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A T E S T O F T H E V A L I D I T Y O F A C O N C R E T E -A B S T R A C T H I E R A R C H Y O F A D D I T I O N F A C T S O N K I N D E R G A R T E N C H I L D R E N b y B R I A N WARD E d . , 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 , 1 9 7 2 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 L M E N T OF THE R E Q U I R E M E N T S FOR THE D E G R E E OF M A S T E R OF A R T S ^ i n THE F A C U L T Y OF 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 ) We 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 THE U N I V E R S I T Y OF B R I T I S H C O L U M B I A J u n e , 1 9 7 9 © B r i a n W a r d 1 9 7 9 I n 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 t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t 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 . D e p a r t m e n t n f OTfotJ The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e DE-6 BP 7 5-5 1 1 E i i A B S T R A C T T h e s t u d y a t t e m p t s t o v a l i d a t e a t a s k - a n a l y t i c h i e r a r c h y f o r t h e n u m b e r f a c t s o f a d d i t i o n b e t w e e n 6 a n d 9 a l o n g a C o n c r e t e - A b s t r a c t c o n t i n u u m . L i t e r a t u r e i s r e v i e w e d i n f o u r a r e a s : T a s k A n a l y t i c h i e r a r -c h i e s i n a r i t h m e t i c ; t h e ; l e a r n i n g t h e o r y w h i c h i s a s s o c i a t e d w i t h t a s k a n a l y t i c h i e r a r c h i e s ; t h e i n t e r a c t i o n o f t a s k a n a l y t i c l e a r n i n g t h e o r y a n d d e v e l o p m e n t a l l e a r n i n g t h e o r y ; a n d a t t e m p t s t o i n d i c a t e b e h a v i o u r s w h i c h s p e c i f y a d e f i n i t i o n o f a c o n c e p t o f n u m b e r s w h i c h c a n b e h i e r a r -c h i c a l l y a r r a n g e d , r e s e a r c h e d a n d d i s c u s s e d . A C o n c r e t e - A b s t r a c t h i e r a r c h y i s p r e s e n t e d w i t h s e v e n T a s k L e v e l s a r r a n g e d i n f o u r C o n c e p t u a l L e v e l s d e s i g n a t e d C o n c r e t e , S e m i - C o n c r e t e , C o n c r e t e - A b s t r a c t a n d A b s t r a c t . Q u e s t i o n s a r e p o s e d w h i c h c o n s i d e r t h e p o s s i b i l i t i e s o f v a l i d a t i n g b o t h a s e v e n - s t a g e h i e r a r c h y a n d a f o u r - s t a g e h i e r a r c h y . T h e t a r g e t p o p u l a t i o n w a s k i n d e r g a r t e n c h i l d r e n b e t w e e n t h e a g e s o f 5 y e a r s 0 m o n t h s a n d 5 y e a r s 1 1 m o n t h s i n o n e s c h o o l d i s t r i c t i n B r i t i s h C o l u m b i a . T h i s e n t i r e k i n d e r g a r t e n p o p u l a t i o n w a s s c r e e n e d a n d t h e o r i g i n a l 1 2 5 c h i l d r e n r e d u c e d t o 4 9 i n o r d e r t o c o n f o r m w i t h p r e t e s t E n t e r i n g B e h a v i o u r s o f a d e v e l o p m e n t a l n a t u r e c o n c e r n e d w i t h t h e c o n c e p t o f n u m b e r s . F r o m t h e s e c h i l d r e n s e v e n s u b j e c t s w e r e c h o s e n a s s u b j e c t s f o r a p i l o t t e s t t o i n v e s t i g a t e w h e t h e r s u b j e c t s w o u l d a n s w e r m o s t o f t h e q u e s t i o n s o n t h e t e s t d e s i g n e d , a n d t o s t a n d a r d i z e t h e t e s t i n g i n s t r u c t i o n s . T w e n t y -e i g h t s u b j e c t s w e r e c h o s e n f o r t h e m a i n s t u d y : 1 4 b o y s a n d 1 4 g i r l s . T w e n t y - o n e o f t h e 2 6 n u m b e r f a c t s b e t w e e n 6 a n d 9 w e r e u s e d i n t h e m a i n s t u d y a n d t h e o t h e r 5 i n t h e p i l o t s t u d y . A G r e c o - L ' a t i n S q u a r e w a s e m p l o y e d t o d e s i g n a b a l a n c e d t e s t w h i c h i n c o r p o r a t e d t h e s e v e n l e v e l s o f t h e h i e r a r c h y w i t h s e v e n s e t s o f i t e m s . E a c h i t e m s e t o f t h r e e n u m b e r f a c t s w a s a l s o b a l a n c e d s o t h a t t h e l e v e l o f d i f f i c u l t y o f t h e s e s e t s w a s , according to established c r i t e r i a , approximately equal. P r i o r to the tes t i n g session each subject was taught three items at each Task Level of the hypothesized hierarchy. This teaching was given to seven groups of chi l d r e n , each group composed of 2 boys and 2 g i r l s . In order to decide the items taught at each l e v e l of the hierarchy, a row of a L a t i n Square was chosen at random and matched with a balanced set of items. When subjected to a Guttman Scalogram Analysis data supported the existence of a f o u r - l e v e l hierarchy. Using a cut-off point of 2 out of 3 items correct a c o e f f i c i e n t of r e p r o d u c i b i l i t y of 0.95 was achieved. C e i l i n g e f f e c t s did occur, however, which made i t d i f f i c u l t to substantiate the hypothesized differences between the Concrete and Semi-Concrete Levels. A possible cause advanced, i s the pretest which eliminated 76 children from the population p r i o r to beginning the study, thus producing a group of subjects who were most l i k e l y high i n i n t e l l i g e n c e and i n t e l l i g e n c e -r e l a t e d behaviours. i v T A B L E OF C O N T E N T S P a g e A b s t r a c t i i T a b l e o f C o n t e n t s i v L i s t o f T a b l e s v i i L i s t o f F i g u r e s v i i i C H A P T E R I N T R O D U C T I O N TO THE P R O B L E M AND A R E V I E W OF THE I . R E L A T E D R E S E A R C H 1 T a s k A n a l y t i c H i e r a r c h i e s . . . . . . 1 A C u m u l a t i v e T h e o r y o f K n o w l e d g e . . . . 3 D e v e l o p m e n t a l L e a r n i n g T h e o r y a s C o n t r a s t e d w i t h C u m u l a t i v e L e a r n i n g T h e o r y . . . . 1 1 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 c e p t s i n P r e - S c h o o l C h i l d r e n . . . . . . . . 1 3 I m p l i c a t i o n s o f t h e R e l a t e d R e s e a r c h . . 1 7 I I . T H E R E S E A R C H P R O B L E M 1 8 C o n c r e t e - A b s t r a c t C o n t i n u u m 1 9 C o n c r e t e l e v e l t a s k s . . . . . . . . 1 9 S e m i - C o n c r e t e l e v e l t a s k s 1 9 C o n c r e t e - A b s t r a c t l e v e l t a s k s 2 0 A b s t r a c t l e v e l t a s k s 2 0 T a s k L e v e l s 2 0 D e f i n i t i o n o f T e r m s 2 1 i\ZZ». F i n a l T a s k 2 1 C r i t e r i o n f o r P e r f o r m a n c e 2 1 I n s t r u c t i o n a l O b j e c t i v e 2 1 E n t e r i n g B e h a v i o u r s 2 1 L e a r n i n g H i e r a r c h y 2 1 T a s k A n a l y s i s 2 1 P a g e R e s e a r c h O b j e c t i v e s 2 3 T a r g e t P o p u l a t i o n 2 3 I I I . METHODOLOGY 2 5 M e a s u r i n g I n s t r u m e n t 2 5 P r e t e s t 2 6 P i l o t T e s t 2 7 S u b j e c t s 2 7 I t e m D i f f i c u l t y 27 T e s t i n g S e q u e n c e 2 9 T e a c h i n g P r o c e d u r e s 3 1 S t a t i s t i c a l A n a l y s i s 3 4 A n a l y s i s o f V a r i a n c e 3 4 D e s c r i p t i v e A n a l y s i s 3 5 G u t t m a n S c a l o g r a m A n a l y s i s 3 5 I V . R E S U L T S OF THE P R E T E S T AND P I L O T T E S T . . . 3 6 P r e t e s t T e s t 3 6 P i l o t T e s t . 4 1 I m p l i c a t i o n s o f t h e P r e t e s t a n d P i l o t T e s t f o r t h e M a i n S t u d y 4 3 V . R E S U L T S OF THE STUDY 4 5 A n a l y s i s o f V a r i a n c e 4 5 D e s c r i p t i v e A n a l y s i s 5 2 G u t t m a n S c a l o g r a m A n a l y s i s 5 3 V I . D I S C U S S I O N , L I M I T A T I O N S AND R E C O M M E N D A T I O N S . 5 7 D i s c u s s i o n 5 7 L i m i t a t i o n s o f t h e S t u d y 6 0 R e c o m m e n d a t i o n s f o r F u t u r e R e s e a r c h . . . 6 2 R E F E R E N C E S 6 4 v i Page APPENDICES . • 67 Appendix A 67 Appendix B 68 Appendix C 70 Appendix D 72 v i i L I S T OF T A B L E S T A B L E S P a g e 1 I t e m s A r r a n g e d b y L e v e l o f D i f f i c u l t y . . . 2 8 2 I t e m s G r o u p e d t o E q u a l i z e D i f f i c u l t y o f S e t s o f T h r e e 2 9 3 T e s t i n g S e q u e n c e a s D e t e r m i n e d b y a G r e c o - L a t i n S q u a r e 3 0 4 I t e m S e t a n d i t s M a t c h e d L e v e l i n t h e H i e r a r c h y 3 2 5 N u m b e r o f S u b j e c t s P a r t i c i p a t i n g i n t h e P r e t e s t 3 6 6 P r e t e s t R e s u l t s 3 8 7 D i s t r i b u t i o n o v e r S e x , S c h o o l s a n d T e s t i n g T i m e s o f S u b j e c t s Who M e t S c r e e n i n g C r i t e r i a . 3 9 8 S u b j e c t s Who F a i l e d t h e P r e t e s t I l l u s t r a t e d A c c o r d i n g t o P r e t e s t C a t e g o r y 4 0 9 R e s u l t s o f t h e P i l o t T e s t A c c o r d i n g t o H i e r a r c h y L e v e l 4 2 1 0 S u m m a r y o f t h e A n a l y s i s o f V a r i a n c e o f t h e S e q u e n c e o f I t e m s W i t h i n t h e T e s t . . . . 4 6 1 1 S u m m a r y o f t h e A n a l y s i s o f V a r i a n c e o f I t e m D i f f i c u l t y A c c o r d i n g t o t h e F o u r E s t a b l i s h e d C r i t e r i a 4 9 1 2 M e a n P e r f o r m a n c e b y S e x a n d I t e m D i f f i c u l t y . 5 0 1 3 A S u m m a r y o f t h e A n a l y s i s o f V a r i a n c e o f I t e m D i f f i c u l t y f o r B a l a n c e d G r o u p s o f I t e m s . . 5 1 1 4 A S u m m a r y o f t h e A n a l y s i s o f V a r i a n c e o f t h e H y p o t h e s i z e d H i e r a r c h y 5 2 1 5 R e s u l t s o f G u t t m a n S c a l o g r a m A n a l y s i s -S e v e n L e v e l s 5 4 1 6 R e s u l t s o f G u t t m a n S c a l o g r a m A n a l y s i s -F o u r L e v e l s 5 6 v i i i LIST OF FIGURES FIGURES Page 1 Hypothesized Hie ra rchy Showing the Four Conceptual L e v e l s and the Seven Task Leve l s . . 22 2 Hypothesized Hie ra rchy f o r A d d i t i o n Facts Showing the Conceptual L e v e l s 1 to 4 and Task L e v e l s 1 to 7 , 48 3 Mean Responses to Item Sets at Each L e v e l of the Hie ra rchy 33 1 Chapter One I n t r o d u c t i o n to the Problem and a Review of the Rela ted Research The f e a s i b i l i t y of a r rang ing conceptua l i n fo rma t ion i n a s e q u e n t i a l order such tha t a h i e r a r c h y of s k i l l s i s cons idered to be e s t a b l i s h e d has been of i n t e r e s t fo r some t ime . The l e a r n i n g of one se t of behaviours has been purported to l ead a u t o m a t i c a l l y t o , and support the l e a r n i n g o f , a r e l a t e d more d i f f i c u l t se t of behav iours . This premise has been the bas i s fo r the des ign of programmed l e a r n i n g and l e a r n i n g experiments which have sought to d i m i n i s h or remove f a i l u r e i n master ing a new behaviour or set of behav iours . This review presents some of the research tha t r e l a t e s to h i e r a r c h i c a l l e a r n i n g i n four major a reas : genera l research and research c o n s i d e r a t i o n s tha t a f f e c t an unders tanding of task h i e r a r c h i e s i n a r i t h m e t i c ; the theory of knowledge tha t emerges from such a d i s c u s s i o n (the cumula t ive l e a r n i n g t h e o r y ) ; the r e l a t i o n s h i p of developmental l e a r n i n g theory to cumula t ive l e a r n i n g theory ; and research on p r e - s c h o o l c h i l d r e n tha t bears most d i r e c t l y upon the research problem and the research ques t ions tha t are posed i n t h i s s tudy . Task A n a l y t i c H i e r a r c h i e s Gagne (1961a) hypothes ized tha t as a l e a r n e r progressed up a h i e r a r c h y h i s r a t e of l e a r n i n g should depend i n c r e a s i n g l y on the a t ta inment or non-at tainment of r e l e v a n t l e a r n i n g se t s (performance of a group of f a c t s at a s p e c i f i e d l e v e l of the h i e r a r c h y ) , and dec r ea s ing ly on r e l e v a n t a b i l i t i e s the i n d i v i d u a l b r i n g s to the s i t u a t i o n . From an experiment designed to 2 analyze the c l a s s of t a sks f o r s o l v i n g l i n e a r equa t ions , Gagne concluded: (a) r e l e v a n t b a s i c a b i l i t i e s are c l e a r l y of more importance i n the master ing of a h i e r a r c h y than i r r e l e v a n t b a s i c a b i l i t i e s ; (b) in s t ances of p o s i t i v e t r a n s f e r to each l e a r