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The attitude and achievement of tenth grade general mathematics students as effected by the use of desk.. 1972

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n THE ATTITUDE AND ACHIEVEMENT OF TENTH GRADE GENERAL MATHEMATICS STUDENTS AS EFFECTED . BY THE USE OF DESK CALCULATORS by LESLIE RAE HUMPHRIES 3. A., U n i v e r s i t y o f B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF MASTER OF ARTS i n the Department o f E d u c a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA NOVEMBER, 1972 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 C o l u m b i a , I ag r ee 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 t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d 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 / 2 The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8, Canada Date ABSTRACT The purpose o f the study was t o determine whether the use o f a desk c a l c u l a t o r i n a grade ten g e n e r a l mathematics c l a s s would produce a p o s i t i v e change i n a p u p i l a t t i t u d e toward mathematics, to see i f " achievement i n q u a n t i t a t i v e t h i n k i n g (problem s o l v i n g ) would be i n c r e a s e d , and t o see i f t h e r e was a c o r r e l a t i o n between any such changes. Two t e a c h e r s , each having two c l a s s e s of grade,ten g e n e r a l mathematics, taught an experimental and a c o n t r o l c l a s s . The e x p e r i m e n t a l c l a s s e s were i n s t r u c t e d i n the use o f desk c a l c u l a t o r s , and encouraged t o use the c a l c u l a t o r s to a r r i v e a t s o l u t i o n s t o a p r e a r r a n g e d program i n mathematics. The c o n t r o l groups were taught i n a s i m i l a r manner, but d i d not have access to desk c a l c u l a t o r s . The c l a s s e s were p r e - t e s t e d and p o s t - t e s t e d f o r a t t i t u d e and achievement. A Mathematics A t t i t u d e Test was c o n s t r u c t e d which y i e l d s (1) a " P r e f e r e n c e f o r C a l c u l a t i o n " s c o r e , (2) a " P r e f e r e n c e f o r Q u a n t i t a t i v e T h i n k i n g " (Problem S o l v i n g ) s c o r e , and (.3) a composite "Mathematics A t t i t u d e " s c o r e . Th t e s t comprised s i x t e e n t h r e e - p a r t s i t u a t i o n s c o n s i s t i n g o f (A), a c a l c u l a t i o n s i t u a t i o n , (B), a q u a n t i t a t i v e t h i n k i n g (problem s o l v i n g ) s i t u a t i o n , and a d i s t r a c t e r , (C), a i i i non-inathematical a c t i v i t y . The p u p i l s were asked to s t a t e a l i k e and a d i s l i k e among the t h r e e o f f e r e d . On item a n a l y s i s u s i n g b i s e r i a l r, a l l but one o f the items proved s a t i s f a c t o r y . When u s i n g the Spearman-Brown prophecy formula f o r e s t i m a t i n g r e l i a b i l i t y , the r e l i a b i l i t y was found to be s u f f i c i e n t f o r d i f f e r e n t i a t i n g between means o f groups, but not s u f f i c i e n t f o r d i f f e r e n t i a t i o n o f i n d i v i d u a l d i f f e r e n c e s . A n a l y s i s o f C o v a r i a n c e was used to compare the groups f o r a s i g n i f i c a n t p o s i t i v e change i n a t t i t u d e , and a l s o f o r a s i g n i f i c a n t improvement i n achievement. The c o r r e l a t i o n o f a s u s p e c t e d p o s i t i v e change i n a t t i t u d e and improvement i n achievement was checked f o r s i g n i f i c a n c e . At the 0.05 l e v e l , t h e r e was no s i g n i f i c a n t change i n a t t i t u d e and no s i g n i f i c a n t change i n achievement. At the 0.08 l e v e l , however, t h e r e was a s i g n i f i c a n t improvement i n achievement. There was no s i g n i f i c a n t c o r r e l a t i o n o f a p o s i t i v e change i n a t t i t u d e and improvement i n achievement. The c o n c l u s i o n drawn was that the use o f a desk c a l c u l a t o r makes no s i g n i f i c a n t change i n a t t i t u d e , but perhaps might change achievement w i t h a c a r e f u l l y planned program. No c o r r e l a t i o n o f a p o s i t i v e change i n a t t i t u d e and achievement was found. The a t t i t u d e t e s t c o n s t r u c t e d may, w i t h m o d i f i c a t i o n s , be o f use to o t h e r experimenters. i V TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 BACKGROUND 1 PURPOSE OF THE STUDY 3 DEFINITION OF TERMS 4 HYPOTHESES 5 2. REVIEW OF THE LITERATURE 6 REVIEW BY AIKEN 6 ATTITUDE 9 ATTITUDE AND ACHIEVEMENT 11 CALCULATORS, ATTITUDE AND ACHIEVEMENT . . . 12 3. DESIGN OF THE STUDY 16 PURPOSE 16 PROCEDURE 16 HYPOTHESES IS TEST SELECTION 19 A r i t h m e t i c A t t i t u d e 19 Q u a n t i t a t i v e T h i n k i n g Achievement . . . . 20 Mathematical A t t i t u d e '. 22 TEST CONSTRUCTION 22 D e f i n i t i o n o f Terms 22 Item S e l e c t i o n 23 S c o r i n g 24 Item A n a l y s i s 25 V Chapter Page R e l i a b i l i t y 25 STATISTICAL DESIGN 26 4. ANALYSIS OF THE RESULTS 28 FIRST HYPOTHESIS 28 SECOND HYPOTHESIS 29 THIRD HYPOTHESIS 29 5. SUMMARY AND CONCLUSIONS 31 LIMITATIONS : 32 IMPLICATIONS FOR FURTHER RESEARCH 35 APPENDICES FOOTNOTES 37 BIBLIOGRAPHY 42 APPENDIX A SAMPLE PROBLEMS 45 APPENDIX B DUTTON'S ATTITUDE SCALE 48 APPENDIX C MATHEMATICS ATTITUDE TEST 50 APPENDIX D IOWA TEST OF QUANTITATIVE THINKING. 55 APPENDIX E ITEM ANALYSIS OF MATHEMATICS ATTITUDE TEST 64 C h a p t e r 1 BACKGROUND One o f t h e p r o b l e m s o f t h e t e a c h e r i s how t o d e a l w i t h t h e low a c h i e v e r i n m a t h e m a t i c s . T h a t low a b i l i t y i s t h e c h i e f d e t e r m i n e r f o r low a c h i e v e m e n t i s n o t n e c e s s a r i l y t h e c a s e . A i k e n i n h i s r e v i e w o f r e s e a r c h on a t t i t u d e s t o w a r d m a t h e m a t i c s p o i n t s o u t " m a t h e m a t i c a l a b i l i t y may be a l e s s i m p o r t a n t d e t e r m i n e r o f t h e a c h i e v e m e n t o f s t u d e n t s h a v i n g e x t r e m e a t t i t u d e s t o w a r d m a t h e m a t i c s t h a n t h o s e h a v i n g more m o d e r a t e a t t i t u d e s . " ' ' ' S t u d i e s have shown t h a t t h e low a c h i e v e r g e n e r a l l y h a s a n e g a t i v e a t t i t u d e w h i l e t h e g o od a c h i e v e r has a p o s i t i v e a t t i t u d e t o w a r d 2 m a t h e m a t i c s . D u t t o n has l i s t e d s e v e r a l a r e a s t h a t c o n t r i b u t e t o t h e n e g a t i v e a t t i t u d e s t o w a r d a r i t h m e t i c some b e i n g t h a t a r i t h m e t i c " t a k e s t o o l o n g " a n d was " b o r i n g o r s t a l e " w i t h 3 " t o o much m e m o r i z a t i o n . " The a c q u i r i n g o f n e g a t i v e a t t i t u d e s c o v e r s a d i v e r s e f i e l d . Some o f t h e c o n t r i b u t i n g f a c t o r s t o w a r d t h e a c q u i r i n g o f a n e g a t i v e a t t i t u d e a r e a s l o w l e a r n i n g r a t e , p a r e n t a l a t t i t u d e , c o u r s e c o n t e n t , a n d t h e t e a c h e r ' s 4 . . a p p r o a c h . The c l a s s r o o m t e a c h e r s c a n n o t hope t o e l i m i n a t e a l l t h e c o n t r i b u t i n g f a c t o r s b u t t h e y m i g h t be a b l e t o m o d i f y t h e e x i s t i n g o n e s . 2 The p u p i l ' s s e l f concept i n r e l a t i o n to mathematics i s p a r t l y developed by h i s e x p e r i e n c e s i n working with mathematics. I f he f a i l s i n a t a s k many times then s e l f c o n f i d e n c e may be reduced and h o s t i l i t y may be i n c r e a s e d . A c c o r d i n g to A i k e n "The t e a c h e r must p r o v i d e f o r success experiences i n the l e a r n i n g . " " ' A l p e r t et a l l e a d us to conclude low achievement generates a n e g a t i v e a t t i t u d e and a n e g a t i v e a t t i t u d e 6 generates low achievement. To change an e x i s t i n g n e g a t i v e a t t i t u d e to a more p o s i t i v e one might be one way o f improving achievement. There a r e i n d i c a t i o n s that such a change can be brought about. ' However, the a t t i t u d e r e v e a l e d by the p u p i l s i s not 9 a unique a t t i t u d e but a m u l t i d i m e n s i o n a l a t t i t u d e . To measure what a t t i t u d e s are p r e s e n t and what a t t i t u d e s can be changed and how they can be changed i s a complex problem. Dutton has g i v e n us i n d i c a t i o n s o f the a t t i t u d e dimensions or f a c e t s . ^ I f a p u p i l were to d i s l i k e a r i t h m e t i c because he f i n d s i t slow and t e d i o u s then any s i t u a t i o n r e q u i r i n g a r i t h m e t i c c o u l d have a n e g a t i v e a t t i t u d e t r a n s f e r r e d t o i t . Perhaps by the removal, of the n e c e s s i t y o f doing t e d i o u s c a l c u l a t i o n s the s i t u a t i o n t i i a t f o r m e r l y would have n e g a t i v e c o n n o t a t i o n s c o u l d have a p o s i t i v e c o n n o t a t i o n . Perhaps the s i t u a t i o n i n i t s e l f might be i n t e r e s t i n g and rewarding which p r e v i o u s l y c o u l d not be because o f the 3 ne g a t i v e a t t i t u d e toward c a l c u l a t i o n s . S e v e r a l such It 12 13 programs have been attempted with v a r y i n g r e s u l t s . " ' ' In order to reduce the work i n v o l v e d i n s o l v i n g q u e s t i o n s posed, desk .-calculators have been used i n home and i n d u s t r y . The manufacturers of desk c a l c u l a t o r s have expressed t h e i r optimism about the machines use i n s c h o o l . Newspapers have c a r r i e d o p i n i o n s from o b s e r v a t i o n s but do not v a l i d a t e the r e s u l t s . H e a d l i n e s such as " C a l c u l a t o r Takes S t i n g Out of Math" have generated q u e s t i o n s about the 14 use of c a l c u l a t o r s i n a mathematical program. PURPOSE OF THE STUDY In t h i s c h apter v a r i o u s sources have been c i t e d which make c l a i m s about the e f f e c t s of a c a l c u l a t o r based program on the a t t i t u d e and achievement of the low a c h i e v e r . From these c l a i m s about improved a t t i t u d e and achievement i t would seem t h a t a c a l c u l a t o r based program might be expected to produce s e v e r a l outcomes f o r the low a c h i e v e r : 1. A p o s i t i v e change i n g e n e r a l a t t i t u d e toward mathematics. 2. A g e n e r a l improvement i n problem s o l v i n g achievement. 3. A p o s i t i v e c o r r e l a t i o n between a p o s i t i v e change i n a t t i t u d e and improvement i n problem s o l v i n g achievement. A p r o p o s a l f o r a study was put f o r t h which would compare the a t t i t u d e s and achievement o f f o u r c l a s s e s 4 o f grade 10 g e n e r a l mathematics s t u d e n t s . Two c l a s s e s u s i n g desk c a l c u l a t o r s and two c o n t r o l c l a s s e s not u s i n g the c a l c u l a t o r s were to be used. Two teachers would be i n v o l v e d each having one c o n t r o l and one experimental c l a s s . The c l a s s e s would be taught as s i m i l a r i l y as was f e a s i b l e . The a t t i t u d e s and achievements would be measured by s e l e c t e d t e s t s b e f o r e and a f t e r the experimental p e r i o d . The measures would be s t a t i s t i c a l l y a n a l y s e d f o r s i g n i f i c a n t changes and c o r r e l a t i o n s . DEFXNITION OF'TERMS (a) Q u a n t i t a t i v e t h i n k i n g : the a s s i m i l a t i o n of s u p p l i e d data together w i t h the a p p l i c a t i o n o f b a s i c o p e r a t i o n s of a r i t h m e t i c t o a r r i v e at a s o l u t i o n t o a q u e s t i o n to which the student has not been t o l d what steps or procedures are t o be taken. (b) g e n e r a l mathematics: a program des i g n e d by the B r i t i s h Columbia Department of E d u c a t i o n to be g i v e n to those students who are not on an a c a d e m i c - t e c h n i c a l program. I t i s n ormally g i v e n to those whose academic achievement i s such, that they would not be a b l e to compete s u c c e s s f u l l y w i t h o t h e r s more a b l e . (c) desk c a l c u l a t o r : a machine such as the O l i v e t t i Underwood Divisi-Suma 24 which i s a ten key machine *- A more d e t a i l e d o u t l i n e i s to be found i n Chapter 3 pages 2(5-2 7. 