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Training children on multiplicative classification Heemskerk, Antonius Jacobus 1972

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TRAINING CHILDREN ON MULTIPLICATIVE CLASSIFICATION by ANTONIUS J. HEEMSKERK B.A.} University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Psychology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1972 In present ing th is thes is in p a r t i a l f u l f i l m e n t o f. the requ i rements for an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying of th is t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is fo r f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion. Department of PSYCHOLOGY The Un ivers i ty of B r i t i s h ' C o l u m b i a Vancouver 8, Canada Date JULY 23, 1972 ABSTRACT Twenty-one Ss received a matrix training task "which made cognitive demands similar to the reclassification test task and 17 Ss received WISC Block Design training which was not cognitively related to the test task. Results supported the hypothesis that cognitively related training significantly improves reclassification performance, and that non-cognitively related training does not. Neither the Matrix training group, nor the Block Design training group generalized to a second reclassification task. The improvement of some Ss and not others i s explained as the result of the variance in the competence and performance level of cognitive structures. i i TABLE OF CONTENTS Page List of Tables i v L i s t of Figures . . . . . v Introduction 1 Background 2 Definition of Concepts 2 Research Background 4 Training 8 Hypotheses 14 Method 15 Results 24 Discussion 28 Future Research 30 Summary 32 Bibliography 34 Appendix A — Raw Scores, Age, Matrix Responses . . . . . . 36 Appendix B — ANOVA.Tables 41 i i i LIST OF TABLES Table Page 1 Raw Scores on Pretest, Posttest and Generalization Task 37-38 2 Mean Scores for Matrix, WISC and Completed Groups . . . 25 3 Analysis of Variance for WISC and latrix Ss for Pre and Post, Sex and Condition 42 4 Analysis of Variance for WISC and Matrix Ss for Pre and Post, Sex and Condition (data from one J5 removed) . . . . 43 5 Analysis of Variance for WISC vs Inproved Matrix S_s for Pre and Post Scores 44 6 Analysis of Variance for WISC, Matrix, and Completed Groups on Generalization Task 45 7 Analysis of Variance for WISC, and Improved and Non-improved Matrix Ss on Generalization Tasks 46 8 Number of Ss Correctly Responding on Matrix Tasks . . . 39 9 llatrix Training Group Analyzed by Age 40 iv LIST OF FIGURES Figure Page 1 Matrix Design for Training Ss . . . . . . . 3 2 Diagram of Cognitive Relations Between Tasks . . . . 5 3 Blocks for Reclassification Task (Pre and Post) . . . 16 4 Matrices with Toys for Matrix Training 17 5 Picture Matrices for Matrix Training . 18 v ACKNOWLEDGEMENT Thanks to Dr. Louis J. Moran and Dr. Chris Tragakis who provided extensive help in the completion of this thesis from i t s i n i t i a l birth to i t s f i n a l maturity. Thanks to Mr. Britton, Principals and Mrs. Seebolt, Grade 2 and 3 teacher, of St. Peter's School; and also to Sister Alexis, Principal, Mrs. Byrne and Mrs. Wilson, grade 2 teachers, of Immaculate Conception School for their help i n providing Sis. Thanks to a l l the children who served as Ss for this thesis. Thanks to Esther who typed this thesis. And fi n a l l y a big thanks to my wife who made the completion of this thesis possible; i t i s to her that this thesis i s dedicated. v i INTRODUCTION This thesis compares the effect of cognitively and non-cogni-tively related experience on the performance of second and third-grade- children on a reclassification task, testing the hypothesis that a 15-minute training period on a cognitively related task w i l l improve posttest performance on a reclassification task, while training on a non-cognitively related task w i l l not. Background The present research Is concerned with the behavior of 7 to 9-year old children i n the period of concrete operations (Piaget, 1950). During this 3tage Piaget (1950, p. 123) notes that "operational grou-pings of thought concerning objects that can be manipulated or known through the senses" develop. The child of this stage i s involved i n a development of cognitive structures and i n a movement towards an integrated system of action, leaving the child i n command of a coherent and integrated cognitive system (Flavell, 1963). An inte-gral part of the child's cognitive development i s the formation of the schema into groupings which allows the child to act upon objects according to the similarities and differences existing between them. What the child attains, then, i s a certain number of logico-mathemati-cal structures to interpret and integrate reality. Definition of Concepts This thesis deals with grouping III called the bi-univocal multiplication of classes, where bi-univocal means that each component in a class is multiplied by each other component of a second class. 2 (Flavell, 1963) "Multiplicative classification consists of classing each element simultaneously i n terms of two additive classes, Bl and B2" (Piaget, 1950, p. 152) and "obtaining a combination of objects from the product of two classes Bl and B2 = B1B2 (A,A + A A 1 + A 1 12 12 1 A*)," (Piaget, 1950, p. 45). For example, the product of two classes, circ l e (B^) and blue (B^ yields four classes: blue c i r c l e (A^A^, non-blue circle (AJA^) , blue non-circle (A^*) , and non-blue non-circle (AJA*). A matrix design best illustrates groupings III (see Figure 1). From Figure 1 i t i s easy to see that the object i n each c e l l i s the result of the product of two additive classes. One of the necessary prerequisites of multiplicative c l a s s i f i -cation i s the ab i l i t y to group objects into classes according to a single common feature such as placing a l l the blue objects in one group and a l l the red in another. An extension of this single c r i t e -rion classification i s reclassification of the those same objects ac-cording to a different criterion, for example, objects sorted as blue and red things on a f i r s t occasion may be regrouped as squares and circles on a second occasion. The reclassification of objects, although not based on the product of two additive groups bears a similarity to multiplicative classification. Like multiplicative classification, reclassification is based on a similar prerequisite