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An exploratory study of the effect of co-operative group learning, involving tutoring, on the achievement… Murphy, Patrick Aloysius 1972

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AN EXPLORATORY STUDY OF THE EFFECT OF CO-OPERATIVE GROUP LEARNING, INVOLVING TUTORING, ON THE ACHIEVEMENT 'AND ATTITUDES OF GRADE EIGHT PUPILS IN 'NEW MATHEMATICS by PATRICK A. MURPHY B . E d . , U n i v e r s i t y of B r i t i s h Columbia, 196l A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF EDUCATION i n the Department of EDUCATION We accept t h i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In presenting t h i s thes i s i n p a r t i a l fu l f i lment of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y ava i l ab le for reference and study. I fur ther agree that permission fo r extensive copying of t h i s thes i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representat ives . I t i s understood that copying or p u b l i c a t i o n of t h i s thes i s f o r f i n a n c i a l ga in s h a l l not be allowed without my wr i t t en permiss ion. P a t r i c k A, Murphy Department of EDUCATION  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date %M^Ur SO: /? 1Z I I ABSTRACT An exploratory inves t i ga t ion into the ef fect of co-operative group learn ing , invo lv ing t u t o r i n g , on the achievement and at t i tudes of 174 grade eight pupi l s i n new mathematics i s descr ibed. Three volunteer teachers and s ix volunteered mathematics classes were involved. Five hypotheses concerning test performance and one concerning a t t i tudes were advanced. Using the scores obtained i n the mathematics sections of the Stanford Achievement Test (Advanced) and an entering behaviour test of p r i o r mathematics l earn ing , to e s t ab l i sh s i m i l a r i t y of the groups, i n s t r u c t i o n was ca r r i ed out over a period of twelve weeks. A re tent ion tes t was given two months l a t e r . Att i tude scores from data co l l ec ted by Semantic D i f f e r e n t i a l before and a f ter the experiment were analysed using a model for multidimensional analys i s of Semantic D i f -f e r e n t i a l a t t i tude data (McKie and Foster , 1972). Achievement i n algebra learning and re tent ion reached the .05 l e v e l of s t a t i s t i c a l s i g n i f i c a n c e , favouring the experimental group. No di f ferences for treatment occurred for in s t ruc tor e f fec t s , in s t ruc tor by group i n t e r a c t i o n or a t t i tudes at the .05 l e v e l of s t a t i s t i c a l s i g n i f i c a n c e . Conclusions for further research and prac t i ce are drawn. Approved: TABLE OF CONTENTS Page LIST OF TABLES I V Chaptert I . INTRODUCTION 1 1. Statement of Problem ^ 2. D e f i n i t i o n of Terms k I I . REVIEW OF THE LITERATURE 7 1. A c h i e v e m e n t . . . . . . . 7 2. At t i tudes 11 3 . Retention of Learning . 14 I I I . DESIGN 16 1. Sampling Procedure 16 •2. Design and Procedures . . . 18 3. M a t e r i a l s . . . . . 18 k. Achievement Pretests 20 5. Achievement Posttests 21 6. At t i tude Measurement 22 7. Development and P i l o t Test of the At t i tude Measuring Instrument 22 8. Admini s t ra t ion and Scoring of Tests 23 9. Treatments 2k 10. Data Analyses 27 11. Hypotheses 27 IV. RESULTS 29 1. Assessment of Classroom S i tuat ions 29 2. Assessment of Teachers* Wri t ten Comments. 30 3. Results of S t a t i s t i c a l A n a l y s e s . . . . , 31 V. SUMMARY AND CONCLUSIONS kk 1. Summary of Findings kk 2. Conclusions and Discuss ion k6 3 . L imi ta t ions -48 k. Further Research 50 BIBLIOGRAPHY 51 APPENDIX A. . 5^ APPENDIX B -. 57 APPENDIX C 6k IV LIST OF TABLES Table Page 1. Results of Spache Diagnost ic Reading S c a l e s . . . . 8 2. Tabular Representation of D e s i g n . . . . 18 3. Table of Pretests and Dates of A d m i n i s t r a t i o n . . 19 4. Table of Posttests and Dates of Admin i s t r a t ion . 19 5. Table of Means, Standard Deviat ions and Inter-c o r r e l a t i o n fo r Algebra A Test and Algebra B Retention Test 21 6. Summary Table of Pretest Means, Standard Deviat ions and In tercor re l a t ions for Stanford (Form W), Board and At t i tude Measures 32 7. Summary Table of Posttest Means, Standard Deviat ions and In tercor re l a t ions fo r Stanford (Form X ) , Algebra and At t i tude M e a s u r e s . . . . . . . . . 33 8. Summary Analys i s of Variance Table fo r Computation Pre te s t : Stanford Achievement (Form W, Advanced)... 35 9. Summary Analys i s of Covariance Table for Computation Pos t te s t : Stanford Achievement (Form X, Advanced) 35 10. Summary Analys i s of Variance Table fo r Concepts Pre te s t : Stanford Achievement (Form W, A d v a n c e d ) . . . . . . 36 11. Summary Analys i s of Variance Table for Concepts Pos t te s t : Stanford Achievement (Form X, Advanced) , 36 12. Summary Analys i s of Variance Table for Appl i ca t ions Pre te s t : Stanford Achievement (Form W, Advanced) 37 13. Summary Analys i s of Variance Table for Appl ica t ions Pos t te s t : Stanford Achievement (Form X, Advanced) 37 14. Summary Analys i s of Variance Table for Board (Dec.) Pretest 39 V LIST OF TABLES (Continued) Table Page 15, Summary Analys i s of Covariance Table for Algebra (Form A) Posttest kO 16. Summary Analys i s of Covariance Table fo r Algebra Retention (Form B) kO 17, Summary Analys i s of Variance for Summated At t i tude Prescores . kZ 18. Summary Analys i s of Covariance Table for Summated At t i tude Postscores h2 ACKNOWLEDGMENTS The w r i t e r wishes to express hi s s incere apprec ia t ion for the generous help and guidance given by Dr. Denis C. Rodgers, advisor for the the s i s , and for the w i l l i n g a s s i s -tance given by Dr. Stephen F. Fos ter , Dr. Robert Conry, Dr . James S h e r r i l l , Dr . Douglas McKie, Dr. Harold R a t z l a f f , Dr . Arthur H. E l l i o t t and Dr. Roland Gray i n the preparat ion r of the t h e s i s . In a d d i t i o n , the w r i t e r wishes to express his thanks to the p r i n c i p a l , Mr. W. L . Bazeley for his f u l l c o - o p e r a t i o n , and to the Department of Research and Services of the Vancouver Pub l i c School System, p a r t i c u l a r l y Dr. N. E l l i s and Mr. A. Moodie, for the encouragement offered and for prov id ing standardized t e s t i n g mater i a l s . The co-operat ion and e f for t s of the volunteer teachers and t h e i r respect ive grade eight mathematics c lasses are also g r a t e f u l l y acknowledged. The w r i t e r also wishes to give s p e c i a l thanks to Dr. Robert Conry, Dr . Douglas McKie and Mr. Robert David Meldrum M.Sc. for t h e i r u n t i r i n g work on the computer programs. F i n a l l y , s incere apprec ia t ion i s offered to my wife , D o r i s , without whose constant encouragement, as s i s tance , personal s a c r i f i c e and devotion to typ ing , the w r i t i n g of t h i s thes i s would have been more d i f f i c u l t . 1 CHAPTER 1 INTRODUCTION Thelen (1968) has stated that perhaps the most compell ing reason fo r using students as tutors i s to change the s o c i a l -p sycho log ica l climate of the school from i n d i v i d u a l competitiveness to concern for each other . Tutor ing through the development of processess of co-operative i n q u i r y i n v o l v i n g students performing responsible tasks on behalf of themselves and others i n the school , seems to be a useful innovative pract ice which may improve achievement i n mathematics and develop more pos i t i ve a t t i tudes toward mathematics. Thelen (1970, P. 19) suggests: As common purpose develops, c r o s s - c u l t u r a l , c ros s -generat iona l , and au thor i ty b a r r i e r s to communication are reduced and to le rab le heterogeneity increased. As processes of co-operative i n q u i r y develop, the s o c i a l -psycholog ica l ' c l i m a t e ' of the school begins to change from competitiveness to concern for each other , and the anxiety which d i s t o r t s c h i l d r e n ' s view o f each other and themselves i s reduced. During the past ten years i n widely separated parts of the U.S. a number of schools have experimented with students teaching each other . Some of the more important programs were "Student Team A c t i o n " , Fleming (1969) , i n Port land , Oregon; "Cadet Teacher C e r t i f i c a t e " , Rogers ( I 9 6 7 ) , i n Overland Park, Kansas; "High School Friend Program", Konikow (undated), Downers Grove, Chicago. These innovative programs arose spontaneously and simultaneously i n many parts of the country. Thelen be l ieves many of them have a r i s en because of what he 2 l i k e s to c a l l * caring". He c i t e s the following experiment conducted by Rosenthal and Jacobson and reported i n The S c i e n t i f i c American, as straightforward evidence that caring r e a l l y matters* A r e l a t i v e l y non-verbal i n t e l l i g e n c e t e s t , that was purported to predict imminent •blooming* or i n t e l l e c t u a l growth, was ad-ministered i n a west coast elementary school. In each of the 18 classes (an average, below average and above average track i n each of the si x grades) about 20% of the children were reported to t h e i r teachers as being l i k e l y to show unusual i n t e l l e c t u a l gains i n the coming year. Actually, the names had been picked by means of a table of random numbers. The ch i l d r e n were re-tested eight months l a t e r , the tests were scored by the investigators, and the change i n IQ f o r each c h i l d was computed. As Rosenthal and Jacobson f i r s t reported i n Psychological Reports i n 1966, f o r the school as a whole the supposed 'bloomers' showed a mean gain of 12,2 points compared with 8.4 f o r the control group. The e f f e c t of expectations was greater i n the lower grades than i n the upper grades. The 'bloomers' i n the f i r s t and second grades gained r e s p e c t i v e l y 15.4 and 9.5 points more than the c o n t r o l children. The e f f e c t was more s t r i k i n g (and more independent of age) on the 'reasoning' than on the 'vocabulary' portions of the t e s t j i t was about the same, r e -gardless of 'track' l e v e l . Teachers were also asked at the end of the year to describe t h e i r p u p i l s . They characterized the 'bloomers' as having a better chance of be-coming successful} as being s i g n i f i c a n t l y more in t e r e s t i n g , curious and happy, and as somewhat more appealing, adjusted and a f f e c t i o n a t e . Curiously, those control-group c h i l d r e n who gained i n IQ were not rated t h i s favorably by t h e i r teachers; i n f a c t , the more the undesignated childre n - p a r t i c u l a r l y the slow-track control children - gained i n IQ, the more they were regarded as being less well adjusted, i n t e r e s t i n g 3 and a f fec t ionate . Rosenthal and Jacobson point out that t h e i r f ind ings , which have been supported i n subsequent s tudies , may-bear importantly on current e f for t s to improve the education of c h i l d r e n i n c i t y slums. ( S c i e n t i f i c American Nov. 1967 P.5*0 Thelen (1968) commenting on Rosenthal ' s experiment says: Old hands at t h i s racket have known for years that the o ld put-down crack ' a l l you're doing i s c a p i t a l i z i n g on the Hawthorne Research' , i s exac t ly what we are t r y i n g to do. The Hawthorne ef fect i s the sense of being cared f o r , and i t ' s the main e f f ec t . The other things we do are just to produce the Hawthorne e f f e c t . (Thelen 1968, P. 8). Thelen be l ieves that the climate of a classroom can be changed through the development of the norm of concern for each other . He wants to subst i tute processes of co-operative inqu i ry fo r the anxious competitiveness which he maintains present ly d i s t o r t s the c h i l d r e n ' s perceptions of each other . Thelen considers that i n the he lp ing r e l a t i o n s h i p knowledge i s the currency of i n t e r a c t i o n . He has stated that t u t o r i n g can increase by a large f ac tor the amount of teaching going on i n a classroom. Ins t ruc t ion could be i n d i v i d u a l i z e d on a one-to-one t u t o r i a l ba s i s . He states that t u t o r i n g can provide remedial resources pinpointed to the p u p i l s when they most need help . " I t i s one th ing to schedule opportunity for youngsters to get he lp , but what about having students ava i l ab le who can help i n a c r i s i s when the teacher has to keep on with the c l a s s ? " (Thelen 1968, P. 12). F i n a l l y he says one of the things that got him started on t u t o r i n g i n the f i r s t place i s the p o s s i b i l i t y of using t u t o r i a l a c t i v i t y as a way to develop li-the c h i l d ' s own insight into the