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Inquiry training in elementary mathematics as related to sixth grade pupils' ability to analyze and solve… Weinstein, Marian S. 1970

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INQUIRY TRAINING IN ELEMENTARY MATHEMATICS AS RELATED TO SIXTH GRADE PUPILS' ABILITY TO ANALYZE AND SOLVE PROBLEMS. by Marian S. Weinstein B.A., Adelphi University, 1968 A THESIS SUBMITTED IN THE REQUIREMENTS MASTER PARTIAL FULFILLMENT OF FOR THE DEGREE OF OF ARTS in the Department of Mathematics Education We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at t h e U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A b s t r a c t T h i s was a study to determine the e f f e c t s o f i n q u i r y t r a i n i n g i n elementary mathematics, i n both group and i n d i v i d u -a l i z e d s i t u a t i o n s . Three experimental c o n d i t i o n s were used. One group o f grade s i x students r e c e i v e d problems on area c a l c u l a t i o n and a r i t h m e t i c r a t e problems i n which not enough i n f o r m a t i o n was gi v e n , r e q u i r i n g them t o a c q u i r e more da t a by a s k i n g q u e s t i o n s on blank cards g i v e n t o them i n d i v i d u a l l y f o r t h a t purpose. They were then g i v e n prepared response cards i n r e p l y . A second group r e c e i v e d the same problems w i t h not enough i n f o r m a t i o n , b ut a l l q u e s t i o n s and a l l responses were made o r a l l y i n a classroom so t h a t a l l c l a s s members c o u l d s i m u l -t a n e o u s l y r e c e i v e the i n f o r m a t i o n which one stu d e n t requested. A t h i r d group r e c e i v e d the same problem sheets, b ut w i t h a l l necessary d a t a p r e s e n t , and no new i n f o r m a t i o n was g i v e n them. Two s h o r t t r a i n i n g s e s s i o n s were h e l d d e a l i n g w i t h area c a l -c u l a t i o n and s o l u t i o n o f a r i t h m e t i c r a t e problems to f a m i l i a r i z e the s u b j e c t s w i t h the nature o f the l e a r n i n g m a t e r i a l s . A t o t a l o f 64 s u b j e c t s p a r t i c i p a t e d i n a l l phases o f the experiment. A l l s u b j e c t s were g i v e n an IQ t e s t and a p r e t e s t , p a r a l l e l to the f i n a l c i r t e r i o n t e s t , c o n s i s t i n g o f problems o f both an a r i t h m e t i c and geometric nature, as w e l l as area c a l c u l a t i o n . The c r i t e r i o n t e s t , a t the c o n c l u s i o n o f the i n s t r u c t i o n a l p e r i o d , c o n s i s t e d o f the same three s u b d i v i s i o n s , i n c l u d i n g items b o t h s i m i l a r and d i s s i m i l a r t o the t r a i n i n g m a t e r i a l s . Each item on both the p r e t e s t and the c r i t e r i o n t e s t was answerable by "not enough i n f o r m a t i o n " as w e l l as by three numerical p o s s i b i l i t i e s . Both p r e t e s t and c r i t e r i o n t e s t s were marked twice, once to g i v e an a c t u a l s c o r e and once t o determine the c o r r e c t use o f the "not enough i n f o r m a t i o n " response. Each o f the th r e e s u b t e s t s - a r i t h m e t i c , geometric, and area c a l c u l a t i o n - was marked s e p a r a t e l y from the t o t a l t e s t , as w e l l , and they were then s c o r e d again t o c o n s i d e r them as t e s t s measuring the c o r r e c t use o f "not enough i n f o r m a t i o n " . Thus a t o t a l o f e i g h t s e p a r a t e c r i t e r i o n s c o r e s w i t h correspond-i n g p r e t e s t scores were i n v o l v e d . An a n a l y s i s o f c o v a r i a n c e was performed u s i n g the IQ and p r e t e s t s c o r e s as c o v a r i a t e s . The r e s u l t s i n d i c a t e d t h a t the group i n q u i r y t r a i n i n g approach was the most s u c c e s s f u l of the t h r e e approaches i n t h a t these students s c o r e d s i g n i f i -c a n t l y h i g h e r marks on.--fiver: o f the e i g h t s c o r e s . No s i g n i f i -c a n t d i f f e r e n c e s were found among the other three s c o r e s . i TABLE OF CONTENTS Page LIST OF TABLES i i i Chapter 1. THE PROBLEM 1 Background 1 The Problem 3 D e f i n i t i o n of Terms 4 J u s t i f i c a t i o n 4 Hypotheses To Be Tested 6 2. A REVIEW OF RELATED LITERATURE . 8 Introduction 8 General Opinions On Discovery and Inquiry . . . 8 Research of General Relevance To The Problem . . 11 Research of S p e c i f i c Relevance To The Problem. . 15 Summary of Research Findings 18 3. PROCEDURE AND DESIGN 20 Design 20 Subjects 21 Controls 22 Instruct i o n a l Materials 23 Measuring Instruments 24 I.Q. Scores 24 Pretest 24 C r i t e r i o n Test 25 i i Chapter Page Procedures 26 S t a t i s t i c a l Procedures 27 4. RESULTS OF THE STUDY 28 Results 28 Discussion of Results 35 5. SUMMARY AND CONCLUSIONS 38 The Problem 38 The Findings 38 Implications 39 Limitations 40 Suggestions For Further Research 41 BIBLIOGRAPHY 43 APPENDICES 45 i i i LIST OP TABLES Table Page 1. Means and Standard Deviations for the Three Experimental Conditions on Total Pretest, C r i t e r i o n Test, and IQ 28 2. Means and Standard Deviations on Arithmetic, Geometry, and Area Pretest and C r i t e r i o n Tests 29 3. Means and Standard Deviations on Total Information Pretest and C r i t e r i o n Tests . . 29 4. Means and Standard Deviations on Arithmetic, Geometry, and Area Information Pretest and C r i t e r i o n Tests 30 5. Regression C o e f f i c i e n t s , Standard Errors, and t-Values for the Two Covariates, Pretest and IQ, on each Test 30 6. Group Means, Adjusted Group Means, and Standard Errors for Eight Tests 31 7. Pair-wise t-Tests for Adjusted Group Means . . 33 8. T-Values for Contrasts i n Adjusted Group Means to Test Individual versus Group and Inquiry Training E f f e c t s 34 Chapter 1 THE PROBLEM Background Few educators would argue with Bruner"s conviction that the greater the p a r t i c i p a t i o n i n the learning process on the part of the student, the greater w i l l be the transfer of t r a i n i n g and the more l i k e l y w i l l be development of i n t u i t i v e thinking.^ Under the present educational system i n North America, students usually learn generalisations which they cannot use e f f e c t i v e l y because these ideas are 2 not t h e i r own and connected to t h e i r own r e a l i t i e s . Attempts have been made to overcome t h i s impediment through the incorporation of various discovery techniques, p a r t i c u l a r l y o over the l a s t f i f t e e n years. 3 However, Maria Montessori began using such techniques i n the f i r s t decade of t h i s century when she attempted to design an educational system to give each c h i l d freedom to develop himself i n a prepared Jerome S. Bruner, On Knowing (Cambridge: Harvard University Press, 1962), p. 85. 2John Holt, How Children F a i l (New York: D e l l Publish-ing Co., 1964), p. 128. 3M.C. Wittrock, "The Learning by Discovery Hypothesis," Learning by Discovery: A C r i t i c a l Appraisal, eds. Lee S. Shulman and Evan R. K e i s l a r (Chicago: Rand McNally and Co., 1966), pp. 38-39. environment.4 In 1910 Dewey stressed the importance of turn-ing from lecturing to passive students to active confrontations. 5 Wertheimer, in 1945, emphasized the importance of altering the teacher's main responsibility from instruction to cooperation. In accord with Bruner's belief that discovery learning is beneficial in increasing intellectual powers, Suchman developed the concept of inquiry training to attempt to increase the quantity of student inquiry by posing problems which provide 7 the student wxth the opportunity to ask for more information. The technique is expected to i n s t i l l autonomy and disciplined thinking, both of which are valuable assets in the problem g solving process. At the same time the technique is meant to afford the teacher a better opportunity to be aware of some of the cognitive processes at work in his students. 9 The student receives practice in hypothesis construction and testing and learns to ask questions more e f f e c t i v e l y . 1 ^ Suchman 4R.C. Orem (ed.), A Montessori Handbook (New York: Capricorn Books, 1966), p. 13. 5John D. Dewey, How We Think (New York: D.C. Heath and Co., 1910), p. 198. M^ax Wertheimer, Productive Thinking (New York: Harper and Row, 1945), p. 276. •7 J. Richard Suchman, "A Model for the Analysis of Inquiry," Analyses of Concept Learning, eds. Herbert J. Klausmeier and Chester W. Harris (New York: Academic Press), 1966, p. 177. 8 J . Richard Suchman, "Inquiry Training: Building Skills for Autonomous Discovery," Merrill-Palmer Quarterly, VTI (July, 1961), 158. 9 I b i d . , p. 152. 1 0 I b i d . 3 hoped to provide new goals for the students in their thinking of the environment as a potential source of information and in their learning to seek out and organize information in productive ways. He also believed these techniques would reduce student passivity.*"'" White pointed out that part of the aim of better instruction i s to reduce excessive levels of tension without 12 promoting boredom. Therefore he claimed that i t i s necessary to promote those types of a c t i v i t i e s , which "although playful and exploratory in character, at the same time show direction, selectivity, and persistence in interacting with the environ-ment. Discovery learning has this quality of exploration as i t s aim, and inquiry training specifically satisfies this requirement for better teaching. If the value of learning by discovery is to be thoroughly tested, i t would seem that inquiry training should be better researched, as learning to ask questions is the crux of discovery work. The Problem The purpose of this study was to explore the effective-ness of inquiry training in both group and individualized settings in producing students better able to analyze and i : LIbid., p. 151. 1 2Robert W. White, "Motivation Reconsidered: The Concept of Competence," Psychological Review, LXVT, 5(1959), 313. 