive capacity of the i n i t i a l test of understanding. Reading ability was a negligible predictor of achievement on the final test of understanding. I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8. C a n a d a . V TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION • i 1 The purpose of the study . 1. Description of the problem 2 Delimitation of the problem 5 Date and locale of the experiment . . . . . . . 5 Instructional materials 5 Teaching programme 5 Testing programme 6 Justification of the problem 7 Factors determining the choice of- subject matter . 10 Factors determining the grade placement of the experiment 11 Limitations of the study 12 Organization of the remainder of the thesis . . . 14 Footnotes 18 •.i l l . REVIEW OF RELATED LITERATURE 20 Changing concepts in arithmetic psychology . . . . 20 Reported research involving the use of manipulative materials in the teaching of arithmetic 25 Footnotes . . . 30 v i CHAPTER PAGE III. THE PLAN AND ADMINISTRATION OF THE EXPERIMENT . . . 32 Steps in planning the experiment 32 Description of the instructional materials . . . . 38 Place value charts 38 Place value cards 43 Rule with movable indicator 44 Flannel board 44 Visualization materials 44 Description of the lessons 45 Selection of classes to participate in the i experiment 48 Assignment of classes to the treatment groups . . 49 Account of orientation and evaluation meetings . . 52 Administration of the testing programme 54 Psychological significance of the limitations imposed on the experiment 55 Reasons for imposing the limitations on the study. 59 Summary 61 Footnotes 63 IV. THE STATISTICAL DESIGN OF THE EXPERIMENT AND DESCRIPTION OF THE MEASURES USED 65 Statistical design of the experiment 65 The general nature and purpose of the statistical method 65 vii-CHAPTER PAGE Statement of the hypothesis 68 Description of the Farquhar test of under-standing of processes with decimal fractions . 70 Data derived from a t r i a l administration of the Farquhar test 71 Method of item analysis 71 Interpretation of data obtained from the t r i a l administration 78 Interpretation of data obtained from the final administration . . .. . . . . 82 Concluding comments about the Farquhar test . . 91 Description of the Decimal Fraction Computation Test . . 92 Interpretation of data obtained from the administration of the decimal fraction computation test 92 Description of the Otis Self-Administering Test of Mental. Ability . . , 100 Description of the Stanford Achievement Test (Advanced Reading Test: Form E) 102 Conclusion 103 Footnotes 106 : y i i i CHAPTER PAGE V. THE STATISTICAL ANALYSIS 107 Introduction 107 An analysis of variance of each of the four independent variables X^ ., X^ , X^ , X^ 112 An analysis of variance of the criterion variable Y 123 Computation of the sums of cross products in deviation form for each pair of variables . . . 126 An examination of the conditions under which an analysis of covariance wil l increase the precision of the test of significance 131 Nature of correlation of means between treatment groups 133 Nature of correlation of individual scores within each treatment group 135 Summary of correlation conditions . . . . . . . 139 Calculation of the coefficients of regression . . 140 Calculation of the coefficients of the total regression equation 142 Calculation of the coefficients of the within groups regression equation 142 Calculation of the sums of squares of residuals . 144 Calculation of F value and application of test of significance 147 i x CHAPTER PAGE Assumptions underlying the derivation of analysis of covariance 149 Statistical test of homogeneity of regression . . 154 Test of the null hypothesis that /3 YE X1E = ^ V l C 1 5 6 Test of the null hypothesis that £ Y E X 4 E = frchc 1 6 2 The use of a binary computer (Alwac III-E) in performing automatic covariance computations . . 167 Footnotes 169 VI. SUMMARY AND CONCLUSIONS ..." 172 Summary 172 Purpose of the experiment 172 Background and justification . . 172 Problems proposed by the investigation . . . . . 172 Procedure 173 Conclusions 175 Summary of results 175 Interpretation of results 177 Summary of conclusions 178 Implications of these conclusions and suggestions for further study 180 BIBLIOGRAPHY 182 X CHAPTER PAGE APPENDICES 186 A. Communication of administrative arrangements to the teachers participating in the study . 186 B. The lessons 199 :G. The pupils' worksheets 284 D. Tests used to measure the criterion variable and the four independent variables 296 E. Samples of the procedures used to determine the suitability of the tests for the study . 307 F. Raw score data 315 G. Supplementary statistical calculations . . . . 321 vsi. LIST OF TABLES TABLE . . . PAGE I. Plan of Lesson Topics with the Numbers of the Farquhar Questions Appropriate to Each Topic . . 34 II. Names of Instructional Materials Used by-Experimental and Control Groups, and the Time Allowed, for Teaching the Objectives of each Lesson 36 III. Number of Pupils in the Classes Assigned to Each Treatment Group . 51 IV. Tests Selected to Measure the Variables . . . . . 69 V. Frequency of Scores in the Trial Administration of Farquhar's Test to Forty Grade VII Pupils in White Rock Elementary School 72 VI. Summary of Statistical Detail Resulting from Trial Administration of Farquhar's Test to Forty Grade VII Pupils in White Rock Elementary School 72 VII. The Difficulties and Validities of Items Resulting from Trial Administration of Farquhar's Test to Forty Grade VII Pupils in White Rock Elementary School 73 VIII. Frequency of Items at the Various Per Cent Levels of Difficulty Resulting from Trial Adminis-tration of Farquhar's Test to Forty Grade VII Pupils in White Rock Elementary School 75 x i i TABLE PAGE IX. Frequency of Items at the Various Per Cent Levels of Validity Resulting from Trial Administration of Farquhar1s Test to Forty Grade VII Pupils in White Rock Elementary School 75 X. Frequency of Items in the Various Validity Coefficient Ranges Resulting from Trial Administration of Farquhar's Test to Forty Grade VII Pupils in White Rock Elementary School . . . 76 XI. Frequency of Scores in Farquhar1s Test Test Administered at the Close of the Experiment to the 147 Participating Subjects . . . . . . . . . 84 XII. Summary of Statistical Detail in Farquhar's Test Administered at the Close of the Experiment to the 147 Participating Subjects 84 XIII. The Difficulties and Validities of Items in Farquhar's Test Administered at the Close of the Experiment to the 147 Participating Subjects 85 XIV. Frequency of Items at the Various Per Cent Levels of Difficulty Resulting from Administration of Farquhar1s Test at the Close of the Experiment to the 147 Participating Subjects 87 XV. Frequency of Items at the Various Per Cent Levels of Validity Resulting from Administration of Farquhar's Test at the Close of the Experiment to the 147 Participating Subjects 87 x i i i TABLE PAGE XVI. Frequency of Items in the Various Validity Coefficient Ranges Resultings from Adminis-tration of Farquhar* s Test at the Close of the Experiment to the 147 Participating Subjects 88 XVII. Comparative Data Obtained from the Administration of Farquhar1 s Test to the Trial Group and to the 147 Participating Subjects 89 XVIII. Frequency of Scores in the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects . . 93 XIX. Summary of Statistical Detail in the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects 93 XX. The Difficulties and Validities of Items in the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects 94 XXI. Frequency of Items at the Various Per Cent Levels of Difficulty Resulting from Administration of • the. Decimal Computation Test at the Beginning of the Experiment to the 147 Participating Subjects. 96 TABLE xiv PAGE XXII. Frequency of Items at the Various Per Cent Levels of Validity Resulting from Administration of the Decimal Computation Test at the Beginning of the Experiment to the 147 Participating Subjects 96 XXIII. Frequency of Items in the Various Validity Coefficient Ranges Resulting from Administration of the Decimal Computation Test at the Beginning of the Experiment to the 147 Participating Subjects 97 XXIV. Frequency of Scores in Stanford Reading Test Administered at the Beginning of the Experiment to the 147 Participating Subjects 104 XXV. Summary of Statistical Detail in Stanford Reading Test Administered at the Beginning of the Experiment to the 147 Participating Subjects . . 104 XXVI. Means and Standard Deviations Obtained by Each Treatment Group in the Criterion Variable and the Four Control Variables 110 XXVII. Sums of Scores in the Five Variables, and Sums of Squares of Scores, Arranged by Classes, for Each Treatment Group and for the Total Sample 118 XXVIII. Sums of squares of Scores in the Five Variables in Deviation form, for the Total Sample, and for V/ithin, and Between, the Treatment Groups . . . 119 XV. TABLE PAGE XXIX. Analysis of Variance of Performance of the Two Treatment Groups on the Independent Variable X 120 XXX. Analysis of Variance of Performance of the Two Treatment Group's on the' Independent Variable X g 120 XXXI. Analysis of Variance of Performance of the Two Treatment Groups on the Independent Variable X, 121 XXXII. Analysis of Variance of Performance' o"f the Two Treatment Groups on the Independent Variable X. 121 4 XXXIII. Analysis of Variance of Performance of the Two Treatment Groups on the Criterion Variable Y . 124 XXXIV. Sums of Cross Products of Scores in the Five Variables, Arranged by Classes, for each Treat-ment Group and for the Total Sample 130 XXXV. Sums of Cross Products of Scores in the Five Variables, in Deviation Form, for the Total Sample, and for V/ithin, and Between, the Treatment Groups . 132 XXXVI. Summary of Reliabilities of Tests Employed to Provide Measures of the Criterion Variable and the Four Independent Variables 137 xvi TABLE PAGE XXXVII. Pearson Product-Moment Coefficient of Correlation and Intrinsic Correlation, Within Groups of the Criterion Variable with Each of the Independent Variables 137 XXXVIII. Regression Coefficients of the Total Regression Equation and the Within Groups Regression Equation 143 XXXIX. Summary of Sums of Squares of Residuals . . . . . 146 XL. Analysis of Covariance of Performance of the Two Treatment Groups on the Criterion Variable Y . 148 XLI. Within Group Variance Accounted for by the Use of Each of FoUr Independent Variables: X1, X2, X3, X 4 155 XLII. Sums of Squares and Cross Products of Variables x^ and y, calculated independently for Experimental and Control Groups 160 XLIII. Sums of Squares and Cross Products of Variables x^ and y, calculated independently for Experimental and Control Groups 163 XLIV. Pearson Product-Moment Coefficients of Correlation between achievement on the criterion test and achievement on each of the independent variables arranged according to treatment groups . . . . 176 x v i i ; LIST OF FIGURES FIGURE PAGE 1. Illustrations of Manipulative Materials 39 2. Illustrations of Visualization Materials . . . . 40 3. Illustrations of Visualization Materials . . . . 41 4. Illustrations of Visualization Materials . . . . 42 x v i i i ACKNOWLEDGMENTS The author wishes to express his indebtedness to various persons who have contributed in important ways to the development of this study. Dr. H. L. Stein of the Faculty of Education, University of British Columbia, under whose direction this investigation proceeded, was always helpful in his assistance at points of difficulty and generous in his encouragement and guidance. Mr. K. B. Woodward, Municipal Inspector, School District No. 36 (Surrey), provided the administrative arrangements for conducting the experiment in the schools. His interest was a stimulus which helped to bring the work to a conclusion. Messrs. R. Beale, P. Carey, R. Crisafio, B. Dacke, and G. Falk, the teachers in charge of the classes, were enthusiastic in their participation and meticulous in their teaching of the prescribed lessons. Finally, Dr. T. Hull of the Department of Mathematics, University of British Columbia, provided the fa c i l i t i e s which made i t possible to process the data through the electronic computer, Alwac III-E. The function of this computer in performing auto-matically the calculations necessary for this study indicates the extent of the assistance which automation may make even now to the needs of research. These contributions are acknowledged with sincere appreciation. CHAPTER I INTRODUCTION I. THE PURPOSE OF THE STUDY The purpose of this study i s to determine experimentally the effectiveness of the group instruction use of certain manipulative materials in contributing to an understanding of particular decimal concepts. Stated in other words, the purpose is to ascertain whether there i s any significant difference in the achievement on a criterion measure of two unselected groups of Grade VII pupils. One group was taught by group demonstration with the use of instructional materials which are concrete and movable. The other group was taught by group demonstration with the use of instructional materials which are merely static representations of the manipulative devices. The study seeks to discover whether the characteristic of manipulability actually contributes to the pupils' understanding of decimal fractions when the particular materials are used in a prescribed manner. The primary concern of this investigation is with the practicality, not the essentiality, of the meaningful approach to teaching arithmetic. No matter how valid the Meaning Theory may be, i t s worth as a trend in arithmetic pedagogy depends upon the discovery of ways of transmitting theory into effective and economical teaching practices. During the past several years manipulative aids have been acclaimed by many competent authorities as effective means of making arithmetic meaningful. This study is an examination of one small area of the foundation for these claims. Its objective i s to add something to the search for materials of instruction that facilitate the communication of arithmetical meanings to children. The interest in teaching materials^it may be emphasized, is only a means to the end of securing better learning. II. DESCRIPTION OF THE PROBLEM This study i s an attempt to determine the effectiveness of certain manipulative materials in contributing to the pupils' understanding of specific decimal fraction concepts when the mani-pulative materials are used exclusively by the teacher in class demonstrations. The manipulative materials are designated: (l) place value charts, (2) place value cards, (3) wall rule with movable indicator, (4) flannel board. The effectiveness of these materials is determined by comparing the achievement on a selected criterion measure of an experimental group composed of 59 subjects in two classes located in different schools, and a control group composed of 88 subjects 3 in three classes also located in different schools. The criterion measure, administered at the end of the experiment, is Farquhar's Test of the Understanding of Processes with Decimal Fractions, which formed a minor part of an un-published Master of Arts Thesis in Education. 1 Since no way has yet been devised to identify the composition of a body of under-standings in arithmetic, Farquhar assumed that "a person's understanding of a process may be revealed by his ability to rationalize the procedure and his insight may become apparent by 2 his grasp of the 'why' behind the performance of the algorism." The specific decimal fraction concepts represented in the test were, according to the author, arbitrarily chosen. The participating classes were selected in accordance with 3 definite c r i t e r i a . .After being matched on the basis of size, they were assigned at random to the experimental or control groups. Suitable tests were administered at the beginning of the experiment to determine the status of the classes in four relevant areas: i n i t i a l understanding of concepts of decimal fractions, computational ability in decimal fractions, mental ability, and 4 reading a b i l i t y . By applying analysis of covariance to the results, the i n i t i a l differences between the groups in these areas were held constant. Both groups were subjected to teaching treatments intended to be identical in a l l details except insofar as the instructional materials are concerned. The experimental group was taught with 4 specified manipulative aids; the control group was taught with static representations of these aids commonly referred to as visual-ization materials. Further, these two types of instructional materials are intended to differ only in their capacity to represent concepts in movable forms. Manipulability, the basic characteristic of mani-pulative aids, constitutes the experimental variable. The extent to which the intended identity in teaching methods and materials used by the two treatment groups actually exists may be judged by examining the lessons contained in Appendix B, and the representations of the materials shown in the Figures on pages 39 to 42, inclusive. The teaching programme was designed to impose rigid controls in the conduct of the experiment, while the analysis of covariance technique was selected to impose statistical controls over the concomitant influences affecting the pupils' responses to the criterion measure. The imposition of these controls enables any difference between the treatment groups in achievement on the criterion measure to be attributed to the experimental variable. The hypothesis to be tested is that there is no difference in the performance of the two groups on the criterion measure which is attributable to the treatments involved. 5 :III. DELIMITATION OF THE PROBLEM Date and locale of the experiment. The administration of the experiment took place in May, 1957, and involved Grade VII classes located in five elementary schools in School District No. 36 (Surrey). Instructional materials. Subsection II of Chapter III contains a description of a l l the teaching aids employed in this study. The judging of the identity of the aids used by the two treatment groups, which was suggested on the previous page, may be facilitated by referring to Table II on page 36 . This table shows the corresponding aids employed by the two groups for the teaching of the particular objectives in each lesson. The extent of the identity in the corresponding aids used for each objective may then be judged by referring to the Figures on pages 39 to 42 , inclusive. No instructional materials of a special nature were used for those objectives which, though essential to the continuity of the lessons, did not refer to concepts included in the Farquhar test. Teaching programme. Subsection III of Chapter III contains a description of the lessons employed in this study. Eleven lessons comprise the total teaching programme. Of these, eight were devoted to the presentation of material new to the experiment, while three were reserved to review material 6 previously taught during the experiment. The schedule of lessons is contained in Appendix A. The duration of each lesson i s one hour. This includes thirty minutes of group instruction, fifteen to twenty minutes of seatwork, and ten to fifteen minutes for the marking of this seatwork "by the class. The entire set of lessons i s contained in Appendix B . The lessons for the experimental group are on pink sheets, while those for the control group are on yellow sheets. Testing programme. Subsection VII of Chapter III contains a brief outline of the testing programme, while subsections II to V, inclusive, of Chapter IV contain a complete, description and evaluation of the four tests used. These tests were administered at the beginning of experiment to measure the independent or concomitant variables in the four areas previously referred to. The results obtained were, obviously, unaffected by the treatments. The Farquhar test, one of the four, performed a dual function in the study. It was administered at the beginning of the experiment to measure one of the independent variables, and, in addition, i t was readministered at the close of the experiment to measure the dependent or criterion variable. 7 .:iV. JUSTIFICATION OF THE PROBLEM The desirability of teaching arithmetic according to the Meaning Theory i s now widely accepted in educational psychology. The best ways to teach meanings, however, are s t i l l a matter of uncertainty. To the question "How are meanings best developed?", Brownell ventured the following statement as part of his answer in an article written ten years ago: The problem (or group of problems) epitomized in the question above arises from the recognition of the fact that concepts, generalizations, etc., i f they are to be of real use, must be more than pat verbalizations. They must be ways of thinking meaningfully about arithmetic relationships. As yet we have l i t t l e exact knowledge with regard to v/ays and means for developing,-those meaningful thought processes. But we must find out. The problem of finding ways and means for developing meaningful thought processes in arithmetic is almost as unsolved, and certainly as urgent, today as i t v/as then. One noticeable development that has taken place since the Meaning Theory gained prominence in educational psychology has been the increased importance placed on the use of teaching aids supplement-ary to the text. Among these aids, manipulative materials rank high in the approval of those who espouse the Meaning Theory. In an article written in 1950, Busv/ell interpreted the Meaning Theory to include the judicious use of manipulative devices. In i t he stated: 8 We are only beginning to realize the important place that manipulative aids can play in learning. We have thought of them, usually, as devices to help pupils to get their answers. A more important use i s to show the thinking which l i e s back of the answers that they got. Used with intelligence and insight, manipulative aids may contribute much to superior thinking.^ In an article written i n 1952 on the subject "A Few Recommendations for the Improvement of the leaching of Mathematics", Lazar said: Let an abacus, or i t s equivalent, serve in the mathematics classroom in the same role as the demonstration models do in the science room - a constant source for the dis-7 covery of new laws and for the confirmation of hunches.' In an article written in 1953 on the subject "How to Make Arithmetic Meaningful in the Junior High School", Stein commented: ...arithmetic can be made meaningful in the Junior High School by u t i l i z i n g concrete situations and by moving gradually from the concrete to the abstract and symbolic. ... It is just as reasonable for Junior High School pupils to use markers, pegs, or an abacus to gain insight into the meanings of the operations as i t i s for them to study plants and animals objectively rather than from pictures in a book. Junior High School teachers should not consider i t beneath their dignity to util i z e concrete materials to develop abstract processes.^ Current educational periodicals show that a wide variety of manipulative aids are being used in the classrooms for the teaching 9 of arithmetic. Judging by the reports contained in these period-icals, i t appears that these aids are being used mainly to develop in pupils an understanding of arithmetical processes, although in some cases they are being used merely for computation purposes. The justification of this study lies in the fact that, while manipulative aids are becoming increasingly prominent in arithmetic 9 teaching, and obviously for the purpose of giving insight into number operations, there i s p r a c t i c a l l y no experimental evidence to prove that these highly recommended instruments of instruction are as effective for this purpose as they are claimed to be. The need for research i n th i s area i s a l l the more important because of certain opinions which suggest that some manipulative aids are used injudiciously. One such view i s that expressed i n 1953 by Van Engen, who i s an advocate of manipulative a c t i v i t y i n arithmetic teaching: Many of the manipulatory a c t i v i t i e s now "going the rounds" i n the world of mathematics instruction do not include those manipulatory a c t i v i t i e s which develop the concept, or concepts, for which they were intended.^ Previous studies which have i n any way involved the use of manipulative aids have not been primarily concerned with invest-igating their effectiveness i n contributing to teaching objectives. Instead, the purpose has been to investigate certain aspects of teaching, such as the computational and problem solving effects of teaching v/ith varying degrees of emphasis on meaning, i n which manipulative materials have been included only in c i d e n t a l l y . As a consequence, the findings of even the most relevant of these studies are not very helpful i n evaluating the effectiveness of these aids i n any phase of arithmetic instruction. The present study, which i s designed exclusively to determine the effectiveness of certain manipulative aids when used i n a prescribed manner and for a definite purpose, i s intended to provide an answer to one small aspect of the s t i l l pertinent question posed 10 ten years ago by Brovmell: "How are meanings best developed?" V. FACTORS DETERMINING THE CHOICE OF SUBJECT MATTER One factor which determined the choice of decimal fractions was the availability of a test designed to measure meanings and understandings. The difficulty of evaluating the development of understandings, which is admitted in the literature, is reflected in the scarcity of suitable tests to meaare this type of learning. The Farquhar test is considered by the present investigator to . be the most suitable of the tests which purport to measure under-standing in arithmetic. This consideration, therefore, was an important factor determining the choice of subject matter. Apart from the influence which the suitability of Farquhar's test had upon the choice of decimal fractions as the material for study in the present investigation, there was one other consideration which reinforced the wisdom of this choice. Decimal fractions is a teaching topic which offers many opportunities for the effective use of both manipulative and visual materials to clarify and extend meanings. References contained in "Teachers' Guide for Thinking with Numbers" indicate pages in the text where manipulative and visual materials may be used."'""'' In comparison to other topics of instruction in the text, the chapter on decimal fractions contains many concepts for the teaching of which these aids are recommended. Decimal fractions, therefore, seems to be a curriculum 11 area where the materials of instruction used in this investig-ation would be subjected to a fair test. VI,. FACTORS DETERMINING THE GRADE PLACEMENT OF THE EXPERIMENT The instructional programme in this study consists of reteaching decimal fractions with exclusive emphasis on conveying to the pupils an understanding of the concepts involved. The factor which determines the approximate level at which a study of this nature should be conducted is the concensus of opinion that the junior high school years are an appropriate time to reteach by using a meaningful approach those concepts previously taught in the elementary grades, often before pupils are mature enough to understand their significance. One such opinion is that expressed by Morton.who, after referring to the desirability of reteaching arithmetical concepts in the junior high school grades, states: ... there should be a carefully planned reteaching program covering what has previously been taught. The term "reteaching" means more than a mere review. It means teaching again, at a higher and more mature level, and more rapidly, what has been taught before.^ A similar opinion is held by Stein, who v/rites the following in an article which emphasizes the desirability of introducing a deliberate effort to teach arithmetic meaningfully in the junior high school grades: 12 ... the junior high school teacher, by using a meaningful approach, can help students to improve their computational s k i l l and to orient their thinking about arithmetic processes by (a) clarifying anew the nature of the number system and (b) teaching the rationale of the arithmetic processes as a basis for review and practice. The factor which determines the exact level at which this study should be conducted is the prevailing opinion that Grade VII is the most appropriate stage to reteach the concepts of decimal fractions with f u l l emphasis on a meaningful approach. For example, Morton advises in another work: "In general, i t should not be necessary to reteach decimals in Grade VIII". On this opinion there is general agreement to the extent of saying that Grade VII is the last grade at which i t should be necessary to reteach the entire f i e l d of decimal fractions to all.the pupils. To facilitate further the effectiveness of the lessons, the experiment was conducted as late as possible in the school year without encountering the usual end-of-the-term classroom interruptions. This ensured that a l l classes had the maximum opportunity to benefit from normal teaching procedures before being confronted with the concentrated teaching for understanding which took place during the experiment. VII.. LIMITATIONS OF THE STUDY Before using the evidence provided by this study for formulating generalizations respecting the contribution which these 13 particular manipulative aids make to the development of meanings and understandings in arithmetic instruction, the limitations of the study must be borne in mind. 1. The manipulation of the materials was'performed exclusively by the teacher as group demonstrations before the class. Most advocates of the use of manipulative aids would insist that their maximum effectiveness in contributing to the pupils' understanding of the concepts taught would depend upon involving the pupils individually in the acts of manipulation. A considerable amount of recently published literature emphasizes the relevance of mental activity to direct motor reaction. This suggests that manipulative aids may not be as effective in conveying the meaning of a concept when the learner's activity is confined to observing another person perform the manipulation. A review and discussion of this literature is contained in Chapter III. Because of the importance of this point of view, i t would be a fallacy to formulate from the evidence presented in this study any generalization concerning the effectiveness of these particular aids in situations where the pupils individually participated in the manipulation. 2. The manipulative materials were used in a limited number of lessons which were devoted exclusively to teaching the rationalization of processes, after the method of performing the processes had been taught. 14 As with the previous limitation, i t would be a fallacy to formulate from the evidence presented in this study any general-ization concerning the effectiveness of these manipulative aids in situations v/here the period of instruction is of longer duration and where i t permits emphasis on the rationalization of the concept to be interpolated with emphasis on the actual performance of the algorism appropriate to the concept. This limitation assumes some importance in view of the so-called HOW-WHY versus WHY-HOW controversy. This involves the question of whether the teaching of HOW the algorism is per-formed should precede or succeed the teaching of the WHY behind the performance of the algorism. The present study follows the HOW-WHY sequence. The effectiveness of the aids used in this experiment may well have been different either i f the sequence had been reversed or i f the rationalization had been presented concurrently with the d r i l l performed in teaching the algorism. Further reference to this controversy is found in Chapter III, which contains also a statement explaining v/hy these two limitations were imposed on the study. VIII., ORGANIZATION OF THE REMAINDER OF THE THESIS Chapter II is composed of two subsections. The f i r s t deals with the trends in arithmetic pedagogy leading to the present 15 popularity of the Meaning Theory. The search for materials of instruction which effectively communicate arithmetic meanings is obviously a worthwhile pursuit only i f the need to make arithmetic meaningful is considered important. The second subsection indicates the research involving manipulative materials which has already been undertaken. Chapter III deals with the planning, organization, and administration of the experiment. This account serves principally to show the extent to which the teaching treatments are the same for both the experimental and control groups, except with respect to the'experimental variable. In view of the fact that the purpose of the experiment is to discover whether the characteristic of manipulability of certain teaching aids actually contributes to the pupils' understanding of decimal' fractions, and thereby to determine the effectiveness of these aids in that regard, i t was necessary to ensure that this characteristic would emerge as the one experimental variable. In other words, the objective of this chapter i s to provide assurance that, as far as the treatments themselves are concerned, any difference in the performance of the two groups on the criterion measure may be attributed to this variable. The purpose of Chapter IV is to extend the objective of the previous chapter in order to provide assurance that any difference in the performance of the two groups on the criterion measure may f i r s t of a l l be attributed to the treatments involved, rather than 16 to any concomitant influences. Unmatched i n i t i a l differences in four areas of capacity and achievement are considered to comprise the total of these influences. The f i r s t subsection of Chapter IV presents an overview of the analysis of covariance technique which was employed to control these influences statistically. Subsequent subsections of the chapter, which contain a description and analysis of the various tests, are intended to serve as bases for evaluating the adequacy of the derived raw data which was subjected to the statistical analysis. Chapter V is devoted to this statistical analysis of the data.. Several steps are involved in the entire analysis. These steps may be grouped into major categories. The fir s t category includes making an analysis of variance of the criterion variable and each of the independent variables. ' The second category includes examining the nature of the correlations among the variables to ascertain whether the analysis of covariance wil l increase appreciably the test of significance. Since the nature of these correlations indicate that i t w i l l , the analysis was continued. The third category includes calculating the sums of squares of residuals and subjecting these residuals to an analysis of covariance, in which the F value is obtained and the final test of significance applied. The fourth category of steps involved in the entire analysis includes ascertaining that the assumptions underlying the application of the analysis of covariance have been satisfied. 17 Finally, the last subsection of Chapter V contains a brief account of the manner in which the raw data obtained in this experiment was processed through the electronic binary computer to result in the automatic performance, within approximately five minutes, of a l l the essential calculations relevant to the analysis of covariance technique. Chapter VI contains a summary of the experiment, the conclusions, reached on the basis of the statistical evidence, and some suggestions for further study. 18 FOOTNOTES • Hugh Ernest Farquhar, "A Study of the Relationship between the Ability to Compute with Decimal Fractions and an Under-standing of the Basic Processes Involved in the Use of Decimal Fractions", Unpublished Master of Arts Thesis in Education, University of British Columbia, 1955. 2 3 4 5 6 7 8 9 10 11 Ibid, p. 7. Infra, p. 48 Infra, p. 54 William A. Brownell, "Making Arithmetic Sensible", Journal of Educational Research, Vol. 40, September, 1946 - May, 1947, pp. 375 - 376. G. T. Buswell, "Study Pupils' Thinking in Arithmetic", Phi Delta Kappan, Vol. 31, 1950, p. 232 Nathan Lazar, "A Few Recommendations for the Improvement of the Teaching of Mathematics in the Secondary School", Progressive Education, Vol. 29, 1952, p. 21 Harry L. Stein, "How to Make Arithmetic Meaningful in the Junior High School", School Science and Mathematics, Vol. 53, 1953, p. 682. For example, the November and December, 1956, issues of "The Arithmetic Teacher" contain four accounts of the classroom use of various manipulative aids. Henry Van Engen, "The Formation of Concepts", Learning of Mathematics: Its Theory and Practice, Twenty-first Yearbook of the National Council of Teachers of Mathematics, (New York: Bureau of Publications, Teachers College, Columbia University, 1953), p. 87. Leo J. Brueckner, Foster E. Grossnickle and Elda L. Merton, Teachers' Guide for Thinking v/ith Numbers, (Toronto: John C. Winston, 1953.) 19 R. L. Morton, "Teaching Arithmetic", No. 2 of Series: "What Research Says to the Teacher", Department of Classroom Teachers of the National Educational Association, 1953, p. 12. Stein, op. c i t . , p. 680 R. L. Morton, Teaching Children Arithmetic, (New York: Silver Burdett Company, 1953), p. 349. 20 CHAPTER II REVIEW OF RELATED LITERATURE Two purposes underlie the review of the literature relevant to this study. The f i r s t purpose is to show the background of the problem by tracing the developments in arithmetic psychology which have led to the present emphasis on meaning and understanding in the teaching of arithmetic. The second purpose i s to reveal the exact research which has already been performed in connection with the use of manipulative materials in the teaching of this subject. I. CHANGING CONCEPTS IN ARITHMETIC PSYCHOLOGY The present experiment is an effort to determine the effectiveness of group instruction use of certain manipulative aids in contributing to an understanding of decimal fraction concepts. Confined to specific manipulative aids used in specified situations, this study deals with only a small portion of one avenue in the search for practical materials which will be helpful to teachers in making arithmetic meaningful to their pupils. To be fruitful this search in general, as well as a l l parts of i t , must be motivated by a knowledge of the trends in arithmetic psychology leading to the present emphasis on meaning, 21 and also an awareness of why each successive stage has yielded to another in the process of this development. This section of the review deals briefly with these two matters. Compendiums of opinion found in various yearbooks and other publications during the past quarter of a century reflect the changing concepts in arithmetic psychology. During this period three stages are evident, though they are not necessarily consecutive. One stage is that marked by the popularity in teaching practices of the d r i l l theory. In the 1930 Yearbook of the National Society for the Study of Education, a chapter written by P. B. Knight proposed methods of teaching arithmetic which clearly shows the application of the prevailing stimulus-response psychology.^" The exclusive purpose of the methods suggested by this author is to present number stimuli repetitively to the pupils in order to facilitate their ability to make correct responses. D r i l l was considered to be the prime factor in teaching; and the accumulation of a repertoire of specific responses was believed to be the major end of learning. There is no question that this was the prevailing psychology for many years prior to 1930. A second stage is that marked by the popularity of the Incidental Learning Theory in the psychology, i f not in the practice, of teaching arithmetic. In the 1935 Yearbook of the National Council of Teachers of Mathematics there is a chapter entitled 2 "The New Psychology of Learning" by R. H. Wheeler. The Gestalt psychology was an important influence upon the Incidental Learning 22 Theory which Wheeler sought to implement in his recommended teaching procedures. He wrote: "The whole purpose of arithmetic is to discover relationships and to be able to reason with 3 numbers." This desirable principle, which shows the application of the Gestalt psychology, had the unfortunate consequence of leading to the Incidental Learning Theory and to the implication that it , was necessary to forget d r i l l and to concentrate instead on projects designed to simulate functional situations. Wheeler expressed the idea thus: "Do not try to teach arithmetic; teach 4 discovery, l i f e , nature, through arithmetic". The proponents of this theory believed that i f a situation involving quantity happened to arise during a project the child would be motivated to grasp, and then to use, the number ideas. A third stage is that marked by the popularity, at least in educational literature, of the Meaning Theory. It is interesting to note that the first articulate presentation of the Meaning Theory is also contained in the same Yearbook in a chapter entitled " Psychological Considerations in;:the Learning and Teaching of 5 Arithmetic" by W. A. Brownell. Brownell's criticism of the Drill Theory is implicit in these words: The teacher need give l i t t l e time to instructing the pupils in the meaning of what he is learning; the ideas and skills involved are either so simple as to be obvious even to the beginner, or else they are so abstruse as to suggest the postponement of explanations until the child is older and is better able to grasp the meaning.^ 23 His criticism of the Incidental Learning Theory is explicit in these words: Incidental learning, whether through "units" or through unrestricted experience, is slow and time consuming. ... Such arithmetic ability as may be developed in these circumstances is apt to be fragmentary, superficial, and mechanical Brownell's own position is between these two extremes. The Meaning Theory stands in marked contrast to the theory which placed such reliance on the d r i l l of isolated number facts. At the same time i t stands opposed to the feasibility of giving up a l l organized learning experience in arithmetic because a prevalent method had been formalistic and mechanical. In his own words Brownell explains that position: This name (the Meaning Theory) is selected for the reason that, more than any other, this theory makes meaning -the fact that children shall see sense in what they learn -the central issue in arithmetic instruction. ... Within the "meaning" theory the virtues of d r i l l are frankly recognized. There i s no hesitation to recommend d r i l l v/hen those virtues are the ones needed in instruction.^ , In the 1941 Yearbook of the National Council of Teachers of Mathematics, published only six years after the Yearbook which contained the two chapters containing the extremely different approaches to the teaching of arithmetic, there is unmistakable evidence that the Meaning Theory was growing in favor. A chapter by T. R. McConnell reaffirmed the place of organization in learning 9 and the concept that learning arithmetic is a meaningful process. By the time the 1951 Yearbook of the National Society for the Study of Education was published (an issue devoted to the 24 teaching of arithmetic) the Meaning Theory was so generally accepted by educationists that Horn wrote: "They (the members of the Yearbook Committee) favor the meaning theory, involving the active processes on the part of the pupils of discovering relation-.. 10 ships, of u t i l i z i n g concrete experiences, and of generalization". A final authorative statement by Dawson and Ruddell in 1955 brings the development up to date: Evidence supporting the meaning theory approach to arithmetic is not complete but i t is impressive when i t is noted that no such evidence is being accumulated to support other theories of instruction.^ Therefore, the place has been reached where the primary question no longer i s : "Should we teach meanings?". The important issues now, certainly from the standpoint of research, are suggested by the questions: "What constitutes the basic arithmetic under-standings?" and "What are the most effective materials for, and methods of, instructing pupils in these understandings?". The fact that the issues suggested by these two questions are not new is indicated by Brownell in an article written almost twenty years after his f i r s t presentation of the Meaning Theory: It is not too much to say that one of the major develop-ments in the past twenty years or so has been the attempt to discover just what this concept of meaningful learning implies for the arithmetic program. One aspect of the development has been the effort to identify the meanings - ideas, principles, relationships, generalizations - that are essential to arithmetic learning. Another aspect of the movement toward meaningful learning is revealed in the search for more effective learning materials and methods of instruction.^ The growing acceptance of the Meaning Theory, which has introduced more serious attempts to implement the theory of meaningful 25 learning into the practice of meaningful teaching, is the circum-stance which makes these two questions of major current importance. The findings of the present study provide some information, positive or negative, with respect to the general area of investigation suggested by part of the second question: "What are the most effective materials for instructing pupils in arithmetic understandings?" Since this experiment is an effort to determine the effective-ness of a specific group of these materials, namely, those which are manipulative in character, the next section of this review contains an account of the reported research which in any way involves the use of manipulative materials in the teaching of arith-metic. II. REPORTED RESEARCH INVOLVING THE USE OF MANIPULATIVE MATERIALS IN THE TEACHING OF ARITHMETIC The 1951 Yearbook of the National Society for the Study of Education contains a chapter entitled, "Proposals for Research on 13 Problems of Teaching and of Learning in Arithmetic". Foster E. Grossnickle 1s contribution to this chapter contains a proposal for research dealing with the use of manipulative materials in the teaching of arithmetic. Research in this area::was represented at that time to be of urgent importance. Since that time, however, the use of manipulative materials 26 has been subjected to extremely l i t t l e experimentation. One experiment somewhat related was reported by Dawson and 14 Ruddell. The purpose was: ... to compare the relative effectiveness of common text-book practices in the introduction of the division of whole numbers with an experimental procedure based on a subtractive approach and a greatly expanded use of visualization devices. One of the several questions to which answers was sought was: Will achievement be affected adversely i f practice through object manipulation and visualization of process replaces much of the paper and pencil drill? ^ The experimental group used counting discs, spool boards, and place value charts. Dawson and Ruddell stated: The data may be interpreted to advocate a teaching procedure which utilized manipulation of representative materials. Higher achievement, greater retention, and an increased ability to solve examples in a new situation were found in the experimental group which devoted time to the development of meanings, principles, and generalizations, through the use of manipulative materials and visualization materials.^ Another study also somewhat related was reported by Martha Norman. The purpose was: ... to investigate the effects of three methods of teaching certain basic division facts to third grade children. The three teaching methods were named the textbook, the con-ventional, and the developmental. Each method was designed to vary in degree of emphasis on meaning.^ The findings of this study are relevant to the present investigation insofar as the developmental method, which possessed the greatest degree of emphasis on meaning, involved the use of such 27 manipulative materials as the number l i n e , counters, and number charts. As the names suggest, the other methods involved the use of various non-manipulative materials. Data obtained from one test, which was used as a pre-test, an immediate r e c a l l test, and a delayed r e c a l l test, were analyzed to compare the effects of the teaching methods used i n the 8 forty minute lessons which comprised the instructional programme. Provision was contained i n the test to measure pupil achievement i n both facts taught, and facts not taught. Among the various conclusions reached, those which are of interest i n the present study may be summarized as follows. F i r s t , there are no significant differences among the three teaching method groups i n the immediate r e c a l l of taught facts. Second, i n the delayed r e c a l l of taught facts there i s a difference, significant between the .05 and .01 l e v e l s , among the three teaching method groups. That i s , while the developmental and conventional methods are each more effective than the textbook method i n the delayed r e c a l l of taught facts, there i s no significant difference between the effectiveness.of the f i r s t two mentioned methods. On the basis of a l l the conclusions reached, th i s f i n a l statement i s reported: This finding implies that conventional procedures may be more effective i n ordinary classroom situations when teachers are given specific directions and when pupils are motivated to learn. Developmental procedures profitably may be used part i c u l a r l y i n the early stages of presenting d i v i s i o n to t h i r d grade children. However, research i s needed to refine the principles and teaching procedures which are outgrowths of the meaning theory.19 28 Several other studies reported in the literature involve the use of manipulative materials in one way or another. Except in minor details, such as grade level, duration of the experiment, number of treatment groups, and so forth, these studies are similar to the ones v/hich have just been discussed. It must be noted that this area of experimentation is not entirely relevant to the present study. Important differences exist. The experimental purpose is the f i r s t way in which the two studies described differ from the present one. These two studies, which parallel each other very closely, were designed for the purpose of testing the role of meaning in teaching to attain certain objectives. This was done by comparing a meaningful method with a control method either in which recognizably conventional procedures were followed or in which d r i l l exercises were emphasized. Manipulative materials were only incidentally introduced into the experimental method because they were regarded by the authors to be the most effective materials by which to teach meaningfully the specified objectives. The present study, on the other hand, was designed for the purpose of testing, deliberately and exclusively, the, effectiveness of the role of manipulative aids in teaching to attain certain objectives. The teaching objectives is the second way in which the two studies described differ from the present one. The objective of the teaching in the former is the pupils' attainment of s k i l l in 29 certain arithmetical operations, while the objective of the teaching i n the present study i s the pupils' attainment of an understanding of certain arithmetical concepts. It i s seen, therefore, that there are important differences betv/een this experiment and previous experiments which have i n any way involved the use of manipulative aids i n the teaching of arithmetic. Previous experiments of that nature have been mainly concerned with the investigation of the computational and problem solving effects of teaching with varying degrees of emphasis on meaning. Manipulative materials have been included quite incidentally i n the teaching aids used i n these experiments, and, as a consequence, the problem of determining their effectiveness i n contributing to the teaching objectives has been i n each case a very subordinate part of the total investigation. The growing acceptance of the Meaning Theory should produce i n the future more experimentation designed exclusively to ascertain the effectivness of teaching materials, especially manipulative aids, i n conveying to pupils an understanding of arithmetical concepts. 30 FOOTNOTES F. B. Knight, "Some Aspects of Modern Thought on Arithmetic", The Teaching of Arithmetic, Twenty-ninth Yearbook of the National Society for the Study of Education, Part I, (Bloomington, I l l i n o i s : Public School Publishing Company-,-, 1930), pp. 145-267. R. H. Wheeler, "The New Psychology of Learning", The Teaching of Arithmetic, Tenth Yearbook of the National Council of Teachers of Mathematics, 1935, pp. 230-263. Ibid, p. 247 Ibid, p. 243 William A. Brownell, "Psychological Considerations i n the Learning and Teaching of Arithmetic", The Teaching of Arithmetic, Tenth Yearbook of the National Council of Teachers of Mathematics, 1935, pp. 1-31. Ibid, p. 2 Ibid, p. 17 Ibid, p. 19 T. R. McConnell, "Recent Trends i n Learning Theory: Their Application to the Psychology of Arithmetic", Arithmetic i n General Education, Sixteenth Yearbook of the National Council of Teachers of Mathematics, 1941, pp. 268-289. ^ ' E r n e s t Horn, "Arithmetic i n the Elementary School Curriculum", The Teaching of Arithmetic, F i f t i e t h Yearbook of the National Society for the Study of Education, Part I I , (Chicago, I l l i n o i s : The University of Chicago Press, 195l), p. 215. 6 7 8 9 11' Dan T. Dawson and Arden K. Ruddell, "The Case for the Meaning Theory i n Teaching Arithmetic", Elementary School Journal, Vol. 55, 1955, p. 394 31 12 William A. Brbwnell, "The Revolution in Arithmetic", The Arithmetic Teacher, Vol. 1, 1954, p. 4. 13 14 15 16 17 18 G-. T. Buswell, "Proposals for Research on Problems of Teaching and of Learning in Arithmetic", The Teaching of Arithmetic, F i f t i e t h Yearbook of the National Society for the Study of Education, Part II, 1951 p. 285. Dan T. Dawson and Arden K. Ruddell, "An Experimental Approach to the Division Idea", The Arithmetic Teacher, Vol. 2, 1955, pp. 6-9. Ibid, p. 6 Ibid Ibid, p. 8 Dissertation Abstracts, Vol. XV, (Ann Arbor, Michigan: University Microfilms, 1955), p. 2134. 