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A test of the validity of a concrete-abstract hierarchy of addition facts on kindergarten children Ward, Brian 1979

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A  TEST  VALIDITY  OF  ABSTRACT ADDITION  OF  THE  A  CONCRETE-  HIERARCHY  FACTS  ON  OF  KINDERGARTEN  CHILDREN  by  B R I A N WARD Ed.,  University  A  THESIS  Columbia,  SUBMITTED IN  FULFILLMENT FOR  of B r i t i s h  THE  OF  THE  REQUIREMENTS  D E G R E E OF OF  PARTIAL  MASTER  ARTS  ^  in THE  FACULTY  OF  GRADUATE  STUDIES (Department  We a c c e p t conforming  THE  to  UNIVERSITY  of  this  Brian  thesis  the required  OF  June,  ©  Education)  as standard  B R I T I S H COLUMBIA  1979  Ward  1979  1972  In p r e s e n t i n g t h i s  thesis in partial  an a d v a n c e d d e g r e e a t the L i b r a r y I further for  shall  the U n i v e r s i t y  make i t  agree that  freely  this  thesis for  It  Department n f  f i n a n c i a l gain shall  OTfotJ  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  Date  BP 7 5-5 1 1 E  I agree  r e f e r e n c e and copying of  this  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  i s understood  permission.  the requirements  B r i t i s h Columbia,  extensive  s c h o l a r l y p u r p o s e s may be g r a n t e d  written  DE-6  of  available for  permission for  by h i s r e p r e s e n t a t i v e s . of  f u l f i l m e n t of  or  publication  be a l l o w e d w i t h o u t  my  i i ABSTRACT The number  study  facts  of  continuum. chies  in  Literature  specify  chically  Levels  is  target  This  both  and  entire  children  are  11  to  49  these  c h i l d r e n seven  test  to  investigate  test  designed,  subjects  Twenty-one study  and  employed the  of the  to  was  and  other  also  with  to  the  balanced  sets so  the  answer testing  which  level  14  6 and  study.  items. the  with  study:  between  test  in  be  four  the  the  5  Columbia. 125  Entering concept  of  boys  The years  of  for the  a  and  questions  14  used  numbers.  pilot  instructions.  incorporated  of  of  original  subjects most  of  British  the  9 were  item  Conceptual  and  ages  A Greco-L'atin  Each  hierar-  hierarchy.  pretest  chosen as  the  theory  behaviours  can  four-stage  with  task  possibilities  s c r e e n e d and  would  main  pilot  that  a  hierar-  Concrete-Abstract  arranged  concerned  were  the  of  which  school d i s t r i c t in  conform  standardize  balanced  seven  to  facts  A  learning  Concrete-Abstract  and  the  Analytic  indicate  numbers  consider  was  subjects  number 5 in  one  nature  for  to  c h i l d r e n between  subjects  chosen  26  in  order  whether  design a  hierarchy  facts  the  which  for  associated with  analytic  Levels  hierarchy  a developmental  were  Task  population  From  of  Task  is  attempts  Semi-Concrete,  months  of  eight  seven  posed  in  task  hierarchy  Concrete-Abstract  areas:  discussed.  kindergarten  kindergarten  reduced  and  a  which  of  a concept  a seven-stage  5 years  four  theory  theory; of  9 along  in  interaction  with  Behaviours  the  reviewed  Concrete,  was  a task-analytic  6 and  r e s e a r c h e d and  presented  population  0 months  the  Questions  validating  is  a definition  designated  Abstract.  between  learning  arranged,  hierarchy  validate  the; l e a r n i n g  hierarchies;  developmental  which  to  addition  arithmetic;  analytic and  attempts  Twentygirls.  in  the  Square  main  was  the  seven  levels  of  three  number  set  difficulty  of  on  these  sets  of  was,  according to established c r i t e r i a ,  approximately equal.  P r i o r t o the  t e s t i n g s e s s i o n each s u b j e c t was taught t h r e e items a t each the h y p o t h e s i z e d h i e r a r c h y .  Task L e v e l o f  T h i s t e a c h i n g was g i v e n t o seven groups o f  c h i l d r e n , each group composed of 2 boys and 2 g i r l s .  In o r d e r t o d e c i d e  the items taught a t each l e v e l o f t h e h i e r a r c h y , a row o f a L a t i n Square was  chosen  a t random and matched with a balanced s e t o f items.  s u b j e c t e d t o a Guttman Scalogram a four-level hierarchy.  When  A n a l y s i s data supported t h e e x i s t e n c e o f  Using a c u t - o f f p o i n t of 2 out o f 3 items  c o r r e c t a c o e f f i c i e n t o f r e p r o d u c i b i l i t y o f 0.95 was a c h i e v e d .  Ceiling  e f f e c t s d i d o c c u r , however, which made i t d i f f i c u l t t o s u b s t a n t i a t e the h y p o t h e s i z e d d i f f e r e n c e s between the Concrete and Semi-Concrete  Levels.  A p o s s i b l e cause advanced, i s the p r e t e s t which e l i m i n a t e d 76 c h i l d r e n from of  the p o p u l a t i o n p r i o r t o b e g i n n i n g the s t u d y , thus p r o d u c i n g a group  s u b j e c t s who were most l i k e l y  related  behaviours.  high i n i n t e l l i g e n c e and i n t e l l i g e n c e -  iv TABLE  OF  CONTENTS Page  Abstract Table  of  i i Contents  iv  List  of  Tables  v i i  List  of  Figures  v i i i  CHAPTER I.  INTRODUCTION  TO  THE  RELATED  RESEARCH  Task  Analytic  A  Cumulative  P R O B L E M AND A R E V I E W  Hierarchies  Developmental  .  .  .  of Knowledge  Learning  Theory  . .  as  THE  of  the Related  .  . .  Concrete  level  Semi-Concrete  Abstract  . . in . .  Research  .  Task  .  .  .  Final  level  .  tasks  tasks  .  17  .  .  .  .  .  19 19 20 20 20  of  Terms  Task  Criterion  13  19  tasks  Levels  Definition  11  .  Continuum  level  level  3  18  tasks  Concrete-Abstract  1 .  RESEARCH PROBLEM Concrete-Abstract  i\ZZ».  .  Contrasted  with Cumulative Learning Theory . The D e v e l o p m e n t o f Number C o n c e p t s Pre-School Children . . . . .  II.  THE 1  Theory  Implications  OF  21 21  f o r Performance  Instructional  Objective  21 21  Entering  Behaviours  21  Learning  Hierarchy  21  Task  Analysis  21  Page  Research Target  III.  Objectives  Population  25 Instrument  26  Pilot  27  Test  Subjects  27  Item  27  Difficulty Sequence  Teaching  29  Procedures  Statistical Analysis  31  Analysis  34  of  Variance  34  Descriptive  Analysis  35  Guttman  R E S U L T S OF Pretest Pilot  Scalogram  THE  PRETEST  Analysis  35  AND P I L O T  TEST  .  .  .  36  Test  R E S U L T S OF  36  Test  Implications for the Main  of the Pretest Study  and P i l o t  41  Test 43  45  of  Variance  45  Descriptive  Analysis  52  Guttman  THE  .  STUDY  Analysis  VI.  25  Pretest  Testing  V.  23  METHODOLOGY Measuring  IV.  23  Scalogram  Analysis  DISCUSSION, LIMITATIONS  53  AND R E C O M M E N D A T I O N S  .  Discussion Limitations  57 of  Recommendations  REFERENCES  57  the Study for Future  60 Research  .  .  .  62  64  vi  Page  APPENDICES  .  •  67  Appendix A  67  Appendix B  68  Appendix C  70  Appendix D  72  vii LIST  OF  TABLES  TABLES 1 2  Page Items Items Sets  3  4  5  Arranged Grouped of  by L e v e l to  of  Equalize  Difficulty  .  Difficulty  .  28  of  Three  29  T e s t i n g Sequence as Determined G r e c o - L a t i n Square  by a  Item Set and i t s Matched Hierarchy  i n the  Number  .  of  Subjects  Level  30  32  Participating  i n the  Pretest  36  6  Pretest  7  Distribution Times  8  Results  of  10  11  Subjects  Who F a i l e d  the Pretest  to Pretest  Summary o f Difficulty Criteria  13  and  Testing Criteria  .  Illustrated  Category Test  39 40  According  to 42 of .  the . .  .  46  t h e A n a l y s i s of Variance of Item According t o t h e Four E s t a b l i s h e d 49 .  50  A Summary o f t h e A n a l y s i s o f V a r i a n c e o f D i f f i c u l t y f o r Balanced Groups of Items  Item . .  51  A Summary o f Hypothesized  the  Performance  Results Seven  16  Schools  Summary o f t h e A n a l y s i s o f V a r i a n c e Sequence of Items W i t h i n t h e Test  Mean  15  Sex,  Who M e t S c r e e n i n g  Results of the P i l o t Hierarchy Level  12  14  over  Subjects  According 9  38  the Analysis Hierarchy  Guttman  of  Scalogram  Variance  of  52 Analysis  -  Levels  Results Four  of  by Sex and Item D i f f i c u l t y  of  Levels  Guttman  54 Scalogram  Analysis  56  viii LIST OF FIGURES  FIGURES 1  Page H y p o t h e s i z e d H i e r a r c h y Showing the Four C o n c e p t u a l L e v e l s and the Seven Task L e v e l s . .  2  3  Hypothesized Hierarchy for A d d i t i o n Facts Showing the C o n c e p t u a l L e v e l s 1 t o 4 and Task L e v e l s 1 t o 7  Mean Responses to Item S e t s at Each L e v e l o f the H i e r a r c h y  22  ,  48  33  1 Chapter One I n t r o d u c t i o n to the Problem and a Review o f t h e R e l a t e d Research  The f e a s i b i l i t y o f a r r a n g i n g c o n c e p t u a l i n f o r m a t i o n i n a s e q u e n t i a l o r d e r such t h a t a h i e r a r c h y o f s k i l l s i s c o n s i d e r e d t o be has been o f i n t e r e s t  f o r some t i m e .  established  The l e a r n i n g o f one s e t o f  behaviours  has been p u r p o r t e d t o l e a d a u t o m a t i c a l l y t o , and s u p p o r t the l e a r n i n g o f , a r e l a t e d more d i f f i c u l t  set of behaviours.  T h i s premise has been  b a s i s f o r the d e s i g n o f programmed l e a r n i n g and l e a r n i n g  the  experiments  which have sought to d i m i n i s h or remove f a i l u r e i n m a s t e r i n g a new b e h a v i o u r or s e t o f b e h a v i o u r s . T h i s r e v i e w p r e s e n t s some o f the r e s e a r c h t h a t r e l a t e s t o h i e r a r c h i c a l l e a r n i n g i n f o u r major a r e a s : g e n e r a l r e s e a r c h and r e s e a r c h that affect  considerations  an u n d e r s t a n d i n g o f t a s k h i e r a r c h i e s i n a r i t h m e t i c ;  the  t h e o r y o f knowledge t h a t emerges from such a d i s c u s s i o n (the c u m u l a t i v e l e a r n i n g t h e o r y ) ; the r e l a t i o n s h i p o f developmental l e a r n i n g t h e o r y  to  c u m u l a t i v e l e a r n i n g t h e o r y ; and r e s e a r c h on p r e - s c h o o l c h i l d r e n t h a t bears most d i r e c t l y upon the r e s e a r c h problem and the r e s e a r c h q u e s t i o n s are posed i n t h i s  that  study.  Task A n a l y t i c H i e r a r c h i e s Gagne (1961a) h y p o t h e s i z e d t h a t as a l e a r n e r p r o g r e s s e d up a h i e r a r c h y h i s r a t e o f l e a r n i n g s h o u l d depend i n c r e a s i n g l y on the a t t a i n m e n t attainment  or non-  o f r e l e v a n t l e a r n i n g s e t s (performance o f a group o f f a c t s  s p e c i f i e d l e v e l o f the h i e r a r c h y ) , and d e c r e a s i n g l y on r e l e v a n t the i n d i v i d u a l b r i n g s t o t h e s i t u a t i o n .  at a  abilities  From an experiment designed  to  2 a n a l y z e the c l a s s o f t a s k s f o r s o l v i n g l i n e a r e q u a t i o n s ,  Gagne c o n c l u d e d :  (a) r e l e v a n t b a s i c a b i l i t i e s are c l e a r l y of more importance i n the m a s t e r i n g o f a h i e r a r c h y than i r r e l e v a n t b a s i c a b i l i t i e s ; (b) i n s t a n c e s  of p o s i t i v e t r a n s f e r  t o each l e a r n i n g s e t  o r d i n a t e r e l e v a n t l e a r n i n g s e t s are found to occur throughout  from s u b the  hierarchy; (c) c o r r e l a t i o n s o f r e l e v a n t b a s i c a b i l i t i e s w i t h r a t e s of  attainment  of l e a r n i n g s e t s a t p r o g r e s s i v e l y h i g h e r l e v e l s o f the h i e r a r c h y show a steeply progressive decrease;  and  (d) c o r r e l a t i o n s o f r a t e o f a t t a i n m e n t  of l e a r n i n g sets with achieve-  ment o f r e l e v a n t s u b o r d i n a t e l e a r n i n g s e t s a r e found t o be s y s t e m a t i c a l l y h i g h e r than w i t h the achievement of i r r e l e v a n t s e t s p a r t i c u l a r l y i n t h e upper l e v e l s o f the  hierarchy.  Gagne' (1962b) then undertook a s t u d y on r e c a l l a b i l i t y and i n t e g r a t i o n presenting subjects  w i t h a programme which had t w e l v e s u b o r d i n a t e  s u p p o r t i n g two r e l a t e d f i n a l t a s k s .  tasks  Low and h i g h a b i l i t y s t u d e n t s were  t e s t e d t o f i n d out how much r e p e t i t i o n of l e a r n i n g s e t s was r e q u i r e d , and how much d e t a i l was t o be p r o v i d e d , i n o r d e r f o r s t u d e n t s t o i n t e g r a t e information.  The s u b j e c t s  the  were 136 s e v e n t h grade c h i l d r e n who were  a s s i g n e d t o groups by achievement  ( t e a c h e r grades f o r a r i t h m e t i c because  the h y p o t h e s i z e d h i e r a r c h i e s were i n a r i t h m e t i c ) .  The experiment showed  a h i g h c o r r e l a t i o n between the achievement o f f i n a l  t a s k s and the number  of s u b o r d i n a t e l e a r n i n g s e t s which were a c q u i r e d .  There was a l s o  evidence t o suggest t h a t s u b o r d i n a t e l e a r n i n g s e t s i n a h i e r a r c h y positive transfer  to a higher l e v e l l e a r n i n g  mediate  set.  Three years l a t e r Gagne'(1965) examined r e t e n t i o n o f i n i t i a l l y knowledge o f n o n - m e t r i c geometry, m a n i p u l a t i n g the i n s t r u c t i o n s  learned  variable  and to g a i n i n f o r m a t i o n t h a t would p e r m i t i n f e r e n c e s t o be made about memory.  A f t e r a n i n e week i n t e r v a l measures were t a k e n on:  (a) achievement on items t h a t were matched f o r c o n t e n t w i t h  the  o r i g i n a l achievement t e s t , and (b) performance on each o f the s u b o r d i n a t e t a s k s t h a t were c o n t r i b u t i n g to the f i n a l  task.  F i v e groups o f s i x t h grade c h i l d r e n , s i x t e e n per g r o u p , worked through a programme w i t h i d e n t i c a l c o n t e n t but composed o f f i v e experimental treatments.  different  P r a c t i c a l examples of each s u b o r d i n a t e  task  had: (a) m i n i m a l v a r i e t y i n c o n t e n t , (b) i n t e r m e d i a t e v a r i e t y , (c) great v a r i e t y , (d) no examples, or (e) u n r e l a t e d  tasks.  The c o r r e l a t i o n between the i n i t i a l achievement t e s t s was +0.64 and the c o r r e l a t i o n between achievement s c o r e s i m m e d i a t e l y a f t e r those o b t a i n e d a f t e r  l e a r n i n g and  a n i n e week r e t e n t i o n i n t e r v a l was +0.62.  C o r r e l a t i o n s between s c o r e s on s u b o r d i n a t e l e a r n i n g h i e r a r c h i e s n i n e weeks, however, were o n l y +0.41 and +0.46.  after  The s u b o r d i n a t e  tasks  n e c e s s a r y t o l e a r n the f i n a l t a s k were not w e l l m a i n t a i n e d ; the q u e s t i o n t h a t a r o s e was whether t h e i r importance d i m i n i s h e d once the f i n a l was l e a r n e d .  task  The l e a r n e r d i d not f o r g e t p a r t i c u l a r p a r t s o f the  h i e r a r c h y ; f o r g e t t i n g was e n t i r e l y random.  The f i n a l  tasks  themselves,  however, were w e l l m a i n t a i n e d . A C u m u l a t i v e Theory of Knowledge Gagneaand Brown (1961b) undertook a p i e c e o f r e s e a r c h which i n f l u e n c e d Gagn^'s t h i n k i n g about both the a c q u i s i t i o n o f knowledge and  4 the importance o f i n s t r u c t i o n s i n a c q u i r i n g t h a t knowledge.  The purpose  of t h e study was t o o b t a i n a measure o f t h e l e a r n i n g e f f e c t i v e n e s s of programmed m a t e r i a l i n terms which would permit a r e a s o n a b l e t h a t understanding had been accomplished. and  10 were randomly assigned  i n d i v i d u a l l y tested. (the teacher  T h i r t y - t h r e e boys i n grades 9  e x p e r i m e n t a l groups and then  The programmes were termed  ' r u l e and example  presented t h e r u l e t h a t 2 + 3 = 3 + 2  s e v e r a l examples);  'guided d i s c o v e r y '  i n the process of discovery acquired  to three  inference  and then t h e c h i l d d i d  (the teacher  and helped t h e c h i l d  was a c t i v e l y i n v o l v e d  re-state  previously  concepts t h a t were r e l e v a n t t o t h e problem); and ' d i s c o v e r y '  ( t r i a l and e r r o r l e a r n i n g where t h e teacher a c t i v e l y involved i n the discovery  provided  process).  