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Transitivity, identity conservation and equivalence conservation of a solid continuous quantity Humphrey, Gary Keith 1975

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TRANSITIVITY, IDENTITY CONSERVATION AND EQUIVALENCE CONSERVATION OF A SOLID CONTINUOUS QUANTITY by GARY KEITH HUMPHREY B.A. (Hons.) Uni v e r s i t y of New Brunswick, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Psychology We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the 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 o f B r i t i s h C o l u m b i a , I ag r ee t ha t t he 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 ag ree 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 pu r po se s may be g r a n t e d by the Head o f my Depar tment 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 . Depar tment o f The U n i v e r s i t y o f B r i t i s h Co l umb i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date X)jUtj .30 J f 7S~ ABSTRACT An i n v e s t i g a t i o n into the d i s t i n c t i o n between i d e n t i t y conservation and equivalence conservation, as presented by Elkind (1967) was examined i n the content area of s o l i d continuous quantity. One group of subjects received the tasks as outlined by E l k i n d (Group I) while another group of subjects received modified versions of the tasks (Group I I ) . Each conservation task was presented at two l e v e l s of transformation; moderate and extreme. In add i t i o n t r a n s i t i v i t y of s o l i d continuous quantity was examined i n r e l a t i o n s h i p to conservation. The sample consisted of 144 subjects; 48 Kindergarten, Grade one and Grade two students. Half of the subjects within each grade l e v e l were assigned to Group I, the other h a l f was assigned to Group I I . Within each group h a l f of the c h i l d r e n were male and h a l f were female. An analysis of variance performed on the conservation tasks indicated that i d e n t i t y and equivalence conservation were of equal d i f f i c u l t y . The main e f f e c t s of Group and Age were s i g n i f i c a n t and the i n t e r a c t i o n of Sex x Grade was s i g n i f i c a n t . The c r i t e r i o n factor of judgment only vs. judgment plus explanation was found to have a s i g n i f i c a n t e f f e c t , with more t r i a l s passed with a judgment only c r i t e r i o n . Data were scored according to two d i f f e r e n t procedures; oneyprocedure required that subjects be consistent i n t h e i r answers i n each phase of the task i n order to receive non-zero scores. This procedure employed a three-point scale with values of 0, 1, and 2. The other procedure used a scale with values ranging from 0 to 6 i n c l u s i v e . Subjects were given a point for each of the s i x questions answered c o r r e c t l y i n the conservation tasks, regardless of the consistency of the answers. The method with the 0, 1, and 2 scale showed that' i d e n t i t y and equivalence conservation were equally d i f f i c u l t , while the method which employed the 0-6 scale showed that i d e n t i t y was easier than equivalence. I t was shown that the l a t t e r method yie l d e d these r e s u l t s because of an a r t i f a c t i n the questions asked. Furthermore i t was shown that scale scores which resulted from an a p p l i c a t i o n of the 0-6 scale were an ambiguous r e f l e c t i o n of the l e v e l of concept attainment. An analysis of variance was performed on the t r a n s i t i v i t y tasks. The main e f f e c t s for Group and Age were s i g n i f i c a n t . The t r a n s i t i v i t y tasks were s i g n i f i c a n t l y easier than a l l conservation tasks at a l l grade l e v e l s . The implications of t h i s and the co-occurrence of i d e n t i t y and equivalence conservation were discussed i n r e l a t i o n to Elkind's (1967) an a l y s i s . Table of Contents Abstract i i Table of Contents i v L i s t of Tables v Acknowledgement v i Introduction 1 Method 11 Results 18 Discussion 44 Footnotes 59 References 60 Appendix A 63 V L i s t of Tables Table Page 1 Frequency and Percentage of Subjects Passing the Conservation Tasks when a C r i t e r i o n of Judgment Only was used. 19 2 Summary of Group x Sex x Grade x Task Type x Level of Transformation Analysis of Variance 21 3 Frequency and Percentage of Subjects Passing the Conservation Tasks when a C r i t e r i o n of Judgment Plus Explanation was Used. 23 4 Number of Subjects Passing Identity and Equivalence Tasks at each Grade Level when a C r i t e r i o n of Judgment Only was Used. 2 6 5 Number of Subjects Passing Identity and Equivalence Tasks at each Grade Level when a C r i t e r i o n of Judgment Plus Explanation was used. 27 6 Frequency and Percentage of Subjects giving Adequate Explanations for a l l Identity Conservation Tasks. 29 7 Frequency and Percentage of Subjects giving Adequate Explanations f o r a l l Equivalence Conservation Tasks. 30 8 Frequency and Percentage of Subjects giving Adequate Explanations f o r a l l Conservation Tasks. 31 9 Frequency and Percentage of Tasks on which Subjects Obtained the Spec i f i e d Values. 33 10 Number and Percentage of Subjects Passing the Four T r a n s i t i v i t y Tasks. 35 11 Summary of Group x Sex x Grade x Type of T r a n s i t i v i t y Task Analysis of Variance. 37 12 Number and Percentage of Subjects Passing Only the Speci f i e d Number of T r a n s i t i v i t y Tasks. 38 13 Number and Percentage of Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and Ident i t y Conservation Tasks ( C r i t e r i o n of Judgment Only). 41 14 Number and Percentage of Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and Equivalence Conservation Tasks ( C r i t e r i o n of Judgment Only). 42 15 Number and Percentage of Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and A l l Conservation Tasks ( C r i t e r i o n of Judgment Only). 43 16 Possible Patterns of Responding 54 Acknowledgement s The author would like to express his gratitude to Dr. Lou Moran and Dr. Tannis Williams for providing assistance and encouragement throughout the project. Thanks are also due to Dr. Ralph Hakstian and Tim McTiernan for advice on s t a t i s t i c a l techniques. The author also wishes to acknowledge the cooperation and assistance of the Vancouver School Board and the principals and teachers of Grenfell and Bruce Elementary Schools. INTRODUCTION 1. The t r a d i t i o n a l conservation task was devised to assess whether c h i l d r e n understand that perceptual deformations of objects do not produce changes i n the quantitative properties of the objects (Piaget, 1952; Piaget and Inhelder, 1974). For example a b a l l of clay may have i t s shape changed to that of a f l a t d i s c , or "pancake", but t h i s leaves quantitative properties of the object, such as i t s mass, weight, or volume, unchanged. In the t r a d i t i o n a l format for assessing conservation a c h i l d i s f i r s t shown two objects (e.g., A and B) that are q u a n t i t a t i v e l y and perceptually s i m i l a r . One of the objects (e.g., B) i s then subjected to a perceptual transformation (e.g., to B"'') . F i n a l l y the c h i l d i s asked to make some quantitative comparison between the standard or untransformed object and the transformed object (e.g., between A and B"*") . In more concrete terms a c h i l d might be presented with two clay b a l l s i d e n t i c a l i n s i z e , shape and quantity of clay. The shape of one of the b a l l s i s changed to that of a pancake. The c h i l d i s then asked such questions as: Is there the same amount of clay i n the pancake as there i s i n the b a l l ? Does one have more clay? The answers given to questions of t h i s type determine whether the c h i l d i s said to conserve. E l k i n d (1967) pointed to c e r t a i n methodological problems i n the t r a d i t i o n a l conservation task. The task was designed to assess children's understanding of the invariance of q u a n t i t a t i v e properties of objects across i r r e l e v a n t perceptual transformations. Success on the task should imply that the invariance i s understood, whereas f a i l u r e should mean i t i s not. However according to Elkind's analysis a c h i l d could f a i l the task 2. for reasons other than a f a i l u r e to grasp the quantitative invariance of objects i n t h i s s i t u a t i o n . A schematic example of the task follows: A=B (The objects are i n i t i a l l y equal on a l l quantitative and perceptual dimensions) B-^B"*" (Object B has i t s shape changed to B"'") A?B^ (The c h i l d i s questioned about the quantitative r e l a t i o n s h i p holding between A and B^) The f i r s t statement, A=B, can be viewed as the f i r s t premise i n a simple deductive argument. The second statement can also be thought of as a premise i n an argument such that B=B^ on the relevant quantitative dimension. F i n a l l y given A=B and B=B^ then i t n e c e s s a r i l y follows that A=B"'", hence the l a s t statement can be seen as a conclusion. Children could f a i l the task because they forgot the f i r s t premise, or because they believe that the relevant q u a n t i t a t i v e property of B changed when i t s shape was changed to B"'". Even i f c h i l d r e n remember that A=B and understand that B=B"'", they may not be able to coordinate t h i s information to reach the conclusion that A=B"'' (see E l k i n d , 1967). That ch i l d r e n f a i l the task because they forget the i n i t i a l e quality of A and B seems u n l i k e l y i n view of Bryant's (1974) observations. Bryant (1974) points out that Bruner, et a l (1966) have shown that c h i l d r e n who would normally f a i l conservation tasks remember the appearance of the display before the transformation. Also, using a task very s i m i l a r to number conservation tasks Bryant (1972) demonstrated that c h i l d r e n who did poorly on these tasks performed w e l l i n a cont r o l condition which required that they remember what the r e l a t i o n s between two rows of counters were before a perceptual transformation. 3. A more l i k e l y reason for f a i l u r e on the t r a d i t i o n a l task, according to Elkind's a n a l y s i s , i s that c h i l d r e n do not understand that B=B"'". This, E l k i n d claims, i s r e a l l y what the task was devised to assess, but does so ambiguously. I f the c h i l d r e n succeed on the task i t can be i n f e r r e d that they understand that B=B"S however f a i l u r e may occur f o r reasons other than the c h i l d ' s understanding of the r e l a t i o n between B and B"*". Elkind proposed that a much more d i r e c t technique be employed to assess whether ch i l d r e n understand that quantitative properties of sin g l e objects remain the same a f t e r perceptual transformations. A c h i l d would be shown a sin g l e object, the shape of the object would then be changed and f i n a l l y the c h i l d would make a judgment about some quantitative r e l a t i o n holding before and a f t e r the transformation (e.g., Is there the same amount of clay i n t h i s pancake as there was when i t was shaped l i k e a b a l l ? ) . E l k i n d has referred to the recognition that quantitative properties of s i n g l e objects remain invariant across perceptual transform-ations as i d e n t i t y conservation. The t r a d i t i o n a l task measures what El k i n d c a l l s equivalence conservation. According to Elkind's analysis a c h i l d must have the concept of i d e n t i t y , as well as the a b i l i t y to reason i n a simple deductive manner, i n order to succeed on the t r a d i t i o n a l task format. He suggested that there may be a developmental lag between i d e n t i t y and equivalence conser-vation with i d e n t i t y conservation being a p r i o r cognitive a c q u i s i t i o n . Hooper (1969b, p. 236) argued s i m i l a r l y ; "since equivalence conservation requires the a d d i t i o n a l deduction sequence, i t should be a l a t e r cognitive achievement than i d e n t i t y conservation." Hooper's study (1969b) was one of the f i r s t published reports 4. to provide evidence which showed that identity conservation occurs prior to equivalence conservation. Identity and equivalence conservation were assessed in the content area of discontinuous quantity (i.e., small seeds placed in glass beakers). Ninety-six subjects of mean ages 6, 7, and 8 years were used in a between subjects design. Hooper (1969b, p. 248) concluded that "identity conservation may be viewed as a necessary but not sufficient prerequisite for adequate equivalence conservation performance. Further evidence that identity conservation precedes equivalence