19 * Ibid 32 CHAPTER III THE PLAN AND ADMINISTRATION OF THE EXPERIMENT This chapter contains an account of the planning involved in organizing the experiment, a description of the administrative arrangements undertaken in connection with its performance, and, finally, a statement of the limitations, and reasons for the limitations, imposed upon the study. I. STEPS IN PLANNING THE EXPERIMENT The f i r s t step in planning the experiment was to select a suitable teaching unit. It v/as considered essential to confine the subject matter to one homogeneous unit, and thereby to restrict the duration of the teaching assignment, in order to maintain adequate controls in the performance of the experiment and to reduce to a minimum the number of materials of instruction required. Subsection >:V of Chapter I contains an account of the two factors which determined the choice of decimal fractions as the unit on which to test the effectiveness of the particular mani-pulative aids used in this study. The second step in planning the experiment was to decide on the grade level most appropriate for the purpose of communicating 33 to pupils an understanding of decimal fractions. Subsection VI of Chapter I contains an account of the factors which determined the grade placement of the experiment. The third step in planning the experiment was to delineate the lesson areas. As a basis for establishing a plan, Farquhar's test was analyzed question by question. Lesson topics were then formulated, f i r s t , in accordance with the existence of Farquhar questions to evaluate each lesson, and, second, in accordance with the teaching practices currently recommended in the published materials which were consulted in preparation for writing the lessons. These published materials, which are listed in the f i r s t section of the bibliography, include the most recent arithmetic texts, books dealing with the teaching of arithmetic, and brochures advertising commercially prepared materials of instruction. The plan of the lessons which evolved from this procedure is presented in Table I. The fourth step in planning the experiment was to subdivide each lesson area into component lesson objectives. Appendix A contains a summary of the lesson objectives in the entire set of eight lessons. In most cases each objective formed a separate entity within the lesson area, although in a few cases more than one objective could be grouped because they lent themselves to common teaching treatment. For the teaching of each objective, or group of objectives, a time allotment was decided upon. The amount of time assigned 34 TABLE "I PLAN OF LESSON TOPICS WITH THE NUMBERS OF THE FARQUHAR QUESTIONS APPROPRIATE TO EACH TOPIC Lesson Number Farquhar Question Number Lesson Topic I n i l Introductory Lesson II 4, 11, 20 Identification and Meaning of Place Names in mixed decimal fractions III 5, 7, 10, 15, 27 Reduction of Decimals to common fractions IV 3, 8, 9 The use of zero as a place holder V 6, 13, 16, 19 Changing the location of the decimal point: i t s effect on the value of the expression VI 18, 21, 22, 28, 30 Rounding decimal fractions VII 12, 14, 17, 23, 24 Division involving decimal fractions VIII 1, 2, 25, 26, 29 Miscellaneous concepts involving decimal fractions (changing common fraction to decimal fraction, add-iti o n and multiplication) 35 depended upon the number of Farquhar questions devoted to the teaching of each objective, or group of objectives, and also upon the evident complexity of the teaching task involved. In the case of Lesson I, which is an introductory lesson area for general orientation purposes, the f i r s t mentioned factor was not a consid-eration because there are no Farquhar questions to evaluate this topic. The f i r s t three columns of Table II contain a summary of the planning involved up to this point. The f i f t h , and last, step in planning the experiment was to select the materials of instruction considered most effective for the teaching of each objective in the entire eight lessons. The materials selected are designated in the last two columns of Table II. These materials may be identified by referring to the representations of the materials shown in '.'Figure's;'"; on pages 39 to 42. The foremost consideration in selecting these materials was the necessity to ensure that the aids used by the two treatment groups embodied as far as possible the same characteristics except the capacity to be manipulated. A classification of arithmetic teaching aids contained in "Teachers' Guide for Thinking with Numbers" by Brueckner, Grossnickle, and Merton, one of the source materials, proved valuable in making the selection. These authors classify the aids into four groups, each of which possesses the characteristics briefly described. 36 TABLE II NAMES OF INSTRUCTIONAL MATERIALS USED BY EXPERIMENTAL AND CONTROL CROUPS, AND THE TIME ALLOWED, FOR TEACHING THE OBJECTIVES OF EACH LESSON Number and Number of Time Instructional Title of Lesson Alot- Materials Used by Lesson Objective ment (Minutes) Exper-imental Control Group ; Groups (No . of Card) Lesson I Objectives i 1 & 2 12 n i l n i l Objective 3 18 P.V. Charts 1 to 6 Lesson II Objectives 1 , 2 15 P.V. Charts 7, 8, ! & 3 Objective 4 15 P.V. Cards 10 Lesson III Objective 1 8 n i l n i l Objective 2 12 P.V. Carts 7, 8, ! Objective 3 10 n i l n i l Lesson IV Objective 1 20 P.V. Charts 9, 10 Objective 2 10 P.V. Cards 10 Lesson V Objective 1 ' 15 P.V. Charts 9, 10 Objective 2 15 P.V. Cards Same Same Lesson VI Objective 1 15 Wall rule 11 Objective 2 8 Wall rule 11 Objective 3 7 n i l n i l Lesson VII Objective 1 8 n i l n i l Objective 2 22 Flannel Board 12 Lesson VIII Objective 1 12 Flannel Board 12 Objective 2 12 Wall rule 11 Objective 3 6 n i l 13 37 1. Manipulative materials. These materials provide the highest level of concreteness in the presentation of an arithmetical idea. The idea is represented in an actual object, capable of being manipulated. 2. Visualization materials. These materials provide the second highest level of concreteness. The idea is not shown in the form of a concrete and movable object; rather, i t is in the form of a representation of the object, drawn on a chart, with arrows to indicate the movement or the thinking necessary to arrive at an answer. 3. Illustration materials. These materials provide the third highest level of concreteness. The component mental processes necessary to formulate the answer are not shown. It is merely the answer which is illustrated. 4. Abstract symbolism. These materials provide the lowest level of concreteness. The symbol is not in any way anchored to it s referrent, except in so far as the learner is capable of providing this link in his own imagination. In order, therefore, to make the instructional materials used by the experimental and control groups as nearly alike as possible in a l l characteristics except the capacity to be manipulated, i t was necessary to choose from the f i r s t two of the above mentioned categories of materials. The selected materials u t i l i z e instructional ideas and principles commonly referred to, though in some cases under different 38 names, in the various sources of reference consulted. These materials, which were constructed by the experimenter, are described in the following subsection. II. DESCRIPTION OF THE INSTRUCTIONAL MATERIALS The f i r s t four materials described are the instructional aids used by the experimental group. These manipulative devices, with the exception of the flannel board, are represented in Figure 1. The next set of materials described are the instructional aids used by the control group. These visualization materials, which are designated by number in an entirely arbitrary manner, are represented in Figures "2,' 3,. and. 4. Place value charts. Made of \ inch plywood, these charts are each one foot square. The decimal point is on a chart which is one foot by six inches in size. Seven charts represent place values extending from thousands to thousandths. The ONES' chart, occupying the central place in our system of notation, is painted red, while the decimal point is a red dot on a white background. This colour arrangement was chosen to emphasize that the primary function of the decimal point is to designate the location of the ONES' digit. In order to present the visual symmetry of the different place values around the ONES', place, the decimal point was, in the actual teaching process, placed in front of the ONES' board and towards the right edge, Place Value Charts Figure 1. Illustrations of Manipulative Materials 40 V i s u a l i z a t i o n Card No. 1. TENS ONES • • V i s u a l i z a t i o n Card No. 3 TENS ONES • • • • • • V i s u a l i z a t i o n Card No. 3 HUNDREDS TENS • V i s u a l i z a t i o n Card No. 4 HUNDREDS TENS ONES ff^^l *M\%m\\ mMmmX Figure 2. I l l u s t r a t i o n s of V i s u a l i z a t i o n Materials V i s u a l i z a t i o n Card No. 5 HUNDREDS TENS ONES minim V i s u a l i z a t i o n Card No. 6 HUNDREDS TE" ONES • V i s u a l i z a t i o n Card No. 7 TENTHS HUNDREDTHS THOUSANDTHS V i s u a l i z a t i o n Card No. 8 TENTHS THOUSANDTHS • ft ^ W W Figure 3 . I l l u s t r a t i o n s of V i s u a l i z a t i o n M a t e r i a l s 42 V i s u a l i z a t i o n Card No. 9 1000's 100's 10's 1 Othfl 1 OOtVis month « M 2 • V i s u a l i z a t i o n Card No. 10 0 V i s u a l i z a t i o n Card No. 11 V i s u a l i z a t i o n Card No. 12 A .1 .1 .1 .1 .1 .1 1 1 1 V i s u a l i z a t i o n Card No. 13 ONES TENTHS HUNDREDTHS THOUSANDTHS 1 1 1 1 • • • 0 0 1 0 1 y .... 1 Figure 4. I l l u s t r a t i o n s of V i s u a l i z a t i o n Materials 43 rather than entirely to the right as shown in the Figure 1. To emphasize further the symmetry of the number system, the corresponding place values on either side of the ONES' place have the same colours. This colour arrangement - blue for TENS and TENTHS, green for HUNDREDS and HUNDREDTHS, and yellow for THOUSANDS and THOUSANDTHS - was followed consistently in the pre-paration of a l l the materials. Each chart holds 30 hooks. Cardboard tickets, 1^ inches by 3 inches, were supplied in the same colours as the charts. Place value cards. Made of \ inch plywood, these cards are each one foot square. They employ the same decimal point arrangement and colour scheme used in the place value charts. These materials are designed to present the actual relation-ship in size of the positional values extending from ONES to THOUSANDTHS. The wider range of place values from THOUSANDS to THOUSANDTHS, besides being'difficult to present within the limitations of a reasonable amount of materials., was considered un-necessary. The idea of the relationship in size of-the positions to the l e f t of ONE is adequately conveyed by the previously described place value charts, where a bundle of ten tickets represents ten, a bundle of one hundred tickets represents one hundred, and so on. Since this procedure could not be-used to the right of ONES' place, the place value cards had to be used to present the actual relationship in size of the positional values extending in this direction. 44 Rule with movable indicator. Made from a board 4 feet long, 3>r inches v/ide, and \ inch thick, this aid includes a movable indicator. The rule involves the same principle found in such pupil materials as the decimal fraction cards and number line, and differs mainly in that i t is designed for group instruction. The entire length of the rule represents one unit (that i s , from the integer marked "1" .to the integer marked "2"). Placing the integer "1" at the beginning of the measurement is to facilitate the explanation of rounding to the nearest whole number. The integers which designate whole numbers appear in red, while the subdivisions into TENTHS appear in blue, and the further subdivisions into HUNDREDTHS are indicated by green markings. Flannel board. Since this is a conventional and widely used type of teaching aid, i t is not illustrated in Figured I. The dimensions of the flannel boards used in the experiment are 4 feet by 2 feet. The manipulative materials attached to the boards are made of lightweight paper. Illustrations of the various materials are shown in the appropriate sections of Lessons VII and VIII of the experimental group. Visualization materials. These materials, used by the control group, are identified merely as Visualization Cards 1 to 13. Heavyweight paper was used for a l l thirteen cards. The cards designated as 1 to 8, inclusive, 12, and 13, are each 36 inches by 24 inches in size; Card 9 is 36 inches by 12 inches; Card 10 '. 45 is 3 6 inches by 18 inches; and Card 11 i s 3 6 inches by 6 inches. As mentioned previously, the characteristic of the visualization materials is that the arithmetical idea is presented in the form of a representation of a concrete object, drawn on a chart, with arrows to indicate the movement or the thinking necessary to arrive at an answer. By suspending these materials from the moulding at the top of the blackboard, the necessary arrows could be drawn on the blackboard in the presence of the class. III. DESCRIPTION OF THE LESSONS In subsection I i t was stated that the third, fourth and f i f t h , steps in planning the experiment were, respectively, to delineate the lesson areas, to subdivide each lesson area into component lesson objectives, and to select the materials of instruction. The selected materials, described in subsection II, were then constructed. When these steps were accomplished the next undertaking was to prepare the lessons. The construction of the eight lessons involved, v/hich are contained in Appendix B., proved to be a major part of the work entailed in organizing the experiment. It was imperative to include in the lessons only carefully planned procedures to which the aids used by both treatment groups are adaptable, and in which both types of aids are provided with opportunities, intended to be 46 as equal as possible, of contributing to the pupils' understanding of decimal fractions. Furthermore,.; since i t is the capacity of the teaching aids to be manipulated which constitutes the single experimental variable, the lessons had to be equalized for the two treatment groups in every detail except those related to this variable. The lessons for the experimental group are on pink sheets, while those for the control group are on yellow sheets. The fi r s t three lessons are accompanied by introductory material v/hich is common to both groups. This material, which is on v/hite sheets, is intended primarily to provide the teachers with a common background knowledge of the Hindu-Arabic system of notation. The format of the lessons, as well as the general instruc-tions and the number of steps involved in the presentation, is identical for the experimental and control groups. In addition to the regular purple lettering, three special colours are employed consistently throughout a l l the lessons for the following purposes: RED lettering indicates the statement of each objective, and the teaching time allowed to achieve i t ; GREEN lettering indicates instructional directions or summaries of a more general nature than is contained in the detailed steps of the lessons; BLACK lettering indicates the generalizations which the pupils are expected to formulate in their own words after the presentation of the whole lesson or part of i t . 47 As mentioned in subsection III (Chapter l):., the entire time for each lesson is one hour. This includes thirty minutes of group instruction, fifteen to twenty minutes of seatwork, and ten to fifteen minutes for the marking of the seatwork by the class. The general psychology of the teaching procedure is to present the various concepts at the lowest level of abstraction permitted by the particular teaching aids used. Emphasis on the meaningful relationship of these ideas is developed through a process of induction which leads to the concluding direction for each objective. This concluding direction indicates that the teachers are to "draw" from the pupils the generalizations which they have formulated, not by pat verbalizations but by their own insight and understanding. These generalizations are the concepts to the understanding of. which the instructional aids are expected to contribute. There i s no question on the Farquhar test which is not covered by a suitable generalization. In cases where there was no Farquhar question to test a concept considered essential to the development of the whole lesson, the concept was taught without the use of the teaching aids. In the schedule of lessons contained in Appendix A- i t w i l l be observed that arrangements were made for three review lessons: one following the third lesson, another following the sixth lesson, and the third review following the eighth lesson. To maintain adequate controls over the use of the teaching aids, these lessons were confined-to'a recapitulation of the lesson worksheets (contained in Appendix C) which accompanied each lesson. The incidental review of concepts previously taught was conducted without 48 using the aids. No formal outline was provided for the teaching of these review lessons. IV. SELECTION OF CLASSES TO PARTICIPATE IN THE EXPERIMENT Q In School District No. 36 (Surrey) there were thirty-seven elementary schools at the time the classes were being selected (January, 1957). The mean number of divisions in each school was six. Since there were no junior high schools, a l l the Grade VII and Grade VIII classes were located in elementary schools, although not a l l the elementary schools had Grade VII and Grade VIII classes. In order of enrollment the six largest schools had nineteen, eighteen, thirteen, twelve, ten, and ten, divisions. There were twenty-eight schools which had Grade VII classes. In nineteen of these schools the Grade VII pupils were either grouped with pupils of another class, or they were divided on a homogeneous grouping basis into two or more classes. Excluding these nineteen schools, which were obviously unsuitable to participate in the study, there were nine schools with completely unselected Grade VII classes. The three main criteria used for the selection of the five classes considered necessary to participate in the study were: (l) the teacher's interest in, and aptitude for, taking part in an educational experiment, (2) the teacher's experience and ability in classroom management, (3) the teacher's normal adherence to 49 reasonably conventional teaching methods.1 The five schools selected are located within a three mile radius centering on Whalley. Comparatively homogeneous socio-economic conditions exist within the area. Appendix A contains the f i r s t communication concerning this study which the experimenter had with the teachers who were to participate. Although two details noted in this letter were later changed, the original choice of teachers and classes remained. The five teachers who took part are male. V. ASSIGNMENT OF CLASSES TO THE TREATMENT GROUPS . The analysis of covariance •> statistical design used in this experiment is valid only i f certain conditions surrounding the conduct of the experiment are satisfied. One condition, which Lindquist says."has perhaps most often been violated with serious 2 consequences" in educational research, concerns the manner of selecting the treatment groups. In a controlled experiment, i f one is safely to conclude from a significant F that the experimental and control treatments have been responsible for producing different results, then i t is necessary, in Lindquist's words, to assume that "the subjects in each treatment group were originally drawn either (!a) at random from the same parent population, or (b) selected from the same parent population on the basis of their X-measures only ..." 50 In this study, the classes were assigned to the treatment groups in a manner intended to satisfy the f i r s t of these alternate assumptions, and at the same time to take into one other consideration. This consideration was the size of the classes. It is understandable that the effectiveness of the instructional aids would be related to the size of the classes in which they were used. For example, in the case of observing a group demonstration, the pupils at the back of a large class would be at a disadvantage in comparison with the pupils at the back of a small class. The size of the classes, therefore, is a concomitant variable which could not be controlled except through the procedure of pairing the classes.' As the f i r s t step in this procedure, the classes with approximately equal enrollments, as they were immediately prior to the experiment, were considered as a unit. Three smaller classes (in General Montgomery, Hjorth Road, and Simon Cunningham Schools) formed one such unit, referred to as Unit A; two larger classes (in Prince Charles and Fleetwood Schools) formed another such unit, referred to as Unit B. As the second step in this procedure, the classes for the experimental and control groups were selected from each of these units by the random method of tossing a coin. It was previously decided that two classes should be in the experimental group and three classes in the control group. From the f i r s t unit containing the three smaller classes, one was selected at random for the experimental group, thus leaving two classes for the control group 51 From the second unit containing the two larger classes, one was selected at random for the experimental group, thus leaving one class for the control group. Table III shows the composition of the treatment groups, together with the class enrollments. TABLE III NUMBER OF PUPILS IN THE CLASSES ASSIGNED TO EACH TREATMENT GROUP School Treatment Group Enrollment prior to Experiment Net Number Studied After Eliminations Unit A General Montgomery Experimental 25 23 Hjorth Road Control 27 25 Simon Cunningham Control 27 26 Unit B Prince Charles Experimental 39 36 Fleetwood Control 42 37 Total in experimental group 64 59 Total in control group 96 88 Total in both groups 160 147 52 Pupils who were absent for any of the eleven prescribed lessons, or who missed any of the tests, were eliminated from the study. This accounts for the withdrawal of the thirteen pupils noted in Table III. VI. ACCOUNT OF ORIENTATION AND EVALUATION MEETINGS The selection of the teachers to participate in this study was made early in January, 1957. Originally scheduled for February, the experiment was eventually held in May. During this time the teachers had an opportunity to orient themselves in a general way to the experimental idea, which was discussed with them by the experimenter during this time. After the selection of the classes for each treatment group, . separate orientation meetings we're held for the teachers of each group. The experimental group teachers had two pre-instructional meetings, and the control group teachers had tv/o similar meetings. At the f i r s t of these tv/o meetings with each group, the experimenter distributed the materials for the fi r s t four lessons. This pair of meetings (one for the teachers of the experimental group, the other for the teachers of the control group) was held on consecutive days immediately prior to the commencement of the experiment. The distributed materials included: the teaching aids, the introductory material to the f i r s t three lessons (white sheets), lessons I to IV (pink and yellow sheets), and the.worksheets I to IV. 53 The experimenter demonstrated the teaching of each lesson to the teachers of each group, and provided an opportunity for a f u l l discussion of any issues that were raised. At the second of the two meetings with each group, the experimenter distributed the materials for the last four lessons. This pair of meetings (one for the teachers of the experimental group, the other for the teachers of the control group) was held on consecutive days during the course of the teaching of the f i r s t four lessons. The procedure noted above, and relevant to lessons V to VIII, was followed. During the two week period from Monday, May 13th to Tuesday, May 28th, when the experiment was in progress, the experimenter visited the teachers at least twice each week, and on the remaining days he contacted them by telephone. They were invited, and encouraged, to communicate with him by telephone in the event of any problem arising. Shortly after the close of the experiment, on Thursday, May 30th, an evaluation meeting was held with a l l five participating teachers. At the outset of the experiment, the teachers had been asked to make preparations for the concluding meeting by f u l f i l l i n g two requests: (l) to keep a diary of their experiences in the teaching of the lessons and (2) to complete an evaluation form which \ was distributed at the second pair of pre-instructional meetings. This form is contained in Appendix A. 54 This information was used in considering the implications of the conclusions reported in Chapter VI. VII. ADMINISTRATION OF THE TESTING PROGRAMME Immediately prior to the commencement of the experiment, the following four tests were administered personally by the experimenter: (it) Farquhar's Test of the Understanding of Processes with Decimal Fractions, (2) A Decimal Fraction Computation Test, (3) Otis Self-Administering Test of Mental Ability, Intermediate Examination, Form A, and (4) The Stanford Achievement Test, Advanced Reading, Form E. The administrations took place, in the absence of the classroom teachers, during the week from Monday, May 6th to Friday, May 10th. Two sittings were held in each school to ensure that conditions of fatigue v/ould be equalized among the classes and reduced to a minimum. By using in the analysis of covariance the results of these tests, which provide measures of the four independent variables considered relevant to the problem, the i n i t i a l unmatched differences between the treatment groups were controlled statistically. Immediately following the experiment, the Farquhar test was re-administered personally by the experimenter. To ensure an equalization of testing conditions, the re-administrations to the five classes involved were held during the mornings only, on 55 Wednesday, May 29th, and Thursday, May 30th. By evaluating the results of this re-administration of the Farquhar test, which provides a measure of the criterion variable, the effectiveness of the teaching aids was judged. Complete details concerning these four tests are given in Chapter IV. In this testing programme about 160 pupils were tested. The marking of approximately 800 papers was undertaken by the experimenter, assisted by his wife. VIII. PSYCHOLOGICAL SIGNIFICANCE OF THE LIMITATIONS IMPOSED ON THE EXPERIMENT . In Chapter I mention was made of two limitations of this study. The f i r s t limitation is that the pupils themselves were given no opportunity of manipulating the instructional materials. The second limitation is that the nature of the experiment dictated a very rigid sequence of instruction in which the pupils experienced a short, intensive encouter with meanings sometime after they had learned the actual performance of the algorisms :. . involved. This latter learning had taken place prior to the experiment during the course of normal classroom instruction. The experiment thus allowed no opportunity to reverse the procedure so as to teach the rationalization of a process before teaching the method of performing the process. Neither did i t allow an opportunity to present the two emphases by teaching rationalization concurrently with method. 56 These details could pass unnoticed, were i t not for the fact that they represent important issues about which a considerable amount of psychological and educational literature has been written. An evaluation of the conclusions reached in Chapter VI requires an awareness of this literature. The issue surrounding the f i r s t limitation is that ofothe learner's own involvement in the manipulative activity. The reports that this involvement facilitates learning is found mainly in psychological literature. Heidbreder, for example, has investigated the manner in which concepts are learned. Her experimentation v/ith adults furnishes some evidence that the ease of attaining a concept, in the case of adults at least, "seems more highly correlated with manipulability 4 than with perceptibility". In a chapter entitled "The Formation of Concepts", contained in a recent yearbook, Van Engen cites an impressive l i s t of authorities to support his conviction that, in the case of children as well as adults, manipulability, or relevance to direct motor reaction, i s an important factor in the learning of arithmetical concepts. He quotes from G-esell: A l l mental l i f e has at i t s roots the actions or mani-pulations performed in a learning situation. ... It i s probable that a l l mental l i f e has a motor basis and a motor origin. ... This behaviour (of motor priority) i s so fundamental that virtually a l l behaviour ontogenetically has a motor origin and aspect.^ 57 He quotes from Werner: To conceive and define things in terms of concrete activity is in complete accordance with the world of action characteristic of the child.^ Finally, he quotes from Piaget, whose work "seems to be resting in an undeserved obscurity": ... i t (childish thought) is nearer to action than ours, g and consists simply of mentally pictured manual operations... In view of the literature which testifies to the importance of action or manipulability in the child's thought processes, i t would seem that the learning outcomes resulting from the instruc-tion offered in the experimental group were likely curtailed as a consequence of the fact that the pupils in that group were not afforded an opportunity to manipulate the instructional materials themselves. The issue surrounding the second limitation concerns the place .in the instructional sequence where emphasis should be laid on rationalization or understanding. This may be referred to as the "HOW-WHY versus WHY-HOW controversy". Though the views on this controversy of the two authors quoted below are not exactly opposed to each other, they serve to show the shades of opinion expressed in the literature. Commenting on one aspect of the controversy, Johnson writes: ... a rationalization of a process in arithmetic is meaning-less unless the HOW to do that process is understood f i r s t , let is be understood that I do not minimize the importance of the role played by rationalization. When rationalization of a process i s understood, the process is better appreciated. 58 ... But what I am trying to say here is that since rationalization of a process is not understood until the HOW of the process is understood, and the HOW is not understood on f i r s t presentation by a l l students, and since i t takes a greater maturity of mind to understand the rationalization than to understand the HOW of a process, many teachers err in trying to rationalize every process upon f i r s t presentation before the HOW of the process is known.^ Later in the same article he states: What could be a better program of teaching than to bring in rationalization of newer and higher orders as the process is reviewed in later grades? The review would then not be a rehash only, but a true review with the process seen in a new light. Research would have to lead the way showing at what mental age the various arithmetic processes could be rationalized.1° Commenting on another aspect of the controversy, Weaver writes: There are numerous persons who advocate exclusive adherence to a HOW-WHY sequence: the HOW of a computational process or s k i l l must precede the WHY. The present writer is not at a l l certain that the HOW of a process or s k i l l must necessarily precede the WHY. No contention has been made or implied that i t is always feasible for WHY to lead to HOW. In some instructional situations i t may seem virtually necessary to present the algorismic form of certain computational skills on the basis of a HOW-WHY sequence. In such instances, when HOW-WHY is selected as the course to be taken, let us be certain that the WHY is coupled with the HOW just as soon as possible or feasible. There is grave danger that WHY may follow HOW at such temporal distance that ultimate rationalization is minimized in effectiveness. A In view of the prominence of these views expressed in the literature, i t is necessary to be aware of the fact that the effectiveness of teaching rationalization of arithmetical concepts and processes may be affected, not only by the instructional aids and the other factors that have been taken into account in this 59 study, but also by the particular temporal sequence employed in the HOW-WHY teaching relationship. The generalizations resulting from this experiment must be drawn with recognition of this fact. IX. REASONS FOR IMPOSING THE LIMITATIONS ON THE STUDY The f i r s t limitation could have been removed by supplying suitable forms of the manipulative aids in sufficient quanitities to permit the pupils to use them either individually or in small groups. Difficulties were evident in this plan. In the f i r s t place, the manipulative materials would have had the added advantage of providing increased motivation through allowing pupils the opportunity of self-participation. Within the design of this experiment i t would have been d i f f i c u l t to equalize this opportunity for the pupils of the other treatment group because visualization materials do not lend themselves to the same degree of pupil participation. The experimental variable would not, therefore, be confined to the characteristic of manipulabili.ty. In the second place, the difficulty of establishing uniformity between the two treatment groups in such things as teaching procedure and teacher competence to manage individual pupil activity would inevitably have been increased. The second limitation could not have been entirely removed. It would always be necessary to make some choice between the HOW-WHY and WHY-HOW sequences. 60 However, an experiment could have been designed to provide a compromise whereby the HOW would precede the WHY in the teaching of some decimal concepts and'processes, and follow i t in the teaching of others, with the intervening temporal distance between the two emphases reduced as much as possible for each concept or process. In fact, such a design would have afforded a more likely usage to which manipulative materials would be put in normal class-room practice. One major difficulty, as usual, presented i t s e l f with this idea. It would have necessitated extending the area of the experiment to include the teaching of decimal fractions in their entirety - the HOW as well as the WHY. Since this assignment comprises a large part of the arithmetic programme normally undertaken in Grades VI and VII, one or other of two major problems would have been encountered. On the one hand, there would have been an enormous problem of. maintaining adequate controls in an experiment which extended over the long period' of time during:which decimal fractions are ordinarily taught. On the other hand, there would have been an awkward problem of arranging to shorten this long period of time by providing for the teaching of the entire area of decimal fractions, uninterrupted by the teaching of other units in the arithmetic syllabus. The nature of these di f f i c u l t i e s , as well as the desirability of using manipulative aids in the type of "carefully planned 61 12 reteaching program" recommended by Morton, encouraged the experimenter to proceed v/ith the present design, notwithstanding the two limitations involved., X. SUMMARY The purpose of this study is to determine the effectiveness of the group instruction use of certain manipulative aids in contributing to an understanding of decimal concepts. The essence of the proposition involved is to determine the effectiveness which results, specifically, from the capacity of these aids to be manipulated, rather than from their capacity, for example, to influence motivation or to be prominently displayed. Chapter III contains a description of the instructional aids and lessons used in the experiment, and an account of the planning and administration undertaken, to ensure that the mani-pulative characteristic of the aids would emerge as the experimental variable. This chapter also contains a discussion of the limitations imposed upon the use of the particular manipulative materials used in the study. The psychological nature of these limitations, as revealed by the literature on the subject, emphasizes the need to proceed v/ith caution in forming generalizations respecting the effectiveness of the particular aids used. While this chapter contains a description of the actual controls 62 exercised in the conduct of the experiment, Chapter V contains, in addition to the test of significance of the achievement of the two treatment groups on the criterion variable, an account of the stati s t i c a l controls imposed upon the independent variables. Complete descriptions and evaluations of the tests used to measure a l l these variables are contained in Chapter IV. 63 FOOTNOTES The third criterion is important in order to avoid the inad-vertent introduction of systematic differences into the experiment, even though the subjects were originally drawn at random from the same normally distributed and homogeneous population. From the standpoint of satisfying one of the basic assumptions underlying the analysis of covariance, i t is necessary to adhere to this criterion. A f u l l discussion is contained in Chapter V, pages 151 and 152. E. F. Lindquist, Design and Analysis of Experiments in Psychology and Education, (Boston: Houghton Mifflin, 1953), p. 328. Ibid, p. 323 E. Heidbreder, "The Attainment of Concepts.:, I. Terminology and Methodology," Journal of General Psychology, Vol. 35, 1946, p. 182. Henry Van Engen, "The Formation of Concepts", Learning of Mathematics: Its Theory and Practice, Twenty-first Yearbook of the National Council of Teachers of Mathematics, 1953, pp. 68-112. Arnold Gesell, Infant Development: The Embryology of Early Human Behaviour, (New York: Harper and Brothers, 1952) p. 58 Heinz Werner, Comparative Psychology of Mental Development, (New York: Harper and Brothers, 1940) p. 272. Jean Piaget, Judgment and Reasoning in the Child, (New York: Harcourt Brace and Company, 1928) p. 146. J. T. Johnson, "What Do We Mean by Meaning in Arithmetic?", The Mathematics Teacher, Vol. 41, 1948, p. 365. Ibid, p. 366 J. Fred Weaver, "Misconceptions about Rationalization in Arithmetic", The Mathematics Teacher,- Vol. 44, 1951, pp. 378-379 Supra, p. 11 65 CHAPTER IV THE STATISTICAL DESIGN OF THE EXPERIMENT AND DESCRIPTION OF THE MEASURES USED I. STATISTICAL DESIGN OF THE EXPERIMENT The General Nature and Purpose of the Statistical Method In Chapter III i t was stated that the classes were matched on the basis of size only. This matching resulted in the formation of two so-called units, referred to in Table III on page 51 as Unit A and Unit B. From Unit A one class was selected at random for the Experimental Group, leaving two classes for the Control Group; and from Unit B one class was selected at random for the Experimental Group, leaving one class for the Control Group. Since size was the only factor taken into account in establish-ing the equivalence of the classes, there were obviously many unmatched individual differences in the classes assigned to the two treatment groups. The relative response of each group to the criterion could conceivably be influenced by these differences. It is apparent that i f these unavoidable concomitant influences were not controlled, any differences between the Experimental and Control Groups on the criterion could not specifically be attributed to the treatments being tested. 66 To provide statistical control over these unmatched individual differences in the Experimental and Control Groups, analysis of covariance was selected as the statistical design to be applied to the data derived from the experiment. The following statements respecting the analysis of covariance technique indicate in general terms i t s suitability for the present study. Further statements, referring to somewhat more technical aspects of i t s appropriateness, are contained in Chapter V. Edwards writes: The analysis of covariance i s applicable to any experiment in which a source of variation, which i t may not be possible to equalize between the various experimental groups prior to the experiment proper, can be measured. An adjustment is then made for this source of variation in the analysis of the out-comes of the experiment.1 Wert, Neidt, and Ahmann write: To provide the investigator with a means of attaining a measure of control of individual differences, the statistical technique known as analysis of covariance was developed. Analysis of covariance incorporates elements of the analysis of variance and of regression. In general, i t will provide tests of significance for the comparison groups whose members may have been stratified and whose members have been measured with regard to one or more variable characteristics other than the criterion.2 Analysis of Covariance has really two purposes. First, by providing for the correlation between the criterion and control scores, i t makes i t possible to determine the relative weight with which each independent variable "enters in" or contributes to the criterion in-dependently of the other variables. Thus, depending on the nature of correlations,- the-precision of the test of significance may be increased 67 considerably, even though extremely small differences exist between the means of the treatment groups i n the various independent variables. Second, by making allowances for the differences that exist, analysis of covariance makes i t possible to exercise a degree of s t a t i s t i c a l control over these independent variables which permits the treatment effect to be evaluated with as much accuracy as i f the variables had been experiment-a l l y controlled by actually matching the groups with respect to these variables. Limitations, as well as p o s s i b i l i t i e s , accompany the use of the covariance technique. I t i s not a magic formula capable of eliminating a l l differences, without reservations, between the means of the treatment groups i n the independent X variables. S t i l l less capable i s i t of eliminating the effects of systematic differences o r i g i n a l l y existing between the groups i n certain characteristics which are independent of the X variables employed. Subsections IX and X of Chapter V contain an account of a l l the limitations imposed by the assumptions underlying the use of analysis of covariance. The same subsections contain also the necessary s t a t i s t i c a l tests to ensure that the analysis i s appropriate to this specific problem. In the present study the concomitant influences are considered to e x i s t , primarily, i n four areas, namely: i n i t i a l understanding of concepts of decimal fractions, computational a b i l i t y i n decimal fractions, mental a b i l i t y , and reading a b i l i t y . Table IV shows the names of the tests selected to measure performance i n these areas. I t also indicates the instrument used to 68 measure the criterion. These tests are contained in Appendix D. The contribution which each of these tests made to the prediction of the criterion is eventually reported in Chapter V (Table XLI, page 155.) Judged by this information, i t is unlikely that additional measurable influences would have an appreciable effect upon the performance of the treatment groups on the criterion. It may be said, therefore, that the application of analysis of covariance to the data removed the possible bias introduced by unmatched individual differences between the Experimental and Control Groups. This is true, at least, to the extent that the four areas referred to represent the differences in question, and, further, to the extent that the differences in these areas are adequately controlled by the tests administered for that purpose. With the removal of this bias, and the imposition of necessary controls in the plan and administration of the experiment, any significant statistical difference in the criterion measures of the experimental and control groups is assumed to be attributable to the treatments used in each group. These treatments, i t may be emphasized again, are intended to be identical .'in every respect except in the use of the materials of instruction, which differ only in the characteristic of manipulability. Statement of the Hypothesis The hypothesis to be tested is that there is no significant difference in the criterion achievement of the two treatment groups whi TABLE IV TESTS SELECTED TO MEASURE THE VARIABLES Classification of Variables Names of Variables Tests selected to measure Each Variable Independent or Concomitant Variables 1. Init i a l Understanding of Concepts of Decimal Fractions Farquhar's Test of the Understanding of Processes with Decimal Fractions (First Administration) 2. Computational Ability in Decimal Fractions Adapted from Unit Test "Making Sure of Deci-mals" contained in Silver Burdett Text "Making Sure of Arithmetic" 3. Mental Ability Otis Self-Administering Test of Mental Ability ^Intermediate Examinations) Form A 4. Reading Ability Stanford Achievement Test (Advanced Reading Test: Form E for Grades 7-9 Dependent or Criterion Variable 1. Final Understanding of Concepts of Decimal Fractions Farquhar's Test of the Understanding of Processes with Decimal Fractions (Second Administration) 70 i s attributable to the treatments involved. Stated in other words, this hypothesis is. that the pupils who are taught with the use of manipulative aids in the manner prescribed in this experiment achieve an understanding of decimal fractions which is not significantly different, after the bias due to unmatched individual differences in each group has been removed, from the understanding achieved by pupils who are taught with the use of "visualization" materials which bear characteristics identical to those of manipulative materials in a l l details except .that-o:f' manipulability.';'-'•/• -II. DESCRIPTION OF THE FARQUHAR TEST OF UNDERSTANDING OF PROCESSES WITH DECIMAL FRACTIONS The Farquhar Test, shov/n in Appendix D, performed a dual function in this study. Immediately prior to the assignment, i t was used to measure one of the independent variables shown in Table IV. Immediately following the teaching assignment i t was used, in a second administration, to measure the dependent or criterion variable. These two functions confer upon the Farquhar Test an importance which necessitates thorough investigation of i t s efficiency for these purposes. This necessity is a l l the greater in view of the fact that the Farquhar test was validated in relation to groups of teachers-in-training. The fact that this validation.took place against educationally more advanced subjects than those participating in this experiment provided the major source of apprehension concerning the suitability of the test for the present investigation. 71 Data derived from a t r i a l administration of the Farquhar Test To provide further data on which to determine its suitability for the present study, the Farquhar Test was administered on April 4th, 1957 by the experimenter to a group of forty representative pupils selected from 2-g- classes of unselected Grade VII pupils in V/hite Rock Elementary School, located outside the proposed experimental area. It was assumed that the results obtained from this t r i a l administration would be substantially the same as those which could be expected from the i n i t i a l administration of the same test to the classes participating in the experiment. Tables V to X, inclusive, contain data derived from the t r i a l administration. Part of this information was obtained from an item analysis of the test undertaken to indicate the effectiveness of individual test items. The items were evaluated on the bases of two internal criteria, namely, their difficulty and their discriminating value or validity. Method of Item Analysis The method employed is based on the simplified item analysis 3 procedure devised by Stanley. Page 308,'-' Appendix E contains the recording sheets, •. page..311 conts,ins the calculation sheets, both of which were used in the present analysis. The technique deals with the top and bottom 21% of the group. Rows (a), (b), and (c) of the calculation sheets merely show, the data obtained from the recording sheet. 72 TABLE V FREQUENCY OF SCORES IN THE TRIAL ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL (Maximum: 30 Items) Score Frequency Score Frequency 20 0 11 5 ?. 19 1 10 5 18 0 9 6 17 1 8 2 16 3 7 . 7 15 1 6 3 14 0 5 0 13 2 4 1 12 3 3 0 TABLE VI SUMMARY OF STATISTICAL DETAIL RESULTING FROM TRIAL ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL Mean Item Difficulty Median Mean Standard Corrected Not Range Reliability Deviation for chance Corrected of Item for chance Validity -18$ 9.7 10.175 3.382 85.6$ 64.9$ to .549 64$ TABLE VII THE DIFFICULTIES AND VALIDITIES OF ITEMS RESULTING FROM TRIAL ADMINISTRATION OF FARQUHAR1S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL Item Per cent of Difficulty Corrected for chance Uncorrected for chance Per cent of Validity Validity Coefficient (Flanagan1 s) WL - . W H Discrimination 1 2 3 4 5 30 62 36 55 127 23 50 27 41 95 27 9 55 64 -9 .38 .10 .75 .66 -.38 3 1 6 7 -1 6 11 8 9 10 97 97 67 68 115 73 73 50 55 86 36 36 64 18 27 .46 .46 .63 .20 .60 4 4 7 2 3 11 12 13 14 15 85 42 85 114 73 64 32 64 91 55 55 27 18 18 36 .60 .30 .20 .51 .38 6 3 2 2 4 TABLE VII (continued) Item Per cent of Difficulty uncorrected Corrected for chance for chance Per cent of Validity-Validity Coefficient (Flanagan1s) W H Discrimination 16 48 36 17 102 82 18 109 82 19 97 73 20 42 32 21 102 82 22 115 86 23 85 68 24 109 82 25 97 73 26 127 95 27 40 32 28 97 77 29 97 77 30 121 91 18 .20 2 0 :00 0 0 00 0 -18 -.23 -2 27 .31 3 36 .66 4 9 .18 1 64 .78 7 36 .66 4 36 .46 4 9 .38 1 45 .54 5 45 .71 5 27 .38 3 0 00 0 TABLE VIII FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF DIFFICULTY RESULTING FROM TRIAL ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL Per Cent Frequency Per Cent Frequency Range of of Items Range of of Items Difficulty Difficulty 0$ _ 46$ - 50$ 2 1% - 5% 51$ - 55$ 2 6$ - 10% 56$ - 60$ 11$ - 15% 61$ - 65$ 2 16$ - 20$ 66$ — 70$ 1 21% - 25$ 1 71$ - 75$ 4 26% - 30$ 1 76$ - 80$ 2 31% - 35$ 3 81$ - 85$ 4 36% - 40$ 1 86$ - ..90$ 2 \1% - 45$ 1 91$ - 95$ 4 TABLE XIX FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF VALIDITY RESULTING FROM TRIAL ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS - IN WHITE ROCK ELEMENTARY SCHOOL Per Cent Range of Validity Frequency of Items Per Cent Range of Validity Frequency of Items 0$ - or less 53 46$ -- 50$ 1 1$ - 5$ 51$ -- 55$ 2 6$ - 10$ 3 56$ -• 60$ 11$ - 15$ 61$ -- 65$ 3 16$ - 20$ 4 66$ -- 70$ 21$ . - 25$ 71$ -- 75$ 26$ - 30$ 5 76$ -- 80$ 31$ - 35$ 81$ -- 85$ 36$ - 40$ 6 86$ -- 90$ 41$ - 45$ 1 91$ -- 95$ TABLE X 76 FREQUENCY OF ITEMS IN THE VARIOUS VALIDITY COEFFICIENT RANGES RESULTING FROM TRIAL ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL (Based on Flanagan's Estimates of Correlation between Individual Items and the Test as a Whole) Validity-Coefficient Range Frequency Validity Coefficient Range Frequency 00 or less .01 -".05 .063- .10 .11 - .15 .16 - .20 5 0 1 0 4 .46 - .50 .51 - .55 .56 - .60 .61 - .65 .66 - .70 3 2 2 1 3 .21 - .25 .26 - .30 .31 - .35 .36 - .40 .41 - .45 0 1 1 4 0 .71 - .75 .76 - .80 .81 - .85 .86 - .90 .91 - .95 2 1 0 0 0 77 Row (d) indicates the per cent difficulty of each item, uncorrected for chance. It is the ratio, converted to per cent, of the total number of incorrect or omitted responses to the total number of possible responses in the top and bottom 21% sections. Expressed algebraically i t is 100c where "c" is the total obtained in Row (c) 2n and "n" is the total number of possible responses in 27$ of the entire group. Rows (e), ( f ) , and (g) deal with the per cent difficulty, corrected for chance. To obtain this, the total number of wrong or omitted responses recorded in row (c) is. multiplied by the correction factor shown in row ( f ) . Expressed algebraically the correction factor is 100 "0" , where "0" is the number of options in each 2n ("0"-l) question and "n" is the total number of possible responses in 21% of the entire group. Rows (h) and (i) deal with the item discriminating value or validity. The discrimination of each item is found by subtracting row (b) from row (a). This value may be converted to a per cent ratio by dividing i t by the maximum discrimination possible and multi-plying by 100. Expressed algebraically the per cent of the discrim-ination or validity of each item is lOOh, where "h" is the discrim-n ination recorded".In row (h) and "n" retains- the representation indicated above. To find the Flanagan validity coefficient i t is necessary, in addition to the foregoing procedure, to compute from rows (a) and (b) the per cent of correct responses in the bottom and top sections, 78 respectively. The validity index is obtained for each item by entering Flanagan's Table with these per cent computations. Interpretation of data obtained from the t r i a l administration Two internal criteria are available by which to evaluate the effectiveness of the individual test items. One of these criteria is that of item difficulty. On this subject Ross and Stanley write: Difficulty alone, therefore, is not a dependable measure of discrimination ... Test experts have usually found, however, that the average difficulty of the items in a test is related to the adequacy of the test as a whole. The rule suggested for the construction of tests to discriminate best among a l l the members of a group i s to make every item of 50 per cent di f f i c u l t y when corrected for chance, so far as possible. This will mean that virtually a l l the items of 0-15 per cent and 85-100 per cent difficulty when corrected for chance w i l l be omitted from the revised form of the test, unless they can be rewritten to make them closer to the 50 per cent difficulty level.4 An examination of Table VII shows that compliance with this suggestion would result in the omission of 19 of the 30 test items. In fact, the difficulty, corrected for chance, of 10 of these items exceeds 100 per cent. This means that fewer pupils answered these items correctly than would be expected on the basis of chance alone. However, as Ross and Stanley say, "Quite a few test experts do 5 not favor correcting item difficulty indexes for 'chance'". If the correction for chance is not made, the mean per cent difficulty of the items is reduced from 85.6 to 64.9 (Table Vi), and, as seen in Table VIII, only six- items have a per cent difficulty greater than 85. In addition, i t will be observed that the distribution 79 of scores is a satisfactory one (Table V), in which the mean slightly exceeds the median, and the standard deviation indicates a reasonable, though small, variability. (As samples, these three calculations are shown in Appendix E.) Nevertheless, the results of the t r i a l administration indicated that Farquhar's test would likely be rather difficult when used as a measure of one of the independent variables at the beginning of the experiment. Despite this realization, the experimenter believed that the difficulty of the test would not be entirely a disadvantage. After an intensive period of instruction on the subject matter covered by the testy j i t was to be used a second time in the.even more important role of measuring the criterion performance. It was anticipated, and hoped, that in this capacity the level of difficulty of the Farquhar items would enable the test to meet the ideal statistical^-requirements. The second of the. criteria by which to evaluate the effective-ness of the individual test items is that of item discrimination or validity. One commonly used standard of validity is that items must show a positive discrimination of as much as 20 per cent. A reference to Table IX reveals that 12 items f e l l at, or below, the 20 per cent level. Another commonly used standard of validity is that items must show a positive validity coefficient, based on Flanagan's Table, of more than .25. A reference to Table X reveals that 10 items f e l l below the .25 coefficient level. One particularly undesirable feature v/hich resulted from this administration is that 3 items have a zero validity and 2 have a 80 negative validity. In 4 of these 5 items the per cent of difficulty, corrected for chance, exceeded 100 per cent. In the case of the f i f t h , question 19, the per cent v/as 97 (Table V i i ) . In view of the difficulty of the test for this sample of pupils, the degree of validity was to be expected. At the 50 per cent level of difficulty an item has the maximum opportunity to discriminate between the top and bottom 27 per cent sections. The results of the trialaadministration indicated, therefore, that Farquhar's test, as well as being rather d i f f i c u l t , would like l y also be rather low in discrimination value when used as a measure of one of the independent variables at the beginning of the experiment. Yet the general levels of difficulty and discrimination, when used for this purpose, were not considered lik e l y to be sufficiently extreme to make the test unsuitable. Furthermore, when used a second time as a measure of the criterion variable at the close of the experiment, i t was believed that the decrease in diffi c u l t y anticipated in nearly a l l the items would be just about the right amount to increase quite substantially the dis-criminating value of these items. While this desire to find a test which, from the standpoint of item effectiveness, would be a satisfactory measure of one of the independent variables and also of the criterion variable v/as a main consideration affecting the experimenter's decision to select the Farquhar test, there were the following additional considerations, though not necessarily in this order of importance. 