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The effects of cognitive style, conceptual tempo and training on problem solving processes of fifth grade… Greer, Ruth Nancy Elizabeth 1974

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THE EFFECTS OF COGNITIVE STYLE, CONCEPTUAL TEMPO, AND TRAINING ON PROBLEM SOLVING PROCESSES OF FIFTH GRADE CHILDREN by RUTH NANCY ELIZABETH GREER B.A. Carleton University, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF • THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS < \i * in the Department of Educational Psychology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1974 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of E d u c a t i o n a l Psychology The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada - f i i i ABSTRACT The purpose of the present study was to explore the nature of the relationships between cognitive style, conceptual tempo, and processes employed by f i f t h grade children during verbal problem solving. In addition, the effectiveness of programmed instruction to train children in the use of analytic and reflective modes of problem solving was investigated. Cognitive style (analytic responses on Denney's Cognitive Style Test), conceptual tempo (errors and latency of response on Kagan's Marching Familiar Figures Test), verbal creative thinking a b i l i t y , and school achievement were determined for eighty-one children in three f i f t h grade classes. Verbal creative thinking a b i l i t y and school achievement were treated .as covariates.. Classes were randomly assigned to three treatment conditions: 1) problem solving training via programmed instruction; 2) programmed instruction of unrelated content; 3) no programmed instruction. The treatment, style, and tempo variables were evaluated in terms of their effects upon ten measures of problem solving which included time spent on four criterion problems, quantity and quality of questions asked, and number of solutions offered. The data indicated that training was successful in increasing time spent on the problems, and quantity and quality of questions asked, but had no effect on number of solutions offered. However, aptitude by treatment interactions between training and both style and tempo indicated that the performance i i i of non-analytic and highly impulsive children was no better than untrained children of similar style and tempo. While conceptual tempo was found to account for a significant amount of the variance in the three measures of time spent on the problems and one measure of question asking, cognitive style was not an important contributor to measures.of problem solving. Interactions were repeatedly found between style and tempo which indicated that these two variables tended to have a moderating effect on each other. Thus, a child who was highly impulsive performed reflectively i f he was also highly analytic. Likewise, a non-analytic child performed analytically i f he was also highly reflective. While school achievement was not related to performance, verbal creative thinking a b i l i t y was found to be positively related to problem solving performance, with verbal originality being a better predictor than either verbal fluency or f l e x i b i l i t y . A limitation of this study arose from differences in teaching style to which the students had been exposed during the eight months preceding this study. As a result of the analyses, i t was concluded that a reappraisal of the effects of style and tempo is warranted, with attention given to the interaction between these two variables. Assessment of the effectiveness of modified versions of the present self-paced training program with non-analytic and highly impulsive children was recommended. iv TABLE OF CONTENTS CHAPTER P A G E I. INTRODUCTION 1 II. REVIEW OF THE LITERATURE 4 Problem Solving 4 Cognitive Style and Conceptual Tempo 22 Effects of Style and Tempo on Problem Solving 33 The Problem 3 5 Research Hypotheses 37 Rationale 39 The Test Problems.. 39 The Training Program 41 III. METHODOLOGICAL CONSIDERATIONS 44 Subjects • 44 Design 44 Description of Covariates 46 Achievement • 46 Creativity 47 Stimulus Materials. . . . 48 Procedure; 50 Pilot Testing 5 0 Pre-Testing Si-Training 52 Post-Testing • • 5 3 IV. RESULTS •• • 5 5 S t a t i s t i c a l Procedures 55 V CHAPTER PAGE Tests of Hypotheses 58 Additional Findings 81 V. DISCUSSION 105 Cognitive Style and Problem Solving 105 Conceptual Tempo and Problem Solving . . . 107 , Training 110 Relationship Between Process and Product . . . 113 Limitations 114 Recommendations for Further Research . 117 BIBLIOGRAPHY 120 APPENDICES 129 A. The Test Problems and Classification of Question Type B. The Conceptual Style Test and The Matching Familiar Figures Test C. The Training Program D. Results of Regression Analysis E. Tables of Mean Residuals for Trained and Untrained Groups with Style and Tempo Trichotomized. v i LIST OF TABLES TABLE TITLE PAGE I COMPOSITION OF EXPERIMENTAL CLASSES. 45 II SYMBOLS USED IN STATISTICAL ANALYSIS. 57 III SUMMARY RESULTS (F-RATIOS) OF REGRESSION ANALYSES FOR 59 ALL DEPENDENT VARIABLES. IV GROUP MEANS FOR TEN MEASURES OF PROBLEM SOLVING. 60 V CORRELATIONS BETWEEN ORGANISMIC VARIABLES AND MEASURES 64 OF PROBLEM SOLVING. VI TESTS (FISHER'S Z) OF SIGNIFICANCE BETWEEN CORRELATIONS 68 FOR TRAINED AND UNTRAINED GROUPS: COGNITIVE STYLE (CST), CONCEPTUAL TEMPO, (MFF ERRORS, MFF TIME) AND MEASURES OF PROBLEM SOLVING. VII COMPOSITION OF NINE CELLS DERIVED VIA TRICHOTIMIZING 83 RESPONSES ON THE CST AND ERRORS ON THE MFF. VIII MEAN RESIDUALS OF FIVE MEASURES OF PROBLEM SOLVING 85 FOR LEVELS OF THE CST BY MFF ERRORS INTERACTION. IX COMPOSITION OF NINE CELLS DERIVED VIA TRICHOTIMIZING 95 LATENCY OF RESPONSE AND ERRORS ON THE MFF. X MEAN RESIDUALS OF TWO MEASURES OF PROBLEM SOLVING 96 FOR LEVELS OF THE MFF ERRORS BY TIME INTERACTION. XI RESULTS OF REGRESSION ANALYSES FOR TORRANCE SUB-TEST 103 SCORES AND PROBLEM SOLVING MEASURES. v i i LIST OF FIGURES FIGURE TITLE PAGE. 1 FACTORS IN PROBLEM SOLVING. 12 2 MEAN RESIDUALS OF MEAN TOTAL QUESTIONS FOR TRAINED (Tl) 7 0 AND UNTRAINED SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (LOW, MEDIUM, HIGH CST). 3 MEAN RESIDUALS OF MEAN INFORMATION-SEEKING QUESTIONS 71 ASKED BY TRAINED (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (CST). 4 MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR TRAINED 72 (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (CST). 5 TORRANCE SUB-TEST SCORES FOR TRAINED (Tl) AND UNTRAINED 75 (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (LOW, MEDIUM, HIGH). 6 MEAN RESIDUALS OF MEAN TOTAL TIME FOR TRAINED (Tl) AND 77 UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (LOW, MEDIUM, HIGH MFF ERRORS). 7 MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR TRAINED (Tl) 78 AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 8 TORRANCE SUB-TEST SCORES FOR TRAINED (Tl) AND UNTRAINED 80 (T23) SUBJECTS AT THREE LEVELS.OF IMPULSIVITY (LOW, MEDIUM HIGH). 9 MEAN RESIDUALS OF MEAN TOTAL TIME FOR SUBJECTS AT THREE 86 LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 10 MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR SUBJECTS AT 87 THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 11 MEAN RESIDUALS OF MEAN TOTAL QUESTIONS FOR SUBJECTS AT 88 THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 12 MEAN RESIDUALS OF MEAN INFORMATION-SEEKING QUESTIONS 89 FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). v i i i List of Figures continued. FIGURE TITLE PAGE 13 MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR SUBJECTS 90 AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 14 MEANS FOR CTBS (A), FLUENCY (B), FLEXIBILITY (C), AND 92 ORIGINALITY (D), FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND IMPULSIVITY (MFF ERRORS). 15 MEAN RESIDUALS OF MEAN PRIOR TIME FOR SUBJECTS AT 97 THREE LEVELS OF ACCURACY (MFF ERRORS) AND THREE LEVELS OF REFLECTIVITY (MFF TIME). 16 MEAN RESIDUALS OF MEAN PRIOR QUESTIONS ASKED BY 98 SUBJECTS AT THREE LEVELS OF ACCURACY (MFF ERRORS) AND THREE LEVELS OF REFLECTIVITY (MFF TIME). 17 MEANS FOR CTBS (A), FLUENCY (B), FLEXIBILITY (C), AND 100 ORIGINALITY (D) FOR SUBJECTS AT THREE LEVELS OF ACCURACY (MFF ERRORS) AND REFLECTIVITY (MFF TIME). ix ACKNOWLEDGEMENTS It i s with, pleasure that the writer acknowledges her gratitude to those who have contributed in various ways to this work: To Dr. Stanley Blank, under whose supervision this work was completed. Throughout the course of the investigation and in : the preparation of the training materials and manuscript, his advice, criticisms and encouragements have been most helpful. To Dr. Robert Conry, for direction and advice during the data analysis and manuscript preparation. To Dr. Nancy Suzuki, whose helpful suggestions and continued interest in the work is gratefully acknowledged. To Darlene Harris, Joyce Fox, and Ju l i a Litwintschik for their enthusiastic assistance during data collection. To Mr. M. Folkman, Mr. D. MacAulay, and the participating principals, teachers, and students of the Chilliwack School System whose continued cooperation and interest made this study possible. Finally, to my husband Galen for empathy. CHAPTER I INTRODUCTION In the past, research in both psychology and education has devoted considerable attention to identifying a variety of individual differences among children, and determining their effects on learning and achievement in the educational setting. Perhaps motivated by the apparently awesome differences in performance of lower and middle class children, recent interest has focused increasingly on individual variations in cognitive processing. Expanding on the early work of Bruner (1961) , Gardner (1953), Witkin (1962, 1964) and others, Kagan and his associates (1963, 1965 a,b; 1966, a,b) have identified stable individual differences in two dimensions of cognition; "style" and "tempo". Style refers to the degree to which one attends to and analyzes stimulus components, while tempo involves the tendency to be reflective or impulsive when processing information. What is of particular importance i s the relationship that has been found to exist between these two response modes and a variety of cognitive processes such as attention, perception, memory, inductive reasoning, and decision-making. In a l l of these areas impulsive, non-analytic individuals have been found interior to their more reflective analytic peers (eg. Kagan and Moss, 1963; Kagan, et a l . , 1964; Kagan, 1965; Seigelman, 1969; Drake, 1970; Odom, et a l . , Ault, et a l . , 1972). In school children these differences result in poorer performance in activities such as reading, concept learning, and general problem solving (Meichenbaum and Goodman, 1969; Davis and Klausmeier, 1970; Davey, 1971; 2 Butler, 1972; Mann, 1973). In a recent review article concerning individual differences in cognitive processing, Kagan and Kogan (1970) present evidence to demonstrate that the impulsive and non-analytic modes of information processing affect the efficiency of problem solving at a l l stages in the search, for solutions. These individuals incompletely perceive the subtleties of the stimulus array i n i t i a l l y , f a i l to ask questions that w i l l c l a r i f y the problem, and then continue to impulsively offer solutions without adequate reflection on their relative merits. In short, the impulsive, non—analytic child i s characterized by his use of Inadequate strategies. It has been demonstrated, however, that these individuals can be trained to use the more productive analytic and reflective response strategies in some isolated aspects of the problem solving process such as question-asking (eg. Ostfeld and Neimark, 1967; Baird and Bee, 1969; Messer, 1970; Briggs and Weinberg, 1973; Denney, 1973). It has further been demonstrated that children can be trained in the use of effective problem-solving strategies via programmed instruction (Covington and Crutchfield, 1965; Blank and Covington, 1965; Ripple and Dacey, 1967; Stokes, 1968). The intent of the present study was to explore the possibility of training fifth-grade children in the use of analytic and reflective strategies when solving problems through the use of programmed instruction. This research addressed i t s e l f to providing further insight into three major issues: 1. jThe nature and direction of the relationship that exists between cognitive style and processes used during verbal problem solving. The nature and direction of the relationship that exists Between conceptual tempo and processes used during verbal problem solving. The relative effectiveness of programmed instruction to train children at different points along the cognitive style and conceptual tempo continuums in the use of analytic and reflective strategies during verbal problem solving. 4 CHAPTER II LITERATURE REVIEW Problem Solving The definition of problem solving and the underlying processes thought to be involved vary according to the theoretical approach of the researcher, whether Cognitive Gestalt, Behaviorist, or Information Processing. Lti the literature review to follow, each of these theoretical approaches w i l l be de-Mt with.in turn.as they relate to problem solving. Cognitive Gestalt; From the viewpoint of Cognitive Gestalt thinking, a problem is a psychological state of discomfort or disequili-brium as sensed by the individual and solving restores the "psychical balance" (Shulman, 1965). As early as 1925 Kbhler referred to this state of "psychic tension" as resulting when a direct route to a goal i s blocked. Kohler proposed that the problem solution requires insight, a cognitive restructuring or reorganizing of thinking and perception. For the Gestaltist, problem solving is not the running off of prior habits but a process of restructuring and reorganizing. Changes occur in perception rather than in memory and what i s transferred from one situation to another are common patterns and configurations, not specific elements. Other Cognitive Gestalt theorists have stressed the importance of a change or recentering of perception as important in problem solving. Maier (1945) emphasized that the transfer of learning from one situation to another i s dependent on the relations perceived. For Maier, problem solving i s not simply t r i a l and error but is dependent on what the person has learned before that can be applied to the new situation. "Reasoning 5 w i l l not be limited to the way we have learned things, but w i l l depend upon the readiness with which the past learning i s subject to modification and reorganization. Before objectively identical elements can aid in solving the problem, i t is necessary for the subject to change his perceptions or memories so as to make for subjective identity." (Maier, 1945). Dunker (1945) spoke of the concept of "functional fixedness" as the in a b i l i t y of the learner to perceive new relations or new uses for elements in the situation due to the fixedness of perception. The process of solution involves a productive reformulation of the problem and of the elements therein. Solving problems such as the candle problem he developed requires a change in the perception of the possible uses of a box containing matches. More recent writing by cognitive theorists have continued to stress the importance of overcoming fixation behavior by the recentering of perception of elements within the problem. Sheerer (1963) has used his "9-dot" problem to demonstrate the d i f f i c u l t i e s experienced by individuals when habits and conventions prevent recognition of the appropriate use of materials to solve problems. Asher (1963) has written that problem solving i s the process of disrupting previously established concepts, which he contends i s just the reverse of learning. A similar approach has been taken by Maier and Burke (1966) with regard to the effect of previous learning and experience. The a b i l i t y to u t i l i z e experience rather than merely to have had i t , in their view, becomes- the c r i t i c a l factor in problem solving. Generalization is a matter of similarity among problem situations not the similarity among solutions. 6 To the Cognitive Gestalt theorist, behavior in the problem solving situation is conscious, strategic and purposive. "The organism perceives, thinks about, and analyzes his environment; he forms tenable hypotheses, tries plausible leads, follows rules, reasons, encodes, deduces, makes predictions, and calculates guesses." (Davis, 1973). This concept of an active organism is further exemplified in the writings of Bruner (1961). In Bruner's view, discovery and problem solving is a matter of rearranging or transforming the evidence that is assembled to yield additional new insights. Strategies are imposed by the learner to simplify the problem, but ultimately human problem solving goes beyond the information given. The principal weakness of this theory li e s in the fact that i t does not specify what w i l l lead the learner to the reformulation of a problem in a particular manner. While i t does emphasize the need for perceptual acuity and thorough analysis of problem elements, for reflection on the implications of past situations to the present problem, and for f l e x i b i l i t y of thought processes, i t offers l i t t l e else of substantive prescriptive value to the educator interested in training students to perform effectively in the problem solving situation. Behaviorist; A second theoretical approach to problem solving is that of the behavibrists. An early proponent of the S-R approach to problem solving, Thorndike (1898) proposed that problem solving is simply a complex.form of what is currently referred.to as operant conditioning. The cats in his puzzle box solved the escape problem by t r i a l and error through the gradual stamping.in of rewarded responses and.extinction of unrewarded responses. Skinner (1966) however, disagrees with Thorndike"s concept of t r i a l and error learning. The problem solving situation 7 occurs, i t i s Skinner's .position, .when the relations between a stimulus, a response, and the contingencies are complex. The solution involves finding the response which satisfies this complex set of contingencies. Thus, to aid problem solving, the contingent-relevant properties must be made more discriminable. Hull (1935) proposed the concept of habit-family hierarchies. Any given stimulus w i l l have associated with i t a series of responses which are arranged in hierarchical order. The response at the top of the hierarchy by definition has the greatest probability of occuring and the shortest latency. If the strongest response is incorrect, then a problem exists. Problem solving is completed when the order of the response hierarchy has so changed that the correct response is now dominant. This change in hierarchical arrangement takes place through t r i a l and error with selective reinforcement of correct responses and.extinction of the incorrect ones. Maltzman (1955) and Kendler and Kendler (1961) have provided further elaboration of this concept. Maltzman stresses that problem solving involves both divergent and convergent mechanisms. Productive thinking and problem solving .is the consequence of the integration of previously unrelated experiences. " A l l thinking involves mediated generalization and hence compounding of previously isolated habit segments" (Maltzman, 1955). Maltzman further states that "Thinking in general, and problem solving in particular may involve the selection of habit family hierarchies as well as the.selection of specific response sequences within the. hierarchy. The entire habit hierarchy w i l l be strengthened or weakened, not just individual, responses." Maltzman has stressed the importance of creativity, foriffpproblem solving is the 8 combining of previously unassociated elements, then the more remote the elements in the new combination the more creative the process of solution. To Hull's concept of habit family hierarchies, Kendler and Kendler (1961) have added the idea of horizontal and vertical processes. While problem solving is the consequence of combining previously independent S-R units, these can involve chaining of responses over time (horizontal) and the simultaneous operation of multiple associations at any one given time (vertical). Both Cofer (1957) and Staats (1966) stressed the role of verbal processes in the f a c i l i t a t i o n of generalized mediation referred to by Maltzman. Cofer contends that verbal processes function to impose restraints on response systems, f a c i l i t a t e discrimination of elements, assist in the formation of mediative or associative links, and are important tools for elaborating hypotheses and judging validity of solutions. Within Staats' mediational model, mediating verbal responses serve to e l i c i t appropriate chains of motor behavior sequences. Harlow's (1949) concept of learning set has also been an important component of behaviorist theory. It is his contention that a solution to a problem does not result from a single learning experience but occurs over multiple learning problems. "Learning to learn transforms the organism from a creature that adapts to a changing environment by t r i a l and error to one that adapts by seeming hypothesis and insight". Both Luchin (1942) and Maier (1945) as Gestaltists, have disagreed with Harlow on the value of sets. Luchin contends that set interferes with optimal performance. In his water-jar problems subjects who have acquired a particular set for solution w i l l perseverate in the use of 9 i t even when more efficient strategies are possible. Maier stresses that i t is direction and not set that i s important to efficient problem solving. "Mental set grows out of past experience and carry over into new situations, but directions in thinking arise under the stress of a problem and when they are new they cannot be traced to previously situations." (Maier, 1945). Thus, while a behaviorist approach would stress the importance of previously learned.responses.and their recombination, the cognitive Gestalt approach stresses the perception of stimuli in each problem situation and the reorganization the solver imposes on them. Hilgard (1966) has offered a c l a r i f i c a t i o n of the differences between the insightful versus t r i a l and error explanations of problem solving. He states that the difference between association theories and Gestalt theories l i e s in the implication of association theories that the possession of the necessary past experience somehow guarantees the solution. "While Gestalt theorists would agree that past experiences w i l l f a c i l i t a t e solution, they object to explanations in terms of previous experience without taking organization into account." For the Gestaltist insight is used to exemplify the applicability of the laws of organization but does not, in and.of i t s e l f , explain a l l problem solving. As one considers more contemporary writings of neo-behaviorists and cognitive psychologists, the schism in theoretical approaches to problem solving become increasingly d i f f i c u l t to discriminate. Gagn£ (1966), as a neo-behaviorist, defines problem, solving as "...an inferred change, in human capability that results in the acquisition of a generalizable rule which is novel to the individual, which cannot have 10 been established by direct r e c a l l , and which can manifest i t s e l f in applicability to the solution of a class of problems.". It i s dependent on the presence of certain previously learned rules. The role of the problem solver is to search through these rules to find ones that are relevant to the situation, and then combine these subrules to formulate an hypothesis which i s verified against reality. Ausubel (1968) as a cognitive psychologist, defines problem solving as "...any activity in which both the cognitive representation of prior experience and the components of a current problem situation are reorganized in order to achieve a designated objective." He emphasizes that existing cognitive structure plays a key role in problem solving and that.the solution of any given problem involves a reorganization of the residue of past experience so as to f i t the parti -cular requirements of the current problem. Ausubel further characterizes problem solving as an organic process in which the learner f i r s t defines the goal and then works backwards from i t in endeavoring to discover means-ends relationships. The increasing similarity between these two theoretical approaches as exemplified by Gagne and Ausubel is marked in comparison to the difference between earlier theorists such as Thorndike and KBhler. Information Processing: The third principal theoretical approach to problem solving is the Information Processing model, which has evolved from attempts to simulate human cognitive activity with a computer. In 1958 Newell, Simon, and Show stated that an adequate theory of problem solving must provide the following: 11 1. Predict performance of in specific tasks. 2. Explain how human problem solving takes place. 3. Define what processes are used. 4. Specify the mechanisms used in performing these processes. 5. Predict incidental phenomena that accompany problem solving. 6. Show how changes, internal and external, change problem solving behavior. 7. Explain how problem solving s k i l l s are learned. 