s t r u c t u r e , but was not  Performance was measured i n  terms o f time taken t o s o l v e a problem and t h e number o f h i n t s Learning  1  occurred  i n a l l three  most l e a r n i n g f o l l o w e d  groups but 'guided d i s c o v e r y '  by ' d i s c o v e r y '  and f i n a l l y  i n t e r e s t i s t h e f a c t t h a t t h e guided d i s c o v e r y  required.  produced the  ' r u l e and example'.  programme not o n l y  Of  used  'small s t e p s ' , as i s common i n programmed l e a r n i n g , but a l s o r e q u i r e d t h e subject  to r e - s t a t e c e r t a i n previously acquired  concepts which were  r e l e v a n t t o t h e problems i n t h e e x p e r i m e n t a l c o n d i t i o n . of c e r t a i n concepts by t h e s t u d e n t s a t t h e request the p o s i t i v e r e s u l t s o b t a i n e d ,  focused  The r e - s t a t i n g  of t h e r e s e a r c h e r , and  a t t e n t i o n upon t h e nature of  i n s t r u c t i o n s , and a l s o upon t h e n e c e s s i t y o f r e c a l l i n g r e l e v a n t l e a r n i n g s e t s i n order  t o o b t a i n a s o l u t i o n t o t h e task  subordinate  presented.  In a t h e o r e t i c a l paper on Knowledge and I n s t r u c t i o n s , Gagne (1962a) /  stated that i f l e a r n i n g sets are e s s e n t i a l f o r p o s i t i v e t r a n s f e r the f o l l o w i n g consequences should  ensue:  5  (a) i f a h i g h e r l e v e l l e a r n i n g s e t l e v e l s e t s must have been  i s passed,  a l l related  lower  passed;  (b) i f one or more of the lower l e v e l s e t s have been f a i l e d ,  the  r e l a t e d h i g h e r l e v e l t a s k s must be f a i l e d ; and (c) i f a higher l e v e l set may have been p a s s e d . attributable  has been f a i l e d , r e l a t e d lower l e v e l  The absence of p o s i t i v e t r a n s f e r  sets  would be  to a d e f i c i e n c y i n i n s t r u c t i o n s .  A t h e o r y was o u t l i n e d which d e f i n e d knowledge as an i n f e r r e d c a p a b i l i t y t h a t makes p o s s i b l e the s u c c e s s f u l performance o f a c l a s s o f t a s k s t h a t c o u l d not be performed b e f o r e l e a r n i n g was  undertaken.  P r o d u c t i v e l e a r n i n g was seen as i n v o l v i n g two v a r i a b l e s : (a) knowledge, and (b)  instructions.  I t i s i m p o r t a n t t o understand  e x a c t l y what Gagne'meant by  and ' s a t i s f a c t o r i l y c o m p l e t i n g a f i n a l not n e c e s s a r i l y t r a d i t i o n a l ones.  'instructions'  t a s k ' , because h i s d e f i n i t i o n s  In o r d e r to have a framework  d i s c u s s i n g the l e a r n i n g c o n d i t i o n s necessary  for enquiry, that  were  for  is  a c t i v i t i e s c h a r a c t e r i z e d by a p r o b l e m - s o l v i n g approach - Gagne'' (1963) delineated  the f i n a l t a s k as the  i n s t r u c t i o n s , more b r o a d l y , as  ' t e r m i n a l c a p a b i l i t y o f the l e a r n e r ' , and  'instructional conditions'.  p r a c t i c e and a s u f f i c i e n t background o f e x p e r i e n c e ; o n l y be s o l v e d by the a c q u i s i t i o n o f knowledge. both components o f i n s t r u c t i o n s ; the same.  E n q u i r y needed  new problems c o u l d  D r i l l and d i s c o v e r y were  however, d i s c o v e r y and e n q u i r y were not  The c o n s t r u c t i o n o f a response by a l e a r n e r was termed  c o v e r y ' , whereas ' e n q u i r y ' was the t e r m i n a l t h i n k i n g p r o c e s s .  'dis-  A series  of r e q u i r e d s t e p s were h y p o t h e s i z e d i n o r d e r f o r a s t u d e n t t o a t t a i n  the  practice of enquiry: (a) competent  performer  ( l e a r n i n g of  skills),  (b) a c q u i s i t i o n o f knowledge, (c) s c i e n t i f i c enquirer  (genuine e n q u i r y ; a b i l i t y t o d i s t i n g u i s h  good and bad i d e a s ; a b i l i t y t o s o l v e problems by means o f  unrestrained  i n d u c t i v e t h i n k i n g ) , and (d) independent  investigation (e.g.  scientist).  E x t e n d i n g h i s t h i n k i n g about i n s t r u c t i o n s f u r t h e r ,  Gagne^ (1964)  noted t h a t the p r o b l e m - s o l v e r came to a g i v e n s i t u a t i o n w i t h p r e v i o u s l y l e a r n e d c a p a b i l i t i e s ; the experimenter  then communicated w i t h the  subject  by way o f i n s t r u c t i o n s i n f o u r ways: (a) by i d e n t i f y i n g new s t i m u l i , (b) by i d e n t i f y i n g the expected form of the t e r m i n a l (c) by h e l p i n g t h e s u b j e c t  performance,  r e c a l l previously acquired c a p a b i l i t i e s ,  (d) by c h a n n e l i n g t h i n k i n g i n a r e l e v a n t  direction.  G a g n e ^ c a l l e d f o r r e s e a r c h which would m a n i p u l a t e the v a r i a b l e  instructions  based upon the h y p o t h e s i s t h a t b e t t e r i n s t r u c t i o n s would f a c i l i t a t e r e c a l l of p r e v i o u s l y acquired relevant concepts. reduced.to  (a) s u b j e c t (b)  Problem s o l v i n g c o u l d be  an e x p e r i m e n t a l s i t u a t i o n where the independent  i n t o t h e broad c a t e g o r i e s  the  variables  fell  of:  c a p a b i l i t y , and  instructions.  Gagne''and B r o w n ' s (1961b) s t u d y opened the way f o r a s e r i e s  of experi  ments which c u l m i n a t e d i n the model o f l e a r n i n g p r e s e n t e d by Gagne''in 1968 These experiments emphasized the n e c e s s i t y of exact communication between t e a c h e r and s t u d e n t .  The i n s t r u c t i o n s , which are q u i t e c l e a r l y  i n a l l forms o f t e a c h e r - p u p i l  important  l e a r n i n g , t a k e on a new d i m e n s i o n when  7  l o o k e d a t from the v i e w p o i n t o f a t a s k h i e r a r c h y .  Each new and h i g h e r x  s t a g e can o n l y be l e a r n e d i f the s t u d e n t understands the  instructions.  A l s o , i n any h y p o t h e s i z e d h i e r a r c h y a h i g h e r l e a r n i n g s e t can o n l y be l e a r n e d when the r e l e v a n t lower s e t s i n the h i e r a r c h y are  mastered.  Gagne''defined communication i n terms o f s t i m u l u s i d e n t i f i c a t i o n , i d e n t i f i c a t i o n o f the form of t e r m i n a l r e s p o n s e ,  r e c a l l of p r e v i o u s l y  a c q u i r e d c a p a b i l i t i e s , and c h a n n e l i n g t h i n k i n g i n a r e l e v a n t  direction.  Q u i t e c l e a r l y , however, the problem i s not t h a t s i m p l e ; P e r s o n a l i t y , m o t i v a t i o n a l , and a d d i t i o n a l c o g n i t i v e v a r i a b l e s must be c o n s i d e r e d . One v a r i a b l e which G a g n e ' d i d i n v e s t i g a t e was memory.  Because many  s u b o r d i n a t e t a s k s w i t h i n h i e r a r c h i e s were f o r g o t t e n ,  he q u e s t i o n e d  whether t h e s e t a s k s were n e c e s s a r y achieved.  A further  once the t e r m i n a l o b j e c t i v e was  problem was posed by the l i m i t e d amount o f i n f o r m a t i o n  a v a i l a b l e about how the human memory s t o r e d and coded i n f o r m a t i o n . apparent problem o f f o r g e t f u l n e s s  The  might be more one o f r e t r i e v a l o f  i n f o r m a t i o n from t h e l o n g - t e r m memory s t o r e . Gagne^s model o f l e a r n i n g (based on a t h e o r y o f c u m u l a t i v e l e a r n i n g ) contrasted  i n a number o f r e s p e c t s w i t h developmental t h e o r i e s  c e n t r a l themes were m a t u r a t i o n , r e a d i n e s s ,  and c o g n i t i v e  He i d e n t i f i e d t h e s t a g e a t which a person c o u l d be as .(a)  relevant c a p a b i l i t i e s possessed,  whose  adaptation.  involving  and  (b) a number o f h i e r a r c h i e s o f c a p a b i l i t i e s t o be a c q u i r e d , such t h a t i t i s p o s s i b l e t o combine s u b o r d i n a t e e n t i t i e s terminal  the  task.  The model took i n t o account t r a n s f e r  o f l e a r n i n g : what was l e a r n e d  c o u l d be combined w i t h o t h e r l e a r n e d e n t i t i e s transfer.  to h e l p a c h i e v e  v i a a mechanism o f l e a r n i n g  8  In the second e d i t i o n o f h i s book Gagne'' (1970) d i s c u s s e d the r e l a t i o n s h i p o f v a r i o u s forms o f l e a r n i n g a n d , i n p a r t i c u l a r , noted t h a t the e x i s t e n c e o f p r i o r c a p a b i l i t i e s - i m p o r t a n t i n drawing d i s t i n c t i o n s among c o n d i t i o n s r e q u i r e d f o r l e a r n i n g - was s l i g h t e d or i g n o r e d by most t r a d i t i o n a l learning prototypes.  He went on to s t a t e t h a t when b u i l d i n g  a h i e r a r c h y the t a s k s s h o u l d be a n a l y z e d such t h a t the l o w e s t ' b o x e s ' i n a hierarchy represented  the k i n d of performance a l l s t u d e n t s i n a group  could already successfully accomplish.  The s t a t e m e n t s i n the  were i n t e n d e d to d e s c r i b e a s i n g l e c a p a b i l i t y to be l e a r n e d ,  'boxes' representing  what the l e a r n e r was a b l e to:';do when l e a r n i n g had been a c c o m p l i s h e d . They were, t h e r e f o r e ,  s t a t e d i n performance t e r m s .  The s u p e r o r d i n a t e  c a p a b i l i t y was more r e a d i l y l e a r n e d i f the s u b o r d i n a t e c a p a b i l i t i e s had been p r e v i o u s l y a c q u i r e d and.were r e a d i l y a v a i l a b l e f o r r e c a l l .  Each  s u b o r d i n a t e c a p a b i l i t y had been i d e n t i f i e d as such because i t was known, o r h y p o t h e s i z e d , to c o n t r i b u t e p o s i t i v e t r a n s f e r superordinate c a p a b i l i t y .  to the l e a r n i n g o f a  Students who had l e a r n e d s u b o r d i n a t e  skills  s h o u l d l e a r n s u p e r o r d i n a t e s k i l l s more e a s i l y than t h o s e who had not l e a r n e d them. Engelmarnwas concerned w i t h t h e m a n i p u l a t i o n o f t e a c h i n g v a r i a b l e s ; l i k e Gagne he was i n t e r e s t e d i n t a k i n g i d e a s , c o n c e p t s , and i n f o r m a t i o n , and a r r a n g i n g them i n a h i e r a r c h i c a l f a s h i o n .  He argued (1969a) t h a t a  concept was a s e t o f c h a r a c t e r i s t i c s shared by a l l ' i n s t a n c e s ' p a r t i c u l a r s e t and o n l y by t h o s e  in a  ' i n s t a n c e s ' , and t h a t concepts were  c l a s s e s and thus amenable t o h i e r a r c h i c a l arrangement.  He c o n s i d e r e d i t  n e c e s s a r y t o r e a l i z e t h a t c o n c e p t s were always dependent on the c o n t e x t o r u n i v e r s e i n which they were p r e s e n t e d ,  and t h a t t o a n a l y z e a concept  so t h a t i t c o u l d be taught was to d e s c r i b e the concept i n terms o f the  9 minimum  set  recognize  e s s e n t i a l d i s c r i m i n a t i o n s the  that  an  'instance'  For  Engelmann,  (a)  the  demonstrations  of  the  concepts,  (b)  the  test  istics  standing He  of  of  the  commented  stration,  or  that  least  for  potential  critical when was  from  to  the  in  what  because  he  a  was  the  concept  (b)  the  rules  (c)  the  rule  the  therefore,  that  concept  two  order  to set.  components:  performs  to  c o u l d be  taught  child  to  the  of  and  had  show  the  demonstrate  been  both  character-  his  under-  a single  demon-  'instances'  should  concentrate  make.  Engelmann  translated  a concept,  into and  to  s p e c i f i c a t i o n s , and i t  was  taught:  task  not  for  he  had  one  noted  that the  range  specific  ways.  unless  the  the  a  might  learn  with load;  on  from  tests  move of  were  determine  fail  responding. to  and  tasks,  p o s s i b l e to  a child  convention  was  a minimum memory  behaviour,  of  in  presentation  with  to  were  a  problem  Basically,  three  when  things:  taught,  responding, responding  important  and is  factor  responding  acknowledged  best  specific  tasks  being  that  present  teacher  testing  had  for  one  act  to  The  the  a particular  the  find  of  concept  necessary  exact  know  (a)  must  a member  to  understanding  child  taught  added  in  performs  understanding  of  not  Engelmann  make  child  concept.  yielded  terms  did  teacher  misunderstanding  test  analysis  exactly  is,  no  was  concept  ways  specified  child  that  possible behaviour  possible Concept  the  a presentation  of  not,  c o n s i s t e d of the  d i s c r i m i n a t i o n s the  tests  was  to  and  task  i t  of  developing  or  had  concept.  non-'instances'  in  teaching  further  and  was,  child  and  worthwhile. not  previously  worthwhile. i t  is  also  The  mentioned:  concept  apparent  that  of  a  student  motivation  Engelmann  is  10 s u g g e s t i n g a r o l e f o r p e r s o n a l i t y v a r i a b l e s t h a t have some e f f e c t student's  upon.a  self-esteem.  U p r i c h a r d and P h i l l i p s  (1977) reviewed t h e r e s e a r c h l i t e r a t u r e o f  the 1 9 6 0 ' s and c o n c l u d e d t h a t e p i s t e m o l o g i c a l c o n s i d e r a t i o n s were i n the d e s i g n o f l e a r n i n g h i e r a r c h i e s .  They c i t e d Gagne's s t a t e m e n t t h a t  the " d e t e r m i n a t i o n o f a h i e r a r c h i c a l sequence o f s u b t a s k s  from s i m p l e s t  most complex was not e a s i l y a c h i e v e d , " and c l a i m e d t h a t l o g i c a l based upon a t h e o r y o f knowledge, was d i f f i c u l t ignored p s y c h o l o g i c a l f a c t o r s .  paramount  to  analysis,  t o v a l i d a t e because i t  Engelmann (1969a) however, had c o n s i d e r e d  p s y c h o l o g i c a l v a r i a b l e s i n emphasizing the r o l e o f m o t i v a t i o n .  The  o r d e r i n g o f a h i e r a r c h y would c l e a r l y be i n f l u e n c e d by p r i o r l e a r n i n g ,  the  i n s t r u c t i o n a l s i t u a t i o n , memory and many o t h e r f a c t o r s .  the  w i l l i n g n e s s of a student to i n t e r a c t  w i t h the l e a r n i n g environment  a l o n g many dimensions and would no doubt a f f e c t ments s e e k i n g the  'perfect'  Additionally,  varies  the outcome o f any e x p e r i -  hierarchy.  C u m u l a t i v e L e a r n i n g t h e o r y has enabled e d u c a t i o n a l r e s e a r c h e r s focus on i n s t r u c t i o n a l o b j e c t i v e s .  to  W h i l e both Gagne''and Engelmann  broached t h e i s s u e , B a r b a r a Bateman (1971) f u r t h e r e d  its  development.  L i k e Gagne''"and Engelmann, she b e l i e v e d i n s t r u c t i o n a l o b j e c t i v e s t o be n e c e s s a r y i n d e t e r m i n i n g the g o a l s i n a l e a r n i n g t a s k .  She suggested  the  f o l l o w i n g c r i t e r i a f o r the s e l e c t i o n of o b j e c t i v e s : (a) do the o b j e c t i v e s i n d i c a t e l e a r n i n g outcomes t h a t are  appropriate  t o the i n s t r u c t i o n a l a r e a ? (b) do the o b j e c t i v e s r e p r e s e n t a l l l o g i c a l l e a r n i n g outcomes of instructional  area?  ( c ) are the o b j e c t i v e s o b t a i n a b l e by t h e s e s t u d e n t s ?  the  11 (d) a r e the o b j e c t i v e s i n harmony w i t h the p h i l o s o p h y o f the s c h o o l i n which the i n s t r u c t i o n i s to be g i v e n ? (e) are the o b j e c t i v e s i n harmony w i t h the t e a c h e r ' s p e r c e i v e d needs f o r the f u t u r e  o f the c h i l d and s o c i e t y ?  Task a n a l y s i s i s c l e a r l y one way o f s p e c i f y i n g o b j e c t i v e s a c c o r d i n g t o the f o r e g o i n g c r i t e r i a ; i t i s a t f i r s t l o g i c a l , then e m p i r i c a l . s t u d y d e s c r i b e d i n the f o l l o w i n g c h a p t e r s h y p o t h e s i z e s t h a t t a s k i s e s s e n t i a l to successful l e a r n i n g .  