5 p e r f o r m i n g a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , a n d d i v i s i o n w i t h p r i n t e d o u t p u t . HYPOTHESES T h r e e h y p o t h e s e s were f o r m u l a t e d . 1. T h e r e w i l l b e a p o s i t i v e c h ange i n g e n e r a l 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 f o r t h e g r a d e t e n G e n e r a l M a t h e m a t i c s s t u d e n t s u s i n g d e s k c a l c u l a t o r s . 2. The a c h i e v e m e n t i n d o i n g q u a n t i t a t i v e t h i n k i n g w i l l show g r e a t e r improvement f o r t h o s e u s i n g d e s k c a l c u l a t o r s t h a n f o r t h o s e n o t u s i n g d e s k c a l c u l a t o r s . 3. Where t h e r e i s a change i n a t t i t u d e t o w a r d q u a n t i t a t i v e t h i n k i n g t h e r e w i l l be a c h a n g e i n t h e same d i r e c t i o n i n t h e a c h i e v e m e n t i n d o i n g q u a n t i t a t i v e t h i n k i n g . Chapter 2 REVIEW OF THE LITERATURE REVIEW BY AIKEN In Aiken's (1969) review o f the r e s e a r c h c o n c e r n i n g a t t i t u d e s toward mathematics he d i s c u s s e s s e v e r a l methods o f measuring a t t i t u d e s even though some ma i n t a i n that t h e r e a r e no v a l i d m e a s u r e s . ^ A i k e n notes that " o b s e r v a t i o n i s s u p e r f i c i a l l y the most o b j e c t i v e measure but ...teacher o b s e r v a t i o n ( i s found) to be inadequate."- 1-^ A second method i s a q u e s t i o n a l re where students are asked to s t a t e t r u e or f a l s e to a statement and the responses are t a b u l a t e d to i n d i c a t e an o v e r a l l a t t i t u d e or, i n another example, a 17 p r e f e r e n c e to a s u b j e c t f i e l d . The Thurstone and L i k e r t a t t i t u d e s c a l i n g t e chniques were found to be popular, but Guttman's scalogram a n a l y s i s was i n f r e q u e n t l y used. A study by N e a l e i g h used p i c t u r e p r e f e r e n c e s to measure a t t i t u d e s and achievement p r o n e n e s s . ^ P i c t u r e p r e f e r e n c e was found to be d i s c r i m i n a t i n g at the t h i r d grade. P h y s i c a l i n d i c a t o r s o f a n x i e t y such as e l e c t r i c a l s k i n r e s i s t a n c e , b r e a t h i n g c h a r a c t e r i s t i c s , b l o o d p r e s s u r e and heart beat r a t e were found by A i k e n t o be used by a few 19 experimenters. 7 When the a t t i t u d e s were measured i t was noted " t h a t v ery d e f i n i t e a t t i t u d e s toward a r i t h m e t i c may be formed as 2 0 e a r l y as the t h i r d grade." I t was f u r t h e r noted that J u n i o r High School was the p e r i o d when a l a r g e percentage 21 o f p r o s p e c t i v e teachers formed t h e i r a t t i t u d e . A i k e n summarizes the s e c t i o n on measuring a t t i t u d e s by s a y i n g : . . . a t t i t u d e s are p r o b a b l y not v e r y s t a b l e i n the e a r l y grades. In a d d i t i o n , the p r e c i s e n e s s with which p u p i l s can express t h e i r a t t i t u d e s v a r i e s with l e v e l o f m a t u r i t y . F i n a l l y i t i s c l e a r that a t t i t u d e s toward d i f f e r e n t a s p e c t s o f a r i t h m e t i c and mathematics are measured by " g e n e r a l a t t i t u d e " instruments a d m i n i s t e r e d at d i f f e r e n t grade l e v e l s . A t t i t u d e toward m a t e r i a l s to be l e a r n e d by r o t e , such as the m u l t i p l i c a t i o n t a b l e , i s not the same v a r i a b l e as a t t i t u d e toward word problems and a l g e b r a i c symbols. Of more importance than the exact frequency o f a t t i t u d e s at d i f f e r e n t grade l e v e l s , however, are the causes and e f f e c t s of these a t t i t u d e s . ^ I t was noted that A l p e r t et a l (1963) made an a n a l y s i s o f the r e l a t i o n s h i p s among a t t i t u d e , e x p e c t a t i o n and performance, they viewed t h e l e v e l o f e x p e c t a t i o n and performance as a k i n d of s e l f p e r p e t u a t i n g c y c l e a f f e c t i n g 23 the c h i l d ' s s e l f concept. Another study by Brown and A b e l l found the " c o r r e l a t i o n o f a t t i t u d e and achievement was h i g h e r f o r a r i t h m e t i c than f o r s p e l l i n g , r e a d i n g or 24 language." Durrance found s i g n i f i c a n t c o r r e l a t i o n between performance i n mathematics and measures o f a t t i t u d e and 25 a n x i e t y . I t was noted by Cech t h a t a c h i e v e r s had a more p o s i t i v e a t t i t u d e toward mathematics than u n d e r a c h i e v e r s . 26 A study by Jackson maintains that i t i s only at 8 "the extremes - h i g h l y p o s i t i v e or h i g h l y n e g a t i v e - that a t t i t u d e a f f e c t s achievement i n any s i g n i f i c a n t way."^ Aiken summarizes h i s review o f the r e l a t i o n s h i p s among a t t i t u d e , e x p e c t a t i o n , and performance by s a y i n g : C o l l e c t i v e l y , the f i n d i n g s o f s t u d i e s r e l a t i n g p e r s o n a l i t y v a r i a b l e to mathematics a t t i t u d e and achievement i n d i c a t e t h a t i n d i v i d u a l s with more p o s i t i v e a t t i t u d e s and high e r achievement tend t o have b e t t e r p e r s o n a l and s o c i a l adjustment than those with n e g a t i v e a t t i t u d e s and low achievement. These r e s u l t s must be kept i n p e r s p e c t i v e , however. The c o r r e l a t i o n s a re r e l a t i v e l y low, and i t i s a t r u i s m that c o r r e l a t i o n does not imply c a u s a t i o n . P e r s o n a l - s o c i a l adjustment, a t t i t u d e s , and achievement not o n l y i n t e r a c t w i t h each other, but they are the e f f e c t s o f other home, s c h o o l , and community v a r i a b l e s . In h i s review Aiken has a l s o noted that " i n o r d e r t o change a p u p i l ' s a t t i t u d e toward mathematics h i s p e r c e p t i o n o f h i m s e l f i n r e l a t i o n t o mathematics m a t e r i a l s must be 29 changed." That p u p i l s who c o n s t a n t l y f a i l l o o s e s e l f c o n f i d e n c e , develop d i s l i k e and h o s t i l i t y was noted by L e r c h 30 and o t h e r s . The p u p i l must be p r o v i d e d with s i t u a t i o n s i n which he can e x p e r i e n c e s u c c e s s . How success experiences can be brought about i s up to many people: the te a c h e r , the t e x t w r i t e r , the home and o t h e r s . At l e a s t one i n v e s t i g a t o r (Tulock) recommended that "games, c o n t e s t s and other a u d i o - v i s u a l a i d s be used t o 31 heighten i n t e r e s t i n mathematics." While N a t k i n suggests f o r the p u p i l " t o a s s o c i a t e mathematics with something p l e a s a n t may a l t e r h i s a t t i t u d e or a n x i e t y toward the s u b j e c t . " 3 2 9 A i k e n expressed h i m s e l f by s a y i n g "the concept o f a g e n e r a l a t t i t u d e toward mathematics should be supplemented w i t h the a t t i t u d e s toward more s p e c i f i c a s p e c t s o f mathematics, f o r example, problem s o l v i n g and r o u t i n e d r i l l . " ATTITUDE The problem o f the p u p i l s ' n e g a t i v e a t t i t u d e s toward mathematics i s not a new one. The s o l u t i o n to the problem has not yet been found, and as such, a study by P o f f e n b e r g e r and Norton has helped to show how n e g a t i v e a t t i t u d e s a r e formed. They f e l t the p r e s e n t l a c k of i n t e r e s t i n mathe- matics i s l a r g e l y a c u l t u r a l phenomenon and that the a t t i t u d e s of some c h i l d r e n a r e c o n d i t i o n e d by the f a m i l y . They s t a t e t h a t : A t t i t u d e s are developed i n the home i n some c a s e s b e f o r e the c h i l d begins s c h o o l . In the f i r s t and second grades he i s a f f e c t e d not o n l y by h i s t e a c h e r and h i s r e a d i n e s s to d e a l w i t h numbers, but a l s o by the a t t i t u d e of h i s parents toward the s u b j e c t matter. He c a r r i e s i n t o h i s h i g h s c h o o l mathematics c l a s s e s a t t i t u d e s t h a t are long i n b u i l d i n g and d i f f i c u l t to change. C e r t a i n l y i t i s l o g i c a l to exnect that the student who goes i n t o a c l a s s with the thought "Here i s another lousy math c l a s s " , i s s e v e r l y handicapped.34 A predetermined a t t i t u d e i s d i f f i c u l t to overcome; as they r e l a t e : Students w i t h an i n i t i a l l y n e g a t i v e a t t i t u d e toward mathematics may go i n t o the classroom with a mental a t t i t u d e s e t a g a i n s t the s u b j e c t which may be maintained even when p o s i t i v e i d e n t i f i c a t i o n with the teacher i s made. I t i s t h e r e f o r e o f c o n s i d e r a b l e importance that parents and t e a c h e r s i n the e a r l y grades make every e f f o r t to g i v e p o s i t i v e e x p e r i e n c e s with a r i t h m e t i c . ^ In order to determine what the a t t i t u d e s o f p u p i l s toward a r i t h m e t i c were, Dutton has developed an a t t i t u d e s c a l e . Dutton's s c a l e c o n s i s t s o f twenty-two q u e s t i o n s which are to be checked i f the person agrees with them. I t i s an attempt to o b j e c t i v e l y measure the a t t i t u d e s o f a r u p i l or t e a c h e r . From the a d m i n i s t r a t i o n o f h i s s c a l e Dutton has concluded t h a t p u p i l s d i s l i k e a r i t h m e t i c because o f the f o l l o w i n g : 1. Lack o f understanding--confused by thought- problems without p r a c t i c a l a p p l i c a t i o n s . 2. A r i t h m e t i c was hard--made poor g r a d e s - - i r r e g u l a r attendance. 3. Poor t e a c h e r s - - p u n i s h m e n t - - f r i g h t e n i n g e x p e r i e n c e s . 4. Not sure o f s e l f - - i n s e c u r e - - m e n t a l b l o c k s — f e a r o f f a i l u r e . 5. Time f a c t o r s - - n o t enough time--takes too l o n g — p r e s s u r e . 6. A r i t h m e t i c was b o r i n g , s t a l e — s l o w l e a r n i n g , b e hind. 37 7. Too much d r i l l — memorization. I t seems from the responses r e c e i v e d by Dutton, then, t h a t i f some o f the reasons f o r d i s l i k i n g a r i t h m e t i c were removed or reduced, improvement i n achievement c o u l d r e s u l t . For again, Dutton s t a t e s : How p u p i l s f e e l toward a r i t h m e t i c i s important. L i k i n g a r i t h m e t i c has a pronounced e f f e c t upon the amount o f work attempted, the e f f o r t expended, and the l e a r n i n g t h a t i s acquired.38 11 He a l s o concludes t h a t : A p p a r e n t l y l a s t i n g a t t i t u d e s toward a r i t h m e t i c a re developed at each g r a d e l e v e l . Grades V and VII were pronounced most c r u c i a l . Some c o n t r a d i c t i o n o f c o n c l u s i o n s occurs f o r a stud y by S t r i g h t concludes that a c o n s i s t e n t l y b e t t e r s c o r e i s made by g i r l s i n g e n e r a l s t u d i e s , w h i l e boys have a s l i g h t edge i n a r i t h m e t i c , geography, and s c i e n c e . The S t r i g h t study found that g i r l s l i k e d a r i t h m e t i c b e t t e r than boys. I t made no attempt to c o r r e l a t e a r i t h m e t i c achievement and 40 a t t i t u d e toward a r i t h m e t i c . ATTITUDE A-ID ACHIEVEMENT I t has been h y p o t h e s i z e d that a t t i t u d e s are formed e a r l y and that they p l a y a r o l e i n the g e n e r a l achievement 41 o f the s t u d e n t . A study by Fedon set out t o determine some o f the a t t i t u d e s h e l d toward a r i t h m e t i c . Fedon then used the s c a l e d a t t i t u d e s toward a mathematical program to e v a l u a t e that program. In h i s summary he s t a t e s : The s c a l e s d i s c u s s e d i n t h i s paper are an attempt to c a s t l i g h t on some a s p e c t s o f the a r i t h m e t i c c u r r i c u l u m that a r e l i k e d and d i s l i k e d . A t t i t u d e s ^ l a y an important p a r t i n t h e success o f the a r i t h m e t i c program. I f we f e e l that they are a v a l i d c r i t e r i a f o r e v a l u a t i n g the e f f e c t i v e n e s s o f our program, then the a p p l i c a t i o n o f t h i s s c a l e w i l l p r o v i d e b e t t e r o p p o r t u n i t i e s to study c h i l d r e n ' s r e a c t i o n s as they e x p e r i e n c e a r i t h m e t i c i n d a i l y l i f e . 