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The relationship between selected teacher variables and growth in arithmetic in grades four, five and… Prekeges, Demitrios Peter 1974

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THE RELATIONSHIP BETWEEN SELECTED TEACHER VARIABLES AND GROWTH IN ARITHMETIC IN GRADES FOUR, FIVE, AND SIX by Demitrios Peter Prekeges B. A., Eastern Washington State College, 1951 M. A., University of Montana, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in the Department of Mathematics Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f Mathematics Education The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date December 19, 1973 ABSTRACT THE RELATIONSHIP BETWEEN SELECTED TEACHER VARIABLES AND GROWTH IN ARITHMETIC IN GRADES FOUR, FIVE, AND SIX by Demitrios Peter Prekeges Problem For many years mathematicians and mathematics educators have been s t a t i n g that teachers of arithmetic need a greater knowledge i n mathematics and methods of teaching mathematics. Many colleges have required more mathematics for t h e i r future elementary teachers. The b e l i e f i s that an i n d i v i d u a l with a stronger mathematical background w i l l better teach mathematics to h i s elementary students. The review of the l i t e r a t u r e as a whole does not agree. Few researchers have found s i g n i f i c a n t r e l a t i o n s h i p s between teacher knowledge and teacher effectiveness. The review of the l i t e r a t u r e further indicates that most researchers did not measure teacher variables p r e c i s e l y . Also, most researchers neither p a r t i t i o n e d nor measured d i r e c t l y student growth. They used standardized tests or administrative ratings to determine teacher effectiveness. Procedures Two instruments were constructed to measure teacher understanding and teacher att i t u d e . The test of understanding was designed to measure i i i i i the mathematical understandings as rela t e d to the arithmetic s e r i e s and syllabus of the two school d i s t r i c t s p a r t i c i p a t i n g i n this study. The att i t u d e inventory was a forced choice inventory which measured the teacher's a t t i t u d e toward contemporary mathematics•as opposed to t r a d i t i o n a l mathematics. Each p a r t i c i p a t i n g teacher also completed a questionnaire giving information about 12 other commonly reported vari a b l e s . These were i n the areas of quarter hours of college mathe-matics, quarter hours of new mathematics, quarter hours of mathematics methods, experience, and p r i n c i p a l ' s ratings as he viewed the teachers. To determine teacher effectiveness, student tests were constructed to d i r e c t l y measure the material of the arithmetic series and syllabus of the two school d i s t r i c t s p a r t i c i p a t i n g i n th i s study. Three tests were constructed for each grade l e v e l ; an understanding t e s t , a problem solving test, and a computation test. The pre-test post-test procedure was used to determine student growth. The population for th i s study was 61 fourth, f i f t h , and s i x t h grade classes and t h e i r 61 teachers. The population was randomly selected from over 400 teachers i n two Washington State school d i s t r i c t s . The d i s t r i c t s used the same arithmetic s e r i e s and a s i m i l a r syllabus, but are i n d i f f e r e n t geographic locations. Results and Conclusions With the minor exception of a s i g n i f i c a n t c o r r e l a t i o n between p r i n c i p a l ' s rating and growth i n computation, there were no s i g n i f i c a n t r e l a t i o n s h i ps between any of the teacher v a r i a b l e s , when taken i n d i v i d u a l l y or i n groups, and student growth i n any of the three areas—understanding, problem solving, and computation—when taken i n d i v i d u a l l y or in groups. In t h i s study, every e f f o r t was made to eliminate the de f i c i e n c i e s of previous studies. Yet the i r r e s u l t s are, i n general, confirmed. If mathematicians and mathematics educators are to p e r s i s t i n t h e i r opinion that the educational background of teachers i s related to student gains, then i t seems that d i f f e r e n t independent variables must be i d e n t i f i e d . It seems highly u n l i k e l y that success would reward any further exploration of those i d e n t i f i e d i n this study, TABLE OF CONTENTS Page ABSTRACT . i i LIST OF TABLES v i i ACKNOWLEDGEMENT . . . . . . . . . . . v i i i > Chapter 1. INTRODUCTION . . . . 1 The Problem 1 2. BACKGROUND AND JUSTIFICATION 9 Secondary School Reviews . . . . . 9 Elementary School Reviews 18 A t t i t u d i n a l Reviews 30 J u s t i f i c a t i o n . of the Study 36 The Variables 40 Hypotheses 40 3. DESIGN OF THE STUDY 42 De f i n i t i o n s 42 The Subjects 42 Construction of Tests . 43 Construction of a Test of Teacher Understanding . . 44 Construction of an Inventory to Measure Teacher Attitude Toward Contemporary Mathematics Opposed to T r a d i t i o n a l Mathematics 46 Construction of Student Tests of Arithmetic . . . . 54 v v i Chapter Page Plan of the Study 62 S t a t i s t i c a l Procedure . 64 4. RESULTS 70 5. CONCLUSIONS . . . . . . . . . . 74 BIBLIOGRAPHY . . . . . . . . 79 APPENDICES . . . . . . . . . . 91 A. Understanding Inventory 91 B. An Inventory to Measure Teacher's A t t i t u d e Toward Contemporary as Opposed to T r a d i t i o n a l Mathematics . 104 C. A r i t h m e t i c Information, Fourth Grade . . . . 128 D. A r i t h m e t i c Information, F i f t h Grade . . . 144 E. A r i t h m e t i c Information, S i x t h Grade 160 F. Teacher Questionnaire 177 G. P r i n c i p a l ' s Rating Sheet 179 LIST OF TABLES Table Page 1. Point B i s e r i a l . C o r r e l a t i o n 51 2. Items Which Have Cor r e l a t i o n C o e f f i c i e n t s , of .20 or Greater.with a Factor . . . . . . . . . . . 52 3. Point B i s e r i a l Correlation . . . . . . . . . . . . . 53 4. Items Which Have Correlation C o e f f i c i e n t s of .20 or Greater with a Factor 54 5. Number of Students Involved f o r Third Administration . 60 6. Test-Retest R e l i a b i l i t y C o e f f i c i e n t s for Each Test and T4, T5, and T6 61 7. Mean Student Score on Each Test 62 8. Number of Teachers P a r t i c i p a t i n g 63 9. Number of Students P a r t i c i p a t i n g 63 10. Correlation C o e f f i c i e n t s Comparing Teacher Variables to Student Variables 71 2 11. R Results When the Teacher Variables Are Compared with Each of the Student Variables . . . . 72 v i i ACKNOWLEDGEMENTS The a u t h o r wishes t o express h i s a p p r e c i a t i o n to the s u p e r i n -t e n d e n t s , p r i n c i p a l s , and t e a c h e r s of the many s c h o o l d i s t r i c t s t h a t c o o p e r a t e d to make t h i s s t u d y p o s s i b l e . The author i s a l s o i n d e b t e d to the f a c u l t y and s t a f f members at the U n i v e r s i t y o f B r i t i s h Columbia. In p a r t i c u l a r , the author wishes to ex p r e s s h i s a p p r e c i a t i o n to the s u p e r v i s o r o f t h i s s t u d y , Dr. E r i c D. MacPherson, f o r h i s i n t e r e s t , encouragement, u n d e r s t a n d i n g , and i n s i g h t , and to the o t h e r committee members, Dr. Donald E. A l l i s o n , Dr. Thomas A. Howitz, Dr. Benjamin N. Moyls, f o r t h e i r c r i t i c a l a n a l y s i s and i n t e r e s t . The author i s d e e p l y i n d e b t e d to h i s w i f e , M a r i l y n , who has been a c o n t i n u o u s s o u r c e o f i n s p i r a t i o n and u n d e r s t a n d i n g throughout the w r i t e r ' s c o u r s e o f s t u d y . The au t h o r a l s o r e a l i z e s the t o l e r a n c e d i s p l a y e d by h i s f o u r sons: P e t e r A l l e n , David James, Donald Lee, and P a u l S c o t t . D e m i t r i o s P. Prekeges 477 North 5th S t r e e t Cheney, Washington 99004 v i i i Chapter 1 INTRODUCTION The Problem For many years mathematicians and mathematics educators have been st a t i n g that teachers of arithmetic need a greater knowledge of mathematics. Morton,^ i n 1939, recommended that every elementary teacher be required to complete 6 to 10 semester hours i n mathematics. 2 Wren, i n 1941, pointed out that the mathematical background of teachers was inadequate, and i t was up to the teacher t r a i n i n g colleges to improve the s i t u a t i o n . In the 10 year period following World War I I , many organi-zations emphasized the mathematical needs of the elementary teachers. The f i r s t such postwar suggestion was made by the U.S. Commission on Post-3 war Plans. This commission recommended that teachers of arithmetic should study a course i n the teaching of arithmetic and one or more courses i n subject matter background. This report was followed by the Manpower "*"R. L. Morton, "Mathematics i n the. Training of Arithmetic Teachers," Mathematics Teacher, 32:106-110, December, 1939. 2 F. L. Wren, "Questions f o r the Teacher of Arithmetic," A r i t h -metic i n General Education, Sixteenth Yearbook of the National Council of Teachers of Mathematics (Washington, D.C: National Council of Teachers of Mathematics, 1941), pp. 290-303. 3 Commission on Postwar Plans (Washington, D.C: National Council of Teachers of Mathematics, 1945). 1 2 Report which.recommended, "A professionalized subject-matter course emphasizing the use of mathematics i n projects undertaken by ch i l d r e n to learn the meaning of concepts i s a minimum requirement.""* More recently the Mathematical Association of America, through i t s Committee • 6 on the Undergraduate Program i n Mathematics, recommended that every teacher of arithmetic should have a minimum of three college courses i n mathematics consisting of four semesters of mathematics and a one-semester methods course i n arithmetic. The Committee also suggested content outlines f or these courses. During the same period of time many contemporary mathematicians and mathematics educators expressed t h e i r viewpoint. In 1948, Wren^ again wrote on the needs of the elementary teacher. He pointed out that functional competence i n arithmetic i s e s s e n t i a l as a character-i s t i c of the educated i n d i v i d u a l . He indicated that f u n c t i o n a l competence i n arithmetic consisted of: 1. P r o f i c i e n c y i n fundamental s k i l l s 2. Comprehension of basic concepts 3. Appreciation of s i g n i f i c a n t meanings 4. Development of desirable attitudes 5. E f f i c i e n c y i n making sound applications 4 Manpower for Research, S c i e n t i f i c and Public P o l i c y , Vol. IV (Washington, D.C.: Government P r i n t i n g O f f i c e , 1947). "*Ibid. , p. 11. ^Recommendations for the Training of Teachers, A Summary (Buffalo, New York: Mathematics Association of America, 1961). ^F. L. Wren, "The Professional Preparation of Teachers of Arithmetic," Supplementary Educational Monographs, No. 66 (Chicago: University of Chicago Press, 1948), pp. 80-90. 6. Confidence in making intelligent and independent 8 interpretations. Wren then considered these as the six major objectives in arithmetic. If these are the objectives, then a teacher must have these compe-tencies. In 1949, Layton surveyed the certification requirements of mathematics teachers and found that most states did not have any requirements. He recommended the development of such requirements. In 1949, Glennon ^tested a group of college freshmen and seniors and found that the mathematical understanding of the freshmen was higher than the mathematical understanding of the seniors. These results seem to indicate that either a loss of mathematical understanding takes place while a student is in college, or the freshmen in 1949 were 11 better prepared than the seniors. In 1951, Layton surveyed the training prescribed by teacher training colleges and found that a majority did not require any mathematics courses for their elementary I 12 teachers. In 1951, Newsom outlined the mathematical background he fel t was needed by elementary teachers. This outline included topics 8Ibid., p. 82. 9 W. I. Layton, "The Certification of Teachers of Mathematics,1  Mathematics Teacher, 42:377-380, December, 1949. "^V. J. Glennon, "A Study in Needed Redirection in the Prepa-ration of Teachers of Arithmetic," Mathematics Teacher, 42:389-396, December, 1949. ''"Hi. I. Layton, "Mathematical Training Prescribed by Teachers Colleges in Preparation of Elementary Teachers," Mathematics Teacher, 44:551-556, December, 1951. 12 C. V. Newsom, 'Mathematical Background Needed by Teachers," The Teaching of Arithmetic, Fi f t i e t h Yearbook of the National Society for the Study of Education, Part II (Chicago: University of Chicago Press, 1951), pp. 232-250. such as h i s t o r i c a l development, the r e a l number system, measurement, 13 and a p p l i c a t i o n s . In 1951, G r o s s n i c k l e surveyed the s t a t e teacher t r a i n i n g c o l l e g e s and received responses from 129 of them. He found that more c o l l e g e s were r e q u i r i n g four years of study to teach than was the case 20 years e a r l i e r . He a l s o found l i t t l e change i n the mathematics requirements of f u t u r e elementary teachers. He recommended that a l l f u t u r e teachers should have a methods course i n the teaching of mathematics and those who had not taken mathematics beyond the eighth grade should have a content course i n mathematics before t h e i r methods course. In 1953, mathematics educators were s t i l l d i s c u s s i n g these 14 same problems. Schaaf, a f t e r w r i t i n g about the l a c k of courses f o r teachers, o u t l i n e d the scope of a course de a l i n g w i t h the subject matter of a r i t h m e t i c . He suggested that a l l f u t u r e teachers need such a course. Phillips'*""' attempted to show the need f o r a mathematics course f o r elementary teachers by recording data about students entering the course " A r i t h m e t i c f o r Teachers" at the U n i v e r s i t y of I l l i n o i s . Two of h i s seven conclusions were: 1. The four major f a c t o r s i n f l u e n c i n g the students' r e a c t i o n to mathematics are method of p r e s e n t a t i o n , o p p o r t u n i t i e s f o r achievement, teacher's p e r s o n a l i t y , and type of problems solved. 13 F. C. G r o s s n i c k l e , "The T r a i n i n g of Teachers of A r i t h m e t i c , " The Teaching of A r i t h m e t i c , F i f t i e t h Yearbook of the N a t i o n a l Society f o r the Study of Education, Part I I (Chicago: U n i v e r s i t y of Chicago Press, 1951), pp. 203-231. "^ W. L. Schaaf, " A r i t h m e t i c f o r A r i t h m e t i c Teachers," School. Science and Mathematics, 53:537-543, October, 1953. "*"^ C. P h i l l i p s , "Background and Mathematical Achievement of Elementary Education Students i n A r i t h m e t i c f o r Teachers," School  Science and Mathematics, 53:48-52, January, 1953. 2 . A c h i e v e m e n t i n t h e m e a n i n g and u n d e r s t a n d i n g o f a r i t h m e t i c i s e x t r e m e l y l o w . O r l e a n s and Wandt"*" ^ w r o t e : I f a r i t h m e t i c i s t o be t a u g h t so t h a t c h i l d r e n a c q u i r e r e a l u n d e r s t a n d i n g o f a r i t h m e t i c p r o c e s s e s and c o n c e p t s , i t w o u l d seem o b v i o u s t h a t t h e t e a c h e r s o f a r i t h m e t i c must p o s s e s s t h e u n d e r -s t a n d i n g t h a t t h e y a r e a t t e m p t i n g t o t r a n s m i t t o t h e i r s t u d e n t s . 19 They t h e n p r e s e n t e d t h e f i n d i n g s o f O r l e a n s f r o m a s t u d y i n w h i c h he a d m i n i s t e r e d a t e s t t o 722 s u b j e c t s . The p u r p o s e o f t h e e v a l u a t i o n was t o d e t e r m i n e t h e u n d e r s t a n d i n g o f t h e p r o c e s s e s and c o n c e p t s o f a r i t h m e t i c . He c o n c l u d e d t h a t f u t u r e and p r a c t i c i n g t e a c h e r s h a v e a l o w u n d e r s t a n d i n g o f t h e s e p r o c e s s e s and c o n c e p t s . O r l e a n s and Wandt t h e n w r o t e : I f t h e u n d e r s t a n d i n g o f a r i t h m e t i c p o s s e s s e d by t e a c h e r s i s t o be i n c r e a s e d , t e a c h e r - t r a i n i n g i n s t i t u t i o n s m u s t make t h i s one o f t h e i r g o a l s . The t e a c h e r - e d u c a t i o n i n s t i t u t i o n s may h a v e o n l y an i n d i r e c t i n f l u e n c e o n t h e p r o g r a m o f number w o r k i n t h e s c h o o l s , b u t t h e y c a n d i r e c t l y i n f l u e n c e t h e p r o s p e c t i v e t e a c h e r ' s k n o w -l e d g e and u n d e r s t a n d i n g o f a r i t h m e t i c and h i s p r e p a r a t i o n f o r c h i s r e s p o n s i b i l i t i e s i n g e t t i n g c h i l d r e n t o l e a r n a b o u t n u m b e r s . 21 I n 1 9 5 6 , S n a d e r , a f t e r r e v i e w i n g t h e l i t e r a t u r e , w r o t e , " T h i s 22 s i t u a t i o n i s d e p l o r a b l e , t o s a y t h e l e a s t . " He s e n t a q u e s t i o n n a i r e " ^ I b i d . , p . 5 1 . 1 7 J . S . O r l e a n s and E . W a n d t , "The U n d e r s t a n d i n g o f A r i t h m e t i c P o s s e s s e d by T e a c h e r s , " E l e m e n t a r y S c h o o l J o u r n a l , 5 3 : 5 0 1 - 5 0 7 , M a y , 1953 . 18 I b i d . , p . 5 0 1 . 19 J . S . O r l e a n s , The U n d e r s t a n d i n g o f A r i t h m e t i c P r o c e s s e s and  C o n c e p t s P o s s e s s e d by T e a c h e r s o f A r i t h m e t i c , P u b l i c a t i o n N o . 12 (New Y o r k : O f f i c e o f R e s e a r c h and E v a l u a t i o n , C o l l e g e o f t h e C i t y o f New Y o r k , 1 9 5 2 ) . 20 O r l e a n s and W a n d t , o p . c i t . , p . 507 . 21 D . S n a d e r , " M a t h e m a t i c a l B a c k g r o u n d f o r T e a c h e r s o f A r i t h m e t i c , " The A r i t h m e t i c T e a c h e r , 3 : 5 9 - 6 5 , M a r c h , 1956 . 2 2 I b i d . , p . 6 1 . t o a r e p r e s e n t a t i v e g r o u p o f s p e c i a l i s t s i n a r i t h m e t i c and f o u n d t h i s g r o u p w o u l d l i k e e l e m e n t a r y s c h o o l t e a c h e r s t o h a v e s t u d i e d m a t h e m a t i c s f o r a minimum o f s i x s e m e s t e r h o u r s . F u r t h e r , t h e m a t h e m a t i c s s t u d i e d s h o u l d n o t be t h e t y p i c a l c o l l e g e m a t h e m a t i c s , b u t i t s h o u l d be m a t h e m a t i c s t h a t i n v o l v e s m a i n l y t h e u n d e r s t a n d i n g o f t h e b a c k g r o u n d s n e e d e d by a t e a c h e r o f a r i t h m e t i c . Such f i n d i n g s , s t a t e m e n t s , and r e c o m m e n d a t i o n s h a v e l e d t o t h e d e v e l o p m e n t o f c o u r s e s f o r f u t u r e e l e m e n t a r y t e a c h e r s . M o s t c o l l e g e s 23 now h a v e a t l e a s t one s u c h r e q u i r e d c o u r s e and some c o l l e g e s h a v e e s t a b l i s h e d a m a j o r e m p h a s i s i n m a t h e m a t i c s f o r e l e m e n t a r y t e a c h e r s . These a r e g e n e r a l l y c l a s s e s d e s i g n e d f o r t h i s p u r p o s e as d i s t i n c t f r o m r e g u l a r f r e s h m a n and sophomore m a t h e m a t i c s p r o g r a m s . The r e c o m m e n d a t i o n s f o r and t h e e x t e n s i v e d e v e l o p m e n t o f t h e s e c o u r s e s a r e b a s e d on t h e b e l i e f t h a t i f t h e t e a c h e r h a s a b e t t e r m a t h e m a t i c a l b a c k g r o u n d , h i s s t u d e n t s w i l l l e a r n and u n d e r s t a n d more 24 a r i t h m e t i c . M e t z n e r has b e e n one o f t h e few t o q u e s t i o n t h i s b e l i e f . I n h i s summary o f a sympos ium a t t h e H a r v a r d G r a d u a t e S c h o o l o f 25 E d u c a t i o n he q u o t e s P r o f e s s o r James Coleman o f J o h n H o p k i n s U n i v e r s i t y : " . . . n o o n e knows enough a b o u t t e a c h e r s ' p e r f o r m a n c e t o b e a b l e t o ,,26 p r e d i c t t h e e f f e c t s o f l o n g e r t e a c h e r p r e p a r a t i o n o n p u p i l a c h i e v e m e n t . A r e v i e w o f e d u c a t i o n a l r e s e a r c h l i t e r a t u r e seems t o s u p p o r t t h e v i e w s o f M e t z n e r and Coleman w h e r e t h e e l e m e n t a r y t e a c h e r i s i n v o l v e d . I n f a c t , 23 E a s t e r n W a s h i n g t o n S t a t e C o l l e g e , C h e n e y , W a s h i n g t o n ; S o u t h -w e s t e r n S t a t e C o l l e g e , W e a t h e r f o r d , O k l a h o m a ; N o r t h e r n M i c h i g a n U n i v e r s i t y , M a r q u e t t e , M i c h i g a n . 24 S. M e t z n e r , "The T e a c h e r P r e p a r a t i o n M y t h : A P h o e n i x Too F r e q u e n t , " P h i D e l t a K a p p a n , L : 1 0 5 - 1 0 7 , O c t o b e r , 1968 . 2 5 I b i d . 2 6 I b i d . , p . 1 0 5 . as i s discussed i n Chapter 2, this researcher f a i l e d to f i n d any studies which, with any confidence, imply that increased mathematical education of elementary teachers or future elementary teachers increases p u p i l achievement. . As w i l l be discussed below, i t i s questionable that as much confidence as the res u l t s i n d i c a t e can be placed on these studies. At the same time, i t i s evident the majority of the people working i n mathematics education f e e l that better t r a i n i n g f o r the elementary teacher i s a necessity and th i s t r a i n i n g should include more work i n mathematics which should be designed to teach mathematical understanding. These same experts seem to f e e l that such t r a i n i n g w i l l lead to better p u p i l achievement i n mathematics even though there i s very l i t t l e good evidence to substantiate t h i s b e l i e f . Some studies have been c a r r i e d out i n an e f f o r t to determine whether or not teacher knowledge has an e f f e c t on student achievement. Two things characterize most of these studies: 1. Indirect measures of teacher a b i l i t y . Most of these studies use college c r e d i t s i n mathematics or some type of arithmetic test. They do not attempt to measure mathematical understanding. 2. Imprecise measures of student performance. Most of these studies use some type of published standardized test. They do not attempt to measure the material of a given text or the material of the school syllabus. The naive b e l i e f that increased teacher understanding has an ef f e c t on student understanding remains strong. As the review of the l i t e r a t u r e indicates i n Chapter 2, there i s c e r t a i n l y no hope of supporting t h i s b e l i e f by r e p l i c a t i n g or expanding on the studies with the two f a i l i n g s noted above. By constructing tests which c a r e f u l l y measure teacher understanding i n and attitu d e toward mathematics and student competencies i n mathematics, i t might be possible to i d e n t i f y some r e l a t i o n s h i p between these v a r i a b l e s . In t h i s study, a very serious attempt i s made to i d e n t i f y and measure p r e c i s e l y those teacher variables most apt to be re l a t e d to s i m i l a r l y i d e n t i f i e d and measured student v a r i a b l e s . This study f i r s t tests c e r t a i n a p r i o r i hypotheses concerning the r e l a t i o n s h i p between these v a r i a b l e s , and then searches, speculatively, for unexpected possible r e l a t i o n s h i p s which might form a foundation for further study. Chapter 2 BACKGROUND AND JUSTIFICATION This review of the literature is separated into three sections. The f i r s t section deals with secondary (including junior high school) teachers' knowledge and how i t is related to student achievement; the second section deals with the elementary (grades Kindergarten through six) teachers' knowledge and how i t is related to student achievement; and the third section deals with'teachers' attitude toward mathematics. Secondary School Reviews The f i r s t postwar report which attempted to determine a relation-ship between teacher variables and student growth seems to be Rostker's^ report of the results of data collected in 1936 and 1937. Rostker tested 350 social studies students in the seventh and eighth grades who were taught by 28 different teachers. He pre-tested and post-tested the students and used student gains as a measure of teaching a b i l i t y . He measured the teachers' subject matter knowledge by using tests covering the material taught. He wrote: These teacher measures are primarily tests of information and indicate no significant relationship between knowledge of subject information and teaching ab i l i t y . L. E. Rostker, "The Measurement of Teaching Ability, Study Number One," The Journal of Experimental Education, 14:6-51, September, 1945. 2 Ibid., p. 45. •9 10 3 4 R o l f e and L a D u k e , a l s o u s i n g s e v e n t h a n d e i g h t h g r a d e s t u d e n t s , f o u n d s i m i l a r r e s u l t s . R o l f e u s e d c i t i z e n s h i p w h i l e L a Duke u s e d t h e c h i l d ' s s e n s e o f r e s p o n s i b i l i t y i n t h e f u n c t i o n i n g o f a d e m o c r a t i c s o c i e t y as s u b j e c t m a t t e r m a t e r i a l . B o t h R o l f e and L a Duke c o l l e c t e d t h e i r d a t a be tween 1937 and 1 9 3 9 , b u t d i d n o t r e p o r t t h e i r r e s u l t s u n t i l 1 9 4 5 . I n 1 9 4 6 , L i n s ^ a t t e m p t e d t o d e t e r m i n e w h e t h e r o r n o t a r e l a t i o n -s h i p e x i s t s b e t w e e n p r e - s e r v i c e e d u c a t i o n and s t u d e n t g a i n s . H i s s a m p l e c o n s i s t e d o f 17 f i r s t y e a r t e a c h e r s and t h e i r 27 c l a s s e s , w h i c h c o m p r i s e d most a r e a s and l e v e l s o f t h e s e c o n d a r y s c h o o l s . The t e a c h e r m e a s u r e s f o r p r e - s e r v i c e e d u c a t i o n w e r e g r a d e s i n c o l l e g e c o u r s e s and r a t i n g s o f p o s s i b l e s u c c e s s i n t e a c h i n g b y t h e i r c o l l e g e p r o f e s s o r s . The s t u d e n t g a i n s w e r e c a l c u l a t e d by p r e - t e s t i n g and p o s t - t e s t i n g t h e c o u r s e m a t e r i a l f o r t h e s e c o n d s e m e s t e r u s i n g s t a n d a r d i z e d t e s t s . L i n s f o u n d t h a t g r a d e s a n d r a t i n g s i n p r e - s e r v i c e e d u c a t i o n , i n c l u d i n g p r a c t i c e t e a c h i n g a r e n o t s i g n i f i c a n t l y r e l a t e d t o t e a c h i n g e f f i c i e n c y as m e a s u r e d by s t u d e n t g a i n s c o r e s . I n 1949 , S n i d e r ^ l o o k e d a t s e v e r a l f a c t o r s w h i c h m i g h t b e r e l a t e d t o s t u d e n t a c h i e v e m e n t i n c o l l e g e . He f o u n d a p o s i t i v e b u t n o t 3 J . F . R o l f e , "The Measurement o f T e a c h i n g A b i l i t y , S t u d y Number T w o , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 1 4 : 5 2 - 7 4 , S e p t e m b e r , 1 9 4 5 . 4 C . V . L a D u k e , "The Measurement o f T e a c h i n g A b i l i t y , S t u d y Number T h r e e , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 1 4 : 7 5 - 1 0 0 , S e p t e m b e r , 1945 5 L . J . L i n s , " T h e P r e d i c t i o n o f T e a c h i n g E f f i c i e n c y , " The J o u r n a l  o f E x p e r i m e n t a l E d u c a t i o n , 1 5 : 2 - 6 0 , S e p t e m b e r , 1946 . ^ H . L . S n i d e r , " R e l a t i o n s h i p s Between F a c t o r s o f H i g h S c h o o l Back-g r o u n d and A c h i e v e m e n t i n C e r t a i n S u b j e c t F i e l d s " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f N e b r a s k a , 1 9 4 9 ) . 11 significant relationship between the college preparation secondary teachers have in their teaching f i e l d and the achievement of their students in this f i e l d when the students attend college. It would seem that the studies reported in the forties did nothing to support the premise that teacher change causes student change of a like kind. It might well be, as noted in Chapter 1, that teacher Variables and student variables were not adequately measured by the standardized tests and other procedures used in these studies. In 1950, Schunert^ compared the f i n a l achievement of algebra and geometry classes whose teachers had less than two years of college mathematics with algebra and geometry classes whose teachers had more than two years of college mathematics. He found no significant d i f f e r -ence, but the results favored those teachers with the lesser amount of college preparation in mathematics. 8 In 1957, Taylor, attempting to find a significant relationship between teacher factors and science students, tested more than 1500 science students with the Essential High School Content Battery and the California Occupational Interest Inventory. He compared these results with four teacher factors: (1) attitude, (2) college credit in pro-fessional education, (3) college credit in science, and (4) years of experience. None of the factors had a significant relationship with ^J. R. Schunert, "The Association of Mathematics Achievement with Certain Factors Resident in the Teacher, in the Teaching, in the Pupil, and in the School" (unpublished Doctoral dissertation, University of Minnesota, 1951). 8 T. W. Taylor, "A Study to Determine the Relationship Between Growth in Interest and Achievement of High School Science Students and Science-Teacher Attitudes, Preparation, and Experience" (unpublished Doctoral dissertation, North Texas State College, 1957). Dissertation  Abstracts, 17:2943-2944, No. 12, 1956/57. 12 t h e d e v e l o p m e n t o f g r e a t e r s c i e n c e a c h i e v e m e n t . When a l l f o u r f a c t o r s w e r e t a k e n as a c o m p o s i t e , a s i g n i f i c a n t p o s i t i v e r e l a t i o n s h i p was f o u n d . T h i s r e l a t i o n s h i p m i g h t i n d i c a t e t h a t an i n t e r a c t i o n o f f a c t o r s i s i n v o l v e d i n a f f e c t i n g s c i e n c e a c h i e v e m e n t . 9 S p a r k s , u s i n g t h e Iowa T e s t o f E d u c a t i o n a l D e v e l o p m e n t , t e s t e d a g r o u p o f h i g h s c h o o l s t u d e n t s i n 1955 and a g a i n i n 1958 . From t h e s e r e s u l t s he d e f i n e d h i g h a c h i e v e m e n t s c h o o l s and l o w a c h i e v e m e n t s c h o o l s . He f o u n d t h a t t e a c h e r s i n h i g h a c h i e v e m e n t s c h o o l s had t a k e n more h o u r s o f m a t h e m a t i c s as u n d e r g r a d u a t e s i n c o l l e g e t h a n t h e t e a c h e r s i n l o w a c h i e v e m e n t s c h o o l s . He a l s o f o u n d t h a t t h e s t u d e n t s i n t h e h i g h a c h i e v e m e n t s c h o o l s r a t e d t h e i r t e a c h e r s h i g h e r i n s u b j e c t m a t t e r k n o w l e d g e and t e a c h e r competency t h a n d i d t h e s t u d e n t s i n l o w a c h i e v e -ment s c h o o l s . The two s t u d i e s i n t h e l a t t e r h a l f o f t h e f i f t i e s seem t o i n d i -c a t e t h a t t h e r e was some r e l a t i o n s h i p b e t w e e n s t u d e n t a c h i e v e m e n t and some c o m p o s i t e o f t e a c h e r f a c t o r s . A g a i n i t m i g h t be t h a t t h e u s e o f s t a n d a r d i z e d t e s t s d i d n o t g i v e a s u f f i c i e n t l y p r e c i s e m e a s u r e o f s t u d e n t a c h i e v e m e n t . Some o t h e r e v a l u a t i o n more d i r e c t l y c o n n e c t e d t o t h e m a t e r i a l t o be l e a r n e d m i g h t p r o d u c e more p o s i t i v e r e l a t i o n s h i p s . I n 1 9 6 0 , S t o n e k i n g " ^ a t t e m p t e d t o d e t e r m i n e w h i c h o f t h e f o u r f a c t o r s — a g e , amount o f t e a c h i n g e x p e r i e n c e , l e v e l o f a c a d e m i c p r e p a -r a t i o n , o r m a t h e m a t i c s b a c k g r o u n d — c o n t r i b u t e s most t o an i n d i v i d u a l ' s u n d e r s t a n d i n g o f s e l e c t e d b a s i c a r i t h m e t i c a l p r i n c i p l e s and 9 J . N . S p a r k s , " A C o m p a r i s o n o f Iowa H i g h S c h o o l s R a n k i n g H i g h and Low i n M a t h e m a t i c a l A c h i e v e m e n t " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f I o w a , 1 9 6 0 ) . 1 0 L . W. S t o n e k i n g , " F a c t o r s C o n t r i b u t i n g t o U n d e r s t a n d i n g o f S e l e c t e d B a s i c A r i t h m e t i c a l P r i n c i p l e s and G e n e r a l i z a t i o n s " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , I n d i a n a U n i v e r s i t y , 1 9 6 0 ) . 13 generalizations. He administered h i s self-constructed instrument to measure basic arithmetical p r i n c i p l e s and generalizations to 1066 examinees. He also obtained a personal data sheet from each examinee to determine which of the four factors they possessed. The examinees were pupils i n grades 8 through 12, students i n a college preparatory course, and p r a c t i c i n g teachers. He found that there was no s i g n i f i c a n t difference i n the scores of the examinees who were p r a c t i c i n g teachers and those who were not p r a c t i c i n g teachers. This would i n d i c a t e that experience as a teacher does not enhance one's understanding of basic arithmetical p r i n c i p l e s and generalizations. Stoneking's r e s u l t s might also be an i n d i c a t i o n that experience as a teacher does not enable one to be a more e f f e c t i v e teacher of mathematics. In 1960, Lindstedt"''"'" compared the scores on the f i n a l examination of ninth grade mathematics students with the number of college mathe-matics courses taken by t h e i r teachers. There was no s i g n i f i c a n t difference i n the scores of students taught by teachers c l a s s i f i e d on the basis of the amount of mathematics preparation. 12 In 1962, Leonhardt, using the Cooperative General Mathematics  Test f o r High School Classes with tenth grade geometry classes, ranked 45 d i f f e r e n t high schools. The ranking was from high to low depending upon the mean score of the student. He then chose 12 schools for h i s analysis: four small schools, four medium sized schools, and four large " S . A. Lindstedt, "Teacher Q u a l i f i c a t i o n arid Grade IX Mathe-matics Achievement," The Alberta Journal, of Education, 6:76-85, June, 1960. 12 E. A. Leonhardt, "An Analysis of Selected Factors Related to High and Low Achievement i n Mathematics," (unpublished Doctoral d i s s e r t a t i o n , U niversity of Nebraska, 1962). schools. Two schools i n each group were high-ranked and two schools i n each group were low-ranked. He chose one teacher from each school. He found that more of the teachers from high-ranked .schools had t h e i r major undergraduate preparation i n mathematics than did those from low-ranked schools. The teacher r a t i o was four to three. He also reported that more of the teachers from high-ranked schools had taken graduate work i n mathematics than had the teachers from low-ranked schools. The teacher r a t i o was two to one. In analyzing the above study and s i m i l a r studies, an important possible compounding v a r i a b l e must be noted. There could be an auto-matic s e l e c t i o n process operating where the schools which are noted f o r t h e i r strong programs a t t r a c t candidates with stronger backgrounds. It i s also possible that these same people desire to continue these programs and t h e i r education; therefore, they attend graduate school to become better prepared. 13 In 1963, Garner pre-tested and post-tested ninth grade algebra students using the Cooperative Algebra Test, Form 1. From the super-vi s o r s of the teachers of these algebra students he obtained the number of hours of college mathematics each of these teachers had taken. He found a s i g n i f i c a n t r e l a t i o n s h i p between the college mathematics prepa-r a t i o n of the teachers and t h e i r p u p i l s ' achievement i n algebra. 13 M. V. Garner, "A Study of the Educational Backgrounds and Attitudes of Teachers Toward Algebra as Related to the Attitudes and Achievements of Their Anglo-American and Latin-American Pupils i n First-Year Algebra Classes of Texas" (unpublished Doctoral d i s s e r t a t i o n , North Texas State University, 1963). D i s s e r t a t i o n Abstracts, 24:189, No. 1, 1963. In 1964, Peskin reported a s i g n i f i c a n t c o r r e l a t i o n between teacher understanding and student achievement. Teacher understanding was measured for the 55 teachers by Glennon's^ Test of Basic Mathe- matical Understandings. Student achievement for the 565 students was measured by the Cooperative Arithmetic Test, Form A and some self-made tests r e l a t e d to the material covered. Also i n 1964, Smith reported on the r e s u l t s of data c o l l e c t e d in 1957-58 concerning the r e l a t i o n s h i p between teacher professional education and student achievement. He used as h i s student c r i t e r i o n the r e s u l t s of the C a l i f o r n i a Achievement Test i n Arithmetic, Inter- mediate Battery, which he administered to 528 students i n the eighth grade. The information on the 28 teachers used i n this study was obtained from personnel records of the schools involved. He found a s i g n i f i c a n t r e l a t i o n s h i p between the c r e d i t s earned i n professional education courses (more than 28 c r e d i t s against less than 28 c r e d i t s ) and student achievement as measured by the C a l i f o r n i a Achievement Test  i n Arithmetic. He further reported that the number of college c r e d i t s i n mathematics and the number of years of teaching experience did not appear to be related to student achievement. """^A. S. Peskin, "Teacher Understanding and Attitude and Student Achievement and Attitude in Seventh Grade Mathematics" (unpublished Doctoral d i s s e r t a t i o n , New York University, 1964). L~*V. J. Glennon, "A Study of the Growth and Mastery of Certain Basic Mathematical Understandings on Seven Educational Levels" (unpublished Doctoral d i s s e r t a t i o n , Harvard University, 1948). 