UBC Theses and Dissertations

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UBC Theses and Dissertations

An analysis of number conservation ability Schreck, Cynthia Ann Nagel 1971

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AN ANALYSIS OF NUMBER CONSERVATION ABILITY by CYNTHIA ANN NAGEL SCHRECK B.A., G r i n n e l l C o l l e g e , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of P s y c h o l o g y We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1971 In presenting th i s thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes i s for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes is f o r f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of Psychology The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 18, 1971 A B S T R A C T Seven hypotheses were d e r i v e d from P i a g e t ' s (a) t a s k a n a l y s i s o f the o r d i n a l c o r r e s p o n d e n c e and .-conservation t a s k s ; (b) d e s c r i p t i o n o f the development from p r e o p e r a t i o n to. c o n c r e t e o p e r a t i o n a l t h i n k i n g ; and (c) i d e a s about the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h i s t r a n s i t i o n . An attempt was a l s o made to s o l v e the m e t h o d o l o g i c a l problems o f Ca) how to determine whether a c h i l d i s a r e a l c o n s e r v e r or a p s e u d o - c o n s e r v e r ; (b) how to determine whether the c h i l d u n d e r s t a n d s the terms i n a P i a g e t i a n t a s k ; and (c) v a r i a t i o n i n performance due to s t i m u l u s v a r i -a b l e s . Twenty-eight f o u r - y e a r - o l d Ss, 36 k i n d e r g a r t e n Ss, and 28 grade one Ss were each g i v e n the f o l l o w i n g seven t e s t s , i n a d d i t i o n to a t e s t o f c o u n t i n g a b i l i t y : (1) a t e s t o f t h e i r a b i l i t y to match the number o f o b j e c t s i n a s t a n d a r d row; (2) a c o n s e r v a t i o n o f number t e s t ; (3) a t e s t of t h e i r a b i l i t y to a n t i c i p a t e and remember the r e s u l t s o f a t r a n s f o r m a -t i o n i n a row of o b j e c t s ; (4) a t e s t o f the c r i t e r i a on which they made judgments o f q u a n t i t y ; (5) a t e s t o f t h e i r a b i l i t y to b u i l d two c o r r e s -ponding s e r i e s of o b j e c t s ; (6) a t e s t o f t h e i r a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s ; and (7) a t e s t o f t h e i r a b i l i t y to m e n t a l l y t r a n s f o r m a s t i m u l u s . F i v e o f the hypotheses were c o n f i r m e d and one h y p o t h e s i s was p a r t i a l l y c o n f i r m e d . The m e t h o d o l o g i c a l problems were p a r t i a l l y s o l v e d . A new approach t o the development o f number c o n s e r v a t i o n a b i l i t y was sugge s t e d by the d a t a . i i TABLE OF CONTENTS Page LIST OF TABLES i v LIST OF FIGURES v i i i INTRODUCTION . . . . 1 METHOD . . . . 42 RESULTS . . . . . . 63 DISCUSSION 90 GENERAL CONCLUSIONS 105 BIBLIOGRAPHY 107 APPENDIXES 113 APPENDIX A. D e f i n i t i o n s 113 APPENDIX B. Chi-Square T a b l e s • • • H 4 APPENDIX C. A n a l y s i s of V a r i a n c e T a b l e s 134 i i i LIST OF TABLES T a b l e Page 1. Cards Used In the Density-Length-Number T e s t (D/L/N). . . . 54 2. P r e d i c t i o n o f the Rank Order o f the Mean Numbers of C o r r e c t Responses on the Density-Length-Number T e s t (D/L/N) '56 3. Performance on the T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) 63 4. The E f f e c t o f S e s s i o n on Performance on the T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) 63 5. Performance on the T e s t f o r C o n s e r v a t i o n of Number (CON) . . 65 6. A n a l y s i s o f V a r i a n c e of Scores on the T e s t f o r C o n s e r v a t i o n o f Number .134 7. Mean Numbers o f C o r r e c t Responses on the T e s t f o r C o n s e r v a t i o n o f Number (CON) . . .: 66 8. Performance on the S e r i a t i o n T e s t (ST) 67 9. E f f e c t o f S e s s i o n on Performance on the S e r i a t i o n T e s t ( S T ) . 67 10. Performance on the O r d i n a l Correspondence T e s t (OCT) . . . . 68 11. A n a l y s i s o f V a r i a n c e of Scor e s on t h e O r d i n a l Correspondence T e s t U s i n g K i n d o f T r a n s f o r m a t i o n as the W i t h i n - S u b j e c t s V a r i a b l e 135 12. A n a l y s i s o f V a r i a n c e o f Scores on the O r d i n a l Correspondence T e s t U s i n g the Number to be Matched as the W i t h i n - S u b j e c t s V a r i a b l e 136 13. Mean Numbers of C o r r e c t Responses on the O r d i n a l Correspondence T e s t (OCT) 69 14. A n a l y s i s o f V a r i a n c e on the Scor e s on the Imagery T e s t With L e v e l o f Development (G) Determined by Performance on the T e s t f o r I n t e n s i v e Q u a n t i t y (Method 1) 137 i v V T a b l e 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Page A n a l y s i s o f V a r i a n c e on the Scor e s on t h e Imagery T e s t With L e v e l o f Development (G) Determined by Performance on the O r d i n a l Correspondence T e s t (Method 2) . 139 A n a l y s i s o f V a r i a n c e on the Scores on the Imagery T e s t With L e v e l of Development (G) Determined by Performance on the T e s t f o r C o n s e r v a t i o n o f Number and the T e s t f o r I n t e n s i v e Q u a n t i t y (Method 3) 141 Mean Numbers o f C o r r e c t Responses From the Pl a n n e d Compari-sons T e s t s on the Density-Length-Number T e s t (D/L/N). . . .73 A n a l y s i s of V a r i a n c e on the Scores on the Density-Length-Number T e s t 143 Mean Numbers o f C o r r e c t Responses From the A n a l y s i s o f V a r i a n c e on the Density-Length-Number T e s t (D/L/N) . . . 74 A n a l y s i s o f V a r i a n c e on the Scores on the T r a n s f o r m a t i o n T e s t With L e v e l o f Development (G) Determined by P e r -formance on the T e s t f o r I n t e n s i v e Q u a n t i t y (Method 1 ) . . 145 Mean Numbers of C o r r e c t Responses From the A n a l y s i s o f V a r i a n c e on the T r a n s f o r m a t i o n T e s t (TT) With G Determined by Method 1 . . . . . . . . . . 77 A n a l y s i s o f V a r i a n c e on the Scores on the T r a n s f o r m a t i o n Test. With L e v e l o f Development (G) Determined by P e r -formance on the O r d i n a l Correspondence T e s t (Method 2 ) . . 146 Mean Numbers o f C o r r e c t Responses From the A n a l y s i s o f V a r i a n c e on the T r a n s f o r m a t i o n T e s t (TT) With G Determined by Method 2 779 A n a l y s i s o f V a r i a n c e on the Scor e s on the T r a n s f o r m a t i o n T e s t With L e v e l o f Development (G) Determined by P e r -formance on the T e s t f o r C o n s e r v a t i o n o f Number and the T e s t f o r I n t e n s i v e Q u a n t i t y (Method 4) 147 T e s t f o r C o n s e r v a t i o n o f Number (.CON) (Method 1) X O r d i n a l Correspondence T e s t (OCT) (Method 1) 118 T e s t f o r C o n s e r v a t i o n o f Number CCON) (Method 4) X O r d i n a l Correspondence T e s t (OCT) (Method 2) . . . . . . 118 T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X S e r i a t i o n T e s t (ST) (Method 2) 119 v i T a b l e Page 28. S e r i a t i o n T e s t (ST) (Method 2) X O r d i n a l Correspondence T e s t (OCT) (Method 1) . . . . . . . . . . . . . . 119 29. T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 2 ) . . . . . . . . 120 30. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 3) X Imagery T e s t ( S t r o n g C r i t e r i o n ) (SI) . . . 120 31. T e s t f o r C o n s e r v a t i o n o f Number (CON)'(Method 3) X Imagery T e s t (Weak C r i t e r i o n ) ' ( C I ) . . 121 32. O r d i n a l Correspondence T e s t (OCT) (Method 1) X Imagery T e s t ( S t r o n g C r i t e r i o n ) ( S I ) . . . . . . . . . . . . 121 33. O r d i n a l Correspondence T e s t (OCT) (Method 1) X Imagery T e s t (Weak C r i t e r i o n ) (CI) 122 34. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 3) X T r a n s f o r m a t i o n T e s t (Weak C r i t e r i o n ) (WTR). . . . . . . . •. 122 35. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 3) X T r a n s f o r m a t i o n T e s t ( S t r o n g C r i t e r i o n ) (STR) 123 36. O r d i n a l Correspondence T e s t (OCT) (Method 1) X T r a n s f o r m a t i o n T e s t ( S t r o n g C r i t e r i o n ) (STR) 123 37. O r d i n a l Correspondence T e s t (OCT) (Method 1) X T r a n s f o r m a t i o n T e s t (Weak C r i t e r i o n ) (WTR) 124 38. T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X T r a n s f o r m a t i o n T e s t (Weak C r i t e r i o n ) (WTR) 124 39. T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X T r a n s f o r m a t i o n T e s t ( S t r o n g C r i t e r i o n ) (STR) 125 40. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 2) X C o u n t i n g A b i l i t y (CT) 125 41. T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X C o u n t i n g A b i l i t y (CT) 126 42. O r d i n a l Correspondence T e s t (OCT) (Method 1) X C o u n t i n g A b i l i t y (CT) 126 43. T e s t f o r C o n s e r v a t i o n o f Number (CON)(Method 1) X Age . . . . 127 v i i T a b l e Page 44. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 2) X Age . . .127 45. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 3) X Age . . .128 46. T e s t f o r C o n s e r v a t i o n o f Number (CON) (Method 4) X Age.. . .128 47. O r d i n a l Correspondence T e s t (OCT) (Method 1) X Age 129 48. O r d i n a l Correspondence T e s t (OCT) (Method 2) X Age 129 49. S e r i a t i o n T e s t (ST) (Method 1) X Age 130 50. S e r i a t i o n T e s t (ST) (Method 2) X Age . . 130 51. T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) X Age 131 52. Imagery T e s t ( S t r o n g C r i t e r i o n ) (SI) X Age 131 . 53. Imagery T e s t (Weak C r i t e r i o n ) (CI) X Age 132 54. T r a n s f o r m a t i o n T e s t (Weak C r i t e r i o n ) (WTR) X Age 132 55. T r a n s f o r m a t i o n T e s t ( S t r o n g C r i t e r i o n ) (STR) X Age 133 56. C o u n t i n g A b i l i t y (CT) X Age 133 57. T e s t f o r C o n s e r v a t i o n o f Number (CON) X O r d i n a l Correspondence T e s t (OCT) ( A l l Ss i n c l u d e d ) . . . . 82 58. T e s t f o r C o n s e r v a t i o n o f Number (CON) X O r d i n a l Correspondence T e s t (OCT) (Only those j5s who un d e r s t o o d i n c l u d e d ) 83 LIST OF FIGURES F i g u r e Page 1. Placement of S t i c k s i n the T e s t f o r C o n s e r v a t i o n o f Number 46 2. Placement o f S t i c k s i n Sample Cards From the Imagery T e s t 50 3. Placement of S t i c k s i n the O r d i n a l Correspondence T e s t . . 60 v i i i A C K N O W L E D G E M E N T S I w i s h to ex p r e s s my g r a t i t u d e f o r the s u p p o r t and a d v i c e o f Dr. Lou Moran and Dr. C h r i s T r a g a k i s i n the p r e p a r a t i o n of t h i s t h e s i s . My deep a p p r e c i a t i o n a l s o goes to the Vancouver S c h o o l Board, p a r t i c u -l a r l y to Mr. C o u s i n s , p r i n c i p a l ; Mrs. M a r i c h , k i n d e r g a r t e n t e a c h e r ; and Mrs. P i t h e r , grade o n e - t e a c h e r a t the Van Home S c h o o l , and t o Mr. Shrimpton, head o f Cyp r e s s House S c h o o l and Mrs. Holden, head o f P e t e r Pan N u r s e r y S c h o o l , f o r p e r m i t t i n g t h e i r s t u d e n t s t o p a r t i c i p a t e i n t h i s s t u d y . I am a l s o g r a t e f u l t o Mrs. Abraham o f L i t t l e Mountain N u r s e r y S c h o o l , and to S i s t e r Muriam, r e c e n t p r i n c i p a l o f Our Lady o f P e r p e t u a l Help S c h o o l , f o r the use o f t h e i r s t u d e n t s as s u b j e c t s i n p i l o t work done f o r t h i s s t u d y . Thanks a r e a l s o due my husband, D a v i d Schreck, f o r the many hours o f programming he d i d to a s s i s t i n the a n a l y -s i s o f t h e r e s u l t s ; I a l s o w i s h to thank Susan Aizenman f o r her e x c e l l e n t j o b o f t y p i n g . i x INTRODUCTION The purpose o f t h i s t h e s i s i s t h r e e f o l d . F i r s t , i t attempts to t e s t P i a g e t ' s t a s k a n a l y s i s o f the a b i l i t i e s o f c o n s e r v a t i o n o f number and o r d i n a l c o r r e s p o n d e n c e . Second, i t attempts t o r e p l i c a t e P i a g e t ' s o b s e r v a t i o n s on the development o f t h e s e two a b i l i t i e s . , And f i n a l l y , i t d e v e l o p s an h y p o t h e s i s on the development p r o c e s s l e a d i n g t o the achievement o f the two a b i l i t i e s . Task A n a l y s i s A c c o r d i n g t o P i a g e t [1964a], the t r a n s i t i o n from p r e o p e r a t i o n a l to o p e r a t i o n a l thought i n v o l v e s a 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 n l y of the q u a l i t a t i v e n a t u r e o f t h i n g s to an u n d e r s t a n d i n g o f q u a n t i t y . I n o r d e r f o r t h i s t o o c c u r , the c h i l d ' s t h i n k i n g must change from s u b l o g i c a l to l o g i c a l . S u b l o g i c a l judgments o f q u a n t i t y a r e based on the s p a t i a l arrangement or g l o b a l a s p e c t s o f the o b j e c t s t o be q u a n t i f i e d , e.g., l e n g t h i n c o n s e r v a t i o n o f number t a s k s . L o g i c a l judgments o f q u a n t i t y , however, a r e based on an u n d e r s t a n d i n g t h a t the o b j e c t s to be q u a n t i f i e d a r e d i s c r e t e u n i t s . U n i t s a r e d e f i n e d by P i a g e t as o b j e c t s t h a t can be simultaneously (a) d i f f e r e n t i a t e d from each o t h e r ; and (b) i n c l u d e d t o -g e t h e r i n a c l a s s . A c c o r d i n g t o h i s t a s k a n a l y s i s o f c o n s e r v a t i o n of number and o r d i n a l c o r r e s p o n d e n c e [ P i a g e t , 1952] b o t h a b i l i t i e s depend on an u n d e r s t a n d i n g t h a t the o b j e c t s i n v o l v e d a r e u n i t s . (A t a s k a n a l y s i s i s an attempt to s p e c i f y the a b i l i t i e s n e c e s s a r y f o r s u c c e s s f u l p e r f o r m -ance on a g i v e n t a s k ) . 1 2 F i r s t , c o n s i d e r the example of a t a s k i n which e i g h t r e d s t i c k s and e i g h t b l u e s t i c k s a r e p l a c e d randomly on a t a b l e and the c h i l d i s asked to make a judgment as to whether t h e r e a r e more r e d s t i c k s , the same number of r e d s t i c k s as b l u e s t i c k s , o r more b l u e s t i c k s . There a r e two p o s s i b l e ways the c h i l d c o u l d a r r i v e a t a c o r r e c t answer. One i s to make ( e i t h e r m e n t a l l y o r p h y s i c a l l y ) a one-to-one c o r r e s p o n d e n c e between the r e d s t i c k s and the b l u e s t i c k s . Another i s to count the number o f r e d s t i c k s and b l u e s t i c k s . C o n s i d e r the f i r s t method. The c h i l d must i s o l a t e ( e i t h e r m e n t a l l y o r p h y s i c a l l y ) a r e d s t i c k from the o t h e r r e d s t i c k s . Then he must i s o l a t e a b l u e s t i c k from the o t h e r b l u e s t i c k s . Then, he must p a i r the i s o l a t e d b l u e s t i c k w i t h the i s o l a t e d r e d s t i c k . T h i s p r o c e d u r e would be r e p e a t e d e i g h t t i m e s . As the c h i l d p r oceeded, he would have to group t o g e t h e r , o r somehow keep t r a c k o f , those s t i c k s he had a l r e a d y p a i r e d so they would not be p a i r e d a g a i n . Otherwise, f o r example, i f he had p a i r e d r e d s t i c k , a, w i t h b l u e s t i c k , b, and then l a t e r f o r g o t t h a t he had a l r e a d y p a i r e d a w i t h a b l u e s t i c k , he might c o n c l u d e t h e r e were more r e d s t i c k s than b l u e ones; he would t h i n k he had an u n p a i r e d r e d s t i c k , a, l e f t o v e r . I f the c h i l d counts the s t i c k s , he must a l s o i s o l a t e an i n d i v i d u a l s t i c k to count i t and then group i t w i t h o t h e r s a l r e a d y counted so as not to count the same s t i c k t w i c e . Not o n l y does he have to i s o l a t e the s t i c k s from each o t h e r , but he must a l s o , a t the same time, group them t o g e t h e r . I t i s t h i s a b i l -i t y t o s i m u l t a n e o u s l y group t o g e t h e r and s e p a r a t e d i s c r e t e o b j e c t s t h a t P i a g e t says u n d e r l i e s c o n s e r v a t i o n a b i l i t y . Now c o n s i d e r the example of a t y p i c a l o r d i n a l c o r r e s p o n d e n c e t a s k i n which t h e r e a r e e i g h t b l u e s t i c k s v a r y i n g i n s i z e from one to n i n e i n -ches arid e i g h t r e d s t i c k s v a r y i n g i n s i z e from e i g h t to f i f t e e n i n c h e s 3 p l a c e d randomly b e f o r e the c h i l d . The c h i l d ' s t a s k i s t o p o i n t to the b l u e s t i c k which goes w i t h the f i f t h l a r g e s t r e d s t i c k (which, o f c o u r s e , i s the f i f t h l a r g e s t b l u e s t i c k ) . A g a i n , the c h i l d can s o l v e the problem by making a one-to-one co r r e s p o n d e n c e o f the l a r g e s t r e d s t i c k w i t h the l a r g e s t b l u e s t i c k , the second l a r g e s t r e d s t i c k w i t h the second l a r g e s t b l u e s t i c k , e t c . Or he can s o l v e the problem by c o u n t i n g . An a n a l y s i s o f t h i s t a s k , s i m i l a r t o the one a p p l i e d t o the c o n s e r v a t i o n o f number t a s k , can be made. When t h a t i s done, i t i s c l e a r t h a t the o r d i n a l correspondence t a s k , t o o , depends on an a b i l i t y t o s i m u l t a n e o u s l y g r o u p . t o g e t h e r and sep-a r a t e d i s c r e t e o b j e c t s . I f P i a g e t ' s t a s k a n a l y s i s i s c o r r e c t , t h e r e ought t o be a c o r r e -l a t i o n between the c h i l d ' s a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s and h i s a b i l i t y t o co n s e r v e number because they b o t h r e q u i r e the a b i l i t y to con-s t r u c t t h i s . As F l a v e l l and W o h l w i l l [1969] have p o i n t e d out, however, most attempts a t f i n d i n g c o r r e l a t i o n s between t a s k s which, a c c o r d i n g t o P i a g e t , depend upon the same a b i l i t y have not been s u c c e s s f u l . P i a g e t h i m s e l f , a f t e r p r e d i c t i n g such a c o r r e l a t i o n , s t a t e s t h a t r e s e a r c h e r s w i l l n ot f i n d i t because of the v a r i a t i o n s a c r o s s the two t a s k s [ P i a g e t , 1952], P i n a r d and Laurendeau [1969] have p o i n t e d out t h a t , a c c o r d i n g to P i a g e t , the d i f f e r e n c e ( h o r i z o n t a l d e c a l a g e ) between performances on two t a s k s i s a t a minimum i f the same o b j e c t s are used i n b o t h t a s k s . I f P i a g e t ' s t a s k a n a l y s i s i s a t a l l t e s t a b l e , the c o r r e l a t i o n ought t o appear i f the same o b j e c t s a r e used a c r o s s t a s k s , something which p r e v i o u s r e s e a r c h e r s have not done. F o r example, D o d w e l l [1960] attempted to r e l a t e the c h i l d ' s a b i l i t y t o co n s e r v e number u s i n g eggs and egg cups as m a t e r i a l s 4 to h i s a b i l i t y to c o n s e r v e number u s i n g r e d and b l u e poker c h i p s as mater-i a l s . An attempt was then made to r e l a t e t h ese a b i l i t i e s t o the a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e between d o l l s and w a l k i n g s t i c k s . He found no r e l a t i o n s h i p between the c o n s e r v a t i o n o f number t a s k s and the o r d i n a l c o r -respondence t a s k s as measured by c o n d i t i o n a l p r o b a b i l i t i e s . The v a r i a b l e of m a t e r i a l s used, however, was o b v i o u s l y n o t h e l d c o n s t a n t . The p r e s e n t s t u d y w i l l attempt t o f i n d a c o r r e l a t i o n between the c h i l d ' s a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s and to c o n s e r v e number, h o l d i n g as many v a r -i a b l e s as p o s s i b l e c o n s t a n t a c r o s s the two t a s k s . T h i s l e a d s to Hypothesis 1: There w i l l be a p o s i t i v e relationship between the child's ability to find ordinal correspondences regardless of- the spatial arrangement of the series and his ability to conserve, if the same materials are used to test for the two abilities. D e s c r i p t i o n o f Development C o n s i s t e n t w i t h h i s t a s k a n a l y s i s , P i a g e t ' s d e s c r i p t i o n o f the b e -h a v i o r which appears b e f o r e and l e a d s up to c o n s e r v a t i o n of number and o r d i n a l c o r r e s p o n d e n c e emphasizes the development away from a r e l i a n c e on what he c a l l s the " q u a l i t a t i v e " a s p e c t s of the s t i m u l u s . I n i t i a l l y , d u r i n g Stage I , when the c h i l d i s asked to match the number of o b j e c t s i n a s t a n d -a r d row, he w i l l s i m p l y match the l e n g t h of the s t a n d a r d row, i g n o r i n g i t s d e n s i t y and the number of o b j e c t s i n i t . A n a l o g o u s l y , the c h i l d i n Stage I b u i l d s o n l y a g l o b a l s e r i e s — he w i l l put o b j e c t s i n two rows i n s p a t i a l c o r r e s p o n d e n c e , i g n o r i n g the r e l a t i o n s (e.g., he w i l l put the f o u r t h l a r g ^ e s t i n one row o p p o s i t e the s m a l l e s t i n the second row). 5 As the r e g u l a t i o n s of the i n t u i t i v e s t a g e d e v e l o p , the c h i l d reaches Stage I I and becomes c a p a b l e of c o - o r d i n a t i n g two r e l a t i o n s ( l e n g t h and den-s i t y i n the case of c o n s e r v a t i o n of number and b i g g e r and s m a l l e r i n the case of a s i z e s e r i a t i o n ) . The a b i l i t y t o b u i l d a s e r i e s and the a b i l i t y t o match the number, o f o b j e c t s i n a s t a n d a r d row b o t h depend on t h i s a b i l -i t y t o c o - o r d i n a t e two r e l a t i o n s , a c c o r d i n g to P i a g e t . T h i s l e a d s to Hypothesis 2: There w i l l be a p o s i t i v e relationship between -the child's ability to match the number of objects in a standard row and his ability to build a series if the same materials are used to test for the two abil*-ities. P i a g e t a l s o contends t h a t when the Stage I I c h i l d i s a b l e to make one s e r i e s he can a l s o b u i l d a second s e r i e s and make the c o r r e s p o n d e n c e between them, a g a i n because c o r r e c t performance on a l l t h r e e t a s k s depends on the a b i l i t y t o c o - o r d i n a t e r e l a t i o n s . T h i s l e a d s to Hypothesis 3: When the child is able to build one series, he w i l l also be able to build a second series and to place the two series in correspondence. The c h i l d a t Stage I I b u i l d s the s e r i e s o n l y by t r i a l and e r r o r . When he r e a c h e s Stage I I I , i . e . , when he i s a b l e t o f i n d the m a t c h i n g ob-. j e c t s o f the two s e r i e s when t h e i r s p a t i a l c o r r e s p o n d e n c e i s d i s r u p t e d and when he can c o n s e r v e number, P i a g e t says he f o l l o w s one of two s y s t e -m a t i c p r o c e d u r e s i n b u i l d i n g the s e r i e s . He e i t h e r (a) matches the b i g g e s t i n one row w i t h the b i g g e s t i n the second row, the second b i g g e s t w i t h the second b i g g e s t and on down; or (b) b u i l d s each s e r i e s s e p a r a t e l y , by f i r s t f i n d i n g the b i g g e s t , t h e n the second b i g g e s t and on down, b e f o r e p l a c i n g them i n c o r r e s p o n d e n c e . T h i s l e a d s to Hypothesis 4:-. There w i l l be a posi^ tive relationship between the ability to find ordinal correspondences with-?-out depending on a spatial one-to-one correspondence between the. objects in 6 the two series and the ability to build the series systematically rather than by trial and error. On the t a s k r e q u i r i n g him to match the number of o b j e c t s i n a s t a n d a r d row, the c h i l d a t Stage I I I s h o u l d n ot need t o make a p h y s i c a l one-to-one co r r e s p o n d e n c e between h i s row and the s t a n d a r d row to p e r f o r m c o r r e c t l y . T h i s l e a d s to Hypothesis 5: The child who conserves w i l l be able to match the number of objects in a standard row without mak- • ing a s p a t i a l one-to-one correspondence. T r a n s i t i o n From Stage I I to Stage I I I P i a g e t i a n t h e o r y , as d e v e l o p e d i n The Child's Conception of Number [1952], does not a d e q u a t e l y e x p l a i n what causes the c h i l d ' s b e h a v i o r to change from n o n c o n s e r v a t i o n t o c o n s e r v a t i o n ; i t p r o v i d e s o n l y a d e s c r i p t i o n o f the a b i l i t i e s t h a t emerge a f t e r the t r a n s i t i o n , i . e . , r e v e r s i b i l i t y , com-p e n s a t i o n , and i d e n t i t y . W a l l a c h [1969] argues t h a t t h e s e t h r e e a b i l i t i e s cannot e x p l a i n the t r a n s i t i o n t o o p e r a t i o n a l thought, as measured by con-s e r v a t i o n a b i l i t y . She p o i n t s out t h a t a l t h o u g h e m p i r i c a l r e v e r s i b i l i t y and p a r t i a l compensation and p s e u d o c o n s e r v a t i o n ( i d e n t i t y of o b j e c t o r su b s t a n c e , not o f q u a n t i t y ; see B e i l i n , 1969, and E l k i n d , 1969) a l l o c c u r b e f o r e c o n s e r v a t i o n , o p e r a t i o n a l r e v e r s i b i l i t y , compensation, and i d e n t i t y a l l presuppose c o n s e r v a t i o n a b i l i t y , and thus they cannot e x p l a i n i t . (See Appendix A f o r d e f i n i t i o n s o f e m p i r i c a l r e v e r s i b i l i t y , p a r t i a l compensation, compensation, p s e u d o c o n s e r v a t i o n , and i d e n t i t y ) . B r a i n e [1962] makes a s i m -i l a r c r i t i c i s m o f P i a g e t i a n t h e o r y when he p o i n t s out t h a t P i a g e t ' s r e s e a r c h t e c h n i q u e s a r e good f o r d i a g n o s i n g whether o r not the c h i l d p o s s e s s e s a c e r -t a i n a b i l i t y b u t not f o r d i s c o v e r i n g the a n t e c e d e n t s to t h a t a b i l i t y . Of 7 c o u r s e , a d e t a i l e d and a c c u r a t e d e s c r i p t i o n o f t h e a b i l i t i e s t h a t emerge w i t h t h e o p e r a t i o n a l s t a g e must be a c h i e v e d b e f o r e an e x p l a n a t i o n o f t h e i r emergence can be a t t e m p t e d . But an a t t e m p t s h o u l d a l s o be made t o e s t a b -l i s h t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r change, s o m e t h i n g w h i c h P i a g e t has n o t done v e r y e x p l i c i t l y f o r t h e t r a n s i t i o n , i n d i c a t e d by con-s e r v a t i o n a b i l i t y , f r o m p r e o p e r a t i o n a l t o o p e r a t i o n a l i n t e l l i g e n c e . A l t h o u g h The Child's Conception of Number [,1952] does no t a d e q u a t e -l y d e a l w i t h t h e c auses o f t h e change f r o m p r e o p e r a t i o n a l t o o p e r a t i o n a l t h o u g h t , some of P i a g e t ' s o t h e r s o u r c e s do, n o t a b l y The Early Growth of Logic in the Child [1964] and Play, Breams, and Imitation in Childhood [ 1 9 6 2 ] . The a n a l y s i s o f c o g n i t i v e development g i v e n i n t h e s e s o u r c e s i s n e v e r d i r e c t l y a p p l i e d t o t h e c o n s e r v a t i o n t a s k by P i a g e t . The f o l l o w i n g d i s c u s s i o n i s an a t t e m p t t o a p p l y P i a g e t ' s a n a l y s i s t o t h e development o f t h e a b i l i t y t o c o n s e r v e number and f i n d o r d i n a l c o r r e s p o n d e n c e s . The e s s e n -t i a l change t h a t o c c u r s , a c c o r d i n g t o t h e s e s o u r c e s , i s t h a t t h e c h i l d a t the o p e r a t i o n a l s t a g e i s no l o n g e r g u i d e d s o l e l y by t h e p r e s e n t . Because r e p r e s e n t a t i o n becomes p o s s i b l e a t t h e end o f t h e s e n s o r i m o t o r s t a g e and t h e b e g i n n i n g of t h e i n t u i t i v e s t a g e , a new k i n d o f r e g u l a t i o n , i n a d d i t i o n t o t h e s e n s o r i m o t o r r e g u l a t i o n s ( s u c c e s s i v e c o - o r d i n a t i o n o f s e n s o r i m o t o r schemes), becomes p o s s i b l e . T h i s new k i n d o f r e g u l a t i o n i s " r e p r e s e n t a -t i v e r e g u l a t i o n . " P i a g e t s a y s , " . . . t h e development f r o m g r a p h i c s t r u c t u r e s t o o p e r a t i o n a l s t r u c t u r e s depends on a complex i n t e r p l a y o f r e t r o a c t i v e and a n t i c i p a t o r y a c t i v i t i e s . These may be grouped t o g e t h e r under t h e h e a d i n g o f ' r e p r e s e n t a -t i v e r e g u l a t i o n s . ' We now know t h a t t h e y p r e p a r e t h e way f o r o p e r a t i o n a l r e v e r s i b i l i t y . " [ P i a g e t , 1964a, p. 2 3 2 ] . 8 R e p r e s e n t a t i v e r e g u l a t i o n s a l l o w the p r e s e n t to be e v a l u a t e d i n terms of the c h i l d ' s r e p r e s e n t a t i o n of the p a s t . D u r i n g the p r e o p e r a t i o n a l s t a g e , the p r e s e n t i s s t i l l overemphasized; i t has not become c o - o r d i n a t e d w i t h the p a s t . The two can become c o - o r d i n a t e d o n l y i f they a r e b o t h seen as p a r t o f the t r a n s f o r m a t i o n from p a s t to p r e s e n t . Thus i t i s t h a t P i a g e t s a y s , " F o l l o w i n g t h i s l i n e o f e x p l a n a t i o n , the t r a n s i t i o n from Stage I I to Stage I I I i s e x a c t l y what one would ex^ p e c t once the a n t i c i p a t i o n and the h i n d s i g h t a r e more de-v e l o p e d . When they f i r s t appear on the scene, they a r e o n l y p a r t i a l and what t h i s means i n p a r t i c u l a r i s t h a t they do not b e a r on transformations as suoh, b u t o n l y on t h e i r i s o l a t e d or s t a t i c r e s u l t s . But we a l r e a d y know t h a t a n t i c i p a t i o n and h i n d s i g h t c o n s t i t u t e a s h u t t l i n g back and f o r t h and we a r e t h e r e f o r e e n t i t l e d t o expect t h a t sooner o r l a t e r the p r o c e s s w i l l r e a c h a s t a t e o f e q u i l i b r i u m . " ( I t a l i c s p r o v i d e d by the a u t h o r ) . [Ibid., p. 288].-In another s o u r c e P i a g e t s a y s , "The c o n n e c t i o n s e s t a b l i s h e d by sensory-motor i n t e l -l i g e n c e l i n k o n l y s u c c e s s i v e p e r c e p t i o n s and movements, w i t h o u t an o v e r a l l r e p r e s e n t a t i o n d o m i n a t i n g the s t a t e s , d i s t i n c t i n time, o f the a c t i o n s thus o r g a n i z e d , and p l a c i n g them s i m u l t a n e o u s l y i n a complete t a b l e . . . Sensory-motor i n t e l l i g e n c e thus f u n c t i o n s l i k e a slow motion f i l m , r e p r e s e n t i n g one s t a t i c image a f t e r a n o t h e r i n s t e a d of a c h i e v i n g a f u s i o n o f the images. . . I f we a c c e p t the f u n c t i o n a l c o n t i n u i t y between s e n s o r y -motor i n t e l l i g e n c e and c o n c e p t u a l thought, and a l s o t h e i r s t r u c t u r e d i s s i m i l a r i t y , as d e f i n e d by these f o u r d i f f e r -e n ces, f o u r c o n d i t i o n s , c a p a b l e o f b e i n g f u l f i l l e d s i m u l -t a n e o u s l y , would seem to s u f f i c e f o r the t r a n s i t i o n from one of t h e s e forms of i n t e l l i g e n c e to the o t h e r . 