1 3 I b i d . , p. 329. 4 solve problems in elementary mathematics. Examined were both the a b i l i t y to solve problems accurately and the a b i l i t y to determine whether a problem provided sufficient information to enable a solution. Problems used were based in several areas of mathematics, including arithmetic and elementary geometry. Definition of Terms Inquiry training shall mean a process of investigating in which students are given problems with insufficient informa-tion to solve, and during which the students are allowed to ask questions to gather the data to arrive at solutions. Group inquiry w i l l involve inquiry training in which any student in the class can ask as many questions as desired, while a l l other students hear the questions and the responses to them. Mass training shall be the designation for those periods, before problem sets have been distributed, during which the experimenter explains to a l l groups the basic instructions of operation for the duration of the study as well as reviews mathematical concepts needed in the course of the study. Justification for the Study Aside from i t s use as a further test of discovery learning and the inquiry training techniques, this study serves two other functions. They are the examination of the role of mathematics as the vehicle for inquiry training and the 5 comparison on in d i v i d u a l and group approaches i n inquiry t r a i n i n g . Blank and Covington, i n a study of inquiry t r a i n -ing, u t i l i z e d a t r a i n i n g program designed to encourage question asking through the use of programmed i n s t r u c t i o n . As the present study does, they looked at the r e s u l t of the t r a i n i n g i n terms of performance on a school-subject 14 oriented t e s t . However, they did not attempt to t r a i n i n a s p e c i f i c content area. Considering the practice of schools, i t i s much more feasible for inquiry t r a i n i n g to be attempted on a subject basis than on the more general type problem basis used i n the Blank and Covington study. Secondly, there i s no reason to believe that p o s i t i v e r e s u l t s due to tr a i n i n g on a generalized program would also r e s u l t from t r a i n i n g i n a subject area, and the present study may give some indications as to how subject material af f e c t s inquiry t r a i n i n g e f f e c t s . I t i s equally important that a study incorporate a comparison between ind i v i d u a l and group approaches to inquiry Stanley S. Blank and Martin Covington, "Inducing Children to Ask Questions i n Solving Problems," Journal of  Educational Research, LIX (September, 1965), 21. 6 t r a i n i n g . I n t r i n s i c a l l y the two approaches are d i f f e r e n t i n t h a t the group approach may imply t h a t many students do not g e t a chance to i n q u i r e , w h i l e on the oth e r hand, i t s e f f i c i e n c y may be so much g r e a t e r than the i n d i v i d u a l approach t h a t t h e r e might be compensation f o r the l a c k o f t o t a l p a r t i c i p a t i o n . Furthermore, many classrooms do not operate on an i n d i v i d u -a l i z e d b a s i s and i t i s important t o see whether these c l a s s -rooms, too, can b e n e f i t from an i n q u i r y t r a i n i n g approach. Hypotheses t o be Tested Four major hypotheses are to be t e s t e d i n t h i s s t u d y : H^: On a t e s t measuring the a b i l i t y to s o l v e s p e c i f i e d mathe-m a t i c a l problems w i t h s u f f i c i e n t i n f o r m a t i o n , those s u b j e c t s t r a i n e d through the use o f problems w i t h i n s u f -f i c i e n t i n f o r m a t i o n w i l l perform s i g n i f i c a n t l y b e t t e r than those not so t r a i n e d . H2: On a t e s t measuring the a b i l i t y to determine whether a g i v e n problem c o n t a i n s enough i n f o r m a t i o n t o pe r m i t s o l u -t i o n , those s u b j e c t s t r a i n e d through the use o f problems w i t h i n s u f f i c i e n t i n f o r m a t i o n w i l l perform s i g n i f i c a n t l y b e t t e r than those not so t r a i n e d . H3: On a t e s t measuring the a b i l i t y t o s o l v e s p e c i f i e d mathe-m a t i c a l problems, group i n q u i r y t r a i n e d s u b j e c t s w i l l perform s i g n i f i c a n t l y b e t t e r than i n d i v i d u a l l y t r a i n e d s u b j e c t s . On a t e s t measuring the a b i l i t y to determine whether a given problem contains enough information to permit solution, group trained subjects w i l l perform s i g n i f i c a n t l y better than i n d i v i d u a l l y trained subjects. 8 Chapter 2 A REVIEW OP RELATED LITERATURE Introduction The use of t r a i n i n g which demands learner p a r t i c i p a -t i o n i n the form of question asking has been examined previously i n educational research l i t e r a t u r e , p a r t i c u l a r l y for use i n examination of problem solving s t r a t e g i e s . Some of t h i s research i s subsumed under the heading of research on the merits of discovery learning and inquiry t r a i n i n g , whereas generally the pertinent studies deal with the question of techniques employed by untrained subjects to solve problems whose content i s not that of t r a d i t i o n a l school subjects. General Opinions on Discovery and Inquiry Training The term discovery learning has been so widely used that at t h i s point i t has assumed many d i f f e r e n t meanings. Most d e f i n i t i o n s agree that discovery learning involves techniques of rearranging data i n a manner which enables the learner to go beyond the existent data to gain new i n s i g h t s . ^ In so doing, the ideas being learned are made more compre-he n s i b l e . 2 I t i s a type of approach which requires " f l e x -Jerome S. Bruner, On Knowing (Cambridge:Harvard University Press, 1962), p. TTT. 2 I b i d . , p. 101. 9 i b i l i t y i n thinking and divergent,..., thought processes." 3 Interest i n discovery learning i s based on numerous d i f f e r e n t motives, but Bruner has pointed out four of the most probable basic benefits to be expected from discovery learning, v i z . , "an increase i n i n t e l l e c t u a l p o t e n c y t h e learning of the h e u r i s t i c s of learning... a s h i f t from e x t r i n s i c to i n t r i n s i c rewards,... and as an aid i n conserving memory." Bruner states that the emphasis on discovery leads to organized thinking designed to d i s c o v e r i r e g u l a r i t y and avoid "information 4 d r i f t , " thus t r a i n i n g good guessers, and so stimulating the " a b i l i t y to go beyond the information given to probable reconstructions of other events." 5 Suchman notes that data from active information gathering as i s involved i n inquiry t r a i n i n g should be more useful to and better retained by the learner than the passive reception of information because of the rewarding experience of information gathering i n i t s e l f , because of the self-confidence engendered i n creation of hypotheses, and because of the p r a c t i c e in the use of l o g i c a l inductive processes.** I t should also be noted that not a l l educators agree with those who advocate discovery learning techniques as the John D. Cunningham, "Rigidity i n Children's Problem Solving," School, Science, and Mathematics, LXVT ( A p r i l , 1966), 377. ^Bruner, op. c i t . , pp. 85-95. ^Bruner, "Going Beyond the Information Given," Contemporary Approaches to Cognition, eds. Jerome S. Bruner, and others (Cambridge: Harvard University Press, 1957), p. 67. 6 J . Richard Suchman, "Inquiry Training: Building S k i l l s for Autonomous Discovery," Merrill-Palmer Quarterly, VTI (July, 1961), 148. 10 primary teaching approach. Newton points out that inquiry t r a i n i n g can be dishonest i n that i t i s "not consonant with the demonstrated needs of adolescents,... not honest prepara-ti o n for the college-bound,... does not honestly r e f l e c t the nature of science,... has not been analyzed adequately,... 7 and i s often i n e f f e c t i v e and i n e f f i c i e n t . " Wittrock asserts that discovery learning i n i t s assumption that each c h i l d can best be taught by one method i s as much at f a u l t as other monistic theories of teaching. In addition, because discovery learning i s frequently time consuming and because the basis of culture i s the transmission of knowledge of other's discover-i e s , there i s reason to consider the value of demanding every i n d i v i d u a l to learn by discovery an hypothesis s t i l l i n need 9 o f t e s t i n g . Some of t h i s testing has been going on i n the work of study groups such as the University of I l l i n o i s Committee on School Mathematics and the School Mathematics Study Group i n mathematics and has gone on i n the past i n the work of 10 11 Montesson and Dewey. David E. Newton, "The Dishonesty of Inquiry Teaching," School, Science, and Mathematics, LXVIII (December, 1968) p. 807. 8M.C. Wittrock, "The Learning by Discovery Hypothesis," Learning by_ Discovery: A C r i t i c a l Appriasal, eds. Lee S. Shulman and Evan R. Ke i s l a r (Chicago: Rand McNally and Co., 1966), p. 36. 9 I b i d . 1 0R.C. Orem (ed.), A Montessori Handbook (New York: Capricorn Books, 1966), p. 13. 1 1 J o h n D. Dewey, How We Think (New York: D.C. Heath and Co., 1910), p. 193. 11 Research of General Relevance to the Problem Studies dealing with the identification and evaluation of problem-solving strategies have been conducted by several researchers. John ut i l i z e d a machine called the PSI (problem solving 12 instrument). The subject's task was to learn the method to achieve a stipulated output from a specified input on the 13 electromechanical c i r c u i t . John designed the machine to try to approximate best the ideal format for studying problem solving strategies using subjects in advanced study in various disciplines; he was giving the subjects a minimum of informa-tion to start and allowing them to structure their behavior 14 without specific content to consider. In comparing technique differences used in solving two problems identical with the exception that the second depended on use of the same pattern in two different ways, variables measured included time to completion, number of questions asked, complexity of questions asked, rate, effort, and redundancy. John found that scientists performed significantly better than students in other disciplines on most measures and that as the problem became more d i f f i c u l t , time needed and number of questions asked increased, as did 15 r i g i d i t y and stereotypy of questions. 