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

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AUTOMATON AND COMPETENCE ASPECTS OF PIAGETIAN LOGICAL CONCEPTS , by NELLY ADELINA TOUSSAINT B.A., University of Tulsa, 1968 M.A., University of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Psychology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t 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 t u d y . 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 p u rposes 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 . I t i s u n d e r s t o o d t h a 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 t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h C olumbia Vancouver 8, Canada 11 Abstract Simultaneous measures were obtained of (a) the Ss ' l e v e l of attainment on automaton variables (foresight and hindsight a b i l i t i e s and size of computing space) t h e o r e t i c a l l y presumed to be necessary to the attainment of l o g i c a l structures, and (b) the Ss' l e v e l of attainment in l o g i c a l tasks belonging to the same conceptual family (m u l t i p l i c a t i o n of classes, multi-p l i c a t i o n of r e l a t i o n s , s e r i a t i o n , and t r a n s i t i v i t y ) . Children from Grade 1 and 2 served as Ss. The task demands of the d i f -ferent l o g i c a l tasks were equated in terms of the competence as tvell as automaton demands. Considerations about the d i s t i n c -tion between operative and f i g u r a t i v e aspects of i n t e l l i g e n c e were incorporated into the testing methods. The results i n d i -cated there was a strong relationship between the Ss 1 performances across a l l the tasks. The o v e r a l l findings support Piaget's theoretical conceptions about the relationships of the variables tested. Some of the theoretical and empirical implications of these findings are discussed. i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES i v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i CHAPTER I. Introduction 1 CHAPTER I I . Theoretical Framework of Diagnostic Procedure 10 CHAPTER I I I . Method 50 CHAPTER IV. Results 85 CHAPTER V. Discussion 110 REFERENCES 133 APPENDIX. FORHIN Sets 137 ADDENDUM 144 iv L i s t of Tables Table Page 1. Correlations between FORHIN scores and Operative (0) and Figurative (F) measures for Gl tnd G2 86 2i Correlations between MOPER scores and Operative (0) and Figurative (F) measures for Gl and G2 86 3* Relationship between the TRAN scores and the scores below and above the medians of the FORHIN, MOPER, MCMUL(O), MRMUL(O), and SERDIM(O) scores for Gl and G2 . . 87 4. Intercorrelations between Operative (1,2,3) and Figurative (4,5,6) measures for Gl and G2... 91 5. Intercorrelations between measures derived from Operative measures (1-5) and those derived from Figurative measures (6-10) 92 6. Intercorrelations between the F i l l - i n and Anticipatory measures and the Operative scores for Gl and G2 93 7. Proportion of Ss obtaining MCFILL and MRFILL scores greater than 2/3 of the maximum possible (12/18) 94 8. Proportion of Ss in Grade 1 and Grade 2 obtaining scores greater than or equal to 4 in the MC and MR Anticipation tasks 96 9. Intercorrelations of the difference between Figurative and Operative scores for the MC, MR, and SER tasks and a l l the Figurative and Operative scores for Gl and G2..... 99 10. Proportion of Ss in Gl and G2 whose FORHIN score f e l l above or below the median and whose MC and MR Reverse or Operative scores increased or decreased itfith reference to t h e i r correspond-ing Copy or Figurative score... 100 11. Intercorrelations between the F i l l - i n and Anticipatory measures and the Figurative scores for Gl and G2 101 12. Means, SD's, differences between the means, t - t e s t s , and significance levels of the measures obtained for Gl and G2 103 13. Number of units of information that Gl Ss were able to handle successfully in the FORHIN, MOPER, MC, MR, and SER tasks. 105 V Table Page 14. Number of units of information that G2 Ss were able to handle successfully in the FORHIN, MOPER, MC, MR, and SER tasks 106 15. Pairwise comparisons of the scores below and above the medians of the MOPER, FORHIN, SERDIM(O), and m u l t i p l i c a t i v e combinations in MR and MC(MUL) scores for Gl and G2 107 v i L i s t of Figures Figure Page 1. M u l t i p l i c a t i o n of Classes matrix (a) and reversed matrix (b). Dimensions remaining constant on the horizontal axis s i z e , diamond, color; and along the v e r t i c a l axis background, thickness , and shape 53 2. M u l t i p l i c a t i o n of Relations matrix (a) and reversed matrix (b). Dimensions remaining constant along horizontal axis s i z e , orientation, color; and along v e r t i c a l axis thickness, brightness,, and l i n e s . . . 61 3. Seriation task. Stimuli varied in height, width, thickness, eyes' rotation, width of frock, and position of t i e . . . . . . . . . . . . 64 4. T r a n s i t i v i t y task. The relevant character-i s t i c s to the t r a n s i t i v i t y judgments were height and color of bottom h a l f ("shirt") of " l i t t l e men" 68 5. Foresight-Hindsight task. Training set..... 72 6. M-Operator task. (a) Stimuli and response units (b) Sample cards of t r a i n i n g set 1 (c) Sample cards of t r a i n i n g set 2 (d) Sample cards of compound stimuli cards 79 Acknowledgements v i i The author would l i k e to express her gratitude to the chairman of the doctoral committee, Dr. Lou Moran, fo r providing constant assistance and encouragement, as well as a most enjoy-able and stimulating i n t e l l e c t u a l atmosphere. Special thanks are due to the other members of the committee, Dr. Chris Tragakis and Dr. Meredith Kimball for t h e i r help and counsel throughout the various phases of this work. The author also wishes to acknowledge the co-operation and assistance of the Vancouver School Board and the p r i n c i p a l and s t a f f members of the General Gordon Elementary School, Our Lady of Perpetual Help School, and York House School. In addition, the author wishes to thank Ms. Bernice Wong for proofreading the di s s e r t a t i o n ; Mr. Jon Moran, for the photography work; Mr. Godfried Toussaint, for the drawing of figures; and Ms. Rosanne Rumley for the excellent typing work. This work has been made possible with the support of a National Research Council of Canada Post-graduate Scholarship to the author. CHAPTER I Introduction Recently psychologists interested i n cognitive development have t r i e d to understand the antecedent psychological processes or mechanisms which make possible the development of the ch i l d ' s a b i l i t y to perform symbolic and abstract operations of the kind outlined i n Piaget's theory. Two of the major questions that have concerned researchers are: (1) What are the mechanisms of t r a n s i t i o n from one Piagetian stage of development to another? With reference to a more s p e c i f i c concept such as conservation, the question i s , what factors f a c i l i t a t e the t r a n s i t i o n from an understanding of the q u a l i t a t i v e aspects of the s t i m u l i (pre-l o g i c a l thinking) to an understanding of the quantitative r e l a -tions as well ( l o g i c a l thinking)? ( 2 ) What generalizes between stage-related concepts, or what can be expected to generalize to related s i t u a t i o n s from the kno\<rledge that the c h i l d has attained a c e r t a i n concept? This l a t t e r question can also be posed i n re c i p r o c a l terms (that r e f l e c t the r e a l i t i e s borne out by the data): what are the reasons for the f a i l u r e to obtain the expected high within-subjects i n t e r - c o r r e l a t i o n s among tasks presumably belonging to the same developmental level? The lack of adequate answers to these problems i s i n part a manifestation of the developmental psychologists' d i f f i c u l t i e s in diagnosing cognitive states. There are very few s a t i s f a c t o r y tools to determine l e v e l s of cognitive structure p r i o r to or during the attainment of l o g i c a l concepts. In turn, the lack of e f f e c t i v e diagnostic procedures based on Piaget's theory i s a r e f l e c t i o n of two pervasive problems: f i r s t , the lack of consensus as to the 2 conceptual meaning of major constructs such as "stage" and a l l i t s related concepts (synchrony, t r a n s i t i o n , inter-item r e l a t i o n -ships, e t c . ) ; and second, a dir e c t consequence of the f i r s t , the lack of consensus as to the operational c r i t e r i a that should be used to render meaningful and comparable empirical measures of the constructs. One goal of the present study was to devise a diagnostic procedure to assess quantitatively the child's attained l e v e l of competence in logico-mathematical operations through procedures which are based on Piaget's t h e o r e t i c a l analysis of how the development of this type of structure occurs. The analysis and construction of the procedure, of course, d i r e c t l y r e f l e c t s Piaget's t h e o r e t i c a l formulations on the two major questions stated at the outset. Before elaborating the s p e c i f i c theoreti-cal and empirical bases of the diagnostic procedure, i t w i l l be p r o f i t a b l e to discuss some of the problems found in the l i t e r a -ture on the various e f f o r t s to deal with questions of assessment of the cognitive concepts defined by Piaget. Piaget's system, as has been pointed out by numerous authors (Braine, 1959; Wohlwill, 1966; F l a v e l l , 1953, e t c . ) , is a d i f f i -c u l t one to operationalize i n unambiguous terms, p a r t i c u l a r l y so the more one i s dealing with more t h e o r e t i c a l l y s i g n i f i c a n t and general constructs (for instance, the s p e c i f i c concept of "con-servation of substance," has been more frequently tested than the construct of "equilibration.") It i s also the case that i n many instances the theory has been seriously misrepresented or inappro-p r i a t e l y applied. To substantiate this assertion, three major types of inadequate treatments of the theory w i l l be i l l u s t r a t e d . 3 F i r s t , one common reason for inappropriately defined opera-t i o n a l c r i t e r i a i s the fact that many authors have dealt with d i f f e r e n t but intimately related aspects of the theory sepa-r a t e l y or i n an isola t e d manner, when i n fact the analysis at hand may require a simultaneous consideration of several types of variables. This problem is re f l e c t e d in the f a i l u r e of many researchers to distinguish the Competence and Performance aspects of the theory. This d i s t i n c t i o n , borrowed from psycholinguists, was f i r s t applied to developmental data by F l a v e l l and Wohlwill (1969). The competence aspects are said to correspond to the formal l o g i c a l operations and structures which Piaget outlines as underlying the di f f e r e n t stages of development, for example, the mathematical groupings which characterize the cognitive develop-ment occurring between seven and eleven years of age. The per-formance aspects correspond to the psychological mechanisms or "automaton"1 functions which permit the competence described by Piaget to be expressed i n any s p e c i f i c s i t u a t i o n , e.g., atten-t i o n a l , memory, perceptual, or other yet undefined non-structural functions. The importance of the Competence-Automaton d i s t i n c -t i o n i s that i t brings into focus the necessity of considering simultaneously (by co n t r o l l i n g or manipulating, as the case might be) both of these aspects of logico-cognitive development, p a r t i c -u l a r l y i n reference to questions concerning the acqui s i t i o n of l o g i c a l concepts. JThe word "automaton" w i l l be used, in preference to the word "performance," to refer to the subject's functions that theoret-i c a l l y are said to make the expression of a given competence possible. The word "performance" i s reserved for actual behavioral instances r e f l e c t i n g the given competence. 4 The significance of the competence-automaton d i s t i n c t i o n becomes more obvious through the i l l u s t r a t i o n of problems that have arisen when the d i s t i n c t i o n i s not made or when only one aspect of i t is taken into account when studying the acquisition of l o g i c a l concepts. The c h i l d who understands conservation of li q u i d s may assert or deduce that the l i q u i d remains the same despite the changes in appearances because nothing has been added or taken away -- addition-subtraction argument; or because i f the water is poured back to the o r i g i n a l container, i t would look the same -- r e v e r s i b i l i t y argument; or because the i n i t i a l equal-i t y i s what is important -- id e n t i t y argument; or because changes i n one dimension are compensated by changes in the other dimension -- compensation argument. According to Piaget, use of these arguments to j u s t i f y the assertion of conservation indicates that the c h i l d has achieved the competence necessary to understand that the quantity of l i q u i d remains invariant because the trans-formations performed are part of a system of co-ordinated and reciprocal operations (Piaget, 1968). Some investigators of the competence of conservation (Smedslund, 1966; Wallace, Wall, and Anderson, 1967; R o l l , 1970) have t r i e d to elucidate the precursor factors i n the development of the grouping structures by bringing about the necessary com-petence in non-conserving children through the tr a i n i n g of empirical procedures based on the postulated operations of the groupings themselves (the subjects are s p e c i f i c a l l y taught opera-tions such as r e v e r s i b i l i t y , addition-subtraction, et c . ) . The results of these procedures have been mixed ( B e i l i n , 1971). How-ever, the value of this approach i s put i n question by data 5 (Wallach, Wall, and Anderson, 196 7; T. rallach, 1969 ; Piaget, 1969; B e i l i n , 1971) that indicate that t r a i n i n g techniques d i r e c t l y based on the operations of the f i n a l grouping structures are i n -s u f f i c i e n t to bring about conservation. In f a c t , a pre-conserva-tion c h i l d may be f u l l y aware (through t r a i n i n g or other means) of a l l or some of the empirical changes involved in the stimuli s i t u a t i o n (e.g., that i f the water i s poured back into the o r i g i n a l container, i t would look the same as before, etc.) and yet not be able to assert that there i s conservation of substance or l i q u i d . In other words, the competence represented in a s p e c i f i c struc-t u r a l concept (grouping) i s the end resu l t or f i n a l convergence of the development of a m u l t i p l i c i t y of automaton functions, among which are those that make possible the connection of related empirical events i n a l o g i c a l (grouping) fashion. Thus, approaches that focus only on the competence aspects of cognitive development when trying to uncover how l o g i c a l concepts are acquired, may provide inadequate and i n s u f f i c i e n t answers to this question. This i s because the attainment of a l o g i c a l competence involves a variety of psychological functions (automaton variables) beyond those that may be immediately suggested by the ch a r a c t e r i s t i c s of the l o g i c a l operations themselves. The fundamental problem i s the precise s p e c i f i c a t i o n of these c r u c i a l automaton functions. A contrasting problem arises when the focus of inquiry is exclusively on automaton variables, as has been the case in certain t r a i n i n g procedures (Gelman, 1969) which have attempted to teach conservation by dealing only with variables such as the child's attention to relevant dimensions. Even though such procedures may indeed help a t t a i n conservation, they s t i l l leave unanswered the 6 question of why the procedure i s e f f e c t i v e . This question i s important because automaton-oriented procedures have been found to be e f f e c t i v e only with children around or past five years of age but not with younger ones ( B e i l i n , 1969). Thus, the ante-cedents of the conservation concept are not explained by such t r a i n i n g procedures; they merely demonstrate some obstacles that must be removed and some procedures that are effe c t i v e i n removing them once the c h i l d has the basic concepts necessary for conserva-tion . A second common reason for the inadequate operational c r i t e r i a of Piagetian concepts is obvious misrepresentations or misunder-standing of the constructs. A t y p i c a l example of Piagetian experiments with basic conceptual confusion are those of Bruner, et a l . (1966). Bruner rejects the explanation from Piaget's theory that compensation and r e v e r s i b i l i t y are responsible for conservation because he contends that the nrn-conserving c h i l d understands such phenomena. However, the non-conserving c h i l d may understand that a physical transformation can be reversed (an'empirical return 1'), r e s u l t i n g i n equality ( i . e . , he may understand that i f the water is poured back i t w i l l be the same as before) without understanding operational r e v e r s i b i l i t y , which en t a i l s the use of such empirical knowledge to j u s t i f y l o g i c a l l y the quantitative transformations involved i n conservation. S i m i l a r l y , there i s a d i s t i n c t i o n between the child's understanding of "functional covariations" --which may allow him to predict, for instance, that the water l e v e l i n a thin glass w i l l be higher than that of the same water poured into a wider glass -- which again i s empirical knowledge, and operational compensation, which also involves an understanding of 7 the l o g i c a l quantitative relations involved i n conservation. A t h i r d problem that has resulted i n inadequate operational c r i t e r i a i s the tendency of many authors to translate Piaget according to the i r own id i o s y n c r a t i c conceptual systems. This has resulted i n two types of problems. F i r s t , Piaget's constructs are interpreted and translated into other conceptual systems i n order to explain the conservation phenomena through other mechan-isms when in fact the necessity and basis for the tra n s l a t i o n i s not warranted or c l e a r l y j u s t i f i e d . And second, Piaget's con-structs are used to deal with behaviors other than the ones inten-ded to be subsumed under the s p e c i f i c constructs. One example of the f i r s t problem, pointed out by B e i l i n (1971a), i s the Halford and Fullerton (1970) experiment i n x^hich a t r a i n i n g technique was used \irhich presumably was based on non-Piagetian constructs but which in fact could very e a s i l y be considered to be a Piagetian-based procedure. In the Halford and Fullerton (1970) experiment the same " r e v e r s i b i l i t y 7 ' procedure used by Wallach (1969) -- a Piagetian-based experiment -- i s described i n terms of "discrim-ination" and 'sets.' 1 The authors attribute the moderately positive success which was evidenced in the post-tests not to r e v e r s i b i l i t y but to the : :sets" induced by the discrimination procedure. An example of the second problem is shown i n the Mehler and Sever (1967) and Sever, Hehler, and F.pstein (1968) studies that reported to have found number conservation in two-year old children, a fact that would be against Piaget's stage theory. However, subsequent research ( B e i l i n , 1968; Rothenberg and Courtney, 1969) has not replicated such results and i n fact the task that they used has been shown not to be a number conservation test. 8 Two conclusions can be drawn from the above examples of inadequate Piagetian-based c r i t e r i a . F i r s t , itfhen dealing with developmental problems such as those involved in determining how a certa i n l o g i c a l competence i s acquired, or in how to assess an attained l e v e l of competence, i t i s necessary to take into account simultaneously, not only the subject's use of the s p e c i f i c compe-tence rule in question, but also his level of attainment i n v a r i -ous other functions (automaton variables) which may indeed make possible the subject's understanding of the competence rule. The necessity for this type of "multivariate" approach to developmental phenomena has become increasingly recognized (Klahr and Wallace, 19 70; F l a v e l l , 1970a; Ayers, 1971). As w i l l be shown l a t e r , an assessment of Piaget's individual constructs based on his o v e r a l l theory demands such a multidimensional approach. Second, i f one wishes to investigate Piagetian constructs on thei r own grounds and within t h e i r ovrn l i m i t s rather than just to make use c f the constructs as a source of hypotheses to be int e r -preted according to one's own conceptual system, i t w i l l be necessary to interpret and operationalize the constructs, according to the c r i t e r i a outlined by P i a g e t . Only through this kind of approach w i l l i t be possible t o devise v a l i d testing procedures which assess the l i m i t s and strengths o f the Piagetian constructs. When the intention is to compare P i a g e t 1 s conceptual explanations with that of others, i t becomes even more necessary to have accurate and clear conceptual d e f i n i t i o n s which permit precise d i f f e r e n t i a -tions between the two contrasting approaches. V/ith the above considerations i n n i n d , the scope of this study can be further elaborated. The major aim was to devise a diagnos-9 t i c procedure to assess the following relationships: (1) the relationship between (a) the s p e c i f i c basic capacities or auto-maton variables that are necessary for the acquisition of s p e c i f i c logico-mathematical concepts, whose s p e c i f i c functional charac-t e r i s t i c s d i r e c t l y bear on the types of mental operations involved in the l o g i c a l operations, and (b) the level of attainment i n those l o g i c a l mathematical concepts; (2) the relationship between (a) a basic automaton variable also postulated as necessary for the acqu i s i t i o n of logico-mathematical concepts (Pascual-Leone, 1970) , but whose s p e c i f i c defining c h a r a c t e r i s t i c s do not d i r e c t l y bear on the c h a r a c t e r i s t i c s of l o g i c a l operations ( i . e . , assess-ment of a construct intended to measure quantitative aspects of cognitive growth independent of the nature or content of the s p e c i f i c concept), and again (b) the l e v e l of attainment i n those logico-mathematical concepts; and (3) the degree of relationship between the levels of attainment in l o g i c a l concepts t h e o r e t i c a l l y presumed to develop concurrently, i . e . , m u l t i p l i c a t i o n of classes and of r e l a t i o n s . CHAPTER II Theoretical Framework of Diagnostic Procedure Four aspects of the diagnostic procedure are considered i n this chapter: 1. The scope of operative knowledge that the procedure was designed to tap. 2. A theoreti c a l analysis of the psychological processes postu-lated as underlying such operative knowledge which allowed to decide which aspects of i t should be measured, 3. A theoretical analysis of the methods and procedures which provided the behavioral measures of the processes delineated above. 4. The s p e c i f i c goals of the diagnostic procedure. Scope The focus of this investigation was on abstract logico-mathematical operations ("groupings") such as multiplications of classes and re l a t i o n s , s e r i a t i o n , etc. for the following reasons. F i r s t , the acquisition of structural concepts such as the con-servations of substance, volume, area, etc., and i n general, concepts that f a l l i n the i n f r a l o g i c a l realm (deal with s p e c i f i c aspects of the physical world) has been found to be influenced by d i f f e r e n t i a l experiences. For example, Price-Williams, Gordon, and Ramirez (1969) indicated that there was better con-servation of substance performance ( p a r t i c u l a r l y in r u r a l areas) among potters' children than among children of the same s o c i a l l e v e l whose parents were in other occupations. The authors suggested that children from pottery-making families had greater experience 11 manipulating clay, and this s k i l l learning, by bringing the cog-n i t i v e mechanisms i n co-ordination with operations, leads to operative development. The implication seems to be that the expression of cognitive capacity through concepts that r e f l e c t physical phenomena may vary as a function of factors external to the c h i l d for which i t might not be possible to provide adequate control across a l l sub-j e c t s . Since the intent of the assessment i s to measure an abstract basic capacity independent of the s p e c i f i c (material) content of the child's experiences, i t is necessary to use measures i n which hopefully the effects of d i f f e r e n t i a l material contents of experiences are minimized. Second, i n f r a l o g i c a l concepts such as the conservations of substance, weight, and volume are acquired at d i f f e r e n t ages despite the fact that t h e o r e t i c a l l y they involve the same l o g i c a l operations. The reasons for these time lags ('decalages') are far from understood. Piaget himself (1970b, 1971b) has hypothesized that the c h i l d ' s e f f o r t s at operationally structuring physical r e a l i t y interact with his notions of physical causality. In other words, operational capacities might not be revealed when measured i n a certain context because the causal relations which the subject may attribute to the objects, present resistances to the structural quantifications required. For example, u n t i l about seven years of age x^eight i s conceived as changing from one moment to another for a single object as soon as i t s s p a t i a l position and causal function are changed, even when i t s shape remains constant. For instance, i n a glass of water the weight of a pebble is seen as varying according to whether i t i s at the bottom or at mid-height, 12 etc. 