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The effect of a year's teacher-training course on the Vancouver Normal School students' understanding… Kilgour, Alma Jean 1953

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THE E F F E C T  OF A Y E A R ' S TEACHER-TRAINING  COURSE  ON T H E V A N C O U V E R NORMAL S C H O O L S T U D E N T S ' UNDERSTANDING OF  ARITHMETIC  by  ALMA J E A N  A THESIS  SUBMITTED IN  KILGOUR  PARTIAL  FULFILMENT  OF  T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F  MASTER O F ARTS  in  the  Department of  EDUCATION  We a c c e p t standard  this  thesis  required degree  from  as  conforming  candidates  o£, MASTER O F  Members  THE UNIVERSITY  of  the  1953  for  the the  ARTS.  OF BRITISH  October,  to  Department  COLUMBIA  of  Education  ABSTRACT  THE EFFECT OF A YEAR'S TEACHER-TRAINING COURSE ON THE VANCOUVER NORMAL SCHOOL STUDENTS' UNDERSTANDING OF ARITHMETIC  The meaning theory of teaching arithmetic requires that those who do the teaching understand the mathematical bases of arithmetic.  This study was concerned with deter-  mining the extent to which one teacher-training institution was successful in raising the level of understanding of arithmetic on the part of i t s students during i t s usual year's programme. The 280 testees were students of the Vancouver Normal School.  The data were obtained through the admini-  stration of Glennon's Test of Basic Mathematical  Understandings  at the beginning and at the end of the school year. The analysis of the data led to the following conclusions : 1.  The teacher-training programme was effective in  bringing about small but significant gains in the students' achievement of the basic mathematical understandings contained in Glennon's Test.  The testees knew an average of 61.5 per  cent of the understandings at the beginning of the study and an average of 66 per cent of them at the end of the study.  2  2. of  notation  to  the  lated the  The u n d e r s t a n d i n g s r e l a t e d and to  students to  on b o t h  of  decimal  fairly  test  of  in  matical  quarter  of  difficulty  Superior  the  number  standing  which  rationale the  made  integers  four  of  known  understandings and p r o c e s s e s , well  for  the  in  the  areas  and p r o c e s s e s , computation.  areas  and p r o c e s s e s  for  at  took  of  attained  5-choice  was  for  quarter  5.  it  system  were w e l l  considerably less  reand  known  of  of  five  The  arithmetic.  remained  areas  of  gains The gains.  arithmetic  essentially  the  the  fractions  showed no s i g n i f i c a n t  gains  in  achievement  u n d e r s t a n d i n g s w e r e made b y  those  by t h e  decimal  in-  same f o r  the  retest.  lowest  than  the  Glennon's Test  and  with  and  equal  4.  the  those  fractions  g a i n s were  notation,  on d e c i m a l s  order  cluded  whereas  c o m p u t a t i o n were  system o f  section The  and p r o c e s s e s  and p r o c e s s e s ,  Significant  processes,  were  tests  the  tests. 3.  and  integers  on both  decimals  rationale  the  to  cases The  this  by in  test  items of  the  for  actual  the  in  items this  tended  to  group o f  6.  83  the  initial  basic  who w e r e  test  in  mathe-  as  the  in  compared  highest  test.  choices per  beginning of  the  students  who w e r e  initial  g r o u p was h i g h e r  per  the  students  the  the  place  cases  the  of  be  testees.  item at  3.5-choice  the  the  As  level  end o f  rather  determined of  the  underyear  than  year.  cent  of  the  rejection  was made  in  favour  of  the  of  the  correct  misleads answers  to  3 the items whereas 17 per cent o f the r e j e c t i o n was made i n favour o f the incorrect a l t e r n a t i v e s .  The l a t t e r finding  indicates that the change (perhaps gain) which takes place i n r e l a t i o n to certain items i s incompletely assessed by the usual s t a t i s t i c a l 7.  procedures.  The mean of the gross changes involved i n the  s h i f t of responses to and from the correct answers i n the t e s t - r e t e s t s i t u a t i o n averaged s i x times the mean of the d i f ferences between the net changes i n the responses.  It i s  evident that minor changes are considered i n present methods of estimating r e l i a b i l i t y , whereas the major changes are obscured and so ignored.  In spite of the apparent inconsistency  of the responses of the group to the test items i n t h i s study, r e l i a b i l i t y was .94, i n d i c a t i n g that further research i s necessary i n the area of test  reliability.  ACKNOWLEDGMENTS  The Director British the of  of  the  of  staff  gratefully  indebted of  his  Education guidance  study.  students  to  of  Dr. at and  J . the  R.  Mcintosh,  University  helpful  The  cooperation  the  Vancouver  of  advice  of  the  Normal  during  members  School  is  acknowledged.  J .  in  allowing  be  used  in  for  this  and  The Vincent  is  School  Columbia,  progress the  author  writer  Glennon his the  feels  of  Test  particularly  Syracuse of  present  Basic study.  indebted  University Mathematical  for  his  to  Dr.  generosity  Understandings  to  T A B L E OF CONTENTS  Chapter I  Page THE  PROBLEM  Background of Related Points  of  9  and  11  complete  statement  of  the  problem  24  PROCEDURE of  the  testees  The  teacher-training  The  method  data  of  the  to  on  programme  the  analysis  incorrect  and on  its  .  of  42  five  the  Glennon's  . .  47  .  54  . . .  56  gains according to the students . . .  incorrect  BIBLIOGRAPHY A.  responses  45  SUMMARY AND I M P L I C A T I O N S  APPENDIX  the  parts  The d i s t r i b u t i o n o f t h e the i n i t i a l status o f The  analyzing  DATA  Glennon's Test  constituent  28  chapter  THE ANALYSIS OF THE Gains  27  s c o r i n g and o f  relating  Summary o f  IV  1  departure  Selection  III  problem  studies  Summary II  the  Test  responses  67  LIST OF TABLES Table I II III IV V VI VII VIII  IX X  XI  Page Comparison o f Mean Test and Retest Test as a Whole  Scores  48  Comparison o f Mean Test and Retest Scores Part A, the Decimal System o f N o t a t i o n . . .  49  Comparison o f Mean Test and Retest Part B, I n t e g e r s and Processes  Scores 50  Comparison o f Mean Test and Retest Part C, F r a c t i o n s and Processes  Scores  Comparison o f Mean Test and Retest Part D, Decimals and Processes  Scores  50 51  Comparison o f Mean Test and Retest Scores Part E, The R a t i o n a l e o f Computation . . . .  52  A Comparison o f t h e Gains ( i n Per Cent) f o r the F i v e S e c t i o n s o f Glennon's Test  52  . . .  A Comparison o f the Order o f D i f f i c u l t y f o r the F i v e Areas o f A r i t h m e t i c on the Test and Retest  53  Mean Test and Retest Scores f o r the Low Group and the High Group  55  A Summary o f the Number o f A c t u a l Choices Per Item f o r the E i g h t y Items i n Glennon's Test  57  The Acceptance-Rejection Patterns f o r the E i g h t y Items o f Glennon's Test as Determined by the Responses to t h e C o r r e c t Answers on the Two T e s t s  61  CHAPTER  'THE  PROBLEM  Background of  The well  be  called  concomitant particular from  the  metic  as  emergence the  Age o f  problems  in  problem with  newer  of  I  an  the  era  Problem  in  Meaning,  both  has  learning  which  this  in  brought  which with  the  Buswell's^" concise  is  might  it  and t e a c h i n g .  study  conception regarding  contained  education,  The  concerned  teaching  of  arises  arith-  statement:  The t e a c h i n g o f a r i t h m e t i c has been moving toward t h e d e v e l o p i n g o f an u n d e r s t a n d i n g o f number relations and o f t h e number s y s t e m a s c o n t r a s t e d w i t h thinking o f a r i t h m e t i c mainly i n terms o f separate combinations and o p e r a t i o n s . The o u t c o m e s o f t h i s e m p h a s i s go much b e y o n d t h e c o m p u t a t i o n a l e f f i c i e n c y w h i c h was a l m o s t the sole c r i t e r i o n of arithmetical a b i l i t y i n the 1920's. The c o n t r i b u t i o n s o f a m e a n i n g f u l a r i t h m e t i c t o q u a n t i t a t i v e t h i n k i n g have i m p l i c a t i o n s o f first-order importance i n a s o c i e t y such as o u r s .  A study arithmetic among the  of  shows t h a t  specialists  teaching  of  in  recent there this  arithmetic  publications is  almost  subject  in  the  universal  regarding  should take.  It  the is  field  of  agreement direction evident  that  Guy T . B u s w e l l , "The P s y c h o l o g y o f L e a r n i n g i n R e l a t i o n to the Teaching o f A r i t h m e t i c , "The T e a c h i n g o f A r i t h m e t i c , p. 154. F i f t i e t h Yearbook of the National S o c i e t y f o r the Study o f Education, Part 11. Chicago: U n i v e r s i t y of Chicago Press, 1951.  1  2 the major conflict over the relative merits of the meaning theory and the d r i l l theory, that arose when the newer ideas were coming to the fore, has been resolved i n favour of the meaning theory.  The agreement that exists on this funda-  mental issue extends to a recognition that the lag between theory and practice i s greater i n the f i e l d of arithmetic than i n other areas of education.  The extent of this lag  is reflected i n the vast amount of criticism that has been levelled at the results of the teaching of arithmetic. 2 Wingo  claims that the sources from which the criticism  comes give i t a j u s t i f i a b i l i t y which cannot be denied: Criticism of the arithmetic program of the American elementary school has also come from people in the profession who have spent much time and effort in studying the psychological foundations of the teaching of arithmetic, the problems involved i n methods of teaching and related fields, and problems of the place of arithmetic i n the curriculum. With few, i f any, exceptions, these investigators have found ground for dissatisfaction with the present status of arithmetic instruction. Administrators, supervisors, and teachers cannot dismiss this criticism l i g h t l y . It i s founded on sober, and sometimes alarming, fact. Stated i n simple language the educators are saying, "We know what should be done. We know i t i s not being done. Why not?"  The search for a solution leads eventually to the  G. Max Wingo, "The Organization and Administration of the Arithmetic Program i n the Elementary School", Arithmetic 1948, p. 69. Supplementary Educational Monographs, No. 66. Chicago: University of Chicago Press, 1948.  3  teachers themselves, for as Robinson^ says: Elementary teachers as a whole are f a i r l y proficient in the manipulation of the mechanical processes of arithmetic but are significantly lacking in their knowledge of the fundamental principles of arithmetic which underlie the mechanical processes. This group lacks mathematical insight when confronted with mathematical situations varying from those of the textbook type. The serious implications of this criticism l i e in the realization that changes in the understanding of learning can bring desirable changes in teaching only to the extent that those who do the teaching are themselves competent in the areas of learning considered to be of c r i t i c a l significance today.  Wingo^ blames the lack of  achievement of teachers in the understanding of arithmetic on the kind of teaching they themselves have received in the past and claims that this has so affected both their knowledge and attitudes that extensive reeducation i s necessary.  He affirms that the one essential i s that  teachers must be led to understand arithmetic through gain: ing fresh insight into the nature of number and that, until this i s accomplished, no improvement in techniques of teaching can be expected.  It i s obvious, then, from such  remarks as these, that attempts to improve methods of ^A. E. Robinson, "Training Elementary School Teachers in the Field of A r i t h m e t i c p p . 475-47o^ Special Survey StucFies, Vol. V. United States Office of Education Bulletin No. 10, 1933. 4  I b i d . , pp. 68-79.  4 t e a c h i n g w i l l be o f l i t t l e v a l u e u n t i l methods o f improving the t e a c h e r s ' own competencies  i n t h e understanding o f  a r i t h m e t i c have been d e v i s e d . A t t e n t i o n t o t h e problem  o f the t e a c h e r ' s own  understanding o f a r i t h m e t i c i s g i v e n i n the F o r t y - f i f t h Yearbook o f the N a t i o n a l S o c i e t y f o r the Study o f Education : O b v i o u s l y , i f t e a c h e r s are to measure (and t o teach) u n d e r s t a n d i n g s , they must f i r s t know what these unders t a n d i n g s a r e . They are not always equipped with t h i s knowledge. Sometimes, indeed, they themselves l a c k the e s s e n t i a l u n d e r s t a n d i n g s . Sometimes, on t h e o t h e r hand, they possess the understandings and a c t u a l l y use them i n t h e i r b e h a v i o r , but do not r e c o g n i z e them as such. Many primary-grade t e a c h e r s , f o r example, t h i n k t h a t the a b i l i t y to f u r n i s h " e i g h t " o r "8" i n response to the q u e s t i o n , "How many are f i v e and t h r e e ? " , i s a l l that i s i n v o l v e d i n knowing t h e f a c t , 5 + 3 8. And why not? Do they not, themselves, t h i n k " e i g h t " at once i n these s i t u a t i o n s , and do they not do so "without t h i n k i n g " , which is to say, without making use o f meani n g s and understandings? Such may seem t o be t h e case, but i t i s not t r u e to f a c t . P a r r o t s can l e a r n t o say " e i g h t " when asked, "How many are f i v e and t h r e e ? " . But p a r r o t s cannot do a r i t h m e t i c . Evidently parrots l a c k something, and t h a t something i s a s t o r e o f r e l a t e d meanings. a  Whether t e a c h e r s l a c k e s s e n t i a l understandings o r have them without being conscious o f the f a c t , the e v a l u a t i o n o f understandings must n e c e s s a r i l y s u f f e r . The f i r s t step..in c o r r e c t i n g t h e e v i l s o f t h e s i t u a t i o n i s t o i d e n t i f y those c r i t i c a l understandings which t e a c h e r s must possess and the presence (and absence) o f which they must be able t o r e c o g n i z e i n the behavior of t h e i r p u p i l s . The above statement  falls  short i n t h a t i t does  not g i v e s u f f i c i e n t emphasis to t h e need f o r and t h e means o f  The Yearbook Committee, "Next Steps", The Measurement o f Understanding, p. 323. F o r t y - f i f t h Yearbook o f the N a t i o n a l S o c i e t y f o r t h e Study o f Education, Part 1. Chicago U n i v e r s i t y o f Chicago Press, 1946.  5 reeducating will is  be  to  teachers  known  be  in  grasp  the  mathematics  of  that  critical  states  the  that  theory  may  be  understandings  Corroboration  Second Report  they  meanings,  the  teachers.  P l a n s ^ which  sufficient  acquire  by t h e  found  Post-War  so t h a t  of  the  on t h i s  Commission on  teachers  s h o u l d have  and background o f  qualified  generalizations  to  point  help  a  elementary  their  pupils  and  appreciations  of  the  teacher's  under-  relationships. The standing the  of  lacking that  but  arithmetic  implementation  and t h a t  is  acknowledgment  such  also  with  serve  brought  to  to  bear  upon the  abilities. to  That the  one  the  the  key  the  Commission on  it  the  remedy not  is  the  only  type  is  of  in  in  the  of  The  action spotlight  to  teachers First  be  teachers  relation of  in  programme  Thus the  abilities  training  claimed  elements  c o n s i d e r e d to  situation.  teachers  own  arithmetic  determining  upon the of  essential  generally  preservice  situation  the  onus o f  training the  of  newer  understanding  brings  will  of  is  that  these holds  Report  of  7 Post-War  Plans:  How s h a l l a r i t h m e t i c b e i m p r o v e d ? By s t r i k i n g at the crux o f the problem—the education o f t e a c h e r s . . . . T h e r i g h t k i n d o f p r o f e s s i o n a l c o u r s e s s h o u l d b e made a v a i l a b l e at once to every p r o s p e c t i v e teacher o f the elementary school.  Plans",  "The S e c o n d R e p o r t o f t h e Mathematics Teacher. XXXVIII  Commission on (May, 1 9 4 5 ) ,  Postwar  195-220.  7 Plans",  "The F i r s t R e p o r t Mathematics Teacher,  of the XXXVII  Commission on (May, 1 9 4 4 ) ,  Postwar  230-231.  6 Confirmation teachers  i n the f i e l d  t h a t the p r e s e r v i c e t r a i n i n g of arithmetic requires  of  improvement  comes from 3 u s w e l l . , the present l a c k o f teacher p r e p a r a t i o n i n t h i s f i e l d i s a l a r m i n g . In no other major c u r r i c u l u m f i e l d are the o p p o r t u n i t i e s f o r s c h o l a r l y p r e p a r a t i o n so meager. The a r i t h m e t i c understandings of most t e a c h e r s o f a r i t h m e t i c r e s t on no content beyond t h a t covered i n t h e i r own eighth-grade course i n a r i t h m e t i c . High-school and c o l l e g e mathematics c o n t r i b u t e s l i t t l e o f a s p e c i f i c nature to the t e a c h e r o f a r i t h m e t i c . The t e a c h e r s ' c o l l e g e courses d e a l with methods but seldom go beyond the eighth-grade l e v e l i n mathematical content. The committee b e l i e v e s t h a t b e t t e r p r e p a r a t i o n o f t e a c h e r s o f a r i t h m e t i c i s a p r e r e q u i s i t e to i n c r e a s e d e f f e c t i v e n e s s o f the a r i t h m e t i c program. Such statements n e c e s s i t a t e some c l a r i f i c a t i o n t h i n k i n g about the purpose of t e a c h e r - t r a i n i n g courses. a teacher-training institution students who,  justified  of Is  i n r e f u s i n g to accept  w h i l e able to produce evidence o f having  met  the minimum academic requirements f o r entrance, are found to be l a c k i n g i n c e r t a i n areas of subject matter background, under the  c o n v i c t i o n that the work o f the  teacher-training  programme i s concerned w i t h the psychology of l e a r n i n g , with methods o f t e a c h i n g ,  and w i t h l i k e t o p i c s r a t h e r than with  the t e a c h i n g  o f s u b j e c t matter?  A r e f u s a l to admit those  students who  were found to ba l a c k i n g i n the  understanding  o f a r i t h m e t i c would e l i m i n a t e most o f the c a n d i d a t e s i f  Guy T. Buswell, " I n t r o d u c t i o n , The Teaching o f A r i t h m e t i c , p. 3. F i f t i e t h Yearbook of the N a t i o n a l S o c i e t y f o r the Study o f Education, Part I I . Chicago: University o f Chicago Press, 1951. n  7  Taylor  and M i l l s  Q  are  correct  in  saying:  M o s t s t u d e n t s who a r e p r e p a r i n g t o t e a c h arithm e t i c a r e h i g h - s c h o o l g r a d u a t e s who h a v e n o t s t u d i e d arithmetic s i n c e the e i g h t h g r a d e and who, c o n s e q u e n t l y , do n o t h a v e a n a d e q u a t e w o r k i n g k n o w l e d g e o f t h e p r o c e s s e s o f a r i t h m e t i c , much l e s s s u c h an u n d e r s t a n d i n g o f t h o s e p r o c e s s e s as i s n e c e s s a r y f o r good t e a c h i n g .  