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A study of the relationship between the ability to compute with decimal fractions and the understanding… Farquhar, Hugh Ernest 1955

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A STUDY OF THE RELATIONSHIP BETWEEN THE ABILITY TO COMPUTE WITH DECIMAL FRACTIONS AND THE UNDERSTANDING OP THE BASIC PROCESSES /  INVOLVED IN THE USE OF DECIMAL FRACTIONS  by HUGH ERNEST FARQUHAR  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n the Department of EDUCATION  We accept t h i s t h e s i s as conforming t o the standard required from candidates f o r the degree a* MASTER OF ARTS.  Members of the Department of Education THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1955  ABSTRACT  A STUDY OF THE RELATIONSHIP BETWEEN THE ABILITY TO COMPUTE WITH DECIMAL FRACTIONS AND'THE UNDERSTANDING OF THE BASIC PROCESSES INVOLVED IN THE USE OF DECIMAL FRACTIONS  Modern theory of arithmetic i n s t r u c t i o n supports the idea that the development of understandings  of basio mathematical p r i n c i p l e s pro-  duces a desirable type of l e a r n i n g .  T h i s i s a r e a o t i o n against the  t r a d i t i o n a l method of i n s t r u c t i o n -which places emphasis upon meohanlcal d r i l l procedures*  devoid of meanings.  T h i s study i s an attempt t o deter-  mine what r e l a t i o n s h i p , i f any, exists between computational a b i l i t y and understanding of fundamental processes*  The i n v e s t i g a t i o n has been  l i m i t e d t o the area of decimal f r a c t i o n s . Two t e s t s -were developed f o r the purpose of the i n v e s t i g a t i o n * The t e s t i n computation was constructed and v a l i d a t e d using pupils of the junior high school l e v e l as testees*  Student-teachers constituted the  personnel f o r the construction and v a l i d a t i o n of the t e s t i n understandings* The investigation, of r e l a t i o n s h i p was Sohool students as testees.  performed using 236 Normal  The t e s t s , whioh had been constructed f o r use  i n the study, -were administered a t the beginning of the school term. The data obtained from the i n v e s t i g a t i o n were analyzed and the following conclusions were formulatedj 1*  There i s a p o s i t i v e c o r r e l a t i o n of considerable magnitude between  the scores on the teat i n computation and the scores on the t e s t i n understandings.  ( r  -  .640 ) .  This i s an i n d i c a t i o n that there i s a  tendency f o r the scores to vary i n the same d i r e c t i o n .  2 2.  "When the f a c t o r of i n t e l l i g e n c e i s held constant, there i s a  net c o r r e l a t i o n of marked magnitude -which i s somewhat less than, t h e apparent c o e f f i c i e n t . This indicates that the common f a c t o r of i n t e l l i g e n c e has an influence upon the r e l a t i o n s h i p between the two v a r i a b l e s * 3.  The magnitude of the r e l a t i o n s h i p between scores i n understand-  ing and i n t e l l i g e n c e t e s t scores i s an i n d i c a t i o n of common elements i n both these t e s t s * 4*  The r e l a t i o n s h i p between, the scores i n computation and the  i n t e l l i g e n c e t e s t scores i s not high*  A high i n t e l l i g e n c e does not appear  t o be a prerequisite f o r high achievement i n computation* 5*  There i s evidence that a b i l i t y i n computation i s not e s s e n t i a l  f o r high achievement i n understandings and v i c e versa* nor do high scores i n one of these faotors guarantee high scores i n the other* 6*  Although a study of the scatter diagram suggests that  sucoess  i n computation i s more probable i f i t i s accompanied by a high degree o f understanding, i t cannot be i n f e r r e d from the data that one variable i s the cause or the e f f e c t of the other.  ACKNOWLEDGMENT S  The assistance and cooperation of many students and teaohers a t various eduoationallevels were necessary i n t h i s investigation*  The w r i t e r i s most appreciative of the help  he has reoeived from these sources*  I t i s impossible t o  name a l l the p r i n c i p a l s and teaohers "who have contributed i n some way t o the development of t h i s study*  Without the  testees, obtained through the permission of the Chief Inspect o r of Sohools of the Greater V i c t o r i a School D i s t r i c t and the P r i n c i p a l s of the P r o v i n c i a l Normal Sohools, t h i s t h e s i s could not have been written* Dr. C. B* Conway of the B r i t i s h Columbia Department of Education gave muoh needed advice i n the e a r l y stages o f the preparation of the t e s t s *  H i s help i s acknowledged with  thanks. The guidance and constant encouragement* generously given by Dr. J. E. Mcintosh of the University of B r i t i s h Columbia, provided the stimulus that was neoessary t o b r i n g the work t o a conclusion.  The w r i t e r wishes t o acknowledge  h i s debt t o Dr. Mcintosh, w i t h sincere appreciation.  TABLE OF CONTENTS  Chapter I  Pag« INTRODUCTION Introductory Statement . . . . . . . . . . . . . .  1  Statement of the Problem • • • . . . • • • . . . *  2  Plan of the Study  . . . . . . . . . . . . . . . .  3  Materials of the Study . . . . . . . . . . . . . .  3  Background of the Problem* . . . . . . . . . . . .  4  The Measurement  7  of Understandings •  C r i t e r i a f o r Measurement Related Studies Summary II  of Understandings • • . •  . . . . . . .  . . . . .  . . . . . . . . . . . . . . . . . . . . .  7 8 11  CONSTRUCTION AND ANALYSIS OF A TEST IN COMPUTATION WITH DECIMAL FRACTIONS Introduction  12  Characteristics of a Good Test  12  Currioular V a l i d i t y Experimental Foxm  •• • • • • • • • •  13 14  Preliminary Form . . . . . . . * . . . . • • • • •  16  Method of Item Analysis Used i n This Study • • • •  15  Reliability  18  F i n a l Form . . . . . . . . .  24  Item V a l i d i t y Indices Based on Flanagan's Tables •  24  R e l i a b i l i t y of F i n a l Form  30  Summary  . . . . o . . . . . . . . . . . . . . . .  30  TABLE OF  III  CONTENTS—Continued  CONSTRUCTION AND ANALYSIS OP A TEST IN UNDERSTANDINGS OF PROCESSES INVOLVING DECIMAL FRACTIONS Introduction . . . . . . . * . . . . . . * . . » •  32  Factors Involved i n the Construction of Test Items  33  Experimental Form  . . . . . . . . . . . . . . . .  34  Preliminary Form . . . . . . . . . . . . . . . . .  35  Reliability  .  40  •  41  Item V a l i d i t y Indioes Based on Flanagan's Tables •  47  R e l i a b i l i t y of F i n a l Form  49  F i n a l Form • • • •  • • • • •  . . . . . . .  Relationship Between Scores on Test i n Understandings and I n t e l l i g e n c e Test Soores . . . . . Summary  17  . . . . . . . . . .  49  . . . . . .  50  The Subjects . . . . . . . . . . . . . . . . . . .  51  Administration of the Tests  52  Analysis of the Results  52  P a r t i a l Correlation • •  55  INVESTIGATION OF THE RELATIONSHIP BETWEEN COMPUTATION AND UNDERSTANDINGS IN THE USE OF DECIMAL FRACTIONS  Summary  •  . . . . . . . . . . . . .  56  TABLE OF  CONTESTS—Continued  Chapter V  Page SUMMARY AND CONCLUSIONS Summary  • • . . • • « . . . . . . . « * . . . » .  Conclusions Suggestions f o r Further Study  57 59  • • • • • • • • » •  61  BIBLIOGRAPHY  65  APPENDIX At Test i n Computation, with Decimal Fraotions  66  APPENDIX Bt Test i n Understanding of Processes with Deolmal Fractions . . . . . . . . .  67  LIST OF TABLES  Table I  II  HI  IV  V  71  VII  VIII  IX  X:  Page The V a l i d i t i e s and D i f f i c u l t i e s i n Terms of Per Cent of the Items of the.Preliminary Form of the Test i n Computation with Decimal Fractions . . . . .  19  C o e f f i c i e n t of R e l i a b i l i t y of Preliminary Form of the Test i n Computation with Decimal Fractions Determined by the Odd-Even Split-Halves Teohnique . . . . •  22  C o e f f i c i e n t of R e l i a b i l i t y of the Preliminary Form of the Test i n Computation w i t h Decimal Fractions Determined by the Kuder-Riohardson Formula . . . . . .  23  The V a l i d i t i e s and D i f f i c u l t i e s i n Terms of Per Cent of the Items of the F i n a l Form of the Test i n Computation with Decimal Fractions . . . . . . . . . .  25  Internal Consistency of the F i n a l Form of the Test i n Computation with Deoimal Fractions. Based on Flanagan's Estimates of C o r r e l a t i o n between Individual Items and the Test as a Ifhole . . . . . . .  29  C o e f f i c i e n t s of R e l i a b i l i t y of the F i n a l Form of the Test i n Computation with Decimal Fractions • • • • • •  30  The V a l i d i t i e s and D i f f i c u l t i e s i n Terms of Per Cent of the Items of the Preliminary Form of the Test i n Understandings of Processes •  37  C o e f f i c i e n t of R e l i a b i l i t y of the Preliminary Form of the Test i n Understandings of Processes Determined by Odd-Even Split-Halves Teohnique • • • •  40  C o e f f i c i e n t of R e l i a b i l i t y of the Preliminary Form of the Test i n Understandings of Processes Determined by the Kuder-Riehardson Formula . . . . . .  41  The V a l i d i t i e s and D i f f i c u l t i e s i n Terms of Per Cent of the Items of the F i n a l Form of the Test i n Understandings of Prooesses • « • 0  XL  Internal Consistency of the F i n a l Form of the Test i n Understandings of Prooesses, Based on Flanagan's Estimates of C o r r e l a t i o n between Individual Items.and the Test as a Whole . . . . . . . . . . . . . . . . .  44  48  LIST OF TABIJES—Continued  Table XII XIII  XIV  XV  Page C o e f f i c i e n t s of R e l i a b i l i t y of the F i n a l Form of the Test i n Understandings of Processes • • • • • •  49  Relationship Between Scores on Test i n Understandings of Processes and Otis Test of Mental A b i l i t y Obtained by 150 Normal Sohool Students . . . . . . . .  50  Relationship between Scores Obtained on Tests i n Computation and Understandings of Processes involved i n the Use of Deoimal Fractions Obtained by 236 Normal School S t u d e n t s » » * • • • • • . .  53  Coeffieients of Correlations Between Test Scores • . •  55  LIST OF FIGURES  Figure I  II  III  IV  Page Graphioal Analysis of Items of Test i n Computation with Decimal Fractions i n Terms of Per Cent of V a l i d i t y and Per Cent of D i f f i c u l t y Preliminary Form . . . . . . . . . . . . . . . . . . .  21  Graphioal Analysis of Items of Test i n Computation with Decimal Fractions i n Terms of Per Cent of V a l i d i t y and Per Cent of D i f f i c u l t y F i n a l Form  27  Graphical Analysis of Items of Test i n Understanding of Processes i n Terms of Per Cent of V a l i d i t y and Per Cent of D i f f i c u l t y Preliminary Form . . . . . . . . . . . . . . . . . . .  39  Graphioal Analysis of Items of Test i n Understanding of Processes i n Terms of Per Cent of V a l i d i t y and Per Cent of D i f f i c u l t y F i n a l Form *  46  CHAPTER  I  INTRODUCTION  Introductory Statement  During the past two or three decades, the theory of arithmetic i n s t r u c t i o n has been subjected t o a close scrutiny because of f a i r l y general d i s s a t i s f a c t i o n with the achievement of the graduates of oar schools•  As a r e s u l t , there has emerged a method of i n s t r u c t i o n known as  the meaning theory, which stresses the d e s i r a b i l i t y of developing understandings of processes i n contrast t o the teaching of the mechanical manipulation of numbers, devoid of meanings*  The advocates of t h i s theory  include such authorities as Brownell, Morton* Wheat, Brueckner, Grossniekle, Spit zer, Buckingham, and many others*  However, i n spite of the weighty  opinions of these experts, much teaching continues t o be of the more t r a d i t i o n a l type—based upon meaningless  d r i l l , r e p e t i t i o n and rote memory*  I f teachers are t o pay more than l i p - s e r v i c e t o the meaning theory i n classroom p r a c t i c e , doubtless i t w i l l be neoessary t o demonstrate conc l u s i v e l y , time and again, that learning proceeds best, and i s more permanent, when a high degree of understanding i s present*  Hot u n t i l they  have been convinced of the e f f i c a c y of the meaningful approach, by the evidence of sound s t a t i s t i c a l Studies, are teachers l i k e l y t o be  concerned  about objectives i n the arithmetic programme other than those that are purely mechanical* F a i l u r e t o produoe a strong case i n support of the meaning theory may w e l l r e s u l t i n a continuation of the status quo as set f o r t h by  2  Wingo  i n the following*  With few* i f any, exceptions, the investigators have found grounds f o r d i s s a t i s f a c t i o n w i t h the present status of arithmetic i n s t r u c t i o n * Administrators* supervisors, and teaohers cannot dismiss the c r i t i c i s m l i g h t l y * I t i s founded on sober, and sometimes alarming f a c t * I t i s directed a t the most important aspect of any teaching problem: the problem of method*  Statement of the Problem How adept at manipulating numbers may students become and yet possess l i t t l e or no understanding of the underlying p r i n c i p l e s involved i n the computations?  W i l l they be more suooessful i n the operation of 2  numbers i f meanings of basic oonoepts are c l e a r t o them?  