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The development of a predictive test for mathematical success Biggs, Marjorie Dudley 1968

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THE DEVELOPMENT OF A PREDICTIVE TEST FOR MATHEMATICAL SUCCESS MARJORIE DUDLEY BIGGS B.Sc. U n i v e r s i t y o f C a l i f o r n i a a t D a v i s , 1 9 6 3 A THESIS SUBMITTED IN PARTIAL FULFILLMENT' 'OF THE REQUIREMENTS FOR THE DEGREE, OF MASTER OF ARTS i n t h e F a c u l t y o f E d u c a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA November, 1 9 6 8 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agre e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f ^f/Ccc.CLgS^^<^C. The U n i v e r s i t y . o f B r i t i s h Columbia Vancouver 8, Canada Date Q e X L , ^ / f % y ABSTRACT I t was t h e p u r p o s e o f t h i s s t u d y ( 1 ) t o r e v i e w t h e l i t e r a t u r e i n r e g a r d s t o t h e p r e d i c t i v e v a l i d i t y o f t h e a v a i l a b l e m a t h e m a t i c a l c o u n s e l l i n g t e s t s ; ( 2 ) t o a n a l y z e t h e s e q u e n c e s o f m a t h e m a t i c s c o u r s e s t h a t have and w o u l d be u n d e r t a k e n by s t u d e n t s ; ( 3 ) t o a n a l y z e t h e c o n t e n t o f t h e s e m a t h e m a t i c s c o u r s e s f o r s k i l l s and c o n c e p t s ; and ( h ) t o d e t e r m i n e t h e c o n t e n t o f t h e new m a t h e m a t i c a l c o u n s e l l i n g t e s t . A l l v e r s i o n s o f t h i s c o u n s e l l i n g t e s t c o n s i s t e d o f two d i s t i n c t s e c t i o n s . P a r t one c o n t a i n e d p r o b l e m s whose s o l u t i o n s were d e p e n d e n t upon t h e s t u d e n t ' - s c o m p r e h e n s i o n o f m a t h e m a t i c a l c o n c e p t s and p r o c e d u r e s t o w h i c h he had been e x p o s e d . The s e c o n d s e c t i o n was b a s e d on new m a t e r i a l p r e s e n t e d i n a t e a c h - t e s t f o r m a t . The t e s t was a d m i n i s t e r e d , e v a l u a t e d , r e v i s e d and a d m i n i s t e r e d a g a i n s e v e r a l t i m e s . On a s m a l l s a m p l e o f s t u d e n t s t h e t e s t s c o r e s f r o m b o t h s e c t i o n s had a c o r r e l a t i o n c o e f f i c i e n t o f j u s t l e s s t h a n 0.70. The c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e two s e c t i o n s has been c l o s e t o 0.50. E a c h s e c t i o n o f t h i s t e s t i s a good p r e d i c t o r - o f a s t u d e n t ' s g r a d e i n m a t h e m a t i c s and m e a s u r e s d i f f e r e n t a s p e c t s n e e d e d f o r m a t h e m a t i c a l c o m p e t e n c e . i i i TABLE OF CONTENTS CHAPTER PAGE I . THE INVESTIGATION • 1 The P r o b l e m 1 S t a t e m e n t o f t h e p r o b l e m 1 I m p o r t a n c e o f t h e s t u d y 2 The T e a c h - T e s t 3 . : i l . REVIEW OF THE LITERATURE 5 L i t e r a t u r e on t h e P r e d i c t i o n o f S u c c e s s i n F i r s t Y e a r C o l l e g e M a t h e m a t i c s 5 I m p l i c a t i o n s o f P r e v i o u s S t u d i e s 9 I I I . THE DESIGN . . : 10 S e l e c t i o n and P r e p a r a t i o n o f T e s t I t e m s 10 The T e s t s 11 S e c t i on one 11 S e c t i o n two 13 I V . AN ANALYSIS OF THE DATA AND CONCLUSIONS l h BIBLIOGRAPHY l 8 APPENDIX 20. i v L I S T OF TABLES TABLE PAGE I . A n a l y s i s o f R e s u l t s 15 CHAPTER I THE INVESTIGATION Many u n i v e r s i t i e s have a d m i n i s t e r e d a v a r i e t y o f t e s t s t o e n t e r i n g s t u d e n t s . The p r i n c i p a l p u r p o s e o f s u c h a t e s t i n g p r o g r a m i s t o i d e n t i f y t h o s e a r e a s i n w h i c h a s t u d e n t i s l i k e l y t o s u c c e e d a n d , i n t u r n , t o p r o v i d e a b a s i s f o r e f f e c t i v e c o u n s e l l i n g . At t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a t h e c o r r e l a t i o n c o e f f i c i e n t d e t e r m i n e d by c o m p a r i n g t h e s c o r e s on t h e m a t h e m a t i c a l c o u n s e l l i n g t e s t a d m i n i s t e r e d t o new s t u d e n t s w i t h t h e f i n a l s c o r e s o b t a i n e d i n f i r s t y e a r m a t h e m a t i c s c o u r s e s has b een d e c r e a s i n g . The M a t h e m a t i c s E d u c a t i o n D e p a r t m e n t a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a was i n v i t e d t o a s s i s t i n t h e p r o c e d u r e s f o r o b t a i n i n g a c o u n s e l l i n g t e s t t h a t w o u l d p r o v i d e a b e t t e r i n d i c a t i o n o f a s t u d e n t ' s f i n a l s c o r e i n f i r s t y e a r u n i v e r s i t y m a t h e m a t i c s c o u r s e s . T h i s s t u d y c o n s i s t e d o f t h e i n i t i a l s t