'Decalages' can occur in the child's manifestation of his understanding of l o g i c a l operations also as a function of the type of physical r e a l i t y that i s used to embody such concepts. For example, a time lag involving just a change i n materials i s i l l u s -trated with the class inclusion problem. When children are pre-sented a bunch of flowers of various kinds, t u l i p s , d a i s i e s , primulas, etc. and are asked to compare a subset or kind with the the whole bunch (Are there more flov/ers or more tulips?) , more than 75% of them successfully solve the problem at around seven or eight years of age. However, as Inhelder and Piaget (1964) have shown, when the class inclusion question deals with classes such as animals (Are there more birds or more animals?) the ques-tion is not successfully answered by the majority of the children u n t i l much l a t e r . Since the goal of the present diagnostic procedure was to assess the child' s l e v e l of op e r a t i v i t y and not how that l e v e l of operat i v i t y might interact with the child's causality notions about a s p e c i f i c physical domain, i t was necessary then to r e s t r i c t the measures to contents whose ch a r a c t e r i s t i c s minimize the poss-i b l e source of 'decalages.' Theoretical Psychological Processes Underlying Development of Operativity This theoretical analysis i s intended to indicate the basis from which i t was decided which aspects of logico-mathematical knowledge were to be assessed. To reach a decision on this matter i t was f i r s t necessary to answer two in t e r r e l a t e d questions. (1) What, in f a c t , is the c h i l d acquiring that makes possible his 13 understanding of logico-mathematical operations? And (2), how and what aspects of that kno\>rledge (competence rule, type of s i t u a t i o n , etc.) are expected to generalize to other related situations? It is thus necessary to review Piaget's theoretical treatments r e l e -vant to these questions. What makes possible the acqui s i t i o n of logico-mathematical concepts? The knowledge the c h i l d acquires during the concrete-operation-a l period has been characterized by Piaget through the l o g i c a l structures which he terms "groupings." These (incomplete) l o g i c a l structures represent Piaget's formalization of the most general co-ordinations of actions and operations of the c h i l d during this period. It i s necessary to analyze further the nature of these structures to be able to deal more s p e c i f i c a l l y with the following question: (1) What kinds of conceptual capacities are the structures supposed to encompass; (2) what is the psychological nature of the actions that make up the structures; (3) How are the actions that make up the structure co-ordinated? Logical structures - t h e i r conceptual range and functions The structures defined by the l o g i c a l groupings are intended to refer to a certain state of co-ordination that can be attained by the child's mental actions: The mental actions become co-ordi nated into "transformational" systems such that the re s u l t i n g organization can be said to have the following c h a r a c t e r i s t i c s . The f i r s t c h a r a c t e r i s t i c , "wholeness," results because, when co-ordinated with each other, the elements of the structure cannot be defined independently of the connections involved among them. 14 that i s , the elements are subordinated to the t o t a l organization. Piaget, i n his bbok Structuralism (1970), s p e c i f i c a l l y states: "...the elements of a structure are subordinated to laws * and i t is in terms of these laws that the structure qua whole or system i s defined. Moreover, the laws governing a structure are not reducible to cumulative one-by-one association of i t s elements; they confer on the whole as such o v e r a l l properties d i s t i n c t from i t s ele-ments." (Piaget, 1970a, p. 7) An example of this c h a r a c t e r i s t i c discussed by Piaget i l l u s -trates i t s meaning. He states that the integers do not exist in i s o l a t i o n from one another nor were they discovered one by one in some accidental sequence and then, f i n a l l y , united into a whole. They appear as ordered and this order of the integers is associ-ated with s t r u c t u r a l properties (of groups, f i e l d s , rings, and the l i k e ) which are quite d i f f e r e n t from the properties of i n d i -vidual numbers, each of which are even or odd, prime or non-prime, and so on. Second, the "transformational" C h a r a c t e r i s t i c results from the operations or regulations which make up the structure or give i t i t s "structuring" as a system. The transformations define a l l possible relationships among the elements themselves; therefore, they are the laws of composition of the structure. For the l o g i c a l groupings these operations are: (1) Combinativity - any two elements of a grouping can be combined and thus produce a new element of the same grouping, i . e . , tiro d i s t i n c t classes may be combined into one comprehensive class (2) R e v e r s i b i l i t y - each operation implies a converse operation, i . e . , subtraction for addition, d i v i s i o n for m u l t i p l i c a t i o n , etc. (3) A s s o c i a t i v i t y -the same outcome may be obtained by the combination of d i f f e r e n t operations (4) General Identity - an operation combined with i t s 15 converse i s annulled, (e.g., + 1 - 1 = 0 ) (5) Tautology or special i d e n t i t i e s - a qu a l i t a t i v e element which i s repeated or combined with i t s e l f does not y i e l d a new element and i t i s thus not transformed; there is a "tautology" i n the case of A + A = A. Third, a structure has a c h a r a c t e r i s t i c of " s e l f - r e g u l a t i o n " or "closure" by virtue of the transformations which preserve the r e l a t i o n a l t o t a l i t y of the whole. The nature of the transforma-t i o n a l operations outlined above i s such that they never engender elements outside of the system. An important point that needs to be stressed from the above conception of structures and operations is that the transformational character of the groupings is the resu l t i n g manifestation of psy-chological processes whose c h a r a c t e r i s t i c s are d i f f e r e n t from those of the operations as above defined. This point i s further supported by Piaget's analysis of the formation of structures (Piaget, 1972; 1971a; 1970a; 1968). The f i n a l co-ordinations embodied in the l o g i c a l groupings are preceded by gradual changes in "self-regulatory" functions which eventually become the opera-tions. Piaget points out that the term "operation", used i n reference to l o g i c a l structure, is reserved for those functions which, from a cybernetic point of view, are "perfect" regulations. That i s , the operational system excludes errors before they are made because every operation has i t s inverse i n the system (e.g., subtraction i s the inverse of addition). Thus, only when a co-ordination such as that between subtraction and addition i s at-tained, can one define the self - r e g u l a t i o n as an operation. The l o g i c a l necessity inherent in the l o g i c a l structures i s the ultimate product which results when the self-regulatory functions have at-tained the highest degree of co-ordination or equilibrium* the f i n a l "operational" state, therefore, i s achieved only after the s e l f - r e g u l a t i o n functions have gone through a long succession of more "imperfect" states i n terms of t h e i r feedback c a p a b i l i t i e s . Piaget has termed the levels of these less perfect s e l f -regulations as just "regulations" -- which are not s t r i c t l y operations because they are not e n t i r e l y reversible; and "rhythms" which constitute s e l f - r e g u l a t i n g mechanisms by virtue of sym-metries and repetitions. In reference to these three types of s e l f - r e g u l a t i n g mechanisms, Piaget (1970a) s p e c i f i c a l l y states: "...these are the three basic mechanisms of s e l f -regulation and s e l f •^maintenance. One may, i f one so desires, view them as the " r e a l " stages of a structure's construction." (p. 16) A more s p e c i f i c example of the difference between regulations and Operations comes from the d i s t i n c t i o n , discussed by Piaget (1968) , bettveen the nature of the " i d e n t i t i e s " the c h i l d can e s t a b l i s h p r i o r to and a f t e r acquisition of conservation. Preoperational, i d e n t i t y involves only q u a l i t a t i v e invariants (e.g., the 4 or 5 year-old c h i l d , in the conservation of l i q u i d s experiment, who maintains that the amount of water has changed, w i l l admit that i t is "the same water," i n the sense that the nature of the matter "water" has not changed even i f the quantity of the matter has changed) . The i d e n t i t y necessary for conservation, hoiveVer, i n -volves quantitative invariants. Pre-conservation i d e n t i t y i s simply the d i s s o c i a t i o n between a permanent quality (the same water') and the variable quality (shape). A number of very important theore t i c a l and empirical implica-tions may be derived from th i s conceptualization on the formation Of structures. 17 F i r s t , the conceptual analysis emphasizes that i n terms of developmental p r i o r i t i e s i t i s the self-regulatory functions which come f i r s t since, i n fa c t , i t is the f i n a l e q u i l i b r a t i o n of these functions \tfhich eventually comes to define the structure. Expound-ing on this point, Piaget (1970a) states: " . . . i t i s the relations among the elements that count. In other words, the l o g i c a l pro-cedures or natural processes by which the whole is formed are primary, not the whole, which i s consequent on the system's laws of composition." (p. 8) If the aim, then, i s to uncover or under-stand how the structures come about, i t is necessary according to this t h e o r e t i c a l p o s i t i o n to f i r s t analyze how the self-regulatory functions evolve and achieve i t s ultimate organization. Second, i t follows from the above, that in order to assess the development of any st r u c t u r a l concept i t xtfill be necessary to spec-i f y what the self-regulatory functions which precede i t develop-mentally are. These preceding functions can be said to constitute the pre-requisite or necessary s k i l l s to atta i n the given struc-tural concept. In any attempt to measure a child's true l e v e l of op e r a t i v i t y several empirical consequences derive from the above: (1) As was stated above, because the nature of these self-regulatory functions i s d i f f e r e n t from the f i n a l operations which make up the l o g i c a l structure, i t is necessary to define these functions for the spe-c i f i c concepts at hand. (2) In the period during which certain l o g i c a l operations are being acquired i t is necessary to make the assessment, not only i n terms of whether the c h i l d possesses the concept or f i n a l i z e d structure (e.g., understands conservation), but also i n terms of the l e v e l of attainment achieved i n the s e l f -18 regulatory functions which precede the acquisition of the structure, In other words, these "self-regulatory functions" are to be treated as "continuous" variables, and not just as thresholds that are passed or not. ( 3 ) Piaget's theoretical conception indicates that these functions grow in a gradual manner, thus, the measures r*ed should r e f l e c t i n some quantitative way the possible various l e v e l s of attainment. To the theoretical question, then, of what i s the c h i l d acquir-ing that allows the understanding of l o g i c a l operations, i t can be said that according to Piaget, the c h i l d i s acquiring "self-regulat-ing" functions or mental actions which eventually a t t a i n a level of development in their feedback capacities which makes possible the kinds of transformational co-ordinations described i n the mathemat-i c a l groupings. A question that remains i s : what is the psycho-l o g i c a l nature of these actions, or what are some of these s e l f -regulating functions? This question i s dealt with i n the next section. Psychological nature of the self-regulatory functions To understand the psychological nature of these self-regulat-ing functions and what they make possible, i t is necessary to look into Piaget's analysis of the changes that occur during the t r a n s i -t i o n period between pre-operational and operational thought. For example, i n a conservation of l i q u i d experiment, i f a c h i l d is aware (or e x p l i c i t l y is made aware through training) of a l l the aspects of the stimuli p r i o r to, during, and after the transforma-tions, what makes i t d i f f i c u l t for him to co-ordinate the r e l a t i o n -ships between these events so that he asserts that there i s no 19 conservation of the amount of l i q u i d to drink? Any conservation experiment constitutes a sequence of temporal events involving transformations of the s t i m u l i . According to Piaget (1962; Piaget and Inhelder, 1964) in order to deduce what aspect remains i n v a r i -ant i n the temporal sequence i t is necessary to take into account simultaneously a l l the aspects of the temporal sequences. What the pre-conservation c h i l d lacks, then, i s the a b i l i t y to consider or "center" simultaneous1y on a l l aspects of the temporal sequence. He tends to focus (and thus base his judgments) on the end s t a t i c results without co-ordinating them with the i n i t i a l condition or with the transformations that brought the change about. To sim-ultaneously co-ordinate past and present information in a contin-uum, i t is necessary, not just to remember the past, but also to know and be able to perform mentally the transformations just carried out on the physical s t i m u l i . Acquisition of the l a t t e r , a b i l i t y then, is what f a c i l i t a t e s the t r a n s i t i o n from pre-opera-t i o n a l to operational thought. There is growing evidence in the l i t e r a t u r e that i t is indeed the a b i l i t y to mentally perform the transformations on the stimuli which allows the c h i l d to conserve (Minichello and Goodnow, 1969; Wallach, 1969; Bearison, 1969, B e i l i n , 1969). The a b i l i t y to mentally perform transformations is (or should be) d i r e c t l y related to the expansion of the child's a b i l i t y to apprehend connected events both forwards and backwards in time. Research on the psychological processes that make apprehension of a wider range of temporal events possible has been somewhat neglec-ted because of the researchers' main concern with the cues of the stimulus s i t u a t i o n that allow the c h i l d to arrive at a conservation 20 solution rather than with the ch a r a c t e r i s t i c s of the child's men-t a l processes that allow him to integrate a l l the relevant cues of the stimuli 4 Piaget himself (Inhelder and Piagst, 1964; Piaget, 1971a; 1972) postulates that the psychological processes or se l f - r e g u l a t -ing mechanisms which make mental transformations possible are the interconnected a b i l i t i e s which he labels "hindsight" and "fore-sight". Hindsight or retroaction involves the a b i l i t y to take into account information used in the past i n r e l a t i o n to activ-i t i e s in the present, e.g., the a b i l i t y to maintain a c l a s s i f i -cation c r i t e r i o n throughout a task rather than continually s h i f t i n g c r i t e r i a depending on what s p e c i f i c object the c h i l d i s immediately attending. Foresight or anticipation involves the a b i l i t y to extend information from the past or present to make judgments about the future, e.g., the a b i l i t y to anticipate the continuation of a series of events from the relations previously observed. Piaget ascribes these two functions a causal role i n the formation of l o g i c a l operations. He s p e c i f i c a l l y states, "...the interest of retroaction and anti c i p a t i o n is that these two notions help us specify the conditions for the i n t e r i o r i z a t i o n "(of l o g i c a l operations)... they arise as a result of the grow-ing coordination between successive actions which eventually overcomes the one-directionality inherent i n a succession and tahes the form of a shuttling from the present to the past which very soon begins to impinge on the future... this kind of s h u t t l i n g is essential to the comparison of elements in a set taken as a whole; we begin to understand why these regulations (anticipation and retroaction) are l i k e l y to end up in the form of operations, since the shuttling i t s e l f i s a primitive form of r e v e r s i b i l i t y . " (Inhelder and Piaget, 1964, pp. 286-287.) In one of his most recent publications, The Pri n c i p l e s of Genetic Epistemology, (1972), Piaget s p e c i f i c a l l y states: 21 "...and so, the fundamental differences separat-ing the behavior of one stage from that of the preceding ones must be conceived as t r a n s i t i o n to a l i m i t . . . In the present case of the knox'/ledge of operations Me encounter an anal-ogous temporal process: the fusion into a single act of anticipations and retrospections -- which is the basis of operational r e v e r s i b i l i t y . ...the l i m i t character of operations as opposed to the simple 'regulations' or e a r l i e r l e v e l s , means that instead of corrections being made after the event, that i s once the action has been carried out p h y s i c a l l y , errors are pre-corrected in virtue of the interplay of dire c t and inverse operations or, in other words, as a result of combination of anticipations and retrospections, or more pre c i s e l y , of a possible anticipations of the retrospections themselves." (Piaget, 1972, pp. 35-36) Co-ordination of hindsight and foresight constitutes, there-fore, the necessary prerequisite to reversible transformational aspects of s t i m u l i . The most direct studies seeking to empirically investigate these a b i l i t i e s have been carried out by Piaget him-s e l f (Inhelder and Piaget, 1364; Piaget and Inhelder, 1971). How-ever, in Inhelder and Piaget's Early Growth of Logic (1964) , the analysis cf these processes i s rather incomplete for various reasons. F i r s t , the analysis of f l e x i b i l i t y in hindsight and fore-sight is made i n the context of the same structural a b i l i t y ( i . e . , c l a s s i f i c a t i o n ) that the hindsight and foresight are presumed to underlie. In other words, one task was the basis of inferences about the two di f f e r e n t (but allegedly related) a b i l i t i e s . If hindsight and foresight are to have any explanatory value for the formation of structures, as Piaget c l e a r l y purports to give them, then t h e i r measurement should be made independently and separately i n context from the measurement of opera t i v i t y as defined in any str u c t u r a l task. Also, i f foresight and hindsight are basic and 22 general psychological functions of relevance to concepts other than those defined i n the l o g i c a l realm, then t h e i r operational d e f i n i t i o n should bear on i t s most important aspects which pre-sumably can be sp e c i f i e d independently of the ch a r a c t e r i s t i c s of a given context where they might be applied. Second, a consequence of the f i r s t , there was not a simultan^ eous and separate within-subjects comparison of levels of fore-sight and hindsight and operative level in Piaget's analysis. Third, the a b i l i t y to perform the mental transformations which the self-regulatory functions are presumed to confer was again inferred from the child's performance i n the c l a s s i f i c a t i o n tasks. Of course, i t is necessary by the d e f i n i t i o n of the clas-s i f i c a t i o n task to assume that i t w i l l involve the a b i l i t y to make mental transformations. The problem i s that a l l aspects of operativity ( a b i l i t y to make mental transformations, achievement of c l a s s i f i c a t o r y schemes, and necessary f l e x i b i l i t y in hindsight and foresight) were inferred from measures cn the same task. To validate these theoretical constructs, i t is necessary to measure them independently through tasks that e x p l i c i t l y require the psychological operations presumably involved i n them. To assess the acquisition of l o g i c a l concepts, i t i s neces-sary to evaluate the basic processes that t h e o r e t i c a l l y bring them about. One i n i t i a l attempt in this d i r e c t i o n can be assessment of the lev e l s of attainment in measures operationally defined exclu-s i v e l y i n terms of the ch a r a c t e r i s t i c s of the underlying s e l f -regulatory mechanisms to determine i t s relationship to the le v e l of attainment on other s t r i c t l y defined l o g i c a l tasks. The methodological basis for the implementation of this approach i s 2 3 described in a subsequent section. Genesis of structures - e q u i l i b r a t i o n model Piaget's t h e o r e t i c a l analysis of the genesis of structures must be discussed further to arrive at s p e c i f i c aspects of the development of l o g i c a l structures that are open to empirical measurement. Piaget outlines several aspects involved in the formation of structures. In his book Structuralism (1970), defending the position that structures are endogenously constructed, Piaget states that: "...through the interplay of r e f l e c t i v e abstrac-t i o n , which furnishes increasingly complex "materials" for construction, and of e q u i l i b r a -tion (self-regulation) mechanisms, which make for i n t e r n a l r e v e r s i b i l i t y , structures -- in being constructed give r i s e to that necessity which a p r i o r i s t theories have always thought i t necessary to posit at the outset. Necessity, instead of being the p r i o r condition for learn-ing, i s i t s outcome." (1970a, p. 62) From the above quotation, and as indicated i n the previous section, there i s f i r s t the gradual growth of the self-regulatory functions; and second, as the self-regulatory functions attain a " c e r t a i n " l e v e l of f l e x i b i l i t y , there i s the occurrence of " r e f l e c t i v e abstraction." The increase in f l e x i b i l i t y of the self-regulatory functions -- hindsight and foresight -- enables the individual to assimilate input information with less d i s t o r t i o n because he increasingly becomes able to center or focus on a l l relevant aspects and not just on the most prominent perceptual c h a r a c t e r i s t i c s . P.eflective abstraction consists in the abstraction of proper-t i e s from the actions, or from the ways of acting on things. This 24 abstraction involves the subject's reconstruction or rearrangement of some actions or operations previously made such that this re-construction leads to operations upon operations or actions beyond the scope 6f the component operations. Again, an example from the conservation of l i q u i d experiment w i l l elucidate. For the non-Conserver, knowing that i f the water is poured into the o r i g i n a l container i t w i l l be the same as before (a fact that a c h i l d might be able to p r e d i c t ) , is not the same as the operational understand-ing of i d e n t i t y i n the conserving c h i l d for whom such a fact i s true by l o g i c a l necessity. The same physical action, for the conserving c h i l d , acquires a meaning beyond the one f i r s t ascribed to the s p e c i f i c content. The relationships present i n the actions (involved in a conservation experiment, for instance) become a l o g i c a l necessity whereas previously they had been considered simply probable. The structure, then, i s attained when a r e f l e c t i v e abstrac-tion of the actions involved i s achieved and, this i s manifested in the l o g i c a l necessity now seen in the relationships. "The feel i n g of necessity comes £rc»z the closure or completion of a structure." (Piaget, 1971b, p. 5). The question that arises from this conceptualization i s : when or how i s r e f l e c t i v e abstraction achieved? The e q u i l i b r a t i o n model that Piaget (1967) postulates as describing the formation of structures ascribes a p r o b a b i l i s t i c basis to the occurrence of r e f l e c t i v e abstraction. Through the continuing operation of the e q u i l i b r a t i o n mechanisms (self-regula-tory functions), the subject is said to develop strategies that allow him to deal more and more successfully (with less distortion) 25 with external input. For any s p e c i f i c s i t u a t i o n such as the con-servation of substance, where a b a l l of clay is changed into a sausage, there are a number of sequential strategies which the subject might use to deal with the stimulus s i t u a t i o n . For example, as described by Piaget (1967), the most probable strategy at the outset i s concentration on just one of the transformed c h a r a c t e r i s t i c s , i . e . , the quantity appears to increase because the object becomes elongated. Once this has occurred, the strategy which then becomes most probable consists of noticing the second transformed c h a r a c t e r i s t i c and supposing that the quantity diminishes because the sausage becomes thinner. Thereafter, a neiv strategy becomes most probable as a function of having attended to the two preceding c h a r a c t e r i s t i c s . This new strategy consists of o s c i l l a t i n g between these c h a r a c t e r i s t i c s and vaguely noticing the interdependence of the sausage's elongation and i t s thinness. This t h i r d reaction leads to the action being placed on the trans-formational pi 3.116 cLS OppOSCG. to the s t a t i c configuration plane alone. A fourth strategy ensues which consists of the discovery of the compensations among the transformations and acceptance of the fact of conservation. "...the f i n a l equilibrium i s thus the product of a compensation for the ambiguous st i m u l i by the a c t i v i t i e s of the subject which in turn are characterized by successive p r o b a b i l i t i e s . " (Piaget, 1967, p. 112) This p r o b a b i l i s t i c model has been c r i t i c i z e d by a number of authors ( F l a v e l l , 1963; B e i l i n , 1971 ) because of the d i f f i c u l t y of empirically testing i t in situations other than those somewhat a r t i f i c i a l l y created when the subject is given s p e c i f i c problems such as conservation. Despite the v a l i d i t y of these c r i t i c i s m s , 26 i f indeed such a model were to be used to investigate questions concerning the l e v e l a c q u i s i t i o n of s p e c i f i c concepts, i t would be necessary for every s p e c i f i c operational concept to deal with problems such as: (1) S p e c i f i c a t i o n and operational d e f i n i t i o n s of the s k i l l s or various strategies that from the subject's point of vieiv ivould be necessary to achieve by means of r e f l e c t i v e abstraction the s p e c i f i c concept. (2) Creation of operational measures of the various s k i l l s such that i t could be possible to d i f f e r e n t i a t e through them, in a quantitative sense, the various " l e v e l s " of the necessary s k i l l s possessed by the subjects. These i n turn could be translated in terms of how l i k e l y they are to bring about the r e f l e c t i v e abstraction. These problems indicate how the e q u i l i b r a t i o n model requires that the nature of the self-regulatory functions be analyzed. Nothing i n the theory indicates within which range of the prob-a b i l i t i e s of occurrence of the strategies, under which conditions or experiences, the r e f l e c t i v e abstraction involved in a concept occurs. The s p e c i f i c a t i o n of the p r o b a b i l i t i e s or certain stra-tegies or the conditions that would make the attainment of the co-ordination involved in a l o g i c a l concept more l i k e l y i s an empirical question that would have to be dealt with i n d i v i d u a l l y for each s p e c i f i c concept. This point is emphasized because of i t s methodological impli-cations. Host attempts in the l i t e r a t u r e aimed at dealing with problems of assessment of the child's operative knoi/ledge are based mainly on whether or not the c h i l d has already achieved the l o g i c a l concept or grouping, rather than on an assessment of the development of the s k i l l s necessary to or leading up to the attain-27 ment of the concept. An example of a testing procedure somewhat r e f l e c t i n g an approach based on assessment of necessary self-regulatory func-tions i s a study by Lefrancois (1968) in which a quasi-training procedure was devised for children who f a i l e d a conservation of substance pre-test. The subjects were presented with a series of tasks ? ordered from simple to d i f f i c u l t , based on a hierar-ch i c a l analysis of the prerequisite s k i l l s necessary to at t a i n conservation. Of 60 non-conservers, age 5 to 6, 56 performed according to a perfect Guttman scale on the nine prerequisite s k i l l s outlined. This measurement procedure was designed to be e s s e n t i a l l y diagnostic since i t attempted to specify the s k i l l s that the subject had already incorporated and those he had yet to acquire. On the basis of the assessment from the scale, subjects were trained on the s k i l l s s t i l l needed to achieve conservation. The t r a i n i n g did not involve actual teaching in the di d a c t i c sense. Subjects were merely asked to observe objects, to notice transformations carried out by the experimenter, and to answer questions regarding the f i n a l state of the object after transformation. Preliminary studies ( l a t e r extended, Ayers, 1971) indicated that 25 out of 40 of the children given t r a i n i n g based on the hierarchy were successful i n acquiring conservation of substance while no subjects i n a matched group of 20 subject did likewise in the absence of t r a i n i n g . Again, t h i s experiment emphasizes the conclusions made in the titfo previous theore t i c a l sections for the need to focus on the "self-regulatory" functions i n order to deal successfully with questions of acquisition of l o g i c a l concepts. The c h i l d ' s 28 '"attainment" of a Certain l e v e l of o p e r a t i v i t y is more accurately assessed when the measurement instruments take into account the child's l e v e l of development i n the Underlying necessary s k i l l s as well as his degree of achievement of the f i n a l structure represented in the l o g i c a l concept. What generalizes between stage-related concepts? In Piagetian theory a cognitive-development stage, such as the concrete-operational stage, i s conceived of as a period during which the c h i l d acquires q u a l i t a t i v e l y d i f f e r e n t behaviors that can be described i n terms of similar conceptual rules, such as the mathematical l o g i c a l groupings. The important aspect of this construct of stage i s that i t s p e c i f i c a l l y conceives of the child's new cognitive acquisitions, not as psychologically i s o l a -ted and unrelated a b i l i t i e s , but rather as interacting with one another i n specified ways, as a unitary system or structure. In view of this t h e o r e t i c a l assumption, i t is l o g i c a l that any diagnostic procedure designed to assess the child's l e v e l of operativity would attempt to simultaneously compare the l e v e l of achievement and degree of relationship between concepts theoret-i c a l l y presumed to be part of one u n i f i e d l o g i c a l system. It i s thus f i r s t necessary to analyze what the nature of the r e l a t i o n -ship between the concepts i n the u n i f i e d system i s meant to be. Pinard and Laurendeau (1969) have recently discussed the problem of the relationship among stage-related concepts i n t h e i r analysis of the construct of "stage." They have made a thorough scrutiny of the f i v e s t r u c t u r a l c r i t e r i a which according to Piaget characterize a stage. B r i e f l y , these are: (1) Hierarchization, 2? the necessity of a fixed order in the succession of the d i f f e r e n t lev e l s that constitute a developmental sequence, ( 2 ) Integration, the necessity that new acquisitions of a given stage be integrated with those of the preceding stage instead of simply substituting for them or juxtaposing their., ( 3 ) Consolidation, the necessity that a stage always involve at once an aspect of achievement of the recently acquired behavior and an aspect of preparation for the behavior at the following l e v e l , ( 4 ) Structuring, (or "struc-ture d'ensemble"), the necessity for the operations of a given l e v e l to be, not simply juxtaposed with one another i n an additive fashion, but organically interconnected by t i e s of implication and reciprocal dependence that unite them and group them into t o t a l structures, and ( 5 ) E q u i l i b r a t i o n , the necessity of a series of equilibrium levels in which the range