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 B r i t i s h Columbia, 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Education W-e accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1958 il 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 e f f e c t i v e l y 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 i s an attempt to determine the effectiveness of group instruction use of certain manipulative aids i n 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 i n decimal fractions. The two groups were given teaching treatments identical except i n so far as the materials of instruction were concerned. group used manipulative aids; these aids. One the other used static representations of These materials were intended to d i f f e r only with respects to the characteristic of manipulability. Since manipulability of concepts i s the most essential property of manipulative aids, i t was iii isolated as the experimental variable. Because the groups were randomly assigned, analysis of covariance was selected to control statistically the i n i t i a l differences between groups in the four variables considered likely to influence achievement on the criterion test: 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. 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. 1 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 i n a teaching procedure i n which the pupils themselves, participated individually i n the manipulative a c t i v i t y . It must not be inferred that any generalization concerning the effectiveness of these specific materials of instruction, which were used i n a.brief teaching assignment devoted exclusively to the rationalization of processes, would be applicable also to the same materials i f they were used i n a teaching assignment of longer duration, and/or a teaching assignment i n 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 i n decimal fractions was the variable most predictive of achievement on the f i n a l test of understanding. Computational a b i l i t y i n decimal fractions and mental a b i l i t y each shared approximately one-half the predictive capacity of the i n i t i a l test of understanding. negligible predictor of achievement Reading a b i l i t y was a on the f i n a l test of understanding. In presenting this the requirements f o r of B r i t i s h Columbia, it freely agree t h a t for Department the the for representative. of t h i s by t h e It is I further thesis of Columbia thesis Head o f my understood for be a l l o w e d w i t h o u t my w r i t t e n The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8. C a n a d a . University extensive copying of t h i s may be g r a n t e d of L i b r a r y s h a l l make f o r r e f e r e n c e and s t u d y . copying or p u b l i c a t i o n g a i n s h a l l not fulfilment degree at I agree t h a t purposes o r by h i s in partial an advanced permission scholarly Department that available thesis financial permission. V TABLE OF CONTENTS CHAPTER I. PAGE 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 J u s t i f i c a t i o n 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 Organization of the remainder of the thesis 12 ... Footnotes •.ill. 14 18 REVIEW OF RELATED LITERATURE 20 Changing concepts i n arithmetic psychology . . . . 20 Reported research involving the use of manipulative materials i n the teaching of arithmetic Footnotes 25 . . . 30 vi PAGE CHAPTER III. THE PLAN AND ADMINISTRATION OF THE EXPERIMENT ... 32 Steps i n planning the experiment 32 Description 38 of the instructional materials . . . . 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 i n 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 IV. 55 Reasons for imposing the limitations on the study. 59 Summary 61 Footnotes 63 THE STATISTICAL DESIGN OF THE EXPERIMENT AND DESCRIPTION OF THE MEASURES USED 65 S t a t i s t i c a l design of the experiment 65 The general nature and purpose of the s t a t i s t i c a l method 65 viiCHAPTER PAGE Statement of the hypothesis 68 Description of the Farquhar test of understanding 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 f i n a l administration . . .. . . . . Concluding comments about the Farquhar test . . 82 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. A b i l i t y . . , 100 Description of the Stanford Achievement Test (Advanced Reading Test: Form E) 102 Conclusion 103 Footnotes 106 :yiii CHAPTER V. PAGE 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 c r i t e r i o n variable Y 123 Computation of the sums of cross products i n deviation form for each pair of variables . . . 126 An examination of the conditions under which an analysis of covariance w i l 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 Summary of correlation conditions 135 . . . . . . . Calculation of the coefficients of regression 139 . . 140 Calculation of the coefficients of the total regression equation 142 Calculation of the coefficients of the within groups regression equation Calculation of the sums of squares of residuals 142 . 144 Calculation of F value and application of test of significance 147 ix CHAPTER PAGE Assumptions underlying the derivation of analysis of covariance 149 S t a t i s t i c a l test of homogeneity of regression . . 154 Test of the null hypothesis that /3 E 1E Y X = ^ V l C 1 5 6 Test of the null hypothesis that £ E 4E Y X frchc = 1 6 2 The use of a binary computer (Alwac III-E) i n performing automatic covariance computations . . Footnotes VI. SUMMARY AND CONCLUSIONS 167 169 ..." 172 Summary 172 Purpose of the experiment Background and j u s t i f i c a t i o n . 172 . 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 f o r further study BIBLIOGRAPHY 180 182 X CHAPTER PAGE APPENDICES A. Communication of administrative arrangements to the teachers participating i n the study . B. The lessons :G. The pupils' worksheets D. 186 199 284 Tests used to measure the c r i t e r i o n variable and the four independent variables E. 186 296 Samples of the procedures used to determine the s u i t a b i l i t y of the tests for the study . 307 F. Raw score data 315 G. Supplementary s t a t i s t i c a l calculations . . . . 321 vsi. LIST OF TABLES TABLE I. . .. Plan of Lesson Topics with the Numbers of the Farquhar Questions Appropriate II. PAGE to Each Topic . . 34 Names of Instructional Materials Used byExperimental and Control Groups, and the Time Allowed, for Teaching the Objectives of each Lesson III. 36 Number of Pupils i n the Classes Assigned to Each Treatment Group IV. V. Tests Selected to Measure the Variables . 51 . . . . . 69 Frequency of Scores i n the T r i a l Administration of Farquhar's Test to Forty Grade VII Pupils i n White Rock Elementary School VI. 72 Summary of S t a t i s t i c a l Detail Resulting from T r i a l Administration of Farquhar's Test to Forty Grade VII Pupils i n White Rock Elementary School VII. 72 The D i f f i c u l t i e s and V a l i d i t i e s of Items Resulting from T r i a l Administration of Farquhar's Test to Forty Grade VII Pupils i n White Rock Elementary School VIII. 73 Frequency of Items at the Various Per Cent Levels of D i f f i c u l t y Resulting from T r i a l Administration of Farquhar's Test to Forty Grade VII Pupils i n White Rock Elementary School 75 xii TABLE IX. PAGE Frequency of Items at the Various Per Cent Levels of V a l i d i t y Resulting from T r i a l Administration of Farquhar s Test to Forty Grade VII Pupils i n 1 White Rock Elementary School 75 X. Frequency of Items i n the Various V a l i d i t y Coefficient Ranges Resulting from T r i a l Administration of Farquhar's Test to Forty Grade VII Pupils i n White Rock Elementary School . . . XI. Frequency of Scores i n Farquhar s Test Test 1 Administered at the Close of the Experiment to the 147 Participating Subjects . . . . . . . . . XII. at the Close of the Experiment to the 147 Participating Subjects 84 The D i f f i c u l t i e s and V a l i d i t i e s of Items i n Farquhar's Test Administered at the Close of the Experiment to the 147 Participating Subjects XIV. 84 Summary of S t a t i s t i c a l Detail i n Farquhar's Test Administered XIII. 76 85 Frequency of Items at the Various Per Cent Levels of D i f f i c u l t y Resulting from Administration of Farquhar s Test at the Close of the Experiment 1 to the 147 Participating Subjects XV. 87 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 xiii TABLE XVI. PAGE Frequency of Items i n the Various V a l i d i t y Coefficient Ranges Resultings from Administration of Farquhar* s Test at the Close of the Experiment to the 147 Participating Subjects XVII. 88 Comparative Data Obtained from the Administration of Farquhar s Test to the T r i a l Group and to 1 the 147 Participating Subjects XVIII. 89 Frequency of Scores i n the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects . . XIX. 93 Summary of S t a t i s t i c a l Detail i n the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects XX. 93 The D i f f i c u l t i e s and V a l i d i t i e s of Items i n the Decimal Computation Test Administered at the Beginning of the Experiment to the 147 Participating Subjects XXI. 94 Frequency of Items at the Various Per Cent Levels of D i f f i c u l t y Resulting from Administration of • the. Decimal Computation Test at the Beginning of the Experiment to the 147 Participating Subjects. 96 xiv PAGE TABLE XXII. Frequency of Items at the Various Per Cent Levels of V a l i d i t y Resulting from Administration of the Decimal Computation Test at the Beginning of the Experiment to the 147 Participating Subjects XXIII. 96 Frequency of Items i n the Various V a l i d i t y 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 i n Stanford Reading Test Administered at the Beginning of the Experiment to the 147 Participating Subjects XXV. 104 Summary of S t a t i s t i c a l Detail i n Stanford Reading Test Administered at the Beginning of the Experiment to the 147 Participating Subjects . . XXVI. 104 Means and Standard Deviations Obtained by Each Treatment Group i n the Criterion Variable and the Four Control Variables XXVII. 110 Sums of Scores i n the Five Variables, and Sums of Squares of Scores, Arranged by Classes, for Each Treatment Group and for the Total Sample XXVIII. 118 Sums of squares of Scores i n the Five Variables i n Deviation form, for the Total Sample, and for V/ithin, and Between, the Treatment Groups . . . 119 XV. TABLE XXIX. PAGE Analysis of Variance of Performance of the Two Treatment Groups on the Independent Variable X XXX. 120 Analysis of Variance of Performance of the Two Treatment Group's on the' Independent Variable X XXXI. 120 g Analysis of Variance of Performance of the Two Treatment Groups on the Independent Variable X, XXXII. 121 Analysis of Variance o f Performance' o"f the Two Treatment Groups on the Independent Variable X. 4 XXXIII. 121 Analysis of Variance of Performance of the Two Treatment Groups on the Criterion Variable Y XXXIV. . 124 Sums of Cross Products of Scores i n the Five Variables, Arranged by Classes, for each Treatment Group and for the Total Sample XXXV. 130 Sums of Cross Products of Scores i n the Five Variables, i n Deviation Form, for the Total Sample, and for V/ithin, and Between, the Treatment Groups XXXVI. . 132 Summary of R e l i a b i l i t i e s of Tests Employed to Provide Measures of the Criterion Variable and the Four Independent Variables 137 xvi TABLE XXXVII. PAGE Pearson Product-Moment Coefficient of Correlation and I n t r i n s i c Correlation, Within Groups of the Criterion Variable with Each of the Independent Variables XXXVIII. 137 Regression Coefficients of the Total Regression Equation and the Within Groups Regression Equation XXXIX. XL. 143 Summary of Sums of Squares of Residuals . . . . . Analysis of Covariance of Performance of the Two Treatment Groups on the C r i t e r i o n Variable Y XLI. 146 . 148 Within Group Variance Accounted for by the Use of Each of FoUr Independent Variables: X, X, X, X 1 XLII. 2 3 155 4 Sums of Squares and Cross Products of Variables x^ and y, calculated independently for Experimental and Control Groups XLIII. 160 Sums of Squares and Cross Products of Variables x^ and y, calculated independently for Experimental and Control Groups XLIV. 163 Pearson Product-Moment Coefficients of Correlation between achievement on the c r i t e r i o n test and achievement on each of the independent variables arranged according to treatment groups . . . . 176 xvii; 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 xviii ACKNOWLEDGMENTS The author wishes to express his indebtedness to various persons who have contributed i n important ways to the development of this study. Dr. H. L. Stein of the Faculty of Education, University of B r i t i s h Columbia, under whose direction this investigation proceeded, was always helpful i n his assistance at points of d i f f i c u l t y and generous i n his encouragement and guidance. Mr. K. B. Woodward, Municipal Inspector, School D i s t r i c t No. 36 (Surrey), provided the administrative arrangements for conducting the experiment i n the schools. His interest was a stimulus which helped to bring the work to a conclusion. Messrs. R. Beale, P. Carey, R. C r i s a f i o , B. Dacke, and G. Falk, the teachers i n charge of the classes, were enthusiastic i n their participation and meticulous i n their teaching of the prescribed lessons. F i n a l l y , Dr. T. Hull of the Department of Mathematics, University of B r i t i s h Columbia, provided the f a 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 i n 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 i n contributing to an understanding of particular decimal concepts. Stated i n other words, the purpose i s to ascertain whether there i s any significant difference i n the achievement on a c r i t e r i o n 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 i n a prescribed manner. The primary concern of this investigation i s with the p r a c t i c a l i t y , not the essentiality, of the meaningful approach to teaching arithmetic. No matter how v a l i d the Meaning Theory may be, i t s worth as a trend i n 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 i s 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 f a c i l i t a t e the communication of arithmetical meanings to children. The interest i n teaching materials^it may be emphasized, i s 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 i n contributing to the pupils' understanding of specific decimal fraction concepts when the mani- pulative materials are used exclusively by the teacher i n class demonstrations. The manipulative materials are ( l ) place value charts, movable indicator, designated: (2) place value cards, ( 3 ) wall rule with ( 4 ) flannel board. The effectiveness of these materials i s determined by comparing the achievement on a selected c r i t e r i o n measure of an experimental group composed of 59 subjects i n two classes located i n different schools, and a control group composed of 88 subjects 3 in three classes also located i n different schools. The c r i t e r i o n measure, administered at the end of the experiment, i s Farquhar's Test of the Understanding of Processes with Decimal Fractions, which formed a minor part of an unpublished Master of Arts Thesis i n Education. yet 1 Since no way has been devised to identify the composition of a body of under- standings i n arithmetic, Farquhar assumed that "a person's understanding of a process may be revealed by his a b i l i t y 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 i n the test were, according to the author, a r b i t r a r i l y chosen. The participating classes were selected i n 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 areas: to determine the status of the classes i n four relevant 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 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 i n these areas were held constant. Both groups were subjected to teaching treatments intended to be i d e n t i c a l i n a l l details except insofar as the instructional materials are concerned. The experimental group was taught with 4 specified manipulative aids; static representations the control group was taught with of these aids commonly referred to as v i s u a l - ization materials. Further, these two types of instructional materials intended to d i f f e r only in their capacity to represent movable forms. are concepts i n Manipulability, the basic characteristic of mani- pulative aids, constitutes the experimental variable. The extent to which the intended identity i n teaching methods and materials used by the two treatment groups actually exists may be judged by examining the lessons contained representations i n Appendix B, and the of the materials shown i n the Figures on pages 39 to 42, i n c l u s i v e . The teaching programme was designed to impose r i g i d controls i n the conduct of the experiment, while the analysis of covariance technique was selected to impose s t a t i s t i c a l 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 i n achievement on the criterion measure to be attributed to the experimental variable. The hypothesis to be tested i s that there i s no difference i n the performance of the two groups on the c r i t e r i o n measure which i s attributable to the treatments involved. 5 :III. DELIMITATION OF THE PROBLEM Date and locale of the experiment. the experiment took place i n May, The administration of 1957, and involved Grade VII classes located i n five elementary schools i n School D i s t r i c t No. 36 (Surrey). Instructional materials. Subsection II of Chapter III contains a description of a l l the teaching aids employed i n 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 f a c i l i t a t e d by referring to Table II on page 36 . shows the corresponding aids employed by the two groups for the teaching of the particular objectives i n each lesson. of the identity i n the corresponding may This table The extent aids used for each objective 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 i n the Farquhar test. Teaching programme. Subsection III of Chapter III contains a description of the lessons employed i n this study. Eleven lessons comprise the t o t a l teaching programme. these, eight were devoted to the presentation of material new the experiment, while three were reserved to review material Of to 6 previously taught during the experiment. The schedule of lessons i s contained i n Appendix A. The duration of each lesson i s one hour. This includes thirty minutes of group instruction, f i f t e e n 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 i n 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 b r i e f 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 areas previously referred to. variables i n the four The results obtained were, obviously, unaffected by the treatments. The Farquhar test, one of the four, performed a dual function i n the study. I t was administered at the beginning of the experiment to measure one of the independent variables, and, i n addition, i t was readministered at the close of the experiment to measure the dependent or c r i t e r i o n variable. 7 .:iV. JUSTIFICATION OF THE PROBLEM The d e s i r a b i l i t y of teaching arithmetic according to the Meaning Theory i s now widely accepted i n 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 i n an a r t i c l e written ten years ago: The problem (or group of problems) epitomized i n 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 f o r developing,-those meaningful thought processes. But we must find out. The problem of finding ways and means f o r developing meaningful thought processes i n arithmetic i s 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 i n educational psychology has been the increased importance placed on the use of teaching aids supplementary to the text. Among these aids, manipulative materials rank high i n the approval of those who espouse the Meaning Theory. In an a r t i c l e written i n 1950, Busv/ell interpreted the Meaning Theory to include the judicious use of manipulative devices. he stated: In i t 8 We are only beginning to realize the important place that manipulative aids can play i n 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 a r t i c l e 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 i n the mathematics classroom i n the same role as the demonstration models do i n the science room - a constant source for the d i s 7 covery of new laws and for the confirmation of hunches.' In an a r t i c l e written i n 1953 on the subject "How to Make Arithmetic Meaningful i n the Junior High School", Stein commented: ...arithmetic can be made meaningful i n 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. ... I t i s 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 i n a book. Junior High School teachers should not consider i t beneath their dignity to u t i l i z e concrete materials to develop abstract processes.^ Current educational periodicals show that a wide variety of manipulative aids are being used i n the classrooms for the teaching 9 of arithmetic. Judging by the reports contained i n these period- i c a l s , i t appears that these aids are being used mainly to develop i n pupils an understanding of arithmetical processes, although i n some cases they are being used merely for computation purposes. The j u s t i f i c a t i o n of this study l i e s i n the fact that, while manipulative aids are becoming increasingly prominent i n arithmetic 9 teaching, and obviously for the purpose of giving i n s i g h t 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 i n s t r u c t i o n are as e f f e c t i v e f o r t h i s purpose as they are claimed to be. The need f o r research i n t h i s area i s a l l the more important because of certain opinions which suggest that some manipulative aids are used i n j u d i c i o u s l y . 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 i n s t r u c t i o n 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 i n t e n d e d . ^ Previous studies which have i n any way involved the use of manipulative aids have not been p r i m a r i l y concerned with investi g a t i n g t h e i r effectiveness i n contributing to teaching objectives. Instead, the purpose has been to investigate c e r t a i n aspects of teaching, such as the computational and problem s o l v i n g effects of teaching v/ith varying degrees of emphasis on meaning, i n which manipulative materials have been included only i n 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 h e l p f u l i n evaluating the effectiveness of these aids i n any phase of a r i t h m e t i c i n s t r u c t i o n . The present study, which i s designed exclusively to determine the effectiveness of certain manipulative aids when used i n a prescribed manner and f o r a d e f i n i t e 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 V. are meanings best developed?" FACTORS DETERMINING THE CHOICE OF SUBJECT MATTER One factor which determined the choice of decimal fractions was the a v a i l a b i l i t y of a test designed to measure meanings and understandings. The d i f f i c u l t y of evaluating the development of understandings, which i s admitted i n the l i t e r a t u r e , i s reflected in the scarcity of suitable tests to meaare this type of learning. The Farquhar test i s considered by the present investigator to . be the most suitable of the tests which purport to measure understanding i n arithmetic. This consideration, therefore, was an important factor determining the choice of subject matter. Apart from the influence which the s u i t a b i l i t y of Farquhar's test had upon the choice of decimal fractions as the material for study i n the present investigation, there was one other consideration which reinforced the wisdom of this choice. Decimal fractions i s a teaching topic which offers many opportunities for the effective use of both manipulative and visual materials to c l a r i f y and extend meanings. References contained i n "Teachers' Guide for Thinking with Numbers" indicate pages i n the text where manipulative and visual materials may be used."'""'' In comparison to other topics of instruction i n 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 i n this investigation would be subjected to a f a i r test. VI,. FACTORS DETERMINING THE GRADE PLACEMENT OF THE EXPERIMENT The instructional programme i n 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 l e v e l at which a study of this nature should be conducted i s 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 i n the elementary grades, often before pupils are mature enough to understand their significance. One such opinion i s that expressed by Morton.who, after referring to the d e s i r a b i l i t y of reteaching arithmetical concepts i n 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. I t means teaching again, at a higher and more mature l e v e l , and more rapidly, what has been taught b e f o r e . ^ A similar opinion i s held by Stein, who v/rites the following i n an a r t i c l e which emphasizes the d e s i r a b i l i t y of introducing a deliberate effort to teach arithmetic meaningfully i n 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) c l a r i f y i n g 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 i s the prevailing opinion that Grade VII i s 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 i n another work: "In general, i t should not be necessary to reteach decimals i n Grade VIII". On this opinion there i s general agreement to the extent of saying that Grade VII i s the last grade at which i t should be necessary to reteach the entire f i e l d of decimal fractions to a l l . t h e pupils. To f a c i l i t a t e further the effectiveness of the lessons, the experiment was conducted as late as possible i n 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 i n arithmetic instruction, the limitations of the study must be borne i n 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 i n s i s t that their maximum effectiveness i n contributing to the pupils' understanding of the concepts taught would depend upon involving the pupils individually i n 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 i n conveying the meaning of a concept when the learner's a c t i v i t y i s confined to observing another person perform the manipulation. A review and discussion of this l i t e r a t u r e i s contained i n Chapter III. Because of the importance of this point of view, i t would be a f a l l a c y to formulate from the evidence presented i n this study any generalization concerning the effectiveness of these particular aids i n situations where the pupils individually participated i n the manipulation. 2. The manipulative materials were used i n 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 l i m i t a t i o n , i t would be a fallacy to formulate from the evidence presented i n this study any generalization concerning the effectiveness of these manipulative aids i n situations v/here the period of instruction i s 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 l i m i t a t i o n assumes some importance i n view of the so-called HOW-WHY versus WHY-HOW controversy. This involves the question of whether the teaching of HOW the algorism i s performed 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 i n 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 i n teaching the algorism. Further reference to this controversy i s found i n Chapter I I I , 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 i s composed of two subsections. The f i r s t deals with the trends i n arithmetic pedagogy leading to the present 15 popularity of the Meaning Theory. The search for materials of instruction which e f f e c t i v e l y communicate arithmetic meanings i s obviously a worthwhile pursuit only i f the need to make arithmetic meaningful i s 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 i s 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 i n 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 i n the performance of the two groups on the c r i t e r i o n measure may be attributed to this variable. The purpose of Chapter IV i s to extend the objective of the previous chapter i n order to provide assurance that any difference i n the performance of the two groups on the c r i t e r i o n 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 i n 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 s t a t i s t i c a l l y . 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 s t a t i s t i c a l analysis. Chapter V i s devoted to this s t a t i s t i c a l analysis of the data.. Several steps are involved i n the entire analysis. steps may be grouped into major categories. These The f i r 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 w i l 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, i n which the F value i s obtained and the f i n a l test of significance applied. The fourth category of steps involved i n the entire analysis includes ascertaining that the assumptions underlying satisfied. the application of the analysis of covariance have been 17 F i n a l l y , the last subsection of Chapter V contains a b r i e f account of the manner i n which the raw data obtained i n this experiment was processed through the electronic binary computer to result i n 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 s t a t i s t i c a l evidence, and some suggestions for further study. 18 FOOTNOTES • Hugh Ernest Farquhar, "A Study of the Relationship between the A b i l i t y to Compute with Decimal Fractions and an Understanding of the Basic Processes Involved i n the Use of Decimal Fractions", Unpublished Master of Arts Thesis i n Education, University of B r i t i s h 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 i n Arithmetic", Phi Delta Kappan, Vol. 31, 1950, p. 232 Nathan Lazar, "A Few Recommendations for the Improvement of the Teaching of Mathematics i n the Secondary School", Progressive Education, Vol. 29, 1952, p. 21 Harry L. Stein, "How to Make Arithmetic Meaningful i n 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: S i l v e r Burdett Company, 1953), p. 349. 20 CHAPTER II REVIEW OF RELATED LITERATURE Two purposes underlie the review of the l i t e r a t u r e relevant to this study. The f i r s t purpose i s to show the background of the problem by tracing the developments i n arithmetic psychology which have led to the present emphasis on meaning and understanding i n the teaching of arithmetic. The second purpose i s to reveal the exact research which has already been performed i n connection with the use of manipulative materials i n the teaching of this subject. I. CHANGING CONCEPTS IN ARITHMETIC PSYCHOLOGY The present experiment i s an effort to determine the effectiveness of group instruction use of certain manipulative aids i n contributing to an understanding of decimal fraction concepts. Confined to specific manipulative aids used i n specified situations, this study deals with only a small portion of one avenue i n the search for practical materials which w i l l be helpful to teachers i n making arithmetic meaningful to their pupils. To be f r u i t f u l this search i n general, as well as a l l parts of i t , must be motivated by a knowledge of the trends i n arithmetic psychology leading to the present emphasis on meaning, 21 and also an awareness of why each successive stage has yielded to another i n the process of this development. This section of the review deals b r i e f l y with these two matters. Compendiums of opinion found i n various yearbooks and other publications during the past quarter of a century reflect the changing concepts i n arithmetic psychology. During this period three stages are evident, though they are not necessarily consecutive. One stage i s that marked by the popularity i n 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 i s to present number stimuli repetitively to the pupils i n order to f a c i l i t a t e their a b i l i t y to make correct responses. considered to be the prime factor i n teaching; D r i l l was and the accumulation of a repertoire of specific responses was believed to be the major end of learning. There i s no question that this was the prevailing psychology for many years prior to 1930. A second stage i s that marked by the popularity of the Incidental Learning Theory i n the psychology, i f not i n the practice, of teaching arithmetic. In the 1935 Yearbook of the National Council of Teachers of Mathematics there i s 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 i t , was necessary to forget d r i l l and to concentrate instead on projects designed to simulate functional situations. expressed the idea thus: Wheeler "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 i s 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 D r i l l 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 i s older and is better able to grasp the meaning.^ 23 His c r i t i c i s m of the Incidental Learning Theory i s e x p l i c i t i n these words: Incidental learning, whether through "units" or through unrestricted experience, i s slow and time consuming. ... Such arithmetic a b i l i t y as may be developed i n these circumstances i s apt to be fragmentary, s u p e r f i c i a l , and mechanical Brownell's own position i s between these two extremes. The Meaning Theory stands i n marked contrast to the theory which placed such reliance on the d r i l l of isolated number f a c t s . At the same time i t stands opposed to the f e a s i b i l i t y of giving up a l l organized learning experience i n arithmetic because a prevalent method had been formalistic and mechanical. In his own words Brownell explains that position: This name (the Meaning Theory) i s selected for the reason that, more than any other, this theory makes meaning the fact that children shall see sense i n what they learn the central issue i n 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 i n 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 i s unmistakable evidence that the Meaning Theory was growing i n favor. A chapter by T. R. McConnell reaffirmed the place of organization i n learning 9 and the concept that learning arithmetic i s 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 r e l a t i o n .. 10 ships, of u t i l i z i n g concrete experiences, and of generalization". A f i n a l authorative statement by Dawson and Ruddell i n 1955 brings the development up to date: Evidence supporting the meaning theory approach to arithmetic i s not complete but i t i s impressive when i t i s noted that no such evidence i s being accumulated to support other theories of i n s t r u c t i o n . ^ Therefore, the place has been reached where the primary question no longer i s : "Should we teach meanings?". issues now, certainly from the standpoint The important of research, are suggested by the questions: "What constitutes the basic arithmetic understandings?" and "What are the most effective materials for, and methods of, instructing pupils i n these understandings?". The fact that the issues suggested by these two questions are not new i s indicated by Brownell i n an a r t i c l e written almost twenty years after his f i r s t presentation of the Meaning Theory: It i s not too much to say that one of the major developments i n 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 i s revealed i n the search for more effective learning materials and methods of i n s t r u c t i o n . ^ 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, i s the circumstance 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 i n arithmetic understandings?" Since this experiment i s an effort to determine the effectiveness of a specific group of these materials, namely, those which are manipulative i n character, the next section of this review contains an account of the reported research which i n any way involves the use of manipulative materials i n the teaching of a r i t h 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 i n Arithmetic". Foster E. Grossnickle s contribution to this chapter contains a proposal for 1 research dealing with the use of manipulative materials i n the teaching of arithmetic. that time to be of urgent Research i n this area::was represented at 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 textbook practices i n the introduction of the division of whole numbers with an experimental procedure based on a subtractive approach and a greatly expanded use of v i s u a l i z a t i o n devices. One of the several questions to which answers was sought was: W i l l achievement be affected adversely i f practice through object manipulation and visualization of process replaces much of the paper and pencil d r i l l ? ^ 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 u t i l i z e d manipulation of representative materials. Higher achievement, greater retention, and an increased a b i l i t y to solve examples i n a new situation were found i n 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 divisio n facts to third grade children. The three teaching methods were named the textbook, the conventional, and the developmental. Each method was designed to vary i n 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 t e s t , which was used as a pre-test, an immediate r e c a l l t e s t , and a delayed r e c a l l t e s t , were analyzed to compare the effects of the teaching methods used i n the 8 f o r t y minute lessons which comprised the i n s t r u c t i o n a l programme. P r o v i s i o n 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 i n t e r e s t i n the present study may be summarized as f o l l o w s . First, there are no s i g n i f i c a n t differences among the three teaching method groups i n the immediate r e c a l l of taught f a c t s . Second, i n the delayed r e c a l l of taught facts there i s a d i f f e r e n c e , s i g n i f i c a n t 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 e f f e c t i v e than the textbook method i n the delayed r e c a l l of taught f a c t s , there i s no s i g n i f i c a n t difference between the effectiveness.of the f i r s t two mentioned methods. On the basis of a l l the conclusions reached, t h i s f i n a l statement i s reported: This finding implies that conventional procedures may be more e f f e c t i v e i n ordinary classroom situations when teachers are given s p e c i f i c d i r e c t i o n s and when pupils are motivated to l e a r n . Developmental procedures p r o f i t a b l y may be used p a r t 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 c h i l d r e n . However, research i s needed to refine the p r i n c i p l e s and teaching procedures which are outgrowths of the meaning theory.19 28 Several other studies reported i n the l i t e r a t u r e involve the use of manipulative materials i n one way or another. Except i n minor details, such as grade l e v e l , duration of the experiment, number of treatment groups, and so forth, these studies are similar to the ones v/hich have just been discussed. I t must be noted that this area of experimentation i s not entirely relevant to the present study. Important differences exist. The experimental purpose i s the f i r s t way i n which the two studies described d i f f e r from the present one. These two studies, which parallel each other very closely, were designed for the purpose of testing the role of meaning i n teaching to attain certain objectives. This was done by comparing a meaningful method with a control method either i n which recognizably conventional procedures were followed or i n 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 i n teaching to attain certain objectives. The teaching objectives i s the second way i n which the two studies described differ from the present one. of The objective the teaching i n the former i s the pupils' attainment of s k i l l i n 29 certain a r i t h m e t i c a l operations, while the objective of the teaching i n the present study i s the pupils' attainment of an understanding of certain a r i t h m e t i c a l concepts. I t i s seen, therefore, that there are important differences betv/een t h i s 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 i n v e s t i g a t i o n of the computational and problem solving e f f e c t s of teaching with varying degrees of emphasis on meaning. Manipulative materials have been included quite i n c i d e n t a l l y i n the teaching aids used i n these experiments, and, as a consequence, the problem of determining t h e i r effectiveness i n contributing to the teaching objectives has been i n each case a very subordinate part of the t o t a l i n v e s t i g a t i o n . The growing acceptance of the Meaning Theory should produce i n the future more experimentation designed exclusively to ascertain the effectivness of teaching materials, e s p e c i a l l y manipulative aids, i n conveying to pupils an understanding of arithmetical concepts. 30 FOOTNOTES F. B. Knight, "Some Aspects of Modern Thought on A r i t h m e t i c " , The Teaching of A r i t h m e t i c , Twenty-ninth Yearbook of the National S o c i e t y f o r the Study of Education, Part I , (Bloomington, I l l i n o i s : P u b l i c School P u b l i s h i n g Company-,-, 1930), pp. 145-267. R. H. Wheeler, "The New Psychology of Learning", The Teaching of A r i t h m e t i c , Tenth Yearbook of the N a t i o n a l Council of Teachers of Mathematics, 1935, pp. 230-263. I b i d , p. 247 I b i d , p. 243 W i l l i a m A. Brownell, " P s y c h o l o g i c a l Considerations i n the Learning and Teaching of A r i t h m e t i c " , The Teaching of A r i t h m e t i c , Tenth Yearbook of the N a t i o n a l Council of Teachers of Mathematics, 1935, pp. 1-31. 