4 2 To r e l a t e a t t i t u d e and achievement, Stephens used a histogram to compare the a t t i t u d e s o f an a c c e l e r a t e d and a remedial c l a s s . The histogram i n d i c a t e d t h a t the a c c e l e r a t e d 12 c l a s s d i d have a more p o s i t i v e a t t i t u d e toward a r i t h m e t i c 43 than the remedial c l a s s . Another study by Lyda and Morse reaches the c o n c l u s i o n s : 1. When meaningful methods o f t e a c h i n g a r i t h m e t i c are used, changes i n a t t i t u d e s toward a r i t h m e t i c take p l a c e . Negative a t t i t u d e s become p o s i t i v e , and the i n t e n s i t y o f p o s i t i v e a t t i t u d e s becomes enhanced. 2. A s s o c i a t e d w i t h meaningful methods o f t e a c h i n g a r i t h m e t i c and changes i n a t t i t u d e are s i g n i f i c a n t g a i n s i n a r i t h m e t i c achievement, that i s , i n a r i t h m e t i c a l computation and reasoning.'*'* T h i s study, then, i n d i c a t e s that methods change a t t i t u d e s . However, the e f f e c t a t t i t u d e has on achievement i s open t o q u e s t i o n as A i k e n and Dreger conclude that "...the h y p o t h e s i s of s i g n i f i c a n t c o n t r i b u t i o n of mathematics a t t i t u d e s t o p r e d i c t i o n o f achievement i s borne out f o r females, but not m a l e s . " 4 5 CALCULATORS, ATTITUDE AND ACHIEVEMENT A study by Fehr, McMeen, and S o b e l , u s i n g hand oper a t e d computing machines at the grade f i v e l e v e l made t h e h y p o t h e s i s : P u p i l s who use computing machines to l e a r n a r i t h m e t i c w i l l g a i n s i g n i f i c a n t l y i n paper and p e n c i l commutations, and i n a r i t h m e t i c r e a s o n i n g over those who do not use the computing machines. T h i s h y pothesis was s u b s t a n t i a t e d by t h e i r experiment. I t was concluded by the experimenters t h a t : 1. Machine-taught students g a i n more i n reasoning a b i l i t y ; 2. Machine-taught students g a i n more i n commutation a b i l i t y ; 13 3. Machine-taught s t u d e n t s l e a r n more a l s o , because they understand machine computation as w e l l as o r d i n a r y a r i t h m e t i c ; 4. The i n t e r e s t o f students and t e a c h e r s i n a r i t h m e t i c i s heightened by the use of machines; 5. The machines f i t i n t o our p r e s e n t c u l t u r e . They g i v e a d d i t i o n a l learning.4*7 The c o n c l u s i o n reached by Fehr, McMeen, and Sobel was not that reached by Durrance i n h i s experiment, f o r he concludes that " . . . t h e r e i s no p r o o f that the use o f the c a l c u l a t o r w i l l s i g n i f i c a n t l y e nable a student to a c h i e v e 48 i n a r i t h m e t i c . " A study was performed by Cech u s i n g desk c a l c u l a t o r s with n i n t h grade g e n e r a l mathematics students to t e s t f o r improvement i n (1) t h e i r a t t i t u d e toward mathematics (2) t h e i r paper and p e n c i l computational s k i l l s (3) the use of c a l c u l a t o r s t o o b t a i n computational r e s u l t s . The PY011 Pro- Math Composite t e s t developed by SMSG was used and as measured by t h i s t e s t showed no s i g n i f i c a n t d i f f e r e n c e s i n a t t i t u d e . T h i s t e s t does not, however, d i f f e r e n t i a t e among a t t i t u d e toward computation, toward q u a n t i t a t i v e t h i n k i n g and toward mathematics i n g e n e r a l . The S t a n f o r d D i a g n o s t i c A r i t h m e t i c T e s t , Test 2, P a r t s A, B and C was used t o measure the computational s k i l l s . The r e s u l t s showed no s i g n i f i c a n t d i f f e r e n c e s i n the s k i l l s o f the experimental and the c o n t r o l groups. The S t a n f o r d t e s t d i d not t e s t f o r improvement i n achievement i n more complex q u a n t i t a t i v e t h i n k i n g . 14 The experiment d i d show s i g n i f i c a n t improvement o f the experimental g roup i n computing u s i n g c a l c u l a t o r s as 49 opposed to the group computing without them. Various grade l e v e l s have been used to t e s t the e f f e c t o f c a l c u l a t o r s or s i m i l a r machines on the achievement o f the p u p i l s . The s t u d i e s by Fehr et a l , Durrance, and Cech g i v e q u i t e d i s s i m i l a r r e s u l t s . I t i s supposed t h a t a t t i t u d e a f f e c t s l e a r n i n g , but not much has been shown that s u b s t a n t i a t e s t h i s c l a i m f o r mathematics. L i t t l e r e s e a r c h has been done at the grade ten l e v e l to see i f a t t i t u d e s can be changed. I t seems t h a t a t t i t u d e s are formed e a r l y , but i t has not been determined that they can be changed a t the l a t e r s t a g e . The a t t i t u d e s o f c h i l d r e n toward a r i t h m e t i c a r e formed e a r l y i n the home. C h i l d r e n ' s a t t i t u d e s can sometimes be moderated by a p o s i t i v e i d e n t i f i c a t i o n w i t h the t e a c h e r and p o s i t i v e e x p e r i e n c e s w i t h a r i t h m e t i c . A c c o r d i n g t o Dutton, the amount o f work attempted, e f f o r t expended, and l e a r n i n g a c q u i r e d i s dependent upon the a t t i t u d e toward a r i t h m e t i c . The r e l a t i o n s h i p o f a t t i t u d e and achievement i s open to q u e s t i o n , f o r Stephens i n d i c a t e d t h a t the r e l a t i o n - s h i p has a p o s i t i v e c o r r e l a t i o n , white A i k e n and Dreger found that the r e l a t i o n s h i p was a p r e d i c t o r f o r g i r l s but not f o r boys. S t u d i e s u s i n g c a l c u l a t o r s are not c o n s i s t e n t i n t h e i r g e n e r a l c o n c l u s i o n s . Fehr et a l found that t h e r e was a 15 s i g n i f i c a n t improvement i n a r i t h m e t i c computation and re a s o n i n g , w h i l e Durrance found no s i g n i f i c a n t change i n achievement. Chapter 3 DESIGN O F THE STUDY PURPOSE The purpose o f the study was to determine whether the use o f a desk c a l c u l a t o r would produce a p o s i t i v e change i n a p u p i l ' s a t t i t u d e toward mathematics, t o see i f a c h i e v e - ment i n q u a n t i t a t i v e t h i n k i n g would be i n c r e a s e d , and t o see i f t here was a c o r r e l a t i o n between any such changes. PROCEDURE Two t e a c h e r s , i n c l u d i n g the experimenter, and f o u r c l a s s e s o f grade t en g e n e r a l mathematics p u p i l s , were i n v o l v e d i n the experiment. Each teacher had two c l a s s e s , one c l a s s u s i n g desk c a l c u l a t o r s , the other u s i n g no c a l c u l a t o r s . A l l c l a s s e s i n v o l v e d were grouped by the s c h o o l a d m i n i s t r a t o r s a t the s t a r t o f the s c h o o l year a c c o r d i n g t o the p u p i l s ' programs, the e l e c t i v e s chosen and c l a s s e s a v a i l a b l e . No steps were made to randomize the p u p i l s . Each c l a s s was pr e - t e s t e d with the Dutton a t t i t u d e t e s t , the experimenter's own mathematics a t t i t u d e t e s t and the Iowa achievement t e s t . The c a l c u l a t o r s were s i t u a t e d i n one room to be used by the experimental group. The c o n t r o l 17 group was taught i n a d i f f e r e n t room to pr event contamination due to the presence o f the c a l c u l a t o r s . The t e a c h e r s met before and d u r i n g the experiment to d i s c u s s the l e s s o n s to be pr esented. Each c l a s s was kept as s i m i l a r as was f e a s i b l e w i t h d i f f e r e n t t e a c h e r s and c l a s s e s . While the experimental group was being taught to use the c a l c u l a t o r s , the c o n t r o l group was taught to f i n d and e l i m i n a t e e r r o r s i n c a l c u l a t i n g . During the f i r s t l e s s o n s the c o n t r o l group was taught some ways o f remembering numbers, copying numbers c o r r e c t l y and adding c e r t a i n number combinations. Groups o f numbers were s u p p l i e d which the p u p i l s were to copy and add. Some e x e r c i s e s were timed and checked f o r a c c u r r a c y . In the second l e s s o n p u p i l s were asked to copy and add numbers w i t h t h r e e d i g i t s i n columns and check. The p u p i l s were grouped i n p a i r s i n which one r e a d and the others c o p i e d , then added. The t h i r d l e s s o n c o n s i s t e d o f e x e r c i s e s f o r q u i c k a d d i t i o n and s u b t r a c t i o n . In the e x p e rimental groups the p u p i l s were grouped i n p a i r s with one machine. They were i n s t r u c t e d i n the f u n c t i o n and o p e r a t i o n o f the machine. While one was u s i n g the machine, the o t h e r would me n t a l l y add and v e r i f y some of the r e s u l t s . The ways o f remembering and copying numbers were e x p l a i n e d . The two remaining l e s s o n s were s i m i l a r with both p u p i l s a l t e r n a t e l y o p e r a t i n g the machine. The o p e r a t i o n s on the machine were g r a d u a l l y broadened i n scone. 18 The l e s s o n s of the experimental and the c o n t r o l groups were as s i m i l a r as was f e a s i b l e . For example, one l e s s o n d e a l t with the weekly earnings o f a number o f employees. The c o n t r o l group was expected to do twenty- computations while the experimental group because of the s p e e d i e r c a l c u l a t i o n s was expected to do t h i r t y computations. The type of q u e s t i o n was not v a r i e d , , only the number of q u e s t i o n s . The s t a n d a r d types of q u e s t i o n s and e x e r c i s e s were u t i l i z e d . The q u e s t i o n s d e a l t w i t h p a y r o l l computations, commission e a r n i n g s , simple i n t e r e s t and i n s t a l l m e n t b u y i n g . The c a l c u l a t i o n s i n v o l v e d the b a s i c o p e r a t i o n s e i t h e r s i n g l y or i n combination. There were s i x t y p u p i l s i n the experimental grouns, and f i f t y - s e v e n i n the c o n t r o l groups. The experiment ran f o r seven weeks d u r i n g February, March and A p r i l of 1968. A l l t h r e e t e s t s were a g a i n a d m i n i s t e r e d to a l l p u p i l s at the completion of the experimental program. HYPOTHESIS Three hypotheses were fo r m u l a t e d . 1. There w i l l be a p o s i t i v e change i n g e n e r a l a t t i t u d e toward mathematics f o r the grade ten General Mathematics students u s i n g desk c a l c u l a t o r s . 2. The achievement i n doing q u a n t i t a t i v e t h i n k i n g w i l l show g r e a t e r improvement f o r those u s i n g c a l c u l a t o r s than f o r those not u s i n g c a l c u l a t o r s . * For f u r t h e r d e t a i l s p l e a s e see the appendix. 3. Where there i s a change i n a t t i t u d e toward q u a n t i t a t i v e t h i n k i n g t h e r e w i l l be a change i n t h e same d i r e c t i o n i n the achievement i n doing q u a n t i t a t i v e t h i n k i n g . TEST SELECTION A r i t h m e t i c A t t i t u d e A t e s t t o determine a p u p i l ' s a t t i t u d e toward 50 a r i t h m e t i c has been c o n s t r u c t e d by Dutton. Dutton had c o l l e g e s t u d e n t s w r i t e statements c o n c e r n i n g t h e i r l i k e s and d i s l i k e s toward a r i t h m e t i c . From these statements, some f o r t y - f i v e statements were r e t a i n e d . These were screened f u r t h e r and twenty-two statements were r e t a i n e d . The statements on examination were found to r e f e r to a r i t h m e t i c a l o p e r a t i o n s or procedures which c o u l d e l i c i t v a r i o u s f e e l i n g s . The f e e l i n g s mentioned appear to be both p o s i t i v e or n e g a t i v e to v a r y i n g degrees. The statements c o u l d be c o n s i d e r e d as having f a c e v a l i d i t y f o r an i n d i c a t i o n o f a person's a t t i t u d e toward a r i t h m e t i c . 