conservation in the area of discontinuous quantity was obtained by Hooper (1969a) in a study of low socio-economic-status subjects aged 5 1/2 to 6 1/2 years. Schwarty and Scholnick (1970) investigated identity and equivalence conservation in 40 children of nursery and kindergarten age. Glasses equal in diameter and differing in diameter were partially f i l l e d with candies. Subjects were required to make direct comparisons, identity judgments and equivalence judgments both when the containers had the same diameter and when they differed. When the glasses were of equal diameter there were no significant differences among identity judgments, equivalence judgments and direct comparisons. However, when the containers differed in diameter identity judgments were easier than both equivalence judgments and direct comparisons. Papalia and Hooper (1971) worked with 60 subjects at ages 4, 5 and 6 years. Each subject was given a battery of tasks designed to measure qualitative identity, quantitative identity and equivalence conservation of discontinuous quantity and number. It should be noted that qualitative identity differs from the notion of identity (i.e., quantitative) developed by Elkind. Qualitative identity refers to a child's recognition that i t is 5. the same clay even though i t s shape has been changed; t h i s does not mean that the c h i l d n e c e s s a r i l y believes that i t i s the same amount. The l a t t e r depends upon the concept of quantitative i d e n t i t y (see Brainerd and Hooper, 1975; Hooper, 1969b; Papalia and Hooper, 1971, Piaget, 1968). Q u a l i t a t i v e i d e n t i t y concepts were found to develop p r i o r to quantitative i d e n t i t y concepts, which i n turn developed p r i o r to equivalence concepts i n the area of discontinuous quantity. There was no conclusive evidence regarding the order of emergence of the various number concepts. El k i n d and Schoenfeld (1972) used 22 four year olds and 22 s i x year olds to investigate the problem i n the content areas of length, l i q u i d , mass and number. Their general conclusion was that i d e n t i t y conservation was easier than equivalence conservation but the d i f f e r e n c e was most pronounced i n the; younger c h i l d r e n . Brainerd and Hooper (1975) investigated the i d e n t i t y - equivalence issue with 60 four year olds, 60 s i x year olds and 60 eight year olds. A l l subjects were given i d e n t i t y and equivalence tasks i n the content areas of length and weight. Identity was acquired p r i o r to equivalence conservation, e s p e c i a l l y at the lower age l e v e l s . In a s i m i l a r study Toniolo and Hooper (1975) investigated the identity-equivalence issue i n the content areas of length and weight. In a d d i t i o n to the tasks from the Brainerd and Hooper (1975) study tasks measuring t r a n s i t i v i t y of length and weight were given to 60 four year olds, 60 s i x year olds and 60 eight year olds. The r e s u l t s supported the notion that i d e n t i t y i s acquired p r i o r to equivalence conservation. The r e s u l t s of other studies have f a i l e d to provide support f o r the developmental p r i o r i t y of i d e n t i t y over equivalence conservation. 6. Northman and Gruen (1970) did not obtain the hypothesized sequence f o r l i q u i d quantity i n 60 second and t h i r d graders. Moynahan and G l i c k (1972) presented 96 kindergarten and f i r s t - g r a d e children with i d e n t i t y and equivalence tasks i n v o l v i n g number, length, continuous quantity and weight. Identity preceded equivalence only under length transformations. On the number, quantity and weight tasks i d e n t i t y and equivalence conservation were found to co-occur. Murray (1970) f a i l e d to f i n d the developmental sequence with 33 kindergarten and f i r s t - g r a d e subjects i n the content areas of weight and number. F i n a l l y , Koshinsky and H a l l (1973) used 72 kindergarten and second-grade subjects i n an experiment r e p l i c a t i n g Hooper's study (1969b) with the exception that a within subject design was used. They f a i l e d to f i n d the sequence. As noted above, according to Elkind's a n a l y s i s , the main reasons for f a i l u r e on the equivalence tasks would be the lack of the i d e n t i t y concept or the i n a b i l i t y to make t r a n s i t i v e inferences. Moynahan and G l i c k (1972) have suggested that the reason for t h e i r not f i n d i n g evidence of the sequence i s that the a b i l i t y to make t r a n s i t i v e inferences was so r e a d i l y a v a i l a b l e to the subjects i n t h e i r study that the i d e n t i t y and equivalence tasks were of equal d i f f i c u l t y , even though the equivalence task has an inference requirement. S i m i l a r l y , Northman and Gruen (1970) proposed that the a b i l i t y to make t r a n s i t i v e inferences occurs at about the same time as the operations necessary f o r i d e n t i t y conservation, hence i d e n t i t y and equivalence tasks are of equal d i f f i c u l t y . Further, the research of Bryant and Trabasso (1971) implies that c h i l d r e n as young as four years of age, which i s below the usual age for attainment of conservation, can make t r a n s i t i v e inferences i n v o l v i n g length provided they are given memory support. 7. Brainerd (1973) has reviewed Piaget's t h e o r e t i c a l p o s i t i o n regarding the order of emergence of t r a n s i t i v i t y , c l a s s i f i c a t i o n and conservation ( i . e . , equivalence). According to Brainerd i t i s not at a l l c l e a r what predictions should be derived from the various statements made by Piaget concerning t r a n s i t i v i t y and conservation. His writings can be interpreted as p r e d i c t i n g that (a) conservation and t r a n s i t i v i t y should emerge synchronously or, (b) conservation should precede t r a n s i t i v i t y developmentally. Brainerd has also reviewed the contradictory r e s u l t s concerning the order of emergence of these two concepts. He draws attention to the methodological i n s e n s i t i v i t i e s i n many of the t r a n s i t i v i t y tasks used to assess the order of emergence of these two concepts. His own data support the order of emergence as being t r a n s i t i v i t y p r i o r to conser-vation. I f i t i s true that t r a n s i t i v i t y emerges p r i o r to equivalence conservation i n a l l or most content areas, then provided that Elkind's analysis i s accepted, several questions regarding the r e s u l t s of the studies supporting the p r i o r i t y of i d e n t i t y over equivalence conservation can be rai s e d . The problems are concerned with the order of a c q u i s i t i o n of t r a n s i t i v i t y and i d e n t i t y conservation. The s p e c i f i c problem which formed the basis of the present study was the order of a c q u i s i t i o n of t r a n s i t i v i t y , i d e n t i t y conservation and equivalence conservation. I t was reasoned that i f t r a n s i t i v i t y precedes equivalence conservation as Brainerd (1973) indicates and i f i d e n t i t y precedes equivalence conservation then t r a n s i t i v i t y would develop p r i o r to, concurrent with, or a f t e r i d e n t i t y conservation. I f i t were found that t r a n s i t i v i t y develops p r i o r to or concurrent with i d e n t i t y conservation then i t would be necessary to question whether equivalence conservation i s a l a t e r cognitive a c q u i s i t i o n than i d e n t i t y conservation, 8. for according to Elkind's analysis (1967) i t i s the lack of the t r a n s i t i v e reasoning a b i l i t y which accounts for f a i l u r e on the t r a d i t i o n a l or equivalence task. I f , however, t h i s i s present as a s k i l l i n subjects who pass i d e n t i t y and f a i l equivalence Elkind's analysis should be s e r i o u s l y questioned. Indeed the only order of emergence that would be congruent with Elkind's analysis and the evidence which suggests i d e n t i t y develops p r i o r to equivalence i s that i d e n t i t y conservation precedes t r a n s i t i v i t y which i n turn precedes equivalence conservation. Then i t could be argued that the reason f o r f a i l u r e on the equivalence task, when the i d e n t i t y task has been passed s u c c e s s f u l l y , i s the lack of the a b i l i t y to form t r a n s i t i v e inferences and i f a l l subjects have been assessed on a l l three tasks t h i s may be supported by empirical evidence. There may however be other reasons for a subject to pass i d e n t i t y tasks and f a i l equivalence tasks. According to Piaget a very important factor i n the assessment of conservation i s that of perceptual c o n f l i c t . Piaget believes that for conservation to be assessed properly there should be a c o n f l i c t between the subject's immediate perceptual experiences and his i n t e l l e c t u a l operations. Thus i n the t r a d i t i o n a l assessment format the objects, a f t e r a perceptual transformation, may look s u f f i c i e n t l y d i f f e r e n t that one object may appear q u a n t i t a t i v e l y d i f f e r e n t from the other. However, i f the c h i l d r e n are reasoning c o r r e c t l y they understand that i f the objects had the same amount to begin with, they s t i l l have the same amount i f only the shape of one of the objects has been changed. Piaget and Inhelder (1974, p. 10) state "the problem of conservation r e f l e c t s a c o n f l i c t between d i r e c t experience or perceptions and r a t i o n a l operations". I t i s c l e a r that t h i s c o n f l i c t should be present within the task s i t u a t i o n i n order to provide a measure of conservation as Piaget conceives i t . In 9. another place he states "the d i r e c t and immediate pouring of a l i q u i d from two i d e n t i c a l glasses, one remaining untouched as a means of comparison i s not the same thing as pouring a l i q u i d from a s i n g l e receptor into others" (Piaget, 1967, p. 533). The i d e n t i t y task, as described previously, removes t h i s perceptual c o n f l i c t between the transformed and untransformed stimulus and as such may remove a source of d i f f i c u l t y encountered by subjects i n the equivalence task. This could also account for the d i f f e r e n c e between the developmental a c q u i s i t i o n of i d e n t i t y and equivalence conservation. The r o l e of perceptual c o n f l i c t has been further emphasized by Piaget i n the case of c h i l d r e n i n a t r a n s i t i o n a l stage of conservation a c q u i s i t i o n . A small perceptual change i n the transformed object may not produce the same amount of perceptual c o n f l i c t as would a more extreme transformation. Piaget and Inhelder (1974, p. 12) state, i n the context of discussing c h i l d r e n i n a t r a n s i t i o n a l state of conservation a c q u i s i t i o n that " i n small scale transformations the c h i l d ' s mind can surmount the perceptible appearances thanks to a grasp of the operations, but as soon as the deformations go beyond a c e r t a i n l i m i t , d i r e c t i n t u i t i o n comes to p r e v a i l over operational i n t e l l i g e n c e and conservation i s again c a l l e d into question". The present study was undertaken to delineate the r e l a t i v e importance of the l o g i c a l (according to Elkind's analysis) and perceptual components i n the s o l u t i o n of i d e n t i t y and equivalence conservation tasks. The order of emergence of t r a n s i t i v i t y , i d e n t i t y and equivalence conser-vation was assessed i n the content area of continuous s o l i d quantity. The i d e n t i t y and equivalence tasks were presented as outlined above with regard to Elkind's a n a l y s i s , and i n addition a modified version of each task was 10. given. Thus there were two identity tasks, one with perceptual conflict absent and one with this conflict present. Similarly there were two equivalence tasks, one with the conflict absent and one with i t present. Each of the conservation tasks was administered under a "moderate" and an "extreme" degree of perceptual transformation. METHOD 11. Sub.j ects Subjects were drawn from kindergarten, f i r s t and second grade i n schools located i n a broadly lower middle class neighbourhood. There were 24 males and 24 females at each grade l e v e l . Four kindergarteners and two f i r s t graders had to be replaced because of f a i l u r e on the pretest. The mean ages for the Kindergarten, Grade one and Grade two groups were 5 years, 11 months (S.D.= 4 months), 6 years, 10 months (S.D.= 4 months), and 7 years, 10 months (S.D.