6 7 8 9 I b i d , p. 2 I b i d , p. 17 I b i d , p. 19 T. R. McConnell, "Recent Trends i n Learning Theory: T h e i r A p p l i c a t i o n to the Psychology of A r i t h m e t i c " , A r i t h m e t i c i n General Education, Sixteenth Yearbook of the N a t i o n a l C o u n c i l of Teachers of Mathematics, 1941, pp. 268-289. ^ ' E r n e s t Horn, " A r i t h m e t i c i n the Elementary School Curriculum", The Teaching of A r i t h m e t i c , F i f t i e t h Yearbook of the N a t i o n a l Society f o r the Study of Education, Part I I , (Chicago, I l l i n o i s : The U n i v e r s i t y of Chicago Press, 1 9 5 l ) , p. 215. 11' Dan T. Dawson and Arden K. Ruddell, "The Case f o r the Meaning Theory i n Teaching A r i t h m e t i c " , Elementary School J o u r n a l , V o l . 55, 1955, p. 394 31 12 William A. Brbwnell, "The Revolution i n Arithmetic", The Arithmetic Teacher, Vol. 1, 1954, p. 4. 13 G-. T. Buswell, "Proposals for Research on Problems of Teaching and of Learning i n Arithmetic", 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 , 1951 p. 285. 14 Dan T. Dawson and Arden K. Ruddell, "An Experimental Approach to the Division Idea", The Arithmetic Teacher, Vol. 2, 1955, pp. 6-9. 15 Ibid, p. 6 16 17 18 Ibid Ibid, p. 8 Dissertation Abstracts, Vol. XV, (Ann Arbor, Michigan: University Microfilms, 1955), p. 2134. 19 * Ibid 32 CHAPTER I I I THE PLAN AND ADMINISTRATION OF THE EXPERIMENT This chapter contains an account of the planning involved i n organizing the experiment, a description of the administrative arrangements undertaken i n connection with i t s performance, and, f i n a l l y , 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 i n planning the experiment was to select a suitable teaching unit. I t v/as considered essential to confine the subject matter to one homogeneous unit, and thereby to r e s t r i c t the duration of the teaching assignment, i n order to maintain adequate controls i n 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 manipulative aids used i n this study. The second step i n 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 i n 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 , i n accordance with the existence of Farquhar questions to evaluate each lesson, and, second, i n accordance with the teaching practices currently recommended i n the published materials which were consulted i n preparation for writing the lessons. These published materials, which are l i s t e d i n 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 i s presented i n Table I. The fourth step i n planning the experiment was to subdivide each lesson area into component lesson objectives. Appendix A contains a summary of the lesson objectives i n the entire set of eight lessons. In most cases each objective formed a separate entity within the lesson area, although i n 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 Lesson Number I APPROPRIATE TO EACH TOPIC Farquhar Question Number nil Lesson Topic Introductory Lesson II 4, 11, 20 Identification and Meaning of Place Names i n 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, addi t i 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 i s an introductory lesson area for general orientation purposes, the f i r s t mentioned factor was not a consideration 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 l a s t , step i n planning the experiment was to select the materials of instruction considered most effective for the teaching of each objective i n the entire eight lessons. The materials selected are designated i n the last two columns of Table I I . These materials may be identified by referring to the representations of the materials shown i n '.'Figure's;'"; on pages 39 to 42. The foremost consideration i n 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 c l a s s i f i c a t i o n of arithmetic teaching aids contained i n "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 b r i e f l y 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 Instructional Materials Used by Number and T i t l e of Lesson Number of Lesson Objective Lesson I Objectivesi 1 & 2 12 Objective 3 18 P.V. Charts 1 to 6 Objectives 1 , 2 & 3 15 P.V. Charts 7, 8, ! Objective 4 15 P.V. Cards 10 Objective 1 8 Objective 2 12 Objective 3 10 Objective 1 20 P.V. Charts 9, 10 Objective 2 10 P.V. Cards 10 Objective 1 ' 15 9, 10 Objective 2 15 P.V. Charts P.V. Cards Same Same Lesson VI Objective 1 Objective 2 Objective 3 15 8 7 Wall rule Wall rule nil 11 11 nil Lesson VII Objective 1 Objective 2 8 22 nil Flannel Board nil 12 Lesson VIII Objective 1 Objective 2 Objective 3 12 12 6 Flannel Board Wall rule nil 12 11 13 Lesson II Lesson III Lesson IV Lesson V Time Alotment (Minutes) Experimental Groups Control Group ; (No . of Card) nil nil P.V. Carts nil nil nil 7, 8, ! nil 37 1. Manipulative materials. These materials provide the highest l e v e l of concreteness i n the presentation of an arithmetical idea. The idea i s represented i n an actual object, capable of being manipulated. 2. Visualization materials. These materials provide the second highest level of concreteness. The idea i s not shown i n the form of a concrete and movable object; rather, i t i s 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. I l l u s t r a t i o n materials. These materials provide the third highest level of concreteness. processes necessary to formulate The component mental the answer are not shown. It i s merely the answer which i s i l l u s t r a t e d . 4. Abstract symbolism. level of concreteness. These materials provide the lowest The symbol i s not i n any way anchored to i t s referrent, except i n so far as the learner i s capable of providing this link i n his own imagination. In order, therefore, to make the instructional materials used by the experimental and control groups as nearly alike as possible i n 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 i n some cases under different 38 names, i n the various sources of reference consulted. These materials, which were constructed by the experimenter, are described i n 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 i n an entirely arbitrary manner, are represented i n Figures "2,' 3,. and. 4. Place value charts. are each one foot square. Made of \ inch plywood, these charts The decimal point i s on a chart which i s one foot by six inches i n size. Seven charts represent place values extending from thousands to thousandths. The ONES' chart, occupying the central place i n our system of notation, i s painted red, while the decimal point i s a red dot on a white background. This colour arrangement was chosen to emphasize that the primary function of the decimal point i s to designate the location of the ONES' d i g i t . In order to present the visual symmetry of the different place values around the ONES', place, the decimal point was, i n the actual teaching process, placed i n 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 ff^^l F i g u r e 2. *M\%m\\ ONES mMmmX 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 Figure 3 . ^ W W 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 42 V i s u a l i z a t i o n Card No. 9 1000's 100's 10's M 2• 1 Othfl 1 OOtVis month « 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 1 1 1 1 F i g u r e 4. • • • TENTHS HUNDREDTHS THOUSANDTHS y 0 0 1 0 1 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 .... 1 43 rather than entirely to the right as shown i n 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 i n the preparation of a l l the materials. Each chart holds 30 hooks. Cardboard tickets, 1^ inches by 3 inches, were supplied i n the same colours as the charts. Place value cards. are each one foot square. Made of \ inch plywood, these cards They employ the same decimal point arrangement and colour scheme used i n the place value charts. These materials are designed to present the actual relationship i n 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 unnecessary. The idea of the relationship i n size of-the positions to the l e f t of ONE i s 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 i n size of the positional values extending i n 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 i n such pupil materials as the decimal fraction cards and number l i n e , and differs mainly i n that i t i s 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 i s to f a c i l i t a t e the explanation of rounding to the nearest whole number. The integers which designate whole numbers appear i n red, while the subdivisions into TENTHS appear i n blue, and the further subdivisions into HUNDREDTHS are indicated by green markings. Flannel board. Since this i s a conventional and widely used type of teaching aid, i t i s not i l l u s t r a t e d i n Figured I. The dimensions of the flannel boards used i n the experiment are 4 feet by 2 feet. The manipulative materials attached to the boards are made of lightweight paper. I l l u s t r a t i o n s of the various materials are shown i n 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 i n size; Card 9 i s 36 inches by 12 inches; Card 10 '. 45 i s 3 6 inches by 18 inches; and Card 11 i s 3 6 inches by 6 inches. As mentioned previously, the characteristic of the v i s u a l i z a t i o n materials i s that the arithmetical idea i s presented i n 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 i n 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 i n 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 i n subsection I I , 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 i n Appendix B., proved to be a major part of the work entailed i n organizing the experiment. It was imperative to include i n the lessons only carefully planned procedures to which the aids used by both treatment groups are adaptable, and i n 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 i s 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 i n 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 f i r s t three lessons are accompanied by introductory material v/hich i s common to both groups. This material, which i s on v/hite sheets, i s 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 instructions and the number of steps involved i n the presentation, i s identical f o r 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 l e t t e r i n g indicates instructional directions or summaries of a more general nature than i s contained i n the detailed steps of the lessons; BLACK l e t t e r i n g indicates the generalizations which the pupils are expected to formulate i n their own words after the presentation of the whole lesson or part of i t . 47 As mentioned i n subsection III (Chapter l):., the entire time for each lesson i s one hour. This includes thirty minutes of group instruction, f i f t e e n 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 i s 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 i s 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 i s 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 i n 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 i n 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 D i s t r i c t No. 36 (Surrey) there were thirty-seven elementary schools at the time the classes were being selected (January, 1957). six. The mean number of divisions i n each school was Since there were no junior high schools, a l l the Grade VII and Grade VIII classes were located i n 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 i n the study, there were nine schools with completely unselected Grade VII classes. The three main c r i t e r i a used for the selection of the five classes considered necessary to participate i n the study were: ( l ) the teacher's interest i n , and aptitude for, taking part i n an educational experiment, (2) the teacher's experience and a b i l i t y i n 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 i n this l e t t e r 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 this experiment •> s t a t i s t i c a l design used i n i s v a l i d only i f certain conditions surrounding the conduct of the experiment are s a t i s f i e d . One condition, which Lindquist says."has perhaps most often been violated with serious 2 consequences" i n educational research, concerns the manner of selecting the treatment groups. In a controlled experiment, i f one i s safely to conclude from a significant F that the experimental and control treatments have been responsible for producing different results, then i t i s necessary, i n Lindquist's words, to assume that "the subjects i n each treatment group were o r i g i n a l l y 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 i n a manner intended to s a t i s f y 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 i s understandable that the effectiveness of the instructional aids would be related to the size of the classes i n which they were used. For example, i n the case of observing a group demonstration, the pupils at the back of a large class would be at a disadvantage i n comparison with the pupils at the back of a small class. The size of the classes, therefore, i s a concomitant variable which could not be controlled except through the procedure of pairing the classes.' As the f i r s t step i n this procedure, the classes with approximately equal enrollments, as they were immediately prior to the experiment, were considered as a unit. Three smaller classes ( i n General Montgomery, Hjorth Road, and Simon Cunningham Schools) formed one such unit, referred to as Unit A; two larger classes ( i n Prince Charles and Fleetwood Schools) formed another such unit, referred to as Unit B. As the second step i n 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. I t was previously decided that two classes should be i n the experimental group and three classes i n 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 Prince Charles Experimental 39 36 Fleetwood Control 42 37 Total i n experimental group 64 59 Total i n control group 96 88 160 147 Unit B Total i n both groups 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 i n Table I I I . VI. ACCOUNT OF ORIENTATION AND EVALUATION MEETINGS The selection of the teachers to participate i n this study was made early i n January, 1957. February, the experiment Originally scheduled for was eventually held i n May. During this time the teachers had an opportunity to orient themselves i n 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 f i 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 experiment. The distributed materials included: of the 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 i n progress, the experimenter v i s i t e d 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 i n the event of any problem a r i s i n g . 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 i n 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 i s contained i n Appendix A. 54 This information was used i n considering the implications of the conclusions reported i n Chapter VI. VII. ADMINISTRATION OF THE TESTING PROGRAMME Immediately prior to the commencement of the experiment, the following four tests were administered experimenter: personally by the (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 A b i l i t y , Intermediate Examination, Form A, and (4) The Stanford Achievement Test, Advanced Reading, Form E. The administrations took place, in the absence of the classroom May 10th. teachers, during the week from Monday, May Two 6th to Friday, s i t t i n g s were held i n each school to ensure that conditions of fatigue v/ould be equalized among the classes and reduced to a minimum. By using i n 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 s t a t i s t i c a l l y . Immediately following the experiment, the Farquhar test re-administered personally by the experimenter. To ensure an equalization of testing conditions, the re-administrations to the f i v e classes involved were held during the mornings only, on was 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 c r i t e r i o n variable, the effectiveness of the teaching aids was judged. Complete details concerning these four tests are given i n 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 i s that the pupils themselves were given no opportunity of manipulating the instructional materials. The second l i m i t a t i o n i s that the nature of the experiment dictated a very r i g i d sequence of instruction i n 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. experiment thus allowed no opportunity to reverse the procedure The 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 l i t e r a t u r e has been written. An evaluation of the conclusions reached i n Chapter VI requires an awareness of this l i t e r a t u r e . The issue surrounding the f i r s t l i m i t a t i o n i s that ofothe learner's own involvement i n the manipulative activity. The reports that this involvement f a c i l i t a t e s learning i s found mainly i n psychological l i t e r a t u r e . Heidbreder, for example, has investigated the manner i n which concepts are learned. Her experimentation v/ith adults furnishes some evidence that the ease of attaining a concept, i n the case of adults at least, "seems more highly correlated with manipulability 4 than with p e r c e p t i b i l i t y " . In a chapter entitled "The Formation of Concepts", contained i n a recent yearbook, Van Engen cites an impressive l i s t of authorities to support his conviction that, i n the case of children as well as adults, manipulability, or relevance to direct motor reaction, i s an important factor i n 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 manipulations performed i n a learning situation. ... It i s probable that a l l mental l i f e has a motor basis and a motor o r i g i n . ... This behaviour (of motor priority) i s so fundamental that v i r t u a l l y a l l behaviour ontogenetically has a motor origin and aspect.^ 57 He quotes from Werner: To conceive and define things i n terms of concrete a c t i v i t y i s i n complete accordance with the world of action characteristic of the c h i l d . ^ F i n a l l y , he quotes from Piaget, whose work "seems to be resting i n an undeserved obscurity": ... i t ( c h i l d i s h thought) i s nearer to action than ours, g and consists simply of mentally pictured manual operations... In view of the literature which t e s t i f i e s to the importance of action or manipulability i n the child's thought processes, i t would seem that the learning outcomes resulting from the instruction offered i n the experimental group were l i k e l y curtailed as a consequence of the fact that the pupils i n 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 l a i d 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 i n the l i t e r a t u r e . Commenting on one aspect of the controversy, Johnson writes: ... a rationalization of a process i n arithmetic i s meaningless unless the HOW to do that process i s understood f i r s t , l e t i s 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 i s better appreciated. 58 ... But what I am trying to say here i s that since rationalization of a process i s not understood u n t i l the HOW of the process i s understood, and the HOW i s 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 i n trying to rationalize every process upon f i r s t presentation before the HOW of the process i s known.^ Later i n the same a r t i c l e he states: What could be a better program of teaching than to bring in rationalization of newer and higher orders as the process i s reviewed i n later grades? The review would then not be a rehash only, but a true review with the process seen i n a new l i g h t . 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 i s 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 i s always feasible for WHY to lead to HOW. In some instructional situations i t may seem v i r t u a l l y necessary to present the algorismic form of certain computational s k i l l s on the basis of a HOW-WHY sequence. In such instances, when HOW-WHY i s selected as the course to be taken, l e t us be certain that the WHY i s coupled with the HOW just as soon as possible or feasible. There i s grave danger that WHY may follow HOW at such temporal distance that ultimate rationalization i s minimized i n effectiveness. A In view of the prominence of these views expressed i n the l i t e r a t u r e , i t i s 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 i n this 59 study, but also by the particular temporal sequence employed i n 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 i n sufficient quanitities to permit the pupils to use them either individually or i n small groups. D i f f i c u l t i e s were evident i n 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 d i f f i c u l t y of establishing uniformity between the two treatment groups i n 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 i n the teaching of some decimal concepts and'processes, and follow i t i n 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 l i k e l y usage to which manipulative materials would be put i n normal classroom practice. One major d i f f i c u l t y , as usual, presented i t s e l f with this idea. I t would have necessitated extending the area of the experiment to include the teaching of decimal fractions i n their entirety - the HOW as well as the WHY. Since this assignment comprises a large part of the arithmetic programme normally undertaken i n 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 i n 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 i n the arithmetic syllabus. The nature of these d i f f i c u l t i e s , as well as the d e s i r a b i l i t y of using manipulative aids i n the type of "carefully planned 61 reteaching program" recommended by Morton, 12 encouraged the experimenter to proceed v/ith the present design, notwithstanding the two limitations involved., X. SUMMARY The purpose of this study i s to determine the effectiveness of the group instruction use of certain manipulative aids i n contributing to an understanding of decimal concepts. The essence of the proposition involved i s to determine the effectiveness which results, s p e c i f i c a l l y , 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 i n the experiment, and an account of the planning and administration undertaken, to ensure that the manipulative 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 i n the study. materials used The psychological nature of these limitations, as revealed by the literature on the subject, emphasizes the need to proceed v/ith caution i n forming generalizations respecting the effectiveness of the particular aids used. While this chapter contains a description of the actual controls 62 exercised i n the conduct of the experiment, Chapter V contains, i n addition to the test of significance of the achievement of the two treatment groups on the c r i t e r i o n variable, an account of the s t a t i 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 i n Chapter IV. 63 FOOTNOTES The third c r i t e r i o n i s important i n order to avoid the inadvertent 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 i s necessary to adhere to this c r i t e r i o n . A f u l l discussion i s contained i n Chapter V, pages 151 and 152. E. F. Lindquist, Design and Analysis of Experiments i n Psychology and Education, (Boston: Houghton M i f f l i n , 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 i n the Child, (New York: Harcourt Brace and Company, 1928) p. 146. J . T. Johnson, "What Do We Mean by Meaning i n Arithmetic?", The Mathematics Teacher, Vol. 41, 1948, p. 365. Ibid, p. 366 J . Fred Weaver, "Misconceptions about Rationalization i n 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 S t a t i s t i c a l Method In Chapter III i t was on the basis of size only. stated that the classes were matched This matching resulted i n the formation of two so-called units, referred to i n 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; from Unit B one class was and selected at random for the Experimental Group, leaving one class for the Control Group. Since size was the only factor taken into account i n establish- ing the equivalence of the classes, there were obviously many unmatched individual differences i n the classes assigned to the two groups. treatment The relative response of each group to the c r i t e r i o n could conceivably be influenced by these differences. It i s apparent that i f these unavoidable concomitant influences were not controlled, any differences between the Experimental and Control Groups on the c r i t e r i o n could not s p e c i f i c a l l y be attributed to the treatments being tested. 66 To provide s t a t i s t i c a l control over these unmatched individual differences i n the Experimental and Control Groups, analysis of covariance was selected as the s t a t i s t i c a l design to be applied to the data derived from the experiment. The following statements respecting the analysis of covariance technique indicate i n general terms i t s s u i t a b i l i t y f o r the present study. Further statements, referring to somewhat more technical aspects of i t s appropriateness, are contained i n Chapter V. Edwards writes: The analysis of covariance i s applicable to any experiment i n 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 i s then made for this source of variation i n the analysis of the outcomes 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 s t a t i s t i c a l 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 w i l l provide tests of significance f o r the comparison groups whose members may have been s t r a t i f i e d and whose members have been measured with regard to one or more variable characteristics other than the c r i t e r i o n . 2 Analysis of Covariance has r e a l l y two purposes. F i r s t , by providing for the correlation between the c r i t e r i o n and control scores, i t makes i t possible to determine the r e l a t i v e weight with which each independent variable "enters i n " or contributes to the c r i t e r i o n i n 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 e x i s t , analysis of covariance makes i t possible to exercise a degree of s t a t i s t i c a l c o n t r o l over these independent variables which permits the treatment effect to be evaluated with as much accuracy as i f the variables had been experimenta l l y controlled by a c t u a l l y matching the groups with respect to these variables. L i m i t a t i o n s , 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 v a r i a b l e s . S t i l l l e s s capable i s i t of e l i m i n a t i n g the e f f e c t s of systematic differences o r i g i n a l l y e x i s t i n g between the groups i n c e r t a i n c h a r a c t e r i s t i c s which are independent of the X variables employed. Subsections IX and X of Chapter V contain an account of a l l the l i m i t a t i o n s 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 t h i s s p e c i f i c problem. In the present study the concomitant influences are considered to e x i s t , p r i m a r i l y , i n four areas, namely: i n i t i a l understanding of concepts of decimal f r a c t i o n s , computational a b i l i t y i n decimal f r a c t i o n s , 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 c r i t e r i o n . These tests are contained i n Appendix D. The contribution which each of these tests made to the prediction of the c r i t e r i o n i s eventually reported i n Chapter V (Table XLI, page 155.) Judged by this information, i t i s unlikely that additional measurable influences would have an appreciable effect upon the performance of the treatment groups on the c r i t e r i o n . It may be said, therefore, that the application of analysis of covariance to the data removed the possible bias introduced unmatched individual differences between the Experimental and Groups. by Control This i s true, at least, to the extent that the four areas referred to represent the differences i n question, and, further, to the extent that the differences i n these areas are adequately controlled by the tests administered for that purpose. With the removal of this bias, and the imposition of necessary controls i n the plan and administration of the experiment, any significant s t a t i s t i c a l difference i n the criterion measures of the experimental and control groups i s assumed to be attributable to the treatments used i n each group. These treatments, i t may be emphasized again, are intended to be identical .'in every respect except i n the use of the materials of instruction, which d i f f e r only i n the characteristic of manipulability. Statement of the Hypothesis The hypothesis to be tested i s that there i s no significant difference i n the c r i t e r i o n 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. I n i t i a l Understanding of Concepts of Decimal Fractions Farquhar's Test of the Understanding of Processes with Decimal Fractions (First Administration) 2. Computational A b i l i t y i n Decimal Fractions Adapted from Unit Test "Making Sure of Decimals" contained i n Silver Burdett Text "Making Sure of Arithmetic" 3. Mental A b i l i t y Otis Self-Administering Test of Mental A b i l i t y ^Intermediate Examinations) Form A 4. Reading A b i l i t y Stanford Achievement Test (Advanced Reading Test: Form E for Grades 7-9 1. Final Understanding of Concepts of Decimal Fractions Farquhar's Test of the Understanding of Processes with Decimal Fractions (Second Administration) Dependent or Criterion Variable 70 i s attributable to the treatments involved. Stated i n other words, this hypothesis is. that the pupils who are taught with the use of manipulative aids i n the manner prescribed i n this experiment achieve an understanding of decimal fractions which i s not s i g n i f i c a n t l y different, after the bias due to unmatched individual differences i n 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 i n 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 i n Appendix D, performed a dual function i n this study. Immediately prior to the assignment, i t was used to measure one of the independent variables shown i n Table IV. Immediately following the teaching assignment i t was used, i n 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 i s a l l the greater i n view of the fact that the Farquhar test was validated i n relation to groups of teachers-in-training. The fact that this validation.took place against educationally more advanced subjects than those participating i n this experiment provided the major source of apprehension concerning the s u i t a b i l i t y 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 i t s s u i t a b i l i t y for the present study, the Farquhar Test was administered on A p r i l 4th, 1957 by the experimenter to a group of forty representative pupils selected from 2-g- classes of unselected Grade VII pupils i n 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 i n 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 c r i t e r i a , namely, their d i f f i c u l t y and their discriminating value or v a l i d i t y . Method of Item Analysis The method employed i s 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 i n 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 ?. 19 18 0 1 0 11 10 9 5 5 6 17 16 15 1 3 1 8 7 . 6 2 7 3 14 13 12 0 2 3 5 4 3 0 1 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 9.7 Mean 10.175 Standard Deviation 3.382 Corrected for chance Not Corrected for chance Range of Item Validity 85.6$ 64.9$ -18$ to 64$ Reliability .549 TABLE 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 1 Per cent of D i f f i c u l t y Item Corrected for chance Uncorrected for chance Per cent of Validity Validity Coefficient (Flanagan s) W - . W Discrimination L H 1 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 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 1 2 3 4 5 6 11 TABLE VII Item Per cent of D i f f i c u l t y Corrected uncorrected for chance for chance (continued) Per cent of Validity- Validity Coefficient (Flanagan s) WH Discrimination 1 16 17 18 19 20 48 102 109 97 42 36 82 82 73 32 18 0 0 -18 27 .20 :00 00 -.23 .31 2 0 0 -2 3 21 22 23 24 25 102 115 85 109 97 82 86 68 82 73 36 9 64 36 36 .66 .18 .78 .66 .46 4 1 7 4 4 26 27 28 29 30 127 40 97 97 121 95 32 77 77 91 9 45 45 27 0 .38 .54 .71 .38 00 1 5 5 3 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 Range of Difficulty 0$ _ 1% - 5% - 10% - 15% 6$ 11$ 16$ 21% 26% 31% 36% \1% Frequency of Items 46$ 51$ 56$ 61$ 66$ — - 20$ - 25$ 30$ 35$ 40$ 45$ Per Cent Range of Difficulty 1 1 3 1 1 71$ 76$ 81$ 86$ 91$ Frequency of Items 50$ 55$ 60$ 65$ 70$ 2 2 - 75$ - 80$ - 85$ - ..90$ - 95$ 4 2 4 2 4 2 1 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 0$ 1$ 6$ 11$ 16$ - or less 5$ 10$ 15$ 20$ 21$ . - 25$ 26$ - 30$ 31$ - 35$ 36$ - 40$ 41$ - 45$ Frequency of Items 53 3 4 5 6 1 Frequency of Items Per Cent Range of Validity 46$ 51$ 56$ 61$ 66$ ---• --- 50$ 55$ 60$ 65$ 70$ 71$ 76$ 81$ 86$ 91$ ------ 75$ 80$ 85$ 90$ 95$ 1 2 3 76 TABLE 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 (Based on Flanagan's Estimates of Correlation between Individual Items and the Test as a Whole) ValidityCoefficient Range Validity Coefficient Range Frequency 5 .46 - .50 3 .01 -".05 0 .51 - .55 2 .063- .10 1 .56 - .60 2 .11 - .15 0 .61 - .65 1 .16 - .20 4 .66 - .70 3 .21 - .25 0 .71 - .75 2 .26 - .30 1 .76 - .80 1 .31 - .35 1 .81 - .85 0 .36 - .40 4 .86 - .90 0 .41 - .45 0 .91 - .95 0 00 or less Frequency 77 Row (d) indicates the per cent d i f f i c u l t y of each item, uncorrected for chance. I t i s the r a t i o , converted to per cent, of the total number of incorrect or omitted responses to the total number of possible responses i n the top and bottom 21% sections. Expressed algebraically i t i s 100c where "c" i s the total obtained i n Row (c) 2n and "n" i s the total number of possible responses i n 27$ of the entire group. Rows (e), ( f ) , and (g) deal with the per cent d i f f i c u l t y , corrected for chance. To obtain this, the total number of wrong or omitted responses recorded i n row (c) is. multiplied by the correction factor shown i n row ( f ) . Expressed algebraically the correction factor i s 100 "0" , where "0" i s the number of options i n each 2n ("0"-l) question and "n" i s the total number of possible responses i n 21% of the entire group. Rows (h) and ( i ) deal with the item discriminating value or validity. The discrimination of each item i s 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 multiplying by 100. Expressed algebraically the per cent of the discrim- ination or v a l i d i t y of each item i s lOOh, where "h" i s the discrimn ination recorded".In row (h) and "n" retains- the representation indicated above. To find the Flanagan v a l i d i t y coefficient i t i s necessary, i n addition to the foregoing procedure, to compute from rows (a) and (b) the per cent of correct responses i n the bottom and top sections, 78 respectively. The v a l i d i t y index i s 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 c r i t e r i a are available by which to evaluate the effectiveness of the individual test items. One of these c r i t e r i a i s that of item d i f f i c u l t y . On this subject Ross and Stanley write: D i f f i c u l t y alone, therefore, i s not a dependable measure of discrimination ... Test experts have usually found, however, that the average d i f f i c u l t y of the items i n a test i s 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 d i f f i c u l t y when corrected for chance, so far as possible. This w i l l mean that v i r t u a l l y a l l the items of 0-15 per cent and 85-100 per cent d i f f i c u l t y 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 d i f f i c u l t y level.4 An examination of Table VII shows that compliance with this suggestion would result i n the omission of 19 of the 30 test items. In fact, the d i f f i c u l t y , 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 d i f f i c u l t y indexes for 'chance'". If the correction for chance i s not made, the mean per cent d i f f i c u l t y of the items i s reduced from 85.6 to 64.9 (Table V i ) , and, as seen i n Table VIII, only six- items have a per cent d i f f i c u l t y greater than 85. In addition, i t w i l l be observed that the distribution 79 of scores i s a satisfactory one (Table V), i n which the mean s l i g h t l y exceeds the median, and the standard deviation indicates a reasonable, though small, v a r i a b i l i t y . (As samples, these three calculations are shown i n Appendix E.) Nevertheless, the results of the t r i a l administration indicated that Farquhar's test would l i k e l y be rather d i f f i c u l t when used as a measure of one of the independent variables at the beginning of the experiment. the Despite this realization, the experimenter believed that d i f f i c u l t y of the test would not be entirely a disadvantage. After an intensive period of instruction on the subject matter covered by the testyj i t was to be used a second time i n the.even more important role of measuring the c r i t e r i o n performance. It was anticipated, and hoped, that i n this capacity the level of d i f f i c u l t y of the Farquhar items would enable the test to meet the ideal statistical^-requirements. The second of the. c r i t e r i a by which to evaluate the e f f e c t i v e ness of the individual test items i s that of item discrimination or validity. One commonly used standard of v a l i d i t y i s 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 v a l i d i t y i s that items must show a positive v a l i d i t y coefficient, based on Flanagan's Table, of more than .25. the A reference to Table X reveals that 10 items f e l l below .25 coefficient level. One p a r t i c u l a r l y undesirable feature v/hich resulted from this administration i s that 3 items have a zero v a l i d i t y and 2 have a 80 negative v a l i d i t y . In 4 of these 5 items the per cent of d i f f i c u l t y , corrected for chance, exceeded 100 per cent. question 19, the per cent v/as 97 In the case of the f i f t h , (Table V i i ) . In view of the d i f f i c u l t y of the test for this sample of pupils, the degree of v a l i d i t y was to be expected. At the 50 per cent level of d i f f i c u l t y 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 l i k e l y also be rather low i n discrimination value when used as a measure of one of the independent variables at the beginning of the experiment. Yet the general levels of d i f f i c u l t y and discrimination, when used for this purpose, were not considered l i k e l y to be s u f f i c i e n t l y extreme to make the test unsuitable. Furthermore, when used a second time as a measure of the c r i t e r i o n variable at the close of the experiment, i t was believed that the decrease i n d i f f i c u l t y anticipated i n nearly a l l the items would be just about the right amount to increase quite substantially the discriminating 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 c r i t e r i o n variable v/as a main consideration a f f e c t i n g the experimenter's decision to select the Farquhar test, there were the following additional considerations, though not necessarily i n this order of importance. 81 The f i 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 i n 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 i n Appendix E. While there are obvious d i f f i c u l t i e s i n the interpretation of test r e l i a b i l i t y , certain minimal requirements have been suggested for the r e l i a b i l i t y coefficients of tests which serve various purposes. Reference i s 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 i n 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 i s to differentiate the achievement or status of individuals, rather than that of a group. Since the Farquhar test was to be used for the l a t t e r 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 i n favor of selecting the test for use i n 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 i s rendered very d i f f i c u l t by the lack of c r i t e r i a for this purpose. The investigator must determine a r b i t r a r i l y those concepts that should be included i n a measuring instrument designed to evaluate understanding of any phase of arithmetic. 7 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 f r 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 t o p i c s of the eight lessons i n v o l v e d i n the teaching procedure. The l e s s o n planning i s described i n Chapter III. The c u r r i c u l a r v a l i d i t y of the t e s t , when used as a measure of one of the independent v a r i a b l e s at the beginning of the experiment, seems assured by the f a c t that the concepts measured by the t e s t are i d e n t i c a l to the concepts emphasized f o r teaching i n the Grade V I I text c u r r e n t l y p r e s c r i b e d by the B r i t i s h Columbia Department of E d u c a t i o n . The c u r r i c u l a r v a l i d i t y of the t e s t , when used as a measure o f the c r i t e r i o n v a r i a b l e at the end o f the experiment, i s more d e f i n i t e l y assured by the f a c t that the l e s s o n s were c a r e f u l l y planned a c c o r d i n g to the s p e c i f i c concepts measured by the t e s t . A f t e r t a k i n g account o f the foregoing c o n s i d e r a t i o n s , that is-, the e f f e c t i v e n e s s of the . i n d i v i d u a l t e s t items when evaluated on the bases of t h e i r d i f f i c u l t y and d i s c r i m i n a t i o n value, the t e s t r e l i a b i l i t y , and the s p e c i f i c nature and c u r r i c u l a r v a l i d i t y o f the concepts measured, i t was decided that Farquhar's test would be s u i t a b l e f o r the present study. I n t e r p r e t a t i o n o f data obtained from the f i n a l administration I t has been stated that Farquhar's test was used p r i o r to the experiment immediately to measure one of the-'ihdependent v a r i a b l e s , and immediately f o l l o w i n g the experiment to measure the c r i t e r i o n v a r i a b l e . 