Weaver  suggests t h a t : (1) It i s quite possible that a person may possess considerable s k i l l i n arithmetic computation but have l i t t l e or no understanding of why he does things i n a manner which has become habituated* (2) I t i s equally possible that a person may have a thorough understanding of the mathematical bases f o r the algorisms which he us 08 but operate a t a r e l a t i v e l y low l e v e l of computational e f f i c i e n c y * Neither a b i l i t y i s prerequisite f o r the other* The attainment of e i t h e r a b i l i t y does not guarantee attainment of the other* The present study w i l l attempt t o examine what r e l a t i o n s h i p , i f any, exists between the a b i l i t y t o manipulate numbers on a meohanioal  level  and the understanding of the processes which underly the number operations i n one phase of arithmetic, namely, decimal f r a c t i o n s * A major aspect of t h i s study w i l l be the construction of  1 G. Max Wingo, "The Organization and Administration of the Arithmetic Program i n the Elementary School", Arithmetic 1948. p* 69. Supplementary Eduoational Monographs, No. 66. Chicago: U n i v e r s i t y of Chicago Press, 1948. • Fred Weaver, "Some Areas of Misunderstanding About Meaning i n Arithmetic", The Elementary Sohool Journal, L I (September, 1950), p.36.  3 appropriate t e s t s f o r the i n v e s t i g a t i o n .  This w i l l e n t a i l t e s t i n g , obser-  v a t i o n and analysis extending over a period of more than a year.  This  task i n i t s e l f , while subordinate t o the main investigation, w i l l c o n s t i tute a study of considerable magnitude.  I t i s hoped that the r e s u l t i n g  t e s t s w i l l provide evaluative instruments -which w i l l have further u s e f u l ness. P l a n of the Study  The background of the study and a statement of the problem are presented i n the introductory ohapter.  T h i s w i l l be followed by a  d e s c r i p t i o n of the development and analysis of a t e s t i n computation w i t h decimal f r a c t i o n s .  Next, the construction and v a l i d a t i o n of a t e s t i n  understandings of basic processes involving decimal f r a c t i o n s w i l l be described.  With the use of these t e s t s , an i n v e s t i g a t i o n w i l l be made t o  determine the degree of r e l a t i o n s h i p , i f any, that exists between mechanics and meanings.  The r e s u l t s w i l l then be analysed and the conclusions  formulated. Materials of the Study  The investigator decided t o work i n the area of deoimal f r a c t i o n s because of the e s s e n t i a l nature of t h i s subject matter and i t s extensive use both i n and out of school, and also because of f a i r l y general c r i t i c i s m of the laok of competence demonstrated arithmetic.  i n the a p p l i c a t i o n of t h i s phase of  When considering the subjects t o be used i n the study, he was  guided by the a v a i l a b i l i t y of s u f f i c i e n t numbers f o r the purpose.  I t was  deoided t o perform the i n v e s t i g a t i o n w i t h Normal Sohool students because, i n t h i s case, t e a c h e r s - i n - t r a i n i n g provided a convenient group w i t h which t o work.  A l s o , i t was thought t h a t , i f the possession of understandings  1  '  •  4 i s a desirable outcome of arithmetic i n s t r u c t i o n f o r the p u p i l s , surely, at the etude nt-teaoher l e v e l , i t must be an even more e s s e n t i a l aspect  of  3 learning*  In the opinion of Wrens  It should be a t r i t e remark t o say that, unless the teacher hims e l f can tread over h i l l and dale through the f i e l d s and f o r e s t s of arithmetic with confidence and assurance that he knows where he i s going and how he i s going to get there, he c e r t a i n l y cannot render a great deal of assistance t o h i s pupils* As Normal Sohool students were t o provide the ultimate group t o be used i n the i n v e s t i g a t i o n , i t was  deemed necessary to use s i m i l a r sub-  j e c t s f o r the purpose of v a l i d a t i n g the t e s t on understandings*  However,  i t was f e l t that the t e s t on computation could be prepared by using groups of unseleoted  students who  had oompleted the work on decimal f r a c -  t i o n s , since the basio material remains the same* at the Grade 7-8-9  any  By working with pupils  l e v e l I n a o i t y school system, the Investigator had  a v a i l a b l e a large number of subjects, thus obtaining more scope f o r the construction of the t e s t *  I t was thought that the r e s u l t i n g t e s t could be  used equally w e l l with groups at higher l e v e l s and that i t would provide a s a t i s f a c t o r y t e s t i n g instrument f o r the purpose of the proposed Investigation* Background of the Problem  A popular conception e x i s t s that the schools are not.adequately preparing the pupils t o handle the basio s k i l l s of arithmetic*  5  P. Lynwood Wren, "The Professional Preparation of Teaohers of A r i t h m e t l o V Arltnmetio 1948. , 82* Supplementary Educational Monographs, No* 66* Chicago: University of Chicago Press, 1948* p  5 Grossniekle  reports that t  A frequent o r i t l o i s m direoted towards publio and elementary schools concerns the f a i l u r e of t h e i r students t o demonstrate adequate preparation i n the fundamental subjects* T h i s p a r t i c u l a r l y holds t r u e i n the f i e l d of arithmetic* As a r e s u l t of the concern over the p l i g h t of arithmetic i n the elementary sohools, there has been extensive reaearoh conducted i n t h i s area during the past f o r t y years*  Psychological study on the prooess of  learning has direoted a t t e n t i o n t o how the o h i l d learns and has  developed  the view that one of the objectives of arithmetic i n s t r u c t i o n should be the development of understandings*  According t o Brownell:  5  From a l l t h i s research and from experimentally oriented teaching emerged the notion that one ingredient i n a f u n c t i o n a l program i n arithmetic i s p r o v i s i o n f o r meaningful learning* T h i s theory of meaning or understanding  impregnates the philosophy of  p r a c t i c a l l y a l l the a u t h o r i t i e s i n the f i e l d of arithmetio today*  Tflhile  there are minor differences i n i n t e r p r e t a t i o n of t h i s theory, most of the experts base t h e i r philosophy upon the basio ideas propounded by Brownell i n the foilowingt The "meaning" theory ooneeives of arithmetio as a o l o s e l y k n i t system of understandable ideas, p r i n c i p l e s , and prooesses* According t o t h i s theory, the t e s t of l e a r n i n g i s not mere mechanical " f i g u r i n g " . The true t e s t i s an i n t e l l i g e n t grasp upon number.relations and the a b i l i t y t o deal with the a r i t h m e t i c a l situations with proper comprehension of t h e i r mathematical as w e l l as t h e i r p r a o t i o a l s i g n i f i c a n c e *  F o s t e r E . G-rossniokle, "Dilemmas Confronting the Teachers of Arithmetic", The Arithmetio Teacher, I (February, 1954), p* 12 % i l l i a m A* Brownell, "The Revolution i n Arithmetio", The Arithmetio Teaoher, I (February, 1954), pp* 3-4* % i l l i a m A* Brownell, "Psychological Considerations i n the Learning and the Teaching of Arithmetio , The Teaching of Arithmetio^ p* 19, Tenth Yearbook of the National.Council of TeaoKers of Mathematics. New York: Teachers' College, Columbia U n i v e r s i t y , 1935. 11  .  6  Brownell goes on t o explain that the meaning theory has been designed t o enoourage the understanding of arithmetic and the most frequent put t o the c h i l d should be, "Why  question  d i d you do that? * 1  T h i s study i s concerned with that aspect of the meaning theory whioh r e l a t e s t o the understanding of basio s k i l l s — t o the a b i l i t y t o r a t i o n a l i s e processes-—to t h a t mathematical phase of arithmetic which provides the "why"  f o r the  algorism*  There i s a convincing body of opinion i n support of the idea that the development of meanings and understandings i s an e s s e n t i a l aspeot of the k i n d of I n s t r u c t i o n that w i l l produoe better r e s u l t s i n arithmetic*  7 Brueokner and Grossnickle  state:  Today there i s almost universal acceptance of the view that c h i l d r e n l e a r n arithmetic more e a s i l y i f they understand what they are learning and i f i t i s mathematioally meaningful t o them* Further support f o r the e f f i o a o y of the meaning theory i s pro8 vided by Wren  i n the f o l l o w i n g :  Appreciation of s i g n i f i c a n t meanings cannot be overemphasized e i t h e r f o r the p u p i l or f o r the teacher •••••••• Meanings not only form a basis f o r more i n t e l l i g e n t use of computational s k i l l s , but they also give a background f o r a better appreciation of arithmetic as part of our c u l t u r a l heritage* Because of opinions suoh as these. Impetus has been given t o the pursuance of a method of i n s t r u c t i o n designed to develop understandings* The emergence of t h i s new approach t o arithmetic i n s t r u c t i o n forms the background of the present study*  Leo J* Brueokner and Foster E, Grossnickle, Making Arithmetic Meaningful, p* i i i * P h i l a d e l p h i a : The John C* Winston Company, 19^3* 'wren, ££• p i t * ,  p. 85*  7 The Measurement of Understandings I f the aforementioned gospel of the proponents of the meaning theory of arithmetio i n s t r u c t i o n i s t o spread, oonvinoing evidence of the s u p e r i o r i t y of t h i s method must be produced*  To do t h i s requires, f i r s t l y ,  some method of evaluating the existence and growth of basio mathematical understandings. The immediate problem becomes one of determining just what constitutes a body of understandings i n arithmetio*  One aspect of the  study of arithmetio during the past twenty years has been an attempt t o i d e n t i f y and i s o l a t e the ideas, p r i n c i p l e s , relationships and generalisations whioh constitute the understandings of the mathematical phase of the subject*  Ag yet^ re searoh has produoed l i t t l e t o guide the investigator  i n t h i s f i e l d and, t o some extent, he i s forced t o grope i n the dark* For the purposes of t h i s study, i t i s assumed that a person's understanding of a process may be revealed by h i s a b i l i t y t o r a t i o n a l i z e the procedure and that h i s i n s i g h t i n t o number operations may become apparent by h i s grasp of the "-why" behind the performance of the algorism* This w i l l form the basis of the measurement of understandings i n t h e present investigation*  C r i t e r i a f o r Measurement of Understandings  The attempt t o measure understandings 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 f o r t h i s purpose*  The  s e t t i n g up of suitable c r i t e r i o n measures by whioh t o evaluate understandings suggests an area f o r f u r t h e r research*  In l i e u of e x i s t i n g c r i t e r i a ,  the investigator must determine a r b i t r a r i l y those concepts that should be  8 included i n a measuring instrumentdesigned t o evaluate understandings of any phase of arithmetic*  (Most researchers have been f o r c e d t o adopt t h i s  procedure*) The mechanics of the t e s t on understandings w i l l be discussed i n a l a t e r ohapter.  I t i s considered that an understanding of the processes  involved i n computation with deoimal f r a c t i o n s should include i n t e l l i g e n t oontrol of the following concepts*  1* Meaning of deoimal f r a c t i o n s 2* Reading and w r i t i n g decimals 3* Value of deoimal fraotions 4* Comparison of deoimal fraotions 5* Function of zero 6. Rounding o f f numbers 7* Accuracy of measurement 8* Effect of moving the decimal point 9* Looation of the decimal point i n a d d i t i o n and subtraction 10* Looation of the deoimal point i n m u l t i p l i c a t i o n 11* Looation of the decimal point i n d i v i s i o n 12* Changing common f r a c t i o n s t o decimals 13* Changing decimals t o common fraotions 14* Relative value of d i g i t s  Related Studies While the l i t e r a t u r e i s replete with material supporting the meaning theory of i n s t r u c t i o n , more studies are needed t o show the r e s u l t of i t s a p p l i c a t i o n ;  and there seems t o have been l i t t l e attempt made t o  9 evaluate the mathematical understandings aoquired by pupils*  Little  research i s available t o lend support t o the opinion that the development of understandings produces a higher l e v e l of achievement* The following studies are i l l u s t r a t i v e of the research that has been conducted i n t h i s area: 9 1* Glennon  oonduoted a f r o n t i e r research study designed t o  disoover t h e degree of mathematical understanding possessed by representat i v e groups on d i f f e r e n t educational l e v e l s * obliged t o construct a s p e c i a l t e s t *  For t h i s purpose he was  H i s study revealed that some pro*  gross i s being made i n the f i e l d o f evaluation i n arithmetio*  However,  he reports that h i s findings "do not o f f e r a favorable picture of our present practices i n teaching meanings and understandings i n arithmetio ' • 1  He was f oroed t o the conclusion that teachers a r e not succeeding i n developing an understanding of mathematical p r i n c i p l e s * 2* K i l g o u r  1 0  used Glennon s t e s t t o oonduot a study on the 1  development of understandings i n a one-year teacher-training programme* She found that small, but s i g n i f i c a n t , gains were made by the students i n t h e i r understandings of basic concepts. 