a b i l i t y , single criterion classification, and requires Ss to see objects as members of groups for various different reasons; once a £ i s able to see objects as members of different groups, he can move towards classifying objects as the product of two additive classes. TOP CIRCLE SQUARE BLUE E J BLUE - CIRCLE BLUE. HON - CIRCLE LEFT RED R V . J W3N-BLUE. CIRCLE ^ON-BLUE NON-CIRCLE FIGURE 1 MATRIX DESIGN FOR TRAINING Ss 4 When two operations are directly related to a necessary prerequisite, they can be said to be cognitively related (see Figure 2). Thus, re-classification i s cognitively related to multiplicative classification. As Piaget (1964, p. 209) says, "Once a child can divide the same ob-jects according to two or three complete dichotomies, i t is but a short step to being able to cross-classify them in accordance with a multiplicative schema." Research Background Inhelder and Piaget (1964) tested the multiplicative c l a s s i f i -cation a b i l i t y of children using matrices. In this study, as i n many others, the i s presented a matrix as displayed in Figure 1 with one of the four cells blank, and must choose a correct solution from a number of relevant alternatives. Also, the is required to construct a matrix from a number of objects so that the objects i n adjacent cells w i l l have one criterion common to them, distinguishing them from the other two cells. (Figure 1 shows how adjacent cells have common attrib-utes. Each arrox? indicates which cells have similar attributes.) In their research, Inhelder and Piaget discovered an increasing ab i l i t y to successfully complete matrix tasks with increasing age. Interestingly, when the Ss ? scores were divided into three age groups, the 4 to 5-year old Ss were more successful than the 6 to 7-year old Ss on three item matrices. (This involves three attributes on each side of the matrix making six cells.) Eight to 9-year olds completed many more matrices correctly than the 6 to 7~year olds and the 4 to 5-year olds. The better performance of 4 to 5-year olds on three item 5 RECLASSIFICATION 7' PREREQUISITE x COGNITIVELY --• RELATED SINGLE PREREQUISITE , MULTIPLICATIVE CLASSIFICATION " - — ' CLASSIFICATION FIGURE 2 Diagram of Cognitive Relations Between Tasks 6 matrices was explained as the result of perceptual factors which leave the matrix open to solution by means of graphic collections. The poorer completion percentage of the 6 to 7-year aids on the more d i f f i c u l t tests was a result of task variables interfering with the application of newly forming cognitive structures. Task variables also seem to affect the 8 to 9-year olds, who have stable cognitive structures on three item matrices, because their rate of success de-creases from their two item matrix performance. Following the extensive research of Inhelder and Piaget have been a number of studies analyzing the effect of perceptual factors in multiplicative classification. Parker, Parker and Day (1971) working with matrices found better matrix completion with increasing age — 41% of 6-year olds, 57% of 7-year olds, 74% of 8-year olds and 79% of 9-year olds completed the matrices. They also found that performance on different types of matrix tasks, perceptual (color), functional (cutting), abstract (fruit) was related to the age of the S^. "The 6-year olds performed more adequately on Perceptual x Perceptual matrices than on a l l other types of matrices. Eight-year olds were as successful with Functional x Functional,as with Perceptual x Per-ceptual matrices and 9-year olds performed equally well on a l l but the Abstract x Abstract matrices." (Parker, Parker and Day, 1971, p. 317). ,:The finding that children can combine certain atrributes and not others at particular ages f i t s Piaget 5s definition of "horizontal decalage" — the a b i l i t y to perform specific logical operations in some situa-tions (or on some materials) before others....It i s possible that some 7 children might be able to identify common attributes and yet not be able to combine them because of less experience with categorizing on a functional and/or abstract level and therefore a failure to gene-ralize the rule used wthh perceptual attributes to functional or abstract attributes." (p. 317) Thus, a possessing an operative schema or a logical structure relevant to a number of attributes may not be able to apply that structure to a l l the relevant situations because the S_ lacks experience in working with certain attributes or situations and cannot apply the necessary operative schema. Overton and Brodzinsky (1972) comparing perceptual and logical factors i n multiplicative classification discovered better matrix completion performance with increased age. Reducing perceptual factors by altering a 2 x 2 matrix to form a 1 x 4 matrix significantly im-proved the performance of 6-year old Ss but not of 4 or 8-year olds on matrix completion. "During the transition phases (6 to 7 years) i t seems that logical operational structures have developed, but their functioning can be partially masked by task variables such as the 2 x 2 perceptual instruction condition...." (Overton and Brodzinsky, 1972, p. 108.) It is hypothesized that an E_ can improve the reclas-s i f i c a t i o n a b i l i t y of an S^  by giving that S^  cognitively-related training through matrix tasks. The training gives the j> experience in applying his cognitive schema to many different task stimuli and so this should improve the ab i l i t y of the J5 i n applying his cognitive structure to different task stimuli. This i s the major hypothesis of this thesis. 