1. A g e n e r a l a c c e l e r a t i o n of movements, s u c c e s s i v e a c t i o n s b e i n g merged i n t o a m o b i l e epitome o f the a c t i o n as a whole — the speeded-up f i l m o f 'the b e h a v i o r thus becom-i n g i n t e r i o r r e p r e s e n t a t i o n , the d r a f t o r p r e l i m i n a r y schema of the a c t i o n . " [ P i a g e t , 1962, p. 238]. I t i s i m p o r t a n t to emphasize t h a t what i s i n v o l v e d i s not s i m p l y - a know-l e d g e t h a t the p a s t has been t r a n s f o r m e d i n t o the p r e s e n t , or c o n v e r s e l y , 9 t h a t the p r e s e n t i s the r e s u l t o f a t r a n s f o r m a t i o n of the p a s t . That would p r o v i d e no continuum o r common dim e n s i o n on the b a s i s of which p a s t and p r e s e n t can be compared. I n s t e a d , the c h i l d must know the d e t a i l s o f the t r a n s f o r m a t i o n i t s e l f so t h a t he can m e n t a l l y (not j u s t p h y s i c a l l y ) p e r -form the t r a n s f o r m a t i o n . Those d e t a i l s were d i s c u s s e d above i n the s e c t i o n e n t i t l e d Task A n a l y s i s — the t r a n s f o r m a t i o n i n v o l v e d i n the c o n s e r v a t i o n of number and o r d i n a l correspondence t a s k s i s a t r a n s f o r m a t i o n of u n i t s . T h i s l e a d s t o Hypothesis 6: The child's ability to conserve and to find ordinal correspondences regardless of the spatial arrangement of the series w i l l be positively related to (a) his ability to anticipate and remember the results of transformations in the spatial arrangements of objects in a row; and also to (b) his ability to perform mental transformations. What causes a c h i l d to change from a r e l i a n c e on p r e s e n t p e r c e p -t u a l cues to a r e l i a n c e on t r a n s f o r m a t i o n a l cues? The cause i s one o f t e n g i v e n i n the l i t e r a t u r e [Gruen* 1965, H a l f o r d , 1970, Smedslund, 1961a] — a c o n f l i c t i n p e r c e p t u a l cues. The c h i l d l e a r n s , i n a s e n s o r i m o t o r f a s h i o n , t h a t i f one row i s l o n g e r , or i f one g l a s s i s t a l l e r than a n o t h e r , i t w i l l have "more," w i t h a l l o f the i d i o s y n c r a t i c meanings of "more" t o the c h i l d ( e . g . , i f they a r e rows of c a n d i e s , i t w i l l take l o n g e r to e a t the l o n g e r row, o r i f t h e g l a s s i s a g l a s s of m i l k , i t w i l l take l o n g e r to d r i n k i t ) . He may a l s o l e a r n t h a t d e n s i t y o r w i d t h can i n d i c a t e amount. I n the case of number, the d a t a [ P i a g e t , 1952] i n d i c a t e t h a t l e n g t h i s more o f t e n the c r i t e r i o n , p r o b a b l y because i t i s more o f t e n a s s o c i a t e d w i t h "more" i n the c h i l d ' s e x p e r i e n c e . But d e n s i t y must a l s o be l e a r n e d — i t seems 10 r e a s o n a b l e to assume t h a t i f the. d e n s i t y d i f f e r e n c e between two rows i s s u f f i c i e n t l y l a r g e t h e c h i l d w i l l c o n s i d e r the denser row to have more, d e s p i t e i t s l e n g t h . As the c h i l d comes to n o t i c e s m a l l e r and s m a l l e r den-s i t y . c d i f f e r e n c e s , a c o n f l i c t between l e n g t h and d e n s i t y c r i t e r i a w i l l b e g i n to occur more and more o f t e n when he i s r e q u i r e d to make judgments of quan-t i t y i n cases where t h e r e a r e e q u a l l e n g t h and d e n s i t y d i f f e r e n c e s . I t i s t h i s c o n f l i c t w h i c h i n d u c e s the change from a r e l i a n c e on p r e s e n t p e r c e p t u a l cues ( s e n s o r i m o t o r thought) to a r e l i a n c e on t r a n s f o r m a t i o n a l cues ( o p e r a -t i o n a l t h o u g h t ) . T h i s l e a d s to Hypothesis 7: For children in Stage I, i.e., those who are unable to correctly match the number of objects in a standard row, the d i f f e r e n t i a l lengths of two rows w i l l be the effective stimulus on which quantity judgments are based. For children in Stage II, i.e., those who can correctly match the number of objects in a standard row but who do not conserve, the effective stimulus w i l l be the density differences between the two rows minus the length difference between the two rows. For children in Stage III, i.e., those who conserve, the effective stimulus for quantity judgments w i l l be the difference between the number of objects in the two rows. (For a more d e t a i l e d d i s c u s s i o n of t h i s p r e d i c t i o n , see the d i s c u s s i o n of the Density-Length-Number T e s t i n the Method s e c t i o n , pp. 55-56). But the change from a r e l i a n c e on p e r c e p t u a l cues to a r e l i a n c e on the t r a n s f o r m a t i o n a l cues c o n s t r u c t e d by the c h i l d cannot o c c u r u n t i l the c h i l d has enough e x p e r i e n c e w i t h p h y s i c a l t r a n s f o r m a t i o n s (and has become c a p a b l e o f r e p r e s e n t i n g t h e p a s t ) to c o n s t r u c t the t r a n s f o r m a t i o n . m e n t a l l y . The v e r y n a t u r e of the c h i l d ' s a c t i v i t y , i n , say, p l a c i n g o b j e c t s i n a box o r row, h e l p s to f o c u s on t h e i d e a of a u n i t b o t h d i f f e r e n t i a t e d from o t h e r 11 u n i t s and i n c l u d e d w i t h o t h e r - u n i t s -— he p l a c e s one, then another (thus d i f f e r e n t i a t i n g them) u n t i l he has a group of them (thus i n c l u d i n g them i n a c l a s s ) . As P i a g e t s a y s , "The number i s a s y n t h e s i s of c l a s s i n c l u s i o n and o r d e r i n g . T h i s s y n t h e s i s i s b e i n g f a v o r e d by the c h i l d ' s own a c t i o n s . You have s e t up a s i t u a t i o n where t h e r e i s an i t e r a t i o n o f one same a c t i o n which c o n t i n u e s and which i s t h e r e f o r e o r d e r e d w h i l e a t the same time b e i n g i n c l u -s i v e . You have, so to speak, a l o c a l i z e d s y n t h e s i s of i n -c l u s i o n and o r d e r i n g which f a c i l i t a t e s the c o n s t r u c t i o n of the i d e a of number. . ." [ P i a g e t , 1964b, p. 17]. Two c o n d i t i o n s a r e thus n e c e s s a r y f o r c o n s e r v a t i o n : (1) s u f f i -c i e n t e x p e r i e n c e w i t h d e n s i t y cues so t h a t they come i n t o c o n f l i c t w i t h l e n g t h cues; and (2) s u f f i c i e n t e x p e r i e n c e w i t h p h y s i c a l t r a n s f o r m a t i o n s so t h a t the c h i l d i s a b l e to m e n t a l l y c o n s t r u c t the t r a n s f o r m a t i o n d f u n i t s . The f i r s t c o n d i t i o n a l l o w s the c h i l d to make a s p a t i a l one-to-one cor r e s p o n d e n c e i n which b o t h l e n g t h and d e n s i t y d i f f e r e n c e s between the two s e t s a r e c o n s i d e r e d and made p e r c e p t u a l l y e q u i v a l e n t , i . e . , a l l o w s him t o match the number o f o b j e c t s i n a s t a n d a r d row. The c h i l d ' s ex-p e r i e n c e w i t h v a r y i n g t r a n s f o r m a t i o n s to and from s p a t i a l one-to-one cor r e s p o n d e n c e s e n a b l e s him to meet the second c o n d i t i o n — i . e . , to con-s t r u c t a t r a n s f o r m a t i o n of u n i t s . M e t h o d o l o g i c a l Problems A f t e r i n i t i a l s u c c e s s f u l attempts a t the r e p l i c a t i o n o f the pheno-menon o f n o n c o n s e r v a t i o n [Dodwell, 1960, E l k i n d , 1964, 1961, Hood, 1962], r e c e n t r e s e a r c h e r s have t u r n e d t h e i r a t t e n t i o n to attempts a t e x p l a i n i n g the t r a n s i t i o n from n o n c o n s e r v a t i o n to c o n s e r v a t i o n . B e f o r e these attempts and the e v i d e n c e r e l a t e d to them a r e c o n s i d e r e d , some d i s c u s s i o n of the m e t h o d o l o g i c a l problems i n a s s e s s i n g c o n s e r v a t i o n i s i n o r d e r . 12 Some o f the c r i t i c i s m s most o f t e n made a g a i n s t P i a g e t ' s r e s e a r c h t e c h n i q u e s a r e t h a t he f a i l s to use l a r g e samples, to r e p o r t h i s r e s u l t s s t a t i s t i c a l l y , and to use s t a n d a r d i z e d p r o c e d u r e s . L a t e r r e s e a r c h e r s , however, have found the phenomenon he d e s c r i b e s to e x i s t i n l a r g e samples of c h i l d r e n u s i n g s t a n d a r d i z e d p r o c e d u r e s [Dodwell, 1960, " E l k i n d , 1961, 1964, Hood, 1962]. T h i s s u c c e s s a t r e p l i c a t i o n c o n f i r m s P i a g e t ' s c a r e i n check-i n g and d o u b l e - c h e c k i n g h i s c o n c l u s i o n s , r u l i n g out a l t e r n a t i v e i n t e r p r e -t a t i o n s through h i s c l i n i c a l method, even though he does not r e p o r t the de-t a i l s o f t h i s p r o c e d u r e v e r y o f t e n . Thus, t h i s c r i t i c i s m i s not c r u c i a l , a l t h o u g h r e s e a r c h e r s s h o u l d c e r t a i n l y c o n t i n u e to attempt to r e p l i c a t e P i a g e t ' s f i n d i n g s b e f o r e they a c c e p t them. A more s e r i o u s c r i t i c i s m , one most emphasized by B r a i n e [1959, 1962], i s t h a t P i a g e t ' s v e r b a l p r e s e n t a t i o n p r o c e d u r e s t e s t the c h i l d ' s u n d e r s t a n d -i n g o f words, not of c o n c e p t s . ( H i s v e r b a l assessment p r o c e d u r e s p r e s e n t a somewhat d i f f e r e n t problem to be d i s c u s s e d b e l o w ) . F o r example, i n the case of q u a n t i t y c o n s e r v a t i o n the c h i l d may be a b l e to make c o r r e c t judgments on the b a s i s of q u a n t i t y but may answer i n c o r r e c t l y because he i n t e r p r e t s the e x p e r i m e n t e r ' s q u e s t i o n c o n c e r n i n g whether one of the rows has "more" or i s "the same" as r e f e r r i n g to the l e n g t h of the rows or the h e i g h t of the l i q u i d , not to the q u a n t i t y . B r a i n e [1959, 1962] s u g g e s t s the use of n o n v e r b a l l e a r n i n g t e c h n i q u e s as a s o l u t i o n to t h i s problem. He has performed some experim e n t s u s i n g non-v e r b a l t e c h n i q u e s o s t e n s i b l y t e s t i n g the c h i l d f o r the p r e s e n c e of P i a g e t ' s o p e r a t i o n s . H i s r e s u l t s i n d i c a t e t h a t younger c h i l d r e n a r e a b l e to p e r f o r m t a s k s p r e s e n t e d n o n v e r b a l l y w h i c h they cannot p e r f o r m i f the t a s k i s p r e -s e n t e d v e r b a l l y . Recent w r i t e r s , [ F u r t h , Y o u n i s s , and Ross, 1970], however, 13 have p o i n t e d to a problem i n a s s e s s i n g a c h i l d ' s grasp o f a c o n c e p t . T h i s problem e x i s t s i n B r a i n e ' s work. F u r t h et. al. f i r s t d i s t i n g u i s h between the e x t e n s i o n and the i n t e n s i o n of a c o n c e p t , c r i t i c i z i n g s t a n d a r d methods o f s t u d y i n g c o n c e p t s because t h e s e s t a n d a r d methods are not d e s i g n e d to uncover the i n t e n s i o n o f the c o n c e p t s they s t u d y . F u r t h et. al. p r e s e n t the f o l l o w i n g example: "However, w i t h o u t c o n c e r n f o r i n t e n s i o n a l c h a r a c t e r i s t i c s , i n v e s t i g a t i o n s of c o n c e p t s a r e bound to remain ambiguous. Con-s i d e r , f o r example, the o p e r a t i o n a l d e f i n i t i o n of a concept o r d i n a r i l y used i n the c o n c e p t - a t t a i n m e n t l i t e r a t u r e : ' e i t h e r r e d , o r square, o r b o t h r e d and s q u a r e . ' Assume t h a t the ex-p e r i m e n t e r has i n t e n d e d t h i s i n c l u s i v e d i s j u n c t i v e c o n cept as the r u l e f o r combining the r e l e v a n t a t t r i b u t e s r e d - c o l o r and square-shape i n a p a r t i c u l a r experiment. Suppose the e x p e r -imenter p r e s e n t s the f o l l o w i n g t w o - a t t r i b u t e s t i m u l i : r e d s q u a r e , r e d c i r c l e , y e l l o w square, and y e l l o w c i r c l e and r e -i n f o r c e s a s u c c e s s i v e s o r t i n g i n t o l e f t (L) and r i g h t (R) p i l e s , L, L, L, R, r e s p e c t i v e l y . While t h i s p a t t e r n c o r r e s -ponds e x a c t l y w i t h the e x t e n s i o n of the d i s j u n c t i v e c o n c e p t , i t e q u a l l y i l l u s t r a t e s more c o n c e p t s than the one i n t e n d e d by the e x p e r i m e n t e r . "One such concept would be the ' d e n i a l o f the c o n j u n c - r t i o n o f y e l l o w and c i r c l e , ' t h a t i s , 'not a y e l l o w c i r c l e . ' N o t i c e t h a t i n t e n s i o n a l l y t h i s concept and the e x p e r i m e n t -e r ' s i n t e n d e d d i s j u n c t i v e concept do not agree even i n terms of r e l e v a n t a t t r i b u t e s . Another case which a l s o adheres to the a f o r e m e n t i o n e d s e t of exemplars i s the s i m p l e c o n d i t i o n -a l : ' I f i t i s y e l l o w , i t must be s q u a r e . ' Here, the e x p e r -imenter and the s u b j e c t agree on the r e l e v a n t a t t r i b u t e s - y e t d i s a g r e e as to the concept t h a t the s u b j e c t i s r e s p o n d i n g t o . Thus, the s u b j e c t might h o l d the concept of c o n j o i n t n e g a t i o n , 'not r e d and n o t s q u a r e , ' f o r which the r e i n f o r c e m e n t v a l u e s would be R, R, R, L. But s i n c e the s u b j e c t m e r e l y s o r t e d a l l items i n t o two c a t e g o r i e s i t would make no d i f f e r e n c e whether he s o r t e d an i n s t a n c e i n t o one c a t e g o r y or the o t h e r so l o n g as he s o r t e d a l l i n s t a n c e s s y s t e m a t i c a l l y . " [1970, p. 37]. The o n l y way to s t u d y the i n t e n s i o n o f a concept i s to p r e s e n t i n s t a n c e s of the concept w h i c h can r u l e out a l t e r n a t i v e i n t e n s i o n s and then, to o b s e r v e 14 the s u b j e c t ' s b e h a v i o r . I n the above example of the a l t e r n a t i v e i n t e n s i o n " d e n i a l o f the c o n j u n c t i o n of y e l l o w and c i r c l e , " one c o u l d p r e s e n t a b l u e c i r c l e . I f the s u b j e c t responded n e g a t i v e l y one c o u l d r u l e out the above a l t e r n a t i v e . In B r a i n e ' s well-known 1959 monograph, f i v e - to s i x - y e a r -o l d s , who a r e not a b l e to s o l v e t r a n s i t i v i t y problems when P i a g e t ' s v e r -b a l p r e s e n t a t i o n t e c h n i q u e s a r e used, were a b l e to s o l v e s i z e t r a n s i t i v -i t y problems. B r a i n e rewarded them f o r c h o o s i n g the " l o n g e r " (the word " l o n g e r " was never used) of two s t i c k s which were compared t o - a measur-i n g r o d i n view of the c h i l d . The d i f f e r e n c e i n l e n g t h was reduced so t h a t the c h i l d r e n had to use t h e measuring r o d to t e l l which of the com-p a r i s o n s t i c k s was the l o n g e s t . B r a i n e c o n c l u d e d t h a t the i n t e n s i o n o f the concept the c h i l d r e n were u s i n g was " i f A i s l o n g e r than B (the meas-u r i n g rod) and B i s l o n g e r than C, then A i s l o n g e r than C." L a t e r i n v e s -t i g a t o r s [Murray and Y o u n i s s , 1968, Y o u n i s s andvMurray, 1970] suggested t h a t the c h i l d r e n i n B r a i n e ' s s t u d y never l a b e l l e d A " l o n g e r " w h i l e they always l a b e l l e d C " l o n g e r , " and i t was on t h i s b a s i s t h a t they chose C. By u s i n g s t i m u l i which f o r c e d the s u b j e c t s .:to l a b e l A and B b o t h " l o n g e r " and " s h o r t e r , " they found t h a t f i v e - and s i x - y e a r - o l d s u b j e c t s d i d not show t r a n s i t i v i t y ; o n l y n i n e - y e a r - o l d s u b j e c t s d i d . Thus, they s u g g e s t e d and demonstrated t h a t B r a i n e ' s s u b j e c t s were u s i n g an i n t e n s i o n d i f f e r e n t than t h a t w i t h which P i a g e t was c o ncerned. (For a n o t h e r example of a s t u d y i n which the i n t e n s i o n of the concept used by c h i l d r e n t r a i n e d to behave o p e r a t i o n a l l y d i f f e r s from t h a t used by " n o r m a l l y " o p e r a t i o n a l c h i l d -r e n , see the s t u d y by Juan P a s c u a l - L e o n e , 1969). A s i m i l a r o b j e c t i o n can be made to W o h l w i l l and Lowe's [1962] p r o c e d u r e . S u b j e c t s were rewarded f o r l o o k i n g f o r a token under the c o r r e c t one o f t h r e e l a b e l l e d doors b o t h be-f o r e and a f t e r the t r a n s f o r m a t i o n of a row of o b j e c t s . The l a b e l s were 15 numbers, one of which, c o r r e s p o n d e d to t h e number of o b j e c t s i n the row. The c h i l d r e n c o u l d have been s o l v i n g t h e problem s i m p l y by g o i n g to the same door t w i c e i n a row. Another example of the c o n f u s i o n i n t h e l i t e r a -t u r e due to a l a c k of c o n c e r n about the i n t e n s i o n of c o n c e p t s t h a t g u i d e b e h a v i o r i s found i n Z i m i l e s ' [1966] d i s c u s s i o n . He d i s c o v e r e d ( i n p i l o t work) t h a t c h i l d r e n c o n s e r v e d i n e q u a l i t y b e t t e r . i f , i n s t e a d o f a s k i n g "which has more," one t o l d a s t o r y about the c a n d i e s o r asked "which s e t of t r u c k s would you l i k e to take home." He t h e r e f o r e i n d i c t s P i a g e t ' s v e r -b a l t e c h n i q u e . But h i s d a t a c o n c e r n c o n s e r v a t i o n of d i f f e r e n c e s which are p e r c e p t u a l [Smedslund, 1966b], not the c o n s e r v a t i o n of e q u a l i t y , which i s an i n f e r e n c e based on o p e r a t i o n s . Rothenberg [1969b] a l s o found t h a t con-s e r v a t i o n of i n e q u a l i t y i s e a s i e r than c o n s e r v a t i o n of e q u a l i t y . These d a t a show o n l y t h a t the i n t e n s i o n of the word "more" i s a s e n s o r i m o t o r scheme f o r t h e s e c h i l d r e n and i s not r e l a t e d to an o p e r a t i o n a l concept o f number. No n v e r b a l methods, the n , i n c r e a s e the number of a l t e r n a t i v e i n -t e n s i o n s p o s s i b l e — a l o n g l e a r n i n g s e s s i o n i s n e c e s s a r y i n which i n s t a n -ces a r e p r e s e n t e d to convey the i n t e n d e d i n t e n s i o n and d u r i n g the s e s s i o n a l t e r n a t i v e i n t e n s i o n s can d e v e l o p . Thus, when u s i n g n o n v e r b a l t e c h n i q u e s of p r e s e n t a t i o n , i f the c h i l d s ucceeds he might succeed by i n t e n s i o n s o t h e r than those d i s c u s s e d by P i a g e t . There i s a c o n t r o v e r s y between the Genevan and H a r v a r d s c h o o l s over the c r i t e r i o n f o r t h e assessment of c o n -s e r v a t i o n , e s p e c i a l l y a f t e r t r a i n i n g - s e s s i o n s . [ I n h e l d e r , 1966 and 1969, and P i a g e t , 1967]. The r e a s o n f o r .the c o n t r o v e r s y i s t h a t t h e H a r v a r d s c h o o l appears to be concerned w i t h t h e problem of t e a c h i n g the c h i l d a 16 p a r t i c u l a r b e h a v i o r — c o n s e r v a t i o n b e h a v i o r — and P i a g e t i s concerned w i t h the development o f t h e i n t e n s i o n t h a t produces c o n s e r v a t i o n b e h a v i o r — an i n t e r n a l m e n t a l s t r u c t u r e . A word evokes a c e r t a i n i n t e n s i o n i mmediately ( t h e r e i s no l e a r n -i n g s e s s i o n i n which a l t e r n a t i v e s can be d e v i s e d ) and the n a t u r e of the language environment l i m i t s the p o s s i b l e i n t e n s i o n i t c o u l d evoke so t h a t a l t e r n a t i v e s can be more e a s i l y r u l e d o u t . Thus, i f the c h i l d s o l v e s the problem when v e r b a l p r e s e n t a t i o n methods a r e used one can be more c e r t a i n he has the i n t e n s i o n which the p r o c e d u r e i s d e s i g n e d to s t u d y . The o r i g i n a l problem B r a i n e p o i n t e d out remains to be s o l v e d — i f the c h i l d does not succeed the r e a s o n f o r h i s f a i l u r e c o u l d s i m p l y be t h a t the p r o p e r i n t e n s i o n n e c e s s a r y to the s o l u t i o n , a l t h o u g h understood by the c h i l d , i s not evoked by the word. Because c h i l d r e n r e c e i v e a g r e a t d e a l of exposure to the a d u l t usage o f the terms "more" and "same" i n n u m e r i c a l s i t u a t i o n s , i f they c o u l d p e r c e i v e n u m e r i c a l r e l a t i o n s as d i s t i n c t from l e n g t h and d e n s i t y , the l a b e l s "more" and "same number" would q u i c k l y be-come l i n k e d to these n u m e r i c a l r e l a t i o n s . The most r e a s o n a b l e i n t e r p r e -t a t i o n o f n o n c o n s e r v a t i o n i s t h a t the c h i l d s i m p l y does not p e r c e i v e numer-i c a l r e l a t i o n s as s e p a r a t e from l e n g t h and d e n s i t y and so i t i s i m p o s s i b l e f o r "more" and "same number" to r e f e r t o something the c h i l d does n o t u n d e r s t a n d . I t has been demonstrated t h a t most c h i l d r e n o f f o u r y e a r s n i n e months to s i x y e a r s two months do u n d e r s t a n d the terms "more," "same," and " l e s s " as they a p p l y to number . [ G r i f f i t h s , Shantz, and S i g e l , 1967]. Seventy-two per c e n t u n d e r s t o o d "more" and "same" and e i g h t y per c e n t u n d e r s t o o d " l e s s . " Thus, t h e amount of n o n c o n s e r v a t i o n due to a l a c k 17 of the u n d e r s t a n d i n g of the terms, i f any, i s q u i t e s m a l l . N e v e r t h e l e s s , i t would c e r t a i n l y be b e s t to a s s e s s t h i s u n d e r s t a n d i n g i n each new e x p e r -i m e n t a l s i t u a t i o n . P i a g e t o f t e n does t h i s i n h i s c l i n i c a l p r o c e d u r e . In the s e r i a t i o n t a s k , he d i s c u s s e s [1952] how the c h i l d can p o i n t to c o r r e s -pondences i f the items a r e taken i n o r d e r and a l s o i f o n l y t h r e e o b j e c t s i n each s e r i e s a r e used [1962]. In the i n s t a n c e o f c o n s e r v a t i o n of num-be r , the c h i l d ' s u n d e r s t a n d i n g of the q u e s t i o n s can be shown by u s i n g the q u e s t i o n s i n a s i t u a t i o n w hich one knows the c h i l d can h a n d l e . Greco [1960, c i t e d i n F l a v e l l , 1963] found t h a t the numbers one through f i v e ^ a r e much e a s i e r f o r the c h i l d to u n d e r s t a n d than numbers g r e a t e r than f i v e . The e x p e r i m e n t e r c o u l d f i r s t ask the q u e s t i o n s u s i n g o n l y two o r t h r e e ob-j e c t s . I f the c h i l d answers c o r r e c t l y , he would be more c o n f i d e n t t h a t the c h i l d u n d e r s t a n d s the v o c a b u l a r y used. T h i s t e c h n i q u e w i l l be attempted i n the p r e s e n t s t u d y . C l o s e l y t i e d to the above problem of a l t e r n a t i v e i n t e n s i o n s i s the q u e s t i o n of v e r b a l assessment of c o n s e r v a t i o n a b i l i t y . The c l i n i c a l p r o -cedure i s u n i q u e l y d e s i g n e d to weed out a l t e r n a t i v e i n t e n s i o n s . The t y p i -c a l s t a n d a r d i z e d p r o c e d u r e s used by N o r t h American p s y c h o l o g i s t s , however, are n o t , and they o f t e n use the adequacy of the c h i l d ' s v e r b a l e x p l a n a t i o n s to check the i n t e n s i o n of the c h i l d ' s c o n c e p t . I t i s i n t e r e s t i n g t h a t many r e s e a r c h e r s have found no d i f f e r e n c e between the r e s u l t s o b t a i n e d when be-h a v i o r a l measures a r e used as the c r i t e r i a and those o b t a i n e d when c o r r e c t j u s t i f i c a t i o n s a r e used, as t h e c r i t e r i a [ E l k i n d , 1961, Pratoomraj and J o h n -son, 1966]. Gruen [1966], however, found t h a t the r e s u l t s d i f f e r s i g n i f i -c a n t l y depending on whether or not one uses c o r r e c t e x p l a n a t i o n s as a c r i -t e r i a . I t i s i n t e r e s t i n g t h a t Gruen's s t u d y was a t r a i n i n g s t u d y and the 18 o t h e r s t u d i e s m e r e l y d i a g n o s e d c o n s e r v a t i o n a b i l i t y . Thus those* c h i l d r e n who c o u l d not g i v e adequate e x p l a n a t i o n s might have b u i l t up i n t e n s i o n s t h r o u g h t h e i r t r a i n i n g w h i c h were d i f f e r e n t than t h e i n t e n s i o n s used by those who had a c q u i r e d c o n s e r v a t i o n " n o r m a l l y . " P i a g e t ' s o b j e c t i o n to B r u n e r ' s work [ P i a g e t , 1967] c a l l e d f o r f u r t h e r i n v e s t i g a t i o n of the i n -t e n s i o n of the concept the c h i l d had a t t a i n e d b y . a s k i n g f o r v e r b a l j u s t i -f i c a t i o n s and by a s k i n g f o r f u r t h e r t e s t s o f h i s u n d e r s t a n d i n g . I n h e l d e r and S i n c l a i r [1969] a l s o emphasize t h a t t e s t s o f the g e n e r a l i z a t i o n and t h e s t a b i l i t y o f a c q u i r e d c o n s e r v a t i o n a b i l i t y be used i n the d e s i g n of t r a i n i n g s t u d i e s . Some method s h o u l d c e r t a i n l y be used to f u r t h e r c l a r i f y the c h i l d ' s i n t e n s i o n i f the p r o c e d u r e s may l e a d to a l t e r n a t i v e i n t e n s i o n s , a l t h o u g h a s s e s s i n g the c h i l d ' s v e r b a l e x p l a n a t i o n s may not be the most v a l i d method because i t may o n l y t e s t the c h i l d ' s v e r b a l f l u e n c y . The b e s t method f o r r u l i n g out a l t e r n a t i v e i n t e n s i o n s r e s p o n s i b l e f o r a be-h a v i o r i n d u c e d by a t r a i n i n g s e s s i o n would be to o b t a i n f u r t h e r b e h a v i o r a l t e s t s of the s u b j e c t ' s u n d e r s t a n d i n g . The most a p p r o p r i a t e t e s t s would be those d i s c o v e r e d to be r e l a t e d to the t r a i n e d - i n a b i l i t y i n the " n o r m a l l y " d e v e l o p i n g ( i . e . , u n t r a i n e d ) c h i l d . One purpose of the p r e s e n t s t u d y i s to attempt to e s t a b l i s h some b e h a v i o r a l t e s t s t h a t a r e r e l a t e d t o number c o n s e r v a t i o n i n the n o r m a l l y d e v e l o p i n g c h i l d . A nother s o u r c e o f d i f f i c u l t y i n a s s e s s i n g c o n s e r v a t i o n a b i l i t y i s the e f f e c t s of t a s k and s t i m u l u s v a r i a b l e s on c o n s e r v a t i o n . I f t h e r e a r e s u c h e f f e c t s , maybe c o n s e r v a t i o n a b i l i t y i s a c t u a l l y dependent on a cong-lo m e r a t e of a b i l i t i e s , (e.g.., a t t e n t i o n a l . or 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 ) , r a t h e r t h a n on a mental s t r u c t u r e . The number o f beads (14 v s . 28) 19 and the amount o f d i s c r e p a n c y between the s t a n d a r d and t h e comparison beakers have no e f f e c t on c o n s e r v a t i o n of d i s c o n t i n u o u s ' q u a n t i t y [ F e i g e n -baum, 1963]. Z i m i l e s [1966] found no d i f f e r e n c e between the use of homo-geneous o r heterogeneous rows of m a t e r i a l s and no d i f f e r e n c e due to the s i g n i f i c a n c e of the m a t e r i a l s t o the c h i l d r e n i n a c o n s e r v a t i o n of number t a s k . ( T h i s l a t e r f i n d i n g was r e p l i c a t e d by R o l l , 1970). Z i m i l e s a l s o found t h a t a l t h o u g h the s i z e of the aggregate (3 v s . 7) had no o v e r a l l e f f e c t on c o n s e r v a t i o n of number, c h i l d r e n d i d b e t t e r i f they were p r e s e n t e d the s m a l l e r a g g r e g a t e f i r s t than i f they were f i r s t p r e s e n t e d the l a r g e r a g g r e g a t e . Z i m i l e s s u g g e s t s t h a t t h e se r e s u l t s a r e due to a tendency i n c h i l d r e n to r e p e a t t h e m s e l v e s . Expanding on t h i s i n t e r p r e t a t i o n , one might c o n c l u d e t h a t because t h r e e i s e a s i e r to conserve than seven [Greco, 1960, c i t e d i n F l a v e l l , 1963] the c h i l d r e n who were g i v e n the a g g r e g a t e of seven f i r s t were more l i k e l y to respond p e r c e p t u a l l y on s u c c e e d i n g problems. T h i s would be e s p e c i a l l y t r u e i f the c h i l d r e n were i n t r a n s i t i o n from non-c o n s e r v a t i o n to c o n s e r v a t i o n . A l t h o u g h most of h i s r e s u l t s were n e g a t i v e ( i . e . , he d i d not f i n d d i f f e r e n c e s due to t a s k and s t i m u l u s v a r i a b l e s ) , Z i m i l e s c o n c l u d e d t h a t c o n s e r v a t i o n i s not a " s p e c i f i c a r t i c u l a t e d c o n c e p t " b u t a " s e t o f s e m i - i n t e r r e l a t e d c o g n i t i v e a t t i t u d e s . " T h i s c o n c l u s i o n i s based on the o r d e r e f f e c t d i s c u s s e d above, on the c o r r e l a t i o n of .37 found between c o n s e r v a t i o n and performance on the embedded f i g u r e s t a s k , and on the f a c t t h a t a comparison between an aggregate of n i n e and an a g g r e g a t e of s i x was e a s i e r t h a n a comparison between seven and n i n e items ( t h e den-s i t y of t h e row of n i n e items was h e l d c o n s t a n t and t h e d e n s i t y of the row of seven was, of c o u r s e , g r e a t e r than t h e d e n s i t y of the row of s i x ) . 20 T h i s l a t e r r e s u l t l e d him to c o n c l u d e t h a t " n u m e r o s i t y , " i n a d d i t i o n to d e n s i t y and l e n g t h a r e f a c t o r s w h i c h the c h i l d must l e a r n to i n t e g r a t e . But the n i n e v s . s i x comparison c o u l d have been e a s i e r than n i n e v s . seven because the d e n s i t y d i f f e r e n c e between the two rows i s g r e a t e r f o r n i n e v s . s i x than f o r n i n e v s . seven. (One c o u l d e a s i l y t e s t t h i s by s y s -t e m a t i c a l l y v a r y i n g the l e n g t h , d e n s i t y , and number d i f f e r e n c e s between the rows to be compared, which w i l l be done i n the p r e s e n t s t u d y i n c o n -n e c t i o n w i t h H y p o t h e s i s 7 ) . The embedded f i g u r e s t e s t i s r e l a t e d to c o g -n i t i v e s t y l e and one i s h a r d l y j u s t i f i e d i n c o n c l u d i n g t h a t c o n s e r v a t i o n i s not a " s p e c i f i c a r t i c u l a t e d c o n c e p t , " as Z i m i l e s does, on t h a t b a s i s . O t h e r w i s e , many o t h e r phenomenon would a l s o have to be so d e s c r i b e d . A l l the c o r r e l a t i o n means i s t h a t c o g n i t i v e s t y l e may be r e l a t e d to c o n s e r v a -t i o n , not t h a t " c o n s e r v a t i o n i s not a v i a b l e mode of b e h a v i o r f o r the ex-p e r i m e n t a l a n a l y s i s of c o g n i t i o n , " the c o n c l u s i o n Z i m i l e s draws. T h i s i s s u e c o n c e r n i n g the e f f e c t of t a s k v a r i a b l e s on c o n s e r v a t i o n b e h a v i o r i s a t h e o r e t i c a l as w e l l as a m e t h o d o l o g i c a l i s s u e . Z i m i l e s ' [1966] p o i n t t h a t c o n s e r v a t i o n depends on a number of c o g n i t i v e a t t i t u d e s and i n f o r m a t i o n - p r o c e s s i n g v a r i a b l e s r e a p p e a r s i n Smedslund's [1966c] d i s -c u s s i o n . But Smedslund adds t h a t a l t h o u g h i n f o r m a t i o n - p r o c e s s i n g d i f f i -c u l t i e s may o b s c u r e the l o g i c a l c h a r a c t e r i s t i c s of a t a s k , t h e r e a r e d i f -f e r e n c e s i n performance (which he found) t h a t can be accounted f o r o n l y by d i f f e r e n c e s i n the l o g i c a l c h a r a c t e r i s t i c s of t h e t a s k s . T h i s s u g g e s t s t h a t i f i n f o r m a t i o n - p r o c e s s i n g v a r i a b l e s a r e h e l d c o n s t a n t , . o n e can see d i f f e r e n c e s i n b e h a v i o r due to d i f f e r e n c e s i n c o n s e r v a t i o n a b i l i t y . F l a -v e l l and W o h l w i l l [1969] have .suggested t h a t t h e r e a r e two a s p e c t s of c o g n i t i v e development, an i n c r e a s e i n the u n d e r s t a n d i n g of l o g i c a l c o n -21 c e p t s and an i n c r e a s e i n i n f o r m a t i o n - p r o c e s s i n g a b i l i t y . They have sug-g e s t e d an a n a l o g y between t h e s e two a s p e c t s o f c o g n i t i v e development and the competence-performance d i s t i n c t i o n i n language development — compe-tenc e c o r r e s p o n d i n g to the o p e r a t i o n s and schemes, and performance t o i n f o r m a t i o n - p r o c e s s i n g v a r i a b l e s . . They have put f o r t h a model o f c o g n i -t i v e development i n which changes i n competence and i n the a p p l i c a t i o n o f o p e r a t i o n s i n competence t o performance o c c u r somewhat s e p a r a t e l y (compe-tence b e i n g a c h i e v e d b e f o r e p e r f o r m a n c e ) . They a r e thus a b l e to p r e d i c t v a r i a n c e s i n performance due to d i f f e r e n c e s i n m a t e r i a l s , v a r a i a n c e s which d e c r e a s e w i t h development due to the c h i l d ' s i n c r e a s i n g a b i l i t y to a p p l y h i s competence to d i f f e r e n t s i t u a t i o n s . T h i s k i n d of a n a l y s i s i s s u p p o r t e d (weakly) by d a t a a t t a i n e d by U z g i r i s [1964] on the c o r r e l a t i o n s among the performances o f c h i l d r e n on f o u r d i f f e r e n t m a t e r i a l s as a f u n c t i o n o f t h e i r a b i l i t y to c o n s e r v e s u b s t a n c e , w e i g h t , and volume. The i n t e r c o r r e l a t i o n s among the performances on the v a r i o u s m a t e r i a l s were h i g h e s t ( a l t h o u g h n ot s i g n i f i c a n t l y ) (a) when most of the c h i l d r e n f a i l e d a l l t h r e e t a s k s ; (b) when most o f t h e c h i l d r e n c o n s e r v e d s u b s t a n c e but f a i l e d to c o n s e r v e weight and volume; and (c) when most o f the c h i l d r e n c o n s e r v e d weight and s u b s t a n c e but f a i l e d t o . c o n s e r v e volume. What t a s k v a r i a b l e s a f f e c t the i n f o r m a t i o n - p r o c e s s i n g p a r t o f t h i s model and how im p o r t a n t the i n f o r m a -t i o n - p r o c e s s i n g p a r t i s (what p o r t i o n o f the v a r i a n c e i t a c c o u n t s f o r ) r e -mains to be seen — e s p e c i a l l y because many i n v e s t i g a t o r s c i t e d above have f a i l e d to f i n d d i f f e r e n c e s due to task, v a r i a b l e s . There a r e o t h e r m e t h o d o l o g i c a l problems w h i c h need to be c o n s i d -e r e d i n s t u d y i n g number c o n s e r v a t i o n . S i e g e l and G o l d s t e i n 11969] found 22 t h a t young c h i l d r e n i n a number c o n s e r v a t i o n t a s k tend to respond posi:-t i v e l y t o the l a s t t h i n g the e x p e r i m e n t e r s a y s . (They say t h a t young c h i l d r e n respond on a " r e c e n c y " r a t h e r than on a " r e l a t i o n a l " b a s i s ) . Thus, i f an e x p e r i m e n t e r , u s i n g the t y p i c a l c o n s e r v a t i o n q u e s t i o n , "Are t h e r e more h e r e , the same i n b o t h , o r more h e r e ? " p u t s " t he same" l a s t i n the q u e s t i o n , t h e r e w i l l be a s p u r i o u s i n c r e a s e i n the number of con-s e r v e r s found. As i s to be e x p e c t e d , the number o f s u b j e c t s r e s p o n d i n g on a r e c e n c y b a s i s d e c r e a s e d and the number of s u b j e c t s r e s p o n d i n g on a r e l a t i o n a l b a s i s i n c r e a s e d w i t h an i n c r e a s e i n age. A t t e m p t i n g to r e -duce the cumbersome s t a n d a r d q u e s t i o n by s p l i t t i n g i t i n t o , "Are t h e r e more i n one group?" and "Are t h e r e the same i n b o t h ? , " Rothenberg [1969a, 1969b] d i s c o v e r e d t h a t young c h i l d r e n w i l l o f t e n c o n t r a c t themselves, answering " y e s " to b o t h q u e s t i o n s ( u s u a l l y ) o r answering "no" to b o t h q u e s t i o n s (sometimes). She i n t e r p r e t s t h i s as d e m o n s t r a t i n g t h a t the c h i l d does not u n d e r s t a n d the terms used. Another i n t e r p r e t a t i o n o f t h i s r e s u l t might be t h a t the tendency e i t h e r t o always respond " y e s " o r to always respond "no" i s so h i g h t h a t the c h i l d does n o t even c o n s i d e r the r e l a t i o n a l n a t u r e o f the q u e s t i o n . The tendency t o c o n t r a d i c t d e c r e a s e s w i t h an i n c r e a s e i n age and s o c i o - e c o n o m i c s t a t u s and some c h i l d r e n who a r e c o n s i s t e n t and t h u s , a c c o r d i n g t o Rothenberg, u n d e r s t a n d the terms, s t i l l do n o t c o n s e r v e . I f q u e s t i o n s which can be answered w i t h a s i m p l e " y e s " o r "no" a r e asked, one can t h e r e f o r e expect a s p u r i o u s i n c r e a s e o r d e c r e a s e i n c o n s e r v a t i o n a b i l i t y . Tbe p r e s e n t study w i l l use the q u e s t i o n , "Are t h e r e more r e d ones, the same number of r e d ones as b l u e ones, or more b l u e ones?" The c h i l d w i l l n o t be a b l e t o answer w i t h a si m p l e yes o r no and r e c e n c y r e s p o n s e s w i l l not s p u r i o u s l y i n c r e a s e the number of c o n s e r v e r s . 