of operations becomes greater and more mobile with successive l e v e l s . In reference to the question of the i n t e r r e l a t i o n s h i p among stage-related items, the c r i t e r i o n of most relevance that needs to be further considered is that of structuring. According to Pinard and Laurendeau, the meaning of this construct is such that i t cannot be reduced "...to a simple form of transfer or of generali-zation between diverse concepts, nor to necessary (but not s u f f i c i e n t ) synchronisms that would r e f l e c t the operation of this c h a r a c t e r i s t i c . The s t r u c t u r a l c h a r a c t e r i s t i c asserts above a l l a complete functional interdependence, an organic connection among the operations (or actions) that can i n fact anply to related concepts." (op. c i t . , p. 137) This c r i t e r i o n requires that once a l o g i c a l concept is acquired there should be a simultaneous mastery or immediate generalization to a l l tasks based on that l o g i c a l operation. Pinard and Laurendeau 30 consider the intraconcept l e v e l as the minimum range of concepts for which this c r i t e r i o n has to be demonstrated (and thus give v a l i d i t y to Piaget's stage construct), mainly because at this l e v e l i t i s presumably easier to reduce the effects of the v a r i -ous confounding variables that might cloud i t s manifestation, i . e . , factors that produce the h o r i z o n t a l ' d e c a l a g e s o r temporal lags between s t r u c t u r a l l y similar concepts. For example, this would mean that "...the construction of the concept of weight xvould simultaneously imply, at about the age of nine years: the mastery of groupings of simple addition and of vicariances (conservation of weight): o f addition of asymmetrical ( s e r i a t i o n of weight) and symmetrical ( t r a n s i t i v i t y of equivalences)relations: of biunivocal m u l t i p l i -cation of relations or of classes (concept of density by the combination of weight and sub-stance) ; of counivocal m u l t i p l i c a t i o n (conserva-ti o n of the weight of p a r t i c l e s of flo u r in the d i l a t e d corn): and f i n a l l y of the corresponding quantitative groupings, by a fusion of classes and r e l a t i o n s . " (op. c i t . , pp. 138-139) Thus, the structuring c r i t e r i o n necessarily implies that, "...to affirm, the psychological existence of an authentic operational grouping bearing on a given content, this content must at the same time e l i c i t not only the whole set o f constituent operations of this grouping but also the ensemble of p a r a l l e l and connected groupings." (op. c i t . , p. 140) The interdependence among the groupings and thus the syn-chrony i n t h e i r development is necessary because on the basis of a l o g i c a l (or epistemological) analysis o f the groupings " . . . a l l are s t r i c t l y isomorphic because a l l are based on the same combination of fundamental operations (direct composition, inverse compo-s i t i o n , a s s o c i a t i v i t y , and so on). The psycho-l o g i c a l analysis of these same groupings reveals the necessary complementarity of the systems of classes, r e l a t i o n s , and numbers." (op. c i t . , p. 139) 31 The data from studies that test this c r i t e r i o n , as revienred by Pinard and Laurendeau (1969) and F l a v e l l and Wohlwill (1969) are rather inconsistent for a variety of methodological and con-ceptual reasons which need not be spec i f i e d here. Pinard and Laurendeau contend that to submit this hypothesis to experimenta-t i o n , i t is necessary to conform as much as possible to the str u c t u r a l c h a r a c t e r i s t i c s assigned by Piaget. Most authors have not done this p a r t l y because of the experimental d i f f i c u l t i e s involved. However, aside from the above c r i t i c i s m , the present author would contend that no clear answers to these problems are yet possible because of the following important reasons. F i r s t , the conceptualization of the structuring c r i t e r i o n i s i n s u f f i c i e n t to allow i t s empirical t e s t i n g . (1) The conception of i t s mean-ing i s s t r i c t l y based on a structural analysis which does not incorporate how, on the basis of the functioning of the s e l f -regulatory mechanisms, such synchrony would be expected to occur. (2) It i s in fact unclear ifhat the term "concurrent development" or "interdependence" i s supposed to imply. Bo these mean that a l l the groupings involved i n a concept are acquired at the same time, or -- put i n functional terms -- that the r e f l e c t i v e abstraction necessary for a l l the concepts i s achieved simultane-ously? Or do these mean that the self-regulatory mechanisms underlying the operations develop concurrently? (3) I f , as Pinard and Laurendeau (1969) state " . . . i t goes without saying that the completed elaboration of 'structure d'ensemble' would not be expected before the end of each of the corresponding developmental l e v e l s " (p. 137), what i s the relationship between subsequently related concepts p r i o r to the end of the corresponding period? 32 ( 4 ) Does the fact that in terms of l o g i c a l or epistemological analysis the operations of the groupings are isomorphic make i t necessary that they be acquired or generalized simultaneously? Or, expressed in other terms, assuming that the set of inter-related groupings involved i n a concept specify a similar level of competence, can one assume that the "automaton" requirements i" • for a l l the groupings are also the same? It is then obvious that from a structural analysis i t i s not possible to answer questions about the meaning of the struc-turing c r i t e r i o n . Unfortunately, most authors who have specif-i c a l l y dealt with this question have formulated t h e i r experimental tests purely on structural considerations (e.g., Kofsky, 1966; Shantz, 1967), when from the above i t is apparent that to deal with the problems of ac q u i s i t i o n of structures functional (auto-maton) as well as structural aspects have to be taken into con-sideration. One of the authors who has s p e c i f i c a l l y addressed himself to these problems i s F l a v e l l (1970a, 1970b, 1971), who has pointed out the need for more of a functional analysis of the a c q u i s i t i o n problems. He postulates that most of the present data indicate that the development of most cognitive operations involves gradual changes over time (rather than abrupt acquisitions as might be implied by l i t e r a l interpretations of the structuring c r i t e r i o n ) such that i t could be possible to distinguish between the i n i t i a l levels of acquisition and the level of "functional maturity." Functional maturity i s defined in terms of the item's evoc a b i l i t y the child's operational a v a i l a b i l i t y of an item, and u t i l i z a b i l i t y -- the child's a b i l i t y , one having sensed an 33 item-to-problem f i t , to u t i l i z e i t e f f e c t i v e l y as a solution pro-cedure. Because the acqu i s i t i o n of stage-specific items is con-ceived as a gradual development over an extended temporal i n t e r v a l rather than as a temporal point, the notion of developmental synchrony or concurrence i s rendered ambiguous. As F l a v e l l (1970a) states i t , "...to say that two stage-specific items "develop concurrently" could mean that they begin th e i r development at the same time or conclude i t (achieve functional maturity) synchronously, or both, or even neither ( i . e . , have developmental courses that show some chronological overlap but only in the middle regions)." (p. 5 2 ) Even i f the meaning of inter-item concurrence i s r e s t r i c t e d to mean synchronous emergence, F l a v e l l points out further that two main issues s t i l l remain. F i r s t , the usual methodological procedures used to test inter-item developmental concurrences, i n terms of the child ' s i n t e r - t e s t consistency of performance in two or more related tasks, presents serious methodological prob-lems. For one, such a procedure assumes that the solution of the tests requires an equal degree of attained functional maturity. If such i s not the case, and i t i s unclear at present how items might be equalized in that respect, one could f i n d an apparent asynchronisn where there is a real synchronism or vice versa. Fut in other terms, i f the genuine synchronous development of two cognitive items of similar "competence" requirements i s tested simultaneously in terms of the presence of i n t e r - t e s t consistency, the concurrency might not appear simply because the "automaton" demands (functional maturity) required might be d i f f e r e n t for both and the tests might not have made allowances for th i s d i f f e r -ence . 34 Second, F l a v e l l contends that a s t r i c t developmental concur-rence is not mandated by Piaget's stage theory. Even i f i t i s accepted as Pinard and Laurendeau (1969) state, that a l l concrete-operational operations are fundamentally the same from one group-ing to another (e.?., a l l groupings specify a dir e c t operation, an a s s o c i a t i v i t y r u l e , etc.) and that they are l o g i c a l l y and psychologically interdependent both within and across groupings and groups, i t can s t i l l be argued that such facts need result in at most only a very loose sort of developmental pa r a l l e l i s m . F l a v e l l states that "...the existence of psychological connections among items does not presuppose any developmental synchrony among them, since those connections derive only from the properties of the items once in existence, i . e . , once acquired.... A l l concrete-operational items exemplify a certain form of thinking, a c e r t a i n l e v e l of cognitive functioning, consequently one would not expect the ensemble of such "same-level" items to show r e a l l y extreme developmental asynchronies, e.g., one item beginning to emerge at age four and another not u n t i l age twelve. On the other hand, i t does not follow that such items must emerge in tight concurrence, that i s , within the same week, mont*...; or even year." ( F l a v e l l , 1971, p. 36) It could be contended that, i f Piaget's vie\</s about the nature of the mechanisms that bring about the structures were incorporated into a structural analysis of stages, the meaning of developmental concurrence might appear somewhat di f f e r e n t from the one most present interpretations assign to i t , and a meaning more congruent with the views of F l a v e l l might emerge. For example, the analysis of the gradual growth of self-regulatory functions implies that the operations develop in gradual fashion. Also, there i s nothing i n the model that requires that the r e f l e c -t i v e abstraction involved i n the various concepts be achieved at 35 the same time. Although related concepts (such as those necessary for the understanding of "weight") may involve the same operations, each concept ( t r a n s i t i v i t y , conservation, s e r i a t i o n , etc.) involves a new combination of these operations, whose individual formation (during the a c q u i s i t i o n phase of the whole concept) may be spread over time, and each concept may be governed by external factors not yet possible to specify. Piaget's lack of knowledge in this regard i s expressed when he states: " . . . i t is neither the elements nor a whole that  comes in a matter one knows not how, but the relations among elements that count." (Piaget, 1970a, pp. 8-9, author's i t a l i c s ) Compounding this i n s u f f i c i e n c y of conceptual understanding of what i s involved i n inter-item generalization, there is the prob-lem of accounting for the presence of horizontal 'de"calages' or temporal lags. As discussed in an e a r l i e r section, according to Piaget the child's l e v e l of o p e r a t i v i t y seems to interact with his notions of physical causality about the s p e c i f i c context involved, which may lead him to make non-operational judgments. A number of other factors have been pointed out by Piaget (Piaget, 1971b; 1970b; Piaget and Szeminslca, 1 9 4 1 ) as possible sources of "decalages:' how the operational questions are formulated, the relationship between the f i g u r a t i v e aspects of the techniques i'.sed and the operative aspects to be solved, the relationship between the instructions and the individual experiences of the c h i l d , the number of elements involved i n the tasks, etc. In view of the above problems, how can one deal with the question of the i n t e r - r e l a t i o n s h i p or generalization between stage related items? Conceptually, as Pinard and Laurendeau (1969) suggest, the analysis should f i r s t encompass concepts whose neces-36 sary operations are not only l o g i c a l l y related but also involve the same conceptual content, e.g., attainment of c l a s s i f i c a t i o n structures. This r e s t r i c t i o n would f a c i l i t a t e the analysis of t h e i r development in terms of the necessary self-regulatory functions which presumably are common to a l l of them. Methodol-o g i c a l l y , the analysis should guarantee that as many of the possible sources of "decalages 1 are minimized or equalized across the various tasks. That i s , the automaton demands should be comparable for a l l the tests. The present study diet not attempt to test for the structuring c r i t e r i o n i n the s t r i c t sense that Pinard and Laurendeau (1969) require (see above quotation 2 on p. 30), but rather i t was l i m i t e d to assessing the degree of relationship betx^een struc-t u r a l l y related tasks ( m u l t i p l i c a t i o n of classes and r e l a t i o n s , s e r i a t i o n , and t r a n s i t i v i t y ) and t h e i r presumed underlying s e l f -regulatory functions, when a number of automaton aspects are also taken into consideration. Such an approach should provide a more comprehensive picture of what is involved in the child's acquisition of l o g i c a l concepts. Methodological Aspects - Theoretical Considerations As stated i n the previous sections this investigation attempted to measure: (1) the relationships among various logico-mathematical concepts (multiplication of classes and of r e l a t i o n s , s e r i a t i o n , and t r a n s i t i v i t y ) that involve i n t e r r e l a t e d groupings which presumably develop concurrently, and (2) the relationship of these concepts to the basic self-regulatory functions, hind-sight and foresight, that t h e o r e t i c a l l y are responsible for t h e i r 37 development. In order to accomplish these goals i t was necessary to attempt to control for as many of the factors that, as discussed above, might produce confounded r e s u l t s . It was thus necessary to indicate what the competence and automaton requirements of each test were in order to be able to specify what the basis for the relationship (or lack of i t ) might be between them. The tests involved three types of tasks: (1) those dealing with the st r u c t u r a l concepts ( 2 ) those dealing with the s e l f -regulatory functions (3) those dealing with another s t r i c t l y automaton variable operationally defined outside of a structural context. A general analysis of each is next considered. Structural Tasks Since a l l of the structural tasks dealt with a set of clo s e l y related conceptual operations, i t is necessary to specify the basis of t h e i r comparability both in t h e i r competence and auto-maton requirements. The competence demands were equated by hav-ing a l l the tasks deal with l o g i c a l groupings from the same conceptual family such as the groupings involved in l o g i c a l c l a s s i f i c a t i o n s k i l l s . The tasks were: (1) a m u l t i p l i c a t i o n of classes task -- biunivocal m u l t i p l i c a t i o n of classes grouping (2) a. m u l t i p l i c a t i o n of relations task -- biunivocal m u l t i p l i c a -t i o n of relations grouping (3) a s e r i a t i o n of relations task --addition of asymmetrical relations grouping (4) a t r a n s i t i v i t y task -- involving a property of the addition of asymmetrical r e l a -tions grouping. A closer analysis of the l e v e l of d i f f i c u l t y of the s p e c i f i c rules involved in each task is made under the s p e c i f i c 38 t a s k a n a l y s i s o f e a c h c o n c e p t . t h e a u t o m a t o n d e m a n d s w e r e e q u a t e d i n t w o w a y s : (1) b y k e e p i n g c o n s t a n t a c r o s s t a s k s t h e q u a n t i t a t i v e a s p e c t s t h a t h a d t o b e d e a l t w i t h , i . e . , n u m b e r o f d i m e n s i o n s , a n d (2) b y m a k i n g t h e p r o c e s s i n g a n d o u t p u t o f r e s p o n s e r e q u i r e m e n t t h e s a m e f o r a l l t h e t a s k s . P r o c e s s i n g r e q u i r e m e n t s r e f e r t o t h e c o n s t r a i n t s s e t b y t h e f o r m o f t a s k i n t h e w a y s i n w h i c h t h e p r o b l e m m a y b e s o l v e d , a n d o u t p u t o r r e s p o n s e r e q u i r e m e n t s r e f e r t o t h e l i m i t a -t i o n s s e t i n t h e w a y s i n w h i c h t h e s u b j e c t may p r o d u c e t h e o u t -c ome o f h i s m e n t a l o p e r a t i o n s . T o f u r t h e r c l a r i f y t h e n e c e s s i t y o f t h e s e t w o t y p e s o f c o n s t r a i n t s , a t h e o r e t i c a l a n a l y s i s f o l l o w s . One c o u l d c o n s i d e r t h a t t r a d i t i o n a l l y P i a g e t i a n r e s e a r c h h a s b e e n o r i e n t e d t o w a r d s f i n d i n g o u t w h a t a r e t h e p r o c e s s e s o r i n t e r n a l m e c h a n i s m s t h a t m a k e p o s s i b l e t h e t r a n s f o r m a t i o n s t h a t o c c u r b e t w e e n a " s t i m u l u s " a n d a " r e s p o n s e " ; t h a t i s , t h e f o c u s h a s b e e n o n t h o s e u n o b s e r v a h l e o p e r a t i o n s t h a t c h a r a c t e r i z e t h e h u m a n " b l a c k b o x . " I n r e c e n t y e a r s , P i a g e t a n d h i s c o l l a b o r a t o r s h a v e b e e n i n t e r e s t e d i n m e m o r y ( e n c o d i n g a n d d e c o d i n g a s p e c t s o f c o g n i t i o n ) a n d p a r t i c u l a r l y i n t h e m e m o r y c o d e s a s s u c h i n a n e f f o r t t o u n c o v e r t h e e x t e n t t o w h i c h m n e m o n i c c o d e s r e f l e c t o p e r a t i v e i n t e l l i g e n c e , t h e s p e c i f i c q u e s t i o n s o f c o n c e r n h a v e b e e n : (1) d o e s t h e m e m o r y o r m n e m o n i c c o d e o f p a s t e v e n t s , p r o -d u c e d b y e i t h e r r e c o g n i t o r y , r e p r o d u c t i v e , o r e v o c a t i v e m e a n s , r e f l e c t t h e c h i l d ' s l e v e l o f o p e r a t i v i t y ? a n a (2) d o e s t h e m n e m o n i c c o d e v a r y o v e r t i m e a c c o r d i n g t o s p e c i f i c m n e m o n i c l a w s o r r a t h e r a c c o r d i n g t o t h e d e v e l o p m e n t a l l a x v s t h a t , ( a s m o d e l e d i n P i a g e t i a n t h e o r y ) g o v e r n t h e c h a n g e s i n c o g n i t i v e s t r u c t u r e s ? T h e s e q u e s t i o n s a r e o f s p e c i a l s i g n i f i c a n c e b e c a u s e o f t h e i r 39 relevance to a very important d i s t i n c t i o n which i s made i n Piaget ' s theory. According to the theory there are two aspects of cogni-t i v e functions: the operative aspect (actions and operations) which bear on the transformations of r e a l i t y -- the operations themselves, and the fig u r a t i v e aspects (perception, imitation, images) I'/hich are limited to the s t a t i c representation of states without reference to the transformations that result in those states. Mnemonic evocation, s p e c i f i c a l l y being possible through r e c a l l of mental images, etc. f a l l s under the figura t i v e realm. Mnemonic evocation thus provides a valuable means of testing the relationship between memory functions and operations, as for example, through the comparison of the r e c a l l of memory for s p e c i f i c configurations involving s t r u c t u r a l contents and the actual understanding of the concepts or operations represented in the configuration. In the actual experiments reported by Inhelder (1969) , memory for configurations of various Icinds ( c l a s s i f i c a t i o n , s e r i a t i o n , outcome of physical events, etc.) were compared with actual operative attainment of the concepts involved. For example, one group of children ( 3 to 8 years of age) was presented with con-figurations such as groups of sticks of 9 to 15 cm in length arranged in series from smallest to longest, and t o l d merely to look c a r e f u l l y at the sticks and that they i^ould have to remember them l a t e r . Another equivalent group was told to construct the series (in order to determine t h e i r l e v e l of o p e r a t i v i t y ) , and yet another group v.;as asked to imagine and draw what the sticks ( a l l in disarray) would look l i k e xvhen put i n order. After a week, the f i r s t two groups were asked to indicate by gesture and 40 by drawing what they remembered. The series were not presented again on this occasion. Six tb eight months l a t e r these children were asked to draw again from memory what they had previously Seen. In each instance after the children finished their memory df'awiiigs, they were given sticks and asked to make the se r i a t i o n themselves so that t h e i r l e v e l of operativity could be determined. Drawings after a week showed differences among the various age levels that could be c l a s s i f i e d in terms of the various levels of o p e r a t i v i t y . In fact, the various types of drawings cor-responded closely with those obtained from the group asked just to imagine and draw (without actually seeing) the configuration of s t i c k s , and with the actual types of seriations obtained from the group asked to materialize his own arrangement of the s t i c k s . After the six to eight months i n t e r v a l , the memory drawings of most children (74% of the group as a whole, and 90% between 5 and 8 years) showed operative progress r e l a t i v e to the f i r s t drawing. This progress was alv/ays gradual, that i s to say, subjects progressed from one sub-stage to the next and never more than that. Inhelder (1969) interpreted these results to mean that "...the memory image is not a simple residue of the perception of the model, but rather a symbol that corresponds to the schemes of the c h i l d . . . the action schemes -- in this p a r t i c u l a r case, the schemes of s e r i a t i o n -- constitute the code for memorizing: this code i s modified during the in t e r v a l and the modified version is used as a new code for the next evocation. At each stage the memory image is symbolized according to the constraints of the corresponding code." (op. c i t . , p. 343) In other words, Inhelder's data seem to indicate that memory does not conform to the perceptual configuration of the model but 41 rather to the manner in which the model is assimilated to the operative schemes of the subject. This t h e o r e t i c a l rationale was further tested in the present study because i t was used as a basis for equating the various s t r u c t u r a l tasks in terms of their processing and output or response demands. S p e c i f i c a l l y , the subject's operative level was measured for a l l structural tasks in two ways. F i r s t , i n terms of his r e c a l l of a s t a t i c configuration that represented a certain s t r u c t u r a l concept -- processing demands in terms of memory --such as the configuration of a n i n e - c e l l matrix representing a m u l t i p l i c a t i o n of relations concept. The r e c a l l was measured in terms of the accuracy of reconstruction -- output demands -- of the configuration i t s e l f . Second, i t was measured i n terms of the subject's accuracy of construction of the same matrix when required to mentally perform transformations on i t not carried out or seen before, i . e . , the subject was required to reverse in two directions the relations present in the matrix after the i n i t i a l configuration is no longer v i s i b l e . In this manner i t was possible to obtain simultaneous measures of operativity based on the subject's understanding of the f i g u r a t i v e aspects of the stimulus s i t u a t i o n and based on his understanding of the actual operations embodied in the s t i m u l i . For every s t r u c t u r a l test (except the t r a n s i t i v i t y task), the same task requirements were made by applying the constraints explained above. The procedure is d i f f e r e n t from Inhelder's i n that the r e c a l l i n t e r v a l in this case is very short; the c h i l d is required to reconstruct the matrix immediately after having just seen i t . The s p e c i f i c administration procedure used for each task i s i n d i c a t e d under the d e s c r i p t i o n and p r o c e d u r e s e c t i o n o f each t e s t . F o r e s i g h t and ?-iindsight F l e x i b i l i t y Task A s d i s c u s s e d i n the p r e v i o u s t h e o r e t i c a l s e c t i o n , h i n d s i g h t and f o r e s i g h t are the b a s i c s e l f - r e g u l a t o r y f u n c t i o n s whose developmenti a c c o r d i n g t o P i a g e t , has a c a u s a l r o l e i n the forma-t i o n o f l o g i c a l s t r u c t u r e s . I t i s these i n t e r r e l a t e d f u n c t i o n s w h i c h make p o s s i b l e the a b i l i t y t o m e n t a l l y p e r f o r m th e t r a n s -f o r m a t i o n s i n v o l v e d i n the l o g i c a l o p e r a t i o n s and thus a t t a i n the r e f l e c t i v e a b s t r a c t i o n r e q u i r e d i n the f o r m a t i o n o f the s t r u c -t u r e s . The competence e x p r e s s e d i n t h e s e a b i l i t i e s does not r e f e r t o the use o f e x p l i c i t r u l e s such as those i n v o l v e d i n the l o g i c a l t a s k s , but r a t h e r t o more g e n e r a l a b i l i t i e s t h a t c h a r a c t e r i z e c o g n i t i v e development and w h i c h s h o u l d thus be m a n i f e s t e d i n c o n t e x t s o t h e r than t h o s e u s u a l l y c h a r a c t e r i z i n g l o g i c a l opera-t i o n s . However, the competence r e q u i r e d i n f o r e s i g h t and h i n d -s i g h t c l e a r l y has t o i n v o l v e or be m a n i f e s t e d i n the u n d e r s t a n d -i n g o f t r a n s f o r m a t i o n a l a s p e c t s o f s t i m u l i . I n v iew o f the above, the p r e s e n t study attempted t o e v a l u a t e t h e s e i n t e r r e l a t e d a b i l i t i e s by means o f a t e s t t h a t t r a n s l a t e d t h e i r c o n c e p t u a l meaning i n such a way t h a t the f o l l o w i n g r e q u i r e -ments were net: ( 1 ) the t a s k r e q u i r e m e n t s e x p l i c i t l y c a l l e d f o r the use o f t r a n s f o i m a t i o n a l a b i l i t i e s ( 2 ) the s p e c i f i c c o n t e n t o f the t e s t was independent and t o t a l l y d i f f e r e n t from th e c h a r a c -t e r i s t i c s o f the c o n t e n t s o f the l o g i c a l t a s k s ( 3 ) the t e s t d i r e c t l y a l l o w e d f o r a q u a n t i t a t i v e d i f f e r e n t i a t i o n between the 43 v a r i o u s l e v e l s o f a t t a i n m e n t . The concep tua l meaning o f what i s i n v o l v e d i n these two r e c i p r o c a l a b i l i t i e s -- h i n d s i g h t , a b i l i t y to take i n t o account i n f o r m a t i o n used i n the immediate past i n r e l a t i o n to a c t i v i t i e s i n the p r e s e n t , and f o r e s i g h t , a b i l i t y to extend i n f o r m a t i o n from the p r e s e n t or immediate pas t to make judgments i n the p r e s e n t about f u t u r e s u c c e s s i v e outcomes -- was t r a n s l a t e d i n t o a t ask w i t h the f o l l o w i n g c h a r a c t e r i s t i c s . The genera l requirement o f the task was to extend and cont inue the t r a n s f o r m a t i o n s i n d i c a t e d i n the s t i m u l i o f the f i r s t two cards o f a s e r i e s ( i n v o l v i n g a t o t a l of s i x ca rds) by t a k i n g i n t o c o n s i d e r a t i o n as w e l l the f i n a l outcome ( s i x t h card) of the t r a n s f o r m a t i o n a l s e r i e s , which was a l s o g i v e n . In o ther words , the s u b j e c t was p r e s e n t e d w i t h the f i r s t two cards and the l a s t one o f a s e r i e s of s i x , and was asked to complete the s e r i e s by ex tend ing the changes i n d i c a t e d i n the f i r s t two s t i m u l i such tha t the c o n t i n u a t i o n o f these changes g r a d u a l l y and s u c c e s s i v e l y l e d to the f i n a l s t i m u l u s c a r d s . S p e c i f i c a l l y , t h i s t ask r e q u i r e d the s u b j e c t to s i m u l t a n e -o u s l y h o l d i n t o c o n s i d e r a t i o n f o r each s u c c e s s i v e s tep the change (or a l l the changes) n o t i c e d i n the f i r s t two s t i m u l i cards such t h a t each s u c c e s s i v e ca rd was d i f f e r e n t from a l l o thers i n the s e r i e s w i t h r e s p e c t to the changing c h a r a c t e r i s t i c ( s ) . To s i m u l t a n e o u s l y h o l d i n a t t e n t i o n a l l the n e c e s s a r y i n f o r m a t i o n f o r each s u c c e s s i v e s t e p , the s u b j e c t had to (a) c o n t i n u o u s l y keep t r a c k o f -- h i n d s i g h t -- a l l the chanees i n d i c a t e d i n the f i r s t two cards (and c o n t i n u o u s l y assess how many o f those he had a l r e a d y taken i n t o account and how many he had y e t to i n c o r p o r a t e ) , as w e l l as (b) a n t i c i p a t e -- f o r e s i g h t -- the degree o f change f o r 44 each next card by assessing the changes indicated in the f i n a l step of the s e r i e s . The requirements for hindsight and foresight a b i l i t i e s x^ere thus combined in one operational task. It xvas decided to fbllox-/ such a procedure because conceptually and also operationally (at least i n t u i t i v e l y ) , both of these functions involve r e c i p r o c a l , i f not the same, mental a c t i v i t y . In t h e i r work* Inhelder and Piaget (1964) found that, according to t h e i r separate measures of these a b i l i t i e s , the l e v e l of one depended on the l e v e l of the other. In this regard Inhelder and Piaget (1964) s p e c i f i c a l l y state: "As soon as an assimilatory schema becomes retro-active, i t also takes an anticipatory character, because one cannot be consistent with the past without eventually making choices and forming intentions as to the future." (op. c i t . , p. 286) The subject's l e v e l of attainment or degree of f l e x i b i l i t y i n his hindsight and foresight a b i l i t i e s (as defined by t h i s task) was measured by the number of changes that could simultaneously and successively be co-ordinated. The series were varied in d i f f i -culty by increasing the number of simultaneously varying dimen-sions from one to f i v e per card. This increase in the number of changes provided a quantitative assessment of hindsight and fore-sight a b i l i t i e s and a parameter that served as a basis for a quantitative comparison between the levels attained in the auto-maton functions and the operative l e v e l s , which were also expressed in quantitative terms. The s p e c i f i c description, scoring, and other methodological aspects of this task are discussed in the procedure section. 