THE DEVELOPMENT AND NORMING OF THE DELTA TESTS OF MATHEMATICS SKILLS By JOHN MARTIN CUMMINS B.A., U n i v e r s i t y of Western Ontario, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n v THE FACULTY OF GRADUATE STUDIES The Measurement, Evaluation and Research Methodology Programme Department of Educational Psychology and Special Education We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1988 cj John Martin Cummins, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of cA t^c^oi^-t r^L cj c A c I o j «* al The University of British Columbia P Vancouver, Canada Date ^&p-\~-1 TO D-6. G r a d e 2 G r a d e 3 G r a d e 4 A d d i t i o n F a c t s .77 .74 .70 A d d i t i o n C o m p u t a t i o n s .72 . 8 1 .66 S u b t r a c t i o n F a c t s .86 .80 .80 S u b t r a c t i o n C o m p u t a t i o n s .73 .80 .80 M u l t i p l i c a t i o n F a c t s .77 .84 M u l t i p l i c a t i o n C o m p u t a t i o n s .76 .80 D i v i s i o n F a c t s .85 D i v i s i o n C o m p u t a t i o n s .59 6 0 I n a d d i t i o n t o c a l c u l a t i n g r e l i a b i l i t y c o e f f i c i e n t s f o r i n d i v i d u a l t e s t s a t e a c h g r a d e l e v e l , e a c h s t u d e n t ' s m a t h e m a t i c s f a c t s s c o r e s a n d m a t h e m a t i c s c o m p u t a t i o n s k i l l s s c o r e s w e r e t o t a l e d f o r e a c h s e s s i o n a n d t e s t - r e t e s t r e l i a b i l i t i e s w e r e c a l c u l a t e d f o r t h e s e t o t a l s c o r e s ( T a b l e s D-7 t o D-9). R e l i a b i l i t y c o e f f i c i e n t s r a n g i n g b e t w e e n .72 a n d .94 w e r e o b t a i n e d . T h e d i s t r i b u t i o n o f 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 s i s s h o w n i n F i g u r e 4. T a b l e 1 0 g i v e s t h e r e l i a b i l i t y c o e f f i c i e n t s f o r e a c h g r o u p i n g w h i c h w e r e o b t a i n e d b y a v e r a g i n g t h e c o r r e l a t i o n s a m o n g t h e s c o r e s o b t a i n e d o n t h e f o u r a d m i n i s t r a t i o n s o f t h e t e s t s . T h e e f f e c t o f t i m e o n t h e m a g n i t u d e o f t h e c o r r e l a t i o n s f o r t h e t o t a l m a t h e m a t i c s f a c t s c o r r e c t a n d t o t a l m a t h e m a t i c s c o m p u t a t i o n s s k i l l s c o r r e c t i s m a d e e v i d e n t i n T a b l e 1 1 . I n a l l c a s e s 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 s o f t h e t o t a l s t a t i s t i c s a r e s e e n t o d e c r e a s e i n s i z e o v e r t i m e . W h i l e t h e d e c r e a s e f o r a n y o n e g r o u p i n g o f t e s t s a t a n y p a r t i c u l a r g r a d e l e v e l ( e . g . T o t a l C o m p u t a t i o n s S k i l l s , G r a d e 4) m a y n o t b e l a r g e e n o u g h t o b e s i g n i f i c a n t b y t h e m s e l v e s , t h e t r e n d f o r a l l g r o u p i n g s f o r a l l g r a d e s w o u l d a p p e a r t o b e t h a t c o r r e l a t i o n c o e f f i c i e n t s d e c r e a s e o v e r t i m e . T h i s p e r m i t s u s t o s a y , f o r e x a m p l e , t h a t t o t a l t e s t r e s u l t s f r o m S e s s i o n 1 a r e b e t t e r a b l e t o p r e d i c t t o t a l t e s t s c o r e s i n S e s s i o n 2 t h a n t h e y a r e i n S e s s i o n 3 a n d b e t t e r a b l e t o p r e d i c t t o t a l t e s t s c o r e s i n S e s s i o n 3 t h a n i n S e s s i o n 4. A n e x a m i n a t i o n o f t h e t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s c a l c u l a t e d b e t w e e n S e s s i o n s 1 a n d 2, 1 a n d 3, a n d 1 a n d 4 f o r 61 FIGURE 4 DISTRIBUTION OF THE TEST-RETEST RELIABILITY COEFFICIENTS CALCULATED BETWEEN SESSIONS 1 AND 2, 2 AND 3 AND 3 AND 4 FOR TOTAL SCORES CORRECT AS REPORTED IN TABLES D-7, D-8 AND D-9. Frequency 8 7 6 - -5 0 0 4 0 0 3 X 0 -2 X X -I X X X 0 .76- .80 .81- .85 .86- .90 .91- .95 C o r r e l a t i o n C o e f f i c i e n t s Grade 2 - 'X* Grade 3 - '0 ' Grade 4 - '-' 62 TABLE 10 RELIABILITY COEFFICIENTS FOR TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS SKILLS CORRECT OBTAINED BY AVERAGING THE CORRELATIONS CALCULATED BETWEEN SESSIONS 1 AND 2, 2 AND 3 AND 3 AND 4 AS REPORTED IN TABLES D-7 TO D-9. G r a d e 2 G r a d e 3 G r a d e 4 T o t a l M a t h e m a t i c s F a c t s C o r r e c t .82 .87 .92 T o t a l C o m p u t a t i o n s S k i l l s C o r r e c t .88 .88 .86 63 TABLE 11 TEST-RETEST RELIABILITY COEFFICIENTS BETWEEN SESSIONS 1 AND 2, 1 AND 3, AND 1 AND 4 SCORES FOR TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS SKILLS CORRECT AS REPORTED IN TABLES D-7 TO D-9. T e s t - R e t e s t R e l i a b i l i t y Between S e s s i o n s Total Mathematics Facts Correct Grade 1 and 2 1 and 3 1 and 4 2 3 4 .79 .85 .88 .74 .84 .84 .72 .80 .81 Total Mathematics Computations S k i l l s Correct Grade 1 and 2 1 and 3 1 and 4 2 3 4 .84 .83 .83 .77 .80 .82 .74 .76 .80 6 4 i n d i v i d u a l f u n c t i o n s ( T a b l e 1 2 ) r e v e a l s t h a t w i t h t h e e x c e p t i o n o f G r a d e 2 A d d i t i o n F a c t s a n d C o m p u t a t i o n S k i l l s , t h e a b i l i t y t o p r e d i c t p e r f o r m a n c e o n t h e i n d i v i d u a l t e s t s o f t h e DTMS f o r G r a d e s 2 a n d 3, d e c r e a s e s o v e r t i m e . T h i s t r e n d i s n o t e v i d e n t f o r t h e i n d i v i d u a l t e s t s a t t h e G r a d e 4 l e v e l . R e l i a b i l i t y and Validity; Discussion A s p r e v i o u s l y n o t e d , " t h e c o n c e p t s o f r e l i a b i l i t y a n d v a l i d i t y a r e c e n t r a l t o t h e t h e o r y a n d p r a c t i c e o f e d u c a t i o n a l a n d p s y c h o l o g i c a l t e s t i n g " ( H o p k i n s a n d S t a n l e y , 1 9 8 1 ; p . 1 1 3 ) . A s d e t a i l e d i n C h a p t e r 3, t h e DTMS w a s c a r e f u l l y c o n s t r u c t e d t o r e f l e c t b o t h t h e B . C . M a t h e m a t i c s C u r r i c u l u m a n d t h e m a n n e r i n w h i c h t h i s c u r r i c u l u m i s e v i d e n c e d i n t h e i n s t r u c t i o n a l p r a c t i c e s o f D e l t a ' s t e a c h e r s . I t s c u r r i c u l a r a n d i n s t r u c t i o n a l v a l i d i t y w a s e n s u r e d b y i n c l u d i n g i n i t o n l y i t e m t y p e s w h i c h h a d b e e n i d e n t i f i e d a n d a p p r o v e d b y e x p e r i e n c e d t e a c h e r s w h o w e r e t h e n e n r o l l i n g c l a s s e s a t t h e g r a d e s i n q u e s t i o n . T h e r e l i a b i l i t y o f t h e DTMS h a s b e e n d e m o n s t r a t e d b y u s e o f t h e t e s t - r e t e s t p r o c e d u r e s a n d r e s u l t s r e p o r t e d i n t h e p r e v i o u s s e c t i o n . A s e v i d e n c e d i n T a b l e 1 0 , t h e t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s f o r t h e t o t a l m a t h e m a t i c s f a c t s c o r r e c t a n d t o t a l m a t h e m a t i c s c o m p u t a t i o n s k i l l s e i t h e r e x c e e d o r a p p r o a c h t h e .90 r e c o m m e n d e d b y H o p k i n s a n d S t a n l e y ( 1 9 8 1 ) f o r s t a n d a r d i z e d a c h i e v e m e n t t e s t s . C o n s i d e r e d i n l i g h t o f t h e a s s e r t i o n b y G h i s e l l i , C a m p b e l l a n d Z e d e c k ( 1 9 8 1 ) t h a t "we e x p e c t l o w e r a n d l o w e r e s t i m a t e s o f r e l i a b i l i t y a s t h e t i m e i n t e r v a l b e t w e e n t h e t e s t i n g o c c a s i o n s i n c r e a s e s " ( p . 2 4 9 ) t h e a c h i e v e m e n t o f t e s t -TABLE 12 TEST-RETEST RELIABILITY COEFFICIENTS BETWEEN SESSIONS 1 AND 2, 1 AND 3, AND 1 AND 4 MATHEMATICS FACTS AND MATHEMATICS COMPUTATIONS SKILLS SCORES FOR EACH FUNCTION TESTED AS REPORTED IN TABLES D-l TO D-6. Mathematics Facts T e s t - R e t e s t R e l i a b i l i t y B e t w e e n S e s s i o n s • a d e F u n c t i o n 1 a n d 2 1 a n d 3 1 a n d 2 A d d i t i o n . 7 5 .70 .70 2 S u b t r a c t i o n . 8 3 .76 .74 3 A d d i t i o n .64 .62 . 5 7 3 S u b t r a c t i o n . 7 9 .76 .75 3 M u l t i p l i c a t i o n . 