8. Define what a problem solver has when he has learned the s k i l l s necessary for problem solving. The viewpoint of these writers was that computer simulation of human problem solving may shed further light on these issues. Within this f i e l d of research complex problem solving behavior Is built up of simple symbols manipulation. An algorithmic computer program (Hunt, 1968) is one which is generated to examine a l l possible solution-combinations in some pre-determined order. Eventually and inevitably, by pure t r i a l and error, i t locates the correct solution alternative. Such_ a system, however, f a l l s far short of imitating the shortcuts seen in human problem solving. This deficiency is overcome by the use of heuristics, which are rules of thumb which are built into the system to define hbw to search in ways that are probably f r u i t f u l and efficient, thus cutting search time, as a l l possibilities are not considered. A heuristic can be conceived of as similar to a set or a readiness to make a specific response to a specific stimulus. While these advances in computer programming sophistication are of interest, i t is questionable how much direct use they are in clarifying human problem solving. Reitman (1965) has cited two particular deficiencies: 1. The sequence is too rigid and does not provide for human d i s t r a c t i b i l i t y ; 2. They assume a perfect memory, uncharacteristic of human problem solvers. They are of value, however, 12 Functions, of Instructions Internal Processes Individual Differences Solution Rule Provide solution-model Verification Provisional i Rule i Matching specific to retained general model -Retaining solution model Guide Thinking-Combining Sub-o r d i n a t e Rules 4-4* Fluency in making new combinations Make cues distinctive— o o Search and • — Selection ^ t t t i t J -Distinguising rele-vant and irrelevant cues Stimulate Recall Recall of "* Subordinate*" Rules .Recall of previously learned rules i l l t U Number of •previously learned rules Figure 1 Factors in Problem Solving (Gagne, 1966) 13 in that such a study requires a precise and complete, statement of a sequentially organized set of problem solving processes that w i l l successfully complete the task, which no other theoretical approach has succeeded in doing to date. While theoretical approaches to problem solving differ somewhat there has been considerable consistency over the years in the stages thought to be involved. Dewey (1910) proposed a five stage sequence: 1. recognition of the problem; 2. location and definition of the problem or the isolation of relevant features: 3. formulation of possible alternative solutions; 4. mulling over or reasoning through the various poss i b i l i t i e s to determine the most l i k e l y one; 5. testing the selected solution. Wallas (1926) proposed the following four stages: 1. preparation; 2. incubation; 3. illumination; 4. verification. Kingsley and Garry (1957) conceived of the following six steps: 1. A d i f f i c u l t y is f e l t ; 2. The problem i s c l a r i f i e d and defined; 3. A search for clues is made; 4. Various suggestions appear and are tried out; 5. A suggested solution is accepted; 6. The solution is tested. Gagne in 1966 developed the following model which allows for the examination of the functions of instructions and the effect of individual differences in past experience, memory, perception, fluency, and evaluative s k i l l , (see Figure 1). From the standpoint of the educator, this model seems of particular prescriptive value. It can be seen from Gagne's model that instructions given to the learner have functional value at each stage of the process. Gagne (1964) has further articulated these functions as follows: 14 1. To identify terminal performance required 2. To identify parts of the problem situation 3. To assist in recall of relevant subordinate performance capabilities 4. To channel thinking in a given direction Colgrove (1970) has further demonstrated the effects of instruction in research with adults. The subjects were told to assume the role of an industrial time-study specialist. Half of the subjects were further instructed that this specialist was a very creative person. The Control group were simply given a description of the problem, which involved developing a worker rotation scheme that would maximize performance. The experimental group given the "creative" instructions produced significantly more original and innovative solutions than did the Control group. Gagne and Smith (1962) gave school aged children the "pyramid problem" to solve, and one half of the subjects were instructed to give reasons for each move made, while the Control group was not required to do so. Instructions to verbalize reasons had the effect of significantly increasing the number of successful solutions over that of the Control group. The authors conclude.that such instructions serve.to force children to analyze steps and to discover general principles involved. In a recentlstudy with both school aged children and adults, Robinson (1973) presented a series of mathematical problems to subjects In a verbal format and instructed one half of the subjects to draw diagrams representing the relations'presented in theT.problems while trying to solve them. A Control group received no such instructions. The results indicate superior performance for the experimental group. Robinson suggests that this superiority i s the result of an increase in 15 the concreteness of the problem, components: when such, drawing Instructions are given, and this aids solution. Other task variables have also been investigated as they affect performance. Burke, Maier, and Hoffman (1966) examined the function of hints to successful solution of the "Hat-Rack" problem. One group of subjects was given the hint at the onset of the problem while the second group was not given i t until twenty minutes of attempts at solving the problem. The results indicate that a greater number of subjects in the former group solved the problem in the ten-minute period following presentation of the hint than in the latter group. The authors conclude that the hint serves to give direction and to stop perseveration on inappropriate methods. Safren (1962) investigated the importance of establishing a set prior to working on a problem. His adults were given a verbal memory task which served to establish a set for looking for words of a particular category prior to solving anagrams. The superior performance of this group over a Control who only worked on the anagrams led Safren to conclude that such a set acts as a heuristic to decrease the number of solutions that must be considered. Hoffman, Burke, and Maier (1963) designed a study to investigate the effect of solving a simpler but related problem on subsequent performance on a test problem. One half of their subjects worked on several versions of the "hat-rack" problem which allowed for the use of additional equipment than just the two sticks and clamp. A Control group had no such prior experience. When performance was examined for both groups on the traditional version of this problem, those without previous experience solved the problem faster than those with such 16 experience. The authors conclude that previous experience may inhibit productive problem solving by providing too many false starts. Attention has also been directed to the examination of individual differences within the problem solver as they effect.performance. Age trends have been examined in a number of different tasks. Weir (1964) , using a probability learning problem found children between seven and ten years of age, unlike adults, respond in highly stereotype patterns and seem unable to discard a simple response pattern when i t does not pay off. He suggests that these young children may be unable to develop complex hypotheses, or cannot make f u l l use of them. Beilin (1967) examined performance in anagram problems. With children from eight to fourteen years of age both solution time and number of problems solved increased with age. It is to be expected that vocabulary and verbal experience are strongly influentlaihlin such a task. A twenty-questions problem format was used by.VanHHdrne.aridBBartz (1968) with six to eight year olds. The use of a constraint-seeking strategy in information-seeking behavior was observed to increase with age. Younger children tended to simply guess unt i l the correct response is located by chance. It was the conclusion of the authors that the younger child lacks the ab i l i t y to impose order on a perceptually disordered environment. Bruner (1961) noted similar patterns of response as Van Horn and Bartz, but among different individuals of the same age. He found differences in the degree to which children of the same age attempt to locate constraints before formulating an hypothesis. He classified children as using either "organized persistence" or "sheer doggedness". 17 In. Bruner's words, "...the child who has flooded himself with disorganized information from unconnected hypotheses w i l l Become discouraged and confused sooner than the child who has shown a certain cunning in his strategy- of getting information - a cunning whose principal component is the recognition that the value of information is not simply in getting It But in Being aBle to use i t . " . A recent study conducted By Ault (19 73) found similar trends. Fir s t , third, and f i f t h grade children were f i r s t divided into two conceptual tempo groups, impulsive and reflective, using Kagan's (1965) Matching Familiar Figures test. These children were then oBserved in a twenty-questions game of proBlem solving. The results indicate that impulsive children at a l l grade levels ask less mature questions and use a constraint-seeking strategy to a much lesser extent than their reflective peers. Score on the twenty-question proBlems also was correlated negatively with time measures from the Matching Familiar Figures test. Shulman (1965) examined differences in strategies used By good and poor proBlem-solvers among adults using a "Teacher In-Basket Technique". It was his conclusion that good proBlem solvers were associatively fluent, cognitively complex, sound interpreters of written passages, reflective, non—anxious, and spent more time than poor performers in inquiry activity. There was no significant relation Between grade point average of these college students and performance on his task. The relationship Between measured intelligence and proBlem-solving performance has also been examined. French (1958) examined the performance of adults on a switch-light problem in which a pattern of lights must be 18 produced by the manipulation of switches, and found that performance was positively- related to measured I.Q. Mendelsohn, Griswold, and Anderson (1966). also found that a positive relation existed between I.Q. and solving of anagram problems In children. With the large verbal component in both of these measures, this finding i s not surprising. A problem solving task was developed by Klausmeier and Laughlin (1961) which involved the manipulation of a specified number of coins and b i l l s to produce a specified sum. They observed that as measured I.Q. increased, a greater incidence in noting and correcting mistakes independently, in varifying solutions, and in the use of a logical approach was noted. As I.Q. decreased, nonpersistence, offering of incorrect solutions, and the use of a random approach was noted. Using the California Test of Mental Maturity as the measure of I.Q., Harootunian and Tate (1960) found a strong positive correlation with- this score and performance on the Differential Aptitude Test of verbal and abstract reasoning, and on the Davis-Eells Game. Sex differences in problem solving a b i l i t y indicate that this variable interacts with problem type. While males tend to outperform females i n the "two-string" problem (Duncan, .1961), and Maier's mathematical "Horse-trading"problem (Maier, 1970), no difference was found between the sexes in anagram (Russell and Sarason, 1965) and inductive reasoning problems (Kagan, Pearson, Welch, 1966). One additional subject variable that has received considerable attention as i t relates to problem solving i s creativity. For example Maier and Janzen (1970) found that those who gave a creative innovative solution on the "Change Worker" problem referred to previously also solved 19 significantly-more objective problems in which, only- one solution is possible. The authors suggest that good problem solvers are also creative. The use of such techniques as brainstorming (Meadow and Parnes, 1959), attribute l i s t i n g (Crawford, 1954), synectics, and Bionics (Davis, 19.73) In problem solving situations outside of the purely educational sphere would attest to the popularity of the relation expressed by Maier and Janzen. Davis (1973) has recently proposed a "creative problem solving model" in which he suggests that the following mental capabilities are necessary for both problem solving and creativity: abstracting, combining, perceiving novel relationships, associating, imagining, f i l l i n g in missing information, transforming, and taking an "intuitive leap". Whether one views problem solving from a cognitive Gestalt, or a behaviorist approach theoretically, i t seems d i f f i c u l t to argue with Davis that these s k i l l s are involved i n successful problem solving. It is the contention of Davis that a creative attitude is a voluntary act that can be learned in a deliberate manner, and the more effort invested, the greater the quantity of novel ideas. To this end, Davis and his associates (Houtman, 1968; Houtman, Warren and Roweton, 1969; Houtman, Warren, Roweton, Mari, and Belcher, 19 72) have developed workbooks for senior elementary and junior high school students which seek to encourage creative problem solving behavior in the following ways: by instructions and illustrations dealing with creative attitudes and techniques; by student exercises that allow him to find new ideas while practicing the strategies; and by example in the stories given. These programmes emphasize four steps in successful problem solving: 1. Clearly understand the problem and define 20 i t in general terms; 2. Think, of different approaches to solving the problem; 3. Think of different specific ideas for each problem approach; 4. Choose the best idea. The authors report that these instructional materials are markedly successful in improving problem solving strategies and outcomes of the students who have been exposed to them. The present writer has not located any research, however, that has evaluated the effect of these materials by an outside agent. Another set of similar instructional materials to those referred to above is the Productive Thinking Program of Covington, Crutchfield and Davies (19.66). This material is prepared in individual lesson booklets using a programmed instruction format that allows for individual student use. Covington and Crutchfield (1965) report significant gains for an instructional group as compared with a non-instructional group of f i f t h and sixth—graders on three types of c r i t e r i a : paper and pencil problem-solving tasks, paper and pencil tests of divergent thinking, and attitude and self evaluation measures. These authors believe the the program acts primarily to sensitize the pupil to the use of s k i l l s he already possesses rather than to help him in developing new problem solving s k i l l s . Further, they note that their results support the notion that there is a great degree of generalization of the s k i l l s strengthened by the program. A number of other studies have confirmed these findings. Ripple and Dacey (1967) employed these materials with eighth-grade students, and then tested performance on Maier's "two-string" problem. Their results indicate that the subjects in the instructional treatment solved this problem s i g n i f i -cantly faster than did those in the noninstructional treatment. Stokes 0-968) adapted the Productive Thinking Programme to the format of 21 computer assisted instruction and found that both, average and high a b i l i t y students receiving this: instructional outperformed similar non-instructed children on three post-tests measuring problem-solving a b i l i t y . Contrary to the above findings, Treffinger and Ripple (1968) found no difference in performance of an instructed and non-instructed group on a paper and pencil test of problem solving a b i l i t y following exposure to the Productive Thinking Programme. Other forms of training have also been investigated as they effect later performance. Blank and Covington (1965) developed programmed instructional materials to induce grade six children to ask questions during problem solving. Those who received this training improved in performance on an oral test of problem solving and on a written science achievement test, while a non-instructed group did not. An interesting project designed to improve problem solving s k i l l s of senior public school students has been reported by Robinson, Tickle and Brison (1972). These authors have developed instructional materials which train children in the use of the various classical experimental models; matched groups, randomization, correlational studies, case studies, and a " r e a l - l i f e " model, and have achieved promising results with children so instructed. Further evaluation is s t i l l in progress. Anderson (1965) was also successful in teaching what he termed "advanced problem solving s k i l l s " to f i r s t grade children through training. The task the children were required to learn were twenty-seven different conjunctive concepts using a selection paradigm. This task requires that the child be able to vary each factor in the stimulus display 22 successively while holding a l l other factors constant, a technique normally very d i f f i c u l t for young children. The training took a programmed instruction format, But functioned as a script used By a trainer in individual sessions with each child. Results indicate that on training tasks presented later, the trained group solved more problems with fewer unnecessary t r i a l s than the control group. The trained group also solved more transfer problems and solved these more effici e n t l y than did the control group. Cognitive Style and Conceptual Tempo Over the past several decades a number of investigators have directed their'efforts toward the identification of individual differences in mode of cognitive functioning. Gardner, in 1953 used the terms "levelers" and "sharpeners" as indicative of contrasting cognitive attitudes, particularly at the perceptual level. He suggested that these attitudes reflect individual differences in adaptive modes or the method of organizing and experiencing the stimulus world. Sharpeners "analyze out" suBtle differences in a stimulus array, their perception of the world Being geared toward "oBjective verity". Levelers, on the other hand, appear not to act upon their awareness of differences, and adopt a more relaxed approach to perception which Gardner terms "adaptive economy". In a suBsequent research report i n 1960, Holtzman and Gardner again used the terms "levelers" and "sharpeners" to descriBe systematic differences in memory- organization. They contend that the leveling-sharpening dimension reflects "...structural principles of personality organization that give a particular cast to an individual's cognitive 23 behavior..." and describe opposite poles- of a dimensional principle of cognitive control concerning the degree of assimilation between perceptual processes and memory traces. Levelling leads to omission of inconsistencies, condensation of elements, and general simplification of reca l l of material due to undifferentiated memory organization. In contrast, sharpening brings about recall dominated by accurate reporting of detail in the original perception brought about by a highly different-iated memory- organization. This difference was further substantiated in a recent study (Ausubel and Schwartz, 1972) which examined retention of prose material for levelers and sharpeners. Sharpeners recalled more specific detail and made fewer errors and misinterpretations of content than did levelers. The work of Witkin and his associates (1962) with reference to field-dependent and field-independent cognitive styles follows logically from this early work of Gardner and Holtzman. Witkin contends that this dichotomy reflects a global versus analytic or highly differentiated way of experiencing and perceiving. Cognitive style refers to "...the specific class of structures comprised of enduring arrangements of cognitive processes that shape the expression of intentions under particular types of environmental conditions." (Witkin, 1964). These structures include cognitive controls, defense mechanisms and intellectual structures. While the fieId-dependent individual makes use of less differentiated processing of information typical of a more rudimentary form of responding of the system as a whole, the field-independent, analytic style brings about greater differentiation and highly specific responding characteristic of a more advanced level of functioning. These two modes of functioning 24 are particularly- important in situations whicfL require the individual to selectively attend to relevant stimuli in the face of other irrelevant features. Witkin further suggests that style of cognitive functioning employed w i l l have far-reaching Implications: "Intellectual problems that c a l l for a high degree of creative activity but do not involve perception directly, often also require that 'parts' be separated from the context in which they are imbedded and brought into new relationships. It is lik e l y that i f a person has this basic a b i l i t y to 'break up' a configuration i t w i l l be manifest not only in straight-forward perceptual situations, but in problem solving situations as well." (Witkin, Dyk, Faterson, Goodenough, Karp, 1962) In a recent .study conducted by J. K. Davis (1973) the effect of global versus analytic cognitive style was assessed with regard to hypothesis testing strategies employed. Witkin's Hidden Figures Test was used to determine cognitive style of university students. In a concept formation task which involved the simultaneous presence of four dimensions, the analytic subjects gave evidence of more proficient hypo-thesis testing behavior. The global subjects frequently made two types of errors: 1. Switching hypotheses in absence of feedback which would indicate one should do so, and 2. Responding with a hypothesis that did not f i t the relevant dimensions given. The author concluded that this could be due to either faulty memory, inadequate attention, or faulty uti l i z a t i o n of feedback. Kagan, as a developmentalist, began to investigate differences in cognitive style in young children in 1963. In an early study with Lee and Rabson (1963) Kagan operationalized analytic versus non-analytic mode of information processing in terms of the classes of responses given in a 25 categorization task. These response categories were "analytic": based on objective elements of similarity that were part of the total stimulus; "relational": involving pairing based on a functional relationship between obvious aspects of the total stimulus; and "inferential-categori-cal": including conceptual pairings that are based on some inferred quality or language convention. He found that children classed as analytic learned an analytic concept in a paired-associate task significantly faster than non-analytic children, and that these non-analytic children learned a relational concept faster. Kagan suggests that i t is the children's differing a b i l i t y to visually analyze presented material into component parts that resulted in these findings. A paper by Kagan and Moss in 1963 added further to the developmental components of this difference between analytic and non-analytic modes of information processing. While these authors noted that there are general developmental trends toward less-global perceptions of the world to more articulated and greater differentiation of perception, they found evidence for stable individual differences in this capacity to analyze which was independent of a l l sub-test scores on the W.I.S.C. except the picture arrangement section. There was also marked difference in the willingness to persevere at a d i f f i c u l t task among these two groups of children. Kagan, Rosman, Day, Albert and Phillips (1964) began to look at an additional behavioral component of the analytic-non-analytic dimension. In addition to the inclination to visually analyze and to differentiate stimuli, they added the component of reflection versus impulsivity of response mode. They found significant negative correlations between the 26 impulsive reporting of a response and the degree of differentiation or visual analysis imposed on the stimuli presented to the children. In addition, by structuring the experimental situation in such a way that a reflective attitude was encouraged, they found an increase in analytic responses. In 1966 Kagan operationalized the dimension of reflection-impulsivity as the decision time to solution in situations of high response uncertainty. He contends that this mode of responding is linked tosome fundamental aspect of the child's personality, not unlike the thinking of Gardner with respect to "levelling-sharpening". "A reflective disposition displays a s t a b i l i t y over time and a generalltyaacross tasks that i s unusual for psychological attributes, and tempts one to conclude that this response tendency must be a basic component of the individual's behavioral organization." (Kagan, 1966). What is of considerable importance in educational terms is the relationship between the impulsive "conceptual tempo" and high error rate in a variety of cognitive tasks, independent of scores on the verbal sub-test of the WISC (Kagan, 1965). For example, Kagan found that impulsive f i r s t and second grade children made more errors in reading both individual words and short passages (Kagan, 1965), had higher error scores on inductive reasoning tests (Kagan, Pearson, and Welch, 1966), reported more errors of commission in a seria l learning task (Kagan, 1966), and showed autonomic patterns (cardiac and respiratory measures) characteristic of less attention than children who were reflective and preferred an analytic conceptual style (Kagan and Rosman, 1964). Kagan suggests that this higher error rate demonstrated by impulsive children is due to their failure to take time to reflect on possible alternative hypotheses and to welgh_ each before responding, being more concerned with providing a quick, but not necessarily correct response. "Each, child i s in an approach-avoidance conflict situation. If the strength of the approach, gradient is stronger (seek quick success) he w i l l be impulsive; i f the strength of the avoidance gradient is stronger (anxiety over making a mistake), he w i l l be reflective." (Kagan, 1966, p. 155). In summary, the difference between analytic and non-analytic individuals, as seen by Kagan is determined by two behavioral components: 1. The tendency to visually analyze stimulus arrays into differentiated components 2. The tendency to be impulsive or reflective in response mode when solving a problem. Since these earlier investigations by Kagan and his associates a considerable number of further studies have been reported by other researchers which are in general agreement with the i n i t i a l proposals of Kagan. Perceptual s k i l l s were examined in four studies (Siegelman, 1969; Drake, 1970; Odom, Mclntyre and Neale, 1971; Ault, Crawford, Jeffrey, 1972) comparing the performance of reflective and impulsive school children using an eye marker apparatus, videotapes of eye movement, or an apparatus that required the child to manipulate a dial to bring stimuli into focus. In a l l cases performance was examined using versions of Kagan's