On t h e into  to  make  uphold  its  vital  lacking.  Such i s  Our  Times:  to  the  course  integrity  provision for  learning,  for  hand,  a teacher-training  wishes to  other  the  acceptance  compels the as  position  of  which  taken  such  students  institution  "educational"  development  good t e a c h i n g , the  an  of  in  those  institution  areas  are  found  the  book  which  of  to  be  Teachers  1 0  A c o l l e g e may p r o p e r l y d e c l i n e t o u n d e r t a k e the p r e p a r a t i o n o f an i n d i v i d u a l f o r t e a c h i n g on t h e g r o u n d t h a t he h a s c e r t a i n w e a k n e s s e s w h i c h a r e irremediable or which the i n s t i t u t i o n , at any r a t e , i s not p r e p a r e d to remedy. But i f i t a c c e p t s and r e t a i n s him i t has the o b l i g a t i o n to p r o v i d e him w i t h t h o s e e d u c a t i o n a l e x p e r i e n c e s b e s t c a l c u l a t e d to d e v e l o p i n him the q u a l i t i e s t h a t w i l l make h i m p r o f e s s i o n a l l y c o m p e t e n t .  The is  educational  concerned are  understandings^ limited  to  this  ^E. H . Teacher-training a n d C o . , 1949.  those of one  related  arithmetic, area.  to  American Washington: "^Glennon's  with  the  which  basic  so t h e  A realistic  T a y l o r and C . N. C l a s s e s , p. i i i .  1 0  Times.  experiences  this  mathematical  discussion will view  of  the  M i l l s , Arithmetic New Y o r k : Henry  C o u n c i l on E d u c a t i o n , T e a c h e r s A m e r i c a n C o u n c i l on E d u c a t i o n , terminology  is  used  study  here.  be  situation,  for Holt  f o r Our i'944'  3 then, nize of  requires the  fact  that that  its  basic mathematical  determine  the  beginning  of  status the  to  evaluate  secured  by  the  end o f  the  an  of  this  is  stated  for  of  its  area  of  kind  the  its  procedure  of  in  Study  the  to is  in  learning in  be  students  programme  measurement  significance  to  may  understandings,  so o b t a i n e d  general its  students  teacher-training  information This  teacher-training  teacher-training  proceed  but  the  the  well  this that  past.  in  the  Forty-fifth  of  Education:  it  this  the  lies  steps  in  of  it  little  of  the  has  to  the  then  the  occurs  its  a beginning  at  it  use  findings at  the  the  accordingly.  evaluation  received  Yearbook  achievement  that  programme in  recog-  regard  light  and t h a t  its  in take  change t h a t  study  That  that in  known  has  lacking  programme,  course, adjust  institution  practices  application attention been  made  National  Society  1 2  The measurement o f meanings and u n d e r s t a n d i n g s i s beginning to creep into research i n arithmetic. Of t w e n t y - s e v e n s t u d i e s e x a m i n e d , e i g h t showed t h a t the a u t h o r was d e l i b e r a t e l y t r y i n g t o m e a s u r e b e y o n d t h e traditional scope o f c o m p u t a t i o n s and problem s o l v i n g . As m i g h t of  the  literature  problem metic  at  of  be  expected  revealed  from  only  two  measuring understanding  the  teacher-training  this  statement,  studies in  the  related  field  of  a to  search the  arith-  level.  12 B e n A . S u e l t z , H . B o y n t o n , a n d I. S a u b l e , "The Measurement o f U n d e r s t a n d i n g i n E l e m e n t a r y - S c h o o l M a t h e m a t i c s " , T h e M e a s u r e m e n t o f U n d e r s t a n d i n g , p . 156. Forty-fifth Yearb o ok~oT~tTTe"~Nlitional~lk7cTeTy for t h e S t u d y o f E d u c a t i o n , P a r t I. Chicago: U n i v e r s i t y o f C h i c a g o P r e s s , 19A-6. —  9 Related  A study  1. prospective  teachers  reported  1938  the  in  Eastern  about  the  by  for  designed to  evaluate  the  concerning arithmetical Taylor. -*  He  1  Illinois  need  Studies  State  tested  Teachers  carrying  on  333  in  of  meanings  was  freshmen  at  College.  research  knowledge  His  this  comments  area  are  pertinent: T h e r e i s much s t a t i s t i c a l e v i d e n c e a s w e l l a s g e n e r a l agreement t h a t h i g h s c h o o l graduates have n e i t h e r reasonable s k i l l nor a c c u r a c y i n the fundamental operations of arithmetic. That they are i g n o r a n t of the meanings o f a r i t h m e t i c a l concepts, operations, and symbols has r e c e i v e d l e s s a t t e n t i o n , although this l a c k o f u n d e r s t a n d i n g i s f u l l y as d i s a s t r o u s to good t e a c h i n g a s l a c k o f s k i l l , a n d much m o r e d i f f i c u l t to remedy.  Taylor by  giving  found  illustrates  some r e s u l t s  of  the  of  the  students  represented 34%  of  the  74%  Taylor  arithmetic  of the and i n  by  given  not  of  meanings"  to  his  group.  know  the  numbers  digits  marked  section  He  of  j+ the  2, 3,  and  ^*  ^ true  | |  4  2304,  in on  a  test,  s t u d e n t s m a r k e d $12 -f $4 = 3 i n c o r r e c t , most c a s e s changed the q u o t i e n t t o $3.  concluded that until  did the  students  true-false  of  test  "ignorance  that 36%  While  this  students  "there  cannot  preparing  to  be  good  teach  get  teaching rid  of  13 ^ E . H. T a y l o r , "Mathematics f o r a Four-Year f o r Teachers i n the Elementary School", School Science M a t h e m a t i c s , X X X V I I I ( M a y , 1938), 499-503.  Course and  10 such e r r o r s as t h e s e " , h i s study d i d not i n c l u d e an e v a l u a t i o n o f the t e a c h e r - t r a i n i n g course to determine the extent to which i t was  a b l e t o h e l p the s t u d e n t s overcome these  deficiencies. 2. of  In 1948  Glennon ^" r e p o r t e d an e x t e n s i v e study 1  "the growth and mastery  of c e r t a i n b a s i c  understandings on seven e d u c a t i o n a l l e v e l s " . Basic Mathematical U n d e r s t a n d i n g s w a s used to gather data.  The 1,139  the f o l l o w i n g l e v e l s :  mathematical A "Test o f  d e v i s e d and  was  t e s t e e s were d i s t r i b u t e d  on  p u p i l s i n the seventh, e i g h t h , n i n t h  and t w e l f t h grades, t e a c h e r s - c o l l e g e freshmen, t e a c h e r s c o l l e g e s e n i o r s and Glennon  teachers-in-service.  a r r i v e d at c e r t a i n c o n c l u s i o n s which have  s i g n i f i c a n c e f o r the p r e s e n t study: A course i n the psychology and t e a c h i n g o f a r i t h metic d i d not b r i n g about growth i n understanding o f b a s i c mathematics on the part o f t e a c h e r s c o l l e g e s e n i o r s . The t e a c h e r s c o l l e g e freshmen t e s t e d have mastered an average of 44 per cent o f the understandings b a s i c t o computational processes taught i n grades one through six. The t e a c h e r s c o l l e g e s e n i o r s t e s t e d have mastered an average of 43 per cent o f the understandings b a s i c to the computational p r o c e s s e s taught i n grades one through s i x .  ^Vincent J . Glennon, "A Study o f the Growth and Mastery o f C e r t a i n Basic Mathematical Understandings on Seven E d u c a t i o n a l L e v e l s " . Unpublished Doctor's d i s s e r t a t i o n . Cambridge, Massachusetts: Graduate School o f Education, Harvard U n i v e r s i t y , 1948. See Appendix A.  11  Glennon examined many t e s t s and found t h a t none o f them was s u i t a b l e f o r measuring b a s i c mathematical understandings. the  Consequently, he found i t necessary  "Test o f Basic Mathematical Understandings".  to c o n s t r u c t Because  t h i s t e s t was c a r e f u l l y c o n s t r u c t e d and v a l i d a t e d i t was decided t o use i t f o r g a t h e r i n g data i n the present It  should be noted t h a t t h i s study  study  study.  i s indebted to Glennon's  f o r a g e n e r a l r e s e a r c h p l a n and, i n p a r t i c u l a r , f o r a  measuring d e v i c e , which, f o r the sake o f b r e v i t y , w i l l be c a l l e d Glennon's Test throughout the remainder o f t h i s r e p o r t . The  p o i n t s o f s i m i l a r i t y between the two s t u d i e s having  noted, the p o i n t s o f departure  been  w i l l now be e x p l a i n e d .  P o i n t s o f Departure Glennon found no g a i n i n achievement o f b a s i c mathematical understandings at t h e end o f the f o u r - y e a r teacher-training period. obtained  H i s f i n d i n g s were based on data  by t e s t i n g a group o f t e a c h e r s - c o l l e g e freshmen at  the beginning  o f the year and by t e s t i n g a group o f t e a c h e r s -  c o l l e g e s e n i o r s at the end o f the f o u r - y e a r t e a c h e r - t r a i n i n g period.  Since t h e r e was no means o f determining  the i n i t i a l  s t a t u s o f the group o f s e n i o r s on the t e s t m a t e r i a l at the beginning  o f t h e i r freshman year i t i s p o s s i b l e that the  apparent l a c k o f g a i n was due to i n i t i a l d i f f e r e n c e s i n the groups s t u d i e d , so i t appears unwise to draw c o n c l u s i o n s on data obtained  i n t h i s manner.  12  The found  first  p o i n t o f d e p a r t u r e , t h e r e f o r e , i s t o be  i n t h e method o f c o m p i l i n g d a t a .  used throughout  the study.  One g r o u p w i l l  The d a t a w i l l  be  be o b t a i n e d b y  t e s t i n g t h e same g r o u p a t t h e b e g i n n i n g a n d a t t h e e n d o f t h e t e a c h e r - t r a i n i n g programme. institution, this  One  teacher-training  t h e Vancouver Normal S c h o o l , w i l l  study but since t h i s i n s t i t u t i o n t r a i n s  t w o - t h i r d s o f t h o s e who p r e p a r e ,  be u s e d i n approximately  e a c h y e a r , t o become e l e m e n -  tary teachers i n the province of B r i t i s h  Columbia,  the  s a m p l e may be c o n s i d e r e d s a t i s f a c t o r i l y r e p r e s e n t a t i v e o f t h e elementary of  student teachers o f t h i s  province.  The  testing  one g r o u p a s d e s c r i b e d above i s f a c i l i t a t e d b y t h e r e l a -  tively  s h o r t t r a i n i n g p e r i o d o f one s c h o o l y e a r ' s d u r a t i o n . The  purpose o f t h e t e s t i n g procedure  may now be  s t a t e d i n q u e s t i o n -form: 1.  To what e x t e n t do V a n c o u v e r N o r m a l  students possess  the b a s i c mathematical  t a i n e d i n Glennon's  understandings  con-  Test  a.  at the beginning of the t e a c h e r - t r a i n i n g  b.  a t t h e end o f t h e t e a c h e r - t r a i n i n g 2.  School  I s t h e r e any s i g n i f i c a n t  ment o f t h e b a s i c m a t h e m a t i c a l  year?  year?  difference i n achieve-  understandings  contained i n  Glennon's Test between t h e Vancouver Normal S c h o o l  students  t e s t e d a t t h e b e g i n n i n g o f t h e s c h o o l y e a r a n d t h e same s t u d e n t s t e s t e d a t t h e end o f t h e s c h o o l  year?  13 The from  a  difference  Glennon's vey  of  study  second point  study  was  the an  is  on t h e  amount  The of  an  about  the  amount  of of  teacher-training  the  total  period,  of  of  the  separate  determined  the  items  of  the  of  relative  tested,  machinery It  is  arithmetic  number  teaching  the  of  will  to  at  one  group  not  changes take  at  sur-  only  end o f  the place.  information  the  end o f  requirements  areas  a  present  bear  the  give  occurs  courses  tested  test  but  in  the  leading  to  necessitate  to  determine  doing  items  are  did  which  of  these,  not  units.  so  in  included in  each  make-up  so t h a t These are  of  the  areas as  he  of  exception, he  likely  did  areas  However,  the  separated.  would be  five  the  these  Neither  one  of  group  they  the  with  conveniently arranged tested  difficulty  programme.  difficulty  teaching  for  relative  test  units  although  considered as  itself.  place  on what  the  a teacher-training  vide  brought  teacher-training  in  be  of  diagnostic  to  well  study  that  the  arithmetic  concerned with  intensive is  arises  studies.  the  scores  separate  two  study  while  also  to  determine  one,  takes  according appear  the  this  changes o c c u r r e d .  Glennon eighty  but  in  levels  then,  change  adjustment  evaluation  wherein  focus,  period  analysis  possible  a more  change t h a t  teacher-training While  purposes of  successive  concerned with  level.  departure  extensive  seven groups on  on one  the  in  of  did the  five and  follows:  might protest areas  the  The d e c i m a l s y s t e m o f n o t a t i o n Basic understandings of integers and p r o c e s s e s Basic understandings of fractions and p r o c e s s e s Basic understandings of decimals and p r o c e s s e s Basic understandings of the rationale of computation  1. 2. 3. 4. 5.  It  is  apparent  satisfactory not of  lend this  to  itself  area  analysis the  of  of  of  that  procedure. item w i l l  partial  It  could  the  However, be  then  scores intended  to  by  extent  areas  the  of  are  the  teacher-training  there  any  significant  understandings contained  School Students  areas the  of  15  items  does  the  purposes  form  a in  answers  at  What  of  basic  each  same  beginning students  period?  period?  of  at  the  in  five  between  of  Glennon's  students  difference  Glennon's Test the  in  teacher-training  to  basic  the  the  achieveareas  Vancouver  teacher-  end o f  the  period? is  arithmetic  testing  in  and the  teacher-training 3.  items  included  School  end o f  Is  the  Normal  the  period  20  contained  at  training  items  understandings  arithmetic  Vancouver of  the  b.  arithmetic  15  questions:  beginning  the  items  provide  the  of  for  be  15  item  assumed t o  will  items  constitute  fifth  at  Normal  by  items  a.  2. ment  four  but  arithmetic.  five  possessed  units  fifth  To w h a t  the  first  this  the  following  each  Test  to  of  1. to  the  teaching  study,  separate the  that  15  the  order  contained  the  in  Vancouver  of  difficulty  of  Glennon's Test Normal  School  the as  five determined  students  15 a.  at  the  beginning of  b.  at  the  end o f  A third a  in  of  the  the  basic  quarter  amount  of  from  of  the  gain  cases  in  quarter  of  be  drawn  on t h e be  from  major  such  of  an  analysis  gain  unequal.  distributed  in  low  achievement  initial  those  who  such  in  such  a way  to  have  the  of  the  least  A given as  middle  a given  initially value  to  to  place  high  in  in  the  ammunition  those  high  held  who  gain  neither value gain  that be  students test  basic  the  mathe-  highest  will  be  equal that  gain  based  a  may  given it  is  relatively  more  than  achievement is  to  gains  showed  that  in  distributed be  education. is  and  conclusions  group w i l l to  achieve-  e d u c a t i o n when  initial  of  group w i l l  argument  affirmation  achievement  to  by  involves  considered In  a  distributed  considered to  similar  in  favour  have  the  education.  A powerful lies  of  in  understanding  in  of  data  be  in  initial  The  the  will  amount  favour  amount  of  that  in  the  test.  value  showed r e l a t i v e l y  understanding.  fashion,  a way  study  gain  statistically  It  has g r e a t e s t  in  students  initial  assumption that  educationally  amount  the  of  achievement  u n d e r s t a n d i n g s made b y in  original  amount  period?  period?  u n d e r s t a n d i n g s made  matical  cases  the  average  mathematical  lowest  average  the  teacher-training  teacher-training  deviation  comparison between  ment  the  the  of  the  that  tested  necessary to  work  to  support  those  who  the  are  understandings out  their  above  assumption  relatively possess  own s a l v a t i o n  high  all  the  in  new  16 situations, of  while  understandings  small  gains  portion  to  unique with  their  of  determining  understandings incorrect  this  be  in  the  are  such  actual  relatively  precarious of  point  of  analyzing  in  achievement  positions that  stability  departure  data.  out  of  an  evaluation the  particular literature,  is  all  the  even  pro-  of  of  the  attention  from  other  a  concerned of  redistribution  situation.  incorrect  although  is  in  achievement  test-retest  use  centred  T h i s method  p o s s i b l e changes i n  recieved  low  size.  responses in  of  have  in  through  instance  scores  who  a measure  fourth  method  found  are  afford  The  of  those  No  responses could  uses of  statisticians  error and  re-  16 searchers.  Guilford  has  this  to  say:  R e c e n t e x p e r i e n c e s show t h a t e r r o r s c o r e s m i g h t w e l l be g i v e n much a t t e n t i o n a s s o u r c e s o f certain kinds of v a r i a n c e that i t i s worth our while to measure. Some A A F f i n d i n g s i n d i c a t e d t h a t a t r a i t o f c a r e f u l n e s s was q u i t e m e a s u r a b l e by u s i n g w r o n g s s c o r e s i n one o f s e v e r a l t e s t s , where t h e number o f r i g h t r e s p o n s e s u s u a l l y f a i l e d to measure i t . F r u c h t e r has more r e c e n t l y found by f a c t o r a n a l y z i n g r i g h t s s c o r e s and wrongs s c o r e s i n t h e same t e s t s t h a t w h i l e t h e two s c o r e s i n t h e s a m e t e s t may m e a s u r e t h e same f a c t o r s ( i n reverse), t h e y do s o t o d i f f e r e n t degrees. He a l s o f o u n d t h a t some f a c t o r s a r e m o r e m e a s u r a b l e b y [ t h e ] w r o n g s s c o r e than o t h e r s . In f a c t , i t i s possible that a certain k i n d o f r e a s o n i n g s h o u l d be m e a s u r e d b y e r r o r s rather t h a n by c o r r e c t s o l u t i o n s . These r e s u l t s have not been v e r i f i e d as y e t , but they are s u g g e s t i v e o f the rich p o s s i b i l i t i e s t h e r e may b e i n f u l l e r u s e a n d w e i g h t i n g o f wrong r e s p o n s e s .  Psychology Book C o . ,  J. P. G u i l f o r d , F u n d a m e n t a l S t a t i s t i c s i n a n d E d u c a t i o n , p . 537. New Y o r k : McGraw-Hill  19~50.  17 The  use o f i n c o r r e c t responses i n t h i s study i s  a s s o c i a t e d with a warning t h a t most s t a t i s t i c i a n s d i r e c t t o test-makers r e g a r d i n g  the c h a r a c t e r i s t i c s t h e m i s l e a d s  17 should have i n a m u l t i p l e - c h o i c e  type o f t e s t .  