16 R. W. Smith, "The Achievement of Eighth Grade Students in Arithmetic with Respect to Selected Patterns of Teacher Preparation" (unpublished Doctoral d i s s e r t a t i o n , University of Oklahoma, 1964). In 1965, Goldberg et a l . 1 7 studied 51 seventh grade classes and t h e i r 1477 pupils i n the Talented Youth Project. By the end of the ninth grade, normal a t t r i t i o n had reduced the numbers to 37 classes and 868 students. Teacher factors such as amount of mathematical prepa-ra t i o n , degrees earned, and experience i n teaching mathematics were found to bear a s i g n i f i c a n t r e l a t i o n s h i p to pupil success at the end of the seventh grade. In aggregate, such factors accounted for about 20 percent of the variance i n p u p i l achievement. However, at the end of the ninth grade, teacher factors appeared to be exerting less influence on p u p i l achievement than i n e a r l i e r grades. When i n i t i a l p u p i l differences f or the ninth graders were co n t r o l l e d , the observed differences were no longer s i g n i f i c a n t . 18 In 1967, Rouse studied the c o r r e l a t i o n between the academic preparation of teachers of arithmetic and the arithmetic achievement of t h e i r students i n kindergarten through grade eight. He measured the academic preparation of the teachers by t o t a l i n g the mathematics courses they had taken i n high school, i n college, and any i n - s e r v i c e courses. He c a l l e d this t o t a l the t o t a l mathematics preparation of the teacher. The measure of student arithmetic achievement was h i s arithmetic scores on the C a l i f o r n i a Achievement Tests. The sample was M. L. Goldberg et a l . , "A Comparison of Mathematics Pre-programs f o r Able Junior High School Students," Summary, Conclusions  and Implications, Teachers College, Columbia University, New York: 1965. 18 W. M. Rouse, J r . , "A Study of the Co r r e l a t i o n Between the Academic Preparation of Teachers of Mathematics and the Mathematics Achievement of Their Students i n Kindergarten through Grade Eight" (unpublished Doctoral d i s s e r t a t i o n , Michigan State U n i v e r s i t y , 1967). 206 students and 273 teachers who had taught these students from kindergarten through grade eight. He found a low negative c o r r e l a t i o n between student achievement i n both arithmetic reasoning and arithmetic fundamentals and the t o t a l mathematics preparation of the teachers responsible for t h e i r arithmetic i n s t r u c t i o n from kindergarten through the middle of grade eight. The studies of the s i x t i e s seem to have added l i t t l e to the knowledge of the r e l a t i o n s h i p e x i s t i n g between teacher knowledge i n mathematics and t h e i r students' knowledge i n mathematics. It might be that the teacher v a r i a b l e related to teacher knowledge cannot accu-r a t e l y be measured by looking at the number of courses taken i n pre-service education. It might be better to measure teacher knowledge 19 i n some more d i r e c t way. Peskin did t h i s and did get a s i g n i f i c a n t r e l a t i o n s h i p . A second reason for fi n d i n g very few s i g n i f i c a n t r e l a t i o n s h i p s between teacher variables and student growth could be that most researchers use n a t i o n a l l y standardized evaluation i n s t r u -ments to measure student growth. More p o s i t i v e r e s u l t s might be possible i f the student achievement evaluation instrument covered that 20 material which was pertinent to that grade. Again, Peskin used some of these for the measure of student achievement and did get a s i g n i f i -cant r e l a t i o n s h i p . To summarize, using the mainly i n d i r e c t techniques of these studies, there i s l i t t l e evidence to ind i c a t e a r e l a t i o n s h i p between teacher knowledge i n mathematics and student achievement i n the secondary school (grades 7 through 12). Some studies do ind i c a t e that Peskin, loc. c i t . 2 0 I b i d . 18 some student achievement may be related to some teacher knowledge, but pr e c i s e l y what i s related to what i s not indicated. Elementary School Reviews Many studies attempting to r e l a t e teacher knowledge to teacher effectiveness have been done at the elementary l e v e l . Few attempt to r e l a t e teachers' knowledge i n a p a r t i c u l a r subject (say arithmetic) to student improvement or gain i n that subject. As was noted for the secondary studies, these studies use mainly i n d i r e c t measures of teacher q u a l i t y and vague measures of student performance. Those that seem to be relevant to th i s study have been published since 1950. 21 The f i r s t such study was car r i e d out by Ryans i n 1951. He worked with 275 teachers i n the t h i r d and fourth grades. He found no s i g n i f i c a n t r e l a t i o n s h i p between the amount of college t r a i n i n g ( i n t o t a l , no p a r t i c u l a r subject area) and a composite evaluation of effectiveness as a teacher. Three trained observers working inde-pendently determined, by observation, the effectiveness of the teacher. Notice again that Ryans used t o t a l hours of college t r a i n i n g as the measure of h i s teacher v a r i a b l e . Further, he used the opinions of observers as his measure of e f f e c t i v e teaching. If e f f e c t i v e teaching means student learning, and most educators accept t h i s d e f i n i t i o n , then one must measure the student learning and not attempt to i n f e r i t . In any case, i t i s not d i f f i c u l t to see how he could have f a i l e d to D. G. Ryans, "A Study of the Extent of Association of Certain Professional and Personal Data with Judged Effectiveness of Teacher Behavior," The Journal of Experimental Education, 20:67-77, September, 1951. 19 determine a r e l a t i o n s h i p between these measures of teacher knowledge and teacher e f f e c t i v e n e s s . 22 In c o n t r a s t , Mork, i n 1953, constructed f i v e d i f f e r e n t s c i e nce t e s t s f o r the students i n 8 grade f i v e and grade s i x c l a s s e s . He pre-tested and p o s t - t e s t e d these students f o r two consecutive years. During the second year of the study, four of the teachers (the experimental group) p a r t i c i p a t e d i n a one year i n - s e r v i c e program w h i l e the other four teachers (the c o n t r o l group) d i d not. The i n - s e r v i c e course d e a l t w i t h o b j e c t i v e s , content, methods, and m a t e r i a l s of science i n s t r u c t i o n . The course met once a month. Some of the gains on the f i v e d i f f e r e n t t e s t s were s i g n i f i c a n t when the r e s u l t s of the second year were compared w i t h the r e s u l t s of the f i r s t year. Because of t h i s , Mork concluded: The n u l l hypothesis was r e j e c t e d w i t h s u f f i c i e n t frequency to i n d i c a t e that teachers, through the given t e s t r e s u l t s of t h e i r p u p i l s , show an increased e f f e c t i v e n e s s i n i n s t r u c t i o n wh^ch i s asso c i a t e d w i t h an i n - s e r v i c e science education program. 24 In 1955, Steinbrook, attempting to determine a r e l a t i o n s h i p between c o l l e g e p r e p a r a t i o n and teacher e f f e c t i v e n e s s , r e c e i v e d from a d m i n i s t r a t i v e personnel a l i s t of 50 teachers who were considered to have had outstanding teaching success and 50 teachers who were 22 G. M. A. Mork, " E f f e c t s of an In-Service Teacher T r a i n i n g Program on P u p i l Outcomes i n F i f t h and S i x t h Grade Science" (unpublished D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y of Minnesota, 1953). D i s s e r t a t i o n  A b s t r a c t s , 13:522-523, No. 4, 1953. 2 3 I b i d . , p. 523. 24 R. S. Steinbrook, "Study of Some D i f f e r e n c e s i n Background, A t t i t u d e , Experience, and P r o f e s s i o n a l P r e p a r a t i o n of Selected Elementary Teachers w i t h C o n t r a s t i n g L o c a l Success Records" (unpublished D o c t o r a l d i s s e r t a t i o n , Indiana U n i v e r s i t y , 1955). D i s s e r t a t i o n A b s t r a c t s , 15:1013, No. 6, 1955. considered to have had the l e a s t successful teaching experiences. He sent each of these 100 teachers a questionnaire asking for a wealth of data. His opinion of the data indicated that the t o t a l amount of college work appears to contribute to teaching effectiveness, but teaching effectiveness at the elementary school appears to be more clo s e l y r e l a t e d to the types of professional preparation experienced by teachers. 25 In 1957, Soper found s i g n i f i c a n t r e s u l t s contrary to 26 Steinbrook. Soper worked with 2656 students and 128 teachers i n the fourth, f i f t h , and s i x t h grades. He separated the teachers into two groups using as h i s c r i t e r i o n f o r separation the amount of general academic and professional t r a i n i n g each teacher had accumulated. He also pre-tested and post-tested t h e i r students using the Stanford  Achievement Test. He found that the students with the higher gains had teachers from the group with le s s t r a i n i n g . It should be noted that Soper measured teacher effectiveness by evaluating student learning. He did not depend upon the opinions of administrative personnel as did Steinbrook. 27 In 1959, McCall and Krause worked with 73 teachers and t h e i r s i x t h grade students. They defined teacher effectiveness as growth 25 E. F. Soper, "A Study of the Relationship Between Certain Teacher-School C h a r a c t e r i s t i c s and Academic Progress, As Measured by Selected Standardized Tests, of Elementary Pupils i n Grades Four, Five, and Six of New York State Public Schools in C i t i e s under 10,000 Population" (unpublished Doctoral d i s s e r t a t i o n , Syracuse U n i v e r s i t y , 1956). 26 Steinbrook, l o c . c i t . 27 W. A. McCall and G. R. Krause, "Measurement of Teacher Merit for Salary Purposes," The Journal of Educational Research, 53:73-75, October, 1959. 21 i n the nine R's—reading, r i t i n , rithmetic ( s i c ) , research, reasoning, reporting, r e l a t i o n s h i p of persons, recreation, reasonable work s k i l l s — m e a s u r i n g t h i s growth by pre-testing and post-testing the students. These r e s u l t s were s t a t i s t i c a l l y analyzed and each of the 73 teachers was given a teacher effectiveness score which ranged from le a s t e f f e c t i v e , 20, to most e f f e c t i v e , 88. They found that the teachers' knowledge of a p a r t i c u l a r subject produced zero c o r r e l a t i o n when compared to teacher effectiveness. They also observed that classes taught by teachers whose average college grades were below 90 percent achieved better growth than did classes whose teachers' average college grades were above 90 percent. 28 In 1959, Smail reported on what seems to be the most complete study to date. He worked with 97 teachers and t h e i r 2438 students i n grades four, f i v e , and s i x . He defined teacher effectiveness as student gain by pre-testing i n the f a l l and post-testing i n the spring. He used the arithmetic tests of The Iowa Tests of Basic S k i l l s as h i s measurement instrument. He c a l l e d t h i s difference the p u p i l s ' mean-gain i n arithmetic. He did not f i n d a s i g n i f i c a n t difference i n p u p i l s ' mean-gain i n arithmetic when the classes of teachers with two years of preparation were compared with classes of teachers with four years of preparation. However, Smail did f i n d a s i g n i f i c a n t p o s i t i v e r e l a t i o n -ship between the number of mathematics methods courses completed by the teacher with four years of preparation and the p u p i l s ' mean-gain i n arithmetic. Smail determined teacher understanding in mathematics by R. W. Smail, "Relationships Between Pupil Mean-Gain i n Arithme- : t i c and Certain Attributes of Teachers: (unpublished Doctoral d i s s e r t a t i o n , University of South Dakota, 1959). 22 29 administering Glennon's Test of Basic Mathematical Understandings, He found that teacher understanding of basic mathematical concepts as measured by Glennon's test and p u p i l s ' mean-gain was not s i g n i f i c a n t . Further, he did not f i n d a s i g n i f i c a n t r e l a t i o n s h i p between the number of college mathematics courses a teacher had completed and p u p i l mean-gain i n arithmetic. This study would indi c a t e that an arithmetic methods course i n the pre-service education of future teachers i s the most important course leading to teacher effectiveness when defined as student learning. In 1960, Barnes, Cruickshank, and J. Foster used p r i n c i p a l s ' ratings of the teachers' mathematics i n s t r u c t i o n as the c r i t e r i o n f o r teacher effectiveness i n teaching mathematics. Their subjects were a l l of the fourth grade teachers from 66 d i f f e r e n t buildings. No s i g n i f i c a n t r e l a t i o n s h i p was found between the number of high school mathematics courses completed by the teachers and the p r i n c i p a l s ' ratings as to t h e i r effectiveness i n teaching mathematics. They also reported no s i g n i f i c a n t r e l a t i o n s h i p between the number of college mathematics courses completed by the teachers and the p r i n c i p a l s ' ratings as to t h e i r effectiveness i n teaching mathematics. 31 32 In 1960, Bassham conducted a study somewhat s i m i l a r to Smail's. He tested 28 s i x t h grade teachers using Glennon's Test of Basic 29 Glennon, loc. c i t . 30 K. Barnes, C. Cruickshank, and J. Foster, "Selected Educational and Experience Factors and Arithmetic Teaching," The Arithmetic Teacher, 7:418-420, December, 1960. 31 H. C. Bassham, "Relationship of Pupil Gain in Arithmetic Achievement to Certain Teacher C h a r a c t e r i s t i c s " (unpublished Doctoral d i s s e r t a t i o n , University of Nebraska, 1960). 32 Smail, loc. c i t . 23 Mathematical Understandings. The teachers' score on th i s test was considered as an i n d i c a t i o n of the l e v e l of the teachers' understanding of arithmetic. The 620 students were pre-tested i n September by the C a l i f o r n i a Achievement Test, Arithmetic, 1951, Form AA. The res u l t s of t h i s t e s t , i n conjunction with other data, allowed Bassham to predict the score of each student on Form BB of the same test when i t was given as a post-test i n A p r i l . Any r e s u l t s that varied from the predicted score was c a l l e d the deviation score of p u p i l gain. A s i g n i f i c a n t r e l a t i o n s h i p between teacher scores on the paper and p e n c i l test and deviation scores of pu p i l gain was reported. Bassham reported that i . teacher understanding as measured by Glennon's test explained approxi-mately one-fourth of the v a r i a t i o n i n the deviation scores of the pupil s . He also reported that t i e s i g n i f i c a n t r e l a t i o n s h i p between teacher understanding and deviation scores existed for p u p i l s with above mean i n t e l l i g e n c e , but not for students with below mean i n t e l l i -gence. In 1960, another study of a s i m i l a r nature was conducted by 33 H e i l , Powell, and F e i f e r . The subjects i n this study were 55 teachers and t h e i r fourth, f i f t h , and si x t h graders. This study was not r e s t r i c t e d to mathematics, but i t compared teacher knowledge with student achievement i n the l i b e r a l a r t s . The l i b e r a l arts knowledge of the teacher was measured by the Teacher Education Examination. Two parts of the examination, Professional Education Knowledge and L i b e r a l Arts Knowledge, were administered. Student achievement was measured by 33 L. M. H e i l , M. Powell, and I. F e i f e r , " C h a r a c t e r i s t i c s of Teacher Behavior and Competency Related to Achievement of Di f f e r e n t Kinds of Children i n Several Elementary Grades," New York: Brooklyn College, 1960. 24 pre-testing and post-testing with the Stanford Elementary and Inter- mediate Achievement Batt e r i e s. Their findings are s i m i l a r to those of most other investigators; that i s , n e g l i g i b l e c o r r e l a t i o n between student achievement and teacher knowledge. They also reported n e g l i -g i b l e c o r r e l a t i o n between student achievement and the teaching effectiveness of the teachers as determined by observers. 34 It should be noted that a l l three s i m i l a r studies, Smail, 35 36 Bassham, and H e i l et a l . , used paper and p e n c i l tests f or t h e i r measurements of teacher knowledge and standardized tests for determining student gain. Again, i t might be asked whether or not paper and p e n c i l tests taken by the teachers r e a l l y measure understanding i n mathematics. One must also again question the use of a standardized test to measure pup i l knowledge. The concern i s whether or not these tests evaluate the syllabus at the given grade l e v e l . If the answer i s no to e i t h e r one or both of these concerns, then enough information could be l o s t to eliminate the p o s s i b i l i t y of s i g n i f i c a n t differences. 37 In a related study, Houston, i n 1961, found by using objective tests that there i s no difference i n change i n mathematics achievement and mathematics i n t e r e s t between two groups of fourth, f i f t h , and s i x t h grade p u p i l s . One group of pupils had teachers who p a r t i c i p a t e d i n an i n - s e r v i c e education series by t e l e v i s i o n while the other group of pupils had teachers who p a r t i c i p a t e d i n a face-to-face l e c t u r e -discussion i n - s e r v i c e education s e r i e s . It seems that these r e s u l t s 34 35 Smail, l o c . c i t . Bassham, l o c . c i t . 36 H e i l et a l . , l o c . c i t . 37 W. R. Houston, "Selected Methods of In-Service Education and the Mathematics Achievement and Interest of Elementary School P u p i l s " (unpublished Doctoral d i s s e r t a t i o n , U n i v e r s i t y of Texas, 1961). 25 were to be expected. If researchers are hardpressed to r e l a t e teacher knowledge to student gains, then i t would seem even more l i k e l y that no s i g n i f i c a n t r e l a t i o n s h i p s would be found i n a study of t h i s sort. 38 In a continuation of the above study, Houston and DeVault, i n 1963, reported that teacher growth increased student growth. They reported a s i g n i f i c a n t r e l a t i o n s h i p between teachers' growth i n the understanding of the mathematics concepts of the i n - s e r v i c e education program and p u p i l s ' growth i n the understanding of those mathematics concepts s p e c i f i c a l l y developed i n this program. The researchers constructed the instruments to measure teachers' growth and p u p i l s ' growth. These instruments were designed to measure the mathematics emphasized i n the i n - s e r v i c e education program. They administered these instruments to both teachers and students as pre-tests and post-test s . They also reported no s i g n i f i c a n c e when teacher scores on the pre-test were compared with p u p i l s ' growth. The above study seems to indi c a t e that teacher growth i n a given area begets student growth i n that area. It also indicates that i n i t i a l teacher knowledge does not r e l a t e to student growth. It should be noted that the instruments were constructed by the researchers to evaluate s p e c i f i c objectives. These evaluations led to the reported s i g n i f i c a n t difference. It might be that i f researchers are to f i n d s i g n i f i c a n t r e l a t i o n s h i p s , they must develop t h e i r own s p e c i f i c i n s t r u -ments to evaluate s p e c i f i c objectives instead of using standardized t e s t s . • W. R. Houston and M. V. DeVault, "Mathematics In-Service Education: Teacher Growth Increases Pupil Growth," The Arithmetic  Teacher, 9:243-247, May, 1963. In 1964, H a l l compared the gain of students taught by 17 f i r s t year c e r t i f i e d teachers with the gain of students taught by 21 college graduates with p r o v i s i o n a l c e r t i f i c a t e s i n t h e i r f i r s t teaching assignment. Student gain scores were derived from the school's administration of the Stanford Achievement Tests each September. The gain i s the difference i n grade l e v e l as calculated from the re s u l t s of the test from one September to the next September. The s i x areas of the test are: (1) paragraph meaning, (2) word meaning, (3) s p e l l i n g , (4) language, (5) arithmetic reasoning, and (6) arithmetic computation. The r e s u l t s favored the c e r t i f i e d teachers i n a l l of the s i x areas, and 40 some of the re s u l t s were s i g n i f i c a n t . H a l l found, as had Smith and others, that there i s a s i g n i f i c a n t r e l a t i o n s h i p between the amount of professional teacher education completed by a teacher and student achievement. In th i s instance, he found a s i g n i f i c a n t r e l a t i o n s h i p e x i s t i n g between professional teacher education and each of the three areas: (1) paragraph meaning, (2) word meaning, and (3) s p e l l i n g . The other three areas had a p o s i t i v e , n onsignificant r e l a t i o n s h i p with professional teacher education. The concern that must be expressed about Hall's study relates to the p o t e n t i a l loss during the summer and the p o s s i b i l i t y that t h i s l o s s i s greater i n one area than another area. 41 In 1964, Watts pre-tested 2121 s i x t h grade pupils using the C a l i f o r n i a Achievement Test, Elementary. He then used a regression ""H. 0. H a l l , "Professional Preparation and Teacher E f f e c t i v e -ness," The Journal of Teacher Education, 15:72-76, March, 1964. 40 Smith, l o c . c i t . 41 G. D. Watts, "A Cor r e l a t i o n Analysis Between Level of Achieve-ment and Certain Teacher C h a r a c t e r i s t i c s i n Selected School Systems" (unpublished Doctoral d i s s e r t a t i o n , Ohio University, 1964). Di s s e r t a t i o n Abstracts, 25:2329-2330, No. 4, 1964/65. equation i n a manner s i m i l a r to Bassham and predicted the post-test score. The difference between the actual score and the predicted score was the " l e v e l of achievement." No s i g n i f i c a n t difference was found between " l e v e l of achievement" and (1) degree held, (2) years of t r a i n i n g , (3) recency of t r a i n i n g , and (4) teachers' q u a l i f i c a t i o n s . 43 In 1965, Moore pre-tested and post-tested the students i n 10 fourth grade classes and 11 s i x t h grade classes with the SRA Arithmetic Series Grades 4-6. He tested the 21 teachers with Glennon's t e s t . He found no s i g n i f i c a n t r e l a t i o n s h i p between teacher understanding and p u p i l gain i n achievement i n arithmetic. 44 In 1965, Shim used a d i f f e r e n t approach. He looked at the cumulative e f f e c t of 87 teachers who taught 214 students while they were i n attendance i n grades one through f i v e . He measured student achievement (in arithmetic, language, and reading) with the C a l i f o r n i a  Achievement Test Form W Elementary. The four teacher variables were: (1) college grade-point average, (2) degree, (3) c e r t i f i c a t e , and (4) experience. He then dichotomized each of these v a r i a b l e s and checked a l l possible hypotheses which r e l a t e teacher variables to student achievement. He concluded: ^"Bassham, l o c . c i t . 43 R. E. Moore, "The Mathematical Understanding of the Elementary School Teacher as Related to Pu p i l Achievement i n Intermediate-Grade Arithmetic" (unpublished Doctoral d i s s e r t a t i o n , Stanford University, 1965). D i s s e r t a t i o n Abstracts, 26:213-214, No. 1, 1965/66. 44 Chung-Phing Shim, A Study of the Cumulative E f f e c t of Four Teacher C h a r a c t e r i s t i c s on the Achievement of Elementary School P u p i l s , " The Journal of Educational Research, 59:33-34, September, 1965. T h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n p u p i l a c h i e v e m e n t t o s u p p o r t t h e i d e a t h a t an e l e m e n t a r y t e a c h e r has t o be a s u p e r i o r s t u d e n t i n c o l l e g e , t o h a v e a d e g r e e , to be f u l l y c e r t i f i e d , o r t o h a v e many y e a r s o f e x p e r i e n c e i n o r d e r t o be s u c c e s s f u l as f a r as m e a s u r a b l e p u p i l a c h i e v e m e n t i s c o n c e r n e d . 46 I n 1 9 6 5 , R a i l s b a c k i n v e s t i g a t e d t h e r e l i a b i l i t y and v a l i d i t y c o e f f i c i e n t s o f two d i f f e r e n t i n s t r u m e n t s w h i c h w e r e d e v e l o p e d t o measure c e r t a i n f a c e t s o f t e a c h e r e f f e c t i v e n e s s . A team o f r a t e r s e v a l u a t e d 25 e l e m e n t a r y t e a c h e r s on b o t h i n s t r u m e n t s . The Iowa T e s t  o f B a s i c S k i l l s was a d m i n i s t e r e d t o t h e s t u d e n t s a t t h e end o f t h e y e a r . A weak n o n s i g n i f i c a n t r e l a t i o n s h i p was f o u n d b e t w e e n r a n k i n g o f e f f e c t i v e n e s s and p u p i l a c h i e v e m e n t s . I n v i e w o f t h e many s t u d i e s w h i c h h a v e shown no s i g n i f i c a n t r e l a t i o n s h i p when r a t i n g s o f t e a c h e r s by o b s e r v e r s as t o t h e i r e f f e c t i v e n e s s i s compared t o a c h i e v e m e n t o r g a i n , i t i s n o t s u r p r i s i n g t h a t R a i l s b a c k f o u n d no s i g n i f i c a n t r e l a t i o n s h i p . 47 I n 1 9 6 7 , H u r s t f a i l e d t o f i n d a r e l a t i o n s h i p b e t w e e n t h e number o f h o u r s o f c o l l e g e m a t h e m a t i c s p o s s e s s e d by a t e a c h e r and s t u d e n t g a i n s c o r e s d e r i v e d f r o m a d m i n i s t r a t i o n o f The M e t r o p o l i t a n  A c h i e v e m e n t T e s t . H i s p o p u l a t i o n was 55 t h i r d g r a d e t e a c h e r s and t h e i r s t u d e n t s . To o b t a i n s t u d e n t g a i n he u s e d t h e same p r o c e d u r e as H a l l ; ^ t h a t i s , he u s e d t h e s c h o o l ' s r e c o r d s and o b t a i n e d s u c c e s s i v e 45 I b i d . , p. 34 . 46 C . E . R a i l s b a c k , " A C o m p a r i s o n o f t h e R e l i a b i l i t y and V a l i d i t y o f Two Types o f C r i t e r i o n M e a s u r e s f o r E v a l u a t i o n o f I n s t r u c t i o n " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f I o w a , 1 9 6 5 ) . D i s s e r t a t i o n A b s t r a c t s , 2 6 : 5 8 2 9 , N o . 6 , 1 9 6 5 / 6 6 . 47 D . H u r s t , " T h e R e l a t i o n s h i p Between C e r t a i n T e a c h e r - R e l a t e d V a r i a b l e s and S t u d e n t A c h i e v e m e n t i n T h i r d G r a d e A r i t h m e t i c " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Oklahoma S t a t e U n i v e r s i t y , 1 9 6 7 ) . 48 H a l l , l o c . c i t . 29 September scores on the te s t . Again, one must c r i t i c a l l y question the summer e f f e c t s on such a procedure and the lack of s p e c i f i t y of the tests used. 49 Rouse, also reporting i n 1967, used a technique s i m i l a r to Shim. He found no r e l a t i o n s h i p between arithmetic achievement of fourth grade students and the t o t a l mathematics preparation of the teachers responsible f o r t h e i r i n s t r u c t i o n from kindergarten through grade s i x . The arithmetic achievement of the students was measured by the C a l i f o r n i a Achievement Tests. The t o t a l mathematics preparation of the teachers was the mathematics courses they had completed i n high school, college, and i n - s e r v i c e . In 1970, Cox"'''" tested t h i r d graders and sixth graders with the SRA Achievement Series, Arithmetic, 1964. She c l a s s i f i e d the teachers of these students as high, average, or low as determined by 52 t h e i r scores on Dr. Leroy Callahan's Test of Mathematical Under- standing. She found no s i g n i f i c a n t r e s u l t s when she made the comparison between teacher knowledge as measured by Callahan's test and p u p i l mean-gain as measured by the pre-test post-test procedure. She did report that f o r the s i x t h graders there was a nonsignificant p o s i t i v e 49 Rouse, l o c . c i t . 5 0 P L . Shim, l o c . c i t . "^*"L. S. Cox, "A Study of Pupil Achievement i n Mathematics and Teacher Competence i n Mathematics" (unpublished Doctoral d i s s e r t a t i o n , University of Kansas, 1970). D i s s e r t a t i o n Abstracts, 31:2767-A, 2768-A, No. 6, 1970/71. 52 L. G. Callahan, "A Study of Knowledge Possessed by Elementary School Teachers, In-Service and In-Training, of C u l t u r a l , Psychological, and Mathematical Foundations of the Elementary School Program" (unpublished Doctoral d i s s e r t a t i o n , Syracuse U n i v e r s i t y , 1966). r e l a t i o n s h i p . Pupils of high c l a s s i f i e d teachers made larger gains than did pupils of average c l a s s i f i e d teachers. Further, students of average c l a s s i f i e d teachers made larger gains than did students of low c l a s s i f i e d teachers. She did not report any r e l a t i o n s h i p s , s i g -n i f i c a n t or not s i g n i f i c a n t , f o r the t h i r d graders. In summarizing the elementary school studies, i t seems they use the same techniques over and over and get the. same r e s u l t s . Even 53 i n 1970 Cox did not change the procedure. If we are to f i n d any re l a t i o n s h i p s we w i l l have to change some ways of obtaining information 54 and some ways of analyzing the information. Houston and DeVault moved toward a more appropriate approach when they developed s p e c i f i c instruments to measure the desired goals. When using the pre-test post-test idea, care must be taken to assure that gain i s being measured. Using r e s u l t s obtained i n successive Septembers leaves the r e s u l t s open to several serious questions. Even testing i n September and May can be questioned because most of September, October, and November are commonly spent i n review. If c e r t a i n understanding i s possessed by the student, then the teacher's understanding or lack of understanding w i l l have l i t t l e e f f e c t on the student during this period. The true e f f e c t of the teacher might better be obtained by pre-testing and post-testing around well i d e n t i f i e d and controlled blocks of novel material. A t t i t u d i n a l Reviews A great amount of time and e f f o r t has gone into e f f o r t s to construct attit u d e scales that give an in d i v i d u a l ' s a t t i t u d e to a Cox, loc. c i t . Houston and DeVault, loc. c i t p a r t i c u l a r matter. Much more time and e f f o r t w i l l be spent with s i m i l a r r e s u l t s ; that i s , r e s u l t s which must be suspect because of the instruments used. In 1956, Poffenburger and Norton"'"' asked 16 college seniors' to complete a questionnaire r e l a t i v e to t h e i r attitudes toward mathematics and when they developed these at t i t u d e s . After reviewing the results', they concluded: 1. Parents determine i n i t i a l attitudes of t h e i r children toward arithmetic. 2. Parents' expectations of th e i r children's performance and the encouragement they give i n regard to the study of arithmetic a f f e c t children's achievement. 3. Arithmetic and mathematics teachers can have strong p o s i t i v e or negative e f f e c t s upon students' attitudes and achievement. Also working with college students, Purcell,"' 7 i n 1964, studied the e f f e c t of c e r t a i n factors on attitu d e change toward elementary mathe-matics i n a group of prospective teachers. The Dutton Arithmetic 58 Attitude Scale was used to determine student atti t u d e . P u r c e l l reported a s i g n i f i c a n t c o r r e l a t i o n between at t i t u d e i n elementary mathematics and understanding of elementary mathematics, but reported a nonsignificant c o r r e l a t i o n when comparing improved understanding " '" 'T. N. Poffenburger and D. D. Norton, "Factors Determining Attitudes Toward Arithmetic and Mathematics," The Arithmetic Teacher, 3:113-116, A p r i l , 1956. 5 6 I b i d . , p. 116 "^ W. J. P u r c e l l , ""Some Factors A f f e c t i n g Attitudes of Pros-pective Teachers Toward Elementary Mathematics" (unpublished Doctoral d i s s e r t a t i o n , Teachers College, Columbia University, 1964). 58 W. H. Dutton, "Measuring Attitudes Toward Arithmetic," Elementary School Journal, 55:24-31, September, 1954. 32 with favorable a t t i t u d e change. He also reported a nonsignificant c o r r e l a t i o n when comparing favorable a t t i t u d e change with a high grade i n course work. This study seems to indic a t e that i f a student under-stands the material he has a favorable a t t i t u d e , but a favorable change i n a t t i t u d e does not assure increased understanding or better grades. 59 O'Donnell, i n 1958, examined 109 college seniors i n elementary education with the C a l i f o r n i a Achievement Test, Mathematics Section, Grades 9 t o 1 4 , Form W, to determine t h e i r arithmetic p r o f i c i e n c y . He also administered H. H. Remmers'^ at t i t u d e scale, Scale to Measure  Attitudes Toward Any School Subject, to f i n d student a t t i t u d e toward arithmetic. He found that a t t i t u d e toward arithmetic showed only a low nonsignificant c o r r e l a t i o n with arithmetical achievement and arithmetical problem solving behavior. White,^ i n 1962, disagreed with O'Donnell af t e r evaluating 92 college students, enrolled i n a methods course f o r elementary school arithmetic, with the Dutton Attitude Scale and Test D: Basic Arithmetic S k i l l s of the Iowa Every-Pupil Tests of  Basic S k i l l s , Advanced Battery. She reported s i g n i f i c a n t p o s i t i v e changes occurred i n students' attitudes toward arithmetic, and ""^J. R. O'Donnell, "Levels of Arithmetic Achievement and Attitude Toward Arithmetic and Problem Solving by Prospective Elementary Teachers" (unpublished Doctoral d i s s e r t a t i o n , Pennsylvania State University, 1958). D i s s e r t a t i o n Abstracts, 19:1300, No. 6, 1958/59. ^H. H. Remmers, N. L. Gage, and J. F. Rummel, A P r a c t i c a l  Introduction to Measurement and Evaluation, (New York: Harper and Brothers, 1960), pp. 285-342. ^M. J. White, "A Study of the Change of Achievement and Attitude Toward Arithmetic by Prospective Elementary School Teachers Under Conditions of T e l e v i s i o n " (unpublished Doctoral d i s s e r t a t i o n , Wayne State U n i v e r s i t y , 1963). D i s s e r t a t i o n Abstracts, 25:2302-2303, No. 4, 1964/65. 33 s i g n i f i c a n t gains were made i n vocabulary and fundamental knowledge, computations, and t o t a l a r i t h m e t i c achievement. 62 63 64 The three s t u d i e s , P u r c e l l , O'Donnell, and White, are i n d i c a t i v e of the s t u d i e s which attempt to r e l a t e a t t i t u d e and achievement i n a r i t h m e t i c as i t r e l a t e s to p r e - s e r v i c e t r a i n i n g of teachers. Since these s t u d i e s and other s i m i l a r s t u d i e s are c o n t r a -d i c t o r y , one must question the procedures. I t i s not c l e a r what i s being measured when standardized t e s t s and a t t i t u d e s c a l e s are being used. I t i s evident that d i f f e r e n t procedures are necessary to determine whether or not there i s a r e l a t i o n s h i p between a t t i t u d e and achievement i n a r i t h m e t i c . Studies at the secondary l e v e l are no more co n c l u s i v e . 