3* Orleans and Tftindt  11  oonduoted a study t o a s c e r t a i n the extent  of understandings possessed by teachers and student-teachers. They designed Vincent J . Glennon* "Testing Meanings i n Arithmetic", Arithmetic 1949, pp. 64-74* Supplementary Educational Monographs, Ho. 70. Chicago:~~ University of Chicago Press, 1949* 10 Alma Jean Kilgour, The E f f e c t of a Year's Teaoher-Training; Course on the Vancouver Normal School Students * Understanding of Arithmetio* Unpublished Master's thesis i n education* U n i v e r s i t y of B r i t i s h Columbia, 1953. 11 Jaoob S. Orleans and Edwin Wandt, "The Understanding of A r i t h metic Possessed by Teaohers", The Elementary School Journal, L I I I (May, 1953), pp. 501-507* " [  t h e i r own t e s t f o r the i n v e s t i g a t i o n s . They concludedj There are apparently few prooesses, concepts, or relationships i n arithmetic which are understood by a large per cent of teachers. 1 2  4. Brownell  reports a study conducted by Spainhour t o deter-  mine the r e l a t i o n s h i p between arithmetic understanding and a b i l i t y i n problem solving and computation. i n grade f o u r and .751 and .756 18 5* Taylor  He found c o r r e l a t i o n s of .665 and  .751  i n grade 6.  t e s t e d college freshmen on mathematical meanings and  revealed a woeful l a c k of understanding of basio concepts.  His f i n d i n g s  lead him t o the conclusion that "students entering teaohers oolleges are d e f i c i e n t i n both the mechanics and the understanding of arithmetic"• These studies reveal, i n the opinion of the w r i t e r , that  little  research has been done i n the area of evaluation of a r i t h m e t i c a l understandings ;  that, generally speaking, students possess very l i t t l e i n s i g h t  i n respect t o basic number operations and that only s l i g h t progress has been made i n developing meanings i n arithmetic; ings may  that growth i n understand-  be an outcome of arithmetic i n s t r u c t i o n ; and that there i s a  s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n between understanding of prooesses and computational a b i l i t y .  Obviously, there i s a d e f i n i t e need f o r f u r t h e r  research i n t h i s f i e l d of arithmetic* 12 William A. Brownell, "The Evolution of Learning i n Arithmetic",. Arithmetic i n General Eduoation, p. 229. Sixteenth. Yearbook of, the National Council of Teachers of Mathematics. New York: Teachers* College, Columbia University,, 1941. 13 E. H. Taylor, "Mathematics f o r a Pour-Year Course f o r Teachers In the Elementary School", Sohool Soienoe and Mathematics, XXXVTII (May, 1938), pp. 499-603. ;  The "new"  theory of arithmetio i n s t r u c t i o n i s based upon, the  psychologists* knowledge of how the c h i l d learns*  While much has been  w r i t t e n about the meaning approach t o the teaching of arithmetio, more evidence i s desirable t o give support t o the opinion that t h i s theory of i n s t r u c t i o n i s l i k e l y t o produce more worthwhile r e s u l t s * the  The purpose of  present study i s t o investigate the r e l a t i o n s h i p , i f any, between a  student's achievement i n a t e s t i n computation with decimal f r a c t i o n s and a t e s t i n understandings of the prooesses used i n the computations*  It i s  an attempt t o supply data which w i l l help t o answer the question: "Does i n s t r u c t i o n designed t o develop meanings r e s u l t i n higher achievement than a method of teaching which stresses mechanical operation of numbers only?" The i n v e s t i g a t i o n necessitates the construction and v a l i d a t i o n of suitable t e s t s f o r the purpose*  The construction of a t e s t on under*  standings i s made d i f f i c u l t because of r e l a t i v e l y l i t t l e research i n t h i s area, because of the d i f f i c u l t y of defining understandings, and because of the  l a c k of adequate c r i t e r i a f o r the purpose of v a l i d a t i o n * The study w i l l be conducted with the a i d of Normal School  students as subjects*  An attempt w i l l be made to analyze the r e s u l t s and  to a r r i v e at conclusions whioh may have significance i n terms of the method of arithmetio i n s t r u c t i o n whioh i s most desirable*  12  CHAPTER I I  CONSTRUCT ION AND ANALYSIS OF A TEST IN COMPUTATION WITH DECIMAL FRACTIONS  Introduction As discussed i n the previous chapter, i t was decided that a neoessary part of the i n v e s t i g a t i o n must be the construction and v a l i d a t i o n of a t e s t on the basic s k i l l s involved i n computation with deoimal fractions*  The material of such a t e s t should consist of items based  upon the subject-matter i n the curriculum of the elementary schools of B r i t i s h Columbia*  As the basic e s s e n t i a l s of computation with deoimal  f r a c t i o n s are re-taught and used throughout the school years beyond the elementary grades, a t e s t on t h i s material should be suitable f o r the evaluation of pupils of any higher grade l e v e l *  Pupils of the Junior  High School grades should possess a high degree of proficiency In comput a t i o n with deoimal f r a c t i o n s , and i t i s l i k e l y that t h i s degree of s k i l l w i l l be maintained or raised during the f o l l o w i n g years*  Thus, i t i s  f e l t t h a t the construction of a t e s t based upon the achievement of subjects at the grade 7-8-9  l e v e l should provide a v a l i d t e s t i n g instrument whioh  oould be used f o r evaluating p r o f i c i e n c y at a more advanced l e v e l , Including high sohool graduation and teacher-training*  A d e s c r i p t i o n of the  preparation and v a l i d a t i o n of the t e s t on computation with decimal f r a c t i o n s w i l l be presented i n t h i s chapter*  Characteristics of a Good Test  In the construction of a t e s t there are a number of f a c t o r s whioh  must receive due consideration*  Greene, Jorgensen and Gerberioh* include  the following distinguishing c h a r a c t e r i s t i c s of good examinations i n t h e i r specificationst  v a l i d i t y , r e l i a b i l i t y , objectivity, administrability,  comparability, economy* u t i l i t y * I n a study i n whioh he examined the worth of a teaohernmade t e s t , 2 Carlile  considered the following c r i t e r i a t o be s i g n i f i c a n t t  validity,  r e l i a b i l i t y , discrimination, l e v e l of d i f f i c u l t y , o b j e c t i v i t y , ease of administration and ease of sooring* In the construction of the present t e s t s a t t e n t i o n w i l l be paid to the aforementioned f a c t o r s i n an endeavour t o produce s t a t i s t i c a l l y sound t e s t i n g instruments* ,  Currioular V a l i d i t y  An examination of the course of study and the current t e x t books was made and a l i s t of the major s k i l l s and concepts involving computation with decimal f r a c t i o n s was compiled*  Items* designed t o t e s t these con-  cepts, were constructed and entered on i n d i v i d u a l cards*  These items were  then put together i n d i s c r i m i n a t e l y and mimeographed copies-were submitted t o a number of teachers f o r o r i t i o i s m and suggestion*  I n t h i s manner  material whioh f i t t e d the curriculum was selected f o r the t e s t and an attempt was made t o assure o u r r i o u l a r v a l i d i t y *  1 Harry A* Greene, Albert ft* Jorgensen, J * Raymond Gerberioh, Measurement and Evaluation i n the Secondary School. Sew York: Longmans, Green and Co** 1943* 2 A. B* C a r l i l e , "An Examination of a Teaoher-aade Test", Educational Administration, and Supervision, 40 ( A p r i l , 1954), pp. 212-218*  14 Experimental Form  On the bases of suggestions received and some i n d i v i d u a l t e s t i n g -which was oarried out, ohanges were made and the items were put i n t o an experimental form*  T h i s form of the t e s t was administered t o 185 pupils  i n grades 7-8-9 i n d i f f e r e n t types of schools i n the Greater V i c t o r i a School D i s t r i c t *  The w r i t e r personally administered some of the t e s t s ,  made c a r e f u l observations t o determine f a o e n  n  v a l i d i t y and i n t e r e s t , and  reoorded the time f a c t o r * The r e s u l t s of the f i r s t experimental run were c a r e f u l l y studied and discussed with classroom teaohers*  C e r t a i n weaknesses I n form, con-  tent and wording were immediately apparent and further r e v i s i o n ensued*  A  orude analysis of the degree of d i f f i c u l t y was made and the t e s t items were rearranged i n what appeared t o be ascending order of d i f f i c u l t y * contained 32 items at t h i s stage of development*  The t e s t  A f t e r consultation w i t h  an expert i n the f i e l d of t e s t i n g , the material was put i n t o a form s u i t able f o r a t r i a l run* The time f a c t o r appeared t o be s a t i s f a c t o r y .  T h i r t y minutes  provided ample time f o r most pupils t o f i n i s h the work, whioh meant that the t e s t could be administered oemfortably i n a normal forty-minute period* As the population upon whioh the f i n a l study was t o be made cons i s t e d of teaohers-in-training, i t was deemed wise t o obtain same i n d i c a t i o n of the performance of such a group on the t e s t *  Therefore, the t e s t  was administered t o 165 students of the V i c t o r i a Normal School*  Although  these students had been studying the t o p i c of deoimal fraotions recently, t h e i r scores were d i s t r i b u t e d over a f a i r l y wide range of achievement* Analysis of t h i s performance l e d t o f u r t h e r r e v i s i o n t o d e a r up  15 ambiguities and t o improve o b j e c t i v i t y *  The opportunity of c l o s e l y  observing r e a c t i o n t o the items at various stages i n the development of the t e s t made i t possible t o r e v i s e and restate the items i n such a way to ensure a high degree of o b j e c t i v i t y *  At t h i s stage It was  as  decided that  a few more items could be added t o advantage, bringing the t o t a l up t o thirty-five*  Preliminary Form The t e s t was now organised i n t o a Preliminary Form, containing 35 items, with a time l i m i t of 30 minutes* 300 pupils i n grades  I t was administered to over  7-8*9 i n various types of schools throughout the  Greater V i c t o r i a area*  A b r i e f l i s t of instructions f o r administering the  t e s t was issued f o r the purpose of ensuring uniformity of procedure* papers were returned and soored and were made ready f o r a n a l y s i s * nature of the sooring was h i g h l y objective, i t was  The As the  possible t o e n l i s t the  assistance of student-teachers f o r the task*  Method of Item Analysis Used i n This Study An accepted method of determining the discriminating power of a t e s t item i s t o compare the performance of the best seotion of the group with that of the poorest section*  Although of obscure o r i g i n , the t e c h -  nique of comparing portions of the group i s widely used and i s described 3 i n some d e t a i l by Long and Sandiford .  They explain:  The idea underlying a l l the Upper and Lower Methods i s that the good item i s one which the good pupils do w e l l , and the poor pupils do poorly* 11  'g  J* A* Long and P. Sandiford, The V a l i d a t i o n of Test Items* B u l l e t i n Ho* 3 of the Department of Educational Research (Toronto, Ontario: The Department of Educational Researoh, U n i v e r s i t y of Toronto, 1935), p. 31*  16 While various f r a c t i o n s of the d i s t r i b u t i o n nay be used, Long and 4 Sandiford  state unequivocally that the Upper and Lower T h i r d s teohnique  gives the best r e s u l t s * Experiments show that Upper and Lower Halves, whioh use a l l the data at the disposal of the examiner, i s not so good a method as Upper and Lower Thirds or the Upper and Lower 27 per oent*  They  go on t o point out that the Upper and Lower Thirds tech-  nique tends t o discriminate i n favour of items of 50$ d i f f i c u l t y . Taking both effectiveness and ease of computation i n t o cons i d e r a t i o n , of those techniques which tend t o seleot 50%, or balanced d i f f i c u l t y , the Upper versus Lower Thirds may be adopted as the preferred technique* The procedure used i n t h i s study i s outlined below: 1* A group of 300 papers was seleoted f o r analysis and arranged i n score order* 2* The papers were divided into three groups of 100 e a c h — an Upper T h i r d — a Lower T h i r d — a Middle T h i r d * The number of papers employed f a c i l i t a t e d the use of per cents* 3* Large sheets of squared paper were prepared and the correct responses t o each item on each t e s t paper were tabulated* 4* Per cents of correct responses on each item were calculated f o r the upper T h i r d and f o r the Lower T h i r d * 5* Discriminatory value or item v a l i d i t y was found by comparing the performance of the Upper T h i r d group with that of the Lower T h i r d group. V a l i d i t y of each item was expressed as a per cent and determined by f i n d i n g the difference between the per cent correct i n the Upper T h i r d and that i n the Lower T h i r d . 