UBC Theses and Dissertations

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UBC Theses and Dissertations

An investigation of the effect of operational level on the arithmetic performance of grade one children Davies, Donna Lynn 1980

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A N I N V E S T I G A T I O N O F T H E E F F E C T O F O P E R A T I O N A L L E V E L O N T H E A R I T H M E T I C P E R F O R M A N C E O F G R A D E O N E C H I L D R E N b y D O N N A L Y N N D A V I E S B . E 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 , 1968 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A R T S i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( D e p a r t m e n t o f E d u c a t i o n ) W e a c c e p t t h i s t h e s i s a s 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 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A M a r c h 1980 (c) Donna Lynn Davies, 1980 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 re 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 Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . 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 underst 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 n f C Z U U S C O ^ L L J - - ? L , The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 DE-6 BP 75-51 I E . i i . A B S T R A C T R e s e a r c h S u p e r v i s o r : D r . P a t r i c i a K e n n e d y A r l i n T h i s s t u d y i n v e s t i g a t e d t h e i n t e r a c t i v e i n f l u e n c e o f o p e r a t i o n a l l e v e l , t y p e o f n u m b e r s e n t e n c e , a n d l e v e l o f d i f f i c u l t y o n g r a d e o n e a r i t h m e t i c p e r -f o r m a n c e . O f s p e c i f i c i n t e r e s t w a s 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 c h i l d ' s o p e r a t i o n a l l e v e l a n d h i s o r h e r a c c u r a c y a n d s o l u t i o n m e t h o d s w h e n s o l v i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s a t t h r e e l e v e l s o f d i f f i -c u l t y . S u b j e c t s o f t h e s t u d y w e r e s i x t y g r a d e o n e s t u d e n t s f r o m a s u b u r b i n t h e l o w e r m a i n l a n d o f B r i t i s h C o l u m b i a . N i n e t y p e s o f P i a g e t i a n t a s k s , i n c l u d i n g s e r i a t i o n , c l a s s i f i c a t i o n , a n d c o n s e r v a t i o n w e r e a d m i n i s t e r e d t o a l a r g e s a m p l e o f c h i l d r e n . T h e c h i l d r e n w e r e t h e n a s s i g n e d t o t h r e e o p e r a t i o n a l l e v e l s : t r a n s i t i o n a l , e a r l y c o n c r e t e , a n d l a t e r c o n c r e t e . T w e n t y s u b j e c t s a t e a c h o f t h e t h r e e o p e r a t i o n a l l e v e l s w e r e s e l e c t e d f o r t h i s s t u d y . S u b -s e q u e n t l y , a n i n e - i t e m a r i t h m e t i c t e s t i n c l u d i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s a t t h r e e l e v e l s o f d i f f i c u l t y w a s a d m i n i s t e r e d i n d i v -i d u a l l y t o t h e s i x t y s u b j e c t s . B o t h t h e c h i l d ' s a n s w e r s t o t h e n u m b e r s e n t e n -c e s a n d h i s o r h e r s o l u t i o n p r o c e d u r e s w e r e r e c o r d e d . T h e r e w a s o n e d e p e n d e n t v a r i a b l e , a r i t h m e t i c p e r f o r m a n c e , a n d t h r e e i n -d e p e n d e n t v a r i a b l e s : o p e r a t i o n a l l e v e l , t y p e o f n u m b e r s e n t e n c e , a n d l e v e l o f d i f f i c u l t y . A r i t h m e t i c p e r f o r m a n c e w a s a n a l y s e d b y a f i x e d e f f e c t s t h r e e -w a y a n a l y s i s o f v a r i a n c e w i t h r e p e a t e d m e a s u r e s o n t h e l a s t t w o f a c t o r s . T h e s o l u t i o n s t r a t e g y s t u d y w a s a n a l y s e d w i t h d e s c r i p t i v e s t a t i s t i c s . B a s e d o n a r e v i e w o f t h e l i t e r a t u r e , s i x h y p o t h e s e s w e r e f o r m u l a t e d a n d t e s t e d . T h e r e s u l t s o f t h e a n a l y s i s o f v a r i a n c e r e v e a l e d a s e c o n d o r d e r i n t e r a c t i o n o f o p e r a t i o n a l l e v e l , t y p e o f n u m b e r s e n t e n c e , a n d l e v e l o f i i i . d i f f i c u l t y ; t w o f i r s t o r d e r i n t e r a c t i o n s , o p e r a t i o n a l l e v e l a n d t y p e o f n u m -b e r s e n t e n c e a n d o p e r a t i o n a l l e v e l a n d l e v e l o f d i f f i c u l t y ; a n d t h r e e m a i n e f f e c t s o f o p e r a t i o n a l l e v e l , t y p e o f n u m b e r s e n t e n c e , a n d l e v e l o f d i f f i -c u l t y ( p < .001). A n a n a l y s i s o f t h e s o l u t i o n s t r a t e g y s t u d y r e s u l t s i n d i c a t e d t h a t t h e c h i l d r e n i n t h i s s t u d y u s e d c o u n t i n g p r o c e d u r e s t o s o l v e t h e n u m b e r s e n t e n -c e s . C o u n t i n g - a l l , p a r t i a l c o u n t i n g - o n , a n d c o u n t i n g - w i t h - t a l l y w e r e t h e i m p l e m e n t e d s t r a t e g i e s f o r s o l v i n g a d d i t i o n a n d m i s s i n g a d d e n d s e n t e n c e s . C o u n t i n g - a l l a n d c o u n t i n g - w i t h - g r o u p i n g w e r e t h e p r o c e d u r e s u s e d t o a n s w e r t h e s u b t r a c t i o n i t e m s . T h e p a r t i a l c o u n t i n g - o n a n d c o u n t i n g - w i t h - t a l l y p r o -c e d u r e s w e r e d e s i g n a t e d a s i n t e r m e d i a t e s t r a t e g i e s b e t w e e n c o u n t i n g - a l l a n d c o u n t i n g - o n . T h e c h i l d ' s o p e r a t i o n a l l e v e l i n f l u e n c e d h i s o r h e r c h o i c e o f s o l u t i o n m e t h o d . L a t e r c o n c r e t e c h i l d r e n u s e d m o r e a d v a n c e d s t r a t e g i e s t h a n e a r l y c o n c r e t e c h i l d r e n w h o , i n t u r n , u s e d m o r e a d v a n c e d s t r a t e g i e s t h a n t r a n s i t i o n a l c h i l d r e n . I t w a s c o n c l u d e d t h a t P i a g e t i a n t a s k s c o u l d b e i n c l u d e d a s o n e a s s e s s -m e n t t o d e t e r m i n e t h e c h i l d ' s r e a d i n e s s f o r t h e g r a d e o n e a r i t h m e t i c p r o g r a m . S p e c i f i c a l l y , t h e c o n s e r v a t i o n o f n u m b e r t a s k m a y s e r v e t o i d e n t i f y w h e t h e r t h e c h i l d h a s t h e c o g n i t i v e m a t u r i t y t o c o m p r e h e n d t h e o p e r a t i o n s o f a d d i t i o n a n d s u b t r a c t i o n ; a n d t h e c l a s s i n c l u s i o n t a s k m a y i n d i c a t e t h e c h i l d ' s r e a d i -n e s s f o r m i s s i n g a d d e n d s e n t e n c e s . I t w a s a l s o r e c o m m e n d e d t h a t t h e c h i l d b e t a u g h t s o l u t i o n m e t h o d s w h i c h a r e a p p r o p r i a t e f o r h i s o r h e r l e v e l o f c o g -n i t i v e d e v e l o p m e n t . T h e t r a n s i t i o n a l p r o c e d u r e s , p a r t i a l c o u n t i n g - o n a n d . c o u n t i n g - w i t h - t a l l y , t h a t w e r e o b s e r v e d i n t h i s s t u d y h a v e p o t e n t i a l i m p l i c a -t i o n s f o r c l a s s r o o m p r a c t i s e a n d f u t u r e r e s e a r c h . i v . T A B L E O F C O N T E N T S P a g e L I S T O F T A B L E S x L I S T O F F I G U R E S x i i A C K N O W L E D G E M E N T x i i i C h a p t e r 1. I N T R O D U C T I O N 1 S t a t e m e n t o f t h e P r o b l e m 2 P u r p o s e o f t h e S t u d y 3 O r g a n i z a t i o n o f t h e T h e s i s . . . . . k D e f i n i t i o n o f t h e V a r i a b l e s U T e r m s A s s o c i a t e d w i t h C o g n i t i v e M a t u r i t y k O p e r a t i o n a l l e v e l . . . h T r a n s i t i o n a l • 5 E a r l y c o n c r e t e o p e r a t i o n a l 5 L a t e r c o n c r e t e o p e r a t i o n a l 6 T e r m s A s s o c i a t e d w i t h A r i t h m e t i c P e r f o r m a n c e . . . . . . . 6 T y p e o f n u m b e r s e n t e n c e 6 L e v e l o f d i f f i c u l t y 7 S o l u t i o n s t r a t e g y 7 S u m m a r y o f C h a p t e r 1 7 2. A R E V I E W O F T H E L I T E R A T U R E 8 N u m b e r i n P i a g e t i a n T h e o r y 8 T h e C o n c e p t o f N u m b e r 9 D e v e l o p m e n t o f t h e N u m b e r C o n c e p t . . . 9 V. C h a p t e r P a g e O p e r a t i o n s o n N u m b e r s 9 C o n c e p t o f N u m b e r C o n t r o v e r s y 1 0 T h e R e l a t i o n s h i p b e t w e e n O p e r a t i o n a l L e v e l a n d A r i t h m e t i c P e r f o r m a n c e 1 1 C o n s e r v a t i o n o f N u m b e r a n d A r i t h m e t i c P e r f o r m a n c e . . . . . 1 2 D e v e l o p m e n t a l s t a g e s o f n u m b e r c o n s e r v a t i o n 1 2 C o n s e r v a t i o n a n d n u m b e r o p e r a t i o n s . 1 2 S t u d i e s i n v e s t i g a t i n g t h e r e l a t i o n s h i p b e t w e e n c o n s e r v a t i o n o f n u m b e r a n d a r i t h m e t i c p e r f o r m a n c e 1 3 C l a s s I n c l u s i o n a n d A r i t h m e t i c P e r f o r m a n c e l £ D e v e l o p m e n t a l s t a g e s o f c l a s s i n c l u s i o n . . . . . . . . . 1 6 C l a s s i n c l u s i o n a n d m i s s i n g a d d e n d s e n t e n c e s 1 7 S t u d i e s i n v e s t i g a t i n g t h e r e l a t i o n s h i p b e t w e e n c l a s s i n c l u s i o n a n d m i s s i n g a d d e n d s e n t e n c e s . . . . . . 17 A F o r m a l S t a t e m e n t o f t h e F i r s t H y p o t h e s i s • 18 ; T h e R e l a t i o n s h i p b e t w e e n T y p e o f N u m b e r S e n t e n c e a n d A r i t h m e t i c P e r f o r m a n c e 1 9 S t u d i e s I n v e s t i g a t i n g t h e R e l a t i o n s h i p b e t w e e n T y p e o f N u m b e r S e n t e n c e a n d A r i t h m e t i c P e r f o r m a n c e 1 9 T a s k A n a l y s i s o f N u m b e r S e n t e n c e s . . 2 0 A F o r m a l S t a t e m e n t o f t h e S e c o n d H y p o t h e s i s . 2 3 T h e R e l a t i o n s h i p b e t w e e n L e v e l o f M f f i c u l t y a n d A r i t h m e t i c P e r f o r m a n c e . . 2 3 S t u d i e s I n v e s t i g a t i n g t h e R e l a t i o n s h i p b e t w e e n L e v e l o f D i f f i c u l t y a n d A r i t h m e t i c P e r f o r m a n c e 2h A F o r m a l S t a t e m e n t o f t h e T h i r d H y p o t h e s i s 2 5 * T h e I n t e r a c t i v e E f f e c t s o f O p e r a t i o n a l L e v e l , T y p e o f N u m b e r S e n t e n c e , a n d L e v e l o f D i f f i c u l t y o n A r i t h m e t i c P e r f o r m a n c e 2 5 * vi. C h a p t e r P a g e T h e I n t e r a c t i v e E f f e c t o f O p e r a t i o n a l L e v e l a n d T y p e o f N u m b e r S e n t e n c e 25 A f o r m a l s t a t e m e n t o f t h e f o u r t h h y p o t h e s i s 26 T h e I n t e r a c t i v e E f f e c t o f O p e r a t i o n a l L e v e l a n d L e v e l o f D i f f i c u l t y . 26 A f o r m a l s t a t e m e n t o f t h e f i f t h h y p o t h e s i s . 27 T h e I n t e r a c t i v e E f f e c t o f O p e r a t i o n a l L e v e l , T y p e o f N u m b e r S e n t e n c e , a n d L e v e l o f D i f f i c u l t y o n A r i t h m e t i c P e r f o r m a n c e . . . 2 7 A f o r m a l s t a t e m e n t o f t h e s i x t h h y p o t h e s i s 27 S o l u t i o n S t r a t e g y S t u d y 27 D e v e l o p m e n t a l S e q u e n c e o f C o u n t i n g S t r a t e g i e s 28 A d d i t i o n c o u n t i n g s t r a t e g i e s 28 S u b t r a c t i o n c o u n t i n g s t r a t e g i e s 29 M i s s i n g a d d e n d c o u n t i n g s t r a t e g i e s . . 30 O b s e r v a t i o n S t u d y o f S o l u t i o n S t r a t e g i e s 33 S t a t e m e n t o f S t a t i s t i c a l H y p o t h e s e s . . . . . 3U 3. M E T H O D 35 S a m p l e 35 S u b j e c t s 35 S e l e c t i o n P r o c e d u r e . . . . . . . . . . 35 P i a g e t i a n A s s e s s m e n t . . . 36 T a s k D e s c r i p t i o n s 36 S i m p l e s e r i a t i o n . . . . . . . . . . . . . 37 D o u b l e s e r i a t i o n . . . . . 37 O n e - w a y c l a s s i f i c a t i o n ( f i r s t f o r m ) . . . 38 O n e - w a y c l a s s i f i c a t i o n ( s e c o n d f o r m ) . . . . . 38 T w o - w a y c l a s s i f i c a t i o n ( f i r s t f o r m ) . . . . . 38 T w o - w a y c l a s s i f i c a t i o n ( s e c o n d f o r m ) 39 v i i . C h a p t e r P a g e T h r e e - w a y c l a s s i f i c a t i o n . . . 39 C l a s s i n c l u s i o n ( f i r s t f o r m ) 39 C l a s s i n c l u s i o n ( s e c o n d f o r m ) hO C o n s e r v a t i o n o f n u m b e r • hO C o n s e r v a t i o n o f c o n t i n u o u s q u a n t i t y Ul C o n s e r v a t i o n o f d i s c o n t i n u o u s q u a n t i t y . Ul T e s t A d m i n i s t r a t i o n Ul S c o r i n g P r o c e d u r e . . . . . . k2 I n t e r - r a t e r R e l i a b i l i t y . . . . U3 A s s i g n m e n t t o O p e r a t i o n a l L e v e l 1*3 L e v e l o n e - t r a n s i t i o n a l U3 L e v e l t w o - e a r l y c o n c r e t e o p e r a t i o n a l hh L e v e l t h r e e - l a t e r c o n c r e t e o p e r a t i o n a l UU A r i t h m e t i c A s s e s s m e n t . . . . h$ A r i t h m e t i c T e s t V a r i a b l e s h$ T y p e o f n u m b e r s e n t e n c e 16 L e v e l o f d i f f i c u l t y US A r i t h m e t i c T e s t I t e m s . U6 A r i t h m e t i c T e s t M a t e r i a l s U6 T e s t b o o k l e t U6 P u p i l m a t e r i a l s U6 I n t e r v i e w r e c o r d i n g f o r m hi S o l u t i o n S t r a t e g y O b s e r v a t i o n hi T e s t A d m i n i s t r a t i o n . . . . . . . hi S c o r i n g P r o c e d u r e U8 v i i i . C h a p t e r P a g e T e s t i t e m s . . . . . . . U3 S o l u t i o n s t r a t e g y . . U8 I n t e r - r a t e r r e l i a b i l i t y U8 D a t a A n a l y s i s 10 S u m m a r y o f C h a p t e r 3 . • h9 i i . R E S U L T S 50 A r i t h m e t i c T e s t R e s u l t s 50 S e c o n d O r d e r I n t e r a c t i o n . . . . 52 T r a n s i t i o n a l 53 E a r l y c o n c r e t e • 53 L a t e r c o n c r e t e 53 F i r s t O r d e r I n t e r a c t i o n s 5U O p e r a t i o n a l l e v e l a n d t y p e o f n u m b e r s e q u e n c e 5u O p e r a t i o n a l l e v e l a n d l e v e l o f d i f f i c u l t y . . . 55 M a i n E f f e c t s • 56 O p e r a t i o n a l l e v e l 56 T y p e o f n u m b e r s e n t e n c e 57 L e v e l o f d i f f i c u l t y 58 S o l u t i o n S t r a t e g y R e s u l t s 58 A d d i t i o n C o u n t i n g S t r a t e g i e s . . . . . 59 S u b t r a c t i o n C o u n t i n g S t r a t e g i e s . . . . 61 M i s s i n g A d d e n d C o u n t i n g S t r a t e g i e s 6h S u m m a r y o f C h a p t e r . U 66 5 . D I S C U S S I O N , C O N C L U S I O N S , A N D I M P L I C A T I O N S 6? A r i t h m e t i c A c h i e v e m e n t . . . . 6 7 S e c o n d O r d e r I n t e r a c t i o n . . . . 68 ix. C h a p t e r P a g e T r a n s i t i o n a l 68 E a r l y c o n c r e t e . . . . . . . . . . . . . 70 L a t e r c o n c r e t e • 70 F i r s t O r d e r I n t e r a c t i o n s . . . . . . . . . . 7 1 I n t e r a c t i v e e f f e c t o f o p e r a t i o n a l l e v e l a n d t y p e o f n u m b e r s e n t e n c e 7 1 I n t e r a c t i v e e f f e c t o f o p e r a t i o n a l l e v e l a n d l e v e l o f d i f f i c u l t y 72 M a i n E f f e c t s 73 O p e r a t i o n a l l e v e l 73 T y p e o f n u m b e r s e n t e n c e 7U L e v e l o f d i f f i c u l t y 7U S o l u t i o n S t r a t e g y S t u d y 76 A d d i t i o n c o u n t i n g s t r a t e g i e s 76 S u b t r a c t i o n c o u n t i n g s t r a t e g i e s . 77 M i s s i n g a d d e n d c o u n t i n g s t r a t e g i e s . 78 L i m i t a t i o n s o f t h e S t u d y 79 I m p l i c a t i o n s f o r C l a s s r o o m s 81 S u g g e s t i o n s f o r F u r t h e r R e s e a r c h • 83 S u m m a r y o f C h a p t e r 5 • • • 85 R E F E R E N C E S 86 A P P E N D I C E S A . T w o - w a y C l a s s i f i c a t i o n M a t r i x G r i d 90 B . T w o - w a y C l a s s i f i c a t i o n M a t r i x G r i d . 9 1 C . T h r e e - w a y C l a s s i f i c a t i o n M a t r i x G r i d 92 D . P i a g e t i a n A s s e s s m e n t • 93 E . A r i t h m e t i c I n t e r v i e w R e c o r d i n g F o r m . . . . . . . . . 95 L I S T O F T A B L E S T a b l e P a g e 2.1 S t a g e s i n t h e D e v e l o p m e n t o f N u m b e r C o n s e r v a t i o n . . . . . . . . 12 2.2 S t a g e s i n t h e D e v e l o p m e n t o f C l a s s I n c l u s i o n • • 16 2 .3 S e q u e n c e o f S k i l l s a n d C o n c e p t s f o r S o l v i n g A d d i t i o n S e n t e n c e s • 21 2,k S e q u e n c e o f S k i l l s a n d C o n c e p t s f o r S o l v i n g S u b t r a c t i o n S e n t e n c e s . . . . . 22 2.5 S e q u e n c e o f S k i l l s a n d C o n c e p t s f o r S o l v i n g M i s s i n g A d d e n d S e n t e n c e s . . . . 23 2.6 A d d i t i o n C o u n t i n g S t r a t e g i e s . 29 2.7 S u b t r a c t i o n C o u n t i n g S t r a t e g i e s . . 30 2 . 8 M i s s i n g A d d e n d C o u n t i n g S t r a t e g i e s . . . 31 3.1 P i a g e t i a n A s s e s s m e n t S c o r i n g P r o c e d u r e . . . h2 3.2 A r i t h m e t i c T e s t I t e m s U6 hml 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 T a b l e O p e r a t i o n a l L e v e l b y T y p e o f N u m b e r S e n t e n c e b y L e v e l o f D i f f i c u l t y 50 h»2 C e l l 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 S u m m a r y T a b l e O p e r a t i o n a l L e v e l b y T y p e o f N u m b e r S e n t e n c e b y L e v e l o f D i f f i c u l t y 51 a . 3 C o u n t i n g S t r a t e g i e s f o r C o r r e c t R e s p o n s e s t o A d d i t i o n S e n t e n c e s 59 h»h S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r C o r r e c t R e s p o n s e s t o A d d i t i o n S e n t e n c e s 60 i t . 5 S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r I n c o r r e c t R e s p o n s e s t o A d d i t i o n S e n t e n c e s 61 U.6 C o u n t i n g S t r a t e g i e s f o r C o r r e c t R e s p o n s e s t o S u b t r a c t i o n S e n t e n c e s 62 lu7 S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r C o r r e c t R e s p o n s e s t o S u b t r a c t i o n S e n t e n c e s . 62 U.8 S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r I n c o r r e c t R e s p o n s e s t o S u b t r a c t i o n S e n t e n c e s 63 T a b l e P a g e U.9 C o u n t i n g S t r a t e g i e s f o r C o r r e c t R e s p o n s e s t o M i s s i n g A d d e n d S e n t e n c e s . . . * . 6k U.10 S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r C o r r e c t R e s p o n s e s t o M i s s i n g A d d e n d S e n t e n c e s • 6 £ l u l l S t r a t e g y F r e q u e n c y a n d P e r c e n t a g e s f o r I n c o r r e c t R e s p o n s e s t o M i s s i n g A d d e n d S e n t e n c e s . 6£ L I S T O F F I G U R E S F i g u r e P a g e l u l G r a p h i c a l D i s p l a y o f t h e M e a n S c o r e s O p e r a t i o n a l L e v e l b y T y p e o f N u m b e r S e n t e n c e b y L e v e l o f D i f f i c u l t y 52 U.2 G r a p h i c a l D i s p l a y o f t h e M e a n S c o r e s O p e r a t i o n a l L e v e l b y T y p e o f N u m b e r S e n t e n c e 5U i t . 3 G r a p h i c a l D i s p l a y o f t h e M e a n S c o r e s O p e r a t i o n a l L e v e l b y L e v e l o f D i f f i c u l t y . 55 x i i i . A C K N O W L E D G E M E N T T h e w r i t e r e x p r e s s e s h e r a p p r e c i a t i o n t o h e r a d v i s o r , D r . P a t r i c i a A r l i n , f o r t r a i n i n g i n t h e P i a g e t i a n a s s e s s m e n t p r o c e d u r e s a n d f o r t h e o p p o r t u n i t y t o c o n d u c t t h i s s t u d y . A p p r e c i a t i o n i s e x t e n d e d t o D r . D o u g l a s O w e n s f o r h i s a s s i s t a n c e a n d e n c o u r a g e m e n t t h r o u g h o u t t h e w r i t i n g o f t h i s m a n u s c r i p t . T h e a u t h o r w o u l d a l s o l i k e t o t h a n k D r . E m i l y O k s a k o v s k i f o r h e r e d i t o r i a l c o m m e n t s . F u r t h e r a p p r e c i a t i o n i s g r a t e f u l l y a c k n o w l e d g e d t o t h r e e p e o p l e w h o h a v e c o n t r i b u t e d i n v a r i o u s w a y s t o t h i s t h e s i s : D r . T o d d R o g e r s f o r h i s g u i d a n c e i n t h e s t a t i s t i c a l a n a l y s i s j D r . P a u l i n e W e i n s t e i n f o r h e r c o u n s e l a n d s u p p o r t j a n d J a n e t W e b s t e r f o r h e r g u i d a n c e a n d i n s p i r a t i o n . 1 C h a p t e r 1 I N T R O D U C T I O N C u r r e n t r e s e a r c h i n e a r l y c h i l d h o o d e d u c a t i o n h a s f o c u s e d o n t h e p r o b l e m o f i d e n t i f y i n g e f f e c t i v e m e a s u r e s f o r p r e d i c t i n g t h e c h i l d ' s p o t e n t i a l s u c c e s s i n u n d e r s t a n d i n g a r i t h m e t i c c o n c e p t s . T r a d i t i o n a l l y , n u m e r o u s t e s t s o f a r i t h -m e t i c r e a d i n e s s a n d i n t e l l i g e n c e h a v e b e e n u s e d t o e v a l u a t e t h e g r a d e o n e c h i l d ' s r e a d i n e s s f o r t h e a r i t h m e t i c c o n t e n t i n t h e f i r s t g r a d e m a t h e m a t i c s p r o g r a m . H o w e v e r , i n a n e f f o r t t o o b t a i n a l t e r n a t e a s s e s s m e n t d e v i c e s , r e c e n t a t t e n t i o n h a s b e e n d i r e c t e d t o w a r d t h e u s e o f P i a g e t ' s l o g i c a l t h i n k i n g t a s k s . K a m i ! (1971) p r o p o s e d t w o r e a s o n s f o r u t i l i z i n g P i a g e t i a n t a s k s t o e v a l u a t e t h e y o u n g c h i l d ' s a c a d e m i c p r o g r e s s . F i r s t , u n l i k e p s y c h o m e t r i c t e s t s , P i a g e t ' s p r o c e d u r e s e x a m i n e n o t o n l y t h e c h i l d ' s a n s w e r s t o q u e s t i o n s b u t a l s o t h e t h i n k i n g p r o c e s s e s u n d e r l y i n g h i s o r h e r r e s p o n s e s . S e c o n d , P i a g e t ' s l o g i c a l t h i n k i n g t a s k s a r e b a s e d o n a t h e o r y w h i c h d e s c r i b e s t h e c o g n i t i v e d e v e l o p m e n t o f c h i l d r e n . I t h a s b e e n r e c o m m e n d e d b y a t l e a s t f o u r e a r l y c h i l d h o o d e d u c a t o r s t h a t P i a g e t i a n t a s k s b e i n c l u d e d a s o n e a s s e s s m e n t t o f o r e c a s t t h e c h i l d ' s s u c c e s s i n u n d e r s t a n d i n g n u m b e r o p e r a t i o n s ( K a m i i , 1971; D i m i t r o v s k y a n d A l m y , 1975; K a m i i a n d D e V r i e s , 1978). H o w e v e r , t h e r e l e v a n c e o f P i a g e t i a n t a s k s t o t h e c h i l d ' s s u c c e s s o n p a r t i c u l a r s k i l l s a n d c o n c e p t s i n t h e g r a d e o n e a r i t h m e t i c p r o g r a m i s n o t c l e a r . S t u d i e s h a v e e x a m i n e d t h e l o g i c a l a b i l i t i e s o f y o u n g c h i l d r e n i n a n a t -t e m p t t o d e t e r m i n e i f a r e l a t i o n s h i p e x i s t s w i t h p e r f o r m a n c e o n v a r i o u s t y p e s o f n u m b e r s e n t e n c e s . S e v e r a l r e s e a r c h e r s h a v e i n v e s t i g a t e d w h e t h e r t h e a b i l -i t y t o c o n s e r v e n u m b e r i n f l u e n c e s p e r f o r m a n c e o n a d d i t i o n a n d s u b t r a c t i o n s e n t e n c e s ( W h e a t l e y , 1967; S t e f f e , 1971; L e B l a n c , 1968). O t h e r r e s e a r c h e r s h a v e c o n s i d e r e d w h e t h e r t h e a b i l i t y t o s o l v e c l a s s i n c l u s i o n p r o b l e m s 2. i n f l u e n c e s p e r f o r m a n c e o n m i s s i n g a d d e n d s e n t e n c e s ( H e w l e t t , 197kj S t e f f e , S p i k e s a n d H i r s t e i n , 1976 j a n d K e l l e h e r , 1977). A l t h o u g h m a n y o f t h e s e s t u d -i e s i n d i c a t e t h a t l o g i c a l a b i l i t i e s a r e a p r e r e q u i s i t e f o r s o l v i n g n u m b e r s e n t e n c e s , i n s o m e i n s t a n c e s , t h e m e t h o d s t h e r e s e a r c h e r s u s e d t o a s s e s s t h e l o g i c a l t h i n k i n g c o n c e p t s o f c o n s e r v a t i o n a n d c l a s s i n c l u s i o n h a v e d e v i a t e d s i g n i f i c a n t l y f r o m P i a g e t ' s p r o c e d u r e s . T h e r e f o r e , t h e r e l a t i o n s h i p b e t w e e n t h e a b i l i t y t o s o l v e P i a g e t ' s t a s k s a n d c o r r e c t l y a n s w e r a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s n e e d s t o b e e x a m i n e d f u r t h e r u s i n g P i a g e t ' s c o n -v e n t i o n a l t a s k s . A s e c o n d p r o b l e m w i t h m a n y o f t h e c u r r e n t s t u d i e s i s t h a t a c c u r a c y o f p e r f o r m a n c e h a s b e e n c o n s i d e r e d , b u t s o l u t i o n m e t h o d s h a v e b e e n i g n o r e d . S t e f f e , S p i k e s , a n d H i r s t e i n (1976) c o n d u c t e d t h e o n l y s t u d y t h a t h a s e x a m -i n e d t h e r e l a t i o n s h i p b e t w e e n t h e g r a d e o n e c h i l d ' s a b i l i t y t o u n d e r s t a n d c o n s e