65 Goldberg et a l . found that a t t i t u d e s of j u n i o r high school students toward a r i t h m e t i c were not c o r r e l a t e d to t h e i r gain i n achievement i n 66 a r i t h m e t i c . This i s i n agreement w i t h O'Donnell at the secondary l e v e l . Goldberg et a l . , wrote: Why the students who showed the greatest gains i n achieve-ment d i d not a l s o show more p o s i t i v e a t t i t u d e s towardgmathematics i s a question which cannot be answered from the data. 68 Peskins working w i t h seventh graders compared teacher a t t i t u d e i n a r i t h m e t i c w i t h student a t t i t u d e i n a r i t h m e t i c and w i t h student achievement i n a r i t h m e t i c . Out of 24 p o s s i b l e c o r r e l a t i o n s between teacher a t t i t u d e and student a t t i t u d e or student achievement, 15 were 62 64 P u r c e l l , l o c . c i t . White, l o c . c i t . 6 3 O'Donnell, l o c . c i t . 65 ^ O ' D o n n e l l , l o c . c i t . 67 Goldberg et a l . , l o c . c i t . Goldberg et a l . , op. c i t . , p. 2 68 Peskins, l o c . c i t . n e g a t i v e and 2 o f t h e s e c o r r e l a t i o n s w e r e s i g n i f i c a n t . These r e s u l t s i n d i c a t e t h a t a t e a c h e r ' s a t t i t u d e t o w a r d m a t h e m a t i c s m i g h t p l a y an i n v e r s e r o l e i n a f f e c t i n g t h e s t u d e n t s ' a t t i t u d e o r a c h i e v e m e n t . A s i g n i f i c a n t p o s i t i v e r e l a t i o n s h i p d i d e x i s t b e t w e e n t h e t e a c h e r s ' u n d e r -69 s t a n d i n g o f a r i t h m e t i c and p u p i l s ' a t t i t u d e t o w a r d a r i t h m e t i c . G a r n e r f o u n d no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n t e a c h e r a t t i t u d e t o w a r d a l g e b r a and s t u d e n t a c h i e v e m e n t i n a l g e b r a . M c C r a d l e , 7 ^ i n 1 9 5 9 , r e p o r t e d , i n what a p p e a r s t o be one o f t h e most c o m p r e h e n s i v e and w e l l d e s i g n e d s t u d i e s t o d a t e , some r e l a t i o n s h i p s b e t w e e n t e a c h e r a t t i t u d e and s t u d e n t a c h i e v e m e n t i n f i r s t y e a r a l g e b r a . H i s p o p u l a t i o n was 29 t e a c h e r s and 1642 s t u d e n t s . He u s e d t h e M i n n e s o t a T e a c h e r A t t i t u d e I n v e n t o r y t o m e a s u r e t h e t e a c h e r s ' a t t i t u d e t o w a r d t e a c h i n g . He t h e n c l a s s i f i e d t h e t e a c h e r s as h i g h , m i d d l e , o r l o w , d e p e n d i n g upon t h e r e s u l t s o f t h e a t t i t u d e i n v e n t o r y . The s t u d e n t s w e r e e v a l u a t e d i n t h r e e a r e a s : (1) q u a n t i -t a t i v e t h i n k i n g , (2) f u n c t i o n a l competence i n m a t h e m a t i c s , and (3) a l g e b r a a c h i e v e m e n t . M c C r a d l e f o u n d t h e s t u d e n t s i n c l a s s e s o f t h e h i g h t e a c h e r g r o u p h a d s i g n i f i c a n t l y l a r g e r g a i n s i n q u a n t i t a t i v e t h i n k i n g and f u n c t i o n a l competence i n m a t h e m a t i c s t h a n d i d t h e s t u d e n t s w i t h t e a c h e r s i n t h e m i d d l e g r o u p o r t h e l o w g r o u p . F u r t h e r , t h e a t t i t u d e o f t h e t e a c h e r was n o t s i g n i f i c a n t l y r e l a t e d t o p u p i l s c o r e s G a r n e r , l o c . c i t . 7 ^ J . H . M c C r a d l e , " A n I n v e s t i g a t i o n o f t h e R e l a t i o n s h i p Be tween P u p i l A c h i e v e m e n t i n F i r s t Y e a r A l g e b r a and Some T e a c h e r C h a r a c t e r i s -t i c s " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f M i n n e s o t a , 1 9 5 9 ) . D i s s e r t a t i o n A b s t r a c t s , 2 0 : 1 6 5 , N o . 1 , 1 9 5 9 / 6 0 . o n t h e a l g e b r a a c h i e v e m e n t m e a s u r e . T a y l o r , 7 ^ t h o u g h , c o u l d f i n d no s i g n i f i c a n t c o r r e l a t i o n s b e t w e e n t e a c h e r a t t i t u d e to p u p i l s and h i g h s c h o o l s c i e n c e g r o w t h . I t seems t h a t t h e e v i d e n c e i s a g a i n i n c o n c l u s i v e . The s t u d y 72 b y M c C r a d l e w o u l d seem t o i n d i c a t e t h a t t h o s e t e a c h e r s w i t h a more p o s i t i v e a t t i t u d e t o w a r d t e a c h i n g do d e v e l o p some c h a r a c t e r i s t i c s i n s t u d e n t s , q u a n t i t a t i v e t h i n k i n g and f u n c t i o n a l competence i n m a t h e -m a t i c s , t h a t o t h e r t e a c h e r s do n o t d e v e l o p . A g a i n , more r e s e a r c h i s n e c e s s a r y t o d r a w s t r o n g c o n c l u s i o n s . 73 Two s t u d i e s seem r e l e v a n t a t t h e e l e m e n t a r y l e v e l . S m a i l a l s o u s e d t h e M i n n e s o t a T e a c h e r A t t i t u d e I n v e n t o r y t o m e a s u r e t e a c h e r a t t i t u d e t o w a r d t e a c h i n g . He f o u n d a s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n t h e a t t i t u d e o f t h e t e a c h e r t o w a r d t e a c h i n g and p u p i l m e a n - g a i n i n a r i t h m e t i c . H i s p o p u l a t i o n was f o u r t h , f i f t h , and s i x t h g r a d e r s . 74 These r e s u l t s somewhat s u p p o r t t h e f i n d i n g s o f M c C r a d l e . B a s s h a m , M u r p h y , and M u r p h y 7 ^ compared s t u d e n t a t t i t u d e t o s t u d e n t a c h i e v e m e n t . They m e a s u r e d s t u d e n t a t t i t u d e u s i n g D u t t o n ' s s c a l e . They s e p a r a t e d t h e s t u d e n t s i n t o two g r o u p s , o v e r - a c h i e v e r s and u n d e r — a c h i e v e r s . They made t h i s g r o u p i n g on t h e b a s i s o f r e s u l t s f r o m K u h l m a n - A n d e r s o n  I n t e l l i g e n c e T e s t s and The Iowa T e s t s o f B a s i c S k i l l s ( R e a d i n g C o m p r e - h e n s i o n ) . They r e p o r t e d a s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n a t t i t u d e and c l a s s i f i c a t i o n as o v e r - a c h i e v e r and u n d e r - a c h i e v e r . They f u r t h e r r e p o r t e d , t h o u g h : 71 72 T a y l o r , l o c . c i t . M c C r a d l e , l o c . c i t . 73 74 S m a i l , l o c . c i t . M c C r a d l e , l o c . c i t . 7 ^ H . B a s s h a m , M . M u r p h y , and K. M u r p h y , " A t t i t u d e and A c h i e v e m e n t i n A r i t h m e t i c , " The A r i t h m e t i c T e a c h e r , 1 1 : 6 6 - 7 2 , F e b r u a r y , 1964 . 36 The wide v a r i a b i l i t y i n weighted achievement at both extremes of the d i s t r i b u t i o n of a t t i t u d e scale scores would indicate that p r e d i c t i o n of achievement on the^basis of a t t i t u d e score f o r i n d i v i d u a l s would be hazardous. These studies seem to indicate that teacher a t t i t u d e and/or student a t t i t u d e toward arithmetic might have some r e l a t i o n s h i p to student learning of arithmetic. By using wide-range attitude scales, most studies possibly l o s t the r e s u l t s necessary f o r s i g n i f i c a n c e . It might be that narrowing the scope of the a t t i t u d e measured could lead to s i g n i f i c a n t r e s u l t s . - 1 ' • ' ; J u s t i f i c a t i o n of the Study As indicated i n the above review, most of the studies did not test teacher knowledge or teacher a t t i t u d e d i r e c t l y . They used p r i n c i p a l s ' r atings, number of college courses taken, and other i n d i r e c t measures. I t might be expected that the findings of these studies would be less r e l i a b l e than from studies i n which these variables are d i r e c t l y measured. H e i l et a l . agreed, concluding: "Observers ratings, per se, are next to worthless as a c r i t e r i o n of 77 78 teacher effectiveness." Medley and M i t z e l , a f t e r reviewing 'conclusions from previous research involving supervisory ratings, .came to a s i m i l a r conclusion. I 7 ^ I b i d . , p. 71. • 77 H e i l et a l . , op. c i t . , p. 66. 78 D. M. Medley and H. E. M i t z e l , "Some Behavioral Correlates of Teacher Effectiveness," Journal of Educational Psychology, 50:239-246, December, 1959. Five studies at the elementary l e v e l d i r e c t l y tested teacher 79 knowledge with a paper and p e n c i l test. H e i l et a l . made no e f f o r t to determine teachers' understanding of arithmetic. They determined teacher knowledge by administering the Teacher Education Examination, 80 81 82 which i s a very general examination. Smail, Bassham, and Moore determined teacher understanding by administering Glennon's Test of 83 Basic Mathematical Understandings constructed i n 1948. This test was designed to determine the understandings basic to computational 84 processes taught i n grades one through s i x at that time. Cox 85' administered Callahan's Test of Mathematical Understanding f o r her c r i t e r i o n of teacher understanding. This test, looks at many aspects of arithmetic and i t i s not clear what i t i s designed to measure, but i t i s not designed to measure only understanding i n arithmetic. In view of recent developments i n elementary school mathematics, i t seems that teaching at the elementary l e v e l now involves more advanced understandings than those basic to the computational processes. It must therefore be concluded that the f a i l u r e of these studies to report s i g n i f i c a n t r e s u l t s may be a t t r i b u t a b l e to the i n p r e c i s i o n of the measures rather than to a lack of any underlying r e l a t i o n s h i p . An attempt to detect any such underlying r e l a t i o n s h i p must now r e s u l t from an e f f o r t to measure p r e c i s e l y those teacher understandings 7 9 H e i l et a l . , l o c . c i t . 8°Smail, l o c . c i t , 8 1Bassham, l o c . c i t . 8 2Moore, l o c . c i t . QT 84 Glennon, l o c . c i t . Cox, l o c . c i t . 85 Callahan, l o c . c i t . related to contemporary mathematics and to the syllabus of a specific elementary program. For the same reasons mentioned above, i t is now necessary to refine the measures of student growth. It might be worthwhile to partition student growth into three parts: (1) computation, (2) problem solving, and (3) understanding. One might expect con-siderable variation in the relationships between specific teacher variables and each of these parts of student growth in arithmetic. This might be especially true when relating teacher understanding to student understanding in arithmetic. None of the five studies mentioned above measured student growth in understanding. A l l five of these studies used nationally normed standardized tests to determine student gain. It is probable that these tests did not adequately evaluate the goals of a given syllabus. More useful data might be obtained i f the tests used to determine student growth were designed for the goals of the syllabus for the grades involved. The review of the literature indicates that present information regarding the relationship between teacher variables, especially understanding in arithmetic, and student achievement and/or growth 86 is inconclusive. Only one study, Bassham, at the elementary level found a significant relationship and that relationship held for only above average students. The literature suggests that this incon-clusiveness may be partially the result of insufficient identification Bassham, loc. c i t . and measurement o f t h e v a r i a b l e s w h i c h a r e l i k e l y to be s i g n i f i c a n t . 87 I n 1 9 4 5 , B a r r s u m m a r i z e d : The s u c c e s s o f t h e t e a c h e r depends no s m a l l p a r t upon t h e e x t e n t t h a t what she h a s t o o f f e r f i t s i n t o t h e e x p e c t a n c y o f p u p i l s , p a r e n t s , and s c h o o l o f f i c i a l s i n t h e community i n w h i c h s h e w o r k s . These ^ d i v i d u a l d e t e r m i n e r s o f t e a c h i n g e f f i c i e n c y n e e d f u r t h e r s t u d y . A t e a c h e r v a r i a b l e w h i c h d e s e r v e s some c o n s i d e r a t i o n i n t o d a y ' s w o r l d i s t h e a t t i t u d e o f t h e t e a c h e r t o w a r d c o n t e m p o r a r y m a t h e m a t i c s . T h i s v a r i a b l e h a s n o t b e e n compared w i t h s t u d e n t g a i n s i n any s t u d y f o u n d b y t h i s r e s e a r c h e r . I t m i g h t be t h a t t h i s a t t i t u d e has a r e l a t i o n s h i p t o t h e l e a r n i n g o f c o n t e m p o r a r y m a t h e m a t i c s . 89 Weaver and G i b b , a f t e r r e v i e w i n g t h e l i t e r a t u r e t o 1 9 6 4 , c o n c l u d e d : I n v e s t i g a t i o n s s u c h as t h e s e , h o w e v e r , l e a v e u n a n s w e r e d t h e q u e s t i o n c o n c e r n i n g " c a u s e and e f f e c t . " E x i s t i n g e v i d e n c e i s c o n s i s t e n t w i t h t h e h y p o t h e s i s t h a t t e a c h e r change b e g e t s p u p i l change o f a l i k e k i n d i n m a t h e m a t i c s . N e v e r t h e l e s s , one must l o o k t o t h e f u t u ^ g f o r r e s e a r c h d e s i g n e d s p e c i f i c a l l y t o t e s t t h i s h y p o t h e s i s . The demands f o r more r e s e a r c h i n t h e a r e a o f t e a c h e r v a r i a b l e s v e r s u s s t u d e n t g a i n s seem t o be g r e a t . They n o t o n l y come f r o m t h e r e v i e w e r s o f t h e l i t e r a t u r e , b u t f r o m D e p a r t m e n t s o f M a t h e m a t i c s who w o u l d l i k e t o a d j u s t t h e i r p r o g r a m s t o meet t h e needs o f p r e - s e r v i c e s t u d e n t s , and f r o m p r a c t i c i n g t e a c h e r s who w o u l d l i k e t o t a k e i n - s e r v i c e c o u r s e s t o b e t t e r e d u c a t e t h e i r s t u d e n t s . 87 A . S . B a r r , " I m p r e s s i o n s , T r e n d s , and F u t u r e R e s e a r c h , " J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 1 4 : 2 0 0 - 2 0 6 , D e c e m b e r , 1 9 4 5 . QQ I b i d . , p . 206 . 89 F . J . Weaver and G . E . G i b b , " M a t h e m a t i c s i n t h e E l e m e n t a r y S c h o o l , " R e v i e w o f E d u c a t i o n a l R e s e a r c h , 3 4 : 2 7 3 - 2 8 5 , J u n e , 1964 . 90 I b i d . , p . 2 8 2 . The Variables Because of the problems, needs, and j u s t i f i c a t i o n discussed above, t h i s study w i l l attempt, i n a more sophisticated way than appears to have been attempted to date, to e s t a b l i s h r e l a t i o n s h i p between selected teacher variables and student growth i n arithmetic. The teacher variables w i l l be: understanding of arithmetic by d i r e c t t e s t i n g , a t t i t u d e toward contemporary mathematics, college courses taken i n mathematics, how long since the l a s t of these mathematics courses was taken, college courses taken i n methods of teaching mathematics, how long ago was the l a s t of these methods courses taken, number of quarter hours of professional education courses, number of years of teaching experience, number of years i n present d i s t r i c t , and p r i n c i p a l ' s r a t i n g . Student growth i n arithmetic w i l l be p a r t i t i o n e d into three parts: (1) computation, (2) problem solving, and (3) understanding. The reason for including teacher variables d i f f e r e n t from teacher understanding and teacher a t t i t u d e i s that some investigators have reported s i g n i f i c a n c e when using some of these va r i a b l e s . Further, other researchers have reported that a composite of these v a r i a b l e s have a s i g n i f i c a n t e f f e c t on student learning. Hypotheses The following n u l l hypotheses w i l l be checked: HI. There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n computation. H2. There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n problem solving. H3. There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n understanding. H 4 . There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n achievement. Since no s p e c i f i c a p r i o r i hypotheses have been selected from among the huge number of possible i n t e r a c t i o n e f f e c t s , any observations made of such interactions w i l l be considered suggestions for further research. C h a p t e r 3 DESIGN OF THE STUDY D e f i n i t i o n s T h e r e i s c o n s i d e r a b l e v a r i a t i o n i n t h e l i t e r a t u r e r e g a r d i n g d e f i n i t i o n s o f t e r m s u s e d i n m a t h e m a t i c s . A l t h o u g h n o t a l l r e s e a r c h e r s w o u l d a g r e e , t h i s s t u d y w i l l a d o p t t h e f o l l o w i n g d e f i n i t i o n s : 1 . E l e m e n t a r y g r a d e s : k i n d e r g a r t e n t h r o u g h s i x t h g r a d e . 2 . S e c o n d a r y s c h o o l : s e v e n t h g r a d e t h r o u g h s e n i o r y e a r i n h i g h s c h o o l . 3 . C o m p u t a t i o n : t h a t p a r t o f a r i t h m e t i c d e a l i n g w i t h t h e a l g o r i t h m s o f t h e r e a l n u m b e r s . 4 . P r o b l e m s o l v i n g : t h a t p a r t o f a r i t h m e t i c d e a l i n g w i t h w o r d e d p r o b l e m s and t h e e s t a b l i s h m e n t o f e q u a t i o n s w h i c h l e a d t o c o r r e c t s o l u t i o n s . 5 . U n d e r s t a n d i n g : t h a t p a r t o f a r i t h m e t i c d e a l i n g w i t h t h e a l g e b r a i c p r i n c i p l e s , t h e p a t t e r n s , t h e f u n d a m e n t a l p r o p e r t i e s o f t h e r e a l n u m b e r s , and t h e n o t a t i o n a l a g r e e m e n t s a c c o m p a n y i n g t h e m . 6 . G r o w t h : The d i f f e r e n c e b e t w e e n p o s t - t e s t and p r e - t e s t . 7. A c h i e v e m e n t : g r o w t h i n p r o b l e m s o l v i n g , c o m p u t a t i o n a l s k i l l s , and u n d e r s t a n d i n g . The S u b j e c t s I n an e f f o r t t o o v e r c o m e some o f t h e d e s i g n p r o b l e m s d i s c u s s e d i n C h a p t e r s 1 and 2 , t h e S p o k a n e , W a s h i n g t o n , and B r e m e r t o n , W a s h i n g t o n , 42 43 s c h o o l d i s t r i c t s w e r e c h o s e n as t h e a r e a s i n w h i c h t o c a r r y o u t t h e s t u d y . They w e r e c h o s e n b e c a u s e : 1 . They b o t h u s e d t h e L a i d l a w M a t h e m a t i c s S e r i e s . " ' " 2 . They had v e r y s i m i l a r p r o g r a m s i n e l e m e n t a r y s c h o o l m a t h e m a t i c s . 3 . T e s t s c o u l d be c o n s t r u c t e d w h i c h m e a s u r e t h e m a t e r i a l o f t h e t e x t and t h e p r o g r a m s . 4 . Spokane i s a m e t r o p o l i t a n a r e a o f 2 0 0 , 0 0 0 p e o p l e and i s a t r a n s p o r t a t i o n c e n t e r i n t h e e a s t e r n p a r t o f t h e s t a t e , w h i l e B r e m e r t o n i s a c i t y o f 3 0 , 0 0 0 a n d i s a n i n d u s t r i a l a r e a i n t h e w e s t e r n p a r t o f t h e s t a t e . A r e a s o n a b l e c r o s s - s e c t i o n o f t h e s t a t e ' s p o p u l a t i o n was p o s s i b l e b y u s i n g b o t h c i t i e s . 5 . The t e a c h e r p o p u l a t i o n was l a r g e enough so t h a t a random s a m p l e o f t h e 400 t e a c h e r s w o u l d e n s u r e v a l i d s t a t i s t i c a l t r e a t m e n t o f t h e d a t a . 6 . N e i t h e r d i s t r i c t g r o u p s s t u d e n t s h o m o g e n e o u s l y . They a r e a s s i g n e d t o t e a c h e r s on a random b a s i s . 7. The t e a c h e r s w e r e t h e r e g u l a r f o u r t h , f i f t h , and s i x t h g r a d e c l a s s r o o m t e a c h e r s , a l l o f whom met t h e s t a t e ' s c e r t i f i c a t i o n r e q u i r e m e n t s . 8 . The s t u d e n t s w e r e f o u r t h , f i f t h , and s i x t h g r a d e s t u d e n t s f r o m t h e s c h o o l s i n t h e two d i s t r i c t s . C o n s t r u c t i o n o f T e s t s A g a i n , t o overcome t h e s t a t e d d e f i c i e n c i e s o f t h e s t u d i e s r e p o r t e d i n C h a p t e r 2 , t e s t s d e s i g n e d to measure t h e m a t e r i a l o f t h e """B. H . G u n d l a c h e t a l . , A r i t h m e t i c ( R i v e r F o r e s t , I l l i n o i s : L a i d l a w B r o t h e r s P u b l i s h e r s , 1 9 6 4 ) . 44 Laidlaw seri e s and the understanding of students and teachers were constructed. Eleven d i f f e r e n t tests were constructed: 1. A test of teacher understanding. 2. An inventory to measure teacher attitu d e toward contemporary as opposed to t r a d i t i o n a l mathematics. 3. Nine student tests of arithmetic: (a) Three problem solving t e s t s , one f o r each grade. (b) Three computation t e s t s , one for each grade. (c) Three understanding t e s t s , one f o r each grade. In an attempt to ensure r e l i a b i l i t y of the t e s t s , extensive use was made of item analysis procedures. On the basis of the item analyses, changes were made i n the various tests which resulted i n an increase i n th e i r i n t e r n a l consistency. Realizing the d i f f i c u l t y encountered i n attempting to v a l i d a t e a measuring device, the primary attempts at v a l i d a t i o n were i n the areas of content v a l i d i t y and grade discrimination. The items included i n the student tests were determined by c a r e f u l examination of the concepts found i n the Laidlaw Arithmetic Series. These items, as we l l as the items i n teacher tests, were then subjected to close scrutiny by experts i n the f i e l d of Mathematics Education. The s p e c i f i c procedures followed i n test construction are discussed i n the following sections. Construction of a Test of Teacher Understanding Because of the nonexistence of a test for p r a c t i c i n g teachers that attempts to measure a l l areas of understanding, a tes t f o r use i n t h i s study was constructed. 45 A s e t o f 114 i t e m s was c o l l e c t e d . S i x t y - t h r e e o f t h e s e i t e m s had b e e n u s e d by o t h e r r e s e a r c h e r s t o m e a s u r e u n d e r s t a n d i n g . F i f t y - o n e o f t h e i t e m s w e r e c o n s t r u c t e d s p e c i f i c a l l y f o r t h i s t e s t . A l l i t e m s were c h o s e n and d e s i g n e d t o m e a s u r e u n d e r s t a n d i n g t h e t e a c h e r s h o u l d p o s s e s s so t h a t s h e c a n t e a c h t h e m a t h e m a t i c a l u n d e r s t a n d i n g s t r e s s e d i n t h e L a i d l a w s e r i e s , t h e s e r i e s u s e d i n t h e two d i s t r i c t s . A l l i t e m s w e r e m u l t i p l e c h o i c e . Some i t e m s h a d t h r e e c h o i c e s , some h a d f o u r c h o i c e s , and some h a d f i v e c h o i c e s . The 114 i t e m s w e r e a d m i n i s t e r e d t o 58 s t u d e n t t e a c h e r s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . I t e m a n a l y s i s l e d t o t h e r e m o v a l o f 16 i t e m s . The r e m a i n i n g 98 i t e m s w e r e a d m i n i s t e r e d t o 75 s t u d e n t s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a who w e r e i n t h e l a s t month o f a m a t h e -m a t i c s c o u r s e f o r e l e m e n t a r y t e a c h e r s . I t e m a n a l y s i s l e d t o t h e r e m o v a l o f s i x more i t e m s . B e c a u s e t h e s u b j e c t s u s e d w e r e c o l l e g e s t u d e n t s , c a r e f u l c o n s i d e r a t i o n was g i v e n t h o s e i t e m s w h i c h seemed t o o d i f f i c u l t o r e a s y . The r e m a i n i n g 92 i t e m s w e r e d i v i d e d i n t o two s u b t e s t s w i t h 12 i t e m s d u p l i c a t e d . E a c h o f t h e s e s u b t e s t s was g i v e n t o 80 s t u d e n t s i n summer s c h o o l a t E a s t e r n W a s h i n g t o n S t a t e C o l l e g e , C h e n e y , W a s h i n g t o n . The m a j o r i t y o f t h e s e s u b j e c t s , 6 8 , w e r e p r a c t i c i n g t e a c h e r s . F o r t y -one o f t h e 68 w e r e t e a c h e r s o f t h e f o u r t h , f i f t h , o r s i x t h g r a d e s . The i t e m a n a l y s i s e l i m i n a t e d 22 i t e m s l e a v i n g 70 i t e m s on t h e t e s t . B e f o r e t h e s e i t e m s w e r e a g a i n u s e d , e a c h i t e m was r e w r i t t e n so t h a t i t h a d f i v e p o s s i b l e answer c h o i c e s , and one o f t h e c h o i c e s was " n o n e o f t h e s e " o r i t s e q u i v a l e n t . T h e s e 70 i t e m s w e r e t h e n a d m i n i s t e r e d t o 164 p r a c t i c i n g e l e m e n t a r y t e a c h e r s a t t e n d i n g summer s c h o o l a t E a s t e r n W a s h i n g t o n S t a t e C o l l e g e . E i g h t y - n i n e o f t h e s e s u b j e c t s w e r e f o u r t h , f i f t h , o r s i x t h g r a d e t e a c h e r s . 46 None o f t h e i t e m s w e r e removed b e c a u s e t h e y were t o o d i f f i c u l t o r t o o e a s y . A l l t h e i t e m s w e r e a n s w e r e d c o r r e c t l y by a t l e a s t 23 p e r c e n t o f t h e s u b j e c t s and none o f t h e i t e m s w e r e a n s w e r e d c o r r e c t l y by more t h a n 81 p e r c e n t o f t h e s u b j e c t s . F u r t h e r i t e m a n a l y s i s i n d i c a t e d t h a t 10 i t e m s w e r e n o t e x c e l l e n t d i s c r i m i n a t o r s when t h e 50 h i g h s c o r e r s w e r e compared w i t h t h e 50 l o w s c o r e r s . The d i f f e r e n c e was 15 p e r c e n t o r l e s s when t h e c o r r e c t p e r c e n t a g e o f t h e t o p 50 on a g i v e n i t e m was compared w i t h t h e c o r r e c t p e r c e n t a g e o f t h e l o w 50 o n t h e same i t e m . H e n c e , t h e s e 10 i t e m s w e r e removed f r o m t h e t e s t . To e s t i m a t e t h e minimum p o s s i b l e r e l i a b i l i t y o f t h e t e s t , t h e 2 K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t was c a l c u l a t e d . The 60 r e m a i n i n g i t e m s h a d a K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t o f . 7 9 . The d a t a f o r t h i s c a l c u l a t i o n was f r o m t h e 164 summer s c h o o l s t u d e n t s i d e n t i f i e d a b o v e . B e c a u s e o f t h e v a l i d a t i o n p r o c e d u r e s and t h e h i g h K u d e r -R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t , t h e s e 60 i t e m s w e r e u s e d as t h e T e s t o f T e a c h e r U n d e r s t a n d i n g f o r t h i s s t u d y . A copy o f t h e i n s t r u -ment i s A p p e n d i x A . C o n s t r u c t i o n o f An I n v e n t o r y t o M e a s u r e T e a c h e r A t t i t u d e Toward  C o n t e m p o r a r y M a t h e m a t i c s Opposed t o T r a d i t i o n a l M a t h e m a t i c s S i n c e 1957 t h e u s e o f ' n e w , ' ' m o d e r n , ' o r ' c o n t e m p o r a r y ' m a t h e -m a t i c s has b e e n on t h e i n c r e a s e t h r o u g h o u t t h e c o n t i n e n t . E v e n t h o u g h t h e s e a r e i n g e n e r a l u s e , i t i s q u e s t i o n a b l e w h e t h e r o r n o t t h e m a j o r i t y o f t e a c h e r s h a v e a p o s i t i v e a t t i t u d e t o w a r d c o n t e m p o r a r y m a t h e m a t i c s c u r r i c u l a . G . F . K u d e r and M . W. R i c h a r d s o n , "The T h e o r y o f t h e E s t i m a t i o n o f T e s t R e l i a b i l i t y , " P s y c h o m e t r i k a , 2 : 1 5 1 - 1 6 0 , S e p t e m b e r , 1937 . I n a r e v i e w o f t h e l i t e r a t u r e c o n c e r n i n g t e a c h e r a t t i t u d e s , no i n v e n t o r y was f o u n d t h a t a t t e m p t e d t o measure t e a c h e r s ' a t t i t u d e t o w a r d c o n t e m p o r a r y as o p p o s e d t o more t r a d i t i o n a l m a t h e m a t i c s c u r r i c u l a . Such an i n v e n t o r y f o r u s e w i t h e l e m e n t a r y t e a c h e r s and e l e m e n t a r y e d u c a t i o n m a j o r s was c o n s t r u c t e d . 3 A n i n s t r u m e n t c o n s t r u c t e d b y R i c e , i n 1 9 6 4 , e v a l u a t e d a t t i t u d e t o w a r d modern m a t h e m a t i c s as w e l l as a t t i t u d e t o w a r d m a t h e m a t i c s . B e c a u s e o f t h e d u a l p u r p o s e o f t h i s i n s t r u m e n t i t was n o t deemed a p p r o p r i a t e f o r t h e p u r p o s e s d e s i r e d . The c o r r e l a t i o n b e t w e e n t h i s i n v e n t o r y and R i c e ' s i n v e n t o r y i s . 7 5 . T h i s was computed f r o m t h e r e s u l t s o f 46 e l e m e n t a r y m a j o r s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . A l i s t o f f a c t o r s w h i c h seem t o r e f l e c t t h e d i f f e r e n c e s be tween t r a d i t i o n a l m a t h e m a t i c s and c o n t e m p o r a r y m a t h e m a t i c s a t t h e e l e m e n t a r y l e v e l was g e n e r a t e d as a r e s u l t o f a q u e s t i o n n a i r e c i r c u l a t e d among a g r o u p o f f i v e a u t h o r i t i e s i n t h e f i e l d . These f a c t o r s w e r e : 1 . T e a c h e r s ' g e n e r a l a n d / o r o v e r a l l r e a c t i o n t o w a r d c o n t e m p o -r a r y m a t h e m a t i c s . 2 . T e a c h e r s ' o p i n i o n s o f c o m p u t a t i o n a l s p e e d a n d / o r c o m p u t a -t i o n a l a b i l i t y i n m a t h e m a t i c s . 3 . T e a c h e r s ' o p i n i o n s o f t h e p l a c e a n d / o r t h e v a l u e o f new t o p i c s i n m a t h e m a t i c s , e . g . , s e t t h e o r y , o t h e r b a s e s . 4 . T e a c h e r s ' o p i n i o n s o f s t u d e n t needs i n m a t h e m a t i c s a n d / o r s t u d e n t r e a c t i o n s t o m a t h e m a t i c s . 5 . T e a c h e r s ' o p i n i o n s o f t h e p l a c e a n d / o r t h e v a l u e o f t h e p r i n c i p l e s o f a r i t h m e t i c i n m a t h e m a t i c s . w j M  J' M ' * i c e > " A S t u d y o f A t t i t u d e s o f E l e m e n t a r y T e a c h e r s Toward 6. T e a c h e r s ' o p i n i o n s o f t h e methods o f t e a c h i n g a r i t h m e t i c . A l i s t o f 49 i t e m s was c o n s t r u c t e d w i t h t h e s e f a c t o r s as t h e g u i d e . E a c h i t e m was a s t a t e m e n t w h i c h was f o l l o w e d by two c h o i c e s f r o m w h i c h t h e s u b j e c t was t o c h o o s e h i s r e s p o n s e . The c h o i c e s r e p r e -s e n t p r e f e r e n c e f o r modern m a t h e m a t i c s c u r r i c u l a o r p r e f e r e n c e f o r t r a d i t i o n a l m a t h e m a t i c s c u r r i c u l a . The f i r s t v e r s i o n ( A p p e n d i x B) o f t h e a t t i t u d e i n v e n t o r y was a d m i n i s t e r e d t o a summer s c h o o l c l a s s o f 18 s t u d e n t s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . M o s t o f t h e s e s t u d e n t s w e r e p r a c t i c i n g e l e m e n t a r y t e a c h e r s . T h e i r s c o r e s r a n g e d f r o m 28 t o 44 w i t h a mean o f 3 5 . 3 9 , a m e d i a n o f 3 4 . 5 , and a s t a n d a r d d e v i a t i o n o f 4 . 5 5 . T h i s c l a s s was a l s o g i v e n t h e o p p o r t u n i t y t o comment on s t a t e m e n t s w h i c h t h e y f o u n d ambiguous o r m i s l e a d i n g . An a n a l y s i s o f t h e s e r e s u l t s and comments l e d to t h e r e m o v a l o f s e v e n i t e m s — 3 , 7, 1 1 , 3 2 , 3 3 , 3 6 , 3 9 — a n d t h e r e w r i t i n g o f s i x t e e n i t e m s . The f i r s t v e r s i o n o f t h i s i n v e n t o r y was a l s o a d m i n i s t e r e d t o n i n e s t u d e n t s a t t h e m a s t e r ' s l e v e l who w e r e t a k i n g a c o u r s e i n m a t h e m a t i c s e d u c a t i o n . These s t u d e n t s w e r e i n s t r u c t e d t o mark t h e c h o i c e f o r e a c h s t a t e m e n t t h a t i n t h e i r o p i n i o n i n d i c a t e d t h e s t r o n g e r a t t i t u d e t o w a r d c o n t e m p o r a r y m a t h e m a t i c s . I f a t l e a s t e i g h t o f t h e n i n e s t u d e n t s a g r e e d on a r e s p o n s e , t h a t r e s p o n s e was assumed t o show t h e more p o s i t i v e a t t i t u d e . These r e s u l t s and t h e comments o f t h e s e s t u d e n t s r e s u l t e d i n t h e r e w r i t i n g o f e l e v e n i t e m s and t h e r e m o v a l o f two i t e m s — 1 8 and 4 9 . To e s t i m a t e t h e minimum p o s s i b l e r e l i a b i l i t y o f t h e i n v e n t o r y , t h e K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t was c a l c u l a t e d . T h i s f i r s t v e r s i o n had a K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t o f . 6 4 . The s e c o n d v e r s i o n o f t h e i n v e n t o r y ( A p p e n d i x B ) , c o n t a i n i n g 40 i t e m s j was a d m i n i s t e r e d t o 18 d i f f e r e n t summer s c h o o l s t u d e n t s , a l s o a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . M o s t o f t h e s e s t u d e n t s w e r e p r a c t i c i n g e l e m e n t a r y t e a c h e r s . T h e i r s c o r e s r a n g e d f r o m 17 t o 36 w i t h a mean o f 2 8 . 1 7 , a m e d i a n o f 3 0 , and a s t a n d a r d d e v i a t i o n o f 5 . 0 3 . T h e K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t f o r t h e s e c o n d v e r s i o n was . 7 3 . The a n a l y s i s o f t h e s e r e s u l t s l e d t o t h e r e m o v a l o f s e v e n i t e m s — - 3 , 4 , 5 , 1 3 , 1 5 , 2 2 , 2 8 — a n d t h e r e w r i t i n g o f e i g h t i t e m s . The t h i r d v e r s i o n o f t h e i n v e n t o r y ( A p p e n d i x B ) , c o n t a i n i n g 33 i t e m s , was a d m i n i s t e r e d t o a c l a s s o f 33 d i f f e r e n t summer s c h o o l s t u d e n t s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . M o s t o f t h e s e s t u d e n t s w e r e p r a c t i c i n g e l e m e n t a r y t e a c h e r s . T h e i r s c o r e s r a n g e d f r o m 9 t o 30 w i t h a mean o f 2 1 . 7 6 , a m e d i a n o f 2 2 , and a s t a n d a r d d e v i a t i o n o f 5 . 1 6 . The K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t was . 7 8 . I t e m a n a l y s i s o f t h e s e r e s u l t s l e d t o t h e r e m o v a l o f e i g h t i t e m s — 4 , 7 , 1 2 , 1 6 , 2 2 , 2 3 , 2 7 , 2 9 . None o f t h e r e m a i n i n g i t e m s w e r e r e w r i t t e n . A t t h e end o f t h e t h i r d v e r s i o n o f t h e i n v e n t o r y t h e t e a c h e r s w e r e a s k e d to r a t e t h e i r a t t i t u d e t o w a r d m o d e r n m a t h e m a t i c s o n a s c a l e f r o m 1 t o 1 1 . T h e i r r a t i n g s r a n g e d f r o m 1 t o 11 w i t h a mean o f 7 . 4 8 , a m e d i a n o f 8 , and a s t a n d a r d d e v i a t i o n o f 2 . 2 4 . The c o r r e l a t i o n b e t w e e n t h e i r s c o r e s on t h e i n v e n t o r y and t h e i r o p i n i o n s was . 6 8 . The f o u r t h v e r s i o n ( A p p e n d i x B ) , c o n t a i n i n g 25 i t e m s , was a d m i n i s -t e r e d t o 137 summer s c h o o l s t u d e n t s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . M o s t o f t h e s e s t u d e n t s w e r e p r a c t i c i n g e l e m e n t a r y t e a c h e r s . T h e s e 137 s t u d e n t s h a d s c o r e s w h i c h r a n g e d f r o m 6 to 24 w i t h a mean o f 1 7 . 3 9 , a m e d i a n o f 1 8 , and a s t a n d a r d d e v i a t i o n o f 3 . 6 8 . The K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t f o r t h i s f o u r t h v e r s i o n was . 6 7 . 50 A s p a r t o f t h e i t e m a n a l y s i s o f t h e 137 r e s u l t s , t h e p o i n t 4 b i s e r i a l c o r r e l a t i o n c o e f f i c i e n t o f e a c h i t e m w i t h t h e w h o l e t e s t was c a l c u l a t e d . A n e x a m i n a t i o n o f T a b l e 1 i n d i c a t e s t h a t t h r e e o f t h e i t e m s -5 , 8 , 1 9 — d i d n o t h a v e s i g n i f i c a n t c o r r e l a t i o n c o e f f i c i e n t s . A s a m a t t e r o f i n t e r e s t a f a c t o r a n a l y s i s was c o n d u c t e d . The f a c t o r a n a l y s i s o f t h e s c o r e s o f t h e above m e n t i o n e d 137 s t u d e n t s was p e r f o r m e d by t h e c o m p u t i n g c e n t e r a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . The p r o g r a m u s e d was t h e f a c t o r a n a l y s i s s a m p l e p r o g r a m f r o m t h e IBM 360 s c i e n t i f i c s u b r o u t i n e package"* and t h e f a c t o r s c o r e s p r o g r a m f r o m C o o l e y and L o h n e s 1 M u l t i v a r i a t e P r o c e d u r e s f o r t h e B e h a v i o r a l S c i e n c e s . ^ I t a l s o u s e s t h e v a r i m a x p r o c e d u r e f o r a n a l y t i c a l o r t h o g o n a l r o t a t i o n . 7 I t s h o u l d be n o t e d t h a t G l a s s and T a y l o r p o i n t o u t t h a t t h i s p r o g r a m g i v e s o n l y a p p r o x i m a t e f a c t o r s c o r e s . 8 B e c a u s e o f t h e c o n c l u s i o n s by J o h n B . C a r r o l , t e t r a c h o r i c c o r r e l a t i o n c o e f f i c i e n t s w e r e u s e d f o r t h e f a c t o r a n a l y s i s i n s t e a d o f P e a r s o n i a n c o e f f i c i e n t s . To v e r i f y some o f t h e p r o b l e m s m e n t i o n e d by C a r r o l , a s p e c i a l r u n u s i n g 23 o f t h e 25 i t e m s a n d P e a r s o n i a n c o r r e l a t i o n c o e f f i c i e n t s was made. The r e s u l t s g a v e e i g h t f a c t o r s w h i c h a c c o u n t e d ~*H.. E . G a r r e t t , S t a t i s t i c s i n P s y c h o l o g y and E d u c a t i o n , (New Y o r k : D a v i d McKay Company, I n c . , 1 9 5 8 ) . "*IBM S y s t e m / 3 6 0 S c i e n t i f i c S u b r o u t i n e P a c k a g e . P r o g r a m m e r ' s G u i d e ( 3 6 0 A - C M - 0 3 X ) . ^W. W. C o o l e y and P . R . L o h n e s , M u l t i v a r i a t e P r o c e d u r e s f o r t h e  B e h a v i o r a l S c i e n c e s , (New Y o r k : J o h n W i l e y and S o n s , 1 9 6 2 ) . 