7 0 . 6 5 . 5 9 4 A d d i t i o n .50 .59 .54 4 S u b t r a c t i o n . 71 . 7 3 . 7 1 4 M u l t i p l i c a t i o n .82 .80 .80 4 D i v i s i o n . 80 .77 . 7 3 Mathematics Computations S k i l l s a d e F u n c t i o n 1 a n d 2 1 a n d 3 1 a n d 2 A d d i t i o n . 6 9 .62 . 6 3 2 S u b t r a c t i o n . 6 5 . 5 9 .54 3 A d d i t i o n . 7 9 .70 . 6 5 3 S u b t r a c t i o n .70 .68 .63 3 M u l t i p l i c a t i o n . 67 . 5 5 .46 4 A d d i t i o n .60 . 5 9 . 5 9 4 S u b t r a c t i o n .74 . 7 3 .67 4 M u l t i p l i c a t i o n .72 .65 .68 4 D i v i s i o n . 50 .38 . 2 5 66 r e - t e s t r e l i a b i l i t y c o e f f i c i e n t s o f t h e m a g n i t u d e o f t h o s e r e p o r t e d a b o v e s u g g e s t s t h a t a s t u d e n t ' s t o t a l m a t h e m a t i c s f a c t s s c o r e a n d h i s / h e r t o t a l m a t h e m a t i c s c o m p u t a t i o n s k i l l s s c o r e m a y b e q u i t e r e l i a b l e . T h e t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s f o r e a c h t e s t o f t h e DTMS a r e r e p o r t e d i n T a b l e 9. F o r t h e m o s t p a r t , t h e s e c o r r e l a t i o n s a r e s e e n t o b e s u b s t a n t i a l , t h o u g h s m a l l e r t h a n t h o s e o b t a i n e d f o r t h e " t o t a l ' s c o r e s . T h i s d e c r e a s e i n r e l i a b i l i t y i s m o s t l i k e l y a t t r i b u t a b l e t o t h e r e d u c e d s i z e o f t h e b e h a v i o r a l s a m p l e a v a i l a b l e w h e n c o n s i d e r i n g t h e r e s u l t s o f o n e t e s t . F o r e x a m p l e , a s t u d e n t ' s s c o r e o n t h e G r a d e 4 d i v i s i o n f a c t s t e s t r e p r e s e n t s o n l y o n e - q u a r t e r o f h i s / h e r t o t a l m a t h e m a t i c s f a c t s s c o r e . T h e e f f e c t o f l e n g t h e n i n g t h e d i v i s i o n t e s t c a n b e e s t i m a t e d u s i n g t h e S p e a r m a n - B r o w n f o r m u l a : r\*K = k r > x / <1 + 6 - 3 - 1 2 - U - 1 1 - ? 1 3 - 7 - 7 - 5 • 1 JI - 7 1 2 - 3 - 1 - 1 - 8 - fi 1 0 - C - 9 - 7 - 1 5 - 7 n - 9 • 1 3 - 5 - K . Ii . q - i - 1 6 - 8 - n . H , 1 7 - 9 - 7 - 1 - ' R . Ii 6 - 5 - . 11 - 8 -132 1 x S - . 5 x 9 -3 x 3 -8 x 9 -6 x 3 -9 x 5 -2 x 3 -Hill Tl PI I fA™" FAfTf\ 9 x 8 - | 2 x 1 -4 x 7 -5 x 2 -7 x. 6 -7 x 7 -3 x 9 -8 x 2 -7 x 9 -6 x 1 -i\ x 1 -5 x 8 -2 x 2 - . 9 x 6 -3 x 6 -7 x 2 -8 x 1 -1 x C -5 x 5 -4 x 3 -2 x 7 -8 x 7 - . 2 x 9 -9 x 3 -3 x 5 " 6 x 9 ' 6 x 7 - . 9 x 2 - . 7 x 1 m q x 5 -5 x 6 " 8 x 3 " 3 x 7 - . 2 x 5 - . 6 x S -H x 1 -5 x 3 -3 x 8 -8 x 6 " 7 x 5 ' 9 x 7 - . 5 x 3 -7 x 8 -6 x 6 -3 x 4 ' 2 x 6 ' 9 x 9 8 x 5 H x t ° -1 x 9 " . 7 x 8 -3 x 2 -5 x 1 -2 x 8 -0 x 7 ' 6 x 5 9 x 1 " 1 x 2 - . 7 x 3 -6 x 2 -5 x 7 -133 1 0 6 3 1 G t 2 2 1 3 2 1 8 1 1 7 2 8 H 8 2 7 5 k 3 0 2 1 1 2 3 C C 3 5 1 2 H 2 5 8 5 «4 D1V1S1M FACTS TO 31 2 1 1 8 2 0 3 0 C 1 0 1 0 1 6 1 2 2 8 6 1 2 1 2 8 1 1 5 6 5 6 4 7 3 6 i 6 2 1 f 7 2 5 ^ 5 2 8 r H 3 5 f 7 1 5 •J 1 8 6 3 2 0 1 0 2 8 1 8 5 M 4 0 2 H 2 7 8 7 2 3 6 1 4 H 5 3 2 4 9 1 6 8 1 8 f 9 r H • 7 8 f 2 i 9 1 3 4 Section III L i s t of Computation S k i l l s Taught at Each Grade Level. G r a d e "2_ A d d i t i o n 1 - C o m p u t e s u m s 2 + 4 + 2 2 - C o m p u t e s u m s 2 4 + 2 3 - M i s s i n g a d d e n d 4 + 4 - M i s s i n g a d d e n d 4 + 5 - N o r e g r o u p i n g - 1 1 2 - d i g i t + 1 - d i g i t + 2 6 - N o r e g r o u p i n g - 1 4 2 - d i g i t + 2 - d i g i t + 24 7 - R e g r o u p i n g l ' s 2 4 2 - d i g i t + 1 - d i g i t + 6 8 - R e g r o u p i n g l ' s 2 7 2 - d i g i t + 2 - d i g i t + 24 G r a d e 2 S u b t r a c t i o n 1 - M i s s i n g n u m e r a l 9 - = 4 - 4 = 5 2 - M i s s i n g n u m e r a l 9 - 5 4 4 3 - N o r e g r o u p i n g -2 - d i g i t - 1 - d i g i t 2 7 - 6 G r a d e 2 S u b t r a c t i o n ( C o n t ' d ) 4 - No r e g r o u p i n g -2 - d i g i t - 2 - d i g i t 5 - R e g r o u p i n g 1 0 * s -2 - d i g i t - 1 d i g i t 6 - R e g r o u p i n g 1 0 ' s G r a d e 3_ A d d i t i o n 1 - C o m p u t e s u m s t o 1 8 2 - C o m p u t e s u m s t o 1 8 3 - C o m p u t e s u m s 18+ 4 - M i s s i n g a d d e n d 5 - W i t h o u t r e g r o u p i n g -2 - d i g i t + 2 - d i g i t 6 - W i t h r e g r o u p i n g l ' s o n l y 2 - d i g i t + 2 - d i g i t 7 - W i t h r e g r o u p i n g - 1 0 ' s o n 2- d i g i t + 2 - d i g i t 8 - R e g r o u p i n g l ' s o r 1 0 ' s 3- d i g i t + 2 / 3 - d i g i t s 9 - R e g r o u p i n g l ' s , 1 0 ' s , 1 0 0 ' 3 - d i g i t + 2 / 3 - d i g i t s i n c o l u m n o f 3 n u m b e r s G r a d e 3 S u b t r a c t i o n 136 1 - N o r e g r o u p i n g 2 - d i g i t - 1/2 d i g i t s 32 - 2 20 - 10 2 - No r e g r o u p i n g 3 - d i g i t s - 1, 2, o r 3 426 313 158 - 17 279 - 8 3 - R e g r o u p i n g 10's 2 - d i g i t - 1 - d i g i t 53 4 4 - R e g r o u p i n g 10's 2 - d i g i t - 2 - d i g i t 36 19 5 - R e g r o u p i n g 10's 3 - d i g i t - 2/3 d i g i t 182 - 69 582 - 166 6 - R e g r o u p i n g 100's 3 - d i g i t - 2 - d i g i t 628 - 84 7 - R e g r o u p i n g 100's 3 - d i g i t - 3 - d i g i t 439 284 8 - R e g r o u p i n g 10's a n d / o r 100's 3 - d i g i t - 2 - d i g i t 735 86 9 - R e g r o u p i n g 10's a n d / o r 100's 3 - d i g i t - 3 - d i g i t 653 284 G r a d e 3_ M u l t i p l i c a t i o n 1 - B a s i c f a c t s t o 50 9 x 5 2 - B a s i c f a c t s ( m i s s i n g f a c t o r ) 8 x G r a d e 3 M u l t i p i i c a t i o n ( C o n t ' d ) 3 - No r e g r o u p i n g 24 1 - d i g i t m u l t i p l i e r x 2 x 2 - d i g i t s 4 - B a s i c f a c t s t o 5 0 5 x 9 = 5 - B a s i c f a c t s ( m i s s i n g f a c t o r ) 8 x G r a d e j4 A d d i t i o n 1 - R e g r o u p i n g l ' s , 10's, 100's 8 6 6 3 - d i g i t + 3 - d i g i t + 2 4 3 2 - R e g r o u p i n g l ' s , 1 0 ' s , 1 0 0 ' s 4 9 3 - d i g i t + 1 , 2, & 3 - d i g i t s 9 6 7 + 8 3 - R e g r o u p i n g l ' s , 1 0 ' s , 1 0 0 ' s 1 2 8 3 - d i g i t + 3 - d i g i t 2 6 4 4 3 2 + 3 5 4 4 - R e g r o u p i n g l ' s , 1 0 ' s , 1 0 0 ' s 2 9 3 - d i g i t + 1, 2, o r 3 - d i g i t 3 5 6 2 1 8 + 9 7 5 - A m i x t u r e o f 1 t o 4 d i g i t s 2 3 4 9 R e g r o u p i n g a m i x t u r e l ' s , + 3 6 8 1 0 ' s , 1 0 0 ' s 6 - A m i x t u r e o f 2 t o 5 d i g i t s 7 3 R e g r o u p i n g a m i x t u r e l ' s , + 5 4 3 2 8 10's, 100's, 1000's G r a d e 4 A d d i t i o n ( C o n t ' d ) 7 - A m i x t u r e o f 2 t o 5 d i g i t s 29644 R e g r o u p i n g a m i x t u r e o f l ' s , 1738 10's, 100's, 1000's 29 + 4667 138 8 - A d d i t i o n o f l i k e f r a c t i o n s N o r e g r o u p i n g , n o r e d u c i n g 1 4 2 + -4 9 - A d d i t i o n o f l i k e f r a c t i o n s _2_ N o r e g r o u p i n g , n o r e d u c i n g 5 JL_ + 5 G r a d e 4 S u b t r a c t i o n 1 - R e g r o u p i n g 10's a n d / o r 100's 808 3 - d i g i t - 2 - d i g i t - 69 2 - R e g r o u p i n g 10's a n d / o r 100's 669 3 - d i g i t - 3 - d i g i t - 378 3 - R e g r o u p i n g a m i x t u r e o f 10*s, 100's 4 - d i g i t - 3 or 4 - d i g i t 9731 6297 5690 - 128 4 - R e g r o u p i n g a m i x t u r e o f 10's, 100's 4 - d i g i t - 3 o r 4 - d i g i t 7574 2651 9698 • 980 5 - R e g r o u p i n g a m i x t u r e o f 10's, 100's, 1000's, 10000's 70230 - 4221 80545 - 39644 G r a d e 4 S u b t r a c t i o n ( C o n t ' d ) 6 - N o r e d u c i n g , s u b t r a c t i o n o f l i k e f r a c t i o n s 7 - N o r e d u c i n g , s u b t r a c t i o n o f l i k e f r a c t i o n s G r a d e 4^ M u l t i p l i c a t i o n 1 - 1 - d i g i t m u l t i p l i e r x 3 - d i g i t s N o r e g r o u p i n g 2 - 1 - d i g i t m u l t i p l i e r x 3 - d i g i t s R e g r o u p i n g l ' s o n l y 3 - 1 - d i g i t m u l t i p l i e r x 3 - d i g i t s R e g r o u p i n g 1 0 ' s o n l y 4 - 1 - d i g i t m u l t i p l i e r x 2 - d i g i t s R e g r o u p i n g l ' s a n d 1 0 ' s 5 - 1 - d i g i t m u l t i p l i e r x 3 - d i g i t s R e g r o u p i n g l ' s , 1 0 ' s 6 - 1 - d i g i t m u l t i p l i e r x 3 - d i g i t s R e g r o u p i n g a m i x t u r e o f l ' s , 1 0 ' s , 1 0 0 ' s 7 - 1 - d i g i t m u l t i p l i e r x 4 - d i g i t s R e g r o u p i n g a m i x t u r e o f l ' s , 1 0 ' s , 1 0 0 ' s 140 Grade 4 M u l t i p l i c a t i o n (Cont'd) 8 - 2 - d i g i t m u l t i p l i e r x 30 2 - d i g i t s x 52 Grade 4 D i v i s i o n 1 - d i g i t d i v i s o r , no r e m a i n d e r 1 -2) 28 2 -4) 448 3 -2) 6448 1 - d i g i t d i v i s o r , w i t h r e m a i n d e r 4 -5) 14 5 -3) 553 6 -2) 8041 141 Section IV The Computations Subtests of the DTMS. *The number i n b r a c k e t s b e s i d e e a c h q u e s t i o n c o r r e s p o n d s t o t h e number o f t h e q u e s t i o n t y p e as i d e n t i f i e d i n S e c t i o n I I I o f t h i s A p p e n d i x . 