e p s i n t h e d e v e l o p m e n t o f t h e d e s i r e d t e s t . I . THE PROBLEM S t at ement o f t h e p r o b l e m . I t was t h e p u r p o s e o f t h i s s t u d y ( l ) t o r e v i e w t h e l i t e r a t u r e i n r e g a r d s t o t h e p r e d i c t i v e v a l i d i t y o f t h e a v a i l a b l e m a t h e m a t i c a l c o u n s e l l i n g t e s t s ; ( 2 ) t o a n a l y z e t h e s e q u e n c e s o f m a t h e m a t i c s c o u r s e s t h a t have been and w o u l d be u n d e r t a k e n by s t u d e n t s ; ( 3 ) t o a n a l y z e t h e c o n t e n t o f t h e s e m a t h e m a t i c s c o u r s e s f o r s k i l l s and c o n c e p t s ; and ( h ) t o d e t e r m i n e t h e c o n t e n t o f t h e new m a t h e m a t i c a l c o u n s e l l i n g t e s t . I m p o r t ance of. t h e s t u d y . U n i v e r s i t y p e r s o n n e l and s t u d e n t s seem t o have assumed on o c c a s i o n t h a t a . s c o r e on t h e m a t h e m a t i c s s e c t i o n o f any a c h i e v e m e n t o r a p t i t u d e t e s t i s a r e l i a b l e e s t i m a t i o n o f t h e s t u d e n t s f i n a l g r a d e i n a m a t h e m a t i c s course."'" F i l e s o f t h e c o u n s e l l i n g c e n t e r a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a show t h a t an E n g l i s h t e s t g i v e n t o s t u d e n t s i n r e c e n t y e a r s was a more r e l i a b l e p r e d i c t o r o f s u c c e s s i n m a t h e m a t i c s c o u r s e s t h a n a m a t h e m a t i c s t e s t g i v e n a t t h e same t i m e . S i m i l a r l i m i t a t i o n s o f s t a n d a r d i z e d c o u n s e l l i n g t e s t s 2 3 have been p o i n t e d out by C o d d i n g t o n and H o l l a n d . C o d d i n g t o n r e p o r t e d a n e g a t i v e c o r r e l a t i o n b e t w e e n t h e s c o r e s f r o m t h e m a t h e m a t i c s p o r t i o n o f t h e S c h o l a s t i c A p t i t u d e T e s t s and f i n a l g r a d e s o b t a i n e d i n f i r s t y e a r m a t h e m a t i c s c o u r s e s a t t h e U n i v e r s i t y o f C a l i f o r n i a a t Los A n g e l e s . H o l l a n d w r o t e a "'"Boyce, R.W. and R.C. P a x s o n , " P r e d i c t i v e V a l i d i t y o f E l e v e n T e s t s a t One S t a t e C o l l e g e , " E d u c a t i o n a l and P s y c h o l o g i c a l  M e a s u r e m e n t , XXV ( W i n t e r 1 9 6 5 ) , 1 1 4 3 . 2 C o d d i n g t o n , E.A., " S c h o l a s t i c A p t i t u d e T e s t s i n M a t h e m a t i c s , " The A m e r i c a n M a t h e m a t i c a l M o n t h l y , LXX ( S e p t e m b e r , 1 9 6 - 3 ) , 7 5 0 - 5 5 -H o l l a n d , J . L . , "The P r e d i c t i o n o f G r a d e s f r o m t h e C a l i f o r n i a P s y c h o l o g i c a l I n v e n t o r y and t h e S c h o l a s t i c A p t i t u d e T e s t s , " J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , L ( A u g u s t , 1 9 5 9 ) , 1 3 5 - 4 2 ; r e g r e s s i o n e q u a t i o n u s i n g t h e s c o r e s f r o m s e v e r a l s e c t i o n s o f t h e S c h o l a s t i c A p t i t u d e T e s t s and t h e C a l i f o r n i a P e r s o n a l i t y -I n v e n t o r y t o o b t a i n a b e t t e r p r e d i c t o r t h a n t h e s c o r e s f r o m t h e S c h o l a s t i c A p t i t u d e T e s t s . He f o u n d t h a t he c o u l d make b e t t e r p r e d i c t i o n s f r o m t h e a v a i l a b l e d a t a by r e s t r i c t i n g h i s p o p u l a t i o n t o t h a t o f a s i n g l e u n i v e r s i t y . As a r e s u l t o f t h e s e r e p o r t s h 5 and s t u d i e s by H e i m e r and B o y c e and P a x s o n , t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a d e c i d e d t o d e v e l o p i t s own m a t h e m a t i c a l c o u n s e l l i n g t e s t w i t h t h e a n t i c i p a t i o n t h a t i t w o u l d more a c c u r a t e l y p r e d i c t a s t u d e n t ' s f i n a l s c o r e i n a f i r s t y e a r m a t h e m a t i c s c o u r s e t h a n t h e m a t h e m a t i c s p o r t i o n o f an A m e r i c a n s t a n d a r d i z e d t e s t s u c h as t h e A m e r i c a n C o l l e g e T e s t s , t h e C o o p e r a t i v e S c h o o l and C o l l e g e A b i l i t y T e s t s , o r t h e S c h o l a s t i c A p t i t u d e T e s t s . I I . THE TEACH-TEST The s e a r c h f o r a t e s t t o measure a s t u d e n t ' s a b i l i t y t o l e a r n ' m a t h e m a t i c s l e d t o t h e d e v e l o p m e n t o f a c o l l e c t i o n o f i t e m s w h i c h w o u l d p r o v i d e a measure o f an i n d i v i d u a l ' s a b i l i t y t o l e a r n m a t h e m a t i c s u n d e r t e s t - l i k e c o n d i t i o n s . These i t e m s H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n