6. Per cent d i f f i c u l t y was calculated on the basis of the t o t a l item-errors i n the Upper and Lower Thirds combined.  4 i b i d * , p. 32 5  i b i d . , p. 118  1? g  7* Per cent d i f f i c u l t y and per cent v a l i d i t y were plotted on graph paper. An a r b i t r a r y curve was drawn* An item f a l l i n g unduly f a r below the curve was considered unsuitable or i n need of r e v i s i o n . Items f a l l i n g above the curve were considered s a t i s f a c t o r y and were retained i n that form* An example or two w i l l serve t o i l l u s t r a t e the procedure*  An  item with a v a l i d i t y of one hundred per cent would be answered c o r r e c t l y by a l l i n the Upper T h i r d and i n c o r r e c t l y by a l l i n the Lower Third} d i f f i c u l t y would thus be f i f t y per cent* be quite u n l i k e l y t o occur*  its  A s i t u a t i o n of t h i s kind would  (Per cent of d i f f i c u l t y refers t o the per  cent of incorrect responses to an item*)  An item with f i f t y per cent  d i f f i c u l t y might be answered c o r r e c t l y by eighty per cent of the upper group and by twenty per oent of the lower group, which would give i t a v a l i d i t y of s i x t y per cent*  This item would l i e w e l l above the curve and  would be considered a s a t i s f a c t o r y item t o r e t a i n *  An item answered  c o r r e c t l y by f o r t y - f i v e per oent of the top group and by t h i r t y - f i v e per cent of the bottom group would have a s i x t y per cent d i f f i c u l t y ( f o r t y per oent ease) but i t s power t o discriminate would be only t e n per oent* It would f a l l w e l l below the curve and would contribute l i t t l e t o the t e s t * In order t o place items along the f u l l range of d i f f i c u l t y , which i s desirable, there w i l l , of course* be same with small value from the point of view of discriminatory power*  This cannot be avoided*  showing negative v a l i d i t y * however, would be discarded  Any item  immediately*  Following the procedure as outlined,, the papers were grouped and analyzed*  Per cents of v a l i d i t y and d i f f i c u l t y were computed using the  6  This graphical technique was devised and i s used by Dr* C* B* Conway, D i r e c t o r , D i v i s i o n of Tests, Standards and Research, Department of Education, V i c t o r i a , B. C*  -8 Upper Third-Lower T h i r d technique.  The r e s u l t s of t h i s procedure appear  i n Table I . Using a prepared g r i d , with the v e r t i c a l axis representing the per cent of v a l i d i t y and the horizontal axis the per oent of d i f f i c u l t y , the e f f i c i e n c y of the items was demonstrated graphically*  An a r b i t r a r y  curve was drawn and items appearing above the curve were considered s a t i s f a c t o r y f o r i n c l u s i o n i n the t e s t , while those f a l l i n g below appeared t o be i n need of r e v i s i o n or deletion*  As w i l l be seen by the graph, shown  i n Figure I, no item f a l l s s e r i o u s l y out of l i n e and i t was decided that a l l might be retained f o r the f i n a l form of the t e s t *  Per cents of  v a l i d i t y range from 14$ t o 70% and a l l are p o s i t i v e *  Thus, each item has  some power t o discriminate between the good and the poor pupils*  Per  cents of d i f f i c u l t y f o r the items extend from 9% t o 81% w i t h an average d i f f i c u l t y of 44$.  This i s i n l i n e w i t h the findings of Hawkes, Lindquist  7 and Mann who report that i n general t e s t authorities *..** are agreed tt  that there should be a range of d i f f i c u l t y from about &  t o 20 per cent  t o 80 t o 95 per oent, and that the average d i f f i c u l t y of a l l items should be about 50 per c e n t *  8  Reliability The r e l i a b i l i t y of a t e s t must be estimated s t a t i s t i c a l l y but i t i s not dependent upon an external c r i t e r i o n *  Thus, i t does not present  the same d i f f i c u l t i e s as are encountered i n e s t a b l i s h i n g the v a l i d i t y of  7  Herbert E. Hawkes, E* F* Lindquist, C. R. Mann, The Construction and Use of Achievement Examinations, p» 32. Boston: Houghton M i f f l i n Company, 1936*  Table I THE VALIDITIES AND DIFFICULTIES IN TERMS OF PER CENT OF THE ITEMS OF THE PRELIMINARY FORM OF THE TEST IN. COMPUTATION WITH DECIMAL FRACTIONS  Item  Per cent of Validity  Per oent of Difficulty  1  47  29  2  24  17  3  20  15  4  21  15  5  35  19  6  49  68  7  43  30  8  18  12  9  22  23  10  52  41  11  28  25  12  36  29  13  62  53  14  63  50  15  37  31  16  54  41  17  26  38  »« While i t i s recognized that the term "per cent of v a l i d i t y " i s one of current usage, i t i s r e a l i z e d that i t i s somewhat of a misnomer and, as pointed out previously, i t represents the difference between the per oent of oorrect responses obtained by the top t h i r d of the group and the bottom t h i r d of the group*  Table I  (continued)  THE VALIDITIES AND DIFFICULTIES IK TERMS OF PER CEHE OF THE HEMS OF THE PRELIMINARY FORM OF THE TEST IN COMPUTATION WITH DECIMAL.FRACTIONS  f  1.  I  Item  Per Cent of Validity  Per cent of Difficulty  18  43  46  19  55  46  20  53  47  21  51  67  22  56  66  23  70  52  24  51  58  25  50  52  26  38  72  27  20  81  28  40  70  29  45  75  50  48  72  51  57  57  32  52  73  33  32  34  36  26  35  14  9  '  42  V A L I D I T Y  21 Figure I GRAPHICAL ANALYSIS OF HEMS OF TEST IN COMPUTATION WITH DECIMAL FRACTIONS IN TERMS OF PER GENT OF VALIDITY AND PER CENT OF DIFFICULTY - PRELIMINARY FORM  1  14. 13 /•* 1? 3Z  10 2-0 /.  7  .JO  L  I?  •29  2.$  " isII-  z  •  N,  / 17  >  / • !  V  10  20  i  1  j 30  40  50  DIFFICULTY  60  \I  70  80  90  100??  22 a test*  For the purpose of determining the r e l i a b i l i t y of t h i s t e s t , a l l  of the 300 papers were included i n the calculations*  The odd-even s p l i t *  halves teohnique was used and the r e s u l t was corrected by the Spearman-  g Brown Prophecy Formula*  Greene, Jorgensen and Gerberleh  point out t h a t ,  while the c o e f f i c i e n t obtained by t h i s method i s l i k e l y t o be spuriously high,  "... t h i s i s one of the most f e a s i b l e methods f o r use with informal  objective examinations f o r which o r d i n a r i l y no second or alternate form i s available*" The r e s u l t of these c a l c u l a t i o n s appears i n Table I I . Table I I COEFFICIENT OF RELIABILITY OF PRELIMINARY FORM OF THE TEST IN COMPUTATION WITH DECIMAL FRACTIONS .DETERMINED BY THE ODD-EVEN SPLIT-HALVES TECHNIQUE  1 Form  Preliminary  1  I  Coefficient  ,780  Corrected Coefficient  #876  Although the c o e f f i c i e n t of c o r r e l a t i o n appeared t o be reasonably satisfactory, i t was decided t o estimate the r e l i a b i l i t y c o e f f i c i e n t by a second method*  Kuder and Richardson have devised a simple formula f o r  determining the r e l i a b i l i t y of a t e s t *  "  The only data used by t h i s method  8 Greene, Jorgensen and Gerberleh, op* o i t * , p. 63  23 are the number of t e s t items, the standard deviation, and the arithmetio mean*  I t i s based upon a number of assumptions whioh are hard t o meet 9  but* nevertheless* Bemmers  believes t h a t :  "....« i t i s probable that the  quick estimate afforded by t h i s formula i s good enough f o r a l l p r a c t i c a l purposes*"  T h i s teohnique u s u a l l y gives an estimate lower than that  obtained by the s p l i t - h a l f method used with the Spearman-Brown Formula* The r e s u l t obtained by the Euder-Richardson method i s given i n Table I I I and, as i s t o be expected* i s s l i g h t l y lower than the previously determined c o e f f i c i e n t *  In terms of r e l i a b i l i t y the t e s t now seemed t o be  acceptable and ready f o r use i n f i n a l form* Table I I I  COEFFICIENT OF RELIABILITY OF THE PRELIMINARY FORM OF THE TEST IN COMPUTATION WTTH DECIMAL FRACTIONS DETERMINED BY THE KDDER-RICHARDSON FORMULA  Form  R e l i a b i l i t y Coefficient  Preliminary  •820  9 H* H. Rammers and N* L* Gage, Educational Measurement and Evaluation, p* 205.  New York: Harper & Brothers, 1943*  24 P i n a l Form  The only r e v i s i o n undertaken f o r the F i n a l Form of the t e s t was to arrange the items i n order of d i f f i c u l t y .  With t h i s , the t e s t was  ready t o he administered f o r a f i n a l run. She t e s t was now given t o over 300 grade 7-8-9 pupils i n a single junior high school. Form.  The same procedure was used as i n the Preliminary  The papers were cheeked objectively, arranged In rank order* and  divided i n t o three sections of 100 each* Correct item responses were t a l l i e d and per oent of v a l i d i t y and per oent of d i f f i c u l t y were calculated as before. results.  Table IV shows the  As before* the material i s presented i n graphical form i n  Figure I I .  I t may be seen that only one item f a l l s s l i g h t l y below the  a r b i t r a r y ourve*  7% t o 79$.  The v a l i d i t y of a l l items i s p o s i t i v e and ranges from  The per oent of d i f f i c u l t y of items runs from 9# t o 87$ with  an average d i f f i c u l t y of 31*53$.  I t w i l l be noted that the t e s t appeared  to be l e s s d i f f i c u l t f o r t h i s group than f o r the group t e s t e d i n the preliminary run*  This may be aooounted f o r , i n part, by the f a c t that  only one sohool was used i n t h i s run and by the f a c t that the t e s t was administered l a t e r i n the sohool year.  I t may a l s o be observed that the  order of d i f f i c u l t y of the items, although not i d e n t i c a l with that found i n the f i r s t run, i s approximately the same*  Item V a l i d i t y Based on Flanagan*s Tables  Although the method above appeared t o provide a s a t i s f a c t o r y estimate of the i n t e r n a l consistency of the items of the t e s t * i t was dsoided t o apply a f u r t h e r cheok of the item v a l i d i t y i n d i c e s . According  26  Table 17  THE VALIDITIES AND DIFFICULTIES IN TERMS OF PER CENT OF THE HEMS OF THE FINAL FORM OF THE TEST IN COMPUTATION WITH DECIMAL FRACTIONS  Item  Per cent of Validity  Per cent of Difficulty  1  7  11  2  9  9  3  7  10  4  13  13  5  20  13  6  27  15  7  19  17  8  22  15  9  15  10  10  30  19  11  21  14  12  29  17  13  22  21  14  29  28  15  36  20  16  51  29  17  14  24  Table IV (continued) THE VALIDITIES AND DIFFICULTIES IN TERMS OF PER CENT OF THE HEMS OF THE FINAL FORM OF THE TEST IN COMPUTATION WITH DECIMAL FRACTIONS  Item  Per cent of Validity  Per cent of Difficulty  18  34  35  19  48  30  20  37  27  21  51  32  22  56  32  22  47  35  24  49  27  25  41  52  26  55  48  27  54  36  28  35  32  29  55  68  30  15  87  31  65  59  32  79  47  33  58  67  34  68  50  35  38  73  27  V A L I D I  Figure I I GRAPHICAL ANALYSIS OF HEMS OF TEST IN COMPUTATION WITH DECIMAL FRACTIONS IN TERMS OF PER CENT OF VALIDITY AND PER CENT OF DIFFICULTY - FINAL FORM  T Y  -  3Z  •  J9ZJ  ; ; •  N  12.  k ii:  9. 4.  /  '/7  30  V  >7  3- 1. / 10  t 20  30  40  ' 50  DIFFICULTY  60  70  80  \ i 90  ,.  100?$  28 t o T h o r n d l k e ^ , -mho favours working with the t o p 27 per oent and the bottom 27 per cent of the t o t a l group: The most s a t i s f a c t o r y item v a l i d i t y index based on the upper and lower 27 per oent i s the estimate of the c o e f f i c i e n t of c o r r e l a t i o n between item and t e s t obtainable from tables prepared by Flanagan. Using the data obtained from the f i n a l administration of the t e s t . per cents succeeding i n the upper 27 per oent and the lower 27 per oent Were calculated.  Estimates of product-moment c o e f f i c i e n t s of c o r r e l a t i o n 11  were found by reference t o t a b l e s prepared by Flanagan appear i n Table 7.  •  The r e s u l t s  I t w i l l be seen that the c o e f f i c i e n t s of c o r r e l a t i o n  range from .120 t o .810 and that a l l are p o s i t i v e .  Items 1 and 2 are not  s i g n i f i c a n t l y greater than zero at the 1% l e v e l and are b a r e l y s i g n i f i c a n t at the Qfo l e v e l .  The rest, of the items are s i g n i f i o a n t l y greater than  zero*  10 fiobert L. Thorndike,, Personnel Selection, p. 242. John Wiley & Sons, I n c , 1949. 11 r i b i d . , pp. 348-351.  Hew  York:  29 Table V INTERNAL CONSISTENCY OP THE FINAL FORM OF THE TEST IN COMPUTATION WTTH DECIMAL FRACTIONS, BASED ON FLANAGAN'S ESTIMATES OF THE CORRELATION BETWEEN INDIVIDUAL HEMS. AND THE TEST AS A WHOLE  Item  Coefficient  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  .120 .140 .215 •243 •400 .700 •280 •420 •370 •580 •505 •520 •320 • 375 • 560 •666 •245 •400 •640 •495 •638 •680 • 510 .795 •535 •600 •605 •460 .600 •345 •680 •810 .715 •690 •470  30  Reliability R e l i a b i l i t y of the F i n a l Form of the t e s t was determined by using the same methods as those used i n the Preliminary Form* given i n Table V I .  The findings are  While the r e l i a b i l i t y c o e f f i c i e n t s have dropped  s l i g h t l y * the f i g u r e s are reasonably close t o those obtained f o r the e a r l i e r form*  Table V I  COEFFICIENTS OF RELIABILITY OF THE FINAL FORM OF THE TEST IN COMPUTATION WITH DECIMAL FRACTIONS 1  .  