= 5 months) re s p e c t i v e l y . A t o t a l of 144 subjects completed the study. Materials Blue clay ; b a l l s (play dough) were used to assess conservation and both blue and red b a l l s of clay were used to assess t r a n s i t i v i t y . White b a l l s and yellow b a l l s were used i n the pretest. The materials were displayed approximately i n the center of a sheet of white cardboard 81 x 61 cm. which was on a small rectangular wooden table. Two small p l a s t i c bowls one yellow and one red, were also used i n the t r a n s i t i v i t y tasks. Only those materials i n use during any p a r t i c u l a r task were i n the subject's s i g h t . Procedure Each c h i l d was taken i n d i v i d u a l l y to the experimental room and was seated at the small rectangular table opposite the experimenter. The experimenter described the s i t u a t i o n as a game i n which some questions about clay b a l l s would be asked. 12. In addition to the equivalence task outlined by Elkind (1967) , a modification of the equivalence task s i m i l a r to Hooper's (1969b) equivalence I task was used. In t h i s task the i n i t i a l equality of the two objects was established. Then p r i o r to any conservation questions or transformations the standard stimulus was removed from the subject's s i g h t . This made the task comparable to the i d e n t i t y task described above i n terms of both the memory requirements and the perceptual information a v a i l a b l e to the subject. Although the source of perceptual c o n f l i c t (as conceived by Piaget) was removed the l o g i c a l requirements for the proper s o l u t i o n of the two equivalence tasks should have been the same. The t r a d i t i o n a l task was referred to as equivalence A and the modified task as equivalence B. A modification of the i d e n t i t y task as outlined by E l k i n d (1967) was also included i n the present study. In t h i s modified task a standard comparison object which was q u a n t i t a t i v e l y and perceptually i d e n t i c a l to the object to be transformed was i n the subject's view during the course of the task presentation. A l l questions asked were i d e n t i c a l to those asked i n the other i d e n t i t y condition. The modified task introduced perceptual c o n f l i c t since there were two objects, one transformed the other untrans-formed, i n the subject's f i e l d of view. This made the task perceptually i d e n t i c a l to the t r a d i t i o n a l (equivalence A) conservation task. However since the questions asked were the same as those asked i n the other i d e n t i t y task i t should not have required the deductive a b i l i t y for proper s o l u t i o n . The modified task was r e f e r r e d to as i d e n t i t y B and the i d e n t i t y task described by E l k i n d as i d e n t i t y A. Each of the conservation tasks was presented under two l e v e l s of transformation, moderate and extreme. The cross s e c t i o n a l diameter of the b a l l used i n a l l conservation tasks was 4.45 cm. Under the moderate 13. transformation the b a l l was fl a t t e n e d changing the diameter to approximately 6.35 cm.; t h i s was ref e r r e d to as a " f a t cookie". The b a l l was pressed into a "pancake" with a diameter of approximately 14 cm. under the extreme transformation. In a l l conservation tasks the moderate transformation preceded the extreme. Four tasks were given to each subject to assess t h e i r a b i l i t y to form t r a n s i t i v e inferences. The tasks can be described i n schematic form using the symbol ">" to stand for more clay and "=" to stand for the same amount of clay. Each of the tasks involved three b a l l s of clay and can be schematically represented as follows: (A) A=B=C, (B) A>B=C, and (C) A=B>C, and (D) A>B>C. These tasks were s i m i l a r to some of the tasks used by Murray and Youniss (1968), Youniss and Murray (1970) and Brainerd (1973). In addition to the tasks described above a pretest was given to each subject i n order to assess understanding of the r e l a t i o n a l terms more, less and same amount. The c h i l d r e n were divided into two groups: Group I received the pretest, the four t r a n s i t i v i t y tasks and i d e n t i t y A and equivalence A under both the moderate and extreme transformations. Group I I received the pretest, the four t r a n s i t i v i t y tasks and i d e n t i t y B and equivalence B under both l e v e l s of transformation. The order i n which the i d e n t i t y , equivalence and t r a n s i -t i v i t y tasks occurred was f u l l y randomized across subjects. Pretest There were four d i f f e r e n t tasks i n the pretest. Two b a l l s of clay d i f f e r i n g i n s i z e were used i n each of the f i r s t two presentations. The degree of dif f e r e n c e i n the amount of clay between the.two b a l l s was less i n 14. the second presentation than in the f i r s t . Subjects were asked the following questions: (1) Do these two balls of clay have the same amount of clay, do they each have just as much? (2) Does one have more clay? (3) Which one has more clay? (The subject was instructed to point.) (4) Does one have less clay? (5) Which one has less clay? The third part of the pretest involved three balls of clay differing in size. The questions asked were: (1) Do these three balls a l l have the same amount of clay, do they each have just as much? (2) Does one b a l l have more clay than the other two? (3) Which one? (4) Does one b a l l have less clay than the other two? (5) Which one? The fourth part of the pretest involved four balls of clay, two differing in size and two identical. The only question asked of the subject was: (1) Which two of these four balls look lik e they have the same amount of clay? Conservation Tasks Each of the conservation tasks included three phases: prediction, judgment and explanation. (See Brainerd & Brainerd, 1972; Brainerd & Hooper, 1975; and Elkind, 196L) Identity A. One blue b a l l of clay was placed on the white card-board in front of the subject. In the prediction phase the experimenter asked the following questions: (1) If I press this b a l l into the shape of a fat cookie (pancake) w i l l i t have the same amount of clay as i t has now? (2) Will i t have more clay than i t has now? (3) Will i t have less clay than i t has now? In the judgment phase the ba l l was transformed and the subject was asked: (1) Does this fat cookie (pancake) have the same amount of clay 15. as i t had when i t was shaped l i k e a b a l l ? (2) Does i t have more clay now than i t had before? (3) Does i t have l e s s clay now than i t had before? (The order of the questions in v o l v i n g the terms more, les s and same was randomized for the p r e d i c t i o n and judgment phase of a l l conservation tasks.) F i n a l l y the c h i l d was asked to explain his/her answer to the judgment phase by answering the questions: Why do you say i t has (more, les s or the same)? How do you know i t has (more, l e s s , or the same)? Identity B. The presentation of t h i s task was i d e n t i c a l to that of i d e n t i t y A except for the fact that two clay b a l l s were i n the subject's view. Also, before the questions were asked the quantitative equivalence of the two b a l l s was established by having the c h i l d agree that the two b a l l s had the same amount of clay. A l l of the questions were i d e n t i c a l to those asked i n the i d e n t i t y A task. Equivalence A. Two b a l l s were placed on the white cardboard sheet and t h e i r i n i t i a l q u antitative equivalence was established by having the c h i l d agree that the two b a l l s had the same amount of clay. In the pre-d i c t i o n phase the experimenter asked: (1) If I press t h i s b a l l (E pointed to one of the b a l l s ) into a f a t cookie (pancake) w i l l t h i s b a l l (E pointed to standard) have the same amount of clay as the f a t cookie (pancake)? (2) W i l l one of them have more clay? (3) W i l l one of them have less clay? Then i n the judgment phase one b a l l was transformed and the following questions were asked: (1) Do the b a l l and the f a t cookie (pancake) have the same amount of clay? (2) Does one have more? (3) Does one have less? F i n a l l y the c h i l d r e n were required to explain t h e i r responses by answering the questions: Why do you say that one has (more, less or the same amount)? 16. Equivalence B. Two b a l l s were placed on the white cardboard sheet and t h e i r i n i t i a l equivalence was established. Before the p r e d i c t i o n phase began one b a l l was placed out of sight behind a screen. The same questions were asked as i n equivalence A except that reference was made to the b a l l behind the screen when necessary. T r a n s i t i v i t y . Two b a l l s A and B were f i r s t placed on the table. The c h i l d was then t o l d that one b a l l (E pointed to A which was always red) had the same amount ( i n cases (A) and (C) of the t r a n s i t i v i t y tasks described above) or more ( i n the cases of (B) and (D) described above) than the other b a l l B which was always blue. The red b a l l (A) was then put under a small yellow or red bowl. Another red b a l l (C) was then placed on the table and the c h i l d was t o l d that the blue b a l l (B) had the same amount ( i n cases (A) and (B) or more than C ( i n cases (C) and (D)). C was then placed under the other bowl and B was removed from s i g h t . The following questions were then asked: (1) Do the b a l l s under the two bowls have the same amount of clay, do they each have j u s t as much? (2) Does one have more? (3) Which one? (4) Does one have less? (5) Which one? The questions were asked i n d i f f e r e n t orders. Scoring Pretest. Subjects were given a score of 1 or 0 depending on t h e i r answers to each of the questions asked during the pretest. Since there were 16 questions i n t o t a l the maximum score was 16. Only subjects who scored 15 or 16 were allowed to proceed i n the study. 17. Conservation. Three questions were asked i n the p r e d i c t i o n phase and three questions were asked i n the judgment phase. I f a c h i l d was correct on a l l three questions i n the p r e d i c t i o n phase he/she was given a score of 1, s i m i l a r l y i f a c h i l d was correct on a l l three questions i n the judgment phase he/she was given a score of 1. These scores could then be added to give a composite score for both phases. Children who were correct on both phases obtained a score of 2, ch i l d r e n correct on only one phase obtained a score of 1, and c h i l d r e n i n c o r r e c t on both phases were given a score of 0. When children's scores depended only on the p r e d i c t i o n and judgment phases the scores were said to be based on a judgment only c r i t e r i o n . When a judgment plus explanation c r i t e r i o n was used a c h i l d was given a score of 2 i f he/she was correct on both p r e d i c t i o n and judgment phases and gave an explanation which could be placed i n an acceptable category (see Appendix A). I f a c h i l d was correct on both the p r e d i c t i o n and judgment phases, but gave an inadequate explanation he/she received a score of 1. A l l explanations were tape recorded. T r a n s i t i v i t y . Children were given a score of 1 i f they answered a l l questions c o r r e c t l y ; otherwise they received a score of 0. 18. Results Conservation The frequency and percentage of subjects passing the various conservation tasks on the basis of a judgment only c r i t e r i o n i s given i n Table 1. I t i s apparent that while very few Kindergarten subjects passed the tasks considerably more Grade 1 and Grade 2 c h i l d r e n were successful. I t i s also evident from Table 1 that c h i l d r e n r e c e i v i n g the i d e n t i t y task as outlined by E l k i n d (1967) and the t r a d i t i o n a l conservation task (Group I) performed better on a l l conservation tasks than those c h i l d r e n who received the modified tasks (Group I I ) . There was very l i t t l e d i f f e r e n c e i n the number of subjects passing i d e n t i t y tasks and equivalence tasks within each group. Twenty-seven subjects i n Group I passed the i d e n t i t y task under both l e v e l s of transformation and 27 passed the equivalence tasks under both l e v e l s of transformation. In Group. II 13 subjects passed both i d e n t i t y tasks and 16 passed both equivalence tasks. Conservation task scores were subjected to a 2x2x3x2x2 analysis of variance i n which the v a r i a b l e s were Group (Group I, Group I I ) , Sex, Age (Kindergarten, Grade 1, Grade 2), Task Type (Identity, Equivalence) and Transformation l e v e l (Moderate, Extreme). As can be seen from Table 2 the only main e f f e c t s to reach s i g n i f i c a n c e were Group (F=6.02; df=l,132, _p_<.025) and Age (F=11.92; df.