n i n g se t from sub-o rd ina t e r e l e v a n t l e a r n i n g se t s are found to occur throughout the h i e r a r c h y ; (c) c o r r e l a t i o n s of r e l e v a n t b a s i c a b i l i t i e s w i t h r a t e s of a t ta inment of l e a r n i n g se t s at p r o g r e s s i v e l y h igher l e v e l s of the h i e r a r c h y show a s t e e p l y p rog re s s ive decrease; and (d) c o r r e l a t i o n s of r a t e of a t ta inment of l e a r n i n g se t s w i th a c h i e v e -ment of r e l e v a n t subordina te l e a r n i n g se t s are found to be s y s t e m a t i c a l l y higher than wi th the achievement of i r r e l e v a n t se t s p a r t i c u l a r l y i n the upper l e v e l s of the h i e r a r c h y . Gagne' (1962b) then undertook a study on r e c a l l a b i l i t y and i n t e g r a t i o n p resen t ing sub jec t s w i th a programme which had twelve subord ina te t a sks suppor t ing two r e l a t e d f i n a l t a s k s . Low and h igh a b i l i t y s tudents were t e s t ed to f i n d out how much r e p e t i t i o n of l e a r n i n g se t s was r e q u i r e d , and how much d e t a i l was to be p r o v i d e d , i n order for s tudents to i n t e g r a t e the i n f o r m a t i o n . The sub jec t s were 136 seventh grade c h i l d r e n who were ass igned to groups by achievement ( teacher grades f o r a r i t h m e t i c because the hypothes ized h i e r a r c h i e s were i n a r i t h m e t i c ) . The experiment showed a h igh c o r r e l a t i o n between the achievement of f i n a l t a sks and the number of subord ina te l e a r n i n g se t s which were a c q u i r e d . There was a l s o evidence to suggest tha t subord ina te l e a r n i n g se t s i n a h i e r a r c h y mediate p o s i t i v e t r a n s f e r to a h igher l e v e l l e a r n i n g s e t . Three years l a t e r Gagne'(1965) examined r e t e n t i o n of i n i t i a l l y l earned knowledge of non-metr ic geometry, man ipu la t ing the i n s t r u c t i o n s v a r i a b l e and to ga in i n fo rma t ion tha t would permit in fe rences to be made about memory. A f t e r a n ine week i n t e r v a l measures were taken on: (a) achievement on i tems tha t were matched fo r content w i th the o r i g i n a l achievement t e s t , and (b) performance on each of the subordina te tasks tha t were c o n t r i b u t i n g to the f i n a l t a s k . F i v e groups of s i x t h grade c h i l d r e n , s i x t e e n per group, worked through a programme wi th i d e n t i c a l content but composed of f i v e d i f f e r e n t exper imenta l t rea tments . P r a c t i c a l examples of each subord ina te task had: (a) minimal v a r i e t y i n con ten t , (b) i n t e rmed ia t e v a r i e t y , (c) great v a r i e t y , (d) no examples, or (e) un re la t ed t a s k s . The c o r r e l a t i o n between the i n i t i a l achievement t e s t s was +0.64 and the c o r r e l a t i o n between achievement scores immediately a f t e r l e a r n i n g and those obta ined a f t e r a n ine week r e t e n t i o n i n t e r v a l was +0.62. C o r r e l a t i o n s between scores on subord ina te l e a r n i n g h i e r a r c h i e s a f t e r nine weeks, however, were on ly +0.41 and +0.46. The subordina te t a sks necessary to l e a r n the f i n a l task were not w e l l main ta ined; the ques t ion tha t arose was whether t h e i r importance d imin i shed once the f i n a l task was l e a r n e d . The l e a r n e r d i d not forge t p a r t i c u l a r pa r t s of the h i e r a r c h y ; f o r g e t t i n g was e n t i r e l y random. The f i n a l tasks themselves , however, were w e l l ma in ta ined . A Cumulat ive Theory of Knowledge Gagneaand Brown (1961b) undertook a p iece of r esea rch which i n f l uenced Gagn^'s t h i n k i n g about both the a c q u i s i t i o n of knowledge and 4 the importance of i n s t r u c t i o n s i n acquiring that knowledge. The purpose of the study was to obtain a measure of the learning effectiveness of programmed material i n terms which would permit a reasonable inference that understanding had been accomplished. Thirty-three boys i n grades 9 and 10 were randomly assigned to three experimental groups and then i n d i v i d u a l l y tested. The programmes were termed 'rule and example 1 (the teacher presented the ru l e that 2 + 3 = 3 + 2 and then the c h i l d did several examples); 'guided discovery' (the teacher was a c t i v e l y involved i n the process of discovery and helped the c h i l d re-state previously acquired concepts that were relevant to the problem); and 'discovery' ( t r i a l and error learning where the teacher provided structure, but was not a c t i v e l y involved i n the discovery process). Performance was measured i n terms of time taken to solve a problem and the number of hints required. Learning occurred i n a l l three groups but 'guided discovery' produced the most learning followed by 'discovery' and f i n a l l y 'rule and example'. Of in t e r e s t i s the f a c t that the guided discovery programme not only used 'small steps', as i s common i n programmed learning, but also required the subject to re-state c e r t a i n previously acquired concepts which were relevant to the problems i n the experimental condition. The r e - s t a t i n g of c e r t a i n concepts by the students at the request of the researcher, and the p o s i t i v e r e s u l t s obtained, focused attention upon the nature of in s t r u c t i o n s , and also upon the necessity of r e c a l l i n g relevant subordinate learning sets i n order to obtain a sol u t i o n to the task presented. In a t h e o r e t i c a l paper on Knowledge and Instructions, Gagne /(1962a) stated that i f learning sets are e s s e n t i a l for p o s i t i v e t r a n s f e r the following consequences should ensue: 5 (a) i f a h igher l e v e l l e a r n i n g set i s passed, a l l r e l a t e d lower l e v e l se t s must have been passed; (b) i f one or more of the lower l e v e l se t s have been f a i l e d , the r e l a t e d h igher l e v e l t a sks must be f a i l e d ; and (c) i f a h igher l e v e l se t has been f a i l e d , r e l a t e d lower l e v e l s e t s may have been passed. The absence of p o s i t i v e t r a n s f e r would be a t t r i b u t a b l e to a d e f i c i e n c y i n i n s t r u c t i o n s . A theory was o u t l i n e d which def ined knowledge as an i n f e r r e d c a p a b i l i t y tha t makes p o s s i b l e the s u c c e s s f u l performance of a c l a s s of t a sks tha t cou ld not be performed before l e a r n i n g was undertaken. P roduc t ive l e a r n i n g was seen as i n v o l v i n g two v a r i a b l e s : (a) knowledge, and (b) i n s t r u c t i o n s . I t i s important to understand e x a c t l y what Gagne'meant by ' i n s t r u c t i o n s ' and ' s a t i s f a c t o r i l y complet ing a f i n a l t a s k ' , because h i s d e f i n i t i o n s were not n e c e s s a r i l y t r a d i t i o n a l ones. In order to have a framework fo r d i s c u s s i n g the l e a r n i n g c o n d i t i o n s necessary fo r enqu i ry , tha t i s a c t i v i t i e s c h a r a c t e r i z e d by a p rob l em-so lv ing approach - Gagne'' (1963) de l i nea t ed the f i n a l task as the ' t e r m i n a l c a p a b i l i t y of the l e a r n e r ' , and i n s t r u c t i o n s , more b r o a d l y , as ' i n s t r u c t i o n a l c o n d i t i o n s ' . Enqui ry needed p r a c t i c e and a s u f f i c i e n t background of exper ience ; new problems c o u l d only be so lved by the a c q u i s i t i o n of knowledge. D r i l l and d i s c o v e r y were both components of i n s t r u c t i o n s ; however, d i s c o v e r y and enquiry were not the same. The c o n s t r u c t i o n of a response by a l e a r n e r was termed ' d i s -c o v e r y ' , whereas ' e n q u i r y ' was the t e r m i n a l t h i n k i n g p rocess . A s e r i e s of r e q u i r e d s teps were hypothes ized i n order fo r a s tudent to a t t a i n the p r a c t i c e o f enqu i ry : (a) competent performer ( l e a r n i n g of s k i l l s ) , (b) a c q u i s i t i o n of knowledge, (c) s c i e n t i f i c enqu i re r (genuine enqu i ry ; a b i l i t y to d i s t i n g u i s h good and bad i deas ; a b i l i t y to s o l v e problems by means of un re s t r a ined i n d u c t i v e t h i n k i n g ) , and (d) independent i n v e s t i g a t i o n ( e . g . s c i e n t i s t ) . Extending h i s t h i n k i n g about i n s t r u c t i o n s f u r t h e r , Gagne^ (1964) noted tha t the p rob lem-so lve r came to a g iven s i t u a t i o n w i th p r e v i o u s l y learned c a p a b i l i t i e s ; the experimenter then communicated w i th the sub jec t by way of i n s t r u c t i o n s i n four ways: (a) by i d e n t i f y i n g new s t i m u l i , (b) by i d e n t i f y i n g the expected form of the t e r m i n a l performance, (c) by h e l p i n g the sub jec t r e c a l l p r e v i o u s l y acqu i red c a p a b i l i t i e s , (d) by channe l ing t h i n k i n g i n a r e l e v a n t d i r e c t i o n . Gagne^cal led fo r research which would manipulate the v a r i a b l e i n s t r u c t i o n s based upon the hypothes is tha t b e t t e r i n s t r u c t i o n s would f a c i l i t a t e the r e c a l l o f p r e v i o u s l y acqu i red r e l e v a n t concepts . Problem s o l v i n g cou ld be reduced. to an exper imenta l s i t u a t i o n where the independent v a r i a b l e s f e l l i n t o the broad c a t e g o r i e s o f : (a) sub jec t c a p a b i l i t y , and (b) i n s t r u c t i o n s . Gagne''and Brown's (1961b) s tudy opened the way fo r a s e r i e s of e x p e r i ments which culminated i n the model of l e a r n i n g presented by Gagne' ' in 1968 These experiments emphasized the n e c e s s i t y of exact communication between teacher and s tuden t . The i n s t r u c t i o n s , which are q u i t e c l e a r l y important i n a l l forms of t e a c h e r - p u p i l l e a r n i n g , take on a new dimension when 7 looked at from the v iewpoin t of a task h i e r a r c h y . Each new andx h igher s tage can on ly be learned i f the s tudent understands the i n s t r u c t i o n s . A l s o , i n any hypothes ized h i e r a r c h y a h igher l e a r n i n g se t can on ly be learned when the r e l e v a n t lower se t s i n the h i e r a r c h y are mastered. Gagne' 'defined communication i n terms of s t imu lus i d e n t i f i c a t i o n , i d e n t i f i c a t i o n of the form of t e r m i n a l response, r e c a l l of p r e v i o u s l y acqu i red c a p a b i l i t i e s , and channe l ing t h i n k i n g i n a r e l e v a n t d i r e c t i o n . Quite c l e a r l y , however, the problem i s not tha t s i m p l e ; P e r s o n a l i t y , m o t i v a t i o n a l , and a d d i t i o n a l c o g n i t i v e v a r i a b l e s must be c o n s i d e r e d . One v a r i a b l e which Gagne 'd id i n v e s t i g a t e was memory. Because many subordina te t a sks w i t h i n h i e r a r c h i e s were f o r g o t t e n , he quest ioned whether these t asks were necessary once the t e r m i n a l o b j e c t i v e was ach ieved . A fu r the r problem was posed by the l i m i t e d amount of i n f o r m a t i o n a v a i l a b l e about how the human memory s to red and coded i n f o r m a t i o n . The apparent problem of f o r g e t f u l n e s s might be more one of r e t r i e v a l of i n fo rma t ion from the long- te rm memory s t o r e . Gagne^s model of l e a r n i n g (based on a theory of cumula t ive l e a r n i n g ) con t ras t ed i n a number of respec ts wi th developmental t h e o r i e s whose c e n t r a l themes were ma tu ra t i on , r e a d i n e s s , and c o g n i t i v e a d a p t a t i o n . He i d e n t i f i e d the stage at which a person cou ld be as i n v o l v i n g . ( a ) r e l e v a n t c a p a b i l i t i e s possessed, and (b) a number of h i e r a r c h i e s of c a p a b i l i t i e s to be a c q u i r e d , such that i t i s p o s s i b l e to combine subord ina te e n t i t i e s to help achieve the t e r m i n a l t a s k . The model took i n t o account t r a n s f e r of l e a r n i n g : what was l ea rned could be combined wi th other l ea rned e n t i t i e s v i a a mechanism of l e a r n i n g t r a n s f e r . 