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Effects of instruction in groups on individual equation writing Underwood, Barry Richard 1971

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THE  EFFECTS OF INSTRUCTION IN GROUPS ON INDIVIDUAL EQUATION WRITING by BARRY RICHARD UNDERWOOD  B. Sc. , U n i v e r s i t y o f Manitoba, 1965 B. Ed., U n i v e r s i t y o f Manitoba, 1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics E d u c a t i o n  We a c c e p t t h i s t h e s i s as conforming required  THE  to the  standard  UNIVERSITY OF BRITISH COLUMBIA J u l y , 1971  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  f u l f i l m e n t o f the requirements f o r  the U n i v e r s i t y of B r i t i s h  make i t  freely available  that permission  for  Columbia,  I agree  r e f e r e n c e and  for extensive copying o f  this  that  study. thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s of  this  written  representatives. thesis  for financial  i s u n d e r s t o o d that gain s h a l l  not  copying o r p u b l i c a t i o n  be allowed without my  permission.  Department o f  fTlA-TH-g ^ yVT_7CS»  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  It  3TVL W ^ I Q 7 /  Columbia  t?7- O et-t-T-Toy\J  ABSTRACT T h i s study was motivated by the w r i t e r ' s b e l i e f t h a t youngsters  do have a tendency  s i t y , no matter  t o group, and t h a t t h i s propen-  how emphemeral and v a c i l l a t i n g i t may be a t  t i m e s , should be taken i n t o account i n the d e s i g n o f teacher strategies. Grade f o u r s t u d e n t s were a s s i g n e d t o two groups a t random, and then, i n one group, subgroups o f f o u r students were randomly made up.  A l l students were i n s t r u c t e d by f i l m  f o r t h r e e days on w r i t i n g an e q u a t i o n f o r a d i v i s i o n On the f o u r t h day o f t h e experiment,  loops  problem.  the students wrote a  c r i t e r i o n t e s t o f t w e n t y - f i v e d i v i s i o n problems.  The i n v e s t i -  g a t i o n o f s t u d e n t - s t u d e n t i n t e r a c t i o n was done by comparing the e f f e c t s o f i n s t r u c t i o n t o groups o f f o u r s t u d e n t s w i t h those o f i n s t r u c t i o n t o the i n d i v i d u a l l y taught A t w o - t a i l e d t - t e s t was used  students.  to t e s t the s i g n i f i c a n c e  between the means o f the two groups and a F - t e s t was employed t o t e s t the d i f f e r e n c e i n the v a r i a n c e s o f the two groups. There was no s i g n i f i c a n t d i f f e r e n c e between the i n d i v i d u a l - t a u g h t group and the group-taught  group i n terms of  e i t h e r mean o r v a r i a n c e . The  c o n c l u s i o n was drawn t h a t the use o f s m a l l groups  t o teach students t o w r i t e e q u a t i o n s f o r d i v i s i o n problems d i d  ii  not  improve the i n s t r u c t i o n .  research and  using  desirable.  But i t was  f e l t that  d i f f e r e n t dependent v a r i a b l e s  further  i s both warranted  iii  TABLE OF CONTENTS CHAPTER I.  Page THE PROBLEM  1  Background  •••••  •  1  Statement o f the Problem  4  Review o f the L i t e r a t u r e Statement o f the Hypothesis II.  .  12  THE DESIGN OF THE STUDY  13  Introduction  13  Formation o f the Groups  15  The P o p u l a t i o n  15  The Sample  15  Assignment o f S u b j e c t s  15  Development  of M a t e r i a l s  16  The I n s t r u c t i o n a l Device  16  The T e s t  17  Instrument  Procedure  18  S t a t i s t i c a l Analysis  21  Data N u l l Hypotheses S t a t i s t i c a l Treatment III.  4  ••>•••  21 .  22 22  ANALYSIS OF THE RESULTS  24  T e s t i n g o f Hypotheses  24  Hypothesis One  25  H y p o t h e s i s Two  25  iv  CHAPTER  IV.  Page Conclusions  25  A n a l y s i s o f A d d i t i o n a l Data  26  Answers on the C r i t e r i o n T e s t  26  E q u a t i o n s and Answers D u r i n g I n s t r u c t i o n ...  26  IMPLICATIONS OF THE STUDY  28  Introduction  28  Discussion of Conclusions  28  The Content o f the Lessons  28  The Method o f I n s t r u c t i o n  29  Length o f Experiment  30  Subgroup Makeup  30  Limitations  o f the Study  31  The F i l m Loops  31  A Broader Sample  32  Q u o t i t i v e - P a r t i t i v e Approach  .. ...  33  Suggestions f o r F u t u r e S t u d i e s  33  Summary  35  FOOTNOTES  36  BIBLIOGRAPHY  39  APPENDIX A APPENDIX B  . .  44 52  APPENDIX C  58  APPENDIX D  63  V  LIST OF TABLES TABLE I.  Page S t a t i s t i c s f o r Equation Writing  24  vi  L I S T OF FIGURES FIGURE 1.  Page The E x p e r i m e n t a l P r o c e d u r e  14  vii  ACKNOWLEDGEMENT  The and  author wishes t o thank the p r i n c i p a l , t e a c h e r s ,  students who p a r t i c i p a t e d i n the experiment and the members  o f h i s t h e s i s c o m m i t t e e — D r . E. D. MacPherson Mr.  (Chairman),  P. O l l e y , and Mr. T. B a t e s — f o r t h e i r c o o p e r a t i o n and  assistance.  CHAPTER I THE I.  PROBLEM  BACKGROUND  Teachers have o c c a s i o n a l l y used c o n c r e t e introduce  o r develop c e r t a i n concepts and  have p r o b a b l y f e l t the  objects  algorithms.  to  They  t h a t t h i s procedure a s s i s t s the student i n  i n t e r n a l i z a t i o n o r a b s t r a c t i o n of a concept because he i s  a b l e t o r e l a t e i t to a r e a l s t a t e or a c t i o n . In d e c i d i n g on a method o f p r e s e n t a t i o n ,  i t seems  r e a s o n a b l e t h a t an i n s t r u c t o r take account o f the f a c t t h a t c l a s s r o o m i s an extremely complex, s h i f t i n g web r e l a t i o n s , and  of  the  interpersonal  t r y t o maximize the components i n v o l v e d which  improve l e a r n i n g . The provided  d i s c i p l i n e s of psychology and  a wide range of i n f o r m a t i o n  mediate l e a r n i n g i n groups o f p e o p l e .  s o c i o l o g y have  about the f o r c e s Studies  that  on the e f f e c t s  1 2 of c l a s s r o o m i n t e r a c t i o n , c l i m a t i c i n f l u e n c e s , discrimina3  tion,  .  4  motivation,  information  5  independence,  and  communica-  t i o n structure** have p r e s e n t e d evidence t h a t s i g n i f i c a n t group factors exist. by  I t seems reasonable t h a t s c h o o l  i n d i v i d u a l s would be  achievement  improved i f group i n f l u e n c e s i n the  c l a s s r o o m were b e t t e r understood and  manipulated.  2.  I t has o f t e n been s t a t e d t h a t students l e a r n b e s t i f they p a r t i c i p a t e a c t i v e l y i n the l e a r n i n g p r o c e s s . argued  I t can  t h a t group work p r o v i d e s an o p p o r t u n i t y f o r  be  involvement.  I n a group s i t u a t i o n , the "onus p r o b a n i " l i e s w i t h the s t u d e n t s . Where they c o o p e r a t i v e l y p r o v i d e the s o l u t i o n to a problem, i t i s most l i k e l y  t h a t o p p o r t u n i t i e s f o r i n t e r a c t i o n and i n v o l v e -  ment are maximized. In a d d i t i o n , an i n c r e a s e i n group work by students l e a d t o the achievement o f an important  g o a l of  may  education—the  development of i n d i v i d u a l s capable of l i v i n g and working w i t h others.  W i t h i n a group, a student has a chance t o e x p e r i e n c e  s o c i a l i n t e r a c t i o n i n v o l v i n g a wide range of p e r s o n a l i t y skill  characteristics.  As Wilhelms  7  and  stated:  The layman and the u n s o p h i s t i c a t e d teacher may and do c o n t i n u e t o t h i n k o f each subgroup as "homogeneous"; the e x p e r t knows i t i s rampantly heterogeneous, c o n c e a l i n g tremendous ranges on a l l but the one v a r i a b l e chosen as a basis for d i v i s i o n . I t a l s o seems l i k e l y t h a t work i n groups p r o v i d e s an o p p o r t u n i t y f o r immediate r e i n f o r c e m e n t not g i v e n i n s t r a i g h t forward d i d a c t i c p r e s e n t a t i o n s v i a l e c t u r e or textbook,  because  the students are a b l e to check t h e i r r e a s o n i n g step by step w i t h g  the o t h e r students i n the group. l a p s e of o n l y a few  And  i f Skinner's  seconds between the response  and  dictum  "the  reinforce-  ment d e s t r o y s most of i t s e f f e c t s " i s even p a r t i a l l y t r u e , then we  have a f u r t h e r reason f o r examining  students i n s m a l l  groups.  the i n s t r u c t i o n of  3.  A d d i t i o n a l l y , Piaget's lend i t s e l f  to the use  9  of groups.  model of l e a r n i n g appears to He maintains t h a t to  acquire  a concept some c o g n i t i v e r e o r g a n i z a t i o n must take p l a c e . readjustment, the e q u i l i b r a t i o n p r o c e s s ,  c e n t e r i n g upon  This the  temporary imbalance between the major f u n c t i o n a l a c t i v i t i e s a s s i m i l a t i o n and invoking  accommodation, i s u s u a l l y conceived  some form of c o n f l i c t .  of  of  as  T h i s c o n f l i c t develops between  the c h i l d ' s p a r t i a l l y e s t a b l i s h e d schemas and  new  environmental  demands which are meaningful to the c u r r e n t l e v e l o f p s y c h o l o g ical one  functioning.  I t seems t h a t peer r e l a t i o n s h i p s may  be  means o f s e t t i n g n a t u r a l c o n f l i c t f o r c e s i n motion to  f a c i l i t a t e reorganization  of the  t h i n k i n g o f the i n d i v i d u a l  participants. Even though group work appears to be v a l u a b l e , question  remains of whether or not i t lends  involving material objects. t h a t they should  itself  In r e p l y , P i a g e t " ^  be combined, f o r he  the  to work  seems t o argue  states:  When I say " a c t i v e , " I mean i t i n two senses. One i s a c t i n g on m a t e r i a l t h i n g s . But the other means doing t h i n g s i n s o c i a l c o l l a b o r a t i o n , i n a group e f f o r t . T h i s leads to a c r i t i c a l frame o f mind, where c h i l d r e n must communicate with each o t h e r . T h i s i s an e s s e n t i a l f a c t o r i n i n t e l l e c t u a l development. C o o p e r a t i o n i s indeed cooperation. I f he d e s i r e s t h a t c h i l d r e n be and  engaged i n the  wishes t h a t " a c t i v e " have the two  he advocates the  concurrent  use  learning  process,  senses, then, seemingly,  of m a t e r i a l o b j e c t s  and  groups.  4. A f u r t h e r j u s t i f i c a t i o n f o r the concomitant employment of group work and work w i t h m a t e r i a l o b j e c t s material objects They are  serve  concrete  as a s t i m u l u s  i s that  f o r group i n t e r a c t i o n .  r e f e r e n t s about which d i s c u s s i o n can  take  place. Thus, i t f o l l o w s o b j e c t s may  provide  an o p p o r t u n i t y  unique a b i l i t i e s and cation.  