=2,132; £.<.001). The mean score for c h i l d r e n i n Group I was s i g n i f i c a n t l y greater than the mean score for Group I I . Duncan's New M u l t i p l e Range S t a t i s t i c indicated that only the d i f f e r e n c e between the scores of c h i l d r e n i n kindergarten and Grade 2 were s t a t i s t i c a l l y s i g n i f i c a n t (p_<.05). No other age d i f f e r e n c e s reached s t a t i s t i c a l s i g n i f i c a n c e . TABLE 1 Frequency and Percentage of Subjects Passing the Conservation Tasks when a C r i t e r i o n of Judgment Only was Used. IDENTITY Moderate Extreme EQUIVALENCE Moderate Extreme Group I No Kindergarten Males 2 Females 1 Combined 3 Grade 1 Males 8 Females 7 Combined 15 Grade 2 Males 4 Females 7 Combined 11 Group I To t a l Males 14 Females 15 Combined 29 % No % No % No % (16.7) 2 (16.7) 1 ( 8.3) 2 (16.7) ( 8.3) 1 ( 8.3) 3 (25.0) 2 (16.7) (12.5) 3 (12.5) 4 (16.7) 4 (16.7) (66.7) 9 (75.0) 8 (66.7) 8 (66.7) (58.3) 7 (58.3) 6 (50.0) 7 (58.3) (62.5) 16 (66.7) 14 (58.3) 15 (62.5) (33.3) 5 (41.7) 5 (41.7) 4 (33.3) (58.3) 8 (66.7) 7 (58.3) 9 (75.0) (45.8) 13 (54.2) 12 (50.0) 13 (54.2) (38.9) 16 (44.4) 14 (38.9) 14 (38.9) (41.7) 16 (44.4) 16 (44.4) 18 (50.0) (40.3) 32 (44.4) 30 (41.7) 32 (44.4) IDENTITY EQUIVALENCE Moderate & Moderate & Extreme Extreme No, % No % 2 (16.7) 1 ( 8.3) 0 ( 0.0) 1 ( 8.3) 2 ( 8.3) 2 ( 8.3) 8 (66.7) 8 (66.7) 7 (58.3) 6 (50.0) 15 (62.5) 14 (58.3) 4 (33.3) 4 (33.3) 6 (50.0) 7 (58.3) 10 (41.7) 11 (45.8) 14 (38.9) 13 (36.1) 13 (36.1) 14 (38.9) 27 (37.5) 27 (37.5) TABLE 1 (cont'd) IDENTITY Moderate Extreme EQUIVALENCE Moderate Extreme Group II Kindergarten Males Females Combined Grade 1 Males Females Combined Grade 2 Males Females Combined Group I I T o t a l Males Females Combined T o t a l Sample Males Females Combined IDENTITY Moderate & Extreme EQUIVALENCE Moderate & Extreme No. % No % No % No % No % No, % 1 ( 8.3) 2 (16.7) 0 ( 0.0) 1 ( 8.3) 1 ( 8.3) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 (0.0) 2 (16.7) 0 ( 0.0) 0 ( 0.0) 1 ( 4.2) 2 ( 8.3) 0 ( 0.0) 3 (12.5) 1 ( 4.2) 0 ( 0.0) 2 (16.7) 3 (25.0) 3 (25.0) 3 (25.0) 2 (16.7) 3 (25.0) 3 (25.0) 2 (16.7) 2 (16.7) 2 (16.7) 2 (16.7) 1 ( 8.3) 5 (20.8) 5 (20.8) 5 (20.8) 5 (20.8) 4 (16.7) 4 (16.7) 3 (25.0) 4 (33.3) 4 (33.3) 4 (33.3) 3 (25.0) 4 (33.3) 5 (41.7) 7 (58.3) 8 (66.7) 9 (75.0) 5 (41.7) 8 (66.7) 8 (33.3) 11 (45.8) 12 (50.0) 13 (54.2) 8 (33.3) 12 (50.0) 6 (16.7) 9 (25.0) 7 (19.4) 8 (22.2) 6 (16.7) 7 (19.4) 8 (22.2) 9 (25.0) 10 (27.8) 13 (36.1) 7 (19.4) 9 (25.0) 14 (19.4) 18 (25.0) 17 (23.6) 21 (29.2) 13 (18.1) 16 (22.2) 20 (27.8) 25 (34.7) 21 (29.2) 22 (30.6) 20 (27.8) 20 (27.8) 23 (31.9) 25 (34.7) 24 (33.3) 31 (43.1) 20 (27.8) 23 (31.9) 43 (29.9) 50 (34.7) 45 (31.3) 53 (36.8) 40 (27.8) 43 (29.9) O Group I - Received the Identity Task as Outlined by Elkind (1967) and the T r a d i t i o n a l Conservation Task referred to as the Equivalence Task by E l k i n d . Group II - Received the Modified Identity and Equivalence Tasks. 21. TABLE 2 Summary of Group x Sex x Grade x Task Type x Level of Transformation Analysis of Variance Source DF Between Subjects Group (A) 1 Sex (B) 1 Grade (C) 2 AB 1 AC 2 BC 2 ABC 2 Error Between 132 Within Subjects Task Type D(A) 2 BD(A) 2 CD(A) 4 BCD(A) ( 4 D x SS w/in gps. 132 Level of Trans. F 1 AF 1 BF 1 CF 2 ABF 1 ACF 2 BCF 2 ABCF 2 F x SS w/in gps. 132 FD(A) 2 BFD(A) 2 BCFD(A) 4 FD(A) x SS w/in gps. 132 MS 14.38 1.09 28.44 0.14 6.48 8.72 0.13 2.39 0.64 0.20 0.95 0.26 0.23 0.43 0.17 0.17 0.11 0.43 0.21 0.14 0.30 0.13 0.29 0.27 0.57 0.14 6.02 ** 0.45 11.92 **** 0.06 2.72 3.65 * 0.05 0.28 0.87 0.41 1.09 0.33 0.01 0.01 0.82 0.33 1.62 1.06 2.34 0.21 1.87 0.40 * P<.05 ** P<.025 **** p<.001 22. The only i n t e r a c t i o n to reach s i g n i f i c a n c e was the i n t e r a c t i o n of Sex x Grade (F=3.65; dF=2,132; p_<.05). A Newman Kuels test indicated that the mean for g i r l s i n Grade 2 was s i g n i f i c a n t l y greater than the mean fo r g i r l s i n Kindergarten (p_<.01). The differe n c e between boys i n Kinder-garten and boys i n Grade 1 was s i g n i f i c a n t (p_<.05) according to Duncan's New Mu l t i p l e Range S t a t i s t i c . Duncan's test was used when the difference f a i l e d to read s t a t i s t i c a l s i g n i f i c a n c e according to the Neuman Kuels t e s t . The frequence and percentage of subjects who passed the various conservation tasks with a c r i t e r i o n of judgment plus-explanation i s shown i n Table 3. Group I can be seen to have performed better than Group I I . Also, fewer chi l d r e n i n Kindergarten than i n Grade 1 or Grade 2 passed the tasks when a judgment plus explanation c r i t e r i o n was used. A comparison of Tables 1 and 3 reveals that when explanations were included i n the c r i t e r i o n fewer subjects passed a l l conservation tasks than when a c r i t e r i o n of judgment only was used. With a c r i t e r i o n of judgment only, 40 subjects i n both groups, passed the i d e n t i t y tasks under both l e v e l s of transformation and 43 subjects passed both equivalence tasks. However with a c r i t e r i o n of judgment plus explanation only 28 subjects passed both i d e n t i t y tasks and 33 passed both equivalence tasks. An a d d i t i o n a l analysis of variance which included c r i t e r i o n (judgment only vs. judgment plus explanation) as a fact o r indicated that the judgment plus explanation c r i t e r i o n resulted i n s i g n i f i c a n t l y lower scores on the conservation tasks (F=30.73; df=1,132; p_<.001). The number of subjects passing i d e n t i t y tasks and equivalence tasks with a c r i t e r i o n of judgment only i s presented i n Table 4 and with TABLE 3 Frequency and Percentage of Subjects Passing the Conservation Tasks When a C r i t e r i o n of Judgment Plus Explanation was Used Group I Group IDENTITY EQUIVALENCE Moderate Extreme Moderate Extreme IDENTITY Moderate & Extreme EQUIVALENCE Moderate & Extreme No, % No % No, % No, % No. % No, % Kindergarten Males 1 ( 8.3) 1 ( 8.3) 1 ( 8.3) 2 (16.7) 1 ( 8.3) 1 ( 8.3) Females 0 (0.0) 1 ( 8.3) 2 (16.7) 1 ( 8.3) 0 ( 0.0) 1 ( 8.3) Combined 1 ( 4.2) 2 ( 8.3) 3 (12.5) 3 (12.5) 1 ( 4.2) 2 ( 8.3) Grade 1 Males 6 (50.0) 7 (58.3) 8 (66.7) 7 (58.3) 5 (41.7) 7 (58.3) Females 4 (33.3) 4 (33.3) 4 (33.3) 5 (41.7) 4 (33.3) 4 (33.3) Comb ined 10 (41.7) 11 (45.8) 12(50.0) 12 (50.0) 9 (37.5) 11 (45.8) Grade 2 Males 3 (25.0) 5 (41.7) 4 (33.3) 3 (25.0) 3 (25.0) 3 (25.0) Females 6 (50.0) 7 (58.3) 7 (58.3) 9 (75.0) 5 (41.7) 7 (58.3) Combined 9 (37.5) 12 (50.0) 11 (45.8) 12 (50.0) 8 (33.3) 10 (41.7) I T o t a l Males 10 (27.8) 13 (36.1) 13 (36.1) 12 (33.3) 9 (25.0) 11 (30.6) Females 10 (27.8) 12 (33.3) 13 (36.1) 15 (41.7) 9 (25.0) 12 (33.3) Combined 20 (27.8) 25 (34.7) 26 (36.1) 27 (37.5) 18 (25.0) 23 (31.9) Moderate No. Group II Kindergarten Males 1 ( 8.3) Females 0 ( 0.0) Combined 1 ( 4.2) Grade 1 Males 2 (16.7) Females 3 (25.0) Combined 5 (20.8) Grade 2 Males 3 (25.0) Females 4 (33.3) Combined 7 (29.2) Group II T o t a l Males 6 (16.7) Females 7 (19.4) Combined 13 (18.1) T o t a l Sample Males 16 (22.2) Females 17 (23.6) Combined 33 (22.9) TABLE 3 (cont'd) Extreme Moderate Extreme Moderate & Moderate & Extreme Extreme No % No % No % No % No % 2 (16.7) 0 ( 0.0) 1 ( 8.3) 1 ( 8.3) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 1 ( 8.3) 0 ( 0.0) 0 ( 0.0) 2 ( 8.3) 0 ( 0.0) 2 ( 8.3) 1 ( 4.2) 0 ( 0.0) 3 (25.0) 2 (16.7) 2 (16.7) 2 (16.7) 2 (16.7) 1 0=8.3) 1 ( 8.3) 1 ( 8.3) 1 ( 8.3) 0 ( 0.0) 4 (16.7) 3 (12.5) 3 (12.5) 3 (12.5) 2 ( 8.3) 4 (33.3) 3 (25.0) 3 (25.0) 3 (25.0) 3 (25.0) 3 (25.0) 5 (41.7) 6 (50.0) 3 (25.0) 5 (41.7) 7 (29.2) 8 (33.3) 9 (37.5) 6 (25.0) 8 (33.3) 9 (25.0) 5 (13.9) 6 (16.7) 6 (16.7) 5 (13.9) -4 :(n.i) 6 (16.7) 8 (22.2) 4 (11.1) 5 (13.9) 13 (18.1) 11 (15.2) 14 (19.4) 10 (13.9) 10 (13.9) 22 (30.5) 18 (25.0) 18 (25.0) 15 (20.8) 16 (22.2) 16 (22.2) 19 (26.4) 23 (31.9) 13 (18.1) 17 (23.6) 38 (26.4) 37 (25.7) 41 (28.5) 28 (19.4) 33 (22.9) 25. a c r i t e r i o n of judgment plus explanation i n Table 5. I t i s evident from a comparison of Tables 4 and 5 that more subjects passed both tasks with a c r i t e r i o n of judgment only than with a c r i t e r i o n of judgment plus explan-ation . T h i r t y - f i v e subjects passed both i d e n t i t y and equivalence tasks at the moderate l e v e l of transformation with a c r i t e r i o n of judgment only, while 25 passed with a c r i t e r i o n of judgment plus explanation. At the extreme l e v e l 44 subjects passed both tasks with a c r i t e r i o n of judgment only and 34 passed with a c r i t e r i o n of judgment plus explanation. When both l e v e l s of transformation are combined i t can be seen that 35 subjects passed a l l tasks with a c r i t e r i o n of judgment only and 23 subjects passed with a c r i t e r i o n of judgment plus explanation. Three subjects passed both i d e n t i t y tasks and f a i l e d both equivalence tasks and 3 subjects passed both equivalence tasks and f a i l e d both i d e n t i t y tasks with a c r i t e r i o n of judgment only as i s evident from Table 4. An examination of Table 5 shows that 2 subjects passed both i d e n t i t y tasks and f a i l e d both equivalence tasks and 1 subject passed both equivalence tasks and f a i l e d both i d e n t i t y tasks with a c r i t e r i o n of judgment plus explanation. I t can also be seen from Table 4 that 14 subjects passed eit h e r the moderate (8 subjects) or extreme (6 subjects) i d e n t i t y tasks, but f a i l e d the corresponding equivalence tasks, while 21 subjects passed e i t h e r the moderate (13 subjects) or extreme (8 subjects) equivalence tasks but f a i l e d the corresponding i d e n t i t y tasks. With a c r i t e r i o n of judgment plus explanation, as i n Table 5, 13 subjects passed ei t h e r the moderate (8 subjects) or extreme (5 subjects) i d e n t i t y tasks but f a i l e d the corre-sponding equivalence tasks and 19 subjects passed either the moderate (12 subjects) or extreme (7 subjects) equivalence tasks and f a i l e d the corre-sponding i d e n t i t y tasks. 26-TABLE 4 Number of Subjects Passing Identity and Equivalence Tasks at Each Grade Level when a C r i t e r i o n of Judgment Only was Used. Passed Both F a i l e d Both Passed Ident. Passed Equiv. F a i l e d Equiv. F a i l e d Ident. M E M&E M E M&E M E M&E M E M&E Group I Kinder-garten 1 3 1 18 20 18 2 0 0 3 1 0 Grade 1 14 14 14 9 8 8 1 2 1 0 0 0 Grade 2 9 12 9 10 10 10 2 1 1 3 1 0 Group I To t a l 24 29 24 37 38 36 5 3 2 6 2 0 Group II Kinder-garten Grade 1 Grade 2 Group II To t a l T o t a l Sample 1 1 1 2 4 2 8 10 8 11 15 11 35 44 35 23 20 20 16 18 16 12 10 10 51 48 46 88 86 82 0 1 0 3 1 1 0 1 0 3 3 1 8 6 3 0 2 0 3 1 1 4 3 2 7 6 3 13 8 3 M = E = Moderate Level of Transformation Extreme Level of Transformation TABLE 5 27. Number of Subjects Passing Identity and Equivalence Tasks at Each Grade Level When a C r i t e r i o n of Judgment Plus Explanation was Used. Passed Both F a i l e d Both Passed Ident. Passed Equiv. F a i l e d Equiv. F a i l e d Ident. M E M&E M E M&E M E M&E M E m Group I Kinder-garten 1 2 1 21 21 20 0 0 0 2 1 0 Grade 1 9 11 8 11 12 11 1 0 0 3 1 l Grade 2 7 10 7 11 10 10 2 2 1 4 2 0 Group I Total 17 23 16 43 43 41 3 2 1 9 4 1 Group II Kinder-garten Grade 1 Grade 2 Group II T o t a l 0 1 0 1 3 1 7 7 6 8 11 7 23 21 21 17 20 17 16 14 14 56 55 52 1 1 0 4 1 1 0 1 0 0 1 2 0 1 2 0 0 0 T o t a l Sample • 25 34 23 99 98 93 8 5 2 12 28. The frequency and percentage of subjects giving adequate explanations on i d e n t i t y conservation, equivalence conservation and a l l conservation tasks are given i n Tables 6, 7 and 8 r e s p e c t i v e l y . The c l a s s i f i c a t i o n scheme for the various types of explanations i s described i n Appendix A. It can be seen from the tables that the number of adequate explanations given by Grade 2 subjects was greater than the number given by Grade 1 subjects which i n turn was greater than the number given by Kindergarten subjects. I t can be seen from Table 8 that the addition-subtraction and statement of operations categories account for approximately 67% of the explanations given by Kindergarten subjects on a l l tasks. Four categories used by Grade 1 subjects account for 88.3% of t h e i r responses; addition-subtraction, statement of operations, reference to previous state and inversion. These same four categories plus the use of the "more than one category" category account for 91.8% of the second graders' explanations. However the "more than one category" category i s composed t o t a l l y of composite explanations from the other four categories. The three categories of r e c i p r o c i t y , sameness (same stimulus) and sameness (same quantity) account for only 9.6% of the explanations o v e r a l l . The category of compensation was never used. Comparison of Tables 6 and 7 reveals that the categories of state-ment of operations, reference to previous state, and inversion were used more frequently on equivalence tasks than on i d e n t i t y tasks. The a d d i t i o n -subtraction category was used more frequently on i d e n t i t y tasks than on equivalence tasks. TABLE 6 Frequency and Percentage of Subjects Giving Adequate Explanations for a l l Identity Conservation Tasks Addition -Subtration Statement of Operations Reference to Previous state Group I Group II Group I and Group II Total KDG. GR. 1 GR. 2 KDG. GR. 1 GR. 2 KDG. GR. 1 GR. 2 No. % No. % No. % No. % No. % No. % No. % No. % No. % No. % 2 (50.0) 8 (38.1) 3 (11.5)2(66.7) 3 (27.3) 4 (22.2) 4 (57.1)11 (34.4) 7 (15.9) 22 (26.5) 0 (0.0) 9 (42.9)10 (38.5)0 (0.0) 3 (27.3) 5 (27.8) 0 (0.0)12 (37.5)15 (34.1) 27 (32.5) 0 (0.0) 0 (0.0) 2 ( 7.7) 0 (0.0) 3 (27.3) 4 (22.2) 0 (0.0) 3 (9.4) 6 (13.6) 9 (10.8) Inversion 0 (0.0) 1 (4.8) 1 ( 3.8)0 (0.0) 0 ( 0.0) 1 (5.6) 0 (0.0) 1 (3.1) 2 (4.5) 3 (3.6) Reciprocity 0 (0.0) 0 (0.0) 0 ( 0.0) 0 (0.0) 1 ( 9.1) 0 (0.0) 0 (0.0) 1 (3.1) 0 (0.0) 1 (1.2) Compensation Sameness (same stimulus) Sameness (same quantity) More than one category No. and % of Explanations 0 (0.0) 0 (0.0) 0 ( 0.0) 0 (0.0) 0 ( 0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 2 (50.0) 1 (4.8) 0 ( 0.0) 0 (0.0) 1 ( 9.1) 0 (0.0) 2 (28.6) 2 (6.3) 0 (0.0) 4 (4.8) 0 (0.0) 2 (9.5) 5 (19.2)0 (0.0) 0 ( 0.0) 0 (0.0) 0 (0.0) 2 (6.3) 5 (11.4) 7 (8. 