8 In the second e d i t i o n of h i s book Gagne'' (1970) d i scussed the r e l a t i o n s h i p of va r i ous forms of l e a r n i n g and, i n p a r t i c u l a r , noted tha t the ex i s t ence of p r i o r c a p a b i l i t i e s - important i n drawing d i s t i n c t i o n s among c o n d i t i o n s r equ i r ed f o r l e a r n i n g - was s l i g h t e d or ignored by most t r a d i t i o n a l l e a r n i n g p ro to types . He went on to s t a t e tha t when b u i l d i n g a h i e r a r c h y the t asks should be analyzed such tha t the lowest ' boxes ' i n a h i e r a r c h y represented the k i n d of performance a l l s tudents i n a group cou ld a l r eady s u c c e s s f u l l y accompl i sh . The statements i n the ' boxes ' were in tended to d e s c r i b e a s i n g l e c a p a b i l i t y to be l e a r n e d , r ep re sen t ing what the l e a rne r was ab le to:';do when l e a r n i n g had been accompl ished. They were, t h e r e f o r e , s t a t ed i n performance terms. The superord ina te c a p a b i l i t y was more r e a d i l y learned i f the subordina te c a p a b i l i t i e s had been p r e v i o u s l y acqu i red and.were r e a d i l y a v a i l a b l e fo r r e c a l l . Each subordina te c a p a b i l i t y had been i d e n t i f i e d as such because i t was known, or hypo thes ized , to c o n t r i b u t e p o s i t i v e t r a n s f e r to the l e a r n i n g of a superord ina te c a p a b i l i t y . Students who had learned subordina te s k i l l s should l e a r n superord ina te s k i l l s more e a s i l y than those who had not learned them. Engelmarnwas concerned wi th the man ipu la t ion of t each ing v a r i a b l e s ; l i k e Gagne he was i n t e r e s t e d i n t a k i n g i d e a s , concep ts , and i n f o r m a t i o n , and a r rang ing them i n a h i e r a r c h i c a l f a s h i o n . He argued (1969a) tha t a concept was a se t of c h a r a c t e r i s t i c s shared by a l l ' i n s t a n c e s ' i n a p a r t i c u l a r se t and on ly by those ' i n s t a n c e s ' , and tha t concepts were c l a s s e s and thus amenable to h i e r a r c h i c a l arrangement. He cons idered i t necessary to r e a l i z e tha t concepts were always dependent on the contex t or un iverse i n which they were presented , and tha t to ana lyze a concept so tha t i t cou ld be taught was to desc r ibe the concept i n terms of the 9 m i n i m u m s e t o f e s s e n t i a l d i s c r i m i n a t i o n s t h e c h i l d h a d t o m a k e i n o r d e r t o r e c o g n i z e t h a t a n ' i n s t a n c e ' w a s , o r w a s n o t , a m e m b e r o f t h a t c o n c e p t s e t . F o r E n g e l m a n n , t e a c h i n g c o n s i s t e d o f t w o c o m p o n e n t s : ( a ) t h e d e m o n s t r a t i o n s t h e t e a c h e r p e r f o r m s t o s h o w t h e c h a r a c t e r -i s t i c s o f t h e c o n c e p t s , a n d ( b ) t h e t e s t o r t a s k t h e c h i l d p e r f o r m s t o d e m o n s t r a t e h i s u n d e r -s t a n d i n g o f t h e c o n c e p t . H e c o m m e n t e d f u r t h e r t h a t n o c o n c e p t c o u l d b e t a u g h t i n a s i n g l e d e m o n -s t r a t i o n , a n d t h a t i t w a s n e c e s s a r y t o p r e s e n t b o t h ' i n s t a n c e s ' a n d n o n - ' i n s t a n c e s ' o f t h e c o n c e p t . T h e b e s t p r e s e n t a t i o n w a s o n e w i t h t h e l e a s t p o t e n t i a l f o r m i s u n d e r s t a n d i n g a n d w i t h a m i n i m u m m e m o r y l o a d ; i n d e v e l o p i n g a p r e s e n t a t i o n t h e t e a c h e r s h o u l d c o n c e n t r a t e o n t h e c r i t i c a l d i s c r i m i n a t i o n s t h e c h i l d h a d t o m a k e . E n g e l m a n n n o t e d t h a t w h e n t e s t s o f c o n c e p t u n d e r s t a n d i n g w e r e t r a n s l a t e d i n t o t a s k s , t h e m o v e w a s f r o m p o s s i b l e b e h a v i o u r t o s p e c i f i c b e h a v i o u r , a n d f r o m a r a n g e o f p o s s i b l e w a y s t o t e s t u n d e r s t a n d i n g o f a c o n c e p t , t o s p e c i f i c w a y s . C o n c e p t a n a l y s i s y i e l d e d t e s t i n g s p e c i f i c a t i o n s , a n d u n l e s s t e s t s w e r e s p e c i f i e d i n t e r m s o f e x a c t t a s k s i t w a s n o t p o s s i b l e t o d e t e r m i n e e x a c t l y w h a t t h e c h i l d h a d b e e n t a u g h t : a c h i l d m i g h t f a i l a p r o b l e m b e c a u s e h e d i d n o t k n o w t h e c o n v e n t i o n f o r r e s p o n d i n g . B a s i c a l l y , w h e n a c h i l d w a s t a u g h t a p a r t i c u l a r t a s k h e h a d t o l e a r n t h r e e t h i n g s : ( a ) t h e c o n c e p t b e i n g t a u g h t , ( b ) t h e r u l e s f o r r e s p o n d i n g , a n d ( c ) t h e r u l e t h a t r e s p o n d i n g i s w o r t h w h i l e . E n g e l m a n n a d d e d o n e i m p o r t a n t f a c t o r n o t p r e v i o u s l y m e n t i o n e d : a s t u d e n t m u s t f i n d t h e a c t o f r e s p o n d i n g w o r t h w h i l e . T h e c o n c e p t o f m o t i v a t i o n i s , t h e r e f o r e , a c k n o w l e d g e d a n d i t i s a l s o a p p a r e n t t h a t E n g e l m a n n i s 10 sugges t ing a r o l e f o r p e r s o n a l i t y v a r i a b l e s tha t have some e f f e c t upon.a s t u d e n t ' s s e l f - e s t e e m . Upr i cha rd and P h i l l i p s (1977) reviewed the research l i t e r a t u r e of the 1960's and concluded tha t e p i s t e m o l o g i c a l c o n s i d e r a t i o n s were paramount i n the des ign of l e a r n i n g h i e r a r c h i e s . They c i t e d Gagne's statement tha t the "de te rmina t ion of a h i e r a r c h i c a l sequence of subtasks from s i m p l e s t to most complex was not e a s i l y a c h i e v e d , " and c la imed tha t l o g i c a l a n a l y s i s , based upon a theory of knowledge, was d i f f i c u l t to v a l i d a t e because i t ignored p s y c h o l o g i c a l f a c t o r s . Engelmann (1969a) however, had cons idered p s y c h o l o g i c a l v a r i a b l e s i n emphasizing the r o l e of m o t i v a t i o n . The o rde r ing of a h i e r a r c h y would c l e a r l y be i n f l u e n c e d by p r i o r l e a r n i n g , the i n s t r u c t i o n a l s i t u a t i o n , memory and many other f a c t o r s . A d d i t i o n a l l y , the w i l l i n g n e s s of a s tudent to i n t e r a c t w i th the l e a r n i n g environment v a r i e s a long many dimensions and would no doubt a f f e c t the outcome of any e x p e r i -ments seeking the ' p e r f e c t ' h i e r a r c h y . Cumulat ive Learn ing theory has enabled educa t iona l researchers to focus on i n s t r u c t i o n a l o b j e c t i v e s . While both Gagne''and Engelmann broached the i s s u e , Barbara Bateman (1971) fu r thered i t s development. L i k e Gagne''"and Engelmann, she b e l i e v e d i n s t r u c t i o n a l o b j e c t i v e s to be necessary i n de termining the goa ls i n a l e a r n i n g t a s k . She suggested the f o l l o w i n g c r i t e r i a f o r the s e l e c t i o n of o b j e c t i v e s : (a) do the o b j e c t i v e s i n d i c a t e l e a r n i n g outcomes tha t are appropr i a t e t o the i n s t r u c t i o n a l area? (b) do the o b j e c t i v e s represent a l l l o g i c a l l e a r n i n g outcomes of the i n s t r u c t i o n a l area? (c) are the o b j e c t i v e s ob ta inab le by these s tudents? 11 (d) are the o b j e c t i v e s i n harmony w i t h the ph i losophy of the s choo l i n which the i n s t r u c t i o n i s to be g iven? (e) are the o b j e c t i v e s i n harmony wi th the t e a c h e r ' s pe rce ived needs fo r the fu ture of the c h i l d and s o c i e t y ? Task a n a l y s i s i s c l e a r l y one way of s p e c i f y i n g o b j e c t i v e s accord ing to the fo rego ing c r i t e r i a ; i t i s at f i r s t l o g i c a l , then e m p i r i c a l . The s tudy desc r ibed i n the f o l l o w i n g chapters hypothes izes tha t task a n a l y s i s i s e s s e n t i a l to s u c c e s s f u l l e a r n i n g . Whi le the b a s i s i s a n a l y t i c a l and h i e r a r c h i c a l , developmental c o n s i d e r a t i o n s w i l l not be i gno red . The view espoused i s tha t i n order to teach w i th purpose, c l e a r goals need to be o u t l i n e d i n order f o r t each ing to be s u c c e s s f u l w i th most c h i l d r e n . Developmental Learn ing Theory as Contras ted w i th Cumulat ive Learn ing Theory Engelmann (1967) was p a r t i c u l a r l y c r i t i c a l of the developmental t h e o r i e s of l e a r n i n g , i n d i c a t i n g tha t developmental i n t e r p r e t a t i o n s , such as those of P i a g e t , are b a s i c a l l y i r r e l e v a n t to t eache r s , and tha t when an i r r e l e v a n t e x p l a n a t i o n i s accepted by a teacher she ceases to f u n c t i o n i n a t each ing c a p a c i t y . I r r e l e v a n t exp lana t ions ' l o c k the door ' on a problem, whereas a teacher seeks to develop and change a c h i l d i r r e s p e c t i v e of t h i s e x p l a n a t i o n . To Engelmann, a developmental e x p l a n a t i o n does not imply a remedy tha t can be achieved through the man ipu la t ion of environmental v a r i a b l e s ; as a consequence i t does not dea l w i th v a r i a b l e s under the t e a c h e r ' s c o n t r o l . He added tha t of ten the developmental exp lana t ion cannot be t r a n s l a t e d i n t o concepts whereas the teacher must t r a n s f e r the c h i l d ' s r e l a t i v e d e f i c i e n c y i n t o terms of concepts because these imply a man ipu la t ion of v a r i a b l e s , no t i ng tha t P iage t cons idered ' c o n s e r v a t i o n ' a concept , yet p laced a host of developmental r e s t r i c t i o n s on the concep t ' s a c q u i s i t i o n . He f e e l s tha t i f i t i s a concept then i t 12 can be taught ; i f i t i n v o l v e s s p e c i f i c con ten t , i t i s c e r t a i n l y r . n o t the product of c o g n i t i v e s t r u c t u r i n g but r a the r the product of environmental c o n s i s t e n c y . Many other authors have agreed wi th Engelmann's admi t t ed ly extreme po in t of v iew. K i l p a t r i c k (1970) i n d i c a t e d tha t the c r e a t o r s of the School Mathematics Study Group programmes were guided by l o g i c a l c o n s i d -e r a t i o n s r a the r than p s y c h o l o g i c a l , c l a i m i n g the e m p i r i c a l to be the on ly d e f e n s i b l e approach. P i a g e t , they argued, was an obse rver , not a t eacher . They cons idered the p r e d i c t i o n s which emerged from h i s developmental model u n j u s t i f i e d as they were based on i n v e s t i g a t i o n s of the unders tand-i n g of c h i l d r e n who had been taught by conven t iona l methods. Karp lus (1970) supported t h i s v iew, no t ing tha t P iage t found tha t formal opera t ions developed wi thout s p e c i f i c i n s t r u c t i o n , and tha t t h i s was not an adequate premise to encompass the r e s u l t s , t h i n k i n g or a t t i t u d e s of modern s c i e n c e . This d i s c u s s i o n i s important f o r many reasons . B u e l l (1970) s t a t ed tha t the c o n t r o v e r s i a l i s s u e as to whether the l e a r n i n g of concepts may be a c c e l e r a t e d i s of c o n s i d e r a b l e s i g n i f i c a n c e i n view of the tendency i n recent years to advance the p r e s e n t a t i o n of s c i e n t i f i c and mathematical m a t e r i a l to e a r l i e r l e v e l s of the c u r r i c u l u m . M i l l e r (1976) , us ing 52 Kinde rga r t en c h i l d r e n from a lower s o c i o -economic background, attempted to f i n d out whether conse rva t i on as def ined by P iage t emerged e a r l i e r i n a c h i l d ' s development i f the t a sks were presented n o n - v e r b a l l y and m o t i v a t i o n was maximized. The c h i l d cou ld p i c k the candies tha t he wanted to eat i n a conse rva t ion of number t a s k , or the j u i c e he wanted to d r i n k i n a conse rva t ion of l i q u i d t a s