81 The fi r s t of these considerations was the r e l i a b i l i t y of the test which, as reported in Table VI, was .549. This r e l i a b i l i t y was computed by applying the Hoyt modification of the Kuder-Richardson Formula to the data obtained from the t r i a l administration. The Hoyt Formula and the calculations involved are shown in Appendix E. While there are obvious di f f i c u l t i e s in the interpretation of test r e l i a b i l i t y , certain minimal requirements have been suggested for the re l i a b i l i t y coefficients of tests which serve various purposes. Reference is made to this suggestion by Ross and Stanley, who write: ".50 ( r e l i a b i l i t y coefficient needed) for determining the status of a group in some subject or group of subjects." They note also, of course, that considerably higher r e l i a b i l i t y coefficients are required where the purpose of the test is to differentiate the achievement or status of individuals, rather than that of a group. Since the Farquhar test was to be used for the latter of these two purposes, i t appeared that the r e l i a b i l i t y coefficient of .549, obtained from the t r i a l administration, was deservedly a consideration in favor of selecting the test for use in the experiment. The second of these considerations was the nature of the concepts covered by the test. Concerning the method of selecting these concepts, Farquhar wrote: The attempt to measure understanding of arithmetic processes is rendered very difficult by the lack of criteria for this purpose. The investigator must determine arbitrarily those concepts that should be included in a measuring instrument 7 designed to evaluate understanding of any phase of arithmetic. 82 Farquhar l i s t e d f i f t e e n of these a r b i t r a r i l y chosen concepts i n the f i e l d of decimal fr a c t i o n s . They proved to be s p e c i f i c concepts around which i t was convenient to plan the topics of the eight lessons involved i n the teaching procedure. The lesson planning i s described i n Chapter I I I . The c u r r i c u l a r v a l i d i t y of the test, when used as a measure of one of the independent variables at the beginning of the experiment, seems assured by the fact that the concepts measured by the test are id e n t i c a l to the concepts emphasized for teaching i n the Grade VII text currently prescribed by the B r i t i s h Columbia Department of Education. The curricular v a l i d i t y of the test, when used as a measure of the c r i t e r i o n variable at the end of the experiment, i s more d e f i n i t e l y assured by the fact that the lessons were car e f u l l y planned according to the sp e c i f i c concepts measured by the test. After taking account of the foregoing considerations, that is-, the effectiveness of the .individual test items when evaluated on the bases of their d i f f i c u l t y and discrimination value, the test r e l i a b i l i t y , and the spe c i f i c nature and cu r r i c u l a r v a l i d i t y of the concepts measured, i t was decided that Farquhar's test would be suitable for the present study. Interpretation of data obtained from the f i n a l administration It has been stated that Farquhar's test was used immediately prior to the experiment to measure one of the-'ihdependent variables, and immediately following the experiment to measure the c r i t e r i o n variable. 83 The interpretation given above referred to the data obtained from the t r i a l administration of the test to a sample group outside the experimental area., A similar study of the data obtained from the i n i t i a l administration has not been undertaken since i t is assumed that these results are substantially the same as those obtained from the t r i a l administration. Such an assumption, however, could not be made regarding the results obtained from the final administration. Therefore, these results have been subjected to an analysis similar to that which was undertaken in connection with the t r i a l administration. Tables XI to XVI, inclusive, which correspond respectively to Tables V to X, inclusive, contain data derived from the final administration. To facilitate making comparisons, a summary of comparative data from the two administrations is presented in Table XVII. An examination of this latter table shows that the level of difficulty, which was the most serious criticism of the test when i t was used with the t r i a l group, was considerably reduced when i t was used as a measure of the criterion variable. Whereas 6 of the 30 items lay beyond the suggested levels of difficulty (uncorrected for chance) in the t r i a l administration, only 1 item was in this position in the final administration. It is interesting to note that this 1 item (Mo. 4) tended to be too easy (15$). The general extent of the reduction in difficulty is shown by the fact that 25 of the 30 items were easier for the pupils in the 84 TABLE XI FREQUENCY OF SCORES IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Maximum: 30 Items) Score Frequency Score Frequency 26 4 14 13 25 0 13 14 24 6 12 11 23 1 11 10 22 4 10 12 21 4 9 5 20 4 8 6 19 5 7 6 18 11 6 4 17 8 5 0 16 8 4 1 15 9 3 1 TABLE XII SUMMARY OF STATISTICAL DETAIL IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Mean Item — — Difficulty Median Mean Standard Corrected Not Range of Reli-Deviation for chance corrected Item ability for chance Validity 13.77 14.33 5.019 66.8$ 51.1$ 8 to .541 65$ TABLE XIII THE DIFFICULTIES AND VALIDITIES OF ITEMS IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING- SUBJECTS Per cent of Difficulty Item Corrected Uncorrected Per cent Validity W_ for chance for chance of Validity Coefficient Discrimination (Flanagan's) 1 28 21 2 58 46 3 72 54 4 20 15 5."> 87 65 6 60 45 7 62 46 8 43 33 9 67 54 10 102 76 11 35 26 12 60 45 13 67 50 14 103 83 15 70 53 28 .42 11 . 42 .43 17 57 .57 23 20 .38 8 65 .74 26 45 .46 18 48 .49 : 19 45 .52 18 33 .34 13 8 .11 3 . 32 .41 13 55 .33 22 35 .36 14 20 .32 8 35 .36 14 CO TABLE XIII (continued) Per cent of Difficulty Item Corrected Uncorrected Per cent Validity WL -for chance for chance of Validity Coefficient Discrimination (Flanagan1s) 16 47 35 17 72 58 18 47 35 19 .63 48 20 77 58 21 63 50 22 82 61 23 88 70 24 72 54 25 82 61 26 92 69 27 52 41 28 70 56 29 83 66 30 80 60 65 .76 26 50 .51 20 35 .39 ' 14 60 .60 24 40 .41 16 45 .45 18 32 .34 13 40 .48 16 43 .44 17 48 .51 19 43 .51 17 32 .34 13 62 .62 25 32 .36 13 35 .37 14 oo CTl 87' TABLE XIV FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF DIFFICULTY RESULTING FROM ADMINISTRATION OF FARQUHAR1S TEST AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Per Cent Range of Difficulty Frequency of Items Per Cent Range of Difficulty Frequency of Items 0$ - 46% - 50% 5. 1$ - 5% 51% - 55% 4 6% - 10% 56% - 60% 3 11% -15% 1 61% - 65% 4 16% - 20% 66% - 10% 3 21% -25% 1 11% - 15% 26% - 10% 1 16% - 80% 1 31% -35% 3 81% - 85% 1 36% - 40% 86% - 90% 41% - 45$ 3 91% - 95% TABLE XV FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF VALIDITY RESULTING FROM ADMINISTRATION OF FARQUHAR1S TEST AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Per Cent Frequency Range of of Items Validity Per Cent Frequency Range of of Items Validity 0% - 46$ - 50$ 3 1% - 5% 51$ - 55$ 1 6% - 10% 1 ., .56$ - 60$ 2 11% - 15% v6l$ - 65$ 3 16% - 20% 2 66$ - 70$ 21% - 25% 71$ - 75$ 26% - 30% 1 76$ - 80$ 31% -35% 9 81$ - 85$ 36% - 40% 2 86$ - 90$ 41% -45% 6 91$ - 95$ TABLE XVI FREQUENCY OF ITEMS IN THE VARIOUS VALIDITY COEFFICIENT RANGES RESULTING FROM ADMINISTRATION OF FARQUHAR1S TEST AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Based on Flanagan* s Estimates of Correlation between ; '• ' Individual Items and the Test as a Whole) Validity Frequency Validity Frequency Coefficient Coefficient Range Range 00 .46 - .50 3 .01 - .05 .51 - .55 4 -.06 - .10 .56 - .60 2 .11 - .15 ' 1 .61 - .65 1 .16 - .20 .66 - .70 .21 .25- .71 - .75 1 .26 - .30 .76 - .80 1 .31 - .35 5 .81 - .85 .36 - .40 6 .86 - .90 .41 - .45 6 .91 - .95 TABLE XVII COMPARATIVE DATA OBTAINED FROM,THE ADMINISTRATION OF FARQUHAR1S TEST TO THE TRIAL GROUP AND TO THE 147 PARTICIPATING SUBJECTS Criteria Correction for chance (Difficulty) Trial Adminis-tration Final Adminis-tration Item Difficulty No. Items below desirable minimum difficulty (l5$ or below) Yes No 0 0 0 1 No. Items above desirable Yes maximum difficulty No (over 85$) 19 6 5 0 Total No. Items beyond desirable difficulty No. Items over 100$ Difficulty when corrected for chance Yes No 19 6 10 5 1 Mean Per Cent Difficulty Yes No 85.6 64.9 66.8 51.1 Item Discrimination No. Items below desirable minimum discrimination (below 20$) 12 No. Items below desirable Coefficient of Validity (Flanagan Coefficient over .25) 10 No. Items v/ith negative discrimination 2 0 90 final administration, and 5 items (No's. 3, 12, 20, 23, 27) were more dif f i c u l t for this group than they were for the pupils in the t r i a l group, who did not have any special instruction in the concepts involved. The extent of the reduction in difficulty i s shown also by the fact that the mean per cent level of difficulty (uncorrected for chance) f e l l to 51.1%. As the test proved to be generally easier, though not too easy, for the pupils in the final administration, so also i t proved to be more discriminating. Table XVII shows that only one item f e l l below the desirable minimum discrimination. This item (No. 10) is one that remained quite d i f f i c u l t (76$). However,tw.o.-other items were at the minimum acceptable level (20$): one of these is item 14, which also remained quite d i f f i c u l t (83$), and the other is item 4, referred to above, which tended to become too easy (l5$). The decrease in d i f f i -culty which was anticipated in nearly a l l the items :proved to be just about the right amount to increase quite substantially the discriminating value of these items. In the case of three (No's. 3, 12, 20) of the five questions which for some reason proved more di f f i c u l t for the group in the final administration than for the t r i a l group, the increase in diffic u l t y actually resulted in an increase in per cent of validity. The validity of the other two items, though decreased in the final administration, remained satisfactory (40$ in No. 23; 32$ in No. 27). When used to measure the criterion variable, the Farquhar test proved to be a suitable measuring instrument in other respects beside 91 item difficulty and discrimination, which have just been discussed. The distribution of scores (Table Xl) is a very satisfactory one, in which the mean slightly exceeds the median, and the standard deviation (Table XIl) indicates greater variability than existed in the results of the t r i a l administration. The r e l i a b i l i t y of the test in this situation is .541, approximately the same as in the previous analysis (.549). The assurance of curricular validity, when the test was used 8 in i t s final role, has already been discussed. Concluding Comments about the Farquhar Test In the planning of this experiment i t was considered necessary to use the same test to measure the independent variable concerned with the pupils' i n i t i a l understanding of decimal fractions, and to measure the criterion variable also. A test suitable for these two purposes was difficult to find. Although the results of the t r i a l administration to the 40 Grade VII pupils in White Rock Elementary School indicate that Farquhar test was probably somewhat more d i f f i c u l t than was desirable when used as a measure of one of the independent variables, i t proved to be an almost ideal instrument by which to measure the criterion variable. The capacity of the test to perform these two functions in this manner indicates its suitability for the present study. 92 I I I . DESCRIPTION OF THE DECIMAL FRACTION COMPUTATION TEST The Decimal Fraction Computation Test, shown i n Appendix D, was used to measure one of the independent variables shown i n Table IV, page 69. I t i s the second of the battery of four tests which was administered by the experimenter immediately prior to the commencement of the experiment. The test was constructed by the experimenter, although i t i s to some extent an adaptation of a diagnostic unit test e n t i t l e d "Making Sure of Decimals", which i s contained i n the S i l v e r Burdett 9 Text "Making Sure of Arithmetic". Tables XVIII. to XXIII, in c l u s i v e , which correspond to the two previous sets of tables, contain data derived from the results of the test which was administered at the beginning of the experiment. This analysis was undertaken to ensure that the test had been a s a t i s -factory instrument by which to measure the pupils' computational a b i l i t y i n decimal fractions. As i n the two previous cases, an item analysis was made to indicate the effectiveness of individual test items. Interpretation of data obtained from the administration of the decimal. f r a c t i o n computation te s t . The effectiveness of the items i s the f i r s t consideration determining i t s s u i t a b i l i t y for the present study. Item d i f f i c u l t y i s one of the two i n t e r n a l c r i t e r i a used to evaluate item effectiveness. TABLE XVIII 93 FREQUENCY OF SCORES IN THE DECIMAL COMPUTATION TEST ADMINISTERED AT THE BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Maximum: 25 Items) Score Frequency Score Frequency 25 3 13 7 24 6 12 8 23 6 11 4 22 9 10 4 21 15 9 2 . 20 13 8 1 19 14'. 7 2 18 10 6 2 17 11 5 2 16 10 4 2 15 8 3 0 14 8 2 0 TABLE XIX SUMMARY OF STATISTICAL DETAIL IN THE DECIMAL COMPUTATION TEST ADMINISTERED AT THE BEGINNING OF THE EXPERIMENT • TO THE 147 PARTICIPATING SUBJECTS Median Mean Standard Mean Item . Range of .Reli-Deviation Difficulty Item ability Validity 17.75 16.98. 4.804 34.28$ 10 to 80$ .821 TABLE XX THE DIFFICULTIES AND VALIDITIES OF ITEMS IN THE DECIMAL COMPUTATION TEST ADMINISTERED AT THE-BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Item Per cent of Per cent of Validity Wj, - WH Difficulty Validity Coefficient Discrimination (Flanagan1 s) 1(a) (b) (c) 2 3 40 31 31 21 14 45 22 42 32 23 .48 .26 .50 .49 .51 18 9 17 13 9 4 5? 65 7(a) (b) 16 24 15 11 26 22 33 10 22 42 .40 .46 .18 .55 .57 9 13 4. 9 17 (c) (d) 8 9 10 23 51 34 4:8 45 35 52 43 60 70 .51 .52 .49 .60 .70 14 21 16 23 28 TABLE XX (continued) Item Per cent of Difficulty Per cent of Validity Validity Coefficient (Flanagan1 s) W L _ WH Discnmination 11(a) '.(b) (c) (d) (e) 29 38 45 20 41 37 55 75 35 68 .46 .59 .75 .60 .70 15 22 30 14 27 ( f ) 12 13 14 15 40 53 60 53 48 75 80 70 60 55 .80 .77 .73 .60 .55> 30 32 28 24 22 V£> VJ1 96 TABLE XXI FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF DIFFICULTY RESULTING FROM ADMINISTRATION OF THE DECIMAL COMPUTATION TEST AT THE BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Per Cent Frequency Per Cent Frequency Range of of Items Range of of Items Difficulty Difficulty 0$ 46$ - 50$ 2 1$ - 5$ 51$ - 55$ 3 6$ - 10$ 56$ - 60$ 1 11$ - 15% 3 61$ - 65$ 16% - 20$ 2 66$ - 70$ 21$ - 25$ 3 71$ - 75$ 26$ - 30$ 2 76$ - 80$ 31$ - 35$ 3 81$ - 85$ 36$ - 40$ 3 86$ - 90$ 41$ - 45$ 3 91$ - 95$ TABLE XXII FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF VALIDITY RESULTING FROM ADMINISTRATION OF THE DECIMAL COMPUTATION TEST AT THE BEGINNING OF'THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Per Cent Frequency Per Cent Frequency Range of of Items Range of Validity Validity 0$ 46$ - 50$ 1$ - 5$ 51$ - 55$ 3 6$ - 10$ 1 56$ - 60$ 2 11$ - 15$ 61$ - 65$ 16$ - 20$ 66$ - 70$ 3 21$ - 25$ 4 71$ - 75$ 2 26$ - 30$ 76$ - 80$ 1 31$ - 35$ 4 81$ - 85$ 36$ - 40$ 1 86$ - 90$ 41$ - 45$ 4 91$ - 95$ 97 TABLE XXIII FREQUENCY OF ITEMS IN THE VARIOUS VALIDITY COEFFICIENT RANGES RESULTING FROM ADMINISTRATION OF THE DECIMAL COMPUTATION TEST AT THE BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Based on Flanagan's Estimates of Correlation between Individual Items and the Test as a Whole) Validity Frequency Validity Frequency Coefficient Coefficient Range Range 00 .01 .06 .11 .16 .05 .10 .15 .20 .21 .26 .51 .36 .41 .25 .30 .35 .40 .45 .46 .51 .56 .61 .66 .50 .55 .60 .65 .70 6 5 5 .71 .76 .81 .86 .91 .75 .80 .85 .90 .95 2 2 98 The suggestion of Ross and Stanley, referred to on page 78, is that virtually a l l items of 0-15$ and 85-100$ difficulty should be omitted to ensure adequate discrimination for the test as a v/hole. Three items -No. 3 (14$), No. 6 (15$), and No. 7(a) (ll$) - f a l l into the former category, while none f a l l into the latter. There are other indications that the test was somewhat easier than was desirable: the mean difficulty of a l l 25 items is 34.28 per cent (Table XIX), and, further, there is a frequency of 3 scores at the maximum (Table XVIII). However, an examination of this table shows that there is a satisfactory range of scores. Likewise, the median, mean, and standard deviation, reported in Table XIX, suggest that the test could not be considered unduly easy for the group to which i t was administered. Item discrimination, the second criterion used to evaluate item effectiveness, should be at least 20 per cent, according to one of the standards of discrimination previously accepted in dealing with the Farquhar test. Table XXII shows that only one item (No. 6 - 10$) f e l l short of this desirable minimum. According to another of the standards of discrimination previously accepted, the Flanagan coefficient of validity of an item should exceed .25- Table XXIII shows that only one item (No. 6, once again) f e l l into this category. The coefficient of this item is .18. It will be seen that two of the three items which have already been regarded as unsatisfactory because they were too easy, nevertheless retained an acceptable discrimination value. Item 6, alone, remains unsatisfactory with respect to both difficulty and discrimination value. 99 The effectiveness of the items on the computation test, judged on the bases of difficulty and discrimination, is considered satisfactory with the exception of this item. The reliability of the test is the second consideration determining i t s suitability for the present study. The r e l i a b i l i t y , calculated by means of the Hoyt Formula, is .821. According to the standard referred to previously (page 8l), this coefficient indicates that the test was satisfactory from the point of view of r e l i a b i l i t y . The curricular validity of the test is the third consideration determining i t s suitability for the present study. The questions contained in the test dealt with the four fundamental processes and with the conversion of common fractions into decimal fractions. These areas of computation are of primary importance in the unit dealing with decimal fractions in the Grade VII Arithmetic Course of Studies for British Columbia. After taking account of the foregoing considerations, that is; the effectiveness of the individual test items when evaluated on the bases of their difficulty and discrimination value, the test r e l i a b i l i t y , and its curricular validity, i t may be concluded that the. decimal fraction computation test v/as a suitable testing instrument by which to measure the second independent variable. 100 IV. DESCRIPTION OF THE OTIS SELF-ADMINISTERING TEST OF MENTAL ABILITY The Otis Self-Administering Test of Mental Ability, Intermediate Examination, Form A, shown in Appendix D, is the testing instrument used to measure the third independent variable. This well-established and widely known test requires only a brief description concerning three matters: the purpose of the test, the criteria used by the author to judge the validity of each item contained in i t , and the reported r e l i a b i l i t y . The purpose of the test, according to the author, i s to predict the rate at which a student can progress through school. The Otis Intelligence Quotient i s , therefore, a relative numerical indica-tion of brightness. In the Manual of Directions there is no statement of the extent to which the Intermediate Examination does, in fact, serve i t s avowed purpose. There is a meagre report concerning the correlation between scores in the Higher Examination and "scholarship". This report is that of the Principal of a High School in Maine who found a correlation of approximately .58 between scores in the Higher Examination and the "scholarship" of about 400 students in Grades 11 and 12. The author states: "The method of standardization is perhaps the best assurance as to the validity of the t e s t s " . ^ In this standardization procedure the criterion used to judge validity was the ability of each item to discriminate between two 101 groups - a so-called "good group" and "poor group". The only distinction between the two groups was that the median age of the good group was over two years less than that of the poor group. . They had reached the same average educational status, therefore, but at different rates. Only those items were included in the test which distinguished between the students who progressed slowly and the ones who progressed rapidly. The entire standardization group was composed of about 2000 high school students in three cities located in California, I l l i n o i s , and Minnesota. ' Finally, the last matter to be described about the test i s the r e l i a b i l i t y . The r e l i a b i l i t y v/as determined by means of correlation between different forms of the same test. For the Intermediate Examination an average correlation of .948 was found between Forms A and B when these two forms were administered to two;;groups composed altogether of 427 cases. In one group Form A was administered f i r s t , while in the other group Form B was administered f i r s t . The probable error of a score in the Intermediate Examination i s reported to be slightly over 2-jg- points in hald the cases. The author states that "this means also that the probable error of an I.Q. is about 2-g- points. The Otis Self-Administering Tests of Mental Ability, Intermediate Examination, is designed for Grades 4 to 9. It i s , therefore, extremely appropriate for Grade 7. Before deciding on the Otis, the experimenter inquired into the number of pupils who had previously written Form A, or any of the other forms, of the Intermediate Examination. Since the 102 elementary schools in Surrey at the time of the experiment included Grade 8, the practice is to administer the Otis test immediately prior to the entrance of the pupils into the High Schools. It was discovered that a negligible number of pupils participating in the experiment had written any form of the Intermediate Examination. V. DESCRIPTION OF THE STANFORD ACHIEVEMENT TEST (ADVANCED READING TEST: FORM E) The Stanford Advanced Reading Test: Form E, shown in Appendix D, is the testing instrument used to measure the fourth independent variable. Composed of two sub-tests: paragraph meaning and word meaning, i t forms part of the Stanford Advanced Battery of Achievement Tests for Grades 7, 8, and 9. Form E is one of five alternate forms available in the 1940 edition which has been superceded by the 1953 revision. Provision is contained in the test for converting the raw score into an equated score which makes possible many interpretations of the test results. However, to avoid implications involving the normative group this conversion was not made. Instead, the final raw score for each pupil in this variable was obtained simply by finding the average of the original raw score in each of the two sub-tests, and disregarding the. fraction where i t occurred. In a l l respects, except in the matter of converting the raw scores into equated scores, the publisher's directions were strictly adhered to. 103 Because the conversion table was not used, the standardization data supplied by the authors i s not presented as a basis for determining the suitability of this test for the present study, except for reporting i t s r e l i a b i l i t y . In place of this standardization data the following data, which pertains to the administration of the test at the beginning of the experiment, is presented for this purpose in Tables XXIV and XXV. It may be concluded from a study of these tables that the test distributed the scores in a satisfactory manner. The split^half r e l i a b i l i t y coefficients of the test, corrected by the usual Spearman-Brown formula based on random samples of pupils from 34 school systems in the standardization population, is reported to be .841 for the Paragraph Meaning and .907 for the Word Meaning. The average of these coefficients, .874, is accepted as the r e l i a b i l i t y of the Stanford Advanced Reading Test when used in the present situation. VI. CONCLUSION Chapter IV contains a description and evaluation of each of the four tests used to measure the five variables involved in this study. Two administrations of Farquhar's test, at the beginning and at the end of the experiment, provide respective measures of one of the independent variables and of the criterion variable. The other tests were admin-istered at the beginning of the experiment to measure the remaining three independent variables. TABLE XXIV 104 FREQUENCY OF SCORES IN STANFORD READING TEST ADMINISTERED AT THE BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Maximum: 47) Score Frequency Score Frequency 38 1 20 5) 37 2 19 17 36 0 18 5 35 3 17 9 34 4 16 5 33 3 15 8 32 2 14 11 31 4 13 9 30 2 12 2 29 4 11 3 28 1 10 3 27 5 9 1 26 3 8 1 25 2 7 2 24 5 6 0 23 9 5 1 22 3 4 1 21 11 3 0 TABLE XXV SUMMARY OF STATISTICAL DETAIL IN STANFORD READING TEST ADMINISTERED AT THE BEGINNING OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Median Mean Standard Deviation 19.24 20.37 7.277 105 These descriptions and evaluations afford assurance that the tests provided efficient measurements of the variables for which they were used. The i n i t i a l differences between the treatment groups in the four independent variables are held constant in the analysis of covariance technique. The numerical extent to which these variables would otherwise have been responsible for the pupils' achievement on the criterion variable is.stated in the multiple regression analysis which follows. No additional variables are considered to have exercised an appreciable influence on this achievement. This statistical control of a l l the important concomitant influences, together with the careful imposition of actual controls in the plan and administration of the experiment, enables any differences between the groups in the criterion variable to be attributed to the treatments involved. Except for the materials of instruction used, the treatments are intended to be identical. The materials of instruction differ only in the characteristic of manipulability. The method of imposing the actual controls in the plan and administration of the experiment i s described in Chapter I I I ; the method of imposing the statistical control over the independent variables is described in Chapter V. 106 FOOTNOTES Allen L. Edwards, Experimental Design in Psychological Research, (New York: Rinehart & Company, Inc., 1950), p. 335-James E. V/ert, Charles 0. Neidt, and J. Stanley Ahmann, Statistical Methods in Educational and Psychological Research, (Mew York: Appleton-Century-Crofts, Inc., 1954), p. 343. C. C. Ross and Julian C. Stanley, Measurement in Today's Schools, (New York: Prentice-Hall, Inc., 1954), pp. 436-453. Ibid, p. 119. J Ibid, p. 440 6 Ibid, p. 125 7 Farquhar, op. c i t . , p. 7 ^ Supra, p. 82 9 / Robert Lee Morton et a l . , Making Sure of Arithmetic, (New York: Silver-Burdett Company, 1955), p. 165. See the Manual of Directions accompanying the Otis Self-Administering Tests of Mental Ability, p. 12. 107 CHAPTER V THE STATISTICAL ANALYSIS I. INTRODUCTION The problem to be analyzed statis t i c a l l y in this study involves two teaching treatment groups. The experimental group, composed of two classes with a net number of 59 subjects, was taught with the use of manipulative aids. The control group, composed of three classes with a net number of 88 subjects, was taught with the use of static represent-ations of the aids used by the experimental group. Referred to as "visualization" materials, these aids are intended to possess character-istics identical to those of the manipulative materials in a l l details except the capacity to be, manipulated. The classes selected were matched on the basis of size. By a method described elsewhere, each class was then assigned at random to i t s treatment group."*" Before the commencement of the experiment tests were administered to a l l subjects to provide measures of the four control variables. At the conclusion of the experiment one of these tests was readministered to provide a measure of the criterion variable. The original data is contained in Appendix F. Broadly stated, the hypothesis to be tested is that there is no significant difference between the achievement of the two treatment groups 108 on the criterion when the i n i t i a l differences, measured by the four control variables, have been removed or held constant. In other words, according to this hypothesis, any difference in the mean scores of the two groups on the criterion, after allowances have been made for chance differences in the mean level of achievement in the control variables, may be accounted for entirely by chance fluctuations in random sampling. The allowances for i n i t i a l differences are to be made in terms of the multiple regression of the criterion measure (Y) on the control measures (X^; X^ ; X^; X^). Analysis of Covariance is the statistical procedure used to test this hypothesis. Commenting on the use of the analysis of covariance in research, Edwards states: In particular, i t is applicable to those situations where the matching of'groups is not feasible prior to the assignment of the subjects to the experimental conditions, but where some measure of i n i t i a l performance may be obtained after the assignment. In experiments of this sort, the analysis of covariance may be effectively used to reduce the error mean square in the test of significance.^ Lindquist states: ... through a purely statistical control we can secure the . same precision in the evaluation of the treatment effect as i f we had experimentally controlled the X-factor by actually matching the groups with reference to X ...^ Garrett states: Covariance analysis is especially useful to experimental psychologists when for various reasons i t is impossible or quite difficult to equate control and experimental groups at the start: ... Through covariance one is able to effect 109 adjustments in final or terminal scores which will allow for differences in some i n i t i a l variable. These statements attest to the suitability of analysis of covariance to the present problem in which no attempt was made to match the groups with reference to any of the control variables. A further explanation of the applicability of covariance analysis may be made with reference to Table XXVI, which shows the means and standard deviations obtained by each of the treatment groups in the criterion variable and the four control variables. It will be observed that the means of the two groups in each variable differ very slightly. Likewise, except in the case of the variability on the Otis Test (X^), the standard deviations of the two groups in each variable differ very slightly. Even though the difference between the group means of the control variables does not seem large enough to influence greatly the difference in the means of the criterion variable, the analysis of covariance, which represents an extension of analysis of variance to allow for the correlation betv/een criterion and control scores, is worthwhile. The correlation of means between-groups, and the correlation of variables within groups, wil l increase the precision of the test of significance through altering the mean square (both within groups and between groups) used as the error term by the regression of Y on each of the X variables. Certain degrees of the correlations referred to could change an insignificant F value obtained in an analysis of variance of the TABLE XXVI MEANS AND STANDARD DEVIATIONS OBTAINED BY EACH TREATMENT GROUP IN THE CRITERION VARIABLE AND THE FOUR CONTROL VARIABLES Criterion I n d e p e n d e n t V a r i a b l e s Variable (X ) (X ) (X ) (X ) (Y) 1 2 3 4 Means Experimental 14.492 7.797 17.559 108.305 20.356 Control 14.227 8.795 16.591 110.375 20.386 Standard Deviations Experimental 4.538 Control 5.315 3.002 3.718 4.022 10.835 6.947 5.301 15.806 7.490 111. criterion into a significant F value, after the application of the analysis of covariance. Needless to say, the adjustment from the analysis of covariance may also give results just the opposite of this. By means of an analysis of these correlations a preliminary examination, as given in Subsection 5 of this chapter, may be undertaken to determine whether the analysis of covariance will prove efficient in detecting differences between the means of the two groups on the criterion Y. In the present problem, where there is l i t t l e difference between the means of the two treatment groups in each variable, the primary purpose of the analysis of covariance is to increase the precision of the test of significance. In both the analysis of variance and the analysis of covariance the F value used in the test of significance of the means is obtained by dividing the mean square between groups by the error mean square within groups. Both the correlation of the means between groups (the tendency for the group with the higher mean on each;of the X variables to have the higher mean on the Y variable) and the correlation of individual scores within groups (the tendency for subjects within each group who achieve high scores in. each of the X variables to achieve high scores also on the Y variable) determine the nature and extent of the adjustment in the numerator and denominator of the F ratio that will result from the application of the analysis of covariance. 112 Using the covariance analysis in the present problem for the purpose of making allowances for differences between the means of the experimental and control groups in the X variables follows unquestion-ably as a secondary objective. II. AN ANALYSIS OF VARIANCE OF EACH OF THE FOUR INDEPENDENT VARIABLES X,,X_,X,,X. 1 c. 3 4 The f i r s t step in the application of the covariance technique to the present problem is to analyze the data for each of the independent variables in the usual manner of an analysis of variance. The purpose of this step is to test the hypothesis that the scores of the two treatment groups in each of the independent variables are in reality random samples drawn from the same normally distributed population and, further, that the means between the groups in each variable differ only through the fluctuations of sampling. The rationale of the analysis of variance, by which this hypothesis i s tested, is stated comprehensively by Lindquist: The basic proposition (of the analysis of variance) is that from any set of r groups of n cases each, we may, on the hypothesis that a l l groups are random samples from the same population, derive two independent estimates of the population variance, one of which is based on the variance of group means, the other on the average variance within groups. The test of this hypothesis then consists of determining whether or not the ratio (F) between these estimates lies below the value in the table for F that corresponds to the ..selected level of signif-icance. 5 113 The variance of the group means, the f i r s t independent estimate of the population variance, is represented by the sum of the square of the deviations of the mean of each group from the general • mean. Each of these squared deviations is weighted or multiplied by the number of subjects in each group in order to put them on a per individual measure basis. The greater the difference in the group means, the larger the sum of squares between the groups. The sum of squares based upon variation of group means for two treatment groups is equal to k (M - M)2 + k (M - M) 2 e e ' c N c ' where k designates the number of subjects, M designates the grand mean for the total sample, and the subscripts designate the treatment group. Mentioning the methods of finding the sum of squares between groups (the deviations of the mean of each group from the general mean),. Wert, Neidt, and Ahmann say: "the f i r s t (shown above) is relatively easy to understand but usually time consuming to compute; the second is mathematically identical and is more generally used".^ This second formula i s %*2 w (XX)2 ^ (IX ) 2 (£X) 2 ^ between groups = e- + c — k k N e c where N refers to the number of subjects in the total sample. The variance within groups, the second independent estimate of the population variance, is derived from the sum of squares of the 114 deviations of each score from the mean of i t s own treatment group. Unlike the variance for the total sample, i t is free from any influence of the difference in the means between the treatment groups. The sum of squares within groups is •(S, - M ) 2 + (S_ - M ) 2 +....+ (S - H ) 2 1 g 2 g y n g' where ; etc. designate the scores of individual subjects in the variable concerned, and the subscript "g" refers to the treatment group to which the subjects belong. Explaining the two methods of finding the sum of squares within groups, Wert, Neidt, and Ahmann write: "Here again, the f i r s t (shown above) is self-explanatory and the second 7 saves time". The second method suggested by these authors is one in which the within sum of squares is not directly computed. It is found by subtracting the sum of squares between groups from the total sum of squares. This sum of squares for the total sample may be found directly from the original measures without f i r s t subtracting the mean. The formula used in this case is total ~ **A N; Thus, the sum of squares within groups is the difference between the sum of squares for the total and the sum of squares between groups. This is shown as follows: f £ x 2 - £ x £ J - J ( £ x j 2 + ( t x ) 2 : - a x ) 2 ) 1 ' • / ( — -t~ — J 115 Therefore, x 2 2 f ' " X 2 " ' ^ within groups = £x -' k k e c The sum of squares between groups and the sum of squares within groups are each divided by the number of degrees of freedom involved. These calculations yield, respectively, independent estimates of the population variance between groups and within groups. On the assumption that the groups making up a total series of measurements are random samples from a single normally distributed population, the two foregoing independent estimates of the population variance may be expected to differ only within the limits of the chance fluctuations that occur from random sampling. To test this null hypothesis the ratio of the variance between groups to the variance within groups is expressed as a quotient, called an F value. This F value is then compared with the .05 and Q .01 points of the variance ratios tabled by Snedecor. The value at .05, given in the table for a particular number of degrees of freedom, is the value which would be exceeded only 5$ of the time as a result of sampling variation i f the null hypothesis were true. Therefore, an F value which equals or exceeds the tabled value at the .05 level has a probability equal to or less than 5$. This means that there i s only 1 chance in 20, or less than 1 chance in 20, that an F value as large as this could be obtained by sampling variation. Consequently, a result that happens as seldom as this by 116 chance would be indicative of systematic differences between treatment effects, and so the null hypothesis would be rejected at the 5$ level of significance. Further, i f the F value equals or exceeds the tabled value at the .01 level, i t means that there is only 1 chance in 100, or less than one chance in 100, that a value as large as this could be obtained by sampling variation. Such a result, said to be significant at the 1% level, would be even more convincing evidence on which to reject the null hypothesis. On the other hand, i f the F value falls short of the tabled value at the .01 level, i t has a probability greater than ifo. This means that there is more than 1 chance in 100 that an F value as large as this could be obtained by sampling variation. Consequently, a result that happens as frequently as this by chance would not be indicative, at the ifo level of significance, of systematic differences between treatment effects, and so the null hypothesis would be con-sidered tenable at this level. Further, i f the F value f a l l s short of the tabled value even at the .05 level, the probability is greater than 5%. This means that there is more than 1 chance in 20 that a value as large as this could be obtained by sampling variation. A result that happens as frequently as this by chance would be even less indicative of systematic differences between treatment effects, and so the null hypothesis would be con-sidered tenable at the 5% level of significance. . 117 In accordance with the procedures presented in the preceding pages of this subsection, an analysis of variance was computed for each variable in the whole battery of test controls used in this experiment, namely: X^ , X2, X^, and X^. In addition, an analysis of variance was computed for the variable used as the criterion measure, namely: Y. In each case the formulae used are those shown on pages 114 and 115. The data required for substitution in these formulae are contained in Table XXVII. It may be noted in passing that the sums of scores and the sums of squares of scores shown in this table, as well as the sums of cross products shown in Table XXXIV (page 130 ) and used in a subsequent step, may a l l be secured in a single operation on an automatic Monroe computing machine. The accumulating sums are carried in the machine, and only the totals recorded. As an example of the procedures employed in applying these data to the formulae referred to, the calculations of the sums of the squares of the variable X.^ , for the total sample and within the sub-groups, are shown below. The calculation of the sum of squares for the total sample: * X 1 2 - £ X 1 2 - ^1 N (1254)2 12142 -12142 -147 1522736 147 12142 - 10358.88435 1783.11565 TABLE XXVII SUMS OF SCORES IN THE FIVE VARIABLES, AND SUMS OF SQUARES OF SCORES, ARRANGED BY CLASSES, FOR EACH TREATMENT GROUP AND FOR THE TOTAL SAMPLE General Prince Total for Fleetwood Hjorth Road Simon Total for Total for Montgomery Charles Experimental Cunningham Control Both Groups Group Group £ x 2 E X . E X **1 s x j 2 x ' r-2 318 537 855 576 333 343 1,252 2,107 176 284 460 359 191 224 774 1,234 357 679 1,036 578 432 450 1,460 2,496 2,498 3,892 6,390 4,143 2,656 2,914 9,713 16,103 434 767 1,201 776 480 538 1,794 2,995 4,778 8,827 13,605 9,912 5,075 5,311 20,298 33,9037 1,518 2,600 4,118 3,911 1,829 2,284 8,024 12,142 : 5,949 13,197 19,146 10,268 8,040 8,388 26,696 45,842 273,822 425,174 698,996 471,773 291,380 330,904 1,094,057 1,793,053 9,042 18,253 27,295 18,206 10,846 12,458 41,510 68,805 119 The calculatipn of the sum of squares within the subgroups: - ^ 12142 12142 = : 12142 1747.87750 The sum of squares between groups was not directly computed. It was found by subtracting the sum of squares within groups from the sum of squares for total. Table XXVIII shows these three sums of squares for each of the five variables involved in the study. TABLE XXVIII SUMS OF SQUARES OF SCORES IN THE FIVE VARIABLES, IN DEVIATION FORM, FOR THE TOTAL SAMPLE, AND FOR WITHIN, AND BETWEEN, THE TREATMENT GROUPS Variable Total Within Groups Between Groups £r\ 3702.66667 3700.20031 2.46636 1783.11565 1747.87750 35.23815 w 3460.93878 3427.81510 33.12368 Ex,* 29062.46259 28911.13347 151.32912 w 7784.42177 7784.38906 0.03271 ( E X }2 V 1C; kc (774) 2 ] 88 J 599076 88 } TABLE XXIX 120 ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TV/O TREATMENT GROUPS ON THE INDEPENDENT VARIABLE X (Farquhar Test of Understanding - First Administration) Source of df Sum of Mean F Variation Squares Square Total 146 1783.11565 Within groups 145 1747.87750 12.05433 Between groups 1 35.23815 35.23815 2.923 F 35.23815 ' ^ " 12.05433 " From Table F .923 df 1/145 F at .05 level = 3.91 F at .01 level = 6.81 TABLE XXX ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TWO TREATMENT GROUPS ON THE INDEPENDENT VARIABLE X 2 (Decimal Fraction Computation Test) Source of df Sum of Mean F Variation Squares Square Total 146 3460.93878 Within groups 145 3427.81510 23.64010 Between groups 1 33.12368 33.12368 1.401 Y 33.12368 ' ^ " 23.64010 ~ From Table F 1.401 df 1/145 F at .05 level = 3.91 F at .01 level = 6.81 TABLE XXXI 121 ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TWO TREATMENT (GROUPS ON THE INDEPENDENT VARIABLE X, 3 (Otis S.A. Test of Mental Ability, Intermediate, Form A) Source of Variation Idf /Sum of Squares Mean F.: Square Total 146 Within groups 145 Between groups 1 29062.46259 28911.13347 151.32912 199.38713 151.32912 0.759 *\ 151.32912 „_Q l j 1 4 5 = 199.38713 = ° * 7 5 9 rFrom Table F df .1/145 F at .05 level = 3-91 F at .01 level = 6.81 TABLE XXXII ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TV/0 TREATMENT iGROUPS ON THE INDEPENDENT VARIABLE X 4 (Stanford Advanced Reading Test: Form E) Source of df Sum of Mean F Variation Squares Square Total 146 7784.42177 Within groups 145 .7784.38906 53.68544 Between groups 1 .03271 .03271 0.0006 F-, = 0.03271 ' 5 53.68544 0.0006 From Table F df -71/145 >F at .05 level = 3.91 F at .01 level = 6.81 122 The summary of the analyses of variance for the independent variables Xn, X0, X_, and X. is recorded in Tables XXIX to XXXII, 1 d 5 4 inclusive. An examination of these tables reveals the partition of the total sum of squares into the two independent' estimates of the population variance referred to in Lindquist 1 s quotation on page 112. One of these estimates i s based on the variance within groups; the other on the variance of the group means (between groups). Along with this partition of the total sum of squares into two parts there is a corresponding partition of the total number of degrees of freedom. This partition may be shown as follows: Sum of Squares General Number of Specific Number of Degrees of Freedom Degrees of Freedom Within groups N - r 145 Between groups r - 1. 1 TOTAL ' N - 1 146 where N i s the total number of subjects and r is the number of treatment groups. In Snedecor's table the F value at the .05 level of signif-icance for 1 and 150 (the tabled value nearest to 145) degrees of freedom is 3.91, while at the .01 level i t is 6.81. The. F values in each of these four analyses of variance f a l l considerably short of the value required for significance at the .05 level. Thus, the null hypothesis is tenable. The difference between 123 the means of the experimental and control groups in each of the independent variables is less than may be expected through the fluctuations of sampling. It may be concluded that the scores of both groups in a l l four independent variables are in reality random samples drawn from the same normally distributed and homogeneous population. III. AN ANALYSIS OF VARIANCE OF THE CRITERION VARIABLE Y The second step in the application of the covariance technique to the present problem is to analyze the data for the dependent or criterion variable in the usual manner of an analysis of variance. It will be remembered that the re-administration of Farquhar1 s test at the close of the experiment provided the measure of the criterion variable Y. The f i r s t administration of this test at the beginning of the experiment supplied the measure of one of the independent variables, namely: .• In accordance with the procedures outlined in the preceding subsection, the analysis of variance was computed and is summarized in Table XXXIII. In this preliminary analysis of the Y-means no allowance has been made for the i n i t i a l differences between the groups. It is seen that the resulting F value (.097) falls far short of significance at the .05 level; i t i s , in fact, considerably less significant than the F value (2.92) obtained in the analysis of the results of the f i r s t administration of Farquhar*s test. 124 TABLE XXXIII ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TWO TREATMENT GROUPS ON THE CRITERION VARIABLE Y (Farquhar Test of Understanding - Final Administration) Source of df Sum of Mean F Variation Squares Square Total 146 3702.66667 Within groups 145 3700.20031 25.51862 Between groups 1 2.46636 2.46636 0.097 p; 1, 145 2.46636 25.51862 = 0.097 From Table F df 1/145 F at .05 level = 3.91. F at .01 level = 6.81 Thepurpose"; of the remaining computations in this statistical treatment i s to make allowance in the analysis of the criterion scores ( Y) for individual differences in the control scores (X^, X^ , X^ , X^) obtained at the beginning of the experiment. The general procedure by which this purpose is accomplished involves the prediction of the criterion variable from known values of the control variables. If the deviation of the control scores of any pupil from the general means of these scores is know* the amount by which the pupil's criterion score would be expected to deviate from the criterion mean may be computed. This expectation, which is based entirely on i n i t i a l performance in the control tests without regard for 125 the methods of teaching the two groups, constitutes the prediction of Y by X^ , X^, X^, and X^. It is otherwise referred to as the regression of Y on these control variables though, as G-arrett points out, the "original meaning of 1 stepping back1 to some stationary average is not 9 necessarily implied". Certain basic assumptions inherent in this prediction or regression procedure are discussed l a t e r . ^ The difference between the predicted sum of squares of the criterion and the actual sum of squares of the criterion is known as the residual sum of squares or the sum of squares of errors of estimate.. The relationship may be shown in this way: Original sum of squares of criterion tfY2) Sum of squares of criterion predic-ted on control variable scores (Sum of squares due to regression) t Adjusted sum of squares of criterion (Sum of squares of residuals or sum of squares of errors of estimate. The residuals or errors of estimate are the sums of squares based upon the variation remaining in Y after that portion which can be attributed to the regression of Y on the X variables has been.,taken into account. In other words, the original sums of squares of the criterion, as shown in Table XXXIII, are adjusted so that the variability con-tributed to these sums of squares by the control scores X., X„, X,, and 1 d $ X^ is removed or held constant. This adjustment, of course, concerns the sum of squares for total, the sum of squares within groups, and the sum of squares between groups. 126 When these adjusted sums of squares are calculated a further analysis, similar to that presented in Table XXXIII, is made of the criterion means to ascertain whether these means between the two treat-ment groups have become significantly different as a result of taking into account the individual differences in the control variables. This analysis is located in Subsection VIII of this chapter.: IV. COMPUTATION OF THE SUMS OF CROSS PRODUCTS IN DEVIATION FORM FOR EACH PAIR OF VARIABLES The third step in the application of the covariance technique to the present problem is to compute the sums of cross products in deviation form for each pair of variables. Four prediction variables and one criterion variables involve ten pairs of cross products. The analysis of covariance represents an extension of the analysis of variance in that i t takes into account the regression of Y on the X variables. The dependence of regression upon the relation-ship between the Y scores and each of the X., X-, X,, X. scores may be I ^ p 4 expressed in Edwards' words: "It is the presence of correlation or association between the two that makes prediction possible, and the efficiency or accuracy of such predictions is a function of the degree or strength of the relationship that exists". 1 1 The formulae used to compute the sums of squares due to regression, or in other words the predicted sums of squares of the criterion, are derived from the correlation formula: Lindquist traces the derivation which results in £xy = £(x-x) (y-y) +£xy It is understood that this summation is for the total sample. Then he writes: Thus we see that the total sum of the PRODUCTS (of deviations) may be analyzed into two components, just as • the total sum of SQUARES (of deviations) may be analyzed for either variable considered alone. The components of the total sum of products (of deviations from the general mean) are the sum of the products of deviations from the group means and n times the sum of the products of the group means (each mean expressed as a deviation from the general mean). The COVARIANCE of two variables for a sample is the mean of the PRODUCTS of their deviations from their means, just as the VARIANCE of a single variable is the mean of the SQUARES of the deviations. 1 2. Stated in other words, i t may be said that the total sum of crossproducts may be analyzed into two components, just as in the analysis of variance i t was possible to analyze the total sum of squares into two components. The f i r s t component,;is the sum of cross products within groups. It i s based upon the deviations of the individual scores from the means of the treatment group of which they are a part. . The second component is the sum of cross products between groups. It is based upon the deviations of the means of each treat-ment group from the general mean of the total sample. These two components correspond, respectively, to the two independent estimates of the population variance, namely: the sum of 128 squares within groups and the sum of squares between groups. Referring to his own derivation of the above formula for the sum of cross products for the t o t a l , Edwards says that i t ... does not represent the most convenient method of calculating the sum of cross products. Instead, i t i s easier to.take the values of X and Y from zero origin and to apply a correction term to the products of the original values. ' The resulting formulae are: £ V t o t a l = £ X 1 Y - E £ N: ^ 1 between groups = 1 + 1 — 1 k k N e . c The sum of cross products within groups i s the difference between the sum of cross products for the t o t a l and the sum of cross products between groups. This i s shown as follows: £ X X Y - S X S Y - ( ( C x ^ e + GBCjE^c - C X E Y - J HT J i T ~ 5ET ~ N J Therefore, 7 t y . ' ,. £ X , Y - f (£x.EY)e (EXCY)C 1 within groups = 1 J 1 • + 1 k k e c In a l l cases the subscripts designate the treatment groups, k designates the number of subjects i n the treatment group referred to, and N designates the number of subjects i n the total-sample. 129 The data required for substitution in these formulae, are contained in Tables XXVII (page 118) and XXXIV (page 130). As an example of the procedures employed in applying these data to the formulae, the calculations of the sums of cross products of the combination of variables X^ Y, for the total sample and within the subgroups, is shown below. This exemplifying parallels that, on pages 117 and 119 for the sums of squares. The calculation of the sum of cross products X^ Y for the total sample: £*l y total = C X 1 Y - ZXP N = 19268 - (1234) ( 2107) 147 = 19268 - 2600038 147 = 19268 - 17687.33333 = 1580.66667 The calculation of the sum of cross products X^ Y within the the subgroups: T/ within groups 1 I 1 .; 1 i k k e c = 1 9 2 6 8 _ (835)(460) + (1252)'(774) 59 88 _ i q 2 6 Q _ 395300 969048 - I K U O 5 9 88 = 19268 - 6666.10169 + 11011.90909 = 19268 - 17678.01078 = 1589.98922 As in the case of the sum of squares, the sum of cross products between groups was not directly computed. It was found by subtracting TABLE XXXIV SUMS OF CROSS PRODUCTS OF SCORES IN THE FIVE VARIABLES, ARRANGED BY CLASSES, FOR EACH TREATMENT GROUPS AND FOR THE TOTAL SAMPLE General Prince Total for Fleetwood Hjorth Road Simon Total for Total for Montgomery Charles Experimental Cunningham Control Both Groups Group Group 2 , 5 5 9 4 , 5 1 8 £ X 2 Y 5 , 1 2 7 1 0 , 3 8 6 £ X 3 Y 3 5 , 1 6 3 5 9 , 3 5 2 £ X 4 Y 6,119 1 2 , 1 8 9 I X 1 X 2 2 , 8 0 3 5 , 4 3 9 C X 1 X 3 1 9 , 3 8 8 3 1 , 3 2 7 3 , 3 6 3 6 , 5 1 2 Cx 2x^ 3 9 , 3 1 0 7 3 , 8 1 1 £ x 2 x 4 6 , 8 1 2 1 4 , 6 6 2 Cx 3x 4 4 7 , 9 6 2 8 5 , 2 5 3 7 , 0 7 7 6 , 0 6 2 2 , 8 5 4 1 5 , 5 1 3 9 , 7 0 8 6 , 1 1 9 9 4 , 5 1 5 6 5 , 7 7 8 3 6 , 8 9 4 1 8 , 3 0 8 1 2 , 7 9 4 6 , 7 8 0 8 , 2 4 2 6 , 0 3 0 3 , 4 8 6 5 0 , 7 1 5 - 4 0 , 9 6 2 2 0 , 7 7 9 9 , 8 7 5 8 , 0 5 1 3 , 9 6 0 1 1 3 , 1 2 1 6 6 , 7 0 0 4 7 , 2 4 5 2 1 , 4 7 4 1 3 , 0 3 8 8 , 6 3 1 1 3 3 , 2 1 5 8 9 , 3 7 6 5 3 , 7 8 2 3 , 2 7 5 1 2 , 1 9 1 1 9 , 2 6 8 6,370 2 2 , 1 9 7 3 7 , 7 1 0 3 9 , 4 4 8 1 4 2 , 1 2 0 2 3 6 , 6 3 5 7 , 6 5 4 2 7 , 2 2 8 4 5 , 5 3 6 4 , 0 4 6 1 3 , 5 6 2 2 1 , 8 0 4 2 5 , 8 6 6 8 7 , 6 2 7 1 3 8 , 3 4 2 5,098 1 7 , 1 0 9 2 6 , 9 8 4 5 1 , 3 9 5 1 6 5 , 3 4 0 2 7 8 , 4 6 1 9,619 3 1 , 2 8 8 5 2 , 7 6 2 6 1 , 9 0 0 2 0 5 , 0 5 8 3 3 8 , 2 7 3 O l o 131 the sum of cross products within groups from the sum of cross products for total. Table XXXV shows these three sums of cross products for each of the ten pairs of variables. An explanation of the reason that deviation scores rather than raw scores are used in both the sums of squares and the sums of cross products is given on page 141. Tables XXVIII (page 119) and XXXV (page 132) contain the essential data to be used in the calculation of the regression coefficients. This calculation is found in Subsection VI. These data, however, are f i r s t used in Subsection V, where a preliminary examination is made of the conditions under which analysis of covariance is worthwhile. V. AN EXAMINATION OF THE CONDITIONS UNDER WHICH AN ANALYSIS OF COVARIANCE WILL INCREASE THE PRECISION OF THE TEST OF SIGNIFICANCE The fourth step in the application of the covariance technique to the present problem is to compute the correlation coefficients of the means between treatment groups and of the individual scores within groups. Though not an actual part of the covariance procedure, a description of these tv/o correlation coefficients should give a good indication of the conditions under which an analysis of covariance will prove efficient in detecting differences between the means of the groups on the Y variable. In addition, this description should be helpful in interpreting the final test of significance. 