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Automaton and competence aspects of Piagetian logical concepts Toussaint, Nelly Adelina 1972

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AUTOMATON AND COMPETENCE ASPECTS OF PIAGETIAN LOGICAL CONCEPTS , by NELLY ADELINA TOUSSAINT B.A., U n i v e r s i t y M.A., U n i v e r s i t y  of T u l s a , 1968  o f B r i t i s h Columbia, 1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Psychology  We accept t h i s required  thesis  as conforming  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1972  In p r e s e n t i n g an the  thesis  advanced degree at Library  I further for  this  shall  agree  scholarly  by  his  of  this  written  the  in p a r t i a l  fulfilment of  University  of  make i t f r e e l y that permission  p u r p o s e s may  representatives. thesis for  be It  financial  available  for  for extensive  g r a n t e d by  the  i s understood gain  permission.  Depa r t m e n t The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  11  Abstract  Simultaneous measures were obtained of automaton v a r i a b l e s  abilities  s i z e of computing space) t h e o r e t i c a l l y presumed  to be necessary to the (b) the the  Ss'  from Grade 1 and  ( m u l t i p l i c a t i o n of c l a s s e s ,  2 served as Ss.  The  automaton demands.  were i n c o r p o r a t e d i n t o the  t e s t i n g methods.  across a l l the theoretical tested.  a strong relationship tasks.  The  overall  conceptions about the  Some of the  these f i n d i n g s  are  theoretical  discussed.  aspects of The  results Ss  1  indi-  performances  support P i a g e t ' s  relationships and  distinc-  intelligence  between the  findings  dif-  competence  C o n s i d e r a t i o n s about the  figurative  cated there was  Children  task demands of the  t i o n between o p e r a t i v e and  to  multi-  transitivity).  l o g i c a l tasks were equated i n terms of the  as tvell as  and  l e v e l of attainment i n l o g i c a l tasks b e l o n g i n g  p l i c a t i o n of r e l a t i o n s , s e r i a t i o n , and  of  hindsight  attainment of l o g i c a l s t r u c t u r e s ,  same conceptual f a m i l y  ferent  and  Ss ' l e v e l  attainment on and  (foresight  (a) the  empirical  o f the  variables  implications  of  iii TABLE OF CONTENTS Page ABSTRACT  i i  LIST OF TABLES  iv  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  v i i  CHAPTER I .  Introduction  CHAPTER I I .  T h e o r e t i c a l Framework o f D i a g n o s t i c Procedure  1  10  CHAPTER I I I . Method  50  CHAPTER IV.  Results  85  CHAPTER V.  Discussion  REFERENCES APPENDIX. ADDENDUM  110 133  FORHIN Sets  137 144  iv L i s t o f Tables Table 1. 2i 3*  Page C o r r e l a t i o n s between FORHIN scores and Operative (0) and F i g u r a t i v e (F) measures f o r Gl t n d G2  86  C o r r e l a t i o n s between MOPER scores and Operative (0) and F i g u r a t i v e (F) measures f o r Gl and G2  86  R e l a t i o n s h i p between the TRAN scores and the scores below and above the medians o f the FORHIN, MOPER, MCMUL(O), MRMUL(O), and SERDIM(O) scores f o r G l and G2 . .  87  4.  I n t e r c o r r e l a t i o n s between Operative (1,2,3) and F i g u r a t i v e (4,5,6) measures f o r G l and G2...  91  5.  I n t e r c o r r e l a t i o n s between measures d e r i v e d from Operative measures (1-5) and those d e r i v e d from F i g u r a t i v e measures (6-10)  92  I n t e r c o r r e l a t i o n s between the F i l l - i n and A n t i c i p a t o r y measures and the Operative scores f o r G l and G2  93  P r o p o r t i o n o f Ss o b t a i n i n g MCFILL and MRFILL scores g r e a t e r than 2/3 o f the maximum p o s s i b l e (12/18)  94  P r o p o r t i o n o f Ss i n Grade 1 and Grade 2 o b t a i n i n g scores g r e a t e r than o r equal to 4 i n the MC and MR A n t i c i p a t i o n tasks  96  I n t e r c o r r e l a t i o n s o f the d i f f e r e n c e between F i g u r a t i v e and Operative scores f o r the MC, MR, and SER tasks and a l l the F i g u r a t i v e and Operative scores f o r G l and G2.....  99  P r o p o r t i o n o f Ss i n G l and G2 whose FORHIN score f e l l above o r below the median and whose MC and MR Reverse o r Operative scores i n c r e a s e d or decreased itfith r e f e r e n c e t o t h e i r corresponding Copy o r F i g u r a t i v e s c o r e . . .  100  I n t e r c o r r e l a t i o n s between the F i l l - i n and A n t i c i p a t o r y measures and the F i g u r a t i v e scores f o r G l and G2  101  Means, SD's, d i f f e r e n c e s between the means, t - t e s t s , and s i g n i f i c a n c e l e v e l s o f the measures obtained f o r G l and G2  103  Number o f u n i t s o f i n f o r m a t i o n that Gl Ss were able to handle s u c c e s s f u l l y i n the FORHIN, MOPER, MC, MR, and SER t a s k s .  105  6.  7.  8.  9.  10.  11.  12.  13.  V  Table 14.  15.  Page Number o f u n i t s o f i n f o r m a t i o n that G2 Ss were able t o handle s u c c e s s f u l l y i n the FORHIN, MOPER, MC, MR, and SER tasks  106  P a i r w i s e comparisons o f the scores below and above the medians o f the MOPER, FORHIN, SERDIM(O), and m u l t i p l i c a t i v e combinations i n MR and MC(MUL) scores f o r Gl and G2  107  vi L i s t of Figures Figure 1.  Page M u l t i p l i c a t i o n of Classes matrix (a) and r e v e r s e d matrix (b). Dimensions remaining constant on the h o r i z o n t a l axis s i z e , diamond, c o l o r ; and along the v e r t i c a l axis background, t h i c k n e s s , and shape  53  M u l t i p l i c a t i o n of R e l a t i o n s matrix (a) and r e v e r s e d matrix (b). Dimensions remaining constant along h o r i z o n t a l axis s i z e , o r i e n t a t i o n , c o l o r ; and along v e r t i c a l axis t h i c k n e s s , brightness,, and l i n e s . . .  61  S e r i a t i o n task. Stimuli varied in height, width, t h i c k n e s s , eyes' r o t a t i o n , width of f r o c k , and p o s i t i o n of t i e . . . . . . . . . . . .  64  T r a n s i t i v i t y task. The r e l e v a n t c h a r a c t e r i s t i c s to the t r a n s i t i v i t y judgments were height and c o l o r of bottom h a l f ( " s h i r t " ) of " l i t t l e men"  68  5.  Foresight-Hindsight  72  6.  M-Operator task. (a) S t i m u l i and response u n i t s (b) Sample cards of t r a i n i n g s e t 1 (c) Sample cards of t r a i n i n g set 2 (d) Sample cards of compound s t i m u l i cards  2.  3.  4.  task.  Training set.....  79  vii Acknowledgements  The  author would l i k e to express  her g r a t i t u d e t o the  chairman o f the d o c t o r a l committee, Dr. Lou Moran, f o r p r o v i d i n g constant  a s s i s t a n c e and encouragement, as w e l l as a most  able and s t i m u l a t i n g i n t e l l e c t u a l atmosphere.  enjoy-  S p e c i a l thanks  are due t o the other members o f the committee, Dr. C h r i s Tragakis and Dr. Meredith  Kimball f o r t h e i r help and counsel  the v a r i o u s phases o f t h i s work.  throughout  The author a l s o wishes t o  acknowledge the c o - o p e r a t i o n and a s s i s t a n c e o f the Vancouver School Board and the p r i n c i p a l and s t a f f members o f the General Gordon Elementary  School, Our Lady o f P e r p e t u a l Help  and York House School.  School,  In a d d i t i o n , the author wishes to  thank Ms. Bernice Wong f o r p r o o f r e a d i n g the d i s s e r t a t i o n ; Mr. Jon Moran, f o r the photography work; Mr. G o d f r i e d T o u s s a i n t , f o r the drawing o f f i g u r e s ; and Ms. Rosanne Rumley f o r the e x c e l l e n t t y p i n g work. the support graduate  T h i s work has been made p o s s i b l e with  o f a N a t i o n a l Research C o u n c i l o f Canada Post-  S c h o l a r s h i p t o the author.  CHAPTER  I  Introduction Recently  psychologists  i n t e r e s t e d i n c o g n i t i v e development  have t r i e d t o understand the antecedent p s y c h o l o g i c a l  processes  o r mechanisms which make p o s s i b l e the development o f the c h i l d ' s ability  to p e r f o r m symbolic and a b s t r a c t o p e r a t i o n s  o u t l i n e d i n Piaget's  theory.  have concerned r e s e a r c h e r s  o f the k i n d  Two o f the major q u e s t i o n s  are:  that  (1) What are the mechanisms o f  t r a n s i t i o n from one P i a g e t i a n stage o f development t o another? With r e f e r e n c e the q u e s t i o n  t o a more s p e c i f i c concept such as  conservation,  i s , what f a c t o r s f a c i l i t a t e the t r a n s i t i o n from an  u n d e r s t a n d i n g o f the q u a l i t a t i v e aspects o f the s t i m u l i ( p r e l o g i c a l thinking)  to an u n d e r s t a n d i n g o f the q u a n t i t a t i v e  rela-  t i o n s as w e l l  (logical thinking)?  ( 2 ) What g e n e r a l i z e s between  stage-related  c o n c e p t s , o r what can be expected t o g e n e r a l i z e to  r e l a t e d s i t u a t i o n s from the kno\<rledge t h a t the c h i l d has a t t a i n e d a c e r t a i n concept?  This l a t t e r question  can a l s o be posed i n  r e c i p r o c a l terms ( t h a t r e f l e c t the r e a l i t i e s borne out by the data):  what are the reasons f o r the f a i l u r e t o o b t a i n the expected  high within-subjects belonging The  i n t e r - c o r r e l a t i o n s among tasks  presumably  t o the same developmental l e v e l ? l a c k o f adequate answers t o these problems i s i n p a r t a  manifestation diagnosing  o f the developmental p s y c h o l o g i s t s '  cognitive states.  There are very  difficulties in  few s a t i s f a c t o r y t o o l s  to determine l e v e l s o f c o g n i t i v e s t r u c t u r e p r i o r t o or d u r i n g the attainment o f l o g i c a l concepts.  In t u r n , the l a c k o f e f f e c t i v e  d i a g n o s t i c procedures based on P i a g e t ' s two  pervasive  problems:  first,  theory  i s a reflection of  the l a c k o f consensus as t o the  2  conceptual  meaning of major c o n s t r u c t s such as "stage"  and a l l  i t s r e l a t e d concepts (synchrony, t r a n s i t i o n , i n t e r - i t e m r e l a t i o n s h i p s , e t c . ) ; and  second, a d i r e c t consequence of the f i r s t ,  the  l a c k of consensus as to the o p e r a t i o n a l c r i t e r i a that should  be  used to render meaningful and comparable e m p i r i c a l measures of the  constructs. One  goal of the present  procedure to assess  study was  to devise  a diagnostic  q u a n t i t a t i v e l y the c h i l d ' s a t t a i n e d l e v e l  competence i n logico-mathematical  operations  through procedures  which are based on P i a g e t ' s t h e o r e t i c a l a n a l y s i s of how development of t h i s type of s t r u c t u r e o c c u r s . c o n s t r u c t i o n of the procedure, of course, Piaget's  t h e o r e t i c a l formulations  s t a t e d at the o u t s e t .  Before  the  The  analysis  directly  reflects  on the two  major  and  questions  e l a b o r a t i n g the s p e c i f i c t h e o r e t i -  c a l and e m p i r i c a l bases o f the d i a g n o s t i c procedure, i t w i l l p r o f i t a b l e to d i s c u s s some o f the problems found i n the t u r e on the v a r i o u s  e f f o r t s to deal w i t h questions  of the c o g n i t i v e concepts d e f i n e d by Piaget's ( B r a i n e , 1959; c u l t one  Wohlwill,  be  litera-  of assessment  Piaget.  been p o i n t e d out by numerous authors  1966;  F l a v e l l , 1953,  etc.), is a  diffi-  to o p e r a t i o n a l i z e i n unambiguous terms, p a r t i c u l a r l y  the more one general  system, as has  of  i s d e a l i n g w i t h more t h e o r e t i c a l l y s i g n i f i c a n t  constructs  ( f o r i n s t a n c e , the  so  and  s p e c i f i c concept of "con-  s e r v a t i o n of substance," has been more f r e q u e n t l y t e s t e d than the construct of " e q u i l i b r a t i o n . " )  I t i s a l s o the case that i n many  i n s t a n c e s the theory has been s e r i o u s l y misrepresented p r i a t e l y applied.  or  inappro-  To s u b s t a n t i a t e t h i s a s s e r t i o n , three major  types of inadequate treatments o f the theory w i l l be  illustrated.  3  F i r s t , one common reason f o r i n a p p r o p r i a t e l y d e f i n e d  opera-  t i o n a l c r i t e r i a i s the f a c t that many authors have d e a l t with d i f f e r e n t but i n t i m a t e l y r e l a t e d aspects  o f the theory  sepa-  r a t e l y o r i n an i s o l a t e d manner, when i n f a c t the a n a l y s i s at hand may r e q u i r e a simultaneous c o n s i d e r a t i o n o f s e v e r a l types o f variables.  T h i s problem i s r e f l e c t e d i n the f a i l u r e o f many  researchers  to d i s t i n g u i s h the Competence and Performance  o f the theory. was f i r s t (1969). formal  aspects  T h i s d i s t i n c t i o n , borrowed from p s y c h o l i n g u i s t s ,  a p p l i e d to developmental data by F l a v e l l and Wohlwill The competence aspects  l o g i c a l operations  underlying  are s a i d to correspond to the  and s t r u c t u r e s which Piaget o u t l i n e s as  the d i f f e r e n t stages  o f development, f o r example, the  mathematical groupings which c h a r a c t e r i z e the c o g n i t i v e development o c c u r r i n g between seven and eleven years formance aspects "automaton"  1  o f age.  The per-  correspond t o the p s y c h o l o g i c a l mechanisms or  f u n c t i o n s which permit the competence d e s c r i b e d by  P i a g e t to be expressed i n any s p e c i f i c s i t u a t i o n , e.g., a t t e n t i o n a l , memory, p e r c e p t u a l , or other y e t undefined functions.  The importance o f the Competence-Automaton  t i o n i s t h a t i t b r i n g s i n t o focus simultaneously be)  u l a r l y i n reference  J  distinc-  the n e c e s s i t y o f c o n s i d e r i n g  (by c o n t r o l l i n g or m a n i p u l a t i n g ,  both o f these aspects  logical  non-structural  as the case might  o f l o g i c o - c o g n i t i v e development, p a r t i c -  to questions  concerning  the a c q u i s i t i o n o f  concepts.  The word "automaton" w i l l be used, i n preference to the word "performance," to r e f e r to the s u b j e c t ' s f u n c t i o n s that t h e o r e t i c a l l y are s a i d t o make the expression o f a given competence p o s s i b l e . The word "performance" i s reserved f o r a c t u a l b e h a v i o r a l i n s t a n c e s r e f l e c t i n g the given competence.  4  The  s i g n i f i c a n c e o f the competence-automaton  distinction  becomes more obvious through the i l l u s t r a t i o n o f problems that have a r i s e n when the d i s t i n c t i o n i s not made or when only one aspect o f i t i s taken i n t o account when s t u d y i n g the a c q u i s i t i o n o f l o g i c a l concepts.  The c h i l d who understands  c o n s e r v a t i o n of  l i q u i d s may a s s e r t or deduce that the l i q u i d remains d e s p i t e the changes i n appearances  because  the same  n o t h i n g has been added  or taken away -- a d d i t i o n - s u b t r a c t i o n argument; o r because i f the water i s poured back to the o r i g i n a l  c o n t a i n e r , i t would look  the same -- r e v e r s i b i l i t y argument; or because  the i n i t i a l equal-  i t y i s what i s important -- i d e n t i t y argument; o r because  changes  i n one dimension are compensated by changes i n the other  dimension  -- compensation  argument.  According to P i a g e t , use o f these  arguments to j u s t i f y the a s s e r t i o n of c o n s e r v a t i o n i n d i c a t e s  that  the c h i l d has achieved the competence necessary t o understand that the q u a n t i t y o f l i q u i d remains formations performed  i n v a r i a n t because  the t r a n s -  are p a r t o f a system o f c o - o r d i n a t e d and  r e c i p r o c a l operations (Piaget,  1968).  Some i n v e s t i g a t o r s o f the competence o f c o n s e r v a t i o n (Smedslund, 1966; Wallace, W a l l , and Anderson,  1967; R o l l , 1970)  have t r i e d t o e l u c i d a t e the p r e c u r s o r f a c t o r s i n the development of the grouping s t r u c t u r e s by b r i n g i n g about petence  the necessary com-  i n non-conserving c h i l d r e n through the t r a i n i n g o f  e m p i r i c a l procedures based on the p o s t u l a t e d o p e r a t i o n s o f the groupings themselves  (the s u b j e c t s are s p e c i f i c a l l y taught opera-  t i o n s such as r e v e r s i b i l i t y ,  addition-subtraction, etc.).  The  r e s u l t s o f these procedures have been mixed ( B e i l i n , 1971). ever, the value o f t h i s approach  i s put i n q u e s t i o n by data  How-  5  (Wallach,  Wall,  and Anderson, 196 7; . a l l a c h , 1969 ; P i a g e t , 1969; T  r  B e i l i n , 1971) that i n d i c a t e that t r a i n i n g techniques based on the operations  o f the f i n a l  directly  grouping s t r u c t u r e s are i n -  s u f f i c i e n t to b r i n g about c o n s e r v a t i o n .  In f a c t , a pre-conserva-  t i o n c h i l d may be f u l l y aware (through t r a i n i n g or other means) o f a l l or some o f the e m p i r i c a l changes i n v o l v e d i n the s t i m u l i situation  (e.g., that i f the water i s poured back i n t o the o r i g i n a l  container,  i t would look the same as b e f o r e ,  able t o a s s e r t that there  i s conservation  etc.)  and y e t not be  o f substance or l i q u i d .  In other words, the competence represented  i n a s p e c i f i c struc-  t u r a l concept (grouping) i s the end r e s u l t or f i n a l  convergence  o f the development o f a m u l t i p l i c i t y o f automaton f u n c t i o n s , among which are those that make p o s s i b l e the connection e m p i r i c a l events i n a l o g i c a l that focus  of related  (grouping) f a s h i o n .  Thus, approaches  only on the competence aspects o f c o g n i t i v e development  when t r y i n g to uncover how l o g i c a l concepts are a c q u i r e d , provide  inadequate and i n s u f f i c i e n t answers to t h i s  may  question.  This i s because the attainment o f a l o g i c a l competence i n v o l v e s a v a r i e t y of psychological  functions  (automaton v a r i a b l e s ) beyond  those that may be immediately suggested by the c h a r a c t e r i s t i c s of the l o g i c a l operations  themselves.  The fundamental problem i s  the p r e c i s e s p e c i f i c a t i o n o f these c r u c i a l automaton f u n c t i o n s . A c o n t r a s t i n g problem a r i s e s when the focus  of inquiry i s  e x c l u s i v e l y on automaton v a r i a b l e s , as has been the case i n c e r t a i n t r a i n i n g procedures conservation  (Gelman, 1969) which have attempted to teach  by d e a l i n g only w i t h v a r i a b l e s such as the c h i l d ' s  a t t e n t i o n t o r e l e v a n t dimensions. indeed help  a t t a i n conservation,  Even though such procedures may they s t i l l  leave unanswered the  6  q u e s t i o n o f why the procedure important because  i s effective.  automaton-oriented  This question i s  procedures have been found  to be e f f e c t i v e only w i t h c h i l d r e n around or past f i v e years o f age but not w i t h younger ones ( B e i l i n , 1969).  Thus, the ante-  cedents o f the c o n s e r v a t i o n concept are not e x p l a i n e d by such t r a i n i n g procedures; they merely demonstrate  some o b s t a c l e s t h a t  must be removed and some procedures that are e f f e c t i v e  i n removing  them once the c h i l d has the b a s i c concepts necessary f o r conservation . A second common reason f o r the inadequate o p e r a t i o n a l  criteria  of P i a g e t i a n concepts i s obvious m i s r e p r e s e n t a t i o n s or misunders t a n d i n g o f the c o n s t r u c t s .  A t y p i c a l example o f P i a g e t i a n  experiments w i t h b a s i c conceptual c o n f u s i o n are those o f Bruner, et a l . (1966).  Bruner r e j e c t s the e x p l a n a t i o n from P i a g e t ' s theory  t h a t compensation because  and r e v e r s i b i l i t y are r e s p o n s i b l e f o r c o n s e r v a t i o n  he contends  phenomena.  that the n r n - c o n s e r v i n g c h i l d understands  However, the non-conserving c h i l d may understand  a p h y s i c a l t r a n s f o r m a t i o n can b e r e v e r s e d ( a n ' e m p i r i c a l resulting i n equality  such that  return '), 1  ( i . e . , he may understand t h a t i f the water  i s poured back i t w i l l b e the same as before) without understanding o p e r a t i o n a l r e v e r s i b i l i t y , which e n t a i l s the use o f such  empirical  knowledge to j u s t i f y l o g i c a l l y the q u a n t i t a t i v e t r a n s f o r m a t i o n s involved i n conservation.  S i m i l a r l y , there i s a d i s t i n c t i o n  between the c h i l d ' s understanding o f " f u n c t i o n a l c o v a r i a t i o n s " -which may allow him to p r e d i c t , f o r i n s t a n c e , that the water l e v e l i n a t h i n g l a s s w i l l be h i g h e r than that o f the same water poured i n t o a wider glass  -- which  o p e r a t i o n a l compensation,  again i s e m p i r i c a l knowledge, and  which  also i n v o l v e s an understanding o f  7  the l o g i c a l q u a n t i t a t i v e r e l a t i o n s i n v o l v e d i n c o n s e r v a t i o n . A t h i r d problem that has r e s u l t e d i n inadequate o p e r a t i o n a l c r i t e r i a i s the tendency o f many authors t o t r a n s l a t e  Piaget  a c c o r d i n g to t h e i r own i d i o s y n c r a t i c conceptual systems. has r e s u l t e d i n two types o f problems.  This  F i r s t , Piaget's constructs  are i n t e r p r e t e d and t r a n s l a t e d i n t o other conceptual systems i n order to e x p l a i n the c o n s e r v a t i o n phenomena through other mechanisms when i n f a c t the n e c e s s i t y and b a s i s f o r the t r a n s l a t i o n i s not warranted or c l e a r l y j u s t i f i e d .  And second, P i a g e t ' s con-  s t r u c t s are used to deal w i t h b e h a v i o r s other than the ones i n t e n ded to be subsumed under the s p e c i f i c c o n s t r u c t s . the f i r s t problem, p o i n t e d out by B e i l i n and F u l l e r t o n  One example o f  (1971a), i s the H a l f o r d  (1970) experiment i n x^hich a t r a i n i n g technique was  used \irhich presumably was based on n o n - P i a g e t i a n c o n s t r u c t s but which i n f a c t c o u l d very e a s i l y be c o n s i d e r e d to be a P i a g e t i a n based procedure.  In the H a l f o r d and F u l l e r t o n  (1970) experiment  the same " r e v e r s i b i l i t y ' procedure used by Wallach (1969) -- a 7  P i a g e t i a n - b a s e d experiment i n a t i o n " and 'sets.'  1  -- i s d e s c r i b e d i n terms o f " d i s c r i m -  The authors a t t r i b u t e the moderately  p o s i t i v e success which was evidenced i n the p o s t - t e s t s not to r e v e r s i b i l i t y but to the s e t s " induced by the d i s c r i m i n a t i o n ::  procedure.  An example o f the second problem i s shown i n the  Mehler and Sever (1967) and Sever, Hehler, and F.pstein (1968) s t u d i e s that r e p o r t e d to have found number c o n s e r v a t i o n i n two-year o l d c h i l d r e n , a f a c t that would be against P i a g e t ' s stage theory. However, subsequent r e s e a r c h ( B e i l i n , 1968; Rothenberg 1969)  and Courtney,  has not r e p l i c a t e d such r e s u l t s and i n f a c t the task that  they used has been shown not to be a number c o n s e r v a t i o n  test.  8  Two c o n c l u s i o n s can be drawn from the above examples o f inadequate  Piagetian-based  developmental  criteria.  problems such as those  F i r s t , itfhen d e a l i n g w i t h i n v o l v e d i n determining how  a c e r t a i n l o g i c a l competence i s a c q u i r e d , or i n how to assess an a t t a i n e d l e v e l o f competence, i t i s necessary  to take i n t o  account  s i m u l t a n e o u s l y , not only the s u b j e c t ' s use o f the s p e c i f i c competence r u l e i n q u e s t i o n , but also h i s l e v e l of attainment ous  in vari-  other f u n c t i o n s (automaton v a r i a b l e s ) which may indeed make  p o s s i b l e the s u b j e c t ' s understanding  o f the competence r u l e .  n e c e s s i t y f o r t h i s type o f " m u l t i v a r i a t e " approach t o phenomena has become i n c r e a s i n g l y recognized 19 70; F l a v e l l , 1970a; Ayers, 1971).  The  developmental  (Klahr and Wallace,  As w i l l be shown l a t e r , an  assessment o f P i a g e t ' s i n d i v i d u a l c o n s t r u c t s based on h i s o v e r a l l theory demands such a m u l t i d i m e n s i o n a l  approach.  Second, i f one wishes to i n v e s t i g a t e P i a g e t i a n c o n s t r u c t s on t h e i r own grounds and w i t h i n t h e i r ovrn l i m i t s r a t h e r than j u s t to make use c f the c o n s t r u c t s as a source o f hypotheses to be i n t e r p r e t e d a c c o r d i n g to one's own conceptual necessary  system, i t w i l l be  to i n t e r p r e t a n d o p e r a t i o n a l i z e the c o n s t r u c t s , a c c o r d i n g  to the c r i t e r i a o u t l i n e d b y P i a g e t . approach w i l l  O n l y through  t h i s k i n d of  i t be p o s s i b l e t o devise v a l i d t e s t i n g  procedures  which assess the l i m i t s and strengths o f the P i a g e t i a n c o n s t r u c t s . When the i n t e n t i o n i s to compare P i a g e t s 1  conceptual  w i t h that o f o t h e r s , i t becomes even more necessary  explanations to have accurate  and c l e a r conceptual d e f i n i t i o n s which permit p r e c i s e d i f f e r e n t i a t i o n s between the two c o n t r a s t i n g approaches. V/ith the above c o n s i d e r a t i o n s i n n i n d , the scope o f t h i s can be f u r t h e r e l a b o r a t e d .  study  The major aim was to devise a diagnos-  9  t i c procedure  to assess the f o l l o w i n g r e l a t i o n s h i p s :  (1) the  r e l a t i o n s h i p between (a) the s p e c i f i c b a s i c c a p a c i t i e s or automaton v a r i a b l e s that are necessary f o r the a c q u i s i t i o n of s p e c i f i c logico-mathematical concepts, whose s p e c i f i c f u n c t i o n a l  charac-  t e r i s t i c s d i r e c t l y bear on the types o f mental o p e r a t i o n s i n v o l v e d i n the l o g i c a l o p e r a t i o n s , and those l o g i c a l mathematical  (b) the l e v e l  concepts;  of attainment i n  ( 2 ) the r e l a t i o n s h i p between  (a) a b a s i c automaton v a r i a b l e a l s o p o s t u l a t e d as necessary f o r the a c q u i s i t i o n of logico-mathematical concepts  (Pascual-Leone,  1970) , but whose s p e c i f i c d e f i n i n g c h a r a c t e r i s t i c s do not  directly  bear on the c h a r a c t e r i s t i c s of l o g i c a l o p e r a t i o n s ( i . e . , assessment o f a c o n s t r u c t intended to measure q u a n t i t a t i v e aspects of c o g n i t i v e growth independent  of the nature or content o f the  s p e c i f i c c o n c e p t ) , and again (b) the l e v e l of attainment logico-mathematical concepts; and between the l e v e l s of attainment presumed to develop and of r e l a t i o n s .  i n those  (3) the degree of r e l a t i o n s h i p i n l o g i c a l concepts  theoretically  c o n c u r r e n t l y , i . e . , m u l t i p l i c a t i o n of c l a s s e s  CHAPTER II T h e o r e t i c a l Framework o f D i a g n o s t i c  Procedure  Four aspects o f the d i a g n o s t i c procedure this 1.  are c o n s i d e r e d i n  chapter: The scope o f o p e r a t i v e knowledge that the procedure was designed to tap.  2.  A t h e o r e t i c a l a n a l y s i s of the p s y c h o l o g i c a l processes  postu-  l a t e d as u n d e r l y i n g such o p e r a t i v e knowledge which allowed to decide which aspects of i t should be measured, 3.  A t h e o r e t i c a l a n a l y s i s o f the methods and procedures  which  p r o v i d e d the b e h a v i o r a l measures of the processes d e l i n e a t e d above. 4.  The s p e c i f i c goals o f the d i a g n o s t i c procedure. Scope The  focus o f t h i s  mathematical  i n v e s t i g a t i o n was on a b s t r a c t l o g i c o -  operations ("groupings") such as m u l t i p l i c a t i o n s o f  c l a s s e s and r e l a t i o n s , s e r i a t i o n , e t c . f o r the f o l l o w i n g reasons. F i r s t , the a c q u i s i t i o n o f s t r u c t u r a l concepts  such as the con-  s e r v a t i o n s o f substance, volume, area, e t c . , and i n g e n e r a l , concepts  that f a l l  i n the i n f r a l o g i c a l realm  (deal with  specific  aspects o f the p h y s i c a l world) has been found to be i n f l u e n c e d by d i f f e r e n t i a l experiences.  For example, P r i c e - W i l l i a m s ,  Gordon, and Ramirez (1969) i n d i c a t e d that there was b e t t e r cons e r v a t i o n o f substance  performance ( p a r t i c u l a r l y i n r u r a l  areas)  among p o t t e r s ' c h i l d r e n than among c h i l d r e n of the same s o c i a l l e v e l whose parents were i n other occupations. that c h i l d r e n from pottery-making  The authors  suggested  f a m i l i e s had g r e a t e r experience  11  m a n i p u l a t i n g c l a y , and  this s k i l l  l e a r n i n g , by b r i n g i n g the  n i t i v e mechanisms i n c o - o r d i n a t i o n w i t h o p e r a t i o n s , operative The  leads  to  development. i m p l i c a t i o n seems to be  that the e x p r e s s i o n  of c o g n i t i v e  c a p a c i t y through concepts that r e f l e c t p h y s i c a l phenomena vary  as a f u n c t i o n of f a c t o r s e x t e r n a l  might not be p o s s i b l e to provide jects.  cog-  Since  the  may  to the c h i l d f o r which i t  adequate c o n t r o l across  a l l sub-  i n t e n t of the assessment i s to measure an  a b s t r a c t b a s i c c a p a c i t y independent of the s p e c i f i c content o f the c h i l d ' s experiences,  (material)  i t i s necessary to use  i n which h o p e f u l l y the e f f e c t s o f d i f f e r e n t i a l m a t e r i a l  measures  contents  of experiences are minimized. Second, i n f r a l o g i c a l concepts such as the conservations substance, weight, and despite  volume are a c q u i r e d  at d i f f e r e n t ages  the f a c t that t h e o r e t i c a l l y they i n v o l v e the  operations.  The  from understood.  same l o g i c a l  reasons f o r these time lags ('decalages') are Piaget  himself  of  (1970b, 1971b) has  far  hypothesized  that the c h i l d ' s e f f o r t s at o p e r a t i o n a l l y s t r u c t u r i n g p h y s i c a l r e a l i t y i n t e r a c t w i t h h i s notions  of p h y s i c a l c a u s a l i t y .  In  other words, o p e r a t i o n a l c a p a c i t i e s might not be r e v e a l e d when measured i n a c e r t a i n context  because the causal  the  subject may  a t t r i b u t e to the o b j e c t s , present  the  s t r u c t u r a l quantifications required.  seven years of age x^eight i s conceived  r e l a t i o n s which resistances  For example, u n t i l  to about  as changing from one moment  to another f o r a s i n g l e o b j e c t as soon as i t s s p a t i a l p o s i t i o n causal  f u n c t i o n are changed, even when i t s shape remains  and  constant.  For i n s t a n c e , i n a glass of water the weight of a pebble i s seen as v a r y i n g  according  to whether i t i s at the bottom or at mid-height,  12  etc.  'Decalages' can occur i n the c h i l d ' s m a n i f e s t a t i o n o f h i s  understanding of  physical  o f l o g i c a l o p e r a t i o n s a l s o as a f u n c t i o n o f the type  r e a l i t y that i s used to embody such concepts.  For  example, a time l a g i n v o l v i n g j u s t a change i n m a t e r i a l s i s i l l u s t r a t e d with the c l a s s i n c l u s i o n problem.  When c h i l d r e n are pre-  sented a bunch o f flowers o f v a r i o u s k i n d s , t u l i p s , p r i m u l a s , e t c . and are asked  daisies,  to compare a subset or k i n d w i t h the  the whole bunch (Are there more flov/ers or more t u l i p s ? ) , more than 75% o f them s u c c e s s f u l l y s o l v e the problem at around or  e i g h t years o f age.  seven  However, as Inhelder and Piaget (1964)  have shown, when the c l a s s i n c l u s i o n q u e s t i o n deals w i t h c l a s s e s such as animals  (Are there more b i r d s or more animals?)  the ques-  t i o n i s not s u c c e s s f u l l y answered by the m a j o r i t y o f the c h i l d r e n u n t i l much  later.  Since the goal o f the present d i a g n o s t i c procedure  was to  assess the c h i l d ' s l e v e l of o p e r a t i v i t y and not how that l e v e l o f o p e r a t i v i t y might i n t e r a c t with the c h i l d ' s c a u s a l i t y n o t i o n s about a s p e c i f i c p h y s i c a l domain, i t was necessary then to r e s t r i c t the measures to contents whose c h a r a c t e r i s t i c s minimize  the poss-  i b l e source of 'decalages.' T h e o r e t i c a l P s y c h o l o g i c a l Processes U n d e r l y i n g Development of This t h e o r e t i c a l  Operativity  a n a l y s i s i s intended to i n d i c a t e the b a s i s  from which i t was decided which aspects o f logico-mathematical knowledge were to be assessed.  