teaching-learning process so that he can co-operate more e f f e c t i v e l y with his own teachers i n meaningful learning a c t i v i t i e s . That i s , i t might contribute to the objective of the c h i l d ' s learning how to learn. Statement of the Problem S p e c i f i c a l l y , t h i s study w i l l attempt to investigate the e f f e c t of co-operative group learning, involving t u t o r i n g , on the achievement and attitudes of grade eight pupils i n algebra. I t i s designed to provide data which may be h e l p f u l i n suggesting answers to following questions. (1) Does a co-operative form of group learning, i n which students help each'other by tutoring, r e s u l t i n better learning as indicated by higher achievement scores and greater retention of the materials learned than occurs i n a t r a d i t i o n a l classroom? (2) Does co-operative group learning bring about a p o s i t i v e change of attitude toward the learning of mathematics? D e f i n i t i o n of terms The term co-operative group l e a r n i n g ' s used i n t h i s study ; means having two students, who have pretest scores above the c l a s s mean tutor and provide help f o r two students, who have scores lower than the class mean. Both pairs w i l l have t h e i r desks pulled together to form a foursome, which w i l l work together. The help provided w i l l be mainly i n the form of tutoring. Tutoring^as referred to i n t h i s study ?means having the 5 student who understands how to do the work, show the student who doesn't know. The intent i s that there w i l l he a one-to-one r e l a t i o n s h i p i n which the stronger p u p i l w i l l be the tu tor and the weaker p u p i l the tutee . The term a t t i t u d e , as used i n t h i s study, i s defined as a learned, emotional ly toned p r e d i s p o s i t i o n to react i n a consis tent way, favourably or unfavourably toward mathematics. Algebra , as re ferred to i n t h i s study, means Unit Four: Algebra i n Introduct ion to Mathematics (Brumfiel , E i c h o l z and Shanks 1962, 175-240). Computation, as used i n t h i s study, i s presumed to measure p r o f i c i e n c y i n computational s k i l l s re l a ted to a d d i t i o n , subt rac t ion , m u l t i p l i c a t i o n and d i v i s i o n . The four operations are extended to include computation with f r a c t i o n s , s o l u t i o n of a number sentence, and s o l u t i o n of per cent examples. Concepts, as used i n t h i s study, i s presumed to measure the understanding of place value, Roman numerals, opera t iona l terms, the meaning of f r a c t i o n s , number s e r i e s , number names, e s t imat ion , average, number sentences, per cent , decimal f r a c t i o n s , common denominator, rounding whole numbers, decimal and common f r a c t i o n equiva lents , reduct ion of f r ac t ions and geometric terms. I t a lso measures knowledge of formulas, propert ies of operat ions , prime numbers and understanding of non-decimal bases. A p p l i c a t i o n s , as used i n th i s study, i s presumed to measure reasoning with problems taken from l i f e experiences. The p u p i l i s required to apply h i s mathematical knowledge and a b i l i t y to think mathematically i n p r a c t i c a l s i tua t ions which concern area,-volume, r a t i o ; , graphs , tables , s ca le s , per cent, business t ransac t ions , averages, problems with c i r c l e s and other geometric f igures and the s e l e c t i o n of mathematical models for problems. 7 CHAPTER TWO REVIEW OF THE LITERATURE Most of the l i t e r a t u r e on t u t o r i n g comes i n the form of a r t i c l e s , l e t t e r s or anecdotal reports wr i t ten on the various tu tor ing pro ject s which have sprung up spontaneously i n many parts o f the U.S. Very few s tudies , with the exception of those i n reading , have s t a t i s t i c a l l y analysed any form of empi r i ca l data. Not a s ing le study invo lv ing t u t o r i n g i n mathematics was found i n which e m p i r i c a l data had even been c o l l e c t e d . Most of the s tudies seem to have been wr i t ten under Thelen ' s assumption that t u t o r i n g 'works* . 'Educators , almost to a man, f e e l that t u t o r i n g works. I can th ink of no other innovat ion which has been so cons i s t en t ly perceived as success ful * (Thelen 1 9 7 0 , P . 1 7 ) . Many of the studies involved older tu tors working^with much younger students and as a r e s u l t cannot be d i r e c t l y compared with the present study. More than h a l f o f the ava i l ab le l i t e r a t u r e i s contained i n a r t i c l e s , papers and working essays w r i t t e n by Thelen ( 1 9 6 8 ) . Many of these are published i n The Thelen C o l l e c t i o n ava i l ab le from the U n i v e r s i t y of Chicago. Achievement Schoeler and Pearson ( 1 9 7 0 ) analyzed the changes occurr ing i n 1 1 5 below average readers enro l led i n a Milwaukee community-sponsored Voluntary Reading Tutor ing Program for periods ranging from 1 to k.6 months. An attempt was made to assess improvements 8 i n reading competence as w e l l as i n a t t i tude towards reading. Before-and-after reading competence was assessed using the Spache Diagnost ic Reading Sca les . A l l e ight parts of the Spache Diagnost ic Reading Scales showed above average ga in during the average J.l month t e s t ing period as shown i n Table I. TABLE I Results o f the Spache Diagnost ic Reading Scales Expected average gain 3.1 months Tes t ing period 3.1 months Months 95 Percent ^ Test Sect ion Mean Gain Confidence In te rva l Word Recognit ion 5.2 4.2 - 6.2 O r a l Reading 3.6 3.7 - <K5 Consonant Sounds 8.9 7.3 10.5 Vowel Sounds 4.3 4.2 - 4.4 Consonant Blends 6.8 5.2 - 8.4 Common S y l l a b l e s 4.4 - 6.4 Blending 4.4 3.3 - 5.5 L e t t e r Sounds 9.2 7.2 -11.2 Note - Table I reproduced from The Reading Teacher, 23« A p r i l 1970, 628. The lowest mean ga in was 3.6 months i n o r a l reading. When confidence i n t e r v a l s were ca lcu la ted at the 95 percent l e v e l of confidence, only the o r a l reading score d id not meet the c r i t e r i o n . In a l l categories pup i l s achieved at a bet ter than average rate despite the fact that they, up to. that point i n t h e i r school careers , had been underachievers i n reading. Subjective assessments of a t t i tude change were achieved through quest ionnaires administered to teachers , parents, pupi l s and student t u t o r s . Evidence of improvement was found i n p u p i l s ' a t t i tudes toward reading , school and themselves. Although evidence about a t t i tude b u i l d i n g i s not conclusive because of i t s present ly subject ive nature, s t a t i s t i c a l l y s i g n i f i c a n t gains i n a t t i tude at the .05 l e v e l were obtained. Only 3 o f the o r i g i n a l 115 pup i l s involved did not wish to continue i n the program. Frager and Stern (1970) conducted a study i n Los Angeles i n v o l v i n g 48 s i x t h grade t u t o r s , both high and low achievers , and 48 k indergarten c h i l d r e n , who had been -tested and found i n need of remedial work. Tutors received two modes of counse l l ing on t u t o r i n g procedures before assuming t h e i r d u t i e s . The kindergarten c h i l d r e n were d iv ided into three t r e a t -ment groups* c h i l d r e n taught by tutors who had received counse l l ing by method one, c h i l d r e n taught by tutors counsel led by method two, and a t h i r d group used as a c o n t r o l . Within each of the two experimental treatments, h a l f the c h i l d r e n were taught by tutors who had made high scores and h a l f by those who made low scores on the Stanford Achievement Tes t . The language-readiness program used by the tutors i n working with the kindergarten c h i l d r e n was the McNeil ABC Learning A c t i v i t i e s (1966). Using the c r i t e r i o n te s t provided with the McNeil ABC Learning A c t i v i t i e s as a pretes t and posttest measure, i t was demonstrated that the kindergarten c h i l d r e n who received t u t o r i n g , whether by the f i r s t or second method, were super ior to the c h i l d r e n who 10 did not receive t u t o r i n g (t = 6.3 and 8.0, p < .001, r e s p e c t i v e l y ) . Not only d id the c h i l d r e n show gains i n l e a r n i n g , but they also looked forward to the t u t o r i n g sessions and absenteeism became v i r t u a l l y non-exis tent . (Frager and Stern 1970, P. 405). In a program developed i n the Oneida Consolidated School D i s t r i c t (New York) , B e l l ? G a r l o c k and C o l e l l a (196l) made a d e f i n i t e attempt to provide a greater measure of i n -d i v i d u a l i z e d i n s t r u c t i o n and student involvement i n the l ea rn ing process; Each of t h i r t y - f o u r elementary students who had l ea rn ing d i f f i c u l t i e s was provided with a volunteer high school t u t o r . No s p e c i a l s e l e c t i o n procedure was used f o r the tutors except that they must have shown some competence i n t h e i r t u t o r i a l subject . At the conclus ion of the program, quest ionnaires were mailed to the elementary teachers who made r e f e r r a l s , to the student tutors and to the parents of the c h i l d r e n who had been tutored . The quest ionnaires asked fo r comments on the t u t o r i n g program, on benef i t s to the students and tutors and on changes observed i n pupi l s due to p a r t i c i p a t i o n i n the program. B e l l , Garlock and C o l e l l a reported the quest ion-naire r e s u l t s as fol lows 1 There were s l i g h t v a r i a t i o n s i n the wording of the quest ionnaires depending on whether they were addressed to parents , tu to r s , or teachers . A l l of the parents who responded were favorably impressed by the program. A l l reported that they had not iced an improvement i n the c h i l d ' s performance i n the t u t o r i a l area. The parents 11 expressed t h e i r b e l i e f t h a t the high s c h o o l students would acquire f e e l i n g s o f accom-plishment and s e l f - c o n f i d e n c e as a r e s u l t of t h e i r r o l e i n the program. Parents a l s o reported t h a t the t u t o r s would l e a r n to be more p a t i e n t and would acquire a g r e a t e r sense of r e s p o n s i b i l i t y . Some of the opinions expressed by the high s c h o o l students were s i m i l a r t o those expressed by the parents. The students i n d i c a t e d t h a t the g r e a t e s t s a t i s f a c t i o n they had d e r i v e d from the program was t h a t o f having helped o t h e r s . They a l s o f e l t t h a t they were of some value i n improving the academic s t a t u s of t h e i r charges. Those t h a t are p r o s p e c t i v e teachers reported i n -creased enthusiasm about a career i n teach-i n g . Some who were undecided as to the grade l e v e l on which they wanted to teach found t h i s experience h e l p f u l i n making t h i s c h o i c e . The teachers, i n t h e i r responses to the q u e s t i o n n a i r e s , noted p a r t i c u l a r l y the i n -fl u e n c e of the program on the high s c h o o l student. They i n d i c a t e d t h a t these students were s i n c e r e , c o n s c i e n t i o u s , f a i t h f u l , and punct u a l . The teachers a l s o i n d i c a t e d t h a t the elementary p u p i l s were very responsive to the enthusiasm o f the high school s t u -dents. The younger c h i l d r e n knew the day of the t u t o r i n g s e s s i o n , and were eager i n t h e i r a n t i c i p a t i o n of the a r r i v a l o f t h e i r t u t o r s . The changes i n the elementary p u p i l s themselves were t w o - f o l d : F i r s t of a l l , the classroom teachers s t a t e d t h a t the p u p i l s being t u t o r e d showed considerable improvement i n t h e i r academic perform-ance, p a r t i c u l a r l y i n rote-type o p e r a t i o n s , e.g., m u l t i p l i c a t i o n , d i v i s i o n . Secondly, the p u p i l s i n v o l v e d had more p o s i t i v e a t t i t u d e s towards school and t h e i r s t u d i e s . ( B e l l , Garlock and C o l e l l a 1 9 6 l , P.243). A t t i t u d e s Jackson and S t r a t t n e r (1964) have suggested t h a f ' a person with a po s i t i ve a t t i tude toward a subject learns more r e a d i l y than does a person with a negative a t t i t u d e . . . " ( P .524) Montessori (1967) has been quoted as having stated that experience showed us that c h i l d r e n had only a s l i g h t i n t e r e s t i n ar i thmet ic i n comparison with the enthusiasm which they had for wr i t t en language. Wilson (1969) h a s s tated that before the advent of progressive education, a r i thmet ic caused more school f a i l u r e s than any other subject . Even i f t h i s i s no longer true Aiken (1970), who has done considerable work i n the f i e l d , has suggested that a t t i tudes about mathematics probably have not changed i n recent years . Given the widespread b e l i e f that c h i l d r e n do, indeed, have a s t ab le , measurable a t t i tude toward the l earn ing of mathematics, and that t h i s a t t i tude has some e f fec t upon t h e i r immediate achievement i n the subject , continued i n v e s t i g a t i o n i n t h i s area i s j u s t i f i a b l e . A search of the l i t e r a t u r e on a t t i tude t e s t i n g , however, q u i c k l y ind ica te s a s c a r c i t y of re levant , non-cognit ive instruments developed and va l ida ted fo r a jun ior -h igh school populat ion. The reasons for t h i s dearth may be that i t i s extremely d i f f i c u l t to construct items fo r a t t i tude measurement that are comprehensible and yet not transparent to young c h i l d r e n . The Semantic D i f f e r e n t i a l , Osgood, Suci and Tannenbaum (1957), which i s a combination of cont ro l l ed a s soc ia t ion and s ca l ing procedure seems to have the most p o t e n t i a l for t h i s 13 i n v e s t i g a t i o n , because i t possesses the required propert ies of s i m p l i c i t y of format, and content; opaqueness regarding responses and s e n s i t i v i t y to the degrees of a t t i t u d i n a l i n t e n s i t y . Accordingly a general model for mult idimensional amalysis of semantic d i f f e r e n t i a l data constructed by McKie and Foster (1972) w i l l be used. Dutton (1968) found that (1) c h i l d r e n have ambivalent f ee l ings toward a r i thmet i c , l i k i n g some aspects and d i s l i k i n g others (2) younger c h i l d r e n have more p o s i t i v e a t t i tudes t o -ward the new mathematics than do o lder c h i l d r e n . He suggested that the f indings be used as a basis for a d d i t i o n a l research on a t t t i tude development and as clues fo r improved i n s t r u c t i o n a l prac t ices i n the teaching of new mathematics. Neale (1969) discussed a t t i t u d i n a l ob ject ives and a t t i tude toward l ea rn ing mathematics. The subjects were 215 grade s i x pup i l s i n eight classrooms i n suburban elementary schools i n Minnesota. He found cor re l a t ions between achievement and a t t i tudes were low - (.2 to .4) and that inc rea s ing ly un-favourable a t t i tudes developed as students went through schoo l . Anttonen (1969) examined the r e l a t i o n s h i p s between mathematics a t t i tude and mathematics achievement over a s i x year per iod from the la te elementary to the la te secondary school l e v e l . The subjects were 607 students from an above average socio-economic suburb of St . Paul , Minnesota. Using a .05 l e v e l of s i gn i f i cance the r e su l t s showed a s i g n i f i c a n t pos i t ive 14 c o r r e l a t i o n of .305 between elementary a t t i tude scores and secondary a t t i tude scores . S i g n i f i c a n t po s i t i ve c o r r e l a t i o n s of .2 to .4 at the .05 l e v e l of s i gn i f i cance also exis ted between a l l measures of a t t i tude and achievement. Retention of Learning Word and Davis (1938) showed that p r i n c i p l e s , genera l iza t ions and app l i ca t ions of p r i n c i p l e s were remembered much be t te r over periods of time as opposed to f a c t u a l m a t e r i a l . S i m i l a r l y Anderson (1949) revealed that knowledge of number facts learned with understanding were reta ined more e f f e c t i v e l y than when the facts were learned i n a mechanical , rote fashion. Bruce and Freeman (1942) s tudied the rates of f o r g e t t i n g for mater ia l s with varying degrees of meaningfulness. They found that i n i t i a l loss was very rapid for both nonsense-syl lables and for poetry . A f t e r t h i r t y days, however, the group that had learned an equivalent amount of poetry reta ined about twice as much of the mate r i a l learned. Underwood (1968) maintains that r e t e n t i o n i s p r i m a r i l y and perhaps only , r e l a t ed to the degree of o r i g i n a l l e a rn ing . Thus i t i s c l ea r that a l i s t of f ive 3 - l e t t e r words, such as CAT, PEN, BUS, FAR, AND ELK, would be learned more r a p i d l y than a l i s t of f ive non-words, such as RZL, DBQ, HFG, BJX, and PCR. T e c h n i c a l l y , we say the words have a higher l e v e l of meaningfulness than do the non-words, but i f both sets are learned to an equal degree, they w i l l be equal ly remembered 24 hours l a t e r . The c r i t i c a l point i s the 15 "equal degree of l e a r n i n g , " because degree of l earn ing i s the one var iab le that does inf luence the amount of fo rge t t ing (Underwood, 1968 P. 538). Concerning i n d i v i d u a l d i f ferences i n rate of l earn ing and r e -t e n t i o n he w r i t e s i A s low-learning student, there fore , may take much longer to achieve a l e v e l of l earn ing a t ta ined by a r a p i d - l e a r n i n g student, but g iven the equivalent degree of l e a r n i n g , f o rge t t ing does not d i f f e r . (Underwood 1968, P. 539).^ CHAPTER THREE DESIGN The present study i s designed to invest igate the e f fec t of co-operative group l e a r n i n g , i n v o l v i n g t u t o r i n g , on the achievement and a t t i tudes of grade eight pup i l s i n new mathematics. This chapter out l ines the procedures to be used i n car ry ing out the i n v e s t i g a t i o n . Spec i f i c a t t ent ion i s g iven to the t o p i c s : sampling procedure, experimental des ign, mater ia l s used, admini s t ra t ion and scor ing of t e s t s , treatments, analyses of data and the hypotheses stated i n the n u l l form. The experiment w i l l be c a r r i e d out from December 14, 1970 to A p r i l 2, 1971. Pretests w i l l be administered during the f i r s t week and posttests during the f i n a l week. A r e t e n t i o n tes t w i l l be g iven two months a f te r the f i n a l pos t tes t . Sampling Procedure The populat ion for the. study w i l l cons i s t of a l l the grade eight students (N=311) i n a Vancouver, B r i t i s h Columbia high school o f 1700 p u p i l s . Three of the four mathematics teachers at the grade e ight l e v e l have volunteered. The volunteers teach three , three and two classes r e s p e c t i v e l y ; however, the teachers having three classes w i l l have one of t h e i r classes randomly e l iminated from the experiment. Thus s i x grade e ight c lasses w i l l remain. Of the two classes being taught hy each teacher, the c lass v/hich receives the exper i -mental treatment w i l l be determined by a co in f l i p . The fourth mathematics teacher at the grade eight l e v e l i s not ava i l ab le to p a r t i c i p a t e i n the study. Teacher 1 i s s i n g l e , i n her twenties and has a B .A. honours i n E n g l i s h . She has been teaching for four years . Teacher 2 i s married and has a B .A. i n P o l i t i c a l Science and Economics and an M.A. i n Hi s to ry . He has been teaching for twelve years . Teacher 3 i s married and has a B . E d . , an M.Ed, and a Ph.D. i n Educat ional Psychology. He has been teaching for ten years . A l l three teachers have mathematics majors and have taught mathematics r e g u l a r l y dur ing t h e i r teaching careers . They are a l l enthus ia s t i c to p a r t i c i p a t e i n the experiment. The teachers were assigned t h e i r c lasses i n Septemberj each c la s s cons i s t s o f pup i l s placed i n i t i n a non-systematic manner by computer. The computer assigns pupi l s to classes according to the number a lready i n the c l a s s . The c lass with the fewest members automat ica l ly receives the next p u p i l . F i n a l c las s s izes are 29, 32, 3^, 33 and 36 p u p i l s . The reason for the va r i a t ions i n c lass s ize i s that not a l l students assigned to each c las s i n June return to enro l i n September. For the purposes of data analyses, tes t scores w i l l be reta ined for only twenty-nine members of each c l a s s . The reason fo r t h i s i s to maintain an orthogonal des ign. In t h i s study the lowest c lass enrollment i s twenty-nine. To get each c e l l down to twenty-nine^score cards w i l l he removed at random from the classes with more than twenty-nine. Those students whose cards are removed w i l l s t i l l " p a r t i c i p a t e " i n the experiment although t h e i r scores w i l l not he analysed. Design and Procedures Data w i l l be c o l l e c t e d from 1 7 4 pup i l s i n s i x grade eight c l a s se s , with each teacher teaching two c las ses . The design w i l l be as shown i n Table 2. Table 2 Tabular Representation of Design  Treatment Teacher Experimental ( 1 ) Contro l (2,) 1 n=29 (29) n=29 (32) 2 n=29 (32) n=29 ( 3 4 ) 3 n=29 ( 3 3 ) n=29 ( 3 6 ) N=87 ( 9 4 ) N=87 ( 1 0 2 ) Notet Real c las s s izes i n parentheses. Mater ia l s The ob ject ive eva luat ion program for t h i s study w i l l involve f i v e p re te s t s , f i v e post tests and a r e t e n t i o n t e s t to be administered according to the date schedule shown i n Table 3 and Table 4 . 19 Table 3 Table of Pretests and Dates o f Adminis t ra t ion  Pretests Date A. Stanford Achievement (Form W. Advanced, 1 9 6 4 ) ( 1 ) Ar i thmet ic Computation Dec, 1 4 , 70 ( 2 ) Ar i thmet ic Concepts Dec. 15, 70 ( 3 ) Ar i thmet ic App l i ca t ions Dec. l 6 , 70 B. Vancouver School Board Mathematics Grade 8 December Test Dec. 17» 70 C. McKie-Foster At t i tude Measurement Scale ( 1 9 7 2 ) Dec. 1 8 , 70 Table 4 Table of Posttests and Dates of Admini s t ra t ion Post tes t s A. Stanford Achievement (Form X. Advanced, 1 9 6 4 ) ( 1 ) Ar i thmet ic Computation March 2 9 , 71 ( 2 ) Ar i thmet ic Concepts March 30, 71 ( 3 ) Ar i thmet ic App l i ca t ions March 3 1 , 71 B. Algebra Test (Form A) A p r i l , 1 , 71 C. McKie-Foster At t i tude Measurement Scale ( 1 9 7 2 ) A p r i l 2 , 71 Retention Test D. Algebra Test (Form B) June 7 , 71 Achievement Pretests The Stanford Achievement Test (Advanced Battery) i s p r i m a r i l y designed for t e s t i n g from the beginning of Grade 7 to the end of Grade 9. The Advanced Battery contains e ight tests and the three of these which dea l with grades 7 to 9 Arithmet ic w i l l be used, namely: Ar i thmet ic Computation, Ar i thmet ic Concepts and Ar i thmet ic A p p l i c a t i o n s . Four forms of t h i s ba t tery W, X, Y and Z are ava i l ab le and the W and X forms w i l l be used as pretests and posttests r e -s p e c t i v e l y . According to the information suppl ied by the authors, K e l l e y , Gardner, Madden and Rudman, (1964, P.3) these tes t s are "matohed fo r content and d i f f i c u l t y ; " they represent "equa l ly good measures of t h e i r respect ive sub-jects and y i e l d d i r e c t l y comparable r e s u l t s . " The pretest (Form W), used mainly as a c o n t r o l var i ab le and i n con-junct ion with the Vancouver School Board Grade E ight t e s t , which has been g iven year ly at Christmas from 1959 to 1965, forms the basis for ranking the students i n the experimental group. The board t e s t , i n conjunct ion with the three Stanford t e s t s , w i l l be used p r i m a r i l y as an enter ing behaviour tes t to e s t a b l i s h the s i m i l a r i t y of the s i x c las ses . The board t e s t i s presumed to measure the competencies required to begin the study of algebra and w i l l be used as a s t a t i s -t i c a l c o n t r o l for the study. Achievement Posttests The Stanford Posttests (Form X) w i l l he used to measure the mathematical development of the classes involved i n the experiment. The algebra post tes t and' the r e t e n t i o n te s t w i l l be constructed by the experimenter from e x i s t i n g p a r a l l e l forms of the Vancouver School Board Grade E ight June mathematics t e s t s . These forms have been used year ly by the School Board i n a l l i t s schools form 1959 through 1968. Ofrly the questions which dea l with the algebra mater i a l to be studied w i l l be used. The two forms A and B thus constructed, conta ining forty-one questions each, w i l l be p i l o t - t e s t e d by administer ing them on successive days to the experimenter's Grade Nine p u p i l s . Means, standard deviat ions and i n t e r c o r r e l a t i o n s for the p i l o t tes t are shown i n Table 5« Table 5 Table of Means, Standard Deviat ions and I n t e r c o r r e l a t i o n for Algebra A T.est and Algebra B Retention Test . ( P i l o t Test) Form Mean S.D Corr . A (Posttest) 30.33 6.52 B (Retention) 32.11 6.02 .89 The i n t e r - t e s t c o r r e l a t i o n i s .89 which i s a high c o r r e l a t i o n and the experimenter assumes both these tes t s are measuring e s s e n t i a l l y the same algebra m a t e r i a l . The s l i g h t r i s e i