12 Erwin Roy John, "Contributions to the Study of the Problem Solving Process," Psychological Review, LXXI, 447 (1957), p. 5. 1 3 I b i d . , p. 6. *^Ibid., p. 5. 1 5 I b i d . , pp. 35-37. Rimoldi, i n a description of his techniques for study-ing problem solving, discussed information seeking questions where the subject was given a choice of possible questions to 16 ask. Measured were three properties of the items: u t i l i t y index, median value, and dispersion of items, and scoring was done i n several ways including scores based on u t i l i t y , number of items asked, the correctness of the f i n a l solution, 17 and a type of q u a l i t a t i v e analysis of strategy. Although th i s study was descriptive rather than experimental i t s techni-ques are of relevance i n examination of problem solving. U t i l i z i n g a twenty-questions-game approach and two types of s i t u a t i o n s , ore i n which the children were shown forty-two pictures of common objects and asked to f i n d the one the experimenter had i n mind and one i n which the children were given a s i t u a t i o n and asked to f i n d a cuase, Mosher and Hornsby investigated problem solving strategies and discovered two basic strategies - constraint seeking and hypothesis 18 scanning. In constraint seeking, the subject assumes equally l i k e l y alternatives' and eliminates by halves, whereas i n hypothesis scanning each guess tests a s e l f s u f f i c i e h t 1 6H.J.A. Rimoldi, "A Technique for the Study of Problem Solving," Educational and Psychological Measurement, XV, (1955), 451. 1 7 I b i d . , p. 454. 1 8 F r e d e r i c k A. Mosher and Joan Rigney Hornsby, "On Asking Questions," Studies i n Cognitive Growth, eds. Jerome S. Bruner, and others (New York: John Wiley and Sons, Inc., 1966), p. 88. 13 hypothesis. Generally constraint seeking, which involves more efficiency than hypothesis scanning and which insures better coverage of a l l p o s s i b i l i t i e s , involves more strain.*^ However, the constraint seeking approach is generally preferred by older 20 children. In his investigation of two types of creativity v a r i -ables, in complex problem solving, one defined in terms of cognitive structure integrative complexity, as measured by scores on the Paragraph Completion Inventory, and one in terms of emphasis on breadth of association, as determined by the Remote Associates Test and labelled associative creativity, Karlins u t i l i z e d an information-seeking approach in which each subject was required to learn about an unfamiliar South Seas 21 island culture in order to solve a situation-problem. Information was made available in a 57-category program deck. Here the subject was active in every phase of the learning process, from active information gathering to processing and ut i l i z a t i o n of data received. Subjects were permitted to ask 22 for information which would be provided i f avilable. He found that the two groups of subjects termed associatively 19 Ibid., p. 89. 2 0 I b i d . , p. 90. 2*Marvin Karlins, "Conceptual Complexity and Remote-Associative Proficiency as Creativity Variables in a Complex Problem Solving Task," Journal of Personality and Social  Psychology, VI, 3 (1967), 268. 2 2 I b i d . , p. 274. 14 creative, but differing in integrative complexity, showed no significant differences in terms of breadth of information search, evenness of information search, or willingness to directly explore, but the integratively complex subjects, regardless of associative creativity scores, did show s i g n i f i -23 cantly better results than noncomplex subjects in a l l areas. In a study to analyze the influence of logical structure and problem language in thinking processes and their relation to age, Rimoldi and his associates presented a problem and requested the subjects to choose from a l i s t of possible questions to ask those they considered crucial to the solution of the problem, continuing u n t i l they either solved the problem or decided to stop. 2^ A l l problems were individually admin-istered without a time limit and content was not based on standard school material. Scoring was based on the number of questions asked, scheme, and correctness of the answer. They found that a significant improvement comes through aging and that there is some significant difference due to inter-action of problem language and logical structure. Also they found that there is increased agreement among subjects' tactics with age and that logical structure and language are not experimentally independent variables. 2 5 2 3 I b i d . , p. 277. ?4 H.J.A. Rimoldi, M. Aghi, and G. Burger, "Some Effects of Logical Structure, Language, and Age in Problem Solving in Children," Journal of Genetic Psychology, CXII (1968), 127. 2 5 I b i d . , p. 142. 15 In the area of the use of problems which differ from standard textbook problems in terms of either overabundance or insufficiency of information for solution, two studies are most relevant. James found that on a test measuring correctness of responses to arithmetic problems, poorer performance resulted from problems with too much information than with standard 2 6 textbook problems. At the other end of the spectrum, O'Brien and Shapiro found that introducing the alternative of "not enough information" in a test designed to examine logical thinking in children significantly lowered marks from the same 27 test without these p o s s i b i l i t i e s . This tends to indicate that a deficiency now exissfcs in recognizing the insufficiency of information for problem solution. Research of Specific Relevance to the Problem One of the most popular devices in the research involv-ing the asking of questions by subjects is the tab item originally developed by Glaser for use in the Air Force to replace more expensive procedures for problem solving observa-28 tion without sacrificing validity in examining the behavior. Jim Butler James, "A Comparison of Performance of Sixth Grade Children in Three Arithmetic Tasks: Typical Text-book Verbal Problems, Revised Verbal Problems Including Irrelevant Data, and Computational Exercises, "Dissertation  Abstracts, 28:2030 B, November, 1964. 2^Thomas C. O'Brien and Bernard J. Shapiro, "The Develop-ment of Logical Thinking in Children, " American Educational  Research Journal, V (November, 1968),.537. 2 8Robert Glaser, Dora E. Damrin, and Floyd M. Gardner, "The Tab Item: A Technique for the Measurement of Proficiency in Diagnostic Problems Solving Tasks," Educational,and Psychologi- cal Measurement, XXV, 2(1954), 283. 16 The t a b item, as a technique i n measuring v a r i a b l e s i n i n f o r m a t i o n g a t h e r i n g i n the course o f problem s o l v i n g , was d e s c r i b e d by G l a s e r r e g a r d i n g i t s use i n d i a g n o s t i c work where the l e a r n e r was f o r c e d t o "perform a s e r i e s o f procedures i n which the r e s u l t s , o f one procedure y i e l d i n f o r m a t i o n to s u p p l y 29 a cue for. the s e l e c t i o n o f the next item." The s u b j e c t chooses a tabbed item, each c o n s i s t i n g o f s e v e r a l pages o f i n f o r m a t i o n i n c l u d i n g b o t h r e l e v a n t , redundant, inadequate, 30 and i r r e l e v a n t types o f i n f o r m a t i o n . S c o r i n g i s based on the weighted s e l e c t i o n e f f i c i e n c y . No comparisons between groups were made by G l a s e r i n t h i s d e s c r i p t i o n . Suchman, i n h i s i n q u i r y t r a i n i n g , showed p h y s i c s experiments on f i l m t o students and presented them w i t h the tas k o f e x p l a i n i n g why the demonstrations i l l u s t r a t e d r e s u l t e d as they d i d . 3 ^ To achieve t h i s end, he i n s t r u c t e d them t o o b t a i n more i n f o r m a t i o n by a s k i n g q u e s t i o n s which c o u l d be answered "yes" or "no" i n a thr e e - s t a g e process o f a n a l y s i s o f the episode, d e t e r m i n a t i o n o f r e l e v a n c e o f v a r i o u s c o n d i t i o n s necessary f o r the outcome, and i d e n t i f i c a t i o n o f c o n s t r u c t s to 32 e x p l a i n the phenomena. u n l i k e the above s t u d i e s which were c a r r i e d * o u t on an i n d i v i d u a l i z e d b a s i s , Suchman had a l l 2 9 I b i d . , p . 284. 3 0 I b i d . , p . 289. J XSuchman, " I n q u i r y T r a i n i n g : B u i l d i n g S k i l l s f o r Autonomous D i s c o v e r y , " p . 151. 3 2 i b i d . , p . 159. questioning done o r a l l y i n a group and each p a r t i c i p a n t was 33 allowed to hold the flo o r as long as desired. During a twenty-four week period Suahman found that questions became both more abundant and more precise as time passed and p r i n -34 c i p l e s were better learned than i n a control group. However, Suchman did not examine the d i r e c t e f f e c t s of t h i s procedure on other types of problem s o l v i n g . U t i l i z i n g the vehicle of programmed i n s t r u c t i o n , Blank and Covington attempted to induce question asking i n solving problems through the use of problems i n which i n s u f f i c i e n t information was provided. The aim of the investigators was to increase question asking among a group of grade s i x students and to determine the e f f e c t s of this information-seeking t r a i n -ing technique on achievement test scores i n an area unrelated to the t r a i n i n g , as well as on both an o r a l and written t e s t 35 si m i l a r to the t r a i n i n g material. Unlike the other studies which were more interested i n observing question asking strategies. Blank and Covington examined the f i n a l e f f e c t s of the questioning. The t r a i n i n g problems were those of a type l i k e l y to occur i n the d a i l y routine, as, for example, a challenge to determine how a truck might get through a tunnel too low for i t . The students f i r s t examined sample problems 3 3 I b i d . , p. 163. 