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Memory performance as related to individual differences with respect to a unified formal-operational… Groves, Muriel Kathleen 1977

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MEMORY PERFORMANCE AS RELATED TO INDIVIDUAL DIFFERENCES WITH RESPECT TO A UNIFIED FORMAL-OPERATIONAL STRUCTURE by MURIEL KATHLEEN CROKER GROVES B.A., U n i v e r s i t y of B r i t i s h Columbia, 1963 M.A., U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY xn THE FACULTY OF GRADUATE STUDIES (Department of Psychology) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1977 © M u r i e l Kathleen Croker Groves, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requ i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f P g y r h n l ogv The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date 21 Dec.- 1 9 7 7 ABSTRACT The major aims of the study were to i d e n t i f y i n d i v i d u a l differences with respect to a u n i f i e d formal-operational structure, independent of age and IQ, and to re l a t e these to predictable differences i n memory performance on a variety of tasks. F i f t y - s i x female grade seven students were adminis-tered the vocabulary test of the Wechsler Intelligence Scale for Children and four Piagetian tasks, the chemical combina-tions, pendulum, balance, and conservation and measurement of volume tasks. In a l a t e r session, they were administered eight memory tasks, each designed to be related both to general formal-operational a b i l i t y and to one or more p a r t i c u l a r Piagetian schemes or concepts. The l a t t e r included the conservation of occupied volume, the understanding of combinations and permutations, and the method of holding variables constant to test the e f f e c t of others. Memory of the displays was tested immediately and four weeks l a t e r . The two hypotheses concerning the u n i f i e d structure of formal-operations were confirmed. F i r s t , even when the eff e c t s of age and IQ were removed s t a t i s t i c a l l y , s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n s were found between performance on each of the four f o r m a l - o p e r a t i o n a l tasks and the average of per-formance on the other three t a s k s . Second, a p r i n c i p a l components a n a l y s i s revealed that the f i r s t component accounted f o r a s u b s t a n t i a l 89 percent of the variance of the assessment t a s k s . The p r i n c i p a l hypothesis concerning memory performance as r e l a t e d to f o r m a l - o p e r a t i o n a l competency was confirmed. Even when the e f f e c t s of age and IQ were removed, average P i a g e t i a n task performance was s i g n i f i c a n t l y c o r r e l a t e d w i t h o v e r a l l memory performance i n the o r i g i n a l (r = .47) and r e t e s t (r = .36) pe r i o d s . Furthermore, average P i a g e t i a n task performance showed p o s i t i v e and, p a r t i c u l a r l y i n the o r i g i n a l t e s t i n g p e r i o d , o f t e n s i g n i f i c a n t c o r r e l a t i o n s w i t h performance on the s p e c i f i c memory ta s k s . Two s u b s i d i a r y hypotheses were not confirmed. In general, performance on p a r t i c u l a r P i a g e t i a n tasks thought to be measuring s p e c i f i c f o r m a l - o p e r a t i o n a l schemes or concepts was not s i g n i f i c a n t l y r e l a t e d to performance on p a r t i c u l a r memory tasks a l s o thought r e l a t e d to the schemes. Secondly, c o n t r a r y to expectations based on a hypothesized considerable d e t e r i o r a t i o n i n memory performance over time on the part of con c r e t e - o p e r a t i o n a l Ss who d i d w e l l i n i t i a l l y , the magnitude of the r e l a t i o n s h i p between P i a g e t i a n task performance and memory performance decreased r a t h e r than increased from the o r i g i n a l to the r e t e s t p e r i o d . P o s s i b l e reasons f o r the l a c k of c o n f i r m a t i o n of these two hypotheses were discussed. A l s o , the d i s c u s s i o n concerned the p o s i t i v e f i n d i n g s as r e l a t e d to the concept of a u n i f i e d f o r m a l - o p e r a t i o n a l s t r u c t u r e , p o s s i b l e design weaknesses i n s t u d i e s not f i n d i n g consistency of performance across formal t a s k s , the s e l e c t i o n of tasks p r o v i d i n g optimal measurement of f o r m a l - o p e r a t i o n a l a b i l i t y , and the d i s t i n c t i o n between the psychometric and P i a g e t i a n concepts of i n t e l l i g e n c e . F i n a l l y , the f i n d i n g of a r e l a t i v e l y high percentage of Ss (42.9) at the f o r m a l - o p e r a t i o n a l stage was discussed i n terms of the methodology of the present study and the p o s s i b i l i t y of the u n i v e r s a l achievement of formal operations. V TABLE OF CONTENTS Page LIST OF TABLES v i i i LIST OF FIGURES x i ACKNOWLEDGEMENTS x i i Chapter 1. INTRODUCTION 1 PIAGET'S THEORY OF COGNITIVE DEVELOPMENT ... 2 Concrete Operations 2 Formal Operations 4 EXPERIMENTS RELATING TO FORMAL-OPERATIONAL THOUGHT 11 Commonly Used Formal-Operational Tasks ... 11 E m p i r i c a l Status of Formal Operations .... 15 HISTORICAL OVERVIEW OF MEMORY RESEARCH 25 S t a t i c Concept of Memory 25 Dynamic Concept of Memory 27 STUDIES RELATING MEMORY TO COGNITIVE DEVELOPMENT 32 General: Memory and Co g n i t i v e Development 32 Memory and Development from Concrete to Formal Operations 40 v i Chapter Page PURPOSE OF THIS STUDY 43 2. METHOD 45 METHOD: GENERAL 45 Subjects 48 Design 48 General Procedure 49 ASSESSMENT TASKS 50 Chemical Combinations of Colorless Liquids 50 Pendulum 54 Conservation and Measurement of Volume .... 59 Equilibrium i n the Balance 61 Vocabulary Test: Wechsler Intelligence Scale for Children 65 MEMORY TASKS 65 Memory Related to Volume Conservation and Measurement: Task 1 66 Memory Related to Combinations: Tasks 2 and 3 70 Memory Related to Permutations: Tasks 4 and 5 78 Memory Related to the L a t t i c e of Propositions: Tasks 6, 7, and 8 87 Overall Memory Performance Measure 97 3. RESULTS 99 RESULTS: ASSESSMENT TASKS 99 v i i Chapter Page Categorizing of Performance on the Piagetian Tasks and WISC Vocabulary Performance 99 Unadjusted and Adjusted Correlation Matrices for Assessment Tasks 102 Pr i n c i p a l Components Analysis of Assessment Tasks 105 RESULTS: MEMORY TASKS 106 RESULTS: RELATIONS BETWEEN ASSESSMENT TASKS AND MEMORY TASKS 109 Overall Memory Performance and Assessment Task Performance 110 Performance on S p e c i f i c Memory Tasks and Average Piagetian Task Performance .. 113 Performance on S p e c i f i c Memory and Spe c i f i c Piagetian Tasks 116 4. DISCUSSION 120 FORMAL OPERATIONS AND MEMORY 120 Unified Structure of Formal Operations ... 120 Formal-Operational Competency and Memory Performance 123 UNIVERSALITY OF FORMAL OPERATIONS 129 REFERENCES 135 APPENDIXES 141 A. Complete Tables of Unadjusted and P a r t i a l Correlations Among Assessment Tasks and between Assessment Tasks and Memory Tasks: Tables 11 to 17 141 B. Minimal Correlations between Performance on Spe c i f i c Piagetian and Memory Tasks: Possible Reasons 149 v i i i LIST OF TABLES Table Page 1. L a t t i c e of propositions 7 2. Sample of combinations i n the o r i g i n a l display and i n a S's reconstruction, with successive pairs scored for horizontal and v e r t i c a l order • 77 3. Sample of permutations in the o r i g i n a l display and i n a S's reconstruction, with successive pairs scored for i n i t i a l members constant (IMC) and v e r t i c a l order 86 4. Percentages of the 56 Ss whose performance on each Piagetian task and average performance on a l l the Piagetian tasks was assigned to each stage and substage 101 5. Product-moment c o r r e l a t i o n matrix and adjusted correlations, with the effects of age and WISC vocabulary scores removed, for the assessment tasks..' 104 6. P r i n c i p a l component loadings for performance on the Piagetian tasks and the WISC vocabulary test 106 7. Maximum possible value of each composite measure and the o r i g i n a l and retest means and standard deviations, expressed as a percentage of the maximum value, for each memory task 108 8. Or i g i n a l and retest unadjusted corr e l a t i o n s , and adjusted correlations, with the effects of age and WISC vocabulary performance removed, between o v e r a l l memory performance and performance on the assessment tasks 112 O r i g i n a l and r e t e s t unadjusted c o r r e l a t i o n s and adjusted c o r r e l a t i o n s , w i t h the e f f e c t s of age and WISC vocabulary performance removed, between average performance on the P i a g e t i a n tasks and memory task performance O r i g i n a l and r e t e s t unadjusted c o r r e l a t i o n s and adjusted c o r r e l a t i o n s , w i t h the e f f e c t s of age and WISC vocabulary performance removed, between memory task performance and performance on p a r t i c u l a r P i a g e t i a n tasks Product-moment c o r r e l a t i o n matrix and p a r t i a l c o r r e l a t i o n s f o r the assessment tasks O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between P i a g e t i a n task performance and o v e r a l l memory performance... O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between P i a g e t i a n task performance and performance on the memory task r e l a t e d to volume conservation. , O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between P i a g e t i a n task performance and performance on the memory tasks r e l a t e d to combinations O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between P i a g e t i a n task performance and performance on the memory tasks r e l a t e d to permutations O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between P i a g e t i a n task performance and performance on the memory tasks r e l a t e d to the l a t t i c e . »* O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between average P i a g e t i a n task performance and performance on the component measures i n each memory task X Table Page 18. Possible reasons for the lack of relationships between performance on s p e c i f i c memory tasks and s p e c i f i c Piagetian tasks 156 x i LIST OF FIGURES Figure Page 1. Apparatus f o r pendulum task 54 2. Apparatus f o r e q u i l i b r i u m i n the balance task 62 3. The top one h a l f of the d i s p l a y of the 15 combinations of four animals, red dog, green dog, red c a t , and green cat 74 4. The bottom one h a l f of the d i s p l a y of the 15 combinations of four v e h i c l e s , t r u c k , C a d i l l a c , Volkswagen, and motorcycle . ... 75 5. The s i x permutations of three people, f a t h e r , mother, and son = 82 6. The f i r s t 6 permutations of the 24 permutations of four people, f a t h e r , mother, son, and daughter 83 7. A s s o c i a t i o n matrix of four v a r i a b l e s , each w i t h two values, w i t h p o s i t i v e (,/) and negative (x) instances shown 92 8. A s s o c i a t i o n matrix of three v a r i a b l e s , each w i t h two or three values, w i t h p o s i t i v e and negative (x) instances shown 94 9. A s s o c i a t i o n matrix of three v a r i a b l e s , each w i t h two values, w i t h p o s i t i v e (,/) and negative (x) instances shown 95 x i i ACKNOWLEDGEMENTS I wish to thank my advisors, Dr. L. J. Moran, Dr. P. K. A r l i n , Dr. R. S. Corteen, and Dr. L. M. Ward, for their assistance and comments. I esp e c i a l l y would l i k e to thank Dr. Moran for his kindness, encouragement, and good temper throughout th i s project and Dr. A r l i n for her invaluable assistance with the Piagetian tasks. I also would l i k e to thank my family for their support. A special thanks to my parents, Flora and Arthur Croker, for their very great help and encouragement and to my l i t t l e sons, John and Stuart, for being such good boys and making i t possible for me to do th i s research. 1 C h a p t e r 1 I N T R O D U C T I O N T h e m a j o r a i m s o f t h e p r e s e n t s t u d y w e r e t o i d e n t i f y i n d i v i d u a l d i f f e r e n c e s w i t h r e s p e c t t o a u n i f i e d f o r m a l -o p e r a t i o n a l s t r u c t u r e a n d t o r e l a t e t h e s e t o p r e d i c t a b l e d i f f e r e n c e s i n memory p e r f o r m a n c e o n a v a r i e t y o f t a s k s . I n o r d e r t o i n t r o d u c e t h e s t u d y a n d e x p l a i n i t s p u r p o s e , t h i s c h a p t e r w i l l p r o v i d e t h e f o l l o w i n g : 1. A r e v i e w o f P i a g e t ' s t h e o r y o f c o g n i t i v e d e v e l o p -m e n t w i t h some e m p h a s i s o n t h e c o n c r e t e - o p e r a t i o n a l s t a g e a n d m a j o r e m p h a s i s o n t h e f o r m a l - o p e r a t i o n a l s t a g e , w h i c h i s o f p r i m e c o n c e r n i n t h i s s t u d y . 2. A d e s c r i p t i o n o f t h e c o m m o n l y r e p l i c a t e d f o r m a l -o p e r a t i o n a l e x p e r i m e n t s w i t h a d i s c u s s i o n o f t h e e m p i r i c a l s t a t u s o f t h e f o r m a l - o p e r a t i o n a l s t a g e . 3. A n o v e r v i e w o f m e m o r y r e s e a r c h i n c l u d i n g t h e t w o h i s t o r i c a l v i e w s o f memory a n d P i a g e t ' s c o n c e p t o f memory; t h e l a t t e r 1 s r e l a t i o n s h i p b o t h t o t h e s e h i s t o r i c a l a p p r o a c h e s a n d t o t h e c u r r e n t l y p r e v a i l i n g v i e w s i n N o r t h A m e r i c a w i l l b e n o t e d . 4. A r e v i e w o f s t u d i e s r e l a t i n g c o g n i t i v e 2 development to memory. 5. A statement of the purpose of the present study. PIAGET'S THEORY OF COGNITIVE DEVELOPMENT Piaget (e.g., Inhelder and Piaget, 1958; Piaget, 1950; Piaget and Inhelder, 1969) has postulated several stages i n the c h i l d ' s development to mature adult thinking, the sensory-motor, preoperational, concrete-operational, and formal-operational stages. The l a t t e r two, which are of concern i n this study, w i l l be discussed below. Concrete Operations According to Piaget, the concrete-operational stage, occurring from approximately 7 to 11 years, involves several substructures or groupings. These enable the c h i l d to organize and understand data from the world i n terms of either c l a s s i f i -cations or r e l a t i o n s . Thus the concrete-operational c h i l d develops many new operations which are unavailable to . the pre-operational c h i l d , only a few of which w i l l be mentioned below. For example, the concrete-operational c h i l d becomes capable of h i e r a r c h i c a l l y c l a s s i f y i n g s t i m u l i i n the environment and understands the re l a t i o n s h i p between classes and subclasses. Also i n t h i s stage, the understanding of the s e r i a t i o n of objects such as sti c k s d i f f e r i n g i n size i s acquired. S i m i l a r l y the c h i l d understands the c l a s s i f i c a t i o n of objects 3 i n terms of more than one dimension to form a m u l t i p l i c a t i v e matrix, such as one based on objects d i f f e r i n g both i n size and color. In addition the c h i l d achieves an understanding of ispacial r e l a t i o n s , including the idea of v e r t i c a l and horizontal, the l a t t e r evidenced by successful prediction of how a l i q u i d would l i e i n a t i l t e d container. The c h i l d ' s imagery becomes less s t a t i c so that he can anticipate the changing positions of objects such as a tr i a n g l e which i s rotated or a f a l l i n g and turning s t i c k . This period i s marked by the ch i l d ' s development of a number of conservations, that i s , understandings that certain properties of objects remain the same despite trans-formations that may change the physical appearance of the objects. These conservations include those pertaining to substance, weight, length, and number. For example, the c h i l d r e a l i z e s that despite changes i n the shape of a b a l l of clay, i t s weight and amount of substance remain the same. Si m i l a r l y , the c h i l d knows that the rearrangement of two st i c k s of i d e n t i c a l length or two rows containing the same number of objects does not a l t e r the equivalence of the objects or sets of objects. These conservations seem cl o s e l y related to the c h i l d ' s new understanding of the r e v e r s i b i l i t y of operations. This occurs through both r e c i p r o c i t y , n e u t r a l i z -ing the operation while leaving i t intact (as i n the application of an equal counterforce), and more p a r t i c u l a r l y negation, a c t u a l undoing of the ope r a t i o n . Despite i t s many achievements, the thought of the con c r e t e - o p e r a t i o n a l c h i l d has c e r t a i n l i m i t a t i o n s . F i r s t l y , as the name of the stage i m p l i e s , the operations are d i r e c t e d toward concrete things and happenings i n the present. Furthermore, the various l o g i c a l groupings of the concrete-o p e r a t i o n a l c h i l d are not i n t e g r a t e d i n t o one u n i f i e d system, which would be r e q u i r e d f o r success at c e r t a i n complex tasks. For example, the co n c r e t e - o p e r a t i o n a l c h i l d , w h i l e possessing the two types of r e v e r s i b l e operations, negation, found i n the c l a s s groupings, and r e c i p r o c i t y , found i n the r e l a t i o n a l groupings, cannot co-ordinate these operations. This l a c k of c o - o r d i n a t i o n i s seen i n the problem i n v o l v i n g a seesaw balance, where the e q u i l i b r i u m i s d i s t u r b e d by the a d d i t i o n of e x t r a weight on one s i d e . The c h i l d may r e a l i z e that the removal of the added weight (negation) or the r e p o s i t i o n i n g of weight on one or both sides ( r e c i p r o c i t y ) may b r i n g the balance once more i n t o e q u i l i b r i u m . He does not know, however, how to co-ordinate these operations i n any p r e c i s e l o g i c a l or mathematical manner. Formal Operations General. The t h i n k i n g of the fo r m a l - o p e r a t i o n a l c h i l d overcomes the shortcomings of the con c r e t e - o p e r a t i o n a l stage. A major achievement of the fo r m a l - o p e r a t i o n a l p e r i o d 5 i s that an adolescent at t h i s stage considers not only the r e a l but a l s o the p o s s i b l e . What a c t u a l l y happens i s a subset of a l l the p o s s i b i l i t i e s which the c h i l d i s capable of envisaging. U n l i k e the c o n c r e t e - o p e r a t i o n a l c h i l d , whose world i s the concrete, the f o r m a l - o p e r a t i o n a l adolescent operates i n the framework of the hypothetico-deductive method. The adolescent i n t r y i n g to determine the cause of c e r t a i n phenomena may e n t e r t a i n a number of hypotheses or p r o p o s i t i o n s from which deductions are made; these deductions are t e s t e d w i t h r e s u l t i n g c o n f i r m a t i o n or d i s c o n f i r m a t i o n of the v a r i o u s hypotheses. Thus the f o r m a l - o p e r a t i o n a l c h i l d becomes capable of s c i e n t i f i c thought. When presented w i t h a d i f f i c u l t problem he i s able to i s o l a t e the r e l e v a n t v a r i a b l e s ; e n v i s i o n a l l the v a r i o u s combinations or s o l u t i o n s to the task; t e s t out the e f f e c t s of the various v a r i a b l e s , o f t e n by h o l d i n g f a c t o r s constant and manipulating others; and conclude c o r r e c t l y on the b a s i s of experimental r e s u l t s . These conclusions are f a c i l i t a t e d by the i n t e g r a t i o n of the operations of r e c i p r o c i t y and negation i n t o the group s t r u c t u r e , which i s described below. D e s c r i p t i v e models. In a d d i t i o n to the general c h a r a c t e r i s t i c s of formal operations given'above, Piaget uses two l o g i c a l models, the l a t t i c e and the group, to describe i n d e t a i l t h e p e r i o d o f f o r m a l o p e r a t i o n s . T o g e t h e r t h e s e s t r u c t u r e s p r o v i d e t h e l o g i c o - m a t h e m a t i c a l p r o p e r t i e s c o n s i d e r e d i n h e r e n t i n a d o l e s c e n t t h o u g h t . 1. L a t t i c e s t r u c t u r e . A s m e n t i o n e d p r e v i o u s l y , t h e f o r m a l - o p e r a t i o n a l a d o l e s c e n t i s c a p a b l e o f i s o l a t i n g t h e r e l e v a n t v a r i a b l e s a n d t h e n c o m b i n i n g t h e m i n a n o r d e r l y a n d e x h a u s t i v e m a n n e r . T h e r e s u l t i n g n e t w o r k o f p o s s i b i l i t i e s i s c a l l e d t h e l a t t i c e . F o r e x a m p l e , o n e c a n c o n s i d e r t h e s i z e o f a n i n d i v i d u a l ( p = f a t a n d p = t h i n ) a n d s t a t e o f m i n d ( q = h a p p y a n d q = s a d ) . B o t h t h e c o n c r e t e - o p e r a t i o n a l a n d t h e f o r m a l -o p e r a t i o n a l c h i l d c a n c o me u p w i t h t h e f o u r p o s s i b l e b a s e a s s o c i a t i o n s , p q , p q , p q a n d p q , t h a t i s , f a t a n d h a p p y , f a t a n d s a d , t h i n a n d h a p p y , t h i n a n d s a d . T h e c o n c r e t e -o p e r a t i o n a l c h i l d , h o w e v e r , c o n s i d e r s t h e f o u r a s s o c i a t i o n s a s c o n c r e t e p h e n o m e n a l e v e n t s . T h e f o r m a l - o p e r a t i o n a l a d o l e s c e n t t r e a t s t h e m a s p r o p o s i t i o n s ; t h e y may b e p o t e n t i a l a n d n o t a c t u a l l y p e r c e i v e d o c c u r r e n c e s . F u r t h e r m o r e , u n l i k e t h e c o n c r e t e - o p e r a t i o n a l c h i l d , t h e a d o l e s c e n t i s c a p a b l e o f g e n e r a t i n g a l l t h e p o s s i b l e c o m b i n a t i o n s o f t h e s e f o u r a s s o c i a t i o n s o r p r o p o s i t i o n s t o f o r m a l a t t i c e o f s i x t e e n p r o p o s i t i o n a l c o m b i n a t i o n s , d e s c r i b e d i n T a b l e 1. T h e a d o l e s c e n t w i t h t h i s l a t t i c e o f p o s s i b i l i t i e s c a n s e t o u t t o d e t e r m i n e w h i c h o f t h e s i x t e e n p o s s i b i l i t i e s a c t u a l l y d o e s o c c u r a n d t h e n c a n f o r m u l a t e h i s c o n c l u s i o n s . T h e t a b l e i n d i c a t e s , i n t h e f o r m o f e i g h t c o m p l e m e n t a r y T a b l e 1. L a t t i c e o f p r o p o s i t i o n s . Name C o m b i n a t i o n s O b s e r v e d Name o f C o m p l e m e n t C o m b i n a t i o n s O b s e r v e d C o m p l e t e a f f i r m a t i o n I n c o m p a t i b i l i t y D i s j u n c t i o n I m p l i c a t i o n ! R e c i p r o c a l i m p l i c a t i o n E q u i v a l e n c e A f f i r m a t i o n o f p A f f i r m a t i o n o f q p q + p q + p q + p q p q + p q + p q p q + p q + p q p q + p q + p q p q + p q + p q p q + p q p q + p q p q + p q N e g a t i o n C o n j u n c t i o n C o n j u n c t i v e n e g a t i o n N o n i m p l i c a t i o n N e g a t i o n o f r e c i p r o c a l i m p l i c a t i o n R e c i p r o c a l e x c l u s i o n N e g a t i o n o f p N e g a t i o n o f q p q p q p q p q p q + p q p q + p q p q + pq: 8 p a i r s o f p r o p o s i t i o n s , t h e name o f e a c h p r o p o s i t i o n a n d t h e c o m b i n a t i o n s o b s e r v e d i f t h e p r o p o s i t i o n i s c o n f i r m e d . F o r e x a m p l e , t h e c h i l d may o b s e r v e t h e c o m b i n a t i o n s p q ( f a t a n d s a d ) a n d p q ( t h i n a n d h a p p y ) , l a b e l l e d a s r e c i p r o c a l e x c l u s i o n i n l i n e 6 , c o l u m n 3 , o f t h e t a b l e . T h e c h i l d may n e v e r o b s e r v e p q ( f a t a n d h a p p y ) a n d p q ( t h i n a n d s a d ) w h i c h i s t h e p r o p o s i t i o n o f " e q u i v a l e n c e , " t h e c o m p l e m e n t o f r e c i p r o c a l e x c l u s i o n ( s e e l i n e 6 , c o l u m n 1, o f t h e t a b l e ) . T h u s t h e c h i l d may c o n c l u d e t h a t o b e s i t y a n d h a p p i n e s s n e v e r c o - o c c u r ; t h e y a r e n e g a t i v e l y c o r r e l a t e d o r i n c o m p a t i b l e . 2 . G r o u p s t r u c t u r e . T h e t h i n k i n g o f t h e f o r m a l -o p e r a t i o n a l c h i l d a l s o m a n i f e s t s t h e p r o p e r t i e s o f t h e g r o u p . T h i s s t r u c t u r e h e l p s t o c a p t u r e t h e e s s e n c e o f how t h e a d o l e s -c e n t m a n i p u l a t e s t h e r e s u l t s o f h i s e x p e r i m e n t s t o come t o c e r t a i n c o n c l u s i o n s b e y o n d t h e d a t a . T h e g r o u p c o n t a i n s f o u r t r a n s f o r m a t i o n s , i d e n t i t y , n e g a t i o n , r e c i p r o c a l , a n d c o r r e l a t i v e ; a . I d e n t i t y ( I ) . T h i s " n u l l " t r a n s f o r m a t i o n c h a n g e s n o t h i n g . T h u s i f t h e p r o p o s i t i o n i s p v q ( f a t a n d / o r h a p p y ) , t h e n I ( p v q ) = P V q - S i m i l a r l y , t h e i d e n t i t y t r a n s f o r m a t i o n o f p . q . ( f a t a n d h a p p y ) i s p . q . b . N e g a t i o n ( N ) . T h i s t r a n s f o r m a t i o n n e g a t e s a l l a s p e c t s o f t h e p r o p o s i t i o n s . A l l c o n j u n c t i v e ( a n d ) s t a t e m e n t s b e c o m e d i s j u n c t i v e ( a n d / o r ) s t a t e m e n t s , a n d v i c e v e r s a , a n d a l l a s s e r -t i o n s b e c o m e n e g a t i o n s , a n d v i c e , v e r s a . T h u s N (pvq) i s p . q , 9 or to provide an example, the negation of f a t and/or happy i s t h i n and sad. c. R e c i p r o c a l (R). This transformation changes a s s e r t i o n s and negations but leaves conjunctions and d i s j u n c -t i o n s unchanged. For example, R (p.q) = p.q, or the r e c i p r o c a l of f a t and happy i s t h i n and sad. d. C o r r e l a t i v e (C). The c o r r e l a t i v e transformation a l t e r s c o n j u n c t i v e and d i s j u n c t i v e p r o p o s i t i o n s , but a s s e r t i o n s and negations are unchanged. Thus C (p.q)' equals p V q ^ o r C ( f a t and happy) i s f a t and/or happy. In order to reach conclusions the S uses the various INRC transformations on h i s data. For example, i f the S f i n d s that a long, l i g h t rod bends and so does a s h o r t , heavy one, he can understand that a long, l i g h t rod i s the r e c i p r o c a l of a s h o r t , heavy rod. In other words, an increase i n weight can be counteracted by a decrease i n length and v i c e versa. Furthermore, through the c o r r e l a t i v e t ransformation, he can conclude that the c o r r e l a t i v e of long i s heavy; that i s , length and weight have the same e f f e c t and are both c o r r e l a t e d w i t h bending. An example of the use of the negative transformation i s provided by Inhelder and Piaget (1958, Chapter 8) i n the conservation of motion task. Success at t h i s task r e q u i r e s f i r s t the d i s c o v e r y that the stopping of a b a l l on a h o r i z o n t a l plane r e s u l t s from a v a r i e t y of f a c t o r s , such as f r i c t i o n and 10 a i r r e s i s t a n c e . The manipulation of t h i s d iscovery by the negative transformation makes p o s s i b l e the c o n c l u s i o n that the absence of these f a c t o r s i n v o l v e s the b a l l not stopping. These two models, then, the l a t t i c e and the group, form the s t r u c t u r e of the f o r m a l - o p e r a t i o n a l p e r i o d . As the under-standing of the 16 p r o p o s i t i o n s develops, the c h i l d becomes aware of t h e i r i n t e r r e l a t i o n s and l e a r n s to transform them through the INRC group; thus the presence of the l a t t i c e pre-supposes the presence of the group and v i c e versa. Formal-operational concepts. From t h i s i n t e g r a t e d t o t a l s t r u c t u r e are developed s u b s t r u c t u r e s , or formal-o p e r a t i o n a l schemes, which are s p e c i a l i z e d f o r c e r t a i n problems. These i n c l u d e : 1. Combinatorial operations, which are systematic procedures f o r generating a l l the p o s s i b l e permutations or combinations of o b j e c t s . 