132 TABLE XXXV SUMS OF CROSS PRODUCTS OF SCORES IN THE FIVE VARIABLES, IN DEVIATION FORM, FOR THE TOTAL SAMPLE, AND FOR WITHIN, AND BETWEEN, THE TREATMENT GROUPS Cross Products Total Within Groups Between Groups £ x i y r x 2 y £ X 3 y £ x 4 y £ X i X 2 C1 X3 £ X 1 X 4 E X 2 X 3 £ X 2 X 4 £ X 3 X 4 1580.66667 1934.00000 5825.33333 2607.66667 851.18367 3164.43537 1842.29932 5038.63265 1908.12245 10188.06803 1589.98922 1924.96148 5844.65254 2607.95070 885.34822 3091.41102 1841.22574 5109.43220 1909.16333 10185.84322 -9.32255 9.03852 -19.31921 -0.28403 -34.16455 73.02435 1.07358 -70.79955 -1.04088 2.22481 133 Nature of correlation of means between treatment groups The f i r s t part of this subsection deals with the correlation of the means between the treatment groups. It will be remembered that this correlation refers to the tendency for the group with the higher mean on each of the X variables to have the higher mean on the Y variable. Where only two groups are involved the correlation must, of course, be either +1 or -1. ' This correlation may be computed by the Pearson product-moment method. The formula used when deviations are taken from the means of the two distributions is rx^y (between) = ^ x l y Substituting the appropriate values from the between groups source of variation in Tables XXVIII (page 119) and XXXV (page 132) results in the following: _ -9.32255 x y (between) = 1 /35.23815 x 2.46636 -9.32255 y"86.9099636340 -9.32255 +9.32255 -1 In Appendix G similar calculations are shown for each of the remaining 14 three correlations x 0y, x,y and x.y. ^ 3 4 An examination of the means in Table XXVI (page 110) reveals that in three of these four pairs of variables there is a negative 134 correlation. In other words, in these cases the group with the higher mean in the one variable has the lower mean in the other. In Table XXXV (page 132) i t is seen that in these three pairs of variables the sums of cross products are negative. In analysis of variance, to produce significant differences in the means of the treatment groups i t is desirable, where there are only two groups, that the sums of cross products between groups be negative, or where there are more than two groups, that the sums of cross products be as near zero as possible. In view of the fact that three of :the sums of cross products between groups in this experiment are negative, i t would seem likely that the sum of squares due to regression within groups will exceed the sum of squares due to regression for total. This anticipation 15 i s justified by subsequent calculations. Such a condition inevitably results in the sum of square's of residuals between groups becoming larger than the original sum of squares between groups. This sum of squares of residuals between groups, when divided by the number of degrees of freedom for that source of variation, becomes the mean square between groups which forms the numerator of the F ratio. The nature of the correlation of the means between the treatment groups is such that i t indicates that the precision of the test of significance will be increased in this problem by the application of the analysis of covariance. 135 Nature of correlation of individual scores within each treatment group The second part of this subsection deals with the correlation of individual scores within groups. I t w i l l be remembered that this correlation refers to the tendency for subjects within each -group who achieve high scores i n each of the X variables to achieve high scores also on the Y variable. The higher the correlations within groups between the c r i t e r i o n variable and each of the independent variables, the larger w i l l be the sum of squares due to regression, and, consequently, the smaller w i l l be the sum of squares of residuals. The error mean square, which i s the variance obtained by dividing the sum of squares of residuals by the number of degrees of freedom within groups, w i l l likewise become smaller. Since this forms the denominator of the F r a t i o , it".'..will be seen that the strengths of the correlations referred to affect d i r e c t l y the precision of the f i n a l test of significance of the means between the two treatment groups. A formula exists by which the extent of the reduction i n the adjusted error variance may be estimated on the basis of the strength of the correlations. These correlations, corrected for attenuation, w i l l now be calculated. The Pearson product-moment formula, shown on page 133, i s used. Four correlations must be calculated, namely: x^y, x 2y, x^y and x^y. The necessary data are found i n the within groups source of variation i n Tables XXVIII (page 118) and XXXV (page 132). As an example of the 136 procedure employed in applying these data to the formula, the calculation of the within group correlation between x^ and y is shown below. In Appendix G similar calculations are shown of the v/ithin group correlations for each of the remaining pairs of variables x^y, x,y, x.y. c 3 4 1589.98922 x,y (within) = 1 1 ^3700.20031 x 1747.87750 1589.98922 y^467496.8673420250 1589.98922 2543.1273 .63 Each of the correlations computed in this way has been corrected for attenuation to give an intrinsic correlation between two series of measures with postulated perfect reliability. The formula used to obtain this correction for attenuation is r r = / r r J x 1x 1x. yy where r and r refer to the r e l i a b i l i t i e s of the tests involved. yy For convenience, a summary of these r e l i a b i l i t i e s previously reported in the various tables of Chapter IV, is reproduced in Table XXXVI. As an example of the procedure employed in applying the data to; the formula, the correction for attenuation of the correlation within groups of x^y i s shown on page 137. In Appendix G similar calculations of corrections for attenuation are shown for each of the remaining correlations x y, x y, and x y. 2 3 4 137 r = *63 4-549 x .541 < .63 ^ .297009 = .63 .545 = greater than unity Table XXXVII summarizes the correlation data obtained from the calculations exemplified above. TABLE XXXVI SUMMARY OF RELIABILITIES OF TESTS EMPLOYED TO PROVIDE MEASURES OF THE CRITERION VARIABLE AND THE FOUR INDEPENDENT VARIABLES Test Variable Reliability Method Farquhar's Test Y. .541 Hoyt modifica-(final admin.) tion of K.R. Formula Farquhar's Test X, .549 same (t r i a l admin.) Decimal Computa- X 2 . .821 same tion Test Otis Test X, .948 comparable-forms Stanford Test X 4 .874 Split-half TABLE XXXVII PEARSON PRODUCT MOMENT COEFFICIENT OF CORRELATION, AND INTRINSIC CORRELATION, WITHIN GROUPS OF THE CRITERION VARIABLE WITH EACH OF THE INDEPENDENT VARIABLES Independent Product-Moment Intrinsic Correlation and Criterion Coefficient of (after correction for Variables Correlation attenuation) X l y .63 greater than unity .54 .81 x3 y .57 .80 X 4 y .49 .71 138 The average intrinsic correlation may be assumed to be approximately .73. ^ This coefficient may now be substituted in the following formula, which was f i r s t referred to on page 135. 17 Sum of squares within groups r (1 - r 2 / .... 0 £ ( n - 1) - 1 X y ( W l t h i n ) 1. Edwards refers to this as a variation of the formula used to calculate the standard error of estimate. (1 - -732) ( l - .5329) .467 In his treatment of this particular point, Lindquist states: "The ratio.between the adjusted error variance and the unadjusted error 2 _8 variance is very nearly equal to ( l - r w ) . " Shown in the form of a proportion, this becomes: Adjusted error variance _ ,457 Unadjusted error variance Substituting the mean square variance within groups reported in Table XXXIII (page 124): Adjusted error variance _ ^g 7 25.51862 Adjusted error variance = approximately 12 139 This reveals that the error mean square used in the final test of significance, after allowances have been made for the regression of the Y variable on each of the X variables, wi l l be approximately 12. This compares to 25.51862, the original mean square used in the f i r s t test of significance before any allowances were made for regression. Since this reduces the denominator of the F ratio by over one-half i t s original value, i t is apparent that the pscision of the experiment wi l l be more than doubled by reason of the within groups correlation alone. Summary of correlation conditions In summary, i t may be said that the foregoing examination of the correlation between groups and the correlation within groups indicates that the F value used in the test of significance will be substantially increased through the application of an analysis of covariance. The f i r s t indication is that the correlation of the means between groups has been found to be such that the numerator of the original variance ratio (2.46636, as shown in Table XXXIII - page 124) will be increased by taking into account the regression of Y on the X variables. The second indication is that the correlation of individual scores within groups is sufficiently high (.73) to permit a considerable reduction in the denominator of the same variance ratio (25.51862). Only an extremely insignificant difference in achievement 140 between the two groups was detected by the analysis of variance of the Y scores alone (F=.097). Because of the nature of the correlations described in this subsection, this difference will inevitably be more pronounced and may indeed be significant when multiple regression is taken.: into account, and the results tested against the mean square • for error in the analysis of covariance. The prospect of this outcome warrants the continuation of the present statistical treatment. VI. CALCULATION OF THE COEFFICIENTS OF REGRESSION The f i f t h step in the application of the covariance technique to the present problem is to calculate the coefficients of regression. This step is necessary in a multiple regression analysis; that i s , where more than one X variable i s involved in the prediction of the criterion variable. In a four variable regression^problem the general regression equation in deviation form is y = + a 2x 2 + a ^ + a ^ When the expression 21 (y - a 1 x 1 ~ a2 X2~ a3 X3~ a4 X4^ 2 is differentiated with respect to a^, a 2, a^, and a^, respectively, and each of the derivatives is set equal to zero the resulting normal 141 equations are 19 E x l y £ x 4 y a l E x i 2 + a 2 ^ x l x 2 + a 3 ^ x i x 3 + a 4 ^ * x i x 4 al£ Xl X2 + a 2 ^ X 2 2 + a 3 ^ x 2 X 3 + a4^- X2 X4 a l E X l X 3 + a 2 ^ X 2 X 3 + a 3 ^ x 3 2 + a 4 ^ X 3 X 4 a l S X l X 4 + a 2 ^ X 2 X 4 + a3^* X3 X4 + a 4 ^ x 4 2 An explanation should now be made of the reason that the deviation form of the general regression equation is used in preference to the raw score form. Wert, Neidt, and Ahmann offer this explanation: As in the case of single variable regression, the deviation score method can be used to calculate the prediction equation for multiple regression. The general equation in deviation form differs from the equation in raw score form in that the C term has again disappeared. Thus the number of normal equations nec-essary has been reduced by one. Whereas the raw score computations require one more normal equation than the number of prediction variables present, the deviation score method requires the same number of normal equations as prediction variables used. This labor-saving aspect is the principal advantage of the deviation score method over the raw score method, particularly as the number of prediction variables increases. 2^ Only the coefficients of the total regression equations and of the within groups regression equations need be calculated. These coefficients provide the necessary data to obtain the sums of squares of residuals for the total sample and for within the treatment groups. The sum of squares of residuals between the treatment groups is the difference between these two sums. The coefficients of the between groups regression equations, therefore, do not need to be calculated. 142 Calculation of the coefficients of the total regression equation To obtain these coefficients i t is necessary to substitute in the four.normal equations the appropriate deviation values of the sums of squares and of cross products for the total sample. Sub-stitution in the normal equations of these values, which are contained in Tables XXVIII (page 119) and XXXV (page 132) yields: 1580.66667 = 1783.11565a]+851.18367a2+3l64.43537a +1842.29932a4 1934.00000 = 851.18367a,+3460.93878ao+5038.63265a_+1908.12245a. 1 d f> 4 5825.33333 = 3164.43537a.,+5038.63265ao+29062.46259a,+10188.06803a, 1 d $ 4 2607.66667 = 1842.29932a,+1908.12245ao+10188.06803a_+7784.42177a. 1 d 5 4 Table XXXVIII indicates the values of the regression coefficients a,, a 0, a,, and a .(for total) which result from the solving of these ± d 5 4 four simultaneous equations. The values were checked by substituting them in the original equations and obtaining identities. Calculation of the coefficients of the within groups regression equation To obtain these coefficients the appropriate deviation values, also contained in Tables XXVIII (page 119) and XXXV (page 132), of the sums of squares and of cross products for within groups are substituted in the normal equations. The equations become: 1589.98922 = 1747.87750a.+885.34822ao+3091.41102a,+1841.22574a, 1 d p 4 1924.96148 = 885.34822a1+3427.81510ao+5109.43220a,+1909.l633a, J. d p 4 5844.65254 = 3091.4110 2a.,+ 5109.43220ao+ 28911.13347'a_+10185.84322a, i d 5 4 2607.95070 = 1841.22574an +1909.1633a_+10185.84322a_+7784.38906a 1 2 3 4 143 TABLE XXXVIII REGRESSION COEFFICIENTS OF THE TOTAL REGRESSION EQUATION AND THE WITHIN GROUPS REGRESSION EQUATION Regression Coefficient Total Regression Equation Within Groups Regression Equation .585892389 .608249961 a, .287384266 .271002746 .07888538 .084037579 ,022633764 .014727310 Table XXXVIII indicates also the values of the regression coefficients a^, a^, a^, and a^ (for within groups) which result from the solving of these four simultaneous equations. The values were checked as in the manner of the previous solutiCns. 144 VII. CALCULATION OF THE SUMS OF SQUARES OF RESIDUALS The sixth step in the application of the covariance technique to the present problem is to calculate the sums of squares of residuals, otherwise known as the errors of estimate. As previously shown, the sum of squares of residuals is obtained by subtracting the sum of squares due to regression from the original sum of squares of the criterion."' This relationship i s 21 represented by the formula: Sum of squares 2 _ „ of residuals = 2»y - (a^ioi^y + a^x^y + a^ux^y + a^x^y) The sum of squares of residuals is based upon the variation remaining in Y after due allowance has been made for the regression of Y on each of the X variables. Through covariance, therefore, a statistical control has been maintained over those unmatched pupil abilities v/hich purport to be measured by the tests selected. Exercising this statistical control over these variables, which are considered to be the most important ones in influencing the pupils' performance on the criterion, permits a precise evaluation of the treatment effects. Substituting in the above equation the appropriate values obtained from Tables XXVIII (page 139'), XXXV (page 132), and XXXVIII' (page 143) results in the following: 145 Sum of = 3702.66667 squares of residuals for t o t a l = 3702.66667 = 3702.66667 = 1702.19132 (.585892389)(l580.66667) C.287384266)(l934.00000) ( .078888583)(5825.33333) (.022633764) (2607.66667) 926.10057149897463 555.801170.44400000 459.55229190637139 59.02131199944588 2000.47535 Sum of squares of residuals for within = 3700.20031 - ( .60824996l)(l589.98922) ( .271002746)(1924.96148) ( .084037579K5844.65254) ( .014727310)(2607.95070) = 3700.20031 - 967.1108810554.2042 521.66984702422408 491.17044955780066 38.40809842361700 = 3700.20031 - 2018.35928 =•1681.84105 Since the co e f f i c i e n t s of regression between groups were not calculated, the sum of squares of residuals between groups are obtained by finding the difference between the two sums of squares already calculated, thus: ISum of squares^ of residuals \ for t o t a l J 'Sum of squares] of residuals } within groups J Sum of squares of residuals ^between groups Substituting the appropriate values produces the following: 1702.19152 - 1681.84105 = 20.55029 Table XXXIX contains a summary of the computation of the sums of squares of residuals. 146 TABLE XXXIX SUMMARY OF SUMS OF SQUARES OF RESIDUALS Source of Variation Sum of Squares of Criterion Sum of Squares Sum of Squares due to of Residuals Regression Total Within 3702.66667 3700.20031 2.46636 2000.47535 2018.35928 1702.19132 1681.84103 20.35029 An examination of this table confirms the general accuracy of the preliminary analysis contained in Subsection V of this chapter. The nature of the correlation between groups has made the sum of squares of residuals between groups considerably larger than the original sum of squares of the criterion. In addition, the degree of the correlation within groups ( .75) has made the sum of squares of residuals v/ithin groups less than one-half as large as the original sum of squares of the criterion. The effect upon the F value of these adjustments in the original sums of squares of the criterion is seen in the summary of the analysis of covariance contained in Subsection VIII. 147 VIII. CALCULATION OF F VALUE AND APPLICATION OF TEST OF SIGNIFICANCE The seventh step in the application of the covariance technique to the present problem i s to calculate the F value and apply the test of significance to the adjusted group means. \ The analysis of covariance of the performance of the two treatment groups on the criterion variable Y is summarized in Table XL. This analysis may be compared with the preliminary analysis of the Y means contained in Table XXXIII (page 124), where no allowance was made for i n i t i a l differences between the groups in the control variables. In presenting a comparison of the two tables, an explanation should be made concerning the change in the degrees of freedom. In the analysis of covariance an additional degree of freedom was lost for each of the four prediction variables through the reduction in vari-ability imposed by the calculation of the regression coefficients for total and within. This reduces the degrees of freedom for each of these sources of variation to 142 and 141 respectively. Since a new regression coefficient was not calculated in obtaining the adjusted sum of squares between groups, no additional degree of freedom i s lost. It was stated previously that in the present problem, where there is l i t t l e difference between the means of the two treatment groups TABLE XL 148 ANALYSIS OF COVARIANCE OF PERFORMANCE OF THE TWO TREATMENT GROUPS ON THE CRITERION VARIABLE Y Source of df Sum of squares Mean F Variation of residuals Square Total 142 1702.19132 Within groups 141 1681.84103 11.9280 Adjusted Means between groups 1 20.35029 20.3503 1.706 From Table F \ 1 4 1 = 20.3503 = 1.706 df l/l41 1 1 , 9 2 8 0 F at .05 level = 3.91 F at .01 level = 6.81 in each variable, the primary purpose of the analysis of covariance r 22 is to increase the precision of the test of significance. An examination of Table XL reveals the extent to which the analysis f u l f i l l e d this purpose both through increasing the mean square between groups, and through decreasing the error mean square within groups. It will be noted that the mean square between groups (the numerator of the variance ratio) has been increased from 2.4664 in the original analysis of variance to 20.3503 in the final analysis of covariance..,. At the same time, the error mean square within groups (the denominator of the variance ratio) has been decreased from 25.5186 to 11.9280. 1 4 9 In the latter case, where a formula exists by which this reduction may be estimated on the basis of known correlations, the extent of the adjustment in the error variance was precisely anticipated. While the resulting F value (1.706) s t i l l falls short of significance at .05, i t represents an increase in the precision of the experiment of over 17 times the F value obtained in the original analysis of the criterion means. However, since the difference in the adjusted criterion means remains insignificant despite the statistical control of the X variables, it can be concluded with reasonable certainty that the difference which does exist is due to sampling fluctuations rather than to a real treat-ment effect. Therefore, the statistical analysis contained in this chapter confirms the null hypothesis. This hypothesis states that pupils who are taught with the use of certain specified manipulative materials in the manner prescribed by this experiment achieve an understanding of decimal fractions that is not significantly different from the achievement of pupils who are taught with the use of visualization materials which bear characteristics similar to those of manipulative aids in a l l details except the capacity to be manipulated. IX. ASSUMPTIONS UNDERLYING THE DERIVATION OF ANALYSIS OF COVARIANCE Several assumptions underlie the application of the analysis of covariance to the present problem. To be able to draw valid conclusions 150 respecting the effect upon the criterion of the teaching treatments, i t is necessary that the assumed conditions actually exist in the design and conduct of the experiment. Wert, Heidt, and Ahmann emphasize this necessity: "The more the data in an investigation depart from the strict fulfillment of the assumptions the more likely is the investigator 23 to reach erronious conclusions". Lindquist l i s t s these assumptions as follows: (1) The subjects in each treatment group were originally drawn either (a) at random from the same parent population, or (b) selected from the same parent population on the basis of their X-measures only - the selection being random with reference to a l l other factors for any given value of X. (2) The X-measures are unaffected by the treatments. (3) The criterion measures for each treatment group are a random sample from those for a corresponding treatment population. (4) The regression of Y on X is the same for a l l treatment populations. (5) This regression is linear. (6) The distribution of adjusted scores for each treatment population is normal. ( 7 ) These distributions have the same variance. (8) The mean of the adjusted scores is the same for a l l treatment populations. 2^ Referring to these conditions which establish the validity6f the procedure, Lindquist writes: Judging by past applications of the method of analysis of covariance in educational and psychological research, the assumptions underlying the test of the hypothesis of equal treatment effects are, in general, in greater need of cr i t i c a l 151 attention than is true with most, i f not a l l , of the designs previously considered. Generally the method has been employed with l i t t l e regard to the conditions under which the test is valid, and instances are numerous in which one or more of the conditions have clearly not been satisfied.^5 The same author then deals specifically with each assumption. Dealing with Assumption 1, he states: The f i r s t condition, concerning the manner of selection of the treatment groups, has perhaps most often been violated with serious consequences.26 Lindquist describes one misconception which contributes to this violation: ... they (experimenters) seem to have assumed that the method eliminates the effects of any systematic differences that may have existed originally among the treatment groups, even though some of these differences may be quite independent of the X variable employed.^ He then presents two examples to illustrate unwise reliance upon analysis of covariance to remove systematic differences. These examples involve the use of analysis of covariance in an experimental comparison of three ways of teaching fourth grade arithmetic. The fi r s t example is one in which ... throughout the f i r s t semester the classes had had different arithmetic teachers, who had not only differed in personal effectiveness but also had used somewhat different methods of teaching arithmetic. Suppose the teacher of the class that was later to use experimental Method A used a method much like Method A, so that when the experiment began the pupils were able at once to use the experimental method with near maximum effectiveness. Suppose, however, that the teacher of the class that was later to use Method B had used a method which conflicted with Method B, so that considerable time was required early in the experiment before the pupils were able to use this method effectively. In this case, no "adjustments" based on i n i t i a l intelligence test scores, or even on i n i t i a l arithmetic achieve-ment test scores, could possibly account for the effects .of these differences upon the final adjusted means of the treatment groups. 28 152 In the present study c a r e f u l precautions were observed i n the class s e l e c t i o n to avoid the inadvertent i n t r o d u c t i o n i n t o the experiment of the inadequately c o n t r o l l e d systematic differences described by L i n d q u i s t . The three main c r i t e r i a f o r the s e l e c t i o n of the f i v e classes 29 were based p r i m a r i l y on the q u a l i f i c a t i o n s of the teachers i n v o l v e d . Among these c r i t e r i a due importance was given to s e l e c t i n g teachers who had been f o l l o w i n g reasonably conventional methods i n t h e i r every-day teaching p r a c t i c e s and who, though e n t h u s i a s t i c , were nevertheless d i s i n t e r e s t e d i n the manipulative and v i s u a l i z a t i o n methods of teaching decimal understanding. The experimenter i s unaware of any c h a r a c t e r i s t i c s i n the performance of the f i v e teachers selected which could possibly work to the advantage or disadvantage of any class p a r t i c i p a t i n g i n the experiment. No need e x i s t e d , therefore, to introduce any of the c o n t r o l v a r i a b l e s f o r the planned or i n c i d e n t a l purpose of imposing i n v a l i d c ontrols over any of the systematic differences noted by Li n d q u i s t i n the foregoing quotations. The second example i s .one.io i n which ... the classes were o r i g i n a l l y selected not at random but so as to d i f f e r markedly with reference to some t r a i t or character-i s t i c r e l a t e d to the c r i t e r i o n v a r i a b l e i n the experiment. Suppose, f o r example, that the classes had been selected according to a b i l i t y and i n t e r e s t , that the abler and more in d u s t r i o u s students had been assigned to one c l a s s and the l e a s t able to another, and that appropriate modifications i n i n s t r u c t i o n had been used with these classes during the f i r s t semester. Suppose then that an i n i t i a l achievement provided the X-measures used i n the a n a l y s i s of covariance. In t h i s case, not only 1 53 would Assumption 1 would be invalid, but differences in regression (Assumption 4) and in variability of adjusted scores (Assumption 7) or even differences in the nature of the regression (Assumption 5) might well be expected. Nevertheless, many applications of this type also may be found reported in the research literature.30 In the present study Subsection II of this chapter contains the report of the analysis of variance which was applied to each of the independent variables X^, X^, and X^. The P values obtained in these analyses, and shown in Tables XXIX to XXXII inclusive, (pages 120 and 12l), are 2.293, 1.401, 0.759, 0.0006, respectively. The F value required for significance with the given number of degrees of freedom is 3.91 at the .05 level. These data support the hypothesis that the scores of both treatment groups in a l l four variables are in reality random samples drawn from the same normally distributed and homogeneous population, and that the means between the groups in each variable differ only through the fluctuations of sampling. Statistical evidence is thus available to assure the validity of Assumption 1. Dealing next with Assumption 2, Lindquist writes: If the X-measures are taken at the beginning of the experiment or before, they could obviously not be affected by the treatments no matter what X may represent.31 In the present study each of the four X-measures was obtained before the administration of the teaching methods. On this account the X-measures are assuredly unaffected by the treatments. Lindquist then concludes his discussion of the importance of the assumptions underlying the test of significance of the -treatment effect: 154 Of the remaining assumptions, perhaps the most c r i t i c a l in practice is the assumption (Condition 4) that the regression of Y on X is the same for a l l treatment populations. Decisions concerning the validity of the other assumptions - linearity of regression, normality of distribution, and homogeneity of variance - must generally represent judgments based on a priori considerations like those discussed in earlier chapters, since available statistical tests of the validity of these assumptions are both low in power and dif f i c u l t to apply. A statistical test of homogeneity of regression, however, is readily available. This statistical test of homogeneity of regression, performed to satisfy Assumption 4, is presented in the following subsection. X. STATISTICAL TEST OF HOMOGENEITY OF REGRESSION Regression refers to a correlation relationship between the criterion variable and each of the independent variables. From such correlations i t is possible to determine the relative weight with which each independent variable "enters in" or contributes to the criterion, independently of the other factors. To be homogeneous this weight which each independent variable contributes to the criterion must be the same within the limits of sampling error for a l l treatment groups. To satisfy Assumption 4 in the present study, homogeneity in this respect between the two treatment groups must be proven for four pairs of variables, namely: X^, X2Y, X^Y, and X^Y. However, to establish this proof i t i s considered sufficient to test the homogeneity of regression between the two treatment groups of only two of these pairs, namely: X..Y arid X I , 155 These pairs were selected because, as shown in Table XLI, they contribute, respectively, the most and the least to the proportion of the entire variance accounted for by the use of the complete battery of four variables. The data in Table XLI is obtained from the calculations of the sum of squares of residuals for within which were reported on page 145. It may be noted that the within group regression variance offers the most nearly unbiased estimate of the regression of Y on the X variables because i t is free from any influence of systematic differences in the mans of the two treatment groups. TABLE XLI WITHIN GROUP VARIANCE ACCOUNTED FOR BY THE USE OF EACH OF FOUR INDEPENDENT VARIABLES: Xg, X , X^ . Variable Name of Variable Variance Proportion X l I n i t i a l Understanding of Concepts of Decimal Fractions 967.111 .479 X2 Computational Ability in Decimal Fractions 521.670 .259 X3 Mental Ability (Otis Test) 491.170 .243 X4 Reading Ability (Stanford Test) 38.408 .019 Total 2018.359 1.000 156 Specifically, the problem involved in this subsection is to determine whether the regression coefficients of the two pairs of variables X^ Y and X^ Y differ significantly in the two treatment groups. This involves the testing of two hypotheses which, stated in terms of beta coefficients, are: >6 YE X 1E =y6 Y c XlC a n d £ Y E X 4 E =/6 YC X 4C If these null hypotheses remain tenable in the test of significance, i t may be said that the values of b and b do not differ y E x 1 E y C x 4 C significantly, and neither do the values of b and b y E x 4 E yC x 4 c . It will follow, then, that the regressions between the two treatment groups are homogeneous, and that Assumption 4 has been f u l f i l l e d . Test of the Null Hypotheses t h a t ^ Y g X ^ =>6YcXic A statement of the purpose of each calculation involved in the testing of this hypothesis is included in the sequence of enumerated 35 steps which follow; 1. Sum of squares of errors of estimate for the variables X^ and Y in the two treatment groups. For the Experimental and Control groups independent calculations are made of the sum of squares based upon the variation remaining in Y after due allowance has been made for the regression of Y on X^. The sum of these two calculations divided by the number of degrees of freedom produces the sum of squares of errors of estimate for 157 the variables X^ and Y in the two treatment groups. These calculations are presented by the following formula, in which the sums of squares and of cross products are expressed in deviation form: V 2 * i E 2 / \ ° E*1C 2 / k + v - 4 E C It may be noted that the sums of squares and cross products in this formula are the basic quantities referred to, respectively, on pages 114 and 128. They are obtained from raw scores in the following manner: £ x , 2 - S c 2 -N' N When these equivalents are substituted in the deviation formula shown above, the raw score formula shown on the f i r s t line of page 159 is obtained. The necessary raw score data is contained in Tables XXVII (page 118) and XXXIV (page 1 3 0 ) . The denominator in the formula refers to the degrees of freedom available for the sum of squares being calculated. Two degrees 158 2 of freedom are lost due to the res t r i c t i o n s imposed by y_ 2 and y c . Two additional degrees are lo s t through the calculations of the regression coefficients b and b y E X l E y C X l C The complete calculations, shown on the next page result i n the following: 3vv 2 = 15.661343 y x l Standard error of estimate. This i s a measure of the average errors of estimate or prediction. I t represents the scatter of the Y values around the regression l i n e , and i s found by taking the square root of the variance obtained i n the foregoing; step. s ^ = /15.661343 3.957 Regression Coefficients of y on x^ for the Experimental and Control Groups. The regression coefficient of y on x^ may be written £x-y b =: =-— At this point certain data obtained i n the calculations of Step 1 are summarised for convenience i n Table XLII. These data are used i n the present step and the one immediately following. Step 1 in the test of the null hypothesis that „ = Q E IE / Y c x i c CALCULATION OF SUM OF SQUARES OF ERRORS OF ESTIMATE FOR THE VARIABLES X AND Y IN THE TWO TREATMENT GROUPS 11 2 -cy.)2 - h . s . - ^ i E E y E _EJ *E 1E-E ZY W i c ' - P^ic) 2 kE + kC - 4 13605 (855)2 ((7077 -5 9 * (460)(855 59 r 4118 - (460)' 59 20298 (1252)2 - (12191 -88 (774)(1252)| 88 J 8024 - (774)' 88 ( 1214.746 - (410.898 531.559" 59 + 88 -4 2485.455 - (1179.091)' 1216.318 143 897.120 } * [ 1342.452 j 143 S 2 = 15.661343 y X l TABLE XLII SUMS OF SQUARES AND CROSS PRODUCTS OF VARIABLES x AND y, CALCULATED INDEPENDENTLY FOR EXPERIMENTAL AND CONTROL GROUPS Sum of Squares Sum of Cross Products £ X 1 E 2 = : 5 3 1 ' 5 5 9 £ XlE yE = 4 1 0 * 8 9 8 Z*1C2 = 1216.318 2xiCyc = 1 1 7 9 * 0 9 1 Using the appropriate data in the regression coefficient formula results in b = 410.898 ^ ^ E 531.559 .773 b = 1179.091 yC XlC 1216.318 .969 4. Standard Error of the Regression Coefficients b y E X l E and b y c x i c This is represented by the formula: S.= 161 Using the approp y E X l E y c x i c iate data i n this 3.957 /53I7559 .172 3.957 /1216.318 .113 formula results i n Standard error of the difference between the regression coefficients b and b y E X l E ' y C X l C This i s represented by the formula: / 2 i 1 2 J y E X l E y C X l C = ^(.172)* I (.113)' .2.06 Test of significance of the difference between the regression coefficients b (.773) and b (.969). V X V X E IE JC 1C The t value i s obtained by dividing the difference between the regression coefficients by the standard error of the difference between the regression coefficients. 162 34 This is represented by the formula: b b yC XlC - y E X l E t = .969 - .773 .206 ( .951 This t value may be evaluated by entering the t table with 35 k g + - 4 = 143 degrees of freedom. For the two-tailed test of significance of the null hypothesis that / ^ y E X l E = y C X l C ' a * o f 1.976 would be required at the .05 level of significance, while 2.609 would be required at the .01 level. Since the observed value of t is only .951 > the null hypothesis remains tenable. It may be said, therefore, that the regression coefficients b and b do not v x v x ^E IE 1C differ significantly. Test of the Null Hypothesis that^ J Y E X 4 E = /5 Y c X4C In the testing of this hypothesis the procedure is identical to that followed in the treatment of the foregoing hypothesis. 1*-. Sum of squares of errors of estimate for the variables X^ and Y in the tv/o treatment groups. The necessary raw score data is contained in Tables XXVII (page 118) and XXXIV (page 130). 163 The complete calculations, shown on the next page, result in the following: s 2 = 19.755776 y x4 2. Standard error of estimate. = 4.445 3. Regression Coefficients of y on x^ for the Experimental and Control Groups. At this point certain data obtained in the calculations of Step 1 are summarized for convenience in Table XLIII. These data are used in the present step and the one immediately following. TABLE. XLIII SUMS OF SQUARES AND CROSS PRODUCTS OF VARIABLES x AND y, CALCULATED INDEPENDENTLY FOR EXPERIMENTAL 'AND CONTROL GROUPS Sum of Squares: Sum of Cross Products £ x 4 E 2 = 2847.525 £ x 4 E y E = 9 ° 3 - 6 7 8 £ x 4 c 2 = 4936.864 £ x 4 C y C = 1 7 0 4 ' 2 7 5 164 CM O X K 05 -p a += CO •rl CO 0 43 o PH >» .S3 a < H O +> CQ CD -P d) 45 C •H CO -t-> CO O SB EH W K EH S3 M X co PH* rH m <: M ft! < > w H c« to O PH PH :=> Q W « EH cjj S EH M S3 EH pq co S W EH «! PH W O « EH CO ft; o w ft; PH o co ft! <t! *3 Of CO p q O a C O PH O o H EH <U t-H O rH <! o CM o X W i o X u o X CM O CM X U4 CM CM O U3 CM I 111 t-3 + CM in CM rH CM CO CO cn D- t— rH rH v y CO CM CM C-CM CO CO O H in CM CM tn CM CO CO CO cn CM o CM p '— 0.J in in CO v • CM CTi ^ . rH in rH o o CM CM rH rH —' —s CO o tn co rH CM in in cn CM t-CM CM in in co <J1 in in o V O tn CM I CO CO cn in tn c-CM VO • CO • O VO t— tn H cn in m • in CO CM CM CO in [- CM VO in • • tn O -=i-cn CO CM VO H CM t— rH rH • CO VO C-t-m in c— • cn tn H tn rH CM C Q X cn in cn • c-CM cn CM X CO CM CO X >> CM C O X CM CO Using the appropriate data i n the regression coefficient formula"results i n J = 9 0 5 . 6 7 8 Y E X 4 E 2 8 4 7 . 5 2 5 .317 b = 1704.275 yC X4C 4956.864 .545 Standard Error of the Regression Coefficients b_ _ and b yC X4C yE X4E Using the appropriate data i n the formula results i n % = 4.445 Y E X 4 E 2 8 4 7 . 5 2 5 .085 sb = 4.445 yC X4C 4956.864 .065 Standard error of the difference between the regression coefficients b and b v x V Y *E 4E JC X4C Using the appropriate data i n the formula results i n \ - * 2 = / ( . 0 8 5 ) 2 + (.065)2 .104 166 6. Test of significance of the difference between the regression coefficients b (.317) and b (.345). yE X4E yC X4C Using the appropriate data i n the formula results i n * = -545 - .317 .104 * .269 This t value may be evaluated by entering the t table with k„ + k - 4 = 143 degrees of freedom. For the two-tailed test of significance of the nu l l hypothesis that ^^YgX^g = Y^X^g, a t of 1.976 would be required at the .05 l e v e l of significance, while 2.609 would be required at the .01 l e v e l . Since the observed value of t i s only .269, the n u l l hypothesis remains tenable. I t may be said, therefore, that the regression coefficients b and v x yC X4C do not d i f f e r s i g n i f i c a n t l y . A non-significant difference between treatment groups has thus been proven for the regression coefficients of X^ Y and X^Y, two of the four pairs of variables involved i n this experiment. It w i l l be recalled that the independent variables X^and X^ made, respectively, the greatest and the least contribution to the prediction of the c r i t e r i o n . Accordingly, the t value involving each of these variables with the c r i t e r i o n are .951 (X^Y) and .269 (X 4Y ). 167 The t values of the other two pairs of variables, X^ Y and X,Y, will l i e between these two extreme t values. Consequently, the difference between the treatment groups of the regression coefficients of X-Y and X,Y, taken separately, are not significant. It may be said, therefore, that the weight which each independent variable contributes to the criterion variable is the same within the limits of sampling error for both of the treatment groups in this experiment. This completes the test of homogeneity of regression. The fulfillment of a l l the assumptions noted by Lindquist makes it' possible to draw valid conclusions respecting the effect upon the criterion of using certain manipulative materials in group instruction. These conclusions are contained in Chapter VI. XI. THE USE OF A BINARY COMPUTER (ALWAC I I I - E ) IN PERFORMING AUTOMATIC COVARIANCE COMPUTATIONS The computations involved in the covariance analysis reported in the preceding subsections of this chapter were performed with the assistance of an automatic Monroe computing machine. The raw data, in addition to being treated in this way, were processed through the electronic binary computer, Alwac III-E. The Alwac III-E operates on the binary counting system. In this system each digit position assumes only two discrete values-, 0 and 1. Numbers of higher value are indicated by increasing the next most 168 significant digit and repeating the sequence. The most time-consuming part of the computational procedure is the i n i t i a l preparation of a programme, containing the technical details by which the computer executes the required processes. Once prepared, however, the programme provides for the treatment of any data. The programme used in the treatment of the present data was written by Dr. T. Hull of the Department of Mathematics, University of British Columbia. It was designed to accommodate a maximum of eight-variables in the performance of the various covariance calculations. As the f i r s t step in dealing with the present data, a Flexowriter was used to punch on tapes the pupils' scores in the five variables. On the tapes these scores appear in the binary form. The next step was to process the punched tapes through another Flexo-writer which operates in conjunction with the computer. The resulting computations, typed by this Flexowriter and completed within approximately five minutes, produced automatically the following results: the means and standard deviations of each of the five variables, a five by five correlation matrix and covariance matrix, and the coefficients for the regression of the f i r s t on the remaining four variables. Except for minor discrepancies which are attributable to differences in the formulae employed, the results obtained in the Alwac computations agree with those that have already been reported in this chapter. The only function performed by the author was to operate the f i r s t mentioned Flexowriter to record the raw scores. 169 FOOTNOTES 1 Supra, p. 50 2 Allen L. Edwards, Experimental Design in Psychological Research, (New York: Rinehart & Company, Inc.,) 1950, p. 355. 3 E. F. Lindquist, Design and Analysis of Experiments in Psychology and Education, (Boston: Houghton Mifflin Company, 1953), p. 318 4 Henry E. Garrett, Statistics in Psychology and Education, (Toronto: Longmans, Green and Co., 1953), p. 289. 5 E. F. Lindquist, Statistical Analysis in Educational Research, (Cambridge, Mass.: The Riverside Press, 1940), p. 91 g James E. Wert, Charles 0. Neidt, and J. Stanley Ahiaann, Statistical Methods in Educational and Psychological Research, (New York: Appleton-Century-Crofts, Inc., 1954), p. 175 7 I b i d Q Most statistic textbooks reproduce this table which is taken from Snedecor: Statistical Methods, Iowa State College Press, Ames, Iowa. q Garrett, op_. c i t . , p. 154 1 0 Infra, pp. 149-154 1 1 Allen L. Edwards, Statistical Analysis for Students in Psychology and Education, (New York: Rinehart & Company, Inc., 1946), p. 261. 12 E. F. Lindquist, Statistical Analysis in Educational Research, pp. 183-184 13 Edwards, Experimental Design in Psychological Research, p. 339. 14 While the correlation of each of the ten combinations of variables is involved in the calculation of the regression coefficients, i t is the correlation of only the criterion variable with each of the independent variables (four pairs in all) which is involved in the calculation of the sum of squares due to regression. It i s the estimation of this sum which is the concern of this subsection. 170 1 5 Infra, p. 145 16 Since the intrinsic cornaLation of x^y is greater than unity, the uncorrected coefficient of .63 was used to arrive at this fraction. 17 Edwards, Experimental Design in Psychological Research, p. 347 18 Lindquist, Design and Analysis of Experiments in Psychology and Education, p. 327. 19 This derivation procedure, in greater detail, i s contained in Wert, Neidt, and Ahmann, _op_. cit •, p. 241. 20 Ibid 21 An explanation, from which the derivation of this formula may he deduced, is contained in Wert, Neidt, and Ahmann, oj>. c i t . , pp. 235 et seq. 22 Supra, p. I l l 23 Wert, Neidt, and Ahmann, _op_. c i t . , p. 183 24 Lindquist, Design and Analysis of Experiments in Psychology and Education, p. 323-2 5 Ibid, p. 328 Ibid, 27 Ibid. 28 Lindquist, Design and Analysis of Experiments in Psychology and and Education. 29 Supra, pp. 48 et seq. 30 Lindquist, Design and Analysis of Experiments in Psychology and Education, pp. 329 et seq. 3 1 Ibid, p. 330 171 32 J Ibid. The general plan followed in the development of this test of homogeneity of regression is taken from: Allen L. Edwards, Statistical Methods for the Behavioral Sciences, (New York: Rinehart & Company, Inc., 1954), pp. 303-312. Since the two-tailed test of significance of this t value is based upon both tails (positive and negative) of the distribution of t, there is the probability of obtaining a positive or negative t. In the formula the two terms of the numerator may be arranged in either order. To obtain a positive numerator in the present case, the b term has been made the minuend. y c x i c The abridged t table contained in most statistics textbooks is from Table IV of Fisher: Statistical Methods fcr Research Workers, (Edinburgh: Oliver & Boyd, Ltd.) Additional entries (over 30 df) are taken from Snedecor: Statistical Methods, (Ames, Iowa: Iowa State College Press). 172 CHAPTER VI SUMMARY AND CONCLUSIONS I. SUMMARY Purpose of the experiment. This experiment was under-taken to secure data upon which to determine the effectiveness of the group instruction use of certain manipulative aids in contrib-uting to an understanding of particular decimal concepts. Since manipulability of a concept is the most essential characteristic of manipulative aids, this study seeks to determine the effectiveness of these particular aids by isolating this characteristic as the experimental variable. Background and .justification. The movement toward meaningful arithmetic learning emphasizes the need to find teaching materials which make effective contributions to the pupils' understanding of con-cepts. This study may justifiably be included in the quest for these materials because, though subjective opinions are common, objective studies involving manipulative aids are not only few in number but are not designed specifically to determine their effectiveness. Problems proposed by the investigation. 1. Do pupils who are taught with the use of certain mani-pulative aids in the manner prescribed by this experiment achieve an 173 understanding of decimal fractions that is significantly different from the achievement of pupils who are taught with the use of visualization materials similar to the manipulative aids in a l l details except manipulability? 2. What is the relative weight with which each of the four independent variables, i n i t i a l understanding of the processes involved in decimal fractions, computational ability in decimal fractions, mental ability, and reading ability, "enters into",.or contributes to, the criterion variable independently of the treatment groups? 3. When the concomitant influences represented by the four independent variables referred to in Problem 2 are held constant by means of analysis of covariance, do pupils taught with the use of manipulative aids achieve an understanding significantly different from the understanding achieved by pupils taught with the use of visual-ization materials? 4. Por which treatment group - experimental or control -is there the higher correlation between achievement on the criterion variable and achievement on each of the independent variables? Procedure. The effectiveness of the manipulative materials was determined by comparing the achievement on a criterion measure of an experimental group of 59 subjects and a control group of 88 subjects. These groups were composed, respectively, of two and three classes, which were f i r s t of a l l selected in accordance v/ith certain criteria, then matched on the basis of size, and finally, assigned at random to each treatment group. 174 Teaching treatments, prescribed by a series of 11 lessons (including 3 review lessons) for each group, were identical except with respect to the materials of instruction. These materials were, in turn, intended to possess similar characteristics except with respect to that of manipulability. This characteristic emerged, therefore, as the experimental variable. The criterion measure is Farquhar* s Test of Understanding of the Processes Involved in Decimal Fractions. The hypothesis tested is that no significant difference exists between the achievement of the two treatment groups on the criterion variable. By means of a battery of four tests (Farquhar*s Test, a Decimal Fraction Computation Test, the Otis Test of Mental Ability, and Stanford Reading Test), measures were obtained of pupil abilities in areas which v/ere considered to influence achievement on the criterion. The efficiency of each test for i t s purpose was fully investigated. An analysis of variance was made of the results of each of these tests. In a l l four cases the differences between the treatment groups were found not to exceed those which could be attributed to fluctua-tions of sampling. These differences were then controlled statistically by the analysis of covariance, which allows for the correlation between criterion and independent variable scores. The resulting F value, though substantially larger than the F value obtained in the original analysis of variance of the criterion 175 variable, remained insignificant at the .05 level, and the null hypothesis v/as sustained. II. CONCLUSIONS Summary of results. Results of the experiment, stated as direct answers to the problems proposed by the investigation, are as follows: 1. There is no significant difference between the achieve-ment on the criterion test of the pupils taught with the use of the manipulative materials and those taught with the use of the visual-ization materials. (The F value obtained in the analysis of variance is 0.097, while an F value of 3.91 is required at the .05 level of significance.) 2. Of the total influence which the four independent variables exerted upon the achievement on the criterion test, the percentage contributed by each variable, independently of treatments, is as follows: (l) i n i t i a l understanding of the processes involved in decimal fractions - 48$; (2) computational ability in decimal fractions -26$; ( 3 ) mental ability - 24$; (4) reading ability - 2$. 3 . When the concomitant influences represented by the four independent variables are held constant by the statistical procedure of analysis of covariance, there is s t i l l no significant difference between the achievement on the criterion test of the pupils taught 176 with the use of manipulative materials and those taught with the use of visualization materials. As a result of holding constant these concomitant influences, however, the F value obtained in the analysis of covariance became 1.706. 4. Table XLIV shows for each treatment group the correlation between achievement on the criterion variable and achievement on each of the independent variables. TABLE XLIV PEARSON PRODUCT MOMENT COEFFICIENTS OF CORRELATION BETWEEN ACHIEVEMENT ON THE CRITERION TEST AND ACHIEVEMENT ON EACH OF THE INDEPENDENT VARIABLES, ARRANGED ACCORDING TO TREATMENT GROUPS Group X 1 Y X2Y V Experimental Control .51137 .67816 .46417 .57482 .65998 .53172 .48592 .48654 It will be noted that the largest difference in correlations between the treatment groups is between X^ and Y (achievement on the i n i t i a l test of understanding and achievement on the final test of understanding). Even this difference, when tested by transforming the r's into Fisher's z-function, was found to be non-significant. 177 (The procedures for calculating the Pearson Coefficients of Correlation and for determining the significance of the difference between correlations are shown in Appendix G.) Neither group, therefore, has a significantly higher correlation between achievement on the criterion variable and achieve-ment on each independent variable. Interpretation of results. The f i r s t interpretation deals with the fact that the F value obtained in the analysis of covariance of the criterion variable (1.706) was larger than the F value obtained in the analysis of variance of the same variable. (0.097). The control group achieved the higher mean in three of the four independent variables (Table XXVI - page 110), namely: i n i t i a l understanding of the processes involved in decimal fractions (F value is 2.923), mental ability (F value is 0.759), and reading abi l i t y (F value is 0.0006). The experimental group achieved the higher mean in the remaining independent variable, namely: computational ability in decimal fractions (F value is 1.40l). Despite this i n i t i a l advantage of the control group (though is no case was the difference between group means significant), the experimental group at the end of the instructional period achieved the higher mean in the criterion test (F value is 0.097). 178 The i n i t i a l understanding of decimal fractions, in which the largest difference between group means existed, was.also the variable which was most predictive of achievement on the criterion, independ-ently of the treatment groups. Therefore, i t was the statistical control of this variable in particular which increased the final F value obtained in the analysis of covariance to 1.706. The second interpretation deals with the correlations for both treatment groups between achievement on the criterion variable and achievement on each of the independent variables. The lack of any significant difference between the various correlations of the two treatment groups shows that the level of the pupils' ability in respect to the four areas considered in no way determined the effectiveness of the manipulative aids in contributing to an understanding of decimal concepts.