7 G . V . G l a s s and P . A . T a y l o r , " F a c t o r A n a l y t i c M e t h o d o l o g y , " R e v i e w o f E d u c a t i o n a l R e s e a r c h , 3 6 : 5 6 6 - 5 8 7 , D e c e m b e r , 1966 . g J . B . C a r r o l , " T h e N a t u r e o f t h e D a t a o r How t o Choose a C o r r e l a t i o n C o e f f i c i e n t , " P s y c h o m e t r i k a , 2 6 : 3 4 7 - 3 7 2 , D e c e m b e r , 1 9 6 1 . f o r 60 p e r c e n t o f t h e v a r i a n c e . The same d a t a u s i n g t e t r a c h o r i c c o r r e l a t i o n c o e f f i c i e n t s gave n i n e f a c t o r s b u t a c c o u n t e d f o r 81 p e r c e n t o f t h e v a r i a n c e . T a b l e 1 P o i n t B i s e r i a l C o r r e l a t i o n I t e m C o e f f i c i e n t 3 1 38 2 . . 2 8 3 . . 2 7 4 . 3 7 5 . 2 0 6 . . 38 7 37 8 . 1 5 9 . 35 10 . . 2 8 11 . 3 0 12 .40 13 52 14 47 15 . 3 0 16 . 4 3 17 33 18 . 4 7 19 . . . 0 1 20 . . 29 21 45 22 .44 23 . 3 8 24 37 25 . 2 8 .22 f o r s i g n i f i c a n c e a t t h e . 0 1 l e v e l . The r e s u l t s o f t h e f a c t o r a n a l y s i s u s i n g a l l 25 i t e m s showed 10 f a c t o r s w h i c h a c c o u n t e d f o r 8 1 . 5 p e r c e n t o f t h e v a r i a n c e . The r o t a t e d m a t r i x was e x a m i n e d i n an e f f o r t t o d e t e r m i n e w h i c h i t e m s c o n t r i b u t e d t o t h e s e f a c t o r s . I t was assumed t h a t a c o r r e l a t i o n o f 52 . 2 0 o r l a r g e r b e t w e e n an i t e m and a f a c t o r w o u l d i d e n t i f y t h o s e i t e m s t h a t compose most o f t h e f a c t o r . T a b l e 2 shows t h e i t e m s w h i c h compose e a c h f a c t o r . B e c a u s e o f t h e . l a c k o f a s i g n i f i c a n t p o i n t b i s e r i a l c o r r e l a t i o n c o e f f i c i e n t , t h e t h r e e i t e m s 5 , 8 , and 19 w e r e removed and t h e t e a c h e r s ' s c o r e s w e r e r e c a l c u l a t e d . The 137 t e a c h e r s ' s c o r e s r a n g e d f r o m 5 t o 22 w i t h a mean o f 1 6 , a m e d i a n o f 1 6 , and a s t a n d a r d d e v i a t i o n o f 3 . 5 9 . The K u d e r - R i c h a r d s o n 20 r e l i a b i l i t y c o e f f i c i e n t f o r t h e s e 22 i t e m s was . 7 1 . I t must be n o t e d t h a t a c o e f f i c i e n t r e c a l c u l a t e d on o r i g i n a l d a t a i n t h i s f a s h i o n may be s p u r i o u s l y h i g h . T a b l e 2 I tems W h i c h Have C o r r e l a t i o n C o e f f i c i e n t s o f . 20 o r G r e a t e r w i t h a F a c t o r F a c t o r s I tems 1 4 , 7 , 1 2 , 1 3 , 1 5 , 1 8 , 3 2 0 , 2 1 , 3 2 2 , a 2 3 , 3 25 2 3 , a 4 , 6 , 1 1 , 1 4 , 1 6 a 1 7 , a 2 3 , 2 4 a 3 l , a 1 1 , 1 3 , 1 5 , 1 6 , 2 2 , 2 5 3 4 1 , 6 , a 7 , 9 , a 1 3 , 1 4 , a 1 6 , 1 8 , 21 5 4 , a 5 , a 7 , 9 , 2 0 , 2 1 , 25 6 2 , a 7 , a 1 2 , 1 3 , 1 5 , 1 6 , 17 7 3 , 4 , 7, 1 0 , a 1 3 , 1 4 , 1 5 , a 20 8 1 , 3 , 4 , 6 , 7, 9 , 1 1 , a 1 2 , 1 4 , 21 9 3 , 4 , 8 , a 1 3 , 2 0 a 10 6 , 7, 1 2 , 1 9 , a 25 A c o r r e l a t i o n g r e a t e r t h a n . 5 . 53 An e x a m i n a t i o n o f T a b l e 3 i n d i c a t e s t h a t a l l o f t h e r e m a i n i n g 22 i t e m s h a d s i g n i f i c a n t p o i n t b i s e r i a l c o r r e l a t i o n s w i t h t h e w h o l e t e s t . T a b l e 4 shows t h e r e s u l t s o f t h e f a c t o r a n a l y s i s on t h e 22 i t e m s . These e i g h t f a c t o r s a c c o u n t e d f o r 7 7 . 8 p e r c e n t o f t h e v a r i a n c e . T a b l e 3 Si P o i n t B i s e r i a l C o r r e l a t i o n I t e m C o e f f i c i e n t 1 5 1 . . 36 2 . 30 3 ' . • . . 29 4 . 37 6 . 3 5 7 . 3 7 9 34 10 . 3 1 11 30 12 45 13 55 14 -15 29 16 43 17 . 35 18 48 20 . 3 3 21 44 22 .46 23 40 24 38 25 . 2 8 22 i t e m s ; 5 , 8 , and 19 r e m o v e d . b . 2 2 f o r s i g n i f i c a n c e a t t h e . 0 1 l e v e l . The d a t a f r o m t h e t h i r d v e r s i o n o f t h e t e s t w e r e t h e n r e a n a l y z e d i n g j u s t t h e s e 22 i t e m s . The c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e 54 t e a c h e r s ' s c o r e s o n t h e s e 22 i t e m s and t h e i r o p i n i o n o f t h e i r a t t i t u d e t o w a r d c o n t e m p o r a r y m a t h e m a t i c s was . 7 9 . B e c a u s e o f t h e v a l i d a t i o n p r o c e d u r e s and t h e r e s u l t s o f t h e a n a l y s e s on t h e 22 i t e m s , i t was d e c i d e d t o u s e t h e s e 22 i t e m s as t h e i n v e n t o r y t o d e t e r m i n e t e a c h e r ' s a t t i t u d e t o w a r d c o n t e m p o r a r y as o p p o s e d t o t r a d i t i o n a l m a t h e m a t i c s . A copy o f t h e i n s t r u m e n t i s i n A p p e n d i x g . T a b l e 4 I tems W h i c h Have C o r r e l a t i o n C o e f f i c i e n t s o f . 20 o r G r e a t e r w i t h a F a c t o r 3 F a c t o r s I tems 1 4 , 7, 1 2 , 1 3 , 1 5 , 1 8 , b 2 2 , b 2 3 , b 25 2 3 , b 4 , 6 , 1 1 , 1 4 , 1 6 , b 1 7 , b 2 3 , 2 4 b 3 2 , b 7 , b 1 2 , 1 3 , 1 5 , 1 6 , 17 4 3 , 4 , 7, 1 0 , b 1 4 , 1 5 , b 16 5 l , b 7, 1 3 , 1 5 , 1 6 , 2 2 , 2 5 b 6 1 , 3 , 4 , b 6 , 9 , 1 1 , b 1 2 , 14 7 1 , 6 , b 7, 9 , b 1 3 , 1 4 , b 1 6 , 21 8 4 , 7, 1 0 , 1 3 , b 1 8 , 2 0 b 22 i t e m s ; 5 , 8 , and 19 r e m o v e d . A c o r r e l a t i o n g r e a t e r t h a n . 5 . C o n s t r u c t i o n o f S t u d e n t T e s t s o f A r i t h m e t i c T h r e e t e s t s w e r e c o n s t r u c t e d f o r e a c h o f t h e t h r e e g r a d e s : f o u r t h , f i f t h , and s i x t h . These t e s t s w e r e t e s t s o f u n d e r s t a n d i n g , p r o b l e m s o l v i n g , and c o m p u t a t i o n . 55 To o b t a i n i t e m s f o r t h e F o u r t h G r a d e C o m p u t a t i o n T e s t , h e r e a f t e r d e n o t e d C 4 , t h e f o u r t h g r a d e t e x t b o o k was s c r u t i n i z e d and compared w i t h t h e t h i r d g r a d e t e x t b o o k . A l l f o r m s o f c o m p u t a t i o n new t o t h e f o u r t h g r a d e w e r e i d e n t i f i e d . F o r t y i t e m s w e r e c o n s t r u c t e d r e p r e s e n t a t i v e o f . t h e s e t y p e s o f c o m p u t a t i o n . S i m i l a r p r o c e d u r e s w e r e u s e d t o o b t a i n i t e m s f o r t h e F i f t h G r a d e C o m p u t a t i o n T e s t and t h e S i x t h G r a d e C o m p u t a t i o n T e s t , h e r e a f t e r known as C5 and C 6 . F o r t y - f o u r i t e m s w e r e c o n s t r u c t e d f o r C5 and 54 i t e m s w e r e c o n s t r u c t e d f o r C 6 . To o b t a i n i t e m s f o r t h e F o u r t h , F i f t h , and S i x t h G r a d e P r o b l e m S o l v i n g T e s t s , t h e t e x t b o o k s f o r t h e s e g r a d e s w e r e s c r u t i n i z e d and compared w i t h t e x t b o o k s f r o m t h e p r e v i o u s g r a d e s . P r o b l e m s o l v i n g p r o c e d u r e s new t o e a c h o f t h e s e g r a d e s w e r e i d e n t i f i e d . T e s t s o f 20 i t e m s e a c h , one t e s t f o r e a c h g r a d e , w e r e c o n s t r u c t e d . The i t e m s w e r e r e p r e s e n t a t i v e o f t h e p r o b l e m s o l v i n g p r o c e d u r e s new t o e a c h g r a d e . These t e s t s w i l l h e r e a f t e r be known a s P 4 , P 5 , and P 6 . To o b t a i n i t e m s f o r t h e t h r e e t e s t s o f u n d e r s t a n d i n g , t h e t e x t b o o k s o f g r a d e s f o u r , f i v e , and s i x w e r e s c r u t i n i z e d and t h e u n d e r s t a n d i n g s w e r e i d e n t i f i e d . One h u n d r e d t h i r t e e n i t e m s w e r e c o n s t r u c t e d w h i c h w e r e r e p r e s e n t a t i v e o f t h e s e u n d e r s t a n d i n g s . These i t e m s w e r e r a n d o m l y d i v i d e d i n t o two s u b t e s t s . S u b t e s t A c o n t a i n e d 57 i t e m s and s u b t e s t B c o n t a i n e d 56 i t e m s . T h e s e 113 i t e m s w e r e e v a l u a t e d on a s c a l e o f one t o s e v e n by n i n e members o f t h e M a t h e m a t i c s E d u c a t i o n D e p a r t m e n t a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . A s c o r e o f one i n d i c a t e d no v a l u e as a m e a s u r e o f u n d e r s t a n d i n g w h i l e a s c o r e o f s e v e n i n d i c a t e d a h i g h v a l u e as a m e a s u r e o f u n d e r s t a n d i n g . Any i t e m w h i c h d i d n o t h a v e a summed s c o r e o f 27 o r h i g h e r was removed f r o m t h e t e s t . One h u n d r e d one i t e m s r e m a i n e d w i t h 50 i n s u b t e s t A and 51 i n s u b t e s t B . 56 The u n d e r s t a n d i n g i t e m s w e r e m u l t i p l e c h o i c e , w h i l e t h e c o m p u t a t i o n and p r o b l e m s o l v i n g i t e m s c a l l e d f o r c o n s t r u c t e d r e s p o n s e s . S i n c e t h e s e t e s t s w e r e f i r s t a d m i n i s t e r e d i n November , i t was assumed t h a t t h e f o u r t h g r a d e r s w o u l d a p p r o x i m a t e f o u r t h g r a d e r s a t t h e b e g i n n i n g o f t h e y e a r , and t h e f i f t h g r a d e r s w o u l d a p p r o x i m a t e f o u r t h g r a d e r s a t t h e end o f t h e y e a r . The p e r c e n t a g e o f f o u r t h g r a d e r s who had an i t e m c o r r e c t was compared w i t h t h e p e r c e n t a g e o f f i f t h g r a d e r s who h a d t h e same i t e m c o r r e c t . T h i s c o m p a r i s o n g a v e an i n d i -c a t i o n as t o w h i c h i t e m s w e r e f o u r t h g r a d e i t e m s i n t h i s t e x t s e r i e s . A s i m i l a r c o m p a r i s o n was made b e t w e e n t h e f i f t h and s i x t h g r a d e r s t o d e t e r m i n e t h o s e i t e m s w h i c h seem t o be f i f t h g r a d e i t e m s i n t h i s t e x t s e r i e s . A s i m i l a r c o m p a r i s o n was made b e t w e e n t h e s i x t h a n d s e v e n t h g r a d e r s t o d e t e r m i n e t h o s e i t e m s w h i c h seem t o b e l e a r n e d i n t h e s i x t h g r a d e i n t h i s t e x t s e r i e s . A l l a d m i n i s t r a t i o n s o f t h e s e t e s t s w e r e i n s c h o o l d i s t r i c t s w h i c h u s e d t h e L a i d l a w s e r i e s . T e s t C4 w i t h 40 i t e m s was a d m i n i s t e r e d t o 17 f o u r t h g r a d e r s a n d 16 f i f t h g r a d e r s . The c h e c k t o d e t e r m i n e w h i c h i t e m s a r e f o u r t h g r a d e i t e m s a n d t h e i t e m a n a l y s i s l e d t o t h e r e m o v a l o f 14 i t e m s . T e s t C4 was a l s o a d m i n i s t e r e d t o 50 d i f f e r e n t f o u r t h and f i f t h g r a d e r s t o d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The 26 i t e m s , as m u l t i p l e c h o i c e i t e m s , w e r e u s e d f o r t h e s e c o n d g i v i n g o f C 4 . These i t e m s w e r e a d m i n i s t e r e d t o 26 f o u r t h g r a d e r s and 24 f i f t h g r a d e r s . I t e m a n a l y s i s e l i m i n a t e d one i t e m l e a v i n g t e s t C4 w i t h 25 i t e m s . The r e l i a b i l i t y o f t h i s t e s t and a l l o t h e r t e s t s i s g i v e n i n T a b l e 6 on page 6 1 . T e s t C5 was a d m i n i s t e r e d t o 14 f i f t h g r a d e r s and 16 s i x t h g r a d e r s . The c h e c k t o d e t e r m i n e w h e t h e r o r n o t t h e i t e m s a r e f i f t h g r a d e i t e m s and t h e i t e m a n a l y s i s l e d t o t h e r e m o v a l o f 18 i t e m s . T e s t C5 was a l s o a d m i n i s t e r e d t o 53 d i f f e r e n t f i f t h and s i x t h g r a d e r s t o d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The r e m a i n i n g 26 i t e m s , as m u l t i p l e c h o i c e q u e s t i o n s , w e r e u s e d f o r t h e s e c o n d g i v i n g o f C 5 . T h i s f o r m was a d m i n i s t e r e d t o 13 f i f t h g r a d e r s and 25 s i x t h g r a d e r s . I t em a n a l y s i s e l i m i n a t e d one i t e m l e a v i n g t e s t C5 w i t h 25 i t e m s . T e s t C6 w i t h 54 i t e m s was a d m i n i s t e r e d t o 15 s i x t h g r a d e r s and 12 s e v e n t h g r a d e r s . The c h e c k t o d e t e r m i n e w h e t h e r o r n o t t h e i t e m s a r e s i x t h g r a d e i t e m s and t h e i t e m a n a l y s i s removed 16 i t e m s . The 54 i t e m s w e r e a l s o g i v e n t o 52 o t h e r s i x t h and s e v e n t h g r a d e r s t o d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The r e m a i n i n g 38 i t e m s , as m u l t i p l e c h o i c e q u e s t i o n s , w e r e u s e d f o r t h e s e c o n d a d m i n i s t r a t i o n . T h i s f o r m was a d m i n i s t e r e d t o e i g h t s i x t h g r a d e r s and 32 s e v e n t h g r a d e r s . I t e m a n a l y s i s e l i m i n a t e d e i g h t i t e m s l e a v i n g 30 i t e m s i n t e s t C 6 . T e s t P4 w i t h 20 i t e m s was a d m i n i s t e r e d t o 13 f o u r t h g r a d e r s and 10 f i f t h g r a d e r s . The c h e c k t o d e t e r m i n e w h e t h e r o r n o t t h e s e p r o b l e m s w e r e f o u r t h g r a d e i t e m s and t h e i t e m a n a l y s i s e l i m i n a t e d f i v e i t e m s . The 20 i t e m s w e r e a l s o a d m i n i s t e r e d t o 50 d i f f e r e n t f o u r t h and f i f t h g r a d e r s t o d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The r e m a i n i n g 15 i t e m s , as m u l t i p l e c h o i c e q u e s t i o n s , w e r e u s e d f o r t h e s e c o n d a d m i n i s t r a t i o n . I t was a d m i n i s t e r e d t o 26 f o u r t h g r a d e r s and 24 f i f t h g r a d e r s . I t e m a n a l y s i s d i d n o t e l i m i n a t e any i t e m s l e a v i n g 15 i t e m s i n t e s t P 4 . T e s t P5 w i t h 20 i t e m s was a d m i n i s t e r e d t o 13 f i f t h g r a d e r s and 14 s i x t h g r a d e r s . The c h e c k t o d e t e r m i n e w h e t h e r o r n o t t h e s e p r o b l e m s 58 w e r e f i f t h g r a d e i t e m s and t h e i t e m a n a l y s i s c a u s e d t h e r e m o v a l o f f i v e i t e m s . The 20 i t e m s w e r e a l s o a d m i n i s t e r e d t o 53 d i f f e r e n t f i f t h and s i x t h g r a d e r s to d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The r e m a i n i n g 15 i t e m s , as m u l t i p l e c h o i c e q u e s t i o n s , w e r e u s e d f o r t h e s e c o n d a d m i n i s t r a t i o n . I t was f i r s t s e p a r a t e d i n t o two s u b t e s t s . One s u b t e s t o f e i g h t i t e m s was a d m i n i s t e r e d t o 20 f i f t h g r a d e r s and 24 s i x t h g r a d e r s . The o t h e r s u b t e s t o f s e v e n i t e m s was a d m i n i s t e r e d to 18 f i f t h g r a d e r s and 23 s i x t h g r a d e r s . I t e m a n a l y s i s d i d n o t e l i m i n a t e any i t e m s l e a v i n g 15 i t e m s i n t e s t P 5 . t e s t P6 w i t h 20 i t e m s was a d m i n i s t e r e d t o 13 s i x t h g r a d e r s and 10 s e v e n t h g r a d e r s . The c h e c k t o d e t e r m i n e w h e t h e r o r n o t t h e s e p r o b l e m s w e r e s i x t h g r a d e i t e m s and t h e i t e m a n a l y s i s d i d n o t remove any o f t h e i t e m s . The 20 i t e m s w e r e a l s o a d m i n i s t e r e d t o 52 d i f f e r e n t s i x t h and s e v e n t h g r a d e r s t o d e t e r m i n e p o s s i b l e m u l t i p l e c h o i c e d i s t r a c t o r s . The 20 i t e m s , as m u l t i p l e c h o i c e q u e s t i o n s , w e r e u s e d f o r t h e s e c o n d a d m i n i s -t r a t i o n . They w e r e f i r s t s e p a r a t e d i n t o two s u b t e s t s . One s u b t e s t o f 10 i t e m s was a d m i n i s t e r e d t o 20 s i x t h g r a d e r s and 31 s e v e n t h g r a d e r s . The o t h e r s u b t e s t o f 10 i t e m s was a d m i n i s t e r e d t o 20 s i x t h g r a d e r s and 32 s e v e n t h g r a d e r s . I t e m a n a l y s i s l e d t o t h e r e m o v a l o f f o u r i t e m s l e a v i n g 16 i t e m s i n t e s t P 6 . V e r s i o n A o f t h e s t u d e n t t e s t o f u n d e r s t a n d i n g was a d m i n i s t e r e d t o 16 f o u r t h g r a d e r s , 37 f i f t h g r a d e r s , 22 s i x t h g r a d e r s , and 30 s e v e n t h g r a d e r s . V e r s i o n B o f t h e s t u d e n t t e s t o f u n d e r s t a n d i n g was a d m i n i s t e r e d t o 33 f o u r t h g r a d e r s , 25 f i f t h g r a d e r s , 33 s i x t h g r a d e r s , and 28 s e v e n t h g r a d e r s . A f t e r a c h e c k t o d e t e r m i n e g r a d e l e v e l and t h e i t e m a n a l y s i s , a 65 i t e m F o u r t h G r a d e T e s t o f U n d e r s t a n d i n g , an 89 i t e m F i f t h G r a d e T e s t o f U n d e r s t a n d i n g , and an 88 i t e m S i x t h G r a d e 59 T e s t o f U n d e r s t a n d i n g w e r e c o n s t r u c t e d . These t e s t s w i l l h e r e a f t e r be r e f e r r e d t o as t e s t U 4 , t e s t U 5 , and t e s t U 6 . T h e r e w e r e i t e m s w h i c h w e r e i n a l l t h r e e t e s t s o f u n d e r s t a n d i n g . The s e c o n d a d m i n i s t r a t i o n o f t e s t U4 was t o 22 f o u r t h g r a d e r s and 20 f i f t h g r a d e r s . I t e m a n a l y s i s e l i m i n a t e d 15 i t e m s l e a v i n g 50 i t e m s i n t e s t U 4 . The s e c o n d a d m i n i s t r a t i o n o f h a l f o f t e s t U5 was t o 20 f i f t h g r a d e r s and 24 s i x t h g r a d e r s , and t h e o t h e r h a l f to 18 f i f t h g r a d e r s and 23 s i x t h g r a d e r s . The i t e m a n a l y s i s removed 25 i t e m s l e a v i n g t e s t U5 w i t h 55 i t e m s . F o r t h e s e c o n d a d m i n i s t r a t i o n , t e s t U6 was d i v i d e d i n t o two s u b t e s t s o f 44 i t e m s e a c h . One s u b t e s t was a d m i n i s t e r e d t o 20 s i x t h g r a d e r s and 31 s e v e n t h g r a d e r s . The o t h e r s u b t e s t was a d m i n i s t e r e d t o 20 s i x t h g r a d e r s and 30 s e v e n t h g r a d e r s . I t e m a n a l y s i s removed 24 i t e m s l e a v i n g 64 i t e m s i n t e s t U 6 . T h i s c o m p l e t e d t h e s e c o n d a d m i n i s t r a t i o n o f e a c h o f t h e n i n e t e s t s . The t h i r d a d m i n i s t r a t i o n h a d two p u r p o s e s : t o p e r m i t a n a d d i t i o n a l a n a l y s i s o f t h e i t e m s . a n d . t o c a l c u l a t e t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s . To f a c i l i t a t e a d m i n i s t r a t i o n , t e s t s U 4 , P 4 , and C4 w e r e c o m b i n e d as o n e t e s t b o o k l e t , T 4 . T e s t s U 5 , P 5 , and C5 w e r e c o m b i n e d as one t e s t b o o k l e t , T 5 . T e s t s U 6 , P 6 , a n d C6 w e r e c o m b i n e d as one t e s t b o o k l e t , T 6 . The t h i r d a d m i n i s t r a t i o n was b e f o r e t h e C h r i s t m a s v a c a t i o n . I t was a g a i n a d m i n i s t e r e d a f t e r t h e C h r i s t m a s v a c a t i o n . A b o u t one month e l a p s e d b e t w e e n t h e two a d m i n i s t r a t i o n s . T a b l e 5 shows t h e number o f s t u d e n t s p a r t i c i p a t i n g i n e a c h a d m i n i s t r a t i o n . B e c a u s e o f t h e d i f f e r e n c e s i n s c o r e s and number o f i t e m s o n e a c h p a r t o f t h e t e s t , a s t a n d a r d i z e d s c o r e was computed f o r each s t u d e n t on t h e r e d u c e d s e t o f i t e m s and t h e t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s 60 T a b l e 5 Number o f S t u d e n t s I n v o l v e d f o r T h i r d A d m i n i s t r a t i o n T4 T5 T6 S t u d e n t s 4 t h 5 t h 5 t h 6 t h 6 t h 7 t h I n v o l v e d 180 210 215 180 252 114 A n s w e r s h e e t s r e t u r n e d b e f o r e C h r i s t m a s 169 193 201 172 237 114 A n s w e r s h e e t s r e t u r n e d a f t e r C h r i s t m a s 3 165 196 203 170 , 238 T a k i n g a l l t h e t e s t s o n b o t h g i v i n g s 131 159 160 151 197 T a k i n g T e s t U b e f o r e C h r i s t m a s 163 186 196 171 228 105 T a k i n g T e s t U a f t e r C h r i s t m a s 159 192 196 169 233 T a k i n g T e s t U on b o t h g i v i n g s 144 170 176 160 210 T a k i n g T e s t P b e f o r e C h r i s t m a s 163 183 196 167 230 110 T a k i n g T e s t P a f t e r C h r i s t m a s 160 193 201 166 230 T a k i n g T e s t P o n b o t h g i v i n g s 145 167 181 155 211 T a k i n g T e s t C b e f o r e C h r i s t m a s 156 184 192 166 230 111 T a k i n g T e s t C a f t e r C h r i s t m a s 162 195 196 166 228 T a k i n g T e s t C on b o t h g i v i n g s 141 170 173 154 208 The s e v e n t h g r a d e r s d i d n o t p a r t i c i p a t e a f t e r C h r i s t m a s . 61 w e r e c o m p u t e d . T a b l e 6 shows t h e r e l i a b i l i t y o f e a c h o f t h e t e s t s and t h e r e l i a b i l i t y o f t e s t b o o k l e t s T 4 , T 5 , and T 6 . T a b l e 6 T e s t - R e t e s t R e l i a b i l i t y C o e f f i c i e n t s f o r E a c h T e s t and T 4 , T 5 , and T6 T e s t G r a d e U P . C T 4 t h .7677 . 5695 .5763 .7966 5 t h .8679 .8017 . . 8191 .9280 4 t h & 5 t h . 8581 .7864 .8176 .9215 5 t h . 7417 . 4485 . 3265 .6664 6 t h . 8073 .6661 . 8145 .8870 5 t h & 6 t h .8089 .6466 .8004 .8795 6 t h .8574 . 5 4 6 1 .7365 .8350 To f u r t h e r e n s u r e t h a t t h e t e s t s a c t u a l l y measure s t u d e n t g r o w t h a t t h e g i v e n g r a d e , t h e mean s c o r e s f o r t h e i n t e n d e d g r a d e l e v e l s and f o l l o w i n g g r a d e l e v e l s w e r e c a l c u l a t e d f o r e a c h o f t h e n i n e t e s t s . T a b l e 7 shows t h e means a t t h e i n t e n d e d g r a d e l e v e l and s i g n i f i c a n t l y h i g h e r means ( a t t h e 1 p e r c e n t l e v e l ) a t t h e f o l l o w i n g g r a d e l e v e l f o r each o f t h e n i n e t e s t s . I t e m a n a l y s i s r e d u c e d t h e number o f i t e m s on e a c h t e s t as f o l l o w s : T e s t U4 removed 8 i t e m s l e a v i n g 42 i t e m s . T e s t P4 removed 2 i t e m s l e a v i n g 13 i t e m s . T e s t C4 removed 1 i t e m l e a v i n g 24 i t e m s . T e s t U5 removed 12 i t e m s l e a v i n g 43 i t e m s . T e s t P5 removed 2 i t e m s l e a v i n g 13 i t e m s . T e s t C5 removed 2 i t e m s l e a v i n g 23 i t e m s . T e s t U6 removed 20 i t e m s l e a v i n g 44 i t e m s . 62 T e s t P6 removed 0 i t e m s l e a v i n g 16 i t e m s . T e s t C6 removed 0 i t e m s l e a v i n g 30 i t e m s These r e d u c e d v e r s i o n s o f t h e t e s t s w e r e u s e d as t h e s t u d e n t t e s t s f o r t h i s s t u d y . . The t h r e e t e s t s f o r each g r a d e l e v e l w e r e p l a c e d i n one b o o k l e t . They c a n be f o u n d i n A p p e n d i c e s C, D, and E . T a b l e 7 Mean S t u d e n t S c o r e on E a c h T e s t T e s t 4 t h 5 t h 6 t h 7 t h U4 1 5 . 2 9 2 1 . 6 9 P4 2 . 9 5 5 . 4 0 — — — C4 6 . 2 2 1 1 . 6 7 U5 — 1 4 . 9 3 1 9 . 9 3 P5 — 3 .19 5 . 2 7 C5 — 4 . 3 9 9 . 5 6 U6 — — 1 9 . 9 0 2 4 . 2 3 P6 — — 4 . 3 7 6 . 7 3 C6 —• — 9 . 4 1 1 4 . 3 3 P l a n o f The S t u d y D u r i n g t h e s p r i n g o f 1 9 6 8 , t h e a p p r o p r i a t e a d m i n i s t r a t o r s o f S p o k a n e , W a s h i n g t o n , and B r e m e r t o n , W a s h i n g t o n , g r a n t e d p e r m i s s i o n t o c o n d u c t t h e s t u d y i n t h e i r s c h o o l d i s t r i c t s . I t was m u t u a l l y d e c i d e d t o c a r r y o u t t h e s t u d y d u r i n g t h e 1968-1969 s c h o o l y e a r . To m a i n t a i n t h e a n o n y m i t y o f t e a c h e r s , each s c h o o l was numbered and e a c h t e a c h e r w i t h i n t h e s c h o o l was n u m b e r e d . A s i x d i g i t n u m e r a l was g i v e n t o e a c h s t u d e n t . The f i r s t two d i g i t s r e p r e s e n t e d t h e s c h o o l , t h e m i d d l e two d i g i t s r e p r e s e n t e d t h e t e a c h e r , and t h e l a s t two d i g i t s r e p r e -s e n t e d t h e s t u d e n t s . The r e s e a r c h e r d i d n o t know t h e names o f t h e p a r t i c i -p a t i n g t e a c h e r s and had no way o f r e l a t i n g them t o t h e d a t a c o l l e c t e d . The d i s t r i b u t i o n o f m a t e r i a l was h a n d l e d i n t e r n a l l y by each s c h o o l d i s t r i c t . 63 N i n e t y - n i n e t e a c h e r s , 33 a t e a c h g r a d e l e v e l , w e r e r a n d o m l y s e l e c t e d t o p a r t i c i p a t e i n t h e s t u d y . T a b l e 8 shows t h e number o f t e a c h e r s c o m p l e t i n g a l l a s p e c t s o f t h e s t u d y . T a b l e 9 shows t h e number o f s t u d e n t s c o m p l e t i n g t h e s t u d y . T a b l e 8 Number o f T e a c h e r s P a r t i c i p a t i n g G r a d e 4 5 6 T o t a l Spokane 14 14 12 40 B r e m e r t o n 7 9 5 21 T o t a l 21 23 17 61 T a b l e 9 Number o f S t u d e n t s P a r t i c i p a t i n g G r a d e 4 5 6 T o t a l Spokane 321 365 403 1089 B r e m e r t o n 164 218 140 522 T o t a l 485 583 543 1611 I n S e p t e m b e r , 1 9 6 8 , t h e p r i n c i p a l s w e r e g i v e n t h e d e t a i l s o f t h e s t u d y . I n O c t o b e r , 1968 , t h e p r i n c i p a l s a d m i n i s t e r e d t h e T e a c h e r T e s t  o f U n d e r s t a n d i n g and t h e a t t i t u d e i n v e n t o r y , A t t i t u d e Toward C o n t e m p o -r a r y M a t h e m a t i c s . Because some researchers reported a possible effect on student learning from a composite of teacher variables, i t was decided to obtain information on other teacher variables: 1. Number of quarter hours taken in college mathematics. 2. Number of quarter hours of hew mathematics. 3. How long since the last of these mathematics courses was taken. 4. Number of quarter hours of mathematics method courses. 5. How long since the last of these method courses was taken. 6. Number of quarter hours of professional education courses. 7. Number of years of teaching experience. 8. Number of years in present d i s t r i c t . 9. Principal's rating of teachers. The principal obtained a l l of this information, except his rating of the teacher, by having each teacher complete a questionnaire (Appendix,F). This information was also collected in October, 1968. On the f i r s t of May, 1969, the principals were asked to rate their teachers (Appendix G). This rating was concerned with the teacher's a b i l i t y to teach mathe-matics using a contemporary approach. Statistical Procedure The relationships between teacher variables and student growth were compared by multiple linear regression. The analysis was performed at the computer center of the University of British Columbia using the 65 " B o t w a r d " v e r s i o n o f l i n e a r r e g r e s s i o n a n a l y s i s . . T h i s v e r s i o n was 9 o r i g i n a l l y p r e s e n t e d by R o b e r t A . B o t t e n b e r g and J o e H . W a r d . 2 M u l t i p l e l i n e a r r e g r e s s i o n p r o d u c e s R , t h e . p e r c e n t a g e o f v a r i a n c e i n t h e s p e c i f i e d c r i t e r i o n t h a t i s a c c o u n t e d f o r by t h e s p e c i f i e d p r e d i c t o r v a r i a b l e s . The s p e c i f i e d c r i t e r i a i n t h i s s t u d y , t h e d e p e n d e n t v a r i a b l e s , a r e : 1 . S t u d e n t g r o w t h i n u n d e r s t a n d i n g . 2 . S t u d e n t g r o w t h i n p r o b l e m s o l v i n g . 3 . S t u d e n t g r o w t h i n c o m p u t a t i o n . 4 . S t u d e n t a c h i e v e m e n t . The s p e c i f i e d p r e d i c t o r v a r i a b l e s , t h e i n d e p e n d e n t v a r i a b l e s , a r e : 1 . The raw s c o r e on t h e T e a c h e r T e s t o f U n d e r s t a n d i n g . 2 . The raw s c o r e on t h e a t t i t u d e i n v e n t o r y , A t t i t u d e Toward  C o n t e m p o r a r y M a t h e m a t i c s . 3 . The c a t e g o r i z a t i o n o f t h e q u a r t e r h o u r s o f c o l l e g e m a t h e -m a t i c s c o m p l e t e d b y e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s 0 q u a r t e r h o u r s . 1 r e p r e s e n t s 1 t o 7 q u a r t e r h o u r s . 2 r e p r e s e n t s 7 t o 13 q u a r t e r h o u r s . 3 r e p r e s e n t s 13 t o 19 q u a r t e r h o u r s . 4 r e p r e s e n t s 19 o r more q u a r t e r h o u r s . 4 . The c a t e g o r i z a t i o n o f t h e q u a r t e r h o u r s o f ' n e w ' m a t h e m a t i c s c o m p l e t e d b y e a c h t e a c h e r s u c h t h a t : 9 R . A . B o t t e n b e r g and J . H . W a r d , A p p l i e d M u l t i p l e L i n e a r R e g r e s s i o n , C l e a r i n g h o u s e f o r F e d e r a l S c i e n t i f i c and T e c h n i c a l I n f o r -m a t i o n , U n i t e d S t a t e s D e p a r t m e n t o f Commerce, T e c h n i c a l D o c u m e n t a r y R e p o r t P R L - T D R - 6 3 - 6 , M a r c h , 1 9 6 3 . 0 r e p r e s e n t s 0 q u a r t e r h o u r s . 1 r e p r e s e n t s 1 t o 7 q u a r t e r h o u r s . 2 r e p r e s e n t s 7 t o 13 q u a r t e r h o u r s . 3 r e p r e s e n t s 13 t o 19 q u a r t e r h o u r s . 4 r e p r e s e n t s 19 o r more q u a r t e r h o u r s . 5 . The c a t e g o r i z a t i o n o f t h e number o f y e a r s s i n c e t h e l a s t m a t h e m a t i c s c o u r s e was c o m p l e t e d b y e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s t h e p a s t y e a r . 1 r e p r e s e n t s 1 t o 2 y e a r s . 2 . r e p r e s e n t s 2 t o 5 y e a r s . 3 r e p r e s e n t s 5 t o 10 y e a r s . 4 r e p r e s e n t s 10 o r more y e a r s . 6 . The c a t e g o r i z a t i o n o f t h e number o f q u a r t e r h o u r s o f mathe m a t i c s methods c o u r s e s c o m p l e t e d by e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s 0 q u a r t e r h o u r s . 1 r e p r e s e n t s 1 t o 4 q u a r t e r h o u r s . 2 r e p r e s e n t s 4 to 9 q u a r t e r h o u r s . 3 r e p r e s e n t s 9 t o 13 q u a r t e r h o u r s . 4 r e p r e s e n t s 13 o r more q u a r t e r h o u r s . 7 . The c a t e g o r i z a t i o n o f t h e number o f y e a r s s i n c e t h e l a s t m a t h e m a t i c s methods c o u r s e was c o m p l e t e d by e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s t h e p a s t y e a r . 1 r e p r e s e n t s 1 t o 2 y e a r s a g o . 2 r e p r e s e n t s 2 t o 5 y e a r s a g o . 3 r e p r e s e n t s 5 t o 10 y e a r s a g o . 4 r e p r e s e n t s 10 o r more y e a r s a g o . 8 . The c a t e g o r i z a t i o n o f t h e number o f q u a r t e r h o u r s o f p r o f e s s i o n a l e d u c a t i o n c o u r s e s c o m p l e t e d by e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s 0 t o 20 q u a r t e r h o u r s . 1 r e p r e s e n t s 20 t o 30 q u a r t e r h o u r s . 2 r e p r e s e n t s 30 t o 40 q u a r t e r h o u r s . 3 r e p r e s e n t s 40 t o 50 q u a r t e r h o u r s . 4 r e p r e s e n t s 50 o r more q u a r t e r h o u r s . 9 . The c a t e g o r i z a t i o n o f t h e number o f y e a r s t e a c h i n g e x p e r i -ence b y e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s 0 y e a r s o f e x p e r i e n c e . 1 r e p r e s e n t s 1 y e a r o f e x p e r i e n c e . 2 r e p r e s e n t s 2 y e a r s o f e x p e r i e n c e 3 r e p r e s e n t s 3 o r 4 y e a r s o f e x p e r i e n c e . ' 4 r e p r e s e n t s 5 o r 6 y e a r s o f e x p e r i e n c e . 5 r e p r e s e n t s 7 t o 10 y e a r s o f e x p e r i e n c e . 6 r e p r e s e n t s 10 t o 15 y e a r s o f e x p e r i e n c e . 7 r e p r e s e n t s 15 t o 20 y e a r s o f e x p e r i e n c e . 8 r e p r e s e n t s 20 o r more y e a r s o f e x p e r i e n c e . 1 0 . The c a t e g o r i z a t i o n o f t h e number o f y e a r s t e a c h i n g e x p e r i -ence w i t h i n t h e d i s t r i c t by e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s 0 y e a r s o f e x p e r i e n c e . 1 r e p r e s e n t s 1 y e a r o f e x p e r i e n c e . 2 r e p r e s e n t s 2 y e a r s o f e x p e r i e n c e . 3 r e p r e s e n t s 3 o r 4 y e a r s o f e x p e r i e n c e . 4 r e p r e s e n t s 5 o r 6 y e a r s o f e x p e r i e n c e . 5 r e p r e s e n t s 7 t o 10 y e a r s o f e x p e r i e n c e . 6 r e p r e s e n t s 10 t o 15 y e a r s o f e x p e r i e n c e . 68. 7 r e p r e s e n t s 15 t o 20 y e a r s o f e x p e r i e n c e . 8 r e p r e s e n t s 20 o r more y e a r s o f e x p e r i e n c e . 1 1 . The c a t e g o r i z a t i o n o f t h e k n o w l e d g e o f c a l c u l u s by e a c h t e a c h e r s u c h t h a t : 0 r e p r e s e n t s no c a l c u l u s c o m p l e t e d i n c o l l e g e . 1 r e p r e s e n t s some c a l c u l u s c o m p l e t e d i n c o l l e g e . 1 2 . The r a t i n g o f t h e t e a c h e r by h i s p r i n c i p a l o n a s c a l e f r o m one t o s e v e n , s e v e n i s s u p e r i o r , as t o t h e a b i l i t y o f t h e t e a c h e r i n g e n e r a l as a t e a c h e r . 1 3 . The r a t i n g o f t h e t e a c h e r b y h i s p r i n c i p a l o n a s c a l e f r o m one t o s e v e n , s e v e n i s s u p e r i o r , as t o t h e a b i l i t y o f t h e t e a c h e r as a m a t h e m a t i c s t e a c h e r . 1 4 . The r a t i n g o f t h e t e a c h e r by h i s p r i n c i p a l on a s c a l e f r o m one t o s e v e n , s e v e n i s s u p e r i o r , as t o t h e amount o f new m a t h e m a t i c s u s e d by t h e t e a c h e r . 2 2 The F r a t i o c o m p a r i n g R f r o m t h e f u l l m o d e l t o R f r o m t h e r e s t r i c t e d m o d e l i s t h e n c a l c u l a t e d . The p r o b a b i l i t y t h a t an F r a t i o t h i s l a r g e o r l a r g e r o c c u r i n g by c h a n c e a l o n e i s t h e n d e t e r m i n e d . I f t h e p r o b a b i l i t y v a l u e i s l e s s t h a n 5 p e r c e n t , t h e n t h e h y p o t h e s i s , no r e l a t i o n s h i p b e t w e e n v a r i a b l e s , w i l l be r e j e c t e d . The f o l l o w i n g h y p o t h e s e s w i l l be c h e c k e d a t t h e 5 p e r c e n t l e v e l o f s i g n i f i c a n c e : H I . T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n c o m p u t a t i o n . H 2 . T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n p r o b l e m s o l v i n g . 69 H3.. There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n understanding. H4. There i s no s i g n i f i c a n t r e l a t i o n s h i p between selected teacher variables and student growth i n achievement. Many other models w i l l also be checked for s i g n i f i c a n c e . This w i l l enable the researcher to determine such things as, "Do years of teaching experience have an e f f e c t upon student growth i n mathematics?" Since none of these models have been hypothesized, any i n d i c a t i o n of e f f e c t on learning must be considered subjects for future research. C h a p t e r 4 RESULTS The f i r s t c o m p a r i s o n o f t h e t e a c h e r v a r i a b l e s w i t h t h e s t u d e n t v a r i a b l e s u s e d P e a r s o n C o r r e l a t i o n C o e f f i c i e n t s . T a b l e 10 g i v e s t h e s e c o r r e l a t i o n s . A c o r r e l a t i o n c o e f f i c i e n t o f 0 . 2 5 0 o r l a r g e r i s r e q u i r e d f o r s i g n i f i c a n c e a t t h e 5 p e r c e n t l e v e l . O n l y one c o r r e l a t i o n , t h e one c o m p a r i n g p r i n c i p a l ' s r a t i n g o f t h e t e a c h e r as a t e a c h e r and g r o w t h i n c o m p u t a t i o n , i s s i g n i f i c a n t . These r e s u l t s seem t o i n d i c a t e , c o n t r a r y t o many e a r l i e r s t u d i e s , t h a t t h e p r i n c i p a l does h a v e some i d e a who h i s ' b e s t t e a c h e r s ' a r e when ' b e s t t e a c h e r s ' a r e d e t e r m i n e d by s t u d e n t g r o w t h i n a r i t h m e t i c c o m p u t a t i o n . M u l t i p l e l i n e a r r e g r e s s i o n e q u a t i o n s w e r e t h e n u s e d t o compare t h e t e a c h e r v a r i a b l e s w i t h e a c h o f t h e s t u d e n t v a r i a b l e s . The d a t a f r o m T a b l e 11 i n d i c a t e s t h e 14 t e a c h e r v a r i a b l e s a c c o u n t e d f o r a p p r o x i m a t e l y 21 p e r c e n t o f t h e v a r i a n c e i n t h e d e p e n d e n t v a r i a b l e , 2 s t u d e n t g r o w t h i n c o m p u t a t i o n . When t h e r e l a t e d R , 0 . 2 1 1 7 , i s 2 compared w i t h R f r o m t h e r e s t r i c t e d m o d e l , 0 , an F - r a t i o o f 0 . 9 7 1 2 i s c o m p u t e d . T h i s i s c l e a r l y n o n s i g n i f i c a n t . The d a t a f r o m T a b l e 11 a l s o i n d i c a t e s t h a t t h e 14 t e a c h e r v a r i a b l e s a c c o u n t e d f o r a p p r o x i m a t e l y 21 p e r c e n t o f t h e v a r i a n c e i n t h e d e p e n d e n t v a r i a b l e , s t u d e n t g r o w t h i n p r o b l e m s o l v i n g . The F - r a t i o o f 0 . 