M a t h e m a t i c a l M o n t h l y , L X X I I I ( O c t o b e r , 1 9 6 6 ) , 8 8 3 - 8 9 . ^ B o y c e , R.W. and R.C. P a x s o n , " P r e d i c t i v e V a l i d i t y o f E l e v e n T e s t s a t One S t a t e C o l l e g e " , E d u c a t i o n a l and P s y c h o l o g i c a l Measurement , XXV ( W i n t e r 1 9 6 5 ) , llk3-hT. •k. a r e s i m i l a r i n f o r m t o t h e ex a m p l e i n t h e a p p e n d i x . I n t h i s r e p o r t t h e t e r m t e a c h - t e s t w i l l he u s e d e x c l u s i v e l y t o r e f e r t o t h e s e i t e m s . I t was a n t i c i p a t e d t h a t s u c h a measure c o u l d make a s i g n i f i c a n t c o n t r i b u t i o n t o t h e p r e d i c t i o n o f a p e r s o n ' s s u c c e s s i n a ' m a t h e m a t i c s c o u r s e . The r e s u l t s g i v e n i n C h a p t e r I V o f t h i s r e p o r t i n d i c a t e t h a t t h e t e a c h - t e s t m e a s u r e s some d i f f e r e n t f a c t o r s n e e d e d f o r s u c c e s s t h a t known p r e v i o u s t e s t s have f a i l e d t o m e a s u r e . CHAPTER I I REVIEW OF THE LITERATURE A l t h o u g h most o f t h e s t u d i e s c o n c e r n e d w i t h p r e d i c t i o n s o f s u c c e s s i n u n i v e r s i t y have b e e n o f a g e n e r a l n a t u r e and d e a l w i t h t o t a l s u c c e s s i n u n i v e r s i t y s t u d y , a few have been c o n c e r n e d w i t h t h e p r e d i c t i o n o f s u c c e s s i n c o l l e g e l e v e l m a t h e m a t i c s . B e c a u s e o f t h e c h a n g e s i n t h e n a t u r e o f t h e c o l l e g e p r e p a r a t o r y m a t h e m a t i c s c u r r i c u l u m s and i n t h e c h a r a c t e r i s t i c s o f u n i v e r s i t y p o p u l a t i o n s , t h e f i n d i n g s o f p r e d i c t i v e s t u d i e s o f t e n become o b s o l e t e . T h i s has b e e n t h e e x p e r i e n c e at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . I . LITERATURE ON THE PREDICTION OF SUCCESS IN FIRST YEAR COLLEGE MATHEMATICS 1 2 The s t u d i e s by H o l l a n d and C o d d i n g t o n showed t h e p r e d i c t i v e i n a d e q u a c y o f t h e m a t h e m a t i c s s e c t i o n o f t h e S c h o l a s t i c A p t i t u d e T e s t s f o r s a m p l e s t h a t d i d n o t r e p r e s e n t t h e u n i v e r s i t y p o p u l a t i o n . H o l l a n d , J . L . , "The P r e d i c t i o n o f C o l l e g e G r a d e s f r o m t h e C a l i f o r n i a P s y c h o l o g i c a l I n v e n t o r y and t h e S c h o l a s t i c A p t i t u d e T e s t s , " J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , L ( A u g u s t , 1959), 135-1+2 . The C o d d i n g t o n , E.A., " S c h o l a s t i c A p t i t u d e T e s t s A m e r i c a n M a t h e m a t i c a l M o n t h l y , LXX ( S e p t e m b e r , i n M a t h e m a t i c s , " 1963) 750-55. W i c k i n h i s s t u d y o f some f a c t o r s a s s o c i a t e d w i t h s u c c e s s i n f i r s t y e a r c o l l e g e m a t h e m a t i c s a r r i v e d a t f o u r c o n c l u s i o n s . 1. F o r t h e p a r t i c i p a n t s i n t h i s i n v e s t i g a t i o n t h e r e a p p e a r s t o be no s i g n i f i c a n t d i f f e r e n c e i n t h e q u a l i t y o f p r e p a r a t i o n f o r f i r s t - y e a r c o l l e g e m a t h e m a t i c s as b e t w e e n t h e e x p e r i m e n t a l (S.M.S.G.) and t h e t r a d i t i o n a l p r o g r a m s i n h i g h s c h o o l m a t h e m a t i c s . T h e r e i s some i n d i c a t i o n t h a t t h e e x p e r i m e n t a l m a t h e m a t i c s m a t e r i a l s do have a c u m u l a t i v e e f f e c t . 2. The h i g h s c h o o l m a t h e m a t i c s r e c o r d was c o n s i s t e n t l y t h e s o u r c e o f t h e b e s t p r e d i c t o r s o f s u c c e s s i n f i r s t - y e a r c o l l e g e m a t h e m a t i c s . The m a t h e m a t i c s a v e r a g e o v e r g r a d e s t e n t o t w e l v e gave t h e h i g h e s t c o r r e l a t i o n w i t h s u c c e s s i n c o l l e g e a l g e b r a c o u r s e s , w h e r e a s t h e a v e r a g e i n g r a d e 12 a p p e a r e d t o be s l i g h t l y more i n d i c a t i v e o f s u c c e s s i n b e g i n n i n g c a l c u l u s . A l l c o r r e l a t i o n s b e t w e e n t h e f a c t o r s a n a l y z e d and c o l l e g e m a t h e m a t i c s g r a d e s a r e b e s t d e s c r i b e d . .,as b e i n g l o w . 