3y u s i n g methods suggested by Thurstone and Chave, a s c a l e o f v a l u e f o r each o f the twenty-two statements was 51 e s t a b l i s h e d . These s c a l e d v a l u e s range from 1 to 10.5. The p u p i l s were asked to check the statements that they agreed w i t h . They c o u l d check as many or as few as they wished. The s c o r e s f o r each statement checked were t a b u l a t e d and t o t a l l e d , and an average response score was a r r i v e d a t . That i s , i f f i v e statements were checked whose 20 values t o t a l l e d t h i r t y , then the average r e s p o n s e s c o r e would be s i x . The r e l i a b i l i t y o f the t e s t u s i n g the average response score method on a t e s t - r e t e s t i s r e p o r t e d to be 0.94. 5 2 Q u a n t i t a t i v e T h i n k i n g Achievement S e l e c t i o n . In s e l e c t i n g a t e s t to measure the achievement a t t a i n e d i n q u a n t i t a t i v e t h i n k i n g , a t e s t was sought which covered m a t e r i a l a p p r o p r i a t e f o r the p u p i l s i n v o l v e d with the g e n e r a l mathematics course at the grade 10 l e v e l , a t e s t which c o u l d be a d m i n i s t e r e d w i t h i n the one hour p e r i o d a l l o t e d f o r t h i s course, a t e s t which c o u l d be a d m i n i s t e r e d and s c o r e d e a s i l y , as w e l l as a t e s t t h a t had s u i t a b l e s t a n d a r d i z a t i o n and s t a t i s t i c a l background. The t e s t chosen was "The Iowa T e s t o f E d u c a t i o n a l Development Test 4: A b i l i t y t o do Q u a n t i t a t i v e T h i n k i n g : Form Y-3S". The q u e s t i o n s a r e concerned with the a b i l i t y o f the student to use b a s i c a r i t h m e t i c i d e a s and p r i n c i p l e s to a r r i v e at s o l u t i o n s to the q u e s t i o n s posed. The q u e s t i o n s i n v o l v e mainly f i n a n c e and measurement s i t u a t i o n s . There i s no attempt, however, to measure the a b i l i t y to do a b s t r a c t t h i n k i n g or t o i d e n t i f y students who have s p e c i a l mathematical a b i l i t y . The t e s t was designed f o r use w i t h grades nine to twelve. There was a c l a s s v e r s i o n c o n s i s t i n g o f t h i r t y - t h r e e m u l t i p l e - c h o i c e q u e s t i o n s that c o u l d be a d m i n i s t e r e d w i t h i n the s i x t y minute p e r i o d . The t e s t d i d not r e q u i r e l a b o r i o u s c a l c u l a t i o n s by the s t u d e n t s , but d i d r e q u i r e an a n a l y s i s of" the q u e s t i o n f o l l o w e d by a s e l e c t i o n o f the c o r r e c t a r i t h m e t i c a l o p e r a t i o n s . The t e s t d i d r e q u i r e t h a t a student have a good a r i t h m e t i c a l sense as the d i s t r a c t e r s i n the m u l t i p l e c h o i c e s were such that the student must know what i s c o r r e c t b e f o r e he c o u l d , except by chance, S3 s e l e c t the c o r r e c t response. For example, one o f the q u e s t i o n s was: In a s h o r t cut method o f m u l t i p l y i n g 4087 x 198, one f i r s t m u l t i p l i e s 4087 x 200. What product must then be s u b t r a c t e d from t h i s r e s u l t i n o r d e r to o b t a i n the c o r r e c t answer? 1) 13 x 2 4) 4087 x 2 2) 4087 x 13 5) Not g i v e n 3) 198 x 2 The student c o u l d know t h a t 4100 - 13 = 4087 and 200 - 2 = 198 but not know what to do with them. They then c o u l d be d i s t r a c t e d to the wrong response. The t e s t c o u l d be machine s c o r e d r e a d i l y by having the student p l a c e h i s responses on a mark sense c a r d . T^e t e s t has been w i d e l y a d m i n i s t e r e d . I t i s r a t e d as good to e x c e l l e n t by the Mental Measurements Y e a r b o o k . 5 4 R e l i a b i l i t y . The r e l i a b i l i t y i s s a i d to oe about 0.91 f o r p u p i l s o f the same g r a d e . 5 5 22 Mathematical A t t i t u d e No t e s t was found by the experimenter t h a t would measure the a t t i t u d e toward c a l c u l a t i n g and a l s o toward q u a n t i t a t i v e t h i n k i n g . I t was then determined that such a t e s t s h o u l d be c o n s t r u c t e d . TEST CONSTRUCTION A t e s t was sought which would d i f f e r e n t i a t e from those who l i k e simple c a l c u l a t i o n s but do not l i k e q u e s t i o n s i n v o l v i n g p l a n n i n g , e v a l u a t i n g or other mental c a p a c i t i e s , as w e l l as a t e s t which would d i f f e r e n t i a t e those who would p r e f e r the l a t t e r but not the former. Three areas were s e l e c t e d which might a l l o w t h i s type o f d i f f e r e n t i a t i n g . D e f i n i t i o n of Terms C a l c u l a t i o n : (A) : the performance o f a s p e c i f i e d o p e r a t i o n or o p e r a t i o n s on a giv e n set of numbers. Q u a n t i t a t i v e T h i n k i n g : ( B ): the a s s i m i l a t i o n o f s u p p l i e d data t o g e t h e r with the a p p l i c a t i o n o f b a s i c o p e r a t i o n s o f a r i t h m e t i c to a r r i v e at a s o l u t i o n to a q u e s t i o n to which the student has not been t o l d what stens or procedures are to be taken. Non-mathematical A c t i v i t y : ( C ) : an a c t i v i t y e i t h e r p h y s i c a l or mental not i n v o l v i n g mathematics. 23 Item S e l e c t i o n A grade ten geometry c l a s s was asked to t h i n k o f one d e s i r a b l e a c t i v i t y and one u n d e s i r a b l e a c t i v i t y i n mathematics and a l s o o u t s i d e mathematics. The author then s u p p l i e d some o t h e r s o f h i s own. Some of the suggested or s u p p l i e d items were then p l a c e d i n the f o l l o w i n g c a t e g o r i e s ; C a l c u l a t i o n (A), Q u a n t i t a t i v e T h i n k i n g (Pi) , and Non-mathematical A c t i v i t y (C) and p r e s e n t e d as a group o f thr e e to the p u p i l s , who were asked to put a check f o r the one that they l i k e d the most and a c r o s s f o r the one they l i k e d the l e a s t . An example was gi v e n as f o l l o w s : 1. (a) Add up a column o f 20 three d i g i t numbers (b) S o l v e the eq u a t i o n 3x + 2 = 20 (c) Wash the b r e a k f a s t d i s h e s There were i n the o r i g i n a l t e s t some t h i r t y s i t u a t i o n s , each w i t h a statement f o r each c a t e g o r y . The o r i g i n a l t e s t was l a b e l l e d Type I, and a d m i n i s t e r e d to two c l a s s e s o f General Mathematics 9. S i n c e the s t u d e n t s were Ge n e r a l Mathematics s t u d e n t s they were assumed to be as near as p o s s i b l e t o the ex p e r i m e n t a l c l a s s e s i n a t t i t u d e and achievement. The responses o f the two General Mathematics 9 c l a s s e s were then t a b u l a t e d f o r each of the c a t e g o r i e s i n the t h i r t y s i t u a t i o n s . From an i n s p e c t i o n o f the resnonses, c e r t a i n statements were r e v i s e d to be more s e l e c t i v e . One o f the o r i g i n a l statements was "Watch t e l e v i s i o n " . T h i s 3 X 24 statement was changed to "Watch Batman on t e l e v i s i o n " . The o r i g i n a l had r e c e i v e d many l i k e but no d i s l i k e responses; thus i t was r e v i s e d so that i t e l i c i t e d both p o s i t i v e and ne g a t i v e responses. The r e v i s e d t h i r t y item t e s t was l a b e l l e d Type I I , and a d m i n i s t e r e d t o another two c l a s s e s o f grade nine g e n e r a l mathematics s t u d e n t s . A t a b u l a t i o n was made o f the responses from t h i s t a b u l a t i o n , many statements were r e j e c t e d or reassembled i n d i f f e r e n t combinations. The f i n a l form c o n s i s t e d o f s i x t e e n s i t u a t i o n s , each c o n t a i n i n g a statement f rom c a t e g o r y A, B and C. T h i s f i n a l r e v i s i o n was a d m i n i s t e r e d to a grade nine a l g e b r a c l a s s t h a t was homogeneously grouped a c c o r d i n g to past marks i n mathematics. Only those students w i t h C+ or b e t t e r were p l a c e d i n t h i s c l a s s . The r e s u l t a n t s c o r e s were compared with the tea c h e r ' s e x p e c t a t i o n s o f student a t t i t u d e and found to i n d i c a t e a s t r o n g p r e f e r e n c e f o r c a l c u l a t i n g f o l l o w e d c l o s e l y b\ a p r e f e r e n c e f o r problem s o l v i n g and a low p r e f e r e n c e f o r non-mathematical a c t i v i t i e s . S c o r i n g A s c o r e f o r each categ o r y was o b t a i n e d by a l l o t t i n g two marks f o r each p o s i t i v e response i n t h a t ca t e g o r y (a check mark), one mark f o r each blank i n t h a t c a t e g o r y , and no marks f o r a ne g a t i v e response (a c r o s s ) i n t h a t c a t e g o r y . Hence, there i s a p o s s i b l e high s c o r e o f 25 t h i r t y - t w o , and a low score o f ^ero. Each o f the three c a t e g o r i e s , A, B, and C, r e c e i v e d s e p a r a t e s c o r e s . Scores were than a v a i l a b l e f o r an a t t i t u d e or p r e f e r e n c e toward c a l c u l a t i o n (A), q u a n t i t a t i v e t h i n k i n g (B), and t' e i n d i c a t e d non-mathematical a c t i v i t i e s ( C ) . A mathematics a t t i t u d e s c o r e was d e r i v e d by a d d i n g the A and B s c o r e s . Item A n a l y s i s An item a n a l y s i s was performed on the t e s t u s i n g methods suggested by G a r r e t t " ^ whereby the top and bottom twenty-seven percent o f the scores were compared. A b i s e r i a l ' r ' was computed f o r each item i n each c a t e g o r y and w i t h one e x c e p t i o n a l l items were found to be above the suggested 0.20 minimum. The e x c e p t i o n was number twelve o f the problem s o l v i n g c a t e g o r y . A complete a n a l y s i s may be found i n the appendix. R e l i a b i l i t y o f A t t i t u d e T e s t A t e s t f o r r e l i a b i l i t y u s i n g the s p l i t - h a l f method was performed. The s i x t e e n items were s p l i t on an odd-even b a s i s . A response o f a check was s c o r e d one, and a blank or a c r o s s s c o r e d z e r o . The odd-even s c o r e s were t a b u l a t e d on t h i s b a s i s f o r each o f the t h r e e c a t e g o r i e s . The c o r r e l a t i o n f o r the category A odd-even was found to be 0.4782. The c o r r e l a t i o n f o r the c a t e g o r y B odd-even was 0.5433, and f o r the c a t e g o r y C odd-even was 26 0.7001. When u s i n g the Spearman-Brown nronhecy formula S 7 f o r e s t i m a t i n g r e l i a b i l i t y from s n l i t h a l v e s , " f i g u r e s o f 0.64 70 f o r A, 0.7043 f o r B, and 0.8236 f o r C were o b t a i n e d . The r e l i a b i l i t y f i g u r e s are above the minimum o f 0.60 which G a r r e t t 5 ^ c o n s i d e r s necessary f o r d i f f e r e n t i a t i n g between means o f groups, but not s u f f i c i e n t f o r the d i f f e r e n t i a t i n g o f i n d i v i d u a l d i f f e r e n c e s . STATISTICAL DESIGN The Mathematics A t t i t u d e Test y i e l d e d two subscores d e s i g n a t e d " A t t i t u d e toward C a l c u l a t i o n " (A) and " A t t i t u d e toward Q u a n t i t a t i v e T h i n k i n g " (B ). Category (C) was used as a d i s t r a c t o r . The score f o r " A t t i t u d e toward Mathematics" was d e r i v e d from the sum o f these two subscores (A + B ) . Data f o r the i n d i v i d u a l s were o b t a i n e d from th e s c o r e s on the Mathematics A t t i t u d e Test and were d e s i g n a t e d A^, B ^ and A., + B-^ f o r the p