r v a t i o n o f q u a n t i t y a n d c l a s s i n c l u s i o n p r o b l e m s a n d h i s o r h e r c o u n t i n g s t r a t e g i e s f o r s o l v i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s . H o w e v e r , t h e a u t h o r s f a i l e d t o f i n d a s i g n i f i c a n t r e l a t i o n s h i p b e t w e e n c l a s s i n c l u s i o n a n d n u m b e r s e n t e n c e p e r f o r m a n c e . T h e r e f o r e , t h e p r o b l e m w h i c h f o r m e d t h e b a s i s o f t h e p r e s e n t s t u d y w a s a n i n v e s t i g a t i o n o f t h e r e l a t i o n -s h i p b e t w e e n t h e g r a d e o n e c h i l d ' s c o g n i t i v e m a t u r i t y a s a s s e s s e d b y P i a g e t ' s s e r i a t i o n , c l a s s i f i c a t i o n , a n d c o n s e r v a t i o n t a s k s a n d h i s o r h e r a c c u r a c y a n d s o l u t i o n m e t h o d s w h e n s o l v i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t -e n c e s . S t a t e m e n t o f t h e P r o b l e m T h i s s t u d y w a s d e s i g n e d t o e x a m i n e t h e r e l a t i o n s h i p b e t w e e n o p e r a t i o n a l l e v e l a n d a r i t h m e t i c p e r f o r m a n c e o f g r a d e o n e c h i l d r e n . T w o a s p e c t s o f t h i s p e r f o r m a n c e a r e o f s p e c i f i c i n t e r e s t : 3. (1) I s t h e c h i l d ' s a b i l i t y t o c o r r e c t l y s o l v e a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s a t v a r i o u s l e v e l s o f d i f f i c u l t y r e l a t e d t o h i s o r h e r o p e r a t i o n a l l e v e l ? ( 2 ) I s t h e c h i l d ' s c o u n t i n g s t r a t e g y f o r s o l v i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s a t v a r i o u s l e v e l s o f d i f f i c u l t y r e l a t e d t o h i s o r h e r o p e r a t i o n a l l e v e l ? T h e r a t i o n a l e f o r a s s o c i a t i n g t h e s e t w o q u e s t i o n s w i t h t h e p r o b l e m w i l l b e c o m e a p p a r e n t f r o m t h e l i t e r a t u r e r e v i e w i n t h e f o l l o w i n g c h a p t e r . P u r p o s e o f t h e S t u d y A s t u d y o f t h i s n a t u r e h a s b o t h a t h e o r e t i c a l a n d p r a c t i c a l p u r p o s e . O f t h e o r e t i c a l i m p o r t a n c e i s a d e l i n e a t i o n o f l o g i c a l t h i n k i n g v a r i a b l e s w h i c h i n f l u e n c e t h e g r a d e o n e c h i l d ' s a r i t h m e t i c p e r f o r m a n c e . T h e p a r t i c u l a r c o n -t r i b u t i o n o f t h i s s t u d y w a s t h e a t t e m p t t o e x t e n d p r e v i o u s w o r k b y i n v e s t i g a t -i n g t h e e f f e c t o f c o g n i t i v e d e v e l o p m e n t a l l e v e l o n t h e a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d p e r f o r m a n c e o f g r a d e o n e c h i l d r e n . W i t h r e s p e c t t o p r a c t i c a l i s s u e s , t h i s s t u d y w a s d e s i g n e d f o r t h e p u r -p o s e o f : (1) i d e n t i f y i n g t h e l o g i c a l a b i l i t i e s w h i c h a r e a s s o c i a t e d w i t h s u c c e s s o n a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s , ( 2 ) i d e n t i f y i n g s p e c i f i c P i a g e t i a n t a s k s w h i c h m a y s e r v e t o a s s e s s t h e g r a d e o n e c h i l d ' s r e a d i n e s s f o r u n d e r s t a n d i n g t h e c o n c e p t s u n d e r l y i n g a d d i -t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s , a n d (3) i d e n t i f y i n g a d e v e l o p m e n t a l s e q u e n c e o f s o l u t i o n s t r a t e g i e s t h a t c o u l d s u g g e s t e i t h e r a t e a c h i n g s e q u e n c e o r t e a c h i n g m e t h o d w h i c h i s a p p r o p -r i a t e f o r p a r t i c u l a r o p e r a t i o n a l p e r f o r m a n c e . T h e p r a c t i c a l c o n t r i b u t i o n s o f t h i s s t u d y a r e c o n c e r n e d w i t h s e l e c t i n g l u a r i t h m e t i c t a s k s f o r c h i l d r e n t h a t a r e c o m p a t i b l e w i t h t h e i r s t a g e o f c o g n i t -i v e d e v e l o p m e n t . O r g a n i z a t i o n o f t h e T h e s i s T h i s t h e s i s c o n s i s t s o f f i v e c h a p t e r s . T h e r e m a i n d e r o f t h i s i n t r o d u c t -o r y c h a p t e r i s d e v o t e d t o d e f i n i t i o n s o f t h e v a r i a b l e s o f i n t e r e s t . C h a p t e r 2 d e a l s p r i m a r i l y w i t h a r e v i e w o f t h e p e r t i n e n t l i t e r a t u r e a n d c u l m i n a t e s w i t h t h e p r e s e n t a t i o n o f t h e s i x h y p o t h e s e s . C h a p t e r 3 c o n t a i n s a d e s c r i p -t i o n o f t h e m e t h o d o l o g y . T h e r e s u l t s a r e p r e s e n t e d i n C h a p t e r l u C h a p t e r 5> c o n t a i n s t h e d i s c u s s i o n a n d t h e c o n c l u s i o n s . D e f i n i t i o n o f t h e V a r i a b l e s T o c o m m u n i c a t e t h e a i m s a n d f i n d i n g s o f t h e s t u d y c l e a r l y , i t i s n e c e s -s a r y t o d e f i n e t h e t e r m s a s s o c i a t e d w i t h t h e v a r i a b l e s o f i n t e r e s t . T h e r e w a s o n e d e p e n d e n t v a r i a b l e , a r i t h m e t i c p e r f o r m a n c e , a n d t h r e e i n d e p e n d e n t v a r i a b l e s : o p e r a t i o n a l l e v e l , t y p e o f n u m b e r s e n t e n c e , a n d l e v e l o f d i f f i -c u l t y . C o g n i t i v e m a t u r i t y w a s r e g a r d e d i n a P i a g e t i a n f r a m e w o r k , t h e r e f o r e c o g n i t i v e m a t u r i t y a n d o p e r a t i o n a l l e v e l a r e s y n o n y m o u s . T h i s s e c t i o n o n d e f i n i t i o n s i s d i v i d e d i n t o t w o p a r t s . T h e f i r s t p a r t s p e c i f i e s t h e m e a n i n g s o f t e r m s a s s o c i a t e d w i t h c o g n i t i v e m a t u r i t y . T h e s e c -o n d p a r t d e a l s w i t h v o c a b u l a r y p e r t i n e n t t o a r i t h m e t i c p e r f o r m a n c e . T e r m s A s s o c i a t e d w i t h C o g n i t i v e M a t u r i t y O p e r a t i o n a l l e v e l . O p e r a t i o n a l l e v e l r e f e r s t o i n t e l l e c t u a l f u n c t i o n i n g a s d e f i n e d b y P i a g e t . I n P i a g e t ' s d e v e l o p m e n t a l a n a l y s i s o f i n t e l l e c t u a l g r o w t h , p r o g r e s s f r o m i n f a n c y t o a d u l t h o o d i s c h a r a c t e r i z e d b y m o v e m e n t t h r o u g h f o u r m a j o r s t a g e s : s e n s o r i m o t o r ( b i r t h t o t w o y e a r s ) ; p r e o p e r a t i o n a l ( t w o t o s e v e n y e a r s ) ; c o n c r e t e o p e r a t i o n a l ( s e v e n t o t w e l v e y e a r s ) ; a n d f o r -m a l o p e r a t i o n a l ( t w e l v e y e a r s a n d b e y o n d ) ( P i a g e t , 1 9 7 U ) . A s t h i s s t u d y w a s 5". l i m i t e d t o g r a d e o n e c h i l d r e n , t h e s t a g e s o f s p e c i f i c i n t e r e s t w e r e t h e p e r i o d o f t r a n s i t i o n b e t w e e n t h e p r e o p e r a t i o n a l a n d c o n c r e t e o p e r a t i o n a l s t a g e s a n d t h e c o n c r e t e o p e r a t i o n a l s t a g e . F o r t h e p u r p o s e o f t h i s s t u d y , t h r e e o p e r a -t i o n a l l e v e l s h a v e b e e n i d e n t i f i e d a n d l a b e l l e d : t r a n s i t i o n a l , e a r l y c o n -c r e t e o p e r a t i o n a l , a n d l a t e r c o n c r e t e o p e r a t i o n a l . T r a n s i t i o n a l . P r e o p e r a t i o n a l t h o u g h t i s c h a r a c t e r i z e d b y i t s a t t e n t i o n t o s t a t e s a n d i t s l a c k o f r e v e r s i b i l i t y ( I n h e l d e r a n d P i a g e t , 196U). T h e t r a n s i t i o n f r o m p r e o p e r a t i o n a l t o c o n c r e t e o p e r a t i o n a l t h o u g h t i s e v i d e n t w h e n t h e c h i l d i s c a p a b l e o f s i m p l e s e r i a t i o n a n d s i m p l e c l a s s i f i c a t i o n . T h e c h i l d ' s f i r s t a t t e m p t s t o o r d e r o b j e c t s t h a t v a r y a l o n g a d i m e n s i o n ( e . g . , l e n g t h , c i r c u m f e r e n c e , h e i g h t ) , a r e a c c o m p l i s h e d b y t r i a l a n d e r r o r , t h a t i s , b y p h y s i c a l l y c o m p a r i n g o n e o b j e c t w i t h a n o t h e r . S p o n t a n e o u s s e r i a t i o n i s m a n i f e s t e d w h e n t h e c h i l d c a n l o o k a t a n u n o r d e r e d s e t o f g r a d e d o b j e c t s , m e n t a l l y o r d e r t h e m , a n d t h e n p h y s i c a l l y a r r a n g e t h e m i n t h e c o r r e c t s e q u e n c e ( I n h e l d e r a n d P i a g e t , 196U). L o g i c a l c l a s s i f i c a t i o n b e g i n s t o e m e r g e w h e n t h e c h i l d c a n i d e n t i f y a p r o p e r t y t h a t i s c o m m o n t o a s e t o f o b j e c t s ( e . g . , c o l o r , s h a p e , s i z e ) . T h e a b i l i t y t o u n d e r s t a n d t h a t t h e s a m e o b j e c t c a n b e r e c l a s s i -f i e d i n t w o o r t h r e e w a y s b y d i f f e r e n t a t t r i b u t e s d e v e l o p s s u b s e q u e n t l y . A c -c o r d i n g t o P i a g e t , a s e r i e s o r a c l a s s c a n n o t b e c o n s t r u c t e d b y p e r c e p t u a l t h o u g h t c h a r a c t e r i s t i c o f t h e p r e o p e r a t i o n a l s t a g e . R a t h e r , t h e c h i l d m u s t b e c a p a b l e o f a b s t r a c t i n g a n d g e n e r a l i z i n g t h a t a n o b j e c t h a s a p a r t i c u l a r p o s i t i o n i n a s e r i e s o r t h a t a s e t o f o b j e c t s b e l o n g t o g e t h e r b e c a u s e t h e y s h a r e a c o m m o n a t t r i b u t e ( I n h e l d e r a n d P i a g e t , 196k), T h e r e f o r e , t h e t r a n s i -t i o n a l c h i l d w i t h h i s o r h e r a b i l i t y t o s u c c e e d o n s i m p l e s e r i a t i o n o r c l a s s i -f i c a t i o n t a s k s h a s d e v e l o p e d i n i t i a l c o n c e p t s t h a t a r e a s s o c i a t e d w i t h c o n -c r e t e o p e r a t i o n a l t h o u g h t ( I n h e l d e r , S i n c l a i r , a n d B o v e t , 197k)• E a r l y c o n c r e t e o p e r a t i o n a l . T h e e a r l y c o n c r e t e o p e r a t i o n a l t h i n k e r , i n 6 contrast to the preoperational child, i s able to disregard potentially mis-leading perceptual information and to deal with such abstract concepts as conservation and class inclusion (Flavell, 1963). Concrete operational children can distinguish between "appearance" and "reality" (McV. Hunt, I96l). Concrete or manipulative materials are s t i l l necessary but the child r e l i e s less on the perceptual saliency of the objects and more on the mental activ-i t y of re v e r s i b i l i t y (Inhelder, 1962). According to Piaget, there are three forms of reve r s i b i l i t y which are associated with the a b i l i t y to conserve: negation of inversion, reciprocity, and identity. I t has been argued that the child bases his or her argument of constancy on negation or inversion (a return to the original form would make them the same); reciprocity (height can compensate for width); or identity (nothing has been added or taken away) (Inhelder, Sinclair, and Bovet, 197U). These forms of reve r s i b i l i t y serve to distinguish the concrete operational thinker (Inhelder, 1972). Later concrete operational. The later concrete operational child has the cognitive maturity to understand the concepts of conservation and class inclus-ion. He or she i s able not only to classify correctly i n the form of additive groupings ( i . e . , hierarchies), but also to recognize and coordinate inclusions implied by that structure (i.e., A + A 1 = B; A = B - A 1; A > B). In other words, the child understands the relationship between the total class and i t s subclasses (Inhelder and Piaget, 196a). Terms Associated with Arithmetic Performance Type of number sentence refers to three equation formats. In particular, addition sentences refer to open addition sentences of the form a + b = Q , subtraction sentences refer to open subtraction sentences of the form a - b = [ and missing addend sentences refer to open addition sentences of the form a + • =b. 7. Level of d i f f i c u l t y refers to the magnitude of the numbers used i n the open sentence. In particular, level one includes basic fact combinations with sums or differences of ten or lessj level two includes basic fact com-binations between eleven and twenty with two one-digit numbers and one two-digit number i n the number sentencej and level three includes combinations between eleven and twenty with one one-digit number and two two-digit numbers i n the number sentence. Solution strategy refers to the counting procedure used to determine the answer to the number sentence. In this study the terms solution strategy and counting strategy are synonymous« Summary of Chapter 1 The purpose of this introductory chapter was to specify the nature of the problem. The problem investigated i n this study was the interactive i n -fluence of operational level, type of number sentence, and level of d i f f i -culty on the arithmetic performance of grade one children. The statement of the problem was followed by the purpose and organization of the present study. The chapter concluded with a discussion of the variables and definitions of the terms used. Chapter 2 i s devoted to a review of the literature which led to the formation of the six hypotheses which were tested. 8. Chapter 2 A REVIEW OF THE LITERATURE This chapter i s devoted to a review of the literature pertaining to cog-nitive maturity and arithmetic performance. As a result of this review six hypotheses were formulated. Main effects of operational level, type of number sentence, and level of d i f f i c u l t y were predicted. First order interactive effects of operational level and type of number sentence, and operational level and level of d i f f i c u l t y were expected. A second order interactive ef-fect of operational level, type of number sentence and level of d i f f i c u l t y was also anticipated. To aid c l a r i t y , the literature review i s divided into three parts. In the f i r s t part, Piaget's theory of number and the associated controversy i s discussed. The second part deals with the research pertinent to the variables related to arithmetic performance. This part i s subdivided into four sections: the relationship between operational level and arithmetic performance! the relationship between type of number sentence and arithmetic performance; the relationship between level of d i f f i c u l t y and arithmetic performance; and, f i n -a l l y , the interactive effects of these variables. The third part i s concerned with the research pertaining to the solution strategy study. The chapter con-cludes with a summary of the six s t a t i s t i c a l hypotheses. Number i n Piagetian Theory This part deals with Piaget's theory of number. Piaget's views on the concept of number, the development of the number concept, and operations on numbers are presented. A controversy regarding Piaget's position on number i s then discussed. 9 The Concept of Number For Piaget and Szeminska (1952) number i s at the same time a class and an asymmetrical relation. In Piaget's words: A cardinal number i s a class whose elements are conceived as 'units' that are equivalent, and yet distinct i n that they can be seriated, and therefore ordered. Conversely, each ordinal number i s a series whose terms though follow-ing one another according to the relations of order that determine their respective positions, are also units that are equivalent and can therefore be grouped as a class. Finite numbers are therefore necessarily at the same time ordinal and cardinal. (Piaget and Szeminska, 1952, p. 157.) According to Piaget and Szeminska ( 1952) , logical classes and numbers have a common basis, namely the additive operation which brings together the elements into a whole or separates the whole into parts. However, the d i f -ference between number and class arises from the fact that i n number the parts are homogeneous units while the parts of a class are only qualified classes that are united by their common properties. Development of the Number Concept Basic to Piaget's and Szeminska's position on the child's development of the concept of number i s the assumption that the growth of logic and num-ber are paral l e l (Piaget and Szeminska, 1952) . From his observations, Piaget states that an understanding of classes and order relations does not precede the development of number, rather, both emerge simultaneously. When the child i s capable of comprehending that classes give rise to hierarchical wholes (i. e . , A + A x = Bj B + B = C) and transitive relations (i.e., A < B, B <. C, then A<.C) to seriations he or she also becomes capable of understanding that number i s neither merely a uniting class nor merely a seriating relation but both a hierarchical class and a series (Piaget and Szeminska, 1952) . Operations on Numbers In relating logical concepts to number, Piaget proposes that the additive 10. composition of classes i s the psychological counterpart of the additive com-position of numbers. For example, to understand addition as an operation the child must recognize that two addends such as 1 + 6 can be expressed as 7 and that i n reverse 7 i s also 3 + It or 5 + 2, At the same time he or she must appreciate that addition and subtraction are two inverse processes where 3 + U = 7, 7 - 3 = U, and 7 - h = 3 ( i . e . , A + A 1 = B, B - A = A 1, and B - A^ * =A), In Piagetian theory, i t i s not until the logical concepts of invariance of number and conservation of numerical wholes are acquired that the child i s capable of comprehending the whole-part relationships inherent i n the operations of addition and subtraction (Piaget and Szeminska, 1952; Copeland, 197U). Concept of Number Controversy There i s a controversy i n the literature regarding Piaget's theory of number and i t s implications for teaching arithmetic. Macnamara (1975, 1976) and Brainerd (1973, 1974, 1976) have analyzed Piaget's theory of number from the standpoint of arithmetic. They argue that the theory i s unacceptable i n two respects. The f i r s t objection i s with Piaget's use of the term "class" and i t s reference to adding numbers. In Macnamara1s view, arithmetic i s involved more with sets than classes. He proposes that sets, unlike classes, are ar-bitrary collections of objects which have no logical hierarchical connection or class inclusion relationships. Therefore, teaching children class logic as a basis for arithmetic i s inadvisable. The second point of dissention i s the objection to Piaget's use of seri-ation to discriminate among the members of a class (Macnamara, 1975; Brainerd, 1976). Macnamara (1975) argues that a group of classified objects does not i n -clude units that are exactly equal from one to another, therefore i t i s i l -l o gical to appeal to order to distinguish the items to be ordered. 11 Researchers who support Macnamara's position claim that Piaget has re-l i e d too heavily on the work of logicians for an explanation of how children learn number (Macnamara, 1975; Brainerd, 1973, 1974, 1976). They hold the view that the concept of number develops through ordering and counting rather than classifying. Therefore, they recommend that teachers base their a r i t h -metic instruction on seriating ( i . e . , ordination) and counting. In response to Macnamara's and Brainerd's argument, the writings of Piaget suggest that counting alone does not develop the concepts of addition and subtraction. In counting, there i s merely awareness of a succession of events, and a more or less vague feeling that the sets i n question are exhausted or increased. At the level of elementary enumerations, there i s no operation by which the child can colligate the units into a real and stable whole. (Piaget and Szeminska, 1952, p. 198.) Piaget sees counting as a s k i l l necessary for quantifying setsj and views logi c a l concepts essential to understanding operations underlying addition and subtraction. Although there i s disagreement among mathematicians and psychologists regarding the child's development of arithmetic competency, neither claim to have a clear understanding of the problem. Macnamara has expressed the situa-tion well: ". . .we know very l i t t l e about the nature of the psychological elements and rules which play a part i n the child's arithmetical operations, to learn about these, we depend upon empirical work. . . (Macnamara, 1975, p. 429.) The Relationship between Operational Level  and Arithmetic Performance In this section, the relationship between logical thinking a b i l i t i e s and arithmetic performance i s examined. The discussion i s presented i n two 12. subsections. The f i r s t subsection deals with conservation of number, and the second subsection with class inclusion. This section concludes with a formal statement of the f i r s t hypothesis. Conservation of Number and Arithmetic Performance Piaget's developmental stages of conservation of number, the relevance of conservation to understanding addition and subtraction, and studies examin-ing this relationship are the topics pursued i n the following discussion. Developmental stages of number conservation. Piaget identified three stages which give rise to the concept of number conservation (Piaget and Szeminska, 1952). These stages and their associated behaviors are outlined i n Table 2.1. Table 2.1 Stages i n the Development of Conservation of Number (Adapted from Piaget and Szeminska, 195>2) Stage Performance Stage 1 The child i s ^ unable to match i n a one-to-one (no 1:1 correspondence) correspondence one set of objects to another. Stage 2 The child i s able to make a numerical one-to-one (transitional) correspondence between two sets of objects, but when one set i s rearranged he or she no longer agrees to the equivalence of the two sets. Stage 3 The child i s able to make a numerical one-to-one (conservation) correspondence between the two sets and agrees to their equivalence when one set is rearranged. Conservation and number operations. A conserver of number has achieved re v e r s i b i l i t y of thought, therefore he or she i s capable of understanding that a particular quantity i s invariant regardless of i t s arrangement. Piaget maintains that mobility of thought i s required to carry out the operations 13. underlying the concept of number, that i s , for combining and separating, or constructing and reconstructing a numerical set (i.e., adding and subtracting) (Irmen, 1974)• Di f f i c u l t i e s of synthesis are solved by the detachment from the child's pre-logical centration on perception and his or her growth toward the log i c a l dynamism of reversibility (Irmen, 1974). Therefore, without the a b i l i t y to conserve number, the operations of addition and subtraction are meaningless. Studies investigating the relationship between conservation of number  and arithmetic performance. Almy, Chittenden, and Miller (1966) were the f i r s t investigators to report that f i r s t grade children who conserve number achieve higher scores on tests of addition and subtraction facts. However, their application of Piaget's number conservation task i s deficient i n three significant ways! i ) only one row of blocks was used, therefore the child was unable to make a one-to-one correspondence, i i ) the child was asked to count the blocks before and after the transformation, therefore his or her judgement was based on counting rather than logic, and i i i ) the child was not asked to justify his response. It i s possible that children who would be assessed as nonconservers on Piaget's tasks were designated as conservers. Therefore, the generalizability of the results of this study i s limited. Wheatley (1967) investigated the relative contributions of conservation, cardination, one-to-one correspondence, and counting a b i l i t y i n learning f i r s t grade mathematics. He assessed these a b i l i t i e s near the beginning of f i r s t grade and compared them with mathematics achievement at the end of f i r s t grade. Conservers scored significantly higher than nonconservers on the Stanford Achievement Test: Arithmetic Section. However, children who could count to twenty-five at the beginning of the year scored no higher on the achievement test than those who were unable to count. Wheatley concluded that teachers liu should be aware of the concept of conservation and i t s important role i n de-veloping such operations as addition of whole numbers. Steffe (1966, 1971) examined the relationship between f i r s t grade child-ren's IQ, their a b i l i t y to conserve number, and their performance i n solving both addition facts and word problems. He found that children i n the lowest of four levels of number conservation scored significantly lower on the addi-tion test than the other three levels. He further divided the children into twelve groups; four levels of conservation by three groups based on