3. The use o f m u l t i p l e r e g r e s s i o n t e c h n i q u e s f o r p r e d i c t i n g s u c c e s s r e p r e s e n t e d a c o n s i d e r a b l e i m p r o v e m e n t o v e r t h e use o f s i n g l e p r e d i c t o r v a r i a b l e s . The r e s u l t i n g m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s were h i g h enough t o w a r r a n t t h e use o f p r e d i c t i o n s b a s e d on t h e m u l t i p l e r e g r e s s i o n e q u a t i o n s f o r s e l e c t i o n and p l a c e m e n t o f s t u d e n t s , and f o r o t h e r g r o u p - c o u n s e l i n g p r o c e d u r e s . h. The use o f p r o b a b i l i t y t a b l e s s u c h as d e s c r i b e d i n t h i s r e p o r t w o u l d a p p e a r t o be an e f f e c t i v e way o f u s i n g p r e d i c t e d g r a d e s when c o u n s e l i n g w i t h i n d i v i d u a l s t u d e n t s . The l i m i t a t i o n s on t h e use o f m u l t i p l e r e g r e s s i o n e q u a t i o n s i n c o u n s e l i n g have l a r g e l y b e e n removed by t h e a d v e n t o f c o m p u t e r f a c i l i t i e s a t most c o l l e g e s . 3 B o y c e and P a x s o n were c o n c e r n e d w i t h t h e p r e d i c t i v e v a l i d i t y o f e l e v e n t e s t s u s e d a t T r o y S t a t e C o l l e g e . The new s t u d e n t s a t t h i s c o l l e g e d u r i n g t h e f i v e p r e v i o u s y e a r s h a d o b t a i n e d s c o r e s w h i c h , t h e y c l a i m e d , a v e r a g e d b e t w e e n t h e W i c k , M.E., " S t u d y o f t h e F a c t o r s i n F i r s t Y e a r C o l l e g e M a t h e m a t i c s , " The (November, 1 9 6 5 ) , 6^7-^8. A s s o c i a t e d w i t h S u c c e s s M a t h e m a t 1 c s T e a c h e r , L V I I 7 . t w e n t y - s i x t h and t h i r t y - e i g h t h p e r c e n t i l e . They w r o t e , The p r e d i c t i v e v a l i d i t y d a t a r e p o r t e d i n a t e s t m a n u a l a r e . o f t e n i n t e n d e d t o be t r u e o r r e p r e s e n t a t i v e o f a n a t i o n a l s a m p l e . S i n c e t h e d i s c r e p a n c i e s i n p r e d i c t i v e v a l i d i t y c o e f f i c i e n t s a r e l i k e l y t o change as a l o c a l c o l l e g e s a m p l e d e p a r t s f r o m a n a t i o n a l s a m p l e , v a l i d i t y s t u d i e s s h o u l d be p e r f o r m e d f o r e a c h c o l l e g e ; ^ H e i m e r r e a l i z e d t h a t c o l l e g e bound s t u d e n t s h a d r e c e i v e d a w i d e v a r i e t y o f p r i o r t r a i n i n g i n m a t h e m a t i c s w i t h v a r i e d d e g r e e s o f u n d e r s t a n d i n g and a c h i e v e m e n t . I n p a r t , t h i s v a r i a b i l i t y was due t o d i f f e r e n c e s i n s t u d e n t m o t i v a t i o n and a b i l i t y . A n o t h e r s o u r c e o f v a r i a b i l i t y was due t o t h e t y p e , e x t e n t , r e g u l a r i t y , and v a l u e o f p r e v i o u s m a t h e m a t i c a l e d u c a t i o n . He c o n s i d e r e d i t o b v i o u s t h a t a c h i e v e m e n t e x a m i n a t i o n s c o u l d n o t d i s t i n g u i s h b e t w e e n s t u d e n t s who d i d n o t l e a r n m a t e r i a l w h i c h was p r e s e n t e d t o them i n a s a t i s f a c t o r y manner and t h o s e who d i d n o t l e a r n t h e m a t e r i a l b e c a u s e i t was p r e s e n t e d p o o r l y o r n o t at a l l . Thus he u n d e r t o o k a p i l o t s t u d y on t h e p r e d i c t i v e v a l i d i t y o f a t e s t w r i t t e n i n a t e a c h - t e s t f o r m a t . H i s s t a t i s t i c a l d a t a s u p p o r t s h i s h y p o t h e s i s t h a t t h e t e a c h - t e s t p r o c e d u r e has a h i g h p o t e n t i a l f o r o b t a i n i n g a v a l i d m easure o f m a t h e m a t i c a l a p t i t u d e . ' ' A n d e r s o n and W h i t t e m o r e ' s s t u d y r e l a t e d o n l y one h i g h s c h o o l and one u n i v e r s i t y m a t h e m a t i c s d e p a r t m e n t . The h i g h B o y c e , R.W. and R.C. P a x s o n , ' " P r e d i c t i v e V a l i d i t y o f E l e v e n T e s t s at One S t a t e C o l l e g e , " Educ a t i o n a l and P s y c h o l o g i c a l  M e a s u r e m e n t , XXV ( W i n t e r 1 9 6 5 ) , 1 1 ^ 3 . ^ H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n M a t h e m a t i c a l  M o n t h l y , L X X I I I ( O c t o b e r , 1 9 6 6 ) , 8 3 3 - 8 9 . 