3 4 I b i d . , p. 167. 3 5 S t a n l e y S. Blank and Martin Covington, "Inducing Children to Ask Questions i n Solving Problems," Journal of Educational  Research, LIX (September, 1965), 21. with pertinent questions which they might have asked to solve them. Then they performed on t h e i r own. The researchers compared a group trained for ten days, f o r t y - f i v e minutes per day, on the above method mentioned, with a group exposed to the same problems with a l l information provided and no opportunity for question asking and with a control group given no exposure to these problems. They found that i n a l l cases the group trained on the data-seeking items s i g n i f i c a n t l y outperformed 3 6 the other groups. Summary of Research Findings In much research, discovery methods have proved to be e f f e c t i v e tools i n the better i n s t r u c t i o n of students from both a motivational viewpoint and from the point of view of better 37 retention. Suchman and Blank and Covington found that s p e c i f i c a l l y the inquiry technique proved to be valuable i n terms of producing students more incl i n e d to ask questions and to perform better on achievement tests i n other areas. The work of James and O'Brien and Shapiro indicates that problems d i f f e r i n g from standard texts present new challenges to students untrained i n an inquiry s e t t i n g . Although the concept of examination of question-asking strategies has been researched by many experimenters, l i t t l e attention has been paid to the outcome of t h i s t r a i n i n g except by Suchman and Blank and Covington. However, Suchman Ibid., p. 25. 37M.C. Wittrock, "The Learning by Discovery Hypothesis," pp. 50-53. did not seek to measure transfer to other areas of study and Blank and Covington did not examine inquiry t r a i n i n g applied to s p e c i f i c content areas nor i n group s i t u a t i o n s . Therefore, i t seems important that research be done i n both group and in d i v i d u a l inquiry t r a i n i n g i n content areas. 20 Chapter 3 PROCEDURE AND DESIGN Pes ign The present study employed three experimental conditions with the three treatment groups labelled as T-^ T-,and T-.. The subjects were trained through individual question asking to seek the additional information required to find solutions to problems stated with insufficient information. subjects were trained as a group on the same problems with insufficient information as in T,, and T subjects were trained on the same 1 3 problems as the other two groups with the exception that the problems were stated with a l l information necessary to solu-tion provided. Scores on a group IQ test and on a criterion pretest were f i r s t gathered for a l l pupils to be used as covariates to insure equality of groups with respect to intelligence, i n i t i a l a b i l i t y in recognizing problems with insufficient information, and i n i t i a l problem solving a b i l i t y in three areas to be tested, viz., finding areas of polygonal figures, solving geometric reasoning problems, and solving arithmetic reasoning problems. A l l three groups then participated in a mass training of two sessions of twenty and forty minutes respectively on the topic of finding areas of polygons using the technique of decomposition. Discussion of what information would be 21 needed to find areas of various general figures was not brought up by the experimenter, but references to t h i s , i f made by subjects, were allowed and b r i e f l y discussed. Groups and T-, were instructed together while group T-, was taught the same material as a group, but separately. Each group met twice, for f o r t y minutes each session, at which time the subjects worked on problem sets involving finding the area of polygons by decomposition. Following the two sessions on area, the groups again pa r t i c i p a t e d i n a mass t r a i n i n g session, i n which examples of verbal rate problems were examined. This session lasted only f i f t e e n minutes because the concepts were fam i l i a r to a l l subjects i n i t i a l l y and required l i t t l e explanation. The session was followed by two separate meetings where the subjects worked on problem sets involving the solution of rate problems. A c r i t e r i o n test to measure a b i l i t y to recognize problems with i n s u f f i c i e n t information and to accurately f i n d solutions to problems i n arithmetic, geometry, and area calcu-l a t i o n was administered to a l l groups on the f i n a l day. Subjects I n i t i a l l y one class of 35 grade s i x children was a r b i t r a r i l y selected to be placed under experimental condition T 2 and 35 students from another class to be randomly assigned to T j and T 3. After students absent for the pretest and/or the c r i t e r i o n t e s t were eliminated, and following the random assign-22 ment of subjects to groups and T 3, the numbers i n each group were fixed at 33 i n T 2, 14 i n T^, and 17 i n T 3. A l l subjects were i n two arithmetic classes i n a Vancouver elementary school. Controls Precautions were taken i n attempting to control for several types of va r i a b l e s : i n t e l l i g e n c e and achievement, experimenter bias, Hawthorne type e f f e c t s , and feedback d i f f e r -ences between the groups. IQ scores and c r i t e r i o n pretest scores were used i n an analysis of covariance on the c r i t e r i o n posttest scores to control for i n i t i a l differences between the groups i n the areas of i n t e l l i g e n c e and problem solving a b i l i t y . No attempt was made to control for reading a b i l i t y due to Balow's and others' findings that when IQ i s controlled, computational a b i l i t y appears more s i g n i f i c a n t than reading a b i l i t y for problem solving.^-To minimize the effects of experimenter bias and teacher differences, the same instructor, the experimenter, was used i n a l l groups, and minimal contact between the experimenter and the subjects during the problem solving a c t i v i t i e s was maintained at a l l times. Although the novelty of a strange instructor under exceptional circumstances was a factor i n the experiment, a l l Irving H. Balow, "Reading and Computational A b i l i t y as Determinants of Problem Solving," The Arithmetic Teacher, XI (January, 1964), 22. 23 groups experienced this same e f f e c t and therefore i t can be assumed that a l l groups were affected equally. Care was taken to see that subjects did not know who was i n which group or who was expected to perform best, and no p a r t i a l results were discussed during the period of the study. F i n a l l y , subjects i n the group T-, were given the oppor-tunity to ask questions to seek information, although no data were given them, merely cards which indicated that they already possessed s u f f i c i e n t data to solve the problem, i n order to control for any advantage to the or the T 2 groups with respect to personal feedback. A l l cards looked a l i k e so that T3 subjects were not aware that t h e i r answers were any d i f f e r -ent than those for T^. The motivation for this approach was derived from Smith and Right's finding that personalized feed-back can markedly improve problem solving e f f i c i e n c y and i n s i g h t . 2 Instructional Materials The f i r s t mass t r a i n i n g sessions, for a l l groups, was used to review the already e x i s t i n g knowledge of how to find area of rectangles. The concepts of finding areas of parallelograms and t r i a n g l e s were then introduced and explained. Following t h i s , several more complicated polygonal figures were discussed i n terms of decomposition into more primitive figures to fi n d ^Ewart E. Smith and Stanford S. Kight, "Effects of Feedback on Insight and Problem Solving E f f i c i e n c y i n Training Groups," Journal of Applied Psychology, XLIII, 3 (1959), 211. 24 their areas. A l l subjects were given opportunities to ask questions at these sessions. The two problem sheets consisted of five problems each. Subjects were presented figures with some but not a sufficient number of dimensions to solve the problems and asked to find the areas by requesting other dimensions i f needed. Only units of measurement familiar to the students and only scale drawings were used. A l l of the dimensions were given as whole numbers, to avoid unnecessary d i f f i c u l t i e s with fractions. After a brief discussion of the solution of an example rate problem, the subjects began work on two sets of arithmetic rate problems sheets, each consisting of six problems. The problems were classifiable in terms of both rate-divisive and rate-multiplicative types, that i s , some requiring division work and others multiplication. Measuring Instruments I.Q. Scores. IQ scores for a l l subjects were available from school records based on the Henmon-Nelson Test of Mental Ability, Revised 1958 edition, published by Houghton-Mifflin Company, and administered throughout the Vancouver School d i s t r i c t . Criterion Pretest. The criterion pretest consisted of twelve multiple-choice problems. The pretest was f i r s t administered to a p i l o t group of 28 students for determination of r e l i a b i l i t y coefficients. A test-retest r e l i a b i l i t y of .81 was obtained. Four items dealt with finding areas of polygons, three were of a nonmetric geometric type and five 25 were a r i t h m e t i c problems. Of the f o u r response c h o i c e s (d) was "not enough i n f o r m a