IQ scores. The children who obtained the highest addition scores were those i n the high-est three levels of conservation and i n the top two IQ groups and those i n the lowest IQ group and highest conservation le v e l . However, children who were i n the lowest conservation level and lowest IQ group obtained the lowest scores on the addition test. The other four cells comprised the middle per-formance group. Steffe concluded that conservation of number i s a readiness variable for understanding the operation of addition and that the a b i l i t y to conserve number i s not necessarily a characteristic of children with high i n -telligence. Similarly, Le Blanc (1968) found comparable results with subtraction problems and the same pattern of levels of number conservation and IQ groups. His findings indicated that the combined effect of IQ and conservation were a better predictor of success i n subtraction problem solving than either taken separately. The findings of Steffe (1966, 1971), and Le Blanc (1968) regarding con-servation a b i l i t y and intelligence were verified by the work of DeVries (197U). She factor analyzed the performance of five to seven year old children on the Stanford Binet Test of Intelligence and a series of Piagetian tasks including conservation of number. The results indicated that these two measures not 15 only overlap to some degree but also assess different aspects of cognitive functioning. Therefore, the a b i l i t y to conserve number i s not necessarily indicative of high intelligence. Although Steffe's and Le Blanc's findings support Piaget's hypothesis, their method of assessing conservation a b i l i t y differed from Piaget's and Szeminska's (1952) procedure. They designed a p i c t o r i a l conservation task which required the child to use one-to-one correspondence to assess invari-ance of number. Since the task did not involve a physical transformation, the child was unable to base his answer on identity (i.e., nothing has been added or taken away). Mpiangu and Gentile (1975) used Piaget's number conservation tasks to determine whether conservers performed better than nonconservers after receiv-ing training i n adding two numbers below ten. The researchers concluded that conservation of number was not necessary to solve simple addition sentences. Steffe, et a l . (1976) took exception to Mpiangu's and Gentile's conclusion be-cause only rote counting or point counting was required i n the addition task, that i s , the children counted the dots on two cards and selected a third card with the equivalent number of dots. Therefore, nonconservers could solve the addition problems from physical knowledge (i . e . , using concrete materials) rather than by mental operations ( i . e . , mentally abstracting a quantity). The research reviewed above, although limited i n terms of methodological discrepancies, does serve to support the hypothesis that children who conserve number perform significantly higher on tests of addition and subtraction than children who f a i l to conserve number. Class Inclusion Piaget's developmental stages of class inclusion, the relevance of class inclusion to understanding missing addend sentences, and studies examining 16. this relationship are the topics pursued i n the following discussion. Developmental stages of class inclusion. Inhelder and Piaget (l961i) view the a b i l i t y to understand part-whole relations as a developmental cog-nitive acquisition and used class inclusion problems to measure this capacity. They stated: I t i s one thing to carry out the union expressed by A + A-'- = B and quite another to understand that, i t i s logically equivalent to i t s inverse A = B - A l which means that the whole, B, retains i t s identity and that the entire relation can be quantitatively expressed i n the form A < B. The conservation of the whole and the quanti-tative comparison of whole and part are the two essential characteristics of genuine class inclusion, (p. 117.) Inhelder and Piaget (196a) observed three stages i n the development of the a b i l i t y to simultaneously recognize the whole and i t s parts. These stages and their associated behaviors are identified i n Table 2.2 below. Table 2.2 Stages i n the Development of Class Inclusion (Adapted from Inhelder and Piaget, 196a) Stage Performance Stage 1 The child does not recognize that the whole i s (no class inclusion) greater than i t s parts, that i s , he or she does not regard class B as resulting from the union of A + A^ and class A as resulting from the subtrac-tion of A l from B (i.e., l o g i c a l addition and sub-traction). Stage 2 (intuitive class inclusion) Stage 3 (class inclusion) By t r i a l and error the child verifies that class B contains more elements than A or A-*-, nonconserva-tion of wholes prevails. The child immediately agrees that class B i s larger than class A or A l because he or she approaches the problem from the point of view of additive composition. He or she recognizes A as a subset of B and can determine A l by comparing A and B. 17. Class inclusion and missing addend sentences. Missing addend sentences logically require a comparison of a given addend to the sum or a part to a whole ( i . e . , A = B - A^). Children with no understanding of the missing ad-dend question often consider the addend and sum as two parts and add them to determine the sum. This error i s conceptual i n nature and may be related to an i n a b i l i t y to recognize the part-whole relationship expressed i n the number sentence. Conversely, the a b i l i t y to successfully solve missing addend sent-ences may reflect an understanding of that relationship. For both additive (i . e . , counting on from the missing addend to the sum), and subtractive (i.e., counting to the sum, separating the given addend and counting the missing addend) approaches the child must be able to simultaneously recognize the addend as part of the sum. Studies investigating the relationship between class inclusion and  missing addend sentences. Howlett (1973) examined the relationship between class inclusion performance and missing addend computation among f i r s t grade students. Performance on a Piagetian class inclusion task was used to class-i f y children as Stage 1 (incapable of solving the class inclusion problem) or Stage 3 (capable of solving the class inclusion problem). Stage 2 or transi-tional subjects were excluded from the study. Findings of the study showed that performance on both computational and verbal missing addend problems was significantly related (p < . 0 5 ) to performance on the class inclusion tasks. Howlett recommended that missing addend sentences be delayed until the child i s able to solve class inclusion tasks. Steffe, Spikes, and Hirstein (1976) investigated the relationship between class inclusion a b i l i t y and missing addend performance of f i r s t year children by using a non-Piagetian pre-test and post-test measure of class inclusion. No children were successful with the class inclusion pre-test, whereas thirty-two percent were successful with the post-test three months later. Results 18. i o f t h e s t u d y i n d i c a t e d t h a t c l a s s i n c l u s i o n c o r r e l a t e d n o n - s i g n i f i c a n t l y w i t h m i s s i n g a d d e n d p e r f o r m a n c e o r a n y f i r s t g r a d e a r i t h m e t i c c o n t e n t . T h e a u t h o r s a n a l y z e d t h e r e a s o n s f o r t h e l o w c o r r e l a t i o n a n d s u g g e s t e d t h a t m a n y c h i l d r e n w h o m a y h a v e b e e n a b l e t o s o l v e t h e c l a s s i n c l u s i o n t a s k f a i l e d t o d o s o b e c a u s e t h e i t e m s w e r e o f a p i c t o r i a l n a t u r e a n d p e r c e p t u a l l y d i s t r a c t i n g . H o w e v e r , a n o t h e r p o s s i b i l i t y m a y b e d u e t o t h e f a c t t h a t o n l y a f e w c h i l d r e n i n t h e s a m p l e w e r e a c t u a l l y c a p a b l e o f s o l v i n g c l a s s i n c l u s i o n p r o b l e m s . A c c o r d i n g t o I n h e l d e r a n d P i a g e t (I96u), t h e a b i l i t y t o c o m p r e h e n d p a r t - w h o l e r e l a t i o n s h i p s d e v e l o p s a t a p p r o x i m a t e l y e i g h t y e a r s o f a g e . S i n c e t h e s a m p l e c o n s i s t e d o f g r a d e o n e s t u d e n t s i t i s u n l i k e l y t h a t m a n y o f t h e c h i l d r e n w o u l d h a v e t h e c o g n i t i v e c o m p e t e n c e t o s o l v e c l a s s i n c l u s i o n p r o b l e m s . K e l l e h e r (1977) e x a m i n e d t h e r e l a t i o n s h i p b e t w e e n m i s s i n g a d d e n d p e r f o r m -a n c e a n d c l a s s i n c l u s i o n a b i l i t y w i t h g r a d e t w o s t u d e n t s . B o t h P i a g e t i a n a n d v e r b a l t a s k s w e r e u s e d t o a s s e s s t h e p a r t - w h o l e c o n c e p t . T h e r e s u l t s o f t h e s t u d y s h o w e d t h a t p e r f o r m a n c e o n t h e c l a s s i n c l u s i o n t a s k s a n d t h e m i s s i n g a d d e n d i t e m s o f t h e t y p e a = b c o r r e l a t e d s i g n i f i c a n t l y ( p < .01). A l t h o u g h t h e r e a r e f e w s t u d i e s r e l a t i n g c l a s s i n c l u s i o n t o m i s s i n g a d d e n d p e r f o r m a n c e , t w o o f t h e t h r e e s t u d i e s c i t e d i n d i c a t e t h a t a s t r o n g l i n k e x -i s t s b e t w e e n t h e a b i l i t y t o c o m p r e h e n d p a r t - w h o l e r e l a t i o n s a n d t h e a b i l i t y t o s o l v e m i s s i n g a d d e n d s e n t e n c e s . I n b o t h i n s t a n c e s , P i a g e t i a n c l a s s i n -c l u s i o n t a s k s w e r e i n c l u d e d a s o n e m e a s u r e o f t h e c h i l d ' s u n d e r s t a n d i n g o f t h e p a r t - w h o l e r e l a t i o n s . A l t h o u g h f u r t h e r r e s e a r c h i s n e e d e d , t h e r e s u l t s s u g g e s t t h a t P i a g e t i a n c l a s s i n c l u s i o n t a s k s a r e v a l i d m e a s u r e s o f p a r t - w h o l e r e l a t i o n s w i t h r e s p e c t t o m i s s i n g a d d e n d s e n t e n c e s . A F o r m a l S t a t e m e n t o f t h e F i r s t H y p o t h e s i s O p e r a t i o n a l l e v e l i s r e l a t e d t o g r a d e o n e a r i t h m e t i c p e r f o r m a n c e „ O n a n a r i t h m e t i c t e s t o f a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d n u m b e r s e n t e n c e s 1 9 a t t h r e e l e v e l s o f d i f f i c u l t y , l a t e r c o n c r e t e o p e r a t i o n a l c h i l d r e n w i l l o b t a i n h i g h e r s c o r e s t h a n e a r l y c o n c r e t e o p e r a t i o n a l c h i l d r e n w h o , i n t u r n , w i l l o b -t a i n h i g h e r s c o r e s t h a n t r a n s i t i o n a l c h i l d r e n . R e l a t i o n s h i p b e t w e e n T y p e o f N u m b e r S e n t e n c e  a n d A r i t h m e t i c P e r f o r m a n c e W h i l e P i a g e t i a n r e s e a r c h e r s h a v e a t t e m p t e d t o a n a l y z e l o g i c a l t h i n k i n g a b i l i t i e s u n d e r l y i n g a d d i t i o n a n d s u b t r a c t i o n , m a t h e m a t i c s e d u c a t o r s a n d c o g -n i t i v e p s y c h o l o g i s t s h a v e e n d e a v o r e d t o i d e n t i f y t h e m a t h e m a t i c a l c o n c e p t s w h i c h a r e n e c e s s a r y f o r s o l v i n g o p e n n u m b e r s e n t e n c e s . I n t h i s s e c t i o n , s t u d -i e s t h a t h a v e b e e n c o n d u c t e d b y m a t h e m a t i c s e d u c a t o r s i n v e s t i g a t i n g t h e r e l a -t i o n s h i p b e t w e e n a r i t h m e t i c a c h i e v e m e n t a n d t y p e o f n u m b e r s e n t e n c e a r e p r e -s e n t e d . A t a s k a n a l y s i s s p e c i f y i n g t h e s k i l l s a n d r e l a t e d m a t h e m a t i c a l c o n -c e p t s a s s o c i a t e d w i t h e a c h t y p e o f n u m b e r s e n t e n c e i s t h e n o u t l i n e d . T h i s s e c t i o n c u l m i n a t e s w i t h a f o r m a l s t a t e m e n t o f t h e s e c o n d h y p o t h e s i s . S t u d i e s I n v e s t i g a t i n g t h e R e l a t i o n s h i p b e t w e e n T y p e o f N u m b e r  S e n t e n c e a n d A r i t h m e t i c P e r f o r m a n c e S e v e r a l s t u d i e s h a v e i n v e s t i g a t e d p e r f o r m a n c e d i f f e r e n c e s o n v a r i o u s t y p e s o f o p e n s e n t e n c e s . B e a t t i e a n d D e i c h m a n n ( 1 9 7 a ) a n a l y z e d t h e t y p e o f e r r o r s g r a d e s o n e a n d t w o c h i l d r e n m a d e w h e n s o l v i n g o p e n n u m b e r s e n t e n c e s . T h e e r r o r s w e r e c o d e d : " p r o c e s s " , " c o m p u t a t i o n a l " , a n d " r a n d o m " . T h e y r e p o r t e d t h a t a p l a c e h o l d e r i n e i t h e r t h e l e f t o r i n t h e m i d d l e o f a n a d d i t i o n o r a s u b t r a c t i o n s e n t e n c e p r o d u c e d h i g h e r p r o c e s s a n d r a n d o m e r r o r r a t e f o r f i r s t g r a d e c h i l d r e n . S i n c e m i s s i n g a d d e n d s e n t e n c e s a p p e a r e d i n t h e a r i t h m e t i c w o r k b o o k s l e s s f r e q u e n t l y t h a n a d d i t i o n a n d s u b t r a c t i o n s e n t e n c e s , t h e r e s e a r c h e r s c o n c l u d e d t h a t t h e e r r o r r a t e w a s d i r e c t l y r e l a t e d t o a m o u n t o f p r a c t i c e . N o c o n s i d e r a t i o n w a s g i v e n t o t h e c o g n i t i v e c o m p e t e n c i e s r e q u i r e d t o s o l v e t h e v a r i o u s t y p e s o f 20. number sentences. Weaver (1973) examined the performance of f i r s t , second, and third grade students on twenty types of open addition and subtraction sentences generated from basic facts with sums between ten and eighteen. His descriptive data i n -dicated that differential achievement effects existed at the three grade lev-el s . These effects were: operation (addition and subtraction); placeholder position; and solvability i n the set of whole numbers. The results showed that addition sentences were correctly answered more frequently than subtraction sentences, which were correctly answered more frequently than missing addend sentences. Weaver suggested that the findings reflect not only the relative amount of workbook practice provided for each sentence format but also the failure or i n a b i l i t y of pupils to use certain mathematical ideas or properties. Suppes1 (1967) research results indicated that grade one students made fewer errors on a + b = D number sentences than on a ± Q = b number senten-ces. He concluded that the difference i n d i f f i c u l t y was due to the transform-ation required for solution. For example, a + b = Q i s i n canonical form and can be solved without transformation, whereas a +0 =b must be transformed to canonical form before solution i s possible (a +0 = b-S'0 = b - a-?-b - a = • ) . Task Analysis of Number Sentences As a response to research evidence indicating that some types of number sentences are more d i f f i c u l t for children to solve, mathematics educators and cognitive psychologists have attempted to analyze the cognitive competencies involved i n solving addition, subtraction, and missing addend sentences. Ac-cording to Gagne (l97u), the acquisition of a complex concept or s k i l l i s a hierarchical process i n which lower order concepts or s k i l l s are gradually integrated to form more complex ones. In applying the procedure of task an-alysis to the learning of addition and subtraction concepts at the grade one 2 1 l e v e l , S p i t l e r a n d M c K i n n o n ( 1 9 7 6 ) h a v e d e s i g n e d a h i e r a r c h y f o r t e a c h i n g t h e n u m b e r f a c t s t o t e n . T h e y p r o p o s e t h a t o p e n s e n t e n c e s b e t a u g h t i n t h e f o l -l o w i n g o r d e r : a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d . A n o u t l i n e o f t h e s k i l l s a s s o c i a t e d w i t h e a c h n u m b e r s e n t e n c e , t h e c o r r e s p o n d i n g m a t h e m a t i c a l c o n c e p t s , a n d t h e h y p o t h e s i z e d l o g i c a l t h i n k i n g a b i l i t i e s i s p r e s e n t e d i n T a b l e s 2 . 3 , 2 . U , a n d 2 . 5 . T a b l e 2 . 3 S e q u e n c e o f S k i l l s a n d C o n c e p t s f o r S o l v i n g A d d i t i o n S e n t e n c e s ( A d a p t e d f r o m S p i t l e r a n d M c K i n n o n , 1 9 7 6 ) S k i l l M a t h e m a t i c a l C o n c e p t s ^ L o g i c a l T h i n k i n g C o n c e p t s C a n s a y t h e n u m b e r n a m e s i n s e q u e n c e t o t e n C a n m a t c h a c o u n t t o a n o b j e c t t o d e t e r m i n e t h e n u m b e r o f e l e m e n t s i n a s e t t o 1 0 C a n m a t c h e q u i v a l e n t s e t s h a v i n g t h e s a m e s p a t i a l a r r a n g e m e n t C a n m a t c h t h e s e t s h a v i n g d i f f e r -e n t s p a t i a l a r r a n g e m e n t s C a n a d d o n o n e s a n d n a m e t h e s e t s t o t e n C a n r e m o v e o n e a t a t i m e f r o m a s e t o f t e n a n d c o u n t b a c k w a r d s f r o m t e n t o o n e C a n n a m e s e t s o n e g r e a t e r t h a n a n d o n e l e s s t h a n t o 1 0 R o t e c o u n t i n g C a r d i n a l n u m b e r O r d i n a l n u m b e r # 1 : 1 c o r r e s p o n d e n c e ( d o u b l e s e r i a t i o n ) ( t r a n s i t i o n a l c o n s e r v a t i o n ) # 1 : 1 c o r r e s p o n d e n c e ( t r a n s i t i o n a l c o n s e r v a t i o n ) # C o n s e r v a t i o n o f n u m b e r ( 1 : 1 c o r r e s -p o n d e n c e w i t h l a s t i n g e q u i v a l e n c e ) O r d e r o f c o u n t i n g n u m b e r s S u b t r a c t i o n c a n " u n d o " a d d i t i o n O r d i n a l n u m b e r C a r d i n a l n u m b e r # C o n s e r v a t i o n o f n u m b e r C a n j o i n t w o s e t s o f o b j e c t s u s i n g t h e a d d i t i o n s y m b o l , e . g . , h + 1 - 5 C a n s t a t e t h e s u m w h e n o n e i s a d d e d t o t h e c o u n t i n g n u m b e r s t o 9 + 1 K n o w l e d g e o f a d d i t i o n s y m b o l C a r d i n a l n u m b e r O r d i n a l n u m b e r # C o n s e r v a t i o n o f n u m b e r K n o w l e d g e o f a d d i t i o n f a c t s 1 + 1 t o 9 + 1 # C o n s e r v a t i o n o f n u m b e r 22 T a b l e 2.3 C o n t ' d . S k i l l M a t h e m a t i c a l C o n c e p t s - ^ L o g i c a l T h i n k i n g C o n c e p t s C a n s t a t e t h e s u m f o r a d d i t i o n f a c t s t o t e n ( n u m b e r f a m i l i e s 5 t o 10), e . g . , • + • = 5 C a n d e m o n s t r a t e a n u n d e r s t a n d i n g o f t h e c o m m u t a t i v e p r o p e r t y o f a d d i t i o n f o r n u m b e r f a c t s t o 10 e . g . , 2 + 3 = 5 3 + 2 = 5 K n o w l e d g e o f a d d i t i o n f a c t s u m s t o 10 - ^ C o n s e r v a t i o n o f n u m b e r • ^ C o n s e r v a t i o n o f n u m b e r T a b l e 2.U S e q u e n c e o f S k i l l s a n d C o n c e p t s f o r S o l v i n g S u b t r a c t i o n S e n t e n c e s ( A d a p t e d f r o m S p i t l e r a n d M c K i n n o n , 1976) S k i l l M a t h e m a t i c a l C o n c e p t s ^ L o g i c a l T h i n k i n g C o n c e p t s C a n s u b t r a c t o n e f r o m t h e c o u n t i n g n u m b e r s t o 10 ( i . e . . r e m o v e a s u b s e t f r o m a s e t ) C o u n t i n g b a c k O r d i n a l n u m b e r C a r d i n a l n u m b e r - ^ C o n s e r v a t i o n o f n u m b e r C a n s u b t r a c t u s i n g m i n u e n d s t o 10 ( i . e . , a - b = c ) e . g . , 5 - 1 = { \ 5 - 2 ={ } 5 - 3 =( } 5 - ii ={ } G i v e n a n a d d i t i o n f a c t t h e c h i l d c a n i d e n t i f y t h e r e l a t e d s u b t r a c t i o n f a c t s K n o w l e d g e o f a d d i t i o n f a c t s t o 10 • ^ C o n s e r v a t i o n o f n u m b e r C o n c e p t o f s u b t r a c t i o n a s t h e i n v e r s e o f a d d i t i o n - ^ C o n s e r v a t i o n o f n u m b e r 23. Table 2.5 Sequence of S k i l l s and Concepts for Solving Missing Addend Sentences (Adapted from Spitler and McKinnon, 1976) S k i l l Mathematical Concepts ^Logical Thinking Concepts Can solve a missing addend sentence with a place holder i n the middle Knowledge of addition and subtraction facts to 10 •^-Conservation of number *Class inclusion ( i . e . , i n a set, i f A + A l = B then B - A = A* and B - A 1 = A) e.g., 3 + • = 5 Can solve a missing addend sentence with a place holder on the l e f t Knowledge of addition and subtraction facts to 10 ^Conservation of number *Class inclusion ( i . e . , i n a set, i f A + A 1 = B then B - A l = A and B - A = A e.g., • + 2 = 5 An examination of the s k i l l s and concepts associated with solving addi-tion, subtraction, and missing addend sentences shows that a dependency rela-tionship exists. The a b i l i t y to solve subtraction sentences requires an un-derstanding of concepts underlying addition. Subsequently, the a b i l i t y to solve missing addend sentences i s based on the knowledge of both addition and subtraction. A Formal Statement of the Second Hypothesis The arithmetic performance of grade one children i s related to the type of number sentence. The number of correct answers for addition sentences w i l l be greater than for subtraction sentences which, i n turn, w i l l be greater than for missing addend sentences. Relationship between Level of D i f f i c u l t y  and Arithmetic Performance In this section, the relationship between arithmetic performance and size of the number constants used i n the number sentence i s examined. Research 2lu studies that have considered the relative d i f f i c u l t y of addition and subtrac-tion combinations i n both the canonical (a + b = Q ) and missing addend (a + • =b) formats are reviewed. This section concludes with a formal statement of the third hypothesis. Studies Investigating the Relationship between Level of  Diff i c u l t y and Arithmetic Performance Washburne and Vogel (1928), and Clapp (1921;) report that the size of the addends i n an addition sentence i s the principal indicator of d i f f i c u l t y . The studies also indicated that sums greater than ten are more d i f f i c u l t to learn than sums less than ten, except for combinations such as doubles (e.g., 8+8 =16). Groen (1967), Suppes and Groen (196?), and Parkman (1971), report that for addition sentences i n canonical form (i.e., a + b = D ), the size of the addend to be added i s the major determiner of response latency. This finding concurs with Groen and Poll (1973) who concluded that the size of the missing addend i s a major determiner of response latencies for missing addend senten-ces. Grouws (1971) found that performance on items from the two-digit domain was not as high as items from the basic fact domain. Analysis of the results showed that errors on the two-digit items were not processing errors. Grouws suggested that the d i f f i c u l t y of the two-digit items involved more than the computational complexity of processing larger numbers. Findings of the research studies reviewed agree that the magnitude of the numbers i s an indicator of the d i f f i c u l t y of the number sentence. In this study, three levels of d i f f i c u l t y were used. Level one included number fact combinations to ten. Level two included number sentences with two one-digit and one two-digit numbers. Level three included number sentences with one one-digit and two two-digit numbers. Therefore, i t was reasonable to assume 25. that the scores on the three levels of d i f f i c u l t y defined i n this study would vary with respect to the size of the numbers i n the number sentence. A Formal Statement of the Third Hypothesis The arithmetic performance of grade one children i s related to the level of d i f f i c u l t y of the number sentence. On an arithmetic test with three lev-els of d i f f i c u l t y , the number of correct answers for level one w i l l be greater than for level two which, i n turn, w i l l be greater than for level three. The Interactive Effects of Operational Level. Type of Number Sentence, and Level of  Difficulty on Arithmetic Performance In this section, two f i r s t order interactions and a second order inter-action are hypothesized. After each interaction i s discussed, i t s accompany-ing research hypothesis i s presented. The Interactive Effect of Operational Level and  Type of Number Sentence In a previous section, a review of the literature indicated that the a b i l -i t y to conserve number and the a b i l i t y to understand part-whole relations i n -fluenced arithmetic performance. Specifically, conservers out-performed non-conservers on tests of addition and subtraction sentences and class includers out-performed nonclass includers on missing addend sentences. It follows then that children at the transitional level (i.e., nonconservers) w i l l obtain lower scores on addition, subtraction, and missing addend sentences. Children at the early concrete level ( i . e . , conservers) w i l l obtain high scores on ad-dition and subtraction number sentences but lower scores on the missing addend sentences. Finally, children at the later concrete level ( i . e . , conservers and class includers) w i l l maintain a high performance on the three types of number sentences. 