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 i s 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 f i n a l administration. Therefore, these results have been subjected to an analysis similar to that which was undertaken i n 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 f i n a l administration. To f a c i l i t a t e making comparisons, a summary of comparative data from the two administrations i s presented i n Table XVII. An examination of this l a t t e r table shows that the level of d i f f i c u l t y , which was the most serious c r i t i c i s m 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 d i f f i c u l t y (uncorrected for chance) i n the t r i a l administration, only 1 item was i n this position in the f i n a l administration. It i s interesting to note that this 1 item (Mo. 4) tended to be too easy (15$). The general extent of the reduction i n d i f f i c u l t y i s shown by the fact that 25 of the 30 items were easier for the pupils i n the 84 TABLE XI FREQUENCY OF SCORES IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS (Maximum: 30 Items) Score Frequency Score Frequency 24 4 0 6 14 13 12 13 14 11 23 22 21 1 4 4 11 10 9 10 12 5 20 19 18 4 5 11 8 7 6 6 6 4 17 16 15 8 8 9 5 4 3 0 1 1 26 25 TABLE XII SUMMARY OF STATISTICAL DETAIL IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS Mean Item —— Difficulty Median Mean Standard Corrected Not Deviation for chance corrected for chance 13.77 14.33 5.019 66.8$ 51.1$ Range of Item Validity Reliability 8 to 65$ .541 TABLE XIII THE DIFFICULTIES AND VALIDITIES OF ITEMS IN FARQUHAR'S TEST ADMINISTERED AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING- SUBJECTS Item Per cent of D i f f i c u l t y Corrected Uncorrected for chance for chance Per cent of Validity Validity Coefficient (Flanagan's) W_ Discrimination 1 2 3 4 28 58 72 20 87 21 46 54 15 65 28 42 57 20 65 .42 .43 .57 .38 .74 11 . 17 23 8 26 6 7 8 9 10 60 62 43 67 102 45 46 33 54 76 45 48 45 33 8 .46 .49 .52 .34 .11 18 : 19 18 13 3 . 11 12 13 14 15 35 60 67 103 70 26 45 50 83 53 32 55 35 20 35 .41 .33 .36 .32 .36 13 22 14 8 14 5."> CO TABLE XIII (continued) Per cent of D i f f i c u l t y Item Corrected for chance Uncorrected for chance Per cent of V a l i d i t y Validity Coefficient (Flanagan s) WL Discrimination 1 16 17 18 19 47 35 58 35 .63 48 20 77 58 21 22 23 63 50 82 61 70 24 25 47 72 88 72 82 54 61 26 27 28 92 29 30 83 56 66 80 60 52 70 69 41 65 50 35 60 40 .76 .51 .39 .60 .41 26 20 ' 14 24 16 45 32 40 43 48 .45 .34 .48 .44 .51 18 13 16 17 19 43 32 62 32 35 .51 .34 .62 .36 .37 17 13 25 13 14 oo CTl 87' TABLE XIV FREQUENCY OF ITEMS AT THE VARIOUS PER CENT LEVELS OF DIFFICULTY RESULTING FROM ADMINISTRATION OF FARQUHAR S TEST AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS 1 Per Cent Range of Difficulty Frequency of Items Per Cent Range of Difficulty 0$ 1$ - 5% 6% - 10% 11% -15% 16% - 20% 1 21% -25% 26% - 10% 1 1 31% -35% 36% - 40% 3 41% 3 - 45$ Frequency of Items 46% - 50% 5. 51% 56% 61% 66% - 55% 60% 65% 10% 4 11% 16% 81% 86% - 15% 80% 85% 90% 91% - 95% 3 4 3 1 1 TABLE 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 1 Per Cent Range of Validity 0% - 1% - 5% 6% - 10% 1 20% 2 11% 16% - 15% 21% 26% - 25% 30% 31% -35% 36% - 40% 41% Frequency of Items -45% 1 9 2 6 Per Cent Range of Validity 46$ 51$ ., .56$ v6l$ 66$ - 50$ 55$ 60$ 65$ 70$ 71$ 76$ 81$ 86$ 91$ - 75$ 80$ 85$ 90$ 95$ Frequency of Items 3 1 2 3 TABLE XVI FREQUENCY OF ITEMS IN THE VARIOUS VALIDITY COEFFICIENT RANGES RESULTING FROM ADMINISTRATION OF FARQUHAR S TEST AT THE CLOSE OF THE EXPERIMENT TO THE 147 PARTICIPATING SUBJECTS 1 (Based on Flanagan* s Estimates of Correlation between ; '• ' Individual Items and the Test as a Whole) Validity Coefficient Range Frequency 00 Validity Coefficient Range Frequency .46 - .50 3 .01 - .05 .51 - .55 4 -.06 - .10 .56 - .60 2 .61 - .65 1 .11 - .15 ' 1 .16 - .20 .66 - .70 .21 .71 - .75 1 .76 - .80 1 .25- .26 - .30 .31 - .35 5 .81 - .85 .36 - .40 6 .86 - .90 .41 - .45 6 .91 - .95 TABLE XVII COMPARATIVE DATA OBTAINED FROM,THE ADMINISTRATION OF FARQUHAR S TEST TO THE TRIAL GROUP AND TO THE 147 PARTICIPATING SUBJECTS 1 Criteria Correction for chance (Difficulty) Trial Administration Final Administration Item D i f f i c u l t y No. Items below desirable minimum d i f f i c u l t y (l5$ or below) Yes No 0 0 0 1 No. Items above desirable maximum d i f f i c u l t y (over 85$) Yes No 19 6 5 0 Total No. Items beyond desirable d i f f i c u l t y Yes No 19 6 5 1 No. Items over 100$ D i f f i c u l t y when corrected for chance Mean Per Cent D i f f i c u l t y 10 Yes No 85.6 64.9 66.8 51.1 Item Discrimination No. Items below desirable minimum discrimination (below 20$) No. Items below desirable Coefficient of V a l i d i t y (Flanagan Coefficient over .25) No. Items v/ith negative discrimination 12 10 2 0 90 f i n a l administration, and 5 items (No's. 3, 12, 20, 23, 27) were more d i f f i c u l t for this group than they were for the pupils i n the t r i a l group, who did not have any special instruction i n the concepts involved. The extent of the reduction i n d i f f i c u l t y i s shown also by the fact that the mean per cent l e v e l of d i f f i c u l t y (uncorrected for chance) f e l l to 51.1%. As the test proved to be generally easier, though not too easy, for the pupils i n the f i n a l administration, so also i t proved to be more discriminating. Table XVII shows that only one item f e l l below the desirable minimum discrimination. remained quite d i f f i c u l t (76$). This item (No. 10) i s one that However,tw.o.-other items were at the minimum acceptable level (20$): one of these i s item 14, which also remained quite d i f f i c u l t (83$), and the other i s item 4, referred to above, which tended to become too easy ( l 5 $ ) . The decrease i n d i f f i - culty which was anticipated i n 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 f o r some reason proved more d i f f i c u l t for the group i n the f i n a l administration than for the t r i a l group, the increase i n d i f f i c u l t y actually resulted i n an increase i n per cent of v a l i d i t y . The v a l i d i t y of the other two items, though decreased i n the f i n a l administration, remained satisfactory (40$ i n No. 23; 32$ i n No. 27). When used to measure the c r i t e r i o n variable, the Farquhar test proved to be a suitable measuring instrument i n other respects beside 91 item d i f f i c u l t y and discrimination, which have just been discussed. The distribution of scores (Table Xl) i s a very satisfactory one, i n which the mean s l i g h t l y exceeds the median, and the standard deviation (Table XIl) indicates greater v a r i a b i l i t y than existed i n 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 i n this situation i s .541, approximately the same as i n the previous analysis (.549). The assurance of curricular v a l i d i t y , when the test was used 8 i n i t s f i n a l 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 c r i t e r i o n variable also. A test suitable for these two purposes was d i f f i c u l t to find. Although the results of the t r i a l administration to the 40 Grade VII pupils i n 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 c r i t e r i o n variable. The capacity of the test to perform these two functions i n this manner indicates i t s s u i t a b i l i t y for the present study. 92 III. DESCRIPTION OF THE DECIMAL FRACTION COMPUTATION TEST The Decimal F r a c t i o n Computation T e s t , shown i n Appendix D, was used to measure one of the independent IV, page 69. was I t i s the second of the b a t t e r y of four t e s t s which administered by the experimenter commencement of the v a r i a b l e s shown i n Table immediately p r i o r to the experiment. The t e s t was constructed by the experimenter, although i t i s to some extent an adaptation of a d i a g n o s t i c u n i t test entitled "Making Sure of Decimals", which i s contained i n the S i l v e r Burdett 9 Text "Making Sure of A r i t h m e t i c " . Tables XVIII. to XXIII, i n c l u s i v e , which correspond to the two previous sets of t a b l e s , c o n t a i n data derived from the r e s u l t s of the t e s t which was administered at the beginning of the T h i s a n a l y s i s was undertaken to ensure that the test had been a s a t i s - f a c t o r y instrument by which to measure the p u p i l s ' a b i l i t y i n decimal f r a c t i o n s . experiment. computational As i n the two previous cases, an item a n a l y s i s was made to i n d i c a t e the e f f e c t i v e n e s s of i n d i v i d u a l test items. I n t e r p r e t a t i o n of data obtained from the a d m i n i s t r a t i o n of the decimal. f r a c t i o n computation test. The e f f e c t i v e n e s s of the items i s the f i r s t c o n s i d e r a t i o n determining i t s s u i t a b i l i t y f o r the present study. i s one of the two i n t e r n a l c r i t e r i a used Item d i f f i c u l t y to evaluate item e f f e c t i v e n e s s . 93 TABLE XVIII 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 25 24 23 3 6 6 13 12 11 7 8 4 22 21 20 9 15 13 10 9 8 4 2 . 1 19 18 17 14'. 10 11 7 6 5 2 2 2 16 15 14 10 8 8 4 3 2 2 0 0 Frequency 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 Deviation Mean Item . Range of Difficulty Item Validity 17.75 16.98. 4.804 34.28$ 10 to 80$ .Reliability .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 Difficulty Per cent of Validity Validity Coefficient (Flanagan s) Wj, W Discrimination H 1 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) 23 51 34 4:8 45 35 52 43 60 70 .51 .52 .49 .60 .70 14 21 16 (d) 8 9 10 23 28 TABLE XX (continued) Item Per cent of Difficulty Per cent of Validity Validity Coefficient (Flanagan s) W _ W Discnmination H L 1 11(a) '.(b) (c) (d) (e) 12 13 14 15 (f) 29 38 45 20 41 37 55 75 35 68 .46 .59 .75 .60 .70 15 22 30 14 27 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 Range of Difficulty 0$ 1$ - 5$ 6$ - 10$ 11$ - 15% 16% - 20$ 21$ - 25$ 26$ - 30$ 31$ - 35$ 36$ - 40$ 41$ - 45$ Frequency of Items Per Cent Range of Difficulty 3 2 46$ 51$ 56$ 61$ 66$ - 50$ - 55$ - 60$ - 65$ - 70$ 3 2 3 3 3 71$ 76$ 81$ 86$ 91$ - 75$ - 80$ - 85$ - 90$ - 95$ Frequency of Items 2 3 1 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 Range of Validity 0$ 1$ 6$ 11$ 16$ - 5$ - 10$ - 15$ - 20$ 21$ 26$ 31$ 36$ 41$ - 25$ - 30$ - 35$ - 40$ - 45$ Frequency of Items 1 4 4 1 4 Per Cent Range of Validity 46$ 51$ 56$ 61$ 66$ - 50$ - 55$ - 60$ - 65$ - 70$ 71$ 76$ 81$ 86$ 91$ - 75$ - 80$ - 85$ - 90$ - 95$ Frequency 3 2 3 2 1 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 Coefficient Range 00 Frequency Validity Coefficient Range Frequency .46 .50 6 .01 .05 .51 .55 5 .06 .10 .56 .60 5 .11 .15 .61 .65 .16 .20 .66 .70 .21 .25 .71 .75 2 .26 .30 .76 .80 2 .51 .35 .81 .85 .36 .40 .86 .90 .41 .45 .91 .95 98 The suggestion of Ross and Stanley, referred to on page 78, i s that v i r t u a l l y a l l items of 0-15$ and 85-100$ d i f f i c u l t y 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) ( l l $ ) - f a l l into the former category, while none f a l l into the l a t t e r . There are other indications that the test was somewhat easier than was desirable: the mean d i f f i c u l t y of a l l 25 items i s 34.28 per cent (Table XIX), and, further, there i s a frequency of 3 scores at the maximum (Table XVIII). However, an examination of this table shows that there i s a satisfactory range of scores. Likewise, the median, mean, and standard deviation, reported i n 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 i n 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 v a l i d i t y of an item should exceed .25- Table XXIII shows that only one item (No. 6, once again) f e l l into this category. this item i s .18. The coefficient of I t w i l l 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 d i f f i c u l t y and discrimination value. 99 The effectiveness of the items on the computation test, judged on the bases of d i f f i c u l t y and discrimination, i s considered satisfactory with the exception of this item. The reliability of the test i s the second consideration determining i t s s u i t a b i l i t y for the present study. The r e l i a b i l i t y , calculated by means of the Hoyt Formula, i s .821. According to the standard referred to previously (page 8 l ) , 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 v a l i d i t y of the test i s the third consideration determining i t s s u i t a b i l i t y for the present study. The questions contained i n 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 i n the unit dealing with decimal fractions i n the Grade VII Arithmetic Course of Studies for B r i t i s h Columbia. After taking account of the foregoing considerations, that i s ; 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 i t s curricular v a l i d i t y , 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 A b i l i t y , Intermediate Examination, Form A, shown i n Appendix D, i s 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 c r i t e r i a used by the author to judge the v a l i d i t y of each item contained i n 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 indication of brightness. In the Manual of Directions there i s no statement of the extent to which the Intermediate Examination does, i n fact, serve i t s avowed purpose. There i s a meagre report concerning the correlation between scores i n the Higher Examination and "scholarship". This report i s that of the Principal of a High School i n Maine who found a correlation of approximately .58 between scores i n the Higher Examination and the "scholarship" of about 400 students i n Grades 11 and 12. The author states: "The method of standardization i s perhaps the best assurance as to the v a l i d i t y of the t e s t s " . ^ In this standardization procedure the c r i t e r i o n used to judge v a l i d i t y was the a b i l i t y of each item to discriminate between two 101 groups - a so-called "good group" and "poor group". The only d i s t i n c t i o n 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 i n 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 i n three c i t i e s located i n California, I l l i n o i s , and Minnesota. ' F i n a l l y , the last matter to be described about the test i s the reliability. 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 i n the other group Form B was administered f i r s t . The probable error of a score i n the Intermediate Examination i s reported to be s l i g h t l y over 2-jg- points i n hald the cases. The author states that "this means also that the probable error of an I.Q. i s about 2-g- points. The Otis Self-Administering Tests of Mental A b i l i t y , Intermediate Examination, i s designed for Grades 4 to 9. appropriate for Grade 7. It i s , therefore, extremely 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 i n Surrey at the time of the experiment included Grade 8, the practice i s to administer the Otis test immediately to the entrance of the pupils into the High Schools. prior It was discovered that a negligible number of pupils participating i n the experiment had written any form of the Intermediate V. Examination. DESCRIPTION OF THE STANFORD ACHIEVEMENT TEST (ADVANCED READING TEST: FORM E) The Stanford Advanced Reading Test: Form E, shown i n Appendix D, i s 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. available i n the 1940 Form E i s one of five alternate forms edition which has been superceded by the 1953 revision. Provision i s contained i n 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 f i n a l raw score for each pupil i n this variable was obtained simply by finding the average of the o r i g i n a l raw score i n each of the two sub-tests, and disregarding the. fraction where i t occurred. In a l l respects, except i n the matter of converting the raw scores into equated scores, the publisher's directions were s t r i c t l y 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 s u i t a b i l i t y of this test for the present study, except for reporting its reliability. In place of this standardization data the following data, which pertains to the administration of the test at the beginning of the experiment, i s presented for this purpose i n Tables XXIV and It may XXV. be concluded from a study of these tables that the test distributed the scores i n a satisfactory manner. The s p l i t ^ h a l f 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 i n the standardization population, i s reported to be .841 for the Paragraph Meaning and average of these coefficients, .874, .907 for the Word Meaning. i s 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 VI. The situation. CONCLUSION Chapter IV contains a description and evaluation of each of the four tests used to measure the five variables involved i n 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 c r i t e r i o n variable. The other tests were admin- istered at the beginning of the experiment to measure the remaining three independent variables. 104 TABLE XXIV 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 37 36 1 2 0 20 19 18 5) 17 5 35 34 33 3 4 3 17 16 15 9 5 8 32 31 30 2 4 2 14 13 12 11 9 2 29 28 27 4 1 5 11 10 9 3 3 1 26 25 24 3 2 5 8 7 6 1 2 0 23 22 21 9 3 11 5 4 3 1 1 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 19.24 20.37 Standard Deviation 7.277 105 These descriptions and evaluations afford assurance that the tests provided e f f i c i e n t measurements of the variables for which they were used. The i n i t i a l differences between the treatment groups i n the four independent variables are held constant i n the analysis of covariance to which these variables technique. The numerical extent would otherwise have been responsible for the pupils' achievement on the c r i t e r i o n variable is.stated i n the multiple regression analysis which follows. No additional variables are considered to have exercised an appreciable influence on this achievement. This s t a t i s t i c a l control of a l l the important concomitant influences, together with the careful imposition of actual controls i n the plan and administration of the experiment, enables any differences between the groups i n the c r i t e r i o n variable to be attributed to the treatments involved. Except for the materials of instruction used, the treatments are intended to be i d e n t i c a l . The materials of instruction d i f f e r only i n the characteristic of manipulability. The method of imposing the actual controls in the plan and administration of the experiment i s described i n Chapter I I I ; method of imposing the s t a t i s t i c a l control over the independent variables i s described i n Chapter V. the 106 FOOTNOTES A l l e n L. Edwards, Experimental Design i n Psychological Research, (New York: Rinehart & Company, Inc., 1950), p. 335James E. V/ert, Charles 0. Neidt, and J . Stanley Ahmann, S t a t i s t i c a l Methods i n Educational and Psychological Research, (Mew York: Appleton-Century-Crofts, Inc., 1954), p. 343. C. C. Ross and Julian C. Stanley, Measurement i n 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 A b i l i t y , p. 12. 107 CHAPTER V THE STATISTICAL ANALYSIS I. INTRODUCTION The problem to be analyzed s t a t i s t i c a l l y i n 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 s t a t i c representations of the aids used by the experimental group. Referred to as "visualization" materials, these aids are intended to possess characteri s t i c s i d e n t i c a l to those of the manipulative materials i n 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. the conclusion of the experiment At one of these tests was readministered to provide a measure of the c r i t e r i o n variable. The original data i s contained i n Appendix F. Broadly stated, the hypothesis to be tested i s that there i s no significant difference between the achievement of the two treatment groups 108 on the c r i t e r i o n 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 i n the mean scores of the two groups on the c r i t e r i o n , after allowances have been made for chance differences i n the mean l e v e l of achievement i n the control variables, may be accounted for e n t i r e l y by chance fluctuations i n random sampling. The allowances for i n i t i a l differences are to be made i n terms of the multiple regression of the c r i t e r i o n measure (Y) on the control measures (X^; X^; X^; X^). Analysis of Covariance i s the s t a t i s t i c a l procedure used to test this hypothesis. Commenting on the use of the analysis of covariance i n research, Edwards states: In particular, i t i s applicable to those situations where the matching of'groups i s 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 e f f e c t i v e l y used to reduce the error mean square i n the test of significance.^ Lindquist states: ... through a purely s t a t i s t i c a l control we can secure the . same precision i n 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 i s especially useful to experimental psychologists when for various reasons i t i s impossible or quite d i f f i c u l t to equate control and experimental groups at the s t a r t : ... Through covariance one i s able to effect 109 adjustments i n f i n a l or terminal scores which w i l l allow for differences i n some i n i t i a l variable. These statements attest to the s u i t a b i l i t y of analysis of covariance to the present problem i n which no attempt was made to match the groups with reference A further explanation to any of the control variables. of the a p p l i c a b i l i t y 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 i n the criterion variable and the four control variables. It w i l l be observed that the means of the two groups i n each variable d i f f e r very s l i g h t l y . Likewise, except i n the case of the v a r i a b i l i t y on the Otis Test (X^), the standard deviations of the two groups i n each variable d i f f e r very s l i g h t l y . Even though the difference between the group means of the control variables does not seem large enough to influence greatly the difference i n 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 c r i t e r i o n and control scores, i s worthwhile. The correlation of means between-groups, and the correlation of variables within groups, w i l 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 i n 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 Variable (Y) I n d e p e n d e n t (X ) (X ) 2 Experimental 14.492 7.797 17.559 108.305 20.356 Control 14.227 8.795 16.591 110.375 20.386 4.538 3.002 4.022 10.835 6.947 3.718 5.301 15.806 7.490 1 V a r i a b l e s (X ) (X ) 3 4 Means Standard Deviations Experimental Control 5.315 111. c r i t e r i o n 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 t h i s . By means of an analysis of these correlations a preliminary examination, as given i n Subsection 5 of this chapter, may be undertaken to determine whether the analysis of covariance w i l l prove e f f i c i e n t i n detecting differences between the means of the two groups on the c r i t e r i o n Y. In the present problem, where there i s l i t t l e difference between the means of the two treatment groups i n each variable, the primary purpose of the analysis of covariance i s to increase the precision of the test of significance. In both the analysis of variance and the analysis of covariance the F value used i n the test of significance of the means i s 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 i n the numerator and denominator of the F ratio that w i l l result from the application of the analysis of covariance. 112 Using the covariance analysis i n the present problem for the purpose of making allowances for differences between the means of the experimental and control groups i n the X variables follows unquestionably 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 i n the application of the covariance technique to the present problem i s to analyze the data for each of the independent variables i n the usual manner of an analysis of variance. The purpose of this step i s to test the hypothesis that the scores of the two treatment groups i n each of the independent variables are i n r e a l i t y random samples drawn from the same normally distributed population and, further, that the means between the groups i n each variable d i f f e r only through the fluctuations of sampling. The rationale of the analysis of variance, by which this hypothesis i s tested, i s stated comprehensively by Lindquist: The basic proposition (of the analysis of variance) i s 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 i s 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 l i e s below the value i n the table for F that corresponds to the ..selected level of s i g n i f icance. 5 113 The variance of the group means, the f i r s t independent estimate of the population variance, i s represented by the sum of the square of the deviations of the mean of each group from the general • mean. the Each of these squared deviations i s weighted or multiplied by number of subjects i n each group i n order to put them on a per individual measure basis. The greater the difference i n 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 i s equal to k (M - M) e e ' 2 + k (M - M) c c ' 2 N 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) i s relatively easy to understand but usually time consuming to compute; i s mathematically i d e n t i c a l and i s more generally used".^ the second This second formula i s %* ^ 2 (XX) e2 w between groups = k e ^+ (IX k ) c c 2 — (£X) 2 N where N refers to the number of subjects i n the total sample. The variance within groups, the second independent estimate of the population variance, i s 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 i s free from any influence of the difference i n the means between the treatment groups. The sum of squares within groups i s •(S, - M ) + (S_ - M ) +....+ (S - H ) 2 1 where 2 2 g ; g n y 2 g' etc. designate the scores of individual subjects i n 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) i s self-explanatory and the second 7 saves time". The second method suggested by these authors i s one i n which the within sum of squares i s not directly computed. It i s 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 d i r e c t l y from the o r i g i n a l measures without f i r s t subtracting the mean. The formula used i n this case i s total ~ ** N; A Thus, the sum of squares within groups i s the difference between the sum of squares for the total and the sum of squares between groups. This i s shown as follows: f £ x 1 2 - £ x £ J - J (£xj '• / ( — 2 + (tx) 2 : - -t~ a x ) 2 ) — J 115 Therefore, x 2 within groups = £x 2 f ' " -' k e X "'^ k c 2 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 y i e l d , 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 d i f f e r 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 i s expressed as a quotient, called an F value. This F value i s then compared with the .05 and Q .01 points of the variance ratios tabled by Snedecor. The value at .05, given i n the table for a particular number of degrees of freedom, i s the value which would be exceeded only 5$ of the time as a result of sampling variation i f the n u l l hypothesis were true. Therefore, an F value which equals or exceeds the tabled value at the .05 l e v e l has a probability equal to or less than 5$. This means that there i s only 1 chance i n 20, or less than 1 chance i n 20, that an F value as large as this could be obtained by sampling variation. Consequently, a result that happens as seldom as this by 116 chance would be indicative of systematic differences between treatment effects, and so the n u l l hypothesis would be rejected at the 5$ level of significance. Further, i f the F value equals or exceeds the tabled value at the .01 l e v e l , i t means that there i s only 1 chance i n 100, or less than one chance i n 100, that a value as large as this could be obtained by sampling variation. Such a result, said to be significant at the 1% l e v e l , would be even more convincing evidence on which to reject the n u l l hypothesis. On the other hand, i f the F value f a l l s short of the tabled value at the .01 l e v e l , i t has a probability greater than ifo. This means that there i s more than 1 chance i n 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 l e v e l of significance, of systematic differences between treatment effects, and so the n u l l hypothesis would be considered tenable at this l e v e l . Further, i f the F value f a l l s short of the tabled value even at the .05 level, the probability i s greater than 5%. This means that there i s more than 1 chance i n 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 n u l l hypothesis would be considered tenable at the 5% level of significance. . 117 In accordance with the procedures presented i n the preceding pages of this subsection, an analysis of variance was computed for each variable i n the whole battery of test controls used i n this experiment, namely: X^, X , X^, and X^. 2 In addition, an analysis of variance was computed for the variable used as the c r i t e r i o n measure, namely: Y. In each case the formulae used are those shown on pages 114 and 115. The data required for substitution i n these formulae are contained i n Table XXVII. It may be noted i n passing that the sums of scores and the sums of squares of scores shown i n this table, as well as the sums of cross products shown i n Table XXXIV (page 130 ) and used i n a subsequent step, may a l l be secured i n a single operation on an automatic Monroe computing machine. The accumulating sums are carried i n the machine, and only the totals recorded. As an example of the procedures employed i n applying these data to the formulae referred to, the calculations of the sums of the squares of the variable X.^, for the t o t a l sample and within the subgroups, are shown below. The calculation of the sum of squares for the total * 1 X 2 - £ X 1 2 ^1 - N 12142 - (1254) 147 12142 - 1522736 147 12142 - 10358.88435 1783.11565 2 sample: 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 Montgomery Prince Charles Total for Experimental Group Fleetwood Hjorth Road Simon Cunningham Total for Control Group Total f o r Both Groups 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 9,912 5,075 5,311 20,298 33,9037 1,518 2,600 13,605 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 sxj 273,822 425,174 698,996 291,380 330,904 1,094,057 1,793,053 2x' 9,042 18,253 27,295 471,773 18,206 10,846 12,458 41,510 68,805 £x 2 EX. EX **1 r-2 119 The calculatipn of the sum of squares within the subgroups: - ( ^ EX V } ; k =: 2 1C c 12142 (774) 88 12142 599076 88 2 ] J } 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 t o t a l . Table XXVIII shows these three sums of squares for each of the five variables involved i n 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 £r \ w w Ex,* Total Within Groups Between Groups 3702.66667 3700.20031 2.46636 1783.11565 1747.87750 35.23815 3460.93878 3427.81510 33.12368 29062.46259 28911.13347 151.32912 7784.42177 7784.38906 0.03271 120 TABLE XXIX ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TV/O TREATMENT GROUPS ON THE INDEPENDENT VARIABLE X (Farquhar Test of Understanding - F i r s t Administration) Source of Variation Sum of Squares df Mean Square Total 146 1783.11565 Within groups 145 1747.87750 12.05433 1 35.23815 35.23815 Between groups F ' ^ 35.23815 " 12.05433 2.923 From Table F df 1/145 .923 " F 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 Variation Sum of Squares df Mean Square Total 146 3460.93878 Within groups 145 3427.81510 23.64010 1 33.12368 33.12368 Between groups Y ' ^ 33.12368 " 23.64010 ~ 1.401 F 1.401 From Table F df 1/145 F at .05 l e v e l = 3.91 F at .01 level = 6.81 121 TABLE XXXI ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TWO TREATMENT (GROUPS ON THE INDEPENDENT VARIABLE X, 3 (Otis S.A. Test of Mental A b i l i t y , Intermediate, Form A) Source of Variation /Sum of Squares Idf Total 146 29062.46259 Within groups 145 28911.13347 1 151.32912 Between groups *\ l j 1 4 5 = 151.32912 199.38713 = „_ °* Mean Square F. : 199.38713 151.32912 0.759 rFrom Table F df .1/145 Q F at .05 level = 3-91 F at .01 level = 6.81 7 5 9 TABLE XXXII ANALYSIS OF VARIANCE OF PERFORMANCE OF THE TV/0 TREATMENT iGROUPS ON THE INDEPENDENT VARIABLE X 4 (Stanford Advanced Reading Test: df Source of Variation Sum of Squares Form E) Mean Square Total 146 7784.42177 Within groups 145 .7784.38906 53.68544 1 .03271 .03271 Between groups -, = F ' 5 0.03271 53.68544 0.0006 F 0.0006 From Table F df -71/145 >F at .05 l e v e l = 3.91 F at .01 level = 6.81 122 The summary of the analyses of variance for the independent variables X , X , X_, and X. i s recorded i n Tables XXIX to XXXII, 1 d 5 4 n 0 inclusive. An examination of these tables reveals the partition of the t o t a l sum of squares into the two independent' estimates of the population variance referred to i n Lindquist s quotation on page 112. 1 estimates i s based on the variance within groups; One of these 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 i s a corresponding partition of the total number of degrees of freedom. This partition may be shown as follows: Sum of Squares Within groups Between groups TOTAL ' General Number of Degrees of Freedom Specific Number of Degrees of Freedom N - r 145 r - 1. 1 N-1 146 where N i s the total number of subjects and r i s the number of treatment groups. In Snedecor's table the F value at the .05 level of s i g n i f icance for 1 and 150 (the tabled value nearest to 145) degrees of freedom i s 3.91, while at the .01 level i t i s 6.81. The. F values i n each of these four analyses of variance f a l l considerably short of the value required f o r significance at the .05 level. Thus, the n u l l hypothesis i s tenable. The difference between 123 the means of the experimental and control groups i n each of the independent variables i s less than may be expected through the fluctuations of sampling. It may be concluded that the scores of both groups i n a l l four independent variables are i n r e a l i t y random samples drawn from the same normally distributed and homogeneous population. III. AN ANALYSIS OF VARIANCE OF THE CRITERION VARIABLE Y The second step i n the application of the covariance technique to the present problem i s to analyze the data for the dependent or c r i t e r i o n variable i n the usual manner of an analysis of variance. It w i l l be remembered that the re-administration of Farquhar s test at 1 the close of the experiment provided the measure of the c r i t e r i o n 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 i n the preceding subsection, the analysis of variance was computed and i s summarized i n 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 i s seen that the resulting F value (.097) f a l l s far short of significance at the .05 l e v e l ; i t i s , i n fact, considerably less significant than the F value (2.92) obtained i n 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 - F i n a l Administration) df Source of Variation Sum of Squares Mean Square Total 146 3702.66667 Within groups 145 3700.20031 25.51862 1 2.46636 2.46636 Between groups p; 1, 145 2.46636 25.51862 = 0.097 F 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 i n this statistical treatment i s to make allowance i n the analysis of the c r i t e r i o n scores ( Y ) for individual differences i n the control scores (X^, X^, X^, X^) obtained at the beginning of the experiment. The general procedure by which this purpose i s accomplished involves the prediction of the criterion variable from known values of the control variables. I f the deviation of the control scores of any pupil from the general means of these scores i s know* the amount by which the pupil's c r i t e r i o n score would be expected to deviate from the criterion mean may be computed. This expectation, which i s based entirely on i n i t i a l performance i n 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 i s otherwise referred to as the regression of Y on these control variables though, as G-arrett points out, the "original meaning of stepping back 1 to some stationary average i s not 1 9 necessarily implied". Certain basic assumptions inherent i n this prediction or regression procedure are discussed later.^ The difference between the predicted sum of squares of the c r i t e r i o n and the actual sum of squares of the c r i t e r i o n i s known as the residual sum of squares or the sum of squares of errors of estimate.. The relationship may be shown i n this way: Original sum of squares of criterion tfY ) 2 Sum of squares of c r i t e r i o n predicted on control variable scores (Sum of squares due to regression) Adjusted sum of squares of c r i t e r i o n (Sum of squares of residuals or sum of squares of errors of estimate. t The residuals or errors of estimate are the sums of squares based upon the variation remaining i n 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 i n Table XXXIII, are adjusted so that the v a r i a b i l i t y contributed to these sums of squares by the control scores X., X„, X,, and 1 X^ i s removed or held constant. d $ 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 i n Table XXXIII, i s made of the c r i t e r i o n means to ascertain whether these means between the two treatment groups have become s i g n i f i c a n t l y different as a result of taking into account the individual differences i n the control variables. This analysis i s located i n Subsection VIII of this chapter.: IV. COMPUTATION OF THE SUMS OF CROSS PRODUCTS IN DEVIATION FORM FOR EACH PAIR OF VARIABLES The t h i r d step i n the application of the covariance technique to the present problem i s to compute the sums of cross products i n deviation form for each pair of variables. Four prediction variables and one c r i t e r i o n variables involve ten pairs of cross products. The analysis of covariance represents an extension of the analysis of variance Y on the X variables. i n that i t takes into account the regression of The dependence of regression upon the r e l a t i o n - ship between the Y scores and each of the X., X-, X,, X. scores may be I ^ p 4 expressed i n Edwards' words: " I t i s the presence of correlation or association between the two that makes prediction possible, and the efficiency or accuracy of such predictions i s a function of the degree or strength of the relationship that e x i s t s " . 1 1 The formulae used to compute the sums of squares due to regression, or i n other words the predicted sums of squares of the c r i t e r i o n , are derived from the correlation formula: Lindquist traces the derivation which results i n £xy = £(x-x) (y-y) +£xy It i s understood that this summation i s 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 i s the mean of the PRODUCTS of their deviations from their means, just as the VARIANCE of a single variable i s the mean of the SQUARES of the deviations. . 12 Stated i n other words, i t may be said that the total sum of crossproducts may be analyzed into two components, just as i n the analysis of variance i t was possible to analyze the t o t a l sum of squares into two components. The f i r s t component,;is the sum of cross products within groups. I t 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 i s the sum of cross products between groups. I t i s 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 w i t h i n groups and the sum of squares between groups. Referring to h i s 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 c a l c u l a t i n g the sum of cross products. Instead, i t i s easier to.take the values of X and Y from zero o r i g i n and to apply a c o r r e c t i o n term to the products of the o r i g i n a l values. ' The r e s u l t i n g formulae are: £ V total = £ 1 X - Y £ E N: ^ 1 between groups = 1 k + e 1 . k c — 1 N 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. £ X X Y - SXSY HT This i s shown as follows: - ( ( C x ^ e + iT~ J - GBCjE^c 5ET CXEY-J ~N J Therefore, 7ty.' 1 ,. w i t h i n groups = £ X , Y - f (£x.EY)e 1 J 1 • k e + (EXCY)C 1 k c In a l l cases the subscripts designate the treatment groups, k designates the number of subjects i n the treatment group referred t o , and N designates the number of subjects i n the total-sample. 129 The data required for substitution i n these formulae, are contained i n Tables XXVII (page 118) and XXXIV (page 130). As an example of the procedures employed i n 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, i s 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 P N - Y ZX = 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 k c e - I K U O _ (835)(460) 59 395300 = 19268 - 6666.10169 = 19268 - 17678.01078 = 1589.98922 = _ 1 9 2 6 i q 2 6 8 Q _ 1 k + 5 9 i (1252)'(774) 88 969048 88 + 11011.90909 As i n the case of the sum of squares, the sum of cross products between groups was not d i r e c t l y computed. I t 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 Montgomery £ X I C X X Total for Experimental Group Fleetwood Hjorth Road Simon Cunningham Total for Control Group Total for Both Groups 2,559 4,518 7,077 6,062 2,854 3,275 12,191 19,268 2 Y 5,127 10,386 15,513 9,708 6,119 6,370 22,197 37,710 3 Y 35,163 59,352 94,515 65,778 36,894 39,448 142,120 236,635 Y 6,119 12,189 18,308 12,794 6,780 7,654 27,228 45,536 2,803 5,439 8,242 6,030 3,486 4,046 13,562 21,804 19,388 31,327 50,715 20,779 25,866 87,627 138,342 3,363 6,512 9,875 8,051 3,960 5,098 17,109 26,984 39,310 73,811 113,121 66,700 47,245 51,395 165,340 278,461 £ X £ X Prince Charles 4 1 X 1 X 2 3 Cx x^ 2 - 40,962 £x x 4 6,812 14,662 21,474 13,038 8,631 9,619 31,288 52,762 Cx x 4 47,962 85,253 133,215 89,376 53,782 61,900 205,058 338,273 2 3 Ol o 131 the sum of cross products within groups from the sum of cross products for t o t a l . 