45 Other Automaton Tasks There have been s e v e r a l attempts i n the l i t e r a t u r e t o r e l a t e the competence d e s c r i b e d by P i a g e t t o o t h e r more g e n e r a l f a c e t s o f c o g n i t i v e development ( G o l d s c h m i d , 1967; Goldschmid, 1968; Dudek, L e s t e r , G o l d b e r g , and Dyer, 1969). These a t t e m p t s , o f c o u r s e , r e f l e c t the developmental p s y c h o l o g i s t s ' e f f o r t s t o u n d e r s t a n d P i a g e t ' s " e p i s t e m i c c h i l d " beyond t h e r e a l m o f l o g i c a l c o n c epts t h a t P i a g e t m a i n l y d e a l s w i t h . I t a l s o r e f l e c t s t h a t e v e r - p r e s e n t i n t e r e s t i n d e f i n i n g i n t e l l i g e n c e i n terms o f g e n e r a l , o v e r a l l a b i l i t i e s w hich are somewhat independent o f the c o n t e n t i n w h i c h they may be a p p l i e d . One such, attempt i s P a s c u a l - L e o n e ' s (P a s c u a l - L e o n e , 1970 ; P a s c u a l - L e o n e and S m i t h , 1969) model t o i n t e r p r e t o r t r a n s l a t e P i a g e t ' s d e v e l o p m e n t a l s t a g e s i n terms o f an i n f o r m a t i o n - p r o c e s -s i n g model w h i c h can e x p r e s s q u a n t i t a t i v e l y the p o s s i b i l i t y i n h e r e n t i n P i a g e t ' s stage n o t i o n o f c o n s t r u c t i n g an o r d i n a l s c a l e o f i n t e l l e c t u a l development. A c c o r d i n g t o t h i s i n t e r p r e -t a t i o n , the d i m e n s i o n o f t h i s o r d i n a l s c a l e c o r r e s p o n d s t o the i n f o r m a t i o n a l c o m p l e x i t y o f the t a s k c o n s i d e r e d from t h e s u b j e c t ' s  p o i n t o f view. From P a s c u a l - L e o n e ' s a n a l y s i s , each stage-model o f o p e r a t i o n a l development would appear t o be a q u a l i t a t i v e s t a tement about "the number" o f d i f f e r e n t schemes ( i . e . , d i f f e r e n t chunks o f i n f o r m a t i o n ) on w h i c h the s u b j e c t can s i m u l t a n e o u s l y o p e r a t e . The a u t h o r proposes t h a t such a c o n t e n t - f r e e , s t r u c t u r a l a n a l y s i s o f i n t e l l e c t u a l development i s suggested by P i a g e t when he s t a t e s : ;'...the r e a l problem i s t h a t o f the a s s i m i l a t i v e mechanisms o f the c o o r d i n a t i o n o f schemes: how 46 the subject comes to coordinate by means of reciprocal assimilations., several behavioral segments into a single superordinate behavior." (Piaget j, 1967b, p. 203 ; Pascual-Leone's trans-lation) PasCual-Lebne's model thus attempts t o measure the quanti-tative increases i n what he labels as the child's "central com-puting space," or F-Operator. This construct is then an automaton aspect characterizing the subject which refers to the increases i n the child's mental or attentional span. One test of this model has been through a task in which the subject i s required to make as many responses as possible to nested compound stimuli of various complexity (from two t o eight nested stimuli) which are presented at b r i e f intervals varying between f i v e to ten seconds. For each stimulus a l l subjects have learned the appro-priate response, therefore, differences in the number of responses made to the compound stimuli are presumed to r e f l e c t the maximum number of discrete "Chunks" o f information the sub-je c t can integrate i n a single act. Successful quantitative predictions have been obtained about the size of the mental span of children of d i f f e r e n t developmental leve l s (Pascual-Leone, 1979). The relevance of the construct of a central computing space to the present study is that i t deals s t r i c t l y with an automaton variable and thus provides a point of comparison between the quantitative of performance demands within the context of struc-tu r a l tasks ( i . e . , Ss' performance in the l o g i c a l tasks) and quantitative performance demands independent of a structural.  Context ( i . e . , Pascual-Leone's task). An analysis of the child's l e v e l of dperativity in terms of his mental span as defined by Pascual-Leone might lead to a better understanding of questions such as, why i s i t that a c h i l d who successfully solves a c l a s s i f i -cation task involving two dimensions (which would imply that he possesses the c l a s s i f i c a t o r y structure) cannot do so when the task involves four dimensions? Or more generally, to which extent is the manifestation of operational schemes a function of the information-processing demands? To x^hich extent is the degree of generalization between operations also a function of these same demands? The diagnostic procedure so far outlined involves an assess-ment of structural and functional aspects of cognitive operations ( l o g i c a l tasks) whose va r i a t i o n across subjects i s expressed i n quantitative terms that presumably can make them comparable with other tests. In addition to these, a mental span test fashioned after Pascual-Leone's (1970) procedure is incorporated to obtain (from the same subjects at the same time) a "purer" measure of their performance or automaton c a p a b i l i t i e s that may provide a further basis for a quantitative comparison. Aside from the p o s s i b i l i t y of a quantitative comparison of the various operational aspects of Piagetian Tasks, the procedure used i n Pascual-Leone 1s (1970) test has another theore t i c a l i n t e r e s t . It allows comparison of the child's memory capacity in a context in which the memory requirements are i n terms of the time l i m i t s set in the exposure of the s t i m u l i , with the child's memory capacity when there are no such time constraints, i . e . , the task requirements of the s t r u c t u r a l task i n terms of reconstructive memory. 4 8 S p e c i f i c Goals of Diagnostic Procedure From the theoreti c a l and methodological considerations pre-sented above, i t is possible to summarize more succinctly what the s p e c i f i c aims of the study were: (1) Measure the relationship between the degree of attain-ment in the self-regulatory functions which are postulated to make possible the development of logico-mathematical operations, test of foresight and hindsight a b i l i t i e s , and the l e v e l of achievement in the l o g i c a l tasks themselves. I f indeed the test of foresight and hindsight a b i l i t i e s measured the prerequisite functions of logico-mathematical operations, there should be a close correspondence between the level s of attainment i n both types of tasks and, i t should be possible to v btain some indicati o n of rvhat l e v e l of attainment in the hindsight - foresight a b i l i t i e s is necessary for the development of l o g i c a l operations. (2) hieasure the degree of relationship between a family of logico-mathematical operations which are presumed to develop concurrently through procedures that make them comparable, both in terms of t h e i r competence and automaton demands, in an attempt to minimize as many sources of'decalages'as possible. If the above, procedures were e f f e c t i v e , then there should be a close correspondence for each subject in his level of attainment in these tasks. (3) Compare levels of attained operativity in the logico-mathematical tasks through procedures which emphasize the fig u r a t i v e aspects of the stimuli versus procedures which emphasize the operative aspects. 49 ( 4 ) Compare the quantitative measures of a l l of the above to an automaton measure which i s presumed to assess the subjects' general information-processing capacity i n quantitative terms. CHAPTER III Method Subjects JJs were 32 fi r s t - g r a d e children with a mean age of 7.02 (range: 6.5 - 3.4) and 32 second-grade children with a mean age of 7.93 (range: 7.4 - 9.0). Half the Ss i n each grade were males, h a l f females. The sample was drawn from a primary school i n a lower-middle class d i s t r i c t . About h a l f of the school pop-u l a t i o n comes from homes where Greek or I t a l i a n were the primary languages. Of the G4 Ss who p a r t i c i p a t e d in t h i s study, 2 7 were Greeks or I t a l i a n s . Design For each group, Grade 1 (Gl) and Grade 2 (G2), a l l measures were i n t e r c o r r e l a t e d , thus obtaining one c o r r e l a t i o n a l matrix for each group. General procedure The tasks were administered i n d i v i d u a l l y in two sessions, with an i n t e r v a l of 2h weeks between the two sessions. Each session took between 30 to 50 minutes. In the f i r s t session, the Foresight-Hindsight (FORHIN) , T r a n s i t i v i t y (TRAIJ) , and the M u l t i p l i c a t i o n of Classes (MC) tasks were administered. The TRAN task was always the middle task, while FORHIN was f i r s t for h a l f the Ss, and MC f i r s t for the other h a l f . In the second session, the M u l t i p l i c a t i o n of Relations (KR) , the 1-T-Operator (MOPER), and the S e r i a t i o n (SER) tasks were administered. MOPER was always the middle task while SER was f i r s t for h a l f the Ss and HR was f i r s t for the otber h a l f . 51 Tasks M u l t i p l i c a t i o n of Glasses Task (MCj The competence rule of this task is represented i n the group-ing of biunivocal m u l t i p l i c a t i o n of classes. A l o g i c a l class can be considered as a union of elements brought together on the basis of an equivalent r e l a t i o n , i . e . , a l l members of class A possess property a, a l l members of class B possess property b . members of a class may have more than one defining property, and therefore i t is possible to simultaneously c l a s s i f y a set of elements i n terms of more than one c r i t e r i o n . The multiple c r i t e r i a of a set of elements can be simultane-ously represented by means of a matrix which s p a t i a l l y symbolizes the equivalences and correspondences that may exist between a l l the members of the classes (See Fig. l a ) . The rule embodied in the matrix i s that each c e l l i s the intersection of the dimen-sions that remain constant across the respective rows and columns. In order to understand the complementary aspects of the matrix, the S has thus to focus simultaneously on the equiva-lences and differences along the two axes. The arrangement of elements used in the present study was made in such a way that perceptually the equivalences between the elements rather than the differences would be emphasized. In this manner, i t was hoped that the task would be maximally d i s s i m i l a r in i t s figura-t i v e aspects from the M u l t i p l i c a t i o n of Relations (MR) task (See Fig. 2a) . Inhelder and Piaget ( 1 9 6 4 ) found that, when the testing of the M u l t i p l i c a t i o n of Classes concept is done by having the S complete one c e l l of 2 x 2 or 2 x 3 matrices, the sp a t i a l configura-t i o n may l e a d t o c o r r e c t s o l u t i o n s which are s i m p l y based on the p e r c e p t u a l symmetries o f the h o r i z o n t a l and v e r t i c a l axes r a t h e r than on an o p e r a t i o n a l u n d e r s t a n d i n g o f the p o s s i b i l i t y o f m u l t i p l e c l a s s i f i c a t i o n s o f o b j e c t s . The p r e s e n t s t u d y a s s e s s e d the c h i l d ' s u n d e r s t a n d i n g o f the m u l t i p l i c a t i o n o f c l a s s e s concept through h i s performance on a m u l t i p l i c a t i o n c f c l a s s e s m a t r i x t a s k . Hoxvever, the t e s t was d e s i g n e d t o m i n i m i z e the p o s s i b i l i t y o f s o l u t i o n s based on the p e r c e p t u a l symmetries o f the m a t r i x . T h i s was done, f i r s t , by i n c r e a s i n g the number o f c e l l s and number o f dimensions the S had t o d e a l w i t h ; and second, by t e s t i n g the S's u n d e r s t a n d i n g o f the m a t r i x i n more than one way: ( 1 ) t h r o u g h the S's c o m p l e t i o n o f t h r e e ( i n s t e a d o f one) o f the d i a g o n a l c e l l s ( F I L L - I U measure), ( 2 ) t h r o u g h the S's a b i l i t y t o reproduce the m a t r i x c o n f i g u r a t i o n i m m e d i a t e l y a f t e r i t i s no l o n g e r v i s i b l e ( C O P Y measure), and ( 3 ) t h r o u g h the S's a b i l i t y t o m e n t a l l y r e v e r s e the s p a t i a l c o n f i g u r a t i o n o f the m a t r i x when a g a i n i t i s no l o n g e r v i s i b l e (Reverse o r REVB measure). M a t e r i a l s - The m a t e r i a l s c o n s i s t e d o f 4 5 i n d i v i d u a l c a r d b o a r d p i e c e s and a 3 0 x 2 3 cm gray c a r d b o a r d which was d i v i d e d i n t o a 3 x 3 - c e l l m a t r i x . The i n d i v i d u a l p i e c e s each had d i f f e r -ent v a l u e s on each o f the f o l l o w i n g d i m e n s i o n s . ( 1 ) Shape --t r i a n g l e s , s q u a r e s , r e c t a n g l e s ( 2 ) s i z e s m a l l ( 2 . 6 x 2 . 6 cm), medium ( 4 . 5 x 4 . 5 cm), l a r g e ( 6 . 4 x 6 . 4 cm) ( 3 ) c o l o r -- b l u e , p i n k , w h i t e ( 4 ) background -- l i n e s , b l a c k d o t s , p l a i n ( 5 ) superimposed diamonds -- 0 , 1 , 2 b l a c k diamonds on c e n t e r o f shape ( 6 ) t h i c k -ness -- t h i n ( o n l y one l a y e r o f 1 . 5 cm t h i c k c a r d b o a r d ) , medium ( 4 l a y e r s ) , t h i c k ( 7 l a y e r s ) . A l l shapes were c o v e r e d v / i t h p l a s t i c c o a t i n g . The c o r r e c t arrangements of the a p p r o p r i a t e o r 5 3 WHITE BLUE PINK THIN THICK MEDIUM 6 PINK BLUE WHITE MEDIUM: • : : •• »• THICK • THIN Fig. 1. M u l t i p l i c a t i o n o£ Classes matrix ( a) and reversed matrix (b). Dimensions remaining constant on the horizontal axis s i z e , diamond, color; and along the v e r t i c a l axis background, thickness, and shape. 54 correct pieces is shown in Fig. l a . Procedure - S_ was presented with the empty matrix board and was shown the three trays adjacent to the board in each of which were the 15 stimuli that corresponded to each one of the three shapes. The S_ was asked to point out a l l the dimensions present in the stimuli by having him describe to h the s i m i l a r i t i e s and differences between the stimuli in the three trays. E proceeded to f i l l in. positions 1, 2, 4, on the matrix (See Fig. la) and immediately asked S_ to indicate the s i m i l a r i t i e s and differences between stimuli in c e l l s 1 and 2, and in 1 and 4. Then E f i l l e d in p o s i t i o n 8 and asked S to indicate s i m i l a r i t i e s and d i f f e r -ences between 2 and 8: then c e l l s 6 and 9 were f i l l e d and E asked for comparisons between 8 and 9, and 4 and 6. Cells 7, 5, and 3 remained empty. This i n i t i a l procedure was followed to make Ss aware of equivalences among the dimensions present i n the pieces. F i l l - i n - After E finished f i l l i n g in the matrix p a r t i a l l y as described above, S_ was asked to look at the whole board and try to see i f he could f i n d from the pieces i n the trays, those that would f i l l c e l l s 3, 5, and 7, such that each piece would f i t with the others in the board both v e r t i c a l l y and horizontally (axes indicated by gesture). After S_ f i l l e d in c e l l s , E asked S to check the whole board just in case he might want to change any one of the p i e c e s 1 . After S_ indicated that he was s a t i s f i e d with his choices, E recorded the choices and immediately gave S feedback as to the appropriateness of his choice by replacing the S's choices with the correct ones, i f necessary, and indica-lFrom now on this f i n a l question w i l l be referred to as Q. 55 ting why the replacement piece was appropriate. Reproduction or Copy - After the correct matrix was completed, E t o l d S that she was going to remove a l l the shapes and mix them vfith the others and that she wanted to see whether the S could f i n d them and put them back in exactly the same places as before. The S was allowed to look at the matrix for as long as he wished and When he indicated that he had looked at i t long enough, a l l the pieces were removed and mixed thoroughly with the rest of the shapes in t h e i r respective trays. A l l these operations were performed in f u l l view of the S. The S proceeded to f i l l the matrix, and was allowed to change as many pieces as he wanted, with no time l i m i t . When S completed the matrix, he was asked Q. The E recorded Ss choices and immediately proceeded to give S feedback as to the appropriateness of his choices by replacing the incorrect pieces with the correct ones while at the same time explaining why the correct choices were the appropriate ones. Reversal - Immediately after the Reproduction task, E said to S that he would have one more chance to f i n d the correct pieces and that again the pieces would be removed. When the S indicated that he had looked at the f i l l e d matrix long enough, a l l pieces were removed except the one in c e l l 7, then S was t o l d that he had to put back the same pieces in exactly the same places as before except that the board would this time be turned around -•• E then turned the board 189° so that c e l l no. 7 now corresponded to c e l l no. 3 (See F i g . l b ) . Anticipation - Before S was allowed to complete the f u l l matrix, he was asked to f i n d which shape should go in position 6. This was asked in order to assess the degree to which the S could 56 anticipate aspects of the reversed matrix before he actually engaged in the task. Such a measure of anticipatory a b i l i t y was obtained to find out how e x p l i c i t the transformations of the t o t a l reversed matrix were for the S, and thus perhaps d i f f e r e n t i a t e between those solutions which might have been achieved simply by t r i a l and error and those which were t r u l y operational. After the S anticipated the shape for position no. 6, he proceeded to f i l l in the whole matrix. When he finished, E asked Q and then proceeded to record S's choices. On this l a s t occasion, no feedback was given, except that the S was t o l d that his choices were " f i n e " whether or not they were correct. Scoring of the M u l t i p l i c a t i o n of Classes task F i l l - i n (MCFILL) - One point was given for each dimension co r r e c t l y solved out of the six t o t a l possible in each of the three diagonal c e l l s of the matrix. Thus the maximum score pos-s i b l e was 18 points (6 dimensions x 3 c e l l s ) . Reproduction (MCCOPY)1 - The rationale underlying t h i s scoring procedure was the following. A S, p a r t i c u l a r l y those who do not yet possess the operative structure necessary to solve the m u l t i p l i c a t i v e matrix, may attempt to solve i t simply by looking at the consistencies across rows only, or at the consist-encies across columns only, but not to both simultaneously. To maximize the chances of d i f f e r e n t i a t i n g between "segmented" vs. " t o t a l " solution strategies that the Ss might have employed, the compilation of the f i n a l score was made in the following way: ^rom here on, the score Reproduction i s referred to as COPY. This was done in order to have an abbreviation for this score e a s i l y distinguishable from that used for the Reverse (REVE) scores. 57 (a) the t o t a l number of dimensions (1 to 6) c o r r e c t l y solved across each and a l l of the three c e l l s in each roxv was obtained and that number was multiplied by three (the number of c e l l s ) . For example, i f a S c o r r e c t l y solved, for each one of the three  c e l l s of a row (as opposed to just one or two c e l l s ) , 3 out of the 6 dimensions, he would then get a score of 9 (3 (dimensions) x 3 ( c e l l s ) = 9)o The t o t a l score from each of the three rows was added. (b) The same procedure if as followed for each column, and a l l the individual column scores were added. Scores (a) and (b) were added together and t h e i r sum indicated how well the S succeeded in the individual rows and columns, or more specif-i c a l l y how well S co-ordinated s p e c i f i c values of a dimension ( i . e . , a l l small shapes placed in the appropriate row). However, the c r i t i c a l feature of a m u l t i p l i c a t i v e matrix is the simultan-eous c r o s s - c l a s s i f i c a t i o n of the objects ( i . e . , correct c l a s s i f i -cation of a l l small, medium, large / t r i a n g l e s , squares, and rectangles) that is manifested over the entire matrix. Thus to the above sum of the (a) and (b) scores, a t h i r d score was added: (c) the number of dimensions corre c t l y solved over the  entire matrix was obtained and multiplied by nine (the t o t a l number of c e l l s ) . This (c) score indicated that the S's cor r e c t l y solved dimensions over the t o t a l matrix as opposed to over individual rows or columns. The f i n a l M u l t i p l i c a t i o n of Classes-Reproduction score (MCCOPY) was made up of the sum of the (a), (b) , and (c) scores. The maximum t o t a l number of points possible for MCCOPY was 162. Derived measures - The t o t a l HCCOPY score just described does not indicate, for the cases i n which the S solves between 2 and 4 58 correct dimensions over the entire matrix, whether or how many-dimensions of this t o t a l involve m u l t i p l i c a t i v e combinations of  dimensions. Looking at Figure 1, i t w i l l be noticed that the matrix involves three overlapping dimensions on each axis (color, s i z e , and diamonds on the horizontal axis, and thickness, back-ground, and shape on the v e r t i c a l one); thus a S_ can solve up to three correct dimensions over the entire matrix and yet not have a single possible m u l t i p l i c a t i v e or intersection combination between these dimensions, i . e . , i f he solves thickness, back-ground and shape. By the same token, another S may only solve txvo dimensions, and have one possible m u l t i p l i c a t i v e combination between the two., e.g., i f S solves thickness and color. A d i s t i n c t i o n was made, therefore, between the t o t a l number of dimensions c o r r e c t l y solved over the entire matrix (range from 0 to 6)--MCDIM, and the t o t a l number of m u l t i p l i c a t i v e combinations possible within that t o t a l -- MCMUL. The importance of the d i s t i n c t i o n between the MCDIM and MCMUL scores i s the following. For the most frequent intermediate cases (Ss who cor r e c t l y solve less than 5 or 6 dimensions over the entire matrix), the MCDIi.-MCMUL d i s t i n c t i o n may d i f f e r e n t i a t e between (1) those Ss who, because they have a good r e c a l l of the percep-tual configuration of the matrix, may cor r e c t l y solve a number of dimensions even though they have not yet attained an operational understanding of the biunivocal m u l t i p l i c a t i o n concept, and ( 2 ) those Ss who possess an understanding of the m u l t i p l i c a t i v e concept, but whose application of i t may be limited i n terms of the number of m u l t i p l i c a t i v e combinations that they can simul-taneously handle at one time. 59 The number of possible m u l t i p l i c a t i v e combinations out of the t o t a l number of dimensions corr e c t l y solved (MCMUL) was obtained in the following way: the number of dimensions c o r r e c t l y solved across the v e r t i c a l axis was multiplied by the number of dimen-sions that were corre c t l y solved across the horizontal axis. It was possible to obtain scores such as 0 -- when no dimensions were Correctly solved or when a l l the dimensions corr e c t l y solved were along one axis only, up to 9 -- when a l l three dimensions along each axis were co r r e c t l y solved. The t o t a l range of possible scores was: 0, 1, 2, 3, 4, 6, and 9. In summary, the two measures derived from the t o t a l MCCOPY score were: (1) MCDIM -- t o t a l number of dimensions corr e c t l y solved over the entire matrix, range, 0 to 6 pts. (2) MCMUL -- t o t a l number of m u l t i p l i c a t i v e combinations possible within the t o t a l number of dimensions c o r r e c t l y solved, range, 0 to 9 pts. Reverse (MCREVE) - A scoring obtained exactly in the same way as for the copy section was obtained for reverse-scores i , except that in th i s case the correct c r i t e r i o n was the reversed matrix (See Fig. l b ) . Also, a score was obtained for anticipa-t i o n (MCANT). Anticipation (MCANT) - One point was given for each of the dimensions, out of the t o t a l s i x possible, c o r r e c t l y solved in c e l l no. 6 of the reversed matrix. The possible range of scores was from 1 - 6 . M u l t i p l i c a t i o n of Relations task (MR) The competence rule of this task i s embodied i n the structure 6 0 of the biunivocal m u l t i p l i c a t i o n of relations grouping. The con-ceptual operations involved in this task are quite similar to those of the MC task except that i t i s the relations or d i f f e r -ences among class elements that are emphasized (as opposed to the i r equivalences) as the elements are put together or seriated on the basis of those differences. The m u l t i p l i c a t i v e matrix arrangement;, however, symbolizes not only the seriated d i f f e r e n -ces, but also the equivalences betxveen the ordinal correspond-ences of sets of seriated elements (See Fig. 2 a ) . Thus the operational or conceptual rule of a m u l t i p l i c a t i o n of relations task as represented in a m u l t i p l i c a t i v e matrix i s very close to that of the MC task: to comprehend the matrix configuration the S_ has to simultaneously focus on the differences and equivalences that occur across rows and columns. Materials - The materials consisted of the same 3 0 x 2 3 cm matrix board used in the MC task and 4 5 cardboard e q u i l a t e r a l t r i a n g l e s , each having d i f f e r e n t values on each of the following s i x dimensions: ( 1 ) color -- pink, gray, blue ( 2 ) brightness --dark, medium, l i g h t ( 3 ) size -- small ( 2 . 6 x 2 . 6 cm), medium ( 4 . 5 x 4 . 5 cm), large ( 6 . 4 x 6 . 4 cm) ( 4 ) thickness -- thin (one layer of 1 . 5 cm cardboard), medium (4 l a y e r s ) , thick ( 7 layers) ( 5 ) number of lines -- 1 , 2 , C T 3 horizontal lines at the base of the triangles (6) orientation -- small superimposed yellow t r i a n g l e pointing to the r i g h t , upwards, or to the l e f t . A l l trian g l e s were covered with p l a s t i c coating. The correct arrange-ment of the correct triangles is shown in Figure 2 . Procedure - The procedure was i d e n t i c a l in every respect to that of the MC task and i t also comprised the following scores: 61 BLUES REDS GRAYS THIN/DARK MEDIUM/MEDIUM. THICK/LIGHT GRAYS REDS BLUES THICK/LIGHT MEDIUM/MEDIUM THIN/DARK w w V Fig. 2. M u l t i p l i c a t i o n of Relations matrix (a) and reversed matrix (b). Dimensions remaining constant along, horizontal axis s i z e , o r ientation, color; and along v e r t i c a l axis thickness, brightness, and l i n e s . 