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Study of the relation between teacher and student understanding of limit concepts taught in grade eight Broadley , George William 1970

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A STUDY OF THE RELATION BETWEEN TEACHER AND STUDENT UNDERSTANDING OF LIMIT CONCEPTS TAUGHT IN GRADE EIGHT by GEORGE WILLIAM BROADLEY B.A., University of B r i t i s h Columbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Faculty of Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb i a , 1 a g ree tha 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 s tudy . 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Education  The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT The purpose of t h i s study was to investigate i f a rela t i o n s h i p exists between the understandings of students and those of t h e i r teachers f o r a s p e c i f i c concept i n mathematics. A review of l i t e r a t u r e revealed that no study-had attempted to examine the r e l a t i o n between student and teacher understanding of a s p e c i f i c concept i n mathematics although several had investigated the r e l a t i o n between teacher and student understanding of general mathematical concepts, usually i n arithmetic. The single concept chosen for the present study was i n t u i t i v e l i m i t concepts as pre-scribed f o r Mathematics 8 students i n B r i t i s h Columbia schools. The following n u l l hypothesis was established and tested: For Mathematics 8 classes of better students there i s no s i g n i f i c a n t c o r r e l a t i o n between teacher under-standing of i n t u i t i v e l i m i t concepts and student under-standing of i n t u i t i v e l i m i t concepts. Measures of understanding were obtained by the use of two t e s t i n g instruments constructed by the investigator, one f o r students and one f o r teachers. The preliminary student test constructed was checked f o r content v a l i d i t y and given a t r i a l use. The r e l i a b i l i t y of the test was calculated and an item analysis made to determine which items to use i n the f i n a l form of the test. The teacher i i i t e s t constructed used hypothetical answers to student test items. Teacher test items were taxonomized according to Bloom. Fourteen classes of Mathematics 8 students of better a b i l i t y and t h e i r teachers were tested using the f i n a l form of each test. Class means f o r student tests were adjusted by analysis of covariance to allow f o r I n i t i a l differences i n i n t e l l i g e n c e and mathematics achievement. Calculation of the c o e f f i c i e n t of co r r e l a t i o n between these adjusted means and teacher scores gave a re s u l t of 0.09. This c o r r e l a t i o n was not s i g n i f i c a n t . Thus the n u l l hypothesis tested was accepted. ACKNOWLEDGEMENT F o r t h e i r c o n t i n u e d encouragement and p a t i e n c e , I w i s h t o t h a n k my a d v i s o r , D r . E r i c D. M a c P h e r s o n , my w i f e , L i l y , a n d my d a u g h t e r s , S h a r o n and D e b o r a h . F o r t h e i r c o -o p e r a t i o n and a s s i s t a n c e i n a r r a n g i n g t h e t e s t i n g p r o g r a m , I w i s h t o t h a n k M r . Horace R. Dawson, M r . H a r r y Downard , M r . J . M. Drummond, M r . D. G. M a c d o n a l d , and M r . D o n a l d S m i t h . F o r t h e i r work i n c a t e g o r i z i n g t e s t i t e m s , I w i s h t o t h a n k M r s . E . Kennedy and D r . Edgar B . H o m e . F i n a l l y , f o r t h e i r w i l l i n g n e s s t o be p a r t o f t h e s t u d y , I w i s h t o t h a n k t h e f o u r t e e n anonymous p a r t i c i p a t i n g t e a c h e r s . George W i l l i a m B r o a d l e y A u g u s t , 1970 TABLE OP CONTENTS CHAPTER PAGE I. THE PROBLEM 1 General Statement of the Problem . 1 Definitions of Terms 1 Mathematics 8 1 Mathematics 8 classes of better students . . 1 Student understanding of l i m i t s 1 Teacher understanding of l i m i t s 1 I n t u i t i v e l i m i t concepts 2 The Question 2 Footnotes 3 I I . REVIEW OF THE LITERATURE 4 Introduction . . . . * 4 Expressed Need f o r Teacher Understanding . . . 5 Mathematical Understandings of Teachers . . . . 6 Teacher Academic Preparation and Pupil Achievement 8 In-service Education and Pupil Achievement . . 10 Teacher Understanding and Pupil Achievement . . 11 Limits as a Topic i n Junior High School . . . . 13 The Hypothesis l 5 Footnotes 16 V CHAPTER PAGE I I I . DESIGN OF THE STUDY 19 Preparation of the Tests 19 Administration of the Tests . . 21 Co l l e c t i o n of Data 22 Treatment of Data 23 Footnotes 25 IV. ANALYSIS OF THE DATA 26 Preliminary Student Test Data 26 Preliminary Teacher Test Data . , 28 Fi n a l Test Data 30 Footnotes 32 V. SUMMARY AND CONCLUSIONS 33 Summary 33 Conclusions 34 BIBLIOGRAPHY 36 APPENDICES 41 LIST OF TABLES TABLE PAGE I. Point B i s e r i a l Correlation and Limit Topic Categorization of Test Items 27 I I . Taxonomization of Teacher Test Items 29 I I I . Teacher Scores and Adjusted Class Means 31 CHAPTER I THE PROBLEM I. GENERAL STATEMENT OP THE PROBLEM The proposition that one cannot teach what one does not understand would appear to be a most reasonable assump-ti o n . Does a teacher need to understand a concept i n math-ematics i n order that his students can develop an under-standing of t h i s same concept? The purpose of t h i s inves-t i g a t i o n i s to make a preliminary enquiry into the v a l i d i t y of t h i s assumption. II . DEFINITIONS OF TERMS Mathematics 8 i s the mathematics course prescribed fo r a l l grade eight students i n B r i t i s h Columbia Schools. 1 Mathematics 8 classes of better students refers to classes i d e n t i f i e d as such by the administration of each p a r t i c i p a t i n g school. Student understanding of l i m i t s i s a measure deter-mined by a test of i n t u i t i v e l i m i t concepts prepared as part of the study. Teacher understanding of l i m i t s i s a measure deter-mined by the accuracy of marking hypothetical answers pre-pared f o r the student test. 2 I n t u i t i v e l i m i t concepts are concepts related to the following ideas developed i n Mathematics 8: 1. The set of r a t i o n a l numbers i s i d e n t i f i e d with the set of repeating decimals. 2. The set of i r r a t i o n a l numbers i s i d e n t i f i e d with the set of i n f i n i t e , non-repeating decimals. 3. Rational and i r r a t i o n a l numbers are dense and can be compared to each other using the r e a l number l i n e . I I I . THE QUESTION An answer to the following question w i l l be sought: When students' r e s u l t s are compared with those of t h e i r teacher, i s there a relationship between student under-standing of i n t u i t i v e l i m i t concepts and teacher under-standing of these concepts? 3 FOOTNOTES .^British Columbia Department of Education, Secondary  School Mathematics, 1966, ( V i c t o r i a : B r i t i s h Columbia Department of Education, 1966), p. 7. 2 Charles F. Brumfiel, Robert E. Eicholz, and M e r r i l l E. Shanks, Introduction to Mathematics, (Reading, Massa-chusetts: Addison-Wesley Publishing Company, 1963), pp. 147-172. CHAPTER I I REVIEW 0 ? T H E L I T E R A T U R E I . INTRODUCTION A s e a r c h o f l i t e r a t u r e r e l e v a n t t o t h e p r o b l e m r e -v e a l s t h a t b o t h e x p e r t s a n d c l a s s r o o m t e a c h e r s h a v e i n d i -c a t e d a n e e d f o r i m p r o v e d u n d e r s t a n d i n g b y t e a c h e r s o f m a t h e m a t i c a l c o n c e p t s . I n v e s t i g a t o r s show c o n c e r n o v e r l a c k o f s u c h u n d e r s t a n d i n g , p r i m a r i l y a t t h e e l e m e n t a r y s c h o o l l e v e l . No i n v e s t i g a t i o n s w e r e f o u n d c o n c e r n i n g 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 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 t e a c h e r s . A v a r i e t y o f s t u d i e s h a v e a t t e m p t 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 e a c h e r a n d p u p i l u n d e r s t a n d i n g s o f m a t h e m a t i c a l c o n c e p t s . Some o f t h e s e h a v e r e l a t e d t h e a c a d e m i c b a c k g r o u n d i n m a t h e m a t i c s o f t e a c h e r s t o t h e a c h i e v e m e n t i n m a t h e m a t i c s o f t h e i r c l a s s e s . O t h e r s h a v e r e l a t e d t e a c h e r g a i n i n u n d e r s t a n d i n g f r o m i n - s e r v i c e e d u c a t i o n p r o g r a m s t o t h e g a i n i n u n d e r s t a n d i n g made b y t h e i r p u p i l s o v e r s e v e r a l m o n t h s . A few h a v e d e s c r i b e d t h e 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 s c o r e s o n t e s t s o f g e n e r a l 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 a n d t h e a c h i e v e m e n t o f t h e i r p u p i l s . No s t u d i e s h a v e a t t e m p t e d t o r e l a t e t h e u n d e r -s t a n d i n g o f a s p e c i f i c m a t h e m a t i c a l c o n c e p t h e l d by a t e a c h e r t o t h a t o f s t u d e n t s who w e r e t o l e a r n t h i s c o n c e p t 5 from that teacher. I I . EXPRESSED NEED FOR TEACHER UNDERSTANDING The assumption that teachers must understand math-ematical concepts as a necessary condition f o r t h e i r pupils to learn mathematics i s widely held by teachers and experts. Instructors of an a i r c r a f t mechanics hydraulics course are reported to rate the effectiveness of fellow instructors on the basis of t h e i r knowledge of subject matter. 