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 i n both the sums of squares and the sums of cross products i s given on page 141. Tables XXVIII (page 119) and XXXV (page 132) contain the essential data to be used i n the calculation of the regression coefficients. This calculation i s found i n Subsection VI. These data, however, are f i r s t used i n Subsection V, where a preliminary examination i s made of the conditions under which analysis of covariance i s 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 i n the application of the covariance technique to the present problem i s 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 w i l l prove e f f i c i e n t i n detecting differences between the means of the groups on the Y variable. In addition, this description should be helpful i n interpreting the f i n a l 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 £ i y 1580.66667 1589.98922 -9.32255 r 2 x y 1934.00000 1924.96148 9.03852 £ 3 X y 5825.33333 5844.65254 -19.31921 £ 4 y 2607.66667 2607.95070 -0.28403 851.18367 885.34822 -34.16455 3164.43537 3091.41102 73.02435 1842.29932 1841.22574 1.07358 5038.63265 5109.43220 -70.79955 1908.12245 1909.16333 -1.04088 10188.06803 10185.84322 2.22481 x x £ i 2 X X C 1 3 X £ 1 4 X X E 2 3 X X £ 2 4 X X £ 3 4 X X 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 w i l l 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 Pearson product-moment method. taken from the means of the two r Substituting x^y be computed by the The formula used when deviations are distributions i s (between) = ^ x l y the appropriate values from the between groups source of variation i n Tables XXVIII (page 119) and XXXV (page 132) results i n the following: _ -9.32255 x y (between) = 1 /35.23815 x -9.32255 2.46636 y"86.9099636340 -9.32255 +9.32255 -1 In Appendix G similar calculations are shown for each of the remaining 14 three correlations x y, 0 ^ x,y and 3 x.y. 4 An examination of the means i n Table XXVI (page 110) reveals that i n three of these four pairs of variables there i s a negative 134 correlation. In other words, i n these cases the group with the higher mean i n the one variable has the lower mean i n the other. In Table XXXV (page 132) i t i s seen that i n these three pairs of variables the sums of cross products are negative. In analysis of variance, to produce significant differences i n the means of the treatment groups i t i s 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 i n this experiment are negative, i t would seem l i k e l y that the sum of squares due to regression within groups w i l l exceed the sum of squares due to regression for t o t a l . This anticipation 15 i s justified by subsequent calculations. Such a condition inevitably results i n 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 r a t i o . The nature of the correlation of the means between the treatment groups i s such that i t indicates that the precision of the test of significance w i l l be increased i n this problem by the application of the analysis of covariance. 135 Nature of c o r r e l a t i o n of i n d i v i d u a l scores within each treatment group The second part of t h i s subsection of i n d i v i d u a l scores w i t h i n groups. deals with the c o r r e l a t i o n I t w i l l be remembered that t h i s c o r r e l a t i o n 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 v a r i a b l e . The higher the correlations w i t h i n groups between the c r i t e r i o n variable and each of the independent v a r i a b l e s , the l a r g e r 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 d i v i d i n g the sum of squares of residuals by the number of degrees of freedom within groups, w i l l l i k e w i s e become smaller. Since this forms the denominator of the F r a t i o , it".'..will be seen that the strengths of the c o r r e l a t i o n s 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 e x i s t s by which the extent of the reduction i n the adjusted error variance may be estimated on the basis of the strength of the c o r r e l a t i o n s . These c o r r e l a t i o n s , corrected for attenuation, will now be calculated. The Pearson product-moment formula, shown on page 133, i s used. Four correlations must be calculated, namely: x^y, x y, x^y and 2 x^y. The necessary data are found i n the within groups source of v a r i a t i o n i n Tables XXVIII (page 118) and XXXV (page 132). As an example of the 136 procedure employed i n applying these data to the formula, the calculation of the within group correlation between x^ and y i s 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, c x.y. 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 i n this way for attenuation has been corrected to give an i n t r i n s i c correlation between two series of measures with postulated perfect reliability. obtain this correction for attenuation The formula used to is r r = J where r and r /r r x x x. 1 1 yy 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 i n the various tables of Chapter IV, i s reproduced i n Table XXXVI. As an example of the procedure employed i n applying to; the formula, the correction for attenuation within groups of x^y i s shown on page 137. calculations of corrections for attenuation remaining correlations x y, x y, and x y. 2 3 4 the data of the correlation In Appendix G similar are shown for each of the 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 ( f i n a l admin.) Y. .541 Farquhar's Test ( t r i a l admin.) X, .549 same Decimal Computation Test X . .821 same Otis Test X, .948 comparable-forms Stanford Test X .874 Split-half 2 4 Hoyt modification of K.R. Formula TABLE XXXVII PEARSON PRODUCT MOMENT COEFFICIENT OF CORRELATION, AND INTRINSIC CORRELATION, WITHIN GROUPS OF THE CRITERION VARIABLE WITH EACH OF THE INDEPENDENT VARIABLES Independent and Criterion Variables X l x 3 X y y 4 y Product-Moment Coefficient of Correlation I n t r i n s i c Correlation (after correction for attenuation) .63 greater than unity .54 .81 .57 .80 .49 .71 138 The average i n t r i n s i c correlation may be assumed to be approximately .73. ^ This coefficient may now be substituted i n the following formula, which was f i r s t referred to on page 135. 17 Sum of squares within groups (1 - r r £ ( n - 1) - 1 X / .... 0 2 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 - -73 ) 2 ( l - .5329) .467 In his treatment of this particular point, Lindquist states: "The ratio.between the adjusted error variance and the unadjusted error _8 2 variance i s very nearly equal to ( l - r ) . " Shown i n the form w of a proportion, this becomes: Adjusted error variance Unadjusted error variance _ ,457 Substituting the mean square variance within groups reported i n Table XXXIII (page 124): Adjusted error variance 25.51862 _ Adjusted error variance = ^g 7 approximately 12 139 This reveals that the error mean square used i n the f i n a l test of significance, after allowances have been made for the regression of the Y variable on each of the X variables, w i l l be approximately 12. This compares to 25.51862, the o r i g i n a l mean square used i n 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 i s apparent that the pscision of the experiment w i 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 i n the test of significance w i l l be substantially increased through the application of an analysis of covariance. The f i r s t indication i s 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 i n Table XXXIII - page 124) w i l l be increased by taking into account the regression of Y on the X variables. T he second indication i s that the correlation of individual scores within groups i s s u f f i c i e n t l y high (.73) to permit a considerable reduction i n the denominator of the same variance ratio (25.51862). Only an extremely insignificant difference i n 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 i n this subsection, this difference w i l l inevitably be more pronounced and may indeed be significant when multiple regression i s taken.: into account, and the results tested against the mean square • for error i n the analysis of covariance. The prospect of this outcome warrants the continuation of the present s t a t i s t i c a l VI. treatment. CALCULATION OF THE COEFFICIENTS OF REGRESSION The f i f t h step i n the application of the covariance technique to the present problem i s to calculate the coefficients of regression. This step i s necessary i n a multiple regression analysis; that i s , where more than one X variable i s involved i n the prediction of the c r i t e r i o n variable. In a four variable regression^problem the general regression equation i n deviation form i s y = When the expression + a x 21 (y - 2 a + 2 x 1 + a ^ ~ 2 2~ 3 3~ 4 4^ a 1 a ^ X a X a X 2 i s differentiated with respect to a^, a , a^, and a^, respectively, 2 and each of the derivatives i s set equal to zero the resulting normal 141 equations are E l x £ x 4 y y 19 a lE i a l£ l 2 + a a lE l 3 + a lS l 4 + a x X X X 2 + X X X a 2^ l 2 + a 3^ i 3 2^ 2 + a 3^ 2 3 + 3^ 3 + x X 2 ^ 2 3 + 2^ 2 4 + X 2 a x X X X x x x a a X x 2 + a x X X a + x 4^- 2 4 X ^ 3 4 X 4 3^* 3 4 X 4^* i 4 a a X 4^ 4 x 2 An explanation should now be made of the reason that the deviation form of the general regression equation i s used i n preference to the raw score form. Wert, Neidt, and Ahmann offer this explanation: As i n the case of single variable regression, the deviation score method can be used to calculate the prediction equation for multiple regression. The general equation i n deviation form differs from the equation i n raw score form i n that the C term has again disappeared. Thus the number of normal equations necessary 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 i s the p r i n c i p a l 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 i s the difference between these two sums. groups regression equations, The coefficients of the between therefore, do not need to be calculated. 142 Calculation of the coefficients of the t o t a l regression equation To obtain these coefficients i t i s necessary to substitute i n the four.normal equations the appropriate deviation values of the sums of squares and of cross products for the total sample. Sub- s t i t u t i o n i n the normal equations of these values, which are contained i n Tables XXVIII (page 119) and XXXV (page 132) y i e l d s : 1580.66667 = 1783.11565a +851.18367a +3l64.43537a +1842.29932a ] 1934.00000 = 2 4 851.18367a,+3460.93878a +5038.63265a_+1908.12245a. o 1 5825.33333 = d f> 4 3164.43537a.,+5038.63265a +29062.46259a,+10188.06803a, o 1 d 4 $ 2607.66667 = 1842.29932a,+1908.12245a +10188.06803a_+7784.42177a. o 1 5 d 4 Table XXXVIII indicates the values of the regression coefficients a,, a , a,, and a .(for total) which result from the solving of these ± 0 d 5 4 four simultaneous equations. The values were checked by substituting them i n the original equations and obtaining i d e n t i t i e s . Calculation of the coefficients of the within groups regression equation To obtain these coefficients the appropriate deviation values, also contained i n Tables XXVIII (page 119) and XXXV (page 132), of the sums of squares and of cross products for within groups are substituted i n the normal equations. The equations become: 1589.98922 = 1747.87750a.+885.34822a +3091.41102a,+1841.22574a, o 1 d 4 p 1924.96148 = 885.34822a +3427.81510a +5109.43220a,+1909.l633a, 1 o d J. p 4 5844.65254 = 3091.4110 2a.,+ 5109.43220a + 28911.13347'a_+10185.84322a, o i d 5 4 2607.95070 = 1841.22574a +1909.1633a_+10185.84322a_+7784.38906a n 1 2 3 4 143 TABLE XXXVIII REGRESSION COEFFICIENTS OF THE TOTAL REGRESSION EQUATION AND THE WITHIN GROUPS REGRESSION EQUATION Regression Coefficient a, Total Regression Equation Within Groups Regression Equation .585892389 .608249961 .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 manner of the previous solutiCns. The values were checked as i n 144 VII. CALCULATION OF THE SUMS OF SQUARES OF RESIDUALS The sixth step i n the application of the covariance technique to the present problem i s to calculate the sums of squares of residuals, otherwise known as the errors of estimate. As previously shown, the sum of squares of residuals i s obtained by subtracting the sum of squares due to regression from the o r i g i n a l sum of squares of the c r i t e r i o n . " ' This relationship i s 21 represented by the formula: Sum of squares of residuals The 2 = 2»y _ - (a^ioi^y + a^x^y „ + a^ux^y + a^x^y) sum of squares of residuals i s based upon the variation remaining i n Y after due allowance has been made for the of Y on each of the X variables. regression Through covariance, therefore, a s t a t i s t i c a l control has been maintained over those unmatched pupil a b i l i t i e s v/hich purport to be measured by the tests selected. Exercising this s t a t i s t i c a l control over these variables, which are considered to be the most important ones i n influencing the pupils' performance on the c r i t e r i o n , permits a precise evaluation of the treatment effects. Substituting i n the above equation the appropriate values obtained from Tables XXVIII (page 139'), XXXV (page 132), and XXXVIII' (page 143) results i n the following: 145 Sum of squares of residuals for total = 3702.66667 (.585892389)(l580.66667) C.287384266)(l934.00000) ( .078888583)(5825.33333) (.022633764) (2607.66667) = 3702.66667 926.10057149897463 555.801170.44400000 459.55229190637139 59.02131199944588 = 3702.66667 2000.47535 = 1702.19132 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 c o e f f i c i e n t s of r e g r e s s i o n between groups were not c a l c u l a t e d , the sum of squares of r e s i d u a l s between groups are obtained by f i n d i n g the d i f f e r e n c e between the two sums of squares already c a l c u l a t e d , thus: I Sum of squares of r e s i d u a l s ^between groups 'Sum of squares] of r e s i d u a l s } w i t h i n groups J Sum of squares^ of r e s i d u a l s \ for t o t a l J S u b s t i t u t i n g the appropriate values produces the f o l l o w i n g : 1702.19152 - 1681.84105 = 20.55029 Table XXXIX contains a summary of the computation of the sums of squares of r e s i d u a l s . 146 TABLE XXXIX SUMMARY OF SUMS OF SQUARES OF RESIDUALS Source of Variation Sum of Squares of Criterion Sum of Squares due to Regression Sum of Squares of Residuals Total 3702.66667 2000.47535 1702.19132 Within 3700.20031 2018.35928 1681.84103 2.46636 20.35029 An examination of this table confirms the general accuracy of the preliminary analysis contained i n 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 o r i g i n a l sum of squares of the c r i t e r i o n . 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 c r i t e r i o n . The effect upon the F value of these adjustments i n the original sums of squares of the c r i t e r i o n i s seen i n the summary of the analysis of covariance contained i n Subsection VIII. 147 VIII. CALCULATION OF F VALUE AND APPLICATION OF TEST OF SIGNIFICANCE The seventh step i n 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 c r i t e r i o n variable Y i s summarized i n Table XL. This analysis may be compared with the preliminary analysis of the Y means contained i n Table XXXIII (page 124), where no allowance was made for i n i t i a l differences between the groups i n the control variables. In presenting a comparison of the two tables, an explanation should be made concerning the change i n 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 i n v a r i a b i l i t y 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 i n obtaining the adjusted sum of squares between groups, no additional degree of freedom i s l o s t . It was stated previously that i n the present problem, where there i s l i t t l e difference between the means of the two treatment groups 148 TABLE XL ANALYSIS OF COVARIANCE OF PERFORMANCE OF THE TWO TREATMENT GROUPS ON THE CRITERION VARIABLE Y Source of Variation df Sum of squares of residuals Mean Square Total 142 1702.19132 Within groups 141 1681.84103 11.9280 1 20.35029 20.3503 Adjusted Means between groups \ 1 4 1 = 20.3503 1 1 , 9 2 8 0 = 1.706 F 1.706 From Table F df l / l 4 1 F at .05 level = 3.91 F at .01 level = 6.81 i n each variable, the primary purpose of the analysis of covariance r 22 i s 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 w i l l be noted that the mean square between groups (the numerator of the variance ratio) has been increased from 2.4664 i n the original analysis of variance to 20.3503 i n the f i n a l analysis of covariance..,. (the At the same time, the error mean square within groups denominator of the variance ratio) has been decreased from 25.5186 to 11.9280. 149 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, i t can be concluded with reasonable certainty that the difference which does exist i s due to sampling fluctuations rather than to a real treatment 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 i s necessary that the assumed conditions actually exist i n the design and conduct of the experiment. Wert, Heidt, and Ahmann emphasize this necessity: "The more the data i n an investigation depart from the s t r i c t fulfillment of the assumptions the more l i k e l y i s the investigator to reach erronious conclusions". 23 Lindquist l i s t s these assumptions as follows: ( 1 ) The subjects i n each treatment group were o r i g i n a l l y 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 c r i t e r i o n measures for each treatment group are a random sample from those for a corresponding treatment population. (4) The regression of Y on X i s the same for a l l treatment populations. (5) This regression i s l i n e a r . (6) The distribution of adjusted scores f o r each treatment population i s normal. (7) These distributions have the same variance. (8) The mean of the adjusted scores i s the same for a l l treatment populations. ^ 2 Referring to these conditions which establish the v a l i d i t y 6 f the procedure, Lindquist writes: Judging by past applications of the method of analysis of covariance i n educational and psychological research, the assumptions underlying the test of the hypothesis of equal treatment effects are, i n general, i n greater need of c r i t i c a l 151 attention than i s 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 i s v a l i d , and instances are numerous i n which one or more of the conditions have clearly not been satisfied.^5 The same author then deals s p e c i f i c a l l y 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 v i o l a t i o n : ... they (experimenters) seem to have assumed that the method eliminates the effects of any systematic differences that may have existed o r i g i n a l l y 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 i l l u s t r a t e unwise reliance upon analysis of covariance to remove systematic differences. These examples involve the use of analysis of covariance i n an experimental comparison of three ways of teaching fourth grade arithmetic. The f i r s t example i s one i n which ... throughout the f i r s t semester the classes had had different arithmetic teachers, who had not only differed i n 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 l i k e 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 l a t e r to use Method B had used a method which conflicted with Method B, so that considerable time was required early i n 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 achievement test scores, could possibly account for the effects .of these differences upon the f i n a l adjusted means of the treatment groups. 28 152 In the present study c a r e f u l p r e c a u t i o n s were observed i n the c l a s s s e l e c t i o n to avoid the i n a d v e r t e n t i n t r o d u c t i o n i n t o the experiment of the i n a d e q u a t e l y c o n t r o l l e d s y s t e m a t i c d i f f e r e n c e s d e s c r i b e d by L i n d q u i s t . The t h r e e main c r i t e r i a f o r the s e l e c t i o n o f the f i v e c l a s s e s 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 t o s e l e c t i n g t e a c h e r s who had been f o l l o w i n g reasonably c o n v e n t i o n a l methods i n t h e i r every-day t e a c h i n g 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 n e v e r t h e l e s s d i s i n t e r e s t e d i n the m a n i p u l a t i v e and v i s u a l i z a t i o n methods of t e a c h i n g 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 s e l e c t e d which c o u l d p o s s i b l y work to the advantage o r disadvantage of any c l a s s 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 , t h e r e f o r e , to i n t r o d u c e 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 o f imposing invalid c o n t r o l s over any of the s y s t e m a t i c d i f f e r e n c e s noted by L i n d q u i s t i n the f o r e g o i n g q u o t a t i o n s . The second example i s .one.io i n which ... the c l a s s e s were o r i g i n a l l y s e l e c t e d not at random but so as to d i f f e r markedly w i t h reference t o some t r a i t or c h a r a c t e r 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, t h a t the c l a s s e s had been s e l e c t e d a c c o r d i n g to a b i l i t y and i n t e r e s t , that the a b l e r and more i n d u s t r i o u s students had been a s s i g n e d to one c l a s s and the l e a s t able to another, and that a p p r o p r i a t e m o d i f i c a t i o n s i n i n s t r u c t i o n had been used w i t h these c l a s s e s during the f i r s t semester. Suppose then t h a t an i n i t i a l achievement p r o v i d e d the X-measures used i n the a n a l y s i s of c o v a r i a n c e . I n t h i s case, not o n l y 1 53 would Assumption 1 would be invalid, but differences i n regression (Assumption 4) and i n v a r i a b i l i t y of adjusted scores (Assumption 7) or even differences i n the nature of the regression (Assumption 5) might well be expected. Nevertheless, many applications of this type also may be found reported i n 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 i n these analyses, and shown i n 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 i s 3.91 at the .05 l e v e l . These data support the hypothesis that the scores of both treatment groups i n a l l four variables are i n r e a l i t y random samples drawn from the same normally distributed and homogeneous population, and that the means between the groups i n each variable d i f f e r only through the fluctuations of sampling. S t a t i s t i c a l evidence i s thus available to assure the v a l i d i t y 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 i s the assumption (Condition 4) that the regression of Y on X i s the same for a l l treatment populations. Decisions concerning the validity of the other assumptions - l i n e a r i t y of regression, normality of distribution, and homogeneity of variance - must generally represent judgments based on a p r i o r i considerations like those discussed i n e a r l i e r chapters, since available s t a t i s t i c a l tests of the v a l i d i t y of these assumptions are both low i n power and d i f f i c u l t to apply. A s t a t i s t i c a l test of homogeneity of regression, however, i s readily available. This s t a t i s t i c a l test of homogeneity of regression, performed to satisfy Assumption 4, i s presented i n the following subsection. X. STATISTICAL TEST OF HOMOGENEITY OF REGRESSION Regression refers to a correlation relationship between the c r i t e r i o n variable and each of the independent variables. From such correlations i t i s possible to determine the relative weight with which each independent variable "enters i n " or contributes to the c r i t e r i o n , independently of the other factors. To be homogeneous this weight which each independent variable contributes to the c r i t e r i o n must be the same within the limits of sampling error for a l l treatment groups. To satisfy Assumption 4 i n the present study, homogeneity i n this respect between the two treatment groups must be proven for four pairs of variables, namely: X^, X Y, X^Y, and X^Y. 2 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 i n 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 i n Table XLI i s 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 i s free from any influence of systematic differences i n the mans of the two treatment groups. TABLE XLI WITHIN GROUP VARIANCE ACCOUNTED FOR BY THE USE OF EACH OF FOUR INDEPENDENT VARIABLES: X , X , X^. g Variable X l X 2 X 3 X 4 Name of Variable Variance Proportion I n i t i a l Understanding of Concepts of Decimal Fractions 967.111 .479 Computational A b i l i t y i n Decimal Fractions 521.670 .259 Mental A b i l i t y (Otis Test) 491.170 .243 38.408 .019 2018.359 1.000 Reading A b i l i t y (Stanford Test) Total 156 S p e c i f i c a l l y , the problem involved i n this subsection i s to determine whether the regression coefficients of the two pairs of variables X^Y and X^Y d i f f e r s i g n i f i c a n t l y i n the two treatment groups. This involves the testing of two hypotheses which, stated i n terms of beta c o e f f i c i e n t s , are: >6 E 1E Y X =y6 c lC Y X a n £ E 4E d Y =/6 C 4C Y X X I f these null hypotheses remain tenable i n the test of significance, i t may be said that the values of b y and b Ex do not d i f f e r y x 1 E C 4 C s i g n i f i c a n t l y , and neither do the values of b y and b E x y 4 E C 4c. x It w i l l 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 ^ =>6 ic Y X c A statement of the purpose of each calculation involved i n the testing of this hypothesis i s included i n the sequence of enumerated 35 steps which follow; 1. Sum of squares of errors of estimate for the variables X^ and Y i n the two treatment groups. For the Experimental and Control groups independent calculations are made of the sum of squares based upon the variation remaining i n 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 f o r 157 the variables X^ and Y i n the two treatment groups. These calculations are presented by the following formula, i n which the sums of squares and of cross products are expressed i n deviation form: V 2*i / 2 E k + v E C \ ° E*1C 2 / -4 It may be noted that the sums of squares and cross products i n this formula are the basic quantities referred to, respectively, on pages 114 and 128. They are obtained from raw scores i n the following manner: £x, 2 - S c 2 - N' N When these equivalents are substituted i n the deviation formula shown above, the raw score formula shown on the f i r s t line of page 159 i s obtained. The necessary raw score data i s contained i n Tables XXVII (page 118) and XXXIV (page 1 3 0 ) . The denominator i n the formula refers to the degrees of freedom available for the sum of squares being calculated. Two degrees 158 of freedom are l o s t due to the r e s t r i c t i o n s imposed by y_ 2 2 and y . Two a d d i t i o n a l degrees are l o s t through the c a l c u l a t i o n s of the regression c o e f f i c i e n t s b and b c y E lE X y C lC X The complete c a l c u l a t i o n s , shown on the next page r e s u l t i n the following: 3 = 2 vv y x l 15.661343 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 C o e f f i c i e n t s of y on x^ f o r the Experimental and Control Groups. The regression c o e f f i c i e n t of y on x^ may be w r i t t e n b =: £x-y =-— At t h i s point c e r t a i n data obtained i n the c a l c u l a t i o n s of Step 1 are summarised for convenience i n Table X L I I . These data are used i n the present step and the one immediately following. Step 1 i n the test of the n u l l hypothesis that „ Q = E IE / Y c ic x CALCULATION OF SUM OF SQUARES OF ERRORS OF ESTIMATE FOR THE VARIABLES X AND Y IN THE TWO TREATMENT GROUPS 11 E 2 -cy.) - h . 1E-E s. -^i _EJ 2 E E y ZY *E Wic'-P^ic) k 13605 (855) 5 9 2 (460)(855 ((7077 59 * (460)' 4118 59 E + k C - 2 4 r 20298 (1252) 88 2 - (12191 8024 - (774)(1252)| 88 J (774)' 88 59 + 88 - 4 ( 1214.746 - (410.898 531.559" 2485.455 - (1179.091)' 1216.318 143 } 897.120 * [ 143 S y 2 X l = 15.661343 1342.452 j TABLE XLII SUMS OF SQUARES AND CROSS PRODUCTS OF VARIABLES x CALCULATED INDEPENDENTLY FOR EXPERIMENTAL AND CONTROL GROUPS Sum of Squares £ 1E X 2 Z* = : ' 5 3 1 Sum of Cross Products £ lE E 5 5 9 X 2 = 1C AND y, y = 4 1 0 2xiC c y 1216.318 = * 8 9 8 1 1 7 9 * 0 9 1 Using the appropriate data i n the regression coefficient formula results i n b = ^^E 410.898 531.559 .773 b y C lC X = 1179.091 1216.318 .969 4. Standard Error of the Regression Coefficients b y and b y c ic x This i s represented by the formula: S.= E lE X 161 Using the approp i a t e data i n t h i s formula results i n y 3.957 /53I7559 E lE X .172 y 3.957 /1216.318 c ic x .113 Standard error of the difference between the regression coefficients b y and b ' C lC E lE X y X This i s represented by the formula: / 1 2 J = 2 y i E lE X ^(.172)* y I C lC X (.113)' .2.06 Test of significance of the difference between the regression coefficients b V (.773) and b X E IE V J (.969). X C 1C The t value i s obtained by d i v i d i n g the difference between the regression c o e f f i c i e n t s by the standard error of the difference between the regression c o e f f i c i e n t s . 162 This i s represented by the formula: b b y t 34 C lC X y E lE X = .969 - .773 .206 ( .951 This t value may be evaluated by entering the t table with 35 kg + - 4 = 143 degrees of freedom. For the two-tailed test of significance of the n u l l hypothesis that /^ E lE y X = y C lC' X a * o f 1.976 would be required at the .05 level 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 .951 > the null hypothesis remains tenable. that the regression coefficients It may be said, therefore, b and b do not v x v x ^E IE 1C differ significantly. Test of the Null Hypothesis t h a t ^ J Y X E 4 E = /5 c 4 C Y X In the testing of this hypothesis the procedure i s i d e n t i c a l to that followed i n the treatment of the foregoing hypothesis. 1*-. Sum of squares of errors of estimate for the variables X^ and Y i n the tv/o treatment groups. The necessary raw score data i s contained i n Tables XXVII (page 118) and XXXIV (page 130). 163 The complete calculations, shown on the next page, result i n the following: s 2 y4 = 19.755776 x 2. Standard error of estimate. = 3. 4.445 Regression Coefficients of y on x^ f o r the Experimental and Control Groups. At this point certain data obtained i n the calculations of Step 1 are summarized for convenience i n Table XLIII. These data are used i n 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: £x 4 £x 4 c 2 E 2 Sum of Cross Products = 2847.525 £ x = 4936.864 £ x E E = y 4 y 4 C C = 9 ° - 6 7 8 ' 2 7 5 1 7 0 4 3 164 CM CM O SB EH in CM rH W K EH CO CO CM S3 M CM o o X X D - rH v y O CO CM CM CCM W i X O X CM u co o PH* rH M ft! c nCO t— CO rH CM tn c - O H in CM VO • CO • O VO t— tn Hc n U4 in m • in CM CM tn CO CM CO CM < > CM H 05 -p a += CO •rl CO O P H PH >» .S3 ft; a o O CM O +> CQ CD -P U3 p W « + EH PH co ft! < t ! o PH H EH <U t-H t > CO CM X CQ CM CO in CM VO in [- • • tn O= i c nCO in c n CM CM t- in in <J1 c o in 111 • CM c n VO CM CM c n in c n c - CM CO C CO rH O O •H tn c o a H CM o in o I VO tn O rH <! o t-3 CM CM CM X CO c n c n in 0.J CO pq H '— in in CO CM v• CTi ^ . rH in rH o o CM CM CM rH rH in —' —s o tn rH tn CO CO o w d ) 4 5 c— • I *3 Of CO <H tm in CM CM Q W « EH cjj S EH M S3 EH pq co S W EH «! CO ft; VO C- c n PH :=> 0 43 o CO c« to PH O CO CO w K t— rH rH • X X m <: CM X >> CO CM X CO CO Using the appropriate data i n the regression c o e f f i c i e n t formula"results i n J = E Y X 4 E 905.678 2847.525 .317 b y = C 4C X 1704.275 4956.864 .545 Standard Error of the Regression Coefficients b_ _ and E 4E b C 4C y y X X Using the appropriate data i n the formula r e s u l t s i n % = Y E X 4 E 4.445 2847.525 .085 = s b y C 4C X 4.445 4956.864 .065 Standard error of the difference between the regression coefficients b and b v x V Y *E 4E C 4C Using the appropriate data i n the formula r e s u l t s i n J \-* 2 = X /(.085) .104 2 + (.065) 2 166 6. Test of s i g n i f i c a n c e of the difference between the regression coefficients b y (.317) and b C 4C E 4E X y (.345). X Using the appropriate data i n the formula r e s u l t s 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 n u l l hypothesis ^^YgX^g = Y^X^g, that a t of 1.976 would be required at the .05 l e v e l of s i g n i f i c a n c e , 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 c o e f f i c i e n t s b v x do not d i f f e r s i g n i f i c a n t l y . C 4C y and X A non-significant difference between treatment groups has thus been proven f o r the regression c o e f f i c i e n t s of X^Y and X^Y, two of the four p a i r s of variables involved i n t h i s experiment. I t w i l l be r e c a l l e d that the independent variables X^and X^ made, respectively, the greatest and the l e a s t contribution to the prediction of the c r i t e r i o n . Accordingly, the t value i n v o l v i n g each of these variables with the c r i t e r i o n are .951 (X^Y) and .269 (X Y ). 4 167 The t values of the other two pairs of variables, X^Y X,Y, w i l l l i e between these two extreme t values. and 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 c r i t e r i o n variable i s the same within the l i m i t s of sampling error for both of the treatment groups i n 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 c r i t e r i o n of using certain manipulative materials i n group instruction. These conclusions are contained i n Chapter VI. XI. THE USE OF A BINARY COMPUTER (ALWAC I I I - E ) IN PERFORMING AUTOMATIC COVARIANCE COMPUTATIONS The computations involved i n the covariance analysis reported i n the preceding subsections of this chapter were performed with the assistance of an automatic Monroe computing machine. addition to being treated i n this way, The raw data, i n were processed through the electronic binary computer, Alwac III-E. The Alwac III-E operates on the binary counting system. In this system each d i g i t 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 i s 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 i n the treatment of the present data was written by Dr. T. Hull of the Department of Mathematics, University of B r i t i s h Columbia. It was designed to accommodate a maximum of eight- variables i n the performance of the various covariance calculations. As the f i r s t step i n dealing with the present data, a Flexowriter was used to punch on tapes the pupils' scores i n the five variables. On the tapes these scores appear i n the binary form. The next step was to process the punched tapes through another Flexowriter which operates i n 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 i n the formulae employed, the results obtained i n the Alwac computations agree with those that have already been reported i n 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 i n Psychological Research, (New York: Rinehart & Company, Inc.,) 1950, p. 355. 3 E. F. Lindquist, Design and Analysis of Experiments i n Psychology and Education, (Boston: Houghton M i f f l i n Company, 1953), p. 318 4 Henry E. Garrett, S t a t i s t i c s i n Psychology and Education, (Toronto: Longmans, Green and Co., 1953), p. 289. 5 E. F. Lindquist, S t a t i s t i c a l Analysis i n Educational Research, (Cambridge, Mass.: The Riverside Press, 1940), p. 91 g James E. Wert, Charles 0. Neidt, and J . Stanley Ahiaann, S t a t i s t i c a l Methods i n Educational and Psychological Research, (New York: Appleton-Century-Crofts, Inc., 1954), p. 175 7 I b i d Q Most s t a t i s t i c textbooks reproduce this table which i s taken from Snedecor: S t a t i s t i c a l 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, S t a t i s t i c a l Analysis for Students i n Psychology and Education, (New York: Rinehart & Company, Inc., 1946), p. 261. 12 E. F. Lindquist, S t a t i s t i c a l Analysis i n Educational Research, pp. 183-184 13 Edwards, Experimental Design i n Psychological Research, p. 339. 14 While the correlation of each of the ten combinations of variables i s involved i n the calculation of the regression coefficients, i t i s the correlation of only the c r i t e r i o n variable with each of the independent variables (four pairs i n a l l ) which i s involved i n the calculation of the sum of squares due to regression. It i s the estimation of this sum which i s the concern of this subsection. 170 1 5 Infra, p. 145 16 Since the i n t r i n s i c cornaLation of x^y i s greater than unity, the uncorrected coefficient of .63 was used to arrive at this fraction. 17 Edwards, Experimental Design i n Psychological Research, p. 347 18 Lindquist, Design and Analysis of Experiments i n Psychology and Education, p. 327. 19 This derivation procedure, i n greater detail, i s contained i n Wert, Neidt, and Ahmann, _op_. c i t •, p. 241. 20 Ibid 21 An explanation, from which the derivation of this formula may he deduced, i s contained i n 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 i n Psychology and Education, p. 3232 5 Ibid, p. 328 Ibid, 27 Ibid. 28 Lindquist, Design and Analysis of Experiments i n Psychology and and Education. 29 Supra, pp. 48 et seq. 30 Lindquist, Design and Analysis of Experiments i n Psychology and Education, pp. 329 et seq. 3 1 Ibid, p. 330 171 32 J Ibid. The general plan followed i n the development of this test of homogeneity of regression i s taken from: Allen L. Edwards, S t a t i s t i c a l 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 i s based upon both t a i l s (positive and negative) of the distribution of t, there i s the probability of obtaining a positive or negative t. In the formula the two terms of the numerator may be arranged i n either order. To obtain a positive numerator i n the present case, the b term has been made the minuend. y c ic x The abridged t table contained i n most s t a t i s t i c s textbooks i s from Table IV of Fisher: S t a t i s t i c a l Methods fcr Research Workers, (Edinburgh: Oliver & Boyd, Ltd.) Additional entries (over 30 df) are taken from Snedecor: S t a t i s t i c a l 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 i n contributing to an understanding of particular decimal concepts. Since manipulability of a concept i s 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 concepts. This study may j u s t i f i a b l y be included i n the quest for these materials because, though subjective opinions are common, objective studies involving manipulative aids are not only few i n number but are not designed s p e c i f i c a l l y to determine their effectiveness. Problems proposed by the investigation. 1. Do pupils who are taught with the use of certain mani- pulative aids i n the manner prescribed by this experiment achieve an 173 understanding of decimal fractions that i s significantly different from the achievement of pupils who are taught with the use of visualization materials similar to the manipulative aids i n a l l details except manipulability? 2. What i s the relative weight with which each of the four independent variables, i n i t i a l understanding of the processes involved i n 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 , "enters into",.or contributes to, the c r i t e r i o n variable independently of the treatment 3. groups? When the concomitant influences represented by the four independent variables referred to i n 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 v i s u a l ization materials? 4. Por which treatment group - experimental or control - i s there the higher correlation between achievement on the c r i t e r i o n 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 i n accordance v/ith certain c r i t e r i a , then matched on the basis of size, and f i n a l l y , 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, i n turn, intended to possess similar characteristics except with respect to that of manipulability. This characteristic emerged, therefore, as the experimental variable. The c r i t e r i o n measure i s Farquhar* s Test of Understanding of the Processes Involved i n Decimal Fractions. The hypothesis tested i s that no significant difference exists between the achievement of the two treatment groups on the c r i t e r i o n variable. By means of a battery of four tests (Farquhar*s Test, a Decimal Fraction Computation Test, the Otis Test of Mental A b i l i t y , and Stanford Reading Test), measures were obtained of pupil a b i l i t i e s i n areas which v/ere considered to influence achievement on the c r i t e r i o n . The efficiency of each test for i t s purpose was f u l l y 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 fluctuations of sampling. These differences were then controlled s t a t i s t i c a l l y by the analysis of covariance, which allows for the correlation between c r i t e r i o n and independent variable scores. The resulting F value, though substantially larger than the F value obtained i n the original analysis of variance of the c r i t e r i o n 175 variable, remained insignificant at the .05 l e v e l , and the n u l l hypothesis v/as sustained. II. Summary of results. CONCLUSIONS Results of the experiment, stated as direct answers to the problems proposed by the investigation, are as follows: 1. There i s no significant difference between the achieve- ment on the c r i t e r i o n test of the pupils taught with the use of the manipulative materials and those taught with the use of the v i s u a l ization materials. (The F value obtained i n the analysis of variance i s 0.097, while an F value of 3.91 i s required at the .05 level of significance.) 2. Of the t o t a l influence which the four independent variables exerted upon the achievement on the c r i t e r i o n test, the percentage contributed by each variable, independently of treatments, i s as follows: ( l ) i n i t i a l understanding of the processes involved i n decimal fractions - 48$; (2) computational a b i l i t y i n decimal fractions 26$; ( 3 ) mental a b i l i t y - 24$; (4) reading a b i l i t y - 2$. 