The  analysis  W h i l e the b a s i s i s a n a l y t i c a l and  h i e r a r c h i c a l , developmental c o n s i d e r a t i o n s w i l l not be i g n o r e d . espoused i s t h a t i n o r d e r t o t e a c h w i t h purpose,  The view  c l e a r g o a l s need to be  o u t l i n e d i n o r d e r f o r t e a c h i n g t o be s u c c e s s f u l w i t h most c h i l d r e n . Developmental L e a r n i n g Theory as C o n t r a s t e d w i t h C u m u l a t i v e L e a r n i n g Theory Engelmann (1967) was p a r t i c u l a r l y c r i t i c a l of the  developmental  t h e o r i e s o f l e a r n i n g , i n d i c a t i n g t h a t developmental i n t e r p r e t a t i o n s , as those o f P i a g e t , are b a s i c a l l y i r r e l e v a n t to t e a c h e r s ,  such  and t h a t when  an i r r e l e v a n t e x p l a n a t i o n i s a c c e p t e d by a t e a c h e r she ceases to f u n c t i o n i n a teaching capacity.  Irrelevant explanations  ' l o c k the d o o r ' on a  p r o b l e m , whereas a t e a c h e r seeks to develop and change a c h i l d of t h i s e x p l a n a t i o n .  To  irrespective  Engelmann, a developmental e x p l a n a t i o n does  not i m p l y a remedy t h a t can be a c h i e v e d through the m a n i p u l a t i o n o f e n v i r o n m e n t a l v a r i a b l e s ; as a consequence i t does not d e a l w i t h under the t e a c h e r ' s c o n t r o l .  He added t h a t o f t e n the  e x p l a n a t i o n cannot be t r a n s l a t e d transfer  variables  developmental  i n t o concepts whereas the t e a c h e r must  t h e c h i l d ' s r e l a t i v e d e f i c i e n c y i n t o terms of concepts  because  these imply a manipulation of v a r i a b l e s , noting that Piaget considered 'conservation' a concept,  yet p l a c e d a host o f developmental  on the c o n c e p t ' s a c q u i s i t i o n .  restrictions  He f e e l s t h a t i f i t i s a concept then  it  12 can be t a u g h t ;  i f i t i n v o l v e s s p e c i f i c c o n t e n t , i t i s c e r t a i n l y r . n o t the  p r o d u c t o f c o g n i t i v e s t r u c t u r i n g but r a t h e r the p r o d u c t of e n v i r o n m e n t a l consistency. Many o t h e r a u t h o r s have agreed w i t h Engelmann's a d m i t t e d l y extreme point of view.  Kilpatrick  (1970) i n d i c a t e d t h a t the c r e a t o r s o f the  S c h o o l Mathematics Study Group programmes were guided by l o g i c a l  consid-  e r a t i o n s r a t h e r than p s y c h o l o g i c a l , c l a i m i n g the e m p i r i c a l t o be the o n l y d e f e n s i b l e approach.  P i a g e t , they a r g u e d , was an o b s e r v e r , not a  teacher.  They c o n s i d e r e d the p r e d i c t i o n s which emerged from h i s d e v e l o p m e n t a l model u n j u s t i f i e d as they were based on i n v e s t i g a t i o n s of the i n g o f c h i l d r e n who had been taught by c o n v e n t i o n a l methods.  understandKarplus  (1970) s u p p o r t e d t h i s v i e w , n o t i n g t h a t P i a g e t found t h a t f o r m a l  operations  developed w i t h o u t s p e c i f i c i n s t r u c t i o n , and t h a t t h i s was not an adequate premise t o encompass the r e s u l t s , t h i n k i n g or a t t i t u d e s  o f modern  science. T h i s d i s c u s s i o n i s i m p o r t a n t f o r many r e a s o n s .  Buell  (1970)  stated  t h a t the c o n t r o v e r s i a l i s s u e as t o whether the l e a r n i n g o f c o n c e p t s may be a c c e l e r a t e d i s o f c o n s i d e r a b l e s i g n i f i c a n c e i n view o f the tendency i n r e c e n t y e a r s t o advance the p r e s e n t a t i o n o f s c i e n t i f i c and m a t h e m a t i c a l m a t e r i a l t o e a r l i e r l e v e l s o f the c u r r i c u l u m . Miller  (1976) , u s i n g 52 K i n d e r g a r t e n c h i l d r e n from a l o w e r s o c i o -  economic background, attempted t o f i n d out whether c o n s e r v a t i o n as d e f i n e d by P i a g e t emerged e a r l i e r i n a c h i l d ' s development i f the t a s k s were p r e s e n t e d n o n - v e r b a l l y and m o t i v a t i o n was maximized.  The c h i l d  could  p i c k the c a n d i e s t h a t he wanted t o eat i n a c o n s e r v a t i o n o f number  task,  o r the j u i c e he wanted t o d r i n k i n a c o n s e r v a t i o n o f l i q u i d t a s k .  Miller  commented t h a t language might w e l l be an i m p o r t a n t f a c t o r because  the  c h i l d r e n seemed t o judge numbers i n terms of l e n g t h , and concept  seemed t o be a c t i v a t e d by the word  ever, t h a t t h e r e was  'more'.  that t h i s  He c o n c l u d e d , how-  no evidence from h i s study t o suggest  that  c o n s e r v a t i o n emerged a t an e a r l i e r c h r o n o l o g i c a l age than t h e o r i z e d by Piaget. I t i s very p o s s i b l e t h a t no s o l u t i o n t o the argument e x i s t s a t present.  Bruner  (1967) took t h i s p o s i t i o n , a s k i n g whether i t was  more  v a l u a b l e t o p o s t u l a t e s t a g e s of growth or t o t h i n k i n terms of g r a d u a l p r o c e s s e s of growth.  He claimed the argument was  fruitless.  The Development of Number Concepts i n P r e - S c h o o l C h i l d r e n Saxe (1977) analyzed c h i l d r e n ' s c o u n t i n g s t r a t e g i e s with a r r a y s of more than f i v e o b j e c t s with the express purpose of a s s e s s i n g whether developmental  changes i n t h e i r s t r a t e g i e s were p a r t i a l l y independent  t h e i r a b i l i t y to count a c c u r a t e l y .  He concluded  that counting  accuracy  and s t r a t e g i e s were i n t e r r e l a t e d  and t h a t they appeared  t o the same c o g n i t i v e p r o c e s s .  The main p o i n t of i n t e r e s t i n t h i s  study was  t h a t a t the P r e - q u a n t i t a t i v e l e v e l s almost  t o be  of  connected  no a c c u r a t e  count-  i n g o c c u r r e d , and a t the Q u a n t i t a t i v e l e v e l s c o u n t i n g a c c u r a c y markedly improved.  However, a t the T r a n s i t i o n a l l e v e l s , between the P r e -  q u a n t i t a t i v e and miscount  the Q u a n t i t a t i v e l e v e l s some c h i l d r e n appeared  'purposely' as a r e s u l t of t r y i n g t o i n t e g r a t e t h e i r  judgments of numbers with the products of t h e i r c o u n t i n g . emphasized the d i f f i c u l t y  of attempting  spatial  This a r t i c l e  to a r b i t r a r i l y d i s t i n g u i s h a  sequence of i n t e r r e l a t e d concepts without c h i l d ' s c o g n i t i v e maturation.  to  taking into consideration a  In some c a s e s , however,  evidence  i n the r e s e a r c h t o s u b s t a n t i a t e the i d e a of t e a c h i n g one concept  appears before  another.  B r a i n e r d (1974) took a group o f p r e - s c h o o l e r s who e v i d e n c e d  no p r o f i c i e n c y w i t h e i t h e r t h e o r d i n a l or c a r d i n a l p r o p e r t i e s o f n a t u r a l numbers and t r a i n e d them t o a c q u i r e t h e s e p r o p e r t i e s . 60 female C a u c a s i a n s u b j e c t s  H i s 60 male and  ranged from 4 years 0 months to 5 years  5 months w i t h a mean age o f 4 years 7 months.  The r e s u l t s o f h i s s t u d y  showed t h a t both p r o p e r t i e s were t r a i n a b l e , but t h a t t h e o r d i n a l p r o p e r t i e s of numbers were much e a s i e r t o t r a i n than the c a r d i n a l properties.  He a l s o noted t h a t the i n f o r m a t i o n l e a r n e d about o r d i n a l  p r o p e r t i e s was more e a s i l y t r a n s f e r r e d  to other l e a r n i n g s i t u a t i o n s  than  the i n f o r m a t i o n l e a r n e d about c a r d i n a l p r o p e r t i e s . The r e s u l t s o f t h i s r e s e a r c h i m p l i e d t h a t the u n d e r s t a n d i n g o f  the  o r d i n a l p r o p e r t i e s o f numbers would precede t h a t of the c a r d i n a l p r o p e r t i e s i n such a way t h a t i t would be f e a s i b l e t o p l a c e them i n a h i e r a r c h i c a l r e l a t i o n s h i p t o one a n o t h e r .  S i e g e l (1971) d e s i g n e d a sequence o f  e x p e r i m e n t a l s i t u a t i o n s i n o r d e r to a n a l y z e whether c e r t a i n a r i t h m e t i c b e h a v i o u r s formed a h i e r a r c h i c a l r e l a t i o n s h i p t o one a n o t h e r .  One o f  t h e s t a g e s o f t h a t h y p o t h e s i z e d h i e r a r c h y was d e s i g n e d t o measure concept o f the o r d i n a l p r o p e r t y o f numbers.  It i s important to  the note,  however, t h a t the t a s k s she d e s i g n e d were n o n - v e r b a l t o remove language as a confounding f a c t o r i n the a n a l y s i s o f t h e r e s u l t s .  Her h y p o t h e -  s i z e d h i e r a r c h y was: (a) magnitude d i s c r i m i n a t i o n s o f s o l i d c o n t i n u o u s a r e a s , (b) magnitude d i s c r i m i n a t i o n s o f d i s c o n t i n u o u s d i s c r e t e (c) e q u i v a l e n c e o f s e t s ( o n e - t o - o n e  areas,  correspondence),  (d) c o n s e r v a t i o n o f numbers ( P i a g e t i a n concept but  tested  non-verbally), (e)  the O r d i n a l p r o p e r t y of numbers f o r example, 1 s t ;  2nd; 3 r d ;  15 ( f ) s e r i a t i o n ( r e c o g n i z i n g the o r d e r o f a s e r i e s o f o b j e c t s  that  a r e a r r a n g e d i n a s t e p - w i s e f a s h i o n ) , and (g) a d d i t i o n e q u a t i o n s  (presented  pictorially).  The s u b j e c t s were 77 m i d d l e - c l a s s c h i l d r e n r a n g i n g from 3 years 0 months t o 4 years 11 months i n age, w i t h boys and g i r l s a p p r o x i m a t e l y equally represented.  S i e g e l attempted t o v a l i d a t e her r e s u l t s by u s i n g  the Guttman Scalogram A n a l y s i s t e c h n i q u e , but f a i l e d  to o b t a i n the  r e q u i r e d v a l u e o f .95 f o r the c o e f f i c i e n t o f r e p r o d u c i b i l i t y . a l l average c o e f f i c i e n t o f r e p r o d u c i b i l i t y was 0.89 and the s i n g l e c o e f f i c i e n t was 0 . 9 3 .  Despite t h i s r e s u l t ,  h y p o t h e s i z e d h i e r a r c h y was o f some i n t e r e s t .  The o v e r -  highest  the o r d e r i n g o f the  There was no d i f f e r e n c e  between the two s e r i e s o f t a s k s t h a t t e s t e d magnitude d i s c r i m i n a t i o n s , and s e r i a t i o n and a d d i t i o n were a t a p p r o x i m a t e l y the same l e v e l o f difficulty.  O r d i n a t i o n o c c u r r e d i n the m i d d l e o f t h e h i e r a r c h y and was  c l e a r l y much e a s i e r than s e r i a t i o n and a d d i t i o n .  C o n s e r v a t i o n was  h a r d e r than one-to-one correspondence but e a s i e r than s e r i a t i o n and addition.  O v e r a l l , the p o s i t i o n o f the t a s k s i n r e l a t i o n to one  another formed a b r a n c h i n g h i e r a r c h y r a t h e r than a l i n e a r h i e r a r c h y . C l e a r l y , i t would be advantageous to have some c l e a r s p e c i f i c a t i o n s o f c u r r i c u l a , or r a t i o n a l e f o r c u r r i c u l a r c h o i c e s , i n o r d e r t o p l a n more p r e c i s e l y f o r the e d u c a t i o n of p r e - s c h o o l c h i l d r e n .  Resnick  (1967, 1970) o u t l i n e d an o p e r a t i o n a l d e f i n i t i o n o f the number concept i n the form o f a s e t o f b e h a v i o u r s w h i c h , t a k e n t o g e t h e r ,  a l l o w a person  t o c o n s i d e r the c h i l d t o have mastered the concept o f numbers: (a) o n e - t o - o n e correspondence t o 5, (b) o n e - t o - o n e correspondence to 10, ( c ) r e c o g n i z i n g the numerals from 0 t o 5,  .  16  (d) r e c o g n i z i n g the numerals from 0 t o 1 0 , (e) comparison o f Sets  ( c o n s e r v a t i o n o f numbers),  ( f ) s e r i a t i o n and O r d i n a l p o s i t i o n ( o r g a n i z i n g m a t e r i a l i n o r d e r o f s i z e and d e s i g n a t i n g i t as 1 s t ,  2nd, 3 r d , e t c . ) ,  and  (g) a d d i t i o n and s u b t r a c t i o n E q u a t i o n s . Wong, R e s n i c k and Boozer (1971) examined the sequence i n which young c h i l d r e n a c q u i r e the elementary m a t h e m a t i c a l b e h a v i o u r s o f c o u n t i n g , one-to-one correspondence and r e c o g n i t i o n o f n u m e r a l s , h y p o t h e s i z i n g t h a t : (a) r e c o g n i t i o n o f numerals i s l e a r n e d e a r l y i n the p r o c e s s o f l e a r n i n g to c o u n t , (b) c o u n t i n g and r e c o g n i t i o n o f numerals f o r q u a n t i t i e s up t o  five  would n o r m a l l y be l e a r n e d b e f o r e e i t h e r c l a s s o f s k i l l s was l e a r n e d f o r q u a n t i t i e s up t o  ten,  ( c ) c o u n t i n g and r e c o g n i t i o n o f numerals a r e l e a r n e d with n e i t h e r c l a s s of tasks p r e r e q u i s i t e to the other,  independently  and  (d) t h e r e i s a p s y c h o l o g i c a l independence of c o u n t i n g and o n e - t o one s k i l l s . The s u b j e c t s were 42 boys and 36 g i r l s between the ages o f 4 y e a r s 6 months and 6 y e a r s 0 months who were a t t e n d i n g k i n d e r g a r t e n .  Sixty-three  p e r c e n t of the c h i l d r e n were b l a c k and t h i r t y - s e v e n p e r c e n t were w h i t e ; p a r e n t s ' work s t a t u s ranged from unemployed t o e x e c u t i v e - p r o f e s s i o n a l . R e s u l t s i n d i c a t e d t h a t command over n u m e r a l s , numerals a c q u i r e d i n a r e g u l a r sequence,  b e g i n n i n g w i t h p e r c e p t u a l matching o f the numerals and  c o n c l u d i n g w i t h t h e a s s o c i a t i o n o f s e t s and n u m e r a l s , i s not o r d i n a r i l y learned u n t i l after  c o u n t i n g o p e r a t i o n s f o r s e t s of the s i z e  by the numerals a r e w e l l e s t a b l i s h e d .  Data a l s o s u p p o r t e d  p s y c h o l o g i c a l d i f f e r e n t i a t i o n o f c o u n t i n g and o n e - t o - o n e  represented the  correspondence.  I m p l i c a t i o n s o f t h e R e l a t e d Research C o n s i d e r i n g e x t a n t r e s e a r c h i n f o r m a t i o n , i t i s p o s s i b l e t o argue t h a t a r i t h m e t i c b e h a v i o u r s can be arranged i n a h i e r a r c h i c a l manner. Whether i n f a c t t h e y can be arranged so t h a t i t i s p o s s i b l e to c l a i m t h a t c e r t a i n b e h a v i o u r s a r e p r e r e q u i s i t e t o o t h e r s i n forming a concept of numbers i s another m a t t e r .  I t does appear r e a s o n a b l e , however, to  h y p o t h e s i z e t h a t a d d i t i o n e q u a t i o n s would be a t the top o f a h i e r a r c h y which i n c l u d e d t h e f a c t o r s taken i n t o c o n s i d e r a t i o n by S i e g e l , Saxe, B r a i n e r d , Wong, R e s n i c k and B o o z e r .  Resnick (1967, 1970) p l a c e d  a d d i t i o n and s u b t r a c t i o n e q u a t i o n s a t the top o f her h y p o t h e s i z e d h i e r a r c h y and S i e g e l (1971) noted t h a t a d d i t i o n e q u a t i o n s  presented  p i c t o r i a l l y come a t the top o f a h i e r a r c h y a l o n g w i t h s e r i a t i o n . B r a i n a r d (1974) found t h a t o r d i n a l numbers appear to be grasped  before  c a r d i n a l numbers, S i e g e l (1971) t h a t c o n s e r v a t i o n appeared e a s i e r  than  s e r i a t i o n and a d d i t i o n , and Wong, R e s n i c k and Boozer (1971) t h a t counting establishes i t s e l f  p r i o r t o the r e c o g n i t i o n o f n u m e r a l s .  I t appears t h a t i t i s r e a s o n a b l e t o h y p o t h e s i z e a h i e r a r c h y o f arithmetic behaviours.  I t seems u n l i k e l y , t h a t t h e s e b e h a v i o u r s  form a l i n e a r r e l a t i o n s h i p to one a n o t h e r ; Resnick and Boozer (1971) comment on the h i e r a r c h i e s formed.  both S i e g e l  will  (1971) and Wong,  ' b r a n c h i n g ' n a t u r e o f the  The Research Problem posed i n Chapter Two w i l l  t a k e i n t o account e x i s t i n g r e s e a r c h i n v o l v i n g the concept of number, the b e h a v i o u r s which appear t o be p r e r e q u i s i t e to t h i s c o n c e p t , and t h e nature of h i e r a r c h i e s .  18 Chapter Two The Research Problem  Chapter One has o u t l i n e d s i g n i f i c a n t r e s e a r c h by R e s n i c k (1970) and Wong, R e s n i c k and Boozer ( 1 9 7 1 ) , who attempted  to present r a t i o n a l  arguments f o r t e a c h i n g a s p e c i f i e d group o f a r i t h m e t i c b e h a v i o u r s as a b a s i s f o r d e v e l o p i n g a concept o f numbers.  