2 . P r o p o r t i o n s , which i n v o l v e s the a b i l i t y to deal w i t h the e q u a l i t y of two r a t i o s , X/Y = X /Y , as i n the under-standing of the balance. 3 . M u l t i p l i c a t i v e compensations, c l o s e l y r e l a t e d to ^proportions, which i n v o l v e s the understanding, as i n the case of volume conservation, that gains i n one dimension can be compensated by changes i n other dimensions; that i s , rectangular b u i l d i n g s of d i f f e r e n t dimensions can be understood and c a l c u l a t e d to have e x a c t l y the same volume. 11 4. Co-ordination of two systems of reference, which, f o r example, i n v o l v e s the understanding of the p o s i t i o n of a person (ih.aterms of an e x t e r n a l frame of reference) who i s walking on a moving sidewalk i n a d i r e c t i o n opposite to that of the sidewalk-'-s movement. 5. The concept of mechanical e q u i l i b r i u m , which i n v o l v e s the understanding of opposing f o r c e s as i n a c t i o n and r e a c t i o n . 6 . The concept of p r o b a b i l i t y , which i n v o l v e s the understanding of the r a t i o of the number of confirming cases to the t o t a l number of e q u a l l y l i k e l y cases, the l a t t e r c a l c u l a t i o n r e q u i r i n g the knowledge of combinations. 7. C o r r e l a t i o n , which i n v o l v e s understanding the degree of r e l a t i o n between v a r i a b l e s . 8. Conservation i n the a b s t r a c t , which i n v o l v e s forms of conservation (such as the conservation of motion) that go beyond d i r e c t e m p i r i c a l discovery or v e r i f i c a t i o n . EXPERIMENTS RELATING TO FORMAL-OPERATIONAL THOUGHT The f o l l o w i n g i n v o l v e s a d e s c r i p t i o n of the more commonly r e p l i c a t e d f o r m a l - o p e r a t i o n a l tasks and a d i s c u s s i o n of the e m p i r i c a l s t a t u s of formal operations. Commonly Used Formal-Operational Tasks The m a j o r i t y of the tasks r e l a t i n g to f o r m a l - o p e r a t i o n a l thought are the 15 simple physical experiments reported i n Inhelder and Piaget (1958). In these the c h i l d i s required to experimentally manipulate variables i n order to reach con-clusions concerning the p r i n c i p l e s involved. To provide an idea of these 15 experiments the 7 most commonly re p l i c a t e d ones w i l l be discussed. In addition, there w i l l be a description of the volume conservation task (see Piaget and Inhelder, 1941; Piaget, Inhelder, and Szeminska, 1960), which has been widely r e p l i c a t e d (e.g., Elkind, 1961b, 1962; Towler and Wheatley, 1971). In the case of the f i r s t seven tasks, Inhelder independently conducted the studies and Piaget afterward described the lo g i c allegedly used. In the description which follows concerning these tasks and the volume conservation task, the l o g i c ( i n terms of the formal-operational structures and schemes) supposedly measured by the tasks i s that of Piaget. The following tasks, then, are some of the most common formal-operational tasks. 1. F l e x i b i l i t y of rods. In thi s task the S i s required to determine which variables are responsible for the f l e x i b i l i t y of rods, the material they are made of, the length, thickness, and/or the form of their cross sections. 2. Pendulum. This task requires the S to determine the e f f e c t s of a number of variables (including the length of st r i n g , the weight of the object fastened to the s t r i n g , the height of the dropping point, and the force of the push) on 13 the frequency of o s c i l l a t i o n of the pendulum. Both the f l e x i b i l i t y and pendulum problems are cl o s e l y related to the l a t t i c e structure. The successful Ss presumably must consider a wide variety of p o s s i b i l i t i e s and determine which of these occur by holding variables constant and manipulating others to see the e f f e c t s of the l a t t e r . 3. Chemical combinations of co l o r l e s s liquids. This study involves combining several chemicals to determine which ones reproduce a yellow color. Success i s related to the l a t -t i c e structure and more p a r t i c u l a r l y to the concept of combinations. The remaining tasks are related to the INRC group and to p a r t i c u l a r formal-operational concepts. 4. Conservation of motion on a horizontal plane. As mentioned previously (p. 9), t h i s task, involving a formal-operational conservation, requires for success the trans-formation of experimental findings by negation to reach a new conclusion; that i s , the S concludes that as certain factors cause a b a l l to stop r o l l i n g , the absence of these factors implies the b a l l w i l l not stop. 5. Correlation. In t h i s task the S i s required to determine i f there i s a r e l a t i o n s h i p between two variables, such as hair and eye color, and what i s the extent of the re l a t i o n s h i p . 6. Equilibrium i n the balance. In both t h i s and the f o l l o w i n g tasks the concept of proportions i s r e q u i r e d . In t h i s task the S i s r e q u i r e d to determine w i t h a seesaw balance the r e l a t i o n s h i p between the magnitudes of weights hung on each side of the fulcrum and the distances from the fulcrum that these weights are hung. 7. P r o j e c t i o n of shadows. Employing r i n g s p l a c e d between a l i g h t source and a screen, t h i s task i n v o l v e s t r y i n g to determine the r e l a t i o n s h i p of the s i z e of the shadows cast both to the diameter of the r i n g s ( d i r e c t proportion) and to the d i s t a n c e between the r i n g s and the l i g h t source ( i n v e r s e p r o p o r t i o n ) . 8. Volume conservation.- This task i n v o l v e s the concepts of both i n t e r i o r volume conservation and the conservation of occupied volume',; which are c l o s e l y r e l a t e d to the INRC group and more p a r t i c u l a r l y to the concept of m u l t i p l i c a t i v e compensations. I n t e r i o r volume conservation i s t e s t e d by having the Ss demonstrate t h e i r understanding of why an object can c o n t a i n the same amount of room or space i n s i d e even when the shape of the object i s changed. The conservation of occupied volume i n v o l v e s r e a l i z i n g that the room occupied by an object (e.g., a model b u i l d i n g under water) w i l l not change when the shape, but not the volume, of the object i s changed. 15 Empirical Status of Formal Operations There would seem to be two major unanswered questions concerning the empirical status of formal-operational thinking. One question concerns whether there exists u n i v e r s a l i t y of achievement of formal-operational thinking among people of normal i n t e l l i g e n c e who are at the age where th i s competency i s supposed to have developed. The other question concerns whether there i s a u n i f i e d formal-operational structure. This would be evidenced by f a i r l y consistent performance across tasks and the presence of one component or factor (produced by p r i n c i p a l components or factor analysis) accounting for a considerable amount of variance i n the tasks. I f , as described by Piaget, formal-operational thinking r e f l e c t s an organized structure of operations, one might expect to f i n d such consistency with properly designed tasks. U n i v e r s a l i t y of formal operations. In the o r i g i n a l statement of t h i s theory (Inhelder and Piaget, 1958), formal-operational thought was considered to develop through the ages 11 to 15 years with equilibrium achieved by 75 percent of adolescents by the age of 15. However, a considerable body of research has not supported t h i s contention, with the percentage of formal-operational Ss r a r e l y i n excess of 55 to 60 percent even i n the case of college students. For example, consider-ation of the conservation of volume, generally characterized 16 as a formal-operational competency that emerges early, reveals considerable lack of u n i v e r s a l i t y . Approximately 25 percent of grade s i x students (Elkind, 1961a; U z g i r i s , 1964), 47 percent of junior and senior high school students (Elkind, 1961b), and 60 percent of college students (Elkind, 1962; Towler and Wheatley, 1971) achieved success on measures of this concept. Such r e s u l t s have led Piaget (1972) to restate his p o s i t i o n concerning t h i s issue and tent a t i v e l y conclude: . . . a l l normal subjects a t t a i n the stage of formal operations or structuring i f not between 11.^ 12 to 14 - 15 years, i n any case between 15 and 20 years. However, they reach t h i s stage i n d i f f e r e n t areas according to their aptitudes and their professional s p e c i a l i z a t i o n s (advanced studies or d i f f e r e n t types of apprenticeship for the various trades): the way i n which these formal structures are used, however, i s not necessarily the same in a l l cases (pp.'9-10). Unif i e d structure of formal operations. Surprisingly few studies have assessed the performance of Ss i n a wide number of formal-operational tasks, and thus the presence of a u n i f i e d formal-operational structure i s not clear. The majority of the l i m i t e d number of studies that have i n v e s t i -gated the question of consistency of performance among these tasks have been interpreted, however, to support a consistency p o s i t i o n . One of the most ambitious studies (Lovell, 1961) employed 5 combinations of 10 of the experiments used by Inhelder and Piaget (1958). These combinations were given to d i f f e r e n t groups of Ss between 8 and 18 years of age , 17 (192 Ss i n a l l ) . The values of Kendall's c o e f f i c i e n t of concordance W, used as an indicator of the relat i o n s h i p between tasks, were s i g n i f i c a n t . They varied from .89 to .52, depending upon the age and the a b i l i t y range of the students who took the p a r t i c u l a r combination of tasks. For example, comparison of the performance of 50 comprehensive students (secondary students offered a variety of vocational and academic courses) on the chemical combinations, pendulum, balance, and shadows tasks produced a W of .73. This i s equivalent to an average Spearman rank-order c o r r e l a t i o n c o e f f i c i e n t (p) of approxi-mately .64. In another study, Jackson (1965) employed two groups of children, average or subnormal i n i n t e l l i g e n c e , who ranged in age from 5 to 15 years. The performance of these Ss on each of s i x formal-operational tasks was assigned to one of s i x substages. Over 70 percent of the Ss i n each group had a l l their responses included within two or fewer substages. When the data from both groups were combined, Jackson found that o v e r a l l performance on a l l the Piagetian tasks showed rank correlations of .61 and .86 with, respectively, age and in t e l l i g e n c e , as measured by Raven's Matrices scores. S i m i l a r l y , Tomlinson-Keasey (1970), using as Ss si x t h -grade g i r l s , college students, and mature women (mean age, 54 years), found s i g n i f i c a n t l y p o s i t i v e correlations between the pendulum, balance, and f l e x i b i l i t y of rods tasks. These 18 ranged from r = .21 (pendulum-flexibility of rods) to r = .45 (pendulum-balance). In another study, Lee (1971) reported a p o s i t i v e c o r r e l a t i o n of .85 between the balance and the shadows tasks, with Ss selected from kindergarten through 12th grade. Two studies by A r l i n (1974, 1977) found p o s i t i v e correlations between the formal-operational tasks used. The f i r s t study involved female college seniors of approximately the same age, randomly selected from a l l the female students enrolled i n t h e i r f i r s t class i n educational psychology. The correlations found between the pendulum, shadows, and chemical combinations tasks were small but s i g n i f i c a n t , ranging from r = .22 (pendulum-shadows) to r = .39 (pendulum-chemical combinations). The second study, which employed male and female students selected randomly from a similar c l a s s , found a s i g n i f i c a n t c o r r e l a t i o n of .58 between the chemical com-binations and pendulum tasks. A number of studies have submitted the performance of Ss to factor analysis and the majority have found that the formal-operational tasks investigated loaded heavily on one factor. L o v e l l (1971) reports a study by Hughes (1965), involving 40 pupils of average and below average a b i l i t y tested yearly from 11+ years to 14+ years. Kendall's c o e f f i c i e n t of concordance r e l a t i n g performance on the balance, pendulum, chemical combinations, and f l e x i b i l i t y of rods tasks varied from W = .39 on the f i r s t testing to W = .57 on the fourth. The l o g i c a l thinking scores on these tasks obtained on the fourth testing were also p o s i t i v e l y correlated with scores on other tasks, such as tests of nonverbal i n t e l l i g e n c e and numerical analogies. A l l the tests showed substantial loadings, ranging from .57 (pendulum) to .81 (chemical combinations) on the f i r s t p r i n c i p a l component yielded by a p r i n c i p a l components analysis. S i m i l a r l y , L o v e l l and Butterworth (1966) found that performance on a number of tasks involving proportions, including the shadows and balance tasks, correlated highly, .79 or more, with the f i r s t p r i n c i p a l component. Another study by Lov e l l and Shields (1967) involved 30 children 8 to 11 years o ld who had verbal IQs on WISC of 140 or higher. Included i n the battery of tests were the balance, chemical combinations, and pendulum tasks; they showed s i g n i f i c a n t loadings, respectively .37, .61, and .72, on the f i r s t axis r e s u l t i n g from a p r i n c i p a l components analysis. When the p r i n c i p a l axes were rotated by the varimax method, the loadings of these tasks on one factor were a l l increased to respectively .83, .72, and .60. Si m i l a r l y Bart (1971), using 90 Ss of ages 13, 16, and 19 years, found evidence for a unifactor underlying formal-operational thought. Intercorrelations between the shadows, balance, and pendulum tasks ranged from .52 to .78 with the 20 e f f e c t of v e r b a l i n t e l l i g e n c e , measured by the Experimental Omnibus Vocabulary Test, s t a t i s t i c a l l y removed through p a r t i a l c o r r e l a t i o n techniques. These P i a g e t i a n tasks and the vocabulary t e s t were seen as u n i f a c t o r , c o r r e l a t i n g from .44 (vocabulary) to .89 (balance) w i t h a f a c t o r r e s u l t i n g from an u n r e s t r i c t e d maximum l i k e l i h o o d f a c t o r a n a l y s i s . Contrary to the m a j o r i t y of the researchers employing f a c t o r a n a l y s i s , however, Ross (1973) reported no evidence of a u n i f i e d f o r m a l - o p e r a t i o n a l s t r u c t u r e . He employed a sample of 65 undergraduates of approximately the same age. Among the t e s t s used were the American Co l l e g e Test, a measure of general i n t e l l i g e n c e , and the balance, pendulum, c o r r e l a t i o n and de n s i t y t a s k s , the l a t t e r task t e s t i n g the understanding of why objects s i n k or f l o a t . A l l the c o r r e l a t i o n s among the formal tasks were i n s i g n i f i c a n t except f o r the c o r r e l a t i o n between the den s i t y and the balance problems, r = .42. The American Co l l e g e Test c o r r e l a t e d s i g n i f i c a n t l y w i t h only the balance (r = .53) and de n s i t y (r = .38) tas k s . Three f a c t o r s , as found by a Varimax r o t a t i o n of the p r i n c i p a l components, were r e q u i r e d to account f o r the m a j o r i t y of variance i n the P i a g e t i a n tasks. A number of other s t u d i e s not employing f a c t o r a n a l y s i s s i m i l a r l y have found no s i g n i f i c a n t l y c o n s i s t e n t performance across a l l the tasks employed. Neimark (1970), w i t h Ss from grades f o u r , f i v e , and s i x , found no s i g n i f i c a n t c o r r e l a t i o n s between the chemical combinations task and a s l i g h t l y modified 21 version of the c o r r e l a t i o n task; the l a t t e r task was found to be of greater d i f f i c u l t y . In addition, Neimark (1975a) reported that Kuhn, Langer, Kohlberg, and Haan (1972), using the pendulum, chemical combinations, and c o r r e l a t i o n tasks, found much more i n t r a - i n d i v i d u a l v a r i a t i o n than did Jackson (1965) and also clear evidence of d i f f e r e n t i a l task d i f f i c u l t y ( i n the order l i s t e d ) . In another study, Neimark (i"975b^ examined at i n t e r v a l s during periods up to as long as nearly four years the perform-ance of Ss i n i t i a l l y chosen from grades three to s i x . Included in the study were tasks measuring the a b i l i t i e s to combine and permute and a variant of the c o r r e l a t i o n task. The permutation task involved having the S provide a l l the permutations of four d i g i t s . The combination task required the S to make a l l possible pairs of coloured squares, each square being one of s i x d i f f e r e n t colours. S i g n i f i c a n t correlations generally were found between the various measures of performance on the permutation and combination tasks. However, the majority of the correlations between these l a t t e r tasks and measures of c o r r e l a t i o n performance were not s i g n i f i c a n t . This was due i n part to the lack of consistent improvement with age of perform-ance on the c o r r e l a t i o n task measures. F i n a l l y , a study by Schwebel (1975), employing two groups of u n i v e r s i t y students, one of 30 males, the other of 30 females, found generally no s i g n i f i c a n t correlations between 22 the f l e x i b i l i t y of rods, balance, and i n c l i n e d plane tasks. The correlations generally were i n the low . 3 0 's, and only the c o r r e l a t i o n i n the men's group between the i n c l i n e d plane and f l e x i b i l i t y of rods tasks was s i g n i f i c a n t , r = . 37 , p < . 05 . While the majority of the preceding studies, with the few exceptions noted (Kuhn et a l . , 1972; Neimark, 1970, 1975b; Ross, 1973; Schwebel, 1975), provide evidence for s i g n i f i c a n t l y consistent performance across the tasks employed, they are open to serious c r i t i c i s m . As none of these studies finding consistency controlled for differences among Ss i n both age and IQ, the obtained p o s i t i v e correlations among tasks may have resulted, at least i n part, from t h i s lack of control. C l e a r l y age i s related to formal-operational thinking and in t e l l i g e n c e also would seem to be (Bart, 1971; Hughes, 1965; Jackson, 1965). In only s i x of the studies showing consistent per-formance across the tasks was some attempt made to control for either age or IQ. In none of these studies were both of these factors controlled. In the case of the IQ variable, employing Ss of d i f f e r e n t ages, Bart (1971) p a r t i a l e d out the effects of verbal i n t e l l i g e n c e , while Jackson (1965) and L o v e l l and Shields (1967) employed groups of Ss having IQs within a f a i r l y narrow range. With regard to age, two studies ( A r l i n , 1974; Hughes, 1965) have d e f i n i t e l y controlled for t h i s 23 variable; moreover, another study ( A r l i n , 1977) has employed Ss that might be expected hot to vary greatly i n age. Hughes employed Ss of the same age i n a longitudinal study while A r l i n (1974) found her sample was f a i r l y homogeneous i n terms of age. Although the ages i n the study of A r l i n (1977) were not reported, i t i s u n l i k e l y that Ss drawn from the same university class would vary widely i n age. In none of the preceding three studies i n which there was at least some control for age were the effects of IQ removed. In the case of the study of Hughes the Ss c l e a r l y varied greatly i n IQ as they were selected to be average or below average i n a b i l i t y . The v a r i a t i o n i n IQ among the univers i t y students employed by A r l i n (1974, 1977) would not be expected to be as great as that among junior or senior high school students, but s t i l l might be considerable. In conclusion then, the preceding studies which showed consistency of performance across tasks provide no evidence for the concept of a u n i f i e d structure of formal operations, independent of age and IQ. In fa c t , of these studies, only three controlled, at least to some extent, for the former variable, and another three had some control for the l a t t e r . None controlled for both. It i s important to consider, furthermore, that several studies c i t e d (Neimark, 1970, 1975b;; Ross, 1973; Schwebel, 1975) did not f i n d that the majority of relationships among tasks were 24 s i g n i f i c a n t l y p o s i t i v e . These findings may be interpreted as running counter to the concept of a u n i f i e d formal-operational structure. However, the v a l i d i t y of t h i s interpretation may be questioned. In the case of the study of Schwebel, th i s lack of s i g n i f i c a n t l y consistent performance across tasks probably was due to the r e l a t i v e l y small number of Ss i n each group. In the case of the other studies, t h i s lack of consistency may have resulted from one or more design weak-nesses. These included the selection of Ss too young to be expected to be at the formal-operational stage; considerable v a r i a t i o n i n d i f f i c u l t y among the Piagetian tasks employed; the selection of tasks that were poor measures of formal-; operational thought; and the use of procedures that deviated considerably from those of Inhelder and Piaget (1958). For example, i n the case of the study of Neimark (1970), the Ss selected were very young, the i r grade l e v e l s ranging from four to s i x . It would be expected that formal-operational thinking would not be achieved by many, i f any, of these Ss. Furthermore, the c o r r e l a t i o n task employed was found to be more d i f f i c u l t than the other Piagetian task used (chemical combinations) and, i n f a c t , may be a poor measure of cognitive l e v e l for any age group. In the l a t e r , longitudinal study Neimark (T.97.5bx) found no consistent age trends i n performance on t h i s task. F i n a l l y , Neimark's (1970) procedure for scoring the chemical combinations task can be c r i t i c i z e d as deviating 25 too far from that of Inhelder and Piaget. Credit was given primarily for how systematic the S's method of generating the combinations was. The S's knowledge cncerning the various chemicals, including which one was neutral and which i n h i b i t e d the yellow colour, seems not to have been taken into account. Inhelder and Piaget considered both of these factors, that i s , method and solution, when describing the level s of performance. S i m i l a r l y , Ross (1973) gave cred i t only for the method i n the chemical combinations task. HISTORICAL OVERVIEW OF MEMORY RESEARCH There have been b a s i c a l l y two approaches to the study and understanding of memory. The f i r s t i s the s t a t i c approach, which was determined larg e l y by the work of Ebbinghaus (1885) and influenced greatly verbal learning research i n North America. The second, the dynamic approach, was determined to a great extent by the Gestalt theorists and Freudians and neo-Freudians. S t a t i c Concept of Memory The study of human learning and memory achieved s c i e n t i f i c status with the work of Ebbinghaus (1885) on the learning and memory of nonsense s y l l a b l e s . His work, combined with the p r e v a i l i n g t h e o r e t i c a l orientation of North American psychologists, namely behaviourism, resulted i n a c l e a r l y 26 defined approach to memory, which was dominant u n t i l the mid-1950's. Underlying t h i s approach were c e r t a i n assumptions. For example, judging by the type of research c a r r i e d on i n t h i s p e r i o d , i t would seem that the le a r n e r was assumed to be, or at l e a s t was t r e a t e d l i k e , a "t a b u l a rasa" upon which experience was w r i t t e n . Learning and memory were t r e a t e d i n a q u a n t i t a t i v e manner w i t h the a c q u i s i t i o n of v e r b a l s t i m u l u s -response u n i t s seen as a f u n c t i o n of t h e i r r e i n f o r c e d contiguous occurrence. D i f f e r e n c e s among people, normals and ^subnormals, a d u l t s and c h i l d r e n , were considered i n q u a n t i t a t i v e terms; that i s , the d i f f e r e n c e s that e x i s t e d were i n t e r p r e t e d , not i n terms of q u a l i t a t i v e d i f f e r e n c e s i n f u n c t i o n i n g , but i n terms of q u a n t i t a t i v e d i f f e r e n c e s i n memory c a p a c i t y . The ba s i c laws of l e a r n i n g and memory f o r humans were considered to be i d e n t i c a l to those of lower animals. I t was thought that more complex processes, of which only humans were capable, could at a l a t e r date be examined and i n a sense could be deriv e d i n an a d d i t i v e manner from the understanding of these laws common to animals, c h i l d r e n , and a d u l t s . Learning and memory were considered as synonymous processes independent of others, such as per c e p t i o n , emotion, t h i n k i n g , and reasoning. Except f o r mo t i v a t i o n , understood i n terms of need red u c t i o n through reinforcement, organismic v a r i a b l e s were neglected. This neglect both r e s u l t e d from 27 and influenced the choice of learning materials and method of presentation. For example, the presentation of simple items such as nonsense s y l l a b l e s i n paired-associate or s e r i a l l i s t s resulted i n learning material with minimal inherent order. Thus the ro l e of the individual's past ex-perience, reasoning, i n t e r e s t , and so fo r t h was reduced. (See Mandler, 1967.) Dynamic Concept of Memory H i s t o r i c a l l y , the dynamic approach concerning learning and memory was basic to two major theories, the Gestalt theory and the psychoanalytic theory. Psychoanalytic view of memory. In the case of the psychoanalytic view, the term dynamic refers to motives and drives and psychic structures which determine behaviour, including learning and memory. In his early theories Freud infe r r e d a "tabula rasa" type of memory i n which p r a c t i c a l l y every event that the person experienced was recorded. These memory events either were available i n l a t e r years or were repressed. The repressed memories, while not conscious, were unaltered. However, Freud l a t e r discovered "screen memories," formerly repressed memories which come to consciousness i n a distorted version of the o r i g i n a l event. He also found that many of his patients' "traumatic" memories were i n a c t u a l i t y only fantasies. These findings effected a