To reach a d e c i s i o n on t h i s  matter  i t was f i r s t necessary to answer two i n t e r r e l a t e d q u e s t i o n s . What, i n f a c t , i s the c h i l d a c q u i r i n g that makes p o s s i b l e h i s  (1)  13  understanding o f logico-mathematical operations?  And ( 2 ) , how and  what aspects o f that kno\>rledge (competence r u l e , type o f s i t u a t i o n , etc.) are expected to g e n e r a l i z e to other i s thus necessary to review P i a g e t ' s vant t o these  related situations?  t h e o r e t i c a l treatments  It  rele-  questions.  What makes p o s s i b l e the a c q u i s i t i o n o f logico-mathematical concepts? The  knowledge the c h i l d acquires  during  a l p e r i o d has been c h a r a c t e r i z e d by P i a g e t s t r u c t u r e s which he terms "groupings." s t r u c t u r e s represent co-ordinations period.  Piaget's  the  concrete-operation-  through the l o g i c a l  These (incomplete)  f o r m a l i z a t i o n o f the most  of a c t i o n s and operations  logical  general  o f the c h i l d d u r i n g  this  I t i s necessary to analyze f u r t h e r the nature o f these  s t r u c t u r e s to be able to deal more s p e c i f i c a l l y w i t h the f o l l o w i n g question:  (1) What kinds  o f conceptual  c a p a c i t i e s are the s t r u c t u r e s  supposed to encompass; (2) what i s the p s y c h o l o g i c a l nature o f the a c t i o n s that make up the s t r u c t u r e s ; (3) How are the a c t i o n s make up the s t r u c t u r e Logical structures The  co-ordinated?  - t h e i r conceptual  range and f u n c t i o n s  s t r u c t u r e s d e f i n e d by the l o g i c a l groupings are intended  to r e f e r to a c e r t a i n s t a t e o f c o - o r d i n a t i o n by the c h i l d ' s mental a c t i o n s : nated i n t o " t r a n s f o r m a t i o n a l " organization The  that  be d e f i n e d  The mental a c t i o n s become c o - o r d i systems such that the r e s u l t i n g  can be s a i d to have the f o l l o w i n g c h a r a c t e r i s t i c s .  first  co-ordinated  that can be a t t a i n e d  c h a r a c t e r i s t i c , "wholeness," r e s u l t s because, when  w i t h each o t h e r ,  the elements o f the s t r u c t u r e cannot  independently o f the connections i n v o l v e d among them.  14  t h a t i s , the elements are subordinated Piaget,  i n h i s bbok S t r u c t u r a l i s m  to the t o t a l  organization.  (1970), s p e c i f i c a l l y  states:  "...the elements o f a s t r u c t u r e are subordinated to laws * and i t i s i n terms of these laws that the s t r u c t u r e qua whole or system i s d e f i n e d . Moreover, the laws governing a s t r u c t u r e are not r e d u c i b l e to cumulative one-by-one a s s o c i a t i o n of i t s elements; they confer on the whole as such o v e r a l l p r o p e r t i e s d i s t i n c t from i t s e l e ments." ( P i a g e t , 1970a, p. 7) An example o f t h i s c h a r a c t e r i s t i c d i s c u s s e d by Piaget t r a t e s i t s meaning.  illus-  He s t a t e s that the i n t e g e r s do not e x i s t i n  i s o l a t i o n from one another nor were they d i s c o v e r e d  one by one i n  some a c c i d e n t a l sequence and then, f i n a l l y , u n i t e d i n t o a whole. They appear as ordered and t h i s order ated w i t h s t r u c t u r a l p r o p e r t i e s the  of the i n t e g e r s  i s associ-  ( o f groups, f i e l d s , r i n g s , and  l i k e ) which are q u i t e d i f f e r e n t from the p r o p e r t i e s o f i n d i -  v i d u a l numbers, each of which are even or odd, prime or non-prime, and  so on. Second, the " t r a n s f o r m a t i o n a l "  the operations it  C h a r a c t e r i s t i c r e s u l t s from  o r r e g u l a t i o n s which make up the s t r u c t u r e or give  i t s " s t r u c t u r i n g " as a system.  The t r a n s f o r m a t i o n s  p o s s i b l e r e l a t i o n s h i p s among the elements themselves; they are the laws o f composition o f the s t r u c t u r e . l o g i c a l groupings these operations two  are:  define a l l therefore,  For the  (1) Combinativity  - any  elements o f a grouping can be combined and thus produce a new  element o f the same grouping, i . e . , tiro d i s t i n c t c l a s s e s may be combined i n t o one comprehensive c l a s s operation  i m p l i e s a converse o p e r a t i o n ,  (2) R e v e r s i b i l i t y  i . e . , subtraction f o r  addition, division f o r m u l t i p l i c a t i o n , etc. the same outcome may be obtained operations  - each  (3) A s s o c i a t i v i t y -  by the combination o f d i f f e r e n t  (4) General I d e n t i t y - an o p e r a t i o n  combined with i t s  15  converse i s a n n u l l e d , special identities combined with  (e.g., + 1 - 1 = 0 )  (5) Tautology or  - a q u a l i t a t i v e element which i s repeated or  i t s e l f does not y i e l d a new element and i t i s thus  not transformed;  there  i s a " t a u t o l o g y " i n the case o f A + A = A.  T h i r d , a s t r u c t u r e has a c h a r a c t e r i s t i c of " s e l f - r e g u l a t i o n " or " c l o s u r e " by v i r t u e of the transformations  which preserve the  r e l a t i o n a l t o t a l i t y o f the whole.  o f the transforma-  t i o n a l operations  The nature  o u t l i n e d above i s such that they never engender  elements o u t s i d e o f the system. An  important  conception  p o i n t t h a t needs to be s t r e s s e d from the above  o f s t r u c t u r e s and operations  i s that the t r a n s f o r m a t i o n a l  c h a r a c t e r o f the groupings i s the r e s u l t i n g m a n i f e s t a t i o n of psyc h o l o g i c a l processes those  whose c h a r a c t e r i s t i c s are d i f f e r e n t  o f the o p e r a t i o n s  supported  as above d e f i n e d .  This point i s f u r t h e r  by P i a g e t ' s a n a l y s i s o f the formation  ( P i a g e t , 1972; 1971a; 1970a; 1968). embodied i n the l o g i c a l  from  The f i n a l  of structures co-ordinations  groupings are preceded by gradual  changes  i n " s e l f - r e g u l a t o r y " f u n c t i o n s which e v e n t u a l l y become the operations.  P i a g e t p o i n t s out that the term " o p e r a t i o n " , used i n  reference to l o g i c a l s t r u c t u r e , i s r e s e r v e d f o r those  functions  which, from a c y b e r n e t i c p o i n t o f view, are " p e r f e c t " r e g u l a t i o n s . That i s , the o p e r a t i o n a l system excludes made because every  e r r o r s before  they are  o p e r a t i o n has i t s i n v e r s e i n the system  s u b t r a c t i o n i s the i n v e r s e o f a d d i t i o n ) .  (e.g.,  Thus, only when a co-  o r d i n a t i o n such as that between s u b t r a c t i o n and a d d i t i o n i s a t t a i n e d , can one d e f i n e the s e l f - r e g u l a t i o n as an o p e r a t i o n . l o g i c a l n e c e s s i t y inherent product  i n the l o g i c a l  The  s t r u c t u r e s i s the u l t i m a t e  which r e s u l t s when the s e l f - r e g u l a t o r y f u n c t i o n s have a t -  t a i n e d the highest  degree o f c o - o r d i n a t i o n  or e q u i l i b r i u m *  f i n a l " o p e r a t i o n a l " s t a t e , t h e r e f o r e , i s achieved  the  only a f t e r the  s e l f - r e g u l a t i o n f u n c t i o n s have gone through a long s u c c e s s i o n more " i m p e r f e c t " Piaget has regulations operations  s t a t e s i n terms of t h e i r feedback  capabilities.  termed the l e v e l s of these l e s s p e r f e c t  as j u s t " r e g u l a t i o n s " -- which are not because they are not  of  self-  strictly  e n t i r e l y r e v e r s i b l e ; and  "rhythms"  which c o n s t i t u t e s e l f - r e g u l a t i n g mechanisms by v i r t u e o f symmetries and  repetitions.  In reference  s e l f - r e g u l a t i n g mechanisms, P i a g e t  to these three  types o f  (1970a) s p e c i f i c a l l y  states:  "...these are the three b a s i c mechanisms o f s e l f r e g u l a t i o n and s e l f •^maintenance. One may, i f one so d e s i r e s , view them as the " r e a l " stages of a structure's construction." (p. 16) A more s p e c i f i c example o f the d i f f e r e n c e between r e g u l a t i o n s Operations comes from the d i s t i n c t i o n , d i s c u s s e d by Piaget  and  (1968) ,  bettveen the nature of the " i d e n t i t i e s " the c h i l d can e s t a b l i s h p r i o r to and  a f t e r a c q u i s i t i o n of c o n s e r v a t i o n .  Preoperational,  i d e n t i t y i n v o l v e s only q u a l i t a t i v e i n v a r i a n t s (e.g., the 4 or 5 y e a r - o l d c h i l d , i n the  conservation  of l i q u i d s experiment,  maintains that the amount of water has it  i s "the  changed, w i l l  admit  who that  same water," i n the sense that the nature o f the matter  "water" has changed) .  not The  changed even i f the q u a n t i t y of the matter i d e n t i t y necessary f o r c o n s e r v a t i o n ,  hoiveVer, i n -  volves  quantitative invariants.  simply  the d i s s o c i a t i o n between a permanent q u a l i t y (the same water')  and  the v a r i a b l e q u a l i t y A number of very  t i o n s may  be  structures.  derived  Pre-conservation  has  identity is  (shape).  important t h e o r e t i c a l and  empirical  from t h i s c o n c e p t u a l i z a t i o n on  the  implica-  formation  Of  17  F i r s t , the conceptual developmental p r i o r i t i e s come f i r s t  a n a l y s i s emphasizes that i n terms o f  i t i s the s e l f - r e g u l a t o r y f u n c t i o n s which  s i n c e , i n f a c t , i t i s the f i n a l  e q u i l i b r a t i o n o f these  f u n c t i o n s \tfhich e v e n t u a l l y comes t o d e f i n e the s t r u c t u r e . ing  on t h i s p o i n t , P i a g e t  (1970a) s t a t e s :  among the elements t h a t count. cedures or n a t u r a l processes  "...it  Expound-  i s the r e l a t i o n s  In other words, the l o g i c a l  pro-  by which the whole i s formed are  primary, not the whole, which i s consequent on the system's laws of composition."  (p. 8)  I f the aim, then, i s to uncover or under-  stand how the s t r u c t u r e s come about, i t i s necessary according t o t h i s t h e o r e t i c a l p o s i t i o n to f i r s t f u n c t i o n s evolve  and achieve  analyze  i t s ultimate  how the s e l f - r e g u l a t o r y organization.  Second, i t f o l l o w s from the above, that i n order to assess the development o f any s t r u c t u r a l concept i t xtfill be necessary to speci f y what the s e l f - r e g u l a t o r y f u n c t i o n s which precede i t developmentally  are.  These preceding  f u n c t i o n s can be s a i d to c o n s t i t u t e  the p r e - r e q u i s i t e o r necessary s k i l l s tural  to a t t a i n the given s t r u c -  concept. In any attempt to measure a c h i l d ' s true l e v e l o f o p e r a t i v i t y  s e v e r a l e m p i r i c a l consequences d e r i v e from the above:  (1) As was  s t a t e d above, because the nature o f these s e l f - r e g u l a t o r y f u n c t i o n s is different  from the f i n a l operations  which make up the l o g i c a l  s t r u c t u r e , i t i s necessary to d e f i n e these f u n c t i o n s f o r the specific  concepts at hand.  l o g i c a l operations  (2) In the p e r i o d during which c e r t a i n  are being  acquired  i t i s necessary to make the  assessment, not only i n terms of whether the c h i l d possesses the concept o r f i n a l i z e d s t r u c t u r e (e.g., understands but  conservation),  a l s o i n terms o f the l e v e l o f attainment achieved  i n the s e l f -  18  regulatory  f u n c t i o n s which precede the a c q u i s i t i o n of the  structure,  In other words, these " s e l f - r e g u l a t o r y f u n c t i o n s " are to be as "continuous" v a r i a b l e s , and not passed or not.  ( 3 ) Piaget's  t h e o r e t i c a l conception  these f u n c t i o n s grow i n a gradual should  reflect  j u s t as t h r e s h o l d s  manner, thus,  i n some q u a n t i t a t i v e way  that  treated are  i n d i c a t e s that  the measures r*ed  the p o s s i b l e v a r i o u s l e v e l s  of attainment. To the t h e o r e t i c a l q u e s t i o n , ing  that allows  the understanding of l o g i c a l o p e r a t i o n s ,  s a i d that according ing"  then, of what i s the c h i l d  to P i a g e t , the  acquir-  i t can  child is acquiring "self-regulat-  f u n c t i o n s or mental a c t i o n s which e v e n t u a l l y  attain a level  development i n t h e i r feedback c a p a c i t i e s which makes p o s s i b l e kinds ical  of t r a n s f o r m a t i o n a l groupings.  co-ordinations  A question  described  that remains i s :  T h i s question  of  the  i n the mathemat-  what i s the psycho-  l o g i c a l nature o f these a c t i o n s , or what are some o f these regulating functions?  be  i s d e a l t w i t h i n the  selfnext  section. P s y c h o l o g i c a l nature of the  s e l f - r e g u l a t o r y functions  To understand the p s y c h o l o g i c a l nature o f these s e l f - r e g u l a t ing  functions  i n t o Piaget's  and what they make p o s s i b l e , i t i s necessary to a n a l y s i s of the changes that occur d u r i n g  t i o n p e r i o d between p r e - o p e r a t i o n a l example, i n a c o n s e r v a t i o n aware (or e x p l i c i t l y  and  the  look  transi-  o p e r a t i o n a l thought.  For  of l i q u i d experiment, i f a c h i l d i s  i s made aware through t r a i n i n g ) of a l l the  aspects o f the s t i m u l i p r i o r t o , d u r i n g , t i o n s , what makes i t d i f f i c u l t  f o r him  ships between these events so that he  and  a f t e r the  to co-ordinate  transformathe  a s s e r t s that there  relationi s no  19  c o n s e r v a t i o n of the amount of l i q u i d to drink?  Any  conservation  experiment c o n s t i t u t e s a sequence of temporal events i n v o l v i n g transformations and  of the s t i m u l i .  Inhelder, 1964)  According  to P i a g e t  i n order to deduce what aspect  ant i n the temporal sequence i t i s necessary simultaneously  c h i l d l a c k s , then,  or " c e n t e r " simultaneous1y tends to focus  r e s u l t s without  account What  i s the a b i l i t y to c o n s i d e r  on a l l aspects of the temporal sequence.  c o - o r d i n a t i n g them with the i n i t i a l  i t i s necessary,  static  c o n d i t i o n or  t h a t brought the change about.  u l t a n e o u s l y c o - o r d i n a t e past and present  know and be  to take i n t o  invari-  (and thus base h i s judgments) on the end  w i t h the t r a n s f o r m a t i o n s  uum,  remains  a l l the aspects of the temporal sequences.  the p r e - c o n s e r v a t i o n  He  (1962; Piaget  To  sim-  information i n a contin-  not j u s t to remember the p a s t , but a l s o to  able to perform  mentally  the t r a n s f o r m a t i o n s  c a r r i e d out on the p h y s i c a l s t i m u l i .  A c q u i s i t i o n of the  a b i l i t y then, i s what f a c i l i t a t e s  the t r a n s i t i o n from  just latter,  pre-opera-  t i o n a l to o p e r a t i o n a l thought. There i s growing evidence  i n the l i t e r a t u r e that i t i s indeed  the a b i l i t y to mentally perform  the t r a n s f o r m a t i o n s  which allows the c h i l d to conserve Wallach, The be)  1969;  B e a r i s o n , 1969,  ( M i n i c h e l l o and  Beilin,  a b i l i t y to mentally perform  d i r e c t l y r e l a t e d to the expansion  on the  stimuli  Goodnow,  1969;  1969).  transformations  i s (or should  of the c h i l d ' s a b i l i t y to  apprehend connected events both forwards and backwards i n time. Research on the p s y c h o l o g i c a l processes a wider range of temporal events  that make apprehension  of  p o s s i b l e has been somewhat neglec-  ted because of the r e s e a r c h e r s ' main concern with the cues of the stimulus s i t u a t i o n that allow the c h i l d to a r r i v e at a c o n s e r v a t i o n  20  s o l u t i o n r a t h e r than w i t h the c h a r a c t e r i s t i c s o f the c h i l d ' s ment a l processes that allow him to i n t e g r a t e a l l the r e l e v a n t  cues  of the s t i m u l i 4 Piaget 1972)  himself  postulates  (Inhelder  and P i a g s t , 1964; P i a g e t ,  that the p s y c h o l o g i c a l processes or s e l f - r e g u l a t -  ing mechanisms which make mental t r a n s f o r m a t i o n s interconnected sight".  p o s s i b l e are the  a b i l i t i e s which he l a b e l s " h i n d s i g h t " and " f o r e -  Hindsight  or r e t r o a c t i o n i n v o l v e s the a b i l i t y to take  i n t o account i n f o r m a t i o n ities  1971a;  i n the p r e s e n t ,  used i n the past  i n r e l a t i o n to a c t i v -  e.g., the a b i l i t y to maintain a c l a s s i f i -  c a t i o n c r i t e r i o n throughout a task r a t h e r than c o n t i n u a l l y s h i f t i n g c r i t e r i a depending on what s p e c i f i c o b j e c t the c h i l d i s immediately a t t e n d i n g .  Foresight  a b i l i t y to extend i n f o r m a t i o n  or a n t i c i p a t i o n i n v o l v e s the  from the past or present  to make  judgments about the f u t u r e , e.g., the a b i l i t y to a n t i c i p a t e the c o n t i n u a t i o n o f a s e r i e s o f events from the r e l a t i o n s p r e v i o u s l y observed. the  Piaget  formation  a s c r i b e s these two f u n c t i o n s  of l o g i c a l operations.  a causal role i n  He s p e c i f i c a l l y  states,  "...the i n t e r e s t o f r e t r o a c t i o n and a n t i c i p a t i o n i s that these two notions help us s p e c i f y the c o n d i t i o n s f o r the i n t e r i o r i z a t i o n "(of l o g i c a l o p e r a t i o n s ) . . . they a r i s e as a r e s u l t o f the growing c o o r d i n a t i o n between s u c c e s s i v e a c t i o n s which e v e n t u a l l y overcomes the o n e - d i r e c t i o n a l i t y i n h e r e n t i n a s u c c e s s i o n and tahes the form o f a s h u t t l i n g from the present t o the past which very soon begins to impinge on the f u t u r e . . . t h i s k i n d o f s h u t t l i n g i s e s s e n t i a l t o the comparison of elements i n a s e t taken as a whole; we begin to understand why these r e g u l a t i o n s ( a n t i c i p a t i o n and r e t r o a c t i o n ) are l i k e l y to end up i n the form of o p e r a t i o n s , s i n c e the s h u t t l i n g i t s e l f i s a p r i m i t i v e form o f r e v e r s i b i l i t y . " (Inhelder and P i a g e t , 1964, pp. 286-287.) In one o f h i s most recent p u b l i c a t i o n s , The P r i n c i p l e s o f Genetic  Epistemology, (1972), Piaget  specifically  states:  21  "...and so, the fundamental d i f f e r e n c e s separati n g the behavior o f one stage from that o f the preceding ones must be conceived as t r a n s i t i o n to a l i m i t . . . In the present case o f the knox'/ledge o f operations Me encounter an a n a l ogous temporal p r o c e s s : the f u s i o n i n t o a s i n g l e act o f a n t i c i p a t i o n s and r e t r o s p e c t i o n s -- which i s the b a s i s o f o p e r a t i o n a l r e v e r s i b i l i t y . ...the l i m i t c h a r a c t e r o f operations as opposed to the simple ' r e g u l a t i o n s ' or e a r l i e r l e v e l s , means that i n s t e a d o f c o r r e c t i o n s being made a f t e r the event, that i s once the a c t i o n has been c a r r i e d out p h y s i c a l l y , e r r o r s are p r e - c o r r e c t e d i n v i r t u e o f the i n t e r p l a y o f d i r e c t and i n v e r s e operations o r , i n other words, as a r e s u l t o f combination o f a n t i c i p a t i o n s and r e t r o s p e c t i o n s , or more p r e c i s e l y , o f a p o s s i b l e a n t i c i p a t i o n s o f the r e t r o s p e c t i o n s themselves." ( P i a g e t , 1972, pp. 35-36) Co-ordination  of hindsight  and f o r e s i g h t c o n s t i t u t e s , there-  f o r e , the necessary p r e r e q u i s i t e to r e v e r s i b l e aspects o f s t i m u l i .  transformational  The most d i r e c t s t u d i e s seeking  to e m p i r i c a l l y  i n v e s t i g a t e these a b i l i t i e s have been c a r r i e d out by Piaget self  (Inhelder  and P i a g e t ,  ever, i n Inhelder  1364; Piaget  and Piaget's  and Inhelder,  E a r l y Growth o f Logic  him-  1971).  How-  (1964) , the  a n a l y s i s c f these processes i s r a t h e r incomplete f o r v a r i o u s reasons. F i r s t , the a n a l y s i s o f f l e x i b i l i t y sight  i s made i n the context  and f o r e -  o f the same s t r u c t u r a l a b i l i t y  c l a s s i f i c a t i o n ) that the h i n d s i g h t underlie.  i n hindsight  (i.e.,  and f o r e s i g h t are presumed to  In other words, one task was the b a s i s o f i n f e r e n c e s  about the two d i f f e r e n t (but a l l e g e d l y r e l a t e d ) a b i l i t i e s .  If  hindsight  and f o r e s i g h t are to have any explanatory  value  formation  o f s t r u c t u r e s , as P i a g e t  t o give them,  c l e a r l y purports  f o r the  then t h e i r measurement should be made independently and s e p a r a t e l y i n context  from the measurement o f o p e r a t i v i t y as d e f i n e d  s t r u c t u r a l task.  A l s o , i f f o r e s i g h t and h i n d s i g h t  i n any  are b a s i c and  22  general p s y c h o l o g i c a l f u n c t i o n s o f relevance  to concepts  other  than those d e f i n e d i n the l o g i c a l realm, then t h e i r  operational  d e f i n i t i o n should bear on i t s most important aspects  which pre-  sumably can be s p e c i f i e d independently a given context  o f the c h a r a c t e r i s t i c s o f  where they might be a p p l i e d .  Second, a consequence o f the f i r s t , eous and separate  within-subjects  s i g h t and h i n d s i g h t and o p e r a t i v e T h i r d , the a b i l i t y  there was not a simultan^  comparison o f l e v e l s o f f o r e level  i n Piaget's a n a l y s i s .  to perform the mental  transformations  which the s e l f - r e g u l a t o r y f u n c t i o n s are presumed to c o n f e r was again  i n f e r r e d from the c h i l d ' s performance i n the c l a s s i f i c a t i o n  tasks.  Of course,  sification  i t i s necessary by the d e f i n i t i o n o f the c l a s -  task to assume that i t w i l l  make mental t r a n s f o r m a t i o n s . operativity  i n v o l v e the a b i l i t y to  The problem i s that a l l aspects o f  ( a b i l i t y to make mental t r a n s f o r m a t i o n s ,  o f c l a s s i f i c a t o r y schemes, and necessary f l e x i b i l i t y and  achievement i n hindsight  f o r e s i g h t ) were i n f e r r e d from measures cn the same task.  To  v a l i d a t e these t h e o r e t i c a l c o n s t r u c t s , i t i s necessary to measure them independently  through tasks t h a t e x p l i c i t l y r e q u i r e the  p s y c h o l o g i c a l operations To assess  the a c q u i s i t i o n o f l o g i c a l concepts, i t i s neces-  sary to evaluate about.  presumably i n v o l v e d i n them.  the b a s i c processes  One i n i t i a l  t h a t t h e o r e t i c a l l y b r i n g them  attempt i n t h i s d i r e c t i o n can be assessment o f  the l e v e l s of attainment i n measures o p e r a t i o n a l l y d e f i n e d s i v e l y i n terms o f the c h a r a c t e r i s t i c s o f the u n d e r l y i n g r e g u l a t o r y mechanisms  self-  to determine i t s r e l a t i o n s h i p to the  l e v e l o f attainment on other methodological  exclu-  s t r i c t l y defined l o g i c a l tasks.  The  b a s i s f o r the implementation o f t h i s approach i s  23  described  i n a subsequent s e c t i o n .  Genesis of s t r u c t u r e s Piaget's  - e q u i l i b r a t i o n model  t h e o r e t i c a l a n a l y s i s of the genesis  must be d i s c u s s e d  of structures  f u r t h e r to a r r i v e at s p e c i f i c aspects o f  development of l o g i c a l s t r u c t u r e s that are open to  the  empirical  measurement. Piaget  o u t l i n e s s e v e r a l aspects i n v o l v e d  structures.  In h i s book S t r u c t u r a l i s m  i n the  formation  (1970), defending  p o s i t i o n that s t r u c t u r e s are endogenously c o n s t r u c t e d , states  of  the  Piaget  that: "...through the i n t e r p l a y of r e f l e c t i v e a b s t r a c t i o n , which f u r n i s h e s i n c r e a s i n g l y complex " m a t e r i a l s " f o r c o n s t r u c t i o n , and of e q u i l i b r a t i o n ( s e l f - r e g u l a t i o n ) mechanisms, which make f o r i n t e r n a l r e v e r s i b i l i t y , s t r u c t u r e s -- i n being c o n s t r u c t e d give r i s e to that n e c e s s i t y which a p r i o r i s t t h e o r i e s have always thought i t necessary to p o s i t at the o u t s e t . Necessity, i n s t e a d of being the p r i o r c o n d i t i o n f o r l e a r n i n g , i s i t s outcome." (1970a, p. 62)  From the  above q u o t a t i o n ,  and  gradual  as i n d i c a t e d i n the  s e c t i o n , there  i s f i r s t the  f u n c t i o n s ; and  second, as the s e l f - r e g u l a t o r y f u n c t i o n s  " c e r t a i n " l e v e l of f l e x i b i l i t y , "reflective The  growth of the s e l f - r e g u l a t o r y  there  attain a  i s the occurrence o f  abstraction."  increase  -- h i n d s i g h t input  previous  and  i n f l e x i b i l i t y o f the  s e l f - r e g u l a t o r y functions  f o r e s i g h t -- enables the  i n d i v i d u a l to a s s i m i l a t e  i n f o r m a t i o n w i t h l e s s d i s t o r t i o n because he i n c r e a s i n g l y  becomes able to center  or focus  on a l l r e l e v a n t aspects and  not  j u s t on the most prominent p e r c e p t u a l c h a r a c t e r i s t i c s . P.eflective a b s t r a c t i o n c o n s i s t s i n the a b s t r a c t i o n of properties  from the  a c t i o n s , or from the ways of a c t i n g on t h i n g s .  This  24  a b s t r a c t i o n i n v o l v e s the s u b j e c t ' s of some a c t i o n s or operations c o n s t r u c t i o n leads  r e c o n s t r u c t i o n or rearrangement  p r e v i o u s l y made such that t h i s r e -  to operations  upon operations  the scope 6 f the component o p e r a t i o n s . conservation  of l i q u i d  or a c t i o n s beyond  Again, an example from the  experiment w i l l e l u c i d a t e .  For the non-  Conserver, knowing that i f the water i s poured i n t o the o r i g i n a l container be  i t w i l l be the same as before  (a f a c t that a c h i l d might  able to p r e d i c t ) , i s not the same as the o p e r a t i o n a l  i n g o f i d e n t i t y i n the conserving true by l o g i c a l n e c e s s i t y . conserving  c h i l d f o r whom such a f a c t i s  The same p h y s i c a l a c t i o n , f o r the  c h i l d , acquires  to the s p e c i f i c content.  a meaning beyond the one f i r s t The r e l a t i o n s h i p s present  (involved i n a conservation  ascribed  i n the a c t i o n s  experiment, f o r i n s t a n c e )  l o g i c a l n e c e s s i t y whereas p r e v i o u s l y they had been simply  understand-  become a  considered  probable.  The  s t r u c t u r e , then, i s a t t a i n e d when a r e f l e c t i v e  t i o n o f the a c t i o n s  i n v o l v e d i s achieved  and, t h i s  abstrac-  i s manifested  i n the l o g i c a l n e c e s s i t y now seen i n the r e l a t i o n s h i p s .  "The  f e e l i n g o f n e c e s s i t y comes £rc»z the c l o s u r e o r completion o f a structure." The  ( P i a g e t , 1971b, p. 5 ) .  question  that a r i s e s from t h i s c o n c e p t u a l i z a t i o n i s :  when or how i s r e f l e c t i v e The  a b s t r a c t i o n achieved?  e q u i l i b r a t i o n model that Piaget  d e s c r i b i n g the formation  (1967) p o s t u l a t e s as  of structures ascribes a p r o b a b i l i s t i c  b a s i s to the occurrence o f r e f l e c t i v e a b s t r a c t i o n . continuing  operation  Through the  o f the e q u i l i b r a t i o n mechanisms ( s e l f - r e g u l a -  t o r y f u n c t i o n s ) , the s u b j e c t  i s s a i d to develop s t r a t e g i e s that  allow him to deal more and more s u c c e s s f u l l y (with l e s s d i s t o r t i o n )  25  with e x t e r n a l i n p u t .  For any  s p e c i f i c s i t u a t i o n such as the con-  s e r v a t i o n of substance, where a b a l l of c l a y i s changed i n t o a sausage, there  are a number of s e q u e n t i a l s t r a t e g i e s which the  s u b j e c t might use  to deal with  the stimulus  example, as d e s c r i b e d by Piaget at  the outset  i s concentration  situation.  For  (1967), the most probable on j u s t one  of the  strategy  transformed  c h a r a c t e r i s t i c s , i . e . , the q u a n t i t y appears to i n c r e a s e because the o b j e c t becomes elongated.  Once t h i s has  occurred,  the  which then becomes most probable c o n s i s t s of n o t i c i n g the transformed c h a r a c t e r i s t i c and diminishes  supposing that the  because the sausage becomes t h i n n e r .  strategy second  quantity Thereafter,  neiv s t r a t e g y becomes most probable as a f u n c t i o n of having to the two  preceding  characteristics.  This new  vaguely n o t i c i n g  the interdependence of the sausage's e l o n g a t i o n and  i t s thinness.  This t h i r d r e a c t i o n leads to the a c t i o n being p l a c e d  alone.  on the  trans-  p i 3.116 cLS OppOSCG. to the s t a t i c c o n f i g u r a t i o n plane  A f o u r t h s t r a t e g y ensues which c o n s i s t s of the  of the compensations among the the f a c t of  attended  strategy consists  o f o s c i l l a t i n g between these c h a r a c t e r i s t i c s and  formational  a  transformations  and  discovery  acceptance o f  conservation.  "...the f i n a l e q u i l i b r i u m i s thus the product of a compensation f o r the ambiguous s t i m u l i by the a c t i v i t i e s of the subject which i n turn are c h a r a c t e r i z e d by s u c c e s s i v e p r o b a b i l i t i e s . " ( P i a g e t , 1967, p. 112) T h i s p r o b a b i l i s t i c model has authors ( F l a v e l l , 1963;  been c r i t i c i z e d by a number of  B e i l i n , 1971  ) because of the d i f f i c u l t y  e m p i r i c a l l y t e s t i n g i t i n s i t u a t i o n s other than those somewhat a r t i f i c i a l l y c r e a t e d when the s u b j e c t such as c o n s e r v a t i o n .  Despite  i s given s p e c i f i c problems  the v a l i d i t y of these  criticisms,  of  26  i f indeed such a model were to be used to i n v e s t i g a t e concerning  questions  the l e v e l a c q u i s i t i o n of s p e c i f i c concepts, i t would  be n e c e s s a r y f o r every s p e c i f i c o p e r a t i o n a l concept to deal w i t h problems such as:  ( 1 ) S p e c i f i c a t i o n and  of the s k i l l s or v a r i o u s  operational d e f i n i t i o n s  s t r a t e g i e s that from the s u b j e c t ' s  o f vieiv ivould be necessary to achieve by means o f a b s t r a c t i o n the s p e c i f i c concept. measures of the v a r i o u s  skills  reflective  (2) C r e a t i o n o f  operational  such that i t could be p o s s i b l e  d i f f e r e n t i a t e through them, i n a q u a n t i t a t i v e sense, the " l e v e l s " of the necessary s k i l l s possessed by i n turn c o u l d be t r a n s l a t e d i n terms of how b r i n g about the  reflective  Nothing i n the theory  occurs.  The  These  l i k e l y they are  to  the e q u i l i b r a t i o n model  requires  analyzed.  i n d i c a t e s w i t h i n which range o f the probs t r a t e g i e s , under which  the r e f l e c t i v e a b s t r a c t i o n i n v o l v e d  conditions  i n a concept  s p e c i f i c a t i o n of the p r o b a b i l i t i e s or c e r t a i n s t r a -  t e g i e s or the c o n d i t i o n s co-ordination  that would make the attainment of  the  i n v o l v e d i n a l o g i c a l concept more l i k e l y i s an  empirical question f o r each s p e c i f i c This p o i n t cations.  various  the s u b j e c t s .  s e l f - r e g u l a t o r y f u n c t i o n s be  a b i l i t i e s o f occurrence of the or experiences,  to  abstraction.  These problems i n d i c a t e how that the nature o f the  point  that would have to be d e a l t w i t h i n d i v i d u a l l y concept.  i s emphasized because of i t s methodological i m p l i -  Host attempts i n the l i t e r a t u r e aimed at d e a l i n g with  problems o f assessment of the c h i l d ' s o p e r a t i v e based mainly on whether or not  the c h i l d has  knoi/ledge  already  are  achieved  l o g i c a l concept or grouping, r a t h e r than on an assessment of  the the  development of the s k i l l s necessary to or l e a d i n g up to the a t t a i n -  27  ment of the concept. An example o f a t e s t i n g procedure somewhat r e f l e c t i n g approach based on assessment of necessary  an  s e l f - r e g u l a t o r y func-  t i o n s i s a study by L e f r a n c o i s (1968) i n which a q u a s i - t r a i n i n g procedure was  devised f o r c h i l d r e n who  substance p r e - t e s t . of t a s k s  ?  ordered  The  f a i l e d a conservation  subjects were presented  from simple  with a s e r i e s  to d i f f i c u l t , based on a h i e r a r -  c h i c a l a n a l y s i s of the p r e r e q u i s i t e s k i l l s necessary conservation.  of  Of 60 non-conservers, age  5 to 6,  to  attain  56 performed  a c c o r d i n g to a p e r f e c t Guttman s c a l e on the nine p r e r e q u i s i t e skills  outlined.  