26. Support for the proposed interaction between operational level and type of number sentence i s further provided by the findings of Steffe, Spikes, and Hirstein (1976). The researchers compared the a b i l i t y of f i r s t grade child-ren to conserve quantity with their performance on addition, subtraction, and missing addend sentences. The results showed that nonconservers scored sig-nificantly lower on a l l types of number sentences than conservers (seventy-one percent, twenty-five percent, and seventeen percent respectively). Conserv-ers, on the other hand, obtained high mean scores on the addition and subtrac-tion sentences (eighty-two percent and eighty percent respectively), but only moderate mean scores (forty-one percent) on the missing addend number senten-ces. In response to the depressed performance of conservers on missing ad-dend sentences, the authors suggested that cognitive a b i l i t i e s other than reversibility of thought are required to solve that type of number sentence. A formal statement of the fourth hypothesis. There w i l l be an interact-ive effect of operational level and type of number sentence on grade one arithmetic performance. The Interactive Effect of Operational Level and  Level of Diff i c u l t y A review of the literature suggested that the size of the numbers i n the number sentence influenced arithmetic performance. Number combinations below ten were reported to be easier than number combinations to twenty. The grade one arithmetic program includes number sentences to ten. Therefore, when the child i s given a number sentence which involves combinations beyond ten, he or she i s required to apply the addition and subtraction s k i l l s to larger, unfamiliar quantities. It seems reasonable to assume that the a b i l i t y to ex-tend arithmetic s k i l l s to higher magnitude numbers i s related to cognitive maturity. However, there i s insufficient research on the relationship between cognitive level and level of d i f f i c u l t y of the number sentence to pre-specify 27. the nature of this relation. A formal statement of the f i f t h hypothesis. There w i l l be an interact-ive effect of operational level and level of d i f f i c u l t y on grade one ar i t h -metic performance. The Interactive Effect of Operational Level, Type of  Number Sentence, and Level of Dif f i c u l t y Due to the complexity of second order interactions, i t i s impossible to pre-specify the exact nature of the combined influence of operational l e v e l , type of number sentence, and level of d i f f i c u l t y on grade one arithmetic per-formance. However, based on the available literature, a general outcome can be anticipated. Since children who conserved number performed better than non-conservers on addition and subtraction sentences (Almy, et a l . , 1966; Steffe, 1966, 1971; Le Blanc, 1968), i t i s reasonable to predict that early and later concrete children w i l l obtain higher scores than transitional children on the addition and subtraction sentences, particularly at the second and third lev-els of d i f f i c u l t y . Similarly, later concrete children, with their a b i l i t y to comprehend part-whole relations, should obtain higher scores than transitional and early concrete children on the missing addend sentences (Hewlett, 1973; Kelleher, 1977). They should be able to transfer their understanding of the meaning implied by the missing addend sentence to numbers beyond the basic facts to ten that are taught i n school. A formal statement of the sixth hypothesis. There w i l l be an interactive effect of operational level, type of number sentence, and level of d i f f i c u l t y on grade one arithmetic performance. Solution Strategy Study This part of the chapter contains a review of the literature dealing with the solution strategies that grade one children use to solve number sentences. 28. A developmental sequence of counting strategies for solving addition, subtrac-tion, and missing addend sentences i s presented f i r s t . Research investigating the relationship between the child's operational level and type of solution strategy i s then discussed. As a result of this literature review, an obser-vation study i s proposed. Developmental Sequence of Counting Strategies Several researchers have analyzed children's solution strategies i n an attempt to determine causes of performance differences i n solving various types of open sentences. Results of studies indicate that f i r s t grade child-ren use counting strategies most frequently (Brownell, 1928$ I l g and Ames, 1951; Steffe, et a l , , 1976), A developmental sequence of increasingly sophisti-cated counting behavior has been observed by some authors. This •'. w i l l be dis-cussed according to sentence format: addition, subtraction, and missing ad-dend counting strategies. Addition counting strategies. Three researchers have identified count-ing strategies that f i r s t grade children used to solve addition sentences. Brownell (1928) interviewed children i n grades one through four to examine the successive processes used to answer additive combinations. I l g and Ames (1951) assessed the number concepts of children from five through nine years of age. Steffe, et al.(1976) trained grade one children to use counting strategies and examined their a b i l i t y to apply the techniques to solve addition number sentences. An analysis of the three studies reveals that the authors share common findings i n regard to the counting methods grade one children use to complete open addition sentences. The specified counting strategies i n their order of development are presented i n Table 2.6. Inspection of Table 2.6 indicates that the a b i l i t y to apply the counting-a l l procedure for solving addition sentences develops before the a b i l i t y to 29. use counting-on. As a result of their research, Steffe, et a l . (1976) sug-gested that the type of strategy the child uses i s indicative of his or her understanding of the operation underlying the number sentence. In other words, children who count-all have not developed the operational concept of addition (i.e., A U B = C). They regard each element of the set as a separ-ate unit rather than a union of two parts into an invariant whole. On the other hand, children who have an operational understanding of addition, solve the number sentence by mentally representing the f i r s t addend and counting-on to reach the sum. Table 2.6 Addition Counting Strategy (Adapted from Brownell, 1928j I l g and Ames, 1 9 5 1 ; Steffe, et a l . , 1976) Strategy Behavior (example: 3 + U = Q ) Counting-all 1. Point count f i r s t addend 2. Point count,second addend 3. Recount from one to the sum Example: 1 2 3 1 2 3 li 1 2 3 It 5 6 7 Counting-on 1. Mentally represent the f i r s t addend 2. Count-on and t a l l y to the sum Eximple: r three li 5 6 7 Say . . . . Think 1 2 3 li Subtraction counting strategies. Two research studies have investigated •the counting strategies grade one children use to solve subtraction sentences. Previously mentioned work by I l g and Ames ( 1 9 5 1 ) and Steffe, et a l . ( 1 9 7 6 ) 30. report similar findings. The identified counting strategies i n their observed sequence of development are reported i n Table 2.7. Table 2.7 Subtraction Counting Strategies (Adapted from I l g and Ames, 195lj Steffe, et a l . , 1976) Strategy Behavior (example: 7 - 3 = D ) Counting-all 1. Point count the minuend 2. Point count the subtrahend 3. Point count the difference Example: 1 2 3 h 5* 6 7 h 3 2 1 3 2 1 Counting-back 1. Mentally represent the minuend 2. Count back the amount of the subtrahend 3. State the difference Example: four 5 6 7 As for addition strategies, counting-all emerges before rational count-ing-back (i.e., the counterpart of rational counting-on). Steffe, et a l . (1976) further suggests that children who do not understand the subtraction operation (i.e., B - A* =A) use a counting-all procedure along with a phys-i c a l removal of one subset to find the difference. However, children who comprehend the operation of subtraction mentally represent the set and count-back to find the difference. Missing addend counting strategies. Three researchers have studied the missing addend performance of grade one children. Howlett (1973) observed that stage three children (i.e., class includers) processed missing addend 31. sentences differently than stage one children ( i . e . , nonclass includers). Stage one children incorrectly transformed the number sentence into a canon-i c a l addition sentence more often than stage three children. However, Howlett did not identify the solution strategies that were used to correctly answer the number sentences. Peck and Jencks (1976) observed that grade one children who used counting strategies were more successful i n solving missing addend sentences than those who did not employ a counting procedure. However, the authors did not specify the counting methods that grade one children used to succeed on missing addend computation. Steffe, et a l . (1976) reported that f i r s t grade children were capable of implementing counting procedures to solve missing addend sentences. These strategies are described i n Table 2 . 8 . Table 2.8 Missing Addend Counting Strategies (Adapted from Steffe, et a l . , 1976) Strategy Behavior (example: 3 + O = 7) Counting-all 1 . Point count the given addend 2. Point count to the sum 3 . Recount and remove the given addend U. Point count the missing addend Example: 1 2 3 h $ 6 7 • • • • • • • 1 2 3 1 2 3 h Counting-on 1 . Mentally represent the given addend 2. Count on and t a l l y the missing addend Example: three h 5* 6 7 . . . • Say 1 2 3 li Think Counting-back 1 . Mentally represent the sum 2. Count backwards from the sum to the given addend while tal l y i n g the decrement Example: Think, seven Say 7, 6, 5", h, "four" 32. As with addition and subtraction sentences, Steffe, et a l . (1976) main-tain that the counting-all procedure for solving missing addend sentences i s a less mature strategy than counting-on and counting-back. Point-counters who use the counting-all method conceive each addend as separate units ( i . e . , ones) rather than two parts (i.e., addends) which constitute an invariant whole. The Relationship between Operational Level and  Choice of Solution Strategy Several studies have suggested that the counting strategy the child uses to solve a particular type of number sentence i s a reflection of his or her cognitive maturity (Brownell, 1928; Ilg and Ames, 195lj Steffe, et a l . , 1976). However, only a recent investigation by Steffe, et a l . , (1976) has considered whether there i s a relationship between the child's cognitive level as asses-sed by Piagetian tasks and the a b i l i t y to learn and apply counting strategies to solve number sentences. In a training study, Steffe, et a l . (1976) attempted to teach grade one children rational counting strategies (i.e., counting-on and counting-back) as a means of solving addition, subtraction and missing addend number senten-ces. The researchers also assessed the student's a b i l i t y to conserve quan-t i t y . Their results indicated that counting instruction successfully raised conservers' counting a b i l i t i e s to the level of rational counting. However, nonconservers were incapable of progressing beyond point-counting. The auth-ors suggested that reversibility of thought which underlies the a b i l i t y to conserve number and quantity i s necessary to mentally represent collections. In conclusion, they recommend that conservation be considered a readiness variable for rational counting. Subsequently, i n the same study Steffe, et a l . (1976) assessed the child-ren's a b i l i t y to apply independently the counting strategies that they had 33. been taught on a test of addition, subtraction, and missing addend sentences. Nonconservers were capable of using point-counting to solve the addition sent-ences, but were unable to implement the point-counting procedures to solve either the subtraction or missing addend sentences. The researchers main-tained that dependency on counting-all indicated that nonconservers had no operational understanding of addition. They further suggested that noncon-servers were unsuccessful at solving the subtraction sentences because they lacked r e v e r s i b i l i t y of thought necessary to apply the point-counting proced-ure for subtraction. Conservers successfully answered the addition sentences by counting-on. However, most conservers resorted to counting-all rather than rational counting-back to solve the subtraction sentences. In general, conservers were only moderately successful at solving the missing addend sentences which were answered using either the counting-all or counting-on procedures. The authors suggested that although a child may demonstrate the a b i l i t y to use a particular counting technique during a teaching situation, he or she may not be able to transfer that knowledge and independently apply the counting procedure when given varied types of number sentences to answer. Observation Study of Solution Strategies An examination of the literature reviewed i n this section indicates that few studies have investigated whether a relationship exists between the child's level of cognitive maturity and his or her strategy for solving number sent-ences. Therefore, i t was the present author's intention to conduct an obser-vation study of grade one children's counting strategies for solving addition, subtraction, and missing addend sentences. Of specific interest were the following two questions: (1) What counting strategies do the grade one children i n this sample use to solve addition, subtraction, and missing addend sentences? 3h (2) Is the child's choice of counting strategy related to his or her operational level? The purpose of this study was to observe trends i n strategy usage which may serve to stimulate questions for future research. Therefore, specific hypotheses were not formulated. Statement of S t a t i s t i c a l Hypotheses As a result of the literature reviewed, six hypotheses were formulated. Three of the hypotheses were related to the main effects of operational level, type of number sentence, and level of d i f f i c u l t y on grade one arithmetic per-formance, and three of these hypotheses pertained to the interaction of these variables. For the purposes of s t a t i s t i c a l analysis, the hypotheses are stated here i n null form: (1) There i s no significant difference i n arithmetic performance between transitional, early concrete operational, and later concrete opera-tional children. (2) There i s no significant difference i n arithmetic performance between addition, subtraction, and missing addend number sentences. (3) There i s no significant difference i n arithmetic performance between the three levels of d i f f i c u l t y . (U) There i s no significant interaction between operational level and type of number sentence. (5) There i s no significant interaction between operational level and level of d i f f i c u l t y . (6) There i s no significant interaction between operational level, type of number sentence, and level of d i f f i c u l t y . The methodology used to test the six hypotheses and to analyze the solu-tion strategies i s described i n the next chapter. 35. Chapter 3 METHOD This study was designed to ascertain the effects of cognitive level, type of number sentence, level of difficulty, and the interaction of these three variables on the arithmetic performance of grade one children. The following discussion of the methodology used in this research is divided into four parts. Information concerning the sample and sampling procedure is provided in the firs t part. It was necessary to determine the child's operational level prior to administering the arithmetic test. Therefore, in the second part, proced-ures for the Piagetian assessment are presented. The third part focuses on the arithmetic assessment procedures. Finally, in part four, the data analy-sis techniques are outlined. Sample Subjects The subjects were sixty grade one children selected from five elementary schools in a suburb adjacent to a large metropolitan area in the lower main-land of British Columbia. The sample consisted of twenty-eight girls and thirty-two boys. The subjects ranged in age from 77 months to 90 months, with a mean age of 83 months. The socioeconomic level of the school catch-ment was lower-middle to upper-middle class. Two of the schools were situ-ated in the older commercial part of the suburb while the other three schools were located in a rapidly expanding neighborhood comprised of modern single family dwellings and condominiums. Selection Procedure The subjects for this study were a subsample of a much larger research 3 6 . project concerning the scaling of Piagetian tasks. One hundred and twenty-one children, the total grade one population of the five schools, were given a battery of nine Piagetian tasks. Each subject was then assigned to one of three operational levels: level one - nonconservers of number or quantity; level two - conservers of number and quantity; and level three - conservers of number and quantity and class includers. To obtain a sample for this study twenty children were randomly selected from each of the three operational lev-el s . This procedure resulted i n a sample of sixty subjects with equal repres-entation of operational levels. Piagetian Assessment In this section, the procedure used to identify the child's operational level i s detailed. F i r s t i s a description of the Piagetian assessment tasks. The test administration, scoring procedure, and decision rules for determining the three operational levels follows. Task Descriptions Nine types of Piagetian tasks were selected for assessing the child's level of cognitive maturity. They represented three sets of concepts and operations: seriation, classification, and conservation. There were two types of seriation tasks (simple seriation and double seriation), four types of classification tasks (one-way, two-way, three-way, and class inclusion), and three types of conservation tasks (number, continuous quantity, and dis-continuous quantity). Each task was designed to comply with the following p r i o r i t i e s : (1) the task must replicate as closely as possible those tasks used by Piaget to assess a particular concept (Piaget and Szeminska, 19!?2j Inhelder and Piaget, 196H). 37. (2) the task must be solved by logical thinking processes rather than perceptual cues. (3) the task language should be adaptable to the child's vocabulary and language comprehension. (h) the task must be scoreable i n both answer and rationale. A description of the materials and procedure for each task follows (ad-apted from Piaget and Szeminska, 1952j Inhelder and Piager, 196U). Simple seriation. Materials: Seven nesting barrels (i.e., half barrels) ranging i n diameter from 2 cm. to 8 cm. Procedure: (1) The researcher presents the barrels i n a scrambled order. (2) The researcher asks, "Can you line up the barrels from biggest to smallest?" (The researcher used his or her hand to indicate where the row of barrels should be placed.) (3) Leave child's row of barrels i n position for the double seriation task. Double seriation. Materials: Barrels (i.e., half barrels), as above, and seven wooden rods graded i n length from 2 cm. to 8 cm. Procedure: The researcher says, "Let's pretend the barrels are cups and the sticks are straws. Can you give each cup a straw that best f i t s i t ? The biggest cup should have the biggest straw and the smallest cup should have the smal-lest straw, and so on." 38. One-way classification ( f i r s t form). Materials: "Setsco" attribute blocks ranging i n size from h cm. to 6 cm. The colors are red, blue, yellow and greenj and the shapes are c i r c l e , square, diamond, triangle, and rectangle. Procedure: (1) The researcher says, "Can you sort the blocks (or put the blocks to-gether i n piles) so that they are alike or the same i n some way?" (2) The child completes sorting the blocks. The researcher points to each group i n turn that the child has formed and asks, "Will you t e l l me how these blocks are alike?" One-way classification (second form). Materials: Wooden animals and trees ranging i n size from 2 cm. to h cm. Procedure: (1) The researcher says, "Here are some wooden toys, can you sort the toys (or put the toys together i n piles) so that they are alike or the same in some way?" (2) When the child has sorted the toys, the researcher points to each group and asks, "Will you t e l l me how these toys are alike?" Two-way classification ( f i r s t form). Materials: Two-by-two matrix (21 cm. by 13 cm.) as shown i n Appendix A. Procedure: (1) The researcher points to each object i n the matrix as he says, "Here i s a large c i r c l e , here i s a small c i r c l e , and here i s a large triangle. What do you think w i l l go here?" 39. (2) The researcher points to the row of pictures below the matrix and asks, "Which one of these pictures belongs i n the empty space? Why do you think i t goes there?" (3) The researcher points to another picture i n the answer row that has only one attribute the same as the child's selection and asks, "Do you think I could put this picture i n the space? Why do you think that?" Two-way classification (second form). Materials: Two-by-three matrix (21 cm. by 13 cm.) as shown i n Appendix B. Procedure: ( l ) Same as for the f i r s t form except the pictures are blue flower, red flower, yellow flower, blue square, and red square. Three-way classification. Materials: Matrix (21 cm. by 13 cm.) as shown i n Appendix C. Procedure: Same as for two-way classification except the pictures are large yellow triangle, small red triangle, and small yellow square. Class inclusion ( f i r s t form). Materials: Six pink plastic roses and two yellow plastic daffodils. Procedure: (1) The researcher holds the flowers i n a bouquet. (2) The researcher says, "I have a bunch of plastic flowers. What color are my flowers?" (The child should agree that the flowers are yellow and pink.) "Do you think I have more pink flowers or more yellow flowers? How do you know?" ko. Class inclusion (second form). Materials: Six white wooden beads and two blue wooden beads. Procedure: The researcher says, "I have some beads. Do you know what my beads are made from?" (The child should agree that the beads are made of wood.) "What colors are my beads?" (The child should agree that there are white beads and blue beads.) "Do you think I have more white beads or more wooden beads? How do you know?" Conservation of number. Materials: Ten blue and ten green wooden 2.5 cm. square cubes. Procedure: (1) The researcher says, "Which color of blocks would you like? Here are some for you." The researcher places three of the child's color choice i n a row i n front of him. The researcher says, "Here are some for me." The researcher places three of his or her colored blocks i n a row opposite the child's blocks. The researcher says, "Do I have more, less, or the same number of blocks as you?" The child should agree that the two rows are equivalent. (2) The researcher adds two blocks to one end of the child's row and asks, "Now do you have more, less, or the same as me?" The researcher adds two blocks to the end of his or her row and asks, "Do I have more, less, or the same as you?" (3) The researcher repeats this procedure of adding two blocks to the end of each row. (U) The researcher then places three blocks at the opposite end of his or her row and asks the child, "Do I have more, less, or the same as you?" The researcher places three blocks at the end of the child's row so that there i s a direct one-to-one correspondence between the two rows and asks, "Do you have more, less, or the same number of blocks as me?" (5) The researcher transforms the child's row by pushing the blocks to-gether and asks, "Now what do you think? Do you have more, less, or the same number of blocks as me? Why do you think that way?" Conservation of quantity (continuous quantity). Materials: Two equal plasticine balls of approximately f? cm. diameter. Procedure: (1) The researcher presents the child with the two balls and asks, "Is there just as much plasticine i n this b a l l as there i s i n this one?" I f the child does not think that they are equal, the child i s asked to make them equal. (2) The researcher says, "Now I w i l l take my b a l l and r o l l i t into a hot dog shape." The researcher then asks, "Do you s t i l l have as much plasticine i n your b a l l as I do i n my hot dog? Do you have more, less, or the same? Why do you think that?" Conservation of quantity (discontinuous quantity). Materials: The materials are the