62 (1) F i l l - i n -- MRFILL (2) Reproduction -- MRCOPY (3) Total number of dimensions c o r r e c t l y solved over the entire matrix -- Figurative MCDIM(F) ( 4 ) number of m u l t i p l i c a t i v e combination of dimensions possible in the MRDIM(F) score -- Figurative MCMUL(F) (5) Reverse -- MCRZVE ( 6 ) Total number of dimensions c o r r e c t l y solved over the entire reverse matrix -- Operative MRDIM(O) (7) t o t a l number of mu l t i p l i c a t i v e combinations poss-ib l e in MR'DIM -- Operative MRMUL(O), and ( 8 ) Anticipation (MRANT). Seriation task (SER) The competence rule of thi s task is based on the structure of the addition of asymmetrical relations groupings. Conceptually the task deals with the asymmetrical relations among elements that can be expressed in terms of ordered differences. That i s , i t deals with relationships that can exist between elements when these are ordered i n terms of the i r increasing or decreasing r e l a t i v e magnitudes. In order for a S to group the differences among a set of elements i n an ordered series, he has to r e a l i z e that any one element i s in reciprocal and simultaneous r e l a t i o n -ship with those elements that follow and precede i t , such that a given element can be at the same time in a "before" and " a f t e r " r e l a t i o n with the succeeding and preceding ones. The usual procedure that has been used to test this concept is to have the S fre e l y attempt to order a set of elements ( i . e . , 10 sticks) and then require him to insert extra elements in the ser i e s . The c r i t e r i o n of operational achievement is defined in terms of the S's construction of the series without t r i a l and error (by immediately and systematically searching for the smallest (and/or biggest) element and thus proceeding with the 63 rest of the remaining elements), and in terms of whether the S can insert new elements i n the series again without t r i a l and error. To keep the task demands congruent with those used in the MR and MC tasks, the child's understanding of this concept was again assessed i n terms of his memory of the figurative aspects of the configuration (reproductive memory) as well as in terms of his a b i l i t y to mentally reverse the relationships involved in the series or the operative aspects of the configuration. The performance or automaton demands were set up also in terms of the number of dimensions in the series that the S was required to co-ordinate. Materials - The stimuli consisted of a set of 36 cardboard " l i t t l e men" made up of 1.5 cm thick cardboard (See Fig. 3) of which only 9 constituted the correct series. The " l i t t l e men" varied on the six following dimensions: (1) Height — t h e stimuli ranged in height from 6.4 cm. to 10.2 cm and the d i f f e r -ence between successive elements was of 4 cm (2) Width -- the width varied from 2.1 cm to 6.4 cm (3) Orientation of eyes --the eyes were rotated at successive angles of 30° st a r t i n g with the eyes oriented straight dox^nward (stimuli 1) , and ending in the l e f t bottom quadrant after a 270° turn (stimuli 9). (4) Location of t i e -- the t i e was lowered from the center horizontal l i n e in successive 6-cm stops (5) Width of frock -- the horizontal lines in the bottom half of l i t t l e men varied i n xtfidth from 3 cm to 2.5 cm (6) Thickness -- only three degrees of thickness xvere used, thin (2 layers of 1.5 cm thick cardboard), medium (4 la y e r s ) , thick (7 lay e r s ) . In the correct series, the F i g . 3 . S e r i a t i o n t a s k . S t i m u l i v a r i e d i n h e i g h t , w i d t h , t h i c k n e s s , eyes' r o t a t i o n , w i d t h o f f r o c k , and p o s i t i o n o f t i e . 65 f i r s t three elements were thin, the middle three were medium, and the l a s t three were thick. A l l stimuli xvere covered with p l a s t i c coating. There was also a board 17.-1/2 * 4-3/4" numbered 1 to 9 along the upper and lower edges. Procedure - The S_ was f i r s t presented with three trays (one with 12 thin l i t t l e men, another one with 12 medium ones, and the l a s t one with the 12 fat ones). The S was made aware of the c h a r a c t e r i s t i c s of the stimuli by having him describe to E the differences between two of the l i t t l e men. Then the E proceeded to make up the series by placing the shortest of the correct stimuli in position #1 of the board. E then nlaced the next stimuli on position #2 and as he proceeded, he indicated how the six dimensions were successively changing. This was continued u n t i l a l l the nine positions x\rere f i l l e d . Reproduction or Copy - After the series was completed, the S was asked to look at i t c a r e f u l l y because the l i t t l e men were going to be put back on the respective trays and the E wanted to see whether he could remember them and put a l l the nine l i t t l e men back i n exactly the same way. After S looked as long as he wanted, the S removed the stimuli and mixed them with the others in t h e i r appropriate trays. S_ was allowed to make up a. series and as before he could change the pieces as many times as he wanted. 'Then S indicated he he had finished E asked 0, and i f no changes were made, E recorded the S's choices and then pro-ceeded to give S feedback as to the accuracy of his choices by replacing the incorrect pieces with the appropriate ones, \/hile at the same time indicating where necessary why his choices were inappropriate. 66 Reverse - When the whole correct series was completed again, the S_ was t o l d that he would have one more chance to f i n d the nine l i t t l e men and to look at them c a r e f u l l y again. When S indicated he had looked enough, the stimuli were removed but this time the board was turned around and S was asked to complete the series but this time st a r t i n g with the l i t t l e men previously in p o s ition #9, xvhich now was placed in no. 1 position. In this way, the S was forced to reverse a l l the relationships. The S was allowed to complete the series, when S indicated he had finished E asked Q and i f no changes were made, E proceeded to record the S's choices. This time no feedback was given as to the appropriateness of the choices and the S_ was simply t o l d that his choices were " f i n e . " Scoring of Seriation Task Reproduction (SERCOPY) - This score was comprised of the following: (a) one point was given for each dimension corr e c t l y solved at each individual position and the t o t a l number of points for each of the nine position was added (b) an o v e r a l l score was also obtained by multiplying the number of dimensions  cor r e c t l y solved across the entire series (range from 0 to 6), times the t o t a l number of individual positions in the series. Scores (a) and (b) were added to make the SERCOPY score. Thus, the t o t a l score or the Seriation task combined how many i n d i -vidual pieces or " l i t t l e men" the S_ c o r r e c t l y arranged as well as how many correct dimensions in a l l the stimuli he co-ordinated c o r r e c t l y across the entire series. Reverse (SEREVE) - The score was obtained in the same way as the SERCOPY except that the correct c r i t e r i a was the series i n 6 7 reversed order. Derived measures - From SERCOPY, the Figurative score, a SERDIM(F) score was obtained, which indicated the t o t a l number of dimensions co r r e c t l y solved over the entire s e r i e s . From SEREVE, the Operative score, a SERDIM(O) score was obtained which indicated the t o t a l number of dimensions correctly solved i n the reversed series . T r a n s i t i v i t y task (TRAN) The competence rule of this task is also based on the group-ing of addition of asymmetrical r e l a t i o n s . The task was admin-istered to test in a direct and d i f f e r e n t way from that of the SER task, the S's understanding of the reciprocal nature of asymmetrical r e l a t i o n s . The task requirements for this task were not exactly the same as for the three previous ones, as w i l l become apparent from the description of the procedure. This measure, therefore, was intended to supplement the one obtained from the SER task and not as an equivalent to the SER task. Materials - The stimuli consisted of four " l i t t l e men" (See Fig. 4 ) which varied in height and i n the color of t h e i r s h i r t s (bottom h a l f of l i t t l e men). The differences in height were small enough that the stimuli had to be d i r e c t l y compared with each other to determine which one was t a l l e r . Stimuli 1 (Sj , purple bottom) was 8 . 3 cm t a l l , S 2 (white polka dotted bot-tom) and S 3 (pink bottom) were the same height - 7 . 6 cm, and Sn (blue bottom) was 7 . 3 cm t a l l . Procedure - The S was f i r s t shown a l l the four l i t t l e men in scrambled order and told that this i f as going to be a memory Fig. 4. T r a n s i t i v i t y task. The relevant character i s t i c s to the t r a n s i t i v i t y judgments were height and color of bottom h a l f ("shirt") of " l i t t l e men." 69 game. E put away a l l stimuli under a board and took out only S i and S2 and asked S to check to see which one of the two was t a l -l e s t . S proceeded to compare S i x^ith S2 and i f he only said something l i k e " t h i s is the biggest," E would, repeat after him "So the purple one is t a l l e r than the white one." This was done to give S verbal labels which, would further d i f f e r e n t i a t e the st i m u l i . E emphasized to S_ that he had to remember what he had just seen (that the purple l i t t l e man was t a l l e r than the white one), and then proceeded to remove the purple one ( S i ) , leaving the white one ( S 2 ) . E took out S 3 (the pink one) and repeated exactly the same procedure. E then removed 82, leaving S 3 out and took out S i * , repeating the previous procedure once again. Thus, S compared sequentially the pairs of s t i m u l i , S1-S2, S 2 - S 3 , and S 3 - S i t . After the l a s t comparison, E told S_ , "Nov; I want to ask you some questions." E aligned together S i and S ^ under a board so that only the bottoms of S i and S 2 showed and no d i f f e r -ences i n height were v i s i b l e and asked S to point out which one of the titfo he thought was the t a l l e s t . .After he indicated xvhich one he thought x-/as t a l l e s t , E removed the board to l e t the S_ see whether he xvas right or xtfrong. When the 5 pointed to the right s t i m u l i , E xvould ask S , "How did you figure that out without looking?" The S ' s j u s t i f i c a t i o n was scored as appropriate i f he based his ansxver on the fact that he remembered that S i was the t a l l e s t of a l l the l i t t l e men, or that S i , was the shortest. E then removed a l l stimuli from the S_'s viex/ and then took out S 3 and S i again in the same way as before and repeated the same pro-cedure. F i n a l l y , E took out S 2 and S i » and again repeated the same procedure as above, recording the S ' s choice and j u s t i f i c a t i o n . 70 Scoring of T r a n s i t i v i t y Task Of the three comparisons asked of S (Si vs. Si», S 3 vs. S i , and S2 vs. S 4 ) , the f i r s t one (Si vs. S^) could be said to involve a simple discrimination task, since Si and Si* were the f i r s t and the l a s t stimuli presented to S. Thus to diminish the p o s s i b i l i t y of including as t r a n s i t i v i t y choices those that might be based on a simple discrimination or primacy-recency e f f e c t , the c r i t e r i o n of whether the TRAN task was passed or not was based on whether the S_ c o r r e c t l y solved a l l three comparisons. Thus, the data obtained from this task were binary (pass-fail) and the comparisons with other tasks were made by means of x 2 tests. Foresight-Hindsight task (FORHIN) As was discussed i n the theoretical section, the conceptual or competence rule of t h i s task was defined in terms of the child ' s a b i l i t y to simultaneously take into account and thus sim-ultaneously extend the various changes indicated in the f i r s t two stimulus cards (of a set of s i x ) , such that the transformations for each successive card would gradually lead to the f i n a l card of the set (which was also i n i t i a l l y given). Materials - These consisted of: ( 1 ) a folding 6 3 . 3 x 4 5 . 9 cm board i n the upper center of which there were six 5 . 1 x 5 . 1 cm boxes drawn close together ( . 2 cm apart) in a horizontal alignment ( 2 ) 1 1 sets of cards (the number of cards i n each set varied from 9 to 1 4 ) , one t r a i n i n g set and 1 0 testing sets. These sets of cards represented changes in the dimensions of various s t i m u l i , the d i f f i c u l t y of the s p e c i f i c set being determined by the number of dimensions that simultaneously changed i n the s t i m u l i . For example, set no. 1 (the t r a i n i n g set) constituted the easiest l e v e l , 71 since only one dimension changed through the entire s e r i e s . This t r a i n i n g set showed i n the f i r s t card (See Fig. 5) a candle-rest-ing i n horizontal p o s i t i o n with the flame towards the l e f t side. The second card showed the candle raised at a 30° angle from the p o s i t i o n in card no. 1 ( C i ) ; the l a s t card showed the candle rest i n g h o r i z o n t a l l y again, but this time the flame was on the right side. The S had to choose from the choice cards, which included some incorrect choices, those that would complete the changes i n i t i a t e d i n Ci and C 2 such that C 3, Ci» , and C 5 would gradually lead to C 6 . Thus i n th i s set only one dimension, o r i e n t a t i o n , changed from Ci to C 6 . The t e s t i n g sets involved the following changes (See Appendix): - Set No. 2, one change in the shape (Sh) - Set Ho. 3, one change in the o r i e n t a t i o n (0) - Set No. 4, two changes, i n brightness (B) and size (Si) - Set No. 5, two changes, i n Sh and 0 - Set Mo. 6, three changes, in Sh, S i , and 0 - Set Ho. 7, three changes, i n Sh, B, and 0 - Set No. 8, four changes, i n 0, Sh, S i , and number (N) - Set No. 9, four changes, in 0, N, S i , and Sh - Set Ho. 10, f i v e changes, in N, 0, 3, S i , and Sh - Set MO. 11, f i v e changes, i n N, 0, B, S i , and Sh The choice cards were made up i n such a way that~<^here were only three cards that would meet the following c r i t e r i a : \1) have the same number of changes in the dimensions indicated in the, f i r s t two cards, and (2) have the degree of change appropriate for the given p o s i t i o n s , i . e . , the correct card for p o s i t i o n no. 5 of the t r a i n i n g s e r i e s , for instance (See Fig. 5) would not be the appro-72 Fig. 5. Foresight-Hindsight task. T r a i n i n g s e t . 73 priate one for position no. 3, even though the dimension changing, orientation, would have changed in the correct d i r e c t i o n . Thus, the incorrect choice cards included incorrect changes i n the dimensions ( i . e . , candle going in the opposite d i r e c t i o n ) , as well as incomplete changes (one dimension changing but not the other, for sets with mors than one dimension changing). The most important aspects of the FORHIN task then were that the S had to remember that a l l the changes indicated in the f i r s t two cards had to occur in each of the three successive cards, and that these changes had to occur i n gradual steps which could successively lead to the l a s t card of the series. The s p e c i f i c stimuli in the sets most frequently consisted of abstract shapes which did not involve representations of fam i l i a r transformations. For example, set no. 2 showed changes in a b a l l that s p l i t s into two equal halves, which i n the l a s t card are shown turned upside down touching each other on the curved sides. Reproductions of the sets are included in the Appendix. The number of choice cards increased as a function of the le v e l of d i f f i c u l t y of the set. This was required because as the number of dimensions that were changing were increased, i t was necessary to increase the number of p o s s i b i l i t i e s that would discriminate that indeed the S was focusing on a l l the changes involved. Procedure - S was f i r s t shown the board, then E t o l d S that this was going to be a card game. E placed the f i r s t card of the t r a i n i n g set on box no. 1 and asked S to describe what he saw. Then E placed card 2 and asked S to indicate the changes 74 t h a t had o c c u r r e d i n Card 2 (from Card 1). E r e c o r d e d ( f o r the t e s t i n g s e t s , p a r t i c u l a r l y those w i t h more than one d i m e n s i o n changing) the number o f c h a n g i n g dimensions t h a t the S spontane-o u s l y p o i n t e d o u t , and i f S missed any change, E would then i n d i c a t e i t t o S. F i n a l l y , E p l a c e d the l a s t (6th) c a r d o f the s e r i e s and a g a i n asked S t o i n d i c a t e a l l the changes he n o t i c e d from the f i r s t two c a r d s t o the l a s t one. A g a i n , i f the S d i d not i n d i c a t e a l l the changes, 5 would p o i n t them out t o him. A f t e r t h i s , E t o l d S t h a t he now had t o t r y t o f i n d w h i c h ca r d s s h o u l d go i n boxes 3, 4, and 5 such t h a t the changes t h a t s t a r t e d i n Card 1 and Card 2 would c o n t i n u e i n l i t t l e s t e p s and end up as i n Card 6. Then E s a i d , ; L e t ' s t r y t o see i f you can do i t w i t h t h e s e c a r d s . " E then proceeded t o p l a c e the c h o i c e cards j u s t below the boxes and d i d not a l l o w 5 t o s t a r t f i l l i n g out the 3, 4, and 5 boxes u n t i l he had i n s p e c t e d a l l o f the c h o i c e c a r d s ('Make sure you've l o o k e d at a l l o f them f i r s t b e f o r e you s t a r t , because some o f them are wrong and so you have t o check them a l l f i r s t v e ry c a r e f u l l y . " ) . For the t r a i n i n g s e t , when S made an e r r o r , he was i m m e d i a t e l y t o l d t h a t the c a r d c o u l d not f i t t h e r e and t h e n gave the s p e c i f i c r e a son why. Then E a l l o w e d S t o s e a r c h a g a i n and i f S_ c o u l d not f i n d t h e r i g h t c a r d , E p r o -ceeded t o p l a c e the c o r r e c t c a r d i n t h e c o r r e c t p l a c e . T h i s p r o c e d u r e was c o n t i n u e d u n t i l a l l t h r e e boxes were f i l l e d . In t h i s t r a i n i n g s e t , the E emphasized t h a t the changes had t o con-t i n u e f o r each o f the c a r d s . A f t e r t h i s t r a i n i n g p a r t , the S was t o l d t h a t from now on he would have t o t r y t o f i n d the c a r d s on h i s own and t h a t the E was not g o i n g t o h e l p . E s t a r t e d Set 2 and f o l l o w e d the same pr o c e d u r e as b e f o r e except t h a t the S was 75 not helped while he made his choices. S_ was allowed to try and change as many cards as he wished. When S expressed that he had the cards he thought f i t t e d best, E asked Q and, i f no changes were made, she recorded the S's choices. Then E proceeded to give S feedback as to the accuracy of his choices and change the incorrect ones while at the same time explaining why they were inappropriate. The sets were given i n order u n t i l the 5 f a i l e d in two consecutive series. A f a i l u r e was defined when two or a l l three of the cards were incorrect. Scoring of Foresight-Hindsight Task This score was compiled in the following way: (a) a point was given for each correct change in the ( 1-5) dimensions i n -volved in the transformational series that was correctly chosen; and ( b ) a point was also given for the correct change placed in the right p o s i t i o n . Thus (a) gives the S credit for extending the change i n the appropriate d i r e c t i o n , and (b) gives S credit for c o r r e c t l y evaluating the degree of change. For example, in the cases where the S might place card 4 or 5 in position 3 , he would get credit for choosing one right card that indicated changes in the appropriate d i r e c t i o n , but none for the position. Scores (a) and (b) were added. The f i n a l d i s t r i b u t i o n of points per set was as follows: Sets 2 and 3 = 6 pts. per set, 12 pts. together ( 3 x 1 (one change per position) + 3 ( 1 point per correct placement = 6 pts. per set) Sets 4 and 5 = 9 pts. each [ ( 3 x 2 ) + 3 ] , 1 8 pts. together Sets 6 and 7 = 12 pts. each [ ( 3 x 3 ) + 3 ] , 24 pts. together Sets 8 and 9 = 15 pts. each [ ( 4 x 3 ) + 3 ] , 30 pts. together 76 Sets 10 and 11 = 13 pts. each [(5*3)+3], 36 pts. together The maximum t o t a l score possible was 120 pts. The FORHIN t o t a l score then consisted of the sum of the points obtained in a l l of the sets on which the S succeeded, as well as the p a r t i a l points on the la s t two sots on which the task was discontinued. The FORHIN score could also be translated simply in terms of a score that indicated the number of changing dimensions that the S could co-ordinate. This was done by tran s l a t i n g each S's FORHIN scoro in terms of how far the score indicated that the S had mastered a certain l e v e l of d i f f i c u l t y . The cutting points for the various levels of d i f f i c u l t y along the FORHIN scale were: | 0-X8j 19-42| 43-72| 73-108] 109-120 | 1 2 3 4 5 Thus, a S who obtained less than 18 pts., i . e . , 12 pts., was considered to be able to handle less than one changing dimension and his score was changed i n terms of the f r a c t i o n , out of the t o t a l maxi ium possible for that l e v e l of d i f f i c u l t y , that the score indicated ( i . e . , 12 pts. equal .6, 29 pts. equal 1.5, 49 pts. equal 2.25, etc.) . The range of points for each level of d i f f i c u l t y ( i . e . , pts. from level 1 to level 2) was obtained by adding to the t o t a l maximum number of points in the FORHIN sets corresponding to each of the levels of d i f f i c u l t y , the maximum number of points the S could obtain in the next level at which the S no longer could co-ordinate a l l the dimensions. For example, tho t o t a l maximum for sets #2 and #3 was 12 pts. Six points were added to this t o t a l since a S could s t i l l obtain three points i n set #4 and three points in #5 even i f he f a i l e d in these two sets. 77 Level 1 0 to 12 + 6 = 0-18 Level 2 19 to 30 + 12 = 19-42 Level 3 43 to 54 + 18 = 43-72 Level 4 73 to 84 + 24 = 73-108 Level 5 109 to 120 + 0 = 109-120 M-Operator taslc (MOPER) This task was modelled after the one devised by Pascual-Leone (1970) in order to test a mathematical model which describes the t r a n s i t i o n rule from one stage of development to another. Pascual-Leone (1970) intended to measure the child's central computing space -- H-Operator, which is defined in terms of the "number" of d i f f e r e n t chunks of information on which the S can simultane-ously operate. In the present study, this construct was operation-al i z e d by means of a task which, just as Pascual-Leone's task, required the S to make as many responses as possible to multiple or compound sets of s t i m u l i (varying in complexity from 2 to 8 stimuli per compound stimuli) which were presented at b r i e f inter-vals from 5 to 8 seconds. The differences between this task and Pascual-Leone's were: (1) the stimuli used were a l l within one dimension (drawn shapes), while Pascual-Leone used various dimensions; (2) the motor responses that the S_ had to make i n -volved a smaller spatio-temporal range, i . e . , a l l responses were signs made with fingers, while Pascual-Leone's task involved motor responses as diverse as raise hand, kick basket, clap hands, etc.; (3) the complexity of the task, as defined by the E, was the same for a l l S s, i . e . , a l l Ss xvere presented the same set of stimuli cards, while Pascual-Leone varied the testing set ac-73 cording to the theoretical requirements of the d i f f e r e n t develop-mental levels concerning the size of the M-Qperator; (4) the set of t o t a l stimuli presented was much smaller than that used by Pascual-Leone. In this task, for both Grade 1 and Grade 2 , a t o t a l Of 3 5 testing compound stimuli cards were presented while Pascual-Leone used 6 2 and 74 cards respectively for the Ss i n his sample of equivalent ages. Materials -• The materials consisted of three d i f f e r e n t sets of cards, two tr a i n i n g sets and one testing set. The tr a i n i n g sets consisted of simple s t i m u l i , while the testing set consisted of compound s t i m u l i . The f i r s t t r a i n i n g set (Set 1 ) contained sixteen 1 5 . 2 x 1 0 . 2 cm cards divided into four equal ( 7 . 6 X 5 . 1 cm) c e l l s . For half (8) of the cards in three of the four c e l l s , there were posit i v e instances (one per c e l l ) of one of the eight signs to which the S was supposed to learn a motor response, and one negative instance of that same stimulus sign. For the remain-ing h a l f , there were three negative instances of one of the signs to be iearned and one positive instance (See Fig. 6 b ) . The second t r a i n i n g set (Set 2) consisted of sixteen 1 5 . 2 x 1 0 . 2 cm cards also divided into four equal c e l l s , except that each of three of the Cells had a dif f e r e n t negative instance of the signs to be learned and the fourth c e l l had a po s i t i v e instance (See Fig. 6c). The testing set (Set 3) consisted of t h i r t y - f i v e 1 0 . 1 7 . 6 cm cards, in which there were 5 cards of each of the seven possible kinds of compound stimuli ( 2 , 3, 4, 5 , 6 , 7 , and 8 compound s t i m u l i ) . The compound stimuli cards were made up only of positive instances of the stimulus signs (See Fig. 6 d ) . The stimuli were not nested within each other to make a single com-STIMULI '. MOTOR .RESPONSE . STIMULI MOTOR RESPONSE INDEX AND MIDDLE (I) FINGERS CROSSED '($) I^ Sg? P A L M FACING ALL FINGERS EXTENDED, (3) win INDEX AND: MIDDLE . FINGERS SEPARATED KNOCK ON TABLE CLOSED FIST, FINGERS FACING OUTWARDS M (?) (8) Maxima :;:r:/ : T ' OUTWARDS I- , ' • • ' DENT THUMB i . INDEX AND MIDDLE FINGERS TOGETHER TOUCHING NOSE INDEX UPWARDS "'•\. JX'/. r ir;;; (b) - 'I • ' • ' y : 1 * ^ (c) (d) Fig. 6. M-Operator task. (a) Stimuli'and response units, (b) Sample cards of t r a i n i n g set 1 , (c) Sample cards of t r a i n i n g ,set 2 , (d) Sample cards of compound st i m u l i cards. :. •<. • \• • 80 pound unit (as were Pascual-Leone's compound stimuli) but rather were spaced in random arrangements close to each other thus attempting to reduce the "scanning" aspect which the nested compound stimuli controls for. The signs were designed in such a way that t h e i r p i c t o r i a l representation would somewhat resemble the motor response which corresponded to i t (See Fig, 6a). This was done to f a c i l i t a t e the learning of the S-R units. For example, Stimuli No. 4 (Si») was designed p i c t o r i a l l y to resemble the response (closed f i s t w i t h open side turned outward) connected to i t . The only exception was S 3, whose motor response was a knock on the table. Procedure - The assessment involved two parts: (a) t r a i n -ing session and (b) testing session. The task was started by t e l l i n g S that this was going to be some sort of "spy game" in w h i c h E was going to send S_ secret messages. However, before d o i n g that, E was going to f i r s t teach him a sign language. Set 1 xtfas started. E introduced the cards by saying, "Here is one sign (E pointed to one of the three i d e n t i c a l p o s i t i v e instances); so whenever you see a sign l i k e t h i s , you are sup-posed to l e t me know that you have received the message by doing t h i s . " (E showed the motor response.) "Let :s see i f you can do i t . And i f you see this one (E pointed to another of the posi-tive instances), what would you do?" E gave feedback as to the accuracy of the S's response. "But i f you see one l i k e t h i s (E pointed to negative instance), you are not supposed to do any-thing." E then shoTved the matching card that had three negative instances and one p o s i t i v e instance, and repeated the same pro-81 cedure, each time giving S feedback as to the accuracy of his responses, either by correcting him and giving him the correct response, or by r e i n f o r c i n g him p o s i t i v e l y when he made the cor-rect response. When E finished with Set. 1, she said, "Let's see i f you remembered a l l of them." Then she started Set 2 (3 d i f f e r e n t negative instances and only one p o s i t i v e instance) and again gave S feedback on each instance. The S was considered to have learned the task when he was able to c o r r e c t l y solve the 16 cards of Set 2 without error. (In no instance was i t neces-sary to run Set 2 more than three times, as the majority of the Ss met the c r i t e r i o n by the second run.) When S attained the correct c r i t e r i o n , E said, "Now I w i l l send you the messages and this time there w i l l be more than one sign per card. Hoxvever, I w i l l only show each card for a short l i t t l e while, and while I show i t , you have to try to remember as many signs i n the card as you can. When I put the card dov/n, you can send the signs back to me. Since I don't know how many signs there are i n the cards, I w i l l wait u n t i l you t e l l me that you have- finished sending signs, before I send you the next message. Remember you can only start sending signs after I put the card dovm." Set 3 was started and each card was presented manually for about 5 to 8 seconds and a l l the responses e l i c i t e d were manually recorded. The various cards with d i f f e r e n t combinations of compound stim-u l i i n Set 3 x^ere a l l i n a random order. However, the f i n a l order of the 35 cards was the same for a l l Ss. Scoring of M-Operator task The number of correct responses to each compound stimuli card were t a l l i e d and from them a frequency d i s t r i b u t i o n was obtained 8 2 of the number of appropriate responses that the S made per sti m u l i card. Thus for each S i t was calculated how many times he made 1, 2, 3, 4, 5, 6, 7, and 8 appropriate responses per card. The data from compound stimuli cards with eight stimuli \rcre elimi-nated because when some Ss saw these cards, even though they might not remember many stimuli from i t , they would tend to con-tinue sending signals simply because they knew that i n that card "there were many." So, the t o t a l score was compiled from only 30 compound stimuli cards. The frequencies of the responses for each type of compound stimuli (2, 3, 4, etc.) were divided by the number of times the S had an opportunity to make such number of responses. Thus the response frequencies 1 (Ri) and 2 (R2) were divided by 3 0 because the S had 30 instances to make just one or two responses; response frequency 3 ( R 3 ) was divided by 25; response frequency 4 (RO, by 20; response frequency 5 ( R 5 ) , by 15; response frequency 6 (Re), by 10; and response frequency 7 ( R 7 ) , by 5. In this manner, each S obtained a frequency dis-t r i b u t i o n of his various response classes (Ri, R2, R 3 , etc.) i n proportion to the number of chances he had to make such a number of responses for any one card. The S's MOPER score was the type of response (or frequency class) with the highest proportion, plus a fr a c t i o n (the proportion) i n the next highest frequency c l a s s . For example, i f a S obtained the following proportions: Ri = .00, R2 = .23, R3 = . 