9 6 7 1 i s a g a i n c l e a r l y n o n s i g n i f i c a n t . F u r t h e r , t h e d a t a f r o m T a b l e 11 i n d i c a t e s t h a t t h e 14 t e a c h e r v a r i a b l e s a c c o u n t e d f o r a p p r o x i m a t e l y 18 p e r c e n t o f t h e v a r i a n c e i n t h e d e p e n d e n t v a r i a b l e , 70 71 T a b l e 10 C o r r e l a t i o n C o e f f i c i e n t s C o m p a r i n g T e a c h e r V a r i a b l e s t o S t u d e n t V a r i a b l e s T e a c h e r V a r i a b l e s S t u d e n t S t u d e n t S t u d e n t G r o w t h i n G r o w t h i n , G r o w t h i n U n d e r s t a n d i n g P r o b l e m S o l v i n g C o m p u t a t i o n S c o r e on t e s t o f u n d e r s t a n d i n g - 0 . 0 1 4 0 S c o r e on a t t i t u d e i n v e n t o r y 0 . 0 6 3 3 Q u a r t e r h o u r s o f c o l l e g e m a t h e m a t i c s 0 . 0 0 9 2 . 0 . 0 6 7 2 0 . 0 8 5 5 0 . 2 2 3 2 - 0 . 0 7 6 4 0 . 0 3 4 6 - 0 . 1 8 4 1 Q u a r t e r h o u r s o f new m a t h e m a t i c s 0 . 0 2 8 0 0 .0994 0 . 0 9 8 8 Y e a r s s i n c e l a s t m a t h e -m a t i c s c o u r s e - 0 . 0 4 9 5 - 0 . 0 0 8 7 - 0 . 1 1 3 6 Q u a r t e r h o u r s o f mathe-m a t i c s methods - 0 . 0 6 8 7 0 . 1 3 0 7 0 . 1 5 6 4 Y e a r s s i n c e l a s t methods c o u r s e 0 . 0 0 3 9 0 . 0 7 5 2 - 0 . 0 1 7 8 Q u a r t e r h o u r s o f p r o f e s s i o n a l e d u c a t i o n - 0 . 0 0 8 6 Y e a r s o f t e a c h i n g e x p e r i e n c e - 0 . 1 3 6 3 Y e a r s o f d i s t r i c t t e a c h i n g e x p e r i e n c e - 0 . 0 1 6 4 Taken c a l c u l u s 0 . 0 3 1 7 0 . 0 4 8 8 0 .0532 0 . 0 3 5 9 0 . 1 0 7 7 - 0 . 1 0 9 1 - 0 . 0 4 2 5 0 . 0 4 5 6 0 . 0 2 0 7 P r i n c i p a l ' s r a t i n g as a t e a c h e r 0 .1932 0 . 2 1 5 5 0 . 3 0 4 1 P r i n c i p a l ' s r a t i n g as a m a t h e m a t i c s t e a c h e r - 0 . 0 4 0 6 0 . 1 0 2 1 0 . 2 0 4 6 P r i n c i p a l ' s r a t i n g as t h e use o f new m a t h e -m a t i c s 0 . 0 7 0 8 0 . 2 2 8 6 0 . 2 4 1 6 s t u d e n t g r o w t h i n u n d e r s t a n d i n g . A g a i n , a n o n s i g n i f i c a n t F - r a t i o o f 0 . 8 0 9 9 was c a l c u l a t e d . L a s t l y , T a b l e 11 i n d i c a t e s t h e 14 t e a c h e r v a r i a b l e s a c c o u n t e d f o r a p p r o x i m a t e l y 18 p e r c e n t o f t h e v a r i a n c e i n t h e d e p e n d e n t v a r i a b l e , s t u d e n t a c h i e v e m e n t i n a r i t h m e t i c . A g a i n , a n o n s i g n i f i c a n t F - r a t i o o f O .7733 was c a l c u l a t e d . T a b l e 11 2 R R e s u l t s When t h e T e a c h e r V a r i a b l e s A r e Compared w i t h Each o f t h e S t u d e n t V a r i a b l e s S t u d e n t V a r i a b l e s R F P r o b a b i l i t y Growth i n c o m p u t a t i o n 0 . 2 1 1 7 0 . 9712 0 . 4924 G r o w t h i n p r o b l e m s o l v i n g 0 . 2 1 1 0 0 . 9671 0 . 4961 G r o w t h i n u n d e r s t a n d i n g 0 . 1 8 3 0 0 . 8099 0 . 6471 G r o w t h i n a c h i e v e m e n t 0 . 1 7 6 2 0 . 7733 0 . 7413 S i n c e none o f t h e f o u r a p r i o r i n u l l h y p o t h e s e s w e r e r e j e c t e d , i t was d e c i d e d t o f u r t h e r examine t h e d a t a by t h e t e c h n i q u e commonly c a l l e d ' d a t a s n o o p i n g ' f o r any p o s s i b l e n o n l i n e a r r e l a t i o n s h i p w h i c h m i g h t be u s e d f o r f u r t h e r r e s e a r c h . I n t h i s t e c h n i q u e , each v a r i a b l e i s p a r t i t i o n e d i n t o a s e t o f i n t e r v a l s , and each i n t e r v a l a c t s , a t f i r s t , as an i n d e p e n d e n t v a r i a b l e i n a r e g r e s s i o n e q u a t i o n . I f any a p p a r e n t s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p i s i n d i c a t e d b y t h i s p r o c e d u r e , p o s s i b l e n o n l i n e a r h y p o t h e s e s r e l a t i n g t h e s e q u e n t i a l i n t e r v a l s a r e t e s t e d . O n l y one p o s s i b l e r e l a t i o n s h i p was f o u n d . When t h e p r i n c i p a l s ' r a t i n g s as t e a c h e r s , as m a t h e m a t i c s t e a c h e r s , and as t e a c h e r s o f modern m a t h e m a t i c s a r e a l l d i c h o t o m i z e d b e t w e e n f o u r and f i v e (on a 73 seven p o i n t s c a l e ) , and the number of y e a r s s i n c e the l a s t mathematics c o u r s e was d i c h o t o m i z e d between one and two y e a r s , then a r e g r e s s i o n e q u a t i o n u s i n g t h e above mentioned v a r i a b l e s and the Teacher T e s t of U n d e r s t a n d i n g , the Teacher A t t i t u d e I n v e n t o r y , number o f q u a r t e r hours o f c o l l e g e mathematics, the number o f q u a r t e r hours o f 'new mathematics,' the q u a r t e r hours o f methods c o u r s e s , and the p r e s e n c e o f a c a l c u l u s c o u r s e i n the t e a c h e r ' s background as independent v a r i a b l e s produced a p o s s i b l y s i g n i f i c a n t r e l a t i o n s h i p (p = 2.21 p e r c e n t ) w i t h t h e dependent v a r i a b l e , s t u d e n t problem s o l v i n g . V e r y l i t t l e c o n f i d e n c e may be p l a c e d i n t h i s r e s u l t . In enough 'data s n o o p i n g ' a s i g n i f i c a n t c o r r e l a t i o n i s almost bound to t u r n up sooner o r l a t e r . No s i n g l e c o r r e l a t i o n i n t h i s e q u a t i o n was h i g h enough to encourage f u r t h e r e x p l o r a t i o n . Chapter 5 CONCLUSIONS The review of the l i t e r a t u r e indicated that most researchers who looked f o r teacher variables which might r e l a t e to teacher e f f e c t ! ness did not measure teacher variables p r e c i s e l y . They used quarter hours of college mathematics and other such measures. Further, most researchers neither p a r t i t i o n e d nor measured d i r e c t l y student growth. They used scores on standardized tests to i n f e r achievement. Many of these researchers used administrative ratings or other such i n d i r e c t measures to determine teacher effectiveness. Therefore, this study, using a more precise d e f i n i t i o n of teacher effectiveness and measuring some of the teacher variables d i r e c t l y , attempted to re l a t e teacher effectiveness to teacher v a r i a b l e s . Two instruments were constructed to p r e c i s e l y measure teacher v a r i a b l e s . One was a test of mathematical understanding. The items were rel a t e d to mathematical concepts taught i n grades four, f i v e , and s i x . The other was an inventory to measure teacher attitude toward contemporary mathematics as d i s t i n c t from t r a d i t i o n a l mathe-matics. To measure student growth, three tests were constructed f o r each grade. These were tests of understanding, tests of computation, and tests of problem solving. The tests were c a r e f u l l y constructed and 74 s u b m i t t e d t o i t e m a n a l y s e s i n an a t t e m p t t o e n s u r e g r a d e d i s c r i m i n a t i o n , c o n t e n t v a l i d i t y , and i n c r e a s e d i n t e r n a l c o n s i s t e n c y . T e a c h e r e f f e c t i v e n e s s was d e t e r m i n e d by p r e - t e s t i n g and p o s t -t e s t i n g t h e s t u d e n t s . S t u d e n t g r o w t h was d e f i n e d as t h e d i f f e r e n c e b e t w e e n s c o r e s on t h e p o s t - t e s t and t h e p r e - t e s t . T e a c h e r e f f e c t i v e n e s s was d e f i n e d as t h e m e a n - g a i n o f t h e s t u d e n t s i n h e r c l a s s . B e c a u s e o t h e r r e s e a r c h e r s r e p o r t e d on a v a r i e t y o f t e a c h e r v a r i a b l e s , i n f o r m a t i o n a b o u t 12 o t h e r commonly r e p o r t e d v a r i a b l e s was o b t a i n e d . These w e r e q u a r t e r h o u r s o f c o l l e g e m a t h e m a t i c s , c a l c u l u s , q u a r t e r h o u r s o f new m a t h e m a t i c s , when was t h e l a s t o f t h e s e m a t h e -m a t i c s c o u r s e s t a k e n , q u a r t e r h o u r s o f m a t h e m a t i c s methods c o u r s e s , when was t h e l a s t o f t h e s e m e t h o d s c o u r s e s t a k e n , q u a r t e r h o u r s o f p r o f e s s i o n a l e d u c a t i o n , y e a r s o f t e a c h i n g e x p e r i e n c e , y e a r s o f e x p e r i -ence i n p r e s e n t d i s t r i c t , p r i n c i p a l ' s r a t i n g o f t h e t e a c h e r as a t e a c h e r , p r i n c i p a l ' s r a t i n g o f t h e t e a c h e r as an a r i t h m e t i c t e a c h e r , and p r i n c i p a l ' s r a t i n g o f t h e t e a c h e r as a t e a c h e r o f new m a t h e m a t i c s . The m a i n p u r p o s e f o r o b t a i n i n g i n f o r m a t i o n on t h e s e v a r i a b l e s was t o d e t e r m i n e w h e t h e r o r n o t t e a c h e r e f f e c t i v e n e s s as d e f i n e d i n t h i s s t u d y w o u l d y i e l d s i g n i f i c a n t r e l a t i o n s h i p s . I n an e f f o r t t o d e t e r m i n e w h e t h e r o r n o t any s u c h r e l a t i o n s h i p s e x i s t e d , 1611 f o u r t h , f i f t h , and s i x t h g r a d e s t u d e n t s and t h e i r 61 t e a c h e r s w e r e s e l e c t e d to p a r t i c i p a t e . The t e a c h e r s w e r e r a n d o m l y s e l e c t e d f r o m o v e r 400 t e a c h e r s i n t h e S p o k a n e , W a s h i n g t o n , and B r e m e r t o n , W a s h i n g t o n , p u b l i c s c h o o l s . These s c h o o l d i s t r i c t s w e r e c h o s e n b e c a u s e t h e y u s e d t h e same a r i t h m e t i c s e r i e s and s i m i l a r s y l l a b u s , b u t t h e y a r e i n d i f f e r e n t g e o g r a p h i c l o c a t i o n s . C o r r e l a t i o n c o e f f i c i e n t s c o m p a r i n g t h e 14 t e a c h e r v a r i a b l e s w i t h t h e t h r e e s t u d e n t v a r i a b l e s w e r e c a l c u l a t e d . O n l y o n e , t h a t c o m p a r i n g p r i n c i p a l s ' r a t i n g s o f t e a c h e r s as g e n e r a l t e a c h e r s and s t u d e n t g r o w t h i n c o m p u t a t i o n , was s i g n i f i c a n t . T h i s i n d i c a t e s t h a t i f s t u d e n t g r o w t h i n c o m p u t a t i o n i s c a r e f u l l y m e a s u r e d by s p e c i f i c p r e - t e s t p o s t - t e s t p r o c e d u r e s , t h e n t h e p r i n c i p a l ' s r a t i n g i s c o r r e l a t e d t o t h e e f f e c t i v e n e s s o f t h e t e a c h e r . T h i s r e s u l t i s c o n t r a r y to t h e f i n d i n g s r e p o r t e d i n most e a r l i e r s t u d i e s , b u t t h i s i s t h e f i r s t d a t a b a s e d o n t e s t s d e s i g n e d t o measure g r o w t h i n c o m p u t a t i o n a t a s p e c i f i c l e v e l and f o r a s p e c i f i c t e x t b o o k and a r i t h m e t i c p r o -g r a m . T h e r e f o r e , i f t e a c h e r e f f e c t i v e n e s s i s p r e c i s e l y m e a s u r e d , t h e p r i n c i p a l ' s r a t i n g o f t h e t e a c h e r seems t o s i g n i f i c a n t l y c o r r e l a t e w i t h t e a c h e r e f f e c t i v e n e s s . N e x t , t h e f o l l o w i n g f o u r n u l l h y p o t h e s e s w e r e t e s t e d : H I . T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n c o m p u t a t i o n . H 2 . T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n p r o b l e m s o l v i n g . H 3 . T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n u n d e r s t a n d i n g . H4. T h e r e i s no s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n s e l e c t e d t e a c h e r v a r i a b l e s and s t u d e n t g r o w t h i n a c h i e v e m e n t . The above f o u r h y p o t h e s e s w e r e t e s t e d b y m u l t i p l e l i n e a r 2 2 r e g r e s s i o n . The F r a t i o s c o m p a r i n g R f r o m t h e f u l l m o d e l t o R f r o m t h e r e s t r i c t e d m o d e l w e r e e x a m i n e d f o r s i g n i f i c a n c e a t t h e 5 p e r c e n t l e v e l o f c o n f i d e n c e . No s i g n i f i c a n t r e l a t i o n s h i p s w e r e f o u n d . T h e s e r e s u l t s seem t o i n d i c a t e t h a t none o f t h e 14 v a r i a b l e s , when t a k e n i n d i v i d u a l l y o r as a g r o u p , w e r e r e l a t e d t o s t u d e n t g r o w t h i n any o f t h e t h r e e a r e a s — u n d e r s t a n d i n g , p r o b l e m s o l v i n g , and compu-t a t i o n o f a r i t h m e t i c . I n t h i s s t u d y , e v e r y e f f o r t was made t o e l i m i n a t e t h e d e f i c i e n c i e s o f p r e v i o u s s t u d i e s . Y e t t h e i r r e s u l t s a r e , i n g e n e r a l , c o n f i r m e d . E v e n t h e v e r y t o l e r a n t s a n c t i o n s o f ' d a t a s n o o p i n g ' p r o d u c e d no a d d i t i o n a l r e l a t i o n s h i p s . On t h e b a s i s o f t h e above r e s u l t s i t seems h i g h l y u n l i k e l y t h a t any f u r t h e r e x p l o r a t i o n o f t h e t e a c h e r c h a r a c t e r i s t i c s as i d e n t i f i e d i n t h i s s t u d y w o u l d be w a r r a n t e d . H o w e v e r , i n any s t u d y w h i c h f a i l s t o y i e l d s i g n i f i c a n t d i f f e r e n c e s , t h e r e i s a p o s s i b i l i t y t h a t s u c h f i n d i n g s a r e a r e s u l t o f i n s e n s i t i v i t y o f t h e t e s t i n g d e v i c e s . I t s h o u l d be o b s e r v e d , t h o u g h , t h a t t h e t e a c h e r s f o r t h i s s t u d y w e r e p r o f e s s i o n a l l y t r a i n e d . I t w o u l d b e a g r o s s o v e r -g e n e r a l i z a t i o n t o s u p p o s e t h a t t h e s e r e s u l t s s u p p o r t t h e h y p o t h e s e s t h a t t h i s p r o f e s s i o n a l t r a i n i n g d i d n o t i n f l u e n c e s u b s e q u e n t b e h a v i o r i n t e a c h i n g , o r t h a t p r o f e s s i o n a l t r a i n i n g i s u n n e c e s s a r y . T h e r e r e m a i n s t h e o p i n i o n o f many c o l l e g e i n s t r u c t o r s who t r a i n f u t u r e t e a c h e r s t h a t t h e r e is_ a r e l a t i o n s h i p b e t w e e n t e a c h e r v a r i a b l e s and t e a c h e r e f f e c t i v e n e s s . I f t h i s i s i n f a c t t h e c a s e , i t seems t h a t d i f f e r e n t i n d e p e n d e n t v a r i a b l e s must b e i d e n t i f i e d o r o t h e r methods o f m e a s u r i n g t h o s e i n t h i s s t u d y must be d e v e l o p e d . P h i l l i p s , " ' ' i n 1 9 7 0 , u s e d a d i f f e r e n t a p p r o a c h i n s t u d y i n g t h e t e a c h e r c h a r a c t e r i s t i c , t e a c h e r a t t i t u d e . He f o u n d t h a t t h e t y p e R. B . P h i l l i p s , " T e a c h e r A t t i t u d e as R e l a t e d t o S t u d e n t A t t i t u d e and A c h i e v e m e n t i n E l e m e n t a r y S c h o o l M a t h e m a t i c s " ( u n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f V i r g i n i a , 1 9 6 9 ) . D i s s e r t a t i o n A b s t r a c t s , 3 0 . 4 3 1 6 - A , 4 3 1 7 - A , N o . 1 0 , 1 9 6 9 / 7 0 . 78 of teacher attitude encountered by the student for two and for three of his past three years was significantly related to his present attitude and to his achievement. This might indicate a compounding effect over 2 a period of years. Flora, in 1972, looked at classroom behavior as a means to determine teacher effectiveness. He developed an instrument to measure teacher classroom behavior. He found a significant relation-ship between teacher effectiveness and teacher behavior. Studies such as the two reported above indicate that some researchers are measuring teacher variables in a different way or are considering different teacher variables. With the results of this 3 study in mind, i t seems that variables such as those used by Ph i l l i p s 4 and Flora might be teacher variables which do relate to teacher effectiveness. 2 B. V. Flora, Jr., "Diagnosing Selected Behavior Characteristics of Teachers of Secondary School Mathematics," Journal for Research in Mathematics Education, 3:7-20, January, 1972. ~ ~ 3 Phillips, loc. c i t . 4Flora, loc. c i t . B i b l i o g r a p h y A i k e n , L . R . " I n t e l l e c t i v e V a r i a b l e s and M a t h e m a t i c s A c h i e v e m e n t : , ' D i r e c t i o n s f o r R e s e a r c h , " J o u r n a l o f S c h o o l P s y c h o l o g y , 9 : 2 0 1 - 2 1 2 , and R . M . D r e g e r . "The E f f e c t o f A t t i t u d e s o n P e r f o r m a n c e v - - -. — - — — v- wj- . 1 1 H . J . U U U C D v t i i e x . j - u L i l i a n ' i n M a t h e m a t i c s , " J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , 5 2 : 1 9 - 2 4 , F e b r u a r y , 1 9 6 1 . A l s c h u l e r , A . S . "The E f f e c t s o f C l a s s r o o m S t r u c t u r e o n A c h i e v e m e n t M o t i v a t i o n and A c a d e m i c P e r f o r m a n c e , " E d u c a t i o n a l T e c h n o l o g y , 9 : 1 9 - 2 4 , A u g u s t , 1 9 6 9 . B a r n e s , K . , C . C r u i c k s h a n k , and J . F o s t e r . " S e l e c t e d E d u c a t i o n a l and E x p e r i e n c e F a c t o r s and A r i t h m e t i c T e a c h i n g , " The A r i t h m e t i c  T e a c h e r , 7 : 4 1 8 - 4 2 0 , D e c e m b e r , 1 9 6 0 . B a r r , A . S . " I m p r e s s i o n s , T r e n d s , and F u t u r e R e s e a r c h , " J o u r n a l o f  E x p e r i m e n t a l E d u c a t i o n , 1 4 : 2 0 0 - 2 0 6 , D e c e m b e r , 1 9 4 5 . Bassham, H . C . " R e l a t i o n s h i p o f P u p i l G a i n i n A r i t h m e t i c A c h i e v e m e n t t o C e r t a i n T e a c h e r C h a r a c t e r i s t i c s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f N e b r a s k a , 1960 . M . M u r p h y , a n d K . M u r p h y . " A t t i t u d e , and A c h i e v e m e n t i n ~1 j - J - l t r m . . . . - - — _ t J „ — — — •—.. » v i i J . b V b U [ C l l l . -A- L1 A r i t h m e t i c , " The A r i t h m e t i c T e a c h e r , 1 1 : 6 6 - 7 2 , F e b r u a r y , 1964 . B a u r , G . R . " A S t u d y o f t h e E f f e c t s o f a C r e a t i v e C l a s s r o o m , C r e a t i v e P r o b l e m s , and M a t h e m a t i c s E d u c a t o r s o n t h e C r e a t i v e A b i l i t y i n M a t h e m a t i c s o f P r o s p e c t i v e E l e m e n t a r y T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , I n d i a n a U n i v e r s i t y , 1 9 7 0 . D i s s e r t a t i o n  A b s t r a c t s , 3 1 A : 5 8 9 5 , M a y , 1 9 7 1 . B e a n , J . E . "The A r i t h m e t i c U n d e r s t a n d i n g o f E l e m e n t a r y S c h o o l T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , 1958 . B r o w n , E . D . " A r i t h m e t i c a l U n d e r s t a n d i n g s and A t t i t u d e s Toward A r i t h m e t i c o f E x p e r i e n c e d and I n e x p e r i e n c e d T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f N e b r a s k a T e a c h e r s C o l l e g e , 1 9 6 1 . B r o w n e l l , W. A . " A r i t h m e t i c a l R e a d i n e s s as a P r a c t i c a l C l a s s r o o m C o n c e p t , " E l e m e n t a r y S c h o o l J o u r n a l , 5 2 : 1 5 - 2 2 , S e p t e m b e r , 1 9 5 1 . B r u n e , I . H . " G e o m e t r y i n t h e G r a d e s , " The A r i t h m e t i c T e a c h e r , 8 : 2 1 0 - 2 1 9 , M a y , 1 9 6 1 . 79 80 B u r b a n k , I . K . " R e l a t i o n s h i p s B e t w e e n P a r e n t a l A t t i t u d e Toward M a t h e m a t i i and S t u d e n t A t t i t u d e Toward M a t h e m a t i c s , and Between S t u d e n t A t t i t u d e Toward M a t h e m a t i c s and S t u d e n t A c h i e v e m e n t i n M a t h e m a t i c s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U t a h S t a t e U n i v e r s i t y , 1968 . D i s s e r t a t i o n A b s t r a c t s , 3 0 A : 3 3 5 9 - 3 3 6 0 , F e b r u a r y , 1 9 7 0 . C a l l a h a n , . L . G . " A S t u d y o f K n o w l e d g e P o s s e s s e d by E l e m e n t a r y S c h o o l T e a c h e r s , I n - S e r v i c e and I n - T r a i n i n g , o f C u l t u r a l , P s y c h o l o g i c a l , and M a t h e m a t i c a l F o u n d a t i o n s o f t h e E l e m e n t a r y S c h o o l P r o g r a m . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , S y r a c u s e U n i v e r s i t y , 1 9 6 6 . C a r p e n t e r , R . " I d e n t i f y i n g C o n c e p t s and P r o c e s s e s i n M a t h e m a t i c s Needed f o r t h e A d e q u a t e P r e p a r a t i o n o f E l e m e n t a r y T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Oklahoma S t a t e U n i v e r s i t y , 1959 . C a r r o l , J . B . . . "The N a t u r e o f t h e D a t a o r How t o Choose a C o r r e l a t i o n C o e f f i c i e n t , " P s y c h o m e t r i k a , 2 6 : 3 4 7 - 3 7 2 , D e c e m b e r , 1 9 6 1 . C a r r o l l . , E . C . " A S t u d y o f t h e M a t h e m a t i c a l U n d e r s t a n d i n g s P o s s e s s e d by U n d e r g r a d u a t e S t u d e n t s M a j o r i n g i n E l e m e n t a r y E d u c a t i o n . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Wayne S t a t e U n i v e r s i t y , 1 9 6 1 . C a t t e l l , R . B . F a c t o r A n a l y s i s . New Y o r k : H a r p e r and B r o t h e r s , 1 9 5 2 . C o o l e y , W. W . , and P . R. L o h n e s . M u l t i v a r i a t e P r o c e d u r e s f o r t h e  B e h a v i o r a l S c i e n c e s . New Y o r k : J o h n W i l e y and S o n s , 1 9 6 2 . C o x , L . S . " A S t u d y o f P u p i l A c h i e v e m e n t i n M a t h e m a t i c s and T e a c h e r Competence i n M a t h e m a t i c s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f K a n s a s , 1970 . D i s s e r t a t i o n A b s t r a c t s , 3 1 A : 2 7 6 7 - 2 7 6 8 , December , 1 9 7 0 . C r i t t e n d e n , W. B . " A S t u d y o f A t t i t u d e s o f E l e m e n t a r y and S e c o n d a r y T e a c h e r s o f M a t h e m a t i c s Toward S e l e c t e d D e t e r r e n t s t o P u p i l P r o g r e s s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f H o u s t o n , 1 9 7 0 . D i s s e r t a t i o n A b s t r a c t s , 3 2 A : 2 8 0 , J u l y , 1 9 7 1 . D a v i s , B . "The E f f e c t o f R e i n f o r c e m e n t i n T e a c h i n g A r i t h m e t i c o n t h e P e r f o r m a n c e o f F i f t h G r a d e S t u d e n t s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a -t i o n , The P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1 9 6 1 . D i s s e r t a t i o n  A b s t r a c t s , 2 2 : 1 7 8 - 1 7 9 , J u l y , 1 9 6 1 . D a v i s , R . B . " T h e ' M a d i s o n P r o j e c t ' o f S y r a c u s e U n i v e r s i t y , " M a t h e m a t i c s  T e a c h e r , 5 3 : 5 7 1 - 5 7 5 , N o v e m b e r , 1960 . De L o p e z , C . R . " A P r o g r a m f o r T r a i n i n g T e a c h e r s f o r t h e P u e r t o R i c a n E l e m e n t a r y S c h o o l s i n t h e T e a c h i n g o f A r i t h m e t i c . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , T e a c h e r s C o l l e g e , C o l u m b i a U n i v e r s i t y , 1961 . D i l l , N . , and E . E . G o t t s . " I m p r o v e m e n t o f A r i t h m e t i c S e l f C o n c e p t T h r o u g h Combined P o s i t i v e R e i n f o r c e m e n t , P e e r I n t e r a c t i o n , and S e q u e n t i a l C u r r i c u l u m , " J o u r n a l o f S c h o o l P s y c h o l o g y , 9 : 4 6 2 - 4 7 2 , D u n n , W. H . "The I n f l u e n c e o f t h e T e a c h e r F a c t o r i n P r e d i c t i n g S u c c e s s i n N i n t h G r a d e A l g e b r a , " J o u r n a l o f E d u c a t i o n a l R e s e a r c h , 3 0 : 5 7 7 - 5 8 8 , A p r i l , . 1 9 3 7 . D u t t o n , W. H . " A t t i t u d e s o f P r o s p e c t i v e T e a c h e r s Toward A r i t h m e t i c , " E l e m e n t a r y S c h o o l J o u r n a l , 5 2 : 8 4 - 9 0 , O c t o b e r , 1 9 5 1 . " M e a s u r i n g A t t i t u d e s Toward A r i t h m e t i c , " E l e m e n t a r y S c h o o l J o u r n a l , 5 5 : 2 4 - 3 1 , S e p t e m b e r , 1954 . " A t t i t u d e o f J u n i o r H i g h S c h o o l P u p i l s Toward A r i t h m e t i c , " S c h o o l R e v i e w , 6 4 : 1 8 - 2 2 , J a n u a r y , 1 9 5 6 . E d w a r d s , A . L . " A T e c h n i q u e f o r I n c r e a s i n g t h e R e p r o d u c i b i l i t y o f C u m u l a t i v e A t t i t u d e S c a l e s , " J o u r n a l o f A p p l i e d P s y c h o l o g y , 4 0 : 2 6 3 - 2 6 5 , A u g u s t , 1956 . T e c h n i q u e s o f A t t i t u d e S c a l e C o n s t r u c t i o n . New Y o r k : /-I _' . * / - l f . * K T~ —t A p p l e t o n - C e n t u r y - C r o f t s , 1957 . and F . P . K i l p a t r i c k . " S c a l e A n a l y s i s and t h e M e a s u r e o f S o c i a l A t t i t u d e s , " P s y c h o m e t r i k a , 1 3 : 9 9 - 1 1 4 , J u n e , 1 9 4 8 . and F . P . K i l p a t r i c k . " A T e c h n i q u e f o r t h e C o n s t r u c t i o n o f A t t i t u d e S c a l e s , " J o u r n a l o f A p p l i e d P s y c h o l o g y , 3 2 : 3 7 4 - 3 8 4 , A u g u s t , 1948 . F a u s t , C . E . " A S t u d y o f t h e R e l a t i o n s h i p Between A t t i t u d e s and A c h i e v e m e n t i n S e l e c t e d E l e m e n t a r y S c h o o l S u b j e c t s , " D i s s e r t a t i o n  A b s t r a c t s , 2 3 : 2 7 5 2 - 2 7 5 3 , F e b r u a r y , 1 9 6 3 . F e d o n , P . J . "The R o l e o f A t t i t u d e i n L e a r n i n g A r i t h m e t i c , " The A r i t h m e t i c T e a c h e r , 5 : 3 0 4 - 3 1 0 , D e c e m b e r , 1 9 5 8 . F e r g u s o n , G . A . "The F a c t o r i a l I n t e r p r e t a t i o n o f T e s t D i f f i c u l t y , " P s y c h o m e t r i k a , 6 : 3 2 3 - 3 2 9 , December , 1 9 4 1 . S t a t i s t i c a l A n a l y s i s i n P s y c h o l o g y and E d u c a t i o n . 2d e d . New Y o r k : M c G r a w - H i l l Book C o . , 1966 F e r g u s o n , L . W. . "The R e q u i r e m e n t s o f an A d e q u a t e A t t i t u d e S c a l e , " P s y c h o l o g i c a l B u l l e t i n , 3 6 : 6 6 5 - 6 7 3 , O c t o b e r , 1939 . " A S t u d y o f t h e L i k e r t T e c h n i q u e o f A t t i t u d e S c a l e C o n s t r u c t i o n , " J o u r n a l o f S o c i a l P s y c h o l o g y , 1 3 : 5 1 - 5 7 F e b r u a r y , 1 9 4 1 . F e y , J . " C l a s s r o o m T e a c h i n g o f M a t h e m a t i c s , " R e v i e w o f E d u c a t i o n a l  R e s e a r c h , 3 9 : 5 3 5 - 5 5 1 , O c t o b e r , 1 9 6 9 . 82 F i t z g e r a l d , W. M . " A S t u d y o f Some o f t h e F a c t o r s R e l a t e d t o t h e L e a r n i n g o f M a t h e m a t i c s i n G r a d e s F i v e , S e v e n , and N i n e . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f M i c h i g a n , 1 9 6 2 . F l o r a , B . V . , J r . . " D i a g n o s i n g S e l e c t e d B e h a v i o r C h a r a c t e r i s t i c s o f T e a c h e r s o f S e c o n d a r y S c h o o l M a t h e m a t i c s , " J o u r n a l f o r R e s e a r c h i n  M a t h e m a t i c s E d u c a t i o n , 3 : 7 - 2 0 , J a n u a r y , 1 9 7 2 . F o u r n e t , F . G . " A S t u d y o f V a r i o u s F a c t o r s R e l a t e d to S u c c e s s i n C o l l e g e G e n e r a l M a t h e m a t i c s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , L o u i s i a n a S t a t e U n i v e r s i t y , 1 9 6 3 . G a g n e , R . M . , e t a l . " F a c t o r s i n A c q u i r i n g K n o w l e d g e o f a M a t h e m a t i c a l T a s k , " P s y c h o l o g i c a l M o n o g r a p h s , 7 6 : 1 - 2 1 , N o . 7 , 1962 (Whole N o . 5 2 6 ) . and N . E . P a r a d i s e . " A b i l i t i e s and L e a r n i n g S e t s i n K n o w l e d g e A c q u i s i t i o n , " P s y c h o l o g i c a l M o n o g r a p h s , 7 5 : 1 - 2 3 , N o . 1 4 , 1961 (Whole N o . 5 1 8 ) . G a r n e r , M . V . " A S t u d y o f t h e E d u c a t i o n a l B a c k g r o u n d s and A t t i t u d e s o f T e a c h e r s Toward A l g e b r a as R e l a t e d t o t h e A t t i t u d e s and A c h i e v e m e n t s o f T h e i r A n g l o - A m e r i c a n and L a t i n - A m e r i c a n P u p i l s i n F i r s t - Y e a r A l g e b r a C l a s s e s o f T e x a s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , N o r t h T e x a s S t a t e U n i v e r s i t y , 1 9 6 3 . D i s s e r t a t i o n  A b s t r a c t s , 2 4 : 1 8 9 , J u l y , 1 9 6 3 . G a r r e t t , H . E . S t a t i s t i c s i n P s y c h o l o g y and E d u c a t i o n . New Y o r k : D a v i d McKay C o . , 1958 . G l a s s , G . V . " A R a n k i n g V a r i a b l e A n a l o g u e o f B i s e r i a l C o r r e l a t i o n : I m p l i c a t i o n s f o r S h o r t - C u t I t e m A n a l y s i s , " J o u r n a l o f E d u c a t i o n a l  M e a s u r e m e n t , 2 : 9 1 - 9 5 , J u n e , B 6 5 . and P . A . T a y l o r . " F a c t o r A n a l y t i c M e t h o d o l o g y , " R e v i e w o f E d u c a t i o n a l R e s e a r c h , 3 6 : 5 6 6 - 5 8 7 , D e c e m b e r , 1966 . G l e n n o n , V . J . " A S t u d y o f t h e G r o w t h and M a s t e r y o f C e r t a i n B a s i c M a t h e m a t i c a l U n d e r s t a n d i n g s o n Seven E d u c a t i o n a l L e v e l s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , H a r v a r d U n i v e r s i t y , 1948 . . " A S t u d y i n Needed R e d i r e c t i o n i n t h e P r e p a r a t i o n o f T e a c h e r s o f A r i t h m e t i c , " M a t h e m a t i c s T e a c h e r , 4 2 : 3 8 9 - 3 9 6 , D e c e m b e r , 1 9 4 9 . 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G . " A S t u d y o f t h e E x t e n t o f A s s o c i a t i o n o f C e r t a i n P r o f e s s i o n a l and P e r s o n a l D a t a w i t h J u d g e d E f f e c t i v e n e s s o f T e a c h e r B e h a v i o r , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 2 0 : 6 7 - 7 7 , S e p t e m b e r , 1 9 5 1 . 88 S c h a a f , W. L . " A r i t h m e t i c f o r A r i t h m e t i c T e a c h e r s , " S c h o o l S c i e n c e and M a t h e m a t i c s , 5 3 : 5 3 7 - 5 4 3 , O c t o b e r , 1 9 5 3 . S c h u l t z , E . W. "The I n f l u e n c e o f T e a c h e r B e h a v i o r and Dyad C o m p a t i b i l i t y on C l i n i c a l G a i n s i n A r i t h m e t i c T u t o r i n g , " J o u r n a l f o r R e s e a r c h  i n M a t h e m a t i c s E d u c a t i o n , 3 : 3 3 - 4 1 , J a n u a r y , 1 9 7 2 . S c h u n e r t , J . R . "The A s s o c i a t i o n o f M a t h e m a t i c s A c h i e v e m e n t w i t h C e r t a i n F a c t o r s R e s i d e n t i n t h e T e a c h e r , i n t h e T e a c h i n g , i n t h e P u p i l , and i n t h e S c h o o l . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f M i n n e s o t a , 1 9 5 1 . S e a s h o r e , R . H . , and K . H e v n e r . " A T i m e - S a v i n g D e v i c e f o r t h e C o n s t r u c t i o n o f A t t i t u d e S c a l e s , " J o u r n a l o f S o c i a l P s y c h o l o g y , 4 : 3 6 6 - 3 7 2 , A u g u s t , 1 9 3 3 . , S h i m , C h u n g - P h i n g . " A S t u d y o f t h e C u m u l a t i v e E f f e c t o f F o u r T e a c h e r C h a r a c t e r i s t i c s on t h e A c h i e v e m e n t o f E l e m e n t a r y S c h o o l P u p i l s , " The J o u r n a l o f E d u c a t i o n a l R e s e a r c h , 5 9 : 3 3 - 3 4 , S e p t e m b e r , 1 9 6 5 . Sh y o c k , A . J . " A S t u d y o f M a t h e m a t i c s B a c k g r o u n d C o u r s e s f o r E l e m e n t a r y T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , S t a t e U n i v e r s i t y o f I o w a , 1962 . S i e g e l , S . N o n p a r a m e t r i c S t a t i s t i c s f o r t h e B e h a v i o r a l S c i e n c e s . New Y o r k : M c G r a w - H i l l Book C o . , 1956 . S m a i l , R . W. " R e l a t i o n s h i p s Between P u p i l M e a n - G a i n i n A r i t h m e t i c and C e r t a i n A t t r i b u t e s o f T e a c h e r s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f S o u t h D a k o t a , 1 9 5 9 . S m i t h , R . W. "The A c h i e v e m e n t o f E i g h t h G r a d e S t u d e n t s i n A r i t h m e t i c w i t h R e s p e c t t o S e l e c t e d P a t t e r n s o f T e a c h e r P r e p a r a t i o n . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f O k l a h o m a , 1964 . S n a d e r , D . " M a t h e m a t i c a l B a c k g r o u n d f o r T e a c h e r s o f A r i t h m e t i c , " The  A r i t h m e t i c T e a c h e r , 3 : 5 9 - 6 5 , M a r c h , 1956 . S n i d e r , H . L . " R e l a t i o n s h i p s Between F a c t o r s o f H i g h S c h o o l B a c k g r o u n d and A c h i e v e m e n t i n C e r t a i n S u b j e c t F i e l d s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f N e b r a s k a , 1949 . S o e t e b e r , W. H . " M a j o r - M i n o r T e a c h i n g A s s i g n m e n t s and R e l a t e d P u p i l A c h i e v e m e n t . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , C o l o r a d o S t a t e C o l l e g e , 1 9 6 9 . D i s s e r t a t i o n A b s t r a c t s , 3 0 A : 4 2 0 5 , A p r i l , 1970 . S o p e r , E . F . " A S t u d y o f t h e R e l a t i o n s h i p Between C e r t a i n T e a c h e r -S c h o o l C h a r a c t e r i s t i c s and A c a d e m i c P r o g r e s s , As M e a s u r e d by S e l e c t e d S t a n d a r d i z e d T e s t s , o f E l e m e n t a r y P u p i l s i n G r a d e s F o u r , F i v e , and S i x o f New Y o r k S t a t e P u b l i c S c h o o l s i n C i t i e s Under 1 0 , 0 0 0 P o p u l a t i o n . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , S y r a c u s e U n i v e r s i t y , 1956 . 89 S p a r k s , J . N . " A C o m p a r i s o n o f Iowa H i g h S c h o o l s R a n k i n g H i g h and Low i n M a t h e m a t i c a l A c h i e v e m e n t . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f I o w a , I 9 6 0 . . S t e i n b r o o k , R . W. " S t u d y o f Some D i f f e r e n c e s i n B a c k g r o u n d , A t t i t u d e , E x p e r i e n c e , and P r o f e s s i o n a l P r e p a r a t i o n o f S e l e c t e d E l e m e n t a r y T e a c h e r s w i t h C o n t r a s t i n g L o c a l S u c c e s s R e c o r d s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , I n d i a n a U n i v e r s i t y , 1955 . D i s s e r t a t i o n  A b s t r a c t s , 1 5 : 1 0 1 3 , D e c e m b e r , 1 9 5 5 . S t o n e k i n g , L . W. " F a c t o r s C o n t r i b u t i n g t o U n d e r s t a n d i n g o f S e l e c t e d B a s i c A r i t h m e t i c a l P r i n c i p l e s and G e n e r a l i z a t i o n s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , I n d i a n a U n i v e r s i t y , 1960 . and R . C . W e l c h . " T e a c h e r s ' and S t u d e n t s ' U n d e r s t a n d i n g o f A r i t h m e t i c , " I n d i a n a U n i v e r s i t y S c h o o l o f E d u c a t i o n B u l l e t i n , 3 7 : 1 - 5 6 , S e p t e m b e r , 1 9 6 1 . T a y l o r , H . R . "The R e l a t i o n s h i p o f E s t i m a t e d T e a c h i n g A b i l i t y to P u p i l A c h i e v e m e n t s i n R e a d i n g and A r i t h m e t i c . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , S t a n f o r d U n i v e r s i t y , 1928 . T a y l o r , T . W. " A S t u d y t o D e t e r m i n e t h e R e l a t i o n s h i p B e t w e e n G r o w t h i n I n t e r e s t and A c h i e v e m e n t o f H i g h S c h o o l S c i e n c e S t u d e n t s and S c i e n c e T e a c h e r A t t i t u d e s , P r e p a r a t i o n , and E x p e r i e n c e . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , N o r t h Texas S t a t e C o l l e g e , 1957 . D i s s e r t a t i o n  A b s t r a c t s , 1 7 : 2 9 4 3 - 2 9 4 4 , J u n e , 1957 . T h u r s t o n e , L . L . P s y c h o m e t r i c M o n o g r a p h N o . 1 . C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1938 . . The Measurement o f V a l u e s . C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1 9 5 9 . T r a v e r s , R . M . W. The Measurement o f S t u d e n t A d j u s t m e n t and A c h i e v e m e n t . Ann A r b o r : U n i v e r s i t y o f M i c h i g a n P r e s s , 1949 . U . S . D e p a r t m e n t o f Commerce. A p p l i e d M u l t i p l e L i n e a r R e g r e s s i o n , by R. A . B o t t e n b e r g and J . H . W a r d . W a s h i n g t o n : C l e a r i n g h o u s e f o r F e d e r a l S c i e n t i f i c and T e c h n i c a l I n f o r m a t i o n , T e c h n i c a l D o c u m e n t a r y R e p o r t P R L - T D R - 6 3 - 6 , M a r c h , 1 9 6 3 . U . S . D e p a r t m e n t o f H e a l t h , E d u c a t i o n and W e l f a r e . R e s e a r c h P r d b l e m s  i n M a t h e m a t i c s E d u c a t i o n . C o o p e r a t i v e R e s e a r c h , M o n o g r a p h N o . 3 , 0 E - 1 2 0 0 8 . W a s h i n g t o n : Government P r i n t i n g O f f i c e , 1 9 6 0 . U . S. S c i e n t i f i c and P u b l i c P o l i c y , V o l . I V . Manpower f o r R e s e a r c h . W a s h i n g t o n : Government P r i n t i n g O f f i c e , 1947 . V a k i l , R. " C l a s s r o o m C l i m a t e , P u p i l A c h i e v e m e n t and A t t i t u d e . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , C a s e W e s t e r n R e s e r v e U n i v e r s i t y , 1 9 7 0 . D i s s e r t a t i o n A b s t r a c t s , 3 2 A : 1 3 5 1 , S e p t e m b e r , 1 9 7 1 . W a t t s , G . D . " A C o r r e l a t i o n A n a l y s i s B e t w e e n L e v e l o f A c h i e v e m e n t and C e r t a i n T e a c h e r C h a r a c t e r i s t i c s i n S e l e c t e d S c h o o l S y s t e m s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , O h i o U n i v e r s i t y , 1964 . D i s s e r t a t i o n A b s t r a c t s , 2 5 : 2 3 2 9 - 2 3 3 0 , O c t o b e r , 1964 . W e a v e r , J . F . " T e a c h e r E d u c a t i o n i n A r i t h m e t i c , " R e v i e w o f E d u c a t i o n a l  R e s e a r c h , 2 1 : 3 1 7 - 3 2 0 , O c t o b e r , 1 9 5 1 . A C r u c i a l P r o b l e m i n t h e P r e p a r a t i o n o f E l e m e n t a r y S c h o o l T e a c h e r s , " E l e m e n t a r y S c h o o l J o u r n a l . 