Guilford's  statement i s a r e p r e s e n t a t i v e one. I t might be p o i n t e d out, i n c i d e n t a l l y , t h a t when examinees are i g n o r a n t o f the answer t o an item, t h e i r h a b i t s o f t a k i n g t e s t s are such t h a t t h e y do not choose among the a l t e r n a t i v e s e n t i r e l y at random. Certain p o s i t i o n s i n a l i s t o f f i v e responses may be f a v o r e d by h a b i t s o f reading o r o f a t t e n t i o n . T h i s i s probably not s u f f i c i e n t l y important i n i t s e l f to overthrow the u s e f u l n e s s o f "chance" s c o r i n g formulas. In the long run, i f t h e p o s i t i o n o f the r i g h t answer i s randomized, the c o r r e c t i o n may work w e l l enough. More s e r i o u s , however, i s the f a c t t h a t many t e s t w r i t e r s , i n p r e p a r i n g f o u r - o r f i v e - c h o i c e items, do not provide "misleads" o r " d i s t r a c t o r s " t h a t are e q u a l l y a t t r a c t i v e . I t i s easy, perhaps, t o t h i n k o f one good wrong answer to an item, but t o t h i n k o f more than one and t o make a l l e q u a l l y a t t r a c t i v e i s a t r y i n g a r t . Many a f o u r o r f i v e - choice item reduces v i r t u a l l y to a t h r e e - o r two-choice item because o f t h i s f a c t . The a p r i o r i s c o r i n g formula as g i v e n above u n d e r c o r r e c t s . We do not know by how much. G u i l f o r d ' s expression  "when examinees are i g n o r a n t  o f the answer t o an i t e m " i m p l i e s an e i t h e r - o r dichotomy i n which an item he one  i s considered  t o be known to t h e examinee i f  chooses the c o r r e c t response and not known i f he s e l e c t s o f the m i s l e a d s .  Any such d i s c u s s i o n , which  overlooks  the p o s s i b i l i t y that an examinee may have some knowledge concerning  t h e item even i f t h i s knowledge i s not s u f f i c i e n t l y  ' J . P. G u i l f o r d , Fundamental S t a t i s t i c s i n Psychology and Education, p. 53^*. New York: McGraw-Hill Book Co., 1950.  18  great  to  permit  doubtful ing  value  for  him in  Brownell  all-or-none  to  select  relation and  to  the  correct  the  measurement  Sims t e l l  affairs,  us  that,  understandings  response, of  "Rather  vary  in  is  of  understandthan  being  degree  of  implications  of  18 definiteness statement the  for  the  possible  at  to  may  be  listed  able  may  In  to  to  completeness" the  testees  has  been  will,  of  — — — — — —  levels  the  reject  permit  the  selection  of  the  the in  of  vary  items  in  no  discrimination  with  of  the  the  degree  the  correct  definiteness  measure  the  of  under-  response. is  con-  and  testee a test  has  or  which  understandings.  testee's  from  the  item  is  resultant  of  of  single  one  alternatives  the  which  any  positions  "degree  a  test.  regard  understanding  to  in have  a  to  this  in  of  regard  individual  found  level  understandings  designed to  course,  that  levels  be  understanding for  certain  upper  to  to  intermediate  the  have  The  answers  At  refer  of  alternatives,  Definition—The sidered  levels  are  understanding  low  guessing.  certainty.  standing  of  The testing  certain  such  among t h e  resorting  with  of  levels  may b e g i n  testees  examinees'  selection  examinees'  item  completness".  multiple-choice  influence  on t h e i r The  and  item  to  level item.  of  understanding Obviously,  while  , M  W. A . B r o w n e l l , a n d V . M . S i m s , "The N a t u r e of U n d e r s t a n d i n g " , The Measurement o f U n d e r s t a n d i n g , p. 31. Forty-fifth Yearbook of the N a t i o n a l S o c i e t y f o r the Study o f E d u c a t i o n , P a r t I. Chicago: U n i v e r s i t y of Chicago P r e s s , 194-6.  19 the  number  of  the  actual  choices in  individuals  listed  and  for  standing  upon which  separate  items.  Guilford's  of  regarding  the  fully  they  This  test  constructed, of  testees  are  about  but  item.  the  in  by  such reductions  need  it  of  does  levels  so f a r are  as  in  not  of the  vary  level  render care  reductions  constant,  of  relation  give  Assuming the  will  the  for  does  such r e d u c t i o n s  higher  of  operating  the  remain  situation  according to  reasoning  items,  item w i l l  testing  implications  choices per  symptomatic  the  groups  statement  struction  actual  choices per  in  for under-  to  the  ineffective the  a broader  conviewpoint  in  the  number  test  has  been  may b e  of  care-  considered to  understanding particular  on t h e  items  be  part  affected  concerned.  19 Weitzman careful  and McNamara  construction  sideration  of  the  of  level  test of  point  items  the  and  testees  out the in  the need  the  need for  for con-  process:  S i n c e one o f the c h i e f aims o f t e s t i n g i s to place students into categories i n keeping with their l e v e l s of c o m p r e h e n s i o n , t e s t items s h o u l d be c o n s t r u c t e d i n s u c h a m a n n e r t h a t t h o s e w i t h some knowledge o f the c o u r s e m a t e r i a l s c o r e h i g h e r than t h o s e who a r e n a i v e , w h i l e t h o s e who t h o r o u g h l y u n d e r s t a n d the e s s e n t i a l s o f the c o u r s e s h o u l d be a b l e to earn s i g n i f i c a n t l y higher scores than those with only an a v e r a g e d e g r e e o f a c h i e v e m e n t . T h i s does not result i f t h e t e s t q u e s t i o n s a r e s u c h t h a t l i t t l e r e a l knowledge i s needed by the t e s t e e because o f the ease o f e l i m i n a t i n g r i d i c u l o u s or remote p o s s i b i l i t i e s i n the incorrect choices.  E. W e i t z m a n , a n d W. J . M c N a m a r a , " A p t U s e o f Inept Choice i n Multiple Choice T e s t i n g s " , Journal of E d u c a t i o n a l R e s e a r c h . X X X I X ( M a r c h , 1946), 517-522. y  the  20 Whether o r not g i v e n i n c o r r e c t c h o i c e s are l o g i c a l o r p r o b a b l e m i s l e a d s depends l a r g e l y upon the background of the t e s t e e . The t e s t maker s h o u l d c e r t a i n l y keep the l e v e l of the students i n mind d u r i n g the process of preparing test items.  These whether they  an  that  misleads."  the  being  items  given  to  be  that  equal  which  in a  a more  was  this low  level  after  a  period  of  by  the  the  attributed  but  equally  the  test  reduction  are  was  students  to  is base  group  a  or  most  growth evident  line,  five-choice  the be  of  to  meet  reasoning would  regard be  group,  to  then or  to  expectation  discarded  number  claim  in  preceding of  that  test  it  it  has  that  is  to  items  in  actuality  would  has  itself,  signifies  achievement  where  would  quotations  choices  the  the  as  corresponding items  defect  in  each  a whole  superior  As t h e  the  to  as  not  containing  the  test  training,  a  did  testing  Should this or  in  who  test  actual  when  satisfactory  in  and the  defensible  achievement  a  item  group  understanding  items.  been  establish  of  group  generally  to  of  misleads might  a  This  particular  by  the  of  group,  deciding  According to  test.  such  material.  for  response selected  found  report  indicate,  superior  a  four-choice  criterion.  mature  certain  is  for  correct  become f e w e r - c h o i c e  it  criterion  percentage  the  this  on  possibilities thus  certain  composing t h i s  same g r o u p  a  Consider a well-constructed  presented  operating  the  a  items  demands o f  gave  a good one  certainty  five-choice  be  is  "a f a i r l y  know w i t h the  item  stated  because  authors  in  the  been  show t h a t for  test  possible  the one  items of  of  the  21  groups  for  of  test  the  whom t h e  test  has been  r e s p o n s e s made  by  designed.  seventh-grade  An  examination  pupils  on  one  20 of  the  the each  items  in  feasibility choice  Glennon's Test, of  the  above  represents  the  will  serve  argument.  frequency  The  of  to  illustrate  number  after  r e s p o n s e by  the  students. IV. from  Which  statement  best  tells  w h y we  carry  251 161 252  the  second column?  A)  T h e sum o f t h e s e c o n d c o l u m n i s 2 3 , w h i c h h a s two f i g u r e s i n i t . We h a v e r o o m f o r t h e 3 o n l y , s o we p u t t h e 2 i n the next column.  43  T h e sum o f t h e t h a n 2 0 , s o we column.  15  B)  second column i s put the 2 i n the  D)  The v a l u e r e p r e s e n t e d by t h e figures i n t h e s e c o n d c o l u m n i s more t h a n 9 t e n s , s o we p u t t h e h u n d r e d s i n t h e next column.  credence  to  the  glance  in  at  the  supposition that  group would  responses  add t h a t  c a r r y t h e 2, t h e than the c o r r e c t  I f we do n o t w i l l be l e s s  A brief  to  13  B e c a u s e we  mature  learned  be  likely  B rather  than  in  to  way.  answer answer.  misleads a  in  superior  concentrate  A or  E,  as  271 ——  more next  C)  E)  2  in  this  (16)  24  item  group or its this  lends a  more  incorrect case.  The  20 Vincent metic", Arithmetic M o n o g r a p h s , Wo~. To.  1949.  J .  Glennon, " T e s t i n g Meanings i n A r i t h p p . 64-74* Supplementary Educational Chicago: U n i v e r s i t y of Chicago Press,  1949,  22  h i g h frequency  o f responses  to A and E might be  considered  as i n d i c a t i v e o f the n a i v e t e o f the group, which i s operat i n g at a low l e v e l of understanding item.  i n r e g a r d to t h i s  A j u s t i f i a b l e assumption would be t h a t , f o r t h i s  same group, a p e r i o d o f t r a i n i n g intended to understandings refinement  develop  i n the f i e l d o f a r i t h m e t i c would e f f e c t a  to the t e s t items on a l a t e r r e t e s t .  There  should be a d i m i n u t i o n o f the o r i g i n a l misapprehensions h e l d by the group which might be i n d i c a t e d by an i n c r e a s e i n the number o f c o r r e c t responses  on the r e t e s t , or by a  r e d i s t r i b u t i o n o f i n c o r r e c t responses  or by both o f t h e s e .  To be c o n s i d e r e d as an i n d i c a t o r o f i n c r e a s e d achievement i n the understandings i n c o r r e c t responses percentage  t e s t e d , the p a t t e r n o f r e d i s t r i b u t i o n o f would have to be such t h a t a  o f i n c o r r e c t responses  would be a l l o c a t e d to  c e r t a i n choices while a correspondingly higher o f i n c o r r e c t responses  lower  percentage  would be a l l o c a t e d to o t h e r  Thus some misleads would have l o s t  their  misleads.  a t t r a c t i v e n e s s to  the group, w h i l e o t h e r s would have gained what these have lost.  There i s no i n t e n t i o n o f a r g u i n g t h a t t h i s i s a  g a i n i n achievement because one m i s l e a d i s b e t t e r than another but r a t h e r t h a t the v e r y c o n d i t i o n s , which a l l o w the students to r e j e c t a d i s t r a c t o r t h a t f o r m e r l y h e l d appeal, s i g n i f y g r e a t e r understanding  concerning  the  23  item even i f t h i s understanding i s not o f a s u f f i c i e n t l y high l e v e l t o permit the s e l e c t i o n o f the c o r r e c t response. The compilation o f data under t e s t - r e t e s t c o n d i t i o n s f a c i l i t a t e s the e v a l u a t i o n of changes i n the s t a t u s o f the t e s t e e s i n r e l a t i o n to the t e s t m a t e r i a l .  I t also permits  the e v a l u a t i o n o f the t e s t m a t e r i a l i n r e l a t i o n to changes i n the students' responses to i t .  Both methods o f a n a l y z i n g  data have t h e i r c o n t r i b u t i o n to make i n the present study. Conclusions concerning the e f f e c t o f the year's teachert r a i n i n g course on the Vancouver Normal School students' understanding o f a r i t h m e t i c w i l l be based on the f i n d i n g s regarding the v a r i a t i o n s i n the students' scores on the two t e s t s and on the f i n d i n g s regarding the v a r i a t i o n s i n the responses assigned to each item i n the t e s t and r e t e s t . The l a t t e r c o n s i d e r a t i o n r e q u i r e s the development o f a means of a n a l y z i n g data t o provide the necessary information to answer the f o l l o w i n g questions: 1.  How many of the eighty f i v e - c h o i c e items  contained i n Glennon's Test are reduced to fewer-choice items on the r e t e s t as compared with t h e i r s t a t u s on the test? 2.  To what extent i s the p a t t e r n o f r e j e c t i o n  f o r the misleads a t t r i b u t a b l e to an i n c r e a s e i n the number of c o r r e c t responses f o r the corresponding items and t o what extent i s i t a t t r i b u t a b l e to the r e d i s t r i b u t i o n o f the i n c o r r e c t responses w i t h i n the corresponding items?  24 Summary  and  The of  present  arithmetic  teachers  can  standings  That  when  value  of  the  study.  sidered  as  the  will  be  points  which  What  is  the  has  the  been  course the  done in  major  previous  of  departure  effect  as  Vancouver Normal  a  to  when of  under-  course assess  has the  these  problem  the  study  may b e  study  by  been  subsidiary  of  con-  Glennon.  noted  and  problems  to  first:  year's  School  teacher-training  students'  understand-  arithmetic? 1.  students contained  To what  possess in  the  extent basic  Glennon's  do  Vancouver  mathematical  Normal  Test  at  the  beginning  b.  at  the  end o f  the  teacher-training  there  any  significant  Is  2. achievement in  students  of  the  Glennon's tested  at  basic Test the  School  understandings  a.  tained  field  developing  have  stated of  the  bases  necessary  the  below  in  mathematical  of  be  Problem  classroom only  present  will  the  the  the  summarized  one,  the  teacher-training  be  Basically,  refinement  on the  of  little  This  several  major  course ing  a  will  the  teacher-training  present  these  in  seldom have  enter  but  of  on u n d e r s t a n d i n g  understand  they  understandings.  However,  effect  they  demonstrated  Statement  emphasis  have  themselves  arithmetic.  been  Complete  of  the  teacher-training  mathematical between  beginning  the of  year?  year?  difference  in  understandings  con-  Vancouver Normal the  school  year  School  and  the  25 same g r o u p  tested  each  of  Glennon's  the  To what  3. to  at  the  five  Test  end o f  extent  areas  possessed  are  of by  the  school  the  understandings  arithmetic the  year?  contained  Vancouver  Normal  basic  in School  students a.  at  the  beginning  b.  at  the  end  4. ment of  of  the  Is  training  5.  by  of  the  significant  in at  to  each  of  Glennon's Test the  same  beginning students  of  of  at  the  is  the  order  contained  the  of  in  Vancouver  end o f  the  teacher-training  there  any  significant  of  gain  understandings the  in  achievement  of  made  by  students  the  initial  in  in  made  initial  gain  the  the  basic  and  in  the  quarter  determined  students period?  between  basic  the  mathe-  lowest  average  of  five  period?  of  mathematical  highest  as  difference  students  test  the  teacher-training  achievement  by  of  School  the  amount  of  Normal  at  in  Vancouver  teacher-  Glennon's Test  b.  cases  areas  end o f  difficulty  beginning  the  achieve-  five  the  the  average  the  in  between  at  Is  period?  period?  arithmetic  testing  basic  period?  difference  a.  matical of  any  What  6. the  there  and the  teacher-training  areas  teacher-training  students  period  teacher-training  the  contained  School  the  of  understandings  arithmetic  Normal  of  quarter  amount  understandings the  cases  in  test? 7.  contained  in  How many  of  the  Glennon's Test  eighty  are  five-choice  reduced to  of  items  fewer-choice  the  26 items  on the  retest  as compared with  their  status  on  the  test? 8. the  misleads  correct extent correct  To w h a t  attributable  responses is  it  extent  for  the  attributable  responses  within  to  is  the  pattern  an i n c r e a s e  corresponding to the  the  in  of  rejection  the  items  number  and t o  redistribution of  corresponding  items?  for  of  what  the  in-  27  CHAPTER  II  PROCEDURE  Selection  The  testees  School.  This  of  students  those  prepare, The  area,  while  of  sample, the  of  ance  c a n be  the  was  due  sample.  to  withdrawals those  included  school  in  standing.  to  the  from  parts  teachers  begin  of  to  British  the  407  reduction  students who  two-thirds  the  was  in  time  the  to  schools  the  were  present  forty-four Entrance  in  number  years to  in  of  were  given  both  tests  and  in  in  testees  the  varied  attend-  280  during  testees  of  original  reduced to  for  The  elementary  were  tests  all  careers.  The  who  school  The  mainland  province.  on the  the  the  lower  of  Columbia.  from  University  the  teachers.  representative  teach  number  sample. to  be  Normal  C o l u m b i a who  teaching  students  The  at  go  their  this  years  Vancouver  British  come f r o m  but  final  the  approximately  other  preparing  of  students  seventeen  background  from  absenteeism of  of  Province of  students  Province  included a l l  at  trains  c o n s i d e r e d to  teachers  the  Testees  students  student  then,  final  from  the  the  become e l e m e n t a r y - s c h o o l  come  province  the  Only  to  the  in  sample  the  these  a few  student  level  of  year,  Upon g r a d u a t i o n parts  were  institution  each  majority  of  and  year. were  in  age  educational  Bachelor of  Arts  28 The  In evaluation programme  Chapter  of to  emphasized.  Teacher-training  the  a  to  particular  course  of  minor  course  will  be  methods  in  of  teachers'  such  to  the  teaching  student-teachers'  the  background f o r  the is  in  a  the  which  in  and  The of  may  their  the  discussion teaching  so t h e  this  the  the a  the  of  relatively through  the  discussion various student-  contributions will  of  the  be  limited  arithmetic  practice of  will  on  students'  arithmetic  understanding  consideration  in  be made  possible  of  example,  demands on  of  involved.  contribution  A description  their  actual  to  arithmetic,  latter.  clearer  phase  recognized that from  those  is  methods  c o n s i d e r e d as  understanding  arithmetic.  mathematical  separated  may b e  arithmetic,  competencies,  for  some a d d i t i o n a l  arithmetic  the  bearing  included  contributions  of  direct  is  the  course  most  which  was  this  in  which  may h a v e  in  course  statistics  concerned with  confined  the  courses,  comparison with  directly  the  the  arithmetic  knowledge  other  course,  of  have  for  findings  studies  assumed t h a t  will  of  that  However,  in  course  is  recognized  understanding  the  upon a  arithmetic  make.  understanding  extent  it  need  teacher-training  large  While  psychology  the  the  of  the  is  of  interpretation  of  partial  contribution  The  nature  it  study  students'  depend to  study,  this  the  regard  teaching  I of  Programme  social  set  present to  although  phase  to  cannot  it be  significance  29 of  arithmetic.  used  for  The  numerical  illustrative  Theories The The response  9+  the  of  the  as  an  arbitrary  of  providing  drill  nature  form  of  a  theory  standing  of  fix  The  be  Arithmetic  12  would of  be  knowing  repetitive  that  number  in  of  add n o t h i n g  Method  plus  a  pupils'  teaching  basis  for  would  are  in  minds.  