This measurement procedure was  designed  e s s e n t i a l l y d i a g n o s t i c s i n c e i t attempted to s p e c i f y the that the s u b j e c t had to  acquire.  On  yet  the b a s i s of the assessment from the s c a l e ,  The  the d i d a c t i c sense. to  skills  already i n c o r p o r a t e d and those he had  s u b j e c t s were t r a i n e d on the s k i l l s s t i l l conservation.  to be  t r a i n i n g d i d not  needed to  achieve  involve actual teaching i n  Subjects were merely asked to observe o b j e c t s ,  notice transformations  c a r r i e d out by the experimenter, and  answer questions  r e g a r d i n g the f i n a l  s t a t e of the o b j e c t  transformation.  Preliminary studies  ( l a t e r extended, Ayers,  to  after 1971)  i n d i c a t e d that 25 out of 4 0 of the c h i l d r e n given t r a i n i n g based on the h i e r a r c h y were s u c c e s s f u l i n a c q u i r i n g c o n s e r v a t i o n  of  substance while no s u b j e c t s i n a matched group o f 20 s u b j e c t d i d l i k e w i s e i n the absence o f t r a i n i n g . Again,  t h i s experiment emphasizes the c o n c l u s i o n s made i n  the titfo previous  t h e o r e t i c a l s e c t i o n s f o r the need to focus  on  the " s e l f - r e g u l a t o r y " f u n c t i o n s i n order to deal s u c c e s s f u l l y w i t h questions  of a c q u i s i t i o n of l o g i c a l concepts.  The  child's  28  '"attainment" of a C e r t a i n l e v e l o f o p e r a t i v i t y i s more a c c u r a t e l y assessed when the measurement instruments take i n t o account c h i l d ' s l e v e l of development i n the U n d e r l y i n g necessary as w e l l as h i s degree of achievement of the f i n a l represented i n the l o g i c a l  the  skills  structure  concept.  What g e n e r a l i z e s between s t a g e - r e l a t e d concepts? In P i a g e t i a n theory a cognitive-development  stage, such as  the c o n c r e t e - o p e r a t i o n a l stage, i s conceived of as a p e r i o d during which the c h i l d a c q u i r e s q u a l i t a t i v e l y d i f f e r e n t behaviors that can be d e s c r i b e d i n terms of s i m i l a r conceptual r u l e s , such as the mathematical  l o g i c a l groupings.  The  important aspect of t h i s  c o n s t r u c t of stage i s that i t s p e c i f i c a l l y conceives o f the c h i l d ' s new  c o g n i t i v e a c q u i s i t i o n s , not as p s y c h o l o g i c a l l y  isola-  ted and u n r e l a t e d a b i l i t i e s , but r a t h e r as i n t e r a c t i n g w i t h  one  another i n s p e c i f i e d ways, as a u n i t a r y system or s t r u c t u r e . In view of t h i s t h e o r e t i c a l assumption, any d i a g n o s t i c procedure o p e r a t i v i t y would attempt achievement and degree  i t is logical  that  designed to assess the c h i l d ' s l e v e l of to simultaneously compare the l e v e l of  of r e l a t i o n s h i p between concepts t h e o r e t -  i c a l l y presumed to be part of one u n i f i e d l o g i c a l  system.  thus f i r s t necessary to analyze what the nature of the ship between the concepts i n the u n i f i e d system  It i s  relation-  i s meant to be.  P i n a r d and Laurendeau (1969) have r e c e n t l y d i s c u s s e d the problem  of the r e l a t i o n s h i p among s t a g e - r e l a t e d concepts  a n a l y s i s o f the c o n s t r u c t of " s t a g e . "  They have made a  in their thorough  s c r u t i n y of the f i v e s t r u c t u r a l c r i t e r i a which a c c o r d i n g to P i a g e t c h a r a c t e r i z e a stage.  B r i e f l y , these are:  (1) H i e r a r c h i z a t i o n ,  2?  the n e c e s s i t y o f a f i x e d order  i n the s u c c e s s i o n  o f the d i f f e r e n t  l e v e l s that c o n s t i t u t e a developmental sequence, ( 2 ) I n t e g r a t i o n , the n e c e s s i t y that new  a c q u i s i t i o n s of a given stage be  with those of the preceding f o r them or juxtaposing  stage i n s t e a d of simply  the r e c e n t l y acquired behavior and the behavior at the  an aspect  following l e v e l ,  ture d'ensemble"), the n e c e s s i t y simply  f a s h i o n , but  o r g a n i c a l l y interconnected  juxtaposed with one  it  to the question  items, the  f u r t h e r considered and  of a  given  and  group them i n t o t o t a l  of the  becomes  levels. i n t e r r e l a t i o n s h i p among  c r i t e r i o n of most relevance i s that of s t r u c t u r i n g .  that needs  According  Laurendeau, the meaning of t h i s c o n s t r u c t  cannot be  (or " s t r u c -  another i n an a d d i t i v e  range of operations  and more mobile with s u c c e s s i v e  In r e f e r e n c e  Pinard  for  ( 5 ) E q u i l i b r a t i o n , the n e c e s s i t y of a s e r i e s of  e q u i l i b r i u m l e v e l s i n which the  to be  of p r e p a r a t i o n  by t i e s o f i m p l i c a t i o n  r e c i p r o c a l dependence that u n i t e them and  stage-related  o f achievement of  f o r the operations  not  greater  necessity  ( 4 ) Structuring,  l e v e l to be,  s t r u c t u r e s , and  substituting  their., ( 3 ) C o n s o l i d a t i o n , the  that a stage always i n v o l v e at once an aspect  integrated  to  i s such that  reduced  " . . . t o a simple form of t r a n s f e r or of g e n e r a l i z a t i o n between d i v e r s e concepts, nor to necessary (but not s u f f i c i e n t ) synchronisms that would r e f l e c t the o p e r a t i o n o f t h i s c h a r a c t e r i s t i c . The s t r u c t u r a l c h a r a c t e r i s t i c a s s e r t s above a l l a complete f u n c t i o n a l interdependence, an organic connection among the operations (or a c t i o n s ) that can i n f a c t anply to r e l a t e d concepts." (op. c i t . , p. 137) T h i s c r i t e r i o n r e q u i r e s that once a l o g i c a l concept i s there  should be  acquired  a simultaneous mastery or immediate g e n e r a l i z a t i o n  to a l l tasks based on that l o g i c a l o p e r a t i o n .  Pinard  and  Laurendeau  30  c o n s i d e r the i n t r a c o n c e p t l e v e l as the minimum range of concepts f o r which t h i s c r i t e r i o n has to be demonstrated  (and thus give  v a l i d i t y to P i a g e t ' s stage c o n s t r u c t ) , mainly because l e v e l i t i s presumably  at t h i s  e a s i e r to reduce the e f f e c t s of the v a r i -  ous confounding v a r i a b l e s that might i . e . , f a c t o r s that produce  the  cloud i t s m a n i f e s t a t i o n ,  h o r i z o n t a l ' d e c a l a g e s o r temporal  lags between s t r u c t u r a l l y s i m i l a r concepts.  For example, t h i s  would mean that "...the c o n s t r u c t i o n of the concept of weight xvould s i m u l t a n e o u s l y imply, at about the age of nine y e a r s : the mastery of groupings o f simple a d d i t i o n and of v i c a r i a n c e s ( c o n s e r v a t i o n of w e i g h t ) : o f a d d i t i o n of asymmetrical ( s e r i a t i o n of weight) and symmetrical ( t r a n s i t i v i t y of equivalences)relations: of biunivocal m u l t i p l i c a t i o n of r e l a t i o n s or of c l a s s e s (concept o f d e n s i t y by the combination of weight and substance) ; of c o u n i v o c a l m u l t i p l i c a t i o n (conservat i o n of the weight of p a r t i c l e s of f l o u r i n the d i l a t e d c o r n ) : and f i n a l l y of the corresponding q u a n t i t a t i v e groupings, by a f u s i o n of c l a s s e s and r e l a t i o n s . " (op. c i t . , p p . 138-139) Thus, the s t r u c t u r i n g c r i t e r i o n n e c e s s a r i l y i m p l i e s  that,  " . . . t o affirm, the p s y c h o l o g i c a l e x i s t e n c e of an a u t h e n t i c o p e r a t i o n a l grouping b e a r i n g on a given content, t h i s content must at the same time e l i c i t not only the whole set o f c o n s t i t u e n t o p e r a t i o n s of t h i s grouping but also the ensemble of p a r a l l e l and connected groupings." (op. c i t . , p. 140) The  interdependence  among the groupings and thus the syn-  chrony i n t h e i r development i s necessary because logical  on the b a s i s o f a  (or e p i s t e m o l o g i c a l ) a n a l y s i s o f the groupings " . . . a l l are s t r i c t l y isomorphic because a l l are based on the same combination of fundamental o p e r a t i o n s ( d i r e c t composition, i n v e r s e compos i t i o n , a s s o c i a t i v i t y , and so on). The psychol o g i c a l a n a l y s i s of these same groupings r e v e a l s the necessary complementarity o f the systems of c l a s s e s , r e l a t i o n s , and numbers." (op. c i t . , p. 139)  31  The data from s t u d i e s that t e s t t h i s c r i t e r i o n , as revienred by P i n a r d and Laurendeau  (1969) and F l a v e l l and W o h l w i l l (1969)  are r a t h e r i n c o n s i s t e n t f o r a v a r i e t y o f methodological and conc e p t u a l reasons which need n o t be s p e c i f i e d here. Laurendeau  contend that to submit t h i s hypothesis to experimenta-  t i o n , i t i s necessary t o conform  as much as p o s s i b l e t o the  s t r u c t u r a l c h a r a c t e r i s t i c s assigned by P i a g e t . not done t h i s p a r t l y because involved.  P i n a r d and  of the experimental  Most authors have difficulties  However, aside from the above c r i t i c i s m , the present  author would contend that no c l e a r answers to these problems are yet p o s s i b l e because  o f the f o l l o w i n g important reasons.  First,  the c o n c e p t u a l i z a t i o n o f the s t r u c t u r i n g c r i t e r i o n i s i n s u f f i c i e n t to allow i t s e m p i r i c a l t e s t i n g .  (1) The conception o f i t s mean-  i n g i s s t r i c t l y based on a s t r u c t u r a l a n a l y s i s which does not i n c o r p o r a t e how, on the b a s i s o f the f u n c t i o n i n g o f the s e l f r e g u l a t o r y mechanisms, such synchrony would be expected to occur. (2) I t i s i n f a c t u n c l e a r ifhat the term "concurrent development" or "interdependence" i s supposed  t o imply.  Bo these mean that  a l l the groupings i n v o l v e d i n a concept are a c q u i r e d at the same time, or -- put i n f u n c t i o n a l terms -- t h a t the r e f l e c t i v e a b s t r a c t i o n necessary f o r a l l the concepts i s achieved simultaneously?  Or do these mean that the s e l f - r e g u l a t o r y mechanisms  u n d e r l y i n g the o p e r a t i o n s develop c o n c u r r e n t l y ? and Laurendeau completed  (1969) s t a t e  "...it  (3) I f , as P i n a r d  goes without s a y i n g that the  e l a b o r a t i o n o f 'structure d'ensemble' would not be  expected b e f o r e the end o f each o f the corresponding developmental l e v e l s " (p. 137), what i s the r e l a t i o n s h i p between subsequently r e l a t e d concepts p r i o r to the end o f the corresponding p e r i o d ?  32  ( 4 ) Does the f a c t that a n a l y s i s the  o p e r a t i o n s o f the  necessary that they be Or,  i n terms of l o g i c a l or  acquired  groupings are  epistemological isomorphic make i t  or g e n e r a l i z e d  simultaneously?  expressed i n other terms, assuming that the  r e l a t e d groupings i n v o l v e d o f competence, can one  set o f i n t e r -  i n a concept s p e c i f y a s i m i l a r l e v e l  assume that the  "automaton" requirements  i" •  for  a l l the It  groupings are  a l s o the same?  i s then obvious that  not p o s s i b l e  from a s t r u c t u r a l a n a l y s i s  i t is  to answer questions about the meaning o f the  turing c r i t e r i o n .  U n f o r t u n a t e l y , most authors who  struc-  have s p e c i f -  i c a l l y d e a l t w i t h t h i s q u e s t i o n have formulated t h e i r experimental tests purely  on s t r u c t u r a l c o n s i d e r a t i o n s  Shantz, 1967), when from the  (e.g., Kofsky,  1966;  above i t i s apparent that to  w i t h the problems of a c q u i s i t i o n o f s t r u c t u r e s  functional  maton) as w e l l as s t r u c t u r a l aspects have to be  deal (auto-  taken i n t o con-  sideration. One  o f the  authors who  these problems i s F l a v e l l out  s p e c i f i c a l l y addressed h i m s e l f  (1970a, 1970b, 1971), who  has  the need f o r more of a f u n c t i o n a l a n a l y s i s of the  problems. that  has  He  postulates  pointed  acquisition  that most of the present data i n d i c a t e  the development of most c o g n i t i v e  operations  involves  gradual changes over time (rather than abrupt a c q u i s i t i o n s might be  implied  by  l i t e r a l i n t e r p r e t a t i o n s o f the  c r i t e r i o n ) such that the  initial  maturity." evocability and  i t could be p o s s i b l e  l e v e l s o f a c q u i s i t i o n and Functional the  utilizability  to  the  maturity i s defined  child's operational  as  structuring  to d i s t i n g u i s h between l e v e l of  "functional  i n terms o f the  a v a i l a b i l i t y of an  -- the c h i l d ' s a b i l i t y , one  item's item,  having sensed  an  33  item-to-problem f i t , cedure.  to u t i l i z e  i t e f f e c t i v e l y as a s o l u t i o n pro-  Because the a c q u i s i t i o n o f s t a g e - s p e c i f i c items i s con-  ceived as a gradual development  over an extended temporal  interval  r a t h e r than as a temporal p o i n t , the n o t i o n of developmental synchrony or concurrence i s rendered ambiguous.  As F l a v e l l  (1970a)  states i t , " . . . t o say that two s t a g e - s p e c i f i c items "develop c o n c u r r e n t l y " could mean that they b e g i n t h e i r development at the same time or conclude i t (achieve f u n c t i o n a l m a t u r i t y ) synchronously, o r both, or even n e i t h e r ( i . e . , have developmental courses that show some c h r o n o l o g i c a l o v e r l a p but o n l y i n the middle r e g i o n s ) . " (p. 5 2 ) Even i f the meaning of i n t e r - i t e m concurrence i s r e s t r i c t e d to mean synchronous  emergence, F l a v e l l p o i n t s out f u r t h e r that  two main i s s u e s s t i l l  remain.  F i r s t , the usual methodological  procedures used to t e s t i n t e r - i t e m developmental  concurrences, i n  terms o f the c h i l d ' s i n t e r - t e s t c o n s i s t e n c y o f performance  in  two or more r e l a t e d t a s k s , p r e s e n t s s e r i o u s methodological problems.  For one, such a procedure assumes that the s o l u t i o n o f the  t e s t s r e q u i r e s an equal degree of a t t a i n e d f u n c t i o n a l m a t u r i t y . I f such i s not the case, and i t i s u n c l e a r at present how might be e q u a l i z e d  items  i n that r e s p e c t , one c o u l d f i n d an apparent  asynchronisn where there i s a r e a l synchronism or v i c e v e r s a . Fut i n other terms, i f the genuine synchronous  development  of  two c o g n i t i v e items o f s i m i l a r "competence" requirements i s t e s t e d s i m u l t a n e o u s l y i n terms o f the presence o f i n t e r - t e s t c o n s i s t e n c y , the concurrency might not appear simply because  the "automaton"  demands ( f u n c t i o n a l m a t u r i t y ) r e q u i r e d might be d i f f e r e n t f o r both and the t e s t s might not have made allowances f o r t h i s ence .  differ-  34  Second, F l a v e l l contends  that a s t r i c t developmental  rence i s not mandated by P i a g e t ' s stage t h e o r y . accepted as P i n a r d and Laurendeau  concur-  Even i f i t i s  (1969) s t a t e , that a l l c o n c r e t e -  o p e r a t i o n a l o p e r a t i o n s are fundamentally the same from one i n g t o another  group-  (e.?., a l l groupings s p e c i f y a d i r e c t o p e r a t i o n ,  an a s s o c i a t i v i t y r u l e , e t c . ) and that they are l o g i c a l l y  and  p s y c h o l o g i c a l l y interdependent both w i t h i n and across groupings and groups, i t can s t i l l be argued that such f a c t s need r e s u l t i n at most only a very loose s o r t of developmental  parallelism.  F l a v e l l s t a t e s that "...the e x i s t e n c e of p s y c h o l o g i c a l connections among items does not presuppose any developmental synchrony among them, s i n c e those connections d e r i v e o n l y from the p r o p e r t i e s of the items once i n e x i s t e n c e , i . e . , once a c q u i r e d . . . . A l l c o n c r e t e - o p e r a t i o n a l items exemplify a c e r t a i n form of t h i n k i n g , a c e r t a i n l e v e l of c o g n i t i v e f u n c t i o n i n g , consequently one would not expect the ensemble of such "same-level" items to show r e a l l y extreme developmental a s y n c h r o n i e s , e.g., one item b e g i n n i n g to emerge at age four and another not u n t i l age twelve. On the other hand, i t does not f o l l o w that such items must emerge i n t i g h t concurrence, that i s , w i t h i n the same week, mont*... or even y e a r . " ( F l a v e l l , 1971, p. 36) ;  It c o u l d be contended  t h a t , i f P i a g e t ' s vie\</s about  the  nature o f the mechanisms that b r i n g about the s t r u c t u r e s were i n c o r p o r a t e d i n t o a s t r u c t u r a l a n a l y s i s of stages, the meaning of developmental  concurrence might  appear somewhat d i f f e r e n t  the one most present i n t e r p r e t a t i o n s a s s i g n to i t , more congruent with the views  of F l a v e l l might  from  and a meaning  emerge.  For  example, the a n a l y s i s of the gradual growth of s e l f - r e g u l a t o r y f u n c t i o n s i m p l i e s that the o p e r a t i o n s develop i n gradual f a s h i o n . A l s o , there i s n o t h i n g i n the model that r e q u i r e s that the  reflec-  t i v e a b s t r a c t i o n i n v o l v e d i n the v a r i o u s concepts be achieved at  35  the same time.  Although  r e l a t e d concepts  f o r the understanding o f "weight") may each concept a new  (transitivity,  combination  (such as those  i n v o l v e the same o p e r a t i o n s ,  c o n s e r v a t i o n , s e r i a t i o n , etc.) i n v o l v e s  of these o p e r a t i o n s , whose i n d i v i d u a l  (during the a c q u i s i t i o n phase o f the whole concept) over time, and each  concept  not yet p o s s i b l e to s p e c i f y . regard i s expressed when he  necessary  may  be  governed  may  formation be  spread  by e x t e r n a l f a c t o r s  P i a g e t ' s l a c k o f knowledge i n t h i s states:  " . . . i t i s n e i t h e r the elements nor a whole t h a t comes i n a matter one knows not how, but the r e l a t i o n s among elements t h a t count." (Piaget, 1970a, pp. 8-9, author's i t a l i c s ) Compounding t h i s  i n s u f f i c i e n c y o f conceptual understanding  of  what i s i n v o l v e d i n i n t e r - i t e m g e n e r a l i z a t i o n , there i s the problem o f accounting f o r the presence temporal  lags.  of  h o r i z o n t a l 'de"calages' or  As d i s c u s s e d i n an e a r l i e r s e c t i o n , a c c o r d i n g to  P i a g e t the c h i l d ' s  l e v e l of o p e r a t i v i t y seems to i n t e r a c t  h i s n o t i o n s of p h y s i c a l i n v o l v e d , which may  c a u s a l i t y about the s p e c i f i c  with  context  l e a d him to make n o n - o p e r a t i o n a l judgments.  A number of other f a c t o r s have been p o i n t e d out by P i a g e t ( P i a g e t , 1971b; 1970b; Piaget and Szeminslca, 1 9 4 1 ) as p o s s i b l e sources of "decalages:'  how  the o p e r a t i o n a l questions are formulated, the  r e l a t i o n s h i p between the f i g u r a t i v e  aspects o f the  i'.sed and the o p e r a t i v e aspects to be s o l v e d , the between the i n s t r u c t i o n s and the i n d i v i d u a l  techniques  relationship  experiences o f the  c h i l d , the number of elements i n v o l v e d i n the t a s k s , e t c . In view o f the above problems, how  can one  deal w i t h the  q u e s t i o n of the i n t e r - r e l a t i o n s h i p or g e n e r a l i z a t i o n between stage related  items?  C o n c e p t u a l l y , as P i n a r d and Laurendeau  suggest, the a n a l y s i s should f i r s t  (1969)  encompass concepts whose neces-  36  sary o p e r a t i o n s are not the  only l o g i c a l l y r e l a t e d but  same conceptual content, e.g.,  structures.  This  attainment of  r e s t r i c t i o n would f a c i l i t a t e  also  involve  classification  the  analysis  of  t h e i r development i n terms o f the necessary s e l f - r e g u l a t o r y functions  which presumably are common to a l l of them.  o g i c a l l y , the possible  analysis  should guarantee that as many of  sources of "decalages  the various  tasks.  present study diet not  (see  regulatory  attempt to t e s t f o r the  2 on p. 30),  Laurendeau (1969)  but  rather  transitivity)  and  it  was  and  relations,  t h e i r presumed u n d e r l y i n g  f u n c t i o n s , when a number of automaton aspects  a c q u i s i t i o n of l o g i c a l  s t a t e d i n the previous s e c t i o n s  mathematical concepts s e r i a t i o n , and  are  i n the  child's  concepts.  M e t h o d o l o g i c a l Aspects - T h e o r e t i c a l  attempted to measure:  self-  Such an approach should provide  a more comprehensive p i c t u r e of what i s i n v o l v e d  (1)  the  Considerations  this investigation  r e l a t i o n s h i p s among v a r i o u s  ( m u l t i p l i c a t i o n of c l a s s e s  t r a n s i t i v i t y ) that  involve  which presumably develop c o n c u r r e n t l y , o f these concepts to the b a s i c s i g h t and  structuring  and  ( m u l t i p l i c a t i o n of c l a s s e s  a l s o taken i n t o c o n s i d e r a t i o n .  As  be  the degree o f r e l a t i o n s h i p betx^een s t r u c -  t u r a l l y r e l a t e d tasks s e r i a t i o n , and  automaton demands should  sense that Pinard  above q u o t a t i o n  l i m i t e d to a s s e s s i n g  across  tests.  c r i t e r i o n i n the s t r i c t require  the  are minimized or e q u a l i z e d  1  That i s , the  comparable f o r a l l the The  Methodol-  and  logico-  of r e l a t i o n s ,  i n t e r r e l a t e d groupings  and  (2) the  relationship  self-regulatory functions,  f o r e s i g h t , that t h e o r e t i c a l l y are  responsible  hind-  for their  37  development. In order to accomplish these goals i t was attempt  necessary to  to c o n t r o l f o r as many of the f a c t o r s t h a t , as d i s c u s s e d  above, might produce  confounded  results.  I t was  thus necessary  to i n d i c a t e what the competence and automaton requirements o f each t e s t were i n order to be able to s p e c i f y what the b a s i s f o r the r e l a t i o n s h i p The  (or l a c k of i t ) might  be between them.  t e s t s i n v o l v e d three types of t a s k s :  with the s t r u c t u r a l concepts  (1) those d e a l i n g  ( 2 ) those d e a l i n g with the  r e g u l a t o r y f u n c t i o n s (3) those d e a l i n g with another  self-  strictly  automaton v a r i a b l e o p e r a t i o n a l l y d e f i n e d o u t s i d e of a s t r u c t u r a l context. Structural  A general a n a l y s i s of each i s next c o n s i d e r e d . Tasks  Since a l l o f the s t r u c t u r a l tasks d e a l t w i t h a set o f c l o s e l y r e l a t e d conceptual o p e r a t i o n s , i t i s necessary to s p e c i f y the b a s i s o f t h e i r c o m p a r a b i l i t y both i n t h e i r competence and maton requirements.  auto-  The competence demands were equated by hav-  i n g a l l the tasks deal w i t h l o g i c a l groupings from the same conceptual f a m i l y such as the groupings i n v o l v e d i n l o g i c a l classification skills.  The tasks were:  (1) a m u l t i p l i c a t i o n o f  c l a s s e s task -- b i u n i v o c a l m u l t i p l i c a t i o n of c l a s s e s  grouping  (2) a. m u l t i p l i c a t i o n o f r e l a t i o n s task -- b i u n i v o c a l  multiplica-  t i o n of r e l a t i o n s grouping (3) a s e r i a t i o n of r e l a t i o n s task -a d d i t i o n of asymmetrical  r e l a t i o n s grouping (4) a t r a n s i t i v i t y  task -- i n v o l v i n g a p r o p e r t y o f the a d d i t i o n of asymmetrical t i o n s grouping.  A c l o s e r a n a l y s i s o f the l e v e l of d i f f i c u l t y  the s p e c i f i c r u l e s i n v o l v e d i n each task i s made under the  relaof  specific  38  task  analysis the  keeping to  be  of  automaton constant  dealt  processing  a l l  the  set  by  the  solved,  come of  set of  internal occur  have  on  been  effort  to  been: duced  by  reflect  either the  mnemonic or  rather  in  Piagetian These  which  to  the  the  clarify  constraints, that  out  make  recent  in  the  the  memory  recognitory, level over  according theory)  questions  to  of  time the  research  transformations  a  "response";  operations years,  the  specific  that  his  codes  as  reproductive,  of or  specific  the  changes  are  special  in  events,  does  (2)  cognitive  of  an  have pro-  means, the  mnemonic  that,  significance  the  reflect  evocative  according  laxvs  in  concern  past  ana  developmental  focus  aspects  such  of  operativity? to  that  collaborators  codes  questions code  the  decoding  mnemonic  mnemonic  i s ,  has  or  characterize  and  and  memory  which  processes  that  Piaget  (encoding  govern of  follows.  the  to  or  analysis  Piagetian  out-  necessity  possible  in  the  the  the  be  limita-  the  memory  extent  the  are  unobservahle In  to  may  what  and  for  constraints  produce  theoretical  traditionally  finding  that  a  refer  had  making  problem  further  of  vary  in  refer  To  child's  code  same  operations.  particularly  does  the  may  intelligence,  (1)  requirement  subject  "stimulus"  uncover  operative  by  the  box."  and  (2)  which  interested  cognition)  ways  that  and  in  ways  those  "black  response  by  (1)  aspects  requirements  consider  a  ways:  response  mechanisms  been  human  or  in  two  dimensions,  the  task  towards  between  of  of  types  oriented  of  in  quantitative  requirements  mental  could  the  number  output  equated  Processing  the  two  One  has  in  were  tasks  i.e.,  output  his  these  been  form  and  demands  and  tasks.  concept.  across  with,  the  tions  each  (as  laws  modeled  structures?  because  of  their  39  relevance to a very important theory.  d i s t i n c t i o n which i s made i n P i a g e t ' s  According to the theory there are two aspects o f c o g n i -  tive functions:  the o p e r a t i v e aspect  ( a c t i o n s and o p e r a t i o n s )  which bear on the t r a n s f o r m a t i o n s of r e a l i t y themselves,  and the f i g u r a t i v e aspects  -- the o p e r a t i o n s  (perception, imitation,  images) I'/hich are l i m i t e d to the s t a t i c r e p r e s e n t a t i o n o f s t a t e s without  r e f e r e n c e to the t r a n s f o r m a t i o n s t h a t r e s u l t  states.  i n those  Mnemonic e v o c a t i o n , s p e c i f i c a l l y being p o s s i b l e through  r e c a l l o f mental images, e t c . f a l l s under the f i g u r a t i v e  realm.  Mnemonic e v o c a t i o n thus p r o v i d e s a v a l u a b l e means o f t e s t i n g the r e l a t i o n s h i p between memory f u n c t i o n s and o p e r a t i o n s , as f o r example, through  the comparison o f the r e c a l l  of memory f o r  s p e c i f i c c o n f i g u r a t i o n s i n v o l v i n g s t r u c t u r a l contents a c t u a l understanding  of the concepts  or o p e r a t i o n s  and the  represented  i n the c o n f i g u r a t i o n . In the a c t u a l experiments r e p o r t e d by Inhelder f o r c o n f i g u r a t i o n s o f v a r i o u s Icinds ( c l a s s i f i c a t i o n ,  (1969) , memory seriation,  outcome o f p h y s i c a l events, etc.) were compared w i t h a c t u a l o p e r a t i v e attainment group o f c h i l d r e n  o f the concepts  involved.  For example, one  ( 3 t o 8 years o f age) was presented with  con-  f i g u r a t i o n s such as groups o f s t i c k s o f 9 to 15 cm i n l e n g t h arranged  i n s e r i e s from s m a l l e s t to l o n g e s t , and t o l d merely to  look c a r e f u l l y at the s t i c k s and that they i^ould have to remember them l a t e r . series  ( i n order to determine t h e i r l e v e l o f o p e r a t i v i t y ) , and  yet another (all  Another e q u i v a l e n t group was t o l d to c o n s t r u c t the  group v.;as asked  t o imagine and draw what the s t i c k s  i n d i s a r r a y ) would look l i k e xvhen put i n o r d e r .  week, the f i r s t  After a  two groups were asked to i n d i c a t e by gesture and  40  by drawing what they remembered. again on t h i s o c c a s i o n .  The s e r i e s were not presented  S i x tb eight months l a t e r these  children  were asked to draw again from memory what they had p r e v i o u s l y Seen.  In each i n s t a n c e a f t e r the c h i l d r e n f i n i s h e d t h e i r memory  df'awiiigs, they were given s t i c k s and asked to make the s e r i a t i o n themselves  so that t h e i r l e v e l o f o p e r a t i v i t y c o u l d be determined.  Drawings a f t e r a week showed d i f f e r e n c e s among the v a r i o u s age l e v e l s that c o u l d be c l a s s i f i e d of o p e r a t i v i t y . responded to imagine  i n terms o f the v a r i o u s l e v e l s  In f a c t , the v a r i o u s types o f drawings cor-  c l o s e l y w i t h those obtained from the group asked  just  and draw (without a c t u a l l y seeing) the c o n f i g u r a t i o n  o f s t i c k s , and w i t h the a c t u a l types o f s e r i a t i o n s obtained from the group asked to m a t e r i a l i z e h i s own arrangement o f the s t i c k s . A f t e r the s i x to eight months i n t e r v a l , the memory drawings o f most c h i l d r e n  (74% of the group as a whole, and 90% between 5  and 8 years) showed o p e r a t i v e progress r e l a t i v e to the f i r s t drawing.  T h i s progress was alv/ays g r a d u a l , that i s t o say,  s u b j e c t s progressed from one sub-stage to the next and never more than t h a t . Inhelder (1969) i n t e r p r e t e d these r e s u l t s to mean that "...the memory image i s not a simple r e s i d u e o f the p e r c e p t i o n o f the model, but r a t h e r a symbol that corresponds to the schemes o f the c h i l d . . . the a c t i o n schemes -- i n t h i s p a r t i c u l a r case, the schemes of s e r i a t i o n -- c o n s t i t u t e the code f o r memorizing: t h i s code i s m o d i f i e d d u r i n g the i n t e r v a l and the m o d i f i e d v e r s i o n i s used as a new code f o r the next e v o c a t i o n . A t each stage the memory image i s symbolized a c c o r d i n g to the c o n s t r a i n t s o f the corresponding code." (op. c i t . , p. 343) In other words, I n h e l d e r ' s data seem to i n d i c a t e that memory does not conform  to the p e r c e p t u a l c o n f i g u r a t i o n o f the model but  41  r a t h e r to the manner i n which the model i s a s s i m i l a t e d to operative  schemes o f the  subject.  This t h e o r e t i c a l r a t i o n a l e was study because i t was s t r u c t u r a l tasks  f u r t h e r t e s t e d i n the  used as a b a s i s f o r equating  i n terms of t h e i r p r o c e s s i n g  response demands.  the  and  S p e c i f i c a l l y , the s u b j e c t ' s  measured f o r a l l s t r u c t u r a l tasks  i n two  the  various  output  operative  ways.  present  or level  was  F i r s t , i n terms  of h i s r e c a l l of a s t a t i c c o n f i g u r a t i o n that represented a c e r t a i n s t r u c t u r a l concept -- p r o c e s s i n g such as the  demands i n terms o f memory --  c o n f i g u r a t i o n of a n i n e - c e l l matrix r e p r e s e n t i n g  m u l t i p l i c a t i o n of r e l a t i o n s concept.  The  terms of the accuracy o f r e c o n s t r u c t i o n the c o n f i g u r a t i o n i t s e l f . the  subject's  required  to mentally  d i r e c t i o n s the  initial was  -- output demands --  Second, i t was  of  measured i n terms of  perform t r a n s f o r m a t i o n s i . e . , the  c o n f i g u r a t i o n i s no  on i t not c a r r i e d  subject was r e q u i r e d to reverse  r e l a t i o n s present  in  i n the matrix a f t e r the  longer v i s i b l e .  In t h i s manner i t  p o s s i b l e to o b t a i n simultaneous measures of o p e r a t i v i t y  based on the s u b j e c t ' s of the  stimulus  understanding of the f i g u r a t i v e aspects  s i t u a t i o n and based on h i s understanding of  a c t u a l operations  embodied i n the  same task  explained  transitivity  requirements were made by a p p l y i n g  above.  The  the  stimuli.  For every s t r u c t u r a l t e s t ( e x c e p t the the  measured i n  accuracy o f c o n s t r u c t i o n of the same matrix when  out or seen b e f o r e , two  r e c a l l was  a  the  task),  constraints  procedure i s d i f f e r e n t from Inhelder's  in  that the r e c a l l i n t e r v a l i n t h i s case i s very s h o r t ; the c h i l d i s required  to r e c o n s t r u c t  seen i t . The  the matrix immediately a f t e r having j u s t  s p e c i f i c administration  procedure used f o r each  task  is  i n d i c a t e d under the  d e s c r i p t i o n and  procedure s e c t i o n of  each  test.  Foresight As and  and  ?-iindsight F l e x i b i l i t y  discussed  f o r e s i g h t are  developmenti  i n the  the b a s i c  according  t i o n of l o g i c a l  previous  Task  theoretical section,  self-regulatory functions  t o P i a g e t , has  structures.  