same as for continuous quantity. Procedure: As for continuous quantity only one b a l l i s broken into five equivalent smaller balls rather than rolled into a hot dog shape. Test Administration The subjects i n this study were individually assessed on a series of U2. Piagetian logical thinking tasks between April 18 and May h, 1979 by a team of four research assistants. The research assistants, including the present author, were trained i n the administration of Piagetian tasks. Training was accomplished by observation of the procedures and by t r i a l demonstrations i n which feedback was given. The children were brought individually to the testing room, usually a library or medical room. The experimenters made every effort to ensure that the children were comfortably seated and that they understood the instruc-tions. Each task was presented i n a game-like manner. The children were told that the experimenter was interested not only i n their answers to the puzzles but also i n the reasons for their answers. Scoring Procedure The child's response to each task was recorded on the Piagetian Assess-ment form (see Appendix D). Then the items were scored as described i n Table 3.1. Table 3.1 Piagetian Assessment Scoring Procedure Task Score Response Simple Seriation G 1 2 incorrect seriation t r i a l and error seriation spontaneous seriation Double Seriation 0 1 2 incorrect double seriation t r i a l and error double seriation spontaneous double seriation One-way Classification 0 1 2 a l l tasks incorrect one task correct two tasks correct Two-way Classification 0 1 2 a l l tasks incorrect one task correct two tasks correct U3. Table 3.1 Cont'd, Task Score Response Three-way Classification 0 incorrect 1 one task correct 2 two tasks correct Class Inclusion 0 a l l tasks incorrect 1 one task correct 2 two tasks correct Number Conservation 0 incorrect answer and explanation 1 correct answer only 2 correct answer and explanation Continuous Quantity 0 incorrect answer and explanation Conservation 1 correct answer only 2 correct answer and explanation Discontinuous Quantity G incorrect answer and explanation Conservation 1 correct answer only 2 correct answer and explanation Inter-rater R e l i a b i l i t y The r e l i a b i l i t y of scoring the nine tasks was computed on a random sample of fifteen out of the sixty cases (i.e., five subjects at each of the three operational levels). The percentage of agreement between the two raters for each task was: single seriation 100; double seriation 100; one-way c l a s s i f i -cation 93.3; two-way classification 86.6; three-way classification 100; class inclusion 100; conservation of number 100; conservation of continuous quantity 100; conservation of discontinuous quantity 100o  Assignment to Operational Level A decision rule for each of the three operational levels was formulated as follows: Level one - transitional. This level included children i n transition between the preoperational and concrete operational stages. Children who hh. were able to spontaneously seriate or classify objects one-way but were un-able to conserve number or quantity were designated as transitional. Accord-ing to Piaget, the transitional child does not comprehend either the notion of r e v e r s i b i l i t y or the logic of classes ( i . e . , class inclusion) which are associated with concrete operational thinking (Inhelder and Piaget, 196Uj Kamii and Radin, 1970). Level two - early concrete operational. This stage included children who gave correct answers and explanations for the conservation of number and quantity tasks but were unsuccessful on the class inclusion tasks. Support for the decision to label children who could conserve as concrete operational i s provided i n the writing of Piaget: "the best criterion of the emergence of operations at the level of concrete structure (towards the age of seven) i s , i n fact, the constitution of invariants or notions of conservation." (Inhelder and Piaget, 196U.) Furthermore, Elkind (1967) views the schema of conservation as a pivotal construct i n the child's cognitive transition from the preoperational to the concrete operational stage. Level three - later concrete operational. This stage included children who gave correct answers and explanations for the conservation and class i n -clusion tasks. As mentioned previously, the entrance to concrete operational thinking i s marked by the child's acquisition of conservation of number. How-ever, the a b i l i t y to comprehend relations between the total class and sub-classes as evidenced by success on the class inclusion tasks i s regarded as a more sophisticated lo g i c a l concept (Piaget and Szeminska, 1952). Piaget's research indicates that the two essential characteristics of genuine class inclusion, conservation of the whole and quantitative comparison of the whole and part, develop later than conservation of number—at about eight years of age (Piaget and Szeminska, 1952; Inhelder and Piaget, 19610. l i * . Arithmetic Assessment An explanation of the procedure used to assess the child's arithmetic performance i s provided i n this part. A description of the test variables, test items, test materials, solution strategy recording scheme, test admin-istration, and scoring procedure i s given. Arithmetic Test Variables Type of number sentence. The arithmetic test consisted of three types of number sentences which form the basis of the grade one arithmetic curricu-lum: addition, subtraction, and missing addend addition (right placeholder position). In order to provide maximum pupil familiarity with the format of the items, the number sentences were presented horizontally, an empty box was used as the placeholder, and the equality sign was to the right of the opera-tion symbol. These characteristics are representative of the number senten-ces most frequently used i n the classrooms of the school d i s t r i c t and i n the workbooks that are supplied by the B.C. Department of Education. Level of d i f f i c u l t y . The arithmetic items were chosen from three levels of d i f f i c u l t y . A range of d i f f i c u l t y levels were used for the purpose of testing the child's comprehension of addition, subtraction, and missing ad-dend computation by working beyond the familiar basic facts to ten. Level one included basic addition or subtraction fact combinations to ten. Level two included basic addition and subtraction fact combinations between eleven and twenty with two one-digit numbers and one two-digit number i n the com-pleted number sentence. Level three included addition and subtraction com-binations between eleven and twenty with one one-digit number and two two-digit numbers i n the completed number sentence. The number sentences were chosen to provide a variety of basic fact combinations. In an effort to standardize the items at each level of d i f f i -culty, the numbers zero and one were excluded from the arithmetic sentences U6. and combinations involving doubles (e.g., 3 + 3 = 6) were avoided. Arithmetic Test Items The arithmetic test consisted of three types of number sentences at three levels of d i f f i c u l t y , t o t a l l i n g nine items. The actual test items are pre-sented i n Table 3.2 below. Table 3.2 Arithmetic Test Items Type of Number Sentence Level of Di f f i c u l t y a + b = • a - b - •_. a = b Level One 7 + 2 10 - li 5 + = 9 Level Two 9 + 5 15 - 7 7 + = 13 Level Three 12 + 8 18 - 6 6 + = 19 Arithmetic Test Materials Test booklet. Each item was printed on a separate sheet, 9.5 cm. by 21.5 cm. The test booklet included ten items: one practice item and nine test items. With the exception of the f i r s t item, the number sentences were randomly ordered and stapled into a booklet. Items were randomized i n an effort to control the effects of order of item presentation. Since the item k * 2 was a practice t r i a l , i t appeared f i r s t i n each booklet. A practice item was provided to ensure that the child knew to read each sentence aloud before completing i t , and knew where to record his or her answer. Pupil materials. Approximately f i f t y colored plastic counting chips, two centimeters i n diameter, were available for the child to use as counters Pencils were provided for recording answers. hi Interview recording form. Each interview was documented on an Interview Recording Form which i s presented i n Appendix E. The form consisted of two parts, an answer section, and a solution strategy section. In the f i r s t c o l -umn of the form, the child's answer for each question was recorded. Details pertaining to the child's solution methods were written adjacent to each test item i n the solution strategy section of the form. Solution Strategy Observation Study Through a review of the related literature, counting strategies for solv-ing addition, subtraction, and missing addend sentences were identified. These strategies included counting-all and counting-on procedures for solv-ing addition and missing addend sentences and counting-all and counting-back procedures for solving subtraction sentences. It was expected that some children would use these solution methods to answer the arithmetic test items. Since the instructional background of the sample was unknown, i t was also anticipated that the children might implement strategies that were not speci-fied i n the literature. Therefore, the researcher chose to observe each child's solution strategy behavior and record his or her counting actions and verbalizations. As the child answered each test item, the observed solution method was entered appropriately i n the solution strategy section of the Interview Recording Form (see Appendix E). Test Administration The Arithmetic Test was administered either immediately or a short time after the child's operational level had been assessed. Each interview took between 15 and 25> minutes to complete. Interviews occurred between A p r i l 1979 and May 1979, and were a l l conducted by the investigator. After accompanying the child to the testing room, the interviewer told the child that he or she would be given some number puzzles to solve. The U8. child was shown the counting materials and given the test booklet. The child was instructed to read the practice equation aloud and then to write his or her answer i n the box. The researcher reminded the child that the counters were available i f he or she needed them. As each item was presented, the child read i t aloud, proceeded to solve i t , and recorded his or her answer. The researcher observed the child's solution strategy and noted how he or she arrived at his or her answer. In cases where no observable behaviors were evident, the question, "How did you get your answer?" was asked by the re-searcher. When the interview was completed, the child was thanked and sent back to class. Scoring Procedure Test items. For ease of computer analysis, each correct item on the arithmetic test was awarded two points and each incorrect answer, one point. Since the arithmetic test consisted of nine items, the total test score was eighteen. Solution strategy. The child's strategy for solving each arithmetic test item was recorded during the test interview. Solution procedures were examined and categorized during the analysis of the data. The number of child-ren using each type of strategy was t a l l i e d . Inter-rater r e l i a b i l i t y . Six solution strategy protocols (i.e., ten per-cent) were rated twice: once by the experimenter, and once by a person who was naive about the purpose of the study. Solution strategy classifications for the f i f t y - f o u r items (six children x nine items) were then compared. In two of the fi f t y - f o u r instances the examiners' ratings differed. The result was ninety-six percent agreement. Data Analysis The answers to the number sentences and the solution methods were analyzed U 9 . separately. The responses to the number sentences were computer analyzed using program BMD08V. The dependent variable, arithmetic performance was analyzed by the fixed effects three-way analysis of variance with repeated measures on the last two factors. Operational level was a between group variable^ type of number sentence and level of d i f f i c u l t y were within group repeated variables. The organismic variable, operational level, was entered first> followed by type of number sentence and level of d i f f i c u l t y . The alpha for a l l tests was set at .0$, The classification data on the solution strategies was treated using descriptive s t a t i s t i c s . Summary of Chapter 3 Chapter 3 was concerned with reporting the methodology involved i n test-ing the six hypotheses and recording the solution strategies. The sample and sampling procedure were described. The Piagetian assessment procedures, scor-ing system, and decision rules for assigning operational levels were presented. Procedures concerned with the design, administration, and scoring of the arith-metic test were also detailed. The chapter concluded with a brief description of the design and the analysis. The results of the arithmetic assessment are presented i n the next chapter. 50. Chapter li RESULTS The purpose of this chapter is to present the results of the study. The results for the arithmetic test and the solution strategy study w i l l be re-ported separately. Arithmetic Test Results The arithmetic test included nine items. There were three types of num-ber sentences (addition, subtraction, and missing addend) and three levels of d i f f i c u l t y . One item was used to measure each c e l l of the test design. Cor-rect answers were awarded a score of two, incorrect answers a score of one, resulting i n a tot a l test score of eighteen. There were twenty observations at three operational levels (transitional, early concrete, later concrete), resulting i n a total of sixty subjects. The results of the arithmetic test were analyzed using three-way ANOVA procedures (operational level x type of number sentence x level of d i f f i c u l t y ) with repeated measures on the last two factors. A summary of the analysis of variance i s presented i n Table U.l. Table l i . l Analysis of Variance Summary Table Operational Level by Type of Number Sentence by Level of Di f f i c u l t y SV SS df MS F Prob. Operational level (0) Error within U6 .U 10.50 2 57 23.21 .18 126.06 .001 Type of number sentence (T) 0 x T Error within 9.05 5.ia lii.65 2 li im ii.52 1.35 .13 35.19 10.52 .001 .001 Level of d i f f i c u l t y (D) 0 x D Error within U.69 2.36 7.37 2 li llU 2.35 0.59 .07 36.21 9.12 .001 .001 T x D 0 x T x D Error within J»3 2.1*5" 12.01 li 8 228 .10 .31 .05 2.0b 5.81 .090 .001 5 1 . The c e l l means and standard deviations for operational level, type of number sentence, and level of d i f f i c u l t y are presented i n Table U.2. Table k.2 Cell Means and Standard Deviations Summary Table Operational Level, Type of Number Sentence, and Level of Di f f i c u l t y Operational Level Level of Difficulty Addition Type of Number Sentence Subtraction Missing Addend X SD X SD X SD Transitional One Two Three 1 . 8 5 ( . 3 6 ) 1 . 6 0 ( . 5 0 ) 1 . 3 5 U 8 ) 1 . 3 0 (.1*7) 1 . 2 5 (.UU) 1 . 1 5 ( . 3 6 ) 1 . 5 0 ( . 3 6 ) 1 . 0 0 ( 0 . 0 ) 1 . 0 0 ( 0 . 0 ) Early concrete One Two Three 2 . 0 0 ( 0 . 0 ) 1 . 9 5 ( . 2 2 ) 1 . 8 5 ( . 3 6 ) 2 . 0 0 ( 0 . 0 ) 1 . 9 0 ( . 3 0 ) 1 . 7 0 (.1*7) 1 . 9 0 ( . 3 0 ) l . 6 o ( . 5 0 ) 1 . 1 5 ( . 3 6 ) Later concrete One Two Three 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 2 . 0 0 ( 0 . 0 ) 1 . 9 5 ( . 2 2 ) As shown i n Table k . l the six s t a t i s t i c a l hypotheses were rejected (p < . 0 0 1 ) . The three main effects, operational level, type of number sent-ence, and level of d i f f i c u l t y were significant. There were two significant f i r s t order interactions: operational level and type of number sentence, and operational level and level of d i f f i c u l t y . Finally, there was a s i g n i f i cant second order interaction: operational level, type of number sentence, and level of d i f f i c u l t y . An inspection of Table h»l also indicates that the variables, type of number sentence, and level of d i f f i c u l t y did not interact (p < .09) at the . 0 5 level of significance. Since the level of d i f f i c u l t y for each type of number sentence was expected to remain constant, an interaction hypothesis 52. for these two variables was not formulated. The results of the s t a t i s t i c a l analysis confirm that this a p r i o r i decision was appropriate. The discussion of the results w i l l proceed from the second order inter-action, to the f i r s t order interactions, to the main effects. This order per-mits the main effects to be interpreted i n relation to the interactions. Second Order Interaction The mean scores recorded i n Table lu2 are graphically displayed by oper-ational level i n Figure l u i . Figure J u l Graphical Display of the Mean Scores Operational Level x Type of Number Sentence x Level of Di f f i c u l t y Operational Level One (Transitional) u o o CO c 2.0 1.5 l.o Addition Subtraction Missing Addend Operational Level Two (Early Concrete) u o o CO c 2.0 1.5 1.0 Addition Subtraction Missing Addend 53. Operational Level Three (Later Concrete) 2.0 Level One Level Two c rt Level Three 1.0 Addition Subtraction Missing Addend Transitional. Inspection of Figure u . l indicates that for transitional children, addition scores were higher at a l l three levels of d i f f i c u l t y rela-tive to the subtraction and missing addend scores. • However, as the level of d i f f i c u l t y increased, the addition scores decreased. Performance was gener-all y low on subtraction sentences, however, as the level of d i f f i c u l t y i n -creased, the subtraction scores decreased. Missing addend scores remained a l -most uniformly low at a l l three levels of d i f f i c u l t y . Early concrete. Children at the early concrete level performed uniformly well on addition and subtraction sentences, with only a slight decrease i n scores as the level of d i f f i c u l t y increased. However, performance on the mis-sing addend sentences greatly decreased as the level of d i f f i c u l t y increased. This d i f f e r e n t i a l effect of level of d i f f i c u l t y on type of number sentence resulted i n an ordinal interaction. Later concrete. For later concrete children, arithmetic performance was uniformly high and independent of type of number sentence and level of d i f f i -culty. A differential effect of operational level, type of number sentence, and level of d i f f i c u l t y on arithmetic performance resulted i n an ordinal interac-tion. Later concrete children succeeded on a l l types of number sentences 5U. regardless of level of d i f f i c u l t y . In contrast, transitional children, with the exception of addition sentences, scored relatively low irrespective of type of number sentence and level of d i f f i c u l t y . Early concrete children scored high on addition and subtraction items but their performance declined on missing addend sentences, particularly at the third level of d i f f i c u l t y . Therefore, the most pronounced interaction between type of number sentence and level of d i f f i c u l t y occurred at the early concrete operational l e v e l . F i r s t Order Interactions Operational level and type of number sentence. The mean scores for the three types of number sentences by the three operational levels are presented i n Figure it.2. Figure U.2 Graphical Display of;the Mean Scores Operational Level x Type of Number Sentence Operational Level Transitional Early Concrete Later Concrete — I , 1 i 1 1 Addition Subtraction Missing Addend Type of Number Sentence Inspection of Figure it.2 reveals that there was less disparity between operational levels on the addition scores than on the subtraction or missing addend scores. The subtraction scores remained consistent with addition scores for early and later concrete children. However, subtraction scores for 55. transitional children were lower than the addition scores. The missing addend performance of later concrete children remained high and consistent with their addition and subtraction scores. On the other hand, the missing addend scores for early concrete children decreased and were relatively low i n comparison to their addition and subtraction scores. While the performance of early con-crete children declined on the missing addend sentences, the transitional children's scores were also low, but not greatly lower than their subtraction scores. An ordinal interaction exists due to the differential effect of type of number sentence on the arithmetic performance of the three operational levels. Later concrete children performed well regardless of the type of number sent-ences. Early concrete children experienced d i f f i c u l t y with missing addend sentences. Transitional children were relatively unsuccessful on both subtrac-tion and missing addend sentences. Operational level and level of d i f f i c u l t y . The mean scores for the three levels of d i f f i c u l t y by the three operational levels are presented i n Figure U.3. Figure U.3 Graphical Display of the Mean Scores Operational Level x Level of D i f f i c u l t y 2.0 Operational Level Transitional Early Concrete ci) 1.0 Later Concrete Level One Level Two Level Three Level of Dif f i c u l t y 56. Inspection of Figure U.3 indicates that at the f i r s t level of d i f f i c u l t y both early concrete and later concrete operational children obtained high arithmetic scores whereas transitional children obtained somewhat lower arith-metic scores. At the second level of d i f f i c u l t y , later concrete children maintained their perfect scores, however, the performance of early concrete and transitional children declined uniformly. At the third level of d i f f i -culty, later concrete children scored high but slightly lower than previously; transitional children scored consistently lower; and early concrete children scored substantially lower. The differential effect of level of d i f f i c u l t y on the arithmetic per-formance of the three operational levels resulted i n an ordinal interaction. Later concrete children performed well regardless of the level of d i f f i c u l t y . Transitional children obtained low scores which consistently declined as the level of d i f f i c u l t y increased. Early concrete children obtained high scores at the f i r s t level of d i f f i c u l t y but lower scores at the third level of d i f f i -culty. Main Effects Operational l e v e l . As shown i n Table U.2, the variable operational level contributed significantly to the total variance (p <.001). Performance d i f -ferences between each operational level were compared using Tukey's test for multiple comparisons (Winer, 1971). There was a significant difference i n arithmetic performance between each of the three operational levels, F (3,57) = »13; (p .05). Later concrete operational children (x =1.99) out-performed early concrete children (x = 1.78) who, i n turn, out-performed transitional children (x =1.29). The difference i n arithmetic performance between the operational levels was attributed to their varying success on the three types of number sentences 57. at the three levels of d i f f i c u l t y . Later concrete children performed well on addition, subtraction, and missing addend sentences at each level of d i f f i -culty, which resulted i n a high arithmetic test score. Early concrete children performed well on addition and subtraction sentences, but achievement declined on missing addend sentences, particularly at the third level of d i f f i c u l t y , which resulted i n a lower arithmetic test score than later concrete children. Transitional children scored lower than early and later concrete operational children on a l l three types of number sentences, particularly on subtraction and missing addend sentences, which resulted i n a low arithmetic test score. Type of number sentence. As shown i n Table U.2, the variable type of num-ber sentence also contributed significantly to the total variance (p <. .001). Performance differences between each type of number sentence were compared us-ing Tukey's test for multiple comparisons (Winer, 1971). The result was a sig-nificant difference i n performance between addition, subtraction, and missing addend sentences, F (3 , l l U ) = .09J p < .05. Scores on addition sentences (x =1.8U) were greater than scores on subtraction sentences (x =1.70) which, i n turn, were greater than scores on missing addend sentences (x = 1.53). The difference i n performance on the three types of number sentences was attributed to the combined effect of operational level, type of number sent-ence, and level of d i f f i c u l t y . A l l three operational levels obtained rela-tively high scores on the addition sentences. However, only the early and later, concrete operational children performed well on the subtraction senten-ces, therefore the