t h a t group work w i t h m a t e r i a l f o r the student to develop  s a t i s f y some broader o b j e c t i v e s o f edu-  I t appears important t h a t we  factors operating  examine c l o s e l y the  i n group l e a r n i n g , so t h a t they can  be  optimized. Statement of the  Problem  Does the  teacher  are handled o n l y by increased  the  strategy  teacher  i n which p h y s i c a l  objects  become more e f f e c t i v e w i t h  student-student i n t e r a c t i o n ? II.  REVIEW OF THE  LITERATURE  A t an e a r l i e r time, many people have been tempted t o a s s e r t t h a t groups are g e n e r a l l y more s u c c e s s f u l than individuals.  A number o f experimenters, Gournee,"^ Klugman,^" 13  Perlmutter  and  de Montmollin,  14 Hudgins,  and  Taylor  and  15 Faust,  have found t h a t groups have produced more c o r r e c t  s o l u t i o n s than comparable s u b j e c t s working as i n d i v i d u a l s .  5.  The g r e a t e r p r o d u c t i v i t y o f g r o u p s , a s c o m p a r e d w i t h i n d i v i d u a l s , was a t t i m e s a t t r i b u t e d  t o t h e most c a p a b l e  student.  That i s ,  g r o u p s w e r e s u p e r i o r , n o t b e c a u s e o f some  "special"  group e f f e c t s , b u t because group performance a c h i e v e d  t h e l e v e l o f i t s most p r o f i c i e n t  member.  16 Gournee,  who s t u d i e d t h e e f f e c t i v e n e s s o f c o l l e c -  t i v e a c t i o n i n t h e absence o f a l l  verbal  found t h a t group s c o r e s were s i m i l i a r student's score.  interstimulation,  t o t h e most c a p a b l e  With f i f t y - t h r e e undergraduates i nthe  i n d i v i d u a l s i t u a t i o n and s i x t y - s i x  s t u d e n t s i n the group  a t i o n , h i s comparison o f i n d i v i d u a l and c o l l e c t i v e  situ-  judgements  r e v e a l e d t h a t g r o u p s c o r e s w e r e a p p r o x i m a t e d by t h e s c o r e s o f t h e b e s t members. 17 P e r l m u t t e r a n d de M o n t m o l l i n ' s  data also  showed  t h a t t h e g r o u p s c o r e was a d u p l i c a t e o f t h e s c o r e o f t h e m o s t p r o f i c i e n t member.  With t w e n t y - t h r e e s m a l l groups  "nonsense" words e i t h e r  i n a group f i r s t  and t h e n  studying individually,  o r i n d i v i d u a l l y a n d t h e n i n a g r o u p , he f o u n d no d i f f e r e n c e between t h e g r o u p - i n d i v i d u a l groups and t h e b e s t groupi n d i v i d u a l p e r s o n i n each group. B u t t h e r e were i n s t a n c e s where groups a t t a i n e d  better  s u c c e s s t h a n a n y c o m p o n e n t member, a c t i n g a l o n e , was a b l e t o d o . 18 For example, Faust  c o n d u c t e d two e x p e r i m e n t s w i t h s t u d e n t s i n  an i n t r o d u c t o r y p s y c h o l o g y c o u r s e .  One s t u d y h a d f i f t e e n  groups  6. of four, and f o r t y - o n e i n d i v i d u a l s , w h i l e the o t h e r had seventeen groups of f o u r and s i x t y - s e v e n i n d i v i d u a l s .  In a d d i t i o n ,  he c r e a t e d nominal g r o u p s — g r o u p s made up by random assignment from people who  had worked i n d i v i d u a l l y , and to which  credit  i s g i v e n f o r a c o r r e c t s o l u t i o n i f one or more o f the i n d i v i d u a l s i n the group s o l v e d the problem. the performance of h i s r e a l four-man the performance of nominal four-man  On f o u r s p a t i a l  problems,  groups was w e l l matched by c o l l e c t i o n s , but on the  anagram problems, the r e a l groups s o l v e d more problems  than  d i d nominal groups. R e s u l t s having s i m i l i a r s i g n i f i c a n c e are p r e s e n t e d 19 by Anderson,  based on a task o f making as many words as  p o s s i b l e from the l e t t e r s " a f l i y b a t " i n a 15 minute time p e r i o d . Anderson's  two and three person groups, composed of j u n i o r h i g h  s t u d e n t s , exceeded the output of the b e s t i n d i v i d u a l s i n comp a r a b l e nominal groups.  The r e a l groups were e q u a l t o the  nominal groups when the l a t t e r was  c r e d i t e d w i t h a l l the d i f -  f e r e n t words produced by the component i n d i v i d u a l s .  It is  noted t h a t t h i s r e s u l t i s e n t i r e l y c o n s i s t e n t w i t h a p o o l i n g view of the r e a l group's performance, i n t h a t the assumption-i f one o r more persons s o l v e a problem then they w i l l be able to c o n v i n c e the o t h e r s — a p p e a r s s a t i s f i e d . The author should l i k e t o a v o i d c o n f u s i o n by why  the p o o l i n g o f complementary  explaining  s k i l l s enables r e a l groups t o  surpass nominal groups i n F a u s t ' s experiment but not i n  7.  Anderson's.  The reason seems t o be r e l a t e d t o the d i f f e r e n c e  between the t a s k s of the two experiments.  In Anderson's  "nominal" groups, c r e d i t was r e c e i v e d f o r every l e g a l word any one of the two o r three i n d i v i d u a l s was able t o d i s c o v e r ; w h i l e F a u s t ' s t a s k r e q u i r e d s o l v i n g each word i n the l i s t b e f o r e b e i n g a l l o w e d t o proceed t o t h e next one.  A person who might  have s o l v e d one o f the l a t e r words never had a chance t o do so if, it  working a l o n e , he missed an e a r l i e r one.  For t h i s reason,  appears t h a t F a u s t ' s method o f g i v i n g c r e d i t t o a nominal  group does not adequately r e f l e c t t h e p o t e n t i a l g a i n s t o be d e r i v e d from p o o l i n g o f r e s o u r c e s i n group  activity.  20 In c o n t r a s t , Davis and R e s t l e  found t h a t group per-  formance was below t h e l e v e l o f the most p r o f i c i e n t member. T h i s was p a r t i c u l a r l y t r u e f o r two problems t h a t were r a t h e r l o n g and r e q u i r e d working through a sequence of ideas i n o r d e r to a r r i v e a t the c o r r e c t answer.  S i m i l a r r e s u l t s emerge from  21 the Lorge and Solomon  r e s e a r c h on the T a r t a g l i a problem.  It  appeared t h a t the group p r o c e s s e s handicapped the most p r o f i c i e n t member. Using the i n f o r m a t i o n p r e v i o u s l y mentioned, L i n d z e y  22 and Aronson  have h y p o t h e s i z e d t h a t the p r o f i c i e n c y o f groups  i n problem s o l v i n g , as compared to i n d i v i d u a l s , depends on the type o f problem undertaken.  They m a i n t a i n t h a t i f the q u e s t i o n s  are such t h a t the answers prove to be e a s i l y v e r i f i e d , are amenable to wide acceptance, p r o v i d e a b a s i s f o r more c o n f i dent advocacy, and tend to be p r e s e n t e d by a competent person,  8.  then t h e most c a p a b l e person w i l l From t h i s  h y p o t h e s i s , i t appears  exert greater  t h a t t h e most c a p a b l e  i n f l u e n c e may b e , i n c e r t a i n s i t u a t i o n s , a particular  type of a c t i v i t y .  was s e l e c t e d  so t h a t i t s a t i s f i e d  Aronson,  Another  student's  c o n t r o l l e d by c h o o s i n g  I n o t h e r words, i f the m a t e r i a l  t h e n i t c a n be a r g u e d  c o u l d be e x t r e m e l y  influence.  the c r i t e r i a  o f L i n d z e y and  t h a t t h e most c a p a b l e  student  influential. approach  h a s b e e n t a k e n t o e x p l a i n why g r o u p  s c o r e s were h i g h e r than i n d i v i d u a l s c o r e s .  I t was m a i n t a i n e d  t h a t t h e s t u d e n t s i n a group perform b e t t e r because t h e d i s a g r e e m e n t among g r o u p members a b o u t a s o l u t i o n review process d u r i n g which critical  ipants  attention i s directed  steps i n the problem.  s u c c e s s f u l problem  initiates  to the various  The g r o u p ' s c o n t r i b u t i o n t o  s o l v i n g was t h a t i t " i n s t r u c t e d "  i n proper problem  a  s o l v i n g procedure.  the p a r t i c -  That i s , the  g r o u p ' s i n f l u e n c e was t h a t i t i n v o k e d a p r o c e s s w h i c h  one w o u l d  not o r d i n a r i l y use. 23  But Hudgms mentioned  argument.  eight f i f t h  grade  found  no e v i d e n c e  t o s u p p o r t t h e above  H i s s u b j e c t s w e r e one h u n d r e d a n d t w e n t y -  students i n four S t . Louis schools.  the C a l i f o r n i a Test o f Mental M a t u r i t y t o c o n t r o l ability,  solving ability,  h i ssubject's  showed t h a t t h e s c o r e s o f s t u d e n t s who s p e c i f i e d t h e  steps they used which  mental  a n d t h e C a l i f o r n i a A r i t h m e t i c T e s t s 4-6 f o r m W t o  measure t h e s t u d e n t ' s problem results  Using  i n s o l v i n g a p r o b l e m — t h i s was t h e a c t i v i t y  Hudgins chose  to simulate the review process of the  9.  group—were who  solved  not s i g n i f i c a n t l y  different  from the scores of those  t h e p r o b l e m s i n t h e u s u a l manner. When s u b j e c t s w o r k e d  alone a f t e r a group  session,  24 D u n n e t t e , C a m p b e l l and J a a s t a d of  found t h a t a l a r g e r  i d e a s o r s o l u t i o n s were produced.  number  A s t u d y by P e r l m u t t e r  and  25 de M o n t m o l l i n  i n w h i c h s t u d e n t s memorized  n o n s e n s e w o r d s a l s o showed t h a t p e o p l e who  two  syllable  f i r s t worked  g r o u p a n d t h e n a l o n e had a b e t t e r r a t e o f r e c a l l who  had o n l y worked  in a  than those  alone.  From t h e l a t t e r  results,  i t appeared that the  indiv-  i d u a l s a c q u i r e d i n f o r m a t i o n w h i l e t h e y were w o r k i n g i n t h e 26 group which they c o u l d apply l a t e r . contradictory evidence.  But Hudgins  produced  I n the second p a r t of the experiment  m e n t i o n e d b e f o r e , h i s r e s u l t s showed t h a t g r o u p w o r k on m a t h e m a t i c a l p r o b l e m s h a d no more e f f e c t on s u b s e q u e n t p e r f o r m a n c e than i n d i v i d u a l  work.  I t i s n o t e d t h a t t h e r e was nonsense likely  syllables  in earlier  an a p p a r e n t h e a v y u s e o f  r e s e a r c h , and i t seems  quite  t h a t such m a t e r i a l might p r e c l u d e group e f f e c t s .  t h o u g h g r o u p s h a v e p e r f o r m e d b e t t e r t h a n i n d i v i d u a l s on  Even such  t a s k s , t h i s m i g h t be e x p l a i n e d by t h e v i e w t h a t a g r o u p i s l e s s likely  t o f o r g e t o r t h a t an i m p l i c i t o r g a n i z a t i o n  w i t h i n t h e group where s t u d e n t s are a l l o t t e d syllables.  