Matching Familiar Figures Test (MFF) items. A l l the above authors came to the same conclusion. Reflective children look longer at each item, and make significantly more comparisons with the standard item before making decisions. Impulsives ignore significantly more alternatives than reflectives and make a decision before a l l alternatives have been considered. Siegelman suggested that impulsives may u t i l i z e an impoverished input such that the f i r s t plausible alternative becomes the preferred 28 candidate while reflectives canvass the array more extensively, sample from i t more impartially and spend longer per look at alternatives. Odom, Mclntyre and Neale also noted that reflectives process distinctive features to a greater extent than impulslves. The relationship between tempo and measures- of academic achievement and a b i l i t y have also been further examined. Ault (1972) found no relation between cognitive tempo and achievement motivation in school. Christos (1972) found tempo was not meaningfully related to standardized school measures of achievement, academic aptitude, age, sex, or social class. Ward (1968) compared scores on theePeabody Picture Vocabulary Test and tempo and found no significant correlation. Bierbryer (1972) compared performance of reflective and impulsive children on the Torrance test of figural creativity and found that the two tempo groups did not significantly differ. Kopfstein (1973) also found no difference between impulsives and reflectives in an experimental situation that allowed him to assess risk-taking behavior. In spite of the insignificant differences between the two groups on the variety of measures mentioned above, impulsive and reflective children continue to be found to differ on a number of cognitive tasks. Both Davey (1971) and Butler (1972) found impulsive children to be markedly inferior to reflective children on measured reading achievement. Meichenbaum and Goodman (1969) found impulsive kindergarten children demonstrated less a b i l i t y to inhibit motor behavior by covert self-instruction than reflective peers. Mann (1973) examined the latency in making decisions of reflective and impulsive children in four settings: toy choice, Mischel decisions, a goal-setting game, and spelling decisions, 29 and found that reflective children took significantly longer in making decisions but no difference existed in the quality of decisions made by the two groups. Ault (1973), as- previously noted, found impulsive children used a less mature and less efficient strategy in solving problems than their reflective peers-. A variety of procedures have been employed to attempt to modify both the analytic-non-analytic cognitive style and the reflective-impulsive conceptual temp. In 1967 Ostfeld and Neimark administered one version of Kagan's Conceptual Style Test (CST) to seven and eight year olds and divided them into analytic and non-analytic groups on this basis. These children were then retested on a second version of the CST, but one half of each cognitive style group was required to delay their response. The authors found that by forcing an increaseiin response latency, an increase in the proportion of analytic responses was seen. No such change occurred in the Control groups. Baird and Bee (1969) examined the effects of reinforcement in the form of candy on the production of analytic or non-analytic responses on the CST. Not unexpected, they found that the category of response that was reinforced occurred to a greater extent than the non-reinforced response mode for these f i r s t and second grade children. Davis and Klausmeier (1970) examined the performance of analytic and non-analytic children in a concept learning task. He then required one half of the children from each cognitive style group to verbalize a l l attributes of each stimulus that was presented to them before responding. The results indicate that this procedure resulted in significantly better performance of the non-analytic experimental children over their l i k e Control group. The authors suggest that this procedure forces the child to differentiate 30 the relevant variable, something that these non-analytic children tend to do inadequately. Denney 0-972) exposed.non-analytic and impulsive second grade children to a female model who displayed an analytic and reflective pattern of behavior on the CST. The result of this exposure was an increase in both latency of response and in number of analytic responses by these children when exposed to a version of the CST. This difference lasted for fourteen days, as delayed post-testing indicated. Attempts to change tempo are also numerous. Messer (1970) induced anxiety over adequacy of intellectual performance in young school children and found that this treatment resulted in an increase in response latency for both reflective and impulsive children. Briggs and Weinberg (1973) found that by training children to delay responses on items similar to the match to standard items on the MFF resulted in an increase in response latency and a decrease in errors on subsequent testing on MFF items. No such change was noted in a Control group not given training. In an interesting cross-modal study conducted by Butter (1971) children were trained to delay responses in match to standard items presented in either the visual or haptic mode. This training resulted in a significant increase in response latency and decrease in errors for both training groups over that of a control group when tested on the visually presented MFF test. It has been noted previously that impulsive children differ from their reflective counterparts in the use of constraint-seeking strategies in a twenty-questions problem format. Denney (1973) differentiates two types of questions used by children in such a task. Hypothesis-seeking questions are those which test specific self-sufficient hypotheses that 31 have no relation to previous questions asked, and result i n pure guessing behavior. This type of question is found to be more typical of impulsive children. Constraint-seeking questions are more general questions relating to previous questions and seek an answer which would eliminate a number of alternatives. These are more typical of the reflective child. Denney found that by instructing seven— and eight-year—olds to delay their responses in this twenty-questions problem, more constraint-seeking questions occur than in a control group not so instructed. Modelling has been used in a number of studies to bring about changes in conceptual tempo. Debus (1970) using a l i v e model with third graders and Ridberg, Parke, and Hetherington (19 71) using a filmed model with, fourth graders found that the observation of a reflective model resulted in more reflective patterns of behavior of these children on subsequent tests with the MPF. No difference occurred in control groups who did not observe a model in either of these studies. An investigation conducted by Yando and Kagan (1968) would suggest that young children are susceptible to very subtle cues concerning tempo in the classroom setting. In this study the tempo of both teachers and children was determined at the beginning of the school year. At the end of the school year children were again tested to determine their tempo. Results indicate that exposure to an experienced reflective teacher resulted in an increase in r e f l e c t i v i t y in the children, while such a difference did not occur for children taught by an impulsive teacher. Meichenbaum and Goodman 0-971) trained impulsive second grade children to talk to themselves on items similar to those used in the MFF test. Retesting on the MFF following training indicated an increase in 32 response latency- and a decrease in errors. The authors report that this difference in response mode also transferred to performance on the Porteus Maze test and on performance scores- on the WISC. Retesting on the MFF one month later indicated that the change in behavior continued to be seen. The deficiencies noted earlier in the perceptual s k i l l s of impulsive children led several investigators to attempt to train such children in more systematic strategies. Egeland (1973) used items similar to those employed in the MFF to train children. During training the child was required to look at each alternative, to break each alter-native down into parts, then to compare each part successively over alternatives, and fi n a l l y to compare each alternative with the standard. The results of performance on the MFF following training indicate significantly longer latencies and lower error scores than a non-trained group of controls. The authors also note that the trained group showed improvement on the vocabulary and comprehension subtests of the Gates-MacGinitie-Reading Test. Roettger (1970) trained a group of impulsive kindergarden children in efficient scanning strategies using items from the Word Discrimination Test. The children were taught to compare each word letter by letter to make the necessary discrimination. An increase in response time and a decrease in errors over that of an untrained control group on MFF items was found. It would appear from the above that both dimensions of the cognitive style variable; degree of analytic responding, and tempooof response are modifiable and that changing a child's style as measured by the CST or the MFF can transfer to a variety of other tasks. 33 Effects of Cognitive Style and Conceptual Tempo on Problem Solving In a recent comprehensive review of the literature concerning individual differences in cognitive processing Kagan and Kogan (1970) have suggested that the analytic-non-analytic and the impulsive-reflective dimensions affect the efficiency of problem-solving at a l l stages in the process. In the f i r s t phase when the problem is being decoded, the thoroughness of the i n i t i a l discrimination of the stimulus array constituting the problem, and the identification of relevant components w i l l effect a l l subsequent stages of the problem solving sequence. Gardner, Holtzman, Witkin, and Kagan have a l l noted stable individual differences in the degree to which subtle differences insstimulus arrays are discriminated and attended to. This finding has further been substantiated by the research studies of Siegelman, Drake, Odom, and Ault concerning differences in perceptual s k i l l s of children differing on cognitive style and conceptual tempo dimensions. Children who are high in this a b i l i t y to "attend and discriminate" should demonstrate more efficient problem solving than impulsive non-analyzers, and such would appear to be the case from the research evidence of Bruner (1961), Ault (1973), and Shulman (1967). The next important step in problem solving is information gathering, either by the recall of previously learned rules, as suggested by Gagne's model, and/or by asking questions whose answers w i l l f i l l in missing data or impose further organization on existing information. It would appear jus t i f i e d to conclude, on the basis of the writing of Holtzman and Gardner with regard to memory organization, and the research of Kagan (1965, 1966), Davey (1971), and Butler (1972) that children at the non-34 a n a l y t i c and impulsive ends, of the s t y l e and tempo continuums are l e s s able to r e c a l l and u t i l i z e p r e v i o u s l y experienced information. In addition, the demonstrated i n f e r i o r i t y of these c h i l d r e n i n question-asking s k i l l s i n a number of studies reviewed above would also lead one to p r e d i c t i n f e r i o r performance at t h i s stage i n the problem so l v i n g process. In the next general stage of problem s o l v i n g when the c h i l d must use the information available to discover possible s o l u t i o n s , the impulsive non-analytic c h i l d i s again at a disadvantage. He i s working from a more impoverished data base, f a i l s to r e f l e c t s u f f i c i e n t l y on the implications of the information derived, and as a consequence i s more l i k e l y to choose a wrong inference. The evidence reported by Kagan (1966) that impulsive ch i l d r e n make more errors of inductive reasoning would support t h i s contention. In the f i n a l stage of the problem so l v i n g process i n which hypothesis v e r i f i c a t i o n against r e a l i t y occurs, the impulsive non-analytic c h i l d i s again at a disadvantage. His perception of r e a l i t y i s geared toward "adaptive economy" as opposed to "objective v e r i t y " (Gardner, 1953) which can be expected to lead to l e s s stringent t e s t i n g of possible hypotheses. In addition, the impulsive q u a l i t y of h i s response tempo would r e s u l t i n the reporting of a quick but not n e c e s s a r i l y correct response w i l l lead to sub-optimal performance. Kagan has suggested that the pattern of behavior t y p i c a l of the impulsive c h i l d when faced with a problem to solve i s as follows: Problem ^ Inadequate discrimination of components Impulsive s e l e c t i o n of a solution, Failure, •> Anxiety Impulsive s e l e c t i o n of a second s o l u t i o n + F a i l u r e 35 Anxiety > Withdrawal from the problem situation. Thus, the child perseverates in the use of an inadequate strategy brought about by his failure to thoroughly analyze the elements in the problem, his failure to ask questions that w i l l c l a r i f y the problem, and by continuing to impulsively offer solutions without adequate reflection on their relative merits. Considerable research evidence exists to suggest, however, that the tendency of the child to employ these inadequate strategies is modifiable. In the literature reviewed, i t was demonstrated that problem solving s k i l l s , cognitive style, and conceptual tempo can be independently changed, given appropriate training procedures. It has also been demonstrated that changing cognitive style and conceptual tempo w i l l result in changes in some isolated aspects of problem solving behavior. For example, i t w i l l be recalled that Denney (1973) increased the level of maturity of question-asking s k i l l s in impulsive children by imposing a reflective response mode on them. The Problem The purpose of the present study was to further investigate the nature of the relationships that may exist between cognitive style, conceptual tempo, and processes used during problem solving. In addition, this study sought to explore the possibility of improving the problem solving s k i l l s of impulsive, non-analytic children through exposure to training materials in the programmed instruction format which were designed to teach, reflective and analytic modes of responding at a l l stages of the problem solving process. Furthermore, the research sought to 36 determine the relative effectiveness of these techniques with children at differing points: along the analytic-non-analyttc and the impulsive-reflecting continuums. These possibilities were explored by- adapting to a printed programmed instruction format materials from several sources. These included a combination of various training techniques found to be successful in previous research designed to change problem solving and/or cognitive style and conceptual tempo, as well as materials developed by the present researcher. To provide an adequate measure of the variety of s k i l l s involved in problem solving, each child was administered a set of four problems in the form of short mysteries to be solved in individual testing sessions following experimental treatment (see Appendix A). These problems were presented with a paucity of information so that the child must have asked questions for solution to occur. In addition, a l l problems were structured in such a manner that more than one solution was possible. Ten measures were employed to assess problem solving performance. These were as follows: A. Time Factors 1. Mean Total Time - Time from presentation of a problem to when the subject indicates by turning over the "Finished" card that he has no further questions to ask or solutions to offer, to a maximum of five minutes. 2. Mean Prior Time - Time from presentation of a problem to when the subject indicates by turning over the "Ready" card to indicate that he is prepared to offer his f i r s t solution. (Did not include those solutions phrased as hypothesis-seeking questions). 3. Mean Residual Time - Time from offering of f i r s t solution un t i l the subject indicates that he has no further questions to ask or solutions to offer, by turning over the "Finished" card. 37 B. , Ques t ion-Askin g.; B ehavio r: 4. Mean. Information-Seeking Questions Asked .-, Those questions which seek factual information about some parameter of a stimulus element in thepproblem. 5. Mean Constraint-Seeking Questions Asked - Those questions which relate to previously asked questions and seek an answer which would eliminate a number of alternative solutions (Denney, 1973). 6. Mean Hypothesis-Seeking Questions Asked - Those questions which test specific self-sufficient hypotheses that have no relation to previously asked questions (Denney, 1973). 7. Mean Total Questions Asked - The mean number of questions asked over four problems, regardless of type of question. 8. Mean Prior Questions Asked - Those questions asked, regardless of type, from the i n i t i a l presentation of a problem until the subject indicates by turning over the "Ready" card that he is prepared to offer his f i r s t solution. 9. Mean Residual Questions Asked - Those questions asked after the f i r s t solution is offered, u n t i l the subject indicates that he has no further questions to ask or solutions to offer, by turning over the "Finished" card. C. Solutions: 10. Total Solutions: The total number of solutions offered by the subject as indicated by his turning over the "Ready" card. (Did not include those stated as Hypothesis-seeking questions.) The performance of three separate groups of fifth-grade children were assessed: those that had received the problem solving training program (TI); those that had received programmed instruction whose content was unrelated to problem solving (T2); those that had received no programmed instruction (T3). The inclusion of the T2 group acted as a control for any influence which might result from exposure to programmed instructional materials, independent of content. The three major substantive hypotheses that were explored were: 1. Problem solving performance of children at differing points along the cognitive style continuum w i l l increase as measured analytic a b i l i t y increases. 38 2. Problem solving performance of children at differing points along the conceptual tempo continuum w i l l increase as measured re f l e c t i v i t y increases. 3. Training in problem solving w i l l be equally effective for children at a l l points along the cognitive style and conceptual tempo continuum and with result in the trained group being superior in problem solving as measured by the ten dependent variables. While this study was principally intended to be exploratory in nature, a number of specific research hypotheses were tested, based on the findings of previous research in this area. These were as follows: 1. In terms of a l l measures of problem solving except Mean Hypothesis-seeking Questions Asked, the trained group (Tl) w i l l spend more time, ask more questions, and provide more solutions than the untrained groups (T2, T3). 2. The two untrained groups (T2, T3) w i l l ask more Hypothesis-seeking Questions than the trained group (Tl). 3. No difference w i l l exist between the two untrained groups (T2, T3) on any of the ten measures of problem solving. 4. The number of analytic responses on the CST w i l l be positively related to the scores on a l l measures of problem solving except mean hypothesis-seeking questions asked for a l l experi-mental groups combined. 5. The number of analytic responses on the CST w i l l be negatively related to the mean number of hypothesis-seeking questions asked for a l l experimental groups combined. 6. The total number of errors on the MFF w i l l be negatively related to the scores on a l l measures of problem solving except mean hypothesis-seeking questions asked for a l l experimental groups combined. 7. T 7. The total number of errors on the MFF w i l l be positively related to the mean number of hypothesis-seeking questions asked for a l l experimental groups combined. 8. The mean latency of response on the MFF w i l l be positively related to the scores on a l l measures of problem solving except mean hypothesis-seeking questions asked for a l l experimental groups combined. 9. The mean latency of response of the MFF w i l l be negatively related to the mean number of hypothesis-seeking questions asked for a l l experimental groups combined. 39 10. An interaction w i l l exist between training and cognitive style such that the relationship between the number of analytic responses on the CST and a l l measures of problem-solving w i l l be positive and greater for the untrained groups (T2, T3) than for the trained group (Tl). 11. An interaction w i l l exist between training and conceptual tempo such that the relationship between the number of errors on the MFF and a l l measures of problem solving w i l l be negative and greater for the untrained groups (T2, T3) than for the trained group (Tl). 12. An interaction w i l l exist between training and conceptual tempo such that the relationship between the mean latency of response on the MFF and a l l measures of problem-solving w i l l be positive and greater for the untrained groups (T2, T3) than for the trained group (Tl). Rationale The rationale for the predictions made were based on two principal assumptions. Fi r s t , i t was assumed that an analytic and reflective child w i l l be better equipped to carry out the processes necessary to solve the four problems given to him. These problems were selected from The Productive Thinking Program developed by Covington, Crutchfield, and Davies (1966), as being particularly suited to assess-ment of the differences in problem solving behavior expected of children who differ in analytic and reflective response modes. For purposes of i l l u s t r a t i o n , one of the problems, t i t l e d "The Mystery of the Missing Jewel" w i l l be described in detail. The problem facing the subject in this mystery story is to determine how a jewel was stolen from a rich widow during a black-out which occurred during a dinner party given by herself for three other guests. The policessearched a l l the guests and the rooms, but the jewel was nowhere to be found. A window was open, however no footprints were to be seen on the muddy ground below. Key 40 clues for the subject to solve this problem are a feather found on the floor and an opened box with perforations which one of the guests had brought with him. The correct solution of the mystery is that one of the guests brought a trained bird in the perforated box into the room which he released after blacking out the lights and opening the window. This guest stole the jewel, attached i t to the trained bird which flew out the window, where i t presumably waited for i t s master, the guest. The strategies needed for the solution to this mystery problem appear related to the cognitive processes assessed by the Conceptual Style Test and the Matching Familiar Figures Test. The situation facing the subject in this problem is to combine the various clues presented to him in such a way that he arrives at the solution: who stole the jewel. At the beginning of the problem he is given the situational context where any of the three guests could have a logical motive for stealing the jewel, and must resist the impulse to designate any one of the three as the thief before careful attention is given to the details of the problem. In addition, he must be able to perceive some facts as relevant clues and other facts as irrelevant to the solution. Success in perceiving the relevant facts or clues by asking information and constraint-rseeking questions prior to offering a solution would seem to indicate an analytic and reflective approach to the elements of the problem. Insufficient attention of the details of the situation in which the theft occurred would result in the various relevant and irrelevant clues appearing 'fused together', thereby hampering solution of the problem. Where such an approach to the problem seemed to occur, as indicated by failure to ask questions prior to offering a solution, one might suspect the subject 41 to be relatively non-analytic and impulsive. He would be unable to separate the elements of the problem and recombine them into new configurations which would have led him to the correct solution. While no research exists to date which tests children on this type of problem material, the wealth of data that does exist suggesting that non-analytic and impulsive children use less efficient strategies at a l l individual stages in the problem solving process would give support to the assumption that similar trends would be found in the present study. The second assumption on which the predictions were based was that the programmed instructional materials to be used for training w i l l teach children analytic and reflective strategies for coping with problems. These materials evolved from a task analysis of problem solving (see Appendix C), and drew heavily on training materials developed by other researchers. The materials were prepared into six separate booklets, each including a number of exercises and activities thought to be necessary to achieve the behavioral objectives to which each task in the analysis aimed. The materials used in the f i r s t booklet were designed to develop s k i l l s in visual analysis and iniidentification and discrimination of problem elements. These exercises included the following: 1. Mazes: French, Ekstrom, and Price (1963) suggest that mazes require a b i l i t y to scan the f i e l d and reject false leads; willingness to find a correct path visually before wasting time in marking the paper. By gradually increasing the d i f f i c u l t y of these items, i t is the contention of the writer that these can serve as appropriate materials for teaching reflective and analytic perceptual s k i l l s . 2. Ambiguous Figures: These items are designed to alert the child to the fact that things are not always what they seem on f i r s t inspection. The intent i s to encourage children to go back and take another careful look when presented with 42 p i c t o r i a l or written material. 3. Match to Standard Items: These include the training items used by Briggs and Weinberg (1973) to develop s k i l l s in making successive comparisons between visual stimuli that are highly similar; in varying only one element at a time to identify subtle differences. 