3. When the concomitant influences represented by the four independent variables are held constant by the s t a t i s t i c a l procedure of analysis of covariance, there i s 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 v i s u a l i z a t i o n materials. As a result of holding constant these concomitant influences, however, the F value obtained i n 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 XY V 2 Experimental .51137 .46417 .65998 .48592 Control .67816 .57482 .53172 .48654 It w i l l be noted that the largest difference i n correlations between the treatment groups i s 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 i n Appendix G.) Neither group, therefore, has a s i g n i f i c a n t l y higher correlation between achievement on the c r i t e r i o n variable and achievement on each independent variable. Interpretation of results. The f i r s t interpretation deals with the fact that the F value obtained i n the analysis of covariance of the c r i t e r i o n variable (1.706) was larger than the F value obtained i n the analysis of variance of the same variable. (0.097). The control group achieved the higher mean i n three of the four independent variables (Table XXVI - page 110), namely: i n i t i a l understanding of the processes involved i n decimal fractions (F value i s 2.923), mental a b i l i t y (F value i s 0.759), and reading a b i l i t y (F value i s 0.0006). The experimental group achieved the higher mean i n the remaining independent variable, namely: computational a b i l i t y i n decimal fractions (F value i s 1.40l). Despite this i n i t i a l advantage of the control group (though i s no case was the difference between group means s i g n i f i c a n t ) , the experimental group at the end of the instructional period achieved the higher mean i n the c r i t e r i o n test (F value i s 0.097). 178 The i n i t i a l understanding of decimal fractions, i n which the largest difference between group means existed, was.also the variable which was most predictive of achievement on the criterion, independently of the treatment groups. Therefore, i t was the s t a t i s t i c a l control of this variable i n particular which increased the f i n a l F value obtained i n the analysis of covariance to 1.706. The second interpretation deals with the correlations for both treatment groups between achievement achievement on the c r i t e r i o n variable and 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' a b i l i t y i n respect to the four areas considered i n no way determined the effectiveness of the manipulative aids i n 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 i s no advantage, or disadvantage, for an unselected group of Grade VII pupils i n 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 s t a t i c representations of these materials, and which are thereby intended to possess similar characteristics i n a l l details except that of manipulability. 2. The manipulative materials used i n this investigation are neither more nor less effective than the s t a t i c representations as 179 media for conveying an understanding of specific decimal fraction concepts to Grade VII pupils of any particular capacity i n the following areas: i n i t i a l understanding of decimal fraction processes, computational a b i l i t y i n decimal fractions, mental a b i l i t y , and reading ability. 3. It must not be inferred that any generalization concern- ing the effectiveness of these specific materials, of instruction, which were used i n this investigation exclusively by the teacher for group demonstration purposes, would be applicable also to similar materials i f they were used i n a teaching procedure i n which the pupils themselves participated individually i n the.manipulative a c t i v i t y . 4. I t must not be inferred that any generalization concerning the effectiveness of the specific manipulative aids used i n this investigation, i n a.brief teaching assignment devoted exclusively to the rationalization of processes, would be applicable also to the same materials i f they were used i n a teaching assignment of longer duration, and/or a teaching assignment i n 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 i n decimal fractions v/as the variable most predictive of achievement test of understanding. on the f i n a l Computational a b i l i t y i n decimal fractions and mental a b i l i t y each shared approximately one-half the predictive 180 capacity of the i n i t i a l test of understanding. Reading a b i l i t y was a negligible predictor of achievement on the f i n a l 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 i n the investigation necessitates that certain reservations be made with respect to these inferences and conclusions. Teacher comments show that the presentation of certain concepts v i a manipulative devices caused d i f f i c u l t i e s i n pupil learning. This indicates a need for further investigation i n 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 s p e c i f i c a l l y i n 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 i s longer and when teachers are trained i n their use i s a problem for further investigation. Conclusions 3 and 4, submitted i n 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 i n a teaching procedure i n which the pupils participated i n the manipulative activity. Another appropriate investigation would be to determine the effectiveness of manipulative aids used i n a teaching procedure i n which the HOW-WHY sequence was varied from that followed i n the present study. The implications relating to these two conclusions were stated i n Subsection VIII of Chapter I I I . 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, 1953Buckingham, Burdette R. Practice. Toronto: Elementary Arithmetic - Its Meaning and 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. and Education. S t a t i s t i c a l Analysis for Students i n Psychology New York: Rinehart and Company, Inc., 1946. Edwards, Allen L. Experimental Design i n Psychological Research. New York: Rinehart and Company, Inc., 1950. Edwards, Allen L. S t a t i s t i c a l Methods for the Behavioral Sciences. New York: Rinehart and Company, Inc., 1954. Garrett, Henry E. Statistics i n Psychology and Education. Longmans, Green and Company, 1953. Gesell, Arnold. Behaviour. Toronto: Infant Development: The Embryology of Early Human New York: Harper and Brothers, 1952. Lindquist, E. F. S t a t i s t i c a l Analysis i n Educational Research. Cambridge, Mass.: The Riverside Press, 1940. Lindquist, E. F. Design and Analysis of Experiments i n Psychology and Education. Boston: Houghton M i f f l i n 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. Burdett Company, 1953. New York: Silver Morton Robert Lee et a l . , Making Sure of Arithmetic - Grade Seven. New York: S i l v e r Burdett Company, 1955. 183 Piaget, Jean. Judgment and Reasoning i n the Child. Harcourt, Brace and Company, 1928. Ross, C. C. and Julian C. Stanley. New York: Prentice-Hall, Inc., New York: Measurement i n Today's Schools. 1954. Spitzer, Herbert. The Teaching of Arithmetic. M i f f l i n Company, 1954. Boston: Houghton Werner, Heinz. Comparative Psychology of Mental Development. New York: Harper and Brothers, 1940. Wert, James E., Charles 0. Neidt, and J . Stanley Ahmann. S t a t i s t i c a l Methods i n Educational and Psychological Research. New York: Appleton-Century-Crofts, Inc., 1954. B. DISSERTATION ABSTRACTS Dissertation Abstracts, Vol. XV. Microfilms, 1955. C. Ann Arbor, Michigan: University 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 i n Arithmetic," Teacher, Vol. 1, 1954, pp. 1-5. Buswell, G. T. "Study Pupils' Thinking i n Arithmetic," Kappen, Vol. 31, 1950, pp. 230-233. The Arithmetic Phi Delta Dawson, Dan T., and Arden K. Ruddell, "The Case for the Meaning Theory i n 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 i n Arithmetic?" The Mathematics Teacher, Vol. 41, 1948, pp. 362-367. Lazar, Nathan. "A Pew Recommendations for the Improvement of the Teaching of Mathematics i n the Secondary School," Progressive Education, Vol. 29, 1952, pp. 21-23Morton, 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 i n the Junior High School," School Science and Mathematics, Vol. 53, 1953, pp. 680-684. Weaver, J . Fred. "Misconceptions about Rationalization i n Arithmetic," The Mathematics Teacher, Vol. 44, 1951, pp. 377-381. D. UNPUBLISHED THESIS Farquhar, Hugh Ernest. "A Study of the Relationship between A b i l i t y To Compute with Decimal Fractions and an Understanding of the Basic Processes Involved i n the Use of Decimal Fractions," Unpublished Master of Arts Thesis i n Education, University of B r i t i s h Columbia, 1955. E. YEARBOOKS Brownell, William A. "Psychological Considerations i n 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 i n Arithmetic," The Teaching of Arithmetic, pp. 279-291. F i f t i e t h Yearbook of the National Society for the Study of Education, Part I I . Chicago: Distributed by University of Chicago Press, 1951. 185 Horn, Ernest. "Arithmetic i n the Elementary School Curriculum," The-Teaching of Arithmetic, pp. 6-21. F i f t i e t h Yearbook of the National Society for the Study of Education, Part I I , 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 i n Learning Theory: Their Application to the Psychology of Arithmetic," Arithmetic i n 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: I t s 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 l e t t e r to the teachers conveying concerning the experiment information 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 i n connection with the experiment i n arithmetic about which Mr. Niedzielski spoke to you early i n 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 i n accordance with a plan. I believe that stage has now been reached. F i r s t , l e t me say that the experiment i s scheduled for the period extending from Thursday, March 21st to Wednesday, A p r i l 10th, instead of the l a t t e r two weeks of February as was originally planned. While i t i s desirable that there should be no extensive treatment of decimals before that time, there i s no reason why classes involved i n the experiment should not proceed with, some of the ordinary computation processes i n decimals. It i s realized that the work i n percentage depends i n part upon f a c i l i t y i n work with some decimal processes. Participation i n the experiment should not impede normal progress i n this phase of the Grade VII arithmetic. Second, you w i l l no doubt be interested i n the way i n which participating i n the experiment would effect you personally. 1. I think you w i l l enjoy.it. and procedure w i l l have been and by the U.B.C. College of that the project w i l l be the and advice from many people. Prior to the experiment the lessons approved by the local authorities Education. You can be assured result of a good deal of thought 2. The experiment i s a matter of current interest i n educational research. It i s 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 i s as yet quite unsupported. This experiment i s an attempt to contribute some s c i e n t i f i c conclusion, however, infinitesimal, to the existing body of knowledge i n this f i e l d . (Page two follows) - 2 - 3. You w i l l probably be interested i n 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 w i l l be composed of two Grade VII classes, and the control group w i l l be composed of two, or possibly three, Grade VII classes. The lessons for each group are complete i n mimeograph form with a l l the teaching materials and work sheets and other tests supplied. No advance preparation i s needed except a thorough reading of the content of each day's lesson as i t i s supplied on the form. To f a c i l i t a t e this preparation i t w i l l 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 w i l l be arranged. The lessons are one hour i n length. Approximately half of this time w i l l be spent i n teaching or i n exercises of a group work nature. Prepared assignments for individual seat work occupy the remaining half of the lesson. These assignments are designed so that they may be marked i n 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 w i l l be administered to each class. The marking of these tests w i l l be done quite independently of the participants i n the experiment. At the conclusion of the experiment another test w i l l be administered which w i l l also be marked and recorded independently of the participants. In the interests of the experiment I feel that this i s 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 i n i t s administration please do not hesitate to l e t 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 i n the experiment. I look forward with anticipation to your cooperation, and, I am quite certain that you w i l l derive some benefit and satisfaction i n the project. Very sincerely yours, George J . Greenaway 189 fflpHfATIOH NOTES The purpose of this experiment i n «hieh you are engaged Is to determine the advantage Cox* disadvantage) f o r an uas©l©eted group of Grad® VII pupils of being taught i a 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 instructional materials used c therefor® i s the type of 9 Otherwise^ the lesson procedures d are identicals Ths various concomitant variables? the pupils* intelligence reading a b i l i t y s 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 these tdJLl be 9 controlled i n the s t a t i s t i c a l analysis of the d&ta* • Ia order to mak© identical for both groups the teaching that results from the prescribed lesson procedures i t i s 9 irapes^tire that the instructions and time limits contained i n 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 F i n a l l y s for the, purpose of evaluating the ssperiment there w i l l b© a joint meeting of both groups immediately following the f i n a l testing* l a preparation for this concluding meeting i t i s necessary to ruake two requests s First s a diary should be kept of your experiences i n the teaching of these lessons Whether reported upon verbally or i n c 190 ta^itisg at the f i n a l ra®efci&g "shis uIXX p*wid© aa i n s t r u c t i v e 9 ®<g«oun% o f your < M persons! experiences l a the teaching o f th©©8 lessens© Shis aseount w i l l make an invaluable ©ontributioa t o the worth o f t h i s study* Second 8 an ©valuation f o m 8 wblen wiXX b© d i s t r i b u t e d near the end o f th© e&periraent^ should be completed,* Thi;s f o r a ^ p r e s e n t s a ste-uetiiiped interrogation i n t o s@se aspects o f youi? ©aperiene© td.th the teaehing o f these lessons* It© purpose 1© fce @asur® a report by eaeh p a r t i c i p a n t sspoa eossaos assess o f isat®r©st sad ©oneern i n th© performance o f the experiment* In® @®apleti@n ©f t h i s f o r a w i l l contribute further t o the value o f t h i s study* 191 EVALUATION FORM The following evaluation form i s designed to obtain opinions upon c e r t a i n fundamental issues involved i n t h i s experiment© The p r a c t i c a l 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 t h i s problem <, You are i n v i t e d therefor© , to be frank i n your anstsers and comments Apart from containing an i n d i c a t i o n of the group t o which you belong t h i s form need reveal no f u r t h e r identification„ 9 0 s This f o r a i s intended to supplement not to replace, whatever notes and observations you have accumulated'in the d i a r y o f your experiences i n the teaching of these lessons« p In Parts One and Tm of t h i s form the GREEK l e t t e r s i n th® column at th® l e f t i d e n t i f y the lesson objectives bearing the same l e t t e r on the accompanying Sussoary o f Lesson Objectives 0 P A R T pm For the purpose o f a t t a i n i n g each objective i n t h i s s e r i e s o f lessens, ho^ e f f e c t i v e 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 o f lessons? Not Dbjeetiv© Suitable Suitable Suitable A' B C D EP4V b o 1 I h L M — r — < ~ 1 '• ' 19k This series o f lessons contains & seiae&hat concentrated e f f o r t to teach the meaning o f c e r t a i n decimal concepts» I f t h i s e f f o r t tser© spread uniformly throughout the whole period o f i n s t r u c t i o n on decimal f r a c t i o n s 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 © E C X M L SYSTEM OF ?JOTATIOH Lesson Objectives o f Part One to Te present a b r i e f h i s t o r y o f the a r t o f reckoning* Z © & To convey an appreciation o f th© s i m p l i c i t y and convenience o f our presently used Hiiidu*»Arabi« decimal system o f notation as compared t o e a r l i e r methods e s p e c i a l l y the Roiaan systems 9 Lesson Objective o f Part Turn." 3* *o show v i s u a l l y the structure o f th© decimal number system* ^ggom XDEMTIPICATIOB AND KEAOTG OF PLAGE KATIES XH I-HXEl) DSGXIiAL FRACTIOHS Lesson Ob.le&tives o f Bart One -To show that the decimal system o f uhole number notation may be-extended t o the r i g h t o f the ONES* place* © ?.« To emphasise that the ONES* place i s the centra o f t h i s B extended system o f notation^ and that the other place names are symmetrical around it© 3 To provide a f a m i l i a r i s a t i o n i&tfo the decimal f r a c t i o n place I* names* 8 ••Lesson Objeetive^of^ Part Two n< 4 2© @ompare the r e l a t i o n s h i p i n s i s e i f the various p o s i t i o n a l 6 . -values ,>• 0 LESSON 111 REJOCTIOM OP ©ECXMftLS TO COB®©!? PACTIONS Lesson,,.Objecti|e;e of nt Part 1« -1?e consider decimals as a s p e c i a l form o f common f r a c t i o n s ]i. having denominators o f 10 100,, 1000 e t e * that i s , any power o f 10. 9 9 Lesson Objective of Part Two m 2* X ^o show ho-tf decimal f r a c t i o n s .indicate the"numerator ' and denominator o f equivalent common f r a c t i o n s 1 9 196 Lesson Objective o f Part, ffhree 3„ To provide p r a c t i c e i n the reading and w r i t i n g o f decimal J fractions© LESSON 17 THE USB OF ZERO AS A PLAGE HOLJ&i Lesson Objective o f Part One 1* ^o demonstrate the us© o f sero as a nlace holder* Lesson Objective o f Part 2 e Two 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 o f Part One 1„ To demonstrate the e f f e c t upon the value of a decimal f r a c t i o n tt d f moving the decimal point* Lesson Objective of Part 2 0 Two To demonstrate the e f f e c t upon the l o c a t i o n of the decimal point of multiplying o r d i v i d i n g a decimal f r a c t i o n by a power o f 10* LESSON VI ROUNJIHG DECIMAL FHACTIONS Lesson Objective of Part One 1« i l l u s t r a t e the s i g n i f i c a n c e o f rounding decimal f r a c t i o n s * 0 Lesson Objective of Part Two 2, To demonstrate various applications o f the rounding S* decimal fractions of Lesson Objective of Part Three 3« To i n d i c a t e why UNLIKE decimal f r a c t i o n s must be changed to Q LIKE decimal f r a c t i o n s (that i s , with the same understood denominator) i n order that they may be added or subtracted,, 197 LESSON ra rirw^mnM i ttmrimi r. •uun .n m DIVISION INVOLVING DECIMAL FRACTIONS Lesson Objective o f Part One l o '^o explain the s i g n i f i c a n c e o f performing d i v i s i o n i n v o l v i n g a deei-rial f r a c t i o n s . Lesson Objective o f Part Two 2» t o demonstrate the s i g n i f i c a n c e o f moving the decimal noint J? i n performing d i v i s i o n s i n v o l v i n g decimal f r a c t i o n s 8 LESSON VIII MISCELLANEOUS CONCEPTS INVOLVING JECBIAL FRACTIONS Lesson 0b 1eetive o f Part One n 1„ To convey the, s i g n i f i c a n c e o f changing a common f r a c t i o n t o •f a declnal f r a c t i o n • Lesson Objective o f Part Two 2 v a To i l l u s t r a t e the reason f o r the placement o f the decimal point i n the product obtained by the m u l t i p l i c a t i o n of decimal f r a c t i o n s j, Lesson Objective o f Part Three J v e To develop an understanding o f the importance i n the addition o f decimal f r a c t i o n s o f a l i g n i n g columns according t o place value*. Note: The GREEN l e t t e r s a t 1he l e f t i n d i c a t e the ©onseeutiv© enumeration o f the various Lesson Objectives involved i n t h i s s e r i e s o f lessons» l a parts One and Two o f the accompanying evaluation f c m the GREEN l e t t e r s i n the column a t the l e f t i d e n t i f y the" lesson objective bearing the same l e t t e r on t h i s summary c 198 SCHEDULE OF LESS0H3 DATE LESSON NUMBER BEMARKS Introductory Lessons "The Hindu«Ar&Mc System o f .dotation" 0 i d e n t i f i c a t i o n and Meaning o f Place Names i n Ki&ed Decimal Fractions™ e ^Reduction &f Decimals t o Common F r a c t i o n s ^ 5 Review o f concepts taught in \ • Lessons I I X XII« plus I r e c a p i t u l a t i o n ' o f Lesson f Exercised • p 9 The Use o f Zero as a Place Holder®* . . ra Lesson V ^Changing the Location @& the Decimal Points I t s e f f e c t on" the value o f the Expression" 0 22 May Lesson ?X "Rounding O f f Declaal Fractions" 23 Slay REVIEW LESSON Cumulative Review o f concepts I previqgly taught i n Lessons ' X t o vX plus r e c a p i t u l a t i o n I o f Lesson E x e r c i s e s I 3 I- G 2 4 Hay Lesson VII "Jivis&oa Involving Decimals* | 2? May Lesson VIII ^Reduction o f Common Fractions to.Decimal Fractions* --j 8 e •;-®lSO SesE© I#4or Concepts Involved In th® M t u t i p l l e a t i o n and s Is I f f o r any reason you ssish t o eonsnuaieat© t*ith m© i n regard t o yonr p a r t i c i p a t i o n i n t h i s £bqp@Fiinent» please phoa® th® Surrey School Bo^rd O f f i c e CGLOVERDALs £*&551 o r 2«l?6i) during the day or ay hcsae (LAKEVXEW 2-2073) during the evening a {• 199 APPENDIX B THE LESSONS PAGE Background material f o r the teaching of Lessons I, I I , 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 b r i e f history of the art of r©skoaiag e 2 0 To convoy en appreciation of the simplicity and eoavsaienee of our presently used Hiadu^Arabio decimal system of sotatlom as Compared to earlier methods ©specially the Boaaa systtiBfo 9 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 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. 0 The l a t t e r 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 an evaluative exercise i n the foraj of a worksheet entitled The Decimal System of notation* should be available fo? distribution at the end of the teaching : ^^sentatione, 9 w 5<SS^^grouadj . (A) DISADVANTAGES OF 2HE ASG1EBT S2STBUS OF H02ATI03S Reckoning i s one of the oldest arts practised by raan 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 i n design and d i f f i c u l t to reproduce* The seven numerals used by the Romans though' 'convenient to wrlte operated i n a very cumbersome system* 0 s s 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 i l l u s t r a t e , a sheep-herder„ desiring to calculate the number o f sheep he may have lost during the day, would place a pebbla i n a p i l e as each sheep l e f t the pasture* When the sheen TeturneC at night he would remove a pebble from the p i l e . The number of'pebbles l e f t over would represent the number of sheep s t i l l out* I f necessary to record his losses, he would resort at this stage to the use of numerals. I t i s true that more refined devices, 8uch as the abaejds, were used f o r 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 a i d i n number thinking. ( ( B j HSSTCaiCAL OEVSLQIMEKr OF THE HXHDUciARABIG DOTATION SYSTEM 'Many centuries passed with no progress i n the art of number. Eventually the f i r s t step was reached i n 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. B ? - . . . . About the tenth century the Sera- was invented to complete the Hindu-Arabic system of notation. Despite the amasing transformation which the 2ero made i n 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. I t mas about tvio centuries l a t e r 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. I t 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 symbols 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. fi The new system extended Its use beyond the recording function to which the e a r l i e r systems were confined- i t could also be used f o r counting and calculating. Unlike Rowan numeration, the Hindu*Arabic system appropriated to i t s e l f the principle o f positional value *toieh f a c i l i t a t e d i n a practical way the ancient process of calculating on the abacus. The longseparated 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 i n 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, w i l l be on a completely mechanical level in accordance with some prescribed rule, . ",, V ^ (D) m s i E m m § " 0 F mz-mzm p 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 i n counting. The number 28, f o r 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 i s called a decimal system* The word "decimal" i s derived from the Latin *d©eimus yhich meaBS tenth and ^decern* which means ten,. Nine is, as f a r as we go i n our'nttmber system without regrouping and starting over again with one. Ten i n any group join to make one i n the position of next higher value, t {11} tt 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 I " Hindu-Arabic Numerals has a value of oj£e, I t i s called one. 1 XX has a value of one and one* It i s called two* i i III has a value of one; one, and one, m i t i s called three* ***** - (This system i s based on an additive, OT subtractive, principle*) ' has a value of aa§. I t i s called one* a value of one ten and one one. It i s called n a a has a vfetS^f one hundred, one ten, and one qae, • . I t i s called one hundred, eleven* (This system 13based on a place value principle*) The- above comparison shows that i s Roman numerals the ;. " I * always has a value of one, regardless of i t s position* In Hindu-Arabic numbers the ^ or any other symbol, has a value which depends on i t s location, v r t 9 ( 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;undreds . no jfcens, eight ones, When the number i s written i n t h i s way« the Heedlessness o f i n d i c a t i n g that there are no hundreds ana no tens i s apparent g e However, when the same number i s expressed i n symbols the denomination o f each numeral (that is2 the f a c t that 2 r e f e r s t o thousands and 8 r e f e r s t o ones) i s indicated only by the p o s i t i o n occupied by the 2 and 80" 'These positions ares Thousands Hundreds 2 0 fens Ones 0 8 The zero by i n d i c a t i n g that a place i s empty serves to keep the numerals i n the proper p l a c e I t has been c a l l e d 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 t o protect the values o f the other numerals which l i e i n the other places to the l e f t i n a whole number, o r other Places t o the r i g h t i n a f r a c t i o n a l number. This protective function o f zero explains why i t i s not necessary i n expressing twenty i n symbols t o use the seros i n tbe following ways s g c 9 s Thousands Hundreds 0 0 Tens Ones 2 0 The f i r s t two seros are unnecessary because, u n l i k e the t h i r d sero i n the ones column, they have no numeral to protect;,. Civ) The number system mav be extended t o the r i g h t o f the ONES* p^aee t o provide a notation o f decimal.fractions, A s i g n i f i c a n t feature o f the Hindu-Arabic number system i s that f r a c t i o n a l parts may be expressed simply by extending the numeral places to the r i g h t o f the ones* place, thus: Pig O v c ONESNSSSSKS TENTHS HUNDREDTHS ' THOUSANDTHS The three p r i n c i p l e s underlying the structure o f the whole number system^ (1) ton-ness (2) place value and (3) place holding^ apply also to the decimal f r a c t i o n 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 t o th® r i g h t of the ONES' place* 2 . © emphasise that the ONES' place i s the centre o f t h i s extended system o f notation " and that the other place names are symmetrical around i t . T 9 3. To provide a f a m i l i a r i s a t i o n with the decimal f r a c t i o n place names. 4. compare the r e l a t i o n s h i p i n s i s e o f the various p o s i t i o n a l values. 2. Lesson Preparation; The procedure o f teaching t o a t t a i n the above four objectives i s divided into two p a r t s . 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 o f the fourth objective. While both parts w i l l contribute i n some measure t o a l l o b j e c t i v e s s i t i s desirable that the primary intent of each part be kept c l e a r l y i n mind as the lesson proceeds. Both parts involve the use o f s p e c i a l teaching m a t e r i a l s 0 These materials are l i s t e d a t the beginning o f each part o f the lesson procedure. Section (&) of the background material recapitulates b r i e f l y that portion o f Lesson I which dealt with the four basic p r i n c i p l e s i n the structure o f the Hindu-Arabic number system. Section (B) contains the background orientation relevant to Objective 1; section (0) i s s i m i l a r l y relevant to Objectives 2 and 3* while section (B) i s relevant t o Objective 4. Finally 9 an evaluative exercise i s provided f o r d i s t r i b u t i o n a t the end o f the teaching presentation. 3o Background; 206 (A) REVIEW OF THE PRINCIPLES UNDERLYING WHOLE NUMBERS Th© backgroundof Lesson I dealt with the four e s s e n t i a l p r i n c i p l e s underlying the structure of the Hindu-Arabic number system: 1; 2, ' 3 " k„ 0 i t i s based on a grouping by tens i t uses p o s i t i o n to determine the value o f a symbol i n a number i t uses aero as a place-holder to keep symbols i n t h e i r appropriate positions i t may be extended to the r i g h t o f ONES' place to provide a * . n o t a t i o n of decimal f r a c t i o n s . Showing i n a meaningful way the application of the f i r s t ttr ee of these p r i n c i p l e s i n whole numbers was one of the objectives of Lesson I (B) DECIMAL FRACTIONS ARE AN INTEGRAL PART OF THE WHOLE NUMBER SYSTEM WITH THE SAME COMMON PRINCIPLES 6 The fourth p r i n c i p l e serves as the means o f introducing decimal f r a c t i o n notation i n Lesson I I , This method of introducing decimal fractions as an i n t e g r a l 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 s p e c i a l form of common fractions having denominators of ten or some power o f 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 f r a c t i o n concepts. I f pupils regard decimals from the outset as a l o g i c a l extension o f the whole number system, they w i l l r e a d i l y recognize the application o f the f i r s t three p r i n c i p l e s i s decimal f r a c t i o n s as well as i n whole numbers. Teaching pupils t« regard decimals i n t h i s way constitutes the f i r s t objective of Lesson I I , 207 _{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 i s 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 i s to designate the location of the ONES' d i g i t . The point occupies no column or place i n 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 i n "Elementary Arithmetic: Its I leaning and Practice" describes as "the purely incidental mark of distinction between ONES and TENTHS", It i s interesting to note that some cultures use identifying marks other than the decimal'point as we know i t , Taylor and Mills state i n "Arithmetic for Teacher-training classes" that: "The number which we write as 1 6 , 3 5 7 has been written i n these forms: l 6 , 3 ' 5 " 7 1 6 ° : 3 » 5 " 7 " » ; 16,(o)3(l)5(2)7; 16)?57« W , M ; l a France and Germany, they point out, i t would be written 1 6 3 5 7 . However the purpose, i f not the 9 /S form, of the identifying mark i s common to a l l cultures. It i s to indicate the position of the ONES' column. A l l other symbols have their values determined i n relation to this column, 2 0 The other positional values are symmetrical around the ONES' place. ,208 ; ihis i s i l l u s t r a t e d 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 ONES place i s to overcome the 1 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. I f 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 i n our number system, and to note the correspondence between the TENS and TENTHS, HUNDREDS and HUNDREDTHS, etc. constitutes the second objective of Lesson I I . In the teaching procedure that follows, the same technique of instruction i s used for the attainment of Lesson Objectives 1 and 2 During the process of this instruction the pupils w i 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 i s intended to give pupils a general understanding of the positional values extending from THOUSANDS 0 209 to THOUSANDTHS. Within the l i m i t a t i o n s o f a reasonable amount o f i n s t r u c t i o n a l materials i t i s d i f f i c u l t t o make an adequate v i s u a l presentation o f the actual r e l a t i o n s h i p i n siae o f such a wide range o f p o s i t i o n a l values. The attainment o f Lesson Objective 4 i s intended t o give pupils a v i s u a l experience with a l i m i t e d range o f p o s i t i o n a l values extending from the ONES' place t o the THOUSANDTHS* p l a c e The 0 conceptual learning obtained therefrom w i l l be r e a d i l y transmitted to the wider range. LESSON I I I - 210 REDUCTION OF DECIMALS TO COM-JON FRACTIONS . Z : : ° L@sse® Objectives? l e To eonsidgr decimals m s. s p e c i a l fom. o f eo&imoa fgf&ctioas haviag deaoMnators o f 10 100, 1000 etc., that i s , any pm@? o f 1 0 , s 2, T© show hcer decimal f r a c t i o n s i n d i c a t e the Numerator sad deacraitsator o f equivalent eomaan f r a c t i o n s o 3, To provide p ^ c t l c e i n the reading and writing frsetloas. o f decimal Lesson Preparation; The procedure o f teaching t o a t t a i n the above three objectives i s divided i n t o three parts: one part f o r each objective. The f i r s t part does not involve the use o f s p e c i a l teaching materials§ the materials to be used i n the secondand t h i r d parts are l i s t e d a t the begi&ning of each part o f the lesson procedure. Sections (A), (B) and (C) o f the Background material are relevant, respectively, to the three lesson objectives. F i n a l l y , an evaluative exercise i s provided for distribution at the end o f the teaching presentation. BACKGROUND; U ) DECIMAL FRACTIONS ARE A SPECIAL FORM OF COMMON FRACTIONS l a the background material o f Lesson I I reference was made to two suggestions f o r the introduction o f decimal f r a c t i o n s given by Brueckner and Grossnickle i n "How t o I lake Arithmetic Meaningful"« One way i s t o consider decimals as an i n t e g r a l part o f our decimal system o f number. This approach was used i n Lesson I I , The other way i s t o consider decimal f r a c t i o n s as a s p e c i a l form o f common f r a c t i o n s having denominators o f any power of 10 This 0 approach i s used i n Lesson I I I 0 Buswell andBrownell reinforce t h i s 211 opinion when they state i n t h e i r manual to teaching "Arithmetic We Heed": "Once pupils understand that decimals are f r a c t i o n s , the denominators of which are not v i s i b l e and are always 10 or a multiple of 10 9 they w i l l have developed a r e a l understand ng of the meaning §f decimals„" Jecimal fractions may, part o f a l l common f r a c t i o n s 5 therefore, .be regarded as a selected namely: those #iose denominators are a power of 10, Any f r a c t i o n which has a denominator of 10 s 100 s 1000 ? etc 0 i s a decimal f r a c t i o n . As Spitser says i n "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 i n which a decimal point i s 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 t h i s way i s h e l p f u l f o r developing meai i n g f u l 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 higher terms ( f o r example, changing 5,1 to 510 hundredths) 4 , the rounding of decimal fractions to•lower terms ( f o r example, changing 0,942 to 9 TENTHS). These points w i l l be i l l u s t r a t e d at the approprleate places i n t h i s , and subsequent, lessons, (B) INTERPRETATIONS OF THE NUMERATOR ANJDENOMINATOR OF A JECIMAL FRACTION In Seetion (A) of t h i s background material we have seen that decimal f r a c t i o n s are simply common fractions having denominators 1 0 , 100 8 1000 s ete of 0 The numerator of a. decimal point i s indicated directly-by the , 1 212 number t o the r i g h t of the decimal point. I t i s read as a whole number. For example, i n .425 the numerator i s 425 and i s read four hundred twenty f i v e ; 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 o f the numerator. The denominator o f a decimal f r a c t i o n , being unwritten, must be interpreted from the name of the last-used decimal place t o the r i g h t . Lesson I I (Objective No. 3) provided famili a r i z a t i o n with the decimal f r a c t i o n place names which w i l l enable pupils to make t h i s i n t e r p r e t a t i o n . Thus, 1.2 i s one and two tenths as a mixed f r a c t i o n while i t i s twelve tenths as an improper f r a c t i o n . S i m i l a r l y , .12 i s twelve HUNDREDTH'S, .012 i s twelve THOUSANDTHS, etc. In each case i t i s the name of the last-used place to the r i g h t o f the decimal point which i n d i c ates the " i n v i s i b l e " denominator. Lesson I I I (Objective No. 2) i s intended to show the reason why the denominator of a decimal f r a c t i o n may be interpreted from the name of the last-used decimal place to the r i g h t . In addition, t h i s objective i s intended to show why the position o f the l a s t d i g i t a f t e r the decimal point a c t u a l l y determines the value o f a decimal f r a c t i o n . For example,ta,12 i t i s the f a c t that the d i g i t 2 i s i n the HUNDREDTHS 1 position which determines the value of the f r a c t i o n . This may be explained i n t h i s way: since the 1 i s i n the f i r s t p o s i t i o n to the r i g h t o f ONES* place i t represents 1TENTH which i s equivalent to 10 HUNDREDTHS. Together with the 2 already i n the HUNDREDTHS 1 position t h i s makes 12 HUNDREDTHS. 213 Certain generalisations w i l l r e s u l t from interpreting the numerator and denominator of a few decimal f r a c t i o n s 0 One such generalization iss i f the last-used decimal place i s one place to the r i g h t of the ONES* place the f r a c t i o n represents'TENTHS; i f i t i s two places to the r i g h t of the 0!IKS» place i t represents HUN ORT^DTHS, e t c . A somewhat more mechanical form of the sane generalization is? the denominator of a common f r a c t i o n w i l l have one zero f o r every figure to the'right of the decimal t>oint i n the equivalent decimal fraction,, (C) 1HE READING AND WRITING OP DECIMAL FRACTIONS The f i r s t objective i n t h i s lesson i s to lead the pupils to understand that decimal fractions are merely common fractions with unvjritten but understood 9 9 denominators. Furthermore, they are a s e l e c t i v e type of common f r a c t i o n s because the denominators always 10 s 8 are 100, 1000,, etc*, that i 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 f r a c t i o n equivalent to the decinal f r a c t i o n . These two objectives should give pupils an understanding that decimals are f r a c t i o n s , the denominators though unwritten, are a power of 10 B of which f According to Buawell, Brownell and Sauble i n the Manual to TEACHING ARITHMETIC WE NEED, t h i s i s the basis f o r developing "a r e a l understanding of the meaiing of decimals" I f these two objectives are attained, t h i s t h i r d object~ ive concerning the reading and writing of decimal f r a c t i o n s w i l l have been achieved as w e l l 0 This procedure follows Spitzer's recommendation of w r i t i n g the common f r a c t i o n of decimals as a means of reading the decimal written with the decimal point, In the decimal f r a c t i o n 036 e 9 for example, i f the p u p i l understands that the unwritten denominator must be a power of 10; that the power i s determined by the place value 0 214 of the 6; and g further 9 that th® numerator i s determined by convert- ing the 3 TENTHS i n t c THOUSANDTHS and adding i t to the e x i s t i n g 6 THOUSANDTHS t o give 36/1000, then he w i l l have a meaningful insight into the reading c f „036 c Procedure of Teaching: £sfol^mm& &g - ksm$£>& S!bMsSiis.m 1 ssd Z (Time: 12 minutes) The contents of Sections ( a h (b), and {4) o f 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 i n general terms* The emphasis should be on the spCnlaneiiy of the presentation rather than on too r i g i d an adherence to minute d e t a i l . I f the objectives of the lesson are reached successf u l l y i t should motivate the pupils to expLore and experiment with decimals i n subsequent lessons. Achievement of, Lesson Objective £ (Time* 18 minutes) Materialsi Three place value charts. 210 tickets of ishieh 200 should be i n groups of 10 held together with an elastic band, and 10 loose tickets. * To i l l u s t r a t e; (i) the ten~ness (or decimal nature) of our number system. ( i l ) positional value, .©r-.thaJtyufc&i* numeral at the right of a whole number hat a 0Nf$* value, the next numeral on the l e f t has a TENS' value, and the next on the l e f t has a HUNDREDS* value. Steps; 1. Set the three place value charts on the bottom ledge of the blackboard. Hote: i n 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 i n these two given positions. 2. Place the Single tickets on the ONES' chart while counting 2.* 2, 3, h 5, , ?t *• % ? 6 1 0 t 3. Explain that t*e must regroup when We reach 10 Provide a , small square Immediately above the Charts as shown i n the i l l u s t r a t i o n at the top of page 216. These squares may be drawn en the blackboard. s k* Explain that i n thtgse squares we customarily write only one figure to indicate the numbed of t i c k e t s on each place value chart* This illustrates the haisd to regroup when 10 has been reached i n 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 i n the email squares. • •jhpttbo: Symbol Symbol SINS ONES xxxxxxxxxx (regroup to form 10)| 6 S&phaslse the f a c t that 1 bundle on the TENS board i s composed o f 10 times as many t i c k e t s as 1 t i e k e t on the • ONES* board, 7 o Continue to replace s i n g l e t i c k e t s 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 d e t a i l o f t h i s regrouping as often as i s ' considered necessary* 9, As i n Step 6, draw attention frequently t o such f a c t s as: (a) 4 on the ONES' chart represents one«fifth as many t i c k e t s as 2 on. the TENS-' chart, (b) 8 on the TENS' chart represents.