I t was  the  i s established nonsense  t h e r e f o r e t h o u g h t t h a t i t w o u l d be o f  interest to test subjects after  a group s e s s i o n i n which  10. m a t e r i a l , b e l i e v e d conducive t o the development of group e f f e c t s , was used. 27 Even though Hudgins  d i s c o v e r e d t h a t no t r a n s f e r  of t r a i n i n g took p l a c e a f t e r s t u d e n t s had worked together i n groups on mathematical problems, i t seems r e a s o n a b l e t o s u s p e c t t h a t i f the a c t i v i t y o f the group i n v o l v e d the l e a r n i n g o f new concepts as w e l l as working w i t h l e a r n e d ones, then group work might produce d i f f e r e n c e s s i g n i f i c a n t l y g r e a t e r than those from i n d i v i d u a l work. When the i n s t r u c t i o n of s m a l l groups i s c o n s i d e r e d , the q u e s t i o n a r i s e s o f whether o r n o t s e l f - p a c i n g i s e s s e n t i a l . 28 Sawris  examined t h i s problem u s i n g programmed  instruction.  He d i v i d e d a sample o f one hundred and twenty-four s t u d e n t s i n the t h i r d  form a t a t e c h n i c a l s c h o o l i n the f o l l o w i n g way:  i n d i v i d u a l s , homogeneous p a i r s , two heterogeneous groups of s i z e e i g h t , and two heterogeneous groups o f s i x t e e n .  Using  ANOVA t o a n a l y z e the d i f f e r e n c e s between p r e t e s t and p o s t t e s t s c o r e s , he found the f o l l o w i n g s c a l e o f s c o r e s p r o c e e d i n g from the h i g h e s t t o the lowest:  homogeneous p a i r s ,  heterogeneous  p a i r s , heterogeneous e i g h t s and s i x t e e n s , and i n d i v i d u a l s . concluded t h a t group l e a r n i n g g i v e s r e s u l t s comparable  He  to indiv-  i d u a l s e l f - p a c e d i n s t r u c t i o n , and t h a t i t was p o s s i b l e t o p r e s e n t programs t o groups o f e i g h t p u p i l s and a c h i e v e r e s u l t s s i m i l a r t o those from a s e l f - p a c e d s t u d e n t , p r o v i d e d the s t u dents are a l l o w e d enough time t o respond.  11.  A d d i t i o n a l i n f o r m a t i o n about s e l f - p a c i n g i s s u p p l i e d by Moore.29  Seventy students  c a t e g o r i z e d by  IQ and  so t h a t the students  of ages twelve and  by speed on a p r e v i o u s  The  worked as i n d i v i d u a l s i n groups.  students  The  determined by a percentage o f students  learned  He  s e l f - p a c e d program,  u s i n g the t e a c h i n g machines were matched  w i t h those working on b o o k l e t s .  answer.  t h i r t e e n were  on the  system  pace of the machine  was  g e t t i n g the c o r r e c t  determined t h a t the groups working on the machines  f a s t e r d u r i n g the f i r s t  working on the b o o k l e t s , and between the s e l f - p a c e d and  three weeks than the  t h a t there was  paced student;  subgroups showing more s i g n i f i c a n t  students  no d i f f e r e n c e  w i t h homogeneous  results.  From the i n f o r m a t i o n a v a i l a b l e on s e l f - p a c i n g , i t seems reasonable  t h a t we  pacing i n studying  may  discount  the e f f e c t s of group l e a r n i n g .  Accordingly,  i t f o l l o w s t h a t groups may  s o l u t i o n s than i n d i v i d u a l s , and to the most capable certain type—or although  the e f f e c t s of f o r c e d  produce more  t h a t t h i s c o u l d be a t t r i b u t e d  student—provided  the problems are of a  to the m o t i v a t i o n a l aspects o f a group.  H u d g i n s ^ found t h a t grade f i v e students  a c q u i r e any more techniques  know whether students  taught  b e t t e r than t h e i r i n d i v i d u a l  d i d not  which c o u l d be a p p l i e d l a t e r  group work than i n d i v i d u a l work, i t i s s t i l l  And  of i n t e r e s t  from to  i n groups would l e a r n a l e s s o n counterparts.  Statement o f Hypothesis I t i s h y p o t h e s i z e d t h a t grade  f o u r students i n  groups o f f o u r , taught a t a s p e c i f i c pace how t o w r i t e p a r t i t i v e and q u o t i t i v e d i v i s i o n e q u a t i o n s , w i l l produce  more c o r  r e c t equations and have a s m a l l e r range o f s c o r e s on a subsequent  task than comparable grade  f o u r s t u d e n t s , taught a t  the same pace, but taught as i n d i v i d u a l s .  CHAPTER I I THE DESIGN OF THE STUDY I.  INTRODUCTION  A s e t o f f i l m loops was used t o teach two groups o f grade f o u r students how t o w r i t e equations lems.  fordivisions  prob-  The treatment method f o r each group v a r i e d o n l y w i t h  r e s p e c t to s t u d e n t - s t u d e n t i n t e r a c t i o n .  One group o f students  was i n s t r u c t e d i n d i v i d u a l l y , w h i l e the subgroups o f the o t h e r group were taught  together.  Four days f o r each group were r e q u i r e d t o o b t a i n the d a t a f o r the experiment.  On the f i r s t day, the students were  shown e i g h t q u o t i t i v e sequences.  On the second  day, the s t u -  dents were i n s t r u c t e d w i t h e i g h t p a r t i t i v e sequences.  The  t h i r d day o f i n s t r u c t i o n c o n s i s t e d o f r e p e a t i n g sequences to e i g h t o f each d i v i s i o n type i n an unordered f o u r t h day, a c r i t e r i o n t e s t was a d m i n i s t e r e d . mental procedure  manner.  five  On the  The e x p e r i -  i s summarized by F i g u r e 1.  The means o f the c r i t e r i o n t e s t scores f o r each group were compared u s i n g a two-sample t - t e s t , w h i l e the v a r i a n c e s o f the c r i t e r i o n t e s t s c o r e s were examined u s i n g a F - t e s t .  FIGURE 1  THE EXPERIMENTAL PROCEDURE  Day 1  QUOTITIVE  Day 2  PARTITIVE  3 CALENDAR DAYS  Day 3 Day 4  QUOTITIVE-PARTITIVE CRITERION TEST  15. II. The  FORMATION OF THE  GROUPS  Population The  elementary  p o p u l a t i o n c o n s i s t e d of grade f o u r students  s c h o o l s i n G r e a t e r Vancouver.  the r e g u l a r B r i t i s h Columbia program. be expected equations  The  from  students were on  Grade f o u r students c o u l d  t o have s u f f i c i e n t background f o r the w r i t i n g o f  f o r d i v i s i o n problems, but would have had  little  o p p o r t u n i t y f o r a c o n c e n t r a t e d study o f e q u a t i o n w r i t i n g . The  Sample The  sample was  from an elementary rooms.  school.  The  students came from two  class-  From the r e s u l t s of a p i l o t study, a sample s i z e o f  f o r t y s t u d e n t s was mental r e s u l t s .  determined  f o r more than one  adequate f o r s i g n i f i c a n t e x p e r i -  A c c o r d i n g l y , the sample s i z e was  A student was  test.  made up of a l l the grade f o u r students  forty-nine.  o m i t t e d from the study i f he was  treatment  or i f he was  away f o r the  absent  criterion  A t o t a l of f o r t y students were used i n the a n a l y s i s of  the d a t a . Assignment of S u b j e c t s The  students were a s s i g n e d to the two groups a t r a n 31  dom,  u s i n g a t a b l e of random numbers.  Random assignment  a l s o employed i n making up the subgroups w i t h i n one of the  was  16. e x p e r i m e n t a l groups.  I t was  f e l t t h a t c l o s e f r i e n d s might have  a tendency to gather together rearrangement would d e s t r o y III. The  i n the subgroups, and  some of the  DEVELOPMENT OF  that  this  randomness.  MATERIALS  I n s t r u c t i o n a l Device Since  presenting g i v e the  the  i t i s h i g h l y u n l i k e l y t h a t e i t h e r two same m a t e r i a l or one  teacher  same l e s s o n , i n s t r u c t i o n was  loop p r o j e c t o r . displayed  screen  a  lesson  g i v e n by means of a  Seven c a r t r i d g e s of 8 mm.  on a 5'x5'  repeating  teachers  film  c o l o r e d f i l m were  by a T e c h n i c o l o r  800  I n s t a n t Movie  Projector. There were e i g h t sequences o f p a r t i t i v e problems and  e i g h t sequences of q u o t i t i v e d i v i s i o n problems.  d e s c r i p t i o n of the p a r t i t i v e sequences i s g i v e n The  As  i n Appendix  q u o t i t i v e sequences were p a r a l l e l t o the p a r t i t i v e The  how  division  to w r i t e the  sequences were designed to i n s t r u c t the equations f o r the problems d e p i c t e d  less e x p l i c i t information  so t h a t the seventh and explicit  information  and  students  i n the  sequences. pro-  explicit direction,  e i g h t h sequences of each type gave  or d i r e c t i o n to the  A.  sequences.  s t u d e n t s moved through the sequences, they r e c e i v e d  gressively  A  students.  no  17. The T e s t  Instrument The  c r i t e r i o n t e s t consisted of twenty-five d i v i s i o n  problems p r e s e n t e d i n words  (see Appendix B ) .  T h i r t e e n o f the  problems were o f the p a r t i t i v e type, and the remaining were q u o t i t i v e .  The two types had been s c a t t e r e d  items  throughout  the t e s t t o i n s u r e t h a t there were not too many s u c c e s s i v e items of  the same type. To a s s u r e the c o n t e n t v a l i d i t y of the t e s t , the items  were e i t h e r  chosen d i r e c t l y from the supplementary  the course o r were m o d i f i c a t i o n s o f such items. the problems was checked altered  textbooks f o r  The format o f  f o r f a m i l i a r i t y and the words were  where necessary t o keep the problems a t an a p p r o p r i a t e  reading l e v e l . To p r o v i d e the r e l i a b i l i t y experiment  data f o r the t e s t , a p i l o t  was r u n one month b e f o r e the experiment.  they wrote the t e s t , s i x t y - t h r e e students i n grade elementary  Before f o u r a t an  s c h o o l i n Vancouver were i n s t r u c t e d by the f i l m loops  f o r two p e r i o d s on separate days.  On the t h i r d day, they wrote  the t e s t w i t h an adequate amount of time f o r completion by all  students. The r e l i a b i l i t y  a KR20.  of the t e s t i s p r e s e n t e d i n terms o f  Because the t e s t was measuring  a certain  s t a t e and not  a change over time, an i n t e r n a l c o n s i s t e n c y measure was  chosen.  In a d d i t i o n , s i n c e the t e s t was composed of p a r t i t i v e and  18. q u o t i t i v e items, and the KR20 i s i n t e n d e d t o be used on homogeneous t e s t s , a KR20 was found  f o r the q u o t i t i v e items and  a s e p a r a t e KR20 was found f o r the p a r t i t i v e The  items.  i t e m a n a l y s i s , g i v e n i n Appendix B, produced the  following results:  the s u b t e s t o f the twelve q u o t i t i v e  had a KR20 o f .74, and a l l items had a p o i n t b i s e r i a l  items  correla-  t i o n above .2 and a p v a l u e — t h e p e r c e n t of people who got the item c o r r e c t — b e t w e e n  .05 and .95.  