00 cn 1 0 a- K-\ L O ST C M K > ^ C M CM cn • cn | oo i—i • u L A + C O • L A 00 oo cn LO CSI O l I I A • LO 0 0 • r o L A CN —^' N K \ N • C M O C M cn L A •—I oo CA L A •» • • OO CO •f " 3 = • 1 cn • cn •» o I f l J l \ 4 o o • O0 L A C M cn 143 to 3_ 5." O CD /5 K> OO cn LA i 10 • • L A OO i n cn rvi L A g 3D CM c r c n CNI LO *—' L A • CA CD LD LTV • oo —I i c n L A v - cn en a-i m cn i • LA CO OO o o I r v . CO CO L A CM OO f A I L A CO co a-• cn »A = • 3 GRADE 3 ADDITION COMPUTATIONS (6) 3 8 43 4 8 7 • 5 (3) (2) 8 -(1) (4) 9 • D 1 8 7 7 2 4 5 7 + 19 8 (9) (8) 3 17 3 2 5 6 1 +24 (5) (7) 3 9 • 7 7 (6)9 5 3 9 5 9 • 2 2 + 8 7 6 (9) (1) • (4) (7) 17 6 5 • 7 9 (3) (8) (2) (5) 5 9 9 + 321 • 7 0 + 15 (2) (A) 6 • 5 • 5 6 *\^\* 1 - 1 3 1 2 7 •12 9 (8) 7 5 • 8 (3) 2 8 + 2 (1) (5) (7) 7 6 • 2 1 8 3 + 67 2 8 6 3 2 5 f g 9 (9) (6) 2 8 • 3 5 (5) 5 3 • 2 5 9 1 + 6 (1) (4) + 2 1 5 5 1 1 3 2 1 • 667 (9) 7 6 + 9 (3) 3 6 + 6 5 2 1 5 + 666 (8) 5 7 • 2 8 (6) • (2) 8 8 8 1 6 y 9 7 (9) (5) 1 2 + 3 7 a- (4) (7) (8) (6) 1 3 7 6 • 5 4 6 0 8 + 96 1 7 + 2 6 (2) 8 9 • 6 (3) 1 9 • 5 (1) GRADE 3 SUBTRACTION COMPUTATIONS ( n (7) (3) (2) '(9) (4) (6) (5) (8) 1 9 u ; 9 1 6 u ; 8 1 v ; 815 923 91 202 223 988 -15 -161 -6 -13 -172 -83 -12 -219 - 8 3 7 0 (3) 9 5 2 (2) 1 0 7 (8) , 5 (i) 6 7 (A> 2 7 2 ( 9 ) 5 7 5 ( 7 ) 6 2 8 ( 6 ) 5 5 7 ( 5 ) -6 -121 -18 -3 -18 -133 -281 - 8 1 - 2 8 1 1 0 (9) 7 3 1 (5) 5 0 7 (7) 9 7 O) 6 7 g (8) , 2 6 (2) 7 2 (4) ? 3 (1) , „ 9 (6) -395 -303 -152 - 9 - 9 9 - 5 -56 -32 -72 5 9 3 (5) 1 0 7 (8) 9 0(3) 7 2 3 (6) 8 1 8 <2) 1 1 8 <7) 1 5 2 (9> 5 1 <*> 9 1 (1> -27 -67 -3 -31 -105 -383 -327 -16 -1 8 3 (4) 5 1 6 (8) 8 0 6 (6) 5 0 0 (7) 1 5 8 (2) 3 19 <9> 5 1 1 <5) 5 3 (3) 8 8 (1> -21 -31 -73 -280 -17 -126 -538 -1 -21 GRADE 3 MULTIPLICATION' COMPUTATIONS (1) Q (2) (4) (5) ' (3) (2) 7 -I I 3 X - 2 1 x I x Q 6 x k x k k x T e 3 6 , „ (3) (4) (1) (5) (1) (3) 3 2 . 1 8 8 3 0 x 3 6 x 7 - x 6 x 6 - 2 «f x 5 x 3 2 x [ ] . ] 8 ( 5 ) e X 6 . [ ] w jff (2) r ] x 8 " 3 2 ( 5 ) £1 (2) i j (1) 2«t 28 S x 5 W 2 3 x 3 (3) 2 4 x 2 (3) 3 JO 2 7 (2) (5) H 5 k x 8 • (4) x 6 (1) 147 c o u r. o V =3 OJ m 6 *—^ in + vo a. > vo ^ «• v D m r - l oo fN r-<-t in in f—t oo GO vo o r» c o H H o v n (N i-H fN r-l vo r» rH fN m + 00 r-t IN I*. 00 VO 00 + —^\ fN r» •v fN m n IN + r-l vo m an in + cn s—' CN >N n | r -— + co cn oo fN 10 m r-t + r-t o 00 in vo an r-t + o in "» O in m r » rH + on n IN r -m v O in vo 00 IN an r-r-t 00 r-l + m cn vo an m r-co f r» o m o vo m on on oo m CN an CN vo oo m r m r » » o cn r- vo r~ m o on en «N r-t fN m r-t 00 + o on m m m vo CN i n m OS vo + 00 «* cn rH CN fN rH «—« v O •—-+ r » CO vo cn an CN m IN 00 rH + rH —' rH on rn r~ VO m + «»• m ao vo cn an fN vo r » O rH fN o CN m a + vo m on oo CM on IN m + CO r - m on 00 n an o on r « m on /•—V + 00 Cn s m f c o ,—* U + rH |m * |C0 rH f-m so SO rH o m ao m co in + 10 r -OB c n m IN r - CO 1 OQ on O co to l r—\ CM CO 00 n m lA • n ut M* r~ «*1 CM CO cn (N 1 I O m V ca rH CO so CN cn 1 r-t cn r-l fN CN n m oa on co r» cn r-l l co r~ oo o\ i k v o | c n i co r» cn r> n i I in. co i n r -o cn O an O o oa in I o r-t o r» *r aa co cn l Imkn i-tUn t o cn CO CN in I co r» M * m r» on cn CN I cD I co an o oa co l co en m r-t\ i 149 «-< x m CM m m CN VO >—-CN m 0\ 00 -3-00 CO *r oo CO X 00 M ' m CO VO CO CO • o CN . rH CN CN CO CN H VO o CN r- vo m CN 00 —^' en r-VO N—' n vo cn CN VO VO o CN vO >—' r» m CN m CN n on X oo 00 •J § U 6 a 5 ro r-Cf» CN X f> CN vo VO CN vo r~ on X cn • CN co vo m cn vo m fN m CN CO r- oo in vo •8 ^ in cn in CN CO CN co VO o Grade 4 Division Computations 3 T e o i r ( 3 ) 6 T 9 3 — W 9jn^~(6) 4TTor- ( 5 ) e T W ^ 4 T 6 8 — ( 1 ) iVn—t" ijen 9imrW 7)1848 ( 3 ) 3 T F 5 ( A ) 77ToT- ( 2 ) SjTtW*^ 4 ) 3 9 1 ( 5 ) &JWfZ~*6^ 7JW (1) (2) • _ (4) (2) (3) (1) • (6) 2751 7)602 8;TS <)348 9)7083 3751 7")5Z28" 9T&6T (6) (A) (1) . (2) (5) i (3) 6)4583 9733 6T5H 9 ) 7 2 9 8 ) 6 3 7 9 ) 6 5 7 0 151 APPENDIX C The instructions given to pupils being administered the Delta Tests of Mathematics S k i l l s . S e c t i o n I D i r e c t i o n s : M a t h F a c t s ( C o m p l e t e a n d S h o r t F o r m ) S e c t i o n I I : D i r e c t i o n s : C o m p u t a t i o n s S k i l l s ( C o m p l e t e a n d S h o r t F o r m ) 152 Section I: Directions: Math Facts (Complete) E x a m i n e r : P l e a s e l e a v e y o u r b o o k l e t s c l o s e d a n d l o o k a t m i n e . W h e n I s a y b e g i n , I w a n t y o u t o s t a r t o n t h e f i r s t q u e s t i o n o n t h e t o p l e f t h a n d c o r n e r o f t h e p a g e . [ E x a m i n e r i n d i c a t e s f i r s t q u e s t i o n o n s a m p l e p r o b e ] O n t h i s p a g e d o t h e q u e s t i o n s d o w n t h e p a g e . I f y o u g e t t o t h e b o t t o m o f t h e c o l u m n s t a r t o n t h e n e x t c o l u m n . [ E x a m i n e r d e m o n s t r a t e s ] D o n o t s k i p q u e s t i o n s . L o o k c a r e f u l l y a t e a c h q u e s t i o n . I f y o u c o m e t o a q u e s t i o n y o u a r e s u r e y o u d o n o t k n o w h o w t o d o y e t , p u t a . b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . [ E x a m i n e r d e m o n s t r a t e s ] E x a m i n e r : O n t h i s p a g e y o u d o t h e q u e s t i o n s d o w n t h e p a g e . W h a t d o y o u d o i f y o u c o m e t o a q u e s t i o n y o u d o n o t k n o w h o w t o d o y e t ? S t u d e n t s : P u t a b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . E x a m i n e r : A l l t h e q u e s t i o n s o n t h i s p a g e a r e ( 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 / d i v i s i o n ) q u e s t i o n s . W h a t k i n d o f q u e s t i o n s a r e y o u d o i n g o n t h i s p a g e ? S t u d e n t s : ( 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 / d i v i s i o n ) q u e s t i o n s . E x a m i n e r : Y o u w i l l h a v e o n e m i n u t e t o w o r k o n t h e s e q u e s t i o n s . Y o u a r e n o t e x p e c t e d t o b e a b l e t o c o m p l e t e t h e p a g e i n t h e t i m e y o u h a v e B U T y o u a r e e x p e c t e d t o w o r k a s f a s t a s y o u c a n . I f , h o w e v e r , y o u d o a l l t h e q u e s t i o n s y o u c a n b e f o r e t i m e i s u p , c h e c k y o u r w o r k o n t h i s p a g e B U T d o n o t t u r n t o a n o t h e r p a g e . A r e y o u e x p e c t e d t o f i n i s h t h e p a g e ? 1 5 3 S t u d e n t s : N o . E x a m i n e r : M a y y o u t u r n t o a n o t h e r p a g e ? S t u d e n t s : N o . E x a m i n e r : A r e y o u e x p e c t e d t o w o r k a s f a s t a s y o u c a n a n d d o y o u r b e s t w o r k ? S t u d e n t s : Y e s . E x a m i n e r : R e m e m b e r , w o r k q u i c k l y a n d d o y o u r b e s t w o r k . D o n o t s k i p q u e s t i o n s . D o e a c h q u e s t i o n o n e a f t e r t h e o t h e r d o i n g d o w n t h e p a g e a n d i f y o u c o m e t o a q u e s t i o n y o u c a n n o t d o y e t p u t a b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . A s w e l l , r e m e m b e r t h i s i s a ( 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 / d i v i s i o n ) p a g e . W h a t k i n d o f p a g e i s t h i s ? S t u d e n t s : ( 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 / d i v i s i o n ) p a g e . E x a m i n e r : P l e a s e t u r n t o p a g e 1 . [ W h e n t h e c h i l d r e n h a v e t u r n e d t o t h e c o r r e c t p a g e ] E x a m i n e r : Y o u m a y b e g i n . 