re v i s i o n 28 of his former "tabula rasa" position to a more active "reconstruction" view. (See R e i f f and Scheerer, 1959.) A further extension of Freud's concept of memory resulted from the neo-Freudian emphasis on the r o l e of the ego and i t s development. According to t h i s view (Hartmann, K r i s , and Loewenstein, 1946; K r i s , 1956) the learner's l e v e l of development, including ego development, determines what i s experienced and retained. Furthermore, even though an event may not be brought to awareness, subsequent memories may influence and change the o r i g i n a l memory so that when i t i s brought to consciousness i t w i l l be i n a changed form. In addition, what i s remembered i s more often a c o n s t e l l a t i o n of events rather than any single unchanged and completely intact trace. Thus r e c o l l e c t i o n of a happening often involves reconstruction of this c o n s t e l l a t i o n of events; t h i s recon-struction sometimes necessitates the aid of a therapist i n the case of a long-forgotten or repressed memory. Gestalt view of memory. Another main contribution to the dynamic approach to memory came from the Gestalt t h e o r i s t s . In t h i s case, the dynamic aspect refers to certain processes which take place i n perception, learning, and memory. The Gestalt modification of the trace theory (Koffka, 1935; Kohler, 1929; Wulf, 1922) postulated that experience i s l a i d down in the brain by some sort of isomorphic process. What i s l a i d down i s determined both by the structure of the material, 29 which i s more than an aggregate of discrete s t i m u l i , and the organizing a c t i v i t i e s of the i n d i v i d u a l . Furthermore, the traces which r e s u l t are subject to modification by two influences; communication with other traces and stresses inherent i n the trace. These influences e f f e c t through processes, such as sharpening and l e v e l l i n g , memory traces having maximum s i m p l i c i t y , symmetry, and good form. Thus, for example, working i n the Gestalt t r a d i t i o n , Wulf (1922) found that v i s u a l l y perceived forms when reproduced at a l a t e r date showed evidence of the processes of sharpening and l e v e l l i n g . Often considered a support for the Gestalt p o s i t i o n i s Ba r t l e t t ' s work on memory (B a r t l e t t , 1932). B a r t l e t t concluded that the perception and memory of experiences are r a r e l y l i t e r a l or precise but are determined by schemas, which are abstractions, s i m p l i f i c a t i o n s , and a r t i c u l a t i o n s of experience. His concept of memory,I while i n the Gestalt t r a d i t i o n , marked a major deviation i n that memory was not considered to be the res u l t of the formation of isomorphic traces;; rather i t re s u l t s from the int e r a c t i o n -of s t i m u l i and an already structured, active organization of schemas. Recall becomes an active construction based upon the schema. While certain "dominant d e t a i l " of the o r i g i n a l stimulus presentation does p e r s i s t , the major component of the o r i g i n a l s i t u a t i o n that remains i s the attitude--broadly conceived--which was involved i n i t . Reproduction then can be 30 understood as an attempt to " j u s t i f y " t h i s attitude by " r a t i o n a l i z a t i o n , " " e f f o r t after meaning," and " f i t . " P i a get 1s theory of memory and i t s r e l a t i o n to other  views. The view of memory of Piaget (Piaget and Inhelder, 1968) i s i n the dynamic t r a d i t i o n and can be considered most cl o s e l y related to that of B a r t l e t t . Piaget stresses that memory of an event i s not a passive recording but i s cl o s e l y bound up with the individual's l e v e l of understanding. Piaget makes a d i s t i n c t i o n between "memory i n the wider sense" and "memory in the s t r i c t sense." The former involves the conservation, of the general schemata i n the form of repeatable processes and operations. The l a t t e r pertains to the recognition, reconstruction, and r e c a l l of situations, events, or objects which have been personally experienced and are l o c a l i z e d i n the past. Memory in the s t r i c t sense i s a store of information i n f i g u r a t i v e form which has been encoded through the trans-formation of stimulus input by the schemata, or memory i n the wider sense. Perception, understanding, and memory of events r e f l e c t the nature of these schemata, which change and develop as the c h i l d matures and interacts with his environment. Piaget's concept of memory d i f f e r s from that of Ba r t l e t t mainly i n terms of the nature of the schemata. According to Piaget, the schemata proposed by B a r t l e t t are b a s i c a l l y mnemonic schemata, which are considerably less 31 general than the structures of Piaget. The precise r e l a t i o n -ship of these structures to B a r t l e t t ' s schemas, however, s t i l l remains to be determined. It i s int e r e s t i n g to note that the t r a d i t i o n a l s t a t i c approach to learning and memory i n North America began to lose i t s dominance i n the mid-1950's with the emergence of a more dynamic and developmental approach to learning and memory. This emergence, while probably not due to Piaget 1s work on memory, presented a favourable environment for his ideas on memory to be explored and extended. This new approach i n North America was i n part a re s u l t of the general d i s s a t i s -f a c t i o n with the t r i v i a l t y and paucity of findings concerning memory after over half a century of research i n the " s t a t i c 7 t r a d i t i o n . The new view both resulted from and helped to produce a number of int e r e s t i n g research findings and modifi-cations of experimental paradigms. Such findings as the discovery of o n e - t r i a l verbal learning (e.g., Estes, 1960; Rock, 1957) undermined the p r i n c i p l e s c o f frequency and contiguity as did evidence for active selection, mediational and organizational strategies on the part of the learner (e.g., Bousfield, 1953; Bugelski, 1962; Underwood, 1963). These strategies obviously were determined by the learner's past experience and were u t i l i z e d to overcome both long and short term memory l i m i t a t i o n s . S i m i l a r l y , such research findings as those concerning the 32 e f f i c a c y of imagery to promote learning (e.g. Paivio, 1969), the q u a l i t a t i v e differences i n memory performance between children and adults (e.g., Bousfield, Esterson, and Whitmarsh, 1958), and the complexity of the supposedly simple paired-associate and s e r i a l paradigms (e.g., Battig, 1968; Jensen and Rohwer, 1965), were but a few of the causes and the re s u l t s of the new Z e i t g e i s t which challenged the t r a d i t i o n a l views. Closely related to t h i s new view were changes i n methodology. Now i n North America,tasks involving such materials as free r e c a l l l i s t s , sentences, and paragraphs with testing by recognition and non-rote methods are commonly used, in addition to s e r i a l and paired-associate tasks. STUDIES RELATING MEMORY TO COGNITIVE DEVELOPMENT This section w i l l provide (a) a general review of studies r e l a t i n g memory to cognitive development and (b) a s p e c i f i c discussion of studies r e l a t i n g memory to development from the concrete-operational to the formal-operational stage. General: Memory and Cognitive Development The following w i l l include a discussion of Piaget's work on the relat i o n s h i p between memory and i n t e l l i g e n c e and the research of others who re p l i c a t e d and extended his work in both similar and d i f f e r e n t paradigms. Piaget and Inhelder (1968) provide a number of studies 33 to support t h e i r view that "memory i n a s t r i c t sense" i s dependent upon the operational structures available during both i n i i t i a l viewing and r e c a l l . Subjects of d i f f e r e n t ages were presented with a number of displays, each seemingly related to certain cognitive operations. Thus each display possibly required that the S possess the rela t e d operations in order to successfully memorize the display. These displays included the following types: 1. S t a t i c presentations, such as an i n c l i n e d bottle p a r t l y f i l l e d with water or a row of sti c k s decreasing i n height... 2. Presentations involving transformations, such as the rotation of a tr i a n g l e through 180 degrees. 3. Presentations of causally related events, such as •'- the. transmission of motion (cone b a l l h i t s a f i x e d object, which transmits the motion to another b a l l touching the object). Piaget and Inhelder reported two major findings. F i r s t , memory performance was found to be p o s i t i v e l y related to the c h i l d ' s assessed operative understanding or to the chil d ' s age, with operative l e v e l i n f e r r e d . Second, i n the case of some Ss, there was improvement i n memory performance from the f i r s t test period to the second, even though' the Ss were shown the memory displays only once. These improvements were considered to r e s u l t from cognitive development during the t e s t - r e t e s t i n t e r v a l , the operations thus developed 34 presumably serving to improve the memory image through correction and reconstruction. A small number of researchers working with memory tasks similar to those of Inhelder and Piaget attempted to r e p l i c a t e or extend th e i r findings. One study (Altemeyer, Fulton, and Berney, 1969), employing kindergarten children, found that i n approximately 40 percent of cases the memory drawings of a seriated array of st i c k s improved over a s i x month period. In another study, Furth, Ross, and Youniss (1974) found that memory drawings of such pictures as a t i l t e d glass of l i q u i d and a f a l l i n g and turning s t i c k depended upon age, and hence (they concluded) operative l e v e l . Certain improve-ments in memory over time also were interpreted by the authors as r e s u l t i n g from cognitive development. Similar r e s u l t s were found by Liben (1974), who with f i f t h grade students found a small but s i g n i f i c a n t r e l a t i o n -ship between assessed understanding of h o r i z o n t a l i t y and memory drawings of a picture of a tipped b o t t l e with water one week (r = .39) and s i x months (r = .29) after viewing. She concluded, however, that the small number of improvements in memory performance could be interpreted best as being due to chance. In a study involving anticipatory imagery (Anooshian and Carlson, 1973), Ss i n each of a number of t r i a l s viewed one of nine nonsense s y l l a b l e s and were asked to recognize 35 from a sheet containing a l l the nine nonsense s y l l a b l e s the one they had just seen. Each of the nine nonsense s y l l a b l e s was presented i n four ways, either i n the p o s i t i o n shown on the recognition sheet or rotated 180 degrees or 90 degrees to the l e f t or to the r i g h t . The test of recognition took place immediately or after 10 seconds. The immediate memory performance correlated neither with IQ scores (Lorge-Thorndike Intelligence Test, Form A, l e v e l s 1 and 2) or operational understanding, as measured by conservation tasks (continuous quantity and length). Recognition scores after 10 seconds correlated s i g n i f i c a n t l y with both IQ and conservation per-formance. When IQ was held as a covariate, however, 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 recognition and conservation performance. Other studies, while interpreting the r e s u l t s i n terms of operational understanding, employed tasks such as paired-associate, free r e c a l l , or sentence recognition; these tasks are f a i r l y d i s s i m i l a r to those of Piaget and Inhelder and are more clos e l y related to North American research paradigms. Wolf and Levin (1972), using a paired-associate task and instructions to form an interacting mental image of the objects i n each p a i r , found that memory performance was superior i n t h i r d grade children as opposed to kindergarten children. These re s u l t s were interpreted by the authors as being due to the older children's a b i l i t y to generate and use 36 dynamic mental images without a d d i t i o n a l support provided, f o r example, by a c t u a l manipulation of the o b j e c t s to form i n t e r a c t i n g p a i r s . The a b i l i t y to produce dynamic mental imagery was presumed to be a c o n c r e t e - o p e r a t i o n a l s k i l l , which the younger c h i l d r e n had not yet achieved. A number of s t u d i e s employing some form of f r e e or modified r e c a l l procedure have r e l a t e d i n c r e a s i n g o r g a n i z a t i o n and r e c a l l of the items to i n c r e a s i n g o p e r a t i o n a l understanding. In a task i n v o l v i n g f r e e r e c a l l of an array of p i c t u r e s f o l l o w e d by s e q u e n t i a l l o c a t i o n r e c a l l , Furth and Milgram (1973) r e p o r t e d evidence supporting an i n c r e a s e , from ages 4 to 12, i n the a b i l i t y to c l a s s i f y items i n t o c a t e g o r i e s to f a c i l i t a t e r e c a l l . This increase was i n t e r p r e t e d as due p r i n c i p a l l y to ether greater o p e r a t i v e understanding of the older Ss. S i m i l a r l y , Tomlinson-Keasey, Crawford, and Miser (1975), who c l a s s i f i e d k i n d e r -garten and f i r s t - g r a d e c h i l d r e n on the b a s i s of c l a s s i n c l u s i o n s k i l l s as c l a s s i f i e r s and n o n c l a s s i f i e r s , found (d e s p i t e no s i g n i f i c a n t d i f f e r e n c e i n age between the two groups) that the c l a s s i f i e r s both r e c a l l e d s i g n i f i c a n t l y more items and showed s i g n i f i c a n t l y more c l u s t e r i n g i n r e c a l l than d i d n o n c l a s s i f i e r s . Another study (Haynes and Kulhavy, 1976) examined f r e e r e c a l l performance of c h i l d r e n i n elementary and j u n i o r high school, who were at one of three developmental l e v e l s defined by t h e i r a b i l i t y to conserve weight, mass, and volume. In the 37 f i r s t of two experiments reported, a s i g n i f i c a n t r e l a t i o n s h i p was found between developmental l e v e l and both r e c a l l and c l u s t e r i n g . The second experiment examined the use of paradigmatic, syntagmatic, c a t e g o r y - i n c l u s i v e , and un r e l a t e d words as cues. I t was found that Ss who conserved volume were more i n c l i n e d to s e l e c t , and hence perhaps use, superordinate infor m a t i o n as an encoding device than were l e s s c o g n i t i v e l y mature c h i l d r e n . One of the most recent of these s t u d i e s i n v o l v i n g some form of f r e e r e c a l l i s that of A r l i n (1977), who employed a group of u n i v e r s i t y students e n r o l l e d i n the f i r s t c l a s s of educational psychology. They were r e q u i r e d to r e c a l l an array of 12 types of obj e c t s a f t e r being asked to r a i s e questions about t h i s problem-rich array. The q u a l i t y of these questions was considered to r e f l e c t problem-finding a b i l i t y , which, according to A r l i n (1975), when i n the superior range i s dependent upon f o r m a l - o p e r a t i o n a l t h i n k i n g but c o n s t i t u t e s the stage beyond. Accuracy of r e c a l l of the items was s i g n i f i c a n t l y and moderately r e l a t e d to o v e r a l l performance on formal-o p e r a t i o n a l tasks assessing c o m b i n a t o r i a l and p r o p o s i t i o n a l t h i n k i n g . The r e l a t i o n s h i p s increased from r = .38 f o r immediate r e c a l l to r = .59 f o r r e c a l l measured a f t e r one month. A l s o , there appeared to be a s i m i l a r but s l i g h t l y stronger r e l a t i o n s h i p i n both the o r i g i n a l and r e t e s t periods between r e c a l l and the q u a l i t y of questions asked .concerning 38 the array. A r l i n concluded that the res u l t s imply that r e c a l l was related to the organization of the material that had taken place. In thi s case, the organization would seem clo s e l y associated with formal-operational thinking i n the Piagetian sense and with problem-finding a b i l i t y . The f i n a l study to be mentioned (Prawat and C a n c e l l i , 1976) examined the tendency to recognize sentences which were not presented i n i t i a l l y but were correct l o g i c a l inferences of the presented statements. The Ss were f i r s t grade children who were c l a s s i f i e d as conservers and nonconservers. The two groups thus formed were equivalent i n age and IQ. A 'significant i n t e r a c t i o n was found. Conservers made s l i g h t l y more errors than nonconservers on true inference sentences, whereas on the other types of sentences conservers made a similar number of errors or fewer'errors than did nonconservers. It must be noted that many of the memory experiments reported by Piaget and Inhelder (1968) i n addition to several of the l a t e r studies c i t e d (e.g., Furth and Milgram, 1973; Furth et a l . , 1974; Wolff and Levin, 1972) explored memory performance as a function of age. As many s k i l l s not c l e a r l y related to operational l e v e l develop with age, the interpre-tat i o n of findings of relationships between age and memory performance i n such studies i s d i f f i c u l t to make. In the majority of the studies c i t e d where operational l e v e l was measured d i r e c t l y and not infer r e d from age, either 39 or both of the possible covariates of operational l e v e l , age and IQ, were not taken into account. Without age and IQ held constant through S selection or s t a t i s t i c s , any finding of 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 operational l e v e l and memory performance i s again d i f f i c u l t to interpret. In the cases of the experiments of Piaget and Inhelder where operative understanding was assessed and the study of Haynes and Kulhavy (1976), there was no control for IQ and the ages of the Ss varied considerably. In the study of A r l i n (1977) the ages and IQ's of the Ss were not reported, but i t might bfe expected that the majority of the university students i n the sample would be approximately the same age. Whether they d i f f e r e d much i n IQ i s not known. Two studies (Liben, 1974; Tomlinson-Keasey et a l . , 1975) c l e a r l y took into account age but not IQ. Liben used Ss from the same grade, while Tomlinson-Keasey found that the two groups, c l a s s i f i e r s and n o n c l a s s i f i e r s , formed from kindergarten and grade one children, did not d i f f e r i n age. In only two studies where operative l e v e l was assessed were both age and IQ taken into account. In the study of Anooshian and Carlson (1973) the ef f e c t s of age and IQ were removed s t a t i s t i c a l l y . On the other hand, Prawat and C a n c e l l i (1976) used Ss from the same grade:and found that the conservers did not d i f f e r from the nonconservers i n age or IQ. 40 Memory and Development from Concrete to Formal Operations Of p a r t i c u l a r relevance to the study reported here i s that the majority of the studies discussed i n the preceding section employed pre-teen Ss and interpreted r e s u l t s i n terms of changes from preoperational to concrete-operational thinking. Very few studies investigated memory as related to the develop-ment i n thinking from the concrete-operational to formal-operational stages. In several of their memory tasks Piaget and Inhelder (1968) employed groups of Ss i n which a few Ss were older than 11 years and thus possibly could be at the formal-operational l e v e l . However, successful performance i n the majority of these tasks would seem to be related p r i n c i p a l l y to achievement of concrete-operational thinking. Such tasks included remembrance of double c l a s s i f i c a t i o n s , remembrance of double s e r i a l correspondences, reconstruction of a geometrical configuration with p a r t l y regular and p a r t l y contingent elements, and remembrance of the movement of a three-sectioned lever fastened to a board by a central b o l t . Only one of the memory tasks of Piaget and Inhelder (1968) seems to be related to formal-operational thought. Employing Ss from 4 to 12 years, t h i s task investigated the memory of the nine arrangements of three objects taken two at a time. It was found that correct memory drawings were made 41 only by the older Ss. One of f i v e 9-year-olds, two of s i x 10-year-olds, and four of seven 11-to 12-year-olds achieved correct memory performance.. Piaget and Inhelder concluded that these successful Ss were i n the formal-operational stage, and thus memory performance was related to formal-operational achievement. A p i l o t study by thi s E indicated, however, that t h i s ^conclusion may be unwarranted. Despite differences i n operational l e v e l , as assessed by the chemical combinations, pendulum, and balance tasks, 18 of 19 grade seven g i r l s made correct memory drawings of the nine arrangements. Another study which investigated memory change as a function of development from the concrete-operational to the formal-operational stage i s that of A r l i n (1977). In this study, r e c a l l was found to be correlated with measures of formal-operational thought and problem-finding a b i l i t y . However, A r l i n (1975) concluded that formal-operational thinking i s necessary but not s u f f i c i e n t for the development of the problem-finding stage. Therefore the rel a t i o n s h i p between r e c a l l and formal-operational thinking may not be di r e c t i n th i s p a r t i c u l a r case. As o v e r a l l performance on the formal-operational tasks covaries with the measure of problem-finding a b i l i t y (r = .43), the rel a t i o n s h i p found between formal-operational performance and memory performance possibly would be mediated to a great extent by problem-finding a b i l i t y . The 42 questions asked by the Ss concerning the array (the qual i t y of which defined problem-finding a b i l i t y ) would serve as strategies to organize i t ; thus increased qu a l i t y of the questions would r e s u l t i n more e f f e c t i v e organizational str a t e -gies and hence better memory performance. Furthermore, while the re s u l t s of th i s study are of considerable i n t e r e s t , the task used i s a very spec i a l i z e d case of free r e c a l l . The Ss were instructed to ask questions, which presumably could be used to organize stbrage and r e c a l l , the qua l i t y of these questions being known to be related to formal-operational thought. Furthermore, as mentioned previously, there was no clear control for age and IQ i n t h i s study. In conclusion, then, i t would seem that there has been very l i t t l e work investigating:;, whether there are changes in memory performance wrought by the achievement of formal-operational thinking. Neither within the Piagetian t r a d i t i o n nor with tasks more c l o s e l y related to North American research paradigms has thi s question been investigated with a variety of memory st i m u l i seemingly related to the various important aspects of the l a t t i c e and INRC group. In fact the two studies c i t e d which have investigated t h i s question, the arrangement study of Piaget and Inhelder and the modified free r e c a l l study of A r l i n , have been questioned concerning weaknesses i n methodology, generality, and/or interpretation. 43 PURPOSE OF THIS STUDY The two major aims of thi s study were to i d e n t i f y i n d i v i d u a l differences with respect to a u n i f i e d formal-operational structure and to rel a t e them to predictable differences i n memory performance on a variety of tasks designed i n the Piagetian t r a d i t i o n of memory research. With regard to the f i r s t aim, two hypotheses were investigated. The f i r s t hypothesis was that each of a representative sample of formal-operational tasks would show s i g n i f i c a n t p o s i t i v e correlations with the averaged performance of a l l the Piagetian tasks (excluding the task being correlated with the average). These s i g n i f i c a n t correlations would be maintained even when the effects of age and a measure of IQ were removed. The second hypothesis was that one component, as indicated by a p r i n c i p a l components analysis, would account for a considerable amount of variance i n the formal-operational tasks. With regard to the second aim, the p r i n c i p a l hypothesis investigated was that both o v e r a l l performance on a wide variety of memory tasks and performance on each of these tasks would be related s i g n i f i c a n t l y to ov e r a l l differences in cognitive maturity, as measured by average performance on a l l the Piagetian tasks. It also was hypothesized that performance on each Piagetian task thought to be measuring a s p e c i f i c 4 4 f o r m a l - o p e r a t i o n a l scheme or concept would be s i g n i f i c a n t l y r e l a t e d to performance on a p a r t i c u l a r memory task or tasks thought to be r e l a t e d to the s p e c i f i c scheme. For example, i t was expected that performance on the volume conservation task would be s i g n i f i c a n t l y r e l a t e d to performance on the memory task presumed to be r e l a t e d to the understanding of volume. A l l the preceding r e l a t i o n s were expected to remain s i g n i f i c a n t even when the e f f e c t s of age and a-measure of IQ were removed. The f i n a l hypothesis p e r t a i n i n g to the second aim was that the magnitude of the c o r r e l a t i o n s between P i a g e t i a n task performance and both o v e r a l l memory performance and performance on s p e c i f i c memory tasks would be greater one month a f t e r p r e s e n t a t i o n of the d i s p l a y s as compared to immediately f o l l o w -i n g . C e r t a i n c o n c r e t e - o p e r a t i o n a l Ss may have memory ( o r g a n i z a t i o n a l ) s t r a t e g i e s s u f f i c i e n t to permit s u c c e s s f u l memory performance over the short term, but not over the long term. S u c c e s s f u l long term memory performance was considered to r e q u i r e s t r a t e g i e s i n v o l v i n g a complete understanding of the task, presumably a f u n c t i o n of fo r m a l - o p e r a t i o n a l thought i n t h i s study. 4 5 Chapter 2 METHOD In order to explain the method of thi s study, t h i s chapter w i l l provide the following: 1. a b r i e f introduction to the general rationale of the method, followed by a more detailed description concerning the selection of Ss, the design, and the general procedure for testing and scoring. 2. a description of the assessment tasks, including the four Piagetian tasks and the vocabulary test of the Wechsler Intelligence Scale for Children, and the scoring c r i t e r i a for these tasks. 