t i o n " f o r a l l problems, and s u b j e c t s were i n s t r u c t e d t o use t h i s p o s s i b i l i t y o n l y i f they f e l t t h a t i n s u f f i c i e n t i n f o r m a t i o n to s o l v e the problem had been p r o v i d e d . Wording o f items was a l t e r e d from an o r i g i n a l v e r s i o n on the b a s i s o f p i l o t runs w i t h 100 grade s i x s t u d e n t s . The p r e t e s t items were chosen from the item p o o l used i n c o n s t r u c t i o n o f the c r i t e r i o n t e s t i n an e f f o r t t o m a i n t a i n e q u i v a l e n c e . The number o f items was r e s t r i c t e d t o twelve due t o the i n v o l v e d nature o f s e v e r a l o f them and the time r e q u i r e d f o r s o l u t i o n . No s t a n d a r d i z e d t e s t i n v o l v i n g geometric problems a t t h i s l e v e l was deemed s u i t a b l e . T h e r e f o r e , the t e s t was experimenter c o n s t r u c t e d . C r i t e r i o n T e s t . At the end o f the experimental p e r i o d , a seventeen item c r i t e r i o n t e s t was a d m i n i s t e r e d , o f which f i v e items d e a l t w i t h area c a l c u l a t i o n , four w i t h non-m e t r i c geometry, and e i g h t w i t h a r i t h m e t i c problems, i n c l u d i n g r a t e problems. Each item had one "not enough i n f o r m a t i o n " response c h o i c e . A t e s t - r e t e s t r e l i a b i l i t y o f .92 was computed based on a p i l o t sample o f 28 s t u d e n t s . As w i t h the c r i t e r i o n p r e -t e s t , items were f i r s t t e s t e d on 100 grade s i x students t o determine c l a r i t y of wording. T h i s t e s t , too, was experimenter c o n s t r u c t e d . Procedure Before the experimental procedures were under way, random assignment t o groups and T^ was made from a s i n g l e c l a s s . The group IQ t e s t had been p r e v i o u s l y a d m i n i s t e r e d t o a l l s u b j e c t s . The p r e t e s t was a d m i n i s t e r e d on the f i r s t day o f the experiment, f o l l o w e d by a twenty minute mass t r a i n i n g review o f p o l y g o n a l area concepts. Next, the concept o f a r e a o f polygons was extended through a f o r t y minute t r a i n i n g s e s s i o n i n a l l groups. Two days were devoted t o two problem s e t s i n which the s u b j e c t s were asked to f i n d the areas o f v a r i o u s p o l y g o n a l f i g u r e s a f t e r the q u e s t i o n a s k i n g procedure was e x p l a i n e d . A l l t h r e e groups were given the same f i g u r e s ; however, o n l y T3 s u b j e c t s were g i v e n a l l o f the necessary i n f o r m a t i o n a t the s t a r t . Each student i n and T 3 was s u p p l i e d w i t h a s e t o f cards 1, blank except f o r the o u t l i n e s o f the f i g u r e s on the worksheets. He was then p e r m i t t e d t o f i n d needed data by i n d i c a t i n g which dimension he r e q u i r e d on the blank c a r d and then handing the c a r d t o the experimenter, who immediately r e t u r n e d t o him a p r e v i o u s l y prepared a p p r o p r i a t e response c a r d . Students i n T^ who asked q u e s t i o n s always r e c e i v e d a ca r d i n d i c a t i n g t h a t s u f f i c i e n t i n f o r m a t i o n was a l r e a d y a v a i l a b l e . No h i n t s were p r o v i d e d t o any s u b j e c t s , o n l y d a t a . The s u b j e c t s met as a c l a s s t o work on t h e i r problem s h e e t s . A l l q u e s t i o n s and answers were o r a l , w i t h answers marked i n l a r g e f i g u r e s a t the f r o n t o f the room; i t may be assumed t h a t a l l group members thus heard and b e n e f i t t e d from each 27 other's questions. Solutions to a l l problems i n a l l three groups were found i n d i v i d u a l l y . Following these two sessions, the groups again p a r t i c i -pated i n a f i f t e e n minute mass t r a i n i n g session reviewing verbal rate problems and solutions. Two problem sets were d i s t r i b u t e d . Again the card system was employed. As before, only T3 subjects i n i t i a l l y received s u f f i c i e n t data to solve the problems. The c r i t e r i o n t e s t was administered to a l l groups on the f i n a l day of the study. In scoring both the c r i t e r i o n pretest and posttests, t o t a l scores were calculated as well as scores i n each of the three topic subdivisions. In addition, t o t a l scores and sub-divison scores were obtained on the basis of the correct or incorrect use of answer (d) - "not enough information". S t a t i s t i c a l Procedures The analysis to follow i s considered i n terms of eight separate marks: t o t a l score, arithmetic score, geometry score, area score, t o t a l information score, arithmetic information score, geometry information score, and area information score, each with i t s corresponding pretest score. A l l data were analyzed at the University of B r i t i s h Columbia Computing Centre using an analysis of covariance program. 3 Health Sciences Computing F a c i l i t y , Department of Preventive Medicine, University of C a l i f o r n i a , "BMDX82-Analysis of Covariance," Los Angeles, C a l i f o r n i a . 28 Chapter 4 RESULTS OF THE STUDY Results The means and standard deviations for the scores on the eight separate pretest and criterion test marks are given in Tables 1, 2, 3, and 4. Table 1 Means and S.D.s for the Three Experimental Conditions on Total Pretest, Criterion Test and IQ Number of Questions Correct Pretest Criterion IQ Mean S.D. Mean S.D. Mean S.D. T l 3.57 1.60 6.36 2.50 112.14 11.47 T2 5.39 1.71 7.67 2.44 11-5.21 10.79 T3 3.82 1.88 5.06 2.30 111.24 12.83 N.B. T^ is the individualized-inquiry trained group; T 2 i s the group-inquiry trained group; T^ is the individualized-non-inquiry trained group. 29 Table 2 Means and Standard Deviations on Arithmetic, Geometry, and Area Pretest and Criterion Tests. Arithmetic Arithmetic Geometry Geometry Area Area - Pretest Criterion Pretest Criterion Pretest Criterion Mean S.D. Mean S^D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. T x 1.50 1.29 2.93 1.64 .64 .84 1.00 .55 1.43 1.16 2.43 1.40 T 2 1.91 1.04 2.52 1.33 .61 .60 1.24 .94 1.21 .96 2.27 1.04 T 3 1.71 1.16 2.24 1.44 .88 .86 .65 .79 1.59 1.23 1.94 1.14 Table 3 Means and Standard Deviations on Total Information Pretest and Criterion Test. Pretest Criterion Mean S.D. Mean S.D. T x 7.79 1.52 8.71 2.33 T 2 8.97 1.19 9.97 2.19 T 3 8.24 1.44 7.47 2.12 30 Table 4 Means and Standard Deviations on Arithmetic, Geometry, and Area Information Pretest and Criterion Tests Arithmetic Geometry Area  Pretest Criterion Pretest Criterion Pretest Criterion Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Tj 3.14 1.17 4.07 1.21 1.14 .95 1.64 .74 3.50 .65 3.07 1.38 T 2 4.09 .95 4.90 1.33 .27 .45 2.03 .92 3.73 .57 3.12 .82 T 3 3.59 1.12 3.71 1.65 .82 .80 1.29 .85 3.82 .39 2.53 1.28 Table 5 Regression Coefficients, Standard Errors, and t-values for the Two Covariates, Pretest and IQ, on each Test. Type of Test Covariate Regr. Coeff. S.E. t-value total test pretest .90 .04 23.80*( .01) IQ .05 .02 2.17*( .02) arith test pretest .43 .13 3.25 IQ .05 .01 4.19*( .01) geom. test pretest .09 .15 .60 IQ -.01 .01 -.98 area test pretest .24 .14 1.72 IQ .03 .01 2.06*( .05) total info. pretest .10 .21 .46 test IQ .03 .02 1.34 arith. info. pretest .21 .17 1.24 test IQ ,00 .02 .21 geom. info. pretest -.17 .17 -1.02 test IQ .00 .01 - .20 area info. pretest .17 .25 .69 .05) test IQ .02 .01 2.28*( Examination of the regression coefficients and t-test values for the two covariates, pretest and IQ, for each of the eight scores presented in Table 5 indicates that IQ was si g n i f i -cantly related to score on both the total and arithmetic tests beyond the .02 level, and pretest score was related to these two marks beyond the .01 level. IQ was also significant on both the area and area information scores at the .05 level. Because the covariates were significant in several instances, the analysis of covariance, rather than analysis of variance, was required. Therefore, adjusted group means were f i r s t calculated for a l l groups for scores on the criterion test. These adjusted scores are indicated in Table 6. Table 6 Group Means, Adjusted Group Means, and Standard Errors for Eight Tests Gp. Mean Adj. Gp. Mean S.E. Gp. Mean Adj. Gp. Mean S.E. Total Test Total : Info. Test T l 6.36 7.34 .59 8.71 8.83 .61 T2 7.67 6.84 .38 9.97 9.87 .40 T3 5.06 5.86 .53 7.47 7.57 .54 Arith Test Arith : Info. Test T l 2.93 3.11 .32 4.07 4.20 .39 T2 2.51 2.36 .21 4.91 4.83 .25 T3 2.24 2.38 .29 3.71 3.75 .34 Geom Test Geom Info. Test T l 1.00 .99 .22 1.64 1.73 .25 T 2 1.24 1.26 .15 2.03 1.98 .16 .65 .61 .20 1.29 1.33 .21 Table 6 (Continued) 32 Group Mean Adj. Mean S.E. Mean Ad j . Mean S.E. Area Test Area Info. Test T l 2 A3 2.44 .29 3 .07 3.14 .29 T2 2.27 2.26 .19 3.12 3.07 .19 1.94 1.94 .26 2.53 2.57 .26 33 Table 7 Pairwise t-Tests for Adjusted Group Means Pair Regular Tests Info. Tests Total A r i t h Geom Area Total A r i t h Geom Area T1" T2 .70 1.97 -1.02 .52 -1.39 -1.31 -.79 .19 T1" T3 1.87 1.73 1.26 1.30 1.57 .89 1.27 1.47 T,-T- 1.48 -.04 2.58* .97 3.36* 2.51* 2.36* 1.59 2 3 (.02) (.01) (.02) (.05) N.B. T^ i s the indi v i d u a l i z e d - i n q u i r y trained group; T 2 i s the group-inquiry trained group; T3 i s the individualized-non-inquiry trained group. The pairwise t-tests for adjusted group means shown i n Table 7 indicates that pairwise differences occurred i n the geometry as well as the information t e s t s , at the .02 l e v e l for the geometry test, at the .01 l e v e l for the t o t a l informa-t i o n test, at the .02 l e v e l for arithmetic information test, and at the .05 l e v e l for the geometry information t e s t . No s i g n i f i c a n t differences occurred i n the area information t e s t . In a l l cases only the T 2- ^comparison i s s i g n i f i c a n t . I t therefore appears that an in t e r a c t i o n e f f e c t of the two factors, amount of information given (the use of inquiry training) and the group versus i n d i v i d u a l factor i s s i g n i f i c a n t i n favor of the group approach with not enough information 34 i n terms of a b i l i t y to determine whether problems provide s u f f i c i e n t information to enable s o l u t i o n . To tes t the main hypotheses themselves, two t-values were computed for each of the eight tests - one to examine group versus i n d i v i d u a l e f f e c t and one to examine the not enough i n i t i a l information versus s u f f i c i e n t i n i t i a l information e f f e c t . Table 8 t-Values for Contrasts In Adjusted Group Means to Test Individual versus Group and Inquiry Training E f f e c t s Contrast Test t-Value Contrast Test t-Value T l & T3VS . To Total - . 