8 Training According to Piaget (1964), experience is one of the four major factors which explain cognitive development from one stage to the next. Experience refers to the effects of the physical environment on an i n -dividual's cognitive structures. Piaget speaks of two kinds of ex-perience; physical experience which "consists of acting upon objects and drawing some knowledge about the objects by abstraction from the objects," (p. 11), such as learning that knives have sharp edges; and logico-mathematical experience which i s 'not drawn from objects, but is drawn from the actions effected on the objects," (p. 12) such as counting a number of beads and discovering no matter what the arran-gement of the beads, the number remains the same. Social transmission, another factor important to cognitive development, can only be effective once a child has developed certain prerequisite structures. Social transmission i s the processing and handing down of experience through education and/or language of a society (Piaget, 1964). Experience and social transmission as factors in cognitive development are relevant to the purpose of this thesis. Through the use of education (training), E gives S^  logico-mathematical experience with objects to strengthen existing structures for further use. Piaget (1964) states that the learning of logico-mathematical structures is possible through training in simpler, more elementary, logico-mathematical structures. "In other words, learning is possible i f you base the more complex structures on simpler structures; that is when there i s a natural relationship and development of structures and not simply an external 9 reinforcement. (Piaget, 1964, p. 17). Complex logico-mathematical structures are developed through the combination of simpler cognitive structures. As an extension of this idea, this thesis proposes i n -creased probability of success on a task, by providing S_ with ex-perience on a second task requiring similar logico-mathematical structures. ' A l l cognitive structures are subject to temporal effects-Any structure, whether perceptual or conceptual, tends to affect any of those that succeed i t , provided there is sufficient degree of relationship between the two (e.g., analogy, nearness in time or space, etc.) 5'. (Inhelder and Piaget, 1964, p. 197). If existent structures affect developing structures, then the application of a structure to one task should affect the future use of that structure on other tasks. A subject by using his cognitive schema to work on certain situations, w i l l increase the probability of successfully recalling and using that schema later as a result of allowing the cognitive structure and be more easily and adequately expressed through performance. A number of studies have been done to determine the effect of training on the performance of Ss engaged in multiplicative c l a s s i f i c a -tion tasks — many concerned with affecting success by training the Sis on prerequisite logico-mathematical structures. Jacobs and Vandeventer (1971) trained f i r s t graders on double classification tasks, by having the Ss identify f i r s t one, and then a second criterion and use both c r i t e r i a simultaneously to arrive at a solution to either a 2 x 2 or 10 3 x 3 matrix. The training lasted for 30 minutes or unt i l a certain level of performance was reached, whichever came f i r s t . The results indicated that the training group, compared to a control group involved in a game, showed more direct learning on a post-matrix task (color and form) and more transfer on a related matrix, but no difference i n transfer resulted on Raven's Matrix problems which are only distantly related to the matrix test task. An extension of this experiment compared regular training (30 minutes) to extended training (1 month), finding significantly more transfer to far and moderately related tasks under the extended conditions. (Jacobs and Vandeventer, 1971). (Transfer across double-classification tasks is defined as the number of similar class categories or attributes contained in the posttest item not encountered in the training.) Transfer i n the above ex-periments was a function of the length of the training period, which means that the degree of transfer i s a function of the level of the cognitive structure with regard to both the use and state of develop-ment of that structure. Resnick and Siegel (1971) investigating the differential effects of an optimal versus a non-optimal learning sequence, maintained that learning a harder task f i r s t (inferring the attributes of a matrix to complete the cells of a matrix) w i l l result in almost immediate learning of an easier task (placing objects in the cells of a matrix with the attributes v i s i b l e ) . Learning the easy task f i r s t (Optimal sequence) should mean that the harder task is learned in fewer t r i a l s than the non-optimal order (learning the harder task f i r s t ) . They found only 11 an indication that learning of the harder task f i r s t resulted in quicker acquisition of ;the easier task and concluded that :,exposure to a task does not guarantee or show a significant difference in doing an inferring task in less t r i a l s " (p. 147). They stated that the acquisition of more complex s k i l l s may be a matter of learning specific prerequisites rather than the result of entering into a general level or stage of development. Not denying the importance of learning prerequisites in the acquisition of complex structures, i t should be noted that exposure alone does not guarantee improvement. It is through specific training that a can be directed to overcome interfering task variables and attain the ab i l i t y to look beyond the immediate task variables to the logico-mathematical relations existing between the stimuli. Parker, Rieff and Sperr (1971) designed a hierarchical arrange-ment of multiplicative classification prerequisites as a training procedure and found an improvement in matrix performance of 6 and 7%-year olds, but not 4% and 5%-year olds on the posttest. Training affected only the older Ss, which suggests that before training can be effective for a j>, that S; must have a certain basic cognitive level — more advanced than the younger group of Ss. This suggests that only those Ss who have a necessary minimum level of competence w i l l profit from a short-term cognitively related training session. Those Sis who improve on posttest reclassification behavior should have a higher pretest test score than those j5s who do not improve on the posttest. It i s likely that the amount of training necessary to cause improvement i n 12 performance varies with the individual -- some needing more than others — and that the amount of training necessary i s a function of the kind of trainings but also the competence and performance level of the j>. An important distinction made by Flavell and Wohlwill (1969) ap-plying to cognitive behavior, i s the competence and performance dis-tinction. "The competence model (refers to) the formal logical repre-sentation of the structure cf the domain; a performance model represents the psychological processes by which the information embodied in com-petence actually gets assessed and utilized in real