k . M i l l e r commented tha t language might w e l l be an important f a c t o r because the c h i l d r e n seemed to judge numbers i n terms of length, and that t h i s concept seemed to be activated by the word 'more'. He concluded, how-ever, that there was no evidence from his study to suggest that conservation emerged at an e a r l i e r chronological age than theorized by Piaget. It i s very possible that no s o l u t i o n to the argument e x i s t s at present. Bruner (1967) took t h i s p o s i t i o n , asking whether i t was more valuable to postulate stages of growth or to think i n terms of gradual processes of growth. He claimed the argument was f r u i t l e s s . The Development of Number Concepts i n Pre-School Children Saxe (1977) analyzed children's counting s t r a t e g i e s with arrays of more than f i v e objects with the express purpose of assessing whether developmental changes i n t h e i r s t r a t e g i e s were p a r t i a l l y independent of t h e i r a b i l i t y to count accurately. He concluded that counting accuracy and s t r a t e g i e s were i n t e r r e l a t e d and that they appeared to be connected to the same cognitive process. The main point of i n t e r e s t i n t h i s study was that at the Pre-quantitative l e v e l s almost no accurate count-ing occurred, and at the Quantitative l e v e l s counting accuracy markedly improved. However, at the T r a n s i t i o n a l l e v e l s , between the Pre-quantitative and the Quantitative l e v e l s some chil d r e n appeared to miscount 'purposely' as a r e s u l t of t r y i n g to integrate t h e i r s p a t i a l judgments of numbers with the products of t h e i r counting. This a r t i c l e emphasized the d i f f i c u l t y of attempting to a r b i t r a r i l y d i s t i n g u i s h a sequence of i n t e r r e l a t e d concepts without taking into consideration a c h i l d ' s cognitive maturation. In some cases, however, evidence appears i n the research to substantiate the idea of teaching one concept before another . B r a i n e r d (1974) took a group of p r e - s c h o o l e r s who evidenced no p r o f i c i e n c y wi th e i t h e r the o r d i n a l or c a r d i n a l p r o p e r t i e s of n a t u r a l numbers and t r a i n e d them to acqu i re these p r o p e r t i e s . H i s 60 male and 60 female Caucasian sub jec t s ranged from 4 years 0 months to 5 years 5 months w i th a mean age of 4 years 7 months. The r e s u l t s of h i s s tudy showed tha t both p r o p e r t i e s were t r a i n a b l e , but tha t the o r d i n a l p r o p e r t i e s of numbers were much e a s i e r to t r a i n than the c a r d i n a l p r o p e r t i e s . He a l s o noted tha t the i n fo rma t ion lea rned about o r d i n a l p r o p e r t i e s was more e a s i l y t r a n s f e r r e d to o ther l e a r n i n g s i t u a t i o n s than the i n fo rma t ion l ea rned about c a r d i n a l p r o p e r t i e s . The r e s u l t s of t h i s research i m p l i e d tha t the understanding of the o r d i n a l p r o p e r t i e s of numbers would precede tha t of the c a r d i n a l p roper t i e s i n such a way tha t i t would be f e a s i b l e to p lace them i n a h i e r a r c h i c a l r e l a t i o n s h i p to one another . S i e g e l (1971) designed a sequence of exper imenta l s i t u a t i o n s i n order to ana lyze whether c e r t a i n a r i t h m e t i c behaviours formed a h i e r a r c h i c a l r e l a t i o n s h i p to one another . One of the stages of tha t hypothes ized h i e r a r c h y was designed to measure the concept of the o r d i n a l p roper ty of numbers. I t i s important to no te , however, tha t the t asks she designed were non-verba l to remove language as a confounding f a c t o r i n the a n a l y s i s of the r e s u l t s . Her hypothe-s i z e d h i e r a r c h y was: (a) magnitude d i s c r i m i n a t i o n s of s o l i d cont inuous a reas , (b) magnitude d i s c r i m i n a t i o n s of d i scon t inuous d i s c r e t e a reas , (c) equ iva lence of se t s (one-to-one correspondence) , (d) conse rva t ion of numbers ( P i a g e t i a n concept but t e s t ed n o n - v e r b a l l y ) , (e) the O r d i n a l p roper ty of numbers f o r example, 1 s t ; 2nd; 3 rd ; 15 ( f ) s e r i a t i o n ( r e c o g n i z i n g the order of a s e r i e s of ob jec t s tha t are arranged i n a s t ep -wise f a s h i o n ) , and (g) a d d i t i o n equat ions (presented p i c t o r i a l l y ) . The sub jec t s were 77 m i d d l e - c l a s s c h i l d r e n ranging from 3 years 0 months to 4 years 11 months i n age, wi th boys and g i r l s approximate ly e q u a l l y represen ted . S i e g e l attempted to v a l i d a t e her r e s u l t s by us ing the Guttman Scalogram A n a l y s i s t echn ique , but f a i l e d to o b t a i n the r equ i r ed value of .95 fo r the c o e f f i c i e n t of r e p r o d u c i b i l i t y . The ove r -a l l average c o e f f i c i e n t of r e p r o d u c i b i l i t y was 0.89 and the h ighes t s i n g l e c o e f f i c i e n t was 0 .93 . Desp i te t h i s r e s u l t , the o r d e r i n g of the hypothes ized h i e r a r chy was of some i n t e r e s t . There was no d i f f e r e n c e between the two s e r i e s of t a sks tha t t e s t ed magnitude d i s c r i m i n a t i o n s , and s e r i a t i o n and a d d i t i o n were at approximate ly the same l e v e l of d i f f i c u l t y . O r d i n a t i o n occurred i n the middle of the h i e r a r c h y and was c l e a r l y much e a s i e r than s e r i a t i o n and a d d i t i o n . Conserva t ion was harder than one-to-one correspondence but e a s i e r than s e r i a t i o n and a d d i t i o n . O v e r a l l , the p o s i t i o n of the tasks i n r e l a t i o n to one another formed a branching h i e r a r c h y r a the r than a l i n e a r h i e r a r c h y . C l e a r l y , i t would be advantageous to have some c l e a r s p e c i f i c a t i o n s of c u r r i c u l a , or r a t i o n a l e f o r c u r r i c u l a r c h o i c e s , i n order to p l an more p r e c i s e l y f o r the educat ion of p r e - schoo l c h i l d r e n . Resnick (1967, 1970) o u t l i n e d an o p e r a t i o n a l d e f i n i t i o n of the number concept i n the form of a se t of behaviours which , taken toge the r , a l l o w a person to cons ider the c h i l d to have mastered the concept of numbers: (a) one-to-one correspondence to 5, (b) one-to-one correspondence to 10, (c) r e c o g n i z i n g the numerals from 0 to 5, . 16 (d) r e c o g n i z i n g the numerals from 0 to 10, (e) comparison of Sets ( conse rva t ion of numbers), ( f ) s e r i a t i o n and O r d i n a l p o s i t i o n ( o r g a n i z i n g m a t e r i a l i n order of s i z e and des igna t i ng i t as 1 s t , 2nd, 3 r d , e t c . ) , and (g) a d d i t i o n and s u b t r a c t i o n Equa t ions . Wong, Resnick and Boozer (1971) examined the sequence i n which young c h i l d r e n acqu i r e the elementary mathematical behaviours of c o u n t i n g , one-to-one correspondence and r e c o g n i t i o n of numerals, hypo thes i z ing t h a t : (a) r e c o g n i t i o n of numerals i s l ea rned e a r l y i n the process of l e a r n i n g to count , (b) count ing and r e c o g n i t i o n of numerals fo r q u a n t i t i e s up to f i v e would normal ly be learned before e i t h e r c l a s s of s k i l l s was l ea rned fo r q u a n t i t i e s up to t e n , (c) count ing and r e c o g n i t i o n of numerals are l ea rned independent ly w i th n e i t h e r c l a s s of t a sks p r e r e q u i s i t e to the o the r , and (d) there i s a p s y c h o l o g i c a l independence of count ing and one - to -one s k i l l s . The sub jec t s were 42 boys and 36 g i r l s between the ages of 4 years 6 months and 6 years 0 months who were a t t end ing k i n d e r g a r t e n . S i x t y - t h r e e percent of the c h i l d r e n were b lack and t h i r t y - s e v e n percent were wh i t e ; pa ren t s ' work s t a tus ranged from unemployed to e x e c u t i v e - p r o f e s s i o n a l . Resu l t s i n d i c a t e d tha t command over numerals, numerals acqui red i n a r e g u l a r sequence, beginning wi th pe rcep tua l matching of the numerals and conc lud ing wi th the a s s o c i a t i o n of se t s and numerals , i s not o r d i n a r i l y learned u n t i l a f t e r count ing opera t ions f o r se t s of the s i z e represented by the numerals are w e l l e s t a b l i s h e d . Data a l s o supported the p s y c h o l o g i c a l d i f f e r e n t i a t i o n of count ing and one-to-one correspondence. I m p l i c a t i o n s of the Rela ted Research Cons ide r ing extant research i n f o r m a t i o n , i t i s p o s s i b l e to argue tha t a r i t h m e t i c behaviours can be arranged i n a h i e r a r c h i c a l manner. Whether i n f a c t they can be arranged so that i t i s p o s s i b l e to c l a i m tha t c e r t a i n behaviours are p r e r e q u i s i t e to o thers i n forming a concept of numbers i s another mat ter . I t does appear reasonable , however, to hypothes ize tha t a d d i t i o n equat ions would be at the top of a h i e r a r c h y which i n c l u d e d the f a c t o r s taken i n t o c o n s i d e r a t i o n by S i e g e l , Saxe, B r a i n e r d , Wong, Resnick and Boozer . Resnick (1967, 1970) p laced a d d i t i o n and s u b t r a c t i o n equat ions a t the top of her hypothes ized h i e r a r c h y and S i e g e l (1971) noted tha t a d d i t i o n equat ions presented p i c t o r i a l l y come at the top of a h i e r a r c h y along wi th s e r i a t i o n . B r a i n a r d (1974) found tha t o r d i n a l numbers appear to be grasped before c a r d i n a l numbers, S i e g e l (1971) tha t conse rva t ion appeared e a s i e r than s e r i a t i o n and a d d i t i o n , and Wong, Resnick and Boozer (1971) tha t count ing e s t a b l i s h e s i t s e l f p r i o r to the r e c o g n i t i o n of numerals. I t appears tha t i t i s reasonable to hypothes ize a h i e r a r c h y of a r i t h m e t i c behav iours . I t seems u n l i k e l y , tha t these behaviours w i l l form a l i n e a r r e l a t i o n s h i p to one another; both S i e g e l (1971) and Wong, Resnick and Boozer (1971) comment on the ' b r a n c h i n g ' nature of the h i e r a r c h i e s formed. The Research Problem posed i n Chapter Two w i l l take i n t o account e x i s t i n g research i n v o l v i n g the concept of number, the behaviours which appear to be p r e r e q u i s i t e to t h i s concept , and the nature of h i e r a r c h i e s . 18 Chapter Two The Research Problem Chapter One has o u t l i n e d s i g n i f i c a n t research by Resnick (1970) and Wong, Resnick and Boozer (1971), who attempted to present r a t i o n a l arguments fo r t each ing a s p e c i f i e d group of a r i t h m e t i c behaviours as a b a s i s f o r deve loping a concept of numbers. Whi le Resnick (1967, 1970) suggested a d e f i n i t i o n of the number concept , Wong, Resnick and Boozer went f u r t he r and d e t a i l e d a hypothes ized h i e r a r c h y of a r i t h m e t i c behaviours i n a u se fu l order f o r c lassroom teachers at the K inde rga r t en and F i r s t Grade l e v e l . Resu l t s of t h e i r study supported the idea tha t some a r i t h m e t i c behaviours are l ea rned i n a c l e a r l y def ined sequence, a l though they were quick to po in t out tha t some behav iours , such as count ing and one-to-one correspondence, appeared to be p s y c h o l o g i c a l l y independent of one another . I f the idea of a h i e r a r c h y i s t enab le then i t i s u n l i k e l y tha t i t would be s t r i c t l y l i n e a r w i th one behaviour l e a d i n g a u t o m a t i c a l l y to the next most d i f f i c u l t behaviour i n a s t e p -wise f a s h i o n . Resnick (1970) o u t l i n e d seven behaviours which c o u l d be cons idered as one p o s s i b l e o p e r a t i o n a l d e f i n i t i o n of a number concept : (a) one-to-one correspondence to 5, (b) one-to-one correspondence to 10, (c) r e c o g n i z i n g the numerals 0 - 5 , (d) r e c o g n i z i n g the numerals 0 - 1 0 , (e) comparison of s e t s , ( f ) s e r i a t i o n and o r d i n a l p o s i t i o n , and (g) a d d i t i o n and s u b t r a c t i o n equa t ions . 