5 6 : 2 5 5 - 2 6 1 , F e b r u a r y , 1956 and G . E . G i b b . " M a t h e m a t i c s i n t h e E l e m e n t a r y S c h o o l . " R e v i e w o f E d u c a t i o n a l R e s e a r c h , 3 4 : 2 7 3 - 2 8 5 , J u n e , 1964 . W e r t , J . E . , C . 0 . N e i d t , and J . S . A l m a n n . S t a t i s t i c a l M e t h o d s i n  E d u c a t i o n a l and P s y c h o l o g i c a l R e s e a r c h . New Y o r k : A p p l e t o n -C e n t u r y - C r o f t s , 1954 . W e s t o n , L . D . " A n E x p l o r a t i o n o f t h e I n t e r r e l a t i o n s h i p s Among C h i l d r e n ' A r i t h m e t i c A c h i e v e m e n t , T h e i r S t y l e s o f L e a r n i n g , T h e i r R e s p o n s i b i l i f o r I n t e l l e c t u a l A c a d e m i c A c h i e v e m e n t , and T h e i r P a r e n t s ' A t t i t u d e s , D i s s e r t a t i o n A b s t r a c t s , 3 0 : 1 0 8 7 - 1 0 8 8 , S e p t e m b e r , 1 9 6 9 . W h i t e , M . J . " A S t u d y o f ' t h e Change o f A c h i e v e m e n t and A t t i t u d e Toward A r i t h m e t i c by P r o s p e c t i v e E l e m e n t a r y S c h o o l T e a c h e r s U n d e r C o n d i t i o n s o f T e l e v i s i o n . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , Wayne S t a t e U n i v e r s i t y , 1 9 6 3 . D i s s e r t a t i o n A b s t r a c t s , 2 5 : 2 3 0 2 - 2 3 0 3 , O c t o b e r , 1964 . W i c k , M . E . " A S t u d y o f t h e F a c t o r s A s s o c i a t e d w i t h A c h i e v e m e n t i n F i r s t - Y e a r C o l l e g e M a t h e m a t i c s . " U n p u b l i s h e d D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f M i n n e s o t a , 1 9 6 3 . Woody, C . W. " A G e n e r a l E d u c a t o r L o o k s a t A r i t h m e t i c R e a d i n e s s , " M a t h e m a t i c s T e a c h e r , 3 0 : 3 1 4 - 3 2 1 , N o v e m b e r , 1937 . W r e n , F . L . The P r o f e s s i o n a l P r e p a r a t i o n o f T e a c h e r s o f A r i t h m e t i c . S u p p l e m e n t a r y E d u c a t i o n a l M o n o g r a p h s , N o . 6 6 . C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1948 . W r i g h t s t o n e , J . W. " I n f l u e n c e o f R e s e a r c h o n I n s t r u c t i o n i n A r i t h m e t i c , ' M a t h e m a t i c s T e a c h e r . 4 5 : 1 8 7 - 1 9 2 , M a r c h , 1 9 5 2 . Appendix A UNDERSTANDING INVENTORY Teacher Number Following i s a set of questions which you are to answer. Place your answer on the blank to the l e f t of each question. You may guess i f you wish. Answer as many as you can i n the 45 minute time l i m i t . You may begin. 3 1. The numeral can also be thought of as A) 3 x 4 B) 4 x 3 C) 3 * 4 D) 4 * 3 E) none of these 2. If 0 i s a binary operation defined i n S and i f for a l l a, b i n S, a 8 b = b 8 a, then 8 i s said to obey the: A) associative law B) commutative law C) d i s t r i b u t i v e law D) i d e n t i t y property E) none of these a b 3. If a, b, i , n are whole numbers d i f f e r e n t from zero, then V — = m n a + b A) B) C) D) m a + b n a + b m + n an + bm mn E) none of these 91 92 X 4 . What i s t h e b e s t r e a s o n f o r p l a c i n g t h e d e c i m a l . 5 2 p o i n t a f t e r t h e 3 i n s t e a d o f some o t h e r p l a c e ? 1 .42 1 > 3 2 : A) t o keep t h e d e c i m a l p o i n t s i n a s t r a i g h t 3 . 2 6 l i n e B) s i x d e c i m a l p l a c e s i n t h e p r o b l e m d i v i d e d b y 3 , t h e number o f a d d e n d s , e q u a l s 2 C) h u n d r e d t h s added t o h u n d r e d t h s e q u a l s h u n d r e d t h s P) b e c a u s e t h e a n s w e r must be l a r g e r t h a n any addends 5 . G i v e n a f r a c t i o n a l n u m b e r , i f t h e d e n o m i n a t o r o f t h e f r a c t i o n a l number i s d e c r e a s e d and t h e n u m e r a t o r i s k e p t t h e same, t h e n t h e new number i s : A) l a r g e r t h a n t h e o l d number B) s m a l l e r t h a n t h e o l d number C) a p p r o a c h i n g one D) t h e same as t h e o l d number E) . unknown i n r e l a t i o n s h i p t o t h e o l d number f r o m t h e i n f o r m a t i o n g i v e n 6 . L o o k a t b v a w h e r e " a " and " b " a r e b o t h w h o l e numbers g r e a t e r t h a n o n e . How does t h e answer compare w i t h " b " ? A) t h e answer i s g r e a t e r t h a n b B) t h e a n s w e r i s s m a l l e r t h a n b C) c a n ' t t e l l u n t i l we s e e b o t h w h o l e numbers D) c a n ' t t e l l u n t i l we s e e b E) c a n ' t t e l l u n t i l t h e d i v i s i o n i s done 7 . The f o l l o w i n g s t a t e m e n t shows a p r o p e r t y o f a r i t h m e t i c 4 x (5 + 6) = (4 x 5) + (4 x 6) W h i c h o f t h e f o l l o w i n g shows t h e same p r o p e r t y ? A) 2 x (3 x 4) = 4 x (2 x 3) B) (6 x 5) x 7 = 6 x (5 x 7) C) (5 x 6) + (3 x 4) = 8 x 10 D) 8 x 6 = (8 x 4) + (8 x 2) E) none o f t h e s e 8 . L o o k a t t h e p r o b l e m 439 x 4 5 0 . How w o u l d t h e a n s w e r b e c h a n g e d i f two z e r o s w e r e p l a c e d t o t h e r i g h t o f 439 and t h e z e r o removed f r o m 450? The a n s w e r w o u l d b e : A) t h e same as t h e o l d a n s w e r B) o n e - t e n t h as l a r g e as t h e o l d a n s w e r C) t e n t i m e s l a r g e r t h a n t h e o l d answer D) o n e - h u n d r e d t h as l a r g e as t h e o l d a n s w e r E) none o f t h e s e 93 9 . P a r t o f t h i s a d d i t i o n p r o b l e m was a c c i d e n t a l l y e r a s e d f r o m t h e c h a l k b o a r d . E a c h " X " shows w h e r e a d i g i t u s e d t o b e . These d i g i t s w e r e n o t n e c e s s a r i l y t h e same. What d i g i t b e l o n g s on t h e q u e s t i o n mark? A) 9 XXXX1 B) 7 + ? C) 5 XXXXXX D) 2 E) none o f t h e s e _10. H e r e i s a s e q u e n c e whose f i r s t t e r m i s 1 2 . From any t e r m i n t h e s e q u e n c e y o u c a n g e t t h e n e x t t e r m by a d d i n g 1 2 : 1 2 , 2 4 , 3 6 , 4 8 , 60 . . . . . What i s t h e 1 0 0 t h t e r m o f t h i s s e q u e n c e ? A) 6000 B) 1212 C) 1200 D) 112 E) none o f t h e s e _ 1 1 . W h i c h o f t h e f o l l o w i n g w o u l d g i v e t h e c o r r e c t a n s w e r t o 2 . 1 x 21? A) t h e sum o f 2 x 21 and 1 x 21 B) t h e sum o f 20 x 21 and . 1 x 21 C) t h e sum o f 10 x 2 . 1 and 20 x 2 . 1 D) t h e sum o f 1 x 2 . 1 and 20 x 2 . 1 E) none o f t h e s e 749 _ 1 2 . I n t h e e x a m p l e v o u m u l t i p l y by t h e 6 , t h e n by t h e 3 . How do t h e two r e s u l t s ( p a r t i a l p r o d u c t s ) compare? A) t h e s e c o n d r e p r e s e n t s a number o n e - h a l f as l a r g e as t h e f i r s t B) t h e s e c o n d r e p r e s e n t s a number t w i c e as l a r g e as t h e f i r s t C) t h e s e c o n d r e p r e s e n t s a number f i v e t i m e s as l a r g e as t h e f i r s t D) t h e s e c o n d r e p r e s e n t s a number t e n t i m e s as l a r g e as t h e f i r s t E) none o f t h e s e _13. G i v e n a f r a c t i o n a l n u m b e r , i f t h e n u m e r a t o r o f t h e f r a c t i o n a l number i s d e c r e a s e d and t h e d e n o m i n a t o r i s k e p t t h e same, t h e n t h e new number i s : A ) l a r g e r t h a n t h e o l d number B) s m a l l e r t h a n t h e o l d number C) a p p r o a c h i n g one D) t h e same as t h e o l d number E) unknown i n r e l a t i o n s h i p t o t h e o l d number f r o m t h e i n f o r m a t i o n g i v e n 94 _14. Given the example 368 x 24, then change 368 to,3680 and 24 to 2.4. The new answer would be: A) the same as the answer for the original example B) one-tenth as large as the answer for the original example C) ten times larger than the answer for the original example D) one hundred times larger than the answer for the original example E) none of these 15. For two sets M and N, the set of elements that are in both M and N is called: A) the union o f M and N B) the intersection of M and N C) the complement of M with respect to N D) the cross product of M and N E) none of these 16. Which of the following w i l l give the same answer as 13 x 23? A) (10 x 20) + (3 x 3) B) (10 x 20) + (10 x 3) C) (13 x 20) + ( 3 x 3 ) D) (13 x 20) + (13 x 3) E) none of these 17. The number, is irrational. So also i s : A) /F x /5 B) /F + /5 C) /5 - /5 D) / J v /h~ E) none of these 18. If * is an operation and i t is replaced by +, then -, then x, then T , then (u * v) * w = u * (v * w) w i l l be true exactly: A) zero times when a l l four replacements are tried B) one time when a l l four replacements are tried C) two times when a l l four replacements are tried D) three times when a l l four replacements are tried E) none of these 19. If a binary operation "*" is defined on any pair of real numbers "c" and "d" such that c * d = 2c + d, then 3 * 4 is equal to: A) 11 B) 10 C) 12 D) 14 E) none of these 95 20. If " r " i s the m u l t i p l i c a t i v e inverse of n, then A) n + r = n B) n x r = n C) n + r = 0 D) n x r = 1 E) none of these 21. Which of the following numbers i s smaller than 2.047? A) 2.111 B) 2.048 C) 2.050 D) 2.1 E) none of these 22. When working 12 x .5 we get 6 as an answer. The best reason for the answer being smaller than 12 i s A) 12 i s larger than .5 B) .5 i s smaller than 12 C) we are fin d i n g how many halves i n 12 D) we are f i n d i n g h a l f of 12 E) we are multiplying by a decimal 23. Peter i s asked to bring 13 bushels of potatoes from the barn to the house. He can carry 3 bushels i n each t r i p . How many t r i p s w i l l Peter make? A) 4 t r i p s B) 4y t r i p s C) 5 t r i p s D) can't be determined from the information given E) none of these 24. When we compare .60 and ^ we f i n d that .60 i s A) larger than ^ B) smaller than C) the same s i z e as 4- i 9 i 5 D) unknown i n s i z e to 25. If every element of a set M i s an element of a set N, then M i s : A) a subset of N B) a proper subset of N C) equivalent to N D) an element of N E) none of these 96 _26. When b o t h t h e n u m e r a t o r and d e n o m i n a t o r o f a f r a c t i o n a r e d i v i d e d b y t h e same n u m b e r , t h e n t h e v a l u e o f t h e new f r a c t i o n i s : A) g r e a t e r t h a n t h e v a l u e o f t h e o l d f r a c t i o n B) l e s s t h a n t h e v a l u e o f t h e o l d f r a c t i o n C) t h e same as t h e v a l u e o f t h e o l d f r a c t i o n D) unknown u n t i l t h e number u s e d t o d i v i d e i s known E) unknown u n t i l t h e f r a c t i o n i s known 2 7 . I n a d i v i s i o n p r o b l e m t h e q u o t i e n t i s t h e same a s t h e d i v i d e n d w h e n : A) t h e d i v i s o r i s l e s s t h a n one B) t h e d i v i s o r i s l e s s t h a n one b u t g r e a t e r t h a n z e r o C) t h e d i v i s o r i s a f a c t o r o f t h e d i v i d e n d D) t h e d i v i s o r i s g r e a t e r t h a n one E) none o f t h e s e 2 8 . T w e n t y - t h r e e a c r e s o f a 55 a c r e f a r m i s u s e d t o r a i s e c o r n . The p a r t o f t h e f a r m u s e d t o . r a i s e c o r n i s s l i g h t l y more t h a n : 3 5 2 3 2 5 1 3 A) B) C) D) E) none o f t h e s e 2 9 . What i s t h e b e s t r e a s o n f o r p l a c i n g t h e d e c i m a l p o i n t b e f o r e t h e 6 i n s t e a d o f some p l a c e e l s e ? 1 . 2 5 A) c o u n t i n g a l l d e c i m a l p l a c e s y o u x . 5 g e t t h r e e . 6 2 5 B) t h e r u l e f o r m u l t i p l y i n g d e c i m a l s m t e l l s us t o p u t i t t h e r e C) t e n t h s t i m e s h u n d r e d t h s e q u a l s t h o u s a n d t h s D) s i n c e . 5 e q u a l s . 5 0 0 , t h i s k e e p s t h e d e c i m a l p o i n t s i n a s t r a i g h t l i n e 3 0 . L o o k a t b * a w h e r e " a " and " b " a r e b o t h p r o p e r f r a c t i o n s . How does t h e a n s w e r compare w i t h " b " ? A) t h e a n s w e r i s l a r g e r t h a n b B) t h e a n s w e r i s s m a l l e r t h a n b C) c a n ; t t e l l u n t i l I s e e t h e numbers D) t h e answer i s t h e same as b E) c a n ' t t e l l u n t i l i t i s w o r k e d 97 3 1 . The b e s t way t o e x p l a i n why m o v i n g t h e d e c i m a l p o i n t does n o t change t h e a n s w e r i n t h e e x a m p l e . 5 ) 3 5 . 5 5 o r 5 . ) 3 5 5 . 5 i s A) when d i v i d i n g b y a d e c i m a l number y o u move t h e d e c i m a l i n t h e d i v i s o r and t h e d i v i d e n d t h e same number o f p l a c e s B) t h e r e i s no way t o d i v i d e by a d e c i m a l w i t h o u t m o v i n g t h e d e c i m a l p o i n t C) t h e r u l e f o r d i v i d i n g d e c i m a l numbers t e l l s us t o move t h e d e c i m a l p o i n t t h e same number o f p l a c e s i n t h e d i v i s o r a n d t h e d i v i d e n d D) m o v i n g t h e d e c i m a l p o i n t i s t h e same as m u l t i p l y i n g t h e n u m e r a t o r and d e n o m i n a t o r o f a f r a c t i o n by t h e same number E) i t i s e a s i e r t o d i v i d e by a w h o l e number t h a n a d e c i m a l number 3 2 . L o o k a t u x v w h e r e " u " and " v " a r e b o t h p r o p e r f r a c t i o n s . How does t h e answer compare w i t h " u " ? A) t h e a n s w e r i s l a r g e r t h a n u B) t h e answer i s s m a l l e r t h a n u C) t h e a n s w e r i s e q u a l t o u D) c a n ' t t e l l u n t i l we s e e t h e f r a c t i o n s E) c a n ' t t e l l u n t i l we do t h e a r i t h m e t i c 3 3 . I f a r e l a t i o n " R " d e f i n e d i n a s e t " S " has t h e p r o p e r t y t h a t f o r a l l a , b , c , and S , i f a R b and b R c , t h e n a R c , i t i s s a i d t o b e : A) c o m m u t a t i v e B) r e f l e x i v e C) a s s o c i a t i v e D) t r a n s i t i v e E) none o f t h e s e 3 4 . P a r t o f t h i s s u b t r a c t i o n p r o b l e m was a c c i d e n t a l l y e r a s e d f r o m t h e c h a l k b o a r d . E a c h " X " shows w h e r e a d i g i t u s e d t o b e . What d i g i t b e l o n g s o n t h e b l a n k ? A) 0 B) 1 C) 6 D) 7 E) none o f t h e s e X X _ 6 - X X 7 7 XX9X 98 _ 3 5 . A r a t i o n a l number e x p r e s s e d as a d e c i m a l f r a c t i o n w i l l n e v e r b e : A) i n f i n i t e and h a v e a r e p e a t i n g d e c i m a l e x p a n s i o n B) i n f i n i t e and h a v e a n o n r e p e a t i n g d e c i m a l e x p a n s i o n C) a t e r m i n a t i n g d e c i m a l e x p a n s i o n D) none o f t h e s e _36 . C o n s i d e r i n g e a c h o f t h e f o l l o w i n g as a s e p a r a t e p r o b l e m , t h e p r o b l e m i n w h i c h t h e l o w e s t common d e n o m i n a t o r w i l l b e one o f t h e d e n o m i n a t o r s i s : A ) 5 + 10 15 c) I + -J- + I U 3 + 12 + 6 °> E) none o f t h e s e _37. The v a l u e o f t h e 4 i n r e l a t i o n t o t h e 2 i n t h e number 4032 i s : A) 1000 t i m e s as l a r g e B) 2 t i m e s a s l a r g e C) 500 t i m e s as l a r g e D) 2000 t i m e s as l a r g e E) n o n e o f t h e s e _38. I f 9 i s a b i n a r y o p e r a t i o n d e f i n e d i n S and i f f o r a l l a , b , c i n S , ( a 6 b ) 8 c = a 8 (b 8 c ) , t h e n 8 i s s a i d t o obey t h e : A) a s s o c i a t i v e l a w B) c o m m u t a t i v e l a w C) d i s t r i b u t i v e l a w D) i d e n t i t y p r o p e r t y E) n o n e o f t h e s e 3 9 . When a n a t u r a l number i s d i v i d e d by a p r o p e r f r a c t i o n how does t h e answer compare w i t h t h e n a t u r a l number? A) t h e a n s w e r . i s l a r g e r t h a n t h e n a t u r a l number B) t h e a n s w e r i s s m a l l e r t h a n t h e n a t u r a l number C) t h e answer i s e q u a l t o t h e n a t u r a l number D) c a n ' t t e l l u n t i l we s e e t h e n a t u r a l number and t h e p r o p e r f r a c t i o n E) c a n ' t t e l l u n t i l we do t h e a r i t h m e t i c 99 8 _40. -JJ * 1 e c l u a l s w h i c h o f t h e f o l l o w i n g ? 2 •6. 3 3 4 _ 7 11 A) B ) C) D) E) none o f t h e s e _ 4 1 . I f r + s = t , t h e n w h i c h o f t h e f o l l o w i n g i s a l s o c o r r e c t ? A) 4 r + 4s = 8 t B) 4 r + 4s = 4 t C) 4 r + s = 5 t D) 4 r + 4s = t x t x t x t E) none o f t h e s e . _42 . When a n a t u r a l number i s m u l t i p l i e d by a p r o p e r f r a c t i o n , how does t h e a n s w e r compare w i t h t h e n a t u r a l number? A) t h e a n s w e r i s g r e a t e r t h a n t h e n a t u r a l number B) t h e a n s w e r i s s m a l l e r t h a n t h e n a t u r a l number C) c a n ' t t e l l u n t i l we s e e t h e numbers D) t h e answer i s t h e same as t h e n a t u r a l number E) c a n ' t t e l l u n t i l i t i s w o r k e d _43. I f " p " i s t h e a d d i t i v e i n v e r s e o f " q " , t h e n : A) p + q = 1 B) p + q = 0 C) p x q = 1 D) p + 0 = q E) n o n e o f t h e s e a b 44 . I f a , b , m, n a r e w h o l e numbers d i f f e r e n t t h a n z e r o , t h e n — * — ' ' ' m n A) B) C) D) ab mn mn ab bm an an bm E) n o n e o f t h e s e 100 _45. W h i c h o f t h e f o l l o w i n g a l s o s t a n d s f o r 4275? A) 42 h u n d r e d s + 75 t e n s B) 427 t e n s + 5 ones C) 4 t h o u s a n d s + 27 h u n d r e d s + 5 t e n s D) 4 t h o u s a n d s + 2 h u n d r e d s + 75 t e n s E) none o f t h e s e _46. G i v e n t h e e x a m p l e 6 . 5 ) 8 4 . 5 , t h e n change 6 . 5 t o . 65 and 8 4 . 5 t o 8 4 5 . The new a n s w e r w o u l d b e : A) t h e same as t h e answer f o r t h e o r i g i n a l e x a m p l e B) o n e - t e n t h as l a r g e as t h e answer f o r t h e o r i g i n a l e x a m p l e C) t e n t i m e s l a r g e r t h a n t h e answer f o r t h e o r i g i n a l e x a m p l e D) one h u n d r e d t i m e s l a r g e r t h a n t h e answer f o r t h e o r i g i n a l e x a m p l e E) n o n e o f t h e s e 4 7 . 5 . 5 i s e q u a l t o : A) f i v e and 50 t e n t h s B) f i v e and 50 h u n d r e d t h s C) f i v e and 5 h u n d r e d t h s D) f i v e and 50 u n i t s E) none o f t h e s e 4 8 . 6 x 7 x 8 = ( 6 x 7 ) x 8 shows a p r o p e r t y o f a r i t h m e t i c . W h i c h o f t h e f o l l o w i n g shows t h e same p r o p e r t y ? A) 6 x 7 x 8 = 6 x (8 x 7) B) 6 x 7 x 8 = 8 x ( 6 x 7 ) C) 6 x 7 x 8 = ( 7 x 6 ) x 8 D) 6 x 7 x 8 = 6 x ( 7 x 8 ) E) none o f t h e s e 4 9 . I f 6 i s a b i n a r y o p e r a t i o n i n t h e s y s t e m o f w h o l e numbers W and i f a 0 0 = 0 6 a = a f o r a l l " a " i n W, t h e n t h i s i s an e x a m p l e o f t h e : A) a s s o c i a t i v e l a w B) d i s t r i b u t i v e l a w C) c o m m u t a t i v e l a w D) i d e n t i t y p r o p e r t y E) n o n e o f t h e s e 101 _50. I f a + b = c , and " d " i s any n u m b e r , t h e n w h i c h o f t h e f o l l o w i n g i s a l w a y s t r u e ? A) a + b + d = c - d ,. B) . b + d = c - (a + d) C) d x ( a + b) = d + c D) d x c = (d x a) + (d x b) E) n o n e o f t h e s e 5 1 . I f f + g > h , w h i c h s t a t e m e n t i s ALWAYS t r u e f o r a l l r e a l numbers f , g , and h? A) 3 f + 3g > 3h B) f + | g > | b •:• C) |f + g < h D) g > h . E ) n o n e o f t h e s e 5 2 . I f t h e q u o t i e n t when d i v i d i n g 6 . 7 b y .04 i s t h e same as d i v i d i n g " n " by 4 . 0 , t h e n " n " i s : A) 6700 B) 6 7 0 . 0 C) . 6 7 D) . 067 E) none o f t h e s e 5 3 . C o n s i d e r i n g e a c h o f t h e f o l l o w i n g as s e p a r a t e p r o b l e m s , t h e p r o b l e m i n w h i c h t h e l o w e s t common d e n o m i n a t o r w i l l b e t h e p r o d u c t o f d e n o m i n a t o r s i s : « W 4 B, I + I + I O I 7 _ 3 L ) 6 + 8 + 10 1 . 1 . 1 D ) 3 + 4 + 6 E) none o f t h e s e 54^ I f G and $ a r e b i n a r y o p e r a t i o n s d e f i n e d on a s e t S and i f f o r a l l a , b , c i n S , a 9 (b <j> c ) = ( a 9 b) <|> (a 6 c ) , t h e n we say A) t h e a s s o c i a t i v e l a w h o l d s i n S B) t h e c o m m u t a t i v e l a w h o l d s i n S C) t h e d i s t r i b u t i v e l a w h o l d s i n S D) t h e i d e n t i t y p r o p e r t y h o l d s i n S E) none o f t h e s e 102 W h i c h f r a c t i o n a l number i s b e t w e e n 2 and 3? B , i E) n o n e o f t h e s e " n " i n t h e p r o b l e m . 7 * n ' = . 007 I s : A) 1000 B) . 0 1 C) 100 D) . 0 0 1 E) none o f t h e s e 23 90 G i v e n t h e p r o b l e m x ^ " ^ y how w o u l d t h e a n s w e r be c h a n g e d i f t h e z e r o w e r e removed f r o m 2 3 . 9 0 ? The new answer w o u l d b e : A) t h e same as t h e o l d a n s w e r B) o n e - t e n t h as l a r g e as t h e o l d a n s w e r C) t e n t i m e s l a r g e r t h a n t h e o l d a n s w e r D) l a r g e r t h a n t h e o l d answer b e c a u s e t h e r e w o u l d b e f e w e r d e c i m a l p l a c e s E) unknown u n t i l we do t h e m u l t i p l i c a t i o n I f y o u c a n add e v e r y number i n a s e t o f numbers to i t s e l f o r t o e v e r y o t h e r number i n t h e s e t , and t h e sum i s a number a l s o i n t h a t s e t , t h e n t h a t s e t o f numbers i s c l o s e d u n d e r t h e o p e r a t i o n o f a d d i t i o n . W h i c h o f t h e f o l l o w i n g s e t s i s NOT c l o s e d u n d e r t h e o p e r a t i o n o f a d d i t i o n ? A) 0 , 1 , 2 , 3 , 4 B) 1 , 2 , 3 , 4 , 5 , C) 0 , 2 , 4 , 6 , 8 , D) 1 , 3 , 5 , 7, 9 , . . . . . . . E) none o f t h e s e P a r t o f t h i s m u l t i p l i c a t i o n p r o b l e m was a c c i d e n t a l l y e r a s e d f r o m t h e c h a l k b o a r d . E a c h " X " shows w h e r e a d i g i t u s e d t o b e . T h e s e d i g i t s w e r e n o t n e c e s s a r i l y t h e same. What d i g i t b e l o n g s o n t h e q u e s t i o n mark? A) 0 XXX? B) 2 XX C) 4 XXXX D) 5 XXX5 E) 7 XXXX0 103 6 0 . A n i r r a t i o n a l number e x p r e s s e d as a d e c i m a l f r a c t i o n w i l l a l w a y s b e : A) i n f i n i t e and h a v e a r e p e a t i n g d e c i m a l e x p a n s i o n B) i n f i n i t e and h a v e a n o n r e p e a t i n g d e c i m a l e x p a n s i o n C) a t e r m i n a t i n g d e c i m a l e x p a n s i o n D) none o f t h e s e Appendix B AN INVENTORY TO MEASURE TEACHER'S ATTITUDE TOWARD CONTEMPORARY AS OPPOSED TO TRADITIONAL MATHEMATICS First Version Following is a multiple choice inventory. Complete each statement so that your choice best reflects your beliefs, opinions, and practices. If you do not know the meaning of some word in a statement, use "D" as your choice. Place your answer on the blank to the l e f t of each statement. Thank you for your cooperation. New mathematics is A) a success and here to stay B) an educational fad which w i l l pass on Students A) dislike arithmetic because i t is a dry subject B) like arithmetic because i t is f u l l of new ideas In the teaching of arithmetic A) we should stress that equal values may have different forms, in other words 3 = 2 + 1 = 1 + 1 + 1 . B) i t i s not necessary to stress that equal values may have different forms There is considerable discussion as to whether fractions should be written as ^ or as 2/3. This is an A) unimportant distinction B) important distinction When teaching multiplication by the number one i t A) should be emphasized as a special property B) need not be emphasized 104 1) 2) 3) 4) 5) 105 6) When I t e a c h a r i t h m e t i c I do so b e c a u s e A) I must t e a c h i t as p a r t o f t h e c u r r i c u l u m B) I e n j o y w a t c h i n g s t u d e n t s l e a r n i n g a r i t h m e t i c 7) When l e a r n i n g a r i t h m e t i c i t i s b e s t t o A) d i s c o v e r t h o s e t h i n g s we s h o u l d l e a r n B) b e t o l d t h o s e t h i n g s we s h o u l d l e a r n 8) I n a new m a t h e m a t i c s p r o g r a m m e m o r i z a t i o n o f t h e a r i t h m e t i c f a c t s i s A) n o t as i m p o r t a n t as i n t h e o l d p r o g r a m s B) j u s t as i m p o r t a n t as i n t h e o l d p r o g r a m s 9) When t e a c h i n g a d d i t i o n and s u b t r a c t i o n , i t i s b e s t t o t e a c h them as A) i n v e r s e o p e r a t i o n s B) s e p a r a t e o p e r a t i o n s 10) S t u d e n t s A) d i s l i k e a r i t h m e t i c b e c a u s e o f t h e r e p e t i t i o u s homework B) l i k e a r i t h m e t i c b e c a u s e o f t h e o p p o r t u n i t y t o t h i n k t h i n g s o u t _11) A l l o p e r a t i o n s o f a r i t h m e t i c a r e d e f i n e d o p e r a t i o n s , A) t h e r e f o r e i t i s n o t n e c e s s a r y t o h a v e u n d e r s t a n d i n g s a t t a c h e d t o them B) b u t i t i s s t i l l i m p o r t a n t t o h a v e u n d e r s t a n d i n g s a t t a c h e d t o them _12) T a b l e s s u c h as t h e m u l t i p l i c a t i o n t a b l e s A) a r e n o t i m p o r t a n t i n new m a t h e m a t i c s b e c a u s e t h e y s t r e s s m e m o r i z a t i o n B) a r e i m p o r t a n t i n new m a t h e m a t i c s b e c a u s e t h e y a s s i s t i n u n d e r s t a n d i n g number r e l a t i o n s h i p s _13) C o n s i d e r i n g t h e w o r l d we l i v e i n , s t u d e n t s s h o u l d s p e n d more t i m e A) s t u d y i n g m a t h e m a t i c s B) s t u d y i n g s u c h t h i n g s as a r t and m u s i c 14) N o n - p o s i t i o n a l n u m e r a t i o n s y s t e m s s u c h as t h e Roman S y s t e m a r e A) u n i m p o r t a n t i n a r i t h m e t i c t o d a y B) i m p o r t a n t i n a r i t h m e t i c t o d a y 106. When teaching arithmetic, the differe n c e between number and numeral i s A) unimportant and should not be stressed B) important and should be stressed In new mathematics checking an answer i s A) important because i t helps lead to better understandings B) unimportant because the c h i l d has the understandings before he does the work The a s s o c i a t i v e property of addition i s A) necessary to understand addition of three or more numbers B) unnecessary to understand addition of three or more numbers Tables such as the m u l t i p l i c a t i o n tables A) are not important i n new mathematics B) are important i n new mathematics When I teach arithmetic I A) enjoy the teaching B) d i s l i k e the teaching Most of the arithmetic taught i s A) of p r a c t i c a l use B) designed to b u i l d background for future study i n mathematics The a b i l i t y to calculate i s A) less necessary i n the new mathematics B) j u s t as important as ever Work i n bases d i f f e r e n t from ten A) should be performed with most students B) should not be performed with most students In new mathematics i t i s A) j u s t as important as before that students calculate rapidly B) not as important as before that students c a l c u l a t e rapidly 101 _24) Student understanding of arithmetic is . A) less necessary today because of. the many machines that do our calculating B) more necessary today because of sci e n t i f i c advancements _25) Words such as commutative, associative, and distributive A) are important words in mathematics and by the end of the third grade, most children should know the words B) represent important ideas in mathematics and by the end.of the third grade, most children should understand these ideas—the words are not important 26) High school algebra is A) a prestige course B) a course most students should and can take 27) In arithmetic the teacher should teach the student to A) do the work and understand i t by practicing B) understand what he is doing 28) Set theory A) fundamentals should be studied with arithmetic B) is a separate branch of mathematics and, therefore, has l i t t l e effect on learning arithmetic 29) From my experiences I A) lik e new mathematics best B) lik e old mathematics best 30) Computational shortcuts i n arithmetic are A) not as important as in the past B) just as important as in the past 31) New mathematics is better for A) a l l students B) college bound students only 32) Understandings in arithmetic should be stressed with A) the college capable student only—the slow learner has to memorize anyway B) a l l students 108 _33) When solving story problems there A) is one best way to do the problem and the teacher should stress this way B) are many ways to do any problem and the teacher should accept any logically correct method _34) New mathematics w i l l train the student to A) ask why B) accept what he is told 35) The number line A) i s for use in understanding algebra and need not be used before grade seven B) can be used to help understand addition of whole numbers 36) Arithmetic should be taught as a A) set of rules for the students to follow B) step by step process where one step builds upon the other 37) When teaching division of whole numbers, i t is best to use A) 234 B) 56)13104 56)13104 112 11200 200 190 1904 168 1680 30 224 224 224 224 4 234 38) For learning and understanding the multiplication algorithm, the understanding of the distributive property i s A) quite important B) relatively unimportant 39) New mathematics A) shows the student the structure of arithmetic B) has the student memorize certain definitions and then asks the student to use these memorized definitions 40) In arithmetic zero is a A) number as is one, two, three, etc. B) placeholder 109 41) When teaching the addition facts A) the commutative property i s presented so students know a + b = b + a B) i t i s not necessary to stress the commutative property because the idea i s simple enough for the students to realize a + b = b + a _42) Realizing there is more than one way to subtract, i t is best to A) teach the student a l l ways B) teach the student only one way so he w i l l not become confused _43) When introducing division of fractions, i t is best to use • } 4 3 4 x 1 4 n\ I • I _ -I • JL = 3 * 4 = 3 * 4 _ , . . 3 ; 4 * 3 ~ 12 ' 12 12 T 12 1 J ' H 4 _44) For learning and understanding the multiplication algorithm, the knowledge of place value is A) quite important B) relatively unimportant _45) When introducing addition of two digit whole numbers, i t is best to use A) 1 3 A ; + ^ 18 B) 13 + 5 = (10 + 3) + 5 = 10 + (3 + 5) = 10 + 8 = 18 46) For learning and understanding the multiplication algorithm, the associative property of multiplication is A) quite important B) relatively unimportant 47) Arithmetic A) i s seldom boring to the student B) usually boring to the student 48) When teaching multiplication and division, i t is best to teach them as A) inverse operations B) separate operations 110 49) For learning and understanding the m u l t i p l i c a t i o n algorithm A) the d i s t r i b u t i v e property i s most important B) knowledge of place value i s most important I l l Second Version Following i s a multiple choice inventory. Complete each statement so that your choice best reflects your beliefs, opinions, and practices. If you do not know the meaning of some word in a statement, use "D" as your choice. Place your answer on the blank to the l e f t of each statement. Thank you for your cooperation. 1) New mathematics is A) a success and here to stay B) an educational fad which, as many have i n the past, w i l l pass on 2) Students A) dislike arithmetic because i t is a dry subject B) like arithmetic because i t i s f u l l of new ideas 3) There is considerable discussion as to whether fractions should be written as ^  or as 2/3. This is an A) unimportant distinction B) important distinction 4) When teaching multiplication by the number one i t A) should be emphasized as a special property B) need not be emphasized 5) When I teach arithmetic I do so because A) I must teach i t as part of the curriculum B) I enjoy watching students learning arithmetic 6) In a new mathematics program early rote memorization of the arithmetic facts is A) not as important as in the old programs B) just as important as in the old programs 7) When teaching subtraction, i t is better to teach i t as A) the inverse operation of addition B) a separate operation 112 _ 8) S t u d e n t s A) d i s l i k e a r i t h m e t i c b e c a u s e o f t h e r e p e t i t i o u s homework B) l i k e a r i t h m e t i c b e c a u s e o f t h e o p p o r t u n i t y t o t h i n k t h i n g s o u t _ 9) T a b l e s s u c h as t h e m u l t i p l i c a t i o n t a b l e s A) a r e n o t i m p o r t a n t i n new m a t h e m a t i c s b e c a u s e t h e y s t r e s s m e m o r i z a t i o n B) a r e i m p o r t a n t i n new m a t h e m a t i c s b e c a u s e t h e y a s s i s t i n u n d e r s t a n d i n g number r e l a t i o n s h i p s _10) C o n s i d e r i n g t h e w o r l d we l i v e i n , e l e m e n t a r y s t u d e n t s s h o u l d s p e n d more t i m e s t u d y i n g A) m a t h e m a t i c s B) s u c h t h i n g s as a r t and m u s i c _11) N o n - p o s i t i o n a l n u m e r a t i o n s y s t e m s s u c h as t h e Roman S y s t e m a r e A) u n i m p o r t a n t i n new m a t h e m a t i c s B) i m p o r t a n t i n new m a t h e m a t i c s _12) When t e a c h i n g a r i t h m e t i c , t h e d i f f e r e n c e b e t w e e n number and n u m e r a l i s A ) u n i m p o r t a n t and s h o u l d n o t be s t r e s s e d B) i m p o r t a n t and s h o u l d be s t r e s s e d _13) I n new m a t h e m a t i c s p r o v i n g an a n s w e r c o r r e c t i s A) as i m p o r t a n t as i n o l d m a t h e m a t i c s B) u n i m p o r t a n t _14) The a s s o c i a t i v e p r o p e r t y o f a d d i t i o n i s A) n e c e s s a r y t o u n d e r s t a n d c o l u m n a d d i t i o n B) u n n e c e s s a r y t o u n d e r s t a n d c o l u m n a d d i t i o n _15) When I t e a c h a r i t h m e t i c I A) e n j o y t h e t e a c h i n g B) d i s l i k e t h e t e a c h i n g 16) M o s t o f t h e a r i t h m e t i c t a u g h t b y e l e m e n t a r y t e a c h e r s s h o u l d be d e s i g n e d A) f o r a p p l i c a t i o n s i n p r a c t i c a l l i f e B) t o b u i l d b a c k g r o u n d f o r f u t u r e s t u d y i n m a t h e m a t i c s 113 The a b i l i t y to c a l c u l a t e with large numbers i s A) l e s s necessary i n new mathematics B) j u s t as important as ever Work i n bases d i f f e r e n t from ten.should A) be performed with most students B) not be performed with most students In l i g h t of the philosophy of new mathematics, c a l c u l a t i n g with great speed i s A) j u s t as important as before B) not as important as before Student; understanding of arithmetic i s A) l e s s necessary today because c a l c u l a t i n g machines are used to do the d i f f i c u l t c a l c u l a t i o n s B) more necessary today Words such as commutative, assoc i a t i v e , and d i s t r i b u t i v e A) are important words i n mathematics and by the end of the t h i r d grade, most children should know these words B) represent important ideas i n mathematics and by the end of the t h i r d grade, most children should understand these i d e a s — t h e words are not important Ninth grade algebra i s a A) prestige course B) course most students should and can take In teaching arithmetic the teacher should teach so that the student A) learns to do the work and then understands i t as he practices or applies i t B) understands what he i s doing Set theory A) should be studied with the early introduction of arithmetic B) i s a separate branch of arithmetic and, therefore, should not be studied with the early introduction of arithmetic 114 25) B e c a u s e o f my e x p e r i e n c e s , I l i k e A ) new m a t h e m a t i c s b e t t e r B) o l d m a t h e m a t i c s b e t t e r _26) The t e a c h i n g o f c o m p u t a t i o n a l s h o r t c u t s i n a r i t h m e t i c i s A ) n o t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s B) j u s t as I m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s . _27) New m a t h e m a t i c s i s b e t t e r f o r A) a l l s t u d e n t s B) c o l l e g e b o u n d s t u d e n t s o n l y _.28) New m a t h e m a t i c s w i l l h e l p t r a i n t h e s t u d e n t s t o A) a s k why t h i n g s a r e h a p p e n i n g i n t h e w o r l d t o d a y B) a c c e p t t h e t h i n g s t h a t a r e h a p p e n i n g i n t h e w o r l d t o d a y 29) The number l i n e i s f o r u s e i n u n d e r s t a n d i n g A) n e g a t i v e numbers i n a l g e b r a and m a t h e m a t i c s b e y o n d a l g e b r a B) a r i t h m e t i c as w e l l as a l g e b r a and m a t h e m a t i c s b e y o n d a l g e b r a 30) When t e a c h i n g d i v i s i o n o f w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s A) 234 B) 56)13104 56)13104 112 11200 200 190 1904 168 1680 30 224 224 224 224 __4 234 31) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e u n d e r s t a n d i n g o f t h e d i s t r i b u t i v e p r o p e r t y i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 32) I n a r i t h m e t i c z e r o i s a A) number as i s o n e , t w o , t h r e e , e t c . B) p l a c e h o l d e r 115 _33) When t e a c h i n g t h e a d d i t i o n f a c t s A) t h e c o m m u t a t i v e i d e a s h o u l d be s t r e s s e d B) i t i s n o t n e c e s s a r y t o s t r e s s t h e c o m m u t a t i v e i d e a a t s u c h an e l e m e n t a r y l e v e l _34) R e a l i z i n g t h e r e i s more t h a n one a l g o r i t h m f o r s u b t r a c t i o n , i t i s b e s t t o A) t e a c h t h e s t u d e n t s e v e r a l a l g o r i t h m s and l e t h i m c h o o s e t h e one he w i s h e s t o u s e B) t e a c h t h e s t u d e n t o n l y one a l g o r i t h m so he w i l l n o t become c o n f u s e d _35) When i n t r o d u c i n g d i v i s i o n o f f r a c t i o n s , t h e a l g o r i t h m I w o u l d u s e i s . 