situations." (Flavell and Wohlwill, 1859, p. 71). This distinction i s important to this thesis because the limited amount of training w i l l have i t s specific effect on the performance variable — the assessment and ut i l i z a t i o n of the structure in real situations. In order to cause a significant change in the competence of an individual which is reflected in performance, much more than a 15-minute training session is necessary. Using the distinction described above, F l a v e l l and Wohlwill constructed a general model describing conceptual development from never-in-competence to always--in-performance involving four stages. Completion of any task is considered to be a function of the product of the level of competence and the level of performance tempered by individual reactions to task variables. In the f i r s t stage, the child w i l l always f a i l because competence equals zero. In the second stage, competence increases from zero to one (where one is the ideal state), 13 but performance has a value of zero or very low, leaving the proba-b i l i t y of task completion at a minimum. In stage three, competence i s close to one and the effect of task variables begins to minimize, so that success is a function of each s a b i l i t y to look beyond the inter-ference of task variables. In stage four, competence equals one, performance equals one and the interference of task variables i s a l -most n i l , so that the probability of success is nearly one. (Flavell and Wohlwill, 1969). Training in a multiplicative classification task should affect specifically those Ss near the end of stage 2 and into stage 3, since at these stages the interference of task variables is the primary reason for lack of success. This study intends to elimi-nate the effect of some task variables through horizontal training — interrelation of cognitive structures at the same levels. : It seems likely that experience in these situations (training), where successful, operated to make functional an operation that probably was close to becoming established already, at the start, so that i t required a certain amount of "priming" from the mediator u t i l i z e d during the training session." (Flavell and Wohlwill, 1969, p. 108). A fifteen minute training session should prime the cognitive structure of a S so that the S^  may use that structure more effectively. In this thesis Ss were divided into two groups.° the cognitively-related training group and the non-cognitively related training group. A l l Ss were given a pre and post reclassification task and a generali-zation task. The J3s given cognitively related training received matrix training; the Ss given npn-cognitively related training received the 14 WISC Block Design test. Subjects were alternatively placed i n the WISC or Matrix condition and those Ss who successfully completed the pretest reclassification (obtaining a score of 5), were immediately given the generalization task, a reclassification task involving several different dimensions than any of the test or training task. Summary of Hypotheses 1. Subjects who receive cognitively-related training should improve on a posttest reclassification task, but ^ s who receive non-cognitively related training should not improve. 2. Subjects who improve under cognitively-related training w i l l have a higher pretest score than those Ss who do not improve ? under the same training. 3. Subjects successfully completing the pretest should have significantly higher scores on the generaliaation task than the WISC or Matrix groups. The Matrix group should have higher scores on the generalization task than the WISC group. 15 METHOD Subjects Subjects were 47 grade 2 students and 12 grade 3 students from private Catholic schools. There were 28 females and 31 males ranging in age from 75 months to 110 months with a mean age of approximately 96 months. Test Materials The reclassification task used for the pre and posttest consis-ted of two sets of nine styrofoam blocks varying according to color (red, yellow, green) and shape (square, c i r c l e , triangle). Five other blocks were also added to the original number for a total of 23 stimuli. These last five blocks consisted of a blue donut c i r c l e , a red odd-shaped form, two small squares (yellow and green) and a small yellow circ l e (see Figure 3). There were seven matrix tasks used for the cognitively-related training group. The c r i t e r i a on the matrices were as follows: (1) object (pens and brushes) X number; (2) object (fishes and birds) X orientation; (3) object (boys and girls) X expression (smiling and sad); (4) number (hearts) X shading; (5) object (beads and flowers) X number; (6) object (bracelets, rings, watches) X color (red and green); (7) object (checkers and sticks) X color (red and black). (See Figures 4 and 5.) The f i r s t , f i f t h , sixth and seventh matrices are constructed with real objects (see Figure 4) and the second, third and fourth matrices are on paper (see Figure 5). The WISC Block Design Test (Wechsler, 1949) was used for the R = Red G = Green Y = Yellow B = Blue FIGURE 3 Blocks for Reclassification Task (Pre and Post) 17 j 3 PENS 1 PEN ASSORTMENT OF PENS AND PENCILS AND BRUSHES 1 BRUSH MATRIX 1 4 BEADS ! ASSORTMENT OF BEADS AND FLOWERS j 4 FLOWERS ! 2 FLOWERS' MATRIX 5 ASSORTMENT OR WATCHES, BRACELETS AND RINGS MATRIX 6 i GREEN BRACELET IRED BRACELET WATCH WATCH RED BRACELET; I RING FOUR RED CHECKERS FOUR BLACK CHECKERS FOUR RED STICKS FOUR BLACK STICKS MATRIX 7 ? MUST BE FILLED BY SUBJECT FIGURE 4 Matrices with Toys for Matrix Training 13 PICTURE MATRICES FOR MATRIX TRAINING non-cdgnitively related training group. The generalization task consisted of 16 cards of people clas-sif i a b l e according to a number of different categories: cartoon versus real, male vs. female, adult vs. child, on telephone vs. not on telephone, running vs. not running, square vs. rectangle, body vs. face, long hair vs. short hair, dressed vs. not dressed, and together vs. alone. Procedure A l l Ss were tested in a private room for a 25-minute session. Throughout a l l phases of the experiment Ss were seated at a desk facing the E. Successive Ss were alternately placed i n either the control (WISC) or training (Matrix) group. Those who successfully completed the pretest (attaining a score of 5) formed a third group who were im-mediately given the generalization task. Subjects in the control condition