W h i l e R e s n i c k (1967, 1970)  suggested a d e f i n i t i o n o f the number c o n c e p t , Wong, R e s n i c k and Boozer went f u r t h e r and d e t a i l e d a h y p o t h e s i z e d h i e r a r c h y o f a r i t h m e t i c b e h a v i o u r s i n a u s e f u l o r d e r f o r c l a s s r o o m t e a c h e r s at the K i n d e r g a r t e n and F i r s t Grade l e v e l .  R e s u l t s of t h e i r study s u p p o r t e d the i d e a t h a t  some a r i t h m e t i c b e h a v i o u r s are l e a r n e d i n a c l e a r l y d e f i n e d  sequence,  a l t h o u g h they were q u i c k t o p o i n t out t h a t some b e h a v i o u r s , such as c o u n t i n g and o n e - t o - o n e c o r r e s p o n d e n c e , independent o f one a n o t h e r .  appeared to be p s y c h o l o g i c a l l y  I f the i d e a o f a h i e r a r c h y i s  tenable  then i t i s u n l i k e l y t h a t i t would be s t r i c t l y l i n e a r w i t h one b e h a v i o u r l e a d i n g a u t o m a t i c a l l y t o the next most d i f f i c u l t  behaviour i n a s t e p -  wise f a s h i o n . R e s n i c k (1970) o u t l i n e d seven b e h a v i o u r s which c o u l d be c o n s i d e r e d as one p o s s i b l e o p e r a t i o n a l d e f i n i t i o n o f a number c o n c e p t : (a) o n e - t o - o n e correspondence t o 5, (b) o n e - t o - o n e correspondence t o 1 0 , ( c ) r e c o g n i z i n g the numerals  0 - 5 ,  (d) r e c o g n i z i n g the numerals  0-10,  (e) comparison o f s e t s , ( f ) s e r i a t i o n and o r d i n a l p o s i t i o n , and (g) a d d i t i o n and s u b t r a c t i o n  equations.  19 The focus o f t h i s s t u d y i s the mastery o f e q u a t i o n s .  Other a s p e c t s  of a number c o n c e p t , i n c l u d i n g f a c t o r s such as c o u n t i n g , c o n s e r v a t i o n o f discrete entities,  and numeration a r e c o n s i d e r e d t o be p r e r e q u i s i t e s  the b e h a v i o u r s i n v o l v e d i n s o l v i n g a d d i t i o n  to  equations.  The r e s e a r c h problem under i n v e s t i g a t i o n i s whether a h i e r a r c h i c a l r e l a t i o n s h i p e x i s t s between c e r t a i n b a s i c number A t e m a j _ a d d i t i o n .  The  n  a d d i t i o n number (li-emsfehosen were t h o s e between 6 and 9 ( f o r 5+1;  4+3;  6+2),  example,  r a t h e r than those between 1 and 5 i n o r d e r  ensure a s u f f i c i e n t degree o f d i f f i c u l t y  f o r the s u b j e c t s .  s i z e d h i e r a r c h y t o be c o n s i d e r e d i n c l u d e s f o u r d i f f e r e n t  to  The h y p o t h e conceptual  l e v e l s - C o n c r e t e , S e m i - C o n c r e t e , A b s t r a c t - C o n c r e t e and A b s t r a c t . terms a r e d e s c r i b e d on the f o l l o w i n g  These  pages.  C o n c r e t e - A b s t r a c t Continuum C o n c r e t e - A b s t r a c t can be d e f i n e d a l o n g a r e a l e n t i t y - a b s t r a c t e n t i t y continuum, where r e a l e n t i t i e s are a c t u a l p a r t s of the e n v i r o n ment and a b s t r a c t e n t i t i e s a r e numeric r e p r e s e n t a t i o n s is  of i t .  (An e n t i t y  a n y t h i n g to which a numeral can be a s s i g n e d such t h a t i t can be  designated  quantitively).  Concrete l e v e l . t a s k s .  C o n c r e t e t a s k s w i l l use o b j e c t s  environment which can both be seen and p i c k e d up.  from the  As a consequence  they  w i l l be u n l i k e the o t h e r l e v e l s t o be d e s c r i b e d i n t h a t they do not r e p r e s e n t r e a l i t y ; they are i n f a c t a p a r t o f r e a l i t y i t s e l f . crayons and b l o c k s w i l l be used a t the c o n c r e t e l e v e l o f the Semi-concrete l e v e l t a s k s .  Semi-concrete l e v e l tasks  h y p o t h e s i z e d t o be more a b s t r a c t than the c o n c r e t e t a s k s  Both hierarchy. are  because  p i c t u r e s of o b j e c t s are used t o r e p r e s e n t the r e a l o b j e c t s .  The  p i c t u r e s seen are o f s h a p e s , f i s h and m a t c h s t i c k s and r e q u i r e the  child  t o a b s t r a c t at a d i f f e r e n t Level  Concret  Tasks. Concrete-Abstract  symbols i n combination  l e v e l tasks. with  to be more a b s t r a c t than any  c o g n i t i v e l e v e l to t h a t o u t l i n e d under  T a l l y Marks.  These t a s k s are  hypothesized  In the p r e v i o u s l e v e l  particular  not  o b j e c t or o b j e c t s i n the environment.  of symbols i s hypothesized  use  (Semi-Concrete)  were used whereas a t t h i s l e v e l the t a l l y marks do  r e p r e s e n t any  of  the Semi-Concrete t a s k s i n t h a t they do not  r e p r e s e n t a t i o n of r e a l i t y .  matchsticks  These t a s k s i n v o l v e the use  The  use  to i n c r e a s e the degree of a b s t r a c t n e s s .  However, i n t h i s combination  the a b s t r a c t n e s s i s decreased  t h a t the t a l l y marks can s t i l l  to the  extent  f u n c t i o n as a c o n c r e t e a i d i n s o l v i n g  an  equation. Abstract l e v e l tasks.  A b s t r a c t t a s k s are those i n which  numerical  symbols a r e used by themselves to r e p r e s e n t e n t i t i e s w i t h i n the  environ-  ment. Task L e v e l s Some l e v e l s of t h e . h i e r a r c h y w i l l as mentioned above. distinguish  These w i l l  the h y p o t h e s i z e d  w i t h i n the f o u r c o n c e p t u a l task l e v e l s - one  levels.  Concrete  t h e r e i s o n l y one  just  At the Concrete  a t h i r d matchsticks.  l e v e l there  are - as a  At the Semi-  s e t s of t a s k s - one  uses shapes,  At the C o n c r e t e - A b s t r a c t  s e t of t a s k s which uses the t a l l y marks, and  A b s t r a c t l e v e l t h e r e i s only one  to  levels  the o t h e r , crayons  l e v e l of the h i e r a r c h y .  tasks  described.  h i e r a r c h y with seven task  l e v e l t h e r e are t h r e e d i f f e r e n t  a second f i s h , and  levels  which uses b l o c k s , and  means t o a s s e s s the c o n c r e t e  s e t s of  be c a l l e d Task L e v e l s i n o r d e r  them from the f o u r c o n c e p t u a l  Figure 1 presents  two  contain d i f f e r e n t  at  s e t of t a s k s which uses numerals.  level the  21 D e f i n i t i o n o f Terms The f o l l o w i n g d e f i n i t i o n s a r e e s s e n t i a l to c l a r i f y the d e s i g n procedures,  r e s u l t s and c o n c l u s i o n s o f t h i s  F i n a l Task:  study:  A d d i t i o n o f two one d i g i t numerals whose sum i s no  g r e a t e r than n i n e . C r i t e r i o n f o r Performance:  C o m p l e t i o n o f the t a s k w i t h i n  l i m i t s o f the s t a n d a r d i z e d i n s t r u c t i o n s Appendix D ) .  (Instructions  I f e r r o r o c c u r s , the s u b j e c t  c o r r e c t i o n without  the  for Teaching,  has t o i n i t i a t e and complete  assistance.  Instructional Objective:  S u c c e s s f u l c o m p l e t i o n of the F i n a l Task  a t the h i g h e s t l e v e l o f t h e h i e r a r c h y ; t h a t i s , a t the most a b s t r a c t level. Entering Behaviours:  Those a r i t h m e t i c b e h a v i o u r s which  are  c o n s i d e r e d p r e r e q u i s i t e t o the u n d e r s t a n d i n g o f b a s i c number i t e m s , and which a l o n g w i t h a demonstrated i l l u s t r a t i v e o f an u n d e r s t a n d i n g  mastery of b a s i c a d d i t i o n i t e m s of the concept o f numerals as  i n g e n t i t i e s w i t h i n the environment which can be q u a n t i t a t i v e l y Learning Hierarchy:  are representdesignated.  The r u l e or problem s o l v i n g t a s k i s a n a l y z e d  i n t o s i m p l e c a p a b i l i t i e s to be l e a r n e d as p r e r e q u i s i t e s .  When such an  a n a l y s i s i s c o n t i n u e d p r o g r e s s i v e l y to the p o i n t of d e l i n e a t i n g an e n t i r e s e t o f c a p a b i l i t i e s h a v i n g an o r d e r e d r e l a t i o n s h i p to each then a L e a r n i n g h i e r a r c h y e x i s t s Task a n a l y s i s :  (Gagne 1 9 7 0 ) .  The p r o c e s s o f i s o l a t i n g , d e s c r i b i n g and  i n g a l l the n e c e s s a r y s u b - t a s k s ,  other,  w h i c h , when they a r e m a s t e r e d ,  enable s u c c e s s f u l mastery o f the i n s t r u c t i o n a l o b j e c t i v e .  sequencwill  (Bateman 1 9 7 1 ) .  22  Level  4  Abstract  Level  3  Abstract-Concrete  < Level  2  Semi-Concrete  Level  1  Concrete  Figure  1  The  Seven H y p o t h e s i z e d  a Four-Level  Hierarchy  Test Levels  Arranged  in  Research O b j e c t i v e s 1.  To a s c e r t a i n whether a l i n e a r r e l a t i o n s h i p , or a b r a n c h i n g  r e l a t i o n s h i p , e x i s t s between the l e v e l s o f t h e h y p o t h e s i z e d h i e r a r c h y . 2.  To a s c e r t a i n whether t a s k s h y p o t h e s i z e d to be a t a g i v e n l e v e l  of the h i e r a r c h y a r e o f e q u a l d i f f i c u l t y r e l a t i o n s h i p to one 3.  and remain i n the same  another.  To a s c e r t a i n whether any t a s k s w i t h i n any l e v e l o f the  chy are o f any g r e a t e r or l e s s e r d i f f i c u l t y t h a t t h e i r r e l a t i o n s h i p t o one another 4.  5.  than any o t h e r t a s k s ,  o f boys and  difference  girls.  To a s c e r t a i n whether any o f the b a s i c number f a c t  between 6 and 9 a r e any more d i f f i c u l t  than any o t h e r  combinations  combinations  following t h e i r i n i t i a l balancing, according to established 6.  such  becomes l i n e a r .  To a s c e r t a i n whether t h e r e i s any s i g n i f i c a n t  between the performances  hierar-  criteria.  To a s c e r t a i n whether any r e l a t i o n s h i p e x i s t s between  l e v e l s o f the h y p o t h e s i z e d h i e r a r c h y and the t a s k l e v e l s o f the  the hierar-  c h y , such t h a t i t i s p o s s i b l e t o say t h a t a h i e r a r c h y e x i s t s which moves from the C o n c r e t e l e v e l t o the A b s t r a c t l e v e l . Target P o p u l a t i o n I n i t i a l l y u s i n g a Grade One p o p u l a t i o n was c o n s i d e r e d but became readi%yapparent designed, too easy.  t h a t t h e s e c h i l d r e n would f i n d the t a s k s ,  it as  A l s o , u n l e s s t h e data c o u l d be c o l l e c t e d immediate  l y the s t u d e n t s e n t e r e d Grade One, the r e s u l t s would be confounded by t h f a c t t h a t the s t u d e n t s would have been exposed t o a r i t h m e t i c  teaching.  Thus, K i n d e r g a r t e n c h i l d r e n were s e l e c t e d as the most s u i t a b l e group use i n t h i s  study.  to  However, g i v e n the f a c t o r s d i s c u s s e d to developmental v a r i a b l e s , i t was  i n Chapter One  necessary  p o p u l a t i o n so as t o i n c l u d e o n l y those  with  to d e l i m i t the  c h i l d r e n who  l e v e l of c o g n i t i v e development i n mathematics, and  had  subject  a sufficient  to exclude  would a l r e a d y be a b l e to answer a l l of the q u e s t i o n s  regard  those  t h a t would  arise  from the seven l e v e l s of the hypothesized  hierarchy.  a p r e t e s t and  the P r e t e s t to e s t a b l i s h  a pilot  t e s t were designed;  whether the s u b j e c t s had  the necessary  Test to e s t a b l i s h whether any L e v e l One  of the h i e r a r c h y  To e f f e c t  E n t e r i n g Behaviours and  who  this  the  Pilot  of the s u b j e c t s would f i n d the t a s k s at  ( A b s t r a c t ) too  easy.  Chapter Three Methodology  In o r d e r t o i n v e s t i g a t e t h e r e s e a r c h Chapter Two i t was necessary standardized  questions  specified i n  t o c o n t r o l f a c t o r s such as item  difficulty,  t e s t i n g and t e a c h i n g i n s t r u c t i o n s , item p r e s e n t a t i o n i n a  t e s t i n g s i t u a t i o n , and present b e f o r e attempting  knowledge o f a r i t h m e t i c  behaviours,  t o i n t e r p r e t data from each l e v e l o f t h e h i e r a r c h y .  I t i s important  t o f i n d out whether the items s e l e c t e d a r e  s u i t a b l e f o r t h e s u b j e c t s chosen.  Standardizing  i n s t r u c t i o n s allows  each s u b j e c t an equal chance t o l e a r n i n t h e t e a c h i n g s i t u a t i o n and an equal chance t o respond i n t h e t e s t i n g s i t u a t i o n ; randomizing the t e s t i n g procedures e s t a b l i s h e s a c o n t r o l t o d e a l with unequal o p p o r t u n i t i e s t o l e a r n when a c t u a l l y being it  i s a l s o important  tested.  P r i o r t o t h e commencement o f t e a c h i n g  t o a s s e s s whichcschildren a r e not  developmentally  ready t o b e n e f i t from the t e a c h i n g , and which c h i l d r e n have a l r e a d y a c q u i r e d t h e knowledge t h a t i s p r e s e n t e d . Measuring  Instrument  As i n d i c a t e d p r e v i o u s l y , t h e items s e l e c t e d were t h e 26 b a s i c a d d i t i o n .items whose sums were no l e s s than s i x , and no g r e a t e r than (Appendix A ) .  nine  To f a c i l i t a t e c o n t r o l o f item d i f f i c u l t y and t o a l l o w t h e  c o n s t r u c t i o n o f a measuring instrument  which would c o n t r o l f o r f a c t o r s  t h a t would a f f e c t t h e t e s t r e s u l t s , f i v e o f these ftemsxwere randomly e l i m i n a t e d , l e a v i n g 21 f a c t s t o be i n c o r p o r a t e d i n t o t h e t e s t i n g and t e a c h i n g procedures.  The f i v e items randomly e l i m i n a t e d were:  26 5 + 1 7 + 1 2 + 7 3 + 6 4+5 These f a c t s were used t o e s t a b l i s h whether any o f the s u b j e c t s a l r e a d y answer the a b s t r a c t  could  q u e s t i o n s a t l e v e l f o u r of the h i e r a r c h y .  Pretest One hundred t w e n t y - f i v e s u b j e c t s were a d m i n i s t e r e d a p r e t e s t i n o r d e r t o e s t a b l i s h whether t h e s u b j e c t s had t h e minimum E n t e r i n g B e h a v i o u r s , and to a s c e r t a i n how many s u b j e c t s had a l r e a d y a c q u i r e d the I n s t r u c t i o n a l O b j e c t i v e o f the h y p o t h e s i z e d h i e r a r c h y .  Those  c h i l d r e n who d i d not have the s p e c i f i e d E n t e r i n g B e h a v i o u r s , and those c h i l d r e n who reached the I n s t r u c t i o n a l O b j e c t i v e , were then e x c l u d e d from the s u b j e c t p o o l . First,  There were t h r e e s p e c i f i e d E n t e r i n g B e h a v i o u r s .  the s u b j e c t had t o i d e n t i f y v e r b a l l y numerals on c a r d s  presented  i n a random o r d e r .  Second, each s u b j e c t had t o r e c i t e t h e numerals  from one t o n i n e .  T h i r d , t h e s u b j e c t was r e q u i r e d to count a group o f  s i x and then f o u r randomly a r r a n g e d b l o c k s (see Appendix C ) .  The f i v e  randomly e l i m i n a t e d i t e m s were used t o a s s e s s whether any c h i l d  could  a l r e a d y r e a c h the I n s t r u c t i o n a l O b j e c t i v e at l e v e l f o u r o f the h i e r a r c h y . Any c h i l d r e s p o n d i n g t o f o u r or f i v e problems c o r r e c t l y w i t h o u t the use of a i d s such as f i n g e r s , paper and p e n c i l , was e x c l u d e d from the pool.  Of the 125 s u b j e c t s a s s e s s e d ,  subject  49 met the r e s e a r c h c r i t e r i a .  (For d e t a i l e d i n f o r m a t i o n , see Chapter F o u r ) .  Pilot  Test The  3  girls)  pilot who  met  study.  This  subjects  in  to  Test  was  the  research  involved  order  to  would  questions see  at  Chapter  be  giving  a random  very  Level  the  of  the  this  study  but  subjects  did  not  of  subjects  the  test,  hierarchy.  B)  and who  that  (For  (4  boys  participate  (Appendix  instructions  sample  the  seven  posttest  s u c c e s s f u l on  Four  to  criteria  standardize  a s c e r t a i n whether  criteria  administered  to  and in  the  these  procedures, met  the  i s ,  and  research  complete  detailed  information,,  Four).  Subjects Subjects comprising  for  both  Squamish  School  criteria  were  take  part  randomly Item  in  District.  