4. Verbal to Visual Identification: Several highly similar visual stimuli are given along with a verbal discription that identifies only one. The child must coordinate two sets of information and attend to fine detail to make the necessary identification. The exercises in the second booklet were designed to promote the a b i l i t y to select relevant cues from a stimulus display in the presence of distracting information, and to use constraint-seeking questions to impose organization on the problem elements. These exercises included: 1. Witkin Embedded Figures Items: These items are proported to test the a b i l i t y to keep one or more definite configurations in mind in spite of perceptual distraction; to search a perceptual f i e l d containing irrelevant materials (French, Ekstrom, and Price, 1963). By gradually increasing the d i f f i c u l t y of a series of such items, i t is the contention of the writer that these items can also be used as appropriate training devices for this a b i l i t y . 2. Twenty-Questions Items: These exercises are similar to those used by Ault (1973) and others, and present the child with a number of items that f a l l into different general categories. The use of constraint-seeking questions is demonstrated and developed as a device in f a c i l i t a t i n g identification, and the ab i l i t y to vary one component while holding a l l others constant. The third booklet contained problems from both the Blank and Covington programme on question-asking (1965) and from the Productive Thinking Programme (Covington, Crutchfield, Davies 1965). These exercises presented problems to a child with inadequatelinformation for solution and guided him in the identification of points where information was lacking, and in the use of information-seeking questions. 4 3 The remaining three booklets were comprised of material taken from the Productive Thinking Programme. These items were designed to promote hypothesis generation in the form of "general p o s s i b i l i t i e s " and "particular ideas". They encouraged the generation of more than on hypothesis and provided guidelines to the child for using the information he had collected to evaluate each solution he generated. Reports by Covington and Crutchfield (1965), Ripple and Dacey (1976), and Stokes (1968)1 would attest to the success of these materials in achieving these goals. Sample exercises from each booklet are provided in Appendix C. The program was of a linear format which provided consistent feed-back of correctness of response, where applicable.. When more than one response was possible, a number of alternatives were provided against which the student could compare his own. The sequence of exercises of any one type showed a gradual increment in d i f f i c u l t y and dimunision of explicitness of prompts. This was done to maximize the possibility of correct responding and thereby attempted to avoid the experiencing of failure that Kagan has suggested is a l l too frequent and inhibiting for the non-analytic impulsive child. Given the previous research data that reports success in the use of isolated components of these materials in promoting efficient problem solving s k i l l s , i t seemed logical to assume that similar f a c i l i t a t i o n would result from a combination of these materials. 44 CHAPTER III METHODOLOGICAL CONSIDERATIONS Subjects Ninety-two fifth, grade children enrolled in three elementary schools within the Chilliwack School System which were not specifically designed for the "gifted" or the "learning disabled" served as the i n i t i a l pool of subjects. Three schools serving comparable lower-middle to middle class residential areas were selected in consultation with the central administrative staff of the Chilliwack School Board. A l l three classes were taught in a traditional non-rotation system, solely by their respective female teachers. None of the subjects had had previous exposure to the Productive Thinking Program. Subject attrition due to repeated absenteeism during the course of the research program (five subjects), technical failure of recording equipment during post-testing (three subjects), or procedural error by the experimenter during post-testing (three subjects) resulted in a f i n a l sample of eighty-one subjects for whom a l l experimental measures were available. The f i n a l composition of each class by sex can be seen in Table I. Design The design involved three between-subjects experimental conditions; a training class (TI) which was exposed to the problem solving programmed materials, a Control class (T2) which worked through programmed 45 TABLE I COMPOSITION OF EXPERIMENTAL CLASSES CHARACTERISTICS T l T2 T3 F p FREQUENCY MALES 13 FEMALES 12 TOTAL 25 MEAN CST 7.60 MEAN MFF ERRORS 8.00 MEAN MFF TIME 11.74 MEAN CTBS 5.21 MEAN FLUENCY 36.64 MEAN FLEXIBILITY 43.86 MEAN ORIGINALITY 45.96 12 15 14 15 26 30 5 . 96 6 . 13 1 . 0759 . 34 8 • 19 6.93 .8881 . 41 13 . 06 14.64 • 1 .3526 . 26 5 . 28 4 . 94 .8849 .41 39 .31 42.77 5 .8385 .004 44 . 19 49 . 63 2 . 7825 .07 42 .62 49 . 03 5 . 1497 . 008 46 instructional materials not specifically designed to f a c i l i t a t e problem solving, and a second Control class ( T 3 ) which received no programmed instruction. Classes were randomly assigned to the treatment and control conditions. Three pre-treatment measures were obtained for each subject 'as measures of cognitive style and conceptual tempo. These included the total number of analytic responses on Denny's Conceptual Style Test (CST), total number of errors on Kagan's Matching Familiar Figure Test (MFF errors) , and mean latency of response on the MFF (MFF time). Sample test items are presented in Appendix B. To determine that the three classes did not differ significantly on these measures of cognitive style and conceptual tempo, a simple analysis of variance was carried out on these scores. As can be seen from Table I, no significant differences existed between the classes on these measures. Description of the Covariates The intent of this study was to examine processes used in problem-solving controlling for the effects of general a b i l i t y and creativity s t a t i s t i c a l l y . Therefore, these measures were treated as covariates. As a measure of general a b i l i t y , the overall score on the Canadian Tests of Basic Skills was employed. This test is a Canadian adaptation of the Iowa Tests of Basic Ski l l s and includes sub-scores for vocabulary, reading comprehension, language (spelling, capitalization, punctuation, usage), work-study s k i l l s (maps, graphs, and tables, reference materials), and mathematics s k i l l s (concepts, problem-solving). "This battery aims at the evaluation of generalized educational s k i l l s and a b i l i t i e s , not content achievement" (Birch, 1972). A l l norms are expressed as grade equivalents and the overall scores used in this study represent the mean of the"fifteen sub-test scores for each child. The treatment group means for this measure of a b i l i t y are presented in Table I. No significant difference existed between the three classes on this measure (F_ = .8849, p < .41). j, /y Three measures of creativity; fluency, f l e x i b i l i t y , and originality were obtained by administering the Torrance Test of Creative Thinking, Verbal Form A to each class prior to conducting this research. In this test "fluency" is defined as the total number of relevant responses given, relevancy, being defined in terms of the requirements of the tasks as set forth in the instructions. " F l e x i b i l i t y " is the number of different categories of responses, as given in the scoring manual. "Originality" i s the sum of credits where routine responses count zero, less common responses get a score of one, and responses too infrequent to be on the l i s t i n the scoring manual (given by less than two percent of the sample on whom the test was standardized) are given a credit of two. Thus, the originality score is based on the s t a t i s t i c a l rarity of the responses. The sum of the raw scores on these measures are converted into T-scores, using the norms given in the scoring manual for f i f t h -grade students. The Verbal Test consists of seven tasks, each battery requiring a total of forty-five minutes in addition to the time necessary for giving an orientation, passing out booklets, and giving instructions. The activities involve: asking questions about a drawing, making guesses about the causeslaf the event pictured, making guesses about the possible 48 consequences of the event, producing ideas for improving a toy so that i t w i l l be more fun for children to play with, thinking of unusual uses of cardboard boxes, asking provocative questions, and thinking of the varied possible ramifications of an improbable event. This test battery was designed to provide a measure of creativity defined by Torrance as "...a process of becoming sensitive to problems, deficiencies, gaps in knowledge, missing elements, disharmonies, and so on: identifying the dif f i c u l t y ; searching for solutions; making guesses; or formulating hypotheses about these deficiencies; testing and retesting these hypo-theses and possibly modifying and retesting.them; and fi n a l l y communicating the results." (Torrance, 1966, p. 6). The similarity of this description to the problem-solving measures selected for examination in the present research makes the Torrance Test scores particularly suited as covariates. The means for each treatment group on these creativity scores are presented on Table I. While no significant difference existed between the three groups on mean f l e x i b i l i t y , this was not the case for mean fluency and originality. Using Dunn's test for pairwise.comparisons among the means for fluency showed that this significant difference was accounted for by the superiority of T3 over Tl (d=6.1267, exceeding c r i t i c a l value of d=4.6146 with alpha .05). For originality the difference was accounted for by the superiority of T3 over T2 (d=4180, exceeding c r i t i c a l value of d=5.1623 with alpha .05). Treatment of these variables as covariates, however, s t a t i s t i c a l l y controlled for any systematic effect. Stimulus Materials The fifteen items constituting Form N of Denny's CST were mimeo-49 graphed and prepared in booklet form for each subject. The order of items was randomized for each subject. One item from Form M of the CST served as a sample item. Each of the twelve items of the MFF were presented in black ink on two sheets of white paper covered by protective plastic. The standard item was drawn in the center of one sheet, and the five variants and one identical item were arranged in two rows of three items on the second sheet. The position of the correct item within these six positions was arranged such that each position was occupied by the correct item twice. Two additional sample items were provided for practice. One set of these items was prepared for each of two female experimenters. Each experimenter was equipped with a stopwatch and a score sheet for recording latency of response, and errors. The four items to be used i n the test of problem-solving a b i l i t y were prepared on a single sheet of paper covered in protective plastic. This included the verbal description of the problem and a p i c t o r i a l representation. Each experimenter was provided with a l i s t of standard responses to be given to questions asked by the child (see Appendix A). Two cards were provided for use by the child, one with the word "READY" printed in large black letters, and.one marked "FINISHED". The problem-solving training program has been described fu l l y in Chapter II. Each booklet was bound in a different colored cover, clearly indicating the number of the booklet. The f i r s t of the six booklets was placed in an envelope marked with each child's name. The teacher was supplied .with the five remaining booklets. On the face of the envelope a chart was provided so that the child could record the page at which he 50 stopped work each day. The control group program was entitled Day and Night and the Four  Seasons: A Study of Earth-Sun Relationships published by the American Institute for Research in the Behavioral Sciences. This program was of a linear, small step format which provided one hundred per cent knowledge of results. Concepts dealt with by the content of the program included rotation, revolution, the significance of certain latitudes, causes of the four seasons, etc. The material in the program had not to this point been ex p l i c i t l y covered in the students' regular curriculum. The lesson booklet and accompanying answer booklet were provided to each child in T2 in a large envelope. A chart was provided on the face of the envelope so that the child could record the page at which he stopped work each day. Procedure Pilot Testing: Formative testing of the problem-solving program was carried out with thirty-six f i f t h grade children within the Chilliwack School System who were not subsequently involved in this research. This pilot testing was undertaken to ensure that the content and instructions in the program were sufficiently unambiguous, and to obtain time measures for completion of each booklet. Each child was given one of the six booklets which constituted the program. Children were requested to work through the booklet without interruption, and to notify the researcher i f there were any instructions which they did not understand. Total time to completion was recorded for each child. No child reported d i f f i c u l t y in understanding instructions or in completing any exercise in the booklets. The mean time to complete booklets one through six were as follows: thirty-seven minutes, thirty-seven minutes, f i f t y - f i v e minutes, f i f t y - s i x minutes, fifty-four minutes, sixty-three minutes. On the basis of this data i t was decided that fourteen thirty-minute periods would allow ample time for a l l children in the training class to complete a l l six booklets. Following completion of the booklets, each child involved in pilot testing was given the four post-test problems intended for use in this research in mimeograph form.. Instructions were given to examine the problem carefully and write down a l l the questions that they would ask i f they were trying to solve the problem. This procedure provided a pool of questions for which standardized answers could be constructed prior to undertaking the testing of the experimental subjects. Pre-testing: Two weeks prior to beginning training, group administration of the CST was undertaken within each experimental class-room. . One week later the Torrance Test of Creative Thinking, Verbal Form A was administered in the same manner. The School Board had administered the Canadian Tests of Basic S k i l l s (CTBS) to a l l fifth-grade children at the beginning of the school year and these scores were collected from the school f i l e s for each child involved in this research. The administration of the MFF was carried out with each child individually by two female experimenters. Each experimenter tested an equal number of subjects of each sex from each class. The child was taken to a quiet room outside of his classroom for testing. The f i r s t sample item was placed on the desk with the standard item above the picture of six variants. The following instructions were read to the child: "I am going to show you some pictures, like these (points to sample). I want you to look at this picture carefully (points to standard). Then look at each of these pictures (points to variants). One of these pictures is exactly the 52 same as the one at the top. Your job is to fine the one that is exactly the same as the one at the top. You may take as long as you need to make your choice. Point to the one that is exactly the same when you find i t . " If the subject made an incorrect selection, the experimenter directed the child to look again. Following successful completion of the sample items, the f i r s t test item was presented. The experimenter started the watch as soon as the item was placed .in .front of the subject and stopped i t when the subject indicated his f i r s t selection. If the selection was incorrect, the experimenter recorded"the error and directed the child to look again. This procedure continued un t i l a l l twelve items had been presented. Training: Prior to the commencement of training, the teacher of Tl was visited by the researcher and given a f u l l description of the intent of the study and the importance of her not providing aid to the students while they worked on the materials beyond making available the booklets. She was cautioned to refrain from discussing the children's progress with the material and was requested to act as a "watch-dog" to ensure that children did not talk among themselves during work sessions. On the f i r s t day of''training the researcher introduced the program to both Tl and T2 with the explanation that these were new books that had been written for f i f t h grade ' students, and they were being asked to use them so that the researcher could see i f students would complete a l l the exercises in the booklets with no help from anyone else. The children were then provided with general instructions on the use of the materials, and on the importance of keeping an accurate record of how many, pages they completed each day. 53 It was arranged with the two teachers that the thirty minute period immediately following lunch would be set aside for work on the materials for.the following fourteen days. The teachers were instructed to collect the envelopes at the end of each work session. If any child completed the program before the alotted time, he was instructed to work on other reading or school projects. Post-Testing: The administration of the tests of problem-solving abil i t y were conducted during the week following training. The test was administered individually to each child by four female experimenters who had received training in administering these materials. The assignment of subjects to experimenters was such that each experimenter tested as equal a number of children of each sex in each.class ast-thettotal number would allow. A l l test sessions were recorded on audio tape. Each child was tested in a quiet room separate from the classroom. The child was seated across the table from the experimenter and read theffollowing instructions: "I am going to.give you some mystery stories to solve. I w i l l show you a picture and read you the story about the mystery. You can follow along on your own copy while I read. Your job is to try to guess at some solutions. You may ask me any questions that you think w i l l help you to find a solution. When you think you are ready to take a guess, turn over this card and say "READY". Then you can t e l l me your guess. If you want to ask me some more questions after you have made your f i r s t guess, you can. Try to think of as many guesses as you can. When you can't think of any more guesses or questions, turn over this card and say "FINISHED" and we w i l l go on to the next mystery." The experimenter then placed the sheet containing the f i r s t problem before the child and read the description. If after two minutes the subject had said nothing, the experimenter asked "Are there any questions you would like to ask me?". If after three minutes the child had not 54 said anything he was asked i f he was finished, and instructed to turn over the FINISK card. A l l questions were answered according to the pre-determined manner. Each'time the subject offered a solution the experimenter said "Yes, that could be what happened. Do you have any more questions or guesses?". If a subject spent more than five minutes on a problem, the experimenter allowed the child one further opportunity to provide a solution and then said "We had better move on to the next one now: or there won't be time to look at a l l of them." This procedure continued u n t i l a l l four problems has been presented. Order of presentation of problems was randomly determined for each child. Transcriptions of a l l test sessions were made and responses scored according to the ten dependent measures. The procedure employed for classification of questions into the three types, information-, constraint-, and hypothesis-seeking, was as follows. A l l questions were collected and classified into the three types by the experimenter. Following this, an independent judge classified a l l questions. Those questions on which there was not i n i t i a l agreement were then discussed until a concensus was reached. Examples of questions of the three types are presented in Appendix A for each problem. 55 CHAPTER IV RESULTS Mean scores for each subject for the four problems were determined for each dependent variable. These were total time, prior time, total questions, prior questions, information-, constraint-, and hypothesis-seeking questions, residual time, residual questions, and the total number of solutions offered. These data were then analyzed using a regression analysis procedure designed to verify the research hypotheses stated in Chapter II. The regression analysis approach employed for this analysis has been described by several writers (Bottenberg and Ward, 1963; Cohen, 1968; Overall and Spiegal, 1969; Walberg, 1971; Draper and Smith, 1966). The advantages of.this method of data analysis over the conventional ANOVA approach have been discussed in detail by these writers. In the present study the use of regression analysis allowed for the effects of the continuous variables Cognitive Style and Conceptual Tempo to be tested without having to arbitrarily level the variables as in ANOVA. In addition, Cronbach and Snow (1969) and Bracht (1970) have recommended the use of the regression analysis method for testing interaction terms in. A.T.I. (Aptitude by Treatment Interaction) studies. In.the present study the i n i t i a l identification of these interactions was undertaken by a regression analysis procedure. Following this, categorization of the continuous organismic variables then allowed for closer examination of those interactions which the regression analysis found to be significant. 56 Separate stepwise regression analyses were performed for each of the ten dependent variables in order to test the hypotheses of this study. These analyses were performed and correlations generated using the BMD02R Stepwise Regression Program (Halm, 1972). Major hypotheses were tested at alpha .05. The hypotheses given at the end of Chapter II were translater into s t a t i s t i c a l terms. The symbols used to represent the variables in this study are given in Table II. An ordering logic was defined in testing the terms in the equation in a stepwise manner, as suggested by Overall and Spiegal (1970). The ordering model was as follows: Y = Covariates + Organismic Variables + Treatment + Interactions The resulting regression equation took the following general form: Y l 10 = X l + X2 + X3 + X4 ( C o v a r i a t e s ) + X 5 + X 6 + X 7 + Xg (Organismic Variables) + Xg + X 1 Q (Treatment) + XgX^ + XgXg + X^Xg (Organismic Interactions) + X 5 X 9 + x 5 x i o + X6X9 + X6 X10 + X7 X9 + X7 X10 + X8 X9 + X8 X10 (Organismic by Treatment Interactions) + e (Error) Thus, the order used involved entering the covariates (School Achievement and Creativity) f i r s t into the regression equation, followed by organismic variables (Sex, Cognitive Style, Conceptual Tempo), in turn followed by treatment terms and then organismic interactions and treatment by organismic interactions terms last. This means that the effects of organismic variables were tested with the effects of the covariates being controlled s t a t i s t i c a l l y . Likewise, treatment and interaction effects were tested with the effects of the covariates and organismic variables 57 TABLE II SYMBOLS USED IN STATISTICAL ANALYSIS SYMBOL VARIABLE Y Mean T o t a l Time Y^ Mean P r i o r Time Y^ Mean T o t a l Questions Y^ Mean P r i o r Questions Y^ Mean Information-Seeking Questions Y Mean C o n s t r a i n t - S e e k i n g Questions o Y^ Mean Hypothesis-Seeking Questions Y Q T o t a l S o l u t i o n s o Yg Mean R e s i d u a l Time Y ^ Q Mean Re s i d u a l Questions X 1 Canadian Test of Basic S k i l l s (CTBS) Torrance Verbal Fluency Torrance V e r b a l F l e x i b i l i t y X^ Torrance V e r b a l O r i g i n a l i t y X 5 Sex X, A n a l y t i c Responses on C o g n i t i v e S t y l e Test (CST) o X^ E r r o r s on Matching F a m i l i a r F i g u r e s Test (MFF e r r o r s ) X 0 Mean Latency of Response on MFF Test (MFF time) o X g F i r s t Treatment Contrast - TI versus T2 + T3 X 1 Q Second Treatment Contrast - T2 versus T3 58 controlled s t a t i s t i c a l l y . Hypothesis Testing In order to test Hypotheses 1, 2, and 3 concerning the effects of the treatment, a series of regression analyses were run, one for each of the ten dependent variables. A summary of the results of these analyses is presented in Table III. A more detailed report of the results of these analyses can be found in Appendix D. The results for Treatment DV1 (Tl vs T23) in Table III indicate that training accounted Ipraa significant* amount of the variance in the scores of four of the dependent variables when the trained group (Tl) is compared to the combined control groups (T23). These measures were mean total time (F C Q = 7.2439, p..<.009), mean prior time (F = 10.3918, p.<.002), mean total questions (F- = 4.8119, p< .03) and mean constraint-l , o y seeking questions (F 1 5 g = 8.3301, p< .005). The means for Tl and T23 are presented in Table IV. From this table i t i s apparent that while mean total time spent on the problems by the trained group was 227.13 seconds, that for the combined control groups was 201.45. The difference between T l and T23 in terms of mean prior time was also highly significant, with means of 72.75 and 46.70 seconds respectively. The trained group also exceeded the control groups on mean total questions asked, the means being 4.82 and 3.28 respectively. In terms of mean number of constraint-seeking questions asked, the mean for Tl of 1.16 also exceeded that of T23 with a mean of .44. throughout this Chapter, the term "significant" denotes " s t a t i s t i c a l l y significant." SUMMARY OF RESULTS OF REGRESSION TABLE ANALYSES III (F-RATIOS) FOR TEN DEPENDENT VARIABLES SOURCE OF VARIATION MEAN TOTAL TIME MEAN PRIOR TIME MEAN TOTAL QUES . MEAN PRIOR QUES . MEAN I . S . QUES . MEAN C . S . QUES . MEAN H. S . QUES . TOTAL SOL. MEAN RES . TIME MEAN RES . QUES . CTBS 6. 