1 , 1 . 1 2 2 A ) 4 ' 3 " 4 X 1 = 4 1 - 1 _ 3 • 4 _ 3 * 4 3 * 4 n . A ^ 3 ' 4 ' 3 12 " 12 1 2 - 1 2 1 * 4 _36) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , k n o w l e d g e o f p l a c e v a l u e i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 37) When i n t r o d u c i n g a d d i t i o n o f two d i g i t w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s A) 1 3  ; +_5 18 B) 13 + 5 = (10 + 3) + 5 - 10 + (3 + 5) = 10 + 8 = 18 _38) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e a s s o c i a t i v e p r o p e r t y o f m u l t i p l i c a t i o n i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t _39) A r i t h m e t i c A) i s s e l d o m b o r i n g t o t h e s t u d e n t B) u s u a l l y b o r i n g to t h e s t u d e n t 40) When t e a c h i n g d i v i s i o n , i t i s b e t t e r t o t e a c h i t as A) t h e i n v e r s e o p e r a t i o n o f m u l t i p l i c a t i o n B) a s e p a r a t e o p e r a t i o n 116 Third Version The following i s an inventory to determine certain teacher attitudes toward elementary school arithmetic. Each statement has two possible completions. Complete the statements so that your choices best reflect your beliefs, opinions, and practices. Place your answer on the blank to the l e f t of each statement. If you do not know the meaning of some word in a statement, use "D" as your choice. Thank you for your cooperation. 1) New mathematics i s A) a success and here to stay B) an educational fad which, as many have in the past, w i l l pass on 2) Students A) dislike arithmetic because i t is a dry subject B) like arithmetic because i t is f u l l of new ideas 3) In a new mathematics program early rote memorization of the arithmetic facts i s A) not as important as in the old programs B) just as important as in the old programs 4) When teaching subtraction, i t i s better to teach i t as A) the inverse operation of addition B) a separate operation 5) Students A) dislike arithmetic because of the repetitious homework B) like arithmetic because of the opportunity to think things out 6) Teaching multiplication by the number one A) is a special property and should be emphasized B) i s easy and need not be emphasized 117 _ 7) C o n s i d e r i n g t h e w o r l d we l i v e i n , e l e m e n t a r y s t u d e n t s s h o u l d spend more t i m e s t u d y i n g , . . A) m a t h e m a t i c s B) s u c h t h i n g s as a r t and m u s i c 8) N o n - p o s i t i o n a l n u m e r a t i o n s y s t e m s s u c h as t h e Roman S y s t e m a r e A ) u n i m p o r t a n t i n new m a t h e m a t i c s B) i m p o r t a n t i n new m a t h e m a t i c s 9) When t e a c h i n g a r i t h m e t i c , t h e d i f f e r e n c e b e t w e e n number and n u m e r a l i s A) u n i m p o r t a n t and s h o u l d n o t b e - s t r e s s e d B) i m p o r t a n t and s h o u l d be s t r e s s e d 10) The a s s o c i a t i v e p r o p e r t y o f a d d i t i o n i s A) n e c e s s a r y t o u n d e r s t a n d c o l u m n a d d i t i o n B) u n n e c e s s a r y t o u n d e r s t a n d c o l u m n a d d i t i o n _11) The m a j o r i t y o f t h e a r i t h m e t i c t a u g h t by e l e m e n t a r y t e a c h e r s s h o u l d be d e s i g n e d A) f o r a p p l i c a t i o n s i n p r a c t i c a l l i f e B) t o b u i l d b a c k g r o u n d f o r f u t u r e s t u d y i n m a t h e m a t i c s _12) The a b i l i t y t o c a l c u l a t e w i t h l a r g e numbers i s A ) l e s s n e c e s s a r y i n new m a t h e m a t i c s B) j u s t as i m p o r t a n t as e v e r 13) Work i n b a s e s d i f f e r e n t f r o m t e n s h o u l d A) be p e r f o r m e d by most s t u d e n t s B) n o t be p e r f o r m e d b y most s t u d e n t s 14) I n l i g h t o f t h e p h i l o s o p h y o f new m a t h e m a t i c s , c a l c u l a t i n g w i t h g r e a t s p e e d i s A) j u s t as i m p o r t a n t as b e f o r e B) n o t as i m p o r t a n t as b e f o r e 15) S t u d e n t u n d e r s t a n d i n g o f a r i t h m e t i c i s A) l e s s n e c e s s a r y t o d a y b e c a u s e c a l c u l a t i n g m a c h i n e s a r e u s e d t o do t h e d i f f i c u l t c a l c u l a t i o n s B) more n e c e s s a r y t o d a y 118 16) Words s u c h as c o m m u t a t i v e , a s s o c i a t i v e , and d i s t r i b u t i v e A) a r e i m p o r t a n t w o r d s i n m a t h e m a t i c s and b y t h e end o f t h e s e c o n d g r a d e , most c h i l d r e n s h o u l d know t h e s e w o r d s B) r e p r e s e n t i m p o r t a n t i d e a s i n m a t h e m a t i c s and b y t h e end o f t h e s e c o n d g r a d e , most c h i l d r e n s h o u l d know t h e s e i d e a s — t h e w o r d s a r e n o t i m p o r t a n t 17) S t u d e n t s A) s h o u l d a l w a y s o b t a i n u n d e r s t a n d i n g s b e f o r e s k i l l s B) somet imes need s k i l l s b e f o r e , u n d e r s t a n d i n g _18) Se t t h e o r y A) s h o u l d be s t u d i e d w i t h t h e i n t r o d u c t i o n o f a r i t h m e t i c i n g r a d e one B) i s a s e p a r a t e b r a n c h o f a r i t h m e t i c a n d , t h e r e f o r e , s h o u l d n o t b e s t u d i e d w i t h t h e e a r l y i n t r o d u c t i o n o f a r i t h m e t i c 19) B e c a u s e o f my e x p e r i e n c e s , I l i k e A) new m a t h e m a t i c s b e t t e r B) o l d m a t h e m a t i c s b e t t e r _20) The t e a c h i n g o f c o m p u t a t i o n a l s h o r t c u t s i n a r i t h m e t i c i s A ) n o t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s B) j u s t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s 21) New m a t h e m a t i c s i s b e t t e r f o r A) most s t u d e n t s B) more a b l e s t u d e n t s 22) The number l i n e i s f o r u s e i n u n d e r s t a n d i n g A) n e g a t i v e numbers i n a l g e b r a and m a t h e m a t i c s b e y o n d a l g e b r a B) g r a d e one a r i t h m e t i c as w e l l as h i g h e r m a t h e m a t i c s .23) When t e a c h i n g d i v i s i o n o f w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s A) 234 56)13104 B) 56)13104 112 190 168 11200 1904 1680 224 224 200 30 224 224 4 234 24) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e u n d e r s t a n d i n g o f t h e d i s t r i b u t i v e p r o p e r t y i s 26) When t e a c h i n g t h e a d d i t i o n f a c t s A ) t h e c o m m u t a t i v e i d e a s h o u l d . b e s t r e s s e d B) i t i s n o t n e c e s s a r y t o s t r e s s t h e c o m m u t a t i v e i d e a a t s u c h an e l e m e n t a r y l e v e l 27) R e a l i z i n g t h e r e i s more t h a n one a l g o r i t h m f o r s u b t r a c t i o n , i t i s b e s t t o A) t e a c h t h e s t u d e n t s e v e r a l a l g o r i t h m s and l e t h i m c h o o s e t h e one he w i s h e s t o u s e B) t e a c h t h e s t u d e n t o n l y one a l g o r i t h m so he w i l l n o t become c o n f u s e d 28) When i n t r o d u c i n g d i v i s i o n o f f r a c t i o n s , t h e a l g o r i t h m I w o u l d u s e i s 29) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , k n o w l e d g e o f p l a c e v a l u e i s A ) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 25) I n a r i t h m e t i c z e r o i s a A) number as i s o n e , t w o , t h r e e , e t c . B) p l a c e h o l d e r A ) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 120 _30) When I n t r o d u c i n g a d d i t i o n o f two d i g i t w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s A) 1 3 A ) +_5_ 18 B) 13 + 5 = (10 + 3) + 5 = 10 + (3 + 5) - 1 0 + 8 = 1 8 31) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e a s s o c i a t i v e p r o p e r t y o f m u l t i p l i c a t i o n i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t _32) A r i t h m e t i c i s u s u a l l y A) e n j o y a b l e t o s t u d e n t s B) u n e n j o y a b l e t o s t u d e n t s 33) When t e a c h i n g d i v i s i o n , i t i s b e t t e r t o t e a c h i t as A) t h e i n v e r s e o p e r a t i o n o f m u l t i p l i c a t i o n B) a s e p a r a t e o p e r a t i o n How do y o u f e e l a b o u t new m a t h e m a t i c s ? L e t 11 be h i g h l y f a v o r a b l e and 1 b e h i g h l y u n f a v o r a b l e t o w a r d new m a t h e m a t i c s . G i v e y o u r s e l f a s c o r e f r o m 1 t o 11 d e p e n d i n g upon y o u r own o p i n i o n o f y o u r s e l f . 121 Fourth Version The following i s an inventory to determine c e r t a i n teacher attitudes toward elementary school arithmetic. Each statement has two possible completions. Complete the statements so that your choices best r e f l e c t your b e l i e f s , opinions, and p r a c t i c e s . Place your answer on the blank to the l e f t of each statement. I f you do not know the meaning of some word i n a statement, use "D" as your choice. Thank you f o r your cooperation. 1) New mathematics i s A) a success and here to stay B) an educational fad which, as many have i n the past, w i l l pass on 2) In a new mathematics program early rote memorization of the arithmetic f a c t s i s A) not as important as i n the old programs B) j u s t as important as i n the old programs 3) Students A) d i s l i k e arithmetic because of the r e p e t i t i o u s homework B) l i k e arithmetic because of the opportunity to think things out 4) Teaching m u l t i p l i c a t i o n by the number one A) i s a s p e c i a l property and should be emphasized B) i s easy and need not be emphasized 5) Non-positional numeration systems such as the Roman System are A) unimportant i n new mathematics B) important i n new mathematics 6) When teaching arithmetic, the difference between number and numeral i s A) unimportant and should not be stressed B) important and should be stressed 122 _ 7) The a s s o c i a t i v e p r o p e r t y o f a d d i t i o n i s A) n e c e s s a r y t o u n d e r s t a n d , c o l u m n a d d i t i o n B) u n n e c e s s a r y t o u n d e r s t a n d c o l u m n a d d i t i o n _ 8) The m a j o r i t y o f t h e a r i t h m e t i c t a u g h t b y e l e m e n t a r y t e a c h e r s s h o u l d be d e s i g n e d A) f o r a p p l i c a t i o n s i n p r a c t i c a l l i f e B) t o b u i l d b a c k g r o u n d f o r f u t u r e s t u d y i n m a t h e m a t i c s _ 9) Work i n b a s e s d i f f e r e n t f r o m t e n s h o u l d A) b e p e r f o r m e d b y most s t u d e n t s B) n o t be p e r f o r m e d b y most s t u d e n t s _10) I n l i g h t o f t h e p h i l o s o p h y o f new m a t h e m a t i c s , c a l c u l a t i n g w i t h g r e a t s p e e d i s A) j u s t as i m p o r t a n t as b e f o r e . B) n o t as i m p o r t a n t as b e f o r e _11) S t u d e n t u n d e r s t a n d i n g o f a r i t h m e t i c i s A) l e s s n e c e s s a r y t o d a y b e c a u s e c a l c u l a t i n g m a c h i n e s a r e u s e d t o do t h e d i f f i c u l t c a l c u l a t i o n s B) more n e c e s s a r y t o d a y _12) S t u d e n t s A) s h o u l d a l w a y s o b t a i n u n d e r s t a n d i n g s b e f o r e s k i l l s B) somet imes n e e d s k i l l s b e f o r e u n d e r s t a n d i n g _13) S e t t h e o r y A) s h o u l d b e s t u d i e d w i t h t h e i n t r o d u c t i o n o f a r i t h m e t i c i n g r a d e one B) i s a s e p a r a t e b r a n c h o f a r i t h m e t i c a n d , t h e r f o r e , s h o u l d n o t b e s t u d i e d w i t h t h e e a r l y i n t r o d u c t i o n o f a r i t h m e t i c 14) B e c a u s e o f my e x p e r i e n c e s , I l i k e A) new m a t h e m a t i c s b e t t e r B) o l d m a t h e m a t i c s b e t t e r 15) The t e a c h i n g o f c o m p u t a t i o n a l s h o r t c u t s i n a r i t h m e t i c i s A) n o t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s B) j u s t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s 123 16) New m a t h e m a t i c s i s b e t t e r f o r A ) most s t u d e n t s B) more a b l e s t u d e n t s 17) S t u d e n t s A) d i s l i k e a r i t h m e t i c b e c a u s e i t i s a d r y s u b j e c t B) l i k e a r i t h m e t i c b e c a u s e i t i s f u l l o f new i d e a s 18) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e u n d e r s t a n d i n g o f t h e : d i s t r i b u t i v e p r o p e r t y i s A ) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 19) I n a r i t h m e t i c z e r o i s a A) number as i s o n e , t w o , t h r e e , e t c . B) p l a c e h o l d e r 20) When t e a c h i n g t h e a d d i t i o n f a c t s A) t h e c o m m u t a t i v e i d e a s h o u l d b e s t r e s s e d B) i t i s n o t n e c e s s a r y t o s t r e s s t h e c o m m u t a t i v e i d e a a t s u c h an e l e m e n t a r y l e v e l _21) When i n t r o d u c i n g d i v i s i o n o f f r a c t i o n s , t h e a l g o r i t h m I w o u l d u s e i s A ) 4 3 4 * 1 4 RN, ±*.L=-A*.JL. 3 * 4 3 * 4 . A , 3 } 4 ' 3 12 - 12 12 * 12 1 J ' 4 4 _22) When i n t r o d u c i n g a d d i t i o n o f two d i g i t w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s A) 1 3 +_5 18 B) 13 + 5 = (10 + 3 ) + 5 = 10 + (3 + 5) = 10 + 8 = 18 _23) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e a s s o c i a t i v e p r o p e r t y o f m u l t i p l i c a t i o n i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 24) A r i t h m e t i c i s u s u a l l y A) e n j o y a b l e t o s t u d e n t s B) u n e n j o y a b l e t o s t u d e n t s When teaching d i v i s i o n , i t i s better to teach i t A) the inverse operation of m u l t i p l i c a t i o n B) a separate operation 125 Final Version The following i s an inventory to determine certain teacher attitudes toward elementary school arithmetic. Each statement has two possible completions. Complete the statements so that your choices best reflect your beliefs, opinions, and practices. Place your answer on the blank to the l e f t of each statement. If you do not know the meaning of some word in a statement, use "D" as your choice. Thank you for your cooperation. 1) New mathematics i s A) a success and here to stay B) an educational fad which, as many have in the past, w i l l pass on , 2) In a new mathematics program early rote memorization of the arithmetic facts is A) not as important as in the old programs B) just as important as in the old programs 3) Students A) dislike arithmetic because of the repetitious homework B) like arithmetic because of the opportunity to think things out 4) Teaching multiplication by the number one A) is a special property and should be emphasized B) is easy and need not be emphasized 5) When teaching arithmetic, the difference between number and numeral is A) unimportant and should not be stressed B) important and should be stressed 6) The associative property of addition is A) necessary to understand column addition B) unnecessary to understand column addition 126 7) Work i n b a s e s d i f f e r e n t f r o m t e n s h o u l d A) b e p e r f o r m e d b y most s t u d e n t s B) n o t be p e r f o r m e d b y most s t u d e n t s _ 8) I n l i g h t o f t h e p h i l o s o p h y o f new m a t h e m a t i c s , c a l c u l a t i n g w i t h g r e a t s p e e d i s A) j u s t as i m p o r t a n t as b e f o r e B) n o t as i m p o r t a n t as b e f o r e 9) S t u d e n t u n d e r s t a n d i n g o f a r i t h m e t i c i s A) l e s s n e c e s s a r y t o d a y b e c a u s e c a l c u l a t i n g m a c h i n e s a r e u s e d t o do t h e d i f f i c u l t c a l c u l a t i o n s B) more n e c e s s a r y t o d a y 10) S t u d e n t s A) s h o u l d a l w a y s o b t a i n u n d e r s t a n d i n g s b e f o r e s k i l l s B) somet imes n e e d s k i l l s b e f o r e u n d e r s t a n d i n g 11) S e t t h e o r y A) s h o u l d b e s t u d i e d w i t h t h e i n t r o d u c t i o n o f a r i t h m e t i c i n g r a d e one B) . i s a s e p a r a t e b r a n c h o f a r i t h m e t i c a n d , t h e r e f o r e , s h o u l d n o t be s t u d i e d w i t h t h e e a r l y i n t r o d u c t i o n o f a r i t h m e t i c 12) B e c a u s e o f my e x p e r i e n c e s , I l i k e A) new m a t h e m a t i c s b e t t e r B) o l d m a t h e m a t i c s b e t t e r 13) The t e a c h i n g o f c o m p u t a t i o n a l s h o r t c u t s i n a r i t h m e t i c i s A) n o t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s B) j u s t as i m p o r t a n t i n new m a t h e m a t i c s as i n o l d m a t h e m a t i c s 14) New m a t h e m a t i c s i s b e t t e r f o r A) most s t u d e n t s B) more a b l e s t u d e n t s 15) S t u d e n t s A) d i s l i k e a r i t h m e t i c b e c a u s e i t i s a d r y s u b j e c t B) l i k e a r i t h m e t i c b e c a u s e i t i s f u l l o f new i d e a s 127 _16) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e u n d e r s t a n d i n g o f t h e d i s t r i b u t i v e p r o p e r t y i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t _17) When t e a c h i n g t h e a d d i t i o n f a c t s A ) t h e c o m m u t a t i v e i d e a s h o u l d b e s t r e s s e d B) i t i s n o t n e c e s s a r y t o s t r e s s t h e c o m m u t a t i v e i d e a a t s u c h a n e l e m e n t a r y l e v e l _18) When i n t r o d u c i n g d i v i s i o n o f f r a c t i o n s , t h e a l g o r i t h m I w o u l d u s e i s A } 4 3 4 x 1 4 wv 1 • 1 3 • * _ • 3- * * _ 3 * 4 _ . _ 3 ' 4 ' 3 " 12 * 12 1 2 * 1 2 1 4 19) When i n t r o d u c i n g a d d i t i o n o f two d i g i t w h o l e n u m b e r s , t h e a l g o r i t h m I w o u l d u s e i s 18 B) 13 + 5 = (10 + 3) + 5 = 10 + (3 + 5) » 10 + 8 = 18 _20) F o r l e a r n i n g and u n d e r s t a n d i n g t h e m u l t i p l i c a t i o n a l g o r i t h m , t h e a s s o c i a t i v e p r o p e r t y o f m u l t i p l i c a t i o n i s A) q u i t e i m p o r t a n t B) r e l a t i v e l y u n i m p o r t a n t 21) A r i t h m e t i c i s u s u a l l y A) e n j o y a b l e t o s t u d e n t s B) u n e n j o y a b l e t o s t u d e n t s _22) When t e a c h i n g d i v i s i o n , i t i s b e t t e r t o t e a c h i t as A) t h e i n v e r s e o p e r a t i o n o f m u l t i p l i c a t i o n B) a s e p a r a t e o p e r a t i o n Appendix C ARITHMETIC INFORMATION, FOURTH GRADE Student Number Following are three different sets of arithmetic questions. These questions W i l l t e l l us what arithmetic you already know and what arithmetic you w i l l learn during this school year. Some of the questions w i l l be easy for you and some of the questions w i l l contain arithmetic you have not had. Each question has four or five possible answers. Choose the answer you believe is correct and mark i t on the answer sheet as the f i r s t three examples have been done. 1) If 4 + n = 9, then n = ?. A) 4 B) 5 C) 6 D) 7 E) none of these Notice that "B" has been marked on the answer sheet for example 1 because 5 i s the correct answer. 2) If 4 x p = 16, then p = ?. F) 2 G) 3 H) 4 I) 5 J) none of these Notice that "H" has been marked on the answer sheet for example 2 because 4 i s the correct answer. 3) If 24 * 3 = n, then n - ?. A) 2 B) 3 C) 4 D) 5 E) none of these Notice that "E" has been marked on the answer sheet for example 3 because the correct answer, 8, i s not given as a choice. 128 Good h e a v y p e n c i l m a r k s h a v e b e e n made i n t h e c o r r e c t s p a c e s . T h i s w i l l be n e c e s s a r y when y o u mark y o u r a n s w e r s . Now, do e x a m p l e s 4 and 5 . 4) I f 9 - p = 4 , t h e n p = ? . F ) 2 G) 3 H) 4 I ) 5 J ) none o f t h e s e 5) I f (1 + 2) + n = 7 , t h e n n = ? . '• A ) 2 B) 3 C) 4 D) 5 E none o f t h e s e Y o u s h o u l d h a v e marked " I " as t h e c o r r e c t a n s w e r f o r e x a m p l e 4 . Y o u s h o u l d h a v e marked " C " as t h e c o r r e c t answer f o r e x a m p l e 5 . Y o u . a r e now r e a d y t o answer t h e q u e s t i o n s as y o u h a v e done t h e e x a m p l e s a b o v e . PART 1 F o l l o w i n g i s a s e t o f 42 q u e s t i o n s w h i c h w i l l t e l l us what a r i t h m e t i c u n d e r s t a n d i n g s y o u know. Do n o t u s e p a p e r t o s o l v e -any o f t h e s e q u e s t i o n s . Y o u s h o u l d s o l v e e a c h o f them i n y o u r h e a d and t h e n mark y o u r answer on t h e a n s w e r s h e e t . Do n o t w r i t e i n t h i s t e s t b o o k l e t o r on t h e a n s w e r s h e e t . Y o u w i l l h a v e 32 m i n u t e s t o c o m p l e t e P a r t 1 . S t o p a t t h e end o f P a r t 1 . Do n o t go on t o P a r t 2 u n t i l y o u a r e i n s t r u c t e d t o do s o . A t t h i s t i m e , a s k any q u e s t i o n s y o u m i g h t h a v e . 1) What w o u l d be t h e n e x t number i n t h e f o l l o w i n g s e t ? 3 , 6 , 9 , . A) 10 1 B) 11 C) 12 D) 13 E) none o f t h e s e 2) I n t h e p r o b l e m a t t h e r i g h t , t h e number 1284 i s 214 " b e s t " e x p l a i n e d as b e i n g x 365 1070 F ) 600 x 214 1284 G) 60 x 214 642 H) 6 x 214 78110 I ) 6 x 214 w i t h o u t a z e r o added J ) 6 x 214 moved t o t h e l e f t one p l a c e 3) The i n v e r s e o p e r a t i o n o f s u b t r a c t i o n i s A) a d d i t i o n B) s u b t r a c t i o n C) m u l t i p l i c a t i o n D) d i v i s i o n E) n o n e o f t h e s e 4) The e x e r c i s e . 8 )24 means F) 8 t i m e s 24 G) how many s u b s e t s o f 24 i n a s e t o f 8 H) how many s u b s e t s o f 8 i n a s e t o f 24 I ) 8 T 24 J ) n o n e o f t h e s e 5) I n t h e e x a m p l e a t t h e r i g h t , w h a t i s t h e b e s t 34 r e a s o n f o r p l a c i n g 68 one p l a c e t o t h e l e f t ? x 23 102 A ) b e c a u s e t h e 8 m u s t b e u n d e r t h e 2 68 B) b e c a u s e t h i s i s t h e r u l e i n m u l t i p l i c a t i o n 782 C) b e c a u s e t h e 68 i s r e a l l y 680 D) b e c a u s e 34 i s a two d i g i t number E) b e c a u s e 23 i s b e l o w 34 i n s t e a d o f above 34 6) The i n v e r s e o p e r a t i o n o f a d d i t i o n i s F) a d d i t i o n G) s u b t r a c t i o n H) d i v i s i o n I ) m u l t i p l i c a t i o n J ) none o f t h e s e 7) The s h a d e d p a r t o f t h e f i g u r e i s what p a r t o f t h e f i g u r e ? 1 2 1 3 1 4 1 5 A) B) C) D) E) none o f t h e s e Which number i s the largest? F) 7000 G) 6999 H) 7001 I) 7010 J) 7100 Look at the two squares. • M The shaded part of the square M N A) l e s s than the shaded part of square N B) more than the shaded part of square N C) equal to the shaded part of square N D) cannot t e l l from the picture In the number 7342, about how many thousands are there? F) two thousands G) four thousands H) three thousands I) seven thousands J) eight thousands In the expression 427 x 638 x 546, how w i l l the answer be changed i f i t i s worked as 546 x 638 x 427? A) the answer B) the answer C) the answer D) can't t e l l A mixed number such F) 3 X 4 G) H) 3 - ! I) 3 * t w i l l be less w i l l be greater w i l l be the same u n t i l i t i s worked out *j3 as 3f- means J) none of these 132 13) The shaded part of this figure is what fractional part of the figure? A) 1 2 B, | « I 1 5 E) 4 + 4 = t, none of these 14) The inverse operation of division i s F) addition G) subtraction 1 H) multiplication I) division J) none of these 15) These statements are true: 4 + 6 = r, 6 + 6 = s, 6 + 4 = u. Which of the following is also true? A) r - s B) r = t C) r = u D) s = u E) none of these 16) The inverse operation of multiplication i s F) addition G) subtraction H) multiplication I) division J) none of these 17) If 7 x t = 0, then " t " is always A) zero B) one C) seven D) ten E) i t i s impossible to t e l l from the information given 133 18) Which shaded figure shows one-half of one-third? F) G) H) I) J) none of these 19) As the denominator of a f r a c t i o n a l number decreases and the numerator remains the same, the number A) becomes larger . . B) becomes smaller C) remains the same D) approaches one E) can't t e l l from!the information given 20) As the numerator of a f r a c t i o n a l number decreases and the denominator remains the game, the number F) becomes smaller G) becomes la r g e r H) remains the same I) gets close to one J) can't t e l l from the information given 21) These statements are true: a + b = d , c + b = e , Which of the following i s also true? A) a = b + d B) c = b + e C) d + e = f D) e = b + c E) none of these 22) If 0 x y = 0, then "y" i s always F) zero G) one H) two I) ten J) any number you choose c + c = f. 134 23) Which o f the f o l l o w i n g i s a n o t h e r numeral f o r 526? A) (5 x 10 x 10) + (2 x 10) +6 B) (5 x 5 x 5) + (2 x 2) + 6 C) (5 x 100) + (2 x 100) +6 D) (5 x 2 x 6 x 100) + (2 x 6 x 10) +6 E) none o f t h e s e 24) A f r a c t i o n such as y means F) c h o o s i n g 7 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 5 p a r t s G) c h o o s i n g 2 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 5 p a r t s H) c h o o s i n g 5 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 7 p a r t s I) c h o o s i n g 2 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 7 p a r t s J ) none o f t h e s e 25) I f a x b = 0, then A) " a " always e q u a l s "b" B) "b" must be z e r o C) " a " must be z e r o D) e i t h e r " a " o r "b" must be z e r o E) none o f these 26) Which f r a c t i o n i s t h e s m a l l e s t ? F) i « f H) I I) 4 3 4 27) I f r x s = r , then " s " i s always A) z e r o B) one C) " r " D) t e n E) i t i s i m p o s s i b l e to t e l l from the i n f o r m a t i o n g i v e n 28) Jimmy's b i k e has a speedometer which shows m i l e s and t e n t h s o f m i l e s , I t l o o k s l i k e t h i s now I 027 I 7 speedometer reads | 033 | Q /'? How f a r w i l l he r i d e b e f o r e the F) more than 5 m i l e s G) l e s s than 5 m i l e s H) e x a c t l y 5 m i l e s I) c a n ' t t e l l from the i n f o r m a t i o n g i v e n I f r + s = r , t h e n " s " I s a l w a y s A) z e r o B) one C) " r " D) t e n E) i t i s i m p o s s i b l e t o t e l l f r o m t h e i n f o r m a t i o n g i v e n To s u b t r a c t i n t h e e x e r c i s e ^ 6 2 ' w e s ^ o u l ^ F ) make t h e 5 ones s m a l l e r G) make t h e 5 ones l a r g e r H) make t h e 4 t e n s s m a l l e r I ) make t h e 4 t e n s l a r g e r J ) none o f t h e s e I f u - v = u , t h e n " v " i s a l w a y s A) z e r o B) one C) " u " D) t e n E) i t i s i m p o s s i b l e t o t e l l I n t h e e x e r c i s e 7 )364 , t h e 5 r e a l l y s t a n d s f o r F ) 5 ones G) 5 t e n s H) 5 h u n d r e d s I ) 5 t e n t h s J ) n o n e o f t h e s e T h i s l i n e segment i s c u t i n t o I \ 1 1 1 1 1 I -I A) s e v e n t h s B) e i g h t h s C) n i n t h s D) t e n t h s E) none o f t h e s e F o u r t h o u s a n d t h r e e h u n d r e d s e v e n i s w r i t t e n F ) 4037 G) 0437 H) 4370 I ) 40003007 J ) none o f t h e s e 136 35) Look at the numeral 85,626. The 6 on the l e f t has a value how many times larger than the 6 on the right? . A) 100 times larger B) 10 times larger C) 1000 times larger D) the same value E) hone of these 937 36) When multiplying in the problem x we move the second partial product, which ve get when we multiply by 8, one place to the l e f t because F) that is the rule in multiplying G) the 8 means 8 tens H) the answer must be larger than 937 I) the top number i s a number larger than ten J) none of these 37) Look at the problem p - q. If "q" i s the identity number for this subtraction, then "q" i s equal to A) zero B) one C) ten D) there isn't one E) none of these 38) The number 4357 i s about F) 4 hundreds G) 43 hundreds H) 435 hundreds I) 4357 hundreds 39) Round off 9766 to the nearest hundred. A) 9770 , B) 9700 C) 9800 D) 10,000 E) none of these 64 40) How would the sum in the problem be changed i f 78 was placed above 64 instead of below it? F) the new sum would be less than the old sum G) the new sum would be the same as the old sum H) the new sum would be greater than the old sum I) i t i s unknown until i t i s worked both ways 13.7 4 1 ) M u l t i p l y i n g 6 and 9 i s the same as A) increasing 6 by 9 B) adding six-ninths C) adding nine-sixths D) adding s i x nines E) none of these 4 2 ) How would the product i n the problem X ^ 4 7 he changed i f 4 7 was placed above 7 6 0 8 instead of below i t ? F) the new product would be the same as the o l d product G) the new product would be greater than the old product H) the new product would be less than the o l d product I) i t i s impossible to multiply when the la r g e r number i s on the bottom and the smaller number on the top J) i t i s unknown u n t i l i t i s worked both ways END OF PART 1 DO NOT TURN THE PAGE 138 PART 2 F o l l o w i n g i s a s e t o f 13 p r o b l e m s w h i c h w i l l t e l l us w h a t a r i t h m e t i c p r o b l e m s y o u c a n w o r k . Y o u may u s e p a p e r t o s o l v e t h e s e p r o b l e m s . P u t t h r e e c l e a n s h e e t s o f p a p e r o n y o u r d e s k t o f i g u r e o n . Do n o t w r i t e i n t h i s t e s t b o o k l e t o r o n t h e a n s w e r s h e e t . Y o u w i l l h a v e 22 m i n u t e s t o c o m p l e t e P a r t 2 . Do n o t go on t o P a r t 3 u n t i l y o u a r e i n s t r u c t e d t o do s o . A t t h i s t i m e , a s k any q u e s t i o n s y o u m i g h t h a v e . 1) J o e p a i d $ 8 . 6 8 f o r 7 b a s e b a l l s . How A) $ 6 0 . 7 5 much d i d e a c h b a s e b a l l c o s t ? B) $ 1 . 2 5 C) $ 1 .24 D) $ 1 .09 E) none o f t h e s e 2) M a r i l y n b o u g h t - | o f a y a r d o f r i b b o n . F) 24 i n c h e s G) 12 i n c h e s How many i n c h e s d i d s h e b u y ? H) 15 i n c h e s I ) 36 i n c h e s J ) none o f t h e s e 3) J o e c o u l d p u t 6 p i c t u r e s on e a c h page A) 72 p a g e s and 5 p i c t u r e s o f h i s p h o t o a l b u m . He h a d 77 p i c - B) 66 pages and 1 p i c t u r e t u r e s . How many f u l l pages c o u l d he C) 462 p a g e s and 0 p i c t u r e s mount and how many p i c t u r e s a r e l e f t ? D) 12 pages and 5 p i c t u r e s E) n o n e o f t h e s e 4) J o h n w i s h e s t o b u y a b i c y c l e w h i c h F) $ 1 5 . 7 5 s e l l s f o r $ 3 7 . 5 0 . He now has $ 1 9 . 7 5 . G) $ 1 7 . 7 5 He w i l l w o r k t o e a r n enough money t o H) $ 1 9 . 7 5 buy t h e b i c y c l e . How much money I ) $ 5 7 . 2 5 must he e a r n ? J ) none o f t h e s e 5) J a n e w a i t e d — o f an h o u r f o r h e r A ) 10 m i n u t e s m o t h e r . How many m i n u t e s d i d s h e B) 12 m i n u t e s C) 15 m i n u t e s w a i t ? D) 20 m i n u t e s E) none o f t h e s e 6) David bought a.pair of pants for $8.95 and a s h i r t f o r $.3.75. How much change did he receive from $20.00? F) $ 7.30 G) $ 8.30 H) $12.70 I) $14.80 J) none of these 7) Peter bought 3 baseballs at $1.25 each, 2 bats at $1.75 each, and one glove at $5.95. How much did Peter spend? A) $ 8.95 B) $ 9.95 C) $12.20 D) $13.20 E) none of these 8) The eight members of the Boys Club F) bought 12 b o t t l e s of pop at 8 c e n t s / G) f o r each b o t t l e and 6 dozen cookies H) at 32 cents for" each dozen. If the I) members share the cost equally, J) how much did each member pay? $ 2.88 '; 40c 44c' 24c none of these 9) Frank had $1.75 to buy school supplies. His father gave him $4.50. While shopping-he spent $5.37. How much money did he have l e f t ? A) $ 6.25 B) 98C C) 88e D) 78c . E) none of these 10) Mrs. Smith needs 375 cookies. She has 22 packages, each containing 11 cookies. How many more cookies does she need? F) 133 cookies G) 342 cookies H) 353 cookies I) 242 cookies J) none of these 11) Mrs. Johnson made 36 cookies on A) Monday, 45 cookies on Tuesday, and B) 27 cookies on Wednesday. By C) Saturday h a l f of the cookies were D) gone. How many cookies did she E) have l e f t ? 108 cookies 54 cookies 44 cookies 40 cookies none of these 12) Farmer Brown sold 355 pounds of hay i n January, 267 pounds of hay i n February, and 216 pounds of hay i n March. On the average, how many pounds of hay did he s e l l each month? F) 212 pounds G) 266 pounds H) 279j pounds I) 838 pounds J) none of these 1 1 J a n e p i c k e d 2 y d o z e n a s t e r s and l y A) 31 f l o w e r s B) 3 d o z e n f l o w e r s d o z e n r o s e s . How many f l o w e r s d i d 2 J a n e p i c k ? C) 3^- d o z e n f l o w e r s D) 4 d o z e n f l o w e r s E ) none o f t h e s e END OF PART 2 DO NOT TURN THE PAGE 141 PART 3 F o l l o w i n g i s ' a s e t o f 24 p r o b l e m s w h i c h w i l l t e l l us what a r i t h m e t i c p r o b l e m s y o u c a n w o r k . Y o u may u s e p a p e r t o s o l v e t h e s e p r o b l e m s . P u t t h r e e c l e a n s h e e t s o f p a p e r on y o u r d e s k t o f i g u r e o n . Do n o t f i g u r e i n t h i s t e s t b o o k l e t o r o n t h e a n s w e r s h e e t . Y o u w i l l h a v e 29 m i n u t e s t o c o m p l e t e P a r t 3 . A t t h i s t i m e , a s k any q u e s t i o n s y o u m i g h t h a v e . 1) 2478 A) 2 1 , 4 2 4 2) 692 F) 245 6002 B) 2 0 , 4 2 4 - 4 5 7 G) 235 6201 C) ""20,224 H) 135 5743 D) 2 0 , 2 4 4 I ) 145 E) none o f t h e s e J ) none o f t h e s e 9 0 , 0 0 6 A) 6 0 , 0 0 2 4) 16 F ) 636 - 3 6 , 7 5 8 B) 5 3 , 2 4 8 x_6 G) 336 C) • 5 3 , 2 3 8 H) 96 D) . 5 3 , 1 4 8 I ) 10 E) none o f t h e s e J ) none o f t h e s e 5) 302 A ) 2486 6) 54 F) 3718 x 8 B) 2416 x6_7 G) 3658 C) 2406 H) 378 D) 406 I ) 324 E) none o f t h e s e J ) n o n e o f t h e s e 7) 6215 A) 2 9 2 , 1 0 5 8) 6075 F) 2 , 6 8 1 , 7 9 5 x 47 B) 2 9 2 , 0 0 5 x 423 G) 2 , 5 6 9 , 7 2 5 C) 4 3 , 5 0 5 H) 2 , 5 6 2 , 7 7 5 D) 2 , 4 8 6 , 5 2 5 I ) 2 , 5 5 5 , 7 2 5 E) none o f t h e s e . J ) n o n e o f t h e s e 9) 7003 A ) 2 , 8 4 3 , 2 1 8 10) 7)84 F) 6 x 406 B) 2 8 , 0 4 3 , 2 1 8 G) 10 O 2 , 4 4 3 , 2 1 8 H) 10 r l D) 3 2 2 , 0 1 8 I ) 12 E) •none o f t h e s e J ) none o f t h e s e 1 4 2 11) 96 * 5 = A) 10 B) 11 r 5 C) 19 D) 19 r l E) none o f t h e s e 12) 7)8216 F ) 112 G) 1000 r l •fl) 1314 r 7 I ) 1173 r 5 J ) none o f t h e s e 13) 3 )4032 A) 1010 r 3 B) 1034 r l C) 1344 D) 1343 r 2 E) none o f t h e s e 14) 1 + 2 - F) 1 G) — H) 19 » i f J ) none o f t h e s e 1 5 ) 2 l + ! = A ) , 2 i f B) 2 ^ O 2 | D) 23 E) none o f t h e s e 1 6 ) 4 I + 2 | = p ) 6 2 | 0) 2| 1) 6 | J ) none o f t h e s e 17) 4 l i + 3 l 3 + 6 l 2 , A) 1 3 g ' . IT c ) ' 1 3 i r E) none o f t h e s e 23 H) 7 | I ) J ) none o f t h e s e W f - i - A j ' j f B) f « % E) . none o f t h e s e 2 0 ) Z j f - J - f) £ H) 2A°-. ' i) J ) none o f t h e s e 143 21) 8 - 5 7 = 4 A) B) C) D) E) 10 4 none of these 22) 6 - - 2 y - F) G) H) 4 31 7 i) 4 J) none of these 23) Change to its' lowest terms. 24) Change j | to i t s lowest terms, « f B) C) . E) none of these G) H) I) 12 18 5_ 9 L5 17 J) none of these END OF PART 3 Appendix D ARITHMETIC INFORMATION, FIFTH GRADE Student Number Following ar.e three different sets of arithmetic questions. These questions w i l l t e l l us what arithmetic you already know and what arithmetic you w i l l learn during this school year. Some of the questions w i l l be easy for you and some of the questions w i l l contain arithmetic you have not had. Each question has four or five possible answers. Choose the answer you believe is correct and mark i t on the answer sheet as the f i r s t three examples have been done. 1) If 4 + n = 9, then n = ?. A) 4 B) 5 C) 6 D> 7 •; E) none of these Notice that "B" has been marked on the answer sheet for example I because 5 is the correct answer. 2) If 4 x p = 16, then p = ?. F) 2 G) 3 H) 4 I) 5 J) none of these Notice that'"H" has been marked on the answer sheet for example 2 because 4 is the correct answer. 3) If 24 * 3 = n, then n = ?. A) 2 B) 3 C) 4 D) 5 E) none of these Notice that "E" has been marked on the answer sheet for example 3 because the correct answer, 8, is not given as a choice. 144 145 Good heavy pencil marks have been made in the correct spaces. This w i l l be necessary when you mark your answers. Now, do examples 4 and 5. 4) If 9 - p = 4, then p = ?. F) 2 G) 3 H) 4 I) 5 J) none of these 5) If (1 + 2) + n = 7, then n = ?. A) 2 B) 3 C) 4 D) 5 E) none of these You should have marked " I " as the correct answer for example 4. You should have marked "C" as the correct answer for example 5. You are now ready to answer the questions as you have done the examples above. PART 1 Following i s a set of 43 questions which w i l l t e l l us what arithmetic understandings you know. Do not use paper to solve any of these questions. You should solve each of them i n your head and then mark your answer on the answer sheet. Do not write in this test booklet or on the answer sheet. You w i l l have 33 minutes to complete Part 1. Stop at the end of Part 1. At this time, ask any questions you might have. 1) In the expression 427 x 638 x 546, how w i l l the answer be changed i f i t is worked 546 x 638 x 427? A) the answer w i l l be less B) the answer w i l l be greater C) the answer w i l l be the same D) can't t e l l until i t is worked out 146 2) A mixed number such: as 3 | means 4 F) 3 x r ' G) 3 + f • H) 3 - | J) none of these 3) The shaded part of this figure i s what fractional part of the figure? B, f C) 1 4 .VAV.V.V.V .