r e - t e s t and A 2, P>2 and A 2 + B2 f o r the p o s t - t e s t . The Dutton A t t i t u d e S c a l e y i e l d e d s c o r e s d e s i g n a t e d D^ and D^ f o r the p r e - t e s t and p o s t - t e s t , w h i l e the Iowa Achievement Test s c o r e s were d e s i g n a t e d r e s p e c t i v e l y P-̂  and P 2 f o r the p r e - t e s t and p o s t - t e s t . A s c o r e was o b t a i n e d from the Mathematics A t t i t u d e Test f o r change i n a t t i t u d e toward Q u a n t i t a t i v e T h i n k i n g by f i n d i n g the d i f f e r e n c e between the p o s t - t e s t and p r e - t e s t s c o r e s o f A t t i t u d e toward Q u a n t i t a t i v e T h i n k i n g ( B 2 - B ^ ) . A score f o r change i n achievement was found by t a k i n g the d i f f e r e n c e between the p o s t - t e s t and p r e - t e s t 27 scores o f the Iowa Achievement T e s t , and was d e s i g n a t e d P 2 " p l - A n a l y s i s o f Covariance was used to e v a l u a t e the scores from Dutton's Test (D-, and D2) , the Mathematics A t t i t u d e Test (A^ + B-̂  and A2 + 8 3 ) , and the Iowa Achievement Test on Q u a n t i t a t i v e T h i n k i n g (P-^ and P2) • The l e v e l s o f 59 s i g n i f i c a n c e were found from a t a b l e by Dixon and Massey. The c o r r e l a t i o n o f the sc o r e s from Change i n Achievement (P2 - P^) and Change i n A t t i t u d e toward Q u a n t i t a t i v e T h i n k i n g (B2 - B-̂ ) were c a l c u l a t e d f o r the experimental and the c o n t r o l groups. These c o r r e l a t i o n s were then checked f o r s i g n i f i c a n c e w i t h the r e s p e c t i v e number o f degrees o f freedom. Chapter 4 ANALYSIS OF THE RESULTS FIRST HYPOTHESIS The h y p o t h e s i s that t h e r e would be a p o s i t i v e change i n g e n e r a l a t t i t u d e toward mathematics f o r students u s i n g c a l c u l a t o r s was t e s t e d u s i n g a n a l y s i s o f c o v a r i a n c e . The number of o b s e r v a t i o n s made was f i f t y - s e v e n f o r the c o n t r o l group and s i x t y - o n e f o r the experimental group f o r a t o t a l o f 118 o b s e r v a t i o n s . Dutton's A t t i t u d e S c a l e y i e l d e d r e s u l t s as f o l l o w s : CONTROL EXPERIMENTAL Mean S. D. Mean S. D. P r e - t e s t 5.376 1.247 5.513 1.413 P o s t - t e s t 5.326 1.127 5.562 1.303 The F - r a t i o was found to be 0.89 which, w i t h 115 degrees of freedom, i s not s i g n i f i c a n t . The r e s u l t s o f the Mathematics A t t i t u d e T e s t u s i n g the subscore " A t t i t u d e toward Mathematics" (A + B) were as f o l l o w s : CONTROL EXPERIMENTAL Mean S. D. Mean S. D. P r e - t e s t 33.70 6.902 32.20 6.623 P o s t - t e s t 31.98 7.308 30.98 6.828 29 The F - r a t i o f o r the " A t t i t u d e toward Mathematics" subscore was found to be 0.00 which w i t h 115 degrees o f freedom i s not s i g n i f i c a n t . The c o n c l u s i o n must be that t h e r e was no s i g n i f i c a n t change i n a t t i t u d e . SECOND HYPOTHESIS The h y p o t h e s i s that the achievement i n doing q u a n t i t a t i v e t h i n k i n g would show g r e a t e r improvement f o r those u s i n g c a l c u l a t o r s than those not u s i n g c a l c u l a t o r s was t e s t e d u s i n g the Iowa Achievement Test r e s u l t s with a n a l y s i s o f c o v a r i a n c e . The r e s u l t s o f the t e s t were as f o l l o w s : CONTROL EXPERIMENTAL Mean S. D. Mean S. D. P r e - t e s t 11.82 4.445 13.08 3.625 P o s t - t e s t 12.74 4.249 14.51 3.404 A F - r a t i o o f 3.33 wit h 115 degrees o f freedom was not s i g n i f i c a n t at the 0.05 l e v e l . I t i s , however, s i g n i f i c a n t a t the 0.08 l e v e l . The c o n c l u s i o n must be that there i s no s i g n i f i c a n t change i n the achievement i n doing q u a n t i t a t i v e t h i n k i n g by u s i n g a c a l c u l a t o r - b a s e d program. THIRD HYPOTHESIS The raw scores were handled by a computer and i t was asked t o e s t a b l i s h the necessary c a l c u l a t i o n s . S i n c e the computer was pre-programmed without p r i o r knowledge o f any 30 r e s u l t s , t h e t h i r d h y p o t h e s i s was t e s t e d a l o n g w i t h t h e f i r s t two. The h y p o t h e s i s t h a t where t h e r e i s a p o s i t i v e c h a n g e i n a t t i t u d e t o w a r d s q u a n t i t a t i v e t h i n k i n g t h e r e w i l l b e a c h a n g e i n t h e a c h i e v e m e n t i n d o i n g q u a n t i t a t i v e t h i n k i n g was t e s t e d by t h e i r c o r r e l a t i o n . The c o r r e l a t i o n b e t w e e n c h a n g e i n a t t i t u d e t o w a r d q u a n t i t a t i v e t h i n k i n g as m e a s u r e d by t h e M a t h e m a t i c s A t t i t u d e T e s t a n d t h e a c h i e v e m e n t i n d o i n g q u a n t i t a t i v e t h i n k i n g as m e a s u r e d by -the Iowa A c h i e v e m e n t T e s t was f o u n d t o b e f o r t h e e x p e r i m e n t a l g r o u p s -0.0492, a n d f o r t h e c o n t r o l g r o u p s , 0.2105. N e i t h e r o f t h e c o r r e l a t i o n f i g u r e s i s s i g n i f i c a n t a t t h e 0.05 l e v e l . The c o n c l u s i o n must t h e n be t h a t t h e r e i s no c o r r e l a t i o n o f c h a n g e i n a t t i t u d e t o w a r d q u a n t i t a t i v e t h i n k i n g a n d c h a n g e i n a c h i e v e m e n t i n d o i n g q u a n t i t a t i v e t h i n k i n g . Chapter 5 SUMMARY AND CONCLUSIONS Research i n t o a t t i t u d e f o r m a t i o n i n d i c a t e s t hat t h e r e i s a p o s i t i v e c o r r e l a t i o n o f a t t i t u d e and achievement i n mathematics. Some o f the a t t i t u d e s toward mathematics are formed i n j u n i o r high s c h o o l . I t was thought that some of the reasons f o r d i s l i k i n g mathematics might be decreased i f a desk c a l c u l a t o r c o u l d be used to work the t e d i o u s c a l c u l a t i o n s . The purpose o f the study was to determine whether the use o f a desk c a l c u l a t o r would change a p u p i l ' s a t t i t u d e toward mathematics, t o see i f achievement i n q u a n t i t a t i v e t h i n k i n g would be i n c r e a s e d , and to see i f th e r e was a c o r r e l a t i o n between any such changes. Four c l a s s e s o f grade 10 G e n e r a l Mathematics p u p i l s were used with two t e a c h e r s . Each t e a c h e r taught one c o n t r o l and one experimental c l a s s u s i n g desk c a l c u l a t o r s . The c l a s s e s were p r e - t e s t e d and p o s t - t e s t e d u s i n g Dutton's a t t i t u d e s c a l e , the Iowa Achievement T e s t and the experimenter's own Mathematics A t t i t u d e T e s t . For the program used, the grade l e v e l t e s t e d , and the t e s t s used to measure p o s s i b l e changes i n a t t i t u d e toward mathematics, i n c r e a s e s i n achievement i n q u a n t i t a t i v e t h i n k i n g and c o r r e l a t i o n between any such changes, t h e r e 3 2 i s no evidence f o r a p o s i t i v e c o r r e l a t i o n o f change i n a t t i t u d e and achievement w h i l e the i n c r e a s e d achievement i n doing q u a n t i t a t i v e t h i n k i n g u s i n g desk c a l c u l a t o r s i s not s i g n i f i c a n t at the 0.05 l e v e l i t i s s i g n i f i c a n t at the 0.08 l e v e l . LIMITATIONS The study was c a r r i e d out w i t h g e n e r a l mathematics students at the grade t en l e v e l . No randomization was done as the groups were not n r e - s e l e c t e d f o r the program as they had been p l a c e d by the a d m i n i s t r a t i o n i n t h e i r r e s p e c t i v e c l a s s e s . Often the placement i s done i n c o n j u n c t i o n w i t h some s p e c i f i c program b e i n g predominant i n a c l a s s . The program predominance was not taken i n t o account. I t was assumed t h a t some e q u i v a l e n c e o f a t t i t u d e and achievement c o u l d be accomodated f o r i n the t e s t r e s u l t s by a n a l y s i s o f c o v a r i a n c e . P o f f e n b e r g e r 1 s study suggested that the a t t i t u d e s o f p u p i l s are e s t a b l i s h e d e a r l y . I t i s only a f t e r s e v e r a l p o s i t i v e mathematical e x p e r i e n c e s have been encountered that a change of any degree c o u l d o c c u r . Because a t t i t u d e s a re formed e a r l y and s e v e r a l p o s i t i v e e x periences must be encountered b e f o r e a change i s made c o u l d mean that any exp e r i e n c e would have to be l o n g - l a s t i n g and o f c o n s i d e r a b l e s t r e n g t h t o s h i f t a t t i t u d e at the grade ten l e v e l . The g e n e r a l mathematics students have not met with a gr e a t d e a l 33 of success i n mathematics and hence may have d i f f e r e n t a t t i t u d e s from students i n r e g u l a r mathematics c l a s s e s . The two grouos were taught i n d i f f e r e n t rooms, and the environment may have had some i n f l u e n c e on t h e i r a t t i t u d e . The c o n t r o l groups were taught i n a r e l a t i v e l y new classroom u s i n g c o n v e n t i o n a l desks. The experimental groups were taught i n an o l d e r b u i l d i n g u s i n g t a b l e s and c h a i r s . Going t o the o l d e r b u i l d i n g n e c e s s i t a t e d a t r i p o u t s i d e , f r e q u e n t l y i n the r a i n . The p h y s i c a l f a c t o r s c o u l d have had a n e g a t i v e e f f e c t on the a t t i t u d e o f the p u p i l s t a k i n g p a r t i n the experimental group. Some room changes were necessary i n order to conduct the study. P u p i l s f r e q u e n t l y do not l i k e changes i n t h e i r e s t a b l i s h e d p a t t e r n s . They l i k e a p a r t i c u l a r seat i n a p a r t i c u l a r row, and any changes t h r e a t e n t h e i r f e e l i n g o f s e c u r i t y . Changes i n s e a t s and rooms may a g a i n have had a s l i g h t n e g a t i v e i n f l u e n c e on a t t i t u d e s and achievement. Other f a c t o r s that may have a f f e c t e d the r e s u l t s o f the study i n c l u d e the n o i s e generated by the c a l c u l a t o r s when o p e r a t i n g . The n o i s e may have been d i s t r a c t i n g to the p u p i l s . The program s e l e c t e d may not have been the best p o s s i b l e f o r any i n t e r a c t i o n between p u p i l , teacher, and c a l c u l a t o r . The d u r a t i o n o f the study was seven weeks, which may not have been long enough to remove a p o s s i b l e Hawthorne e f f e c t or t o overcome any deep s e a t e d n e g a t i v e f e e l i n g s toward mathematics. 