23 Bruner's H y p o t h e s i s C o n c e r n i n g the O r i g i n s of C o n s e r v a t i o n The r e m a i n i n g l i t e r a t u r e t o be reviewed concerns attempts to d i s -c o v e r the a n t e c e d e n t s o f c o n s e r v a t i o n e i t h e r by d i r e c t assessment or by t r a i n i n g t e c h n i q u e s . T h i s r e s e a r c h i s , o f c o u r s e , guided by the r e s e a r c h ^ e r ' s b i a s about the o r i g i n s o f c o n s e r v a t i o n . Bruner [1966] proposes t h a t c o n s e r v a t i o n a r i s e s from a c o n f l i c t between s y m b o l i c (language) modes and i k o n i c ( p e r c e p t u a l ) modes of r e p r e -s e n t a t i o n s . A c c o r d i n g to him the c h i l d has a p r i m i t i v e n o t i o n of same^ ness o f su b s t a n c e t h a t i s a t f i r s t u n d i f f e r e n t i a t e d . "But the f o u r - and f i v e - y e a r o l d has p l a i n l y n ot d i f f e r e n t i a t e d the i d e a o f i d e n t i t y i n t o a form t h a t c o n t a i n s a base (what the water r e a l l y i s ) and s u r f a c e (what i t l o o k s l i k e ) o r i n t o a m e t r i c (amount) and con-^ t u i n u i t y ( i t s p e r d u r i n g sameness)." [Bruner, 1966, p. 205]. A c c o r d i n g t o Bruner, "how t h i n g s l o o k " and "how t h i n g s r e a l l y a r e " a r e n o t s e p a r a t e d . They become s e p a r a t e d and the d i f f e r e n t i a t i o n of s u b s t a n c e dis*-c u ssed above o c c u r s because o f c o n f l i c t s between the modes of r e p r e s e n t a ^ t i o n . He c i t e s d a t a which show t h a t the c h i l d ' s a b i l i t y t o make judgments c o n c e r n i n g the i d e n t i t y o f s u b s t a n c e ( i n a t a s k i n which ducks and t h e i r ponds were t r a n s f e r r e d to c o n t a i n e r s v a r y i n g i n s i z e ) precedeS"his^abi-li.ty t o make judgments c o n c e r n i n g the i d e n t i t y o f amount of s u b s t a n c e , which p r e c e d e s the a b i l i t y to c o n s e r v e . A l s o , a c c o r d i n g to h i s d a t a , a p r o c e d u r e which causes a c o n f l i c t between the two judgments i n d u c e s c o n s e r v a t i o n . ( S i x t y - f i v e p e r cent of those who d i d not con s e r v e i n the p r e t e s t c o n s e r v e d i n the p o s t t e s t ; t h e r e was no c o n t r o l group which r e c e i v e d j u s t the p r e t e s t and the p o s t t e s t w i t h o u t t r a i n i n g ) . T h i s same mechanism the c o n f l i c t 24 between p e r c e p t u a l and s y m b o l i c modes of r e p r e s e n t a t i o n — i s used to account f o r the well-known s u c c e s s of the Frank [1964] s c r e e n i n g t r a i n i n g p r o c e d u r e . I n t h i s p r o c e d u r e , the c h i l d compares the amount of l i q u i d be-f o r e and a f t e r i t i s t r a n s f e r r e d from one beaker to a n o t h e r t h a t v a r i e s i n one o r two d i m e n s i o n s . D u r i n g the t r a n s f e r , the beakers a r e c o v e r e d w i t h a s c r e e n which e l i m i n a t e s the i r r e l e v a n t p e r c e p t u a l cue. Bruner e x p l a i n s the s u c c e s s of the p r o c e d u r e by p o i n t i n g to the c o n f l i c t between the c h i l d ' s s y m b o l i c judgment of sameness b e f o r e the t r a n s f e r (and a f t e r the t r a n s f e r w h i l e the beakers a r e s t i l l c o v e r e d by the s c r e e n ) and h i s p e r -c e p t u a l judgment of d i f f e r e n c e a f t e r the removal of the s c r e e n . F u r t h e r e v i d e n c e he c i t e s c oncerns the c o n s e r v a t i o n o f s o l i d s (mass). In t h i s ex-periment the s c r e e n i n g p r o c e d u r e was i n e f f e c t i v e and o n l y a m a n i p u l a t i o n ( i n which the c h i l d r a t h e r than the e x p e r i m e n t e r makes the t r a n s f e r ) com-b i n e d w i t h a l a b e l l i n g p r o c e d u r e ( i n which the e x p e r i m e n t e r p o i n t s out t h a t a l t h o u g h one o f the o b j e c t s i s l o n g e r , i t i s a l s o t h i n n e r ) was e f f e c t i v e . B runer says t h i s p r o c e d u r e c r e a t e s a c o n f l i c t between the p e r c e p t u a l mode and the e n a c t i v e ( p r o v i d e d by m a n i p u l a t i o n ) and s y m b o l i c ( p r o v i d e d by l a b e l -l i n g ) modes t o g e t h e r . P i a g e t [1967] p o i n t e d out, i n s p e c i f i c r e f e r e n c e to Frank's [1964] st u d y ( a l t h o u g h i t c o u l d a l s o be a p p l i e d to the d u c k - i n - t h e - p o n d s t u d y ) , t h a t what Bruner p r o b a b l y i n d u c e d was " p s e u d o - c o n s e r v a t i o n . " The i n t e n s i o n o f the concept t h a t was i n d u c e d s h o u l d have been checked f u r t h e r . There was no t e s t i n g f o r t r a n s f e r t o o t h e r s i t u a t i o n s and no t e s t i n g a few weeks l a t e r f o r the s t a b i l i t y of the concept. There was a l s o no c o n t r o l group. I t i s h i g h l y l i k e l y t h a t the t r a i n i n g p r o c e d u r e i n the duck^in-pond and i n the Frank experiment s i m p l y primed the c h i l d r e n to say "same" on the p o s t t e s t because 25 they had o f t e n s a i d same d u r i n g the t r a i n i n g . When asked d u r i n g t r a i n i n g i f i t was the same water a f t e r the t r a n s f e r o f the duck and h i s pond to another c o n t a i n e r the s u b j e c t would have no d i f f i c u l t y i n s a y i n g i t was the same. And when the beakers were h i d d e n by a s c r e e n and they were asked to compare the amounts i n them, a g a i n the c h i l d r e n would have no d i f f i c u l t y i n s a y i n g "same." I t i s a l s o p o s s i b l e t h a t on the p o s t t e s t the s u b j e c t s i n t e r p r e t e d the q u e s t i o n c o n c e r n i n g amount as a q u e s t i o n c o n c e r n i n g the identity of the substance i n the two comparison b e a k e r s . Support f o r t h i s l a t t e r i n t e r p r e t a t i o n comes from B r u n e r ' s f i n d i n g t h a t n o n c o n s e r v i n g c h i l d -r e n answered the q u e s t i o n c o n c e r n i n g amount c o r r e c t l y much more o f t e n i f i t was p r e s e n t e d . f i r s t . Even Bruner a c c e p t s the e x p l a n a t i o n t h a t the q u e s t i o n c o n c e r n i n g amount, i f p r e s e n t e d f i r s t , was m i s i n t e r p r e t e d as r e f e r r i n g to s u b s t a n c e . Only i f the q u e s t i o n c o n c e r n i n g amount was p r e s e n t e d second d i d the c h i l d r e a l i z e i t concerned amount and thus answered i n c o r r e c t l y . A f t e r the t r a i n i n g e x p e r i e n c e i t appears l i k e l y t h a t the c h i l d i n t e r p r e t e d the ques-t i o n i n the p o s t t e s t as r e f e r r i n g to the i d e n t i t y o f s u b s t a n c e . A t l e a s t t h i s p o s s i b i l i t y cannot be r u l e d out. The importance o f c h e c k i n g the i n t e n -s i o n o f a concept i s h i g h l i g h t e d by P i a g e t ' s [1970] and F l a v e l l and W o h l w i l l ' s [1969] s u g g e s t i o n t h a t the t r a i n i n g of c o n s e r v a t i o n may n o t , i n f a c t , speed up development. I n s t e a d , t h e r e may be a t r a d e - o f f between the amount of time taken to l e a r n a concept and the e x t e n t of i t s g e n e r a l i t y . Data g i v e n by I n h e l d e r and S i n c l a i r [1969] s u p p o r t t h i s i d e a . The l o n g e r the s u b j e c t s took i n t h e i r t r a i n i n g p r o c e d u r e , the more they t r a n s f e r r e d t h e i r c o n s e r ^ v a t i o n l e a r n i n g to o t h e r t a s k s ; Thus, Bruner f a i l e d to e s t a b l i s h what the i n t e n s i o n used by the s u b j e c t s i n these two s t u d i e s was. 26 The s u c c e s s of the m a n i p u l a t i o n and l a b e l l i n g p r o c e d u r e c o u l d be e x p l a i n e d by the P i a g e t i a n i d e a , d i s c u s s e d e a r l i e r , t h a t c o n s e r v a t i o n o c c u r s because the c h i l d i s a b l e to c o n s t r u c t the t r a n s f o r m a t i o n . The l a b e l l i n g and m a n i p u l a t i o n p r o c e d u r e a l l o w s the c h i l d n o t o n l y to f o c u s on the t r a n s f o r m a t i o n but a l s o on the n a t u r e o f the t r a n s f o r m a t i o n ( d i s -placement o f u n i t s o f e q u a l w i d t h and l e n g t h ) and not s i m p l y on the co-v a r i a n c e o f the dimensions b u t on the f a c t t h a t the c o v a r i a n c e i s a r e -s u l t o f the t r a n s f o r m a t i o n . B r u n e r ' s example o f the p r o c e d u r e makes t h i s c l e a r : " A f t e r the s u b j e c t had made a p e n c i l , he was asked, "Which one i s the l o n g e s t ? " I f he i n d i c a t e d t h a t the pen-c i l was the l o n g e r , the ex p e r i m e n t e r then s a i d , 'Now t e l l me which one i s the f a t t e s t ? " I f the c h i l d i n d i c a t e d the b a l l , the ex p e r i m e n t e r s a i d , 'O.K., the p e n c i l i s l o n g e s t , but the b a l l i s f a t t e s t ; now t e l l me, does one o f them have more c l a y than the o t h e r o r do they have the same?' A f t e r the c h i l d had g i v e n h i s judgment, the e x p e r i m e n t e r s a i d , 'O.K;, you t o l d me the p e n c i l i s the l o n g e s t and the b a l l i s the f a t t e s t . Now I want you t o take t h i s l o n g pen-c i l and make i t j u s t as f a t as the b a l l f o r me.' As the c h i l d r o l l e d the p e n c i l back i n t o a b a l l (which he u s u a l l y d i d to make i t j u s t as f a t ) , the expe r i m e n t e r asked him s e v e r a l t i m e s , 'Is i t as f a t y e t ? ' and 'Is i t s t i l l l o n g e r ? ' When the c h i l d a s s e r t e d t h a t i t was j u s t as f a t , he was asked to judge the r e l a t i v e amounts of the two p i e c e s a g a i n . In the second h a l f o f th e s e t r i a l s , when the s u b j e c t made a p e n c i l from the o t h e r b a l l , he was'asked t o 'make the f a t b a l l j u s t as l o n g as the p e n c i l , ' i n s t e a d of the o t h e r way around as i n the f i r s t h a l f . " [1966, p. 205]. Bruner d i s a g r e e s on two p o i n t s w i t h P i a g e t ' s t h e o r y o f c o n s e r v a t i o n development. F i r s t , he r e j e c t s the i d e a t h a t compensation and r e v e r s i b i l i t y a r e 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 because the n o n c o n s e r v i n g c h i l d u n d e r s t a n d s them. The n o n c o n s e r v i n g c h i l d does u n d e r s t a n d t h a t the t r a n s f o r m a t i o n can be r e v e r s e d , r e s u l t i n g i n e q u a l i t y . T h i s , however, i s not the o p e r a t i o n a l 27 r e v e r s i b i l i t y P i a g e t d i s c u s s e s (see Appendix A ) . Knowing t h a t i f a t r a n s f o r m a t i o n from e q u a l i t y i s r e v e r s e d , e q u a l i t y w i l l be r e g a i n e d s h o u l d not be e x p e c t e d to be s u f f i c i e n t to e n a b l e the c h i l d to c o r r e c t l y judge the amounts before the t r a n s f o r m a t i o n i s r e v e r s e d , which i s the judgment demanded i n a c o n s e r v a t i o n t a s k . Bruner's a s s e r t i o n t h a t com--p e n s a t i o n i s u n d e r s t o o d by a n o n c o n s e r v i n g c h i l d i s based on d a t a which show t h a t n o n c o n s e r v i n g c h i l d r e n can p i c k the c o r r e c t one of two b e a kers t h a t w i l l f i l l an empty g l a s s e q u a l to a s t a n d a r d 50 p e r cent of the time. T h i s i s not v e r y s t r o n g e v i d e n c e , however, because such performance would be e x p e c t e d on chance a l o n e and because sometimes the c h o i c e s v a r i e d from the s t a n d a r d on o n l y one d i m e n s i o n . Secondly, Bruner c l a i m s t h a t the con-f l i c t which c r e a t e s c o n s e r v a t i o n i s a c o n f l i c t between s y m b o l i c p r o c e s s e s and p e r c e p t u a l modes of r e p r e s e n t a t i o n , P i a g e t says t h a t c o n s e r v a t i o n r e -s u l t s from a c o n f l i c t o f p e r c e p t u a l schemes ( a l t h o u g h i t seems r e a s o n a b l e to assume t h a t once o p e r a t i o n a l thought: emerges, t h e r e may be a. p e r i o d of c o n f l i c t between i t and p e r c e p t u a l schemes). Bruner c i t e s d a t a i n which c h i l d r e n " l e a r n e d " t h a t i n narrower b e a k e r s water goes up and i n w i d e r ones i t goes down but t h i s d i d not improve t h e i r c o n s e r v a t i o n performance. T h i s p r o c e d u r e , however, c o n t r a r y to B r u n e r ' s assessment, would not be ex-p e c t e d to b r i n g c o n f l i c t between p e r c e p t u a l schemes s i n c e a judgment of amount was never r e q u i r e d of the s u b j e c t s (they s i m p l y poured water from one beaker t o a n o t h e r and then a n o t h e r , e t c . , t h a t v a r i e d i n h e i g h t and w i d t h ) . Frank's p r o c e d u r e c o u l d be i n t e r p r e t e d as a c o n f l i c t between p a s t p e r c e p t u a l and p r e s e n t p e r c e p t u a l i n t e r p r e t a t i o n s s i n c e the c h i l d sees the c o n t r a s t between h i s p r e d i c t i o n of the water l e v e l and the a c t u a l 28 water l e v e l . The p r e s e n t s t u d y w i l l attempt to p r o v i d e e v i d e n c e t h a t i n -t r a p e r c e p t u a l c o n f l i c t i s a p r e r e q u i s i t e t o c o n s e r v a t i o n i n c o n n e c t i o n w i t h H y p o t h e s i s 7. E l k i n d [1967] argues f o r the l o g i c a l and d e v e l o p m e n t a l p r i o r i t y o f i d e n t i t y c o n s e r v a t i o n over e q u i v a l e n c e c o n s e r v a t i o n . I f the c h i l d i n i t i a l l y a grees to the e q u a l i t y o f A and B, and B i s t r a n s f o r m e d i n t o B", then the c h i l d must r e a l i z e t h a t B=B''before he can r e a l i z e t h a t A=B'. The r e a l i z a t i o n t h a t B=B' i s termed i d e n t i t y c o n s e r v a t i o n . I f B r u n e r ' s a n a l y s i s i s c o r r e c t , i . e . , i f c o n s e r v a t i o n i s based on the i d e n t i t y o f s u b s t a n c e which becomes d i f f e r e n t i a t e d i n t o i d e n t i t y of amount, then one would expect i d e n t i t y c o n s e r v a t i o n to precede e q u i v a -l e n c e c o n s e r v a t i o n . I f , however, c o n s e r v a t i o n ( o f b o t h i d e n t i t y and e q u i v a l e n c e ) depends on the c o n s t r u c t i o n o f u n i t s , then i d e n t i t y con-s e r v a t i o n ought to emerge s i m u l t a n e o u s l y w i t h e q u i v a l e n c e c o n s e r v a t i o n . Hooper [1969] found t h a t 66.7 p e r cent of the J3s g i v e n an i d e n t i t y t a s k c o n s e r v e d compared to 43.1 per cent of the j3s g i v e n an e q u i v a l e n c e t a s k i n a b e t w e e n - s u b j e c t d e s i g n , the c h i - s q u a r e b e i n g s i g n i f i c a n t f o r type o f c o n s e r v a t i o n a t the .05 l e v e l w i t h the g r a d e - l e v e l a n a l y s i s on type of c o n s e r v a t i o n s i g n i f i c a n t o n l y f o r male k i n d e r g a r t e n e r s . Northman and Grune [1970] found no d i f f e r e n c e between c h i l d r e n ' s performance on i d e n -t i t y v s . e q u i v a l e n c e t a s k s i n a w i t h i n - s u b j e c t d e s i g n . They found t h a t c h i l d r e n c o n s e r v e i n an a l l - o r - n o n e f a s h i o n , w i t h 0-%l or 5^ -6 c o r r e c t , the chi-^square s i g n i f i c a n t a t p < .001. The second study used second and t h i r d g r a d e r s , however, which might account f o r the d i f f e r e n c e . There i s the problem, d i s c u s s e d e a r l i e r , of p s e u d o - c o n s e r v a t i o n , , i . e . , the i d e n t i t y 29 c o n s e r v e r s c o u l d be c o n s e r v i n g s u b s t a n c e and not amount. I t would be u s e f u l to use a p r o c e d u r e i n which a q u e s t i o n o f i d e n t i t y o f su b s t a n c e p r e c e d e d the q u e s t i o n o f i d e n t i t y o f amount to emphasize the amount. Thus, the q u e s t i o n of whether o r not i d e n t i t y c o n s e r v a t i o n precedes equi<-v a l e n c e c o n s e r v a t i o n i s n o t y e t r e s o l v e d . A t t e n d i n g to the R e l e v a n t Dimension Some i n v e s t i g a t o r s [Gelman, 1969, Watson, 1960] have s u g g e s t e d t h a t the o n l y d i f f i c u l t y the p r e o p e r a t i o n a l c h i l d has i s one o f a t t e n d i n g to the r e l e v a n t cue. These c h i l d r e n r e a l l y have the c a p a c i t y to c o n s e r v e , they argue, i f o n l y they a r e s h i e l d e d from i r r e l e v a n t p e r c e p t u a l cues or a r e g i v e n feedback so t h a t r e l e v a n t " q u a n t i t y c u e s " can come to c o n t r o l t h e i r b e h a v i o r . An e x t r e m e l y s u c c e s s f u l experiment a l o n g t h i s l i n e [Gelman, 1969] used a t r a i n i n g p r o c e d u r e i n which on some t r i a l s c h i l d r e n were to p i c k which two o f t h r e e rows of d i s c r e t e o b j e c t s v a r y i n g i n l e n g t h and den-s i t y had the same number of o b j e c t s . On o t h e r t r i a l s they were to p i c k which two of t h r e e l i n e s v a r y i n g i n placement were the same l e n g t h . B l o c k s of s i x number t r i a l s were i n t e r s p e r s e d w i t h b l o c k s of s i x l e n g t h t r i a l s . The s u b j e c t s , twenty k i n d e r g a r t e n e r s , . , a l t h o u g h they had f a i l e d on a l l of e i g h t , c o n s e r v a t i o n p r e t e s t s ( l e n g t h , number, l i q u i d , mass, two v e r s i o n s o f each) r a p i d l y l e a r n e d to s o l v e these problems o f l e n g t h and number and t r a n s f e r r e d almost 100 per cent t o immediate and one-week d e l a y e d p o s t t e s t i n number and l e n g t h c o n s e r v a t i o n , w i t h over 50 p e r c e n t t r a n s f e r t o imme-d i a t e l i q u i d q u a n t i t y and mass c o n s e r v a t i o n t a s k s , w i t h t r a n s f e r i m p r o v i n g on the l a t t e r two t a s k s a f t e r the week's d e l a y . The c o n t r o l group who r e ^ c e i v e d o n l y the p r e t e s t and p o s t t e s t made s i g n i f i c a n t l y l e s s p r o g r e s s . 30 The a u t h o r c o n c l u d e d t h a t because they l e a r n e d the problems so f a s t the c h i l d r e n had the concept o f q u a n t i t y b e f o r e the experiment and s i m p l y d i d n o t u n d e r s t a n d the t a s k r e q u i r e m e n t s . Gelmon goes on to say, however, t h a t we a r e s t i l l f a c e d w i t h the problem o f t r a c i n g and e x p l a i n i n g the development of' the concept o f num-b e r which o b v i o u s l y took p l a c e b e f o r e t h e s e c h i l d r e n e n t e r e d the e x p e r i -mental s e s s i o n . F r a n k ' s p r o c e d u r e , which c o u l d a l s o be i n t e r p r e t e d as a s s i s t i n g the c h i l d t o a t t e n d to t h e r e l e v a n t d i m e n s i o n was i n e f f e c t i v e w i t h f o u r - y e a r - o l d s . B e i l i n ' s feedback p r o c e d u r e f o r t r a i n i n g c o n s e r v a -t i o n o f a r e a [1969] was s u c c e s s f u l when grade one and two c h i l d r e n were used as s u b j e c t s but u n s u c c e s s f u l when k i n d e r g a r t e n e r s were used as sub-j e c t s . Thus, a n t e c e d e n t s of c o n s e r v a t i o n performance a r e not r e a l l y tapped by t h e s e t r a i n i n g p r o c e d u r e s — they demonstrate some o b s t a c l e s t h a t must be removed and some p r o c e d u r e s which a r e e f f e c t i v e i n removing t h e s e o b s t a c l e s once the c h i l d has the b a s i c concepts n e c e s s a r y f o r con-s e r v a t i o n . B e i l i n ; [1969] a l s o p o i n t s to the p o s s i b i l i t y t h a t when a feedback p r o c e d u r e i s used an a l g o r i t h m f o r s o l v i n g the problem, hot the b a s i c con-c e p t , may have been a c h i e v e d by the c h i l d — i . e . , the i n t e n s i o n may not be the same as t h a t used by " n o r m a l l y " c o n s e r v i n g c h i l d r e n . T h i s may be t r u e of B e i l i n ' s p r o c e d u r e . He found no t r a n s f e r to o t h e r k i n d s o f con^-s e r v a t i o n tasks, than t h a t on which the c h i l d r e n had d i r e c t t r a i n i n g and t h e r e was no d e l a y e d p o s t t e s t i n h i s d e s i g n . A l s o , he d i s c o v e r e d t h a t the a b i l i t y o f h i s s u b j e c t s to c o n s e r v e an i n e q u a l i t y d e c r e a s e d as a r e -s u l t of feedback. (The i n e q u a l i t y c o n s e r v a t i o n t a s k i s n o r m a l l y q u i t e 31 easy. I t was randomly mixed w i t h t h e e q u a l i t y c o n s e r v a t i o n t a s k i n t h e s t u d y ) . The i n t e r p r e t a t i o n i s t h a t f e e d b a c k i n c r e a s e s random r e s p o n d i n g b ecause t h e s u b j e c t s s t o p u s i n g l e n g t h as t h e i r c r i t e r i o n . T h i s i n -c r e a s e s the number of c o r r e c t r e s p o n s e s on t h e d i f f i c u l t e q u a l i t y con-s e r v a t i o n t a s k and d e c r e a s e s t h e number o f c o r r e c t r e s p o n s e s on t h e easy i n e q u a l i t y c o n s e r v a t i o n t a s k , w h i c h can be c o r r e c t l y s o l v e d u s i n g p e r c e p -t u a l c r i t e r i a . F l e i s h m a n , G i l m o r e , and G i n s b u r g [1967] f o u n d t h a t a l t h o u g h f e e d b a c k was somewhat e f f e c t i v e i n i n d u c i n g c o n s e r v a t i o n , o f t h e n i n e who c o n s e r v e d on t h e p o s t t e s t o n l y two gave c o r r e c t answers on a n o n - c o n s e r -v a t i o n t r i a l . But the c r i t i c i s m t h a t t h e i n t e n s i o n by w h i c h t h e s u b j e c t s s ucceeded was n o t t h a t o f t h e " n o r m a l l y " c o n s e r v i n g c h i l d i s weaker when a p p l i e d t o t h e Gelman s t u d y because she d e m o n s t r a t e d the g e n e r a l i t y and s t a b i l i t y o f t h e t r a i n e d c o n c e p t . There was, however, a g r e a t d e a l o f o v e r t r a i n i n g g i v e n i n t h e Gelman s t u d y . The c h i l d r e n were r e s p o n d i n g c o r r e c t l y 90 p e r c e n t o f t h e t i m e by t h e s i x t h t r i a l , b u t a t o t a l o f 96 t r i a l s were g i v e n . T h i s might have s t r e n g t h e n e d any a l g o r i t h m they were u s i n g ( e . g . , r o t e c o u n t i n g ; Gelman e l i m i n a t e d c h i l d r e n who c o u l d n o t count f r o m h e r s t u d y ) s u f f i c i e n t l y t o a l l o w t r a n s f e r and r e t e n t i o n . I t w o u l d be i n t e r e s t i n g t o r e p e a t the s t u d y u s i n g f o u r - y e a r - o l d s as s u b j e c t s and t r a i n -i n g them o n l y u n t i l t h e y r e a c h e d c r i t e r i o n . What i s t h e " q u a n t i t y cue" t o w h i c h the c h i l d must a t t e n d ? L e n g t h i s p o s s i b l y a p e r c e p t u a l cue b u t , as B e i l i n [1969] and Smedslund [1966b] p o i n t o u t , t h e r e i s no o s t e n s i v e , p e r c e p t u a l , " p o i n t - a t - a b l e " d e f i n i t i o n o f l i q u i d and mass q u a n t i t y . Even number i s a c o n c e p t w h i c h i s r e l a t i v e l y i n d e p e n d e n t o f p e r c e p t u a l c u e s , a l t h o u g h t h e r e i s more p e r c e p t u a l s u p p o r t 32 f o r i t than f o r l i q u i d or mass q u a n t i t y . B e i l i n [1969] p o i n t s out t h a t c o n s e r v a t i o n t a s k s demand a c t i v i t y — t r a n s f o r m a t i o n s by the s u b j e c t . T h i s i s e s p e c i a l l y t r u e i n the q u a s i - c o n s e r v a t i o n t a s k i n which the c h i l d must compare two q u a n t i t i e s which a r e p e r c e p t u a l l y u n equal b u t a c t u a l l y e q u a l w i t h o u t the b e n e f i t o f s e e i n g them o r i g i n a l l y p e r c e p t u a l l y e q u a l and then t r a n s f o r m e d . ( T h i s can be done w i t h number, l e n g t h , and a r e a c o n c e p t s b u t not w i t h concepts of l i q u i d and mass q u a n t i t y ) . When B e i l i n says t h a t i t i s the c h i l d ' s m ental t r a n s f o r m a t i o n which l e a d s to c o n s e r -v a t i o n (and t h a t the feedback p r o c e d u r e primes the c h i l d to p e r f o r m t h i s t r a n s f o r m a t i o n i f i t i s i n h i s c a p a c i t y to do so) he i s e s s e n t i a l l y s a y i n g the same t h i n g P i a g e t says when he i n s i s t s t h a t an o p e r a t i o n i s a me n t a l a c t i v i t y and t h a t o p e r a t i v i t y i s a n e c e s s a r y c o n d i t i o n f o r c o n s e r v a t i o n . Thus, t h r e e o b j e c t i o n s can be made to the r e l e v a n t cue h y p o t h e s i s . 1. The p r o c e d u r e s d e s i g n e d to teac h the s u b j e c t s to a t t e n d t o the r e l e v a n t cue may be i n d u c i n g o n l y an a l g o r i t h m f o r s o l v i n g the problem. 2. The r e l e v a n t cue i s not p e r c e p t u a l , b u t i s c o n s t r u c t e d by a s u b j e c t ' s mental a c t i v i t y . 3. T r a i n i n g i n a t t e n d i n g to the r e l e v a n t cue may o n l y be s u c c e s s f u l i n removing c e r t a i n o b s t a c l e s i n the way o f c h i l d r e n who a l r e a d y a r e a b l e to c o n s t r u c t the r e l e v a n t cue. The R e v e r s i b i l i t y H y p o t h e s i s Some r e s e a r c h e r s have attempted t o use P i a g e t ' s t a s k a n a l y s i s as 33 a b a s i s f o r r e s e a r c h t o uncover the p r e r e q u i s i t e s of c o n s e r v a t i o n and of o p e r a t i v e thought. They have assumed t h a t r e v e r s i b i l i t y , i n p a r t i c u l a r , i s not a r e s u l t b u t a cause o f o p e r a t i o n a l thought, and have attempted to i n d u c e c o n s e r v a t i o n by r e v e r s i b i l i t y t r a i n i n g . R o l l [1970] and W a l l a c h , W a l l and Anderson [1967] found t h a t r e v e r s i b i l i t y t r a i n i n g ( i n which the c h i l d i s r e p e a t e d l y exposed t o the r e s u l t s of a t r a n s f o r m a t i o n back to one-to-one correspondence) was e f f e c t i v e i n i n d u c i n g c o n s e r v a t i o n o f num-b e r on b o t h an immediate and d e l a y e d p o s t t e s t , a l t h o u g h t h e r e was no d i r e c t t r a n s f e r t o c o n s e r v a t i o n of l i q u i d q u a n t i t y . Only o n e - h a l f of the s u b j e c t s c o n s e r v i n g on the p o s t t e s t t r a n s f e r r e d t o l i q u i d q u a n t i t y a f t e r a t r a n s f e r s e r i e s which g r a d u a l l y i n c r e a s e d the c o n t i n u i t y o f the m a t e r i a l s . Thus, the problem of i n t e n s i o n a g a i n p r e s e n t s i t s e l f , a l t h o u g h not as s t r o n g l y as i n o t h e r s t u d i e s . The C o n f l i c t H y p o t h e s i s Guided by P i a g e t i a n i d e a s on e q u i l i b r a t i o n , t r a i n i n g t e c h n i q u e s i n v o l v i n g c o g n i t i v e c o n f l i c t have been used i n an attempt to i n d u c e con-s e r v a t i o n . Gruen [1965] found t h a t a c o n f l i c t p r o c e d u r e ( i n which he sub-t r a c t e d c h i p s from the p i l e o f c h i p s which the c h i l d thought c o n t a i n e d more) only, when combined w i t h a v e r b a l p r e t r a i n i n g p r o c e d u r e ( i n which an attempt was made to get the s u b j e c t to respond t o number r a t h e r than l e n g t h by v a r y i n g the s i z e o f the items i n each row, w h i c h i s a p r o c e d u r e to i n d u c e a t t e n d i n g to the r e l e v a n t dimension) was e f f e c t i v e i n i n d u c i n g number c o n s e r v a t i o n w i t h t r a n s f e r t o l e n g t h and s u b s t a n c e c o n s e r v a t i o n . A nother c o n f l i c t p r o c e d u r e [Smedslund, 1961a and 1961b] i n w h i c h the 34 substance was both changed perceptually and added to or subtracted from, was e f f e c t i v e i n inducing conservation only among those who were dominated by the addition-subtraction cue,.during the t r a i n i n g . L i t t l e c o n f l i c t was induced and those who conserved gave the reason that nothing was added or taken away, allowing one to suspect that the c h i l d had simply become con-t r o l l e d by the addition and subtraction cue. Winer [1968], using a num-ber rather than a substance procedure, also found no c o n f l i c t was produced. He demonstrated that conservation behavior could be induced merely by ex-posure to a number of addition-subtraction items which presumably induced a set to respond by addition-subtraction cues i n the subjects. In these studies the subject may not have been operating by the intension of the conservation concept as defined by Piaget. This p o s s i b i l i t y i s strong because no further exploration of intension was undertaken i n the form of transfer tasks or delayed posttests i n the Smedslund study and because Winer found v i r t u a l l y no transfer to a continuous quantity conservation task. The Addition-Subtraction Hypothesis Smedslund [1961b] had demonstrated that the a b i l i t y to perform s u c c e s s f u l l y i n s i t u a t i o n s r e q u i r i n g the understanding of the compensa-tory nature of addition and subtraction preceded conservation a b i l i t y , and thus he postulated that i t i s the i n t e r n a l i z a t i o n of the operation of addition and subtraction that produces conservation. However, Feigen~ baum and Sulken [1964] were unsuccessful, using an addition-subtraction procedure, i n inducing conservation. - Wohlwill and Lowe [1962] also found 35 no s i g n i f i c a n t e f f e c t on c o n s e r v a t i o n of an a d d i t i o n - s u b t r a c t i o n t r a i n i n g p r o c e d u r e ( a l t h o u g h i t was more e f f e c t i v e than the o t h e r p r o c e d u r e s they t r i e d ) . H a l f o r d ' s Theory A r e c e n t r e v i e w of the t r a i n i n g s t u d i e s on c o n s e r v a t i o n [ B r a i n e r d and A l l e n , 1971] c o n c l u d e s t h a t t h o se which a r e s u c c e s s f u l attempt to dem-o n s t r a t e the r e v e r s i b l e n a t u r e of the t r a n s f o r m a t i o n . As has been d i s c u s s e d above, however, even a t r a n s f o r m a t i o n which produces i n e q u a l i t y can be r e -v e r s e d t o produce e q u a l i t y . Thus, the c h i l d must a l s o u n d e r s t a n d the de-t a i l s o f the t r a n s f o r m a t i o n to c o r r e c t l y judge i t s r e s u l t . He must under-s t a n d t h a t the i n c r e a s e i n one d i m e n s i o n i s compensated f o r by a d e c r e a s e i n the o t h e r d i m e n s i o n . I t i s t h i s l a t t e r p r e r e q u i s i t e t h a t H a l f o r d [1969, 1970] d i s c u s s e s . H a l f o r d s u g g e s t s t h a t what i s i n v o l v e d i n the development o f con-s e r v a t i o n i s the c o n s t r u c t i o n of a t r u t h t a b l e r e l a t i n g two k i n d s of cues,, a d d i t i o n - s u b t r a c t i o n and l e n g t h - d e n s i t y . The c h i l d l e a r n s e m p i r i c a l l y , as one example of an e n t r y