total test score for subtraction was lower than for addi-tion. Later concrete children responded well to the missing addend sentences, early concrete children were less successful i n answering the missing addend sentences, particularly at level of d i f f i c u l t y three, and transitional children fa i l e d to correctly answer the missing addend sentences regardless of level of d i f f i c u l t y . Therefore, performance on missing addend sentences was lower than 58. on subtraction sentences which was lower than on addition sentences. Level of d i f f i c u l t y . As shown i n Table u.2, the variable level of d i f f i -culty also contributed significantly to the total variance ( p < .001). Per-formance differences between each level of d i f f i c u l t y were compared using Tukey's test f o r multiple comparisons (Winer, 1971). The result was a s i g -nificant difference i n performance between the three levels of d i f f i c u l t y . F (3,llu) - «06j p <.05. Performance on number sentences at level of d i f f i c u l t y one (x = 1.80) was superior to performance on number sentences at level of d i f f i c u l t y two (x = 1.70) which, i n turn, was superior to perform-ance on number sentences at level of d i f f i c u l t y three (x =1.57). Performance differences at the three levels of d i f f i c u l t y were attrib-uted to the combined effect of operational level, type of number sentence, and level of d i f f i c u l t y . At the f i r s t level of d i f f i c u l t y , early and later concrete operational children obtained high scores on a l l types of number sentences, whereas transitional children obtained high scores only on addition sentences. At the second level of d i f f i c u l t y , later concrete children contin-ued to obtain perfect scores, early concrete children continued to score high on addition and subtraction but lower on missing addend sentences, and transi-tional children dropped i n both addition and subtraction performance. The re-sult was a lower test score on level of d i f f i c u l t y two than on level of d i f f i -culty one. At the third level of d i f f i c u l t y , later concrete children contin-ued to perform well. The performance of early concrete children dropped slightly on subtraction and greatly on missing addend sentences and the per-formance of transitional children continued to decline on addition sentences. The result was a lower test score on level of d i f f i c u l t y three than on level of d i f f i c u l t y two than on level of d i f f i c u l t y one. Solution Strategy Results During the individual interviews, the procedure the child used to solve 59. each type of number sentence was recorded. The strategies for both correct and incorrect answers were then examined and categorized. The frequency of each type of strategy was tabulated for the three opera-tional levels. Twenty subjects at each operational level times three levels of d i f f i c u l t y resulted i n a to t a l of sixty responses for each of the three types of number sentences. The percent of usage was then calculated (fre-quency times 100 divided by 60). The results of the solution strategy observation w i l l be presented by type of number sentence: addition, subtraction, and missing addend. Addition Counting Strategies Three types of counting strategies for correctly solving addition senten-ces were identified from the observational data. For the purpose of this study, the strategies have been labelled counting-all, partial counting-on, and count-ing- with-tally. These three strategies are defined by example i n Table U.3 Table U.3 Counting Strategies for Correct Responses to Addition Sentences Strategy Application of Strategy (example: 3 + h =0 ) Counting-all 1. Point count the f i r s t addend 2. Point count the second addend 3. Recount from one to the sum Example: 1 2 3 1 2 3 U . . . . . . . 1 2 3 h 5 6 7 Partial 1. Point count the f i r s t addend Counting-on 2. Point count the second addend 3. Count on from the f i r s t addend to the sum Example: 1 2 3 1 2 3 U 'three' h 5 6 7 6 0 . Table U.3 Cont'd. Strategy Application of Strategy (example: 3 + h — • ) Counting- 1 . Point count the f i r s t addend with-Tally 2. Point count and t a l l y the second addend Example: 1 2 3 h 5 6 7 Say ... .... 1 2 3 U Think (i. e . , t a l l y ) The strategy frequency for correct and incorrect responses to addition sentences are reported i n Tables l t . lt and l t . 5 respectively. Table h,k Strategy Frequency and Percent for Correct Responses to Addition Sentences Strategy Level of Operational Level Diff i c u l t y Transitional Early Concrete Later Concrete Counting-all One 17 ( 2 8 . 3 * ) 10 ( 1 6 . 6 * ) 6 (10.0*) Two 12 (20.250 lit (23.3*) 12 (20.0*) Three 8 (13.3*) 13 (21 .6*) 13 (21 .6*) Partial One 0 ( 0 * ) 5 ( 8 . 5 * ) It ( 6 . 6 * ) Counting-on Two 0 ( 0* ) 3 ( 5.0*) 7 (11.6*) Three 0 ( 0 * ) 3 ( 5.0*) 5 ( 8.3*) Countings One 0 ( 0 * ) 5 ( 8.3*) 10 (16.6*) with-tally Two 0 ( 0 * ) 2 ( 3.3*) 1 ( 1 . 6 * ) Three 0 ( 0 * ) 1 ( 1 . 6 * ) 2 ( 3.3*) Total 37 ( 6 1 . 8 * ) 56 ( 93.lt*) 60 (100* ) 61 Table h.5 Strategy Frequency and Percent for Incorrect Responses to Addition Sentences Strategy Level of Operational Level D i f f i c u l t y Transitional Early Concrete Later i Concrete Incorrect One 3 ( 5.0%) 0 ( 0 $ ) 0 ( 0$ ) Counting-all Two 8 (13.3$) 1 ( 1.6$) 0 ( 0$ ) Three 11 (18.3$) 3 ( 5.0$) o ( 0$ ) No attempt One 0(0%) 0 ( 0 $ ) o ( 0$ ) Two 0(0%) 0 ( 0 $ ) o ( 0$ ) Three 1 ( 1.6%) 0 ( 0 $ ) o ( 0$ ) Total 23 (38.2$) U ( 6.6$) o ( 0$ ) Inspection of Tables h.h and h»5 indicates that strategies for solving addition sentences varied according to operational l e v e l . Transitional child-ren solved addition sentences by a counting-all strategy. No instances of partia l counting-on or counting-with-tally were observed. Incorrect answers occurred when the children miscounted during the counting-all procedure. Early concrete children also tended to use counting-all most frequently at a l l three levels of d i f f i c u l t y . Partial counting-on and counting-with-tally strategies were used less often but at approximately the same frequency. A few errors i n addition were made when the counting-all solution method was applied. The counting strategies implemented by later concrete children var-ied according to the d i f f i c u l t y of the addition sentence. At level of d i f f i -culty one, addition sentences were solved by using the count-with-tally strat-egy. At levels of d i f f i c u l t y two and three, counting-all and partial counting-on were the favored procedures. Subtraction Counting Strategies Two types of counting strategies for correctly solving subtraction sent-ences were identified from the observational data. For the purpose of this 62. study, the strategies are referred to as counting-all and counting-with-group-ing. These strategies are defined by example i n Table 1*.6. Table 1*.6 Counting Strategies for Correct Responses to Subtraction Sentences Strategy Counting-all Counting-with-grouping Application of Strategy (example: 7 - 3 = • ) 1. Point count the minuend 2. Point count the subtrahend 3. Point count the difference Example: 1 2 3 k 5 6 7 1 2 3 1 2 3 It 1. Point count the minuend 2. Count the subtrahend or the difference using addition facts Example: 1 2 3 h 5 6 7 2 * 1*= 3 U 3 2 1 The strategy frequency for correct and incorrect responses to subtraction sentences are reported i n Tables I*.7 and 1*.8 respectively. Table l*.7 Strategy Frequency and Percent for Correct Responses to Subtraction Sentences Strategy Level of Operational Level Diff i c u l t y Transitional Early Concrete Later Concrete _ - I — T 1 - . 1 — r - , - r T - n — r — r - m — r — — — — i———•—————— 1 I, \ . i I 1 " 1 Counting-all One Two Three 8 (13.3*) 5 ( 8.3*) 3 ( 5.0*) 15 (25.0*) 16 (26.6*) 13 (21.6*) 11 (18.1**) 18 (30.0*) 18 (30.0*) C o unting-with-grouping One Two Three 0 ( 0.0*) 0 ( 0.0*) 0 ( 0.0*) 5 ( 8.1**) 2 ( 3.3*) 1 ( 1.6*) 9 (15.0*) 2 ( 3.3*) 2 ( 3.3*) Total 16 (26.6*) 52 (86.7*) 60 ( 100*) 63. Table U.8 Strategy Frequency and Percent for Incorrect Responses to Subtraction Sentences Strategy Level of Dif f i c u l t y Transitional Operational Level Early Concrete Later Concrete Incorrect counting-all One Two Three 6 (10.0$) 8 (13.3$) k ( 6.6$) 0 ( 0.0$) 1 ( 1.6$) 6 (10.0$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) Addition counting-all One Two Three 6 (10.0$) 7 (11.6$) 11 (I8.h$) 0 ( 0.0$) 1 ( 1.6$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) No attempt One Two Three 0 ( 0.0$) 0 ( 0.0$) 2 ( 3.3$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) 0 ( 0.0$) Total hh (73.4$) 8 (13.3$) 0 ( 0.0$) Inspection of Tables u.7 and H.8 indicates that strategies for solving subtraction sentences varied according to operational l e v e l . Transitional children had d i f f i c u l t y applying a counting strategy to successfully solve subtraction sentences. For the correct responses, counting-all was the only strategy implemented. A similar number of errors resulted from either incor-rectly counting-all or applying the counting-all strategy for addition senten-ces. Early concrete children used counting-all more frequently than counting-with-grouping at a l l levels of d i f f i c u l t y . Errors occurred from incorrect counting-all, particularly at level of d i f f i c u l t y three. Later concrete child ren solved correctly a l l of the subtraction items by correctly applying the counting procedures. Counting-all and counting-with-grouping were used to solve subtraction sentences at level of d i f f i c u l t y one whereas counting-all was the predominant strategy applied at levels of d i f f i c u l t y two and three. 6k. Missing Addend Sentences Three types of counting strategies for correctly solving missing addend sentences were identified from the observational data. For the purpose of this study, the strategies are referred to as counting-all, partial counting-on and counting-with-tally. These strategies are defined by example i n Table 4.9. Table lu9 Strategy Classification for Correct Responses to Missing Addend Sentences Strategy Application of Strategy (example: 3 + O = 7) Counting-all 1. Point count the given addend 2. Point count on to the sum 3. Recount the given addend i l . Count the missing addend Example: 1 2 3 1 2 3 it 5 6 7 • • . . 1 2 3 U Partial counting-on 1. Point count the f i r s t addend 2. Point count on to the sum 3. Count the missing addend Example: 1 2 3 it 5" 6 7 . . . . 1 2 3 4 Counting-with-tally 1. Point count the given addend 2. Count on to the sum and t a l l y the increment Example: 1 2 3 U 5 6 7 • • . . 1 2 3 h Say Think The strategy frequency for correct and incorrect responses to missing addend sentences are reported i n Tables it.10 and it.11 respectively. 65 Table U.10 Strategy Frequency for Correct Responses to Missing Addend Sentences Strategy Level of Operational Level D i f f i c u l t y Transitional Early Concrete Later Concrete Counting-all One 2 ( 3.3%) h ( 6.6*) 3 ( 5.0*) Two 0 ( 0.0%) 6 (10.0*) 7 (11.6*) Three 0 ( 0.0%) 1 ( 1.6*) 10 (16.6*) Partial One 2 ( 3.3%) 10 (16.6*) k ( 6.6*) counting-on Two 0 ( 0.0%) 6 (10.0*) 13 (21.6*) Three 0 ( 0.0%) 5 ( 8.3*) 9 (15.0*) Counting- One 0 ( 0.0%) k ( 6.6*) 13 (21.6*) with-tally Two 0 ( 0.0%) 0 ( 0.0*) 0 ( 0.0*) Three 0 ( 0.0%) 0 ( 0.0*) 0 ( 0.0*) Total h ( 6.6%) 36 (59.7*) 59 (98.U*) Table l u l l Strategy Frequency for Incorrect Responses to Missing Addend Sentences Strategy Level of Operational Level Difficulty Transitional Early Concrete Later Concrete Incorrect One 0 ( o.o*) 0 ( o.o*) 0 ( o.o*) counting-all Two 0 ( o.o*) 5 ( 8.3*) 0 ( o.o*) Three 0 ( o.o*) k ( 6.6*) 1 ( 1.6*) Incorrect One 9 (i5.o*) 1 ( 1.6*) 0 ( o.o*) transformation Two 8 (13.3*) 1 ( 1.6*) 0 ( o.o*) Three 8 (13.3*) 6 (10.0*) 0 ( o.o*) Count One 9 (15.0*) 1 ( 1.6*) 0 ( o.o*) and Guess Two 10 (16.6*) 2 ( 3.3*) 0 ( o.o*) Three 10 (16.6*) h ( 6.6*) 0 ( o.o*) No Attempt One 0 ( 0.0*) 0 ( 0.0*) 0 ( o.o*) Two 0 ( 0.0*) 0 ( 0.0*) 0 ( o.o*) Three 2 ( 3.3*) 0 ( 0.0*) 0 ( o.o*) Total 56 (93.49*) 2h (U0.3*) 1 ( 1.6*) 66. Inspection of Tables li.10 and u . l l indicates that strategies for solving missing addend sentences varied according to operational l e v e l . Transitional children attempted to use their counting s k i l l s to answer the missing addend sentences but were unable to apply the correct strategies. For the four items (6.6$) that were answered correctly, counting-all and counting-on strategies were used equally. Incorrect answers resulted almost equally from either transforming the missing addend sentence to a canonical addition sentence and applying the counting-all strategy or counting either the given addend or the sum and then guessing the answer. Early concrete children were moderately successful i n applying the correct counting strategies. For correct answers, par t i a l counting-on was the method used most frequently, followed by counting-a l l . Errors resulted almost equally from incorrect counting-all, incorrect transformation, and count and guess. Later concrete children correctly applied the appropriate counting strategies. The preferred strategy for level of d i f -f i c u l t y one was counting-with-tally. For levels of d i f f i c u l t y two and three, partial counting-on and counting-all were the predominant strategies. Summary of Chapter h Chapter k dealt with the presentation of the results. The v a r i a b i l i t y i n grade one arithmetic performance was attributed to the combined effect of operational level, type of number sentence, and level of d i f f i c u l t y . F i r s t order interactions of operational level and type of number sentence, and opera-tional level and level of d i f f i c u l t y were also significant. The significant main effects of operational level, type of number sentence, and level of d i f -f i c u l t y were interpreted i n l i g h t of the interactions. Observation of the counting procedures used by grade one children to solve the number sentences indicated that operational level influenced choice of solution strategy. A discussion of these results i s presented i n the succeeding chapter. 67 Chapter S> DISCUSSION, CONCLUSIONS, AND IMPLICATIONS This study was conducted to test hypotheses related to the influence of operational level, type of number sentence, and level of d i f f i c u l t y on grade one arithmetic performance and to observe the solution strategies used to solve specific number sentences* The dependent variable, arithmetic performance, was analyzed using a fixed effects three-way analysis of variance (operational level x type of number sentence x level of d i f f i c u l t y ) with repeated measures on the last two factors. Results of the analysis revealed a significant sec-ond order interactive effect of operational level, type of number sentence, and level of d i f f i c u l t y ; two f i r s t order interactive effects, operational level and type of number sentence and operational level and level of d i f f i c u l t y ; and three main effects of operational lev e l , type of number sentence, and lev e l of d i f f i c u l t y . Counting procedures were identified i n the solution strategy an-alysis that related to operational l e v e l . The purpose of this chapter i s to discuss the results, conclusions, and implications of the study. The results and conclusions of the arithmetic test and solution strategy study w i l l be discussed f i r s t . The limitations of this study, some implications for classrooms, and suggestions for further research w i l l follow. Arithmetic Test The major purpose of this study was to determine i f a relationship exists o between the grade one child's operational level and his or her performance on addition, subtraction, and missing addend sentences at three levels of d i f f i -culty. Three operational levels were included i n this study: transitional (nonconservers of number and quantity), early concrete (conservers of number 68. and quantity), and later concrete (conservers of number and quantity and class includers). To investigate the research problem, six hypotheses were formu-lated and tested. A discussion of the interpretations and conclusions w i l l commence with the second order interaction which w i l l be followed by the two f i r s t order interactions, and f i n a l l y the three main effects. Second Order Interaction This study demonstrated that there was a significant interactive effect of operational lev e l , type of number sentence, and level of d i f f i c u l t y . The f o l -lowing discussion of this interaction i s organized by operational l e v e l . Transitional. An examination of Figure U.1 (A Graphical Display of the Mean Scores, Operational Level x Type of Number Sentence x Level of Difficulty) illustrates that the arithmetic performance scores of transitional children were lower than those of early concrete and later concrete children on a l l types of number sentences at each level of d i f f i c u l t y . Transitional children's addi-tion scores were higher than their subtraction scores which, i n turn, were higher than their missing addend scores. Table k.h (Strategy Frequency and Percent for Correct Responses to Addi-tion Sentences) shows that 61.8 percent of the addition questions were answered correctly. The higher scores on the addition sentences occurred because of the children's success at the f i r s t level of d i f f i c u l t y . Since the f i r s t level of d i f f i c u l t y involved basic facts to ten, i t i s l i k e l y that the number sentence on the arithmetic test was answered by a rote application of the solution pro-cedure that was taught i n school. The overall success of the transitional children on the addition items may be attributed to the fact that addition sentences can be solved by merely applying the point-counting procedure (i . e . , point-counting each addend and recounting to reach the sum). However, as the level of d i f f i c u l t y increased, counting errors occurred which resulted i n i n -correctly answered addition sentences. 69 Inspection of Table U.7 (Strategy Frequency and Percent for Correct Res-ponses to Subtraction Sentences) reveals that transitional children answered only 25 percent of the subtraction items correctly. Low scores on the subtrac-tion questions occurred due to a combination of counting errors and the incor-rect interpretation of the subtraction sentence as a simple addition sentence. Generally, transitional children had d i f f i c u l t y translating the subtraction operation as the decomposition of a set. Transitional children were also unsuccessful at correctly responding to the missing addend items regardless of the level of d i f f i c u l t y . An examina-tion of Table U.10 (Strategy Frequency and Percent for Correct Responses to Missing Addend Sentences) indicates that only 6.6 percent of the missing ad-dend sentences were answered correctly. The missing addend sentence was either incorrectly transformed into a simple addition sentence or the given addend was point-counted and the answer guessed. Both of these solution pro-cedures suggest that the transitional children did not comprehend the part-whole relationship implied by the missing addend sentence. It i s apparent that the transitional children did not understand the operations underlying the addition, subtraction, and missing addend sentences. They tended to treat the number sentences at each level of d i f f i c u l t y a l ike. That i s , they frequently applied the point-counting procedure for addition regardless of the type of number sentence. Because of the indiscriminate use of the addition counting-all strategy, i t i s questionable whether the opera-tion of addition was understood even though the addition scores were somewhat higher than the subtraction and missing addend scores. Transitional children made several conceptual errors on both the subtraction and missing addend questions. That i s , they incorrectly interpreted the operation underlying the open sentence. Therefore, i t can be concluded that transitional children 70. did not comprehend the meaning of these sentences. The i n a b i l i t y of transi-tional children to understand the operations of addition and subtraction and the associated number sentences may i n part be related to an i n a b i l i t y to either conserve number or work with part-whole relations. Early concrete. An examination of Figure h . l reveals that early concrete children succeeded on the addition and subtraction sentences; but they experi-enced d i f f i c u l t y with the missing addend sentences, particularly as the magni-tude of the numbers increased. Tables luU, U.7, and h.10 show that early con-crete children correctly answered 93.U percent of the addition items, 86.6 percent of the subtraction items, and $9.7 percent of the missing addend items. Incorrect responses for the addition and subtraction sentences resulted from counting errors. High{scores for the missing addend sentences at the f i r s t level of d i f f i c u l t y were most l i k e l y due to memory of the facts to ten that were taught i n school. However, as the level of d i f f i c u l t y increased, so did the frequency of incorrect transformations. In other words, number combina-tions beyond the facts to ten obscured the meaning of the missing addend sent-ence for early concrete children. The a b i l i t y to conserve number may be associated with the success of early concrete children on the addition and subtraction sentences and,: as well, may have assisted them on the missing addend sentences. However, Figure u . l also indicated that children who were able to solve class inclusion problems (i . e . , later concrete children) correctly answered the missing addend items. This results implies that an understanding of part-whole relationships Is beneficial for understanding the meaning inherent i n the missing addend sent-ence. Therefore, without the understanding of part-whole relations the con-cept underlying the missing addend sentence i s unstable for early concrete children. Later concrete. Further observation of the graphical display of mean 71. scores i n Figure 4.1 indicates that later concrete children obtained high scores on a l l three types of number sentences at each level of d i f f i c u l t y . Inspection of Tables k.h, 4.7, and U.10 reveals that later concrete children correctly answered 100 percent of the addition and subtraction items and 98.U percent of the missing addend items. The fact that later concrete children applied correct strategies to solve the three types of number sentences sug-gests that they understood the underlying operations. Thus, i t could be con-cluded that the performance of later concrete children supports the hypothesis that the a b i l i t y to conserve number i s associated with success on addition and subtraction sentences and the a b i l i t y to understand part-whole relations i s an advantage when solving missing addend sentences. First Order Interactions Interactive effect of operational level and type of number sentence. In-spection of Figure U.2 (Graphical Display of the Mean Scores, Operational Level x Type of Number Sentence) reveals that success on a particular number sentence i s related to operational l e v e l . Children who conserved number (early concrete and later concrete) performed better on addition and subtrac-tion sentences than nonconservers (transitional). These results are consist-ent with previous findings of Almy, et a l . (1966), Wheatley (1967), Steffe (1966 & 1971), and Le Blanc (1968), but inconsistent with the findings of Mpiangu and Gentile (1975). Children who conserved number also obtained higher scores on the missing addend sentences than nonconservers. These re-sults are i n agreement with the findings of Steffe, et a l . (1976) who found that conservers of substance out-performed nonconservers on the missing ad-dend sentences. Children who were able to solve class inclusion problems (later concrete operational) obtained higher scores on the missing addend sentences than nonclass includers (transitional and early concrete operational). This finding i s consistent with the results of both Howlett (1974) and Kelleher 72. (1977), but inconsistent with the work of Steffe, et a l . (1976). The results of the present study indicate that as the child's logical a b i l i t i e s develop, he or she becomes capable of understanding more complex mathematical concepts which are associated with the various types of number sentences. The results of the present study support the suggestion by early child-hood educators (Kamii, 1971j Dimitrovsky and Almy, 1975j Kamii and DeVries, 1978) that Piaget's l o g i c a l thinking tasks be included as one measure to ass-ess the child's readiness for arithmetic s k i l l s and concepts i n the grade one mathematics program. Specifically, the conservation of number task may serve to identify whether the child has the cognitive maturity to comprehend the operations of addition