w h i c h make p o s s i b l e t h e  whose  r o l e i n the  forma-  I t i s these i n t e r r e l a t e d functions  ability  i n the  a causal  hindsight  formations  involved  reflective  abstraction required  to mentally  logical  perform the  operations  i n the  and  formation  trans-  thus a t t a i n the of the  struc-  tures . The to the  competence e x p r e s s e d i n t h e s e a b i l i t i e s  use  t a s k s , but  of e x p l i c i t  r u l e s s u c h as  r a t h e r t o more g e n e r a l  c o g n i t i v e d e v e l o p m e n t and contexts tions. sight  other  those i n v o l v e d  abilities  which should  However, the c l e a r l y has  aspects of  (1)  task  of t r a n s f o i m a t i o n a l  directly  of the allowed  opera-  i n f o r e s i g h t and  means o f a t e s t t h a t  ments w e r e net:  t e s t was  in  hind-  understand-  stimuli.  m e a n i n g i n s u c h a way  teristics  manifested  m a n i f e s t e d i n the  t h e i r conceptual  the  logical  characterize  above, the p r e s e n t study attempted to  t h e s e i n t e r r e l a t e d a b i l i t i e s by  o f the  t h u s be  competence r e q u i r e d  t o i n v o l v e o r be  I n view o f the  use  that  refer  i n the  than those u s u a l l y c h a r a c t e r i z i n g l o g i c a l  ing of t r a n s f o r m a t i o n a l  the  does n o t  t h a t the  evaluate  translated  following require-  requirements e x p l i c i t l y c a l l e d f o r  abilities  i n d e p e n d e n t and c o n t e n t s o f the  (2)  totally logical  the  specific  content  d i f f e r e n t from the tasks  (3)  the  charac-  test  f o r a q u a n t i t a t i v e d i f f e r e n t i a t i o n between  the  43  various  levels  of  attainment.  The c o n c e p t u a l reciprocal  abilities  information i n the  used i n  present,  the p r e s e n t about with the  or  future the  stimuli of  six  final  outcome  asked to in  the  changes  of  the  cards)  all  gradually  that  the  the  two c a r d s taken well  as  (b)  extend  to make judgments --  into the  task  noticed  hold  the  in  attention  step,  hindsight  account  the of  the --  all  the  the  the  first  --  to  chanees  foresight  --  the  change  cards  such  in  the  To  information  continuously  a s s e s s how many o f to  the  others  indicated  cards.  simultane-  characteristic(s).  (a)  and was  these  step  the n e c e s s a r y  with  stimulus  two s t i m u l i  from a l l  had to  which  indicated  of  final  subject  and how many he had y e t  anticipate  six,  changes  a  the  series,  of  of  indicated  as w e l l  each s u c c e s s i v e  all  a task  (involving  continuation  l e d to  changing  subject  into  was p r e s e n t e d  a series  from  present  requirement  a series  subject  one o f  the  activities  the  transformational  the  account  transformations  c a r d was d i f f e r e n t to  in  The g e n e r a l  r e q u i r e d the  in  to  was t r a n s l a t e d  the  for  into  two  information  consideration  such that  (and c o n t i n u o u s l y  into  to  last  consideration  respect  each s u c c e s s i v e --  ability  s e r i e s by e x t e n d i n g  this  changes)  simultaneously  of  relation  and s u c c e s s i v e l y  each s u c c e s s i v e  series with  track  and t h e  two s t i m u l i  into  of  take  in  two c a r d s  by t a k i n g  to  i n these  past  In other words,  Specifically,  (or  first  ( s i x t h card)  complete  ously hold  past  involved  ability  characteristics.  two c a r d s  first  immediate  is  e x t e n d and c o n t i n u e  given.  first  the  s u c c e s s i v e outcomes  total  the  hindsight,  and f o r e s i g h t ,  following  was a l s o  --  immediate  t a s k was t o  i n the  meaning o f what  in  the  keep first  t h o s e he had incorporate),  degree of  for  change  already as for  44  each next card by a s s e s s i n g step o f the s e r i e s .  The  the changes i n d i c a t e d i n the  requirements f o r h i n d s i g h t  a b i l i t i e s x^ere thus combined i n one decided  (at l e a s t  o p e r a t i o n a l task.  i n t u i t i v e l y ) , both of these  i n v o l v e r e c i p r o c a l , i f not work* Inhelder separate  and f o r e s i g h t  to fbllox-/ such a procedure because c o n c e p t u a l l y  operationally  and Piaget  I t xvas and  also  functions  the same, mental a c t i v i t y .  In  their  (1964) found t h a t , according  to  their  measures of these a b i l i t i e s , the l e v e l of one  on the l e v e l of the other.  final  In t h i s regard  Inhelder  and  depended Piaget  (1964) s p e c i f i c a l l y s t a t e : "As soon as an a s s i m i l a t o r y schema becomes r e t r o a c t i v e , i t a l s o takes an a n t i c i p a t o r y c h a r a c t e r , because one cannot be c o n s i s t e n t with the past without e v e n t u a l l y making choices and forming i n t e n t i o n s as to the f u t u r e . " (op. c i t . , p. 286) The  s u b j e c t ' s l e v e l of attainment or degree of  i n h i s h i n d s i g h t and was  foresight a b i l i t i e s  flexibility  (as d e f i n e d by t h i s  measured by the number of changes t h a t c o u l d simultaneously  s u c c e s s i v e l y be c u l t y by sions  co-ordinated.  The  from one  to f i v e per card.  sight a b i l i t i e s  This i n c r e a s e  v a r y i n g dimeni n the number of  a q u a n t i t a t i v e assessment of h i n d s i g h t and  a parameter that served  and  and  the o p e r a t i v e  expressed i n q u a n t i t a t i v e terms. other methodological  procedure s e c t i o n .  aspects  The  fore-  as a b a s i s f o r a  q u a n t i t a t i v e comparison between the l e v e l s a t t a i n e d i n the maton f u n c t i o n s  and  s e r i e s were v a r i e d i n d i f f i -  i n c r e a s i n g the number of simultaneously  changes provided  and  task)  auto-  l e v e l s , which were also specific description, scoring,  of t h i s task are d i s c u s s e d  i n the  45  Other Automaton There  Tasks  have b e e n s e v e r a l  attempts i n the l i t e r a t u r e  to  relate  t h e c o m p e t e n c e d e s c r i b e d by P i a g e t t o o t h e r more g e n e r a l f a c e t s o f c o g n i t i v e development  ( G o l d s c h m i d , 1967;  Dudek, L e s t e r , G o l d b e r g , and D y e r , 1 9 6 9 ) . course, r e f l e c t  Goldschmid,  1968;  These a t t e m p t s , o f  the d e v e l o p m e n t a l p s y c h o l o g i s t s ' e f f o r t s  to  u n d e r s t a n d P i a g e t ' s " e p i s t e m i c c h i l d " beyond the r e a l m o f  logical  concepts that P i a g e t mainly deals w i t h .  that  ever-present interest general, overall content  in defining  such, a t t e m p t  Pascual-Leone  intelligence  i n terms  be  and S m i t h , 1969)  applied.  model t o i n t e r p r e t o r  s t a g e s i n terms  scale of i n t e l l e c t u a l tation,  the dimension o f t h i s  informational point  of view.  ordinal  From P a s c u a l - L e o n e ' s  s t a t e m e n t about  interpre-  s c a l e corresponds t o the  would  a n a l y s i s , each  appear  t o be  " t h e number" o f d i f f e r e n t  subject's  stage-model  a qualitative  schemes ( i . e . , d i f f e r e n t  o f i n f o r m a t i o n ) on w h i c h t h e s u b j e c t c a n s i m u l t a n e o u s l y  operate.  The  author proposes  analysis of intellectual he  According to t h i s  ordinal  c o m p l e x i t y o f the t a s k c o n s i d e r e d from the  o f o p e r a t i o n a l development  chunks  possibility  s t a g e n o t i o n o f c o n s t r u c t i n g an development.  translate  o f an i n f o r m a t i o n - p r o c e s -  s i n g model w h i c h can e x p r e s s q u a n t i t a t i v e l y the i n Piaget's  of the  i s P a s c u a l - L e o n e ' s ( P a s c u a l - L e o n e , 1970 ;  Piaget's developmental  inherent  of  a b i l i t i e s w h i c h a r e somewhat i n d e p e n d e n t  i n w h i c h t h e y may  One  I t also r e f l e c t s  that such a c o n t e n t - f r e e ,  development  structural  i s s u g g e s t e d by P i a g e t when  states: '...the r e a l problem i s t h a t o f the a s s i m i l a t i v e mechanisms o f t h e c o o r d i n a t i o n o f schemes: how ;  46  the s u b j e c t comes to c o o r d i n a t e by means of r e c i p r o c a l assimilations., several behavioral segments i n t o a s i n g l e superordinate b e h a v i o r . " (Piaget j, 1967b, p. 203 ; Pascual-Leone's t r a n s lation) PasCual-Lebne's model thus attempts t o tative  quanti-  i n c r e a s e s i n what he l a b e l s as the c h i l d ' s " c e n t r a l com-  p u t i n g space," or F-Operator. aspect  measure the  T h i s c o n s t r u c t i s then an automaton  c h a r a c t e r i z i n g the s u b j e c t which r e f e r s to the  i n the c h i l d ' s mental or a t t e n t i o n a l span.  One  increases  test of this  model has been through a task i n which the s u b j e c t i s r e q u i r e d to make as many responses as p o s s i b l e to nested of v a r i o u s complexity are presented seconds.  (from two  to  e i g h t nested  compound  stimuli  s t i m u l i ) which  at b r i e f i n t e r v a l s v a r y i n g between f i v e to ten  For each stimulus  a l l s u b j e c t s have learned the  appro-  p r i a t e response, t h e r e f o r e , d i f f e r e n c e s i n the number of responses made to the compound s t i m u l i are presumed to r e f l e c t the maximum number of d i s c r e t e "Chunks" o f j e c t can i n t e g r a t e i n a s i n g l e a c t . p r e d i c t i o n s have been obtained  i n f o r m a t i o n the  sub-  Successful quantitative  about the s i z e of the mental span  o f c h i l d r e n of d i f f e r e n t developmental l e v e l s  (Pascual-Leone,  1979). The  relevance  to the present v a r i a b l e and  o f the c o n s t r u c t of a c e n t r a l computing space  study  i s that i t deals s t r i c t l y w i t h an automaton  thus provides  a p o i n t of comparison between the  q u a n t i t a t i v e o f performance demands w i t h i n the context t u r a l tasks  ( i . e . , Ss' performance i n the l o g i c a l  of s t r u c -  tasks)  and  q u a n t i t a t i v e performance demands independent of a s t r u c t u r a l . Context  ( i . e . , Pascual-Leone's t a s k ) .  An a n a l y s i s of the  child's  l e v e l of d p e r a t i v i t y i n terms of h i s mental span as d e f i n e d by  Pascual-Leone might lead to a b e t t e r understanding of such as, why  i s i t that a c h i l d who  c a t i o n task i n v o l v i n g two  questions  s u c c e s s f u l l y solves  a classifi-  dimensions (which would imply that  he  possesses the c l a s s i f i c a t o r y s t r u c t u r e ) cannot do so when the task i n v o l v e s f o u r dimensions? i s the m a n i f e s t a t i o n  Or more g e n e r a l l y , to which  of o p e r a t i o n a l  information-processing  demands?  schemes a f u n c t i o n of  To x^hich extent  o f g e n e r a l i z a t i o n between operations  i s the  extent  the  degree  a l s o a f u n c t i o n o f these same  demands? The  d i a g n o s t i c procedure so f a r o u t l i n e d i n v o l v e s an  ment of s t r u c t u r a l and  f u n c t i o n a l aspects o f c o g n i t i v e  ( l o g i c a l tasks) whose v a r i a t i o n across  subjects  assess-  operations  i s expressed i n  q u a n t i t a t i v e terms that presumably can make them comparable w i t h other  tests.  In a d d i t i o n to these,  a f t e r Pascual-Leone's (from the  a mental span t e s t  (1970) procedure i s i n c o r p o r a t e d  same subjects  fashioned to  obtain  at the same time) a "purer" measure of  t h e i r performance or automaton c a p a b i l i t i e s that may  provide  a  f u r t h e r b a s i s f o r a q u a n t i t a t i v e comparison. Aside from the p o s s i b i l i t y of a q u a n t i t a t i v e comparison of the various  o p e r a t i o n a l aspects o f P i a g e t i a n Tasks, the  used i n P a s c u a l - L e o n e s 1  interest.  I t allows  i n a context  (1970) t e s t has  procedure  another t h e o r e t i c a l  comparison of the c h i l d ' s memory c a p a c i t y  i n which the memory requirements are i n terms of  the time l i m i t s set i n the exposure of the c h i l d ' s memory c a p a c i t y when there  are no  s t i m u l i , with  such time c o n s t r a i n t s ,  i . e . , the task requirements of the s t r u c t u r a l task r e c o n s t r u c t i v e memory.  the  i n terms of  48  S p e c i f i c Goals o f D i a g n o s t i c  Procedure  From the t h e o r e t i c a l and methodological  c o n s i d e r a t i o n s pre-  sented above, i t i s p o s s i b l e to summarize more s u c c i n c t l y what the s p e c i f i c aims o f the study were: (1) Measure the r e l a t i o n s h i p between the degree o f a t t a i n ment i n the s e l f - r e g u l a t o r y f u n c t i o n s which are p o s t u l a t e d to make p o s s i b l e the development o f logico-mathematical o p e r a t i o n s , test of foresight  and h i n d s i g h t a b i l i t i e s ,  and the l e v e l o f  achievement i n the l o g i c a l tasks themselves.  I f indeed the t e s t  o f f o r e s i g h t and h i n d s i g h t a b i l i t i e s measured the p r e r e q u i s i t e f u n c t i o n s o f logico-mathematical c l o s e correspondence  o p e r a t i o n s , there should be a  between the l e v e l s of attainment  i n both  types o f tasks and, i t should be p o s s i b l e to v b t a i n some i n d i c a t i o n o f rvhat l e v e l o f attainment  i n the h i n d s i g h t - f o r e s i g h t  abilities  i s necessary f o r the development o f l o g i c a l o p e r a t i o n s . ( 2 ) hieasure the degree o f r e l a t i o n s h i p between a f a m i l y o f logico-mathematical  operations which are presumed to develop  c o n c u r r e n t l y through procedures  that make them comparable, both  i n terms o f t h e i r competence and automaton demands, i n an attempt to minimize  as many sources of'decalages'as p o s s i b l e .  I f the  above, procedures were e f f e c t i v e , then there should be a c l o s e correspondence  f o r each s u b j e c t i n h i s l e v e l o f attainment  i n these  tasks. (3) Compare l e v e l s o f a t t a i n e d o p e r a t i v i t y i n the l o g i c o mathematical  tasks through procedures which emphasize the f i g u r a t i v e  aspects o f the s t i m u l i versus procedures o p e r a t i v e aspects.  which emphasize the  49  ( 4 ) Compare the q u a n t i t a t i v e measures o f a l l of the above to an automaton measure which i s presumed to assess the s u b j e c t s ' general i n f o r m a t i o n - p r o c e s s i n g  c a p a c i t y i n q u a n t i t a t i v e terms.  CHAPTER I I I Method Subjects JJs were 32 f i r s t - g r a d e (range: 6.5 - 3.4) o f 7.93  w i t h a mean age o f  7.02  and 32 second-grade c h i l d r e n w i t h a mean age  (range: 7.4  - 9.0).  males, h a l f females. in  children  H a l f the Ss i n each grade were  The sample was  a lower-middle c l a s s d i s t r i c t .  drawn from a primary s c h o o l  About h a l f o f the s c h o o l  pop-  u l a t i o n comes from homes where Greek or I t a l i a n were the primary languages.  Of the G4 Ss who  participated  i n t h i s study, 2 7 were  Greeks o r I t a l i a n s . Design For  each group, Grade 1 (Gl) and Grade  were i n t e r c o r r e l a t e d , for  2 (G2), a l l measures  thus o b t a i n i n g one c o r r e l a t i o n a l m a t r i x  each group.  General procedure The tasks were a d m i n i s t e r e d i n d i v i d u a l l y i n two  sessions,  w i t h an i n t e r v a l o f 2h weeks between the two s e s s i o n s . s e s s i o n took between 30 t o 50 minutes. the  Foresight-Hindsight  (MC) tasks were a d m i n i s t e r e d .  always the middle task, w h i l e FORHIN was  h a l f the Ss, and MC  first  f o r the other h a l f .  s e s s i o n , the M u l t i p l i c a t i o n o f R e l a t i o n s (MOPER), and the S e r i a t i o n was  first  The f i r s t for  In the second  (KR) , the 1-T-Operator  (SER) tasks were a d m i n i s t e r e d .  always the middle task while SER was  and HR was  session,  (FORHIN) , T r a n s i t i v i t y (TRAIJ) , and the  M u l t i p l i c a t i o n of Classes TRAN task was  In the f i r s t  Each  f o r the o t b e r h a l f .  first  MOPER  f o r h a l f the Ss  51  Tasks M u l t i p l i c a t i o n of Glasses Task (MCj The ing  competence r u l e of t h i s task i s represented i n the group-  of biunivocal multiplication of classes.  A logical  class  can be c o n s i d e r e d as a union of elements brought together on the b a s i s o f an e q u i v a l e n t r e l a t i o n , i . e . , a l l members o f c l a s s A possess p r o p e r t y a, a l l members o f c l a s s B possess  property b .  members o f a c l a s s may have more than one d e f i n i n g p r o p e r t y , and t h e r e f o r e i t i s p o s s i b l e to simultaneously c l a s s i f y a set o f elements i n terms o f more than one c r i t e r i o n . The m u l t i p l e c r i t e r i a o f a s e t o f elements can be simultaneo u s l y represented by means of a matrix which s p a t i a l l y the equivalences  and correspondences  t h a t may e x i s t between a l l  the members o f the c l a s s e s (See F i g . l a ) . the matrix  i s that each c e l l  sions t h a t remain constant columns.  symbolizes  The r u l e embodied i n  i s the i n t e r s e c t i o n of the dimen-  across the r e s p e c t i v e rows and  In order to understand  the complementary aspects o f the  matrix, the S has thus to focus simultaneously on the equivalences and d i f f e r e n c e s along the two axes. elements used i n the present  The arrangement o f  study was made i n such a way that  p e r c e p t u a l l y the equivalences between the elements r a t h e r than the d i f f e r e n c e s would be emphasized.  In t h i s manner, i t was  hoped that the task would be maximally d i s s i m i l a r i n i t s f i g u r a t i v e aspects from the M u l t i p l i c a t i o n o f R e l a t i o n s (MR) task (See Fig.  2a) . Inhelder and P i a g e t  (1964)  found t h a t , when the t e s t i n g of  the M u l t i p l i c a t i o n o f Classes concept complete one c e l l  i s done by having the S  o f 2 x 2 or 2 x 3 m a t r i c e s , the s p a t i a l c o n f i g u r a -  t i o n may  lead to correct  perceptual  symmetries  s o l u t i o n s which  o f the h o r i z o n t a l  are s i m p l y based and v e r t i c a l  on  axes  the  rather  t h a n on an o p e r a t i o n a l u n d e r s t a n d i n g o f t h e p o s s i b i l i t y  of  multiple  assessed  classifications  of objects.  The  present study  the c h i l d ' s understanding of the m u l t i p l i c a t i o n concept  through h i s performance  matrix task. possibility  by  T h i s was  way:  t h e t e s t was  done, f i r s t ,  by  measure),  (COPY measure),  m e n t a l l y r e v e r s e the s p a t i a l a g a i n i t i s no Materials  t o d e a l w i t h ; and  - The  and  (2)  t h r o u g h t h e S's  immediately a f t e r  ( 3 ) t h r o u g h t h e S's  ( R e v e r s e o r REVB of 45  materials consisted a 3 0 x 2 3 cm  a 3 x 3 - c e l l matrix.  The  white  ( 4 ) background  d i a m o n d s -- 0 , 1 , ness  -- t h i n  large --  to  cm)  l a y e r of 1.5  (7 layers) .  plastic  The  cm  divided differ-  ( 1 ) Shape  small (2.6x2.6  size (3)  color  l i n e s , black dots, plain  correct  ability  i n d i v i d u a l p i e c e s each had  (6.4x6.4  (4 l a y e r s ) , t h i c k coating.  i t i s no  individual  -cm),  -- b l u e , p i n k , (5)  2 b l a c k d i a m o n d s on c e n t e r o f shape  ( o n l y one  of  ability  g r a y c a r d b o a r d w h i c h was  squares, rectangles (2) cm),  one  measure).  e n t v a l u e s on e a c h o f t h e f o l l o w i n g d i m e n s i o n s .  medium ( 4 . 5 x 4 . 5  second,  c o n f i g u r a t i o n o f t h e m a t r i x when  longer v i s i b l e  c a r d b o a r d p i e c e s and  triangles,  of  i n c r e a s i n g t h e number o f  t h e S had  the m a t r i x c o n f i g u r a t i o n  longer v i s i b l e  the  c o m p l e t i o n o f t h r e e ( i n s t e a d o f one)  (FILL-IU  the d i a g o n a l c e l l s  into  designed to minimize  u n d e r s t a n d i n g o f t h e m a t r i x i n more t h a n  ( 1 ) t h r o u g h t h e S's  to reproduce  classes  on t h e p e r c e p t u a l s y m m e t r i e s  and number o f d i m e n s i o n s  t e s t i n g t h e S's  classes  on a m u l t i p l i c a t i o n c f  of s o l u t i o n s based  the m a t r i x . cells  Hoxvever,  of  superimposed (6)  thick-  t h i c k c a r d b o a r d ) , medium  A l l shapes were c o v e r e d v/ith arrangements  of the a p p r o p r i a t e o r  53  THIN  THICK  MEDIUM  WHITE  6  BLUE  PINK  MEDIUM:  THICK  • THIN  PINK  BLUE  WHITE  • : : •• »•  F i g . 1. M u l t i p l i c a t i o n o£ Classes matrix ( ) and r e v e r s e d matrix ( b ) . Dimensions remaining constant on the h o r i z o n t a l axis s i z e , diamond, c o l o r ; and along the v e r t i c a l axis background, t h i c k n e s s , and shape. a  54  c o r r e c t p i e c e s i s shown i n F i g . l a . Procedure was  - S_ was  presented with the empty matrix board  and  shown the three t r a y s adjacent to the board i n each o f which  were the 15 s t i m u l i that corresponded to each one o f the three shapes. in  The S_ was  asked to p o i n t out a l l the dimensions present  the s t i m u l i by having him d e s c r i b e to h the s i m i l a r i t i e s  d i f f e r e n c e s between the s t i m u l i i n the three t r a y s . to  fill  in. p o s i t i o n s 1, 2,  and  E proceeded  4, on the matrix (See F i g . l a ) and  immediately asked S_ to i n d i c a t e the s i m i l a r i t i e s  and  differences  between s t i m u l i i n c e l l s 1 and 2, and i n 1 and 4.  Then E  i n p o s i t i o n 8 and asked S to i n d i c a t e s i m i l a r i t i e s  and d i f f e r -  ences between 2 and 8: then c e l l s 6 and 9 were f i l l e d asked f o r comparisons 5, and 3 remained  between 8 and 9, and 4 and 6.  empty.  This i n i t i a l  filled  and E Cells  7,  procedure was f o l l o w e d  to make Ss aware o f e q u i v a l e n c e s among the dimensions present in  the p i e c e s . Fill-in  - After E finished  as d e s c r i b e d above, S_ was  filling  i n the matrix p a r t i a l l y  asked to look at the whole board  and  t r y to see i f he could f i n d from the p i e c e s i n the t r a y s , those that would f i l l fit  c e l l s 3, 5, and 7, such that each p i e c e would  w i t h the others i n the board both v e r t i c a l l y and  (axes i n d i c a t e d by g e s t u r e ) .  A f t e r S_ f i l l e d  horizontally  i n c e l l s , E asked  S to check the whole board j u s t i n case he might want to change any one of the p i e c e s . 1  A f t e r S_ i n d i c a t e d that he was  satisfied  with h i s c h o i c e s , E recorded the c h o i c e s and immediately gave S feedback as to the appropriateness of h i s choice by the l  S's c h o i c e s with the c o r r e c t ones,  From now  replacing  i f necessary, and  indica-  on t h i s f i n a l q u e s t i o n w i l l be r e f e r r e d to as Q.  55  t i n g why  the replacement  Reproduction E  p i e c e was  appropriate.  or Copy - A f t e r the c o r r e c t matrix was  t o l d S that she was  completed,  going to remove a l l the shapes and mix  them  vfith the others and that she wanted to see whether the S c o u l d f i n d them and put them back i n e x a c t l y the same p l a c e s as b e f o r e . The  S was  allowed to look at the matrix f o r as long as he wished  and When he i n d i c a t e d that he had  looked at i t long enough, a l l  the p i e c e s were removed and mixed thoroughly w i t h the r e s t of the shapes i n t h e i r r e s p e c t i v e t r a y s . performed  i n f u l l view of the S.  matrix, and was  The S proceeded  to f i l l  the  allowed t o change as many p i e c e s as he wanted,  w i t h no time l i m i t .  When S completed  The E recorded Ss c h o i c e s and feedback  A l l these o p e r a t i o n s were  the matrix, he was  immediately  asked  Q.  to give S  proceeded  as to the appropriateness of h i s c h o i c e s by  replacing  the i n c o r r e c t p i e c e s w i t h the c o r r e c t ones while at the same time e x p l a i n i n g why  the c o r r e c t c h o i c e s were the a p p r o p r i a t e ones.  Reversal - Immediately a f t e r the Reproduction to  task, E s a i d  S that he would have one more chance to f i n d the c o r r e c t p i e c e s  and t h a t again the p i e c e s would be removed. that he had  When the S i n d i c a t e d  looked at the f i l l e d matrix long enough, a l l p i e c e s  were removed except  the one  in c e l l  7, then S was  t o l d that he  had to put back the same p i e c e s i n e x a c t l y the same p l a c e s as before except that the board would t h i s time be turned around -•• E then turned the board to  189° so t h a t c e l l no.  7 now  corresponded  c e l l no. 3 (See F i g . l b ) . A n t i c i p a t i o n - Before S was  allowed to complete the  full  m a t r i x , he was  asked to f i n d which shape should go i n p o s i t i o n  T h i s was  i n order to assess the degree to which the S c o u l d  asked  6.  56  a n t i c i p a t e aspects  o f the reversed matrix before he a c t u a l l y  engaged i n the task. obtained  Such a measure o f a n t i c i p a t o r y a b i l i t y was  to f i n d out how e x p l i c i t  the t r a n s f o r m a t i o n s  r e v e r s e d matrix were f o r the S, and thus perhaps between those trial  o f the t o t a l  differentiate  s o l u t i o n s which might have been achieved  simply by  and e r r o r and those which were t r u l y o p e r a t i o n a l . A f t e r the S a n t i c i p a t e d the shape f o r p o s i t i o n no. 6, he  proceeded to f i l l  i n the whole matrix.  When he f i n i s h e d , E  asked Q and then proceeded t o r e c o r d S's c h o i c e s .  On t h i s  last  o c c a s i o n , no feedback was given, except that the S was t o l d  that  h i s choices were " f i n e " whether or not they were c o r r e c t . S c o r i n g o f the M u l t i p l i c a t i o n of Classes Fill-in  task  (MCFILL) - One p o i n t was given f o r each dimension  c o r r e c t l y s o l v e d out o f the s i x t o t a l p o s s i b l e i n each o f the three diagonal  c e l l s o f the matrix.  s i b l e was 18 p o i n t s Reproduction  Thus the maximum score pos-  (6 dimensions x 3 c e l l s ) .  (MCCOPY)  1  - The r a t i o n a l e u n d e r l y i n g  s c o r i n g procedure was the f o l l o w i n g .  this  A S, p a r t i c u l a r l y  who do not y e t possess the o p e r a t i v e s t r u c t u r e necessary the m u l t i p l i c a t i v e matrix,  those to solve  may attempt t o s o l v e i t simply by  l o o k i n g at the c o n s i s t e n c i e s across  rows only, o r at the c o n s i s t -  encies across columns o n l y , but not to both simultaneously.  To  maximize the chances o f d i f f e r e n t i a t i n g between "segmented" v s . " t o t a l " s o l u t i o n s t r a t e g i e s that the Ss might have employed, the c o m p i l a t i o n o f the f i n a l  score was made i n the f o l l o w i n g way:  ^ r o m here on, the score Reproduction i s r e f e r r e d to as COPY. This was done i n order to have an a b b r e v i a t i o n f o r t h i s score e a s i l y d i s t i n g u i s h a b l e from that used f o r the Reverse (REVE) s c o r e s .  57  (a) the t o t a l number o f dimensions (1 to 6) c o r r e c t l y across each and and  a l l of the three c e l l s  that number was  i n each roxv was  m u l t i p l i e d by three  (the number of  For example, i f a S c o r r e c t l y s o l v e d , f o r each one c e l l s of a row  (as opposed to j u s t one  or two  The  (b) The  t o t a l score  and  a l l the i n d i v i d u a l column scores were added.  and  (b) were added together  (dimensions)  and t h e i r sum  well S co-ordinated  Scores  i n d i c a t e d how  eous c r o s s - c l a s s i f i c a t i o n o f the o b j e c t s  row).  i s the  (a) and  obtained  number of c e l l s ) .  simultan-  and  over the e n t i r e matrix.  Thus to  (b) s c o r e s , a t h i r d score  and m u l t i p l i e d by nine  This (c) score  was  i n d i v i d u a l rows or columns. Reproduction  score  (MCCOPY) was  (c) s c o r e s .  f o r MCCOPY was  The  The  (the  total  i n d i c a t e d that the S's  s o l v e d dimensions over the t o t a l matrix  correctly  as opposed to over  f i n a l M u l t i p l i c a t i o n of C l a s s e s made up of the sum  of the ( a ) ,  maximum t o t a l number of p o i n t s p o s s i b l e  162.  Derived measures - The not  However,  (c) the number of dimensions c o r r e c t l y s o l v e d over the  e n t i r e matrix was  (b) , and  the  (i.e., correct c l a s s i f i -  c a t i o n of a l l s m a l l , medium, l a r g e / t r i a n g l e s , squares,  of the  well  s p e c i f i c values o f a dimension  f e a t u r e o f a m u l t i p l i c a t i v e matrix  r e c t a n g l e s ) that i s manifested  (a)  columns, or more s p e c i f -  ( i . e . , a l l small shapes p l a c e d i n the a p p r o p r i a t e  added:  of  same procedure if as f o l l o w e d f o r each column,  S succeeded i n the i n d i v i d u a l rows and  the above sum  three  from each of the three rows  added.  the c r i t i c a l  cells).  c e l l s ) , 3 out  was  i c a l l y how  obtained  of the  the 6 dimensions, he would then get a score of 9 (3 x 3 (cells) = 9)o  solved  i n d i c a t e , f o r the cases  t o t a l HCCOPY score j u s t d e s c r i b e d  does  i n which the S s o l v e s between 2 and 4  58  c o r r e c t dimensions dimensions  over the e n t i r e matrix, whether or how many-  of this t o t a l  dimensions.  i n v o l v e m u l t i p l i c a t i v e combinations of  Looking at Figure 1, i t w i l l be n o t i c e d that the  matrix i n v o l v e s three o v e r l a p p i n g dimensions  on each axis  (color,  s i z e , and diamonds on the h o r i z o n t a l a x i s , and t h i c k n e s s , background,  and shape on the v e r t i c a l one); thus a S_ can solve up  to three c o r r e c t dimensions  over the e n t i r e matrix and yet not  have a s i n g l e p o s s i b l e m u l t i p l i c a t i v e or i n t e r s e c t i o n between these dimensions, ground  and shape.  txvo dimensions,  combination  i . e . , i f he s o l v e s t h i c k n e s s , back-  By the same token, another S may only s o l v e  and have one p o s s i b l e m u l t i p l i c a t i v e  combination  between the two., e.g., i f S s o l v e s t h i c k n e s s and c o l o r . A d i s t i n c t i o n was made, t h e r e f o r e , between the t o t a l number of dimensions  c o r r e c t l y s o l v e d over the e n t i r e matrix  (range  from 0 to 6)--MCDIM, and the t o t a l number of m u l t i p l i c a t i v e combinations p o s s i b l e w i t h i n that t o t a l  -- MCMUL.  The  importance  of the d i s t i n c t i o n between the MCDIM and MCMUL scores i s the following.  For the most frequent i n t e r m e d i a t e cases (Ss who  c o r r e c t l y s o l v e l e s s than 5 or 6 dimensions  over the e n t i r e  m a t r i x ) , the MCDIi.-MCMUL d i s t i n c t i o n may d i f f e r e n t i a t e between (1) those Ss who, because  they have a good r e c a l l of the percep-  t u a l c o n f i g u r a t i o n o f the matrix, may c o r r e c t l y s o l v e a number o f dimensions  even though  they have not y e t a t t a i n e d an o p e r a t i o n a l  understanding o f the b i u n i v o c a l m u l t i p l i c a t i o n concept, and ( 2 ) those Ss who possess an understanding o f the m u l t i p l i c a t i v e concept, but whose a p p l i c a t i o n o f i t may be l i m i t e d i n terms o f the number o f m u l t i p l i c a t i v e combinations taneously handle at one time.  that they can simul-  59  The  number o f p o s s i b l e m u l t i p l i c a t i v e combinations out o f the  t o t a l number o f dimensions c o r r e c t l y s o l v e d i n the f o l l o w i n g way: across  the v e r t i c a l  (MCMUL) was  obtained  the number o f dimensions c o r r e c t l y s o l v e d  axis was m u l t i p l i e d by the number o f dimen-  sions t h a t were c o r r e c t l y solved across was p o s s i b l e to o b t a i n scores  the h o r i z o n t a l a x i s .  It  such as 0 -- when no dimensions  were C o r r e c t l y solved or when a l l the dimensions c o r r e c t l y s o l v e d were along one axis o n l y , up to 9 -- when a l l three along each a x i s were c o r r e c t l y s o l v e d . p o s s i b l e scores was:  The t o t a l  dimensions  range o f  0, 1, 2, 3, 4, 6, and 9.  In summary, the two measures d e r i v e d from the t o t a l MCCOPY score were: (1) MCDIM -- t o t a l number o f dimensions c o r r e c t l y s o l v e d over the e n t i r e matrix,  range, 0 to 6 p t s .  (2) MCMUL -- t o t a l number o f m u l t i p l i c a t i v e combinations p o s s i b l e w i t h i n the t o t a l number o f dimensions c o r r e c t l y s o l v e d , range, 0 to 9 p t s . Reverse (MCREVE) - A s c o r i n g obtained way as f o r the copy s e c t i o n was obtained  e x a c t l y i n the same  f o r reverse-scoresi,  except t h a t i n t h i s case the c o r r e c t c r i t e r i o n was the reversed matrix  (See F i g . l b ) .  A l s o , a score was obtained  for anticipa-  t i o n (MCANT). Anticipation  (MCANT) - One p o i n t was given  the dimensions, out o f the t o t a l  s i x p o s s i b l e , c o r r e c t l y solved  i n c e l l no. 6 o f the r e v e r s e d matrix. scores was from  The p o s s i b l e range o f  1-6.  