8 . s c h o o l f r o m w h i c h t h e d a t a was o b t a i n e d t a u g h t modern m a t h e m a t i c s t o some c l a s s e s a n d ' t r a d i t i o n a l m a t h e m a t i c s t o o t h e r s . S i g n i f i c a n t c o r r e l a t i o n s were f o u n d b e t w e e n modern m a t h e m a t i c s g r a d e s a n d ' t h e A m e r i c a n C o l l e g e T e s t i n g P r o g r a m m a t h e m a t i c s s c o r e , and b e t w e e n modern m a t h e m a t i c s g r a d e s and t h e c o l l e g e g r a d e p o i n t a v e r a g e i n m a t h e m a t i c s . However no s i g n i f i c a n t c o r r e l a t i o n s were o b t a i n e d i n v o l v i n g t r a d i t i o n a l m a t h e m a t i c s 6 g r a d e s . Wampler s t u d i e d t h e f e a s i b i l i t y o f u s i n g several:'.measures o f a p t i t u d e w h i c h c o u l d be u s e d t o p r e d i c t p e r f o r m a n c e i n c o l l e g e m a t h e m a t i c s . An i n i t i a l s e l e c t i o n o f e l e v e n t e s t s m e a s u r i n g e i g h t d i f f e r e n t a p t i t u d e f a c t o r s was made, and t h e s e s c o r e s made up a p o o l o f e l e v e n p r e d i c t o r v a r i a b l e s . The c r i t e r i o n v a r i a b l e was t h e s c o r e made on a s t a n d a r d i z e d c a l c u l u s a c h i e v e m e n t t e s t . The b e s t c o m b i n a t i o n o f f i v e p r e d i c t i o n v a r i a b l e s was c h o s e n . The m u l t i p l e c o e f f i c i e n t was 0 . 9 5 f o r t h e t w e n t y - t w o s u b j e c t s who h a d c o m p l e t e d t e n s e m e s t e r h o u r s o f c a l c u l u s w i t h a n a l y t i c g e o m e t r y . The f i v e v a r i a b l e s , were t h e s c o r e s on t h e L o c a t i o n s T e s t s , D i v i s i o n T e s t , I n f e r e n c e 7 T e s t , Wide Range V o c a b u l a r y . T e s t , and.Cube C o m p a r i s o n T e s t . H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n Mathemat i c a l M o n t h l y , L X X I I I , ( O c t o b e r , 1 9 6 6 ) , 8 3 3 - 8 9 . Wampler, J o e F., " P r e d i c t i o n o f A c h i e v e m e n t i n C o l l e g e M a t h e m a t i c s , " The M a t h e m a t i c s T e a c h e r , L I X ( A p r i l , 1 9 6 6 ) , 3 6 4 - 9 . From a s t u d y o f 1,2hk s t u d e n t s K i n z e r and K i n z e r f o u n d t h a t t h e l a r g e s t c o e f f i c i e n t o f m u l t i p l e c o r r e l a t i o n when t h e g r a d e i n an a d v a n c e d a l g e b r a c o u r s e was u s e d as t h e c r i t e r i o n and t h e p r e d i c t o r v a r i a b l e s were g r a d e s i n t h e s i x p r e r e q u i s i t e c o u r s e s and t h e p e r c e n t i l e s c o r e on t h e O h i o S t a t e P s y c h o l o g i c a l g E x a m i n a t i o n . T h i s c o e f f i c i e n t has t h e v a l u e 0-.75. I I . IMPLICATIONS OF PREVIOUS STUDIES S i n c e t h e p r e d i c t i v e s t u d i e s have d e m o n s t r a t e d t h a t t h e s c o r e u s e d t o p r e d i c t s u c c e s s i n m a t h e m a t i c s i s d e p e n d e n t upon many v a r i a b l e s , i n c l u d i n g t h e u n i v e r s i t y at- w h i c h t h e s t u d e n t i s e n r o l l e d , t h e C o u n s e l l i n g D e p a r t m e n t o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a d e c i d e d t o i n v e s t i g a t e t h e p r e d i c t i v e v a l i d i t y o f t h e c o e f f i c i e n t o b t a i n e d f r o m a t e s t t h a t w o u l d be w r i t t e n s p e c i f i c a l l y f o r s t u d e n t s e n t e r i n g t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . I t was a n t i c i p a t e d t h a t t h i s c o e f f i c i e n t w o u l d be an i m p o r t a n t f a c t o r i n t h e m u l t i p l e r e g r e s s i o n e q u a t i o n t h a t w o u l d be u s e d f o r t h e p r e d i c t i o n o f s u c c e s s i n m a t h e m a t i c s . K i n z e r , J.R. and L.G. K i n z e r , " P r e d i c t i n g G r a d e s i n A d v a n c e d M a t h e m a t i c s , " J o u r n a l o f . A p p l i e d P s y c h o l o g y , X X X V I I ( J u n e , 1 9 5 3 ) , 1 8 2 - 8 U . CHAPTER I I I THE DESIGN A l l v e r s i o n s o f t h i s c o u n s e l l i n g t e s t c o n s i s t e d o f two d i s t i n c t s e c t i o n s . P a r t one c o n t a i n e d p r o b l e m s whose s o l u t i o n s were d e p e n d e n t upon t h e s t u d e n t : ' s c o m p r e h e n s i o n o f m a t h e m a t i c a l c o n c e p t s and p r e c e d u r e s t o w h i c h he had been e x p o s e d and f o u r i t e m s t h a t were t o d e t e r m i n e i f t h e s t u d e n t had an u n d e r -s t a n d i n g o f t h e t e a c h - t e s t c o n t e n t . These i t e m s w i l l be r e f e r r e d t o as t h e p r e - t e a c h - t e s t i t e m s . The s e c o n d s e c t i o n was b a s e d on new m a t e r i a l p r e s e n t e d i n a t e a c h - t e s t f o r m a t . The t e s t was a d m i n i s t e r e d i n s e v e r a l s e c o n d a r y s c h o o l s t o s t u d e n t s i n M a t h e m a t i c s 1 1 , M a t h e m a t i c s 1 2 , and M a t h e m a t i c s l.P-3 . and a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a t o s t u d e n t s p l a n n i n g t o e n r o l l i n t h e autumn o f 1 9 6 8 . S i n c e t h e r e l i a b i l i t y and v a l i d i t y o f t h e c o u n s e l l i n g t e s t w o u l d be d e c r e a s e d i f i t s p r e c i s e c o n