-forty times as many t i c k e t s as 2 on the ONES' chart, etc. 6 ; 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 t o the HUNDREDS' posi-bion, 2n other words rfc illustrates situations ?Mch requis © two successive 9 • f ^ e u p i a g s . Step 11 emphasises the r e l a t i o n s h i p 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 t i c k e t s on the TENS' chart and 15 s i n g l e t i c k e t s on the ONES * chart. Emphasize the fact that i n the squares above each chart we write only one f i g u r e t o i n d i c a t e the number o f t i c k e t s on that p a r t i c u l a r chart, and emphasise also that i t takes 10 i n one p o s i t i o n t o equal 1 i n the adjacent p o s i t i o n on the l e f t . Following the emphasis on these d e t a i l s proceed t o perform the regrouping t o • obtain 1 t i c k e t 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 f a c t s as: (a). 1 on-the HUNDREDS' chart represents 20 times as many t i c k e t s as 5 on the ONES' chart. (b) 5 on the ONES' chart represents 1/6 as many t i c k e t s as 3 on the TENS' chart.. ? 12. Draw from pupils* out o f the experience they have had With the foregoing r e l a t i o n s h i p s generalisations framed around the following: 3 (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 i n the number* Each "place" has a value ten times as mush as the "place* immediately t o the r i g h t , or one*tenth as much as the p o s i t i o n immeoTately t o the l e f t . 13. At t h i s point a very b r i e f comparative description may be .made o f the p r i n c i p l e s underlying the Roman numeral system o f notation (see heading (d) ( i i ) o f the background m a t e r i a l ) . To i l l u s t r a t e : ( i l l ) the use o f sere, o r cipher, as a place holder. Steps: li Show 9 bundles on the TENS' chart and 9 t i c k e t s on the ONES' chart. Then add one t i c k e t to the ONES' chart. Regroup. ymbol 11—i HUNDREDS 2. 3; 4, Symbol l— TENS i—r Symbol ONES In the space on the blackboard above each chart write the appropriate symbol. Explain that a f i g u r e must be written t o ©how each place i n the three place number, even though the charts i n two o f the places are empty. .This i s the PLACE HOLDING FUNCTION OF ZERO i n our number system. Show that aero has a protecting r o l e to keep 1 i n the t h i r d space from the r i g h t , or on the HUNDREDS' chart. • Add some s i n g l e - t i c k e t s , say 7, t o the ONES' chart, While referring; to the place value charts deal with the number shown under three headings: (a) How i t reads • one hundred seven, (Note: t h i s i s a . convenient point at which t o explain that the use o f "And* i n the reading o f a number i s reserved e x c l u s i v e l y to i n d i c a t e the connection betv/een the ^ S S ^ p I a c e and the TENTHS' place. I t i s never used either" i n a whole number o r i n a f r a c t i o n a l number* (b) What ,it means* • one, hundred, no tens, seven ones. JLJ (c) ffbwTK i s written - kc Prne:-.i-,ue of Teaching; s 218 Achic^emant o f Lesson Objectives .1.;. 2. and 3 (Times 1?.minutes) MaJ/grials'; Seven place value charts* Chart i n d i c a t i n g the decimal point. Twentyf i v e sards i n each of seven d i f f e r e n t colors, (Motes t h i s s u b s t i t u t i o n f o r the bundles o f cards used with the place value charts i n Lesson I i s necessitated by the i m p r a c t i c a b i l i t y of using cards smaller than the ONES' cards to represent units l e s s than ONE* 4^3^ S^iiJ?03L«£;S' ( ~ — ™ - * • . JSmMSSSS^St^J^nr — - — . — A l l three steps sfcoumft contribute tc* «h* attainment o r 1 0 Set the place value chart to represent ONES on the bottom ledge o f •the' blackboard. Place the decimal point to the r i g h t of the ONES* chart, dentify the "19 methods ..~w« — d i g i i — has been located a l l other d i g i t s obtain t h e i r values from the position they occupy i n r e l a t i o n to the ONES' place, A number, l i k e 34#?3, i s quite meaningless unless we knew which d i g i t stands f o r unity, 2, Place a d d i t i o n a l charts to secure the arrangenant shown below: "oHK AJLJ.IAA XV vv.i; vy vv 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 t o t&rm the structure of whole number system, appjy,also t o decimal f r a c t i o n s : (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 2 and then assign successive p a i r s of pupils t o come forward to place"cherts i n positions which"are symmetrical around the ONES' c h a r t as shown i n the diagrams. 9 ? 'the- lowest r o w o f c h a r t s shows the arrangement when a l l have been placed o n the bottom ledge of the blackboard a r a w l i n e s o n the blackboard above the charts, as shown i n the i l l u s t r a t i o n above, to emphasise the symmetry aro-und the ONES' place.. J While the charts are i n t h i s position* discussion should be :';sld which points o u t the following: (a) the ..antral p o s i t i o n 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 r i g h t , one hundred times as large as the second place to i t on the r i g h t , e t c I l l u s t r a t e these ^ l a t i o n s h i p s with s p e c i f i c examples shown on the charts,, e,g : c (i } a 4 i n the OSES' place i s ten times, as large as 4 i n the TENTHS' place ( i i ) a 7 i n 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 o n e - f i f t i e t h as much as 1 i n the TENS* place (that i 3 , the 1 i n the TENS' place a c t u a l l y represents 100 TENTHS , which i s f i f t y times l a r g e r than 2 TENTHS), 4 G 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 b e d r a w n tram p u p i l s a t t h i s s t a g e : (a) s three generalisations should T h e 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 a l s o 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 p a r t i c u l a r value* a 220 ill } XffflWfrgftf! * "^slu® assigned to a <ft'git i n one position i s tan times larger than the valuta assigned to i t i n the position next to i t on the right, ©te 0 ( i l l ) Hlae^hold^n^ function of -%erp « i n order to ^protect* the value bT numerals oy keeping them i n the required positions« seros are needed to record whatever empty positions exist BETWEEN the decimal and numerals i n the most 'extreme positions to the l e f t or right of the decimal point. : Nofcs: i t may be mentioned i n passing that seres f i l l another function quite apart .from a placebol&ug function. This function, m wall as the pjle&*«hoiding; function, «111 be dealt witk more f u l l y 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 l e f t - and third i rom. the ONES plase on the right are THOUSANDS and THOUSANDTHS respectively. v 1 (i« ) the position which i s 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 l e f 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 f i f 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 Thinking with Numbers" contains a drawing, shown at the l e f t , which may be presented on the blackboard to pupils to emphasise that one must learn to check conclusions i n arithmetic. In comparing decimal fractions, as i n comparing the lengths of these 30 inch lines, "You cannot always be sure". tt 221 Choose' pupils each to c a r r y a place value card and take t h e i r positions facing the class while holding'the card i n f u l l view* Start with the ONE c a r d g followed by the decimal p o i n t 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 i n the coi*responding position t o the l e f t o f 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 o f 100 cards. This arrangement may be represented on the blackboard;, thus: VJhile pupils are i n t h i s position discuss the manner i n which we would arrange the following i n order o f 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 f r a c t i o n s , e.g., #25 and ,3 i n t h i s way: M M 1*1 M X X X X X X X O O O Assign pupils to take up t h e i r positions behind the card bearers as shown i n the diagram; The x's represent , 2 5 and the o's represent . 3 . Pupils representing ,25 may be referred t o as Team (a), while those representing .3 may be referred t o as Team (b). Discuss why Team (b) represents a l a r g e r f r a c t i o n than Team ( a ) . The same procedure may be followed i n showing the reasoning involved i n arranging the following according to s i z e : (a) .5 (b) , 0 5 Cc) 5.5 (d) . 0 5 5 (e) . 5 5 222 To toaelode. t h i s portiaa o f the lesson, the following generalisation should he dissta from pupils a f t e r those standing have resisaed t h e i r seats© B®tsiJaal fractions eaa be ranked i n order of sis© by comparing the absolute value of the digits i s the corresponding places, thusg w (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' . •place e (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; - PAST v ONE - JZjL.jmi-11-Jt- -J*- JIWmi i m m WIIII i w m i i i i ' i B i 1 " .. Achievement. o i . , t ^ i s s ^ , O M ^ M g ^ - l (Tia*: 8 minutes) > T© consider decimals as a s p e c i a l form o f common f t a c t i o n s having: denominators o f 1 0 , 100, 1000 etc ., that -ley assy power o f 10. Materials: •• No s p e c i a l materials required. Steps; 1. Write the following series of common f r a c t i o n s on the blackboard: 2. Verbal Explanations: (a) Explain what i s meant by "a power of 10 » Obviously, i t i s be§rond the scope o f the pupils* comprehension at t h i s stage to explain that i t means "the index o f ? 10 «Consequently i t w i l l s u f f i c e to explain that i n e f f e c t i t means tt rt 9 10 multiplied by i t s e l f any number o f times, or 10 by i t s e l f , thus: 10, 100, 1000, e t c . The meaning o f "a power of 10" should be made d i s t i n c t from the meaning of "a multiple o f l o * which ineahS 10 m u l t i p l i e d , not by i t s e l f any nuifber Of times, but by any sw^b®r, f o r example: 5* 8- 12, 20, 30, e t c , to give these respective multiples of 10: 50, 80, 120, 200, 300 e t c . (b) Explain that wMle a l l the f r a c t i o n s written on Common f r a c t i o n s , those with a denominator of a may also be regarded as decimal f r a c t i o n s , even customary practice i n writing decimal f r a c t i o n s the denominator andto indicate i t i n d i r e c t l y by decimal point. the board are power of 10 though i t i s to omit w r i t i n g the use o f a v 3. Form two columns on the blackboard, and at the top of each write headings as follows: Fractions which may be.considered only as common f r a c t i o n s a ;— — ,——•.,.„. .„ Fractions which may be considered as Decimal f r a c t i o n s „ Under the appropriate heading enter each of the f r a c t i o n s already written on the blackboard. To shew hbw decimal fr&stiess' iadleafce the atraerator and denominator o f emiiw&leafc eoasBoa f r a c t i o n s Materials: Four place value charts, namely: ONES'• TENTHS' 5 HUNDREDTHS*j THOUSANDTHS' 2 5 yellow t i c k e t s , 15 each o f blue and green t i c k e t s , and 5 red t i c k e t s . steps: $g&es The -two points stated below should be c l e a r l y emphasised a f t e r each o f the following three r G p r l l ^ i t l t l o ^ c o n t H h e d i n PartrWo o f t h i s Lesson. X« The p o s i t i o n o f the l a s t d i g i t a f t e r the decimal point determines the value the value o f a decimal f r a c t i o n * That is» each o f the d i g i t s i n the decimal positions preceding the l a s t place may ^ j ^ J ^ c ^ y e r t f d t o the place value o f the l a s t position after thedeclSaX point. The number so obtained determines the NUMERATOR o f the equivalent common f r a c t i o n « — 0 At the same time the p a r t i c u l a r place value o f the l a s t occupied p o s i t i o n Indicates the DENOMINATOR o f the equivalent common f r a c t i o n . 2. When a decimal f r a c t i o n i s changed t o a common f r a c t i o n , the denominator h&s ONE ZERO f o r every figure t o the r i g h t o f the decimal point. 1. ?er£otm. representations o f 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 t i c k e t s from the Hundredths* Board and substitute 2 0 yellow t i c k e t s on the THOUSANDTH-'*' board, thus! ZL. ONES TENTHS xxxxxxxxxx xxxxx Sraphasize c l e a r l y the two points stated.above i« s'b) Represent 12' on the place valoa charts, writing the number above the «har"ts as shown j 225 t) DO ONES TENTHS HUNDREDTHS •THOUS iNDTHS XX Remove the blue t i c k e t from the T E N T H S ' board and substitute 10 green t i c k e t s on the. HUNOTirUTHS' board, thus: 53 ONES HUNDREDTHS XXXXXXXXXX THOUSANJTHS Emphasize c l e a r l y the two points stated above i n green. (c) Represent 2.3 on the place value charts, repeating the procedure as i n (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 f r a c t i o n In this ease, of course, i t i s the position o f the terminating sero which determines the value of the improper fraction. . 2 0 . . . . . . ] Perform representations o f the following as indicated: (a) Represent 2»0 on the place value charts. Remove the two red t i c k e t s , and stibstitute 20 blue t i c k e t s on the TENTHS* chart, thus: ONES .TENTHS HUNDREDTHS THOUSANDTHS XXXXXXXXXX (b) Though the manipulation involved i n the following need not be undertaken, proceed t o explain, nevertheless, that 2.00 would be shown as 200 green t i c k e t s on HUNDREDTHS' chartj 2 „000 would be shown as 2000 yellow t i c k e t s on the THOUS ANTHS* chart. Emphasise, as i n Step 1, the s i g n i f i c a n c e o f the last-used p o s i t i o n to the r i g h t o f the decimal point i n determining the numerator and the denominator o f the improper f r a c t i o n , ^ o r example, i s 200 HUNDREDTHS« 2,000 i s 2000 THOUSANDTHS. PAR T8 'f R & S •'. " •Aehigygaeat o f Leases Objective ft (Time? XO minutes)" v . 2 26 provide practice i n the reading and writing" o f decimal f r a c t i o a s Materials: No s p e c i a l materials required. Steps:' Notes The achievement o f Lesson Objective 2 w i l l enable pupils to v i s u a l i s e the common f r a c t i o n equivalent o f a decimal f r a c t i o n I t i s t h i s a b i l i t y to v i s u a l i s e the ©ommoa f r e c t i form x^hichj according to Spitser, provides a good procedure f o r the reading of decimals. Therefore, the f i r s t step belov presents, at a more abstract l e v e l , the same method used i n the achievement of Lesson Objective 2. 0 •1. Write the decimal f r a c t i o n 0.256 on the blackboard, Then explain the meanings f o r t h i s decimal that are shown below: 0.256 means 0;200 (200 THOUSANDTHS) 0.0$0 ( 50 THOUSANDTHS) o;oo6 (• 6 0.256 THOUSANDTHS: 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 1 1 5 . 2 3 1 we connect the whole number and the f r a c t i o n by "AND", In the reading of decimals the word "AND" i s reserved f o r t h i s purpose and i s never, used,-with one exception, i n either the i n t e g r a l or f r a c t i o n a l portion or the mixed decimal. Tims, i s read "one hundred f i f t e e n AND THOUSANDTHS". . 115.231 two hundred t h i r t y one ,847 i s read "eight hundred forty-seven thousandths"., 800.047 i s read "eight hundred AND '. forty-seven THOUSANDTHS, The exception i s i n the reading of a decimal f r a c t i o n containing a common f r a c t i o n , f o r example: 4 . 1 2 ^ i s read " f o u r AND twelve and one-half HUNDREDTHS", • 0 . 0ok * *i s read "one seventh of a TENTH". 3 . Explain that in"reading a NON-TERMINATING or INFINITE decimal f r a c t i o n l i k e 3 . 1 4 1 6 i t i s common usage to read t h i s as a telephone number thus: f 3.1416 may be read "three DECIMAL (or POINT) One-four-one-six? .4c Explain that i n reading a TERMINATING or FINITE'decimal f r a c t i o n such as might be obtained as a measurement by the use of a micrometer, f o r example , 0 5 0 0 , would be read" f i v e hundred TENTHOUSAI-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 n 6 Lesson IV (I-'age 1) 4 iitiit'jw*(.^.«rn-^" 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: Decimal place value c h a r t s . Four charts to represent place values: one w h o l e 3 one t e n t h , the one h u n d r e d t h , one following thousandth, 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 zero symbol. A l s o , d e c i m a l p l a c e value c a r d s as shown o n n e x t the page. Steps: Note: Steps 1 , 2 , and 3 demonstrate v i s u a l l y the use of aero as a place"holder. 1. S e t t h e f i v e c a r d s blackboardo shown b e l o w o n the b o t t o m ledpre o f the 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 f o r 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 s y m m e t r y 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 a n d THOUSANDTH'S c a r d s w i l l b e l o c a t e d o n e and two places r e s p e c t i v e l y t o the r i g h t of the decimal 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 b e c o n s i d e r e d t o r e p r e s e n t TENTHS a n d 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 r e m o v e 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 "to protect" the values of these cards. Accordingly, insert the card bearing the sero i n the empty place. 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 a n d 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 a n d f o l l o w t h e p r o c e d u r e as i n S t e p 2 0 Lesson 17 (Page 2} 228 Notes Steps k 5, and 6 demonstrate v i s u a l l y the e f f e c t upon the value o f a mixed decimal f r a c t i o n o f Inserting s. zero immediately a f t e r the Secimal point. 9 Set the four cards shovm below on the bottom ledge of the blackboard. (Note: follow the i n s t r u c t i o n contained i n Step 1 above i n regard to the plaeenehtof the board containing the decimal point) (a) • 1 1 (b) Figure 1 (c) Then Insert the ZERO iramediately a f t e r the decimal point, thus: Figure 2 Since, however, the second and t h i r d 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) Figure } (f) By comparing the arrangement shown i n Figure 1 with that sho^n i n figure 3 i t should be pointed out that we have, i n e f f e c t , t a k e n - ^ - o f card (b) t o give us card ( f ) , and we have taken "1_ of card (c) t o give us card (g). ^ 1 Since we have not, o f course, i n any way altered the ONE'S card, i t cannot be said that we have taken one-tenth of the o r i g i n a l mixed"'"decimal expression. A l l that can be said i s that i n s e r t i n g the zero immediately a f t e r the decimal point has the e f f e W T J f "reducing the value o f the mixed decimal expression. Note: Steps ? and 8 demonstrate v i s u a l l y the e f f e c t upon the value o f a simple decimal f r a c t i o n Of i n s e r t i n g a sero immediately a f t e r the decimal point. Set the three cards shown below on the bottom ledge o f the blackboard? 1'hei: i n s e r t the ZERO immediately a f t e r the decimal point thus i 9 TBT However, as i n Step 6, Cards (A) and {b5 must be changed to give* • Unli.ke the previous example,, t h i s i l l u s t r a t i o n shows that i n s e r t i n g the sero immediately a f t e r the decimal point i n a simple f r a c t i o n has the effect of making the value o f the new f r a c t i o n EXACTLY ONE-TENTH o f the value o f the o r i g i n a l fraction,, o 0 '?o conclude this portion o f the lesson be dra^a fro® pupils a t t h i s stages two generalisations should 9 (a) I f a s«3*© i s inserted after the desires! point i n a mixed decimal expression I t has th© effect o f reducing the value of the expression. (h) I f a aero i s inserted after the-decimal point i n a simple de^JUiliv expression i t makes the value ONE-»TSNTH as much as i t was originally. aetftevuaent o f 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 t h i r d 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 r i g h t o f the TEKTH^S card thus: «3w^AU&IM< 3 C {.".ma* W_:Jir*arjw» I in mmf Draw attention o f pupils to the following two points? Lessen IF (Page /*) (a) a Terminal Zero, unlike a place l i d d i n g zero, i s annexed to the end of a decimal f r a c t i o n . .(b) a Terminal ssero does not change the actual value o f adecimal fraction™ but i t does change the SIGNIFICANCE o f it,, This change i n SIGNIFICANCE o r MEANING which r e s u l t s from adding a Terminal Zero w i l l be discussed i n Lesson 71 0 At t h i s point i t w i l l be s u f f i c i e n t t o point out that adding the aero i n the above example enables the f r a c t i o n to be read "ONE and TEN HUNDREDTHS" instead of "ONE and ONE TENTH*» This indicates that the decimal f r a c t i o n i s accurate t o the nearest HUNDREDTH. Without the terminal ZERO i t i s accurate onTylJo the nearest TENTH. 4. To conclude t h i s portion o f the lesson, the f o l l o w i n g geaeraligatios should be drawn from p u p i l s a t t h i s stages The addition o f a terminal sero t o a decimal f r a c t i o n does not change the value o f the f r a c t i o n but i t does change the s i g n i f i c a n c e o f the f r a c t i o n . _?rogedure , of „ Teaching: 231 PABf ONE *-> deiaoE3tjrate 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, ! Materials: T h r e e place v a l u e " c h a r t s : CUES'» T E N S ' ; H U N D R E D S ' C h a r t i n d i c a t i n g ' the decimal point."TicSets; 1 single.. 1 p a c k e t o f 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 N o t e s S t o p s 1 a n d Z demonstrate 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 moving t h e d e c i m a l p o i n t t o t h e left. 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 the decimal f r a c t i o n o f moving the decimal 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 l s Set the three blackboard. a 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 I • IIIHIIIIIM ; £ 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 o n t h e TENS' c h a r t , a n d a p a c k e t o f 1 0 0 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 o n e p l a c e t o " t h e l e f t 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 , as i a d i c a t e d b y the E x p l a i n : Since the place immediately to the l e f t o f the decimal p o i n t m u s t a l w a y s b e 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 present•shown on t h e T E N S ' chart h a s , i n e f f e c t , been reduced 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 b e 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 t w o 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 diagram. Repeat 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 given i n step 1 must Set th« board. shree c a r d s 232 shown b e l o w o n vho bottom l e 1 1 1 i u 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, t h i s makes i t aecessary to consider that'the representation of' OKE-TEMTH (immediately to the Ifeft o f the new l o c a t i o n of the decimal point) has, i n e f f e c t , been Increased to ONE. Likewise the representations shown on adjacent cards must be increased to ten times the original, s i s e la order to maintain the p r i n c i p l e o f TEN-NESS. s I-Iove the decimal point twc places to the righg of the o r i g i n a l l o c a t i o n as indicated by the green arrow a t the bottom o f 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 i s moved t£o thejff numberji* has' "the effect o f multiplying the nurao°ern^r^j0 That i s , i f the decimal point i s 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 i s moved to the laffl i n a number i t has "the effect of dividing the number'oY' BgCTTnat i s i f the. decimal point i s moved onenlece to the lrftry t n * number i s reduced t© 0KE«5 EHTIt i t s c ^ g l n e X v l l u e i if i t i s moved fewe places to WeljSfB'*'th& number i s reduced to pKEe4^JRED'gI i'iis original valus, etc. ; s - m& TWO AcMageaattt of Lessen Objective 2 (Times 15 minutes) *f Materials: Same as f o r Part One, Steps: s 233 *art» 81 to tnsrGXt>re« a r e s a r a l l s X uiioso contained 111 t md 2 demonstrate v i s u a l l y i&e effect upon the of the decimal point o f 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 10, ' 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* Set the three charts -blackboard. shown b e l o w o n t h e b o t t o m l e d g e o f the P l a c e a s i n g l e c a r d o r t i c k e t o n t h e OSES c h a r t , a packet of 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 - , 11 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, a n d s o we g e t : ; m 10 t i c k e t * 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 be i d e n t i f i e d b y t h e d e c i m a l point i t i s , consequently, necessary to adjust the 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 o n e p l a c e t o t h e l e f t , a s shown by the red 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* e a c h e a c h p l a c e b y 100, a n d s o we g e t : 1 H U N D R E D S " 1 I ticket here This time divide 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 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 arrow* of f the he bottom Set the tar. blackboard: 234 0 1 1 ( M u l t i p l y by This represents .111. by 10, t h u s : 1 10} 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 fraction 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 h e . r i g n t . S e e r e d a r r o w . t 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 t o 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. T o 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 s h o u l d be drawn f r o m p u p i l s a t t h i s s t a g e : generalisation {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 , ete , (that is 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 place to the r i g h t f o r every sero i n the m u l t i p l i e r ^ ( t f ( 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 . 1 0 0 0 e t c . , , that i s some p o w e r o f 1 0 ) t h e d e c i m a l p o i n t i s ' m o v e d o n e place to the l e f t f o r every zero i n the d i v i s o r 9 0 235 LESSON V I Page 1 ROUNDING JECII-IAL FRACTIONS I i i 5? ONE Achievemoat o f Lesson Ob.1egt3.TO 1 (Time; 15 s i n u t s s ) To lllustrat® the s i g n i f i c a n c e o f rounding decimal f r a c t i o n s 3 Materials: Wall r o l e with movable i n d i c a t o r . Steps: Note: The s i g n i f i c a n c e o f rounding decimal f r a c t i o n s i s shown by c o m p a r i n g the v a r i a t i o n i n a measurement rounded o n l y to UNITS to the v a r i a t i o n s i n measurements rounded successively t o TENTHS and HUND2EDTHS. 1 . The scale indicated below represents a longer section o f the wall r u l e used i n t h i s lesson. Draw t h i s representation on the blackboard. (a) Explain that ?jhen we say that a l i n e i s 2 inches long we s i g n i f y by t h i s i n d i c a t i o n merely that the length i s CLOSER TO 2 INCHES THAN IT IS TO 1 INCH o r 3 INGRES. The rather considerable amount o f v a r i a t i o n i n length permitted i s indicated by the area marked i n RED. I t should be evident that i n order t o round a measurement number t o the NEAREST u n i t i t i s necessary t o know a t l e a s t the number o f TENTHS involved i n the measurement. (b) Explain that when we say that a l i n e i s 2 0 inches long we s i g n i f y by t h i s i n d i c a t i o n that the length t h i s time i s CLOSER TO 2 . 0 INCHES THAN IT IS TO 1 , 9 INCHES or 2 . 1 INCHES. The 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 length permitted by t h i s designation i s indicated by the area marked i n PURPLE W 4 I t should be evident i n t h i s case that i n order to 236 round a. meaurement number to t h e NEAREST TENTH i t i s necessary to know at l e a s t the mmber o f 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 2 ,0G inches itlon t h a t the length t h i s time i s IS TD l $ 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 iihe v e r y i t m a l l a r e a m a r k e d i n GREEN t> C c I t should be evident i n t a t s case that i n order 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 rule, 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 l &7 u n i t s , . T h i s means t h a t t h i s measurement i s r o u n d e d t o t h e n e a r e s t HUNDREJTH, a n d t h a t i n order t o be able t o e f f e c t t h i s degree o f rounding i t i s n e c e s s a r y t o k n o w 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 a and 1«674« I n t h e c h a r t a b r v e , t h © s m a l l a r e a s h a d e d 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 enlargement 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 , , lOths 100th s 1 7 1000th 100th* t h e i n d i c a t o r o n 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 v a r i a t i o r i n l e n g t h t h a t eoufcd b e 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 described a s 1«67 u n i t s . llorr P o i n t o u t o n 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 o f t h e l i n e r e p r e s e n t e d b y t h e m e a s u r e m e n t * T h a t i s t o say, point out t h a t i f t h i s l i n e ware r o u n d e d t o the n e a r e s t TENTH i t would be Lesson VI (PageJ) 1 . 7 , and show on the w a l l n i l s that t h i s ' description af' i t s length would e n t i t l e i t t o be plaeed between l.oj? and 1 . 7 4 . This.variation i n enlarged form i s shown on the chart below: F i n a l l y , point out that i f t h i s l i n e were rounded to th® nearest UNIT i t would be 2 , and show on the wall r u l e that finis description of the length would e n t i t l e i t to be placed between 1©5 and 2 . 4 . In a l l these cases o f rounding„ i f the f r a c t i o n i s equal t o , or greater than, one«half o f the f r a c t i o n a l i n t e r v a l , the f r a c t i o n w i l l be raised to the next highest i n t e r v a l . 3» Repeat Step .2 with other i l l u s t r a t i o n s on the w a l l r u l e e Assume„ f&r example, that the length o f a l i n e i s 1 . 3 2 * Show the v a r i a t i o n i n length permitted'when t h i s l i n e i s rounded successively t o : (a) HUNDREDTHS (b) TENTHS (c) UNITS, Show that i n rounding a number t o HUNDREDTHS i t i s necessary to know the number of THOUSANDTHS' i n rounding to TENTHS i t i s necessary t o know the number of HUNDREDTHS% and. i n rounding to .UNITS i t i s necessary t o know the number o f TENTHS. 4© To conclude t h i s portion o f £he lesson^ three ' generalisations should be drawn from pupils a t . t h i i stages (®5 Ih rounding a mixed decimal f r a c t i o n to th© nearest whole number, i f the number o f TENTHS i s 5 o r greaTJerTsEETT: to the whole number. . In rounding a decimal, f r a c t i o n t o the nearest TENTH, i f the number o f HUNDREDTHS i s 5 or greater add 1 t o the number o f TENTHS e t c . 9 9 (b) Jn rmm&'km a Mm& dactiwal f r a c t i c s #c tba ne*rcia&' uttole swoftbttr 1* i«: naaeasarsr t o tawtf the, -&uub«r o f #EK5H$ In Vli&ding & kiuafear *ft the nearest M f R i t i s necessary to know the number o f HUNDREDTHS* tf (c) After 'rounding has been completed, the place occupied by the l a s t d i g i t o r sere indicates the accuracy o f the me&surementrr^oT example,, 2.060 i s accurate to the nearest THOUSANDTH,. Lesson VI (Page 4) PART TWO Achievement of Lesson Ob.iective 2 (Time: 8 minutes) , 238 To demonstrate various applications of the rounding of decimal f r a c t i o n s . Materials: Wall r u l e with movable i n d i c a t o r . Steps: Mote: Decimal fractions are frequently expressed to a degree ©f accuracy beyond that required f o r a p a r t i c u l a r purpose. The following steps show v i s u a l l y how approximations o f such decimal f r a c t i o n s may be made by various applications o f rounding. 1. Assume the length o f a l i n e to be 1.837. Indicate on the Wall rule the very small v a r i a t i o n i n length that would be permitted by .'this very accurate description o f length. 2. For convenience we may round t h i s mixed decimal expression to : . HUNDREDT1S i f the.purpose for"which the measurement was being • used warranted i t , and report i t as. 1.84 OR. 184 HUNDREDTHS. v Remind pupils of the point that was emphasized i n Part Two of Lesson III concerning the- importance of the last-used p o s i t i o n a f t e r 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 t h i s measurement, 1.84 o r 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 i n the diagram above, point out on the - wall r u l e that t h i s may be rouncfia to 1.8 or 18 TENTHS. / Repeat the various points made i n Step 2 above., 4. Demonstrate on the blackboard how 1.596 could be expressed as: (a) 1.60 (read "one and s i x t y hundredths") or 160 HUNDREDTHS. (b) 1.6 or 16 TENTHS. Lesson VI (Pag® 5) P A R T .THREE •/Acfoleven^fr of .Lesson Objective 2 5 9 (Tim®? 7 minutes) To indisate tshy UNLIKE decimal fractions must be changed to LIKE decimal fractions (that i s , with the same understood denominator) i n 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 e n t i t i e s a Step 2 refers to measurement numbers, that i s . those specifically indicated i n 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 0;65 2*222 change to OiSOO 0.650 0.239 2© When the numbers represent measurements, as i n the examples below, I t i s neeessary to find the mimber with the fewest decimal places and round a l l the othernumbers to that number of plaees, f o r example: 08 0;65 0*239 o change to OiS 0.7 0.2 Note: i t i s understood that tljese numbers refer 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 w i l 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 f r a c t i o n * . Material*: No s p e c i a l 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 o f seros t o the dividend. I t may be explained however i f the need a r i s e s t h a t the same p r i n c i p l e holds i n the case of NOB^TERKDUTIHQ o r INFINITE quotients.where s & The above examples may be worked cut by d i f f e r e n t pupils on the blackboard„ 2 . When the quotients have been obtained demonstrate by means of d i v i s i o n s involving common f r a c t i o n s that i n the case of: Example (a) HUNDREDTHS divided by TENTHS i s Tenths. [b) THOUSANDTHS divided by HUNDREDTHS^gFTenths (c) THOUSANDTHS divided by TENTHS i s Hundredths 0 0 241 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 f r a c t i o n s • Materials: Flannel Board with supply o f paper prepared with the appropriate design e Steps: Note: Steps 1 to 4, i n c l u s i v e , r e f e r t o examples where a whole number i s divided by a decimal. 1. Write on the blackboard the following d i v i s i o n question: .5) 3 Point out that when a whole number i s divided by a simple f r a c t i o n the answer i s LARGER than the dividend. This may r e v o l u t i o n i s e somewhat the concept c h i l d r e n 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 t h i s , apply Sheet 1 o f the accompanying materials to the f l a n n e l board. DIVIDEND The quotient i s l a r g e r than the i n t e g r a l dividend when the d i v i s o r i s l e s s than one. DIVISOR Sheet 1 Supply the answer to the d i v i s i o n question already written on the blackboard. 2 0 Write on the blackboard-th© seeond d5.vision question: .125nS With the p a r t i c i p a t i o n o f the p u p i l s , i d e n t i f y .125 decimal equivalent of 1/8. •••••• as the When t h i s has been done apply Sheet ft to the f l a n n e l board, DIVIDEND DIVISOR Sheet 2 Lesson VII (Page 3 ) 2 4 2 Supply the answer to the division question already written on the blackboard e Out c f the above two steps pupils should have gained an understanding o f the fact that when <> r-hole number i s divided by a divisor less than ONE, the quotient w i l l be a ^ t e r than the dividend,; 3o Apply Sheet 3 to the flannel board. This i s shown below: DIVIDEND , • — DIVISOR f M ^ ^ ^ ^ M ^s^s^i $ To i l l u s t r a t e that 24*2/3 t i s equal to Notes Circles rather than rectangles are suppliediy 6«§» 2 The d i f f i c u l t y i n t h i s division i s obvious. Let us multiply the DIVISOR by 3 i n order to make i t a whole number. This gives us a new DIVISOR o f 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, l e t $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 i n order to remove the decimal fraction, and also to multiply the DIVIDEND by 1 0 In order to compensate f o r the change i n the DIVISOR. Out o f 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 i s 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 i n Step 3 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. 9 Apply the representations shown below, and contained on Sheet 5, Sheet 6 Alongside and to fee r i g h t o f Sheet 5» apply t o th© f l a n n e l board the contents attached to Sheet 6. ? This i s intended t o i l l u s t r a t e thF.t i n the d i v i s i o n : ishen th© D17&S0R i s m u l t i p l i e d by 10 t o make i t 4. and the DIVIDEND m u l t i p l i e d by 10 also, the d i v i s i o n process becomes much e a s i e r t o performs I t i l l u s t r a t e s B too- that when the DIVIDEND,, as w o l l as th© DIVISOR, i s m u l t i p l i e d by the same amount, the quotient remains unchanged. Stated i n another way, i t may be s a i d 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 i n 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. I l l u s t r a t e the d i v i s i o n •1)"'J2 ' cn the blackboard, and point out thatthat i f the decimal point were moved t o the RIGHT i n the DIVIDEND and t o th© LEFT in. the DIVISOR the answer would be 100 times l a r g e r than i t should be. 9 S i m i l a r l y , 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 t o the RIGHT i n the DIVISOR. Using Sheet 7, i l l u s t r a t e the d i v i s i o n . l f T ~ " " on the flannel board t o Show that the answer would be only 1/100 of what i t should be. 2hh Lessonthree VII (Baas 5) *o ee&elude t h i s lesson,- the following generalisations should he drawn fro©, p u p i l s : (a) When a Whole number is? divided by a simple f r a c t i o n th© quotient (answer), w i l l be l a r g e r than the • dividenda (b) In d i v i d i n g with dselmals, the d i v i s o r ma$r be made a whole number by m u l t i p l y i n g i t by a given aaount provided the dividend a l s o i s m u l t i p l i e d by the same amourafc* e (e) In d i v i d i n g with decimals* i f the decimal point i s moved OSSE PLACE' IN OPPOSITE DIRECTIONS i n the DIVIDEND and 'the DIVISOR the answer w i l l be EITHER 100 times greater than o r l/lOO as great a s i t should be. n 9 9 Page 1 245 LESSON V I I I 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 s i g n i f i c a n c e o f changing a common f r a c t i o n to a decimal f r a c t i o n . Materials: Flannel Board with supply o f paper prepared with the appropriate designso Steps: 1, Set the Flannel Board on the bottom ledge o f the blackboard* Apply the symbols attached t o Sheet 7 t o the l e f t side o f the Flannel Board, as shown below: Not©: Materials are Numerator l a %he u f y S N P * _so supplied f o r I l l u s t r a t i n g 3/8 Denominator i s the DIVISOR Explain t h a t a common f r a c t i o n merely Indicates an unperformed d i v i s i o n , and that the changing o f t h i s common f r a c t i o n into a decimal f r a c t i o n INVOLVES THE PERFORMANCE OF THIS DIVISION. In t h i s d i v i s i o n the numerator o f the f r a c t i o n 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 t o TENTHS. Apply the appropriate paper to the Flannel Board t o represent t h i s conversion. Since 10 i s not evenly d i v i s i b l e by 4, i t i s necessary t o convert the 10 t o HUNDREDTHS. Apply the appropriate paper t o the Flannel o a r d t o represent t h i s conversion. B Since t h i s numerator i s now d i v i s i b l e by 4, perform the d i v i s i o n by w r i t i n g 25 HUNDREDTHS on the blackboard t o the r i g h t of the Flannel Board, and then express1 "tnls as' a decimal f r a c t i o n , . 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 F r a c t i o n Cfomge t o « — _ — - Y a n t h a | |2 C h a n g © . . ; v t!5^tw^3^» J^^^J^S ........ . . . . . ,^£Mk FraeHon . o 5 3« $ 0 conclude t h i s portion of the lesson, two generalisations should be d r a m from pupils a t t h i s stages (a) Converting a common f r a c t i o n t o a decimal, f r a c t i o n involves a d i v i s i o n i n i M c h the Numerator o f the • f r a c t i o n becomes the d i v i d e n d ^ and the denominator becomes the d i v i s o r * (b) Before performing t h i s d i v i s i o n i t i s necessary t o add seres t o the numerator. Adding 1 ^ amounts t o c«snverting the KUI^ERATOE from OKES t o TENTHS HUNDREDTHS g, THOUSANDTHS or whatever smaller u n i t i s i?equired t o obtain a s u i t a b l e decimal f r a c t i o n equivalent* s g P A R ? f ¥ G Multiplication Involving Deeia&l Fractions Achi@vem.eBls of'l«eson Objective 3£'(Wines 12'minutes) f® i l l u s t r a t e the reason f o r the placement o f the decimal point i n the .product obtained by the multiplication o f decimal fractions« - Wall r u l e with movable i n d i c a t o r . Steps; 1. Hang the Wall Rule from the moulding a t the top o f the blackboard. Regard the distance between 1 and 2 on the r u l e as 1 who}.© u n i t . Show cn the r u l e by means o f moving the i n d i c a t o r : (a) 1/10 o f (which means times) 1 whole u n i t i s 1 TENTH 9 or 9 i n other words That i s 9 9 .1 times 1 equals o l TENTHS TIKES UNITS EQUALS TENTHS. 247 Lesson V I I I (Page 3) (b) 1/10 times I/IO equals or, i n other words 0 1/100 »1 times .1 equals ,01 That i s , TENTHS TI.ES TENTHS EQUALS HUNdil^THS. 2 In the same way explain that i n the question: e 19«8 times 7o6 s 150*48 the decimal point I s located i n t h i s place i n the answer because TENTHS TIMES TENTHS XS HUNDREDTHS, % a Point out how the value of t h i s product would be altered i f the decimal i n the f i r s t number were changed two places t o the l e f t , f o r example, and changed one, place t o therlgfat' i n the second number* Thus, instead o f 19 ©8 times 7«6 .198 we would now have times 76, This product would have t o be expressed i n THOUSANDTHS, because THOUSANDTHS times ONES (76 ones) equals THOUSANDTHS. The o r i g i n a l was expressed i n HUNDREDTHS. Therefore, the act o f changing the decimal points as we d i d had the e f f e c t o f making the value o f the f r a c t i o n exactly 1/10 o f what i t was at f i r s t . 