3. a description of the memory tasks and the i r scoring c r i t e r i a . METHOD: GENERAL In the case of the f i r s t aim of the study, to i d e n t i f y i n d i v i d u a l differences with respect to a u n i f i e d formal-operational structure, each S was assessed through the chemical combinations, pendulum, balance, and volume conservation tasks. These assessment tasks were so chosen that together they were 46 thought to measure a l l the important aspects of formal-operational thinking; thus they presumably could provide a s o l i d estimate of operational l e v e l . The f i r s t two tasks are considered more c l o s e l y related-to the l a t t i c e properties of formal-operational thought where the S must v e r i f y hypotheses by the systematic manipulation of variables. The p a r t i c u l a r schemes thought to be tested by these tasks were the combinatorial operations, i n the case of the chemical combinations task, and the a b i l i t y to hold variables constant to investigate the e f f e c t s of others, in the case of the pendulum task. The l a t t e r two tasks are considered more cl o s e l y related to the INRC group, with the balance task presumably requiring the concept of proportions and the volume., task, the concept of m u l t i p l i c a t i v e compen-sations . With regard to the method of the present study, care was taken to avoid problems which could reduce the consistency of performance across the formal-operational tasks chosen. The Ss were selected from grade seven, because i t was thought that t h i s grade would be a time of t r a n s i t i o n to formal-operational thinking; thus a wide range of responses, including responses at the formal-operational l e v e l , might be expected. In addi-tio n , the tasks were presented i n such a manner that they would e l i c i t among the Ss a wide range of responses. Thus a l l the Ss would not f i n d a task too d i f f i c u l t or too easy but 47 would vary i n their performance l e v e l . An example of this attempt to produce assessment tasks of equivalent d i f f i c u l t y which also e l i c i t a wide range of performance among the Ss was the selection of the apparatus for the balance task. Inhelder and Piaget (1958, ChapterIML) provided a number of d i f f e r e n t models of the balance apparatus, d i f f e r i n g , for example, i n such features as the number of hooks on each side from which weights could be suspended. The type of apparatus selected for the study was the model which p i l o t work indicated would produce the desired range of responses. F i n a l l y , whenever possible, the method followed, p a r t i c u l a r l y in.the case of scoring, c l o s e l y resembled that of Inhelder and Piaget i n the ess e n t i a l s . In the case of the second aim (to re l a t e formal-operational competency to memory performance')', each memory task was designed to r e l a t e c l o s e l y to the various aspects of the l a t t i c e and/or INRC structures. Thus when considered together, the memory tasks would seem to be related to a l l the major aspects of formal-operational thought. In order to make clear statements concerning whether the two aims were achieved i n the present study, an attempt •-was made to control the two possible confounding variables, age and IQ. A l l the Ss were from the same grade i n school, and any differences i n age that remained were controlled by s t a t i s t i c a l procedures. A measure of IQ, the vocabulary test 48 of the Wechsler Intelligence Scale for Children (Wechsler, 1949), was used, and any differences among Ss i n t h i s variable were taken into account. Subjects The Ss were 5 6 female grade seven students for whom E received parental permission to take part i n the three sessions of the study. One of these Ss la t e r became unavail-able for the l a s t session. Forty-seven of the Ss were from elementary schools where the p r i n c i p l e s of the balance had not been taught. Three of the remaining Ss were obtained through a community centre and six through acquaintances of the E; these g i r l s were ones who stated that they had no experience with the balance. The average age of the Ss was 13.1 years. While exact d e t a i l s are not known, i t would seem that the majority of the Ss came from working class and lower middle-class backgrounds. Design Each of- the Ss was administered (a) f i v e assessment tasks, including four Piagetian tasks to assess l e v e l of understanding and the vocabulary test of the Wechsler i Intelligence Scale for Children to estimate IQ, and (b) eight membry tasks. 49 General Procedure One E administered the f i v e assessment tasks during the f i r s t session, while another E was responsible for the memory tasks i n the second and t h i r d session. During the second session, occurring usually a day after the f i r s t , the eight. memory displays were presented and each display was tested. The f i n a l session, which took place four weeks after the second, involved only the testing of the memory of the displays with no further presentation of them. In t h i s session the E, for each memory task except Task 1, both reviewed how each display had been introduced i n the l a s t session p r i o r to the viewing period and re-explained the testing procedure. In the case of Task 1, the S was questioned i n a nonleading manner as to what had taken place i n the l a s t session and then memory performance was tested. In a l l sessions the S was tested i n d i v i d u a l l y , and an attempt was made to ensure that the S was relaxed before testing began. The S was encouraged to speak f r e e l y concerning what she was doing and why, and i f any doubts existed concerning these points, she was questioned. The assessment and memory tasks were administered to a l l Ss i n the order they are presented i n thi s chapter. The tasks were scored by the E who had administered them. In order to evaluate i n t e r r a t e r r e l i a b i l i t y of the scoring of the Piagetian tasks, 10 response protocols were randomly selected i n the case of each task. These protocols then were scored by a person not involved i n the study. The co r r e l a t i o n c o e f f i c i e n t s between t h i s person's scoring of the protocols and the E's scoring of them were as follows: chemical combinations .86; pendulum .94; volume conservation .99; equilibrium i n the balance .96. ASSESSMENT TASKS The following provides a description of the assessment tasks, including the four Piagetian tasks and the vocabulary test of the Wechsler Intelligence Scale for Children, and the scoring c r i t e r i a for these tasks. In the' case of the scoring of each Piagetian task, the basic performance measures w i l l be discussed and then the method of integrating these measures to form the substages of concrete-operational and formal-operational performance w i l l be described. Three of the Piagetian tasks, chemical combinations of col o r l e s s l i q u i d s , pendulum, and equilibrium i n the balance, are described i n Inhelder and Piaget (1958). The fourth, the conservation and measurement of volume task, i s from Piaget, Inhelder, and Szeminska (1960). Chemical Combinations of Colorless Liquids  In t h i s task the Ss were presented with f i v e small bottles with droppers; each b o t t l e contained a,colorless 51 l i q u i d and was l a b e l l e d 1, 2, 3, 4, or g. The Ss also were presented with a box of test tubes and two test tubes containing clear l i q u i d into which the E had added several drops from the g bo t t l e . In one of the two test tubes to which the E had added l i q u i d from bo t t l e g, a color change took place. I n i t i a l l y , the l i q u i d took on a yellow tone which gradually turned brown. The S was t o l d that these l a t t e r two test tubes both contained l i q u i d s taken i n some way from the bott l e s . The S's task was to reproduce the color using l i q u i d s from the bottles as she wished and using as many of the test tubes as she wished. At any time when the S .indicated., that ..she. had solved the problem or could not think of anything else, she was asked i f there was anything else that she could do. When the S said she was fi n i s h e d and did not wish to continue, even aft e r the preceding questions, she was questioned concerning the way(s) of making the yellow color, the roles of l i q u i d s 2 and 4, and what combinations she considered i n determining the roles of 2 and 4. Scoring Basic Measures. The two basic measures, which related to how systematic the S's method was and to what extent the S determined the solutions to the problem, were as follows: 1. Method measure. The score on thi s measure was the number of d i f f e r e n t combinations the S made minus the number 52 o f r e p e t i t i o n s w h i c h t h e S d i d n o t a p p e a r t o k n o w w e r e r e p e t i t i o n s w h e n a s k e d b y t h e E why s h e h a d made t h e p a r t i c u l a r c o m b i n a t i o n . I f t h e S w o r k e d s y s t e m a t i c a l l y u n t i l s h e f o u n d o n e c o m b i n a t i o n t h a t made y e l l o w a n d t h e n w e n t o n t o t e s t t h e r o l e o f t h e m e m b e r s o f t h e c o m b i n a t i o n , a p r o c e d u r e t h a t o f t e n w o u l d p r o d u c e r e p e t i t i o n s , s h e s t i l l w a s c o n s i d e r e d t o h a v e a n e x c e l l e n t s c o r e o n t h e m e t h o d m e a s u r e . 2. S o l u t i o n m e a s u r e . T h e S ' s s o l u t i o n s c o r e was b a s e d o n h e r a n s w e r s t o t h e E ' s q u e s t i o n s c o n c e r n i n g t h e w a y ( s ) o f m a k i n g t h e y e l l o w c o l o r , t h e r o l e s o f l i q u i d s 2 a n d 4, a n d t h e c o m b i n a t i o n s s h e c o n s i d e r e d i n d e t e r m i n i n g t h e r o l e s o f 2 a n d 4. T h e s e m e a s u r e s w e r e s c o r e d a s f o l l o w s w i t h a r e s u l t i n g maximum s c o r e o f s e v e n : a . O ne p o i n t f o r e a c h c o m b i n a t i o n f o u n d a n d c o r r e c t l y s t a t e d a s m a k i n g t h e y e l l o w c o l o r . b . O n e p o i n t f o r s t a t i n g t h e c o r r e c t c o m b i n a t i o n s c o m p a r e d t o d e t e r m i n e t h e r o l e o f 2 ( l + 2 + 3 + g a n d 1 + 3 + g ) ; i f t h e p r e c e d i n g c o m p a r i s o n was c o r r e c t , o n e p o i n t f o r c o r r e c t l y s t a t i n g t h e r o l e o f 2. c . O n e p o i n t f o r s t a t i n g t h e c o r r e c t c o m b i n a t i o n s c o m p a r e d t o d e t e r m i n e t h e r o l e o f 4 ( 1 + 3 + g a n d 1 + 3 + 4 + g a n d / o r 1 + 2 + 3 + g a n d 1 + 2 + 3 + 4 + g ) ; i f t h e p r e c e d i n g c o m p a r i s o n w a s c o r r e c t , o n e p o i n t f o r s t a t i n g c o r r e c t l y t h e r o l e o f 4. d . One p o i n t f o r c o r r e c t l y s t a t i n g t h e r o l e o f 2 53 as being d i f f e r e n t from that of 4. Substaqes of performance. In order to integrate the method and solution measures-to form substages of performance, the scores on these measures were c l a s s i f i e d as follows: C l a s s i f i c a t i o n Method measure: no. of Solution-measure: d i f f e r e n t combinations no. of points minus no. of repetitions poor = 7 = 2 average 8 - 1 1 3 - 4 good 1 2 - 1 3 5 - 6 excellent 1 4 - 1 5 6 - 7 The f i v e substages of performance (and their point values) formed from considering the c l a s s i f i c a t i o n s obtained on both the method measure and the solution measure are outlined below. The achievement required for each substage i s l i s t e d to the r i g h t . 1. Substage 2 A, 3 points, poor method and poor solution; 2. Substage 2 B, 4 points, average method and poor or low average solution or average solution and poor or low average method; 3. Substage 2 B+, 4.5 points, poor method and good solution or good method and poor solution; 4. Substage 3 A, 5 points, average method and good or excellent solution or good or excellent method and average solution; 54 5. Substage 3 B, 6 p o i n t s , good or excellent method, and good or e x c e l l e n t s o l u t i o n . While the majority of performances f e l l within the preceding substages, the few that d i d not f i t p r e c i s e l y were c l a s s i f i e d with these substages i n mind. Pendulum i n the pendulum problem. It consisted of two s t r i n g s of equal length, four d i f f e r e n t weights of equal volume, l a b e l l e d 5 oz., 10 oz., 15 oz., and 20 oz., and a support on which to t i e these s t r i n g ( s ) and hang the weights. The pendulum, the weights, and the s t r i n g ( s ) were presented to the S. With one s t r i n g , the S was shown how to t i e the s t r i n g to produce the various lengths. The other s t r i n g was pointed out as i t As seen i n Figure 1, a simple apparatus was used Figure 1. Apparatus f o r pendulum task. 55 lay at the base of the support. The S was asked to determine which factor, factors, or their combination(s) determined the frequency of the o s c i l l a t i o n , or more simply, the time i t takes for the pendulum to make one complete round t r i p . The possible factors, weight, length of s t r i n g , and amplitude, were explained to the S. While experimenting, the S was asked throughout what she was doing and finding out. If the S f a i l e d to test the e f f e c t of one or more variables and said she had solved the problem, she was asked i f there was anything else that might have an e f f e c t . If the S could not remember, she was reminded of the untested v a r i a b l e ( s ) . After f i n i s h i n g experimenting, the S was asked to write down her solutions to the problem. If she f a i l e d to mention one or more variables i n her solution, she was asked i f there was anything more to her solution. If she s t i l l did not mention a l l three variables, the role of the neglected variable(s) : was questioned. After the S f i n i s h e d writing down her solution, she was questioned concerning any portions of i t which were d i f f i c u l t to understand. Then, i f the S had not used both strings simultaneously while experimenting, she was instructed to do so and to test again the effects of the three variables. Scoring Basic measures. The two basic measures, which related 56 to the S's method and to her s o l u t i o n , were as f o l l o w s : 1. Method measure. The method measure was based p r i m a r i l y on the S's t e s t i n g procedure p r i o r to the w r i t i n g of her s o l u t i o n . For those Ss who d i d not use both s t r i n g s simultaneously during the i n i t i a l experimentation, the method during the p e r i o d of experimentation f o l l o w i n g the w r i t i n g of the s o l u t i o n was taken i n t o account only when (a) there was some doubt as to whether the S was hold i n g v a r i a b l e s constant i n her t e s t i n g of one or more v a r i a b l e s i n the i n i t i a l p e r i o d , sometimes the case, f o r example, when the S had very few t r i a l s , or (b) the c l a s s i f i c a t i o n of the S's performance was unclear. There was a maximum of one p o i n t f for each v a r i a b l e , which was given i n the f o l l o w i n g manner: a. One po i n t f o r ho l d i n g constant everything but the v a r i a b l e under c o n s i d e r a t i o n . F u l l c r e d i t was given i f there was evidence that a f t e r some t r i a l s of i n c o r r e c t t e s t i n g the S seemed to "catch on" and began to t e s t c o r r e c t l y . b. One point f o r r e v e r s i n g v a r i a b l e s which were not h e l d constant. For example, i n the case of the length v a r i a b l e , f u l l c r e d i t was given i f the S_, using two s t r i n g s of d i f f e r e n t lengthsito t e s t the r o l e of len g t h , put one weight on one s t r i n g and a d i f f e r e n t weight on the second, t e s t e d what happened, and then repeated the procedure w i t h the weights and s t r i n g s used before but w i t h each weight 57 placed on the other s t r i n g . 2. S o l u t i o n measure. The s o l u t i o n measure was based on the S's w r i t t e n statement concerning the rolesa'f the three v a r i a b l e s , weight, length of s t r i n g , and amplitude, on the frequency of o s c i l l a t i o n . There was a maximum of one p o i n t f o r each v a r i a b l e , which was given i n the f o l l o w i n g manner: a. One p o i n t f o r c o r r e c t l y s t a t i n g the r o l e of the v a r i a b l e i f the e f f e c t s of the v a r i a b l e had been t e s t e d and a s i m i l a r c o n c l u s i o n reached during the t e s t i n g p e r i o d . b. No p o i n t f o r s t a t i n g c o r r e c t l y the r o l e of a v a r i a b l e i f i t s e f f e c t had not been t e s t e d or a d i f f e r e n t c o n c l u s i o n had been reached during the t e s t i n g p e r i o d . c. One-half c r e d i t f o r i n c o r r e c t l y s t a t i n g the r o l e of the v a r i a b l e i f c o n s i s t e n t l y c o r r e c t conclusions had been reached during the t e s t i n g p e r i o d . Substages of performance. In order to i n t e g r a t e the method and s o l u t i o n measures to form substages of performance, the scores r e c e i v e d on the method and s o l u t i o n measures were c l a s s i f i e d as e x c e l l e n t i f they were three p o i n t s , good, i f two p o i n t s , average, i f one p o i n t , and poor, i f no p o i n t s . The f i v e c a t e g o r i e s of performance ( i n c l u d i n g p o i n t values) formed from c o n s i d e r i n g the c l a s s i f i c a t i o n obtained on each measure are o u t l i n e d below w i t h the achievement r e q u i r e d f o r each category l i s t e d to the r i g h t : 58 1. S u b s t a g e 2 A, t h r e e p o i n t s , p o o r m e t h o d a n d a v e r a g e o r p o o r s o l u t i o n o r a v e r a g e o r p o o r m e t h o d a n d p o o r s o l u t i o n ; 2. S u b s t a g e 2 B, f o u r p o i n t s , a v e r a g e m e t h o d a n d a v e r a g e s o l u t i o n o r g o o d m e t h o d a n d p o o r s o l u t i o n ; 3. S u b s t a g e 2 B+, 4.5 p o i n t s , g o o d o r e x c e l l e n t m e t h o d a n d a v e r a g e s o l u t i o n o r a v e r a g e m e t h o d a n d g o o d o r e x c e l l e n t s o l u t i o n ; 4. S u b s t a g e 3 A, 5 p o i n t s , g o o d m e t h o d a n d g o o d s o l u t i o n ; 5. S u b s t a g e 3 B, 6 p o i n t s , e x c e l l e n t m e t h o d a n d g o o d o r e x c e l l e n t s o l u t i o n o r g o o d o r e x c e l l e n t m e t h o d a n d e x c e l l e n t s o l u t i o n . I f t h e S ' s p e r f o r m a n c e b o r d e r e d b e t w e e n t w o c l a s s i -f i c a t i o n s , g r e a t e r w e i g h t w a s g i v e n t o m e t h o d t h a n t o c o r r e c t s o l u t i o n ; t h a t i s , t h e c l a s s i f i c a t i o n o f S s w h o s e m e t h o d was a c c e p t a b l e t e n d e d t o b e m o v e d u p w a r d , w h i l e t h e c l a s s i f i c a t i o n w a s m o v e d d o w n w a r d i f t h e m e t h o d was n o t a c c e p t a b l e . A l s o , i n t h e c a s e o f t h e b o r d e r l i n e S s , i f t h e i r m e t h o d s s e e m e d a c c e p t a b l e b u t t h e r e w e r e v e r y f e w t r i a l s t o j u d g e p r o p e r l y , t h e c l a s s i f i c a t i o n t e n d e d t o b e m o v e d d o w n w a r d . I f a n y p e r f o r m a n c e s d i d n o t f i t e x a c t l y i n t o t h e s e s u b s t a g e s , t h e y w e r e a s s i g n e d w i t h t h e s e s u b s t a g e s i n m i n d . 59 Conservation and Measurement of Volume  Five subtasks were involved i n t h i s assessment. The procedures for these tasks were as follows: 1. The S was shown a s o l i d wooden model 3 X 3 X 4 centimetres and was asked to reproduce with p l a s t i c bricks, each 1 cubic centimetre, an i d e n t i c a l building some distance away from the model. The S's construction was halted when the S had shown what she considered to be the number of bricks required for the base and for the height. 2. The S again was shown the wooden model ( 3 X 3 X 4 centimetres) and was asked to construct from the 1 cubic centimetre p l a s t i c bricks a number of buildings having the same volume as the model. However, the buildings were to be constructed on d i f f e r e n t - s i z e d p l o t s of land which included 2 X 2 centimetres, 2 X 3 centimetres, 3 X 4 centimetres, 1 X 2 centimetres, and 1 X 1 centimetre. Generally, once the S indicated her f i n a l idea of how high the building would be, she was not required to continue construction. 3. The E b u i l t with the p l a s t i c bricks a building 3 X 4 X 3 centimetres and then with the same bricks constructed a building 2 X 2 X 9 centimetres. The S was questioned as to whether the two buildings had the same or d i f f e r e n t volumes. 4. The S was asked to compare the volumes of s i x pairs of wooden models. She was asked to determine i f the 60 members of each pair had the same or d i f f e r e n t volumes and was to explain her conclusion for each p a i r . The S was provided with s i x p l a s t i c bricks and was advised that she could use them to help her solve the problem. The members of the f i r s t two pairs had the same dimensions, but one member of the pair was placed d i f f e r e n t l y so that the height of the two members d i f f e r e d . The members of the other pairs had d i f f e r e n t dimensions. The dimensions of the s i x pairs were (a) 1 X 2 X centimetres, 1 X 1 X 2 centimetres, (b) 1 X 3 X 1 centimetres 1 X 1 X 3 centimetres, (c) 2 X 2 X 3 centimetres, 1 X 1 X 12 centimetres, (d) 2 X 3 X 3 centimetres, 1 X 2 X 9 centimetres (e) 1 X 2 X 9 centimetres, 4 X 3 X 2 centimetres, and (f) 4 X 3 X 2 centimetres, 12 X 2 X 1 centimetres. 5. The E:'built with metal blocks, each 1 cubic centimetre, a house, 3 X 3 X 4 centimetres, i n a.glass dish having a base of 10 inches by 4% inches and a height of 3 inches. The dish was f i l l e d two-thirds f u l l of water. The S was asked to predict whether there would be any change i n water l e v e l i f the bricks were rearranged by cutting the building v e r t i c a l l y and separating the two parts or by spread ing a l l the bricks along the bottom of the dish. If the S was successful i n these predictions, she was further questioned as to whether there was any way the bricks could be rearranged under the water to af f e c t the water l e v e l . 61 Scoring Basic measures. Performance on the f i v e subtasks provided f i v e measures of performance. Substages of performance. The following four sub-stages (including their point values) are based on success i n the number of subtasks noted to the right of the substage; 1. Substage 2 A, 3 points, two tasks; 2. Substage 2 B, 4 points, three tasks; 3. Substage 3 A, 5 points, four tasks; 4. Substage 3 B, 6 points, f i v e tasks. The t o t a l point value of the S's c l a s s i f i c a t i o n was modified s l i g h t l y i n two cases. F i r s t , .25 was subtracted for each task i n which the S was f i n a l l y successful but had a considerable amount of d i f f i c u l t y . Second, i n the case of subtask 4, .25 was subtracted i f , i n comparing the volumes of the pairs of models, the S, i n a majority of t r i a l s , did a good deal of placing one model against the other rather than using the bricks to measure. Equilibrium i n the Balance As seen i n Figure 2, t h i s apparatus involved a simple balance with 11 equidistant hooks on each side and a set of weights. These weights included a pair marked 5 oz. and three i n d i v i d u a l ones marked respectively 10 o>z. , 15 oz. , and 20 oz. The S was required to t r y to understand the 62 Figure 2. Apparatus f o r e q u i l i b r i u m i n the balance task. p r i n c i p l e of the balance so that i n a t e s t i n g s i t u a t i o n she could balance a l l the s i x combinations of the four d i f f e r e n t weights with one member of each combination placed by the E. The S f i r s t was given the 5 oz. and 10 oz. weights and was asked to put the balance i n e q u i l i b r i u m with equal weights on each side and then with unequal weights on each side. The S was required to balance the 5 oz. and the 10 oz. weights at at l e a s t three d i f f e r e n t places. The S then was encouraged to experiment with the remaining f i v e combinations of weights. While the S was fre e to choose the order the combinations were t r i e d , she was encouraged to t r y each combination of weights at at l e a s t two d i f f e r e n t places. I f the S f a i l e d to t r y a l l combinations, the neglected ones were suggested. Throughout the t r i a l s the S was asked why the 6 3 w e i g h t s w e r e i n b a l a n c e . A f t e r t h e l e a r n i n g p e r i o d , t h e S was t e s t e d b y b e i n g r e q u i r e d t o b a l a n c e p a i r s o f w e i g h t s w i t h o n e member o f t h e p a i r b e i n g p l a c e d b y t h e E . E a c h o f t h e s i x c o m b i n a t i o n s o f w e i g h t s w a s t e s t e d t w o t i m e s w i t h t h e w e i g h t s a t d i f f e r e n t p l a c e s . T h e S w a s q u e s t i o n e d a f t e r e a c h b a l a n c i n g a s t o why t h e w e i g h t s w e r e i n e q u i l i b r i u m . S u b j e c t s who e x p e r i e n c e d d i f f i c u l t y i n a c h i e v i n g e q u i l i b r i u m w e r e a s k e d o n s e v e r a l t e s t t r i a l s , o n c e e q u i l i b r i u m w a s a c h i e v e d , t o r e a c h i e v e e q u i l i b r i u m b y r e v e r s i n g t h e w e i g h t s ( m o v i n g e a c h w e i g h t t o t h e c o r r e c t p l a c e o n t h e o p p o s i t e s i d e o f t h e b a l a n c e ) . When t h e t e s t i n g o f t h e c o m b i n a t i o n s o f w e i g h t s w a s c o m p l e t e d , t h e S w a s a s k e d t o s t a t e a g e n e r a l p r i n c i p l e t o c o v e r a s many c a s e s a s p o s s i b l e . S c o r i n g B a s i c m e a s u r e s . T h e b a s i c m e a s u r e s o f t h e S ' s p e r f o r m a n c e d u r i n g t h e t e s t i n g p e r i o d i n c l u d e d a s s e s s m e n t s o f t h e f o l l o w i n g a b i l i t i e s o u t l i n e d b r i e f l y b e l o w i n a s c e n d i n g o r d e r o f d i f f i c u l t y : 1. t h e a b i l i t y t o move w e i g h t s i n t h e a p p r o p r i a t e d i r e c t i o n w h e n b a l a n c i n g t h e m ; 2. t h e a b i l i t y t o r e a c h i e v e e q u i l i b r i u m w i t h d i f f e r e n t w e i g h t s a l r e a d y i n b a l a n c e b y r e v e r s i n g t h e m ; 3. t h e a b i l i t y t o u n d e r s t a n d t h e b a l a n c i n g o f a f e w 64 pairs of weights including (a) at least one pair where the members have a two-to-one r a t i o and '(b) at least one other combination not involving a two-to —one r a t i o ; 4. the a b i l i t y to balance c o r r e c t l y the s i x combina-tions, but with no understanding, i n terms of proportion, as to why the weights are i n balance; 5. the a b i l i t y to understand the balancing of a l l the simpler combinations (5 oz. and 10 oz., 5 oz. and 15 oz., 5 oz. and 20 oz., 10 oz. and 20 oz.) where the r a t i o of the weights does not involve a mixed number; 6. the a b i l i t y to provide a general rule to explain the balancing of the simpler combinations; 7. the a b i l i t y to understand the balancing of the more complex combinations (10 oz. and 15 oz., 15 oz. and 20 oz.) where the r a t i o of the weights involves a mixed number; 8. the a b i l i t y to provide a general rule to explain the balancing of a l l the combinations. Substages of performance. The following substages (including their point values) are based on the assessed achievement of the a b i l i t i e s l i s t e d to the right of the category: 1. Category 2 A, 3 points, none of the above a b i l i t i e s ; 2. Category 2 A-2B, 3.5 points, a b i l i t y 1 or 2; 3. Category 2 B, 4 points, a b i l i t i e s 1 and 2; 4. Category 2 B-3 A, 4.5 points, a b i l i t i e s 1, 2, and 3 or 1, 2 and 4; 5. Category 3 A, 5 points, a b i l i t i e s 1, 2, 3, and 5; 6. Category 3 A+, 5.25 points, a b i l i t i e s 1, 2, 3, 5, and 6; 7. Category 3 B-, 5.75 points, a b i l i t i e s 1, 2, 3, 5, 6, and 7; 8. Category 3 B, 