154 Directions: Math Facts (Short Form) E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : [ W h e n t h e E x a m i n e r : T h e n e x t p a g e i s a ( 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 / d i v i s i o n ) p a g e . O n t h i s p a g e d o t h e q u e s t i o n s d o w n t h e p a g e . Y o u w i l l h a v e o n e m i n u t e t o w o r k o n i t . A r e y o u e x p e c t e d t o f i n i s h t h e p a g e ? N o . M a y y o u t u r n t o a n o t h e r p a g e ? N o . A r e y o u e x p e c t e d t o w o r k a s f a s t a s y o u c a n a n d d o y o u r b e s t w o r k ? S t u d e n t s : Y e s . E x a m i n e r W h a t d o y o u d o w h e n y o u c o m e t o a q u e s t i o n y o u c a n ' t d o y e t ? P u t a b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . W h a t k i n d o f t e s t i s t h i s ? ( 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 d i v i s i o n ) p a g e . P l e a s e t u r n t o p a g e . c h i l d r e n h a v e t u r n e d t o t h e c o r r e c t p a g e ] Y o u m a y b e g i n . 155 Section II; Directions; Computation S k i l l s (Complete) E x a m i n e r : P l e a s e l e a v e y o u r b o o k l e t s c l o s e d and l o o k a t m i n e . When I s a y b e g i n , I want y o u t o s t a r t on t h e f i r s t q u e s t i o n on t h e t o p l e f t hand c o r n e r o f t h e p a g e s . [ E x a m i n e r i n d i c a t e s f i r s t q u e s t i o n on s a m p l e p r o b e ] On t h i s page do t h e q u e s t i o n s a c r o s s t h e p a g e . I f y o u g e t t o t h e end o f t h e row s t a r t on t h e n e x t r o w . [ E x a m i n e r d e m o n s t r a t e s ] Do n o t s k i p q u e s t i o n s . Look c a r e f u l l y a t e a c h q u e s t i o n . I f y o u come t o a q u e s t i o n y o u a r e s u r e y o u do n o t know how t o do y e t , p u t a b i g " X " on i t and go on t o t h e n e x t q u e s t i o n . [ E x a m i n e r d e m o n s t r a t e s ] E x a m i n e r : On t h i s page y o u do t h e q u e s t i o n s a c r o s s t h e p a g e . What do y o u do i f y o u come t o a q u e s t i o n y o u do n o t know how t o do y e t ? S t u d e n t s : P u t a b i g " X " on i t and go on t o t h e n e x t q u e s t i o n . E x a m i n e r : A l l t h e q u e s t i o n s on t h i s page a r e ( 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 / d i v i s i o n ) q u e s t i o n s . What k i n d o f q u e s t i o n s a r e y o u d o i n g on t h i s p a g e ? S t u d e n t s : ( 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 / d i v i s i o n ) q u e s t i o n s . E x a m i n e r : You w i l l h a v e f i v e m i n u t e s t o work on t h e s e q u e s t i o n s . You a r e n o t e x p e c t e d t o be a b l e t o c o m p l e t e i n t h e t i m e y o u h a v e BUT y o u a r e e x p e c t e d t o w o r k a s f a s t a s y o u c a n . I f , h o w e v e r , y o u do a l l t h e q u e s t i o n s y o u c a n b e f o r e t ime, i s u p , c h e c k y o u r wo rk on t h i s page BUT do n o t t u r n t o a n o t h e r p a g e . A r e y o u e x p e c t e d t o f i n i s h t h e p a g e ? S t u d e n t s : N o . E x a m i n e r : May y o u t u r n t o a n o t h e r p a g e ? S t u d e n t s : N o . 156 E x a m i n e r : A r e y o u e x p e c t e d t o w o r k a s f a s t a s y o u c a n a n d d o y o u r b e s t w o r k ? S t u d e n t s : Y e s . E x a m i n e r : R e m e m b e r , w o r k q u i c k l y a n d d o y o u r b e s t w o r k . D o n o t s k i p q u e s t i o n s . D o e a c h q u e s t i o n o n e a f t e r t h e o t h e r g o i n g a c r o s s t h e p a g e a n d i f y o u c o m e t o a q u e s t i o n y o u c a n n o t d o y e t p u t a b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . A s w e l l , r e m e m b e r t h i s i s a 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 / d i v i s i o n ) p a g e . W h a t k i n d o f p a g e i s t h i s ? S t u d e n t s : ( A d d i t i o n / s u b t r a c t i o n / r o u l t i p l i c a t i o n / d i v i s i o n ) p a g e . E x a m i n e r : P l e a s e t u r n t o P a g e 2. [ W h e n t h e c h i l d r e n h a v e t u r n e d t o t h e c o r r e c t p a g e ] E x a m i n e r : Y o u m a y b e g i n . 157 D i r e c t i o n s : C o m p u t a t i o n S k i l l s ( S h o r t F o r m ) E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : S t u d e n t s : E x a m i n e r : [ W h e n t h e E x a m i n e r : T h e n e x t p a g e i s a J 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 / d i v i s i o n ) p a g e . O n t h i s p a g e d o t h e q u e s t i o n s a c r o s s t h e p a g e . Y o u w i l l h a v e f i v e m i n u t e s t o w o r k o n i t . A r e y o u e x p e c t e d t o f i n i s h t h e p a g e ? N o . M a y y o u t u r n t o a n o t h e r p a g e ? N o . A r e y o u e x p e c t e d t o w o r k a s f a s t a s y o u c a n a n d d o y o u r b e s t w o r k ? Y e s . W h a t d o y o u d o w h e n y o u c o m e t o a q u e s t i o n y o u c a n ' t d o y e t ? P u t a b i g "X" o n i t a n d g o o n t o t h e n e x t q u e s t i o n . W h a t k i n d o f t e s t i s t h i s ? ( 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 d i v i s i o n ) q u e s t i o n s . P l e a s e t u r n t o p a g e . c h i l d r e n h a v e t u r n e d t o t h e c o r r e c t p a g e ] Y o u m a y b e g i n . 158 APPENDIX D Descriptive S t a t i s t i c s of Individual Functions. S e c t i o n I : T e s t - R e t e s t R e l i a b i l i t y C o e f f i c i e n t s B e t w e e n S e s s i o n s . S e c t i o n I I ; T h e S e s s i o n a l M e a n s , S t a n d a r d D e v i a t i o n s a n d N u m b e r o f V a l i d C a s e s o f t h e I n d i v i d u a l F u n c t i o n s T e s t e d w i t h t h e DTMS a t G r a d e s 2, 3 a n d 4. S e c t i o n I I I : T h e A n a l y s i s o f V a r i a n c e S u m m a r i e s a n d T u k e y ' s H S D ( H o n e s t l y - S i g n i f i c a n t - D i f f e r e n c e ) f o r t h e DTMS T e s t S c o r e s , S e s s i o n s 1 t o 4. S e c t i o n I V : T r e n d A n a l y s i s S u m m a r i e s a n d r , t h e P r o p o r t i o n o f T o t a l V a r i a t i o n A c c o u n t e d f o r b y S i g n i f i c a n t T r e n d s f o r t h e DTMS S e s s i o n a l M e a n s . Section I: Test-ReteBt R e l i a b i l i t y Coefficients Between Sessions. The t e s t - r e t e s t r e l i a b i l i t y c o e f f i c i e n t s b e t w e e n s e s s i o n s we re c a l c u l a t e d u s i n g t h e " P e a r s o n C o r r ' command o f t h e U . B . C . S t a t i s t i c a l P a c k a g e f o r t h e S o c i a l S c i e n c e s e x t e n d e d v e r s i o n ( S P S S x ) . TABLE D-l GRADE 2 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION AND SDBTRACTION FACTS TESTS OF THE DTMS. 160 Addition Facts SESSION 1 2 3 4 1 1.00 .75 .70 .70 2 1.00 .78 .75 3 1.00 .77 4 1.00 Subtraction Facts SESSION 1 2 3 4 1 1.00 .83 .76 .74 2 1.00 .89 .82 3 1.00 .86 4 1.00 TABLE D-2 GRADE 2 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION AND SUBTRACTION COMPUTATION SKILLS TESTS OF THE DTMS. 161 A d d i t i o n Computation S k i l l s SESSION 1 2 3 4 1 1.00 .69 .62 .63 2 1.00 .73 .74 3 1.00 .73 4 1.00 S u b t r a c t i o n Computation S k i l l s SESSION 1 2 3 4 1 1.00 .65 .59 .54 2 1.00 .74 .66 3 1.00 .80 4 1.00 162 TABLE D-3 GRADE 3 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION, SDBTRACTION AND MULTIPLICATION FACTS TESTS OF THE DTMS. A d d i t i o n F a c t s SESSION 1 2 3 4 1 1 .00 .64 .62 . 57 2 1 .00 . 7 9 .74 3 1 .00 . 78 4 1 .00 S u b t r a c t i o n F a c t s SESSION 1 2 3 4 1 1 .00 . 79 .76 . 7 5 2 1 .00 .81 . 77 3 1 .00 . 7 9 4 1 .00 M u l t i p l i c a t i o n F a c t s SESSION 1 2 3 4 1 1 .00 .70 . 65 . 5 9 2 1 .00 . 79 . 75 3 1 .00 . 