Form  1  Reliability Coefficient  Method  Final .  Split-halves Kuder-Riohardson  *853 (corrected) *812 .  Summary  In the opinion of the w r i t e r , the analysis has demonstrated that the F i n a l Form of the t e s t i s a s a t i s f a c t o r y instrument f o r use i n t h i s study*  The t e s t i s almost self-administering; i t can be given i n a  normal olassroom period;  i t i s e a s i l y scored and i s h i g h l y objective;  i t i s economical i n terms of cost; e n t i r e l y upon the curriculum;  the material of the t e s t i s based  i n construction of the t e s t , l o g i c a l and  "face" v a l i d i t y have received consideration;  the items meet the require-  ments of v a l i d i t y and d i f f i c u l t y and suoceed i n discriminating between the good and the poor p u p i l s ;  on the b a s i s of two s t a t i s t i c a l techniques the  t e s t i s found t o possess a f a i r l y high degree of r e l i a b i l i t y *  In the  l i g h t of the foregoing arguments* the t e s t i s considered suitable f o r i t s intended purpose and w i l l be used i n the investigation* The F i n a l Form of the Test In Computation with Decimal Fractions appears i n Appendix A*  32  CHAPTER I I I  5  CONSTRUCT ION OF A TEST IN UNDERSTANDINGS OF PROCESSES INVOLVING DECIMAL FRACTIONS  Introduction  The construction of a t e s t i n understandings of processes involves not only the same problems as those encountered i n the preparation of the t e s t i n computation* but also* a d d i t i o n a l d i f f i c u l t i e s due t o t h e nature of such a t e s t *  A discussion of understandings and meanings  presented i n an e a r l i e r chapter*  ms  The problem now a r i s i n g i s one of  securing a t e s t designed t o evaluate a pupil*s understanding of the basic processes inherent i n computation w i t h decimal f r a c t i o n s *  That l i t t l e  research has been c a r r i e d out i n t h i s area i s indicated by Glennon ' i n the 3  following: The paucity of research studies i n the area of t e s t i n g f o r meanings j u s t i f i e s the conclusion that t h i s i s one of the most neglected educational problems of the day* An examination of Glennon s t e s t i n mathematical understandings 1  revealed that, by i t s nature, i t would not s u i t the purpose of t h i s study* In addition, no data r e l a t i v e t o i t s v a l i d i t y were available*  As no other  tests which would f i t the requirements of t h i s study could be located, i t was decided that i t would be necessary t o construct a t e s t i n understandings of basic processes Involved i n the use of decimal f r a c t i o n s *  1 Glennon, op* p i t * , p. 68  33  Factors Involved i n the Construction of Test Items In constructing the t e s t items the test-maker had t o be conscious of a number of f a c t o r s that -mere not relevant to the computational t e s t * The items must not involve computation, otherwise they would duplioate the f u n c t i o n of the other t e s t *  The material must be based upon the material  of the companion t e s t i n computation--it must t r y t o evaluate the subject's understanding of h i s use of the mechanics whioh involve the same basio concepts*  Verbalism must be minimized so that the t e s t does not become  an evaluation of reading comprehension*  Wording of the items must  receive c a r e f u l consideration so that ambiguity may be avoided*  A suit-  able form of item must be selected t o ensure a high degree of o b j e c t i v i t y * With an awareness of these requirements i n mind, the w r i t e r set about t o prepare the material f o r the t e s t *  The l a c k of a c r i t e r i o n  against which t o v a l i d a t e the t e s t made i t imperative t o construct i t with the greatest care so that i t might be e f f i c i e n t i n terms of o u r r i c u l a r or  2 analytical v a l i d i t y .  Both C a r l i l e  S and Glennon  depended upon t h i s  aspeot of v a l i d a t i o n i n t h e i r t e s t s * The questions on the computational t e s t were taken as a basis on whioh t o work and an attempt was made t o design items t o t e s t understandings of the prooesses used i n these computations.  To obtain some idea of  the manner i n whioh students are l i k e l y t o describe t h e i r thought prooesses when performing computations, some subjective questions were given inform2 Carlile,  Off*  oit„  p. 214  3 Glennon, op* p i t * ,  p. 70  34 a l l y t o several groups.  A series of questions was then prepared on  Individual cards. The form of item decided upon was the multiple-choice. cussing the use of t h i s s t y l e of question, Ross  Dis-  has t h i s t o says  The multiple-choice type of item i s u s u a l l y regarded as the most valuable and most generally applicable of a l l t e s t forms. Lee regards i t as "one of the best means f o r t e s t i n g judgment that i s a v a i l a b l e " . Lindquist asserts t h a t i t i s " d e f i n i t e l y superior t o other-types" f o r measuring sueh educational object* ives as " i n f e r e n t i a l reasoning, reasoned understanding, or sound judgment and d i s c r i m i n a t i o n on the part of the pupils" • Rammers  and others support t h i s opinion*  I n addition, t h i s s t y l e of  item ensures o b j e c t i v i t y of scoring, whioh i s an e s s e n t i a l c h a r a c t e r i s t i c for this test* Experimental Form  An exploratory group of twenty-seven items was prepared and t h e material was submitted t o a number of competent educationalists f o r o r i t i o i s m and suggestion*  The combined judgments of these people ensured  some degree of v a l i d i t y and made i t possible t o prepare an experimental form of the t e s t * The t e s t was administered t o over one hundred grade 7-8-9 pupils i n the Greater V i c t o r i a School D i s t r i c t *  Some of the t e s t i n g was dons  personally by the w r i t e r t o observe p u p i l r e a c t i o n and t o c a l c u l a t e the time f a c t o r *  I n addition, some t e s t i n g of i n d i v i d u a l subjects was per-  formed*  Results were c a r e f u l l y analyzed *TV* a orude comparison was made  "~~  *  '  C, C. Ross, Measurement i n Today's Schools, p* 145* Hew York: Prentice-Hall, Ino*, 1941* 5, Rammers, OP* p i t . ,  p* 167  35 with the performance of the same pupils on the t e s t I n computation* At t h i s point i t was decided that there was l i t t l e t o be gained by v a l i d a t i n g the t e s t at t h i s grade l e v e l because an important f a c t o r *  reading a b i l i t y appeared t o be suoh  As the study was t o be performed ultimately w i t h a  group of student-teachers* i t appeared that f u r t h e r t e s t i n g should be oonduoted with a s i m i l a r group* The Experimental Form was next administered t o a group of f i r s t year college students under the personal supervision of the writer*  In  a l l stages of preparation of the t e s t comments and suggestions were i n v i t e d from the students so that r e v i s i o n might be made with a view t o securing greater "face" v a l i d i t y *  A c a r e f u l analysis of i n d i v i d u a l working times  was made and i t was found that the t e s t oould be completed w i t h ease i n f i f t e e n t o twenty minutes*  The r e s u l t s of t h i s group were c a r e f u l l y  analyzed i n terms of a l t e r n a t i v e s selected and a crude comparison was made with t h e i r general achievement standing i n mathematics *  As a r e s u l t of  these observations* i t was now possible t o revise the t e s t and prepare a Preliminary Form*  Preliminary Form  The Preliminary Form of the t e s t was made up of twenty-seven multiple ohoioe items of f o u r or f i v e a l t e r n a t i v e s *  The time f a c t o r was  set a t f i f t e e n minutes or u n t i l a large per oent of the group had completed the paper.  Simple directions were prepared and a marking key was provided*  The paper was administered t o over three hundred teaohers-in-training who were i n v i t e d t o comment upon t h e i r r e a c t i o n t o the various items. The items were e a s i l y and o b j e c t i v e l y scored by competent students*  36 Three hundred completed papers were selected and arranged i n so ore order.  They -were then divided i n t o three p i l e s of one hundred each,  thus providing an upper and a lower t h i r d .  Item responses were tabulated  on large sheets of squared paper and, using the Upper Third-Lower T h i r d technique, per cent of v a l i d i t y and per cent of d i f f i c u l t y were calculated* The analysis i s given i n Table VII* As i n the t e s t i n computation, the per cents of v a l i d i t y and of d i f f i c u l t y were p l o t t e d upon a g r i d and an a r b i t r a r y curve was drawn* i s i l l u s t r a t e d i n Figure I I I *  This  I t w i l l be seen t h a t only one item f a l l s  below the ourve t o any appreciable degree*  A l l items show a p o s i t i v e  v a l i d i t y ranging from 10% t o 52$ with only a few much below 20$. they meet the requirement set up by C a r l i l e  i n h i s study*  Thus  He reported:  Items whioh d i d not show a positive d i s c r i m i n a t i o n of as much as twenty per oent were considered lacking i n the power o f discrimination* At the same time, he pointed out that there i s a tendency t o include some easy and same d i f f i c u l t items whioh w i l l have l i t t l e discriminative v a l u e . Per oent of d i f f i c u l t y of the items ranges from 7% t o 69# with an average d i f f i c u l t y of 33%.  The t e s t , as a -whole, i s easier than desirable  but perhaps that i s inevitable i n a t e s t of t h i s type.  A t the same time,  i t must be remembered that the t e s t was administered near the end of a year's consideration of the content.  6 C a r l i l e , 0 £ . p i t . , p. 215  Table V I I THE VALIDITIES AM) DIFFICULTIES IN TEEMS OF PER CENT OF THE HEMS OF THE PRELIMINARY FORM OF THE TEST IN UNDERSTANDING OF PROCESSES  Item  Per oent of Validity  Per oent of Difficulty  1  25  17  2  13  33  3  32  34  4  43  30  5  19  79  6  25  19  7  18  10  8  31  31  9  10  7  10  30  35  11  33  36  12  18  10  13  19  11  14  25  17  15  42  35  Table V I I (oontinued)  THE VALIDITIES AMD DIFFICULTIES IN TERMS OF PER CENT OF THE. ITEMS OF THE PRELIMINARY FORM OF THE TEST IN. UNDERSTANDING OF PROCESSES.  Item  Per oent of Validity  Per cent of Difficulty  16  52  37  17  44  55  18  29  32  19  34  24  20  52  36  21  14  17  22  35  76  23  31  22  24  27  60  25  41  36  26  28  20  27  24  76  V A L I . D I T Y  39  FIGURE I I I GRAPHICAL ANALYSIS OF HEMS OF TEST IN UNDERSTANDING OF PROCESSES IN TERMS OF PER CENT OF VALIDITY AND PER CENT OF DIFFICULTY — PRELIMINARY FORM  .  90  80  70.  60  17  k 40  zz  (1 23  30  f '  3 /  '/  10  /  i& IH  20 7 I'z 10  '<?  /  0  V  10  / y  s •  •  i 20  30  j  40  \ i  50  DIFFICULTY  60  70  80  90  100$  40 Reliability  The c o e f f i c i e n t of r e l i a b i l i t y was calculated by dividing t h e t e s t into chance-halves, using the odd and even scores.  The Pearson  Product-Moment method was employed t o f i n d the c o e f f i c i e n t of the half test.  The r e s u l t was corrected by the Spearman-Brown Prophecy Formula  and appears i n Table T i l l .  Table Y I I I  COEFFICIENT OF RELIABILITY OF TEE PRELIMINARY FORM OF THE TEST IN UNDERSTANDING OF PROCESSES DETERMINED BY ODD-EVEN SPLIT-HALVES TECHNIQUE  Form  Preliminary  Coefficient  ,566  Corrected Coefficient  •723  41  The Kuder-Richardson Formula was then, applied with the r e s u l t shown i n Table XX.  Table IX COEFFICIENT OF RELIABILITY OF THE PRELIMINARY FORM.OF.THE TEST IN UNDERSTANDING OF PROCESSES DETERMINED BY THE K0DER-RICBARDSON FORMULA  Form  Reliability Coefficient  Preliminary  .574  The r e l i a b i l i t y c o e f f i c i e n t s obtained f o r the Preliminary Form of the t e s t were disappointingly low.  The F i n a l Form  The r e s u l t s of the Preliminary Form were subjected t o a most c a r e f u l scrutiny and a number of changes i n the wording of the items were made. difficulty. thirty.  The questions were then arranged i n approximate order of Three a d d i t i o n a l items were added* bringing the t o t a l up t o  The t e s t was now prepared i n f i n a l form. In Chapter I there was set f o r t h a l i s t of the concepts whioh,  i t i s believed, forms the basis of the understandings involved i n the use  of decimal f r a c t i o n s *  I t i s the function of the present t e s t t o eval-  uate the students' understanding of these concepts*  Following i s a n  analysis of the t e s t which indioates the items that are designed t o measure the various concepts: Concept  Item  Meaning of decimal f r a c t i o n s  27  Reading and w r i t i n g of decimals  7  Value of decimal f r a c t i o n s  5,  Comparison of decimal f r a c t i o n s  4  Function of zero  3,  Rounding o f f numbers  21,  22, 28  Accuracy of measurement  18,  30  E f f e c t of moving deoimal point  6,  10, 21, 28  8, 9  8,  14, 19,  23,  25  Location of point i n m u l t i p l i c a t i o n  16,  25, 26  Location of point i n d i v i s i o n  12,  13, 14, 17,  23,  24  Looation of point i n a d d i t i o n  )  Looation of point i n subtraction |  Changing common f r a c t i o n s t o decimals Changing decimals t o common fractions Relative value of d i g i t s  2,  29  15 11,  20  The revised t e s t was now administered t o 150 teachers-intraining.  