On the p a r t i t i v e s u b t e s t ,  the KR20 was .86, and as w i t h the q u o t i t i v e items, no items f a i l e d t o meet the c r i t e r i a  f o r r and p mentioned above.  From the r e s u l t s o f the item a n a l y s i s on t h e data from the p i l o t experiment, i n t h i s experiment i n the p i l o t  i t was d e c i d e d t h a t the t e s t  should be e x a c t l y the same as the one used  experiment. IV.  The and  used  experimenter  PROCEDURE  met w i t h the p a r t i c i p a t i n g  t e a c h e r s two weeks b e f o r e the study began.  the nature o f the experiment times were scheduled  principal  A t t h a t time,  was d i s c u s s e d and the a p p r o p r i a t e  f o r the e x p e r i m e n t a l s e s s i o n s .  i t was d e c i d e d t h a t the experiment  In a d d i t i o n ,  would take p l a c e i n a  separate room i n s t e a d o f a r e g u l a r classroom to a l l o w f o r the e f f i c i e n t s e p a r a t i o n o f the two groups. On each day o f the experiment,  the students who were  taught i n d i v i d u a l l y were i n the experimental room from 9:00 A.M.  19. to  9:45  of  f o u r were i n i t from 9:45  of  the students i n both groups were s u p e r v i s e d by the e x p e r i -  menter.  A.M.,  The  and  the students who A.M.  were taught i n the groups t o 10:30  A.M.  The  activities  t e a c h e r s of the c l a s s e s were o n l y o b s e r v e r s .  On Day  1 (Wednesday), a f t e r i n t r o d u c i n g h i m s e l f t o  each group, the experimenter  informed  the students t h a t f o r  the next t h r e e p e r i o d s they would be taught by a movie projector.  He a l s o s a i d t h a t the p r o j e c t o r was  going t o teach  them how  t o w r i t e an e q u a t i o n f o r a d i v i s i o n problem.  The  s t u d e n t s were then t o l d t h a t i n a f o u r t h p e r i o d they would w r i t e a t e s t , so t h a t we a good t e a c h e r .  c o u l d see i f the movie p r o j e c t o r was  They were t o l d t h a t today they were going t o  see e i g h t examples on w r i t i n g an e q u a t i o n f o r a d i v i s i o n lem, to  and  although they would be a l s o asked  prob-  to f i n d the answer  the problem, they s h o u l d make sure t h a t they c o u l d w r i t e  the e q u a t i o n c o r r e c t l y . menter would work through  They were then t o l d t h a t the e x p e r i the f i r s t example w i t h them and t h a t  they s h o u l d use the answer sheet p r o v i d e d and should work a c c o r d i n g to the experimenter's t o l d t h a t they would p r o b a b l y because the e q u a t i o n was  instructions.  f i n d the f i r s t  They were a l s o few  examples  easy  g i v e n to them, but they should work  a l o n g a c c o r d i n g t o the i n s t r u c t i o n s because i n the  later  examples a l l of the e q u a t i o n would not be g i v e n to them. As the experimenter  went through  the f i r s t  w i t h the s u b j e c t s , he read out i n s t r u c t i o n s .  The  example  instructions  are g i v e n i n Appendix C.  The  s t u d e n t s who  always worked  i n d i v i d u a l l y r e c e i v e d the I n d i v i d u a l - t a u g h t I n s t r u c t i o n s , w h i l e the s t u d e n t s who  worked i n the subgroups were g i v e n the  Group-taught I n s t r u c t i o n s .  The  sentences  c o n t a i n e d i n the  b r a c k e t s under each i n s t r u c t i o n i n Appendix C d e s c r i b e what happened b e f o r e the students r e c e i v e d the next The  next day  (Day  instruction.  2, Thursday), both groups were t o l d  t h a t they were going t o see another e q u a t i o n f o r a d i v i s i o n problem.  e i g h t f i l m s on w r i t i n g  They were a l s o t o l d t h a t the  examples would be s i m i l a r to the ones they saw  yesterday,  t h a t the q u e s t i o n s today would have d i f f e r e n t e q u a t i o n s , t h a t they should watch c l o s e l y f o r the d i f f e r e n c e . then t o l d t h a t the experimenter  experimenter  o u t l i n e d i n Day  except and  They were  would work through the  example w i t h them i n the same way The  an  first  as he d i d b e f o r e .  proceeded  i n the same manner as  1, and showed the e i g h t p a r t i t i v e  sequences.  Because the p a r t i t i v e and q u o t i t i v e f i l m loops were o r g a n i z e d i n the same way,  o n l y the problems and the type of e q u a t i o n were  d i f f e r e n t from those on Day On Day  1.  3, the f o l l o w i n g Monday, the students were  reminded t h a t they had  seen f i l m s on two  f o r d i v i s i o n problems, and  t h a t now  different  equations  they were going t o see  some of the same f i l m s a g a i n , but t h i s time the two k i n d s of problems would be mixed up. the f i l m s i n the same way  They were then t o l d t o work  as they d i d b e f o r e .  through  21.  The way  as was  experimenter went through an example i n the same  done on Day  were p r e s e n t e d was the f o l l o w i n g : Partitive  1,  but the order i n which the loops  different.  The o r d e r of p r e s e n t a t i o n was  Quotitive #5, P a r t i t i v e #7, Quotitive #8,  # 5 , P a r t i t i v e # 6 , Q u o t i t i v e # 6 , Q u o t i t i v e # 7 , and  P a r t i t i v e #8. On the next day was  administered.  (Day 4 , Tuesday),  the c r i t e r i o n t e s t  Before the students wrote the t e s t ,  they  were t o l d t h a t they should w r i t e the e q u a t i o n and answer f o r each problem  i n the space p r o v i d e d on the t e s t paper.  were a l s o t o l d t h a t they had t e s t , but i f they f i n i s h e d  They  the whole p e r i o d to w r i t e the  e a r l y they c o u l d hand i n t h e i r  papers. V.  STATISTICAL ANALYSIS  Data F o r each s t u d e n t , the c o r r e c t number of equations the c r i t e r i o n t e s t was  recorded.  Although the s t u d e n t s were  asked to f i n d the answers t o the problems,  these r e s u l t s  not a n a l y z e d because the o b j e c t i v e of the i n s t r u c t i o n teach the s t u d e n t s t o w r i t e equations t o d i v i s i o n The  s t u d e n t s were asked  were  was  to  problems.  to f i n d answers because i t was  t h a t i t would be d i f f i c u l t  on  felt  to keep the problems i n a form w i t h  which the s t u d e n t s were f a m i l i a r  i f t h i s was  (a) and P a r t (b) i n Appendix D l i s t  not done.  Part  the e x p e r i m e n t a l d a t a .  22. Null  Hypotheses Hi.  There i s no s i g n i f i c a n t d i f f e r e n c e between the mean of the group o f students who were taught  individually  and the mean o f the group o f students who were taught i n groups o f f o u r . H2.  There i s no s i g n i f i c a n t d i f f e r e n c e between the v a r i ance o f the group o f students who were taught  indiv-  i d u a l l y and the v a r i a n c e o f the group o f s t u d e n t s who were taught i n groups o f f o u r . Statistical  Treatment  The f o l l o w i n g s t a t i s t i c s were c a l c u l a t e d from the s c o r e s on the c r i t e r i o n  test: Individual-taught  Group-taught  Mean Score  X  Standard D e v i a t i o n  s  T  s_  Number i n Group  n  T  n  Hypothesis One.  5L,  T  The two-sample t v a l u e was d e t e r -  mined by:  X  - Xj.  (n  2  t =  where s  ^7T—TT—T  p  l)sg +  -  (n  = n  c  + n  - 1) s ~  2  i"  2  Spd/rij. + l/n ) G  and compared of  freedom.  w i t h the t a b u l a t e d v a l u e f o r n^, + n_ - 2 degrees  Hypothesis s Lar er F = ^-x— 9 s Smaller  Two.  The F r a t i o was computed  thus:  2  ar  2  er  where m = n L  T a  r  *  e  r  -1 and n = n_ , .. -1 Smaller  and compared w i t h the t a b u l a t e d v a l u e f o r (m,n) degrees o f freedom. The means were compared u s i n g a t w o - t a i l e d t e s t , as what was b e i n g t e s t e d was whether t h e r e was a d i f f e r e n c e between the groups. tailed  The v a r i a n c e s were a l s o compared u s i n g a two-  t e s t because t h e r e was no evidence  o n e - t a i l e d t e s t c o u l d be employed.  to indicate that a  CHAPTER I I I ANALYSIS OF THE RESULTS I. Hypothesis  TESTING OF HYPOTHESES  One Hypothesis  One was t h a t there would be no  statisti-  c a l l y s i g n i f i c a n t d i f f e r e n c e between the mean o f the group o f s t u d e n t s who were taught i n d i v i d u a l l y of  students who were taught  summarizes the r e s u l t s of  the c r i t e r i o n  and the mean of the group  i n groups o f f o u r .  The t a b l e below  o b t a i n e d on the t w e n t y - f i v e  questions  test. TABLE I STATISTICS FOR EQUATION WRITING  Individual-taught Mean Score Standard  Deviation  Number i n Group  The  13.75  13.80  3.54  3.46  20  critical  Group-taught  20  v a l u e of t a t the .05 l e v e l o f s i g n i f -  i c a n c e w i t h 38 degrees o f freedom and a t w o - t a i l e d t e s t was 2.02.  Since the t v a l u e o b t a i n e d was .044, which d i d not  exceed the c r i t i c a l  v a l u e , i t was concluded  t h a t there was no  statistically  s i g n i f i c a n t d i f f e r e n c e between the groups as  regards the number o f equations c o r r e c t l y w r i t t e n . Hypothesis  Two Hypothesis  Two was t h a t there would be no s i g n i f i c a n t  d i f f e r e n c e between the v a r i a n c e o f the group o f students who were taught i n d i v i d u a l l y and the v a r i a n c e of the group o f students who were taught i n groups of f o u r . The  r e s u l t s g i v e n i n Table I were used t o o b t a i n a  F v a l u e o f 1.05.  T h i s o b t a i n e d v a l u e was compared w i t h the  c r i t i c a l value of F with  (19,19) degrees  l e v e l o f s i g n i f i c a n c e equal t o 2.6.  of freedom a t t h e .05  S i n c e the o b t a i n e d v a l u e  was l e s s than the c r i t i c a l v a l u e , i t was concluded  that there  was no s i g n i f i c a n t d i f f e r e n c e i n the v a r i a n c e s of the two groups. II.  CONCLUSIONS  There was no s t a t i s t i c a l l y  significant  difference  between the group o f students who were taught i n d i v i d u a l l y and the group o f students who were taught i n groups o f f o u r as regards t h e i r performance and v a r i a b i l i t y on the c r i t e r i o n test.  That both groups averaged  showed t h a t the a b i l i t y of  scores around f i f t y  percent  t o d i s t i n g u i s h between the two types  d i v i s i o n problems had g e n e r a l l y not been s u c c e s s f u l l y  taught, f o r the students c o u l d have merely  guessed  a t each  26. q u e s t i o n o r w r i t t e n one  type of e q u a t i o n f o r a l l the q u e s t i o n s .  