82 4 1 .00 163 TABLE D-4 GRADE 3 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION, SUBTRACTION AND MULTIPLICATION COMPUTATION SKILLS TESTS OF THE DTMS. A d d i t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1.00 .79 .70 .65 2 1.00 .82 .79 3 1.00 .82 4 1.00 S u b t r a c t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1.00 .70 .68 .63 2 1.00 .83 .80 3 1.00 .86 4 1.00 M u l t i p l i c a t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1.00 .67 , .55 .46 2 1.00 .81 .72 3 1.00 .81 4 1.00 TABLE D-5 GRADE 4 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION, SDBTRACTION, MULTIPLICATION AND DIVISION FACTS TESTS OF THE DTMS. A d d i t i o n F a c t s S E S S I O N 1 2 3 4 1 1.00 .50 .59 .54 2 1.00 .79 .75 3 1.00 .81 4 1.00 S u b t r a c t i o n F a c t s S E S S I O N 1 2 . 3 4 1 1.00 .71 .73 .71 2 1.00 .86 .81 3 1.00 .84 4 1.00 M u l t i p l i c a t i o n F a c t s S E S S I O N 1 2 3 4 1 1.00 .82 .80 .80 2 1.00 .84 .82 3 1.00 .87 4 1.00 D i v i s i o n F a c t s S E S S I O N 1 2 3 4 1 1.00 .80 .77 .73 2 1.00 .88 .85 3 1.00 .86 4 1.00 TABLE D-6 GRADE 4 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES, ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION COMPUTATION SKILLS TESTS OF THE DTMS. A d d i t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1 . 0 0 .60 . 5 9 . 5 9 2 1 . 0 0 .68 .62 3 1 . 0 0 .71 4 . 1 . 0 0 S u b t r a c t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1 . 0 0 .74 . 7 3 . 6 7 2 1 . 0 0 .84 . 7 9 3 1 . 0 0 .82 4 1 . 0 0 M u l t i p l i c a t i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1 . 0 0 .72 .65 .68 2 1 . 0 0 .84 .82 3 1 . 0 0 . 8 3 4 1 . 0 0 D i v i s i o n C o m p u t a t i o n S k i l l s S E S S I O N 1 2 3 4 1 1 . 0 0 .50 .38 . 2 5 2 1 . 0 0 . 6 9 .48 3 1 . 0 0 .57 4 1 . 0 0 166 TABLE D-7 GRADE 2 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES FOR TOTAL MATHEMATICS FACTS AND TOTAL MATHEMATICS COMPUTATION SKILLS SCORES ON THE DTMS. Total Mathematics Facts SESSION 1 2 3 4 1 1.00 .79 .74 .72 2 1.00 .83 .80 3 1.00 .85 4 1.00 Total Mathematics Computations S k i l l s SESSION 1 2 3 4 1 1.00 .84 .77 .74 2 1.00 .90 .84 3 1.00 .89 4 1.00 1 6 7 TABLE D-8 GRADE 3 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES FOR TOTAL MATHEMATICS FACTS AND TOTAL MATHEMATICS COMPUTATION SKILLS SCORES ON THE DTMS. T o t a l M a t h e m a t i c s F a c t s S E S S I O N 1 2 3 4 1 1 . 0 0 .85 .84 .80 2 1 . 0 0 .88 .84 3 1 . 0 0 .87 4 1 . 0 0 T o t a l M a t h e m a t i c s C o m p u t a t i o n s S k i l l s S E S S I O N 1 2 3 4 1 1 . 0 0 . 8 3 .80 .76 2 1 . 0 0 . 8 9 .86 3 1 . 0 0 .91 4 1 . 0 0 168 TABLE D-9 GRADE 4 TEST-RETEST RELIABILITY COEFFICIENTS FOR SESSIONS 1 TO 4 SCORES FOR TOTAL MATHEMATICS FACTS AND TOTAL MATHEMATICS COMPUTATION SKILLS SCORES ON THE DTMS. T o t a l M a t h e m a t i c s F a c t s SESSION 1 2 3 4 1 1 .00 . 88 .84 .81 2 1 .00 . 94 .91 3 1 .00 . 9 3 4 1 .00 T o t a l M a t h e m a t i c s C o m p u t a t i o n s S k i l l s SESSION 1 2 3 4 1 1 .00 . 83 . 82 . 80 2 1 .00 .90 . 8 5 3 1 .00 .86 4 1 .00 1 6 9 Section I I : T h e S e s s i o n a l M e a n s , S t a n d a r d D e v i a t i o n s a n d N u m b e r o f V a l i d C a s e s o f t h e I n d i v i d u a l F u n c t i o n s T e s t e d w i t h t h e DTMS a t G r a d e s 2, 3 a n d 4. T h e m e a n s a n d s t a n d a r d d e v i a t i o n s w e r e c a l c u l a t e d u s i n g t h e " F r e q u e n c i e s ' c o m m a n d o f t h e U.B.C. S t a t i s t i c a l P a c k a g e f o r t h e S o c i a l S c i e n c e s e x t e n d e d v e r s i o n ( S P S S x ) . T o f a c i l i t a t e c o m p u t a t i o n s r e q u i r i n g e q u a l n ' s t h e h a r m o n i c m e a n o f t h e n u m b e r o f v a l i d c a s e s w a s c a l c u l a t e d u s i n g t h e f o r m u l a : = 4 / 1 / n , + l / n a + l / n 3 + l / n ^ w h e r e 4 = n u m b e r o f s e s s i o n s n = n u m b e r o f v a l i d c a s e s i n e a c h s e s s i o n ( F e r g u s o n , 1 9 8 1 ) 170 TABLE D-10 DESCRIPTIVE STATISTICS OF THE ADDITION AND SDBTRACTION FACTS AND COMPUTATIONS SKILLS SCORES AT GRADE 2 IN TESTING SESSIONS 1 THROUGH 4. T e s t i n g S e s s i o n A d d i t i o n F a c t s Mean 8.8 11.0 11.3 12.3 S t d . D e v . 4.7 5.0 5.5 5.9 V a l i d C a s e s * 285 304 308 304 A d d i t i o n C o m p u t a t i o n s Mean 9.4 12.7 14.3 16.0 S t d . D e v . 7.0 8.1 8.7 9.0 V a l i d C a s e s * 286 304 308 303 S u b t r a c t i o n F a c t s Mean 7.4 8.3 9.0 9.3 S t d . D e v . 4.5 4.6 5.1 5.3 V a l i d C a s e s * 286 305 308 304 S u b t r a c t i o n C o m p u t a t i o n s Mean 4.5 6.5 7.4 7.7 S t d . D e v . 4.0 4.6 5.0 5.4 V a l i d C a s e s * 286 305 308 304 *Number o f v a l i d c a s e s e q u a l s number o f s t u d e n t s w r i t i n g t e s t a t e a c h s e s s i o n . TABLE D - l l DESCRIPTIVE STATISTICS OF THE ADDITION AND SUBTRACTION FACTS AND COMPUTATIONS SKILLS SCORES AT GRADE 3 IN TESTING SESSIONS 1 THROUGH 4. 1 A d d i t i o n F a c t s M e a n 17.6 S t d . D e v . 6.8 V a l i d C a s e s * 322 A d d i t i o n C o m p u t a t i o n s M e a n 18.3 S t d . D e v . 7.4 V a l i d C a s e s * 322 S u b t r a c t i o n F a c t s M e a n 14.6 S t d . D e v . 6.6 V a l i d C a s e s * 322 S u b t r a c t i o n C o m p u t a t i o n s M e a n 10.8 S t d . D e v . 6.7 V a l i d C a s e s * 322 T e s t i n g S e s s i o n 2 3 4 18.7 19.9 19.3 7.0 7.3 7.4 319 327 327 21.6 23.1 23.6 7.8 8.1 8.9 319 326 327 15.5 15.9 16.3 6.8 7.0 7.2 319 327 327 13.9 14.7 14.8 9.1 9.3 9.8 319 327 327 * N u m b e r o f v a l i d c a s e s e q u a l s n u m b e r o f s t u d e n t s w r i t i n g t e s t e a c h s e s s i o n . 172 TABLE D-12 DESCRIPTIVE STATISTICS OF THE MULTIPLICATION FACTS AND COMPUTATIONS SKILLS SCORES AT GRADE 3 IN TESTING SESSIONS 1 THROUGH 4. 1 M u l t i p l i c a t i o n F a c t s Mean 4.5 S t d . D e v . 3.8 V a l i d C a s e s * 322 T e s t i n g S e s s i o n 2 3 4 6.0 6.7 7.4 4.8 5.1 5.0 319 313 327 M u l t i p l i c a t i o n C o m p u t a t i o n s Mean 5.9 S t d . D e v . 5.1 V a l i d C a s e s * 320 7.7 10.0 11.9 6.0 7.0 7.6 319 326 327 *Number o f v a l i d c a s e s e q u a l s number o f s t u d e n t s w r i t i n g t e s t a t e a c h s e s s i o n . TABLE D-l3 DESCRIPTIVE STATISTICS OF THE ADDITION AND SDBTRACTION FACTS AND COMPUTATIONS SKILLS SCORES AT GRADE 4 IN TESTING SESSIONS 1 THROUGH 4. 1 A d d i t i o n F a c t s Mean 22.4 S t d . D e v . 7.6 V a l i d C a s e s * 282 A d d i t i o n C o m p u t a t i o n s Mean 8.6 S t d . D e v . 3.0 V a l i d C a s e s * 283 S u b t r a c t i o n F a c t s Mean 20.8 S t d . D e v . 7.7 V a l i d C a s e s * 282 S u b t r a c t i o n C o m p u t a t i o n s Mean 7.9 S t d . D e v . . 5.0 V a l i d C a s e s * 283 T e s t i n g S e s s i o n 2 3 4 23.1 25.3 25.9 7.3 8.2 9.0 281 283 279 10.6 11.3 11.4 3.6 3.8 4.0 280 284 279 21.3 22.3 23.1 7.0 8.0 8.7 281 284 279 9.8 10.8 12.1 6.1 6.9 6.8 281 284 279 *Number o f v a l i d c a s e s e q u a l s number o f s t u d e n t s w r i t i n g t e s t a e a c h s e s s i o n . TABLE D-14 DESCRIPTIVE STATISTICS OF THE MULTIPLICATION AND DIVISION FACTS AND COMPUTATIONS SKILLS SCORES AT GRADE 4 IN TESTING SESSIONS 1 THROUGH 4. 1 M u l t i p l i c a t i o n F a c t s M e a n 19.0 S t d . D e v . 10.5 V a l i d C a s e s * 283 M u l t i p l i c a t i o n C o m p u t a t i o n s M e a n 12.8 S t d . D e v . 8.2 V a l i d C a s e s * 283 D i v i s i o n F a c t s M e a n 12.4 S t d . D e v . 8.8 V a l i d C a s e s * 283 D i v i s i o n C o m p u t a t i o n s M e a n .9 S t d . D e v . 2.2 V a l i d C a s e s * 283 T e s t i n g S e s s i o n 2 3 4 19.7 26.9 21.5 9.4 9.3 10.0 281 284 279 15.1 15.9 16.6 8.0 8.3 8.4 281 284 279 14.9 16.2 17.3 8.4 8.8 9.0 280 284 278 1.5 2.7 4.4 2.4 3.2 4.3 281 284 277 * N u m b e r o f v a l i d c a s e s e q u a l s n u m b e r o f s t u d e n t s w r i t i n g t e s t e a c h s e s s i o n . 175 S e c t i o n I I I : The A n a l y s i s o f V a r i a n c e Summar ies and T u k e y ' s HSD ( H o n e s t l y - S i g n i f i c a n t - D i f f e r e n c e ) f o r t h e DTMS T e s t S c o r e s , S e s s i o n s 1 t o 4. 