4 3 T i & T3 T 5 vs. Total Info. - 2 . 8 0 * (.01) T i & T 3 T 2 vs. Total 1.89 T i & T 3 T 2 vs. Total Info. 2.62* (.05) T & T 2 T 3 vs. A r i t h . 1.27 T, & T 2 T 3 vs. A r i t h . Info. - 2 . 2 6 * (.05) T X & T 3 T 2 vs. A r i t h . 1.03 T x & T 3 T 2 vs. A r i t h . Info. 1.75 T i & T 2 T 3 vs. Geom. - 2 . 1 8 * (.05) T i & T 2 T 3 vs. Geom. Info. -1 .81 T l &  T 3 T 2 vs. Geom. X .14* ( .05) T, & T 3 T 2 vs. Geom. Info. 1.82 T i & T 2 T 3 vs. Area - . 2 6 T i & T 2 T 3 vs. Area Info. - . 8 2 T, & T3 T 2 vs. Area 1.30 T, & T3 T x 2 vs. Area Info. 1.60 Table 8 indicates only flare c s i g n i f i c a n t differences among the sixteen contrasts tested, three of which confirm the hypothesis that students trained as a group outperform students not so trained and two of which favor inquiry trained students. Scores i n group T 0 were s i g n i f i c a n t l y better than scores i n groups and T 3 on the geometry t e s t at the .05 l e v e l , the t o t a l information t e s t at the .01 l e v e l , and the arithmetic information test at the .05 l e v e l . Scores i n groups T^ and T 2 were s i g n i f i c a n t l y greater than scores i n group T3 on the geometry t e s t and on the t o t a l information t e s t at the .05 l e v e l of s i g n i f i c a n c e . Discussion of Results The data presented i n the present research indicate that group t r a i n i n g i s preferable to individual t r a i n i n g , e s p e c i a l l y i n combination with the inquiry approach, i n terms of improve-ment i n a b i l i t y to a r r i v e at accurate solutions to s p e c i f i e d problems and to determine s u f f i c i e n c y of information i n these problems. Hypothesis H^ which predicted s i g n i f i c a n t differences i n a b i l i t y to a r r i v e at accurate solutions to elementary mathe-matics problems as a r e s u l t of inquiry t r a i n i n g was confirmed only i n the case of the geometry subtest. This may be a t t r i b u t -able to the i n t e r a c t i o n with the group factor, however, i n that no s i g n i f i c a n t differences were found when the group factor was removed. Therefore, no differences based s o l e l y on inquiry t r a i n i n g were found. The short duration of the study as compared to the longer period of time i n which patterns of problem solving behavior are established may be a factor i n the lack of s i g n i f i c a n t changes. Hypothesis H 2 which predicted s i g n i f i c a n t differences i n a b i l i t y to c o r r e c t l y determine the s u f f i c i e n c y of information i n a given problem due to inquiry t r a i n i n g was confirmed only on the t o t a l information test, although not on any of the subtests. This seems to indicate merely an accumulation of nonsignificant scores on the subtests and therefore i s not s i g n i f i c a n t i n i t s e l f . Perhaps, here too, long term patterns r e s i s t a n t to a l t e r a t i o n or too l i m i t e d exposure was responsible for lack of change. Hypothesis H^ which predicted higher marks for group-trained subjects than i n d i v i d u a l l y - t r a i n e d ones on tests measuring a b i l i t y to a r r i v e at problem solutions was confirmed i n part. S i g n i f i c a n t l y higher marks did r e s u l t on the geometry te s t , although not on the arithmetic or area c a l c u l a t i o n portions. Perhaps t h i s may be accounted for i n that the geometry content was the one l e a s t f a m i l i a r to the subjects and so most amenable to change, p a r t i c u l a r l y when information gathering became more e f f i c i e n t , as can be presumed to be the case i n the group se t t i n g and when inquiry t r a i n i n g was involved. Hypothesis H^ which predicted higher marks for group-trained subjects than i n d i v i d u a l l y - t r a i n e d ones on a t e s t measuring a b i l i t y to determine the s u f f i c i e n c y of information i n a given problem was also confirmed i n part. S i g n i f i c a n t l y higher scores did r e s u l t on the arithmetic information subtest and the t o t a l information t e s t . Perhaps the kind of in t e r a c t i o n involved i n group information gathering proved to be more useful to the i n d i v i d u a l group members' a b i l i t y to judge s u f f i c i e n c y of information i n arithmetic problems as compared to geometry or area c a l c u l a t i o n ones because of the better working vocabulary of the students with t h i s type of problem so allowing for faster benefits from this t r a i n i n g . The higher marks on the t o t a l information t e s t can be accounted for sub-s t a n t i a l l y by the higher marks on the arithmetic portion; geometry and area score differences also favoured the group approach, although not s i g n i f i c a n t l y . While no s i g n i f i c a n t differences resulted d i r e c t l y from inquiry t r a i n i n g , the pairwise comparisons did indicate that group inquiry t r a i n i n g i s superior to an in d i v i d u a l non-inquiry approach, but not s i g n i f i c a n t l y superior to an ind i v i d u a l - i n q u i r y approach. Part of the cause for this may r e s t with the fact that i n c o n t r o l l i n g for the advantage due to feedback i n the ind i v i d u a l s i t u a t i o n , the subjects i n the con t r o l group experienced some of the same conditions connected with information seeking, although not a l l of them, as the experimental group. Therefore, i t appears that inquiry t r a i n i n g may have a useful r o l e to play i f combined with the more e f f i c i e n t group approach. These res u l t s do not confirm those of Blank and Covington i n t h e i r study where i n d i v i d u a l l y -trained subjects were more successful than individuals not so trained, a fact which may be attri b u t a b l e to the programmed learning aspect of the t r a i n i n g or to the nonsubject-oriented content. 1 Stanley S. Blank and Martin Covington, "Inducing Children to Ask Questions i n Solving Problems," Journal of  Educational Research, LIX (September, 1965), 27. 38 Chapter 5 SUMMARY AND CONCLUSIONS The Problem The purpose of t h i s study was to determine whether an inquiry t r a i n i n g approach would be b e n e f i c i a l to students in either a group or ind i v i d u a l i z e d setting in terms of increasing a b i l i t y to analyze s p e c i f i e d mathematical problems with respect to competence i n determining whether a problem provides s u f f i c i -ent information for solution, and in proceeding to solution where possibl e . The Findings The r e s u l t s of the data analysis indicate that the group inquiry approach i s more successful than the i n d i v i d u a l approach i n producing students better able to ar r i v e at solutions to problems of a novel nature and better able to examine the adequacy of information provided in problems not previously examined i n t h i s l i g h t . Furthermore, pairwise comparisons indicate that the group inquiry approach i s s i g n i f i c a n t l y superior i n four out of eight comparisons to the in d i v i d u a l non-inquiry approach. I t also may be that a short inquiry t r a i n i n g period had no s i g n i f i c a n t e f f e c t s on the a b i l i t y to improve performance on types of material already very f a m i l i a r to the students. Implications As a consequence of the findings of this study, several implications can be drawn. F i r s t , inquiry t r a i n i n g appears to be e f f e c t i v e i n producing better problem solvers i f employed with a group approach and not used on an in d i v i d u a l basis. This r e s u l t bears out Suchman's findings;* i t does not completely agree with those 2 of Blank and Covington. Since no substantial differences i n duration e x i s t between the present study and Blank and Coving-ton's study, i t would appear that the differences between the two findings must r e s u l t from one of two causes: the in t e r a c t i o n e f f e c t s of inquiry t r a i n i n g and programmed learning or the use of generalized as opposed to s p e c i f i c content-oriented t r a i n i n g problems. I f the cause i s the former, then i t i s apparent that through the use of programmed, material, inquiry t r a i n i n g can be made e f f e c t i v e i n i n d i v i d u a l l y structured classrooms, and could be implemented i n many settings. I f , on the other hand, the cause i s the content i n the t r a i n i n g , perhaps what i s indicated i s the need to t r a i n students to recognize and solve problems i n t h e i r everyday experiences f i r s t , before introducing problems with i n s u f f i c i e n t data i n subject area which might merely confound and frustrate the subjects. The f a c t that the group inquiry approach was favoured over the in d i v i d u a l inquiry approach seems to indicate that J . Richard Suchman, "Inquiry Training: Building S k i l l s for Autonomous Discovery," Merrill-Palmer Quarterly, VTI (July, 1961), 152. 2 Blank and Covington, l o c . c i t . 