4* I f time permits, repeat t h i s procedure contained i n Steps 2 and 3 with the following examples The product o f 4.86 i s THOUSANDTHS). and 6.9 i s 33*534 (HUNDREDTHS times TENTHS Point out i n what way the value of t h i s product would be affected i f the decimals were moved i n t o the following p o s i t i o n s : (a) 48.6 (b) times ,69 «4$6 times 6.9 (Answer remains unchanged) (Answer i s 1/10 o f what i t was). M ^ ^ ^ f f i , is&vpXviiig .deo&ua^., f r a c t i o n s &^^J^LM^mL^k^til^^^^ & minutes) . To develop m understanding o f the importance i n the a d d l t t a o f decimal fraction® o f aligning coXussas according t o place value* .kffiscfl Materials; (Page 248 Ko s p e c i a l materials required, Note: Pupils often f a i l to l i n e up decimal points when they write decimals i n addition problems. Errors r e s u l t i n g from t h i s may hot be detected because ©f the f a i l u r e to recognize usfcat the decimal i n the sum must mean. Writing the sum c o r r e c t l y should be r a t i o n a l i z e d i n terms of place value. 1. Copy the following chart on the blackboard: ONES TENTHS HUNDREDTHS THOUSANDTHS • 1 1 1 1 • 0 1 1 1 . 0 0 1 1 » a 1 0 This v i s u a l i s a t i o n should be used t o Impress upon p u p i l s the f a c t that the necessity t o a l i g n the decimals under one another I s merely t o ensure that numbers with s i m i l a r place values w i l l be added, be added In an addition involving decimal f r a c t i o n s i t i s no more correct to add a 1 i n the 8ENTHS place t o a 1 i n the HUNDREDTHS place than I t i s t o add 1/10 and 1/100 without changing them t o a common denominator. 2. I t should be pointed out that where there i s an addition i n v o l v i n g decimals f e r i v e d FROM MEASUREMENTS, as i n the case a t the r i g h t , these quantities should not, from a p r a c t i c a l p o i n t o f view a t l e a s t , be added as they stand® me number ^*Y"does not necessarily mean 12.30. I t may mean anything from 12.25 t o 12.34 i n c l u s i v e . I f such measurements have been obtained, and they are t o be added, the only sensible thing t o do i s TO ROUND ALL TO TENTHS, that i s , round so that a l l the measurements" are expressed t o the same number o f places* 5 12*3 inches 8*65 • 144059 6*4 " tt (in practical work round t h i s t o TENTHS) TENTHS) Emphasise the f a c t that i n an example such as the one shown i n Step 2» the answer w i l l be accurate ONLY t o the nearest TENTH* '.tafcsos* f i l l (Psg© 5) 249 3© Ite conclude this p o r t i o n of the i e s s o a two. generalisations should h% dram, from pupils at this stager 9 (a) S t the ©ass of. the addition'of measurement numbers involving decimal fraetioas, the suss w i l l "fee aeeurate only as f a r ®& the last-used place value of the HKiaDereoixtainiis.g the f e i ^ s t .number of decimal places* (h) 30a the addition o f decimal f r a c t i o n s a l l f i g u r e s 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) o f the background material may be aiscusseu within approximately the time. In order to achieve.the above objectives i t i s suggested that the material provided should be referred to i n general terms. The emphasis should be on the spontaneity of the presentation rather than on too r i g i d an adherence to minute d e t a i l . I f the objectives of the lesson are reached success* f u l l y i t should motivate the pupils to explore and experiment with decimals i n 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 i l l u s t r a t e: ( i ) the ten-ness (or decimal nature) of our number system. ( i i ) positional value, or the idea that the numeral at the right of a whole number has a ONES' value, the next numeral on the l e f t has a TENS' value, and the next on' the l e f t has a HUNDREDS* value. 1. Hang Cards 1 and 2 from the moulding at the top of the blackboard and i n the positions shown i n the diagrams below. Koto: i a acfciKg the two illusia-atrloss acted above, • Steps 2 to $ inclusive shot? the situatioks ?shichrequire a p^groiipiag fsxai the G2P2S* position to th© petition. Step 9 crsphasiaGS the relationship i a actucl valr.a betrays?, various digits i a thcs& givoa pos'.tSono. 2 Point to each of the block i n the f i r s t row of the ONES' column while counting 1, 2, 3 4 , 5, 0, 7, «, 9 , 1 Q 10. 2 . . . 251 3« Explain that we must regroup when we reach 10* Provide . a small square immediately below the Cards as shown i n the illustration. at the top of the next page* These squares may be drawn on the blackboard* Uo Explain that i n these squares we customarily write only one figure to indicate the number of blocks i n any one column position* This illustrates the need to regroup when 10 has been reached* ONES TEWS <2j3ii o a n n rilTpp^ cunoonnmiii non n Symbol • • Symbol : tma ..DDan "Do Symbol Symbol Chart No^ 2 5* Draw a line on the blackboard, as shown, connecting this f i r s t row of 10 blocks with the equivalent representation i n the TENS' column i n Chart No* 2« 6* Emphasise the fact that 1 block i n the TENS* column represents 10 times as many blocks as 1 block i n the ONES* column* ?* Continue to count to 20, pointing this time to each of the blocks i n the seconc row i n the ONES' column* 8* As i n Step 3, explain that we must regroup when w© reach another group of 10 blocks. Once, again, draw a l i n e on the blackboard connecting this second row of 10 blocks with the equivalent representation i n the TENS' column i n Chart No* 2. Write symbols i n squares* .252 9o As i n 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 i n actual tralue botween various digits l a these three positions? Hang Cards 3 and 4 i n the same position formerly occupied by Cards 1 and 2« Card 3 has 12 blocks i n the TENS* column and 15 blocks i n the 0NES column. Emphasise the fact that i n 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 i n one position to equal 1 i n 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, 0 X0 o T 11. As i n Steps 6 and 9 , draw attention to such facts as: (a) 1 i n the HUNDREDS* position (on Card No. 4) represents 20 times as many blocks as 5 i n the ONES* position. (b) 5 i n the ONES* position (on Card No. 4) represents 1/6 as many blocks as 3 on the TENS' chart. 12. Draw frcss pupils, out of the experience they have had with the foregoing relationships, generalisations framed around the following: (a) The number system i s based on a grouping by tens* (b) The number system has place value. This means that each numeral i n a number possesses a value assigned by the plae<&® i t occupies i n the number. Each place- has a value ten times as much as the . 2place immediately to the right, or ose«tenth as much as the position Xmme^iaTely to the l e f t . . l, B w 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) ( i i ) of the background material). 253 To i l l u s t r a t e : ( H i ) the use o f aero, o r 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 p o s i t i o n write the appropriate symbol. 2. Explain that a f i g u r e must be written t o show each place i n the three place number, even though the columns i n two o f the places are empty. This i s the PLACE HOLDING FUNCTION OF ZEHO i n our number system. Show that zero has a protecting r o l e t o keep 1 i n the t h i r d space from the r i g h t , o r the HUNDREDS' column. 3„ Draw on the blackboard a representation o f Card No. 6 and show on t h i s representation one block i n the HUNDREDS* column, and seven blocks i n the ONES* column. 4. While r e f e r r i n g t o 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: t h i s i s a convenient point a t which t o explain that the use o f "and" i n the reading o f a number i s reserved e x c l u s i v e l y to i n d i c a t e the connection between t W ^ S § ^ p l a s e and the TENTHS' place. I t i s never used e i t h e r i n a whole number o r i n a f r a c t i o n a l 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» 2 . and 1 ( T i m e : 15 m i n u t e s ) S Materials: Three v i s u a l i s a t i o n cards numbered 7 on the reverse ? 8 S 9o (Numbers are inuicated 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 A l l three steps should contribute to the attainment o f Objective 3* $ 0 Steps: MMMMM 1, Hang Cards No 7 and 8 from the moulding at the top of the " blackboard and i n the positions shown i n the diagrams below s e •lamiiii J-J ' i ' s '£•; cl M BE e$ B a Card No. 7 HUNDREDTHS TENTHS ONES ii 4 - ] THOUSANDTHS t HIIII • Card No. 8 — •• • • i • mm**. Point to the decimal point. Explain that the purpose o£ the decimal point i i to ident i f y the ONES d i g i t . I t mav be of interest t o draw attention to the methods explained oh the t&ird page o f ' t h i s lesson by which people e l s e where i d e n t i f y the ONES' d i g i t . Explain that when the ONES' d i g i t has been located a l l other d i g i t s obtain t h e i r values from the position they occupy i n r e l a t i o n to the ONES' place. A number, l i k e 34873, i s quite aeaaiagless tmless we know which digitsstands f o r unity, 5 2 0 In pointing to, aad explaining, the regrouping shown on V i s u a l i z ation Cards 7 and 8, emphasise the fact that the following three p r i n c i p l e s which were shown i n Lesson I to form the structure of the whole number system- apply also to decimal f r a c t i o n s : (1) tenness; ( 2 ) place value; (3j place holding "Tune t ion o f -zero. § .1 255 l&k® an outline o f Cards 7 andS$ on the blackboard and represesSfc on Card 7 such other f r a c t i o n s as (a) 25 HUNDREDTHS (b) 25 TENTHS. Perform on the representation o f Card 8 on the blackboard the necessary regrouping i n order t o emphasize f u r t h e r the three p r i n c i p l e s noted immediately above. 3* Remove &&s ds 7 and 8 and replace with Card 9, shown below: ? THOUSANDS HUNDREDS TENS 1 2 X ONES TENTHS HUNDREDTHSTHOUSANJTHS Z 2,. T Draw l i n e s on the blackboard below the charts, as shown i n the i l l u s t r a t i o n above, t o emphasise the symmetry around the ONES' place,, While the charts are i n t h i s p o s i t i o n , discussion should be points out the following: held which (a) the c e n t r a l p o s i t i o n occupied by the ONES' place* (b) the symmetry o f 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 t o i t on the rights onehundred times as large as the second place to i t : on the rightsTeteT I l l u s t r a t e these relationships with s p e c i f i c 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 i n 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 o n e - f i f t i e t h as much as a 1 i n the TENS' place (that i s . the 1 i n the TENS' place a c t u a l l y represents 100 TENTHS', which i s f i f t y times larger than 2 T8NTHS)» 4. To conclude t h i s portion o f the lessonm, two generalisations should Be drawn from pupils a t t h i s stage: (a) following p r i n c i p l e s which underlie the Whole number apply a l s o to decimal f r a c t i o n s (Objective 1): (i ) Place value - each p o s i t i o n assigns t o a d i g i t & p a r t i c u l a r value. ( i l ) ¥en~»ess «> the value assigned t o a d i g i t i n one p o s i t i o n i s t e n times l a r g e r than the value assigned t o i t i n the p o s i t i o n nest t o i t on the r i g h t . etc, ( i l l ) Plaee«<holding function o f aero » i n order t o "protect* the value o f numerals by keeping them i n the required p o s i t i o n s seres are needed to record whatever amntv positions e s i s t 9 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 to the l e f t or r i g h t o f t h e d e c i m a l point. positions 256 Note: i t n a y he mentioned i n passing that seros f i l l a n o t h e r function quite apart frcma place-holding function, ^his function, as w e l l 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 o f positions around the ONES' place i s symmetrical (Objective 2 ) : (i ) the p o s i t i o n tshich i s t h i r d from the ONES' place (fourth from the decimal) on the l e f t , and t h i r d from the ONES' place on the r i g h t are THOUSANDS and THOUSANDTHS r e s p e c t i v e l y . ( i i ) the p o s i t i o n which i s second from the ONES' ( t h i r d from the decimal) on the l e f t , and second from the ONES' place on the r i g h t are HUNDREDS sad HUNDREDTHS r e s p e c t i v e l y . ( i i i ) the p o s i t i o n which i s next t o the ONES* place (second from the decimal) on the l e f t , and next t o the- ONES' place on th® r i g h t are TENS and TENTHS r e s p e c t i v e l y . Achievement o f Lesson Objective 4 (Time: 15 minutes) Materials: V i s u a l i s a t i o n 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 formulate a meaningful generalisation respecting*the comparison o f 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 h a v e become a c c u s t o m e d t o m a k i n g c o m p a r i s o n s o n w h o l e n u m b e r s 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 o n 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 o n e m u s t 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 comparing decimal f r a c t i o n s , as i n comparing 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 a t t h e top o f the blackboard m 100 100G 1 1 10 To 100 1000 Card No. 10 Sjfifr & *o?lows- i f h 5 0N 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 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 to ( a ; t h e TENTH chart « e x p l a i n t h a t i f t h e chart located i n the corresponding position to t h e l e f t of the ONE were shown i t would cover an area 10 tines l a r g e r than the area o f the ONE chart. This would be i n the TENS' p o s i t i o n , (b) the HUNDREDTH chart - explain that i f the chart located i n the corresyjonding position t o the l e f t o f the ONE were shown i t would Cover an area 100 times l a r g e r than the area o f the ONE chart. This would be i n the HUNDREDS' p o s i t i o n . (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 i n the THOUSANDS' p o s i t i o n . A representation o f Card NO. 10 should be put on the blackboard, together rrith the extensions to the l e f t o f the ONES' place, as shown on the diagram at the bottom o f the previous page. With the assistance of the Card and diagram on the board, discuss the manner i n which we would arrange the following i n order o f s i z e , beginning with the largest: (a) 1.1 (b) .011 Cc) 11 (d) .11 (e) 1.11 2, Proceed t o compare two decimal f r a c t i o n s , e.g., .25 and ,3 i n t h i s way: a b X X o a b x x x O O X X Under the TENTHS'and HUNDREDTHS' columns'of Card No. 10, as i n the example above, use x's t o represent .25 and o's to'represent . 3 . By r e f e r r i n g to the v i s u a l i z a t i o n explain why the .3 i s l a r g e r than the .25. 3 . The same procedure may be followed i n showing the reasoning involved i n arranging the following according to s i z e : • (a) .5 9 (b) .05 * (c) 5.5 . (d) ,055 • (e) ,55 4. To conclude t h i s portion o f the lesson, t h e following generalisation should be drawn frtas pupils a f t e r the completion o f the above. ^Decimal fraatiojus can be ranked i n order of *±m by comparing the absolute value o ffcfeed i g i t * i n the corresponding places thus; 258 (&} th© largest o f several decimal fractions w i l l be the one with the largest figure I n the TENTHS place* 9 (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 i n 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 i n the THOUSANDTHS place. 5 Procedure o f 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 s p e c i a l fossa o f common f r a c t i o n s having denominators o f 10, 100, 1000 e t c , that i s , any power o f 10* Materials: No s p e c i a l materials required. Steps: 1. Write the following series o f common f r a c t i o n s on the blackboard: (e) I (a) J tbl^B U)j£ ( J ) ^ (k)l^oo 2. Verbal U) » (e g )T (1>5$ (f) | ( g ) ^ ("»-| ( n ) ^ (o) f§ ( p ) i Explanations: (a) Explain what i s meant by "a power o f 10". Obviously, i t i s be&ond the scope o f the pupils' comprehension a t t h i s stage t o explain that i t means "the index o f 10". Consequently, i t w i l l s u f f i c e t o explain that i n e f f e c t i t means 10 m u l t i p l i e d by i t s e l f any number o f times, o r 10 by i t s e l f , thus: 10, 100, 1000, e t c . The meaning o f "a power o f 10" should be made d i s t i n c t from the meaning o f "a multiple o f 10" x-shich means 10 m u l t i p l i e d , not by i t s e l f any number Of times, but by any riiisaber, f o r example: 5, 8. 12. 20, 30, etc., t o give these respective multiples o f 10: 50, 80, 120, 200, 300 e t c . (b) Explain that waile a l l the f r a c t i o n s written on Common f r a c t i o n s , those with a denominator o f a may also be regarded as decimal f r a c t i o n s , even customary practice i n w r i t i n g decimal f r a c t i o n s the denominator andto indicate i t i n d i r e c t l y by decimal point. the board are power o f 10 though i t i s t o omit w r i t i n g the use o f a 3. Form two columns on the blackboard, and a t the top o f each write headings as follows: Fractions which may be considered only as common f r a c t i o n s • — I' • •. Fractions which may be considered as Decimal f r a c t i o n s the appropriate heading~—-renter each o f 'the f r a c t i o n s already written on the blackboard. fUnder Achievement ®£ Lesson Objective 2 (Tiaaes 12 atefces} show how d e c i m a l f r a c t i o n a 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* Materials5 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 o n the r e v e r s e side) Steps: f e t e : The two point* stated below should be c l e a r l y emphasised a f t e r each o f the following threa representations contained i n Step 1 o f t h i s Lesson procedure 1« She position o f the l a s t d i g i t a f t e r the decimal point determines the value o f a decimal fraction,. That i s . ©sch of the d i g i t s i n the decimal positions preceding the la3t place may j a j t o r a Jpe. cenvertad t o the "place value o f th« l a s t p o s i t i o n a f t e r fa© "decimal point* The number so obtained determines the NUMERATOR o f the equivalent common f r a c t i o n * 1 At the same time the p a r t i c u l a r olace -salvi© •£ the l a s t occupied p o s i t i o n indicates the DENOMINATOR of the equivalent common fraction,, 2 G When a decimal f r a c t i o n i s changed t o a common f r a c t i o n , the denominator has ONE ZERO f o r every figure to t h e r i g h t o f the decimal p o i n t a 1 0 Provide representations o f (a) t h e f o l l o w i n g three fractions a s indi* • R a n g C a r d s 7 a n d 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 (lower diagram) and e x p l a i n how 2 HUNDREDTHS a n d 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, a n d a d d e d 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 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 the decimal point i n determining the value o f a decimal f r a c t i o n . Emphasise (b) c l e a r l y t h e two p o i n t s s t a t e d sbove i n green„ 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 a n d 8 a n d t h e n i l l u s t r a t e «,12 o n 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 : """ m , IfUNPREIfT?S~" an THOUSANDTHS j T E N T H S 1 "HuU&B&TIIS 261 " ISI1IIIII1 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 t h e importance o f the p o s i t i o n of the l a s t d i g i t a f t e r the"decimal 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 . Emphasize c l e a r l y t h e two p o i n t s s t a t e d (c) above i n g r e e n . 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 procedure o u t l i n e d i n ( b ) . and f o l l o w t h e N o t e ? Step 2 below i s merely an extension o f (c) above a n d s h o w s that the two points noted above may b e u s e d t o explain t h e conversion of a n i n t e g r a l number i n t o a n improper f r a c t i o a I n t h i s ease o f course,, i t i s the p o s i t i o n o f t h e terminating e e r o which determines the value o f t h e improper f r a c t i o n . s 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) Refer t o the s e c t i o n o f t h i s chart ONES shown b e l o w : TENTHS I m a g i n e the numbers on t h i s s e c t i o n t o b e a s r e p r e s e n t e d above. E x p l a i n t h a t i f the t w o 0N3S w e r e c o n v e r t e d t o TENTHS t h e r e be 2 0 TENTHS. w>uld (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 Imagine 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 above c E x p l a i n t h a t i f t h e t w o 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 b e 2 0 0 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 b e 2 0 0 0 THOUSANDTHS. In e a c h c a s e t h e t w o " p o i n t s n o t e d i n g r e e n o n t h e p r e v i o u s should be emphasized* page c P ii R •' f f H R B E . M&S&B&kotJ4BmSMf$0&® 2 (Times 10rn&mtm)• 2 62 t o provide p r a c t i c e i n the reading and w r i t i n g o f decimal f r a c t i o n s . • Materials: No s p e c i a l materials required. Steps: Note: The achievement o f Lesson Objective 2 w i l l enable pupils . t o v i s u a l i s e the common f r a c t i o n equivalent o f a decimal f r a c t i o n . I t is. t h i s a b i l i t y t o v i s u a l i s e the esommon f r a c t i o n f o r a which, according t o Spltser, provides a good procedure f o r the reading o f decimals. Therefore, the f i r s t step belcrcr presents a t a more abstract l e v e l , the same method used i n the achievement o f Lesson Objective 2. 9 1 . Write the decimal f r a c t i o n 0.256 on the blackboard. Then explain the meanings f o r t h i s decimal that are shown below: 0.256 means 0;200 0;0£0 0:006 lf 2i6 fl (200 THOUSANDTHS) I 50 THOUSANDTHS) ( 6 THOUSANDTHS) {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 f r a c t i o n by "AND". In the reading o f decimals the word "AND" i s reserved f o r t h i s purpose and i s never used, with one exception i n e i t h e r the i n t e g r a l o r f r a c t i o n a l portion of the mixed decimal. 9 Thus, 115.231 i s read "one hundred f i f t e e n AND two hundred t h 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 o f a decimal f r a c t i o n containing a common f r a c t i o n , f o r example: 4*12| is. read "four AND twelve and one-half HUNDREDTHS". 0.0^,is read "one seventh of a TENTH". 3. Explain that i n "reading a N0N«TERI IIN ATIN G or INFINITE decimal f r a c t i o n l i k e 3.1416 i t i s common usage t o read t h i s as a telephone number, thus: 3.1416 may be read "three DECIMAL (or POINT) One«four-one-six i f 4. Explain that i n reading a TERMINATING or FINITE d e c i m a l f r a c t i o n such as might be obtained 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 . f o r example .0500, w o u l d b e r e a d " f i v e h u n d r e d TENTHOUSANDTHS "* i n ' s u c h c a s e s a s N o ' s . 3 a n d 4 i t i s custom; rather tin;,, r u l e , which determines t h e . m o s t a c c e p t a b l e m e t h o d o f reading'. Page: 1 ) are .of. Teaching; PART 263 GiSS 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 * Materials: C a r d s 9 and 10* A l s o two p i e c e s o f b l a n k paper t o be used i n c o v e r i n g u p 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 b e u s e d w i t h C a r d 10* Steps? Note: Steps X, 2, and 3 demonstrate v i s u a l l y 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 * Cover up 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 , thus l e a v i n g exposed t h e p a r t shown b e l o w : TENTHS ONES 2 2. C o v e r t h e 2 i n t h e TENTHS' HUNDREDTHS THOUSANDTHS 2 place, 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' a n d THOUSANDTHS' p l a c e s w i l l b e 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 of the decimal 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 m u s t 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 a n d l e a v e t h e 2 Hundredths and 2 Thousandths 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 Hundredths and 2 Thousandths. A c c o r d i n g l y , hang up i n the a p p r o p r i a t e p l a c e the sheet b e a r i n g the aero. 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 , as i n S t e p 2 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 as 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 . 5 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 o f a m i x e d d e c i m a l f r a c t i o n o f i n s e r t i n g a s e r o immediately a f t e r 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 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 s s h o w n 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 ( s h o w n i n F i g u r 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. blackboard.• arrangements draw on 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 p o i n t a n d t h e TENTH. the 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 a n d t h e HUNDREDTH, a s shown i n F i g u r e 2. of the the decimal TENTH 6. S i n c e , h o w e v e r , t h e s e c o n d a n d t h i r d p o s i t i o n s f r o m t h e O N E ' S p l a c e m u s t b e 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 w h i c h s h o u l d a l s o be d r a w n o n 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. S 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 ) , a n d we h a v e t a k e n >/l0 o f p o s i t i o n (c) t o g i v e us 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 a n y way a l t e r e d t h e O N E ' 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 b e 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 expression. A l l t h a t can 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 the decimal p o i n t has the e f f e c t o f r e d u c i n g the v a l u e o f the mixed decimal 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 v i s u a l l y t h e e f f e c t upon the value o f a simple decimal f r a c t i o n of i n s e r t i n g a zero immediately a f t e r t h e 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 i n Figure fc, 8. Then i n s e r t the ZERO immediately a f t e r the decimal p o i n t Show t h i s by drawing on the blackboard immediately under Card 10 the representation shown i n Figure 5. 0 This figure shows that the i n s e r t i o n of the ZERO causes a displacement o f the TENTH and HUNDREDTH. As i n step 6, since the second and t h i r d positions from the ONE'S place must be HUNDREDTHS and THOUSANDTHS respectively, i t i s necessary to make the appropriate a l t e r a t i o n , shown i n Figure 6 which should also be drawn on the blackboard immediately under Figure 5« S 1 t Figure ,4 Figure 5 Figure 6 Unlike the previous example (ddscribed i n Steps 4, 5, and 6 ) , t h i s i l l u s t r a t i o n shows that i n s e r t i n g the zero immediately a f t e r the decimal point IN A SIMPLE FRACTION has the e f f e c t - o f making the value of the new f r a c t i o n EXSCTLY ONE-TENTH of the value of the o r i g i n a l f r a c t i o n . As shown by the arrows, i n s e r t i n g a zero immediately a f t e r the decimal point i n a simple f r a c t i o n causes a displacement which reduces each place to 1/10 i t s o r i g i n a l value. 9„ To tto&elud* tihis paraxon of the ies&ca, ^ be arawxi Xros pupils a t t h i s atagei gi^er^liswvio&i should (a) I f a aero i s inserted a f t e r the decimal point i n a mixed decimal expression i t has the e f f e c t o f reducing the value of the expressions (b) I f a sero i s inserted a f t e r the decimal decimal expression i t makes the value 01 as i t was o r i g i n a l l y . i n t 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 o f paper bearing a sero symbol* Steps: 1* Hang Card 10 from the moulding a t the top o f the blackboard. ^over up the HUNDREDTH*s and THOUSANDTH'S representations, l e a v i n g the arrangement shown i n Figure 1. 2, Immediately under t h i s portion o f Card No. 10, draw on the blackboard the arrangement shown i n Figure 2, which shows that ZERO has been annexed immediately t o the r i g h t o f the TENTH'S place* Figure 2 3. Draw attention o f pupils t o the following points: (a) a Terminal Zero, unlike a place holding sero, i s annexed to the end o f a decimal f r a c t i o n . (b) a Terminal Zero does not change the actual value o f a decimal f r a c t i o n , but i t does change the s i g n i f i c a n c e o f i t * Thie change i n SIGNIFICANCE o r IIEANING which r e s u l t s #rom a adding a Terminal Zero w i l l be discussed i n Lesson VI* At t h i s point i t w i l l be s u f f i c i e n t t o point out that adding the sero'in the above example enables the f r a c t i o n t o be read "ONE and TEN HUNDREDTHS" instead o f "ONE and ONE TENTH". This indicates that the decimal f r a c t i o n i s accurate t o the nearest HUNDREDTH, Without the terminal sero i t i s accurate only t o 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 of the decimal fraction of moving the decimal xext. Steps 3 and h demonstrate visually the effect of the decimal fradtion of moving the decimal right. upon the value point fio the upon the value point to the Step 5 i s the f i n a l step i n 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 V i s u a l i s a t i o n 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 /fc - J l Emphasise the point that the number represented 2 HUNDREDS, 2 TENS, and 2 ONES. i s composed of As indicated above, draw an arrow {red,in t h i s i l l u s t r a t i o n ) 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 o f the decimal point must always be the ONES* place, t h i s makes i t necessary to consider that the o r i g i n a l 2 TENS have now, i n e f f e c t , been reduced to 2 ONES. Likewise, the oth&? 2*s shown i n adjacent positions must be reduced t o one-tenth the o r i g i n a l place value i n order to maintain the p r i n c i p l e o f 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 o f the decimal point two places t o the l e f t of the o r i g i n a l l o c a t i o n . Repeat the appropriate explanation given i n step 1. L g s s o a 7 {Page 2 ) * 269 Hang V i s u a l i a a t i o n Card 10 from the m o l d i n g a t the top o f the blackboardo Cover up the ONE. This lesves; 3 3 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 t o indicate the movement of the decimal point one place t o the r i g h t . Explain: Since the place immediately t o the l e f t o f the decimal point must always be the ONES* place, t h i s makes i t necessary to consider that the representation o f 0NE«TENTH (immediately t o the l e f t o f the new l o c a t i o n o f the decimal point) has, i n e f f e c t , been increased t o ONE. * Likewise, the representations shown on adjacent places (that i s , the TENTH and HUNDREDTH places) must be increased t o ten times the o r i g i n a l s i s e i n order t o maintain the p r i n c i p l e o f 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 t o i n d i c a t e the movement o f the decimal point two places to the r i g h t . Repeat the appropriate explanation given i n step 3. 5 . To conclude t h i s portion o f the lesson, the following generalisation should be drawn from pupils a t t h i s stage: (a) For every Place that a decimal point i s moved t o the r i g h t i n a number, i t has the e f f e c t o f multiplying the number by TEN. That i s . i f the decimal, point i s moved one place t o the r i g h t , the msaber becomes 10 times l a r g e r : i f i t i s moved two places to the r i g h t , the number oecomeslOO times l a r g e r , e t c . (b) For every place that a decimal point i s moved ,ftq the in a numb"er/*it nas the e f f e c t o f d i v i d i n g the number by 1Q, Thafc i s , i f the decimal point i s moved one place t o 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 t o tn© TeWr'the' number i s reduced t o 0NE~ HUNDREDTH i t s o r i g i n a l value, e t c . PART, tm i ^ M S E S H L ^ ^ ^ ^ ^ S & J ^ ^ l E S , ^ (Tims: 15 minutes) To demonstrate the e f f e c t upon the location of the decimal point of isultiplying o r d i v i d i n g a decimal f r a c t i o n by a poster c f 10. Materials. 270 Same as f o r Part One. Nets: 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 O N E . The s t e p s in 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 those c o n t a i n e d in t h e f i r s t p a r t . Steps 1 and 2 demonstrate v i s u a l l y t h e e f f e c t upon the 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 p o w e r o f 10. Steps 3 and 4 demonstrate visually t h e e f f e c t upon 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 n u m b e r b y a p o w e r o f 10. Step 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 , and 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 of t h e i r experience with tho f i r s t four steps. 1. Hang V i s u a l i s a t i o n Card 9 from the moulding at the top o f the blackboard. Coyer up the same portion o f the Card as i n Part ONE, leaving the following: HUNDREDS • 200 ONES 2 2 20 Figure 1 2 Line (a) Emphasise the point that the number 222 i s composed o f 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 o f these by 10. This, too, may be written on the blackboard under Line l a ) , as shown below: 20 (2 TENS) 2 2 MasaJll 18 (2 ONES) (2 TENTHS) r 0 » 3*ttcs,fthftops, must b,e ^den^ifled, h r .the decimal,point, i t i s , consequently, necessary to adjust t h e l o e a t l c n -of t h e decimal oint from i t s o r i g i n a l p o s i t i o n (Figure 1) t o one place to the e f t , as shown by the red arrow i n Line ( b ) . f 2. Erase L i n e from the blackboard, and proceed t o develop from Line ( a ) , t h i s time t o show what happens to th© p o s i t i o n o f the decimal point when the number i s divided by 100. v Line (b),therefore, becomes: 271 Lesson V •2 0 2 TOo* IB (2 ONES) (Page 4) (2 TENTHS) Line (b^) (2 HUNDREDTHS 0 Since the ONES must be i d e n t i f i e d by the decimal p o i n t , i t i s consequently, necessary to adjust the l o c a t i o n o f the~decimal p o i n t by moving i t from i t s o r i g i n a l p o s i t i o n t o two places t o the l e f t , a3 shown by the green arrow i n L i n e ( b ^ ) . 8 3. Hang V i s u a l i s a t i o n Card 10 from the moulding at the top o f the blackboard, Dover up the ONE. This leaves: ii ( M u l t i p l y by 10) This represents .111. L e t 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 r e p r e s e n t a t i o n shown i n F i g u r e 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 t o adjust the l o c a t i o n o f the decimal p o i n t i n order t o put i t beside the c a r d t h a t stands f o r ONE. That i s when the number i s m u l t i p l i e d by 10 i t i s necessary t o move the decimal p o i n t one place t o the r i g h t . See red arrow, which should a l s o be drawn oh the blacIcF6ara ' i n the appropriate p l a c e . 5 r 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 t o move the decimal p o i n t two places t o 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 p o r t i o n o f the l e s s o n * the f o l l o w i n g g e n e r a l i s a t i o n should be drawn from p u p i l s a t 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 10 100, 1000, e t c . , ( t h a t i s , some power o f 10) the decimal p o i n t 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 . lb) When a decimal f r a c t i o n i s d i v i d e d by 10. 100. 1000 1000 eet c . . ( t h a t i s , some power o f 10) the decimal p o i n t i s istoved one p l a c e t o the l e f t f o r every sero i n the d i v i s o r . 9 272 ROUoiJIKG DECIMAL FR ACTIONS P A R | 5 M T 0 §E OF LESSON OBJECTIVE 1 \?&mt 15 miKutas) To i l l u s t r a t e th.© significance o f rousdi&g deeiisal fractious? ^tarialsi Visualisation Card No, 11. 3teps: 1* Draw the following scale on the blackboard: (a) Bzplaln that when 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 i t i s to & inch or 3 inches. The rather considerable amount of variation i n Isn^ih permitted i s indicated by the RED erea. It should be evident that i n order to round a measurement number to the nearest"unit ifc i s necessary to kna# at least the number of TEliWnnvolved* i n the rasasurementa (b) Exnlaln that when we say the.^ a l i n e i s 2*0 inches long v.& signify by tills indication that the longish this time i s closer to 2 u inches than i t i s to 1.9 inches or to 2.1 inches„ The more restricted: easnaefc ef w $ & $ l a a i n length permitted by this da?ig?iatlon i s indicated by tbs PURPLE axe*« 0 It Should be evident i n this case that i n order to ro.jnd a measttrement iraaber to the nearest TENTH i t i s necessary to know at least the number of HUNDREDTHS involved in the •measurement. (a)finally, arplain that when we say that a l i n e i s 2.00 inches long we signify by this indication that the length this time i s closer to 2o00 than i t i s to 1.99 or to 2,01 inches. The mven more restricted amouatt of variation i n length permitted by this designation i s indicated by the GREEN area, It should be evident i n this case that i n order to round a measurement number to the nearest HUNDREDTH i t i s v Pae© 2 necessary t o know at least the number o f THOUSANDTHS involved i n the measurement*. 2o Rang Visualisation Card No* 11 from the moulding at the top of the blackboard. Show diagrararaatically how t h i s represents only a portion of the blackboard i l l u s t r a t i o n shown i n Step 1* Let us say that that the length of a l i n e Is 1.67 waits This means that this measurement i s rounded to the nearest HUNDREDTH, and ghat i n ardor to he able to effect this degree of rounding i t i s necessary to know the length of the l i n e i n THOUSANDTHSj or, i n other words; to know that the length I l e a somewhere between 1*765 and Q 1*774* Point out on this chart that as we successively reduce tthe m&a^ey of rounding we increase the variation i n the length of the l i n e represented by the measurement * That i s to say, point out that is* this l i n e 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 I f this l i n e were rounded to the nearest UNIT i t would be 2 and show that this variation would e n t i t l e i t 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 l i n e i s 1*^ or again, 1*32* Repeat the same procedure as In . step a* 4o To conclude t h i s 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, I f the number o f 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 l a s t DIGIT or ZERO indicates the accuracy of the measurement* For example, 2*060 i s accurate to the nearest THOUSANDTH* PART VW 0 Page 3 274 To demonstrate various applications o f the rounding o f decimal f r a c t i o n s . Materials: Visualisation Card No* 11* Stena: Decimal fractions are frequently expressed to a degree o 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 l i n e to be 1.837. Indicate on the Visualisation Card the very small variation i n 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 i n Part Two o f 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 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 i i & s may be rounded to 1.8 OR 18 TENTHS. Repeat Repeat the various points made i n 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 275 Achievement d f Lesson Qbleetive 3 (Time: ? minutes) To indicate why UKLIKE" decimal f r a c t i o n s must' be changed to LIKS decimal f r a c t i o n s (that is$ with the same understood denominator) i n order that they may be added or subtracted* Ksterimlss No s p e c i a l materials required, Notes- Stgp 1 refers'to ndne*iieaeurem@nt numbers which may . b@'dolS!Eea a s ^ i s o r e S s , '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, f o r example: < 1 $ 0,8 0.65 change to 0*800 0*650 2* When the numbers represent measurements, as i n the example belowm i t i s necessary to f i n d the number with the fewest decimal places and round a l l the other numbers to that number of p l a c e s f o r example: e 0*8 0*65 change to 0*8 0*7 0*2 Note: i t i s understood that these numbers r e f e r to inches, pounds, etc* .3* To conclude t h i s portion o f the l e s s o n tao following goneralisatioa should be drawn from pupils at t h i s stages s w The sum o r difference o f measurement numbers w i l l be accurate only to the f r a c t i o n a l u n i t o f the number that has the fewest decimal p l a c e s * B DIVISION INVOLVING DECIMAL FRACTIOUS PAR? 276 0 N IS foieyemenfe o f Leasea Ob.leetiVQ X 'T&ees 8 minutes) T® explain the significant© o f i n f o r m i n g d i v i s i b a i n v o l v i n g 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 i s to interpret the reason for the placement of the |vv -decimal point i n a quotient. : The division of common fractions && used as a means of developing this interpretation. . m The time l i m i t 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 r e s t r i c t 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 rofitable a more complete presentation OUTSIDE THE REA OF THIS EXPSRCTNT. S 1. Present the following examples en-fcfeeMm&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. I t may be explained, however I f the need arises, that the same principle holds i n 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 i n 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 kE T _ Achievement of Lesson OMedtive 2 (Timet 22 miautes) To demonstrate the s i g n l f i s a a e e o f moving the deelm'-l point i n performing <*i'££sloas involving decimal fractions,, Eiateriaist Visualisation Card No 12. For use i n this lesson, each section should be regarded as 1/10 o f the dividend and of the d i v i s o r Steps; 0 0 Motes Steps 1 to k incltisii'-e r e f e r to examples where a whole number i s divided by a decimal. lo Write on the blackboard the division o l H T * and i l l u s t r a t e the answer on Visualization Card No. 12„ shown below: f ol ol E IDS ol 1 H ol D .1 ol ol ol ol 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 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 I t 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 i s 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: 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 w i l l be the same. 1 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 i n c l u s i v e r e f e r to examples where a decimal f r a c t i o n i s divided by a decimal f r a c t i o n . 5 . As i n Step 2* use V i s u a l i s a t i o n Card 12 to explain that when the d i v i s o r i s a fraction^ the d i v i s i o n i s more"easily performed i f the d i v i s o r xs made a whole number* I l l u s t r a t e t h i s with such an example es the following; .4) 3.2 i s more e a s i l y divided v&en changed to 4ri*2T The v i s u a l i z a t i o n shown below of t h i s example alaould be drawn on the blackboard and used to supplement the v i s u a l i z a t i o n medium contained on Card 12. UXTIDES $?