6 points, a b i l i t i e s 1, 2, 3, 5, 6, 7, and 8. Vocabulary Test: Wechsler Intelligence Scale for Children The method of administering the vocabulary test, including the procedures for scoring the responses and scaling the t o t a l score according to age, were as prescribed i n the WISC test manual (Wechsler, 1949). This vocabulary test has been found by i t s developers to have a co r r e l a t i o n of .78 with the f u l l scale of WISC i n the case of children 13% years of age. This i s the age having correlations reported which i s closest to the average age of the Ss i n the present study. The f u l l scale of WISC involves a composite of the res u l t s of f i v e verbal tests, including the vocabulary test, and f i v e performance tests. MEMORY TASKS The eight memory tasks can be grouped into four • 56 c a t e g o r i e s i n v o l v i n g m emory d i s p l a y s d e s i g n e d t o r e l a t e t o f o u r a s p e c t s o f f o r m a l - o p e r a t i o n a l u n d e r s t a n d i n g : v o l u m e c o n s e r v a t i o n , c o m b i n a t i o n s , p e r m u t a t i o n s , a n d t h e l a t t i c e o f p r o p o s i t i o n s . F o r e a c h c a t e g o r y o f memory d i s p l a y , t h e f o l l o w i n g p r o v i d e s a n i n t r o d u c t i o n t o t h e c a t e g o r y , a d e s c r i p t i o n o f t h e m a t e r i a l s o f e a c h t a s k i n t h e c a t e g o r y , a n d t h e s c o r i n g c r i t e r i a . T h e m e t h o d o f a c h i e v i n g a n o v e r a l l m e m o r y s c o r e w h i c h i n c l u d e s t h e p e r f o r m a n c e s o n a l l t h e memory t a s k s a l s o w i l l b e d e s c r i b e d . M e m o r y R e l a t e d t o V o l u m e C o n s e r v a t i o n a n d M e a s u r e m e n t : T a s k 1  T a s k 1 i s r e l a t e d t o t h e c o n s e r v a t i o n o f o c c u p i e d v o l u m e , w h i c h , a s d e s c r i b e d p r e v i o u s l y , i n v o l v e s t h e u n d e r -s t a n d i n g t h a t t h e r o o m o r v o l u m e o c c u p i e d b y a n o b j e c t r e m a i n s t h e same w h e n t h e s h a p e , b u t n o t t h e v o l u m e , o f t h e o b j e c t i s m o d i f i e d . T h i s p a r t i c u l a r c o n s e r v a t i o n i s c o n s i d e r e d a f o r m a l - o p e r a t i o n a l a c h i e v e m e n t i n v o l v i n g t h e c o n c e p t o f m u l t i p l i c a t i v e o p e r a t i o n s . I n t h i s t a s k t h e S w a t c h e d w h i l e t w o b a l l s o f c l a y o f e q u a l s h a p e a n d v o l u m e w e r e p l a c e d i n i d e n t i c a l c o n t a i n e r s , e a c h h a v i n g e q u a l a m o u n t s o f w a t e r . T h e b a l l s w e r e r e m o v e d f r o m t h e w a t e r , o n e was r e s h a p e d i n t o t h e f o r m o f a s a u s a g e , a n d t h e n t h e y b o t h w e r e p l a c e d a g a i n i n t h e c o n t a i n e r . T h e 67 S w a s a s k e d t o l o o k a t t h e c o n t a i n e r s o t h a t s h e c o u l d l a t e r r e m e m b e r w h a t s h e saw. T h e p o s s i b i l i t i e s i n v e s t i g a t e d i n T a s k 1 w e r e t h a t t h e S ' s r e c a l l a n d r e c o g n i t i o n o f t h e w a t e r l e v e l s w o u l d b e r e l a t e d b o t h t o g e n e r a l f o r m a l - o p e r a t i o n a l u n d e r s t a n d i n g a n d t o l e s s g e n e r a l c o m p e t e n c i e s . T h e l a t t e r i n c l u d e d t h e u n d e r -s t a n d i n g o f v o l u m e c o n s e r v a t i o n i n g e n e r a l a n d c o n s e r v a t i o n o f o c c u p i e d v o l u m e i n p a r t i c u l a r . T h e p o s s i b l e r e l a t i o n s h i p b e t w e e n v o l u m e c o n s e r v a t i o n i n g e n e r a l a n d m e m o r y p e r f o r m a n c e p r o b a b l y w o u l d b e m e d i a t e d p r i n c i p a l l y b y t h e u n d e r s t a n d i n g o f t h e c o n s e r v a t i o n o f o c c u p i e d v o l u m e . T h i s c o n s e r v a t i o n w o u l d s e e m t h e m o s t c l o s e l y r e l a t e d t o t h e memory t a s k a n d f o r m s p a r t o f t h e g e n e r a l u n d e r s t a n d i n g o f v o l u m e . G e n e r a l f o r m a l - o p e r a t i o n a l a b i l i t y w o u l d b e i n d i c a t e d b y a v e r a g e p e r f o r m a n c e o n t h e P i a g e t i a n t a s k s . U n d e r s t a n d i n g o f v o l u m e c o n s e r v a t i o n i n g e n e r a l a n d c o n s e r v a t i o n o f o c c u p i e d v o l u m e i n p a r t i c u l a r w o u l d b e i n d i c a t e d r e s p e c t i v e l y b y o v e r a l l p e r f o r m a n c e o n t h e v o l u m e c o n s e r v a t i o n a n d m e a s u r e m e n t t a s k a n d s p e c i f i c p e r f o r m a n c e o n s u b t a s k 5. I n t h i s s u b t a s k t h e S w a s r e q u i r e d t o p r e d i c t w h e t h e r t h e w a t e r l e v e l w o u l d s t a y t h e same o r w o u l d c h a n g e w h e n t h e m e t a l b r i c k s o f a b u i l d i n g c o n s t r u c t e d u n d e r w a t e r w e r e r e a r r a n g e d . M e t h o d M a t e r i a l s ; D i s p l a y 1. T h e m a t e r i a l s i n c l u d e d t w o 68 clay b a l l s , each of 50 grams, two amber-colored glasses, approximately f i v e inches i n height with a diameter of two and three-quarters inches in height at the top tapering to two inches at the base, a graduated cylinder, and a pair of ton:gs. Procedure• The two glasses were placed approximately two feet apart i n front of the S. The two b a l l s of clay were shown to the S, and their i d e n t i c a l nature i n terms of shape, volume, and weight was emphasized. In each glass was put one b a l l of clay and then 200 m i l l i l i t r e s of water, measured c a r e f u l l y and obviously i n the graduated cylinder by the E. The S was asked concerning the equality of the water l e v e l s , and, when the E was assured that the S considered the levels equal, both b a l l s of clay were removed with tongs from the glasses. One was put back i n one glass and the other was reshaped into the form of a sausage and then returned to the other glass. The S was permitted to view the two glasses and their contents for 10 seconds and was advised to remember what she saw. One hour after viewing, the S was presented with a drawing of the outlines of both glasses and was asked to draw what she had seen. If the S f a i l e d to draw the clay and/or water l e v e l s , she was prompted by the question "Anything else?" u n t i l she did so. Her drawing then was removed, and she was asked to select from three drawings the drawing closest to 69 what she had seen. Each drawing was of two glasses, one containing the sausage and the other, a b a l l of clay. The drawings were i d e n t i c a l except for the water l e v e l s of the glasses. In one drawing, the leve l s were i d e n t i c a l . In the second, the water l e v e l of the glass with the sausage was s l i g h t l y lower than the l e v e l of the glass with the b a l l . In the t h i r d , the reverse was true; the water l e v e l of the glass containing the sausage was s l i g h t l y higher than that of the glass containing the b a l l . After the S made her choice, the drawings were removed and she was shown her previous drawing and asked why she had drawn the l e v e l s the way she had. If the S r e p l i e d that she simply had remembered them that way, she was asked i f there were any other reasons. Scoring. The measures of performance included two basic measures and a t h i r d measure which was a composite of the f i r s t two. The f i r s t involved whether the S drew the water l e v e l s as equal or unequal. The second concerned whether the S selected as similar to what she had seen before either the drawing containing glasses with equal water leve l s or one of the two drawings of glasses having unequal water l e v e l s . For each measure, the incorrect response was awarded 0 points, the correct response, one point. The t h i r d measure, the composite score, was the sum of the points achieved on 70 the f i r s t two measures. , Memory Related to Combinations: Tasks 2 and 3 As mentioned previously, the achievements of the formal-operational stage include, among other developments, the understanding of the l a t t i c e of propositions and the attainment of a number of formal-operational concepts. The l a t t e r include combinatorial operations, which are of interest here. These operations involve the a b i l i t y to make in a systematic manner a l l the combinations and/or permutations of a set of objects. According to Inhelder and Piaget (1958) the under-standing of the organized l a t t i c e structure and the combina-t o r i a l operations appear together and are c l o s e l y related. At the point of time where children f i r s t show evidence of reasoning i n terms of the propositional combinatorial system, they also spontaneously (as indicated by the experiment involving combinations of co l o r l e s s l i q u i d s ) begin to make systematic one-by-one, two-by-two, three-by-three, and four-by-four combinations. Inhelder and Piaget conclude: The combinatorial operations do not a c t u a l l y belong to the set of propositional operations and do not derive from them; on the contrary, they are the prerequisite condition of t h e i r development (and as such they are quite d i f f e r e n t ) . (p. 313) Piaget and Inhelder (1975) provide a detailed descrip-tion of the stages i n the development of the combinatorial 71 operations; these stages correspond to the preoperational, concrete-operational, and formal-operational stages. In the case of combinations they include: 1. Stage one, up to 7 years of age, which involves the empirical discovery of combinations without system and simply by means of groping (e.g., by looking to see what might be missing). 2. Stage two, 7 to 11 years, where some combinations are made by rudimentary systems, while the remainder are determined empirically by groping. 3. Stage three, from 11 or 12 years, which involves the discovery of a system to generate a l l the combinations. The displays of the following two tasks involve the presentation of complete sets of combinations. Task 2 tested the reconstruction of the 15 possible combinations of the four base associations, red dog, green dog, red cat, green cat. These associations were formed from the m u l t i p l i c a t i o n of two variables (type of animal and c o l o r ) , each having two values (dog and cat or red and green). While any task involving combinations presumably i s related to the understanding of the l a t t i c e structure i n ad-d i t i o n to the understanding of the combinatorial operations, i t was thought that t h i s task was p a r t i c u l a r l y c l o s e l y related to the former. The l a t t i c e , as described previously i n terms of the example involving the variables of size (fat or thin) 72 and state of mind (happy or sad), consists of a l l the possible combinations of four base associations. These associations, in t h i s case, f a t and happy, fat and sad, thin and happy, thin and sad, are formed from the m u l t i p l i c a t i o n of two v a r i -ables, each having two values. Thus the s i m i l a r i t y between the display of Task 2 and the l a t t i c e can be seen; both involve, a l l the combinations of the base elements formed by the multi-p l i c a t i o n of two variables each having two values. Task 3 tested the children's reconstruction of the 15 possible combinations of four d i f f e r e n t objects, truck, C a d i l l a c , Volkswagen, and motorcycle. The display of t h i s task, while presumably not as c l o s e l y related to the l a t t i c e structure as that of Task 2, s t i l l would be expected to be related to the understanding of the l a t t i c e of propositions in addition to the understanding of combinatorial operations. In conclusion, then, the p o s s i b i l i t i e s investigated were that performance on Tasks 2 and 3 would be related to the understanding of both the l a t t i c e and the combinatorial operations. A good indicator of the former general under-standing would be the average of a l l the performances on the Piagetian tasks. The l a t t e r s p e c i f i c competency presumably would be related to performance on the chemical combinations task and more s p e c i f i c a l l y to performance on the basic method measure of t h i s task. 7 3 Method Materials. The materials for the two displays were as follows: 1 . Display 2 , Combinations of Animals. Figure 3 shows the top one half of the display of the 15 combinations of four animals, red dog, green dog, red cat, and green cat. The animals, approximately three-quarters inch i n height, had a sticky back surface which adhered to the cardboard sheet measuring 12 inches by 18t:'inches. 2 . Display 3 , Combinations of Vehicles. Figure 4 shows the bottom one half of the display of the 1 5 combinations of four vehicles, truck, C a d i l l a c , Volkswagen, and motorcycle, which were stuck on a cardboard sheet 12 inches by 22 inches. The vehicles varied i n length from approximately one inch (truck) to one-half inch (motorcycle). -Procedure. In both Tasks 2 and 3 , each S was t o l d that she was to try to remember the t o t a l display i n both the correct horizontal and v e r t i c a l order. For each display the S was asked to determine the rule or order involved i n the display i n order to help her remember i t . The S was permitted to view each display for four minutes. After each viewing the S was given a cardboard sheet i d e n t i c a l to the one on which the display had been mounted. To the l e f t of the cardboard sheet were placed four p i l e s of cutouts, one p i l e for each of 1* £ 7 ft ML F i g u r e 3. The top one h a l f of the d i s p l a y of the 15 combinations o f f o u r animals, red dog, green dog, red c a t , and green c a t . F i g u r e 4. The bottom one h a l f o f the. d i s p l a y o f the 15 c o m b i n a t i o n s o f f o u r v e h i c l e s , t r u c k , C a d i l l a c , Volkswagen, and m o t o r c y c l e . 76 the four d i f f e r e n t kinds of animals or vehicles. The S was advised that she had been given more cutouts than she needed and was asked to place the cutouts on the cardboard in exactly the same way as she had seen them. Scoring. The measures of performance included three basic measures and a fourth measure which was a composite of the f i r s t three. The f i r s t basic measure was the number of d i f f e r e n t combinations reconstructed. If a combination was repeated, c r e d i t was given for only one of these combinations. For t h i s measure, the horizontal order of the combination did not matter; for example, 132 (truck, Volkswagen, and Cadillac) would be given c r e d i t even though the correct horizontal order was 123 (truck, C a d i l l a c , and Volkswagen). The second and t h i r d basic measure evaluated the extent to which reconstruction r e f l e c t e d the system used to generate the displays. These measures can be explained i n terms of Table 2, which shows part of the display of combinations of the four kinds of animals and a record of some of the combina-tions reconstructed by one S. The scoring of the preceding two sets of combinations i n terms of the two systems measures i s i l l u s t r a t e d . In Table 2, the four types of animals, red dog, green dog, red cat, green cat, are indicated respectively by 1, 2, 3, 4. The horizontal order measure provided one point for Table 2. Sample of combinations i n the orxgxnal dxsplay and xn a S's reconstruction, with successive pairs scored for horizontal and v e r t i c a l order. Display Recall of S Combinations Horizontal V e r t i c a l Combinations Horizontal V e r t i c a l order order order order points points points points 2 3 1 2 3 1 ) 1 ) 1 2 4 1 2 4 1 ) 1 ) 1 3 4 1 4 3 0 ) 1 ) 1 1 2 3 1 1 2 3 1 ) 1 ) 1 1 2 4 1 1 2 4 1 ) 1 ) 0 1 3 4 1 2 3 4 1 ) 1 ) 0 2 3 4 1 1 4 3 0 Note: Numbers 1, 2, 3, and 4 refer respectively to red dog, green dog, red cat, and green cat. 78 each combination i n the correct horizontal order. The v e r t i c a l order measure involved comparing successive pairs of combinations (as indicated by the curved lines). One point was given for each pair i n the correct v e r t i c a l order regard-less of whether the.' horizontal order of the members of the pair was correct or not. In order to obtain the composite score, the v e r t i c a l order sc.orte?, which had a maximum of 14 points , was m u l t i p l i e d by 1.07 to make i t comparable to the two other basic measures both having a maximum of 15 points. Scores on the three basic measures then were added to form the composite score. This composite score was such that the greatest c r e d i t was given to combinations reconstructed systematically i n the correct v e r t i c a l order, lesser c r e d i t was given to combinations reconstructed somewhat systematically i n either the correct horizontal order or the correct v e r t i c a l order, and least c r e d i t was given to combinations where both the horizontal and v e r t i c a l orders were incorrect. Memory Related to Permutations: Tasks 4 and 5 The a b i l i t y to systematically make a l l the permutations of a number of objects i s another manifestation of the combinatorial operations, which, as mentioned previously, are c l o s e l y related to the understanding of the l a t t i c e . Piaget and Inhelder (1975) have outlined the following three stages 79 i n the development of the understanding of permutations: 1. Stage one, up to 7 or 8 years, which involves permutations found by groping with the absence of any systems. 2. Stage two, 7 or 8 to 11 or 12 years, which involves p a r t i a l systems to generate some permutations with some others found by groping. 3. Stage three, after 12 years, which involves the progressive discovery of a system to generate a l l permutations The following two tasks involve displays which showed permutations of three or four items. Task 4 tested the c h i l d ' memory of the s i x permutations of three people (father, mother and son) seated on a c h e s t e r f i e l d . Task 5 involved the 24 combinations of four people (father, mother, son, and daughter s i m i l a r l y seated. The display of permutations i n both tasks was derived by holding constant the i n i t i a l member(s) of a permutation while varying the l a s t members'. For example, the second permutation, 1243, i s derived from the f i r s t , 1234, by holding 1 and 2 constant and changing the pos i t i o n of 3 and 4. A modified testing procedure was followed i n which the S was required to reconstruct the display by working from top to bottom and was permitted to view only the l a s t per-mutation she reconstructed. It was thought that t h i s method would reduce the l i k e l i h o o d of the S's: finding missing permutations by groping; t h i s method might increase the pr o b a b i l i t y that reconstruction of the permutations would be 80 related to understanding of the rule inherent i n the display. j The p o s s i b i l i t i e s investigated were that success i n Tasks 4 and 5 would be related to the understanding of the l a t t i c e and to more s p e c i f i c competencies. The l a t t e r included combinatorial operations involving permutations and the s c i e n t i f i c method of holding variables constant while manipulat-ing others to test t h e i r e f f e c t s . Presumably the understanding of the l a t t i c e would be indicated by o v e r a l l formal-operational understanding, as evidenced by the average of the performances on a l l of the Piagetian tasks. Combinatorial operations ^concerning permutations might be assessed by the chemical combinations task and more s p e c i f i c a l l y by the basic method measure of t h i s task. However, t h i s assessment might be somewhat i n d i r e c t . According to Piaget and Inhelder, the a b i l i t y to permute, while re l a t e d to the a b i l i t y to make combinations, i s not i d e n t i c a l and develops at a l a t e r age; the l a t t e r a b i l i t y presumably i s more c l o s e l y related to per-formance on the chemicals task than i s the former. The method of holding variables constant while manipu-l a t i n g others might be assessed by the pendulum task and more s p e c i f i c a l l y by the basic method measure of t h i s task. It must be noted, however, that t h i s experimental method of holding variables constant i s analogous but c e r t a i n l y not i d e n t i c a l to the method of generating the display; the l a t t e r method involved holding the f i r s t member(s) constant while 81 changing the pos i t i o n of the l a s t two. Method Materials. The materials for the two displays were as follows: 1. Display 4, Permutations of Three People. As shown i n Figure 5, th i s display involved s i x permutations of three people, father, mother, and son. Cutouts of the people, ranging from approximately two inches (father) to one and three-quarters inches (son), were stuck on chest e r f i e l d s which were drawn on a cardboard sheet approximately 12 inches by 18 inches. 2. Display 5, Permutations of Four People. Figure 6 shows the f i r s t s i x permutations of the display of the 24 permutations of four cutout people stuck on ch e s t e r f i e l d s . The display involved two columns, each with 12 chest e r f i e l d s containing people. The cutout figures ranged from approximately one and one-quarter inches (father) to three-quarters inch (daughter), and the cardboard sheet containing the chester-f i e l d outlines measured 12 inches by 18 inches. Procedure. The procedure i n the case of Tasks 4 and 5 was i d e n t i c a l to that followed i n the memory task involving combinations with the following exceptions. Before viewing each display, the S was advised that i n the testing 8 2 F i g u r e 5 . The s i x permutations o f t h r e e p e o p l e , f a t h e r , mother, and son. F i g u r e 6. The f i r s t 6 permutations of the 24 permutations of f o u r people, f a t h e r , mother, son, and daughter. .84 period she would have to reconstruct the display by working from top to bottom. She would be permitted to see only the one c h e s t e r f i e l d above the one on which she was placing people, and, i f she skipped a c h e s t e r f i e l d , she would not be permitted to go back. Scoring. The measures involved three basic measures and a fourth measure which was a composite of the f i r s t three. The f i r s t measure was the number of d i f f e r e n t correct per-mutations reconstructed. If a permutation was repeated, cr e d i t was given for only one of these permutations. The second and t h i r d measure took into account the system by which the permutations were reconstructed. These measures evaluated the extent to which reconstruction was based on the system used to generate the display. These measures can be explained i n terms of Table 3. It shows the f i r s t eight permutations of the display of four people, a record of the f i r s t eight permutations reconstructed by one S, and the scoring of the two preceding sets of permutations in terms of the two systems measures. In Table 3, the four people, father, mother, son, and daughter, are indicated respectively by 1, 2, 3, and 4. For both of these system measures, pairs of permutations were compared successively (as indicated by the curved l i n e s ) with a l l but the f i r s t and l a s t permutations involved i n two 8 5 comparisons. The f i r s t of these measures, the i n i t i a l members constant score (IMC)"!" involved the number of i n i t i a l members of each permutation i n the comparison pair which were constant (same members i n the same order); the t o t a l IMC score was the sum found by considering a l l comparison p a i r s . The other system measure, the v e r t i c a l order measure, provided c r e d i t only for pairs i n which the permutations were i d e n t i c a l and i n the same v e r t i c a l order as those i n the o r i g i -nal display. This was a more stringent measure of the systematic nature of reconstruction. For example, i n the case of the pair containing 1342 followed by 1324, two points of IMC cr e d i t were given, but no cr e d i t was given for order, as the correct order was 1324 followed by 1342. Thus for a pair to receive order c r e d i t a necessary but not s u f f i c i e n t pre-r e q u i s i t e was that i t receive the maximum possible IMC c r e d i t . Depending on the pair compared, IMC cr e d i t varied from 0 to 2 i n the case of permutations of four and from 0 to 1 i n the case of permutations of three. In order to obtain the composite score, the v e r t i c a l order score, which had a maximum of three points in the case of the permutations of three task and 32 points i n the case of the permutations of four task, was transformed. In the former This measure i s based on the IMC ( i n i t i a l marks held constant) measure of Leskow and Smock (1970). T a b l e 3. S a m p l e o f p e r m u t a t i o n s i n t h e o r i g i n a l d i s p l a y a n d i n a S ' s r e c o n s t r u c t i o n ; ? w i t h s u c c e s s i v e p a i r s s c o r e d f o r i n i t i a l m e m b e r s c o n s t a n t ( I M C ) a n d v e r t i c a l o r d e r . D i s p l a y R e c a l l o f £ P e r m u t a t i o n s IMC p o i n t s V e r t i c a l o r d e r p o i n t s P e r m u t a t i o n s IMC p o i n t s V e r t i c a l o r d e r p o i n t s 1 2 3 4 2 1 1 2 3 4 2 1 1 2 4 3 1 1 1 2 4 3 1 0 1 3 2 4 2 1 1 3 4 2 2 0 1 3 4 2 1 1 1 3 2 4 1 0 -1 4 2 3 2 1 1 4 2 3 2 1 1 4 3 2 0 1 1 4 3 2 0 1 2 1 3 4 2 1 2 1 3 4 0 0 2 1 4 3 4 1 2 3 N o t e : N u m b e r s 1, 2, 3, 4 r e f e r r e s p e c t i v e l y t o f a t h e r , m o t h e r , s o n a n d d a u g h t e r . 