40 the e f f i c i e n c y o f i n f o r m a t i o n g a t h e r i n g i n the group s i t u a t i o n outweighs any disadvantage t h a t c e r t a i n group members do not f u l l y p a r t i c i p a t e i n the i n q u i r i n g . N o t i c i n g t h a t mean sco r e s were, i n . g e n e r a l , h i g h e r on the i n f o r m a t i o n p r e t e s t s than on the cor r e s p o n d i n g p r e t e s t s i n d i c a t e s t h a t students a l r e a d y possess some a b i l i t y t o analyze whether a problem c o n t a i n s s u f f i c i e n t i n f o r m a t i o n to s o l v e i t , b u t a p p a r e n t l y t h i s a b i l i t y can be f u r t h e r improved through t r a i n i n g . T h i s study tends t o support the c o n t e n t i o n t h a t i n q u i r y t r a i n i n g can be a v a l u a b l e t o o l i n mathematics e d u c a t i o n i n d i s t i n c t s i t u a t i o n s . L i m i t a t i o n s The b a s i c l i m i t a t i o n s o f t h i s study f a l l i n t o two c a t e g o r i e s : the d u r a t i o n o f the r e s e a r c h and the v a l i d i t y o f the measuring instruments. I n terms o f d u r a t i o n , such a short-term study can a t b e s t g i v e o n l y i n d i c a t i o n s o f the e f f e c t s o f r e l e v a n t v a r i a b l e s , which r e q u i r e l o n g e r s t u d i e s f o r f u r t h e r examination. Secondly, because two o f the measuring instruments were exp e r i m e n t e r - c o n s t r u c t e d , and because r e l i a b i l i t y c o -e f f i c i e n t s were based on a r e l a t i v e l y s m a l l sample, some doubt can c o n c e i v a b l y be c a s t on the accuracy o f the sco r e s and t h e i r s i g n i f i c a n c e . U n f o r t u n a t e l y , no s t a n d a r d i z e d t e s t a t t h i s l e v e l d e a l s w i t h a l l o f the m a t e r i a l i n v o l v e d i n the study. C e r t a i n l y , t h i s should be r e c t i f i e d once the study o f geometry a t the elementary l e v e l becomes more widespread and perhaps 4 1 then the experiment can be r e r u n u s i n g the new measuring instruments. One l a s t a l t e r a t i o n o f t h i s d e s i g n which might prove u s e f u l would be t o examine the same students s e v e r a l months l a t e r t o determine the r e t e n t i o n e f f e c t s , i f any, o f the i n q u i r y t r a i n i n g . Suggestions f o r Future Research Of obvious m e r i t would be a l o n g term study d e a l i n g w i t h the same i s s u e s as are i n v o l v e d i n t h i s r e s e a r c h p r o j e c t . Only then c o u l d one make a t r u e e v a l u a t i o n o f the m e r i t s o f t h i s t r a i n i n g system on anything other than a s h o r t term b a s i s . T h i s experiment c o u l d s t i l l encompass group and i n d i v i d u a l approaches as w e l l as i n q u i r y t r a i n e d and n o n i n q u i r y t r a i n e d methods. In a d d i t i o n , a study u s i n g programmed l e a r n i n g i n q u i r y t r a i n i n g as compared t o some o t h e r i n d i v i d u a l i z e d l e a r n i n g i n q u i r y t r a i n i n g s i t u a t i o n s hould be conducted t o determine whether the programmed l e a r n i n g a s p e c t i s an important one i n i n d i v i d u a l i n q u i r y t r a i n i n g . To determine the e x t e n t to which s u b j e c t c o n t e n t i n f l u e n c e s the experimental outcomes, a study t o compare i n q u i r y t r a i n i n g i n v o l v i n g g e n e r a l i z e d problems as compared t o a program w i t h s p e c i f i c mathematical content might be conducted, where two treatments employ programmed m a t e r i a l and two treatments u t i l i z e a d i f f e r e n t form o f i n d i v i d u a l i z e d t r a i n i n g . 4 2 Studies to determine whether various group-training approaches d i f f e r from a group-inquiry approach in both general-ized settings and s p e c i f i c content settings might be devised to decide whether the i n t e r a c t i o n e f f e c t s of inquiry t r a i n i n g and group learning are more powerful than group learning v i a other techniques. 43 BIBLIOGRAPHY Balow, Irving H. "Reading and Computation Abi l i t y as Determin-ants of Problem Solving," The Arithmetic Teacher, XI (January, 1964), 18-22. Blank, Stanley S., and Martin Covington. "Inducing Children to Ask Questions in Solving Problems," Journal of Educational  Research, LIX (September, 1965), 21-27. Bruner, Jerome S. On Knowing. Cambridge: Harvard University Press, 1962. , and others. Contemporary Approaches to Cognition. Cambridge: Harvard University Press, 1957. , and others. Studies in Cognitive Growth. New York: John Wiley and Sons, Inc., 1966. Cunningham, John D. "Rigidity in Children's Problem Solving," School, Science, and Mathematics, LXVT (April, 1966), 377-387. Dewey, John D. How We Think. New York: D.C. Heath and Co., 1910. Glaser, Robert, Dora E. Damrin, and Floyd M. Gardner. "The Tab Item: a Technique for the Measurement of Proficiency in Diagnostic Problem Solving Tasks," Educational and  Psychological Measurement, XIV, (1954), 283-293. Holt, John. How Children F a i l . New York: Dell Publishing Co., 1964. James, Jim Butler. "A Comparison of Performance of Sixth Grade Children in Three Arithmetic Tasks: Typical Textbook Verbal Problems, Revised Verbal Problems, Including Irrelevant Data, and Computational Exercises," Dissertation  Abstracts, 28:2030 B, November, 1967. John, Erwin Roy. "Contributions to the Study of the Problem Solving Process," Psychological Monographs, LXXJ, 447 (1957). Karlins, Marvin. "Conceptual Complexity and Remote-Associative Proficiency as Creativity Variables in a Complex Problem-Solving Task," Journal of Personality and Social Psychology, VT, 3 (1967), 264-278. Kirk, Roger E. Experimental Design: Procedures for the Behavioral  Sciences. Belmont, C a l i f . , : Brooks/Cole Publishing Co., 1968. 43 Klausmeieer, Herbert J., and Chester w. Harris. Analyses of  Concept Learning, New York: Academic Press, 1966. Newton, David E. "The Dishonesty of Inquiry Teaching," School, Science, and Mathematics, LXVTII (December, 1968), 807-810. O'Brien, Thomas C , and Bernard J. Shapiro. "The Development of Logical Thinking in Children," American Educational  Research Journal, V (November, 1968), 531-541. Orem, R.C. (ed.) A Montesorri Handbook. New York: Capricorn Books, 1966. Rimoldi, H.J.A. "A techniques for the study of Problem Solving, Educational and Psychological Measurement, XV, (1955), 450-461. , M. Aghi, and G. Burger. "Some Effects of Logical Structure, Language, and Age in Problem Solving in Children, The Journal of Genetic Psychology, CXII (1968), 127-143. Shulman, Lee S. and Evan R. Keisler (eds.). Learning by Discovery: A C r i t i c a l Appriasal. Chicago: Rand McNally and Co., 1966. Smith, Ewart E., and Stanford S. Kight. "Effects of Feedback on Insight and Problem Solving Efficiency in Training Groups," Journal of Applied Psychology, XLIII, 3 (1959), 209-2111. Suchman, J. Richard. "Inquiry Training; Building Skills for autonomous Discovery," Merrill-Palmer Quarterly, VII (July, 1961), 147-169. Wertheimer, Max. Productive Thinking. New York: Harper and Row, 1945. White, Robert W. "Motivation Reconsidered: The Concept of Competence," Psychological Reviewf LXVI, 5 (1959), 297-331. APPENDIX A CRITERION PRETEST 46 Pretest INSTRUCTIONS: FOR EACH PROBLEM, CIRCLE THE ANSWER THAT YOU THINK IS MOST CORRECT. CHOOSE ONLY ONE ANSWER FOR EACH PROBLEM. USE ANSWER (d) ONLY IF YOU THINK THE PROBLEM DOES NOT GIVE YOU ENOUGH INFORMATION TO FIND AN ANSWER. THERE IS NO TIME LIMIT, SO TRY TO ANSWER AS MANY OF THE QUESTIONS AS YOU CAN. 1. Find the area of the figure below. 3 f t . 4 f t . (a)7 sq. f t . (b)12 sq. f t . (c)6 sq. f t . (d) not enough information 2. These two circles touch at two points. If there are two circles, one with a radius of 2 yards and i one with a radius of 1 yard, at how many points do they touch? (a) 0 (b) 1 (c) 2 (d) not enough information 3. Using four points, at most how many lines can be drawn connecting them two at a time? (a) 4 (b) 8 (c) 6 (d) not enough information 4. In a group o f c h i l d r e n , 3 out o f 5 are g i r l s . I f there are 50 boys, how many g i r l s are t h e r e ? (a) 25 (b) 75 (c) 30 (b) not enough i n f o r m a t i o n S it 5. F i n d the area o f the f i g u r e below. / " ' 'i] •• ? 5 ft/ (a) 15 s q . f t . (b)25 s q . f t . (c)20 s q . f t . (d) not enough i n f o r m a t i o n 6. Two sandwiches and a soda together c o s t 85C. One sandwich and a soda c o s t 30C. How much does a sandwich c o s t ? (a) 30* (b) 35* (c) 25$ (d) not enough i n f o r m a t i o n 7. In a c o l l e c t i o n o f c o i n s worth 93$ made up o f n i c k e l s , dimes, and p e n n i e s , there are 13 pennies and twice as many n i c k e l s as dimes. How many n i c k e l s are t h e r e ? (a) 4 (b) 6 (c) 8 (d) not enough i n f o r m a t i o n 8. Here i s the way t o draw a t r i a n g l e on three p o i n t s . Using f i v e p o i n t s , how many t r i a n g l e s can be drawn? (a) 3 (b) 5 (c) 10 (d) not enough i n f o r m a t i o n 9. F i n d the area o f the f i g u r e below: — P l i f t . (a) 10 s q . f t . (b)8 s q . f t . ( c ) l 2 s q . f t . (d) not enough i n f o r m a t i o n 43 10. There were 100 trees i n an orchard. Half of the trees were pear trees. of the trees i n the orchard died. At most, how many pear trees died? (a) 50 (b) 40 (c) 35 (d) not enough information 11. Which of the figures below has the greatest area? c r Aft. J3 (a) i (b) i i (c) the same (d) not enough information 12. A jar contains 2 quarts of milk and 6 quarters of water. The mixture i s completely blended. % of the mixture i s poured out and the jar i s r e f i l l e d with milk. How much milk i s i n the j a r now? (a) 4 qts. (b) 6 q t s . (c) 5 qts. (d) not enough information APPENDIX B INQUIRY TRAINING WORKSHEETS-ARITHMETIC AND GEOMETRIC 54 T3 Problem II - Sheet I. 1. John had 15 boxes of 30 pencils each and 23 boxes of 60 pencils each. How many pencils did John have? 2. On a t r i p of 180 miles. B i l l s * father drove 60 miles at a speed of 30 miles per hour, and the rest of the distance at a speed of 60 miles per hour. How long did the tr i p take? 3. For each 5 acres of beans and 8 acres of corn Mr. Jones planted, Mr. Smith grew only 3 acres of beans and 6 acres of corn. If Mr. Jones planted 90 acres of beans and 240 acres of corn, how much of each was Mr. Smith growing? 