\ may be charged to 32 Upl HI JlVIBOkj may Dechanged to As an 3tep 3 , i l l u s t r a t e on the V i s u a l i s a t i o n Card that i f the DIVIDEND and the DIVISOR are each multiplied by the same number, the quotient- remains unchanged Stated i n another way 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. 5 Discuss i n what way the answer would be altered ±t 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. t I l l u s t r a t e the d i v i s i o n 1}"72 on the v i s u a l i s a t i o n card to show that the answer would be 100 times larger than i t should be,. 0 Lesson VII (Page 4) • _/ 279 Illustrate the division *1) '*2* on the Visualisation Card to show that the answer wot&d be JjOO times larger than i t should be* L 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 i n 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 V I I I 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 i n this lesson each space In the upper section (dividend) and i n 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 i n 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 t h i s 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 i l l u s t r a t i o n . 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 i n Lesson III, Part Two, Step 2) into a smaller denomination which w i l l be d i v i s i b l e by 1. Illustrate that changing the 1 into 10 TENTHS does not permit I t to be divided by 4* Consequently i t i s necessary to change i t into 100 HUNDREDTHS* 2«Copy other illustrations, such as the following, on the board: Common Fraction i Change t o Tenths Change to Change to Hundredths Thousandths • Decimal Fraction - 5 .375 Lesson VXU (Pag© 2) $ 9 "£Q CONCLUDE THIS portion of the lesson two generalla^tioas should bo drawn fie© pupils a t this stages 281 (a) Converting a common fraction t o a decimal fraction involves a division In which th® numerator of the fraction becomes the dividend* and the denominator becomes the divisor* 9 (b) Before performing this:41visiem.it i s necessary to add seros to the numerator«Addlng these seres r e a l l y amounts to converting the NUMERATOR from ONES toTENTHS HtHSDREDTHS, THOUSANDTHS, or Whatever smaller unit i s required t o obtain & suitable decimal fraction equivalent* p P A R T TWO sent o f Lesson Objective 2 (Time: 12 minutes) To i l l u s t r a t e the rcaswsa f o r the placement o f the decimal dot- i $ t h e product dtrtaStisd Jftgr 3te, i M l t i p i i e a t d e s o f decimal Materials: Visualisation Card 11 (As used i n Lesson VI) Steps: 1* Hang Visualisation Card 11 from the moulding at the top o f 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, i n other words, *1 times •! equals *01 TENTHS TIKES TENTHS EQUA&S HUNDREDTHS* 2* In the same way explain that i n the question 19*8 times 7*6 the'.decimal point i s located i n this place i n 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 , f o r example, and ©hanged one place t o the right i n 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 i n what way the value of this product would be affected i f the decimals were moved into the following positions: 48»6 times »69 (b) .486 times 6*9 PAR (Answer remains unchanged) (Answer i s 1/10 of what i t was). 1 T B RES ^ y e l y l g ^ d ^ i ^ a l f&g M t To develop an understanding o f the importance i n t h e addition o f dseSmaXfraetions o f a l i g n i n g columns according t o pise© valsss^ Visualisation Card 13* fcei Sotes Pupils often f a i l t o l i n e up decimal points when they writ© decimals i a addition problems* E r r o r s r e s u l t i n g from t h i s may not ha detected because o f the f a i l u r e to recognise vhat the decimal i n the sum must mean* w r i t i n g the sum c o r r e c t l y should be r a t i o n a l i s e d i n terms o f place value* Lesson YIXX (Pago 4) 283 Xo Hang V i s u a l i s a t i o n Sard 13 from the moulding a t the top o f the blackboard* pass, TENTHS HUNDREDTHS i THOUSANDTHS 1 c X X X X o & X 1 1 . 0 o 1 X . X X 6 1 the fact that the necessity to align the decimals under one another i s merely to'ensure that numbers with similar place values m i l l be added* 3B an addition involving decimal fractions I t i s no more correct to add a 1 i n 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* 2. Xt should be pointed out that where there Is a 12*3 inches division involving decimals derived from measurements, as i n the ease at the right, these 8.65 • quantities should not* FROM A PRACTICAL POINT OF 14.059 * VIEW AT LEAST, be added as they stand* The number 12o3 does not necessarily'mean 1£*30* I t may mean anything from 12*25 to 12*34» inclusive* I f such measurements have been obtained and they are to (in practical be added, the only sensible thing to do i s TO work round ROUND ALL TO TENTHS* that i s , to round so that a l l t h i s to the measurements are espressed to the same number or places* 9 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* 3« To eoaclude t h i s portion of the lesson, * M > generalisations shouM be drs&sn from pupils at t h i s stage: (a) In the ease of the addition of measurement numbers involving decimal fractions, the summ w i l l be accurate ©sly as f a r 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 " " 8 . 294 295 Worksheet No. I 285 THE DECIMAL SYSTEM OF NOTATION Write the l e t t e r of the best answer on the answer sheets provided. 1. Which of the following i s the largest? (A) 2. 3. 1346 (B) 6341 (C) 1000 (D) 5999 ( E ) 2997 Which one of the following i s represented by the 7 i n ( A ) seven hundred ( B ) seven-tenths (c) seven thousand (D) seventy thousand ( E ) seven 37829? I f you changed the number 7 3 0 6 9 so that the 3 was i n the 9 ' s place and the 9 was i n the 3's place, how would the new number compare with 7 3 0 6 9 ? ( A ) I t would be larger (B) I t would be smaller (c) I t would be the same size (D) Can t t e l l ( E ) I t can't be done 1 4. 5. 6. 7. 8. 9. I f you re-arranged the figures i n the number 5 3 4 2 9 , which of the following arrangements v/ould give the largest number? (A) 95,324 ( B ) (D) 95,234 ( E ) 95,432 (c) 59,432 95,243 Which of the following numbers i s the smallest? (A) 11890 (B) (D) 17999 ( E ) 10999 (c) 19000 18999 I f you re-arranged the figures i n the number 4 3 , 1 2 5 , which of the following arrangements would give the smallest number? (A) 54,321 (B) 21345 (D) 14,532 ( E ) 13,245 (c) 12.345' I f the figures i n 8 6 , 4 7 3 were re-arranged, which of the following would place the largest figure i n the thousand's place? (A) 73,648 (B) 38,467 (D) 87,643 ( E ) 86,734 (c) 76,483 • I f the figures i n 2 3 , 4 6 9 were re-arranged, which of the following would place the smallest figure i n the tens' place? (A) 46,932 ( B ) (D) 34,629 ( E ) 96,432 (c) 69,234 92,346 Which of the following has a 3 i n the hundreds' place? (A) 23,069 ( B ) 86,231 (D) 39,043 ( E ) 42,304 (c) 49,563 Worksheet No. I cont. 286 10. V/hich of the f o l l o w i n g 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 r i g h t represents a number how many times as l a r g e 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 f o l l o w i n g statements best t e l l s why we w r i t e 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 t h i r t y - n i n e " i f we d i d not write the zero. ( B ) W r i t i n g the zero helps us to remember the number c o r r e c t l y . (c) W r i t i n g 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 i s 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 i t s size 3. The value of the 1 in 2.41 i s 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 i s : (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 .72 ( A ) 100 . ( B ) 1/100 (C) 10 (D) 1/1000 (E) 1000 8. Which of the following numbers i s the largest? ( A ) .3248 ( B ) .4 (C) .3249 (D) .329 • (E) .3328 9. 10. In (a)(b) the digit marked (a) is how many times the digit .0 8 4 marked (b)? Which of the following numbers i s the greatest? (A) .3 ( B ) .295 (C) .11 (D) .101 (E) .301 288 Worksheet No. 2 c o n t . 11. The the (A) (B) (C> (D) 12. The main purpose o f the decimal p o i n t i s t o i n d i c a t e the digit i n : (A) HUNDREDS' p l a c e ( B ) HUNDREDTHS' place (c) ONES' p l a c e (D) TENTHS' p l a c e ( E ) TENS' p l a c e . 13. The f o l l o w i n g numbers: .0163; . 0 2 ; . 1 ; .0897; .0911, when arranged i n o r d e r o f s i z e from l a r g e s t to s m a l l e s t would be: (A) .0911 ; .1; (B) .0911 ; .0897; (c) .1; (D) (E) 14. l a r g e s t o f s e v e r a l decimal f r a c t i o n s w i l l be the one w i t h largest d i g i t s i n the TENTHS' place the HUNDREDTHS' place the THOUSANDTHS' place any p l a c e .0897; .0911; .02; .0163 ; .02; .0163; .0897; .0163; .02; .0163 .02 .1; .02; .0163 .0897; .0911 .0897; .0911 ; .1 The l a r g e s t e x p r e s s i o n o f the f o 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 289 WRITING OF DECIMAL FRACTIONS 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 5 (B)_9 17 (C)_7_ 100 (D)_8 50 (E)j3 10 2. In every decimal fraction there i s an unwritten denominator which i s 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 i n words: (a) 0.362 (b) 0.0375 (c) 200.007 (d) 0.0 1 (e) 0.120 (f) 5.75489 4. 9 .15 = __15 100 (a) .031 (b) 2.02 (c) .875 Change the following mixed decimals to improper fractions as i n example: Example: 6. (h) 0.34f Write these decimals with common fractions, as i n example: Example: 5. (g) 0.0560 1.25 i s 125 HUNDREDTHS or 125/100 (a) 4.75 • (b) 10.00. (c) 1.05 The unwritten denominator of a decimal fraction i s understood to possess one zero f o r : (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 (a) (b) (c) (d) unwritten denominator of a deciaml fraction i s determined by: the number of zeros after the decimal point the place value of the last-used decimal place the size of the largest digit after the decimal point the size of the f i r s t digit after the decimal point 8. The (a) (b) (c) numerator of a decimal fraction i s determined by: the position of the last digit after the decimal point the number of d i g i t s after the decimal the size of the f i r s t digit after the decimal point. Worksheet No. 4 290 THE FUNCTIONS OF ZERO IN DECIMAL FRACTIONS Write the l e t t e r of the best answer on the answer sheets provided. 1. Adding two zeros to the right of a whole number i s 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 l i k e 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 i n 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 / l O (D) Adding 10 to the expression ( E ) Increasing the value of the expression 5. Inserting a zero BETWEEN THE DECIMAL POINT AND THE 5 i n 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 / l O ( D ) Adding 10 to the expression ( E ) Increasing the value of the expression Worksheet No. 4 cont. 291 6. I f the length of a board i s measured to the NEAREST FOOT, say 7 feet, i t i s 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 i n some other way ( E ) It gives an unwarranted degree of accuracy to the measurement. 7. The function of zero as a "place-holder" i n a decimal fraction is tor (A) "Hold" each numeral i n the fraction i n the required position when no digit i s present to perform this function ( B ) Give to the fraction a greater degree of accuracy (c) Spread the d i g i t s out to make reading easier ( D ) Indicate the number of zeros i n the unwritten denominator of the f r a c t i o n . Worksheet No. 5 292 CHANGING- THE LOCATION OF THE DECIMAL POINT: ITS EFFECT ON THE VALUE OF THE EXPRESSION Write the l e t t e r of the best answer on the answer sheets provided. 1. Which of the following numbers i s 1 as large as 32.78? 100 (A) 2. .3278 (B) (c) 3.278 327.8 (D) 3278 I f the number . 0 8 5 7 i s changed to 8 5 . 7 i t becomes: ( A ) 1 as large ( B ) _ 1 as large (c) 1 0 times larger 100 ( D ) 1 0 0 times large 1 0 ( E ) 1 0 0 0 times larger 3. When a number i s divided by 1 0 0 0 the decimal point i s moved: ( A ) 2 places to the l e f t ( B ) 3 places to the l e f t (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 i s multiplied by 1 0 0 the decimal point i s moved: ( A ) 2 places to the l e f t ( B ) 3 places to the l e f t (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. I f a decimal fraction i s divided by 1 0 the decimal point i s moved 1 place to the l e f t because: ( A ) the number i s increased by 1 0 ( B ) the number i s decreased by 1 0 (c) the number becomes 1 0 times as large ( D ) the number becomes l / l O as large 293 Worksheet No. 6 ROUNDING DECIMAL FRACTIONS Write the l e t t e r 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 3. Round each of these to nearest HUNDREDTH: (A) .536 4. (c) 7.931 (B) 4.175 (C) 5.7.82 Assuming that the following numbers have already been rounded indicate fractional unit to which each one i s accurate: Example: (A) .490 2.30 i s accurate to the nearest HUNDREDTH. (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 i s 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 l e t t e r of the best answer, or the answer i t s e l f , on the answer sheets provided. 1. When a whole number i s 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 i s 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 ) 4.564 4. In question 3 the answer comes out to HUNDREDTHS because: ( A ) There i s one figure before the decimal point i n the dividend and devisor ( B ) There are two figures i n the divisor (c) Thousandths divided by tenths are hundredths ( D ) Tenths times tenths i s hundredths. 5. Divicte .42 ) .7392 6. In question 5 the decimal point may be moved 2 places to the right i n the divisor, because: ( A ) i t must be placed after the two .. ( B ) i t i s moved 2 places to the right i n 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) 8. .28 ) .4564 ( B ) 4.2 ) 7.392 In the division 1 . 6 ) 9 . 2 8 i f the decimal points are moved into these positions: .16 ) 9 2 . 8 the answer w i l l 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. 8 295 MISCELLANEOUS CONCEPTS INVOLVING DECIMAL FRACTIONS Write the l e t t e r of the best answer, or the answer i t s e l f , on the answer sheets provided. 1. Which statement best t e l l s why we arrange numbers i n addition the way we do? (A) I t i s an easy way to keep the numbers i n straight columns ( B ) I t helps us to add correctly (c) I t helps us to add only those numbers i n the same position ( D ) I t helps us to carry correctly from one column to another ( E ) I t would be harder to add i f the numbers were mixed. 2. What (A) (C) (E) i s the product of: TENTHS and ONES TENTHS and HUNDREDTHS TENS and TENTHS ( B ) TENTHS and TENTHS (D) TENS and HUNDREDTHS 3. When a whole number i s multiplied by a number larger than 1 , the product i s : (A) larger (B) smaller (c) unchanged 4. When a whole number i s multiplied by a number smaller than 1 , the product i s : (A) larger ( B ) smaller (c) unchanged 5. 6. 7. I f 2 . 3 9 8 times 8 7 . 2 equals the answer to: (A) 2 3 9 . 8 times 8 . 7 2 (B) 2 3 . 9 8 times .872 (C) . 2 3 9 8 times 8 7 . 2 209.1056 without multiplying find In the question 2 . 3 9 8 times 8 7 . 2 , i f the decimal point i s moved 2 places to the right i n the f i r s t number and one place to the l e f t i n the second number the answer i s : (A) l / l O as large ( B ) 1 0 times as large (c) 1 0 0 times as large ( D ) l/lOO as large (E) unchanged To change a fraction l i k e 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 Decimal Fraction Computation Test 297 301 Otis Self-Administering Test of Mental A b i l i t y , Intermediate Examination, Form A Stanford Advanced Reading Test: Form E 302 304 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 i s important to arrange the numbers so that: A. the last figures of a l l numbers are i n the same column B. a l l figures with the same place value are i n the same column C. the f i r s t figures of a l l numbers are i n 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 / l O as much C. makes the value 10 more D. does not change the value. 4. The the A. C. 5. The number 6.00 has a value of: A. 6 hundreds C. 6 hundredths largest of several decimal fractions w i l l be the one with largest figure i n : tenths place B. hundredths place thousandths place D. any place. B. D. 600 hundreds 600 hundredths. 6. I f a decimal point i s moved two places to the l e f t the number becomes: B. ten times as large A. one-tenth as large D. one hundred times as large. C. one-hundredths as large 7. The A. B. C. D. 8. Changing .645 to .0645 A. does not change the value B. makes value 10 times as much C. makes value l / l O as much D. makes value l/lOO as much. number .0170 should be read: seventeen hundredths one hundred seventy ten-thousandths one hundred seventy thousandths seventeen thousandths. (page 2) 9. 10. 11. 12. I f the number 42.56 i s 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. The A. B. C. D. value of a decimal fraction i s determined by: the size of the f i r s t digit after the decimal point the position of the last digit after the decimal point the position of the largest digit after the decimal point the position of the f i r s t d i g i t , not including zerosy after the decimal point. ; Which of the following numbers has the figure "6" i n the thousandths place: A. 4695.5417 B. 6495.1724 C. 4325.2164 D. 4175.6000 32 In the question .5) 16 the answer i s larger than the number divided because: A. 16 i s more than .5 B. i t i s the same as multiplied by -gC. dividing a number always gives an answer larger than the number D. i t i s the same as finding how many •§•* s i n 16. 13. I f a decimal fraction i s divided by 1000, the decimal point i s moved three places to the l e f t because: A. the number becomes 1000 times as large B. the number i s increased by 1000 C. the number becomes l/lOOO as large D. the number i s decreased by 1000. 14. In the question 1.6) 620.54 i f the decimal point i s moved one place to the right i n the devisor, and one place to'-the l e f t i n the dividend, the answer w i l l 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 i s changed to a common fraction (not reduced), the denominator w i l l 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 l e f t 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 i s more convenient D. the point i n the quotient must be d i r e c t l y above the point in the dividend E. the value of a fraction i s unchanged when both terms are multiplied by the same quantity. 18. The measurement 1.050 inches i s 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 B. multiplying the number by 1000 C. dividing the number by 100 D. none of these (a) (b) In the number: 5 5 5.55 20. 21. 22. 23. 24. A. B. C. D. digit digit digit digit (a) (a) (a) (a) is is is is 100 times digit (b) 10 times d i g i t (b) l/lO of digit (b) l/lOO of d i g i t (b) The A. C. E. number .6925 has a value of about: .69 hundredths B. 9 hundredths D. 692 hundredths. 2 hundredths 69 hundredths I f a number i s 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. In one A. C. " E. the question: 1.25) 642.3 i f the decimal point were located place to the right i n both numbers the answer would be: ten times as large B. one-tenth as large one hundred times as large D. one hundredth as large unchanged. I f no zeros are added to the dividend, the answer to the question: 4.2) 69.735 w i l l be a two-place decimal, because: A. B. C. D. thousandths divided by tenths i s hundredths there are two figures i n the divisor tenths times tenths i s hundredths there are two places before the point i n the dividend. (page 4) 300 25. In the question: 6.42 x 15.7 i f the decimal point were located one place to the right i n the f i r s t number and two places to the l e f t i n 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 i s located at this place i n the answer because: A. one and two are three B. hundredths times tenths i s thousandths C. tens times hundreds i s thousands D. there are three places to the l e f t of the point i n the numbers multiplied. 27. A "decimal" i s a fraction with an unwritten, but understood, denominator which w i l l 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 D. 213 tenths E. 2.1 tenths. C. 21 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 w i l l be accurate to the nearest: A. inch B. tenth inch C. hundredth inch D. thousandth inch. 301 School: Name: TEST ON DECIMAL FRACTIONS 1. Find the sum of: (a) 1.0687 9.9345 8.9784 5.8459 7.7956 2. (b) 387.85 100.97 89.59 5.74 983.68 4. From take (c) 8.975 34.878 19.479 6.970 98.826 3. From take 7. Multiply each of the following: 8. 124.40 87.85 27.08 15.17 6. From take (b) .203 x .3 = (c) 100 x 8.5 (d) .08 x 25 x i = 9. Multiply: 10. Multiply: 94.72 19.88 Multiply: 94.36 8.7 78.4 .961 Divide each of the following: (d) . .37375 (b) (c) .12)3 .11)1.342 (e) .ATT7T Divide: (f) .2)~nr 13. What i s 3 8 Answer: of 6.4? 1.25) 62.5 Divide: 8.9) 708.44 .834) 91.74 14. 150.000 72.239 4 x .2 = (a) 12. 5. From take 176.062 89.875 (a) 7.8 6.4 11. Subtract: 1 5« Express 1 8 Answer: as a decimal, OTIS SELF-ADMINISTERING TESTS OF M E N T A L ABILITY ARTHUR S. OTIS,,PH.D. By Formerly Development Specialist with Advisory Board, General Staff, United States War Department INTERMEDIATE EXAMINATION: FORM A For Grades .302 4-9 Score 20 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. name, age, birthday, etc. Write plainly. F i l l these blanks, giving your Name Age last birthday Birthday First name, Month initial, and last name Teacher. ..'. Day Grade Date. , School 10- , City. T h i s is a test to see how well you can think. It contains questions of different kinds. a sample question already answered correctly. Notice how the question is answered: Sample: years Here is W h i c h one of the five words below tells what an apple is? 1 flower, 2 tree, - 3 vegetable, 4 fruit, 5 animal ( J/. ) . T h e 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 N o . 4 ; so a figure 4 is placed in the parentheses at the end of the dotted line. T h i s is the way you are to answer the questions. T r y this sample question yourself. put its number i n the parentheses: D o not write the answer; just draw a line under it and then Sample: W h i c h 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 i n 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 i n 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 i n the parentheses without underlining anything. M a k e 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,• Published b y World Book Company, Yonkers-on-Hudson, -New York, a n d 2126Prairie Avenue, This test isC ocopyrighted. mimeograph, p y r i g h t 1 9 2 2 b y WThe o r l d reproduction B o o k C o m p a n y . ofC oany p y r i part g h t r eof n e wit e d by 195 0. Copyright way, whether the reproductions All , Chicago i n hectograph, Great Britain rights sold reserved, P R Ifurnished N T E D I N U . S .free A. O Sfor A T M Ause, : IE: A are or are is- 8a3 violation * or in any other 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 f . ^ !?*? 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.) (D n writ sedot ^\ :^. ,i ) ( 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, Do 5 Itnot is easily stop. broken Go on [2} with the next page. 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 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-------0 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 35 40 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 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 [3] with the next page. S. A. Intermediate; t 52. If a man has walked west from his home 9 blocks and then walked east 4 blocks, how manyblocks 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 vestpocket-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 [4]and make sure that every answer is right. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ( ) ( ) \ ) ( ) ( ) 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 Name R E A D I N G TEST: FORM , E Age. Boy or girl 3 0 g 4 '.Grade. Name of school.... < City : State.. TEST .... .Date...! ~~ AGE EQUIV. GRADE EQUIV. SCORE Parag. Mean. t Word Mean. Average Read. Grade, Equiv. i 1 • 1 4.0 1 ..J I 1 1 J — 1 1 -1 4.5 l _i i i 5.0 i i 5.5 6.0" 6.5 7.0 8.0 i i i i i i i i , i i i i i' • ' t i i i I i i i i I i i • i I I I I 9.0 I I II i II 10.0 30 25 , .1. ,. I 20 ~ l T 8° i 1 40 35 , ,,1 , i. i 9° 8 1 • '. , . , 1 ,,, 6 it i l l 45 V, r , 1 , I 9 1i 1 1 I • I 10° 6 ,i 50 1I i I i i 1 I I 55 I 1 ii 10 11° i i i i i 6 I i i i i I 1 i i ll 6 60 1i 1 I 65 i 1 1.1 I 1 U J J . 12° 12 13° 6 ,75 70 i i , 1 , 414° 1111111111111111111111 u 111 1 1 III. i i II!L * 15° Published b y W o r l d Book Company, Yonkers-on-Hudson, N e w Y o r k , and Chicago, Illinois 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. 16 IIIII.IIIIMIMI J L V a l u e s e x t r a p o l a t e d above t h i s p o i n t . ^, 11.0 111 1 1 1 1 1 1 1 1 i Equated I , Score, iiquiv. 3.5 '3.0 J , . ' s a 6 TEST. 2. READING: ' WORD MEANING {Confd) - ^ . A A V . ^ . - . ^ 6 7 ' 8 9 10 :: :: :: 2 3 4 5 :: :: 7 :: 8 9 :: :! 10 2 3 4 5 7 8 9 10 2 3 4 5 :: 24Interpretation means— 25 Kindred means — 6 petition 7 explanation 8 humility 9 pressure 10 failuresjj 1 delicate 2 gracious 3 humble 4 curious 5 related W 25 ' 6 26To prosper.is.to — 27 Nimble means— 6 endure 7 grieve 8 entertain 1 practical 2 active 3 costly 4 modest 5 dull. ••-H 27 6 selecting 7 removing 8 observing 9 connecting 10 protecting j j 28 Conservation means — 28 29Dubious means- 1 doubtful 3 0 9 forgive 10flourish.. . : . « j j A pavilion is an open — 2 apparent 6 boat 3 desolate 7 forest 4 inferior 8 building ,9store 5 unusual - - H 29 10 valley ? soij 6 kettle jar. 7 hush 8 9 link 10 jj \\ jj M : A g \\ j j H . H 31 To be punctual is to be - 1 bored 2 prompt 3 ashamed 4 worthy 5 determined si jj 32 lullisa— g J.j | jj y_ 32 lOlining 6 7 8 9 10 2 3 4 5 2 * 3 4 5 8 doubtful 9 apparent5 generosity 10 suitable 34 j j 33Liberality means- 1 gravity 2 havoc 3 impunity 4 hospitality . .33 1 3 sincere 34 Obvious means— .6 remote". 7 reasonable 35 Competent means — 1 careless '4 capable 8 shameful 2 useless 5 cunning... .35 9 zealous 10 subtle . . 3 6 • 36 Enthusiastic means — 7 singular 3 critical 6 lusty 37 Conclusive means — . 1 4 decisive 5 compulsory.. .37 ii M- I L passive 2 variable 6 3 8 To amass is to — 6 allay 3 9 Reputable means— 4 0 To indorse is to — 4 1 To quail is t o — 4 2 To reprove is t o — 8 verify 7 accumulate 1 cordial 2 solemn I 3 honorable 7 7 magnify 1 4 fortunate 8 disobey 9 affirm- 2 3 mourn 4 tremble 7 rebuke 9 replace To obstruct is t o — 1 advance 2 check 3 occupy 4 owe 1 10 export Congenial means — T o contend is t o — 1 stroke 2 fasten . 6 empty 7 cruel 8 exact 9 fierce An impediment is a n — 1 agreement 2 obstacle 3 idiot 10 useful score 0 1 2 3 4 3 4 5 6 5 8 9 10 2 3 4 5 7 8 9 10 2 3 4 5 7 9 8 10 1 2 3 ;49ji 6 . 4 8 7 8 9 10 II 5 jj jj 7 9 6 unwholesome 7 impetuous 8 ruthless 9 magnetic 10 monotonous 50 j j End of Test 2. Look over your work. Morbid means— Equated 2 4 outline 5 utterance. .47 ii [j Equity means— . 1- fashion 2 advantage 3 exchange 4 knowledge 5 justice N U M B E R RIGHT M 6 agreement 7 clashes 8 desolation 9 inspiration 10 reality' 48 jj A mediator brings— _ 5 0 9 7 1 > 4 9 5 aeji 6 4 8 4 5 struggle.... .45 ii , 4 7 10 10 refined.. .44 ii 6 .46 Void means — 3 8 1 6 3 pardon -4 exchange 2 7 1 45 9 43 ii 5 pity 6 original 7 universal 8 successful 9 agreeable 8 42 ii - .1 4 4 5 ji 5-trap - 43 7 6 8 regulate 4 40 ii . 6 preside 10 ii i i , 2 attack 3 5 prosperous 39 ii 10 disclose - 1 quarrel 9 jj 6 6 adjoin 8 10 inscribe..... .38 H 9 gamble 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 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 18 jj Stanf. A d v . R e a d . : F o r m TEST 2. E 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. 1 . A rose-is a — 1 box , 2 flower 3 home 4 month . 5 river . . . .AH- J 2 A • 6 A roof is found on a — B 6 book 7 person 8 rock 9 house 10 word B ii 1 Bread is something t o — c 1 catch : 2 drink 3 throw 4wear . 5 eat. 4 5 9 10 N n ii. ii 2 3 4 5 Injury means — 1 haste 2 charm 2 To arise is to — 6 answer 3 pride 7 stand 4 praise 2 3 4. 5 8 sit 9 rest 7 8 9 10 2 3 4 .6 8 9 10 5 harm... 10 carry .2 ,1 Unoccupied means — 3 1 unjust A.peg is usually made o f — 4 2 useless 6 wood . 7 paper To omit is to—-. 1 bore ^ 2 neglect 5 7 Envious means — 1 shallow 8 A scoundrel is a — 6 circus 9 To reject is to — 5 recover.. 6 7 1 2 3 4' 5 7 8 9 10 1 2 3 4 5 . 6 7 8 9 10 2 3 4 5 7 8 2 3 4 5 7 8 9 10 4 5 5 jealous 9 chronicle 10 loom 7 ii. 8 1 1 engage 2 refuse 3 hasten 4 forbid 5 mourn 9 ii. 6 To forewarn is to — 1 0 6 caution 7"recoil 8 moisten 9 contemplate 10 lengthen 10-j] 1 A literary person is a — 12 To prohibit means t o — 1 3 Stern means— 1 painter 6 forbid 2 monarch 7 permit 3 writer 8 assist / 10 expose ....... . 6 4 enormous 8 villain 5 ignorant... 3' 10 sand 9 overcome 3 refined 7 shipment 9 ice 4 control 8 betray 2 social 4 haunted 8.rock 3 concern To defeat is to — . 6 abuse ^ 7 assign 6 3 vacant 7 C;i; 1 1 1 1 3 : 4 rival 9 boast 5 coward... .11 10 deserve .... 12 3 . 4 1 splendid 2 severe 3 joyful , 4 wicked 7 safety 8 appearance 5 eager... 6 , Conduct means ^14 15 6 effort Exterior means— . 1 outer" 1 6 To violate is to — 1 7 A chart is a — 18 An alien is a — 6 abuse 1 card 2 vague „ 3 ignoble 7 appeal 2 flag 3 map 9 actions 4 indoors 8,reward - 9 summon 4 bowl 1 9 2 0 2 1 2 2 2 3 Uneasy means =— 7 candidate 5 fickle 10 tempt 5 debt....' 8 foreigner 9 fortress - - Seriousness means — A prologue is a kind of — 2 comfortable 3 ashamed 8 9 10 2 3 4 5 7 8 9 10 7 8 9 10 2 3 4 5 7 8 9 10 •••H l i . 16 . 17 10 novelty . 1 anxious 7 10 features .... 14 6 6 captive is H 1 4 unhappy 5 foolish.. 19 jj 6 fidelity 7 suffrage 8 refinement 9 solemnity -10 displeasure 20 1\ \ 1 knoll 2 meteor 3 introduction 4 pathway ' 5 platter.. 21 A haven is a — 6 breeze 7 package A witty person is — 1 silly 2 timid 5 13 6 8 reward 9 verse 10 refuge .22 1 3 clever 4 meek 5 sly 23 Go right on to the next page. 2 4 3 5 7 8 9 10 2 3 4 5 2 TEST 1. READING: PARAGRAPH MEANING Stanf - Adv Read - :FormE 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. T h e n 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. SAMPLE. Answer D i c k and Tom were playing ball in the field. Dick was throwing AB -the — A — and — B — was trying to catch it. B Most hawks live on insects, small rodents such as rats, mice, and 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 1 - 2 f 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_. 3 - 4 ^ Benjamin Franklin was one.of,the most versatile of our great men. He was a statesman, philosopher, writer, publisher, and scientist. 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.. 6 7-8-9 i 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-n 10-11-12 The principal diamondfieldsof 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 thefinest—12— produced in Africa, Brazil, or Australia. • 12Go right on to the next page. ' ' : ^stanf.Adv.Read.: F o r m E 1. TEST READING: PARAGRAPH MEANING 13-14^15 Demosthenes was a Greek orator who lived about 200 B.C. {Cont'd) ' He 306 was determined to be an orator although his lungs were weak and his 13 pronunciation faulty. other — 1 3 — . , H e persevered until at length he surpassed all 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. T h e y live in small caravans and earn a livelihood as 1 6 _ _ _ _ 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 — 18__ of living. 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. 2 2 21 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. M a n y of the most popular airs are sung only when used as an accompaniment to — 2 2 — . 23-24 ' - 2 2 — T h e 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: ested in M r . Green's scheme"; " M r . Smith — 2 3 - — that he was inter- 2 3 — or, in another case, " T h e man — 2 4 — . from her remarks that she was not going to be there." 25-26-27 Desert plants solve in many ways the problem of scarcity of water. T h e long roots of certain plants penetrate downward to the permanent water layer. 24_- 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 t h a i 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 r e d u c i n g — 2 7 — . 27 . - G o right on to the next page. 3 «,« TEST 1. READING: PARAGRAPH MEANING (Cont'd) 4 - stanf AdvRead :FormE 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 28-29 are—28— and —29—. 29. 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 3 1 . others they wish for — 3 1 — , and on the other hand, after a long period of seclusion they develop — 3 2 — interests. . 3 2 . 30-31-32 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. 3 3 - 3 4 35-36 Bacteria have greater resistance to injurious influences than any other known organisms. However, most bacteria are killed like any other — 3 5 — by a brief exposure to — 3 6 — of 60°-65° centigrade. 35. 36. 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 3 7 . good driving habits is better than a traffic — 3 8 — where poor drivers are fined or otherwise punished. 38. 37-38 "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. 39-40 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. 41-42-43 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. 44-45 N U M B E R 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 2 8 2 9 30 31 32 33 34 35 36 37 3 8 39 40 41 42 4 3 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 i n 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 i s based on the results obtained from the t r i a l administration of the Farquhar Test to forty Grade VII pupils i n White Rock Elementary School. (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 ra u CD u CD Top H •H co OJ 9 i 10 CD X 11 12 r-i o O 27$ of 40-pupils = 11 pupils 13 G 14 15| 16 18' 17 19 20 21 22 23 , 24 26 27 28! 29 ] . 30 Q U E S T I O N S ft trj crj CO 3 !H tH rH O O O d CO _ 3 . ' co T 3 •H CO , ID d 10 > rH | M > rH •H -H Cd -H -H id ft 3 T3 ft T3 a" a fl 3 CO x: x 289 x 16 256- X 16 256 X 16 256 X 15 225 x: X X X x. xc X x X 169 13 169 12 144 x 12 144 x 12 144 X X 11 17 X x: 10 361 xc 1 x 19 X X x X x x x x x x: X X o CO. f-l Middle 46$ of 40 pupils = 18 pupils rH C5 •H > •H (Part 2) % u o C CQ rH -H t3 C • , (D h ft 3 O, H O H (j O CO 01 ^3 01 OJ -H U > Cj -H rH -H 3 * Pi ^ O vO u Bottom 27$ of 40 pupils = 11 pupils •*0 . rH <H as 3 03 O 3 H 03 CO CO -H ( P > t-1 <U (H > -H -H PH OS -H a H d S o a a j h m ra H 3 a r t CO 1 ffi ' 5) o 03 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 6. 6 8 10 10 10 9 7 11 10 5 8 11 8 a No. wrong or omitted responses i n bottom 27$ section 4 b No. wrong or omitted responses i n top 27$ section 1 5 c' Total no. wrong or omitted'responses (bottom and top sections) 5 11 d 23$ 50$ Per cent d i f f i c u l t y (uncorrected 0 1 11 6 6 2 5 6 9 21 16 16 11 12 19 14 7 14 20 12 27$ 41$ 95$ 73$ 73$ 50$ 55$ 86$ 64$ 32$ 64$ 91$ 55$ 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 85 85 42 85 27$ 55$ 27$ 18$ for chance) 4 5 4 4 4 4 4"' e No. option questions f g Correction factor Per cent d i f f i c u l t y (corrected f o r chance) (c) x ( f ) h V a l i d i t y or Discrimination (a) - (b) 3 1 6 7 -1 4 i Per cent discrimination 27$ 9$ 55$ 64$ -9$ 36$ 97 67 115 4 7 ' 2 36$ 64$ 18$ 114 18$ H H 73 36$ 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 2) 1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a No. wrong or omitted responses i n bottom 27$ secti on 5 9 9 7 5 11 10 11 11 10 11 6 11 10 10 b No. wrong or omitted responses i n top 27$ section 3 9 9 9 2 7 9 4 7 6 10 1 6 7 10 Total no. wrong or omitted responses (bottom and top sections) 8 18 18 16 7 18 19 15 18 16 21 7 17 17 20 d Per cent d i f f i c u l t y (uncorrected for chance) 36$ 82$ 82$ 73$ 82$ 86$ 68$ 82$ 73$ 95$ 32$ 77$ 77$ 91$ e No. option questions 4 5 4 5 5 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 d i f f i c u l t y (corrected for chance) 48 4 4 32$ 4 102 109 97 0 -2 5 42 .102 4 5 115 85 4 4 109 97 127 40 97 5 4 97 121 3 0 27$ 0 (c)x.(f) h V a l i d i t y or Discrimination (a) - (b) i Per cent discrimination 2 18$ 0 0 0 -18$ -27$ 3 36$ 4 1 9$ 7 64$ 4 36$ 4 36$ 1 5 5 9$ 45$ 45$ H ro CALCULATION OF MEDIAN, MEAN, AND STANDARD DEVIATION, OF RESULTS OBTAINED FROM THE TRIAL ADMINISTRATION OF FARQUHAR S TEST TO FORTY GRADE VII PUPILS IN WHITE ROCK ELEMENTARY SCHOOL 1 Median; Median (50th Centile Point) = 9 . 5 + | ( l ) = 9.5+.2 = 9.7 Mean; Mean = —7- = 10.175 Standard Deviation; - v~ [4599 - 407 * 40 40) ^114.975 - 103.530625 \ f l l . 444375 3.382 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: r = " N N-1 x kS s + S. - T(T+k) 1 kS - T* • s k represents number of pupils N represents number of items S represents sum of squares of pupils' scores g S^ represents sum of squares of item scores T represents t o t a l of scores for pupils or items Substituting the figures derived from the calculation sheet, r = 30 x (40 x 4599) - (407) 2 9 30 29 40(4599) + 7689 - 401(407 + 40) x 2 183,960 + 7689 - 181,929 183,960 - 165,649 291,600 531,019 .549 H APPENDIX F RAW SCORE DATA The scores obtained by the 147 p a r t i c i p a t i n g subjects i n the tests used to measure the f i v e 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. 4 Stanford Achievement Test, Advanced Reading, Form E GENERAL MONTBOMERY SCHOOL (EXPERIMENTAL GROUP) Y X l X 2 316 X 3 X 4 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 19 318 176 357 2498 434 SUMS 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 SUMS 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 38 537 284 679 3892 767 HJORTH ROAD SCHOOL (CONTROL GROUP) Y X 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 10 333 191 432 2656 480 SUMS l X 2 X 3 X 4 SIMON CUNNINGHAM SCHOOL (CONTROL GROUP) Y l X X 2 319 h X 4 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 6 3 5 4 86 12 24 15 21 134 27 7 18 11 111 18 8 17 12 19 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 31 343 224 450) 2914 538 SUMS FLEETWOOD SCHOOL (CONTROL GROUP) Y 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 37 SUMS 1 19 7 5 12 13 6 7 7 5 9 8 10 8 11 11 5 11 16 16 6 10 15 9 17 12 9 7 12 9 9 8 7 10 7 17" 5 13 10 576 359 . 9 10 17 15 10 7 10 7 16 7 18 11 17 17 9 18 24 13 12 19 19 19 24 18 15 11 22 18 15 12 17 22 14 26 16 23 320 x„ 2 3 4 19 17 20 13 6 16 12 4 18 5 15 9 20 17 6. 22 21 17 9 10 22 14 20 25 18 5 16 22 18 7 17 17 21 110 111 121 116 81 106 95 92 129 105 106 94 121 108 122 105 120 99 88 88 121 101 123 123 144 96 124 127 129 115 91 116 23 ' 15 21 21 133 110 125 125 14 16 26 24 13 19 15 9 27 14 19 19 32 23 22 17 18 21 13 14 33 18 34 20 14 16 24 26 37 13 14 19 30 31 15 35 22 578 4143 776 123 321 APPENDIX G SUPPLEMENTARY STATISTICAL CALCULATIONS PAGE Calculation of the correlation of the means between the treatment groups of the c r i t e r i o n variable with each of the independent variables 322 Calculation of the within groups correlation between the c r i t e r i o n variable and each of the independent variables .< 323 Calculation of the within groups correlation, corrected for attenuation, between the c r i t e r i o n variable and each of the independent variables . . . 324 Calculation of the Pearson product-moment coefficient of correlation, for the experimental group, between the c r i t e r i o n 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 c o r r e l a t i o n of the mean of mean of y. r x y 2 /..••..-. x (.between) = with the 9.03852 /33.12368 x 2.46636 • 9.05852 { 81.69149194048 9.03852 9.03852 1.0 (ii) Between groups c o r r e l a t i o n of the mean of x^ with the mean o f y. r x y (between) = -19.31921 ^151.32912 x 2.46636 -19.31921 N/373. 2320884032 - 1 9 . 3 1 9 2 1 1 9 . 3 1 9 2 1 -1.0 (iii) Between groups c o r r e l a t i o n of the mean of x^ with the mean of y. r x y 4 (between) = -0.28403 {.03271 x 2.46646 -0.28403 f. 0 8 0 6 7 4 6 3 5 6 - 0 . 2 8 4 0 3 0.28403 -1.0 CALCULATION OF THE WITHIN GROUPS CORRELATION BETWEEN ' THE CRITERION VARIABLE AND EACH OF THE " INDEPENDENT VARIABLES (i) Within groups correlation between r / N = x y vwithin; 2 and y. 1924.96148 ^3700.20031 x 3427.81510 1924.96148 /12683602.4956426810 1924.96148 3561.4045 .54 (ii) Within groups correlation between x^ and y. r , / ... . , = x y {within) 5844.65254 • -• • — /3700.20031 x 28911.13347 5844.65254 /106976986.0281453757 .57 (iii) Within groups correlation between x^ and y. r / . .\ = x.y twithin) 2607.95070 . ^3700.20031 x 7784.38906 2607.95070 {28803798.. 8129 7 26 086 2607.95070 5366.9171 .49 323 CALCULATION OF THE WITHIN GROUPS CORRELATION CORRECTED FOR ATTENUATION, BETWEEN THE CRITERION VARIABLE AND EACH OF THE INDEPENDENT VARIABLES (i) Within groups correlation, corrected for attenuation, between and y. r = .54 J.821 x .541 = .54 J.444161 .54 .666 .81 (ii) Within groups correlation, corrected for attenuation, between x_ and y. 3 r = .57 J.948 x .541 .57 /.512868: .57 .716 .80 (iii) Within groups correlation, corrected for attenuation, between x. and y. 4 r = .49 /.874 x .541 .49 /.472834 .49 .687 .71 324 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 {<%) C *) 8 ? = XY ^ - (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 i n the experimental group i s .51137; the r between X^ and Y i n the control group i s .67816. Is the relationship between X^ and Y s i g n i f i c a n t l y higher i n the control group than i n the experimental group? Pearson r Fisher* s z .51137 .67816 .56 .83 Standard Error of the difference between 2 coefficients ~ 2 z •A ~ V -N - 3 { 59-3 N - 3 88-3 /. 01786 +' .01176 /.02962 .172 C r i t i c a l Ratio = _1 Z 2 l " 2 Z .83 - .56 .172 1.57. This CR of 1.57 i s below the .05 l e v e l of 1.96;(Table of t for use i n determining the r e l i a b i l i t y of s t a t i s t i c s ) . It may be concluded, therefore, that the relationship between and Y i s not s i g n i f i c a n t l y higher i n the control group than i n 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. |
DOI | 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 |
Aggregated Source Repository | DSpace |
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