proceeded to work on the WISC Block Design Test for 15 minutes and Sis in the training condition proceeded to work on the matrix tasks after the pretest. For the pretest, 18 blocks (see Figure 3) were placed i n a ran-dom arrangement on the desk in front of the child. The child was asked to arrange the blocks into three groups so that each block in each group would have something the same as the other blocks i n that group. The experimenter c l a r i f i e d his meaning by arranging a random assort-ment of pens, pencils., and brushes into three groups, explaining that in each group (e.g., pens) a l l had something the same (e.g., they were a l l pens). Subjects were then instructed to begin. On completing the f i r s t classification and/or when the indicated he was finished, the _E rearranged the blocks and asked the to sort the blocks into three groups again. This time the j> was asked to arrange the blocks in a different way than he had done in the previous turn, remembering that the blocks in each group must have something the same as a l l the other blocks in the group (like the pens, pencils and brushes). After making three groups or some semblance thereof, the blocks were again mixed together and a block added. If the last classification was color, a donut circ l e was added before the odd-shaped form, and i f the last classification was form, the odd-shaped form before the donut ci r c l e . This meant that with the addition of an extra block, the J> had to change his criterion of classification. The donut circ l e f i t t e d only the shape c r i t e r i a and the odd-shaped form, only the color c r i t e r i a . With the addition of either of the two blocks, was asked to make three groups. Three small blocks were added for a f i n a l sorting — either by color or shape. If failed to correctly classify the blocks on any two consecutive attempts, the pretest was terminated. Matrix Training Subjects assigned to receive cognitively-related training pro-ceeded to the matrix tasks after the pretest. The f i r s t training task was divided into three parts: (1) S_ f i l l e d in the fourth c e l l of a matrix by considering the attributes to the l e f t of the matrix. (Figure 1 outlines attributes on both the l e f t and top side of the ma-trix.) (2) S^  f i l l e d in the fourth c e l l of a matrix by considering at-tributes on top of the matrix. (3) Si f i l l e d in the fourth c e l l of a 21 matrix by considering both the attributes to the l e f t and on top of the matrix. For this task and for the f i f t h and sixth matrices fJ was instructed to place objects selected by E in three cells of the matrix and to choose the objects for the fourth c e l l from a number of alter-natives. For example, with the f i r s t matrix was asked to place three pens in the bottom-left-hand c e l l of the matrix, one pen in the bottom-right-hand c e l l and, then, to choose the correct solution for the last c e l l from the pens and brushes beside the matrix. For the next three matrices, S_ made his.choice by "X-ing" one of the several alternatives. The experimenter asked the whether two or three alternatives to the one chosen would be correct, after which _E would point out the correct solution ( i f had not already decided) and explained the two c r i t e r i a necessary for the solution. The subject was asked to justify his choice for the f i f t h and sixth matrix completion by explaining to the jE which two c r i t e r i a the solution had to meet. If S_ chose the incorrect objects or did not give the proper explanation, E pointed out the two c r i t e r i a which the solution must meet. For the seventh matrix^ S_ was f i r s t asked to make four groups of things that go together or have something the same (four red check-ers, four black checkers, four red sticks and four black sticks). These four groups were arranged on a f l a t piece of cardboard with four ce l l s , so that adjacent cells had one common attribute. Once the matrix was completed, was asked to explain the attributes common to adjacent c e l l s . If the solution was incorrect, E solved the matrix 22 and pointed out the common attributes. After completion of this task, Ss proceeded to the posttest administered in the same manner as the pretest. WISC Training Subjects assigned to this group received the WISC Block Design Test immediately after the pretest i n the manner prescribed by the manual (Wechsler, 1949, p. 77). If S^had finished the task before 15 minutes had elapsed, _E helped ;S to complete one of the designs by placing one or two of the blocks in the correct perspective according to the example pictures. In this way, S^ was kept at his task for a f u l l 15 minutes. After completion of this task, a l l Ss were given the posttest. A l l JSs were restricted to a 15-minute training session. Generalization Task For the generalization task, the 16 cards were displayed on the desk before j3 who was asked to make two groups of cards so that the cards i n each group went together with a l l the other cards i n that group; that i s , they a l l had something the same. Hie experimenter i l -lustrated how two groups could be formed by showing S^  ten white cards and making two groups, explaining that the cards i n each group had something the same as the other cards i n that group. At the completion of any task, whether successful or unsuccess-f u l , F. responded with a "fine" or "okay" to equalize reinforcement across Ss. Scoring On the block sorting task, j[ was given one point for each 23 correct classification for a maximum of five. Likewise on the matrix training tasks received one point for each correct solution for a maximum possible of three. (Reason for a maximum possible of three w i l l be explained in the discussion). On the generalization task one point was given for every correct classification with a possible maximum of 10. \ 24 RESULTS Twenty-one Ss completed the i n i t i a l task, 17 underwent the WISC condition and 21 Ss received matrix training. Table 1, Appendix A, and Table 2 give information on the raw data and means of the various groups for scores on the pre and post and generalization tasks. An analysis of variance was computed on the pre and posttest scores taking into account the Condition (WISC or Matrix) and the Sex of the S^. The posttest score (X = 2.18) was significantly greater F = 9.89, df = 1,34, p < 0.01 than the pretest score (X = 1.72). No