t e n t were made a v a i l a b l e , none o f t h e i t e m s u s e d have been i n c l u d e d i n t h i s ' p a p e r . I . SELECTION AND PREPARATION OF TEST ITEMS At t h e u n d e r g r a d u a t e l e v e l n e a r l y a l l p e r s o n s s e e k i n g , e n t r a n c e t o t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a i n t h e n e x t few y e a r s c a n be c a t e g o r i z e d by h a v i n g been e n r o l l e d i n one o f t h r e e c o l l e g e p r e p a r a t o r y m a t h e m a t i c s p r o g r a m s . Most o f 11. t h e new s t u d e n t s w i l l have t a k e n m a t h e m a t i c s c o u r s e s c u r r e n t l y -o f f e r e d i n B r i t i s h C o l u m b i a s e c o n d a r y s c h o o l s . Some p e r s o n s w i l l h ave c o m p l e t e d m a t h e m a t i c s c o u r s e s i n t h e p r e v i o u s B r i t i s h C o l u m b i a m a t h e m a t i c s p r o g r a m . ' Those i n d i v i d u a l s c o m i n g f r o m b e y o n d B r i t i s h C o l u m b i a w i l l h ave h ad a d i f f e r e n t s e r i e s o f m a t h e m a t i c s c o u r s e s . W i t h t h e e x c e p t i o n o f i t e m s b a s e d on t r i g o m e t r i c c o n c e p t s and t h e p r e - t e a c h - t e s t i t e m s i n t h e f i r s t s e c t i o n , t h e i t e m s c h o s e n i n v o l v e d b a s i c s k i l l s t h a t c o u l d have been a c q u i r e d by s t u d e n t s i n any o f t h e t h r e e above m a t h e m a t i c s p r o g r a m s . Many o f t h e i t e m s t e s t e d s k i l l t h a t a r e n e e d e d f o r r e a d i l y c o m p r e h e n d i n g t h e c o n c e p t s p r e s e n t e d i n f i r s t y e a r m a t h e m a t i c s c o u r s e s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . I I . THE TESTS S e c t i o n one. T h i s t e s t was d e v e l o p e d by a d m i n i s t e r i n g v e r s i o n s o f t h e t e s t and m o d i f y i n g t h e t e s t a f t e r a n a l y z i n g t h e r e s u l t s o b t a i n e d i n e a c h c a s e . F o r t h e f i r s t a d m i n i s t r a t i o n , f i f t y - s e v e n i t e m s were p r e p a r e d . F o r t y - f i v e s t u d e n t s e n r o l l e d i n M a t h e m a t i c s 11 and f o r t y - f i v e s t u d e n t s i n M a t h e m a t i c s 12 a t a s e c o n d a r y s c h o o l i n B r i t i s h C o l u m b i a w r o t e t h i s e x a m i n a t i o n . A l l o f t h e s t u d e n t s ' c a l c u l a t i o n s and a n s w e r s were, c o l l e c t e d . F o r e a c h p r o b l e m two c o m p a r i s o n s were made. One was b e t w e e n t h e number o f s t u d e n t s i n e a c h c o u r s e who o b t a i n e d a c o r r e c t 12 . s o l u t i o n . The o t h e r was "between t h e t o p t w e n t y - o n e s t u d e n t s and t h e b o t t o m t w e n t y - o n e s t u d e n t s i n M a t h e m a t i c s 12 who w r o t e t h e t e s t . The s t u d e n t s ' a n s w e r s were u s e d t o p r o v i d e most o f t h e a l t e r n a t i v e s s e l e c t e d f o r t h e m u l t i p l e c h o i c e f o r m o f t h e s e q u e s t i o n s . A l l s u b s e q u e n t f o r m a t s were m u l t i p l e c h o i c e . F o r t h e s e c o n d a nd t h i r d a d m i n i s t r a t i o n , s e v e n c h o i c e s i n c l u d i n g 'not g i v e n ' were p r e s e n t e d w i t h e a c h i t e m . The i n s t r u c t i o n s s t a t e d ' t h a t t h e s t u d e n t was t o p r o v i d e t h e c o r r e c t a nswer i n t h e d e s i g n a t e d s p a c e when h i s m u l t i p l e c h o i c e s e l e c t i o n was 'not g i v e n ' . F o r t y - s i x m u l t i p l e c h o i c e i t e m s were i n c l u d e d i n t h e s e c o n d v e r s i o n w r i t t e n by e i g h t y - o n e s t u d e n t s a t a s e c o n d a r y s c h o o l i n B r i t i s h C o l u m b i a . T h i r t y - t w o q u e s t i o n s were g i v e n t o t h e o n e - h u n d r e d - s e v e n t e e n ' p u p i l s who w r o t e t h e t h i r d v e r s i o n . F o r b o t h o f t h e s e a d m i n i s t r a t i o n s t h e p a p e r s whose s c o r e s were i n t h e u p p e r o r l o w e r 27% g r o u p s were a n a l y z e d . A f o u r t h f o r m o f t h e t e s t was c o n s t r u c t e d . T h e r e were f o r t y - t w o m a t h e m a t i c a l i t e m s . F o r e a c h q u e s t i o n f i v e a l t e r n a t i v e s were p r o v i d e d i n c l u d i n g 'not g i v e n ' w i t h t h e same i n s t r u c t i o n s as m e n t i o n e d a b o v e . F o r most o f t h e i t e m s t h e f i v e a l t e r n a t i v e s were s e l e c t e d f r o m t h e s e v e n p r e v i o u s l y u s e d c h o i c e s . C h o i c e s n e v e r s e l e c t e d by t h e s t u d e n t s were e l i m i n a t e d . I f n e c e s s a r y , a new c h o i c e was i n s e r t e d f r o m among t h e wrong a n s w e r s g i v e n by t h e s t u d e n t s . 