4 0 , = .55, R5 = .13, then he was given a MOPER score of 4.1. In the cases where the S obtained two consecutive equal proportions, he was given the score of the mean frequency between the two consecutive frequency classes, i ; e . , i f the S obtained the following proportions, Ri = . 0 0 , R2 = .44, R3 83 = . 4 4 , R„ - .25, R5 = .10, then he would get a MOPER score of 2.5. Summary of the measures Foresight-Hindsight (FORHIN), range possible 0-120 and 0-5 M-Operator (MOPER), range possible 0-7 Mu l t i p l i c a t i o n of Classes (MC) Operative (0) measures: Reverse (MCREVE), range possible 0-162 Total number of correct dimensions (MCDIM(O)), range possible 0-6 Total number of m u l t i p l i c a t i v e combinations (MCMUL(O)), range possible 0-9 Anticipation (MCANT), range possible 1-6 F i l l - i n (MCFILL), range possible 1-18 Figurative (F) measures: Reproduction or Copy (MCCOPY), range possible 0-162 MCDIh(F) , range possible 0-6 MCMUL(F), range possible 0-9 Mu l t i p l i c a t i o n of Relations (MR) Operative (0) measures: Reverse (MRREVE), range possible 0-162 MRDIM(O), range possible 0-6 MRMUL(O), range possible 0-9 MRANT, range possible 1-6 MRFILL, range possible 1-18 Figurative (F) measures: MRCOPY, range possible 0-162 MRDIM(F), range possible 0-6 MRMUL(F), range possible 0-9 8 4 Seriations (SER) Operative measures: Reverse (SEREVE), range possible 0-108 Total number of dimensions c o r r e c t l y solved (SERDIM(O)), range possible 0-6 Figurative (F) measures: Reproduction or Copy (SERCOPY), range possible 0-108 SERDIM(F), range possible 0-6 T r a n s i t i v i t y (TRAN), range possible 0-1 35 CHAPTER IV Results Relationship between Foresight-Hindsight (FORHIN) scores and the  scores on the l o g i c a l tasks FOREHIN and Operative measures - As indicated in the Method section, Operative measures refer to the Reverse (REVE) scores in the l o g i c a l tasks as well as to the F i l l - i n (FILL) and Antic-ipatory (ANT) scores. As shown in Table 1, for both Grade 1 (Gl) and Grade 2 (G2), almost a l l of the Operative measures correlated very highly with the FORHIN scores. They ranged from r=.41 with MRFILL to r=.85 with MCMUL(0). The exceptions were MRANT, MCFILL, and MRFILL scores for G2 Ss which did not s i g n i f i c a n t l y correlate with FORHIN. The lack of relationship between FORHIN and MRANT, for example, was mainly due to the fact that there was not very much v a r i a t i o n among the MRANT scores (See Table 7) and thus no si g n i f i c a n t correlations were possible. The sama was true for the F i l l - i n scores of <J2 in which, as shown in Table 6, 21/32 of the 32 Ss obtained scores in both MC and MR greater than 2/3 of the t o t a l possible (more than 12 pts. out of 18 pts. possible). Thus the scores were clustered towards one end of the d i s t r i b u t i o n . Of the Operative measures, the ones that had the highest correlations with FORHIN were the Reverse scores (as opposed to the Anticipatory or F i l l - i n scores), ranging from r=.66 with MRREVE to r=.85 with MCMUL(0). Also, whether or not the TRAN task was passed was also s i g n i f i c a n t l y related to the FORHIN score. A x 2 test was carried out between the frequency d i s t r i b u -t i o n of the Gl and G2 FORHIN scores above and below the FORHIN median of both groups combined (30 p t s . ) , and the Pass-Fail f r e -quencies for each grade in the TRAN task. These x 2 tests indicated TABLE 1 CORRELATIONS BETWEEN FORHIN SCORES AND OPERATIVE(O) AND FIGURATIVE(F) ' MEASURES FOR Gl AND G2 FORHIN/ MC MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT MOPER 0 Gl F . 82'*** .49** .84*** .70*** .75*** .55** .82*** .48* .82*** .38* .82*** . .68*** . gi *** 58*** .80*** .60*** .50** .41* .44* .61*** .44* 0 G2 F .80*** . 37* .66*** .29 .68*** .55*** .80*** .40* . 85*** .48** .75*** . .21 21 .70*** .07 .51** . 31 .45* .26 .59*** TABLE 2 CORRELATIONS BETWEEN MOPER SCORES AND OPERATIVE(0) AND FIGURATIVECF) : MEASURES . FOR Gl AND G2 MOPER/ MC MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN 0 Gl F .55** .27 . 49** .44* .53** . 31 . 54** .15 .47** .13 .46* .42* 57*** 46* .41* .35* .27 .22 .20 .25 .44* 0 G2 F .63*** .27 .64*** . 37* .64*** .23 .63*** .24 7 2 * * * . 30 .71*** . .27 59*** 18 .64*** .25 .23 . 48** .31 .39* . 59*** * p<.05 ** p<.01 *** p<.001 oo TABLE 3 RELATIONSHIP BETWEEN THE TRAN SCORES AND THE SCORES BELOW AND ABOVE THE MEDIANS OF THE FORHIN, MOPER, MCMUL(O), MRMUL(O), AND SERDIM(O) SCORES FOR Gl AND G2 Gl G2 FORHIN <30 >30. <30 >30 PASS 0 TRAN FAIL 16 12 _4 — 12 20 3 11 14 = _3 = 17 15 16 16 15 17 X2=22. 53, p< . 001 X 2 = 12 .24, p< .001 MOPER <3.1 >3. 1 <3.1 >3.1 PASS 0 TRAN FAIL 18 12 _2 12 20 2 10 15 = _5 = 17 15 18 14 12 20 x2=24. 68, 001 x 2=io .24, p< .001 MCMUL(0) <2 >2 <2 >2 PASS 1 TRAN FAIL 18 11 _2 = 12 20 1 12 16 = _3 = 17 15 19 13 13 19 X2=21. 34, P<- 001 X 2 = 17 .95, p< .001 MRMUL(F) <2_ >2 <2 >2 PASS 2 TRAN FAIL 17 10 _3 = 12 20 3 1_2 14 = _3 = 17 15 19 13 15 x2=14. 69 , P<- 001 X 2 = 12 .44, p< .001 SERDIM(O) <2_ >2 <2 >2 PASS 2 TRAN FAIL 19 10 _1 = 12 20 4 11 13 = _4 = 17 15 21 11 15 17 X2=20. 40, p<. 001 X 2 = 7. 93, p<. 005 8 8 a s i g n i f i c a n t relationship between these two scores (x2=22.53, df=l, p<.0001 for Gl; and x 2 =15.20, df=l, p<.001. See Table 3). The FORHIN median for both groups combined was the same as for each i n d i v i d u a l group. However, the FORHIN scores were combined in the f i r s t place because i f a minimum level of attainment i n FORHIN was required to atta i n a minimum level of opera t i v i t y in the l o g i c a l tasks, this minimum requirement should presumably be the same for both Gl and G 2 i f indeed these two groups represented a developmental continuum. FORHIN and Figurative measures - Table 1 also shows that some of the Figurative measures were s i g n i f i c a n t l y correlated with the FORHIN scores. However, these correlations are not as high as the ones between FORHIN and the Operative measures; they range from r=.37 with the MCCOPYs of G2 to r=.70 with FUCOPY of Gl. In f a c t , most of the correlations of FORHIN with the Fig-urative scores were s i g n i f i c a n t l y lower than the correlations between FORHIH and the corresponding Operative scores. This was indicated by Hotelling t-tests of difference between c o e f f i c i e n t s that are correlated. (For the present samples, any difference between two correlation c o e f f i c i e n t s equal to, or greater than, 14 was s i g n i f i c a n t with a p<.01.) For Gl a l l the Figurative measures were s i g n i f i c a n t l y correlated with FORHIN, while for G2 the Figurative scores MRCOPY, MRDIM(F) and MRMUL(F) did not s i g n i f i c a n t l y correlate with FORHIN. Relationship of MOPER to Figurative and Operative measures MOPER and Operative measures - Table 2 indicates that for both grades the MOPER scores were highly related to a l l the Reverse 89 scores (as opposed to the F i l l - i n or Anticipatory scores) of the MC, MR, and SER tasks (range, r=.53 with SER to r=.72 with MCMUL). With the exception of MRFILL and the MRANT correlations for G2, which had s i g n i f i c a n t correlations with MOPER (r=.48, and r=.39, res p e c t i v e l y ) , a l l other F i l l - i n and Anticipatory scores were not correlated with MOPER. The TRAN p a s s - f a i l score was also highly related to the MOPER score. A x 2 test was computed between the frequencies in MOPER. scores below and above the 3.1 pt. MOPER median and the TRAN p a s s - f a i l frequencies. This x 2 indicated a s i g n i f i c a n t relationship for both grades (x2=20.08, df=1, p<.0001, for Gl; and x2=9.41, df=l, p<.01s for G2. See Table 3). The MOPER scores for both groups xvere combined to obtain a common median (3.1 p t s . ) . This was done based on the assumption that i f indeed a minimum l e v e l of attainment in the MOPER scores was required to atta i n a minimum l e v e l of opera t i v i t y in the l o g i c a l tasks, this minimum l e v e l should be the same for both groups i f the two groups i n fact represented a developmental continuum. MOPER and Figurative measures - Table 2 also shows that the correlations between MOPER and the Figurative scores were almost a l l not s i g n i f i c a n t . The exceptions were the MR scores of Gl, which shox^ed s i g n i f i c a n t correlations with MOPER (MRCOPY, r=.44; MRDIM(F), r=.42; and MRMUL(F), r=.46), and MRCOPY score of G2 (r=.37). MOPER and FORHIN - The corr e l a t i o n of MOPER to FORHIN was included in Table 2, i n the row of correlations of the Operative measures. The correlations are r=.44 ( s i g n i f i c a n t with a p<.01) for Gl, and r=.59 (p<.001) for G2. 90 Consistency between the scores of the l o g i c a l tasks  Relationship between Operative measures Reverse Scores - Table 4 shows the int e r c o r r e l a t i o n s of the t o t a l Operative Reverse scores (1, 2, 3) and the Figurative or Reproduction scores (4, 5, 6). Common to both Gl and G2 were the high in t e r c o r r e l a t i o n s between the Operative Reverse scores of the l o g i c a l tasks. For Gl, MCREVE/MRREVE r=.86, MC RE VS / S E ME VE r=.72, and MR RE VE / S P. RE VE r=.75. For G2 , MCREVE/MRREVE r=.75, MCREVE/ SEREVE r=.70, and MRREVE/SE PE VE r=.66. The int e r c o r r e l a t i o n s between the measures derived from the Reverse scores (MCDIM(O) , MCMUL(0) , etc.) are presented i n Table 5 (Operative measures are 1 to 5) , which shows that for Gl a l l these intercorrelations were s i g n i f i c a n t (range, MCMUL(0)/MCDIM(O), r=.64 to HRDIM(0)/MRMUL(0), r=.91). A sim i l a r pattern of inter-correlations between the measures derived from the operative (Reverse) scores was also obtained for G2 (range, MRMUL(O)/ SERDIM(O) r=.75 to MRDI.M(O) /MP.MUL (0) r=.95). These high inter-correlations necessarily follow from the high correlations between a l l the t o t a l Operative Reverse scores of the l o g i c a l tasks and indicate that in both groups Ss performed i n a consis-tent manner in a l l the Reverse sections of the l o g i c a l tasks. An important finding was that Ss i n G2 showed a s i g n i f i c a n t l y higher c o r r e l a t i o n between the MCDIU(O) scores and the MCMUL(0) scores than Gl Ss. For Gl MCDIM(O)/MCMUL(0) r=.64 and for G2 r=.95. The difference between these two correlations was s i g n i f -icant (Fisher's Z=4.3, p .001). This difference in the consistency of Gl and G2 between the t o t a l number of dimensions solved (MCDIM(O)) and the number of m u l t i p l i c a t i v e combinations possible TABLE INTERCORRELATIONS BETWEEN OPERATIVE (1,2,3) AND FIGURATIVE (4,5,6) MEASURES FOR Gl AND G2 (1) MCREVE (2) MRREVE (3) SEREVE (4) MCCOPY (5) MRCOPY (6) SERCOPY Gl (1) .86*** .72*** .56*** .66*** .53** (2) .75*** . 7 3 * * * .73*** .63*** (3) .60*** .56*** .43** (4) .43** .56** (5) .63*** (6) CD (2) MCREVE MRREVE (3) SEREVE (4) MCCOPY (5) MRCOPY (6) SERCOPY CD 75 * * * . 70*** .38* .34 .61*** (2) .66*** . 46** .44* .50** (3) .21 .36* .25 (4) .61*** .29 (5) .54** G2 (6) * p<.05 ** p<.01 *** p<.001 TABLE 5 INTERCORRELATIONS BETWEEN MEASURES DERIVED FROM OPERATIVE MEASURES (1-5) AND THOSE DERIVED FROM FIGURATIVE MEASURES (6-10) Gl (1) MCDIM(O) (2) MCMUL(0) (3) MRDIM(O) (4) MRMUL(0) (5) SERDIM(O) (6) MCDIM(F) (7) MCMUL(F) (8) MRDIM(F) (9) MRMUL(F) (10) SERDIM(F) (1) .64*** .82*** . 86*** .73*** .60*** .49** 64*** . 56*** . 54** (2) .85*** .86*** y 7 * * * .62*** .54** #64 *** .59*** .50** (3) _ g i * * * . 76*** .63*** . 54** ^ 72 * * * .61*** .54** (4) .78*** . 66*** . 59*** .76*** . 70*** . 66*** (5) .59*** .47** .54** .47** .68*** (6) .88*** .46** .45** .59*** (7) . 56*** .56*** .20 (8) .87*** .46** (9) .30 (10) G2 (1) MCDIM(O) (2) MCMUL(0) (3) MRDIM(O) (4) MRMUL(0) (5) SERDIM(O) (6) MCDIM(F) (7) MCMUL(F) (8) MRDIM(F) (9) MRMUL(F) (10) SERDIM(F) (1) .95*** .78*** _ 7 7* * * .69*** .41* .52** .35 .33 .57*** (2) .85*** . 84*** .71*** . 48** .56** . 33 .28 . 55*** (3) .88*** . 76*** . 39* .48** . 38* .25 .49** (4) 7 5 * * * .43* .49** .37* .25 .60*** (5) .20 .29 .26 .22 .56*** (6) .93*** .58*** .53** .16 (7) .48** .51** .27 (8) .75*** . 10 (9) • .20 (10) * p<.05 ** p<.01 * * * r x . n m TABLE 6 INTERCORRELATIONS BETWEEN THE FILL-IN AND ANTICIPATORY MEASURES AND THE OPERATIVE SCORES FOR Gl AND G2 Gl (1) MCFILL (2) MRFILL (3) MCANT (4) MRANT (5) MCREVE (6) MCDIM(O) (7) MCMUL(0) (8) MRREVE (9) MRDIM(O) (10) MRMUL(O) (ID SEREVE (12) SERDIM(O) CD .39* . 35* . 36* .55*** .52** .58*** .68*** . 72*** .70*** .53** .50** (2) . 12 . 31 . 39* .41* .47* .52** .46** . 5 7 * * * .47** .47** (3) .42** .60*** .56*** .43** .53** . 53** .48** .46** .44* (4) .68*** .67*** .66*** . 66*** .65*** .61*** .46** .46** G2 (1) (2) MCFILL MRFILL (3) MCANT (4) MRANT (5) MCREVE (6) MCDIM(O) (7) MCMUL(0) (8) MRREVE (9) MRDIM(O) (10) MRMUL(O) (11) SEREVE (12) SERDIM(O) (1) .33 .08 .19 . 26 .25 .28 .23 .25 .21 .33 .26 (2) .19 . 38* . 39* .43* . 43* .66*** .61*** .42* . 46** .47** (3) .19 . 64*** .60*** . 56*** .44* .47** .45** .35* . 33 (4) .44** .45** .47** . 50** .43* . 30 .54** . 55*** * p<.05 ** p<.01 *** p<.001 94 (MCMUL(O)) within that t o t a l indicates that, even when Gl Ss remembered c o r r e c t l y many dimensions from the matrix, those dimensions did not necessarily involve m u l t i p l i c a t i v e combina-tions. In the MR reverse scores, both groups had high i n t e r -c o r r e l a t i o n s between the MRDIM(0) and MRMUL(O) scores. F i l l - i n scores - As shown by Table 6, the i n t e r c o r r e l a t i o n s between the MR and MC F i l l - i n scores to other Operative measures varied with each grade as well as with each type of task (MC or MR). For Gl the only non-significant correlations were between MRFILL and both of the anticipatory measures. A l l other c o r r e l a -tions were s i g n i f i c a n t and ranged between r=.35 (MCFILL/MCANT) to r=.72 (MCFILL/MRDIM(0)). For G2, MCFILL did not correlate s i g n i f i c a n t l y with any other Operative measure while MRFILL correlated s i g n i f i c a n t l y with a l l other Operative measures except with MCANT and MRFILL (range r=.38, MRFILL/MRANT, to 4=.61, MRFILL/MRDIM(0)). A possible reason for the lack of s i g n i f i c a n c e between the MCFILL to other Operative measures could be p a r t l y the r e s u l t that 2/3 of the Ss obtained scores above 2/3 of the t o t a l pos-s i b l e (more than 12 pts. out of 18 pts. p o s s i b l e ) , which means that the scores clustered at one end of the d i s t r i b u t i o n . See Tabic 7 below. TABLE 7 PROPORTION OF Ss OBTAINING MCFILL AND MRFILL SCORES GREATER THAN 2/3 OF THE MAXIMUM POSSIBLE (12/18) Gl G2 <12 >12 <12 >12 MCFILL 18/32 14/32 11/32 21/32 MRFILL 14/32 18/32 11/32 21/32 95 However, factors other than s t a t i s t i c a l ones must have pro-duced the present results since MRFILL with a simi l a r d i s t r i b u -tion of scores did correlate s i g n i f i c a n t l y with most Operative measures. The consistency between the two F i l l - i n scores was low for both groups as shown in the cor r e l a t i o n between MCFILL and MRFILL. For G2 this c o r r e l a t i o n was not s i g n i f i c a n t , while for Gl MCFILL/ MRFILL r=.39 was s i g n i f i c a n t at the .05 l e v e l . A n t i c i p a t i o n scores - The intercorrelations of these measures to other Operative scores varied for each grade. As shown in Table 6, for Gl MRANT and MCANT correlated s i g n i f i c a n t l y with a l l other Operative measures, except with MRFILL (range r=.35, MCFILL/MCANT, to r*.67, MRANT/MCDIM(0)). For G2, the pattern i s less consistent. The MCANT score was not s i g n i f i c a n t l y correlated with MCFILL or MRFILL, or with SERDIM(O), but was s i g n i f i c a n t l y correlated with a l l the other scores. MRANT did not correlate s i g n i f i c a n t l y with MCFILL and with MRMUL. The consistency between the two Anticipatory scores also varied for both grades. For Gl, MCANT/MRANT r=.42, while for G2, MCANT/MRANT r=.19. The inconsistency between the G2 intercorrelations i s pa r t l y ex-plained by the fact that for this group the d i s t r i b u t i o n of the MRANT scores was clustered at the high end of the d i s t r i b u t i o n . From a possible range of scores from 1 to 6 pts., 25 out of 32 Ss in G2 obtained scores of 4 and above, while for Gl there were 16/32 Ss with scores higher than 4 pts. See Table 8 on the next page. The means of the d i s t r i b u t i o n of the MRANT for Gl and G2 were also s i g n i f i c a n t l y d i f f e r e n t (t=2.97, df=62, p<.004. See Table 12). 96 There was thus a c e i l i n g e f f e c t in the MRANT scores for G2 that rendered i t s correlations with a l l other measures not s i g n i f i c a n t . TABLE 8 PROPORTION OF Ss IN GRADE 1 AND GRADE 2 OBTAINING SCORES GREATER THAN OR EQUAL TO 4 IN THE MC AND MR ANTICIPATION TASKS MCANT>4 MRANT>4 Gl 20/32 16/32 G2 20/32 25/32 T r a n s i t i v i t y scores - Chi-square tests were carried out between the TRAN p a s s - f a i l frequency scores and the scores below and above the MCMUL(0), MRMUL(O), and SERDIM(O) medians. These derived measures of the Reverse scores were used instead of the to t a l Reverse scores because they indicated more succinctly the Ss' attained level of operativity in each of these l o g i c a l tasks which made them easier to compare to the dichotomous TRAN scores. The median for each of the tasks turned out to be 2 points. For each task (MC, MR and SER), the scores for both groups were com-bined to obtain a common median. This was done for the same reason that the FORHIN and MOPER scores were combined. The x 2 tests yielded s i g n i f i c a n t results (See Table 3). For Gl, MCMUL (0) /TRAN x 2 = =21.34, df=1, p<.001} M.RMUL ( 0 ) / T RAN X 2 = 14.69, df=l, p<.001; and SERDIM(O)/TRAN x2=20.40, df=l, p<.001. For G2 MCMUL(0)/TRAN x2=17.95, df=l, p<.001; MRMUL(0)/TRAN x2=12.44, df=l, p<.001; and SERDIM(O)/TRAN x 2 =7.93, df=l, p<.005. Thus, these tests indicated that the Ss' performance on the TRAN task was highly related to t h e i r performance on the Operative sections of the l o g i c a l tasks. 97 Relationship between Figurative measures Reproduction or Copy scores - As shown i n Table 4, the inter-correlations between the Figurative scores (4 to 6 ) for both groups were s i g n i f i c a n t with the exception of MCCOPY/SERCOPY corr e l a t i o n for G2 which was not s i g n i f i c a n t . However, for both groups, these inte r c o r r e l a t i o n s were s i g n i f i c a n t l y lower than the i n t e r c o r r e l a -tions among the corresponding Operative measures (1 to 3) as i t was shown by Hotelling t-tests of difference between correlated c o e f f i c i e n t s (any difference between within-groUp correlations greater than or equal to .14 was s i g n i f i c a n t with p<.01). The exceptions were the SER Operative and Figurative i n t e r c o r r e l a -tions 'whose differences were not s i g n i f i c a n t l y d i f f e r e n t i n either Gl or G2. Table 5 shows the intercorrelations betx^een the scores derived from the Figurative measures (6 to 10), which confirm the findings of Table 4 that the intercorrelations between the Figurative measures are s i g n i f i c a n t l y lower than the inter c o r r e l a t i o n s between the corresponding Operative measures. Relationship between Figurative and Operative measures Reverse and Copy scores - Table 4 indicates that the inter-correlations between Figurative and Operative measures were a l l s i g n i f i c a n t for Gl (range SEREVE/SERCOPY r=.43 to MCCOPY/MRREVE r=.73). For G2 the intercorrelations were somewhat lower than for Gl (range MCREVE/MCCOPY r=.38 to MCREVE/SERCOPY r=.61) and 4/9 correlations were not s i g n i f i c a n t . The greater number of s i g n i f i c a n t i n t e r c o r r e l a t i o n s between Figurative and Operative measures for Gl would indicate that Gl Ss were performing more s i m i l a r l y across the Figurative and Operative sections of the l o g i c a l tasks than G2 Ss. 98 Hotelling t-tests of significance between correlated means were carried out within each group betxveen the corresponding Fig-urative and Operative scores of MC and MR tasks ( i . e . , the MCREVE and MCCOPY scores for Gl were compared) which indicated that xvithin each group these scores did not d i f f e r s i g n i f i c a n t l y from each other. However, the differences between Figurative (F) and Operative (0) scores ( i . e . , MCREVE-MCCOPY) were correlated with other Figurative and Operative measures (See Table 9). The pat-tern that emerged indicated, that these differences were s i g n i f -i c a n t l y correlated most frequently with Operative measures, p a r t i c u l a r l y for Gl, and very infrequently with Figurative meas-ures. In the cases where the Operative minus Figurative d i f f e r -ences correlated with the Operative measures, i t indicated which of the measures, Figurative or Operative, was higher within a s p e c i f i c l o g i c a l task. The greatest differences between Operative and Figurative scores were for G2, i n which group the Operative scores increased s i g n i f i c a n t l y higher over the Figurative scores than for Gl as is shown by the high negative correlations between Figurative and Operative scores (MCREVE-MCCOPY/MCREVE r = - . 5 7 , MRREVE-MRCOPY/MRREVE r=-.59, S E P.RE VE - S h RC 0 PY / S E RRE VE r=-.47), while for Gl these correlations were a l l not s i g n i f i c a n t . The Ss' increase or decrease between Figurative to Operative scores was then related to t h e i r l e v e l of ope r a t i v i t y . This conclusion is further confirmed by the contingency table below (Table 10) between the frequency d i s t r i b u t i o n of Ss whose Operative scores increased or decreased from i t s Figurative score and the frequency d i s t r i b u t i o n of Ss whose scores f e l l below or above the FORHIN median of 30 points. TABLE 9 INTERCORRELATIONS OF THE DIFFERENCES BETWEEN FIGURATIVE AND OPERATIVE SCORES FOR THE MC, MR AND SER TASKS AND ALL THE FIGURATIVE AND OPERATIVE SCORES FOR Gl AND G2 MC(O)-MC(F)/ MC MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN 0 .71*** .50** .34 .66***.54** .45** .44* .34 .02 .05 .42 .59***.55*** Gl F .19 .41** .26 -.13 -.14 .44* .34 .23 - - -0 .54** .25 .44* .47** .37* .35* .29 .44** -.19 .12 .58***.06 .37* G2 F .57***-.24 .32 -.56***.37* -.29 -.21 -.39* - - . . . MR(O)-MR(F)/ MC MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN 0 .53** .65*** .48** .53** .55** .66***.51** .47** .39* .27 .29 .59***.46** n F .41* -.04 .31 .41* .22 .04 .003 .28 - - . . . O .32 .46** .23 .28 .31 .46** .32 .28 -.30 .23 .61***.22 .31 G2 F -.19 -.59*** .16 -.15 -.01 -.47**-.50** .20 - - -SER(O)-SER(F)/ MC MR SER MCDIM MCMUL MRDIM MRMUL SERDIM MCFILL MRFILL MCANT MRANT FORHIN O .41* .35* .67***.42* .43* .34 .29 .54** .32 .04 .31 .17 .43* G 1 F .34 .17 -.16 .32 .34 .14 .25 -.14 - - -O .06 .13 .44* .10 .13 .25 .12 .42* .10 .31 .07 .19 .11 G2 F -.05 .06 -.47**-.06 -.10 .15 -.06 -.52** - - . . . * p<.05 ** p<.01 *** p<.001 100 TA3LE 10 PROPORTION OF Ss IN Gl AND G2 WHOSE FORHIN SCORE FELL ABOVE OR BELOW THE I IED I AN AND WHOSE MC AND m REVERSE OR OPERATIVE SCORES INCREASED OR DECREASED WITH REFERENCE TO THEIH L CORRESPONDING COPY OR FIGURATIVE SCORE MCCOPY vs. MCREVE MRREVE vs. MRCOPY FORHIN FORHIN <30 >30 <30 >30 Increase 1 13 = 14 2 16 » 18 G l Decrease 15 3 18 14 0 = 14 16 16 16 16 Increase 4 17 = 21 5 17 = 22 G2 Decrease 11 0 = 11 10 0 = 10 15 17 15 17 In conclusion then, the i n t e r c o r r e l a t i o n s between Figurative and Operative measures indicated that Gl Ss had greater consist-ency across t h e i r performances i n the Figurative and Operative sections of the l o g i c a l tasks than G2 Ss. Also, for both grades, the increase or decrease from Figurative to Operative measures was related to the l e v e l of attainment i n a l l other l o g i c a l tasks as well as to the FORHIN scores. F i l l - i n and Anticipatory measures - Table 11 indicates the i n t e r c o r r e l a t i o n s between the F i l l - i n and Anticipatory measures and the Figurative scores for Gl and G2. These results show that the Gl F i l l - i n measures were almost a l l (15/16) s i g n i f i c a n t l y correlated to the Figurative scores (the exception was SERDIM(F)). The correlations ranged from r=.37 in MCFILL/SERCOPY, to r=.79 i n MCFILL/MCCOPY. The G2 F i l l - i n scores, on the other hand, show fewer s i g n i f i c a n t c o rrelations to the Figurative scores (7/16), TABLE 11 INTERCORRELATIONS BETWEEN THE FILL-IN AND ANTICIPATORY MEASURES AND THE FIGURATIVE SCORES FOR Gl AND G2 C D (2) C3) C4) (5) (6) (7) (8) (9) (10) (11) (12) Gl MCFILL MRFILL MCANT MRANT MCCOPY MCDIM(F) MCMUL(F) MRCOPY MRDIM(F) MRMUL(F) SERCOPY SERDIM(F) (1) .39* .35* .36* .79*** .64*** .56*** .54** .53** .51** .37* .33 (2) .12 .31 .49** .51** .46** .44** .46** .44** .59** .55*** (3) .42* .33 .29 .21 .45** .46** .32 .29 .29 (4) .24 .24 .16 .41* .41* .34 .38* .42* C D (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) G2 MCFILL MRFILL MCANT MRANT MRCOPY MCDIM(F) MCMUL(F) MRCOPY MRDIM(F) MRMUL(F) SERCOPY SERDIM(F) ( D .29 .08 .19 .47** .41* .39* .50** .39 * . 32 .22 .23 (2) .19 . 38 .25 .26 .30 . 37* .36* . 34 .15 .08 C3) .19 .00 .03 .14 - .22 - .16 - .11 .27 .29 (4) .36* . 32 .36* .23 .20 . 18 . 34 .32 * p<.05 ** p<.01 *** p<.001 102 and these s i g n i f i c a n t c o rrelations occurred mainly.with the MC Figurative scores. The correlations ranged from r=.36, MRFILL/ MRDIM(F), to r=.50, MCFILL/MRCOPY. The Anticipatory measures for Gl correlated s i g n i f i c a n t l y with only 6/16 of the Figurative measures (MCANT/MRCOPY r=.45, MCANT/MRDIM(F) r=.46, MRANT/MRCOPY r=.41, MRANT/MRDIM(F) r=.41, MRANT/SERCOPY r=.38, and MRAiNT/SERDIM(F) r=.42). For G2 , only one Figurative score s i g n i f i c a n t l y correlated to an Anticipatory score, MRANT/MCMUL(F) r=.36. As in the Reverse measures, Gl appeared to perform more s i m i l a r l y across the F i l l - i n and Antic-ipatory measures than G2. Differences between the two Developmental levels One-way analyses of variance were performed on a l l the scores of Gl and G2 divided on the basis of sex. These indicated that there were no s i g n i f i c a n t differences between the scores when these were dichotomized on the basis of sex. H o t e l l i n g t-tests for correlated means were ca r r i e d out between the mean scores of Gl and G2 i n a l l the tasks. The results are shown i n Table 12 which indicates that Gl and G2 d i f f e r e d s i g n i f i c a n t l y from each other i n t h e i r performances on only the Operative measures and not i n the Figurative ones. The s p e c i f i c measures in which they d i f f e r e d s i g n i f i c a n t l y were the MC and SER Reverse scores and a l l t h e i r derived scores (MCREVE t«2.70, df=62, p<.009; SEREVE t=2.74, df=62, p<.008) MCFILL and MRFILL, and in MOPER scores (t=2.117, df=62, p<.04). Grade 1 and Grade 2 did not d i f f e r s i g n i f i c a n t l y from each other i n the FORHIN scores.** A x 2 test between the p a s s - f a i l frequencies of the TRAN *See Addendum (2). **See Addendum (3). 103 TABLE 12 MEANS, SD's, DIFFERENCES BETWEEN THE MEANS, t-TESTS AND SIGNIFICANCE LEVELS OF THE MEASURES OBTAINED FOR Gl AND G2 Gl G2 1 MEAN SD 1 1 MEAN SD 1 DIFF t P FORHIN 31. 78 16. 41 34. 03 12 . ,67 2, .25 * ,61 . 5 MCREVE 86. ,50 48. ,66 115. ,47 36. ,00 28. .9 2. ,70 .009 MCDIM(O) 2. 47 1. , 86 3. 69 1. , 75 1. , 2 2 , 7 .009 MCMUL(0) 2. 19 2. , 52 3. , 78 3. ,33 1, .6 2. ,2 .03 MCANT 4. 16 1. .61 4. , 53 1. , 41 .38 ,99 . 33 MCCOPY 96. 69 34. ,98 102 . ,91 36. ,92 6, .2 ,69 . 50 MCDIM(F) 2. ,47 1. , 59 3, ,03 1. , 84 ,5 1. , 31 .19 MCMUL(F) 1. ,75 2. , 30 2. , 84 2 , 69 1, .09 1. , 75 .08 MCFILL 11. , 31 3, .85 13, . 78 3. ,03 2, .46 2, .85 .006 MRREVE 102. ,66 50. , 71 115, ,53 37, ,28 12, . 87 1, .16 .25 MRDIM(O) 3. ,19 2. , 36 3, .47 1. ,68 .28 .55 .59 MRMUL(0) 3. ,44 3, .97 2. ,97 3. , 20 .47 . 52 .61 MRANT 3. , 72 1. ,65 4. , 88 1. , 45 1, .16 2. .97 .004 MRCOPY 94. ,91 38. ,44 10 2. , 31 40, ,94 7, .41 . 75 .46 MRDIM(F) 2. ,63 1, .74 3, .19 2, .04 .56 1, .18 .24 MRMUL(F) 1, .88 2. , 51 2 , .66 2, .47 . 78 1, .25 .21 MRFILL 11. , 78 3, . 52 13, .78 3, . 20 2, .00 2, . 38 .02 SEREVE 55. , 75 21, . 75 68, .91 16 , .31 13 .16 2, . 74 . 008 SERDIM(O) 1, . 72 1. , 35 2, .44 1, . 37 .72 2 , .12 . 04 SERCOPY 60, . 56 16 .43 71, .28 17, . 10 10 . 72 2, .56 .02 SERDIM(F) 1, ,91 1, .28 2, . 72 1, . 46 .81 2 . 36 .02 MOPER 2, .93 5, . 24 3, . 24 6 .50 .31 2, . 