5000* 1. 7010 2 . 5940 1. ,8269 2 . 9807 2 . 2162 ,0097 ,5275 2 . ,9696 1. . 4953 TORRANCE TESTS 3. 5813* 3. 0206* 4. 1881* 4. , 2532* 2. 7756* 3 . , 3851* 6. ,7115* 1. .2047 1. . 1481 2 , . 2585 SEX . 0244 . 0412 , 0990 .2884 5576 ,8543 ,2095 2 . 0078 . 0000 . 0000 CST . 0365 7319 1. , 6831 . 2115 1. , 9613 2 . ,7087 .5428 1. .3307 . 6363 2 , .0747 MFF ERRORS 3 . 7560* 5. 8969* ,1287 3, .5000 ,3942 . 0097 . 0097 .6535 . 0808 .7570 MFF TIME 3. 4024 7010 1. . 5544 2 . 2692 ,4134 3 . 9611* 1. .3142 1, .0629 6 , .3030* .3738 TREATMENT DV1 7. 2439* 10. 3918* 4. .8119* 2 , .7500 2 . ,9135 8. .3301* 1, .6952 . 0078 .2626 3 .2243 ( T l vs T23) TREATMENT DV2 1463 1. 8454 .8218 5, .4615* 1. . 0480 .8738 . 1612 .2126 1, .8586 .2056 (T2 vs T3) CST x MFF ERRORS 15. 8048* 1. ,7113 5. .8316* .5384 5, .5576* 2 , . 1165 3, .4095 . 0393 10. .2727* 7 .6728* CST x MFF TIME 1097 ,2164 1, .1980 . 6442 .7403 1, .5048 .5333 .1338 . 0000 .8317 MFF ERRORS x MFF TIME 2. 7805 6. .2164* . 6930 4 .3750* 1 . 0961 . 0485 . 0190 .4409 . 0000 .1401 SEX x TREATMENT 2683 .7268 . 8713 . 2788 1 . 0288 ,.7 67 0 2 . 0285 1 .3070 1, .2576 1, .1916 CST x DV1 .2926 1. .0103 3 .7425* .3942 6 .4711* .7184 . 1809 .1968 1, . 6962 4, . 7943* CST x DV2 2. .8048 .0309 .4257 . 0384 1 . 0384 . 3592 1 .8190 .1732 3, .4747 . 5420 MFF ERRORS x DV1 5. .2682* .4226 .0297 .2019 . 1250 . 1165 . 3238 1 .6535 3, . 6969* .3364 MFF ERRORS x DV2 .0121 .4432 . 0990 . 0576 .1057 . 0097 .0285 .1574 .1212 .3551 MFF TIME x DV1 1, . 9756 . 0412 . 3465 . 1153 . 1442 1 .2524 . 0476 .3543 2 , .5858 1. . 1121 MFF TIME X DV2 , 0000 1 . 0309 .1188 .0096 .1442 .0873 . 6476 1, .1259 .5959 .1588 * p < . 05 60 TABLE IV GROUP MEANS FOR TEN MEASURES OF PROBLEM SOLVING VARIABLE T123* ' T l T2 T3 T23** MEAN TOTAL TIME 209.94 227 . 13 109 . 09 212 .82 201. 45 MEAN PRIOR TIME 54.59 72 . 75 49 . 75 43 .65 46 . 70 MEAN TOTAL QUESTIONS 3.76 4 . 82 3 . 19 3 . 38 3 . 28 MEAN PRIOR QUESTIONS 1.66 2 . 11 1. 72 1 . 24 1. 48 MEAN INFORMATION-SEEKING QUESTIONS 2 . 64 3 . 16 2 . 39 2 . 43 2 . 41 MEAN CONSTRAINT-SEEKING QUESTIONS . 66 1. 16 • 46 .43 • 44 MEAN HYPOTHESIS-SEEKING QUESTIONS .46 • 52 • 34 . 53 • 43 TOTAL SOLUTIONS 10.85 11. 04 9 . 96 11 .47 10 . 71 MEAN RESIDUAL TIME 155.34 154. 38 140. 33 169 . 16 154. 74 MEAN RESIDUAL QUESTIONS 2 . 10 2 . 72 1. 47 2 . 13 1. 80 *T123=ALL EXPERIMENTAL GROUPS POOLED **T2-3= TWO CONTROL GROUPS POOLED 61 There was further evidence of superiority of the trained group over that of the combined control groups which was marginally reliable for three additional measures. These were mean prior questions (Tl = 2.11, T23 = 1.48; F^ ^  = 2.7500, p< .09), mean information-seeking questions (Tl = 3.16, T23 = 2.41; F c_ = 2.9135, p< .08), and mean residual x , questions (Tl = 2.72, T23 = 1.80; F.. = 3.2243, p< .07). J->->9 The results for Hypothesis 1 stated in Chapter II can be summarized as follows: Hypothesis 1: In terms of mean total time, mean prior time, mean total questions, and mean constraint-seeking questions, there was a significant difference between scores of T l and T23 with the scores of Tl being greater. The difference between the scores of trained and control groups approaches s i g n i f i -cance for mean prior questions, mean information-seeking questions, and mean residual questions. No significant difference existed between the scores of the trained and the control groups for total solutions and mean residual time. It was predicted in Hypothesis 2 that the combined control groups would ask more hypothesis-seeking questions than the trained group. An examination of the variance accounted for by Treatment DV1 (Tl vs T23) in Table III for this type of question indicates that this difference was not significant (F., c o = 1.6952, p< .19). In fact, as shown on Table IV, the trend was i n the reverse direction with the mean for T l being .52 and for T23 .43. The results for Hypothesis 2 can be stated as follows: 62 Hypothesis 2: No significant difference existed between the mean number of hypothesis-seeking questions asked by the trained and the control groups. Hypothesis 3 was designed to test for any significant differences in the scores of the two control groups. As can be seen by examining the results for Treatment DV2 (T2 vs T3) on Table III, the scores of these groups differed significantly on only one of the ten dependent measures, mean prior questions (F^ ^  = 5.4615, p<.02). The mean for T2 exceeded that of T3 on this measure (T2 = 1.72, T3 = 1.24), as shown on Table IV. The results for Hypothesis 3 can be stated as follows: Hypothesis 3: No significant difference existed between the scores of the two control groups on any of the ten measures of problem solving except mean prior questions asked, where T2 exceeded T3. Hypotheses 4 and 5 predicted that a positive relationship would exist between the organismic variable "cognitive style" and a l l measures of problem solving, except mean hypothesis-seeking questions asked, for which a negative relationship was predicted. Examination of the values for CST (mean number of analytic responses on the CST) on Table III indicate that no such relationship was found between this variable and any of the ten measures of problem solving. The results for Hypotheses 4 and 5 can be summarized as follows: Hypothesis 4: No significant relationship exists between the number of analytic responses on the CST and measures of problem solving for a l l experimental groups combined. 63 Hypothesis 5: No significant relationship exists between the number of analytic responses on the CST and mean hypothesis-seeking questions asked for a l l experi-mental groups combined. Hypotheses 6 and 7 predicted that a negative relationship would exist between the organismic variable "conceptual tempo" as measured by the errors on the MFF test, and a l l measures of problem solving except mean hypothesis-seeking questions asked, for which a positive relationship was predicted. Examination of the results for the variable MFF errors on Table III indicate that a significant amount of the variance in two problem solving measures, mean total time (F^ ^  = 3.7560, p<.05) and mean prior time (F.. C Q = 5.8969, p <.01) was accounted for by this l,oy measure of conceptual tempo. It should further be noted that the influence of this variable approaches significance for mean prior questions (F.. C Q = 3.5000, p< .06). ±,->y In order to determine the direction of the relationship between this measure of conceptual tempo and the three measures of problem solving, the correlations presented in Table V were examined. From this table i t can be seen that the correlation between MFF errors and mean total time was -.24, between MFF errors .and mean prior time was -.25, and between MFF errors and mean prior questions was -.21. The results for Hypothesis 6 can be summarized as follows: Hypothesis 6: A significant negative relationship exists between total errors on the MFF and mean total time and mean prior time. The relationship approaches significance for mean prior questions. No H H n td s n o < Tl < < H g H T I H Tl CO pd CO pel f W tr1 1—I S T I S Tl H pd Tl H M pd W Pd t n ?d S M O o W w W Tl M X pd X • > • > a > pd CO n f pa K ! 1-3 H H H H H H H H H H H H H rt H H H H H M ho M No M H to 1—1 h-1 NO I—1 h-1 to t—1 to M M to h-1 NO U) to LO to LO NO LO to LO ho LO to LO U) 1 LO LO 1 U) LO LO LO o NO NO h-1 I—1 O O o LO LO to M M to to Lo to LO l-1 Ui Ul 0 0 to -J LO NO NO o to to -p- 0 0 O N h-1 -p- Lo -P- -P-1 O o 1 1 o H* 1 I-1 1 1 1 Ul O -p-1 O 1 O h-1 O o O h-1 O 0 0 -P- O • ^ J O LO 0 0 LO 4> O Ul -P- Ui O N O N 1 o 1 h-1 1 o O 1 o 1 O to LO •P- to M to O to LO to 1 o O 1 M LO VO LO Lo LO M h-' O NO Ul O to -P- O O N to 0 0 LO to 1 h-1 1 o 1 o 1 o O 1 o 1 O 1 O 1 M NO LO to O O 1 O Lo 1 o O O O 0 0 o Ui Ul ON 0 0 LO h-1 vo h-1 LO to I-1 0 0 M Lo I—1 Lo o o 1 to o O h-1 M 1 o NO NO •P- M M to M to LO to 1 o O O O N O N NO NO to LO O to NO vo M Ul -P- Ul O h-1 LO Lo t—1 h-1 1 I- 1 1 o 1 I-1 1 o h-1 1 o H NO LO to O M O O h-1 to 1 t—1 O to vo LO NO Ul Ul O N Ul 0 0 to LO Ul O LO Lo LO VO Ul Ul to to o o 1 1—1 1 o 1 M 1 LO 1 M 1 o 1 O LO •P-O LO 1 to to •e-1 o 1 M 1 o 1 to Ui Ul LO vo - O Ul VO •p- Ul Ul VO to O LO 0 0 Lo -p~ to O N -o 1 O o 1 NO M h-1 LO O o 1 o h-> LO to N) o M 1 h-1 1—1 LO 0 0 -f> Ul 0 0 Ui Ul 0 0 0 0 VO O O N to to 0 0 o Ul to O NO 1 LO h-1 NO O o NO to 1 M h-1 1—1 h-1 to o ho to NO O N o Ui NO O N VO O N O N to O N O 0 0 LO to VO O LO -P- 0 0 Ul 1 o h-1 1 to O 1 O 1 o M o LO IO -p- O M LO O to LO I—1 1 O o 1 M o NO 4> NO h-1 O 0 0 Ul 0 0 to 0 0 0 0 LO O VO o VO to VO PI n n CO <! O Pd CO H W > pa Pd H w X pa o o CO M f> pd > 3 CO W H f CO w s CO M O H H H H H H H H H H H H GROUPING h-1 to M 1—1 to I—1 h-1 NO h-1 t—1 NO H to LO to LO NO LO NO LO LO LO LO LO 1 to 1 O 1 O 1 O O NO NO NO M h-1 TOTAL -p- 0 0 LO -o LO O O N NO NJ TIME 1 to 1 1 LO 1 O 1 o 1 to h-1 O O O NO PRIOR Ul O -O ~ ~ J to ~-J LO Ul NO Ul O TIME o 1 M h-1 1 o to M NO o O O H-1 TOTAL V O LO LO to 0 0 O N LO NO vo •P- LO QUESTIONS 1 to 1 to 1 to O o i o t-1 1 o O O I-1 PRIOR M -P- O N Ui LO o 4> 0 0 0 0 -P- 1—1 Lo QUESTIONS 1 M 1 h-1 1 1—' h-1 1 o -p- t—1 NO o O o O I . S . • NO t-1 0 0 Ul o 0 0 NO 4> h-1 Ui QUESTIONS 1 O 1 h-1 1 o M h-' I—1 LO 1 o M o NO C . S . Lo Ui 1—' V O h-1 0 0 Ul V O -O QUESTIONS 1 O 1 o o 1 I-1 1 o 1 NO o O 1 o M O o H . S . LO LO t—1 V O NO h-1 NO O vo -p- QUESTIONS 1 h-1 i O 1 to M M NO I-1 tNO O 1 h-1 1 o NO TOTAL O N O N V O Lo O LO 0 0 vo O Ui LO SOLUTIONS 1 o o 1 Lo O 1 o LO M H* LO O H" 1 O RESIDUAL V O Ul O 0 0 Ul O N -o LO O N V O LO Lo TIME O o O h-1 1 o LO 1—1 NO O h-1 O M RESIDUAL to Ui to -P- -p> V O LO NO LO O •P- O QUESTIONS n o pa-pa w t - 1 > H S M M O > CO CO <=! po w M M CO H s: o w a > Pd o o pa f w o f > M < s M CO CO s o H n < 1—1 < > o pa M > c-1 M CO > o O N 65 significant relationship exists between total errors on the MFF and the remaining measures of problem solving, for a l l experimental groups combined. MFF errors did not account for a significant amount of the variance in mean hypothesis-seeking questions asked ( F , , C Q = .0097, p < .88), as shown on Table III, indicating that no support was found for Hypothesis 7. In addition, the correlation between mean hypothesis-seeking questions asked of -.03 demonstrates that the relationship was not in the positive direction that had been predicted. The results for Hypothesis 7 can be stated as follows: Hypothesis 7: No significant relationship exists between total errors on the MFF and mean hypothesis-seeking questions asked, for a l l experimental groups combined. Hypotheses 8 and 9 predicted that a positive relationship would exist between the organismic variable "conceptual tempo" as measured by mean latency of response on the MFF test and a l l measures of problem solving except mean hypothesis-seeking questions asked, for which a negative relationship was predicted. An examination of the results of the variable MFF time on Table III indicate that this measure of conceptual tempo accounted for a significant amount of the variance in two measures of problem solving, mean constraint-seeking questions asked (F, ,-Q = 3.9611, p<.04) and mean residual time i , D y (F^ = 6.3030, p< .01). A marginally reliable relationship was found between this organismic variable and mean total time (F 1 c o = 3.4024, p< .06). i , jy The correlations presented in Table V indicate that the direction of the relationship between mean latency of response on the MFF and mean total time (r = +.24), and mean residual time (v ~ +.24) was in the positive direction predicted, but that the reserve was found for mean constraint-seeking questions asked (r = -.15). It can be seen from the data on Table III that the mean latency of response on the MFF did not account for a significant amount of the variance in the mean hypothesis-seeking questions asked (F _ = 1.3142, 1 , 3 9 p <.25), as predicted by Hypothesis 9. From Table V i t can be seen, however, that the relationship between these two variables was in the negative direction predicted with r = -.12. The results for Hypotheses 8 and 9 can be summarized as follows: Hypothesis 8: A positive relationship exists between mean latency of response on the MFF and mean residual time, and a marginally reliable positive relationship exists between mean latency of response on the MFF and mean total time. A negative relationship exists between MFF time and mean constraint-seeking questions asked. No significant relationship exists between this measure of conceptual tempo and mean prior time, mean total questions, mean prior questions, mean information-seeking questions, total solutions, and mean residual questions for a l l experimental groups combined. Hypothesis 9: No significant relationship exists between the mean latency of response on the MFF and mean hypothesis-seeking questions asked for a l l experimental groups combined. In Hypothesis: 10 i t was predicted that a significant interaction would exist Between training and cognitive style such that the relation-ship Between the numBer of analytic responses on the CST and a l l measures of proBlem-solving would Be positive and greater for the untrained groups (T23) than for the trained group (Tl). Examination of the results for the interaction term "CST x DV1" on Table III indicate that this inter-action of treatment with cognitive style accounted for a significant amount of the variance in three of the ten dependent variables. These were mean total questions (F = 3.7425, p<.05), mean information-l , o y seeking questions (F^ ^  = 6.4711, p< .01), and mean residual questions (F = 4.7943, p< .03). In order to determine the direction of these relationships, the correlations between mean analytic responses on the CST and these three dependent variables were examined for T l and T23. From Table V i t can be seen that the correlation between the CST score and mean total questions for Tl is +.28 and for T23 is -.02; for mean information-seeking questions Tl = +.40 and T23 = -.05; for mean residual questions Tl = +.39 and T23 = -.05. Fisher's r to Z transformation was used to test for significance of differences between these pairs of correlation coefficients. These values are presented on Table VI. None of the differences between the pairs of correlation coefficients reached the c r i t i c a l Z value of 1.96 to be significant at .05 alpha. It should be noted, however, that the direction of these relationships was in the reverse of what had been predicted. It was expected that training would decrease the effect of cognitive style on the problem-solving measures. Hence, the correlations for T l were predicted to be lower than those for 68 TABLE VI TESTS (FISHER'S Z) OF SIGNIFICANCE BETWEEN CORRELATIONS FOR TRAINED AND UNTRAINED GROUPS: COGNITIVE STYLE (CST), CON-CEPTUAL TEMPO (MFF ERRORS, MFF TIME) AND MEASURES OF PROBLEM SOLVING. PROBLEM SOLVING MEASURES CST MFF ERRORS MFF TIME MEAN TOTAL TIME . 5507 1.3889 . 7889 MEAN PRIOR TIME . 9861 . 6706 . 3944 MEAN TOTAL QUESTIONS 1 . 1834 . 0788 .5917 MEAN PRIOR QUESTIONS . 1183 . 0788 . 0708 MEAN I.S. QUESTIONS 1 .7751 .2761 . 1183 MEAN C.S. QUESTIONS .2761 . 5522 . 9467 MEAN H.S. QUESTIONS .5128 . 1577 . 8284 TOTAL SOLUTIONS . 5128 . 9072 .7100 MEAN RESIDUAL TIME 1 . 6173 1. 3806 .5128 MEAN RESIDUAL QUESTIONS 1 .7357 . 1183 .8284 69 T23, between cognitive style and measures of problem-solving. In the three cases cited above the correlations between these measures were consistently positive and in the direction of being higher for T l than for T23, although the Z transformation test indicates that no significant differences exist. To allow for a further examination of these interactions, scores on the CST were trichotimized so that the lowest third of the total sample whose scores were between 0 and 3 were designated as LOW, the middle third whose scores were between 4 and 8 were designated as MEDIUM, and the top third whose scores were between 9 and 15 were designated as HIGH analytic. For each of the three interactions which were significant in the regression analysis, the residuals remaining for each subject after the effect of the covariates, organismic main effects, treatment, and organismic interactions had been removed were collected, and the means calculated for the LOW, MEDIUM, and HIGH analytic subjects within Tl and T23. Hence, the residuals represent the raw scores adjusted for the effects of the terms which entered the regression model prior to the interaction terms of interest, and are therefore the appropriate values to use in this type of detailed examination into the specific nature of the interactions. These group means are presented in Figures 2, 3, and 4. CA table of the means can be found in Appendix E). In Figure 2 the relationship between training and cognitive style can be seen for mean total questions. The trained group asked more questions than the control groups i f they were classed as MEDIUM or HIGH analytic. The control groups, however, asked more questions than the trained group i f they were classed as LOW analytic. A similar trend can be seen in Figure 3 70 T l . . T23 + 1.20 GROUP LOWER LIMIT UPPER LIMIT i LOW 0 3 I MEDIUM 4 HIGH 9 15 COGNITIVE STYLE GROUPING FIGURE 2 MEAN RESIDUALS OF MEAN TOTAL QUESTIONS FOR TRAINED ( T l ) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (LOW, MEDIUM, HIGH CST). 71 ,T1 T23 + .75 + .50 + .25L 1-5 < P H W ! 3 w - . 25L - .50 - -75L -1. 00 -1.25--1.50. MEDIUM HIGH 4 9 8 15 COGNITIVE STYLE GROUPING FIGURE 3 MEAN RESIDUALS OF MEAN INFORMATION-SEEKING QUESTIONS ASKED BY TRAINED (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (CST) . GROUP LOW LOWER LIMIT 0 UPPER LIMIT 3 t T l _ _. T23 +1.00 P + . 75 V + .50 1" + . 25 C/3 < P H w 25 53 <« - .50 w 75 -1. 00 -1. 25 GROUP LOW LOWER LIMIT 0 UPPER LIMIT 3 MEDIUM 4 8 COGNITIVE STYLE GROUPING HIGH 9 15 FIGURE 4 MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR TRAINED (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF ANALYTIC COGNITIVE STYLE (CST) for mean, number of information-seeking questions asked. With, regard to mean residual questions asked (Figure 4), the trained group exceeded the untrained group only i f they were highly analytic. If they were moderately analytic, the scores were virtually the same for the two treatment groups. Consistent with the other two dependent variables of concern, the control groups who were LOW analytic asked more residual questions than did the LOW analytic trained group. To gain further insight Into the question of why the LOW analytic trained group did not appear to benefit from training, unlike their MEDIUM and HIGH analytic classmates, a further comparison was performed. As the training program was completely self-paced, the student was free to work through the materials as quickly or as slowly as they wished (to a maximum of 14 30-minute sessions). For this reason i t seemed possible that the LOW analytic child would work through the materials quickly, and hence l i k e l y derive l i t t l e benefit from theiinstruction provided therein. To test for this possibility, a correlation was calculated between the child's score on the CST and the number of days which he spent on the training materials. The resulting coefficient was .45 with a Z of 2.3247 (p< .05). This finding would offer support for the explanation offered above as to why LOW analytic trained children did not appear to benefit from the training. S t i l l unexplained, however, was the consistent finding that low analytic untrained children not only exceeded the trained children of like cognitive style, but also exceeded their moderately and highly analytic classmates on the three dependent variables shown in Figures 2, 3, and 4. From Table V i t can be seen that the correlations between the Torrance measures of creativity and mean total questions, mean information-74 seeking questions, and mean residual questions were positive and consistently higher for the untrained than for the trained group. Because of this trend, i t was f e l t that an examination of the relation-ship between conceptual style and Torrance sub-test scores within Tl and T23 might shed some further light on the unexpected finding for the LOW analytic untrained group. To this end, the mean fluency, f l e x i b i l i t y , and originality scores were calculated for each of the three conceptual style groups within Tl and T23. These means are shown graphically in Figure 5: A, B, and C. As can be seen from these figures, the untrained group consistently exceeded their trained peers in terms of both fluency and f l e x i b i l i t y , but notcon originality. Moreover, the low analytic untrained children proved to be more fluent and flexible than their MEDIUM and HIGH analytic classmates. Given these additional facts concerning the nature of the children who f a l l into this untrained LOW analytic classification, a possible explanation for their unexpectedly high performance on the three dependent variables of concern might be their higher measured verbal fluency and f l e x i b i l i t y . In summary, the results for Hypothesis 10 can be stated as follows: Hypothesis 10: A significant interaction exists between training and cognitive style such that the relationship between number of analytic responses on the CST and mean total questions, mean information-seeking questions, and mean residual questions is positive and greater for the trained than for the untrained subjects. No significant interaction exists between cognitive style and training for the remaining seven dependent measures. FLUENCY H O > o W C O w H W C O H C O o n o pa > M C O > f i-< o H pa H n H ps o > o M o a w - M o H M < H M M S ' C O 1—1 H > CD a f a a —s H O • PS s: ' > M 21 S W M a a M ^ — , a H N J C O H C O t u w C h • M H C O > H H a pd w w r< < m f C O o CO H CO On i\ 1— CO CO •P-CO C/1 FLEXIBILITY n CO H •p- •O CO 1 1 / / / / -p- C/i Cn Co -I H CO ORIGINALITY J> 4>- -P- 4*-CO -P" Ln O n -P- 4> 0 0 n CO H -r 76 In Hypothesis 11 i t was predicted that a significant interaction would exist between training and conceptual tempo (as measured by errors on the MFF), such that the relationship between the number of errors on the MFF and a l l measures of problem-solving would be negative and greater for the untrained groups (T23) than for the trained group (Tl). Examination of the values for the interaction of MFF errors with Training DV1 (Tl vs T23) on Table III indicate that the variance accounted for by this term was significant in two of the ten dependent variables. These were mean total time (F^ ^ = 5.2685, p < .02) and mean residual time (F __ = 3.6969, p< .05). To determine the direction of these i ,i>y relationships, the correlations between errors on the MFF and these two dependent variables for Tl and T23 were examined (see Table V). Contrary to expectation, the negative relationship between these variables was greater for the trained than for the untrained group; for mean total time they were T l = -.47, T23 = -.08; for mean residual time Tl = -.30 and ..i T23 = +.05. The differences between these pairs of coefficients were not s t a t i s t i c a l l y significant, however (see Table VI). To allow for further examination of these interactions, trichotimization of the continuous variable "MFF errors" was undertaken in a manner similar to that used to trichotimize CST. Subjects with 0 to 5 erros were classified as LOW impulsive, 6 to 9 errors as MEDIUM impulsive, and 10-16 errors were considered HIGH impulsive. The means of the residuals for mean total time and for mean residual time for each level of impulsivity within Tl and T23 were then calculated by the same procedure used in the previous section. These mean are shown graphically in Figures 6 and 7 (the means are presented in table form in Appendix E). T l • • T23 + 20, + 151 + 10fr + 5 CO Q M erf - 1 0 -15 -20 -25 -30 GROUP LOWER LIMIT UPPER LIMIT LOW MEDIUM HIGH 0 6 10 5 9 16 IMPULSIVITY GROUPING FIGURE 6 MEAN RESIDUALS OF MEAN TOTAL TIME FOR TRAINED ( T l ) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (LOW, MEDIUM, HIGH MFF ERRORS). 78 T l T23 + 25 r-+ 20 U + 15 + 10 < o M C O w + 5 - 5 -10 -15 -20 JL. _1_ GROUP LOW LOWER LIMIT 0 UPPER LIMIT 5 MEDIUM HIGH 6 10 9 16 IMPULSIVITY GROUPING FIGURE 7 MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR TRAINED ( T l ) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 79 In both, figures i t can be seen that LOW and MEDIUM impulsive children in the trained group exceeded the untrained group on both of these dependent measures, but that the reverse was true for HIGH impulsive children. Because i t was suspected that HIGH impulsive children may have worked through the training materials quickly, as had occurred with LOW analytics, the correlation between days spent on the programme and errors on the MFF was calculated. The resulting coefficient wass-,40, Z = 1.986, p< .05. This finding would support the contention that HIGH impulsive children do not benefit from the training because they do not take the time necessary for benefit to derive while working through the materials. Because the correlations between the creativity sub-test scores and mean total time and mean residual time tended to be higher for the untrained than for the trained subjects (see Table V), the means for the three impulsivity groups on these creativity measures were calculated for each treatment group. The resulting means are presented graphically in Figure 8: A, B, and C. It w i l l be noted that HIGH impulsive subjects in T23 achieved higher scores on the fluency and f l e x i b i l i t y sub-tests than did HIGH impulsive subjects in the trained group. This original difference between these two groups may have contributed to the superior-i t y of the untrained group on the two problem-solving measures of concern. The results for Hypothesis 11 can be summarized as follows: Hypothesis 11: A significant interaction exists between training and conceptual tempo such that the relationship between the number of errors on the MFF and mean total time and mean residual time is negative and F L U E N C Y o H o pd PC) > a o w C O <=! td I H M CO H C O o O pd M C O f W C O pd i—i pd < O H pd H CO K | EC C O C O C O 4>» 4> 4> U l M C O U l 1 1 i 1 M o S Pd hd a H tr 1 Pd CO > M M < a H w H u K! ' N .