87 task, the v e r t i c a l order score was mul t i p l i e d by 2; i n the l a t t e r task, by .75. These transformations were used to make the maximum points possible on the v e r t i c a l order measure equal to the maximum of the other two basic measures; these other basic measures had a maximum of 6 points i n the case of the former task and 24 points i n the case of the l a t t e r . The composite score then was found i n each task by adding the scores on the basic measures. The composite score was such that greatest c r e d i t was given to permutations reconstructed systematically and exactly i n the manner of the display, lesser c r e d i t was given to permutations reconstructed somewhat systematically, not exactly as i n the display but with the i n i t i a l member(s) constant; and least c r e d i t was given to permutations reconstructed i n a manner not related to the system of the display. Memory Related to. the La t t i c e of Propositions: Tasks 6, 7, and 8 •  As mentioned previously, the understanding of the l a t t i c e of propositions or hypothetical p o s s i b i l i t i e s i s a formal-operational achievement. When presented with a problem, the formal-operational c h i l d can envisage a l l the p o s s i b i l i t i e s and sets out to determine which of the p o s s i b i l i t i e s a c t ually does occur. The c h i l d i s o l a t e s the relevant variables andrtests out the eff e c t s of the various variables often by holding 88 variables constant and manipulating others. The following three tasks were designed to tap t h i s general understanding of the l a t t i c e and the s p e c i f i c s k i l l s involving the i s o l a t i o n of variables and the method of holding variables constant while manipulating others. For each task the S was shown a display involving a l l the associations formed by the m u l t i p l i c a t i o n of a number of variables having two or three values. For example, i n the case of Task 6, the S was shown a display of 16 screws and bolts produced by the m u l t i p l i c a t i o n of four variables each with two values (top, round or f l a t ; bottom, pointed or f l a t ; color, copper or gray; length, long or short). Two examples of the 16 associa-and tions were In each display some of the associationsvw .erte indicated as p o s i t i v e instances of a c e r t a i n concept; others, as negative instances. For example, in the case of the screws and bolts, the p o s i t i v e instances, those with either a round top and f l a t bottom or a f l a t top and pointed bottom, were an example of the proposition of r e c i p r o c a l exclusion, one of the propositions of the l a t t i c e . Negative instances, those with either a round top and pointed bottom or a f l a t top and f l a t bottom, provided an example of the proposition of equivalence, the complement of r e c i p r o c a l exclusion. The S was required to r e c a l l the t o t a l display, including which of the members of the display were p o s i t i v e 89 and which, negative. I t was thought that Ss p o s s i b l y could r e c a l l each a s s o c i a t i o n of the d i s p l a y and i t s d e s i g n a t i o n ( p o s i t i v e or negative) by memorizing i n a f a i r l y r o t e f a s h i o n each of the p o s i t i v e and negative i n s t a n c e s . However, success-f u l r e c a l l , p a r t i c u l a r l y over a long term, was considered p o s s i b l y to r e q u i r e that the Ss r e a l i z e that a matrix of a s s o c i a t i o n s was i n v o l v e d . Furthermore, i t might n e c e s s i t a t e that the S i s o l a t e the v a r i a b l e s , determine which v a r i a b l e s were r e l e v a n t and i r r e l e v a n t to the concepts i n v o l v e d , and determine what the concepts a c t u a l l y were. R e c a l l of t h i s c r u c i a l i n f o r m a t i o n , presumably a l l that would be needed to generate the d i s p l a y , would seem to i n v o l v e f a r fewer u n i t s to be s t o r e d than would the r e c a l l of each a s s o c i a t i o n and i t s d e s i g n a t i o n . The l a t t e r r e c a l l probably would be a very d i f f i c u l t f e a t p a r t i c u l a r l y over the long term. Thus i t would seem that s u c c e s s f u l r e c a l l of each d i s p l a y p o s s i b l y would be r e l a t e d to a complete understanding of what was i n v o l v e d i n the d i s p l a y . This understanding p o s s i b l y would depend p r i n c i p a l l y on the understanding of the l a t t i c e of p r o p o s i t i o n s . Each of the concepts i n the d i s p l a y s was an example of one of these p r o p o s i t i o n s . In a d d i t i o n , the p o s i t i v e and negative concepts i n each d i s p l a y provided an example of complementary p r o p o s i t i o n s . Thus i t would seem that to determine those concepts and to understand t h e i r complementary nature would r e q u i r e considerable knowledge of the l a t t i c e . The understanding of the d i s p l a y might be f u r t h e r r e l a t e d , a l b e i t to a l e s s e r extent, to the scheme of hol d i n g v a r i a b l e s constant to determine the r o l e of other v a r i a b l e s . One method of determining the concepts i n each d i s p l a y would be to compare n e a r l y i d e n t i c a l p o s i t i v e and negative instances to determine the f a c t o r ( s ) r e s p o n s i b l e f o r t h e i r d i f f e r e n t d e s i g n a t i o n s . This approach would seem to be at l e a s t analog-co.us to the scheme of holding v a r i a b l e s constant. While probably not necessary f o r s u c c e s s f u l understanding of the concepts, t h i s approach would seem to be the most d i r e c t and e f f i c i e n t , p a r t i c u l a r l y i n the type of d i s p l a y used i n these tasks where the concepts i n v o l v e d are not immediately obvious. In summary, the p r i n c i p a l p o s s i b i l i t y i n v e s t i g a t e d was that success i n Tasks 6, 7, and 8 would be r e l a t e d to the understanding of the l a t t i c e . This understanding presumably would be measured by the average of the performances on a l l the P i a g e t i a n t a s k s . The other p o s s i b i l i t y t e n t a t i v e l y o f f e r e d was that success i n these memory tasks would i n v o l v e the method of comparing n e a r l y i d e n t i c a l p o s i t i v e and negative instances to determine the f a c t o r ( s ) r e s p o n s i b l e f o r t h e i r d i f f e r e n t d e s i g n a t i o n s . This method might be r e l a t e d , at l e a s t to some extent, to performance on the pendulum task and more s p e c i f i c a l l y to performance on the b a s i c method measure of t h i s task; the l a t t e r t e s t e d the a b i l i t y to hold v a r i a b l e s constant while manipulating others. Method Materials and concepts. The materials from the three tasks were as follows: 1. Display>'6, Screws and Bolts. As seen i n Figure 7 the display involved the 16 associations r e s u l t i n g from the m u l t i p l i c a t i o n of four variables each with two values (top, round or f l a t ; bottom, pointed or f l a t ; color, copper or gray; and length, long or short). The concepts involved i n t h i s task, t h e i r propositional symbols, and the propositionstt:6 which they refer were as follows: (To understand the r e l a t i o n s h i p between the concepts and the l a t t i c e of propositions i n t h i s and the next two tasks, reference can be made to Table 1, page 7.) (a) Round top and f l a t bottom (pq) and f l a t top and pointed bottom (pq) were p o s i t i v e , r e c i p r o c a l exclusion; (b) Round top and pointed bottom (pq) and f l a t top and f l a t bottom (pq) were negative, proposition of equivalence, which i s the complement of r e c i p r o c a l exclusion; (c) Length and color were unrelated to whether p o s i t i v e or negative, proposition of complete affirmation; i n the case of length (with the propositional symbols re-assigned) , t a l l p o s i t i v e (pq) , short p o s i t i v e (pq), 'fea'Q-Jl negative (pq), and short negative (pq) , a l l were i n the display. 92 F i g u r e 7. A s s o c i a t i o n m a t r i x of f o u r v a r i a b l e s , each w i t h two v a l u e s , w i t h p o s i t i v e (>/) and n e g a t i v e (x) i n s t a n c e s shown. Note: Reduced to approximately 50 p e r c e n t o f d i s p l a y s i z e . 93 2. Display 7, Flowers. As seen i n Figure 8, the display involves the 12 associations of three variables, leaf type, color, and stem width. The f i r s t variable had three values, one-pronged, two-pronged, or three-pronged; the second, two values, l i g h t or dark green; and the t h i r d , two values, thick or th i n . The propositions or concepts involved i n the display were as follows: (a) Dark and one-or three-pronged were p o s i t i v e , proposition of conjunction; (b) Light and one-, two-, or three-pronged or dark and two-pronged were negative, proposition of incompatibility; (c) Width unrelated to whether p o s i t i v e or negative, proposition of complete affirmation. 3. Display 8, J o l l s . Figure 9 shows the eight associations r e s u l t i n g from the m u l t i p l i c a t i o n of three variables each with two values (nose position, \ y o r / \ ; number of eyebrows on one side, two or three; and foot p o s i t i o n , up or down). The concepts and the propositions to which they refer were as follows: (a) Two eyebrows and shoulders down were p o s i t i v e , proposition of conjunction; (b) Three eyebrows and shoulders up or down or two eyebrows and shoulders up were negative, proposition of ) X 1 X ) O c o o F i g u r e 8 . A s s o c i a t i o n m a t r i x o f t h r e e v a r i a b l e each w i t h two or three v a l u e s , w i t h p o s i t i v e - ^ ) and negative (x) i n s t a n c e s shown. Note: Reduced to approximately 50 percent o f d i s p l a y s i z e . 95 F i g u r e 9. A s s o c i a t i o n m a t r i x o f t h r e e v a r i a b l e s , each w i t h two v a l u e s , w i t h p o s i t i v e (y/) and n e g a t i v e (x) i n s t a n c e s shown. Note: Reduced t o a p p r o x i m a t e l y 80 p e r c e n t o f d i s p l a y s i z e . 96 i n c o m p a t i b i l i t y ; (c) Nose p o s i t i o n u n r e l a t e d to whether p o s i t i v e or negative, p r o p o s i t i o n of complete a f f i r m a t i o n . Procedure. A l l three tasks were introduced i n a s i m i l a r manner. The Ss were t o l d a small s t o r y concerning why c e r t a i n members of the d i s p l a y were l i k e d , as i n d i c a t e d by a and c e r t a i n members, d i s l i k e d , as i n d i c a t e d by a X. In the case of Task 6, a carpenter was s a i d to l i k e c e r t a i n screws and/or b o l t s to f i x a c e r t a i n t a b l e and not to l i k e others. In the case of Task 7, a c e r t a i n bug was s a i d to l i k e to eat only a c e r t a i n k i n d or kinds of flower(s,). In the l a s t task, i n v o l v i n g the imaginary J o l l s , a c e r t a i n J o l l named A l b e r t was s a i d to l i k e only a c e r t a i n k i n d or kinds of J o l 1( sj)na!ndcto d i s l i k e o thers. In each task the S was asked to f i g u r e out which k i n d or kinds were l i k e d and which k i n d or kinds were not l i k e d , as she would be r e q u i r e d to remember the whole d i s p l a y , i n c l u d i n g the des i g n a t i o n of each member as l i k e d or d i s l i k e d . Before the viewing p e r i o d , which f o r each d i s p l a y l a s t e d f i v e minutes, the t e s t i n g method was explained to the Ss. In Tasks 6 and 7, the method i n v o l v e d the S f i r s t drawing and c o l o r i n g a l l the l i k e d instances and then a l l the d i s l i k e d i n s t a n c e s . In the case of Task 8, the S was given a sheet c o n t a i n i n g e i g h t J o l l s w i t h the noses, f e e t , and eyebrows on one s i d e missing; the missing f e a t u r e s were those that i n the 97 o r i g i n a l display varied from one J o l l to the next. The S was asked to reproduce the o r i g i n a l display by drawing i n the missing features and indi c a t i n g whether each of the J o l l s produced was l i k e d or d i s l i k e d . Scoring. The measurescdf performance included two basic r e c a l l measures and a t h i r d measure which was a com-posite of the f i r s t two. The f i r s t r e c a l l measure was the number of drawings c o r r e c t l y drawn and designated by the S as l i k e d or d i s l i k e d . If drawings were repeated, c r e d i t was given only once. The second r e c a l l measure was the number of d i f f e r e n t members of the o r i g i n a l display matrix which were drawn; whether or not the designation as l i k e d or d i s l i k e d was correct was not taken into account. The composite score was the sum of the points achieved on the f i r s t two measures. It was such that greater cre d i t was given to drawings i c o r r e c t l y drawn and designated and lesser c r e d i t , to drawings co r r e c t l y drawn but i n c o r r e c t l y designated. Overall Memory Performance Measure In order to achieve an o v e r a l l measure of memory performance, a p r i n c i p a l components analysis was performed on the composite scores achieved by the Ss on the eight memory tasks. The f i r s t p r i n c i p a l component i s known to provide for a set of variables the single l i n e a r composite 98 having maximal i n t e r n a l consistency. I f a S was missing one of the composite scores, she was given the average score obtained by a l l Ss on that composite measure. The f i r s t p r i n c i p a l component f a c t o r score, the measure of o v e r a l l memory performance, then was derived f o r each S. 99 Chapter 3 RESULTS This chapter w i l l provide the res u l t s concerning (a) the assessment tasks, (b) the memory tasks, and (c) the rel a t i o n s between assessment tasks and memory tasks. RESULTS: ASSESSMENT TASKS This section w i l l provide information concerning the following: 1. The categorizing of the Ss' performance on the Piagetian tasks and a summary of WISC vocabulary performance; 2. the unadjusted c o r r e l a t i o n matrix and the adjusted correlations, with the effects of age and WISC vocabulary performance removed, for performance on the assess-ment tasks; 3. the p r i n c i p a l components analysis of the assess-ment tasks. Categorizing of Performance on the Piagetian Tasks and WISC Vocabulary Performance The re s u l t s i n t h i s section are based on the perform-ance of 56 Ss except i n the case of the WISC vocabulary test, 100 where two Ss did not do the test. Table 4 indicates the percentages of Ss whose performances were assigned to the various substages and stages of concrete and formal operations. These percentages are provided for each of the Piagetian tasks and for o v e r a l l average performance on a l l the Piagetian tasks (found by summing the points obtained on each task and then finding the average). The concrete-operational stage consisted of the substages 2A, 2A-2B, 2B, and 2B-3A, while the formal-operational stage involved the substages 3A, 3A-3B, and 3B. For each substage the maximum range of points i s noted. Substages not used i n certain tasks to categorize performance are indicated by a dash. In the case of the balance task categories 3A, 3A+, and 3B, 3B- were collapsed to form respectively substages 3A and 3B. As indicated by the table, performance at the formal-operational stage i n the chemical combinations, pendulum,volume, and balance tasks was achieved by respectively 35.7, 39.3, 66.1, and 46.4 percent of the Ss. An o v e r a l l average per-formance at the formal-operational l e v e l was achieved by 42.9 percent of the Ss. In the case of the WISC vocabulary test, the average of the vocabulary scores, scaled according to age, was 9.8; the standard deviation was 2.7. The scores ranged from 5 to 17. A scaled score of 10 approximates an IQ of 100. Table 4. Percentages of the 56 Ss whose performance on each Piagetian task and average performance on a l l the Piagetian tasks was assigned to each stage and substage. Stage Substage Maximum Point • TASKS Average of Formal Range Chemicals Penc ulum Volume Balance Tasks 2A 2.75-3.24 25.0% 12. 5% 14.3% 5.4% 5.4% Concrete Opera-tions 2A-2B 2B 2B-3A 3.25-3.74 3.75-4.24 4.25-4.74 a 26.8 12.5 64.3% a 32.1 1 16.1 60.7% a 19.6 a 33.9% 8.9 32.2 7.1 53.6% 8.9 10.7 32.1 57.1% 3A 4.75-5.24 23.2 26.8 28.6 39. 3 b 26.8 Formal 3A-3B 5.25-Opera- 5.74 a 35.7 39.3 3. 66.1 3. 46.4 14.3 42.9 tions 3B 5.75-6.00 12. 5 12.5 37.5 7.1 b 1.8 This substage was not used when categorizing performance. b Substages 3A, 3A+ and 3B-, 3B were collapsed to form respectively substages 3A and 3B. 102 Unadjusted and Adjusted Correlation Matrices for Assessment Tasks Table 5 provides the main points of Table 11, Appendix A, which shows for the assessment tasks the Pearson product-moment^correlation matrix and the p a r t i a l c orrelations. In the case of the l a t t e r , the eff e c t s of either age, WISC vocabulary test performance, or both age and WISC vocabulary performance were removed. When a single task was correlated with the o v e r a l l average to which i t contributed, t h i s average was considered contaminated. The uncontaminated o v e r a l l average resulted from the removal of the contribution of the single task (with which i t was correlated); the res u l t i n g r thus was corrected for spuriousness. Table 5 shows only the unadjusted c o r r e l a t i o n matrix and the p a r t i a l correlations where the eff e c t s of both age and WISC vocabulary performance are removed. These re s u l t s are based on the performance of 56 Ss except i n the case of the p a r t i a l correlations and the one c o r r e l a t i o n involving WISC vocabulary performance; these exceptions involved 54 Ss due to two Ss not doing the WISC vocabulary t e s t . """While the scoring of the Piagetian tasks and, to a much lesser extent, the scoring of the memory tasks did not provide s t r i c t l y i n t e r v a l data, i t was decided to use the Pearson product-moment test i n analyzing these data; t h i s test i s s u f f i c i e n t l y robust to deal with data possessing.less than i n t e r v a l strength. 103 As seen i n Table 5 a l l the unadjusted c o r r e l a t i o n 2 c o e f f i c i e n t s between the Piagetian tasks were s i g n i f i c a n t . These c o e f f i c i e n t s ranged from .35, p <.01 (chemicals-pendulum) to .59, p <..001 (volume-balance), t h i s l a t t e r c o r r e l a t i o n c o e f f i c i e n t being considerably greater than the next highest, r = .41, p«<.01 (pendulum-balance). Each of the Piagetian tasks correlated s i g n i f i c a n t l y with the uncontaminated average of the formal tasks; the correlations involving the balance (r = .62, p<.00l) and volume (r = .59, p<.00l) tasks were considerably stronger than those involving the pendulum (r = .49, p<.00l) and chemicals (r = .45, p<.00l) tasks. Performance on the WISC vocabulary test was related s i g n i f i -cantly to performance on each of the Piagetian tasks and ov e r a l l average performance; these correlations ranged from .32, p <.01 '('chcmicals-WISC vocabulary) to .44, p<.001 (overall average-WISC vocabulary). With the eff e c t s of age and WISC vocabulary scores removed, a l l s i x correlations between the Piagetian tasks were reduced mainly due to the elimination of the ef f e c t of WISC vocabulary performance (see Table 11, Appendix A). A l l but one p a r t i a l c o r r e l a t i o n , involving the chemicals and pendulum In t h i s study OC = .05 was the l e v e l of significance adopted. However, the p r o b a b i l i t y of correlations achieving significance when the n u l l hypothesis i s true also i s indicated. Table 5. Product-moment c o r r e l a t i o n matrix and adjusted c o r r e l a t i o n s , w i t h the ^ef f e c t s of age and WISC vocabulary scores removed, f o r the assessment t a s k s . TASKS Pendulum Volume Balance Average of Uncontaminated WISC Formal Tasks Average of Vocabulary Formal Tasks Unadj. Adj . Unadj. Adj. Unadj. Adj. Unadj. Adj. Unadj. Adj . Unadj. Adj. Chemicals .35 b .26 .38 b .30° .38 b .30° . 7 i a .67 a a .45 .37 b c .32 Pendulum b .40 .32° .41 b .33° a .70 a .65 .49 a .40 b b .37 Volume .59 a a .54 a .82 .79 a .59 a .52a c .34 Balance .77 a .74 a .62 a .55 a c .34 Average of .44 a Formal Tasks a p <.001. b p < . 01. °p<.05. d Uncontaminated average r e s u l t e d from removing from the average of the formal tasks the c o n t r i b u t i o n of the s i n g l e task being c o r r e l a t e d w i t h the average. 105 tasks(r = .26), were s i g n i f i c a n t , however. These s i g n i f i c a n t p a r t i a l correlations ranged from .30, p<.05 (chemicals-volume; chemicals-balance) to .54, p <.001 (volume-balance); t h i s l a t t e r p a r t i a l c o r r e l a t i o n again was considerably greater than the next strongest, r = .33, p<.05 (pendulum-balance). With adjustment a l l the correlations between each Piagetian task and the uncontaminated average of the formal tasks were reduced, but the p a r t i a l correlations were s i g n i f i c a n t ; again the p a r t i a l correlations involving the balance (r = .55, p<.00l) and volume, (r = .52, p<.00l) tasks were stronger than those involving the pendulum (r = .40, p<.0l) and chemicals (r = .37, p < .01) tasks. P r i n c i p a l Components Analysis of Assessment Tasks  Table 6 shows the p r i n c i p a l component loadings for the Piagetian tasks and the WISC vocabulary scaled scores. As can be seen, a l l the assessment tasks loaded heavily on one component. The volume and balance tasks showed the highest loadings of respectively .76 and .72, while the WISC vocabulary test showed the lowest loading, .51. The f i r s t p r i n c i p a l component accounted for approximately 89 percent of the variance of the assessment tasks, while the second accounted for approximately 11 percent. The eigenvalue of the second component was .26, considerably less than the value of 1 generally required for including a component. This minimal 106 e i g e n v a l u e t h u s p r e c l u d e d r o t a t i o n o f p r i n c i p a l component a x e s . T a b l e 6. P r i n c i p a l component l o a d i n g s f o r p e r f o r m a n c e on t h e P i a g e t i a n t a s k s and t h e WISC v o c a b u l a r y t e s t . T a s k s F i r s t P r i n c i p a l Component C h e m i c a l s .59 Pendulum .65 Volume .76 B a l a n c e .72 WISC .51 V o c a b u l a r y RESULTS: MEMORY TASKS T h i s s e c t i o n w i l l p r o v i d e i n f o r m a t i o n c o n c e r n i n g t h e p e r f o r m a n c e o f t h e Ss i n t h e v a r i o u s memory t a s k s . T a b l e 7 p r o v i d e s f o r e a c h memory t a s k t h e maximum v a l u e o f t h e c o m p o s i t e measure and t h e o r i g i n a l a nd r e t e s t means and s t a n d a r d d e v i a t i o n s o f e a c h c o m p o s i t e measure, e x p r e s s e d a s a p e r c e n t a g e o f t h e maximum v a l u e . The r e s u l t s r e p o r t e d i n T a b l e 7 a r e b a s e d on t h e p e r f o r m a n c e o f t h e u s u a l 56 a n d 55 Ss i n r e s p e c t i v e l y t h e o r i g i n a l a nd r e t e s t p e r i o d s e x c e p t i n t h e c a s e o f t h e volume, c o m b i n a t i o n s o f a n i m a l s , and 107 permutations o f fou r t a s k s , which i n v o l v e d 54 Ss i n the r e t e s t p e r i o d . In the case of the volume task, three Ss i n the r e t e s t s e s s i o n c o u l d not remember when drawing whether or how the shape was changed. when computing the composite score f o r the volume task, i t was decided not to e l i m i n a t e these Ss but to g i v e them drawing c r e d i t ; a l l of them chose the c o r r e c t drawing i n the r e c o g n i t i o n task and none of the other Ss who r e c o g n i z e d c o r r e c t l y drew i n c o r r e c t l y . A measure of r e c a l l or r e c o n s t r u c t i o n performance can be c o n s i d e r e d to be of average d i f f i c u l t y i f mean per-formance on t h i s measure l i e s between 45 and 65 percent of the maximum p o s s i b l e s c o r e . In the case of the volume task, where the c o r r e c t drawing c o u l d be chosen or drawn c o r r e c t l y one t h i r d o f the time by chance, a measure of average d i f f i -c u l t y c o u l d be c o n s i d e r e d one that y i e l d s average scores between 55 and 80 per c e n t . Furthermore, a low v a r i a t i o n of scores on a measure can be c o n s i d e r e d to be i n d i c a t e d by a standard d e v i a t i o n o f l e s s than 15 percent of the maximum score p o s s i b l e . As can be seen i n Table 7, i n the o r i g i n a l t e s t i n g p e r i o d a l l but the composite measure i n the animals and permutations of four tasks y i e l d e d mean performance above the range o f average d i f f i c u l t y . In the case o f the measures of r e c o n s t r u c t i o n or r e c a l l , performance g e n e r a l l y f e l l w i t h i n 15 p e r c e n t of the 65 percent c u t o f f p o i n t ; i n the case of the T a b l e 7 . M a x i m u m p o s s i b l e v a l u e o f e a c h c o m p o s i t e m e a s u r e a n d t h e o r i g i n a l a n d r e t e s t m e a n s a n d s t a n d a r d f o r e a c h m e m o r y t a s k . d e v i a t i o n s e x p r e s s e d a s a p e r c e n t a g e > o f t h e m a x i m u m v a l u e , T A S K S O r i g i n a l M a x i m u m R e t e s t X cr X cr V o l u m e 8 2 . 2 % 3 4 . 9 % 2 8 0 . 6 % 3 9 . 4 % C o m b i n a t i o n s o f A n i m a l s 4 5 . 7 1 4 . 0 4 5 3 2 . 7 1 9 . 0 C o m b i n a t i o n s o f V e h i c l e s 7 5 . 6 1 7 . 6 4 5 5 7 . 1 2 0 . 5 P e r m u t a t i o n s o f T h r e e 8 3 . 3 2 1 . 4 1 8 6 7 . 8 2 8 . 6 P e r m u t a t i o n s o f F o u r 5 9 . 2 1 9 . 7 7 2 5 0 . 7 2 2 . 8 S c r e w s a n d B o l t s 7 7 . 3 2 0 . 8 3 2 6 4 . 2 2 4 . 0 F l o w e r s 7 3 . 3 2 3 . 4 2 4 5 8 . 1 2 3 . 0 J o l l s 7 2 . 4 <• 2 6 . 2 1 6 4 0 . 0 2 6 . 5 10 9 volume measure, mean performance was approximately two percent above the 80 percent c u t o f f p o i n t . In the r e t e s t p e r i o d average performance on the tasks g e n e r a l l y f e l l w i t h i n the range o f average d i f f i c u l t y . The exceptions were the volume and permutations o f three t a s k s , which y i e l d e d means of l e s s than one and three percent, r e s p e c t i v e l y , above the c u t o f f p o i n t , and the animals and J o l l s t a s k s , which had average performances of r e s p e c t i v e l y 32.7 and 40 percent of the maximum score p o s s i b l e . With r e g a r d to the o r i g i n a l and r e t e s t v a r i a t i o n s i n performance, o n l y the animals task i n the o r i g i n a l t e s t i n g p e r i o d y i e l d e d a standard d e v i a t i o n l e s s than 15 percent of the maximum score p o s s i b l e . RESULTS: RELATIONS BETWEEN ASSESSMENT TASKS AND MEMORY TASKS T h i s s e c t i o n w i l l p resent the o r i g i n a l and r e t e s t c o r r e l a t i o n s , both unadjusted and ad j u s t e d , with the e f f e c t s of age and WISC vocabulary performance removed, between the f o l l o w i n g : 1. o v e r a l l memory performance as i n d i c a t e d by the f a c t o r scores and performance on the assessment tasks;, 2. s p e c i f i c memory performance as i n d i c a t e d by the composite measure of memory i n each memory task and o v e r a l l average performance on a l l the P i a g e t i a n tasks; 3. s p e c i f i c memory performance on the v a r i o u s memory 110 : tasks and performance on p a r t i c u l a r P i a g e t i a n tasks thought to be r e l a t e d to the memory task i n v o l v e d . O v e r a l l Memory Performance and Assessment Task Performance  Table 8 summarizes the o r i g i n a l and r e t e s t c o r r e l a t i o n s and p a r t i a l c o r r e l a t i o n s , with the e f f e c t s o f age and WISC voca b u l a r y performance removed, between the assessment tasks and o v e r a l l memory performance as i n d i c a t e d by the f a c t o r s c o r e s . T h i s i n f o r m a t i o n i n a d d i t i o n to the p a r t i a l c o r r e l a -t i o n s , where the e f f e c t s o f age or WISC vocabulary performance alone were removed, are shown i n Table 12, Appendix A. F i f t y -s i x Ss were i n v o l v e d i n the o r i g i n a l c o r r e l a t i o n s ; 55, i n the r e t e s t c o r r e l a t i o n s . These numbers were f u r t h e r reduced by two i n the p a r t i a l c o r r e l a t i o n s , where the e f f e c t s o f WISC voca b u l a r y performance were removed, due to two Ss not doing the WISC vo c a b u l a r y t e s t . As seen i n Table 8, o v e r a l l average performance on the P i a g e t i a n tasks c o r r e l a t e d s i g n i f i c a n t l y with o v e r a l l memory performance i n the o r i g i n a l , as = .49, p-<.001, and r e t e s t , a* = .32, p<..05, p e r i o d s . When these c o r r e l a t i o n s were a d j u s t e d , the former was decreased to .47, p<.001, while the l a t t e r was i n c r e a s e d to .36, p<.01. As seen i n Table 12, Appendix A, the s l i g h t r e d u c t i o n and i n c r e a s e i n r e s p e c t i v e l y the o r i g i n a l and r e t e s t p a r t i a l c o r r e l a t i o n s r e s u l t e d from the I l l opposing e f f e c t s of the removal of age and the removal of WISC vocabulary performance. The removal of the former increased s l i g h t l y the percentage of variance accounted for by the cor r e l a t i o n s . The removal of the l a t t e r decreased th i s variance; t h i s decrease was less than four and two percent, respectively, i n the o r i g i n a l and retest periods. In order to gain some understanding of why the cor-r e l a t i o n s between o v e r a l l memory performance and average Piagetian task performance decreased from the o r i g i n a l to the retest period, the test- r e t e s t r e l i a b i l i t y c o e f f i c i e n t between the factor scores i n the o r i g i n a l testing period and a new set of retest factor scores was computed. The l a t t e r factor scores, derived to ensure comparability between the o r i g i n a l and retest factor scores, resulted from weighting the scores on the various memory tasks i n the retest period i n the same way these scores were weighted i n the o r i g i n a l testing period. The re s u l t i n g test-retest r e l i a b i l i t y co-e f f i c i e n t was .60, p<. 