19 The focus of t h i s study i s the mastery of equa t ions . Other aspects of a number concept , i n c l u d i n g f a c t o r s such as c o u n t i n g , conse rva t ion of d i s c r e t e e n t i t i e s , and numeration are cons idered to be p r e r e q u i s i t e s to the behaviours i n v o l v e d i n s o l v i n g a d d i t i o n equa t ions . The research problem under i n v e s t i g a t i o n i s whether a h i e r a r c h i c a l r e l a t i o n s h i p e x i s t s between c e r t a i n bas i c number Atema j _ n a d d i t i o n . The a d d i t i o n number (li-emsfehosen were those between 6 and 9 ( for example, 5 + 1 ; 4 + 3 ; 6 + 2 ) , r a the r than those between 1 and 5 i n order to ensure a s u f f i c i e n t degree of d i f f i c u l t y f o r the s u b j e c t s . The hypothe-s i z e d h i e r a r c h y to be cons idered i n c l u d e s four d i f f e r e n t conceptua l l e v e l s - Concre te , Semi-Concrete , Abs t r ac t -Conc re t e and A b s t r a c t . These terms are desc r ibed on the f o l l o w i n g pages. Conc re t e -Abs t r ac t Continuum Concre t e -Abs t r ac t can be def ined along a r e a l e n t i t y - a b s t r a c t e n t i t y continuum, where r e a l e n t i t i e s are a c t u a l pa r t s of the e n v i r o n -ment and a b s t r a c t e n t i t i e s are numeric r ep re sen t a t i ons of i t . (An e n t i t y i s anyth ing to which a numeral can be ass igned such tha t i t can be designated q u a n t i t i v e l y ) . Concrete l e v e l . t a s k s . Concrete t a sks w i l l use ob jec t s from the environment which can both be seen and p i cked up. As a consequence they w i l l be u n l i k e the other l e v e l s to be desc r ibed i n tha t they do not represent r e a l i t y ; they are i n f a c t a par t of r e a l i t y i t s e l f . Both crayons and b locks w i l l be used a t the concre te l e v e l of the h i e r a r c h y . Semi-concrete l e v e l t a s k s . Semi-concrete l e v e l t a sks are hypothes ized to be more a b s t r a c t than the concre te t a sks because p i c t u r e s of ob jec t s are used to represent the r e a l o b j e c t s . The p i c t u r e s seen are of shapes, f i s h and matchs t i cks and r e q u i r e the c h i l d to abstract at a d i f f e r e n t cognitive l e v e l to that outlined under Concret Level Tasks. Concrete-Abstract l e v e l tasks. These tasks involve the use of symbols i n combination with T a l l y Marks. These tasks are hypothesized to be more abstract than the Semi-Concrete tasks i n that they do not use any representation of r e a l i t y . In the previous l e v e l (Semi-Concrete) matchsticks were used whereas at t h i s l e v e l the t a l l y marks do not represent any p a r t i c u l a r object or objects i n the environment. The use of symbols i s hypothesized to increase the degree of abstractness. However, i n t h i s combination the abstractness i s decreased to the extent that the t a l l y marks can s t i l l function as a concrete aid i n solving an equation. Abstract l e v e l tasks. Abstract tasks are those i n which numerical symbols are used by themselves to represent e n t i t i e s within the environ-ment. Task Levels Some l e v e l s of the.hierarchy w i l l contain d i f f e r e n t sets of tasks as mentioned above. These w i l l be c a l l e d Task Levels i n order to d i s t i n g u i s h them from the four conceptual l e v e l s just described. Figure 1 presents the hypothesized hierarchy with seven task l e v e l s within the four conceptual l e v e l s . At the Concrete l e v e l there are two task l e v e l s - one which uses blocks, and the other, crayons - as a means to assess the concrete l e v e l of the hierarchy. At the Semi-Concrete l e v e l there are three d i f f e r e n t sets of tasks - one uses shapes, a second f i s h , and a t h i r d matchsticks. At the Concrete-Abstract l e v e l there i s only one set of tasks which uses the t a l l y marks, and at the Abstract l e v e l there i s only one set of tasks which uses numerals. 21 D e f i n i t i o n of Terms The f o l l o w i n g d e f i n i t i o n s are e s s e n t i a l to c l a r i f y the des ign procedures , r e s u l t s and c o n c l u s i o n s of t h i s s tudy: F i n a l Task: A d d i t i o n of two one d i g i t numerals whose sum i s no grea ter than n i n e . C r i t e r i o n f o r Performance: Complet ion of the task w i t h i n the l i m i t s of the s t andard ized i n s t r u c t i o n s ( I n s t r u c t i o n s fo r Teaching, Appendix D ) . I f e r r o r o c c u r s , the subjec t has to i n i t i a t e and complete c o r r e c t i o n wi thout a s s i s t a n c e . I n s t r u c t i o n a l O b j e c t i v e : Success fu l comple t ion of the F i n a l Task a t the h ighes t l e v e l of the h i e r a r c h y ; tha t i s , a t the most a b s t r a c t l e v e l . En t e r ing Behav iours : Those a r i t h m e t i c behaviours which are cons idered p r e r e q u i s i t e to the understanding of b a s i c number i t ems , and which a long wi th a demonstrated mastery of bas ic a d d i t i o n i tems are i l l u s t r a t i v e of an understanding of the concept of numerals as r ep re sen t -i n g e n t i t i e s w i t h i n the environment which can be q u a n t i t a t i v e l y des igna ted . Learn ing H i e r a r c h y : The r u l e or problem s o l v i n g task i s analyzed i n t o s imple c a p a b i l i t i e s to be l ea rned as p r e r e q u i s i t e s . When such an a n a l y s i s i s cont inued p r o g r e s s i v e l y to the p o i n t of d e l i n e a t i n g an e n t i r e se t of c a p a b i l i t i e s having an ordered r e l a t i o n s h i p to each o the r , then a Learn ing h i e r a r c h y e x i s t s (Gagne 1970). Task a n a l y s i s : The process of i s o l a t i n g , d e s c r i b i n g and sequenc-i n g a l l the necessary s u b - t a s k s , which , when they are mastered, w i l l enable s u c c e s s f u l mastery of the i n s t r u c t i o n a l o b j e c t i v e . (Bateman 1971). 22 L e v e l 4 A b s t r a c t L e v e l 3 A b s t r a c t - C o n c r e t e L e v e l 2 Semi-Concrete L e v e l 1 C o n c r e t e < F i g u r e 1 The Seven H y p o t h e s i z e d T e s t L e v e l s A r r a n g e d i n a F o u r - L e v e l H i e r a r c h y Research Ob jec t i ve s 1. To a s c e r t a i n whether a l i n e a r r e l a t i o n s h i p , or a branching r e l a t i o n s h i p , e x i s t s between the l e v e l s of the hypothes ized h i e r a r c h y . 2. To a s c e r t a i n whether tasks hypothes ized to be at a g iven l e v e l of the h i e r a r c h y are of equal d i f f i c u l t y and remain i n the same r e l a t i o n s h i p to one another . 3. To a s c e r t a i n whether any t asks w i t h i n any l e v e l of the h i e r a r -chy are of any grea te r or l e s s e r d i f f i c u l t y than any other t a s k s , such tha t t h e i r r e l a t i o n s h i p to one another becomes l i n e a r . 4. To a s c e r t a i n whether there i s any s i g n i f i c a n t d i f f e r e n c e between the performances of boys and g i r l s . 5. To a s c e r t a i n whether any of the b a s i c number f a c t combinat ions between 6 and 9 are any more d i f f i c u l t than any other combinat ions f o l l o w i n g t h e i r i n i t i a l ba lanc ing , accord ing to e s t a b l i s h e d c r i t e r i a . 6. To a s c e r t a i n whether any r e l a t i o n s h i p e x i s t s between the l e v e l s of the hypothes ized h i e r a r c h y and the task l e v e l s of the h i e r a r -chy , such tha t i t i s p o s s i b l e to say tha t a h i e r a r c h y e x i s t s which moves from the Concrete l e v e l to the A b s t r a c t l e v e l . Target P o p u l a t i o n I n i t i a l l y us ing a Grade One p o p u l a t i o n was cons idered but i t became readi%yapparent tha t these c h i l d r e n would f i n d the t a s k s , as des igned , too easy. A l s o , unless the data cou ld be c o l l e c t e d immediate l y the s tudents entered Grade One, the r e s u l t s would be confounded by th f a c t tha t the s tudents would have been exposed to a r i t h m e t i c t e a c h i n g . Thus, K inde rga r t en c h i l d r e n were s e l e c t e d as the most s u i t a b l e group to use i n t h i s s tudy . However, given the factors discussed i n Chapter One with regard to developmental v a r i a b l e s , i t was necessary to delimit the subject population so as to include only those c h i l d r e n who had a s u f f i c i e n t l e v e l of cognitive development i n mathematics, and to exclude those who would already be able to answer a l l of the questions that would a r i s e from the seven l e v e l s of the hypothesized hierarchy. To e f f e c t t h i s a pretest and a p i l o t t e s t were designed; the Pretest to e s t a b l i s h whether the subjects had the necessary Entering Behaviours and the P i l o t Test to e s t a b l i s h whether any of the subjects would f i n d the tasks at Level One of the hierarchy (Abstract) too easy. Chapter Three Methodology In order to investigate the research questions s p e c i f i e d i n Chapter Two i t was necessary to control factors such as item d i f f i c u l t y , standardized t e s t i n g and teaching i n s t r u c t i o n s , item presentation i n a te s t i n g s i t u a t i o n , and present knowledge of arithmetic behaviours, before attempting to i n t e r p r e t data from each l e v e l of the hierarchy. It i s important to f i n d out whether the items selected are sui t a b l e for the subjects chosen. Standardizing i n s t r u c t i o n s allows each subject an equal chance to learn i n the teaching s i t u a t i o n and an equal chance to respond i n the t e s t i n g s i t u a t i o n ; randomizing the t e s t i n g procedures establishes a control to deal with unequal opportunities to learn when a c t u a l l y being tested. Prior to the commencement of teaching i t i s also important to assess whichcschildren are not developmentally ready to benefit from the teaching, and which ch i l d r e n have already acquired the knowledge that i s presented. Measuring Instrument As indicated previously, the items selected were the 26 basic addition . i tems whose sums were no l e s s than s i x , and no greater than nine (Appendix A). To f a c i l i t a t e c o n t r o l of item d i f f i c u l t y and to allow the construction of a measuring instrument which would co n t r o l f o r factors that would a f f e c t the test r e s u l t s , f i v e of these ftemsxwere randomly eliminated, leaving 21 f a c t s to be incorporated into the t e s t i n g and teaching procedures. The f i v e items randomly eliminated were: 26 5 + 1 7 + 1 2 + 7 3 + 6 4 + 5 These f a c t s were used to e s t a b l i s h whether any of the sub jec t s cou ld a l r eady answer the a b s t r a c t ques t ions at l e v e l four of the h i e r a r c h y . P r e t e s t One hundred t w e n t y - f i v e sub jec t s were admin i s t e red a p r e t e s t i n order to e s t a b l i s h whether the sub jec t s had the minimum E n t e r i n g Behav iour s , and to a s c e r t a i n how many sub jec t s had a l ready acqu i red the I n s t r u c t i o n a l Ob jec t i ve of the hypothes ized h i e r a r c h y . Those c h i l d r e n who d i d not have the s p e c i f i e d E n t e r i n g Behav iou r s , and those c h i l d r e n who reached the I n s t r u c t i o n a l O b j e c t i v e , were then excluded from the sub jec t p o o l . There were three s p e c i f i e d E n t e r i n g Behav iour s . F i r s t , the sub jec t had to i d e n t i f y v e r b a l l y numerals on cards presented i n a random order . Second, each sub jec t had to r e c i t e the numerals from one to n i n e . T h i r d , the sub jec t was r e q u i r e d to count a group of s i x and then four randomly arranged b locks (see Appendix C ) . The f i v e randomly e l i m i n a t e d i tems were used to assess whether any c h i l d cou ld a l r eady reach the I n s t r u c t i o n a l Ob jec t i ve at l e v e l four of the h i e r a r c h y . Any c h i l d responding