-Summary of conclusions. The data obtained from the invest-igation, leads to the following inferences and conclusions. 1. There is no advantage, or disadvantage, for an unselected group of Grade VII pupils in being taught the rationalization of specific decimal fraction concepts by group demonstration through the media of certain instructional materials which, are concrete and movable as opposed to certain other materials which are static representations of these materials, and which are thereby intended to possess similar characteristics in a l l details except that of manipulability. 2. The manipulative materials used in this investigation are neither more nor less effective than the static representations as 179 media for conveying an understanding of specific decimal fraction concepts to Grade VII pupils of any particular capacity in the following areas: i n i t i a l understanding of decimal fraction processes, computational ability in decimal fractions, mental ability, and reading ab i l i t y . 3 . It must not be inferred that any generalization concern-ing the effectiveness of these specific materials, of instruction, which were used in this investigation exclusively by the teacher for group demonstration purposes, would be applicable also to similar materials i f they were used in a teaching procedure in which the pupils themselves participated individually in the.manipulative activity. 4. It must not be inferred that any generalization concerning the effectiveness of the specific manipulative aids used in this investigation, in a.brief teaching assignment devoted exclusively to the rationalization of processes, would be applicable also to the same materials i f they were used in a teaching assignment of longer duration, and/or a teaching assignment in which the emphasis on the WHY of the processes was taught concurrently with, or preceded, the emphasis on the HOW of the processes. 5. Independently of treatment groups, the achievement on the i n i t i a l test of understanding of the processes involved in decimal fractions v/as the variable most predictive of achievement on the final test of understanding. Computational ability in decimal fractions and mental ability each shared approximately one-half the predictive 180 capacity of the i n i t i a l test of understanding. Reading ability was a negligible predictor of achievement on the final test of understanding. Implications of these conclusions and suggestions for further study. While these inferences and conclusions were warranted by the data, the complex nature of the teaching and learning assignments in the investigation necessitates that certain reservations be made with respect to these inferences and conclusions. Teacher comments show that the presentation of certain concepts via manipulative devices caused difficulties in pupil learning. This indicates a need for further investigation in actual learning situations of better manipulative ways to represent arithmetical ideas simply and clearly. Indirectly, teacher comments indicate that the manipulative materials may be more effective when teachers are trained specifically in the concomitant philosophy and instructional procedures relevant to this medium of conveying meanings. Conclusions 1 and 2 should be accepted with the reservation that i f the teachers and pupils had been more accustomed to this method of presenting'.:the concepts the results may have been different. How much more effective these materials may be when the instructional period is longer and when teachers are trained in their use is a problem for further investigation. Conclusions 3 and 4, submitted in the form of precautions against unwise inferences, may profitably be made the subjects for 181 future experimentation. One appropriate investigation would be to determine the effectiveness of manipulative aids used in a teaching procedure in which the pupils participated in the manipulative activity. Another appropriate investigation would be to determine the effectiveness of manipulative aids used in a teaching procedure in which the HOW-WHY sequence was varied from that followed in the present study. The implications relating to these two conclusions were stated in Subsection VIII of Chapter III. 182 BIBLIOGRAPHY A. BOOKS Brueckner, Leo J., Foster E. Grossnickle, and Elda L. Merton. Thinking with Numbers. Toronto: John C. Winston Company, 1953. Brueckner, Leo J., Foster E. Grossnickle, and Elda L. Merton. Teachers' Guide for Thinking with Numbers. Toronto: John C. Winston Company, 1953-Buckingham, Burdette R. Elementary Arithmetic - Its Meaning and Practice. Toronto: Ginn and Company, 1947. Buswell, Guy T., William A. Brownell, and Irene Sauble. Arithmetic We Need - Grade Seven. Toronto: Ginn and Company, 1955. Edwards, Allen L. Statistical Analysis for Students in Psychology and Education. New York: Rinehart and Company, Inc., 1946. Edwards, Allen L. Experimental Design in Psychological Research. New York: Rinehart and Company, Inc., 1950. Edwards, Allen L. Statistical Methods for the Behavioral Sciences. New York: Rinehart and Company, Inc., 1954. Garrett, Henry E. Statistics in Psychology and Education. Toronto: Longmans, Green and Company, 1953. Gesell, Arnold. Infant Development: The Embryology of Early Human Behaviour. New York: Harper and Brothers, 1952. Lindquist, E. F. Statistical Analysis in Educational Research. Cambridge, Mass.: The Riverside Press, 1940. Lindquist, E. F. Design and Analysis of Experiments in Psychology and Education. Boston: Houghton Mifflin Company, 1953. McSwain, E. T., Louis E. Ulrich, and Ralph J. Cooke. Understanding Arithmetic - Grade Seven. River Forest, I l l i n o i s : Laidlow Brothers, 1955. Morton, R. L. Teaching Children Arithmetic. New York: Silver Burdett Company, 1953. Morton Robert Lee et a l . , Making Sure of Arithmetic - Grade Seven. New York: Silver Burdett Company, 1955. 183 Piaget, Jean. Judgment and Reasoning in the Child. New York: Harcourt, Brace and Company, 1928. Ross, C. C. and Julian C. Stanley. Measurement in Today's Schools. New York: Prentice-Hall, Inc., 1954. Spitzer, Herbert. The Teaching of Arithmetic. Boston: Houghton Mifflin Company, 1954. Werner, Heinz. Comparative Psychology of Mental Development. New York: Harper and Brothers, 1940. Wert, James E., Charles 0. Neidt, and J. Stanley Ahmann. Statistical Methods in Educational and Psychological Research. New York: Appleton-Century-Crofts, Inc., 1954. B. DISSERTATION ABSTRACTS Dissertation Abstracts, Vol. XV. Ann Arbor, Michigan: University Microfilms, 1955. C. PERIODICALS Brownell, William A. "Making Arithmetic Sensible," Journal of Educational Research, Vol. 40, September, 1946 - May, 1947, pp. 375-376. Brownell, William A. "The Revolution in Arithmetic," The Arithmetic Teacher, Vol. 1, 1954, pp. 1-5. Buswell, G. T. "Study Pupils' Thinking in Arithmetic," Phi Delta Kappen, Vol. 31, 1950, pp. 230-233. Dawson, Dan T., and Arden K. Ruddell, "The Case for the Meaning Theory in Teaching Arithmetic," Elementary School Journal, Vol. 55, 1955, pp. 393-399. Dawson, Dan T., and Arden K. Ruddell. "An Experimental Approach to the Division Idea," The Arithmetic Teacher, Vol. 2, 1955, pp. 609. 184 Heidbreder, E. "The Attainment of Concepts: I. Terminology and •Methodology," Journal of General Psychology, Vol. 35, 1946, • pp. 173-189. Johnson, J. T. "What Do We Mean by Meaning in Arithmetic?" The Mathematics Teacher, Vol. 41, 1948, pp. 362-367. Lazar, Nathan. "A Pew Recommendations for the Improvement of the Teaching of Mathematics in the Secondary School," Progressive Education, Vol. 29, 1952, pp. 21-23-Morton, R. L. "Teaching Arithmetic," No. 2 of Series "What Research Says to the Teacher", Department of Classroom Teachers of the National Education Association, 1953. Stein, Harry L. "How to Make Arithmetic Meaningful in the Junior High School," School Science and Mathematics, Vol. 53, 1953, pp. 680-684. Weaver, J. Fred. "Misconceptions about Rationalization in Arithmetic," The Mathematics Teacher, Vol. 44, 1951, pp. 377-381. D. UNPUBLISHED THESIS Farquhar, Hugh Ernest. "A Study of the Relationship between Ability To Compute with Decimal Fractions and an Understanding of the Basic Processes Involved in the Use of Decimal Fractions," Unpublished Master of Arts Thesis in Education, University of British Columbia, 1955. E. YEARBOOKS Brownell, William A. "Psychological Considerations in the Learning and Teaching of Arithmetic," The Teaching of Arithmetic, pp. 1-31. Tenth Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University, 1935. Buswell, G. T. "Proposals for Research on Problems of Teaching and of Learning in Arithmetic," The Teaching of Arithmetic, pp. 279-291. Fift i e t h Yearbook of the National Society for the Study of Education, Part II. Chicago: Distributed by University of Chicago Press, 1951. 185 Horn, Ernest. "Arithmetic in the Elementary School Curriculum," The-Teaching of Arithmetic, pp. 6-21. Fi f t i e t h Yearbook of the National Society for the Study of Education, Part II, 1951. Knight, F. B. "Some Aspects of Modern Thought on Arithmetic," The Teaching of Arithmetic, pp. 145-267. Twenty-ninth Yearbook of the National Society for the Study of Education, Part I. Bloomington, I l l i n o i s : Public School Publishing Company, 1930. McConnell, T. R. "Recent Trends in Learning Theory: Their Application to the Psychology of Arithmetic," Arithmetic in General Education, pp. 268-289. Sixtieth Yearbook of the National Council of Teachers of Mathematics, 1941. Van Engen, Henry. "The Formation of Concepts," Learning of Mathematics: Its Theory and Practice, pp. 68-112. Twenty-first Yearbook of the National Council of Teachers of Mathematics, 1953. Wheeler, R. H. "The New Psychology of Learning," The Teaching of Mathematics, pp. 230-263. Tenth Yearbook of the National Council of Teachers of Mathematics, 1935. 186 APPENDIX A COMMUNICATION OF ADMINISTRATIVE ARRANGEMENTS TO THE TEACHERS PARTICIPATING IN THE STUDY PAGE I n i t i a l letter to the teachers conveying information concerning the experiment 187 Orientation notes 189 Evaluation form 191 Summary of lesson objectives 195 Schedule of lessons 198 187 School Board Office, Box 66, Cloverdale, B.C., February 11, 1957. Dear At last I am able to write to you in connection with the experiment in arithmetic about which Mr. Niedzielski spoke to you early in January. Meanwhile I have been continuing the preparations for i t . Before writing to you I wanted to see the preparatory work reach a stage of completion where I could be reasonably sure that the arrangements would proceed in accordance with a plan. I believe that stage has now been reached. First, let me say that the experiment is scheduled for the period extending from Thursday, March 21st to Wednesday, April 10th, instead of the latter two weeks of February as was originally planned. While i t i s desirable that there should be no extensive treat-ment of decimals before that time, there is no reason why classes involved in the experiment should not proceed with, some of the ordinary computation processes in decimals. It is realized that the work in percentage depends in part upon f a c i l i t y in work with some decimal processes. Participation in the experiment should not impede normal progress in this phase of the Grade VII arithmetic. Second, you will no doubt be interested in the way in which participating in the experiment would effect you personally. 1. I think you will enjoy.it. Prior to the experiment the lessons and procedure will have been approved by the local authorities and by the U.B.C. College of Education. You can be assured that the project will be the result of a good deal of thought and advice from many people. 2. The experiment is a matter of current interest in educational research. It is concerned with the method of teaching decimal fractions meaningfully. The necessity of teaching for meaning has now been established by research, but the method of doing so is as yet quite unsupported. This experiment is an attempt to contribute some scientific conclusion, however, infinitesimal, to the existing body of knowledge in this f i e l d . (Page two follows) - 2 -3. You will probably be interested in the labor involved from your point of view. The experiment consists of approximately 12-15 daily lessons to be taught to classes comprising two sections: one an experimental group arid the other a control group. The experimental, group will be composed of two Grade VII classes, and the control group will be composed of two, or possibly three, Grade VII classes. The lessons for each group are complete in mimeograph form with a l l the teaching materials and work sheets and other tests supplied. No advance preparation is needed except a thorough reading of the content of each day's lesson as i t is supplied on the form. To facilitate this preparation i t will be necessary to hold two orientation meetings with each group by i t s e l f . Times which are mutually convenient for a l l will be arranged. The lessons are one hour in length. Approximately half of this time w i l l be spent in teaching or in exercises of a group work nature. Prepared assignments for individual seat work occupy the remaining half of the lesson. These assign-ments are designed so that they may be marked in class by the pupils themselves. Prior to the experiment a group intelligence test, an achievement test, and another test relevant to the area of the experiment wil l be administered to each class. The marking of these tests wi l l be done quite independently of the participants in the experiment. At the conclusion of the experiment another test will be administered which wil l also be marked and recorded independently of the participants. In the interests of the experiment I feel that this is a l l the detail I can supply at this time. However, i f you have any questions or comments concerning your own personal involvement in i t s administration please do not hesitate to let me know and I shall be very glad to reply to them. May I thank you for the expression given to Mr. Niedzielski of your willingness to participate in the experiment. I look forward with anticipation to your cooperation, and, I am quite certain that you will derive some benefit and satisfaction in the project. Very sincerely yours, George J. Greenaway 189 fflpHfATIOH NOTES The purpose of this experiment in «hieh you are engaged Is to determine the advantage Cox* disadvantage) for an uas©l©eted group of Grad® VII pupils of being taught ia group situations with the us© of instructional materials ufcich are concrete and .moveable as opposed to those ishich are merely static represents ations of these manipulative deviceso fh© on®, experimental variable c therefor® 9 is the type of instructional materials usedd Otherwise^ the lesson procedures are identicals Ths various concomitant variables? the pupils* intelligence s reading ability 2 i n i t i a l computational s k i l l and i n i t i a l understanding of decimal fractions 9 these tdJLl be controlled in the statistical analysis of the d&ta* • Ia order to mak© identical for both groups the teaching that results from the prescribed lesson procedures9 i t i s irapes^tire that the instructions and time limits contained in the lesson plans be observed closely„ For the purpose of establishing uniform familiarity with these instructions tisro pre^iastruetion?;! meetings for each group i-d.ll b© heldo Finally s for the, purpose of evaluating the ssperiment there w i l l b© a joint meeting of both groups immediately following the final testing* l a preparation for this concluding meeting i t is necessary to ruake two requests s First s a diary should be kept of your experiences in the teaching of these lessons c Whether reported upon verbally or in 190 ta^itisg at the f i n a l ra®efci&g9 "shis uIXX p*wid© aa instructive ®<g«oun% of your < M persons! experiences l a the teaching of th©©8 lessens© Shis aseount w i l l make an invaluable ©ontributioa to the worth of this study* Second8 an ©valuation f o m 8 wblen wiXX b© distributed near the end of th© e&periraent^ should be completed,* Thi;s fora ^presents a ste-uetiiiped interrogation into s@se aspects of youi? ©aperiene© td.th the teaehing of these lessons* It© purpose 1© fce @asur® a report by eaeh participant sspoa eossaos assess of isat®r©st sad ©oneern i n th© performance of the experiment* In® @®apleti@n ©f this fora w i l l contribute further to the value of t h i s study* EVALUATION FORM 191 The following evaluation form i s designed to obtain opinions upon certain fundamental issues involved i n this experiment© The practical experience you have had i n the aetua! teaching of these lessons w i l l make fovia opinions valuable i n th® f i n a l assessment of this problem <, You are invited 9 therefor© , to be frank i n your anstsers and comments 0 Apart from containing an indication of the group to which you belong s this form need reveal no further identification„ This fora i s intended to supplementp not to replace, whatever notes and observations you have accumulated'in the diary of your experiences i n the teaching of these lessons« In Parts One and Tm of this form the GREEK letters i n th® column at th® l e f t identify the lesson objectives bearing the same l e t t e r on the accompanying Sussoary of Lesson Objectives 0 P A R T pm For the purpose of attaining each objective i n this series of lessens, ho^ effective were the teaching materials placed at 192 Ptom the standpoint of the pupils* maturity and aaetnraiatloa of @3qs©ri©ns© with arithmetical coaeep&s, i s the ©nd of Grad® VII a suitable t i r o to teaah e&ah ©f the objectives isi this Eer*l©s of lessons? Dbjeetiv© Suitable Suitable Not Suitable A ' B C D E-P-4V b o 1 I 1 ' • ' h L M — r — < ~ 19k This series of lessons contains & seiae&hat concentrated effort to teach the meaning of certain decimal concepts» I f this effort tser© spread uniformly throughout the whole period of instruction on decimal fractions i n Grades VI and ¥11 j, how do you think the f i n a l outeome &ould be effected? 195 • OF LESSOR OBJECTIVE.^ LISSOM 1 iSJIEOJUOTIOKs THE HlfOIWffiABXG ©ECXML SYSTEM OF ?JOTATIOH Lesson Objectives of Part One to Te present a br i e f history of the art of reckoning* Z& To convey an appreciation of th© simplicity and convenience © of our presently used Hiiidu*»Arabi« decimal system of notation as compared to earlier methods9 especially the Roiaan systems Lesson Objective of Part Turn." 3* *o show visually the structure of th© decimal number system* ^ g g o m XDEMTIPICATIOB AND KEAOTG OF PLAGE KATIES XH I-HXEl) DSGXIiAL FRACTIOHS Lesson Ob.le&tives of Bart One -To show that the decimal system of uhole number notation may © be-extended to the right of the ONES* place* ?.« To emphasise that the ONES* place i s the centra of this B extended system of notation^ and that the other place names are symmetrical around it© 3 8 To provide a familiarisation i&tfo the decimal fraction place I* names* ••Lesson Objeetive^of^ Partn<Two 4 0 2© @ompare the relationship i n sise i f the various positional 6 . -values ,>• LESSON 111 REJOCTIOM OP ©ECXMftLS TO COB®©!? PACTIONS Lesson,,.Objecti|e;entof Part 1« -1?e consider decimals as a special form of common fractions ]i. having denominators of 10 9 100,, 1000 ete* 9 that i s , any power of 10. Lesson Objective mof Part Two 2* ^ o show ho-tf decimal fractions .indicate the"numerator1' and X denominator of equivalent common fractions 9 196 Lesson Objective of Part, ffhree 3„ To provide practice i n the reading and writing of decimal J fractions© LESSON 17 THE USB OF ZERO AS A PLAGE HOLJ&i Lesson Objective of Part One 1* ^o demonstrate the us© of sero as a nlace holder* Lesson Objective of Part Two 2 e To demonstrate the use of sero as a terminal cipher* CHANGING THE LOC \TION OF THE JECIIIAL POINT: ITS EFFECT ON THE VALUE OF THE EXPRESSION Lesson Objective of Part One 1„ To demonstrate the effect upon the value of a decimal fraction tt df moving the decimal point* Lesson Objective of Part Two 2 0 To demonstrate the effect upon the location of the decimal point of multiplying or dividing a decimal fraction by a power of 10* LESSON VI ROUNJIHG DECIMAL FHACTIONS Lesson Objective of Part One 1« i l l u s t r a t e the significance of rounding decimal fractions* 0 Lesson Objective of Part Two 2, To demonstrate various applications of the rounding of S* decimal fractions Lesson Objective of Part Three 3« To indicate why UNLIKE decimal fractions must be changed to Q LIKE decimal fractions (that i s , with the same understood denominator) i n order that they may be added or subtracted,, 197 LESSON ra irw^ mniMtmirmi r. •uu.nn m DIVISION INVOLVING DECIMAL FRACTIONS Lesson Objective of Part One lo '^o explain the significance of performing division involving a deei-rial fractions. Lesson Objective of Part Two 2» to demonstrate the significance of moving the decimal noint J? i n performing divisions involving decimal fractions 8 LESSON VIII MISCELLANEOUS CONCEPTS INVOLVING JECBIAL FRACTIONS Lesson 0bn1eetive of Part One 1„ To convey the, significance of changing a common fraction to •f a declnal fraction • Lesson Objective of Part Two 2a To i l l u s t r a t e the reason for the placement of the decimal point v i n the product obtained by the multiplication of decimal fractions j, Lesson Objective of Part Three J e To develop an understanding of the importance i n the addition v of decimal fractions of aligning columns according to place value*. Note: The GREEN letters at 1he l e f t indicate the ©onseeutiv© enumeration of the various Lesson Objectives involved in this series of lessons» l a parts One and Two of the accompanying evaluation f c m the GREEN letters i n the column at the l e f t identify the" lesson objective bearing the same l e t t e r on this summaryc 198 DATE SCHEDULE OF LESS0H3 LESSON NUMBER BEMARKS Introductory Lessons "The Hindu«Ar&Mc System of .dotation" 0 i d e n t i f i c a t i o n and Meaning of Place Names i n Ki&ed Decimal Fractions™ e ^Reduction &f Decimals to Common Fractions 5^ Review of concepts taught in \ • Lessons I p IX 9 XII« plus I recapitulation'of Lesson f Exercised • raThe Use of Zero as a Place Holder®* . . I-Lesson V ^Changing the Location @& the Decimal Points Its effect on" the value of the Expression" 0 22 May Lesson ?X "Rounding Off Declaal Fractions" 23 Slay 2 4 Hay REVIEW LESSON Cumulative Review of concepts I previqgly taught i n Lessons ' X to vX3 plus recapitulation I of Lesson Exercises G I Lesson VII "Jivis&oa Involving Decimals*8 e | 2? May Lesson VIII ^Reduction of Common Fractions to.Decimal Fractions* --j •;-®lSO {• sSesE© I#4or Concepts Involved In th® Mtutiplleation and Is I f for any reason you ssish to eonsnuaieat© t*ith m© i n regard to yonr participation i n this £bqp@Fiinent» please phoa® th® Surrey School Bo^rd Office CGLOVERDALs £*&551 or 2«l?6i) during the day or ay hcsae (LAKEVXEW 2-2073) during the evening a 199 APPENDIX B THE LESSONS PAGE Background material for the teaching of Lessons I, II, and III"(WHITE SHEETS - distributed to teachers of both treatment groups) 200 Lessons I to VIII, inclusive (PINK SHEETS -distributed to teachers of the experimental group. 215 Lessons I to VIII, inclusive (YELLOW SHEETS -distributed to teachers of the control group) . . 250 200 LESSOR I IMTRODUCTIOJM! THE HIHDU«.ARABIC SYSTEI-i OF NOTATION * * 1* To present* a brief history of the art of r©skoaiage 20 To convoy en appreciation of the simplicity and eoavsaienee of our presently used Hiadu^Arabio decimal system of sotatlom as Compared to earlier methods9 ©specially the Boaaa systtiBfo 3» ^o show visually the struetura of the deeiaal amber systssSo 2* Lesson Preparation: "• 'T. n i l I ia*ii>»aw<MM* , The procedure of teaching to attain the above three objectives i s divided into two parts» The f i r s t partideals with $he achievement of the f i r s t two objectives) the second deals ..-with the achievement of the third objective* The former involves the use of no special teaching materials 0 Sections (a), (b)„ and (c) of the background material contains the necessary information for the, verbal presentation of this part of the lessono These sections should be carefully studied l a advance for that purpose. The latter does involve the use of special teaching materials« These materials are listed at the beginning of the second oart of the lesson procedure,* Section (a) of the background material contains the theoretical information necessary for "the meaningful presentation of the Second part of the lesson. F i n a l l y 9 an evaluative exercise in the foraj of a work-sheet entitled wThe Decimal System of notation* should be available fo? distribution at the end of the teaching : ^ ^sentatione, 5<SS^^grouadj . (A) DISADVANTAGES OF 2HE ASG1EBT S2STBUS OF H02ATI03S Reckoning i s one of the oldest arts practised by raan0 The number systems used by the early civilisations had some Very great disadvantages*, For example! the cuneiform or weoge^shaped characters which formed the notation of the Babylonians were complicated in design and difficult to reproduce* The seven numerals used by the Romanss though' 'convenient to wrlte s operated in a very cumbersome system* The consequences of these disadvantages were that the ancients did not use their numerals for counting and calculating purposes*, They used thaa only for recording that tahich'had been previously counted or calculated by other means« To illustrate, a sheep-herder„ desiring to calculate the number of sheep he may have lost during the day, would place a pebbla in a pile as each sheep le f t the pasture* When the sheen TeturneC at night he would remove a pebble from the pile. The number of'pebbles l e f t over would represent the number of sheep s t i l l out* If necessary to record his losses, he would resort at this stage to the use of numerals. It i s true that more refined devices, 8uch as the abaejds, were used for calculating. But always the number system worked on prise isles different from those of the calculating device* The arts of calculating and recording were distinctly separate. The numerals were simply not devised' to aid in number thinking. ( ( B j HSSTCaiCAL OEVSLQIMEKr OF THE HXHDUciARABIG DOTATION SYSTEM 'Many centuries passed with no progress i n the art of number. Eventually B the f i r s t step was reached in the development of, a more sensible and more easily managed symbolism. In about the f i f t h century of the Christian era the Hindus developed the nine numerals of the present system. Without the zero, however, the nine symbols were an unsystematic disarray of numerals, possessing the same disadvantages of the earlier systems. Nevertheless, the use of the Hindu numerals spread to Arabia. ? - . . . . About the tenth century the Sera- was invented to complete the Hindu-Arabic system of notation. Despite the amasing trans-formation which the 2ero made in the simplicity and' u t i l i t y of the system, the tfindu~Arabic notation Has slow to replace the ponderous Roman system. It mas about tvio centuries later that i t was brought into western Europe, and i t was not'until the sixteenth century that i t came into general use. ( 6 ) ADVANTAGES OF SHE HXHDU~ARABIC EOTATIOU SYSTEM The importance of the invention of the zero i n this development should not be overlooked. It has been v e i l said that "aero i s the catalyst that bfings together static numeral signs into a dynamic system of number thinking"* By enabling the existing symbolsfi 1 to 9 inclusive, to possess a place value as well as a face value, the sero gave the Hindu-Arabic system a function no previous system ever possessed. The new system extended Its use beyond the recording function to which the earlier systems were confined- i t could also be used for counting and calculating. Unlike Rowan numeration, the Hindu* Arabic system appropriated to i t s e l f the principle of positional value *toieh facilitated i n a practical way the ancient process of calculating on the abacus. The long-separated arts of calculating and recording were united at last into a single whole system. Furthermore, i f the structure of the number system, including the principle of place value, i s understood, the numerals are an aid in the number thinking that accompanies the 202 performance of the calculation. I f the structure of the number system i s act understood, the performance of the calculation even .though possible, will be on a completely mechanical level in accordance with some prescribed rule, . " , , V ( D) m s p i E m m § " 0 F mz-mzm wm. ^ aignber system is; based on. a fspoupiag by tens It i s presumed that the base of our number system i s ten because primitive man used his fingers in counting. The number 28, for example, came to mean that a l l the fingers had been used two times [2 groups of 10 fingers), and that 8 of them had been used a third time* 'The ten«nessof the system i s why i t is called a decimal system* The word "decimal" i s derived from the Latin t*d©eimustt yhich meaBS tenth and ^decern* which means ten,. Nine is, as far as we go in our'nttmber system without regrouping and starting over again with one. Ten in any group join to make one in the position of next higher value, {11} The suimber system foas Place value Each position has a value ten times as much as the position Immediately to the right, or one-tenth as much as the position immediately to the l e f t , A comparison of our number system with the Roman system, which does hot have place value, illustrates the significance of this principle,< Roman Numerals " Hindu-Arabic Numerals I has a value of oj£e, 1 has a value of aa§. It i s called one. It i s called one* XX has a value of one and one* i i n a a a value of one ten It i s called two* and one one. It i s called III has a value of one; one, and one, m has a vfetS^f one i t is called three* ***** - hundred, one ten, and one qae, • . It i s called one hundred, eleven* (This system is based on an (This system 13based on a additive, OT subtractive, place value principle*) principle*) ' The- above comparison shows that is Roman numerals the ;. "I* always has a value of one, regardless of its position* In Hindu-Arabic numbers the v r^ t 9 or any other symbol, has a value which depends on i t s location, ( i i i ) The number ayatem uses aero* or,cipher,, ss a place, holder list us look 4t the number two thousand eight* This means 203 two tftogsand. no h;undredsg. no jfcens, eight ones, When the number i s written i n this way« the Heedlessness of indicating that there are no hundreds ana no tens i s apparent e However, when the same number i s expressed i n symbols the denomination of each numeral (that is2 the fact that 2 refers to thousands and 8 refers to ones) i s indicated only by the position occupied by the 2 and 80" 'These positions ares Thousands Hundreds fens Ones 2 0 0 8 The zero s by indicating that a place i s emptyg serves to keep the numerals i n the proper place c It has been called the place holder because i t f i l l s an empty place i n a number written i n symbols i n order to protect the values of the other numerals which l i e i n the other places to the l e f t i n a whole number, or other Places to the right in a fractional number. This protective function of zero explains why i t i s not necessary 9 i n expressing twenty in symbols s to use the seros i n tbe following ways Thousands Hundreds Tens Ones 0 0 2 0 The f i r s t two seros are unnecessary because, unlike the third sero i n the ones column, they have no numeral to protect;,. Civ) The number system mav be extended to the right of the ONES* p^aee to provide a notation of decimal.fractions, A significant feature of the Hindu-Arabic number system i s that fractional parts may be expressed simply by extending the numeral places to the right of the ones* place, thus: Pig O v c ONESNSSSSKS TENTHS HUNDREDTHS ' THOUSANDTHS The three principles underlying the structure of the whole number system^ (1) ton-ness (2) place value and (3) place holding^ apply also to the decimal fraction notation 0 t • }' ISOD^ HAIjBS IN MIXED JEGXI-IAL FRACTIONS lo. £»©sson Objectives: Is, .To show that the decimal system of x^ iol© number notation say be extended to th® right of the ONES' place* 2 . T© emphasise that the ONES' place is the centre of this extended system of notation 9" and that the other place names are symmetrical around i t . 3. To provide a familiarisation with the decimal fraction place names. 4. compare the relationship i n sise of the various positional values. 2. Lesson Preparation; The procedure of teaching to attain the above four objectives i s divided into two parts. The f i r s t part deals with the achievement of the f i r s t three objectives; the second part deals with the achievement of the fourth objective. While both parts w i l l contribute i n some measure to a l l objectives s i t i s desirable that the primary intent of each part be kept clearly in mind as the lesson proceeds. Both parts involve the use of special teaching materials 0 These materials are l i s t e d at the beginning of each part of the lesson procedure. Section (&) of the background material recapitulates b r i e f l y that portion of Lesson I which dealt with the four basic principles i n the structure of the Hindu-Arabic number system. Section (B) contains the background orientation relevant to Objective 1; section (0) i s similarly relevant to Objectives 2 and 3* while section (B) i s relevant to Objective 4. Fin a l l y 9 an evaluative exercise i s provided for distribution at the end of the teaching presentation. 206 3o Background; (A) REVIEW OF THE PRINCIPLES UNDERLYING WHOLE NUMBERS Th© backgroundof Lesson I dealt with the four essential principles underlying the structure of the Hindu-Arabic number system: 1; i t i s based on a grouping by tens 2, i t uses position to determine the value of a symbol i n a ' number 3 0 i t uses aero as a place-holder to keep symbols i n their " appropriate positions k„ i t may be extended to the right of ONES' place to provide a * . notation of decimal fractions. Showing in a meaningful way the application of the f i r s t ttr ee of these principles in whole numbers was one of the objectives of Lesson I 6 (B) DECIMAL FRACTIONS ARE AN INTEGRAL PART OF THE WHOLE NUMBER SYSTEM WITH THE SAME COMMON PRINCIPLES The fourth principle serves as the means of introducing decimal fraction notation i n Lesson II, This method of introducing decimal fractions as an integral part of our decimal system of number i s one of two ways suggested by Brueokner and Grossnickle i n "How to Make Arithmetic Meaningful"„ The other way suggested by them i s to consider decimals as a special form of common fractions having denominators of ten or some power of ten. This l a t t e r approach w i l l be used i n a subsequent lesson to reinforce the pupils' development of an understanding of decimal fraction concepts. If pupils regard decimals from the outset as a logical extension of the whole number system, they w i l l readily recognize the application of the f i r s t three principles i s decimal fractions as well as in whole numbers. Teaching pupils t« regard decimals i n this way constitutes the f i r s t objective of Lesson II, 2 0 7 _{C) SIGNIFICANT FEATURES IN THE RELATIONSHIP BETWEEN THE INTEGRAL AND FRACTIONAL FARTS OF A MIXED DECIMAL EXPRESSION Treating decimal fractions as an integral part of the decimal system establishes the need to show clearly the relationship between the whole and fractional parts of a mixed decimal expression. This necessitates emphasizing the following: 1, The ONES* place is the centre of our system of notation, The prominence of the decimal point should not ve allowed to detract from the importance of the ONES' place. The primary function of the decimal point is to designate the location of the ONES' digit. The point occupies no column or place in the number system. As a matter of fact i t might be more logical to place the decimal point, or some other identifying mark such as a bar, either above the ONES' digit or below i t . This would remove from the point the feature -which Buckingham in "Elementary Arithmetic: Its I leaning and Practice" describes as "the purely incidental mark of distinction between ONES and TENTHS", It is interesting to note that some cultures use identifying marks other than the decimal'point as we know i t , Taylor and Mills state in "Arithmetic for Teacher-training classes" that: "The number which we write as 1 6 , 3 5 7 has been written in these forms: l 6 , 3 ' 5 " 7 , M ; 16°:3» 5 " 7 " »; 16,(o)3(l)5(2)7; 16)?57« W la France and Germany, they point out, i t would be written 16 9 357. However/S the purpose, i f not the form, of the identifying mark is common to a l l cultures. It is to indicate the position of the ONES' column. All other symbols have their values determined in relation to this column, 2 0 The other positional values are symmetrical around the ONES' place. ,208 ;ihis i s illustrated in the following schematic representation; THOUSANDS HUNDREDS t HUNDREDTHS f THOUSANDTHS t It shou3.d be noted that the importance of emphasising the symmetry of positional values around the ONES1 place is to overcome the disadvantage that derives from the incidental distinction given to the decimal point by being placed between the ONES and TENTHS„ Emphasizing the position of the decimal rather than the position of the ONES' place leads to sueh apparent discrepancies as the following; four places are needed to represent THOUSANDS wheregs only three places are needed to represent THOUSANDTHS. If positional values are looked upon as being symmetrical around the ONES' place then the more logical generalization can be n E a d e that three places on EITHER side of the ONES' place represents THOUSANDS and THOUSANDTHS. Teaching pupils to observe the centrallty of ONES in our number system, and to note the correspondence between the TENS and TENTHS, HUNDREDS and HUNDREDTHS, etc. constitutes the second objective of Lesson II. In the teaching procedure that follows, the same technique of instruction is used for the attainment of Lesson Objectives 1 and 2 0 During the process of this instruction the pupils wi l l have ample opportunity to become familiar with the decimal fraction place names, thus providing for the attainment of Lesson Objective 3. (B) fHE VISUAL RELATIONSHIP XH SIZE OF DIGIT POSITIONS The teaching procedure designed for the attainment of the f i r s t three objectives is intended to give pupils a general under-standing of the positional values extending from THOUSANDS 209 to THOUSANDTHS. Within the limitations of a reasonable amount of instructional materials i t i s d i f f i c u l t to make an adequate visual presentation of the actual relationship i n siae of such a wide range of positional values. The attainment of Lesson Objective 4 i s intended to give pupils a visual experience with a limited range of positional values extending from the ONES' place to the THOUSANDTHS* place 0 The conceptual learning obtained therefrom w i l l be readily transmitted to the wider range. LESSON III -210 REDUCTION OF DECIMALS TO COM-JON FRACTIONS . Z : :° L@sse® Objectives? l e To eonsidgr decimals m s. special fom. of eo&imoa fgf&ctioas haviag deaoMnators of 10s 100, 1000 etc., that i s , any pm@? of 1 0 , 2, T© show hcer decimal fractions indicate the Numerator sad deacraitsator of equivalent eomaan fractions o 3, To provide p ^ c t l c e i n the reading and writing of decimal frsetloas. Lesson Preparation; The procedure of teaching to attain the above three objectives i s divided into three parts: one part for each objective. The f i r s t part does not involve the use of special teaching materials§ the materials to be used i n the secondand third parts are list e d at the begi&ning of each part of the lesson procedure. Sections (A), (B) and (C) of the Background material are relevant, respectively, to the three lesson objectives. Finally, an evaluative exercise i s provided for distribution at the end of the teaching presentation. BACKGROUND; U) DECIMAL FRACTIONS ARE A SPECIAL FORM OF COMMON FRACTIONS l a the background material of Lesson II reference was made to two suggestions for the introduction of decimal fractions given by Brueckner and Grossnickle i n "How to I lake Arithmetic Meaningful"« One way i s to consider decimals as an integral part of our decimal system of number. This approach was used in Lesson II, The other way i s to consider decimal fractions as a special form of common fractions having denominators of any power of 100 This approach i s used in Lesson I I I 0 Buswell andBrownell reinforce this 211 opinion when they state in their manual to teaching "Arithmetic We Heed": "Once pupils understand that decimals are fractions, the denominators of which are not visi b l e and are always 10 or a multiple of 10 9 they w i l l have developed a real understand ng of the meaning §f decimals„" Jecimal fractions may, therefore, .be regarded as a selected part of a l l common fractions 5 namely: those #iose denominators are a power of 10, Any fraction which has a denominator of 10 s 100 s 1000? etc 0 i s a decimal fraction. As Spitser says in "The Teaching of 'Arithmetic": ",<,, i t i s the fflCt that the denominators of decinals are a l l powers of tens which makes decimals unique, and not the use of the decimal point". Although general usage has established decimal fractions as those fractions in which a decimal point is u3ed, i t should not obscure the fac& that thasy are simply common fractions with unwritten, but understood^ denominators of some power of 10 o Consideration of decimal fractions i n this way i s helpful for developing meai ingful insights into areas of decimal work such as: l o the reading and writing of decimal fractions 2, the reduction of common fractions to decinals , 3„ the changing of measurement or terminating decimals to 1 higher terms (for example, changing 5,1 to 510 hundredths) 4 , the rounding of decimal fractions to•lower terms (for example, changing 0,942 to 9 TENTHS). These points w i l l be illustrated at the approprleate places in this, and subsequent, lessons, (B) INTERPRETATIONS OF THE NUMERATOR ANJDENOMINATOR OF A JECIMAL FRACTION In Seetion (A) of this background material we have seen that decimal fractions are simply common fractions having denominators of 1 0 , 1008 1000s ete 0 The numerator of a. decimal point is indicated directly-by the 212 number to the right of the decimal point. It i s read as a whole number. For example, in .425 the numerator i s 425 and is read four hundred twenty five; i n .000425 i t i s also 425 and i s read the same way. The position of the decimal point does not change the value of the numerator. The denominator of a decimal fraction, being unwritten, must be interpreted from the name of the last-used decimal place to the right. Lesson II (Objective No. 3) provided famil-iarization with the decimal fraction place names which w i l l enable pupils to make this interpretation. Thus, 1.2 i s one and two tenths as a mixed fraction while i t i s twelve tenths as an improper fraction. Similarly, .12 i s twelve HUNDREDTH'S, .012 is twelve THOUSANDTHS, etc. In each case i t i s the name of the last-used place to the right of the decimal point which indic-ates the "inv i s i b l e " denominator. Lesson III (Objective No. 2) i s intended to show the reason why the denominator of a decimal fraction may be interpreted from the name of the last-used decimal place to the right. In addition, this objective i s intended to show why the position of the last digit after the decimal point actually determines the value of a decimal fraction. For example,ta,12 i t i s the fact that the digit 2 i s i n the HUNDREDTHS1 position which determines the value of the fraction. This may be explained i n this way: since the 1 i s in the f i r s t position to the right of ONES* place i t represents 1TENTH which i s equiv-alent to 10 HUNDREDTHS. Together with the 2 already i n the HUNDREDTHS1 position this makes 12 HUNDREDTHS. 213 Certain generalisations w i l l result from interpreting the numerator and denominator of a few decimal fractions 0 One such generalization iss i f the last-used decimal place i s one place to the right of the ONES* place the fraction represents'TENTHS; i f i t i s two places to the right of the 0!IKS» place i t represents HUN ORT^ DTHS, etc. A somewhat more mechanical form of the sane generalization is? the denominator of a common fraction w i l l have one zero for every figure to the'right of the decimal t>oint in the equivalent decimal fraction,, (C) 1HE READING AND WRITING OP DECIMAL FRACTIONS The f i r s t objective in this lesson i s to lead the pupils to understand that decimal fractions are merely common fractions with unvjritten 9 but understood9 denominators. Furthermore, they are a selective type of common fractions because the denominators are always 10 8 100, 1000,, etc*, that i s s some power of 10, The second objective i s to lead pupils to understand the significance of the last-used decimal place to the right of the ONES' place i n determining the numerator and denominator of the common fraction equivalent to the decinal fraction. These two objectives should give pupils an understanding that decimals are fractions, the denominators of whichf though unwritten, are a power of 10B According to Buawell, Brownell and Sauble i n the Manual to TEACHING ARITHMETIC WE NEED, this i s the basis for developing "a real understanding of the meaiing of decimals" 0 I f these two objectives are attained, this third object~ ive concerning the reading and writing of decimal fractions w i l l have been achieved as well 0 This procedure follows Spitzer's recommendation of writing the common fraction of decimals as a means of reading the decimal written with the decimal point, In the decimal fraction e036 9 for example, i f the pupil understands that the unwritten denominator must be a power of 10; that the power i s determined by the place value 214 of the 6; andg further 9 that th® numerator i s determined by convert-ing the 3 TENTHS intc THOUSANDTHS and adding i t to the existing 6 THOUSANDTHS to give 36/1000, then he w i l l have a meaningful insight into the reading cf „036c Procedure of Teaching: -£sfol^mm& &g ksm$£>& S!bMsSiis.m 1 ssd Z (Time: 12 minutes) The contents of Sections (ah (b), and {4) of the background material may be discussed within approximately the allotted time* In order to achieve the above tibj&^trteit. i s Suggested that the material provided should be re^*r©d to in general terms* The emphasis should be on the spCnlaneiiy of the presentation rather than on too rigid an adherence to minute detail. If the objectives of the lesson are reached success-fully i t should motivate the pupils to expLore and exper-iment with decimals in subsequent lessons. Achievement of, Lesson Objective £ (Time* 18 minutes) Materialsi Three place value charts. 210 tickets of ishieh 200 should be in groups of 10 held together with an elastic band, and 10 loose tickets. * To illustrate; (i) the ten~ness (or decimal nature) of our number system. (il) positional value, .©r-.thaJtyufc&i* numeral at the right of a whole number hat a 0Nf$* value, the next numeral on the lef t has a TENS' value, and the next on the le f t has a HUNDREDS* value. Steps; 1. Set the three place value charts on the bottom ledge of the blackboard. Hote: in making the two illustrations noted above, Steps ITSo 8 inclusive show the situations which require a regrouping from the ONES' position to the TENS' position* Step 9 emphasises the relationship In actual value between various digits in these two given positions. 2. Place the Single tickets on the ONES' chart while counting 2.* 2, 3, ht 5, 6, ?t *• % 1 0 ? 3. Explain that t*e must regroup when We reach 10 s Provide a , small square Immediately above the Charts as shown in the illustration at the top of page 216. These squares may be drawn en the blackboard. k* Explain that in thtgse squares we customarily write only one figure to indicate the numbed of t ickets on each place value chart* This illustrates the haisd to regroup when 10 has been reached in any one position. 5. Remove the 10 single tickets and substitute one bundle of 10 tickets by placing i t on the TENS' board, iwrite the symbols in the email squares. •jhpttbo: • Symbol Symbol SINS ONES x x x x x x x x x x (regroup to form 10)| 6 S&phaslse the fact that 1 bundle on the TENS6 board i s composed of 10 times as many tickets as 1 tieket on the • ONES* board, 7 o Continue to replace single tickets on the ONES' chart '; while counting 11, 12, 13, 14, 15, 16, 17, IS, 19, 20. .. 8«,.As i n step 5, regroup from ONES' place to TENS' place, Repeat the detail of this regrouping as often as i s ' considered necessary* 9, As i n Step 6, draw attention frequently to such facts as: (a) 4 on the ONES' chart represents one«fifth as many tickets as 2 on. the TENS-' chart, (b) 8 on the TENS' chart represents.-forty times as many tickets as 2 on the ONES' chart, etc. S^PP -2-0 shows a situation which requires, a Togrouping i ^ £ S & , * & « aK,P«*4JiS„to the M ' ' .posmoa and t&e Tkm* posxtioa to the HUNDREDS' posi-bion, 2n other words rfc illustrates situations ?Mch requis9© two successive •f^eupiags. Step 11 emphasises the relationship l a actual va$u& bettreea serious digit© i n these thro© gives 10?0a the place value charts show 135 as follows: 12 bundles of 10 tickets on the TENS' chart and 15 single tickets on the ONES * chart. Emphasize the fact that in the squares above each chart we write only one figure to indicate the number of tickets on that particular chart, and emphasise also that i t takes 10 i n one position to equal 1 in the adjacent position on the l e f t . Following the emphasis on these details proceed to perform the regrouping to • obtain 1 ticket on the HUHDRB&S' chart, 3 on the TANS' • chart, and 5 on the ONES' chart. 11- A© i n Steps 6 and 9, draw attention to such facts as: (a) . 1 on-the HUNDREDS' chart represents 20 times as many tickets as 5 on the ONES' chart. (b) 5 on the ONES' chart represents 1/6 as many tickets as 3 on the TENS' chart.. 12. Draw from pupils* out of the experience they have had With the foregoing relationships 3 generalisations framed around the following: (a) The number system i s based on a grouping by teas. 217 ib) The number system has place value. This means that each maaeral i n a number possesses a value assigned by the ttplace** i t occupies in the number* Each "place" has a value ten times as mush as the "place* immediately to the right, or one*tenth as much as the position immeoTately to the l e f t . 13. At this point a very brief comparative description may be .made of the principles underlying the Roman numeral system of notation (see heading (d) ( i i ) of the background material). To i l l u s t r a t e : ( i l l ) the use of sere, or cipher, as a place holder. Steps: l i Show 9 bundles on the TENS' chart and 9 tickets on the ONES' chart. Then add one ticket to the ONES' chart. Regroup. ymbol Symbol 11—i l— i — r Symbol HUNDREDS TENS ONES In the space on the blackboard above each chart write the appropriate symbol. 2. Explain that a figure must be written to ©how each place in the three place number, even though the charts i n two of the places are empty. .This i s the PLACE HOLDING FUNCTION OF ZERO i n our number system. Show that aero has a protecting role to keep 1 i n the third space from the right, or on the HUNDREDS' chart. • 3; Add some single-tickets, say 7, to the ONES' chart, 4, While referring; to the place value charts deal with the number shown under three headings: (a) How i t reads • one hundred seven, (Note: this i s a .JLJ convenient point at which to explain that the use of "And* i n the reading of a number i s reserved exclusively to indicate the connection betv/een the ^SS^pIace and the TENTHS' place. I t i s never used either" i n a whole number or in a fractional number* (b) What ,it means* • one, hundred, no tens, seven ones. (c) ffbwTK i s written -kc Prne:-.i-,use of Teaching; 218 Achic^emant of Lesson Objectives .1.;. 2. and 3 (Times 1?.minutes) MaJ/grials'; Seven place value charts* Chart indicating the decimal point. Twenty-five sards in each of seven different colors, (Motes this substitution for the bundles of cards used with the place value charts i n Lesson I i s necessitated by the impracticability of using cards smaller than the ONES' cards to represent units less than ONE* 4^3^ (S^iiJ?03L«£;S' ~ — ™ - * • . — - — . — JSmMSSSS^St^J^nr A l l three steps sfcoumft contribute tc* «h* attainment or 1 0 Set the place value chart to represent ONES on the bottom ledge of •the' blackboard. Place the decimal point to the right of the ONES* chart, dentify the "19 methods . . ~ w « — — d i g i i has been located a l l other digits obtain their values from the position they occupy in relation to the ONES' place, A number, li k e 34#?3, i s quite meaningless unless we knew which di g i t stands for unity, 2, Place additional charts to secure the arrangenant shown below: "oHK AJLJ.IAA XV v v . i ; v y v v v Represent 25 THOUSANDTHS ©a-the charts, and th'ea,'by performing ' the necessary regrouping point out that the following three principles,which were shown i n Lesson X to t&rm the structure of whole number system, appjy,also to decimal fractions: (I) ten-ness* , (2) place value*(3) place holding function of zero, •'• The same treatment may be applied to other representations, such as j (a) 25 HUNDREDTHS; (b) 25 TENTHS 3« Remove the charts used i n Step 29 and then assign successive pairs of pupils to come forward to place"cherts i n positions which"are symmetrical around the ONES' chart ? as shown i n the diagrams. 'the- lowest r o w o f c h a r t s shows the arrangement when a l l have been placed on the bottom ledge of the blackboard a J r a w lines on the blackboard above the charts, as shown in the il l u s t r a t i o n above, to emphasise the symmetry aro-und the ONES' place.. While the charts are in this position* discussion should be :';sld which points o u t the following: (a) the ..antral position occupied b y the ONSS' place-. (b) The oximetry o f the other place valines around the ONES' place, (c) the various value relationships whereby each place represents a value tec times as large as the place next t o i t on the right, one hundred times as large as the second place to i t on the right, e t c Illustrate these ^lationships with specific examples shown on the charts,, e,g c: ( i } a 4 in the OSES' place i s ten times, as large as 4 in the TENTHS' place ( i i ) a 7 in the TENS' place i s one thousand times as ferge 7 i n the HUJTOflEJTHS' place ' ( i i i ) a 2 i n the TENTHS' place i s i s one-fiftieth as much as 1 i n the TENS* place (that i 3 , the 1 i n the TENS' place actually represents 100 TENTHS , which i s f i f t y times larger than 2 TENTHS), 4G To c o n c l u d e t h i s p o r t i o n o f the l e s s e n s t h r e e g e n e r a l i s a t i o n s s h o u l d b e drawn tram p u p i l s a t t h i s s t a g e : (a ) The f o l X o ^ i j^r p r i n c i p l e s w h i c h u n d e r l i e t h e w h o l e number s y t e m a p p l y also t o decimal f r a c t i o n s { O b j e c t i v e 1): ( i ) Place v a l u a • each p o s i t i o n a s s i g n s t o a d i g i t a p a r t i c u l a r v a l u e * 220 ill } XffflWfrgftf! * "^ slu® assigned to a <ft'git in one position i s tan times larger than the valuta assigned to i t in the position next to i t on the right, ©te0 ( i l l ) Hlae^hold^n^ function of -%erp: « i n order to ^ protect* the value bT numerals oy keeping them in the required positions« seros are needed to record whatever empty positions exist BETWEEN the decimal and numerals in the most 'extreme positions to the l e f t or right of the decimal point. Nofcs: i t may be mentioned in passing that seres f i l l another function quite apart .from a place-bol&ug function. This function, m wall as the pjle&*«hoiding; function, «111 be dealt witk more fully i n Lesson 17. (b) $h$ arrangement of positions around the 0NE3» gftaee l a symmetrical (Objective 2)i it ) the position which i s third from the ONES' place (fourth from the decimal} on the left- and third ivrom. the ONES1 plase on the right are THOUSANDS and THOUSANDTHS respectively. (i« ) the position which is second from the ONSS* (third from the decimal) on the l e f t , and second from the 01*33* place on the right are HUNDREDS and HUNDREDTHS respectively. ( i l l ) the position which i s next to the 0NES8 plaC« (second from the decimal) on the lef t * and next to the, 0N&3* place on the riftht are TENS and TENTHS respectively* Achievement of Lesson Objective 4(Time; 15 minutes) Materials: Decimal place value cards. Fpur cards to represent the following place values: one whole, one tenth, one hundredth, one thousandth. A fif t h card bears the decimal point. Note; The achievement of this objective should enable pupil, to formulate a meaningful generalisation respecting th* Comparison of decimal fractions, e.g.: which i s larger -.379 or .38? Pupils who have become accustomed to making comparisons of whole numbers only may find the comparison of decimal fractions less obvious than i t f i r s t appears to them. The Winston textbook ttThinking with Numbers" contains a drawing, shown at the le f t , which may be presented on the blackboard to pupils to emphasise that one must learn to check conclusions in arithmetic. In comparing decimal fractions, as in comparing the lengths of these 30 inch lines, "You cannot always be sure". 