12 .04 104 task for both groups indicated that Gl and G2 did not d i f f e r s i g n i f i c a n t l y from each other on this measure either. Quantitative Correspondences - A quantitative comparison among the scores obtained for each S on the various tasks was made in terms of the number of "chunks" of information or S-R units ( i . e . , dimensions or stimuli-response units) that the S was able to handle successfully in each task. The FORHIN scores were translated i n terms of the number of dimensions that the S could successfully co-ordinate by the procedure indicated at the end of the Method section. Also, the t o t a l Reverse scores were not used, but rather the comparisons were made with the derived scores from i t , MCDIM(O), MRDIM(O), MCMUL(O), MRMUL(0), and SERDIM(O), since these more succinctly indicated the number of dimensions (and p a r t i c u l a r l y m u l t i p l i c a t i v e combinations) that each S had handled in the respective tasks. This quantitative comparison was only made for the Operative scores since i t was on these scores that the two grades d i f f e r e d s i g n i f i c a n t l y from each other. The various scores for each S are presented in Tables 12 and 13. A x 2 analysis was made between scores below and above the FORHIN median and scores below and above the MOPER median, which indicated that for both Gl and G2 there was a s i g n i f i c a n t relationship between these two scores. Most Ss who obtained a MOPER score greater than or equal to 3.1, obtained a FORHIN score greater than or equal to 1.5 (x 2=24.8, df=l, p<.001, for Gl; and x 2 =21.72, df=l, p<.001, for G2. See Table 14). Also, X 2 tests were made between the scores below and above the MOPER and FORHIN median and the MCREVE and MRREVE scores which had a minimum of one m u l t i p l i c a t i v e combination of dimensions i n each 105 TABLE 13 NUMBER OF UNITS OF INFORMATION THAT Gl Ss WERE ABLE TO HANDLE SUCCESSFULLY IN THE FORHIN, MOPER, MC, MR, AND SER TASKS Ss FORHIN MOPER MCDIM(O) MRDIM(0) MCMUL(0) MRMUL(0) SERDIM(O) 1 1.3 2.3 0 2 0 0 1 2 . 7 2.1 1 2 0 0 1 3 1.7 3.1 5 6 6 9 2 4 1.1 2.4 0 0 0 0 0 5 .7 2.4 1 0 0 0 2 6 2.8 3.4 5 6 6 9 3 7 2.4 3.2 5 4 6 3 3 8 1. 7 3.4 5 5 6 6 3 9 1.0 2.5 1 0 0 0 1 10 2.6 3.2 4 5 4 2 3 11 1.2 3.0 1 2 0 0 0 12 .6 2.5 1 0 0 0 0 13 1.1 2.3 2 2 0 1 1 14 2.7 3.4 5 6 6 9 3 15 1.1 2.1 0 2 0 0 2 16 .8 2.5 0 3 0 0 1 17 1.7 • 3.0 3 5 2 6 2 18 1.9 3.2 5 6 6 9 3 19 2.2 4.0 3 4 6 9 3 20 1.6 2.6 3 3 2 2 1 21 1.2 2.5 2 0 0 0 0 22 2.0 3.4 3 1 6 9 . 3 23 1.6 3.2 4 4 4 4 0 24 1.5 4.0 3 5 3 6 2 25 . 5 2.2 0 3 0 0 0 26 1.2 2.2 1 2 0 0 1 27 1.4 3.0 2 1 0 0 1 28 . 7 2.5 0 0 0 0 1 29 2.2 3.3 4 6 4 9 3 30 1.5 3.3 4 3 4 -7 3 31 • 2.2 3.5 5 6 6 9 3 32 o .9 2.4 1 0 0 0 0 106 TABLE 14 NUMBER OF UNITS OF INFORMATION THAT G2 Ss WERE ABLE TO HANDLE SUCCESSFULLY IN THE FORHIN, MOPER, MC, MR, AND SER TASKS Ss FORHIN MOPER MCDIM(O) MRDIM(0) MCMUL(0) MRMUL(0) SERDIM(O) 1 1.9 4.4 5 4 6 3 3 2 1.4 . 3.0 4 3 4 0 2 3 2.9 4.1 6 6 9 9 4 4 1.3 2.5 3 3 0 3 3 5 1.6 4.1 3 3 2 0 3 6 2.0 3.3 5 4 6 3 4 7 2.4 3.8 6 6 9 9 5 8 1.4 3.2 5 5 6 6 2 9 1.6 4.3 5 4 6 3 3 10 .9 2.5 2 0 0 0 2 11 1.8 3.7 4 4 4 4 4 12 1.4 3.0 0 2 0 0 1 13 1.5 3.3 4 5 4 6 3 14 2.6 3.2 6 6 9 9 4 15 1.2 2.4 2 3 0 0 0 16 1.2 . 3.3 2 3 1 2 3 17 1.7 4.0 6 5 9 6 2 18 1.4 3.0 2 4 1 4 3 19 1.4 2.4 2 0 0 0 0 20 1.7 3.1 4 4 4 4 3 21 2.3 3.8 6 5 9 6 3 22 2 . 3 4.1 6 6 9 9 . 5 23 1.7 3.3 4 3 4 0 2 24 . 7 3.0 1 2 0 0 0 25 1-1 2.3 2 0 0 0 0 26 1.9 3.1 4 3 3 0 1 27 1.4 2 . 2 4 2 2 0 2 28 1.1 2.2 2 2 1 0 1 29 1.3 3.3 1 3 0 0 ' 2 30 2.0 3.4 5 6 4 3 3 31 1.7 4.0 6 5 9 6 3 32 1.4 2.5 2 2 1 0 2 TABLE 15 107 PAIRWISE COMPARISONS OF THE SCORES BELOW AND ABOVE THE MEDIANS OF THE MOPER, FORHIN, SERDIM(O), AND MULTIPLICATIVE COMBINATIONS IN MR AND MC (MUL) SCORES FOR Gl AND G2 Gl G2 MOPER <3.1 i3.1 <3.1 >3.1 FORHIN ^1.5 2 14 = 16 0 17 = 17 <1.5 16 _0 = 16 12 _3 = 15 18 14 12 20 X2=24. 83, p<. 001 X2=21 . 72 > P< .001 MOPER <3.1 >3.1 <3.1 >3.1 MUL >1 2 14 = 16 1 16 = 17 (MC and MR) <1 16 _0 = 16 11 _4 = 15 18 14 12 20 X2=24. 78 , p<. 001 X 2 = 15 . 45, p< .001 MOPER <3.1 - 2.3.1 <3.1 i.3.1 SERDIM(O) >2 3 13 = 16 6 19 = 25 <2 15 _1 = 16 _6 _1 = ' 7 18 14 12 20 X2=18. 28, p<. 001 X 2 = 8. 88, p<. 005 MUL <2 >2 <2 >2 FORHIN >_1. 5 0 16 = 16 3 14 = 17 <1.5 16 _0 = 16 1_2 _3 = 15 16 16 15 17 X2=32. 00 , p<. 001 X2 = 12 . 30 , p< .001 SERDIM(O) <2 >2 <2 >2 FORHIN >1.S 2 14 = 16 1 16 = 17 <1. 5 14 _2_ = 16 6 _9 = 15 16 16 7 25 X2=17. 99, p<. 001 X 2 = 5. 42, p<. 025 108 of the MC and MR Reverse scores. This c r i t e r i o n of at least one mu l t i p l i c a t i v e combination out of the t o t a l number of dimensions correct in each of the MC and MR Reverse scores was thus set as the minimum l e v e l of operativity at which the S was said to pos-sess an understanding of the mu l t i p l i c a t i v e concepts. The x 2 tests indicated that most Ss (33/64, both groups combined) had to have obtained a minimum of 3.1 in the MOPER task and 1.5 in the FORHIN task before they could attain at least one m u l t i p l i -cative combination of dimensions in each task (MOPER and MUL, X2=24.78, df=l, p<.001, for Gl; and x2=15.45, d f = l 5 p<.001, for G2; FORHIN and MUL, x 2 =32.0, df=l, p<.001, for Gl; and x 2 =12.30, df=l, p<.001, for G2). F i n a l l y , x 2 tests were carried out between the scores below and above the FORHIN and MOPER medians and the SERDIM scores greater than or equal to two. Thus, the minimum c r i t e r i o n of operativity for the SER task was defined i n terms of whether or not the S could successfully co-ordinate at least two dimensions out of the t o t a l six. There were again s i g n i f -icant relationships between the levels attained i n the MOPER and FORHIN scores and whether or not the Ss solved a minimum of two correct dimensions in the t o t a l SEREVE score (MOPER and SERDIM(O) X 2 =18.28, df=l, p<.001, for Gl; and x 2 =8.88, df=l, p<.005; FORHIN and SER, x 2 = 1 7 . 9 9 , df=l, p<.001, for Gl; and =5.42, df=l, p<.02). A Kendall c o e f f i c i e n t of concordance W was run also on the data in Tables 12 and 13 in order to determine the corr e l a t i o n between the Ss' rankings across a l l these tasks. The results indicated that there was a high degree of correspondence among the rankings the Ss obtained in a l l these measures. This i n d i -cated that the Ss showed a high degree of consistency across tasks in terms of the number of units of information tl\at th handled successfully in each task (W-.82, x 2~177.87, df=31, p<.001 f o r , G2, and "-.83, x 2 = 179..54? df=3i, p<.001 for Gl) 110 CHAPTER V Discussion Relationship between Automaton and Logical Tasks If indeed, as postulated by Piaget, the foresight and hind-sight a b i l i t i e s are necessary requirements i n the development of l o g i c a l structures, then i t would be expected that performance on the Foresight-Hindsight (FORHIN) task, would be highly related to performance on a l l Operative (as opposed to the Figurative) measures of the l o g i c a l tasks. The present data supported this p r e d i c t i o n . High p o s i t i v e correlations were obtained between the FORHIN and Operative measures as opposed to the s i g n i f i c a n t l y lower, and i n many instances not s i g n i f i c a n t , c o rrelations between FORHIN and the Figurative measures. Only Grade 1 Ss had some s i g n i f i c a n t correlations between FORHIN and some of the Figura-t i v e measures (MRCOPY, MRDIM(F), MRMUL(F)). This group tended to perform more s i m i l a r l y i n both the Operative and Figurative sections of the l o g i c a l tasks than the Grade 2 Ss whose operative scores were higher than t h e i r Figurative scores. It is important to point out again that these high correla-tions were obtained even though the foresight and hindsight a b i l i t i e s were measured in a context and in a manner t o t a l l y separate and d i f f e r e n t from those used to measure the l o g i c a l concepts. These findings could support the view that these se l f - r e g u l a t o r y functions are processes of a general nature which are manifested i n contexts other than those s t r i c t l y dealing with l o g i c a l tasks, as i t could be concluded from Piaget's own studies of foresight and hindsight. An important aspect of the relationships between FORHIN and the Operative scores i s that there appears to be a minimum l e v e l I l l o f a t t a i n m e n t i n the FORHIN s c o r e s n e c e s s a r y t o a c h i e v e the m i n i -mum c r i t e r i o n o f o p e r a t i v i t y i n each o f the l o g i c a l t a s k s (See T a b l e s 13, 14, and 15 ) . T h i s minimum l e v e l o f a t t a i n m e n t i n the FORHIN s c o r e s o f g r e a t e r than o r e q u a l t o 1.5 appears t o be the same f o r a l l l o g i c a l t a s k s , and f o r b o t h age groups. Thus, even when from the p r e s e n t s t u d y i t cannot be c o n c l u d e d t h a t f o r e -s i g h t and h i n d s i g h t have a c a u s a l r o l e i n the a c q u i s i t i o n o f l o g i c o - m a t h e m a t i c a l c o n c e p t s , i t can be s a i d t h a t at l e a s t a minimum l e v e l o f a t t a i n m e n t i n f o r e s i g h t and h i n d s i g h t appears t o be n e c e s s a r y b e f o r e a minimum, l e v e l o f o p e r a t i v i t y i s mani-f e s t e d . The f a c t t h a t the FORHIN t a s k c o n c e p t u a l l y r e q u i r e d the S to d e a l w i t h the t r a n s f o r m a t i o n a i a s p e c t s o f the s t i m u l i (the changes i n the v a r i o u s d i m e n s i o n s ) , adds su p p o r t t o the view o f P i a g e t and o t h e r authors ( i . e . , W a l l a c h , 1969; B e i l i n , 1969) t h a t i t i s the u n d e r s t a n d i n g o f the changes t h a t o c c u r i n the s t i m u l i , m a n i f e s t e d i n the a b i l i t y t o m e n t a l l y p e r f o r m such t r a n s f o r m a t i o n s , t h a t b r i n g s about the a t t a i n m e n t o f o p e r a t i v i t y . T h i s p o i n t i s a l s o i n d i r e c t l y s u p p o r t e d by a comparison of the c o r r e l a t i o n s between the o t h e r automaton v a r i a b l e , MOPER, and the O p e r a t i v e measures, v e r s u s the c o r r e l a t i o n s between the FORHIN s c o r e s and the same O p e r a t i v e measures, as e x p l a i n e d below. The K-Operator r e f e r s t o the S's mental span o r computing s p a c e , which i n the p r e s e n t s t u d y has been o p e r a t i o n a l l y d e f i n e d i n terms o f the number or "chunks" o f i n f o r m a t i o n ( s t i m u l i - r e s p o n s e u n i t s ) , t h a t the S can s i m u l t a n e o u s l y a t t e n d t o and c o - o r d i n a t e . The FORHIN t a s k , on the o t h e r hand, d i f f e r s c o n c e p t u a l l y from t h e MOPER t a s k i n t h a t i t s p e c i f i c a l l y r e q u i r e s the 5 t o s i m u l t a n e -Ill ously co-ordinate changing aspects of the s t i m u l i . Thus, the MOPER task can be said to refe r to the S's multi-channel informa-tion-processing device (as Pascual-Leone (1970) refers to i t ) which i s needed to simultaneously include and co-ordinate a l l the schemes involved in the understanding of any cognitive task. The FORHIN task, on the other hand, can be said to refer to the S's schemes s p e c i f i c a l l y necessary and essential for co-ordinations such as those required in l o g i c a l tasks. It could be expected then, (1) that the S's capacity or computing space would develop before the acquisition of s e l f -regulatory foresight and hindsight s k i l l s ; (2) that a computing space of a certain size would be a prerequisite for the develop-ment of the foresight and hindsight s k i l l s needed to obtain operativity; and (3) that the relationship between the l e v e l of attainment in the l o g i c a l tasks and the FORHIN and MOPER scores would be strongest with the FORHIN scores since this constitutes the automaton variable of most dire c t relevance in the develop-ment of l o g i c a l operations. The empirical findings confirmed a l l these expectancies. The MOPER scores s i g n i f i c a n t l y correlated with a l l the Operative measures (except with the Anticipatory and F i l l - i n measures) and did not correlate s i g n i f i c a n t l y with the Figurative measures (See Table 3). This finding confirms the prediction that follows from Pascual-Leone's (1970) M-Operator model that the size of the com-puting space should be related to S's le v e l of operativity. How-ever, these correlations are a l l s i g n i f i c a n t l y lower (p<.01) than those between FORHIN and the Operative measures, which also con-firms the expected finding (no. 3) that the automaton variable 11S conceptualized in the FORHIN task should have the strongest relationship to the Operative measures. The lower correlations between MOPER and the Operative scores partly resulted from the fact that in a number of instances (See Tables 13 and 14), Ss who attained the minimum MOPER score that for most Ss corresponded to the minimum level of ope r a t i v i t y , did not in fact attain that minimum l e v e l of operativity. That i s , as would be expected, i t was possible for a number of Ss to have the capacity necessary to deal successfully with the l o g i c a l tasks, and yet not have achieved an operational understanding of the l o g i c a l task. This finding, of course, confirmed the above no. 1 expectancy. F i n a l l y , very few Ss (two i n Grade 1 and none in Grade 2) who attained the minimum l e v e l in FORHIN that corresponded to the minimum l e v e l of ope r a t i v i t y in the l o g i c a l tasks had a MOPER score below the minimum MOPER score that corresponded to the minimum c r i t e r i o n of operativity. That i s , very few Ss in both Grade 1 and Grade 2 who attained a score in FORHIN greater than or equal to 1.5, obtained a score in MOPER lower than 3.1. Thus, the majority of Ss had to be able to handle at least 3.1 stimuli of the MOPER task before they could handle more than 1.5 changing dimensions of the FORHIN task. This finding confirms the expectancy (no. 2) that a MOPER score of a certain minimum size would be a prerequisite for the attainment of the minimum lev e l in the foresight and hindsight s k i l l s that may allow operational understanding of the tasks. In conclusion, the findings that could be expected from the automaton tasks and the l o g i c a l tasks on the basis of an analysis of t h e i r conceptual d i s t i n c t i o n s were obtained in these data. 114 From the point of view of Piagetian theory, the high degree of empirical re l a t i o n s h i p between the FORHIN scores and the Operative scores of the l o g i c a l tasks gives support to Piaget's view that i t i s the child's foresight and hindsight a b i l i t i e s and the con-sequent a b i l i t y to mentally perform the transformations i n the s t i m u l i which these se l f - r e g u l a t o r y functions make possible, that brings about the understanding of logico-mathematical con-cepts. A finding of conceptual s i g n i f i c a n c e i s the fact that there were no instances of bimodal d i s t r i b u t i o n s i n any of the scores of the automaton or l o g i c a l tasks. The way i n which the scores were obtained, then, allowed to measure aspects of the Ss behavior that have been t r a d i t i o n a l l y measured i n all-or-none or discrete ways, i n a more continuous manner. Consistency across l o g i c a l tasks I f , as postulated by Piaget, the child's understanding of l o g i c a l concepts should be manifested across families of related and s t r u c t u r a l l y equivalent concepts (groupings), then i t should be possible to obtain high intertask consistency among tasks that meet this requirement (structuring c r i t e r i o n of stages). The obtained data indicated that for both groups the i n t e r c o r r e l a -tions between a l l the Operative measures of the l o g i c a l tasks were high, which indicates that the S_s were consistent in t h e i r per-formance across tasks of common and related conceptual structure. Thus, t h i s f inding lends strong empirical support to the above 115 t h e o r e t i c a l proposition about the developmental relationships between s t r u c t u r a l l y related concepts. This finding i s of p a r t i c ular importance because i n the present i n v e s t i g a t i o n the l o g i c a l tasks were equated both i n terms of automaton and competence demands. Thus, the obtained results strongly indicate that when l o g i c a l tasks of equivalent and related conceptual structures are equated on more than just s t r u c t u r a l grounds, i t is possible to f i n d the intertask consistency that Piaget's structuring c r i t e r i o n requires. It should be noted, however, that the evidence from th i s study cannot be considered s u f f i c i e n t to give the necessary v a l i d i t y and f u l l support to this c h a r a c t e r i s t i c of the stage construct, mainly because i t did not sample a l l the possible groupings underlying the c l a s s i f i c a t i o n concepts. Therefore, t h i s study did not meet the s t r i c t e s t c r i t e r i o n necessary to test this t h e o r e t i c a l requirement of the stage construct. None-theless, the data do lend considerable support to the view that i t i s necessary to take into consideration simultaneously struc-t u r a l (or competence) as well as automaton factors of cognitive development in dealing with this t h e o r e t i c a l problem. This l a s t conclusion is further supported by the high Kendal c o e f f i c i e n t s of concordance W, obtained for both groups, on the data expressing the number of units of information that the Ss successfully handled in each task, including the MOPER and FORHIN tasks (See Tables 13 and 14). This c o e f f i c i e n t indicates a high degree of s t a b i l i t y of i n d i v i d u a l differences in the number of units of information that the Ss were capable of operating upon 116 simultaneously. The large Kendall c o e f f i c i e n t shows that i n d i v i d -uals maintained e s s e n t i a l l y the same rank position over a l l of the tasks. In other words, these Ss successfully handled a constant number of units of information across a l l of the tasks. Thus, these findings confirm the necessity of equating l o g i c a l tasks on the basis of t h e i r automaton requirements as well as s t r u c t u r a l requirements when attempting to obtain equivalent measures of the S s ' attained l e v e l of o p e r a t i v i t y . The intercorrelations between the Figurative scores were s i g n i f i c a n t l y lower (p<.01) than the intercorrelations between the Operative measures. This indicates that the Ss were less consistent in t h e i r performance on the Figurative sections than on the Operative ones. This finding would be expected i f indeed the perceptual differences among the l o g i c a l tasks gave r i s e to a variety of strategies i n the Ss' attempts to reproduce the dif f e r e n t symbolic configurations of the tasks. This inter-pretation i s , of course, congruent with what could be expected on the basis of the d i s t i n c t i o n that Piaget makes between Oper-ative and Figurative aspects of i n t e l l i g e n c e (which is elaborated more extensively in the next section). The Operative procedure emphasized the conceptual rules underlying the tasks, which were the common and related aspects between the tasks. The Fig-urative procedure, on the other hand, emphasized the configura-t i o n a l or representational components, which were the d i f f e r e n t i -ating aspects between the tasks. Figurative vs. Operative Measures of Logical Tasks Piaget's d i s t i n c t i o n between operative aspects -- mental 117 actions and operations, and the figura t i v e aspects -- perceptual, representational, and symbolic aspects of in t e l l i g e n c e i s of overriding importance because i t emphasizes that the child's understanding of concepts cannot merely be considered to be determined by the cha r a c t e r i s t i c s of the stimuli in which the various concepts might be embodied. According to Piaget i t is the S s ' operative a c t i v i t i e s , how he transforms and co-ordinates the s t i m u l i , that represent the most s i g n i f i c a n t c h a r a c t e r i s t i c of i n t e l l i g e n c e . The representational a c t i v i t i e s , e ssential to the development of i n t e l l i g e n c e , nonetheless are considered to be subordinate to the operative functions. The present study u t i l i z e d this conceptual d i s t i n c t i o n be-tween fi g u r a t i v e and operative aspects as a methodological basis to measure operativity in more than one way. In the Figurative Reproduction or Copy sections of the tasks, the Ss had to re-construct the configuration of the stimuli as just seen. In one of the Operative (Reverse) sections, the 5_ had to reconstruct the same configuration except that a l l the relationships between the elements had to be reversed. This required the S to mentally make a series of transformations on the elements while at the same time keeping constant the relationships previously e x i s t i n g between them. Comparisons between the means of the Operative and Figurative scores of the MC, MR, and SER tasks showed that for either group these means did not d i f f e r s i g n i f i c a n t l y from each other. This could be interpreted to indicate that the Operative and Figura-tive measures did not tap d i f f e r e n t aspects of the S's behavior. However, whether a S improved his Operative score or not i n 118 reference to his Figurative score, appeared to be a sensitive indicator of the S's overall l e v e l of operativity. In other words, since for a l l tasks the Figurative section was adminis-tered before the Operative one, i t was possible to obtain a difference score which indicated whether, within that s p e c i f i c task, the Operative score increased or decreased over i t s correspondent Figurative score. It was found that these d i f f e r -ences were s i g n i f i c a n t l y correlated to the Operative measures, including those from other l o g i c a l tasks (See Table 9 ) , and not to the Figurative measures; and p a r t i c u l a r l y , these differences were s i g n i f i c a n t l y correlated to the S's l e v e l of attainment in the FORHIN task. Thus, the pattern of interrelationships of the differences between the Reproduction scores and Reverse scores for the M C , MR, and SER tasks indicated that the Figura-t i v e and Operative procedures i n fact tapped d i f f e r e n t approaches in the Ss' responses to the same l o g i c a l tasks. The above empirical findings have a number of s i g n i f i c a n t implications. In the f i r s t place, the procedure used, in which continuous feedback was given to the Ss after each of the various sections of the l o g i c a l tasks ( F i l l - i n , Reproduction, Reverse), could be viewed as maximizing the S's chances to do better i n the Reverse or Operative section since this part was the l a s t one administered in a l l the l o g i c a l tasks. However, as shown by the data, the increase or decrease of the Operative scores over the Figurative scores was not a function of the order in which the various sections were administered (as could have been inter-preted i f the Operative scores would have been s i g n i f i c a n t l y higher across a l l tasks and for both groups), but rather i t was 119 a function of the S's l e v e l of operativity r e f l e c t e d in the S's FORHIN scores and the Operative scores in the l o g i c a l tasks. Thus 5 i f the feedback given to the S can be thought of as having had a learning e f f e c t , then the degree to which the Ss learned about the task from his exposures to i t through the F i l l - i n and Reproduction sections i^as a function of his attained l e v e l of o p e r a t i v i t y . This finding i n d i r e c t l y supports the view that the effects of t r a i n i n g on l o g i c a l operations are dependent on the S's already attained l e v e l of operativity and not exclusively on the t r a i n i n g i t s e l f (Inhelder and S i n c l a i r , 1969). A second implication of these empirical findings i s that they lend support to the usefulness of Piaget's d i s t i n c t i o n between figur a t i v e and operative aspects of i n t e l l i g e n c e . At the same time, they suggest the p o s s i b i l i t y of this conceptual d i s t i n c t i o n being used as the methodological basis for more accurately assessing the child's operativity. The results discussed in the three sections above indicate then, that the findings that could be expected from Piaget's thepry with respect to the possible relationships between the variables that were investigated, were in fact obtained. Thus, the empirical findings lend considerable support to Piaget's t h e o r e t i c a l views on what factors are involved in the acquisition of logico-mathematical concepts, what relationships can be expec-ted between various related l o g i c a l concepts, and on the d i s t i n c -tion between figur a t i v e and operative aspects of i n t e l l i g e n c e . Before discussing more general implications of these findings and the p o s s i b i l i t i e s for future research, other aspects of the data w i l l be discussed. 120 Differences between the two Developmental Groups The two age groups were chosen in order to obtain measures from groups which, although close enough i n terms of t h e i r attained o p e r a t i v i t y , would s t i l l be s u f f i c i e n t l y d i f f e r e n t such that t h e i r corresponding data could allow examination of develop-mental trends. The results from the various tasks indicated that the two groups d i f f e r e d s i g n i f i c a n t l y from each other in most of the t e s t s . Each one of the tasks i s discussed next. MOPER - The MOPER scores for Grade 1 and Grade 2 were s i g -n i f i c a n t l y d i f f e r e n t from each other, with Grade 2 Ss obtaining the higher mean score of the two groups. This f i n d i n g i s i n congruence with the predic t i o n that follows from Pascual-Leone's (1970) M-Operator model that the size of the computing space should be larger for the older group since i t i s assumed that the M-Operator capacity grows as a function of age in normal subjects. FORHIN - The mean scores for Grade 1 and Grade 2 in the FORHIN task were not s i g n i f i c a n t l y d i f f e r e n t from each other.* This lack of sig n i f i c a n c e would not be expected in view of the fact that the two groups did d i f f e r s i g n i f i c a n t l y from each other in most of the operative measures. Further t e s t i n g of the FORHIN task across a larger number of developmental levels w i l l have to be carr i e d out to determine i t s v a l i d i t y on a more complete basis, and i t s relationships to other operative tasks. HC and MR - The results indicated that Grade 1 and Grade 2 did not d i f f e r s i g n i f i c a n t l y from each other in terms of the mean scores obtained i n the Figurative measures, but that Grade 1 and Grade 2 did d i f f e r s i g n i f i c a n t l y from each other in almost *See Addendum ( 3 ) . 