to four or f i v e problems c o r r e c t l y wi thout the use of a i d s such as f i n g e r s , paper and p e n c i l , was excluded from the subjec t p o o l . Of the 125 sub jec t s assessed , 49 met the research c r i t e r i a . (For d e t a i l e d i n f o r m a t i o n , see Chapter F o u r ) . P i l o t T e s t T h e p i l o t T e s t w a s a d m i n i s t e r e d t o s e v e n s u b j e c t s ( 4 b o y s a n d 3 g i r l s ) w h o m e t t h e r e s e a r c h c r i t e r i a b u t d i d n o t p a r t i c i p a t e i n t h e s t u d y . T h i s i n v o l v e d g i v i n g t h e p o s t t e s t ( A p p e n d i x B ) t o t h e s e s u b j e c t s i n o r d e r t o s t a n d a r d i z e t h e i n s t r u c t i o n s a n d p r o c e d u r e s , a n d t o a s c e r t a i n w h e t h e r a r a n d o m s a m p l e o f s u b j e c t s w h o m e t t h e r e s e a r c h c r i t e r i a w o u l d b e v e r y s u c c e s s f u l o n t h e t e s t , t h a t i s , c o m p l e t e q u e s t i o n s a t L e v e l F o u r o f t h e h i e r a r c h y . ( F o r d e t a i l e d i n f o r m a t i o n , , s e e C h a p t e r F o u r ) . S u b j e c t s S u b j e c t s f o r t h i s s t u d y w e r e k i n d e r g a r t e n c h i l d r e n . C l a s s e s c o m p r i s i n g b o t h b o y s a n d g i r l s w e r e c h o s e n f r o m p u b l i c s c h o o l s i n t h e S q u a m i s h S c h o o l D i s t r i c t . S e v e n o f t h e 4 9 s u b j e c t s w h o m e t t h e r e s e a r c h c r i t e r i a w e r e u s e d i n t h e P i l o t T e s t , l e a v i n g 4 2 s u b j e c t s e l i g i b l e t o t a k e p a r t i n t h e s t u d y . F r o m t h i s g r o u p 1 4 b o y s a n d 1 4 g i r l s w e r e r a n d o m l y s e l e c t e d . I t e m D i f f i c u l t y T h e 2 1 i t e m s w h i c h m a d e u p t h e m e a s u r i n g i n s t r u m e n t w e r e a r r a n g e d i n t h r e e r o w s a c c o r d i n g t o t h e i r d e g r e e o f d i f f i c u l t y i n o r d e r t o b a l a n c e e a c h c e l l o f t h e d e s i g n a n d t h u s e q u a l i z e p r o b l e m d i f f i c u l t y . . T h e c r i t e r i a ( M a c h a t c h y 1 9 3 3 ; W a s h b u r n e ; V o g e l 1 9 2 8 ; W h e e l e r 1 9 3 9 ) f o r a r r a n g i n g t h e t w e n t y - o n e i t e m s i n t o t h r e e l e v e l s o f d i f f i c u l t y w i t h s e v e n i t e m s i n e a c h r o w w e r e : 1 . A n a d d i t i o n c o m b i n a t i o n a n d i t s ' r e v e r s e ' f o r m t e n d t o b e o f e q u a l d i f f i c u l t y . 2 . S i z e o f a d d e n d i s t h e p r i n c i p a l i n d i c a t o r o f d i f f i c u l t y , r a t h e r t h a n t h e s i z e o f t h e s u m . 3. Combinations with a common addend appear to be of s i m i l a r but not equal d i f f i c u l t y ( 4 + 4 ; 3 + 3 ) . 4. The doubles i n addition and those i n which 1 i s added with a greater number appear to be easiest i n addition. According to these c r i t e r i a , the items can be c l a s s i f i e d as follows: A : Easy items B : Moderaly D i f f i c u l t items C : Hard^ items These 21 items were then divided into seven groups of three; according to the c r i t e r i a outlined, a l l the problems i n column A would be the easiest, those i n column B more d i f f i c u l t , and those i n column C the hardest. Table 1 presents r e s u l t s of t h i s c l a s s i f i c a t i o n ; a l l items within each of the columns are considered to be of approximately the same degree of d i f f i c u l t y . Table 1 Items Arranged by Level of D i f f i c u l t y A B C Easy Items Moderately D i f f i c u l t Hard Items Items . 1 + 5 8 + 1 2 + 6 1 + 6 4 + 3 6 + 2 1 + 7 3 + 4 7 + 2 1 + 8 2 + 5 6 + 3 6 + 1 5 + 2 5 + 4 3 + 3 2 + 4 5 + 3 4 + 4 4 + 2 3 + 5 V 1 "hard 1 i s used synonomously with the word ' d i f f i c u l t ' . To control for item d i f f i c u l t y , these 21 items were then divided into seven groups of three with one item i n each group randomly selected from A, one from B, and one from C u n t i l a l l the items were exhausted. The item groups yielded by t h i s procedure are shown i n Table 2. Table 2 Items Grouped to Equalize D i f f i c u l t y of Sets of Three A g C Name of Easy Moderately Hard Item Group litems D i f f i c u l t Items Items a 1 + 5 3 + 3 2 + 6 b 4 + 4 4 + 3 7 + 2 c 3 + 3 4 + 2 6 + 3 d 1 + 7 8 + 1 6 + 2 e 1 + 8 2 + 4 5 + 3 f 6 + 1 5 + 2 3 + 5 g 1 + 6 2 + 5 5 + 4 Testing Sequence Grouping the items to control for t h e i r d i f f i c u l t y made possible the creation of a t e s t i n g sequence for assessing the subjects' responses to the teaching session. The t e s t i n g sequence i t s e l f was created by employing a Greco-Latin Square design that involved placing the 28 subjects at seven d i f f e r e n t l e v e l s , four subjects per l e v e l . Level one, for instance, would involve the f i r s t four subjects i n answering questions at the hypothesized hierarchy l e v e l and item set indicated. Table 3 shows the l e v e l s of the hierarchy combined with the re-grouped items. Looking at (for example) 6e, "6" would be the task l e v e l i n the hypothesized hierarchy and 'e' would be the group of three items 1 + 8 ; 2 + 4; 5 + 3 (see Table 2). Table 3 Testing Sequence as Determined by a Greco-Latin Square T-.ask Subjects Sequence Level 1 1 - 4 *6e 4b 2 f 7c 5g 3d l a 2 5 - 8 7f 5c 3g Id 6a 4e 2b 3 9 - 12 i g 6d 4a 2e 7b 5f 3c 4 13 - 16 2a 7e 5b 3f l c 6g 4d 5 17 - 20 3b I f 6c 4g 2d 7a 5c 6 21 - .24 4c 2g 7d 5a 3e l b 6f 7 25 - 28 5d 3a l e 6b 4f 2c 7g * i n each combination the number refer s to the l e v e l of the hierarchy (Fig. 1) and the l e t t e r r e f e r s to the item set (Table 2) The format for questioning the subjects^was formalized when the p i l o t t e s t was administered. Each subject was administered three questions at each task l e v e l thus requiring a t o t a l of 21 questions to be answered by each subject. (For detailed information concerning the Testing Instructions see Appendix B). 31 Teaching Procedures P r i o r to the t e s t i n g s e s s i o n each sub jec t was taught three i tems at each Task L e v e l of the hypothes ized h i e r a r c h y ; i n s t r u c t i o n was g iven to seven groups of c h i l d r e n , each group composed of two boys and two g i r l s . Each group learned th ree problems i n v o l v i n g b l o c k s , th ree i n v o l v i n g p i c t u r e s of f i s h , th ree i n v o l v i n g shapes, three i n v o l v i n g m a t c h s t i c k s , th ree i n v o l v i n g numerals and t a l l y marks, and three i n v o l v i n g on ly numerals. In order to dec ide which three i tems were to be taught at each l e v e l of the hypothes ized h i e r a r c h y , the f o l l o w i n g row was chosen a t random from a 7 by 7 L a t i n Square. The r e s u l t i n g sequence was: 5, 6, 4 , 7 , 3 , 1, 2. This row was then matched wi th the i tems as grouped i n Table 2 i n such a way tha t the "a" group of i tems (1 + 5; 3 + 3 ; 2 + 6 ) was matched wi th the number 5, the "b" group of problems wi th number 6 and so on . Resu l t s of t h i s procedure are presented i n Table 4 . Table 4 Item Set and i t s Matched Level i n the Hierarchy Item Name of Moderately Hierarchy Set Item Easy D i f f i c u l t Hard Level Group Items Items Items 5 a 1 + 5 3 + 3 2 + 6 1 6 b 4 + 4 4 + 3 7 + 2 2 4 c 3 + 3 4 + 2 6 + 3 3 7 d 1 + 7 8 + 1 6 + 2 4 3 e 1 + 8 2 + 4 5 + 3 5 1 f 6 + 1 5 + 2 3 + 5 6 2 g 1 + 6 2 + 5 5 + 4 7 The hypothesized hierarchy from l e v e l 7 to l e v e l 1 i s shown i n Figure 2. L e v e l 4' NUMERALS A b s t r a c t L e v e l 3 Abs t r ac t -Conc re t e L e v e l 2 Semi-Concrete L e v e l 1 Concrete X FISH 5 X jr CRAYONS NUMERALS AND TALLY MARKS SHAPES 4 MATCH-STICKS 3 = 1 BLOCKS 6 F igure 2. Hypothesized H ie ra r chy f o r A d d i t i o n Facts Showing the Conceptual L e v e l s 1 to 4 and Task L e v e l s 1 to 7 F igu re 2 shows the seven Task L e v e l s of the h i e r a r c h y . The t e a c h -i n g i tems were presented to the sub jec t s at Task L e v e l seven f i r s t , then Task L e v e l s i x , then Task L e v e l f i v e and so on , u n t i l a l l the sub jec t s had been exposed to twenty-one problems. At the Concrete l e v e l - task l e v e l s s i x and seven - two d i f f e r e n t se t s of concre te t a sks were g iven and at the Semi-Concrete l e v e l three d i f f e r e n t se t s of t a sks were presented , to assess whether a se t of t a sks at the same l e v e l (b locks ) posed any more d i f f i c u l t y fo r sub jec t s than another se t of t a sks a t tha t l e v e l ( c r ayons ) . The numbering of the task l e v e l s from seven to one i s , i n f a c t , designed to a s c e r t a i n (1) whether the h igher numbered task l e v e l s are easier than the lower numbered task l e v e l s , and (2) whether a l i n e a r r e l a t i o n s h i p exists between these task l e v e l s as d i s t i n c t from a branching r e l a t i o n s h i p . It i s possible to conceive, for instance, that problems involving "Matchsticks" are more abstract than problems involving " F i s h " . "Matchsticks" c l o s e l y resemble " T a l l y Marks" which are hypothesized to be more abstract than any tasks on the Semi-Concrete l e v e l . The same r a t i o n a l e applies to comparisons between "Fi s h " and "Shapes" (shapes may be more a b s t r a c t ) , Matchsticks and Shapes (match-s t i c k s may be more abstract) and "Blocks" and "Crayons" (blocks may be more ab s t r a c t ) . The i n s t r u c t i o n s for the teaching session were standardized so that no subjects gained any advantage over any other subject. Once the subjects were s e t t l e d , relaxed and attending, i n s t r u c t i o n s were i d e n t i c a l for each of the groups (for d e t a i l s of these i n s t r u c t i o n s see Appendix D). The only variance to the procedures as outlined i n Appendix D concerned support and encouragement of the subjects, and praise for correct responses. S t a t i s t i c a l Analysis An analysis of variance was c a r r i e d out i n order to consider: (a) possible differences i n the performance of the items when boys are compared with g i r l s , (b) possible variances i n the d i f f i c u l t y l e v e l s of c e r t a i n items when compared with other items, (c) possible variances i n the d i f f i c u l t y l e v e l s of groups of problems a f t e r having been balanced according to the l e v e l s of d i f f i c u l t y c r i t e r i a , and 35 (d) possible variances of d i f f i c u l t y l e v e l between i n d i v i d u a l problems within any group, which a l t e r s the balance of the group of problems. A d e s c r i p t i v e analysis was undertaken to e s t a b l i s h whether: (a) the hierarchy exhibits evidence of a l i n e a r r e l a t i o n s h i p between one Level and another, and one Task Level and another, or a branching r e l a t i o n s h i p between these Levels, (b) the tasks hypothesized to be at any one Level of the hierarchy remained at that l e v e l , (c) the degree of d i f f i c u l t y of a task at the Concrete or Semi-Concrete Levels i s any harder than any other task at the same Level, and (d) any r e l a t i o n s h i p e x i s t s between the Levels of the hypothesized hierarchy, and the Task Levels of the hierarchy, such that i t i s possible to say that a hierarchy e x i s t s which moved from the Concrete to the Abstract Level. A Guttman Scalogram Analysis was effected to attempt a s t a t i s t i c a l v a l i d a t i o n of the hierarchy. This technique analyzes the data according to the hypothesized Levels of the hierarchy, according to the Task Levels, and according to the number of items c o r r e c t l y answered by subjects at any given Level or Task Level. 