1 Yet, f o r the 3.000 students and 121 i n s t r u c t o r s involved, the investigators found no s i g n i f i c a n t relationship between student gains i n achievement adjusted f o r i n i t i a l d i f f e r -ences and i n s t r u c t o r knowledge of hydraulics. Outlining h i s views on the minimum mathematical background needed by teachers, Newsom states, " A l l too frequently teachers i n the elementary grades are hardly a jump ahead of t h e i r a l e r t students . . . 1 , 2 A 1958 survey of secondary mathe-matics teachers i n three states found that 85 per cent of those p a r t i c i p a t i n g wanted more workshops and 33 per cent Indicated that advanced mathematics courses would help them i n t h e i r teaching.3 Arguing the need f o r greater emphasis upon the measurement of pup i l understanding of mathematical concepts, Dutton concludes, "Then, and only then, w i l l many classroom teachers begin to examine t h e i r own understand-ings of mathematical concepts . . . A review of recent research by Brown and Ab e l l l e d them to conclude, "Many prospective teachers of elementary school mathematics do not have the understanding of basic mathematical concepts that experts agree they should h a v e . T h e r e i s no doubt about the emphasis on the need f o r the teacher to under-stand mathematical concepts. Does research support t h i s need? I I I . MATHEMATICAL UNDERSTANDINGS OF TEACHERS In one of the f i r s t investigations of mathematical understandings possessed by student teachers and teachers of elementary school, Glennon prepared his own eighty-item test of basic mathematical understandings and found that the average student teacher knew about 43 per cent and the average teacher about 55 per cent of the understandings tested.^ Yet these understandings were basic to the compu-t a t i o n a l processes taught i n grades one to six. Glennon described his findings as presenting a not very optimistic p icture. Using t h e i r own eighteen-item test of arithmetic understandings with 322 teachers and student teachers at-tending summer sessions at three u n i v e r s i t i e s , Orleans and 7 Wandt found that few of the concepts tested were understood by a large percentage of the group.'' Glennon 1 s test was l a t e r used by Weaver who reports si m i l a r conclusions to Q those of Glennon on the basis of his findings. Fulkerson tested students i n an arithmetic methods class with a forty-item test of the knowledge he thought prospective teachers of arithmetic should possess.9 He reports, 11. . . f a r too many of the 158 prospective elementary teachers studied . . . have an i n s u f f i c i e n t knowledge of 1 n arithmetic to teach the subject e f f e c t i v e l y . " Using a f i f t y - i t e m test based p a r t l y on Glennon's t e s t , Kenney found the median f o r 356 teachers who took the test to be 11 29.7. He indicates that a higher degree of mastery of understanding i s needed by these teachers. Kipps used her own c a r e f u l l y designed test of basic mathematical under-standings and obtained a mean of 68 per cent f o r the 310 elementary teachers who wrote the t e s t . 1 2 On the basis of her findings, she f e e l s i t i s necessary f o r teachers to im-prove t h e i r knowledge of the concepts tested. The studies c i t e d share the claim of Orleans and Wandt that f o r children to acquire r e a l understanding of arithmetic, ". . . i t would seem obvious that the teachers of arithmetic must possess the understandings that they are 8 transmitting to t h e i r s t u d e n t s . H o w e v e r the caution given "by Sparks must not be ignored. In 196l, he observed that no research was available to . . indicate that a better comprehension of mathematical concepts on the part of the elementary school teacher r e s u l t s i n better achieve-ment on the part of students." 1^ A si m i l a r statement could be applied to high school teachers. A recommendation f o r further research given by Sparks asks, "What i s the r e l a -tionship between pupi l achievement and teacher knowledge?" 1 IV. TEACHER ACADEMIC PREPARATION AND PUPIL ACHIEVEMENT Several studies have investigated the relat i o n s h i p between the academic preparation i n mathematics of teachers and pup i l achievement i n mathematics. An extensive study of school organizations with and without s p e c i a l i s t teach-ers i n science and mathematics i s reported by Gibb and 1 £ 1 M a t a l a . 1 0 They found no evidence that children learned mathematics more e f f e c t i v e l y with than without a s p e c i a l i s t teacher. Leonhardt compared grade ten geometry classes ranking high i n mathematical achievement with those ranking low and found that mathematics teachers i n the high-ranking schools had studied more undergraduate mathematics courses 9 than those i n the low-ranking s c h o o l s . H e also found that the former usually held a major i n mathematics whereas the l a t t e r did not. In addition, students i n high-ranking classes believed that t h e i r teachers knew the subject matter better. N e i l l used three c r i t e r i o n measures to assess the performance of classes of academically talented grade seven pupils being given one of f i v e selected mathe-matics programs. 1^ While he found that pup i l character-i s t i c s contributed more to the variance i n p u p i l perform-ance than teacher c h a r a c t e r i s t i c s , the l a t t e r did make a contribution with the length of academic preparation of the teacher contributing most. The effects of teacher variables were also less marked than the effects of the d i f f e r e n t programs used. In another study comparing student achieve-ment i n arithmetic reasoning and computation over a nine-year period (kindergarten to grade eight) with the high school and college mathematics preparation of t h e i r teachers, Rouse computed multiple regression s t a t i s t i c s f o r the v a r i -ous combinations but found no high correlations between student achievement and the academic background of t h e i r t e a c h e r s . ^ Thus, the evidence available i s inconclusive. 10 V. IN-SERVICE EDUCATION AND PUPIL ACHIEVEMENT One method of obtaining evidence which supports the need f o r the teacher to understand mathematical concepts hi s students are to learn i s to compare the achievement of students whose teachers have taken in-service education courses i n mathematics with those students whose teachers have not taken such courses. Houston and DeVault used a voluntary thirteen-hour in-service education program on mathematical concepts re l a t e d to the elementary school pro-gram which was provided for 102 intermediate grade teach-ers. Both teachers and t h e i r pupils were tested before and a f t e r the in-service program. Although no s i g n i f i c a n t r e l a t i o n s h i p had been found between the i n i t i a l teacher scores f o r understanding and the change i n pup i l scores f o r understanding, a s i g n i f i c a n t relationship (.01) was found between f i n a l teacher scores f o r understanding and change i n p u p i l scores f o r understanding. A s i g n i f i c a n t r e l a t i o n -ship (.01) was also found between change i n teacher scores f o r understanding and change i n pup i l scores f o r under-standing. Dickens compared mean changes i n mathematical understanding f o r grade four, f i v e , and six pupils whose teachers participated i n a sixteen-hour in-service educa-t i o n program with those whose teachers had no organized i n -service education program. x Although teachers from the in- s e r v i c e education program made s i g n i f i c a n t gains i n mathematical understanding, the comparison between the two groups of pupils showed a s i g n i f i c a n t difference at grade s i x , but not fo r grade four or f i v e . Studies i n t h i s d i -r e c t i o n again give inconclusive evidence concerning the re-la t i o n s h i p between the mathematical understanding of teach-ers and t h e i r pupils. VI. TEACHER UNDERSTANDING AND PUPIL ACHIEVEMENT Three studies have avoided interference from i n -service education i n making a direc t examination of the re-la t i o n s h i p between measures of teachers' mathematical under-standings and measures of pupils' mathematical understand-ings. Bassham sought objective evidence to support or re-fute the assumption that good teacher understanding of bas-i c mathematical concepts i s a necessary condition f o r the promotion of sa t i s f a c t o r y p u p i l growth i n arithmetic. Using twenty-eight sixth-grade teachers and t h e i r classes he compared a measure of teacher understanding of basic mathematical concepts with the arithmetic scores of t h e i r pupils controlled f o r i n i t i a l differences i n arithmetic achievement, reading achievement, mental a b i l i t y , and i n t e r -est i n arithmetic. The co r r e l a t i o n computed between these two sets of data was found to be s i g n i f i c a n t (.05). The co r r e l a t i o n between the same two sets of data r e s t r i c t e d to those i n each class above the mean i n t e l l i g e n c e score was highly s i g n i f i c a n t (.01). However, the cor r e l a t i o n f o r those i n each class below the mean i n t e l l i g e n c e score was not s i g n i f i c a n t . Thus, Bassham found the relationship to be dependent upon the l e v e l of pupil i n t e l l i g e n c e . A simi-l a r study by Lampela involving seventy teachers of grades four, f i v e , and six found no s i g n i f i c a n t relationship be-tween either teacher understanding or change i n teacher un-derstanding of mathematical concepts and the change i n pu-p i l understanding of mathematical concepts over a f i v e -month period. 