i n the t r u t h t a b l e , t h a t when n o t h i n g ; i s added or taken away i n a t r a n s f o r m a t i o n and one dimension changes, the o t h e r d i m e n s i o n can change i n o n l y one d i r e c t i o n . H a l f o r d s u g g e s t s t h a t the c h i l d comes t o use q u a n t i t y cues ( a d d i t i o n and s u b t r a c t i o n cues) r a t h e r than d i m e n s i o n a l cues s i n c e q u a n t i t y cues y i e l d a unique s o l u t i o n when combined, w h i l e d i m e n s i o n a l cues do n o t . ( T a l l e r combined w i t h s h o r t e r , when b r e a d t h i s i g n o r e d can y i e l d l e s s , e q u a l , o r more). The d i f f i c u l t y w i t h h i s a n a l y s i s , however, i s something he h i m s e l f p o i n t s out — q u a n t i t y cues do not always l e a d to unique s o l u t i o n s . F o r example, a d d i n g and then 36 s u b t r a c t i n g c o u l d y i e l d more, e q u a l , o r l e s s . We must a g a i n r e a c h the con-c l u s i o n t h a t the judgment must be based on u n i t s . A l t h o u g h h i s e m p i r i c a l s t u d i e s do seem to show t h a t c h i l d r e n l e a r n e m p i r i c a l l y the r e l a t i o n be-tween dimensions and a d d i t i o n - s u b t r a c t i o n cues, t h i s development cannot l e a d them a l l the way to c o n s e r v a t i o n — u l t i m a t e l y the two cues must be combined and the o n l y way t h i s can be done i s through the c o n s t r u c t i o n of the t r a n s f o r m a t i o n of u n i t s d i s c u s s e d above. H a l f o r d s a y s , " q u a n t i t y cues y i e l d the same r e s u l t as h e i g h t and b r e a d t h cues i f the l a t t e r a r e c o n s i d e r e d j o i n t l y . . ." But h e i g h t and b r e a d t h cues can be c o n s i d e r e d j o i n t l y o n l y i f u n i t s of e q u a l h e i g h t and b r e a d t h a r e p u t i n t o q u a n t i -t a t i v e o r o p e r a t i o n a l one-to-one correspondence.' P i a g e t r e c o g n i z e s t h i s and thus p o s t u l a t e s t h a t what i s i n v o l v e d i n h i s g r o u p i n g of 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 i s the u n d e r s t a n d i n g of one-to-one c o r r e s p o n d e n c e s . As H a l f o r d s u g g e s t s , the u n c e r t a i n t y ( o r c o n f l i c t ) i n v o l v e d i n the use o f d i m e n s i o n cues may w e l l be the m o t i v a t i n g f o r c e b e h i n d the c h i l d ' s con-s t r u c t i o n o f the t r a n s f o r m a t i o n of u n i t s . H a l f o r d d e a l t w i t h l i q u i d quan-t i t y i n which i t i s i m p o s s i b l e f o r the c h i l d t o use h e i g h t and b r e a d t h cues to j udge q u a n t i t y . In judgments of n u m e r i c a l v a l u e , however, i t i s p o s s i b l e to compare q u a n t i t i e s on l e n g t h and d e n s i t y because i t i s p o s s i b l e to c o n s t r u c t u n i t s which can be m e n t a l l y t r a n s f o r m e d i n such a way t h a t l e n g t h and d e n s i t y a r e e q u a l . Thus, the o n l y way l e n g t h - d e n s i t y and a d d i -t i o n - s u b t r a c t i o n cues can be c o r r e l a t e d i s i f l e n g t h - d e n s i t y cues a r e t r a n s -formed by the s u b j e c t i n t o u n i t s . E v i d e n c e f o r P i a g e t ' s View From P i a g e t ' s view, d i s c u s s e d above, one would p r e d i c t t h a t the 37 c o n f l i c t s c r e a t e d i n j u d g i n g l i q u i d o r mass q u a n t i t y a re r e s o l v e d t h r o u g h t h e m e d i a t i o n o f t h e c o n c e p t o f t r a n s f o r m a t i o n o f u n i t s , p r e v i o u s l y a c -q u i r e d i n number and d i s c o n t i n u o u s q u a n t i t y c o n s e r v a t i o n e x p e r i e n c e s . I t i s w e l l known t h a t d i s c o n t i n u o u s q u a n t i t y o r number c o n s e r v a t i o n p r e c e d e s c o n t i n u o u s q u a n t i t y c o n s e r v a t i o n [ E l k i n d , 1961, G o l d s m i d t and B e n t l e r , 1968, 1969, and Smedslund, 1966b]. A l o n g i t u d i n a l s t u d y by W o h l w i l l [1969] demon-s t r a t e d t h a t f u t u r e c o n s e r v a t i o n a b i l i t y c o u l d be p r e d i c t e d on t h e b a s i s o f p r e s e n t a b i l i t y i n p e r f o r m i n g management o p e r a t i o n s . Some r e s e a r c h e r s [ B e a r i s o n , 1970, I n h e l d e r and S i n c l a i r , 1969, a n d . W a l l a c h , W a l l and A n d e r -s o n , 1967] have been q u i t e s u c c e s s f u l i n t e a c h i n g l i q u i d q u a n t i t y c o n s e r -v a t i o n by r e l y i n g on the m e d i a t i n g a b i l i t y o f the c o n c e p t s d e v e l o p e d i n number c o n s e r v a t i o n . A l s o , c o n s e r v a t i o n a b i l i t y ( o f a l l t y p e s , n o t j u s t o f number) i s p o s i t i v e l y r e l a t e d t o a r i t h m e t i c a b i l i t y [ G o l d s m i d t , 1969]. F u r t h e r e v i d e n c e f a v o r i n g the s u g g e s t i o n t h a t c o n s e r v a t i o n depends upon t h e c h i l d ' s a b i l i t y t o c o n s t r u c t a t r a n s f o r m a t i o n o f u n i t s and a t t e n d t o i t as t h e r e l e v a n t cue i s t h a t an a c t i o n p r o c e d u r e a i d s k i n d e r g a r t e n c h i l d r e n b u t has no e f f e c t on grade one c h i l d r e n i n t h e c o n s e r v a t i o n o f d i s c o n t i n u o u s q u a n t i t i e s and a i d s grade one c h i l d r e n , b u t has no e f f e c t on k i n d e r g a r t e n c h i l d r e n , i n t h e c o n s e r v a t i o n o f c o n t i n u o u s q u a n t i t i e s [ M i n i -c h i e l l o , 1 9 6 9 ] . B e i l i n [1969] a l s o found t h a t a p r o c e d u r e f o r t e s t i n g f o r c o n s e r v a t i o n o f a r e a , i n w h i c h a t r a n s f o r m a t i o n was a c t u a l l y made, was e a s i e r t h a n t h e q u a s i - c o n s e r v a t i o n p r o c e d u r e i n w h i c h a s t a t i c c o m p a r i s o n had t o be made. A l l o w i n g the c h i l d t o p e r f o r m t h e r e v e r s i n g a c t i o n h i m s e l f a i d s younger c h i l d r e n i n c o n s e r v a t i o n p e r f o r m a n c e (presumably because t h e y 38 have not y e t c o n s t r u c t e d the t r a n s f o r m a t i o n and the p r o c e d u r e a i d s i n t h a t c o n s t r u c t i o n ) w h i l e p e r c e p t u a l s u p p o r t s , i n which the o b j e c t s were c o l o r - c o d e d to a i d i n one-to-one c o r r e s p o n d e n c e was h e l p f u l o n l y f o r o l d e r c h i l d r e n (who presumably were a b l e to c o n s t r u c t t h e t r a n s f o r m a t i o n but who had to be a i d e d i n it's a p p l i c a t i o n ) [Whiteman and P e r s a c h , 1970]. Pratoomraj [1966] found t h a t among younger c h i l d r e n the p r e d i c t i o n of the r e s u l t s o f a t r a n s f o r m a t i o n was b e t t e r than the judgement of the r e s u l t s once the t r a n s f o r m a t i o n was performed. T h i s would seem t o i n d i c a t e t h a t the younger c h i l d r e n had a f a l s e e x p e c t a t i o n of the d imensions o f the l i q u i d a f t e r the t r a n s f o r m a t i o n , i . e . , were not a b l e to c a r r y out the t r a n s f o r m a t i o n m e n t a l l y i n d e t a i l . When t h a t f a l s e e x p e c t a t i o n was v i o -l a t e d they b a s e d ' t h e i r judgment on the p e r c e p t u a l cues of the p r e s e n t . I n -h e l d e r ' s work on imagery [1964] a l s o s u p p o r t s the n o t i o n t h a t the c h i l d i s a b l e to c o n s t r u c t t r a n s f o r m a t i o n s o n l y when he r e a c h e s the c o n c r e t e o p e r -a t i o n a l s t a g e o f development. W a l l a c h [1969] a l s o emphasizes t h a t i t i s the c h i l d ' s a b i l i t y t o view the p r e s e n t p e r c e p t u a l s i t u a t i o n as one s t a t e i n a t r a n s f o r m a t i o n t h a t a l l o w s him t o c o n s e r v e . She s u g g e s t s t h a t the c h i l d l e a r n s to con-s e r v e because he l e a r n s t h a t : ". . . i f something — the same t h i n g . — were done to each of two q u a n t i t i e s to be compared, they would be the same w i t h r e s p e c t to a p e r c e p t i b l e p r o p e r t y , sameness of which i n d i c a t e s e q u a l i t y and d i f f e r e n c e i n e q u a l i t y , we j u dge the two q u a n t i t i e s as e q u a l ; i f we know t h a t they would be d i f f e r e n t w i t h r e s p e c t to such a p r o p e r t y , we j u dge them as unequal;' . ." [1969, p. 217]. As H a l f o r d [1970] p o i n t s out, however, t h i s argument a v o i d s the q u e s t i o n , 39 r e f e r r i n g t o the ph r a s e "sameness o f which ( p r o p e r t y ) i n d i c a t e s e q u a l i t y and d i f f e r e n c e i n e q u a l i t y " and p o i n t i n g out t h a t " s u r e l y the s e l e c t i o n o f p r o p e r t i e s which i n d i c a t e t h i s i s the e s s e n c e o f the problem. How does a c h i l d know which p r o p e r t i e s i n d i c a t e e q u a l i t y ? " A l s o , u n l e s s the c h i l d has the concept o f u n i t he would have no way of knowing the r e s u l t o f something done t o the q u a n t i t i e s , even i f he d i d have some i n d i c a t o r p r o -p e r t y i n mind on which to base a judgment of q u a n t i t y . He would have to l e a r n e m p i r i c a l l y the r e s u l t o f every s i n g l e p o s s i b l e t r a n s f o r m a t i o n on eve r y s i n g l e p a i r of q u a n t i t i e s t o be compared. C o n c l u s i o n Two t h i n g s seem c l e a r from t h i s r e v i e w o f the l i t e r a t u r e on quan-t i t y c o n s e r v a t i o n . F i r s t , a c l e a r d e f i n i t i o n o f the i n t e n s i o n o f the con-ce p t o f c o n s e r v a t i o n needs to be de v e l o p e d a l o n g w i t h a method f o r a s s e s s -i n g t h a t i n t e n s i o n (because d e l a y e d p o s t t e s t and t r a n s f e r t a s k s c o u l d s t i l l be s u c c e s s f u l i f f a l s e s o l u t i o n s to c o n s e r v a t i o n t a s k s had been f o u n d ) . The b e s t way to a n s w e r • t h i s q u e s t i o n , as d i s c u s s e d e a r l i e r , i s to d i s c o v e r what c a p a b i l i t i e s the n a t u r a l l y c o n s e r v i n g c h i l d p o s s e s s e s — i . e . , p e r f o r m ance on what t a s k s c o r r e l a t e h i g h l y w i t h c o n s e r v a t i o n performance. The p r e se n t study was d e s i g n e d i n the hope t h a t i t might a s s i s t i n d e f i n i n g the i n t e n s i o n o f c o n s e r v a t i o n i n the n a t u r a l l y c o n s e r v i n g c h i l d by d e m o n s t r a t i n g c o r r e l a t i o n among the c h i l d ' s performance on the v a r i o u s t a s k s used t o t e s t the seven hypotheses d i s c u s s e d i n t h i s i n t r o d u c t i o n . S e c o n d l y , there.Jtias been a p e r s i s t e n t and growing emphasis i n the l i t e r a t u r e [ B e i l i n , 1969, B r a i n e r d and Allen,111970, Watson, 1960, and W a l l a c h , 1969] t h a t i t i s the 40 c h i l d ' s a b i l i t y to m e n t a l l y t r a n s f o r m the s t i m u l u s which a l l o w s him to c o n s e r v e . There has a l s o been a growing r e a l i z a t i o n i n the l i t e r a t u r e [ H a l f o r d , 1970, W a l l a c h , 1969] t h a t the dimensions i n v o l v e d i n q u a n t i t y a r e n o t m i s l e a d i n g but r a t h e r need to be taken i n t o account by the c h i l d . These two t r e n d s i n the l i t e r a t u r e make i t p r o f i t a b l e to c o n s i d e r the o r i g i n a l P i a g e t i a n e x p l a n a t i o n of c o n s e r v a t i o n •— t h a t c o n f l i c t i n p e r -c e p t u a l schemes l e a d s to the c o n s t r u c t i o n , through r e p r e s e n t a t i v e r e g u -l a t i o n , o f the t r a n s f o r m a t i o n of u n i t s . The f o l l o w i n g seven hypotheses w i l l be t e s t e d ; H y p o t h e s i s 1; There w i l l be a p o s i t i v e r e l a t i o n s h i p between the c h i l d ' s a b i l i t y to f i n d o r d i n a l c o rrespondences r e g a r d l e s s of the s p a t i a l arrange^-ment of the s e r i e s and h i s a b i l i t y t o c o n s e r v e , i f the same m a t e r i a l s a r e used to t e s t f o r the two a b i l i t i e s . H y p o t h e s i s 2; T h e r e w i l l be a p o s i t i v e r e l a t i o n s h i p between the c h i l d ' s a b i l i t y t o match the number of o b j e c t s i n a s t a n d a r d row and h i s a b i l i t y * t o b u i l d a s e r i e s i f the same m a t e r i a l s are used to t e s t f o r the two a B i l ^ -i t i e s . H y p o t h e s i s 3: When the c h i l d i s a b l e to b u i l d one s e r i e s , he w i l l a l s o be a b l e t o b u i l d a second s e r i e s and to p l a c e the two s e r i e s i n c o r r e s ^ pondence. Hypo thes i s 4: There w i l l be a p o s i t i v e r e l a t i o n s h i p between the a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s w i t h o u t depending on a s p a t i a l one^to-rone 41 c o rrespondence between the o b j e c t s i n the s e r i e s and the a b i l i t y to b u i l d the s e r i e s s y s t e m a t i c a l l y r a t h e r than by t r i a l and e r r o r . H y p o t h e s i s 5: The c h i l d who c o n s e r v e s w i l l be a b l e t o match the number of o b j e c t s i n a s t a n d a r d row w i t h o u t making a . s p a t i a l one-to-one c o r r e s -pondence. H y p o t h e s i s 6; The c h i l d ' s a b i l i t y t o c o n s e r v e and to f i n d o r d i n a l c o r r e s -pondences d e s p i t e the s p a t i a l arrangement of the s e r i e s w i l l be p o s i t i v e l y r e l a t e d t o (a) h i s a b i l i t y t o a n t i c i p a t e and remember the r e s u l t s of t r a n s -f o r m a t i o n s i n the s p a t i a l arrangements of o b j e c t s i n a row; and a l s o to (b) h i s a b i l i t y t o p e r f o r m mental t r a n s f o r m a t i o n s . H y p o t h e s i s 7; F o r those c h i l d r e n i n Stage I, i . e . , t h o se who a r e u n able to c o r r e c t l y match the number o f o b j e c t s i n a s t a n d a r d row, the d i f f e r e n -t i a l l e n g t h s of the two rows w i l l be the e f f e c t i v e s t i m u l u s on which quan-t i t y judgments are based. For c h i l d r e n i n Stage I I , i . e . , t h o se who can c o r r e c t l y match the number of o b j e c t s i n a s t a n d a r d row but who cannot c o n s e r v e , the e f f e c t i v e s t i m u l u s w i l l be the d e n s i t y d i f f e r e n t i a l minus the l e n g t h d i f f e r e n t i a l o f the two rows. For c h i l d r e n i n Stage I I I , i . e . , t h o s e who can c o n s e r v e , the e f f e c t i v e s t i m u l u s f o r q u a n t i t y judgments w i l l be the d i f f e r e n c e between the number o f o b j e c t s i n the two rows. METHOD S u b j e c t s N i n e t y - f o u r c h i l d r e n were used i n the p r e s e n t s t u d y , t w e n t y - e i g h t p r e s c h o o l e r s (mean age, 54 months), t h i r t y - e i g h t k i n d e r g a r t e n e r s (mean age, 66 months), and t w e n t y - e i g h t grade one c h i l d r e n (mean age, 77 months). JE spent t h r e e o n e - h a l f days as an a s s i s t a n t t e a c h e r i n each o f the p r e -s c h o o l s b e f o r e t e s t i n g of the p r e s c h o o l e r s began. Proc e d u r e Each ^ was g i v e n seven t e s t s i n two s e s s i o n s , a p p r o x i m a t e l y twen-t y minutes each. T e s t s 1 through 3, g i v e n i n o r d e r , t o g e t h e r w i t h a t e s t of c o u n t i n g a b i l i t y , formed one b l o c k o f t e s t s . T e s t s 5 through 7, g i v e n i n o r d e r , formed the second b l o c k o f t e s t s . One b l o c k was g i v e n p e r s e s -s i o n w i t h the o r d e r o f the b l o c k s c o u n t e r b a l a n c e d a c r o s s s u b j e c t s . T e s t 4 was d i v i d e d i n t o two p a r t s as w i l l be e x p l a i n e d below, one p a r t g i v e n a t each s e s s i o n . One to t h r e e days s e p a r a t e d the two s e s s i o n s . T e s t 1 — T e s t f o r I n t e n s i v e Q u a n t i t y (NQ). T h i s t e s t i s d e s i g n e d t o determine i f S^  can put out the same number of o b j e c t s as a r e i n a s t a n d -a r d row. MATERIALS. Two s e t s o f 1" x 6" p a i n t e d h a r d b o a r d s t i c k s were used. Set 1 c o n t a i n e d t h r e e y e l l o w s t i c k s and e i g h t b l a c k s t i c k s and was used i n P a r t A. Set 2, used i n P a r t s B and C, c o n t a i n e d e i g h t r e d 42 43 s t i c k s and 16 b l u e s t i c k s . Two s e t s were used to a v o i d the d i r e c t t r a n s -f e r o f r e s p o n s e s from P a r t A to P a r t B o f the t e s t . PART A. The t h r e e y e l l o w s t i c k s a r e l i n e d up a t one end of a l o n g t a b l e where J 3 i s s e a t e d . The s t i c k s a r e p l a c e d p a r a l l e l t o each o t h e r , a p p r o x i m a t e l y 1" a p a r t , and h o r i z o n t a l l y w i t h r e s p e c t to the c h i l d . The e i g h t b l a c k s t i c k s a r e i n a shoebox, a l s o s i t t i n g on the t a b l e . The e x p e r i m e n t e r s a y s , "Here are some y e l l o w s t i c k s and h e r e (showing the c h i l d the shoebox) a r e some b l a c k s t i c k s . I'd l i k e you to take out enough', b l a c k s t i c k s and put them on the t a b l e so t h e r e a r e the same number of b l a c k s t i c k s as y e l l o w s t i c k s . Take out enough b l a c k s t i c k s so t h e r e a r e the same number of b l a c k s t i c k s as y e l l o w s t i c k s . " A f t e r jS does t h i s , IS asks i n a n e u t r a l tone, " I s t h a t r i g h t ? " and S_ i s g i v e n a chance to change h i s answer i f he w i s h e s . I f J 3 performs c o r r e c t l y , E_ c o n t i n u e s to P a r t B. I f he performs i n c o r r e c t l y , E_ c o n t i n u e s to T e s t 2 because j 3 does not appear t o u n d e r s t a n d the p r o c e d u r e . PART B. The s u b j e c t moves to a c h a i r a t the o p p o s i t e end of the t a b l e where t h e r e a r e e i g h t r e d s t i c k s l i n e d up l i k e the y e l l o w s t i c k s were l i n e d up i n P a r t A. S i x t e e n b l u e s t i c k s a r e i n the shoebox. The e x p e r i m e n t e r s a y s , "Now, I'd l i k e you to do the same t h i n g w i t h these r e d s t i c k s and these b l u e s t i c k s . Take out enough b l u e s t i c k s and p u t them on the t a b l e so t h e r e a r e the same number of b l u e s t i c k s as r e d s t i c k s . " A g a i n , a f t e r jS does t h i s , E, asks, " I s t h a t r i g h t ? " i n a n e u t r a l tone and S> i s g i v e n a chance to change h i s answer i f he w i s h e s . I f performs c o r r e c t l y , E! c o n t i n u e s to P a r t C. I f he performs i n c o r r e c t l y , _E r e c o r d s whether o r n o t he matches the l e n g t h o f the s t a n d a r d row and c o n t i n u e s to 44 T e s t 2. I f he i s a b l e to p e r f o r m c o r r e c t l y w i t h o u t making a s p a t i a l one-to-one c o r r e s p o n d e n c e between the r e d and b l u e s t i c k s , IS c o n t i n u e s to T e s t 2, s k i p p i n g P a r t C because _S has a l r e a d y demonstrated h i s a b i l i t y to p e r f o r m c o r r e c t l y w i t h o u t depending on the s p a t i a l arrangement of the s t i c k s . PART. C. The b l u e s t i c k s a r e put back i n the shoebox and _E s a y s , "Now I'd l i k e you to do t h a t a g a i n o n l y t h i s time put the s t i c k s l i k e t h i s (E demonstrates, p u t t i n g out f o u r s t i c k s , p a r a l l e l to each o t h e r and v e r t i c a l w i t h r e s p e c t to S_, and then _E p u t s the s t i c k s back i n the box) . Remember, we s t i l l want the same number of b l u e s t i c k s as r e d s t i c k s . " A g a i n , a f t e r _S does t h i s , E_ says i n a n e u t r a l tone, " I s t h a t r i g h t ? " and J3 i s g i v e n a chance to change h i s answer i f he w i s h e s . The s u b j e c t ' s p e r -formance i s r e c o r d e d . Then E s a y s , " I ' d l i k e you to do i t one more time o n l y t h i s time put t h e c b l u e s t i c k s i n t h i s box." (The box i s 7-5"• ~x."10%" x l V ) > "Remember, we s t i l l want the same number of b l u e s t i c k s as r e d ones." A g a i n , J3 i s g i v e n a chance to change h i s answer, and h i s response i s r e c o r d e d . T e s t 2 — T e s t f o r C o n s e r v a t i o n of Number (CON). T h i s i s a s t a n d a r d t e s t f o r c o n s e r v a t i o n of number. MATERIALS. The m a t e r i a l s f o r t h i s t e s t a r e the same as f o r T e s t 1. PART A ( l ) . The s u b j e c t r e t u r n s t o the f i r s t c h a i r . The e x p e r i -menter l i n e s up two rows o f t h r e e s t i c k s each, one of y e l l o w s t i c k s and one o f b l a c k s t i c k s . Each row of s t i c k s i s p l a c e d as the row of y e l l o w 45 s t i c k s was p l a c e d i n T e s t 1. There i s a p p r o x i m a t e l y an i n c h between the two rows. (See F i g u r e 1 f o r the arrangement of the s t i c k s i n T e s t 2 ) . The e x p e r i m e n t e r s a y s , "Can you t e l l me, a r e t h e r e more y e l l o w ones, the same number of y e l l o w ones as b l a c k ones, or more b l a c k ones?" (E p o i n t s a p p r o p r i a t e l y w h i l e s p e a k i n g ) . ( T h i s q u e s t i o n w i l l h e r e a f t e r be r e f e r r e d to as Q, w i t h " y e l l o w " changed to " r e d " and " b l a c k " changed to " b l u e " where a p p r o p r i a t e ) . I f S/s answer i s c o r r e c t , E_ c o n t i n u e s t o P a r t A ( 2 ) . I f the answer i s i n c o r r e c t , 13 r e c o r d s i t and c o n t i n u e s to T e s t 3, be-cause JS does not appear to u n d e r s t a n d the p r o c e d u r e . I f the answer i s ambiguous, the q u e s t i o n i s r e p e a t e d ; i f i t i s s t i l l ambiguous, JE con-t i n u e s to T e s t 3, a g a i n because jS does not appear to u n d e r s t a n d the p r o c e d u r e . ^ PART A ( 2 ) . The e x p e r i m e n t e r t h e n s p r e a d s out the row of b l a c k s t i c k s so t h a t the end s t i c k s a r e a p p r o x i m a t e l y 1" beyond the end s t i c k s o f the row o f y e l l o w s t i c k s . The e x p e r i m e n t e r asks Q. I f the answer i s c o r r e c t E_ c o n t i n u e s to P a r t B ( l ) . I f t h e answer i s i n c o r r e c t JS c o n t i n -ues to T e s t 3 because S_: does not appear to u n d e r s t a n d the p r o c e d u r e . PART B ( l ) . One of the b l a c k s t i c k s i s put back i n the shoebox, l e a v i n g one row of t h r e e y e l l o w s t i c k s and one row of two b l a c k s t i c k s The normal placement of o b j e c t s i n rows h o r i z o n t a l to the c h i l d , i n which one i s always c l o s e r to the c h i l d , c r e a t e s a problem because c h i l d r e n tend to choose the row c l o s e s t t o them as h a v i n g "more" (Rothen-b e r g and Courtney, 1969). In the p r e s e n t arrangement, the rows a r e e q u a l l y f a r away from the c h i l d . 2 O n ly one J3 gave an ambiguous answer a f t e r r e p e t i t i o n and t e s t i n g f o r him was d i s c o n t i n u e d . 46 FIGURE 1 Placement o f S t i c k s i n the T e s t f o r C o n s e r v a t i o n of Number PART A ( l ) : y e l l o w b l a c k PART A ( 2 ) : y e l l o w b l a c k PART B ( l ) : y e l l o w b l a c k PART B (2 ) : y e l l o w b l a c k PART C: S t a r t r e d b l u e T r a n s f o r m a t i o n (1) (2) (3) r e d b l u e r e d b l u e r e d b l u e 47 i n one-to-one c o r r e s p o n d e n c e , except f o r the m i s s i n g s t i c k which i s the one which would have been f a r t h e s t from J3. E _ asks C^ , i f _S answers c o r -r e c t l y , E c o n t i n u e s t o P a r t B(2). I f S_ answers c o r r e c t l y , E _ c o n t i n u e s to T e s t 3. PART B(2). The e x p e r i m e n t e r spreads out the two b l a c k s t i c k s u n t i l they a r e each about an i n c h beyond the ends o f the row o f y e l l o w s t i c k s and asks.Q. I f j S ' s answer i s c o r r e c t , E c o n t i n u e s t o P a r t C. I f J3 answers i n c o r r e c t l y , E c o n t i n u e s t o T e s t 3. PART C. The s u b j e c t moves to the c h a i r a t the o p p o s i t e end o f the t a b l e where t h e e i g h t r e d s t i c k s a r e s t i l l l i n e d up from T e s t 1. The e x p e r i m e n t e r l i n e s up e i g h t b l u e s t i c k s i n s p a t i a l one-to-one c o r r e s -pondence w i t h the e i g h t r e d s t i c k s , w i t h a p p r o x i m a t e l y 1" between the rows. The e x p e r i m e n t e r asks Q. I f the answer i s c o r r e c t , E _ c o n t i n u e s to the t r a n s f o r m a t i o n s . ^ The e x p e r i m e n t e r performs t h r e e t r a n s f o r m a t i o n s o f the b l u e s t i c k s , s a y i n g , " L e t ' s put them l i k e t h i s and see what you t h i n k . " The ex p e r i m e n t e r e i t h e r (1) spreads the b l u e s t i c k s out e v e n l y so they extend t h r e e s t i c k s beyond the r e d ones a t the upper end and remain f l u s h w i t h the r e d s t i c k s a t the lower end; o r (2) he pushes the b l u e s t i c k s c l o s e r t o g e t h e r so t h a t the r e d s t i c k s extend about two s t i c k s beyond the b l u e ones a t bo t h ends; or (3) he p l a c e s the b l u e s t i c k s i n a p i l e n e x t to the mi d d l e o f the row of r e d s t i c k s . The o r d e r o f the t h r e e t r a n s f o r m a t i o n s was c o u n t e r b a l a n c e d a c r o s s s u b j e c t s . A f t e r each t r a n s f o r m a t i o n E asks Q. No _S answered t h i s q u e s t i o n i n c o r r e c t l y . 48 I f the c h i l d answers c o r r e c t l y E c o n t i n u e s immediately t o the next t r a n s -f o r m a t i o n . I f the c h i l d answers i n c o r r e c t l y , _E r e c o r d s whether he chose the l o n g e r or the denser row as h a v i n g more. Then, to r e e s t a b l i s h t he. e q u i v a l e n c e o f the rows w i t h o u t g i v i n g the c h i l d feedback about the r e v e r s i b i l i t y of the t r a n s f o r m a t i o n , E f i r s t p u t s the b l u e s t i c k s back i n the box and then l i n e s them up a g a i n i n one-to-one c o r r e s p o n d e n c e 4 w i t h the r e d s t i c k s . The e x p e r i m e n t e r asks Q. I f S answers c o r r e c t l y , 15 c o n t i n u e s to the n e x t t r a n s f o r m a t i o n . A f t e r the first t r a n s f o r m a t i o n i f the c h i l d answers c o r r e c t l y , JE s a y s , "Show me what you would do to make i t so t h e r e a r e the same num-be r o f b l u e ones as r e d ones." The e x p e r i m e n t e r r e c o r d s whether the c h i l d p u t s the s t i c k s i n one-to-one c o r r e s p o n d e n c e a g a i n o r adds to the s h o r t e s t row. I f the c h i l d p uts the s t i c k s i n one-to-one c o r r e s p o n d e n c e i t i s , of c o u r s e , u n n e c e s s a r y f o r E to r e e s t a b l i s h the e q u i v a l e n c e of the rows and he t h e r e f o r e p r o c e d e s d i r e c t l y t o the n e x t t r a n s f o r m a t i o n . I f jS adds s t i c k s , E r e e s t a b l i s h e s e q u i v a l e n c e as o u t l i n e d i n the p r e v i o u s p a r a g r a p h . T e s t 3 — Imagery T e s t ( I T ) . T h i s t e s t i s d e s i g n e d to determine whether J3 can a n t i c i p a t e and remember the r e s u l t s o f a t r a n s f o r m a t i o n . There were two c o n d i t i o n s , Memory (MEM) and A n t i c i p a t i o n (ANT), and two t r a n s f o r m -a t i o n s , p u s h - t o g e t h e r (PT) and s p r e a d - a p a r t (SA). There were a l s o two c o l o r s — e i t h e r the r e d row was t r a n s f o r m e d or the b l u e row was t r a n s -formed. Thus each c h i l d was g i v e n f o u r imagery t e s t s — a PT-ANT, a PT-MEM, a SA-ANT, and a SA-MEM t e s t . ANT always p r e c e d e d MEM. The r e d No answered t h i s q u e s t i o n i n c o r r e c t l y . 49 t r a n s f o r m a t i o n was always used i n the f i r s t two t e s t s and the b l u e t r a n s -f o r m a t i o n was always used i n the l a s t two t e s t s . The o r d e r of PT and SA was c o u n t e r b a l a n c e d a c r o s s s u b j e c t s . MATERIALS. Four s e t s of t h r e e w h i t e c a r d b o a r d c a r d s , 4" x 9", c a l l e d c h o i c e c a r d s , were used. On each c a r d i s p a s t e d one row of l i t t l e ' ' s t i c k s o f r e d c a r d b o a r d and one row of l i t t l e s t i c k s o f b l u e c a r d b o a r d . (The s t i c k s were x I V ) . (See F i g u r e 2 ) . The rows are 1" a p a r t . Set 1, used under the PT-red t r a n s f o r m a t i o n c o n d i t i o n , con-t a i n e d c a r d s w i t h one row of 10 b l u e s t i c k s and one row of e i t h e r s i x , e i g h t , or 10 r e d s t i c k s . The c a r d w i t h s i x r e d s t i c k s i s l a b e l l e d 1, the c a r d w i t h e i g h t r e d s t i c k s i s l a b e l l e d 2, and the c a r d w i t h t e n r e d s t i c k s i s l a b e l l e d 3. The row o f b l u e s t i c k s i s 9" l o n g and the row of r e d s t i c k s i s always 5" l o n g . In c a r d 1, the d e n s i t y of the row o f r e d s t i c k s i s the same as the d e n s i t y of the row of b l u e s t i c k s . In c a r d 2, i t i s g r e a t e r but not g r e a t enough f o r the numbers i n the two rows' to be e q u a l . In c a r d 3, of c o u r s e , the d e n s i t y of the row of r e d s t i c k s i s i n c r e a s e d enough to be e q u a l i n number t o the row of b l u e s t i c k s . Set 2 i s the same as Set 1, o n l y the b l u e s t i c k s a r e r e d and the r e d s t i c k s a r e b l u e , and i t i s used under the P T - b l u e t r a n s f o r m a t i o n c o n d i t i o n . Set 3, used under the SA-blue t r a n s f o r m a t i o n c o n d i t i o n c o n t a i n e d c a r d s w i t h one row of 10. r e d s t i c k s and one row o f e i t h e r 18, 13 o r 10 L i t t l e s t i c k s had to be used o r the s t i m u l i would have been too l a r g e f o r the s u b j e c t ' s v i s u a l span. 50 F i g u r e 2 Placement o f S t i c k s i n Sample Cards From the Imagery T e s t (Sets 1 and 3) Set 1 b l u e r e d b l u e r e d b l u e r e d Set 3 b l u e r e d b l u e r e d b l u e r e d 51 b l u e s t i c k s . The c a r d w i t h 18 b l u e s t i c k s i s l a b e l l e d 1, the c a r d w i t h 13 b l u e s t i c k s i s l a b e l l e d 2, and the c a r d w i t h 10 b l u e s t i c k s i s l a b e l -l e d 3. The row of 10 r e d s t i c k s was always 5" l o n g w h i l e the row of b l u e s t i c k s was always 9" l o n g . Set 4 was the same as Set 3 o n l y the r e d s t i c k s were b l u e and the b l u e s t i c k s were r e d . I t was used under the SA-red t r a n s f o r m a t i o n c o n d i t i o n . Twenty l o o s e s t i c k s o f c a r d b o a r d , 10 r e d and 10 b l u e , a l s o V ' x 1", were used, a l o n g w i t h a p l a i n w h i t e p i e c e of c a r d b o a r d , 9" x 4". One red hardboard s t i c k , 9" x 1", was. used under SA, and one r e d hardboard s t i c k , 5" x 1", was used under PT. PART A. The p l a i n w h i t e c a r d was p l a c e d on the t a b l e v e r t i c a l l y w i t h r e s p e c t to the c h i l d . On i t were seven of the 10> l o o s e r e d c a r d -board s t i c k s and seven o f the 10 l o o s e b l u e c a r d b o a r d s t i c k s p l a c e d i n two rows i n one-to-one c o r r e s p o n d e n c e . Under c o n d i t i o n SA each row was a p p r o x i m a t e l y 5" l o n g , p l a c e d i n the c e n t e r o f the c a r d . Under c o n d i t i o n PT each row c o v e r e d the l e n g t h of the c a r d , 9". The c h i l d i s asked Q to s e n s i t i z e him to the importance of number as a cue i n the t a s k . The ex-p e r i m e n t e r p l a c e s the a p p r o p r i a t e r e d hardboard s t i c k a l o n g the row to be t r a n s f o r m e d and a s k s , "Can you s p r e a d these out (push these t o g e t h e r ) un-t i l they a r e j u s t as l o n g ( s h o r t ) as t h i s s t i c k ? " I f S^  cannot p e r f o r m c o r r e c t l y s p o n t a n e o u s l y , E_ demonstrates and g i v e s S_ a n o t h e r t r y . ^ When performs c o r r e c t l y , E_ c o n t i n u e s to P a r t B. A l l c h i l d r e n were a b l e to p e r f o r m c o r r e c t l y a f t e r a d e m o n s t r a t i o n , i f not s p o n t a n e o u s l y . 