221 Choose' pupils each to c a r r y a place value card and take their positions facing the class while holding'the card in f u l l view* Start with the ONE card g followed by the decimal point s then follow: (a) with the TENTH card - explain that i f the card on the l e f t were shown i t would be represented by 1 bundle of 10 cards. This would be i n the TENS' position. (b) with the HUNDREDTH card - explain that i f the card located in the coi*responding position to the l e f t of the ONE were shown i t would be represented by 10 bundles of 10 cards. (c) with the THOUSANDTH card - explain that i f the card located i n the corresponding position to the l e f t of the ONE were shown i t would be represented by 10 bundles of 100 cards. This arrangement may be represented on the blackboard;, thus: VJhile pupils are in this position discuss the manner i n which we would arrange the following in order of sise*, beginning with the largest: (a) 1.1 (b) .011 (c) 11 (d) .11- (e) 1.11 Let the" pupils holding the ONE card and the THOUSANDTH card be seated. Proceed to compare two decimal fractions, e.g., #25 and ,3 i n this way: M M 1*1 M X X O X X O X X O X Assign pupils to take up their positions behind the card bearers as shown in the diagram; The x's represent ,25 and the o's represent . 3 . Pupils representing ,25 may be referred to as Team (a), while those representing .3 may be referred to as Team (b). Discuss why Team (b) represents a larger fraction than Team (a). The same procedure may be followed i n showing the reasoning involved in arranging the following according to size: (a) .5 (b) ,05 Cc) 5.5 (d) .055 (e) .55 222 To toaelode. t h i s portiaa of the lesson, the following generalisation should he dissta from pupils after those standing have resisaed their seats© wB®tsiJaal fractions eaa be ranked i n order of sis© by comparing the absolute value of the digits i s the corresponding places, thusg (a) the largest of ssver&l decimal fractions w i l l he the one with tH« largest figure l a the TENTHS* plaee 0 (b) i f the figures In the TENTHS' pflUee are equal, then the largest fraction H i l l be the one with the largest figure l a the HUNDREDTHS' . •placee (e) i f the figures i n the HUNDREDTHS* place are equal, thea the largest fradtloa w i l l be the one with the largest figure l a the THOUSANDTHS* place. I Procedure of Teaching; -v P A S T O N E - JZjL.jmi-11-Jt- -J*- JIWmi i m m WIIII iwmii i i ' iBi 1 " .. Achievement. o i > . , t ^ i s s ^ , O M ^ M g ^ - l (Tia*: 8 minutes) T© consider decimals as a special form of common ftactions having: denominators of 1 0 , 100, 1000 etc ., that -ley assy power of 10. Materials: •• No special materials required. Steps; 1. Write the following series of common fractions on the blackboard: 2. Verbal Explanations: (a) Explain what i s meant by "a power of 10tt» Obviously, i t i s be§rond the scope of the pupils* comprehension at this stage to explain that i t means "the index of ?10rt«Consequently9 i t w i l l suffice to explain that i n effect i t means 10 multiplied by i t s e l f any number of times, or 10 by i t s e l f , thus: 10, 100, 1000, etc. The meaning of "a power of 10" should be made distinct from the meaning of "a multiple of l o * which ineahS 10 multiplied, not by i t s e l f any nuifber Of times, but by any sw^b®r, for example: 5* 8- 12, 20, 30, e t c , to give these respective multiples of 10: 50, 80, 120, 200, 300 etc. (b) Explain that wMle a l l the fractions written on the board are Common fractions, those with a denominator of a power of 10 v may also be regarded as decimal fractions, even though i t i s customary practice i n writing decimal fractions to omit writing the denominator andto indicate i t indirectly by the use of a decimal point. 3. Form two columns on the blackboard, and at the top of each write headings as follows: Fractions which may be.considered Fractions which may be considered only as common fractions as Decimal fractions a ;— — ,——•.,.„. .„ „ Under the appropriate heading enter each of the fractions already written on the blackboard. To shew hbw decimal fr&stiess' iadleafce the atraerator and denominator of emiiw&leafc eoasBoa f r a c t i o n s Materials: Four place value charts, namely: ONES'• TENTHS' 5 HUNDREDTHS*j THOUSANDTHS' 25 yellow tickets, 15 each of blue and green tickets, and 5 red tickets. steps: $g&es The -two points stated below should be c l e a r l y emphasised after each of the following three rGprll^itltlo^ c o n t H h e d i n PartrWo of this Lesson. X« The position of the last digit after the decimal point determines the value the value of a decimal fraction* That is» each of the digits i n the decimal positions preceding the last place may ^ j ^ J ^ c ^ y e r t f d to the place value of the last position after thedeclSaX point. The number so obtained determines the NUMERATOR of the equivalent common fraction 0 « — At the same time the particular place value of the l a s t occupied position Indicates the DENOMINATOR of the equivalent common fraction. 2. When a decimal fraction i s changed to a common fraction, the denominator h&s ONE ZERO for every figure to the right of the decimal point. 1. ?er£otm. representations of the following three fractions as indicated; (&'} Represent .025 on the place value charts, writing the number above the charts as shown: Remove the tw> green tickets from the Hundredths* Board and substitute 2 0 yellow tickets on the THOUSANDTH-'*' board, thus! ONES TENTHS ZL. x x x x x x x x x x x x x x x Sraphasize clearly the two points stated.above i« s'b) Represent t)12' on the place valoa charts, writing the number above the «har"ts as shown j D O 225 ONES TENTHS HUNDREDTHS X X •THOUS iNDTHS Remove the blue ticket from the T E N T H S ' board and substitute 10 green tickets on the. HUNOTirUTHS' board, thus: 5 3 ONES HUNDREDTHS XXXXXXXXXX THOUSANJTHS Emphasize clearly the two points stated above in green. (c) Represent 2.3 on the place value charts, repeating the procedure as in (a) and (b) above* Not*; step 2 below i s merely an extension of (c) above and shows that the two points noted above may be used to explain the conversion of an integral number into an improper fraction In this ease, of course, i t is the position of the terminating sero which determines the value of the improper fraction. . . . . . . . ] 2 0 Perform representations of the following as indicated: (a) Represent 2»0 on the place value charts. Remove the two red tickets, and stibstitute 20 blue tickets on the TENTHS* chart, thus: ONES . T E N T H S X X X X X X X X X X HUNDREDTHS THOUSANDTHS (b) Though the manipulation involved in the following need not be undertaken, proceed to explain, nevertheless, that 2.00 would be shown as 200 green tickets on HUNDREDTHS' chartj 2 „000 would be shown as 2000 yellow tickets on the THOUS ANTHS* chart. Emphasise, as in Step 1, the significance of the last-used position to the right of the decimal point in determining the numerator and the denominator of the improper fraction, ^or example, i s 200 HUNDREDTHS« 2,000 i s 2000 THOUSANDTHS. P A R 'f T 8 R & S •'. " •Aehigygaeat of Leases Objective ft (Time? XO minutes)"v . 2 26 provide practice i n the reading and writing" of decimal fractioas Materials: No special materials required. Steps:' Notes The achievement of Lesson Objective 2 w i l l enable pupils to visualise the common fraction equivalent of a decimal fraction 0 It i s this a b i l i t y to visualise the ©ommoa f recti form x^hichj according to Spitser, provides a good procedure for the reading of decimals. Therefore, the f i r s t step belov presents, at a more abstract level, the same method used i n the achievement of Lesson Objective 2. •1. Write the decimal fraction 0.256 on the blackboard, Then explain the meanings for this decimal that are shown below: 0.256 means 0;200 (200 THOUSANDTHS) 0.0$0 ( 50 THOUSANDTHS) o;oo6 (• 6 THOUSANDTHS: 0.256 i s read "two hundred f i f t y - s i x thousandths". Explain that i n reading a mixed decimal l i k e 115.231 we connect the whole number and the fraction by "AND", In the reading of decimals the word "AND" i s reserved for this purpose and i s never, used,-with one exception, in either the integral or fractional portion or the mixed decimal. Tims, 115.231 i s read "one hundred fifteen AND two hundred thirty one THOUSANDTHS". . ,847 i s read "eight hundred forty-seven thousandths"., ' . 800.047 i s read "eight hundred AND forty-seven THOUSANDTHS, The exception i s in the reading of a decimal fraction containing a common fraction, f o r example: 4.12^ i s read "four AND twelve and one-half HUNDREDTHS", • 0.0* i s read "one seventh of a TENTH". ok * 3. Explain that in"reading a NON-TERMINATING or INFINITE decimal fraction l i k e 3.1416 i t i s common usage to read this as a telephone numberf thus: 3.1416 may be read "three DECIMAL (or POINT) One-four-one-six? .4c Explain that i n reading a TERMINATING or FINITE'decimal fraction such as might be obtained as a measurement by the use of a micrometer, for example ,0500, would be read" n f i v e hundred TEN-THOUSAI-UTHS". In such .cases as No* s 3 and 4 i t i s oust am, rstuar ruler, which determines the most acceptable method of reading 6 Lesson IV (I-'age 1) 4 i i t i i t ' j w * ( . ^ . « r n - ^ " 227 Achievement of L f g y m Objective 1 (Time? 2 0 siisutss; To dafluoatrate the use of sero as a place holder, r>]<a.l>3X"iais: D e c i m a l p l a c e v a l u e c h a r t s . F o u r c h a r t s t o r e p r e s e n t t h e f o l l o w i n g p l a c e v a l u e s : one w h o l e 3 one t e n t h , one h u n d r e d t h , one t h o u s a n d t h , A f i f t h c h a r t h e a r s t h e d e c i m a l p o i n t , and a s i x t h c h a r t b e a r s the z e r o s y m b o l . A l s o , d e c i m a l p l a c e value c a r d s a s shown o n n e x t p a g e . S t e p s : Note: Steps 1 , 2 , and 3 demonstrate visually the use of aero as a place"holder. 1. S e t t h e f i v e c a r d s shown b e l o w on the b o t t o m ledpre o f t h e b l a c k b o a r d o The i l l u s t r a t i o n i n d i c a t i n g t h e p o s i t i o n o f the d e c i m a l point t o the r i g h t o f t h e 0I3E c a r d i s for d i a g r a m m a t i c c o n v e n i e n c e onlv I n a c t u a l l y s e t t i n g o u t these c a r d s i t w i l l p r e s e n t t h e v i s u a l symmetry o f t h e d i f f e r e n t p l a c e v a l u e s more e f f e c t i v e l y i f t h e d e c i m a l p o i n t i s p l a c e d i n f r o n t o f t h e red b o a r d t o w a r d t h e r i g h t e d g e , i n s t e a d o f b e i n g p l a c e d e n t i r e l y t o the r i g h t a s i s done i n t h e d i a g r a m . ~ 2 . Remove t h e TENTH'S c a r d . E x p l a i n t h e n e c e s s i t y t o f i l l t h e empty p l a c e , o t h e r w i s e t h e HUNDREDTH'S and THOUSANDTH'S c a r d s w i l l be l o c a t e d one and two p l a c e s r e s p e c t i v e l y t o t h e r i g h t o f t h e d e c i m a l p o i n t * A c c o r d i n g t o t h e g e n e r a l i z a t i o n l e a r n e d i n 4 (b) o f L e s s o n I I t h e s e c a r d s o u s t now be c o n s i d e r e d t o r e p r e s e n t TENTHS and HUNDREDTH r e s p e c t i v e l y . T h e r e f o r e , i f i t i s i n t e n d e d m e r e l y t o remove t h e TENTH'S c a r d and l e a v e t h e HUNDREDTH'S ana! THO&SANDTH'S c a r d s w i t h t h e i r o r i g i n a l v a l u e , t h e n a s e r o must be u s e d t o f i l l t h e empty p l a c e " t o p r o t e c t " t h e v a l u e s o f t h e s e c a r d s . A c c o r d i n g l y , i n s e r t t h e c a r d b e a r i n g t h e s e r o i n t h e empty p l a c e . 3 G R e s t o r e t h e c a r d s t o t h e i r o r i g i n a l p o s i t i o n s and t h i s t i m e remove b o t h t h e TENTH'S AND HUNDREDTH'S c a r d s and f o l l o w t h e p r o c e d u r e a s i n S t e p 2 0 Lesson 17 (Page 2} 228 Notes Steps k9 5, and 6 demonstrate visually the effect upon the value of a mixed decimal fraction of Inserting s. zero immediately after the Secimal point. Set the four cards shovm below on the bottom ledge of the black-board. (Note: follow the instruction contained in Step 1 above in regard to the plaeenehtof the board containing the decimal point) 1 • 1 (a) (b) (c) Figure 1 Then Insert the ZERO iramediately after the decimal point, thus: Figure 2 Since, however, the second and third cards from the ONES' place must be HUNDREDTH;? and THOUSANDTHS respectively, these two cards must be replaced to give the follovring arrangement: r 0 <e) (f) Figure } By comparing the arrangement shown in Figure 1 with that sho^n in figure 3 i t should be pointed out that we have, in effect, taken-^-of card (b) to give us card ( f ) , and we have taken "1_ of card (c) to give us card (g). 1 ^ Since we have not, of course, i n any way altered the ONE'S card, i t cannot be said that we have taken one-tenth of the original mixed"'"decimal expression. A l l that can be said i s that inserting the zero immediately after the decimal point has the effeWTJf "reducing the value of the mixed decimal expression. Note: Steps ? and 8 demonstrate visually the effect upon the value of a simple decimal fraction Of inserting a sero immediately after the decimal point. Set the three cards shown below on the bottom ledge of the black-board? 1'hei: insert the ZERO immediately after the decimal point 9 thus i TBT However, as i n Step 6, Cards (A) and {b5 must be changed to give* • Unli.ke the previous example,, this i l l u s t r a t i o n shows that inserting the sero immediately after the decimal point in a simple fraction has the effect of making the value of the new fraction EXACTLY ONE-TENTH of the value of the original fraction,, o0 '?o conclude this portion of the lesson 9 two generalisations should be dra^a fro® pupils at this stages (a) If a s«3*© i s inserted after the desires! point in a mixed decimal expression It has th© effect of reducing the value of the expression. (h) If a aero i s inserted after the-decimal point in a simple de^JUiliv expression i t makes the value ONE-»TSNTH as much as i t was originally. aetftevuaent of Lesson Obiactjve 2 (TJam 1 0 minutes) 3b &^oz?.i*tmk% th© us© of sere as a terminal ci'phor. Hffie, r i a l s : Decimal place value cards. Two cards to represent the following place values: one ^ article, one tenth. A third card bears the decimal point, and a fourth card bears the acre symbol• Steps : 1. Set the three cards shown, below on the bottom ledge of the blackboard. B M W LX3 2* Then annex the ZERO immediately to the right of the TEKTH^S card thus: « 3 w ^ A U & I M < { . " . m a * W_:Jir*arjw» I in mmf 3 C Draw attention of pupils to the following two points? Lessen IF (Page /*) (a) a Terminal Zero, unlike a place lidding zero, i s annexed to the end of a decimal fraction. .(b) a Terminal ssero does not change the actual value of a-decimal fraction™ but i t does change the SIGNIFICANCE of it,, This change i n SIGNIFICANCE or MEANING which results from adding a Terminal Zero w i l l be discussed in Lesson 71 0 At this point i t w i l l be sufficient to point out that adding the aero in the above example enables the fraction to be read "ONE and TEN HUNDREDTHS" instead of "ONE and ONE TENTH*» This indicates that the decimal fraction i s accurate to the nearest HUNDREDTH. Without the terminal ZERO i t i s accurate onTylJo the nearest TENTH. 4. To conclude this portion of the lesson, the following geaeraligatios should be drawn from pupils at this stages The addition of a terminal sero to a decimal fraction does not change the value of the fraction but i t does change the significance of the fraction. _?rogedure , of „ Teaching: PABf ONE 231 *!-> deiaoE3t jrate t h e e f f e c t upon t h e value o f a decimal fraction •of moviag t h e decimal volute, M a t e r i a l s : T h r e e place v a l u e "charts: CUES'» T E N S ' ; H U N D R E D S ' C h a r t indicating ' the decimal point."TicSets; 1 single.. 1 p a c k e t of 1 0 ; 1 p a c k e t of 1 0 0 , T h r e e place value cards: TENTH, HUNDREDTH. THOUSANDTH, Stegss Notes S t o p s 1 and Z demonstrate visually t h e effect u p o n t h e v a l u e o f t h e d e c i m a l fraction o f moving t h e d e c i m a l p o i n t t o t h e l e f t . St©??s 3 a n d 4 d e m o n s t r a t e v i s u a l l y t h e e f f e c t u p o n t h e v a l u e o f t h e d e c i m a l f r a c t i o n o f m o v i n g t h e d e c i m a l p o i a t t o t h e S t e p 5_is t h e f i n a l s t e p i n t h e i n d u c t i o n a n d c o n t a i n s a l s S e t t h e t h r e e c h a r t s ; shown b e l o w o n t h e b o t t o m l e d g e o f t h e b l a c k b o a r d . I • I I I H I I I I I M ; £ P l a c e a s i n g l e c a r d o r t i c k e t on t h e O N E S ' c h a r t , a p a c k e t of 1 0 on t h e TENS' c h a r t , and a p a c k e t o f 100 on t h e HUNDREDS' c h a r t , Move t h e d e c i m a l p o i n t one p l a c e t o " t h e l e f t a s i a d i c a t e d b y t h e r e d a r r o w a t t h e t o p of™"t;he d i a g r a m , E x p l a i n : S i n c e t h e p l a c e i m m e d i a t e l y t o t h e l e f t o f t h e d e c i m a l p o i n t must a l w a y s be t h e OTJES* p l a c e , t h i s makes i t n e c e s s a r y t o c o n s i d e r t h a t t h e p a c k e t o f 10 a t p r e s e n t • s h o w n on t h e T E N S ' c h a r t h a s , i n e f f e c t , b e e n r e d u c e d t o 1 s i n g l e t i c k e t s L i k e w i s e , t h e t i c k e t s shown o n t h e a d j a c e n t c h a r t s must be r e d u c e d t o o n e - t e n t h t h e o r i g i n a l amount i n o r d e r t o m a i n t a i n t h e p r i n c i p l e o f TEN-NESS* 2, Move t h e d e c i m a l p o i n t two p l a c e s t o t h e l e f t o f t h e o r i g i n a l l o c a t i o n a s i n d i c a t e d b y "the g r e e n a r r o w a t t h e b o t t o m o f t h e d i a g r a m . R e p e a t t h e a p p r o p r i a t e e x p l a n a t i o n g i v e n i n s t e p 1 Set th« board. shree c a r d s shown b e l o w on v h o b o t t o m l e 232 i 1 1 u 1 1*. 1 J 1-Iove the decimal point one plage to the-right as indicated by red arrow at the top of the diagram. Explain: Since the place immediately to the l e f t of the decimal point must always be theONES' place, this makes i t aecessary to consider that'the representation of' OKE-TEMTH (immediately to the Ifeft of the new location of the decimal point) has, i n effect, been Increased to ONE. Likewise s the representations shown on adjacent cards must be increased to ten times the original, sise la order to maintain the principle of TEN-NESS. I-Iove the decimal point twc places to the righg of the original location as indicated by the green arrow at the bottom of the diagram. Repeat the appropriate explanation given i n step 3. To conclude- this portion of the lesson, t-ha following g®&&ralisf&&«9> should be drawn from pupils at this stage: (a) For eyery place that a decimal point is moved t£o thejff numberji* has' "the effect of multiplying the nurao°ern^r^j0 That i s , i f the decimal point is moved one place to the rl^fa^^Sh* number becomes y> times larger; i f "it "is moved ^wo'places to the a£gl&,the- number 'becomes %00 times larger. eTc. (b) For every, place that a decimal point is moved to the laffl i n a number s i t has "the effect of dividing the number'oY'; BgCTTnat i s s i f the. decimal point i s moved onenlece to the lrftry tn* number is reduced t© 0KE«5-EHTIt i t s c^glneXvlluei if i t is moved fewe places to WeljSfB'*'th& number i s reduced to pKEe4^JRED'gI i'iis original valus, etc. m& T W O AcMageaattt of Lessen Objective 2 (Times 15 minutes) * f Materials: Same as for Part One, Steps: 233 81 *art» tnsrGXt>re« a r e s a r a l l s X to u i ioso contained 111 t md 2 demonstrate visually i&e effect upon the of the decimal point of dividing a number by a power Steos 3 a n d 4 demonstrate v i s u a l l y the effect upon the l o c a t i o n o f t h e decimal point o f multiplying a number by a power o f 1 0 , ' S t e p 5 i s the f i n a l step i n the induction, and c o n t a i n s a g e n e r a l i s a t i o n which should be-drasm froa pupils a s a result o f t h e i r e x p e r i e n c e wit& t h e f i r s t four steps* S e t t h e t h r e e c h a r t s shown b e l o w on t h e b o t t o m l e d g e o f t h e - b l a c k b o a r d . P l a c e a s i n g l e c a r d o r t i c k e t on t h e OSES 1 1 c h a r t , a p a c k e t o f 10 o n t h e TENS' c h a r t , a n d a p a c k e t o f 100 o n t h e HUNDREDS- c h a r t - , D i v i d e t h i s number s h o w n , t h a t i s 111,1 , b y 10, T h i s means d i v i d i n g e a c h p l a c e b y 3.0, and s o we g e t : ; m 10 ticket* here 1 ticket here S i n c e t h e one t i c k e t o r I u n i t must b e i d e n t i f i e d b y t h e d e c i m a l p o i n t i t i s , c o n s e q u e n t l y , n e c e s s a r y t o a d j u s t t h e l o c a t i o n o f t h e d e c i m a l ' p o i n t b y m o v i n g i t one p l a c e t o t h e l e f t , a s shown b y t h e r e d a r r o w . R e p l a c e t h e t i c k e t s i n o r d e r t o i n d i c a t e 111* T h i s t i m e d i v i d e e a c h each p l a c e b y 100, and s o we g e t : 1 H U N D R E D S " 1 t i c k e t I h e r e E x p l a i n t h e n e c e s s i t y t o make t h e a d j u s t m e n t i n t h e l o c a t i o n o f t h e d e c i m a l p o i n t a s shown by t h e g r e e n a r r o w * Set the tar. blackboard: he bottom f the 234 0 1 1 1 ( M u l t i p l y b y 10} T h i s r e p r e s e n t s . 1 1 1 . L e t u s now m u l t i p l y t h i s d e c i m a l f r a c t i o n b y 1 0 , t h u s : 1 I t i s now n e c e s s a r y t o a d j u s t t h e l o c a t i o n o f the d e c i m a l p o i n t i n o r d e r t o p u t i t b e s i d e t h e c a r d t h a t s t a n d s for ONE. T h a t i s , when t h e number i s m u l t i p l i e d b y 1 0 i t i s n e c e s s a r y t o move t h e d e c i m a l p o i n t one p l a c e ~ t o t t h e . r i g n t . See r e d a r r o w . 4o R e p e a t t h e i l l u s t r a t i o n g i v e n i n s t e p 3 : a p p l y i n g i t t h i s t i n e to d e m o n s t r a t e t h e n e e d t o move t h e d e c i m a l p o i n t two p l a c e s t o the r i g h t when t h e number i s m u l t i p l i e d b y 1 0 0 . 5. To c o n c l u d e t h i s p o r t i o n o f t h e l e s s o n , t h e f o l l o w i n g g e n e r a l i s a t i o n s h o u l d b e drawn f r o m p u p i l s a t t h i s s t a g e : {a} VJhen a d e c i m a l f r a c t i o n i s m u l t i p l i e d b y 1 0 ( 1 0 0 , 1 0 0 0 , e t e 0 , ( t h a t i s t f some p o w e r o f 10} t h e d e c i m a l p o i n t i s moved one p l a c e t o t h e r i g h t f o r e v e r y s e r o i n t h e m u l t i p l i e r ^ (b) When a d e c i m a l f r a c t i o n i s d i v i d e d b y 1 0 , 1 0 0 . 1000 e t c . , , t h a t i s 9 some power o f 10) t h e d e c i m a l p o i n t i s ' m o v e d one p l a c e t o t h e l e f t f o r e v e r y z e r o i n t h e d i v i s o r LESSON VI 235 Page 1 ROUNDING JECII-IAL FRACTIONS I i i 5? O N E Achievemoat of Lesson Ob.1egt3.TO 1 (Time; 15 sinutss) To lllustrat® the s i g n i f i c a n c e of rounding decimal fractions 3 Materials: Wall role with movable indicator. Steps: Note: The significance of rounding decimal fractions i s shown by c o m p a r i n g the variation i n a measurement rounded o n l y to UNITS to the variations i n measurements rounded successively t o TENTHS and HUND2EDTHS. 1. The scale indicated below represents a longer section of the wall rule used i n this lesson. Draw this representation on the blackboard. (a) Explain that ?jhen we say that a l i n e i s 2 inches long we signify by this indication merely that the length i s CLOSER TO 2 INCHES THAN IT IS TO 1 INCH or 3 INGRES. The rather considerable amount of variation i n length permitted i s indicated by the area marked i n RED. It should be evident that i n order to round a measurement number to the NEAREST unit i t i s necessary to know at least the number of TENTHS involved i n the measurement. (b) Explain that when we say that a l i n e i s 2 W 0 inches long we signify by this indication that the length this time i s CLOSER TO 2.0 INCHES THAN IT IS TO 1 , 9 INCHES or 2.1 INCHES. The more restricted amount of variation i n length permitted by this designation i s indicated by the area marked i n PURPLE4 It should be evident i n this case that i n order to 236 round a. meaurement number to the NEAREST TENTH i t i s necessary to know at least the mmber of HUNDREDTHS involved i n the Cc) F i n a l l y , e x p l a i n that when v» say that a l i n e i s 2C,0G inches itlon t h a t the length this time i s IS TD l c $ Q mcHES o r 2o01 INCHES, The e v e n more r e s t r i c t e d amount o f v a r i a t i o n i n l e a g t h p e r m i t t e d b y t h i s d e s i g n a t i o n i s i n d i c a t e d b y i ihe v e r y i t m a l l a r e a marked i n GREEN t> I t s h o u l d b e e v i d e n t i n t a t s c a s e t h a t i n o r d e r t o r o u n d a measurement number t o t h e NEAREST HUNDREDTH i t i s n e c e s s a r y t o know a t l e a s t t h e number o f THOUSANDTHS i n v o l v e d i n t h s measuremento 2„ Refer t o t h e w a l l r u l e , P o i n t o u t how t h i s r e p r e s e n t s o n l y a p o r t i o n o f t h e b l a c k b o a r d i l l u s t r a t i o n shown i n S t e p 1* L e t u s s a y t h a t t h e l e n g t h o f a l i n e i s la&7 u n i t s , . T h i s means t h a t t h i s measurement i s rounded t o t h e n e a r e s t HUNDREJTH, and t h a t i n o r d e r t o b e a b l e t o e f f e c t t h i s d e g r e e o f r o u n d i n g i t i s n e c e s s a r y t o know t . i e l e n g t h o f t h e l i n e i n THOUSANDTHS, o r * i n o t h e r w o r d s , t o k r o w t h a t t h e l e n g t h l i e s somewhere b e t w e e n 1*665 and 1«674« I n t h e c h a r t a b r v e , th© s m a l l a r e a shaded I n GREEN i n d i c a t e s t h e a r e a o f t h i s v a r i a t i o n ; a n d , i n t h e c h a r t b e l o w , an e n l a r g e m e n t o f t h i s same a r e ? o f v a r i a t i o n i s r e p r o d u c e d , , 1 l O t h s 100th s 7 1000th 100th* llorr t h e i n d i c a t o r on t h e w a l l r u l e t o show t h e v e r y s m a l l variatior i n l e n g t h t h a t eoufcd be p e r m i t t e d when t h e l e n g t h o f a l i n e i s d e s c r i b e d a s 1«67 u n i t s . P o i n t o u t on t h e w a l l r u l e t h a t a s we s u c c e s s i v e l y r e d u c e the a c c t r a c y o f r o u n d i n g we i n c r e a s e t h e v a r i a t i o n i n t h e l e n g t h of t h e l i n e r e p r e s e n t e d b y t h e measurement* T h a t i s to say, point out t h a t i f this l i n e ware r o u n d e d to the n e a r e s t TENTH i t would be Lesson VI (PageJ) 1.7, and show on the wall n i l s that this' description af' i t s length would entitle i t to be plaeed between l.oj? and 1.74. This.variation in enlarged form i s shown on the chart below: Finally, point out that i f this line were rounded to th® nearest UNIT i t would be 2 , and show on the wall rule that finis description of the length would entitle i t to be placed between 1©5 and 2 . 4 . In a l l these cases of rounding„ i f the fraction i s equal to, or greater than, one«half of the fractional interval, the fraction w i l l be raised to the next highest interval. 3» Repeat Step .2 with other illustrations on the wall rul e e Assume„ f&r example, that the length of a lin e i s 1 . 3 2 * Show the variation i n length permitted'when th is line i s rounded successively to: (a) HUNDREDTHS (b) TENTHS (c) UNITS, Show that in rounding a number to HUNDREDTHS i t i s necessary to know the number of THOUSANDTHS' i n rounding to TENTHS i t i s necessary to know the number of HUNDREDTHS% and. i n rounding to .UNITS i t i s necessary to know the number of TENTHS. 4© To conclude this portion of £he lesson^ three ' generalisations should be drawn from pupils a t . t h i i stages (®5 Ih rounding a mixed decimal fraction to th© nearest whole number, i f the number of TENTHS i s 5 or greaTJerTsEETT: to the whole number. . In rounding a decimal, fraction to the nearest TENTH, i f the number of HUNDREDTHS i s 5 or greater 9 add 1 to the number of TENTHS9 etc. (b) Jn rmm&'km a Mm& dactiwal f r a c t i c s #c tba ne*rcia&' uttole swoftbttr 1* i«: naaeasarsr to tawtf the, -&uub«r of #EK5H$tf In Vli&ding & kiuafear *ft the nearest M f R i t i s necessary to know the number of HUNDREDTHS* (c) After 'rounding has been completed, the place occupied by the l a s t d i g i t or sere indicates the accuracy of the me&surementrr^oT example,, 2.060 i s accurate to the nearest THOUSANDTH,. Lesson VI (Page 4) P A R T TWO Achievement of Lesson Ob.iective 2 (Time: 8 minutes) , 238 To demonstrate various applications of the rounding of decimal fractions. Materials: Wall rule with movable indicator. Steps: Mote: Decimal fractions are frequently expressed to a degree ©f accuracy beyond that required for a particular purpose. The following steps show visually how approximations of such decimal fractions may be made by various applications of rounding. 1. Assume the length of a line to be 1.837. Indicate on the Wall rule the very small variation in length that would be permitted by .'this very accurate description of length. 2. For convenience we may round this mixed decimal expression to : . HUNDREDT1S i f the.purpose for"which the measurement was being • v used warranted i t , and report i t as. 1.84 OR. 184 HUNDREDTHS. Remind pupils of the point that was emphasized in Part Two of Lesson III concerning the- importance of the last-used position after the.decimal point. Thus, i n 1*84, when we convert everything to the position.occupied by the 4 we get 184 HUNDREDTHS. Point out that this measurement, 1.84 or 184 HUNDREDTHS i s accurate to the nearest HUNDREDTH:, and that IN ORDER.TO OBTAIN; THIS DEGREE OF ACCURACY WE MUST. FIRST, BEFORE ROUNDING, KNOW ALSO THE NUMBER OF THOUSANDTHS. 3 . For even greater convenience,.1.837 may be rounded to TENTHS. 10 t«s As shown in the diagram above, point out on the - wall rule that this may be rouncfia to 1.8 or 18 TENTHS. / Repeat the various points made in Step 2 above., 4. Demonstrate on the blackboard how 1.596 could be expressed as: (a) 1.60 (read "one and sixty hundredths") or 160 HUNDREDTHS. (b) 1.6 or 16 TENTHS. Lesson VI (Pag® 5) P A R T . T H R E E 2 5 9 •/Acfoleven^fr of .Lesson Objective (Tim®? 7 minutes) To indisate tshy UNLIKE decimal fractions must be changed to LIKE decimal fractions (that i s , with the same understood denominator) in crd*r that they may be added or subtracted„ Materials: No special materials required. Steps: Note: Step 1 refers to non»measurement numbers which may be counted as discrete, non-continuous entities a Step 2 refers to measurement numbers, that i s . those specifically indicated in a problem or situation to represent inches, pofends, or some other unit of' measurement which can never be "entirely" exact« lo When the numbers do NOT mean inches, or some other measurement, f i l l the empty spaces with seros, for example: 0i8 OiSOO 0;65 change to 0.650 2*222 0.239 2© When the numbers represent measurements, as in the examples below, It i s neeessary to find the mimber with the fewest decimal places and round a l l the othernumbers to that number of plaees, for example: 0o8 OiS Note: i t i s understood 0;65 change to 0.7 that tljese numbers refer 0*239 0.2 to Inches, pounds, etc. 3» To conclude this portion of the lesson, the following generalisation should be drawn from pupils at this stage: "The sum or difference of measurement numbers wil l be accurate only to the fractional unit of the number that has the fewest decimal places*" 240 DIVISION INVOLVING DECIMAL PRACTIONS AcMjevjre«Kt j o ^ ^ (Time: S m&mtes) To explain the significance of pis*£o*Mlng division iits decimal fraction*. Material*: No special materials required. Steps; 1. Present the following examples «n tk© i®&ie££*©a®!d: (a) 8 4 6)!SX (b) .14)T^~ (c) 9.8) 7.05& The above examples have been selected "because none of them requires the addition of seros to the dividend. It may be explained s however i f the need arises &that the same principle holds i n the case of NOB^TERKDUTIHQ or INFINITE quotients.where The above examples may be worked cut by different pupils on the blackboard„ 2 . When the quotients have been obtained demonstrate by means of divisions involving common fractions that in the case of: Example (a) HUNDREDTHS divided by TENTHS i s Tenths. [b) THOUSANDTHS divided by HUNDREDTHS^gFTenths0 (c) THOUSANDTHS divided by TENTHS i s Hundredths0 241 Lesaon VII (Page 2) 3PA, R I I I T I I I I . T . W 0 Aehlevementof Lessoa Objective 2 {Times 22 mioutea) To demonstrate the significance of moving the decimal point i n performing divisions involving decimal fractions• Materials: Flannel Board with supply of paper prepared with the appropriate design e Steps: Note: Steps 1 to 4, inclusive, refer to examples where a whole number i s divided by a decimal. 1. Write on the blackboard the following division question: .5) 3 Point out that when a whole number i s divided by a simple fraction the answer i s LARGER than the dividend. This may revolutionise somewhat the concept children may have gained i n previous grades i n which i t was believed that i f a number were divided i t would automatically mean that the quotient would be smaller than the dividend. To i l l u s t r a t e this, apply Sheet 1 of the accompanying materials to the flannel board. DIVIDEND DIVISOR The quotient i s larger than the integral dividend when the divisor i s less than one. Sheet 1 Supply the answer to the division question already written on the blackboard. 2 0 Write on the blackboard-th© seeond d5.vision question: .125nS With the participation of the pupils, identify .125 as the decimal equivalent of 1/8. When this has been done apply Sheet ft to the flannel board, DIVIDEND DIVISOR •••••• Sheet 2 Lesson VII (Page 3 ) 2 4 2 Supply the answer to the division question already written on the blackboarde Out cf the above two steps pupils should have gained an understanding of the fact that when <> r-hole number i s divided by a divisor less than ONE, the quotient wil l be a ^ t e r than the dividend,; 3o Apply Sheet 3 to the flannel board. This i s shown below: DIVIDEND DIVISOR To illustrate that 24*2/3 , • — f M ^ ^ ^ ^ M ^s^s^i $ t i s equal to Notes Circles rather than rectangles are suppliediy 6«§» 2 The difficulty in this division i s obvious. Let us multiply the DIVISOR by 3 in order to make i t a whole number. This gives us a new DIVISOR of 2. By means of various SIDE-EXAMPLES on the blackboard show that the quotient remains unchanged When the DIVISOR and the DIVIDEND are each multiplied by the same number. Accordingly, let $s multiply the DIVIDEND by 3 also. THIS gives us a new dividend of 6. Frcm Sheet 4 take representations of whole units and apply these TO the flannel board to the right of the original problem. hi Write the division . 4 ) z on the blackboard, and discuss with pupils the need to multiply the DIVISOR by 1 0 in order to remove the decimal fraction, and also to multiply the DIVIDEND by 1 0 In order to compensate for the change in the DIVISOR. Out of Steps 3 and pupils should have gained an understanding of the following two facta: (a) A division i s made easier when the DIVISOR i s multiplied by a quantity which w i l l make i t a whole number. (b) The quotient remains unchanged when the DIVIDEND is multiplied by the same amount as the DIVISOR. Note* . „ Steps 5 to 8 , inclusive, refer to examples where a decimal fraction i s divided by a decimal fraction. 5 . lis in Step 3 9 use the flannel board to explain that when the divisor i s a fraction, the division i s more easily performed i f the divisor i s made a whole number. Apply the representations shown below, and contained on Sheet 5, Sheet 6 Alongside ? and to fee right of Sheet 5» apply to th© flannel board the contents attached to Sheet 6. This i s intended to i l l u s t r a t e thF.t i n the division: ishen th© D17&S0R i s multiplied by 10 to make i t 4. and the DIVIDEND multiplied by 10 also, the division process becomes much easier to performs It illustratesB too- that when the DIVIDEND,, as woll as th© DIVISOR, i s multiplied by the same amount, the quotient remains unchanged. Stated i n another way, i t may be said that i f the decimal point i s moved th© same number of'places, AND IB THE SAKE DIRECTION, i n th© DIVIDEND and the DIVISOR, the answer remains unchanged. Discuss i n -*hst -*>~y the answer would be altered i f . instead of moving the decimal point the same way in both the DIVIDEND and the DIVISOR, th® point were movedHSE PLACE TO THE LEFT EI THE DIVISOR and 3NE PLACE TO THE RIGHT 18 THE DIVIDED. Illustrate the division •1)"'J2 ' cn the blackboard, and point out that-that i f the decimal point were moved to the RIGHT in the DIVIDEND and to th© LEFT in. the DIVISOR9 the answer would be 100 times larger than i t should be. Similarly, discuss i n what way the answer w^uld be altered i f , instead cf* moving, the decimal point the SAI3S way i n both th© DIVIDEND and the DIVISOR, the point were moved ONE FLAJE TO THE LEFf l a the DIVIDEND and to the RIGHT i n the DIVISOR. Using Sheet 7, i l l u s t r a t e the division . l f T ~ " " on the flannel board to Show that the answer would be only 1/100 of what i t should be. 2hh Lesson VII (Baas 5) *o ee&elude this lesson,- the following three generalisations should he drawn fro©, pupils: (a) When a Whole number is? divided by a simple fraction th© quotient (answer), w i l l be larger than the • dividenda (b) In dividing with dselmals, the divisor ma$r be made a whole number by multiplying i t by a given aaounte provided the dividend also i s multiplied by the same amourafc* (e) In dividing with decimals* i f the decimal point i s moved OSSE PLACE' IN OPPOSITE DIRECTIONS i n the DIVIDEND and 'the DIVISORn the answer w i l l be EITHER 100 times greater than 9 or l/lOO as great as 9 i t should be. LESSON VIII Page 1 245 MISCELLANEOUS CONCEPTS INVOLVING DECI-tAL F R A J T O M P. A. R T 0 K Achievement of.Lesson phlectiv© 1 (Time: 1 2 mia&te®} To convey the significance of changing a common fraction to a decimal fraction. Materials: Flannel Board with supply of paper prepared with the appropriate designso Steps: 1, Set the Flannel Board on the bottom ledge of the blackboard* Apply the symbols attached to Sheet 7 to the l e f t side of the Flannel Board, as shown below: Not©: Materials are Numerator l a %he u f y S N P * Denominator i s the DIVISOR _so supplied for Illustrating 3/8 Explain that a common fraction merely Indicates an unperformed division, and that the changing of this common fraction into a decimal fraction INVOLVES THE PERFORMANCE OF THIS DIVISION. In this division the numerator of the fraction becomes the DIVIDEND and the denominator becomes the DIVISOR. Since 1 i s not evenly d i v i s i b l e by 4, i t i s necessary to convert the 1 to TENTHS. Apply the appropriate paper to the Flannel Board to represent this conversion. Since 10 i s not evenly d i v i s i b l e by 4, i t i s necessary to convert the 10 to HUNDREDTHS. Apply the appropriate paper to the Flannel Board to represent this conversion. Since this numerator i s now divisible by 4, perform the division by writing 25 HUNDREDTHS on the blackboard to the right of the Flannel Board, and then express1 "tnls as' a decimal fraction, . 2 5 . 2« Copy on the blackboard other i l l u s t r a t i o n s , such as the followaHgT"~ Lesson VIIJ {Page 2 ) 2 4 6 Corsmon Fraction Cfomge to C h a n g © . . ; v ,^£Mk « — _ — - Y a n t h a J^ ^^ J^ S t!5^tw^3^» FraeHon | |2 . . . . . . . . . . . . . . o 5 3« $ 0 conclude this portion of the lesson, two generalisations should be dram from pupils at th i s stages (a) Converting a common fraction to a decimal, fraction involves a division i n i M c h the Numerator of the • fraction becomes the dividend^ and the denominator becomes the divisor* (b) Before performing this division i t i s necessary to add seres to the numerator. Adding 1 ^ amounts to c«snverting the KUI^ ERATOE from OKES to TENTHS sHUNDREDTHS g, THOUSANDTHS g or whatever smaller unit i s i?equired to obtain a suitable decimal fraction equivalent* P A R ? f ¥ G Multiplication Involving Deeia&l Fractions Achi@vem.eBls of'l«eson Objective 3£'(Wines 12'minutes) f® illustrate the reason for the placement of the decimal point in the .product obtained by the multiplication of decimal fractions« -Wall rule with movable indicator. Steps; 1. Hang the Wall Rule from the moulding at the top of the blackboard. Regard the distance between 1 and 2 on the rule as 1 who}.© unit. Show cn the rule by means of moving the indicator: (a) 1/10 of (which means times) 1 whole unit i s 1 TENTH9 or 9 i n other words9 .1 times 1 equals ol That i s 9 TENTHS TIKES UNITS EQUALS TENTHS. 247 Lesson VIII (Page 3) (b) 1/10 times I/IO equals 1/100 or, i n other words0 »1 times .1 equals ,01 That i s , TENTHS TI.ES TENTHS EQUALS HUNdil^THS. 2e In the same way explain that i n the question: 19«8 times 7o6 s 150*48 the decimal point Is located i n th i s place i n the answer because TENTHS TIMES TENTHS XS HUNDREDTHS, % a Point out how the value of this product would be altered i f the decimal i n the f i r s t number were changed two places to the l e f t , f or example, and changed one, place to therlgfat' i n the second number* Thus, instead of 19 ©8 times 7«6 we would now have .198 times 76, This product would have to be expressed i n THOUSANDTHS, because THOUSANDTHS times ONES (76 ones) equals THOUSANDTHS. The original was expressed i n HUNDREDTHS. Therefore, the act of changing the decimal points as we did had the effect of making the value of the fraction exactly 1/10 of what i t was at f i r s t . 4* I f time permits, repeat this procedure contained i n Steps 2 and 3 with the following examples The product of 4.86 and 6.9 i s 33*534 (HUNDREDTHS times TENTHS i s THOUSANDTHS). Point out i n what way the value of this product would be affected i f the decimals were moved into the following positions: (a) 48.6 times ,69 (Answer remains unchanged) (b) «4$6 times 6.9 (Answer i s 1/10 of what i t was). M ^ ^ ^ f f i , is&vpXviiig .deo&ua^ ., fractions &^^J^LM^mL^k^til^^^^ & minutes) . To develop m understanding of the importance i n the a d d l t t a of decimal fraction® of aligning coXussas according to place value* .kffiscfl (Page Materials; Ko special materials required, 248 Note: Pupils often f a i l to line up decimal points when they write decimals i n addition problems. Errors resulting from this may hot be detected because ©f the failure to recognize usfcat the decimal i n the sum must mean. Writing the sum correctly should be rationalized i n terms of place value. 1. Copy the following chart on the blackboard: ONES TENTHS HUNDREDTHS THOUSANDTHS a • 1 1 1 1 • 0 1 1 1 . 0 0 1 1 1 » 0 This visualisation should be used to Impress upon pupils the fact that the necessity to align the decimals under one another Is merely to ensure that numbers with similar place values w i l l be added, be added In an addition involving decimal fractions i t i s no more correct to add a 1 i n the 8ENTHS place to a 1 i n the HUNDREDTHS place than I t i s to add 1/10 and 1/100 without changing them to a common denominator. 2. It should be pointed out that where there i s an addition involving decimals ferived FROM MEASUREMENTS, as i n the case at the right, these quantities should not, from a practical point of view at least, be added as they stand® me number ^*Y"does not necessarily mean 12.30. I t may mean anything from 12.25 to 12.345 inclusive. I f such measurements have been obtained, and they are to be added, the only sensible thing to do i s TO ROUND ALL TO TENTHS, that i s , round so that a l l the measurements" are expressed to the same number of places* 12*3 inches 8*65 • 144059 tt 6*4 " (in practical work round this to TENTHS) TENTHS) Emphasise the fact that i n an example such as the one shown i n Step 2» the answer w i l l be accurate ONLY to the nearest TENTH* '.tafcsos* f i l l (Psg© 5) 249 3© Ite conclude this portion of the iessoa 9 two. generalisations should h% dram, from pupils at this stager (a) S t the ©ass of. the addition'of measurement numbers involving decimal fraetioas, the suss w i l l "fee aeeurate only as far ®& the last-used place value of the HKiaDereoixtainiis.g the fei^st .number of decimal places* (h) 30a the addition of decimal fractions a l l figures with the g«§e place value should be placed i n the same 250 Procedure of ^ caching: 1« Achievement of. Lesson Objectives £ and £ (Time: 12 minutes) The contents of Sections (a), (b), and (e) of the back-ground material may be aiscusseu within approximately the time. In order to achieve.the above objectives i t is suggested that the material provided should be referred to in general terms. The emphasis should be on the spontan-eity of the presentation rather than on too rigid an adherence to minute detail. If the objectives of the lesson are reached success* fully i t should motivate the pupils to explore and experiment with decimals in subsequent lessons. 2 0 Achievement of Lesson Objective 3 (Time: 18 minutes) Materials: Six visualisation cards numbered 1. 2, 3, 4, 5, 6. (Numbers are indicated on the reverse side,) TO illustrate: ( i ) the ten-ness (or decimal nature) of our number system. (ii) positional value, or the idea that the numeral at the right of a whole number has a ONES' value, the next numeral on the left has a TENS' value, and the next on' the left has a HUNDREDS* value. 1. Hang Cards 1 and 2 from the moulding at the top of the blackboard and in the positions shown in the diagrams below. Koto: i a acfciKg the two illusia-atrloss acted above, • Steps 2 to $ inclusive shot? the situatioks ?shich-require a p^groiipiag fsxai the G2P2S* position to th© petition. Step 9 crsphasiaGS the relationship i a actucl valr.a betrays?, various digits ia thcs& givoa pos'.tSono. 2 Q Point to each of the block in the f i r s t row of the 1 ONES' column while counting 1, 2, 3 2 4 , 5, 0, 7, «, 9 , 10. . . . 251 3« Explain that we must regroup when we reach 10* Provide . a small square immediately below the Cards as shown in the illustration. at the top of the next page* These squares may be drawn on the blackboard* Uo Explain that in these squares we customarily write only one figure to indicate the number of blocks in any one column position* This illustrates the need to regroup when 10 has been reached* TEWS ONES <2j3ii o a n n rilTpp^ cunoonnmiii non n Symbol • • Symbol : tma "Do ..DDan Symbol Chart No^ 2 Symbol 5* Draw a line on the blackboard, as shown, connecting this f i r s t row of 10 blocks with the equivalent representation in the TENS' column in Chart No* 2« 6* Emphasise the fact that 1 block in the TENS* column represents 10 times as many blocks as 1 block in the ONES* column* ?* Continue to count to 20, the blocks in the seconc pointing this time to each of row in the ONES' column* 8* As in Step 3, explain that we must regroup when w© reach another group of 10 blocks. Once, again, draw a line on the blackboard connecting this second row of 10 blocks with the equivalent representation in the TENS' column in Chart No* 2. Write symbols in squares* .252 9o As in Step 6 j draw attention frequently to such facts as: (a) 4 on the OWES* chart represents one-fifth<as many blocks as 2 on the TENS' chart* .{b} 8 on the TENS* chart represents forty times as many blocks as 2 on the ONES* chart, etc* Notes Step 10 shows a situation which requires a regrouping from BOTH the ONES' position to the TENS? position' and the""TllfS* pos£££on to the HUNJffDS^ ' position^ In other words i t illustrates sHuaijfoas frjhich require tvio successive regroupings. Step 11 emphasises the relationship in actual tralue botween various digits l a these three positions? 0X0o Hang Cards 3 and 4 in the same position formerly occupied by Cards 1 and 2« Card 3 has 12 blocks in the TENS* column and 15 blocks in the 0NEST column. Emphasise the fact that in the squares below each card we write only one figure to indicate the number of blocks on that particular card, and emphasize also that i t takes 10 in one position to equal 1 in the adjacent position on the l e f t . Following the emphasis on these details proceed to periorra the regrouping to obtain the result shown on Card 4: I block on the HUNDREDS*, card. 3 blocks on the TENS* card and 5 blocks on the ONES* card, 11. As in Steps 6 and 9 , draw attention to such facts as: (a) 1 in the HUNDREDS* position (on Card No. 4) represents 20 times as many blocks as 5 in the ONES* position. (b) 5 in the ONES* position (on Card No. 4) represents 1/6 as many blocks as 3 on the TENS' chart. 12. Draw frcss pupils, out o f the experience they have had with the foregoing relationships, generalisations framed around the following: (a) The number system is based on a grouping by tens* (b) The number system has place value. This means that each numeral in a number possesses a value assigned by the l,plae<&® i t occupies in the number. Each Bplace- has a value ten times as much as the . 2placew immediately to the right, or ose«tenth as much as the position Xmme i^aTely to the l e f t . . 13 o At this point a very brief comparative description may be mad© of the principles underlying the Roman numeral system of notation (see heading (d) (ii) of the background material). 253 To i l l u s t r a t e : ( H i ) the use of aero, or cipher, as a place holder. Steps: Hang Cards 5 and 6 i n the same positions formerly occupied by Cards 3 and 4. Card No. 5 has 9 blocks i n the TENS* column and 10 blocks i n the ONES' column. This represents 100 and involves a regrouping as shown on Card No* 6. ONES Symbol 1 L _ J In the space on the blackboard below each column position write the appropriate symbol. 2. Explain that a figure must be written to show each place i n the three place number, even though the columns i n two of the places are empty. This i s the PLACE HOLDING FUNCTION OF ZEHO i n our number system. Show that zero has a protecting role to keep 1 i n the third space from the right, or the HUNDREDS' column. 3„ Draw on the blackboard a representation of Card No. 6 and show on this representation one block i n the HUNDREDS* column, and seven blocks i n the ONES* column. 4. While referring to the blackboard drawing described i n Paragraph 3 above deal with the number shown under three headings: {&.} How i t reads « one hundred seven. (Note: this i s a convenient point at which to explain that the use of "and" i n the reading of a number i s reserved exclusively to indicate the connection between t W ^ S § ^ p l a s e and the TENTHS' place. I t i s never used either i n a whole number or i n a fractional number. . (b) What i t means • one hundred, no tens, seven ones. (c) How i t i s written ~ 254 A c h i e v e m e n t of L e s s o n O b j e c t i v e s 1» 2S. and 1 ( T i m e : 15 m i n u t e s ) Materials: Three visualisation cards numbered 7 ? 8 S 9o (Numbers are inuicated on the reverse side) Note: Steps I and 2 which follow are intended to meet Lesson Objective 1 $ while Step 3 i s intended to meet Objective 2 0 A l l three steps should contribute to the attainment of Objective 3* Steps: MMMMM 1, Hang Cards Nos 7 and 8 from the moulding at the top of the " blackboard and in the positions shown in the diagrams belowe ONES • l a m i i i i J-J ' i 's '£•; e$ cl M BE B a Card No. 7 TENTHS HUNDREDTHS THOUSANDTHS t i i 4 - ] HIIII • — • • • • i • mm**. .1 Card No. 8 Point to the decimal point. Explain that the purpose o£ the decimal point i i to ident i f y the ONES5 d i g i t . It mav be of interest to draw attention to the methods explained oh the t&ird page of'this lesson by which people else-where identify the ONES' di g i t . Explain that when the ONES' digit has been located a l l other digits obtain their values from the position they occupy i n relation to the ONES' place. A number, like 34873, i s quite aeaaiagless tmless we know which digitsstands for unity, 2 0 In pointing to, aad explaining, the regrouping shown on Visualiz-ation Cards 7 and 8, emphasise the fact that the following three principles § which were shown in Lesson I to form the structure of the whole number system- apply also to decimal fractions: (1) ten-ness; ( 2 ) place value; (3j place holding "Tune t ion of -zero. 255 l&k® an outline of Cards 7 andS$ on the blackboard and represesSfc on Card 7 such other fractions as (a) 25 HUNDREDTHS (b) 25 TENTHS. Perform on the representation of Card 8 on the blackboard the necessary regrouping i n order to emphasize further the three principles noted immediately above. 3* Remove &&s?ds 7 and 8 and replace with Card 9, shown below: THOUSANDS HUNDREDS TENS1 TENTHS HUNDREDTHSTHOUSANJTHS ONES 2 2,. Z X T Draw lines on the blackboard below the charts, as shown in the ill u s t r a t i o n above, to emphasise the symmetry around the ONES' place,, While the charts are i n this position, discussion should be held which points out the following: (a) the central position occupied by the ONES' place* (b) the symmetry of the other place values around the ONES' place. (c) the various value relationships whereby each place represents a value ten times as large as the place next to i t on the rights onehundred times as large as the second place to i t : on the rightsTeteT Illustrate these relationships with specific examples written on the blackboard i n the appropriate place under the s$rd, e.g.: ( i } i n the number 4.4 « a 4 in the ONES* place i s ten times as large as 4 i n the TENTHS' place, ( i i ) i n the number 77.77 - a 7 i n the TENS' place i s one thousand times as'large as 7 i n the IIUNDREDfSS' place ( i i i ) i n the number 212.2 • a 2 i n the TENTHS* place i s one-fiftieth as much as a 1 i n the TENS' place (that i s . the 1 i n the TENS' place actually represents 100 TENTHS', which i s f i f t y times larger than 2 T8NTHS)» 4. To conclude this portion of the lessonm, two generalisations should Be drawn from pupils at this stage: (a) following principles which underlie the Whole number apply also to decimal fractions (Objective 1): ( i ) Place value - each position assigns to a digit & particular value. ( i l ) ¥en~»ess «> the value assigned to a d i g i t i n one position i s ten times larger than the value assigned to i t i n the position nest to i t on the right. etc, ( i l l ) Plaee«<holding function of aero » i n order to "protect* the value of numerals by keeping them i n the required positions 9 seres are needed to record whatever amntv positions esist BETWEEN the d e c i m a l a n d n u m e r a l s i n t h e meet e x t r e m e p o s i t i o n s to the l e f t or right o f t h e d e c i m a l point. 