M u l t i p l i c a t i o n o f R e l a t i o n s task The  f o r each o f  (MR)  competence r u l e o f t h i s task i s embodied i n the s t r u c t u r e  60  of the b i u n i v o c a l m u l t i p l i c a t i o n o f r e l a t i o n s grouping.  The  con-  c e p t u a l operations i n v o l v e d i n t h i s task are q u i t e s i m i l a r to those o f the MC task except  that i t i s the r e l a t i o n s or d i f f e r -  ences among c l a s s elements that are emphasized  (as opposed to  t h e i r equivalences) as the elements are put together or on the b a s i s of those  differences.  arrangement;, however, symbolizes  The m u l t i p l i c a t i v e  not only the s e r i a t e d  ces, but a l s o the equivalences betxveen the o r d i n a l ences of sets of s e r i a t e d elements (See F i g . 2 a ) .  seriated matrix differen-  correspondThus the  o p e r a t i o n a l or conceptual r u l e of a m u l t i p l i c a t i o n of r e l a t i o n s task as represented i n a m u l t i p l i c a t i v e matrix i s very c l o s e to that of the MC task: S_ has  to comprehend the matrix c o n f i g u r a t i o n the  to simultaneously focus on the d i f f e r e n c e s and  that occur across rows and  equivalences  columns.  M a t e r i a l s - The m a t e r i a l s c o n s i s t e d of the same 3 0 x 2 3 matrix board used i n the MC task and 4 5 cardboard  cm  equilateral  t r i a n g l e s , each having d i f f e r e n t values on each o f the f o l l o w i n g ( 1 ) color  s i x dimensions:  dark, medium, l i g h t (4.5x4.5  cm),  (3)  ( 2 ) b r i g h t n e s s --  -- pink, gray, blue  size  -- small  l a r g e ( 6 . 4 x 6 . 4 cm)  (4)  (2.6x2.6  cm),  t h i c k n e s s -- t h i n  l a y e r of 1 . 5 cm cardboard), medium ( 4 l a y e r s ) , t h i c k (5)  number of l i n e s  of the t r i a n g l e s  medium (one  ( 7 layers)  -- 1 , 2 , C T 3 h o r i z o n t a l l i n e s at the base  ( 6 ) o r i e n t a t i o n -- small superimposed yellow  t r i a n g l e p o i n t i n g to the r i g h t , upwards, or to the l e f t . t r i a n g l e s were covered with p l a s t i c c o a t i n g .  The  All  c o r r e c t arrange-  ment of the c o r r e c t t r i a n g l e s i s shown i n Figure 2 . Procedure  - The procedure  that of the MC task and  was  identical  i t a l s o comprised  i n every respect to  the f o l l o w i n g s c o r e s :  61  THIN/DARK  MEDIUM/MEDIUM.  THICK/LIGHT  BLUES  REDS  GRAYS  THICK/LIGHT MEDIUM/MEDIUM THIN/DARK  GRAYS  REDS  BLUES  w  w  V  F i g . 2 . M u l t i p l i c a t i o n o f R e l a t i o n s matrix (a) and r e v e r s e d matrix ( b ) . Dimensions remaining constant along, h o r i z o n t a l axis s i z e , o r i e n t a t i o n , c o l o r ; and along v e r t i c a l a x i s t h i c k n e s s , b r i g h t n e s s , and l i n e s .  62  (1) F i l l - i n  -- MRFILL  number o f dimensions  ( 2 ) Reproduction -- MRCOPY  c o r r e c t l y s o l v e d over the e n t i r e matrix ( 4 ) number of m u l t i p l i c a t i v e  -- F i g u r a t i v e MCDIM(F) of  (3) T o t a l  combination  dimensions p o s s i b l e i n the MRDIM(F) score -- F i g u r a t i v e  MCMUL(F)  (5) Reverse  ( 6 ) T o t a l number o f dimensions  -- MCRZVE  c o r r e c t l y s o l v e d over the e n t i r e r e v e r s e matrix -- Operative MRDIM(O)  (7) t o t a l number of m u l t i p l i c a t i v e combinations  poss-  i b l e i n MR'DIM -- Operative MRMUL(O), and ( 8 ) A n t i c i p a t i o n (MRANT). S e r i a t i o n task (SER) The competence r u l e of t h i s task i s based on the s t r u c t u r e of  the a d d i t i o n of asymmetrical  r e l a t i o n s groupings.  the task d e a l s w i t h the asymmetrical  r e l a t i o n s among  Conceptually elements  t h a t can be expressed i n terms o f ordered d i f f e r e n c e s . it  That i s ,  deals with r e l a t i o n s h i p s that can e x i s t between elements when  these are ordered i n terms o f t h e i r i n c r e a s i n g or d e c r e a s i n g r e l a t i v e magnitudes.  In order f o r a S t o group the d i f f e r e n c e s  among a s e t o f elements that any one element  i s i n r e c i p r o c a l and simultaneous  s h i p with those elements given element  i n an ordered s e r i e s , he has to r e a l i z e  that f o l l o w and precede  relation-  i t , such that a  can be at the same time i n a " b e f o r e " and " a f t e r "  r e l a t i o n with the succeeding and p r e c e d i n g ones. The u s u a l procedure is  that has been used t o t e s t t h i s  t o have the S f r e e l y attempt  t o order a set o f elements  10 s t i c k s ) and then r e q u i r e him to i n s e r t e x t r a elements series.  concept  The c r i t e r i o n o f o p e r a t i o n a l achievement  (i.e.,  i n the  i s defined i n  terms o f the S's c o n s t r u c t i o n o f the s e r i e s without t r i a l and e r r o r (by immediately smallest  and s y s t e m a t i c a l l y s e a r c h i n g f o r the  (and/or b i g g e s t ) element  and thus proceeding with the  63  r e s t o f the remaining elements), and i n terms o f whether the S can  i n s e r t new elements i n the s e r i e s again without t r i a l and  error. To keep the task demands congruent w i t h those used i n the MR and MC t a s k s , the c h i l d ' s understanding o f t h i s concept was again  assessed i n terms o f h i s memory o f the f i g u r a t i v e aspects  of the c o n f i g u r a t i o n  (reproductive  of h i s a b i l i t y to mentally in  reverse  the s e r i e s or the o p e r a t i v e  The  memory) as w e l l as i n terms the r e l a t i o n s h i p s i n v o l v e d  aspects o f the c o n f i g u r a t i o n .  performance or automaton demands were s e t up also i n terms  of the number o f dimensions i n the s e r i e s that the S was r e q u i r e d to c o - o r d i n a t e . Materials "little  - The s t i m u l i c o n s i s t e d o f a s e t o f 36 cardboard  men" made up o f 1.5 cm t h i c k cardboard  which o n l y 9 c o n s t i t u t e d the c o r r e c t s e r i e s . v a r i e d on the s i x f o l l o w i n g dimensions: s t i m u l i ranged i n height ence between s u c c e s s i v e  (See F i g . 3) of The " l i t t l e  men"  (1) Height — t h e  from 6.4 cm. to 10.2 cm and the d i f f e r elements was o f 4 cm  (2) Width -- the  width v a r i e d from 2.1 cm to 6.4 cm  (3) O r i e n t a t i o n o f eyes --  the eyes were r o t a t e d at s u c c e s s i v e  angles o f 30° s t a r t i n g with  the eyes o r i e n t e d s t r a i g h t dox^nward ( s t i m u l i 1) , and ending i n the  l e f t bottom quadrant a f t e r a 270° turn  Location line  ( s t i m u l i 9 ) . (4)  o f t i e -- the t i e was lowered from the center h o r i z o n t a l  i n successive  6-cm  stops  (5) Width o f f r o c k -- the  h o r i z o n t a l l i n e s i n the bottom h a l f o f l i t t l e xtfidth from 3 cm t o 2.5 cm  men v a r i e d i n  (6) Thickness -- only three  degrees  o f t h i c k n e s s xvere used, t h i n (2 l a y e r s o f 1.5 cm t h i c k cardboard), medium (4 l a y e r s ) , t h i c k (7 l a y e r s ) .  In the c o r r e c t s e r i e s , the  Fig. 3. S e r i a t i o n task. Stimuli varied in height, w i d t h , t h i c k n e s s , eyes' r o t a t i o n , width of f r o c k , and p o s i t i o n of t i e .  65  first  three elements were t h i n , the middle  and the l a s t three were t h i c k . p l a s t i c coating.  There was  A l l s t i m u l i xvere covered w i t h  a l s o a board  17.-1/2 * 4-3/4"  numbered 1 to 9 along the upper and lower Procedure  - The  w i t h 12 t h i n l i t t l e  S_ was men,  three were medium,  first  edges.  presented with three t r a y s  (one  another one with 12 medium ones, and  the l a s t one with the 12 f a t ones).  The  S was  made aware of the  c h a r a c t e r i s t i c s of the s t i m u l i by having him d e s c r i b e to E the d i f f e r e n c e s between two to  of t h e l i t t l e  men.  Then the E  make up the s e r i e s by p l a c i n g the s h o r t e s t of the  s t i m u l i i n p o s i t i o n #1 of the board.  proceeded  correct  E then n l a c e d the next  s t i m u l i on p o s i t i o n #2 and as he proceeded,  he i n d i c a t e d how  six  T h i s was  dimensions  until  were s u c c e s s i v e l y changing.  a l l t h e nine p o s i t i o n s x\rere Reproduction  S was  asked  the  continued  filled.  or Copy - A f t e r the s e r i e s was  completed,  to look at i t c a r e f u l l y because the l i t t l e  men  the were  going to be put back on the r e s p e c t i v e t r a y s and the E wanted to see whether he c o u l d remember them and put a l l the nine men  back i n e x a c t l y the same way.  little  A f t e r S looked as long as he  wanted, the S removed the s t i m u l i and mixed them w i t h the others in  t h e i r appropriate trays.  S_ was  allowed to make up a. s e r i e s  and as before he could change the p i e c e s as many times as he wanted.  'Then S i n d i c a t e d he he had  f i n i s h e d E asked 0, and i f  no changes were made, E recorded the S's ceeded to g i v e S feedback  c h o i c e s and then pro-  as to the accuracy o f h i s choices by  r e p l a c i n g the i n c o r r e c t p i e c e s w i t h the a p p r o p r i a t e ones, \/hile at  the same time i n d i c a t i n g where necessary why  inappropriate.  h i s c h o i c e s were  66  Reverse - When the whole c o r r e c t s e r i e s was the S_ was  completed  t o l d that he would have one more chance to f i n d  nine l i t t l e  men  and to look at them c a r e f u l l y again.  again, the  When S  i n d i c a t e d he had looked enough, the s t i m u l i were removed but t h i s time the board was  turned around and S was  the s e r i e s but t h i s time  s t a r t i n g w i t h the l i t t l e  i n p o s i t i o n #9, xvhich now way, was  the S was  asked  was  p l a c e d i n no.  to complete  men p r e v i o u s l y  1 position.  In t h i s  f o r c e d to reverse a l l the r e l a t i o n s h i p s .  allowed to complete the s e r i e s ,  f i n i s h e d E asked Q and to  r e c o r d the S's  to  the appropriateness  The  when S i n d i c a t e d he  S  had  i f no changes were made, E proceeded  choices.  This time no  feedback was  of the choices and the S_ was  given as  simply  told  that h i s choices were " f i n e . " S c o r i n g of S e r i a t i o n Task Reproduction following:  (SERCOPY) - This score was  (a) one p o i n t was  comprised  of the  given f o r each dimension  correctly  s o l v e d at each i n d i v i d u a l p o s i t i o n and the t o t a l number of p o i n t s for  each of the nine p o s i t i o n was  added  was  a l s o o b t a i n e d by m u l t i p l y i n g the number of dimensions  c o r r e c t l y s o l v e d across the e n t i r e s e r i e s  (b) an o v e r a l l  (range  from 0 to 6),  times the t o t a l number of i n d i v i d u a l p o s i t i o n s i n the Scores  (a) and  the t o t a l score  series.  (b) were added to make the SERCOPY s c o r e . or the S e r i a t i o n task combined how  v i d u a l p i e c e s or " l i t t l e as how  score  men"  c o r r e c t l y across the e n t i r e Reverse (SEREVE) - The as the SERCOPY except  many i n d i -  the S_ c o r r e c t l y arranged  many c o r r e c t dimensions i n a l l the s t i m u l i he  Thus,  as w e l l  co-ordinated  series. score was  o b t a i n e d i n the same  t h a t the c o r r e c t c r i t e r i a was  way  the s e r i e s i n  6  reversed  7  order.  Derived measures - From SERCOPY, the F i g u r a t i v e s c o r e , a SERDIM(F) score was of  obtained, which i n d i c a t e d the t o t a l number  dimensions c o r r e c t l y s o l v e d over the e n t i r e s e r i e s .  SEREVE, the Operative  score, a S E R D I M ( O ) score was  From  obtained  which  i n d i c a t e d the t o t a l number of dimensions c o r r e c t l y s o l v e d i n the reversed s e r i e s . T r a n s i t i v i t y task The ing  (TRAN)  competence r u l e of t h i s task i s a l s o based on the group-  o f a d d i t i o n of asymmetrical  i s t e r e d to t e s t i n a d i r e c t SER  relations.  and d i f f e r e n t way  t a s k , the S's understanding  asymmetrical  relations.  The  The  task was  from t h a t of the  of the r e c i p r o c a l nature  task requirements  admin-  for this  of task  were not e x a c t l y the same as f o r the three previous ones, as w i l l become apparent from the d e s c r i p t i o n of the This measure, t h e r e f o r e , was obtained  procedure.  intended to supplement the  one  from the S E R task and not as an e q u i v a l e n t to the S E R  task. M a t e r i a l s - The Fig.  4 ) which  shirts  s t i m u l i c o n s i s t e d of four " l i t t l e  v a r i e d i n height and  (bottom h a l f of l i t t l e  men).  d i f f e r e n c e s i n height  to be d i r e c t l y  with each other to determine which one was (Sj , purple bottom) was tom)  8 . 3 cm t a l l ,  S  2  taller.  7 . 3 cm  Procedure - The  S was  compared  Stimuli 1  (white p o l k a d o t t e d bot-  and S 3 (pink bottom) were the same height  (blue bottom) was  (See  i n the c o l o r of t h e i r  The  were small enough that the s t i m u l i had  men"  - 7 . 6 cm,  and  Sn  tall. first  shown a l l the four l i t t l e  i n scrambled order and t o l d that t h i s if as going to be  men  a memory  F i g . 4. T r a n s i t i v i t y t a s k . The r e l e v a n t c h a r a c t e r i s t i c s t o the t r a n s i t i v i t y judgments were h e i g h t and c o l o r of bottom h a l f ( " s h i r t " ) o f " l i t t l e men."  69  E put away a l l s t i m u l i under a board and took out o n l y S i  game.  and S2 and asked S t o check to see which one o f the two was t a l S proceeded to compare S i x^ith S2 and i f he only s a i d  lest.  something  like "this  i s the b i g g e s t , " E would, repeat a f t e r him  "So the p u r p l e one i s t a l l e r than the white one." This was done to  give S v e r b a l l a b e l s which, would f u r t h e r d i f f e r e n t i a t e the  stimuli.  E emphasized  to S_ that he had t o remember what he had  j u s t seen (that the p u r p l e l i t t l e  man was t a l l e r than the white  one), and then proceeded to remove the purple one ( S i ) , l e a v i n g the  white one ( S  2  ) .  E took out S 3 (the pink one) and repeated  e x a c t l y the same procedure.  E then removed 82, l e a v i n g S  3  out  and took out S i * , r e p e a t i n g the previous procedure once again. Thus, and  S  S  3  compared s e q u e n t i a l l y the p a i r s of s t i m u l i , S1-S2,  - S i t .  A f t e r the l a s t comparison, E t o l d  ask you some q u e s t i o n s . "  2  - S  3  ,  "Nov; I want to  E a l i g n e d together S i and S ^ under a  board so that only the bottoms o f S i and S ences i n h e i g h t were v i s i b l e of  S_,  S  2  showed and no d i f f e r -  and asked S to p o i n t out which one  the titfo he thought was the t a l l e s t .  .After he i n d i c a t e d xvhich  one he thought x-/as t a l l e s t , E removed the board to l e t the S_ see whether he xvas r i g h t or xtfrong.  When the 5 p o i n t e d to the r i g h t  s t i m u l i , E xvould ask S , "How d i d you f i g u r e that out without looking?"  The S ' s j u s t i f i c a t i o n was scored as a p p r o p r i a t e i f he  based h i s ansxver on the f a c t that he remembered that S i was the tallest  o f a l l the l i t t l e  men, or that S i , was the s h o r t e s t .  E  then removed a l l s t i m u l i from the S_'s viex/ and then took out S 3 and S i again i n the same way as b e f o r e and repeated the same procedure.  F i n a l l y , E took out S  2  and S i » and again repeated the same  procedure as above, r e c o r d i n g the S ' s choice and j u s t i f i c a t i o n .  70  S c o r i n g of T r a n s i t i v i t y Task Of the three comparisons asked vs. S i , and S2 vs. S 4 ) , the f i r s t to  one  of S (Si vs. Si», S 3  (Si vs. S^)  c o u l d be  said  i n v o l v e a simple d i s c r i m i n a t i o n t a s k , s i n c e Si and Si* were the  first  and  the l a s t s t i m u l i presented to S.  Thus to d i m i n i s h the  p o s s i b i l i t y of i n c l u d i n g as t r a n s i t i v i t y choices those t h a t might be based on a simple d i s c r i m i n a t i o n or primacy-recency the c r i t e r i o n of whether the TRAN task was  passed  effect,  or not was  on whether the S_ c o r r e c t l y s o l v e d a l l three comparisons.  based  Thus,  the data obtained from t h i s task were b i n a r y ( p a s s - f a i l ) and comparisons with other tasks were made by means of x  2  the  tests.  F o r e s i g h t - H i n d s i g h t task (FORHIN) As was  d i s c u s s e d i n the t h e o r e t i c a l s e c t i o n , the  or competence r u l e of t h i s task was child's ability  stimulus cards for of  d e f i n e d i n terms o f the  to simultaneously take i n t o account  u l t a n e o u s l y extend  conceptual  and thus  the v a r i o u s changes i n d i c a t e d i n the f i r s t  (of a set of s i x ) , such that the  the set (which was  board  also i n i t i a l l y  - These c o n s i s t e d o f :  two  transformations  each s u c c e s s i v e card would g r a d u a l l y l e a d to the f i n a l  Materials  sim-  card  given). (1)  a f o l d i n g 6 3 . 3 x 4 5 . 9 cm  i n the upper center of which there were s i x 5 . 1 x 5 . 1 cm boxes  drawn c l o s e together  ( . 2 cm apart) i n a h o r i z o n t a l alignment  (2)  (the number of cards i n each set v a r i e d from  1 1 sets o f cards  9 to 1 4 ) , one  t r a i n i n g set and 1 0 t e s t i n g s e t s .  These s e t s o f  cards represented changes i n the dimensions of v a r i o u s s t i m u l i , the d i f f i c u l t y o f the s p e c i f i c set being determined of  by the number  dimensions that simultaneously changed i n the s t i m u l i .  example, set no.  For  1 (the t r a i n i n g set) c o n s t i t u t e d the e a s i e s t  level,  71  s i n c e o n l y one dimension changed through the e n t i r e s e r i e s . t r a i n i n g s e t showed i n the f i r s t ing  card  This  (See F i g . 5) a c a n d l e - r e s t -  i n h o r i z o n t a l p o s i t i o n w i t h the flame towards the l e f t  side.  The second card showed the candle r a i s e d at a 30° angle from the p o s i t i o n i n card no. 1 ( C i ) ; the l a s t  card showed the candle  r e s t i n g h o r i z o n t a l l y again, but t h i s time the flame was right  side.  on the  The S had to choose from the choice cards, which  i n c l u d e d some i n c o r r e c t choices, those that would complete the changes i n i t i a t e d  i n Ci and C  g r a d u a l l y l e a d to C 6 . o r i e n t a t i o n , changed f o l l o w i n g changes  2  such that C , 3  Ci» , and C 5 would  Thus i n t h i s s e t o n l y one dimension,  from Ci to C 6 .  The t e s t i n g sets i n v o l v e d the  (See Appendix):  - Set No. 2, one change  i n the shape  - Set Ho.  i n the o r i e n t a t i o n  3, one change  (Sh)  - Set No. 4, two changes, i n b r i g h t n e s s  (0)  (B) and s i z e ( S i )  - Set No.  5, two changes, i n Sh and 0  - Set Mo.  6, three changes, i n Sh, S i , and 0  - Set Ho.  7, three changes, i n Sh, B, and 0  - Set No.  8, four changes, i n 0, Sh, S i , and number  - Set No.  9, four changes, i n 0, N, S i , and Sh  - Set Ho.  10, f i v e changes, i n N, 0, 3, S i , and Sh  - Set MO.  11, f i v e changes, i n N, 0, B, S i , and Sh  The c h o i c e cards were made up i n such a way  (N)  that~<^here were  o n l y three cards that would meet the f o l l o w i n g c r i t e r i a : the  same number of changes  \1)  have  i n the dimensions i n d i c a t e d i n the, f i r s t  two c a r d s , and ( 2 ) have the degree o f change  a p p r o p r i a t e f o r the  given p o s i t i o n s , i . e . , the c o r r e c t c a r d f o r p o s i t i o n no. 5 o f the training series,  f o r instance  (See F i g . 5) would not be the appro-  72  Fig.  5.  F o r e s i g h t - H i n d s i g h t task.  Training  set.  73  p r i a t e one f o r p o s i t i o n no. 3, even though the dimension changing, o r i e n t a t i o n , would have changed i n the c o r r e c t d i r e c t i o n . the  i n c o r r e c t choice  cards i n c l u d e d  Thus,  i n c o r r e c t changes i n the  dimensions ( i . e . , candle going i n the opposite  d i r e c t i o n ) , as  w e l l as incomplete changes (one dimension changing but not the other,  f o r sets w i t h mors than one dimension changing).  The  most important aspects o f the FORHIN task then were that the S had  to remember that a l l the changes i n d i c a t e d i n the f i r s t  cards had to occur i n each o f the three  successive  that these changes had t o occur i n gradual  two  c a r d s , and  steps which c o u l d  s u c c e s s i v e l y l e a d to the l a s t card o f the s e r i e s . The  s p e c i f i c s t i m u l i i n the sets most f r e q u e n t l y  consisted  of a b s t r a c t shapes which d i d not i n v o l v e r e p r e s e n t a t i o n s o f f a m i l i a r transformations. in  a b a l l that s p l i t s  card are shown turned curved s i d e s .  For example, s e t no. 2 showed changes  i n t o two equal h a l v e s , which i n the l a s t upside down touching  each other  Reproductions o f the sets are i n c l u d e d  on the i n the  Appendix. The  number o f choice  cards i n c r e a s e d  as a f u n c t i o n o f the  l e v e l o f d i f f i c u l t y o f the s e t . T h i s was r e q u i r e d because as the number of dimensions that were necessary to increase discriminate  changing were i n c r e a s e d ,  i t was  the number o f p o s s i b i l i t i e s that would  that indeed the S was f o c u s i n g on a l l the changes  involved. Procedure - S was f i r s t  shown the board, then E t o l d S that  t h i s was going to be a card game. the saw.  E placed  the f i r s t  card o f  t r a i n i n g set on box no. 1 and asked S to d e s c r i b e what he Then E placed  card 2 and asked S to i n d i c a t e the changes  74  t h a t had testing  occurred  i n C a r d 2 ( f r o m C a r d 1).  sets, particularly  those  E recorded  w i t h more t h a n  one  dimension  c h a n g i n g ) t h e number o f c h a n g i n g d i m e n s i o n s t h a t t h e ously pointed indicate series  and  again  first  two  cards  go  w i t h these just the  S t h a t he 4,  cards  4,  cards."  and  and  Then E s a i d , E then  b e l o w t h e b o x e s and 3,  now  had  ;  Again,  Let's  d i d not he  in l i t t l e  you've l o o k e d  very  made an e r r o r , he fit  there  and  S to search  was  then  again  ceeded to p l a c e  immediately  gave t h e and  the  to  him.  and  i f you the  so y o u  For the told  i f S_ c o u l d n o t  correct card  noticed  can  t h a t the  i n the  before  have t o  Then E  correct place.  told  t h a t f r o m now  his  own  and  and  f o l l o w e d the  a l l t h r e e boxes were f i l l e d .  After this  training  on he w o u l d h a v e t o t r y t o f i n d  t h a t t h e E was  not  allowed  going  to help.  same p r o c e d u r e as b e f o r e  pro-  This  s e t , t h e E e m p h a s i z e d t h a t t h e c h a n g e s had cards.  not  card, E  this  f o r each o f the  check  card could  right  you  s e t , when S  continued  tinue  out  choice  p r o c e d u r e was training  cards  filling  why.  f i n d the  up  do i t  choice  training  reason  cards  end  i n s p e c t e d a l l o f the  specific  until  steps  a t a l l o f them f i r s t  carefully.").  the  i f the S d i d  allow 5 to s t a r t  s t a r t , b e c a u s e some o f them a r e w r o n g and them a l l f i r s t  card of  changes t h a t s t a r t e d  t r y t o see  had  then  to t r y to f i n d which  proceeded to place  5 boxes u n t i l  ('Make s u r e  one.  (6th)  spontane-  c h a n g e s he  5 such t h a t the  Card 2 would continue  as i n C a r d 6.  last  c h a n g e s , 5 w o u l d p o i n t them o u t  i n b o x e s 3,  C a r d 1 and  t o the l a s t  S  change, E would  asked S t o i n d i c a t e a l l the  this, E told  should  i f S m i s s e d any  F i n a l l y , E p l a c e d the  i n d i c a t e a l l the  After  in  and  i t t o S.  from the not  out,  ( f o r the  In  to  con-  p a r t , the S the  cards  E s t a r t e d Set  except t h a t the  S  was on 2  was  75  not helped while he made h i s c h o i c e s . change as many cards  as he wished.  S_ was allowed  t o t r y and  When S expressed  that he had  the cards he thought f i t t e d b e s t , E asked Q and, i f no changes were made, she recorded  the S's c h o i c e s .  give S feedback as to the accuracy i n c o r r e c t ones while inappropriate.  Then E proceeded to  o f h i s choices and change the  at the same time e x p l a i n i n g why they were  The s e t s were given i n order u n t i l  i n two consecutive  series.  the 5 f a i l e d  A f a i l u r e was d e f i n e d when two or  a l l three o f the cards were i n c o r r e c t . Scoring of Foresight-Hindsight This score was compiled  Task  i n the f o l l o w i n g way:  (a) a p o i n t  was given f o r each c o r r e c t change i n the ( 1 - 5 ) dimensions i n v o l v e d i n the t r a n s f o r m a t i o n a l s e r i e s that was c o r r e c t l y and  chosen;  ( b ) a p o i n t was a l s o given f o r the c o r r e c t change p l a c e d i n  the r i g h t p o s i t i o n .  Thus (a) gives the S c r e d i t  the change i n the a p p r o p r i a t e  f o r extending  d i r e c t i o n , and (b) gives S c r e d i t  f o r c o r r e c t l y e v a l u a t i n g the degree o f change.  For example, i n  the cases where the S might place card 4 or 5 i n p o s i t i o n 3 , he would get c r e d i t  f o r choosing  one r i g h t card t h a t i n d i c a t e d  changes i n the a p p r o p r i a t e d i r e c t i o n , but none f o r the p o s i t i o n . Scores  (a) and (b) were added.  The f i n a l d i s t r i b u t i o n o f p o i n t s  per s e t was as f o l l o w s : Sets  2 and 3 = 6 p t s . per s e t , 1 2 p t s . t o g e t h e r ( 3 x 1 (one change per p o s i t i o n ) + 3 ( 1 p o i n t p e r c o r r e c t placement = 6 p t s . per s e t )  Sets 4 and 5 = 9 p t s . each Sets 6 and 7 = 1 2 p t s . each Sets  8  [(3x2)+3],  [(3x3)+3],  and 9 = 1 5 p t s . each [ ( 4 x 3 ) + 3 ] ,  18  pts.  together  2 4 pts.  together  30  together  pts.  76  Sets  10 and 11 = 13 p t s . each [(5*3)+3], The  total  maximum t o t a l  36 p t s .  together  score p o s s i b l e was 120 p t s .  The FORHIN  score then c o n s i s t e d o f the sum o f the p o i n t s obtained i n  a l l o f the s e t s on which the S succeeded, as w e l l as the p a r t i a l p o i n t s on the l a s t two s o t s on which the task was d i s c o n t i n u e d . The  FORHIN score could a l s o be t r a n s l a t e d simply  i n terms o f a  score that i n d i c a t e d the number o f changing dimensions t h a t the S could co-ordinate. FORHIN scoro  This was done by t r a n s l a t i n g each S's  i n terms o f how f a r the score i n d i c a t e d that the S  had mastered a c e r t a i n l e v e l o f d i f f i c u l t y . f o r the v a r i o u s l e v e l s of d i f f i c u l t y  along the FORHIN s c a l e were:  | 0-X8j  19-42|  43-72|  73-108]  1  2  3  4  Thus, a S who obtained considered  The c u t t i n g p o i n t s  109-120 | 5  l e s s than 18 p t s . , i . e . , 12 p t s . , was  to be able to handle l e s s than one changing dimension  and h i s score was changed i n terms o f the f r a c t i o n , out o f the t o t a l maxi ium p o s s i b l e f o r that l e v e l o f d i f f i c u l t y , that the score i n d i c a t e d ( i . e . , 12 p t s . equal  .6, 29 p t s . equal  1.5, 49  p t s . equal 2.25, e t c . ) . The  range of p o i n t s f o r each l e v e l o f d i f f i c u l t y  from l e v e l  (i.e.,pts.  1 to l e v e l 2) was obtained by adding to the t o t a l  maximum number o f p o i n t s i n the FORHIN sets corresponding  to each  of the l e v e l s o f d i f f i c u l t y , the maximum number o f p o i n t s the S c o u l d o b t a i n i n the next l e v e l o r d i n a t e a l l the dimensions. sets #2 and #3 was 12 p t s . s i n c e a S could s t i l l  at which the S no longer c o u l d coFor example, tho t o t a l maximum f o r  S i x p o i n t s were added to t h i s  total  o b t a i n three p o i n t s i n s e t #4 and three  p o i n t s i n #5 even i f he f a i l e d i n these  two s e t s .  77  Level 1  0 to 12 + 6 =  0-18  Level 2  19 to 30 + 12 =  Level 3  43 to 54 + 18 = 43-72  Level 4  73 to 84 + 24 = 73-108  Level 5  109 to 120 + 0 = 109-120  19-42  M-Operator taslc (MOPER) This task was  modelled a f t e r the one d e v i s e d by  Pascual-Leone  (1970) i n order to t e s t a mathematical model which d e s c r i b e s the t r a n s i t i o n r u l e from one  stage o f development  to another.  Leone (1970) intended to measure the c h i l d ' s c e n t r a l space  -- H-Operator, which  Pascual-  computing  i s d e f i n e d i n terms o f the "number"  of d i f f e r e n t chunks o f i n f o r m a t i o n on which the S can simultaneously o p e r a t e .  In the present study, t h i s c o n s t r u c t was o p e r a t i o n -  a l i z e d by means of a task which, j u s t as Pascual-Leone's t a s k , r e q u i r e d the S to make as many responses as p o s s i b l e to m u l t i p l e or compound sets of s t i m u l i  ( v a r y i n g i n complexity from 2 to 8  s t i m u l i per compound s t i m u l i ) which were presented at b r i e f v a l s from 5 to 8 seconds. Pascual-Leone's were: dimension  inter-  The d i f f e r e n c e s between t h i s task and  (1) the s t i m u l i used were a l l w i t h i n  one  (drawn shapes), w h i l e Pascual-Leone used v a r i o u s  dimensions;  (2) the motor responses that the S_ had to make i n -  v o l v e d a s m a l l e r spatio-temporal range, i . e . , a l l responses were signs made w i t h f i n g e r s , w h i l e Pascual-Leone's task i n v o l v e d motor responses as d i v e r s e as r a i s e hand, kick basket, c l a p hands, e t c . ; (3) the complexity of the t a s k , as d e f i n e d by the E, was  the same f o r a l l S s, i . e . , a l l Ss xvere presented the same set  of s t i m u l i c a r d s , while Pascual-Leone v a r i e d the t e s t i n g s e t ac-  73  c o r d i n g to the t h e o r e t i c a l requirements mental l e v e l s concerning  of  the d i f f e r e n t  develop-  the s i z e o f the M-Qperator; (4) the s e t  o f t o t a l s t i m u l i presented was much s m a l l e r than that used by In t h i s task, f o r both Grade 1 and Grade 2 , a  Pascual-Leone.  t o t a l Of 3 5 t e s t i n g compound s t i m u l i cards were presented  while  Pascual-Leone used 6 2 and 74 cards r e s p e c t i v e l y f o r the Ss i n h i s sample o f e q u i v a l e n t ages. Materials  -• The m a t e r i a l s c o n s i s t e d of three d i f f e r e n t sets  of c a r d s , two t r a i n i n g sets and one t e s t i n g s e t . The t r a i n i n g sets c o n s i s t e d of simple of compound s t i m u l i .  s t i m u l i , while  The f i r s t  the t e s t i n g s e t c o n s i s t e d  t r a i n i n g s e t (Set 1 ) contained  s i x t e e n 1 5 . 2 x 1 0 . 2 cm cards d i v i d e d i n t o four equal cells.  For h a l f  cm)  (7.6 5.1 X  ( 8 ) of the cards i n three of the four c e l l s ,  there were p o s i t i v e instances  (one per c e l l ) o f one o f the e i g h t  signs to which the S was supposed to l e a r n a motor response, one negative  i n s t a n c e o f that same stimulus s i g n .  i n g h a l f , there were three negative  and  For the remain-  instances o f one o f the  signs to be i e a r n e d and one p o s i t i v e i n s t a n c e (See F i g . 6 b ) . The  second t r a i n i n g s e t (Set 2 ) c o n s i s t e d of s i x t e e n 1 5 . 2 x 1 0 . 2  cm cards a l s o d i v i d e d i n t o four equal c e l l s , except three of the C e l l s had a d i f f e r e n t negative  that each of  instance o f the  signs to be l e a r n e d and the f o u r t h c e l l had a p o s i t i v e  instance  (See F i g . 6 c ) . The t e s t i n g s e t (Set 3) c o n s i s t e d o f t h i r t y - f i v e 10.1  7 . 6 cm c a r d s , i n which there were 5 cards o f each of the  seven p o s s i b l e kinds of compound s t i m u l i compound  stimuli).  ( 2 , 3, 4, 5 , 6 , 7 , and 8  The compound s t i m u l i cards were made up o n l y  o f p o s i t i v e i n s t a n c e s of the stimulus signs (See F i g . 6 d ) . The s t i m u l i were not nested w i t h i n each other to make a s i n g l e  com-  STIMULI '. MOTOR .RESPONSE INDEX AND MIDDLE  (I)  FINGERS  CROSSED  '($)  . STIMULI  I^Sg?  P  INDEX AND: MIDDLE . FINGERS  SEPARATED  ALL FINGERS EXTENDED, FACING OUTWARDS A  L  M  I- , ' •  M  MOTOR RESPONSE  DENT THUMB  • '  i .  (3)  KNOCK ON TABLE  win  (?)  INDEX AND MIDDLE FINGERS TOGETHER TOUCHING NOSE  CLOSED FIST, FINGERS FACING OUTWARDS  ;:r/  :  Maxima  :  INDEX UPWARDS  (8)  JX'/. r  (b)  ir;;; - 'I  • ' •  '  y  :  (c)  :  T  '  1 * ^  "'•\.  (d)  F i g . 6. M-Operator t a s k . (a) S t i m u l i ' a n d response u n i t s , (b) Sample cards o f t r a i n i n g s e t 1 , (c) Sample cards o f t r a i n i n g ,set 2 , (d) Sample cards o f compound s t i m u l i cards. :. •<. • \• •  80  pound u n i t (as were Pascual-Leone's compound s t i m u l i ) but were  spaced i n random arrangements c l o s e to each other  attempting to reduce the "scanning" aspect  which the  rather  thus  nested  compound s t i m u l i c o n t r o l s f o r . The  signs were designed i n such a way  that t h e i r  pictorial  r e p r e s e n t a t i o n would somewhat resemble the motor response which corresponded to i t (See  Fig, 6a).  