I t was made up of t h i r t y multiple-choice items and the time  43 f a c t o r was set at twenty minutes* students i n the a l l o t t e d time*  The t e s t was completed by nearly a l l Interest appeared t o be f a i r l y high and  ambiguity had been l a r g e l y eliminated*  The test was r e a d i l y scored and  was completely objective. The r e s u l t s were tabulated and analyzed as i n the Preliminary Form*  Table X gives the per oent of v a l i d i t y and d i f f i c u l t y of the items*  In Figure 17 the r e s u l t s are depicted graphically* run from 8$ t o 72%»  Per cents of v a l i d i t y  D i f f i c u l t y of the items ranges from 4/£ t o 77% with  an average d i f f i c u l t y of 33/S, the same as f o r the Preliminary Form*  The  graph shows that only a few items f a l l below the curve but none so f a r as to cause any r e a l concern*  The three a d d i t i o n a l items proved t o be quite  s a t i s f a c t o r y , having the following r a t i n g s :  No*  Per oent of Validity  Per cent of Difficulty  5  66  59  7  26  55  14  68  48  Table X  THE VALIDITIES AND DIFFICULTIES IN TERMS OF PER CENT OF THE ITEMS OF THE FINAL FORM OF THE TEST,IN UNDER. STANDING OF PROCESSES  Item  Per oent of Validity  Per cent of Difficulty  1  10  5  2  8  4  3  14  9  4  12  8  5  66  59  6  40  34f  7  26  55  6  34  19  9  16  14  10  30  67  11  22  13  12  26  13  13  32  16  14  68  48  15  16  24  Table X  (continued)  THE VALIDITIES AMD DIFFICULTIES IN TERMS OF PER CENT OF THE. ITEMS OF THE FINAL FORM OF THE TEST IN UNDER. STANDING OF PROCESSES  Item  Per cent of Validity  Per cent of Difficulty  16  38  25  17  18  29  16  34  31  19  32  30  20  42  23  21  50  29  22  32  32  23  56  32  24  48  40  25  58  45  26  44  32  27  44  50  28  72  58  29  34  77  30  30  59  V A j  46 FIGURE 17  5 jT  GRAPHICAL ANALYSIS OF HEMS OF TEST IN UNDERSTANDING OF PROCESSES IN TERMS OF PER CENT OF VALIDITY AND PER CENT OF DIFFICULTY — FINAL FORM  100?$ 90  80  •  70 it-.  S  60  21  50  16  • 27  'I  40 ./* 13 •  if  30  II.  20  id 30  /  \  7  9 10  0  10  /  K  <  / 1  1  20  30  40  50  DIFFICULTY  60  70  80  90  100%  47 Item V a l i d i t y Indices Based on Flanagan's Tables  Following the same procedure as that used i n the t e s t on computation, v a l i d i t y c o e f f i c i e n t s were determined on the basis of the upper and lower twenty-seven per cent of the group tested* Flanagan's tables gives values as shown i n Table XI. positive and range from .150 t o .820.  Reference t o  A l l c o e f f i c i e n t s are  The c o e f f i c i e n t s of a l l items, with  the exception of one, are s i g n i f i c a n t l y greater than zero. of item Ho. 15 i s too small t o be indicative of any r e a l  The c o e f f i c i e n t  correlation*  48  Table XI INTERNAL CONSISTENCY OP THE FINAL FORM OF THE TEST IN UNDERSTANDING OF PROCESSES. BASED ON FLANAGAN*S. . ESTIMATES OF CORRELATION BETWEEN INDIVIDUAL. . HEMS AND THE TEST AS A WHOLE  Item  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  Coeff i o i e n t  .430 •600 •415 •415 .700 •485 .280 •600 .380 .280 • 505 •665 .700 .725 •150 •550 •205 •415 •540 .780 •646 •415 •805 .665 .730 .560 .450 .820 •645 .720  49 Reliability The r e l i a b i l i t y c o e f f i c i e n t of the F i n a l Form of the t e s t was calculated by the same methods as were used previously and the r e s u l t s are shown i n Table X I I .  The c o e f f i c i e n t of r e l i a b i l i t y was estimated t o be  considerably higher than that found f o r the Preliminary Form and was considered t o be s a t i s f a c t o r y *  Table XII  COEFFICIESTS OF RELIABILITY OF THE FINAL FORM OF THE TEST IN UNDERSTANDINGS OF PROCESSES  Form  Method  Final  Split-halves  "  Kuder-Richardson  Reliability Coefficient  *809 (corrected) .717  Relationship Between Scores on Test on Understandings and Intelligence Test Scores  In an endeavour t o discover more c l e a r l y the nature of the t e s t on understandings, the scores on t h i s t e s t were compared with the scores obtained by the same students on the Otis Test of Mental A b i l i t y .  The  c o e f f i c i e n t of c o r r e l a t i o n was determined by the Pearson Product-Moment  50 Method with the r e s u l t shown i n Table XIII. than the usual  4 •  This i s somewhat higher  *45 (approx.) found between i n t e l l i g e n c e t e s t scores  and achievement scores at t h i s l e v e l *  A c o r r e l a t i o n of t h i s magnitude  i s c l a s s i f i e d by most writers as denoting substantial or marked r e l a t i o n ship*  We may conclude that a f a i r l y strong r e l a t i o n s h i p exists between  the two sets of measures*  Based upon l o g i c a l considerations* t h i s may  suggest the influence of such f a c t o r s as reading comprehension or the existence of s i m i l a r i t i e s i n the two t e s t s , but the oause-and-effect r e l a t i o n s h i p oannet be determined from the data* Table XIII RELATIONSHIP BETWEEN SCORES ON TEST IN UNDERSTANDINGS OF PROCESSES AND OTIS TEST OF MENTAL ABILITY OBTAINED BY 150 NORMAL SCHOOL STUDENTS  r  • .585  Summary A f t e r considering the foregoing f a c t o r s , i . e . , c u r r i c u l a r and "face" v a l i d i t y , discriminatory value, degree of d i f f i c u l t y , o b j e c t i v i t y , p r a c t i c a l i t y , etc., i t was decided that a l l the items i n the t e s t should be retained and that the t e s t might be considered an acceptable instrument of evaluation t o be used i n the present study.  The Test i n Understandings  of Basio Processes Involved i n the Use of Deoimal Fractions w i l l be found i n Appendix B.  51  CHAPTER 17  INVESTIGATION OP THE RELATIONSHIP BETWEEN COMPUTATION AND UNDERSTANDINGS IN THE USE OF DECIMAL FRACTIONS  The Subjects  The subjects selected f o r the i n v e s t i g a t i o n of the r e l a t i o n s h i p between computational a b i l i t y and understanding of basio processes involved i n the use of deoimal f r a c t i o n s consisted of students e n r o l l e d i n the oneyear teacher-training course a t the V i c t o r i a Normal School.  A l l of these  students possessed the basio q u a l i f i c a t i o n s required f o r admission i n t o the Normal Schools of the Province of B r i t i s h Columbia, i . e . , graduation from high school on the U n i v e r s i t y Entrance Programme (or i t s equivalent). Many had acquired a d d i t i o n a l c r e d i t s extending from f i r s t year u n i v e r s i t y standing through t o the Bachelor of A r t s Degree.  While the m a j o r i t y of  the students were graduates of the high schools of B r i t i s h Columbia, same had acquired t h e i r basic education i n other parts of Canada and Europe. The range of I.Q. scores of the group measured by the Otis Test of Mental A b i l i t y , extended from 91 t o 137, -with a standard deviation of 8,7. ages of the testees ranged from 17 t o over 40.  The  While the students formed  a d i v e r s i f i e d group, they were t y p i c a l of the people who teaoh school i n B r i t i s h Columbia*  I t may be assumed that, having f u l f i l l e d the r e q u i r e -  ments f o r entrance i n t o Normal School, a l l had received basio i n s t r u c t i o n i n the fundamentals  of computation with decimal f r a c t i o n s *  However, i t  i s l i k e l y that there would be considerable v a r i a t i o n i n the methods by  52 whioh they had received t h e i r Instruction* Administration of the Tests  The t e s t s which have been described i n the previous chapters, and whioh appear i n Appendix A and Appendix B, were used f o r the i n v e s t i gation of r e l a t i o n s h i p *  S h o r t l y a f t e r the students assembled f o r the  f a l l term, and before any i n s t r u c t i o n had taken place, the t e s t s were administered t o the e n t i r e student-body, c o n s i s t i n g of two hundred f o r t y students*  Thus, the testees received no b e n e f i t from the course of  i n s t r u c t i o n at the Normal Sohool i n which the meaning theory i s emphasised* The students answered the papers, equipped w i t h the knowledge and under- . standing acquired from t h e i r previous education and experience* The t e s t i n computation with decimal f r a c t i o n s was administered f i r s t and a t i m e - l i m i t of t h i r t y minutes was allowed* ample time f o r most students t o f i n i s h with ease*  This proved t o be  The t e s t i n under-  standing of processes was given immediately f o l l o w i n g the f i r s t t e s t and a time-limit of twenty minutes provided long enough f o r the majority t o complete a l l the items*  T h i s order of presenting the t e s t s appeared t o  be the l o g i c a l one as the i n t e n t i o n was t o discover the subjects' understanding of the processes employed i n the computation.  Had the reverse  order been used, i t i s possible that the items i n understandings might have provided d u e s t o the s o l u t i o n o f some of the questions i n computation.  Analysis of the Results  The scoring of the papers, under the supervision of the w r i t e r , was completely objective.  In the t e s t i n computation the scores ranged  from 23$ t o 100$ correct with a mean of 74$ oorreot*  I n the t e s t i n  53 understandings the scores were d i s t r i b u t e d from 18% correct t o 100% correct w i t h a meaa of 61% correot.  There i s a wide range i n achieve-  ment shown i n the r e s u l t s of both t e s t s — a n  outcome which might be a n t i c i -  pated from the range i n i n t e l l i g e n t quotient scores previously  stated*  Two hundred t h i r t y - s i x papers were selected and the Pearson Product-Moment method was employed t o caloulate the c o e f f i c i e n t of c o r r e l a t i o n between the r e s u l t s of the t e s t i n computation and the t e s t i n understandings*  The r e s u l t i s shown i n Table XIV.  The data indicates  the existence of a p o s i t i v e r e l a t i o n s h i p of considerable magnitude between the scores obtained on the t e s t s i n computation and understandings*  Table XI7  RELATIONSHIP BETWEEN SCORES OBTAINED ON TESTS IN COMPUTATION AND UNDERSTANDING OF PROCESSES INVOLVED IN THE USE OF DECIMAL FRACTIONS BY 236 NORMAL SCHOOL STUDENTS  r  =  .670  A study of the scatter diagram reveals the following points of i n t e r e s t : 1*  High scores i n understandings tend to be accompanied by high scores i n computation.  2*  Low soores i n computation are generally aooompanied by low soores i n understandings*  3*  High soores i n computation are found throughout a f a i r l y wide range of soores i n understandings*  4*  There i s a f a i r l y wide range i n scores i n computation accompanying low soores i n understandings*  However,  and  Two examples of extreme scores may be oited*  Case A succeeded  i n answering 86$ of the items i n computation c o r r e o t l y while possessing only 23$ of the understandings of the other t e s t * r e l a t i v e l y low I* Q. r a t i n g * only 28$ i n computation*  This i n d i v i d u a l had a  Case B scored 54$ i n the understandings but  These cases, however* were i s o l a t e d and not  indicative of the general trend of r e l a t i o n s h i p * In a further attempt t o analyze the r e s u l t s and t o secure more data on the i n t e r - r e l a t i o n s h i p s of the t e s t s , c o e f f i c i e n t s of c o r r e l a t i o n between the r e s u l t s of the t e s t s and i n t e l l i g e n c e t e s t soores were determined*  The r e l a t i o n s h i p between I*Q* and soores on the t e s t i n  understanding was found t o be the c o r r e l a t i o n of  r  =  *585  r "2  *547.  (This i s reasonably close t o  referred t o i n the l a s t ohapter .) 1  r e l a t i o n s h i p between computation and I* Q. was calculated t o be  r  A composite statement of these correlations i s given i n Table X7.  1 See p. 50  The r:  .481.  55 Table XV  COEFFICIENTS OP CORRELATION BETWEEN TESTS  I. Q.  I. Q. Computation  .481  Understandings  .547  Computation  Understandings  .481  .547 .670  .670  P a r t i a l Correlation  The influence  of the f a c t o r of i n t e l l i g e n c e , which has a common  r e l a t i o n s h i p t o the variables obscure the true r e s u l t s .  of computation and understandings* tends t o  The differences  among i n d i v i d u a l s , introduced  by the f a c t o r of i n t e l l i g e n c e , can be eliminated by using the method of 2 partial correlation.  Using the technique described by Garrett,  the  net c o r r e l a t i o n between computation and understandings, with i n t e l l i g e n c e " p a r t i a l l e d out", was calculated. r  —  The r e s u l t obtained was: .554  2 Henry E. Garrett, S t a t i s t i c s i n Psychology and Education, pp. 378*405. New York: Longmans, Green and Company* ~1953.  56  Summary The t e s t s i n computation and understandings i n the f i e l d of decimal f r a c t i o n s * were administered t o a group of student-teaohers* While the testees formed a d i v e r s i f i e d group, i t was assumed that they had a common background i n the area covered b y the t e s t s * preceded the t e s t i n g programme*  Ho i n s t r u c t i o n  The r e s u l t s of the i n v e s t i g a t i o n i n d i -  cated the existence of a p o s i t i v e r e l a t i o n s h i p of considerable magnitude between computation and understandings* was found t o be  r ~  .670.  