13. D u r i n g t h e week o f s p r i n g v a c a t i o n t h r e e h u n d r e d s t u d e n t s who c u r r e n t l y were e n r o l l e d i n M a t h e m a t i c s 12 t o o k t h i s t e s t . An a p p r o p r i a t e p r o g r a m a n d ' t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a c o m p u t i n g c e n t e r were u s e d f o r an i t e m a n a l y s i s . To g i v e an i n i t i a l i n d i c a t i o n t h a t t h i s t e s t d i d measure t h e l i k e l i h o o d o f s u c c e s s i n a c o l l e g e l e v e l m a t h e m a t i c s c o u r s e t h i s t e s t was a d m i n i s t e r e d t o n i n e t e e n p u p i l s i n M a t h e m a t i c s 113 a t a s e c o n d a r y s c h o o l i n B r i t i s h C o l u m b i a . The c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e s c o r e o b t a i n e d on t h i s t e s t and t h e s t u d e n t s ' s p r i n g g r a d e was computed. S e c t i o n t w o . The t e a c h - t e s t p o r t i o n c o n s i s t e d o f f o u r b r i e f e x p l a n a t o r y a r t i c l e s f o l l o w e d by a s e r i e s o f q u e s t i o n s f o r w h i c h t h e c o r r e c t s o l u t i o n was d e p e n d e n t upon t h e s t u d e n t ' s u n d e r s t a n d i n g o f t h e w r i t t e n e x p l a n a t i o n s . The s t u d e n t s whose r e s p o n s e s i n p a r t one w h i c h i n d i c a t e d t h a t t h e y h a d some k n o w l e d g e o f t h e s u b j e c t m a t t e r i n c l u d e d i n t h e t e a c h - t e s t i t e m s were o m i t t e d f r o m f u r t h e r a n a l y s i s . A l l o f t h e s t u d e n t s ' c a l c u l a t i o n s and a n s w e r s were c o l l e c t e d . The o n l y a l t e r a t i o n on t h e t e a c h - t e s t p o r t i o n o c c u r r e d b e t w e e n t h e t h i r d and f o u r t h a d m i n i s t r a t i o n . The s e q u e n c e i n w h i c h t h e a r t i c l e s a p p e a r e d was c h a n g e d . F o r t h e s e c o n d and t h i r d a d m i n i s t r a t i o n s a n d ' t h e M a t h e m a t i c s 113 g r o u p c o r r e l a t i o n c o e f f i c i e n t s were computed b e t w e e n t h e s c o r e s o b t a i n e d on s e c t i o n one and t w o . F o r t h e l a t e r g r o u p a c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e s p r i n g g r a d e and t h e t e a c h -t e s t s c o r e a l s o was o b t a i n e d . CHAPTER I V AN ANALYSIS OF THE DATA AND CONCLUSIONS F o r t h e f i r s t s e c t i o n T a b l e I p r o v i d e s a c o m p l e t e n u m e r i c a l d e s c r i p t i o n o f t h e i t e m a n a l y s i s and s u b s e q u e n t a c t i o n t a k e n . The c o r r e l a t i o n c o e f f i c i e n t f o r t h e r e l a t i o n -s h i p b e t w e e n t h e s c o r e s o b t a i n e d by t h e M a t h e m a t i c s 113 s t u d e n t s and t h e i r s p r i n g g r a d e was . 6 8 . The KR20 o f t h e f u l l t e s t i s 0.819. The c o m p u t e r a n a l y s i s o f the. t e s t s w r i t t e n a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a and by t h e M a t h e m a t i c s 113 s t u d e n t s r e s u l t e d i n t h e d e c i s i o n t o i n c l u d e f o r t y i t e m s on t h e m a t h e m a t i c a l c o u n s e l l i n g t e s t . F o r a b o u t h a l f o f t h e q u e s t i o n s a change was made i n t h e c h o i c e s o f f e r e d f o r a n s w e r s . I n t h e f u t u r e t h e number o f i t e m s f o r w h i c h t h e c o r r e c t a n s w e r i s 'not g i v e n ' w i l l be d e c r e a s e d . F o r t h e s e c o n d s e c t i o n . The c o r r e l a t i o n c o e f f i c i e n t . b e t w e e n s e c t i o n s one and two f o r t h e s e c o n d and t h i r d a d m i n i s t -r a t i o n s was .52 and .56 r e s p e c t i v e l y . F o r M a t h e m a t i c s 113 t h e c o r r e l a t i o n was .48. .The c o r r e l a t i o n b e t w e e n t h e s p r i n g g r a d e s and p a r t two was . 6 7 . TABLE I - ANALYSIS OF RESULTS 15. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Run #2 • Run #3 Math 12 27#=21.8J • 27% = 31. 59 <D ft ft & ' 0 o o fr- CM t— a u =tfc H =«= oj o o OJ OJ OJ o o pq n H H -=f H 00 H fH u S-i IH CU +J cd CQ UN H „ H H . i H OJ o • (U H Q) H CU H CD H o • P3 -H J- -p -=t -P Vlcvi II CO A| || cd o is OJ ft CM is ro ft CO cd fl -H !H 0) pq II cd II cd O II O II O fl H CU O Ii ft II ;0 II ft II H U S O 0) -p . S3 S 0 S • =te a fl =te PL, CQ J (3 D a PH CO PS ft to H 1 -.8 37 5 2 20 1 12 20 - 2 . 3 4 6 2 7 27 10 1 16 2 9 19 11 30 3 .372 3 32 . 34 11 3 20 10 11 18 - 10 . 310 4 hi 2 9 11 2 16 4 18 18. - D 5 21 31 !3 0 18 6 6 17 • 8 28 6 • 459 6 28 2 9, 11 0 18 8 . 