the  l e a r n i n g of the S-R  units.  was  designed p i c t o r i a l l y to resemble the response  w i t h open side turned exception  was  S,  s e s s i o n and  going to be  going to f i r s t  1 xtfas s t a r t e d .  E introduced  one  sign  to one  (E p o i n t e d  it.  teach him  see  know that you  i f you  see  t h i s one  do?"  pointed  i n s t a n c e ) , you  However, before  a s i g n language.  "Let s :  "But  see  one  are  sup-  i f you  doing  can  E gave feedback as to i f you  are not  p o s i t i v e i n s t a n c e , and  do  to another of the p o s i -  see one  like this  supposed to do  E then shoTved the matching card that had and  game" i n  identical positive  (E p o i n t e d  response.  instances  s t a r t e d by  a s i g n l i k e t h i s , you  accuracy of the S's  thing."  train-  have r e c e i v e d the message by  t i v e i n s t a n c e s ) , what would you  to negative  table.  the cards by s a y i n g , "Here i s  (E showed the motor response.)  And  fist  only  (a)  some s o r t of "spy  of the three  i n s t a n c e s ) ; so whenever you posed to l e t me  (closed  parts:  task was  going to send S_ s e c r e t messages.  Set  this."  The  4 (Si»)  a knock on the  assessment i n v o l v e d two  (b) t e s t i n g s e s s i o n .  d o i n g t h a t , E was  facilitate  outward) connected to i t . The  t e l l i n g S that t h i s was w h i c h E was  done to  For example, S t i m u l i No.  whose motor response was  3  Procedure - The ing  This was  three  the (E  anynegative  repeated the same pro-  81  cedure, each time g i v i n g S feedback as to the accuracy responses,  e i t h e r by c o r r e c t i n g him  and  of h i s  g i v i n g him the c o r r e c t  response, or by r e i n f o r c i n g him p o s i t i v e l y when he made the r e c t response. i f you  When E f i n i s h e d with Set. 1, she  remembered a l l of them."  d i f f e r e n t negative  Then she  to  have l e a r n e d the task when he was  16 cards o f Set 2 without  error.  s a i d , "Let's  p o s i t i v e instance) The  S was  and  considered  able to c o r r e c t l y s o l v e  (In no  see  s t a r t e d Set 2 (3  i n s t a n c e s and only one  again gave S feedback on each i n s t a n c e .  cor-  i n s t a n c e was  the  i t neces-  sary to run Set 2 more than three times, as the m a j o r i t y o f the Ss met  the c r i t e r i o n by the second run.)  c o r r e c t c r i t e r i o n , E s a i d , "Now t h i s time there w i l l  I will  be more than one  When S a t t a i n e d the  send you  the messages  s i g n per c a r d .  w i l l only show each card f o r a short l i t t l e  Hoxvever, I  w h i l e , and w h i l e I  show i t , you have to t r y to remember as many s i g n s i n the as you can.  When I put the card dov/n, you  back to me.  Since I don't know how  cards, I w i l l wait u n t i l you sending  s i g n s , before  can only s t a r t was  sending  s t a r t e d and  8 seconds and  tell  me  and  can  send the  card  signs  many s i g n s there are i n the that you have- f i n i s h e d  I send you the next message.  Remember you  signs a f t e r I put the card dovm." Set 3  each card was  presented  manually f o r about 5 to  a l l the responses e l i c i t e d were manually  recorded.  The  v a r i o u s cards with d i f f e r e n t combinations of compound stim-  uli  i n Set 3 x^ere a l l i n a random o r d e r .  order of the 35 cards was  However, the  final  the same f o r a l l Ss.  S c o r i n g of M-Operator task The number o f c o r r e c t responses to each compound s t i m u l i were t a l l i e d and  from them a frequency  d i s t r i b u t i o n was  card  obtained  82  of the number o f a p p r o p r i a t e responses that the S made per s t i m u l i card.  Thus f o r each S i t was c a l c u l a t e d how many times he made  1, 2, 3, 4, 5, 6, 7, and 8 a p p r o p r i a t e responses per c a r d .  The  data from compound s t i m u l i cards w i t h e i g h t s t i m u l i \rcre e l i m i nated because when some Ss saw these c a r d s , even though they might not remember many s t i m u l i from i t ,  they would tend to con-  t i n u e sending s i g n a l s simply because they knew that i n that card "there were many."  So, the t o t a l  compound s t i m u l i cards.  score was compiled from only 30  The f r e q u e n c i e s o f the responses f o r  each type o f compound s t i m u l i  (2, 3, 4, e t c . ) were d i v i d e d by  the number o f times the S had an o p p o r t u n i t y to make such number of responses.  Thus the response  f r e q u e n c i e s 1 (Ri) and 2 (R2)  were d i v i d e d by 3 0 because the S had 30 i n s t a n c e s to make j u s t one or two responses; response 25; response  frequency  by 15; response 7 ( R 7 ) , by 5.  frequency 3 ( R 3 ) was d i v i d e d by  4 ( R O , by 20; response  frequency 5  frequency 6 (Re), by 10; and response  frequency  In t h i s manner, each S obtained a frequency  t r i b u t i o n o f h i s v a r i o u s response  ( R i , R2, R 3 ,  classes  (R5),  dis-  etc.) i n  p r o p o r t i o n to the number of chances he had to make such a number o f responses o f response  f o r any one c a r d .  (or frequency c l a s s ) w i t h the h i g h e s t p r o p o r t i o n ,  plus a f r a c t i o n class.  The S's MOPER score was the type  (the p r o p o r t i o n ) i n the next h i g h e s t frequency  For example, i f a S obtained the f o l l o w i n g p r o p o r t i o n s :  Ri = .00, R  2  = .23, R  3  = . 4 0 ,  given a MOPER score o f 4.1.  = .55, R  5  = .13, then he was  In the cases where the S obtained two  consecutive equal p r o p o r t i o n s , he was given the score o f the mean frequency between the two consecutive frequency c l a s s e s , i ; e . , i f the S obtained the f o l l o w i n g p r o p o r t i o n s , Ri = . 0 0 , R2 = .44, R3  83  =  . 4 4 , R„ - .25, R  5  = .10, then he would get a MOPER score o f 2.5.  Summary o f the measures Foresight-Hindsight M-Operator  (FORHIN), range p o s s i b l e 0-120 and 0-5  (MOPER), range p o s s i b l e 0-7  M u l t i p l i c a t i o n o f Classes (MC) Operative (0) measures: Reverse (MCREVE), range p o s s i b l e 0-162 T o t a l number o f c o r r e c t dimensions (MCDIM(O)), range p o s s i b l e 0-6 T o t a l number o f m u l t i p l i c a t i v e combinations (MCMUL(O)), range p o s s i b l e 0-9 Anticipation Fill-in Figurative  (MCANT), range p o s s i b l e 1-6  (MCFILL), range p o s s i b l e 1-18 (F) measures:  Reproduction o r Copy (MCCOPY), range p o s s i b l e 0-162 MCDIh(F) , range p o s s i b l e 0-6 MCMUL(F), range p o s s i b l e 0-9 M u l t i p l i c a t i o n of Relations Operative Reverse  (MR)  (0) measures: (MRREVE), range p o s s i b l e 0-162  MRDIM(O), range p o s s i b l e 0-6 MRMUL(O), range p o s s i b l e 0-9 MRANT, range p o s s i b l e 1-6 MRFILL, range p o s s i b l e 1-18 Figurative  (F) measures:  MRCOPY, range p o s s i b l e 0-162 MRDIM(F), range p o s s i b l e 0-6 MRMUL(F), range p o s s i b l e 0-9  84  Seriations  (SER)  Operative  measures:  Reverse (SEREVE), range p o s s i b l e  0-108  T o t a l number of dimensions c o r r e c t l y s o l v e d  (SERDIM(O)),  range p o s s i b l e Figurative  (F) measures:  Reproduction or Copy (SERCOPY), range p o s s i b l e SERDIM(F), range p o s s i b l e Transitivity  (TRAN),  0-6  range p o s s i b l e  0-1  0-108  0-6  35  CHAPTER IV Results R e l a t i o n s h i p between F o r e s i g h t - H i n d s i g h t (FORHIN) scores and the scores on the l o g i c a l  tasks  FOREHIN and Operative measures - As i n d i c a t e d i n the Method s e c t i o n , Operative measures r e f e r to the Reverse (REVE) scores in  the l o g i c a l tasks as w e l l as to the F i l l - i n  ipatory and  (ANT)  scores.  (FILL) and A n t i c -  As shown i n Table 1, f o r both Grade 1 (Gl)  Grade 2 (G2), almost  a l l o f the Operative measures c o r r e l a t e d  very h i g h l y w i t h the FORHIN s c o r e s . MRFILL to r=.85 with MCMUL(0).  They ranged  from r=.41 w i t h  The exceptions were MRANT, MCFILL,  and MRFILL scores f o r G2 Ss which d i d not s i g n i f i c a n t l y with FORHIN. for  correlate  The l a c k o f r e l a t i o n s h i p between FORHIN and MRANT,  example, was mainly due to the f a c t that there was not very  much v a r i a t i o n among the MRANT scores (See Table 7) and thus no s i g n i f i c a n t c o r r e l a t i o n s were p o s s i b l e . the F i l l - i n  The sama was t r u e f o r  scores o f <J2 i n which, as shown i n Table 6, 21/32 o f  the 32 Ss obtained scores i n both MC and MR g r e a t e r than 2/3 o f the t o t a l p o s s i b l e (more than 12 p t s . out o f 18 p t s . p o s s i b l e ) . Thus the scores were c l u s t e r e d towards one end o f the d i s t r i b u t i o n . Of the Operative measures, the ones that had the h i g h e s t c o r r e l a t i o n s w i t h FORHIN were the Reverse scores (as opposed to the A n t i c i p a t o r y or F i l l - i n  s c o r e s ) , ranging from r=.66 w i t h  MRREVE t o r=.85 with MCMUL(0).  A l s o , whether o r not the TRAN  task was passed was also s i g n i f i c a n t l y r e l a t e d to the FORHIN score.  A x  2  t e s t was c a r r i e d out between the frequency  distribu-  t i o n o f the Gl and G2 FORHIN scores above and below the FORHIN median o f both groups combined quencies  (30 p t s . ) , and the P a s s - F a i l  f o r each grade i n the TRAN t a s k .  These x  2  tests  fre-  indicated  TABLE CORRELATIONS BETWEEN MC  FORHIN/  Gl  MR  1  FORHIN SCORES AND OPERATIVE(O) AND FIGURATIVE(F) ' MEASURES SER  MCDIM  MCMUL  MRDIM  MRMUL SERDIM MCFILL MRFILL MCANT MRANT MOPER  0  . 82'*** .84*** .75*** .82*** .82*** .82*** . gi *** .80***  F  .49**  .70*** .55**  0  .80***  .66***  .68*** .80***  . 85*** .75*** .  F  . 37*  .29  .55*** .40*  .48**  G2  .48*  .38*  FOR G l AND G2  .60***  .41*  .44* .61*** .44*  . 31  .45* .26  .68*** . 58*** .50**  .21 TABLE  .70*** .07 21  .59***  .51**  2  CORRELATIONS BETWEEN MOPER SCORES AND OPERATIVE(0) AND FIGURATIVECF) :MEASURES . FOR Gl AND G2 MOPER/  Gl  G2  MC  MR  SER  MCDIM  MCMUL  MRDIM  MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN  0  .55**  . 49**  .53**  . 54**  .47**  .46*  57*** .41*  F  .27  .44*  . 31  .15  .13  .42*  46*  0  .63***  .64*** .64***  .63***  F  .27  . 37*  .24  * p<.05 ** p<.01 *** p<.001  .23  .35*  .27  18  .20  .25  .44*  . 48**  .31  .39*  . 59***  .27  7 2 * * .71*** * . 59*** .64*** .25 . 30  .22  .23  oo  TABLE  3  RELATIONSHIP BETWEEN THE TRAN SCORES AND THE SCORES BELOW AND ABOVE THE MEDIANS OF THE FORHIN, MOPER, MCMUL(O), MRMUL(O), AND SERDIM(O) SCORES FOR G l AND G2  Gl FORHIN  G2  <30  >30.  PASS  0  12  FAIL  16 16  TRAN  2 X  MOPER  _4  0  FAIL  18 18 2 x  2 X  10  _5  =  15  12  20  14  2  10  FAIL  17  _3  19  13  2  <2  >2  12  1  20  12  16 _3  13  19  =  2 X  12 20  2 X  >2  10 _1  = =  17 15  = 17.95, p< .001 <2  >2  3 1_2  14  =  17  _3  =  15  15  = 1 4 . 69 , P<- 001  19 21 2  =  x = i o .24, p< .001  >2  PASS  X  001  = 2 1 . 34, P<- 001  FAIL  = 12.24, p< .001  20  13  TRAN  17  _2  19  2  15  17  18  PASS  15  =  FAIL  SERDIM(O) <2_  =  15  11 _2  2  _3  2  1  x  11  12  PASS  TRAN  17  <3.1 >3.1  = 2 4 . 68,  MRMUL(F) <2_  =  12  >2  2  14  16  MCMUL(0) <2  X  3  20  <3.1 >3. 1  PASS  >30  12  = 2 2 . 53, p< .001  TRAN  TRAN  —  <30  = 12.44, p< .001 <2  =  >2  12  4  13  =  17  20  11  _4  =  15  15  17  11  = 2 0 . 40, p<. 001  X = 7.93, p<. 005 2  88  a s i g n i f i c a n t r e l a t i o n s h i p between these two scores (x =22.53, 2  d f = l , p<.0001 f o r G l ; and x 1 5 . 2 0 , d f = l , p<.001. 2=  See Table 3 ) .  The FORHIN median f o r both groups combined was the same as f o r each i n d i v i d u a l group.  However, the FORHIN scores were combined  i n the f i r s t p l a c e because i f a minimum l e v e l o f attainment i n FORHIN was r e q u i r e d  t o a t t a i n a minimum l e v e l o f o p e r a t i v i t y i n  the l o g i c a l t a s k s , t h i s minimum requirement  should presumably be  the same f o r both Gl and G 2 i f indeed these two groups r e p r e s e n t e d a developmental  continuum.  FORHIN and F i g u r a t i v e measures - Table 1 also shows that some o f the F i g u r a t i v e measures were s i g n i f i c a n t l y c o r r e l a t e d w i t h the FORHIN s c o r e s .  However, these c o r r e l a t i o n s are not as  high as the ones between FORHIN and the Operative measures; they range Gl.  from r=.37 w i t h the MCCOPYs o f G2 t o r=.70 with FUCOPY o f In f a c t , most o f the c o r r e l a t i o n s o f FORHIN with the F i g -  u r a t i v e scores were s i g n i f i c a n t l y lower than the c o r r e l a t i o n s between FORHIH and the corresponding Operative s c o r e s .  T h i s was  i n d i c a t e d by H o t e l l i n g t - t e s t s o f d i f f e r e n c e between c o e f f i c i e n t s that  are c o r r e l a t e d .  (For the p r e s e n t samples, any d i f f e r e n c e  between two c o r r e l a t i o n c o e f f i c i e n t s equal t o , or g r e a t e r than, 14 was s i g n i f i c a n t w i t h a p<.01.)  For Gl a l l the F i g u r a t i v e  measures were s i g n i f i c a n t l y c o r r e l a t e d with FORHIN, while f o r G2 the F i g u r a t i v e  scores MRCOPY, MRDIM(F) and MRMUL(F) d i d not  s i g n i f i c a n t l y c o r r e l a t e w i t h FORHIN. Relationship  o f MOPER t o F i g u r a t i v e  and Operative measures  MOPER and Operative measures - Table 2 i n d i c a t e s that f o r both grades the MOPER scores were h i g h l y  r e l a t e d t o a l l the Reverse  89  scores (as opposed t o the F i l l - i n  o r A n t i c i p a t o r y scores) o f the  MC, MR, and SER tasks (range, r=.53 w i t h SER t o r=.72 w i t h MCMUL). With the exception o f MRFILL and the MRANT c o r r e l a t i o n s f o r G2, which had s i g n i f i c a n t c o r r e l a t i o n s with MOPER (r=.48, and r=.39, r e s p e c t i v e l y ) , a l l other F i l l - i n c o r r e l a t e d w i t h MOPER.  and A n t i c i p a t o r y scores were not  The TRAN p a s s - f a i l score was a l s o h i g h l y  r e l a t e d t o the MOPER score.  A x  2  t e s t was computed between the  f r e q u e n c i e s i n MOPER. scores below and above the 3.1 p t . MOPER This x  median and the TRAN p a s s - f a i l f r e q u e n c i e s . significant for  r e l a t i o n s h i p f o r both grades  G l ; and x =9.41, d f = l , p<.01 2  s  2  indicated a  (x =20.08, df=1, p<.0001,  f o r G2.  2  See Table 3).  The  MOPER scores f o r both groups xvere combined to o b t a i n a common median (3.1 p t s . ) .  T h i s was done based on the assumption that i f  indeed a minimum l e v e l o f attainment  i n the MOPER scores was  r e q u i r e d t o a t t a i n a minimum l e v e l o f o p e r a t i v i t y i n the l o g i c a l t a s k s , t h i s minimum l e v e l should be the same f o r both groups i f the two groups i n f a c t r e p r e s e n t e d a developmental  continuum.  MOPER and F i g u r a t i v e measures - Table 2 a l s o shows that the c o r r e l a t i o n s between MOPER and the F i g u r a t i v e scores were a l l not s i g n i f i c a n t .  almost  The exceptions were the MR scores o f G l ,  which shox^ed s i g n i f i c a n t c o r r e l a t i o n s w i t h MOPER (MRCOPY, r=.44; MRDIM(F), r=.42; and MRMUL(F), r=.46), and MRCOPY score o f G2 (r=.37). MOPER and FORHIN - The c o r r e l a t i o n o f MOPER to FORHIN was i n c l u d e d i n Table 2, i n the row o f c o r r e l a t i o n s o f the Operative measures. for  The c o r r e l a t i o n s are r=.44 ( s i g n i f i c a n t w i t h a p<.01)  G l , and r=.59 (p<.001) f o r G2.  90  Consistency between the scores o f the l o g i c a l R e l a t i o n s h i p between Operative Reverse Scores  scores  measures  - Table 4 shows the i n t e r c o r r e l a t i o n s o f the  t o t a l Operative Reverse scores Reproduction  (4, 5, 6 ) .  (1, 2, 3) and the F i g u r a t i v e or Common to both Gl and G2 were the  high i n t e r c o r r e l a t i o n s between the Operative l o g i c a l tasks.  tasks  Reverse scores o f the  For G l , MCREVE/MRREVE r=.86, MC RE VS / S E ME VE r=.72,  and MR RE VE / S P. RE VE r=.75.  For G2 , MCREVE/MRREVE r=.75, MCREVE/  SEREVE r=.70, and MRREVE/SE PE VE r=.66. The i n t e r c o r r e l a t i o n s between the measures d e r i v e d from the Reverse scores  (MCDIM(O) , MCMUL(0) , e t c . ) are presented  i n Table  5 (Operative measures are 1 to 5) , which shows that f o r Gl a l l these  i n t e r c o r r e l a t i o n s were s i g n i f i c a n t  r=.64 to HRDIM(0)/MRMUL(0), r=.91).  (range, MCMUL(0)/MCDIM(O),  A similar pattern of inter-  c o r r e l a t i o n s between the measures d e r i v e d from the o p e r a t i v e (Reverse) scores was a l s o obtained  f o r G2 (range, MRMUL(O)/  SERDIM(O) r=.75 to MRDI.M(O) /MP.MUL (0) r=.95).  These high  inter-  c o r r e l a t i o n s n e c e s s a r i l y f o l l o w from the high c o r r e l a t i o n s between a l l the t o t a l Operative  Reverse scores o f the l o g i c a l  tasks and i n d i c a t e that i n both groups Ss performed i n a c o n s i s tent manner i n a l l the Reverse s e c t i o n s of the l o g i c a l An important  tasks.  f i n d i n g was t h a t Ss i n G2 showed a s i g n i f i c a n t l y  higher c o r r e l a t i o n between the MCDIU(O) scores and the MCMUL(0) scores than Gl Ss. r=.95. icant  For Gl MCDIM(O)/MCMUL(0) r=.64 and f o r G2  The d i f f e r e n c e between these two c o r r e l a t i o n s was ( F i s h e r ' s Z=4.3, p .001).  signif-  This d i f f e r e n c e i n the c o n s i s t e n c y  of Gl and G2 between the t o t a l number o f dimensions s o l v e d (MCDIM(O)) and the number of m u l t i p l i c a t i v e combinations p o s s i b l e  TABLE INTERCORRELATIONS BETWEEN OPERATIVE (1,2,3) AND FIGURATIVE (4,5,6) MEASURES FOR G l AND G2 (1) MCREVE (1)  (2) MRREVE  (3) SEREVE  (4) MCCOPY  (5) MRCOPY  (6) SERCOPY  .86***  .72***  .56***  .66***  .53**  .75***  . 7 3 * * *  .73***  .63***  .60***  .56***  .43**  .43**  .56**  (2) Gl  (3) (4) (5)  .63***  (6)  CD  MCREVE  CD  75 * * *  (2) G2  (2) MRREVE  (3) (4) (5)  (3) SEREVE  (4) MCCOPY  (5) MRCOPY  (6) SERCOPY  . 70***  .38*  .34  .61***  .66***  . 46**  .44*  .50**  .21  .36*  .25  .61***  .29 .54**  (6)  * p<.05 ** p<.01 *** p<.001  TABLE  5  INTERCORRELATIONS BETWEEN MEASURES DERIVED FROM OPERATIVE MEASURES (1-5) AND THOSE DERIVED FROM FIGURATIVE MEASURES (6-10) (3) (2) (5) (4) (6) (7) (10) (1) (8) (9) MCDIM(O) MCMUL(0) MRDIM(O) MRMUL(0) SERDIM(O) MCDIM(F) MCMUL(F) MRDIM(F) MRMUL(F) SERDIM(F) 64*** .82*** .64*** . 86*** .73*** .60*** .49** . 56*** . 54** (1) y7*** 64 *** .85*** .86*** .62*** (2) .54** .59*** .50** _ g i * * * ^ 72 * * * . 76*** (3) .63*** . 54** .61*** .54** #  .78***  (4) Gl  (5)  . 66***  . 59***  .76***  . 70***  . 66***  .59***  .47**  .54**  .47**  .68***  .88***  .46**  .45**  .59***  . 56***  .56***  .20  .87***  .46**  (6) (7) (8) (9)  .30  (10) (2) (3) (4) (5) (6) (7) (8) (1) (9) (10) MCDIM(O) MCMUL(0) MRDIM(O) MRMUL(0) SERDIM(O) MCDIM(F) MCMUL(F) MRDIM(F) MRMUL(F) SERDIM(F) .95***  (1) (2) (3) (4) G2  (5) (6) (7) (8) (9)  .78***  _ 7 7* * *  .69***  .41*  .52**  .35  .33  .57***  .85***  . 84***  .71***  . 48**  .56**  . 33  .28  . 55***  .88***  . 76***  . 39*  .48**  . 38*  .25  .49**  75***  .43*  .49**  .37*  .25  .60***  .20  .29  .26  .22  .56***  .93***  .58***  .53**  .16  .48**  .51**  .27  .75***  . 10 • .20  (10) * p<.05 ** p<.01 *** rx.nm  TABLE  6  INTERCORRELATIONS BETWEEN THE FILL-IN AND ANTICIPATORY MEASURES AND THE OPERATIVE SCORES FOR Gl AND G2 (12) (7) (9) (10) (6) (8) (5) (2) (3) (4) (ID (1) SEREVE MRDIM(O) MRMUL(O) SERDIM(O) MRREVE MCANT MRANT MCREVE MCMUL(0) MRFILL MCDIM(O) MCFILL  CD Gl  .39*  (2)  . 35* . 36* .55***  .52**  .58***  .68***  . 72***  .70***  .53**  .50**  . 12 . 31  . 39*  .41*  .47*  .52**  .46**  .57***  .47**  .47**  .60***  .56***  .43**  .53**  . 53**  .48**  .46**  .44*  .68***  .67***  .66***  . 66***  .65***  .61***  .46**  .46**  .42**  (3) (4)  (12) (7) (8) (9) (10) (6) (3) (4) (5) (11) (1) (2) MCFILL MRFILL MCANT MRANT MCREVE MCDIM(O) MCMUL(0) MRREVE MRDIM(O) MRMUL(O) SEREVE SERDIM(O) (1) G2  .33  (2) (4) * p<.05 ** p<.01 *** p<.001  .28  .23  .25  .21  .33  .26  .43*  . 43*  .66***  .61***  .42*  . 46**  .47**  . 64***  .60***  . 56***  .44*  .47**  .45**  .35*  . 33  .44**  .45**  .47**  . 50**  .43*  . 30  .54**  . 55***  .19  .19  . 38* . 39* .19  (3)  .25  . 26  .08  94  (MCMUL(O)) w i t h i n  that t o t a l  i n d i c a t e s t h a t , even when Gl Ss  remembered c o r r e c t l y many dimensions dimensions tions.  d i d not n e c e s s a r i l y  In the MR  from the m a t r i x ,  involve  those  m u l t i p l i c a t i v e combina-  r e v e r s e s c o r e s , both groups had h i g h i n t e r -  c o r r e l a t i o n s between the MRDIM(0) and MRMUL(O) s c o r e s . Fill-in  scores - As shown by Table 6, the i n t e r c o r r e l a t i o n s  between the MR  and MC  Fill-in  scores to other O p e r a t i v e measures  v a r i e d w i t h each grade as w e l l MR).  as w i t h each type of t a s k (MC  For Gl the o n l y n o n - s i g n i f i c a n t  c o r r e l a t i o n s were between  MRFILL and both o f the a n t i c i p a t o r y measures. t i o n s were s i g n i f i c a n t and  ranged  to r=.72 (MCFILL/MRDIM(0)).  or  A l l other c o r r e l a -  between r=.35 (MCFILL/MCANT)  For G2, MCFILL d i d not  correlate  s i g n i f i c a n t l y w i t h any other O p e r a t i v e measure w h i l e MRFILL correlated  s i g n i f i c a n t l y w i t h a l l other O p e r a t i v e measures except  w i t h MCANT and MRFILL (range r=.38, MRFILL/MRANT, to 4=.61, MRFILL/MRDIM(0)). A possible  reason f o r the l a c k o f s i g n i f i c a n c e between the  MCFILL to o t h e r O p e r a t i v e measures c o u l d be p a r t l y the r e s u l t that sible that  2/3  of the Ss o b t a i n e d scores above 2/3  of the t o t a l pos-  (more than 12 p t s . out o f 18 p t s . p o s s i b l e ) , which means the scores c l u s t e r e d  at one  end of the d i s t r i b u t i o n .  See T a b i c 7 below. TABLE  7  PROPORTION OF Ss OBTAINING MCFILL AND MRFILL SCORES GREATER THAN 2/3 OF THE MAXIMUM POSSIBLE (12/18) Gl  G2  MCFILL  <12 18/32  >12 14/32  <12 11/32  >12 21/32  MRFILL  14/32  18/32  11/32  21/32  95  However, f a c t o r s  other than s t a t i s t i c a l ones must have pro-  duced the present r e s u l t s  s i n c e MRFILL with a s i m i l a r  t i o n o f scores d i d c o r r e l a t e  distribu-  s i g n i f i c a n t l y with most Operative  measures. The  c o n s i s t e n c y between the two F i l l - i n  both groups as shown i n the c o r r e l a t i o n For G2 t h i s c o r r e l a t i o n  scores was low f o r  between MCFILL and MRFILL.  was not s i g n i f i c a n t , while f o r Gl MCFILL/  MRFILL r=.39 was s i g n i f i c a n t at the .05 l e v e l . Anticipation  scores - The i n t e r c o r r e l a t i o n s  to other Operative scores v a r i e d  f o r each grade.  Table 6, f o r Gl MRANT and MCANT c o r r e l a t e d  MCFILL/MCANT, to r*.67, MRANT/MCDIM(0)).  As shown i n  s i g n i f i c a n t l y with a l l  o t h e r Operative measures, except with MRFILL  less consistent.  o f these measures  (range r=.35,  For G2, the p a t t e r n i s  The MCANT score was not s i g n i f i c a n t l y  correlated  with MCFILL o r MRFILL, o r w i t h SERDIM(O), but was s i g n i f i c a n t l y correlated  with  a l l the other scores.  MRANT d i d not c o r r e l a t e  s i g n i f i c a n t l y w i t h MCFILL and with MRMUL. between the two A n t i c i p a t o r y  scores also  The c o n s i s t e n c y varied  f o r both  grades.  For G l , MCANT/MRANT r=.42, while f o r G2, MCANT/MRANT r=.19. The i n c o n s i s t e n c y between the G2 i n t e r c o r r e l a t i o n s  i s p a r t l y ex-  p l a i n e d by the f a c t that f o r t h i s group the d i s t r i b u t i o n o f the MRANT scores was c l u s t e r e d  at the high end o f the d i s t r i b u t i o n .  From a p o s s i b l e range o f scores from 1 t o 6 p t s . , 25 out o f 32 Ss i n G2 obtained scores o f 4 and above, while f o r Gl there were 16/32 Ss with scores h i g h e r than 4 p t s . See Table 8 on the next page. The means o f the d i s t r i b u t i o n o f the MRANT f o r Gl and G2 were also  significantly different  (t=2.97, df=62, p<.004.  See Table 12).  96  There was thus a c e i l i n g e f f e c t i n the MRANT scores f o r G2 that rendered  i t s c o r r e l a t i o n s with a l l other measures not s i g n i f i c a n t . TABLE  8  PROPORTION OF Ss IN GRADE 1 AND GRADE 2 OBTAINING SCORES GREATER THAN OR EQUAL TO 4 IN THE MC AND MR ANTICIPATION TASKS MCANT>4  MRANT>4  Gl  20/32  16/32  G2  20/32  25/32  T r a n s i t i v i t y scores  - Chi-square  between the TRAN p a s s - f a i l  frequency  t e s t s were c a r r i e d out scores and the scores below  and above the MCMUL(0), MRMUL(O), and SERDIM(O) medians.  These  d e r i v e d measures o f the Reverse scores were used i n s t e a d o f the t o t a l Reverse scores because they i n d i c a t e d more s u c c i n c t l y the Ss' a t t a i n e d l e v e l of o p e r a t i v i t y i n each o f these l o g i c a l  tasks  which made them e a s i e r t o compare to the dichotomous TRAN s c o r e s . The median f o r each of the tasks turned out to be 2 p o i n t s . each task  For  (MC, MR and SER), the scores f o r both groups were com-  b i n e d to o b t a i n a common median.  This was done f o r the same  reason t h a t the FORHIN and MOPER scores were combined. tests yielded s i g n i f i c a n t results MCMUL (0) /TRAN x  2==  21.34,  (See Table 3).  df=1, p<.001  }  The x  For G l ,  M.RMUL ( 0 ) / T RAN  2= X  d f = l , p<.001; and SERDIM(O)/TRAN x =20.40, d f = l , p<.001. 2  MCMUL(0)/TRAN x =17.95, d f = l , p<.001; MRMUL(0)/TRAN 2  14.69, For G2  x =12.44, 2  d f = l , p<.001; and SERDIM(O)/TRAN x 7 . 9 3 , d f = l , p<.005. 2=  2  Thus,  these t e s t s i n d i c a t e d that the Ss' performance on the TRAN task was h i g h l y r e l a t e d to t h e i r performance on the Operative s e c t i o n s of the l o g i c a l  tasks.  97  R e l a t i o n s h i p between F i g u r a t i v e measures Reproduction  or Copy scores - As shown i n Table 4, the i n t e r -  c o r r e l a t i o n s between the F i g u r a t i v e scores (4 to 6 ) f o r both groups were s i g n i f i c a n t with the e x c e p t i o n o f MCCOPY/SERCOPY c o r r e l a t i o n f o r G2 which was not s i g n i f i c a n t .  However, f o r both groups, these  i n t e r c o r r e l a t i o n s were s i g n i f i c a n t l y lower than the i n t e r c o r r e l a t i o n s among the corresponding Operative measures (1 to 3) as i t was shown by H o t e l l i n g t - t e s t s of d i f f e r e n c e between c o r r e l a t e d coefficients  (any d i f f e r e n c e between within-groUp  correlations  g r e a t e r than or equal to .14 was s i g n i f i c a n t with p<.01).  The  exceptions were the SER Operative and F i g u r a t i v e i n t e r c o r r e l a t i o n s 'whose d i f f e r e n c e s were not s i g n i f i c a n t l y d i f f e r e n t i n e i t h e r Gl or G2.  Table 5 shows the i n t e r c o r r e l a t i o n s betx^een  the scores d e r i v e d from the F i g u r a t i v e measures (6 to 10), which c o n f i r m the f i n d i n g s o f Table 4 that the i n t e r c o r r e l a t i o n s between the F i g u r a t i v e measures are s i g n i f i c a n t l y lower  than the  i n t e r c o r r e l a t i o n s between the corresponding Operative measures. R e l a t i o n s h i p between F i g u r a t i v e  and Operative measures  Reverse and Copy scores - Table 4 i n d i c a t e s that the i n t e r c o r r e l a t i o n s between F i g u r a t i v e  and Operative measures were a l l  s i g n i f i c a n t f o r Gl (range SEREVE/SERCOPY r=.43 to MCCOPY/MRREVE r=.73).  For G2 the i n t e r c o r r e l a t i o n s were somewhat lower  than  f o r Gl (range MCREVE/MCCOPY r=.38 to MCREVE/SERCOPY r=.61) and 4/9 c o r r e l a t i o n s were not s i g n i f i c a n t . significant  i n t e r c o r r e l a t i o n s between F i g u r a t i v e  measures f o r Gl would i n d i c a t e that s i m i l a r l y across the F i g u r a t i v e logical  The g r e a t e r number of  tasks than G2 Ss.  Gl Ss were  and Operative  performing more  and Operative s e c t i o n s o f the  98  H o t e l l i n g t - t e s t s of s i g n i f i c a n c e between c o r r e l a t e d means were c a r r i e d out w i t h i n each group betxveen u r a t i v e and Operative scores o f MC  and MR  the corresponding  Fig-  tasks ( i . e . , the MCREVE  and MCCOPY scores f o r Gl were compared) which i n d i c a t e d that xvithin each group these scores d i d not d i f f e r s i g n i f i c a n t l y each other.  However, the d i f f e r e n c e s between F i g u r a t i v e  from  (F) and  Operative (0) scores ( i . e . , MCREVE-MCCOPY) were c o r r e l a t e d w i t h other F i g u r a t i v e and Operative measures (See Table 9 ) .  The  pat-  t e r n t h a t emerged indicated, that these d i f f e r e n c e s were s i g n i f i c a n t l y c o r r e l a t e d most f r e q u e n t l y w i t h Operative measures, p a r t i c u l a r l y f o r G l , and very i n f r e q u e n t l y with F i g u r a t i v e measures.  In the cases where the Operative minus F i g u r a t i v e  differ-  ences c o r r e l a t e d w i t h the Operative measures, i t i n d i c a t e d which of the measures, F i g u r a t i v e or O p e r a t i v e , was s p e c i f i c l o g i c a l task.  The  higher w i t h i n a  g r e a t e s t d i f f e r e n c e s between Operative  and F i g u r a t i v e scores were f o r G2,  i n which group the Operative  scores i n c r e a s e d s i g n i f i c a n t l y h i g h e r over the F i g u r a t i v e scores than f o r Gl as i s shown by the h i g h n e g a t i v e c o r r e l a t i o n s between F i g u r a t i v e and Operative scores (MCREVE-MCCOPY/MCREVE r = - . 5 7 , MRREVE-MRCOPY/MRREVE r=-.59, S E P.RE VE - S h RC 0 PY / S E RRE VE  r=-.47),  while f o r Gl these c o r r e l a t i o n s were a l l not s i g n i f i c a n t .  The  Ss' i n c r e a s e or decrease between F i g u r a t i v e to Operative scores was  then r e l a t e d to t h e i r l e v e l of o p e r a t i v i t y .  This conclusion  i s f u r t h e r confirmed by the contingency t a b l e below (Table 10) between the frequency d i s t r i b u t i o n of Ss whose Operative scores i n c r e a s e d or decreased  from i t s F i g u r a t i v e score and the  frequency  d i s t r i b u t i o n of Ss whose scores f e l l below or above the FORHIN median of 30 p o i n t s .  TABLE  9  INTERCORRELATIONS OF THE DIFFERENCES BETWEEN FIGURATIVE AND OPERATIVE SCORES FOR THE MC, MR AND SER TASKS AND ALL THE FIGURATIVE AND OPERATIVE SCORES FOR Gl AND G2 MC(O)-MC(F)/  MC  MR  0  .71*** .50**  F 0  .19 .54**  F  .57***-.24  SER .34  MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN .66***.54**  .45** .44*  .34  .02  .05  .42  .59***.55***  .44* .34 .35* .29  .23 .44** -.19  .12  .58***.06  Gl .41** .26 -.13 -.14 .25 .44* .47** .37*  .37*  G2 MR(O)-MR(F)/  MC  MR  .32 SER  -.56***.37* -.29  -.21  -.39*  -  -  .  .  .  MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN  0  .53**  .65*** .48** .53** .55** .66***.51**  F  .41* -.04  .31  .41*  .22  .04  O  .32  .23  .28  .31  .46** .32  .47**  .39*  .27  .29  .59***.46**  n .46**  .003  .28 .28  -.30  .23  .  .  .61***.22  . .31  G2 F -.19 SER(O)-SER(F)/ MC G 1  -.59*** .16 -.15 -.01 -.47**-.50** .20 MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN  O  .41*  .35*  .67***.42*  F O  .34 .06  .17 .13  -.16 .44*  F  -.05  .06  -.47**-.06  .32 .10  .43*  .34  .29  .54**  .32  .04  .31  .17  .43*  .34 .13  .14 .25  .25 .12  -.14 .42*  .10  .31  .07  .19  .11  -.10  .15  -.06  -.52**  -  -  .  .  G2 * p<.05 ** p<.01 *** p<.001  .  100  TA3LE  10  PROPORTION OF Ss IN Gl AND G2 WHOSE FORHIN SCORE FELL ABOVE OR BELOW THE I IED I AN AND WHOSE MC AND m REVERSE OR OPERATIVE SCORES INCREASED OR DECREASED WITH REFERENCE TO THEIHL CORRESPONDING COPY OR FIGURATIVE SCORE MRREVE vs. MRCOPY FORHIN >30 <30  MCCOPY vs. MCREVE FORHIN >30 <30 Increase  1  13  =  Decrease  15  3  16  16  Increase  4  17  =  Decrease  11  0  =  15  17  14  2  16  »  18  18  14  0  =  14  16  16  21  5  17  =  22  11  10  0  =  10  15  17  Gl  G2  In c o n c l u s i o n then, the i n t e r c o r r e l a t i o n s between  Figurative  and O p e r a t i v e measures i n d i c a t e d t h a t Gl Ss had g r e a t e r c o n s i s t ency across t h e i r performances i n the F i g u r a t i v e and O p e r a t i v e s e c t i o n s o f the l o g i c a l tasks than G2 Ss. the i n c r e a s e o r decrease  A l s o , f o r both  grades,  from F i g u r a t i v e to Operative measures  was r e l a t e d t o the l e v e l of attainment  i n a l l other l o g i c a l  tasks  as w e l l as to the FORHIN s c o r e s . Fill-in  and A n t i c i p a t o r y measures - Table 11 i n d i c a t e s the  i n t e r c o r r e l a t i o n s between the F i l l - i n  and A n t i c i p a t o r y measures  and the F i g u r a t i v e scores f o r Gl and G2. t h a t the Gl F i l l - i n  measures were almost  These r e s u l t s show a l l (15/16) s i g n i f i c a n t l y  c o r r e l a t e d t o the F i g u r a t i v e scores (the e x c e p t i o n was SERDIM(F)). The c o r r e l a t i o n s ranged MCFILL/MCCOPY.  