used the  21  in  were  Seven  the  Pilot  study.  items  rows  cell  criteria  of  of  chosen  from  the  subjects  Test,  From t h i s  according the  the  design  in  items  1.  An  each  equal  difficulty.  2.  Size  addition  of  size  made to and  1933;  49  leaving  group  row  their thus  items  14  public  42  schools  who  met  subjects  boys  and  Classes  14  the  in  the  research  eligible girls  to  were  the  measuring  degree  of  equalize  into  difficulty problem  Vogel  three  instrument  order  difficulty..  1928;  levels  in  were  Wheeler of  1939)  difficulty  arranged to  balance The for  with  were:  combination  addend  is  of  sum.  the  up  Washburne;  twenty-one  seven  the  which  (Machatchy  arranging  than  girls  children.  Difficulty  three  each  and  kindergarten  selected.  The in  boys  were  the  and  its  'reverse'  principal indicator  form  of  tend  to  be  difficulty,  of  rather  3. Combinations  with a common addend appear  but not e q u a l d i f f i c u l t y  (4+4;  to be of s i m i l a r  3+3).  4. The doubles i n a d d i t i o n and those i n which 1 i s added with a g r e a t e r number appear  t o be e a s i e s t i n a d d i t i o n .  A c c o r d i n g t o these c r i t e r i a , the items can be c l a s s i f i e d A :  Easy  B :  Moderaly  C :  Hard^  as f o l l o w s :  items Difficult  items  items  These 21 items were then d i v i d e d i n t o seven groups o f t h r e e ; a c c o r d i n g t o the c r i t e r i a  o u t l i n e d , a l l the problems i n column A would  be the e a s i e s t , those i n column B more d i f f i c u l t , and those i n column C the h a r d e s t .  Table 1 p r e s e n t s r e s u l t s of t h i s c l a s s i f i c a t i o n ; a l l  items w i t h i n each of the columns a r e c o n s i d e r e d to be o f a p p r o x i m a t e l y the same degree o f  difficulty. Table 1  Items Arranged by L e v e l of  Easy A Items  1  "hard  1  Difficulty  B Difficult Moderately Items .  Hard C Items  1 + 5  8 + 1  2 + 6  1 + 6  4 + 3  6 + 2  1 + 7  3 + 4  7 + 2  1 + 8  2 + 5  6 + 3  6 + 1  5 + 2  5 + 4  3 + 3  2 + 4  5 + 3  4 + 4  4 + 2  3 + 5  i s used synonomously with the word  V  'difficult'.  To c o n t r o l f o r item d i f f i c u l t y , these 21 items were then d i v i d e d i n t o seven groups of t h r e e with one from A, one The  item  from B,  and  one  item i n each group randomly s e l e c t e d  from C u n t i l a l l the items were exhausted.  groups y i e l d e d by t h i s procedure are shown i n Table  Table  2.  2  Items Grouped to E q u a l i z e D i f f i c u l t y  of  Sets of Three  Name of Item Group  A Easy litems  g Moderately D i f f i c u l t Items  C Hard Items  a  1 + 5  3 + 3  2 + 6  b  4 + 4  4 + 3  7+2  c  3 + 3  4+2  d  1 + 7  8 + 1  6+2  e  1 + 8  2 + 4  5 + 3  f  6 + 1  5+2  3 + 5  g  1 + 6  2+5  5 + 4  6 + 3  T e s t i n g Sequence Grouping the items to c o n t r o l f o r t h e i r d i f f i c u l t y made p o s s i b l e the c r e a t i o n of a t e s t i n g sequence f o r a s s e s s i n g responses t o the t e a c h i n g created  session.  by employing a G r e c o - L a t i n  The  the  subjects'  t e s t i n g sequence i t s e l f  Square d e s i g n  that involved placing  the 28 s u b j e c t s a t seven d i f f e r e n t l e v e l s , f o u r s u b j e c t s per L e v e l one,  f o r i n s t a n c e , would i n v o l v e the f i r s t  answering q u e s t i o n s indicated.  Table  at the hypothesized  was  level.  four subjects i n  h i e r a r c h y l e v e l and  item  3 shows the l e v e l s of the h i e r a r c h y combined  set with  the re-grouped i t e m s .  Looking  task l e v e l i n the h y p o t h e s i z e d t h r e e items  1+8;  2 + 4;  at  ( f o r example) 6e,  h i e r a r c h y and  5 + 3 (see Table  Table  "6" would be  the  'e' would be the group of 2).  3  T e s t i n g Sequence as Determined by a Greco-Latin  T-.ask Level  Square  Subjects  Sequence  1  1 - 4  *6e  4b  2f  7c  5g  3d  la  2  5 - 8  7f  5c  3g  Id  6a  4e  2b  3  9 -  12  ig  6d  4a  2e  7b  5f  3c  4  13 -  16  2a  7e  5b  3f  lc  6g  4d  5  17 -  20  3b  If  6c  4g  2d  7a  5c  6  21 - .24  4c  2g  7d  5a  3e  lb  6f  7  25 -  5d  3a  le  6b  4f  2c  7g  28  * i n each combination the number r e f e r s to the l e v e l of the h i e r a r c h y ( F i g . 1) and the l e t t e r r e f e r s to the item s e t (Table 2)  The pilot  format f o r q u e s t i o n i n g the subjects^was f o r m a l i z e d when the  t e s t was  administered.  Each s u b j e c t was  administered  three  q u e s t i o n s a t each task l e v e l thus r e q u i r i n g a t o t a l of 21 q u e s t i o n s  to  be answered by each s u b j e c t .  the  (For d e t a i l e d i n f o r m a t i o n c o n c e r n i n g  T e s t i n g I n s t r u c t i o n s see Appendix B ) .  31 Teaching P r o c e d u r e s P r i o r t o the t e s t i n g s e s s i o n each s u b j e c t was taught  t h r e e items  at  each Task L e v e l o f the h y p o t h e s i z e d h i e r a r c h y ; i n s t r u c t i o n was g i v e n t o seven groups of c h i l d r e n , each group composed o f two boys and two g i r l s . Each group l e a r n e d t h r e e problems i n v o l v i n g b l o c k s , t h r e e i n v o l v i n g pictures of f i s h ,  three i n v o l v i n g shapes,  three i n v o l v i n g matchsticks,  t h r e e i n v o l v i n g numerals and t a l l y marks, and t h r e e i n v o l v i n g o n l y numerals. In o r d e r t o d e c i d e which t h r e e items were t o be taught a t each l e v e l of the h y p o t h e s i z e d h i e r a r c h y , the f o l l o w i n g row was chosen a t random from a 7 by 7 L a t i n Square. 3,  1,  The r e s u l t i n g sequence was: 5,  6,  4,  7,  2. T h i s row was then matched w i t h t h e items as grouped i n T a b l e 2 i n  such a way t h a t the " a " group o f items (1 + 5;  3+3;  2+6)  was  matched w i t h the number 5, t h e " b " group o f problems w i t h number 6 and so o n .  R e s u l t s o f t h i s procedure are p r e s e n t e d i n T a b l e 4 .  Table 4 Item Set and i t s Matched  Level  i n the H i e r a r c h y  Item Set  Name o f Item Group  Easy Items  Moderately Difficult Items  Hard Items  Hierarchy Level  5  a  1 + 5  3 + 3  2 + 6  1  6  b  4 + 4  4 + 3  7 + 2  2  4  c  3 + 3  4 + 2  6 + 3  3  7  d  1 + 7  8 + 1  6 + 2  4  3  e  1 + 8  2 + 4  5 + 3  5  1  f  6 + 1  5 + 2  3 + 5  6  2  g  1 + 6  2 + 5  5 + 4  7  The h y p o t h e s i z e d h i e r a r c h y from l e v e l 7 t o l e v e l 1 i s shown i n F i g u r e 2.  NUMERALS  L e v e l 4' Abstract  NUMERALS AND  Level 3  TALLY MARKS  Abstract-Concrete X  SHAPES  FISH  Level 2 Semi-Concrete  MATCHSTICKS 3  4  5 X  = 1  jr  BLOCKS  CRAYONS  Level 1  6  Concrete  F i g u r e 2.  H y p o t h e s i z e d H i e r a r c h y f o r A d d i t i o n F a c t s Showing the C o n c e p t u a l L e v e l s 1 t o 4 and Task L e v e l s 1 t o 7  F i g u r e 2 shows the seven Task L e v e l s o f the h i e r a r c h y . ing  items were p r e s e n t e d t o the s u b j e c t s a t Task L e v e l seven f i r s t ,  Task L e v e l s i x , t h e n Task L e v e l f i v e and so o n , u n t i l a l l the had  The t e a c h -  been exposed to twenty-one p r o b l e m s .  then  subjects  A t the C o n c r e t e l e v e l - t a s k  l e v e l s s i x and seven - two d i f f e r e n t s e t s o f c o n c r e t e t a s k s were g i v e n and  a t the S e m i - C o n c r e t e l e v e l t h r e e d i f f e r e n t s e t s of t a s k s were  presented,  t o a s s e s s whether a s e t o f t a s k s a t the same l e v e l  posed any more d i f f i c u l t y that l e v e l  (crayons).  one i s , i n f a c t ,  (blocks)  f o r s u b j e c t s than another s e t o f t a s k s  at  The numbering o f t h e t a s k l e v e l s from seven to  d e s i g n e d t o a s c e r t a i n (1) whether t h e h i g h e r numbered  task l e v e l s are e a s i e r than the lower numbered task l e v e l s , and  (2)  whether a l i n e a r r e l a t i o n s h i p e x i s t s between these task l e v e l s as from a branching  relationship.  I t i s p o s s i b l e to c o n c e i v e , f o r i n s t a n c e ,  t h a t problems i n v o l v i n g " M a t c h s t i c k s " are more a b s t r a c t than involving "Fish". are hypothesized level.  The  problems  " M a t c h s t i c k s " c l o s e l y resemble " T a l l y Marks" which to be more a b s t r a c t than any  t a s k s on the Semi-Concrete  same r a t i o n a l e a p p l i e s to comparisons between " F i s h "  "Shapes" (shapes may s t i c k s may  distinct  be more a b s t r a c t ) , M a t c h s t i c k s  be more a b s t r a c t ) and  " B l o c k s " and  and  and  Shapes (match-  "Crayons" ( b l o c k s may  be  more a b s t r a c t ) . The  i n s t r u c t i o n s f o r the t e a c h i n g s e s s i o n were s t a n d a r d i z e d so t h a t  no s u b j e c t s gained  any  advantage over any  s u b j e c t s were s e t t l e d , r e l a x e d and  other s u b j e c t .  Once the  a t t e n d i n g , i n s t r u c t i o n s were i d e n t i c a l  f o r each of the groups ( f o r d e t a i l s of these i n s t r u c t i o n s see Appendix The  o n l y v a r i a n c e to the procedures  support  and  D).  as o u t l i n e d i n Appendix D concerned  encouragement of the s u b j e c t s , and  praise for correct  responses. Statistical  Analysis  An a n a l y s i s of v a r i a n c e was  c a r r i e d out i n order to c o n s i d e r :  (a) p o s s i b l e d i f f e r e n c e s i n the performance of the items when boys a r e compared with  girls,  (b) p o s s i b l e v a r i a n c e s i n the d i f f i c u l t y when compared with other (c)  criteria,  and  been balanced  items  items,  p o s s i b l e v a r i a n c e s i n the d i f f i c u l t y  a f t e r having  l e v e l s of c e r t a i n  according  l e v e l s of groups of problems  to the l e v e l s of  difficulty  35 (d) p o s s i b l e v a r i a n c e s  of d i f f i c u l t y  l e v e l between i n d i v i d u a l  problems w i t h i n any group, which a l t e r s t h e balance o f the group o f problems. A d e s c r i p t i v e a n a l y s i s was undertaken t o e s t a b l i s h whether: (a) t h e h i e r a r c h y e x h i b i t s evidence o f a l i n e a r r e l a t i o n s h i p between one L e v e l and another, and one Task L e v e l and another, or a branching  r e l a t i o n s h i p between these L e v e l s ,  (b) the t a s k s h y p o t h e s i z e d remained a t t h a t  t o be a t any one L e v e l o f t h e h i e r a r c h y  level,  (c) t h e degree o f d i f f i c u l t y  o f a task a t t h e Concrete or Semi-  Concrete L e v e l s i s any harder than any other  task a t the same L e v e l , and  (d) any r e l a t i o n s h i p e x i s t s between the L e v e l s o f t h e  hypothesized  h i e r a r c h y , and t h e Task L e v e l s o f t h e h i e r a r c h y , such t h a t i t i s p o s s i b l e t o say t h a t a h i e r a r c h y e x i s t s which moved from t h e Concrete t o the A b s t r a c t  Level.  A Guttman Scalogram A n a l y s i s was e f f e c t e d t o attempt a s t a t i s t i c a l v a l i d a t i o n of the h i e r a r c h y . to t h e hypothesized  T h i s t e c h n i q u e analyzes  Levels of the hierarchy, according  L e v e l s , and a c c o r d i n g  the data  according  t o the Task  t o t h e number o f items c o r r e c t l y answered by  s u b j e c t s a t any g i v e n L e v e l or Task L e v e l .  36 Chapter Four R e s u l t s o f the P r e t e s t  and P i l o t  T h i s c h a p t e r w i l l p r e s e n t the r e s u l t s t e s t and d i s c u s s t h e s e r e s u l t s  Test  o f the p r e t e s t and t h e . p i l o t  and t h e i r p o s s i b l e i m p l i c a t i o n s .  The  p r e t e s t , as mentioned i n Chapter T h r e e , was designed t o f i n d out i f subjects  the  had the minimum E n t e r i n g B e h a v i o u r s as w e l l as how many c h i l d r e n  had a l r e a d y a c q u i r e d the I n s t r u c t i o n a l O b j e c t i v e of the hierarchy.  hypothesized  The two t e s t s i d e n t i f i e d a p o o l o f s u b j e c t s  who would be  a t a p p r o x i m a t e l y the same c o n c e p t u a l l e v e l as f a r as t h e i r o f numbers was c o n c e r n e d .  The o r i g i n a l s u b j e c t  understanding  p o o l encompassed a l l o f  the k i n d e r g a r t e n c h i l d r e n i n Squamish, a t o t a l of 125.  Table 5  o u t l i n e s the breakdown o f t h e s e c h i l d r e n by s c h o o l and by s e x .  Table 5 Number o f S u b j e c t s P a r t i c i p a t i n g i n the  School  .Boys  .  .  Pretest  Girls  Total  1  8  12  20  2  11  15  26  3  25  15  40  4  19  20  _39  Total  63  62  125  The was to  results  a s s e s s e d as the  of  having  pretest either  established criteria.  Behaviours  (Conservation  and  Counting  the  Instructional  Table  the  these  from  are  1-9)  of  and  Objective  designated  are  passed The  Number the  presented (P)  or  results with  five  (highest  show  Blocks,  level  Table  failed  problems  'Additional  in  of  both  an  the  item  the  were  at  subject  according  Entering  Recognizing  that  Problems  (F)  Each  6.  Numerals,  the  hierarchy).  level In  of the  Table 6 Pretest C o n s e r v a t i o n of Number W i t h Blocks School  Boys  Recognizing Numerals  Girls  Boys  Results #  Additional Problems  Counting 1-9  Girls  Boys  Girls  Boys  Total  Girls  Boys  F  P  F  P  F  P  F  P  F  P  F  P  F  P  F  P  F  1.  1  7  2  10  4  4  5  7  1  7  2  10  8  0  11  1  14  2.  3  8  5  10  5  6  8  7  2  9  4  11  10  1  15  0  3.  8  17  4  11  16  9  4  11  2  23  2  13  22  3  15  4.  _5  14  _6  14  15  _4  10  10  3  16  1  19  19  0  Total  17  46  17  45  40  23  27  35  8  55  9  53  59  4  F = Failed  Girls F  P  18  20  28  20  24  32  28  0  48  52  25  35  19  1  42  34  36  44  60  2 124 128 113 135  P  P = Passed  CO  An a n a l y s i s of the r e s u l t s presented i n T a b l e 6 showed t h a t 49 s u b j e c t s s u c c e s s f u l l y met  the r e s e a r c h c r i t e r i a .  s u b j e c t s a c c o r d i n g to s c h o o l i s p r e s e n t e d i n Table  A summaryffof  these  7.  Table 7 D i s t r i b u t i o n over Sex, Schools and T e s t i n g Times of S u b j e c t s Who Met S c r e e n i n g C r i t e r i a  Boys School 1  Morning  Girls Afternoon  4  Morning  Afternoon  6  Total 10  2  3  2  3  4  12  3  4  2  9  0  15  4  _4  0  _5  3  12  Total  15  4  23  7  49  40 Of the o r i g i n a l 125 s u b j e c t s , 76 were not s u i t a b l e a c c o r d i n g t o the e s t a b l i s h e d c r i t e r i a . the f i r s t  Table 8 shows where s u b j e c t s f a i l e d i n  three sub-tests of the P r e t e s t .  Table 8 S u b j e c t s Who F a i l e d the P r e t e s t I l l u s t r a t e d According  to Pretest  Category  CATEGORY  BOYS  Conservation Recognizing Counting  o f Number with Numerals  Blocks  1 - 9  1-9  Total  Most s u b j e c t s who f a i l e d of  17  17  40  27  _8  _7  65  51  t h e p r e t e s t were unable  t h e numerals from 1 t o 9 when v i s u a l l y presented  t h r e e c h i l d r e n who f a i l e d  GIRLS  to recognize a l l  t o them.  the t e s t with t h e b l o c k s passed  which demanded t h e r e c o g n i t i o n o f numerals.  Counting  Only  the t e s t  from 1 t o 9 proved  to be o f no a d d i t i o n a l v a l u e i n the p r e t e s t : a l l the c h i l d r e n who  failed  the c o u n t i n g s u b - t e s t a l s o f a i l e d  e i t h e r the s u b - t e s t on b l o c k s or the  s u b - t e s t on numeral r e c o g n i t i o n .  I t i s i n t e r e s t i n g t o note t h a t f a r  more boys than g i r l s f a i l e d  t h e s u b - t e s t on number r e c o g n i t i o n ; on t h e  other two s u b - t e s t s t h e r e were no s i g n i f i c a n t d i f f e r e n c e s . Those s u b j e c t s who s u c c e s s f u l l y completed the f i r s t  three pretests  were a d m i n i s t e r e d  a f o u r t h t e s t t o f i n d out i f they c o u l d a l r e a d y  solve  the problems t h a t were t o be placed a t the t o p o f t h e h y p o t h e s i z e d hierarchy.  S i x o f t h e s u b j e c t s were a b l e t o c o r r e c t l y answer s u f f i c i e n t  problems - f o u r or more - t o be excluded  from t h e r e s e a r c h p o o l .  