23 Peskin investigated a number of r e l a t i o n -ships between teacher understanding and attitude and stu-dent understanding and attitude i n regular seventh-grade 24 mathematics classes. She used three c r i t e r i o n measures i n arithmetic f o r both teachers and students and three i n geometry. F i f t y - f i v e teachers and 565 of t h e i r students chosen at random were involved from nine junior high schools. Using p a r t i a l c o r r e l a t i o n techniques to remove the effects of i n i t i a l differences i n mathematics achieve-ments of the groups, s i g n i f i c a n t correlations (.05) were 13 f o u n d b e t w e e n t e a c h e r u n d e r s t a n d i n g s c o r e s a n d s t u d e n t a c h i e v e m e n t s c o r e s i n a r i t h m e t i c a n d g e o m e t r y . P e s k i n f o u n d t h a t t e a c h e r s w i t h h i g h a t t i t u d e a n d u n d e r s t a n d i n g s c o r e s h a d s t u d e n t s who a c h i e v e d h i g h e s t b u t t h a t t e a c h e r s w i t h l o w u n d e r s t a n d i n g s c o r e s h a d s t u d e n t s w i t h t h e n e x t b e s t r e s u l t s . F u r t h e r m o r e , t h o s e t e a c h e r s w i t h h i g h u n d e r -s t a n d i n g b u t l o w a t t i t u d e s c o r e s h a d s t u d e n t s whose a c h i e v e -ment was p o o r e s t . O b v i o u s l y t h e r e i s much more t o l e a r n a b o u t t h e 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 u n d e r s t a n d i n g a n d s t u d e n t a c h i e v e m e n t i n t h e j u n i o r h i g h s c h o o l . V I I . L I M I T S AS A T O P I C I N JUNIOR HIGH SCHOOL B i n g p o i n t s o u t , " T h e n o t i o n o f l i m i t i s a v e r y i m -p o r t a n t o n e i n m a t h e m a t i c s . . . . A s t u d e n t may do w e l l i n a r i t h m e t i c , a l g e b r a , a n d e v e n g e o m e t r y . . . w i t h o u t u n d e r -s t a n d i n g l i m i t s , b u t h e mus t l e a r n t h i s c o n c e p t i n o r d e r t o 2 go f a r i n m a t h e m a t i c s . " J S i n c e l i m i t I s s u c h a n i m p o r t a n t t o p i c i n m a t h e m a t i c s , t h e s t u d y o f i n t u i t i v e l i m i t c o n c e p t s i s r e c o m m e n d e d f o r t h e s e c o n d a r y s c h o o l c o l l e g e p r e p a r a t o r y p r o g r a m by t h e C o m m i s s i o n o n M a t h e m a t i c s o f t h e C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d . H i g h s c h o o l a n d c o l l e g e t e a c h e r s s u r v e y e d by L e i s s a a n d F i s h e r a r e h i g h l y f a v o r a b l e t o t h i s r e c o m m e n d a t i o n . 2 ' ' 7 T h e p r e s e n t c o u r s e p r e s c r i b e d 14 f o r M a t h e m a t i c s 8 i n c l u d e s a n i n t u i t i v e i n t r o d u c t i o n t o l i m i t c o n c e p t s s i m i l a r t o t h a t recommended "by t h e Commis-28 s i o n on M a t h e m a t i c s . Two s t u d i e s g i v e e v i d e n c e t h a t i n t u i t i v e l i m i t c o n -c e p t s c a n be l e a r n e d s u c c e s s f u l l y by j u n i o r h i g h s c h o o l s t u d e n t s . S m i t h p r o v i d e d t h r e e h o u r s o f i n s t r u c t i o n i n l i m i t c o n c e p t s f o r s t u d e n t s i n g r a d e s s e v e n , n i n e , and e l e v -en 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 p u p i l s i n v o l v e d c o u l d b e n e f i t f rom t h i s e x p e r i e n c e . 2 ^ He f o u n d t h a t t h e y c o u l d on t h e b a s i s o f s c o r e s f r o m a s p e c i a l l y p r e p a r e d l i m i t s t e s t . E q u a t i n g g r o u p s on t h e b a s i s o f m e n t a l age he f o u n d t h e mean s c o r e s f o r g r o u p s who had t h e s p e c i a l i n s t r u c t i o n t o be s i g n i f i c a n t l y h i g h e r t h a n t h o s e f o r g r o u p s w h i c h h a d no i n s t r u c t i o n . D e s s a r t i n v e s t i g a t e d t h e f e a s i b i l i t y o f t e a c h i n g some a s p e c t s o f c o n v e r g e n c e a n d d i v e r g e n c e o f i n -f i n i t e s e r i e s t o s u p e r i o r g r a d e e i g h t s t u d e n t s . - ^ 0 D i f f e r e n t p r e s e n t a t i o n s were made t o d i f f e r e n t g r o u p s but a l l were k e p t i n t u i t i v e a n d p r e c i s e d e f i n i t i o n s a v o i d e d . The m a j o r -i t y o f s t u d e n t s showed a s a t i s f a c t o r y g a i n i n u n d e r s t a n d i n g r e g a r d l e s s o f t h e p r e s e n t a t i o n u s e d . Thus g r a d e e i g h t s t u -d e n t s o f s u p e r i o r a b i l i t y c a n l e a r n c o n c e p t s o f c o n v e r g e n t a n d d i v e r g e n t i n f i n i t e s e r i e s . V I I I . T H E HYPOTHESIS 15 S i n c e i n t u i t i v e l i m i t c o n c e p t s a r e a new t o p i c i n t h e g r a d e e i g h t m a t h e m a t i c s c u r r i c u l u m i n B r i t i s h C o l u m b i a , many t e a c h e r s a t t h i s l e v e l h a v e h a d no p r e v i o u s e x p e r i e n c e w i t h t e a c h i n g t h e s e i d e a s . S i n c e t h e t o p i c d o e s n o t o c c u r i n a n y e a r l i e r m a t h e m a t i c s c o u r s e i t i s a s s u m e d t h a t s t u -d e n t s w i l l h a v e h a d no p r e v i o u s e x p e r i e n c e w i t h t h e t o p i c . I f t h e t e a c h e r h a s l i t t l e o r no u n d e r s t a n d i n g o f t h e c o n -c e p t s t o be p r e s e n t e d , i t i s p o s s i b l e t h a t s u c h a t e a c h e r m i g h t p r o v i d e no o p p o r t u n i t y f o r b e t t e r s t u d e n t s t o d e v e l o p a n u n d e r s t a n d i n g o f i n t u i t i v e l i m i t c o n c e p t s . T h i s t o p i c , t h e r e f o r e , p r o v i d e s a u n i q u e o p p o r t u n i t y t o m e a s u r e t h e 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 u n d e r s t a n d i n g a n d s t u d e n t u n d e r -s t a n d i n g o f a s i n g l e m a t h e m a t i c a l c o n c e p t . T h e n u l l h y p o t h e s i s t e s t e d w i l l b e : F o r M a t h e m a t i c s 8 c l a s s e s o f b e t t e r s t u d e n t s t h e r e i s no s i g n i f i c a n t c o r r e -l a t i o n b e t w e e n t e a c h e r u n d e r s t a n d i n g o f i n t u i t i v e l i m i t c o n c e p t s a n d s t u d e n t u n d e r s t a n d i n g o f i n t u i t i v e l i m i t c o n c e p t s . 16 FOOTNOTES i J o s e p h E . M o r s h , G e o r g e G . B u r g e s s , a n d P a u l N . S m i t h , " S t u d e n t A c h i e v e m e n t a s a M e a s u r e o f I n s t r u c t o r E f f e c t i v e n e s 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 , 4 7 : 8 6 , F e b r u a r y , 1956. C . V . Newsom, " M a t h e m a t i c a l B a c k g r o u n d N e e d e d b y T e a c h e r s , " T h e T e a c h i n g o f A r i t h m e t i c , F i f t i e t h Y e a r b o o k o f t h e N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , P a r t I I ( 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 , 1951). p . 2 3 2 . -^Kenneth E . B r o w n , " T e a c h i n g L o a d and . Q u a l i f i c a t i o n s o f M a t h e m a t i c s T e a c h e r s , " T h e M a t h e m a t i c s T e a c h e r , 53-9 . J a n u a r y , i 9 6 0 . ^ W i l b u r H . D u t t o n , E v a l u a t i n g P u p i l s 1 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 ( E n g l e w o o d C l i f f s , New J e r s e y : P r e n t i c e H a l l I n c o r p o r a t e d , 1964) , p . 104 . - ' K e n n e t h E . Brown a n d T h e o d o r e L . A b e l l . , " R e s e a r c h i n t h e T e a c h i n g o f H i g h S c h o o l M a t h e m a t i c s . " T h e M a t h e m a t i c s  T e a c h e r , 59=56, J a n u a r y , 1966. ^ V i n c e n t J . G l e n n o n , " A S t u d y i n N e e d e d 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 , " T h e M a t h e -m a t i c s T e a c h e r , 4 2 : 3 9 3 , D e c e m b e r , 1949. ? J a c o b S . O r l e a n s a n d E d w i n W a n d t , " T h e U n d e r s t a n d -i n g s 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 , 53:507, M a y , 1953. Q J . F r e d W e a v e r , " 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 , 56:260, F e b r u a r y , 1956. ^ E . P u l k e r s o n , "How W e l l Do 158 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 Know A r i t h m e t i c ? , " T h e A r i t h m e t i c T e a c h e r , 7 : 1 4 1 , M a r c h , i 9 6 0 . 