4) 0 (0.0) 0 (0.0) 4 (19.2) 1 (33.3) 0 ( 0.0) 4 (22.2) 1 (14.3) 0 (0.0) 9 (20.5) 10 (12.0) 4(100.0) 2L(100.0)26(100.0) 3(100.0) 11(100.0)18(100,0) 7(100.0^2.(100.0X44.(100._0.) . 83(100-.0) TABLE 7 Frequency and Percentage of Subjects Giving Adequate Explanations for a l l Equivalence Conservation Tasks GROUP I ' GROUP II GROUP I AND GROUP II TOTAL KDG. GR. 1 GR. 2 KDG. GR. 1 GR. 2 KDG. GR. 1 GR. 2 No. % No. % No. % No. % No. % No. % No. % No. % No. % No. % Addition-Subtraction 2 (28.6) 6 (25.0) 3 (12.0) 1 (25.0) 1 (10.0) 3 (17.6) 3 (27.3) 7 (20.6) 6 (14.3)16 (18.4) Statement of Operations 2 (28.6) 7 (29.2) 9 (36.0) 3 (75.0) 5 (50.0) 6 (35.3) 5 (45.5)12 (35.3)15 (35v7)32 (36.8) Reference to Previous state 2 (28.6) 5 (20.8) 5 (20.0) 0 ( 0.0) 3 (30.0) 6 (35.3) 2 (18.2) 8 (23.5)11 (26.2)21 (24.1) Inversion 0 ( 0.0) 3 (12.5) 2 ( 8.0) 0 ( 0.0) 1 (10.0) 1 ( 5.9) 0 ( 0.0) 4 (11.8) 3 ( 7.1) 7 (16.7) Reciprocity 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) Compensation 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) Sameness (same stimulus) 0 ( 0.0) 1 ( 4.2) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 1 ( 2.9) 0 ( 0.0) 1 ( 2.4) Sameness (same quantity) 1 (14.3) 2 ( 8.3) 2 ( 8.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 1 ( 9.1) 2 ( 5.9) 2 ( 4.8) 5 (11.9) More than one category 0 ( 0.0) 0 ( 0.0) 4 (16.0) 0 ( 0.0) 0 ( 0.0) 1 ( 5.9) 0 ( 0.0) 0 ( 0.0) 5 (11.9) 5 (11.9) No. and % of Explanations 7(100.0)24(100.0)25(100.0) 4(100.0)10(100.0)17(100.0)11(100.0)34(100.0)42(100.0)87(100.0) o TABLE 8 Frequency and Percentage of Subjects Giving Adequate Explanations for a l l Conservation Tasks Addition -Subtraction GROUP I KDG. GR. 1 GR. 2 KDG. No. % No. % No. % No. % GROUP II GR. 1 GR. 2 No. % No. % GROUP I AND GROUP II KDG. GR. 1 GR. 2 TOTAL No. % No. % No. % No. % 4 (36.4)14 (31.2) 6 (11.8) 3 (42.9) 4 (18.2) 7 (20.6) 7 (38.9)18 (26.9)13 (15.3)38 (22.4) Statement of Operations 2 (18.2)16 (35.6)19 (37.3)23 (42.9) 8 (36.4)11 (32.4) 5 (27.8)24 (35.9)30 (35.3)59 (34.7) Reference to Previous state 2 (18.2) 5 (11.1) 7 (13.8) 0 ( 0.0) 6 (27.3)10 (29.5) 2 (11.1)11 (16.5)17 (20.0)30 (17.6) Inversion 0 ( 0.0) 4 ( 8.9) 3 ( 5.9) 0 ( 0.0) 2 ( 9.1) 1 ( 3.0) 0 ( 0.0) 6 ( 9.0) 4 ( 4.7)10 ( 5.9) Reciprocity 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 1 ( 4.6) 0 ( 0.0) 0 ( 0.0 1 ( 1.5) 0 ( 0.0) 1 ( 0.6) Compensation 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) Sameness (same stimulus) 2 (18.2) 2 ( 4.4) 0 ( 0.0) 0 ( 0.0) 1 ( 4.6) 0 ( 0.0) 2 (11.1) 3 ( 4.5) 0 ( 0.0) 5 ( 2.9) Sameness (same quantity) 1 ( 9.1) 4 ( 8.9) 7 (13.8) 0 ( 0.0) 0 ( 0.0) 0 ( 0.0) 1 ( 5.6) 4 ( 6.0) 7 ( 8.3)12 ( 7.1) More than one category 0 ( 0.0) 0 ( 0.0) 9 (17.7) 1 (14.3) 0 ( 0.0) 5 (14.7) 1 ( 5.6) 0 ( 0.0)14 (16.5)15 ( 8.8) No. and % of Explanations 11(100.0)45(100.0)51(100.0) 7(100.0)22(100.0)34(100.0)18(100.0)67(100.0)85(100.0)170(L00.0) 32. Scoring with a 0-6 scale The frequency and percentage of subjects scoring from 0-6 on the various conservation tasks i s given i n Table 9. The scores were derived by assigning subjects a 1 for each correct answer involving the r e l a t i o n a l terms "same", "more" and " l e s s " used i n both the p r e d i c t i o n and judgment phases. This i s the scoring scheme employed by Brainerd and Hooper (1975) and Toniolo and Hooper (1975). Subjects were given an 0 for each incorrect answer. Since there were s i x questions i n a l l , excluding explanations, a maximum score of 6 was possible. As can be seen from the table, subjects received a score of 0 on 69 (24%) of the i d e n t i t y tasks and on 124 (44.3%) of the equivalence tasks. There were 57 (19.8%) tasks on which subjects received a score of 1 and 26 (9.0%) tasks on which subjects received a score of 2 on i d e n t i t y conservation. On the equivalence conservation tasks subjects received a score of 1 on 32 (11.1%) of the tasks and a score of 2 on 9 (3.1%) of the tasks. Subjects received a score of 3 on 13 (4.5%) of both the i d e n t i t y and equivalence conservation tasks. There were 16 (5.6%) i d e n t i t y conservation tasks on whichvsubjects received a score of 4, and 14 (4.9%) tasks on which subjects received a score of 5. On the equivalence conservation tasks subjects received a score of 4 on 5 (1.7%) tasks and a score of 5 on 5 (1.7%) tasks. Ninety-three (32.3%) i d e n t i t y conservation tasks were answered p e r f e c t l y and 100 (34.7%) equivalence conservation tasks were answered p e r f e c t l y . An a d d i t i o n a l analysis of variance performed on the data scored with a 0-6 scale i n the manner presented i n Table 9 indicated that performance on the equivalence tasks was s i g n i f i c a n t l y better than perform-Score Obtained Group I Identity Moderate Ident i t y Extreme Group I I Ident i t y Moderate Identity Extreme T o t a l f o r Identity Group I Equivalence Moderate Equivalence Extreme Group II Equivalence Moderate Equivalence Extreme T o t a l f o r Equivalence TABLE 9 Frequency and Percentage Which Subjects Obtained 1 2 No. % No. % 11 (15.3) 1 8 (11.1) 7 ( 9.7) 6 ( 8.3) of Tasks On the S p e c i f i e d Values 3 4 No. % No. % 2 ( 2.8) 1 5 ( 6.9) 1 2 ( 2.8) 4 ( 5.6) 0 No. % 13 (18.1) 19 (26.4) 15 (20.8) 22 (30.6) 69 (24.0) 24 (33.3) 30 (41.7) 31 (43.1) 39 (54.2) 124 (44.3) 21 (29.2) 18 (25.0) 57 (19.8) 11 (15.3) 3 ( 4.2) 11 (15.3) 7 ( 9.7) 32 (11.1) 6 ( 8.3) 6 ( 8.3) 26 ( 9.0) 1 ( 1.4) 3 ( 4.2) 3 ( 4.2) 2 ( 2.8) 9 ( 3.1) 7 ( 9.7) 2 ( 2.8) 13 ( 4.5) 5 ( 6.9) 2 ( 2.8) 5 ( 6.9) 1 ( 1.4) 13 ( 4.5) 7 ( 9.7) 0 ( 0.0) 16 ( 5.6) 0 ( 0.0) 1 ( 1.4) 3 ( 4.2) 1 ( 1.4) 5 ( 1.7) 5 No. % 1 4 ( 5.6) 2 ( 2.8) 2 ( 2.8) 6.( 8.3) 14 ( 4.9) 1 ( 1.4) 1 ( 1.4) 2 ( 2.8) 1 ( 1.4) 5 ( 1.7) 6 No. % '29 (40.3) 32 (44.4) 14 (19.4) 18 (25.0) 93 (32.3) 30 (41.7) 32 (44.4) 17 (24.6) 21 (29.2) 100 (34.7) 34. ance on i d e n t i t y tasks (F=8.85; dF=l,132; p_<.01). T r a n s i t i v i t y The general performance pattern on the t r a n s i t i v i t y tasks for each grade within each group i s presented i n Table 10. The t o t a l number of subjects who passed each task varied l i t t l e among the four tasks. Considering the t o t a l sample 76 subjects passed task A, 74 passed task B, 84 passed task C, and 77 subjects passed task D. Within group t o t a l s on each of the four tasks were very s i m i l a r , however more subjects i n Group I passed tasks A, B and D and more i n Group II passed task C. T r a n s i t i v i t y tasks were subjected to a 2x2x3x4 analysis of variance i n which the v a r i a b l e s were Group (Group I, Group I I ) , Sex, Age (Kindergarten, Grade 1, Grade 2), and Task Type (A, B, C, and D). As can be seen from Table 11 the only main e f f e c t s to reach s i g n i f i c a n c e were the main e f f e c t s f or group (F=6.84; dF=l,132; p_<.01) and grade (F=22.30; dF=2,132; jK.001)"'". Group I performed s i g n i f i c a n t l y better than Group I I . Newman Kuels tests performed on the grade means showed that Grade 2 c h i l d r e n scored s i g n i f i c a n t l y higher than c h i l d r e n i n Grade 1 (p_<.01) and Kinder-garten (p_<.01), who did not d i f f e r . The frequency and percentage of subjects passing s p e c i f i e d numbers of t r a n s i t i v i t y tasks i s given i n Table 12. As can be seen from the table only a very small percentage (4.2% of the t o t a l sample) f a i l e d a l l t r a n s i t i v i t y tasks. At the other extreme only 11.1% of the t o t a l sample passed a l l t r a n s i t i v i t y tasks. The percentage of subjects who passed a number of tasks between these extremes decreases from 95% who passed 1 TABLE 10 Number and Percentage of Subjects Passing the Four T r a n s i t i v i t y Tasks 35. \ Task A Task B Task C Task D Task Schema A= =B=C A>B=C A= =B>C A> B>C No. I r No. % No. % No. % Group I Kindergarten Males 3 (25. 0) 8 (66. 7) 7 (58. 3) 3 (25. 0) Females 5 (41. 7) 6 (50. 0) 8 (66. 7) 9 (75. 0) Combined 8 (33. 3) 14 (58. 3) 15 (62. 5) 12 (50. 0) Grade 1 Males 10 (83. 3) 3 (25. 0) 3 (25. 0) 7 (58. 3) Females 6 (50. 0) 5 (41. 7) 7 (58. 3) 7 (58. 3) Combined 16 (66. 7) 8 (33. 3) 10 (41. 7) 14 (58. 3) Grade 2 Males 7 (58. 3) 12 (100. 0) 9 (75. 0) 9 (75. 0) Females 11 (91. 7) 9 (75. 0) 7 (58. 3) 9 (75. 0) Combined 18 (75. 0) 21 (87. 5) 16 (66. 7) 18 (75. 0) Group 1 Tot a l Males 20 (55. 6) 23 (63. 9) 19 (52. 8) 19 (52. 8) Females 22 (61. 1) 20 (55. 6) 22 (61. 1) 25 (69. 4) Combined 42 (58. 3) 43 (59. 7) 41 (56. 9) 44 (61. 1) 36. TABLE 10 (cont'd) Group II T a s k A T a s k B T a s k c Kindergarten Task D Males 5 (41 .7) 4 (33 .3) 4 (33 .3) 2 (16 • 7) Females 2 (16 .7) 4 (33, .3) 8 (66 .7) 3 (25 .0) Combined 7 (29 • 2) 8 (33, .3) 12 (50 .0) 5 (20 • 8) Grade 1 Males 6 (50 .0) 4 (33. .3) 6 (50, .0) 5 (41, .7) Females 3 (25 • 0) 5 (41. • 7) 7 (58, .3) 8 (66, .7) Comb ined 9 (37 .5) 9 (37. • 5) 13 (54. .2) 13 (54, • 2) Grade 2 • Males 9 (75 • 0) 9 (75. 0) 9 (75. • 0) 7 (58. .3) Females 9 (75 .0) 5 (41. 7) 9 (75. 0) 8 (66. .7) Combined 18 (75 • 0) 14 (58. 3) 18 (75. • 0) 15 (62. 5) Group II T o t a l Males 20 (55, • 6) 17 (47. 2) 19 (52. 8) 14 (38. 9) Females 14 (38, .9) 14 (38. 9). 24 (66. 7) 19 (52. 8) Combined 34 (47, .2) 31 (43. 1) 43 (59. 7) 33 (45. 8) Tot a l Sample Males 40 (55, .6) 40 (55. 6) 38 (52. 8) 33 (45. 8) Females 36 (50. .0) 34 (47. 2) 46 (63. 9) 44 (61. 1) Comb ined 76 (52. .8) 74 (51. 4) 84 (58. 3) 77 (53. 5) 37. TABLE 11 Summary of Group x Sex x Grade x Type of T r a n s i t i v i t y Task Analysis of Variance SOURCE DF Between subj ects Group (A) 1 Sex (B) 1 Grade (C) 2 AB 1 AC 2 BC 2 ABC 2 Error Between 132 MS F 1.46 6.84 *** 0.85 0.40 4.76 22.30 «.*** 0.14 0.66 0.23 1.08 0.17 0.79 0.68 0.32 0.21 i Within subjects Task Type (D) AD BD CD ABD ACD BCD ABCD Error Within 3 3 3 6 3 6 6 6 396 0.13 0.28 0.57 0.45 0.20 0.29 0.41 0.30 0.23 0.58 1.25 2.50 1.97 0.86 1.28 1.81 1.34 *** P<.01 **** p<.001 38. TABLE 12 Number and Percentage of Subjects Passing Only the Spec i f i e d Number of T r a n s i t i v i t y Tasks 0 Tasks One or Two or Three or Four Tasks More Tasks More Tasks More Tasks No. % No. % No. % No. % No. % Group I Kindergarten Males 0 ( 0.0) 12(100.0) Females 0 ( 0.0) 12(100.0) Combined 0 ( 0.0) 24(100.0) Grade 1 Males 1 ( 8.3) 11 (91.7) Females ?. 1 ( 8.3) 10 (83.3) Combined 2 ( 8.3) 21 (87.5) Grade 2 Males 0 ( 0.0) 12(100.0) Females 0 ( 0.0) 12(100.0) Combined '0 ( 0.0) 24(100.0) 6 (50.0) 3 (25.0) 0 ( 0.0) 9 (75!0) 4 (33.3) 3 (25.0) 15 (62.5) 7 (29.2) 3 (12.5) 8 (66.7) 2 (16.7) 1 ( 8.3) 8 (66.7) 5 (41.7) 1 ( 8.3) 16 (66.7) 8 (33.3) 2 ( 8.3) 11 (91.7) 9 (75.0) 5 (41.7) 12(100.0) . 10 (83.3) 2 (16.7) 23 (95.8) 19 (79.2) 7 (29.2) Group I T o t a l Males 1 ( 2, .8)'! 35 (97. 2) 25 (69 .4) 15 (41. • 7) 6 (16, .7) Females 1 ( 2. .8) 34 (94. 4) 29 (80 .6) 19 (52. • 8) 6 (16, .7) Combined 2 ( 2, .8) 69 (95. 8) 54 (75 • 0) 34 (47. • 2) 12 (16, .7) 39. Table 12 (cont'd) 0 Tasks One or Two or Three or Four Tasks More Tasks More Tasks More Tasks No. % No. % No. % No. % No. % Group II Kindergarten Males 2 (16. 7) 10 (83.3) 4 (33.3) .2 (16.7) 0 ( 0.0) Females 1 ( 8. 3) 11 (91.7) 5 (41.7) 0 ( 0.0) 0 ( 0.0) Combined 3 (12. 