34 I t has been argued that w h i l e the exp e r i m e n t a l group was l e a r n i n g t o operate the machines, the c o n t r o l group c o u l d have been doing some i r r e l e v a n t e x e r c i s e s . It was thought by the experimenter that the procedure used was f a i r to both groups i n that the experimental groups would not have an edge over the c o n t r o l groups i n r e a d i n g , remembering, and copying numbers. Reading, remembering, and copying were thought t o be separate from machine o p e r a t i o n i n f l u e n c e . With two teachers there may have been a b i a s f o r one teacher f o r a p a r t i c u l a r c l a s s . I t was hoped by the experimenter that t e a c h e r b i a s was kept t o a minimum. An a t t i t u d e t e s t i s open to s u s p i c i o n i n that i t i s not known i f the p u p i l can r e l a t e what h i s f e e l i n g s a re toward a s u b j e c t . He may put down what he t h i n k s he i s expected t o put down, r a t h e r than what he a c t u a l l y f e e l s . We may not have measured what we should have measured. A t e s t which uses m u l t i p l e c h o i c e responses f o r s e l e c t i o n o f the c o r r e c t response may be q u i t e d i f f e r e n t i n r e s u l t s from one which i s open-ended. Hence the Iowa achievement t e s t shows onl y what can be done on that t e s t , not what a p u p i l might do on an open-ended t e s t q u e s t i o n such as he would f i n d i n a r e a l l i f e problem s i t u a t i o n . The concepts i n v o l v e d i n problem s o l v i n g or i n q u a n t i t a t i v e t h i n k i n g a re not c l e a r l y understood. Perhaps what was measured was not that which was i n f l u e n c e d or changed. The o r g a n i z a t i o n o f a p u p i l ' s t h i n k i n g was not 3 5 t e s t e d , and i t i s t h i s o r g a n i z a t i o n , perhaps, which i s more a p t l y shaped by the c a l c u l a t o r , r a t h e r than what was t e s t e d . A study was done with grade ten g e n e r a l mathematics p u p i l s at a s c h o o l whose socio-economic l e v e l i s one of lower middle c l a s s , s t a b l e , home s i t u a t i o n s . I t has s t r o n g European c u l t u r a l background. The c u l t u r a l , s o c i a l and economic i n f l u e n c e s p r e s e n t make i t d i f f i c u l t to g e n e r a l i z e to other grades, types of c l a s s e s , c u l t u r a l , s o c i a l or economic s i t u a t i o n s . IMPLICATIONS FOR FURTHER RESEARCH The study was w i t h grade ten g e n e r a l mathematics p u p i l s whose a t t i t u d e s are u s u a l l y f i r m l y entrenched. The study was of a r e l a t i v e l y s h o r t d u r a t i o n and, as such, i t may not have been o f s u f f i c i e n t l e n g t h to overcome the i n e r t i a o f the deep-seated a t t i t u d e s . A study i n v o l v i n g younger p u p i l s f o r a longer p e r i o d o f time might y i e l d s i g n i f i c a n t changes. There are now on the market newer, q u i e t e r machines which may not have the d i s t r a c t i n g n o i s e l e v e l o f the machines used. A study using e l e c t r o n i c c a l c u l a t o r s might prove o f s i g n i f i c a n t worth. The r e s u l t s o f p a r t i c u l a r students were not a n a l y s e d . What k i n d o f a t t i t u d e or achievement i s changed? Would the program have been the same f o r p u p i l s w i t h an i n i t i a l l y h i gher l e v e l o f achievement, but poor a t t i t u d e ? 36 During the o p e r a t i o n of the c a l c u l a t o r , the p u p i l s had to know what numbers and what o p e r a t i o n s were to be used. However, some would use a t r i a l and e r r o r method i n s o l v i n g the q u e s t i o n . I f m u l t i p l i c a t i o n d i d not work, then they would t r y d i v i s i o n . Perhaps i t might be a d v i s a b l e to t e a c h o r g a n i z a t i o n o f the data by means o f a flow c h a r t . That i s to say, that students would be r e q u i r e d to show what numbers and what o p e r a t i o n s were to be performed b e f o r e the c a l c u l a t i o n s were done by the machine. The flow c h a r t procedure might l e a d to a b e t t e r i n s i g h t i n t o how to s o l v e a problem. The r o l e o f the c a l c u l a t o r i n f a c i l i t a t i n g the p u p i l to achieve b e t t e r i s not c l e a r l y understood. Whether the a d d i t i o n a l e x p e r i e n c e i n working w i t h the a i d o f the c a l c u l a t o r to do more work i n a g i v e n time, whether the r e q u i r e d o r g a n i z a t i o n o f the s i t u a t i o n b e f o r e the c a l c u l a t i o n s a r e performed or whether some o t h e r f a c t o r s a re e n a b l i n g the p u p i l to g r e a t e r achievement are q u e s t i o n s yet to be i n v e s t i g a t e d . The teacher needs i n f o r m a t i o n about a p u p i l ' s a t t i t u d e i n order to conduct remedial p r o c e d u r e s . The mathematics a t t i t u d e t e s t c o n s t r u c t e d f o r the study seems to be a v a l i d and r e l i a b l e t e s t . With some m o d i f i c a t i o n s , the t e s t c ould be u s e f u l to o t h e r s . FOOTNOTES 38 "•"Lewis R. Aiken J r . , "Attitudes Toward Mathematics," Studies i n Mathematics T V o l . XIX, ed. James W. Wilson and L. Ray Carry, SMSG, Stanford Univ., 196-9, p.12. 2 P . D. C r i s t a n t i e l l o , "Attitude Toward Mathematics and the Pr e d i c t i v e V a l i d i t y of a Measure of Quantitative Aptitude," Journal of Educational Research. LV (December- January, 1962) pp.184-186. 3wilbur H. Dutton, "Measuring Attitudes Toward Arithmetic," Elementary School Journal T September, 1954, p29. "'"Thomas Poffenberger and Donald Norton, "Factors i n the Formation of Attitudes Toward Mathematics," Journal of Educational Research. Vol. 52, January 1959, pp.171-176. 5 Lewis R. Aiken J r . , op. c i t . , p. 38. Lewis R. Aiken J r . , op. c i t . , c i t i n g R. A l p e r t , G. Slettwagen, and D. Becker, "Psychological Factors i n Mathematics Education, " Report summary i n Newsletter No. l">f SMSG, Stanford Uni v e r s i t y , 1963. ^Wesley J . Lyda and Evelyn Clayton Morse. "Attitude, Teaching Methods and Arithmetic Achievement," The Arithmetic Teacher T March 1963, pp.136-138. o Joseph P h i l i p Cech, "The E f f e c t of the use of Desk Calculators has on Attitude and Achievement i n Ninth-Grade General Mathematics Classes/''Dissertation Abstracts, December 1970, 31A, p.2784-A. ^Lewis R. Aiken J r . , op. c i t . , p.4. 1 0 W i l b u r H. Dutton, "Attitudes of Junior High School Pupils Toward Arithmetic," The School Review T 1956, pp.18-22. """""Victor Rodney Durrance, "The E f f e c t of the Rotary Calculator on Arithmetic Achievement i n Grades Six, Seven, and Eight," D i s s e r t a t i o n Abstracts. 25, p.6307. 12 Joseph P. Cech, op. c i t . ^Howard F. Fehr, George McMeen, Max Sobel. "Using Hand-Operated Computing Machines i n Learning Arithmetic," The Arithmetic Teacher T October 1956, pp.145-150. "^Hobart (Okla.) Democrat-Chief, Wednesday, March 31, 1965. 1 5 Lewis R. Aiken J r . , op. c i t . , p.2. 39 -^Ibid, p.2, citing Morrisett, L. M. and Vinsonhaler, J., (Eds.). "Mathematical Learning," Monographs of the Society for Research In Child Development, XXX, No. 1 (1965). 1 7 I b i d , p.3, citing Brown, K. E., and Abell, T. L. "Research i n the Teaching of Elementary School Mathematics," Arithmetic Teacher. XII (November, 1965), pp. 5^7-9+9* •^Ibid, p.5, citing Nealeigh, T. R. Development and Validation of a Non-verbal Attitude and Achievement Index for Mathematics, unpublished doctoral dissertation, Ohio State University, 1967. (Dissertation Abstracts, XXVIII-A, p.3567). 1 9 I b i d , p.5. 2 0 I b i d , p.6, citing Fedon, J. P. The Role of Attitude i n Learning Arithmetic," A r i t h m e t i c Teaf>hpr 7 V (December, 1958), pp.304-310. 2 1 I b i d , p.6, citing Reys, R. E., and Delon, F. G. "Attitudes of Prospective Elementary School Teachers Towards Arithmetic," Arjtftmetlc Teacher, XV (April, 1968), pp.363-366. 2 2 I b l d , p.8. 2 ^ I b l d , p.8, citing Alpert, R., Stellwagon, G., and Becker, D. "Psychological Factors i n Mathematics Education," Report summary i n Newsletter No. 15, School Mathematics Study Group, Stanford University, 1963. 2 l fIbid, p.9, citing Brown and Abell. 2 5 l b i d , p.10, citing Alpert et a l . *°Ibid, p.10, citing Degnan, J. A. "General Anxiety and Attitudes Toward Mathematics in Achievers and Underachievers in Mathematics," Graduate Research in Education and Related Disciplines, III (1967), pp.49-62. 27 'Ibid, p.12, citing Jackson, P. W. Life i n Classrooms (New York: Holt, Rlnehart & Winston, I968). 2 8 I b i d , p.22. 2 q I b i d , p.37, citing Bassham, H., Murphy, M., and Murphy, Katherine. "Attitude and Achievement in Arithmetic," Arithmetic TeacherT XI (February, 1964), pp.66-72. 3°Ibid, p.37, citing Lerch, H. H., "Arithmetic Instruction Changes Pupils' Attitudes Toward Arithmetic," Arithmetic Teacher, VIII (March, 1961), pp.117-119. 31lbid, p.37, citing Tulock, Mary K. "Emotional Blocks in Mathematics," Mathematics Teacher, L (December, 1957), pp.572-576. 40 ^ 2 I b i d , p.3 8 , c i t i n g Natkin, G. L. The Treatment of Mathematical Anxiety Through Mediated Transfer of Attitude Toward Mathematics, unpublished doctoral d i s s e r t a t i o n , Indiana University, 1966. ( D i s s e r t a t i o n Abstracts, XXVTI-A, p.4 1 3 7 ) . 3 3 I b i d , p.40. ^Thomas Poffenberger and Donald Norton, "Factors l n the Formation of Attitude toward Mathematics," Journal of Educational Research T V o l . 5 2 , No. 5 , January, 1 9 5 9 , p.175* 3 ^ I b i d , p.174. i l b u r H. Dutton, "Measuring Attitudes Toward Arithmetic," Elementary School Journal T September, 1 9 5 4 , pp.24-31. 3 7 I b i d , p.2 9 . 3 8 W i l b u r H. Dutton, "Attitudes of Junior HighSchool Pupils Toward Arithmetic," The School Review T 1 9 5 6 , pp. 1 8 - 2 2 . 3 9 I b i d , p.2 2 . ^ V i r g i n i a M. S t r i g h t , "A Study of the Attitudes Toward Arithmetic of Students and Teachers i n the Third, Fourth, and Sixth Grades," The Arl^hmotin T^rbav, October, i 9 6 0 , pp.2P0-286. "''-'-Aiken, p.5 , c i t i n g L. M. Morrisett and J . Vinsonhaler Eds. "Mathematical Learning," Monographs of the Society for Research i n C h i l d Development, XXX, No. 1 , 1965, p.132. 42 J . Peter Fedon, "The Role of Attitude i n Learning Arithmetic," The Arithmetic Teacher, December, 1 9 5 8 , p.3 1 0 . ^ L o i s Stephens, "Comparison of Attitude and Achievement Among Junior High School Mathematics Classes," The Arithmetic Teacher, November, i 9 6 0 , pp.3 5 1 - 3 5 6 . "•Lesley J . Lyda and Evelyn C. Morse, "Attitudes, Teaching Methods and Arithmetic Achievement," The Arithmetic Teacher, March, 1 9 6 3 , pp.136-138. ^ L e w i s R. Aiken and R. M. Dreger, "The E f f e c t of Attitudes on Performance i n Mathematics," Journal of Educational Psychology, Vol. 5 2 , 1 9 6 1 , p.2 6 . ^^Howard F. Fehr, George McMeen and Max Sobel, "Using Hand-Operated Computing Machines i n Learning Arithmetic," The Arithmetic Teacher, October, 1956, pp. 145 -150. ^ I b i d , p.148. 41 ^ V i c t o r R. Durr-ance, "The E f f e c t of the Rotary Calculator on Arithmetic Achievement i n Grades Six, Seven and Eight," D i s s e r t a t i o n Abstracts. 2 5 , p. -6307. ^ J o s e p h P. Cech, "The E f f e c t the use of Desk Calculators has on Attitude and Achievement i n Ninth-Grade General Mathematics Classes," D i s s e r t a t i o n Abstracts, 31A, December 1970, p.2784A. 5°Wilbur H. Dutton, "Measuring Attitudes Toward Arithmetic," Elementary School Journal. September, 1 9 5 4 , pp.24-31 . L. Thurstone and E. J . Chave, The Measurement of Attitu d e , Chicago: U n i v e r s i t y of Chicago, 1948. 5 2Dutton, "Measuring A t t i t u d e s , " p.26. 53E. F. Lindquist, The Iowa Tests of Educational Development General Manual. Chicago: Science Research Associates, March, 1969, P - 7 . ^*P. A. Lappan J r . , "Review of 'The Iowa Tests of Educational Development 1," Mental Measurements Yearbook, V o l , VI pps. 872-873. ^Sheldon S. Meyers, "Annotated Bibliography of Mathematics Tests," Evaluation i n Mathematics, Washington: NCTM 2 6 t h Yearbook, 1 9 6 I , p.2 0 1 . ^Henry E. Garrett, S t a t i s t i c s i n Psychology and Education, David McKay Co. Inc., New York, 1958, p.3 3 9 . *?Ibid, P.3 3 9 . 