from r=.37 i n MCFILL/SERCOPY, to r=.79 i n  The G2 F i l l - i n  s c o r e s , on the o t h e r hand, show  fewer s i g n i f i c a n t c o r r e l a t i o n s to the F i g u r a t i v e scores  (7/16),  TABLE 11 INTERCORRELATIONS BETWEEN THE FILL-IN AND ANTICIPATORY MEASURES AND THE FIGURATIVE SCORES FOR G l AND G2  Gl (1)  CD (2) C3) C4) (5) (6) (7) (8) (9) (10) (11) (12) MCFILL MRFILL MCANT MRANT MCCOPY MCDIM(F) MCMUL(F) MRCOPY MRDIM(F) MRMUL(F) SERCOPY SERDIM(F) .39*  (2)  .35*  .36*  .79***  .64***  .56***  .54**  .53**  .51**  .37*  .12  .31  .49**  .51**  .46**  .44**  .46**  .44**  .59**  .42*  .33  .29  .21  .45**  .46**  .32  .29  .29  .24  .24  .16  .41*  .41*  .34  .38*  .42*  (3) (4)  G2 (D  .33 .55***  CD (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) MCFILL MRFILL MCANT MRANT MRCOPY MCDIM(F) MCMUL(F) MRCOPY MRDIM(F) MRMUL(F) SERCOPY SERDIM(F) .29  .08 .19  (2) C3) (4)  * p<.05 ** p<.01 *** p<.001  .19  .47**  .41*  .39*  .50**  .39 *  . 32  .22  .23  . 38  .25  .26  .30  . 37*  .36*  . 34  .15  .08  .19  .00  .03  .14  - .22  - .16  - .11  .27  .29  .36*  . 32  .36*  .23  .20  . 34  .32  . 18  102 and these s i g n i f i c a n t c o r r e l a t i o n s o c c u r r e d m a i n l y . w i t h the MC F i g u r a t i v e scores.  The c o r r e l a t i o n s ranged from r=.36, MRFILL/  MRDIM(F), to r=.50, MCFILL/MRCOPY. The A n t i c i p a t o r y measures f o r Gl c o r r e l a t e d w i t h o n l y 6/16  significantly  o f the F i g u r a t i v e measures (MCANT/MRCOPY r=.45,  MCANT/MRDIM(F) r=.46, MRANT/MRCOPY r=.41, MRANT/MRDIM(F) r=.41, MRANT/SERCOPY r=.38, and MRAiNT/SERDIM(F) r=.42). one F i g u r a t i v e score  For G2 , o n l y  s i g n i f i c a n t l y c o r r e l a t e d to an A n t i c i p a t o r y  s c o r e , MRANT/MCMUL(F) r=.36.  As i n the Reverse measures, Gl  appeared to perform more s i m i l a r l y across the F i l l - i n  and A n t i c -  i p a t o r y measures than G2. D i f f e r e n c e s between the two Developmental l e v e l s One-way a n a l y s e s o f v a r i a n c e were performed on a l l the s c o r e s o f Gl and G2 d i v i d e d on the b a s i s o f sex.  These i n d i c a t e d  t h a t 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 between the scores when these were d i c h o t o m i z e d on the b a s i s of sex. Hotelling t-tests  f o r c o r r e l a t e d means were c a r r i e d out  between the mean s c o r e s o f Gl and G2 i n a l l  the t a s k s .  The  r e s u l t s are shown i n Table 12 which i n d i c a t e s that Gl and G2 differed significantly  from each o t h e r i n t h e i r performances on  o n l y the O p e r a t i v e measures and not i n the F i g u r a t i v e ones.  The  s p e c i f i c measures i n which they d i f f e r e d s i g n i f i c a n t l y were the MC and SER Reverse scores and a l l  t h e i r d e r i v e d scores  (MCREVE  t«2.70, df=62, p<.009; SEREVE t=2.74, df=62, p<.008) MCFILL and MRFILL, and i n MOPER scores  (t=2.117, df=62, p<.04).  Grade 2 d i d not d i f f e r s i g n i f i c a n t l y scores.** A x  2  Grade 1 and  from each o t h e r i n the FORHIN  t e s t between the p a s s - f a i l f r e q u e n c i e s o f the TRAN  *See Addendum ( 2 ) . **See Addendum ( 3 ) .  103 TABLE 12 MEANS, SD's, DIFFERENCES BETWEEN THE MEANS, t-TESTS AND SIGNIFICANCE LEVELS OF THE MEASURES OBTAINED FOR Gl AND G2  Gl 1 MEAN  G2 SD 1  1 MEAN  SD 1  DIFF 2,.25  FORHIN  31. 78  16. 41  34. 03  12 . ,67  MCREVE  86.,50  48.,66  115.,47  36.,00  28..9  t ,61  *  P .5  2.,70  .009  MCDIM(O)  2. 47  1., 86  3. 69  1., 75  1., 2  2, ,7  .009  MCMUL(0)  2. 19  2., 52  3., 78  3.,33  1,.6  2.,2  .03  MCANT  4. 16  1..61  4., 53  1., 41  96. 69  34.,98  MCCOPY  102 . ,91  36.,92  1., 75 .08  3.,03  2,.46  2,.85  1., 84  MCMUL(F)  1.,75  2., 30  2., 84 13,. 78  102.,66  . 50  1,.09  3,,03  MRREVE  ,69  2 ,69 ,  1., 59  3,.85  . 33  1., 31 .19  2.,47  11., 31  6,.2  ,99  ,5  MCDIM(F)  MCFILL  .38  50., 71 115,,53  37,,28  .006  12,. 87 1,.16  .25 .59  MRDIM(O)  3.,19  2., 36  3,.47  1.,68  .28  .55  MRMUL(0)  3.,44  3,.97  2.,97  3., 20  .47  . 52 .61  MRANT  3., 72  1.,65  4., 88  1., 45  1,.16  10 2., 31 40,,94  2..97 . 75  7,.41  .004 .46  94.,91  38.,44  MRDIM(F)  2.,63  1,.74  3,.19  2,.04  .56  1,.18  .24  MRMUL(F)  1,.88  2., 51  2 .66 ,  2,.47  . 78 1,.25  .21  MRFILL  11., 78  3,. 52  13,.78  SEREVE  55., 75 21,. 75  68,.91  MRCOPY  SERDIM(O) SERCOPY  1,. 72  1., 35  60,. 56 16 .43  2, .44  71,.28  3,. 20 16 .31 , 1,. 37  2,.00 13 .16 .72  2,. 38 .02 2,. 74 . 008 2 .12 ,  17,. 10 10 . 72 2,.56 . 36  . 04 .02 .02  SERDIM(F)  1,,91  1,.28  2,. 72  1,. 46  .81  2  MOPER  2,.93  5,. 24  3,. 24  6 .50  .31  2,. 12 .04  104  task f o r both  groups i n d i c a t e d that Gl and G2 d i d not d i f f e r  significantly  from each other on t h i s measure e i t h e r .  Q u a n t i t a t i v e Correspondences - A q u a n t i t a t i v e comparison among the scores o b t a i n e d  f o r each S on the v a r i o u s tasks was  made i n terms o f the number o f "chunks" o f i n f o r m a t i o n or S-R units  ( i . e . , dimensions or s t i m u l i - r e s p o n s e u n i t s ) that the S  was able to handle s u c c e s s f u l l y i n each t a s k .  The FORHIN scores  were t r a n s l a t e d i n terms o f the number o f dimensions that the S could s u c c e s s f u l l y c o - o r d i n a t e by the procedure i n d i c a t e d at the end o f the Method s e c t i o n .  A l s o , the t o t a l Reverse scores were  not used, but r a t h e r the comparisons were made with the d e r i v e d scores from i t ,  MCDIM(O), MRDIM(O), MCMUL(O), MRMUL(0), and  SERDIM(O), s i n c e these more s u c c i n c t l y i n d i c a t e d the number of dimensions  (and p a r t i c u l a r l y m u l t i p l i c a t i v e combinations) that  each S had handled  i n the r e s p e c t i v e t a s k s .  comparison was only made f o r the Operative  This q u a n t i t a t i v e scores s i n c e i t was  on these scores that the two grades d i f f e r e d s i g n i f i c a n t l y  from  each other. The and  13.  v a r i o u s scores A x  f o r each S are presented  i n Tables 12  a n a l y s i s was made between scores below and above  2  the FORHIN median and scores below and above the MOPER median, which i n d i c a t e d that f o r both r e l a t i o n s h i p between these  Gl and G2 there was a s i g n i f i c a n t  two s c o r e s .  Most Ss who obtained a  MOPER score g r e a t e r than or equal t o 3.1, o b t a i n e d a FORHIN score g r e a t e r than or equal to 1.5 (x =24.8, d f = l , p<.001, 2  f o r G l ; and x 2 1 . 7 2 , d f = l , p<.001, f o r G2. 2=  X  2  See Table  14).  Also,  t e s t s were made between the scores below and above the MOPER  and FORHIN median and the MCREVE and MRREVE scores which had a minimum o f one m u l t i p l i c a t i v e combination  of dimensions i n each  105 TABLE 13 NUMBER OF UNITS OF INFORMATION THAT G l Ss WERE ABLE TO HANDLE SUCCESSFULLY IN THE FORHIN, MOPER, MC, MR, AND SER TASKS Ss 1 2  FORHIN 1.3 .7  MOPER  MCDIM(O)  MRDIM(0)  MCMUL(0)  MRMUL(0)  SERDIM(O)  2.3  0  2  0  0  1  2.1  1  2  0  0  1  3  1.7  3.1  5  6  6  9  2  4  1.1  2.4  0  0  0  0  0  5  .7  2.4  1  0  0  0  2  6  2.8  3.4  5  6  6  9  3  7  2.4  3.2  5  4  6  3  3  8  1. 7  3.4  5  5  6  6  3  9  1.0  2.5  1  0  0  0  1  10  2.6  3.2  4  5  4  2  3  11  1.2  3.0  1  2  0  0  0  12  .6  2.5  1  0  0  0  0  13  1.1  2.3  2  2  0  1  1  14  2.7  3.4  5  6  6  9  3  15  1.1  2.1  0  2  0  0  2  16  .8  2.5  0  3  0  0  1  17  1.7  • 3.0  3  5  2  6  2  18  1.9  3.2  5  6  6  9  3  19  2.2  4.0  3  4  6  9  3  20  1.6  2.6  3  3  2  2  1  21  1.2  2.5  2  0  0  0  0  22  2.0  3.4  3  1  6  9 .  3  23  1.6  3.2  4  4  4  4  0  24  1.5  4.0  3  5  3  6  2  2.2  0  3  0  0  0  25  . 5  26  1.2  2.2  1  2  0  0  1  27  1.4  3.0  2  1  0  0  1  2.5  0  0  0  0  1  3.3  4  6  4  9  3 3  28 29  .7 2.2  30  1.5  3.3  4  3  4  -7  31  • 2.2  3.5  5  6  6  9  3  .9  2.4  1  0  0  0  0  32 o  106 TABLE 14 NUMBER OF UNITS OF INFORMATION THAT G2 Ss WERE ABLE TO HANDLE SUCCESSFULLY IN THE FORHIN, MOPER, MC, MR, AND SER TASKS Ss  FORHIN  MOPER  MCDIM(O)  MRDIM(0)  MCMUL(0)  MRMUL(0)  SERDIM(O)  1  1.9  4.4  5  4  6  3  3  2  1.4 .  3.0  4  3  4  0  2  3  2.9  4.1  6  6  9  9  4  4  1.3  2.5  3  3  0  3  3  5  1.6  4.1  3  3  2  0  3  6  2.0  3.3  5  4  6  3  4  7  2.4  3.8  6  6  9  9  5  8  1.4  3.2  5  5  6  6  2  9  1.6  4.3  5  4  6  3  3  10  .9  2.5  2  0  0  0  2  11  1.8  3.7  4  4  4  4  4  12  1.4  3.0  0  2  0  0  1  13  1.5  3.3  4  5  4  6  3  14  2.6  3.2  6  6  9  9  4  15  1.2  2.4  2  3  0  0  0  16  1.2  . 3.3  2  3  1  2  3  17  1.7  4.0  6  5  9  6  2  18  1.4  3.0  2  4  1  4  3  19  1.4  2.4  2  0  0  0  0  20  1.7  3.1  4  4  4  4  3  21  2.3  3.8  6  5  9  6  3  22  2. 3  4.1  6  6  9  9 .  5  23  1.7  3.3  4  3  4  0  2  3.0  1  2  0  0  0  2.3  2  0  0  0  0  26  1-1 1.9  3.1  4  3  3  0  1  27  1.4  2.2  4  2  2  0  2  28  1.1  2.2  2  2  1  0  1  29  1.3  3.3  1  3  0  0 '  2  30  2.0  3.4  5  6  4  3  3  31  1.7  4.0  6  5  9  6  3  32  1.4  2.5  2  2  1  0  2  24 25  .7  107 TABLE 15 PAIRWISE COMPARISONS OF THE SCORES BELOW AND ABOVE THE MEDIANS OF THE MOPER, FORHIN, SERDIM(O), AND MULTIPLICATIVE COMBINATIONS IN MR AND MC (MUL) SCORES FOR G l AND G2  Gl MOPER FORHIN  <3.1  G2 <3.1  i3.1  ^1.5  2  14  =  16  0  17  =  17  <1.5  16  _0  =  16  12  _3  =  15  18  14  12  20  X =21 . 72> P< .001  X =24. 83, p<. 001 2  <3.1  MOPER MUL (MC and MR)  2  >3.1  <3.1  2  14  =  16  1  16  =  17  <1  16  _0  =  16  11  _4  =  15  18  14  12  20  X  <3.1  <3.1 - 2.3.1  MOPER  i.3.1  >2  3  13  =  16  6  19  =  25  <2  15  _1  =  16  _6  _1  =  ' 7  18  14  12  20  2 X  X = 8.88, p<. 005  = 1 8 . 28, p<. 001  MUL  <2  >_1. 5 <1.5  2  >2  <2  16  =  16  3  14  =  17  16  _0  =  16  1_2  _3  =  15  16  16  15  17  X = 12 . 30 , p<.001  2  SERDIM(O) <2  >2  >1.S  2  14  =  <1. 5  14  _2_  =  16  16  2 X  >2  0  X =32. 00 , p<. 001  FORHIN  = 15 . 45, p<.001  2  2  FORHIN  >3.1  >1  X =24. 78 , p<. 001  SERDIM(O)  >3.1  = 1 7 . 99, p<. 001  2  <2  >2  16  1  16  =  17  16  6  _9  =  15  7  25  X = 5.42, p<. 025 2  108  of the MC and MR Reverse s c o r e s . m u l t i p l i c a t i v e combination  This c r i t e r i o n o f at l e a s t one  out of the t o t a l number o f dimensions  c o r r e c t i n each of the MC and MR Reverse scores was thus s e t as the minimum l e v e l of o p e r a t i v i t y at which the S was s a i d to possess an understanding  o f the m u l t i p l i c a t i v e concepts.  The x  2  t e s t s i n d i c a t e d that most S s (33/64, both groups combined) had to have obtained a minimum o f 3.1 i n the MOPER task and 1.5 i n the FORHIN task before they could a t t a i n at l e a s t one m u l t i p l i c a t i v e combination  o f dimensions i n each task (MOPER and MUL,  X =24.78, d f = l , p<.001, f o r G l ; and x =15.45, d f = l 2  2  5  p<.001, f o r  G2; FORHIN and MUL, x 3 2 . 0 , d f = l , p<.001, f o r G l ; and x 1 2 . 3 0 , 2 =  d f = l , p<.001, f o r G2).  2=  Finally, x  2  t e s t s were c a r r i e d out between  the scores below and above the FORHIN and MOPER medians and the SERDIM scores g r e a t e r than or equal t o two. Thus, the minimum c r i t e r i o n o f o p e r a t i v i t y f o r the SER task was d e f i n e d i n terms o f whether or not the S could s u c c e s s f u l l y co-ordinate at l e a s t two  dimensions out o f the t o t a l s i x .  There were again  signif-  i c a n t r e l a t i o n s h i p s between the l e v e l s a t t a i n e d i n the MOPER and FORHIN scores and whether or not the Ss s o l v e d a minimum o f two  c o r r e c t dimensions i n the t o t a l SEREVE score (MOPER and  SERDIM(O)  2 = X  1 8 . 2 8 , d f = l , p<.001, f o r G l ; and  p<.005; FORHIN and SER, x  2 =  2 = x  8 . 8 8 , df=l,  1 7 . 9 9 , d f = l , p<.001, f o r G l ; and  =5.42, d f = l , p<.02). A K e n d a l l c o e f f i c i e n t of concordance W was run a l s o on the data i n Tables  12 and 13 i n order to determine the c o r r e l a t i o n  between the Ss' rankings  across a l l these t a s k s .  The r e s u l t s  i n d i c a t e d that there was a high degree o f correspondence among the rankings the Ss o b t a i n e d i n a l l these measures.  This  indi-  cated that the Ss showed a high degree o f c o n s i s t e n c y  across  tasks i n terms o f the number o f u n i t s o f i n f o r m a t i o n tl\at t h handled s u c c e s s f u l l y i n each task p<.001  (W-.82, x ~177.87, df=31, 2  f o r , G2, and "-.83, x 179..54 = 2  ?  d f = 3 i , p<.001 f o r Gl)  110 CHAPTER  V  Discussion R e l a t i o n s h i p between Automaton and L o g i c a l Tasks I f indeed, as p o s t u l a t e d by P i a g e t , the f o r e s i g h t sight a b i l i t i e s  are necessary  requirements  and h i n d -  i n the development of  l o g i c a l s t r u c t u r e s , then i t would be expected  t h a t performance  on the F o r e s i g h t - H i n d s i g h t (FORHIN) task, would be h i g h l y r e l a t e d to performance on a l l O p e r a t i v e  (as opposed to the F i g u r a t i v e )  measures of the l o g i c a l t a s k s .  The  prediction.  present data supported  High p o s i t i v e c o r r e l a t i o n s were o b t a i n e d between  the FORHIN and O p e r a t i v e measures as opposed to the lower, and  this  significantly  i n many i n s t a n c e s not s i g n i f i c a n t , c o r r e l a t i o n s between  FORHIN and the F i g u r a t i v e measures. significant  Only  Grade 1 Ss had some  c o r r e l a t i o n s between FORHIN and some o f the F i g u r a -  t i v e measures (MRCOPY, MRDIM(F), MRMUL(F)).  This group  to perform more s i m i l a r l y i n both the O p e r a t i v e and s e c t i o n s o f the l o g i c a l  tended  Figurative  tasks than the Grade 2 Ss whose o p e r a t i v e  s c o r e s were h i g h e r than t h e i r F i g u r a t i v e s c o r e s . It  i s important  to p o i n t out again t h a t these h i g h  t i o n s were o b t a i n e d even though the f o r e s i g h t a b i l i t i e s were measured i n a context and separate and d i f f e r e n t concepts.  correla-  and h i n d s i g h t  i n a manner t o t a l l y  from those used to measure the  logical  These f i n d i n g s c o u l d support  the view t h a t these  s e l f - r e g u l a t o r y f u n c t i o n s are processes  of a g e n e r a l nature  which are manifested  i n contexts other than those  strictly  d e a l i n g w i t h l o g i c a l t a s k s , as i t c o u l d be concluded P i a g e t ' s own  studies of foresight  An important  from  and h i n d s i g h t .  aspect o f the r e l a t i o n s h i p s between FORHIN and  the O p e r a t i v e scores i s t h a t there appears  to be a minimum l e v e l  Ill  of attainment  i n t h e FORHIN s c o r e s n e c e s s a r y  mum  of operativity  criterion  Tables  1 3 , 1 4 , and 1 5 ) .  FORHIN s c o r e s  t o achieve  i n each o f t h e l o g i c a l t a s k s (See  T h i s minimum l e v e l  o f greater than o r equal  o f attainment  sight  study  i t cannot be concluded  and h i n d s i g h t have a c a u s a l r o l e  logico-mathematical minimum l e v e l  concepts,  o f attainment  t o be n e c e s s a r y  before  i n the  t o 1.5 a p p e a r s t o be t h e  same f o r a l l l o g i c a l t a s k s , and f o r b o t h age g r o u p s . when f r o m t h e p r e s e n t  the mini-  Thus, even  that  fore-  i n the acquisition of  i t can be s a i d t h a t a t l e a s t a  i n f o r e s i g h t and h i n d s i g h t  a minimum, l e v e l  of operativity  appears i s mani-  fested. The  f a c t t h a t t h e FORHIN t a s k c o n c e p t u a l l y  to deal w i t h t h e t r a n s f o r m a t i o n a i aspects changes i n t h e v a r i o u s Piaget that  and o t h e r  dimensions),  authors  (i.e.,  i t i s the understanding  s t i m u l i , manifested transformations, This point  adds s u p p o r t  Wallach,  i n the a b i l i t y  that brings  supported  FORHIN s c o r e s  and t h e same O p e r a t i v e  The  such  of operativity.  by a comparison of the  the correlations  between t h e  m e a s u r e s , as e x p l a i n e d  below.  t o t h e S's m e n t a l s p a n o r c o m p u t i n g study  has been o p e r a t i o n a l l y d e f i n e d  i n t e r m s o f t h e number o r " c h u n k s " o f i n f o r m a t i o n units),  i n the  a u t o m a t o n v a r i a b l e , MOPER, and  measures, versus  space, which i n the present  perform  about t h e attainment  between t h e o t h e r  K-Operator refers  t o t h e view o f  1 9 6 9 ; B e i l i n , 1969)  to mentally  the Operative  The  of the s t i m u l i (the  o f t h e changes t h a t o c c u r  i s also indirectly  correlations  required the S  that the S can simultaneously  (stimuli-response  a t t e n d t o and c o - o r d i n a t e .  FORHIN t a s k , on t h e o t h e r h a n d , d i f f e r s  conceptually  from t h e  MOPER t a s k i n t h a t i t s p e c i f i c a l l y r e q u i r e s t h e 5 t o s i m u l t a n e -  Ill  ously c o - o r d i n a t e changing  aspects o f the s t i m u l i .  Thus, the  MOPER task can be s a i d t o r e f e r to the S's m u l t i - c h a n n e l t i o n - p r o c e s s i n g d e v i c e (as Pascual-Leone  informa-  (1970) r e f e r s to i t )  which i s needed t o simultaneously i n c l u d e and c o - o r d i n a t e a l l the schemes i n v o l v e d i n the understanding  o f any c o g n i t i v e task.  The  FORHIN t a s k , on the other hand, can be s a i d t o r e f e r t o the S's schemes s p e c i f i c a l l y necessary  and e s s e n t i a l f o r c o - o r d i n a t i o n s  such as those r e q u i r e d i n l o g i c a l t a s k s . It  could be expected  then, (1) that the S's c a p a c i t y o r  computing space would develop before the a c q u i s i t i o n o f s e l f r e g u l a t o r y f o r e s i g h t and h i n d s i g h t s k i l l s ;  (2) t h a t a computing  space o f a c e r t a i n s i z e would be a p r e r e q u i s i t e f o r the development o f the f o r e s i g h t and h i n d s i g h t s k i l l s operativity; attainment  needed t o o b t a i n  and (3) that the r e l a t i o n s h i p between the l e v e l o f  i n the l o g i c a l tasks and the FORHIN and MOPER scores  would be s t r o n g e s t with the FORHIN scores s i n c e t h i s  constitutes  the automaton v a r i a b l e of most d i r e c t relevance i n the development o f l o g i c a l o p e r a t i o n s . The  e m p i r i c a l f i n d i n g s confirmed a l l these e x p e c t a n c i e s .  The MOPER scores s i g n i f i c a n t l y c o r r e l a t e d with a l l the Operative measures (except with the A n t i c i p a t o r y and F i l l - i n measures) and did  not c o r r e l a t e s i g n i f i c a n t l y w i t h the F i g u r a t i v e measures (See  Table 3).  T h i s f i n d i n g confirms the p r e d i c t i o n t h a t f o l l o w s from  Pascual-Leone's p u t i n g space  (1970) M-Operator model that the s i z e o f the com-  should be r e l a t e d t o S's l e v e l o f o p e r a t i v i t y .  ever, these c o r r e l a t i o n s are a l l s i g n i f i c a n t l y  lower  (p<.01) than  those between FORHIN and the Operative measures, which a l s o firms the expected  finding  How-  con-  (no. 3) t h a t the automaton v a r i a b l e  11S  c o n c e p t u a l i z e d i n the FORHIN task should have the s t r o n g e s t r e l a t i o n s h i p t o the Operative measures.  The lower  correlations  between MOPER and the Operative scores p a r t l y r e s u l t e d from the f a c t t h a t i n a number o f i n s t a n c e s (See Tables  13 and 14), Ss  who a t t a i n e d the minimum MOPER score that f o r most Ss corresponded to the minimum l e v e l of o p e r a t i v i t y , d i d not i n f a c t a t t a i n that minimum l e v e l o f o p e r a t i v i t y .  That  i s , as would be expected, i t  was p o s s i b l e f o r a number o f Ss to have the c a p a c i t y necessary to deal s u c c e s s f u l l y w i t h the l o g i c a l t a s k s , and y e t not have achieved an o p e r a t i o n a l understanding f i n d i n g , o f course, confirmed  o f the l o g i c a l t a s k .  This  the above no. 1 expectancy.  F i n a l l y , very few Ss (two i n Grade 1 and none i n Grade 2) who a t t a i n e d the minimum l e v e l i n FORHIN that corresponded t o the minimum l e v e l o f o p e r a t i v i t y i n the l o g i c a l tasks had a MOPER score below the minimum MOPER score that corresponded t o the minimum c r i t e r i o n o f o p e r a t i v i t y .  That  i s , very few Ss i n  both Grade 1 and Grade 2 who a t t a i n e d a score i n FORHIN g r e a t e r than o r equal t o 1.5, obtained a score i n MOPER lower Thus, the m a j o r i t y o f Ss had t o be able to handle s t i m u l i o f the MOPER task before they c o u l d handle changing  dimensions of the FORHIN t a s k .  the expectancy  than 3.1.  at l e a s t 3.1 more than 1.5  This f i n d i n g  confirms  (no. 2) that a MOPER score o f a c e r t a i n minimum  s i z e would be a p r e r e q u i s i t e f o r the attainment l e v e l i n the f o r e s i g h t and h i n d s i g h t s k i l l s o p e r a t i o n a l understanding  o f the minimum  that may allow  o f the t a s k s .  In c o n c l u s i o n , the f i n d i n g s that c o u l d be expected  from the  automaton tasks and the l o g i c a l tasks on the b a s i s of an a n a l y s i s of t h e i r conceptual  d i s t i n c t i o n s were o b t a i n e d i n these  data.  114  From the p o i n t o f view o f P i a g e t i a n t h e o r y , the h i g h degree o f e m p i r i c a l r e l a t i o n s h i p between the FORHIN scores and the s c o r e s of the l o g i c a l tasks gives support it  i s the c h i l d ' s  to P i a g e t ' s view t h a t  f o r e s i g h t and h i n d s i g h t a b i l i t i e s  sequent a b i l i t y to m e n t a l l y perform  Operative  and the con-  the t r a n s f o r m a t i o n s i n the  s t i m u l i which these s e l f - r e g u l a t o r y f u n c t i o n s make p o s s i b l e , t h a t b r i n g s about the understanding  of l o g i c o - m a t h e m a t i c a l  con-  cepts. A f i n d i n g o f conceptual were no  s i g n i f i c a n c e i s the f a c t that there  i n s t a n c e s o f bimodal d i s t r i b u t i o n s  o f the automaton or l o g i c a l t a s k s .  i n any o f the scores  The way  i n which the  scores  were o b t a i n e d , then, allowed to measure aspects o f the Ss b e h a v i o r that have been t r a d i t i o n a l l y measured i n a l l - o r - n o n e or d i s c r e t e ways, i n a more continuous Consistency  across l o g i c a l  manner.  tasks  I f , as p o s t u l a t e d by P i a g e t , the c h i l d ' s understanding logical  concepts  s h o u l d be manifested  and s t r u c t u r a l l y e q u i v a l e n t concepts  across f a m i l i e s of  of  related  ( g r o u p i n g s ) , then i t should  be p o s s i b l e to o b t a i n h i g h i n t e r t a s k c o n s i s t e n c y among tasks t h a t meet t h i s The  requirement  (structuring criterion  o b t a i n e d data i n d i c a t e d that f o r both  of s t a g e s ) .  groups the  intercorrela-  t i o n s between a l l the O p e r a t i v e measures of the l o g i c a l  tasks were  h i g h , which i n d i c a t e s t h a t the S_s were c o n s i s t e n t i n t h e i r formance across tasks of common and Thus, t h i s  per-  r e l a t e d conceptual s t r u c t u r e .  f i n d i n g lends s t r o n g e m p i r i c a l support  to the above  115  t h e o r e t i c a l p r o p o s i t i o n about the  developmental r e l a t i o n s h i p s  between s t r u c t u r a l l y r e l a t e d concepts.  T h i s f i n d i n g i s of p a r t i c  u l a r importance because i n the present  i n v e s t i g a t i o n the  tasks were equated both i n terms o f automaton demands.  Thus, the obtained  l o g i c a l tasks  of e q u i v a l e n t  and  logical  competence  r e s u l t s s t r o n g l y i n d i c a t e that when and  r e l a t e d conceptual  structures  are equated on more than j u s t s t r u c t u r a l grounds, i t i s p o s s i b l e to f i n d the criterion  i n t e r t a s k consistency  study cannot be and  be noted, however, t h a t the considered  full  sufficient  support to t h i s  groupings u n d e r l y i n g t h i s study d i d not  the  to give the  c l a s s i f i c a t i o n concepts.  meet the  strictest  this  necessary  sample a l l the  stage  possible Therefore,  c r i t e r i o n necessary  t e s t t h i s t h e o r e t i c a l requirement of the t h e l e s s , the data do lend c o n s i d e r a b l e  to  stage c o n s t r u c t .  None-  support to the view that  i s n e c e s s a r y to take i n t o c o n s i d e r a t i o n  tural  evidence from  c h a r a c t e r i s t i c o f the  c o n s t r u c t , mainly because i t d i d not  it  structuring  requires.  I t should  validity  that P i a g e t ' s  simultaneously  struc-  (or competence) as w e l l as automaton f a c t o r s of c o g n i t i v e  development i n d e a l i n g w i t h t h i s t h e o r e t i c a l problem. This  last  coefficients  conclusion  i s f u r t h e r supported by  o f concordance W,  data e x p r e s s i n g  obtained  tasks  (See  f o r both groups, on  the number o f u n i t s of i n f o r m a t i o n  s u c c e s s f u l l y handled i n each t a s k , Tables 13  degree of s t a b i l i t y u n i t s of i n f o r m a t i o n  and  14).  the high  that the  i n c l u d i n g the MOPER and  This c o e f f i c i e n t  Kendal the Ss FORHIN  i n d i c a t e s a high  of i n d i v i d u a l d i f f e r e n c e s i n the number of that  the Ss were capable of o p e r a t i n g  upon  116  simultaneously.  The  large Kendall  c o e f f i c i e n t shows that  individ-  uals maintained e s s e n t i a l l y the same rank p o s i t i o n over a l l o f the t a s k s . constant  In other words, these Ss s u c c e s s f u l l y handled a  number of u n i t s of i n f o r m a t i o n  across  a l l of the  Thus, these f i n d i n g s confirm the n e c e s s i t y of equating  tasks.  logical  tasks on the b a s i s of t h e i r automaton requirements as w e l l as s t r u c t u r a l requirements when attempting  to o b t a i n  equivalent  measures of the S s ' a t t a i n e d l e v e l of o p e r a t i v i t y . The  i n t e r c o r r e l a t i o n s between the F i g u r a t i v e scores were  significantly  lower (p<.01) than the  the Operative  measures.  i n t e r c o r r e l a t i o n s between  T h i s i n d i c a t e s that the Ss were l e s s  c o n s i s t e n t i n t h e i r performance on the on the Operative the p e r c e p t u a l  ones.  F i g u r a t i v e s e c t i o n s than  This f i n d i n g would be expected i f indeed  d i f f e r e n c e s among the  l o g i c a l tasks  gave r i s e  a v a r i e t y of s t r a t e g i e s i n the Ss' attempts to reproduce d i f f e r e n t symbolic  c o n f i g u r a t i o n s of the t a s k s .  p r e t a t i o n i s , of course,  This  to  the  inter-  congruent with what c o u l d be  expected  on the b a s i s of the d i s t i n c t i o n that Piaget makes between Opera t i v e and  F i g u r a t i v e aspects  of i n t e l l i g e n c e  more e x t e n s i v e l y i n the next s e c t i o n ) . emphasized the conceptual were the common and  The  rules underlying  r e l a t e d aspects  (which i s e l a b o r a t e d Operative  the t a s k s , which  between the t a s k s .  u r a t i v e procedure, on the other hand, emphasized the  between the  F i g u r a t i v e vs. Operative Piaget's  The  Fig-  configura-  t i o n a l or r e p r e s e n t a t i o n a l components, which were the a t i n g aspects  procedure  differenti-  tasks. Measures of L o g i c a l Tasks  d i s t i n c t i o n between o p e r a t i v e  aspects  -- mental  117  a c t i o n s and o p e r a t i o n s , and the f i g u r a t i v e aspects -- p e r c e p t u a l , r e p r e s e n t a t i o n a l , and symbolic aspects o f i n t e l l i g e n c e i s of o v e r r i d i n g importance understanding determined  because i t emphasizes that the c h i l d ' s  o f concepts  cannot  merely  be c o n s i d e r e d t o be  by the c h a r a c t e r i s t i c s o f the s t i m u l i  v a r i o u s concepts  might be embodied.  i n which the  According to Piaget i t i s  the S s ' o p e r a t i v e a c t i v i t i e s , how he transforms  and c o - o r d i n a t e s  the s t i m u l i , that represent the most s i g n i f i c a n t  characteristic  of i n t e l l i g e n c e .  The r e p r e s e n t a t i o n a l a c t i v i t i e s , e s s e n t i a l to  the development of i n t e l l i g e n c e , nonetheless  are c o n s i d e r e d to  be subordinate to the o p e r a t i v e f u n c t i o n s . The  present study u t i l i z e d t h i s conceptual d i s t i n c t i o n be-  tween f i g u r a t i v e and o p e r a t i v e aspects as a methodological b a s i s to measure o p e r a t i v i t y i n more than one way. Reproduction  In the F i g u r a t i v e  or Copy s e c t i o n s of the t a s k s , the Ss had t o r e -  c o n s t r u c t the c o n f i g u r a t i o n o f the s t i m u l i as j u s t seen. one  o f the Operative  (Reverse)  the same c o n f i g u r a t i o n except  In  s e c t i o n s , the 5_ had to r e c o n s t r u c t that a l l the r e l a t i o n s h i p s between  the elements had to be r e v e r s e d .  T h i s r e q u i r e d the S to mentally  make a s e r i e s o f t r a n s f o r m a t i o n s on the elements while at the same time keeping  constant the r e l a t i o n s h i p s p r e v i o u s l y e x i s t i n g  between them. Comparisons between the means of the Operative and F i g u r a t i v e scores o f the MC, MR, and SER tasks showed that f o r e i t h e r group these means d i d not d i f f e r s i g n i f i c a n t l y  from each other.  This  c o u l d be i n t e r p r e t e d to i n d i c a t e that the Operative and F i g u r a t i v e measures d i d not tap d i f f e r e n t  aspects o f the S's behavior.  However, whether a S improved h i s Operative score or not i n  118  r e f e r e n c e to h i s F i g u r a t i v e s c o r e , appeared to be a s e n s i t i v e i n d i c a t o r of the S's  o v e r a l l l e v e l of o p e r a t i v i t y .  In other  words, s i n c e f o r a l l tasks the F i g u r a t i v e s e c t i o n was t e r e d before the Operative one,  i t was  adminis-  p o s s i b l e to o b t a i n a  d i f f e r e n c e score which i n d i c a t e d whether, w i t h i n t h a t s p e c i f i c t a s k , the Operative score i n c r e a s e d or decreased correspondent  F i g u r a t i v e score.  I t was  over i t s  found that these  differ-  ences were s i g n i f i c a n t l y c o r r e l a t e d to the Operative measures, i n c l u d i n g those to  from other l o g i c a l tasks  the F i g u r a t i v e measures; and p a r t i c u l a r l y , these d i f f e r e n c e s  were s i g n i f i c a n t l y c o r r e l a t e d to the S's in  the FORHIN task.  scores f o r the M C , MR,  and  SER  t i v e and Operative procedures the Ss' responses The  l e v e l of  attainment  Thus, the p a t t e r n of i n t e r r e l a t i o n s h i p s  the d i f f e r e n c e s between the Reproduction  in  (See Table 9 ) , and not  scores and Reverse  tasks i n d i c a t e d t h a t the F i g u r a i n f a c t tapped  different  approaches  to the same l o g i c a l t a s k s .  above e m p i r i c a l f i n d i n g s have a number of s i g n i f i c a n t  implications. continuous  In the f i r s t  feedback was  p l a c e , the procedure  used, i n which  given to the Ss a f t e r each of the v a r i o u s  s e c t i o n s o f the l o g i c a l tasks  (Fill-in,  c o u l d be viewed as maximizing the S's  Reproduction,  administered  i n a l l the l o g i c a l t a s k s .  the data, the i n c r e a s e or decrease the F i g u r a t i v e scores was  Reverse),  chances to do b e t t e r i n  the Reverse or Operative s e c t i o n s i n c e t h i s p a r t was one  of  the  last  However, as shown by  o f the Operative scores  over  not a f u n c t i o n of the order i n which  the v a r i o u s s e c t i o n s were administered  (as c o u l d have been i n t e r -  p r e t e d i f the Operative scores would have been s i g n i f i c a n t l y h i g h e r across a l l tasks and  f o r both groups), but r a t h e r i t was  119  a f u n c t i o n of the S's FORHIN scores Thus 5 i f the had  and  l e v e l of o p e r a t i v i t y r e f l e c t e d i n the  the Operative  scores  i n the  logical  tasks.  feedback given to the S can be thought of as having  a l e a r n i n g e f f e c t , then the degree to which the Ss  about the task  learned  from h i s exposures to i t through the F i l l - i n  Reproduction s e c t i o n s i^as a f u n c t i o n of h i s a t t a i n e d l e v e l operativity.  This  already  and of  f i n d i n g i n d i r e c t l y supports the view that  e f f e c t s of t r a i n i n g on l o g i c a l operations S's  S's  are dependent on  the  the  a t t a i n e d l e v e l of o p e r a t i v i t y and not e x c l u s i v e l y on  the t r a i n i n g i t s e l f  (Inhelder  and  Sinclair,  1969).  A second i m p l i c a t i o n of these e m p i r i c a l f i n d i n g s i s that they lend support to the usefulness between f i g u r a t i v e and  operative  of Piaget's  distinction  aspects o f i n t e l l i g e n c e .  the same time, they suggest the p o s s i b i l i t y of t h i s d i s t i n c t i o n being accurately The  used as the methodological b a s i s  assessing  the  i n the  f o r more  three s e c t i o n s above i n d i c a t e  f i n d i n g s that c o u l d be  thepry with respect  conceptual  child's operativity.  