n the mean average may be a t t r ibu tab le to prac t i ce e f f ec t . At t i tude Measurement To c o l l e c t data on a t t i tudes toward mathematical concepts, a general a t t i tude towards mathematics dimension w i l l be derived from a model for the mult idimensional ana lys i s of a t t i tude data obtained "by the Semantic D i f f e r e n -t i a l , Osgood, Suci and Tannenbaum ( 1 9 5 7 ) . The model to be used w i l l be adapted from work done by McKie and Foster ( 1 9 7 2 ) . Ten mathematical concepts} Mathematics text book; Operations with whole numbers; Percentage; Operations with f r a c t i o n s ; Ar i thmet ic i n bases other than ten ; So lv ing equations; Union and i n t e r s e c t i o n of se t s ; So lv ing problems; Algebra and Negative numbers w i l l be rated on each of s ix b i p o l a r , seven-point ad ject ive scales namely; good-bad; beau t i fu l -ug ly ; c o l o r f u l - c o l o r l e s s ; va luable-worthless ; en joyable-d i s tas te-f u l ; happy-sad. Development and P i l o t Test of the At t i tude Measuring Instrument. The ten p a r t i c u l a r concepts were chosen by the exper i -menter because they were prominent i n the students ' recent mathematical experiences and re la ted to the in t roduc t ion of algebra to the students. The s ix b i p o l a r ad ject ives used, were be l ieved to i n t e r c o r r e l a t e h igh ly while loading on a s ingle evaluat ive f ac to r . Ten concepts dea l ing with a t t i tudes toward non-mathematical s chool - re la ted topics were a l ternated w i t h the t e n concepts d e a l i n g w i t h a t t i t u d e s toward mathematical concepts to help prevent a "mental s e t " . The a t t i t u d e measurement s c a l e was p i l o t t e s t e d by a d m i n i s t e r i n g i t to n i n e t y grade e i g h t students who were not i n v o l v e d i n the experiment. An e v a l u a t i v e f a c t o r was iden-t i f i e d by means of f a c t o r a n a l y s i s . Four of the s i x b i p o l a r a d j e c t i v e s namely; good-bad, v a l u a b l e - w o r t h l e s s , enjoyable-d i s t a s t e f u l , happy-sad, proved to be e v a l u a t i v e . An e v a l u a t i o n score was obtained on each concept f o r each person. To t e s t whether i t was j u s t i f i e d to t r e a t the t e n mathematics concepts as a cohesive homogeneous ( g l o b a l ) c l u s t e r , a f a c t o r a n a l y s i s was performed on the Persons X Concepts m a t r i x of the e v a l u a t i o n scores. R e s u l t s o f t h i s a n a l y s i s i n d i c a t e d t h a t a l l ten con-cepts loaded on a g l o b a l f a c t o r ( c l u s t e r ) . Therefore a g l o b a l a t t i t u d e score f o r each p u p i l was obtained by summing scores f o r each o f the t e n concepts over the f o u r e v a l u a t i v e b i p o l a r a d j e c t i v e s . A d m i n i s t r a t i o n and S c o r i n g of Tests. A l l teachers w i l l .score the t e s t s they admin i s t e r . Accuracy of s c o r i n g w i l l be checked when the experimenter re-checks marks and t o t a l s . Scores w i l l then be t a b u l a t e d f o r a n a l y s i s . A l l the p r e t e s t s and p o s t t e s t s w i l l be time l i m i t t e s t s . The Stanford t e s t s w i l l be administered i n accordance w i t h the i n s t r u c t i o n s and time l i m i t s g i ven i n the i n s t r u c t i o n a l manual. The time l i m i t s f o r the board and algebra tes t s w i l l be provided as matters of adminis t ra t ive convenience ra ther than fo r the purpose of p l a c i n g any premium upon speed of work. The time l i m i t s w i l l be c a l -culated to give p r a c t i c a l l y a l l pup i l s s u f f i c i e n t time to attempt a l l quest ions . The in tent i s that the tes t s w i l l be power tes t s and not speed t e s t s . In the e ight week per iod between the admini s t ra t ion of the algebra post tes t and the algebra r e t e n t i o n tes t no algebra w i l l be taught thus ensuring that the pupi l s w i l l get no formal prac t ice i n these s k i l l s . During t h i s per iod the students w i l l begin work on the next sec t ion of t h e i r text which deals almost e n t i r e l y with geometry. Treatments The c o n t r o l group students w i l l be seated a l p h a b e t i c a l l y and i n d i v i d u a l l y i n a f ive-row s ty le c las s room. Those students who have v i s u a l or auditory problems w i l l be iden-t i f i e d and seated near the f ront of the room. A l l students w i l l be required to work i n d i v i d u a l l y , desks not moved. Each teacher w i l l present the lesson mater i a l i n h i s normal teach-ing manner and i n the sequence i n which i t occurs i n the tex t . He w i l l continue to use the s ty le of presentat ion already f a m i l i a r to his students. V/hen help i s required by a student i n the c o n t r o l group, the teacher w i l l administer i t at h i s desk or at the student 's desk whichever i s more convenient. I f the teacher fee l s a problem has occurred, which i s causing d i f f i c u l t y for many of the students, he w i l l s o l i c i t the a t t ent ion of the ent i re c l a s s , go to the blackboard and exp la in f u l l y whatever seems to be causing the d i f f i c u l t y . The experimental group w i l l be d iv ided into foursomes as far as po s s ib l e . No attempt w i l l be made to form any s p e c i a l groupings according to sex. Two groups of f i ve w i l l be necessary because two of the three experimental c las s s izes are not mul t ip le s of four . Each foursome w i l l be comprised of two pa i r s es tabl i shed as fo l lows . In each experimental c lass the students w i l l be ranked highest to lowest according to the t o t a l sum of raw scores each obtains on the four pre te s t s . To ensure that each p a i r has a h igh-scor ing p u p i l and a corresponding low-scoring p u p i l the pa i r s w i l l be formed as f o l l ows : (1, 17), ( 2 , 18), ( 3 , 1 9 ) — ( 1 6 , 3 2 ) . Each p a i r chosen w i l l be l i s t e d on a s l i p and placed i n a conta iner . Two pa i r s w i l l be pu l l ed from the container i n sequence to form a foursome. Swapping of pa i r s w i l l be permitted i f some students are not happy with t h e i r o r i g i n a l placement. The composition of the group w i l l s t i l l be by p a i r s , so that the o r i g i n a l idea of having two weak students and two strong students per group w i l l be maintained. During the experiment each foursome or group w i l l s i t together and work together . The only time they w i l l be seated i n d i v i d u a l l y w i l l be when tes t s are being administered. The teacher ' s presentat ion of lesson mater i a l w i l l be the same as for hi s c o n t r o l group except for the seat ing arrangement of the p u p i l s . When the teaching part of the lesson i s over, the teacher w i l l encourage each group to work together with the two stronger pup i l s prov id ing help fo r the two weaker p u p i l s . Pupi l s w i l l be encouraged to discuss problems with each other i n t h e i r own groups. A p u p i l who understands w i l l be expected to provide help i n h i s group for the p u p i l who i s having d i f f i c u l t y . On occasions when members of one group have grasped a concept more q u i c k l y than members of another group, they w i l l be encouraged to help the members of the l a t t e r group. When help i s required by students i n the experimental group the teacher w i l l administer ass istance at h i s desk or at the group's s t a t i o n whichever i s more convenient. I f the same d i f f i c u l t y seems to be occurr ing for a number of students the teacher w i l l b r i n g the problem to the a t t en t ion of a l l groups and expla in i t to the ent i re c l a s s . During the course of the experiment the experimenter w i l l v i s i t each of the s i x classrooms at leas t once a week to see that in s t ruc t ions are being fo l lowed, and to check on each teacher ' s rate of progress i n covering the m a t e r i a l . The fac t that the experiment w i l l be conducted i n the ex-perimenter ' s home school w i l l allow him to maintain t i g h t contro l s on the i n v e s t i g a t i o n . Teachers w i l l keep notes on how each experimental group funct ions , making note of any p e c u l i a r or unusual occurrences. The experimenter w i l l , i f po s s ib l e , obtain mental te s t 27 scores for p u p i l s , but t h i s may not be poss ible as I .Q. tes t s have not been administered by the Vancouver School Board as a matter of p o l i c y since 1968. Data Analyses Means, standard deviat ions and i n t e r c o r r e l a t i o n s w i l l be obtained for the pre tes t , post tes t and r e t e n t i o n achievement scores and the a t t i tude measurement prescores and postscores . Analyses of variance or covariance, where requ i red , w i l l be performed on data from a l l the achievement tes t s and a t t i tude measurements. S t a t i s t i c a l s i gn i f i cance l eve l s of ,05 and beyond w i l l be accepted as evidence against the n u l l hypotheses. Hypotheses The hypotheses for t h i s study, s tated i n the n u l l form w i l l be as fol lows 1 Hypothesis One No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur for l e a r n i n g achieve-ment i n computation. Hypothesis Two No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur for l ea rn ing achievement i n concepts. Hypothesis Three No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur for l ea rn ing achievement 28 i n a p p l i c a t i o n s . Hypothesis Four No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur fo r learning achievement i n a lgebra . Hypothesis Five No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur for r e ten t ion of l ea rn ing i n a lgebra . Hypothesis S ix No s i g n i f i c a n t d i f ference between the experimental group and the c o n t r o l group w i l l occur i n a t t i tudes towards mathematics. CHAPTER FOUR RESULTS • Results are based on the experimenter's classroom observations, teachers* anecdotal records and p u p i l s ' t e s t scores . I n i t i a l l y a summary i s g iven of the experimenter's notes r e l a t i v e to the classrooms i n which the study was conducted. This i s followed by a summary of the notes and remarks submitted by the co-operat ing teachers . Summaries of s t a t i s t i c a l analyses fo l low these summary statements. Assessment of Classroom S i tuat ions Classroom atmosphere and organiza t ion var ied considerably among the s i x p a r t i c i p a t i n g classrooms. The researcher v i s i t e d each classroom weekly to check on progress and to see that the study was being c a r r i e d on as planned. Although a l l the teachers tended to use a "ques t ioning" technique i n approaching t h e i r lessons , the researcher f e l t that Teacher 2 tended to use more of a " l e c t u r e " approach. Boredom or d i s -in te re s t on the part of the students or teachers was not evident to the researcher during h i s v i s i t s . Interest l e v e l seemed to be high throughout the c la s ses . An attempt was made to secure mental te s t scores for the subjects , but these scores proved d i f f i c u l t to obta in . The small number of scores which were obtained were not usable as they were from d i f f e rent tes t s and they had been 30 administered at d i f f e r e n t ages. Hence the groups could be compared only on the basis of pretest scores . At the s t a r t of the experiment some pup i l s i n Teacher l ' s experimental c lass which, according to that teacher, was a very f r i e n d l y one, were made unhappy by being placed i n the groups the experimenter had assigned to them. They wanted t h e i r "democratic" r i g h t to set up new groups wi th in the l i m i t a t i o n of two strong and two weak students us ing the pa i r s a lready se lec ted . This was done a f t e r some d i s cus s ion , and the experiment continued as planned. It was f e l t that making t h i s change;, would remove any poss ible l ack of co-operat ion on t h e i r par t . Assessment of Teachers ' Wri t ten Comments The general r eac t ion of the teachers was that the pup i l s i n both experimental and c o n t r o l c lasses enjoyed the work and that most lessons had gone w e l l and smoothly. The lesson mater i a l as presented i n the text seemed to have been adequate and e a s i l y followed by both teachers and students. The l o g i c a l and sequent ia l development of the study uni t was h e l p f u l to the teachers i n the preparat ion of t h e i r lessons . Although no formal d a i l y schedule was set the three teachers seemed to be about the same spot i n the t ex t , when the experimenter made his weekly check. A de ta i l ed account of the teachers ' comments on group work i s . g i v e n i n Appendix A, Teacher 3 was absent for the s i x t h and seventh weeks of the experiment. With the co-operat ion of the p r i n c i p a l and the 31 Vancouver School Board a su i tab le subst i tute teacher was employed to take over. The experimenter met with t h i s teacher and f u l l y explained a l l re levant d e t a i l s of the experiment. During the two weeks t h i s subst i tute teacher was employed the experimenter made d a i l y v i s i t s to t h i s classroom, thus ensuring that the experiment could continue as planned. Results of s t a t i s t i c a l analyses Table 6 presents the means, standard deviat ions and i n t e r -c o r r e l a t i o n s for the Stanford Computation, Concepts, A p p l i c a t i o n s , Board and At t i tudes pretests for a l l students. Table 7 presents the means, standard deviat ions and i n t e r c o r r e l a t i o n s for the Stanford Computation, Concepts, A p p l i c a t i o n s , Algebra A and At t i tudes pos t te s t s . In te rcor re l a t ions among the Stanford Pretests range from .70 to .73 and among the Stanford post tes t s from ,6 l to .74. The Board pretest corre la te s with the Stanford pretests i n the range of .61 to .73, and the Algebra A posttest corre la te s with the Stanford post tes t s i n the range of .64 to .72. The c o r r e l a t i o n s between a t t i tudes and a l l achievement tes t s are a l l p o s i t i v e and range from .11 to ,21. Computation. In order to compare the experimentals and contro l s with regard to computation, Table 8 presents the ana lys i s of variance for the Stanford Computation pretest (Form VV). There i s a s i g n i f i c a n t d i f ference (°6 = .05) between the exper i -mental and the cont ro l treatment groups. The experimental mean:, i s 23.54 and the c o n t r o l mean i s 20.62| therefore the experimental group was superior to the c o n t r o l group i n computation before the experiment began. TABLE 6 Summary Table of Pretes t Means, Standard Deviat ions and I n t e r c o r r e l a t i o n s fo r Stanford (Form W), Board and A t t i t u d e Measures (N=174).* Var iab le Mean S .D. Computation Concepts A p p l i c a t i o n s Board A t t i t u d e s 2 2 . 0 8 2 4 . 9 5 1 8 . 3 7 ^ 7 . 3 9 4 2 . 3 5 8 . 2 2 6 . 9 8 4 . 7 9 1 8 . 4 3 1 0 . 4 0 C o r r e l a t i o n s * * Computation Concepts A p p l i c a t i o n s Board A t t i t u d e s Comp, 7 3 70 73 1 9 Con. 70 72 16 App, 6 l 15 Board 16 For s i gn i f i cance @*<= .05 and d/f=173, r=. l48 Decimals omitted. TABLE 7 Summary Table of Post tes t Means, Standard Deviat ions and I n t e r c o r r e l a t i o n s fo r Stanford (Form X) , Algebra and At t i tude Measures (N=174).* Var i ab le Mean S . D . Computation Concepts A p p l i c a t i o n s Algebra (Form A) A t t i t u d e s 23.75 24.75 19.56 27 .55 42.83 9.29 7.17 5.67 7.08 11.92 Corre l a t ions * * Computation Concepts A p p l i c a t i o n s Algebra (Form A) A t t i t u d e s Comp. 71 61 67 17 Con. A P P v 74 72 64 18 11 Algebra 21 For s i gn i f i cance @ oC = .05 and d/f=173, r=. l48 Decimals omitted. 34 There i s no s i g n i f i c a n t i n s t r u c t o r e f fect and there i s no s i g n i f i c a n t i n t e r a c t i o n e f fect between treatment and i n -s t r u c t o r . Comparing the groups i n terms of computation, a f te r the experiment, Table 9 presents an analys i s of covariance for the Stanford Computation post tes t us ing the Stanford Computation pretes t scores as the covar ia te . There i s a s i g n i f i c a n t i n s t r u c t o r e f fec t p l a c i n g the teachers i n the fo l lowing order (1,3,2) with respect ive means of 25.34, 23.96 and 21.93. This means that the d i f f e r e n t i a l c h a r a c t e r i s t i c s of these teachers were important causing variance i n the dependent var i ab le which i s the Stanford Computation pos t te s t . There i s no s i g n i f i c a n t d i f ference (oC = .05) fo r treatment or i n t e r a c t i o n between t r e a t -ment and i n s t r u c t o r . Concepts. In terms of concepts mastery Tables 10 and 11 present analyses of variance r e s u l t s for the Stanford Concepts pretes t s and pos t tes t s . There i s no s i g n i f i c a n t d i f ference (oC = .05) fo r treatment, i n s t r u c t o r , or i n t e r a c t i o n between i n -s t ruc tor and treatment i n e i t h e r v a r i a b l e . A p p l i c a t i o n s . Comparing the groups i n terms of app l i ca t ions before the experiment. Table 12 presents an analys i s of variance for the Stanford Appl ica t ions (Form W) pre te s t . There i s no s i g n i f i c a n t d i f ference (<=< = ,05) fo r treatment, i n s t r u c t o r or i n t e r a c t i o n between i n s t r u c t o r and treatment. Thus, for app l i ca t ions the groups were s i m i l a r at the s t a r t of the experiment. TABLE 8 Summary Ana ly s i s of Variance Table for Computation Pre te s t : Stanford Achievement (Form W, Advanced) (N=174). Source df SS MS F P Treatment 1 370.78 370.78 5.62 .02* Ins t ruc tor 2 185.53 92.76 l . 4 i .25 T r e a t , x I n s t r . 2 52.91 26.45 0.40 .68 E r r o r 168 11078.00 65.94 T o t a l 173 11687.00 Experimental Contro l Treatment means 23.54 20.62 * S i g n i f i c a n t @ ot = .05 TABLE 9 Summary Ana ly s i s of Co-variance Table for Computation Post-t e s t : Stanford Achievement (Form X, Advanced) (N=174). Source df SS MS F P Treatment 1 18.94 18.94 .72 .40 Ins t ruc tor 2 337.30 I68.65 6.40 .002** T r e a t , x In s t . 2 14.29 7.14 .27 .77 E r r o r I67 4403.60 26.37 T o t a l 172 4774.13 1 2 1 Ins t ruc tor Means 25.34 21.93 23.96 S i g n i f i c a n t @°C = .01 TABLE 10 Summary Ana lys i s of Variance Table for Concepts Pre te s t : Stanford Achievement (Form W, Advanced) (N=174). Source df SS MS F P Treatment 1 178.02 178.02 3.69 .054 Ins t ructor 2 30.01 15.01 .31 .74 Trea t . X In s t r . 2 115.25 57.63 1.19 .31 E r r o r 168 8110.30 48.28 T o t a l 173 8433.60 TABLE 11 Summary Analys i s of Variance Table fo r Concepts Pos t tes t : Stanford Achievement (Form X, Advanced) (N=174). Source d f SS MS Treatment 1 143.47 143.47 Ins t ructor 2 5.53 2.76 Treat , x Inst . 2 38.22 19.11 E r r o r 168 8717.70 51.89 T o t a l 173 8904.90 F P 2.76 .09 .05 .94 .37 .70 TABLE 12 Summary Ana ly s i s of Variance Table fo r A p p l i c a t i o n s Pre-t e s t : Stanford Achievement (Form W, Advanced) (N=174), Source df SS MS F P Treatment 1 21.39 21.39 .93 .34 Ins t ruc tor 2 22.70 11.35 .49 .62 Trea t , x In s t r . 2 52.15 26. 08 1.13 .33 E r r o r 168 3872.50 23.05 T o t a l 173 3968.70 TABLE 13 Summary Ana lys i s of Variance Table f o r A p p l i c a t i o n s Post, t e s t : Stanford Achievement (Form X, Advanced) (N=174). Source df SS MS F Treatment 1 148.97 148.97 4.70 Ins t ruc tor 2 2.49 1.25 .04 Trea t , x Ins t . 2 73.60 36.80 1.16 E r r o r 168 5325.90 31.70 T o t a l 173 5550.90 .03* .95 .'32 Experimental C o n t r o l Treatment means 20.48 18.63 * S i g n i f i c a n t @ oL = . 0 5 38 Comparing the groups i n terms of a p p l i c a t i o n s , a f t e r the experimental per iod , Table 43 presents an analys i s of variance fo r the Stanford Appl i ca t ions (Form X) pos t te s t . There i s a s i g n i f i c a n t di f ference for treatment ( = .05). The exper i -mental group performed bet ter on the post tes t than the c o n t r o l group. There i s no s i g n i f i c a n t d i f ference fo r i n s t r u c t o r or i n t e r a c t i o n between treatment and i n s t r u c t o r . Grade E ight Mathematics. Comparing the groups i n terms of the amount of mathematics l ea rn ing which had occurred i n grade eight mathematics i n three months, Table 14 presents an ana lys i s of variance fo r the Board pre tes t , which i s an enter ing behaviour t e s t of p r i o r mathematics l ea rn ing from September to December. There i s no s i g n i f i c a n t d i f ference = .05) for treatment or i n t e r a c t i o n meaning that the experimental and c o n t r o l groups are s i m i l a r at the beginning of the experiment. There i s a s i g n i f i c a n t d i f ference ( <^ = .05) fo r i n s t r u c t o r . According to the New Duncan M u l t i p l e Range Test (Halm and Le, 1972) there are two homogeneous subsets (2,3) and ( 3 » l ) . Thus, in s t ruc tor s 1 and 2 d i f f e r e d s i g n i f i c a n t l y . The d i f f e r e n t i a l c h a r a c t e r i s t i c s of these two teachers were sys temat ica l ly re l a ted to variance the Board pre te s t . TABLE 14 Summary Ana lys i s of Variance Table f o r Board (Dec.) P re te s t . (N=174). Source df SS MS F P Treatment 1 1018.60 1018.60 3.12 .08 Ins t ruc tor 2 2444.80 1222.40 3.74 .03* Trea t , x In s t r . 2 385.74 192.87 .59 .56 E r r o r 168 54920.00 326.98 T o t a l 173 58769.00 1 2 2 Ins t ruc tor means 51.33 42.34 48.48 * S i g n i f i c a n t @ oC = . 05 Algebra . In terms of algebra mastery, Table 15 presents an analys i s of covariance for the Algebra A posttest us ing the Board pretes t scores as the covar ia te . There i s a s i g n i f i c a n t d i f ference ( o L = .05) f o r treatment; thus the experimental group d id s i g n i f i c a n t l y bet ter than the cont ro l group on the Algebra A pos t tes t . There i s a lso a s i g n i f i c a n t d i f ference for i n s t r u c t o r . According to the Duncan New M u l t i p l e Range Test (1972) there i s one homogeneous subset (1,2), meaning that Teachers 1 and 3 and Teachers 2 and 3 d i f f e r s i g n i f i c a n t l y . Thus 7 the d i f f e r e n t i a l c h a r a c t e r i s t i c s o these teachers were sys temat ica l ly re la ted to variance i n the Algebra A pos t te s t . TABLE 15 Summary Ana lys i s Post tes t (N=174). of Covariance Table fo r Algebra (Form A) Source df SS MS F P Treatment Ins t ruc tor Trea t , x In s t r . E r r o r T o t a l 1 2 2 16? 172 100.37 230.41 19.63 3158,10 3508.51 100.37 115.21 9.82 18.91 5.31 .02* 6.09 .003** .52 .60 Experimental Cont ro l Treatment means 28.31 26.77 1 2 2 I n s t r u c t o r means 28.84 27.75 26.04" y * S i g n i f i c a n t @ * * S i g n i f i c a n t @ oL = .05 .01 • — - -< TABLE 16 Summary Ana ly s i s (Form B) . (N=l?4) of Covariance • Table for Algebra Retention Source df SS MS F P Treatment In s t ruc tor Trea t , x In s t r . E r r o r T o t a l 1 2 2 I67 172 231.52 85.47 28.23 3966.40 4311.62 23.52 42.73 14.11 23.75 9.75 .002** 1.80 .17 .59 .56 Experimental Contro l Treatment means 27.39 25.06 * * S i g n i f i c a n t @ .01 1 41 Retent ion. Comparing the groups i n terms of r e t e n t i o n of a lgebra a f ter a period of two months, Table 16 presents an analys i s of covariance for the Algebra B Retention tes t^ using the Board pretest scores as the covar ia te . There i s a s i g n i f i c a n t d i f ference ( oC = . 0 5 ) fo r treatment. The experimental group d id s i g n i f i c a n t l y be t ter than the cont ro l group on the re ten t ion t e s t . No s i g n i f i c a n t di f ference for i n s t r u c t o r or i n t e r a c t i o n between treatment and i n s t r u c t o r ex i s t s for the r e t e n t i o n t e s t . A t t i t u d e s . Comparing the groups on a t t i tudes toward mathematics before the experiment began, Table 17 presents an analys i s of variance for the summated a t t i tude prescores . There i s no s i g n i f i c a n t di f ference (^C = . 0 5 ) for treatment, which means that both the experimental and c o n t r o l groups are s i m i l a r . There i s a s i g n i f i c a n t d i f ference for i n s t r u c t o r . According to the Duncan New M u l t i p l e Range Test ( 1 9 7 2 ) there i s one homogeneous subset, ( 3 , 1 ) . ' T h u s Teacher 2 d i f f e r s from Teachers 1 and 3 . The d i f f e r e n t i a l c h a r a c t e r i s t i c s of these teachers inasmuch as Teacher 2 was more a u t h o r i t a r i a n than Teachers 1 and 3 i were sy s temat ica l ly re l a ted to variance i n the summated at t i tude prescore. There i s a lso a s i g n i f i c a n t i n t e r a c t i o n di f ference between treatment and i n s t r u c t o r . There i s one homogeneous subset of f i ve classes ( 6 , 4 , 3 , 1 , 2 ) . Teacher 2 ' s c o n t r o l group i s s i g n i f i c a n t l y d i f f e r e n t from the other f ive classes and has the highest mean score 4 9 . 2 2 . TABLE 17 Summary Ana lys i s of Variance fo r Summated At t i tude Pre-scores . (N=174). Source df SS MS F P Treatment 1 21.97 21.97 .22 .64 Ins t ruc tor 2 1349.70 674.87 6.83 .002** Trea t , x I n s t r . 2 640.31 320.16 3.24 .04* E r r o r 168 16619.00 98.87 T o t a l 173 18622.00 Ins t ruc tor means 41.07 46.16 39.68 T r e a t . x I n s t r . Ins tructor 1_ 2 2 Treatment E C E G E C Means 42.23 39.91 43.10 49.22 40.51 38.84 * S i g n i f i c a n t @ °L = .05 * * S i g n i f i c a n t @ o< = . 