256 Note: i t n a y he mentioned in passing that seros f i l l a n o t h e r function quite apart frcma place-holding function, ^his function, as well as the place-holding function, w i l l he dealt with more f u l l y i n Lesson IV. (o) The arrangement of positions around the ONES' place i s symmetrical (Objective 2 ) : ( i ) the position tshich i s third from the ONES' place (fourth from the decimal) on the l e f t , and third from the ONES' place on the right are THOUSANDS and THOUSANDTHS respectively. ( i i ) the position which i s second from the ONES' (third from the decimal) on the l e f t , and second from the ONES' place on the right are HUNDREDS sad HUNDREDTHS respectively. ( i i i ) the position which i s next to the ONES* place (second from the decimal) on the l e f t , and next to the- ONES' place on th® right are TENS and TENTHS respectively. Achievement of Lesson Objective 4 (Time: 15 minutes) Materials: Visualisation card No. 10* (Number i s indicated on the reverse side) Note: The a c h i e v e m e n t o f t h i s o b j e c t i v e s h o u l d e n a b l e p u p i l s t o f o r m u l a t e a m e a n i n g f u l g e n e r a l i s a t i o n r e s p e c t i n g * t h e c o m p a r i s o n of d e c i m a l f r a c t i o n s , e . g . : w h i c h i s l a r g e r - .379 o r .33? P u p i l s who have become a c c u s t o m e d to m a k i n g c o m p a r i s o n s o n w h o l e numbers o n l y may f i n d t h e c o m p a r i s o n o f d e c i m a l f r a c t i o n s less o b v i o u s t h a n i t f i r s t a p p e a r s t o t h e m . The W i n s t o n t e x t b o o k " T h i n k i n g w i t h Numbers ' ' c o n t a i n s a d r a w l ? i g , s h o b n a t t h e l e f t , w h i c h may b e p r e s e n t e d on t h e b l a c k b o a r d t o p u p i l s to e m p h a s i s e t h a t one must l e a r n t o c h e c k c o n c l u s i o n s i n a r i t h m e t i c . I n c o m p a r i n g d e c i m a l f r a c t i o n s , a s i n c o m p a r i n g t h e l e n g t h s o f t h e s e 30 i n c h l i n e s , " Y o u c a n n o t a l w a y s b e s u r e * . Steps: 1. Hang Card NO. 10 from the moulding at t h e top o f the blackboard 100G 100 10 m 1 To 1 100 Card No. 10 1000 Sjfifr & ihf 0N5 f l o w e d by t h e d e c i m a l p o i n t . Then c o n t i n u e t o * o ? l o w s - P S t 0 t h e r i ? h t o f t h s d e c i n a I a s ( a ; t h e TENTH chart « e x p l a i n t h a t i f t h e chart located in the corresponding position to t h e l e f t of the ONE were shown i t would cover an area 10 tines larger than the area of the ONE chart. This would be in the TENS' position, (b) the HUNDREDTH chart - explain that i f the chart located i n the corresyjonding position to the l e f t of the ONE were shown i t would Cover an area 100 times larger than the area of the ONE chart. This would be i n the HUNDREDS' position. (c) the THOUSANDTH chart - explain that i f the chart located i n the corresponding position to the l e f t of the ONE were shown i t would cover an area 1000 times larger thatn the area"of the ONE chart. This would be in the THOUSANDS' position. A representation of Card NO. 10 should be put on the blackboard, together rrith the extensions to the l e f t of the ONES' place, as shown on the diagram at the bottom of the previous page. With the assistance of the Card and diagram on the board, discuss the manner in which we would arrange the following in order of size, beginning with the largest: (a) 1.1 (b) .011 Cc) 11 (d) .11 (e) 1.11 2, Proceed to compare two decimal fractions, e.g., .25 and ,3 i n this way: a b a b x x o x X O X X O X Under the TENTHS'and HUNDREDTHS' columns'of Card No. 10, as in the example above, use x's to represent .25 and o's to'represent . 3 . By referring to the visualization explain why the .3 i s larger than the .25. 3. The same procedure may be followed i n showing the reasoning involved i n arranging the following according to size: • 9 * . • (a) .5 (b) .05 (c) 5.5 (d) ,055 (e) ,55 4. To conclude th i s portion of the lesson, the following generalisation should be drawn frtas pupils after the completion of the above. ^Decimal fraatiojus can be ranked i n order of *±m by comparing the absolute value of fcfee d i g i t * in the corresponding places thus; 258 (&} th© largest of several decimal fractions w i l l be the one with the largest figure In the TENTHS9 place* (b) I f the figures i n the TENTHS* place are equal, then the largest fraction w i l l be the one with the largest figure in the HUNDREDTHS* place* Cc) i f the figures i n the HUNDREDTHS' place are equal, then the largest fraction w i l l be the one with the largest figure in the THOUSANDTHS5 place. Procedure of Teaching: 259 P A R T i 0 H. E Aefile^ergent of; lesson Cft> jectlv© 1. (Time: 8 minutes) To consider decimals as a special fossa of common fractions having denominators of 10, 100, 1000 e t c , that i s , any power of 10* Materials: No special materials required. Steps: 1. Write the following series of common fractions on the blackboard: (a) J t b l ^ B (e) I U) » (e)Tg (f) | ( g ) ^ U)j£ ( J ) ^ (k ) l ^ o o (1>5$ ("»-| ( n ) ^ (o) f§ ( p ) i 2. Verbal Explanations: (a) Explain what i s meant by "a power of 10". Obviously, i t i s be&ond the scope of the pupils' comprehension at this stage to explain that i t means "the index of 10". Consequently, i t w i l l suffice to explain that i n effect i t means 10 multiplied by i t s e l f any number of times, or 10 by i t s e l f , thus: 10, 100, 1000, etc. The meaning of "a power of 10" should be made distinct from the meaning of "a multiple of 10" x-shich means 10 multiplied, not by i t s e l f any number Of times, but by any riiisaber, for example: 5, 8. 12. 20, 30, etc., to give these respective multiples of 10: 50, 80, 120, 200, 300 etc. (b) Explain that waile a l l the fractions written on the board are Common fractions, those with a denominator of a power of 10 may also be regarded as decimal fractions, even though i t i s customary practice i n writing decimal fractions to omit writing the denominator andto indicate i t indirectly by the use of a decimal point. 3. Form two columns on the blackboard, and at the top of each write headings as follows: Fractions which may be considered Fractions which may be considered only as common fractions as Decimal fractions f •-— I' • • . ~—-r- ' Under the appropriate heading enter each of the fractions already written on the blackboard. Achievement ®£ Lesson Objective 2 (Tiaaes 12 atefces} show how d e c i m a l fractiona indicate t&e" numerator and d e n o m i n a t o r o f e q u i v a l e n t common fractions* M a t e r i a l s 5 T h r e e v i s u a l i s a t i o n c a r d s numbered 7, 8, 9= (Numbers a r e i n d i c a t e d on the r e v e r s e s i d e ) S t e p s : fete: The two point* stated below should be clearly emphasised after each of the following threa representations contained i n Step 1 of this Lesson procedure 1« She position of the last d i g i t after the decimal point determines the value of a decimal fraction,. That i s . ©sch of the digits i n the decimal positions preceding the la3t place may jajtora Jpe. cenvertad to the "place value of th« last position after fa© "decimal point* The number so obtained determines the NUMERATOR of the equivalent common fraction* 1 At the same time the particular olace -salvi© •£ the l a s t occupied position indicates the DENOMINATOR of the equivalent common fraction,, 2 G When a decimal fraction i s changed to a common fraction, the denominator has ONE ZERO for every figure to t h e right of the decimal point a 1 0 P r o v i d e r e p r e s e n t a t i o n s o f t h e f o l l o w i n g t h r e e f r a c t i o n s a s indi* • (a) Rang C a r d s 7 and 8 from t h e m o u l d i n g a t t h e t o p o f t h e b l a c k b o a r d P o i n t t o t h e r e p r e s e n t a t i o n on C a r d 8 ( l o w e r d i a g r a m ) and e x p l a i n how 2 HUNDREDTHS and 5 THOUSANDTHS may b e c o n v e r t e d t o t h e r e p r e s e n t a t i o n shown o n C a r d 7 ( u p p e r d i a g r a m ) » I n o t h e r w o r d s , when t h e 2 HUNJRE )THS h a v e b e e n c o n v e r t e d ;o THOUSANDTHS, and added t o t h e 5 THOUSANDTHS a l r e a d y t h e r e , i t shows t h e i m p o r t a n c e o f the p o s i t i o n o f t h e l a s t d i g i t a f t e r t h e d e c i m a l p o i n t i n d e t e r m i n i n g t h e v a l u e o f a d e c i m a l f r a c t i o n . E m p h a s i s e c l e a r l y t h e two p o i n t s s t a t e d s b o v e i n g r e e n „ (b) Draw o n t h e b l a c k b o a r d r e p r e s e n t a t i o n s o f C a r d s 7 and 8 and t h e n i l l u s t r a t e «,12 on t h e s e r e p r e s e n t a t i o n s a s shown b e l o w : """ ,IfUNPREIfT?S~" m an THOUSANDTHS j 261 T E N T H S 1 "HuU&B&TIIS " I S I 1 I I I I I 1 I n o t h e r w o r d s , when t h e 1 TENTH h a s b e e n c o n v e r t e d t o HUNDREDTHS, e n d a i d e d t o t h e 2 HUNDREDTHS a l r e a d y t h e r e , i t shows the i m p o r t a n c e o f t h e p o s i t i o n o f t h e l a s t d i g i t a f t e r t h e " d e c i m a l p o i n t i n d e t e r m i n i n g t h e value o f t h e d e c i m a l f r a c t i o n . E m p h a s i z e c l e a r l y t h e two p o i n t s s t a t e d above i n g r e e n . (c ) I l l u s t r a t e 2 . 3 on t h e b l a c k b o a r d r e p r e s e n t a t i o n and f o l l o w t h e p r o c e d u r e o u t l i n e d i n ( b ) . Note? Step 2 below i s merely an extension of (c) above a n d shows that the two points noted above may b e u s e d t o explain t h e conversion of a n integral number into a n improper fractioa c I n this ease s o f course,, i t i s the position o f t h e terminating e e r o which determines the value of t h e improper fraction. 2 f Hang C a r d 9 f r o m t h e m o u l d i n g a t t h e t o p o f t h e b l a c k b o a r d , (a) R e f e r t o t h e s e c t i o n o f t h i s c h a r t shown b e l o w : ONES TENTHS I m a g i n e the numbers on t h i s s e c t i o n t o be as r e p r e s e n t e d a b o v e . E x p l a i n t h a t i f the two 0N3S were c o n v e r t e d t o TENTHS t h e r e w>uld be 20 TENTHS. (b) R e f e r then t o t h e s e c t i o n o f t h i s c h a r t shown b e l o w : ONES .TENTHS .HUNDREDTHS THOUSANDTHS I m a g i n e t h e numbers on t h i s s e c t i o n t o b e ^ r e p r e s e n t e d a b o v e c E x p l a i n t h a t i f t h e two ONES i f e r e c o n v e r t e d t o HUN:EEDTHS t h e r e wou3>d be 200 HUNDREDTHS; O r , i f c o n v e r t e d t o THOUSANDTHS t h e r e w o u l d be 2000 THOUSANDTHS. In e a c h c a s e t h e two " p o i n t s n o t e d i n g r e e n on t h e p r e v i o u s page s h o u l d be e m p h a s i z e d * P i i R •' f f H R B E . M&S&B&kotJ4BmSMf$0&® 2 (Times 10 rn&mtm) • 262 to provide practice i n the reading and writing of decimal fractions. • Materials: No special materials required. Steps: Note: The achievement of Lesson Objective 2 w i l l enable pupils . to visualise the common fraction equivalent of a decimal fraction . It is. this a b i l i t y to visualise the esommon fraction fora which, according to Spltser, provides a good procedure for the reading of decimals. Therefore, the f i r s t step belcrcr presents 9 at a more abstract level, the same method used i n the achievement of Lesson Objective 2. 1 . Write the decimal fraction 0.256 on the blackboard. Then explain the meanings for this decimal that are shown below: 0.256 means 0;200 (200 THOUSANDTHS) 0;0£0 I 50 THOUSANDTHS) 0:006 ( 6 THOUSANDTHS) lffl2i6 {256 ftOUSTOfHST 0.256 i s read "two hundred f i f t y - s i x thousandths". 2. Explain that i n reading a mixed decimal l i k e 115.231 we connect the whole number and the fraction by "AND". In the reading of decimals the word "AND" i s reserved for this purpose and i s never used, with one exception 9 i n either the integral or fractional portion of the mixed decimal. Thus, 115.231 i s read "one hundred fifteen AND two hundred th i r t y one THOUSANDTHS". — " " - * .847 i s read "eight hundred forty-seven thousandths". 800.047.is read "eight.hundred AND forty-seven THOUSANDTHS. The exception i s i n the reading of a decimal fraction containing a common fraction, for example: 4*12| is. read "four AND twelve and one-half HUNDREDTHS". 0.0^,is read "one seventh of a TENTH". 3. Explain that in "reading a N0N«TERI IIN ATIN G or INFINITE decimal fraction l i k e 3.1416 i t i s common usage to read this as a telephone number, thus: 3.1416 may be read "three DECIMAL (or POINT) One«four-one-sixfi 4. Explain that in reading a TERMINATING or FINITE d e c i m a l fraction s u c h as m i g h t be o b t a i n e d a s a measurement b y t h e u s e o f a m i c r o m e t e r. for example .0500, w o u l d be r e a d "five h u n d r e d TEN-THOUSANDTHS "* i n ' s u c h c a s e s as N o ' s . 3 and 4 i t i s custom; rather tin;,, rule, which determines the . most a c c e p t a b l e method o f reading'. Page: 1) are .of. Teaching; P A R T G i S S 263 Achievsessat of Losses Objective 1 (Time: 2Q minutes) " To d e m o n s t r a t e t h e use o f BSTO a s a p l a c e h o l d e r * M a t e r i a l s : C a r d s 9 and 1 0 * A l s o two p i e c e s o f b l a n k p a p e r t o b e u s e d i n c o v e r i n g up c e r t a i n s p a c e s on G a r d 9 , " a n d a p i e c e o f p a p e r b e a r i n g a s s r o s y m b o l t o be u s e d w i t h C a r d 10* S t e p s ? Note: Steps X, 2, and 3 demonstrate visually the use of sero os a place holder* 1* H a n g C a r d 9 f r o m t h e m o u l d i n g a t ' t h e t o p o f t h e b l a c k b o a r d * C o v e r u p t h e t h r e e s e c t i o n s a t t h e l e f t , t h u s l e a v i n g e x p o s e d t h e p a r t shown b e l o w : ONES TENTHS HUNDREDTHS THOUSANDTHS 2 2 2. C o v e r t h e 2 i n t h e TENTHS' p l a c e , E x p l a i n t h e n e c e s s i t y t o f i l l t h e empty s p a c e , o t h e r w i s e t h e 2* s i n t h e HUNDREDTHS' and THOUSANDTHS' p l a c e s w i l l b e l o c a t e d o n e and two p l a c e s r e s p e c t i v e l y t o t h e r i g h t o f t h e d e c i m a l p o i n t * A c c o r d i n g t o t h e g e n e r a l i s a t i o n l e a r n e d i n 4 (b) o f L e s s o n I I t h e s e 2 ' s must n o w ' b e c o n s i d e r e d t o r e p r e s e n t TENTHS* and HUNDREDTHS' r e s p e c t i v e l y . T h e r e f o r e , i f i t i s i n t e n d e d m e r e l y t o remove t h e 2 T e n t h s and l e a v e t h e 2 H u n d r e d t h s a n d 2 T h o u s a n d t h s i n t h e i r o r i g i n a l p l a c e s , t h e n a a e r o m u s t b e u s e d t o f i l l t h e empty s p a c e " t o p r o t e c t " t h e o r i g i n a l p l a c e v a l u e o f t h e 2 H u n d r e d t h s and 2 T h o u s a n d t h s . A c c o r d i n g l y , h a n g u p i n t h e a p p r o p r i a t e p l a c e t h e s h e e t b e a r i n g t h e a e r o . 3. C o v e r t h e 2 i n t h e HUNDREDTHS' p o s i t i o n a s ' w e l l , t h u s l e a v i n g only t h e 2 i n t h e THOUSANDTHS' p o s i t i o n e x p o s e d . E x p l a i n , a s i n S t e p 2 5 t h e n e c e s s i t y t o i n s e r t s e r o s a s p l a c e h o l d e r s " t o p r o t e c t " t h e v a l u e o f t h e 2 i n t h e THOUSANDTHS' P l a c e . Lesson 17 (Page 2} N o t e : S t e p s 4 . 5» a a d 6 d e m o n s t r a t e v i s u a l l y t h e e f f e c t upon t h e value of a m i x e d d e c i m a l f r a c t i o n of i n s e r t i n g a s e r o immediately after t h e d e c i m a l point* • * 4. H a n g C a r d N o * 10 f r o m t h e m o u l d i n g a t t h e t o p o f t h e b l a c k b o a r d . • C o v e r u p t h e THOUSANDTH r e p r e s e n t a t i o n , l e a v i n g t h i s a r r a n g e m e n t s a s shown i n F i g u r e 1. 5. I m m e d i a t e l y toddr C a r d N o . 10 ( shown i n F i g u r e ! ) , d r a w o n t h e b l a c k b o a r d t h e r e p r e s e n t a t i o n shown i n F i g u r e 2. T h i s shows t h a t a z e r o h a s b e e n i n s e r t e d b e t w e e n t h e d e c i m a l p o i n t a n d t h e TENTH. The i n s e r t i o n o f t h i s ZERO c a u s e s a d i s p l a c e m e n t o f t h e TENTH and t h e HUNDREDTH, a s shown i n F i g u r e 2. 6. S i n c e , h o w e v e r , t h e s e c o n d and t h i r d p o s i t i o n s f r o m t h e ONE'S p l a c e must be HUNDREDTHS AND THOUSANDTHS r e s p e c t i v e l y , i t i s n e c e s s a r y t o make t h e a p p r o p r i a t e a l t e r a t i o n , shows} i n F i g u r e 3 S w h i c h s h o u l d a l s o be drawn on t h e b l a c k b o a r d ' i m m e d i a t e l y u n d e r F i g u r e 2. B y c o m p a r i n g t h e a r r a n g e m e n t s h o r n i n F i g u r e 1 w i t h t h a t shown i n F i g u r e 3 i t s h o u l d b e p o i n t e d o u t t h a t we h a v e , i n e f f e c t , t a k e n 1/10 o f p o s i t i o n (b) t o g i v e u s p o s i t i o n ( f ) , and we h a v e t a k e n > / l 0 o f p o s i t i o n ( c ) t o g i v e u s p o s i t i o n ( g ) . See a r r o w s i n d i c a t i n g t h i s . S i n c e we h a v e n o t , o f c o u r s e , i n any way a l t e r e d t h e ONE'S p o s i t i o n , ( p o s i t i o n (a) s t i l l r e m a i n s a s p o s i t i o n (d) ) , i t c a n n o t be s a i d t h a t we h a v e t a k e n o n e - t e n t h o f t h e o r i g i n a l m i x e d d e c i m a l e x p r e s s i o n . A l l t h a t c a n be s a i d i s t h a t i n s e r t i n g t h e z e r o i m m e d i a t e l y a f t e r t h e d e c i m a l p o i n t h a s t h e e f f e c t o f r e d u c i n g t h e v a l u e o f t h e m i x e d d e c i m a l e x p r e s s i o n . Rote2 S t e p s 7 a n d 8 d e m o n s t r a t e visually t h e effect upon the value of a simple decimal fraction of inserting a zero immediately after the decimal point* 265 7© Continue to use Card No. 10. Cover up the ONE'S place and the THOUSANDTH'S place, leaving the arrangement es shown in Figure fc, 8. Then insert the ZERO immediately after the decimal point 0 Show this by drawing on the blackboard immediately under Card 10 the representation shown in Figure 5. This figure shows that the insertion of the ZERO causes a displacement of the TENTH and HUNDREDTH. As i n step 6, since the second and third positions from the ONE'S place must be HUNDREDTHS and THOUSANDTHS respectively, i t i s necessary to make the appropriate alteration, shown in Figure 6 S which should also be drawn on the blackboard immediately under Figure 5« 1 1 t Figure ,4 Figure 5 Figure 6 Unlike the previous example (ddscribed in Steps 4, 5, and 6 ) , this i l l u s t r a t i o n shows that inserting the zero immediately after the decimal point IN A SIMPLE FRACTION has the effect-of making the value of the new fraction EXSCTLY ONE-TENTH of the value of the original fraction. As shown by the arrows, inserting a zero immediately after the decimal point i n a simple fraction causes a displacement which reduces each place to 1/10 i t s original value. 9„ To tto&elud* tihis paraxon of the ies&ca, ^ gi^er^liswvio&i should be arawxi Xros pupils at this atagei (a) I f a aero i s inserted after the decimal point i n a mixed decimal expression i t has the effect of reducing the value of the expressions (b) I f a sero i s inserted after the decimal decimal expression i t makes the value 01 as i t was originally. int i n a simple TENTH S3 much Lesaon I? {Page 4 ) 266 Achievement of Lesson Objective 2 (Time: 10 minutes) To demonstrate tho use of aero as a terminal cipher* Materials: Card 10, and a piece of paper bearing a sero symbol* Steps: 1* Hang Card 10 from the moulding at the top of the blackboard. ^over up the HUNDREDTH*s and THOUSANDTH'S representations, leaving the arrangement shown in Figure 1. 2, Immediately under this portion of Card No. 10, draw on the blackboard the arrangement shown i n Figure 2, which shows that ZERO has been annexed immediately to the right of the TENTH'S place* Figure 2 3. Draw attention of pupils to the following points: (a) a Terminal Zero, unlike a place holding sero, i s annexed to the end of a decimal fraction. (b) a Terminal Zero does not change the actual value of a decimal fraction, but i t does change the significance of i t * Thie change i n SIGNIFICANCE or IIEANING which results #rom a adding a Terminal Zero w i l l be discussed i n Lesson VI* At this point i t w i l l be sufficient to point out that adding the sero'in the above example enables the fraction to be read "ONE and TEN HUNDREDTHS" instead of "ONE and ONE TENTH". This indicates that the decimal fraction i s accurate to the nearest HUNDREDTH, Without the terminal sero i t i s accurate only to the nearest TENTH. 267 Leon-era XV (Pago 5) «V» conclude this portion of the lesson, the following generalisation should be drawn from pupils at this stage: The addition of a terminal sero to a decimal fraction does not change the value of the fraction but i t does change the significance of the fraction. So demonstrate the effect upon the value of a decimal fraefci«* of moving the decimal point. 1-HaterialrS; Steps: Koto: Steps 1 and 2 demonstrate visually the effect upon the value of the decimal fraction of moving the decimal point fio the xext. Steps 3 and h demonstrate visually the effect upon the value of the decimal fradtion of moving the decimal point to the right. Step 5 is the final step in the induction, and contains a generalisation which should he drawn from pupils as a result of their experience with the f i r s t four steps• i„ Hang Visualisation Card 9 from the moulding at the top of the blackboard. Cover up the following 2* s: Thousands, Tenths. Hundredths, Thousandths, thus leaving the portion of the Card shown below. IffltiftSOS z / f c - J l Emphasise the point that the number represented i s composed of 2 HUNDREDS, 2 TENS, and 2 ONES. As indicated above, draw an arrow {red,in this illustration) to indicate the movement of the decimal point one place to the l e f t . Explain: Since the place immediately to the l e f t of the decimal point must always be the ONES* place, this makes i t necessary to consider that the original 2 TENS have now, i n effect, been reduced to 2 ONES. Likewise, the oth&? 2*s shown i n adjacent positions must be reduced to one-tenth the original place value i n order to maintain the principle of TEN-NESS. 2l As indicated by the green arrow i n the i l l u s t r a t i o n above, draw an arrow on the blackboard to indicate the movement of the decimal point two places to the l e f t of the original location. Repeat the appropriate explanation given i n step 1. L g s s o a 7 {Page 2 ) 3 3 * 269 Hang Visualiaation Card 10 from the molding at the top of the blackboardo Cover up the ONE. This lesves; As indicated by the red arrow i n the i l l u s t r a t i o n above, draw an arrow on the blackboard to indicate the movement of the decimal point one place to the right. Explain: Since the place immediately to the l e f t of the decimal point must always be the ONES* place, this makes i t necessary to consider that the representation of 0NE«TENTH (immediately to the l e f t of the new location of the decimal point) has, i n effect, been increased to ONE. Likewise, the representations shown on adjacent places (that i s , the TENTH and HUNDREDTH places) must be increased to ten times the original sise i n order to maintain the principle of TEN-NESS. * 4. As indicated by the green arrow i n the i l l u s t r a t i o n above, draw an arrow on the blackboard to indicate the movement of the decimal point two places to the right. Repeat the appropriate explanation given i n step 3. 5 . To conclude this portion of the lesson, the following generalisation should be drawn from pupils at this stage: (a) For every Place that a decimal point i s moved to the right i n a number, i t has the effect of multiplying the number by TEN. That i s . i f the decimal, point i s moved one place to the right, the msaber becomes 10 times larger: i f i t i s moved two places to the right, the number oecomeslOO times larger, etc. (b) For every place that a decimal point i s moved ,ftq the i n a numb"er/*it nas the effect of dividing the number by 1Q, Thafc i s , i f the decimal point i s moved one place to the l e f t , the Busker i s reduced to OjfB.«Tffl~TH its~orTg1EnaT value; i f i t i s moved two places to tn© TeWr'the' number i s reduced to 0NE~ HUNDREDTH i t s original value, etc. PART, tm i ^ M S E S H L ^ ^ ^ ^ ^ S & J ^ ^ l E S , ^ (Tims: 15 minutes) To demonstrate the effect upon the location of the decimal point of isultiplying or dividing a decimal fraction by a poster cf 10. Materials. Same as for Part One. 270 N e t s : P a r t Two o f t h i s L e s s o n i s t h e c o n v e r s e t o P a r t ONE. The s t e p s i n t h i s p a r t s t h e r e f o r e , a r e p a r a l l e l t o t h o s e c o n t a i n e d i n t h e f i r s t p a r t . S t e p s 1 a n d 2 d e m o n s t r a t e v i s u a l l y t h e e f f e c t u p o n t h e l o c a t i o n o f t h e d e c i m a l p o i n t o f d i v i d i n g a number b y a power o f 10. S t e p s 3 a n d 4 d e m o n s t r a t e visually t h e e f f e c t u p o n t h e l o c a t i o n o f t h e d e c i m a l p o i n t o f m u l t i p l y i n g a number b y a power o f 10. S t e p 5 i s t h e f i l i a l s t e p i n t h e i n d u c t i o n , a n d c o n t a i n s a g e n e r a l i s a t i o n w h i c h s h o u l d b e drawn f r o m p u p i l s a s a r e s u l t o f t h e i r e x p e r i e n c e w i t h t h o f i r s t f o u r s t e p s . 1. Hang Visualisation Card 9 from the moulding at the top of the blackboard. Coyer up the same portion of the Card as i n Part ONE, leaving the following: HUNDREDS ONES • 2 2 200 20 2 Figure 1 Line (a) Emphasise the point that the number 222 i s composed of 2 HUNDREDS, (or 200}*2 TENS (or 20)% and 2 ONES (or 2). These may be written i n the appropriate places on the blackboard as shown above. Divide eaeh of these by 10. This, too, may be written on the blackboard under Line l a ) , as shown below: 20 (2 TENS) 2 (2 ONES) 2 18 (2 TENTHS) 0 r » M a s a J l l 3*ttcs, fthft ops, must b,e ^den^ifled, hr .the decimal,point, i t i s , consequently, necessary to adjust theloeatlcn -of the decimal foint from i t s original position (Figure 1) to one place to the eft, as shown by the red arrow i n Line (b). 2. Erase Line from the blackboard, and proceed to v develop from Line (a), this time to show what happens to th© position of the decimal point when the number i s divided by 100. Line (b),therefore, becomes: 271 Lesson V (Page 4) •2 0 2 I B TOo* (2 ONES) (2 TENTHS) (2 HUNDREDTHS 0 Line (b^) Since the ONES must be i d e n t i f i e d by the decimal point, i t i s 8 consequently, necessary to adjust the l o c a t i o n of the~decimal point by moving i t from i t s o r i g i n a l p o sition to two places to the l e f t , a3 shown by the green arrow i n Line (b^). 3. Hang V i s u a l i s a t i o n Card 10 from the moulding at the top of the blackboard, Dover up the ONE. This leaves: ii (Multiply by 10) This represents .111. Let us now m u l t i p l y t h i s decimal f r a c t i o n by 10j thus: # 1 The representation shown i n Figure 1 should be drawn on the board d i r e c t l y underneath V i s u a l i z a t i o n Card 10. I t i s now necessary to adjust the l o c a t i o n of the decimal point i n order to put i t beside the card that stands f o r ONE. That i s 5 when the number i s m u l t i p l i e d by 10 i t i s necessary to move the decimal point one place to the r i g h t . See red arrow, which should also be drawn oh the blacIcF6arar' i n the appropriate place. 4. Repeat the i l l u s t r a t i o n given i n Step 3: applying i t t h i s time to demonstrate the need to move the decimal point two places to the r i g h t when the number i s m u l t i p l i e d by 3.00. 5. To conclude t h i s portion of the lesson* the following generalisation should be drawn from pupils at t h i s stage: (a) When a decimal f r a c t i o n i s m u l t i p l i e d by 109 100, 1000, etc., (that i s , some power of 10) the decimal point i s moved one place to*the l e f t f o r every sero i n the d i v i s o r . 1000 etc.. lb) When a decimal f r a c t i o n i s divided by 10. 100. 1000 e (that i s , some power of 10) the decimal point i s istoved place to the l e f t f o r every sero i n the d i v i s o r . one ROUoiJIKG DECIMAL FR ACTIONS 272 P A R | 0 § E 5 M T OF LESSON OBJECTIVE 1 \?&mt 15 miKutas) To i l l u s t r a t e th.© significance of rousdi&g deeiisal fractious? ^ t a r i a l s i Visualisation Card No, 11. 3teps: 1* Draw the following scale on the blackboard: (a) Bzplaln that when we say that a line is 2 inches long we signify by this indication merely that the length is closer to 2 inches than i t is to & inch or 3 inches. The rather considerable amount of variation in Isn^ih permitted is indicated by the RED erea. It should be evident that in order to round a measurement number to the nearest"unit ifc i s necessary to kna# at least the number of TEliWnnvolved* in the rasasurementa (b) Exnlaln that when we say the.^ a line is 2*0 inches long v.& signify by tills indication that the longish this time is closer to 2 0u inches than i t is to 1.9 inches or to 2.1 inches„ The more restricted: easnaefc ef w$&$laa i n length permitted by this da?ig?iatlon i s indicated by tbs PURPLE axe*« It Should be evident in this case that in order to ro.jnd a measttrement iraaber to the nearest TENTH i t is necessary to know at least the number of HUNDREDTHS involved in the •measurement. (a)finally, arplain that when we say that a line is 2.00 inches long we signify by this indication that the length this time is closer to 2o00 than i t is to 1.99 or to 2,01 inches. The mven more restricted amouatt of variation in length permitted by this designation is indicated by the GREEN area, It should be evident in this case that in order to round a measurement number to the nearest HUNDREDTH i t is v Pae© 2 necessary to know at least the number of THOUSANDTHS involved in the measurement*. 2o Rang Visualisation Card No* 11 from the moulding at the top of the blackboard. Show diagrararaatically how this represents only a portion of the blackboard illustration shown in Step 1* Let us say that that the length of a line Is 1.67 waits Q This means that this measurement i s rounded to the nearest HUNDREDTH, and ghat in ardor to he able to effect this degree of rounding i t i s necessary to know the length of the line in THOUSANDTHSj or, in other words; to know that the length Ilea somewhere between 1*765 and 1*774* Point out on this chart that as we successively reduce tthe m&a^ey of rounding we increase the variation in the length of the line represented by the measurement * That is to say, point out that is* this line were rounded to the nearest TENTH i t would be 1*$ and show that this variation wousd enti&&e&t to be blaced between 1*75 and 1.84» And further, point out that If this line were rounded to the nearest UNIT i t would be 2 and show that this variation would entitleit to be placed between 1*5 and 2*4. In a l l these cases <gg rounding, i f the fraction i s equal to or greater than one-half of the fractional Interval, the fraction w i l l be raised to the next highest interval* 3 * Repeat with other Illustrations* ^ Assume,for example, that the length of a line i s 1*^ or again, 1*32* Repeat the same procedure as In . step a* 4o To conclude this portion of the lesson, three generalisations should he drawn from pupils at this stages (a) 2a rounding a mixed decimal fraction to the nearest whole number, i f the number of TENTHS i s 5 or greater»aoa 1 to the whole number* In rounding a decimal fraction to the nearest TENTH, If the number of HUNDREDTHS Is 5 or greater* add 1 to the number of TENTHS, etc* (b) In rounding a mixed decimal fraction to the nearest'whole number i t i s necessary to snow the number of TENTHS* In rounding a number to the nearest TENTH i t Is necessary to know the number of HUNDREDTHS* (c) After rounding has been completed, the place occupied by the last DIGIT or ZERO indicates the accuracy of the measurement* For example, 2*060 i s accurate to the nearest THOUSANDTH* P A R T V W 0 Page 3 274 To demonstrate various applications of the rounding of decimal fractions. Materials: Visualisation Card No* 11* Stena: Decimal fractions are frequently expressed to a degree of accuracy beyond that required for a particular purpose. The following steps, show visually how approximations of such decimal fractions may be made by various applications of . rounding. 1» Assume the length of a line to be 1.837. Indicate on the Visualisation Card the very small variation in length that would be permitted by this very accurate description. 2. For convenience we may round this mixed decimal expression to HUNDREDTHS, and report i t as 1.84 OR 184 HUNDREDTHS. Remind pupils of the point that was emphasised in Part Two of Lesson III concerning the importance of the last«*used position after the decimal point. Thus, in 1.84, when we convert everything to the position occupied by the 4 we get 184 HVHBilBDfSllS'* a Point out that this measurement, 1.84 or 184 BUS9J>SE©T&S , i s accurate to the nearest HUNBREBTH. and that IN OR33ER TO OBTAIN THIS DEGREE OP ACCURACY VJE IIUST FIRST, BEFORE ROUNDING, KNOW ALSO THE HUI-iBER OF THOUSANDTHS. 3o For even greater convenience, 1.837 may be rounded to TENTHS. As shown In the diagram above, point out on the Visualisation Card that ii&s may be rounded to 1.8 OR 18 TENTHS. Repeat Repeat the various points made in Step 2 above. 4. Demonstrate on the blackboard how 1.596 could be expressed as: (a) 1.60 (read "one and sixty hundredths") or 160 HUNDRED TcrST* (b) 1.6 or 16 TENTHS. P- A E T T II S E E Achievement df Lesson Qbleetive 3 (Time: ? minutes) 275 To indicate why UKLIKE" decimal fractions must' be changed -to LIKS decimal fractions (that is$ with the same understood denominator) i n order that they may be added or subtracted* Ksterimlss No special materials required, Notes- Stgp 1 refers'to ndne*iieaeurem@nt numbers which may . <b@'dolS!Eea as^isoreSs, 1 'non^^Ga^fnuous'entities. Step 2 refers to measurement numbers* 1* When the numbers do NOT mean inches $ or some other measurement, f i l l the empty spaces with zeros, for example: 0,8 0 . 6 5 0*800 change to 0*650 2* When the numbers represent measurements, as i n the example belowm i t i s necessary to find the number with the fewest decimal places and round a l l the other numbers to that number of places e for example: 0*8 0*65 change to 0*8 0*7 0*2 Note: i t i s understood that these numbers refer to inches, pounds, etc* .3* To conclude this portion of the lesson s tao following goneralisatioa should be drawn from pupils at this stages wThe sum or difference of measurement numbers w i l l be accurate only to the fractional unit of the number that has the fewest decimal places* B 276 DIVISION INVOLVING DECIMAL FRACTIOUS P A R ? 0 N IS foieyemenfe of Leasea Ob.leetiVQ X 'T&ees 8 minutes) T® explain the significant© of informing divisiba involving Nospecial materials required. Steps: Notes. Xn division involving decimal fractions frequently the placement of the'decimal point i s governed only by meaningless rule, ^ e purpose of Part One of this lesson is to interpret the reason for the placement of the ; | v v : -decimal point in a quotient. The division of common fractions && used as a means of developing this interpretation. The time limit devoted to Part One Imposes very great restrictions on the thoroughness with which this topic may be discussed. For this reason i t i s necessary to restrict the examples shown, and deal only with ones . such as the following 4'visions. Such curiosity may be aroused by this incomplete presentation as w i l l make Srofitable a more complete presentation OUTSIDE THE REA OF THIS EXPSRCTNT. m 1. Present the following examples en- fcfee Mm&fa&m&i u ) M J B T (b) tc) 9.s) %m The above examples have been selected because none of them requires the addition of zeros to the dividend. It may be explained, however I f the need arises, that the same principle holds in the case of NONjiTERKINATING or INFINITE quotients .where She above Examples may be worked out by different pupils on the blackboard. 2. When the quotients have been obtained demonstrate by means of divisions Involving common fractions that in the case of: Example U) HUNDREDTHS divided by TENTHS i s Tenths. lb) THOUSANDTHS divided by HUlJDREDTHTl^^nths. (e) THOUSANDTHS divided by TENTHS i s Hundredine. P k E T _ Achievement of Lesson OMedtive 2 (Timet 22 miautes) To demonstrate the signlfisaaee of moving the deelm'-l point i n performing <*i'££sloas involving decimal fractions,, Eiateriaist Visualisation Card No0 12. For use in this lesson, each section should be regarded as 1/10 of the dividend and of the divisor 0 Steps; Motes Steps 1 to k incltisii'-e refer to examples where a whole number i s divided by a decimal. lo Write on the blackboard the division o l H T * and illustrate the answer on Visualization Card No. 12„ shown below: o l f I D S H D o l o l 1 o l .1 ol o l o l E ol Point out that when a whole number i s divided by a simple fraction the answer is larger than the dividend. This may revolutionise somewhat the concept children may have gained in previous grades in which i t was believed the* i f a SMSs&sr .^tere divided i t would automatically mean that th@\-quotient would be smaller than the dividend. 2 0 Though It Is not easy to Illustrate visually, explain that when the divisor (lower section of the Visualisation Card) Is a erection, the division'is more easily performed i f the divisor is made a whole number o Illustrate this with such an example as the following: 6 divided by 3/10 i s not as easy to divide as 60 divided by 3o 3o Refer on the Visualisation Card to the division: 1 f - B — Show that i f the divisor Is multiplied by 10 to give 1} and i f the dividend i s also multiplied by 10 to give 10, the quotient wil l be the same. 4. 4*i this point Sue generalisations should be drawn from 278 (a) Wkea a isaole number i s divided by a simple fraction the quotient (answer) w i l l be larger than the dividend. (b) Whee both the dividend end the divisor are multiplied by the same &Ef&$r the quotient remains the same. Notes Steps 5 t o 8 inclusive refer to examples where a decimal fraction i s divided by a decimal fraction. 5 . As i n Step 2* use Visualisation Card 12 to explain that when the divisor i s a fraction^ the division i s more"easily performed i f the divisor xs made a whole number* Illustrate this with such an example es the following; .4) 3.2 i s more easily divided v&en changed to 4ri*2T The visualization shown below of this example alaould be drawn on the blackboard and used to supplement the visualization medium contained on Card 12. UXTIDES $?\ may be charged to 32 JlVIBOkj Upl HI may De-changed to As an 3tep 3 , i l l u s t r a t e on the Visualisation Card that i f the DIVIDEND and the DIVISOR are each multiplied by the same number, the quotient- remains unchanged Stated i n another way5 i t may be said that i f the decimal point i s moved the same number of places, AND IN THE SAME DIRECTION, i n the'DIVIDEND AND the DIVISOR the answer remains unchanged. Discuss i n what way the answer would be altered ±tt instead of moving the decimal point the same way i n both the DIVIDEND and the DIVISOR, the Point were moved ONE PLACE TO THE LEFT XS THE DIVISOR and ONE PLAGE TO THE RIGHT IN THE DIVIDEND. Illu s t r a t e the division 01}"72 on the visualisation card to show that the answer would be 100 times larger than i t should be,. Lesson VII (Page 4) • _/ 279 Illustrate the division *1) '*2L* on the Visualisation Card to show that the answer wot&d be JjOO times larger than i t should be* 7* Similarly* discuss In what way the answer would be altered i f , instead ox moving the decimal point the same way i n both the DIVIDEND and the DIVISOR, the point were moved ONE PLACE TO THE RIGHT IN THE DIVISOR and ONE PLACE TO THE'LEFT IN THE DIVIDEND* Illustrate the division * & f 2 on the Visualisation ®ard to show that the answer would be only 1/100 of what i t should be* jfeig point two further generalisations should be drawn fipem pupilst (a) In dividing with decimals, the divisor may be made a whole number by multiplying i t by a given amount, provided the dividend also i s multiplied by the same amount* (b) l a dividing with decimals* i f the decimal point i s moved OBIS PLACE XN OPPOSITE DIRECTIONS in the DIVIDEND and the DIVISOR, the answer w i l l be EITHER 1 0 0 times Sweater than, or 1 / 1 0 0 as great as, i t should be* LESSON VIII MSCELLANEOUS CONCEPTS INVOLVING JBOIEIAL FRACTIONS :£A.ft? O H Reductlaa off Gfflag^^^^-fcioas to, Decimal .Ffraetioga Achievement of Lesson OJb^&^&v© 1 {Times 12 minutes) To cesavey th© significance of ehaagiag a eozsaes fraction to a deeam&l fraction* Materials; Visualisation Card 12• (Note: For use in this lesson each space In the upper section (dividend) and in the lower section of this @ffird should be regarded as one whole unit instead of 1/10 of a unit, as was the ease when the card was used in Lesson VII} Steps; lo Rang Visualisation Card 12 front the moulding at the top of the blackboard* While referring to this visualisation, explain that a common fraction merely indicates an unperformed division, and that the changing of this common fraction into'a decimal fraction iavoll^s the .perf of this division* In this division the numerator of the fraction becomes the dividend and the denominator becomes the divisor* Demonstrate the conversion of the fraction \ to a decimal fraction* Point out one section on the upper part of the illustration. Let this represent the numerator of 1* Since 1 i s NOT evenly divisible by 4 i t i s necessary to convert the 1 whole, (as shown in Lesson III, Part Two, Step 2) into a smaller denomination which w i l l be divisible by 1. Illustrate that changing the 1 into 10 TENTHS does not permit It to be divided by 4* Consequently i t is necessary to change i t into 100 HUNDREDTHS* 2«Copy other illustrations, such as the following, on the board: Common Fraction Change to Change to Change to Decimal Tenths Hundredths Thousandths Fraction i • -5 .375 Lesson VXU (Pag© 2) $9 "£Q CONCLUDE THIS portion of the lesson 9 two generalla^tioas should bo drawn fie© pupils at this stages 281 (a) Converting a common fraction to a decimal fraction involves a division In which th® numerator of the fraction becomes the dividend* and the denominator becomes the divisor* (b) Before performing this:41visiem.it i s necessary to add seros to the numerator«Addlng these seres really amounts to converting the NUMERATOR from ONES toTENTHSp HtHSDREDTHS, THOUSANDTHS, or Whatever smaller unit i s required to obtain & suitable decimal fraction equivalent* P A R T TWO sent of Lesson Objective 2 (Time: 12 minutes) To i l l u s t r a t e the rcaswsa f o r the placement of the decimal dot- i $ the product dtrtaStisd Jftgr 3te, iMltipiieatdes of decimal Materials: Visualisation Card 11 (As used in Lesson VI) Steps: 1* Hang Visualisation Card 11 from the moulding at the top of the blackboard* Regard the distance between 1 and 2 on the card as 1 whole unit. Show on the card that: (a) 1/10 of (which means times) 1 whole unit i s 1 TENTH* or, .in other words, *1 times 1 equals »1 TENTHS TIMES UNITS EQUALS TENTHS (b) 1/10 times 1/10 equals 1/100 or, in other words, *1 times •! equals *01 TENTHS TIKES TENTHS EQUA&S HUNDREDTHS* 2* In the same way explain that in the question 19*8 times 7*6 the'.decimal point i s located in this place in the answer, 150*48 because TENTHS TIH33 TENTHS IS HUNDREDTHS, 3* Point out how the value of this product would be altered i f LessKm ¥1X1 (Pag© 3} 282 the decimal l a the f i r s t number were changed two places to the l e f t , for example, and ©hanged one place to the right in the second number* . Thus, instead of 19*8 times 7*6, we would now have •19$ times 76. This product would have to be expressed i n THOUSANDTHS, because THOUSANDTHS times ONES (76 ones) equals THOUSANDTHS. The original was expressed l a HUNDREDTHS. Therefore, the act of changing the decimal points as we did had the effect of making the value of the fraction exactly 1/10 of what i t was at f i r s t . Um I f time permits, repeat this procedure contained In Steps 2 and 3 with the following examples • • • The product of 4*86 and 6*9 i s 33*534 (HUNDREDTHS times TENTHS i s THOUSANDTHS). Point out in what way the value of this product would be affected i f the decimals were moved into the following positions: 48»6 times »69 (Answer remains unchanged) (b) .486 times 6*9 (Answer i s 1/10 of what i t was). P A R 1 T B R E S ^yelylg^ M d ^ i ^ a l t f&g To develop an understanding of the importance i n the addition of dseSmaXfraetions of aligning columns according to pise© valsss^ Visualisation Card 13* fcei Sotes Pupils often f a i l to l i n e up decimal points when they writ© decimals i a addition problems* Errors resulting from this may not ha detected because of the failure to recognise vhat the decimal i n the sum must mean* writing the sum correctly should be rationalised i n terms of place value* Lesson YIXX (Pago 4) 283 Xo Hang Visualisation Sard 13 from the moulding at the top of the blackboard* pass, 1 c TENTHS X HUNDREDTHS i THOUSANDTHS X X X o & X 1 1 . 0 o 1 X . X X 6 1 2. the fact that the necessity to align the decimals under one another i s merely to'ensure that numbers with similar place values mill be added* 3B an addition involving decimal fractions It i s no more correct to add a 1 in the TENTHS place ^ o & i i a the HUNDREDTHS* place than i t i s to add 1/10 and 1/100 wSthout changing them to a common denominator* Xt should be pointed out that where there Is a division involving decimals derived from measurements, as in the ease at the right, these quantities should not* FROM A PRACTICAL POINT OF VIEW AT LEAST, be added as they stand* The number 12o3 does not necessarily'mean 1£*30* It may mean anything from 12*25 to 12*34» inclusive* If such measurements have been obtained9 and they are to be added, the only sensible thing to do i s TO ROUND ALL TO TENTHS* that i s , to round so that a l l 12*3 inches 8.65 • 14 .059 * (in practical work round this to the measurements are espressed to the same number or places* Emphasise the fact that in an example such as the one shown in Step 2, the answer w i l l be accurate ONLY to the nearest TENTH* 3« To eoaclude this portion of the lesson, *M> generalisations shouM be drs&sn from pupils at this stage: (a) In the ease of the addition of measurement numbers involving decimal fractions, the summ w i l l be accurate ©sly as far as the 3&sfc*uBed place value of the number containing the fewest number of decimal places* (b) In the addition of decimal fractions a l l figures with the same place value should be placed i n the same column* 284 APPENDIX C THE PUPILS' WORKSHEETS PAGE Worksheet No. 1 . 285 " " 2 287 " " 3 • . 289 " " 4 . . . 290 " " 5 292 " " 6 293 " " 7 . 294 " " 8 295 Worksheet No. I THE DECIMAL SYSTEM OF NOTATION 285 Write the letter of the best answer on the answer sheets provided. 1 . Which of the following is the largest? ( A ) 1 3 4 6 ( B ) 6 3 4 1 (C) 1 0 0 0 ( D ) 5 9 9 9 ( E ) 2 9 9 7 2 . Which one of the following is represented by the 7 in 3 7 8 2 9 ? ( A ) seven hundred ( B ) seven-tenths (c) seven thousand (D) seventy thousand ( E ) seven 3 . If you changed the number 7 3 0 6 9 so that the 3 was in the 9's place and the 9 was in the 3's place, how would the new number compare with 7 3 0 6 9 ? ( A ) It would be larger (B) It would be smaller (c) It would be the same size (D) Can1t t e l l ( E ) It can't be done 4 . If you re-arranged the figures in the number 5 3 4 2 9 , which of the following arrangements v/ould give the largest number? ( A ) 9 5 , 3 2 4 ( B ) 9 5 , 4 3 2 (c) 5 9 , 4 3 2 ( D ) 9 5 , 2 3 4 ( E ) 9 5 , 2 4 3 5 . Which of the following numbers i s the smallest? ( A ) 1 1 8 9 0 ( B ) 1 0 9 9 9 (c) 1 9 0 0 0 ( D ) 1 7 9 9 9 ( E ) 1 8 9 9 9 6 . If you re-arranged the figures in the number 4 3 , 1 2 5 , which of the following arrangements would give the smallest number? ( A ) 5 4 , 3 2 1 ( B ) 2 1 3 4 5 (c) 1 2 . 3 4 5 ' ( D ) 1 4 , 5 3 2 ( E ) 1 3 , 2 4 5 7 . If the figures in 8 6 , 4 7 3 were re-arranged, which of the following would place the largest figure in the thousand's place? ( A ) 7 3 , 6 4 8 ( B ) 3 8 , 4 6 7 (c) 7 6 , 4 8 3 • ( D ) 8 7 , 6 4 3 ( E ) 8 6 , 7 3 4 8 . If the figures in 2 3 , 4 6 9 were re-arranged, which of the following would place the smallest figure in the tens' place? ( A ) 4 6 , 9 3 2 ( B ) 9 6 , 4 3 2 (c) 6 9 , 2 3 4 ( D ) 3 4 , 6 2 9 ( E ) 9 2 , 3 4 6 9 . Which of the following has a 3 in the hundreds' place? ( A ) 2 3 , 0 6 9 ( B ) 8 6 , 2 3 1 (c) 4 9 , 5 6 3 ( D ) 3 9 , 0 4 3 ( E ) 4 2 , 3 0 4 Worksheet No. I cont. 286 10. V/hich of the following has a 4 i n the ten-thousands' place? (A) 423,104 (B) 643,142 (c) 438,116 (D) 374,942 (E) 763,420 11. In the number 3,944 the 4 on the right represents a number how many times as large as the 4 on the l e f t ? (A) l / l O (B) l/2 (C) 1 ( D ) 5 (E) 10 12. Which of the following statements best t e l l s why we write a zero i n the number 4039 when we want i t to say "four thousand thirty-nine? (A) Because the number v/ould say "four hundred thirty-nine" i f we did not write the zero. ( B ) Writing the zero helps us to remember the number correctly. (c) Writing zero t e l l s us that there are no hundreds i n the number 4039. ( D ) Because the number would be wrong i f we l e f t the zero out. 13. About how many tens are there i n 6452? (A) 6.5 (B) 65 l/2 ( D ) 6,540 (E) 65,000 (C) 654 287 Worksheet Wo. 2 IDENTIFICATION AND MEANING OF PLACE NAMES IN MIXED DECIMAL FRACTIONS Write the letter of the best answer on the answer sheets provided. 1. The value of 2 in .024 is how many times the value of the 4? ( A ) 20 ( B ) l/2 (C) 10 (D) 5 (E) 50 2. Which of the following methods is best for determining the value of the 7 in 3748? (A) Its position in the number (B) Its size when compared with other figures in the number (c) Its size when compared with the whole number 3748 (D) Its size among the numerals from 1 to 9 (E) Its position in the number and its size 3. The value of the 1 in 2.41 is what fractional part of the value of the 2? ( A ) 1/2 ( B ) 1/100 (c) 1/50 ( D ) 1/200 (E) .05 4. The value of 3 written two places to the right of ONES' place is: ( A ) .3 ( B ) .03 (C) 30 (D) .003 (E) 300 5. Which of the following numbers has the figure 4 written in the HUNDREDTHS' place? ( A ) 4486.453 ( B > 3682.474 (c) 3271.043 • (D) 34444.424 6. The value of 6 in the number 1.683 is how many times the value of the 3? ( A ) 100 • ( B ) 1/200 (C) 2 (D) 200 (E) l/2 7. Digit (a), as marked in the following number, is how many times • digit (b): (a) (b) 3 2 5 . 7 2 ( A ) 100 . ( B ) 1/100 (C) 10 (D) 1/1000 (E) 1000 8. Which of the following numbers is the largest? ( A ) .3248 ( B ) .4 (C) .3249 (D) .329 • (E) .3328 9. In (a)(b) the digit marked (a) is how many times the digit .0 8 4 marked (b)? 10. Which of the following numbers is the greatest? (A) .3 ( B ) .295 (C) .11 (D) .101 (E) .301 Worksheet No. 2 cont. 2 8 8 1 1 . The large s t of several decimal f r a c t i o n s w i l l be the one with the largest d i g i t s i n (A) the TENTHS' place (B) the HUNDREDTHS' place (C> the THOUSANDTHS' place ( D ) any place 12 . The main purpose of the decimal point i s to i n d i c a t e the d i g i t i n : (A) HUNDREDS' place (B) HUNDREDTHS' place (c) ONES' place (D) TENTHS' place (E) TENS' place. 13 . The f o l l o w i n g numbers: .0163; .02 ; . 1 ; .0897; .0911, when arranged i n order of size from l a r g e s t to smallest would be: (A) .0911 ; . 1 ; .0897; .02 ; .0163 ( B ) .0911 ; .0897; .0163; . 1 ; .02 (c) . 1 ; .0911; .0897; .02 ; .0163 (D) . 02 ; .0163; .0897; .0911 (E) .0163 ; . 02 ; .0897; .0911 ; .1 14. The l a r g e s t expression of the fo l l o w i n g i s (A) .16 ( B ) 1.6 (C) .016 (D) .0016 (E ) 16.0 Worksheet No. 3 THE READING AND WRITING OF DECIMAL FRACTIONS 289 Write the letter of the best answer on the answer sheets provided. 1. Out of the following common fractions select those which may also be regarded as decimal fractions: (A) 2 (B)_9 (C)_7_ (D)_8 (E)j3 5 17 100 50 10 2. In every decimal fraction there is an unwritten denominator which is alv/ays: (A) 10 (B) 50 (C) a multiple of 10 (D) a power of 10 ( E ) 100 3. Express each of the following decimal fractions in words: (a) 0.362 (b) 0.0375 (c) 200.007 (d) 0.0 1 (e) 0.120 (f) 5.75489 (g) 0.0560 (h) 0.34f 9 4. Write these decimals with common fractions, as in example: Example: .15 = __15 (a) .031 (b) 2.02 (c) .875 100 5. Change the following mixed decimals to improper fractions as in example: Example: 1.25 is 125 HUNDREDTHS or 125/100 (a) 4.75 • (b) 10.00. (c) 1.05 6. The unwritten denominator of a decimal fraction is understood to possess one zero for: (a) every figure to the right of the decimal point (b) every zero to the right of the decimal point (c) every figure, except the zeros, to the right of the point. 7. The unwritten denominator of a deciaml fraction is determined by: (a) the number of zeros after the decimal point (b) the place value of the last-used decimal place (c) the size of the largest digit after the decimal point (d) the size of the f i r s t digit after the decimal point 8. The numerator of a decimal fraction is determined by: (a) the position of the last digit after the decimal point (b) the number of digits after the decimal (c) the size of the f i r s t digit after the decimal point. Worksheet No. 4 THE FUNCTIONS OF ZERO IN DECIMAL FRACTIONS 290 Write the letter of the best answer on the answer sheets provided. 1. Adding two zeros to the right of a whole number is the same as: (A) Adding 10 to the number ( B ) Adding 100 to the number (c) Multiplying the number by 10 ( D ) Multiplying the number by 100 ( E ) Dividing the number by 100 2. Crossing off a zero from the right side of a whole number has the same effect as: (A) Subtracting 10 from the number ( B ) Subtracting 100 from the number (c) Multiplying the number by 10 (D ) Multiplying the number by 1 (E) Dividing the number by 10 3i Adding two zeros to the right of a mixed decimal expression like 8.53 has the same effect upon the value of the expression as: (A) Adding 10 to the expression ( B ) Adding 100 to the expression (c) Leaving the expression unchanged ( D ) Multiplying the expression by 10 ( E ) Multiplying the expression by 100 4. Inserting a zero BETWEEN THE DECIMAL POINT AND THE 5 in the mixed decimal expression 8.53 has the effect of: ( A ) Multiplying the expression by 10 ( B ) Reducing the value of the expression (c) Multiplying the expression by l/lO (D) Adding 10 to the expression ( E ) Increasing the value of the expression 5. Inserting a zero BETWEEN THE DECIMAL POINT AND THE 5 in the decimal expression .53 has the effect of: (A) Multiplying the expression by 10 ( B ) Reducing the value of the expression (C) Multiplying the expression by l/lO ( D ) Adding 10 to the expression ( E ) Increasing the value of the expression Worksheet No. 4 cont. 291 6. If the length of a board is measured to the NEAREST FOOT, say 7 feet, i t is not correct to write this measurement as 7 . 0 0 because: (A) It multiplies the length of the board by 1 0 0 ( B ) It multiplies the length of the board by 1 0 (C) It adds 1 0 0 to the length of the board ( D ) It changes the measurement in some other way ( E ) It gives an unwarranted degree of accuracy to the measurement. 7 . The function of zero as a "place-holder" in a decimal fraction is tor (A) "Hold" each numeral in the fraction in the required position when no digit is present to perform this function ( B ) Give to the fraction a greater degree of accuracy (c) Spread the digits out to make reading easier ( D ) Indicate the number of zeros in the unwritten denominator of the fraction. Worksheet No. 5 292 CHANGING- THE LOCATION OF THE DECIMAL POINT: ITS EFFECT ON THE VALUE OF THE EXPRESSION Write the letter of the best answer on the answer sheets provided. 1 . Which of the following numbers is 1 as large as 3 2 . 7 8 ? 1 0 0 (A) . 3 2 7 8 (B) 3 . 2 7 8 (c) 3 2 7 . 8 (D) 3 2 7 8 2 . If the number . 0 8 5 7 is changed to 8 5 . 7 i t becomes: (A) 1 as large (B) _ 1 as large (c) 1 0 times larger 1 0 0 1 0 ( D ) 1 0 0 times large ( E ) 1 0 0 0 times larger 3 . When a number is divided by 1 0 0 0 the decimal point is moved: ( A ) 2 places to the left ( B ) 3 places to the left (c) 2 places to the right (D) 3 places to the right 4 . In writing an answer a boy makes the mistake of putting his decimal point two places too far to the l e f t . As a result, his answer i s : (A) l/lO of what i t should be (B) l/lOO of what i t should be (c) 1 0 times what i t should be (D) 1 0 0 times what i t should be 5 . When a number is multiplied by 1 0 0 the decimal point is moved: ( A ) 2 places to the l e f t ( B ) 3 places to the left (lC) 2 places to the right (D) 3 places to the right 6 . Moving a decimal point three places to the right has the effect of: (A) multiplying the number by 1 0 0 0 ( B ) dividing the number by 1 0 0 0 (c) multiplying the number by 1 0 0 ( D ) dividing the number by 1 0 0 7 . If a decimal fraction is divided by 1 0 the decimal point is moved 1 place to the left because: ( A ) the number is increased by 1 0 ( B ) the number is decreased by 1 0 (c) the number becomes 1 0 times as large (D ) the number becomes l/lO as large Worksheet No. 6 293 ROUNDING DECIMAL FRACTIONS Write the letter of the best answer on the answer sheets provided. 