52 PART B (ANT). The e x p e r i m e n t e r r e t u r n s the s t i c k s t o t h e i r o r i g i -n a l p o s i t i o n on the p l a i n w h i t e c a r d o n l y uses a l l 10 o f the b l u e and a l l 10'. o f the r e d s t i c k s , s a y i n g , "Now l e t ' s put them back and add t h e s e . " The e x p e r i m e n t e r then s a y s , " T h i s time I want you to pretend we a r e g o i n g to s p r e a d these out (push t h e s e t o g e t h e r ) u n t i l they a r e as l o n g ( s h o r t ) as t h i s s t i c k — j u s t p r e t e n d , don't r e a l l y do i t . T e l l me what i t w i l l l o o k l i k e i f we do t h a t . W i l l i t l o o k l i k e t h i s , o r t h i s , o r t h i s ? " (The e x p e r i m e n t e r p u t s out the a p p r o p r i a t e s e t o f t h r e e c h o i c e c a r d s i n a row a l o n g s i d e the s t a n d a r d c a r d . The o r d e r i n which the t h r e e c a r d s a r e p l a c e d on the f i r s t a n t i c i p a t i o n t e s t g i v e n the c h i l d i s always 1, 2, 3. On the second a n t i c i p a t i o n t e s t the c h i l d i s g i v e n , the o r d e r i s always 1, 3, 2 ) . _E r e c o r d s JS's answer and p i c k s up the t h r e e c h o i c e c a r d s , c o n t i n u i n g to P a r t C. PART C (MEM). The e x p e r i m e n t e r s a y s , "Now I'm g o i n g to show you what i t l o o k s l i k e and l e t ' s see i f you can remember what i t l o o k s l i k e . " E_ p erforms the a p p r o p r i a t e t r a n s f o r m a t i o n and s a y s , "See what i t l o o k s l i k e ? " The e x p e r i m e n t e r c o v e r s the s t a n d a r d w i t h a w h i t e p i e c e o f c a r d -b o a r d and a s k s , "Now can you t e l l me, d i d i t l o o k l i k e t h i s , o r t h i s , or t h i s ? " (The e x p e r i m e n t e r p u t s out the a p p r o p r i a t e s e t of t h r e e c a r d s i n a row a l o n g s i d e the s t a n d a r d c a r d . The o r d e r i n which the t h r e e c a r d s a r e p l a c e d on the f i r s t memory t e s t g i v e n the c h i l d i s always 2, 1, 3, and on the second memory t e s t the o r d e r i s always 3, 2, 1 ) . The e x p e r i -menter r e c o r d s S/s answer, p i c k s up the t h r e e c a r d s , and s a y s , "Now l e t ' s do i t a n o t h e r way." The e x p e r i m e n t e r p r o c e e d s to r e p e a t P a r t s A, B, arid C w i t h the o t h e r t r a n s f o r m a t i o n . 53 TEST OF COUNTING ABILITY (CT) . A f t e r a l l f o u r imagery t r i a l s have been g i v e n , S_ i s asked to count the 10 l o o s e s t i c k s t h a t a r e sp r e a d out over the e n t i r e l e n g t h of the c a r d . T e s t 4 — Density-Length-Number-Test (D/L/N). T h i s t e s t i s d e s i g n e d to determine the c r i t e r i o n on which c h i l d r e n a t v a r i o u s s t a g e s base t h e i r judgment of q u a n t i t y . MATERIALS. A t o t a l o f 20 w h i t e cardboard c a r d s (4" x 11%") were used) , w i t h two rows o f c a r d b o a r d s t i c k s (h;w x I V ) , one row of b l u e and one row of r e d s t i c k s , p a s t e d on them. The c o l o r of the row w i t h the most s t i c k s and the s i d e on which the row w i t h the most s t i c k s was p l a c e d were randomly determined. The l e n g t h of the rows, and the number of s t i c k s i n each row f o r each of the 20 c a r d s a r e p r e s e n t e d i n T a b l e 1. For each c a r d i n Groups 1, 2, and 3, a number r a t i o (number i n rowl/number i n row 2 ) , a l e n g t h r a t i o ( l e n g t h of row 2 / l e n g t h o f row 1 ) , and a d e n s i t y ratio... ( d e n s i t y o f row 1, i . e . , number i n row 1 / l e n g t h o f row 1, d i v i d e d by d e n s i t y o f row 2, i . e . , number i n row 2 / l e n g t h o f row 2) was c a l c u l a t e d and i s shown i n T a b l e 1. As the l e v e l s i n T a b l e l i n c r e a s e , the a b s o l u t e v a l u e s of the number r a t i o , l e n g t h r a t i o , d e n s i t y r a t i o , and DR-LR a l s o i n c r e a s e . The r e a s o n the c a r d s were so c o n s t r u c t e d w i l l be d i s c u s s e d below. PROCEDURE. The 20 c a r d s were randomly d i v i d e d i n t o f o u r s e t s of f i v e c a r d s each. (These s e t s bear no r e l a t i o n s h i p to the Groups i n T a b l e 1.. Two s e t s were g i v e n d u r i n g the f i r s t s e s s i o n and two were 54 TABLE 1 CARDS USED IN THE DENSITY-LENGTH-NUMBER TEST (D/L/N) NUMBER IN LENGTH OF NUMBER LENGTH DENSITY DENSITY RATIO ROW 1 & 2 ROW 1 & 2 RATIO RATIO RATIO MINUS LENGTH (LR) (DR) RATIO (DR-LR) Group 1 l e v e l 1 7 6 9 10 7/6 10/9 1.3 .2 l e v e l 2 8 6 8 10 8/6 10/8 1.7 .3 l e v e l 3 9 6 7 10 9/6 10/7 2.1 .7 l e v e l 4 10 6 6 10 10/6 10/6 2.8 1.1 Group 2 l e v e l 1 7 6 6 10 7/6 10/6 1.9 .3 l e v e l 2 8 6 5 10 8/6 10/5 2.7 .7 l e v e l 3 9 6 4 10 9/6 10/4 3.8 1.3 l e v e l 4 10 6 3 10 10/6 10/3 5.6 2.3 Group 3 l e v e l 1 9 6 9 10 9/6 10/9 1.3 .6 l e v e l 2 10 6 8 10 10/6 10/8 2.1 .8 l e v e l 3 11 6 7 10 11/6 10/7 2.6 1.7 l e v e l 4 12 6 6 10 12/6 10/6 3.3 1.7 Group 4 l e v e l 1 7 6 10 10 7/6 10/10 1.2 . .2 l e v e l 2 8 6 10 10 8/6 10/10 1.3 .3 l e v e l 3 9 6 10 10 9/6 10/10 1.5 .5 l e v e l 4 10 6 10 10 10/6 10/10 . 1.7 .7 Group 5 l e v e l 1 8 7 10 8 8/7 8/10 l e v e l 2 8 7 10 7 8/7 7/10 l e v e l 3 8 6 10 6 8/6 6/10 l e v e l 4 • 8 6 10 8 8/6 8/10 55 g i v e n d u r i n g the second s e s s i o n w i t h the s e s s i o n o r d e r c o u n t e r b a l a n c e d a c r o s s s u b j e c t s . W i t h i n each s e s s i o n , one s e t was g i v e n a t the b e g i n n i n g of the s e s s i o n and one a t the end, w i t h the w i t h i n - s e s s i o n o r d e r a l s o c o u n t e r b a l a n c e d a c r o s s s u b j e c t s . The e x p e r i m e n t e r p r e s e n t e d the c a r d v e r t i c a l l y to S_. A f t e r the p r e s e n t a t i o n of the f i r s t c a r d the c h i l d was asked, "Can you t e l l me which ones a r e b l u e ? " Which ones a r e r e d ? " These q u e s t i o n s were o m i t t e d i n a l l l a t e r p r e s e n t a t i o n s . A f t e r each p r e s e n t a t i o n , S_ was asked Q. In Groups 1, 2, and 3, the row w i t h the l a r g e s t number of o b j e c t s i s always the s h o r t e s t row. C h i l d r e n i n Stage I tend to use l e n g t h as t h e i r o n l y c r i t e r i o n f o r judgment, and a c o r r e c t answer r e q u i r e s t h a t they overcome t h e i r b i a s to choose the l o n g e s t as h a v i n g the most. Because the average l e n g t h r a t i o o f the c a r d s i n Group 2 i s g r e a t e r than the average l e n g t h r a t i o o f the c a r d s i n Groups 3 arid l , i t s h o u l d be h a r d e r f o r c h i l d ^ - . r e n i n Stage I to p e r f o r m c o r r e c t l y when p r e s e n t e d w i t h those c a r d s i n Group 2 than when p r e s e n t e d w i t h the c a r d s i n Groups 3 o r 1. Stage I I c h i l d r e n , however, c o n s i d e r b o t h l e n g t h and d e n s i t y dimensions i n t h e i r judgments of q u a n t i t y . The l a r g e r the d i f f e r e n c e between the d e n s i t y r a t i o and the l e n g t h r a t i o ( i . e . , the more the d e n s i t y cue dominates the l e n g t h cue) the e a s i e r the t a s k s h o u l d be f o r the Stage I I c h i l d r e n . The average DR-LR v a l u e s i n c r e a s e s t e a d i l y from Group 1 to 2 to 3 ; . t h e r e f o r e , the performance of Stage I I c h i l d r e n s h o u l d a l s o i n c r e a s e s t e a d i l y from Group 1 to 2 to 3. C h i l d r e n i n Stage I I I , however, use number cues, p r i m a r i l y , i n t h e i r judgments of q u a n t i t y . Because the average number r a t i o i n Group 1 i s e q u a l to the average number r a t i o i n Group 2 and b o t h a r e l e s s than the average number r a t i o i n Group 3, the mean p e r f o r m -56 ance of Stage I I I c h i l d r e n on Group 3 c a r d s would be expected to be g r e a t -e r than t h e i r mean performance on Group 1 and 2 c a r d s , which s h o u l d not d i f f e r . c a r d s , Stage I c h i l d r e n s h o u l d tend to answer Q by s a y i n g t h a t the rows a r e e q u a l i n number, w h i l e Stage I I c h i l d r e n s h o u l d do q u i t e w e l l be-cause they c o n s i d e r b o t h l e n g t h and d e n s i t y i n t h e i r judgments. Thus, when Group 4 c a r d s a r e used, c h i l d r e n i n Stage I I would be expected to p e r f o r m b e t t e r than Stage I c h i l d r e n and as w e l l as Stage I I I c h i l d r e n . On the c a r d s i n Group 5, the row w i t h the l a r g e s t number o f o b j e c t s i s a l s o always the l o n g e s t row. T h i s s h o u l d be q u i t e easy f o r a l l groups and no d i f f e r e n c e i n performance o f the c h i l d r e n a t d i f f e r e n t s t a g e s s h o u l d be expected w i t h Group 5 c a r d s . These p r e d i c t i o n s , which a r e r e -l a t e d t o H y p o t h e s i s 6, a r e summarized i n T a b l e 2. The l e v e l s were i n -tr o d u c e d t o d i s c o v e r i f the a b s o l u t e v a l u e s , i . e . , the s a l i a n c e of the s t i m u l u s had an e f f e c t upon performance. The c a r d s i n Group 4 a l l have rows o f e q u a l l e n g t h . On these TABLE 2 PREDICTIONS OF THE RANK ORDER OF THE MEAN NUMBERS OF .CORRECT RESPONSES ON THE DENSITY-LENGTH-NUMBER TEST (D/L/N) 1. Stage I Group 1 = Group 3 > Group 2 2. Stage I I 3. Stage I I I Group 1 < Group 2 « Group 3 Group 1 = Group 2 < Group 3 4. Group 4 5. Group 5 Stage I = Stage I I = Stage I I I Stage I < Stage I I = Stage I I I 57 T e s t 5 — S e r i a t i o n T e s t (ST) , T h i s t e s t i s d e s i g n e d to determi n e whether S can b u i l d two c o r r e s p o n d i n g s e r i e s of o b j e c t s and to determine t h e man-ner i n which he b u i l d s them. MATERIALS. Two s e t s of 1" wide p a i n t e d hardboard s t i c k s were u s e d . Set 1 was used f o r P a r t A and c o n t a i n e d t h r e e y e l l o w s t i c k s , 5%", 6%", and 7Y 1 i n l e n g t h . Set 2 was used f o r P a r t B and c o n t a i n e d 16 r e d s t i c k s , two of each l e n g t h , v a r y i n g i n l e n g t h by 1" from 2%" to 9%", and 16 b l u e s t i c k s , two of each l e n g t h , v a r y i n g i n l e n g t h by 1" from 7%" to 14%". Be-cause the c h i l d r e n were to match the r e d s t i c k s w i t h the b l u e ones on the b a s i s of r e l a t i v e s i z e ( e . g . , the 4th l o n g e s t b l u e s t i c k w i t h the 4 t h l o n g e s t r e d s t i c k ) i t was e s s e n t i a l t h a t the o v e r l a p between the range of l e n g t h s of the b l u e and r e d s t i c k s be m i n i m a l . O t h e r w i s e , the c h i l d -r e n c o u l d match the s t i c k s on the b a s i s o f a b s o l u t e r a t h e r than r e l a t i v e s i z e . To a v o i d the p o s s i b i l i t y o f two s t i c k s b e i n g c o r r e c t l y matched on the b a s i s of some u n c o n t r o l l e d random cue (e . g . , a d i r t s p o t o f f i n g e r -p r i n t on the s t i c k s ) , 16 s t i c k s o f each c o l o r were made but o n l y e i g h t of each c o l o r were randomly chosen f o r the t a s k . PART A. The t h r e e y e l l o w s t i c k s and t h r e e b l a c k s t i c k s a r e r a n -domly s c a t t e r e d a t one end o f a l o n g t a b l e where S^  i s s e a t e d . E_ s a y s , "(S's name), we a r e g o i n g to p l a y a game. In t h i s game the b i g g e s t b l a c k s t i c k goes w i t h the b i g g e s t y e l l o w s t i c k , the next b i g g e s t b l a c k s t i c k goes w i t h the next b i g g e s t y e l l o w s t i c k and the l i t t l e s t b l a c k s t i c k goes w i t h the l i t t l e s t y e l l o w s t i c k . " (_E dem o n s t r a t e s , p l a c i n g the s t i c k s as they were p l a c e d i n T e s t 2, P a r t A, w i t h the s m a l l e s t s t i c k s c l o s e s t to the c h i l d ) . E_ then s a y s , "Now, I'm go i n g to mix them up and then w e ' l l see i f you can do i t . Are you r e a d y ? " E_ mixes up the s t i c k s and s a y s , 58 " L e t ' s see i f you can put them back j u s t the way I had them." A f t e r S_ does t h i s , E a s k s , i n a noncommittal tone, " I s t h a t r i g h t ? " and i s g i v e n a chance t o change h i s answer i f he w i s h e s . I f the c h i l d p e r -forms i n c o r r e c t l y , E_ demonstrates once more. Then, i f the c h i l d i s c o r r e c t , E c o n t i n u e s to P a r t B; i f S_ i s s t i l l i n c o r r e c t , E_ c o n t i n u e s to T e s t 6. PART B. The c h i l d moves to the c h a i r a t the o p p o s i t e end o f the t a b l e where the r e d and b l u e s t i c k s a r e randomly s c a t t e r e d . The ex p e r i m e n t e r s a y s , "Now we a r e g o i n g to p l a y the same k i n d of game — the b i g g e s t r e d s t i c k goes w i t h the b i g g e s t b l u e s t i c k , the second b i g -g e s t r e d s t i c k goes w i t h the second b i g g e s t b l u e s t i c k , and on down, un-t i l you g e t to the l i t t l e s t r e d s t i c k which goes w i t h the l i t t l e s t b l u e s t i c k . " A f t e r S_ attempts to put the s t i c k s i n o r d e r , E_ a s k s , i n a neu-t r a l tone, " I s t h a t r i g h t ? " and i s g i v e n a chance to change h i s answer i f he w i s h e s . I f S. performs c o r r e c t l y , E_ r e c o r d s whether he succeeded by t r i a l and e r r o r o r s y s t e m a t i c a l l y (as d i s c u s s e d i n the I n t r o d u c t i o n , p. 5), and c o n t i n u e s to T e s t 6. I f ^ p e r f o r m s i n c o r r e c t l y , E_ b u i l d s the two s e r i e s , mixes them up, and asks S to copy the f i n i s h e d s e r i e s . I f the c h i l d s t i l l cannot p e r f o r m c o r r e c t l y , E_ r e c o r d s whether he made one s e r i e s o r two s e p a r a t e s e r i e s o r was c o m p l e t e l y i n c a p a b l e of b u i l d -i n g the s e r i e s and c o n t i n u e s to T e s t 6. T e s t 6 — O r d i n a l Correspondence T e s t (OCT). T h i s t e s t was d e s i g n e d to determin e whether the c h i l d c o u l d match s t i c k s i n t h e two s e r i e s on the b a s i s of r e l a t i v e s i z e even when the s e r i e s a r e not i n one-to-one c o r r e s -pondence. 59 MATERIALS. The m a t e r i a l s used f o r t h i s t e s t were the same as those used i n T e s t 5. PART A. The ex p e r i m e n t e r p l a c e s the y e l l o w and b l a c k s t i c k s i n two p a r a l l e l s e r i e s w i t h the y e l l o w s t i c k s on S_'s l e f t and w i t h the s m a l l e s t s t i c k s c l o s e s t to S^ i f they a r e not a l r e a d y a r r a n g e d t h a t way from T e s t 5. (See F i g u r e 3 f o r the arrangement of t h e s t i c k s i n T e s t 6). The e x p e r i m e n t e r a s k s , p o i n t i n g to each o f the t h r e e b l a c k s t i c k s i n o r d e r from b i g g e s t t o s m a l l e s t and then to the m i d d l e s t i c k , "Which y e l l o w s t i c k goes w i t h t h i s b l a c k one?" The ex p e r i m e n t e r r e p e a t s t h i s q u e s t i o n p o i n t i n g to each of the t h r e e b l a c k s t i c k s i n o r d e r from b i g -g e s t to s m a l l e s t and then to the m i d d l e s t i c k , a f t e r p u s h i n g the b l a c k s t i c k s t o g e t h e r so t h a t the b i g g e s t b l a c k s t i c k i s o p p o s i t e the m i d d l e y e l l o w s t i c k . ^ PART B. The s u b j e c t moves to the c h a i r a t the f a r end of the t a b l e . E_ p u t s the b l u e and r e d s t i c k s i n two p a r a l l e l s e r i e s w i t h the re d s t i c k s on S/s l e f t and the s m a l l e s t s t i c k s c l o s e s t to S^ . E_ a s k s , p o i n t i n g to each b l u e s t i c k i n o r d e r from b i g g e s t to s m a l l e s t , "Which g red s t i c k goes w i t h t h i s b l u e one?" The ex p e r i m e n t e r then performs t h r e e t r a n s f o r m a t i o n s , e i t h e r (1) r e v e r s i n g the b l u e s e r i e s , i . e . , p u t t i n g the b i g g e s t b l u e s t i c k o p p o s i t e the s m a l l e s t r e d s t i c k , e t c . ; or (2) p u s h i n g the b l u e s t i c k s c l o s e r t o g e t h e r so t h a t the b i g g e s t b l u e s t i c k i s o p p o s i t e the t h i r d b i g g e s t r e d s t i c k w h i l e the s m a l l e s t b l u e ^ A l l c h i l d r e n answered these e i g h t q u e s t i o n s c o r r e c t l y , g Two c h i l d r e n were unable t o answer c o r r e c t l y and t e s t i n g f o r them on T e s t 6 was d i s c o n t i n u e d . P a r t A: F i g u r e 3 Placement of S t i c k s i n the O r d i n a l Correspondence T e s t y e l l o w b l a c k y e l l o w b l a c k P a r t B: S t a r t T r a n s f o r m a t i o n (1) r e d b l u e r e d b l u e T r a n s f o r m a t i o n (2) T r a n s f o r m a t i o n (3) r e d b l u e r e d b l u e 61 s t i c k remains o p p o s i t e the s m a l l e s t r e d s t i c k ; o r (3) p u s h i n g the b l u e s t i c k s away from the c h i l d u n t i l the s m a l l e s t b l u e s t i c k i s above the l a r g e s t red s t i c k . A f t e r each t r a n s f o r m a t i o n E_ a s k s , p o i n t i n g to each b l u e s t i c k i n o r d e r from the b i g g e s t to the s m a l l e s t , "Which red s t i c k 9 goes w i t h t h i s b l u e s t i c k ? " I f the c h i l d answers c o r r e c t l y , E_ p o i n t s to e i t h e r the 4 t h , 5 t h or 6 t h s m a l l e s t b l u e s t i c k and a s k s , "Which r e d s t i c k goes w i t h t h i s b l u e s t i c k ? " The e x p e r i m e n t e r r e c o r d s S/s answer and c o n t i n u e s to the next t r a n s f o r m a t i o n u n t i l a l l t h r e e t r a n s f o r m a t i o n s a r e performed. The o r d e r of the t r a n s f o r m a t i o n s and the o r d e r of the number ( 4 t h , 5 t h , or 6th) of the m i d d l e s t i c k chosen were b o t h c o u n t e r -b a l a n c e d . The c o m b i n a t i o n s of the t r a n s f o r m a t i o n s and numbers of the m i d d l e s t i c k s were v a r i e d randomly. T e s t 7 — T r a n s f o r m a t i o n T e s t (TT) . T h i s t e s t measures S_*s a b i l i t y to p e r f o r m a mental t r a n s f o r m a t i o n . MATERIALS. One w h i t e c a r d b o a r d c a r d (11%" x 4 " ) , h e r e a f t e r c a l l e d the p i c t u r e c a r d , w i t h f i v e c o l o r e d p i c t u r e s p a s t e d on i t was used. The p i c t u r e s were a r r a n g e d v e r t i c a l l y from top to bottom as f o l l o w s : c a r , l i t t l e g i r l , cup, shoe, f l o w e r . Four c a r d s , h e r e a f t e r c a l l e d the c o l o r c a r d s , a l s o 11%" x 4", were used. Each c o l o r c a r d was d i v i d e d i n t o f i v e e q u a l s e c t i o n s of d i f f e r e n t c o l o r s . The o r d e r of the c o l o r s , from the top to the bottom of the c a r d was (a) f o r Card 1, p u r p l e , g r e e n , w h i t e , b l u e , r e d ; (b) f o r Card 2, g r e e n , w h i t e , r e d , A l l c h i l d r e n answered c o r r e c t l y . 62 p u r p l e , b l u e ; (c) f o r Card 3, r e d , gr e e n , b l u e , w h i t e , p u r p l e ; and (d) f o r Card 4, b l u e , w h i t e , green, p u r p l e , r e d . PART A. The ex p e r i m e n t e r asks the c h i l d , "Can you show me the .. p i c t u r e of the l i t t l e g i r l ? Where's the f l o w e r ? And the shoe? And the c a r ? And the cup?" Card 4 o f the c o l o r c a r d s i s p l a c e d next to the p i c t u r e c a r d and E a s k s , p o i n t i n g to each c o l o r from top to bottom, "What c o l o r i s this?""'"^ Then E a s k s , p o i n t i n g to Card 4. "Can you t u r n t h i s c a r d u p s i d e down f o r me?" I f the c h i l d cannot, E_ demonstrates and then asks the c h i l d t o do i t a g a i n . PART B. The e x p e r i m e n t e r s a y s , p l a c i n g each c o l o r c a r d , i n o r d e r o f t h e i r numbers, next to the p i c t u r e c a r d , " T h i s time (or a g a i n I'd l i k e you to pretend we're going to t u r n t h i s c a r d ( p o i n t i n g ) u p s i d e down. What c o l o r w i l l be next to the c a r i f we t u r n t h i s c a r d u p s i d e down? What c o l o r w i l l be next to the f l o w e r i f we t u r n t h i s c a r d up-s i d e down? What c o l o r w i l l be n e x t to the l i t t l e g i r l (or shoe on Cards 2 and 3) i f we t u r n t h i s c a r d u p s i d e down?" The exp e r i m e n t e r r e c o r d s S/s answers t o each of the t h r e e q u e s t i o n s on each o f the f o u r c a r d s . ^ E v e r y was a b l e to answer: these q u e s t i o n s c o r r e c t l y . ^ E v e r y S was a b l e to do t h i s e i t h e r b e f o r e o r a f t e r the demon-s t r a t i o n . RESULTS C o u n t i n g A b i l i t y S i x t y - s e v e n Ss were a b l e to c o r r e c t l y count the s t i c k s and 27 were u n a b l e to c o r r e c t l y count them. T e s t 1 — T e s t f o r I n t e n s i v e Q u a n t i t y (NQ). TABLE 3 -PERFORMANCE ON THE TEST FOR (NQ) INTENSIVE QUANTITY CATEGORY NUMBER OF Ss 1. F a i l e d P a r t A 7 2. F a i l e d P a r t B 11 3. F a i l e d b o t h items i n P a r t C 32 4. F a i l e d one it e m i n P a r t C 15 5. Passed b o t h items i n P a r t C 29 TABLE 4 THE EFFECT OF SESSION ON PERFORMANCE ON THE TEST FOR INTENSIVE QUANTITY (NQ) PROPORTION OF Ss SESSION 1 SESSION 2 Who passed P a r t A 42/46 45/48 Who passed P a r t B 36/46 40/48 Who passed a t l e a s t one i t e m i n P a r t C 17/46 12/48 63 64 The performance o f Ss on the T e s t f o r I n t e n s i v e Q u a n t i t y i s shown i n T a b l e .3. T a b l e 4 shows the e f f e c t o f s e s s i o n o r d e r on p e r -formance on the NQ.test. None o f the d i f f e r e n c e s i n p r o p o r t i o n s a c r o s s s e s s i o n s was s i g n i f i c a n t . The number of Ss (34) p a s s i n g the i t e m i n P a r t C which r e q u i r e d p u t t i n g the s t i c k s i n a box d i d not d i f f e r s i g n i f i c a n t l y from the number (35) p a s s i n g the i t e m which r e q u i r e d p u t t i n g the s t i c k s i n a h o r i z o n t a l row. T e s t 2 — T e s t f o r C o n s e r v a t i o n of Number (NQ). The performance of Ss on the T e s t f o r C o n s e r v a t i o n o f Number i s 12 r e p o r t e d i n T a b l e 5. A l l except two of the n o n c o n s e r v a t i o n r e s p o n s e s were r e s p o n s e s i n which S_ chose the l o n g e s t row as h a v i n g more. TABLE 5 PERFORMANCE ON THE TEST FOR CONSERVATION OF NUMBER (CON) CATEGORY NUMBER OF Ss 1. F a i l e d P a r t A ( l ) 34 2. F a i l e d P a r t A(2) 28 3. F a i l e d P a r t B ( l ) 1 4. F a i l e d P a r t B(2) 1 One o f the two Ss f a i l e d t o c o n s e r v e on o n l y one t r a n s f o r m -a t i o n ; the o t h e r f a i l e d on a l l three-fe-ransformations. TABLE '5 (Continued) CATEGORY NUMBER OF Ss 5. F a i l e d a l l t h r e e t r a n s f o r m a t i o n s and when asked to make the rows e q u a l : (a) added s t i c k s (b) r e s t o r e d the one-to-one cor r e s p o n d e n c e 6. F a i l e d two t r a n s f o r m a t i o n s 7. F a i l e d one t r a n s f o r m a t i o n 8. Passed a l l t r a n s f o r m a t i o n s An a n a l y s i s o f v a r i a n c e u s i n g the l e a s t - s q u a r e s method [Winer, 1962], shown i n T a b l e 6,. Appendix C, was performend on the number of c o r r e c t r e s p o n s e s on the T e s t f o r C o n s e r v a t i o n of Number. The f a c t o r s were c o u n t i n g a b i l i t y (CT), s e s s i o n o r d e r ( S ) , the o r d e r of the t r a n s f o r m a t i o n s ( 0 ) , and the k i n d o f t r a n s f o r m a t i o n ( T ) . The s u b j e c t s who were not g i v e n the t h r e e t r a n s f o r m a t i o n s because they c o u l d n ot p e r f o r m c o r r e c t l y i n P a r t s A and B were g i v e n s c o r e s o f 0 on each o f the t h r e e t r a n s f o r m a t i o n s and i n c l u d e d i n the a n a l y s i s . The mean number o f c o r r e c t r e s p o n s e s on the CON t e s t a r e r e -p o r t e d i n T a b l e 7. Those who c o u l d count d i d s i g n i f i c a n t l y b e t t e r t h a n those who c o u l d not count (F=4.12, d f = l , 7 0 ; p < .05). There was a l s o a s i g n i f i c a n t T e f f e c t (F=4.10, df=2,140; p <.025). The S c h e f f 1 7 0 8 18 66 [Hays, p. 484] r e v e a l e d no s i g n i f i c a n t d i f f e r e n c e s between Ss p e r f o r m -ance on the f i r s t and t h i r d t r a n s f o r m a t i o n s . But Ss average performance on the second t r a n s f o r m a t i o n was s i g n i f i c a n t l y worse than on e i t h e r the f i r s t or the t h i r d t r a n s f o r m a t i o n . TABLE 7 .. MEAN NUMBER OF CORRECT RESPONSES ON THE TEST FOR CONSERVATION OF NUMBER (CON) Those Who Could Count Those Who Could Not Count .90 TRANSFORMATION 1 2 3 .33 AVERAGE PERFORMANCE .28 .20 .26 T e s t 5 — S e r i a t i o n T e s t (ST) . The performance of Ss on the S e r i a t i o n T e s t i s r e p o r t e d i n T a b l e 8. Four S_s b u i l t one s e r i e s s u c c e s s f u l l y b u t n o t the o t h e r . In a l l f o u r c a s e s i t was the r e d s e r i e s which was s u c c e s s f u l l y b u i l t . I n one o t h e r c a s e , S_ c o r r e c t l y c o n s t r u c t e d one s e r i e s b u t b u i l t the second s e r i e s w i t h one m i s p l a c e d i t e m and c o u l d not make the c o r r e s p o n d -ence between the two s e r i e s . A l l f i v e of these Ss a r e i n c l u d e d i n C a t e -g o r y 2. Whenever the Scheff£ i s used and the r e s u l t i s r e p o r t e d as s i g n i f i c a n t , p < .05. 67 TABLE 8 -PERFORMANCE ON THE SERIATION TEST (ST) CATEGORY NUMBER OF Ss 1. F a i l e d P a r t A 20 2. F a i l e d P a r t B 34 3. B u i l t the s e r i e s by t r i a l and e r r o r 25 4. B u i l t the s e r i e s s y s t e m a t i c a l l y 15 The e f f e c t o f s e s s i o n o r d e r on performance on the ST t e s t i s shown i n T a b l e 9. The p r o p o r t i o n who passed P a r t B was s i g n i f i c a n t l y h i g h e r i f the S e r i a t i o n T e s t was g i v e n d u r i n g the second s e s s i o n t h a n i f i t was g i v e n d u r i n g the f i r s t s e s s i o n (z = 2.26, p < .03). The o t h e r d i f f e r e n c e s i n p r o p o r t i o n s a c r o s s s e s s i o n s were not s i g n i f i c a n t . TABLE 9. EFFECT OF SESSION ON PERFORMANCE ON THE SERIATION TEST (ST) PROPORTION OF Ss SESSION 1 SESSION 2 Who passed P a r t A 38/48 36/46 Who passed P a r t B 15/48 25/46 Who b u i l t the s e r i e s s y s t e m a t i c a l l y 5/48 10/46 68 T e s t 6 — O r d i n a l Correspondence T e s t (OCT). TABLE 10 PERFORMANCE ON THE ORDINAL CORRESPONDENCE TEST (OCT) CATEGORY NUMBER OF Ss 1. Were u n a b l e t o f i n d the co r r e s p o n d e n c e b e f o r e t h e t r a n s f o r m a t i o n 2 2. F a i l e d a l l t h r e e t r a n s f o r m a t i o n s 15 3: F a i l e d two t r a n s f o r m a t i o n s 34 4. F a i l e d one t r a n s f o r m a t i o n 28 5. Passed a l l t h r e e t r a n s f o r m a t i o n s 15 The performance o f Ss on the O r d i n a l Correspondence T e s t i s shown i n T a b l e 10. An a n a l y s i s o f v a r i a n c e , shown i n T a b l e 11, Appendix C, was performed on the number of c o r r e c t r e s p o n s e s on t h i s t e s t w i t h c o u n t i n g a b i l i t y (CT) , s e s s i o n o r d e r (S) , o r d e r of the t r a n s f o r m a t i o n s ( 0 ) , and k i n d of t r a n s f o r m a t i o n (T) as the main f a c t o r s . Another a n a l y -s i s o f v a r i a n c e (see T a b l e il2", Appendix C) was done on the number of c o r r e c t r e s p o n s e s w i t h CT, S, the o r d e r of the numbers o f the m i d d l e s t i c k used ( 0 ) , and the number of the m i d d l e s t i c k used (N), as f a c t o r s . The mean numbers o f c o r r e c t r e s p o n s e s under v a r i o u s groups and c o n d i t i o n s a r e r e p o r t e d i n T a b l e 13. Both a n a l y s e s r e v e a l e d a s i g n i f i -c a n t CT e f f e c t (F = 6.64 and 6.93, df = l f 7 0 ; p < .025) w i t h Ss who c o u l d count p e r f o r m i n g b e t t e r than those who c o u l d not co u n t . There was 69 a l s o a s i g n i f i c a n t T e f f e c t (F = 5 #59, df = 2,140; p < .01), the S c h e f f g r e v e a l i n g t h a t t r a n s f o r m a t i o n 1 was s i g n i f i c a n t l y h a r d e r than e i t h e r t r a n s f o r m a t i o n 2 o r t r a n s f o r m a t i o n 3 which d i d not d i f f e r . The T x CT i n t e r a c t i o n was a l s o s i g n i f i c a n t (F = 4 #45, df = 2,140; p < .025), t h e S c h e f f e r e v e a l i n g t h a t t r a n s f o r m a t i o n 1 was s i g n i f i c a n t l y h a r d e r o n l y f o r those who c o u l d count; t h e r e was no s i g n i f i c a n t d i f f e r e n c e a c r o s s the t r a n s f o r m a t i o n s f o r those who c o u l d n ot co u n t . There was a l s o a s i g n i f i c a n t N e f f e c t (F = 7 #96, df = 2,140; p < .01), the '.Scheffe r e -v e a l i n g t h a t matching the 4 t h s t i c k was s i g n i f i c a n t l y h a r d e r than matching the 5 t h s t i c k which was s i g n i f i c a n t l y h a r d e r than matching the 6 t h s t i c k . TABLE 13 MEAN NUMBER OF CORRECT RESPONSES ON THE ORDINAL CORRESPONDENCE TEST (OCT) T R A N S F O R M A T I O N COUNTING ABILITY Could count Could n o t count O v e r a l l .34 .41 .36 .57 .30 .49 .69 .33 .59 T o t a l Score 1.60 1.04 1.44 MATCHING 4 t h STICK .35 MATCHING 5 t h STICK .47 MATCHING 6 t h STICK .62 Of the 58 _S_s who answered i n c o r r e c t l y on t r a n s f o r m a t i o n 1, 40, or 68 per c e n t , chose the s t i c k o p p o s i t e the one p o i n t e d t o by E_. Of 70 the 46 Ss who answered i n c o r r e c t l y on t r a n s f o r m a t i o n 2, 31, o r 67 per c e n t , chose the s t i c k o p p o s i t e the one p o i n t e d to by E_. T h i s type of r e s p o n s e was, of c o u r s e , i m p o s s i b l e on t r a n s f o r m a t i o n 3. T e s t 3 — Imagery-Test ( I T ) . Two c r i t e r i a were used to measure performance on t h i s t e s t . The f i r s t c r i t e r i o n s i m p l y d i v i d e d Ss i n t o those who performed c o r r e c t l y on t h r e e of the f o u r t r i a l s , and those who d i d n o t . By t h a t c r i t e r i o n , 74 f a i l e d and 20 p a s s e d . The c h o i c e c a r d s were l a b e l l e d 1, 2, o r 3 (see Method s e c t i o n ) . S u b j e c t s who chose Card 1, i t was h y p o t h e s i z e d , were e x p e c t i n g the den-s i t y to change to compensate f o r the l e n g t h change because they were not y e t a b l e to c o n s t r u c t the t r a n s f o r m a t i o n of u n i t s . The s u b j e c t s who chose Card 3, however, d i d so because they had c o n s t r u c t e d the t r a n s f o r -m a t i o n of u n i t s and c o u l d thus judge how much the d e n s i t y had to change to compensate f o r the l e n g t h change. The second c r i t e r i o n c a t e g o r i z e d Ss as r e s p o n d i n g e i t h e r p r e d o m i n a t e l y (two or more of the f o u r t i m e s ) , w i t h a Card 1 c h o i c e , p r e d o m i n a t e l y w i t h a Card 2 c h o i c e , o r predomin-a t e l y w i t h a Card 3 c h o i c e . ( I f the r e s p o n s e s were two Card 1 c h o i c e s 14 and two Card 3 c h o i c e s , S_ was u n c l a s s i f i e d . ) U s i n g t h i s c r i t e r i o n , 38 Ss responded p r e d o m i n a t e l y w i t h a Card 1 c h o i c e , 16 responded p r e -d o m i n a t e l y w i t h a Card 3 c h o i c e . Two Ss were u n c l a s s i f i a b l e . 71 Ea c h S/s f o u r r e s p o n s e s c o u l d be d i v i d e d i n t o two p a i r s , one p a i r f o r ea c h t r a n s f o r m a t i o n , e i t h e r spread a p a r t (SA) or push t o g e t h e r ( P T ) , w i t h each p a i r composed o f an a n t i c i p a t i o n (ANT) t r i a l and a mem-o r y (MEM) t r i a l . Out of 146 of such p a i r s of res p o n s e s i n which a t l e a s t one o f the two res p o n s e s was i n c o r r e c t , S_ chose, on the second (MEM) t r i a l , the c a r d i n the same p o s i t i o n as the c a r d he chose on the f i r s t (ANT) t r i a l 72 t i m e s , d e m o n s t r a t i n g t h a t Ss' i n c o r r e c t c h o i c e s on the memory t r i a l s were not random (chance performance would be 43 t r i a l s ) but were o f t e n determined by p o s i t i o n . The s u b j e c t s r e p e a t e d the same c a r d on o n l y 29 o f the 146 p a i r s , d e m o n s t r a t i n g they were g e n e r a l l y n o t c o n s i s t e n t . I f S/s c h o i c e was c o n s i s t e n t l y e i t h e r based on an e x p e c t a t i o n o f same d e n s i t y , d i f f e r e n t d e n s i t y but unequal number, o r same number, even i f he i s i n c o r r e c t , he ought to choose on the mem-or y p o r t i o n of the p a i r the same c a r d as he chose on the a n t i c i p a t i o n p o r t i o n . Three a n a l y s e s of v a r i a n c e were performed on Ss' s c o r e s on the Imagery T e s t . They d i f f e r e d i n the c r i t e r i o n used to c l a s s i f y Ss i n t o l e v e l s of development (G). Method 1 c l a s s i f i e d the Ss i n t o the f i v e c a t e g o r i e s l i s t e d i n T a b l e 3", by t h e i r performance on the T e s t f o r I n t e n s i v e Q u a n t i t y . Method 2 c l a s s i f i e d the Ss i n t o the f i v e c a t e g o r i e s l i s t e d i n T a b l e 7, by t h e i r performance on the T e s t f o r O r d i n a l C o r r e s -pondence. A c c o r d i n g to method 3, i f S^was i n Category 1, 3, o r 4 of the T e s t f o r C o n s e r v a t i o n o f Number ( T a b l e 5.) , he was put i n G^, i . e . , he d i d n o t u n d e r s t a n d the p r o c e d u r e . I f he was i n Category 2, 5a, 5b, o r 6 of the CON t e s t and i n Category 1 o r 2 of the T e s t f o r I n t e n s i v e Quan-t i t y ( T a b l e 3)> he was put i n G„, i . e . , he c o u l d n e i t h e r c o n s e r v e nor 72 match the s t a n d a r d row but d i d understand the p r o c e d u r e . I f he was i n C a t e g o r y 2, 5a, 5b, or 6 o f CON and Category 3, 4, o r 5 o f NQ, he was put i n i . e . , he c o u l d match the s t a n d a r d row but c o u l d not c o n -s e r v e . I f he was i n Category 7 of CON he was put i n G^, i . e . , he c o u l d c o n s e r v e by the weak c r i t e r i a o f two of t h r e e t r a n s f o r m a t i o n s c o r r e c t but not by the s t r o n g c r i t e r i a of a l l t h r e e t r a n s f o r m a t i o n s c o r r e c t . I f he was i n C a t e g o r y 8 o f CON he was put i n G^, i . e . , he c o u l d c o n s e r v e by the s t r o n g c r i t e r i a . The f i r s t a n a l y s i s of v a r i a n c e ( T a b l e 14, Appendix C) was performed on S5s s c o r e s on the Imagery T e s t . The f a c t o r s were l e v e l o f development determined by Method 1 (G), s e s s i o n o r d e r ( S ) , t r a n s -f o r m a t i o n , e i t h e r PT o r SA ( T ) , t r a n s f o r m a t i o n o r d e r ( 0 ) , c o n d i t i o n , e i t h e r MEM or ANT ( C ) , and c o u n t i n g a b i l i t y (CT) . C o n t r a r y to expec-t a t i o n , no s i g n i f i c a n t G e f f e c t was f o u n d . The-only s i g n i f i c a n t e f f e c t was C (F = 6 #99, d f = 1,54; p < .025). The mean number of c o r r e c t r e s p o n s e s under the MEM c o n d i t i o n was .88 and the mean number of c o r r e c t r e s p o n s e s under the ANT c o n d i t i o n was .65. S i m i l a r a n a l y -ses u s i n g Methods 2 and 3 to determine G ( T a b l e s 15 and 16,. Appendix C, r e s p e c t i v e l y ) a l s o f a i l e d to y i e l d s i g n i f i c a n t G e f f e c t . I t was not s u r p r i s i n g t h a t no G e f f e c t was found. A G - e f f e c t would o n l y be s i g n i f i c a n t i f S/s c r i t e r i o n f o r judgment was c o n s i s t e n t even when he was i n c o r r e c t . As was s t a t e d above, t h i s c o n s i s t e n c y was not found. I t appears t h a t the t a s k f a i l e d to determine the bases on w hich c h i l d r e n remember and a n t i c i p a t e t r a n s f o r m a t i o n s of rows of d i s -c r e t e o b j e c t s ; 73 T e s t 4 — Density-Length-Number T e s t (D/L/N). The mean number of c o r r e c t r e s p o n s e s o f Ss by Stage and by Groups 1 t h r o u g h 3 i s r e p o r t e d i n T a b l e 17. TABLE 17 MEAN NUMBER OF CORRECT RESPONSES FROM THE PLANNED COMPARISONS TESTS ON THE DENSITY-LENGTH-NUMBER TEST C D / L / N ) STAGE GROUP 1 GROUP 2 GROUP 3 I I I I I I . 