1 7 6 T h e A n a l y s i s o f V a r i a n c e w e r e c a l c u l a t e d u s i n g t h e ' U s e r p r o c A n o v a r ' c o m m a n d o f t h e U.B.C. s t a t i s t i c a l p a c k a g e f o r t h e S o c i a l S c i e n c e s e x t e n d e d v e r s i o n ( S P S S ) . D i f f e r e n c e s b e t w e e n a d j a c e n t m e a n s ( X ^ - X , , X 3 - X x , X + - X 3 ) w e r e c o m p a r e d w i t h t u k e y ' s H S D ( H o n e s t l y - S i g n i f i c a n t - D i f f e r e n c e ) a t t h e .05 l e v e l t o i d e n t i f y m e a n s w h i c h w e r e ( h o n e s t l y ) s i g n i f i c a n t l y d i f f e r e n t . w h e r e q = v a l u e o f s t u d e n t i z e d r a n g e s t a t i s t i c «C = p r o b a b i l i t y o f a T y p e I e r r o r dfvt/ = d e g r e e s o f f r e e d o m f o r M S W k = n u m b e r o f g r o u p s n = n u m b e r o f s u b j e c t s w i t h i n g r o u p s T u k e y ' s H S D = q <«c, d f k ) ( S h a v e l s o n , 1 9 8 1 ) 1 7 7 T A B L E D - 1 5 A N A L Y S I S O F V A R I A N C E S U M M A R Y , A N D T U K E Y ' S HSD ( H O N E S T L Y - S I G N I F I C A N T - D I F F E R E N C E ) FOR G R A D E 2 T O T A L M A T H E M A T I C S F A C T S C O R R E C T A N D T O T A L M A T H E M A T I C S C O M P U T A T I O N S S K I L L S . T o t a l M a t h e m a t i c s F a c t s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 3 5 9 1 . 0 1 1 9 7 . 2 5 9 . 0 4 0 . 0 0 1 R e s i d u a l 1 4 5 3 9 . 1 7 1 7 2 0 . 3 D i f f e r e n c e s b e t w e e n M e a n s Xx - X,= 3 . 1 5 X j - Xx= 0 . 9 3 X ^ - X,= 1 . 1 3 T u k e y ' s H S D = 1 . 0 6 T o t a l M a t h e m a t i c s C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f S u m o f D e g r e e s o f V a r i a t i o n S q u a r e s F r e e d o m M e a n S q u a r e F P r o b a b i l i t y T r e a t m e n t 1 3 4 3 7 . 5 4 4 7 9 . 2 1 6 1 . 6 0 0 . 0 0 1 R e s i d u a l 1 9 8 7 3 . 8 7 1 7 2 7 . 7 D i f f e r e n c e s b e t w e e n M e a n s Xx - X,= 5 . 5 6 X 3 - X z = 2 . 5 1 X H - X 3 = 1 . 8 7 T u k e y ' s H S D = 1 . 2 3 1 7 8 TABLE D-l6 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 3 TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS SKILLS. Total Mathematics Facts A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 6 5 7 7 . 3 2 1 9 2 . 4 6 0 . 5 0 0 . 0 0 1 R e s i d u a l 2 8 8 0 9 . 0 7 9 5 3 6 . 2 D i f f e r e n c e s b e t w e e n M e a n s X x - X,= 4 . 0 4 X 3 - X a = 1 . 7 1 XH - Xy 0 . 6 3 T u k e y ' s H S D = 1 . 3 4 Total Mathematics Computations S k i l i s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 3 7 6 1 5 . 1 1 2 5 3 8 . 4 2 1 2 . 6 0 0 . 0 0 1 R e s i d u a l 4 6 8 8 7 . 0 7 9 5 5 9 . 0 D i f f e r e n c e s b e t w e e n M e a n s X z - X x= 8 . 9 6 X 3 - X v = 4 . 1 1 X H - X 3 = 2 . 5 3 T u k e y ' s H S D = 1 . 7 1 179 TABLE D-17 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 4 TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS SKILLS. Total Mathematics Facts A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 25851.8 3 8617.3 R e s i d u a l 71370.0 711 100.4 85.85 0.001 D i f f e r e n c e s b e t w e e n M e a n s X a - X,= 4.77 X , - Xx= 5.73 X 4 - X 3 = 3.03 T u k e y ' s H S D = 2.36 Total Mathematics Computations S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 2 6 4 5 2 . 4 3 8 8 1 7 . 5 R e s i d u a l 3 3 4 1 5 . 0 7 1 1 4 7 . 0 1 8 7 . 6 0 0 . 0 0 1 D i f f e r e n c e s b e t w e e n M e a n s X z - X ( = 6 . 5 9 X 3 - X a = 3 . 8 9 X 1 * - X 3 = 3 . 7 8 T u k e y ' s H S D = 1 . 6 1 180 TABLE D-18 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 2 ADDITION FACTS CORRECT AND ADDITION COMPUTATION SKILLS CORRECT. Addition Facts A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 1343.4 447.8 57.14 0.001 R e s i d u a l 5595.4 714 7.8 D i f f e r e n c e s b e t w e e n M e a n s X a - X,= 2.13 X , - X a = 0.14 X H - X 3 = 0.98 T u k e y ' s H S D = 0.66 Addition Computation S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 5576.9 1859.0 143.00 0.001 R e s i d u a l 9279.6 714 13.0 D i f f e r e n c e s b e t w e e n M e a n s X ,= 3.51 X 3 - X*= 1.54 X H - X 3= 1.43 T u k e y ' s H S D = 0.85 181 T A B L E D - 1 9 A N A L Y S I S O F V A R I A N C E S U M M A R Y , A N D T U K E Y ' S HSD ( H O N E S T L Y - S I G N I F I C A N T - D I F F E R E N C E ) FOR G R A D E 2 S U B T R A C T I O N F A C T S C O R R E C T A N D S U B T R A C T I O N C O M P U T A T I O N S K I L L S C O R R E C T . S u b t r a c t i o n F a c t s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 533.1 177.7 22.67 0.001 R e s i d u a l 5621.4 717 7.8 D i f f e r e n c e s b e t w e e n M e a n s X j. - X, = 0.95 X 3 - Xx= 0.78 X 4 - X3= 0.15 T u k e y ' s H S D = 0.66 S u b t r a c t i o n C o m p u t a t i o n S k i l l s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S u m o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 1738.6 579.5 74.10 0.001 R e s i d u a l 5606.4 717 7.8 D i f f e r e n c e s b e t w e e n M e a n s X 3 L-X X=2.06 X,-Xj=0.97 X H - Xj= 0.47 T u k e y ' s H S D = 0.65 1 8 2 TABLE D-20 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 3 ADDITION FACTS CORRECT AND ADDITION COMPUTATION SKILLS CORRECT. Addition Facts S o u r c e o f V a r i a t i o n T r e a t m e n t Sum o f S q u a r e s 8 2 5 . 5 A n o v a S u m m a r y D e g r e e s o f F r e e d o m M e a n S q u a r e 2 7 5 . 2 P r o b a b i l i t y R e s i d u a l 1 1 6 3 4 . 7 7 9 5 1 4 . 6 1 8 . 8 0 0 . 0 0 1 D i f f e r e n c e s b e t w e e n M e a n s X, - X^ = 0 . 6 2 X x - X,= 1 . 4 3 X 7 - Xx= 0 . 9 7 T u k e y ' s H S D = 0 . 8 5 Addition Computation S k i l l s S o u r c e o f V a r i a t i o n T r e a t m e n t S u m o f S q u a r e s 4 6 2 9 . 9 A n o v a S u m m a r y D e g r e e s o f F r e e d o m M e a n S q u a r e 1 5 4 3 . 3 1 0 5 . 2 6 P r o b a b i l i t y 0 . 0 0 1 R e s i d u a l 1 1 6 5 6 . 3 7 9 5 1 4 . 7 D i f f e r e n c e s b e t w e e n M e a n s X JL - X t= 3 . 5 9 X 7 - Xj= 1 . 1 2 XH - Xj= 0 . 6 3 T u k e y ' s H S D = 0 . 8 5 183 TABLE D-21 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 3 SUBTRACTION FACTS CORRECT AND SUBTRACTION COMPUTATION SKILLS CORRECT. Subtraction Facts Anova Summary S o u r c e o f V a r i a t i o n Sum o f S q u a r e s D e g r e e s o f F reedom Mean S q u a r e P r o b a b i l i t y T r e a t m e n t 4 0 9 . 5 136.5 1 2 . 8 6 0 . 0 0 1 R e s i d u a l 8 4 3 6 . 8 795 10.6 D i f f e r e n c e s be tween Means X x - X,= 0.94 X 3 " X a = 0 . 3 4 X u - X ,= 0 . 4 0 / T u k e y ' s HSD = 0 . 7 3 Subtraction Computation S k i l l s Anova Summary S o u r c e o f V a r i a t i o n Sum o f S q u a r e s D e g r e e s o f F reedom Mean S q u a r e P r o b a b i l i t y T r e a t m e n t 3 1 9 5 . 1 1 0 6 5 . 0 5 5 . 7 4 0 .001 R e s i d u a l 1 5 1 9 0 . 9 795 1 9 . 1 D i f f e r e n c e s be tween Means X j - X|— 3 . 3 8 X 3 - X t = 0 . 8 0 X ^ - X,= 0 . 0 2 T u k e y ' s HSD = 0.97 184 TABLE D-22 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 3 MULTIPLICATION FACTS CORRECT AND MULTIPLICATION COMPUTATION SKILLS CORRECT. Multiplication Facts S o u r c e o f V a r i a t i o n T r e a t m e n t R e s i d u a l Sum o f S q u a r e s 1166.0 5046.7 Anova Summary D e g r e e s o f F reedom 759 Mean S q u a r e 388.7 6.6 58.45 P r o b a b i l i t y 0.001 D i f f e r e n c e s b e t w e e n Means X 4 - X3= 0.54 X a - X,= 1.64 X 3 - X;= 0.67 T u k e y ' s HSD = 0.59 Multiplication Computation S k i l l s Anova Summary S o u r c e o f V a r i a t i o n T r e a t m e n t R e s i d u a l Sum o f S q u a r e s 5407.4 11260.7 D e g r e e s o f F reedom 786 Mean S q u a r e 1802.5 14.3 125.81 P r o b a b i l i t y 0.001 D i f f e r e n c e s be tween Means X j, - X ,= 1.83 X 3 - Xx= 2.29 X ^ — Xj= 1.86 T u k e y ' s HSD = 0.85 185 TABLE D-23 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 4 ADDITION FACTS CORRECT AND ADDITION COMPUTATION SKILLS CORRECT. Addition Facts S o u r c e o f V a r i a t i o n T r e a t m e n t Sum o f S q u a r e s 2 0 1 1 . 