01 TABLE 18 Summary Ana ly s i s o f Covariance Table f o r Summated A t t i t u d e Postscores . (N=174). Source d f SS. MS F P Treatment 1 321.67 213.67 2.62 .10 Ins t ructor 2 84.60 42.30 .52 .60 Trea t , x In s t r . 2 189.02 94.51 1.16 .32 E r r o r 167 13640.00 81.68 T o t a l 172 14127.00 Comparing the groups with regard to a t t i tudes towards mathematics a f ter the three month treatment per iod , Table 18 presents an analys i s of covariance for the summated at t i tude postscores using the summated a t t i tude prescores as covar ia te . There i s no s i g n i f i c a n t d i f ference ( <^  = ,05) for treatment, for i n s t r u c t o r or for i n t e r a c t i o n between i n s t r u c t o r and treatment. According to the Duncan New Mul t ip l e Range Test (1972) there i s one homogeneous subset for i n s t r u c t o r (3,1,2) and one homogeneous subset for i n t e r a c t i o n (6 , 4 ,2 ,3»5»1 ) . In summary^signif icant"differences ex i s ted for computation before the s t a r t of the experiment. An ana lys i s of covariance performed on the computation postscores , using the computation prescores as covar ia te , showed that no s i g n i f i c a n t d i f ference occurred for treatment (<?<.= .05). S i g n i f i c a n t d i f ferences however, favouring the experimental group occurred i n the Appl i ca t ions and Algebra A post tests and Algebra B re ten t ion t e s t . No s i g n i f i c a n t d i f ferences for treatment, for i n s t r u c t o r or for i n t e r a c t i o n between i n s t r u c t o r and treatment occurred for a t t i t u d e s . 44 CHAPTER FIVE SUMMARY AND CONCLUSIONS This study was designed to invest igate the e f fect of co-operative group l ea rn ing on the achievement and a t t i tudes of grade eight pupi l s i n the algebra sec t ion of new mathematics. Three volunteer teachers each taught both an experimental and a c o n t r o l c l a s s . Four''teacher - administered pretests were given to e s t a b l i s h the s i m i l a r i t y of the groups. Four post-te s t s and a r e t e n t i o n t e s t were g iven to e s t a b l i s h the r e l a t i v e ef fect iveness of the respect ive l ea rn ing approaches i n a f f ec t ing achievement and to measure the mathematical development of the c la s ses . An a t t i t u d i n a l survey was administered before and a f te r the experimental per iod to determine a t t i tude toward mathematics scores for each student before and a f ter the t r ea t -ment. Analyses of variance or covariance when necessary were employed to s t a t i s t i c a l l y te s t the mean group di f ferences for s i gn i f i cance of both pretes t s and pos t tes t s . The .05 l e v e l of s t a t i s t i c a l s i gn i f i c ance was used throughout. Summary of Findings (1) Learning achievement for computation was not s i g n i -f i c a n t l y enchanced by the co-operative group learning method. On the computation pretest the experimental group d id s i g n i f i c a n t l y bet ter at .05 l e v e l of s t a t i s t i c a l s i g n i f i c a n c e . When t h i s i n i t i a l d i f ference was s t a t i s t i c a l l y removed by an ana lys i s of covariance, us ing the computation prescores as covar ia te , 45 no s i g n i f i c a n t di f ference for treatment was found. As a r e su l t of t h i s f i n d i n g the n u l l hypothesis was accepted for Hypothesis One. (2) Learning achievement for concepts was not s i g n i f i c a n t l y enhanced by the co-operative group learn ing method. On the concepts pretest there was no s i g n i f i c a n t di f ference between the experimental and the c o n t r o l groups. No s i g n i f i c a n t d i f ference at the .05 l e v e l of s t a t i s t i c a l s i gn i f i cance occurred for treatment on the pos t tes t . As a r e s u l t of the posttest f i n d i n g the n u l l hypothesis was accepted for Hypothesis Two. (3) Learning achievement for app l i ca t ions was s i g n i f i c a n t l y enhanced by the co-operative group learn ing method. No s i g n i -f i c a n t d i f ference between the experimental and the c o n t r o l group exi s ted before the treatment per iod . On the app l i ca t ions posttest a s i g n i f i c a n t d i f ference (c(<,03) occurred i n favour of the experimental group. Because of t h i s posttest f i n d i n g the n u l l hypothesis was re jected fo r Hypothesis Three, (4) Learning achievement for algebra was s i g n i f i c a n t l y enhanced by the co-operative group learn ing method. A s i g n i -f i c a n t d i f ference .02) occurred for treatment i n the Algebra A pos t te s t . As a r e s u l t o f t h i s f i n d i n g the n u l l hypothesis was re jected for Hypothesis Four. (5) Retention of l e a r n i n g was s i g n i f i c a n t l y enhanced by the co-operative group l e a r n i n g method, A s i g n i f i c a n t d i f ference (^<.002) occurred for treatment i n the Algebra B re tent ion » 46 t e s t . The n u l l hypothesis was therefore re jected fo r Hypothesis F i v e . (6) At t i tudes towards mathematics were not s i g n i f i c a n t l y enhanced by the co-operative group method. No s i g n i f i c a n t d i f ference at the .05 l e v e l of s t a t i s t i c a l s i gn i f i cance was found fo r treatment at the conclus ion of the experiment. The n u l l hypothesis was therefore accepted fo r Hypothesis S ix . Conclusions and Discuss ion The experiment was concerned with the amount of l earn ing and r e t e n t i o n which occurred during a per iod of twelve weeks i n the study of a lgebra as presented i n Unit Four of Introduct ion  to Mathematics (Brumfiel , E i c h o l z and Shanks, 1962). The Algebra A posttest and the Algebra B r e t e n t i o n te s t were the two tes t s which d i r e c t l y measured the amount of l ea rn ing and r e t e n t i o n which took p lace . Both of these te s t s showed e m p i r i c a l l y that the experimental group d id bet ter than the cont ro l group, w e l l beyond the .05 s i gn i f i cance c r i t e r i o n , which had been set for the experiment. The r e su l t s of both tes t s i n the opinion of the experimenter lend support to the use of co-operative group l e a r n i n g , i n v o l v i n g t u t o r i n g , i n the study of introductory a lgebra . Regarding the Stanford Achievement (Form X, Advanced) post-tes t r e s u l t s , only i n the case of app l i ca t ions was there a s i g n i f i c a n t d i f ference favouring the experimental group. The 4 7 three standardized posttests were given to f ind out whether or not mathematical s k i l l s were being maintained during the course of the experiment. I t i s the op in ion of the experimenter that the s i g n i f i c a n t d i f ference which showed up i n the app l i ca t ions pos t te s t , occurred p r i m a r i l y , because there was a s i m i l a r i t y between the types of questions that appeared on the app l i ca t ions t e s t and the algebra mater i a l s tudied during the treatment p e r i o d . At t i tudes towards mathematics, as measured, were not s i g n i f i c a n t l y enhanced by the co-operative group learn ing method. The experimenter f e l t that maybe the summated g loba l score used to represent a general a t t i t u d i n a l f ac tor was not a true i n -d i c a t o r of a s tudent ' s a t t i tude towards mathematics. From an inspec t ion of the group means and the large pretest i n s t r u c t o r e f fec t i t seems p laus ib le to argue that teachers make a b ig d i f ference i n students ' a t t i tudes and that one of the things one might t r y to do, i s to f i t students into classes taught by the type of teacher who best f i t s that s tudent 's p a r t i c u l a r needs. From the teachers ' comments on group work i t seems that for Teachers I and 3 who were l e s s s tructured i n t h e i r teaching than Teacher 2, any grouping which had three boys and a g i r l or three g i r l s and a boy, d id not work w e l l as a u n i t . With Teacher 2 t h i s type of grouping d id not cause as many problems. Perhaps the students i n Teacher 2's c lass were more concerned 48 about fo l lowing in s t ruc t ions s ince they were i n a more s tructured classroom. Teacher 2 d i d n ' t f e e l that i t made much di f ference whether a group was unevenly mixed. Teacher I was more worried about per sona l i ty clashes which arose i n groups that had a three and one format. As a r e s u l t of these experiences i t would seem p laus ib le i n future group work to form only groups of a l l boys, a l l g i r l s or of two boys and two g i r l s . A sociogram conducted before any groupings were made would probably help to e l iminate many of the c lashes , which occurred mostly with the three and one groupings. L imi ta t ions This study was l i m i t e d to the use of volunteered grade e ight mathematics c la s ses . Insofar as the teachers and pupi l s of these c lasses were representat ive of a l a rger populat ion of grade eight mathematics teachers and grade eight students, genera l i za t ions may be made of the r e su l t s of t h i s study. At best one should general ize with caut ion . Genera l i za t ion of r e s u l t s to other than grade eight mathematics classes i s not warranted. In fact the only g e n e r a l i z a t i o n poss ible i s to l ea rn ing s i t u a t i o n s i n which Unit Four (Algebra) of Introduct ion  to Mathematics (Brumfield, E i c h o l z andShanks, 1962) i s s tudied . The three-month teaching per iod which occurred before the experiment began may have introduced teacher-student i n t e r a c t i o n contaminants which could not e a s i l y be el iminated or overlooked. This may have been one of the causes of the s i g n i f i c a n t d i f ference for i n s t r u c t o r which occurred i n the a t t i tudes and board pretests and the Algebra A pos t te s t . The s t y l i s t i c teaching di f ferences of the three teachers e s p e c i a l l y Teacher 2 may have caused the variance for i n -s t ruc tor i n these three dependent v a r i a b l e s . I t seems poss ible since Teacher l ' s groups were allowed to rearrange t h e i r pa i r s to form foursomes on the basis of f r iendships and e x i s t i n g l o y a l t i e s that t h i s made Teacher l ' s group work harder. I t would be i n t e r e s t i n g to f i n d out what would happen i n an experiment using two experimental groups and one c o n t r o l group. One of two experimental groups would be allowed to work according to t h e i r own choice of pa i r s or based on sociogram r e s u l t s and the other experimental group would be set up by the experimenter, us ing pretest scores . I t i s also d i f f i c u l t to assess what e f fect the two week period •-r-•• : taught by the subst i tute teacher, when Teacher 3 was i l l , \* had on the r e s u l t s . The subst i tute teacher was very competent and as f a r as the experimenter could determine only the i n i t i a l day of the two v/eeks was s e r i o u s l y af fected while t h i s teacher was being or iented to the experiment s i t u a t i o n and there were some necessary interrupt ions i n both the experimental and cont ro l c la s ses . It i s also d i f f i c u l t to assess the e f fec t of g i v i n g four mathematics tests to students p r i o r to having them complete an a t t i tudes towards mathematics measurement sca le , which i n the 50 opinion of the experimenter proved to he too long . I t may have "been that the a t t i tudes were af fected hy the previous t e s t s . Other l i m i t a t i o n s were that in t ac t classes had to be used and that the experiment was conducted i n only one school . Further Research A r e p l i c a t i o n and extension of t h i s research with random assignment of experimental subjects to treatment groups or an increased number of c lasses per treatment group would tend to make the r e su l t s more r e l i a b l e by increas ing sample s i z e . Along with t h i s , a more r i g i d equating of subjects , teachers and other important var iab le s would r e s u l t i n a bet ter con-t r o l l e d i n v e s t i g a t i o n . The study of the usefulness of co-operative group learn ing needs to be extended to the teaching of subject matter other than mathematics. I t may prove more useful for the teaching of course mater ia l s other than those involved i n t h i s study. A beginning has been made i n attempting to show the e f fec t of co-operative group learn ing i n the classroom teaching s i t u a t i o n . The method has l ent some support to claims for the ef fect iveness of t h i s type of approach. Further research i s needed to v e r i f y t h i s r e s u l t and to extend the i n -v e s t i g a t i o n beyond the area of grade eight mathematics. 51 BIBLIOGRAPHY 1. A iken , Lewis R. , J r . , "At t i tudes toward Mathematics." Review of Educat ional Research, 40, 1970, 551-96. 