5 8 l b i d , p.351. Dixon and F. J . Massey J r . , Introduction to S t a t i s t i c a l Analysis 2d ed., McGraw-Hill Book Company, New York, 1957, pp.3 0 6 - 3 1 0 . BIBLIOGRAPHY 43 Aiken, Lewis R. J r . , "Attitudes toward Mathematics," Studies i n Mathematics, Vol. XIX, ed. James Wilson and L. Ray Carry, SMSG, Stanford U n i v e r s i t y , 1969. Cech, Joseph P., " The E f f e c t the use of Desk Calculators has on Attitude and Achievement i n Ninth-Grade General Mathematics Classes," D i s s e r t a t i o n Abstracts, 31A December 1970. C r i s t a n t i e l l o , P. D., "Attitude Toward Mathematics and the Predictive V a l i d i t y of a Measure of Quantitative Aptitude," Journal of Educational Research, LV (December-January 1 9 6 2 ) . Dutton, Wilbur H., "Attitudes of Junior High School Pupils Toward Arithmetic," The School Review, 1956. , "Measuring Attitudes Toward Arithmetic," Elementary School Journal, September, 1954. Dixon, W. J . and Massey, F. J . J r . , Introduction to S t a t i s t i c a l Analysis 2nd ed., McGraw-Hill Book Company, New York, 1957. Durrance, Victor R., "The E f f e c t of the Rotary Calculator on Arithmetic Achievement i n Grades Six, Seven and Eight," D i s s e r t a t i o n Abstracts, 2 5 . Fedon, J . Peter, "The Role of Attitude i n Learning Arithmetic," The Arithmetic Teacher. December, 1958. Fehr, Howard F., George McMeen and Max Sobel, "Using Hand- Opera ted Computing Machines i n Learning Arithmetic," The Arithmetic Teacher, October, 1956. Garrett, Henry E., S t a t i s t i c s i n Psychology and Education, David McKay Co. Inc., New York, 1958. Hobart, (Oklahoma) Democrat-Chief, Wednesday, March 31* 1965« Lappan, P. A. J r . , "Review of 'The Iowa Tests of Educational Development 1," Mental Measurements Yearbook, Vol. VI. Lindquist, E. F., The Iowa Tests of Educational Development General Manual, Chicago: Science Research Associates, March, 1959. Lyda, Wesley J . , and Evelyn C. Morse, "Attitudes, Teaching Methods and Arithmetic Achievement," The Arithmetic Teacher, March, 1963. Meyers, Sheldon S., "Annotated Bibliography of Mathematics Tests," Evaluation i n Mathematics, Washington: NCTM 2 6 t h Yearbook, 1961. PofrenDerger, Thomas and Donald Norton, "Factors In the Formation of Attitudes toward Mathematics," Journal of Educational, Research, Vol. 5 2 , No. 5 5 January, 1 9 5 9 . Stephens, Lois, "Comparison of Attitudes and Achievement Among Junior High School Mathematics Classes," The Arithmetic Teacher, November, i 9 6 0 , N S t r i g h t , V i r g i n i a M., "A Study of the Attitudes Toward Arithmetic of Students and Teachers i n the Third, Fourth, and Sixth Grades," The Arithmetic Teacher T October, i 9 6 0 . Thurstone, L. L. and E. J . Chave, The Measurement of A t t i t u d e T Chicago: U n i v e r s i t y of Chicago, 1 9 4 8 . APPENDIX A SAMPLE PROBLEMS Some of the questions asked were s i m i l a r to the following. A. P a y r o l l Computation Gross Pay Income Tax Pension Net Pay Name Hours Rate per hour a 45 3-25 b 42 3 . 0 0 c 40 2.75 d 40 2.75 e 43 2.75 f 36 2 .50 g 32 2 . 5 0 TOTALS B. Commission 1. Calculate the commission received by the agent i n each of the following: Sales Rate of Commission Commission a) 1560.00 15# b) 750.00 22% 2 . Calculate the rate of commission i n each of the following: Sales Commission Rate a) 4000 .00 340 .00 b) 7 0 , 0 0 0 . 0 0 4200 .00 3 . Calculate the amount of sales i n each of the following: Commission Rate Sales a) 320 .00 h% b) 4 6 0 . 0 0 15% C. Simple Interest 1. Calculate the i n t e r e s t i n each of the following: P r i n c i p a l Rate Time Interest a) 450.00 5% - 2 yr. b) 1250.00 3%% 3 i y r . 2 . Find the rate of i n t e r e s t i n each of the following: P r i n c i p a l Interest Rate Time a) $ 500.00 3 0 . 0 0 1 y r . b) 1200.00 9 0 . 0 0 l i yr. 3 . Calculate the p r i n c i p a l that must be invested at 6% to y i e l d $ 6 0 . 0 0 i n t e r e s t i n 8 months. 4. Calculate the time i n each of the following: 4 7 P r i n c i p a l Interest Rate Time a) $750.00 40 .00 h% b) 350.00 19 .25 5\% D. Installment Buying 1. Tom Barrow borrowed $100. He promised to repay the loan i n 10 installments of $10 each at l~k% monthly on the unpaid balance. Make a table showing: Payments on p r i n c i p a l Balance due Interest charges Each monthly payment Total i n t e r e s t charges Total of repayments and i n t e r e s t APPENDIX B DUTTON'S ATTITUDE SCALE 49 1. I think about arithmetic problems outside of school and like to work them out. 2. I don't feel sure of myself in arithmetic. 3. I enjoy seeing how rapidly and accurately I can work arithmetic problems. h, I like arithmetic, but I l i k e other subjects just as well. 5 . I like arithmetic because i t i s practical. 6. I don't think arithmetic i s fun, but I want to do well ln i t . 7. I am not enthusiastic about arithmetic, but I have no real dislike for i t either. 8. Arithmetic i s as important as any other subject. 9. Arithmetic i s something you have to do even though i t i s not enjoyable. 10. Sometimes I enjoy the challenge presented by an arithmetic problem. 11. I have always been afraid of arithmetic. 12 . I would li k e to spend more time i n school working arithmetic. 13. I detest arithmetic and avoid using i t at a l l times. Ih. I enjoy doing problems when I know how to work them well. 1 5 . I avoid arithmetic because I am not very good with figures. 1 6 . Arithmetic t h r i l l s me, and I l i k e i t better than any other subject. 17. I never get tired of working with numbers. 18. I am afraid of doing work problems. 19. Arithmetic i s very interesting. 20. I have never liked arithmetic. 21. I think arithmetic i s the most enjoyable subject I have taken. 22. I can't see much value i n arithmetic. APPENDIX C MATHEMATICS ATTITUDE TEST 51 NAME BLOCK DATE ATTITUDE TEST FOR MATHEMATICS You w i l l be presented with 16 situations i n which you have three choices (A), (B), (C). From these three choices you are asked to se l e c t the one you l i k e the l e a s t and the one you l i k e the most. Put a check ( N / ) i n the square f o r the one you l i k e the best. Put a cross ( X ) i n the square for the one you l i k e the l e a s t . EXAMPLE 1 . a. Add up a column of 2 0 three d i g i t numbers. b. Solve the equation 3 x + 2 = 2 0 . c. Wash the breakfast dishes. A B C V X SCORES ft 3 C X Add up a s e r i e s o f f r a c t i o n s . F i n d the number when f o u r more than the number i s three l e s s than three times the number. Watch Batman on TV. Add up a column o f 15 numbers. Determine the number o f n i n e inch t i l e s needed to cover t h i s f l o o r . L e a r n to w a l t z . M u l t i p l y a s e r i e s o f f r a c t i o n s . F i n d the le n g t h o f a r e c t a n g l e when the peri m e t e r and width are known. Pai.nt a p i c t u r e o f some c o l o r f u l stones. M u l t i p l y f i v e , two d i g i t numbers. Two men together can do a p i e c e o f work i n s i x days. One man working a l o n e can do the job i n t e n d a y s . How long would i t take the second man h i m s e l f ? Type out t h i s page. F i n d the square o f 3.2. You p a i d $60 f o r a r a d i o a t a s a l e which a d v e r t i s e d that a l l p r i c e s were reduced by one t h i r d . What was the o r i g i n a l p r i c e ? Dust the l i v i n g room. Work some s u b t r a c t i o n q u e s t i o n s i n v o l v i n g d e c i m a l s . Mary i s ten and her aunt i s f o r t y . When was Mary one-seventh as o l d as her aunt? Give an o r a l r e p o r t i n S o c i a l S t u d i e s . 7. a. F i n d the square r o o t o f 169. b. There are 120 a p p l e and peach t r e e s i n an o r c h a r d . There are 2/3 as many peach t r e e s as apple t r e e s . How many t r e e s o f each are t h e r e ? c. Run l a p s d u r i n g P. E. 8. a. F i n d out what percent one number i s o f another. b. The head o f a f i s h i s 10 inches l o n g , the t a i l i s as long as the head p l u s one h a l f o f the body; the body i s as long as the head and the t a i l t o g e t h e r . How long i s the f i s h ? c . Rake up the l e a v e s on the lawn. 9. a. F i n d out what percentage one number i s o f another. b. A b a s e b a l l team won 25 more games than i t l o s t . I f i t won 3/5 o f i t s games, how many d i d i t p l a y ? c. Make a drawing o f a c o f f e e t a b l e t h a t you a r e going to have b u i l t . 10. a. Round o f f seventy numbers t o the nearest hundred. b. A toy t r a i n c o n s i s t s o f an engine, a d i n i n g car and a caboose. The whole t r a i n i s 26 inches l o n g . The r e l a t i o n o f the l e n g t h s o f the d i n i n g c a r and the engine i s (3,2) and the r e l a t i o n o f the l e n g t h s o f the caboose and the d i n i n g c a r i s ( 1 , 2 ) . What i s the l e n g t h o f each s e c t i o n ? c. Mow the lawn. 11. a. Work out f i v e d i v i s i o n q u e s t i o n s i n v o l v i n g numbers o f seven d i g i t s . b. A policeman chases a t h i e f . The policeman takes two s t e p s w h i l e the t h i e f takes t h r e e . But each step the policeman takes covers as much d i s t a n c e as two o f the t h i e f ' s s t e p s . How many ste p s w i l l the policeman t a k e b e f o r e he c a t c h e s the t h i e f ? c. Design a desk f o r your homework. ft 8 53 C 54 fi B G 12. a. Determine the weight o f 990 bags weighing s e v e n t y - f i v e pounds each. b. Determine the c o s t o f making your own t e n t . c. L i s t e n t o an o r a l r e p o r t by a classmate about weeds. 13. a. Determine the volume o f a f i s h tank. b. On a remote i s l a n d t h e r e are two t r i b e s , The Tee's and the L ' s . The Tee's always t e l l the t r u t h . The L's always l i e . A s t r a n g e r to the i s l a n d comes to a group o f t h ree n a t i v e s who a l l look a l i k e . He asks the f i r s t what t r i b e he belongs to but can't make out the answer so he asks the second n a t i v e what he s a i d and gets the r e p l y "He s a i d he was a Tee." The t h i r d says the second was a l i e r . To what t r i b e d i d the t h i r d man b e l o n g ? c. See a stage p l a y . 14. a. C a l c u l a t e the p r i c e of an a r t i c l e i f i t i s to be reduced by l / 3 . b. You have planned to take a boat t r i p f o r 20 people, two were unable t o go. You each had to pay an a d d i t i o n a l t h r e e d o l l a r s . How much was the boat r e n t a l ? c. W rite a l e t t e r o f thanks f o r a b i r t h d a y g i f t . 15. a. Compute the weight o f 1000 yards o f w i r e at ten pounds f o r each 1000 f e e t . b. S o l v e an a l g e b r a i c e q u a t i o n . c. Go f o r a c a r r i d e w i t h your grandparents. 16. a. F i n d the common d i v i s o r o f f o u r numbers. b. Graph an a l g e b r a i c e q u a t i o n . c. S i n g i n a mixed c h o i r . APPENDIX D IOWA TEST OF QUANTITATIVE THINKING 56 ABILITY TO DO QUANTITATIVE THINKING DIRECTIONS: In working these problems, pay no a t t e n t i o n to the suggested answers u n t i l a f t e r you have got your own answer. I f your answer agrees with one of the suggested answers, mark the corresponding box on the answer sheet. I f your answer does not agree with any of the answers given, mark the l a s t box i n the corresponding row on the answer sheet. There are many problems to which the correct answer i s not given. Do not rework a problem simply because your answer i s not among those suggested. Instead mark the f i f t h box ("Not given" or "None of these") and go on to the next problem. The three sample problems have been marked c o r r e c t l y on the answer sheet. Notice how they are marked, and mark the remaining problems s i m i l a r l y . Samples: 0 . What i s the cost of 3 pounds of butter at 500 per pound? 1) $1 .15 4) $1 .65 2) $ 1 . 3 5 5) Not given. 3) $1 .50 0 0 . Mrs. Smith, In paying f o r 600 worth of groceries, gave the cl e r k a d o l l a r b i l l . How much change should she receive? 1) 25£ 4) 600 2) 300 5) Not given. 3) 50^ 0 0 0 . I f one gallon of paint w i l l cover 200 square feet, approximately how many gallons of paint w i l l be needed to cover 380 square feet? 1) 1 gallon h) k gallons 2) 2 gallons 5) 5 gallons. 3) 3 gallons Ĵ C Sjt 3JC ]fc }|C jjc 3JC ĴC 3̂C jjc 3̂C NOTE: Do your f i g u r i n g on scratch paper. Make no marks on these pagesJ 1. A three-pound beef roast contains f i v e ounces of bone. The weight of the bone i s what part of the t o t a l weight? 1) 3/80 h) 15/16 2) 5 A 8 5) Not given. 3) 5/19 2 . The ingredients for making 2k chocolate marshmallow b a l l s , as given i n a p a r t i c u l a r recipe, are: 2 squares (2 oz.) unsweetened chocolate 1 cup evaporated milk 1/2 cup granulated sugar 12 marshmallows, halved 1 cup f i n e l y chopped walnuts How many ounces of chocolate would be needed i n order t o - 5 7 make 3 dozen marshmallow balls? 1) 2 1 A 4 ) 3 1/2 2 ) 2 1/2 5) Not given. 3 ) 3 1/3 3 . In a short-cut method of multiplying 4087 x 198, one f i r s t multiplies 4087 x 2 0 0 . What product must then be subtracted from this result i n order to obtain the correct answer? 1) 13 x 2 4 ) 4 0 8 7 x 2 2 ) 4087 x 13 5) Not given. 3 ) 198 x 2 4 . How many pencils selling at 2 for 5£ can be bought for 600? 1) 30 4 ) 6 2 ) 15 5) Not given. 3 ) 12 5 . The map of a summer camp was drawn according to the scale 1 inch = 1/8 mile. What is the length of the camp ground in miles i f the corresponding distance on the map measures 10 inches? 1) . 