7 21 14 31 8 . 4 4 8 7 2 . 13 5 1 7 M7 13 18 - 7 .189 D 8 9 42 19 2 21 M3 2 . 14 14 30 4' .458 M5 3 15 - 5 .378 9 6 22 5 3 14 9 16 21 - 9 .460 10 20 22 8 3 11 11 11 18 - 11 . 343 11 0 0 0 0 0 15 0 0 0 0 15 -. 019 12 0 0 0 0 0 20 0 0 0 0 20 .079 13 0 2 0 0 2 34 0 0 0 0 31 • 277 14 0 0 0 0 0 39 0 0 0 ' 0 36 .242 15 1 3 1 1 1 D D = i t e m D r o p p e d M C n i t e m M o d i f i e d C o r r e c t number o f s u b j e c t s TABLE I (Continued) 1 6 . 1 2 3 k 5 6 7 8 9 1 0 1 1 1 2 1 3 1 6 1 9 2 9 1 1 3 1 5 23. 1+ 5 - D 1 7 k 3 0 0 3 1*5 0 0 2 1 8 k2 .1*62 1 8 1 0 0 0 0 3 0 . 0 k 0 13 2 7 . 332 19 3 6 1 0 5 D 2 0 0 2 0 0 2 D 2 1 0 0 0 0 0 D 2 2 2 0 0 0 0 M22 . 1 1 3 5 1 0 . 2 2 . 1 7 5 2 3 1 0 0 0 0 Mkk 0 0 2 1 5 i n . 1*07 2k 5 13 2 0 1 1 1 7 •lU 1 6 1 2 . 2 8 1 7 . 3 9 1 2 5 0 2 2 6 2 Ik 1 8 5 1 5 8 2 9 1 8 .1*16 2 6 0 6 1 0 5 M31 ' 0 7 1 2 k 2 8 . 3 7 6 M37 0 3 1 8 2 6 3k . 3 8 0 2 7 0 3 0 0 3 M27. 3 1 1 - 2k .511* 28. 0 0 0 0 0 D D 2 9 0 8 2 0 6 M2 9. 1 1 6 5. 2 0 2 6 .3.1*8 30 7 . 5 0 1 k M U 3 0 5 1 0 1 2 1*0 . 1 1 0 31 lU 2 9 1 0 1 1 8 1 6 13 1 7 - 1 6 . 2 1 2 32 1 3 1 1 1 32 0 2 3 ik 29. . 1*17 3 3 2 8 0 2. 6 13 8 1 6 3; •17 13 .1*1*0 3 k 1 3 0 1 2 MlU 0 8 - lU .H-97 35 1 0 13 0 2 1 1 1 9 1 1 1 8 2 9 19 . 3 52 D = item Dropped M = item M o d i f i e d C = Correct n = number of sub j e c t s TABLE I ( C o n t i n u e d ) 17 1 2 3 4 5 6 7 8 9 10 11 12 1 3 36 1 5 0 1 • 4 33 0 10 4 25 30 .382 37 0 3 0 0 3 D 38 2 8 0 0 8 38 2 12 1 3 31 3 5 . 298 39 37 25 4 2 19 12 18 21 19 29 1 2 .158 4o 8 14 3 1 10 26 2 15 8 22 23 .388 4 l 0 3 1 1 1 2 4. 0 6 D 42 2 1 33 13' 1 2 1 2 1 5 16 12 30 M21* . 209 h3 3 15 0 1 . 14 46 0 5 11 23 • 43 .314 44 0 15 2 0 11 4 i 0 . 6 6 22 38 .403 45 9 37 16 3 18 D 46 1 22 1 2 2 2 28 8 19 M25* .258 47 2 2 0 0 . 2 D 48 3 36 15 3 18 D 49 0 . 1 0 0 0 10 36 0 . • 4 5 2 1 33 .367 50 . 3 11 5 0 6 D 51 2 19 0 "3 16 2 5 1 6 D 52 0 26 1 2 2 12 D 53 ' 0 0 0 0 0 . M35 0 5 7 26 32 .340 •54 0 0 0 0 0 M4O 0 . 1 0 .13 •37 . 329 5 5 0 - 4 0 0 ' 4 D 56 0 3 0 0 3 D 5 7 0 1 9 0 0 19 • 42 0 . 3 1 11 39 ,447 D M * C n i t e m D r o p p e d i t e m M o d i f i e d a c c i d e n t a l l y m o d i f i e d C o r r e c t number o f s u b j e c t s 1 8 . BIBLIOGRAPHY A n d e r s o n , James and R o b e r t G. W h l t t m o r e . " P r e d i c t i v e U t i l i t y o f C e r t a i n C r i t e r i a f o r A d v a n c e d F r e s h m a n M a t h e m a t i c s C o u r s e s , " The M a t h e m a t i c s T e a c h e r , LX ( O c t o b e r , 1 9 6 - 7 ) , 6 1 9 - 2 0 . B o y c e , R.W. and R.C. P a x s o n . " P r e d i c t i v e V a l i d i t y o f E l e v e n T e s t s a t One S t a t e C o l l e g e , " E d u c a t i o n a l and P s y c h o l o g i c a l M e a s u r e m e n t , XXV ( W i n t e r 1 9 6 - 5 ) , l l l * 3 - l * 7 . C o d d i n g t o n , E.A. " S c h o l a s t i c A p t i t u d e T e s t s i n M a t h e m a t i c s , " The A m e r i c a n M a t h e m a t i c a l M o n t h l y ; LXX ( S e p t e m b e r , 1 9 6 3 ) , 7 5 0 - 5 5 . K i n g , W.H. " D i f f e r e n c e s i n M a t h e m a t i c a l A c h i e v e m e n t R e l a t e d t o T ypes o f S e c o n d a r y S c h o o l s and T h e i r G e o g r a p h i c a l L o c a t i o n s , " E d u c a t i o n a l . R e s e a r c h , V I I I (November, 1 9 - 6 - 5 ) , 7 ^ - 8 0 . K i n z e r , J.R. and L.G. K i n z e r . " P r e d i c t i n g G r a d e s i n A d v a n c e d C o l l e g e M a t h e m a t i c s , " J o u r n a l o f A p p l i e d P s y c h o l o g y , X X X V I I ( J u n e , 1 9 5 3 ) , 182-81*. Wampler, J o e F. " P r e d i c t i o n o f A c h i e v e m e n t i n C o l l e g e M a t h e m a t i c s , " The M a t h e m a t i c s T e a c h e r , L I X ( A p r i l , 1 9 6 - 6 ) , 3 6 U - 9 . APPENDIX 20. A TEACH-TEST ITEM R a d i c a l s o f t h e same o r d e r w h i c h have t h e same r a d i c a n d when e x p r e s s e d i n t h e s i m p l e s t f o r m a r e c a l l e d l i k e r a d i c a l s . Hence- 7 / 3 and - 2 / 3 a r e l i k e r a d i c a l s . The sum o f two l i k e r a d i c a l s c a n be w r i t t e n w i t h a s i n g l e r a d i c a l . The sum o f t w o . u n l i k e r a d i c a l s , e a c h o f w h i c h i s w r i t t e n i n i t s s i m p l e s t f o r m , c a n n o t be e x p r e s s e d w i t h one r a d i c a l . E x a m p l e s: ( i ) 5 / 2 + 7 / 2 = ( 5 + 7 ) / 2 = 1 2 / 2 ( i i ) 7 / 2 7 - 3 / 3 +• U-/75 = 7 * 3 / 3 - 3 / 3 + I+ . 5 / 3 = 2 1 / 3 - 3/1 + 2 0 / 3 = ( 2 1 - 3 + 2 0 ) / 3 = 3 8 / 3 ( i i i ) 7 / 3 + 5 / 2 = 7 / 3 + 5>/2 ( i v ) 2 / 3 + 2 / 5 " = 2 ( / 3 + / 5 ) . S i m p l i f y - t h e f o l l o w i n g s t a t e m e n t s : 1 ) / 2 + 3 / 2 3 ) 2 / 2 8 " - 5/"i+T + 5 / 6 3 " 2 ) - 7 / 3 + 2 / 1 2 

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