The c o e f f i c i e n t of c o r r e l a t i o n  When differences i n i n t e l l i g e n c e had been  allowed f o r * the net c o r r e l a t i o n was found t o be somewhat less than the apparent r e l a t i o n s h i p , i . e *  r —  .554.  57  CHAPTER V  SUMMARY AM) CONCLUSIONS  Summary  Purpose of the study* T h i s study -mas undertaken i n an attempt t o discover what r e l a t i o n ship, i f any, e x i s t s between a subject's a b i l i t y t o perform mechanical computations and h i s understanding of the inherent mathematical i n the area of decimal f r a c t i o n s *  principles,  Modern theory of arithmetic i n s t r u c -  t i o n places great emphasis upon the a c q u i s i t i o n of mathematical meanings i n the learning process.  The writer's purpose i n conducting t h i s study-  was t o investigate the v a l i d i t y of the olaims set f o r t h by the  proponents  of the meaning theory and t o add some evidence t o the slowly accumulating body of knowledge concerning the place of understandings i n arithmetic instruction*  Materials of the study* The t o p i c of decimal f r a c t i o n s was chosen as the area of invest i g a t i o n because of the u n i v e r s a l i t y of i t s content and the e s s e n t i a l nature of i t s m a t e r i a l i n our society*  The subjeots chosen f o r the  i n v e s t i g a t i o n were student-teachers, because the t o p i c of the investigat i o n has p a r t i c u l a r s i g n i f i c a n c e f o r teaohers and a l s o because Normal Sohool students provided a convenient group f o r the conduct of the study*  58 Procedure of the study* The pursuance of the i n v e s t i g a t i o n depended upon the use of s u i t able t e s t i n g instruments.  As no t e s t s "which met the rather r i g i d r e q u i r e -  ments of the study could be obtained, i t beoame necessary to undertake the construction and v a l i d a t i o n of o r i g i n a l materials suitable f o r the s p e c i f i o purpose.  The preparation of the t e s t s i n computation and  understandings  became a major phase of the study. Much preliminary t e s t i n g took place and the t e s t s were r e v i s e d several times i n an e f f o r t t o meet the s p e c i f i c a t i o n s of sound, acceptable t e s t i n g instruments.  Pupils of the grade 7-8-9  l e v e l provided the sub-  j e c t s f o r the establishment of the v a l i d i t y and r e l i a b i l i t y of the t e s t i n computation.  The t e s t i n understandings was developed using student-  teachers as t e s t e e s . The i n t e r n a l consistency of t e s t items was determined by using a technique based upon the discriminating power and the degree of d i f f i c u l t y of the items, using the upper and lower t h i r d s of the groups.  A  further check of i n t e r n a l consistency was made by reference t o t a b l e s prepared by Flanagan, using the upper and lower twenty-seven per cent of the groups.  R e l i a b i l i t y was estimated by the s p l i t - h a l v e s teohnlque  and a f u r t h e r estimate was made using the Kuder-Ric hards on method.  These  s t a t i s t i c a l r e s u l t s indicated that the f i n a l forms of the t e s t s could be used f o r the purpose of the i n v e s t i g a t i o n with, some confidence i n t h e i r efficiency.  The f i n i s h e d tests should have f u r t h e r usefulness beyond  the purpose of t h i s study.  Results of the investigations The i n v e s t i g a t i o n was donduoted using student-teachers as subjects.  The group t e s t e d was d i v e r s i f i e d i n respect t o age, background,  59 academic status,  intelligence.  the use of deoimal f r a c t i o n s . administration of the t e s t s .  A l l had i n common some knowledge of  No i n s t r u c t i o n or explanation preceded the The t e s t i n computation (Appendix A) was  f i r s t given and was immediately followed by the t e s t i n understandings (Appendix B ) .  Two hundred t h i r t y - f i v e papers were used t o estimate t h e  degree of r e l a t i o n s h i p between computational a b i l i t y and understandings of mathematical p r i n c i p l e s .  The c o e f f i c i e n t of c o r r e l a t i o n was found  by the Pearson Product-Moment method t o be:  r =  .640.  Further  c o r r e l a t i o n c o e f f i c i e n t s were computed whioh produced the following 1.  Understandings  -  2.  Computation  Intelligences  -  Intelligence:  r =  .547  r =  .481  data:  The common factor of i n t e l l i g e n c e was " p a r t i a l l e d out" and the net c o r r e l a t i o n between computation and understandings was found t o be: r  —  .554.  Conclusions  The data obtained from the i n v e s t i g a t i o n leads t o the following inferences and conclusions: 1.  There i s a p o s i t i v e c o e f f i c i e n t of c o r r e l a t i o n between soores on the t e s t i n computation and soores on the t e s t i n understandings i n decimal f r a c t i o n s .  T h i s indicates that there i s a  tendency f o r the soores t o vary i n the same d i r e o t i o n .  High  scores i n one tend t o aooompany high soores i n the other* while low soores i n one are u s u a l l y found along with low scores i n the other.  The s i z e of the c o r r e l a t i o n c o e f f i c i e n t ( r = .640)  i s of substantial magnitude*  60 2*  The camnion f a c t o r of i n t e l l i g e n c e has an influence upon the r e l a t i o n s h i p between the two v a r i a b l e s *  When i n t e l l i g e n c e i s  held constant, the p a r t i a l c o e f f i c i e n t i s less than the apparent c o e f f i c i e n t , whioh indioates that r e l a t i o n s h i p i s due, i n part, t o the common dependence of both v a r i a b l e s upon the i n t e l l i g e n c e factor* 3*  The net c o r r e l a t i o n i s of marked magnitude ( r s-  .554) •  The magnitude of the r e l a t i o n s h i p between understandings and i n t e l l i g e n c e i s i n d i c a t i v e of oommon elements i n both ( r — ' *547).  However, a high I . Q. i s not a guarantee of a  :  high l e v e l of understanding* 4*  The r e l a t i o n s h i p between i n t e l l i g e n c e and computational a b i l i t y i s positive but not high  ( r —  *43l).  Computational compe-  tence i n decimal f r a c t i o n s seems t o be possible with a r e l a t i v e l y low I* Q* 5*  ( i n terms of the group used i n t h i s i n v e s t i g a t i o n ) *  While the trend i s that increase or decrease i n one v a r i a b l e i s accompanied by increase or deorease i n the other, there i s considerable evidence that neither i s e s s e n t i a l f o r the other, and that high soores i n one do not guarantee high soores i n the other*  6*  Although i t appears, from a study of the scatter diagram, that one who i s aware of the mathematical p r i n c i p l e s involved i n the use of deoimal f r a c t i o n s has a greater l i k e l i h o o d of success i n computation, the suggestion of causal influence must be rejected* It cannot be i n f e r r e d from the data that the concomitance i s an i n d i c a t i o n that understandings insure better computation, or v i c e versa*  61  Suggestions f o r Further Study  As the study progressed, lack of s u f f i c i e n t researoh i n c e r t a i n areas became evident*  The following points are suggested as f i e l d s f o r  further i n v e s t i g a t i o n :  1.  I t immediately became apparent that there i s need f o r a c l e a r l y  defined statement concerning what constitutes a body of understandings i n arithmetic*  While volumes have been w r i t t e n on understandings* there  seems t o be reason f o r Van Engen's  1  statement that*  Judged by i t s c r u c i a l importance i n determining methods of i n s t r u c t i o n * currioulum content* and supervisory praotices* the precise nature of meaning has received r e l a t i v e l y l i t t l e a t t e n t i o n i n the educational l i t e r a t u r e dealing with the outstanding problems of arithmetic i n the elementary schools* Failure to make more precise the nature of meaning i n arithmetic has r e s u l t e d i n confusion and controversy* This lack of a s p e c i f i c statement of the nature of understandings presented a d i f f i c u l t y t o the writer and points the way t o needed studies* 2*  A f t e r deciding upon the nature of the investigation* i t became  necessary to secure suitable t e s t i n g instruments*  It became apparent a t  once that s a t i s f a c t o r y t e s t s were not a v a i l a b l e and would have t o be constructed.  While there are many good t e s t s i n computation with deoimal  f r a c t i o n s * none could be found that e n t i r e l y met the s p e c i f i c a t i o n s demanded by the proposed study*  I t was found that* while a small beginning has  been made i n the measurement of understandings* there i s need f o r much more researoh i n t h i s area of evaluation*  1 H. Van Engen, "An Analysis of Meaning i n Arithmetic. I", The Elementary Sohool Journal, ZLIX (February* 1949), p. 321. .  62 3.  Having embarked upon, the task of constructing a t e s t on under-  standings, the w r i t e r was confronted with the problem posed by l a c k of adequate and v a l i d c r i t e r i o n measures. i n t e r n a l consistency techniques  Thus, i t was necessary t o employ  i n v a l i d a t i n g the t e s t items.  It i s  obvious that there i s great need f o r researoh i n t h i s area. 4.  Although much has been w r i t t e n about the d e s i r a b i l i t y of de-  veloping mathematical understandings, more studies are needed t o reveal the r e s u l t s which accrue from the use of the m A a - n ^ g theory of i n s t r u c t i o n . More investigations, based upon the evaluation of the outcomes of d i f f e r e n t methods of i n s t r u c t i o n , are needed t o indicate the r e s u l t s of meaningful instruction. 5.  Many more studies, similar t o the present one, should be con-  ducted i n other areas of arithmetic and at other educational l e v e l s t o provide data on the r e l a t i o n s h i p of computation and understandings. Only by d i l i g e n t a p p l i c a t i o n t o these relevant problems, can data be accumulated t o add t o the body of knowledge concerning arithmetic i n s t r u c t i o n and the place i n i t of understandings.  63  BIBLIOGRAPHY  Broom, M. E. Educational Measurements i n the Elementary Sohool. McGraw-Hill Book Company, Inc., 1939*  Hew  York:  Brownell, William A. "Psychological Considerations i n the Learning and the Teaching of Arithmetic", The Teaching of Arithmetio. Tenth Yearbook of the National Council of Teachers of Mathematics• New York: Teachers College, Columbia University, 1935* --——"The Evolution of Learning i n Arithmetio", Arithmetic i n General Education. Sixteenth Yearbook of the National Council of Teachers of Mathematics. New York: Teachers College, Columbia U n i v e r s i t y , 1941. _ - . _ » — "The Place of Meaning i n the Teaching of Arithmetic", Elementary School,Journal, XLYII (January, 1947), 256-265. -~—— "The Revolution i n Arithmetio", The Arithmetic Teacher, I (February, 1954), 1-5. Brueokner, Leo J . and Grossniokle, Foster E. Making Arithmetio Philadelphia: The John C. Winston Company, 1953.  Meaningful.  Buswell, G. T. "Methods of Studying P u p i l s ' Thinking i n Arithmetio", Arithmetic 1949. Supplementary Monographs, No. 70. Chicago: , U n i v e r s i t y of Chicago Press, 1949. 55-63. C a r l i l e , A. B. "An Examination of a Teacher-made Test", Educational Administration and Supervision, 40 ( A p r i l , 1954). Baltimore: Warwick & York, Inc. 212-218. Cronbach, Lee J . E s s e n t i a l s of Psychological T e s t i n g . & Brothers, Publishers, 1949.  New York:  Garrett, Henry E. S t a t i s t i c s i n Psychology and Education. Longmans, Green and Co., 1953.  New  Harper  York:  Glennon, Vincent J . "Testing Meanings i n Arithmetic", Arithmetio 1949. Supplementary Eduoational Monographs, No. 70. Chicago: U n i v e r s i t y of Chioago Press, 1949. 64-74. Greene, Harry A., Jorgensen, Albert N. and Gerberleh, J . Raymond "Measurement and Evaluation i n the Secondary Sohool. New York: Longmans, Green and Co.*, 1943. Grossniokle, Foster E. "Dilemmas Confronting the Teachers of Arithmetic", The Arithmetio Teacher. I (February, 1954), 12-15. Hawkes, Herbert E., Lindquist, E. F., Mann, C.R. The Construction and Use of Achievement Examinations. Boston: Houghton M i f f l i n Company, 1936.  64  BIBLIOGRAFHY--Continued Kilgour, Jean Alma. The E f f e c t of a Year's Teacher-Training Coarse on the Vancouver Normal Sohool Students' Understanding of Arithmetic. Unpublished Master's t h e s i s i n education. University of B r i t i s h Columbia, 1953* Lindquist, E. F. A F i r s t Course i n S t a t i s t i c s . Company, 1938. ~" .  