2. Anderson, G. L. "Quantitat ive Thinking as Developed Under Connectionist and F i e l d Theories of L e a r n i n g . " Learning Theory i n School S i t u a t i o n s , U n i v e r s i t y of Minnesota Press, 1949, 40-73. 3. Anttonen, Ralph G. "A Longi tud ina l Study i n Mathematics A t t i t u d e , " Journal of Educat ional Research, 62, 1969* 467-471. 4. B e l l , S tanley , Garlock, N. and C o l e l l a . S. "Students as T u t o r s , " The C lea r ing House, December 1969, 44, 242-244. 5. Bruce, Wi l l i am F. and Freeman, Frank S. Development and  Learning, Houghton M i f f l i n C o . , Boston. , Boston: 1942. 6. Brumfie l , Charles F . , E i c h o l z , Robert E . , Shanks, M e r r i l l E . , Introduct ion to Mathematics, Addison -Wesley Publ i sh ing C o . , I n c . , Reading, Massachusetts: 1962. 7. Dutton, Wilbur H. "Another Look at At t i tudes of Junior High School Pupi l s Toward A r i t h m e t i c . "Elementary  School Journa l . LXVIII , 1968, 265-68. 8. Ferguson, G. A. S t a t i s t i c a l Analys i s i n Psychology. New York: McGraw-Hil l , 1966, 157-158. ' 9. Fleming, C a r l . " P u p i l Tutors and Tutees L e a r n . " Today's Educat ion. October 1969, 22-24. 10. Frager, Stanley and Stern, Carolyn. "Learning by Teaching . " The Reading Teacher, 23, 1970, 403-405. 11. Halm, Jason and Le , Chinh D . , UBC., MFAV, Analys i s of Variance/Covariance. Computing Centre, UBC Vancouver. B.C. 1962, 26-30. 12. Jackson, P h i l l i p W. and S t r a t tner , Nina, "Meaningful Learning and Retention? Noncognitive V a r i a b l e s . " Review of Educat ional Research, 1964, ^4, 513-529. 52 13. Konikow, E l l a , "High School Friend Program" (Mimeographed) Downers Grove, I l l i n o i s , (undated). 14. K e l l e y , Truman L . , Madden, Richard, Gardner E r i c F . , Rudman, Herbert C , The Stanford Achievement Tes t . (Advanced Bat tery ) . Harcourt , Brace and World I n c . , New York:1964. 15. McKie, Douglas and Foster , Stephen F. "A General Model for Mult idimensional Analys i s of Semantic D i f f e r e n t i a l At t i tude D a t a . " Paper presented at 80th Annual Meeting of the American Psycholog ica l A s s o c i a t i o n , Honolulu, Hawaii, September, 1972. 16. McNei l , J . ABC Learning A c t i v i t i e s , New Yorkt American Book Company, 1966. 17. Montessor i , Mar ia , The Discovery of the C h i l d . Fides Pub l i sher s , I n c . , Notre Dame, Indiana: 1967. 18. Neale, Danie l C. "The Role of At t i tudes i n Learning Mathematics," The Ar i thmet ic Teacher 16 19694 631-640. 19. Osgood, Charles E . , S u c i , George J . and Tannebaum, Percy, The Measurement of Meaning. U n i v e r s i t y of I l l i n o i s Press, Urbana: 1957. 20. Rogers, John G. "A School Where Kids Are Teachers" , Parade j J u l y 2,1967, 8-9. 21. Rosenthal,Robert and Jacobson,Lenore, "Science and the C i t i z e n " , S c i e n t i f i c American, Nov. 1967, 217. 22. Schoeler , Arthur W. and Pearson, David A. "Bet ter Reading Through Volunteer Reading T u t o r s . " The  Reading Teacher. 2T3, A p r i l 1970. 625-650. 23. Thelen, Herbert , "Tutor ing by Students , " School  Review. LXXVII fi969 229-244 24. Thelen Herbert A. "The Thelen C o l l e c t i o n " Selected Working Papers i n E x p l i c a t i o n of a Rationale fo r Classroom P r a c t i c e s . 1968, B .C . Teachers ' Federat ion, Vancouver, B .C. 25. Thelen, Herbert A. "Tutor ing by Students , " The  Education Diges t ; February 1970, 17-20. 53 26. Underwood, Benton J . "As soc ia t ion Transfer i n Verbal Learning as a Function of Response S i m i l a r i t y and Degree of F i r s t - L i s t L e a r n i n g . " Journal of  Experimental Psychology, 1951, 42, 44-54. 27. Wilson, G. M . , "Why Do Pupi l s Avoid Math i n High School?" The Ari thmet ic Teacher, 1969, VI I I . 168-171. 28. Word, Aubrey H. & Davis , Robert A. " Ind iv idua l Dif ferences i n Retention of General Science Subject Matter i n the Case of Three Measurable O b j e c t i v e s . " Journal of Ex- perimental Educat ion. 1938, 7, 24-30. 1 54 APPENDIX A Teachers' Comments on Group work. Teacher 1 reported on her groups as f o l l o w s J Group 1t Group H i Group U l i Group IV: Group V t Group V I t Group V I I t Four hoys who enjoyed working together. One hoy seemed t o increase both h i s i n t e r e s t and h i s marks i n t h i s group. Four g i r l s who worked extremely w e l l together although one g i r l may have gone down i n marks, as the type of person who needed s t r u c t u r e d t e aching. Another g i r l rose t o an "A" under the h e l p f u l guidance of her two best f r i e n d s . Four g i r l s - some d i s c i p l i n e problems here which i n t e r f e r e d w i t h l e a r n i n g from each other. F i v e g i r l s - worked w e l l - one g i r l b e n e f i t t e d the most. Lots o f g i g g l i n g a t times i n t h i s group. Three g i r l s , one boy. This group d i s i n t e g r a t e d . The boy worked on h i s own. One g i r l was u s u a l l y absent. The remaining two g i r l s d i d work together. Four boys. Broke i n t o two groups each having a weak and a st r o n g student. The groups d i d not help each other. Four g i r l s . Worked w e l l together. The two stron g students helped the two weak ones. 55 Group 1 1 i Teacher l ' s o v e r a l l impression was favou r a b l e . Except f o r the p e r s o n a l i t y c l a s h e s , which seemed to a f f e c t groups 5 and 6 , the p u p i l s seemed to enjoy and p r o f i t from h e l p i n g each other. Teacher 2's Comments. Group 1 t Three boys and one g i r l . Worked w e l l as a group. One boy was r a t h e r slap-happy and gossipy. The g i r l was the le a d e r . Three boys and one g i r l . The g i r l tended to be on her own. One of the boys was a chatterbox. Three boys and one g i r l . Good cooperation at a l l times i n t h i s group. Two boys and two g i r l s . One boy absent o f t e n . The other boy and the two g i r l s worked w e l l together. Three g i r l s and one boy. The boy was a very weak student. A l l three g i r l s helped him. Two boys and two g i r l s . Worked w e l l together. Two boys and two g i r l s . One boy was absent o f t e n . The remaining three worked w e l l together. Two boys and two g i r l s . Good cooperation and i n t e r a c t i o n . Teacher 2 reported t h a t a l l h i s groups, seemed to work w e l l together. He d i d n ' t f e e l t h a t i t made much d i f f e r e n c e t h a t the groups were mixed. He thought t h a t h i s good students s u f f e r e d some. He a l s o s a i d t h a t the p u p i l s l i k e d to s i t Group 111i Group IV i Group Vt Group V I i Group VI1: Group V l l l > 56 together, Group A: Group Bt Group Ci Group Di Group E i Group F i Group Gi Group Hi Teacher ^'s Comments Two g i r l s and two boys; g e n e r a l l y working as two p a i r s ; some group i n t e r a c t i o n . S i m i l a r t o group A. Four boys; good group i n t e r a c t i o n and group work. Three g i r l s and one boy; boy appeared t o be accepted, but tended to work alone; three g i r l s showed good i n t e r a c t i o n and group work. Three g i r l s and one boy; boy g e n e r a l l y i s o l a t e d and tended to work alone; g i r l s made l i t t l e e f f o r t to i n c l u d e him; boy a weak student to begin w i t h . Four g i r l s ; good i n t e r a c t i o n and group work. Two boys and two g i r l s . S i m i l a r to Groups A and B, Three boys and two g i r l s ; good i n t e r a c t i o n w i t h one g i r l accepted and p a r t i c i p a t i n g i n group work. The second g i r l spent most of her time w i t h Group G or working on her own. APPENDIX B 57 *NOTE: ALL TESTS AND SURVEY SHEETS HAVE BEEN REDUCED IN SIZE FROM ORIGINALS TO FIT AN 8£ x 11 FORMAT. SURVEY INSTRUCTIONS The purpose of t h i s survey i s to f ind out how you rate some of the words or concepts you may have come across i n S o c i a l S tudies , Mathematics and Science. For Example t SCIENCE TEXTBOOK THIS YEAR 1) good -•• - i x t : s : i bad 2) b e a u t i f u l x t : : j : s ugly The above example means that t h i s student considers t h i s year ' s textbook i s quite good and that he l i k e s i t . IMPORTANT 1) Work r a p i d l y - give your f i r s t impressions - do not d a l l y over an-:l item. 2) Try to make each judgement a separate and independent one, 3) Place your X i n a space as shown above - NOT BETWEEN SPACES. MY SCIENCE TEXT BOOK THIS YEAR 1) good i i i t : • • t bad 2) b e a u t i f u l t i : : ••- ugly 3) valuable • i s i : i worthless 4) enjoyable * * t : «» : • • d i s t a s t e f u l 5) c o l o r f u l t i : : • « c o l o r l e s s 6) happy * • : s t • * sad 58 MATHEMATICS TEXT BOOK 1) good 2) beautiful 3 ) valuable 4 ) enjoyable 5) colorful 6) happy bad ugly : worthless distasteful colorless sad SOCIAL STUDIES TEXT-BOOK 1) good 2) beautiful 3) valuable 4) enjoyable 5) colorful 6) happy bad ugly . • worthless distasteful colorless sad 1) good — 2) beautiful 3) valuable 4 ) enjoyable 5 ) 'colorful-6) happy. . -^RATIONS WITH .WHOLE NU#BF,RS bad ugly worthless distasteful colorless sad SOCIAL STUDIES MAP WORK 1) good 2) :beautiful 3) valuable 4 ) enjoyable 5) colorful 6) happy 1) good 2) beautiful 3) valuable enjoyable . colorful happy PERCENTAGE bad ugly worthle distas co'lorl sad bad ugly worthies distastej colorless sad SCIENCE LAB WORK good beautiful valuable enjoyable colorful. lappy bad ugly worthless distas tefii colorless sad 60 OPERATIONS WITH FRACTIONS 1) good 2) beautiful 3) valuable 4) enjoyable 5) '"'colorful 6) happy bad . ugly • worthless •di.stastefi • rcolorless sad SOCIAL STUDIES READING 1) good 2) b e a u t i f u l 3) valuable 4) enjoyable 5) colorful 6) happy .... 1) good 2) beautiful 3) valuable 4) enjoyable '5) colorful 6) happy ARITHMETIC IN BASES OTHER THAN TEN bad ugly worthless distastefu colorless sad-bad ugly l;'. '" ' worthless ' ' distastefu.' '_ colorless ' L.'j sad ... continued 1 ) good 2) beautiful 3) valuable 4 ) enjoyable 5 ) colorful 6) happy !SW__.READING b a d u g l y w o r t h i e s d i s t a s t e o o l o r l e s s a d SOLVING EQUATIONS 1) good 2) beautiful 3) valuable ) enjoyable colorful" h a p p y b a d Ugly worthless d i s t a s t e fu. c o l o r l e s s s a d SCHOOL good b e a u t i f u l v a l u a b l e " j o y a b l e o l o r f u l appy bad u g l y ., w o r t h l e s s d i s t a s t e f u c o l o r l e s s s a d UNION AND^INTERSUCTION OF SETS ESSAY W R I T ENC 1) good 2 ) beautiful 3) valuable 4 ) enjoyable 5) colorful 6) happy 1) good 2 ) beautiful 3) valuable 4 ) enjoyable 5) colorful 6) happy 1) good 2) beautiful 3 ) valuable 4 ) enjoyable 5) colorful 6) happy 1) good 2 ) beautiful 3 ) valuable 4 ) enjoyable 5) colorful ALGEBRA ASSIGNED HOMEWORK NEGATIVE NUMBERS 63 bad ugly worthless distasteful colorless sad bad ugly worthless distasteful colorless sad bad ugly worthless distasteful colorless sad bad ugly worthless distastful colorless APPENDIX C 5^ M A T H E M A T I C S 8 NAME D E C E M B E R CLASS // . READ EACH QUESTION CAREFULLY BUT DO NOT SPEND TOO MUCH TIME ON ANY YOU DO NOT UNDERSTAND. NOTE CAREFULLY THE FORM REQUIRED FOR THE ANSWER AND PLACE IT IN THE SPACE PROVIDED. THERE IS ONE MARK FOR EACH ANSWER. S E C T I O N " A " WRITE THE LETTER CORRESPONDING TO THE CORRECT ANSWER TN THE SPACE PROVIDED FOR ON THE RIGHT. 1) Which statement is true? a) 5^  = 4+ 4+ 4 + 4 + 4 b) 5^ « 4. 4. 4. 4. 4 c) 4 3 - 4. 4. 4 d) 2 * 2 . 3 1- C e) none of these 2) 42 written aa the product of prime factors i s : a) 1 x 6 x 7 b) 37 + 5 J c) 3 x 14 d) ..2 x 3 x 7 2= ^ e) a l l are correct 3) In what base are the numerals written i f 2 x Ji «* 10? a) base tw<? b) base three ^ c) base four d) base five 3» ^ e) none of these 4) Which numeral represents the largest number,? a) J^teo .fa) "four n c) "two d) 2 0seven 4= ^ e) 3twelve 5) The sum of two odd numbers is always i# the set of: a) even nunbers b) odd numbers c) multiples of three d) prime numbers 5° e) squared numbers 6) Every natural number has at least th£ following divisors: _a) zero and one b) zero and itself c) one and itself d) itself and two 6-e) none of these 7) Which of the following is not a prime? a) 7 b) 41 c) 73 d) 87 7-e) 97 CL cL 65 8) The set of a l l factors of 12 is: a) (1,2,3,4,8,12) c) (1,2,3,4,6) e) (2,2,3) b) (1,2,3,4,6,12) d) (2,3,4,6) 9) Which is the complete factorization of 36? a) 4 x 9 c) 3 x 12 o) 13 + 23 10) The operations which are commutative are: a) addition and subtraction c) subtraction and division b) 2 x 3 x 6 d) 2 x 2 x 3 x 3 b) addition and'multiplication d> addition only 1 0 " e) a l l four operations 11) An example of a natural number i s : . a) h b) 3.06 c) 5/4 d)3 11-12) An example of a repeating decimal i s : a) c) e) 5.75 " T T none of these b) d) 1.12112U12... JT 12-13) An example of a prime number i s : a) c) e) 0 6 none of these b) d) 1 35 13-14) The LCM of 12 and 24 i s : a) c) e) 2 24 48 b) d> 12 36 14-15) The greatest common divisor of 7 and 8 i s -a) c) e) 0 7 56 b) d) 1 8 15-16) Which of the following numerals i s incorrect? a) c) e) two b) d) ^ e i g h t three 16-cL C 66 17) The number written in base two is an even number i f : a) i t ends in zero only b) i t ends in 1 c) i t ends in zero or 2 d) the sura of the digits in an / ) e) there is no simple rule even number 17»x "0 -6 18) If 6 is divided by zero the answer i s : a) 6 b) 0 c) 1 d) any number 18"° e) i t is impossible S E C T I O N " B " TRUE OR FALSE READ EACH STATEMENT CAREFULLY. IF IT IS TRUE, WRITE THE WORD "TRUE" IN THE SPACE AT THE RIGHT. IF IT IS FALSE WRITE THE WORD "FALSE". 