1. Round each of these to nearest whole numbers (A) 6.5 (B) .68 2. Round each of these to nearest TENTH: (A) .36 (B) 4.029 (c) 7.931 3. Round each of these to nearest HUNDREDTH: (A) .536 (B) 4.175 (C) 5.7.82 4. Assuming that the following numbers have already been rounded indicate fractional unit to which each one is accurate: Example: 2.30 is accurate to the nearest HUNDREDTH. (A) .490 (B) .70 (C) 1.87 5. In order to. express the length of a line accurately to the nearest TENTH of an inch i t is necessary to measure i t to what fraction of an inch? 6. Round the following numbers to whole numbers, tenths, hundredths: 1.089 2.008 6.509 .7829 13.72 Worksheet No. 7 29k DIVISION INVOLVING DECIMAL FRACTIONS Write the letter of the best answer, or the answer itsel f , on the answer sheets provided. 1 . When a whole number is divided by a number larger than 1 the quotient i s : . ( A ) larger than (B) smaller than (c) the same as, the dividend. 2 . When a whole number is divided by a number smaller than 1 the quotient i s : ( A ) larger than (B) smaller than (c) the same as, the dividend. 3 . Divide: 2 . 8 ) 4 . 5 6 4 4 . In question 3 the answer comes out to HUNDREDTHS because: ( A ) There is one figure before the decimal point in the dividend and devisor ( B ) There are two figures in the divisor (c) Thousandths divided by tenths are hundredths ( D ) Tenths times tenths is hundredths. 5 . Divicte . 4 2 ) . 7 3 9 2 6. In question 5 the decimal point may be moved 2 places to the right in the divisor, because: ( A ) i t must be placed after the two .. ( B ) i t is moved 2 places to the right in the dividend also (c) the answer must come out to HUNDREDTHS ( D ) i t makes the division easier. 7 . Look back at questions 3 and 5 and without dividing write the answers to these divisions:. ( A ) . 2 8 ) . 4 5 6 4 ( B ) 4 . 2 ) 7 . 3 9 2 8 . In the division 1 . 6 ) 9 . 2 8 i f the decimal points are moved into these positions: . 1 6 ) 9 2 . 8 the answer will be: ( A ) the same ( B ) 1 0 times as large (c) one-tenth as large (D ) 1 0 0 times as large ( E ) one-hundredth as large. Worksheet No. 8 MISCELLANEOUS CONCEPTS INVOLVING DECIMAL FRACTIONS 295 Write the letter of the best answer, or the answer it s e l f , on the answer sheets provided. 1 . Which statement best t e l l s why we arrange numbers in addition the way we do? (A) It is an easy way to keep the numbers in straight columns ( B ) It helps us to add correctly (c) It helps us to add only those numbers in the same position ( D ) It helps us to carry correctly from one column to another ( E ) It would be harder to add i f the numbers were mixed. 2 . What is the product of: (A) TENTHS and ONES ( B ) TENTHS and TENTHS (C) TENTHS and HUNDREDTHS (D) TENS and HUNDREDTHS (E) TENS and TENTHS 3 . When a whole number is multiplied by a number larger than 1 , the product i s : (A) larger (B) smaller (c) unchanged 4 . When a whole number is multiplied by a number smaller than 1 , the product i s : (A) larger ( B ) smaller (c) unchanged 5 . If 2 . 3 9 8 times 8 7 . 2 equals 2 0 9 . 1 0 5 6 without multiplying find the answer to: (A) 2 3 9 . 8 times 8 . 7 2 ( B ) 2 3 . 9 8 times . 8 7 2 (C) . 2 3 9 8 times 8 7 . 2 6 . In the question 2 . 3 9 8 times 8 7 . 2 , i f the decimal point is moved 2 places to the right in the f i r s t number and one place to the le f t in the second number the answer i s : (A) l/lO as large ( B ) 1 0 times as large (c) 1 0 0 times as large ( D ) l/lOO as large (E) unchanged 7 . To change a fraction like 7 / 8 to a decimal which comes out evenly, before dividing by 8 we must think of 7 as: (A) 7 0 TENTHS (B) 7 0 HUNDREDTHS (C) 7 0 0 0 THOUSANDTHS (D) 7 0 0 THOUSANDTHS APPENDIX D TESTS USED TO MEASURE THE CRITERION VARIABLE AND THE FOUR INDEPENDENT VARIABLES PAGE Farquhar Test of Understanding of Processes with Decimal Fractions 1 . 297 Decimal Fraction Computation Test 301 Otis Self-Administering Test of Mental Ability, Intermediate Examination, Form A 302 Stanford Advanced Reading Test: Form E 304 This test was used to measure both the criterion variable, and one of the independent variables. UNDERSTANDING OF PROCESSES WITH DECIMAL FRACTIONS 297 Choose the most suitable answer for each question. DO NOT MABK'JTHIS SHEET. INDICATE YOUR ANSWERS ON ANSWER SHEET PROVIDED. 1. In addition of mixed decimal fractions i t is important to arrange the numbers so that: A. the last figures of a l l numbers are in the same column B. a l l figures with the same place value are in the same column C. the fi r s t figures of a l l numbers are in the same column D. none of these. 2. To change a common fraction to a decimal fraction one must know that a common fraction indicates: A. multiplication B. enumeration C. addition D. division E. subtraction. 3. Adding a zero to the end of a decimal fraction: A. makes the value 10 times as much B. makes the value l/lO as much C. makes the value 10 more D. does not change the value. 4. The largest of several decimal the largest figure in: A. tenths place C. thousandths place 5. The number 6.00 has a value of: A. 6 hundreds C. 6 hundredths 6. If a decimal point is moved two becomes: A. one-tenth as large C. one-hundredths as large fractions will be the one with B. hundredths place D. any place. B. 600 hundreds D. 600 hundredths. places to the left the number B. ten times as large D. one hundred times as large. 7. The number .0170 should be read: A. seventeen hundredths B. one hundred seventy ten-thousandths C. one hundred seventy thousandths D. seventeen thousandths. 8. Changing .645 to .0645 A. does not change the value B. makes value 10 times as much C. makes value l/lO as much D. makes value l/lOO as much. (page 2) 9. If the number 42.56 is changed to 42.056 by inserting a zero after the decimal point, the value becomes: A. unchanged B. less C. greater D. ten times greater E. one-tenth as much. 10. The value of a decimal fraction is determined by: A. the size of the f i r s t digit after the decimal point B. the position of the last digit after the decimal point C. the position of the largest digit after the decimal point D. the position of the f i r s t digit, not including zerosy ;after the decimal point. 11. Which of the following numbers has the figure "6" in the thousandths place: A. 4695.5417 B. 6495.1724 C. 4325.2164 D. 4175.6000 32 12. In the question .5) 16 the answer is larger than the number divided because: A. 16 is more than .5 B. i t is the same as multiplied by -g-C. dividing a number always gives an answer larger than the number D. i t is the same as finding how many •§•* s in 16. 13. If a decimal fraction is divided by 1000, the decimal point i s moved three places to the le f t because: A. the number becomes 1000 times as large B. the number is increased by 1000 C. the number becomes l/lOOO as large D. the number is decreased by 1000. 14. In the question 1.6) 620.54 i f the decimal point is moved one place to the right in the devisor, and one place to'-the l e f t in the dividend, the answer will be: A. one hundred times as great B. ten times as great C. one hundredths as great D. one tenth as great E. unchanged 15. When a decimal fraction is changed to a common fraction (not reduced), the denominator will have one zero for: A. every figure to the right of the decimal point B. every figure, except zeros, to the right of the point C. every zero to the right of the point D. none of these 16. Multiplying a decimal by 1000 moves the decimal point: A. two places to the right B. three places to the l e f t C. two places to the left D. three places to the right. (page 3) ^99 17. In division with decimals the divisor may be made a whole number before dividing because: A. you can't divide by a decimal B. moving the point does not change the value of a number C. i t is more convenient D. the point in the quotient must be directly above the point in the dividend E. the value of a fraction is unchanged when both terms are multiplied by the same quantity. 18. The measurement 1.050 inches is accurate to the nearest: A. tenth of an inch B. hundredth of an inch C. thousandths of an inch D. ten thousandth :of an inch. 19. Moving a decimal point 2 places to the right has the same effect as: A. multiplying the number by 10 C. dividing the number by 100 (a) (b) 20. In the number: 5 5 5 . 5 5 B. multiplying the number by 1000 D. none of these A. digit (a) is 100 times digit (b) B. digit (a) is 10 times digit (b) C. digit (a) is l/lO of digit (b) D. digit (a) i s l/lOO of digit (b) 21. The number .6925 has a value of about: A. .69 hundredths B. C. 9 hundredths D. E. 692 hundredths. 22. If a number is to be expressed accurately to the nearest hundredth i t must be found to at least: A. one place after the decimal point B. two places after the decimal point C. three places after the decimal point D. four places after the decimal point. 23. In the question: 1.25) 642.3 i f the decimal point were located one place to the right in both numbers the answer would be: A. ten times as large B. one-tenth as large C. one hundred times as large D. one hundredth as large " E. unchanged. 24. If no zeros are added to the dividend, the answer to the question: 4.2) 69.735 will be a two-place decimal, because: A. thousandths divided by tenths is hundredths B. there are two figures in the divisor C. tenths times tenths is hundredths D. there are two places before the point in the dividend. 2 hundredths 69 hundredths (page 4) 300 25. In the question: 6.42 x 15.7 i f the decimal point were located one place to the right in the first number and two places to the l e f t in the second number the answer would be: A. ten times as large B. one-tenth as large C. one hundred times as large D. one-hundredth as large. 26. In the question: 6.92 x 74.5 = 514.156 the decimal point is located at this place in the answer because: A. one and two are three B. hundredths times tenths is thousandths C. tens times hundreds is thousands D. there are three places to the left of the point in the numbers multiplied. 27. A "decimal" is a fraction with an unwritten, but understood, denominator which will always be: A. one B. ten C. any multiple of ten D. any power of ten E. none of these 28. The number 2.134 has a value of about: A. 1 tenth B. 13 tenths C. 21 tenths D. 213 tenths E. 2.1 tenths. 29. To change a fraction, such as 4 , to a two place decimal we divide the numerator by the denominator and we must think of the numerator as: A. 3 hundreds B. 3 hundredths C. 300 hundredths D. 30 hundredths E. none of these. 30. The sum of: 16.17, 459.4, 142.167, and 2.130 inches will be accurate to the nearest: A. inch B. tenth inch C. hundredth inch D. thousandth inch. Name: School: 301 TEST ON DECIMAL FRACTIONS 1. Find the sum of: 2. Subtract: (a) 1.0687 (b) 387.85 (c) 8.975 176.062 9.9345 100.97 34.878 89.875 8.9784 89.59 19.479 5.8459 5.74 6.970 7.7956 983.68 98.826 3. From 124.40 4. From 27.08 5. From 150.000 6. From 94.72 take 87.85 take 15.17 take 72.239 take 19.88 7. Multiply each of the following: (a) 4 x .2 = (c) 100 x 8.5 8. Multiply: 7.8 6.4 (b) .203 x .3 = (d) .08 x 25 x i = 9. Multiply: 78.4 .961 10. Multiply: 94.36 8.7 11. Divide each of the following: (a) . (b) .37375 .11)1.342 (c) .12)3 (d) .ATT7T (e) .2)~nr (f) 1.25) 62.5 12. Divide: 13. Divide: .834) 91.74 8.9) 708.44 14. What is 3 of 6.4? 8 1 5« Express 1 as a decimal, 8 Answer: Answer: OTIS SELF-ADMINISTERING TESTS OF MENTAL ABILITY B y ARTHUR S. OTIS,,PH.D. Formerly Development Specialist with Advisory Board, General Staff, United States War Department I N T E R M E D I A T E E X A M I N A T I O N : F O R M A .302 For Grades 4-9 20 Score Read this page. Do what it tells you to do. Do not open this paper, or turn it over, until you are told to do so. F i l l these blanks, giving your name, age, birthday, etc. Write plainly. Name Age last birthday years First name, initial, and last name Birthday Teacher. ..'. Date. Month Day , , 10-Grade School C i t y . This is a test to see how well you can think. It contains questions of different kinds. Here is a sample question already answered correctly. Notice how the question is answered: Sample: Which one of the five words below tells what an apple is? 1 flower, 2 tree, - 3 vegetable, 4 fruit, 5 animal ( J/. ) . The right answer, of course, is " f r u i t " ; so the word " f r u i t " is underlined. A n d the word " f r u i t " is No. 4; so a figure 4 is placed in the parentheses at the end of the dotted line. This is the way you are to answer the questions. T r y this sample question yourself. D o not write the answer; just draw a line under it and then put its number in the parentheses: Sample: Which one of the five things below is round? 1 a book, 2 a brick, 3 a ball, 4 a house, 5 a box , . ( ) The answer, of course, is " a b a l l " ; so you should have drawn a line under the words " a b a l l " and put a figure 3 in the parentheses. T r y this one: -Sample: A foot is to a man and a paw is to a cat the.same as a hoof is to a — what? 1 dog, 2 horse, 3 shoe, 4 blacksmith, 5 saddle ( ) ' The answer, of course, is " h o r s e " ; so you should have drawn a line under the word " h o r s e " and put a figure 2 in the parentheses. T r y this one: Sample: A t four cents each, how many cents will 6 pencils cost? ( ) The answer, of course, is 24, and there is nothing to underline; so just put the^4 in the parentheses. If the answer to any question is a number or a letter, put the number or letter in the parentheses without underlining anything. Make all letters like printed capitals. The test contains 75 questions. Y o u are not expected to be able to answer all of them, but do the best you can. Y o u will be allowed half an hour after the examiner tells you to begin. T r y to get as many right as possible. Be careful not to go so fast that you make mistakes." D o not spend too much time on any one question. N o questions about the test will be answered by the examiner after the test begins. L a y your pencil down. Do not turn this page until you are told to begin. ,. _— 1 , • , * P u b l i s h e d b y W o r l d B o o k C o m p a n y , Y o n k e r s - o n - H u d s o n , - N e w Y o r k , a n d 2 1 2 6 P r a i r i e A v e n u e , C h i c a g o C o p y r i g h t 1 9 2 2 b y W o r l d B o o k C o m p a n y . C o p y r i g h t r e n e w e d 1 9 5 0 . C o p y r i g h t i n G r e a t B r i t a i n All rights reserved, P R I N T E D I N U . S . A . O S A T M A : I E : A- 8 3 This test is copyrighted. The reproductio of an part of it by mime grap , hectograph, or in any other way, whether the reproductions a e sold or are furnished free for use, is a violation of the copyright law. S. A. Intermediate EXAMINATION BEGINS HERE. 1. Which one of the five things below does not belong with the others? i potato, 2 turnip, 3 carrot, 4 stone, 5 onion ( Df . n ^ w r i t !?*? s e d o t ^\ , i :^. ) 2. Which one of the five words below tells best what a saw is? 1 something, 2 tool, 3 furniture, 4 wood, 5 machine 3. Which one of the five words below means the opposite of west? 1 north, 2 south, 3 east, 4 equator, 5 sunset 4. A hat is to a head and a glove is to a hand the same as a shoe is to what ? 1 leather, 2 a foot, 3 a shoestring, 4 walk, 5 a toe 5. A child who knows he is guilty of doing wrong should feel (?) - 1 bad, 2 sick, 3 better, 4 afraid, 5 ashamed 6. Which one of the five, things below is the smallest ? 1 twig, 2 limb, 3 bud, 4 tree, 5 branch 7. Which one of the five things below is most like these three: cup, plate, saucer ? 1 fork, 2 table, 3 eat, 4 bowl, 5 spoon 8. Which of the five words below means the opposite of strong? 1 man, 2 weak, 3 small, 4 short, 5 thin 9. A finger is to a hand the same as a toe is to what ? 1 foot, 2 toenail, 3 heel, 4 shoe, 5 knee. . : ( 10. Which word means the opposite of sorrow? 1 sickness, 2 health, 3 good, 4 joy, 5 pride 11. Which one of the ten numbers below is the smallest ? (Tell by letter.) A 6084, B 5160, C 4342, D 6521, E 9703, F 4296, G 747s, H 2657, J 8839, K 3918 12. Which word means the opposite of pretty? 1 good, 2 ugly, 3 bad, 4 crooked, 5 nice 13. Do what this mixed-up sentence tells you to do. number Write the the in 5 parentheses 14. If we believe some one has committed a crime, but we are not sure, we have a (?) , 1 fear, 2 suspicion, 3 wonder, 4 confidence, 5 doubtful 15. A book is to an author as a statue is to (?) 1 sculptor, 2 marble, 3 model, 4 magazine, 5 man 16. Which is the most important reason that words in the dictionary are arranged alphabetically? 1 That is the easiest way to arrange them. 2 It puts the shortest words first. 3 It enables us to find any word quickly. 4 It is. merely a custom. 5 It makes the printing easier .. 17. Which one of the five things below is most like these three: plum, apricot, apple? 1 tree, 2 seed, 3 peach, 4 juice, 5 ripe ; 18. At 4 cents each, how many pencils can be bought for 36 cents? 19. If a person walking in a quiet place suddenly hears a loud sound, he is likely to be (?) 1 stopped, 2 struck, 3 startled, 4 made deaf, 5 angered 20. A boy is to a man as a (?) is to a sheep. 1 wool, 2 lamb, 3 goat, 4 shepherd, 5 dog 21. One number is wrong in the following series. What should that number be? (Just write the correct number in the parentheses.) 1 6 2 6 3 ' 6 4 6 5 6 7 6 22. Which of the five things below is most like these three: horse, pigeon, cricket? 1 stall, 2 saddle, 3 eat, 4 goat, 5 chirp ~ 23. If the words below were rearranged to make a good sentence, with what letter would the last word of the sentence begin ? (Make the letter like a printed capital.) nuts from squirrels trees the gather .' 24. A man who betrays his country is called a (?) 1 thief, 2 traitor, 3 enemy, 4 coward, 5 slacker 25. Food is to the body as (?) is to an engine. 1 wheels, 2 fuel, 3 smoke, 4 motion, 5 fire. 26. Which tells best just what a pitcher is ? / 1 a vessel from which to pour liquid, 2 something to hold milk, 3 It has a handle, 4 It goes on the table, 5 It is easily broken Do not stop. G o with the next page. [2} S.A. Intermediate: A 27. If George is older than Frank, and Frank is older than James, then George is (?) James. 1 older than, 2 younger than, 3 just as old as, 4 (cannot say which) 28. Count each 7 below that has a 5 next after it. Tell how many 7's you count. 7 5 3 0 9 7 3 7 85 7 4"2 1 7 5 7 3 2 4 7 ° 9 3 7 S 5 7 2 3 5 7 7 5 4 7--------29. If the words below were rearranged to make a good sentence, with what letter would the last word of the sentence begin ? (Make the letter like a printed capital.) leather shoes usually made are of. . : 30. An electric light is to a candle as a motorcycle is to (?) 1 bicycle, 2 automobile, 3 wheels, 4 speed, 5 police • 31. Which one of the words below would come first in the dictionary? 1 march, 2 ocean, 3 horse, 4 paint, 5 elbow, 6 night, 7 flown 32. The daughter of my mother's brother is my (?) 1 sister, 2 niece, 3 cousin, 4 aunt, 5 granddaughter 33. One number is wrong in the following series. What should that number be? 3 4 5 4 3 '4 5 4 3 5 34. Which of the five things below is most like these three: boat, horse, train ? . 1 1 sail, 2 row, 3 motorcycle, 4 move, 5 track 35. If Paul-is taller than Herbert and Paul is shorter than Robert, then Robert is (?) Herbert. 1 taller than, 2 shorter than, 3 just as tall as, 4 (cannot say which) 36. What is the most important reason that we use clocks ? 1 to wake us up in the morning, 2 to regulate our daily lives, 3 to help us catch trains, 4 so that children will get to school on time, 5 They are ornamental. 37. A coin made by an individual and meant to look like one made by the government is called(?) 1 duplicate, 2 counterfeit, 3 imitation; 4 forgery, 5 libel 38. A wire is to electricity as (?) is to gas.' 1 a flame, 2 a spark, 3 hot, 4 a pipe, 5 a stove 39. If the following words were arranged in order, with what letter would the^middle word begin? „ Yard Inch Mile Foot Rod 40. One number is wrong in the following series. What should that number be? 5 1 0 15 20 '25 29 35 40 45 50....... 41. Which word means the opposite of truth ? 1 cheat, 2 rob, 3 liar, 4 ignorance, 5 falsehood 42. Order is to confusion as (?) is to war. ' 1 guns, 2 peace, 3 powder, , 4 thunder, 5 army '. .' . 43. In a foreign language, good food — Bano Naab good water = Heto Naab The word that means good begins with what letter?.. .• 44. The feeling of a man for his children is usually (?) 1 affection, 2 contempt, 3 joy, 4 pity, 5 reverence 45. Which of the five things below is most like these three: stocking," flag, sail? 1 shoe, 2 ship, 3 staff, 4 towel, 5 wash '. 46. A book is to information as (?) is to money. 1 paper, 2 dollars, 3 bank, 4 work, 5 gold 47. If Harry is taller than William, and William is just as tall as Charles, then Charles is (?) Harry. 1 taller than, 2 shorter than, 3 just as tall as, 4 (cannot say which) . . . . . . . . . . . . 48. If the following words were arranged in order, with what letter would the middle word begin? Six Ten Two Eight Four 49. If the words below were rearranged to make a good sentence, with what letter would the third word of the sentence begin ? (Make the letter like a printed capital.) men high the a wall built stone - . . . / . 50. If the suffering of another makes us suffer also, we feel (?) 1 worse, 2 harmony, 3 sympathy, 4 love, 5 repelled 51. In a foreign language, grass = Moki green grass = Moki Laap The word that means green begins with what letter? Do not stop. Go on with the next page. [3] S. A. Intermediate; t 52. If a man has walked west from his home 9 blocks and then walked east 4 blocks, how many-blocks is he from his home? ( ) .53. A pitcher is to milk as (?) is to flowers. ' 1 stem, 2 leaves, 3 water, 4 vase, 5 roots ( ) 54. Do what this mixed-up sentence tells you to do. sum three Write two the four and of ( ) 55. There is a saying, "Don't count your chickens before they are hatched." This means (?) 1 Don't hurry. 2 Don't be too sure of the future. 3 Haste makes waste. 4 Don't gamble ( ) 56. Which statement tells best just what a fork is?. • 1 a thing to carry food to the mouth, 2 It goes with a knife, 3 an instrument with prongs at the end, 4 It goes on the table, 5 It is made of silver ( ) 57. Wood is to a table as (?) is to a knife. " 1 cutting, 2 chair, 3 fork, 4 steel, 5 handle. ( ) 58. Do what this mixed-up sentence tells you to do. sentence the letter Write last this i n . . . i ( ) 59. Which one of the words below would come last in the dictionary ? 1 alike, 2 admit, 3 amount, 4 across, 5 after, 6 amuse, ,7 adult, 8 affect ( ) 60. There is a saying, "He that scatters thorns, let him go barefoot." This means (?) 1 Let him who causes others discomforts bear them himself also. 2 Going barefoot toughens the feet. 3 People' should pick up what they scatter. 4 Don't scatter things around ( ) 61. If the following words were arranged in order, with what letter would the middle word begin? Plaster Frame Wallpaper Lath Foundation ( ) 62. In a foreign language, many boys"= Boka Hepo many girls = Marti Hepo many boys and girls = Boka Ello Marti Hepo The word that means and begins with what letter?.. ( ) 63. A statement which expresses just the opposite of that which another statement expresses is said to be a (?) • 1 lie, 2 contradiction, 3 falsehood, 4 correction, 5 explanation.— \... ( ) 64. There is a saying, "Don't look a gift horse in the mouth." .This means.(?) 1 It is not safe to look into the mouth of a horse. 2 Although you question the value of a gift, accept it graciously. 3 Don't, accept a horse as a gift. 4 You cannot judge the age of a gift horse by his teeth. ( ) 65. .Which one of the words below would come last in the dictionary? 1 hedge, 2 glory, 3 label,: 4 green, 5 linen, 6 knife, 7 honor ( ) 66. Which statement tells best just what a watch is? 1 It ticks, 2 something to tell time,, 3 a small, round object with a chain, 4 a vest-pocket-sized time-keeping instrument, 5 something with a face and hands ( ) 67. Ice is to water as water is to what? 1 land, 2 steam, 3 cold, 4 river, 5 thirst... • ( ) 68. Which statement tells best just'what a window is? • 1 something to see through, 2 a glass door, 3 a frame with a glass in it, 4 a glass opening in the wall of a house, 5 a piece of glass surrounded by wood ( ) 69. Which of the five words below is most like these three: large, red, good? 1 heavy, 2 size, 3 color, 4 apple, 5 very.......; '•' v ( ) 70. Write the letter that follows the letter that comes next after M in the alphabet ( ) 71. One number is wrong in the following series. What should that number be? 1 2 4 8 16 24 64 ( ) 72. An uncle is to an aunt as a son is to a (?) 1 brother, 2 daughter, 3 sister, 4 father, 5 girl. .' : . . . . ( ) • 73. If I have a large box with 3 small boxes in it and 4 very small boxes in each of the small boxes, how many boxes are there in all ? •. • \ ) 74., One number is wrong in the following series. What should that number be? 1 2 4 5 7 8 10 11 12 14 ( ) 75. There is a saying, "Don't ride a free horse to death." This means (?) 1 Don't be cruel. 2 Don't abuse a privilege. 3 Don't accept gifts. 4 Don't be reckless. ( ) // you finish before the time is up, go back and make sure that every answer is right. [4] STANFORD A C H I E V E M E N T TEST By TRUMAN L. KELLEY, GILES M. RUCH, and LEWIS M. TERMAN Adv. Read. ADVANCED READING TEST: FORM E 3 0 4 g Name , Age. '.Grade. Boy or girl Name of school.... < City : State.. .... .Date.. . ! TEST ~~ SCORE GRADE EQUIV. AGE EQUIV. Parag. Mean. t Word Mean. Average Read. Grade, Equiv. Equated Score, iiquiv. '3.0 3.5 4.0 4.5 5.0 5.5 6.0" 6.5 7.0 8.0 9.0 10.0 11.0 J 1 20 I , i • 1 I 25 , . 1 . , 30 . I V ..J 1 1 1— 35 -, r , 1 , J 1 1 -1 l _ 40 , , , 1 , i i i i i i i i i i i i i , i i i i i i 45 50 55 i . i I 1 1 I • I 1 I I I I 1 i ' • ' t i i i I i i i i I i i • i I I I I I I 60 65 i i i 1 i 1 I i 1 i i I I i I I 111 70 , 1 , 1 1 1 1 1 1 1 1 1.1 I 1 U J J . ,75 * ~ 8° l T i 1 i t i l l 86 1 • '. , . 9° , 1 , , , , 96 10° 106 11° i 1 i i I i i 1 i i i i i I i i i i I 1 i i l l 6 12° 126 13° 1111111111111111111111 u 111 414° 1 1 I I I . I I ! 15° 16 L I I I I I . I I I I M I M I J L V a l u e s extrapolated above this po int . P u b l i s h e d b y W o r l d B o o k C o m p a n y , Y o n k e r s - o n - H u d s o n , N e w Y o r k , a n d C h i c a g o , I l l i n o i s Copyright 1940 b y W o r l d B o o k C o m p a n y C o p y r i g h t i n G r e a t B r i t a i n . All rights reserved, SET: ADV.: JHO PRINTED IN U.S.A. ^ , 8 ^ ™ This test is copyrighted. The reproduction of any part of it by mimeograph, hectograph, or in any other , . way, whether the reproductions are sold or are furnished free for use, is a violation of the copyright law. ' as 6 - TEST. 2. READING: ' WORD MEANING {Confd) ^ . A A V . ^ . - . ^ ^ 6 24Interpretation means— 6 petition 7 explanation 8 humility 9 pressure 10 failuresjj 25 Kindred means — 1 delicate 2 gracious 3 humble 4 curious 5 related 2 5 W 6 26To prosper.is.to — 6 endure 7 grieve 8 entertain 9 forgive 10 flourish. . . : . « j j 27 Nimble means— 1 practical 2 active 3 costly 4 modest 5 dull. • • -27H 28 Conservation means — 6 selecting 7 removing 8 observing 9 connecting 10 protecting 2 8 j j 29Dubious means- 1 doubtful 2 apparent 3 desolate 4 inferior 5 unusual - -29 H ? g g 10 3 0 A pavilion is an open — 6 boat 7 forest 8 building ,9store 10 valley s o i j \\ j j H . H 31 To be punctual is to be - : 1 bored 2 prompt 3 ashamed 4 worthy 5 determined si jj jj \\ jj M 3 2 A l u l l i s a — 6 kettle 7 jar. 8 hush 9 link lOlining 32J.j || jj y_ 33Liberality means- 1 gravity 2 havoc 3 impunity 4 hospitality 5 generosity . .33 j j 34 Obvious means— .6 remote". 7 reasonable 35 Competent means — 1 careless 2 useless 36 Enthusiastic means — 6 lusty 7 singular 37 Conclusive means — L passive 2 variable 3 8 To amass is to — 6 allay 7 accumulate 3 9 Reputable means— 1 cordial 2 solemn 3 honorable 4 fortunate 5 prosperous 39 ii ii i i 6 7 8 9 10 4 0 To indorse is to — 6 adjoin 7 magnify 8 disobey 9 affirm- 10 disclose 40 ii - , 1 2 3 4 5 4 1 To quail is t o — 1 quarrel 2 attack 3 mourn 4 tremble 5-trap ji . 6 7 8 9 M 4 2 To reprove is t o — 6 preside 7 rebuke 8 regulate 9 replace 10 export 42 ii - - . 1 2 3 4 5 4 3 To obstruct is t o — 1 advance 2 check 3 occupy 4 owe 5 pity 43 ii .1 6 7 8 9 10 4 4 Congenial means — 6 original 7 universal 8 successful 9 agreeable 10 refined.. .44 ii 1 2 3 4 5 4 5To contend is t o — 1 stroke 2 fasten 3 pardon -4 exchange 5 struggle.... .45 ii 6 7 8 9 10 .46 Void means — 6 empty 7 cruel 8 exact 9 fierce 10 useful aeji , 1 2 3 4 5 4 7 An impediment is an— 1 agreement 2 obstacle 3 idiot 4 outline 5 utterance. .47 ii [j 6 7 8 9 10 4 8 A mediator brings— 6 agreement 7 clashes 8 desolation 9 inspiration 10 reality' 48 jj > 1 2 3 . 4 5 4 9 Equity means— . 1- fashion 2 advantage 3 exchange 4 knowledge 5 justice ; 4 9 j i jj jj _ 6 7 8 9 18 5 0 Morbid means— 6 unwholesome 7 impetuous 8 ruthless 9 magnetic 10 monotonous 50 j j jj End of Test 2. Look over your work. NUMBER RIGHT 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29130 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Equated score 45 46 47 48 49 50 51 52 52 53 54 55 56 57 58 58 59 60 61 62 63 64 65 66 67 67 68 69 70 71J72 73 73 74 75 76 77 78 79 81 83 84 85 86 87 88 90 91 92 93 84 7 ' 8 9 10 :: :: :: :: 2 3 4 5 :: :: :: :: : ! 7 ' 8 9 10 2 3 4 5 7 8 9 10 2 3 4 5 6 7 8 9 10 1 2 3 4 5 8 doubtful 9 apparent 10 su able 34 3 sincere '4 capable 5 cunning... .35 8 shameful 9 zealous 10 subtle . . 3 6 • . 1 2 * 3 4 5 3 critical 4 decisive 5 compulsory.. .37 ii M- Ii 6 7 8 9 10 8 verify 9 gamble 10 inscribe..... . 3 8 H jj j= I 1 2 3 4 5 Stanf. A d v . R e a d . : F o r m E TEST 2. READING: WORD MEANING DIRECTIONS. In each exercise one of the five numbered words will complete the sentence correctly. Note the number of this word. Then mark the answer space at 3Q>j the right which is numbered the same as the word you have selected. SAMPLES. . A A rose-is a :— 1 box , 2 flower 3 home 4 month . 5 river . . . . A H - J • 6 7 B A roof is found on a — 6 book 7 person 8 rock 9 house 10 word 1 2 3 4 5 c Bread is something t o — : 1 catch 2 drink 3 throw 4wear . 5 eat. 9 10 B ii N n i i . ii 1 2 3 4 5 C;i; 1 2 3 4 . 5 1 Injury means — 1 haste 2 charm 3 pride 4 praise 5 harm... 2 To arise is to — 6 answer 7 stand 8 sit 9 rest 10 carry .2 3 Unoccupied means — 1 unjust 2 useless 3 vacant 4 haunted 5 ignorant... 3' 4 A.peg is usually made o f — 6 wood . 7 paper 8.rock 9 ice 10 sand 4 5 To omit is to—-. 1 bore ^ 2 neglect 3 concern 4 control 5 recover.. 5 6 To defeat is to — . 6 abuse ^ 7 assign 8 betray 9 overcome 10 expose ....... . 6 7 Envious means — 1 shallow 2 social 3 refined 4 enormous 5 jealous 7 8 A scoundrel is a — 6 circus 7 shipment 8 villain 9 chronicle 10 loom 9 To reject is to — 1 engage 2 refuse 3 hasten 4 forbid 5 mourn 7 8 9 10 . 6 8 ii. 1 9 i i . ,1 2 3 4 .6 6 7 / 8 9 10 1 2 3 4 ' 5 7 8 9 10 1 2 3 4 5 7 8 9 10 2 3 4 5 6 7 8 1 0 To forewarn is to — 6 caution 7"recoil 8 moisten 9 contemplate 10 lengthen 10-j] 1 1 A literary person is a — 1 painter 2 monarch 3 writer 4 rival 5 coward... .11 1 2 To prohibit means t o — 6 forbid 7 permit 8 assist 9 boast 10 deserve .... 12 1 3 Stern means— 1 splendid 2 severe 3 joyful , 4 wicked 5 eager... 13 ,14 Conduct means ^ - 6 effort 7 safety 8 appearance 9 actions 10 features .... 14 15Exterior means— . 1 outer" 2 vague „ 3 ignoble 4 indoors 5 fickle •••liH 1 6 To violate is to — 6 abuse 7 appeal 8,reward - 9 summon 10 tempt 1 7 A chart is a — 1 card 2 flag 3 map 4 bowl 5 debt....' 1 8 An alien is a — 6 captive 7 candidate 8 foreigner 9 fortress 10 novelty is H - - - . 1 1 9 Uneasy means =— 1 anxious 2 comfortable 3 ashamed 4 unhappy 5 foolish.. 19 jj 2 0 Seriousness means — 6 fidelity 7 suffrage 8 refinement 9 solemnity -10 displeasure 20 \ \ 2 1 A prologue is a kind of — 1 knoll 2 meteor 3 introduction 4 pathway ' 5 platter.. 21 2 2 A haven is a — 6 breeze 7 package 8 reward 9 verse 10 refuge .22 . 16 1 2 3 4 5 7 8 9 10 3 . 4 5 6 7 8 9 10 2 3 4 5 7 8 9 10 . 17 6 7 8 9 10 2 3 4 5 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 2 3 A witty person is — 1 silly 2 timid 3 clever 4 meek 5 sly 23 Go right on to the next page. 2 TEST 1. READING: PARAGRAPH MEANING S t a n f - A d v R e a d - : F o r m E D I R E C T I O N S . In the paragraphs below, each number shows where a word has been left out. Read each paragraph carefully, and wherever there is a number decide what word has been left out. Then write the missing word in the answer column at the right, as shown in the sample. Write J U S T O N E W O R D on each line. Be sure to write each answer on the line that has the same number as the number of the missing word in the paragraph. S A M P L E . Answer A B Dick and Tom were playing ball in the field. Dick was throwing -the —A— and — B — was trying to catch it. B 1 - 2 Most hawks live on insects, small rodents such as rats, mice, and f ' ' : squirrels, and other destructive animals. Hawks are not particularly fond of chickens and other birds, but some farmers do not realize this. Whenever they see —l—, they want to shoot them because they do l not understand that most of their food consists of animals that are —2— to farm crops. 2 3 - 4 Trolls are dwarfs in Norse mythology. They are portrayed as squatty, misshapen figures with evil powers and malevolent, natures. They were inclined to thieving and were fond of carrying off children. Sometimes a troll would substitute one of its own offspring for the 3_. —3— of a human mother. It was a most unfortunate person who incurred the ill will of a —4—. 4_. ^ 6 Benjamin Franklin was one.of,the most versatile of our great men. He was a statesman, philosopher, writer, publisher, and scien-tist. Jn his role of —5— he not only held public office in the United States but also represented the United States in both England and 5.1 France. As a —6— he is best known for his identification of lightning with electricity. , 6 . . 7-8-9 i n general, insects may be divided into two classes. The group that lives on solid foods has biting mouth parts. The group that lives on liquid foods has long, hollow, sucking mouth parts. 7 . . The butterfly visits flowers, drawing up its food with its long sucking tube in — 7 — form. Grasshoppers do untold damage to grain and 8_. other farm crops. Because the grasshopper eats — 8 — food, its mouth parts are of/the—9—type.. 9 - -10-11-12 The principal diamond fields of the world are in Africa, Brazil, and Australia. ' Few persons know, however, that —io— are also found 10. in Arkansas. It is estimated that more than 10,000 of these stones have-been taken from the soil of that state. Experts have pro- 11. nounced the —n— gems equal to the finest —12— produced in Africa, Brazil, or Australia. • 12-Go right on to the next page. s^tanf.Adv.Read.: F o r m E T E S T 1. R E A D I N G : P A R A G R A P H M E A N I N G {Cont'd) ' 3 13-14^ 15 Demosthenes was a Greek orator who lived about 200 B.C. He 306 was determined to be an orator although his lungs were weak and his 13 , pronunciation faulty. He persevered until at length he surpassed all other — 1 3 — . Turning to political life, he devoted his eloquence to 14 speeches opposing the designs on Greece of Philip of Macedon. These famous — 1 4 — against Philip by — 1 5 — are known as'his "Philippics!" 15 16-17-18 Gypsies are a peculiar vagabond race, now found in many parts of the world. They live in small caravans and earn a livelihood as 16____ fortune tellers, tinkers, makers and sellers of basket ware, etc. The — 1 6 — can be distinguished from the — 1 7 — among whom they rove by 17 their physical appearance and their language as well as by their — 1 8 — of living. 18__ 19-20-21 Our term "white elephant" for something superfluous or something we do not know what to do with comes from a Siamese custom. In Siam, the white elephant is considered sacred, and anyone possessing one must keep it in a royal and consequently expensive 19: style. Therefore, in the olden days when the king of Siam wished to destroy the fortunes of one of his courtiers he would have a — 1 9 — — 2 0 — 20 j given to the person, who was then obliged to spend so much on its — 2 1 — that he usually ruined himself financially. 21 2 2 In no other country is dancing so interwoven with folk music as in Spain. The favored dances are the solea, the tango, and the sequidilla. Many of the most popular airs are sung only when used as an accom-paniment to — 2 2 — . ' - 2 2 — 23-24 The word " infer " means to surmise or conclude from facts or premises, while " imply " means to. express indirectly or to hint. For example, one might say: " M r . Smith —23-— that he was inter- 2 3 — ested in M r . Green's scheme"; or, in another case, " The man — 2 4 — . from her remarks that she was not going to be there." 24_-25-26-27 Desert plants solve in many ways the problem of scarcity of water. The long roots of certain plants penetrate downward to the permanent water layer. Short-rooted plants like the cactus may have . hollow stems for the storage of water. Other plants conserve their meager water supply by leathery leaves thai prevent water losses by 25:. evaporation. Thus we see three ways that desert plants are adapted to inadequate water supplies; namely, by long — 2 5 — , the — 2 6 — of 26 - water supplies during the brief rainy periods, and the possession of structures reducing—27—. 27 «,« . - Go right on to the next page. 4 TEST 1. READING: PARAGRAPH MEANING (Cont'd) stanf-Adv Read :FormE 28-29 The Smiths bear the predominant surname in the United States. The Browns and the Williamses are exceeded only by them and the Johnsons. Next in order come Jones, Miller, Davis, Anderson, 28. Wilson, and Moore. The two most common American surnames are—28— and — 2 9 — . 29. 30-31-32 Gregariousness, or the desire to be with people, and solitariness are two opposite traits of character. Though there are people who are 30. almost wholly gregarious and others who much prefer solitude, most people possess both — 3 0 — . When satiated with the company of 31. others they wish for — 3 1 — , and on the other hand, after a long period of seclusion they develop — 3 2 — interests. . 3 2 . 3 3 - 3 4 A nineteenth-century poet has said, " Rags are royal raiment when worn for virtue's sake." In other words, it is more noble to do 33. without luxuries and comforts than to — 3 3 — them at the — 3 4 — of one's ideals and honor. 34. 35-36 Bacteria have greater resistance to injurious influences than any 35. other known organisms. However, most bacteria are killed like any other — 3 5 — by a brief exposure to — 3 6 — of 60°-65° centigrade. 36. 37-38 Although the driver is recognized as the prime factor in traffic accidents, little has been done to teach correct driving habits and skills. For many drivers a traffic — 3 7 — where ignorant drivers may be taught 37. good driving habits is better than a traffic — 3 8 — where poor drivers are fined or otherwise punished. 38. 39-40 "Has not your teacher explained to you that if you do not know your arithmetic in this grade what is the chance for success in the next grade?" The preceding sentence as it stands is incorrect, but it can 39. be made into a correct sentence by substituting " — 3 9 — have — 4 0 — " for "what is the." ^ 40. 41-42-43 Charcoal has several properties that make it useful — among which are its resistance to chemical action, its black color, and its 41. ability to absorb large volumes of gases and colored substances. It has been found that charcoal made from peach pits — 4 1 — more poison- 42. ous. gas than does charcoal from other sources. For this reason — 4 2 — charcoal is used in making gas — 4 3 — for use in wartime. 43. 44-45 One advantage of rural life is the close contact with nature which • country people enjoy. The children can roam about over the fields picking flowers and hunting for new and strange scenes. Boys can 44. hunt, fish, and swim. Much of our best literature describes the joy of this — 4 4 — with — 4 5 — which country life provides. 45. End of Test 1. Look over your work. N U M B E R R I G H T 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3 2 3 3 34 3 5 36 37 38 39 40 41 42 43 44 45 Equated score 40 41 42 44 45 46 47 49 50 51 52 53 54 56 57 58 59 60 61 63 65 66 67 68 70 71 72 73 74 75 76 78 79 80 81 83 84 85 86 87 88 89 90 91 91 92 APPENDIX E SAMPLES OF THE PROCEDURES USED TO DETERMINE THE SUITABILITY OF THE TESTS FOR THE STUDY PAGE Recording sheets for the preparation of data used in the item analysis of test results^ 308 Calculation sheets for the preparation of data used in the item analysis of test results 311 Calculation of mean, median, and standard deviation . 313 Calculation of test r e l i a b i l i t y by using the Hoyt modification of the Kuder-Richardson Formula . . . 314 Each of these samples is based on the results obtained from the t r i a l administration of the Farquhar Test to forty Grade VII pupils in White Rock Elementary School. u CD CD H X •H G ft trj 10 11 (Part l) RECORDING SHEETS FOR THE PREPARATION OF DATA USED IN THE ITEM ANALYSIS OF RESULTS OBTAINED FROM THE ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE.VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL Top 27$ of 40-pupils = 11 pupils ra CD u o 1 x X X 9 i 10 x: xc X X xc X X 11 12 13 14 Q U E S T I O N S x: x x: 15| 16 x 17 x 18' x X X 19 X X 20 21 x x 22 23 x. 24 x x: r-i O tH rH O crj CO O d CO , . 3 _ 3 . o co CO. OJ !H O X 26 X X 27 28! 2 9] x X X 30 x X T3 ' co T3 •H CO , ID d 10 > rH | M > rH •H -H Cd -H -H i d ft x x 19 17 16 16 16 15 13 12 12 12 3 T3 ft a CO a" fl 3 361 289 256-256 256 225 169 169 144 144 144 f-l Middle 46$ of 40 pupils = 18 pupils (Part 2) % u o rH CH H O C5 O (j CO • , 01 ^3 -•H CQ OJ -H 01 > rH (D U > rH •H -H h Cj -H -H t 3 ft 3 * P i C 3 O, ^ O vO u Bottom 27$ of 40 pupils = 11 pupils as 3 < H rH O 03 3 H 03 CO CO -H ( P a r t 5) . > t-1 <U (H > CO ' •*0 -H -H PH OS -H d S a 3 o a 1 a j o H h m ra H ffi 0 3 H O CALCULATION SHEETS FOR THE PREPARATION OF DATA USED IN THE ITEM ANALYSIS OF RESULTS OBTAINED FROM THE ADMINISTRATION OF FARQUHAR'S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL (Section l) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a No. wrong or omitted responses in bottom 27$ section b No. wrong or omitted responses in top 27$ section c' Total no. wrong or omitted'responses (bottom and top sections) d Per cent difficulty (uncorrected for chance) e No. option questions f Correction factor g Per cent difficulty (corrected for chance) (c) x (f) h Validity or Discrimination (a) - (b) i Per cent discrimination 4 6. 6 8 10 10 10 9 7 11 10 5 8 11 8 1 5 0 1 11 6 6 2 5 5 11 6 9 21 16 16 11 12 19 14 7 14 20 12 23$ 50$ 27$ 41$ 95$ 73$ 73$ 50$ 55$ 86$ 64$ 32$ 64$ 91$ 55$ 4 5 4 4 4 4 4"' 4 5 4 4 4 4 5 4 6.06 5.68 6.06 6.06 6.06 6.06 6.06 6.06 5.68 6.06 6.06 6.06 6.06 5.68 6.06 30 62 36 55 127 97 97 67 115 85 85 42 85 114 73 3 1 6 7 -1 4 4 7 ' 2 27$ 9$ 55$ 64$ -9$ 36$ 36$ 64$ 18$ 27$ 55$ 27$ 18$ 18$ 36$ H H CALCULATION SHEETS FOR THE PREPARATION OF DATA USED IN THE ITEM ANALYSIS OF RESULTS OBTAINED FROM THE ADMINISTRATION OF FARQUHAR1S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL (Section 2) 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a No. wrong or 5 9 9 7 5 11 10 11 11 10 11 6 11 10 10 omitted responses in bottom 27$ secti on b No. wrong or 3 9 9 9 2 7 9 4 7 6 10 1 6 7 10 omitted responses in top 27$ section Total no. wrong or 8 18 18 16 7 18 19 15 18 16 21 7 17 17 20 omitted responses (bottom and top sections) d Per cent difficulty 36$ 82$ 82$ 73$ 32$ 82$ 86$ 68$ 82$ 73$ 95$ 32$ 77$ 77$ 91$ (uncorrected for chance) e No. option questions 4 5 4 4 4 5 4 5 4 4 4 5 5 5 4 f Correction factor 6.06 5.68 6.06 6.06 6.06 5.68 6.06 5.68 6.06 6.06 6.06 5.68 5.68 5.68 6.06 g Per cent difficulty 48 102 109 97 42 .102 115 85 109 97 127 40 97 97 121 (corrected for chance) ( c ) x . ( f ) h Validity or 2 0 0 -2 3 4 1 7 4 4 1 5 5 3 0 Discrimination (a) - (b) i Per cent discrimination 18$ 0 0 -18$ -27$ 36$ 9$ 64$ 36$ 36$ 9$ 45$ 45$ 27$ 0 H ro CALCULATION OF MEDIAN, MEAN, AND STANDARD DEVIATION, OF RESULTS OBTAINED FROM THE TRIAL ADMINISTRATION OF FARQUHAR1S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL Median; Median (50th Centile Point) Mean; Mean = —7- = 10.175 Standard Deviation; - v~ [4599 * 40 ^114.975 - 103.530625 \ f l l . 444375 3.382 = 9 . 5 + | (l) = 9.5+.2 = 9 . 7 - 407 40) Calculation of test r e l i a b i l i t y by using the Hoyt modification of the Kuder-Richardson Formula: RELIABILITY OF FARQUHAR TEST, BASED ON THE RESULTS OF TRIAL ADMINISTRATION TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL The formula: kS + S. - T(T+k) r = N x s 1 " N-1 kS - T* • s k represents number of pupils N represents number of items S g represents sum of squares of pupils' scores S^ represents sum of squares of item scores T represents total of scores for pupils or items Substituting the figures derived from the calculation sheet, r = 30 x 40(4599) + 7689 - 401(407 + 40) 2 9 (40 x 4599) - (407) 2 30 x 183,960 + 7689 - 181,929 29 183,960 - 165,649 291,600 531,019 .549 H APPENDIX F RAW SCORE DATA The scores obtained by the 147 participating subjects i n the tests used to measure the five variables Variable Name of Test Y Farquhar's Test of Understanding of Processes with Decimal Fractions (Final administration at close of experiment) X^ Farquhar's Test of Understanding of Processes with Decimal Fractions ( F i r s t administration at beginning of experiment) Xg Decimal Fraction Computation Test X Otis Self-Administering Test of Mental A b i l i t y , Intermediate Examination, Form A X. Stanford Achievement Test, Advanced 4 Reading, Form E GENERAL MONTBOMERY SCHOOL (EXPERIMENTAL GROUP) 316 Y X l X2 X3 X4 1 11 6 16 101 19 2 13 13 8 107 24 3 11 2 11 115 23 4 12 6 11 99 16 5 15 10 14 109 14 6 14 5 13 89 17 7 21 7 13 121 34 8 16 12 20 122 26 9 13 5 18 107 19 10 18 8 19 107 14 11 10 6 19 112 20 12 18 13 22 120 19 13 10 6 10 112 21 14 12 8 22 120 31 15 8 7 12 92 19 16 14 8 19 114 24 17 13 5 16 101 15 18 11 8 14 93 7 19 8 7 10 94 13 20 13 6 16 103 11 21 17 7 21 118 15 22 14 9 12 115 14 23 26 12 21 127 19 SUMS 318 176 357 2498 434 PRINCE CHARLES SCHOOL (EXPERIMENTAL GROUP) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 18 7 16 12 17 24 26 12 13 14 11 15 15 14 6 16 20 12 15 20 9 21 20 18 13 10 17 14 12 10 10 12 8 18 24 18 10 10 7 8 8 13 10 5 3 6 8 13 4 6 6 11 14 7 8 6 7 9 8 10 5 3 3 10 10 3 4 9 6 7 15 12 21 11 20 13 13 23 24 14 21 21 19 19 18 20 19 21 24 24 19 20 16 21 15 19 14 21 21 18 22 20 15 18 17 19 23 16 104 93 109 108 102 127 117 103 114 90 112 118 102 106 89 108 121 113 106 120 100 122 127 116 103 118 94 109 98 80 104 100 103 115 127 114 21 15 17 21 15 32 29 23 27 14 19 24 8 27 15 19 28 23 23 20 17 29 34 29 18 23 10 16 19 7 21 17 13 21 35 38 SUMS 537 284 679 3892 767 HJORTH ROAD SCHOOL (CONTROL GROUP) Y X l X2 X3 X4 1 10 5 18 107 11 2 13 6 20 121 19 3 8 3 12 118 17 4 13 2 14 82 5 5 4 6 5 57 4 6 8 6 20 77 14 7 24 11 22 133 30 8 14 9 19 108 19 9 14 10 13 117 27 10 11 8 19 122 37 11 13 14 24 92 10 12 7 9 13 97 21 13 16 8 17 98 13 14 11 10: 12 82 21 15 19 14 24 121 34 16 14 6 20 114 19 17 16 10 14 112 23 18 11 2 15 102 13 19 10 2 25 121 21 20 15 11 18 107 20 21 11 5 15 118 29 22 13 5 15 122 23 23 22 10 23 125 21 24 26 16 23 130 19 25 10 3 12 73 10 SUMS 333 191 432 2656 480 SIMON CUNNINGHAM SCHOOL (CONTROL GROUP) 319 Y X l X2 h X4 1 13 10 21 123 23 2 6 5 14 108 17 3 7 7 10 108 18 4 16 6 22 95 22 5 3 5 4 86 12 6 24 15 21 134 27 7 18 11 19 111 18 8 17 12 22 129 25 9 10 7 15 102 19 10 6 9 19 113 17 11 15 8 11 108 25 12 20 12 24 128 33 13 8 3 18 109 21 14 14 9 . 23 123 20 15 21 4 16 100 13 16 22 20 20 126 33 17 13 9 21 111 13 18 9 5 14 104 15 19 12 13 12 114 35 20 21 8 25 126 31 21 6 5 17 98 12 22 12 8 17 117 17 23 14 7 20 121 14 24 13 9 16 83 11 25 9 6 12 112 16 26 14 11 17 125 31 SUMS 343 224 450) 2914 538 FLEETWOOD SCHOOL (CONTROL GROUP) Y x„ 1 2 3 4 1 9 7 19 110 14 2 10 5 17 111 16 3 17 12 20 121 26 4 15 13 13 116 24 5 10 6 6 81 13 6 7 7 16 106 19 7 10 7 12 95 15 8 7 5 4 92 9 9 16 9 18 129 27 10 7 8 5 105 14 11 18 10 15 106 19 12 11 8 9 94 19 13 17 11 20 121 32 14 17 11 17 108 23 15 9 5 6. 122 22 16 18 11 22 105 17 17 24 16 21 120 18 18 13 16 17 99 21 19 12 6 9 88 13 20 19 10 10 88 14 21 19 15 22 121 33 22 19 9 14 101 18 23 24 17 20 123 34 24 18 12 25 123 20 25 15 9 18 144 14 26 11 7 5 96 16 27 22 12 16 124 24 28 18 9 22 127 26 29 15 9 18 129 37 30 12 8 7 115 13 31 17 7 17 91 14 32 22 10 17 116 19 33 14 7 21 123 30 34 26 17" 23 ' 133 31 35 16 5 15 110 15 36 23 13 21 125 35 37 19 10 21 125 22 SUMS 576 359 . 578 4143 776 320 APPENDIX G 321 SUPPLEMENTARY STATISTICAL CALCULATIONS PAGE Calculation of the correlation of the means between the treatment groups of the criterion variable with each of the independent variables 322 Calculation of the within groups correlation between the criterion variable and each of the independent variables .< 323 Calculation of the within groups correlation, corrected for attenuation, between the criterion variable and each of the independent variables . . . 324 Calculation of the Pearson product-moment coefficient of correlation, for the experimental group, between the criterion variable (Y) and one of the independent variables (X^ 325 Method of determining the significance of the difference between two r's 325 " 322 CALCULATION OF THE CORRELATION OF THE MEANS BETWEEN THE TREATMENT GROUPS OF THE CRITERION VARIABLE WITH EACH OF THE INDEPENDENT VARIABLES ( i ) Between groups correlation of the mean of with the mean of y. r /..••..-. x = 9 . 0 3 8 5 2 x 2y (.between) /33.12368 x 2.46636 • 9.05852 { 81.69149194048 9.03852 9.03852 1.0 ( i i ) Between groups correlation of the mean of x^ with the mean of y. r x y (between) = -19.31921 ^151.32912 x 2.46636 -19.31921 N / 3 7 3 . 2320884032 - 1 9 . 3 1 9 2 1 1 9 . 3 1 9 2 1 - 1 . 0 ( i i i ) Between groups correlation of the mean of x^ with the mean of y. r x 4 y (between) = - 0 . 2 8 4 0 3 {.03271 x 2.46646 -0.28403 f. 0 8 0 6 7 4 6 3 5 6 - 0 . 2 8 4 0 3 0 . 2 8 4 0 3 -1.0 CALCULATION OF THE WITHIN GROUPS CORRELATION BETWEEN ' THE CRITERION VARIABLE AND EACH OF THE " INDEPENDENT VARIABLES 323 (i) Within groups correlation between and y. r / N = 1924.96148 x 2y vwithin; ^3700.20031 x 3427.81510 1924.96148 /12683602.4956426810 1924.96148 3561.4045 .54 ( i i ) Within groups correlation between x^ and y. r , / ... . , = 5844.65254 x y {within) • - • • — /3700.20031 x 28911.13347 5844.65254 /106976986.0281453757 .57 ( i i i ) Within groups correlation between x^ and y. r / . . \ = 2607.95070 x.y twithin) . ^3700.20031 x 7784.38906 2607.95070 {28803798.. 8129 7 26 086 2607.95070 5366.9171 .49 CALCULATION OF THE WITHIN GROUPS CORRELATION CORRECTED FOR ATTENUATION, BETWEEN THE CRITERION VARIABLE AND EACH OF THE INDEPENDENT VARIABLES 324 (i) Within groups correlation, corrected for attenuation, between and y. r = .54 J.821 x .541 = .54 J.444161 .54 .666 .81 ( i i ) Within groups correlation, corrected for attenuation, between x_ and y. 3 r = .57 J.948 x .541 .57 /.512868: .57 .716 .80 ( i i i ) Within groups correlation, corrected for attenuation, between x. and y. 4 r = .49 /.874 x .541 .49 /.472834 .49 .687 .71 CALCULATION OF THE PEARSON PRODUCT-MOMENT COEFFICIENT 325 OF CORRELATION, FOR THE EXPERIMENTAL!. GROUP, BETWEEN THE CRITERION VARIABLE (Y) AND ONE OF THE INDEPENDENT VARIABLES (X^ {<%) C8*) ?X Y = ^ - (7.7966)(14.4915 (3.0016) (4.5375) METHOD OF DETERMINING THE SIGNIFICANCE OF THE DIFFERENCE BETWEEN TWO r's The r between X^ and Y in the experimental group is .51137; the r between X^ and Y in the control group is .67816. Is the relationship between X^ and Y significantly higher in the control group than in the experimental group? Pearson r Fisher* s z .51137 .56 .67816 .83 Standard Error of the difference between 2 coefficients •A ~ z2 ~ V -N - 3 N - 3 { 59-3 88-3 /. 01786 +' .01176 /.02962 .172 Critical Ratio = _1 2 Z l " Z2 .83 - .56 .172 1.57. This CR of 1.57 i s below the .05 level of 1.96;(Table of t for use in determining the r e l i a b i l i t y of statistics). It may be concluded, therefore, that the relationship between and Y is not significantly higher in the control group than in the experimental group.
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An experimental study to determine the effectiveness of group instruction use of certain manipulative… Greenaway, George James 1958
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Title | An experimental study to determine the effectiveness of group instruction use of certain manipulative materials in contributing to an understanding of decimal concepts. |
Creator |
Greenaway, George James |
Publisher | University of British Columbia |
Date Issued | 1958 |
Description | The increasing emphasis on teaching arithmetic meaningfully intensifies the search for materials of instruction which can effectively communicate arithmetical understandings to children. Though manipulative aids are widely approved as effective teaching media for achieving this purpose, most of the endorsements are subjective opinions rather than objective evaluations based on experimentation. This study is an attempt to determine the effectiveness of group instruction use of certain manipulative aids in teaching decimal fraction concepts to Grade VII pupils. The effectiveness was determined by comparing the achievement of two unselected groups, randomly assigned, on a test of understanding of the processes involved in decimal fractions. The two groups were given teaching treatments identical except in so far as the materials of instruction were concerned. One group used manipulative aids; the other used static representations of these aids. These materials were intended to differ only with respects to the characteristic of manipulability. Since manipulability of concepts is the most essential property of manipulative aids, it was isolated as the experimental variable. Because the groups were randomly assigned, analysis of covariance was selected to control statistically the initial differences between groups in the four variables considered likely to influence achievement on the criterion test: initial understanding of the processes involved in decimal fractions, computational ability in decimal fractions, mental ability, and reading ability. The data obtained from the investigation were analyzed and the following conclusions reached. 1. The pupils taught by means of group instruction with the manipulative aids used in this investigation did not acquire a significantly better understanding of decimal fractions than did the pupils taught with static representations of these aids. In other words, the manipulation of the concepts, performed by using the manipulative aids in group demonstrations, was not effective in contributing to the pupils’ understanding of these concepts. 2. A study of the correlations for both treatment groups between achievement on the criterion variable and achievement on each of the independent variables indicates that the manipulative aids proved to be neither more nor less effective than the static representations as media for conveying an understanding of decimal fractions to pupils of any particular ability in the areas represented by the independent variables. 3. It must not be inferred that any generalization concerning the effectiveness of these specific materials of instruction, used exclusively by the teacher for group demonstration purposes, would be applicable also to similar materials if they were used in a teaching procedure in which the pupils themselves, participated individually in the manipulative activity. It must not be inferred that any generalization concerning the effectiveness of these specific materials of instruction, which were used in a brief teaching assignment devoted exclusively to the rationalization of processes, would be applicable also to the same materials if they were used in a teaching assignment of longer duration, and/or a teaching assignment in which the emphasis on the WHY of the processes was taught concurrently with, or preceded, the emphasis on the HOW of the processes. 5. Independently of treatment groups, the achievement on the initial test of understanding of the processes involved in decimal fractions was the variable most predictive of achievement on the final test of understanding. Computational ability in decimal fractions and mental ability each shared approximately one-half the predictive capacity of the initial test of understanding. Reading ability was a negligible predictor of achievement on the final test of understanding. |
Subject |
Decimal system -- Study and teaching |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0106092 |
URI | http://hdl.handle.net/2429/40029 |
Degree |
Master of Arts - MA |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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