1 0 I b i d . , p . 146 . 11 R u s s e l l . A . K e n n e y , " 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 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 , " T h e A r i t h m e t i c T e a c h e r , 12:433, O c t o b e r , 1965. 17 1 2 C a r o l Kipps, "Elementary Teachers' A b i l i t y to Understand Concepts Used i n New Mathematics," The Arithme-t i c Teacher, 15:368, A p r i l , 1968. 1 ^ ^Orleans and Wandt, OJD. c i t . , p. 501. 14 Jack N. Sparks, "Arithmetic Understandings Needed by Elementary School Teachers," The A r i t h m e t i c Teacher, 8:402, December, 1961, l 5 I b i d . , p. 403. l 6 E . G. Gibb and D. M. Matala, "Study of the Use of S p e c i a l Teachers i n Science and Mathematics i n Grades Five and S i x , " School Science and Mathematics, 61:569-572, No-vember, 1961; 62 :565-585, November, 1962. • ^ E a r l A l b e r t Leonhardt, "An A n a l y s i s of S e l e c t e d Factors i n R e l a t i o n to High and Low Achievement i n Mathe-matics" (unpublished D o c t o r a l t h e s i s , The U n i v e r s i t y of Nebraska, 1962), p. 228. l 8 R o b e r t Dudley N e i l l , "The E f f e c t s of S e l e c t e d Teacher V a r i a b l e s on the Mathematics Achievement of Academically Talented J u n i o r High School P u p i l s " (unpub-l i s h e d D o c t o r a l t h e s i s , Columbia U n i v e r s i t y , 1966), p. 2. 19 'W. M. Rouse, "A Study of the C o r r e l a t i o n between the Academic P r e p a r a t i o n of Teachers of Mathematics and the Mathematics Achievement of t h e i r Students i n Kinder-garten through Grade E i g h t " (unpublished D o c t o r a l t h e s i s , Michigan State U n i v e r s i t y , 1967). 20W. R. Houston and M. V. DeVault, "Mathematics I n - s e r v i c e Education: Teacher Growth Increases P u p i l Growth," The A r i t h m e t i c Teacher, 10:243, May, 1963. 21 •^Charles H. Dickens, " E f f e c t s of I n - s e r v i c e T r a i n -i n g i n Elementary School Mathematics on Teachers' Under-standing and Teaching of Mathematics" (unpublished D o c t o r a l t h e s i s , Duke U n i v e r s i t y , 1966). 22 ^ H a r r e l l Bassham, "Teacher Understanding and P u p i l E f f i c i e n c y i n Mathematics: A Study of R e l a t i o n s h i p , " The  A r i t h m e t i c Teacher, 9 :383, November, 1962. 18 -^Roland M i t c h e l l L a n i p e l a , "An 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 b e t w e e n T e a c h e r U n d e r s t a n d i n g a n d Change i n P u p i l U n d e r s t a n d i n g o f S e l e c t e d C o n c e p t 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 " ( u n p u b l i s h e d D o c t o r a l t h e s i s , U n i v e r s i t y o f C a l i f o r n i a , L o s A n g e l e s , 1966). A n n e S t e r n P e s k i n , " T e a c h e r U n d e r s t a n d i n g a n d A t t i -t u d e a n d S t u d e n t A c h i e v e m e n t a n d A t t i t u d e i n S e v e n t h G r a d 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 t h e s i s , New Y o r k U n i v e r -s i t y , 1964), p . 2. 2 % . H . B i n g , " P o i n t S e t T o p o l o g y , " I n s i g h t s i n t o  M o d e r n M a t h e m a t i c s , T w e n t y - t h i r d Y e a r b o o k o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s ( W a s h i n g t o n : T h e N a t i o n -a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s , 1957). P . 314. 26 C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d , R e p o r t o f t h e  C o m m i s s i o n o n M a t h e m a t i c s , A p p e n d i c e s (New Y o r k : C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d , 1959). p p . 6 4 - 7 3 . 2 ^ A r t h u r W. L e i s s a a n d R o b e r t C . F i s h e r , " A S u r v e y o f T e a c h e r s ' O p i n i o n s o f a R e v i s e d M a t h e m a t i c s C u r r i c u l u m , " T h e M a t h e m a t i c s T e a c h e r , 53:116, F e b r u a r y , i 9 6 0 . 2 Pi B r i t i s h C o l u m b i a D e p a r t m e n t o f E d u c a t i o n , 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 , 1966 ( V i c t o r i a : B r i t i s h C o l u m b i a D e p a r t m e n t o f E d u c a t i o n , 1966), p . 7. 2 9 L e h i T . S m i t h , " C o u l d We T e a c h L i m i t s ? , " T h e M a t h -e m a t i c s T e a c h e r , 54:344, May , 1961. 3 ° D o n a l d J o s e p h D e s s a r t , " A S t u d y o f P r o g r a m m e d L e a r n i n g w i t h S u p e r i o r E i g h 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 t h e s i s , U n i v e r s i t y o f M a r y l a n d , 1961). CHAPTER I I I DESIGN OP T H E STUDY I . PREPARATION OP T H E T E S T S A p r e l i m i n a r y u n s p e e d e d t e s t o f i n t u i t i v e l i m i t c o n -c e p t s was p r e p a r e d a n d t h e f o r t y i t e m s c l a s s i f i e d by a m a t h e m a t i c i a n i n t o t h e f o l l o w i n g c a t e g o r i e s d e t e r m i n e d by t h e r e s e a r c h e r : l i m i t o f a s e q u e n c e , l i m i t o f a s e r i e s , l i m i t o f a f u n c t i o n , l e a s t u p p e r b o u n d , a n d g r e a t e s t l o w e r b o u n d . T h i s t e s t was g i v e n t o two M a t h e m a t i c s 8 c l a s s e s o f b e t t e r s t u d e n t s i n a l a r g e u r b a n s c h o o l d i s t r i c t o f B r i t i s h C o l u m b i a . B o t h c l a s s e s h a d b e e n t a u g h t t h e l i m i t c o n c e p t s p r e s c r i b e d f o r M a t h e m a t i c s 8. A d m i n i s t r a t i o n o f t h e t e s t was by t h e r e g u l a r m a t h e m a t i c s t e a c h e r who was p r o v i d e d w i t h d i r e c t i o n s t o f o l l o w . A maximum t i m e o f f o r t y m i n -u t e s was a l l o w e d a s a m p l e t i m e f o r mos t s t u d e n t s t o com-p l e t e t h e t e s t . T e a c h e r s w e r e a s k e d t o n o t e i f t h i s t i m e was i n a d e q u a t e . E a c h i t e m was m a r k e d a s r i g h t o r w r o n g . 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 IBM 7044 c o m p u t e r was u s e d t o c o m p l e t e a n i t e m a n a l y s i s o f t h e t e s t a n d t o c a l -c u l a t e t h e r e l i a b i l i t y o f t h e t e s t u s i n g K u d e r - R l c h a r d s o n f o r m u l a 20. 1 S i n c e t h e t e s t i s u n s p e e d e d , t h i s m e t h o d o f r a t i o n a l e q u i v a l e n c e i s a p p r o p r i a t e t o u s e i n d e t e r m i n i n g r e l i a b i l i t y . 2 20 S e v e r a l h y p o t h e t i c a l s t u d e n t answers were p r e p a r e d f o r each o f t h e i t e m s on t h e p r e l i m i n a r y t e s t f o r s t u d e n t s . Some were c o r r e c t , o t h e r s i n c o r r e c t . I n c o r r e c t answers were p r e p a r e d t o a p p e a r c o r r e c t t o a p e r s o n u n f a m i l i a r w i t h l i m i t c o n c e p t s . P o u r o r more h y p o t h e t i c a l answers were u s u a l l y g i v e n f o r each i t e m . These s e t s o f answers were t h e n u s e d i n c o n j u n c t i o n w i t h t h e p r e l i m i n a r y s t u d e n t t e s t as a 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 o f l i m i t s . The f o r t y i t e m s f o r t e a c h e r s o b t a i n e d i n t h i s manner were t a x o n o m i z e d a c c o r d i n g t o B loom by an e x p e r t i n m a t h e m a t i c s e d u c a t i o n . - ^ To e s t a b l i s h a s u i t a b l e m a r k i n g scheme, t e n t e a c h e r s o f M a t h e m a t i c s 8 i n two l a r g e u r b a n s e c o n d a r y s c h o o l s t o o k t h e t e s t . They were t o l d t h a t some o f t h e c h o i c e s g i v e n were i n c o r r e c t a n d a s k e d t o mark a l l answers w i t h w h i c h t h e y w o u l d a g r e e . Two m a r k i n g schemes were t r i e d . One gave z e r o f o r each i t e m on t h e s t u d e n t t e s t i f any i n c o r r e c t answer was c h o s e n and one mark i f a l l c o r r e c t answers were c h o s e n . The o t h e r gave z e r o f o r each i t e m i f any i n c o r r e c t answer was c h o s e n , one i f o n l y some c o r r e c t answers were c h o s e n , a n d tx^o i f a l l c o r r e c t answers were c h o s e n . W i t h e i t h e r scheme, q u e s t i o n s one t o s i x were g i v e n one i f c o r -r e c t , o t h e r w i s e z e r o . The m a r k i n g scheme c h o s e n f o r t h e f i n a l t e s t was t h e one g i v i n g t h e g r e a t e r d i s t r i b u t i o n o f s c o r e s . A f i n a l f o r m o f t h e s t u d e n t t e s t o f i n t u i t i v e l i m i t c o n c e p t s was c o n s t r u c t e d u s i n g a l l i t e m s o f t h e p r e l i m i n a r y t e s t h a v i n g a p o i n t b i s e r i a l g r e a t e r t h a n 0 . 