r e s u l t s discussed  then, that the  At  expected from  Piaget's  to the p o s s i b l e r e l a t i o n s h i p s between the  v a r i a b l e s that were i n v e s t i g a t e d , were i n f a c t obtained. the e m p i r i c a l f i n d i n g s l e n d c o n s i d e r a b l e t h e o r e t i c a l views on what f a c t o r s are  support to  Thus,  Piaget's  i n v o l v e d i n the a c q u i s i t i o n  of logico-mathematical concepts, what r e l a t i o n s h i p s can be ted between various  r e l a t e d l o g i c a l concepts, and  t i o n between f i g u r a t i v e and  operative  Before d i s c u s s i n g more general and  the p o s s i b i l i t i e s  data w i l l be  distinc-  aspects of i n t e l l i g e n c e .  i m p l i c a t i o n s o f these f i n d i n g s  for future research,  discussed.  on the  expec-  other  aspects of  the  120 D i f f e r e n c e s between the two Developmental  Groups  The two age groups were chosen i n o r d e r t o o b t a i n measures from groups which, although c l o s e enough i n terms o f t h e i r a t t a i n e d o p e r a t i v i t y , would s t i l l  be s u f f i c i e n t l y d i f f e r e n t  such  t h a t t h e i r c o r r e s p o n d i n g d a t a c o u l d allow examination o f developmental t r e n d s .  The r e s u l t s  from the v a r i o u s tasks  t h a t the two groups d i f f e r e d s i g n i f i c a n t l y most o f the t e s t s .  different  1 and Grade  from each o t h e r , w i t h Grade  the h i g h e r mean score o f the two groups. congruence with  the p r e d i c t i o n  (1970) M-Operator  from each o t h e r i n  Each one o f the tasks i s d i s c u s s e d n e x t .  MOPER - The MOPER scores f o r Grade nificantly  indicated  that  2 were s i g -  2 Ss o b t a i n i n g  This f i n d i n g i s i n  f o l l o w s from Pascual-Leone's  model t h a t the s i z e o f the computing  space  s h o u l d be l a r g e r f o r the o l d e r group s i n c e i t i s assumed that the M-Operator  c a p a c i t y grows as a f u n c t i o n o f age i n normal  subjects. FORHIN - The mean scores f o r Grade FORHIN t a s k were not s i g n i f i c a n t l y  1 and Grade  different  2 i n the  from each o t h e r . *  T h i s l a c k o f s i g n i f i c a n c e would not be expected i n view o f the f a c t t h a t the two groups d i d d i f f e r  significantly  o t h e r i n most o f the o p e r a t i v e measures.  from each  Further t e s t i n g o f  the FORHIN task across a l a r g e r number o f developmental  levels  w i l l have t o be c a r r i e d out to determine i t s v a l i d i t y on a more complete b a s i s , and i t s r e l a t i o n s h i p s HC and MR - The r e s u l t s d i d not d i f f e r s i g n i f i c a n t l y  to other o p e r a t i v e  i n d i c a t e d t h a t Grade  1 and Grade 2  from each o t h e r i n terms o f the  mean scores o b t a i n e d i n the F i g u r a t i v e measures, but that 1 and Grade  2 did d i f f e r s i g n i f i c a n t l y  *See Addendum  (3).  tasks.  Grade  from each other i n almost  121  all  the mean O p e r a t i v e  scores o f the MC task  (except  i n the MCANT  s c o r e s ) , and only i n MRANT and MRFILL f o r the MR t a s k . r e s u l t s confirmed mainly  the e x p e c t a t i o n s  i n terms o f how they  of the m u l t i p l i c a t i v e The no  t h a t the Ss should  d e a l t with the O p e r a t i v e  These differ  sections  tasks.  data a l s o showed t h a t , w i t h i n each group, t h e r e were  significant  d i f f e r e n c e s between the mean scores  MR t a s k s , which would i n d i c a t e that no group improved t h e i r scores task a d m i n i s t e r e d .  i n the MC and  significantly  i n the MR t a s k , the second  However, the MRREVE scores  multiplicative  f o r Grade 1  i n c r e a s e d enough over t h e i r MCREVE score t o be comparable t o the MRREVE scores overall  lowest  f o r Grade 2.  Thus, Grade 1, the group w i t h the  l e v e l o f o p e r a t i v i t y , showed only a s l i g h t  improvement i n the O p e r a t i v e  scores o f the second  multiplicative  t a s k , b u t no improvement i n the scores o b t a i n e d w i t h i n each task ( i n the O p e r a t i v e  over the F i g u r a t i v e s c o r e s ) .  Grade 2, on the  other hand, a l r e a d y a t t a i n i n g the h i g h e s t O p e r a t i v e not s i g n i f i c a n t l y  improve these scores on the second  c a t i v e t a s k , but i t s i g n i f i c a n t l y its  scores, d i d multipli-  improved i t s O p e r a t i v e  F i g u r a t i v e scores w i t h i n each t a s k .  over  These f i n d i n g s then,  not only c o n f i r m P i a g e t ' s p r e d i c t i o n s as t o the d i f f e r e n c e s i n operative levels  f o r the two developmental groups, but a l s o  c o n f i r m the dependency o f success  o f t r a i n i n g methods  feedback procedure o f MC and MR tasks) level of operativity  (i.e.,  on the a l r e a d y a t t a i n e d  o f the s u b j e c t s .  A second important  aspect  o f the data has t o do w i t h the  d i s t i n c t i o n between the t o t a l number o f dimensions s o l v e d (MCDIM  122  and MRDIM f o r both O p e r a t i v e and F i g u r a t i v e s e c t i o n s o f the tasks) and t h e t o t a l number o f m u l t i p l i c a t i v e combinations within that t o t a l The  possible  (MCMUL and MRMUL, O p e r a t i v e and F i g u r a t i v e ) .  d i s t i n c t i o n was made t o d i f f e r e n t i a t e between Ss who might  c o r r e c t l y s o l v e a c e r t a i n number o f o v e r l a p p i n g dimensions 3) which d i d not i n v o l v e m u l t i p l i c a t i v e combinations Ss who might have s o l v e d c o r r e c t l y fewer dimensions  (up to  versus  those  which none-  t h e l e s s ivere r e l a t e d t o each other i n a m u l t i p l i c a t i v e f a s h i o n . The presence  o f m u l t i p l i c a t i v e combinations  i s , o f c o u r s e , the  most a p p r o p r i a t e c r i t e r i o n as t o whether a S has the m u l t i p l i c a t i v e concept  or not.  S i n c e the two measures were generated  the same data s e t , i t c o u l d be expected n i f i c a n t l y r e l a t e d t o each o t h e r .  from  t h a t they would be s i g -  T h i s , indeed, was the case.  However, the degree o f r e l a t i o n s h i p s i g n i f i c a n t l y v a r i e d f o r each g T a d e .  The c o r r e l a t i o n s between MCDIM and MRMUL i n the  MCREVE scores f o r Grade 2 were s i g n i f i c a n t l y h i g h e r than f o r Grade 1.  This result  those  i n d i c a t e s t h a t the two measures d i d  d i f f e r e n t i a t e between the two l e v e l s o f o p e r a t i v i t y , arid i t a l s o suggests  t h a t the d i s t i n c t i o n between the measures was an appro-  p r i a t e one s i n c e i n f a c t t h e r e were Ss who were able t o o b t a i n moderately  h i g h t o t a l scores i n the matrices and y e t not a t t a i n  the minimum c r i t e r i o n o f o p e r a t i v i t y . The  other O p e r a t i v e measures o f the MC and MR t a s k  (Fill-in  and A n t i c i p a t i o n ) are d i s c u s s e d next. MCFILL and MRFILL - The mean scores f o r these measures c a t e d t h a t Grade I and Grade 2 d i f f e r e d s i g n i f i c a n t l y from  indieach  o t h e r i n both MCFILL and MRFILL, w i t h Grade 2 o b t a i n i n g the h i g h e r  123  scores.  These f i n d i n g s would be expected  i f i n f a c t the F i l l - i n  scores measured the Ss' l e v e l of o p e r a t i v i t y .  However, the  p a t t e r n o f i n t e r c o r r e l a t i o n s with the other F i g u r a t i v e and Opera t i v e measures r a i s e s some q u e s t i o n as to whether those tasks e x c l u s i v e l y gauged the S s  1  operativity.  MCFILL and MRFILL s i g n i f i c a n t l y and Operative s c o r e s . was  significantly  For the Grade 1 Ss,  c o r r e l a t e d with a l l F i g u r a t i v e  For Grade 2 Ss, on the other hand, MCFILL  c o r r e l a t e d only to the F i g u r a t i v e s c o r e s ,  w h i l e MRFILL was s i g n i f i c a n t l y c o r r e l a t e d with a l l the Operative scores and with very few (and almost no s i g n i f i c a n t of  the F i g u r a t i v e s c o r e s .  correlations)  In other words, i n the group with the  lowest l e v e l of o p e r a t i v i t y , both of the F i l l - i n measures were significantly  c o r r e l a t e d with both the F i g u r a t i v e and Operative  measures, while f o r the more advanced group the r e l a t i o n s h i p s v a r i e d as a f u n c t i o n of the task.  As was  apparent  throughout  the data on the l o g i c a l t a s k s , Grade 1 scores appeared  to be  the most c o n s i s t e n t across a l l t a s k s , which might i n d i c a t e that these Ss d e a l t with most tasks i n a s i m i l a r manner. t h i s f a c t , i t c o u l d be expected  In view of  then that Grade 1 F i l l - i n  scores  would c o r r e l a t e with both Operative and F i g u r a t i v e s c o r e s . Grade 2 Ss, on the other hand, appeared responses  as a f u n c t i o n of the t a s k s .  that s i n c e MCFILL was the f i r s t  to have changed t h e i r I t could be expected  s e c t i o n of the f i r s t  task administered, t h i s score could r e f l e c t in  the Ss' responses  strategies.  and consequently  the most  then  logical variability  the Ss' use of non-operative  This i n t e r p r e t a t i o n i s supported by the f a c t t h a t  MCFILL s i g n i f i c a n t l y  c o r r e l a t e d only w i t h F i g u r a t i v e s c o r e s .  consequent improvement i n the MRFILL score i n d i c a t e s the Ss'  The  ISA  f a m i l i a r i t y with the task requirement and of o p e r a t i v e s t r a t e g i e s . f a c t t h a t the MRFILL was Operative It  t h e i r consequent  T h i s i n t e r p r e t a t i o n i s supported significantly  c o r r e l a t e d only  by  with  procedure can give r i s e  more than j u s t o p e r a t i v e s t r a t e g i e s i n the approaches to s o l u t i o n of the matrix t a s k , even i n those advanced l e v e l of o p e r a t i v i t y .  methodological  implications.  As  Ss who  This has  have reached important  d i s c u s s e d i n the s e c t i o n about  i n the c o n f i g u r a t i o n o f the  r a t h e r than based on an o p e r a t i v e s t r a t e g y .  ably the procedure was  a more d i f f i c u l t  commonly used which simply  one  r e q u i r e s to f i l l  Thus, presum-  than the one i n one  s o l u t i o n s , i t was  not u n e q u i v o c a l l y  still  d i s c r i m i n a t e the Ss'  i s a l s o p o s s i b l e t h a t , s i n c e only two i s t e r e d , the S s not tapped.  1  from  found that t h i s task d i d l e v e l of o p e r a t i v i t y . Fill-in  It  tasks were admin-  most common approach to t h i s type  of t e s t  was  In t h i s case, the complex i n t e r r e l a t i o n s h i p s obtained  atypical.  Nonetheless, t h i s procedure does not seem to  measure the Ss' o p e r a t i v e understanding as d i s t i n c t l y  cell.  difficult  and presumably more i n favor of d i s c r i m i n a t i n g o p e r a t i v e non-operative  most  missing  Thus, even when the task requirements were made more  would be  was  to minimize the chances of c o r r e c t s o l u t i o n s based on  the p e r c e p t u a l symmetries present matrix  to  the  the t h e o r e t i c a l b a s i s of the methods, the present procedure designed  the  scores.  appears then that the F i l l - i n  a fairly  use  of m u l t i p l i c a t i v e  matrices  as the Reverse procedure, f o r i n s t a n c e .  MCANT and MRANT - With respect to the A n t i c i p a t o r y measures, Grade 1 and Grade 2 d i f f e r e d s i g n i f i c a n t l y MR  task  (MRANT), but not on the MC  task.  from each other i n the As with the  Fill-in  s c o r e s , the group with the h i g h e s t l e v e l of o p e r a t i v i t y , Grade was  the one  that improved i t s performance across t a s k s .  as w i t h the F i l l - i n t a s k s , the f a c t that there was  no  2,  Also,  significant  d i f f e r e n c e between Grade 1 and Grade 2 i n MCANT c o u l d have res u l t e d from the Ss' u n f a m i l i a r i t y with the task  requirements.  However, the p a t t e r n of i n t e r c o r r e l a t i o n s more c o n c l u s i v e l y cates that the A n t i c i p a t o r y tasks measured mainly operativity.  aspects of the Ss  Both groups, p a r t i c u l a r l y Grade 2, had  significant  c o r r e l a t i o n s with a l l the Operative s c o r e s , and had very (only one) In  significant  few  c o r r e l a t i o n s with the F i g u r a t i v e s c o r e s .  terms of the conceptual meaning of t h i s t a s k , what \^as  apparent  from the data, i s that many s u b j e c t s who  could not  c o r r e c t l y s o l v e the whole matrix, c o u l d nonetheless  anticipate  c o r r e c t l y the t r a n s f o r m a t i o n asked before a c t u a l l y engaging the task. initially plicit  indi-  Thus, t h i s measure d i d not d i f f e r e n t i a t e , as intended, between those Ss who  in  was  might have had an  ex-  idea of a l l the t r a n s f o r m a t i o n s i n the matrix b e f o r e  a c t u a l l y s o l v i n g the t a s k , versus those Ss who r e c t s o l u t i o n s o f the matrix by t r i a l a c t u a l l y engaged i n the task.  c o u l d a t t a i n cor-  and e r r o r and thus while  Although  initially  i t might appear  c o n t r a d i c t o r y that Ss might be able to a n t i c i p a t e a transformat i o n and y e t not be able to a c t u a l l y s o l v e the matrix, t h i s ing of  i s i n f a c t congruent  find-  with P i a g e t ' s i d e a of gradual development  s t r u c t u r e s i n which i t i s p o s s i b l e f o r a S to have a p a r t i a l  understanding  of the operations i n v o l v e d i n a concept  and y e t not  have achieved the " r e f l e c t i v e a b s t r a c t i o n " which allows the coo r d i n a t i o n of a l l the a c t i o n s or operations i n t o a complete t r a n s f o r m a t i o n a l system.  126  SER icantly  - For t h i s t a s k , Grade 1 and Grade 2 d i f f e r e d  from each other i n both the F i g u r a t i v e and Operative  aspects of the t a s k s . was  signif-  The  d i f f e r e n c e between the two  procedures  g r e a t e s t f o r Grade 2•as shown by the l a c k of s i g n i f i c a n t  c o r r e l a t i o n between the SEREVE and SERCOPY. c o r r e l a t i o n between SEREVE and SERCOPY was p<.0 5).  The  For Grade 1, the significant  (r=.43,  SEREVE and the SERDIM(O) scores f o r both grades were  highly correlated  (range r=.66 to r=.75) to the other Operative  measures i n the other l o g i c a l t a s k s , which i n d i c a t e s that these scores were measuring a common aspect of the S's behavior measured i n the other Operative s c o r e s .  On the other hand, the  F i g u r a t i v e s c o r e s , SERCOPY and SERDIM(F) f o r Grade 2 had few s i g n i f i c a n t  also  very  c o r r e l a t i o n s with the other F i g u r a t i v e measures,  w h i l e f o r Grade 1 SERCOPY and SERDIM(F) had many more s i g n i f i c a n t c o r r e l a t i o n s to other F i g u r a t i v e measures. SER  The  r e s u l t s of the  task again i n d i c a t e s that Grade Ss v a r i e d more i n t h e i r per-  formance as a f u n c t i o n of the type of task than Grade 2 Ss performed  who  more c o n s i s t e n t l y across both F i g u r a t i v e and Operative  s e c t i o n s of the l o g i c a l t a s k s . TRAN - As was quencies  i n d i c a t e d by a x  f o r each grade, the two  i c a n t l y from each other. s i g n i f i c a n c e were apparent.  test  2  on the p a s s - f a i l  groups d i d not d i f f e r  No p o s s i b l e reasons  fre-  signif-  f o r t h i s l a c k of  However, the s i m p l i c i t y of the tech-  nique and the f i n d i n g that t h i s measure was  s i g n i f i c a n t l y corre-  l a t e d to a l l the Operative measures of the l o g i c a l tasks MOPER and FORHIN scores would warrant  and  f u r t h e r r e s e a r c h on i t .  In c o n c l u s i o n , the data on most tasks i n d i c a t e d t h a t the groups were s i g n i f i c a n t l y d i f f e r e n t  from each other i n terms of  127  t h e i r a t t a i n e d l e v e l of o p e r a t i v i t y . General  I m p l i c a t i o n s o f the Study  The  r e s u l t s o f t h i s i n v e s t i g a t i o n give general support t o  s e v e r a l of P i a g e t ' s t h e o r e t i c a l views. to P i a g e t ' s s t r u c t u r a l concept  F i r s t , support  o f stage.  A stage  i s given  i s viewed as  a p e r i o d during which the c h i l d ' s performance on d i f f e r e n t conc e p t u a l tasks can be d e s c r i b e d i n terms o f u n d e r l y i n g common i n v a r i a n t p r o p e r t i e s (conceptual s t r u c t u r e c h a r a c t e r i s t i c s ) . . The high i n t e r t a s k c o n s i s t e n c y t h a t , according to the stage s t r u c t , would be expected  i n the Ss' performances across  o f common conceptual s t r u c t u r e was obtained. give support  con-  tasks  Second, the r e s u l t s  to P i a g e t ' s views on the processes  (e.g., s e l f -  r e g u l a t o r y f u n c t i o n s , h i n d s i g h t and f o r e s i g h t ) that might underl i e the development of o p e r a t i v i t y as manifested understanding  o f logico-mathematical  a l s o provide some support o p e r a t i o n a l concepts none event.  concepts.  i n the c h i l d ' s T h i r d , they  f o r the conception of development of  as a gradual process  F i n a l l y , they lend support  r a t h e r than an a l l - o r -  to Piaget's  distinctions  between o p e r a t i v e and f i g u r a t i v e aspects o f i n t e l l i g e n c e .  There  are s e v e r a l conceptual and e m p i r i c a l i m p l i c a t i o n s that f o l l o w from the above. Stages  and Continuous Development  With respect to the problem o f how developmental  stages  should be c o n c e p t u a l i z e d , both i n terms o f the behavior that i t should encompass and i n terms of how these behaviors i s apparent  develop, i t  that the stage c o n s t r u c t needs to be e l a b o r a t e d beyond  i t s present scope.  T h i s i s o b v i o u s l y necessary because o f the  128  numerous conceptual  issues that remain unresolved  t o r s are i n v o l v e d i n the h o r i z o n t a l i s suggested by  the present  ( i . e . , what fac-  'decalages'?) .  However, what  f i n d i n g s i s that expansions of the  stage c o n s t r u c t might f r u i t f u l l y  go beyond c o n s i d e r a t i o n  l o g i c a l s t r u c t u r e s to i n c l u d e more of Piaget's  general  of  concept-  u a l i z a t i o n s about developmental f u n c t i o n s . For example, the aspect  of the stage c o n s t r u c t that seems  to have been most emphasized, c o n c e p t u a l l y the stepwise development of s u c c e s s i v e interest questions  i s manifested  stages.  i n the numerous e f f o r t s  This  pervasive  to deal with  of what makes p o s s i b l e the t r a n s i t i o n between  However, according stages  and e m p i r i c a l l y , i s  stages.  to P i a g e t , t h i s s u r f a c e d i s c o n t i n u i t y of  i s the r e s u l t of u n d e r l y i n g  f u n c t i o n s inherent  to the organism.  continuous s e l f - r e g u l a t o r y Yet, these continuous under-  l y i n g s e l f - r e g u l a t o r y f u n c t i o n s have been themselves very specifically  investigated.  continuous f u n c t i o n s may put  e v e n t u a l l y d i m i n i s h i n importance  the t r a n s i t i o n between stages.  on (or  such as what makes p o s s i b l e  In f a c t , the presence or absence  i n the development of c o g n i t i v e s t r u c t u r e s may  more the researcher's  seldom  A type of a n a l y s i s t h a t focuses  i n d i f f e r e n t terms) a q u e s t i o n  of stages  the  reflect  a r b i t r a r y c a t e g o r i z a t i o n of content  of  knowledge than the s u b j e c t s ' continuous development of s e l f regulations. In regard competence and i n t r o d u c e d by it  to the above problems, the d i s t i n c t i o n between automaton aspects  of c o g n i t i v e development,  F l a v e l l and Wohlwill  (1969) , i s important because  attempts to i n c o r p o r a t e the continuous and  of development i n t o one  framework.  first  discontinuous  This d i s t i n c t i o n was  aspects  used i n  129  the present  study as the b a s i s f o r d e s i g n i n g  used i n the d i a g n o s t i c procedure.  the e q u i v a l e n t  The p o s i t i v e r e s u l t s c l e a r l y  i n d i c a t e the e m p i r i c a l s i g n i f i c a n c e o f t h i s conceptual tion.  When i n v e s t i g a t i n g questions  t o focus  simultaneously  r u l e s , i t appears thus essen-  on the nature o f the competence  r u l e s and on the automaton f u n c t i o n s the  given  competence p o s s i b l e .  t i n c t i o n gives  rise  that make the expression o f  This competence-automaton  to a number o f new p o s s i b i l i t i e s  c o g n i t i v e development i s to be c o n c e p t u a l i z e d . emphasizes the gradual  distinc-  about the a c q u i s i t i o n and  course o f development o f conceptual tial  tasks  dis-  as to how  For example, i t  development that may be i n v o l v e d i n the  attainment of c o g n i t i v e concepts.  "Attainment," according to  t h i s d i s t i n c t i o n , i s not merely d e f i n e d not the c h i l d has the f i n a l  i n terms o f whether or  form o f the concept, but a l s o i n  terms o f how f a r along he i s i n the attainment o f p r e r e q u i s i t e abilities. that there  Such an approach may i n f a c t u l t i m a t e l y i n d i c a t e are very  can be conceived  few c o g n i t i v e concepts whose a c q u i s i t i o n s  as a l l - o r - n o n e  events.  A l s o , the emphasis on automaton f u n c t i o n s brings  into  the n e c e s s i t y o f l o o k i n g at the a c q u i s i t i o n o f a b s t r a c t ceptual  r u l e s as made p o s s i b l e through p s y c h o l o g i c a l  focus  con-  processes  whose c h a r a c t e r i s t i c s are i n f a c t q u i t e d i f f e r e n t from the conceptual  r u l e s themselves.  In other words, the competence r u l e s  are seen as s p e c i f i c a b s t r a c t i o n s  from complex p s y c h o l o g i c a l  pror  cesses which are separate and d i f f e r e n t from the p o s t u l a t e d l o g i c a l s t r u c t u r e s themselves.  Thus, the study of automaton  t i o n s emphasizes the most b a s i c and general acquisitions.  func-  aspects o f c o g n i t i v e  130  In t h i s regard  i t i s important to p o i n t out the conceptual  s i g n i f i c a n c e o f the M-Operator c o n s t r u c t .  This construct i s  t h e o r e t i c a l l y presumed to be an i n d i c a t o r o f the degree o f maximum complexity can handle.  which, i n terms o f u n i t s o f i n f o r m a t i o n , the c h i l d  As p r e d i c t e d by the M-Operator model, when the  l o g i c a l and automaton tasks were q u a n t i t a t i v e l y equated and compared i n terms o f the number o f i n f o r m a t i o n a l u n i t s , the degree of i n f o r m a t i o n a l complexity r e l a t i v e l y constant confirm  across  the need to analyze  non-structural This l a s t question  the Ss c o u l d handle was found to be tasks.  Thus, these f i n d i n g s f u r t h e r  l o g i c a l tasks  a l s o i n terms o f t h e i r  aspects. c o n c l u s i o n , o f course, bears on the t h e o r e t i c a l  o f what g e n e r a l i z e s between r e l a t e d concepts.  The com-  petence-automaton d i s t i n c t i o n makes c l e a r that high i n t e r - t a s k consistency  or s i g n i f i c a n t  r e l a t e d tasks  g e n e r a l i z a t i o n between  conceptually  can only be expected to occur when the tasks are  equated both i n terms of t h e i r competence requirements and automaton requirements. Foresight The general  and H i n d s i g h t f o r e s i g h t and h i n d s i g h t a b i l i t i e s  are presumed t o be  s e l f - r e g u l a t o r y f u n c t i o n s that make p o s s i b l e the develop-  ment o f o p e r a t i v i t y .  I t would be important t o i n v e s t i g a t e these  f u n c t i o n s t o f i n d out t h e i r p o s s i b l e r o l e i n the development o f concepts other than those i n the l o g i c a l realm.  I t might even-  t u a l l y be p o s s i b l e to d i f f e r e n t i a t e between the s p e c i f i c degree of a s s i m i l a t i o n that they may make p o s s i b l e with  reference  to a  s p e c i f i c area o f knowledge, i . e . , l o g i c a l concepts, and t h e i r general  o v e r a l l l e v e l o f a s s i m i l a t o r y c a p a c i t y as manifested  across  131  a number o f d i f f e r e n t c o g n i t i v e c o n t e n t s . With s p e c i f i c r e f e r e n c e to the r o l e o f f o r e s i g h t - h i n d s i g h t abilities  i n the development o f logico-mathematical concepts,  there are a number of e m p i r i c a l p o s s i b i l i t i e s  that could help  c l a r i f y t h e i r r e l a t i o n s h i p to the a c q u i s i t i o n o f these For example, v a r i o u s authors Gellerman,  concepts.  (Smedslund, 1966; B e i l i n , 1969;  1969: etc.) have attempted  t o a c c e l e r a t e the c h i l d ' s  attainment o f l o g i c a l concepts through procedures  that  either  emphasize the competence or s p e c i f i c l o g i c a l o p e r a t i o n s i n v o l v e d i n the l o g i c a l task or by procedures  that emphasize automaton  f a c t o r s such as t e a c h i n g the S to a t t e n d t o c e r t a i n cues.  Even  i f moderate success has been a t t a i n e d by e i t h e r type o f procedure, i n most i n s t a n c e s the most c r i t i c a l  c r i t e r i o n o f whether  the c h i l d has a c q u i r e d a s t r u c t u r e , the S's g e n e r a l i z a t i o n o f the concept t o other s t r u c t u r a l l y r e l a t e d t a s k s , has not been met. In t h i s r e g a r d , a procedure be used f o r two purposes:  such as the FORHIN task could  F i r s t , to i n d i r e c t l y assess the S's  o v e r a l l l e v e l of o p e r a t i v i t y which might i n d i c a t e to what extent the S might i n f a c t b e n e f i t  from a t r a i n i n g procedure.  This  p o s s i b i l i t y o f the FORHIN task i s , o f course, suggested by the f i n d i n g i n the present study that the S's improvement i n a task was r e l a t e d to h i s performance procedure  i n the FORHIN t a s k .  Second, a  such as that of the FORHIN task could a l s o be used as  a t r a i n i n g procedure  i t s e l f s i n c e i t presumably i n v o l v e s a c t i v -  i t i e s which are b a s i c and common to f a m i l i e s o f r e l a t e d concepts.  logical  I t would be important to f i n d out i f the t e a c h i n g o f  such a c t i v i t i e s would l e a d to g r e a t e r g e n e r a l i z a t i o n across  132  conceptually r e l a t e d tasks.  This outcome c o u l d be  s i n c e a FORHIN t r a i n i n g would not s p e c i f i c l o g i c a l operations  expected  i n v o l v e the teaching of the  t h a t may  be i n v o l v e d i n the  t a s k , i . e . , a d d i t i o n - s u b t r a c t i o n , r e v e r s i b i l i t y , but  given  rather  the c o - o r d i n a t i o n of t r a n s f o r m a t i o n s , which presumably i s a b a s i c and  common a c t i v i t y to a l l c o n c r e t e - o p e r a t i o n a l  t e s t i n g of g e n e r a l i z a t i o n would be c a r r i e d out  tasks. through  The tasks  that would be made e q u i v a l e n t both i n t h e i r competence and automaton demands, as was Operative One  done i n the present  their  study.  vs. F i g u r a t i v e Aspects last  important  i m p l i c a t i o n has  to do with  the methodo-  l o g i c a l problems i n v o l v e d i n the measurement of developmental phenomena.  In the present  f i n d i n g how  the Ss d e a l t with  account conceptual  aspects  i n v e s t i g a t i o n i t proved f r u i t f u l , i n l o g i c a l s t i m u l i , to take i n t o  about the nature  of i n t e l l i g e n c e , i . e . ,  o p e r a t i v e vs. f i g u r a t i v e d i s t i n c t i o n , i n the design of the methods themselves.  In other words, the measurement  were not merely means to t e s t the s p e c i f i c conceptual of i n t e r e s t but  a l s o had  questions  i n c o r p o r a t e d i n t o t h e i r format consid-  e r a t i o n s based on the theory cluded  instruments  of i n t e l l i g e n c e .  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Wallach, L. , W a l l , J . , and Anderson, W. Number c o n s e r v a t i o n : The r o l e o f r e v e r s i b i l i t y , a d d i t i o n - s u b t r a c t i o n , and m i s l e a d i n g p e r c e p t u a l cues. C h i l d Development, 1967, 38, 425. W o h l w i l l , J . F . Piaget's theory o f the development o f i n t e l l i g e n c e i n the concrete operations p e r i o d . American J o u r n a l o f Mental D e f i c i e n c i e s Monograph Supplement, 1966, 7j0» 57-83.  Set  3  FORHIN S e t s .  S e t s 4 and 5.  Set  t  Set 7  FORHIN'Sets .  Sets 6 and 7  Set  Set  FORHIN  Sets.  140  8  Sets  9  8  and  9.  Set  FORHIN  Sets.  Sets  141  10  10  and  11.  142 The Set No.  correct 2:  c h o i c e cards  Change i n one  f o r the FORHIN s e t s were:  dimension,  Position  3:  Third  Position  4:  Second c a r d , f i r s t  Position  5:  First  Set No.  3:  shape.  c a r d , second row  of c h o i c e  cards  row  of c h o i c e  cards  c a r d , second row  of c h o i c e  cards  Changes i n one  dimension,  orientation.  Position  3:  Third  c a r d , second row  Position  4:  First  card, f i r s t  Position  5:  First  c a r d , second row  Set No.  4:  Changes i n two  Position  3:  row  dimensions,  Second c a r d , f i r s t  of choice of c h o i c e  row  of c h o i c e  Position  Third card, last  5:  Changes i n two  Position  size.  cards  of choice  cards  row  of choice  cards  dimensions,  shape and  orientation.  Third card, f i r s t  row  of choice  cards  P o s i t i o n 4:  First  card, f i r s t  row  o f choice  cards  Position  Third  c a r d , second row  Set No.  6:  3:  cards  b r i g h t n e s s and  Second c a r d , second row  Set No.  cards  of choice  P o s i t i o n 4: 5:  cards  5:  of choice  Changes i n three dimensions,  shape, o r i e n t a t i o n ,  Position  3:  Fourth c a r d , second row  Position  4:  Fourth c a r d , f i r s t  Position  5:  Third  Set No.  7:  cards  of choice  cards  row  of choice  cards  c a r d , second row  of choice  cards  Changes i n t h r e e dimensions,  shape, b r i g h t n e s s ,  orientation. Position  3:  Fourth c a r d , second row  Position  4:  Third  Position  5:  Second c a r d , second row  card, f i r s t  row  of c h o i c e  of c h o i c e  cards  cards  of c h o i c e  cards  size.  143  Set No. 8:  Changes i n four dimensions, s i z e , shape,  orientation,  number. Position  c a r d , second row o f c h o i c e  cards  P o s i t i o n 4:  F i f t h c a r d , second row o f choice  cards  Position  Third card, f i r s t  Set No. 9:  3:  5:  First  row o f c h o i c e  cards  Changes i n four dimensions, s i z e , shape, o r i e n t a t i o n , number.  Position  3:  row o f choice  cards  P o s i t i o n 4:  S i x t h c a r d , second row of c h o i c e  cards  Position  Fifth  5:  Second c a r d , f i r s t  card, f i r s t  row o f choice  cards  Set No. 10: Changes i n f i v e dimensions, s i z e , b r i g h t n e s s , o r i e n t a t i o n , shape, number. Position  3:  T h i r d c a r d , second row o f c h o i c e  cards  Position  4:  S i x t h c a r d , second row o f choice  cards  Position  5:  Second c a r d , second row o f c h o i c e  cards  Set No. 11: Changes i n f i v e dimensions, s i z e , b r i g h t n e s s , o r i e n t a t i o n , shape, number. Position  3:  Second c a r d , f i r s t  row of choice  cards  P o s i t i o n 4:  Second c a r d , second row o f choice  Position  First  NOTE:  5:  card, f i r s t  row o f c h o i c e  cards  cards  D i s t o r t i o n s i n the photographs do not show a c c u r a t e l y the d i f f e r e n c e s i n b r i g h t n e s s .  144 ADDENDUM In the MC,  MR,  and SER  tasks each o f the choice p i e c e s  s p e c i f i c d e s c r i p t i o n of m a t e r i a l s i n pages 52, respectively), constituted a different  62,  combination  o f each o f the s i x dimensions i n v o l v e d .  (see  and  63  of values  Thus, f o r a l l three  tasks there were only nine p o s i t i v e i n s t a n c e s or c o r r e c t p i e c e s , the r e s t The  scores  of the p i e c e s c o n s t i t u t e d n e g a t i v e  from a l l Ss were s e p a r a t e d  r a t h e r than age each grade was  (see page 102).  on the b a s i s o f grade  T h i s was  f a i r l y homogeneous w i t h  instances.  done so because  r e s p e c t to the  age  range t h a t i t encompassed  ( f o r G l , range = 6.5-8.4, mean =  7.02  months; f o r G2,  y e a r s , and SD = 6.53  mean = 7.93 thus  y e a r s , and SD = 4.82  t r e a t e d as r e p r e s e n t i n g two  range = 7.4-9.0,  months).  Both grades were  different  developmental  levels. Although  the FORHIN scores  significantly o f the two  1.89  from each o t h e r , the d i f f e r e n c e s i n v a r i a n c e s  The  f o r Gl and  significant and  differ  scores showed a t r e n d which i n d i c a t e d s t a t i s t i c a l  significance. scores  f o r G l and G2 d i d not  at the  r a t i o of the v a r i a n c e s between the FORHIN G2 was .05  1.67  and  1.61 r e s p e c t i v e l y .  .10  with  df = 31/31.  l e v e l s with  The  F ratios  df = 30/30 were  

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