based on  students'  twelve"  necessity  repetitions  the  the  automatic  d r i l l .  and three  of  in  to  an  r e c o g n i z i n g the  number  methods  as  competencies  "nine  fact  mathematical  Drill  learned  student-teachers'  the  would the  3°"  sufficient  A course drill  Teaching  a s s o c i a t i o n and o f  for  to  of  method  Demands o n be  will  s  Theory:  fact  through  3 * 12  purposes.  and Methods Drill  9+  example  the  own  under-  arithmetic.  Supertficial  Coating  of  "Meaning": The  number  fact  would  be  learned  it  to  the  student-teacher  would  prove  the  use o f  under  the  in  objects,  accordance with The  as  before  objects are  the  to  twelve"  have  demonstrate is  meaning  student-teacher  and would  true.  illusion  to  that  the  as before pupils  that  he  but  through  was  teaching  theory. would  be the  able  have to  to  count  statement  know t h e out  "nine  fact  twelve and  three  30 S i n c e t h i s method d i f f e r s o n l y s u p e r f i c i a l l y  from  the d r i l l method, knowledge o f i t would add v e r y l i t t l e , i f anything, t o the s t u d e n t s ' own understanding  of arithmetic.  The Meaning Theory i n R e l a t i o n to a S i n g l e Number Fact T h i s method o f t e a c h i n g has been d e s c r i b e d by K i n s e l l a and Carnahan,^ who summarized Van Engen's p r o cedures f o r p r o v i d i n g meaning i n the l e a r n i n g process as " d i r e c t experience f o l l o w e d by v i s u a l i z a t i o n which i n t u r n is  f o l l o w e d by a b s t r a c t i o n " .  The method p r o v i d e s f o r a  s e r i e s o f e x p e r i e n c e s r e l a t e d to one number f a c t , i n which the experiences proceed from the use o f concrete m a t e r i a l s through  i n c r e a s i n g l y a b s t r a c t m a t e r i a l s , from the immature  l e v e l o f c o u n t i n g through grouping and r e g r o u p i n g  processes  to  i n the  t h e a b s t r a c t i o n o f the g e n e r a l i z a t i o n i m p l i c i t  statement  t h a t "nine and three are twelve".  c o n s i d e r e d t o be a meaningful number f a c t develop  i s concerned,  T h i s can be  process i n so f a r as t h e s i n g l e  but t h e r e i s no attempt  here t o  r e l a t i o n s h i p s between numbers beyond the s p e c i f i c  r e l a t i o n s h i p i n v o l v e d i n t h e one number f a c t . The statement zation.  s t u d e n t - t e a c h e r would have t o know t h a t t h e  "nine and t h r e e are t w e l v e " r e p r e s e n t s a g e n e r a l i He would have to know what g e n e r a l i z a t i o n s are and  J . J . K i n s e l l a , and W. H. Carnahan, " P u t t i n g Meaning i n t o Geometric Concepts", School Science and Mathem a t i c s . X L V I I I (October, 1948), p. 543.  I  31  how t h e y  are  procedure counting  to  is  the as  basis  a  discriminate  abstractness  the  learning  use  the  in  tribution  to  the  to  processes  The  individual  Meaning  Theory  the adds  might  be  width.  selves  to  addition  the  have of  make  the  above that  may to  be  be  concreteness  development  know when  familiar  achievement facts  Relation is  instance,  the  following  sets  give  of and  with the  of  a n d how  maximum  to  con-  to  this  method  of  understandings the  and g u i d i n g  an  fact  upon the  extension  depth  to  pupils  9 + 3  related  t h o s e number such a s : 7 + 8+ 9 +  the  arithmetical  S  into  which  of  Related  Facts  previous  one  it.  previous  The  understanding  whereas  arranging  relationships  number  Groups o f  arithmetic  consists of  of  to  dependent  b a s i s of  number  a.  in  said to  It  groups the  have would  became  c o n s i d e r e d as  mathematical  related  recognize  levels  used i n  the  process.  number  method  s h o u l d be  study  to  apply  involved.  This and  to  He w o u l d  various  they  learning  in  have  process.  He w o u l d  who  able  and t h a t  materials  so t h a t  should gain  be  addition  among t h e the  A student  to  He w o u l d  of  process.  materials  teaching  order  regrouping  and  related  in  intelligently.  considered able  learned  this  number  facts  method into  discovering for exist  1 2 would  of  them-  among t h e m .  appear  in  For  the  facts: facts 5 - 1 2 4 12 3 - 1 2 M  belonging to  the  number  12;  32  b.  t h o s e number f a c t s w h i c h c o u n t i n g by t h r e e s ; such  are as  connected  with  3 + 3 - 6 6 + 3 = 9 9 +  c.  t h o s e number the addends;  3=" 1 2  f a c t s which such as  9 as  involve  one  of  9 + 1 - 1 0  9+ 2 = 1 1 9 + 3= 1 2 d.  the higher-decade which involve the three; such as  addition facts with bridging common e n d i n g s o f n i n e a n d  (9+ 3 - 1 2 ) 3= 2 2 29 + 3 32  1 9 +  3  To must  be  make  able  to  use  arrange  which  a useful  which  generalizations  ways to  they  the  this number  be  will value.  of  desired  to  this  siderable  between  be  He m u s t  of  has  teaching of  the  numbers,  those  drawn.  of m o s t  who  student-teacher  into  in  and  k n o w how t o from  been  the  number  the  what  proceed  pupils. intro-  acquired  system, basic  from  know  successfully  s h o u l d have  and o f  groups  He m u s t  value  generalizations  understanding  lationships of  method  the  facts may  be  A student-teacher duced  method  generalization  will  elicit  of  of  the  conre-  generalizations  arithmetic. It  methods  of  familiarize teaching  is  evident  teaching  from  arithmetic  student-teachers  arithmetic  the  will  above  that  with  a  the  demand m o s t  descriptions course  designed  meaning in  the  of  theory  way  of  to of  student  33 abilities for  and at  the  contributing  mathematical  basis  possibilities, two  of  which  students' later the  to  of  section;  to  and and  programme.  The  the  of  time  will  students'  will  be  The  the the  extent  which  practical factor  arithmetic  will  in  be  of  be  factors, the  covered  in  be made  such as  the  these  discussed  must  in  of  force;  a one-year  operates  cannot  possibilities  upon o t h e r  deterrent  compromise which  latter  greatest  dependent  considerable learn,  have  own u n d e r s t a n d i n g  arithmetic.  have  ability  subject  the  however,  may  desirable  same  a  between  teacher-training a way  that  completely  2 as  some e x p e r t s  would  recommend.  Forest  suggests:  We m u s t p r o v i d e t h e b a c k g r o u n d o f m a t h e m a t i c s . F i r s t i t must p r o v i d e f o r i n s i g h t i n t o m a t h e m a t i c s and i t s methods and purpose and s p i r i t . In the second p l a c e i t must p r o v i d e p r o f e s s i o n a l i z e d s u b j e c t m a t t e r courses i n which the f i e l d of a r i t h m e t i c itself is covered slowly, thoroughly, intensively, extensively so as t o l a y a g a i n t h e b a s i c c o n c e p t s and p r e p a r e an u n d e r s t a n d i n g o f the p r o c e s s e s , t e c h n i q u e s and meaning of arithmetic as a m a t h e m a t i c a l subject.  The matter view two  would  of when  the  this  is  replete  seem t o close  the  in  suggested d i v i s i o n  with  is  both  by  theory  is  no means  similar  J . W. F o r e s t , A r i t h m e t i c " , Mathematics  methods  and  subject  u n n e c e s s a r y and u n d e s i r a b l e  relationship  meaning  regard  be  between  which  must  involved. an  isolated  s u g g e s t i o n s from  exist  between  in the  Forest's  suggestion  one;  literature  other  the  authorities.  "Training Teachers o f High School T e a c h e r , XXXIV ( M a r c h , 1 9 4 1 ) , 121.  34 It  would  appear  accepted process their this  the in  thinking  training would  theory  on the  spent  on  the  field  of  arithmetic  developed  in  the  metic.  of  an  in  The g e n e r a l  teacher-training Since in  the  occurs  as  a  the  w a s made  of  studies  at  the  general  course  courses  for  similar  courses.  one,  in  effect  not  courses in is  the  the  available  teacher-  for  methods  arithmetic  course  elementary of  of  methods  related  which study  the  particular is  concerned.  concerned with arithmetic  programme.  The  School  that  differentiation  The  primary  groups;  no  all  arithmetic  course i s  s e c t i o n which  and p a r t  s e c t i o n which  to  All  students  of  three;  both  take  and the work  sections.  the  given  includes  includes the take  comprises  students  the  the which  programme  Vancouver Normal is  arith-  teacher-training  usual teacher-training the  be  material  teaching  study  of  to  could  and  in  to  this is  school  method  theory  about  there  wherein  understandings basic  pattern  control  and-senior eight.  the  understanding  the  to  different  two  consideration of  now b e  present  of  attempt  grades  extended  meaning  remarks  students'  result  the  yet  course with  no  sections;  as  intensive  for  the  relates have  specific  courses w i l l  change  teaching  time  interwoven  implicit  arithmetic  the  whole-heartedlythe  methods  Whatever  so  to  a  include  number  is  it  to  a greater  which  as  school  has  be  who h a v e  elementary  field.  best  those  meaning  the  theory  that  in  course a of same  or  two  work  for  intermediatefor  grades  three  No t e x t b o o k  is  35 used f o r t h e c o u r s e but t h e s t u d e n t s a r e r e f e r r e d textbooks which approach  are a v a i l a b l e  to arithmetic  i n the school library.  i n this  course  coverage  of the specific  developed but  programme o b v i a t e a  o f these  a sampling  of arithmetic  c a n be s a i d t h a t  time  factor  restrictive  The  The degree  factor;  Students' A b i l i t y  one t o r e c e i v e  common t o most l e a r n i n g students  limit  a s m e a s u r e d by i n t e l l i g e n c e  by t h e  on t h e amount o f deal with the to learn.  t o Learn i s dependent t o a l a r g e the students' a b i l i t y to  attention  situations;  o f such  mathematical  imposed  the students' a b i l i t y  s u c c e s s o f any course  The f i r s t  o f t h e work,  o f basic  section w i l l  upon t h e f a c t o r s w h i c h a f f e c t  learn.  part  The r e s t r i c t i o n s  The n e x t  mathematical  t o measure t h e e f f e c t  however, p u t an u p p e r  change t o be e x p e c t e d . second  grades  understandings.  i s an i n t e g r a l  t h e attempt  is•justified.  will,  s h o u l d be  of the basic  a programme on t h e s t u d e n t s ' a c h i e v e m e n t understandings  comprehensive  i s c e n t r e d i n t h e meaning t h e o r y o f t e a c h i n g  and b e c a u s e  understandings  require-  teaching f o r the elementary  i t does a l l o w f o r a sampling  arithmetic  The p r a c t i c a l  understandings which  i n the arithmetic  Because t h e c o u r s e  it  training  The  i s i n keeping with t h e  meaning t h e o r y o f t e a c h i n g a r i t h m e t i c . ments o f t h e o n e - y e a r  to several  i s one w h i c h i s  the learning tests.  ability of  T h e mean I Q f o r  the  s t u d e n t s o f t h e V a n c o u v e r Normal S c h o o l a s m e a s u r e d by  the  Otis  Self-administering  Test o f Mental  Ability,  Higher  36 Form  A was  mean  for  in  115.5  the  British  School  with  Higher  parison  with  Thus  it  is  will  exert  Form C o f  may b e  the  is  to  a  influence  the  understanding  have  powerful  were  taught  and  subtle  through  both  from  for  Grade  Since XI  form  s e l e c t e d group  a of  that  the  on t h e of  factor  of  findings  of  previous  and a t t i t u d e  That  based on the about  in  com-  students. intelligence this  study. on  the  arithmetic  most  drill  their  Normal  training  towards  ramifications.  comments  Vancouver  high-school  the  pupils  the  influence of  9.2.  of  105.7,  expect  methods  their  test  population  reasonable positive  deviation  this  said to  general  Secondly,  evident  standard  Columbia schools  students  students'  a  may  students  theory  early  is  training  3  and  from  such  statements  as  this  one  by  Carter:^  The m a j o r t a s k i s t h e r e o r i e n t a t i o n o f t e a c h e r s in regard to the "meaning" t h e o r y . Many l e a r n e d arithmetic from a course of i n s t r u c t i o n which l a r g e l y ignored meanings.  It  remains  taught  to  be  through  such  understandings, grades, students ' with  which will  seen to  have  arithmetic  methods  basic  are  what  to  made  and w i l l  have  the  contained  extent  students  been  able  arithmetic in  more have  of  to  who h a v e attain  the  attained  use  of  the  elementary  Glennon's Test.  effective  been  That their  a higher  some experiences  degree  of  'Taul Carter, "From a M e c h a n i s t i c t o a M e a n i n g f u l Program o f A r i t h m e t i c I n s t r u c t i o n : a Suggested Approach", S c h o o l S c i e n c e a n d M a t h e m a t i c s . X L V I I ( O c t o b e r , 1947), 605.  37 understanding  than  scores  initial  low  on t h e  scores  standing  cannot  for  as  others  be  will  test. said  Brownell  be  reflected  However,  to and  be  those  entirely  Sims^  in  tell  relatively  who make  lacking  in  high  relatively under-  us:  S i n c e u n d e r s t a n d i n g i n some m e a s u r e a c c o m p a n i e s a l l l e a r n i n g , t h e p r o b l e m o f e d u c a t i o n c a n n o t be d e f i n e d a s the creation of understanding. A s h a s b e e n s t a t e d , some degree of understanding i s i n e v i t a b l e if a child learns at a l l . The p r o b l e m o f e d u c a t i o n , r a t h e r , i s t o make s u r e t h a t c h i l d r e n possess the k i n d s and degrees o f understanding which are e s s e n t i a l to i n t e l l i g e n t behavior consistent with t h e i r needs.  The  two  groups basic  previous were  to  entered  low  the the  initial  basic  in  their  arithmetic  this  status  understandings  subject  of  meagre of  itself  of  the  He  found that  was  attitudes  of  lack  the also  the  this  field  of  the  elementary  found  that  their  understandings  grades  when  they  course.  contribution  of  student-teachers affects  indicated  by  prospective of  in  achievement  teacher-training  That the  investigators  their  the  in  findings  was  training  to  achievement  of  attitudes  teachers  understanding  early  of  toward one  of  towards  Dutton's  the study  arithmetic. the  seven  most  H/. A . B r o w n e l l , a n d V . M. S i m s , "The N a t u r e of U n d e r s t a n d i n g " , The Measurement o f U n d e r s t a n d i n g , p. 32. Forty-fifth Yearbook of the N a t i o n a l S o c i e t y f o r the Study E d u c a t i o n , P a r t I. Chicago: U n i v e r s i t y of Chicago Press,  of  this  Toward  195D,  H. report.  Taylor  and Vincent  J .  Glennon.  See  °W. H. B u t t o n , "Attitudes of Prospective Arithmetic", Elementary School J o u r n a l , LII 85.  0  of 1946.  Chapter  Teachers (October,  I  38 frequently  mentioned  unfavourable receptivity  attitudes to  new  Another learnings who  in  deterrent  remain  loyal  to  the  educating  them.  a  involve  desire  factors  to  in  this  force  field;  a  method  discussion to  are  the  of  of  of  is  previous  is  is,  of  students'  of  new  well  to  those  the  tendency  which  of  was  of  students  used  no  and o t h e r  such  study.  gaining  training  in  ego-involvement,  known,  this  the  known  phenomenon w o u l d ,  the  of  to  the  Such  acceptance  function  scope  barrier  to  the  this  the  An a d d i t i o n a l  affect  teaching  security  beyond the  related  to  that  attitudes.  area.  courses  An e x p l a n a t i o n  cling  which  standings  course,  teacher-training this  unfavourable  of  learnings  in  doubt,  for  will,  worked  to  have  reasons  in  of  an  new  under-  indirect  manner.  7 Wheat  points  arithmetic it  and  makes  so makes  concerning  the It  methods  out  has  experience  is  that it  it  hard  number  for  provided  for  them  such  that  acquisition  of  teachers to  the  whether  acquire  arithmetic. a handicap  new  early  students  considered desirable  have  familiarity  of to  the be  new  material  critical  of of  understandings  system.  doubtful  of  the  very  difficult  generalizations found  the  learnings  with  for  training the  the  background of  derivation  Nevertheless, does in  not this  b a s e d on  some  of  varied basic  authorities  necessarily area.  the  drill  prevent  A note  of  H. G. Wheat, "The T e a c h e r ' s Self-improvement", How t o T e a c h A r i t h m e t i c , p . 356. Evanston, I l l i n o i s : Row, P e t e r s o n and Company, 1951.  39  optimism i s injected when Forest  says:  It i s always a source of stimulus to me as a teacher to see the lack of concepts and understanding and grasp of quantitative relations in students preparing to teach arithmetic in the elementary grades. I begin to think of the consequences i f these people go out to teach arithmetic without considerable improvement. Likewise i t i s a source of stimulus and satisfaction to see the eagerness with which the great majority of them enter into the program of improvement when the purpose of mathematics i s made a l i t t l e more clear, when the processes begin to be understood and when they begin to feel their ability in this area grow. Morton  discusses the part that mental maturity plays in the  learning sequence: Explanations which were not understood last year may be grasped this year. Even though a pupil gains a satisfactory understanding of what i s taught at one grade level, he w i l l [be l i k e l y to] gain a richer and f u l l e r understanding of the same topic at a later grade level when he is mentally more mature. His remark concerning the contribution of mental maturity to understanding seems to indicate a favourable prognosis in relation to the mature group represented i n teacher-training classes. A third potential which may affect the receptivity of normal-school students to the meaning theory of teaching arithmetic i s the influence of the supervising classroom Forest, J. W., "Training Teachers of High School Arithmetic", Mathematics Teacher, XXXIV (March, 1941), 120. R. L. Morton, "The Place of Arithmetic i n Various Types of Elementary-school Curriculums", Arithmetic 1949, p. 14Supplementary Educational Monographs, No. 70. Chic ago: University of Chicago Press, 1949.  40 teachers It  was  with  pointed  teachers the  whom t h e out  themselves  understanding  reflected  in  theory  the  this  in  in  students the  are  of  first  come  chapter  generally  classroom practices The  contact  during  the  of  report  that  this  considered to  arithmetic.  field.  in  This  lack  of  which  have  not  Measurement  of  be  year.  lacking  the in  understanding kept  pace  Understanding  is  with  contains  statement: ^ 1  In a word, e d u c a t i o n a l prominence to the n e c e s s i t y through instruction.  