2 0 . ^ A c o p y o f t h i s t e s t i s i n c l u d e d i n A p p e n d i x A . C o r r e s p o n d i n g i t e m s o f t h e p r e l i m i n a r y t e a c h e r t e s t w e r e u s e d t o c o n s t r u c t t h e f i n a l t e a c h e r t e s t . A c o p y o f t h i s t e s t i s i n c l u d e d i n A p p e n d i x B . I I . A D M I N I S T R A T I O N OF T H E T E S T S A p p r o v a l t o g i v e t h e f i n a l t e s t t o a l l M a t h e m a t i c s 8 c l a s s e s o f b e t t e r s t u d e n t s i n a s e c o n d l a r g e u r b a n s c h o o l d i s t r i c t o f 3 r i t i s h C o l u m b i a was o b t a i n e d f r o m i t s D i s t r i c t S u p e r i n t e n d e n t o f S c h o o l s . By c o n t a c t i n g t h e p r i n c i p a l s o f e a c h s c h o o l h a v i n g c l a s s e s o f M a t h e m a t i c s 8 , f o u r t e e n c l a s s e s f r o m f o u r s c h o o l s w e r e i d e n t i f i e d a s M a t h e m a t i c s 8 c l a s s e s o f b e t t e r s t u d e n t s . A t i m e t a b l e was t h e n e s t a b -l i s h e d t o p e r m i t a l l c l a s s e s t o be t e s t e d i n a s i n g l e w e e k . A r r a n g e m e n t s w e r e made w i t h e a c h p r i n c i p a l f o r t h e r e g u l a r m a t h e m a t i c s t e a c h e r t o r e m a i n i n t h e c l a s s r o o m w h i l e t h e t e s t s w e r e a d m i n i s t e r e d by two e x a m i n e r s e x p e r i e n c e d i n c l a s s r o o m w o r k . T o e n s u r e u n i f o r m i t y i n t h e a d m i n i s t r a t i o n o f t h e f i n a l t e s t s , t h e e x a m i n e r s w e r e i n s t r u c t e d t o g e t h e r 22 i n i t s a d m i n i s t r a t i o n . W r i t t e n i n s t r u c t i o n s w e r e a l s o p r o v i d e d b o t h e x a m i n e r s . T h e s e a r e i n c l u d e d i n A p p e n d i x C . C l a s s r o o m s e t s o f t e s t s , i n c l u d i n g a t e a c h e r t e s t , w e r e p r o v i d e d t h e e x a m i n e r s i n u n m a r k e d e n v e l o p e s a t t h e s t a r t o f t h e week c h o s e n f o r t e s t i n g i n K a y , 1969. T h e t e s t i n g was d o n e i n t h e r e g u l a r m a t h e m a t i c s c l a s s r o o m w i t h t h e m a t h e m a t i c s t e a c h e r a s s i s t i n g i n t h e d i s t r i b u t i o n o f m a t e r i a l s . On t h e b a s i s o f e x p e r i e n c e w i t h t h e p r e l i m i n a r y t e s t , f o r t y m i n u t e s was a l l o w e d f o r s t u d e n t s t o w r i t e t h e f i n a l t e s t . S h o r t l y a f t e r t h e s t u d e n t t e s t h a d b e g u n , t h e e x a m i n e r a s k e d t h e m a t h e m a t i c s t e a c h e r t o a n s w e r t h e s h e e t o n n o t a t i o n a l a g r e e m e n t s ( t h e f i n a l t e a c h e r t e s t ) w i t h o u t r e f e r e n c e t o a n y t e x t b o o k . I f t h e t e a c h e r a s k e d w h e t h e r h e was w r i t i n g a t e s t , he was t o l d , " Y e s . " E a c h t e a c h e r was a s s u r e d t h a t t h e r e was no way by w h i c h h e , h i s c l a s s , o r h i s s c h o o l c o u l d be i d e n t i f i e d i n t h e i n v e s t i g a t i o n . W h i l e t h e t e s t s w e r e b e i n g w r i t t e n , t h e e x a m i n e r p r e p a r e d a c l a s s l i s t o f t h o s e w r i t i n g t h e s t u d e n t t e s t . A f t e r c o l l e c t i o n , t h e t e s t s w e r e r e t u r n e d t o t h e u n m a r k e d e n v e l o p e t o g e t h e r w i t h t h e c l a s s l i s t . I I I . C O L L E C T I O N OF DATA A f t e r l e a v i n g t h e c l a s s r o o m , t h e e x a m i n e r e i t h e r u s e d a l i s t o f t h e r e q u i r e d d a t a p r e p a r e d by t h e s c h o o l s t a f f o r u s e d t h e p e r m a n e n t r e c o r d c a r d o f e a c h s t u d e n t who w r o t e t h e t e s t t o r e c o r d o n t h e c l a s s l i s t t h e s t u d e n t ' s s c h o o l d i s t r i c t s t a n i n e s c o r e s f o r b o t h t h e L o r g e - T h o r n d i k e V e r b a l I n t e l l i g e n c e T e s t ( F o r m E) a n d t h e S c h o o l D i s t r i c t M a t h e m a t i c s T e s t ( G r a d e S e v e n ) . T h e s e t e s t s w e r e w r i t t e n i n N o v e m b e r , 19&7, a n d J u n e , 1968, r e s p e c t i v e l y . T h e c l a s s l i s t was r e t u r n e d t o t h e e n v e l o p e w h i c h was t h e n s e a l e d . A t t h e e n d o f t h e o n e - w e e k t e s t i n g p e r i o d , t h e f o u r t e e n u n m a r k e d e n v e l o p e s w e r e r e t u r n e d t o t h e i n v e s t i g a t o r f o r m a r k i n g . A s c o r e o f s t u d e n t u n d e r s t a n d i n g o f l i m i t was o b -t a i n e d b y m a r k i n g e a c h i t e m a s r i g h t o r w r o n g a n d r e c o r d i n g t h e t o t a l number r i g h t f o r e a c h s t u d e n t . A s c o r e o f t e a c h -e r u n d e r s t a n d i n g o f l i m i t was o b t a i n e d by t o t a l l i n g t h e m a r k s o b t a i n e d f o r e a c h i t e m u s i n g t h e t w o - o n e - z e r o m a r k -i n g scheme t r i e d w i t h t h e p r e l i m i n a r y t e s t . T h i s d a t a was r e c o r d e d o n t h e c l a s s l i s t , i n c l u d i n g t h e t e a c h e r ' s s c o r e . I V . TREATMENT OF DATA A n a l y s i s o f c o v a r i a n c e was u s e d t o c a l c u l a t e a d -j u s t e d means o f s t u d e n t u n d e r s t a n d i n g o f l i m i t f o r e a c h c l a s s u s i n g d a t a f r o m t h e L o r g e - T h o r n d i k e a n d S c h o o l D i s -t r i c t Mathematics tests as covariates. By using analysis of covariance, the class mean scores f o r the student test were adjusted to allow f o r i n i t i a l class differences i n i n -t e l l i g e n c e and mathematics achievement. The calculations were made by the University of B r i t i s h Columbia IBM /360 computer using the MPACO program prepared by Dempster and Starkey.^ The product-moment c o r r e l a t i o n was then calcu-l a t e d between adjusted class means and teacher understand-ing of l i m i t scores using the formula:^ ZXY NM M x y r = The c o e f f i c i e n t of co r r e l a t i o n obtained was then checked f o r significance. 25 FOOTNOTES 1 H e n r y 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 a n d  E d u c a t i o n (New Y o r k : D a v i d McKay Company , 1958), p . 3 4 1 . 2 I b i d . , p . 353. -^Benjamin S . B l o o m ( e d . ) , Taxonomy o f E d u c a t i o n a l  O b j e c t i v e s , t h e C l a s s i f i c a t i o n o f E d u c a t i o n a l G o a l s , H a n d -b o o k I: C o g n i t i v e D o m a i n ( N e w ~ Y o r k : D a v i d McKay Company , i"95o"), p p . 201-207. ^ G a r r e t t , OJD. c i t . , p . 368. ^ J . R . H . D e m p s t e r a n d G . E . S t a r k e y , MFACO: A n a l y -s i s o f C o v a r i a n c e ( V a n c o u v e r : 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 C o m p u t i n g C e n t r e , 1968). / • G a r r e t t , OJD. c i t . , p . 142 . CHAPTER I V A N A L Y S I S OP T H E DATA I . PRELIMINARY STUDENT T E S T DATA T a b l e I i n d i c a t e s t h e c a t e g o r i z a t i o n o f i t e m s a c -c o r d i n g t o l i m i t t o p i c s f o r t h e p r e l i m i n a r y a n d f i n a l s t u -d e n t t e s t s . A s a t i s f a c t o r y d i s t r i b u t i o n o f i t e m s among t h e f i v e l i m i t t o p i c s i s i n d i c a t e d f o r b o t h t e s t s . I n t h e p r e -l i m i n a r y t e s t , t h e c o n c e p t o f a l i m i t o f a s e q u e n c e i s i n -c l u d e d i n n i n e i t e m s , l i m i t o f a s e r i e s i n t w e n t y - o n e i t e m s , l i m i t o f a f u n c t i o n i n f o u r i t e m s , l e a s t u p p e r b o u n d i n s e v e n t e e n i t e m s , a n d g r e a t e s t l o w e r b o u n d i n s i x t e e n i t e m s . N i n e t e e n o f t h e s e i t e m s i n c l u d e two l i m i t c o n c e p t s a n d f o u r i t e m s i n c l u d e t h r e e c o n c e p t s . T h e i t e m a n a l y s i s o f t h e p r e l i m i n a r y t e s t i n d i c a t e d t h i r t y - f i v e i t e m s w i t h a p o i n t b i s e r i a l c o r r e l a t i o n g r e a t e r t h a n 0.20 w h i c h w e r e u s e d t o make t h e f i n a l t e s t . T h e c o r r e l a t i o n f o r e a c h p r e l i m i n a r y t e s t i t e m i s g i v e n i n T a b l e I . I n t h e f i n a l t e s t , l i m i t o f a s e q u e n c e i s i n c l u d e d i n s e v e n i t e m s , l i m i t o f a s e r i e s i n n i n e t e e n i t e m s , l i m i t o f a f u n c t i o n i n f o u r i t e m s , l e a s t u p p e r b o u n d i n s i x t e e n i t e m s , a n d g r e a t e s t l o w e r b o u n d i n f o u r t e e n i t e m s . S e v e n t e e n o f t h e f i n a l t e s t i t e m s a r e c a t -e g o r i z e d u n d e r two t o p i c s a n d f o u r u n d e r t h r e e t o p i c s . K u d e r - R i c h a r d s o n f o r m u l a 20 g a v e a r e l i a b i l i t y c o e f f i c i e n t TABLE I 27 POINT BISERIAL CORRELATION AND LIMIT TOPIC CATEGORIZATION OP TEST ITEMS Item number Point preliminary bi s e r i a l Limit Limit Limit Least Greatest test c o r r e l a - of a of a of a upper lower ( f i n a l test) t i o n sequence series function bound bound 1(1) 0.21 X 2(2) 0.21 X 3(3) 0.54 X 4 0.15 X 5(4) 0.29 X 6(5) 0.37 X 7(6) 0.68 X 8(7) 0.66 X 9(8) 0.58 X 10(9) 0.52 X 11(10) 0.66 X 12 0.17 X 13 0.12 X X 14(11) 0.47 X X 15(12) 0.39 X X 16(13) 0.32 X X 17(14) 0.33 X X 18 0.00 •v X 19(15) 0.39 X 20(16) 0.31 X 21(17) 0.46 X 22(18) 0.62 X X 23(19) 0.55 X X 24(20) 0.44 X X 25(21) 0.38 X X 26(22) 0.25 X X 27(23) 0.36 X X X 28(24) O.36 X X X 29(25) 0.37 X X X 30(26) 0.44 X X 31(27) 0.55 X X 32(28) 0.37 X X X 33(29) 0.28 X 34(30) O.56 X X 35(31) 0.30 X X 36 - 0 . 