and subtraction. Similarly, the class inclusion task may serve to identify those children who understand the concept of part-whole relations which i s associated with the meaning implied by the missing addend sentence• Although an understanding of number conservation and class inclusion was an asset f o r correctly solving the three types of number sentences, the study indicated that counting s k i l l s were also important. Children who solved the number sentences accurately regardless of level of d i f f i c u l t y , possessed both the appropriate logical and counting a b i l i t i e s . This observation suggests that while logical a b i l i t i e s may be necessary qualities they are not s u f f i c -ient factors on which to base a decision of the child's readiness for ar i t h -metic instruction. Interactive effect of operational level and level of d i f f i c u l t y . The graphical display of the mean scores i n Figure U.3 (Graphical Display of the Mean Scores Operational Level x Level of Difficulty) i l l u s t r a t e s that arith-metic performance at each of the three levels of d i f f i c u l t y was influenced 73. by operational l e v e l . Transitional children's scores were low at the f i r s t level of d i f f i c u l t y and declined slightly as the level of d i f f i c u l t y increased. Early concrete children's scores were high at the f i r s t level of d i f f i c u l t y , but as the level of d i f f i c u l t y increased, their performance decreased, particu-l a r l y at the third level of d i f f i c u l t y . Later concrete children's scores were uniformly high and therefore unaffected by the magnitude of the numbers. The level of d i f f i c u l t y factor had the least influence on the arithmetic achievement of transitional and later concrete children and the greatest i n f l u -ence on the arithmetic achievement of early concrete children. This result may be attributed to differences i n both cognitive maturity as assessed by Piagetian tasks and counting a b i l i t y . Transitional children lacked the log-i c a l concepts and counting s k i l l s that are associated with success i n answer-ing the number sentences and therefore performed poorly regardless of level of d i f f i c u l t y . On the contrary, later concrete children possessed both the cognitive a b i l i t i e s and the counting accuracy to correctly solve the number sentences. They were able to transfer their understanding of the operations implied by the number sentences to correctly solve questions which contained number constants of greater magnitude than those taught i n the grade one arith-metic program. Early concrete children showed less s t a b i l i t y i n their number sentence comprehension and counting s k i l l s and, as a result, they experienced d i f f i c u l t y as the size of the numbers increased. Main Effects Operational l e v e l . The results of this study indicated that operational level functioned effectively i n the discrimination of differences i n grade one arithmetic performance. In terms of to t a l arithmetic test score, later con-crete children out-performed early concrete children who, i n turn, out-per-formed transitional children. The low performance scores of transitional children may i n part be due to their i n a b i l i t y to conserve number or comprehend 7U. part-whole relations. It i s l i k e l y that early concrete children's a b i l i t y to conserve number enhanced their performance on the arithmetic test, particu-l a r l y for the addition and subtraction items. The almost perfect scores ob-tained by the later concrete children may be attributed to their understanding of both invariance of number and part-whole relations. These findings support the view that the child's cognitive maturity as assessed by Piagetian tasks contributes to his or her success on grade one arithmetic number sentences. Type of number sentence. The results of this study also indicated that the type of number sentence influenced grade one arithmetic performance. An order of number sentence d i f f i c u l t y was identified. This order from least to most d i f f i c u l t i s as follows: addition, subtraction, missing addend. This sequence may be attributed to the fact that number sentences d i f f e r i n con-ceptual complexity. Addition sentences, which were correctly answered most frequently, could be solved simply by point-counting. However, this proced-ure does not necessarily demonstrate an understanding of the underlying opera-tion (Steffe, et a l . , 1976). Subtraction sentences require the construction of a given quantity and the removal of a subset; an operation which supposedly involves r e v e r s i b i l i t y of thought (Piaget and Szeminska, 1952). Missing ad-dend sentences, which posed the greatest d i f f i c u l t y , not only involve reversi-b i l i t y of thought but also the additive composition of classes (Inhelder and Piaget, 196U). The observed order of number sentence d i f f i c u l t y i n this study i s consistent with previous research findings and arithmetic s k i l l hierarch-ies (Suppes, 1967; Weaver, 197U; Beattie and Deichmann, 197aj Steffe, et a l . , 1976; Spitler and McKinnon, 1976). Level of d i f f i c u l t y . The results of this study indicated that the child's arithmetic performance was also influenced by the level of number sentence d i f f i c u l t y . Three levels of d i f f i c u l t y which evolved from a review of the 75. l i t e r a t u r e w e r e i n c l u d e d i n t h e p r e s e n t r e s e a r c h s t u d y , l e v e l o n e i n c l u d e d b a s i c f a c t c o m b i n a t i o n s t o t e h . L e v e l t w o i n c l u d e d c o m b i n a t i o n s b e t w e e n e l e v e n a n d t w e n t y w i t h t w o o n e - d i g i t a n d o n e t w o - d i g i t n u m b e r s i n t h e n u m b e r s e n t e n c e . L e v e l t h r e e i n c l u d e d c o m b i n a t i o n s b e t w e e n e l e v e n a n d t w e n t y w i t h o n e o n e - d i g i t n u m b e r a n d t w o t w o - d i g i t n u m b e r s i n t h e n u m b e r s e n t e n c e . A s h y p o t h e s i z e d , n u m b e r s e n t e n c e s a t t h e f i r s t l e v e l o f d i f f i c u l t y w e r e e a s i e r t h a n n u m b e r s e n t -e n c e s a t t h e s e c o n d l e v e l o f d i f f i c u l t y w h i c h , i n t u r n , w e r e e a s i e r t h a n n u m -b e r s e n t e n c e s a t t h e t h i r d l e v e l o f d i f f i c u l t y . S u p e r i o r a c h i e v e m e n t a t t h e f i r s t l e v e l o f d i f f i c u l t y m a y b e a t t r i b u t e d t o t h e f a c t t h a t n u m b e r c o m b i n a -t i o n s t o t e n a r e i n c l u d e d i n t h e g r a d e o n e m a t h e m a t i c s p r o g r a m . T h e r e f o r e , t h e c h i l d r e n w e r e m o s t l i k e l y f a m i l i a r w i t h t h e n u m b e r s e n t e n c e s a t t h e f i r s t l e v e l o f d i f f i c u l t y . I t i s l i k e l y t h a t t h e i t e m s w e r e a n s w e r e d f r o m r o t e m e m -o r y r a t h e r t h a n f r o m t h e s y s t e m a t i c a p p l i c a t i o n o f s t r a t e g i e s a n d t h e u n d e r -s t a n d i n g o f t h e s p e c i f i c o p e r a t i o n s i n v o l v e d i n t h e s u c c e s s f u l s o l u t i o n o f t h e s e p r o b l e m s . A d e c l i n e i n p e r f o r m a n c e a t l e v e l s t w o a n d t h r e e r e s p e c t i v e l y s u g g e s t s t h a t a s t h e m a g n i t u d e o f t h e n u m b e r c o n s t a n t s i n c r e a s e d , t h e c h i l d w a s c h a l l e n g e d t o a p p l y b o t h h i s o r h e r c o u n t i n g s k i l l s a n d k n o w l e d g e o f t h e o p e r a t i o n a l m e a n i n g i m p l i e d b y t h e n u m b e r s e n t e n c e . O n l y t h e c h i l d w h o p o s s e s -s e d b o t h o f t h e s e a b i l i t i e s w a s h i g h l y s u c c e s s f u l o n t h e a r i t h m e t i c t e s t . T h e r e s u l t s o f t h e p r e s e n t s t u d y r e g a r d i n g t h e e f f e c t o f t h e s i z e o f t h e n u m b e r s o n g r a d e o n e a r i t h m e t i c p e r f o r m a n c e a r e i n a g r e e m e n t w i t h p r e v i o u s r e s e a r c h . I t w a s r e p o r t e d t h a t s u m s g r e a t e r t h a n t e n a r e m o r e d i f f i c u l t t o l e a r n t h a n s u m s b e l o w t e n ( C l a p p , 192uj W a s h b u r n e a n d V o g e l , 1928). H o w e v e r , t h e f i n d i n g s o f t h i s s t u d y a l s o i n d i c a t e d t h a t n u m b e r s e n t e n c e s a t t h e s e c o n d l e v e l o f d i f f i c u l t y w e r e e a s i e r t h a n n u m b e r s e n t e n c e s a t t h e t h i r d l e v e l o f d i f f i c u l t y . I n o t h e r w o r d s , f o r n u m b e r c o m b i n a t i o n s t o t w e n t y , n u m b e r s e n t -e n c e s w i t h o n e t w o - d i g i t n u m b e r a r e e a s i e r t h a n n u m b e r s e n t e n c e s w i t h t w o 76. t w o - d i g i t n u m b e r s . F o r e x a m p l e , m o r e c h i l d r e n o b t a i n e d c o r r e c t a n s w e r s o n t h e q u e s t i o n s : 9 + 5 =•..., 15* - 7 = • , a n d 7 + • = 13 t h a n o n 12 + 8 = • , 18 - 6 = Q f a n d 6 + Q = 1 9 . T h i s f i n d i n g c o n c u r s w i t h t h e l i t e r a t u r e t h a t s u g g e s t s t h e s i z e o f t h e a d d e n d t o b e a d d e d i s t h e g r e a t e s t d e t e r m i n e r o f n u m -b e r s e n t e n c e s u c c e s s ( G r o e n , l ° 6 7 j P a r k m a n , 1971). S o l u t i o n S t r a t e g y S t u d y T h e p u r p o s e o f t h e s o l u t i o n s t r a t e g y s t u d y w a s t o o b s e r v e t h e c o u n t i n g s t r a t e g i e s t h a t w e r e u s e d t o s o l v e t h e n u m b e r s e n t e n c e s a n d t o d e t e r m i n e w h e t h e r a r e l a t i o n s h i p e x i s t e d b e t w e e n t h e c h i l d ' s o p e r a t i o n a l l e v e l a n d h i s o r h e r s o l u t i o n p r o c e d u r e . T h e i d e n t i f i e d c o u n t i n g s t r a t e g i e s a n d t h e i r r e l e v -a n c e t o o p e r a t i o n a l l e v e l w i l l b e d i s c u s s e d f o r e a c h t y p e o f n u m b e r s e n t e n c e . A d d i t i o n c o u n t i n g s t r a t e g i e s . T h r e e d i s t i n c t c o u n t i n g s t r a t e g i e s w e r e u s e d b y t h e c h i l d r e n i n t h i s s t u d y : c o u n t i n g - a l l , p a r t i a l c o u n t i n g - o n , a n d c o u n t i n g - w i t h - t a l l y . T h e f e a t u r e s w h i c h d i s t i n g u i s h t h e s e s o l u t i o n m e t h o d s a r e : 1) t h e d e g r e e o f m e n t a l r e p r e s e n t a t i o n o f s e t s , a n d 2) t h e n u m b e r o f s t e p s i n v o l v e d i n t h e p r o c e d u r e . C o u n t i n g - a l l i s a t h r e e - s t e p p r o c e d u r e a t t h e p h y s i c a l , c o n c r e t e l e v e l . E a c h a d d e n d i s p o i n t - c o u n t e d a n d t h e n r e c o u n t e d t o r e a c h t h e s u m . P a r t i a l c o u n t i n g - o n i s a l s o a t h r e e - s t e p p r o c e d u r e b u t i t r e q u i r e s s o m e m e n t a l r e p r e s e n t a t i o n . T h e r e a r e t h e i n i t i a l s t e p s o f p o i n t -c o u n t i n g e a c h a d d e n d . H o w e v e r , t o f i n d t h e s u m t h e f i r s t a d d e n d i s t h e n m e n t -a l l y a b s t r a c t e d a n d t h e a n s w e r i s o b t a i n e d b y c o u n t i n g - o n . C o u n t i n g - w i t h -t a l l y i s a t w o - s t e p p r o c e d u r e i n w h i c h t h e f i r s t a d d e n d i s p o i n t - c o u n t e d a n d t h e s e c o n d a d d e n d i s t h e n t a l l i e d m e n t a l l y a s t h e c h i l d p o i n t - c o u n t s t o r e a c h t h e s u m . I t c o u l d b e a r g u e d t h a t t h e s e t h r e e s t r a t e g i e s f o r m a s e q u e n c e o f c o u n t i n g b e h a v i o r s b e c a u s e i n t h e o r d e r g i v e n , t h e r e i s a n i n c r e a s e i n m e n t a l r e p r e s e n t a t i o n a n d a d e c r e a s e i n t h e n u m b e r o f s t e p s f o r s o l u t i o n . F o r t h i s s a m p l e , t h e r e w e r e n o i n s t a n c e s o f c o u n t i n g - o n ( m e n t a l l y r e p r e -s e n t i n g t h e f i r s t a d d e n d a n d c o u n t i n g - o n t o t h e s u m ) . A c c o r d i n g t o t h e 77. literature, counting-on develops after counting-all (Brownell, 1928; I l g and Ames, 1951; Steffe, et a l . , 1976). I t i s possible, then, that the strategies observed during this study are the transitional strategies to counting-on. Support for the proposal of intermediate strategies between counting-all and counting-on i s provided by the strategy frequency data for correct respon-ses to addition sentences (Table lu3). Transitional children used counting-a l l exclusively to solve addition sentences. However, early concrete and later concrete children displayed partial counting-on and counting-with-tally strategies particularly at the f i r s t two levels of d i f f i c u l t y . These results indicate that the two new strategies identified i n this study are more ad-vanced than counting-all and that children who conserve number used more ad-vanced solution methods than nonconservers. The fact that conservers did not show any instances of counting-on may be due to a lack of classroom instruc-tion. However, readiness for counting-on may be demonstrated by their choice of transitional counting procedures which involve some mental representation. Subtraction counting strategies: Two strategies for solving subtraction sentences were observed: counting-all and counting with grouping. The count-ing - a l l strategy involves point-counting the minuend, point-counting and re-moving the subtrahend, and point-counting the difference. Counting-with-group-ing i s a similar strategy except that addition facts are used to determine the difference. The counting-all strategy was used most frequently, however, at the lower levels of d i f f i c u l t y early and later concrete children applied the counting-with-grouping procedure. Such a trend suggests that using addition facts to count the difference may be a more advanced strategy than simply counting-all. None of the children i n this sample implemented the counting-back strat-egy (i.e., mentally represent the minuend, count backwards the amount of the 78. subtrahend, and state the difference). However, this finding i s consistent with Steffe's results (Steffe, et a l . , 1976). He reported that grade one children who were capable of using the counting-back strategy failed to s e l -ect this method when solving subtraction sentences independently. Missing addend counting strategies. The strategies used by these child-ren to solve missing addend sentences were: counting-all, par t i a l counting-on, and counting-with-tally. As for simple addition sentences, the amount of men-t a l representation and the number of steps i n the solution procedure are the salient features which distinguish the three strategies. Counting-all i n -volves point-counting the given addend, continued point-counting to the sum, a recount and separation of the given addend, and point-counting the remain-ing quantity which i s identified as the missing addend. Partial counting-on involves a point-counting of the given addend, continued point-counting to the sum, mentally representing the given addend, and recounting only the missing addend. Counting-with-tally involves point-counting the given addend and con-tinued counting to the sum while mentally tallying the missing addend. These observed procedures appear to form a sequence of increasingly more mature counting behaviors. Contrary to the literature, the grade one children i n this sample f a i l e d to display the additive counting-on strategy for missing addend sentences (Steffe, et a l . , 1976). Additive counting-on refers to a mental representa-tion of the given addend and counting-on to the sum while tallying the incre-ment as the missing addend. I t may be posited that the strategy i n this study labelled counting-with-tally i s a step prior to counting-on. Both of these strategies are two-step procedures. However, the difference between counting-with-tally and counting-on i s that i n the former strategy the given addend i s point-counted, whereas i n the lat t e r strategy i t i s represented mentally. 79 Steffe, et a l . (1976) report that counting-on succeeds counting-all. It i s suggested i n the present study that there i s a possible sequence of transi-tional strategies that precede the development of the counting-on strategy. Similarly, there was no evidence of the subtractive counting-back strat-egy. This strategy requires a transformation of the missing addend sentence into a subtraction sentence and an application of the counting-back procedure. However, the absence of the subtractive strategy i s consistent with the re-sults of Steffe, et a l . (1976). The authors reported that children who were capable of applying the counting-back technique during instruction chose to use the additive counting-on strategy when their missing addend performance was assessed i n a testing situation. For correct responses to the missing addend sentences, early and later concrete children used a l l of the three strategies identified i n this study. More instances of counting-with-tally occurred with later concrete children, particularly at the f i r s t level of d i f f i c u l t y ; and at the second level of d i f f i c u l t y , more instances of partial counting-on. These results not only support the suggestion that the observed strategies form a sequence of i n -creasingly more mature counting behaviors, but also indicate that later con-crete children used more advanced strategies than early concrete children. Limitations of the Study This study was limited i n terms of subjects and number of items used to measure arithmetic achievement. The subjects were 60 lower-middle to upper-middle class grade one child-ren who were selected from a single school d i s t r i c t i n the lower mainland of Bri t i s h Columbia. Although the B.C. Department of Education provides a mathe-matics program and curriculum guide, the emphasis on content and choice of instructional methods depends on the preferences of individual school d i s t r i c t s 80. a n d c l a s s r o o m t e a c h e r s . I t i s p o s s i b l e , t h e n , t h a t t h e c h i l d r e n t h r o u g h o u t t h e p r o v i n c e m a y v a r y i n e x p e r i e n c e w i t h d i f f e r e n t t y p e s o f o p e n s e n t e n c e s a n d c o u n t i n g s t r a t e g i e s . T h e r e f o r e , t h e a r i t h m e t i c p e r f o r m a n c e o f t h e c h i l d -r e n i n t h i s s t u d y i s n o t n e c e s s a r i l y t y p i c a l o f t h o s e c h i l d r e n i n o t h e r s c h o o l d i s t r i c t s . A s e c o n d f a c t o r w h i c h l i m i t s t h e g e n e r a l i z a b i l i t y o f t h e f i n d i n g s i s t h e n u m b e r o f i t e m s u s e d t o a s s e s s a r i t h m e t i c p e r f o r m a n c e . O n e q u e s t i o n w a s g i v e n t o t e s t a n u n d e r s t a n d i n g o f e a c h o f t h e t h r e e t y p e s o f n u m b e r s e n t e n c e s a t e a c h o f t h e t h r e e l e v e l s o f d i f f i c u l t y , r e s u l t i n g i n a n i n e - i t e m t e s t ( i . e . , t h r e e t y p e s o f n u m b e r s e n t e n c e s t i m e s t h r e e l e v e l s o f d i f f i c u l t y ) . T h e d e c i s i o n f o r a s i n g l e q u e s t i o n t o m e a s u r e e a c h v a r i a b l e w a s b a s e d o n b o t h a c o n s i d e r a t i o n o f t h e g r a d e o n e c h i l d ' s a t t e n t i o n s p a n a n d t h e d e s i r e t o o b s e r v e t h e s o l u t i o n p r o c e d u r e . T h e a p p r o x i m a t e d d u r a t i o n o f t h e a r i t h m e t i c e v a l u a t i o n w a s b e t w e e n f i f t e e n a n d t w e n t y m i n u t e s , a t i m e w h i c h s e e m e d r e a s o n a b l e f o r t h e c h i l d ' s o p t i m a l c o n c e n t r a t i o n . S u b s e q u e n t r e a d i n g h a s l e d t h e p r e s e n t r e s e a r c h e r t o f i n d a n a l t e r n a t e t e c h n i q u e f o r o b t a i n i n g a s i n g l e i t e m m e a s u r e t h a t h a s b e e n d e s c r i b e d b y C l a r k ( 1 9 7 3 ) . H i s m e t h o d i n v o l v e s g e n e r a t i n g a l i s t o f s e v e r a l i t e m s w h i c h c o r r e s p o n d t o a p a r t i c u l a r c a t e g o r y a n d r a n d o m l y a s s i g n i n g o n e o f t h e m t o e a c h s u b j e c t . F o r e x a m p l e , t h e a d d i t i o n q u e s t i o n s a t t h e f i r s t l e v e l o f d i f f i c u l t y ( i . e . , b a s i c f a c t s t o t e n ) c o u l d i n c l u d e t h e f o l l o w i n g t h r e e n u m b e r s e n t e n c e s : 5* + 3 =• , U + 5 = • , a n d 6 + 2 = • . S i m i l a r l y , t h r e e c o m p a r a b l e n u m b e r f a c t s c o u l d b e i d e n t i f i e d f o r t h e r e m a i n i n g n u m b e r s e n t e n c e s a t t h e v a r i o u s l e v e l s o f d i f f i c u l t y . I f t h i s m e t h o d w e r e i m p l e m -e n t e d , t h e t o t a l n u m b e r o f q u e s t i o n s f o r t h e p r e s e n t s t u d y w o u l d b e t w e n t y -s e v e n ( i . e . , t h r e e t y p e s o f n u m b e r s e n t e n c e s t i m e s t h r e e l e v e l s o f d i f f i c u l t y , t i m e s t h r e e c h o i c e s o f n u m b e r s e n t e n c e s ) . O n e i t e m t o f i t e a c h c a t e g o r y ( i . e . , t h r e e t y p e s o f n u m b e r s e n t e n c e s a t t h r e e l e v e l s o f d i f f i c u l t y ) c o u l d 81 t h e n b e r a n d o m l y s e l e c t e d a n d s t a p l e d i n t o a t e s t b o o k l e t . T h e a d v a n t a g e o f t h i s t e c h n i q u e i s t h a t i m p l i c a t i o n s o f t h e r e s u l t s c o u l d b e a p p l i e d t o a c l a s s o f i t e m s r a t h e r t h a n a s i n g l e i t e m , t h u s e n h a n c i n g t h e g e n e r a l i z a b i l i t y o f t h e s t u d y . T h e r e f o r e , i t i s r e c o m m e n d e d t h a t r e s e a r c h e r s w h o r e q u i r e a o n e - i t e m m e a s u r e c o n s i d e r C l a r k ' s (1973) t e c h n i q u e . I m p l i c a t i o n s f o r C l a s s r o o m s T h e r e s u l t s o f t h i s s t u d y h a v e i m p l i c a t i o n s f o r c l a s s r o o m p r a c t i s e . S u g -g e s t i o n s f o r i d e n t i f y i n g r e a d i n e s s t o s o l v e a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s , f o r s e q u e n c i n g t h e t e a c h i n g o f n u m b e r s e n t e n c e s , f o r s e l e c t -i n g t h e s i z e o f n u m b e r s , a n d f o r d e v e l o p i n g s t r a t e g i e s a p p r o p r i a t e t o t h e c h i l d ' s o p e r a t i o n a l l e v e l a r e p r o v i d e d i n t h e f o l l o w i n g d i s c u s s i o n . T h e s u i t a b i l i t y o f P i a g e t i a n t a s k s a s o n e a s s e s s m e n t o f a c h i l d ' s r e a d i -n e s s f o r s o l v i n g a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s i s a s u p p o r t e d o u t c o m e o f t h i s s t u d y . C h i l d r e n w h o c o n s e r v e n u m b e r o u t - p e r f o r m e d n o n c o n s e r v e r s o n a l l t h r e e t y p e s o f n u m b e r s e n t e n c e s . N o n c o n s e r v e r s f r e q u e n t l y m a d e c o n c e p t u a l e r r o r s o n b o t h t h e s u b t r a c t i o n a n d m i s s i n g a d d e n d i t e m s . T h e n u m b e r c o n s t a n t s i n t h e s u b t r a c t i o n s e n t e n c e s w e r e a d d e d r a t h e r t h a n s u b t r a c t e d , i n d i c a t i n g a m i s i n t e r p r e t a t i o n o f t h e s u b t r a c t i o n o p e r a t i o n . I n t h e c a s e o f t h e m i s s i n g a d d e n d q u e s t i o n s , t h e n u m b e r s e n t e n c e s w e r e i n c o r r e c t l y t r a n s f o r m e d i n t o s i m p l e a d d i t i o n s e n t e n c e s . O n e i m p l i c a t i o n t h a t c a n b e d r a w n f r o m t h e f a c t t h a t t h e n o n c o n s e r v e r s a p p l i e d t h e a d d i t i o n c o u n t i n g - a l l p r o c e d u r e t o s o l v e a l l t y p e s o f n u m b e r s e n t e n c e s i s t h a t t h e y a l s o h a d a n i m m a t u r e u n d e r -s t a n d i n g o f s i m p l e a d d i t i o n . S i m i l a r l y , c h i l d r e n w h o s u c c e e d e d o n t h e c l a s s i n c l u s i o n p r o b l e m s o u t - p e r f o r m e d c o n s e r v e r s o n t h e m i s s i n g a d d e n d p r o b l e m , p a r -t i c u l a r l y a t t h e t h i r d l e v e l o f d i f f i c u l t y . I t a p p e a r s t h a t t h e e m p i r i c a l e v i d e n c e p r e s e n t e d i n t h i s s t u d y i n d i c a t e s t h a t a g r e a t e r u n d e r s t a n d i n g o f t h e 8 2 . m i s s i n g a d d e n d s e n t e n c e i s p o s s i b l e w i t h t h e a b i l i t y t o c o m p r e h e n d p a r t - w h o l e r e l a t i o n s . A s u g g e s t i o n t h a t c a n b e m a d e f r o m t h i s s t u d y i s t h a t t h e c o n s e r v a -t i o n o f n u m b e r t a s k b e u s e d a s o n