5) 21 (87.5) 9 (37.5) 2 ( 8.3) 0 ( 0.0) Grade 1 Males 1 ( 8.3) 11 (91.7) 8 (66.7) 2 (16.7) 0 ( 0.0) Females 0 ( 0.0) 12(100.0) 8 (66.7) 3 (25.0) 0 ( 0.0) Combined 1 ( 4.2) 23 (95.8) 16 (66.7) 5 (20.8) 0 ( 0.0) rade 2 Males 0 ( Q.0) 12(100.0) 12(100.0) 8 (66.7) 2 (16.7) Females 0 ( 0.0) 12(100.0) 10 (83.3) 7 (58.3) 2 (16.7) Combined 0 ( 0.0) 24(100.0) 22 (91.7) 15 (62.5) 4 (16.7) Group II T o t a l Males 3 ( 4.2) 33 (91.7) 24 (66. 7) 12 (33.3) 2 ( 5 .6) Females 1 ( 1.4) 35 (97.2) 23 (63. 9) 10 (27.8) 2 ( 5 • 6) Combined 4 ( 2.8) 68 (94.4) 47 (65. 3) 22 (30.6) 4 ( 5 .6) T o t a l Sample Males 4 ( 5. 6) 68 (94. 4) 49 (68.1) 27 (37.5) 8 (11.1) Females 2 ( 2. 8) 69 (95. 8) 52 (72.2) 29 (40.3) 8 (11.1) Combined 6 ( 4. 2) 137 (95. 1) 101 (70.1) 56 (38.9) 16 (11.1) 40. or more, to 70.1% who passed 2 or more, to 38.9% who passed 3 or more. The decline i n the number of subjects who passed 1 or more to the number who passed 3 or more was less i n the case of Grade 2 than i n Kindergarten or Grade 1. The frequency and percentage of subjects who passed or f a i l e d t r a n s i t i v i t y tasks and passed or f a i l e d i d e n t i t y conservation tasks, equivalence conservation tasks and a l l conservation tasks are given i n Tables 13, 14, and 15 re s p e c t i v e l y . The o v e r a l l r e s u l t s i n d i c a t e that many more subjects passed t r a n s i t i v i t y tasks and f a i l e d conservation tasks than f a i l e d t r a n s i t i v i t y tasks but passed conservation tasks. Summing the number of subjects who passed 2, 3 or 4 t r a n s i t i v i t y tasks y i e l d s the number who passed 2 or more tasks. S i m i l a r l y summing the number of subjects who f a i l e d 2, 3 or 4 t r a n s i t i v i t y tasks gives the number of subjects who f a i l e d 2 or more t r a n s i t i v i t y tasks. When t h i s i s done i t can be seen from Table 13 that 65 subjects (45.1%) passed 2 or more t r a n s i t i v i t y tasks but f a i l e d the i d e n t i t y conservation tasks, but only 15 subjects (10.4%) f a i l e d 2 or more t r a n s i t i v i t y tasks and passed the i d e n t i t y tasks. The comparable figures from Table 14 are 64 subjects (44.4%) who passed 2 or more t r a n s i t i v i t y tasks and f a i l e d the equivalence conservation tasks and 20 subjects (13.9%) who f a i l e d 2 or more t r a n s i t i v i t y tasks but passed the equivalence conservation tasks. An examination of Table 15 shows that 70 subjects (48.6%) passed 2 or more t r a n s i t i v i t y tasks and f a i l e d both types of conservation task, while only 13 subjects (9.0%) passed both types of conservation task and f a i l e d 2 or more t r a n s i t i v i t y tasks. TABLE 13 Number and Percentage pf Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and Identity Conservation Tasks. ( C r i t e r i o n of Judgment Only). Number of subjects passing the conser^ Number of subjects f a i l i n g the conser-vation tasks but f a i l i n g the s p e c i f i e d vation tasks but passing the s p e c i f i e d number of T r a n s i t i v i t y tasks. number of T r a n s i t i v i t y tasks. No. of T r a n s i t i v i t y 0 1 2 3 4 0 1 2 3 4 Tasks No. % No. % No. % No, % No. % No. % No. % No. % No. % No. % Group I Kindergarten 0 (0.0) 1 (4.2) . 0 (0.0) 1 (4.2) 0 (0.0) 0 (0.0) 9(37.5) 7(29.2) 3(12 .5) 3(12.5) Grade 1 2 (8.3) 5(20.8) 5(20.8) 3(12.5) 0 (0.0) 2 (8.3) 3(12.5) 3(12.5) 1 (4 .2) 0 (0.0) Grade 2 4(16.1) 6(25.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 1 (4.2) 4(16.7) 6(25 .0) 3(12.5) T o t a l 6 (8.3) 12(16.7) 5 (6.9) 4 (5.6) 0 (0.0) 2 (2.8) 13(18.1) 14(19.4) 10(13 .9) 6 (8.3) Group II Kindergarten 0 (0.0) 1 (4.2) 0 (0.0) 0 (0.0) 0 (0.0) 3(12.5) 12(50.0) 7(29.2) 1 (4 .2) 0 (0.0) Grade 1 0 (0.0) 1 (4.2) 2 (8.3) 1 (4.2) 0 (0.0) 1 (4.2) 6(25.0) 9(37.5) 4(16 .7) 0 (0.0) Grade 2 3(12.5) 2 (8.3) 3(12.5) 0 (0.0) 0 (0.0) 0 (0.0) 2 (8.3) 4(16.7) 9(37 .5) 1 (4.2) To t a l 3 (4.2) 4 (5.6) 5 (6.9) 1 (1.4) 0 (0.0) 4 (5.6) 20(27.8) 20(27.8) 14(19 .4) 1 (1.4) To t a l Sample 9 (6.3) 16(11.1) 10 (6.9) 5 (3.5) 0 (0.0) 6 (4.2) 33(22.9) 34(23.6) 24(16 • 7) 7 (4.9) TABLE 14 Number and Percentage of Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and Equivalence Conservation Tasks ( C r i t e r i o n of Judgment Only). Number of subjects passing the conser- Number of subjects f a i l i n g the conser-vation tasks but f a i l i n g the s p e c i f i e d vation tasks but passing the s p e c i f i e d number of t r a n s i t i v i t y tasks number of t r a n s i t i v i t y tasks 0 1 2 3 4 0 1 2 3 4 No. of t r a n s i t i v i t y No. % No. % No. % No, % No, % No. % No. % No. % No. % No. % tasks Group I Kindergarten 0 (0.0) 0 (0.0) 1 (4.2) 1 (4.2) 0 (0.0) 0 (0.0) 9(37.5) 6(25.0) 4(16. 7) 3(12.5) Grade 1 2 (8.3) 4(16.7) 5(20.8) 3(12.5) 0 (0.0) 2 (8.3) 3(12.5) 3(12.5) 2 (8. 3) 0 (0.0) Grade 2 4(16.7) 5(20.8) 2 (8.3) 0 (0.0) 0 (0.0) 0 (0.0) 1 (4.2) 2 (8.3) 7(29. 2) 3(12.5) T o t a l 6 (8.3) 9(12.5) 8(11.1) 4 (5.6) 0 (0.0) 2 (2.8) 13(18.1) 11(15.3) 13(18. 1) 6 (8.3) Group II Kindergarten 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 3(12.5) 12(50.0) 7(29.2) 2 (8. 3) 0 (0.0) Grade 1 0 (0.0) 1 (4.2) 1 (4.2) 1 (4.2) 1 (4.2) 0 (0.0) 6(25.0) 10(41.7) 4(16. 7) 0 (0.0) Grade 2 3(12.5) 4(16.7) 4(16.7) 1 (4.2) 0 (0.0) 0 (0.0) 1 (4.2) 3(12.5) 7(29. 2) 1 (4.2) T o t a l 3 (4.2) 5 (6.9) 5 (6.9) 2 (2.8) 1 (1.4) 3 (4.2) 19(26.4) 20(27.8) 13(18. 1) 1 (1.4) Total Sample 9 (6.3) 14 (977) 13 (9.0) 6 (4.2) 1 (0.7) 5 (3.5) 32(22.2) 31(21.5) 26(18. 1) 7 (4.9) TABLE 15 Number and Percentage of Subjects Passing and F a i l i n g T r a n s i t i v i t y Tasks and a l l Conservation Tasks ( C r i t e r i o n of Judgment Only) Number of subjects passing the conser- Number of subjects f a i l i n g the conser-vation tasks but f a i l i n g the s p e c i f i e d vation tasks but passing the s p e c i f i e d number of t r a n s i t i v i t y tasks. number of t r a n s i t i v i t y tasks. No. of T r a n s i t i v i t y Tasks 0 No. % 1 No. % 2 No. % No, 3 % No, 4 % 0 No. % 1 No. % 2 No. % 3 No. % 4 No. % Group I Kindergarten 0 (0.0) 0 (0.0) 0 (0.0) 1 (4.2) 0 (0.0) 0 (0.0) 9(37. 5) 7(29.2) 4(16.7) 3(12.5) Grade 1 2 (8.3) 4(16.7) 5(20.8) 3(12.5) 0 (0.0) 2 (8.3) 3(12. 5) 3(12.5) 2 (8.3) 0 (0.0) Grade 2 4(16.7) 5(20.8) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 1 (4. 2) 4(16.7) 7(29.2) 3(12.5) T o t a l 6 (8.3) 9(12.5) 5 (6.9) 4. (5.6) 0 (0.0) 2 (2.8) 13(18. 1) 14(19.4) 13(18.1) 6 (8.3) Group II Kindergarten 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 3(12.5) 12(50. 0) 7(29.2) 2 (8.3) 0 (0.0) Grade 1 . 0 (0.0) 1 (4.2) 1 (4.2) 0 (0.0) 0 (0.0) 1 (4.2) 7(29. 2) 10(41.7) 4(16.7) 0 (0.0) Grade 2 3(12.5) 2 (8.3) 3(12.5) 0 (0.0) 0 (0.0) 0 (0.0) 2 (8. 3) 4(16.7) 9(37.5) 1 (4.2) T o t a l 3 (4.2) 3 (4.2) 4 (5.6) 0 (0.0) 0 (0.0) 4 (5.6) 21(29. 2) 21(29.2) 15(20.8) 1 (4.0) To t a l Sample 9 (6.3) 12 (8.3) 9 (6.3) 4 (2.8) 0 (0.0) 6 (4.2) 34(23. 6) 35(24.3) 28(19.4) 7 (4.9) 44. Discussion Conservation Taken together the r e s u l t s of the present study are i n general agreement with the r e s u l t s of studies which i n d i c a t e that i d e n t i t y conservation and equivalence conservation co-occur i n a developmental sense. Performance on i d e n t i t y conservation tasks was not s i g n i f i c a n t l y d i f f e r e n t from performance on equivalence conservation tasks. There was very l i t t l e d i f f e r e n c e i n the number of subjects who passed i d e n t i t y conservation tasks and equivalence conservation tasks within each group. Also of importance i s the f a c t that there were not more subjects who passed i d e n t i t y conservation tasks and f a i l e d equivalence conservation tasks than the number who passed equivalence conservation tasks and f a i l e d i d e n t i t y conservation tasks. S i g n i f i c a n t l y more Grade 2 than Kindergarten subjects passed the conservation tasks. Very few Kinder-garten subjects passed ei t h e r i d e n t i t y conservation or equivalence conservation tasks and there were no Kindergarten subjects who passed both i d e n t i t y conservation tasks and f a i l e d equivalence conservation tasks. The e f f e c t of the two l e v e l s of transformation was i n s i g n i f i c a n t . This r e s u l t i s i n agreement with other studies which have used more than one l e v e l of transformation. Hooper (1969a) and Koshinsky and H a l l (1973) found that varying the l e v e l of transformation of discontinuous objects i n glass cylinders did not have a s i g n i f i c a n t e f f e c t . This would seem to contradict Piaget and Inhelder's (1974) view of the importance of the l e v e l of transformation, although i t i s also possible that the d i f f e r e n t i a l 45. between the moderate and extreme was not great enough. The group which received the i d e n t i t y task outlined by Elkind (1967) and the t r a d i t i o n a l conservation tasks (Group I) performed co n s i s t e n t l y better than the group which received the modification (Group II) for a l l conservation tasks. I t i s p o s s i b l e , however, that t h i s resulted from some o v e r a l l d i f f e r e n c e i n the a b i l i t y of the groups rather than the d i f f e r e n t i a l e f f e c t s of the tasks. This i n t e r p r e t a t i o n i s based on the fa c t that Group II also was i n f e r i o r i n performance on the t r a n s i t i v i t y tasks even though they were the same tasks administered to Group I. I t i s not c l e a r what factors could account for the superior performance of Group I since many precautions were taken to insure that assignment to the groups was done randomly. Subjects were chosen randomly from t h e i r classes, with the r e s u l t that subjects i n d i f f e r e n t classes at the same grade l e v e l were equally l i k e l y to be placed i n Group I :as i n Group I I . There was random t e s t i n g across time of day to ensure that one group would not be tested at any s i n g l e time period. The number of subjects who were successful on the t r a n s i t i v i t y tasks was the same whether the tasks were administered before any conservation tasks, between conservation tasks ( i . e . , between i d e n t i t y and equivalence tasks) or a f t e r the conservation tasks. This was true within each group, hence a d i f f e r e n t i a l multiple treatment interference e f f e c t (Campbell and Stanley, 1963; p. 6) cannot account for the d i f f e r e n c e between groups. The most frequent explanation categories used when j u s t i f y i n g responses were addition-subtraction, statement of operations, reference to previous state and more than one category. Toniolo and Hooper (1975) 46. found that addition-subtraction was the s i n g l e most frequently used category. In the present study the statement of operations category was used most frequently with the addition-subtraction category being the second most frequently used category. Hooper (1969a) and Toniolo and Hooper (1975) found that addition-subtraction was the category most frequently used on i d e n t i t y conservation, whereas Papalia and Hooper (1971) found that reference to the previous state was used most frequently. The most frequently used category on the equivalence tasks was reference to the previous state i n both the Hooper (1969a) and Papalia and Hooper (1971) studies. Toniolo and Hooper (1975) found that addition-subtraction was the most frequently used category on equivalence tasks. In the present study statement of operations was the most frequently used category on both types of tasks. The second most frequently used explanation category f or i d e n t i t y tasks d i f f e r e d from that used f or equivalence tasks. The statement of operations category was most frequent for i d e n t i t y tasks and the reference to previous state category was most frequent f or equivalence tasks. This i s i n agreement with Toniolo and Hooper (1975) who found the same pattern of r e s u l t s with the second most frequently used category f or the d i f f e r e n t conservation tasks. In agreement with Hooper (1969a), Papalia and Hooper (1971) and Toniolo and Hooper (1975) the present study indicates that inversion and r e c i p r o c i t y are categories that are used very infrequently. The category of compensation was never used i n the present study, which i s i n agreement with the r e s u l t s of Toniolo and Hooper (1975) , who obtained only one explanation of t h i s type. 47. A comparison of Tables 1 and 3, Tables 4 and 5, and Tables 13 and 14 supports Brainerd's (1973) contention that a judgment plus explanation c r i t e r i o n i s much more stringent than a judgment only c r i t e r i o n . The number of subjects considered to have passed the conservation tasks with a c r i t e r i o n of judgment plus explanation i s c o n s i s t e n t l y less than the number who passed with a c r