4. Mr. Adams paid Bob $6.40 for washing. i28 windows and re-painting a window s i l l . Bob's rate for window washing was $1 for each five windows. How much did Bob charge for repainting the window s i l l ? 5. Out of each $8 that he earned, Jim put away $2 for saving. He usually earned $10 a week. How many weeks would have he to work to pay for a $75 television set, i f he didn't use his savings? 6. Mr. Smith drove 208 miles on 16 gallons of gasoline. How many gallons would he use at this rate to go 338 miles? 5§ T i & T 2 Problem II - Sheet I 1. John had some boxes of 30 pencils each and some boxes of 60 pencils each. How many pencils did John have? 2. On a c e r t a i n t r i p . B i l l ' s father drove 60 miles at a speed of 30 miles per hour, and the r e s t of the distance at a faster speed. How long d i d the t r i p take? 3. For each 5 acres of beans and 8 acres of corn Mr. Jones planted, Mr. Smith grew only 3 acres of beans and a few more acres of corn. I f Mr. Jones planted some beans and 240 acres of corn, how much of each was Mr. Smith growing? 4. Mr. Adams paid Bob $6.40 for washing windows and repaint-ing a window s i l l . Bob's rate for window washing was $1 for each f i v e windows. How much did Bob charge f o r repainting the window s i l l ? 5. Out of each $8 that he earned, Jim put away some of i t for saving. He usually earned $10 a week. How many weeks would he have to work to pay for a t e l e v i s i o n set i f he didn't use his savings? 6. Mr. Smith drove 208 miles. How many gaBons of gasoline would he use at the rate of hi s o r i g i n a l t r i p to go 338 miles? 56 T 3 Problem I I - Sheet I I 1. A r e c t a n g l e i s twice as long as i t i s wide. The area i s 162 s q . u n i t s . How long i s the r e c t a n g l e ? 2. The number o f square inches i n a c e r t a i n square's area i s thr e e times the number o f inches around the square. How lo n g i s a s i d e o f the square? 3. One p i p e can f i l l a tank i n 3 hours and another p i p e can f i l l the tank i n 6 hours. How l o n g w i l l i t take t o f i l l the tank i f b o t h p i p e s are used a t the same time? 4. I f Jim can mow a lawn i n 20 minutes and Bob can mow the lawn i n 30 minutes, how long w i l l i t take the two boys working t o g e t h e r ? 5. I f a bank charges 16C a week on each $100 borrowed, and a man borrws $175 and pays i t back two weeks l a t e r , how much w i l l he be charged? 6. There are 40 p u p i l s i n Jane's c l a s s - 16 boys and 24 g i r l s . I f % o f the boys are i n the math c l u b and 2/3 o f the g i r l s are i n the music c l u b , how many c l a s s members do not belong to these c l u b s ? 57 T l & T 2 Problem I I - Sheet I I 1. A r e c t a n g l e i s l o n g e r than i t i s wide. The ar e a i s 162 square u n i t s . How l o n g i s the r e c t a n g l e ? 2. The number o f square inches i n a c e r t a i n square *s area i s a m u l t i p l e o f the number o f inches around the square. How lo n g i s a s i d e o f the square? 3. One p i p e can f i l l a tank i n a few hours and another p i p e can f i l l i t even f a s t e r . How l o n g w i l l i t take t o f i l l the tank i f b o t h p i p e s are used a t the same time? 4. I f Jim can mow a lawn i n p a r t o f an hour and Bob can mow the lawn i n a l i t t l e b i t l o n g e r amount o f time, how l o n g w i l l i t take the two boys working t o g e t h e r ? 5. I f a bank charges i n t e r e s t each week on each $100 borrowed, and a man borrows over $100 and pays i t back i n two weeks, how much w i l l he be charged? 6. There are 40 p u p i l s i n Jane's c l a s s , both boys and g i r l s . I f some o f the boys are i n the math c l u b and 2/3 o f the g i r l s a re i n the nrasic c l u b , how many c l a s s members do not belong t o these c l u b s ? APPENDIX C CRITERION TEST 5.9 C r i t e r i o n Test INSTRUCTIONS; FOR EACH PROBLEM, CIRCLE THE ANSWER THAT YOU THINK IS MOST CORRECT. CHOOSE ONLY ONE ANSWER FOR EACH PROBLEM. USE ANSWER (d) ONLY IF YOU THINK THE PROBLEM DOES NOT GIVE YOU ENOUGH INFORMATION TO FIND AN ANSWER. THERE IS NO TIME LIMIT, SO TRY TO ANSWER AS MANY OF THE QUESTIONS AS YOU CAN. 1. What i s the diameter of the largest c i r c l e that can be f i t into a square with an area of 4 square feet? (a) 4 f t . (b) 1 f t . (c) 2 f t . (d) not enough information 2. A bathtub can f i l l i n 15 minutes and empty i n 5 minutes. The tub holds 30 gallons of water. How much water w i l l be l e f t i n the tub a f t e r three minutes i f the tub starts out f u l l and the water drains a t the same time that more i s coming in? (a) 24 g a l . (b) 12 g a l . (c) 18 g a l . (d) not enough information 3. Sue had two quarts of honey. A l i c e took % of t h i s , and then B i l l took h a l f of what Sue had l e f t . How much did B i l l get? (a) 3/4 qt. (b) 1 qt. (c) ±h qt. (d) not enough information 4. Find the area of the figure below. (a)10 sq. f t . (b)14 sq. f t . (c)8 sq. f t . (d) not enough information 5. Find the area of the figure below. / , 60 (a)33 sq. f t . (b)42 sq. f t . (c)30sq. f t . (d) not enough informa-tion 6. These two circles have three separate regions, or pieces How many separate regions are there contained in three circles of the same size which each touch the other two at exactly two points? (a) 6 (b) 7 (c) 5 (d) not enough information 7. Joe's age is three times Susan's. Four years from now Joe w i l l be only twice as old as Susan. How old i s Joe now? (a) 12 (b) 6 (c) 15 (d) not enough information 8. I f a man can drive 84 miles on 6 gallons of gasoline and the tr i p costs S3.06, how many miles can he drive for $16.83? (a) 462 mi. (b) 235 mi. (c) 182 mi. (d) not enough informa-tion 9. Find the area of the figure below. (a) 80 sq. f t . (b) 128 sq. f t . (c) 120 sq. f t . (d) not enough information 10. Find the area of the figure below. (a) 108 sq. f t . (b) 100 sq. f t . (c) 96 sq. f t . (d) not enough information 11. At a pet shop there were 10 more dogs than cats. The shop sold 2 dogs and 2 cats. Now there were twice as many dogs as cats. How many dogs does the shop have now? (a) 12 (b) 10 (c) 20 (d) not enough information 12. One side of a tria n g l e i s twice as long as another, and the t h i r d side i s three inches less than the sum of the other two. I f the perimeter of the tria n g l e i s 39 inches what i s the length of the t h i r d side? (a) 21 (b) 18 (c) 12 (d) not enough information 13. Find the area of the figure below. 4 ft. (a) 36 s q . f t . (b) 38 sq. f t . (c) 34 sq. f t . (d) not enough information 14. John, who had $1 and some nickels had twice as much money as B i l l who had three times as many nickels as John. How much money did B i l l have? (a) 50$ (b) 45<r (c) 60* (d) not enough information 62 15. Four points A,B,C, and D are on a l i n e and B i s between A and C and B i s between D and C, then (a) A-D-C (b) D-B-C (c) B-D-A (d) dot enough information 16. I f four points A,B,C, and D are drawn on a piece of paper with points A and B, B and C, C and D, and D and A a l l four inches apart, the r e s u l t i n g figure ABCD i s (a) a square (b) a l i n e (c) a parallelogram (d) not enough information 17. At 50* an hour how many hours w i l l B i l l have to work to earn enough bo buy a set of t r a i n s , where each t r a i n costs $3.? (a) 6 (b) 12 (c) 9 (d) not enough information APPENDIX D SAMPLE RESPONSE CARDS Problem I I - Sheet I q u e s t i o n 2-Smith's acres o f beans - 54 Jones' a c r e s o f beans - 90 6 a c r e s o f corn f o r each 8 r a t i o f o r corn - 4/3 t o t a l a cres o f Smith - 234 t o t a l acres of Jones - 330 Smith's acres o f corn - 180 Problem I I - Sheet I I q u e s t i o n 3-f i r s t p i p e - 3 hours to f i l l second p i p e - 6 hours t o f i l l average f i l l i n g time - 4% hours r a t e o f f i r s t p i p e - 200 g a l l o n s per hour r a t e o f second p i p e - 100 g a l l o n s per hour c a p a c i t y o f tank - 600 g a l l o n s APPENDIX E ANALYSES OF VARIANCE FOR ADJUSTED GROUP MEAN SCORES Analysis of Variance for Total Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted Group Means 2 18.23 9.12 1.9062 Zero Slope 2 2819.34 1409.67 294.78*(.01 Error 59 282.15 4.78 Equality of Slopes 4 33.59 8.40 1.86 Error 55 248.56 4.52 Analysis of Variance for Arithmetic Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted Group Means 2 5.92 2.96 2.15 Zero Slope 2 42.97 21.48 15.60*(.01 Error 59 81.26 1.38 Equality of Slopes 4 1.91 .48 .3305 Error 55 79.35 1.44 68 Analysis of Variance for Geometry Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted 2 4.63 2.32 3.43*(.05) Group Means Zero Slope 2 1.03 .52 .74 Error 59 40.91 .69 Equality of Slopes 4 3.31 .83 1.21 Error 55 37.60 .68 Analysis of Variance for Area Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted 2 2.03 1.02 .89 Group Means Zero Slope 2 13.70 6.85 6.01*(.01) Error 59 67.21 1.14 Equality of Slope 4 3.86 .96 .84 Error 55 63.36 1.15 69 Analysis of Variance for Total Info. Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted Group Means 2 55.04 27.52 5.66*( .0 Zero Slope 2 9.19 4.59 .95 Error 59 286.87 4.86 Equality of Slopes 4 17.14 4.28 .87 Error 55 269.74 4.90 Analysis of Variance for Arithmetic Info. Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality o f Adjusted Group Means 2 12.69 6.35 3.22*( .0 Zero Slope 2 3.06 1.53 .78 Error 59 116.12 1.97 Equality of Slopes 4 5.19 1.30 .64 Error 55 110.93 2.02 70 Analysis of Variance for Geometry Info. Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted Group Means 2 4.26 2.13 2.80 Zero Slope 2 .79 .39 .52 Error 59 44.92 .76 Equality of Slopes 4 6.07 1.52 2.15 Error 55 38.85 .71 Analysis of Variance for Area Info. Test Scores Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Equality of Adjusted Group Means 2 3.40 1.70 1.51 Zero Slope 2 6.37 3.19 2.83 Error 59 66.31 1.12 Equality of Slopes 4 1.58 .40 .34 Error 55 64.72 1.18 

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