e r e a d i n e s s a s s e s s m e n t f o r a l l t h r e e t y p e s o f n u m b e r s e n t e n c e s a n d t h a t c l a s s i n c l u s i o n p r o b l e m s a l s o b e u s e d t o d e t e r -m i n e t h e t i m i n g o f t h e i n t r o d u c t i o n o f m i s s i n g a d d e n d s e n t e n c e s . T h e r e s u l t s o f t h i s s t u d y v e r i f i e d a n o r d e r o f n u m b e r s e n t e n c e d i f f i c u l t y . T h i s o r d e r w a s : a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s . T h e r e -f o r e , t e a c h e r s a n d c u r r i c u l u m w r i t e r s s h o u l d c o n s i d e r t h i s s e q u e n c e w h e n o r -g a n i z i n g t h e g r a d e o n e a r i t h m e t i c p r o g r a m o r d e s i g n i n g i n s t r u c t i o n a l u n i t s . T h e l e v e l s o f d i f f i c u l t y u s e d i n t h i s s t u d y w e r e n o t b e y o n d t h e c a p a b i l i -t i e s o f s o m e g r a d e o n e c h i l d r e n . C o n s e r v e r s w e r e a b l e t o c o m p l e t e a d d i t i o n a n d s u b t r a c t i o n s e n t e n c e s w i t h n u m b e r c o m b i n a t i o n s t o t w e n t y ; c l a s s i n c l u d e r s w e r e a b l e t o a n s w e r a l l s e n t e n c e t y p e s w i t h f a c t s t o t w e n t y . H o w e v e r , t h e g r a d e o n e a r i t h m e t i c p r o g r a m e m p h a s i z e s o n l y t h e b a s i c f a c t s t o t e n . B a s e d o n t h e r e s u l t s o f t h i s s t u d y , i t i s r e c o m m e n d e d t h a t n u m b e r s k i l l s b e e x t e n d e d b e y o n d t h e b a s i c f a c t s t o t e n a n d i n c l u d e t h e c o m b i n a t i o n s t o t w e n t y , f o r t h o s e c h i l d r e n w h o h a v e t h e c o g n i t i v e a b i l i t i e s . E x p e r i e n c e w i t h a b r o a d e r r a n g e o f n u m b e r s w o u l d a l l o w g r e a t e r a p p l i c a t i o n t o f u r t h e r d e v e l o p a n d r e i n f o r c e t h e c o n c e p t s u n d e r l y i n g t h e n u m b e r o p e r a t i o n s . C o u n t i n g s t r a t e g i e s w e r e u s e d b y t h e s a m p l e i n t h i s s t u d y t o s o l v e n u m b e r s e n t e n c e s . C o u n t i n g - a l l p r o c e d u r e s w e r e i d e n t i f i e d a s t h e m o s t i m m a t u r e s t r a t e g i e s , w h e r e a s t h o s e s t r a t e g i e s w h i c h h a d s o m e o f t h e c h a r a c t e r i s t i c s o f c o u n t i n g - o n w e r e t h o u g h t t o b e m o r e a d v a n c e d . T h e r e f o r e , i t i s s u g g e s t e d t h a t c o u n t i n g - a l l p r o c e d u r e s a r e a p p r o p r i a t e a s a n i n i t i a l t e a c h i n g s t r a t e g y f o r n u m b e r s e n t e n c e s . F r o m t h e r e s u l t s o f t h i s s t u d y i t w a s s u g g e s t e d t h a t t h e a b i l i t y t o c o u n t o n m a y b e r e l a t e d t o d e v e l o p m e n t a l c o g n i t i v e c o m p e t e n c i e s . H o w e v e r , n o n e o f t h e c h i l d r e n i n t h i s s t u d y u s e d t h e c o u n t i n g - o n p r o c e d u r e . T h i s r e s u l t ( c o u l d 83. i m p l y t h a t c o u n t i n g - o n s t r a t e g i e s a r e n o t n e c e s s a r i l y a c q u i r e d i n t h e a b s e n c e o f i n s t r u c t i o n . T h e r e f o r e , a r e c o m m e n d a t i o n o f t h i s s t u d y i s t h a t c o u n t i n g s t r a t e g i e s w h i c h a r e a p p r o p r i a t e t o t h e c h i l d ' s c o g n i t i v e d e v e l o p m e n t a l c o m p e t e n c e b e t a u g h t i n f o r m a l a r i t h m e t i c l e s s o n s . W h e t h e r t h e i n t e r m e d i a t e s t r a t e -g i e s w h i c h w e r e o b s e r v e d i n t h i s r e s e a r c h a r e s u i t a b l e a s t r a n s i t i o n a l s t e p s l e a d i n g t o c o u n t i n g - o n r e m a i n s t o b e a n s w e r e d b y f u r t h e r i n v e s t i g a t i o n s . S u g g e s t i o n s f o r F u r t h e r R e s e a r c h S e v e r a l q u e s t i o n s , b a s e d o n t h e f i n d i n g s o f t h i s s t u d y , c a n b e p o s e d f o r f u r t h e r r e s e a r c h . I t i s a p p a r e n t f r o m t h i s s t u d y t h a t o p e r a t i o n a l l e v e l h a s a n e f f e c t o n g r a d e o n e a r i t h m e t i c p e r f o r m a n c e . S p e c i f i c a l l y , c o n s e r v e r s o f n u m b e r ( e a r l y c o n c r e t e a n d l a t e r c o n c r e t e o p e r a t i o n a l l e v e l s ) p e r f o r m e d b e t t e r o n a d d i t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s t h a n n o n c o n s e r v e r s j a n d c l a s s i n -c l u d e r s p e r f o r m e d b e t t e r o n m i s s i n g a d d e n d s e n t e n c e s t h a n n o n c l a s s i n c l u d e r s . H o w e v e r , d u e t o t h e c r o s s - s e c t i o n a l d e s i g n o f t h i s s t u d y , a c a u s e - a n d - e f f e c t r e l a t i o n s h i p c a n n o t b e e s t a b l i s h e d . T h e r e f o r e , t h e f i r s t r e c o m m e n d a t i o n f o r f u t u r e r e s e a r c h i s a l o n g i t u d i n a l s t u d y w h i c h t r a c e s t h e d e v e l o p m e n t o f b o t h l o g i c a l a b i l i t i e s a n d n u m b e r s e n t e n c e c o n c e p t s a s t h e y e m e r g e f r o m a g e s f o u r t o s e v e n . T h e p u r p o s e o f s u c h a s t u d y w o u l d b e t o d e m o n s t r a t e w h e t h e r t h e a c q u i s i t i o n o f c o g n i t i v e a b i l i t i e s p r e c e d e s , f a c i l i t a t e s , a n d / o r a l l o w s t h e d e v e l o p m e n t o f n u m b e r s e n t e n c e c o m p r e h e n s i o n . C h i l d r e n i n t h i s s t u d y u s e d s o l u t i o n s t r a t e g i e s t h a t a p p e a r t o b e t r a n s i ' t i o n a l f r o m c o u n t i n g - a l l t o t h e m o r e m a t u r e c o u n t i n g - o n . T h e r e f o r e , a s e c o n d s u g g e s t i o n f o r f u r t h e r r e s e a r c h i s a n e x p e r i m e n t a l s t u d y a i m e d a t t e a c h i n g c o u n t i n g - o n s t r a t e g i e s . S u c h a n i n v e s t i g a t i o n c o u l d a s s e s s w h e t h e r t h e m e d i a t e s t r a t e g i e s i d e n t i f i e d i n t h i s s t u d y , p a r t i a l c o u n t i n g - o n a n d c o u n t i n g w i t h - t a l l y , a r e p r o d u c t i v e t r a n s i t i o n a l p r o c e d u r e s f o r e s t a b l i s h i n g c o u n t i n g -8k. s t r a t e g i e s . T h e r e s u l t s o f t h e p r e s e n t s t u d y s u p p o r t p r e v i o u s r e s e a r c h f i n d i n g s w h i c h i n d i c a t e t h a t g r a d e o n e c h i l d r e n i n f r e q u e n t l y a p p l y c o u n t i n g - b a c k s t r a t e g i e s w h e n s o l v i n g s u b t r a c t i o n a n d m i s s i n g a d d e n d s e n t e n c e s ( S t e f f e , e t a l . , 1976). C o u n t i n g - b a c k r e f e r s t o a m e n t a l r e p r e s e n t a t i o n o f a q u a n t i t y a n d a c o u n t i n g -b a c k i n t h e o r d i n a l n u m b e r c h a i n u n t i l t h e a n s w e r i s o b t a i n e d . F r o m t h e a v a i l a b l e l i t e r a t u r e , i t i s n o t c l e a r w h e t h e r t h e p r e f e r e n c e f o r c o u n t i n g f o r w a r d i s d u e t o a l a c k o f e x p e r i e n c e o r t o d i f f i c u l t i e s w i t h c o n c e p t u a l i z -i n g t h e c o u n t i n g - b a c k s t r a t e g y . C o n s e q u e n t l y , a t h i r d r e s e a r c h p r o b l e m i s t o i d e n t i f y t h e c o g n i t i v e a b i l i t i e s w h i c h a r e a s s o c i a t e d w i t h c o m p e t e n t c o u n t i n g -b a c k a n d t o d e f i n e a p o s s i b l e s e q u e n c e o f t r a n s i t i o n a l s t r a t e g i e s t h a t m a y u l t i m a t e l y l e a d t o a s p o n t a n e o u s a p p l i c a t i o n o f t h e c o u n t i n g - b a c k s t r a t e g y . I t i s e v i d e n t f r o m t h e f i n d i n g s o f t h i s s t u d y t h a t a m o r e a c c u r a t e a s s e s s -m e n t o f t h e c h i l d ' s a r i t h m e t i c c o n c e p t s i s p o s s i b l e w h e n t h e m e t h o d s u s e d t o s o l v e t h e n u m b e r s e n t e n c e s a r e r e c o r d e d a n d a n a l y z e d . B y o b s e r v i n g t h e c h i l d ' s s o l u t i o n p r o c e d u r e t h e i n v e s t i g a t o r w a s a b l e t o d e t e r m i n e w h e t h e r t h e c h i l d u n d e r s t o o d t h e o p e r a t i o n i m p l i e d b y t h e n u m b e r s e n t e n c e . I n a d d i t i o n t o e v a l u -a t i n g w h e t h e r t h e c h i l d c o n c e p t u a l i z e d t h e q u e s t i o n c o r r e c t l y , t h e r e s e a r c h e r w a s a l s o a b l e t o a s s e s s t h e m a t u r i t y o f t h e c h i l d ' s s o l u t i o n p r o c e d u r e . W h e n s o l v i n g t h e a r i t h m e t i c i t e m s , t h e c h i l d r e n a t t h e t h r e e o p e r a t i o n a l l e v e l s i m p l e m e n t e d d i f f e r e n t t y p e s o f s t r a t e g i e s . F o r e x a m p l e , l a t e r c o n c r e t e c h i l d -r e n o f t e n u s e d m o r e a d v a n c e d p r o c e d u r e s t h a t i n v o l v e d p a r t i a l m e n t a l r e p r e s e n t -a t i o n o f q u a n t i t i e s , w h e r e a s t r a n s i t i o n a l c h i l d r e n a p p l i e d t h e l e s s m a t u r e c o u n t i n g - a l l s t r a t e g i e s . T h e r e f o r e , i n c o n c l u s i o n , i t i s r e c o m m e n d e d t h a t f u t u r e r e s e a r c h w h i c h i n v e s t i g a t e s w h e t h e r a r e l a t i o n s h i p e x i s t s b e t w e e n p a r -t i c u l a r l o g i c a l a b i l i t i e s a n d m a t h e m a t i c a l c o n c e p t s o b s e r v e n o t o n l y t h e c h i l d ' s r e s p o n s e s b u t a l s o h i s o r h e r s o l u t i o n p r o c e d u r e s . 85. S u m m a r y o f C h a p t e r 5 C h a p t e r 5 d e a l t w i t h a d i s c u s s i o n o f t h e i n t e r p r e t a t i o n s a n d c o n c l u s i o n s o f t h e r e s u l t s . O n t h e f i n d i n g s o f t h i s s t u d y i t w a s r e c o m m e n d e d t h a t P i a g e t ' s l o g i c a l t h i n k i n g t a s k s , ' c o n s e r v a t i o n o f n u m b e r a n d c l a s s i n c l u s i o n . b e c o n s i d -e r e d a s o n e m e a s u r e t o a s s e s s t h e c h i l d ' s r e a d i n e s s f o r l e a r n i n g t o s o l v e a d d i -t i o n , s u b t r a c t i o n , a n d m i s s i n g a d d e n d s e n t e n c e s . A n a l y s i s o f t h e s o l u t i o n s t r a t e g i e s r e v e a l e d t h a t t h e c h i l d ' s o p e r a t i o n a l l e v e l i s i n d i c a t i v e o f h i s o r h e r a b i l i t y t o i m p l e m e n t p a r t i c u l a r s o l u t i o n p r o c e d u r e s . C h i l d r e n w h o w e r e m o r e c o g n i t i v e l y m a t u r e w e r e c a p a b l e o f u s i n g m o r e a d v a n c e d s o l u t i o n m e t h o d s . T h e m o s t i n t e r e s t i n g o u t c o m e o f t h e s t u d y w a s t h e d i s c o v e r y o f t h e s o l u t i o n p r o c e d u r e s , p a r t i a l c o u n t i n g - o n a n d c o u n t i n g - w i t h - t a l l y . I t a p p e a r s t h a t t h e s e s t r a t e g i e s a r e t r a n s i t i o n a l s t e p s b e t w e e n c o u n t i n g - a l l a n d c o u n t -i n g - o n . T h i s f i n d i n g h a s p o t e n t i a l f o r b o t h c l a s s r o o m p r a c t i s e a n d f u r t h e r r e s e a r c h . 86 R E F E R E N C E S A l i n y , M . , C h i t t e n d e n , E . , a n d M i l l e r , P . Y o u n g c h i l d r e n ' s t h i n k i n g . N e w Y o r k : T e a c h e r s C o l l e g e P r e s s , C o l u m b i a U n i v e r s i t y , 1966. B e a t t i e , I . D . , a n d D e i c h m a n n , J . W . E r r o r t r e n d s i n s o l v i n g n u m b e r s e n t e n c e s  i n r e l a t i o n t o w o r k b o o k f o r m a t a c r o s s f i r s t a n d s e c o n d g r a d e s . 1971*. ( E R I C D o c u m e n t E D 061* 170) B r a i n e r d , C . M a t h e m a t i c a l a n d b e h a v i o r a l f o u n d a t i o n s o f n u m b e r . T h e J o u r n a l  o f G e n e r a l P s y c h o l o g y . 1973, 88, 221-281. B r a i n e r d , C . I n d u c i n g o r d i n a l a n d c a r d i n a l r e p r e s e n t a t i o n s o f t h e f i r s t f i v e n a t u r a l n u m b e r s . J o u r n a l o f E x p e r i m e n t a l C h i l d P s y c h o l o g y , 197a, 18, 520-53U. B r a i n e r d , C . C o n c e r n i n g M a c n a m a r a ' s a n a l y s i s o f P i a g e t ' s t h e o r y o f n u m b e r . C h i l d D e v e l o p m e n t . 1976, 1*7, 893-896. B r o w n e l l , W . A . T h e d e v e l o p m e n t o f c h i l d r e n ' s n u m b e r i d e a s i n t h e p r i m a r y g r a d e s . S u p p l e m e n t a r y E d u c a t i o n a l M o n o g r a p h s , 1928, N o . 35. C l a p p , F . L . T h e n u m b e r c o m b i n a t i o n s : T h e i r r e l a t i v e d i f f i c u l t y a n d t h e f r e q u e n c y o f t h e i r a p p e a r a n c e i n t e x t b o o k s . M a d i s o n , W i s c o n s i n : U n i v e r s i t y o f W i s c o n s i n , 192a, p p . 22-29 ( B u r e a u o f E d u c a t i o n a l R e s e a r c h B u l l e t i n N o . 2 ) . C l a r k , H . T h e l a n g u a g e - a s - f i x e d - e f f e c t s f a l l a c y : a c r i t i q u e o f l a n g u a g e s t a t i s t i c s . J o u r n a l o f V e r b a l L e a r n i n g a n d V e r b a l B e h a v i o r , 1973^ 12, 335-359. C o p e l a n d , R . H o w c h i l d r e n l e a r n m a t h e m a t i c s . L o n d o n : C o l l i e r M a c m i l l a n , 197k. D e V r i e s , R . R e l a t i o n s h i p s a m o n g P i a g e t i a n , I Q , a n d a c h i e v e m e n t a s s e s s m e n t s . C h i l d D e v e l o p m e n t , 197k, 1*5, 71*6-756. D i m i t r o v s k y , L . , a n d A l m y , M . E a r l y c o n s e r v a t i o n a s a p r e d i c t o r o f a r i t h m e t i c a c h i e v e m e n t . T h e J o u r n a l o f P s y c h o l o g y , 1975, 91, 65-70. E l k i n d , D . E g o c e n t r i s m i n a d o l e s c e n c e . C h i l d D e v e l o p m e n t , 1967, 38, 1025-1031*. *~ F l a v e l l , J . H . T h e d e v e l o p m e n t a l p s y c h o l o g y o f J e a n P i a g e t . N e w Y o r k : D . V a n N o s t r a n d , 1963. F l o u r n o y , F . U n d e r s t a n d i n g r e l a t i o n s h i p : a n e s s e n t i a l f o r s o l v i n g e q u a t i o n s . E l e m e n t a r y S c h o o l J o u r n a l , 1961*, 61*, 2ll*-217. G a g n e , R . , a n d B r i g g s , L . P r i n c i p l e s o f i n s t r u c t i o n a l d e s i g n . N e w Y o r k : H o l t , R i n e h a r t a n d W i n s t o n , 197u. 87. G r o e n , G . H . , a n d P o l l , M . S u b t r a c t i o n a n d t h e s o l u t i o n o f o p e n s e n t e n c e p r o b l e m s . J o u r n a l o f E x p e r i m e n t a l C h i l d P s y c h o l o g y . 1973, 16, 292-302. H o w l e t t , K . D . A s t u d y o f t h e r e l a t i o n s h i p b e t w e e n P i a g e t i a n c l a s s i n c l u s i o n t a s k s a n d t h e a b i l i t y o f f i r s t g r a d e c h i l d r e n t o d o m i s s i n g a d d e n d c o m p u t a t i o n a n d v e r b a l p r o b l e m s . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , S t a t e U n i v e r s i t y o f N e w Y o r k , B u f f a l o , 1973. I l g , F . , a n d A m e s , L . B . D e v e l o p m e n t a l t r e n d s i n a r i t h m e t i c . J o u r n a l o f  G e n e t i c P s y c h o l o g y . 1951, 79, 3-28. I n h e l d e r , B . S o m e a s p e c t s o f P i a g e t ' s g e n e t i c a p p r o a c h t o c o g n i t i o n . I n ¥ . K e s s e n a n d C . K u h l m a n ( E d s . ) , T h o u g h t i n t h e Y o u n g C h i l d . C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s , 1962. I n h e l d e r , B . , a n d P i a g e t , J . E a r l y g r o w t h o f l o g i c i n t h e c h i l d . N e w Y o r k ^ H a r p e r a n d R o w , 196k. I n h e l d e r , B » , S i n c l a i r , H . , a n d B o v e t , M . L e a r n i n g a n d t h e d e v e l o p m e n t o f  c o g n i t i o n . H a r v a r d U n i v e r s i t y P r e s s , 197k. I r m e n , A . A p p l i c a t i o n o f P i a g e t ' s t h e o r y t o t h e t e a c h i n g o f e l e m e n t a r y  n u m b e r c o n c e p t s : a t h e o r y - i n t o - p r a c t i c e a p p r o a c h . 197k. ( E R I C D o c u m e n t E D 100 70k) K a m i i , C . E v a l u a t i o n o f l e a r n i n g i n p r e s c h o o l e d u c a t i o n : s o c i o - e m o t i o n a l , p e r c e p t u a l , m o t o r a n d c o g n i t i v e d e v e l o p m e n t . I n B . B l o o m , J . H a s t i n g s , a n d G . M a d a u s ( E d s . ) , H a n d b o o k o n F o r m a t i v e a n d S u m m a t i v e  E v a l u a t i o n o f S t u d e n t L e a r n i n g . N e w Y o r k : M c G r a w H i l l , 1971. K a m i i , C , a n d R a d i n , N . L . A f r a m e w o r k f o r a p r e s c h o o l c u r r i c u l u m b a s e d o n s o m e P i a g e t i a n c o n c e p t s . I n I . A t h e y a n d D . R u b a d e a u ( E d s . ) , E d u c a t i o n a l I m p l i c a t i o n s o f P i a g e t ' s T h e o r y . T o r o n t o : G u n n a n d C o . , 1970. K a m i i , C , a n d D e V r i e s , R . P i a g e t , c h i l d r e n , a n d n u m b e r . N a t i o n a l A s s o c i a -t i o n f o r t h e E d u c a t i o n o f Y o u n g C h i l d r e n , 1978. K e l l e h e r , H . J . T h e p e r f o r m a n c e o f s e c o n d y e a r p r i m a r y c h i l d r e n o n m i s s i n g a d d e n d s e n t e n c e s . U n p u b l i s h e d m a s t e r s t h e s i s , 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 , V a n c o u v e r , 1977. L e B l a n c , J . F . T h e p e r f o r m a n c e o f f i r s t g r a d e c h i l d r e n i n f o u r l e v e l s o f c o n s e r v a t i o n o f n u m e r o u s n e s s a n d t h r e e I Q g r o u p s w h e n s o l v i n g a r i t h m e t i c s u b t r a c t i o n p r o b l e m s . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , T h e U n i v e r s i t y o f W i s c o n s i n , M a d i s o n , 1968. M a c n a m a r a , J . A n o t e o n P i a g e t a n d n u m b e r . C h i l d D e v e l o p m e n t , 1975, k6, k2k-k29. M a c n a m a r a , J . A r e p l y t o B r a i n e r d . C h i l d D e v e l o p m e n t , 1976, k7, 897-898. M c V . H u n t , J . I n t e l l i g e n c e a n d e x p e r i e n c e . N e w Y o r k : R o n a l d P r e s s C o . , 196l. 88 M p i a n g u , B . , a n d G e n t i l e , R . J . I s c o n s e r v a t i o n o f n u m b e r n e c e s s a r y f o r m a t h e m a t i c a l u n d e r s t a n d i n g ? J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n . 1975, 179-191. P e c k , D . M . , a n d J e n c k s , S . M . M i s s i n g a d d e n d p r o b l e m s . S c h o o l S c i e n c e a n d  M a t h e m a t i c s . 1976, 8, 6k7-661. P i a g e t , J . J u d g e m e n t a n d r e a s o n i n g i n t h e c h i l d . L o n d o n : R o u t l e d g e a n d K e g a n P a u l L t d . , 192b. P i a g e t , J . , a n d S z e m i n s k a , A . T h e c h i l d ' s c o n c e p t i o n o f n u m b e r . N e w Y o r k : W . W . N o r t o n a n d C o . , 19%2~l P i a g e t , J . P s y c h o l o g y a n d e p i s t e m o l o g y . M i d d l e s e x : P e n g u i n B o o k s L t d . , 1972. S p i t l e r , G . , a n d M c K i n n o n , H . T h e c o s d a r k i t . U n p u b l i s h e d m a n u s c r i p t , 1976. S t e f f e , L . T h e p e r f o r m a n c e o f f i r s t g r a d e c h i l d r e n i n f o u r l e v e l s o f c o n s e r v a t i o n o f n u m e r o u s n e s s a n d t h r e e I Q g r o u p s w h e n s o l v i n g a r i t h m e t i c a d d i t i o n p r o b l e m s . T e c h n i c a l R e p o r t N o . l k , R e s e a r c h a n d D e v e l o p m e n t C e n t e r f o r L e a r n i n g a n d R e - e d u c a t i o n , U n i v e r s i t y o f W i s c o n s i n , 1966. S t e f f e , L . T h e r e l a t i o n s h i p o f c o n s e r v a t i o n o f n u m e r o u s n e s s t o p r o b l e m s o l v i n g a b i l i t i e s o f f i r s t - g r a d e c h i l d r e n . J o u r n a l f o r R e s e a r c h i n  M a t h e m a t i c s E d u c a t i o n , 1971. S t e f f e , L . , S p i k e s , W . , a n d H i r s t e i n , J . Q u a n t i t a t i v e c o m p a r i s o n s a n d c l a s s i n c l u s i o n a s r e a d i n e s s v a r i a b l e s f o r l e a r n i n g f i r s t g r a d e a r i t h m e t i c a l  c o n t e n t . T h e G e o r g i a C e n t r e f o r t h e S t u d y o f L e a r n i n g a n d T e a c h i n g M a t h e m a t i c s , U n i v e r s i t y o f G e o r g i a , 1976, S u p p e s , P . S o m e t h e o r e t i c a l m o d e l s f o r m a t h e m a t i c s l e a r n i n g . J o u r n a l o f  R e s e a r c h a n d D e v e l o p m e n t i n E d u c a t i o n , 1967, 1 (1), 5-22. S u y d a m , M . N . , a n d W e a v e r , J . F . A d d i t i o n a n d s u b t r a c t i o n w i t h w h o l e n u m b e r s , s e t B . I n U s i n g R e s e a r c h : A K e y t o E l e m e n t a r y S c h o o l M a t h e m a t i c s . ( E R I C I n f o r m a t i o n A n a l y s i s C e n t e r ) , D e c e m b e r , 1975. W a s h b u r n e , C, a n d V o g e l , M . A r e a n y n u m b e r c o m b i n a t i o n s i n h e r e n t l y d i f f i c u l t ? J o u r n a l o f E d u c a t i o n a l R e s e a r c h , 1928, 17, 235-255. W e a v e r , J . F . S o m e f a c t o r s a s s o c i a t e d w i t h p u p i l s p e r f o r m a n c e l e v e l s o n s i m p l e o p e n a d d i t i o n a n d s u b t r a c t i o n s e n t e n c e s . A r i t h m e t i c T e a c h e r , 1971, 18, 513-519. W e a v e r , J . F . T h e s y m m e t r i c p r o p e r t y o f t h e e q u a l i t y r e l a t i o n a n d y o u n g c h i l d r e n ' s a b i l i t y t o s o l v e o p e n a d d i t i o n a n d s u b t r a c t i o n s e n t e n c e s . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 1973, k ( l ) , k5-k6. W h e a t l e y , G . H . C o n s e r v a t i o n , c o u n t i n g , a n d c a r d i n a t i o n a s f a c t o r s i n m a t h e m a t i c s a c h i e v e m e n t a m o n g f i r s t g r a d e s t u d e n t s . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f D e l a w a r e , 1967. 89. Winer, B.J. S t a t i s t i c a l p r i n c i p l e s i n experimental design. New York: McGraw-Hill Inc., 1971. A P P E N D I X A. T w o - w a y C l a s s i f i c a t i o n M a t r i x G r i d 91. A P P E N D I X B . T w o - w a y C l a s s i f i c a t i o n M a t r i x G r i d 92. A P P E N D I X C. T h r e e - w a y C l a s s i f i c a t i o n M a t r i x G r i d A 93 APPENDIX D. Piagetian Assessment Name: School: Identif ication Number: Date: TASK RESPONSE Simple Seriation (a) incorrect sequence (b) t r i a l and error (c) correct sequence (d) other Double Seriation (a) incorrect ordering of second sequence to f i r s t . (b) t r i a l and error (c) correct ordering of second sequence to f i r s t , (d) other Simple Classi f icat ion Attribute blocks (a) unable to sort shapes (b) sorted by colour (c) sorted by shape (d) other Animals (a) unable to sort animals (b) sorted by (c) other Two-way Classification Circ le-tr iangle matrix (a) incorrect choice (b) correct choice; incorrect explanation (c) correct choice; correct explanation (d) other Flower-square matrix (a) incorrect choice (b) correct choice; incorrect explanation (c) correct choice; correct explanation (d) other Three-way Classi f icat ion Triangle-sq uare matrix (a) incorrect choice (b) correct choice; incorrect explanation (c) correct choice; correct explanation (d) other A P P E N D I X D . ( C o n t ' d . ) C l a s s I n c l u s i o n : F l o w e r s ( a ) m o r e p i n k f l o w e r s t h a n f l o w e r s (6 p i n k v s . 2 y e l l o w ( b ) m o r e f l o w e r s t h a n p i n k f l o w e r s ( i . e . , a l l a r e f l o w e r s ( c ) o t h e r W o o d e n B e a d s ( a ) m o r e g r e e n / w h i t e b e a d s t h a n b e a d s ( b ) m o r e b e a d s t h a n g r e e n / w h i t e b e a d s ( c ) o t h e r C o n s e r v a t i o n : N u m b e r ( a ) m o r e b l o c k s ( i . e . , r o w i s l o n g e r o r ( b ) l e s s b l o c k s ( i . e . , r o w i s s q u a s h e d o r ( c ) s a m e ; r e a s o n ( d ) o t h e r C o n t i n u o u s Q u a n t i t y ( a ) m o r e c l a y i n b a l l ; l e s s c l a y i n s n a k e ( i . e . , b a l l i s b i g g e r ) ( b ) m o r e c l a y i n s n a k e ; l e s s c l a y i n b a l l ( i . e . , s n a k e i s l o n g e r ) ( c ) s a m e ; r e a s o n ( d ) o t h e r D i s c o n t i n u o u s Q u a n t i t y ( a ) m o r e c l a y i n s i n g l e b a l l ; l e s s i n s m a l l b a l l s ( i . e . , s i n g l e b a l l i s b i g g e r ) ( b ) m o r e c l a y i n s m a l l b a l l s ; l e s s i n s i n g l e b a l l ( i . e . , f i v e s m a l l b a l l s v s . o n e b i g b a l l ) ( c ) s a m e ; r e a s o n ( d ) o t h e r A d d i t i o n a l C o m m e n t s : 95. APPENDIX E. Arithmetic Interview Recording Form Name: School: Identification Number: Operational Level: Item Solution Strategy 7 * 2 = | | 9 * 5 " n 1 2 - 8 = • l o - U - • 1 5 - 7 = Q 1 8 - 6 = Q 5 • • - 9 7 - | ] = 1 3 

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