36 Chapter Four Resu l t s of the P r e t e s t and P i l o t Test This chapter w i l l present the r e s u l t s of the p r e t e s t and t h e . p i l o t t e s t and d i scus s these r e s u l t s and t h e i r p o s s i b l e i m p l i c a t i o n s . The p r e t e s t , as mentioned i n Chapter Three, was designed to f i n d out i f the sub jec t s had the minimum En te r ing Behaviours as w e l l as how many c h i l d r e n had a l ready acqu i red the I n s t r u c t i o n a l Ob jec t i ve of the hypothes ized h i e r a r c h y . The two t e s t s i d e n t i f i e d a poo l of sub jec t s who would be a t approximate ly the same conceptua l l e v e l as f a r as t h e i r unders tanding of numbers was concerned. The o r i g i n a l sub jec t poo l encompassed a l l o f the k inde rga r t en c h i l d r e n i n Squamish, a t o t a l of 125. Table 5 o u t l i n e s the breakdown of these c h i l d r e n by schoo l and by sex . Table 5 Number of Subjects P a r t i c i p a t i n g i n the P r e t e s t School .Boys . . G i r l s T o t a l 1 8 12 20 2 11 15 26 3 25 15 40 4 19 20 _39 T o t a l 63 62 125 T h e r e s u l t s o f t h e p r e t e s t a r e p r e s e n t e d i n T a b l e 6. E a c h s u b j e c t w a s a s s e s s e d a s h a v i n g e i t h e r p a s s e d ( P ) o r f a i l e d ( F ) a n i t e m a c c o r d i n g t o t h e e s t a b l i s h e d c r i t e r i a . T h e r e s u l t s s h o w b o t h t h e E n t e r i n g B e h a v i o u r s ( C o n s e r v a t i o n o f N u m b e r w i t h B l o c k s , R e c o g n i z i n g N u m e r a l s , a n d C o u n t i n g f r o m 1-9) a n d t h e f i v e p r o b l e m s t h a t w e r e a t t h e l e v e l o f t h e I n s t r u c t i o n a l O b j e c t i v e ( h i g h e s t l e v e l o f t h e h i e r a r c h y ) . I n t h e T a b l e t h e s e a r e d e s i g n a t e d ' A d d i t i o n a l P r o b l e m s Table 6 P r e t e s t Resu l t s School Conserva t ion of Number With B l o c k s Recogniz ing # Numerals Counting 1 - 9 A d d i t i o n a l Problems T o t a l Boys G i r l s Boys G i r l s Boys G i r l s Boys G i r l s Boys G i r l s F P F P F P F P F P F P F P F P F P F P 1. 1 7 2 10 4 4 5 7 1 7 2 10 8 0 11 1 14 18 20 28 2. 3 8 5 10 5 6 8 7 2 9 4 11 10 1 15 0 20 24 32 28 3 . 8 17 4 11 16 9 4 11 2 23 2 13 22 3 15 0 48 52 25 35 4 . _5 14 _6 14 15 _4 10 10 3 16 1 19 19 0 19 1 42 34 36 44 T o t a l 17 46 17 45 40 23 27 35 8 55 9 53 59 4 60 2 124 128 113 135 F = F a i l e d P = Passed CO An analysis of the r e s u l t s presented i n Table 6 showed that 49 subjects s u c c e s s f u l l y met the research c r i t e r i a . A summaryffof these subjects according to school i s presented i n Table 7. Table 7 D i s t r i b u t i o n over Sex, Schools and Testing Times of Subjects Who Met Screening C r i t e r i a Boys G i r l s School Morning Afternoon Morning Afternoon Total 1 4 6 10 2 3 2 3 4 12 3 4 2 9 0 15 4 _4 0 _5 3 12 Total 15 4 23 7 49 40 Of the o r i g i n a l 125 subjects, 76 were not s u i t a b l e according to the established c r i t e r i a . Table 8 shows where subjects f a i l e d i n the f i r s t three sub-tests of the Pretest. Table 8 Subjects Who F a i l e d the Pretest I l l u s t r a t e d According to Pretest Category CATEGORY BOYS GIRLS Conservation of Number with Blocks 17 17 Recognizing Numerals 1 - 9 40 27 Counting 1 - 9 _8 _7 Total 65 51 Most subjects who f a i l e d the pretest were unable to recognize a l l of the numerals from 1 to 9 when v i s u a l l y presented to them. Only three c h i l d r e n who f a i l e d the t e s t with the blocks passed the tes t which demanded the recognition of numerals. Counting from 1 to 9 proved to be of no ad d i t i o n a l value i n the pretest: a l l the chil d r e n who f a i l e d the counting sub-test also f a i l e d either the sub-test on blocks or the sub-test on numeral recognition. I t i s i n t e r e s t i n g to note that f a r more boys than g i r l s f a i l e d the sub-test on number recognition; on the other two sub-tests there were no s i g n i f i c a n t d i f f e r e n c e s . Those subjects who su c c e s s f u l l y completed the f i r s t three pretests were administered a fourth t e s t to f i n d out i f they could already solve the problems that were to be placed at the top of the hypothesized hierarchy. Six of the subjects were able to c o r r e c t l y answer s u f f i c i e n t problems - four or more - to be excluded from the research pool. Of these subjects four were boys and two were g i r l s . P i l o t Study The p i l o t study was ca r r i e d out by administering one row of the f i n a l test to four g i r l s and three boys randomly selected from the research pool. The major purpose for the p i l o t t e s t was to ascertain whether the subjects could answer most of the questions on the Posttest. I f the subjects could answer a s i g n i f i c a n t number of questions p r i o r to teach-ing, then i t was quite l i k e l y that the pool of subjects selected for the study would also be able to answer the questions. A look at the structure of the test shows the t o t a l possible at any given l e v e l of the tes t was 3 points, and the maximum possible score f o r any subject was 21 points. None of the seven ch i l d r e n was successful on any of the items at l e v e l s one and two of the hierarchy; f i v e of the seven c h i l d r e n did not score at l e v e l three; three children did not score at l e v e l s four and f i v e ; and two children did not score at l e v e l s s i x and seven. Most success was achieved at the Concrete l e v e l of the hierarchy, but even here the subjects were able to score only 8 and 11 points r e s p e c t i v e l y - only a f r a c t i o n of the maximum possible of 21. If the scores at each l e v e l of the hierarchy are averaged the r e s u l t s would be 0, 0, 5.66 and 9.5; a score of 10.5 would be required to achieve a 50 percent l e v e l of accuracy. One boy and one g i r l scored w e l l on l e v e l s 3 , 4 , 5 , 6 and 7 . With the excep t ion of these two s u b j e c t s , the most common score was 0 and 1 . The r e s u l t s , o u t l i n e d i n Table 9 , p rov ide support fo r the b e l i e f tha t there i s room f o r va r i ance on the independent v a r i a b l e s . Table 9 Resu l t s of the P i l o t Study Accord ing to H ie ra rchy L e v e l TASK LEVEL OF THE HIERARCHY SUBJECTS 7 6 5 4 3 2 1 TOTAL 1 ( g i r l ) 0 1 0 0 0 0 0 1 2 ( g i r l ) 2 2 2 1 2 0 0 9 3 ( g i r l ) 1 0 1 1 0 0 0 3 4 (boy) 0 3 1 0 0 0 0 4 5 (boy) 2 3 2 3 3 0 0 13 6 ( g i r l ) 1 0 0 1 0 0 0 2 7 (boy) 2 2 0 a 0 0 0 _ 4 T o t a l 8 11 6 6 5 0 0 36 The maximum t o t a l p o s s i b l e i n each c e l l of the t e s t .design"; was 3 . 0 43! I m p l i c a t i o n s of the P r e t e s t and P i l o t Test fo r the Main Study Seven ty - s ix sub jec t s were r e j e c t e d because they e i t h e r d i d not have the d e s i r e d E n t e r i n g Behaviours or because they cou ld a l r eady answer the ques t ions at l e v e l one of the h i e r a r c h y , reduc ing the sub jec t pool from 125 to 49 . T h i r t y of the 49 c h i l d r e n who met the research c r i t e r i a were g i r l s , and n ine teen were boys. The f a c t tha t more g i r l s met the research c r i t e r i a may be s i g n i f i c a n t , but was not a f a c t o r i n t h i s study because 14 boys and 14 g i r l s were chosen at random to form the f i n a l poo l of s u b j e c t s . Resu l t s of the igp i lo t t e s t l end some support to the idea of a Concre t e -Abs t r ac t continuum us ing a r i t h m e t i c f a c t s between 6 and 9. With the excep t ion of l e v e l s s i x and seven there appeared to be a l i n e a r p rogres s ion from the Concrete to the A b s t r a c t l e v e l of the h i e r a r c h y . The sub jec t s c l e a r l y found the more concre te t a sks e a s i e r than the most a b s t r a c t t a s k s . However, i f the r e s u l t s are averaged accord ing to the four l e v e l s hypo thes ized , the h i e r a r c h y appears even more c l e a r l y ( 9 . 5 , 5 .66, 0 . 0 ) . This data suggest tha t no d i f f e r e n c e s are l i k e l y to be found between problems presented a t the va r ious Task L e v e l s . For i n s t a n c e , at the Semi-Concrete l e v e l the number of c o r r e c t responses f o r " F i s h " , "Shapes" and " M a t c h s t i c k s " was 6, 6, and 5 r e s p e c t i v e l y . Another f a c t o r of i n t e r e s t , a l though i t f a l l s ou t s i de the range of the research ques t i ons , i s the impact of t each ing on the sub jec t s chosen f o r the s tudy . I f the r e s u l t s of the p i l o t t e s t are used as a gu ide -l i n e , and i f i t i s reasonable to g e n e r a l i z e from t h i s group of seven to the l a r g e r group of 28, then i t would be reasonable to expect the average score of each subject to r i s e . If the teaching does have an impact, does i t a f f e c t a l l of the items at each l e v e l of the hierarchy or only some items at some l e v e l s ? Are the questions at the more abstract l e v e l s teachable? W i l l the subjects c e i l i n g out on the questions at the Concrete l e v e l ? These questions, and those s p e c i f i c a l l y formulated from the design, w i l l be addressed i n Chapter Six. 45 Chapter Five Results of the Study The r e s u l t s of the main study w i l l be presented and analyzed i n t h i s chapter. F i r s t l y , the analyses of variance w i l l be discussed; secondly, whether a hierarchy i s tenable given the r e s u l t s obtained i n the study w i l l be considered; and t h i r d l y , the evidence concerning s t a t i s t i c a l v a l i d a t i o n of the hypothesized hierarchy of s k i l l s w i l l be examined. The f i r s t s e r i e s of analyses set out to es t a b l i s h whether or not ce r t a i n basic assumptions underlying the t e s t were tenable. As the subjects were not nested by sex within the l e v e l of the t e s t , i t was important that each set of items within each l e v e l of the hierarchy that was attempted by each group of chi l d r e n be of the same l e v e l of d i f f i c u l t y . I f some items proved to be harder to answer than some others, i t would not be possible to assess the impact of sex as a var i a b l e . I f for instance, items i n one row proved to be easier than items i n the other rows, and four g i r l s answered the questions on that row, i t would be impossible to determine whether the g i r l s found some items easier than did the boys, or whether the items themselves were ac t u a l l y more d i f f i c u l t . As well as analyzing the items within each row, i t was necessary to analyze the items within each column to es t a b l i s h whether order e f f e c t s were present. I f questions are presented i n a ce r t a i n order, i t may make i t easier to answer the questions that follow than i f the questions are presented i n another order. 