•.v.v.v.v.v %%\Vf.ti.rA.< E) none of these 4) The inverse operation of division i s F) addition G) subtraction H) multiplication I) division J) none of these 5) These statements are true: 4 + 6 = r, 6 +6 = s, 4 + 4 = t, 6 + 4 = u. Which of the following is also true? A) r = s B) r = t C) r = u D) s = u E) none of these Which shaded f i g u r e shows one-half of one-third? J) hone of these If 7 x t = 0, the " t " i s always A) zero ' B) one C) 7 D) ten ; E) i t i s -impossible to t e l l from the information given The inverse operation of m u l t i p l i c a t i o n i s F) addition G) subtraction H) m u l t i p l i c a t i o n I) d i v i s i o n J) none of these As the denominator of a f r a c t i o n a l number decreases and the numerator remains the same, the number A) becomes larger B) becomes smaller C) remains the same D) approaches one E) can't t e l l from the information given As the numerator of a f r a c t i o n a l number decreases and the denominator remains the same, the number F) becomes smaller G) becomes l a r g e r H) remains the same I) gets close to one J) can't t e l l from the information given 148 11) T h e s e s t a t e m e n t s a r e t r u e ; a + b = d , c + b = e , c + c = f . W h i c h o f t h e f o l l o w i n g i s a l s o t r u e ? A) a = b + d B) c = b + e C) d + e = f D) e = b + c E) n o n e o f t h e s e 12) I f 0 x y = 0 , t h e " y " i s a l w a y s F ) z e r o G) one H) two I ) t e n J ) any number y o u c h o o s e 13) W h i c h o f t h e f o l l o w i n g i s a n o t h e r n u m e r a l f o r 526? A) (5 x 10 x 10) + (2 x 10) 4>6 B) (5 x 5 x 5) + (2 x 2) + 6 C) (5 x 100) + (2 x 100) + 6 D) (5 x 2 x 6 x 100) + (2 x 6 x 10) + 6 E) none o f t h e s e 14) A f r a c t i o n s u c h as y means F) c h o o s i n g 7 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 5 p a r t s G) c h o o s i n g 2 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 5 p a r t s H) c h o o s i n g 5 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 7 p a r t s I ) c h o o s i n g 2 p a r t s a f t e r d i v i d i n g an o b j e c t i n t o 7 p a r t s J ) none o f t h e s e 15) I f a x b = 0 , t h e n A ) " a " a l w a y s e q u a l s " b " B) " b " must be z e r o C) " a " m u s t b e z e r o D) e i t h e r " a " o r " b " must b e z e r o E) n o n e o f t h e s e 16) W h i c h f r a c t i o n i s t h e s m a l l e s t ? F ) G) H) I ) J ) 1 3 2 3 1. 4 3 4 1^  2 V 149 17) If r x s = r, then "s" i s always A) zero B) one C) " r " D) ten E) i t is •impossible to t e l l from the information given 3 now 027 7 033 0 ? How far w i l l he ride F) more than 5 miles G) less than 5 miles H) exactly 5 miles I) can't t e l l from the information given 19) If r + s = r, then "s" is always A) zero B) one C) " r " D) ten . E) i t i s impossible to t e l l from the information given 645 20) To subtract in the exercise » w e should F) make the 5 ones smaller G) make the 5 ones larger H) make the 4 tens smaller I) make the 4 tens larger J) none of these 21) If u - v = u, the "v" is always A) zero B) one C) "u" D) ten E) i t i s impossible to t e l l from the information given 5_ 22) In the exercise 7) 364, the 5 really stands for F) 5 ones G) 5 tens H) 5 hundreds I) 5 tenths J) none of these 150 23) This line segment is cut into I I I I I I I 1 • 1 A) sevenths B) eighths C) ninths D) tenths* E) none of these 24) Four thousand three hundred seven i s written ; F) 4037 * G) 0437 H) 4370 I) 40003007 J) none of these 25) Look at the numeral 85,626. The 6 on the l e f t has a value how many times larger than the 6 on the right? A) 100 times larger B) 10 times larger C) 1000 times larger D) the same value E) none of these 937 26) When multiplying in the problem we move the second partial product, which we get when we multiply by 8, one place to the l e f t because F) that Is the rule in multiplying G) the 8 means 8 tens H) the answer must be larger than 937 I) the top number is a number larger than ten J) none of these 27) Look at b * a where "a" and "b" are both whole numbers greater than one. How does the answer compare with "b"? A) the answer is greater than b B) the answer is smaller than b C) the answer is the same as b D) can't t e l l u n t i l we see the whole number E) can't t e l l until the division is done 28) Three-thirds plus four-fourths is F) seven-sevenths G) twelve-twelfths * i I) 2 J) none of these 151 29) W h i c h o f t h e f o l l o w i n g shows i n a n o t h e r f o r m ? A ) 4 B) 4 C > i f D) 4 E) n o n e 30) The s h a d e d p a r t o f t h e f i g u r e i s what p a r t o f t h e f i g u r e ? i) i 1 'AVAVAVA C'AWAWA VA"WAVA • i H M M J ) n o n e o f t h e s e 31) I n t h e d i v i s i o n e x a m p l e 463)5217468 , t h e f i r s t f i g u r e i n t h e q u o t i e n t w i l l be w r i t t e n i n w h a t co lumn? A ) t e n s B) h u n d r e d s C) t h o u s a n d s D) t e n t h o u s a n d s E) n o n e o f t h e s e 32) When f i n d i n g t h e sum o f s e v e r a l numbers o f t h e same s i z e , t h e o p e r a t i o n t h a t w i l l g i v e us t h e a n s w e r most q u i c k l y i s F.) a d d i t i o n G) s u b t r a c t i o n H) m u l t i p l i c a t i o n I ) d i v i s i o n J ) none o f t h e s e 33) W h i c h o f t h e f o l l o w i n g w i l l g i v e t h e same a n s w e r a s 13 x 23? A) (13 x 20) - 3 B) (13 x 20) + 3 C) (13 x 20) + (13 x 3) D) (10 x 20) + (3 x 3) E) none o f t h e s e 152 34) O n e - f o u r t h o f t h e s e t o f x ' s t o t h e r i g h t b e l o w i s F ) xx /'xxxxxx-) G) xxx <J > H) xxxx ( xxxxxx) I ) xxxxx J ) n o n e o f t h e s e 35) L o o k a t t h e p r o b l e m r * s . I f " s " i s t h e i d e n t i t y number f o r t h i s d i v i s i o n , t h e n " s " i s e q u a l t o A) z e r o B) one C) t e n D) t h e r e i s n ' t one \ E) n o n e o f t h e s e 36) * 1 e q u a l s w h i c h o f t h e f o l l o w i n g ? o « i . .... o, | • H , | J ) none o f t h e s e 37) To r e d u c e a f r a c t i o n t o l o w e s t t e r m s we A) d i v i d e t h e n u m e r a t o r b y t h e d e n o m i n a t o r B) d i v i d e . t h e d e n o m i n a t o r b y t h e n u m e r a t o r C) d i v i d e t h e n u m e r a t o r a n d t h e d e n o m i n a t o r b y z e r o D) d i v i d e t h e n u m e r a t o r and t h e d e n o m i n a t o r b y a common d i v i s o r E) n o n e o f t h e s e 38) What n u m e r a l i s t h e same as t e n a n d o n e - t e n t h ? F ) 1 0 0 . 1 0 G) 1 0 0 . 0 1 H) 1 0 . 0 1 0 I ) 1 0 . 0 1 J ) none o f t h e s e 39) L o o k a t u x v w h e r e " u " and " v " a r e b o t h w h o l e numbers g r e a t e r t h a n o n e . How does t h e a n s w e r compare w i t h " v " ? A) t h e a n s w e r i s g r e a t e r t h a n v B) t h e a n s w e r i s s m a l l e r t h a n v C) t h e a n s w e r i s t h e same a s v D) c a n ' t t e l l u n t i l I s e e t h e w h o l e numbers E) c a n ' t t e l l u n t i l I do t h e m u l t i p l i c a t i o n 153 W h i c h f r a c t i o n a l number i s b e t w e e n 2 and 3? F ) G) H) I ) J ) none o f t h e s e 2 2 2 3 11 5 13 4 L o o k a t t h e p r o b l e m u + v w h e r e " u " and " v " a r e b o t h w h o l e numbers g r e a t e r t h a n z e r o . . I f t h e i r sum i s a n odd n u m b e r , t h e n A) b o t h u and v a r e e v e n numbers B) b o t h u and v a r e o d d numbers C) one number i s e v e n and one number i s odd D) v i s a l w a y s t w i c e as l a r g e as u E) none o f t h e s e The one i n t h e n u m e r a l . 0 5 1 3 i s i n t h e F) ones p l a c e G) t e n t h s p l a c e H) h u n d r e d t h s p l a c e I ) t h o u s a n d t h s p l a c e J ) n o n e o f t h e s e To f i n d t h e a n s w e r t o 34)238 we c o u l d A) m u l t i p l y t h e answer and 34 B) d i v i d e 8 by 4 C) add 238 t h i r t h - f o u r t i m e s and u s e t h e sum as t h e a n s w e r D) f i n d o u t how many 3 4 ! s c a n be s u b t r a c t e d f r o m 238 and u s e t h i s number as t h e a n s w e r E) none o f t h e s e END OF PART 1 DO NOT TURN THE PAGE 154 PART 2 F o l l o w i n g i s a s e t o f 13 p r o b l e m s w h i c h w i l l t e l l us what a r i t h m e t i c p r o b l e m s y o u c a n w o r k . Y o u may u s e p a p e r t o s o l v e t h e s e p r o b l e m s . P u t t h r e e c l e a n s h e e t s o f p a p e r on y o u r d e s k t o f i g u r e o n . Do n o t w r i t e i n t h i s t e s t b o o k l e t o r o n t h e a n s w e r s h e e t . Y o u w i l l h a v e 22 m i n u t e s t o c o m p l e t e P a r t 2 . Do n o t go on t o P a r t 3 u n t i l y o u a r e i n s t r u c t e d t o do s o . A t t h i s t i m e , a s k any q u e s t i o n s y o u m i g h t h a v e . 1) The g r o c e r h a d 137 pounds o f a p p l e s . A t t h e end o f t h e day he h a d 49 pounds o f a p p l e s . How many pounds o f a p p l e s d i d he s e l l d u r i n g t h e day? A) 111 pounds B) 78 pounds C) 88 pounds D) 98 pounds E) none o f t h e s e 2) A f t e r e a r n i n g 35<? a day f o r 14 d a y s J o e s t i l l n e e d s $ 2 . 4 7 t o buy a p r e s e n t . How much does t h e p r e s e n t c o s t ? F) $ 4 . 9 0 G) $ .49 H) $ 2 . 8 2 I ) $ 7 . 3 7 J ) none o f t h e s e 3) J a n e h a d a p i e c e o f r i b b o n 4 y a r d s l o n g . She c u t i t i n t o ; 8 p i e c e s o f e q u a l l e n g t h . What was t h e m e a s u r e i n i n c h e s o f e a c h p i e c e ? A) 72 i n c h e s B) 18 i n c h e s C) 16 i n c h e s D) 32 i n c h e s E) none o f t h e s e 4) F r a n k had 17 m o d e l s a t t h e b e g i n n i n g o f t h e y e a r . Now he h a s 36 m o d e l s . A t $ 1 . 3 5 each what i s t h e v a l u e o f t h e m o d e l s added t o h i s c o l l e c t i o n ? F) $ 2 5 . 6 5 G) $ 4 8 . 6 0 H) $ 2 6 . 6 5 I) $ 7 1 . 5 5 J ) none o f t h e s e 5) A t t h e c l u b p i c n i c t h e r e w e r e 7 g a l l o n s o f i c e c r e a m . I f one q u a r t o f i c e c ream s e r v e s 8 p e o p l e , how many p e o p l e c a n be s e r v e d ? A) 56 p e o p l e B) 112 p e o p l e C) 168 p e o p l e D) 224 p e o p l e E) none o f t h e s e 155 6) The Boys Club collected $9.80 selling popcorn, at 5 cents a bag. How many bags of popcorn did they sell? F) 4900 bags G) 110 bags H) 196 bags I) 112 bags J) none of these 7) Jane made scores of 25, 22, 18, 21, 15, and 25 on arithmetic tests. What was Jane's average score? A) 126 points B) 105 points C) 21 points P) 15 points E) none of these 8) Jaine bought 4 records at $1.89 each. The tax on a l l 4 records together was 3 4 c . How much change should she get from $10.00? F) $1.08 G) $2.10 $2.44 $7.56 H) I) J) none of these 9) Jim has four small rabbits. They weigh X 3 y 12- oz., 13^ - oz., 11- oz., and lCr| oz. How much do a l l four rabbits weigh? A) 4&|| oz. B) 48^ oz. o C) 48yjoz. D) 49^| oz. E) none of these 10) Phil had saved 8^ - dollars. He earned 4 j dollars. He then spent 9j dollars. How much did he have left? F) $3.00 G) $3.50 H) $4.00 I) $4.50 J) none of these 11) Mr. Smith needs 1200 sq. f t . of storage A) 945 sq. f t . space. He rented one building that was B) 255 sq. f t . 35 f t . by 27 f t . How many more sq. f t . C) 260 sq. f t . of space does he need? D) 23 f t . by 27 f t . more E) none of these 156 12) J e a n , J a n e , and M a r y w e i g h 194^- l b s . F) 115-3- l b s . 4 t o g e t h e r . I f J a n e w e i g h s 6 0 j l b s . and G > 79y l b s . 4 J e a n w e i g h s 55^- l b s . , how much does H) 79 l b s . M a r y w e i g h ? I ) 7 8 | l b s . J ) none o f t h e s e 13) M r . S m i t h d r o v e 456 m i l e s i n 8 h o u r s . He A) 24 m i l e s e x p e c t s t o d r i v e 3 more h o u r s a t t h e same B) 1368 m i l e s a v e r a g e s p e e d . How many more m i l e s does C) 459 m i l e s h e e x p e c t t o d r i v e ? D) 171 m i l e s E) none o f t h e s e END OF PART 2 DO NOT TURN THE PAGE 157 PART 3 F o l l o w i n g i s a s e t o f 23 p r o b l e m s w h i c h w i l l t e l l us what a r i t h m e t i c p r o b l e m s y o u c a n w o r k . Y o u may u s e p a p e r t o s o l v e t h e s e p r o b l e m s . P u t t h r e e c l e a n s h e e t s o f p a p e r o n y o u r d e s k t o f i g u r e o n . Do n o t f i g u r e i n t h i s t e s t b o o k l e t o r o n t h e a n s w e r s h e e t . Y o u w i l l h a v e 28 m i n u t e s t o c o m p l e t e P a r t 3 . A t t h i s t i m e , a s k any q u e s t i o n y o u m i g h t h a v e . 1) 7003 A ) 3 , 0 6 7 , 3 1 4 x 438 B) • 3 , 0 5 7 , 3 1 4 C) 2 , 9 6 8 , 3 1 4 D) 256 ,114 E) , none o f t h e s e 2) 6 )4302 F) 700 r l 0 2 G) 850 r 2 H) 717 I ) 1203 J ) none o f t h e s e 3) 5 6 ) 3 3 6 , 8 9 6 A) 601 r 8 4 0 B) 1 616 C) *' 5305 r 3 6 D) 6016 E) none o f t h e s e 4) 3 0 2 ) 4 , 1 7 3 , 0 5 7 F) 1 3 , 4 5 3 r 2 5 1 G) 1361 r 3 8 6 H) 1 3 , 8 1 8 r 2 1 I ) 1411 r 3 0 2 J ) none o f t h e s e 16 5) Change t o i t s l o w e s t t e r m s A) 4 « ! E) . , none o f t h e s e 21 6) Change y j t o i t s l o w e s t t e r m s « i • • H, | f I ) I j J ) none o f t h e s e 15$ 19 7) Change —j t o a m i x e d number i n l o w e s t t e r m s A, ^ B) 8 | C) 3 | D) 4 . E) none o f t h e s e 8) Change 4 j t p an i m p r o p e r f r a c t i o n F) 4 j G) 5 | H) 4 ± i) 4 J ) n o n e o f t h e s e 9 , A, 30 C) 41 D) 5 E) none o f t h e s e 10) y + ! = F ) y H ) i f J ) none o f t h e s e ID f + f - A) | B> i f «=> k D) i f E) none o f t h e s e 1 2 , H) 30 » i f J ) none o f t h e s e »>.f-V- « 3 ? « ! « I » f E) none o f t h e s e U ) f - i - F) i « I? « ! J ) n o n e o f t h e s e 15? 15) 4 - 3 5 A) 4 | B) -3| C) 4 E) none o f t h e s e 16) 24-A : •15-F) •*£ G) H) I ) J ) 17-11. none o f t h e s e 17) 1 0 l T -4 A ) B> C) D) ^ 5 . 80 4 5 5 -8 ^ 5 6 5 5 18) 2 . 3 + 4 . 6 F) 30 G) 6 . 9 H) 8 . 9 I ) 8 . 1 8 J ) n o n e o f t h e s e E) none o f t h e s e 19) 8 . 4 + . 4 = 20) 1 5 . 6 3 + 4 . 7 2 + 2 . 5 = A) 26 F) 2 1 . 1 4 B) 1 2 . 4 G) 2 2 . 8 5 C) 8 . 8 H) 2 1 . 1 8 5 D) : 8 . 0 I ) 2 1 . 1 4 1 E) none o f t h e s e J ) none o f t h e s e 21) 8 . 6 2 - 6 . 4 1 = A) B) C) D) E) 22) 2 2 . 7 4 -730 .116 2 . 2 1 230 1 4 . 1 0 3 none o f t h e s e 9 . 8 = F) G) H) I ) J ) 2 1 . 8 6 ' 7 5 . 2 6 1 2 . 9 4 1 3 . 9 4 n o n e o f t h e s e 23) 1 4 . 2 1 - .4 = A) 1 4 . 2 3 B) 1 0 . 2 1 C) 1 4 . 1 7 D) 1 3 . 8 1 E) none o f t h e s e END OF PART 3 Appendix E-ARITHMETIC INFORMATION, SIXTH GRADE Student Number Following are three different sets of arithmetic questions. These questions w i l l t e l l us what arithmetic you already know and what arithmetic you w i l l learn during this school year. Some of the questions w i l l be easy for you and some of the questions w i l l contain arithmetic you have not had. Each question has four or five possible answers. Choose the answer you believe is correct and mark i t on the answer sheet as the f i r s t three examples have been done. 1) If 4 + n = 9, then n = ?. A) 4 B) 5 C) 6 D) 7 E) none of these Notice that "B" has been marked on the answer sheet for example 1 because 5 is the correct answer. 2) If 4 x p = 16, then p = ?. F) 2 G) 3 H) 4 I) 5 J) none of these Notice that "H" has been marked on the answer sheet for example 2 because 4 i s the correct answer. 3) • If 24 T 3 = n, then n = ?. A) 2 B) 3 C) 4 D) 5 E) none of these Notice that "E" has been marked on the answer sheet for example 3 because the correct,answer, 8, i s not given as a choice. 160 161 . Good heavy, pencil marks have been made in the correct spaces. This w i l l be necessary when you mark your answers. Now, do examples 4 and 5. 4) If 9 - p = 4, then p = ?. F) 2 G) 3 H) 4 I) 5 J) none of these (1 + 2) +.n •-' 7, then A) 2 B) '3 : C) 4 D) 5-E) .none of these You should have marked " I " as the correct answer for example 4. You should have marked "C" as the correct answer for example 5. You are now ready to answer the questions as you have done the examples above. PART 1 Following is- a set of 44 questions which w i l l t e l l us what arithmetic understandings you know. Do not use paper to solve any of these questions. You should solve each of them in your head and then mark your answer on the answer sheet. Do not write in this test booklet or on the answer sheet. You w i l l have 34 minutes to complete Part 1. Stop at the end of Part 1. Do not go on to Part 2 until you are instructed to do so. At this time, ask any questions you might have. 1) As the denominator of a fractional number decreases and the numerator remains the same, the number A) becomes larger B) becomes smaller C) remains the same D) approaches one E) can't t e l l from the information given 162 2) As the numerator of a fractional number decreases and the denominator remains the same, the number F) becomes smaller G) becomes larger H) remains the same I) gets close to one J) can't t e l l from the information given 3) These statements are true: a + b = d, c + b = e, c + c = f . Which of the following is also true? A) a = b + d B) c = b + e C) d. + e = f D) e = b + c E) none of these 4) If 0 x y = 0, then "y" is always F) zero G) one H) two I) ten J) any number you choose 5) Which of the following i s another numeral for 526? A) (5 x 10 x 10) + (2 x 10) + 6 B) (5 x 5 x 5) + (2 x 2) + 6 C) (5 x 100) + (2 x 100) + 6 D) (5 x 2 x 6 x 100) + (2 x 6 x 10) + 6 E) none of these 6) A fraction such as y means F) choosing 7 parts after dividing an object into 5 parts G) choosing 2 parts after dividing an object into 5 parts H) choosing 5 parts after dividing an object into 7 parts I) choosing 2 parts after dividing an object into 7 parts J) none of these 7) If a x b = 0, then A) "a" always equals "b" B) "b" must be zero C) "a" must be zero D) either "a" or "b" must be zero E) none of these 163 8) Which fraction i s the smallest? » t • • • • 9) If r x s = r, then "s" i s always A) zero B) one C) " r " D) ten E) i t i s impossible to t e l l from the information given 10) Jimmy's bike has a speedometer which sh,ows miles and tenths of miles. It looks like this now I 027 I 7 13) How far w i l l he ride before the speedometer reads F) more than 5 miles G) less than 5 miles H) exactly 5 miles I) can't t e l l from the information given 11) If j + s = r, then "s" i s always A) zero B) one C) " r " D) ten E) i t i s impossible to t e l l from the information given 645 12) To subtract in the exercise » w e should F) make the 5 ones smaller G) make the 5 ones larger H) make the 4 tens smaller I) make the 4 tens larger J) none of these - V = u . then " v " is always A) zero B) one C) "u" D.) ten E) i t is impossible to t e l l 14) In the exercise 7)364 , the 5 really stands for F) 5 ones G) 5 tens H) 5 hundreds I) 5 tenths J) none of these 15) This line segment is cut into 1 •  h-1 -4 I I I I—I A) sevenths B) eighths C) ninths D) tenths E) none of these 16) Four thousand three hundred seven is written F) 4037 G) 0437 H) 4370 • I) 40003007 J) none of these 17) Look at the numeral 85,626. The 6 on the l e f t has a value how times larger than the 6 on the right? A) 100 times larger B) 10 times larger C) 1000 times larger D) the same value E) none of these 937 18) When multiplying in the probelm R^ we move the second partial product, which we get when we multiply by 8, one place to the l e f t because F) that is the rule in multiplying G) the 8 means 8 tens H) the answer must be larger than 937 I) the top number is a number larger than ten J) none of these 19) Which of the following w i l l give the same answer as 13 x 23? A) (13 x 20) - 3 B) (13 x 20) + 3 C) (13 x 20) + (13 x 3) D) (10 x 20) + (3 x 3) E) none of these 165 20) O n e - f o u r t h o f t h e s e t o f x ' s t o t h e r i g h t i s F ) xx (xxxxxx1 G) xxx } \ H) xxxx (xxxxxx) I ) xxxxx J ) n o n e o f t h e s e 21) L o o k a t t h e p r o b l e m r * s . I f " s " i s t h e i d e n t i t y number f o r t h i s d i v i s i o n , t h e n " s " i s e q u a l t o A ) z e r o B) one C) t e n D) t h e r e i s n ' t one E) n o n e o f t h e s e 22) , | T 1 e q u a l s w h i c h o f t h e f o l l o w i n g ? o " f « ! J ) n o n e o f t h e s e 23) To r e d u c e a f r a c t i o n t o t h e l o w e s t t e r m s we A ) d i v i d e t h e n u m e r a t o r b y t h e d e n o m i n a t o r B) d i v i d e t h e d e n o m i n a t o r by t h e n u m e r a t o r C) d i v i d e t h e n u m e r a t o r and t h e d e n o m i n a t o r b y z e r o D) d i v i d e t h e n u m e r a t o r and t h e d e n o m i n a t o r b y a common d i v i s o r E) none o f t h e s e 24) What n u m e r a l i s t h e same as t e n and o n e - t e n t h ? F) 1 0 0 . 1 0 G) 1 0 0 . 0 1 H) 1 0 . 0 1 0 I ) 1 0 . 0 1 J ) none o f t h e s e 25) L o o k a t u x v w h e r e " u " and " v " a r e b o t h w h o l e numbers g r e a t e r t h a n o n e . How d o e s t h e a n s w e r compare w i t h " v " ? A) t h e a n s w e r i s g r e a t e r t h a n v B) t h e a n s w e r i s s m a l l e r t h a n v C) t h e a n s w e r i s t h e same as v D) c a n ' t t e l l u n t i l I s e e t h e w h o l e numbers E) c a n ' t t e l l u n t i l I do t h e m u l t i p l i c a t i o n 166 26) W h i c h f r a c t i o n a l number i s b e t w e e n 2 and 3? ' . F ) G) H) i) J ) none o f t h e s e 3 2 2 3 11 '< 5 13 4 27) L o o k a t t h e p r o b l e m u + v w h e r e " u " and " v " a r e b o t h w h o l e numbers g r e a t e r t h a n z e r o . I f t h e i r sum i s an odd n u m b e r , t h e n A) b o t h u and v a r e e v e n numbers B) b o t h u and v a r e odd numbers C) one number i s e v e n and one number i s odd D) v i s a l w a y s t w i c e as l a r g e as u E) none o f t h e s e 28) The one i n t h e n u m e r a l . 0513 i s i n t h e F) o n e s p l a c e G) t e n t h s p l a c e H) h u n d r e d t h s p l a c e I ) t h o u s a n d t h s p l a c e J ) none o f t h e s e 29) What p a r t o f t h e f i g u r e i s shaded? A) 0 . 0 3 B) 3 . 0 0 C) 0 . 0 7 D) 7 .00 E) none o f t h e s e 30) What does h% mean? F) 100 x h H) 100 I ) J ) d i v i d e " h " i n t o 100 p a r t s none o f t h e s e To f i n d t h e a n s w e r t o 34 )238 we c o u l d A) m u l t i p l y t h e a n s w e r and 34 B) d i v i d e 8 by 4 C) add 238 t h i r t y - f o u r t i m e s and use t h e sum o f t h e answer D) f i n d o u t how many 3 4 ' s c a n be s u b t r a c t e d f r o m 238 and u s e t h i s number as t h e a n s w e r E) n o n e o f t h e s e W h i c h number i s l a r g e r t h a n 4 . 0 3 5 ? F ) 4 . 0 3 4 G) 4 . 0 2 9 H) 4 . 1 I ) 4 . 0 0 0 J ) none o f t h e s e 754 To o b t a i n t h e a n s w e r t o t h e e x e r c i s e > we must change t h e f o r m o f one n u m b e r . What number i s i t ? A) 7 ones B) 11 t e n s C) 8 h u n d r e d s D) 9 h u n d r e d s E) n o n e o f t h e s e I f " a " r e p r e s e n t s an odd n u m b e r , t h e n e x t l a r g e r odd number c a n b e r e p r e s e n t e d as F ) 2 x a G) (2 x a) + 1 H) a + 1 I ) a + 2 J ) none o f t h e s e I f " u " i s any number d i f f e r e n t f r o m z e r o , t h e n u * u i s e q u a l t o A ) z e r o B) one C) t e n D) V E) none o f t h e s e How many e v e n w h o l e numbers a r e t h e r e b e t w e e n 35 and 39? F) none G) one H) two I ) t h r e e J ) none o f t h e s e 168 W h i c h f r a c t i o n i s t h e l a r g e s t ? A) B) C) D) E) 1 3 2 3 1 4 3 4 1 2 L o o k a t t h e p r o b l e m a x £ w h e r e " a " i s a w h o l e n u m b e r , b i g g e r t h a n z e r o and ^ i s a p r o p e r f r a c t i o n . How does t h e a n s w e r compare w i t h t h e w h o l e number " a " ? F) t h e answer i s l a r g e r t h a n t h e w h o l e number G) t h e a n s w e r i s s m a l l e r t h a n t h e w h o l e number H) t h e answer i s t h e same as t h e w h o l e number I ) we c a n ' t t e l l u n t i l we s e e t h e numbers J ) we c a n ' t t e l l u n t i l t h e p r o b l e m i s w o r k e d L o o k a t t h e p r o b l e m a * — w h e r e " a " i s a w h o l e number b i g g e r t h a n b c z e r o and — i s an i m p r o p e r f r a c t i o n d i f f e r e n t f r o m o n e . How does b t h e a n s w e r compare w i t h t h e w h o l e number " a " ? A) t h e answer i s l a r g e r t h a n t h e w h o l e number B) t h e a n s w e r i s s m a l l e r t h a n t h e w h o l e number C) t h e answer i s t h e same as t h e w h o l e number D) we c a n ' t t e l l u n t i l we s e e t h e numbers E) we c a n ' t t e l l u n t i l t h e p r o b l e m i s w o r k e d 427 How w o u l d t h e a n s w e r t o ^ g b e c h a n g e d i f we c h a n g e d 427 t o 4270 and 58 t o 5 . 8 ? F ) t h e new answer w o u l d be t h e same as t h e o l d a n s w e r G) t h e new a n s w e r a n s w e r w o u l d be one- - t e n t h as , l a r g e as t h e o l d H) t h e new a n s w e r answer w o u l d be t e n t i m e s l a r g e r t h a n t h e o l d I ) t h e new answer w o u l d be one h u n d r e d t i m e s l a r g e r t h a n t h e o l d a n s w e r J ) none o f t h e s e 169 7 4 . 9 0 How w o u l d t h e answer t o , * 0 be c h a n g e d i f we removed t h e z e r o x 6 5 . 8 ° f r o m 7 4 . 9 0 ? A) t h e new a n s w e r w o u l d be e q u a l t o t h e o l d a n s w e r B) t h e new a n s w e r w o u l d b e o n e - t e n t h as l a r g e as t h e o l d a n s w e r C) t h e new answer w o u l d be o n e - h u n d r e d t h as l a r g e as t h e o l d a n s w e r D) t h e new answer w o u l d be l a r g e r t h a n t h e o l d a n s w e r b e c a u s e t h e r e w o u l d b e f e w e r d e c i m a l p l a c e s i n t h e new a n s w e r E) we c a n ' t t e l l u n t i l we do t h e m u l t i p l i c a t i o n How w o u l d t h e a n s w e r t o 950)63650 b e c h a n g e d i f t h e z e r o s i n t h e two numbers w e r e removed? F ) t h e new a n s w e r w o u l d be e q u a l t o t h e o l d a n s w e r G) t h e new answer w o u l d be one h u n d r e d t i m e s l a r g e r t h a n t h e o l d a n s w e r H) t h e new a n s w e r w o u l d be o n e - h u n d r e d t h as l a r g e as t h e o l d a n s w e r I ) t h e new a n s w e r w o u l d b e t e n t i m e s l a r g e r t h a n t h e o l d answer J ) none o f t h e s e W h i c h number i s s m a l l e r t h a n 2 . 0 4 7 ? A) 2 . 1 1 1 B) 2 . 0 4 8 C) 2 . 0 5 0 D) 2 . 1 E) none o f t h e s e 658 How does t h e a n s w e r t o ^ compare w i t h 658? F) t h e answer i s t e n t i m e s l a r g e r t h a n 658 G) t h e a n s w e r i s t h i r t y t i m e s l a r g e r t h a n 658 H) t h e answer i s 658 t i m e s l a r g e r t h a n 658 I ) t h e answer i s 39 t i m e s l a r g e r t h a n 658 J ) none o f t h e s e END OF PART 1 DO NOT TURN THE PAGE 170 PART 2 Following i s a set of 16 problems which w i l l t e l l us what arithmetic problems you can work. You may use paper to solve these problems. Put three clean sheets of paper on your desk to figure on. Do not write in this test booklet or on the answer sheet. You w i l l have 27 minutes to complete Part 2. Do not go on to Part 3 until you are instructed to do so. At this time, ask any questions you might have. 1) After delivering i t s 3^ ton load of hay, a A) 756 lbs. truck weighed 7256 pounds. How much did the B) 7,259T lbs. truck and i t s load weigh? 4 c) 13,756 lbs. D) 10,500 lbs. E) none of these 2) A hay dealer sold 124 tons of hay in F) •,/2 14— tons October, 133 tons in November, 147 tons 5 in December, 156 tons in January, and G > 110 tons 142 tons in February. What was the average amount of hay sold per month? H) 2 136-JT tons I) 14of tons J) none of these 2 3) Joe walks r of a mile to school. Frank A) 100 feet •J 1 B) 440 feet walks — as far. How many feet does Frank C) 880 feet D) 4840 feet walk to school? E) none of these 4) Mr. Carpenter, the grocer, had 12— lbs. of peanuts. He put them into bags containing -j pound each. How many bags of peanuts did he have? F) 5 bags G) 36 bags H) 38 bags I) 89 bags J) none of these 171 5) M r . J o n e s d r o v e h i s t r u c k 385 m i l e s i n 7 h o u r s . The maximum s p e e d l i m i t was 60 m i l e s p e r h o u r . How much b e l o w t h e maximum s p e e d was h i s a v e r a g e speed? 6) J a c k , a r a c e c a r d r i v e r , a v e r a g e d 93 m i l e s F ) 4 h o u r s 28 p e r h o u r i n a 400 m i l e r a c e . To t h e n e a r e s t m i n u t e s t e n t h o f an h o u r how l o n g d i d i t t a k e J a c k G) 4 . 3 . h o u r s t o c o m p l e t e t h e r a c e ? H) 4 h o u r s I ) 2 h o u r s J ) none o f t h e s e 7) M r s . S m i t h u s e s 12 cans o f w a t e r w i t h 3 A) 20 o z . c a n s o f f r o z e n j u i c e . I f e a c h c a n h o l d s B) 30 o z . 6 o u n c e s , how many o u n c e s o f w a t e r w i l l C) 120 o z . s h e u s e w i t h 5 c a n s o f f r o z e n j u i c e ? D) 150 o z . E) n o n e o f t h e s e 8) The s p e e d l i m i t i n a German c i t y i s F ) 40 k i l o m e t e r s p e r h o u r . To t h e n e a r e s t t e n t h i n m i l e s p e r h o u r , what i s t h e G) s p e e d l i m i t i n t h a t c i t y ? ( .62 m i = 1 km) H) I ) J ) A) 3 m i l e s p e r h o u r B) 5 m i l e s p e r h o u r C) 10 m i l e s p e r h o u r D) 55 m i l e s p e r h o u r E) none o f t h e s e 2 . 4 m i l e s p e r h o u r 2 4 . 8 m i l e s p e r h o u r 55 m i l e s p e r h o u r 70 m i l e s p e r h o u r none o f t h e s e 9) J o e d e l i v e r e d 85% o f h i s 160 p a p e r s . How A) 120 p a p e r s many p a p e r s d i d he d e l i v e r ? B) 135 p a p e r s C) 136 p a p e r s D) 188 p a p e r s E) none o f t h e s e 10) J a n e s p e l l e d 90% o f t h e w o r d s c o r r e c t l y . F ) 19 w o r d s She s p e l l e d 18 w o r d s c o r r e c t l y . How many G) 20 w o r d s w o r d s w e r e o n t h e t e s t ? H) 28 w o r d s I ) 108 w o r d s J ) n o n e o f t h e s e 11) J e a n t o o k a s p e l l i n g t e s t o f 80 w o r d s . A) 16% She s p e l l e d 64 o f t h e w o r d s c o r r e c t l y . B) 64% What p e r c e n t a g e o f t h e w o r d s d i d s h e C) 80% s p e l l c o r r e c t l y ? D) 85% E) none o f t h e s e 172 12) Joe took a 40 problem a r i t h m e t i c t e s t . F) 5 problems He c o r r e c t l y worked 80% of the problems. G) 8 problems How many of the problems d i d he miss? H) 10 problems I ) 20 problems J) none of these 13) In Woodland School 8% of the p u p i l s made A) 176 students a p e r f e c t score on an a r i t h m e t i c t e s t . B) 200 students I f 22 p u p i l s made a p e r f e c t score, how C) 275 students many p u p i l s attend Woodland School? D) 2024 students . E) none of these 14) . A b a s k e t b a l l team l o s t 40% of the 25 games F) 10 games i t played. How many games d i d the team G) 11 games win? H) 15 games I) 20 games J) none of these 15) The Jones f a m i l y had an income of $6500. A) 59% They saved $600 of t h i s money. To the B) 50% nearest whole percent, what percent of C) 10.5% the income d i d they save? D) 9% E) none of these 16) When buying a new car Mr. Woods paid a F) $160 sa l e s tax of 4-r-% on the purchase p r i c e . G) $170 . H) $180 The purchase p r i c e was $4000. How much I) $888.89 sa l e s tax d i d Mr. Woods pay? J) none of these END OF PART 2 DO NOT TURN THE PAGE 173 PART .3 F o l l o w i n g i s a s e t o f 30 p r o b l e m s w h i c h w i l l t e l l us w h a t a r i t h m e t i c p r o b l e m s y o u c a n w o r k . You may u s e p a p e r t o s o l v e t h e s e p r o b l e m s . P u t t h r e e c l e a n s h e e t s o f p a p e r on y o u r d e s k t o f i g u r e o n . Do n o t f i g u r e i n t h i s t e s t b o o k l e t o r on t h e a n s w e r s h e e t . You w i l l h a v e 36 m i n u t e s t o c o m p l e t e P a r t 3 . A t t h i s t i m e , a s k any q u e s t i o n s y o u m i g h t h a v e . 1) 80402 x 728 A) 5 8 , 7 3 2 , 6 5 6 B) 5 8 , 5 3 2 , 6 5 6 C) 5 8 , 5 3 9 , 6 5 6 D) 5 6 , 0 9 2 , 6 5 6 E) none o f t h e s e 2) 42 )91358 F) 2175 r 8 G) 2079 r 4 0 H) 2314 I ) 2172 r 3 4 J ) none o f t h e s e 3) 409)46782 A) 114 r l 5 6 B) 111 r 2 8 3 C) 1110 r 2 8 3 D) 114 r 5 6 E) none o f t h e s e 4) i + l ^ - 5 -J 3 5 6 F) G) H) I ) _9 14 _9 30 i l l 30 J ) none o f t h e s e 5) 4 "10 3 A) B) 1 6 l 9 1 6 6 0 6) 4 -11 D) I S 20 E) none o f t h e s e F ) G) H) I ) J ) " 1 1 , 7 3-A none o f t h e s e 174 7) " i t A) 30 B) C) IBf D) 1 7 - ^ 30 E) none 8) 1 0 - i 11 •4 F) G) H) I ) J ) 4 ' 5 5 -27 >55 . 5 >10 none o f t h e s e 9) . - l x 2 ; 10 x 6 A) B) C) D) 54 60 50 54 3 4 1, 3 10) | x 10 _ 1 20 E) n o n e o f t h e s e G) H) I ) 20 J ) n o n e o f t h e s e I D 4 | x 3 f A) :12f B) C) D) 22 27 15 16i E) none o f t h e s e ' 4 7 A) B) C) D) E) 15 28 20 21 2 ^ 12 none o f t h e s e 12) |x4X Tf = F) G) H) I ) 41 4 4 J ) none o f t h e s e 14) 10 * i - F ) 40 G) H) I ) J ) none o f t h e s e J L 40 10 4 _4 10 173 15) £ * 10 - A) 40 E) • none of these H) 5 6 I) 29 } 31 J) none of these 17) 15.63 +.2.55 = A) 13.08 B) : 18.18 C) 36.18 D) 13.18 E) none of these 18) 4.62 + .403 + 27.2 = F) 32.223 G) 11.37 H) 31.223 I) 12.12 J) none of these 19) 15.21 - 2.3 = A) 13 B) 12.86 C) 17.58 D) 12.84 E) ' none of these 20) 201.3 - 48.004 = F) 153.304 G) 721.34 H) 153.296 I) 720.34 J) none of these 21) 2 . 2 x 8 = A) 1.76 B) 176 C) 16.16 D) .176 E) none of these 22) 13 x .43 = F) 52.39 G) 52.29 H) 5.69 I) 5.59 J) none of these 23) 22.34 A) .160.848 x 7.2 B) 1608.48 C) 180.848 D) 1808.48 E) none of these 24) 42 F) .000168 x .0004 G) .0168 H) .00168 I) .168 J) none of these 25) To the nearest tenth 4.31)71.216 A) 16.0 B) 16.5 C) 16.6 D) 16.7 E) none of these 26) .005)47.2 F) 94500 G) 945 H) 94.5 I) 9450 J) none of these 176 27) To the nearest tenth of a percent 57 = % of 112 A) 50.9% B) 50% C) 19.9% D) 19.8% E) none of these 29) 45% = what fraction in lowest terms? , , « i V B, I c) D) _9 20 1 125 E) none of these 28) 192 is what percent of 256? F) 13% G) 14% H) 75% I) 15% J) none of these 30) 58% = what fraction in lowest terms? F) G) H) I) J) _5 8 2£ 50 8 5 1 160 none of these END OF PART 3 Appendix F TEACHER QUESTIONNAIRE Please complete the following question- Semester Quarter naire and return i t to your p r i n c i p a l . A l l Hours Hours hours are quarter hours. If your courses were • semester hours, use the table at the r i g h t to 3 U=-convert to quarter hours. C i r c l e the appropriate response. 4 6 Teacher number 5 7y 1) Number of quarter hours of college mathematics courses, including i n s e r v i c e salary c r e d i t course. (Do not include methods courses i n _ irJL mathematics.) none 1 to 7 7 to 13 13 to 19 19 or more 2 8 12 Were any of these courses calculus? yes no 9 i o l How many of these quarter hours were new mathematics? 10 15 none 1 to 7 7 to 13 13 to 19 19 or more ^ ^ 12 18 2) How long ago was the l a s t of these mathematics — w vuiu I- -1_ O courses taken? 1 1 1 Q 1 J-J ~2 14 21 past year 1 to 2 years 2 to 5 years 5 to 10 years 10 or more years 3) Number of quarter hours concerned with the teaching of mathematics, (Methods courses i n the teaching of arithmetic or mathematics. Do not include courses counted i n 1) above.) none 1 to 4 4 to 9 9 to 13 13 or more 4) How long ago was the l a s t of these methods courses taken? past year 1 to 2 years 2 to 5 years 5 to 10 years 10 or more years 177 Number o f q u a r t e r h o u r s o f p r o f e s s i o n a l e d u c a t i o n c o u r s e s . ( T h i s i n c l u d e s a l l e d u c a t i o n c o u r s e s and some p s y c h o l o g y c o u r s e s . A c l o s e a p p r o x i m a t i o n i s s u f f i c i e n t . ) 0 t o 20 20 t o 30 30 to 40 40 t o 50 50 o r more Number o f y e a r s o f t e a c h i n g e x p e r i e n c e . (Do n o t c o u n t t h i s y e a r o r s u b s t i t u t i n g e x p e r i e n c e . ) none 1 2 3 o r 4 5 o r 6 7 t o 10 10 t o 15 15 t o 20 20 o r more Number o f y e a r s o f t e a c h i n g e x p e r i e n c e i n t h e Spokane D i s t r i c t * (Do n o t c o u n t t h i s y e a r o r s u b s t i t u t i n g e x p e r i e n c e . ) n o n e 1 2 3 o r 4 5 o r 6 7 t o 10 10 t o 15 15 t o 20 20 o r more A p p e n d i x G P R I N C I P A L ' S RATING SHEET R a t e t h e t e a c h e r numbered on a s c a l e o f 1 t o 7. A r a t i n g o f 1 i n d i c a t e s a p o o r r a t i n g and a r a t i n g o f 7 i n d i c a t e s an e x c e l l e n t r a t i n g . C i r c l e t h e a p p r o p r i a t e r a t i n g . C o m p l e t e and r e t u r n w i t h y o u r answer s h e e t s . T e a c h e r number 1) C o n s i d e r i n g a l l a s p e c t s o f t e a c h i n g , t h i s t e a c h e r i s : 1 . 2 - 3 4 5 6 7 2) C o n s i d e r i n g a l l a s p e c t s o f a r i t h m e t i c t e a c h i n g , t h i s t e a c h e r i s : 1 2 3 4 5 6 7 3) C o n s i d e r i n g t h e newer methods and i d e a s i n a r i t h m e t i c , t h i s t e a c h e r b e l i e v e s i n and u s e s t h e s e i d e a s : 1 2 3 4 5 6 7 179 

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