In a d d i t i o n , the sameness o f the r e s u l t s seem to i n d i c a t e t h a t n e i t h e r method taught the students v e r y w e l l .  E v i d e n t l y , the  t e a c h i n g of these students i n groups o f f o u r was handicap matter  neither a  nor a h e l p t o them v i s - a - v i s mastery of the s u b j e c t  taught. III.  ANALYSIS OF ADDITIONAL DATA  Answers on the C r i t e r i o n T e s t I n a d d i t i o n t o w r i t i n g the e q u a t i o n f o r each problem on the c r i t e r i o n t e s t , the students found the answer t o the problem o r , i n o t h e r words, the v a l u e f o r which the frame i n the e q u a t i o n stood.  The mean and  the standard d e v i a t i o n f o r  the i n d i v i d u a l - t a u g h t group were 22.95 and  2.06,  respectively,  w h i l e the c o r r e s p o n d i n g v a l u e s f o r the group-taught 22.55 and  2.67.  The data can be seen i n Appendix  group were D.  As the t e s t c o n t a i n e d t w e n t y - f i v e items, i t appeared t h a t the students d i d not f i n d the numbers w i t h which they  had  to work above t h e i r l e v e l o f knowledge. Equations  and Answers During  Instruction  In Appendix D the responses  made by the students  the problems p r e s e n t e d by the f i l m loops are summarized. each item the f i r s t p o s i t i o n i n d i c a t e s whether the e q u a t i o n was  w r i t t e n , and  the c o r r e c t answer was  on For  correct  the o t h e r p o s i t i o n s i g n i f i e s whether  given.  27. The  group-taught  d a t a was  arranged  i n the same o r d e r ,  as the students were randomly a s s i g n e d t o t h e i r r e s p e c t i v e subgroups a t the b e g i n n i n g o f the experiment.  But because  of absent s t u d e n t s , not a l l subgroups were made up of f o u r students.  The  t a b l e below o u t l i n e s the number and  the  students  i n each subgroup. Subgroup #  Student  1  1-3  2  4-7  3  8-11  4  12-13  5  14-16  6  17-20  Upon examining  #  (inclusive)  the r e s u l t s of the day  to day  instruc-  t i o n , i t seems t h a t the students were able t o f o l l o w the f i l m loops i n most cases.  But the d a t a i n d i c a t e s t h a t there were  i n s t a n c e s where the students i n the subgroups f a i l e d to  agree  on a group answer b e f o r e they r e c e i v e d the c o r r e c t answer the f i l m . different  from  S p e c i f i c a l l y , i n some subgroups the students have answers and  equations.  CHAPTER IV IMPLICATIONS OF THE STUDY I. The  INTRODUCTION  h y p o t h e s i s i n v o l v e d i n t h i s study was t h a t  i n s t r u c t i o n a l use o f subgroups o f four students would i c a n t l y improve achievement among the s t u d e n t s .  signif-  A non-signif-  i c a n t d i f f e r e n c e between the performances o f the groups was found.  I t was n e c e s s a r y , t h e r e f o r e , t o c o n s i d e r  the p o s s i b l e  reasons f o r the s i m i l a r i t y o f the means and v a r i a n c e s  o f the  two groups i n the experiment. II. The  DISCUSSION OF THE CONCLUSIONS  Content o f the Lessons In c h o o s i n g the m a t e r i a l  menter p a i d  f o r i n s t r u c t i o n , the e x p e r i -  s p e c i a l a t t e n t i o n as t o whether i t i n v o l v e d a broad  range o f i d e a s , so t h a t i t would lend i t s e l f between the students.  I t was f e l t  go i n t o making up an e q u a t i o n ,  to i n t e r a c t i o n  t h a t a l a r g e range o f ideas  and, a c c o r d i n g l y ,  m a t e r i a l would be conducive t o group d i s c u s s i o n .  that  this  But i t may  have been t h a t the e q u a t i o n w r i t i n g f o r these p a r t i c u l a r problems d i d not tend t o produce i n t e r a c t i o n between the students.  29.  The Method of  Instruction  The  e f f e c t i v e n e s s o f the group method may  depend  g r e a t l y on making people adopt a work or performance g o a l or a change i n t h e i r b e h a v i o r . important  Because i t was  considered  to g i v e the students immediate r e i n f o r c e m e n t ,  the  c o r r e c t e q u a t i o n and answer were p r o v i d e d a f t e r each problem. I t may  have been t h a t w i t h the c o n t i n u a l r e l i a n c e of the  on the s c r e e n f o r the c o r r e c t e q u a t i o n and  students  answer, they may  not  have f e l t the need to work along v e r y c l o s e l y w i t h t h e i r f e l l o w members.  The  l a r g e range of s c o r e s i n some of the subgroups  i n d i c a t e s t h a t the weaker students may f o r the c o r r e c t answer and Although group e q u a t i o n and  have r e l i e d on the s c r e e n  equation.  the students were i n s t r u c t e d to agree on a answer b e f o r e they r e c e i v e d the c o r r e c t ones,  there were some of the members of the same subgroup who d i f f e r e n t equations.  Here a g a i n , i t may  had  have been t h a t the  students depended on the s c r e e n f o r t h e i r answers and not  their  f e l l o w group members. The m a t e r i a l was  l e v e l of the a r i t h m e t i c f a c t s i n the low,  so t h a t there would be l i t t l e  chance of the  students m i s s i n g an e q u a t i o n because they were not w i t h the numbers i n the problem.  But i t may  these numbers d i s c o u r a g e d i n t e r a c t i o n . the students informed too easy.  The  the experimenter  students maintained  instructional  familiar  have been t h a t  In a number of i n s t a n c e s , t h a t the q u e s t i o n s were  t h a t these numbers were f o r  30. the grade ones and twos. f a c t s may understood  Thus, the l e v e l of the a r i t h m e t i c  have l e a d the weaker students to t h i n k t h a t they the m a t e r i a l and d i d not need any h e l p from  the  other students. Length of the  Experiment  I t has been argued group i n t e r a c t i o n to develop. ment  was  t h a t s u f f i c i e n t time i s needed f o r To account f o r t h i s , the e x p e r i -  s e t up w i t h the students working  subgroups f o r three days.  i n their respective  A d d i t i o n a l l y , w i t h examples done f o r  the students and the c o r r e c t e q u a t i o n and answer g i v e n immedia t e l y a f t e r each problem, i t was  expected  would have a good i d e a of what was  t h a t the students  b e i n g taught and,  conse-  q u e n t l y , would q u i c k l y and c o n f i d e n t l y partake i n group d i s cussions.  But i t i s p o s s i b l e t h a t even w i t h these  adjustments,  the groups needed more time t o g e t h e r . Subgroup Makeup Although random assignment have e q u i v a l e n t subgroups,  i t may  was  necessary i n o r d e r to  have been t h a t the l a c k of  " n a t u r a l n e s s " i n the subgroups prevented the development of group c o h e s i v e n e s s .  Specifically,  there were a number of  subgroups which c o n t a i n e d students who  d i d not wish t o work  w i t h the o t h e r students i n t h e i r subgroup. the p a r t of some o f the students who  saw  The resentment  their friends i n  on  "good" groups may  have caused a n e g a t i v e a t t i t u d e toward group  work; l e a d i n g these s t u d e n t s t o work as i n d i v i d u a l s . In a d d i t i o n , because o f absent s t u d e n t s , the number of  s t u d e n t s i n some of the subgroups v a r i e d from day to day.  On c e r t a i n days, there were o n l y two students i n some subgroups.  Consequently, some o f the s t u d e n t s , even by b e i n g i n  a subgroup, were not e x p e r i e n c i n g the f u l l e f f e c t s o f group work. III. The F i l m  LIMITATIONS OF THE STUDY  Loops The f i l m loops used t o c o n t r o l the teacher v a r i a b l e  were s i l e n t movies. the  To g e t the s t u d e n t s s t a r t e d each day,  experimenter d i d one example w i t h the s t u d e n t s .  Then he  remained s i l e n t d u r i n g the showing of the o t h e r sequences. T h i s meant t h a t any q u e s t i o n s which the students had about the l e s s o n p r e s e n t a t i o n c o u l d not be answered.  O b v i o u s l y , there  might be d i f f e r e n t r e s u l t s i f sound were t o accompany the movies. In  the f i r s t  two days of the experiment, the s t u d e n t s  were shown q u o t i t i v e and p a r t i t i v e sequences, r e s p e c t i v e l y . Because o n l y a c e r t a i n number of sequences were a v a i l a b l e to the experimenter, some of the sequences were r e p e a t e d on the t h i r d day.  I t i s f e l t t h a t i t would have been b e t t e r to have  had day, the  some d i f f e r e n t e x a m p l e s t o show t h e as  t h e y w e r e l e a r n i n g a new  two  d i d not  but  allow  daylight viewing.  the  p i c t u r e emitted the  by  room was  picture,  was  the  and  appears d e s i r a b l e  and  not  experiment.  a s c r e e n were used. that the  k e p t l i g h t e n o u g h so  students,  f i l m f o r the  that  This  could  hence the  t o have a d a y l i g h t  was  projector,  large To  enough  circumvent Accordingly,  students could  t h a t the  they  sequences  f i t into a daylight  s i z e used i n t h i s  what they were w r i t i n g .  formance of  f i l m l o o p s was  t h i s p r o j e c t o r was  d a r k e n e d enough so  and  the The  i n c a r t r i d g e s which could  t h i s problem, a projector  see  third  t a s k — d i s t i n g u i s h i n g between  f u r t h e r l i m i t a t i o n of  f o r a group of  the  the  types of d i v i s i q n problems. A  placed  s t u d e n t s on  see  students  have a f f e c t e d  the  could  the  per-  r e s u l t s obtained.  It  projector.  A B r o a d e r Sample The school.  As  ability,  the  sample of t o see  this  school  generally  grade four p o p u l a t i o n .  i f s i m i l a r r e s u l t s w o u l d be I t i s quite possible  c a r r i e d out  has  a  representative  I t w o u l d be  obtained  with  interesting less  t h a t group work o f the  able type  i n t h i s e x p e r i m e n t i s l e s s s u i t a b l e f o r more  students.  same  s t u d e n t s o f above average  p a r t i c i p a n t s were p r o b a b l y not  the  students.  capable  s t u d e n t s i n t h i s e x p e r i m e n t a l l came f r o m t h e  Quotitive-Partitive The  Approach  approach taken  i n the i n s t r u c t i o n  ment was t o w r i t e two d i f f e r e n t  equations  types o f d i v i s i o n problems which e x i s t  of the e x p e r i -  f o r t h e two  different  i n the r e a l w o r l d .  The  a d o p t i o n o f t h i s p o i n t o f v i e w i s now l e f t up t o t h e t e a c h e r s in had  t h e s c h o o l s , and t h e s c h o o l i n w h i c h t h e e x p e r i m e n t n o t been s t r i c t l y  the students  f o l l o w i n g t h i s approach.  