9 Anova Summary D e g r e e s o f F reedom Mean S q u a r e 6 7 0 . 6 P r o b a b i l i t y 3 0 . 1 6 0 . 0 0 1 R e s i d u a l 1 5 7 4 2 . 4 708 2 2 . 2 D i f f e r e n c e s be tween Means X* - X,= 0 . 6 7 X a - X a= 2 . 2 2 X* - X 3 = 0 . 5 8 T u k e y ' s HSD = 1 . 1 1 Addition Computation S k i l l s A n o v a Summary S o u r c e o f Sum o f D e g r e e s o f Mean V a r i a t i o n S q u a r e s F reedom S q u a r e F P r o b a b i l i t y T r e a t m e n t 1 1 8 5 . 2 3 3 9 5 . 1 R e s i d u a l 3 4 7 4 . 8 708 4 . 9 8 0 . 4 9 0 . 0 0 1 D i f f e r e n c e s b e t w e e n Means X x - X ,= 1 .99 Xj-Xa.= 0 . 7 0 X * - X3= 0 . 0 9 T u k e y ' s HSD = 0 . 5 2 186 TABLE D-24 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 4 SUBTRACTION FACTS CORRECT AND SUBTRACTION COMPUTATION SKILLS CORRECT. Subtraction Facts A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S u m o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 772.3 257.4 18.10 0.001 R e s i d u a l 10068.5 708 14.2 D i f f e r e n c e s b e t w e e n M e a n s X a - X , = 0 . 6 9 X 8 - X x = 1 . 0 9 X „ - X 3 = 0 . 5 2 T u k e y ' s H S D = 0.89 Subtraction Computation S k i l l s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S u m o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e P r o b a b i l i t y T r e a t m e n t 2405.9 802.0 83.26 0.001 R e s i d u a l 6847.9 711 9.6 D i f f e r e n c e s b e t w e e n M e a n s X a - X v= 2.03 X 3 - X*= 1.16 X * - X,= 1.11 T u k e y ' s H S D = 0.73 187 TABLE D-25 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 4 MULTIPLICATION FACTS CORRECT AND MULTIPLICATION COMPUTATION SKILLS CORRECT. Multiplication Facts A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 995.2 3 331.7 R e s i d u a l 12698.4 711 17.9 1 8 . 5 8 0 .001 D i f f e r e n c e s b e t w e e n M e a n s X A - X x= 0 . 7 1 X 3 - X x = 1 .10 X * - X 3 = 0 . 8 6 T u k e y ' s H S D = 1 .00 M u l t i p l i c a t i o n Computation S k i l l s A n o v a S u m m a r y S o u r c e o f S u m o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 1 8 1 0 . 6 3 6 0 3 . 5 R e s i d u a l 1 2 1 4 0 . 2 711 1 7 . 1 3 5 . 3 5 0 .001 D i f f e r e n c e s b e t w e e n M e a n s X a - X i= 2 . 0 9 X 3 - Xj,= 0 . 7 2 X a - X 3= 0 . 9 3 T u k e y ' s H S D = 0 . 9 7 188 TABLE D-26 ANALYSIS OF VARIANCE SUMMARY, AND TUKEY'S HSD (HONESTLY-SIGNIFICANT-DIFFERENCE) FOR GRADE 4 DIVISION FACTS CORRECT AND DIVISION COMPUTATION SKILLS CORRECT. D i v i s i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 3477.4 3 1159.1 R e s i d u a l 10395.1 705 14.7 78.61 0.001 D i f f e r e n c e s b e t w e e n M e a n s X l - Xx = 2.73 X 3 - Xa= 1.31 X * - Xj = 1.10 T u k e y ' s H S D = 0.91 D i v i s i o n C o m p u t a t i o n S k i l l s A n o v a S u m m a r y S o u r c e o f S u m o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F P r o b a b i l i t y T r e a t m e n t 1686.3 3 562.1 R e s i d u a l 4011.7 705 5.7 98.78 0.001 D i f f e r e n c e s b e t w e e n M e a n s X 2 - X x= 0.54 X 3 - Xa= 1.26 X 4. - X,= 1.68 T u k e y ' s H S D = 0.56 189 Section IV: Trend Analysis Summaries and r , the Proportion of Total V a r i a t i o n Accounted f o r by S i g n i f i c a n t Trends f o r the DTMS Sessional Means. 190 TABLE D-27 TREND ANALYSIS SUMMARY AND r x , THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY SIGNIFICANT TRENDS FOR GRADE 2 TOTAL MATHEMATICS FACTS CORRECT. S e s s i o n a l D a t a 2 3 300 1 6 . 2 4860 300 1 9 . 2 5760 300 2 0 . 3 6090 300 2 1 . 5 6450 O r t h o g o n a l P o l y n o m i a l s C j c 3 ; - 3 1 -1 -1 -1 3 1 -1 - 3 3 1 1 20 4 20 A n o v a S u m m a r y S o u r c e o f V a r i a t i o n L i n e a r Q u a d r a t i c C u b i c Sum o f S q u a r e s T l * / n £ c (5100 ) / 3 0 0 ( 2 0 ) ( -540 ) / 3 0 0 ( 4 ) (600 ) / 3 0 0 ( 2 0 ) D e g r e e s o f F r e e d o m 1 1 1 M e a n S q u a r e 4335 243 60 [MS/MSvJ 2 1 3 . 7 8 1 1 . 7 0 2 . 9 6 T e s t S t a t i s t i c F ( . 0 5 , 1 , 4 0 0 ) = 3 . 8 6 T h e C a l c u l a t i o n o f r = A/SS / S S T 0 . 9 7 * - 0 . 9 3 r = JSS a) / S S T = 0 . 2 3 r * = 0 . 0 5 1 9 1 TABLE D-28 TREND ANALYSIS SUMMARY AND r x, THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY SIGNIFICANT TRENDS FOR GRADE 2 TOTAL MATHEMATICS COMPUTATIONS SKILLS CORRECT. A n o v a S u m m a r y S o u r c e o f V a r i a t i o n L i n e a r Q u a d r a t i c C u b i c Sum o f S q u a r e s 1 5 3 6 0 . 0 8 6 7 . 0 6 0 . 0 D e g r e e s o f M e a n F r e e d o m S q u a r e 1 1 1 1 5 3 6 0 . 0 8 6 7 . 0 6 0 . 0 F 5 5 4 . 1 5 3 1 . 2 8 2 . 1 7 T e s t S t a t i s t i c F = 3 . 8 6 r X L i n e a r Q u a d r a t i c C u b i c r l = 0 . 9 4 r 1 = 0 . 0 5 1 9 2 TABLE D-29 TREND ANALYSIS SUMMARY AND r l , THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY SIGNIFICANT TRENDS FOR GRADE 3 TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS SKILLS CORRECT. Total Mathematics Facts A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 7 0 7 6 . 3 1 7 0 7 6 . 3 1 9 5 . 2 7 Q u a d r a t i c 5 9 0 . 5 1 5 9 0 . 5 1 6 . 3 0 C u b i c 1.5 1 1 . 5 0 . 0 4 T e s t S t a t i s t i c F = 3 . 8 6 r x L i n e a r Q u a d r a t i c C u b i c r * = 0 . 9 2 r x = 0 . 0 8 Total Mathematics Computations S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 6 7 1 8 7 . 2 1 6 7 1 8 7 . 2 1 1 3 9 . 2 1 Q u a d r a t i c 9 4 4 7 . 8 1 9 4 4 7 . 8 1 6 0 . 2 0 C u b i c 5 4 5 . 0 1 5 4 5 . 0 9 . 2 4 T e s t S t a t i s t i c F = 3 . 8 6 r L i n e a r Q u a d r a t i c C u b i c r x = 0 . 8 7 r x = 0 . 1 2 r * = 0 . 0 1 193 TABLE D-30 TREND ANALYSIS SUMMARY AND r x , THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 4 TOTAL MATHEMATICS FACTS CORRECT AND TOTAL MATHEMATICS COMPUTATIONS S K I L L S CORRECT. T o t a l M a t h e m a t i c s F a c t s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e F L i n e a r 28679.5 1 28679.5 285.71 Q u a d r a t i c 158.6 1 158.6 1.58 C u b i c 153.5 T e s t F 1 S t a t i s t i c = 3.86 153.5 1.53 L i n e a r Q u a d r a t i c C u b i c r a = 0.99 T o t a l M a t h e m a t i c s C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f V a r i a t i o n S um o f S q u a r e s D e g r e e s o f F r e e d o m M e a n S q u a r e F L i n e a r 29835.6 1 29835.6 634.84 Q u a d r a t i c 552.7 1 552.7 11.76 C u b i c 225.6 T e s t F 1 S t a t i s t i c = 3.86 225.6 4.80 L i n e a r Q u a d r a t i c C u b i c r *• = 0.97 r x = = 0.02 r x = 0.01 194 TABLE D-31 TREND ANALYSIS SUMMARY AND r x, THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 2 ADDITION FACTS CORRECT AND ADDITION COMPUTATIONS S K I L L S CORRECT. A d d i t i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1749.6 1 1749.6 223.25 Q u a d r a t i c 108.0 1 108.0 13.78 C u b i c 14.1 1 14.1 1.80 T e s t S t a t i s t i c F = 3.86 r 1 L i n e a r Q u a d r a t i c C u b i c r l = 0 . 9 3 r z = 0.06 A d d i t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 6869.4 1 6869.4 528.54 Q u a d r a t i c 192.0 1 192.0 14.77 C u b i c 48.6 1 48.6 3.74 T e s t S t a t i s t i c F = 3.86 L i n e a r Q u a d r a t i c C u b i c r x = 0.97 r x = 0.03 195 TABLE D-32 TREND ANALYSIS SUMMARY AND r \ THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 2 SUBTRACTION FACTS CORRECT AND SUBTRACTION COMPUTATIONS S K I L L S CORRECT. S u b t r a c t i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 614.4 1 614.4 78.37 Q u a d r a t i c 27.0 1 27.0 3.44 C u b i c 0.6 1 0.6 0.08 T e s t S t a t i s t i c F = 3.86 l r L i n e a r Q u a d r a t i c C u b i c r 1 = 0.95 S u b t r a c t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1653.7 1 1653.7 211.50 Q u a d r a t i c 216.7 1 216.7 27.72 C u b i c 3.7 1 3.7 0.48 T e s t S t a t i s t i c F = 3.86 r_ L i n e a r Q u a d r a t i c C u b i c r *•= 0 . 