i t e r i o n of judgment only. These r e s u l t s are i n agreement with those of Brainerd and Hooper (1975) and Toniolo and Hooper (1975) i n that the c r i t e r i o n chosen affected performance on the conservation tasks to a s t a t i s t i c a l l y s i g n i f i c a n t degree. I t i s important at t h i s point to discuss two recent studies which support the thesis that i d e n t i t y conservation i s acquired p r i o r to equivalence conservation. Both Brainerd and Hooper (1975) and Toniolo and Hooper (1975) present evidence to show that i d e n t i t y conservation of length and weight precede equivalence conservation of length and weight. These two studies were very s i m i l a r to the present one and hence deserve close discussion p a r t i c u l a r l y since i t i s possible that the r e s u l t s of these studies are a r t i f a c t u a l . In both studies a scoring technique which assigned values from 0 to 6 on the conservation tasks was used. Three questions were asked i n the p r e d i c t i o n phase employing the terms "more", " l e s s " and "same" and three questions employing these same three r e l a t i o n a l terms were asked i n the judgment phase. Three terms were employed i n each phase i n order to insure that c h i l d r e n had to both agree and disagree with the experimenter i n order to be correct. Rothenberg (1969) has reported on a tendency for c h i l d r e n to agree with what an experimenter says more frequently than they disagree. Each time one of the questions was answered c o r r e c t l y a score 48. of 1 was assigned and each inco r r e c t response was scored 0. Since there were s i x questions on each task a maximum score of 6 was possible and a l l values between 0 and 6 could be obtained. It i s very important to note that there was a di f f e r e n c e between the questions employed i n i d e n t i t y conservation tasks and equivalence conservation tasks. This d i f f e r e n c e may account f o r the supposed develop-mental p r i o r i t y of i d e n t i t y conservation over equivalence conservation. To i l l u s t r a t e l e t us assume that there i s a subject who co n s i s t e n t l y believes that a perceptual transformation changes the relevant quantitative feature of a clay b a l l causing i t to weigh more. The same point would apply to subjects who co n s i s t e n t l y believe that an object weighs l e s s . The following questions are taken from the conservation of weight tasks employed by Toniolo and Hooper (1975) : Identity Format 1. P r e d i c t i o n : P l acing the green clay b a l l i n the middle of the table 8 - 1 0 inches from the S^ the E asks the following questions: a. I f I were to r o l l t h i s clay b a l l into a hotdog would the piece of clay s t i l l have the same weight? Yes ' No Q I don't know No response b. I f I were to r o l l t h i s clay b a l l into a hotdog would the piece of clay weigh more? Yes Q No I don't know No response c. I f I were to r o l l t h i s clay b a l l into a hotdog would the piece of clay weigh less? Yes No 1 I don't know No response 49. 2. Deformation: The E then r o l l s the b a l l into a hotdog, and asks the following questions: a. Does t h i s piece of clay weigh the same as before? Yes No 0 I don't know No response b. Does t h i s piece of clay weigh more than before? Yes 0 No I don't know No response c. Does t h i s piece of clay weigh less than before? Yes No 1 I don't know No response Equivalence Format 1. P r e d i c t i o n : Taking the b a l l s from the S and plac i n g them on the table side-by-side 8 - 1 0 inches from the S, the E asks the following questions while pointing to one of the s t i m u l i : a. I f I were to f l a t t e n t h i s clay b a l l into a pancake, would the two pieces of clay s t i l l have the same weight? Yes No 0 I don't know No response b. I f I were to f l a t t e n t h i s clay b a l l into a pancake would one of the pieces of clay weigh more? Yes 0 No I don't know No response c. I f I were to f l a t t e n t h i s clay b a l l into a pancake would one of the pieces of clay weigh less? Yes 0 No I don't know No response 2. Deformation: The E then f l a t t e n s the clay b a l l into a pancake and asks the following questions: a. Do these two pieces of clay weigh the same as before? Yes No 0 I don't know No response 50. b. Does one of the pieces weigh more than before? Yes 0 No I don't know No response c. Does one of the pieces weigh less than before? Yes 0 No I don't know No response Notice that the c h i l d would receive a score of 2 for these answers on the i d e n t i t y task. This i s because the c h i l d believes that the b a l l w i l l have more clay when i t i s r o l l e d into a hotdog. Consequently i n the p r e d i c t i o n and judgment (deformation) phase when asked i f the b a l l w i l l have les s when i t i s a hotdog the c h i l d answers "no" for he/she believes i t w i l l have more. However a score of 1 w i l l be given for each phase because, i n one sense, the answer i s correct i t w i l l not have l e s s . The analogous questions i n the equivalence task do not lead to such scores. When the subject i s asked i n the p r e d i c t i o n phase i f one object w i l l have le s s he/she r e p l i e s that one w i l l , i . e . , the b a l l . S i m i l a r l y i n the judg-ment phase the subject w i l l reply that one object has l e s s . subjected to an analysis of variance by Brainerd and Hooper (1975) and by Toniolo and Hooper (1975). A s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e be-tween i d e n t i t y conservation and equivalence conservation was obtained and a t t r i b u t e d to the developmental p r i o r i t y of the i d e n t i t y conservation concept over the equivalence conservation concept. My preceding analysis which shows that scores on i d e n t i t y conservation tasks would be greater than scores on equivalence conservation tasks because of the differences i n questions employed on the two tasks suggests that the s i g n i f i c a n t d ifferences obtained i n the analysis of variance are a r t i f a c t u a l . Scores obtained i n t h i s manner, employed the 0-6 scale, were It i s important to note that the differences observed i n the 51. Toniolo and Hooper (1975) study between i d e n t i t y conservation and equivalence conservation were much less pronounced when a dichotomous pass/ f a i l c r i t e r i o n was used. According to t h i s c r i t e r i o n subjects were said to have passed a task i f they were correct on a l l questions i n both the p r e d i c t i o n and judgment phases. Any other pattern of responses was counted as a f a i l u r e . Preschool, kindergarten and t h i r d grade subjects were assessed and when an analysis of variance of the scores obtained with a 0-6 scale was done a s i g n i f i c a n t main e f f e c t was obtained for the Identity X. Equivalence factor i n both length and weight measures. A s i g n i f i c a n t Grade Level x Conservation Task Type i n t e r a c t i o n was also obtained. However when the p a s s / f a i l c r i t e r i o n was used, only the kindergarten and t o t a l sample weight cases indicated that i d e n t i t y conservation was easier than equivalence conservation. Unfortunately the same kinds of questions were used for the conservation tasks i n the present study as i n Brainerd and Hooper (1975) and Toniolo and Hooper (1975) , as were several of the other methodological p r e s c r i p t i o n s of Brainerd and Hooper (1975). However the scoring technique used i n the present study obviates the c r i t i c i s m s directed at the other studies. A three-point scale with values of 0, 1 and 2 was employed. This scale r e f l e c t s the l e v e l of concept attainment more c l e a r l y than does a 0-6 scale. To i l l u s t r a t e l e t us assume that there are two hypothetical subjects X and Y who both obtain scores of 3 using the 0-6 scale. I f these numerical magnitudes accurately r e f l e c t a subject's l e v e l of concept attainment then subject X and subject Y should be considered to be at the same l e v e l of conceptual development on the p a r t i c u l a r task. Suppose, however, that the d i s t r i b u t i o n of scores for the 0-6 scale i s as follows: 52. Sub. X Sub. Y Pr e d i c t i o n Same More Less 1 1 1 1 0 0 Judgment Same More Less 0 0 0 1 0 1 To t a l 3 3 On the scoring technique used i n the present study the scores would be as follows: P r e d i c t i o n Sub. X Judgment Same More Less 0 0 To t a l Sub. Y 0 0 0 1 0 On the 0-6 scoring scheme subject X and Y would be judged equal i n terms of concept attainment, however, on the 0, 1, 2 scoring scheme subject X shows a higher l e v e l of concept attainment than subject Y. The 0, 1, 2 scoring scheme demands that subjects be consistent i n t h e i r answers within each phase i n order to obtain scores other than 0. This i s done because the questions i n v o l v i n g the three r e l a t i o n a l terms are not independent. Given that a subject believes a transformed object to have the same amount of clay as before the transformation he/she should also b e l i e v e that i t does not have more or l e s s , i n order to be consistent. This constraint of consistency i n responding i s absent i n the case of the 0-6 scale as subjects can obtain part values f o r t h e i r answers. To i l l u s t r a t e , the importance of t h i s consistency constraint, consider a l l 53. possible patterns of respondings to one phase of the equivalence conservation task. An example i s presented i n Table 16. Since t h i s example i s concerned with only one phase of the conservation task scores can range from 0 to 3 i n c l u s i v e . Only two patterns of responding, those to the l e f t of the dotted l i n e , can be considered to be consistent patterns of responses. A l l s i x other possible patterns, to the r i g h t of the dotted l i n e , are inconsistent. What t h i s means w i l l become clearer below, but f i r s t l e t us consider another type of example, one i n which numerals are assigned to responses. In many types of mental measurement si t u a t i o n s part-values are assigned to a subject's responses i f some questions are answered i n c o r r e c t l y . For instance, i f a subject i s asked to define the meaning of 10 words and gives 7 acceptable d e f i n i t i o n s , we may assign the number 7 to those responses. Assuming, of course, that the words have been assigned equal weights. Another subject who gave 5 appropriate d e f i n i t i o n s would be assigned a score of 5. We can say (based on the r e l a t i v e magnitude of the numerals assigned to the responses) that the subject scoring 7 performed better ( i . e . , gave more correct d e f i n i t i o n s ) than the subject scoring 5. Furthermore, a l l other subjects can be ordered accordingly. Now consider what happens i n the conservation s i t u a t i o n s outlined above i n Table 16. Three of the inconsistent patterns of responding w i l l be assigned a 1 and three w i l l be assigned a 2. Following the l o g i c of the previous example, i t would appear, within an order of e r r o r , that the three patterns assigned a 2 would evidence the same l e v e l of conceptual development which would i n turn be considered to show more evidence of. the concept than patterns being assigned a 1. There i s one important d i f f e r e n c e , TABLE 16. Possible Patterns of Responding More N Y ' N N N Y Less N Y J Y Y N N Assigned Score 3 0 J 1 2 2 2 Y = Yes N = No 55. however, between the conservation s i t u a t i o n and the example of word d e f i n i t i o n s . The three questions asked i n the conservation task employ r e l a t i o n a l terms which are dependent on each other i n order to lead to a consistent response pattern. However, the word d e f i n i t i o n s are not dependent i n the same sense. To make t h i s point c l e a r e r , consider the f i r s t pattern of responses to the r i g h t of the dotted l i n e i n Table 16. A subject giving t h i s set of responses would say that both b a l l s are not the same weight, one b a l l does not weigh more and one b a l l does weigh l e s s . C l e a r l y t h i s subject i s being inconsistent since he/she i s asserting that mutually exclusive propositions are holding about the same object. In f a c t , a l l of the patterns of responding which have been labeled inconsistent are of t h i s nature. Even those subjects which have been assigned a 2 have been inconsistent i n t h i s manner, but these subjects are supposedly r e l a t i v e l y close to having the concept i n question. The problem with the assignment of a l l of these part-values i s that i n one sense the subject i s p a r t i a l l y correct i n that he/she has