c o u l d have had a d i f f i c u l t  was r u n  Consequently,  time a d j u s t i n g t o the  different presentation. IV. In possibility It  forming  FOR FUTURE  STUDIES  t h e s u b g r o u p s b y random a s s i g n m e n t , t h e  o f s u b g r o u p s w i t h i n c o m p a t i b l e members was c r e a t e d .  seems a d v i s a b l e t h a t t h e r e s h o u l d b e a d i a g n o s i s o f t h e  individual to  SUGGESTIONS  needs and a b i l i t i e s ,  groups i n such  and then assignment o f s t u d e n t s  a way t h a t e a c h g r o u p c o n t a i n s t h e n e c e s s a r y  r a n g e o f s k i l l s w i t h a minimum o f d u p l i c a t i o n i n d i f f e r e n t individuals.  T h i s w o u l d mean t h a t s o c i o l o g i c a l  factors,  as  w e l l a s a c h i e v e m e n t , s h o u l d be u s e d i n p l a c i n g t h e s t u d e n t s i n their  r e s p e c t i v e subgroups, so t h a t the group s i t u a t i o n i s  more l i k e l y  to provide personal relations  more i n l i n e  with  w h a t t h e members o f t h e g r o u p d e s i r e . The ability  c r i t e r i o n v a r i a b l e i n t h i s experiment  o f the student  was t h e  t o r e c o g n i z e whether a d i v i s i o n  problem  34. was  p a r t i t i v e o r q u o t i t i v e , and to w r i t e the  equation.  However, i t may  corresponding  be t h a t s t u d e n t - s t u d e n t  interaction  has no s t a t i s t i c a l l y d i s c r i m i n a b l e e f f e c t on the a c q u i r i n g of the above-mentioned a b i l i t y .  S i n c e i t i s s a f e t o say t h a t  s t u d e n t - s t u d e n t i n t e r a c t i o n does have a decided e f f e c t on v a r i a b l e s as p e r c e p t i o n o f an o b j e c t i v e s t i m u l u s , l i k i n g s t u d e n t s , and  s p o n t a n e i t y of b e h a v i o r , i t i s suggested  such other  t h a t any  f u t u r e work with t h i s t e a c h i n g s t r a t e g y should i n c l u d e the measurement of v a r i a b l e s most l i k e l y  t o i n v o l v e the above.  Improved group performance may  e n t a i l the development  of working r e l a t i o n s w i t h a c e r t a i n group of people. t h a t grade f o u r students do not possess the s o c i a l necessary In  skills  t h i s r e s p e c t , i t i s recommended t h a t an o l d e r group of  The  i n future studies.  answer was  g i v e n a f t e r each problem because i t  f e l t t h a t t h i s immediate r e i n f o r c e m e n t would h e l p the  students l e a r n .  But i t seemed t h a t the students tended  to s i t  back and w a i t f o r the answer to appear on the s c r e e n , i n s t e a d of  be  to e s t a b l i s h the s o c i a l o r g a n i z a t i o n o f the group.  students be used  was  I t may  a s k i n g the o t h e r students i n t h e i r group.  I t would be of  i n t e r e s t t o know t h a t i f the students worked t o g e t h e r f o r a c e r t a i n l e n g t h o f time b e f o r e they r e c e i v e d any c o r r e c t  solu-  t i o n s , whether the weaker students would depend more on  the  o t h e r students from whom v a l u a b l e problem s o l v i n g might be  obtained.  techniques  V.  SUMMARY  T h i s study was designed t o i n v e s t i g a t e the e f f e c t s o f i n s t r u c t i o n i n s m a l l groups.  F i n d i n g s by p s y c h o l o g i s t s and  s o c i o l o g i s t s i n d i c a t e d t h a t s i g n i f i c a n t group f o r c e s e x i s t which c o u l d improve l e a r n i n g .  The p r e s e n t study examined the  a b i l i t y o f s t u d e n t s to w r i t e e q u a t i o n s f o r d i v i s i o n problems a f t e r they had been taught i n groups o f f o u r . The  s t u d e n t s were i n s t r u c t e d f o r three days.  In  order to have e q u i v a l e n t i n s t r u c t i o n f o r the two groups, loops p r e s e n t e d the m a t e r i a l o f the l e s s o n s . all  film  In one group,  the students were taught i n d i v i d u a l l y , w h i l e i n the o t h e r  group,  the students were taught i n groups o f f o u r . The  students were g i v e n a c r i t e r i o n t e s t o f twenty-  f i v e d i v i s i o n problems t o which they wrote the e q u a t i o n and answer.  A l l items i n the t e s t had s a t i s f i e d the r e q u i r e d  c r i t e r i a d u r i n g an item a n a l y s i s r e s u l t i n g from a p i l o t r u n i n the p r e v i o u s month. A t w o - t a i l e d t - t e s t was used t o t e s t the s i g n i f i c a n c e between the mean o f the i n d i v i d u a l - t a u g h t group and the mean o f the group-taught  group.  A t w o - t a i l e d F - t e s t was used to t e s t  the s i g n i f i c a n c e between the v a r i a n c e s o f the two groups. The  r e s u l t s showed t h a t there was no s i g n i f i c a n t  f e r e n c e between the i n d i v i d u a l - t a u g h t group and the grouptaught group w i t h r e s p e c t to mean and v a r i a n c e .  dif-  FOOTNOTES  E v e r e t t W. Bovard, "The Psychology o f Classroom I n t e r a c t i o n , " The J o u r n a l o f E d u c a t i o n a l Research, 45:215-224, 1951. 2 H.V.  P e r k i n s , " C l i m a t i c I n f l u e n c e s o f Group L e a r n i n g , "  The J o u r n a l of E d u c a t i o n Research,  45:115-119,  1951.  3 B.O. Bergum, and D.J. Fehr, " E f f e c t s of A u t h o r i t a r i a n i s m on V i g i l a n c e Performance," The J o u r n a l of A p p l i e d Psychology, 47:75-77, 1963. 4 R.B. Zajonc, and S.M. S a l e s , " S o c i a l F a c i l i t a t i o n o f Dominant and Subordinate Responses," The J o u r n a l of E x p e r i mental S o c i a l Psychology, 2:160-168, 1966. ^E.E. Jones, and H.B. Gerard, Foundations of S o c i a l Psychology (New York: Wiley, 1967), pp.1-154. ^ J . Macy J r . , L.S. C h r i s t i e , and R.D. Luce, "Coding Noise i n a Task O r i e n t e d Group," The J o u r n a l of Abnormal S o c i a l Psychology, 48:401-409, 1953. 7 F r e d T. Wilhelms, and Dorothy Westby-Gibson, "Grouping Research O f f e r s Leads," E d u c a t i o n a l L e a d e r s h i p , 18:410-413, 1961. 8  B.F. S k i n n e r , The Technology of Teaching (New York: Century P s y c h o l o g i c a l S e r i e s , 1968), p.16. 9 I r v i n g E. S i g e l , "The P i a g e t i a n System and the World o f E d u c a t i o n , " S t u d i e s i n C o g n i t i v e Development: Essays i n the Honor of Jean P i a g e t , David E l k i n d and John H. F l a v e l l , E d i t o r s (New York: Oxford U n i v e r s i t y P r e s s , 1969), p.473. "^Jean P i a g e t , "Three L e c t u r e s , " P i a g e t Rediscovered, R.E. R i p p l e and V.N. R o c k c a s t l e , e d i t o r s ( I t h a c a , New York: C o r n e l l U n i v e r s i t y P r e s s , 1964), p.4. "''"'"Herbert Gournee, "A Comparison of C o l l e c t i v e and I n d i v i d u a l Judgements of F a c t , " The J o u r n a l of E x p e r i m e n t a l Psychology, 21:106-112, 1937. 12 Samuel F. Klugman, "Cooperative Versus I n d i v i d u a l E f f i c i e n c y i n Problem S o l v i n g , " The J o u r n a l of E d u c a t i o n a l Psychology, 35:91-100, 1944.  37. 13  H.V. P e r l m u t t e r , and Germaine de Montmollin, "Group L e a r n i n g of Nonsense S y l l a b l e s , " The J o u r n a l o f Abnormal S o c i a l Psychology, 47:762-769, 1953. 14  " Bryce B. Hudgins, " E f f e c t s of Group E x p e r i e n c e on I n d i v i d u a l Problem S o l v i n g , " The J o u r n a l of E d u c a t i o n a l Psychology, 51:37-42, 1960. 15 D.W. T a y l o r , and W.L. F a u s t , "Twenty Q u e s t i o n s : E f f i c i e n c y i n Problem S o l v i n g as a F u n c t i o n of S i z e of Groups," The J o u r n a l of E x p e r i m e n t a l Psychology, 44:360-368, 1952. 16 . , Ibid. T V  17 ., Ibid. T V  18 W.L. F a u s t , "Group Versus I n d i v i d u a l Problem S o l v i n g , " The J o u r n a l o f Abnormal S o c i a l Psychology, 59: 68-72, 1959. 19 N.H. Anderson, "Group Performance i n an Anagram Task," The J o u r n a l o f S o c i a l Psychology, 55:67-75, 1961. 20 James H. Davis and Frank R e s t l e , "The A n a l y s i s of Problems and P r e d i c t i o n of Group Problem S o l v i n g , " The J o u r n a l of Abnormal and S o c i a l Psychology, 66:103-116, 1963. 21 I. Lorge and H. Solomon, " I n d i v i d u a l Performance and Group Performance i n Problem S o l v i n g R e l a t e d to Group S i z e and P r e v i o u s Exposure to the Problem," The J o u r n a l of Psychology, 48:107-114, 1959. 22 Gardner L i n d z e y and E l l i o t Aronson, "Group P s y c h o l ogy and Phenomena of I n t e r a c t i o n , " The Handbook of S o c i a l Psychology, 1-694, 1969. 23 Ibid. 24 M.D. Dunnette, J . Campbell, and Kay J a a s t a d , "The E f f e c t of Group P a r t i c i p a t i o n on B r a i n s t o r m i n g E f f e c t i v e n e s s f o r Two I n d u s t r i a l Samples," The J o u r n a l of A p p l i e d Psychology, 47:30-37, 1963. 25 , . , Ibid. T  38.  26 I,b.i,d . T  27 ^'ibid. 28 M.J. S a w i r i s , "An E x p e r i m e n t a l and Group L e a r n i n g Using a L i n e a r Geometry L e a r n i n g , 3:146-153, 1966. 29 D.L. Moore, "Group Teaching by t i o n , " Programmed L e a r n i n g and E d u c a t i o n a l 1967. Ibid.  Study o f I n d i v i d u a l Program," Programmed Programmed I n s t r u c Technology, 4:37-46,  3 0  Roger E. K i r k , E x p e r i m e n t a l Design: Procedures f o r the B e h a v i o r a l S c i e n c e s , (Belmont: Brooks Cole P u b l i s h i n g Company, 1968), pp.520-521.  BIBLIOGRAPHY  Anderson, N.H. "Group Performance i n an Anagram Task," The J o u r n a l o f S o c i a l Psychology, 55:67-75, 1961. Balow, I r v i n g H. "The E f f e c t s o f Homogeneous Grouping i n Seventh Grade A r i t h m e t i c , " The A r i t h m e t i c Teacher, 11: 186-191, 1964. Banghart, F.W., and H.S. Spraker. 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"Twenty Questions: Efficiency i n Problem S o l v i n g as a F u n c t i o n of S i z e o f Groups," The J o u r n a l o f E x p e r i m e n t a l Psychology, 44:360-368, 1952. Thelen, H e r b e r t A. "Group Dynamics i n I n s t r u c t i o n : P r i n c i p l e o f L e a s t Group S i z e , " The School Review, 58:139-148, 1949. Thomas, E . J . , and C F . F i n k . "The E f f e c t s of Group S i z e , " The P s y c h o l o g i c a l B u l l e t i n , 60:371-384, 1963. T h o r n d i k e , Robert L. "The E f f e c t o f D i s c u s s i o n Upon the C o r r e c t n e s s o f Group D e c i s i o n s , When the F a c t o r of M a j o r i t y I n f l u e n c e i s Allowed f o r , " The J o u r n a l o f S o c i a l Psychology, 9:343-362, 1938.  