other significant effects were found. The summary table for this analysis is in Table 3, Appendix A. An examination of the data indicated more Ss improved from pre to posttest under the Matrix training than under the WISC condition (a= 10 and 2, respectively). A subsequent chi-squared analysis on these data was significant (x2 = 5.32, df = 1, £ < 0.05) ill u s t r a t i n g the positive effect of matrix training. Further observation revealed that one S_ in the WISC condition had increased her pretest score of one to a posttest score of four. This S/s improvement may have resulted from the random arrangement of blocks on the posttest — E_ noticed that one _S had begun her c l a s s i f i -cation on the posttest that five red blocks were beside one another (this arrangement led to a color classification by S^  which she had not done previously). Omitting the data of this S_, an analysis of variance was again performed on the same pre and posttest scores with Condition (WISC and Matrix) and Sex as between S factors (see Table 4, 25 TABLE 2 MEAN SCORES FOR MATRIX, WISC AND COMPLETED GROUPS - • • — 1 PRETEST POSTTEST GENERALIZATION TASK WISC (17) 1.64 1.88 1.64 MATRIX (21) 1.80 2.47 1.76 (1) IMPROVED (10) 2.60 4.00 2.09 (2) NON-IMPROVED (11) 1.09 1.09 1.45 COMPLETED (21) - - 3.05 26 Appendix A). The posttest score (X = 2.09) was significantly greater (F = 10.82, df = 1,33, £ < 0.01) than the pretest score (X = 1.72). The Pre-Post X Condition interaction was also significant this time (F = 5.96, df = 1,33, p < 0.05). An orthogonal polynomial analysis of the P X C interaction showed the posttest score to be significantly larger than the pretest score under the matrix training (X = 2.47, 1.80 respectively), (F = 16.67, p < 0.01), but not under the WISC condition (X = 1.71, 1.64 respectively). An analysis of variance was done comparing Ss who improved un-der matrix training and ^ s in the WISC condition with Pre vs. Post scores as the dependent variable. An orthogonal analysis of the sig-nificant Pre-Post X Condition interaction (F = 12.52, df = 1,25, 2_ < 0.01) showed a significantly higher posttest than pretest score for the improved matrix Ss (X = 4.0, 2.60 respectively), (F = 28.82, df - 1,25, £ < 0.01) but no difference for control Ss (X = 1.88, 1.64 respectively) (see Table 5, Appendix B). To test the hypothesis that Ss improving with matrix training have a significantly higher pretest (X = 2.6) score, than Ss not im-proving (X = 1.09) a t~test was computed. The result was significant (t = 2.36, df = 19, £ < 0.025) supporting the hypothesis. An analysis of variance was also computed for Ss responses on the generalization task, taking into account the Sex and Condition of S^. The completed group was significantly batter than the WISC and Matrix training (F = 10.38, df = 2,53, £ < 0.01) on the number of responses given. The WISC group did not di f f e r significantly from 27 Matrix group (see Table 6, Appendix B). A further analysis of variance comparing WISC, improved Matrix, and non-improved Matrix Ss revealed no significant difference (see Table 7, Appendix B), although the mean of the improved Matrix group i s greater than the mean of both the WISC and the non-improved Matrix group (X = 2.09, 1.64, 1.45 respectively). 28 DISCUSSION A review of training experiments by Flavell and Wohlwill (1969) revealed that generally one-half of the Ss benefitted or improved from training. This was the case in this experiment; matrix training improved the reclassification performance of 10 out of 21 Ss (48%). The number of Ss improving under the matrix training condition was significantly greater than under the WISC training condition and an i n i t i a l analysis of variance showed a significant Pre- Posttest d i f -ference, but a non-significant Pre-Post Condition interaction. A possible explanation for this non-significant interaction was the ceiling effect of the posttest score; seven of ten Ss who improved with matrix training received a maximum of 5 on the posttest. Subjects, trained on the matrices, who improved, had much higher pretest scores (X = 2.6) than Ss who did not improve (X = 1.09). This finding was as predicted, because only Sis who had a relatively stable competence structure (as indicated by their pretest score) should improve from a 15-minute training session. That i s , the matrix training administered should be effective i n reducing task variable interference thus improving performance, but should not be sufficient to effect a major alteration prematurely i n a gradually developing competence structure. A second possible explanation for some Ss improving on the post-test and not other Ss, was that improving Ss correctly understood the matrix tasks. If this were true, these Ss should have had s i g n i f i -cantly higher matrix completion scores. Although Ss who improved 29 did show better matrix completion scores, the differences were not significant (X 2 = 5.62, df = 1, £>0.10) (See Table 8, Appendix A). (Only the scores for correct responses on the last three matrices were compared because the f i r s t four matrices were meant to instruct S^  i n Tfhat was required of him in completing a matrix.) A third possible reason for some Ss improving under matrix training was their age. Table 3, (Appendix A) shows Ss divided into two groups according to age, and illustrates that the older group was only slightly more successful than the younger group on the matrix tasks, suggesting that age was an unimportant factor in determining which Ss improved. The pretest scores of Ss who improved indicated that the level of a £'s logico-mathematical structure rather than his age or his understanding of the matrix task requirements determined imp.rovement. Those Ss who had a more developed competence structure were those who improved. Matrix training did not significantly affect the performance of Ss on the Generalization task. The mean of improved matrix Ss was higher, but not significantly so than that of WISC trained or non-improved matrix trained Ss. The lack of generalization may be a result of the limited amount of training received by each S_. Jacobs and Vandeventer (1971) discovered extended training (1 month) was much more effective in showing transference than a regular 30-minute training session. That the generalization task is related to the pre and post re-30 classification task can be seen from the scores of the Ss who success-fully