121 a l l the mean Operative scores of the MC task (except i n the MCANT scores), and only i n MRANT and MRFILL for the MR task. These results confirmed the expectations that the Ss should d i f f e r mainly i n terms of how they dealt with the Operative sections of the m u l t i p l i c a t i v e tasks. The data also showed that, within each group, there were no s i g n i f i c a n t differences between the mean scores i n the MC and MR tasks, which would indicate that no group s i g n i f i c a n t l y improved t h e i r scores i n the MR task, the second m u l t i p l i c a t i v e task administered. However, the MRREVE scores for Grade 1 increased enough over t h e i r MCREVE score to be comparable to the MRREVE scores for Grade 2. Thus, Grade 1, the group with the ov e r a l l lowest l e v e l of o p e r a t i v i t y , showed only a s l i g h t improvement i n the Operative scores of the second m u l t i p l i c a t i v e task, but no improvement i n the scores obtained within each task (in the Operative over the Figurative scores). Grade 2, on the other hand, already atta i n i n g the highest Operative scores, did not s i g n i f i c a n t l y improve these scores on the second m u l t i p l i -cative task, but i t s i g n i f i c a n t l y improved i t s Operative over i t s Figurative scores within each task. These findings then, not only confirm Piaget' s predictions as to the differences in operative levels for the two developmental groups, but also confirm the dependency of success of t r a i n i n g methods ( i . e . , feedback procedure of MC and MR tasks) on the already attained l e v e l of ope r a t i v i t y of the subjects. A second important aspect of the data has to do with the d i s t i n c t i o n between the t o t a l number of dimensions solved (MCDIM 122 and MRDIM for both Operative and Figurative sections of the tasks) and the t o t a l number of m u l t i p l i c a t i v e combinations possible within that t o t a l (MCMUL and MRMUL, Operative and Fi g u r a t i v e ) . The d i s t i n c t i o n was made to d i f f e r e n t i a t e between Ss who might c o r r e c t l y solve a cer t a i n number of overlapping dimensions (up to 3) which did not involve m u l t i p l i c a t i v e combinations versus those Ss who might have solved c o r r e c t l y fewer dimensions which none-theless ivere r e l a t e d to each other i n a m u l t i p l i c a t i v e fashion. The presence of m u l t i p l i c a t i v e combinations i s , of course, the most appropriate c r i t e r i o n as to whether a S has the mu l t i p l i c a -t i v e concept or not. Since the two measures were generated from the same data set, i t could be expected that they would be s i g -n i f i c a n t l y r e l a t e d to each other. This, indeed, was the case. However, the degree of rel a t i o n s h i p s i g n i f i c a n t l y varied for each gTade. The correlations between MCDIM and MRMUL i n the MCREVE scores for Grade 2 were s i g n i f i c a n t l y higher than those for Grade 1. This res u l t indicates that the two measures did d i f f e r e n t i a t e between the two levels of o p e r a t i v i t y , arid i t also suggests that the d i s t i n c t i o n between the measures was an appro-p r i a t e one since i n fact there were Ss who were able to obtain moderately high t o t a l scores i n the matrices and yet not a t t a i n the minimum c r i t e r i o n of ope r a t i v i t y . The other Operative measures of the MC and MR task ( F i l l - i n and Anticipation) are discussed next. MCFILL and MRFILL - The mean scores for these measures i n d i -cated that Grade I and Grade 2 d i f f e r e d s i g n i f i c a n t l y from each other i n both MCFILL and MRFILL, with Grade 2 obtaining the higher 123 scores. These findings would be expected i f in fact the F i l l - i n scores measured the Ss' l e v e l of operativity. However, the pattern of intercorrelations with the other Figurative and Oper-ative measures raises some question as to whether those tasks exclusively gauged the Ss 1 operativity. For the Grade 1 Ss, MCFILL and MRFILL s i g n i f i c a n t l y correlated with a l l Figurative and Operative scores. For Grade 2 Ss, on the other hand, MCFILL was s i g n i f i c a n t l y correlated only to the Figurative scores, while MRFILL was s i g n i f i c a n t l y correlated with a l l the Operative scores and with very few (and almost no s i g n i f i c a n t correlations) of the Figurative scores. In other words, i n the group with the lowest l e v e l of ope r a t i v i t y , both of the F i l l - i n measures were s i g n i f i c a n t l y correlated with both the Figurative and Operative measures, while for the more advanced group the relationships varied as a function of the task. As was apparent throughout the data on the l o g i c a l tasks, Grade 1 scores appeared to be the most consistent across a l l tasks, which might indicate that these Ss dealt with most tasks in a sim i l a r manner. In view of this fact, i t could be expected then that Grade 1 F i l l - i n scores would correlate with both Operative and Figurative scores. Grade 2 Ss, on the other hand, appeared to have changed t h e i r responses as a function of the tasks. It could be expected then that since MCFILL was the f i r s t section of the f i r s t l o g i c a l task administered, this score could r e f l e c t the most v a r i a b i l i t y in the Ss' responses and consequently the Ss' use of non-operative strategies. This interpretation is supported by the fact that MCFILL s i g n i f i c a n t l y correlated only with Figurative scores. The consequent improvement in the MRFILL score indicates the Ss' I S A f a m i l i a r i t y with the task requirement and t h e i r consequent use of operative strategies. This interpretation is supported by the fact that the MRFILL was s i g n i f i c a n t l y correlated only with Operative scores. It appears then that the F i l l - i n procedure can give rise to more than just operative strategies in the approaches to the solution of the matrix task, even in those Ss who have reached a f a i r l y advanced le v e l of operativity. This has important methodological implications. As discussed i n the section about the t h e o r e t i c a l basis of the methods, the present procedure was designed to minimize the chances of correct solutions based on the perceptual symmetries present in the configuration of the matrix rather than based on an operative strategy. Thus, presum-ably the procedure was a more d i f f i c u l t one than the one most commonly used which simply requires to f i l l in one missing c e l l . Thus, even when the task requirements were made more d i f f i c u l t and presumably more in favor of discriminating operative from non-operative solutions, i t was s t i l l found that this task did not unequivocally discriminate the Ss' level of o p e r a t i v i t y . It is also possible that, since only two F i l l - i n tasks were admin-ist e r e d , the Ss 1 most common approach to this type of test was not tapped. In this case, the complex interrelationships obtained would be at y p i c a l . Nonetheless, this procedure does not seem to measure the Ss' operative understanding of m u l t i p l i c a t i v e matrices as d i s t i n c t l y as the Reverse procedure, for instance. MCANT and MRANT - With respect to the Anticipatory measures, Grade 1 and Grade 2 d i f f e r e d s i g n i f i c a n t l y from each other in the MR task (MRANT), but not on the MC task. As with the F i l l - i n scores, the group with the highest l e v e l of o p e r a t i v i t y , Grade 2, was the one that improved i t s performance across tasks. Also, as with the F i l l - i n tasks, the fact that there was no s i g n i f i c a n t difference between Grade 1 and Grade 2 in MCANT could have re-sulted from the Ss' unfamiliarity with the task requirements. However, the pattern of intercorrelations more conclusively i n d i -cates that the Anticipatory tasks measured mainly aspects of the Ss operativity. Both groups, p a r t i c u l a r l y Grade 2, had s i g n i f i c a n t correlations with a l l the Operative scores, and had very few (only one) s i g n i f i c a n t correlations with the Figurative scores. In terms of the conceptual meaning of this task, what \^ as apparent from the data, is that many subjects who could not corre c t l y solve the whole matrix, could nonetheless anticipate correctly the transformation asked before actually engaging in the task. Thus, this measure did not d i f f e r e n t i a t e , as was i n i t i a l l y intended, between those Ss who might have had an ex-p l i c i t idea of a l l the transformations in the matrix before actually solving the task, versus those Ss who could attain cor-rect solutions of the matrix by t r i a l and error and thus while actually engaged in the task. Although i n i t i a l l y i t might appear contradictory that Ss might be able to anticipate a transforma-tion and yet not be able to actually solve the matrix, this find-ing is in fact congruent with Piaget's idea of gradual development of structures in which i t is possible for a S to have a p a r t i a l understanding of the operations involved in a concept and yet not have achieved the " r e f l e c t i v e abstraction" which allows the co-ordination of a l l the actions or operations into a complete trans-formational system. 126 SER - For this task, Grade 1 and Grade 2 d i f f e r e d s i g n i f -i c a n t l y from each other in both the Figurative and Operative aspects of the tasks. The difference between the two procedures was greatest for Grade 2•as shown by the lack of s i g n i f i c a n t c o r r e l a t i o n between the SEREVE and SERCOPY. For Grade 1, the cor r e l a t i o n between SEREVE and SERCOPY was s i g n i f i c a n t (r=.43, p<.0 5). The SEREVE and the SERDIM(O) scores for both grades were highly correlated (range r=.66 to r=.75) to the other Operative measures in the other l o g i c a l tasks, which indicates that these scores were measuring a common aspect of the S's behavior also measured in the other Operative scores. On the other hand, the Figurative scores, SERCOPY and SERDIM(F) for Grade 2 had very few s i g n i f i c a n t correlations with the other Figurative measures, while for Grade 1 SERCOPY and SERDIM(F) had many more s i g n i f i c a n t correlations to other Figurative measures. The results of the SER task again indicates that Grade Ss varied more in t h e i r per-formance as a function of the type of task than Grade 2 Ss who performed more consistently across both Figurative and Operative sections of the l o g i c a l tasks. TRAN - As was indicated by a x2 test on the p a s s - f a i l fre-quencies for each grade, the two groups did not d i f f e r s i g n i f -i c a n t l y from each other. No possible reasons for this lack of significance were apparent. However, the s i m p l i c i t y of the tech-nique and the finding that this measure was s i g n i f i c a n t l y corre-lated to a l l the Operative measures of the l o g i c a l tasks and MOPER and FORHIN scores would warrant further research on i t . In conclusion, the data on most tasks indicated that the groups were s i g n i f i c a n t l y d i f f e r e n t from each other in terms of 127 t h e i r attained l e v e l of opera t i v i t y . General Implications of the Study The results of this investigation give general support to several of Piaget's t h e o r e t i c a l views. F i r s t , support is given to Piaget's structural concept of stage. A stage is viewed as a period during which the child's performance on di f f e r e n t con-ceptual tasks can be described in terms of underlying common invariant properties (conceptual structure characteristics).. The high intertask consistency that, according to the stage con-st r u c t , would be expected in the Ss' performances across tasks of common conceptual structure was obtained. Second, the results give support to Piaget's views on the processes (e.g., s e l f -regulatory functions, hindsight and foresight) that might under-l i e the development of operativity as manifested in the child's understanding of logico-mathematical concepts. Third, they also provide some support for the conception of development of operational concepts as a gradual process rather than an a l l - o r -none event. F i n a l l y , they lend support to Piaget's d i s t i n c t i o n s between operative and figura t i v e aspects of i n t e l l i g e n c e . There are several conceptual and empirical implications that follow from the above. Stages and Continuous Development With respect to the problem of how developmental stages should be conceptualized, both in terms of the behavior that i t should encompass and in terms of how these behaviors develop, i t is apparent that the stage construct needs to be elaborated beyond i t s present scope. This is obviously necessary because of the 128 numerous conceptual issues that remain unresolved ( i . e . , what fac-tors are involved in the horizontal 'decalages'?) . However, what is suggested by the present findings i s that expansions of the stage construct might f r u i t f u l l y go beyond consideration of l o g i c a l structures to include more of Piaget's general concept-ualizations about developmental functions. For example, the aspect of the stage construct that seems to have been most emphasized, conceptually and empirically, is the stepwise development of successive stages. This pervasive interest is manifested in the numerous e f f o r t s to deal with the questions of what makes possible the t r a n s i t i o n between stages. However, according to Piaget, this surface discontinuity of stages is the result of underlying continuous self-regulatory functions inherent to the organism. Yet, these continuous under-lyi n g self-regulatory functions have been themselves very seldom s p e c i f i c a l l y investigated. A type of analysis that focuses on continuous functions may eventually diminish in importance (or put in d i f f e r e n t terms) a question such as what makes possible the t r a n s i t i o n between stages. In f a c t , the presence or absence of stages in the development of cognitive structures may r e f l e c t more the researcher's arbitrary categorization of content of knowledge than the subjects' continuous development of s e l f -regulations. In regard to the above problems, the d i s t i n c t i o n between competence and automaton aspects of cognitive development, f i r s t introduced by F l a v e l l and Wohlwill (1969) , is important because i t attempts to incorporate the continuous and discontinuous aspects of development into one framework. This d i s t i n c t i o n was used in 129 the present study as the basis for designing the equivalent tasks used in the diagnostic procedure. The positive results c l e a r l y indicate the empirical significance of this conceptual d i s t i n c -tion. When investigating questions about the acquis i t i o n and course of development of conceptual rules, i t appears thus essen-t i a l to focus simultaneously on the nature of the competence rules and on the automaton functions that make the expression of the given competence possible. This competence-automaton dis-t i n c t i o n gives r i s e to a number of new p o s s i b i l i t i e s as to how cognitive development is to be conceptualized. For example, i t emphasizes the gradual development that may be involved in the attainment of cognitive concepts. "Attainment," according to this d i s t i n c t i o n , i s not merely defined in terms of whether or not the c h i l d has the f i n a l form of the concept, but also in terms of how far along he is i n the attainment of prerequisite a b i l i t i e s . Such an approach may in fact ultimately indicate that there are very few cognitive concepts whose acquisitions can be conceived as all-or-none events. Also, the emphasis on automaton functions brings into focus the necessity of looking at the acquisition of abstract con-ceptual rules as made possible through psychological processes whose cha r a c t e r i s t i c s are in fact quite d i f f e r e n t from the con-ceptual rules themselves. In other words, the competence rules are seen as s p e c i f i c abstractions from complex psychological pror cesses which are separate and d i f f e r e n t from the postulated l o g i c a l structures themselves. Thus, the study of automaton func-tions emphasizes the most basic and general aspects of cognitive acquisitions. 130 In t h i s regard i t is important to point out the conceptual significance of the M-Operator construct. This construct is t h e o r e t i c a l l y presumed to be an indicator of the degree of maxi-mum complexity which, in terms of units of information, the c h i l d can handle. As predicted by the M-Operator model, when the l o g i c a l and automaton tasks were quantitatively equated and com-pared in terms of the number of informational units, the degree of informational complexity the Ss could handle was found to be r e l a t i v e l y constant across tasks. Thus, these findings further confirm the need to analyze l o g i c a l tasks also in terms of t h e i r non-structural aspects. This l a s t conclusion, of course, bears on the the o r e t i c a l question of what generalizes between related concepts. The com-petence-automaton d i s t i n c t i o n makes clear that high inter-task consistency or s i g n i f i c a n t generalization between conceptually related tasks can only be expected to occur when the tasks are equated both in terms of the i r competence requirements and auto-maton requirements. Foresight and Hindsight The foresight and hindsight a b i l i t i e s are presumed to be general self-regulatory functions that make possible the develop-ment of opera t i v i t y . It would be important to investigate these functions to fi n d out th e i r possible role in the development of concepts other than those in the l o g i c a l realm. It might even-t u a l l y be possible to d i f f e r e n t i a t e between the s p e c i f i c degree of assimilation that they may make possible with reference to a s p e c i f i c area of knowledge, i . e . , l o g i c a l concepts, and th e i r general overa l l l e v e l of assimilatory capacity as manifested across 131 a number of diff e r e n t cognitive contents. With s p e c i f i c reference to the role of foresight-hindsight a b i l i t i e s in the development of logico-mathematical concepts, there are a number of empirical p o s s i b i l i t i e s that could help c l a r i f y t h e i r relationship to the acq u i s i t i o n of these concepts. For example, various authors (Smedslund, 1966; B e i l i n , 1969; Gellerman, 1969: etc.) have attempted to accelerate the child's attainment of l o g i c a l concepts through procedures that either emphasize the competence or s p e c i f i c l o g i c a l operations involved in the l o g i c a l task or by procedures that emphasize automaton factors such as teaching the S to attend to certain cues. Even i f moderate success has been attained by either type of pro-cedure, in most instances the most c r i t i c a l c r i t e r i o n of whether the c h i l d has acquired a structure, the S's generalization of the concept to other s t r u c t u r a l l y related tasks, has not been met. In this regard, a procedure such as the FORHIN task could be used for two purposes: F i r s t , to i n d i r e c t l y assess the S's ove r a l l l e v e l of operativity which might indicate to what extent the S might i n fact benefit from a t r a i n i n g procedure. This p o s s i b i l i t y of the FORHIN task i s , of course, suggested by the finding in the present study that the S's improvement in a task was related to his performance in the FORHIN task. Second, a procedure such as that of the FORHIN task could also be used as a tra i n i n g procedure i t s e l f since i t presumably involves activ-i t i e s which are basic and common to families of related l o g i c a l concepts. It would be important to find out i f the teaching of such a c t i v i t i e s would lead to greater generalization across 132 conceptually related tasks. This outcome could be expected since a FORHIN tra i n i n g would not involve the teaching of the s p e c i f i c l o g i c a l operations that may be involved i n the given task, i . e . , addition-subtraction, r e v e r s i b i l i t y , but rather the co-ordination of transformations, which presumably i s a basic and common a c t i v i t y to a l l concrete-operational tasks. The testing of generalization would be carried out through tasks that would be made equivalent both in t h e i r competence and their automaton demands, as was done in the present study. Operative vs. Figurative Aspects One last important implication has to do with the methodo-l o g i c a l problems involved i n the measurement of developmental phenomena. In the present investigation i t proved f r u i t f u l , i n finding how the Ss dealt with l o g i c a l s t i m u l i , to take into account conceptual aspects about the nature of i n t e l l i g e n c e , i . e . , operative vs. fig u r a t i v e d i s t i n c t i o n , in the design of the methods themselves. In other words, the measurement instruments were not merely means to test the s p e c i f i c conceptual questions of interest but also had incorporated into t h e i r format consid-erations based on the theory of i n t e l l i g e n c e . What can be con-cluded from this is that when dealing with the problems of measuring developmental phenomena i t might be p r o f i t a b l e i f con-ceptual guidelines about the nature of development (and not just purely methodological, experimental, or normative considerations) are incorporated into the methods themselves. 133 References Ayers, J.D. Assessing Cognitive Development via,Measures of Optimal Performances. In D.R. Green, M.P. Ford, and G.B. Flamer (Eds.), Measurement and Piaget. McGraw H i l l , 1971. Beaver, T.G., Mehler, J . . and Epstein, J. What children do i n spite of what they know. Science, 1968, 162 , 921-924. Bearison, D. Role of measurement operations in the acquisition of conservation. Developmental Psychology, 1969, 16, 653-660. B e i l i n , H. Cognitive capacities of young children: A r e p l i c a t i o n . Science ,^1968, 162 , 920-921. B e i l i n , h. Stimulus and Cognitive Transformation i n Conservation. In D. Elkind and J. F l a v e l l (Eds.), Essays in Honor of  Jean Piaget. New York; Oxford University Press, 1969. B e i l i n , K. The Training and Acquisition of Logical Operations. In M.F. Rosskopf, L.P. Steffe, and S. Taback (Eds.), Piagetian Cognitive-Development Research and. Mathematical  Education. Washington: National Council of Teachers of Mathematics, 19 71(a). Beilin., H. Developmental Stages and Developmental Processes. In D.R. Green, M.P. Ford, and G.B. Flamer (Eds.), Measurement and Piaget. McGraw H i l l , 1971(b). Braine, M.D.S. The Ontogeny of Certain Logical Operations: Piaget's Formulations Examined by Non-verbal Methods. Psychological Monographs, 19 59, Whole No. 4 75. Bruner, J.S. On the Conservation of Liquids. In J.S. Bruner, R.R. Oliver, and P. Greenfield (Eds.), Studies i n Cognitive Growth. New York: Wiley § Sons, Inc. , 1966. Dudek, S.Z., Lester, E.P., and Goldberg, J.S. Relationship of Piaget measures to standard i n t e l l i g e n c e and motor scales. Perceptual and Motor S k i l l s , 1969 , 28_, 351-362. F l a v e l l , J.H. The Developmental Psychology of Jean Piaget. New York: D. Van Nostrand Co., 1963. F l a v e l l , J.H. Stage-related properties of cognitive development. Unpublished Mimeo, 1970(a). F l a v e l l , J.H. Concept Development. In P.M. Mussen (Ed.), Carmichael's Manual of Child Psychology, (Vol. 1). New York: Wiley, 19 70(b). F l a v e l l , J.H. An analysis of cognitive developmental sequences. Unpublished Mimeo, 19 71. 1 3 4 F l a v e l l , J.H. and Wohlwill, J.F. Formal and Functional Aspects of Cognitive Development. In D. Ellcind and J.H. F l a v e l l (Eds.), Essays in Honor of Jean Piaget. Mew York: Oxford University Press, 1969. Gelman, R. Conservation ac q u i s i t i o n : A problem of learning to attend to relevant attributes. Journal of Experimental  Child Psychology, 1969, 38, 1229-1246. Goldschmid, M.L. Different types of conservation and non-conservation and the i r r e l a t i o n to age, sex, IQ, MA, and vocabulary. Child Development, 1967, 3_8 , 1229-1246. Goldschmid, M.L. The r e l a t i o n of conservation to emotional and environmental aspects of development. Child Development, 1968, 39, 579-589. Halford, G.S. and T.J. Fullerton. A discrimination task which induces conservation of number. Child Development, 1970, 41, 205-214. Inhelder, B. Memory and Intelligence in the Child. In D. Elkind and J.H. F l a v e l l (Eds.), Essays in Honor of Jean Piaget. New York: Oxford University Press, 1969. Inhelder, B. and Piaget, J. The Early Growth of Logic. The Norton Library, 1964. Inhelder, B. and S i n c l a i r , H. Learning Cognitive Structures. In P. Mussen, L. Langer, and M. Covington (Eds.), Trends  and Issues in Developmental Psychology. Holt, Rinehard, and Winston, Inc., 1969. Klahr, D. and Wallace, J.G. An information processing analysis of some Piagetian experimental tasks. Cognitive Psychology, 1970, 1, 358-387. Kofsky, E.A. A scalogram study of c l a s s i f i c a t o r y development. Child Development, 1966 , 3M7 , 191-204. Lefrancois, G.R. A treatment of hierarchy for the acceleration of conservation of substances. Canadian Journal of Psychology, 1968, 2_2, 277-284. Mehler, J. and Bever, T.F. Cognitive capacity in very young children. Science, 1967, 158, 141-142. Minichello, M. and Goodnow, J. Effect of an "action" cue on conservation of amount. Psychonomic Science, 1969 , 16_, 200-201. Pascual-Leone, J. A mathematical model for the t r a n s i t i o n rule in Piaget's developmental stages. Acta Psychologica, 1970 , 32 , 301-345. 135 Pascual-Leone, J and Smith, J . The encoding and decoding of symbols by children: A new experimental paradigm and a neo-Piagetian model. Journal of Experimental Child  Psychology. 1969, 8, 328-355. Piaget, J. Play, Dreams, and Imitation in Childhood. Mew York: W.W. Norton and Company, 1962. Piaget, J. Genesis and structure in the psychology of i n t e l l i -gence. In D. Elkind (Ed.), Six Psychological Studies by  Jean Piaget. New York: Random House, 196 7(a). Piaget, J. Biologie et Connaissance. Paris: Gallimard, 1967(b). Piaget, J. Memory and Identity. Clark University Press, 1968. Piaget, J. Structuralism. New York: Haper Torchbooks, 1970(a). Piaget, J. Introduction of Laurendeau., M. and Pinard, A. The  Development of the Concept of Space in the Child. International Universities Press, 1970(b), Piaget, J. Biology and Knowledge. The University of Chicago Press, 1971(a). Piaget, J. The Theory of Stages in Cognitive Development. In D.R. Green, M.P. Ford„ and G.P. Flamer (Eds.), Measurement  and Piaget, 1971(b), 1-11. Piaget, J. The Principles of Genetic Epistemology. London: Routledge and Kegan Paul, 1972. Piaget, J. and Inhelder, B. I n t e l l e c t u a l Operations and t h e i r Development. In P. Fraisse and J. Piaget (Eds.), Experiment-al Psychology: Its Scope and Method. Vol. VII Intelligence. London: Routledge and Kegan Paul, 1969. Piaget, J. and Inhelder, B. Mental Imagery in Children. Basic Books, 1971. Piaget, J. and Szeminska, A. La genese du nombre chez 1'enfant. NeuchHtel et Paris: Delachaux et Niestle", 1941. Pinard, A. and Laurendeau, M. Stage in Piaget's Cognitive-Developmental Theory: Exegis of a Concept. In Elkind and F l a v e l l (Eds.), Essays in Honor of Jean Piaget. New York: Oxford University Press, 1969. Price-Williams, D., Gordon, W., and Ramiriz I I I , M. S k i l l and conservation: A study of pottery-making children. Developmental Psychology, 1969 , 1_, 769. R o l l , S. R e v e r s i b i l i t y t r a i n i n g and stimulus d e s i r a b i l i t y as factors in conservation of number. Child Development, 1970, 41, 501-507. 136 Rothenberg, B. and Courtney, R. A developmental study of non-conservation choices in young children. Merrill-Palmer  Quarterly, 1969, 15, 363-373. Shantz, C. A developmental study of Piaget's theory of l o g i c a l m u l t i p l i c a t i o n . Merrill-Palmer Quarterly, 1967, 13, 121-137. Smedslund, J. Microanalysis of concrete reasoning: I. The d i f f i c u l t y of some combinations of addition and subtraction of one unit. Scandinavian Journal of Psychology, 1966, 7_, 145-146. Wallach, L. On the Basis of Conservation. In D. Elkind and J. F l a v e l l (Eds.), Essays i n Honor of Jean Piaget. New York: Oxford University Press, 1969. Wallach, L. , Wall, J . , and Anderson, W. Number conservation: The role of r e v e r s i b i l i t y , addition-subtraction, and misleading perceptual cues. C h i l d Development, 1967, 38, 425. Wohlwill, J.F. Piaget's theory of the development of i n t e l l i g e n c e i n the concrete operations period. American Journal of  Mental Deficiencies Monograph Supplement, 1966, 7j0» 57-83. S e t 3 FORHIN S e t s . Sets 4 and 5. S e t t Set 7 FORHIN'Sets . Sets 6 and 7 S e t 8 140 S e t 9 FORHIN S e t s . S e t s 8 and 9 . Set 10 141 F O R H I N S e t s . S e t s 10 a n d 1 1 . 142 The correct choice cards for the FORHIN sets were: Set No. 2: Change i n one dimension, shape. Positi o n 3: Third card, second row of choice cards Posit i o n 4: Second card, f i r s t row of choice cards Po s i t i o n 5: F i r s t card, second row of choice cards Set No. 3: Changes i n one dimension, o r i e n t a t i o n . Position 3: Third card, second row of choice cards Position 4: F i r s t card, f i r s t row of choice cards Po s i t i o n 5: F i r s t card, second row of choice cards Set No. 4: Changes in two dimensions, brightness and s i z e . P o s i t i o n 3: Second card, f i r s t row of choice cards Po s i t i o n 4: Second card, second row of choice cards Posit i o n 5: Third card, l a s t row of choice cards Set No. 5: Changes in two dimensions, shape and o r i e n t a t i o n . P o s i t i o n 3: Third card, f i r s t row of choice cards Posit i o n 4: F i r s t card, f i r s t row of choice cards Position 5: Third card, second row of choice cards Set No. 6: Changes i n three dimensions, shape, o r i e n t a t i o n , s i z e . Position 3: Fourth card, second row of choice cards Position 4: Fourth card, f i r s t row of choice cards Position 5: Third card, second row of choice cards Set No. 7: Changes i n three dimensions, shape, brightness, or i e n t a t i o n . Position 3: Fourth card, second row of choice cards Position 4: Third card, f i r s t row of choice cards Position 5: Second card, second row of choice cards 143 Set No. 8: Changes i n four dimensions, s i z e , shape, o r i e n t a t i o n , number. Po s i t i o n 3: F i r s t card, second row of choice cards P o s i t i o n 4: F i f t h card, second row of choice cards Posi t i o n 5: Third card, f i r s t row of choice cards Set No. 9: Changes i n four dimensions, s i z e , shape, o r i e n t a t i o n , number. Pos i t i o n 3: Second card, f i r s t row of choice cards Posi t i o n 4: Sixth card, second row of choice cards P o s i t i o n 5: F i f t h card, f i r s t row of choice cards Set No. 10: Changes in five dimensions, s i z e , brightness, o r i e n t a t i o n , shape, number. Positi o n 3: Thir d card, second row of choice cards Position 4: Sixth card, second row of choice cards Posi t i o n 5: Second card, second row of choice cards Set No. 11: Changes i n f i v e dimensions, s i z e , brightness, o r i e n t a t i o n , shape, number. Position 3: Second card, f i r s t row of choice cards P o s i t i o n 4: Second card, second row of choice cards P o s i t i o n 5: F i r s t card, f i r s t row of choice cards NOTE: Distortions in the photographs do not show accurately the differences i n brightness. 144 ADDENDUM In the MC, MR, and SER tasks each of the choice pieces (see s p e c i f i c description of materials i n pages 52, 62, and 63 r e s p e c t i v e l y ) , constituted a d i f f e r e n t combination of values of each of the s i x dimensions involved. Thus, for a l l three tasks there were only nine p o s i t i v e instances or correct pieces, the rest of the pieces constituted negative instances. The scores from a l l Ss were separated on the basis of grade rather than age (see page 102). This was done so because each grade was f a i r l y homogeneous with respect to the age range that i t encompassed (for Gl, range = 6.5-8.4, mean = 7.02 years, and SD = 6.53 months; for G2, range = 7.4-9.0, mean = 7.93 years, and SD = 4.82 months). Both grades were thus treated as representing two d i f f e r e n t developmental l e v e l s . Although the FORHIN scores for Gl and G2 did not d i f f e r s i g n i f i c a n t l y from each other, the differences i n variances of the two scores showed a trend which indicated s t a t i s t i c a l s i g n i f i c a n c e . The r a t i o of the variances between the FORHIN scores for Gl and G2 was 1.67 with df = 31/31. The F ratios s i g n i f i c a n t at the .05 and .10 levels with df = 30/30 were 1.89 and 1.61 respectively. 

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