1.28 1.52 3.31 1.11 1.68 3.38 1.44 1.90 3.50 I t was p r e d i c t e d t h a t performance by Ss i n Stage I (th o s e who c o u l d n ot match the s t a n d a r d row i n the T e s t f o r I n t e n s i v e Q u a n t i t y ) would be p o o r e s t on Group 2 c a r d s as compared w i t h t h e i r performance on the c a r d s i n Groups 3 and 1 (see T a b l e 2 f o r p r e d i c t i o n s ) . A planned comparisons t e s t between the means s u p p o r t e d t h i s p r e d i c t i o n .(p < .05). I t was p r e d i c t e d t h a t the performance of Stage I I Ss ( t h o s e who c o u l d match the s t a n d a r d row on the NQ t e s t but c o u l d n ot c o n s e r v e on the CON t e s t ) would improve from Group 1 to Group 2 t o Group 3 c a r d s . A planned comparisons t e s t between the means s u p p o r t e d t h i s p r e d i c t i o n (p < .01) . 74 I t was p r e d i c t e d t h a t the performance of Stage I I I Ss ( t h o s e who conserved) would be b e s t on Group 3 c a r d s w i t h no d i f f e r e n c e between t h e i r performance on Group 1 and Group 2 c a r d s . The p l a n n e d comparisons t e s t r e v e a l e d no s i g n i f i c a n t d i f f e r e n c e s . TABLE 19 MEAN NUMBER OF CORRECT RESPONSES FROM THE ANALYSIS OF VARIANCE ON THE DENSITY-LENGTH-NUMBER TEST (D/L/N) STAGE I STAGE I I STAGE I I I T o t a l S c o re On Group 4 c a r d s On Group 5 c a r d s 8.94 2.00 3.11 11.30 2.56 3.64 16.88 3.04 3.65 Those Who Could Count 13.37 Those Who Could Not Count 9.96 An a n a l y s i s o f v a r i a n c e (see T a b l e 18, Appendix C) was p e r -formed on Ss s c o r e s on the D/L/N t e s t w i t h Stage as d e f i n e d i n the p r e -c e d i n g t h r e e p a r a g r a p h s , to be r e f e r r e d t o h e r e a f t e r as Method 4 (G), c o u n t i n g a b i l i t y ( C T ) , s e s s i o n o r d e r ( S ) , w i t h i n - s e s s i o n o r d e r ( B E ) , c a r d Group (N), and l e v e l w i t h i n c a r d Group (A) as f a c t o r s . The mean numbers of c o r r e c t responses on the D/L/N t e s t a r e shown i n T a b l e 19. Stage was s i g n i f i c a n t (F = 14.17, df = 2,70; p < .01) w i t h Stage I Ss p e r f o r m i n g 75 s i g n i f i c a n t l y p o o r e r than Stage I I I S_s by the S c h e f f e . C o u n t i n g a b i l i t y was a l s o s i g n i f i c a n t (F = 7.94, df = 1.70; p < .01) w i t h the mean p e r -formance of Ss who c o u l d count h i g h e r than the mean performance of Ss who c o u l d not count. The N e f f e c t was a l s o s i g n i f i c a n t (F = 36.84, df = 4,280; p < .01) w i t h the mean number of c o r r e c t r e s p o n s e s i n c r e a s i n g from to N,. and w i t h a l l comparisons except t h a t between (Group 1) and N£ (Group 2) s i g n i f i c a n t , a c c o r d i n g to the S c h e f f e . As was to be e x p e c t e d from the above planned comparisons t e s t , the N x G i n t e r a c t i o n was a l s o s i g n i f i c a n t (F = 5 . 8 1 , df = 8,280; p < .01). The S c h e f f 6 r e v e a l e d no s i g n i f i c a n t d i f f e r e n c e s o f G w i t h i n N,. (Group 5 c a r d s ) , as was p r e d i c t e d ( T a b l e 2 ) . On t h e c a r d s i n Group 4, Stage I S_s performed s i g n i f i c a n t l y p o o r e r than Stage I I and Stage I I I Ss. There was no d i f f e r e n c e between the performances of Stage I I and Stage I I I Ss. These r e s u l t s were p r e d i c t e d ( T a b l e 2 ) . T e s t 7 - - T r a n s f o r m a t i o n T e s t ( T T ) . A s t r o n g and weak c r i t e r i a were used to a s s e s s Ss performance on TT. As d i s c u s s e d i n the Method s e c t i o n , each S_ was asked two q u e s t i o n s on e a c h c a r d c o n c e r n i n g which c o l o r would be next to an end p i c t u r e i f t h e c o l o r c a r d was t u r n e d u p s i d e down. He was a l s o asked one q u e s t i o n on e a c h c a r d c o n c e r n i n g which c o l o r would be next to a m i d d l e p i c t u r e i f the c o l o r c a r d was t u r n e d u p s i d e down. A c c o r d i n g to the s t r o n g c r i t e r i o n , i f S^  got l e s s than s i x of the e i g h t end q u e s t i o n s c o r r e c t , and l e s s than t h r e e of the f o u r m i d d l e q u e s t i o n s c o r r e c t he was c l a s s i f i e d i n Group A, i . e . , he was judged to be u n a b l e to p e r f o r m a mental t r a n s f o r m a t i o n . I f S^  answered 76 c o r r e c t l y on s i x or more of th e e i g h t end q u e s t i o n s b u t ranswered c o r r e c t l y l e s s than t h r e e o f t h e f o u r m i d d l e q u e s t i o n s , he was c l a s s i f i e d i n Group B, i . e . , he was judged a b l e to p e r f o r m c o r r e c t l y on the end t r a n s f o r m a -t i o n s but not on the m i d d l e ones. F i n a l l y , i f answered s i x o r more of the e i g h t end q u e s t i o n s c o r r e c t l y and t h r e e o r more of the f o u r m i d d l e q u e s t i o n s c o r r e c t l y , he was c l a s s i f i e d i n Group C, i . e . , he was judged as c a p a b l e of p e r f o r m i n g a mental t r a n s f o r m a t i o n . ( I f S_ answered l e s s than s i x o f the e i g h t end q u e s t i o n s c o r r e c t l y b u t t h r e e o r more of the f o u r m i d d l e q u e s t i o n s c o r r e c t l y , he was unclassifiable.'*""') By t h i s c r i t e r i o n 39 Ss were i n Group A, 23 were i n Group B, and 31 were i n Group C. A weaker c r i t e r i o n .was a l s o used to c l a s s i f y the Ss. A c c o r d -i n g to t h i s c r i t e r i o n a s u b j e c t had to pass l e s s than f o u r of the e i g h t end q u e s t i o n s and l e s s than two of the f o u r m i d d l e q u e s t i o n s to be c l a s s i -f i e d i n Group A. An S had t o pass o n l y f o u r (or more) of the e i g h t end q u e s t i o n s and l e s s than two o f the f o u r m i d d l e q u e s t i o n s to be c l a s s i f i e d i n Group B. An S had to pass f o u r o r more of the e i g h t end q u e s t i o n s and o n l y two o r more of the f o u r m i d d l e q u e s t i o n s to be c l a s s i f i e d i n Group C. ( I f S^  answered l e s s than f o u r o f the e i g h t end q u e s t i o n s c o r r e c t l y b u t two 16 or more o f the f o u r m i d d l e q u e s t i o n s c o r r e c t l y , he was u n c l a s s i f i a b l e . ) A c c o r d i n g t o t h i s c r i t e r i o n , 26 Ss were c l a s s i f i e d i n Group A, 30 were c l a s s i f i e d i n Group B, and 36 were c l a s s i f i e d i n Group C. 15 Only one £[ was u n c l a s s i f i a b l e . 16 Only two Ss. were u n c l a s s i f i a b l e . 77 An a n a l y s i s of v a r i a n c e ( T a b l e 20, Appendix C) was performed on ^Ss s c o r e s ^ on t h e TT t e s t . The f a c t o r s were l e v e l of development, Method 1 (G), s e s s i o n o r d e r ( S ) , c a r d ( C ) , and e i t h e r m i d d l e o r end p o s i t i o n ( P ) . TABLE 21 MEAN NUMBER OF CORRECT RESPONSES FROM THE ANALYSIS OF VARIANCE ON THE TRANSFORMATION TEST (TT) WITH G DETERMINED BY METHOD 1 G, G„ G„ G, G, OVERALL End q u e s t i o n 1.29 1.50 2.52 2.93 3.19 2.60 M i d d l e q u e s t i o n .43 .45 .72 2.60 2.34 1.47 O v e r a l l 1.71 1.95 3.23 5.53 5.53 4.07 The G e f f e c t was s i g n i f i c a n t (F = 9.55, df = 4,84; p < .005). The mean numbers of c o r r e c t r e s p o n s e s on the TT t e s t f o r t h i s a n a l y s i s a r e r e p o t e d i n T a b l e 21. Groups 1 and 2 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 o t h e r but Groups 1 and 2 d i f f e r e d s i g n i f i c a n t l y from Group 3. Group 3 d i f f e r e d s i g n i f i c a n t l y from Group 4 and Groups 4 and 5 d i d not d i f f e r from e a c h o t h e r . Thus, performance on the TT d i s c r i m i n a t e d those who c o u l d e i t h e r not u n d e r s t a n d the T e s t f o r I n t e n s i v e Q u a n t i t y p r o c e d u r e or who c o u l d not match the number i n the s t a n d a r d row as r e q u i r e d by the NQ t e s t from those who c o u l d match the s t a n d a r d row. Performance on t h e Because two end q u e s t i o n s were asked per c a r d and o n l y one m i d d l e q u e s t i o n was asked p e r c a r d , Ss s c o r e on the end q u e s t i o n s was d i v i d e d by two f o r the a n a l y s i s . 78 TT t e s t a l s o d i s c r i m i n a t e d t h o s e who c o u l d match the s t a n d a r d o n l y by depending on a s p a t i a l c o r r e s p o n d e n c e from those who d i d not have to depend on the s p a t i a l c o r r e s p o n d e n c e . The P e f f e c t was a l s o s i g n i f i c a n t (F =69.65, d f = 1,252; p < .005). The s u b j e c t s performed b e t t e r on the end q u e s t i o n s than they d i d on the m i d d l e q u e s t i o n s . A l t h o u g h the i n t e r a c t i o n , P x C, was a l s o s i g -n i f i c a n t (F = 5.38, d f = 3,252; p < .005), the S c h e f f S demonstrated t h a t performance on t h e end q u e s t i o n s was s i g n i f i c a n t l y s u p e r i o r to performance on the m i d d l e q u e s t i o n s f o r a l l l e v e l s of C. An a n a l y s i s of s i m p l e e f f e c t s [ K i r k , 1968] r e v e a l e d a C e f f e c t under V^, i . e . , f o r the end q u e s t i o n s . Thus, the P x C i n t e r a c t i o n was due t o an improvement over t r i a l s i n p e r -formance on the m i d d l e q u e s t i o n s but n o t on the e a s i e r end q u e s t i o n s . There was a l s o a C x G i n t e r a c t i o n CF = 3.81, df = 12,252; p < .005), which, by the S c h e f f e , was due t o an i n c r e a s e i n performance from Card 1 to Card 2 f o r Group 5 but not f o r any o t h e r Group. The P x G i n t e r a c t i o n was a l s o s i g n i f i c a n t CF = 4.00, df = 4,84; p < .005). A c c o r d i n g to the Scheff£, performance on the end q u e s t i o n s by Group 1 and 2 was s i g n i f i c a n t l y i n f e r i o r t o the performance o f Groups 3, 4, and 5 combined. Performance on the m i d d l e q u e s t i o n s by Group 1, 2, and 3 d i f f e r e d s i g n i f i c a n t l y from the performance of Groups 4 and 5 on the m i d d l e q u e s t i o n s . Thus, performance on the end q u e s t i o n s . d i s c r i m i -n a t e d t hose Ss who c o u l d not match, the s t a n d a r d row i n the NQ t e s t from t h o s e who c o u l d . Performance on the m i d d l e q u e s t i o n s d i s c r i m i n a t e d t h o s e Ss who c o u l d match the s t a n d a r d row o n l y by a dependence on a s p a t i a l c o r r e s p o n d e n c e from those Ss who d i d not need to..depend on t h e s p a t i a l 79 correspondence t o p e r f o r m c o r r e c t l y . TABLE 23 MEAN NUMBER OF CORRECT RESPONSES FROM THE ANALYSIS OF VARIANCE ON THE TRANSFORMATION TEST (TT) WITH G DETERMINED BY METHOD 2 G l G 2 G 3 G 4 G 5 OVERALL End q u e s t i o n 1.25 2.17 2.19 3.00 3.27 2.60 M i d d l e q u e s t i o n .50 .73 1.09 1.46 3.20 1.47 O v e r a l l 1.75 2.90 3.28 4.47 6.47 4.07 A second a n a l y s i s of v a r i a n c e (see T a b l e 22, Appendix C) was done on £>s s c o r e s on TT, t h i s time u s i n g Method 2 t o determine i n t o w h i c h of the f i v e groups S^  was c l a s s i f i e d . T a b l e 23 shows the mean number o f c o r r e c t r e s p o n s e s f o r t h i s a n a l y s i s . A G e f f e c t was a g a i n found (F = 5.52, df = 4,84; p < .005) w i t h the performance o f Groups 1, 2, and 3 s i g n i f i c a n t l y i n f e r i o r by t h e S c h e f f e t o the performance of Group 4, wh i c h was s i g n i f i c a n t l y i n f e r i o r t o the performance of Group 5. The C x G i n t e r a c t i o n was a g a i n s i g n i f i c a n t w i t h the S c h e f f e r e v e a l i n g a s i g n i f i c a n t improvement i n performance by Group 5 Ss from Card 1 t o Card 2 b ut no s i g n i f i c a n t change i n . the performance by Groups 1, 2, 3, o r 4 a c r o s s c a r d s . The P x G i n t e r a c t i o n was a g a i n s i g n i f i c a n t . In t h i s a n a l y s i s , performance on t h e end q u e s t i o n s by Groups 1, 2, 3, was s i g n i f i c a n t l y i n f e r i o r t o the performance o f Groups 4 and 5 by the 80 S c h e f f e . The d i f f e r e n c e i n performance on the m i d d l e q u e s t i o n s between Groups 1 and 2 combined and Groups'3'.and 4 combined was s i g n i f i c a n t . Performance on t h e m i d d l e q u e s t i o n by Groups 3 and 4 combined was s i g -n i f i c a n t l y i n f e r i o r t o the performance of Group 5. Thus, performance on the end q u e s t i o n d i s c r i m i n a t e d those who c o u l d pass the OCT t e s t by the weak c r i t e r i o n of two out of t h r e e t r a n s f o r m a t i o n s c o r r e c t from t h o s e who c o u l d n o t . Performance on the m i d d l e q u e s t i o n d i s c r i m i n a t e d t h o s e who c o u l d pass the OCT t e s t by the s t r o n g c r i t e r i o n o f a l l t h r e e t r a n s f o r m a t i o n s c o r r e c t from those who c o u l d n o t . A t h i r d a n a l y s i s of v a r i a n c e (see T a b l e 24, Appendix C) was performed on Ss s c o r e s on t h e TT t e s t . Method 4 was used to c l a s s i f y Ss i n t o one of t h r e e G l e v e l s . The mean number of c o r r e c t r e s p o n s e s f o r Ss i n Stage I , I I , and I I I were 1.86, 4.01, and 5.64 r e s p e c t i v e l y . The G e f f e c t was a g a i n s i g n i f i c a n t (F = 12.14, df = 2,88; p < .055) w i t h Group l ' s performance s i g n i f i c a n t l y . i n f e r i o r to t h a t of Group 2 w h i c h was s i g n i f i c a n t l y i n f e r i o r t o t h a t o f Group 3. The C x G i n t e r a c t i o n was a g a i n s i g n i f i c a n t CF = 4.45, df = 6,264; p < .01) due t o an i m p r o v e -ment from Card 1 to Card 2 by Group 3. In t h i s a n a l y s i s , the P x G i n -t e r a c t i o n was not s i g n i f i c a n t . R e l a t i o n s h i p Between the T e s t s The c h i - s q u a r e a n a l y s e s on Ss performance on v a r i o u s combina-t i o n s o f t e s t s a r e r e p o r t e d i n T a b l e s 25 through 42 i n Appendix B. There was no s i g n i f i c a n t r e l a t i o n s h i p between Ss performance on t h e Imagery T e s t and Ss performance on the T r a n s f o r m a t i o n T e s t and they a r e n o t p r e s e n t e d . 81 T a b l e s 43 through. 56 i n Appendix B show the c h i - s q u a r e a n a l y s e s f o r the r e l a t i o n s h i p between each of t h e t e s t s , u s i n g a l l c l a s s i f i c a t i o n b r e a k -downs d i s c u s s e d i n Appendix B, and Ss age, e i t h e r 48 t o 59, 60 to 71, o r 72 and above months. The t a b l e s a l s o show the t r a n s f o r m a t i o n of c h i - s q u a r e t o <)> (Cramer's s t a t i s t i c [Hays, pp. 604-606]) and T a b l e s 25 t h r o u g h 42 i n c l u d e the <j) v a l u e s a f t e r age i s p a r t i a l l e d out OK). The <K i s then c o n v e r t e d back to a c h i - s q u a r e v a l u e to t e s t f o r the s i g n i -f i c a n c e of t h e r e l a t i o n s h i p w i t h age h e l d c o n s t a n t . As can be seen from the t a b l e s , a l l r e l a t i o n s h i p s a r e s i g n i f i c a n t except t h o s e between the T r a n s f o r m a t i o n T e s t , weak c r i t e r i o n and Age; the Imagery T e s t , com-p l e x c r i t e r i o n and Age; the Imagery T e s t , s i m p l e c r i t e r i o n and Age; t h e T e s t f o r O r d i n a l Correspondence and C o u n t i n g A b i l i t y ; the Imagery T e s t , complex c r i t e r i o n , and the T e s t f o r C o n s e r v a t i o n o f Numbers; and the Imagery T e s t , s i m p l e c r i t e r i o n , and the T e s t f o r C o n s e r v a t i o n o f Number. H y p o t h e s i s 1. There w i l l be a p o s i t i v e r e l a t i o n s h i p between t h e c h i l d ' s a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n -ces r e g a r d l e s s of the s p a t i a l arrangement of the s e r i e s and h i s a b i l i t y t o c o n s e r v e , i f the same m a t e r i a l s are used t o t e s t f o r the two a b i l i t i e s . The r e l a t i o n s h i p between Ss' a b i l i t y to co n s e r v e and Ss' a b i l -i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s i s shown i n T a b l e 25 (which i n c l u d e s a l l Ss) and T a b l e 26 (which e x c l u d e s those Ss i n C a t e g o r y 1 o f T a b l e 5, i . e . , t h o s e who d i d n o t u n d e r s t a n d the CON p r o c e d u r e ) . The r e l a t i o n -2 s h i p between t h e two a b i l i t i e s i s h i g h l y s i g n i f i c a n t (x = 18.75, d f = 2, 2 p < .001 and x = 10.02, d f = 2; p <.01, r e s p e c t i v e l y ) and the h y p o t h e -s i s i s c o n f i r m e d . There a r e e x c e p t i o n s t o the h y p o t h e s i s , however. I f the c r i -t e r i o n f o r p a s s i n g the OCT t e s t i s two out o f t h r e e t r a n s f o r m a t i o n s 82 c o r r e c t , t h e r e a r e 23 Ss (13 i n T a b l e 26) who pass the OCT t e s t but f a i l t o c o n s e r v e and s i x s u b j e c t s who pass t h e CON t e s t b u t f a i l t o f i n d o r d i -n a l c o r r e s p o n d e n c e s . The OCT t e s t t h u s appears t o be p a s s e d e a r l i e r t h a n the CON t e s t i f t h e weak c r i t e r i o n f o r p a s s i n g OCT i s used. I f , however, the c r i t e r i o n f o r p a s s i n g the OCT t e s t i s a l l t h r e e t r a n s f o r m a t i o n s c o r -r e c t , t h e r e a r e o n l y f i v e Ss ( t h r e e i n T a b l e 26) who pass the OCT t e s t and f a i l t o c o n s e r v e and 16 Ss who pas s the CON t e s t b u t f a i l to f i n d o r d i n a l c o r r e s p o n d e n c e s . With t h i s s t r o n g e r c r i t e r i o n , the CON t e s t appears t o be passe d e a r l i e r than the OCT t e s t . TABLE 57 TEST FOR CONSERVATION OF NUMBER (CON) X ORDINAL CORRESPONDENCE TEST (OCT) ( A l l Ss I n c l u d e d ) CON FAIL PASS TOTAL OCT F a i l 70 9 79 Pass 6 9 15 T o t a l 76 18 94 I f a s t r i c t c r i t e r i o n ( t h r e e t r a n s f o r m a t i o n s c o r r e c t ) i s used f o r b o t h t e s t s , T a b l e s 57 and 58 r e s u l t . The e x p e c t e d f r e q u e n c i e s a r e too s m a l l f o r a c h i - s q u a r e a n a l y s i s . N o n e t h e l e s s , the p o s i t i v e r e l a t i o n -s h i p between performance on the two t e s t s a g a i n a p p e a r s . There a r e s t i l l e x c e p t i o n s t o the h y p o t h e s i s . Nine Ss who passed the CON t e s t f a i l e d t h e 83 OCT t e s t . S i x of t h o s e Ss who e i t h e r d i d not u n d e r s t a n d o r f a i l e d the CON t e s t , p a s s e d t h e OCT t e s t . Four o f those Ss who un d e r s t o o d the CON t e s t b u t f a i l e d i t , p a s s e d t h e OCT t e s t . TABLE 58 TEST FOR CONSERVATION OF NUMBER (CON) X ORDINAL CORRESPONDENCE. TEST (OCT) (Only Those Who Und e r s t o o d I n c l u d e d ) CON FAIL PASS TOTAL OCT F a i l . 70 9 79 Pass 6 9 15 T o t a l 76 18 94 H y p o t h e s i s '2. There w i l l be a p o s i t i v e r e l a t i o n -s h i p between the c h i l d ' s a b i l i t y t o match the number of o b j e c t s i n a s t a n d a r d row and h i s a b i l i t y t o b u i l d a s e r i e s i f the same m a t e r i a l s a r e used to t e s t f o r the two a b i l i t i e s . The r e l a t i o n s h i p between the c h i l d ' s a b i l i t y t o match a s t a n d -a r d row. and h i s a b i l i t y t o b u i l d a s e r i e s i s r e p o r t e d i n Table. 26 and i s s i g n i f i c a n t (x = 29.62, df = 3; p < .001). But 38 Ss were a b l e t o match a s t a n d a r d row and unable t o b u i l d a s e r i e s w h i l e o n l y two were a b l e t o b u i l d the s e r i e s b u t were u n a b l e t o match the s t a n d a r d row. Thus, b u i l d -84 i n g t h e s e r i e s was a much more d i f f i c u l t t a s k than matching the s t a n d a r d row and the h y p o t h e s i s i s not c o n f i r m e d . The s i g n i f i c a n t c h i - s q u a r e r e -s u l t e d from the r e l a t i o n s h i p between b u i l d i n g the s e r i e s and t h e much more d i f f i c u l t t a s k of matching the s t a n d a r d row w i t h o u t depending on a s p a t i a l one-to-one c o r r e s p o n d e n c e . H y p o t h e s i s 3: When.the c h i l d i s a b l e t o b u i l d one s e r i e s , he w i l l a l s o be a b l e t o b u i l d the o t h e r and t o p l a c e the two s e r i e s i n c o r r e s p o n d e n c e . T h ere were f i v e s u b j e c t s who d i d not p e r f o r m as p r e d i c t e d by the h y p o t h e s i s . In g e n e r a l , the h y p o t h e s i s was c o n f i r m e d . H y p o t h e s i s 4: There w i l l be a p o s i t i v e r e -l a t i o n s h i p between the a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s w i t h o u t depending on a s p a t i a l one—to—one co r r e s p o n d e n c e between the o b j e c t s i n the s e r i e s and the a b i l i t y to b u i l d the s e r i e s s y s t e m a t i c a l l y r a t h e r than by t r i a l and e r r o r . The r e l a t i o n s h i p between the c h i l d ' s a b i l i t y to b u i l d a s e r i e s s y s t e m a t i c a l l y and h i s a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s , r e p o r t e d i n T a b l e 28, i s s i g n i f i c a n t (x = 13.21, df = 2; p < .005), and the hypo-t h e s i s i s c o n f i r m e d . There a r e e x c e p t i o n s to the h y p o t h e s i s , however. Of t h e 42 S>s who pass e d t h e OCT t e s t by the weak c r i t e r i o n of two of the t h r e e t r a n s -f o r m a t i o n s c o r r e c t , 31 were u n a b l e t o b u i l d the s e r i e s s y s t e m a t i c a l l y . Of t h o s e 15 Ss who pass e d the OCT t e s t by the s t r o n g c r i t e r i o n , e i g h t were un-a b l e to b u i l d the s e r i e s s y s t e m a t i c a l l y . Of the 15 s u b j e c t s who b u i l t the s e r i e s s y s t e m a t i c a l l y , f o u r f a i l e d the OCT t e s t by the weak c r i t e r i o n and e i g h t f a i l e d i t by the s t r o n g c r i t e r i o n . 85 P i a g e t ' s w r i t i n g s [1952] l e a d one to exp e c t the a b i l i t y to b u i l d a s e r i e s to p r e c e d e t h e a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s . I f a weak, c r i t e r i o n f o r p a s s i n g OCT i s used, t h i s e x p e c t a t i o n was v i o l a t e d i n the p r e s e n t s t u d y , i n 14 o r o n e - t h i r d . of.' t h e 42 c a s e s . I f a s t r o n g c r i -t e r i o n i s used, i t was v i o l a t e d i n o n l y one c a s e . H y p o t h e s i s 5: The c h i l d who c o n s e r v e s w i l l be a b l e t o match the number of o b j e c t s i n a s t a n d a r d row w i t h o u t making a s p a t i a l one-to-one c o r r e s p o n d e n c e . The r e l a t i o n s h i p between the a b i l i t y t o co n s e r v e and the a b i l i t y t o match a s t a n d a r d row w i t h o u t depending on a s p a t i a l one-to-one c o r r e s -2 pondence i s r e p o r t e d i n T a b l e 29. The r e l a t i o n s h i p i s s i g n i f i c a n t (x = 35.65, df = 6; p < .001). The m a j o r i t y o f those who c o u l d c o n s e r v e (21 of 26) c o u l d a l s o match the s t a n d a r d row w i t h o u t depending on a s p a t i a l one-to-one c o r r e s p o n d e n c e . Thus, the h y p o t h e s i s i s c o n f i r m e d . The co n -v e r s e , however, does not h o l d . Twenty-three o f the 44 Ss who c o u l d match the s t a n d a r d row w i t h o u t depending on a s p a t i a l one—to-one cor r e s p o n d e n c e c o u l d not c o n s e r v e . These d a t a suggest t h a t the a b i l i t y t o p h y s i c a l l y c a r r y out a t r a n s f o r m a t i o n p r e c e d e s the a b i l i t y t o c o n s e r v e . H y p o t h e s i s 6: The c h i l d ' s a b i l i t y to co n s e r v e and to f i n d o r d i n a l c o r r e s p o n d e n c e s d e s p i t e the s p a t i a l arrangement o f the s e r i e s w i l l be p o s i t i v e l y r e l a t e d to (a) h i s a b i l i t y to a n t i c i p a t e and remember the r e -s u l t s of t r a n s f o r m a t i o n s i n the s p a t i a l arrangements of o b j e c t s i n a row; and a l s o t o (b) h i s a b i l i t y t o p e r f o r m m e n t a l t r a n s f o r m a t i o n s . The r e l a t i o n s h i p between the c h i l d ' s a b i l i t y t o c o n s e r v e and h i s a b i l i t y t o remember and a n t i c i p a t e the r e s u l t s of a t r a n s f o r m a t i o n , 86 as measured by the s t r o n g c r i t e r i o n on the Imagery T e s t (IM) i s shown i n T a b l e 30. The r e l a t i o n s h i p between the c h i l d ' s a b i l i t y t o c o n s e r v e and h i s performance on the IM t e s t , as measured by the weak c r i t e r i o n on t h e IM t e s t , i s shown i n T a b l e 31. The r e l a t i o n s h i p s are not s i g n i f i c a n t . T h i s i s c o n s i s t e n t w i t h f a i l u r e t o f i n d a G e f f e c t on the IM a n a l y s i s of v a r i a n c e when G was det e r m i n e d by a c o n s e r v a t i o n c r i t e r i o n . C e r t a i n -l y , t h e p r e s e n t d a t a do not s u p p o r t the f i r s t p o r t i o n of the h y p o t h e s i s as i t r e l a t e s to the c h i l d ' s a b i l i t y to c o n s e r v e . 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 the a b i l i t y to p e r f o r m s u c c e s s f u l l y on t h e OCT t e s t and t h e a b i l i t y t o p e r f o r m s u c c e s s f u l l y on th e IM t e s t , measured e i t h e r by the weak or s t r o n g c r i t e r i o n , i s r e p o r t e d i n T a b l e s 32 and 33 ( x 2 = 11.13, d f = 2; p < .005 and x 2 = 24.63, df = 4; p < .005, r e s p e c t i v e l y ) . These d a t a s u p p o r t the f i r s t p o r t i o n of the h y p o t h e s i s as i t r e l a t e s t o t h e a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s . The f a i l u r e t o f i n d a G e f f e c t on t h e IM a n a l y s i s of v a r i a n c e when G was dete r m i n e d by performance on the OCT t e s t was p r o b a b l y due to the i n c o n -s i s t e n c y o f Ss i n r e s p o n d i n g on the IM t e s t and t h e r e f o r e t o the l a r g e amount of e r r o r v a r i a n c e on the IM t e s t . There a r e e x c e p t i o n s , however, to the p o s i t i v e r e l a t i o n s h i p found between s u c c e s s f u l performance on t h e OCT t e s t and on t h e IM t e s t . A c c o r d i n g t o t h e d a t a r e p o r t e d i n T a b l e 33, f i v e , o r o n e - t h i r d of t h o s e 15 j>s who pas s e d the OCT t e s t by t h e s t r o n g c r i t e r i o n d i d n o t - p e r f o r m as e x p e c t e d , i n Group C on the IM t e s t , i . e . , they d i d not choose predom-i n a t e l y t h e c a r d w i t h t h e c o r r e c t number of o b j e c t s i n t h e t r a n s f o r m e d row; and 22 of the 43 Ss who pas s e d the OCT t e s t by the weak c r i t e r i o n d i d not 87 p e r f o r m , as e x p e c t e d , i n Group.C on t h e IM. t e s t . C o n v e r s e l y , of t h e 38 j>s who performed a t l e v e l C o f the LM t e s t and were t h e r e f o r e e x p e c t e d to pass t h e OCT t e s t , o n l y 10 pass e d t h e OCT t e s t by the s t r o n g c r i t e r i o n and o n l y 21 by the weak c r i t e r i o n . The d a t a i n T a b l e 33 r e v e a l t h a t a lmost h a l f (seven o f 15) of those who passe d the OCT t e s t by t h e s t r o n g c r i t e r i o n and ov e r h a l f (30 o f 53) of those who passe d the OCT t e s t by the weak c r i t e r i o n , f a i l e d t h e IM t e s t by the s t r o n g c r i t e r i o n . Converse-, l y , of those who passe d t h e IM t e s t by the s t r o n g c r i t e r i o n , a p p r o x i m a t e -l y o n e - t h i r d (seven o f 20) f a i l e d the OCT t e s t by the weak c r i t e r i o n w h i l e t h r e e - f i f t h s (12 of 20) f a i l e d by the s t r o n g c r i t e r i o n . A l l f o u r of T a b l e s 34 t h r o u g h 37 show s i g n i f i c a n t r e l a t i o n s h i p s between performance on the T r a n s f o r m a t i o n T e s t (TT) by e i t h e r c r i t e r i o n 2 2 and performance on b o t h CON and OCT (x = 21.87, df = 4; p < .001 and, x = 21.99, d f = 4; p < .001). These d a t a , t o g e t h e r w i t h the s i g n i f i c a n t G e f f e c t found i n the a n a l y s i s of v a r i a n c e on the TT t e s t b o t h when G was dete r m i n e d by performance on CON and when G was determined by p e r -formance on OCT, s u p p o r t the (b) p o r t i o n of the h y p o t h e s i s , t h a t t h e . a b i l -i t y t o p e r f o r m a me n t a l t r a n s f o r m a t i o n i s r e l a t e d t o Stage I I I b e h a v i o r . I f t h r e e t r a n s f o r m a t i o n s c o r r e c t i s c o n s i d e r e d t h e c r i t e r i o n f o r p a s s i n g OCT, i t appears from a l l f o u r t a b l e s t h a t the a b i l i t y to p e r f o r m a ment a l t r a n s f o r m a t i o n precedes and t h e r e f o r e p o s s i b l y mediates the a b i l i t y to c o n s e r v e and to. f i n d o r d i n a l c o r r e s p o n d e n c e s . CA h i g h e r p r o p o r t i o n o f s u b j e c t s f a i l CON b u t pass TT than f a i l TT but pass CON). A c c o r d i n g t o the d a t a i n T a b l e s 32 and 33, t h e a b i l i t y to a n t i c i p a t e and remember the r e s u l t s o f a t r a n s f o r m a t i o n a l s o p r e c e d e s and t h e r e f o r e p o s s i b l y mediates 88 the a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s . A l t h o u g h the c h i - s q u a r e a n a l y s i s on the d a t a i n T a b l e 31 d i d not r e v e a l a s i g n i f i c a n t r e l a t i o n -s h i p , i t does appear t h a t the a b i l i t y t o a n t i c i p a t e and remember the r e -s u l t s o f a t r a n s f o r m a t i o n a l s o p r e c e d e s and t h e r e f o r e p o s s i b l y mediates the a b i l i t y to c o n s e r v e . Thus, the d a t a i n T a b l e s 32 t h r o u g h 37 suggest t h a t the a b i l i t y to p e r f o r m a mental t r a n s f o r m a t i o n p r e c e d e s the a b i l i t y to c o n s e r v e and to f i n d o r d i n a l c o r r e s p o n d e n c e s . I t was co n c l u d e d above, from the p o s i t i v e r e l a t i o n s h i p between performance on the T e s t f o r I n t e n -s i v e Q u a n t i t y and performance on the T e s t f o r C o n s e r v a t i o n o f Number (Hypo*-t h e s i s 5 ) , t h a t the a b i l i t y t o p e r f o r m a physical t r a n s f o r m a t i o n p r e c e d e s the a b i l i t y to c o n s e r v e . The d a t a i n T a b l e s 38 and 39 i n d i c a t e t h a t the a b i l i t y to p e r f o r m a p h y s i c a l t r a n s f o r m a t i o n precedes the a b i l i t y t o men-t a l l y t r a n s f o r m a s t i m u l u s . A de v e l o p m e n t a l sequence i s sug g e s t e d : the c h i l d f i r s t l e a r n s how t o p h y s i c a l l y t r a n s f o r m a s t i m u l u s ; he then l e a r n s how t o m e n t a l l y t r a n s f o r m i t ; and he then l e a r n s to a p p l y t h a t t r a n s f o r m ^ a t i o n a b i l i t y to the c o n s e r v a t i o n t a s k (and to the o r d i n a l c o r r e s p o n d e n c e t a s k as a means of r e s o l v i n g the c o n f l i c t between l e n g t h and d e n s i t y cues. H y p o t h e s i s 7: For those c h i l d r e n i n Stage I , i . e . , t h o s e who a r e un a b l e to c o r r e c t l y match the number o f ob-^ j e c t s i n a s t a n d a r d row, the d i f f e r e n t i a l l e n g t h s of the two rows w i l l be the e f f e c t i v e s t i m u l u s on which q u a n t i t y judgments a r e based. F o r c h i l d r e n i n Stage I I , i . e . , those who can c o r r e c t l y match the number o f o b j e c t s i n a s t a n d a r d row but who cannot c o n s e r v e , the e f f e c t i v e s t i m u l u s f o r q u a n t i t y judgments w i l l be the d i f f e r e n c e between the num*-be r o f o b j e c t s i n the two rows. F o r c h i l d r e n i n Stage I I I , i . e . , those who can con s e r v e , the e f f e c t i v e s t i m u l u s f o r q u a n t i t y judgments w