theory has c o n s i s t e n t l y given of c u l t i v a t i n g understandings  T h e same s t a t e m e n t c a n n o t be made, h o w e v e r , with regard to p r a c t i c e i n classroom i n s t r u c t i o n . While there i s plenty of evidence o f a growing r e c o g n i t i o n o f the role of understanding i n the l e a r n i n g process, i t i s a f a c t t h a t i n o u r t e a c h i n g we a r e s t i l l p r o n e t o p a y l i p - s e r v i c e t o u n d e r s t a n d i n g s as e d u c a t i o n a l a i m s .  While  some  rooms  in  meaning  students  which theory  so  fortunate.  of  having  methods in  to  their  teaching  choose  course  have  of  in  will  be  arithmetic,  between  will  what  their  be  they  arithmetic  inexperienced  certain  directness,  fortunate  students  teaching  To t h e a  be  practices  Those  practice.  methods  the  will  appeal ease  in  keeping many  placed are  be  others in  the  learning  and what  of  their  application  in  with  they  student-teacher  because of  enough t o  classthe  will  not  be  position in  their  see  done  drill  simplicity,  and because  they  The Y e a r b o o k C o m m i t t e e , "Next S t e p s * , The M e a s u r e ment o f U n d e r s t a n d i n g , p. 3 2 1 . Forty-fifth Yearbook of the N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , P a r t I. Chicago: U n i v e r s i t y o f C h i c a g o P r e s s , 194-6.  41 make  few  based and  demands  on the  meaning  extremely  reject  the  resultant of  such  lead  meaning  The  study  which  may  of  is  of  the  taken  what  the  testees,  taken  to  school had of  year the  as  far  be  on the  would  as  the  to  the  this  to  and  to  analyze  the  has  and  been  administered  by  such  students the  effectiveness of  further  Scoring of  picture  it  course  was  place  of  the was  in  retest  as  members  initial of  the  with  which the  factors The  explanation  data  the  the  to  of  determine  Tests  initial study,  first of  status  care  week  of  teaching  keeping  near  administered  contained  programme. an  the  methods  considered in  Both the the  during  has  effective.  the  As i t  the  devoted  test  term.  the  some o f  be  unbiased  the  of  will  the  before  chapter  of  administer  school  method  complicated,  arithmetic,  discussion  of  possible,  of  programme  effectiveness  programme  an  of  teacher-training  gather  the  to  part  the  Should  acquisition  limitations  study  complex,  l e s s e n the  the  term  contrast  competencies.  teaching  section  chapter  get  to  within  started. the  so  Administration  To  of  In  concerned.  the  the  extent  The  tendency  c o n c e r n e d and a  remainder  to  teacher  attitude  in  influence  steps  appears  theory  preceding  description  this  is  on a  unfavourable  understandings  a  to  a programme  abilities.  theory  demanding  considerations to  on t e a c h e r  with of  during  the  last  Vancouver  purpose  the  tests  Normal  the  the  end  final  was  arithmetic  the  and the  of  school  week were  School  of  42  staff  to  all  structions the  first  page  items  avoid time  the  testees  at  the  deemed n e c e s s a r y were  Precautions test  of  of  were in  that  the  The  test  taken  discussing during  the  the  any  same  to  time.  ask  carefully  the  The  students  before  invalidation  of  members  of  were  items  staff  of  these  with  of  s c o r i n g u s e d by  the  the  in-  to  beginning  against  the  only  read  the  test.  specific  requested  students  at  to  any  year.  method  Glennon i n  his  study  W  was  followed;  that  with  two  answers  that  the  five  were  to  be  will  be  used to  is,  were  areas  testees  in  these  correct  responses had  section  of  The  five  of  this  its  of  developed  S c o r i n g and o f  the  the  evaluation  the  redistribution of  of  to  the  for  in  be  Glennon's  made  of  items  recalled Test  procedure  by  each  of  s c o r i n g the  use  in  this  contained  in  the  rejection  the  gain  of  determining  the the  in  the  in-  study. next  the  the  of to  difference it  lost  Data  Responses  m i s l e a d s was  understanding  amount  value  responses which  of  incorrect  a d o p t e d was  representing  A n a l y z i n g the  Incorrect  The method f i n a l l y  initial  scores  is  will  same s c o r i n g  The method  development  Relating  means  be  u s e d and  chapter.  Method  Since  to  The  partial  areas.  It  contained  separately. the  R - •£ w a s  incorrect.  arithmetic  compute  S=  formula  counted  of  evaluated  A description  the  as  a s s i g n to between  on t h e  key  an  had to  item, be  retest  number and the  by a  devised.  each mislead  the  to  represented  responses for rejection  the  of  a its  number  43 of  responses which  mislead  A for  Item  on t h e r e t e s t indicated retest mislead  it  19  original  It  of  other  -3  while  the  loss  of  rejection  of  from to  the  other  rejection was  -4  (for  to  correct  answer.  misleads  In  B for the  value  represents  responses.  The  for mislead  shift  pattern.  which  showed  a  a change  of  the of  66  retest  mislead  a s -3«  the  the  incorrect 7  showed t h a t  went  to  the  retest  B lost them. Of t h i s  of the correct in  in  rejection  the  on the  8 r e s p o n s e s from  by t h e r e d i s t r i b u t i o n  pattern  the  to  B.  difference  assess  Item  gained  Similarly,  by the  lost  was t o  amount  on the  rejection  a rejection  the r e d i s t r i b u t i o n  lost  the  item  f o r the  each m i s l e a d  B was computed  was c o n t r i b u t e d  16  a  same  an a c c e p t a n c e  B for  words,  than  pattern  was a t t r i b u t a b l e  and gained  for mislead  other  mislead  rejection  +43  of  responses on the  rather  f o r mislead it  118 r e s p o n s e s  pattern  8 of the responses i t  contributed  while  test.  r e s p o n s e s and t h e  The d a t a  while  a  example,  The d i f f e r e n c e  i n the procedure  which  the responses which  answer  initial  a negative  step  was a t t r i b u t a b l e  responses.  original  43 m o r e  received  represents  the  correct  which  that  next  number  its  For  2? r e s p o n s e s a n d g a i n e d  The  pattern)  retest.  responses.  hand,  value  rejection of  of  and t h e g a i n  a positive  amount  21  64  indicated  s h o u l d be n o t e d  between  on the  an a c c e p t a n c e  On t h e  difference  lost  d i d on t h e  showed  its  66  the mislead  pattern. of  gained  and gained  that  than  it  correct came  12 r e s p o n s e s Thus  the  amount  +1  responses  incorrect  responses.  44 A further  use  responses  was  to  lation  the  number  and  to  on t h e  standard whether was  used.  item  to  those,  who  did  25  per  item would  that  the  allowance  per  was  the  tabulation received  the  of  arbitrary  nature  ^See  of  fairly  equal the  re-  test the  determining item,  multiple-choice  percentage correct  of  answer,  if  each o f  Since  it  misleads the  is  distributed  employing t h i s  arbitrarily,  a  of  10  choices per  per  cent  of  the  of  of  item  evenly  criterion  item  of  received  10  included  in  while  results on the  s h o u l d c a u s e no  report.  among  the  incorrect  S i n c e the  choices per  this  the  unlikely  criterion  r e s p o n s e s were  tabulation.  I  that  item,  standard.  Those m i s l e a d s which  number  Chapter  in  incorrect  than  of  four  responses for  ideal. be  the  considerations necessitated  number  this  compare the  for  considered a  certainty  leeway  the  fewer  adaptation  a multiple-choice  incorrect  used.  the  excluded from  used to  the  decided,  of  as  responses would  certain  more  on the  theoretically  would be  cent  were  or  of  practical  It  per  which  be  a  item  an  in  misleads.  cent  of  cent  a  know w i t h  incorrect items  choices per  and McNamara^  w o u l d mean t h a t  that  four  when  incorrect  items  purpose  Weitzman  the  test  a n d McNamara  the  of  of  the  satisfactory  not  each o f  received  actual  satisfactory  This  10  is  evaluation  status  For this  Weitzman  be  selected  the  of  suggested by item  the  compare the  retest.  an  for  those responses were two  tests  concern  45 regarding  the  interpretation  comparison of  the  the  items  number  items  of  results  according to  for  which this  kind, course  training  in  be  to  and  regard  to  the  of  the  Glennon's Test effectiveness  and the  retest  A  revealed  fewer-choice  be  Chapter  expected of  theory by  of  of  analysis  and the the  study of  and  a  factor. of  the  methods  through  The data,  has  this  arithmetic,  operating  which  consequent  a  teaching  forces  change  in  testees  basic mathematical  of  so a f f o r d e d .  reduced to  time-limit  the  amount  the  test  the  group  restricted  through  the  may  meaning  concerned with  achievement in  be  of  selected  b a s e d on the  considered  will  a  the  results  criterion.  gains which  involving  the  had been  Summary  The  of  taken  next the  previous  chapter findings  place  understandings  in  programme  the  contained  conclusions relating  teacher-training  are  to  concerned.  46 CHAPTER  THE  The information training  be a  include  given whole  to  will  according  to  be  the  results  of  the  will  given. It  study  These  will  be  will  be  the  is  in  on  drawn.  In  each in  terms  to  the  First,  of  the  errors  of  (D^),  the  between  the  means  ( S E ^ ^ ) ,  and the  t  T h e mean The  the  of  the  of  the  the  the  means  specific  the  errors  will  also  between  the  gains  problems  with I.  table  form  in  the  conclusions  gains  the  will  deviations  differences differences  correlations be  the  Chapter  standard  of  Then  responses  the  (SE^),  as  Finally,  consequent  to  will  test  the  in  summary  test-retest  scores  of  incorrect  (M),  analysis  test.  students.  The  standard  differences  of  pertaining  means  the  a  this  on the  summarized  with  of  consideration  gains  the  them.  data  means  percentages.  organization  that  together  the  of  students'  c o n c e r n e d were  between  ratio.  School  distribution  study  recalled  case  standard  Normal  sections  the  status  regard  be  presented  (SD),  be  restated  findings  then  initial  will  this  the  centred  provide  teacher-  to  separate  statistical  which  of  five  to  year's  The  related  intended  the  sections.  findings  DATA  is  of  Vancouver  main  and on the  data  effect  arithmetic.  three the  the  the  on t h e  of  attention  be  of  regarding  understanding will  ANALYSIS OF THE  analysis  course  III  given  in  means w i l l  (r), terms be  47  formulated  in  terms  are  included  because  the  standard  errors  formula  used f o r  correlated The  percentages.  i f of  means;  the  that of  is,  The  test-retest  necessary to differences the S  E  latter  d i f f  correlation  product-moment  Gains  was  computing  coefficients  Pearson  of  V  =  compute  between was  3 F ,  were  M  that  2  +  correlations  them  the  to  means.  required 2  "  2  find  r  3  computed through  %  The  for S E  M *  the  method.  on G l e n n o n ' s  Test  and  on  its  Five  r  Constituent  The first  problems  to  the  test  as  a whole  are  given  consideration. 1.  possess  the  Glennon's  To w h a t basic  of  extent  the  mathematical  Normal  understandings  a.  at  the  beginning  b.  at  the  end o f  Is  there  basic  Test  tested  at  the  beginning  tested  at  the  end o f  The  findings  in  between  Table  any  the of  the  I.  of  the  the  School  contained  students in  school  difference  understandings  Vancouver the  teacher-training  teacher-training  significant  mathematical  Glennon's  tabulated  do V a n c o u v e r  Test  2. ment  related  Parts  school  Normal year  and  year?  in  achieve-  contained  School the  in  students same  group  year?  concerning these  two  year?  questions  are  48 TABLE Comparison  of  Test  M  Mean  I  Test  and  as  a Whole  SD  SE  M  Retest  (N-  280)  1  °M  r  Scores  !  Test  49.2(61.5%)  j 13.881.83  Retest  52.8(66.0%)  13.40 . 8 0 j . 8 5 J 3 . 6 ( 4 . 5 % ) j  The cent  of  the  students  eighty  in  Glennon's  of  the  year  the  66  per  cent  of  size  (280),  level. so.the the  The  the  a  degree gain  of  basic at  the  these t  ratio  the  of  of  i j  an  8.00  .45  average  of  61.5  per  understandings  contained  beginning  the  the  of  2.59  is  of  8.00  that  there  is  no  and  cent,  For  a  is  tests  Therefore, although  well  at  the  small,  an  is  end  average of  per  this  cent  limit  difference  rejected  between  with  conclusion is is  of  this  1  the  above  significant  final  At  sample  significant  ratio  initial  study.  sampled had mastered  t  per  t  d i f f  mathematical  confidence.  4.5  E  i  understandings.  obtained  for  had mastered  population  hypothesis  means  high  Test  i i i  S  nevertheless  a  that sig-  nificant. To to  the  determine  following  the  questions  To what  3. each  of  five  Test  possessed  by  wherein  the  were  extent  areas  of  such changes o c c u r r e d ,  are  sought: the  understandings  arithmetic  Vancouver  answers  contained  Normal  a.  at  the  beginning  b.  at  the  end o f  the  of  School the  in  basic  to  Glennon's  students  teacher-training  teacher-training  period?  period?  49  4.  I s t h e r e any s i g n i f i c a n t  ment o f the understandings  d i f f e r e n c e i n achieve-  b a s i c t o each o f the f i v e areas o f  a r i t h m e t i c c o n t a i n e d i n Glennon*s Test between t h e Vancouver Normal School  s t u d e n t s at the b e g i n n i n g  o f the t e a c h e r - t r a i n i n g  p e r i o d and the same students at t h e end o f the t e a c h e r - t r a i n i n g period? TABLE I I Comparison o f t h e Mean Test and Retest Part A, t h e Decimal System o f N o t a t i o n  1  SD j S E  M  I  j1  M  r  j  °M  Scores ( N - 280)  S E  diff  t  j 1 1 . 4 3 ( 7 6 . 2 % )  j Test  !  0  3.02J.18  S  {  ! Retest 1 2 . 3 4 ( 8 2 . 3 % ) ) 2 . 6 7 ! . 1 6  .61  . 9 1 ( 6 . 1 % )  .15  6.06  !  Of the 15 items on the t e s t which d e a l t with t h e decimal  system o f n o t a t i o n the p o p u l a t i o n sampled had achieved  an average o f 76.2 per cent on t h e i n i t i a l test  The f i n a l  r e s u l t s i n d i c a t e d t h a t t h e s t u d e n t s had mastered an average  o f 82.3 per cent o f t h e understandings g a i n o f 6.1 per cent i s s i g n i f i c a n t of  test.  i n t h i s area.  i s supported  That t h e  by t h e t r a t i o  6 . 0 6 .  C o n t i n u i n g , the data f o r the second s e c t i o n o f the test  are summarized i n Table I I I .  50  TABLE I I I Comparison o f the Mean Test and Retest Part B, I n t e g e r s and Processes  M Test  SE  11.85(79.0%) 2.59  S E  M  r  (N * 280) 3  ! 1 %  .16  Retest 12.48(83.2%) 2.28 .14 .63 The  :  Scores  j  .63(4.2%)j  S E  t  |  diff  .13  it  4.84  j i  i n i t i a l t e s t r e s u l t s showed an average a c h i e v e -  ment f o r the group o f 79 per cent on the 15 items  comprising  the s e c t i o n which t e s t e d the b a s i c understandings  of integers  and p r o c e s s e s .  The g a i n o f 4.2 per cent brought t h e f i n a l  average t o 83.2 per cent.  As i n the p r e v i o u s cases the t  r a t i o was s u f f i c i e n t l y l a r g e (4.84) to permit the c o n c l u s i o n that the gain i s s i g n i f i c a n t . Table IV i n c l u d e s the f i n d i n g s f o r the t h i r d  area  of a r i t h m e t i c t e s t e d i n Glennon's T e s t . TABLE IV Comparison o f the Mean Test and Retest Part C, F r a c t i o n s and Processes  M Test  SD  7 . 3 2 ( 4 8 . 8 % ) 3.01  Retest 8.26(55.1%) 3.16  S E  M  Scores  (N- 280)  r  S E  diff  t  .18 .19 .88 .94(6.3%)  .09  10.33  51 In t h e a r e a o f the t e s t concerned with the b a s i c understandings  o f f r a c t i o n s and processes the students a t t a i n e d  an average o f 4 8 . 8 per cent o f t h e 15 items, on t h e i n i t i a l test. 55.1  The g a i n o f 6 . 3 per cent brought the f i n a l average t o per cent.  Once again the s i g n i f i c a n c e o f the o b t a i n e d  g a i n was confirmed by t h e l a r g e t r a t i o o f 1 0 . 3 3 . The  data o b t a i n e d f o r the s e c t i o n o f the t e s t on  decimals and processes are summarized i n Table V. TABLE V Comparison o f the Mean Test and Retest Part D, Dec:Lmals and Proc esses  SD  M Test  S E  M  1 1 . 0 6 ( 5 5 . 3 % ) 5.44  .33  5.56  .33  Retest 1 1 . 4 6 ( 5 7 . 3 % )  (N= 280)  r  .80  Scores  S E  .40(2.0%  t  diff  .21  1.91  1  T h i s area o f a r i t h m e t i c which t e s t e d the b a s i c understandings The  o f decimals and processes c o n t a i n e d 20 items.  average o f these known by t h e t e s t e e s i n t h e i n i t i a l  was 55.3 per cent, w h i l e the average known on the f i n a l was 5 7 . 3 per c e n t .  As the t r a t i o o f 1.91  test test  i s below the 1.97  r e q u i r e d f o r the 5 per cent l e v e l o f confidence, i t must be concluded t h a t d i f f e r e n c e s as g r e a t as the o b t a i n e d might occur by chance.  difference  52  TABLE Comparison Part  E,  of  The  the  Mean  Rationale  M  SD  Test of  S E  7.31(48.7%)  3.17  .19  Retest  8.34(55.6%)  3.35  .20  basic  understandings  achievement  of  constituting a gain ratio  initial  of of  6.9 6.06  an  test  them. the  the  have  .63  of  the  testees  rationale 48.7  part  of  the  per  cent  to  an a v e r a g e  been VII  the  gives  sections of  the  of  per  test.  findings  discussed  280)  The of  in  of  6.06  .17  regard  55.6  obtained  concerning the  c o m p a r i s o n s may the  the  was  the  items  status  per  gain  to  15  the  final  t  diff  E  computation  cent  a comparison of  the  (N=>  1.03(6.9%)  this  that  Scores  S  of  indicates  Table  five  of  Retest  r  average  Now t h a t the  status  and  Computation  M  Test  The  VI  cent.  is  The  t  significant.  five  areas  b e made  gains  showed  of  among  achieved  for  test. TABLE  VII  A COMPARISON O F T H E G A I N S ( I N P E R C E N T ) FOR T H E F I V E SECTIONS OF GLENNON'S TEST Means Test A.  Decimal  system of  B.  Integers  C.  Fractions  D.  Decimals  E.  Rationale  notation  and p r o c e s s e s and p r o c e s s e s and p r o c e s s e s of  computation  65  in  per cent Retest  Gain i n per cent  82.3  6.1  79.0  83.2  4.2  48.8  55.1  6.3  55.3  57.3  2.0  48.7  55.6  6.9  0  2  53 A further sections the  o f the test,  following  as given  i n Table  f o r the  VIII,  five  i s related  to  problem: What  5. areas  comparison o f the r e s u l t s  i s the order  of arithmetic  by t h e t e s t i n g  contained  of difficulty  f o r the  i n Glennon's Test  o f t h e Vancouver Normal  School  as  determined  students  a.  at  the beginning  b.  at  the end o f the t e a c h e r - t r a i n i n g TABLE  five  of the teacher-training  period?  period?  VIII  A C O M P A R I S O N O F T H E ORDER O F D I F F I C U L T Y FOR THE  FIVE  AREAS O F ARITHMETIC  ON T H E T E S T  AND R E T E S T  Test  1.  