completed the pretest. These Ss scored significantly higher than either Matrix or WISC trained JSs on the generalization task. These £>s have a competence structure close to one (on a scale from zero to one) with a performance value approaching one (Flavell and Wohlwill, 1969) so that the probability of completing a task is close to one. Thus the training time was too short to effectively remove the task variable interference of the generalization task. Perhaps the materials (magazine clippings on cards) used for the Generaliza-tion task may have been sufficiently different from the other tasks to confuse Ss. These materials probably present more task interference than the posttest so that the $_ could not overcome them as he did in the posttest. Also, the Generalization task was classifiable accor-ding to ten c r i t e r i a ; this great variety of pos s i b i l i t i e s may be a c r i t i c a l task interference factor. Future Research The present study suggests the possibility for further research in the area of training on multiplicative classification. A compar-ison should be made of a pre- and post-matrix task with reclassifica-tion training and a pre- and post-reclassification task with matrix training. This would show i f the training procedures have reciprocal effects, that i s , i f matrix training causes an equal amount of impro-vement on reclassification tasks as reclassification training does on matrix tasks. This would help to cla r i f y the nature of the relation-ship between the reclassification task and the matrix task. The 31 results should be reciprocal since the tasks are closely related. Reclassification requires the £ to classify a group of objects accor-ding to one criterion remembering but ignoring a previous criterion, and multiplicative classification requires the to classify objects according to two c r i t e r i a simultaneously. In each case S_ must be aware of the fact that objects can be grouped according to two di f -ferent c r i t e r i a . A study should be Conducted in which the ceiling effect was eliminated on the posttest reclassification task. Matrix trained Ss may attain a higher posttest score than five (the maximum possible in this study) i f given a wider range of reclassification p o s s i b i l i t i e s thus tapping the f u l l effect of training. Further, i t i s suggested that a similar study be done using three types of generalization tasks. One with dimensions closely related to those of the reclassification task, a second with one dimension related to the reclassification task, and a third with dimensions not similar to the reclassification task. Two groups of Ss should be tested one receiving 15-minute training and the other receiving 30~minute training. Thus the effect of the amount of training on a short-term basis could be analyzed as a function of the posttest performance and the amount and kind of transfer. The effect of reclassification p o s s i b i l i t i e s x^ithin one group of objects should be analyzed for possible task variable interference in the generalization task. A design requiring one group of Ss to classify each of five groups of objects two ways and another group 32 of Ss to classify one set of objects ten ways. This could validate the possibility of a large amount of task variable interference i n the generalization task used i n the present investigation. A suggestion arising from this thesis is that the cognitive level of i> can be determined not only by his pretest score but also by the amount of training necessary to cause a determined increment in the posttest scores. An experiment could be designed i n which E_ repeats a 15-minute training session un t i l _S. reaches a specified posttest criterion level. This would give E an indication of the cognitive structure of JS prior to training; the more training required to reach criterion, the less functional the cognitive level of S. Although in training _S_, E may alter the cognitive structure as well as removing the task variable interference, the performance of the S^  w i l l give a general indication of his level of cognitive development. SUMMARY It was expected that j5s who received cognitively related training would improve on a posttest reclassification task, but j3s receiving non-cognitively related training would not improve. This hypothesis was not confirmed by an i n i t i a l analysis of variance using pre- vs. posttest scores as the dependent variables. But a s i g n i f i -cantly larger number of Ss showed higher posttest scores under Matrix training than under WISC training. When the data of one improving S in the WISC condition were removed, an analysis of the Pre- vs. Posttest scores resulted i n a significant Pre vs. Post x Condition 33 interaction in the predicted direction. Subjects who improved under cognitively related training were predicted to have a higher pretest score than those Ss who did not improve under the same training. This hypothesis was confirmed with significantly higher pretest scores for improved vs. non-improved matrix ^s. It was also expected that Ss successfully completing the pre-test would have a significantly higher score on the Generalization task than the WISC or Matrix group , and the Matrix group would have higher scores on the Generalization task than the WISC group. This hypothesis was partially confirmed i n that the Completed group had a significantly higher score than either the Matrix or WISC groups on the Generalization task, but the llatrix group did not have a sig-nificantly higher score than the WISC group. 34 BIBLIOGRAPHY Baldwin, A. L. Theories of child development. New York: John Wiley and Sons Ltd.,1967. Denny, Nancy Wadsworth. A developmental study of free classification in children. Child Development, March, 1972. Vol. 43, No. 1, pp. 221-232. Fl a v e l l , J. H. The developmental psychology of Jean Piaget. New York: D. Van Nostrand Co. Ltd., 1964, pp. 164-184. Fl a v e l l , J. H.