Of  these s u b j e c t s f o u r were boys and two were g i r l s .  Pilot  Study The  pilot  study  was c a r r i e d out by a d m i n i s t e r i n g one row o f t h e  f i n a l t e s t t o f o u r g i r l s and t h r e e boys randomly s e l e c t e d from t h e research  pool.  The major purpose f o r t h e p i l o t t e s t was t o a s c e r t a i n whether the s u b j e c t s c o u l d answer most o f t h e q u e s t i o n s  on the P o s t t e s t .  s u b j e c t s c o u l d answer a s i g n i f i c a n t number o f q u e s t i o n s i n g , then i t was q u i t e l i k e l y  I f the  p r i o r to teach-  t h a t t h e pool o f s u b j e c t s s e l e c t e d f o r  the study would a l s o be a b l e t o answer the q u e s t i o n s . A look a t the s t r u c t u r e o f t h e t e s t shows t h e t o t a l p o s s i b l e a t any g i v e n l e v e l o f t h e t e s t was 3 p o i n t s , and the maximum p o s s i b l e s c o r e f o r any  s u b j e c t was 21 p o i n t s .  None o f t h e seven c h i l d r e n was s u c c e s s f u l  on any o f t h e items a t l e v e l s one and two of the h i e r a r c h y ; f i v e o f the seven c h i l d r e n d i d not s c o r e a t l e v e l t h r e e ; t h r e e c h i l d r e n d i d not s c o r e a t l e v e l s f o u r and f i v e ; and two c h i l d r e n d i d not s c o r e a t l e v e l s s i x and seven.  Most success  was achieved  a t the Concrete l e v e l o f t h e  h i e r a r c h y , but even here the s u b j e c t s were a b l e t o s c o r e o n l y 8 and 11 p o i n t s r e s p e c t i v e l y - o n l y a f r a c t i o n o f t h e maximum p o s s i b l e o f 21. I f t h e s c o r e s a t each l e v e l o f t h e h i e r a r c h y a r e averaged t h e r e s u l t s would be 0, 0, 5.66 and 9.5; a s c o r e o f 10.5 would be r e q u i r e d t o a c h i e v e a 50 percent  l e v e l o f accuracy.  One boy and one g i r l  scored  w e l l on l e v e l s 3 , 4 , 5 , 6 and 7 . subjects,  With the e x c e p t i o n o f t h e s e two  the most common s c o r e was 0 and 1 .  Table 9 , p r o v i d e s u p p o r t on the independent  The r e s u l t s ,  outlined in  f o r t h e b e l i e f t h a t t h e r e i s room f o r  variance  variables. Table 9  R e s u l t s o f the P i l o t Study A c c o r d i n g t o H i e r a r c h y L e v e l  TASK LEVEL OF THE HIERARCHY SUBJECTS 1  7  6  5  4  3  2  1  TOTAL  0  1  0  0  0  0  0  1  2  2  2  1  2  0  0  9  1  0  1  1  0  0  0  3  0  3  1  0  0  0  0  4  2  3  2  3  3  0  0  13  1  0  0  1  0  0  0  2  2  2  0  a  0  0  0  _4  8  11  6  6  5  0  0  36  (girl) 2  (girl) 3  (girl) 4  (boy) 5  (boy) 6  (girl) 7  (boy) Total  The maximum t o t a l p o s s i b l e i n each c e l l o f t h e t e s t .design"; was 3 . 0  43! I m p l i c a t i o n s o f the P r e t e s t and P i l o t Test f o r t h e Main Study S e v e n t y - s i x s u b j e c t s were r e j e c t e d because they e i t h e r d i d not have the d e s i r e d E n t e r i n g B e h a v i o u r s or because they c o u l d a l r e a d y answer the q u e s t i o n s a t l e v e l one o f the h i e r a r c h y , r e d u c i n g the s u b j e c t from 125 t o 4 9 .  T h i r t y o f the 49 c h i l d r e n who met the  c r i t e r i a were g i r l s , and n i n e t e e n were b o y s .  pool  research  The f a c t t h a t more g i r l s  met the r e s e a r c h c r i t e r i a may be s i g n i f i c a n t , but was not a f a c t o r i n t h i s study because 14 boys and 14 g i r l s were chosen a t random to form the f i n a l p o o l o f  subjects.  R e s u l t s o f t h e i g p i l o t t e s t l e n d some s u p p o r t to the i d e a o f a C o n c r e t e - A b s t r a c t continuum u s i n g a r i t h m e t i c f a c t s between 6 and 9. With the e x c e p t i o n o f l e v e l s s i x and seven t h e r e appeared to be a l i n e a r p r o g r e s s i o n from the C o n c r e t e t o the A b s t r a c t l e v e l of the h i e r a r c h y . The s u b j e c t s c l e a r l y found the more c o n c r e t e t a s k s e a s i e r than the most abstract  tasks.  However, i f t h e r e s u l t s a r e averaged a c c o r d i n g t o  the  f o u r l e v e l s h y p o t h e s i z e d , the h i e r a r c h y appears even more c l e a r l y (9.5,  5.66,  0.0).  T h i s data suggest t h a t no d i f f e r e n c e s a r e  likely  t o be found between problems p r e s e n t e d a t the v a r i o u s Task L e v e l s . For i n s t a n c e , a t the S e m i - C o n c r e t e l e v e l the number o f c o r r e c t responses f o r " F i s h " , "Shapes" and " M a t c h s t i c k s " was 6, 6, and 5 respectively. Another f a c t o r o f i n t e r e s t ,  a l t h o u g h i t f a l l s o u t s i d e the range o f  the r e s e a r c h q u e s t i o n s , i s the impact o f t e a c h i n g on the s u b j e c t s f o r the s t u d y .  chosen  I f the r e s u l t s o f the p i l o t t e s t a r e used as a g u i d e -  l i n e , and i f i t i s r e a s o n a b l e t o g e n e r a l i z e from t h i s group o f seven to the l a r g e r group o f 28, then i t would be r e a s o n a b l e t o expect t h e  average  s c o r e o f each s u b j e c t t o r i s e . does i t a f f e c t  a l l o f t h e items a t each l e v e l  some items a t some l e v e l s ? levels the  I f t h e t e a c h i n g does have an impact,  teachable?  Concrete l e v e l ?  Will  of the h i e r a r c h y or o n l y  Are t h e q u e s t i o n s a t t h e more a b s t r a c t  the s u b j e c t s c e i l i n g  out on t h e q u e s t i o n s a t  These q u e s t i o n s , and those s p e c i f i c a l l y  from the d e s i g n , w i l l be addressed i n Chapter S i x .  formulated  45  Chapter  Five  R e s u l t s o f t h e Study The r e s u l t s o f t h e main study w i l l be p r e s e n t e d and a n a l y z e d i n t h i s chapter.  Firstly,  the analyses of variance w i l l  be d i s c u s s e d ;  s e c o n d l y , whether a h i e r a r c h y i s t e n a b l e g i v e n the r e s u l t s o b t a i n e d i n the study w i l l be c o n s i d e r e d ; and t h i r d l y , t h e evidence statistical  concerning  v a l i d a t i o n o f t h e h y p o t h e s i z e d h i e r a r c h y o f s k i l l s w i l l be  examined. The  first  s e r i e s o f a n a l y s e s s e t out t o e s t a b l i s h whether or not  c e r t a i n b a s i c assumptions u n d e r l y i n g t h e t e s t were t e n a b l e .  As the  s u b j e c t s were not nested by sex w i t h i n t h e l e v e l o f t h e t e s t , i t was important  t h a t each s e t o f items w i t h i n each l e v e l o f t h e h i e r a r c h y  t h a t was attempted difficulty.  by each group of c h i l d r e n be o f the same l e v e l o f  I f some items proved  t o be harder t o answer than some  o t h e r s , i t would not be p o s s i b l e t o assess t h e impact variable.  I f f o r i n s t a n c e , items i n one row proved  items i n t h e other rows, and f o u r g i r l s row,  i t would be i m p o s s i b l e t o determine  o f sex as a  t o be e a s i e r  answered t h e q u e s t i o n s on t h a t whether t h e g i r l s  found some  items e a s i e r than d i d t h e boys, o r whether t h e items themselves a c t u a l l y more d i f f i c u l t . it  than  were  As w e l l as a n a l y z i n g the items w i t h i n each row,  was necessary t o a n a l y z e t h e items w i t h i n each column t o e s t a b l i s h  whether o r d e r e f f e c t s were p r e s e n t .  I f questions are presented  ina  c e r t a i n o r d e r , i t may make i t e a s i e r t o answer t h e q u e s t i o n s t h a t f o l l o w than i f t h e q u e s t i o n s a r e presented i n another  order.  46 R e s u l t s presented i n Table 10 i n d i c a t e t h a t n e i t h e r the items w i t h i n row or columns were s i g n i f i c a n t l y d i f f e r e n t t o one another. However, t h e r e was a s i g n i f i c a n t i n t e r a c t i o n between the items i n the rows and the columns. Table 10 Summary o f the A n a l y s i s of V a r i a n c e o f t h e Sequence o f Items W i t h i n the Test  Degrees o f Freedom  Sum of Squares  Sequence (S)  6  2.946  0.491  1.257  Groups o f Items (G)  6  0.946  0.158  0.780  Persons (P:S)  21  8.202  0.391  -  Items (I:G)  14  3.167  0.226  2.771  S. x G.  36  30.030  0.834  3.960 *  S. x I:G  126  26.548  0.210  -  G. x P:S  84  8.167  0.972  1.190  P:S x I:G  292  24.000  0.816  -  Total  585  104.006  Source  Mean Square  F  Sequence = Sequence o f items attempted by s u b j e c t s Item Sets = Sets o f t h r e e items  * p < .05  T h i s i n t e r a c t i o n appears most l i k e l y t o have been caused by the low s c o r e s o b t a i n e d a t L e v e l 1 ( A b s t r a c t l e v e l ) o f the s k i l l  hierarchy.  Both the row and column means were h i g h , but w i t h i n each row and column  47 one s c o r e - i n v a r i a b l y f o r t h e problem below t h a t o f the o t h e r s .  r e p r e s e n t i n g L e v e l 1 - dropped  F i g u r e 3 i l l u s t r a t e s t h a t even t h e h i g h e s t  mean o b t a i n e d (0.667 on column 5) was below t h e lowest mean s c o r e f o r any row o r column. The second a n a l y s i s o f v a r i a n c e was c a r r i e d the  out t o a s s e s s whether  e s t a b l i s h e d l e v e l o f d i f f i c u l t y c r i t e r i a f o r the items had been  validated.  The items were p l a c e d i n one o f t h r e e c a t e g o r i e s ,  Moderately D i f f i c u l t , or Hard,  (see Table 1 ) .  Results presented i n  Table 11 showed t h a t l e v e l o f d i f f i c u l t y was s i g n i f i c a n t . no s i g n i f i c a n t  The mean s c o r e s f o r t h e items were  a c c u r a c y on t h e Easy Items,  moderate d i f f i c u l t y , and  68?o  78?o  a c c u r a c y on t h e items o f  a c c u r a c y on t h e items c o n s i d e r e d d i f f i c u l t ) .  As i n d i c a t e d i n T a b l e 12, t h e s u c c e s s r a t e f o r a l l l e v e l s high.  group  on the b a s i s o f t h e f o u r e s t a b l i s h e d c r i t e r i a (see  Chapter Three, Item D i f f i c u l t y ) . (82%  There was  d i f f e r e n c e between sexes and the items w i t h i n each  were d i f f e r e n t i a t e d  similar  Easy,  o f items was  T h i s f a c t c o u l d account f o r t h e s i m i l a r i t y of t h e easy and  moderately d i f f i c u l t  items.  With so few e r r o r s made, those items  answered i n c o r r e c t l y  have a s i g n i f i c a n t  ship of the three sets of items.  effect  upon t h e r e l a t i v e  relation-  Another p o s s i b l e reason f o r t h e  s i m i l a r i t y noted between these items e x i s t s .  Item d i f f i c u l t y was  d e f i n e d a c c o r d i n g t o f o u r c r i t e r i a ; having grouped  t h e 21 items i n t h r e e  rows a c c o r d i n g t o these c r i t e r i a - , i t i s c o n c e i v a b l e t h a t the t h r e e rows might not be a b s o l u t e l y d i s t i n c t from one a n o t h e r .  An item c l a s s e d as  easy, c o u l d q u i t e w e l l have been put with t h e items o f moderate d i f f i c u l t y . T h i s d i d i n f a c t happen with one problem  -8+1.  A l s o , i t must be  remembered t h a t o n l y a s m a l l sampling o f t h e items o c c u r r e d .  Often a  00 •3-  Sequence 1 VSequence 2  co CD CO  Sequence 3  Q.  Sequence 4  c o  co CD  Sequence 5  t-i  CD  Sequence 6  c  co  CD  Sequence 7  Item Sets Figure  3.  Mean responses to Item Sets at each l e v e l of the h i e r a r c h y (  Task L e v e l One o f the h i e r a r c h y )  49  small  sample  results  in  only  slight  differences  Table Summary Difficulty  L e v e l of Item Difficulty (L)  Item  to  the  (C) (I)  L  are  of  Item  of  Variance  Four  Established  Sum o f Squares  0.612  0.612'  2  1.962  0.981  26  12.034  0.463  -  18  2.418  0.134  0.768  2  0.133  0.663  0.823  -  C  x  L:S  52  4.190  0.806  S x  I:L  18  3.949  0.219  C x  I:SL  468  81.918  0.175  587  107.216  = sex  of  child of  F  1  x  Sex  discernible.  Criteria  Mean Square  S  Total  not  11  Analysis  Degrees of Freedom  (S)  Child  the  According  Source  Sex  of  that  Level  = Level  Item  Child Items  = Boy o r G i r l = Three A d d i t i o n  0.132 *12.177  1.253  -  * Difficulty Items  p <  .05  Table 12 Mean Performance by Sex and Item  L e v e l of D i f f i c u l t y  Female  Easiest  0.847  Items  Difficulty  Male  ( 1 )  (by  Means difficulty)  0.80D  0.821  Moderately Difficult Items  0.800  0.765  0.781  Hard  0.673  0.694  0.684  0.772  0.752  Items  Means (by gender)  *  '*  0.762  Grand Mean (1)  There were 7 items i n each group and t h e s c o r e s under 'Male' and 'Female' r e p r e s e n t t h e mean of these s c o r e s  51 The t h i r d a n a l y s i s o f v a r i a n c e was undertaken t o e s t a b l i s h whether any p a r t i c u l a r s e t o f i t e m s was any more d i f f i c u l t other s e t ,  once they had been b a l a n c e d .  item o f moderate d i f f i c u l t y ,  t o answer than any  Each s e t had one easy i t e m , one  and one hard i t e m (see T a b l e 2 ) .  The  r e s u l t s o f t h i s a n a l y s i s which a r e p r e s e n t e d i n Table .12 i n d i c a t e t h a t no group o f i t e m s proved t o be o f any g r e a t e r d i f f i c u l t y  than any o t h e r  group. Table 13 A Summary o f the A n a l y s i s of V a r i a n c e o f Item Difficulty  Source  f o r B a l a n c e d Groups o f Items  Degrees of Freedom  Sum o f Squares  Sex (S)  1  0.184  0.184  0.132  Items ( I )  6  2.286  0.381  0.355  26  36.102  1.389  6  6.102  1.017  C x I:S  156  167.327  1.073  Total  195  212.001  Child  (C)  S x I  Mean Square  F  0.948  52 The f o u r t h a n a l y s i s o f v a r i a n c e was e f f e c t e d t o determine i f t h e r e was s t a t i s t i c a l  evidence  o f a h i e r a r c h y of any type  the h i e r a r c h y as h y p o t h e s i z e d ) .  (not n e c e s s a r i l y  The r e s u l t s o f t h i s a n a l y s i s a r e  d e t a i l e d i n Table 14. Table 14 A Summary of  t h e A n a l y s i s of  Hypothesized  Source  Degrees of Freedom  V a r i a n c e of t h e  Hi e r a r c h y  Sum o f Squares  Mean Square  F  Sex (S)  1  0.184  0.184  Level of Hierarchy (li')  6  68.071  11.345  26  36.102  1.389  -  6  2.316  0.386  0. 5718  C x L:S  156  105.327  Total  195  212.000  C h i l d (C) S x L  0. 132 16. 804 *  *  p<..05  The r e s u l t s o f the a n a l y s i s p r o v i d e d a s t r o n g i n d i c a t i o n o f t h e e x i s t e n c e of  a hierarchy of s k i l l s . Further analyses considered  the hypotheses as they r e l a t e d  t o the  s e v e n - l e v e l l i n e a r h i e r a r c h y , and a l s o as they r e l a t e d t o t h e f o u r - l e v e l branching  hierarchy.  The r e s u l t s expressed  as mean responses t o the  item a t t h e seven t a s k l e v e l s of the h i e r a r c h y and moving from  task  l e v e l seven t o t a s k l e v e l one were: 2.07;  0.93.  2.71;  2.61;  2.57;  2.68;  With the e x c e p t i o n o f l e v e l 4 t h e r e appeared  progression of d i f f i c u l t y  t o be a  from l e v e l one t o l e v e l s e v e n , where l e v e l  was the most a b s t r a c t and l e v e l seven the most c o n c r e t e . difference  2.43;  However,  this  was s t r o n g l y e v i d e n t o n l y when l e v e l s one and two were compared  w i t h the o t h e r f i v e l e v e l s .  The i t e m s at t h e s e l e v e l s were  apparently  much harder than the items a t any o f the o t h e r f i v e l e v e l s , and the a t l e v e l one were a p p a r e n t l y  much more d i f f i c u l t  chy: A b s t r a c t , 0 . 9 3 ; Concrete, 2.66.  Abstract-Concrete, 2.07;  hierar-  S e m i - C o n c r e t e , 2 . 5 5 ; and  As w i t h the seven s t a g e l i n e a r h i e r a r c h y  to be a p r o g r e s s i o n o f d i f f i c u l t y  items  than those a t l e v e l two.  A second a n a l y s i s compared the f o u r l e v e l s o f t h e b r a n c h i n g  appeared  one  from the l e v e l s  there hypothesized  t o be A b s t r a c t to those h y p o t h e s i z e d to be C o n c r e t e , w i t h the more A b s t r a c t l e v e l s p r o v i n g t o be t h e more d i f f i c u l t . A b s t r a c t - C o n c r e t e l e v e l s were c l e a r l y d i f f e r e n t different  The A b s t r a c t and  to one a n o t h e r , and  from the S e m i - C o n c r e t e and Concrete l e v e l s .  