Boston: Houghton M i f f l i n  Long, J . A. and Sandiford, P. The V a l i d a t i o n of Test Items. B u l l e t i n Ho. 3 of the Department of Educational Research. Toronto, Ontario: The Department of Educational Research, University of Toronto, 1935. McConnell, T. R. "Recent Trends i n Learning Theory: T h e i r A p p l i c a t i o n t o the Psychology of Arithmetic", Arithmetic i n General Education. Sixteenth Yearbook of the national Council of Teachers of Mathematics. Hew York: Teachers College, Columbia U n i v e r s i t y , 1941. Measurement of Understanding, The. The F o r t y - f i f t h Yearbook of the National Society f o r the Study of Education, Part I . Chicago: U n i v e r s i t y of Chicago Press, 1946. Morton, Robert Lee. Teaching Arithmetic i n the Elementary School Volume I I . New York: S i l v e r Burdett Company, 1938. — - — - Teaching Children A r i t h m e t i c 1955.  New York:  S i l v e r Burdett Company,  Orleans, Jacob S. and Wandt, Edwin. "The Understanding of Arithmetic Possessed by Teaohers", Elementary School Journal, L I I I (May, 1953), 501-507. . . . Remmers, H. H. and Gage, N. L. Educational Measurement and Evaluation. New York: Harper & Brothers, 1943. Ross, C. c. Measurement i n Today's Schools. In©«» 1941. -  New York: -  Spitzer, Herbert F. The Teaching of Arithmetic. Press, 1948.  Prentice-Hall,  Cambridge:  Stokes, C. Newton. Teaching the Meanings o f Arithmetic. Appleton-Century-Crofts, Inc., 1951.  The Riverside  New York:  Storm, W. B. "Arithmetical Meanings That Should be Tested", Arithmetic 1948. Supplementary Educational Monographs, No. 66. Chicago: University of Chicago, 1948. 26-31.  65  BIBLIOGRAFHJT-- Cont inued  Sueltz, Ben A, "Measuring the New Aspects of Functional Arithmetio", Elementary School Journal, XLVH (February, 1947), 323-330. Taylor, E. H. "Mathematics f o r a Four-Year Course f o r Teachers i n the Elementary School", School Science and Mathematics, XXXVIII (May, 1938), 499-503. Thorndike, Robert L. 1949.  Personnel S e l e c t i o n . New York:  Wiley & Sons, Inc.,  Tiegs, Ernest W. Tests and Measurements i n the Improvement of Learning. Boston: Houghton M i f f 1 i n Company, 1939. Van Engen, H. "An Analysis of Meaning i n Arithmetio. School Journal, XLIX (February, 1949), 321-329.  I",  Elementary  — — — "An Analysis of Meaning i n Arithmetio. I I " , Elementary School Journal, XLIX (March, 1949), 395-400. Weaver, J . Fred. "Some Areas of Misunderstanding About Meaning i n Arithmetic", Elementary School Journal, L I (September, 1950), 35-41. Weitzman, E l l i s and MoNamara, Walter J . Constructing Classroom Examinat i o n s . Chicago: Science Research Associates, 1949. Wingo, G. Max. "The Organization and Administration of the Arithmetio Program i n the Elementary School", Arithmetio 1948. Supplementary Educational Monographs, No. 66. Chicago: U n i v e r s i t y of Chicago Press, 1948. 68-79. Wren, F. Lynwood. "The Professional Preparation of Teachers of Arithmetic", Arithmetio 1948. Supplementary Educational Monographs, No. 66. Chicago: U n i v e r s i t y of Chicago Press, 1948. 80-90.  APPENDIX  A  TEST IN COMPUTATION WITH DECIMAL FRACTIONS  APPENDIX  A  SCORE DECIMAL  T i m e : 30 m i n u t e s  FRACTIONS  SCHOOL  NAME  GRADE  DATE P l a c e answers i n spaces a t t h e r i g h t ,  1.  .09  4  2. 3.  .06  +  .4  x  .2  $1.67 + $ 4 +  4.  $ x  .07  $ .03 f $ 1 2 . 0 0 - h $ 2 . 0 5  .78 56  4v 5.  1000 x  6.  Which i s l a r g e s t : 1.01;  7..  8.  19.62  101;  5.  11.001;  .012 x 2.45  .2.5) '  9.  6.5 .001 x  10. 4)  '  .010;  100.1  6._ 7..  8. 1.01  9«  . 0 0 1 -f 1  10,  .0104  11,  - 2 $15.67 from  12.  Subtract  13.  Prom  14.  6 9 . 7 4- 1 4 5 . 9 6 2 4-  9674.196  take  $4671  12,  362.8074  13,  .0346  4- 1 . 0 0 2 - f  18.11  14..  .61  15.  16.  ^ 0  15.  W r i t e as a d e c i m a l  1 7 #  .61  18.  .7 + 2/3  fraction:  f i f t e e n hundredths  16.  T0126  17._  18._  .9  19.  Find  20.  W r i t e i n words:  of  $3.15  19..  ( a s common f r a c t i o n )  .09  21.  Express  •  20.  as a m i x e d number:  21,  8.031 22.  F i n d the d i f f e r e n c e between . 0 4 4 and . 2 2 22,  23.  Divide to f i n d . t h e value o f : 3.69 12.3  24.  Express  25.  W r i t e as a m i x e d  as a common f r a c t i o n :  .0017  23..  24.  decimal:  s i x h u n d r e d t h i r t y and s e v e n t e e n  thousandths 25.  - 3 26.  F i n d the average o f :  95; 27.  103;  Divide  9.02  (answer 28.  90.5; by  I05f;  100  26.  1000 27.  I n decimal form)  W r i t e i n words: (as mixed number) 107.029  29.  ( g i v e answer t o t h e ) (nearest hundredth )  28.  Which i s l a r g e s t :  .1764;  .2;  .199;  .003;  .21' 29.  3°'  45}  98  (Correct to 2 places of decimals) 30.  31.  What w i l l b e t h e c o s t o f 6-§- g a l l o n s o f gasoline at 40.4/ per gallon? (nearest cent)  31. 32.  Express t o the nearest hundredth:  (as a decimal)  0.106 33*  E x p r e s s as a d e c i m a l f r a c t i o n c o r r e c t nearest hundredth:  6/7 34.  t o the.  33  v  Express to the nearest thousandth(as a decimal)  1706.17428 35  32.  $57.00 -r 6 0 ^  34. 35  APPENDIX  B  TEST IN UNDERSTANDING OF PROCESSES WITH.DECIMAL FRACTIONS  APPENDIX  UNDERSTANDING  B  OF PROCESSES WITH DECIMAL FRACTIONS  SCHOOL  NAME DATE  '  CHOOSE THE MOST SUITABLE ANSWER FOR EACH QUESTION. 1.  I n a d d i t i o n of mixed decimal f r a c t i o n s i t i s important to a r r a n g e t h e numbers so t h a t : A. B. C. D.  2.  To change a common f r a c t i o n t o a d e c i m a l f r a c t i o n one must know t h a t a common f r a c t i o n I n d i c a t e s : A. B. C. D. E.  3.  6.00  has a v a l u e o f :  I s moved  two p l a c e s t o t h e l e f t  o n e - t e n t h as l a r g e t e n t i m e s as l a r g e o n e - h u n d r e d t h as l a r g e one h u n d r e d t i m e s a s l a r g e .0170  s h o u l d be r e a d :  seventeen hundredths One h u n d r e d s e v e n t y ten-thousandths one h u n d r e d s e v e n t y t h o u s a n d t h s seventeen thousandths  Changing A. B. C. D.  be t h e one -  6 hundreds 600 h u n d r e d s 6 hundredths 600 h u n d r e d t h s  The number A. B. C. D.  8.  10 t i m e s a s much 1/10 a s much 1 0 more the value  tenths place hundredths place thousandths p l a c e any p l a c e  I f a decimal point number b e c o m e s : A. B. .C. D.  7.  makes t h e v a l u e makes t h e v a l u e makes t h e v a l u e does n o t change  The number: A. B. C. D.  6.  a z e r o t o t h e end o f a d e c i m a l f r a c t i o n :  The l a r g e s t o f s e v e r a l d e c i m a l f r a c t i o n s w i l l w i t h the l a r g e s t f i g u r e i n : A. B. C. D. .  5.  multiplication enumeration addition division. subtraction  Adding A. B. C. D.  4.  t h e l a s t f i g u r e s o f a l l numbers a r e . i n t h e same c o l u m n a l l f i g u r e s w i t h t h e same p l a c e v a l u e a r e i n t h e same c o l u m n t h e f i r s t f i g u r e s o f a l l numbers a r e i n t h e same c o l u m n none o f t h e s e  .645 t o  .0645:  does n o t c h a n g e t h e v a l u e makes v a l u e 10 t i m e s a s much makes v a l u e 1/10 as much makes v a l u e 1/100 a s much  the .  - 2 ~ 9.  I f t h e number 42.56 i s changed t o 42.056, by i n s e r t i n g a z e r o a f t e r the d e c i m a l p o i n t , t h e v a l u e becomes: A. B. C. D. E.  10.  The v a l u e o f a d e c i m a l f r a c t i o n i s d e t e r m i n e d A. B. C. D.  11,  unchanged less greater t e n times g r e a t e r o n e - t e n t h a s much  t h e s i z e o f the f i r s t d i g i t a f t e r t h e d e c i m a l p o i n t the .position of the l a s t d i g i t a f t e r the decimal p o i n t the p o s i t i o n of the largest d i g i t a f t e r the decimal p o i n t the p o s i t i o n of t h e f i r s t d i g i t , not i n c l u d i n g zeros, a f t e r the decimal point.  Which of the f o l l o w i n g thousandths p l a c e :  A. . 4695. 5417 B. 6495.1724 C. 4325.2163 D. 4175.6000 • 12,  D. 13.  than  l  the the the the  number number number number  becomes 1 0 0 0 t i m e s a s l a r g e I s I n c r e a s e d b y 1000 becomes 1/1000 as l a r g e i s d e c r e a s e d b y 1000  one h u n d r e d t i m e s as g r e a t t e n times as g r e a t one h u n d r e d t h as g r e a t one t e n t h a s g r e a t unchanged  every f i g u r e t o the r i g h t o f the d e c i m a l p o i n t every f i g u r e , except z e r o s , to the r i g h t of the p o i n t e v e r y z e r o t o the r i g h t o f the p o i n t none o f t h e s e  Multiplying A. B. C. D.  .7.  16 I s more t h a n . 5 i t i s t h e same as m u l t i p l y i n g b y d i v i d i n g a number a l w a y s g i v e s a n a n s w e r l a r g e r t h e number I t i s t h e same as f i n d i n g how many |r s i n 16  than  When a d e c i m a l f r a c t i o n i s c h a n g e d t o a common f r a c t i o n ( n o t r e d u c e d ) t h e d e n o m i n a t o r w i l l h a v e one z e r o f o r : A. B. C. D.  .6.  the answer i s l a r g e r  I n t h e q u e s t i o n : 1.6) 6 2 0 . 5 4 i f the decimal point i s moved one p l a c e t o t h e r i g h t i n t h e d i v i s o r and one p l a c e t o t h e l e f t I n t h e d i v i d e n d , t h e a n s w e r w i l l be-: A. B. C. D. E.  15.  32 T6~  "6" i n t h e  I f 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 by 1000 the d e c i m a l p o i n t i s moved t h r e e p l a c e s t o t h e l e f t b e c a u s e : A. B. C. D.  14.  numbers h a s t h e f i g u r e  In the question .57 t h e number d i v i d e d b e c a u s e : A. B. C.  by:  a d e c i m a l b y 1 0 0 0 moves t h e d e c i m a l p o i n t  two p l a c e s t o t h e r i g h t three places to the l e f t two p l a c e s t o t h e l e f t three places t o the r i g h t  I n d i v i s i o n w i t h d e c i m a l s t h e d i v i s o r may be made a w h o l e number b e f o r e d i v i d i n g b e c a u s e : A. y o u c a n ' t d i v i d e by a d e c i m a l B. m o v i n g t h e p o i n t does n o t change t h e v a l u e o f a number C. I t i s more c o n v e n i e n t D. t h e p o i n t I n t h e q u o t i e n t must be d i r e c t l y a b o v e t h e p o i n t i n the d i v i d e n d E. t h e v a l u e o f a f r a c t i o n i s u n c h a n g e d when b o t h t e r m s a r e m u l t i p l i e d b y t h e same q u a n t i t y .  c  - 3~ 18.  19.  1.050  The  measurement  inches  A. B. C. D.  t e n t h .of a n i n c h h u n d r e d t h o f an i n c h t h o u s a n d t h o f an i n c h t e n t h o u s a n d t h o f an I n c h  M o v i n g a d e c i m a l p o i n t two effect as:  i s accurate to the n e a r e s t :  p l a c e s t o t h e r i g h t has  the  same  m u l t i p l y i n g t h e number b y 1 0 m u l t i p l y i n g t h e number by 1 0 0 0 d i v i d i n g t h e number b y 1 0 0 none o f t h e s e (a) (b) I n t h e number: 5 5 5.5 5  A. B. C. D. 20..  21.  22.  Digit Digit Digit Digit  The  number  A. B. C. D. E.  .69 hundredths 2 hundredths 9 hundredths 69. h u n d r e d t h s 692 h u n d r e d t h s  a v a l u e of about:  l t 2  i f the decimal p o i n t i n b o t h numbers the  t e n t i m e s as l a r g e o n e - t e n t h as l a r g e one h u n d r e d t i m e s as l a r g e o n e - h u n d r e d t h as l a r g e unchanged  I f no  A. B. C. D.  nearest  one p l a c e a f t e r t h e d e c i m a l p o i n t two p l a c e s a f t e r the. d e c i m a l p o i n t t h r e e p l a c e s a f t e r the d e c i m a l p o i n t four places a f t e r the decimal point  z e r o s a r e added t o t h e d i v i d e n d , t h e a n s w e r t o  question: 4.2) decimal because:  25.  has  I n the q u e s t i o n : 5l 64273 were l o c a t e d one p l a c e t o t h e r i g h t answer would be: A. B. C. D. E.  24.  .6925  I f a number i s t o be e x p r e s s e d a c c u r a t e l y t o t h e h u n d r e d t h I t m u s t be f o u n d t o a t l e a s t : A. B. C. D.  23..  (a) (a) (a) (a)  i s 100 times d i g i t (b) Is 1 0 t i m e s d i g i t (b) i s 1/10 o f d i g i t (b) i s 1/10.0 o f d i g i t (b)  A. B. C. D.  69-735  will  be a  the  two-place  thousandths d i v i d e d by t e n t h s i s h u n d r e d t h s t h e r e a r e two f i g u r e s i n t h e d i v i s o r tenths times tenths i s hundredths t h e r e a r e two p l a c e s b e f o r e t h e p o i n t I n t h e d i v i d e n d  I n the q u e s t i o n : 6.42 x 15.7 i f the d e c i m a l p o i n t were located one p l a c e t o t h e r i g h t i n t h e f i r s t number and two p l a c e s t o t h e l e f t i n t h e s e c o n d number t h e a n s w e r w o u l d A. B. C. D.  ten one one one  t i m e s as l a r g e t e n t h as l a r g e h u n d r e d t i m e s as l a r g e h u n d r e d t h as l a r g e  be:  _ 4 26.  In the q u e s t i o n : 6 . 9 2 x 7 4 . 3 = 514.156 the decimal p o i n t i s l o c a t e d thus i n the answer because: A. B. C. D.  27.  A "decimal" i s a f r a c t i o n w i t h an u n w r i t t e n , but understood denominator which w i l l always be:  A.  B. C. D. E. 28.  2.134  has a v a l u e of about:  1 tenth 1 3 tenths 2 1 tenths 2 1 3 tenths 2 . 1 tenths  To change a f r a c t i o n , such as 3 / 4 , to a two-place decimal we d i v i d e the numerator by the denominator and we must t h i n k of the numerator a s : A. B. C. D. E.  30.  one ten any m u l t i p l e of ten any power of t e n none of these  The number: A. B. C. D. E.  29.  one and two are three hundredths times tenths i s thousandths tens times hundreds i s thousands there are three p l a c e s to the l e f t of the p o i n t i n the numbers m u l t i p l i e d .  3 hundreds 3 hundredths 300 hundredths 30 hundredths none of these  The sum o f : 16.17", 459=4", 142.167", 2.130" w i l l be a c c u r a t e to the n e a r e s t : A. B. C. D.  inch tenth i n c h hundredth i n c h thousandth i n c h  

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