1) 10* means 10 x 10 x 10 x 10 1) /lu£ 2) In the symbol 5 3 the exponent is 3 2) ~7%u£ 3) In base five, the numeral before 40 is 39 3) ffitse 4) The numeral 8 represents the same number ia base ten as i t does in base four. 4) F^t-Sg 5) In base four we use only the symbols 0, 1, 2, 3, 4. 5) fft<-s£ 6) When we carry, In addition, the value of the carried digit depends upon the base. ' 6) " f ^ £ 7) Rational numbers!.are numbers, which can. be,Represented as fractions 7«* 8) There is a one - to - one correspondence between the natural numbers 8- FrttSE less than ten and the whole numbers less than ten. 9) Every natural•• nu&ber is a rational number. 9=« "77?>Jg-10) The product of any whole number and zero is that whole.; number I O f ^ t - S c 11) 4 3 ^ 3 4 11- TJuE 12) 5< ( 6 + 5 ): • 0 12- F.ILSE 13) 162< 16-15 13- FAiSe 6? 14) 6 base seven has the same value as 6 base ten 14- 1 / s K E 15) The Egyptian number system did not use place value 15-. 16) We inherit our numerals from the Arabs 16- Tl - \HL r . 17) Every rational number can be represented as a repeating decimal 17= S E C T I O N " C '* WRITE YOUR ANSWERS IN THE SPACES PROVIDED AT THE RIGHT 1) Write the following numerals in expanded notation. 0)(a> 3'.ax> + UOO +• )• .0 + X, .(example: 2 3 f o u r - 2 x 4 + 3) . 3L-£ + 3-0.. + -3-a) 3812 ten 3 ;. b) 2 3 3 f o u r * c> . 2) Change to. common fraction i n lowest terms . a) 0.666... ! c) 0.111... b) 0.33... d) 0.0202... a) b) c) d) 3) Factor completely: 275 4) Carry out the operations indicated. a) base four: 3+3=" c) base four-:--3.3=" . " e) base seven: 513.100=« b) base twelve: 5+8= 4. d) base-two: 111+1011~ a) b) c) d) e) 5) If a, b and c represent any rational numbers, complete the following statement of the distributive--principle. , a), a. • . ( ,b +• c ) . 6) Write the number 36 as the sum of two prime numbers 7) Express the fraction h as a "decimal" i n base twelve 8) Work out the following problems in your head i f you can. In any 8. case do as l i t t l e writing as possible.. a) 3 (1/3- 2/5) c) 5(3/8- 8/5) e) 6(5/8+0/4) :b) l/6+(3/5+5/6) d) 3/4+0/5 f) 7*l/3+7'2/3 3*. 3).. 5*- S- /| IL. \oo\ o 5 | 3 Q D 5) Cb°&-ra-L 6) -7+^ 7) . 3 a) b) c) 3 d) e) f) -1 68 9) 1839 = 6 10) a) 3/4 f 3/4 b) 5/8 f c) 12 d) 1/3 -1/3 h -U) a) Write a l l the divisors of 30 b) Write a l l the divisors of 75 c) Write the greatest common divisor of 30 and 75 d) Write one other common divisor of 30 and 75 9) Mo?> 15/16 10. a) J k b) ^ 1/3 c) 3fe, d) _±i 11. a) |,2,S,g/?<l;o, b) l,3,<,/S,lS  c> d) 5" 12) a) What is the least common multiple of 8 and 12? 12, a) XU-b) Name one other common multiple of 8 and 12. b) £f g 13) In the following questions use the abbreviations given for the basic ; principles l i s t e d below: E.U. Existence and uniquetess principle CP. Commutative principle A.P. Associative principle D.P. Distributive principle M. one Multiplication propeity of one M. zero Multiplication propeity of zero A. zero Addition property of zero INDICATE WHICH PRINCIPLE IS USED IN EACH QUESTION IN THE SPACE PROVIDED FOR , ON THE RIGHT., a) (6-3).5=6-(3-5) b) 5+6= 11 13. a) (\.p. c) a'(b+c)=a.b+a«c d) 9'(8-8)» 0 b) EM-e) 1/3-4/5- +2/3-4/5=(l/3+2/3)'*/5 f) x«l» x c) j),p g) 2/3+(3/5+l/3) = (3/5+l/3)+2/3 d) n z*Ao f ) M. D^£ g) JLR 14) The following sets are formed by dividing numbers by 3. 6 9 0 - (0,3,6,9, ...) T - (1,4,7,10, ...) T - (2,5,8,11,...) a) Why do we c a l l the last set — ? 2 14. a) I f HAS RZMtitJpZt? itLA Hint* 1^"?^ By, 3 .  15) What is the one-hundredth number in the set for modulo 8? 0 a) -792-16) CARRY OUT THE OPERATIONS INDICATED AND WRITE YOUR ANSWER IN THE SIMPLEST FORM IN THE COLUMN AT THE RIGHT. a) (3+2)x9 b) 2/5x0/4 16. a) c) 5+4x3 d) 12T4+2 b) 0 c) 187(9-6) f) 7 + 9 c) 17 d) 8) 32 h) e) i ) 3/7+(2/5+4/7) j) 9*(5/8-l/9) f) LL k) 0 . Cs+3/4) 1) 5-(1/5+0/5) a) <f 3 h) Vw ±\ i) *K k) O 1) 1 70 Algebra A Post-Test Read each statement carefully. If the statement Is true, write the word "True" in the space provided. If i t is false, write the word "False". Ft 1. The set of integers does not contain zero. 1. 2. There i s no last number in the set of whole numbers. 2. 3. The empty set is a subset of every set. 3. 4. The solution set of a sentence contains a l l the 4. replacement values which give true statements. 5. A proper subset can be the same as the given set. 5. 6. Every set has at least one member. 6. Multiple Choice INSTRUCTIONS: Write the letter corresponding to the correct answer in the space at the right of each question. USE CAPITAL LETTERS FOR ANSWERS11 7. What operation is the same as subtracting negative 7? A. subtracting 7 B. adding 7 C. adding 0 D. subtracting 0 E. adding (-7) 8. S - {0, 1, (-1), 2, (-2), . . . 10, (-10)} A number in the solution set for 3 jjd + (-7)J = 0 is A. 7 B. 0 C. 1 D. 3 E. (-7) 7. B 8. A 9. S » {1,2,3,4,5,6,7,8,9} A number pair in the solution set for x* + y^" = 25 is A. (0,5) D. (1,5) B. (5,5) E. (-4,3) C (3,4) 9. C 10. S = {0,1,2,3,4,5,6,7} Find the solution set for the sentence x<6 and x>4 A. {0,1,2,3,7} B. {4,5,6} C. {5,6,7} D. {3,7} E. {5} 10. 11. S = {0,l,2,3,4,5j Which of the graphs contains the solution set for the sentence x + y = 2? Starf «— f 5f ~4» 11. 12. The inverse of dividing by 2/7 is A. dividing by 2/7 B. multiplying by 7/2 C. subtracting by 7/2 D. subtracting 2/7 v — i -4 -1 — •> 1-1 r= Select the sentence which best represents the stated situation in questions 13, 14 and 15. 71 13. The sum of 3 times a number and 5 is 77. A. y3 + 5 - 77 B. 3«y + 15 = 77 C. 3«y + 5 - 77 D. 3»y =77+5 E. 3(y + 5) - 77 13. 14. The sum of two consecutive odd numbers is 196. A. x + (x + 1) = 196 B. x + (x + 2) = 196 C. x«(x + 1) = 196 D. x + 2 => 196 E. x + y = 196 14. 15. Bobby weighs 15 pounds more than Alan. Together they weigh 98 pounds. Find the weight of each boy. If x represents Bobby's weight, which is correct? A. x + (x + 15) =98 B. x + 15 = 98 C. x + (x - 15) = 98 D. 2x = 98 - 15 E. x - x + 15 • 98 15. 16. The sum of the ages of two children is 11 years. In five years one child will be twice as old as the other. How old are thev now? Select the number pair. A. (7,4) ' B. (6,5) C. (9,2) D. (10,1) E. <5H, 5k) 16. 17. Name the point, A, B, C, D, or E, not in the solution set of the equation xy = (-12) b 4 1 -7, ~4 1A \ f * 18. The sum of two numbers is 27. One-third of the smaller numher is the same as one-sixth of the larger. Find the numbers. 17, 18. 19. The population of a certain city is 5,000 more than double what is was 50 years ago. The population now is 88,000. What was the population 50 years ago? 19. 2 0 . In a certain class 3 out of 7 pupils were girls. Of the girls 2 out of 3 were married two years after they left school. (a) Write a number pair comparing the number of girls in the class with the number of boys. 20. B C ft ^1 600 In a certain grade eight class, 3 out of 5 pupils were g i r l s . Of the boys, 5 out of 8 completed grade twelve. 72 21. Write a: number pair comparing the number of girls in the 22. 23. 24. class with the number of boys. If there were 40 students in the class, how many boys completed grade twelve? Set S has 8 elements and set T has 10. Set S{J!P has only 15 elements. How many elements are in Sf^ C? A man is 6 f t . t a l l . At a certain time of the day he casts a shadow of 4 f t . At the same time a tree casts a shadoxj- of 20 f t . How high is the tree? 21. 22. 23. 24. lo 3 3o Use the symbols >,'<, and =• to show the correct relation between the following number pairs. Write only the symbol in the space provided on the right. Do not change the order of the given numbers. 25. 26. 27. 28.' 29. 30. 31. 32. 33. 34. 35. 36. 5.687 (-6) _ 5.6865 5 16 25. 26. 27. Solve the following equations for set S « {0,1,2,3,4,5,6,7,8,9,10}. 14/a - 2 28. )7) x«9 = 0 29. JOJ 4(7 - k) - 16 30. \$\ Perform the following operations. 8 - 11 + 7 -(-6) • 3 = ( -14) * (-2) = 8 - (-3) -(-5) + (-4) = (-10) - 0 -Give the coordinates of the points marked A,B,C,D, and E. 1 f 0 \ 31. 32. 33. 34. \\\\ 35. H i 36. 37. A= < 1 (0,0) 38. B= ( 0 , 3 ) 39. C= 40. !> 41. E= ( - Z - 3 ) 73 Algebra B Retention Test Read each statement carefully. If the statement is true, write the word "True" in the space provided. If it is false write the word "False". 1. Subtraction is a commutative operation. 1. Fqlst  2. The empty set is a subset of each set. 2. 3. The sum of an odd number and an even number is always ^ an odd number. 3. ' HUL 4. Every natural number is a rational number. 4. I Mie, 5. 10 3 means 10 x 3. 5. f-Jbu. 6. If set S = {1,2,3,4,} and set T = {3,6,9} then sVr = {3}. 6. FcdUc  Multiple Choice INSTRUCTIONS: Write the letter corresponding to the correct answer in the space at the right of each question. USE CAPITAL LETTERS FOR ANSWERS!! 7. What operation is the same as subtracting negative 7? A. subtracting 7 B. adding 7 C. adding 0 D. subtracting 0 E. adding (-7) 7. B 8. Which of the following is not a subset of the set {1,2,3,4,5}? A. {1,3,5} B. {2,4} C. { } D. {1,2,3,4,5} E. {1,3,5,7} 8. 9. S = {0, 1,(-1), 2,(-2),.. . . 10,(-10)} A number in the solution set for 3 (d + (-7)J = 0 is A. 7 B. 0 C. 1 D. 3 E. (-7) 9. A 10. S = {1,2,3,4,5,6,7,8,9} A number pair in the solution set for x + y = 25 is A. (0,5) B. (5,5) C. (3,4) D. (1,5) E. (-4,3) 10. 11. S = {0,1,2,3,4,5,6,7} Find the solution set for the sentence x<6 and x>4 A. {0,1,2,3,7} B. {4,5,6} C. {5,6,7}-D. {3,7} E. {5} 11. 12. S - {0,1,2,3,4,5 } Wiich of the graphs contains the solution set for the sentence x-+ v =» 2? 12. Select the sentence which best represents the stated situation in questions 13, 14, and 15. 74 13. The difference between 3 times a number and 5 is 31. A. y 3 - 5 = 31 B. 3»y - 15 - 31 C. 3«y - 5 = 31 D. 3-y = 3 1 - 5 E. 3(y - 5) = 31 13. 14. The sum of two consecutive odd numbers is 216. A. x + (x + 1) = 216 B. x + (x + 2) = 216 C. x'(x + 1) = 216 D. x + 2 - 216 E. x + y = 216 14. D. 2x - 141 - 11 E. x - x + 11 = 141 16. The sum of the ages of two children i s 11 years. In five years one child w i l l be twice as old as the other. How old are they now? Select the number pair. A. (7,4) B. (6,5) C. (9,2) D. (10,1) E. (5*5,55s) 17. Name the point, A, B, C, D, or E, not in the solution set of the equation xy = (-12). 16. 5 15. Bobby weighs 11 pounds more than Alan. Together they weigh 141 pounds. Find the weight of each boy. If x represents Bobby's weight, which is correct? A. x + (x + 11) - 141 B. x + 11 - 141 C. x + (x - 11) = 141 15. ~2--4 • 1 3 ( > t — 1 -4 -2 o £ 4-18. The sum of two numbers is 20. One-half of the larger number is the same as two times the smaller number. Find the two numbers. 19. My monthly salary is 75 dollars more than twice what i t was ten years ago. I now earn 585 dollars a month. What was my salary ten years ago? 17. 18. 19. 1+ j \^ 20. Set S has 8 elements and set T has 10. Set SUT has only 15 elements. How many elements are in SflT? 21. A boy is 5 f t . t a l l . At a certain time of the day he casts a shadow of 3 f t . At the same time a building casts a shadow 21 st. How high is the building? Use the symbols and = to show the correct relation between the following number pairs. Write only the symbol in the space provided on the right. Do not change the order of the given numbers. 20. 21. 3 35 75 22. (a) 5.687 23. (b) (-6) 24. (c) 4 6 5.6865 5 16 3 Solve the following equations for set S - {0,1,2,3,4,5,6,7,8,9,10}. 25. (a) 6a - 12 26. (b) t/7 = 2 27. (c) 24 = 3(5 + d) 28. (d) 1/3 a = 2 Perform the following operations 29. (a) (-7) + 2 + 9 = 30. (b) (-5) - 3 -31. (c) (-7) - (-3) => 32. (d) 0 *(-4) = 33. (e) (-5) * ( - l ) -34. (f) 0 - (-4) -35. (g) -4 - (+3) = 22. (a) > 23. (b) < 24. (c) 25. (a) 26. (b) 27. (c) 28. (d) i t 29. (a) 4-30. (b) -tf 31. (c) -u-32. (d) o 33. (e) 34. (f) 4 35. ( g ) - 7 Give the coordinates of the points marked A, B, C, D and.E. ^ I I* — t M S c . 0 1 The inverse of dividing by 3/5 is A. dividing by 3/5 B. multiplying by 5/3 D. subtracting 3/5 E. subtracting 5/3 36. (a) A" (0,0) 37. (b) B= 38. (c) C= 40. (e) E-A-i.,.-XL 39. (d) (-2,u.) C. multiplying by 3/5 4 1 . 

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