the Semi-Concrete l e v e l appeared  to be harder  t h e r e was v e r y l i t t l e d i f f e r e n c e  i n the r e s u l t s  Both the r e s u l t s  However, a l t h o u g h  than the C o n c r e t e  level,  of these l e v e l s .  of the a n a l y s i s o f v a r i a n c e t h a t c o n s i d e r e d  h i e r a r c h y , and the r e s u l t s  o f the d e s c r i p t i v e a n a l y s i s supported  p o s s i b i l i t y t h a t a h i e r a r c h y was f e a s i b l e .  In order to t e s t  h y p o t h e s i s s t a t i s t i c a l l y the d a t a was s u b j e c t e d  t o a Guttman Scalogram  The a n a l y s i s  c o n s i d e r e d the number of items c o r r e c t l y answered out o f the p o s s i b l e a t each t e s t l e v e l .  In e f f e c t ,  the  the  A n a l y s i s which c o n s i d e r e d the p o s s i b i l i t y o f both a s e v e n - l e v e l h i e r a r c h y and a f o u r - l e v e l b r a n c h i n g h i e r a r c h y .  the  linear  also three  t h i s meant t h a t t h e c u t - o f f  54 p o i n t f o r p a s s i n g a Task L e v e l o f t h e h i e r a r c h y was a r b i t r a r i l y s e t a t 1 out o f 3, 2 out o f 3, or 3 out o f 3 items. c u t - o f f p o i n t s was c o n s i d e r e d  Each o f these P a s s - F a i l  f o r t h e s e v e n - l e v e l h i e r a r c h y andtthe f o u r -  l e v e l hierarchy respectively. Of t h e s i x a n a l y s e s a viable hierarchy.  undertaken only two produced evidence  At t h e c u t - o f f l e v e l o f two both the s e v e n - l e v e l  h i e r a r c h y and the f o u r - l e v e l branching of r e p r o d u c i b i l i t y  h i e r a r c h y had high  (0.92 and 0.95 r e s p e c t i v e l y ) .  the s e v e n - l e v e l l i n e a r h i e r a r c h y , however, r e v e a l e d l e v e l s as h y p o t h e s i z e d analyzed.  t o support  coefficients  C l o s e r examination o f that the various  task  d i d not emerge i n t h e same o r d e r when s t a t i s t i c a l l y  The r e s u l t s o f t h i s a n a l y s i s a r e presented  i n Table 15.  Table 15 R e s u l t s o f Guttman Scalogram A n a l y s i s Seven L e v e l s  Hypothesized Hierarchy  Re-Ordered Hierarchy (seven l e v e l s )  Number o f Subjects Who Passed  Symbols  Symbols  10  Tallies  Tallies  20  Matches  Fish  Shapes  Number o f Subjects Who F a i l e d 18  Percentage Percentage of S u b j e c t s of S u b j e c t s Who Passed Who F a i l e d 36  64  8  71  29  25  3  89  11  Matches  25  3  89  11  Fish  Crayons  26  2  93  7  Blocks  Blocks  27  1  96  4  Crayons  Shapes  27  1  96  4  Number o f S u b j e c t s  = 28  C o e f f i c i e n t o f R e p r o d u c i b i l i t y 0.918  Task l e v e l s one and two appeared two  i n the h y p o t h e s i z e d o r d e r ; Task l e v e l s  t o seven, however, d i d n o t .  t a s k s , d i d not remain sized.  Examination  " F i s h " and "Matches", both  Semi-Concrete  i n the same r e l a t i o n s h i p t o one another as hypothe-  o f t h e data showed t h a t 25 s u b j e c t s passed  each  level  and 3 f a i l e d ;  the d i f f e r e n c e between the t a s k s a t t h e s e two l e v e l s ,  then,  was s l i g h t .  "Shapes" was h y p o t h e s i z e d t o be a Semi-Concrete  a c c o r d i n g t o the t h e o r e t i c a l premises proposed, "Crayons"  should have been e a s i e r .  task;  " F i s h " , " B l o c k s " and  The s u b j e c t s d i d , i n f a c t , f i n d t h e  t a s k s presented with "Shapes" as easy as those p r e s e n t e d with " B l o c k s " . The s t r o n g e s t evidence t o s u p p o r t the h y p o t h e s i s o f a h i e r a r c h y o c c u r r e d when the c u t - o f f p o i n t o f 2 was used with t h e f o u r - l e v e l branching h i e r a r c h y . Table 16.  Examination  The r e s u l t s o f t h i s a n a l y s i s a r e p r e s e n t e d i n of t h e data i n d i c a t e t h a t the h i e r a r c h y was  ordered i n t h e manner h y p o t h e s i z e d - Concrete t o A b s t r a c t .  The d i f f e r e n c e  between t h e A b s t r a c t and A b s t r a c t - C o n c r e t e l e v e l s , and t h e A b s t r a c t Concrete and Semi-Concrete percentage  l e v e l s was c l e a r - c u t ; 36, 71, and 89  o f s u b j e c t s r e s p e c t i v e l y passed  each o f t h e s e t h r e e l e v e l s .  However, t h e d i f f e r e n c e between the Semi-Concrete although apparent, was not as s t r i k i n g r e s p e c t i v e l y passed  these two l e v e l s .  and Concrete  - 89 and 96 percentage  presented a t these l e v e l s , and t h i s their  relationship.  of subjects  As the r e s u l t s of t h e seven-  l e v e l l i n e a r h i e r a r c h y showed, a c l e a r d i s t i n c t i o n between and Concrete t a s k s was not apparent.  levels,  Few s u b j e c t s f a i l e d  Semi-Concrete tasks  ' c e i l i n g ' e f f e c t may have a f f e c t e d  Table 16 R e s u l t s o f Guttman Scalogram  Analysis:  Four L e v e l s  Hypothesized Hierarchy  Guttman Scalogram Hierarchy  Abstract  Abstract  10  18  36  64  AbstractConcrete  AbstractConcrete  20  8  71  29  SemiConcrete  SemiConcrete  25  3  89  11  Concrete  27  1  96  4  Concrete  Number o f Subjects Who Passed  Number o f S u b j e c t s = 28 Coefficient  of R e p r o d u c i b i l i t y :  0.95  Number o f Subjects Who F a i l e d  Percentage Percentage of Subjects of Subjects Who Passed Who F a i l e d  57 Chapter Six D i s c u s s i o n , L i m i t a t i o n s and  Recommendations  Research Questions posed i n Chapter Two questions  as w e l l as  pertinent  a r i s i n g from the P i l o t T e s t w i l l be addressed i n t h i s  chapter  i n the l i g h t of t h e o r e t i c a l assumptions o u t l i n e d i n Chapter Three, r e s u l t s presented of v a r i a n c e  i n Chapters Four and  Five.  Results  analyses  i n d i c a t e d t h a t f a c t o r s such as the sex o f the c h i l d ,  d i f f i c u l t y , order  item  e f f e c t w i t h i n the t e s t , and|the r e l a t i v e d i f f i c u l t y  groups of items w i t h i n the t e s t , were not s i g n i f i c a n t ; concerning  of the  and  thus,  questions  the v a l i d a t i o n of a h i e r a r c h y of s k i l l s supported by  n i f i c a n t v a r i a t i o n s i n success  a t each l e v e l w i l l be  of  sig-  considered.  Discussion A l i n e a r h i e r a r c h y of s k i l l s does not appear d e f e n s i b l e . o f v a r i a n c e . i n d i c a t e d a s i g n i f i c a n t amount of v a r i a n c e i n the  Analysis data  a t t r i b u t a b l e to the h i e r a r c h y , but gave no i n d i c a t i o n as to the of h i e r a r c h y r e s p o n s i b l e . showed t h a t t h e r e was  Examination of the means f o r each  indeed  "Symbols".  v a r i a t i o n was.noted between Task l e v e l t h r e e and the mean.for Task l e v e l f o u r was six.  The  higher  However, very  Task l e v e l seven  than t h a t of Task l e v e l s  Guttman Scalogram A n a l y s i s supported the i d e a , o f a  h i e r a r c h y but re-arranged ( T a b l e . 14).  level  a l i n e a r p r o g r e s s i o n moving from Task  l e v e l seven "Crayons" to Task l e v e l one  and  type  The  the items from the order  f a c t t h a t so few  of the  to argue f o r the  of a l i n e a r r e l a t i o n s h i p between these t a s k s .  and five linear  hypothesized.  s u b j e c t s f a i l e d any  from t h r e e to seven makes i t very d i f f i c u l t  little  levels existence  While i t i s p o s s i b l e t h a t  the  linear  pattern  occurred  in  tests  not  do  these  hierarchy same  five  have  A second  does  a  of  to  Taken  individually,  point  in  The a  is  one Task  hierarchy  on  the  "Shapes"  cut-off  of  two  correct  tasks,  "Blocks"  and  data  pooled,  the  are  the  hypothesized  discussion subjects  there Task  of  is  levels to  three  seven  to  Guttman  between  the  Task  is  is  at  the  two.  of  evidence  Scalogram 2 out  Level  Abstract  of  and  one  at  a  items  are  at  they  analyzed  automatically Guttman  on  at  far  out  three  is  employed.  In  terms  of  levels,  for  use  The  hierarchy,  in  and the  i t  branching  could of  the  the  tasks  far  Concrete  with be  the  in  the  the  way  as  with  too  levels  the  few making  Certainly  that  hierarchy,  the  if  exception  argued  when  both  however,  that  same  The  than  difficulty, is  the  easiest  discussion.  hierarchy,  fact,  the  harder  of  the  Analysis.  by  be  the  statistically.  Scalogram  to  of  maintain  remain  appeared  these  level  be  of  that  a l l .  given  that  effect  to  factor the  the  another  tasks  Semi-Concrete  in  'ceiling'  appear  support  with  of  there Task  is levels  level.  to  support  Analysis 3 items  (Table  that  they  the  linear  In  further  be  constructed.  c o n s i d e r e d one  the  a  hand,  confounding  and  strongest  failing  other  the  difficulty  not  responses  linearity  The  point  do  of  to  not  when  sub-test  seven-stage  support  cut-off  levels  hierarchy  clear  evidence  another  some  a  or  equal  there  items  one  whether  tasks.  effects no  the  well  relationship  "Shapes"  the  failed  'ceiling'  from  on  may  according to  items  "Crayon"  i t  sufficiently  relationship  the  a r i s e because  tests,  linear  question  are  not  15).  of  the  concept  the  Four-level  correct There  of  as  a  basis  was  a  very  Abstract-Concrete,  and  a  hierarchy  hierarchy for  passing  clear  between  arose  using or  distinction  the  Abstract-  a  59 Concrete and the Semi-Concrete  levels.  as c l e a r between the Semi-Concrete of the reasons proposed hierarchy.  However, the d i s t i n c t i o n was  and Concrete l e v e l s , perhaps  i n the d i s c u s s i o n of the seven-stage  The C o e f f i c i e n t  enough to p r o v i d e s t a t i s t i c a l A review of the p i l o t  of R e p r o d u c i b i l i t y of 0.95  not  f o r one  linear  i s marginally high  validation.  t e s t l e d t o support f o r the i d e a s of both a  l i n e a r p r o g r e s s i o n from Cpncrete to A b s t r a c t and the p o s s i b i l i t y of a four-level hierarchy. both of these i d e a s .  Data  from the main study s u p p o r t s , to some e x t e n t ,  I t i s important t o note t h a t i t i s not p o s s i b l e t o  g e n e r a l i z e from the r e s u l t s of the p i l o t  t e s t t o the main study  o n l y seven c h i l d r e n were i n v o l v e d i n the p i l o t  test.  v a l u a b l e i d e a s arose which were worth c o n s i d e r i n g . d i f f e r e n c e s between the r e s u l t s of the p i l o t  because  However, some One  of the major  t e s t and the main study  was  the f a c t t h a t so many c h i l d r e n answered most of the q u e s t i o n s on the main study on items a t Task L e v e l s 3 t o 7.  On the p i l o t  t e s t no s u b j e c t  r e c e i v e d more than 11 out of 21 items c o r r e c t on the " B l o c k s " s u b - t e s t . For a l l the o t h e r s u b - t e s t s the mean s c o r e s were w e l l below 50 p e r c e n t of the t o t a l p o s s i b l e .  As the seven s u b j e c t s chosen  from the r e s e a r c h pool of 49 c h i l d r e n who  f o r the P i l o t  qualified  Test were  f o r the study, i t seems  u n l i k e l y t h a t the c h a r a c t e r i s t i c s o f t h i s group would be d i f f e r e n t t o those chosen  f o r the main study.  The major d i f f e r e n c e i n the two  situations  i s t h a t the s u b j e c t s i n the main study were t a u g h t , w h i l e those i n the p i l o t t e s t were not.  The percentage of s u b j e c t s who  l e v e l of the h i e r a r c h y , comparing noteworthy: 96%,  93%,  l e v e l one:  passed a t each  the Main Study and the P i l o t T e s t ,  main study from task l e v e l seven to t a s k l e v e l one:  89%, 89%, 71%, 36%; 38%,  pilot  was  96%,  t e s t from task l e v e l seven t o t a s k  52.4%, 28.6%, 28.6%, 23,8%, 0%,  0%.  60 I t appears t h a t t e a c h i n g  was  e f f e c t i v e at a l l l e v e l s of the  p a r t i c u l a r l y at task l e v e l two t o 11%.  The  most important  " T a l l i e s " , where the change was  task l e v e l s one  and  two  At 100%,  and,  the change i s i n f i n i t e because the p i l o t One  deserves mention: i t i s d i f f i c u l t  to a s s e s s  f a c t o r other  than  the extent  at test  teaching, to which  s u b j e c t s understood the i n s t r u c t i o n s on the-.-pilot t e s t .  The  the  pilot  test  i n p a r t , used as a v e h i c l e f o r s t a n d a r d i z i n g the i n s t r u c t i o n s f o r  the main study. was  results.  f i v e the change i s over 200%;  b a s e l i n e s were 0% i n each case.  was,  task,  seven the percentage change i s approximately  at task l e v e l s t h r e e , f o u r and  0%  from  p o i n t i s t h a t the more a b s t r a c t the  the g r e a t e r i s the percentage i n c r e a s e i n the main study task l e v e l s s i x and  hierarchy,  I f comprehension of the i n s t r u c t i o n s on the p i l o t  poor, t h i s may  from the p i l o t  have depressed the s c o r e s .  t e s t was  Abstract-Concrete  Another q u e s t i o n  and  Results i n d i c a t e d that  s u b j e c t s were most l i k e l y ready f o r the type of symbolic i n these  emerging  whether or not the t a s k s a t the A b s t r a c t  L e v e l s were t e a c h a b l e .  test  teaching  the involved  tasks.  L i m i t a t i o n s of the  Study  From the d i s c u s s i o n thus f a r , i t i s apparent t h a t the high r a t e of success  on the more c o n c r e t e  t a s k s , which r e s u l t e d i n many s u b j e c t s  f i n d i n g the items a t s e v e r a l of the Task L e v e l s too easy, i s a cant f a c t o r i n t h i s study.  The  f a c t o r s of t e a c h i n g , and  signifi-  the l a c k of  a c t u a l d i f f e r e n c e i n the t a s k s themselves, have a l r e a d y been mentioned; it  i s p e r t i n e n t to c o n s i d e r a few  p o o l was  125  additional points.  s u b j e c t s , e v e n t u a l l y reduced to 49,  f o r both the p i l o t  t e s t and  the main study  p o p u l a t i o n were e l i m i n a t e d from the study.,  The  initial  from which the  were chosen., The  subject  subjects  Thus 60% of  s e l e c t i o n of the  49  the  subjects  was a r r i v e d a t - with t h e e x c e p t i o n  o f s i x who c o u l d  already  answer f o u r or f i v e q u e s t i o n s a t the most a b s t r a c t l e v e l of the h i e r a r chy  - by g i v i n g a p r e t e s t which has many developmental c h a r a c t e r i s t i c s  to i t .  Many 5 year o l d c h i l d r e n , f o r i n s t a n c e ,  are unable t o conserve  d i s c r e t e e n t i t i e s which they w i l l have no d i f f i c u l t y  conserving  year l a t e r .  It i s conceivable  who formed the  research  had c h a r a c t e r i s t i c s which would d i s t i n g u i s h them' from the  pool  t h a t the 49 s u b j e c t s  one  76 not chosen, o t h e r than the a b i l i t y t o s u c c e s s f u l l y complete the t a s k s p r e s e n t e d on the p r e t e s t .  Two v a r i a b l e s which c o u l d  would be s o c i o - e c o n o m i c s t a t u s and i n t e l l i g e n c e .  be  considered  It i s possible  that  the c h i l d r e n who formed the r e s e a r c h  p o o l were from a h i g h e r  socio-  economic c l a s s than those who f a i l e d  the p r e t e s t ; i t i s a l s o  possible  t h a t they were more i n t e l l i g e n t  than those who f a i l e d the p r e t e s t .  The  f a c t o r of i n t e l l i g e n c e i s p a r t i c u l a r l y r e l e v a n t t o t h e i s s u e of ' c e i l i n g o u t ' on t h e t a s k s  designed.  I t may be, f o r i n s t a n c e ,  that  b r i g h t e r c h i l d r e n do not r e q u i r e t a s k s p r e s e n t e d i n a very c o n c r e t e form such as " B l o c k s "  or "Crayons" and t h a t any form o f c o n c r e t e  e i t h e r two-dimensional, or t h r e e - d i m e n s i o n a l , w i l l s u f f i c e .  presentation Many  c h i l d r e n who e x p e r i e n c e d i f f i c u l t y with c a r d i n a l numbers r e s o r t t o t h e use  of c o n c r e t e a i d s such as f i n g e r s .  