-—s H S N i—i r 1 1—1 M s o • — ' o s : in » > pd t"1 a M M CO s o t d pd l-H s w 0 0 Pd < et o H 1—1 a Pd H a H CO K! s pd — ' ** Al a • a H M o a F L E X I B I L I T Y 4>- 4> - P - ON •p-oo U l O U l K5 r N * — r a T " l I. • H H C O H N3 CO C O a w C-4 M n H CO •> H H a Pd W < M C O / - N |—I tr1 M C O pO H Pd < O H pd H co K! O R I G I N A L I T Y •P--P~ •P-U l •P> ON - P - -P~ 0 0 •p-tr" L a . V - r o 08 81 greater for the trained than for the untrained group. No significant interaction exists for the remaining eight measures of problem-solving. In Hypothesis 12 i t was predicted that a significant interaction would exist between training and conceptual tempo such that the relation-ship between mean latency of response on the MFF and the ten measures of problem-solving w i l l be positive and greater for the untrained than for the trained group. Examination of the interaction between MFF time and Training DV1 (Tl vs T23) on Table III reveal that this interaction term did not account for a significant amount of the variance in any of the ten dependent variables. It w i l l further be noted that the correlations between MFF time and measures of problem solving (see Table V) for Tl and T23 show no consistent pattern. The results for Hypothesis 12 can be summarized as follows: Hypothesis 12: No significant interaction exists between training and conceptual tempo as measured by mean latency of response on the MFF for any of the ten dependent variables. Additional Findings It w i l l be recalled that the present study was principally intended to be exploratory in nature. While a number of specific hypotheses were put forward for examination concerning the effects of cognitive style, conceptual tempo, and training upon problem-solving processes, of equal interest were any additional significant findings which were uncovered by the regression analysis. In the following section these additional findings w i l l be presented and summarized. 82 To allow for the examination of the interaction between cognitive style and conceptual tempo, the following three terms were generated and entered into the regression equations for each dependent variable: XgX^ - Analytic responses on CST x Total errors on MFF XgXg - Analytic responses on CST x Mean latency of response on MFF X.,X0 - Total errors on MFF x Mean latency of response on MFF / o Interaction of CST with MFF errors: The results presented in Table III indicate that the interaction between CST and MFF errors accounted for a significant amount of the variance in five of the measures of problem-solving. There were mean total time (F-j^  5 g = 15.80, p«i .0003), mean total questions (F 1 5 g = 5.8316, p< .01), mean information-seeking questions (F^ ,_g - 5.5576, p < .02) , mean residual time (F.. C Q = 10.2727, p < .002) and mean residual questions i,oy (F. _ = 7.6728, p < .007). i,jy In order to examine further the effect of the interaction between style and tempo on each of these five problem-solving measures, the two continuous organismic variables (CST and MFF errors) were treated as trichotomous variables, as defined in the reporting of Hypotheses 10 and 11 in the preceding section. Each subject within the total sample of 81 was then assigned to one of the cells of the resulting 3 x 3 matrix. The composition of each of these nine cells in terms of school achievement, creativity, and treatment group membership are presented in Table VII. For each of the five dependent variables for which this interaction was significant, the residuals which remained for each subject when the effects of the covariates, organismic, and treatment main effects had TABLE VII COMPOSITION OF NINE CELLS DERIVED VIA TRICHOTIMIZING RESPONSES ON THE CST AND ERRORS ON THE MFF M F F E R R O R S C . S . LOW MEDIUM HIGH CTBS 5 .66 4 .62 4 .73 FLUENCY 45 . 66 39 . 00 41.16 L FLEXIBILITY 55 .83 45.75 47.33 ORIGINALITY 49 . 50 46.58 47.66 0 T l n: 1 4 3 w T2 n: 3 % 1 T3 n: 2 4 2 T o t a l 6 12 6 CTBS 5 . 45 5 . 38 4.80 M FLUENCY 40.70 38 . 88 39 . 80 E FLEXIBILITY 48 . 30 45 . 11 42 .80 D ORIGINALITY 47 . 90 43 . 55 43.80 I T l n: 2 1 1 U T2 n: 2 4 5 M T3 n: 6 4 4 T o t a l 10 9 10 CTBS 5 . 40 4.82 5 .36 T T FLUENCY 41.81 36.00 36.50 a FLEXIBILITY 48 . 90 41. 85 41.70 I ORIGINALITY 49.18 44 . 00 42 . 80 T l n : 6 2 5 T2 n: 2 1 4 H T3 n: 3 4 1 T o t a l 11 7 10 84 been s t a t i s t i c a l l y removed were obtained. The means of these residuals were then calculated for each of the nine cells in the matrix. These means are presented in Table VIII and are plotted graphically in Figures 9, 10, 11, 12, and 13. For the two dependent measures which deal with time, mean total time (Figure 9) and mean residual time (Figure 10), a number of consistent trends w i l l be noted. These can be summarized as follows: 1. For low analytic children, time spent on the problem generally decreases as impulsivity increases. 2. For moderately analytic children time spent on the problem generally remains constant as impulsivity increases. 3. For highly analytic children time spent on a problem decreases for moderately impulsives, and then increases for highly impulsive children. 4. For low impulsive children, time spent on a problem increases as analytic a b i l i t y decreases. 5. For moderately impulsive children, highly analytic children spend less time on the problem than either moderately or low analytic children, who differ?very l i t t l e from one another. 6. For highly impulsive children, time spent on a problem increases as analytic a b i l i t y increases. For the three measures of question-asking behavior, total questions (Figure 11), mean information-seeking questions (Figure 12), and mean residual questions (Figure 13), very similar trends were found to those concerning time. These can be summarized as follows: 1. For low analytic children, questions asked during problem-solving decreases as impulsivity increases. 2. For moderately analytic children, questions asked during problem-solving increases for moderately impulsives and then decreases again for the highly impulsives to a level similar to the low impulsives. 3. For the highly analytic children, the number of questions asked during problem-solving decreases slightly from low to moderately 85 TABLE VIII MEAN RESIDUALS OF FIVE MEASURES OF PROBLEM SOLVING FOR LEVELS OF THE CST BY MFF ERRORS INTERACTION MFF ERRORS DEPENDENT VARIABLE LOW MEDIUM HIGH C . X TOT . TIME + 13 . 9666 + 24 . 8500 -65 . 5205 L X TOT . QUES . + 1. 7987 - .3902 - 1. 8714 0 X I . S . : QUES. + 1. 3231 - .4703 - 1 . 1990 w X RES . TIME + 20 . 2134 + 8 .7527 -44 . 6417 X RES . QUES . + 1. 7278 — . 1270 - 1. 3152 X TOT . TIME - 6. 7803 + 6 . 0550 + 12 •  2045 M X .TOT . QUES . - 4186 + .8402 - 1070 E X I . S . QUES . - 0854 + . 6012 + 0567 D . X RES . TIME + 6. 5034 + 8 .2976 + 4. 0467 X RES . QUES . — 4240 + .4211 0815 X TOT . TIME -15 . 9156 -40 .7877 +36 . 2921 H X TOT . QUES . - 4943 - .8757 + 1. 1380 I X I . S . QUES . - 2942 - .6690 + 7693 G X • RES . TIME -14. 0976 -48 . 2757 +39 . 9105 H X RES . QUES . - 5985 - .7211 + 1. 1947 + 40 _ + 30 + 20 + 10 C O i—* Q S -20 Pi <! w -30 S -40 -50 -60 -70 LOW GST MEDIUM CST HIGH CST GROUP LOW LOWER LIMIT 0 UPPER LIMIT 5 MEDIUM 6 9 IMPULSIVITY GROUPING •I HIGH 10 16 FIGURE 9 MEAN RESIDUALS OF MEAN TOTAL TIME FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 87 _a LOW CST MEDIUM CST ^ ^ HIGH CST GROUP L 0 W LOWER LIMIT 0 UPPER LIMIT 5 MEDIUM 6 . 9 IMPULSIVITY GROUPING HIGH 10 16 FIGURE 10 MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 88 C O < !=> Q I—I CO W Pi <! W a +2.0. • +1.5 L + 1. 0 + . 5 - . 5 -1.0 -1.5 r-2 . 0 . LOW CST l MEDIUM CST A HIGH CST GROUP LOW LOWER LIMIT 0 UPPER LIMIT 5 MEDIUM 6 9 IMPULSIVITY GROUPING HIGH 10 16 FIGURE 11 MEAN RESIDUALS OF MEAN TOTAL QUESTIONS FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 89 +1.40 _ + 1.20 + 1. 00 + .80 + .60 + .40 co ^+ -20 n C O w Pi 0 53 3- .20 40 60 - .80 • • LOW CST H. | MEDIUM CST A HIGH CST -1. 00 -1.20 I I I • GROUP LOW MEDIUM HIGH LOWER LIMIT 0 6 10 UPPER LIMIT 5 9 16 IMPULSIVITY GROUPING FIGURE 12 MEAN RESIDUALS OF MEAN INFORMATION-SEEKING QUESTIONS FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). + 1.75 +1.50 + 1.25 + 1. 00 + .75 + .50 S+ .25 P 0 a - .25 w a - .50 - .75 -1. 00 -1.25 -1.50 LOW CST -a MEDIUM CST • A HIGH CST 90 I GROUP LOW LOWER LIMIT 0 UPPER LIMIT 5 MEDIUM 6 9 IMPULSIVITY GROUPING HIGH 10 16 FIGURE 13 MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND THREE LEVELS OF IMPULSIVITY (MFF ERRORS). 91 impulsive, but increases for the highly impulsives. 4. For low impulsive children the number of questions asked during problem-solving increases as analytic a b i l i t y decreases. 5. For moderately impulsive children, those who are also moderately analytic ask most questions, and those who are highly analytic ask fewest questions, with the low analytics f a l l i n g in between. 6. For highly impulsive children, the number of questions asked during problem-solving increases as analytic a b i l i t y increases. Because the classification of the eighty-one subjects into nine cells via trichotomization of the two continuous organismic variables had been performed on a post hoc basis, i t appeared beneficial to analyze the composition of the cells in terms of school achievement, creativity, and training group, as presented in Table VII. From Figure 14 i t can be seen that school achievement, as measured by the CTBS generally decreases as impulsivity increases. For highly analytic subjects, however, more impulsive children score only slightly lower than reflective (LOW impulsive) children on this measure of school achievement. Recalling the general superiority of this group on the time and question-asking measures that were presented above, this additional information provides further explanation for this finding. In terms of the creativity measures (Figure 14: B, C, and D), two general trends emerge: 1. Low impulsive children tend to score higher on these creativity measures while moderately and highly impulsive children differ very l i t t l e ; 2. Low analytic children tend to score higher on measured verbal fluency, f l e x i b i l i t y and originality than moderately or highly analytic, particularly at the two extremes of the impulsivity scale. It w i l l be recalled that low analytic-low impulsive children consistently exceeded moderately and highly analytic children of low 5 . 8 r-5 . 6 \ -5 . 4 c o 5 . 2 m H u ^ 5 . 0 <! w a 4 . 8 4 . 6 H H H CQ M XI w <i W LOW CST ^ MEDIUM CST ^ HIGH CST 46 44 £ 42 w 1-1 * 40 w a 92 LOW MEDIUM HIGH IMPULSIVITY A. 56 «. 52 48 L 44 L 40 LOW MEDIUM IMPULSIVITY C . HIGH 38 U 36 <U M O I—I Pi o W a \ \ LOW. MEDIUM HIGH IMPULSIVITY B . 50 „ H 48 46 44 42 LOW MEDIUM IMPULSIVITY D . HIGH FIGURE 14 MEANS FOR CTBS (A), FLUENCY (B), FLEXIBILITY (C), AND ORIGINALITY (D) FOR SUBJECTS AT THREE LEVELS OF ANALYTIC STYLE (CST) AND IMPULSIVITY (MFF ERRORS). impulsivity on the five measures of time and question-asking behavior reported above. The demonstrated superiority of this low analytic group, in terms of Fluency and F l e x i b i l i t y , may partially account for their out-performing moderately and highly analytic peers on these problem-solving measures. Figures 9 through 13 consistently show a U-shaped pattern in both time spent and questions asked during problem-solving for the highly analytic children over the three levels of impulsivity. It w i l l be noted in Table VII that six of the eleven children in the low impulsive-high analytic group were from the trained class (Tl). Five of the ten children in the high impulsive-high analytic group were trained. Only two of the seven children in the moderately impulsive-high analytic group (who consistently scored lowest of the three groups) were trained. This difference in c e l l composition could, in part, account for the poorer performance of this moderately impulsive group, particularly in light of the fact that training had the effect of significantly increasing a l l but one (mean residual time) of the five dependent variables under consideration at this point. Interaction of CST with MFF Time Table III indicates that this interaction term did not account for a significant amount of the variance in any of the ten dependent variables. 94 Interaction of MFF errors with MFF Time The interaction of these two measures of conceptual tempo accounted for a significant amount of the variance in two measures of problem-solving (see Table III). These were mean prior time (F. _ = 6.2164, p< .01) and mean prior questions (F.^  ^  = 4.3750, p*.03). To allow for a closer examination of these interactions, a similar procedure of trichotomization was used for the two continuous organismic variables as had been done for the interactions between CST and MFF errors, placing each of the eighty-one subjects in one of the cells of the resulting 3 x 3 matrix. The composition of each of the resulting nine cells in terms of school achievement, creativity, and treatment group membership is presented in Table IX. It should be recalled by the reader at this point that a high score on the MFF errors measure implies impulsivity while a high schore on the MFF time measure implies the opposite, re f l e c t i v i t y . For the purpose of simplifying the following presentation, differences on the MFF errors w i l l be referred to as "accurate" versus "inaccurate", and differences on the MFF time measure w i l l be referred to as "slowness" versus "fastness". Mean residuals were calculated for each of the nine c e l l s , using the same procedure described in the preceding section. These data are presented in Table X and are plotted graphically in Figures 15 and 16. Inspection of these two figures reveals slightly different trends. For the measure of prior time, the following six trends are noted: 1. For accurate children, as "fastness" increases, time spent prior to offering a solution increases. 2. For moderately accurate children, as "fastness" increases, time spent prior to offering a solution decreases. 95 TABLE IX COMPOSITION OF NINE CELLS DERIVED VIA TRICHOTIMIZING LATENCY OF RESPONSE AND ERRORS ON THE MFF MFF TIME HIGH MEDIUM LOW M F F E R R 0 R S CTBS FLUENCY H FLEXIBILITY I ORIGINALITY G T i n : H T2 n: T3 n: T o t a l CTBS M FLUENCY E FLEXIBILITY D ORIGINALITY I T l n: U T2 n: M T3 n: T o t a l 4 . 98 36 . 00 39.60 41.20 0 3 2 5 4.81 36 . 85 42.57 44. 57 1 3 3 7 5 . 09 40.30 45 .40 47.10 3 3 4 10 5 . 96 38 . 08 44 . 00 45 . 66 4 3 5 12 4 .92 3 8.81 43.36 43 . 18 6 4 1 11 4 .81 39 . 44 46.88 44 . 33 2 3 4 9 CTBS FLUENCY L FLEXIBILITY 0 ORIGINALITY W T l n: T2 n: T3 n: T o t a l 5 .43 41.50 46.92 47 ,50 4 3 7 14 5 .60 41.36 48 . 90 48 .71 5 3 3 11 5 . 05 52.50 67 . 00 57.50 0 1 1 2 c 96 TABLE X MEAN RESIDUALS OF TWO MEASURES OF PROBLEM SOLVING FOR LEVELS OF THE MFF ERRORS BY TIME INTERACTION M F F E R R 0 R S H I G H M E D . L 0 W DEPENDENT VARIABLE X PRIOR TIME X PRIOR QUES. X PRIOR TIME X PRIOR QUES. X PRIOR TIME X PRIOR QUES. HIGH +15.6890 + .9627 +14.0285 - .3895 -11.3392 - .1653 MFF TIME MEDIUM -14 .5244 - .4555 +10.6112 + .2291 + 2.1857 + .1271 LOW - 9.9 34 7 - .4140 + .2238 + .2530 +41.6172 + 1.3235 + 45 + 40 + 35 + 30 + 25 + 20 w +15 <! S3 o M +10 C O w Pi a + 5 w a o - 5 -10 -15 -20 LOW MFF ERRORS 97 I MEDIUM MFF ERRORS A HIGH MFF ERRORS X GROUP HIGH LOWER LIMIT 16 UPPER LIMIT 37 MEDIUM 9 15 REFLECTIVITY GROUPING LOW 1 8 FIGURE 15 MEAN RESIDUALS OF MEAN PRIOR TIME FOR SUBJECTS AT THREE LEVELS OF ACCURACY (MFF ERRORS) AND THREE LEVELS OF REFLECTIVITY (MFF TIME). 98 + 1.2 + 1. 0 + w + . 6 <3 !=> P H + . 4 C O w erf a + . 2 <u w s 0 - . 2 - .4 LOW MFF ERRORS MEDIUM MFF ERRORS HIGH MFF ERRORS J l GROUP HIGH LOWER LIMIT 16 UPPER LIMIT 37 MEDIUM 9 15 REFLECTIVITY GROUPING LOW 1 FIGURE 16 MEAN RESIDUALS OF MEAN PRIOR QUESTIONS ASKED BY SUBJECTS AT THREE LEVELS OF ACCURACY (MFF ERRORS) AND THREE LEVELS OF REFLECTIVITY (MFF TIME). 99 3. For inaccurate children, as "fastness" increases, time spent prior to offering a solution decreases and then increases slightly. 4. For slow children, as accuracy decreases, time spent prior to offering a solution Increases. 5. For moderately slow children time spent prior to offering a solution i s lowest for the inaccurate, highest for the moderately accurate, with the highly accurate f a l l i n g midway. 6. For fast children, as accuracy increases, time spent prior to offering a solution increases. A number of slightly different trends can be seen in Figure 16 with regard to question-asking behavior: 1. For accurate children, as "fastness" increases, number of questions asked prior to offering a solution increases. 2. For the moderately accurate children, as "fastness" increases, number of questions asked prior to offering a solution increases. 3. For highly inaccurate children, as "fastness" increases, the number of questions asked prior to offering a solution decreases. 4. For slow children, number of questions asked prior to offering a solution i s highest for the inaccurate, with the moderately and highly accurate differing very l i t t l e . 5. For moderately slow children, the inaccurate ask fewest questions prior to offering a solution, with the moderately and highly accurate differing very l i t t l e . 6. For fast children, as accuracy increases, the number of questions asked prior to offering a solution increases. To gain additional insight into the composition of each of the nine cell s , the data presented in Table IX was examined further. As can be seen from Figure 17A children who were classified as accurate by their number of MFF errors tend also to be high scorers on measures of school achievement, particularly i f they were assessed as being highly or moderately reflective on the MFF time measure. It w i l l be further noted from Figure 17B, C, and D that the highly accurate children tended to 5 . 6r 5.4 era PQ H U w S 5 . 2 5 . 0 4.8 LOW MFF ERRORS MEDIUM MFF ERRORS HIGH MFF ERRORS 56 52 48 100 HIGH MEDIUM REFLECTIVITY (MFF TIME) LOW 68 EH H 60 M X 52 w 5s 44 «3 W S 36 HIGH MEDIUM REFLECTIVITY (MFF TIME) LOW U 55 W PJ 44 40 S3 <! K s 36 X HIGH MEDIUM REFLECTIVITY (MFF TIME) B . LOW 6 1 -57 >* H 531. >J <! is i—i o 49 H O a 45 < w a 41 L HIGH MEDIUM LOW REFLECTIVITY (MFF TIME) C D. FIGURE 17 MEANS FOR CTBS (A), FLUENCY ( B ) , FLEXIBILITY (C), AND ORIGINALITY (D) FOR SUBJECTS AT THREE LEVELS OF ACCURACY (MFF ERRORS) AND REFLECTIVITY (MFF TIME). 1 0 1 score higher on the Torrance measures of creativity, particularity at the fast end of the MFF time scale. The consistent superiority of this quick-accurate group on both prior time and prior questions could be ju s t i f i e d in part by this superiority in creativity. It should be noted, however, that only two subjects f e l l into this quick—accurate c l a s s i f i -cation, neither of whom were from the training group. The very small size of this particular group, of necessity, places severe restrictions on the generalizability of the performance of this group. Summary of Effects of Organismic Interactions* In summary, i t would appear that the effects of the organismic variables discussed above tend to modify each other. If a child is highly impulsive but analytic, he w i l l spend a longer time on the problem and ask more questions than would normally be predicted on the basis of impulsivity alone. Likewise, i f he i s not analytic but reflective, he w i l l spend a longer time in problem solving and ask more questions than would be expected on the basis of being non-analytic alone. If a child i s highly impulsive but accurate, he w i l l spend a longer time and ask more questions prior to offering a solution than would be expected on the basis of impulsivity only, and i f he is highly inaccurate but reflective, he w i l l also spend more time and ask more questions than his level of inaccuracy alone would lead one to predict. Creativity and Problem-Solving The principal purpose foroobtaining the three Torrance sub-test scores for the eighty-one children who participated in this study was to allow for the examination of problem-solving with the effects of 102 creativity s t a t i s t i c a l l y controlled. The rationale for this decision, based on the research literature in the area of creativity, has been additionally j u s t i f i e d in the present study. As demonstrated oil Table III, the combined effects of the three Torrance sub-test scores did account for a significant amount of the variance in a l l but three of the ten dependent variables. Given the consistency of the influence of fluency, f l e x i b i l i t y , and originality when combined as covariates, i t became of some interest to the present researcher to determine the degree to which each of the three individual Torrance scores related to the problem solving measures examined in this study. To allow for this type of examination, ten separate regression analyses were performed, one for each dependent variable, which entered each Torrance Sub-test score as a separate variable, with the C.T.B.S. scores of school achievement treated as a covariate. A summary of the results of these analyses are presented in Table XI. From these data i t becomes apparent that Fluency did not significantly predict any of the ten problem solving measures, and that F l e x i b i l i t y accounted for a significant amount of the variance in four of the ten measures. Originality proved to be the most powerful predictor of performance in that i t accounted for a significant amount of the variance in six of the ten problem-solving measures, and approached significance in predicting two further measures. From the correlations presented on Table V i t can further be seen that the direction of these relationships are consistently in the positive direction. Thus, while the results of the principal regression analyses, which treated Torrance scores as covariates, indicated that a relationship did exist between this combined multi-dimensional measure of creativity, 103 TABLE XI RESULTS OF REGRESSION ANALYSES FOR TORRANCE SUB-TEST SCORES AND PROBLEM SOLVING MEASURES DEPENDENT VARIABLE TORRANCY SUB-TEST FLUENCY FLEXIBILITY ORIGINALITY ob s ob s ob s X TOTAL TIME .5892 .45 1.7410 .18 5.5357 .02 X PRIOR TIME .6581 .42 5.1111 .02 1.7521 .18 X TOTAL QUES. .2972 .59 3.5045 .06 7.6306 .007 X PRIOR QUES. .7657 .38 7.5135 .007 3.6846 .05 X I.S. QUES. .3793 .54 1.3706 .24 5.7155 .01 XC.S.QUES. .4473 .51 5.7280 .01 3.0000 .08 X H.S. QUES. 3.0194 .08 4.7961 .02 12.7087 .0008 TOTAL SOLUTIONS .9338 .33 2.3305 .12 .5289 .47 X RES. TIME. .0894 .75 .0000 .95 2.6829 .10 X RES. QUES. .0168 .86 .3529 .56 5.7226 .01 104 the additional information presented in Table XI makes i t apparent that i t is not simply being able to generate many responses (fluency) that w i l l relate to performance on the types' of problem used in this research. While the ab i l i t y to provide responses which f a l l into several different categories ( f l e x i b i l i t y ) is of some influence, i t is principally the child's a b i l i t y to generate and willingness to report unique responses (originality) that is of the greatest predictive value. 105 CHAPTER V DISCUSSION The present study was designed to explore three major issues: 1. The nature of the relationship between cognitive style and processes employed during problem solving. 2. The nature of the relationship between conceptual tempo and processes employed during problem solving. 3. The effectiveness of training via programmed instruction to f a c i l i t a t e the use of analytic and reflective modes of problem solving. The following discussion w i l l deal with the results of this study as they relate to each of these issues. Cognitive Style and Problem Solving The results of the present study indicate no s t a t i s t i c a l l y significant relationship between measured cognitive style and the ten measures of problem-solving, contrary to prediction. The rationale for the prediction that analytic children would spend morettimeoon the problems presented, ask more questions, and make fewer irrelevant guesses was based on the previous research of Gardner, Holtzman, Witkin, and Kagan who repeatedly found analytic children more capable than non-analytic in the use of such s k i l l s as information-gathering, hypothesis generation and testing, and selective attention to relevant as opposed to irrelevant information. The failure of the present study to find any marked differences among these children can be explained on several bases. 