001. As seen i n Table 8, i n the o r i g i n a l testing period, performance on the s p e c i f i c Piagetian tasks, with the exception of the pendulum task, showed s i g n i f i c a n t p o s i t i v e correlations with o v e r a l l memory performance. The strongest correlations involved the balance, r = .54, p<.001, and volume, r = .46, p<.001, tasks, while those involving the chemicals^arid pendulum tasks were respectively .30, p<.05, and .20, Table 8. O r i g i n a l and r e t e s t unadjusted c o r r e l a t i o n s and a d j u s t e d c o r r e l a t i o n s , w ith the e f f e c t s of age and WISC vocabulary performance removed, between o v e r a l l memory performance and performance on the assessment t a s k s . TASKS O v e r a l l Memory: O r i g i n a l O v e r a l l Memory: Retest Unadj. Adj . Unadj. Adj . Average of Formal Tasks .49 a .47 a .32° .36 b Chemicals .30° .26 .16 .15 Pendulum .20 .13 .13 ..09 a a c „ c Volume .46 .45 .27 .33 Balance .54 a .52 a .44 a ' .49 a WISC Vocabulary .22 .14 a p <r.ooi. p <. 01. C p <.05. 113 p = .14. With adjustment the correlations were s l i g h t l y reduced due to the elimination of the ef f e c t of WISC vocabulary performance (see Table 12, Appendix A), and the one involving the chemicals task, r; = .26, became not s i g n i f i c a n t . Per-formance on the WISC vocabulary test did not correlate s i g n i f i c a n t l y with o v e r a l l memory performance i n either the o r i g i n a l (r = .22) or retest (r = .14) period. The same pattern of r e l a t i o n s , although somewhat reduced, was found i n the retest period between performance on s p e c i f i c Piagetian tasks and ov e r a l l memory performance. In the retest period, however, while a l l unadjusted correlations were p o s i t i v e , only those involving the balance task, r = .44, p<C.001, and the volume task, r = .27, p< .05, were s i g n i f i c a n t . With adjustment these two correlations increased to respectively .49, p<.001, and .33, p<.05, due to the removal of the effe c t s of age (see Table 12, Appendix A). The other two correlations were reduced s l i g h t l y . Performance on S p e c i f i c Memory Tasks and Average Piagetian Task Performance Table 9 presents the o r i g i n a l and retest correlations and p a r t i a l correlations, with the eff e c t s of age and WISC vocabulary performance removed, between performance on the s p e c i f i c memory tasks, indicated by the composite measure i n each task, and o v e r a l l average performance on the Piagetian 114: tasks. F i f t y - s i x Ss were involved i n the o r i g i n a l correlations; 55, i n the retest correlations except i n the case of the volume, combinations of animals, and permutations of four tasks, which involved 54 Ss. These numbers were further reduced by two i n the p a r t i a l correlations due to the two Ss not doing the WISC vocabulary test. The information of Table 9 i s included i n Tables 13, 14, 15, and 16 i n Appendix A. These tables show a l l the unadjusted and p a r t i a l correlations between Piagetian task performance and performance on the memory tasks related respectively to volume, combinations, permutations, and the l a t t i c e . Table 17, Appendix A, provides a l l the unadjusted and adjusted correlations between average Piagetian task performance and scores on the component measures which were the bases of the composite measure i n each memory task. Comparison of Tables ^ 8 ,and 9 indicates that i n the o r i g i n a l testing period o v e r a l l Piagetian performance correlated more strongly with o v e r a l l memory performance than i t did with memory on any s p e c i f i c memory task; i n the retest period with a few exceptions (unadjusted and adjusted permu-tations of four; unadjusted screws and bolts) a similar pattern was found. As seen i n Table 9 a l l the unadjusted and p a r t i a l correlations i n the o r i g i n a l testing period were p o s i t i v e , ranging from .22 to .39; 12 of the 16 were s i g n i f i c a n t . In Table 9. Original and retest unadjusted correlations and adjusted correlations, with the eff e c t s of age and WISC vocabulary performance removed, between average performance on the Piagetian tasks and memory task performance. MEMORY TASKS Correlations with Average of Formal Tasks Or i g i n a l Retest Unadj. Adj . Unadj. Adj . Volume .28C .24 .28° .18 Combinations of Animals .38 b .33° .02 .02 Combinations of Vehicles .27° .24 .16 .21 Permutations of Three .38 b .39 b .19 .25 Permutations of Four .29° .29° .32° .40 b Screws and Bolts .33° .26 c .32 .27 Flowers .31 C .22 . 28° .22 J o l l s .26C .33° .17 .13 a p < .001. b p <.01. ° p < .05. 116 the retest period, they also were p o s i t i v e . However, i n the animals, vehicles, permutations of three, and the J o l l s tasks the values of both the unadjusted and p a r t i a l correlations were less than their counterparts i n the o r i g i n a l testing period; i n fact they were considerably less i n the case of the l a t t e r three tasks. In the retest period four correlations and one p a r t i a l c o r r e l a t i o n were s i g n i f i c a n t . Performance on S p e c i f i c Memory and S p e c i f i c Piagetian Tasks  Table 10 presents the unadjusted correlations and p a r t i a l c o r r e l a t i o n s , with the eff e c t s of age and WISC vocabulary performance removed, between the composite measure of memory performance i n each memory task and performance on the p a r t i c u l a r Piagetian task(s) thought to be related to each memory task. The correlations between the basic method 3 measure i n each of the chemicals and pendulum tasks and the s p e c i f i c memory tasks postulated to be related to these Piagetian tasks were omitted; none was s i g n i f i c a n t and nearly a l l were less than the correlations involving complete per-formance on the Piagetian tasks. The information of Table 10 in addition to the p a r t i a l correlations, where the effects of age and WISC vocabulary performance alone were removed, are In each of the chemicals and pendulum tasks a basic method measure and solution measure were integrated to form the substages of performance. 11:7-p r o v i d e d i n Ta b l e s 13, 14, 15, and 16; these t a b l e s d e a l r e s p e c t i v e l y with memory tasks r e l a t e d to volume, combinations, permutations, and the l a t t i c e o f p r o p o s i t i o n s . The unadjusted c o r r e l a t i o n s were based on the performance of the us u a l 56 and 55 Ss i n r e s p e c t i v e l y the o r i g i n a l and r e t e s t p e r i o d s except i n the case o f the volume, combinations o f animals, and permutations of fou r t a s k s , which i n v o l v e d 54 Ss i n the r e t e s t p e r i o d . In the case of the p a r t i a l c o r r e l a t i o n s , where the e f f e c t of WISC vocabulary performance was removed, these numbers were reduced by two due to two Ss not doing the WISC vocabulary t e s t . Comparison of Tables 9 and 10 i n d i c a t e that g e n e r a l l y performance on each memory task c o r r e l a t e d more s t r o n g l y with average P i a g e t i a n task performance than with performance on p a r t i c u l a r P i a g e t i a n t a s k ( s ) thought to be r e l a t e d to the p a r t i c u l a r memory task. The onl y c l e a r e x c e p t i o n i n v o l v e d the memory task r e l a t e d to volume, where the o r i g i n a l and r e t e s t c o r r e l a t i o n s and p a r t i a l c o r r e l a t i o n s between memory performance and volume c o n s e r v a t i o n were a l l higher than t h e i r c o u n t e r p a r t s ( i n terms o f t e s t i n g time and whether a d j u s t e d or not) i n v o l v i n g average P i a g e t i a n task performance. As seen i n Table 10, the m a j o r i t y o f the unadjusted and p a r t i a l c o r r e l a t i o n s were p o s i t i v e but, wit h the exception o f those i n v o l v i n g the memory task r e l a t e d to volume and the volume c o n s e r v a t i o n task, they were not s i g n i f i c a n t . I t i s 118 Table 10. O r i g i n a l and r e t e s t unadjusted c o r r e l a t i o n s and adjusted c o r r e l a t i o n s , with the e f f e c t s of age and WISC vocabulary performance removed, between memory task performance and performance on p a r t i c u l a r Piagetian tasks. PIAGETIAN TASKS MEMORY TASKS Volume Original Retest Unadj. Adj. Unadj. Adj. Velum* Subtask 5 .34b .32° .11 .11 .35b .29° .17 .16 Chemicals Combinations of Animals Combinations of Vehicles Original Retest Original Retest Unadj. Adj. Unadj. Adj. Unadj. Adj. Unadj. Adj. .20 .14 -.09 -.11 .17 .13 .20 .23 Chemicals Pendulum Permutations of Three Permutations of Four Original Retest Original Rete St Unadj. Adj. Unadj. Adj. Unadj. Adj. Unadj. Adj. .26 .17 .24 .14 -.07 -.06 .06 .07 .19 .17 .09 .06 .11 .13 .10 .11 Pendulum Screws and Bolts Flowers JoV Is Original Retest Original Rete S t Original Retest Unadj.1 Adj. Unadj. Adj. Unadj.j Adj. Unadj. Adj. Unadj.j Adj. Unadj. Adj. .12 .02 .14 .04 .26 .18 .20 .14 -.03 - 08 .10 .06 ap C .001. bp 4.01. Cp <.05. 119 i n t e r e s t i n g to note that the volume and/or balance task p r e d i c t e d performance best on many of the memory tasks not po s t u l a t e d to be c l o s e l y r e l a t e d to these P i a g e t i a n tasks; many of these c o r r e l a t i o n s were s i g n i f i c a n t . (See Tables 13, 14, 15, and 16, Appendix A.) 1207 Chapter 4 DISCUSSION FORMAL OPERATIONS AND MEMORY Results of the study j u s t i f y the statement that i t was successful i n achieving both of i t s major aims. Present findings permit i d e n t i f i c a t i o n of ind i v i d u a l differences with respect to a u n i f i e d formal-operational structure; they also allow these differences to be related to predictable differences i n memory performance on a variety of tasks designed i n the Piagetian t r a d i t i o n of memory research. Discussion of findings relevant to the u n i f i e d structure of formal operations w i l l be followed by consideration of the relationships found between formal-operational competency and memory. Unified Structure of Formal Operations Two hypotheses concerning the u n i f i e d structure of formal operations were investigated and both were confirmed by the obtained data. F i r s t , s i g n i f i c a n t p o s i t i v e c o r r e l a -tions were found between performance on each of the four 121 f o r m a l - o p e r a t i o n a l tasks and the average of performance on the other three t a s k s . Second, a p r i n c i p a l component a n a l y s i s r evea l ed that the f i r s t p r i n c i p a l component accounted fo r a s u b s t a n t i a l 89 percent of the va r i ance o f the assess -ment t a s k s . The present f i n d i n g of cons i s t ency o f performance across the P i a g e t i a n tasks i s i n agreement w i t h the r e s u l t s of the m a j o r i t y o f s t u d i e s c i t e d ( e . g . , A r l i n , 1974, 1977; Hughes, 1965; L o v e l l , 1961; Tomlinson-Keasey, 1970) which a l s o found such c o n s i s t e n c y . U n l i k e the present s tudy , however, these p rev ious ones f i n d i n g cons i s t ency f a i l e d to c o n t r o l fo r both age and IQ. In f a c t , of these s tud i e s mentioned, on ly three ( A r l i n , 1974, 1977; Hughes, 1965) c o n t r o l l e d , at l e a s t to some ex ten t , fo r age, w h i l e another three ( B a r t , 1971; Jackson , 1965; L o v e l l and S h i e l d s , 1967) had some c o n t r o l f o r IQ. As a r e s u l t o f the f a i l u r e o f these p rev ious s t u d i e s to c o n t r o l fo r these v a r i a b l e s , the present f i n d i n g s p rov ide the p r i n c i p a l support fo r the con t en t ion that observed cons i s t ency of performance i s due to ope ra t i ve unders tanding r a the r than to s k i l l s u n r e l a t e d to formal opera t ions but r e l a t e d to age and/or IQ. Thus the concept of a u n i f i e d s t r u c t u r e o f formal o p e r a t i o n s , independent o f age or IQ, i s supported by the present s tudy . The p o s i t i v e f i n d i n g s o f the present study a l s o support the sugges t ion tha t r epor t ed f a i l u r e s ( e . g . Neimark, 122 1970; Ross, 1973) to f i n d s i g n i f i c a n t relationships across Piagetian tasks may be attributed to methodological weak-nesses i n these investigations. These weaknesses included use of subjects too young to be expected to be at the formal-operational stage; use of procedures that deviated consider-ably from those of Inhelder and Piaget; considerable v a r i a t i o n i n d i f f i c u l t y among the Piagetian tasks employed; and selection of tasks that were poor measures of formal-operational thought. Present findings have important implications for the selection of tasks which are es p e c i a l l y suitable measures of formal-operational a b i l i t y . The substantial c o r r e l a t i o n found between the volume and balance tasks indicates that they are measuring the same competency, according to Inhelder and Piaget (1958), the INRC group. This finding furthermore suggests that either task i s a suitable measure of t h i s competency. Other re s u l t s argue that the balance and volumes tasks also are the best indices of general formal-operational a b i l i t y , described by Inhelder and Piaget i n terms of the integrated INRC group-lattice structure. In general, t h i s o v e r a l l a b i l i t y presumably would be most adequately assessed by average Piagetian task performance and f i r s t p r i n c i p a l component scores. Thus the findings that the volume and balance tasks showed the strongest correlations with average Piagetian task performance and 123 loaded the most heavily on the f i r s t p r i n c i p a l component argues that, of the four tasks employed, they are the best indices of general formal-operational a b i l i t y . This argument i s further supported by other findings showing that these r e s u l t s were not due simply to the substantial c o r r e l a t i o n between the balance and volume tasks. These tasks also were related s i g n i f i c a n t l y to the chemicals and pendulum tasks, which moreover showed the lowest intertask c o r r e l a t i o n . Formal-Operational Competency and Memory Performance  With regard to the second major aim of the study, concerning the rel a t i o n s h i p between formal-operational competency and memory performance, obtained data provide evidence to support the p r i n c i p a l hypothesis investigated. Average Piagetian task performance accounted for approximately 25 percent of the variance i n ov e r a l l memory performance in the o r i g i n a l testing period. In the retest period, i t accounted for about 10 percent of thi s variance. These percentages changedjonly s l i g h t l y to 22 percent and 13 percent when the effects of age and IQ were removed. Overall performance on the Piagetian tasks also was related to performance on the s p e c i f i c memory tasks. A l l the correlations were p o s i t i v e , and, p a r t i c u l a r l y i n the o r i g i n a l testing period, many of them were s i g n i f i c a n t . 124 Together these data suggest that both o v e r a l l memory performance and performance on s p e c i f i c memory tasks were related to differences i n general formal-operational a b i l i t y , presumably measured by average performance on a l l the Piagetian tasks. Further support for the importance of volume and balance task performance as indices of formal-operational development comes from the findings that the largest contribution to predictable variance i n both o v e r a l l memory performance and performance on s p e c i f i c memory tasks was made by the balance and volume tasks. Balance and volume task performance, when compared with o v e r a l l Piagetian task performance, showed, respectively, s l i g h t l y higher and lower correlations with o v e r a l l memory performance. Either or both of these tasks predicted best, and often s i g n i f i c a n t l y , performance on many s p e c i f i c memory tasks including those not hypothesized to be c l o s e l y related to the s p e c i f i c schemes tapped by these tasks. In general, these findings might be best interpreted as further evidence of the predic-t i v e power of general formal-operational a b i l i t y . The volume and balance tasks and the various memory tasks generally shared i n common only their hypothesized r e l a t i o n s h i p to general formal-operational a b i l i t y . Thus th i s a b i l i t y , rather than less general competencies not considered to be related to most of these tasks, probably mediated the majority 125 of the relationships found between memory performance and the balance and volume tasks. Obtained data did not confirm the hypothesis that performance on p a r t i c u l a r Piagetian tasks thought to be measuring s p e c i f i c formal-operational schemes or concepts would be s i g n i f i c a n t l y related to performance on p a r t i c u l a r memory tasks also thought related to the schemes. The correlations between the s p e c i f i c Piagetian tasks and the supposedly related memory tasks generally were p o s i t i v e . But, s i g n i f i c a n t correlations were obtained only i n the case of the re l a t i o n s h i p between the memory task related to volume and the volume conservation task. This s i g n i f i c a n t r e l a t i o n -ship, however, cannot be interpreted as exemplifying the mediation of memory performance by the understanding of s p e c i f i c schemes. There i s no evidence that the conser-vation of occupied volume, the s p e c i f i c concept hypothesized to mediate th i s r e l a t i o n s h i p , i n fact did so. Performance on subtask 5"^ of the volume conservation task, considered to be the most di r e c t measure of thi s scheme, showed only a minimal c o r r e l a t i o n with performance on the memory task related to volume. The s i g n i f i c a n t r e l a t i o n s h i p , then, might Subtask 5 required the S to predict whether the water l e v e l would remain the same or would change when the metal bricks of a building constructed under water were rearranged. 126 be best considered as simply another example of the indiscrim-inate p r e d i c t i v e power of the balance and volume tasks with regard to performance on s p e c i f i c memory tasks. As such i t probably would be best interpreted as being mediated by general formal-operational a b i l i t y . Thus i n the present study, there was no evidence that the understanding of s p e c i f i c l o g i c a l schemes, such as combinatorial operations, the conservation of occupied volume, and the method of holding variables constant, mediated performance on the s p e c i f i c memory analogues of these schemes. Possible reasons for this lack of p o s i t i v e findings are detailed i n Appendix B. Data did not confirm the f i n a l hypothesis, which was that the magnitude of the correlations between Piagetian task performance and both o v e r a l l memory performance and performance on s p e c i f i c memory tasks would be greater one month after presentation of the displays rather than immediately following the presentation. In fa c t , the relat i o n s h i p between average Piagetian task performance and o v e r a l l memory performance was c l e a r l y reduced from the o r i g i n a l to the retest period. In addition, i n only .the permutations of four task was there an increase i n both the unadjusted and p a r t i a l correlations over time. The interpretation of these findings, however, i s somewhat equivocal due to the limi t e d retest r e l i a b i l i t y of the memory tasks, as indicated by the test-retest c o e f f i c i e n t 127. between the Ss' o v e r a l l factor scores ( r t t = .60). There are at least two possible reasons for thi s reduced r e l i a b i l i t y which also would explain the decrease in the r e l a t i o n s h i p between Piagetian task performance and memory performance. F i r s t , some of the displays may have been poor items for testing long term memory. Two of the four memory tasks which showed retest reductions i n both the unadjusted and p a r t i a l correlations involving o v e r a l l Piagetian task performance yielded average retest performance below the psychometrically optimal range of means for test items. Thus in the case of these tasks (the only two showing such low mean performance) there would have been a f l o o r e f f e c t , which probably was at least p a r t l y responsible for the observed reduction i n corr e l a t i o n s . Second, on each memory task, a f a i r l y high percentage of Ss (ranging from 7 to 31 percent) showed improvement. Improvement occurred considerably more often among concrete-operational Ss than among t r a n s i t i o n a l Ss (2B - 3A) and formal-operational Ss. Thus the ov e r a l l e f f e c t of these improvements would be to reduce the retest c o r r e l a t i o n between Piagetian task performance and memory performance. The explanation for these improvements cannot be determined from the data available. One p o s s i b i l i t y i s that they resulted from cognitive development on the part of some of these Ss during the tes t - r e t e s t i n t e r v a l (see Piaget and Inhelder, 1968). Another p o s s i b i l i t y i s that some Ss may 128 have discussed the memory tasks after the i n i t i a l testing period. Overall, however, the explanatory role of cognitive development, as proposed by Piaget, was extended successfully i n the present study. Differences i n memory performance supposedly related to the development from preoperational to concrete-operational thought have been studied f a i r l y exten-s i v e l y (e.g., Furth and Milgram, 1973; Piaget and Inhelder, 1968; Prawat and C a n c e l l i , 1976; Tomlinson-Keasy et a l . , 1975). The present study, however, represents an advance i n that i t shows that changes i n memory performance i n a variety of tasks also are related to the achievement of formal operations. Furthermore, by c o n t r o l l i n g for the effects of age and IQ, the present investigation permits s p e c i f i c a t t r i b u t i o n of these phenomena to changes i n operative l e v e l . Moreover, the res u l t s lend some support to the often-made d i s t i n c t i o n between the Piagetian and the psychometric concepts of i n t e l l i g e n c e (e.g., Furth, 1973; Kohlberg and DeVries, 1974; Kuhn, 1976). Performance on the formal-operational tasks was found to be s i g n i f i c a n t l y related to assessed IQ. This finding i s i n agreement with the results of other studies (e.g., Bart, 1971; Jackson, 1965; Kuhn, Langer, Kohlberg, and Haan, 1972). On the other hand, average performance on the Piagetian tasks correlated more highly with o v e r a l l memory performance than did WISC vocabulary scores. In fact, for both the o r i g i n a l and retest periods, only nonsignificant correlations were found between the WISC vocabulary scores and ov e r a l l memory performance. Further-more, when the eff e c t s of WISC vocabulary performance (which correlates .78 with the WISC f u l l scale) were removed, the variance i n o r i g i n a l and retest memory performance accounted for by average Piagetian task performance was reduced by less than four and two percent, respectively. Thus the present study demonstrated that Piaget's concept of i n t e l l i g e n c e , when defined operationally, had pred i c t i v e v a l i d i t y that not only exceeded but was almost independent of that of the psycho-metric concept of i n t e l l i g e n c e , also defined operationally. UNIVERSALITY OF FORMAL OPERATIONS On the basis of research conducted i n Geneva, Inhelder and Piaget (1958) concluded that formal-operational thought develops through the ages 11 to 15 years. Equilibrium was considered achieved by the age of 15 years by 75 percent of adolescents; t h i s percentage was considered to indicate u n i v e r s a l i t y (Piaget, 1952). A considerable body of research (e.g., Jackson, 1965; Tomlinson-Keasey, 1970), however, did not support the Inhelder and Piaget conclusion. In fact a number of studies (e.g., A r l i n , 1974; Elkind, 1962) found that fewer than 75 percent of even college students showed such achievement. The high success rate reported by Inhelder and 130. Piaget was generally considered (e.g., L o v e l l , 1961) to be due to their selection of an unrepresentative sample of adolescents from p r i v i l e g e d schools. Piaget (1972) has acknowledged th i s lack of random sampling and the dearth of evidence supporting his and Inhelder's findings and ten t a t i v e l y restated his p o s i t i o n . A l l normal people now were considered to achieve formal operations, i f not by 15 years, i n any case by 20 years, but i n di f f e r e n t areas according to thei r aptitudes and professional s p e c i a l i z a t i o n s . Thus the t r a d i t i o n a l assessment tasks, which are s c i e n t i f i c a l l y oriented, would underestimate formal-operational achievement in the case of individuals not having much aptitude or s p e c i a l i z a t i o n i n science. In the present study, almost one-half the Ss performed at the formal-operational l e v e l even though they were r e l a t i v e l y young (average age, 13.1 years) and generally came from a working class or lower middle-class background. This proportion of formal-operational Ss i s higher than that normally found with such young Ss (e.g., Elkind, 1961b) or even with older Ss of a higher socio-economic l e v e l (e.g., D u l i t , 1972). The disparate findings of the present study may be due to the fact that t h i s study, as compared to many others, administered the tasks i n a manner more cl o s e l y related to that of Inhelder and Piaget (1958). A general overview of the various studies investigating 131 formal operations indicates that the methodology of many of them deviated i n one or more s i g n i f i c a n t ways from that of Inhelder and Piaget (1958). Subjects sometimes were given very l i t t l e time to experiment (Lee, 1971) or were not questioned during their experimentation i n the manner of Inhelder and Piaget (Tomlinson-Keasey, 1970). In some cases the method of scoring d i f f e r e d considerably from :Inhelder and Piaget's; f i r s t , considerable available information was not taken into account and/or second, the scoring c r i t e r i a were too s t r i c t . An example of the f i r s t difference i n scoring occurred i n the cases where the S's performance during experimentation was ignored, and only the S's f i n a l conclusion was scored ( A r l i n , 1974j 1977). Another example occurred i n the assessment of volume conservation (Elkind, 1961b, 1962) where the categories of performance involved no d i f f e r e n t i a t i o n between i n t e r i o r volume conservation and the conservation of occupied volume, both of which had been tested. Both narrow-ness and s t r i c t n e s s i n scoring were evident i n the chemicals task when a performance had to include the complete set of 15 combinations to be c l a s s i f i e d as 3B, and no cre d i t was given for understanding of the roles of the various chemicals (Dulit, 1972). A l l the preceding deviations from the methodology of Inhelder and Piaget might serve to reduce the percentage of Ss who appeared to perform at the formal-operational l e v e l . 