8 8 r X = 0.12 196 TABLE D-33 TREND ANALYSIS SUMMARY AND r a , THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 3 ADDITION FACTS CORRECT AND ADDITION COMPUTATIONS S K I L L S CORRECT. A d d i t i o n F a c t s Anova Summary Source of Sum of Degrees of Mean V a r i a t i o n Squares Freedom Square F Linear 643.5 1 643.5 43.97 Quadratic 235.1 1 235.1 16.06 Cubic 58.6 1 58.6 4.00 Test S t a t i s t i c F = 3.86 Linear Quadratic Cubic r * = 0.69 r 1 = 0.25 r * = 0.06 A d d i t i o n C o m p u t a t i o n s S k i l l s Anova Summary Source of Sum of Degrees of Mean V a r i a t i o n Squares Freedom Square F Linear 4889.2 1 4889.2 333.46 Quadratic 632.5 1 632.5 43.14 Cubic 10.5 1 10.5 0.71 Test S t a t i s t i c F = 3.86 „ x Linear r 1 = 0.88 Quadratic r x = 0.11 Cubic 1 9 7 TABLE D-34 TREND ANALYSIS SUMMARY AND r S THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 3 SUBTRACTION FACTS CORRECT AND SUBTRACTION COMPUTATIONS S K I L L S CORRECT. S u b t r a c t i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 4 9 0 . 6 1 4 9 0 . 6 4 6 . 2 3 Q u a d r a t i c 2 0 . 5 1 2 0 . 5 1 . 9 3 C u b i c 4 . 0 1 4 . 0 0 . 3 8 T e s t S t a t i s t i c F = 3 . 8 6 r 1 L i n e a r Q u a d r a t i c C u b i c r *• = 0 . 9 5 S u b t r a c t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 2 6 5 4 . 0 1 2 6 5 4 . 0 1 3 8 . 8 9 Q u a d r a t i c 7 3 0 . 5 1 7 3 0 . 5 3 8 . 2 3 C u b i c 4 1 . 6 1 4 1 . 6 2 . 1 8 T e s t S t a t i s t i c F = 3 . 8 6 2-L i n e a r r x = 0 . 7 7 Q u a d r a t i c r x = 0 . 2 1 C u b i c 198 TABLE D-35 TREND ANALYSIS SUMMARY AND r z , THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 3 MULTIPL ICATION FACTS CORRECT AND MULTIPL ICATION COMPUTATIONS S K I L L S CORRECT. M u l t i p l i c a t i o n F a c t s A n o v a S u m m a r y S o u r c e o f S u m o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1413.8 1 1413.8 212.63 Q u a d r a t i c 51.2 1 51.2 7.70 C u b i c 10.2 1 10.2 1.54 T e s t S t a t i s t i c F = 3.86 r x L i n e a r Q u a d r a t i c C u b i c r r = 0.96 r x = 0.03 M u l t i p l i c a t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 6655.5 1 6655.5 464.54 Q u a d r a t i c 0.8 1 0.8 0.06 C u b i c 13.1 1 13.1 0.91 T e s t S t a t i s t i c F = 3.86 L i n e a r Q u a d r a t i c C u b i c TABLE D-36 TREND ANALYSIS SUMMARY AND r \ THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY SIGNIFICANT TRENDS FOR GRADE 4 ADDITION FACTS CORRECT AND ADDITION COMPUTATIONS SKILLS CORRECT. Addition Facts A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 2267 .8 1 2267 .8 101 .99 Q u a d r a t i c 0 .7 1 0 .7 0 .03 C u b i c 134 .7 1 134 .7 6 .06 T e s t S t a t i s t i c F = 3 .86 r 1 -L i n e a r Q u a d r a t i c C u b i c r * = 0 .94 r i = 0 . 0 6 A d d i t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1160 .7 1 1160 .7 236.48 Q u a d r a t i c 253 .7 1 253 .7 51 .69 C u b i c 7 .0 1 7 .0 1.42 T e s t S t a t i s t i c F = 3 .86 r l L i n e a r Q u a d r a t i c C u b i c r = 0 .82 r *" = 0.18 2 0 0 TABLE D-37 TREND A N A L Y S I S SUMMARY AND r*V THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 4 SUBTRACTION FACTS CORRECT AND SUBTRACTION COMPUTATIONS S K I L L S CORRECT. S u b t r a c t i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 8 7 6 . 1 1 8 7 6 . 2 6 1 . 6 1 Q u a d r a t i c 6 . 4 1 6 . 4 0 . 4 5 C u b i c 6 . 9 1 6 . 9 0 . 4 9 T e s t S t a t i s t i c F = 3 . 8 6 L i n e a r Q u a d r a t i c C u b i c r 1 = 0 . 9 9 S u b t r a c t i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 2 6 0 6 . 3 1 2 6 0 6 . 3 2 7 0 . 6 2 Q u a d r a t i c 2 5 . 6 1 2 5 . 6 2 . 6 6 C u b i c 2 0 . 3 1 2 0 . 3 0 . 1 0 T e s t S t a t i s t i c F = 3 . 8 6 r * L i n e a r Q u a d r a t i c C u b i c 201 TABLE D-38 TREND ANALYSIS SUMMARY AND r \ THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY SIGNIFICANT TRENDS FOR GRADE 4 MULTIPLICATION FACTS CORRECT AND MULTIPLICATION COMPUTATIONS SKILLS CORRECT. M u l t i p l i c a t i o n Facts A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1 0 6 7 . 7 1 1 0 6 7 . 8 5 9 . 7 8 Q u a d r a t i c 0 . 7 1 0 . 7 0 .04 C u b i c 1 7 . 3 1 1 7 . 3 0 . 9 7 T e s t S t a t i s t i c F = 3 . 86 L i L i n e a r Q u a d r a t i c C u b i c r 1 = 0 . 9 8 M u l t i p l i c a t i o n Computations S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 2 0 9 6 . 9 1 2 0 9 6 . 9 1 2 2 . 8 1 Q u a d r a t i c 1 8 0 . 3 1 1 8 0 . 3 1 0 . 5 6 C u b i c 2 7 . 4 1 2 7 . 4 1 .60 T e s t S t a t i s t i c F = 3 . 8 6 r 1 L i n e a r Q u a d r a t i c C u b i c r 1 = 0 .91 r x = 0 . 0 8 202 TABLE D-39 TREND A N A L Y S I S SUMMARY AND r \ THE PROPORTION OF TOTAL VARIATION ACCOUNTED FOR BY S IGNIF ICANT TRENDS FOR GRADE 4 D I V I S I O N FACTS CORRECT AND D I V I S I O N COMPUTATIONS S K I L L S CORRECT. D i v i s i o n F a c t s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 3596.8 1 3596.8 2 4 3 . 9 3 Q u a d r a t i c 138.1 1 138.1 9.37 C u b i c 14.1 1 14.2 0.96 T e s t S t a t i s t i c F = 3.86 r*-L i n e a r Q u a d r a t i c C u b i c r 1 = 0.96 r3- = 0.04 D i v i s i o n C o m p u t a t i o n s S k i l l s A n o v a S u m m a r y S o u r c e o f Sum o f D e g r e e s o f M e a n V a r i a t i o n S q u a r e s F r e e d o m S q u a r e F L i n e a r 1923.3 1 1923.3 338.01 Q u a d r a t i c 85.0 1 85.0 14.94 C u b i c 0.1 1 0.1 0.02 T e s t S t a t i s t i c F = 3.86 L i n e a r Q u a d r a t i c C u b i c 203 APPENDIX E S e c t i o n I_: T h e B e s t F i t t i n g C u r v e s I n d i c a t e d b y T r e n d A n a l y s i s f o r T o t a l M a t h e m a t i c s S c o r e s o n t h e D T M S . * S e c t i o n I I : C u m u l a t i v e P e r c e n t a g e P o l y g o n s f o r T o t a l M a t h e m a t i c s S c o r e s o n t h e D T M S . S e c t i o n I I I : R e a l a n d P r o j e c t e d M e a n s a n d S t a n d a r d D e v i a t i o n s f o r T o t a l M a t h e m a t i c s S c o r e s o n t h e D T M S . * T h e s e b e s t f i t t i n g c u r v e s w e r e c a l c u l a t e d a s f o l l o w s : L i n e a r y , = x . . + £ c , • T j / n £ c , j * ( x - 2 . 5 ) Q u a d r a t i c y a = y , + £ c 4 j T j / n £ c ^ j 1 ( x x - 5 x + 5 ) C u b i c y 3 = yx + $ c3j T j / n £c3j X ( x 3 - 7 . 5 x x + 1 6 . 7 x - 1 0 . 204 TABLE E - l : THE BEST FITTING CURVES INDICATED BY TREND ANALYSIS FOR GRADE 2 TOTAL MATHEMATICS FACTS CORRECT SCORES. 25 • 20 -15 -Sessional Means Linear Curve Quadratic Curve 2 Sessions 205 TABLE E-2: THE BEST FITTING CURVES INDICATED BY TREND ANALYSIS FOR GRADE 2 TOTAL COMPUTATIONS SKILLS CORRECT SCORES. Sessions 206 TABLE E-3: THE BEST FITTING CURVES INDICATED BY TREND ANALYSIS FOR GRADE 3 TOTAL MATHEMATICS FACTS CORRECT SCORES. 45 40 U o u to 35 Sessional Means Linear Curve Quadratic Curve —i— 1 Sessions 207 TABLE E-4: THE BEST FITTING CURVES INDICATED BY TREND ANALYSIS FOR GRADE 3 TOTAL COMPUTATIONS SKILLS CORRECT SCORES. 50 -45 -<: • • ^ / 40 -(LI U O U CO 35 7 30 -Sessional Means Linear Curve Quadratic Curve Cubic Curve "T" 4 Sessions 208 Sessions 209 TABLE E-6: THE BEST FITTING CURVES INDICATED BY TREND ANALYSIS FOR GRADE 4 TOTAL COMPUTATIONS SKILLS CORRECT SCORES. 45 -40 -35 -30 Sessional Means Linear Curve Quadratic Curve Cubic Curve —r-2 Sessions 210 Section I I : Cumulative Percentage Polygons f o r Total Mathematics Scores on the DTMS. Total Score FIGURE E-8: CUMULATIVE PERCENTAGE POLYGON FOR TOTAL MATHEMATICS COMPUTATIONS, CORRECT, GRADE 2. Total Score FIGURE: E - 9: CUMULATIVE PERCENTAGE POLYGON FOR TOTAL MATHEMATICS FACTS CORRECT, GRADE 3 . Total Score FIGURE E - l l : CUMULATIVE PERCENTAGE POLYGON FOR TOTAL MATHEMATICS FACTS CORRECT, GRADE 4. FIGURE E-12: CUMULATIVE PERCENTAGE POLYGON FOR TOTAL MATHEMATICS COMPUTATIONS CORRECT, GRADE 4. 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Total Score 217 Section I I I : Real and Projected Means and Standard Deviations f o r Total Mathematics Scores on the DTMS. FIGURE E-13: GRADE 2 TOTAL MATHEMATICS COMPUTATIONS SKILLS SESSIONAL MEANS AND STANDARD DEVIATIONS. Months 219 FIGURE E-14: GRADE 3 TOTAL MATHEMATICS FACTS SESSIONAL MEANS AND STANDARD DEVIATIONS. 80 H 75 4 70 -65 . Months FIGURE E-15: GRADE 3 TOTAL MATHEMATICS COMPUTATIONS SKILLS SESSIONAL MEANS AND STANDARD DEVIATIONS.