106 F i r s t l y , the nature of the testing situation may have been such that i t decreased the probability of a non-analytic style of responding being measured. Specific attempts were made to develop a testing atmosphere that was as free from pressure as possible. By virtue of the fact that at least one solution was offered in 355 of the total of 364 problems presented (four each to eighty-one subjects), i t appears that the children interpreted the situation as one that required at least one solution to be generated and reported.. Because the non-analytic child tends to lack the a b i l i t y to discriminate important components of a problem and extract meaningful information, he would have to spend a longer time on the problem and ask more questions to meet this requirement of "at least one solution". The analytic child, on the other hand, is capable of performing the necessary information-gathering and hypothesis-testing s k i l l s on his own and would need to ask fewer questions and spend less time on the problem than he is optimally capable of, to meet this c r i t e r i a of "at least one solution". As a consequence, the measured performance of children at the two extremes of the cognitive style continuum do not differ, but this may have come about for very different reasons. While i t ' i s not possible to substantiate this speculation on the basis of the present study, i t could prove a f r u i t f u l area of investigation for future research. By more clearly providing the child with a "No Solution" response alternative, analogous to the use of the "Ready" card in this study, the relative tendencies of the two cognitive style groups to use this form of response could be assessed. A second possible explanation for the failure to find an effect due to cognitive style may be seen in the results of the interaction between cognitive style and conceptual tempo. It w i l l be recalled that 107 on five of the ten dependent measures, the interaction between style and tempo was significant. This interaction was such that measured impulsivity tended to moderate the effects of cognitive style, which resulted in the non-analytic child spending as much time and asking as many questions as the more analytic child i f he was either low or moderately impulsive. The moderating effect of conceptual tempo resulted in decreasing the predicted effect of cognitive style. A third explanation for the failure of cognitive style of reliably predict processes used in problem solving may be found in the data relating to creative thinking factors. It was found that measured creative thinking a b i l i t y positively related to problem solving. It was further found that non-analytic children tended to score higher than moderately or highly analytic children on tests of creative thinking ab i l i t y . Given these relationships, i t seems plausible to conclude that the higher measured creative thinking a b i l i t y of the non-analytic child functioned to decrease the performance differences between the non-analytic and analytic children. Conceptual Tempo and Problem Solving The results of the present study indicate that a significant relationship exists between the two measures of conceptual tempo (MFF errors and time) and four of the measures of problem solving. It is of particular interest to note that conceptual tempo predicted to a l l three of the time measures, but predicted to only one of the measures of question-asking (Mean constraint-seeking questions). The more reflective child appears willing to spend a longer total time on a 108 problem, and a longer period both prior to and following the offering of the f i r s t solution, but he does not ask more questions or generate more solutions than does the impulsive child in his shorter period of time with a problem. The pattern of response for the impulsive group is a quick succession of questions and solutions with very brief silent periods in between. The reflective child asks the same number of questions and offers an equal number of solutions, but intersperses these verbalizations with silent periods, during which he is presumably "reflecting". Interpreted in terms of Kagan's (1966) approach-avoidance theory, the impulsive child seeks quick success, so he spends l i t t l e time between responses to evaluate their merits, while the reflective child i s more anxious over making a mistake and considers each response he makes, whether i t be a question or a solution, before he offers i t . This interpretation could also be applied to the finding that impulsive children asked more constraint-seeking questions, but did not ask more hypothesis-seeking questions (i.e. make more irrelevant guesses) than reflective children. Perhaps during the silent periods the reflective child asks himself the constraint-seeking questions which allow him to avoid the tendency to take the risk of making irrelevant guesses, while the impulsive child asks these constraint-seeking questions overtly. Again i t should be noted that cognitive style and conceptual tempo tended to modify each other. The highly impulsive child w i l l behave in a reflective manner i f he i s also highly analytic. This finding could account for the lack of effect due to conceptual tempo on the majority of problem solving measures. In addition, the finding that greater measured creative thinking a b i l i t y is demonstrated by impulsive thinkers could also have functioned to diminish the main effect of 109 conceptual tempo. In summary, the consistency of the relationships between cognitive style, conceptual tempo, and performance on a variety of tasks found in previous research did not manifest i t s e l f in the majority of problem solving measures collected in the present study. While reflective children were found to spend a greater amount of time during problem solving, they were not found to differ from more impulsive peers in general question-asking behavior or in the number of solutions generated. In addition, measured analytic a b i l i t y was not found to be significantly related to any of the measures of problem solving. Perhaps what is more important than this lack of effect, however, is the interactions which were found to exist between style and tempo. With the exceptionsof the earily work of Kagan, research in this area has tended to deal with either style or tempo, and has not directed i t s e l f to the examination of the interaction between the two variables. The present study indicates that each of these dimensions is sufficiently powerful to moderate the other, and suggests an area that warrants further research. By employing a total sample size that would insure a larger number of children to be classified into each of the nine cells of a style by tempo matrix than was possible in theppcesent study, this interaction could be more thoroughly examined. In addition, a larger sample would allow for the examination of the three-way interaction of training, style, and tempo, which was not possible in the present study due to the existance of several "empty" cell s . 110 Training and Problem Solving The third major issue to which this study addressed i t s e l f was the f e a s i b i l i t y of teaching children analytic and reflective methods of problem solving via programmed instruction. The results indicate that this form of training was effective in increasing both the total time and the time prior to offering the f i r s t solution, the total number of questions asked, and the number of constraint-seeking questions asked. The training was also marginally reliable in bringing about superior performance on three additional measures: prior questions and information-seeking questions asked, and time spent following theooffering of the f i r s t solution. The lack of difference between the scores of the two control groups on a l l but one of the problem solving measures would atest to the fact that i t was the content of the training program and not simply exposure to programmed instructional materials which brought about this increase in performance. The superiority of the trained group is additionally meaningful in light of the fact that one of the control groups consistently scored higher on measured fluency and originality as determined by the Torrance Verbal Creativity Test. It w i l l be recalled from studies reviewed in Chapter II that measured creative thinking a b i l i t y has been found to bear a positive relationship to problem solving (Maier and Janzen, 1970; Davis, 1973; Curtchfield and Covington, 1963). This positive relationship was also found to exist with the measures of problem solving employed in the present study. On this basis, one could assume that the control groups, by virtue of their being more fluent and original, were predisposed to be superior problem solvers. In spite of this bias in favor of the control groups, the training program was I l l sufficiently effective to bring about superior performance of the trained group on at least four of the measures of problem solving. Of particular interest i s the finding that this form of training was effective iniincreasing not only the total number, but also the quality of the questions asked, as evidenced by the increased in constraint-seeking questions. Several authors (Ault, 1973; Anderson, 1965; Denney, 1973) have characterized the use of these questions as evidence of a more advanced, mature and reflective approach to problem-solving. To have brought about a qualitative change of this nature through what might be characterized as a rather minor intervention technique, suggests a promising area for further investigation into the remediational possibilities of this type of material in coordination with other instructional procedures such as classroom discussion. The need for this type of further investigation is made more apparent by the attribute-treatment interaction which was found to exist between both style and tempo with training on five of the problem-solving measures. These interactions indicated that while both moderately and highly analytic and moderately and highly reflective children benefitted from the training materials, the non-analytic and highly impulsive children did not. It was further found that impulsive and non-analytic and highly impulsive children did not. It was further found that impulsive and non-analytic children tended to spend a shorter period of time completing these self-paced materials than did the more analytic and reflective children. A number of authors have written extensively with regard to this type of interaction (Cronbach, 1965; Cronbach, 1967; Carroll, 1967; Cronbach and Snow, 1969; Hunt and Sullivan, 1974; Bracht, 1970). Of the 108 aptitude-treatment interaction 112 studies reviewed by Bracht (1970) however, only five provided s t a t i s t i c a l evidence of the existence of a disordinal interaction. On the basis of this low incidence, i t was anticipated in the present study that the training program would be equally effective for children at a l l points along the style and tempo continuums. The interaction results on several of the measures (total time, total questions, information-seeking questions, residual time, and residual questions) indicate that this method of training was not as effective for children who f e l l at the low end of the style and tempo continuums. It should be noted, however, that the strength of these interactions was partly due to the unusually high performance of the low analytic and low reflective children in the combined control groups. A partial explanation for this finding can be found in the fact that this group of control children consistently out-scored a l l other groups on some measures of creative thinking a b i l i t y . It s t i l l remains apparent, however, that the benefit derived by children of differing analytic and reflective a b i l i t y was not the same, given one style of instruction. While the more highly analytic and reflective child was capable of working through the programmed instructional materials on his own, at a pace that would result in the desired behavioral objectives being acquired, this form of instructionawas inadequately structured for the more impulsive and non-analytic. It is possible that the imposition of more structure in the form of constant checks on thoroughness of completion of each of the exercises, or a teacher-set pace rate would bring about similar beneficial results for these impulsive, non-analytic children. A replication of the present study along with an additional training group exposed to a more externally controlled presentation of the program would allow for this attribute-113 treatment interaction to be more carefully examined. In summary, the programmed instructional materials were successful in bringing about superior performance for the trained group in the majority of the problem solving measures, while simple exposure to programmed instructional material of unrelated content was not. The existence of the attribute-treatment interactions make i t clear,.however, that these materials were not beneficial in their present form for highly impulsive, low analytic children. Additional research using modified forms of presentation of the training program w i l l be required to determine the extent to which these materials can benefit the very impulsive, non-analytic child. Relationship Between Process and Product It is clear from the results of the data analysis that none of the independent variables selected for examination in this study accounted for a significant amount of the variance in total numberoof of solutions offered, in spite of the fact that children were found to be using different processes. It is possible that this may be due to the procedure used in scoring solutions, for no attempt was made to differentiate solutions by their quality or comprehensiveness. The research on problem-solving reviewed in Chapter II tended to employ solely product-related dependent variables with very l i t t l e attention given to those which assess process, beyond measures of time or tr i a l s to solution. The results of this study suggest, however, that a great deal of meaningful information may be gained concerning problem solving i f the major emphasis is placed on the examination of thepprocesses the 114 child uses as opposed to his a b i l i t y to generate acceptable solutions only. Limitations of This Study Over a succession of v i s i t s to each of the three classrooms during 1the course of this research, i t became very apparent to this writer that the children had been exposed to very different teaching styles during the preceding eight months of the school year. The teacher of the class exposed to the training program, who had many more years of teaching experience than the teachers of the two control groups, consist-ently employed a very teacher-centered, authoritarian teaching style. Children were required to remain non-participative during lesson presentation, and work s i l e n t l y through assigned seat-word with no discussion permitted among peets. The contrast between the atmosphere in this classroom and that in the remaining two was very marked. Both of the teachers of the control classes were observed to rely heavily on a student—centered fcrm of instruction which emphasized participation on the part of the children. Students were frequently observed working in small groups on several different aspects of some topic the teachers had presented, or in presenting oral reports of projects to their classmates while the teacher sat at the back of the classroom and observed. In both classes, these teachers created an. atmosphere which encouraged students to participate, ask questions, and searchoout information both independently and in small groups. These differences between classrooms could be interpreted to imply that the control groups were being trained by their teachers in the use 115 of the types of problem-solving s k i l l s this study was designed to examine. The trained group, on the other hand, had been exposed to a teaching style that tended to minimize independent inquiry behavior. While this additional information tends to increase the significance of the affect of the training program, i t also makes i t apparent that care should have been taken, prior to beginning the research, to determine that a l l classrooms involved were similar not only in student character-i s t i c s , but also on this teaching style variable. It was noted previously that the number of solutions provided by the students bore no s t a t i s t i c a l relationship to any of the independent variables, or to the processes employed by students in problem-solving. The scoring procedure used for enumeration of solutions may have been inadequate. Any response the child indicate to be a solution by turning over the "READY" card was accepted by the researcher, unless i t was stated as a question. These solutions differed considerably in quality and complexity, but a l l were given an equal score of one. In retrospect, i t appears that a more stringent c r i t e r i a should have been used. One way this could be accomplished would be to have an independent panel of judges rate solutions on a four or five-point scale in terms of quality and complexity. The use of such a procedure would result in more meaningful data being derived with regard to the relationship between process and product in problem-solving. This study found that conceptual style did not account for a significant amount of the variance in any of the problem-solving measures. The fifteen items of Denney's (1971) Form A version of the CST were used to measure analytic a b i l i t y , resulting i n scores which ranged from 0 to 15. 116 Kagan's (1963) earlier work with, cognitive style employed a thirty-item CST, thus allowing for a greater range of scores. Had the range of possible CST scores in the present study been increased to thirty by using both Form A and Form B of Denney's CST, the predictive power of this style variable may have been greater due to an increase in the a b i l i t y of the CST to discriminate more finely between levels of analytic a b i l i t y . The post-test problems and many of the problems used in the training materials a l l came from the same source, The Productive Thinking Programme (Covington, Crutchfield, Davis and Olton, 1972). A l l of these problems involved a "mystery or crime to be solved" format. The similarity between the post-test problems and a portion of the training materials may have accounted for the superiority of the training group in solving only this one specific class of problems. Since content may act to moderate the effects of other variables, the generalizability of the results is limited to problems having similar structure. While Covington and Crutchfield (1965) contend that the problem solving s k i l l s acquired through working with their programmed instructional materials have a great degree of generalizability, the possibility that this would also occur with the present training materials cannot be assessed without further research. One further limitation is the short term nature of this study. The differences that were found to exist for the training group immediately following the training period may not have been observed over a longer period of time. The novelty of the use of the programmed instructional materials by the training class that was accustomed to a 117 teacher-centered style of instruction may also have affected the results. A further limitation of the results of this study derived from problems associated with the use of regression analysis. One such problem is the absence of well defined procedures for controlling the overall error rate when (a) one is working with correlated dependent variables, and (b) there i s overlap in variance attributed to independent variables which, are not truly independent of each other, even though attempts are made to partial out the variance by controlling the order of entry of terms into the regression model. Thus, the r i s t of a Type I error cannot be precisely estimated, imposing some limitations on the confidence with which "significant" results can be interpreted. Secondly, to examine interactions which are found to be significant in regression analysis i s not possible without the a r t i f i c i a l levelling of continuous variables. The present "state of the art" of regression analysis suggests no alternatives, however. In studies such as the present one, in which interactions among continuous variables are of importance, this s t a t i s t i c a l restriction could impose limitations on the information that can be obtained from the data. Recommendations for Further Research Several areas which require further research have been mentioned throughout this discussion. A systematic analysis of the interaction between cognitive style and conceptual tempo that was found to exist in this study is warranted. Examination of the effectiveness of modified versions of the training program with impulsive, non-analytic children in needed. The transferability of the effects of training over both time and 118 problem types could prove to be a f r u i t f u l area of research. Finally, in the present study, a reasonably large number of predictor variables were examined as they relate to problem-solving. The results of the regression analyses demonstrate that the total amount of variance accounted for by the sum of these variables never exceeded fifty-one percent, with the average being approximately thirty-five percent. It is clear that some additional.factors, or combination of factors must relate significantly to problem-solving. Other learner variables should be examined, such as achievement motivation, verbal aptitude, and. deductive reasoning a b i l i t y to attempt to account for thella^ge amount of error variance. Final Comment This study was not intended, nor has i t succeeded in providing decisive answers to the question of the effects of cognitive style and conceptual tempo on problem solving processes, or to the optimal procedure for training a l l children to be effective problem-solvers. However, i t has served to highlight several issues of important to educators whose goal i t is to f a c i l i t a t e the optimization of performance of every child. F i r s t , children of similar levels of ab i l i t y w i l l differ in their effectiveness as problem-solvers, and these differences w i l l in part be due to the degree to whibht'they are analytic and reflective. Second, the processes used by a child in problem-solving are subject to modification, and this modification can come about by short-term training in the form of programmed instruction. 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Child Development, 1968, 39, 27-34. 129 APPENDICES r 130 APPENDIX A THE TEST PROBLEMS AND C L A S S I F I C A T I O N OF QUESTION TYPES P r o b l e m 1 ........... .131 P r o b l e m 2 . . 1 3 3 P r o b l e m 3 • 135 P r o b l e m 4 I 3 7 131 Escape in an Elevator One night a small boy captured the secret plans of a gang of spies. When he got the plans, he was to go to the roof of a nearby building, where a helicopter would be waiting to help him escape from the gang. He ran to the building as fast as he could and jumped into the elevator. The gang ran into the building just as the elevator door closed. Hoping they could head the boy off, they started to run up .the stairs. Even though the elevator was the fastest way to the roof, the boy got out at the third floor and ran the rest of the way up the stairs to the roof! Shown at the right is a picture of the eight-story building. It shows the eleva-tor stopped at the third floor, the boy running up the stairs with the gang after him, and the helicopter waiting on the roof. Your problem is to figure out why the boy got out at the third floor and ran up the stairs to the roof instead of taking the elevator all the way. 132 Sample Questions Problem 1 "Escape in an Elevator" In f o rma t ion—S eekin ^ Questions: How many spies were there? (Three.) How old was the boy? (About your age.) Is this a door on the roof? (Yes.) I • ,-How t a l l was the building? (Eight stories high.) Constraint-Seeking Questions: Was there a power failure at the time of the escape? (No.) Was anything wrong with the elevator? (No.) Was the boy a very fast runner? (No, about average.) Is this the kind of elevator that stops at every floor? (Only i f you press the button for that floor.) Hypothesis-Seeking Questions: Why did he get out at the third floor? (That's what I want you to guess.) How did he escape from the gang? (That's what I want you to guess.) Did the elevator break down? (Are you ready to make that a guess?) Did he think i t was faster to run up the stairs? (Are you ready to make that a guess?) The Mystery of the Stolen Jewel 133 Mrs. Mnney, a r i c h widow, had a birthday party l a s t night. Three peopl<=» were at. the partv. The f i r s t two ppople w^re Mr. and Mrs. Smith. Mr. Smith owned the pet store i n town. In the l a s t few years, he was not able to s e l l many pets, so he needed a large sum of money to keep the pet store running. The other person at the partv was Mr. Lion, the famous A f r i c a n hunter. Mr. Lion did not l i k e Mrs. Money because t^o years ago she had not loaned him enough money to go on a hunting t r i p . Mr. Lion decided to come to the party because he l i k e d Mr. and Mrs. Smith. A l l three guests brought presents f o r Mrs. Money. As Mrs. Money was opening the presents, the l i g h t s i n the l i v i n g room went out. There was a scream. Twenty seconds l a t e r , Mrs. Money found one of the l i g h t switches. The only thing missing from the room was the jewel which had been around Mrs. Money's neck. The p o l i c e , who were i n v e s t i g a t i n g the mystery, could not f i n d the jewel i n the room. They looked everywhere. They even looked outside around the only open window i n the l i v i n g room. The ground was muddy and there were no f o o t p r i n t s outside. No one had hidden the jewel i n t h e i r clothes. Your f i r s t problem i s to f i n d out how the jewel had been removed„from the l i v i n g room when the l i g h t s were out. Work on t h i s problem f i r s t . There were 4 things found i n the room which might be clues to solvine t h i s . They were: 1. A rose from Mrs. Smith's present 2. A small box 3 . A feather 4. Mr. Smith's umbrella which had been opened 134 Sample Questions Problem 2 "The Mystery of the Stolen Jewel" Information-Seeking Questions: Who screamed? (Mrs. Money) Whose umbrella was it? (Mr. Smith's) How big was theijewel? (About the size of a peanut) What color was the feather? (Grey) Constraint-Seeking Questions: Was the box a present? (No.) Was something in the box? (Yes.) Why are there holes in the box? (To let air in.) Could the umbrella handle come off? (No, i t was nailed in place.) Hypothesis-Seeking Questions: Was the jewel in the rose (box, umbrella)? (Are you ready to make that a guess?) How could the thief get out? (That's what I want you to try to guess.) What is the feather for? (That's what I want you to try to guess.) Was Mr. Smith the thief? (Are you ready to make that a guess?) The tracks i n the Snow 135 After a snowstorm, a man entered a small clearing surrounded by trees. In the clearing he noticed these three d i f f e r e n t kinds of animal tracks: B C Most of these tracks led to a small pond i n the center of the clearing where the ice had melted. Here i s a picture of what the man saw: The problem i s to explain what happened i n the clearing to cause such a puzzling pattern of tracks. 136 Sample Questions Problem 3 "The Tracks in the Snow" Information-Seeking Questions: How big is the pond? (About as big as this room.) What kind of tracks are these? (A-racoon, B-deer, C-bear.) What time of day was it? (About lunch time.) What, direction do the tracks go? (Which one do you want to know about? Specifies direction by pointing.) Constraint-Seeking Questions: Can racoons (bears, deer) swim? (Yes, i f they have to._)_ Are the tracks man-made ? (No.) Were there any shovel marks around? (No.) Did any of the animals die? (Yes.) Hypothesis-Seeking Questions: How come there are no deer tracks going away? (That's what I want you to guess)) What happened in the clearing? (That's what I want you to guess.) How did the bear get over here? (That's what I want you to guess.) How come the racoon went straight across? (That's what I want you to guess). 137 The Man in the Pit A man, wearing only his hiking shorts, gets lost during a summer hike and falls into a pit. The pit is circular. It is about thirty feet across and twenty feet deep. The walls of the pit are of smooth, hard stone and rise straight up. There are no handholds or footholds for climbing. The bottom of the pit is also hard stone. A narrow stream of water flows over the edge of the pit and runs straight down the wall; the water then disappears into a small hole in the floor of the pit. The pit is completely empty except for three things: • Exactly in the middle of the pit is the stump of a dead tree sticking straight up. • Near the tree stump is a loose flat rock. • Two boards are also lying near the tree stump. Your problem is to figure out how the man escapes from the pit. 1 Sample Questions Problem 4 "The Man in the P i t " Information-Seeking Questions: How deep is the pit? (Twenty feet.) How wide i s the pit? (Thirty feet.) How big is the rock? (About the size of this tape recorder.) How big is the hole? (About half the size of this tape recorder.) Constraint-Seeking Questions: Can the man l i f e the rock? (Yes.) How long would i t take the pit to f i l l with water i f the hole was plugged? (About an hour.) Will the log float? (Yes.) Is there a ranged station near by? (No.) Hypothesis-Seeking Questions: How did he get out? (That's what I want you to guess.) How can he fasten the two boards together? (That's what I want you to guess.) Why doesn't he just c a l l for help? (Are you ready to make that a guess Does he use the rock to make foot holes? (Are you ready to make that a guess?) 