i l l be the d i f f e r e n c e between the num-ber of o b j e c t s i n the two rows. T h i s h y p o t h e s i s r e s u l t e d i n the f i v e p r e d i c t i o n s shown i n T a b l e 2. These p r e d i c t i o n s , e x c e p t f o r p r e d i c t i o n 3, were c o n f i r m e d ( T a b l e s 89 17 and 19 I n the R e s u l t s e c t i o n ) , and t h e h y p o t h e s i s i s c o n f i r m e d . R e l a t i o n s h i p Between CT and CON, NQ, and OCT In T a b l e s 40 t h r o u g h 42, Appendix B, a r e shown the r e l a t i o n s h i p of C o u n t i n g A b i l i t y (CT) to performance on the T e s t f o r C o n s e r v a t i o n of Number (CON), the T e s t f o r I n t e n s i v e Q u a n t i t y (NQ) and the O r d i n a l C o r r e s -pondence T e s t (OCT). The r e l a t i o n s h i p between CT and CON ( T a b l e 40) was s i g n i f i c a n t ( x 2 = 12.25, df = 2; p < .005). N o n e t h e l e s s , 44 of the 67 Ss who c o u l d count c o u l d not c o n s e r v e . The r e l a t i o n s h i p between NQ and CT 2 ( T a b l e 40) was a l s o s i g n i f i c a n t (x = 18.80, df = 3; p <.001). But t h e r e were many S_s (28 of 67) who c o u l d count but c o u l d not match a s t a n d a r d row w i t h o u t depending on a s p a t i a l one-to-one c o r r e s p o n d e n c e . There was no s i g n i f i c a n t r e l a t i o n s h i p between CT and OCT ( T a b l e 41). Of the 67 Ss who c o u l d count, 32 f a i l e d the OCT t e s t by the weak pass c r i t e r i o n . From th e s e d a t a i t appears t h a t c o u n t i n g a b i l i t y g e n e r a l l y p r e c e d e s the a b i l -i t i e s t e s t e d by the CON, NQ, and OCT t e s t s . DISCUSSION M e t h o d o l o g i c a l I s s u e s The f i r s t m e t h o d o l o g i c a l i s s u e r a i s e d i n the, I n t r o d u c t i o n was t h e problem o f d e t e r m i n i n g the i n t e n s i o n of the concept the c h i l d uses t o c o n s e r v e . I t was hoped t h a t the p r e s e n t s t u d y would f i n d a s t r o n g r e l a t i o n s h i p between c o n s e r v a t i o n a b i l i t y and the a b i l i t y t o p e r f o r m on the o t h e r t a s k s used. I f s u c h a r e l a t i o n s h i p had been found, then the o t h e r t a s k s c o u l d be used a f t e r t r a i n i n g s e s s i o n s on c o n s e r v a t i o n of number t o determine i f the t r a i n e d - i n i n t e n s i o n i s the same as the i n t e n s i o n used by " n o r m a l l y " c o n s e r v i n g c h i l d r e n . To meet t h i s o b j e c t i v e , a r e l a t i o n s h i p - e q u i v a l e n t to a c o r r e l a t i o n of .90 o r b e t t e r would be needed. U n f o r t u n a t e l y , because such a s t r o n g r e l a t i o n s h i p was n o t found, and because t h e r e were so many Ss who performed w e l l on the T e s t f o r C o n s e r v a t i o n of Number but p o o r l y on the o t h e r t e s t s o r v i c e v e r s a , t h e t e s t s i n the p r e s e n t s t u d y cannot be used to determine i f th e i n t e n s i o n used by c h i l d r e n t o c o n s e r v e a f t e r t r a i n i n g i s the same as t h a t used by " n o r m a l l y " c o n s e r v i n g c h i l d r e n . N e v e r t h e l e s s , i n any group of Ss t r a i n e d i n c o n s e r v a t i o n , r e l a t i o n s h i p s o f the magnitude found i n t h i s s t u d y between c o n s e r v a t i o n and t h e t e s t s used i n t h i s s t u d y ought t o appear. The second m e t h o d o l o g i c a l i s s u e r a i s e d was whether or not sub-j e c t s i n c o n s e r v a t i o n experiments u n d e r s t a n d the terms used i n the t a s k . I t was hoped t h a t by u s i n g the terms i n s i t u a t i o n s which the c h i l d u nder-s t o o d , the c h i l d ' s u n d e r s t a n d i n g of the terms c o u l d be v e r i f i e d . T h i s 90 91 t e c h n i q u e was s u c c e s s f u l i n t h e S e r i a t i o n T e s t , the O r d i n a l C o r r e s p o n d -ence T e s t , and the T e s t f o r I n t e n s i v e Q u a n t i t y . I t was not s u c c e s s f u l on t h e T e s t f o r C o n s e r v a t i o n of Number f o r two r e a s o n s . F i r s t , 34 o f the 94 Ss would n o t a s s e r t the e q u a l i t y of two rows of t h r e e s t i c k s e a c h even when they were l i n e d up i n one-to-one c o r r e s -pondence. T h i s c a s t s some doubt on the v a l i d i t y o f the normal c o n s e r -v a t i o n p r o c e d u r e , e s p e c i a l l y when f o u r - y e a r - o l d s a r e s u b j e c t s . U s u a l l y E_ g e t s S_sto agree t o the i n i t i a l e q u a l i t y of number o f o b j e c t s i n the two rows. In l i g h t o f the p r e s e n t f i n d i n g s , the s u c c e s s o f such a p r o -cedure i s d o u b t f u l . I f i t i s s u c c e s s f u l , t h e r e i s always the p o s s i b i l -i t y t h a t S_ has merely d e c i d e d E_ wants him to say "same." What i s to p r e -v e n t t h a t r e sponse t o E_'s wishes from g e n e r a l i z i n g t o the q u e s t i o n a f t e r the t r a n s f o r m a t i o n ? On the o t h e r hand, i f the c h i l d i s not c o n v i n c e d o f the e q u a l i t y of t h e rows b e f o r e the t r a n s f o r m a t i o n , i t i s i m p o s s i b l e t o judge whether he conserves or n o t . T h i s c r e a t e s a s e v e r e m e t h o d o l o g i c a l problem f o r c o n s e r v a t i o n s t u d i e s , a problem f o r which the author has no s o l u t i o n . When asked t o judge which of the two rows has more i n the Density-Length-Number T e s t , t h e s e 34 c h i l d r e n e i t h e r judge by l e n g t h or l e n g t h and d e n s i t y combined. ( T h i s i s shown by t h e i r performance on the Density-Length-Number T e s t ) . E i t h e r one of th e s e c r i t e r i a s h o u l d l e a d t o a c o r r e c t judgment o f e q u a l i t y on P a r t A ( l ) o f the T e s t f o r Conser-v a t i o n o f Number. That i t does not s u p p o r t s Smedslund's [1966b] hypo-t h e s i s t h a t a judgment of e q u a l i t y i s an i n f e r e n c e based on l o g i c a l o p e r -a t i o n s w h i l e i n e q u a l i t y judgments (which a r e r e q u i r e d i n the D/L/N t e s t ) can be based on p e r c e p t u a l c r i t e r i o n . When l e n g t h or d e n s i t y c r i t e r i a cannot l e a d t o a judgment of "more" s a t i s f a c t o r y t o the c h i l d and when.the 92 c h i l d does not c o n s e r v e , he appears to choose one row as h a v i n g more o b j e c t s e i t h e r by g u e s s i n g o r on some i d i o s y n c r a t i c b a s i s , e.g., c o l o r . S e c o n d l y , t h e r e were o n l y seven Ss who f a i l e d t o c o n s e r v e when e i g h t items were used but who c o n s e r v e d when t h r e e items were used. T h i s f i n d i n g d i f f e r s from Greco's [1960, c i t e d i n F l a v e l l , 1963] f i n d i n g t h a t numbers l e s s t h a n f i v e a r e u n d e r s t o o d e a r l i e r t h a n numbers g r e a t e r than f i v e . F o r t h e s e two r e a s o n s , t h e attempt i n t h i s s t u d y t o a s s e s s the c h i l d ' s u n d e r s t a n d i n g of the q u e s t i o n s by u s i n g them i n s i t u a t i o n s i n w hich the c h i l d performs w e l l f a i l e d . The t h i r d m e t h o d o l o g i c a l problem r a i s e d was the e f f e c t of s t i m -u l u s v a r i a b l e s on performance on P i a g e t i a n t a s k s . Data from the p r e s e n t s t u d y i n d i c a t e t h a t the k i n d of t r a n s f o r m a t i o n used i n the O r d i n a l C o r -respondence T e s t do a f f e c t performance. The number of the s t i c k w h i c h the c h i l d i s asked to match i n the OCT t a s k a l s o a f f e c t s h i s performance. I f s t i m u l u s v a r i a b l e s a f f e c t performance as shown by the p r e s e n t s t u d y and by U z g i r i s [1964], i t would be u s e f u l t o d i s c o v e r what s t i m u l u s di-mensions a f f e c t performance. Then any t a s k c o u l d be p l a c e d on t h a t dimen-s i o n and one c o u l d equate d i f f e r e n t t a s k s (e.g., c o n s e r v a t i o n and o r d i -n a l c o r r e s p o n d e n c e t a s k s ) on t h e s e s t i m u l u s d i m e n s i o n s , a l l o w i n g t h e " h i d d e n " v a r i a b l e of mental s t r u c t u r e to emerge, i f i t e x i s t e d . F o r example, i f p a s s i n g Task -A and B i n d i c a t e s the p r e s e n c e of "the same mental s t r u c t u r e , a c c o r d i n g to P i a g e t i a n t h e o r y , and p a s s i n g Task C i n d i c a t e s t h e p r e s e n c e of a d i f f e r e n t , more de v e l o p e d , s t r u c t u r e , and i f the s t i m u l u s v a r i a b l e s a r e h e l d c o n s t a n t a c r o s s t a s k s , P i a g e t would 93 p r e d i c t t h a t Ss would tend to always p a s s A i f they p a s s e d B and v i c e v e r s a , but would not always pass C i f they p a s s e d A and B. The p r e s e n t s t u d y , however, does not suggest the dimensions of s t i m u l u s v a r i a b l e s t h a t would have to be h e l d c o n s t a n t . I t may be d i f f i c u l t to f i n d t h e s e dimensions because many t a s k s s i m p l y may not i n v o l v e t h e same di m e n s i o n s . I t may even be d i f f i c u l t t o f i n d s t i m u l u s v a r i a b l e s t h a t c o n s i s t e n t l y a f f e c t performance because of i n d i v i d u a l d i f f e r e n c e s a c r o s s Ss. D i s c u s s i o n of Hypotheses There were t h r e e purposes w h i c h t h i s s t u d y was d e s i g n e d to f u l -f i l l . The f i r s t purpose was to t e s t P i a g e t ' s t a s k a n a l y s i s of the c o n -s e r v a t i o n o f number and o r d i n a l c o r r e s p o n d e n c e t a s k s . A c c o r d i n g to t h i s a n a l y s i s , the a b i l i t y t o p e r f o r m s u c c e s s f u l l y on b o t h t a s k s depends on the a b i l i t y t o m e n t a l l y t r a n s f o r m the two rows i n the c o n s e r v a t i o n of number t a s k and the two s e r i e s i n t h e o r d i n a l c o r r e s p o n d e n c e t a s k by u n i t s . A u n i t was d e f i n e d as an o b j e c t t h a t can be grouped t o g e t h e r w i t h o t h e r o b j e c t s . T h i s t a s k a n a l y s i s l e d to H y p o t h e s i s 1: There w i l l be a p o s i t i v e r e l a t i o n s h i p between the c h i l d ' s a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s r e g a r d l e s s of t h e s p a t i a l arrangement of the s e r i e s and h i s a b i l i t y to c o n s e r v e , i f the same m a t e r i a l s a r e used to t e s t f o r the two a b i l i t i t e s . T h i s h y p o t h e s i s was'confirmed. There were a number of e x c e p t i o n s to t h e h y p o t h e s i s , however. I t i s t h e a u t h o r ' s o p i n i o n , from o b s e r v i n g the S s , t h a t a c r i t e r i o n o f t h r e e c o r r e c t f o r the OCT t e s t does not p r o v i d e enough a l l o w a n c e f o r l a p s e s i n a t t e n t i o n and m o t i v a t i o n , and more a t t e n t i o n and m o t i v a t i o n a r e r e q u i r e d i n the OCT t e s t than i n the CON t e s t . However, a c r i t e r i o n of two of t h r e e t r a n s f o r m -94 a t i o n s c o r r e c t i s too l e n i e n t ; many of the Ss. appeared to narrow down t h e i r c h o i c e on a p e r c e p t u a l b a s i s Ce.g., to t h e m i d d l e p o s i t i o n s o f the s e r i e s ) w h i c h g r e a t l y i n c r e a s e d t h e i r chance o f g u e s s i n g c o r r e c t l y . Thus, i t i s t h e a u t h o r ' s o p i n i o n t h a t some of the Ss who a t t a i n e d two of the t h r e e t r a n s f o r m a t i o n s c o r r e c t d i d so by o p e r a t i o n a l means and o t h e r s d i d so by a comb i n a t i o n of p e r c e p t u a l judgment and g u e s s i n g . A p p r o x i m a t e l y o n e - t h i r d of the Ss who answered i n c o r r e c t l y d i d not merely p o i n t t o the s t i c k o p p o s i t e the one p o i n t e d t o by E_. These d a t a i n d i c a t e t h a t the su b -j e c t s were a t t e m p t i n g ( u n s u c c e s s f u l l y ) t o f i n d the c o r r e c t s t i c k by o t h e r means and t h e o t h e r means c o u l d have been a c o m b i n a t i o n o f p e r c e p t u a l judgment and g u e s s i n g . F o r example, a c c o r d i n g t o T a b l e 25, Appendix C, t h e r e a r e 10 Ss who c o n s e r v e d and answered c o r r e c t l y on two of the t h r e e t r a n s f o r m a t i o n s on the O r d i n a l Correspondence T e s t and 18 Ss who d i d not cons e r v e and answered c o r r e c t l y on OCT. P o s s i b l y , the 10 Ss who con s e r v e d answered c o r r e c t l y on the two of t h r e e t r a n s f o r m a t i o n s by u s i n g o p e r a t i o n -a l c oncepts w h i l e the 18 Ss who d i d not con s e r v e answered c o r r e c t l y on the two t r a n s f o r m a t i o n s by u s i n g p e r c e p t u a l c r i t e r i o n combined w i t h g u e s s i n g . A new p r o c e d u r e must be d e v i s e d i n which a l o n g e r s e r i e s i s used, e.g., one o f 11 s t i c k s , e n a b l i n g a g r e a t e r number of OCT t r i a l s t o d i s t i n g u i s h between Ss who use o p e r a t i o n a l means and t h o s e who use p e r c e p t u a l c r i -t e r i o n combined w i t h g u e s s i n g . When i t i s , t h e number of e x c e p t i o n s to the h y p o t h e s i s might be d e c r e a s e d . The second p u r p o s e o f t h e s t u d y was to attempt to r e p l i c a t e P i a e t ' s d e s c r i p t i o n o f the development toward c o n s e r v a t i o n i n the p r e -o p e r a t i o n a l p e r i o d . A c c o r d i n g t o t h i s d e s c r i p t i o n , the t r a n s i t i o n from Stage I t o Stage I I , i . e . , the emergence o f the a b i l i t y t o b u i l d a s e r i e s and of the a b i l i t y t o c o r r e c t l y match the number of o b j e c t s i n a s t a n d a r d 95 row (by depending on a s p a t i a l c o r r e s p o n d e n c e ) , depends on the a b i l i t y t o c o - o r d i n a t e r e l a t i o n s . T h i s l e d us t o H y p o t h e s i s . 2 : There w i l l be a p o s i t i v e r e l a t i o n s h i p between t h e c h i l d ' s a b i l i t y to match the number of o b j e c t s i n a s t a n d a r d row and h i s a b i l i t y t o b u i l d a s e r i e s i f t h e same m a t e r i a l s a r e used to t e s t f o r t h e two a b i l i t i e s . The d a t a do not c o n f i r m t h i s h y p o t h e s i s ; matching the number o f o b j e c t s i n a s t a n d a r d row i s by f a r t h e e a s i e r of t h e two t a s k s . T h ere are t h r e e p o s s i b l e ex-p l a n a t i o n s o f t h i s f i n d i n g . F i r s t , c h i l d r e n may be more f a m i l i a r w i t h matching the number o f o b j e c t s i n a s t a n d a r d row than t h e y a r e w i t h b u i l d i n g two s e r i e s and making c o r r e s p o n d e n c e s between them. S e c o n d l y , P i a g e t [1962] proposes t h a t c h i l d r e n i n Stage I I depend upon s t a t i c images i n b u i l d i n g s e r i e s and i n matching s t a n d a r d rows. The d e v e l o p -ment of an a c c u r a t e image o f two c o r r e s p o n d i n g s e r i e s based on s i z e r e -l a t i o n s does seem more d i f f i c u l t than the development of an a c c u r a t e image of two rows ba s e d o n l y on the s p a t i a l r e l a t i o n s of l e n g t h and den-s i t y . The t h i r d p o s s i b l e e x p l a n a t i o n i s t h a t the p r o c e d u r e d i f f e r e d a c r o s s the two t a s k s . To determine whether or n o t the c h i l d could.match the s t a n d a r d row, t h e e x p e r i m e n t e r showed him a standard.row to copy. But to determine whether o r not the c h i l d c o u l d b u i l d the two s e r i e s , the e x p e r i m e n t e r d i d not g i v e t h e c h i l d a model; the e x p e r i m e n t e r o n l y de-. s c r i b e d t o the c h i l d what he was supposed to do. The c h i l d r e n u n d e r s t o o d the v e r b a l i n s t r u c t i o n s , as demonstrated by t h e i r a b i l i t y t o f o l l o w them when the two s e r i e s o n l y c o n t a i n e d t h r e e o b j e c t s each. N o n e t h e l e s s , t h e absence of a model when the c h i l d was asked to b u i l d t h e s e r i e s c o u l d a c c o u n t f o r t h e f i n d i n g t h a t b u i l d i n g t h e s e r i e s i s more d i f f i c u l t t h a n m a t c h i n g t h e s t a n d a r d row. U n f o r t u n a t e l y , the absence of a model In the 96 S e r i a t i o n T e s t was n e c e s s a r y to t e s t H y p o t h e s i s 3. P i a g e t ' s e x p l a n a t i o n of t h e t r a n s i t i o n from Stage I to Stage I I a l s o l e a d s to H y p o t h e s i s 3: When., the c h i l d i s a b l e to b u i l d one s e r i e s , he w i l l a l s o be a b l e to b u i l d a second s e r i e s and to p l a c e t h e two s e r i e s i n c o r r e s p o n d e n c e . T h i s h y p o t h e s i s was c o n f i r m e d by the d a t a . P i a g e t says the a b i l i t y to c o - o r d i n a t e r e l a t i o n s e n a b l e s the t r a n s i t i o n from Stage I to Stage I I . b e h a v i o r and e n a b l e s the c h i l d b o t h t o b u i l d t h e s i m p l e s e r i e s and to b u i l d the c o r r e s p o n d i n g s e r i e s . The f i n d i n g t h a t b u i l d i n g two c o r r e s p o n d i n g s e r i e s i s not more d i f f i c u l t t h a n b u i l d i n g one s e r i e s s u p p o r t s t h i s t h e o r y . P i a g e t [1952] o b s e r v e d t h a t when c h i l d r e n c o u l d f i n d o r d i n a l c o r r e s p o n d e n c e s even when the two s e r i e s were d i s a r r a n g e d they u s u a l l y b u i l t t h e s e r i e s s y s t e m a t i c a l l y . T h i s l e d to H y p o t h e s i s 4: There w i l l be a p o s i t i v e r e l a t i o n s h i p between the a b i l i t y t o f i n d o r d i n a l c o r r e s -pondences w i t h o u t depending on a s p a t i a l one-to-one cor r e s p o n d e n c e be-tween the o b j e c t s i n the s e r i e s and the a b i l i t y t o b u i l d t h e s e r i e s s y s t e m a t i c a l l y r a t h e r than by t r i a l and e r r o r . The h y p o t h e s i s .was c o n -f i r m e d by the d a t a . There were a number of e x c e p t i o n s which c o u l d be reduced i n e i t h e r o f two ways. The f i r s t way would be to d e v i s e an o r d i -n a l c o r r e s p o n d e n c e t e s t , as d i s c u s s e d above, which would b e t t e r d i s c r i -m i n a t e between Ss who answered c o r r e c t l y on the OCT by a c o m b i n a t i o n of p e r c e p t u a l judgments and g u e s s i n g and t h o s e who answered c o r r e c t l y b e-cause they were a t Stage I I I , i . e . , c o u l d c o n s t r u c t t h e t r a n s f o r m a t i o n of u n i t s . The number of e x c e p t i o n s to H y p o t h e s i s 3 c o u l d a l s o be r e -duced by t h e use of a p r o c e d u r e which communicated to the c h i l d t h a t he 97 was to.'ib.utld t h e s e r i e s s y s t e m a t i c a l l y ' i . i f he c o u l d — most of the e x c e p -t i o n s to t h e h y p o t h e s i s were Ss who p a s s e d the OCT but d i d not b u i l d the s e r i e s s y s t e m a t i c a l l y . G i v e n t h e d i f f i c u l t i e s i n the p r o c e d u r e of the two t e s t s , b o t h of which would work a g a i n s t the h y p o t h e s i s , the s i g n i f i -cant r e l a t i o n s h i p found between the a b i l i t y to f i n d o r d i n a l c o r r e s p o n d -ence- :and t o b u i l d a s e r i e s s y s t e m a t i c a l l y i s e v i d e n c e i n f a v o r of Hypo-t h e s i s 4. In a d d i t i o n t o r e p l i c a t i n g P i a g e t ' s o b s e r v a t i o n , the c o n f i r m a -t i o n o f H y p o t h e s i s 4 a l s o s u p p o r t s P i a g e t ' s t a s k a n a l y s i s . I t w i l l be r e c a l l e d t h a t a c c o r d i n g to the t a s k a n a l y s i s o f the o r d i n a l c o r r e s p o n d -ence task,' t h e a b i l i t y w hich a l l o w s s u c c e s s f u l performance i s the a b i l -i t y to u n d e r s t a n d the concept o f a u n i t , d e f i n e d as an o b j e c t t h a t can be s i m u l t a n e o u s l y s e p a r a t e d from o t h e r o b j e c t s and grouped t o g e t h e r w i t h o t h e r o b j e c t s . B u i l d i n g a s e r i e s s y s t e m a t i c a l l y , i . e . , i s o l a t i n g the l a r g e s t (or s m a l l e s t ) o f the r e m a i n i n g s t i c k s , a l s o appears to demand an u n d e r s t a n d i n g o f t h e u n i t . The s u b j e c t must i s o l a t e the l a r g e s t s t i c k and y e t , t o d i s c o v e r which i s the l a r g e s t , he must c o n s i d e r the s t i c k a member of the group of r e m a i n i n g s t i c k s . The r e l a t i o n s h i p b e -tween the a b i l i t y to f i n d o r d i n a l c o r r e s p o n d e n c e s and to b u i l d a s e r i e s s y s t e m a t i c a l l y c o u l d be due to t h e f a c t t h a t b o t h depend upon an u n d e r -s t a n d i n g o f number. The f i n a l h y p o t h e s i s i s d e r i v e d from P i a g e t ' s d e s c r i p t i o n o f the development of c o n s e r v a t i o n of number a b i l i t y i s H y p o t h e s i s 5: The c h i l d who conserves w i l l be a b l e to match the number of o b j e c t s i n a s t a n d a r d row w i t h o u t making a s p a i t a l one-to-one c o r r e s p o n d e n c e . The h y p o t h e s i s 98 was c o n f i r m e d . A c c o r d i n g t o t h e d a t a , matching the number of o b j e c t s i n a s t a n d a r d row w i t h o u t making a s p a t i a l c o rrespondence i s e a s i e r than c o n s e r v i n g . T h i s s u g g e s t s t h a t the a b i l i t y t o p h y s i c a l l y c a r r y out a t r a n s f o r m a t i o n p r e c e d e s the a b i l i t y to co n s e r v e , a s u g g e s t i o n t h a t w i l l be f u r t h e r e l a b o r a t e d below. The t h i r d purpose o f the p r e s e n t s t u d y was to t e s t h ypotheses d e r i v e d from P i a g e t ' s [1964, 1962] e x p l a n a t i o n of the t r a n s i t i o n from Stage I I to Stage I I I . A c c o r d i n g t o h i s e x p l a n a t i o n , the t r a n s i t i o n depends on the c h i l d ' s a b i l i t y to m e n t a l l y t r a n s f o r m a s t i m u l u s . The m o t i v a t i o n t o a p p l y t h i s m ental t r a n s f o r m a t i o n a b i l i t y to the c o n s e r -v a t i o n t a s k a r i s e s i n the c o n f l i c t between the p e r c e p t u a l cues o f den-s i t y and l e n g t h . The f i r s t h y p o t h e s i s to be d e r i v e d from t h i s t h e o r y was H y p o t h e s i s 6: The c h i l d ' s a b i l i t y t o c o n s e r v e and t o f i n d o r d i n a l c o r r e s -pondence d e s p i t e the s p a t i a l arrangement o f the s e r i e s w i l l be p o s i t i v e l y r e l a t e d t o (a) h i s a b i l i t y t o a n t i c i p a t e and remember the r e s u l t s of t r a n s f o r m a t i o n s i n the s p a t i a l arrangements o f o b j e c t s i n a row; and a l s o to (b) h i s a b i l i t y to p e r f o r m mental t r a n s f o r m a t i o n s . P a r t (b) o f t h i s h y p o t h e s i s was c o n f i r m e d by the d a t a . P a r t (a) was o n l y p a r t i a l l y c o n f i r m e d ; the a b i l i t y t o a n t i c i p a t e and remember the r e s u l t s o f a t r a n s -f o r m a t i o n , t e s t e d by the Imagery T e s t (IM), was r e l a t e d t o t h e a b i l i t y t o f i n d o r d i n a l c o r r e s p o n d e n c e s b u t not to t h e a b i l i t y t o c o n s e r v e . Because the O r d i n a l Correspondence T e s t d i d not d i s c r i m i n a t e w e l l between o p e r a t i o n -a l and p r e o p e r a t i o n a l Ss, and because the Ss d i d not p e r f o r m c o n s i s t e n t l y on t h e IM t e s t , t h e s i g n i f i c a n t r e l a t i o n s h i p found between performance on the IM t e s t and on the OCT t e s t i s d i f f i c u l t t o i n t e r p r e t . The f i n d i n g 99 i s e s p e c i a l l y d i f f i c u l t t o i n t e r p r e t . b e c a u s e w h i l e IM was s i g n i f i c a n t l y r e l a t e d t o OCT, and OCT was s i g n i f i c a n t l y r e l a t e d to CON, IM i s not s i g -n i f i c a n t l y r e l a t e d to OCT. The most, l i k e l y e x p l a n a t i o n f o r t h e se f i n d -i n g s i s t h a t two f a c t o r s c o n t r i b u t e to performance on OCT and IM. and o n l y one t o performance on CON. S u c c e s s f u l performance on a l l . t h r e e t e s t s de-pends on an u n d e r s t a n d i n g of number, which i s the f i r s t f a c t o r . But c h o o s i n g which o f t h r e e c a r d s i s c o r r e c t , as demanded i n the IM t e s t and c h o o s i n g which of e i g h t s t i c k s i s c o r r e c t , as demanded by the OCT t e s t , r e q u i r e s more d i l i g e n c e on the p a r t of the s u b j e c t than merely answering "same" on the CON t e s t . Thus, s u c c e s s f u l performance on OCT and IM, but not on CON depend on h i g h m o t i v a t i o n and a t t e n t i o n , w h i c h i s the second f a c t o r . I f t h i s a n a l y s i s i s c o r r e c t , the r e l a t i o n s h i p be-tween CON and OCT depends on the common f a c t o r of u n d e r s t a n d i n g number. But the IM t e s t d i d not t e s t t h i s f a c t o r , . a s demonstrated by £>s1 i n c o n -s i s t e n t r e s p o n s e s , and thus t h e r e was no r e l a t i o n s h i p between CON and IM. The o n l y f a c t o r a v a i l a b l e to produce a r e l a t i o n s h i p between IM and OCT i s the m o t i v a t i o n a l f a c t o r . G i v e n the i n c o n s i s t e n c y of Ss on the IM . t e s t , t h i s i s a p l a u s i b l e e x p l a n a t i o n . Thus, the (a) p o r t i o n of the h y p o t h e s i s can be n e i t h e r c o n f i r m e d nor r e j e c t e d because the IM t e s t f a i l e d t o t e s t the c h i l d ' s u n d e r s t a n d i n g . o f number. The second h y p o t h e s i s t o be d e r i v e d from P i a g e t ' s e x p l a n a t i o n of the t r a n s i t i o n from Stage L I to Stage I I I i s H y p o t h e s i s . 7 . ; For, those c h i l d r e n i n Stage I , i . e . , t h o s e who.are u n a b l e to c o r r e c t l y match the number of o b j e c t s i n a s t a n d a r d row, t h e d i f f e r e n t i a l l e n g t h s of the two rows w i l l be the e f f e c t i v e s t i m u l u s on w hich q u a n t i t y judgments are based. 100 For c h i l d r e n i n Stage I I , i . e . , those who can c o r r e c t l y match the number of o b j e c t s i n a s t a n d a r d row,but who cannot c o n s e r v e , the e f f e c t i v e s t i m -u l u s w i l l be the d e n s i t y d i f f e r e n t i a l minus the l e n g t h d i f f e r e n t i a l o f the two rows. F o r c h i l d r e n i n Stage I I I , i . e . , those who can c o n s e r v e , the e f f e c t i v e s t i m u l u s f o r q u a n t i t y judgments w i l l be the d i f f e r e n c e between the number of o b j e c t s i n the two rows. Much p r e v i o u s r e s e a r c h on c o n s e r v a t i o n of number has been based on the assumption t h a t c h i l d r e n i n the c o n s e r v a t i o n t a s k e i t h e r do not co n s e r v e and base t h e i r judgments^of q u a n t i t y on the l e n g t h o f the row or they c o n s e r v e , and base t h e i r q u a n t i t y judgments on number. The p r e -sent study was d e s i g n e d to demonstrate t h a t c h i l d r e n pass through two s t a g e s b e f o r e they a r e a b l e to c o n s e r v e . D u r i n g the f i r s t , the c h i l d bases h i s q u a n t i t y judgments on l e n g t h . I n the second s t a g e he n o t i c e s b o t h l e n g t h and d e n s i t y which e v e n t u a l l y l e a d s to a c o n f l i c t , a c o n f l i c t r e s o l v e d i n Stage I I I by the c o n s t r u c t i o n of the t r a n s f o r m a t i o n o f u n i t s and the consequent judgment of q u a n t i t y on the b a s i s of number. The p r e -d i c t i o n s shown i n T a b l e 2, except f o r p r e d i c t i o n 3, were c o n f i r m e d ( T a b l e s 17 and 19 i n the R e s u l t s e c t i o n ) . P r e d i c t i o n 3 was t h a t f o r Stage I I Ss, performance on Group 1 and 2 c a r d s would be i n f e r i o r t o performance on Group 3 c a r d s . The d i f f e r e n c e was i n the expe c t e d d i r e c t i o n and approached s i g n i f i c a n c e w i t h p < .11. Thus, the p l a n n e d comparisons t e s t s on d a t a from the D/L/N t e s t demonstrated the e x i s t e n c e o f t h r e e s t a g e s — one i n which the e f f e c t i v e s t i m u l u s i s l e n g t h , another i n which i t i s d e n s i t y and 101 l e n g t h , and a t h i r d i n w h i c h i t i s number., thus c o n f i r m i n g t h e h y p o t h e s i s . Z i m i l e s ' 11966] h y p o t h e s i s t h a t t h e . c h i l d a t t e n d s .to n u m e r o s i t y b e f o r e he c o n s e r v e s i s not s u p p o r t e d by t h e s e d a t a . These d a t a show t h a t Stage I I c h i l d r e n a t t e n d t o b o t h l e n g t h and d e n s i t y dimensions thus s u p p o r t i n g P i a g e t ' s 11952] h y p o t h e s i s t h a t a c o n f l i c t between p e r c e p t u a l cues l e a d s t o c o n s e r v a t i o n . I t was c o n c l u d e d from the r e l a t i o n s h i p between C o u n t i n g A b i l i t y and Performance on the T e s t f o r C o n s e r v a t i o n o f Number (CON), the T e s t f o r I n t e n s i v e Q u a n t i t y (NQ), and the O r d i n a l Correspondence T e s t (OCT), t h a t c o u n t i n g a b i l i t y p recedes the a b i l i t i e s t e s t e d by CON, NQ and OCT. T h i s i s p r o b a b l y because i n our c u l t u r e , c h i l d r e n are b e i n g c o n s t a n t l y asked t o count t h i n g s . The d a t a show t h a t c o u n t i n g a b i l i t y i s not s u f f i -c i e n t t o produce s u c c e s s f u l performance on the CON, NQ, and OCT t e s t s . Thus, performance on the t a s k s i n t h i s s t u d y cannot be f u l l y accounted f o r by jSs c o u n t i n g a b i l i t y . A Developmental Sequence The r e s u l t s suggest a new h y p o t h e s i s c o n c e r n i n g the development of number c o n s e r v a t i o n a b i l i t y . The c h i l d f i r s t l e a r n s , through, e x p e r -i e n c e , t o t a k e b o t h l e n g t h and d e n s i t y i n t o account i n h i s judgments o f q u a n t i t y . T h i s e n a b l e s him t o match the number of o b j e c t s i n a s t a n d a r d row by c o - o r d i n a t i n g the s p a t i a l . r e l a t i o n s o f l e n g t h and d e n s i t y . (The a b i l i t y t o c o - o r d i n a t e s i z e r e l a t i o n s i s more d i f f i c u l t and o c c u r s l a t e r i n d evelopment). Once the c h i l d , i s a b l e t o match t h e number of o b j e c t s i n a s t a n d a r d row by depending on the s p a t i a l one-to-one c o r r e s p o n d e n c e , 102 he e v e n t u a l l y becomes c a p a b l e , through, e x p e r i e n c e , of p e r f o r m i n g a physi-r c a l t r a n s f o r m a t i o n , i . e . , he can match the s t a n d a r d row w i t h o u t depending on a s p a t i a l one-to-one c o r r e s p o n d e n c e . E x p e r i e n c e i n p e r f o r m i n g p h y s i -c a l t r a n s f o r m a t i o n s on rows o f o b j e c t s , combined w i t h the a b i l i t y t o ment-a l l y r e p r e s e n t o b j e c t s which emerged at the end of the s e n s o r i m o t o r p e r -i o d [ P i a g e t , 1962], e n a b l e s t h e c h i l d to m e n t a l l y t r a n s f o r m rows of ob-j e c t s by u n i t s . F i n a l l y , t h e c o n f l i c t between l e n g t h and d e n s i t y c r i -t e r i a l e a d t h e c h i l d t o a p p l y h i s a b i l i t y to m e n t a l l y t r a n s f o r m rows of o b j e c t s by u n i t s to the c o n s e r v a t i o n t a s k . ( N o t i c e t h a t a c c o r d i n g to t h i s h y p o t h e s i s c o n f l i c t between p e r c e p t u a l cues i s not s u f f i c i e n t t o i n d u c e o p e r a t i o n a l thought as i t i s i n Gruen's, 1966, and Smedslund's, 1961a, c o n f l i c t h y p o t h e s e s . C o n f l i c t o n l y p r o v i d e s the m o t i v a t i o n to a p p l y a p r e v i o u s l y a c q u i r e d a b i l i t y t o a p a r t i c u l a r s i t u a t i o n ) . I n t h i s a n a l y s i s o f the development of c o n s e r v a t i o n a b i l i t y , the s t i m u l u s , b o t h p a s t and p r e s e n t becomes l e s s i m p o r t a n t because i t i s the p o s s i b i l i t y of t r a n s f o r m a t i o n , w h i c h i s c o n s t r u c t e d m e n t a l l y by the c h i l d , w hich g i v e s the s i t u a t i o n i t s meaning. T h i s i n c r e a s e i n the p s y c h o l o g i c a l d i s t a n c e between the s u b j e c t and t h e s t i m u l u s i s one of P i a g e t ' s c r i -t e r i a o f an i n c r e a s e i n i n t e l l i g e n c e . F u t u r e R e s e a r c h As d i s c u s s e d above, i n the s e c t i o n on H y p o t h e s i s 1, another s t u d y s h o u l d be conducted to t e s t H y p o t h e s i s 1, i . e . , to determine the r e l a t i o n s h i p between OCT and CON. That, s t u d y s h o u l d have a l a r g e enough s e r i e s i n the OCT t e s t , one of 11 items or more, and thus ask enough 103 q u e s t i o n s to s e p a r a t e t h o s e who a r e p e r f o r m i n g on t h e b a s i s o f a combi-n a t i o n o f p e r c e p t u a l judgments and g u e s s i n g from those who a r e r e s p o n d -i n g o p e r a t i o n a l l y . The r e l a t i o n s h i p between performance on the