> & Wohlwill, J. F. Formal and functional aspects of cognitive development. Studies i r cognitive development. Elkind, D. and Flavell, J.H. (Eds} , 1969. New York: Oxford University Press, pp. 70— 113. Hays, W. L. Statistics for psychologists. New York: Holt, Rhinehart & Winston, 1966. Inhelder. Barbel and Piaget, J. The early growth of logic i n the  child. Lunzer, E. A., and Papert, D. (Translator) London: Routladge and Kegan Paul, 1964. Jacobs, P. I., and Vandeventer, M. The learning and transfer of double classification s k i l l s : A replication and extension. J. of  Experimental Child Psychology, 1971. Vol. 12, No. 2, pp. 240-257. Jacobs, P.I., and Vandeventer, M. The learning and transfer of double classification s k i l l s by f i r s t graders. Child Development, 1971, 42, pp. 149-159. Kofsky, E. A scalogran study of classificatory development. Child  Development, 1966. Vol. 37, pp. 191-203. Mackay, C. K., Fraser, Joan and Ross, Isabel. Matrices, three by three: Classification and Seriation. Child Development, 1970, 41, pp. 787-797. Overton, W.F., and Brodzinsky, D. Perceptual and logical factors in the development of multiplicative classification. Developmental  Psychology, 1972. Vol. 6, No. 1, pp. 104-109. Overton, W.F., Wager, Janis,, and Dolinsky, Harriet. Social-class d i f -ferences and task variables in the development of multiple c l a s s i f i -cation. Child Development, 1971, 42, pp. 1951-1958. Overton, W. F., and Jordon, R. Stimulus preference and multiplicative classification in children. Developmental Psychology, 1971. Vol. 5, No. 3, pp. 505-510. 35 Parker, ?,.K. s and Day, Mary C. The use of perceptual, functional and abstract attributes in multiple classification. Developmental  Psychology, 1971. Vol. 5, #2, pp. 312-319. Parker, R. K.> Fief f , Margery L., and Sperr, S. J , Teaching multiple classification to young children. Child Development., 1971, 42, pp. 1779-1789. Piaget, J. The Psychology of Intelligence. London: Routledge & Regan Paul Ltd., 1950. Piaget, J. Development and Learning. Hippie, E. R., and Rockcastle, V.N. (Eds.). Piaget Rediscovered. New York? Ithaca. School of Education, Cornell University, March, 1964, pp. 7-21. Resnick, L. B., Siegel, W. A., and Krssh, Ester. Transfer and sequence in learning double classification. J . of Experimental  Child Psychology, 1971. Vol. 11, No. 1, pp. 139-149. Shantz, Carolyn. A developmental study of Piaget's theory of logical .multiplication. Merill-Palmer Quarterly, 1967, JL3, pp. 121-137. Siegel, S. Nonparametric statistics for the behavioral sciences. N.Y.: McGraw-Hill Book Co., 1956, pp. 42-47. Siegel, A. W. , and Kresh, Estfir. Children's abi l i t y to operate within a matrix: A developmental study. Developmental Psychology, 1971, Vol. 4, #2, pp. 232-239. Siegel, I. E., Anderson, M., and Shapiro, H. Categorization behavior of loxirer and middle-class negro preschool children: Differences in dealing with representation of familiar objects. The Journal of  Negro Education'V, 35, 1966, p. 218-229. Sigel, I. E., Rooper, Annemarie, and Hooper, F.H. A training procedure  for acquisition of Piaget's conservation of quantity; A p i l o t study  and i t s replication. Logical Thinking in Children, Sigel, I.E., (Ed.), Hooper, Frank K. New York: Holt. Shinehart and Winston Inc., 1968, pp. 295-308. Wechsler, D. Weschler Intelligence Scale for Children (manual). New York: Psychological Corporation, 1349, pp. 77-79. Wei, I. T. D., Lavatelli, C., and Jones R. Piaget's concept of clas-sif i c a t i o n : A comparative study of socially disadvantaged and middle-class young children. Child Development, 1971, 425 pp. 919-927. Winer, B.J. St a t i s t i c a l principles in experimental design. New York: McGraw-Hill Book Company, 1962. 36 APPENDIX A Table 1 Raw Scores on Pretest, Posttest, and Generalization 8 Number of Ss Correctly Responding on Matrix Tasks 9 Matrix Training Group Analyzed by Age TABLE 1 RAW SCORES ON PRETEST, POSTTEST, AND GENERALIZATION MALE WISC FEMALE MALE MATRIX FEMALE PRE POST GEN. SI 0 0 1 S2 0 1 1 S4 3 3 4 S l l 4 4 3 S12 4 4 3 S13 1 1 2 S16 0 0 0 S3 4 4 3 S5 4 4 3 S6 1 1 0 S7 1 1 2 S8 1 1 1 S9 1 1 0 S10 1 1 0 S14 1 1 2 S15 1 4 3 S17 1 1 0 S20 1 1 1 S21 1 1 2 S22 1 1 2 S24 1 1 0 S26 1 1 1 S27 1 1 ' 4 S28 1 1 2 S29 1 3 3 S32 0 i a. 1 S33 4 5 1 S34 4 5 1 S18 0 0 0 S19 3 3 1 S23 1 1 1 S25 1 1 0 S30 4 5 2 S31 4 5 2 S35 4 5 4 S36 1 5 5 S37 4 5 2 S38 0 1 2 TABLE 1 (continued) S41 S42 S45 S46 S48 S49 MALE S50 S52 S54 S55 S56 S53 S59 COMPLETED S39 S40 S43 FEMALE S44 S47 S51 S53 S57 PRE POST GEN. 3 1 4 4 2 4 1 5 3 4 2 4 - 2 4 2 1 3 6 4 4 1 39 TABLE 8 NUMBER OF Ss CORRECTLY RESPONDING ON MATRIX TASKS GROUP (Ss) MATRIX MATRIX MATRIX TWO OR THREE 5 6 7 MATRICES CORRECT TRAINING NON-IMPROVEMENT (11) 5 6 3 4 18 TRAINING IMPROVEMENT (10) 8 7 9 9 33 X 2 = 5.62, df = 3, p > 0.10 TABLE 9 MATRIX TRAINING GROUP ANALYZED BY AGE GROUPS (Ss) % % CORRECT MEAN RESPONSE MEAN POSTTEST IMPROVING ON MATRIX ON G.T. SCORE AGE 78-92 40% 53% 1.30 2.50 (10) AGE 93-114 54% 56.66 2.18 2.46 (ID 41 APPENDIX B ANOVA TABLES Table 3 Analysis of Variance for WISC and MATRIX Ss for Pre and Post, Sex and Condition 4 Analysis of Variance for WISC and Matrix Ss''" for Pre and Post, Sex and Condition 5 Analysis of Variance for WISC vs. Improved Matrix for Pre and Post Scores 6 Analysis of Variance for WISC, Matrix and Completed Groups on Generalization Task 7 Analysis of Variance for WISC Improved and Non-Improved on Generalization Tasks 42 TABLE 3 ANALYSIS OF VARIANCE FOR WISC AND MATRIX Ss FOR PRE AND POST, SEX AND CONDITION SOURCE df US F CONDITION (C) 1 2.68 0.68 SEX (A) 1 4.32 1.10 A X C 1 5.51 1.40 Ss/A X C 34 172.44 PRE-POST (P) 1 4.26 10.92* P X C 1 0.87 2.23 P X A 1 0.34 0.87 P X A X C 1 0.24 0.61 P X Ss/A X C 34 13.29 MSe BETWEEN =3.91 MSe WITHIN =0.39 * p_ < 0.01 43 TABLE 4 ANALYSIS OF VARIANCE FOR WISC AND MATRIX Ss FOR PRE AND POST, SEX AND CONDITION SOURCE df MS F CONDITION (C) 1 3.26 0.62 SEX (A) 1 3.93 0.76 A X C 1 6.00 1.16 Ss/A X C 33 171.19 PRE-POST (P) 1 3.03 10.82** P X C 1 1.67 5.96* P X A 1 0.10 0.35 P X A X C 1 0.46 1.64 P X Ss/A X C 33 9.24 MSe BETWEEN =5.18 MSe WITHIN =0.28 * p_ 0.05 ** p 0.01 Data of one For reasons j> has been removed from the WISC condition, see results section. 44 TABLE 5 ANALYSIS OF VARIANCE FOR WISC vs. IMPROVED MATRIX FOR PRE AND POST SCORES SOURCE df MS F CONDITION (C) 1 29.68 6.11* Ss/C 25 121.32 PRE-POST (P) 1 6.00 17.64** P X C 1 4.26 12.52** P X Ss/C 25 8.74 MSe BETWEEN =4.85 MSe WITHIN =0.34 * £ 0.05 ** £ 0.01 45 TABLE & ANALYSIS OF VARIANCE FOR WISC, MATRIX AND COMPLETED GROUPS ON GENERALIZATION TASK SOURCE > df MS F CONDITION (C) 2 19.41 3.75* SEX (A) 1 0.22 0.08 C X A 2 2.25 0.43 Ss/C X A 53 137.04 MSe =2.58 * p_ < 0.05 TABLE 7 ANALYSIS OF VARIANCE FOR WISC IMPROVED AND NON-IMPROVED ON GENERALIZATION TASKS SOURCE df MS F CONDITION (C') Ss/C 2 35 5.64 1.49 66.18 


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