139 APPENDIX B COGNITIVE STYLE TEST I n s t r u c t i o n s .140 Sample Item 141 MATCHING FAMILIAR FIGURES TEST I n s t r u c t i o n s 142 Sample Item. .143 140 COGNITIVE STYLE TEST INSTRUCTIONS "On each page i n t h i s booklet you w i l l see three p i c t u r e s . Below each p i c t u r e i s a l e t t e r , e i t h e r A, B, or C. I want you to look very c a r e f u l l y at the three p i c t u r e s . Then choose two of the p i c t u r e s that you think are a l i k e or go together i n some way. When you have made your choice of two p i c t u r e s , c i r c l e the l e t t e r s under the two that you have chosen. Then turn the page and do the same t h i n g with the next set of p i c t u r e s . Keep going u n t i l you have chosen two p i c t u r e s on each page. Take as much time as you want. There i s no need to hurry. Let's t r y the f i r s t one f o r p r a c t i c e to make sure you know what to 141 142 MATCHING FAMILIAR FIGURES TEST INSTRUCTIONS "I am going to show you some p i c t u r e s , l i k e these ones. Look very c a r e f u l l y at t h i s p i c t u r e that i s by i t s e l f . One of the s i x p i c t u r e s down here i s e x a c t l y the same as the one at the top. Your job i s to f i n d the p i c t u r e down'here that i s e x a c t l y the same as the one at the top. Take as much time as you want, and look at a l l the p i c t u r e s very c a r e f u l l y . When you have found the one that i s e x a c t l y the same as the one at the top, point to i t so that I can see which one you have chosen. Let's t r y t h i s one f o r p r a c t i c e to make sure that you know what to do." 143 145 A P P E N D I X C T A S K A N A L Y S I S O F P R O B L E M S O L V I N G 146 T H E T R A I N I N G P R O G R A M B o o k l e t 1, S a m p l e E x e r c i s e s . . . ...147 B o o k l e t 2, S a m p l e E x e r c i s e s 149 B o o k l e t 3, S a m p l e E x e r c i s e s 151 B o o k l e t 4, S a m p l e E x e r c i s e s 153 B o o k l e t 5, S a m p l e E x e r c i s e s . 155 B o o k l e t 6, S a m p l e E x e r c i s e s . . . . . .157 146 Task Analysis of Problem Solving Overall Objective: On completion of the program, the student w i l l use analytic and reflective strategies in problem solving: i.e. w i l l look at and think about a l l stimulus elements in the problem material presented, w i l l ask constraint-seeking questions to aid in identification of relevant cues-, w i l l ask. information-seeking questions when information is Inadequate, w i l l offer several solutions, and w i l l take the time to think about and give reasons for selecting his f i n a l solution among the several other possible solutions he has hypothesized. Task 1: The student w i l l be able to visually analyze the elements in the stimulus population to such a degree that the parts as well as the whole are identified and discriminated. Task 2: The student w i l l be able to identify and select those elements in the stimulus display presented to him that w i l l assist him in solving a given problem through the use of constraint-seeking questions. Task 3: The student w i l l be able to identify where information is Inadequate in a presented problem and w i l l ask information-seeking questions to obtain that needed for problem solution. Task 4: The student, when presented a problem, w i l l be able to provide more than one possible hypothesis for solution. Task 5: When more than one solution i s provided, the student w i l l think about and be able to provide reasons for the relative merits of each solution. Task 6: When given a problem, the student w i l l be able to provide more than one solution, w i l l think about each solution, and w i l l select one solution which he thinks best f i t s the facts, and w i l l be able to provide reasons for his selection of that one solution over his other possible solutions. 147 Page 10. Our eyes help us to find out what i s happening i n the world around us. But we have to remember to look carefully or we may not notice some things. People like scientists and detectives who solve problems a l l the time are always on the lookout for clues that w i l l help them. They have learned to look carefully at what i s i n the world around them. They know that i f you look at something too quickly you may miss some information. Here are some exercises that w i l l show you how impor-tant i t i s to take a good look at what i s i n the world around you. Look very carefully at this picture. There are > things to be seen i n i t . Can you find 3 things? Write down what you see i n this picture. Then turn the page. 148 P a g e 2 3 . I a m t h i n k i n g o f o n e o f t h e s e f l o w e r s . H e r e a r e t h e c l u e s : I t h a s f i v e l a r g e p e t a l s . I t h a s a l o n g s t e m . I t h a s t h r e e s m o o t h l e a v e s . H e r e a r e t h e s t e p s t o f o l l o w t h a t w i l l h e l p y o u t o f i n d t h e f l o w e r . 1 . C r o s s o f f a n y f l o w e r t h a t d o e s n o t h a v e 5 l a r g e p e t a l s . 2. C r o s s o f f a n y f l o w e r t h a t d o e s n o t h a v e a l o n g s t e m . 3. C r o s s o f f a n y f l o w e r t h a t d o e s n o t h a v e 3 s m o o t h l e a v e s . N o w y o u s h o u l d b e l e f t w i t h o n l y o n e f l o w e r . W h i c h o n e i s t h e c o r r e c t f l o w e r ? A B C D E F 149 Page 30. H e r e i s a p i c t u r e o f a b o a t . H i d d e n s o m e w h e r e i n t h i s b o a t i s t h e f i g u r e y o u j u s t d r e w . R e m e m b e r , t h e f i g u r e i s m a d e u p o f 2 p a r t s : a s m a l l t r i a n g l e / \ A l a r g e r e c t a n g l e I f y o u s t a r t b y l o o k i n g f o r o n e o f t h e s e s h a p e s , i t w i l l h w l p y o u f i n d t h e f i g u r e . T r y t o f i n d t h e t r i a n g l e s o m e w h e r e i n t h e b o a t . N o w s e e i f t h e r e i s a l a r g e r e c t a n g l e u n d e r n e a t h i t . W h e n y o u h a v e f o u n d t h e f i g u r e , g o o v e r i t w i t h y o u r p e n c i l . N o w t u r n t o t h e n e x t p a g e . 150 Page 43. Here i s another 20 questions game. I am thinking of one of the objects below. You can ask me questions, but I can only answer YES or NO. Find the object I am thinking of. You should only have to ask 3 questions. Look at the 8 pictures. Can you divide them into 2 big groups? Try to ask a question that w i l l l e t you cross off one half (4) of the pictures. What i s your f i r s t question? Write i t here and turn the page. C A T p»Wfc T U L I P € L . 6 P V 4 A N T 151 Page 6 6 , L o o k a t t h i s m a p . Y o u a r e i n t h e t o w n o f R i c e a n d y o u w a n t t o g o t o t h e t o w n o f E l d o n b y t h e q u i c k e s t r o u t e . T o f i n d t h e a n s w e r t o t h i s p r o b l e m y o u w i l l n e e d t o a s k s o m e q u e s t i o n s t o g e t a l l t h e n e c e s s a r y i n f o r m a t i o n . S t u d y t h e m a p c a r e f u l l y . T h e r e a r e s e v e r a l d i f f e r e n t r o u t e s y o u c a n u s e t o g e t f r o m t h e t o w n o f R i c e t o t h e t o w n o f E l d o n . O n e r o u t e i s m a r k e d o n t h e m a p . RICE T o f i n d o u t i f t h i s i s t h e q u i c k e s t r o u t e y o u w o u l d n e e d t o a s k t h e s e f o u r q u e s t i o n s : 1 . H o w f a r i s i t f r o m R i c e t o B o w R o a d a l o n g H i g h w a y 1 0 ? 2 . W h a t i s t h e s p e e d l i m i t a l o n g h i g h w a y 1 0 ? 3 . H o w f a r I s i t a l o n e B o w R o a d t o t h e H i g h w a y 1 0  t u r n - o f f t o E l d o n ? 4 . W h a t i s t h e s p e e d l i m i t a l o n g B o w R o a d ? W i t h t h e a n s w e r s t o t h e s e 4 q u e s t i o n s y o u c o u l d f i g u r e o u t h o w l o n g i t w o u l d t a k e t o g e t f r o m R i c e t o E l d o n a l o n g t h i s r o u t e . N o w t u r n t o t h e n e x t p a g e . ! 152 P a g e 7 1 . A s k i n g t h e r i g h t q u e s t i o n s h e l p s y o u t o g e t t h e f a c t s y o u n e e d t o s o l v e a p r o b l e m . L e t ' s s e e i f y o u c a n a s k t h e n e c e s s a r y q u e s t i o n s t o s o l v e t h i s p r o b l e m . T w o g o o d f r i e n d s h a v e d e c i d e d t h a t t h e y w o u l d l i k e t o g o t o s e e a m o v i e o n S a t u r d a y a f t e r n o o n . T h e y w o u l d e a c h l i k e t o b u y s o m e p o p c o r n t o e a t w h i l e t h e y a r e t h e r e . T o e a r n e n o u g h m o n e y f o r t h e m o v i e a n d t h e p o p c o r n , t h e p a r e n t s o f t h e 2 f r i e n d s s a i d t h a t t h e y c o u l d c o l l e c t a l l t h e p o p b o t t l e s t h a t a r e i n t h e i r h o u s e s a n d c a s h t h e m i n a t t h e g r o c e r y s t o r e . T h e y c a n k e e p a l l t h e m o n e y t h e y g e t a n d u s e i t f o r t h e m o v i e a n d p o p c o r n . T h e f r i e n d s w i l l t h e m p u t a l l t h e m o n e y t o g e t h e r a n d s p l i t i t e v e n l y b e t w e e n t h e m . W i n t . f r e y h a v e e n o u g h : m o n e y ? W r i t e a l l t h e q u e s t i o n s y o u w i l l n e e d t o a s k t o s o l v e t h i s p r o b l e m . W h e n y o u h a v e f i n i s h e d , t u r n t o t h e n e x t p a g e . l . _ _ 2 . 153 Page 8 5 . T h e I d e a T r e e W h e n y o u a r e t r y i n g t o s o l v e a p r o b l e m i t i s i m p r o t a n t t o f i r s t t h i n k o f a l l t h e d i f f e r e n t i d e a s y o u c a n t o s o l v e t h e p r o b l e m . T h e n i f l a t e r y o u f i n d t h a t o n e o f y o u r i d e a s d o e s n o t s o l v e t h e p r o b l e m y o u s t i l l h a v e o t h e r s t o w o r k o n a n d a r e n o t l e f t w i t h o u t a n y s o l u t i o n s a t a l l . I t w i l l h e l p y o u t o t h i n k u p m a n y i d e a s i f y o u a r e p l a n f u l . O n e g u i d e t o b e i n g p l a n f u l i s t o t h i n k u p g e n e r a l i d e a s f i r s t , a n d t h e n t o t h i n k o f p a r t i c u l a r i d e a s n e x t . F i r s t , y o u t h i n k o f b r o a d g e n e r a l p o s s i b i l i t i e s t o s o l v e t h e p r o b l e m . T h e n y o u e x p l o r e e a c h o n e o f t h e s e g e n e r a l p o s -s i b i l i t i e s f o r a l l t h e p a r t i c u l a r i d e a s t h a t i t m i g h t s u g g e s t . T h i s g i v e s y o u a p l a n , w h i c h h e l p s i n y o u r s e a r c h f o r a l l t h e p o s s i b l e i d e a s . H o w t h i s g u i d e w o r k s c a n b e s h o w n i n a n o t h e r w a y , t o o . I t i s s o m e t h i n g l i k e a t r e e - a n i d e a t r e e . T h e f e w b i g l i m b s o n t h e t r e e r e p r e s e n t t h e b o r a d , g e n e r a l p o s s i b i l i t i e s . T h e s m a l l b r a n c h e s o n e a c h l i m b s t a n d f o r t h e p a r t i c u l a r i d e a s g r o w i n g o u t o f e a c h o f t h e g e n e r a l p o s s i b i l i t i e s . Y o u w i l l n o w l e a r n t o u s e t h i s g u i d e i n a p l a n f u l s e a r c h f o r s o l u t i o n s t o d i f f e r e n t p r o b l e m s . T o b e g i n , y o u w i l l n e e d t o b e a b l e t o r e c o g n i z e t h e d i f f e r e n c e b e t w e e n a G e n e r a l P o s s i b i l i t y a n d a P a r t i c u l a r I d e a t h a t b r a n c h e s o f f f r o m i t . L e t ' s s t a r t w i t h s o m e e x a m p l e s . T u r n t o t h e n e x t p a g e . 154 P a g e 8 6 . S a v i n g t h e W h o o p i n g C r a n e T o d a y t h e w h o o p i n g c r a n e , a h u g e w h i t e b i r d n a t i v e t o N o r t h A m e r i c a i s i n d a n g e r o f d y i n g o u t , p a r t l y b e c a u s e o f m a n ' s c a r e l e s s n e s s . A s m a n c o n q u e r e d t h e w i l d e r n e s s , h e d e s t r o y e d m a n y o f t h e c r a n e ' s n a t u r a l f e e d i n g a n d n e s t i n g g r o u n d s . A f e w y e a r s a g o s c i e n t i s t s b e g a n t o e x p l o r e w a y s o f s a v i n g t h e c r a n e . S i n c e s o f e w b i r d s w e r e l e f t , n o i d e a c o u l d b e o v e r l o o k e d . T h u s , t h e s c i e n t i s t s ' s e a r c h c a l l e d f o r a p l a n f u l a p p r o a c h , w h i c h l e d t h e m t o t w o g e n e r a l p o s s i b i l i t i e s f o r s a v i n g t h e c r a n e s . G e n e r a l P o s s i b i l i t y A : F i n d w a y s t o i n c r e a s e t h e n u m b e r o f c r a n e s b o r n e a c h y e a r . G e n e r a l P o s s i b i l i t y B : F i n d w a y s t o d e c r e a s e t h e n u m b e r o f c r a n e s k i l l e d e a c h y e a r . N e x t , t h e s c i e n t i s t s t o o k e a c h G e n e r a l P o s s b i l i t y a n d e x p l o r e d i t c a r e f u l l y t o s e e w h a t P a r t i c u l a r I d e a s i t m i g h t s u g g e s t . B e l o w a r e 5 p a r t i c u l a r i d e a s f o r s a v i n g t h e w h o o p i n g c r a n e . D e c i d e w h i c h o f t h e s e i d e a s b e l o n g s u n d e r e a c h g e n e r a l p o s s i b i l i t y . P u t e i t h e r A o r B i n f r o n t o f e a c h p a r t i c u l a r i d e a t o 3how w h i c h g e n e r a l p o s s i b i l i t y i t b e l o n g s t o . T h e f i r s t o n e i s d o n e f o r y o u . W h e n y o u h a v e f i n i s h e d , t u r n t o t h e n e x t p a g e . B P a s s l a w s a g a i n s t h u n t i n g w h o o p i n g c r a n e s -K e e p p l a n e s f r o m f l y i n g t o o n e a r t h e n e s t i n g a r e a s s o t h a t t h e c r a n e s w i l l n o t b e f r i g h t e n e d a w a y f r o m t h e i r e g g s . P r o v i d e f o o d i n w i n t e r s o t h a t t h e ^ c r a n e s w i l l n o t n e e d t o f l y t h o u s a n d s o f d a n g e r o u s m i l e s s o u t h . H a t c h e x t r a e g g s i n a l a b o r a t o r y s o t h a t t h e c h i c k s w i l l h a v e a b e t t e r c h a n c e t o l i v e . S e t t r a p s t o c a t c h t h e c r a n e ' s n a t u r a l e n e m i e s s u c h a s t h e w e a s e l . 155 P a g e 106 T h e T u n n e l P r o b l e m A l a r g e t r u c k l o a d e d w i t h b o x e s f u l l o f h e a v y m a c h i n e r y i s d r i v i n g t h r o u g h t h e m o u n t a i n s . T h e d r i v e r c o m e s t o a 200 f o o t l o n g t u n n e l i n t h e r o a d . H e f i n d s t h a t h i s t r u c k i s _ i n c h t o o t a l l t o p a s s t h r o u g h t h e t u n n e l . T h e r e i s a s e r v i c e s t a t i o n a t t h e o t h e r e n d o f t h e t u n n e l . T h e d r i v e r s t o p s t h e t r u c k a n d t r i e s t o f i g u r e o u t h o w h e c a n g e t t h r o u g h t h e t u n n e l . H e d o e s n ' t w a n t t o t u r n a r o u n d t o d r i v e 5 0 m i l e s b a c k t o w h e r e a n o t h e r r o a d w i l l t a k e h i m t h r o u g h t h e m o u n t a i n s , b e c a u s e h e i s i n a g r e a t h u r r y . H e c a n ' t f i g u r e o u t h o w h e c a n g e t t h r o u g h t h i s l o n g t u n n e l . S e v e r a l c a r s p u l l u p b e h i n d h i m , a n d t h e d r i v e r s s t a r t g i v i n g h i m s u g g e s t i o n s a b o u t h o w h e c a n s o l v e h i s p r o b l e m . O n e d r i v e r s a y s : Y o u c a n s o l v e y o u r p r o b l e m b y t a k i n g s o m e o f t h e a i r  o u t o f v o u r t i r e s . T h i s w i l l m a k e t h e t r u c k 1 i n c h  l o w e r . T o d e c i d e i f t h i s i s a g o o d s o l u t i o n , y o u m u s t l o o k a t a l l P a g e 107. T o d e c i d e i f t h e s o l u t i o n o n p a g e 106 i s a g o o d o n e y o u m u s t c o n s i d e r t h e f a c t s . R e a d t h e s t o r y a g a i n a n d a n s w e r t h e s e q u e s t i o n s . 1 . H o w m u c h l o w e r m u s t t h e t r u c k b e t o f i t t h r o u g h t h e t u n n e l ? ; 2 . H o w w i l l t h e d r i v e r b e a b l e t o f i l l u p t h e t i r e s a g a i n w h e n h e g e t s t h r o u g h t h e t u n n e l ? F r o m t h e f a c t s t h a t y o u h a v e c o l l e c t e d , d o y o u t h i n k t h i s i s a g o o d s o l u t i o n t o t h e t r a c k d r i v e r * s p r o b l e m ? Y E S N O N o w t u r n t o t h e n e x t p a g e . 157 Page 125. H e r e i s y o u r f i r s t p r o b l e m . A b o y i s p l a y i n g a g a m e o f p i n g - p o n g i n t h e b a c k y a r d o f a f r i e n d ' s h o u s e . H e m i s s e s t h e b a l l a n d i t b o u n c e s a c r o s s t h e l a w n a n d r o l l s i n t o a s m a l l b u t d e e p h o l e . T h e h o l e b e n d s t o o n e s i d e . H e r e i s y o u r p r o b l e m : H o w d o e s t h e b o v g e t t h e b a l l o u t o f t h e h o l e ? T o f i n d t h e s o l u t i o n t o t h i s p r o b l e m , y o u w i l l h a v e t o a s k q u e s t i o n s f i r s t t o g e t a l l t h e f a c t s . W r i t e a l l t h e Q u e s t i o n s y o u w i l l h a v e t o a s k , t h e n t u r n t o t h e n e x t p a g e . 158 Page 126. Here i s a l i s t of questions that you w i l l need to ask to solve this problem, and the answers to these questions. Question 1 . How deep i s the hole? 2. How big around i s the hole? 3 . How long i s the boy's arm? 4. How big around i s the boy's arm? 5. Are these any tools i n the yard that could be used by the boy? 6. Is i t alright i f the boy digs up the lawn? 7 . W i l l the stick f i t down the hole and around the bend? Answer 26 inches 6 inches 20 inches 3 inches Yes. There i s a shovel, a long stick, and a water hose. No. The friend's mother says that the boy i s not to dig up the lawn. No, a stick w i l l not pass by the bend i n the hole. You have a l o t more information that w i l l help you to solve this problem. Read the problem again, and pay careful attention to the l i s t of tools that the boy could use. How  does he get the b a l l out of the hole? Write your solution below, then turn the page. The boy gets the b a l l out of the hole by_ APPENDIX D SUMMARY TABLES OF RESULTS OF REGRESSION ANALYSES Mean T o t a l Time..... 160 Mean P r i o r Time.... .-161 Mean T o t a l Questions... ..162 Mean P r i o r Questions 163 Mean Information-Seeking Questions 164 Mean C o n s t r a i n t - S e e k i n g Questions 165 Mean Hypothesis-Seeking Questions 166 T o t a l S o l u t i o n s 167 Mean R e s i d u a l Time .....168 Mean R e s i d u a l Questions 169 APPENDIX D • RESULTS OF REGRESSION ANALYSIS FOR MEAN TOTAL TIME (Yl) SOURCE OF VARIATION AR 2 df F , o b s . P< X l . 0533 1 6.5000 . 01 (x 2 ,x 3 ,x 4 ) * . 0881 3 3.5813 . 01 X5 . 0002 1 . 0244 . 84 X6 . 0003 1 . 0365' .82 X7 . 0308 1 3 .7560 . 05 X8 .0279 1 3.4024 . 06 X 9 .0594 . 1 7.2439 . 009 X10 . 0012 1 . 1463 . 70 X 6 X 7 . 1296 1 15.8048 .0003 X 6 X 8 . 0009 • 1 . 1097 .73 X 7 X 8 . 0228 1 2.7805 .09 ( x 5 x 9 , x 5 x 1 0 ) * . 0044 2 .2683 .76 X 6 X 9 . 0024 1 .2926 .59 X 6 X 1 0 . 0230 1 2 .8048 .09 X 7 X 9 . 0432 1 5 . 2682 . 02 X 7 X 1 0 . 0001 1 . 0121 .87 X 8 X 9 . 0162 1 1.9756 .16 x g x 1 0 . 0000 1 . 0000 .95 TOTAL . 5038 21 ERROR . 4962 59 * I n d i c a t e s that terms i n parentheses were entered as a s e t . 161 A P P E N D I X D R E S U L T S O F R E G R E S S I O N A N A L Y S I S F O R M E A N P R I O R T I M E ( Y 2 ) S O U R C E O F V A R I A T I O N A R 2 d f F , o b s . P < X l . 0165 1 1. 7 0 1 0 .19 ( X 2 j j X ^ ) . 0880 3 3.0206 .03 X 5 . 0004 1 . 0412 . 82 X 6 . 0 0 7 1 1 .7319 . 40 X 7 . 0572 1 5 .8969 . 01 X 8 . 0068 1 .7010 . 41 X 9 . 1008 1 10 .39.18 . 002 X 1 0 . 017 9 1 1.8454 . 17 X 6 X 7 . 0166 1 1.7113 . 19 X 6 X 8 . 0 0 2 1 1 .2164 .64 X 7 X 8 . 0603 1 6.2164 . 01 ( X ^ X g , X ^ X ^ Q ) . 0 1 4 1 2 .7268 .49 X 6 X 9 . 0098 1 1.0103 .32 X 6 X 1 0 . 0003 1 . 0309 . 83 X 7 X 9 . 0041 1 . 4 2'2 6 .52 X 7 X 1 0 . 0043 1 .4432 . 51 X 8 X 9 . 0004 1 . 0 4 1 2 .82 X 8 X 1 0 .0100 1 1.0309 . 31 T O T A L .4166 21 E R R O R . 5834 59 162 APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR MEAN TOTAL QUESTIONS (Y3) SOURCE OF VARIATION A R 2 df F , obs . P< X l . 0262 1 2 . 5940 .10 (^ 2> Xg»X^) . 1269 3 4 . 1881 . 009 X 5 . 0010 1 . 0990 . 74 X6 . 0170 1 1.6831 . 19 X7 . 0013 1 . 1287 . 71 X8 . 0157 1 1. 5544 . 21 X 9 . 0486 1 4 .8119 . 03 X10 . 0083 1 .8218 .37 X 6 X 7 4:05 8 9 1 5 .8316 . 01 X 6 X 8 . 0121 1 1.1980 .27 X 7 X 8 . 0070 1 . 6930 .41 (X^Xg , X^X^ Q ) . 0176 2 .8713 .42 X 6 X 9 . 0378 1 3.7425 .05 X 6 X 1 0 . 0043 1 .4257 .52 X 7 X 9 . 0003 1 . 0297 .84 X 7 X 1 0 . 0010 1 . 0990 .74 X 8 X 9 . 0035 1 . 3465 . 56 X 8 X 1 0 .0012 1 . 1188 .72 TOTAL . 3887 21 ERROR .6113 59 APPENDIX D RESULTS OF REGRESSION ANALYSIS (Y4) FOR MEAN PRIOR QUESTIONS SOURCE OF VARIATION A R 2 df F , ob s . P < X l . 0190 1 1.8269 . 17 (X2 »X^,X^) . 1428 3 4 .2532 . 008 X 5 . 0030 1 . 2884 .59 X 6 . 0022 1 .2115 .65 X7 . 0364 1 3 .5000 . 06 X8 . 0236 1 2.2692 . 13 X 9 . 0286 1 2.7500 . 09 X10 .0568 1 5.4615 .02 X 6 X 7 . 0056 1 .5384 .47 X 6 X 8 . 0067 1 . 6442 . 43 X 7 X 8 . 0455 1 4.3750 . 03 ( x 5 x g , x 5 x 1 0 ) . 0058 2 .2788 .76 X 6 X 9 . 0041 1 .3942 • 53 X 6 X 1 0 -.0004 1 . 0384 .82 X 7 X 9 . 0021 1 . 2019 . 65 X 7 X 1 0 . 0006 1 . 0576 .79 X 8 X 9 . 0012 1 . 1153 .73 X 8 X 1 0 , . 0001 1 . 0096 .88 TOTAL .3744 21 ERROR . 6256 59 164 APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR MEAN INFORMATION-SEEKING QUESTIONS (Y5) SOURCE OF VARIATION A R 2 df F , o b s . P< X l . 0310 1 2.9807 . 08 s >X^) . 0866 3 2.7756 . 04 X5 . 0058 1 . 5576 . 46 X6 . 0204 1 1.9613 . 16 X7 . 0041 1 .3942 .53 X8 . 0043 1 .4134 .53 X 9 . 0303 1 2 .9135 . 08 X10 . 0109 1 1. 0480 .31 X 6 X 7 .0578 1 5 . 5576 . 02 X 6 X 8 . 0077 1 .7403 .39 X 7 X 8 . 0114 1 1.0961 .30 : X 5 X 9 ' X 5 X 1 0 ) . 0214 2 1.0288 .36 X 6 X 9 . 0673 1 6.4711 . 01 X 6 X 1 0 . 0108 1 1.0384 . 31 X 7 X 9 • . 0013 1 .1250 .72 X 7 X 1 0 . 0011 1 . 1057 .74 X 8 X 9 . 0015 1 . 1442 .70 X 8 X 1 0 . 0015 1 . 1442 .70 TOTAL .3753 21 ERROR .6247 59 165 APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR MEAN CONSTRAINT-SEEKING QUESTIONS (Y6). SOURCE OF VARIATION A R 2 df F , O D S . P < X l .0219 1 2.1262 . 14 (^ 2 >X^  >X^) . 1046 3 3.3851 . 02 X5 . 0088 1 .8543 . 36 X 6 . 0279 1 2.7087 . 10 X7 . 0001 1 . 0097 .88 X8 . 0408 1 3 .9611 .04 X 9 .0858 1 8.3301 . 005 X10 . 0090 1 .8738 .35 X 6 X 7 . 0218 1 2.1165 . 14 X 6 X 8 . 0155 1 1.5048 . 22 X 7 X 8 . 0005 1 . 0485 .81 ( x 5 x g , x 5 x 1 Q ) . 0158 2 .7670 . 47 X 6 X 9 . 0074 1 .7184 .40 X 6 X 1 0 . 0037 1 . 3592 . 55 X 7 X 9 . 0012 1 .1165 .73 X 7 X 1 0 . 0001 1 . 0097 . 88 X 8 X 9 . 0129 1 1.2524 . 26 X 8 X 1 0 .0009 1 . 0873 .76 TOTAL .3785 21 ERROR .6215 59 166. APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR MEAN HYPOTHESIS-SEEKING QUESTIONS (Y7). SOURCE OF VARIATION A R' df ob s (%2 ' X-j » X^ ) X^ x, X. X, X, X 10 X 6 X 7 X 6 X 8 X 7 X 8 (x 5 x g ,x 5 x 1 Q ) X 6 X 9 X 6 X 1 0 X 7 X 9 X 7 X 1 0 X 8 X 9 X 8 X 1 0 TOTAL ERROR . 0001 .2114 . 0022 . 0057 . 0001 . 0138 . 0178 . 0017 . 0358 . 0056 . 0002 . 0426 . 0019 . 0191 . 0034 . 0003 . 0005 . 0068 .3691 . 6309 1 3 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 59 .0097 6 .7115 .2095 . 5428 . 0097 1. 3142 1.6952 .1612 3 .4095 .5333 .0190 2 . 0285 .1890 1.8190 . 3238 .0285 . 0476 . 6476 88 0006 65 47 88 .25 , 19 .69 . 06 ,47 .86 . 13 .67 . 17 .57 .84 .81 . 42 167 APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR TOTAL SOLUTIONS (Y8) SOURCE OF VARIATION A R df ob s P< ( X 2 > x^»x^) x^ X. xi X, X 10 X 6 X 7 X 6 X 8 X 7 X 8 ( X 5 X 9 » X 5 X i o ^ X 6 X 9 X 6 X 1 0 X 7 X 9 X 7 X 1 0 X 8 X 9 X 8 X 1 0 TOTAL ERROR . 0321 . 0459 .0255 .0169 . 0083 . 0135 . 0001 . 0027 . 0005 . 0017 . 0056 . 0332 .0025 . 0022 . 0210 . 0020 . 0045 . 0143 . 2325 . 7675 1 3. 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 59 2 .5275 1.2047 2.0078 1.3307 . 6535 1.0629 . 0078 .2126 .0393 . 1338 . 4409 1.3070 . 1968 . 1732 1. 6535 . 1574 .3543 1.1259 11 31 15 25 42 30 89 ,65 ,82 ,71 , 51 .27 . 66 . 68 .20 .69 . 56 .29 A P P E N D I X D R E S U L T S OF R E G R E S S I O N A N A L Y S I S F O R M E A N R E S I D U A L T I M E ( Y 9 ) . 168 S O U R C E OF V A R I A T I O N A R df ob s P < X . ( X ^ , X ^ ) X^ X , X . X , X , X 10 X 6 X 7 X 6 X 8 X 7 X 8 ( X 5 X 9 » X 5 X 1 0 ^ X 6 X 9 X 6 X 1 0 X 7 X 9 X 7 X 1 0 X 8 X 9 X 8 X 1 0 T O T A L E R R O R . 0 2 9 4 . 0 3 4 2 . 0 0 0 0 . 0 0 6 3 . 0 0 0 8 . 0 6 2 4 . 0 0 2 6 . 0 1 8 4 . 1 0 1 7 . 0 0 0 0 . 0 0 0 0 . 0 2 4 9 . 0 1 6 1 . 0 3 4 4 . 0 3 6 6 . 0 0 1 2 . 0 2 5 6 . 0 0 5 9 . 4 0 0 4 . 5 9 9 6 1 3 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 21 59 2 . 9 6 9 6 1 . 1 4 8 1 . 0 0 0 0 . 6 3 6 3 . 0 8 0 8 6 . 3 0 3 0 . 2 6 2 6 1 . 8 5 8 6 1 0 . 2 7 2 7 . 0 0 0 0 . 0 0 0 0 1 . 2 5 7 6 1 . 6 2 6 2 3 . 4 7 4 7 3 . 6 9 6 9 . 1 2 1 2 2 . 5 8 5 8 . 5 9 5 9 08 33 95 43 76 01 . 6 1 • . 17 , 0 0 2 . 95 . 95 . 2 9 . 2 0 . 06 . 0 5 . 7 2 . 10 . 44 169 APPENDIX D RESULTS OF REGRESSION ANALYSIS FOR MEAN RESIDUAL QUESTIONS (Y10) . SOURCE OF VARIATION _ K 2 df F , ob s . X l . 0160 1 1.4953 . 22 9X^ > ) , 0725 3 2 .2585 . 08 X5 . 0000 1 . 0000 . 95 X6 . 0222 1 2.0747 . 15 X7 . 0081 1 .7570 • 39 X8 .0040 1 .3738 .55 X 9 . 0345 1 3.2243 . 07 X10 . 0022 1 .2056 .65 X 6 X 7 . 0821 1 7.6728 . 007 X 6 X 8 . 0089 1 . 8317 . 36 X 7 X 8 . 0015 1 . 1401 .70 ( X 5 X 9 ' X 5 X 1 0 ) . 0255 2 1.1916 . 31 X 6 X 9 . 0513 1 4.7943 . 03 X 6 X 1 0 . 0058 1 . 5420 .47 X 7 X 9 . 0036 1 .3364 .57 X 7 X 1 0 .0038 1 . 3551 .56 X 8 X 9 . 0119 1 1. 1121 .29 X 8 X 1 0 . 0017 1 . 1588 .69 TOTAL . 3558 21 ERROR .6442 59 170 APPENDIX E MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR TRAINED ( T l ) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF COGNITIVE STYLE (CST) 171 MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR TRAINED (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (MFF ERRORS).....172 171 APPENDIX E MEAN RESIDUALS OF MEAN RESIDUAL QUESTIONS FOR TRAINED ( T l ) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF COGNITIVE STYLE (CST). DEPENDENT VARIABLE CST T l T23 MEAN TOTAL QUESTIONS MEAN I.S. QUESTIONS M E A N R E S I D U A L Q U E S T I O N S L M H L M H L M H -1.5275 + .8078 + .6911 -1.2470 + .4676 + .6235 -1.0856 - .0019 + .6686 + .4729 + .0202 - .5506 + .3453 + .0886 r- . 5205 + .6281 - .0920 - .5189 172 APPENDIX E MEAN RESIDUALS OF MEAN RESIDUAL TIME FOR TRAINED (Tl) AND UNTRAINED (T23) SUBJECTS AT THREE LEVELS OF IMPULSIVITY (MFF ERRORS). DEPENDENT VARIABLE IMPULSIVITY T l T23 MEAN TOTAL TIME H -26.9955 + 16 . 3435 M +18.2701 - 6.3983 L +12.7831 -10 . 6031 MEAN RESIDUAL TIME H -16 .6180 +18 .9627 M +22.6722 - 8.7247 L - 1.0177 - 8.2443 

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