answered one or two questions c o r r e c t l y but how can t h i s be interpreted? The questions are not independent i n the sense that answers can stand on t h e i r own without reference to the other answers. Moreover, since a reasoning s k i l l ( i n a broad sense) i s being measured consistency i n reasoning i s important. The c r i t i c i s m s of the Brainerd and Hooper (1975) and Toniolo and Hooper (1975) studies can be summarized as follows. Greater care should be taken to insure that the questions on each of the tasks are equivalent i n terms of d i f f i c u l t y . I t i s also important that the questions should be structured so that answers can be scored i n an unambiguous 56. manner. The scale of 0-6 leads to an ambiguous r e f l e c t i o n of the l e v e l of concept attainment l a r g e l y because of the d i f f i c u l t y i n i n t e r p r e t i n g score values of 1 through 5. The present method of scoring using a 0, 1, 2 scale i s , unfortunately, only s l i g h t l y more refined than a p a s s / f a i l c r i t e r i o n , but i t i s not open to the problems of i n t e r p r e t a t i o n that occur with the 0-6 technique. The importance of these c r i t i c i s m s can be appreciated when the data from the present study are scored with the 0-6 scale. There were many more subjects who obtained scores of 1 or 2 on the i d e n t i t y tasks than on the equivalence tasks. The dif f e r e n c e between i d e n t i t y and equivalence tasks was also s t a t i s t i c a l l y s i g n i f i c a n t when an analysis of variance of scores obtained with t h i s a:ale was run. % T r a n s i t i v i t y Subjects i n Kindergarten and Grade 1 performed equally w e l l on the t r a n s i t i v i t y tasks, while subjects i n Grade 2 performed better than both Kindergarten and Grade 1 subjects. As on the conservation tasks Group I performed better than Group II on the t r a n s i t i v i t y tasks. There was no differe n c e i n performance on the various t r a n s i t i v i t y tasks. A study by Murray and Youniss (1969) has indicated that some t r a n s i t i v i t y paradigms are easier than others. They presented tasks of the form A>B=C, A=B>C and A>B>C to Kindergarten, Grade 1 and Grade 2 ch i l d r e n . They found that the A>B>C paradigm was easier than e i t h e r of the other two paradigms, however the A>B=C, and A=B>C forms of the task did not d i f f e r s i g n i f i c a n t l y . Their tasks d i f f e r e d from those used i n 57. the present study i n that none of the objects were hidden and t r a n s i t i v i t y of length was investigated. Almost a l l of the subjects i n the present study passed 1 or more of the t r a n s i t i v i t y tasks. A large proportion of subjects passed 2 or more tasks. Many subjects passed t r a n s i t i v i t y tasks, but f a i l e d conservation tasks, while very few subjects passed conservation tasks and f a i l e d t r a n s i t i v i t y tasks. These r e s u l t s are i n agreement with those . of Brainerd (1973) and Toniolo and Hooper (1975) regarding the a c q u i s i t i o n of t r a n s i t i v i t y and conservation. Conclusion The r e s u l t s of the present study and those of Toniolo and Hooper (1975) cast considerable doubt on Elkind's (1967) analysis about the r o l e of t r a n s i t i v i t y i n the t y p i c a l conservation task. T r a n s i t i v i t y , as a mental operation, develops p r i o r to both i d e n t i t y conservation and equivalence conservation, at least i n the content areas of s o l i d continuous quantity, length and weight, hence the absence of t h i s concept cannot be involved i n an explanation of f a i l u r e on the t r a d i t i o n a l task. This would only be the case, however, i f i t i s shown that i d e n t i t y precede equivalence conservation developmentally. I f , however, as the present study i n d i c a t e s , i d e n t i t y and equivalence conservation co-occur then Elkind's analysis could s t i l l hold true i n the sense that t r a n s i t i v i t y i s indeed important for the proper s o l u t i o n of the t r a d i t i o n a l conservation task but since a l l subjects who pass i d e n t i t y tasks have the concept, they can also pass equivalence tasks. I t i s d i f f i c u l t to see how Elkind's analysis could be investigated e m p i r i c a l l y unless a p a r t i c u l a r content area was found i n which subjects 58. do not have t r a n s i t i v i t y , but can pass i d e n t i t y conservation tasks. I f subjects passed i d e n t i t y tasks but f a i l e d equivalence conservation tasks t h i s would support the a n a l y s i s . I f , however subjects passed i d e n t i t y tasks and passed equivalence conservation tasks t h i s would contradict the a n a l y s i s . I t i s highly u n l i k e l y that such a content area w i l l be found considering the r e s u l t s of the present study and those of Bryant and Trabasso (1971), Brainerd '(1973) , and Toniolo and Hooper (1975) which show t r a n s i t i v i t y to be a r e l a t i v e l y p r i m i t i v e operation which develops p r i o r to conservation. Moreover even i f such a content area were found Elkind's o r i g i n a l analysis would be considerably weakened as i t was meant to apply to a l l conservation tasks i r r e s p e c t i v e of content area. 59. Footnotes *These e f f e c t s should be treated with caution since a "two-point scale" (0,1) was used i n the a n a l y s i s . In an analysis of the e f f e c t s of the length of a score scale on the s i g n i f i c a n c e l e v e l of the F-test Hsu and Feldt (1969) point out that when a two-point scale i s involved a sample si z e of 50 or more should be used. However since the group e f f e c t i s the r e s u l t of a sample s i z e of 72 and the grade e f f e c t a sample s i z e of 48 t h i s p r e s c r i p t i o n has not been se r i o u s l y v i o l a t e d . Further, considering the robustness of the e f f e c t s i t i s highly u n l i k e l y that these factors did not have a s i g n i f i c a n t influence. 2 This c l a s s i f i c a t i o n scheme has been adopted from Toniolo and Hooper (1975). 60. References Brainerd, C.J. Judgments and explanations as c r i t e r i a f o r the presence of cognitive structures. Psychological B u l l e t i n , 1973, _7_9» 172-179. (a) Brainerd, C.J. Order of a c q u i s i t i o n of t r a n s i t i v i t y , conservation, and c l a s s - i n c l u s i o n of length and weight. Developmental Psychology, 1973, 8, 105-116.(b) Brainerd, C.J. & Brainerd, S.H. Order of a c q u i s i t i o n of number and l i q u i d quantity conservation. C h i l d Development, 1972, 43, 1401-1406. Brainerd, C.J. & Hooper, F.H. A methodological analysis of developmental studies of i d e n t i t y conservation and equivalence conservation. Psychological B u l l e t i n , 1975, j52_, i n press. Bruner, J.S., Olver, R.R., Greenfield, P.M. et a l . Studies i n cognitive  growth. New York: John Wiley, 1966. Bryant, P.E. Perception and understanding i n young c h i l d r e n : An  experimental approach. New York: Basic Books, Inc. 1974. Bryant, P.E.- The Understanding of invariance by very young c h i l d r e n . Canadian Journal of Psychology, 1972, 26, 78-96. Bryant, P.E. & Trabasso, T. T r a n s i t i v e inferences and memory i n young chi l d r e n . Nature, 1971, 232, 456-458. Campbell, D.T. & Stanley, J.C. Experimental and Quasi-experimental Designs  for Research. Chicago: Rand McNally, 1963. El k i n d , D. Children's discovery of the conservation of mass, weight, and volume: Piaget r e p l i c a t i o n study I I . The Journal of Genetic  Psychology, 1961, 98, 219-2271. E l k i n d , D. Piaget's conservation problems. C h i l d Development, 1967, 38, 15-27. El k i n d , D. & Schoefeld, E. Identity and equivalence conservation at two age l e v e l s . Developmental Psychology, 1972, 6_, 529-533. F l a v e l l , J.H. Stage r e l a t e d properties of cognitive development. Cognitive Psychology, 1971, 2, 421-453. Furby, L. Cumulative learning and cognitive development: Elaboration and implications of a p r e t h e o r e t i c a l model. Human Development, 1972, 15, 265-286. Halford, G.S. A c q u i s i t i o n of conservation. Psychological Review, 1970, J77, 302-316. 61. Hooper, F.H. Piaget's conservation tasks: The l o g i c a l and developmental p r i o r i t y of i d e n t i t y conservation. Journal of Experimental C h i l d  Psychology, 1969, 8, 234-249. (a) Hooper, F.H. The Appalachian c h i l d ' s i n t e l l e c t u a l c a p a b i l i t i e s — deprivation or d i v e r s i t y . 1969 Yearbook of the Journal of Negro  Education, 1969, 224-235. (b) Hsu, T. & F e l d t , L.S. The e f f e c t of l i m i t a t i o n s on the number of c r i t e r i o n score values on the s i g n i f i c a n c e l e v e l of the F-test. American  Educational Research Journal, 1969, 6_, 515-527. Koshinsky, C. & H a l l , A.E. The developmental r e l a t i o n s h i p between i d e n t i t y and equivalence conservation. Journal of Experimental  C h i l d Psychology, 1973, 15, 419-424. Moynahan, E. & G l i c k , J . Relation between i d e n t i t y conservation and equivalence conservation within four conceptual domains. Develop- mental Psychology, 1972, 6, 247-251. Murray, J.P. & Youniss, J . Achievement of i n f e r e n t i a l t r a n s i t i v i t y and i t s r e l a t i o n to s e r i a l ordering. C h i l d Development, 1968, 39(4) , 1259-1268. Murray, F.B. Stimulus mode and the conservation of weight and number. Journal of Educational Psychology, 1970, 61, 287-291. Northman, G. & Gruen, C. Relationship between i d e n t i t y and equivalence conservation. Developmental Psychology, 1970, ^ ( 2 ) , 311. Papa l i a , D.E. & Hooper, F.H. A developmental comparison of i d e n t i t y and equivalence conservations. Journal of Experimental C h i l d Psychology, 1971, 12, 347-361. Piaget, J . The c h i l d ' s conception of number. New York: Norton L i b r a r y , 1965. Piaget, J . Cognitions and conservations: Two views. Contemporary  Psychology, 1967, 12, 530-533. Piaget, J . On the development of memory and i d e n t i t y . Barre, Mass.: Clark U n i v e r s i t y Press, 1968. Piaget, J . & Inhelder, B. The c h i l d ' s construction of q u a n t i t i e s : Conservation and atomism. London: Routledge & Kegan Paul, 1974. Rothenberg, B.B. Conservation of number among four and f i v e - y e a r - o l d c h i l d r e n : Some methodological considerations. C h i l d Development, 1969, 40, 383-406. 62. Schwartz, M.M. & Scholnick, E.K. Scalogram analysis of l o g i c a l and perceptual components of conservation of discontinuous quantity. Ch i l d Development, 1970, 41, 695-705. Toniolo, T. & Hooper, F.H. Micro-analysis of l o g i c a l reasoning r e l a t i o n s h i p s : Conservation and t r a n s i t i v i t y . Technical Report No. 326. Madison: Wisconsin Research and Development Center for Cognitive Learning, 1975. Youniss, J . & Murray, J.P. T r a n s i t i v e inferences with non-transitive solutions c o n t r o l l e d . Developmental Psychology, 1970, 2^(2), 169-175. 63. Appendix A*" Explanation Categories f o r Conservation Tasks 1) Addition-Subtraction: nothing has been added to or subtracted from the transformed stimulus. 2) Statement of Operations: a s s e r t i o n that the transformation did not a f f e c t the quantity i n question. Example: You ju s t f l a t t e n e d i t down ( i t ' s s t i l l the same amount). 3) Reference to Previous State: standard stimulus and transformed stimulus have the same amount because the standard stimulus and comparison stimulus had the same amount beforetthe transformation. Example: They (the objects) were the same amount before, so they are s t i l l the same now. 4) Inversion: when object can be returned to i t s state p r i o r to trans-formation. Example: You can r o l l i t back into a b a l l and i t w i l l have the same amount. 5) Reciprocity: when standard stimulus can be made to resemble the transformed stimulus. Example: You can f l a t t e n that (the standard) and they w i l l have the same amount. 6) Compensation: one dimension of the transformed stimulus i s compensated by the other dimension. Example: The pancake i s bigger around, but i t i s also f l a t t e r . Sameness (same stimulus): assertion that stimulus as a whole entity is the same piece of clay. Example: It is s t i l l just the same clay as before. Sameness (same quantity): assertion that the stimulus has the same amount as before. Example: It s t i l l has just the same amount of clay. More than one category: use of two or more of the above categories in a composite explanation. Example of 1) and 2): You just flattened i t down, you didn't take any away. 

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