8 Mile 4 ) 80 miles 2 ) 1 . 2 5 miles 5) Not given. 3 ) 8 . 2 miles NOTE: If you do not recognize quickly how a problem should be worked, skip i t and go on to the next. You may come back to the harder problems later i f time permits. 6 . What i s the next term i n the series 2 0 , 6 . 1 . 8 , . 5 4 , . 1 6 2 ? 1) . 0 1 6 2 2 ) . 0 4 8 6 3 ) . 0 5 4 0 4 ) . 0 8 1 0 5) Not given. 7 . The following quotation i s from an automobile club pamphlet. "If you are i n an accident while your car i s traveling under 40 miles an hour, there i s one change i n 4 4 that someone w i l l be k i l l e d . If an accident occurs while your car i s traveling over 40 miles an hour, there i s one chance i n 19 that someone w i l l be k i l l e d . " Which of the following statements best interprets the meaning of this quotation? 1) A car traveling faster than 40 miles an hour w i l l k i l l 2 5 more people than w i l l a car traveling under 40 miles an hour. 2 ) If a l l people drove faster than 40 miles an hour, one i n 19 would be k i l l e d . 3 ) If a l l people drove at 3 0 miles an hour, nobody would be k i l l e d . 4 ) If people drive more than 40 miles an hour, the danger of f a t a l accidents increases. 5) Out of 1,000 people traveling by car, 19 w i l l be k i l l e d by fast drivers, and 44 by slow drivers. 8. Exactly how many hundreds are contained i n the number 58 63,517? 1) 63.517 2) 517 3) 635.17 4) 63,500 5) Not given. 9 . To make bookshelves, a board 8 feet 9 inches long was cut into three equal sections. How long was each section? 1) 2 feet, 6 inches 2) 2 feet, 11 inches 3) 3 feet, 5 inches 4) 3 feet, 10 inches 5) Not given. 10. An agent received commissions at the rate of 2 .5$ on a sale of ^2,500, 3% on a sale of $3,000, 3.2$ on a sale of $3}200, 3«5# on a sale of $3>500, and so on. According to t h i s pattern, how much money would his commission amount to on a sale of $4,000? 1) $16 2) $40 3) $160 4) $400 5) Not given. 1 1 . A family bought a two-family house, planning to rent one apartment and l i v e rent-free i n the other. They were not interested i n making any further p r o f i t . Yearly expenses on the house were estimated at $140 for taxes; $340 f o r insurance, upkeep and depreciation; and $200 for other expenses. What monthly rent should the family charge the tenant? 1) $29 2) $55 3) $57 4) $64 5) $68. Problems 12 and 13 are based on the c i r c l e graph below, representing the budget plan of a family. 12 . Which of these conclusions i s supported by t h i s graph? 1) The family spends too much for food. 2) The family i s wealthy. 3) The family has a very low income. 4) The family i s extravagant. 5) None of the conclusions given above can be drawn. 1 3 . On the basis of t h i s budget plan, how much would be a l l o t e d for rent i f the family's income were $4,000? 1) $400 2) $480 3) $1 ,000 4) $1 ,600 5) Not given. 14. Height i n Inches 52 53 54 55 56 57 58 59 60 Over 60 No. of Students 2 1 0 2 3 1 5 12 8 6 The table above gives, to the nearest inch, the heights of the students i n a c e r t a i n c l a s s . The table shows that of the 40 students i n the c l a s s , 2 students are 52 inches t a l l , 1 student i s 53 inches t a l l , etc. Exactly one-eighth of the members of the class are the same height (to the nearest in c h ) . How t a l l are the students i n that group? 1) 55 inches 2) 56 inches 3) 58 inches 4) 60 inches 5) Not given. 1 5 . I t has been recommended that a person watching t e l e v i s i o n s i t at l e a s t 6 feet away from the set i f the screen measures 10 inches across, and 7 inches farther away fo r each a d d i t i o n a l inch of screen. What i s the minimum distance s a t i s f y i n g t h i s requirement for a 20-inch screen? 1) 11 feet, 8 inches 2) 11 feet, 10 inches 3) 12 feet 4) 17 feet, 8 inches 5) Not given. 1 6 . A housewife purchased linens from a wholesaler for resale from her home. She invested $104.00 i n merchandise, and made sales amounting to $79*50. Her inventory then showed that she s t i l l had merchandise on hand for which she had paid $42.00. What was her gross p r o f i t , to date? 1) $17.50 2) $24 .50 3) $37.50 4) $66.50 5) Not given. 17. Many f i r e insurance p o l i c i e s operate under an 80$ clause. This clause provides that the owner of a house insured f o r 80$ of i t s value w i l l receive the complete cost of damages (up to the face value of the p o l i c y ) i n case of f i r e , while the owner of property insured for l e s s than 80$ of the value w i l l receive only 60/80 of damage costs r e s u l t i n g from the f i r e . A house valued at $15,000 was insured f o r $ 7 , 5 0 0 . Damages caused by f i r e cost $2,000. How much should the company pay the owner? 1) $1,000 2) $1 ,500 3) $1 ,600 4) $2,000 5) Not given. Problems 18 to 20 are based on the following graph. 60 FEDERAL BUDGET: RECEIPTS AND EXPENDITURES (In B i l l i o n s of Dollars) I 0 0 io 10 io 5° Vo 3<? ko io o mo L \ 1 \ / F/c It? two / l/S f/ft / I / - \ I IV •-jf--.,. .-. / / cei iS. 'HI 'Vi "/3 'TO 'ft esr 18. What i s the t o t a l of the estimated receipts for the years 1952 and 1953 combined? 1) 32 b i l l i o n s 2) 70 b i l l i o n s 3) 132 b i l l i o n s 4) 155 b i l l i o n s 5) 287 b i l l i o n s . 19. For which one of the following years were the receipts approximately the same as the expenditures? 1) 19^1 2) 1944 3) 1949 4) 1952 5) None of these. 2 0 . Assuming the population of the United States to be 150 m i l l i o n , what i s the estimated per capita tax for 1953? 1) $470 2) $570 3) $700 4) $1 ,050 5) SI,330. 2 1 . People with checking accounts i n Bank A pay the bank 100 for every check issued but nothing for deposits made, while those dealing with Bank T> pay 50 for each check issued and 50 f o r each deposit made. What kind of depositor w i l l save money oy dealing with Bank B? 1) A l l depositors 2) Those who issue checks more frequently than they make deposits. 3) Those who make deposits more frequently than they issue checks. 4) Those who make deposits just as frequently as they issue checks. 5) There i s no difference, since i n the long run each person deposits as much as he withdraws. 2 2 . In playing a game, John makes a score of - 3 5 and Henry makes a score of +65. What i s the difference i n t h e i r scoras? 1) 30 2) 35 3) 90 4) 100 5) Not given. 2 3 . During World War II, about 9i% of g i r l high school 6 1 graduates were r e c r u i t e d f o r the Army Cadet Nurse Corps. This i s equivalent to saying that the Nurse Corps r e c r u i t e d which of the following? 1) 9 out of 15 g i r l graduates. 2) 9 out of 50 g i r l graduates. 3) 10 out of 95 g i r l graduates. 4) 19 out of 100 g i r l graduates. 5) 19 out of 200 g i r l graduates. 24. Mr. N o l l accepted a temporary job for a week. At the end of the week he was paid $ 6 0 . He was t o l d that he had completed 5/6 of the work, and could return the next week to complete the job at the same rate of pay. How much would he be paid to complete the job? 1) $6 2) $10 3) $12 4) $50 5) Not given. 2 5 . The annual premium for an insurance p o l i c y i s figured at the rate of 54# per $100. For premiums paid f o r 3-year periods i n advance, the 3-year premium i s set at 2 1/2 times the 1-year premium. What would be the average yearly rate on a 3-year premium paid i n advance? 1) 36<z> per $100 2) 45^ per $100 3) 5 V per $100 4) 64^ per $100 5) It would depend upon the face value of the p o l i c y . 2 6 . The hundreds d i g i t of a t h r e e - d i g i t number i s Jh, the tens d i g i t i s t, and the u n i t s d i g i t i s u.. Which of the . following expressions represents t h i s number? 1) h x t x u 2) h + t + u 3) 100 h + lOt + u 4) ( 1 0 0 h ) ( 1 0 t)(u) 5) None of these. 27. / 5 yrf». To determine the distance between points A and B on opposite sides of a pond, l i n e BC i s l a i d o f f perpendicular to the l i n e of sight between A and B, as shown i n the diagram above. The distances BC and CA are then measured. What i s the distance between A and B i n yards? ( Do not tr y to estimate the distance from the f i g u r e , since i t i s not drawn to scale.) 1) 18 2) 20 3) 21 4) 23 5) Not given. 2 8 . 62 2 9 . 3 0 . 3 1 . One commonly used method of a l l o w i n g f o r annual d e p r e c i a t i o n i n value c o n s i s t s of deducting the same per cent each year of the value a t the beginning of that year. According to t h i s method, what i s the value a t the end of two years of a car c o s t i n g $ 2 , 0 0 0 new i f the r a t e of d e p r e c i a t i o n i s 25% per year? 1) $ 1 , 0 0 0 2 ) $ 1 , 1 2 5 3 ) $ 1 , 5 0 0 40 $ 1 , 2 5 0 5) Not g i v e n . The energy requirement of a d u l t s 2 0 - 6 0 years o l d f o r 1 hour of s l i g h t e x e r c i s e i s .4- c a l o r i e per pound of body weight. The energy requirement f o r a d u l t s 6 0 - 7 0 years o l d i s 10% l e s s . What i s the corresponding c a l o r i e requirement per pound of body weight f o r these o l d e r persons? 1) . 0 9 6 2 ) . 0 6 3 ) . 0 3 40 . 3 6 5) Not g i v e n . An automobile dealer determines the t r a d e - i n value o f a c e r t a i n make and model of car by means of the f o l l o w i n g t a b l e . Y = Number of Years 1 2 3 Since Purchased T = Trade-in Value 2 , 0 0 0 1 , 7 0 0 1,4-00 1 , 1 0 0 ^ i n D o l l a r s Which of the f o l l o w i n g formulas expresses the r e l a t i o n s h i p between t r a d e - i n value (T) and the age o f the car (Y)? 1 ) T = 2 , 0 0 0 - 150Y 2 ) T = 2 , 5 0 0 - 500Y 3 ) T = 1 , 1 0 0 + 6 0 0 Y 40 T = 2 , 3 0 0 - 300Y 5) None of these. What i s the area i n square f e e t of the c i t y l o t shown i n the diagram above? 1) 4-,800 2 ) 7 , 6 0 0 3 ) 8 , 0 0 0 40 8 , 1 2 5 5) Not g i v e n . Use the following information l n solving problems ^2 and ^ 3 . Let A = the cash price of an a r t i c l e of merchandise, I = the installment p r i c e , i . e . , the t o t a l cost when purchasing on the installment plan. D = the down payment, n = the number of monthly payments. Then r , the approximate rate of i n t e r e s t paid when purchasing on the installment plan, i s given by the formula: 24(1 - A) r = (A - D)(n + 1) 3 2 . Which of the following expressions represents the amount of each payment? I - D A - D A I 1) n 2) n 3 ) n 4) n 5) None of these. 3 3 . A t e l e v i s i o n set s e l l s f o r $300 cash, or $50 down and $40 per month for 7 months. What i s the approximate rate of i n t e r e s t paid by an i n d i v i d u a l purchasing t h i s set on the installment plan? 1) 7% 2) 10% 3) 17% 4) 26% 5) 3 6 $ . APPENDIX E ITEM ANALYSIS OF MATHEMATICS ATTITUDE TEST 65 A - A t t i t u d e Toward C a l c u l a t i o n Top 27% Bottom 27% Q u e s t i o n Number Number o f Responses Percent o f Responses r b i s P ercent o f Responses Number o f Responses 1 22 69 49 22 7 2 24 75 48 25 8 3 14 44 53 6 2 4 18 56 34 22 7 5 8 25 50 - 0 6 23 72 37 37 12 7 15 47 45 10 3 8 26 81 71 10 3 9 15 47 45 10 3 10 20 62 37 28 9 11 10 32 30 10 3 12 20 62 33 28 9 13 12 37 61 - 0 14 20 62 64 6 2 15 21 65 29 37 12 16 28 87 63 22 7 66 B - Attitude Toward Quantitative Thinking Top 27% Bottom 27% Question Number Number of Responses Percent of Responses r b i s Percent of Responses Number of Responses 1 15 h6 66+ - 0 2 15 h6 4-5 9 3 3 15 he 36 16 5 4 16 50 68+ - 0 5 28 88 63 25 8 6 20 63 68 3 1 7 22 69 26 44 14 8 15 4-6 66 - 0 9 12 37 50 3 1 10 Ih 44 65 - 0 11 18 56 60 6 2 12 12 37.5 09* 31 10 13 15 4-6 64 3 1 14 17 53 4-9 13 4 15 8 25 35 6 2 16 11 34 34- 9 3 + the v a l u e i s g r e a t e r than the i n d i c a t e d number. * the va l u e i s too low to be a c c e p t a b l e . 67 C - Attitude Toward Non Mathematical Activity- Top 27% Bottom 27% Question Number Number of Responses Percent of Responses r b i s Percent of Responses Number of Responses 1 20 77 63 15 4 2 18 69 72 4 1 3 22 85 72 12 3 4- 18 69 65 8 2 5 21 81 82 + - 0 6 5 19 34 4 1 7 12 46 6 6 + - 0 8 19 73 79+ - 0 9 22 85 50 38 10 10 17 65 62 8 2 11 23 88 68 19 5 12 6 23 39 4 1 13 26 100 7 7 + 31 8 1 4 21 81 8 2 + - 0 15 21 81 79+ 4 1 16 19 73 79+ - 0 + the v a l u e i s g r e a t e r than the i n d i c a t e d number.

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