Wallen, Norman E., and Robert 0. Vowles. "The E f f e c t of I n t r a c l a s s A b i l i t y Grouping on A r i t h m e t i c Achievement," The J o u r n a l of E d u c a t i o n a l Psychology, 51:159-163, 1960. Watson, Goodwin B. "Do Groups Think More E f f e c t i v e l y Than I n d i v i d u a l s , " The J o u r n a l of Abnormal and S o c i a l P s y c h o l ogy, 23:328-336, 1929. Wert, James E., C h a r l e s 0. N e i d t , and J . S t a n l e y Ahmann. S t a t i s t i c a l Methods i n E d u c a t i o n a l And P s y c h o l o g i c a l Research. New York: A p p l e t o n - C e n t u r y - C r a f t s , Inc., 1954. West, J e f f , and C a l l i e S i e v e r s . "Experiments i n C r o s s Grouping," The J o u r n a l o f E d u c a t i o n a l Research, 54: 70-71, 1960. Wilhelms, F r e d T., and Dorthy Westby-Gibson. "Grouping: Research O f f e r s Leads," E d u c a t i o n a l L e a d e r s h i p , 18: 410-413, 1961. Zajonc, R.B., and Dominant and Experimental  S.M. S a l e s . " S o c i a l F a c i l i t a t i o n of Subordinate Responses," The J o u r n a l of S o c i a l Psychology, 2:160-168, 1966.  Zimmerman, Donald. E d u c a t i o n , 85:  "Teaching 364-370,  T h i r t y , L i k e Teaching 1965.  One,"  A P P E N D I C E S  APPENDIX A THE INSTRUCTIONAL DEVICE  44. PARTITIVE SEQUENCE 1 Pictorial  Description  B LOC K S  1. The words "EIGHT BLOCKS" were pointed a t . 2. The "8" was brought down. 3. The b l o c k s were p u t i n the box.  ooag oi BLOCKS ARE: IN E>ICH CUP  How  Description  6  E I G H T  T H E Sft/^E. NUMBER  Word  6-*- • 4. Four b l o c k s were p l a c e d i n each cup, a l t h o u g h the viewer of the f i l m c o u l d not observe how many. 5. The words on the l e f t were changed. 6. The v Q was brought down.  MKNV  7, The =2" was brought down. 8, The words on the l e f t were changed. The p i c t u r e on the l e f t below appeared on the s c r e e n u n t i l the students had a c h i e v e d a s o l u t i o n , then the p i c t u r e on the r i g h t below was shown. 11  EACH C U P ?  EQUATIO  N  < 3 t° = ^ ' NuhBf-R  o-f  "BLOCKS = ?  EQUAT/ON NuM6£*  o f B LOCKS - 4 "  45. PARTITIVE SEQUENCE 2 Pictorial  Description  T W E N T Y - ONE.  21  .STRAWS  Word  Description  1. The words "TWENTY-ONE STRAWS* were p o i n t e d a t . 2. The "21" was brought down.  F_ACH BuNDLC H-_os THE. S/^fAe: NUMBER  3. The straws were p u t i n t o three bundles. 4. The words on the l e f t were changed. 5. The T O was brought i n t o place.  STRAWS  How MANY STRAWS IN E>CH  B-WD-E ?  EQ  2  1  3  6. The " = 3 " was brought down. 7. The words on the l e f t were changed. 8. The p i c t u r e below on the l e f t appeared on the s c r e e n w h i l e the students a c h i e v e d a s o l u t i o n ; then the p i c t u r e below on the r i g h t was shown.  EIQ  UAT!  ON  ei -• = 5 •  N ^ B E L R o f S T R A W S - ? N U M B E R «f S T R A W S  -7  46. PARTITIVE SEQUENCE 3 Pictorial  20  PENN IES  EACH CUPHOUDS THE. S A . P \ C Ny/-\i3fR  PENNIES  o-F  20 ?  How AA.MY TCN/MIES IN EACH  2.0 CUP ?  ?-4  N  = 4 •P PEN  Description  1. The words "TWENTY PENNIES" were p o i n t e d a t . 2. The "20" was brought down. 3. The pennies were p l a c e d i n t o a bag, so t h a t they were out o f s i g h t .  F i v e pennies were put i n each cup w i t h o u t the students knowing how many. 5, The words on the l e f t were changed. 6, The "?" was brought down.  7 8  The "=4" was brought down. The words on the l e f t were changed. The p i c t u r e below on the l e f t appeared on the s c r e e n w h i l e the students achieved a s o l u t i o n ; then the p i c t u r e on the r i g h t was shown.  EQUAT/O/V  LTQUAT/ON 20  Word  Description  20 + • - 4 NuM6ElR  o ? PENNIES-  5  47.  PARTITIVE SEQUENCE 4 P i c t o r i a l Description  O U T E E  1. The words "SIXTEEN INCHES OF RIBBON" were p o i n t e d a t . 2. The "16" was brought down.  N  INCHES < RIBBON  C U T  INTO  Pieces EIQU^VL.  How A"\ INCHES . EACH  Word D e s c r i p t i o n  F O U R  3. The r i b b o n was c u t i n t o four equal pieces. 4. The words on t h e l e f t were changed. 5. The "?" was brought down.  °f  Le.NG.TH  ANY  lb  IN  RE.CE.  ?  ??  ft a  6. The second "?" was brought down. 7. The words on the l e f t were changed. 8. The p i c t u r e below on the l e f t appeared on the s c r e e n w h i l e the students achieved a s o l u t i o n ; then the p i c t u r e on the r i g h t was shown.  E Q U r\T/ O N  E-QuAT/O/V /b  lb NunQE.R  o ' T  I N C H E S  ^  NUMBEJR  a  =  I  N  C  H  E  S  4  4-  48. PARTITIVE SEQUENCE 5 Pictorial  Description  T~/cK£TS  PUT THE SAME NUMBER »f TICKETS  EACH  27?  PiuE  How fWyTICKETS  Is EACH  27  AUE ?  27 ? ?  Word D e s c r i p t i o n  1. The words "TWENTY-SEVEN TICKETS" were p o i n t e d a t . 2. The "27" was brought down.  3. The t i c k e t s were put i n t o three p i l e s . 4. The words on the l e f t were changed. 5. The "?" was brought down.  6. The second "?" was brought down. 7. The words on the l e f t were changed. 8. The p i c t u r e on the l e f t below appeared on the s c r e e n w h i l e the students a c h i e v e d a s o l u t i o n ; then the p i c t u r e below on the r i g h t was shown.  UrVT/ON  UAT/ON  27  NUMBER of IICKETS= ?  Muri8e:R  «f  TICKETS^ 9  49. PARTITIVE SEQUENCE 6 P i c t o r i a l Description  (6  EJG,HT££/S/ Ounces of  WATER  Word D e s c r i p t i o n  1. The words "EIGHTEEN OUNCES OF WATER" were p o i n t e d a t . 2. The "18" was brought down.  0 BCHTEEN OUNCES O?  / 6 ?  WATER  0 i3 0 HOW^IANV of  OUNCES  3. Three g l a s s e s were f i l l e d w i t h water. 4. The "?" was brought down.  5. The second "?" was brought down. 6. The words on the l e f t were changed. 7. The p i c t u r e on the l e f t below appeared on the s c r e e n u n t i l the students achieved a s o l u t i o n ; then the s i g n below was shown.  WATER  /N £TACW G-ASS  EQUAT/ON / Q  NunBER  of  OUNCES-?  A/QMSER  +  • o f  =  OUNCES  3  -  6  50.  PARTITIVE SEQUENCE 7 Pictorial  Description  Word D e s c r i p t i o n  THIRTY  1. The words "THIRTY BOOKS" were p o i n t e d a t .  BOOKS  THIRTY 2. The books were d i v i d e d i n t o t h r e e equal p i l e s .  BOOKS  800KS EACH  PILE  ?  — UAT/ON  NUMQER  of  BOOKS  ,  3. The words on the l e f t were changed. 4 . The p i c t u r e below on the l e f t appeared on the screen u n t i l the students achieved a s o l u t i o n ; then the s i g n below was shown.  U A T 7 0/V  3 0-•= 3 NUMBER B°OKS=10 « f  51. PARTITIVE SEQUENCE 8 P i c t o r i a l Description  Word  Description  1. The words "FIFTEEN OUNCES OF CHEESE" were p o i n t e d at,  3> of  CHEESE  2. The cheese was c u t i n t o t h r e e equal p i e c e s .  r/ow M M y Ou^C£.S EACH  IN  RECE. ?  t  3. The words on the l e f t were changed. 4. The p i c t u r e below on the l e f t appeared on the screen u n t i l the students achieved a s o l u t i o n ; then the p i c t u r e below was shown.  QUKT/ ON  NUM8ER  of  OUNCES"?  EQU/\T70/\7 15 +  a  -  3  NUM6E:R °f OUNCES-5  APPENDIX B The T e s t  Instrument  P a r t i t i v e and Q u o t i t i v e Quotitive  Items  Partitive  Items  Items  52. (a)  THE TEST INSTRUMENT  In the spaces p r o v i d e d , w r i t e the e q u a t i o n and the answer f o r each problem. 1) Mary bought 24 f l o w e r s , she put them i n t o bunches o f 8. How many bunches of f l o w e r s d i d she make? 2) B i l l s o l d 20 cents worth o f t i c k e t s f o r the c l a s s r a f f l e . The t i c k e t s were 5 c e n t s each. How many d i d B i l l sell? 3) Mrs. Jones had 24 cabbages i n her p a t c h . There were 6 equal rows i n the p a t c h . How many cabbages were i n each row? 4) J i m caught 15 f i s h . He p u t the same number of f i s h i n each o f 5 bags. How many f i s h i n each bag? 5) Joan has 27 peanuts i n a bag. Joan wants t o g i v e the same number o f peanuts t o each o f her f r i e n d s . How many peanuts does each f r i e n d get i f she has three f r i e n d s ? 6) Mrs. Smith has 24 eggs. Her f a m i l y eats 6 eggs a t each meal. How many meals of eggs can her f a m i l y have? 7) There are 36 boys p l a y i n g b a s e b a l l i n the park w i t h 9 boys on a team. How many teams a r e p l a y i n g ? 8) A mother t o l d her 3 sons t h a t she would d i v i d e 12 apples evenly between them. How many apples d i d each get? 9) 28 in of in  c h i l d r e n went to a hockey game 4 c a r s c o n t a i n i n g the same number children. How many c h i l d r e n were each car?  10) A r e s t a u r a n t cook has 32 ounces o f hamburger meat. He uses 4 ounces of meat f o r every hamburger t h a t he makes. How many hamburgers can he make w i t h t h i s meat?  11) John went to the s t o r e t o buy toy cars. He had 32 cents w i t h him. The c a r s each c o s t 8 c e n t s . How many c a r s d i d he buy? 12) A vase has 12 f l o w e r s i n i t . I f the same number o f f l o w e r s are put i n t o 4 s m a l l e r vases, how many f l o w e r s are i n each vase? 13) A farmer owns 30 p i g s . He d e c i d e s to p u t 5 p i g s i n each pen. How many pens does he have? 14) The 20 c h i l d r e n i n Miss Smith's c l a s s formed 4 r e l a y teams o f the same s i z e . How many c h i l d r e n were on each r e l a y team? 15) I n a garden, t h e r e are 5 rows. Each row has the same number o f c o r n p l a n t s . I f there a r e 15 corn p l a n t s i n the garden, how many c o r n p l a n t s are there i n each row? 16) F a t h e r w i l l have 21 days f o r h i s h o l i d a y s t h i s year. How many weeks of h o l i d a y s does he have? 17) Bob uses 3 wheels f o r each a i r p l a n e . I f he has 24 wheels, how many a i r planes can he make? 18) A g r o c e r had 30 c a n d i e s i n a bag. To s e l l the c a n d i e s , he put them i n s m a l l e r bags h o l d i n g 5 candies each. How many s m a l l bags o f candy d i d he have t o s e l l ? 19) F i v e g i r l s p i c k e d the same number of q u a r t s o f s t r a w b e r r i e s f o r a farmer. I f the farmer had 35 q u a r t s of s t r a w b e r r i e s , how much d i d each g i r l p i c k ? 20) A g r o c e r p u t 26 cans of beans on 2 shelves. I f the s h e l v e s each h e l d the same number o f cans, how many cans were on each s h e l f ? 21) C l a r a gathered 36 r o s e s i n t o bunches. She p u t 6 r o s e s i n each bunch. How many bunches d i d she make?  22) John had 16 p i c t u r e s t o put i n an album. He used 4 pages and p l a c e d the same number o f p i c t u r e s on each page. How many p i c t u r e s d i d he put on each page? — 23) Some cent girl were  g i r l s shared the c o s t o f a 25 bag o f potato c h i p s . I f each p a i d a n i c k e l , how many g i r l s there?  24) Jean d i v i d e d 18 t a r t s e q u a l l y among the 6 g i r l s a t her p a r t y . How many t a r t s d i d she g i v e t o each g i r l ? 25) J i m wanted t o p u t h i s f i s h i n t o 4 f i s h bowls. I f J i m had 28 f i s h , how many d i d he p u t i n each f i s h bowl,so t h a t each bowl had the same number o f f i s h i n i t ?  