0 8 X 37(32) 0.71 X X 38(33) 0.66 X X 39(34) 0.61 X X 40(35) 0.50 X X 28 f o r t h e p r e l i m i n a r y t e s t o f O . 8 7 . S i n c e t h e p u r p o s e o f t h e t e s t r e s u l t s i s t o d i s t i n g u i s h b e t w e e n t h e means o f s i m i l a r c l a s s e s , t h e r e l i a b i l i t y c o e f f i c i e n t e x c e e d s t h e c r i t e r i a o f G a r r e t t who s t a t e s t h a t r e l i a b i l i t y c o e f f i c i e n t s o f 0.50 o r 0.60 a r e a d e q u a t e f o r s u c h p u r p o s e s . T e a c h e r s n o t e d t h a t t h e f o r t y m i n u t e t i m e a l l o t m e n t was s u f f i c i e n t f o r s t u d e n t s t o c o m p l e t e t h e t e s t . T h u s , t h e p r e l i m i n a r y s t u -d e n t t e s t was shown t o i n c l u d e t h e l i m i t t o p i c s i n t e n d e d , p r o v i d e a s u f f i c i e n t number o f i t e m s f o r a f i n a l t e s t , be m o r e t h a n a d e q u a t e i n r e l i a b i l i t y , a n d p e r m i t v i r t u a l l y a l l s t u d e n t s t o a n s w e r e v e r y i t e m . I I . PRELIMINARY T'EACHER T E S T DATA T h e t a x o n o m i z a t i o n o f t h e p r e l i m i n a r y t e a c h e r t e s t i n d i c a t e d n i n e t e e n i t e m s a t t h e l e v e l o f k n o w l e d g e , e i g h t e e n a t t h e l e v e l o f c o m p r e h e n s i o n a n d t h r e e a t t h e l e v e l o f a p p l i c a t i o n . No i t e m s w e r e a s s i g n e d t o t h e t h r e e h i g h e s t e d u c a t i o n a l g o a l s d e s c r i b e d by B l o o m , a n a l y s i s , s y n t h e s i s , a n d e v a l u a t i o n . T h e c o m p l e t e t a x o n o m i z a t i o n a p p e a r s i n T a b l e I I . I t i n d i c a t e s t h a t t h e p r e l i m i n a r y t e s t r e q u i r e s a t e a c h e r t o r e c o g n i z e o r r e c a l l t h e l i m i t c o n c e p t s p r e v i o u s l y c a t e g o r i z e d a n d t o r e s p o n d t o t e s t i t e m s by t r a n s l a t i n g , i n t e r p r e t i n g , o r e x t r a p o l a t i n g f r o m T A B L E I I TAXONOMIZATION OP TEACHER T E S T I T E M S * I t e m number p r e l i m i n a r y t e s t K n o w l e d g e C o m p r e h e n s i o n A p p l i c a t i o n ( f i n a l t e s t ) 1(1) X 2(2) X 3(3) X 4 X 5(4) 6(5) ' X 7(6) X . 8(7) X 9(8) X 10(9) X 11(10) X 12 X 13 X 14(11) X 15(12) X 16(13) X 17(14) X 18 X 19(15) X 20(16) X 21(17) X 22(18) X 23(19) X 24(20) X 25(21) X 26(22) X 27(23) X 28(24) X 29(25) X 30(26) X 31(27) X 32(28) X 33(29) X 34(30) X 3 5 ( 3 D 36 X 37(32) 38(33) X 39(34) X 40(35) X * N o i t e m s w e r e c l a s s i f i e d a t t h e l e v e l o f a n a l y s i s , s y n t h e s i s , o r e v a l u a t i o n . t h e i n f o r m a t i o n g i v e n . On t h i s b a s i s , t h e t e a c h e r t e s t was j u d g e d t o be a s u i t a b l e i n s t r u m e n t f o r m e a s u r i n g t e a c h e r u n d e r s t a n d i n g o f l i m i t c o n c e p t s . T h e two m a r k i n g s y s t e m s t r i e d f o r t h e t e a c h e r t e s t s h o w e d l i t t l e d i f f e r e n c e . B e c a u s e t h e t w o - o n e - z e r o s y s t e m g a v e a s l i g h t l y g r e a t e r d i s t r i b u t i o n o f m a r k s a n d a v o i d e d t i e s , i t was u s e d f o r t h e f i n a l t e a c h e r t e s t . I I I . F I N A L T E S T DATA I n t h e f i n a l t e s t i n g p r o g r a m , c o m p l e t e d a t a was g a t h e r e d f o r a l l m a t h e m a t i c s t e a c h e r s o f t h e f o u r t e e n c l a s s e s t e s t e d a n d f o r 332 o f t h e 462 s t u d e n t s who w r o t e t h e l i m i t s t e s t . T a b l e I I I g i v e s t e a c h e r s c o r e s a n d t h e c o r r e s p o n d i n g c l a s s means a d j u s t e d by t h e MFACO p r o g r a m f o r i n i t i a l c l a s s d i f f e r e n c e s i n i n t e l l i g e n c e a n d m a t h e -m a t i c s a c h i e v e m e n t . T e a c h e r s c o r e s r a n g e f r o m 22 t o 59 w i t h a mean o f 4 2 . 7 . A d j u s t e d c l a s s mean s c o r e s r a n g e f r o m 1 2 . 4 t o 19.0 w i t h a mean f o r t h e f o u r t e e n c l a s s e s o f 15.5 . T h e p r o d u c t - m o m e n t c o r r e l a t i o n c a l c u l a t e d b e t w e e n t e a c h e r s c o r e s a n d a d j u s t e d c l a s s means i s 0 . 0 9 . 2 U s i n g T a b l e X X V o f G a r r e t t w i t h t w e l v e d e g r e e s o f f r e e d o m t h i s c o r r e l a t i o n i s n o t s i g n i f i c a n t . TABLE III TEACHER SCORES A N D ADJUSTED CLASS MEANS Class Teacher score Adjusted class mean A 22 15 .9 B 37 18 .9 C 55 1 2 . 4 D 41 1 3 . 3 E 57 1 9 . 0 P 46 1 6 . 1 G 32 1 3 . 3 H 43 -16.1 I 53 17 .9 J 59 15.7 K 39 13 .9 L 33 1 5 . 3 M 55 14.7 N 27 15.8 Range 2 2 - 5 9 1 2 . 4 - 1 9 . 0 Mean 42 .7 1 5 . 5 FOOTNOTES i H e n r y 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 a n d  E d u c a t i o n (New Y o r k : D a v i d McKay Company , 1 9 5 8 ) , p . 351 2 S e e A p p e n d i x D f o r c o m p u t a t i o n . CHAPTER V SUMMARY AND CONCLUSIONS I . SUMMARY A p r e l i m i n a r y s t u d e n t t e s t o f i n t u i t i v e l i m i t c o n -c e p t s was c o n s t r u c t e d , c h e c k e d f o r c o n t e n t v a l i d i t y , a n d g i v e n a t r i a l u s e . T h e r e l i a b i l i t y o f t h e t e s t vras c a l c u -l a t e d a n d a n i t e m a n a l y s i s made t o d e t e r m i n e w h i c h i t e m s t o vise i n a f i n a l f o r m o f t h e t e s t . A 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 was c o n s t r u c t e d u s i n g s t u d e n t t e s t i t e m s t o g e t h e r w i t h h y p o t h e t i c a l a n s w e r s a n d i t s i t e m s w e r e t a x o n o m i z e d a c c o r d i n g t o B l o o m . F o u r t e e n c l a s s e s o f M a t h e m a t i c s 8 s t u d e n t s o f b e t t e r a b i l i t y a n d t h e i r t e a c h e r s w e r e t e s t e d f o r u n d e r s t a n d i n g o f l i m i t c o n c e p t s u s i n g t h e f i n a l f o r m s o f t h e two t e s t s c o n s t r u c t e d . A f t e r t e s t s w e r e m a r k e d , c l a s s means w e r e a d j u s t e d by a n a l y s i s o f c o v a r i a n c e t o a l l o w f o r i n i t i a l d i f f e r e n c e s i n i n t e l l i g e n c e a n d m a t h e m a t -i c s a c h i e v e m e n t . A c o e f f i c i e n t o f c o r r e l a t i o n was c a l c u -l a t e d b e t w e e n t h e s e a d j u s t e d means a n d t h e t e a c h e r s c o r e s . T h e r e s u l t i n g c o r r e l a t i o n o f 0.09 was n o t s i g n i f i c a n t . T h u s t h e n u l l h y p o t h e s i s was a c c e p t e d : F o r M a t h e m a t i c s 8 c l a s s e s o f b e t t e r s t u d e n t s t h e r e i s no s i g n i f i c a n t c o r r e -l a t i o n b e t w e e n t e a c h e r u n d e r s t a n d i n g o f i n t u i t i v e l i m i t c o n c e p t s a n d s t u d e n t u n d e r s t a n d i n g o f i n t u i t i v e l i m i t 34 c o n c e p t s . I I . CONCLUSIONS A c c e p t a n c e o f t h e n u l l h y p o t h e s i s i n t h i s i n v e s t i -g a t i o n s u g g e s t s t h a t s t u d e n t s a r e n o t p r e v e n t e d f r o m u n d e r -s t a n d i n g i n t u i t i v e l i m i t c o n c e p t s i f t h e i r t e a c h e r does n o t u n d e r s t a n d t h e t o p i c w e l l . S i m i l a r l y , a t e a c h e r ' s u n d e r -s t a n d i n g o f i n t u i t i v e l i m i t c o n c e p t s g i v e s no a s s u r a n c e t h a t h i s s t u d e n t s w i l l a t t a i n a g r e a t e r u n d e r s t a n d i n g o f t h e t o p i c t h a n s t u d e n t s o f a t e a c h e r w i t h l e s s u n d e r s t a n d -i n g o f t h e same t o p i c . A l t h o u g h n o t a b l e t o t e a c h t h e i n t u i t i v e l i m i t c o n c e p t because o f h i s own l a c k o f u n d e r -s t a n d i n g , a t e a c h e r majr be v e r y a b l e t o c r e a t e an a tmos-p h e r e w h i c h f o s t e r s t h e l e a r n i n g o f t h i s c o n c e p t by h i s s t u d e n t s f rom t e x t b o o k s , s u p p l e m e n t a r y p u b l i c a t i o n s , a n d o t h e r m a t e r i a l and human r e s o u r c e s . S h o u l d t h e s e c o n c l u -s i o n s be s u b s t a n t i a t e d by f u r t h e r s t u d i e s , t h e r e a r e d e f -i n i t e i m p l i c a t i o n s r e g a r d i n g t h e s u b j e c t - m a t t e r m a s t e r y e x p e c t e d o f t e a c h e r s - i n - t r a i n i n g . B e f o r e c o n c l u d i n g t h a t no c o r r e l a t i o n e x i s t s between t e a c h e r and s t u d e n t u n d e r s t a n d i n g o f t h e same t o p i c i n m a t h e m a t i c s , c e r t a i n l i m i t a t i o n s o f t h e i n v e s t i g a t i o n s h o u l d be n o t e d . F i r s t , t h i s s t u d y c o n c e r n s o n l y a s i n g l e 35 t o p i c i n a s i n g l e g r a d e — i n t u i t i v e l i m i t c o n c e p t s i n g r a d e e i g h t . F u r t h e r i n v e s t i g a t i o n s seem w a r r a n t e d f o r o t h e r t o p i c s a t d i f f e r e n t gre.de l e v e l s . S e c o n d l y , because i n t u -i t i v e l i m i t c o n c e p t s i s a t o p i c n o t p r e s c r i b e d f o r any m a t h e m a t i c s c o u r s e p r i o r t o g r a d e e i g h t , t h e p r e s e n t s t u d y d i d n o t a t t e m p t t o measure 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 o f t h e t o p i c i n g r a d e e i g h t . However , t h e use o f p r e - t e s t s a n d p o s t - t e s t s f o r s t u d e n t s m i g h t now be w a r r a n t e d as a c h e c k on t h e a s s u m p t i o n made i n t h i s s t u d y . F i n a l l y , t h e u se o f o n l y i n t e l l i g e n c e and m a t h e m a t i c s a c h i e v e m e n t s c o r e s as c o v a r i a t e s may be i n s u f f i c i e n t . I n p a r t i c u l a r , a meas-u r e o f s t u d e n t a t t i t u d e t o m a t h e m a t i c s m i g h t be i n c l u d e d as a c o v a r i a t e i n f u t u r e i n v e s t i g a t i o n s . A l t h o u g h beyond t h e scope o f t h e p r e s e n t s t u d y , t h e r o l e o f t e a c h e r u n d e r s t a n d -i n g o f a s p e c i f i c t o p i c , t e a c h e r a t t i t u d e t o t h e s p e c i f i c t o p i c , t e a c h e r a t t i t u d e t o s t u d e n t s , and t e a c h e r s t r a t e g i e s u s e d t o d e v e l o p t h e t o p i c a r e f o u r i n t e r a c t i n g v a r i a b l e s w h i c h r e q u i r e f u r t h e r i n v e s t i g a t i o n as t o t h e i r r e l a t i o n t o s t u d e n t u n d e r s t a n d i n g o f t h e s p e c i f i c t o p i c . BIBLIOGRAPHY BIBLIOGRAPHY B a r r , A . S . " S e c o n d R e p o r t o f t h e C o m m i t t e e o n C r i t e r i a o f T e a c h e r E f f e c t i v e n e s s , " 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 , 46:641-58, M a y , 1953. B a s s h a m , H a r r e l l . " T e a c h e r U n d e r s t a n d i n g a n d P u p i l E f f i c i e n c y i n M a t h e m a t i c s : A S t u d y o f R e l a t i o n s h i p , " T h e A r i t h m e t i c T e a c h e r , 9 :383-87, N o v e m b e r , 1962. B i n g , R . II. " P o i n t S e t T o p o l o g y , " I n s i g h t s i n t o M o d e r n M a t h e m a t i c s , p p . 306-335. T w e n t y - t h i r d Y e a r b o o k o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s . W a s h i n g t o n : T h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h -e m a t i c s , 1957. B l o o m , B e n j a m i n S . ( e d . ) . Taxonomy o f E d u c a t i o n a l O b j e c -t i v e s , t h e C l a s s i f i c a t i o n o f E d u c a t i o n a l G o a l s , H a n d -b o o k I: C o g n i t i v e D o m a i n . New Y o r k : D a v i d McKay Company , I n c . , 195&~. B r i t i s h C o l u m b i a D e p a r t m e n t o f E d u c a t i o n . 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 , 1966. V i c t o r i a : B r i t i s h C o l u m b i a D e p a r t -ment o f E d u c a t i o n , 1966. B r o w n , K e n n e t h E . " T e a c h i n g L o a d a n d Q u a l i f i c a t i o n s o f M a t h e m a t i c s T e a c h e r s , " T h e M a t h e m a t i c s T e a c h e r , 5 3 : 2-11, J a n u a r y , i 9 6 0 . , a n d T h e o d o r e L . A b e l l . " R e s e a r c h i n t h e T e a c h i n g o f H i g h S c h o o l M a t h e m a t i c s , " T h e M a t h e m a t i c s T e a c h e r , 59-53-57, J a n u a r y , 1966. B r u m f i e l , C h a r l e s F . , R o b e r t E . E i c h o l z , a n d M e r r i l l E . S h a n k s . I n t r o d u c t i o n t o M a t h e m a t i c s . R e a d i n g , M a s s a c h u s e t t s : A d d i s o n - W e s l e y P u b l i s h i n g Company , 1963. C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d . R e p o r t o f t h e C o m m i s -s i o n o n M a t h e m a t i c s , P r o g r a m f o r C o l l e g e P r e p a r a t o r y  M a t h e m a t i c s . New Y o r k : C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d , 1959. . A p p e n d i c e s t o t h e R e p o r t o f t h e C o m m i s s i o n o n M a t h e m a t i c s , P r o g r a m f o r C o l l e g e P r e p a r a t o r y M a t h e -m a t i c s . New Y o r k : C o l l e g e E n t r a n c e E x a m i n a t i o n B o a r d , 1959. 38 D e m p s t e r , J . R . H . , a n d G . E . S t a r k e y . MFACO: A n a l y s i s o f  C o v a r i a n c e , V a n c o u v e r : 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 C o m p u t i n g C e n t r e , 1968 . D e s s a r t , D o n a l d J o s e p h . " A S t u d y o f P r o g r a m m e d L e a r n i n g w i t h S u p e r i o r E i g h 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 t h e s i s , 'The U n i v e r s i t y o f M a r y l a n d , 1 9 6 l . D i c k e n s , C h a r l e s H e n d e r s o n . " E f f e c t o f I n - s e r v i c e T r a i n i n g 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 o n T e a c h e r s ' U n d e r -s t a n d i n g a n d T e a c h i n g o f 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 t h e s i s , Duke U n i v e r s i t y , 1966 . D u t t o n , W i l b u r H . E v a l u a t i n g P u p i l 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 . E n g l e w o o d C l i f f s , New J e r s e y : P r e n t i c e H a l l , I n c . , 1964. F u l k e r s o n , E . "How W e l l Do 158 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 Know A r i t h m e t i c ? , " T h e A r i t h m e t i c T e a c h e r , 7:141-46, M a r c h , i 9 6 0 . G a r r e t t , H e n r y E . S t a t i s t i c s i n P s y c h o l o g y a n d E d u c a t i o n . F i f t h e d i t i o n . New Y o r k : D a v i d McKay Company , I n c . , 1958. G i b b , E . G . , a n d D . M . M a t a l a . " S t u d y o f t h e U s e o f S p e c i a l T e a c h e r s i n S c i e n c e a n d M a t h e m a t i c s i n G r a d e s F i v e a n d S i x , " S c h o o l S c i e n c e a n d M a t h e m a t i c s , 6 l : 569-72, N o v e m b e r , 1961; 62:565rB3, N o v e m b e r , 1962. G l e n n o n , V i n c e n t J . " A S t u d y i n N e e d e d 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 , " T h e M a t h e m a t i c s T e a c h e r , 42:389-96, D e c e m b e r , 1949. H o u s t o n , V/. R . , a n d M . V . D e V a u l t . " M a t h e m a t i c s I n - s e r v i c e E d u c a t i o n : T e a c h e r G r o w t h I n c r e a s e s P u p i l G r o w t h , " T h e A r i t h m e t i c T e a c h e r , 10:243-47, May , 1963. K e n n e y , R u s s e l l A . " 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 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 , " T h e A r i t h m e t i c T e a c h e r , 12: 431-42, O c t o b e r , 1965. K i p p s , C a r o l . " E l e m e n t a r y T e a c h e r s ' A b i l i t y t o U n d e r s t a n d C o n c e p t s U s e d i n New M a t h e m a t i c s , " T h e A r i t h m e t i c  T e a c h e r , 15:367-71, A p r i l , 1968. 39 L a m p e l a , R o l a n d M i t c h e l l . " 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 b e t w e e n T e a c h e r U n d e r s t a n d i n g a n d Change i n P u p i l U n d e r s t a n d i n g o f S e l e c t e d C o n c e p t 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 . " U n p u b l i s h e d D o c t o r a l t h e s i s , T h e U n i v e r s i t y o f C a l i f o r n i a , L o s A n g e l e s , 1966. L e i s s a , A r t h u r W . , a n d R o b e r t C . F i s h e r . " A S u r v e y o f T e a c h e r s ' O p i n i o n s o f a R e v i s e d M a t h e m a t i c s C u r r i c u l u m , " T h e M a t h e m a t i c s T e a c h e r , 53:113-18, F e b r u a r y , i 9 6 0 . L e o n h a r d t , E a r l A l b e r t . " A n A n a l y s i s o f S e l e c t e d F a c t o r s i n R e l a t i o n t o H i g h a n d Low 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 t h e s i s , T h e U n i v e r s i t y o f N e b r a s k a , 1962. M o r s h , J o s e p h E . , G e o r g e C . B u r g e s s , a n d P a u l N . S m i t h . " S t u d e n t A c h i e v e m e n t a s a M e a s u r e o f I n s t r u c t o r E f f e c -t i v e n e s 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 , 4 7 : 7 9 - 8 8 , F e b r u a r y , 1956. N e i l l , R o b e r t D u d l e y . " T h e E f f e c t s o f S e l e c t e d T e a c h e r V a r i a b l e s o n t h e M a t h e m a t i c s A c h i e v e m e n t o f A c a d e m i -c a l l y T a l e n t e d J u n i o r H i g h S c h o o l P u p i l s . " U n p u b l i s h e d D o c t o r a l t h e s i s , C o l u m b i a U n i v e r s i t y , 1966. Newsom, C . V . " M a t h e m a t i c a l B a c k g r o u n d N e e d e d by T e a c h e r s , " T h e T e a c h i n g o f A r i t h m e t i c , p p . 232-49. F i f t i e t h Y e a r -b o o k o f t h e N a t i o n a l S o c i e t y f o r t h e S t u d y o f E d u c a t i o n , P a r t I I . 