from t h i s study by the p r e t e s t . previous research,  These c h i l d r e n were  As mentioned i n t h e d i s c u s s i o n of  i t i s not p o s s i b l e t o d i s c u s s  c h i l d r e n without c o n s i d e r i n g  eliminated  skill  hierarchies for  developmental c h a r a c t e r i s t i c s .  This i s  p a r t i c u l a r l y t r u e f o r c h i l d r e n a t 5 years o f age where c o g n i t i v e i n g and s t r a t e g i e s a r e r a p i d l y changing.  think-  Recommendations f o r Future The  Research  Guttman Scalogram A n a l y s i s v a l i d a t e d the f o u r l e v e l  u s i n g two  hierarchy  out of t h r e e items as the p a s s - f a i l c u t - o f f p o i n t .  c o e f f i c i e n t of r e p r o d u c i b i l i t y was significant. f i n d i n g has  The any  question  relevance  0.95  which was  statistically  t h a t a r i s e s i s the extent  to e d u c a t i o n .  The  The  to which  this  f o r m u l a t i o n of an  opera-  t i o n a l d e f i n i t i o n of numbers, as o u t l i n e d by Resnick (1970) suggests t h a t a d d i t i o n equations  are at the top of the h i e r a r c h y - t h a t i s , at  the h i g h e s t l e v e l of d i f f i c u l t y . present  The  r e s u l t s of the p r e s e n t  the p o s s i b i l i t y of t a k i n g a d d i t i o n equations  and  study  presenting  them at t h r e e d i f f e r e n t l e v e l s of d i f f i c u l t y b e f o r e a c t u a l l y p r e s e n t i n g them a b s t r a c t l y .  R e s u l t s f o r the h i e r a r c h y as p r e s e n t l y c o n s t i t u t e d  would need r e p l i c a t i n g b e f o r e i t c o u l d j u s t i f i a b l y be used i n an educational s e t t i n g .  A second i s s u e which a f f e c t s the arrangement of  items i n the form of a h i e r a r c h y i s item d i f f i c u l t y . some support  f o r the c r i t e r i a of d i f f i c u l t y of a d d i t i o n items as  l i n e d i n Chapter Three. 4 l e v e l h i e r a r c h y with Further research t h i s study.  R e s u l t s showed  I t would be p o s s i b l e , then,  out-  to c o n c e i v e  3 l e v e l s of item d i f f i c u l t y a t each  of a  level.  i s i n d i c a t e d g i v e n the i n f o r m a t i o n a v a i l a b l e from  R e p l i c a t i o n of the r e s u l t s u s i n g a l a r g e r p o p u l a t i o n  kindergarten  c h i l d r e n would be v a l u a b l e , to determine i f the Semi-  Concrete and  Concrete l e v e l s c o u l d be d i s t i n g u i s h e d simply  of  by i n c r e a s i n g  the number of s u b j e c t s .  A l a r g e r p o p u l a t i o n of s u b j e c t s would a l s o  a l l o w the v a r i a b l e of age  t o be s t u d i e d i n more d e t a i l .  It i s possible,  f o r i n s t a n c e , g i v e n the r e s e a r c h a v a i l a b l e about developmental v a r i a b l e s t h a t younger k i n d e r g a r t e n  c h i l d r e n may  perform d i f f e r e n t l y than o l d e r  kindergarten c h i l d r e n . socio-economic  Future r e s e a r c h might a l s o c o n s i d e r the  s t a t u s and  intelligence.  Other p o s s i b l e areas  i n v e s t i g a t i o n i n c l u d e s t u d i e s of grade one  of  c h i l d r e n p r i o r to exposure  t o f o r m a l t e a c h i n g , comparisons of e a r l y grade ones with and  role  kindergarten  comparisons of r e s u l t s from these areas with other r e s e a r c h t h a t  p r o v i d e s data r e l e v a n t to the h i e r a r c h i c a l and s t a n d i n g of the concept  of numbers.  developmental under-  REFERENCES Bateman, B. D.  E s s e n t i a l s of t e a c h i n g  San R a f a e l ,  California:  Dimensions, 1971. Bereiter, C ,  & Engelmann, S.  Englewood C l i f f s , N.J.: Brainserd, C. J . first  Inducing  Teaching disadvantaged c h i l d r e n i n p r e s c h o o l . P r e n t i c e - H a l l , 1966.  o r d i n a l and c a r d i n a l r e p r e s e n t a t i o n s  f i v e n a t u r a l numbers.  Psychology  J o u r n a l of E x p e r i m e n t a l C h i l d  1974, 18, 520-534.  Bruner, J . S., O l v e r , R. R., & G r e e n f i e l d , P. M. C o g n i t i v e Growth. and  of the  In J . S. Bruner ( E d . ) , New  Studies i n York:  John W i l e y  Sons, 1967.  B u e l l , R. R.  P i a g e t i a n theory  & D. C. Rubadeau theory.  (Eds.), Educational  Toronto, O n t a r i o :  Edwards, A. L.  i m p l i c a t i o n s of P i a g e t ' s  i n psychological research.  New  & Winston, 1968.  R e l a t i o n s h i p between p s y c h o l o g i c a l t h e o r i e s and t h e a c t  of t e a c h i n g . Engelmann, S.  In J . I . Athey  Xerox, 1970.  Experimental design  York: H o l t , Rinehart Engelmann, S.  into inquiry action.  J o u r n a l o f School  Psychology, 1967.  Conceptual l e a r n i n g .  San R a f a e l ,  California:  Dimensions 1969a. Engelmann, S. Science  Preventing  Research A s s o c i a t e s  Gagne', R. M. & P a r a d i s e , ledge  f a i l u r e i n the primary grades.  acquisition.  (Whole No. 518).  N. E.  Chicago:  1969b. A b i l i t i e s and l e a r n i n g s e t s i n know-  P s y c h o l o g i c a l Monographs, 1961a, 75,-  65 Gagne'', R. M., & Brown, L. T. conceptual 62,  learning.  Some f a c t o r s i n t h e programming o f  J o u r n a l o f Experimental  Psychology, 1961b,  313-321.  Gagne , R. M. /  69,  The a c q u i s i t i o n o f knowledge, P s y c h o l o g i c a l Review, 1962a,  355-365.  Gagne*, R. M., Mager, J . R.,  Garstens,  H. L., & P a r a d i s e , N. E.  F a c t o r s i n a c q u i r i n g knowledge of a mathematical  task.  P s y c h o l o g i c a l Monographs, 1962, 76, 7_ (Whole No. Gagne'', R. M.  The l e a r n i n g requirements f o r e n q u i r y .  526). Journal  Research  i n s c i e n c e t e a c h i n g , 1963, 1, 144-153. Gagne*, R. M.  Problem s o l v i n g . I n A. W. Melbon ( E d . ) , C a t e g o r i e s o f  human l e a r n i n g . Gagne', R. M.  New York:  Some f a c t o r s i n l e a r n i n g non-metric geometry.  Development, 1965, Gagne*, R. M.  Academic P r e s s , 1964.  30,  Child  42-49.  C o n t r i b u t i o n s o f l e a r n i n g t o human development.  P s y c h o l o g i c a l Review, 1968, 75, 177-191. Gagne'', R. M.  The c o n d i t i o n s o f l e a r n i n g .  Holt, Rinehart  & Winston,  1970. Gronlund, H. E.  C o n s t r u c t i n g an achievement t e s t .  Englewood  Cliffs,  New J e r s e y : P r e n t i c e - H a l l , 1970. Karplus,  R.  Ashley  The s c i e n c e c u r r i c u l u m :  Improvement study.  In I . J .  & D. C. Rubadeau ( E d s . ) , E d u c a t i o n a l i m p l i c a t i o n s o f P i a g e t ' s  theory. Kilpatrick, J. programme.  Toronto, O n t a r i o : C o g n i t i v e theory In I . J . Ashley  Serox, 1970. and t h e s c h o o l mathematics study & D. C. Rubadeau ( E d s i ) ,  i m p l i c a t i o n s of Piaget's theory.  Toronto, O n t a r i o :  group  Educational Xerox, 1970.  Mager, R.  F.,  & C l a r k , C.  instruction. M i l l e r , S. A.  Explorations i n student-controlled  P s y c h o l o g i c a l Reports, 1963,  13,  Non-verbal assessment of c o n s e r v a t i o n  Development, 1976, Resnick, L. B.  47,  of number.  Resnick, L. B.  Development C e n t r e ,  Learning  U n i v e r s i t y of P i t t s b u r g h ,  Behaviour a n a l y s i s i n c u r r i c u l u m  design:  sequenced i n t r o d u c t o r y mathematics c u r r i c u l u m . Research and  Child  722-728.  Design of an e a r l y l e a r n i n g c u r r i c u l u m .  Research and  Learning  71-76.  Development Centre,  1967.  A hierarchically  Monograph  U n i v e r s i t y of  2,  Pittsburgh,  1970. Saxe, G. B.  A developmental a n a l y s i s of n o t a t i o n a l c o u n t i n g .  Development, 1977, S i e g e l , L. S.  The  in preschool Uprichard,  48,  C,  Developmental Psychology 1971,  A. E. & P h i l l i p s , E. R.  An i n t r a c o n c e p t  A v a l i d a t i o n study.  Mathematics E d u c a t i o n , Wong, M.  1512-1520.  Sequence of development of c e r t a i n number concepts  children.  number a d d i t i o n :  Resnick, L. B.,  1977,  Q_,  1767-1778.  5_, 357-361.  a n a l y s i s of  rational  J o u r n a l f o r Research i n  7-16.  & Boozer, R. F.  of some e a r l y mathematics behaviours. 42,  Child  The  sequence of development  C h i l d Development,  1971,  APPENDIX A  MEASURING INSTRUMENT  B a s i c a d d i t i o n f a c t s . ( u s i n g t h e numbers one t o n i n e sums a r e no l e s s than s i x and no g r e a t e r  than n i n e .  (1)  1 + 5  (5)  5 + 1  (2)  1 + 6  (6)  6 + 1  (3)  1 + 7  (7)  7 + 1  (4)  1 + 8  (8)  8 + 1  (9)  2 + 4  (13)  4 + 2  (10)  2 + 5  (14)  5 + 2  (11)  2 + 6  (15)  6 + 2  (12)  2 + 7  (16)  7 + 2  (17)  3 + 3  (21)  4 + 3  (18)  3 + 4  (22)  5 + 3  (19)  3 + 5  (23)  6 + 3  (20)  3 + 6  (24)  4 + 4  (26)  5 •+ 4  (25)  4 + 5 Total:  26 F a c t s  inclusively)  APPENDIX B INSTRUCTIONS  Task L e v e l 7. E t o S:  FOR THE POSTTEST  (Three items were g i v e n ) .  Here a r e two s e t s o f crayons  (E p o i n t s ) .  J o i n these two s e t s of crayons t o make one s e t . How many crayons a r e t h e r e now? S responds: T h i s w i l l be repeated f o r items 2 and 3. Task L e v e l 6.  (Three items w i l l be g i v e n ) .  Same as Task L e v e l 7. Task L e v e l 5. E t o S:  S u b s t i t u t e blocks f o r crayons.  (Three i t e m s ) .  Here a r e two s e t s o f f i s h two s e t s o f f i s h  (E p o i n t s ) .  I f you j o i n e d  t o make one new s e t , how many f i s h  these  will  t h e r e be? S responds: Task L e v e l s 4 and 3 w i l l be t h e same. matchsticks f o r f i s h . Task L e v e l 2. E t o S:  and then  (Three items a t each t a s k l e v e l ) .  (Three i t e m s ) .  Here a r e two numbers which I want you t o add t o g e t h e r (E p o i n t s ) . you.  I f you wish you may use the t a l l y marks t o h e l p  When you add these numbers t o g e t h e r , how many a r e  t h e r e now? S responds:  S u b s t i t u t e shapes  69  Task L e v e l 1. E t o S:  (Three i t e m s ) .  Here a r e two numbers t h a t I want you t o add t o g e t h e r . How many do they make when you add them? begins t o use h i s f i n g e r s , t h e experimenter t h a t t h i s i s not a l l o w e d ) .  S responds:  ( I f the subject should  indicate  APPENDIX PRETEST  Each  child  These  8,  Each  child  it  were  2,  Each  with  presented  7,  9,  was shown  verbally.  numbers  The  was p r e s e n t e d  numbers  3,  ENTERING  1,  6,  the test  had t o  experimenter  If  the subject  by  giving  orally said  recite  to  the  series.  If  complete  E:  Would (No  E.  hesitated,  either  could  The  order  5  4, at  inclusively.  a time  a child  and asked  had to  to  identify  identify  a l l nine  correctly.  child  No  one t o n i n e  i n a random  help error  he asked  then  count  placed  these  Subject  counts  blocks.  of  the subject  i n order  inclusively  nine  help  up  whether to  again  subject  numbers  the  repeat  to  the  the f i r s t  in  subject the  series.  please.  time) to  pass.  s i x blocks before  blocks.  you.  nine  count  could  was u n s u r e  up t o t h e number  was p e r m i t t e d  one t o  Please  or both  i s permissible at this  experimenter Please  second  the experimenter  you count  f o r me.  the experimenter  the f i r s t ,  the task  t h e numbers  subject:  I would l i k e you to count t h e number n i n e .  the  BEHAVIOURS  t h e numbers  one number  To p a s s  C  Count  the  out loud  subject.  so that  I  can hear  71  E.  How many a r e t h e r e ?  Subject The  responds.  examiner then spread the b l o c k s out so t h a t they covered  surface area. E.  the question  again.  How many a r e t h e r e now?  Subject E.  Then the examiner asked  a greater  responds.  Are t h e r e more b l o c k s now?  Subject  responds.  (The task was repeated b l o c k s were used.  i f error.occurred.  The next  time, however, f o u r  I f the s u b j e c t got t h e c o r r e c t answer, one more item  was g i v e n with seven b l o c k s t o make sure the second response guess.)  was not a  APPENDIX D INSTRUCTIONS FOR  Task L e v e l 7 ( r e a l On  was  4+4).  i n the o t h e r . E:  objects)  the t a b l e were two  (item one  The  s e t s of crayons i n f r o n t of each There were f o u r p e n c i l s i n one  experimenter s a i d t o the s e t s of p e n c i l s (E  Here are two pencils  ( p o i n t s ) and  Watch me  TEACHING  while  Subjects E:  Now  (E  Now  (E  I count t h i s s e t of p e n c i l s . side.)  Now  p o i n t s to the s e t on the s u b j e c t s ' l e f t  watch w h i l e (E  set.  I am  going  to each and Now  T h i s i s one  s e t of  t h i s i s another s e t of p e n c i l s ( p o i n t s ) .  I count the other  counts you  count  this  side).'  you  set.  Now  you  count your  p o i n t s to each s u b j e c t ' s r i g h t h a n d to take t h i s s e t  upside-down to the s u b j e c t s ) and  E:  four  responded.  other E:  s e t and  subjects:  points).  demonstration s e t on h i s r i g h t h a n d set  child  (E's  s e t i n turn)..  l e f t h a n d s e t because i t  j o i n i t to t h i s s e t .  (E  was  points  then j o i n s them-.).!  take t h i s s e t  (subjects' righthand  s e t ) and  j o i n i t to  this set. Subjects E: Ei  responded.  Now, obtained  p e n c i l s and  how  many crayons are t h e r e  altogether?  a v e r b a l response from each s u b j e c t .  E.  gave the c o r r e c t answer to the s u b j e c t s .  when applicable,)).  then counted h i s ( P r a i s e was  used  73  The procedure E:  demonstrations was  Here  as  are  two  sets  (points)  Now  many  left))  omitted  from  items  two  and  three.  The  one  of  follows:  crayons how  were  Now  of and  crayons how  crayons  (E  this  another  are  many  in  is  points).  there? this  set  (E  This of  crayons  points  set?  (E  is  to  points  set to  set  (E on  set  points). subjects'  on  subjects',  right.)). E:  Now  you  join  this  set  (on  subjects'  right)  to  this  set  (on  subjects'  left): Subjects E:  responded.  How  Subjects  E:  There  Level  This  was  Task  Level  E:  are  Here  were are  E:  to  crayons  task  (E  shown  two  points  First,  altogether?  altogether.  (Praised  if  appropriate).  level  except  7  crayons  were  replaced  by  the  card  with  of  fish  (E  sets  to  group  points count  on  left  the  group  this  set  of  item  points).  of  to  the  on  on  i t .  This  is  card)  and  this  the  right  of  is the  one  set  another  of  blocks.  fish  set  card).  fish.  responded.  Now,  Subjects  there  5  fish  E:  seven  identical  subjects  Subjects  are  6  (E  E:  crayons  responded.  Task  The  many  count  this  set  of  fish.  responded.  Now  if  this  how  many  set  fish  is  are  added there  to  this  set  altogether?  (E  points  to  each  set),  of  Subjects  E.  responded.  counts  (Repeat Task by  them  supplies correct  for  items  two  Levels  3 and  A were  'shapes'  Task  and  Level  and  E:  We a r e  going  number  on  hand can  card  the  5 except  'fish'  were  together  (E  replaced  f i r s t  want  the  solve  for  Task  Level  E.to  Subjects:  add to  and  the  lefthand these  under  the  number  two  points  and  numbers  numbers.  two  and  these  two  numbers  help  you  get  give  using the  items  numbers  us add  marks  marks  then  to  the  together,  (E  each rightwe  demonstrates  their  tally  together.  an answer  to  I the  want  you  to  question.  answers.  marks  and  gives  the  correct  answer.  three).  1 Do  you  we u s e d t a l l y together look  to  items  numbers  help  two  number).  you  the  first  tally  tally  subjects.  add. t h e s e  To  the  use  you  same a s  the  card,  the  I  (Repeated  to  count  Now  the  before  number).  with  E- a d d s  the  2  placed the  Subjects  three).  'matchsticks'.  E.  E:  and  answer.  at add  marks  these this i t  to  remember  two  number this  to  that  help  numbers (E  us? and  points  number  last  (E  time This  tell to  the  points  we a d d e d time  me t h e  I  the  want  you  answer.  lefthand to  numbers  number)  other  together  to  First, and  number).  add you then How  75  many do they make a l t o g e t h e r ? Subjects  responded.  E-adds the numbers the second.  out l o u d by c o u n t i n g  (e.g.  5 + 3 - 5-6,7,8)-  (Repeated f o r items two and t h r e e ) .  on from the f i r s t answer  8)  number t o  

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