132-In the case of the chemicals task, for example, the simple question asked i n the present study "Is there anything else you could do?" produced for many Ss a considerable increase i n the number of combinations made. That only one question produced such a f a c i l i t a t i n g e f f e c t suggests that the above-mentioned studies would have c l a s s i f i e d many more Ss as formal-operational i f procedures more similar to those of Inhelder and Piaget had been followed. The present argument i s not that the use of procedures similar to those of Inhelder and Piaget would r e s u l t necessarily in universal performance at the formal-operational stage on the part of normal adult Ss. Rather, the argument i s that there i s no basis for the current o v e r a l l pessimism that only a small proportion of adolescents or even adults are capable of operating at the formal-operational l e v e l on t r a d i t i o n a l assessment tasks. It becomes unclear whether these tasks, when administered i n the manner of Inhelder and Piaget, under-estimate formal-operational achievement as seriously as Piaget would seem to suggest they do. In the present study, cer-t a i n l y not a l l of the r e l a t i v e l y high percentage of Ss showing formal-operational performance on these tasks would be expected to have considerable aptitude or s p e c i a l i z a t i o n i n science; i n fa c t , c e r t a i n of the Ss c l a s s i f i e d as formal-operational indicated by the i r comments a lack of interest i n science and hence, possibly i n some cases, minimal s c i e n t i f i c 133 aptitude. Comparative studies of performance on tasks sp e c i a l i z e d according to the individual's aptitudes and professional s p e c i a l i z a t i o n and on the t r a d i t i o n a l tasks administered i n the manner of Inhelder and Piaget would seem to be required. Such studies would shed l i g h t on the issue of whether or not the l a t t e r tasks thus administered would produce substantial underestimation of formal-operational achievement. Such studies also would indicate i f u n i v e r s a l i t y of performance at the formal-operational l e v e l would appear among adults when either s p e c i a l i z e d or t r a d i t i o n a l tasks measured cognitive l e v e l . Also bearing on thi s issue of u n i v e r s a l i t y i s the observation made i n thi s study that many p i l o t Ss who did poorly i n i t i a l l y i n the Piagetian tasks caught on quickly when the correct procedure for experimenting was explained to them. This observation suggests, contrary to Piaget's position, that special t r a i n i n g procedures might be e f f i c i e n t i n teaching formal-operational thinking. In fact such t r a i n i n g procedures might be far more e f f e c t i v e than those designed to produce performance at the concrete-operational l e v e l . The environment by nature of the structure of the physical world would seem to "force" the development of concrete operations upon every normal human being. 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TASKS Pendulum Volume Balance Effects removed oft Effects removed of: Effect 3 removed of i Unadj. WISC Age WISC and age Unadj. wise Age WISC and age Unadj. WISC Age WISC and age Chemicals .35b (54) .26 (51) .35b (53) .26 (50) .38b (54) .30° (51) .37b (53) .30° (50) .38b (54) .30C (51) .37b (53) .30° (50) Pendulum .40b (54) .32? (51) .39b (53) .32C (50) .41b (54) .33C (51) .41b (53) .33C (50) Volume .59a (54) .54a (51) .58a (53) .54a (50) Balance Average of Formal Tasks Average of Formal Tasks Effects removed ofi h Unadj. .71 (54) .70a (54) .82a (54) .77" (54) .67 (51) ,65a (n) .79a (51) .74a (51) .70 (53) .70a (53) .Bl* (53) .77a (53) .67 (50) .65a (50) .79a (50) • 74a (50) Uncontaminated Average of Formal Tasks Unadj. Effects removed ofi .45a !.37b .44a (54) (51) (53) .49" .40b .48a (54) (51) (53) .59a .52a .58a (54) (51) (53) .62a .55a . 6 i a (54) f5l) (53) .37 (50) .40b (50) .52a (50) .55a (50) Unadj. .37 (52) .34C (52) .34° (52) .44a (52) WISC Vocabulary Effects removed of: .36 (51) .29° (51) .32C (51) .41b (51) Notei ( ) indicates degrees of freedom. p < .001. bp <.01. p< .05 A , , „ t h - aVeraae of the formal tasks the contribution of the single task being correlated with the average. Uncontaminated average resulted from removing from the average oi tne rorrn APPENDIX A Table 12. O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between Piagetian task performance and o v e r a l l memory performance. 1 1 1 1 • - '•• Overall Memory t Original Over all Memory I Retest Effects removed of i Effects removee J oft TASKS Unadj. WISC Aga WISC and age Unadj. WISC Age WISC and age Average of Formal .49a (54) .45* (51) .52* (53) .47* (50) .32C (53) .29° (50) .42b (52) .36b (49) Tasks Chemicals .30° (54) .25 (51) .31C (53) .26 (50) .16 (53) .13 (50) .21 (52) .15 (49) Pendulum .20 (54) .13 (51) .20 (53) .13 (50) .13 (53) .08 (50) .17 (52) .09 (49) Volume .46a (54) .42b (51) .49* (53) .45* (SO) .27° (53) .23 (50) .39b (52) .33° (49) Balance .54* (54) .5la (51) .55* (53) .52* (50) .44* (53) .42b (50) .53* (52) .49* (49) Volumes Subtask 5 .28C (54) .28° (51) .29° (53) .30° (50) .11 (53) .10 (SO) .15 (52) .15 (49) Note« ( > i n d i c a t e * d e g r e e s 0 / f r e e d o m . *p< .OOl. bp<.01. ep<.05» APPENDIX A Table 13. O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between Piagetian task performance and performance on the memory task r e l a t e d to volume conservation. Original Memory R e t e s t Memory Effects removed o f i Effects removed of: TASKS Unadj. wise Age WISC and age Unadj. WISC Age WISC and age Average ot Formal .28° (54) .23 (51) .28° (53) .24 (50) .28C (52) .17 (49) .28° (51) .18 (48) Tasks Chemicals -.02 (54) -.08 (51) -.01 (53) -.07 (50) .06 (52) -.04 (49) .05 (51) -.04 (48) Pendulum . 2 7 ° (54) .23 (51) .28° (53) .23 (50) .23 (52) .13 (49) .23 (51) .13 (48) Volume .34b (54) .31° (51) .35b (53) .32° (50) .35b (52) .28° (49) .35° (51) .29° (48) Balance .23 (54) .18 (51) .23 (53) .19 (so) .19 (52) .10 (49) .19 (51) .10 (48) Note i ( ) i n d i c a t e s d e g r e e s ot f r e e d o m * *p< .001. bP<.01. cp<.05. APPENDIX A Table 14. O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between Piaget i a n task performance and performance on the memory tasks r e l a t e d to combinations. ANIMALS WHltt.ES f t r l . j l i i . i l Memory Rctt'it Hcm.iry O r i g i n a l Memory Rolost Memory B f f a o t s removed o f i E f f e c t * removal o f i B / f a o t a removed o f i E f f a c t a rcrcovuil o f i TASKS U i u d j . WISC Ag* WISC • nd •ga Un*dj. WISC Ag* WISC •nd •ga UlMdj. WISC Ag* WISC •nd •0« Un*dj. WZSC Ag* WISC •nd &g« Avar •<;• o f f o c u l 7*atia) (54) .32 e (SI) .39 b (33) .33° (SO) .02 (32) -.01 (49) • OT (31) .02 (48) .27° (34) .24 (31) .28° (53) .24 (30) .18 (33) .18 (SO) .24 (32) .21 (49) C h c o i c a l o • JO (34) .14 (31) .70 (33) .14 (SO) -.00 (32) -.11 (49) -.00 (31) -.11. (48) .17 (34) .13 (31) .17 (33) .13 (SO) .30 (S3) .20 (SO) .25 (32) .33 (49) Fandulim .10 (54) .02 (31) .11 (33) .02 (so) -.03 (32) -.OS (49| -.03 (51) -.09 (40) .16 (34) .12 (31) .10 (53) .12 (50) .08 (33) .07 (SO) .11 (52) .07 (49) VOIUM* (34) .31° (31) .30 b (33) .33° (SO) .04 (52) .02 (49) .11 (31) .08 (48) .14 (54) .12 (31) .17 (33) .13 (50) .02 (S3) .OI (so) .11 (52) .08 (49) D*l<&/vc« .47* (34) . 4 3 b (31) .48* (S3) .44* (»0) .16 (S3) .IS (49) .21 (S I ) .IT (48) .3» b (54) .37* (S I ) .40* (S3) .37* (90) .19 (S3) .30 (50) .28 (52) .33 (49) Hotel ( ) Indicate* Oagraaa of fiaadoa. •p<.O0l. **(><.01. APPENDIX A Table 15. O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between Piaget i a n task performance and performance on the memory tasks r e l a t e d to permutations. Permutation* of three Pornutationa o f four 7 ASKS O r i g i n a l Memory Kctest Memory O r i g i n a l Memory Rotcat Honory E f f a c t a removed o f t E f f e c t * removed o f i B f f a c t e ramovad o f i E f f e c t i i remove 1 o f i Unadj. WISC Ag« wise and aga Unadj. WISC Aga wisc and aga Unadj. WISC Aga WISC and aga Unadj. WISC Agm wise and aga Avaraga of r o r m l Task* .3B b (S4> .33" (SI) .4Qb (33) .3-/> (SO) .19 (33) .21 (50) .26 (53) (49) .29° (34) .20 e (31) . 3 l C (S3) .29° (so) .32 C (32) .37b (49) .30 b (31) .40 b (4S) C h a n l c a l a .24 (54) .24 ( 3 D .24 (S3) .24 (SO) -.07 (S3) -.07 (50) -.04 (32) -.04 (49) .19 (34) .17 (31) .20 (S3) .17 (so) .11 (52) .13 (49) .14 (51) .13 (40) Fandulua. .17 (34) .14 (»D .17 (S3) .14 (SO) .00 (53) .07 (50) .09 (52) .07 (49) .09 (34) .04 (SI) .10 153) .06 (so) .10 (32) .11 (49) .12 (31) .11 (IB) Voluna .41 b (54) .40 b ( 3 D .42* (S3) .41" (50) .24 (33) .26 (30) • 32 C (52) .32 C (49) .2« C (34) .36 (31) .31° (33) .39° (SO) .31" (32) .34° (49) .38 b (3D .3ub (48) tValanca .30° (34) .89° (31) .31° (S3) .39° (30) .36 b (33) .30 b (SO) .41" (93) .43 b (49) .31° (34) .30° (31) .3>° (S3) .31 C (30) .30* (33) .33* (49) .34* (31) .36* (43) Notai ( ) lndlcataa dagraaa of freadoa. *p<.001. " j x . o i . *P<.03. 1 4 6 APPENDIX A Table 16. the l a t t i c e . Orig i n a l and r e t e s t unadjusted and partxal c o r r e l a t i o n s between Piagetian task performance and performance on the memory tasks r e l a t e d to TASKS Average of Formal Tasks Chemicals Pendulum Volume Balance Screws and B o l t s Unadj. .33 (54) .32° (54) .12 (54) .27C (54) .31C (54) O r i g i n a l Memory Effects removed ofi WISC .25 (51) .25 (51) .02 (51) .19 (51) .24 (51) .35 (53) .32° (53) .13 (53) • 29C (53) .32° (53) .26 (50) .25 (50) .02 (50) .22 (SO) .25 (SO) Unadj. .32 (53) .26 (53) .14 (53) .28= (53) .32° (53) Retest Memory E f f e c t s removed o f : .23 (50) .19 (50) .04 (50) .21 (50) .25 (50) .38 (52) .29° (52) .16 (52) .35b (52) .36b (52) .27 (49) .20 (49) .04 (49) .27 (49) .28C (49) O r i g i n a l Memory Effects removed ofi Unadj. .31 (54) .19 (54) .26 (54) .26 (54) .22 (54) WISC .22 (51) .12 (51) .18 (51) .19 (51) .15 (51) Age .30 (53) .19 (53) .25 (53) .25 (53) .22 (53) WISC and age .22 (50) .12 (50) .18 (50) .19 (50) .15 (50) Unadj. .28 (53) .28° (53) .20 (53) .16 (53) .25 (53) Retest Memory E f f e c t s removed o f i WISC .21 (.50) .23 (SO) .14 (50) .09 (50) .19 (SO) Age .28 (52) .28° (52) .20 (52) .16 (52) .25 (52) WISC and age .22 (49) .23 (49) .14 (49) .10 (49) .19 (49) O r i g i n a l Memory Unadj. .26 (54) .17 (54) -.08 (54) .32C (54) .37b (54) Effects removed ofi WISC .30 (51) .18 (51) -.08 (51) .35b (51) .40b (51) Age WISC and age R e t e s t Memory Unadj. .31 (53) .19 (53) -.07 (53) .38b (53) .40b (53) .33 (50) .19 (50) -.08 (50) .39b (50) .42b (50) .17 (53) .12 (53) .10 (53) .14 (53) .16 (53) Effects removed of: WISC .13 (50) .09 (SO) .06 (50) .11 (50) .13 (50) Age .16 (52) .11 (52) .09 (52) .13 (52) .16 (52) WISC and age .13 (49) .09 (49) .06 (49) .11 (49) .13 (49) Notei ( ) indicates degrees of freedom. p< .001. °p <.oi. p< .05. \ 147 APPENDIX A Table 17. O r i g i n a l and r e t e s t unadjusted and p a r t i a l c o r r e l a t i o n s between average Piagetian task performance and performance on the component measures i n each memory task. Correlations with Averaae Piagetian Performance Original Retest Effects removed of: Effects removed o f i TASKS Measure Unadj. WISC Age wise a n d a g e Unadj. WISC Age WISC a n d a g e d Volume Levels drawn correctly Levels recognized correctly .19 (54) .31C (54) .15 (51) .26 (51) .19 (S3) .32C (53) .16 (50) .27 (50) .25 (49) .33C (52) .15 (46) .22 (49) .24 (48) • 32C (51) .16 (45) .23 (48) No. combina -tions .23 (54) .17 (51) .26 (53) .19 (50) .05 (52) -.02 (49) .06 (51) -.00 (48) Animals6 Horizontal order .42b (54) .38b (51) .43* (53) .39b (50) .01 (52) .03 (49) .07 (51) .01 (48) Vertical order .34C (54) .28C (51) .33C (53) .28° (50) .01 (52) .01 (49) .07 (51) .06 (48) No. combina -tions .24 (54) .21 (51) .25 (53) .22 (50) .12 (53) .10 (50) .19 (52) .14 (49) e Vehicles Horizontal order .26 (54) .23 (51) .27C (53) .24 (50) .16 (53) .16 (50) .24 (52) .21 (49) Vertical order .27C (54) .23 (51) .27C (53) .23 (SO) .14 (53) .16 (50) .22 (52) .20 (49) No. permuta -tions .24 (54) .24 (51) .23 (53) .23 (50) .12 (53) .12 (50) .14 <52) .13 (49) Permuta -tions { o f three Initial Members Constant .33C (54) .29C (51) .31C (S3) .28° (50) .13 (53) .12 (50) .16 (52) .14 (49) Vertical Order b .38 (54) .41b (51) . 4 3 A (53). .44* (50) .21 (53) .27C (50) .29° (52) .33C (49) N o t e i ( ) i n d i c a t e s d e g r e e s o f f r e e d o m . *p<.001. bp<.01. Cp<.0S. d, e, f, and gSee p a g e 8 Og_7o, 76-78, 84-87, and 97, respectively, for detailed explana-tions of the component measures of these tasks. 148 APPENDIX A Table 17. (continued) Correlations with Average Piaoetian Performance Original Retest Effects removed oft Effects removed oft TASKS Measure Unadj. WISC Age WISC and age Unadj. WISC Age WISC and ag* No. permuta -tions .31C (54) .28c (51) .32= (53) .29= (50) .28= (52) .28= (49) .31= (51) .30c (48) Permuta. -tions f of four Initial Members Constant .32° (54) .31° (51) .35b (53) .32= (50) .33= (52) .39b (49) .39b (51) .42b (48) Vertical Order .20 (54) .20 (51) .23 (53) .22 (50) .14 (52) .21 (49) .19 (51) .24 (48) Screws and Bolts9 No. members matrix drawn No. correct liked and disliked .37b (54) .26 (54) .29= (51) .17 (51) .38b (53) __c .27 (S3) .30= (50) .19 (50) .37b (53) .23 (53) .29= (50) .15 (50) .39b (52) .22 (52) .31° (49) .14 (49) Flowers9 No. members matrix drawn No. correct liked and disliked drawn .35b (54) .23 (54) .26 (SI) .17 (51) .36b (53) .24 (53) .27 (50) .18 (50) .31= (53) .21 (53) .25 (50) .15 (SO) .31= (52) .20 (52) .24 (49) .14 (49) Jolls9 No. members matrix drawn Na correct liked anc disliked drawn .26 (54) .25 (54) .32C (51) .28° (51) .26 (53) .25 (53) .32= (50) .27 (50) .21 (53) .10 (53) (18) (50) .07 (50) .21 (52) .10 (52) .17 (49) .07 (49) Notei ( ) indicates degrees of freedom. *p< .001. bp<.01. ep<.03. d, e, f, and gSce p a g e s 6o_70. 76-78, 84-87, and 97, tions of the component measures of these tasks. respectively, for detailed explana-149. APPENDIX B. Minimal Correlations between Performance on S p e c i f i c Piagetian and Memory Tasks: Possible Reasons There would seem to be two basic reasons why-performance on s p e c i f i c Piagetian tasks postulated to be related to s p e c i f i c memory tasks did not predict well performance on these memory tasks. F i r s t , the s p e c i f i c schemes or understandings may have been assessed inadequately by the pa r t i c u l a r Piagetian tasks. Secondly, the a b i l i t i e s required to figure out or generate the displays may be i n some way di f f e r e n t from the Piagetian schemes thought related to the displays. Table 18 (p. 155) provides an outline of the p r i n c i p a l relationships predicted between memory task performance and performance on s p e c i f i c Piagetian tasks. The possible reasons for the lack of relationships found are indicated by a t i c k . These reasons include the f i r s t basic explanation, concerning the possibly inadequate assessment of the s p e c i f i c Piagetian schemes, and four subcategories of the second basic explanation. These subcategories w i l l be explained in greater d e t a i l when their relevance to the various memory tasks i s discussed. In the case of the f i r s t reason, the p o s s i b i l i t y exists that a l l the s p e c i f i c schemes were assessed inadequately. These schemes included the conservation of occupied volume, the a b i l i t i e s to combine and permute, and the method of holding variables constant to test the ef f e c t s of others; these 150 schemes were considered to be assessed most d i r e c t l y by, r e s p e c t i v e l y , the volume conservation, chemical combinations, and pendulum tas k s . As the inadequacy of assessment would seem most apparent i n the case of the conservation of occupied volume and the com b i n a t o r i a l operations, only the problems w i t h t h e i r assessment w i l l be discussed. In the memory task r e l a t e d to volume, there was only a mimimal r e l a t i o n s h i p between memory performance and per-formance on subtask 5. On the other hand, there was a strong r e l a t i o n s h i p between memory performance and the nature of the Ss' responses when the Ss were questioned a f t e r completion of the drawing and r e c o g n i t i o n t e s t s . A high percentage of Ss who d i d one or both of these t e s t s i n c o r r e c t l y explained that the weight or volume of the c l a y was changed when the shape was modified. An e q u a l l y high percentage of Ss who drew and recognized the d i s p l a y c o r r e c t l y s a i d the volume (or weight) had not changed. I t i s recognized that the E's questioning may have prompted post hoc reasoning on the part of some Ss; such reasoning presumably would not have mediated memory performance. However, i t i s p o s s i b l e that i n some cases the Ss 1 responses to these questions r e f l e c t e d an understanding or l a c k of understanding which was r e s p o n s i b l e f o r memory performance. The question then becomes why performance on subtask • 5 d i d not assess adequately the understanding or l a c k of T 5 1 understanding of these l a t t e r Ss. P o s s i b l y a more adequate assessment might have occurred i f the Ss had been given c r e d i t only i f they j u s t i f i e d t h e i r d e c i s i o n s and p e r s i s t e d w i t h the d e c i s i o n despite counterarguments on the part of the E. I t was the E's op i n i o n that a number of the Ss might have been dissuaded i n e i t h e r d i r e c t i o n . I t also.l.is p o s s i b l e that assessing the conservation of occupied volume by the method of rearranging blocks under water might y i e l d d i f f e r e n t r e s u l t s than a method i n v o l v i n g changing the shape of the c l a y . Some l i m i t e d postexperimental t e s t i n g i n d i c a t e d that the r e l a t i o n s h i p between the r e s u l t s of these two methods was not as high as would have been expected. I t may be, as suggested by P. K. A r l i n (personal communication), that the apparent success of some Ss when they have been assessed by the method i n v o l v i n g c l a y was based, not on volume conservation, but on the conservation of weight. This suggestion i s supported by the f i n d i n g i n the present study of some Ss who s a i d the water l e v e l s i n the glasses were the same because the weight of the c l a y remained the same. S i m i l a r l y , i n the case of the memory tasks r e l a t e d to combinations, i t i s p o s s i b l e that the assessment of com b i n a t o r i a l a b i l i t y through the chemical combinations task and i t s b a s i c method measure was inadequate. The i n s t r u c t i o n s i n the chemicals task to reproduce the yel l o w c o l o r by using 152 the l i q u i d s from the bottles may have been misleading. The instructions seem to have suggested to the Ss to simply reproduce the color and not to f i n d a l l the ways to make the color. Thus many Ss stopped after making the color one way and often did not recommence afte r the E's questioning whether there was anything else they could do. It was the E's opinion that many of them would have done better i f they had been instructed to f i n d a l l the ways of making the yellow color. It may be argued that the chemicals task, with i t s instructions as provided, may have tapped general formal-operational thinking, including the a b i l i t y to operate within the framework of the hypothetico-deductive method and to consider not only the r e a l but also the possible. This thinking presumably should not be influenced greatly by minor changes i n in s t r u c t i o n s . However, i t also can be argued that the chemicals task may not have tapped adequately the combinatorial scheme, p a r t i c u l a r l y i f one questions the v a l i d i t y of the postulated connection between t h i s scheme and the l a t t i c e structure. The possible inadequacy of the chemicals task i n assessing combinatorial a b i l i t y also might be responsible for the lack of rel a t i o n s h i p found between the chemicals task and the memory tasks related to permutations. In addition, even i f combinatorial a b i l i t y had been assessed adequately by the chemicals task, the rel a t i o n s h i p between th i s task 153 and the memory tasks related to permutations might be i n d i r e c t . According to Piaget and Inhelder (1975), the a b i l i t y to permute, while c l o s e l y related to the a b i l i t y to make combinations, i s not i d e n t i c a l and develops at a l a t e r age. It would seem that the l a t t e r a b i l i t y i s more clos e l y related to performance on the chemicals task than i s the former. Possibly, the use of assessment tasks more c l o s e l y related to the a b i l i t y to combine or to permute might res u l t i n a f a i r l y strong r e l a t i o n s h i p between these tasks and the r e c a l l or reconstruction of combinations or permutations. Examples of such tasks might be found in Piaget and Inhelder (1975). For example, Ss might be required to make a l l the permutations or pair-wise combinations of n d i f f e r e n t objects. The four subcategories of the second basic explanation for the lack of s i g n i f i c a n t relationships include the follow-ing. F i r s t , the f i g u r a t i v e component of the display possibly could have been r e l a t i v e l y simple. Thus higher order mnemonic schemes might not have been required to remember the display. This p o s s i b i l i t y would seem to pertain p a r t i c u l a r l y to the memory task related to volume. Operations concerning the conservation of occupied volume might not have been activated, or i f activated, the r e s u l t i n g understanding might have been negated due to the simple nature of the display. Furthermore, the f i g u r a t i v e aspect of the display might have been emphasized by the procedure, which required the S during the i n i t i a l 1 5 4 d i s p l a y to note twice that the water l e v e l s were equal. Secondly, the f i g u r a t i v e component of the d i s p l a y might not have been n e c e s s a r i l y simple. However, c e r t a i n higher-order mnemonic schemes thought o p t i m a l l y e f f i c i e n t f o r memory may not have been used or r e q u i r e d f o r s u c c e s s f u l memory performance. In the case of the memory tasks r e l a t e d to the l a t t i c e , the method of comparing n e a r l y i d e n t i c a l p o s i t i v e and negative instances of a concept to determine the f a c t o r s r e s p o n s i b l e f o r t h e i r d e s i g n a t i o n may not have been used by many, i f any, Ss. This method, which p o s s i b l y was r e l a t e d to the P i a g e t i a n operation of hol d i n g v a r i a b l e s constant w h i l e manipulating others, had been considered to be o p t i m a l l y e f f i c i e n t but not necessary f o r s u c c e s s f u l memory performance. T h i r d l y , i t i s p o s s i b l e that operations assessed by s p e c i f i c P i a g e t i a n tasks are somewhat d i f f e r e n t from and p o s s i b l y r e q u i r e more on the p a r t of the Ss than do seemingly s i m i l a r mnemonic a b i l i t i e s . This might be the case i n the memory tasks r e l a t e d to combinations and permutations. The a b i l i t i e s r e q u i r e d to spontaneously generate a l l the p o s s i b l e combinations and permutations might d i f f e r from the a b i l i t i e s needed to "catch on" to the system of a d i s p l a y of combina-t i o n s or permutations presented to the S. These former a b i l i t i e s may i n v o l v e operations at a higher l e v e l than do t h e i r memory task counterparts, which may r e q u i r e mnemonic s k i l l s which are somewhat t r i v i a l i n comparison. 1 5 5 There i s some very i n d i r e c t evidence supporting the d i s t i n c t i o n between the a b i l i t i e s to combine and permute and their mnemonic counterparts. According to Piaget and Inhelder (1975), the a b i l i t y to combine develops e a r l i e r than the a b i l i t y to permute. Thus, i f these operations are related to mnemonic a b i l i t i e s , i t would seem that the a b i l i t y to r e c a l l or reconstruct a display of combinations might develop e a r l i e r than the a b i l i t y to r e c a l l or reconstruct a display of permutations. In the case of the present study, i t might be expected that Ss who do well on the memory tasks related to permutations also should do well on the tasks related to combinations. Subjects who do well on the l a t t e r tasks need not do well on the former. The r e s u l t s , however, did not support t h i s expectation. This possible d i s t i n c t i o n between the a b i l i t i e s to combine and permute and their mnemonic counterparts may have been due to the l a t t e r a b i l i t i e s being somewhat t r i v i a l . However, there i s a fourth reason why the a b i l i t i e s required to figure out or generate the displays might be somewhat di f f e r e n t from the Piagetian schemes thought related to the displays. It i s possible that the mnemonic schemes used to generate or figure out the displays were not necessarily t r i v i a l but were only s u p e r f i c i a l l y analogous to the opera-tions assessed by the s p e c i f i c Piagetian tasks. This explanation would seem relevant to the lack of APPENDIX B Table 18. Possible reasons f o r the lack of r e l a t i o n s h i p s between performance on s p e c i f i c memory tasks and s p e c i f i c Piagetian tasks. MEMORY TASKS RELATED TO: Volume Combinations Permutations Lattice PIAGETIAN TASKS Volumei Subtask 5 Chemicals Chemicals Pendulum Pendulum Explanations Piagetian operations inadequately assessed >/ >/ Memory display figuratively simple >/ Higher order mnemonic schemes efficient but not necessary s/ Mnemonic schemes t r i v i a l in comparison to Piagetian counterparts V >/ Mnemonic schemes only analogous to Piagetian counterparts N / Notes Relevant reasons indicated by>/« H Ln 157 r e l a t i o n s h i p s found between the pendulum task and the memory tasks r e l a t e d to permutations and the l a t t i c e of p r o p o s i t i o n s . The method of hol d i n g v a r i a b l e s constant w h i l e manipulating others to t e s t t h e i r e f f e c t s was presumed to be assessed by the pendulum task and, more p a r t i c u l a r l y , by i t s b a s i c method measure. This experimental method might be only s u p e r f i c i a l l y s i m i l a r to the method of generating the d i s p l a y s of permu-t a t i o n s by h o l d i n g the f i r s t member(s) constant w h i l e changing the p o s i t i o n of the l a s t two. S i m i l a r l y , t h i s method might be only analogous to the method of comparing n e a r l y i d e n t i c a l p o s i t i v e and negative instances to determine the f a c t o r ( s ) r e s p o n s i b l e f o r t h e i r d i f f e r e n t d e s i g n a t i o n s . This l a t t e r method had been p r e d i c t e d to be o p t i m a l l y e f f i c i e n t i n " f i g u r i n g out" the concepts i n v o l v e d i n the d i s p l a y s r e l a t i n g to the l a t t i c e . 

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