55. (b) PARTITIVE AND QUOTITIVE ITEMS  Item No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Point  Biserial  0.1550 0.2862 0.4455 0.5341 0.3286 0.2741 0.1815 0.5054 0.3096 0.3046 0.3240 0.4565 0.1842 0.5663 0.5993 0.4033 0.2581 0.1863 0.4163 0.5580 0.4325 0.5568 0.3953 0.5794 0.4277  P  Variance  0.7302 0.6825 0.3402 0.3968 0.4921 0.7619 0.7143 0.3175 0.4127 0.8095 0.7619 0.3016 0.7937 0.3175 0.3968 0.4444 0.6667 0.6508 0.2381 0.3810 0.7143 0.3016 0.5556 0.2698 0.2698  0.1970 0.2167 0.2273 0.2394 0.2499 0.1814 0.2041 0.2167 0.2424 0.1542 0.1814 0.2106 0.1638 0.2167 0.2394 0.2469 0.2222 0.2273 0.1814 0.2358 0.2041 0.2106 0.2469 0.1970 0.1970  ***The Mean i s 12 .7302 ***The KR20 i s 0 . 7661 R e s u l t s o f Task Number 3 Items Omitted I f Any. 1 18 13 7 6 2 11 17 10 21  New KR20 0.7706 0.7790 0.7857 0.7959 0.8020 0.8102 0.8180 0.8288 0.8359 0.8471  56.  (c) QUOTITIVE ITEMS  Item No. 1 2 6 7 10 11 13 16 17 18 21 23  Point  Biserial 0.6194 0.5396 0.5165 0.5868 0.4745 0.6514 0.5496 0.2891 0.4144 0.5217 0.6886 0.3353  ***The Mean i s ***The KR20 i s  P  Variance  0.7302 0.6825 0.7619 0.7143 0.8095 0.7619 0.7937 0.4444 0.6667 0.6508 0.7143 0.5556  0.1970 0.2167 0.1814 0.2041 0.1542 0.1814 0.1638 0.2469 0.2222 0.2273 0.2041 0.2469  8. 2857 0. 7413  R e s u l t s o f Task Number 3 Items Omitted I f Any. Nil  New KR20 0.7413  57. (d) PARTITIVE ITEMS  Item No. 3 4 5 8 9 12 14 15 19 20 22 24 25  Point  Biserial  0.6532 0.7465 0.3762 0.6853 0.5567 0.5612 0.7319 0.6423 0.3928 0.6860 0.6816 0.7133 0.5217  P  Variance  0.3492 0.3968 0.4921 0.3175 0.4127 0.3016 0.3175 0.3968 0.2381 0.3810 0.3016 0.2698 0.2698  0.2273 0.2394 0.2499 0.2167 0.2424 0.2106 0.2167 0.2394 0.1814 0.2358 0.2106 0.1970 0.1970  ***The Mean i s 4 .4444 ***The KR20 i s 0 .8606 R e s u l t s of Task Number 3 Items Omitted I f Any. Nil  New KR20 0.8606  APPENDIX C (a) The D a i l y Work Sheet (b) The I n d i v i d u a l - t a u g h t (c) The Group-taught  Instructions  Instructions  (a) THE DAILY WORK SHEET Name Room  Equation  Answer  59. (b) THE INDIVIDUAL-TAUGHT 1.  INSTRUCTIONS  "Watch c l o s e l y as the problem i s g i v e n t o you on the s c r e e n . " (Six eggs, the words "SIX EGGS" and a "6" were shown;  then  the sequence was paused.) 2.  "Write down the '6' on your paper a f t e r the number one." (The eggs were p a i r e d , the words "PAIRED THE EGGS" were shown, and a "T 2" was p l a c e d i n the e q u a t i o n ; then the sequence was paused.)  3.  "Put the ' T - 2' i n i t s c o r r e c t p o s i t i o n . " (The words "HOW MANY PAIRS ARE THERE?" were shown, and the "=CJ" was brought down; then the sequence was paused.)  4.  "Place the ' = • *  i n your e q u a t i o n , and then w r i t e what the  frame stands f o r i n the l a s t column o f your paper." (The e q u a t i o n "6 -f 2 = • " and the words "NUMBER OF PAIRS=?" appeared on the screen; then the sequence was paused.) 5.  "At t h i s p o i n t i n each f i l m , we w i l l have a s h o r t pause. In the f i l m s t h a t a r e coming, you may need e x t r a time t o change your e q u a t i o n o r answer; i f so, do i t d u r i n g  this  pause." (The e q u a t i o n "6 * 2 = • " and the words "NUMBER OF PAIRS=3" were shown; then the sequence was paused.)  60. 6.  "Mark whether you were r i g h t o r wrong, and get ready f o r the  next sequence.  You are going t o work through i t by  y o u r s e l f , and you should work i n the same way as we did." (The f i l m was p r o g r e s s e d t o the next sequence.)  just  (The e q u a t i o n "6 * 2 = • " and the words "NUMBER OF PAIRS =3" were shown, then the sequence was paused.) "Mark whether your group was r i g h t or wrong, and g e t ready f o r the next f i l m .  You are g o i n g t o work through i t by  y o u r s e l v e s , and you should work i n the same way as we did." (The f i l m was p r o g r e s s e d t o the next sequence.)  just  APPENDIX D (a) E q u a t i o n s f o r I n d i v i d u a l - t a u g h t Group (b) E q u a t i o n s f o r Group-taught Group (c) Answers f o r I n d i v i d u a l - t a u g h t  Group  (d) Answers f o r Group-taught Group (e) Day One  Individual-taught  Group  (f) Day Two  Individual-taught  Group  (g) Day Three  Individual-taught  (h) Day One  Group-taught Group  (i) Day Two  Group-taught Group  (j) Day Three  Group-taught Group  Group  (a) EQUATIONS FOR INDIVIDUAL-TAUGHT GROUP  SCORES FOR EACH ITEM 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 18 20  1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1  1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0  1 0 1 1 0 1 1 0 1 1 1. 0 1 0 1 1 1 0 1 1 1 1 0 0 1  1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 0 0 1 1 1 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0  1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14 . 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.  STUDENT NUMBER  0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 0  1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 1  0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 0 1  1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0  1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  15 18 11 17 13 12 16 25 13 11 16 13 12 11 11 11  1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 0  1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1  1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0  1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0  9 15 13 13  (b) EQUATIONS FOR GROUP-TAUGHT GROUP  SCORES FOR EACH ITEM 1 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.  1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0  STUDENT NUMBER 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20  1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 0 0  1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 1  0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1  1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0  1 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0  1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1  1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 0 0 1  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 1 1 0 0 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1  1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0  1 1 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0  1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1  1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0  16 14 14 18 11 11 12 12 10 18 17 12 15 24 13 11 11 14 10 13  (c) ANSWERS FOR INDIVIDUAL-TAUGHT GROUP  SCORES FOR EACH ITEM 1 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  STUDENT NUMBER 3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20  1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1  1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1  1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1  1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1  1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1  1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1  25 24 22 23 25 21 25 25 21 21 23 25 21 25 23 23 18 25 20 24  (d) ANSWERS FOR GROUP-TAUGHT GROUP  SCORES FOR EACH  STUDENT NUMBER  :tem  1  2  3  l. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1  1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20  1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1  1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 0  1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1  1 1 1 1 1 1 1 ., 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1  25 24 23 22 23 23 22 25 18 25 23 24 23 25 25 14 21 23 21 22  67. (e) DAY ONE  INDIVIDUAL-TAUGHT GROUP  STUDENT JMBER 1. 2. Jo. 4. 5. 6. 7. * 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  ITEM NUMBER 1  2  3  4  5  6  7  8  11 11  11 11  11 11  11 11  11 11  01 11  11 11  11 00  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 01 01 11 01 11 11 11  11 11 11 11 11 11 01 11 11 01 11 11 11 11 11 00 11  11 11 00 11 11 11 11 11 11 11 11 11 11 11 11 11 11  (f) DAY TWO  INDIVIDUAL-TAUGHT GROUP  STUDENT NUMBER 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  ITEM NUMBER 1 01 11 11 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11  2 11 11 11 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11  3 11 11 11 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11  4 11 11 11 11 11 11 11 11 11 01 11 11 01 11 11 11 11 11 11 11  5 01 01 11 00 11 11 11 11 11 11 01 11 01 11 11 11 11 11 11 11  6 11 11 00 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11  7  8  11 01 11 00 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11  11 11 11 11 11 01 11 11 11 11 11 11 01 11 11 11 11 11 11 11  (g) DAY THREE  STUDENT NUMBER 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  INDIVIDUAL-TAUGHT GROUP  ITEM NUMBER 1  2  3  4  5  6  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 01 11 11 01 11 01 11 11 01 11 11 11 11 11 11 11 11 11 11  11 00 11 11 11 11 11 11 11 01 11 11 11 11 11 11 11 11 11 11  11 11 11 11 01 11 11 11 11 11 11 11 11 11 11 11 11 11 11 01  11 00 11 01 01 11 11 11 11 11 11 01 11 11 11 11 11 11 11 01  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 00 01  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 01 01 11 01 11 11 11 11 01 11 01 11 11 11 11 01 11 00 01  (h) DAY ONE  GROUP-TAUGHT GROUP  STUDENT NUMBER 1. 2. 3. 4. 5. 6.  •7 / •  8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  ITEM NUMBER 1  2  3  4  5  6  7  8  11 11 11 11 11 11  11 11 11 11 11 11  11 11 11 11 11 11  11 11 11 11 11 01  11 11 11 11 11 11  11 11 11 11 11 11  11 11 11 11 11 11  11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 01  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 01  11 11  11 11  11 11  11 11  11 11  11 11  11 11  11 11  71.  (i) DAY TWO  GROUP-TAUGHT GROUP  STUDENT NUMBER 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  ITEM NUMBER 1  2  3  4-5  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11  11  11  11  6  7  8  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 01  11 11 11 01 11 11 11 11 11 11 11 01 11 11 11 11 11 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 01 11  11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11  11  11  11  (j) DAY THREE  GROUP-TAUGHT GROUP  STUDENT  ITEM NUMBER  NUMBER 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  1 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11  11 11 11 11 11 11 11 01 11 01 11 01 11 11 11 11 11 11 11 11  2 11 11 11 01 11 10 10 01 11 01 11 11 11 11 11 11 11 11 11 11  3 11 11 11 11 11 11 11 11 00 11. 11 01 11 11 01 11 11 01 11 11  4  5 11 11 11 11 11 11 11 11 11 11 11 01 11 11 11 11 11 11 01 01  6 11 11 11 11 00 10 11 11 11 01 01 11 11 11 11 11 11 11 11 11  7 11 11 11 01 00 10 11 11 01 11 11 11 11 11 11 11 11 11 11 11  8 11 11 11 00 00 11 01 11 11 11 11 01 11 11 01 11 11 11 11 11  

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