Integers  2.  Decimal  3.  Decimals  4.  Fractions  and processes  5.  Rationale  o f computation  and processes  (79.0%)  system o f notation and processes  (76.2%)  (55.3%) (48.8%) (48.7%)  Retest  1.  Integers  2.  Decimal  3.  Decimals  4.  Rationale  of computation  5.  Fractions  and processes  Tables  VII  and processes  (83.2%)  system o f n o t a t i o n and processes  and VIII  (82.3%)  (57.3%) (55.6%) (55.1%)  are interpreted  i n the  following  manner: The test  order  i s essentially  of difficulty  f o r the five  t h e same f o r t h e f i r s t  test  sections of the and t h e  retest.  54 (Table one  VIII)  since  worthy  it  levels.  t h e means  lower  than  to  T h e means  f o r the last  those  population related and  sampled.  to decimals  are fairly  test  area  which  this  difficulty as  cance  even  items  into  been  hand,  Initial Consideration  will  whole  I  and VIII.)  and processesand known  over  gains.  were those  byt h e  and processes,  i t s position  i t  gain fails  well  a c h i e v e d ont h e four  areas o f  The exception  and processes.  (Table  i s the  VII.)  A l -  i n the order o f was v e r y  smalland,  to achieve  signifi-  level.  o f Gains  Status  as a  the understandings  which  i t s obtained  Distribution  definite  are considerably less  distributed  at t h e 5 p e r cent  note-  two s e c t i o n s o f t h e  to fractions  The g a i n s  demonstrated,  two  (Tables  to the integers  decimals  More  f o r the test  as a whole.  showed s i g n i f i c a n t  i s a minor  sections are considerably  o f computation  on the retest  The  of  three  section maintained  has already  differences.  those  and processes,  evenly  concerned with  though  than  On t h e o t h e r  known b y t h e same g r o u p .  the  appears  s y s t e m may b e s a i d t o b e w e l l  to the rationale  retest  which  f o r the f i r s t  f o r the test  understandings related the decimal  small  of the five  are considerably higher  while  The  i n order  i s based on very  i s the separation  difficulty test  The one change  According to the  o f the Students now b e g i v e n  to the sixth  problem  our study. 6.  average  amount  Is there of gain  any s i g n i f i c a n t i n achievement  difference o f basic  between t h e  mathematical  55 understandings made by students i n the lowest quarter of the cases i n the i n i t i a l test and the average amount of gain i n achievement of basic mathematical understandings made by students i n the highest quarter of the cases in the i n i t i a l test? Table IX gives a summary of the data related to this problem. TABLE IX Mean Test and Retest Scores for the Low Group ( N * 70) 3  and the High Group (N=» 70) Low Group (N 70) a  M  SD  Test  30.9  7.62  .92  Retest  38.0  9.93  1.19  SE  M  r  .66  °M  S E  diff  .90  7.1  t  7.89  High Group (N - 70) M  SD  SE  M  Test  66.2  4.42  .53  Retest  67.5  5.18  .62  r  .61  °M  1.3  S E  diff  .51  t  2.55  The data included i n the above table lead to the following conclusions: The students who were i n the lowest quarter of the cases i n the i n i t i a l test had a mean score of 30.9.  This same  56 group of  had  a mean  is  7.1  significant  The cases  in  the  retest  this  points  is  level  of  the  students initial  group  view  of  the  final fact  test.  that  The  the  t  gain  ratio  gain while of  is  fact  initial  test  highest  quarter  stated  score  of  but  not  by  of  quarter  at  of  On  66.2. The  67.5.  hypothesis  occur  made  group  the  when for  the  gain  of  1.3  1 per  cent  that  gains  as  not,  the  the  the  c h a n c e may  previously  a  very  and  small  two  the  are  compared w i t h  the  are  great  then,  be  significant  gain,  lowest  the  the  between  the  signifi-  quarter  on  students the  the  in  means  the  on  the  compared.  Incorrect  problems  section of as  in  a s much a s  each group  of  Responses  this  report.  study  will  These have  receive been  follows:  How m a n y Test  substantial  differences  final this  made  times  5.5  The in  a  students  Analysis of  Glennon»s as  null  might  The  7.  test  5,  highest  doubtful.  gained  retest  attention  at  group  high  which  and  score  the  the  a mean  a mean  The  gain  low  the  In  test  had  in  rejected.  The  cance  had  confidence.  obtained  who w e r e  test  significant  confidently  in  in  on the  38.0  7.89.  was  as  score of  of  the  eighty  reduced to their  five-choice  fewer-choice  status  on the  items  items  test?  contained  on the  re-  57 8. the  To what  misleads  correct extent rect  attributable  responses for is  it  the  responses within  choices  per  the  item  the  cent  of  number  method  item  described  record  in  number  plus  of  actual  of  the  of  which the  of  the  one  are  the  pattern  an  increase  the  each  the  summarized  of  the  the  The  320  that  ITEMS  of  of  what  the  of  incor-  actual  from  first  the  more  item  than  was  the  was  For 10  X.  X,  IN  Test  CHOICES PER  GLENNON'S TEST Retest  1- c h o i c e  items  1  0  2- c h o i c e  items  7  8  3- c h o i c e  items  28  36  4- c h o i c e  items  30  25  5- c h o i c e  items  14  11  per  the  eighty  ITEM  to  each  found.  determined  for  testees  step  misleads.  answer  A SUMMARY O F T H E NUMBER O F A C T U A L FOR T H E E I G H T Y  The  results  Table  TABLE  and to  number  received  correct  in  number  for  items?  chapter.  which  item.  the  consideration  responses for  for  in  rejection  redistribution  determining received  of  items  corresponding  misleads  choices per  test  to  previous  incorrect  is  corresponding  the  responses for  the  to  attributable  The  was  extent  This number items  58 The conclusions are based partly on the data i n Table X and p a r t l y on additional findings included below: On the i n i t i a l test 45 per cent of the items were either 4- or 5-choice items f o r the group as a whole. On the f i n a l test 35 per cent o f the items f e l l into t h i s category. According to the assumptions of the study the students were operating on a moderate l e v e l of understanding f o r t h i s t e s t . The mean of the number of choices per item on the i n i t i a l test was found to be 3.61. On the f i n a l test t h i s mean was reduced to 3.49.  I t might be concluded that the test  tended to be a 3.5-choice rather than a 5-choice test f o r t h i s p a r t i c u l a r group of students. The number of items which reduced to fewer-choice items was 20. Of these 16 were reduced by one choice while the other 4 were reduced by two choices.  Ten of the items gained  one choice each and 2 of them gained two choices each.  Thus  there were 24 reductions and 14 gains i n the choices per item on the r e t e s t .  The greater number of reductions i n choices  per item may well indicate that the students were operating on a higher l e v e l of understanding on the retest than they were on the f i r s t  test.  The findings i n r e l a t i o n to the r e j e c t i o n of the misleads are summarized and interpreted as follows: The t o t a l amount of rejection f o r the 179 rejected misleads was 1238. Of t h i s amount 1027 or 83 per cent was  59 found to  be  attributable  responses. in  an  The  increase  retest.  which  indicates  the  per  tribution ordinary data.  which  of  the  are  is was  Since was  attributed apparent  to  of the  to  197  equally  in  the  been  a  which  heretofore  relative  merits  responses  as  of  attributable  to  incorrect  understanding  no  be  or  has  that  the  correct change,  rejection  compared with  that  be  in in  to  statistical  the  of  drawn  such  the  mis-  to  the  it  of  the  that  is in  ignored.  concerning  of  in  of  a gain,  completely  favour  made  compared w i t h  responses,  favour  of  redis-  is  extent  perhaps  been  the  result  rejection  rejection  responses as  has  usual  consideration  a greater  seen  end  No p r o v i s i o n  the  c o n c l u s i o n s can the  the  remaining  cases the  there  realized  of  for  course,  correct  underlying  obscured in  responses.  of  r e s p o n s e s on  than  that  is  correct  much g r e a t e r  also  number  of  are  completely  change  is,  the  the  the  of  the  shown l a t e r ,  procedures  attributed  redistribution  rejection  number  found  of  in  be  incorrect  4.7  in  increase  this  true  statistical  an  total  place  these This  cent  of  as w i l l  take  and  procedures.  leads  in  However,  changes  17  effect  to  It  the  incorrect  the  correct  responses. The method or the  rejection test  between which  for  involved the  acceptance  each a  initial  deserves or  of  determining  of  the  study  of  and f i n a l  attention rejection  was index  the  amount  of  acceptance  alternatives  for  the  patterns  shift  tests.  in  One o f  concerned with which  was  the  the the  computed  items of  in  response  findings size for  of  the  each  of  60 the  alternatives  400  small  index  ponses.  For  responses that  a  number  on the  straight retest  on t h e  first  one  the  34  large  who  needed changes items, other  that  as  to  to  the the  less  whether  obvious XI  of  similar  for  it  appears  achieved.  gain the  answer  it  is  the  of  answer retest other  test  three  made  res-  responses of on the  rejected losses)  However,  On t h e  a m i s l e a d on  who  it.  can  It  that  changes  are  of  understanding in  in  the  result regard test  to  evident rather  research  the  and  is  involve  unlikely  the  retest  chance f u r t h e r  reliability  illustrates actual  correct  information  surface  is  actual the  itself,  test  or  to  factors.  between  the  correct  retest.  between  res-  of  misleads  The  a  number  been  on the  Since  they  of  on the  (or  by  number  the  correct  gains  cases  correct  retest.  could occur  students'  the  many  s e l e c t e d the  selected  students  small  responses involved. data  item  responses.  lack  of  In  large  On t h e  had  difference  31  the  these  relationships  the  the  very  Table  of  who h a d  determine  in  the  accepted the  numbers o f  large  who  s e l e c t e d one  represents  this  those  omitted  responses of  from  as  test  23  item  233•  a  3 responses has  r e s p o n s e on the  students  the  of  students  correct  ponses  30  of  was  test.  involved  on t e s t  retest gain  student  34  hand,  actually  example,  on the  and  comprising the  While  the  gains this  answers  c o u l d be  it  findings (or  concerning  losses)  summary h a s s h o u l d be  provided  for  the  and the been  the number  limited  realized misleads.  that  to  61 TABLE XI THE ACCEPTANCE-REJECTION PATTERNS FOR THE EIGHTY ITEMS OF GLENNON»S TEST AS DETERMINED BY THE RESPONSES TO THE CORRECT ANSWERS ON THE TWC) TESTS Correct Responses on the Initial Test  Correct Responses D i f on the ference Retest  Acceptance (New Responses Gained)  Rejection (Former Responses Lost)  Sum o f Respon! Involvi i n Shi:  1.  279  260  -19  1  20  21  2.  272  268  - 4  7  11  18  3.  267  271  + 4  8  4  12  4.  262  +19  43  239  - 8  31 22  12  5.  243 247  30  52  6.  247  264  +17  30  13  43  7.  263  263  0  13  13  26  8.  127  234  +107  111  115  9.  255  263  • 8  19  4 11  10. 11.  245  258  26  13  39  211  232  +13 +21  47  26  73  12.  186  188  + 2  26  24  50  13.  219  227  + 8  32  24  56  14.  181  213  +32  68  36  104  15.  163  162  - 1  42  43  85  16.  280  275  - 5  0  5  17.  267  276  + 9  11  5 2  13  18.  251  267  +16  22  6  28  19.  235  264  +29  36  7  43  20.  198  223  +25  46  21  67  21.  247 237  262  +15  25  10  35  242  + 5  24  19  43  23. 24.  230  233  + 3  34  31  65  212  227  +15  42  27  69  25.  268  259  - 9  5  14  19  22.  30  62 TABLE  26.  122  27. 28.  176  130 190  277 268  275 270  234 271 129 277 146 246  233 267 165 279 159  29. 30.  31. 32. 33. 34. 35. 36. 37. 38.  135 127 165  39. 40.  64 168  41. 42.  169  43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.  64 96  231 198 183 160 214 117 145 166 160 252  XI—Continued  • 8  45  +14 - 2  44 2  + 2  7  - 1  23 6  - 4 +36 + 2 +13  73 3 59  + 3  25  +31 +24 + 6  65  - 3 +12 +21  32  104 242  +20 + 8 +11  55 33 35  218 211  +20 +28  204 233 I46  249 166 151 171 61 180 190 84  143 174 135 257  5455. 56.  217 181 247  209 187 240  57.  163  177  54 56 52 57  37 30  4 5 24 10 37 1 46 22 34 30  50 35 40 36  82 74 6 12 47 16  110  4 105 47 99 84 106 67 92  25  93 90 58  45 50  24 25 22  59 70 72  +44 +19 +29 - 2 + 8  69  25  94  37  18 26  55 81 84 82  -25  43 23  43 37 68  55  41 45  35  18  111  + 5 - 8 + 6  40 42  48  41 88  36  78  - 7  20  47  +14  34  27 20  54  63 TABLE  53.  206  XI—Continued  - 7 + 6 -12 - 1  213 146  152  235 91 116  223 90 122  133 231  123 232  -10 + 1  187 119 111  177  -10  35 130  -34 +19  69. 70.  151 210 168  155 228 225  • 4 +18  71. 72.  220 61  235  +15 +22  73.  207 120  193 193 240 102  -14 +73 + 5  194 225 250 132  59. 60. 61. 62. 63. 64. 65. 66. 67. 68.  74. 75. 76.  235 68  77. 73.  185 218  79. 80.  243 179  83  • 6  +57  33 40 28  51 46  40  73  34 40 52  74 68  40  31 34 42  41 33 52  25 57 42  59 33 38  37 78  19 21  37 46 26  22  103 86 72 67 94 84 95 80 56 99  59 70 66  97 26  24 40 24 21  +34  50  16  47 66  + 9 + 7 + 7  35 32  26  61 57  29  25 22  + 3  45  42  121  51 87  64 From the differences the  test  total each  of  the  responses for  retest  was  found  is  been  reliability apt  to  occur  However, number test not  of  of  answers  was  found  to  be  great  of  the  great  is  a  test as  were  standard item  fully  large  as  considerable items  suggest  that  amount  very  of  both  the  group  responsible  and  test  tests.  may  affect  for  the  the  means  administrations  the  of  have the  guessing would  difficult  deviations,  former  there  course,  from the  two  these  of  and  degree  on the  on  the  the  question.  would,  the  Since  open to  that  for  63.7.  is  for  indicate  as  test  A large  test  difficulty  guessing is  times  responses to  choices per to  there  test.  means,  serve  six  guessing which  the  mean o f  responses to  as  the  if  the  all  that  of  The  of  answers  of  reliability  considerable  correct  10.4«  be  mean  shift  the  Changes  the  the  the  approximately  in  to  XI  in  concluded that  inconsistency the  Table  the  correct  mean  in  between  and  can be  Thus  provided  responses involved  latter it  data  was  of  examinees. of  of  as  It  is  for  changes  the  understanding  the  the  moderate  concerned.  be  but unlikely  great  as  those  found. Perhaps group  in  relation  change must as w e l l the  gains  points as  as  take  real  to  and  the  or  63.7.  test  form  losses the  change  the  considerable  while  large,  a  It  items  was  occurred.  considerable  loss  gain  because  the  the  correct  answers  for  mean  of  in  sum o f is  the  losses  unlikely  that  mean  in  the  so,  was  loss  in  this  understanding  difference  and g a i n s a  If  of  only was  between  10.4 six  times  understanding  65 would must  ensue be  after  period of  conflicting  an e s t i m a t e  of  the  formula  high  of  coefficient  an o b t a i n e d  the  scores  less  while  more  than  given  to  puting  the  Such  findings  of  test  to  be  imply of  that  Clarification  is  source of that  other  test  concerned with of  the  words,  in  g r o s s changes which  tuations  which  is  to  the  is  error  s h o u l d be .  methods  required  s h o u l d be  the  have  test-retest  taken  total  changes.  of  com-  as t o  considered  place;  underlying  pattern the  of  does  changes which  their is  the  in  given  to  receive result  responses to  situation.  the  from test  changes  in  No a t t e n t i o n  is  show t h e  extent  of  the  in  words,  no  consideration  other  changes which  Should differences  obscure the  consideration of  net  consideration  to  the  the  group  given  obviously  and to  c o n s i s t e n c y which  in  net  in  reliability.  scores  above  or  points.  reconsideration  reliability  a  of  points be  is  error  two-thirds  expected to  9.99  l e s s than  which  standard  3-33  by  general  .94,  the  that  error  may b e  total  given  in  The  of  Moreover,  c o n s i s t e n c y which  fluctuations in  but  definition  The  items;  explanation  computation  test.  indicating  third  reliability.  estimating  attention  remaining  the  the  a reliability  3-33,  expected  points  test  sources  the  so a n o t h e r  required of  reliability.  3.33  the  gave  s c o r e was  may b e  evidence  reliability  Kuder-Richardson  of  training,  sought. The  of  a  result  fluc-  in  in  test  totals,  changes,  take  precedence  underlying  changes which  the  which over  determine  66 these  totals?  reliability i t e m who  are  of  the  test?  in  the  area  Should weight  to  the  proportion  consistent Such of  test  in  be  given  in  methods  of  of  those  passing or  doing  so on  successive  questions  form  reliability.  the  basis  for  computing  failing  an  applications  further  research  67 CHAPTER  SUMMARY AND  The ments the  of  mathematical  bases of  arithmetic  in  the the  literature paucity  present  evaluation  its  effectiveness  students from  a  1  of  study  study  several In  initial  the  one to  correct  bringing  bringing  data  and f i n a l  year's  as w e l l  as  The  data  1.  The  See  in  of  the  of  in  basis  for  gave  justification  the  problem,  to  determine in  had r e c e i v e d of  the  the attention  arithmetic.  present  research  noted.  presented  c o m p a r i s o n s w e r e made  status  the  of  arithmetic,  students within  justify  teacher-training  I  this  the  test  in  was the  to  limits  of  was of  in-  results.  following  gains  report.  time  evaluation  the  between  relation  An a t t e m p t  programme  significant  of  the  responses i n  seemed t o  but  of  in  the  programme.  effectiveness  Chapter  enthusiastic  improvement  field  of  sources.  programme  an  the  area  phase o f  arithmetic,  the  correct  small  this  achieve-  understanding  had r e c e i v e d  about  teacher-training  determine  about  studies  d i v e r g e n c e s were  understanding  their  teacher-training  formed  1  to  f r o m many a u t h o r i t a t i v e  investigator  Glennon's  their  the in  regard  The p a r t i c u l a r  understanding  previous  but  of  study.  the  made  and i m p r o v i n g the  in  the  the  determining  teachers  Moreover,  the  for  IMPLICATIONS  student  support  to  need  IV  conclusions:  effective students'  in  68 achievement by  in  basic  Glennon's Test.  Both  students  in  this  obtained  by  Glennon.  paribility and the make of  the  the  took  have  place  training one  gains  of  obtained  programme group  of  had taken  contributions of  to the  the  led  to  students'  and no  decimals  arithmetic  taken  comparing  decimal  However,  of  measure  the  putation.  