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 , 1951. O r l e a n s , J a c o b S . , a n d E d w i n W a n d t . " T h e U n d e r s t a n d i n g s 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 , 53:501-7, M a y , 1953. P e s k i n , A n n e S t e r n . " T e a c h e r U n d e r s t a n d i n g a n d A t t i t u d e a n d S t u d e n t A c h i e v e m e n t a n d A t t i t u d e i n S e v e n t h G r a d 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 t h e s i s , New Y o r k U n i v e r s i t y , 1964. R o u s e , W i l l i a m M o r r i s o n . " A S t u d y o f t h e C o r r e l a t i o n b e -t w e e n t h e A c a d e m i c P r e p a r a t i o n o f T e a c h e r s o f M a t h e m a t -i c s a n d t h e M a t h e m a t i c s A c h i e v e m e n t o f T h e i r S t u d e n t s i n K i n d e r g a r t e n t h r o u g h G r a d e E i g h t . " U n p u b l i s h e d D o c t o r a l t h e s i s , M i c h i g a n S t a t e U n i v e r s i t y , 1967. 40 S m i t h , L e h i T . " C o u l d We T e a c h L i m i t s ? , " T h e M a t h e m a t i c s  T e a c h e r , 54:344-45, M a y , 1961. S p a r k s , J a c k N . " A r i t h m e t i c U n d e r s t a n d i n g s N e e d 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 , " T h e A r i t h m e t i c T e a c h e r , 8:395-403, D e c e m b e r , 1961. W a l k e r , H e l e n M . , a n d J o s e p h L e v . S t a t i s t i c a l I n f e r e n c e . New Y o r k : H e n r y H o l t a n d Company,- 1953. W e a v e r , J . F r e d . " 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 , 56:255-61, F e b r i i a r y , 1956. APPENDICES i APPENDIX A FINAL STUDENT TEST \ MATHEMATICS''8 NAME: ; DIVISION:' DIRECTIONS: This test i s to determinejyour understanding of an idea i n mathematics. You w i l l probably f i n d that some of the questions are d i f f e r e n t from those you have seen before, Try them anyway. By thinking and experimenting you w i l l probably be able to answer most questions. NOTE: If a question has no correct answer write NONE In the. answer space. PART A: Use the symbol f o r greater than ( ^ > ), less than ( <C )» or equals (=) to make each of the following true. 1. 0 . 6 6 6 . . . • 0.666 2. 0 . 6 6 6 . . . 0.666666 J. 0 . 6 6 6 . . . | 4. 0.667 I , 5. 0.666 3 PART B: 6. ADD', 7 . ADD 8. ADD 0 , 9 9 9 9 . . . 0 . 3 9 9 9 9 . . . 0 .472222... +0.9999... +0.49999... +0.318888... 9. SUBTRACT 5 . 0 0 0 0 . . . -4.9999... PART C: A number l i n e may help you with these questions. Consider each set of numbers as continuing i n the pattern shown. In the f i r s t two l i s t s the numbers get closer and closer to 1. For each l i s t , what i s the smallest number which the numbers keep getting closer to? 11. 0 . 9 , 0.99, 0.999, 0.9999, ... 11. ': 1 2 , v h i c ? 32' . •*•.• 1 2 * ; — - — In the next two l i s t s the numbers get smaller and smaller. What i s the largest number that i s s t i l l smaller than any number i n each l i s t ? 10. SUBTRACT 3.000000. . ; . "0.262626.. . MATHEMATICS 8 PAGE 2 NAME: PART D: Think of each of the following as an endless l i s t of additions (or subtractions) i n which the pattern continues as shown. Give the sum (or difference) of each endless l i s t . 18. 0.6 + 0.06 + 0.006 + 0.0006 : + . . . 18. .. 19. 9!'+ 0.9 + 0.09 + 0.009 + 0 . 0 0 0 9 + . . . 19. 20. 1 + 2 + 4 + 6 + 8 + 16 + . . . 20. 21. 1:+ 1 + 1 + 1 + 1 + 21. :: 22. 1 + I + i + i + i + 2 3 Ti 5 S 22. . SUBTRACT » • 23. 1 - 0.9 - 0.09 - 0.009 - 0.0009 - . . . 23. ": 24. 1;;- 1 1 1 1 • 2 " 5 " "8* 16~ 24. r o . J.,, 1 0 , 1 0 0 1 0 0 0 ioooo.| *•* ->*-PART S: Think of the set of a l l r e a l numbers les s than 5. i ' • = 26. Give a number i n the set which i s greater than 4.9 but le s s than 5« 26._ 27. Give a number i n the set which i s greater than 4.9999 "but les s than 5. 27.. 28. Give a number i n the set which i s greater than 4 . 9 9 9 . . . but less than. 5. ^ 28._ 29. What i s the large s t number i n the set of a l l v r e a l numbers les s than 5? 29«. PART F: Think of the set of a l l r e a l numbers greater than. 15. 30. Give a number i n the set which i s greater than 15 but le s s than 15.1 30._ i . • ; • 31. Give a number i n the set which i s greater than 15 |but less than 15.111... 31._ PART G: If you replace /\ with larger and larger numbers then: 32. the value of gets closer and closer to y Z\ what number? 32.. 33* the value of gets closer |and:closer to A what number? 33._ 34. the value of -— gets closer and closer to 5 x /\ what number? 34.. 35. the value of ( 2 x A) +• 1 gets closer and A closer to ^ what number? 35« END .. A P P E N D I X . B F I N A L T E A C H E R T E S T ; M T S F R O M ^ i p P S ^ M T • C U A S S S S t«i?Ci P r M S , l > £ £ 3 > A VA.TO.ETY O P W O T A T I O ^ L A t , R £ E M £ s l T S T o >a^ £ R M t i^E T"tiElR. A^SV0&R.SSI: S O M E -v ' • i W-A&e. I M C O R ^ E C I . P L E A S E , CIRCUE. A L L O F - T M E "F-G'^&VJ A ^ S W E R 5 vurru WHICH X o o ACa"R'£.E. „ " . /. > - > •3.. = • * > •3-O0O...-j. O.Q996 o. 79999... o- 6999... O.S9999...& - . 0-9 S. o. 79/n... 6. 191IIO... 0.8000 ... , 7. 0 'O.OOQt... 0. in)... /o. 731 sis...i4 ^.7373 73... a. 7313 ?4... . / / . O. 99999 0.999... / © IX. 6 3 0.4-999... - t 7 <3.S" //OA'S • 3. i ^ /ooo... /4. -?.///... o i o?. ooo. ../ /. 999-.. \/JOA/£ /sr. 0 . 6> & •£> C> ^, "7 J S 4. UI... . .3.999... /7. 0. O. III... 1 O. III... 9 O. IIIIS '8. 0.7 /<?. 9. 999... -9.9999 10 9.999... 9 AfOA/B 10 . /OOO...0 69 999...? AtONB 2.1. rS 2-75 2~ /. 999... -XX- / t /.9'?9... 13. 0.111... O. COO/ 0.000.^./ Vi. 1 ./& 0 / / 32-ni... 0 1 ' ; 1 IS. 1000... /ooo... iOOQ o A/OA/B Mom AlOA/S 2X0. 4.9111... V. 999. * * 2.7. 4. 919989999... 4.9999... 4.9999s- ' 4.m...9 V. 9999mi... 4 999993 4.999...-? 4.999... 4. 99999... '30. is. 1/1,,» /5VO///... ..'f-0999... A/'OAlZ js.0199... JS.I /&. O/JI... 0.999.. \ 0. II/.'. O-OOO... / t.O O. A/OA/-E 33. s,o O (9. ///•... 4- 999... 0-00O...J *iO*/B 34. 0 0. i999.o • 0. I l l . . . 0.000... J W^V£ 35'. 0 0> ood... /. 999... 2.^ 3-«0 A/a\'S APPENDIX C DIRECTIONS FOR A D M I N I S T R A T I O N OF F I N A L T E S T S 1. I n t r o d u c t i o n : " I ' m h e r e t o d a y t o g i v e y o u a t e s t w h i c h i s p a r t o f a U n i v e r s i t y o f B . C . s t u d y o f how w e l l g r a d e e i g h t s t u d e n t s u n d e r s t a n d a c e r t a i n i d e a i n m a t h e m a t i c s . " 2 . D i s t r i b u t i o n : D i s t r i b u t e f o o l s c a p a n d t e s t p a p e r s f a c e -down. "When y o u r e c e i v e y o u r c o p y o f t h e t e s t , l e a v e i t f a c e - d o w n o n y o u r d e s k . T h e f o o l s c a p i s f o r r o u g h w o r k b u t w i l l be c o l l e c t e d . " C h e c k t h a t e v e r y o n e h a s a t e s t p a p e r . 3 . Name a n d d i v i s i o n : "When y o u t u r n t h e t e s t p a p e r o v e r p l e a s e PRINT y o u r f u l l name a n d d i v i s i o n number a t t h e t o p o f p a g e o n e . Do t h i s n o w . " ( p a u s e ) "Now t u r n t o page two a n d a g a i n p r i n t y o u r name a t t h e t o p . " 4 . D i r e c t i o n s : " T u r n t o t h e d i r e c t i o n s on p a g e one a n d r e a d them t o y o u r s e l f w h i l e I r e a d them a l o u d . " ( R e a d f r o m y o u r c o p y o f t e s t , i n c l u d i n g N O T E . ) " T i m e w i l l be p r o v i d e d f o r mos t o f y o u t o f i n i s h b u t i t w i l l be no l o n g e r t h a n 40 m i n u t e s . I f y o u do f i n i s h , c h e c k y o u r s o l u t i o n s b e f o r e p l a c i n g y o u r p a p e r f a c e - d o w n u n t i l t i m e i s u p . Once y o u b e g i n , no 46 q u e s t i o n s w i l l be a n s w e r e d . I s t h e r e a n y o n e n o t s u r e o f what h e i s t o d o ? " ( A n s w e r f r o m d i r e c t i o n s i f n e c e s s a r y . ) " B E G I N . " ( N o t e t i m e p l u s 40 m i n u t e s t o " S T O P " . ) 5. N o t a t i o n : W h i l e t h e s t u d e n t s w r i t e t h e t e s t , h a v e t h e i r m a t h e m a t i c s t e a c h e r c o m p l e t e t h e s p i r i t - s t e n c i l l e d s h e e t o n n o t a t i o n a l a g r e e m e n t s w i t h o u t u s i n g a t e x t -b o o k . A s s u r e h i m t h a t t h e r e w i l l be no way o f i d e n -t i f y i n g s c h o o l , c l a s s , o r t e a c h e r , b u t h i s c o m p l e t i o n o f t h e a n s w e r s i s a k e y p a r t o f t h e s t u d y . A S K HIS COOPERATION I N NOT D I S C U S S I N G T H E T E S T WITH HIS C O L L E A G U E S . 6. C o l l e c t i o n : When 40 m i n u t e s h a s e l a p s e d , a n n o u n c e , " S T O P . " C o l l e c t t e s t p a p e r s f i r s t , t h e n f o o l s c a p . P l a c e a l l i n e n v e l o p e w i t h t e a c h e r a n s w e r s . DO NOT I D E N T I F Y E N V E L O P E . A n y u n u s e d p a p e r s must be b r o u g h t away i n t h e " E X T R A " e n v e l o p e . APPENDIX D COMPUTATION OP PRODUCT-MOMENT CORRELATION BETWEEN TEACHER SCORES (X) AND ADJUSTED CLASS MEANS (Y) £XY - NMxMy r = — • - — — A / (2JX 2 - NM2 )(£Y 2 - NM2,,) 9 3 7 5 . 0 - 14(42 . 8 ) ( 1 5 . 6 )  J (27451 - l 4 ( 4 2 . 8 ) 2 ) ( 3 4 5 6 . 9 - 1 4 ( 1 5 . 6 ) 2 ) 9 3 7 5 . 0 - 9 3 4 7 . 5 V ( 2 7 4 5 1 - 2 5 6 4 5 . 8 ) ( 3 4 5 6 . 9 - 3 4 0 7 ) 2 j \ 5 ^ 1 8 0 5 . 2 ) ( 4 9 . 9 ) 2 7 . 5 A/ 9 0 0 7 9.48 2 7 . 5 3 0 0 . 1 • = 0 . 0 9 

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