T r a n s f o r m a t i o n T e s t and performance on the T e s t f o r I n t e n s i v e Q u a n t i t y , t h e O r d i n a l C o r r e s -pondence T e s t , and the T e s t f o r C o n s e r v a t i o n , o f Number, I n d i c a t e s t h a t t h e a b i l i t y t o m e n t a l l y t r a n s f o r m the s t i m u l u s i s r e l a t e d t o t h e a b i l i t y t o c o n s e r v e and f i n d o r d i n a l c o r r e s p o n d e n c e s . T h i s s u g g e s t s t h a t i f a b e t t e r Imagery T e s t p r o c e d u r e were d e v i s e d , i t too might be r e l a t e d t o c o n s e r v a t i o n and o r d i n a l c o r r e s p o n d e n c e performance. The £!s i n t h i s s t u d y seemed q u i t e c o n f u s e d d u r i n g the t a s k and the r e s u l t s i n d i c a t e d they were not c o n s i s t e n t i n t h e i r c h o i c e s . I t would be h e l p f u l to i n c l u d e i n the t e s t a p r e - t r a i n i n g p r o c e d u r e on s t i m u l i u n r e l a t e d t o number t o g i v e the s u b j e c t s p r a c t i c e on a n t i c i p a t i n g and remembering t r a n s f o r m a t i o n s ( e . g . , E_ might ask the S_ what i t would l o o k l i k e i f a coat was put on a d o l l and to remember what i t l o o k e d l i k e ) . P i a g e t [1964] p r e d i c t s t h a t a n t i c i p a t i o n s h o u l d be no h a r d e r than memory because t h e y b o t h depend on the mental o p e r a t i o n s o f the s u b j e c t . T h i s p r e d i c t i o n i s c o n t r a d i c t e d by the p r e s e n t s t u d y , because MEM was e a s i e r than ANT. P r e v i o u s r e s e a r c h on imagery [ I n h e l d e r , 1966] has t e s t e d memory over a week t o six-month p e r i o d . But the p r e s e n t s t u d y t e s t e d memory over a p e r i o d o f l e s s than a minute. I f another imagery, s t u d y were done i t would be i n t e r e s t i n g to m a n i p u l a t e the v a r i a b l e of time between p r e s e n t a t i o n and r e c a l l . I f t h e e f f e c t o f t h e v a r i a b l e were s i g n i f i c a n t , and i f a n t i c i p a t i o n a b i l i t y was r e l a t e d to l o n g - t e r m 104 memory a b i l i t y , t h e r e s u l t s would be f u r t h e r e v i d e n c e t h a t memory over a s h o r t p e r i o d and memory over a l o n g p e r i o d depend on d i f f e r e n t p r o c e s s e s . One o f the mOst i n t e r e s t i n g r e s u l t s of the p r e s e n t s t u d y was the s uggested d e v e l o p m e n t a l sequence,found and d i s c u s s e d i n c o n n e c t i o n w i t h H y p o t h e s i s 6. The d a t a from t h i s s t u d y i n d i c a t e t h a t the. a b i l i t y to p e r f o r m a p h y s i c a l t r a n s f o r m a t i o n p r e c e d e s and mediates the a b i l i t y t o p e r f o r m a mental t r a n s f o r m a t i o n , w h i c h precedes and t h e r e f o r e p o s s i b l y m e diates the a b i l i t y t o c o n s e r v e . Data from the D/L/N t e s t ( i . e . , the d i s c o v e r y of a s t a g e i n w h i c h l e n g t h and d e n s i t y are i n c o n f l i c t ) , i n d i -c a t e t h a t the a b i l i t y to p e r f o r m a mental t r a n s f o r m a t i o n , combined w i t h a c o n f l i c t i n l e n g t h and d e n s i t y cues may p r o v i d e the n c e s s a r y and s u f f i -c i e n t c o n d i t i o n s f o r c o n s e r v a t i o n . Two s t u d i e s s h o u l d be done t o f o l l o w up t h i s h y p o t h e s i s . F i r s t a l o n g i t u d i n a l s t u d y s h o u l d be done t o determine i f the sequence i s t r u l y d e v e l o p m e n t a l and not j u s t due t o d i f f e r e n t i a l d i f f i c u l t y o f the t a s k s . A t r a i n i n g s t u d y s h o u l d a l s o be done t o determine i f e x p e r i e n c e w i t h p h y s i c a l t r a n s f o r m a t i o n s t r u l y mediate or merely p r e c e d e s the a b i l i t y to p e r f o r m mental t r a n s f o r m a t i o n s and I f the a b i l i t y to p e r f o r m mental t r a n s -f o r m a t i o n s and a c o n f l i c t between l e n g t h and d e n s i t y cues t r u l y mediate c o n s e r v a t i o n a b i l i t y . GENERAL CONCLUSIONS T h i s s t u d y t e s t e d s even hypotheses drawn from (a) P i a g e t ' s t a s k a n a l y s i s of t h e c o n s e r v a t i o n and. o r d i n a l c o r r e s p o n d e n c e t a s k s ; (b) d e s c r i p t i o n o f the development from p r e o p e r a t i o n a l to c o n c r e t e o p e r a t i o n -a l t h i n k i n g ; and (c) i d e a s about the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r t h i s t r a n s i t i o n . A c c o r d i n g t o P i a g e t ' s t a s k a n a l y s i s , performance on b o t h the o r d i n a l c o r r e s p o n d e n c e t a s k and the c o n s e r v a t i o n t a s k demands an under-s t a n d i n g t h a t the o b j e c t s i n b o t h t a s k s a r e u n i t s , i . e . , a r e c a p a b l e of b e i n g s i m u l t a n e o u s l y s e p a r a t e d from each o t h e r . a n d grouped t o g e t h e r . E v i d e n c e i n s u p p o r t of t h i s t a s k a n a l y s i s was p r o v i d e d by the r e s u l t s of t h i s s t u d y ( H y p o t h e s i s 1 and H y p o t h e s i s 4 ) . C e r t a i n a s p e c t s of P i a g e t ' s r e p o r t e d d e s c r i p t i o n of the deve-lopment from p r e o p e r a t i o n a l to o p e r a t i o n a l thought were r e p l i c a t e d , i . e . , b u i l d i n g two s e r i e s w i t h a c o r r e s p o n d e n c e between them i s no more d i f f i -c u l t . than the b u i l d i n g of one s e r i e s ( H y p o t h e s i s 3 ) } t h e r e i s 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 the a b i l i t y t o b u i l d a s e r i e s s y s t e m a t i c a l l y and the a b i l i t y t o f i n d o r d i n a l c orrespondences ( H y p o t h e s i s 4 ) , and a c h i l d who i s a b l e to c o n s e r v e can a l s o match a s t a n d a r d row w i t h o u t de-p e n d i n g on a s p a t i a l one-to-one correspondence ( H y p o t h e s i s 5 ) . The p r e s e n t s t u d y f a i l e d , t o r e p l i c a t e one o f P i a g e t ' s o b s e r v a t i o n s , how-e v e r . M a t c h i n g the number of o b j e c t s i n a s t a n d a r d row is.much e a s i e r t h a n b u i l d i n g the s e r i e s , d e s p i t e t h e i r supposed.dependence on the same concept — 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 ( H y p o t h e s i s 2 ) . A number 105 106 of e x p l a n a t i o n s of t h i s f a i l u r e t o r e p l i c a t e a r e g i v e n . Hypotheses d e v e l o p e d from P i a g e t ' s e x p l a n a t i o n of t h e t r a n s -i t i o n from p r e o p e r a t i o n a l t o o p e r a t i o n a l thought were c o n f i r m e d . The c h i l d does pass t h r o u g h a s t a g e i n which b o t h l e n g t h and d e n s i t y a r e c o n s i d e r e d ( H y p o t h e s i s 6) b e f o r e he c o n s e r v e s . And the a b i l i t y t o p e r -form a mental t r a n s f o r m a t i o n i s r e l a t e d to the a b i l i t y to c o n s e r v e and to f i n d o r d i n a l c o r r e s p o n d e n c e s ( H y p o t h e s i s 7 ) . Because the p r e s e n t s t u d y f a i l e d t o f i n d t a s k s h i g h l y c o r r e -l a t e d w i t h c o n s e r v a t i o n performance,- the s e a r c h f o r b e h a v i o r a l measures to t e s t f o r the i n t e n s i o n of the concept which a l l o w s c o n s e r v a t i o n must be c o n t i n u e d . The p r e s e n t s t u d y a l s o f a i l e d t o s o l v e the m e t h o d o l o g i -c a l problem c r e a t e d by t h e e f f e c t o f s t i m u l u s v a r i a b l e s on performance i n P i a g e t i a n t a s k s and to answer the m e t h o d o l o g i c a l q u e s t i o n of whether the c h i l d u n d e r s t a n d s the terms used i n P i a g e t i a n t a s k s . The d a t a s u p p o r t t h e h y p o t h e s i s . t h a t the c h i l d f i r s t l e a r n s to a t t e n d to b o t h l e n g t h and d e n s i t y cues w h i c h e n a b l e s him t o d e v e l o p from Stage I to Stage I I . 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Rothenberg, B a r b a r a and Courtney, R o s a l e a . A d e v e l o p m e n t a l study of non-c o n s e r v a t i o n c h o i c e s i n young c h i l d r e n . Merrill-Palmer Quarterly, 1969, 15, 363-373. S i e g e l , L i n d a A. and G o l d s t e i n , A l v i n G. C o n s e r v a t i o n of number i n young c h i l d r e n : Recency v s . r e l a t i o n a l r e s ponse s t r a t e g i e s . Developmental Psychology, 1969, 1, 128-130. S i g e l , I r v i n g E., Roeper, Annemarie, and Hooper, F r a n k H.. A t r a i n i n g pro^. cedure f o r a c q u i s i t i o n o f P i a g e t ' s c o n s e r v a t i o n o f quantity^: A p i l o t s tudy and i t s r e p l i c a t i o n . British Journal of Educational Psychology-, 1966, 36, 301-311. I l l Smedslund, J a n . The a c q u i s i t i o n o f c o n s e r v a t i o n of s u b s t a n c e and weight i n c h i l d r e n V. 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APPENDIX A DEFINITION OF TERMS Operational reversibility i s the a b i l i t y t o m e n t a l l y r e v e r s e an o p e r a t i o n t h a t has j u s t been performed. Empirical reversibility i s the a b i l i t y t o p r e d i c t the result o f a physical r e v e r s a l o f an o p e r a t i o n . I t depends on e x p e r i e n c e i n a c t u a l l y s e e i n g the r e s u l t . I t does not n e c e s s a r i l y Imply, as does o p e r a t i o n a l r e v e r s i b i l i t y , an u n d e r s t a n d i n g o f the d e t a i l s o f the r e v e r -s a l o r an a b i l i t y t o . c o r r e c t l y c a r r y out the r e v e r s a l m e n t a l l y even i n the absence of e x p e r i e n c e i n a c t u a l l y s e e i n g the r e s u l t of the p h y s i -c a l r e v e r s a l . Compensation i s the a b i l i t y t o determine whether o r not a change i n one d i m e n s i o n of an o b j e c t i s compensated by a change i n another d i m e n s i o n o f the o b j e c t . Partial compensation i s t h e u n d e r s t a n d i n g t h a t i f one dimen-s i o n changes, so does a n o t h e r . I t does not n e c e s s a r i l y i m p l y as does compensation,,an u n d e r s t a n d i n g o f t h e e x a c t r e l a t i o n s h i p between the change i n one d i m e n s i o n and the change i n an o t h e r . Identity i s the a b i l i t y t o see t h a t an o b j e c t may remain t h e same i n quantity d e s p i t e changes i n i t s 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 . Pseudoconservation i s the a b i l i t y t o see t h a t an o b j e c t I s th e same o b j e c t d e s p i t e changes i n i t s 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 . I t does n o t n e c e s s a r i l y i m p l y an u n d e r s t a n d i n g t h a t the quantity remains t h e same. I t i s a l s o sometimes c a l l e d identity of substance. 113 APPENDIX B CHI-SQUARE TABLES F o l l o w i n g i s a d i s c u s s i o n of the methods of c l a s s i f i c a t i o n on the v a r i o u s t e s t s used i n the v a r i o u s c h i - s q u a r e a n a l y s e s . Sometimes more than one method i s used, e i t h e r t o a v o i d e x p e c t e d f r e q u e n c i e s t h a t a r e t o o s m a l l o r f o r purposes o f c l e a r e r , i n t e r p r e t a t i o n . T e s t f o r I n t e n s i v e Q u a n t i t y S_ was c l a s s i f i e d i n Group A i f he was i n e i t h e r C a t e g o r y 1 o r 2 of the groups d e s c r i b e d i n T a b l e 3, i . e . , i f he c o u l d n ot match t h e s t a n d a r d row. He was c l a s s i f i e d i n Group B i f he was i n C a t e g o r y 3, i . e . , i f he c o u l d match t h e s t a n d a r d row o n l y by depending on a s p a t i a l o n e - t o -one c o r r e s p o n d e n c e . He was c l a s s i f i e d i n Group C i f he was i n Ca t e g o r y 4, i . e . , i f he succeeded i n one of the two items i n P a r t C r e q u i r i n g him to match the s t a n d a r d row w i t h o u t depending on^a s p a t i a l c o r r e s p o n d e n c e . S_was c l a s s i f i e d i n Group D i f he was i n Category 5, i . e . , i f he passed a l l items on the NQ t e s t . T e s t f o r C o n s e r v a t i o n of Number (Method 1) I f S^  d i d not c o n s e r v e on at l e a s t two o f the t h r e e t r a n s f o r m -a t i o n s he was p l a c e d i n Group A. Ot h e r w i s e , he was p l a c e d i n Group B. T e s t f o r C o n s e r v a t i o n of -Number (Method.2) I f S_ was i n Ca t e g o r y 1, 3, 4, o r 5a of the groups d e s c r i b e d i n 114 115 T a b l e 5, he was c o n s i d e r e d i n Group A. I f S_was i n C a t e g o r y 2, 5b, o r 6, he was c o n s i d e r e d i n Group B. I f S_ was i n C a t e g o r y 8 or 9, he was c o n s i d e r e d i n Group C, i . e . , he c o n s e r v e d . T e s t f o r C o n s e r v a t i o n o f Number. (Method 3) I f S^  was i n C a t e g o r y 1 o r 3 o f T a b l e 5 o r i f he was i n C a t e g o r y 2 o f T a b l e 2 and Category 2, 5a, 5b, 6 or 4 of T a b l e 3, he was c o n s i d e r e d i n Group A, i . e . , e i t h e r he d i d n o t und e r s t a n d the c o n s e r v a t i o n t a s k o r he f a i l e d t h e NQ t a s k . Group B c o n s i s t e d o f those i n Ca t e g o r y 2, 4, 5a or 5b of T a b l e 5 and i n C a t e g o r y 1, 3, 4, o r 5 of T a b l e 3, i . e . , t h o s e who were a b l e t o match the s t a n d a r d row but un a b l e t o c o n s e r v e . Group C c o n s i s t e d o f those i n C a t e g o r y 7 o r 8 of T a b l e 5, i . e . , those who were a b l e t o c o n s e r v e . T e s t f o r C o n s e r v a t i o n o f Number (Method 4) T h i s method was the same as Method 1 except those who are i n C a t e g o r y 1 o f T a b l e 5, i . e . , those who d i d not un d e r s t a n d the CON p r o c e d u r e , a r e e l i m i n a t e d from the a n a l y s i s . S e r i a t i o n Test.(Method 1) I f S^  c o u l d not b u i l d a s e r i e s he was.placed i n Group.A. I f he b u i l t t h e s e r i e s by t r i a l and e r r o r he was p l a c e d i n Group.B. I f he b u i l t the s e r i e s s y s t e m a t i c a l l y he was p l a c e d i n Group C. 116 S e r i a t i o n T e s t (Method 2) Groups B and C from Method 1 were added t o g e t h e r to form Group B. Group A from Method 1 formed Group A. C o u n t i n g A b i l i t y Those Ss who c o u l d count were p l a c e d i n Group A; those who c o u l d n o t were p l a c e d i n Group B. Imagery T e s t ( S t r o n g C r i t e r i o n ) The two groups were formed as d e s c r i b e d i n the d i s c u s s i o n of the r e s u l t s of the I T , those f a i l i n g the c r i t e r i a p l a c e d i n Group A and t h o s e p a s s i n g i t p l a c e d i n Group B. Imagery T e s t (Weak C r i t e r i o n ) The t h r e e groups were formed as d e s c r i b e d i n the d i s c u s s i o n of t h e r e s u l t s o f the I T , t h o s e p r e d o m i n a t e l y r e s p o n d i n g w i t h Card 1 c h o i c e s p l a c e d i n Group A, w i t h Card 2 c h o i c e s p l a c e d i n Group B,^.and w i t h Card 3 c h o i c e s p l a c e d i n Group C. O r d i n a l Correspondence T e s t (Method 1) Group A c o n s i s t e d of t h o s e Ss who e i t h e r d i d not u n d e r s t a n d o r who performed c o r r e c t l y on none of the t r a n s f o r m a t i o n s or who performed c o r r e c t l y on o n l y one of the t r a n s f o r m a t i o n s . Group B c o n s i s t e d of those £>s who performed c o r r e c t l y on two of the t h r e e t r a n s f o r m a t i o n s , and Group C c o n s i s t e d of those Ss who performed c o r r e c t l y on a l l t h r e e of t h e t r a n s -f o r m a t i o n s . 117 O r d i n a l Correspondence T e s t (Method 2) T h i s method i s t h e same as Method 1 except those i n C a t e g o r y 1 of T a b l e 5, i . e . , those who d i d not u n d e r s t a n d t h e CON t e s t a r e e l i m i n -a t e d from t h e a n a l y s i s . T r a n s f o r m a t i o n T e s t ( S t r o n g C r i t e r i o n ) The t h r e e groups were formed as d e s c r i b e d i n the d i s c u s s i o n o f the r e s u l t s o f the TT, u s i n g the s t r o n g c r i t e r i a . Those who d i d p o o r l y on b o t h the end and m i d d l e q u e s t i o n s formed Group A. Those who d i d p o o r -l y on the m i d d l e q u e s t i o n s b u t w e l l on the end q u e s t i o n s formed Group B. Those who d i d w e l l on b o t h t h e end and the m i d d l e q u e s t i o n s formed Group C. ' T r a n s f o r m a t i o n T e s t (Weak C r i t e r i o n ) The t h r e e groups were formed e x a c t l y l i k e the t h r e e g r o u p s . i n STR e x c e p t the weak c r i t e r i a , as d i s c u s s e d i n the R e s u l t s e c t i o n under T r a n s f o r m a t i o n T e s t , was used. Age Those who were from 48 to 59 months o l d were p l a c e d i n Group A. Those who were from 60 to 71 months o l d were p l a c e d i n Group B. Those who were 72 o r more months.old were p l a c e d i n Group C. TABLE 25 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 1) X ORDINAL CORRESPONDENCE TEST (METHOD. 1 )(OCT) A B TOTAL CON A 45 6 51 B 18 10 28 C 5 10 15 T o t a l 68 26 94 2 '•> X = 18.75, P < .001 45, * A = .48, 2 X = 21.74, i TABLE 26 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD-4) X ORDINAL CORRESPONDENCE TEST (METHOD 2)(OCT) A B TOTAL CON A 19 6 25 B 10 10 20 OCT C 3 10 13 T o t a l 32 26 58 df = 2, X 2 = 10.02, p <• .01 * = .42, * A = -44, x 2 = 11. .23, p<.005. TABLE 27 TEST FOR INTENSIVE QUANTITY(NQ) X SERIATION TEST (METHOD ' 2) (ST) A B C D TOTAL NQ A 16 25 7 6 54 ST B 2 7 8 23 40 T o t a l 18 32 15 29 94 d f = 29. 62, P < .001 <j> = .56, * A = . 44,; 2 •:x = 18.12, P < .001 TABLE 28 SERIATION TEST(ST) (METHOD 2) X ORDINAL CORRESPONDENCE TEST (METHOD 1) (OCT) A B C TOTAL OCT A. 48 23 8 79 ST B 4 4 7 15 T o t a l 52 27 15 94 df = 2, x 2 = 13. 21, P < .005 * = .37, <f>A = . 35, 2 X = 11.49, p < . TABLE 29 TEST FOR INTENSIVE QUANTITY(NQ) X TEST FOR CONSERVATION OF NUMBER (METHOD 2) (CON) A B C D TOTAL NQ A 13 16 6 2 37 B 5 11 4 11 31 C 0 5 5 16 26 T o t a l 18 32 15 29 94 df = 6, X = 35.65, P < • .001 <f> = .51, * A = -59, 2 X = 32.63, p < .001 TABLE 30 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 3) X IMAGERY TEST (STRONG CRITERION) (SI) A B C TOTAL CON A 32 24 18 74 sr. B 7 5 8 20 T o t a l 39 29 26 94 df = 1.94, P < .05 = ;14 TABLE 31 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 3) X IMAGERY TEST (WEAK CRITERION) (CI) A B C TOTAL CON A 19 14 5 38 B 7 2 7 16 CI C 12 12 14 38 T o t a l 38 28 26 92 df - 4, x 2 - 9.22, P < .05 + = .22 TABLE 32 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 1) X IMAGERY TEST (STRONG CRITERION) (SI) A B C TOTAL OCT A 44 23 7 74 B '. 7 5 8 20 T o t a l 51 28 15 94 df = 2, 2 X = 11.13, p < .005 <J> = .34, $ = .37, x 2 = 12.55, p <.005 TABLE 33 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 1) X IMAGERY TEST (WEAK CRITERION) (CI) A B C TOTAL CI OCT A 26 9 3 38 B 6 8 2 16 C 17 11 10 38 T o t a l 49 28 15 92 = 24 .63, p < .001 • .36, cj> A = .40,,x2 = 15.12, P < TABLE 34 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 3) X TRANSFORMATION TEST (WEAK CRITERION) (WTR) A B C TOTAL A 19 B 14 WTR C 6 T o t a l 39 d f = 4, X 2 = 21. .34, 4 CON 5 2 26 10 6 30 14 16 36 29 24 92 P < .001 X 2 = 15.47, p < .005 TABLE 35 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 3) X TRANSFORMATION TEST CSTRONG CRITERION) (STR) A B c TOTAL CON A 26 7 6 39 B 8 10 5 23 STR C 4 12 15 31 T o t a l 38 29 26 93 d f = 23.51, P < .001 cj, = •36, ^ A = .39, 2 X = 27.86, p < .001 TABLE 36 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 1) X TRANSFORMATION TEST (STRONG CRITERION) (STR) A B C TOTAL STR A B C T o t a l 28 11 11 50 OCT 8 12 8 28 3 0 12 15 39 23 31 93 d f = 4, x 2 = 24.77, p < .001 <j> = .36, <j> = -40, x 2 = 27.73, p < .001 124 TABLE 37 ORDINAL CORRESPONDENCE TEST (OCT) (METHOD 1) X TRANSFORMATION TEST (WEAK CRITERION) (WTR) A B C TOTAL OCT A 21 4 1 26 B 17 12 1 30 C 12 12 12 36 T o t a l 50 28 14 92 df = 4, x 2 = 21.99, p < .001 <j> = .35, <j> = -69, x 2 = 88.86, p < .001 TABLE 38 TEST FOR INTENSIVE QUANTITY(NQ) X TRANSFORMATION TEST (WEAK CRITERION) (WTR) TOTAL NQ A 12 9 2 3 26 B 5 17 2 6 30 WTR C 1 6 10 19 36 T o t a l 18 32 14 28 92 df = 6 , x 2 = 38.41, P < .001 $ = .46, <j>A = .54, x 2 = 53.06, p < .001 125 TABLE 39 TEST FOR INTENSIVE QUANTITY(NQ) X TRANSFORMATION TEST (STRONG CRITERION) (STR) A B C D TOTAL NQ A 14 13 6 6 39 B 3 13 0 7 23 STR C 0 6 9 16 31 T o t a l 17 32 15 29 93 df = 6, X 2 = 32.16, P < .001 .42, « A = -47, 2 X = = 41. 79, p < .001 TABLE 40 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 2) X COUNTING ABILITY(CT) A B c TOTAL CON A 19 25 23 67 CT B 18 6 3 27 T o t a l 37 31 26 94 df 12. 25, P .005 * = .36, <j)A = . 36, 2 X = 12.33, p < .005 TABLE 41 TEST FOR INTENSIVE QUANTITY(NQ) X COUNTING ABILITY(CT) A B C D TOTAL NQ A 7 21 11 28 67 CT B 11 11 4 1 27 T o t a l 18 32 15 29 94 df = 3, x = 18. 80, P < .001 * = -45, (j >A= • 30, 2 X = 18.80, p < .001 TABLE 42 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 1) X_ COUNTING ABILITY(CT) A B C TOTAL OCT A 32 22 13 67 CT B 21 6 2 27 T o t a l 51 28 15 94 df - 2 . . x 2 - 5.74, p < .05 <j> = .25 TABLE 43 TEST FOR CONSERVATION OF NUMBERCCON) (METHOD 1) X AGE A B TOTAL CON A 28 0 28 B 24 12 36 AGE c 16 14 30 T o t a l 68 26 94 df = 2, X 2- 16.70, p < .001 (j, = .42 TABLE 44 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 2) X AGE A B C TOTAL CON 20 11 6 37 8 13 10 31 AGE C 0 12 14 26 T o t a l 28 36 30 94 df = 4, x 2 = 23.24, p < .001 128 TABLE 45 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 3) X AGE A B C TOTAL CON A 20 8 0 28 B 14 10 12 36 AGE C 5 11 14 30 T o t a l 39 29 26 94 df = 4, x 2 = 23.13, P < -.001 <f> = .35 TABLE 46 TEST FOR CONSERVATION OF NUMBER(CON) (METHOD 4) X AGE A B TOTAL CON A 8 0 8 B 13 12 25 C 11 14 25 T o t a l 32 26 58 df = 2, x 2 = 7.86, p < .025 .37 TABLE 47 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 1) X AGE A B C TOTAL OCT A 20 8 0 28 B 21 13 2 36 C 10 7 13 30 T o t a l 51 28 15 94 df = 4 , x 2 = 25.97, p < .001 <j> = .37 TABLE 48 ORDINAL CORRESPONDENCE TEST(OCT) (METHOD 2) X AGE A B C TOTAL AGE A 5 13 7 25 B 3 11 6 20 OCT C 0 1 12 13 T o t a l 8--- 25 25 58 df = 4, x 2 = 16.83, p < .005 cf> = .38 TABLE 49 SERIATION TESTCST) (METHOD 1) X AGE A B C TOTAL ST A 25 2 1 28 B 25 9 2 36 AGE C 4 14 12 30 T o t a l 54 25 15 94 df = 4, x 2 = 40.57, p < .001 <j> = .46 TABLE 50 SERIATION TEST(ST) (METHOD 2) X AGE A B TOTAL ST A 25 3 28 B 25 11 36 AGE C '4 26 30 T o t a l 54 40 94 df =.2, xZ = 37.61, P < .001 <j> = .63-TABLE 51 TEST FOR INTENSIVE QUANTITY(NQ) X AGE A B C D TOTAL NQ A 11 13 2 2 28 B 6 13 8 9 36 AGE C 1 6 5 18 30 T o t a l 18 32 15 29 94 df = 6, x 2 = 29.03, P < .001 <J> • 39 TABLE 52 IMAGERY TEST (STRONG CRITERION) (SI) X AGE A B TOTAL SI A 26 2 28 B 28 8 36 C 20 10 30 T o t a l 74 20 94 df = 2, x 2 = 5.96, p < .05 ct = .25 TABLE 53 IMAGERY TEST (WEAK CRITERION) (CI) X AGE A B C TOTAL CI A 17 4 6 27 B 13 6 16 35 C 8 6 16 30 T o t a l 38 16 38 92 df = 4, x 2 = 8.59, p < .05 <j) = .22 TABLE 54 TRANSFORMATION TEST (WEAK CRITERION) (WTR) X AGE A B C TOTAL WTR A , 12 11 5 28 B 8 12 15 35 AGE C 6 7 16 29 T o t a l 26 30 36 92 df = 9.38, p < .05 * = .23 TABLE 55 TRANSFORMATION TEST (STRONG CRITERION) (STR) X AGE A B C TOTAL A 14 B 16 AGE C 9 T o t a l 39 df = 4, x 2 = 10.46, <j> = .24 STR 10 .4 28 8 11 35 5 16 30 23 31 93 p < .025 TABLE 56 COUNTING ABILITY(CT) X AGE A B TOTAL CT A 13 15 28 B 28 8 36 AGE C 26 4 30 T o t a l 67 27 94 df = 12.66, P < .005 .37 APPENDIX C ANALYSIS OF VARIANCE TABLES TABLE 6 ANALYSIS OF VARIANCE OF SCORES ON THE TEST FOR CONSERVATION OF NUMBER SOURCE DF SS MS F T o t a l 281 52. 12 Between 93 C o u n t i n g A b i l i t y (CT) 1 2. 03 2 .03 4. ,12 Order o f T r a n s f o r m a t i o n :(o) 5 3. 70 0 .74 1. ,50 S e s s i o n o r d e r (S) 1 0. 74 0 .74 1. ,51 CTXO 5 2. 14 0 .43 0. .87 CTXS 1 1. 38 1 .38 2. ,80 OXS 5 1. 79 0 .36 0. ,73 CTXOXS 5 -0. 14 -0 .03 -0. ,06 Ss/CTXOXS 70 34. 48 0 .49 W i t h i n 188 K i n d o f T r a n s f o r m a t i o n (T) 2 0. 28 0 .14 4, .10 TXSs 186 TXCT 2 0. 11 0 .06 1, .65 TXO 10 0. 28 0 .03 0. .82 TXS 2 0. 13 0 .06 1, .90 TXCTXO 10 0. 07 0 .01 0. .22 TXCTXS - 2 0. 10 0 .05 1. .41 TXOXS 10 0. 29 0 .03 0, .87 TXCTXOXS 10 0. 02 0 .002 0, .06 TXSs/CTXOXS 140 4. 73 0 .03 134 135 TABLE ANALYSIS OF VARIANCE OF SCORES ON USING KIND OF TRANSFORMATION AS 11 THE ORDINAL CORRESPONDENCE TEST THE WITHIN-SUBJECTS VARIABLE SOURCE DF SS MS T o t a l 281 70. ,37 Between 93 C o u n t i n g A b i l i t y (CT) 1 2. ,01 2. .01 6. .64 Order o f T r a n s f o r m a t i o n (0) 5 1. ,71 0. ,34 1. ,13 S e s s i o n O rder (S) 1 0. ,12 0. ,12 0. ,40 CTXO 5 0. .89 0. ,18 0. ,59 CTXS 1 0. ,10 0, ,10 0, ,34 OXS •5 1. ,69 0, .34 1, .12 CTXOXS 5 1. ,32 0. .26 0. ,87 Ss/CTXOXS 70 21. ,19 0. .30 W i t h i n 188 K i n d o f T r a n s f o r m a t i o n CT) 2 2, .36 1, .18 5. .59 TXCT 2 1. ,88 0, .94 4, .45 TXO 10 1. .70 0, .17 0. .80 TXS 2 0. .98 0, .49 2, .32 TXCTXO 10 1. .89 0, .19 0. .89 TXCTXS 2 -0. .22 -0, .11 -0. .53 TXOXS 10 0. .37 0. .04 0. .17 TXCTXOXS 10 2, .82 0, .28 1, .34 TXSs/CTXOXS 140 29, .56 0, .21 .025 .01 .025 136 TABLE 12 ANALYSIS OF VARIANCE OF SCORES ON THE ORDINAL CORRESPONDENCE TEST USING THE NUMBER TO BE MATCHED AS THE WITHIN-SUBJECTS VARIABLE SOURCE DF SS MS F T o t a l 281 70. 37 Between 93 C o u n t i n g A b i l i t y (CT) 1 2. 01 2. .01 6. 93 Order o f Number (0) 5 2. 85 0. .57 1. 97 S e s s i o n Order (S) 1 0. 12 0. .12 0. 42 CTXO 5 2. 12 0, .42 1. 46 CTXS" 1 0. 10 0. .10 0. 35 OXS 5 0. 46 0, .09 0. 32 CTXOXS 5 1. 05 0, .21 0. 72 Ss/CTXOXS 70 20. 32 0, .29 W i t h i n 188 Number Matched (N) 2 3. 34 1. .67 7. 96 NXCT 2 0. 88 0, .44 2. 10 TXO 10 2. 31 0, .23 1. 10 TXS 2 0. 40 0. .20 0. 95 TXCTXO 10 1. 80 0. .18 0. 86 TXCTXS 2 -0. 15 -0, .07 -0. 36 TXOXS 10 1. 88 0, .19 0. 90 TXCTXOXS 10 1. 49 0, .15 0. 71 TXSs/CTXOXS 140 29. 38 0. .21 .025 .01 TABLE 14 ANALYSIS OF VARIANCE ON THE SCORES ON THE IMAGERY TEST WITH LEVEL OF DEVELOPMENT (G) DETERMINED-BY PERFORMANCE ON THE TEST FOR INTENSIVE QUANTITY (METHOD 1) SOURCE DF SS MS F T o t a l 376 88 .85 Between 93 L e v e l o f Development (G) 4 2 .05 . 51 0. 37 C o u n t i n g A b i l i t y (CT) 1 0 .91 0. 91 0. 65 S e s s i o n Order (S) 1 0 .06 0. 06 0. 05 Order of T r a n s f o r m a t i o n (0) 1 0 .17 0. 17 0. 12 GXCT 4 -0 .39 -o, 10 -0. 07 GXS 4 2 .25 0. 56 0. 40 GXO 4 1 .05 0. 26 0. 19 CTXS 1 0 .17 0. 17 0. 12 CTXO 1 0 .05 0. 05 0. 03 SXO 1 0 .63 0. 63 0. 45 Ss/CTXSXGXO 54 75 .78 1. 40 W i t h i n 283 C a r d (C) 1 1 .29 1. 29 6. 99 T r a n s f o r m a t i o n (T) 1 0 .01 0. 01 0. 03 CXT 1 0 .04 0. 04 2. 47 CXSs 93 CXG 4 0 .62 0. 16 0. 85 CXCT 1 0 .28 0. 28 1. 55 CXS 1 0 .24 0. 24 1. 33 CXO 1 0 .45 0. 45 2. 42 CXGXCT 4 0 .92 0. 23 1. 25 CXSs/GXCTXSXO 54 9 .94 0. 18 TXSs 93 TXG 4 0 .30 0. 08 0. 20 TXCT 1 0 .002 0. 002 0. 01 TXS 1 0 .09 0. 09 0. 26 TXO 1 0 .38 0. .38 1. 03 TXGXCT 4 0 .19 0. 05 0. 13 TXSXO 1 0 .34 0. ,34 0. 92 138 TABLE 14 (Continued) SOURCE DF SS MS F p TXSs/GXCTXSXO 54 19 .88 0 .37 TXCXSs 93 TXCXG 4 1 .17 0 .29 2. 24 TXCXCT 1 0 .0003 0 .0003 0. 002 TXCXS 1 0 .04 0 .04 0. 31 TXCXO 1 0 .54 0 .54 4. 08 TXCXGXCT 4 0 .76 0 .19 1. 44 CXTXSs/GXCTXSXO 54 7 .09 0 .13 TABLE 15 ANALYSIS OF VARIANCE ON THE SCORES ON THE IMAGERY TEST WITH. LEVEL OF DEVELOPMENT CG) DETERMINED BY•PERFORMANCE ON THE ORDINAL CORRESPONDENCE TEST (METHOD 2) SOURCE DF SS MS F p T o t a l 376 88 .85 Between 93 L e v e l o f Development (G) 4 4 .12 1 .03 0 .73 C o u n t i n g A b i l i t y (CT) 1 0 .91 0 .91 0 .65 S e s s i o n Order (S) 1 0 .06 0 .06 0 .05 Order o f T r a n s f o r m a t i o n (0) 1 0 .17 0 .16 0 .12 GXCT 4 -0 .004 -0 .001 -0 .0008 GXS 4 0 .97 0 .24 0 .17 GXO 4 0 .50 0 .12 0 .09 CTXS 1 0 .17 0 .17 0 .12 CTXO 1 0 .05 0 .05 0 .03 SXO 1 0 .63 0 .63 0 .45 Ss/CTXSXGXO 54 75 .92 1 .41 W i t h i n 283 Card (C) 1 1 .29 1 .29 7 .24 T r a n s f o r m a t i o n (T) ' 1 0 .01 0 .01 0 .03 CXT 1 0 .04 0 .04 0 .28 CXSs 93 CXG" 4 0 .36 0 .09 0 .51 CXCT 1 0 .28 0 .28 1 .60 CXS 1 0 .24 0 .06 0 .34 CXO 1 0 .45 0 .45 2 .51 CXGXCT 4 1 .07 0 .27 1 .51 CXSs/GXCTXSXO 54 9 .61 0 .78 TXSs 93 TXG 4 0 .86 0 .22 0 .54 TXCT 1 0 .002 0 .002 0 .006 TXS 1 0 .09 0 .09 0 .24 TXO 1 0 .38 0 .38 0 .96 TXGXCT 4 0 .25 0 .05 0 .14 TXSXO 1 0 .34 0 .34 0 .86 140 TABLE 15 (Continued) SOURCE DF SS MS F p TXSs/GXCTXSXO 54 21, .32 0 .39 TXCXSs 93 TXCXG 4 0, .36 0 .09 0 .59 TXCXCT 1 0, .0003 0 .0003 0 .002 TXCXS 1 0, .04 0 .04 0 .27 TXCXO 1 0, .54 0 .54 3 .53 TXCXGXCT 4 1, .22 0 .31 2 .02 CXTXSs/GXCTXSXO 54 8, .19 0 .15 TABLE 16 ANALYSIS OF VARIANCE ON THE SCORES ON THE IMAGERY TEST WITH LEVEL OF DEVELOPMENT (G) DETERMINED BY PERFORMANCE ON THE TEST FOR CONSERVATION OF NUMBER AND THE TEST FOR INTENSIVE QUANTITY (METHOD 3) SOURCE DF SS MS F p T o t a l 376 88. 85 Between 93 L e v e l o f Development (G) 4 3. 47 0. 87 0. 62 C o u n t i n g A b i l i t y (CT) 1 0. 91 0. 91 0. 65 S e s s i o n Order (S) 1 0. 06 0. 06 0. 05 Order of T r a n s f o r m a t i o n (0) 1 0. 17 0. 17 0. 12 GXCT 4 0. 15 0. 04 0. 03 GXS 4 0. 25 0. 06 0. 04 GXS 4 1. 79 0. 45 0. 32 CTXS 1 0. 17 0. 17 0. 12 CTXO 1 0. 05 0. 05 0. 03 SXO 1 0. 63 0. 63 0. 45 Ss/CTXSXGXO 54 75. 90 1. 41 W i t h i n 283 Card (C) 1 1. 29 1. 29 6. 25 T r a n s f o r m a t i o n (T) 1 0. 01 0. 01 0. 03 CXT 1 0. 04 0. 04 0. 27 CXSs 93 0. 20 0. 05 0. 25 CXG 4 0. 28 0. 28 1. 38 CXCT 1 0. 24 0. 06 0. 29 CXS 1 0. 45 0. 45 2. 17 CXO 1 0. 10 0. 03 0. 12 CXGXCT 4 -0. 04 -0. 04 -0. 20 CXSs/GXCTXSXO 54 11. 12 0. 21 TXSs 93 TXG 4 1. 24 0. 31 0. 91 TXCT""-' 1 0. 002 0. 002 0. 01 TXS 1 0. 09 0. 09 0. 28 TXO 1 0. 38 0. 38 1. 11 TXGXCT 4 0. 25 0. 06 0. 18 TXSXO 1 0. 34 0. 34 1. 00 142 TABLE 16 (Continued) SOURCE DF SS MS TXSs/GXCTXSXO 54 18. .40 0 .34 TXCXSs 93 TXCXG 4 0, .39 0 .10 0. .61 TXCXCT 1 0. .003 0 .003 0, .002 TXCXS 1 0. .04 0 .04 0. .26 TXCXO 1 0. .54 0 .54 3, .37 TXCXGXCT 4 0. .41 0 .10 0. .65 CXTXSs/GXCTXSXO 54 8, .56 0 .16 TABLE 18 ANALYSIS OF VARIANCE ON THE SCORES ON THE DENSITY-LENGTH-NUMBER TEST SOURCE DF SS MS F P Between 93 L e v e l of Development (G) 2 39.92 19.96 14.17 .01 C o u n t i n g A b i l i t y (CT) 1 11.19 11.19 7.94 .01 S e s s i o n O r d e r (S) 1 0.76 0.76 0.54 W i t h i n - s e s s i o n Order (BE) 1 0.22 0.22 0.15 GXCT 2 -6.01 -3.01 -2.13 GXS 2 5.11 2.55 1.81 GXBE 2 0.16 0.08 0.06 CTXS 1 3.35 3.35 2.38 CTXBE 1 0.01 0.01 0.01 SXBE 1 0.11 0.11 0.08 Ss/CTXSXBEXG 70 98.62 1.41 W i t h i n 1786 Card Group (N) 4 38.63 9.66 36.84 Card L e v e l (A) 3 1.89 0.63 2.40 NXA 12 4.92 0.41 1.56 NXSs 372 NXG 8 12.18 1.52 5.81 NXCT 4 1.86 0.47 1.78 NXS 4 1.05 0.26 1.00 NXBE 4 1.52 0.38 1.45 NXGXCT 8 1.92 0.24 0.91 .NXSXBE 4 0.37 0.09 0.35 NXSs/GXCTXSXBE 280 73.40 0.26 AXSs 279 AXG 12 0.57 0.05 0.54 AXCT 3-1 0.42 0.14 1.60 AXS 3 0.15 0.05 0.55 AXBE 3 0.24 0.08 0.89 -.AXGXCT 12 0.03 0.003 0.03 AXSXBE 3 0.54 0.18 2.04 144 TABLE 18 (Continued) SOURCE DF SS MS F p AXSs/GXCTXSXBE 210 18.43 0.09 AXNXSs 1116 AXNXG 24 1. ,65 0, .07 .78 AXNXCT 12 0, .78 0. .07 .78 AXNXS 12 1. .55 0. .13 1.44 AXNXBE 12 1. .18 0. .10 1.11 AXNXGXCT 24 1, .51 0, .06 .67 AXNXSs/GXCTXSXBE 840 72.65 0.09 TABLE 20 ANALYSIS OF VARIANCE ON THE SCORES ON THE TRANSFORMATION TEST WITH LEVEL OF DEVELOPMENT CG) DETERMINED BY PERFORMANCE ON THE TEST FOR INTENSIVE QUANTITY (METHOD 1) SOURCE DF SS MS F p T o t a l 751 168 .22 Between 93 S e s s i o n O rder (S) 1 0 .18 0 .18 0, .27 L e v e l o f Development (G) 4 25 .59 6 .40 9. ,55 .005 SXG 4 7 .01 1 .75 2. .62 .05 Ss/SXG 84 56 .29 0 .67 W i t h i n 658 Card (C) 3 0 .60 0 .20 2, .57 P o s i t i o n (P) 1 14 .52 14 .52 69. .65 .005 CXP 3 0 .96 0 .32 5. .38 .005 CXSs 279 CXG 12 3 .56 0 .30 3. .81 .005 CXS 3 0 .10 0 .03 0, .41 CXSXG 12 0 .91 0 .08 0. .97 CXSs/SXG 252 19 .62 0 .08 PXSs 93 PXS 1 0 .05 0 .05 0. .25 PXG 4 3 .33 0 .83 4. .00 .005 PXSXG 4 1 .17 0 .29 1, .40 PXSs/SXG 84 17 .51 0 .21 PXCXSs • 279 CXPXS 3 0 .11' 0 .04 0, .61 CXPXG 12 0 .75 0 .06 1, .05 CXPXSXG 12 1 .04 0 .09 1, .47 CXPXSs/SXG 252 14 .93 0 .06 146 TABLE 22 ANALYSIS OF VARIANCE ON THE SCORES ON THE TRANSFORMATION TEST WITH LEVEL OF DEVELOPMENT (G) DETERMINED BY PERFORMANCE ON THE ORDINAL CORRESPONDENCE TEST (METHOD 2) SOURCE DF SS MS F P T o t a l 751 168 .22 Between 93 S e s s i o n O rder (S) r 0 .18" 0. 18 0. ,22 L e v e l of Development (G) 4 17 .88 4. 47 5. ,52 .005 SXG 4 2 .99 0. 75 0. ,93 Ss/SXG 84 68 .02 0. 81 W i t h i n 658 Card (C) 3 0 .60 0. 20 2. ,37 P o s i t i o n (P) 1 14 .52 14. 52 69. ,57 .005 CXP 3 0 .96 0. 32 5. ,16 .005 CXSs 279 CXS 3 0 .10 0. 03 0. .38 CXG 12 2 .07 0. 17 2. ,05 .05 CXSXG 12 0 .78 0. 06 0. .77 CXSs/GXS 252 21 .23 0. 08 PXSs 93 PXS 1 0 .05 0. 05 0. .25 PXG 4 2 .90 0. 73 3. .48 .025 PXSXG 4 1 .58 0. 40 1, .89 PXSs/SXG . 84 17 .53 0. 21 CXPXSs 279 CXPXS 3 0 .11 0. 04 0, .58 CXPXG 12 1 .06 0. 09 1, .43 CXPXSXG 12 0 .10 0. .01 0, .14 CXPXSs/SXG 252 15 .56 0. .06 147 TABLE 24 ANALYSIS OF VARIANCE ON THE SCORES ON THE TRANSFORMATION TEST WITH LEVEL OF DEVELOPMENT (G) DETERMINED BY PERFORMANCE ON THE TEST FOR CONSERVATION OF NUMBER AND THE TEST FOR INTENSIVE QUANTITY (METHOD 4) SOURCE DF SS MS F P T o t a l 751 168.22 Between 93 S e s s i o n Order (S) 1 0.18 0.18 0.23 L e v e l o f Development (G) 2 18.95 9.48 12.14 .005 SXG 2 1.25 0.63 0.80 Ss/SXG 88 68.69 0.78 W i t h i n 658 Card (C) 3 0.60 0.20 2.49 P o s i t i o n (P) 1 14.52 14.52 61.79 .005 CXP 3 0.96 0.32 5.39 .01 CXSs 279 CXS 3 0.10 0.03 0.40 CXG 6 2.15 0.36 4.45 .01 CXSXG 6 0.72 0.12 1.50 CXSs/GXS 264 21.22 0.08 PXSs 93 PXS 1 0.05 0.05 0.22 PXG 2 0.44 0.22 0.94 PXSXG 2 0.90 0.45 1.91 PXSs/SXG 88 20.68 0.24 PXCXSs 279 CXPXS 3 0.11 0.04 0.61 CXPXG ' 6 0.26 0.04 0.73 CXPXSs/SXG 6 0.84 0.14 2.38 CXPXSs/SXG 264 15.61 0.06 

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