w e r e made w e r e  the  in  results the  com-  two the  studies two  relation  studies to  the  teacher-training to  programme  different  groups,  place.  In  the  initial  the  conclusion  of  and  pro-  measure by  he  the  effectiveness  teacher-training  fractions  of  of  the of  the  programmes  Glennon attempted  two  had  the  of  com-  concluded  present the  teacher-  final that  results significant  place.  processes,  area  end  When  results  findings  measured  regarding  in  teacher-training  gains  by  the  as  status  with  evidence  testees  the  the  for  testees  The  2.  the  the  attempt.to  areas  at  final  teacher-training  the  However,  significant  the  of  the of  and  favourably  lack  significance.  results  no  study  the  training  effectiveness  that  The  understandings  initial  compare  testees,  value.  which  paring  the  c o m p a r i s o n s between  doubtful  gramme  study  previous  direct  gains  for  of  mathematical  fairly  processes,  significant  and p r o c e s s e s .  concerned.  made  understanding  system of  evenly  programme  of  notation, and the  arithmetic  integers rationale  contributions The  distributed  significant in  and of  com-  w e r e made  in  contributions  which  over  areas  the  four  69 3. of  notation  The  understandings related  and t o  the  known t o  the  were the  understandings related  The in  order  testees  integers  of  on b o t h  difficulty  Glennon's Test  of  remained  the  and p r o c e s s e s  the  the  to  initial to  the  five  were  system  much  better  and  final  tests  than  other  three  areas  tested.  areas  essentially  decimal  of  the  arithmetic  included  same t h r o u g h o u t  the  study. The  4. programme was achievement students initial  the  who w e r e  who w e r e  in  as the  5. per  further  of  test  item  in  considered  When t h e  as  a  test.  Moreover,  at  end o f  and f i n a l retest were  lowest  of  the the  items tests.  those  the of  of  the  Of  the  the  the  level  level  in  the  initial  items  which  of  actual choices  was  understanding found that  upon  of  understanding for  than  the items  test  a greater  number were  reduced to  expanded to  more-choice  items.  was  the  tended to  status  fewer-choice  this  higher  at  on both t h e  changed i n  the  a little it  test.  choices  was  of  5-choice  the  it  period  items  in  students  listed  of  the  the  number of  cases  in  by  u n d e r s t a n d i n g was  The than  of  cases  number  teacher-training  rather  superior gains  u n d e r s t a n d i n g s by  was w o r k i n g ,  level  period.  the  attained  of  the  on a moderate the  of  criterion  testees  in  teacher-training  quarter  quarter  determiner  was w o r k i n g  the  mathematical  comparison with  group  3.5-choice  the  highest  a group of  beginning  basic in  of  demonstrated  compared w i t h  which  the  effectiveness  be initial  on  items  the than  70 An i n c r e a s e  6. to  an  item  certain the  of  study  mislead,  that was  made  in  misleads the  in  in  is  area  The of by  correct  The mean  of  responses  under  the the  for  further  incorrect findings  correct  need  in  per  research  responses to  concerning  of was  research  consideration  of  involved  mean the the  the  answer of  the  correct shift  indirectly  for  area items  shift  each  the  in  test  correct  answers  of  study  the  in  the  items  the  in  responses involved  times  so  recog-  cent  further  of  from  and  rejection  this  for  place  giving  number  responses for  the  place  17  took  procedures.  for  Since  correct  showed  items  test  the  The  the  the  the  indicated.  six  tests.  misleads.  of  which  statistical  by  testees  study  to  and  of  to  procedures  the  the  of  particular  the  misleads  answers  took  by  favour  of  of  assumption  that  in  revealed  which  approximately number  of  usual  misleads  be  a mislead  findings  customary  other  A major  made  responses  rejection  answers  the  to  was  The  other  correct  concerning  understanding  need  of  correct  misleads  of  evaluation  the  in  of  item.  rejection  the  seems t o  substantiated to  made  of  7. the  of  favour  favour  this  rejection  the  of  corresponding  rejection  recognition  rejection  made  of  number  that  mislead.  greater in  for  a  understanding  favour  due  to  the  the  cent  provision  nition  that  another  per  83  misleads  whether  received No  the  greater  or  the  accompanied by  was  signified  answer  is  in  of  answers  a  was  responses  test-retest the  in  situation.  shift  item  differences  of  of  of  was  between the  two  responses to  and  consideration  of  from the  71 corresponding the  items.  obtained  shift  It  for  misleads.  That  has been  these  must  correct there  from  While of  that  would  is  room  for  tribution  student-teachers  the  limitations  which  require  those  which  the  it  quarter  is  to  to  most  of  be  the  heeded  the  as  well  as  s u g g e s t e d by t h e  of  arithmetic  research  is  in  areas  these  of  by  of  to of  the  areas  been  known  now.  gains.  of  It  for  is  the  in  the  have  been  discover to  what  extent  contribute  since  arithmetic.  as w e l l  evident has  in  need  made  the  to  as  that  been  statistical  teachers  processes  con-  low  consider  procedures  Research  and weakness  attention  the  study.  Further  can  the  arithmetic  who w e r e  Finally,  this  of  indicated,  responses in of  in  significant  understanding  1  have  the  findings  weakness  and  a  students  Particular  decimals  the  shown t h a t  since  correct  student  has  understanding  strength  needed to  understanding. study  of  for  responses  students'  Suggestions  Areas  of  those  interpretation  study  study  well  showed s u b s t a n t i a l  incorrect  The  c o n c e r n e d makes  improvement  appear  contribution  where  was  most  obtained  much i m p r o v e m e n t  this  programme  Within  be  to  for  study.  understandings  the  alternatives  similar  patterns  study.  teacher-training to  incorrect  results  definite  this  further there  the  answers  are  shown i n  await  achievement  and  seems l i k e l y  the  items  to  determined.  to  definite further  s h o u l d be this  understanding  area  given  Further training  gains to  showed  the no  in  72 s i g n i f i c a n t gain i n the present study. The need f o r further research i n the f i e l d of test r e l i a b i l i t y has already been mentioned.  Does the extensive  s h i f t i n responses to and from correct and incorrect answers to items occur i n most t e s t s or was i t more prevalent i n t h i s test than would o r d i n a r i l y occur?  Is such a condition more charac-  t e r i s t i c of tests of understanding than of other types of tests? How  can such a s h i f t be given the necessary weight i n determining  test r e l i a b i l i t y ? Another question which arises from the findings i n regard to the s h i f t i n responses i s that concerned with the degree of understanding which i s being tested i n Glennon's Test.  Could the obtained changes r e f l e c t a condition i n which  students answer c o r r e c t l y without having a r e a l understanding of the item being tested?  In other words, are the basic mathe-  matical understandings r e a l l y basic? i  Is i t possible to devise  another test i n which the understandings are more basic and i n which the students would have to have such a thorough understanding of them that the resultant certainty i n regard to the test items would obviate or lessen such s h i f t i n g i n responses?  APPENDIX A Name  A BASIC  TEST  MATHEMATICAL  OP UNDERSTANDINGS  Directions; T h i s i s a t e s t t o s e e how Y o u do n o t h a v e t o do a n y  w e l l you understand arithmetic. w r i t t e n work to f i n d the a n s w e r s .  Read each statement c a r e f u l l y and d e c i d e which of the a n s w e r s i s the c o r r e c t answer._ w r i t e the l e t t e r f o r this a n s w e r on t h e l i n e a t t h e r i g h t o f t h e example. Sample  item:  '•Vhich o f  .the  following  A.  B.  9  23  C.  45 i s t h e c o r r e c t the r i g h t . .  numbers  35  L.  answer  has 45  s o we  go  all  the  Shaded p i c t u r e s  Remember Now  begin  -at  way  are  through  read  top  of  value?  11  D on  J the  line  the  test  WORK TO F I N D  the  next  without  kind  at  stopping.  THE ANT-7ERS,  page.  Copyright 1947 by Vincent J . Glennon r e p r o d u c e d by  )  t o o l o n g on a n y o n e a n s w e r y o u may go on  thus:  DO NO WRITTEN the  largest  E.  write  T r y e a c h e x a m p l e b u t do n o t s t a y example, If you c a n n o t f i n d the to the n e x t example. Y o u may  the  permission  of  the  author  9.  "ifhich o f t h e place? A. 423,102 D. 3 7 4 , 9 4 2  following  has B. E.  10. If the f i g u r e s in 8 6 , 4 7 3 of the f o l l o w i n g would p l a c e thousands' place? A, 73,648 . B. D. 8 7 , G 4 3 E.  -  2  -  a  4  in  the  ten  thousands*  643,142 763,420  C.  were a r r a n g e d d i f f e r e n t l y , the l a r g e s t f i g u r e in the 33,467 86,734  C.  11.  In t h e n u m b e r 3 , 9 4 4 t h e 4 o n t h e v a l u e how many t i m e s a s l a r g e e s A. 1/10 B. 1/2 D . 1 (same v a l u e ) E . 10  12.  In t h e number 5 , 4 9 2 t h e 4 r e p r e s e n t s t i m e s a s lar.-';e a s t h e 2? A. 2 " B. 10 D. ICO E. 200  a  A b o u t hcv! A. 3-1/2 D . 3,500  34,820?  13.  14.  438,116  man" h u n d r e d s a r e t h e r e B. 35 35,000  right the 4  in  which  70,483  represents a cn the loft? C. 5  value  how  many  C.  20  C.  350  "thich of the f o l l o w i n g methods i s the b e s t f o r d e t e r m i n i n g the v a l u e of a f i g u r e in a number? f o r e x a m p l e , the v a l u e of the 7 i n 3 7 4 6 . A. Its p o s i t i o n i n the number. B . I t s v a l u e when c c m p a r e o v i t h o t h e r f i g u r e s i n t h e number. C . I t s v a l u e i n t h e o r d e r f r o m 1 to 9. D. I t s v a l u e when c o m p a r e d w i t h t h e w h o l e o f t h e numbe . E . I t s p o s i t i o n i n t h e number a n d i t s v a l u e , -  15.  In t h e n u m b e r 7 , 8 4 3 t h e 4 r e p r e s e n t s t i m e s ns larcre a s t h e 3 ? A. 1/10 . B. 1/20 D. 2 E . 20  a  value  ho'.v many C.  1/2  Section  II  Ba s i c  understandings  of  Integer?  and  processes.  1.  "f y o u h a d a b a g o f 365 m a r b l e s t o w h i c h " x u l d b e t h e q u i c k e s t way t o A. c o u r t i n g B. adding D. m u H i p l y i n g E. nividing  be s h a r e d determine  e q u a l l y by 5 b o y s , each b o y ' s share? 0. subtracting  2.  "hen whole number i s m u l t i p l i e d by how d o e s t h o -answer c o m p a r e w i t h t h e A. larger B. s m a l l e r P . 10 t i m e s s s l a r g e E . c a n ' t t e l l  3.  ^'foen a w h o l e n u m b e r i s d i v i d e d b y a w h o l e n u m b e r o t h e r t h a n how d o e s t h e a n s w e r c o m p a r e w i t h t h e w h o l e n u m b e r d i v i d e d ? A. larrer B. s m a l l e r C . same D. o n e - h a l f a s l a r g e S . c a n ' t t e l l  4.  ^ h i c h of the f o l l o w i n g i s the q u i c k e s t s e v e r a l n u m b e r s o f t h e same s i z e ? A . by c o u n t i n g E . by a d d i n g D. b y m u l t i p l y i n g E . by d i v i d i n g  a whole number o t h e r than. whole number m u l t i p l i e d ? C . same  way  to  C.  by  find  the  sum  1,  1,  of  subtracting  5;  I f t h e z e r o s i n t h e two how w o u l d t h e a n s w e r b e A. The a n s w e r w o u l d b e B. The a n s w e r w o u l d b e C. The a n s w e r w o u l d be D. The a n s w e r w o u l d b e E. The a n s w e r w o u l ; ; n o t  6.  H e r e i s an e x a m p l e i n - u b t r a c t i o n in which l e t t e r s instead of f i g u r e ? . "hich statement is true? A . A P G B a n d C X U a d d e d t o g e t h e r a c u a l TIVLY. B . CXU a n d M added t o g e t h e r e q u a l APGB. C . AFGB a n d T ' O T a d d e d t o g e t h e r e q u a l C X U . D» T " ? T s u b t r a c t e d f r o m CXU e q u a l s A P G B * E . C X U s u b t r a c t e d f r o m T™KY e q u a l s A P G B .  n u m b e r s i n t h i s e x a m p l e were l e f t off, changed? 6 0 / 3720 ten t i m e s es larjge. one h u n d r e d 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 . o n e - h u n d r e d t h as l a r g e . change. . have  been  1 !  -  APGB CXU TTMY  ;  7.  How w o u l d t h e a n s w e r t o t h i s e x a m p l e be c h a n g e d , i f a z e r o were a d d e d to t h e r i g h t o f e s c h number? 364 A . T h e a n s v e r w o u l d bo t e n t i m e s a s L a r g e . 2936 B . The a n s w e r w o u l d be o n e h u n d r e d t i n e s a s l a r g e . 14 C . The a n s w e r w o u l d n o t c h a n g e . 438 D. C a n n o t t e l l u n t i l y o u 3dd b o t h w a y s . E . ^htf a n s w e r w o u l d be o n e t h o u s a n d t i r r e s a s l a r g e .  8.  A d d i n g two z o r o s t o t h e r i g h t o f a whole number effect as: A d d i n g ten to tha number, B. A d d i n g one h u n d r e d to the n u m b e r . C. M u l t i p l y i n g the number by t e n . D. M u l t i p l y i n g the number by one h u n d r e d , E. D i v i d i n g t h e number by one h u n d r e d .  has  the  same  used  -  10,  11.  12.  4  -  •Vhat w o u l d b e t h e e f f e c t o n t h e a n s w e r i f y o u a d d e d t o 439 a n J t o o k a"n;/ the z o r o f r o m 450? A. Trie a n s w e r w o u l d by t e n t i m - j s a a l a r g o , P. The a n s w e r w o u l d be o n e h u n d r e d t i m e s a s l a r g e , 0. The a n s w e r w o u l d r e m a i n t h e s a m e , D. The a n s w e r w o u l d b e o n e - t e n t h a s l a r g e , !?. The a n s w e r ' " o u l d b e o n e - h u n d r e d t h a s l a r g e .  two  Crossing off a zero from some e f f e c t as: A . S u b t r a c t i n g ten C . M u l t i p l y i n g by ten E . D i v i d i n g by t e n .  has  the  right B, D.  side  of  Subtracting Multiplying  a numbor  7  14.  15.  439 x450  the  one h u n d r e d by one  "Vhat w o u l d b e t h e e f f e c t o n t h e a n s w e r i f y o u a d d e d ( a n n e x e d ) two z e r o s t o 98 a n d c h a n g e d 4 5 0 0 t o 4 5 0 ? A . T h e a n s w e r w o u l d be t e n t i m e s a s "V 3 r g e . 9 2 / 4500 '3, Tii6 a n s w e r w o u l d bo o n e - t e n t h a s l a r g e . C . T h e a n s w e r w o u l d be o n e h u n d r e d t i m e s lar.f/j, D. The a n s w e r w e a l d be o n e - h u n d r e d t h a s l a r g e . E . The a n s w e r w o u l d b e o n e - t h o u s a n d t h a s l a r g e . "Jhich o n e o f t h e f o l l o w i n g m e t h o d s c o u l d be u s e d t o f n d t h o answer to t h i s example? A . m u l t i p l y 17 b;- t h o q u o t i e n t . / * B . A d d 17 s i x h u n d r e d t w e l v e t i m e s . Answer w o u l d be the s u m . . C . S u b t r a c t 17 f r o m 612 a s many t i m e s as c o s s i b l o . A n s w e r v . c u i d be n u m b e r o f t i m e s y c u w t - r e a b l e t o subtract. D . A d d 612 s e v e n t e e n t i m e s . Answer w o u l d be the sum. E . M u l t i p l y 17 b y 6 1 2 . A n s v e r v c u l d be t h e p r o d u c t ,  T  ?  1  T  zeros  7  b  l  If t h e n u n b o r n "n a l a r g e a d d i t i o n • e x a m p l e w e r e c h a n g e d , s c t'y.wt the t o p number w9!» p l a c e d s t t h e b o t t o m a n d the bott^-r. number was p l a c e d a t t h e t o p ; how w o u l d t h e a n s w e r be nffectv'i? A. Answer would be l a r g e r . B. A n s w e r w o u l d be s m a l l e r . : C. Answer would not change, L. C o u l d n o t do t h e example! E. C a n n o t t e l l u n t i l y o u a d d b o t h ways a n d c o m p a r e . . How w o u l d t h e e x a m p l e b e a f f e c t e d i f y o u p u t t h e 29 A. The a n s w e r w o u l d be l a r g e r . B. The a n s w e r w o u l d be s m a l l e r . C. The a n s w e r w o u l d be t h e s a m e . D. C a n n o t t e l l u n t i l you m u l t i p l y b o t h way... E. Y o u c a n n o t do t h e e x a m p l e when t h e l a r g e n u m b e r on t h e b o t t o m a n d t h e s m a l l n u m b e r on t o p .  above 4306? 4306 x 29  is  "•'hat w o u l d b e t h e e f f e c t o n t h e a n s w e r i f y o u a d d e d ( a n n e x e d ) t'.vo z e r o s t o 3 9 ? A. The a n s w e r w o u l d be o n e h u n d r e d t i m e s a s l a r g e . 3 0 / 359 B. The a n s w e r w o u l d be o n e - h u n d r e d t h a s l a r g - : . C. Tl.e a n s w e r w o u l d b e o n e - t h o u s a n d t h a s l a r g e . D. The a n s w e r w o u l d n o t c h a n g e . E. Y o u c o u l d n o t do t h e e x a m p l e .  Section 1.  III  Basic  5  and  u n d e r s t a n d i n g s of f r a c t i o n s  W h i c h of t h e f o l l o w i n g A . 1/7 B. 5/7 C. 3/7  Tactions D. 1 1 / 7  processes.  i s the l a r g e s t ? E . 6/7  _  W h i c h o f t h e s e s t a t e m e n t s b a s t t e l l s why we c a n n o t s a y t h a t the unshaded p a r t s of t h i s p i c t u r e r e p r e s e n t s 5 ' e i g h t h s ' ? A . B e c a u s e more than 5 / 8 o f i t i s u n s h a d e d , 7A B. B e c a u s e the u n s h a d e d p a r t s are n o t together, C . B e c a u s e a l l t h e u n s h a d e d p a r t s a r e n o t same s i z e . D. B e c a u s e l e s s t h a n 5 / 8 o f i t i s u n s h a d e d . E . B e c a u s e t h e p a r t s a r e n o t t h e same s h a p e . 3.  "Thich o f the f o l l o w i n g A . 1/9 B.. . 1 / 5 C . 1/2  4.  Which p i c t u r e the numerator  5*  When a w h o l e how d o e s t h e A . l a r g e r B.  6.  'Vhich p i c t u r e the numerator A.  IF  'Vhich  pic ture  smallest?  s h o w s how t h e r e s u l t w o u l d l o o k a n d d e n o m i n a t o r o f 1 0 / 8 b y 2? B. C.  if  you  divided  n u m b e r i s m u l t i p l i e d b y a common ( p r o p e r ) a n s w e r compslre w i t h t h e w h o l e n u m b e r ? s m a l l e r C . same D. c a n n o t t e l l E . h a l f a s s h o w s how t h e r e s u l t w o u l d o f t h i s f r a c t i o n by 2? —  best  look  if  you  fraction large  divided  1  B.  D.  7.  f r a c t i o n s i s the D. 1/7 E. 1/3  0  shows t h e B.  example  4 x 2  LT. C.  8.  wm um  ••••  E.  t i  V/A  y/2  "Then a common ( p r o p e r ) f r a c t i o n i s d i v i d e d b y a common f r a c t i o n how d o e s t h e a n s w e r c a m p a r e w i t h t h e f r a c t i o n d i v i d e d ? A. larger B. s m a l l e r C . same D. c a n n o t t e l l E . twice as l a r g e  9*  - 6 Which p i c t u r e shows how t h e r e s u l t would l o o k the n u m e r a t o r and d e n o m i n a t o r o f 3/5 by 2?  i f you  multiplied  -TT-  B  A  _  D  10.  E  ^Vhich p i c t u r e shows how t h e r e s u l t w o u l d l o c k i f you m u l t i p l i e d t h e d e n o m i n a t o r o f t h i s f r a c t i o n by 2? — »  c.  B.  A.  j _ E.  D.  "jhen a whole number i s d i v i d e d by a common' ( p r o p e r ) f r a c t i o n how does t h e answer compare w i t h t h e whole number? A. larger B. s m a l l e r C. same D. c a n n o t t o l l E. v a r i e s 12.  "faich.of  the p i c t u r e s  looks  like  t h i s example?  3—1/2  C.  13.  Which  sentence best  tells  6  why  ^! =4  the answer  is larger  t h a n the  A. B e c a u s e i n v e r t i n g the d i v i s o r t u r n e d tho 3/4 u p s i d e down. B. Because m u l t i p l y i n g a l w a y ? makos t ' 9 answer l a r g o r . C. Because t h e d i v i s o r 3/4 i s l e s s t h a n 1. D. Because d i v i d i n g by p r o p e r and i m p r o p e r f r a c t i o n s makes the a n s v e r l a r g e r t h a n tho number d i v i d e d . E . I n v e r t i n g a f r a c t i o n p u t s t h e l a r g e r number on t o p . 1  '•'hich s o n t o n c o i s shown by  this  picture?  2fc  A. F r a c t i o n s v-'ith comn.on d e n o m i n a t o r s may bs added, 3 . The v a l u o of a f r a c t i o n i s c r a n g o d i f a number i s s u b t r a c t e d from the numerator and denominator. C. D i v i d i n g t h e n u m e r a t o r and d e n o m i n a t o r o f a f r a c t i o n by the san-o number doas n o t change t h o v a l u e of t h e f r a c t i o n . D. F r a c t i o n s w i t h t h e 'same d e n o m i n a t o r s a r o e q u a l . 15,  'Vhen a common ( p r o p e r ) f r a c t i o n i s m u l t i p l i e d by a common f r a c t i o n how d o e s the answer compare w i t h the f r a c t i o n m u l t i p l i e d ' ; A. l a r g e r  B,  smaller  C. same D.  cannot t e l l  E.  varies  Section 1. How A. 2. How A. B. C. D. E.  IV E a 3 i c  understandings  7 o f d e c i m a I s and  processes.  s h o u l d you w r i t e tho d e c i m a l ' e i g h t y and e i g h t h u n d r e d t h s ' ? .8008 B. 80.800 C. 80.08 D. 80.003 E . 3008.03 s h o u l d you r e a d t h i s d e c i m a l ? .0309 Three and n i n e h u n d r e d t h s . Three h u n d r e d n i n e t h o u s a n d t h s . Tnree h u n d r e d n i n e t e n - t h o u s a n d t h s . Thirty-nine thousandths. Three h u n d r e d n i n e h u n d r e d t h s .  3. 'Vhich d e c i m a l t e l l s how l o n g l i n o Y i s when compared line X . . . i . . . > line Y J A. .5 B. .625 C. 1.25 D. .75 E . .33 4. About hov; many t e n t h s a r e t h e r e A, .13 B. 1.3 C. 13  in  1.25? D. 125  5. About how many h u n d r e d t h s a r e t h e r e i n A. 1/2 B. 6.35 C. 63.5 D. 635 6.  7.  line  X?  1250  .635? E. 6350  Vhat would be the e f f e c t on the answer i f you d r o p p e d the z e r o f r o m 23.90? A. The answer w o u l d have the same v a l u e , 23.90 B. The answer w o u l d be o n e - t e n t h as l a r g e . x 2.75 C. Tho answer w o u l d be t e n t i m e s as l a r g o . D. You w o u l d p o i n t o f f t h r e e p l a c e s , E . I t would be the same as s u b t r a c t i n g z e r o f r o m t h e answer,  1!  How w o u l d t h e 84.5 t o 845? A. Tho answer B. The a n s w e r C. The a n s w e r D. The answer E. The answer  answer would would would would would  bo be ba bo bo be  changed  A.  i f you c h a n g e d 6,5  t o .65  and  t h e same. t e n t i m e s as l a r g o . 6.5/34,5 one h u n d r e d t i m e s as l a r g e . o n e - t e n t h as l i r g o . o n e - h u n d r e d t h as l a r g e .  8. 'Vhich  9.  E.  with  seems to be the c o r r e c t answer to t h i s ton d i v i d e d by f i v e - t e n t h s 1/2 B. 2 C. 10 D. 20 E . 50  example?  h i c h d e c i m a l t a i l s how l o n g l i n e Y i s when compared w i t h l i n e X? line X , , , . , , l i n e Y, , , , u^-, , , , , . • t A. 1.25 B. 1.50 C, 2 D. 2.40 E . 2.50 _____  ,,?  10. 'Vhich o f t h e f o l l o w i n g d e c i m a l s A. 30,3 E. 30.03 C. 30.0333  has tho l a r g e s t v a l u e ? D. 30.303 E . 30.003  11. 'Vhat w o u l d be the e f f e c t on the answer i f you c h a n g o d 368 to 3630 and 24 to 2.4? A. Tho a n s w e r w o u l d be s m a l l e r . 363 B. I t . w o u l d n o t c h a n g o d the answer. x 24 C. I t w o u l d be tho samo as a d d i n g a z e r o to tho answer. D. Tho answer w o u l d be o n e - t o n t h as l a r g o . E. Cannot t e l l u n t i l you do the example b o t h ways.  -  12,  13,  8  -  Which d e c i m a l h a s t h e s m a l l e s t v a l u e ? A. .3 B. .09 C. .048 D. ,0693 E. How one A, B. G. D, E,  .0901  would the answer be a f f e c t e d i f you moved t h e p o i n t p l a c e to the l e f t i n b o t h numbers? The answer w o u l d be o n e - t e n t h as l a r g e . 43,5 The answer would be o n e - h u n d r e d t h as l a r g e , x' 4 , 8 The answer would be one h u n d r e d t i m e s .as l a r g e . I t o u l d be the same as s u b t r a c t i n g 100 f r o m the a n s w e r . The answer would have t h e same v a l u e . w  14, Fow would the answer be c h a n g e d i f you moved the p o i n t two p l a c e s to the r i g h t i n b o t h numbers? A. The answer would h a v e - t h e same v a l u e . 43.6 B. The answer would, be one t h o u s a n d t i m e s as 13 r g o , x 2 • 45 C. You would p o i n t o f f d i f f e r e n t l y . D. You c a n n o t move t h e p o i n t i n t h e t o p number two p l a c e s , E. The answer would be 10,000 t i m e s as l o r g o , 15. How in A. B. C. D. E.  w o u l d the answer be a f f e c t e d i f you moved the p o i n t 485.3 one p l a c e to the r i g t ? The answer would t e n ' t'men r:s l a r g e . 62/ 485,3 The answer would bo 10 l a r g e r * Tiie answer would bo. o n e - t e n t h us l a r g e . The answer w o u l d h i v e a s e r o i t the r i g h t ; The v a l u e o f t h e answer w o u l d be the stsmy,  16. Fow and A. B. G» B. • E*  w o u l d t h e answer bo a f f e c t e d i f you c h i n g j d 7.3 t o 75 1390 t o 13.90? 7.3/1390 The answer would be one h u n d r e d t i m e s . a s . l a r g o , The answer would bo o n e ^ t e n t h as l a r g e . The answer would be one t h o u s a n d t i m e s as l a r g e , The answer would bo o n e - h u n d r e d t h as l a r g e i The -mswer would be o n e - t h o u s a n d t h as l a r g e .  u  17. About how many t e n t h s a r e t h e r e A. 0 B. 1/3 C. 5 D. 10 E,  in 50  ,055?  18, About how many t h o u s a n d t h s sr'a t h e r e i n 16.5? A. 1.7 B. 17 C. 170 D. 1,700 E . 17,000 19. "toy Is t h o answer s m a l l e r than the t o p number? . 3 A. Because 8 i s more than .5 x. 5 B. Because you a r e f i n d i n g how m'ny .5's i n 3. 4,0 C. Because .5 i a l e s s than 8. D. 'Then you m u l t i p l y by a d e c i m a l t h e answer i s a l w a y s s m a l l e r t h a n the t o p number. E. Because m u l t i p l y i n g by ,5 i s the same as f i n d i n g . h a l f c f the number, 20. How A. 3. C. D. E.  would the answer be c h a n g e d i f you c h a n g o d 1.47 t o 147? You w o u l d g e t the same answer, The answer would be t e n t i m e s as l a r g e . 1.47/34.75 Tho answer would bs one h u n d r e d t i m e s as l a r g e . The answer w o u l d be o n e - t e n t h as 1 i r g o , The answer w o u l d be o n e - h u n d r e d t h as l a r g e .  - 9 Section V Basic 1.  u n d e r s t a n d i n g s o f the r a t i o n a l e  of computation.  "fay do v.'9 f i n d a common d e n o m i n a t o r when a d d i n g f r a c t i o n s u n l i k e denominators? A. You c a n n o t add t o g e t h e r t h i n g s t h a t a r e d i f f e r e n t . B. I t i s e a s i e r t o add f r a c t i o n s when t h e y havo a common denominator. C. The d e n o m i n a t o r s have t o bo t h e same i n o r d e r to a d d . D . ' e l e a r n e d t o a d d u n l i k e f r a c t i o n s t h a t way. E. So t h a t a l l the f r a c t i o n s w i l l have t h e same v a l u e ,  with  v  2.  'vhen d i v i d i n g by a d e c i m a l , why do we move t h e p o i n t t o t h e right? A. M u l t i p l y i n g by a m u l t i p l e o f t e n i s a q u i c k way o f c h a n g i n g a d e c i m a l to a " h o l e nur^sr. B. I t p l a c e s the d e c i m a l p o i n t i n t h e q u o t i e n t c o r r e c t l y . G. You c a n o n l y d i v i d e by a whole number. D. To make t h e d i v i s o r e q u a l t o t h e d i v i d e n d . E. I t i s e a s i e r t o d i v i d e by a whole number t h a n a d e c i m a l .  3.  'Which one of tho f o l l o w i n g w o u l d g i v e t h i s example? 2,1'x 21 A. The sum o f 1 x 2.1 and 21 x 2.1 B. The sum of 10 x 2.1 and 2 x 2.1 C. The sum o f 10 x 2.1 and 20 x 2.1 D. The sum o f 1 x 2.1 a n d 20 x 2.1 E. The sum o f 1 x 2.1 and 2 x 2.1  the c o r r e c t  answer t o  4. "Thich s t a t e m e n t b e s t t o l l s why we ' i n v e r t t h e d i v i s o r and m u l t i p l y ' when d i v i d i n g a f r a c t i o n by a f r a c t i o n ? A. I t i s i n easy method o f f i n d i n g a common d e n o m i n a t o r and a r r a n g i n g the numerators i n m u l t i p l i c a t i o n form, P. I t i s an easy method f o r d i v i d i n g t h e d e n o m i n a t o r s and m u l t i p l y i n g t h e n u m e r a t o r s o f the 2 f r a c t i o n s . C. I t i s a q u i c k , e a s y and a c c u r a t e method o f a r r a n g i n g two f r a c t i o n s i n m u l t i p l i c a t i o n form. D. D i v i d i n g by a f r a c t i o n i s t h e same as m u l t i p l y i n g by the r e c i p r o c a l o f the f r a c t i o n . E. I t Is a q u i c k method o f f i n d i n g t h e r e c i p r o c a l s o f both f r a c t i o n s and r e d u c i n g t o l o w e s t terms ( c a n c e l l i n g ) . 5.  "by do wo move t h e s e c o n d p a r t i a l p r o d u c t one p l a c e t o t h e l e f t when we m u l t i p l y by the 6? 729 A. Because t h e answer has t o be l a r g e r t h a n 72P. x 68 B. B e c a u s e t h e s i x means s i x t e n s , C. Because 6 i s t h e s e c o n d f i g u r e i n 68. D. Because wo l e a r n e d t o m u l t i p l y t h a t way. E. Because t h e 6 r e p r e s e n t s a g r e a t e r v a l u e t h a n t h e 8 represents.  6. ''Jhich s t a t e m e n t b e s t t e l l s why *VQ a r r a n g e numbers i n a d d i t i o n the ".-ay t h a t wo do? A. I t i s an easy way t o k e e p the numbers i n s t r a i g h t c o l u m n s . B. I t h e l p s u s t o s d d c o r r e c t l y , C. I t h e l p s u s add o n l y t h o s e numbers i n t h e same p o s i t i o n . D. I t h e l p s us t o c a r r y c o r r e c t l y f r o m one column t o a n o t h e r . E. I t would be h a r d e r t o a d d i f t h e numbers were m i x e d ,  - 10  -  7.  -'/hen you m u l t i p l y by tho 4 i n 48 you " ' i l l g a t a number t h a t i s how l a r g e compared w i t h tho f i n a l answer? 435 A. O n e - t w e l f t h as l a r g o . x 48 B. O n e - t o n t h as l a r g e . C. . Ono -h.a I f as' l a r g o . D. P i v e - s i x t h s as l a r g e . E . Tw i c e a s 1 a r g o . ______  8.  The answer t o t h i s oxamDle w i l l bo how l a r g e when compared w i t h the 69? A. Twice as l a r g e . 827 B. S i x t y - n i n e t i m a s as l a r g o . x 69 C. On o s i x t y - n i n t h as I T r g o . D. E i g h t h u n d r e d t w o n t y - s o v e n t i m o s as l a r g s . E. 1 -as l a r g o . . 327  9.  VShich s t a t e m e n t b e s t t e l l s »'hy i t i s n o c o s s a r y to 'borrow' i n t h i s example? 567 A. Because the t o p number i s s m a l l e r than tho b o t t o m -392 number. B. You c a n n o t s u b t r a c t 92 f r o m 67. C. You c a n n o t s u b t r a c t 9 t o n s f r o m 6 t o n s . D. You c a n n o t s u b t r a c t 39 t e n s f r o m 56 t e n s . E . You c a n n o t s u b t r a c t 9 f r o m 6.  10.- "'"faich s t a t e m e n t h o s t t o l l s why wo c a r r y 2 f r o m the second column? A. Tho sum o f tho s e c o n d column i s 23 w h i c h h a s two 251 f i g u r e s i n i t . "<e havo room f o r tho 3 o n l y , so 161 wo put tho 2 i n the n e x t c o l u m n . 252 B. The sum of tho s e c o n d column I s more t h a n 20, so 271 we put tho 2 i n tho n o x t c o l u m n . C. Because wo l e a r n e d t o ?idd t h a t way. D. Tho v a l u e r e p r e s e n t e d by t h e f i g u r e s i n tho socond. column i s mors than 9 t e n s , so "'o put t h o h u n d r e d s i n t h o n e x t column. E. I f wo do not c a r r y tho 2, tho answer " ' i l l bo 20 l o s s t h a n t h : c o r r e c t answer.11.  In t h i s example you m u l t i p l y by tho 6, thon by the 3. How do tho two r e s u l t s ( p a r t i a l p r o d u c t s ) comparo? 749 A. Tho s e c o n d r e p r o s o n t s a numbor o n o - h a l f as l a r g e x 36 a s tho f i r s t . E. Tho socon-d r e p r e s e n t s a number t w i c e as l a r g o as tho f i r s t . C. Tho sooond r e p r e s e n t s a numbo.- f i v o t i m e s as l a r g e as the  firsts  D.  The s e c o n d r e p r e s e n t s a numbor ton t i m e s as l a r g o i s the first. E . The s e c o n d r e p r e s e n t s a number twenty t i ^ o s as l a r g o as tho first. 12. *?hich w o u l d A. M u l t i p l y B. M u l t i p l y C. M u l t i p l y D. M u l t i p l y E. M u l t i p l y  give tho 439 x 3; 4.39 x 3 j 563 x 9; 563 x 9; 439 x 3;  c o r r e c t answer t o 439 x 563? 439 x 6; 439 x 5 - then add a n s w e r s . 439 x 63; 439 x 563 - t h e n add a n s w e r s . 563 x 3; 563 x 4 - t h e n add a n s w e r s . 563 x 39; 563 x 439 - t h e n add a n s w e r s , 439 x 60; 439 x 500 - t h e n add a n s w e r s .  13.  11 -  "faich s t a t e m e n t b a s t e x p l a i n s tho 4 i n t h e answer? A. Tho 4 moans t h a t t h e r e a r o f o r t y - e i ^ h t 2G's i n 1243. " B. Tho 4 i n t h e a n s v e r moans t h a t t h e r e a r o f o u r 26's i n 1243. C. Tho 4 moans t h a t 2 goes i n t o 12 f o u r t i m e s , and 5 would bo t o o l a r g o . B, Tho 4 moans t h a t t h e r e a r o a t l e a s t f o r t y 26's i n 1243, 1. Tho 4 moans t h a t tho answer w i l l come o u t o v e n .  43 26/1248 104 208 208  TJ  14.  Here ia- an oxarrplo i n s u b t r a c t i o n o f m i x e d numbers i n 'vhich i t i s n e c e s s a r y t o 'borrow', -"fhich s t a t e m e n t b e s t e x p l a i n s the b o r r o w i n g . , A. You c a n n o t s u b t r a c t 5/8 f r o m 3/8, so you t a k e 1 5 | from t h o 5 and put i t i n f r c m t o f tho 3 making ^ 13. -2 | B. You c a n n o t s u b t r a c t 5/8 f r o m 3/8, so y o u add t h o ° 3 a n d tho 8 making 11/8, G. You c a n n o t s u b t r a c t 5/3 f r o m 3/8, so you t u r n them a r o u n d and s u b t r a c t 3/3 f r o m 5/3. D. You c a n n o t s u b t r a c t 5/8 f r o m 3/8, so y o u t a k e 1 f r o m t h o 5 and nod i t t o 3/3 making i t 4/3. 3. You c a n n o t s u b t r a c t 5/3 f r o m 3/8, so you take 1 f r o m t h o 5 and change i t t o 8/8; then add t h e 8/8 t o 3/3 m a k i n g 11/3,  15.  TJhich s t a t e m e n t b a s t e x p l a i n s what h a p p e n s when you r e d u c e a f r a c t i o n to l o w o s t terms? A. Tho s i z e o f t h o terms a n d t h e v a l u o o f t h o f r a c t i o n become s m a l l e r . 3. Tho valu3- of t h e f r a c t i o n doo-s n o t c h a n g e . Tho s i z o o f tho p a r t r e p r o s o n t e d by t h o now d e n o m i n a t o r i s s m a l l e r , and "the numb o r o f p a r t s r e p r e s e n t e d by t h o now n u m e r a t o r is less. G. Tho v a l u e o f tho f r a c t i o n doos n o t c h y n g u . Too terras aro s m a l l e r . D. Tho v a l u o o f t h e f r i c t i o n doos n o t c h a n g e , b u t tho p a r t s o f t h e f r a c t i o n r e p r e s e n t e d by t h o now numbers become f e w e r i n numb or a n d s m a l l e r i n s i z e . E. The v a l u o of t h o f r a c t i o n c h a n g e s b o c a u s o t h o now numbers aro s m a l l e r .  End  BIBLIOGRAPHY American Council on Education. Teachers f o r Our Times. Washington: American Council on Education, 1944* Brownell, W. A., and Sims, V. M. "The Nature of Understanding", The Measurement of Understanding. 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: University of Chicago Press, 1946. Buswell, Guy T. "The Psychology of Learning i n Relation to the Teaching of Arithmetic", The Teaching of Arithmetic. F i f t i e t h Yearbook of the National Society f o r tKe Study of Education, Part I I . Chicago: University of Chicago Press, 1951. Carter, Paul. "From a Mechanistic to a Meaningful Program of Arithmetic Instruction: a Suggested Approach", School Science and Mathematics. XLVII (October, 1947), 604-608. Dutton, W. H. "Attitudes of Prospective Teachers Toward Arithmetic", Elementary School Journal. LII (October, 1951), 84-90. "The F i r s t Report of the Commission on Postwar Plans", Mathematics Teacher. XXXVII (May, 1944), 226-32. Forest, J . W. "Training Teachers of High School Arithmetic", Mathematics Teacher. XXXIV (March, 1941), 119-23. Glennon, Vincent J . "A Study of the Growth and Mastery of Certain Basic Mathematical Understandings on Seven Educational Levels". Unpublished Doctor's d i s s e r t a t i o n . Cambridge, Massachusetts: Graduate School of Education, Harvard Univers i t y , 1948. — "Testing Meanings i n Arithmetic", Arithmetic 1949. Supplementary Educational Monographs, No. 70. Chicago: University o f Chicago Press, 1949. G u i l f o r d , J . P. Fundamental S t a t i s t i c s i n Psychology and Education. New York: McGraw-Hill BooF~Co., 1950. K i n s e l l a , J . J . , and Carnahan, W. H. "Putting Meaning into Geometric Concepts", School Science and Mathematics. XLVIII (October, 1948), 541-59"! The Measurement of Understanding. 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: University of Chicago Press, 1946.  BIBLIOGRAPHY—Continued  M o r t o n , R. L . "The P l a c e o f A r i t h m e t i c i n Various Types o f E l e m e n t a r y - s c h o o l C u r r i c u l u m s " , A r i t h m e t i c 1949* Supplementary E d u c a t i o n a l M o n o g r a p h s , N o . 70. Chicago: University of Chicago Press, 1949. Robinson, A. E. "Training Elementary School Teachers i n Field of Arithmetic. Special Survey Studies, V o l . V. S t a t e s O f f i c e o f E d u c a t i o n B u l l e t i n N o . 10, 1933.  the United  "The S e c o n d R e p o r t o f t h e C o m m i s s i o n o n P o s t w a r P l a n s " , M a t h e m a t i c s T e a c h e r . X X X V I I I ( M a y , 1945), 195-220. S u e l t z , B e n A . , B o y n t o n , H . , a n d S a u b l e , I. "The Measurement o f U n d e r s t a n d i n g i n E l e m e n t a r y - s c h o o l M a t h e m a t i c s " , The Measurement o f U n d e r s t a n d i n g . Forty-fifth Yearbook o f the N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , P a r t I. Chicago: University of Chicago Press, 1946. T a y l o r , E . H. "Mathematics f o r a F o u r - y e a r i n the Elementary School", School Science X X X V I I I ( M a y , 1938), 499-503.  Course f o r Teachers and Mathematics.  — — — , and M i l l s , C . N. Arithmetic for Teacher-training Classes. New Y o r k : H e n r y H o l t a n d C o . , 1949. The T e a c h i n g o f A r i t h m e t i c . Fiftieth S o c i e t y f o r the Study o f Education, U n i v e r s i t y o f Chicago Press, 1951.  Yearbook Part II.  of the National Chicago:  Weitzman, E . , a n d M c N a m a r a , W. J . "Apt Use o f t h e Inept Choice in Multiple Choice Testings", Journal of Educational Research. X X X I X ( M a r c h , 1 9 4 6 ) , 517-22. " Wheat, H. G . "Why n o t b e S e n s i b l e T e a c h e r . X X X V I I I ( M a r c h , 1945), —  How t o Peterson  Teach  Arithmetic.  ancTCompany,  About  Meaning?"  Mathematics  99-102. Evanston,  Illinois:  Row,  1951.  Wingo, G . Max. "The O r g a n i z a t i o n and A d m i n i s t r a t i o n o f t h e Arithmetic Program i n t h e Elementary S c h o o l " , Arithmetic 1948« S u p p l e m e n t a r y E d u c a t i o n a l M o n o g r a p h s , N o . 6 6 . c E i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1948.'  

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