46 Results presented i n Table 10 in d i c a t e that neither the items within row or columns were s i g n i f i c a n t l y d i f f e r e n t to one another. However, there was a s i g n i f i c a n t i n t e r a c t i o n between the items i n the rows and the columns. Table 10 Summary of the Analysis of Variance of the Sequence of Items Within the Test Degrees of Sum of Mean Source Freedom Squares Square F Sequence (S) 6 2.946 0.491 1.257 Groups of Items (G) 6 0.946 0.158 0.780 Persons (P:S) 21 8.202 0.391 -Items (I:G) 14 3.167 0.226 2.771 S. x G. 36 30.030 0.834 3.960 * S. x I:G 126 26.548 0.210 -G. x P:S 84 8.167 0.972 1.190 P:S x I:G 292 24.000 0.816 -Total 585 104.006 Sequence = Sequence of items attempted by subjects Item Sets = Sets of three items * p < .05 This i n t e r a c t i o n appears most l i k e l y to have been caused by the low scores obtained at Level 1 (Abstract l e v e l ) of the s k i l l hierarchy. Both the row and column means were high, but within each row and column 47 one score - i n v a r i a b l y for the problem representing Level 1 - dropped below that of the others. Figure 3 i l l u s t r a t e s that even the highest mean obtained (0.667 on column 5) was below the lowest mean score for any row or column. The second analysis of variance was car r i e d out to assess whether the established l e v e l of d i f f i c u l t y c r i t e r i a for the items had been validated. The items were placed i n one of three categories, Easy, Moderately D i f f i c u l t , or Hard, (see Table 1). Results presented i n Table 11 showed that l e v e l of d i f f i c u l t y was s i g n i f i c a n t . There was no s i g n i f i c a n t d i f f e r e n c e between sexes and the items within each group were d i f f e r e n t i a t e d on the basis of the four established c r i t e r i a (see Chapter Three, Item D i f f i c u l t y ) . The mean scores f o r the items were si m i l a r (82% accuracy on the Easy Items, 78?o accuracy on the items of moderate d i f f i c u l t y , and 68?o accuracy on the items considered d i f f i c u l t ) . As indicated i n Table 12, the success rate f o r a l l l e v e l s of items was high. This f a c t could account f o r the s i m i l a r i t y of the easy and moderately d i f f i c u l t items. With so few errors made, those items answered i n c o r r e c t l y have a s i g n i f i c a n t e f f e c t upon the r e l a t i v e r e l a t i o n -ship of the three sets of items. Another possible reason for the s i m i l a r i t y noted between these items e x i s t s . Item d i f f i c u l t y was defined according to four c r i t e r i a ; having grouped the 21 items i n three rows according to these c r i t e r i a - , i t i s conceivable that the three rows might not be absolutely d i s t i n c t from one another. An item classed as easy, could quite well have been put with the items of moderate d i f f i c u l t y . This did i n f a c t happen with one problem - 8 + 1 . Also, i t must be remembered that only a small sampling of the items occurred. Often a 00 •3-co CD CO c o Q. co CD t-i CD c co CD F i g u r e 3. Item Sets Mean responses to Item Sets at each l e v e l of the hierarchy Sequence 1 VSequence 2 Sequence 3 Sequence 4 Sequence 5 Sequence 6 Sequence 7 ( Task Level One of the hierarchy) 4 9 s m a l l s a m p l e r e s u l t s i n o n l y s l i g h t d i f f e r e n c e s t h a t a r e n o t d i s c e r n i b l e . T a b l e 1 1 S u m m a r y o f t h e A n a l y s i s o f V a r i a n c e o f I t e m D i f f i c u l t y A c c o r d i n g t o t h e F o u r E s t a b l i s h e d C r i t e r i a S o u r c e D e g r e e s o f F r e e d o m Sum o f S q u a r e s M e a n S q u a r e F S e x ( S ) 1 0 . 6 1 2 0 . 6 1 2 ' 0 . 1 3 2 L e v e l o f I t e m D i f f i c u l t y ( L ) 2 1 . 9 6 2 0 . 9 8 1 * 1 2 . 1 7 7 C h i l d ( C ) 2 6 1 2 . 0 3 4 0 . 4 6 3 -I t e m ( I ) 1 8 2 . 4 1 8 0 . 1 3 4 0 . 7 6 8 S x L 2 0 . 1 3 3 0 . 6 6 3 0 . 8 2 3 C x L : S 5 2 4 . 1 9 0 0 . 8 0 6 -S x I : L 1 8 3 . 9 4 9 0 . 2 1 9 1 . 2 5 3 C x I : S L 4 6 8 8 1 . 9 1 8 0 . 1 7 5 -T o t a l 5 8 7 1 0 7 . 2 1 6 S e x = s e x o f c h i l d * p < . 0 5 L e v e l = L e v e l o f I t e m D i f f i c u l t y C h i l d = B o y o r G i r l I t e m s = T h r e e A d d i t i o n I t e m s Table 12 Mean Performance by Sex and Item D i f f i c u l t y Level of D i f f i c u l t y Female Male Means (by d i f f i c u l t y ) Easiest Items 0 . 8 4 7 ( 1 ) 0.80D 0.821 Moderately D i f f i c u l t Items 0.800 0.765 0.781 Hard Items 0.673 0.694 0.684 Means (by gender) 0.772 0.752 '* 0.762 * Grand Mean (1) There were 7 items i n each group and the scores under 'Male' and 'Female' represent the mean of these scores 51 The t h i r d a n a l y s i s of va r i ance was undertaken to e s t a b l i s h whether any p a r t i c u l a r se t of i tems was any more d i f f i c u l t to answer than any other s e t , once they had been ba lanced. Each se t had one easy i t e m , one i tem of moderate d i f f i c u l t y , and one hard i tem (see Table 2 ) . The r e s u l t s of t h i s a n a l y s i s which are presented i n Table .12 i n d i c a t e that no group of i tems proved to be of any grea ter d i f f i c u l t y than any other group. Table 13 A Summary of the A n a l y s i s of Var iance of Item D i f f i c u l t y fo r Balanced Groups of Items Source Degrees of Sum of Mean F Freedom Squares Square Sex (S) 1 0.184 0.184 0.132 Items ( I ) 6 2.286 0.381 0.355 C h i l d (C) 26 36.102 1.389 S x I 6 6.102 1.017 0.948 C x I :S 156 167.327 1.073 T o t a l 195 212.001 52 The fourth analysis of variance was effected to determine i f there was s t a t i s t i c a l evidence of a hierarchy of any type (not necessarily the hierarchy as hypothesized). The r e s u l t s of t h i s analysis are detail e d i n Table 14. Table 14 A Summary of the Analysis Hypothesized H of Variance ierarchy of the Source Degrees of Freedom Sum of Squares Mean Square F Sex (S) 1 0.184 0.184 0. 132 Level of Hierarchy ( l i ' ) 6 68.071 11.345 16. 804 * Child (C) 26 36.102 1.389 -S x L 6 2.316 0.386 0. 5718 C x L:S 156 105.327 Total 195 212.000 * p < . . 0 5 The r e s u l t s of the analysis provided a strong i n d i c a t i o n of the existence of a hierarchy of s k i l l s . Further analyses considered the hypotheses as they related to the seven-level l i n e a r hierarchy, and also as they related to the f o u r - l e v e l branching hierarchy. The re s u l t s expressed as mean responses to the item at the seven task l e v e l s of the hierarchy and moving from task l e v e l seven to task l e v e l one were: 2 .71 ; 2 .61 ; 2 .57; 2 .68; 2 .43; 2 .07; 0 .93 . With the excep t ion of l e v e l 4 there appeared to be a p rog res s ion of d i f f i c u l t y from l e v e l one to l e v e l seven , where l e v e l one was the most a b s t r a c t and l e v e l seven the most conc re t e . However, t h i s d i f f e r e n c e was s t r o n g l y ev ident on ly when l e v e l s one and two were compared wi th the other f i v e l e v e l s . The i tems at these l e v e l s were apparen t ly much harder than the i tems a t any of the other f i v e l e v e l s , and the i tems a t l e v e l one were apparen t ly much more d i f f i c u l t than those at l e v e l two. A second a n a l y s i s compared the four l e v e l s of the branching h i e r a r -chy: A b s t r a c t , 0 .93 ; A b s t r a c t - C o n c r e t e , 2 .07; Semi-Concrete , 2 .55; and Concre te , 2 .66 . As wi th the seven stage l i n e a r h i e r a r c h y there appeared to be a p rog res s ion of d i f f i c u l t y from the l e v e l s hypothes ized to be A b s t r a c t to those hypothes ized to be Concre te , w i th the more A b s t r a c t l e v e l s p rov ing to be the more d i f f i c u l t . The A b s t r a c t and Abs t r ac t -Conc re t e l e v e l s were c l e a r l y d i f f e r e n t to one another , and d i f f e r e n t from the Semi-Concrete and Concrete l e v e l s . However, a l though the Semi-Concrete l e v e l appeared to be harder than the Concrete l e v e l , there was very l i t t l e d i f f e r e n c e i n the r e s u l t s of these l e v e l s . Both the r e s u l t s of the a n a l y s i s of va r i ance tha t cons idered the h i e r a r c h y , and the r e s u l t s of the d e s c r i p t i v e a n a l y s i s supported the p o s s i b i l i t y tha t a h i e r a r c h y was f e a s i b l e . In order to t e s t the hypothes is s t a t i s t i c a l l y the data was subjected to a Guttman Scalogram A n a l y s i s which cons idered the p o s s i b i l i t y of both a s e v e n - l e v e l l i n e a r h i e r a r chy and a f o u r - l e v e l branching h i e r a r c h y . The a n a l y s i s a l s o cons idered the number of i tems c o r r e c t l y answered out of the three p o s s i b l e a t each t e s t l e v e l . In e f f e c t , t h i s meant tha t the c u t - o f f 54 point for passing a Task Level of the hierarchy was a r b i t r a r i l y set at 1 out of 3, 2 out of 3, or 3 out of 3 items. Each of these Pass-Fail cut-off points was considered for the seven-level hierarchy andtthe four-l e v e l hierarchy r e s p e c t i v e l y . Of the s i x analyses undertaken only two produced evidence to support a viable hierarchy. At the cu t - o f f l e v e l of two both the seven-level hierarchy and the f o u r - l e v e l branching hierarchy had high c o e f f i c i e n t s of r e p r o d u c i b i l i t y (0.92 and 0.95 r e s p e c t i v e l y ) . Closer examination of the seven-level l i n e a r hierarchy, however, revealed that the various task l e v e l s as hypothesized did not emerge i n the same order when s t a t i s t i c a l l y analyzed. The r e s u l t s of t h i s analysis are presented i n Table 15. Table 15 Results of Guttman Scalogram Analysis -Seven Levels Hypothesized Hierarchy Re-Ordered Hierarchy (seven l e v e l s ) Number of Subjects Who Passed Number of Subjects Who Fa i l e d Percentage of Subjects Who Passed Percentage of Subjects Who F a i l e d Symbols Symbols 10 18 36 64 T a l l i e s T a l l i e s 20 8 71 29 Matches Fish 25 3 89 11 Shapes Matches 25 3 89 11 Fish Crayons 26 2 93 7 Blocks Blocks 27 1 96 4 Crayons Shapes 27 1 96 4 Number of Subjects = 28 Co e f f i c i e n t of Reproducibility 0.918 Task l e v e l s one and two appeared i n the hypothesized order; Task l e v e l s two to seven, however, did not. "F i s h " and "Matches", both Semi-Concrete tasks, did not remain i n the same r e l a t i o n s h i p to one another as hypothe-sized. Examination of the data showed that 25 subjects passed each l e v e l and 3 f a i l e d ; the difference between the tasks at these two l e v e l s , then, was s l i g h t . "Shapes" was hypothesized to be a Semi-Concrete task; according to the t h e o r e t i c a l premises proposed, " F i s h " , "Blocks" and "Crayons" should have been easier. The subjects d i d , i n f a c t , f i n d the tasks presented with "Shapes" as easy as those presented with "Blocks". The strongest evidence to support the hypothesis of a hierarchy occurred when the cut-off point of 2 was used with the f o u r - l e v e l branching hierarchy. The r e s u l t s of t h i s analysis are presented i n Table 16. Examination of the data ind i c a t e that the hierarchy was ordered i n the manner hypothesized - Concrete to Abstract. The differ e n c e between the Abstract and Abstract-Concrete l e v e l s , and the Abstract-Concrete and Semi-Concrete l e v e l s was clear-cut; 36, 71, and 89 percentage of subjects re s p e c t i v e l y passed each of these three l e v e l s . However, the differ e n c e between the Semi-Concrete and Concrete l e v e l s , although apparent, was not as s t r i k i n g - 89 and 96 percentage of subjects res p e c t i v e l y passed these two l e v e l s . As the r e s u l t s of the seven-l e v e l l i n e a r hierarchy showed, a c l e a r d i s t i n c t i o n between Semi-Concrete and Concrete tasks was not apparent. Few subjects f a i l e d tasks presented at these l e v e l s , and t h i s ' c e i l i n g ' e f f e c t may have affected t h e i r r e l a t i o n s h i p . Table 16 Results of Guttman Scalogram Analysis: Four Levels Hypothesized Guttman Number of Number of Percentage Percentage Hierarchy Scalogram Subjects Subjects of Subjects of Subjects Hierarchy Who Passed Who Fai l e d Who Passed Who Fai l e d Abstract Abstract 10 18 36 64 Abstract-Concrete Abstract-Concrete 20 8 71 29 Semi-Concrete Semi-Concrete 25 3 89 11 Concrete Concrete 27 1 96 4 Number of Subjects = 28 C o e f f i c i e n t of Reproducibility: 0.95 57 Chapter Six Discussion, Limitations and Recommendations Research Questions posed i n Chapter Two as well as pertinent questions a r i s i n g from the P i l o t Test w i l l be addressed i n t h i s chapter i n the l i g h t of t h e o r e t i c a l assumptions outlined i n Chapter Three, and r e s u l t s presented i n Chapters Four and Five. Results of the analyses of variance indicated that factors such as the sex of the c h i l d , item d i f f i c u l t y , order e f f e c t within the t e s t , and|the r e l a t i v e d i f f i c u l t y of groups of items within the t e s t , were not s i g n i f i c a n t ; thus, questions concerning the v a l i d a t i o n of a hierarchy of s k i l l s supported by s i g -n i f i c a n t v a r i a t i o n s i n success at each l e v e l w i l l be considered. Discussion A l i n e a r hierarchy of s k i l l s does not appear defensible. Analysis of variance .indicated a s i g n i f i c a n t amount of variance i n the data a t t r i b u t a b l e to the hierarchy, but gave no i n d i c a t i o n as to the type of hierarchy responsible. Examination of the means for each l e v e l showed that there was indeed a l i n e a r progression moving from Task l e v e l seven "Crayons" to Task l e v e l one "Symbols". However, very l i t t l e v a r i a t i o n was.noted between Task l e v e l three and Task l e v e l seven and the mean.for Task l e v e l four was higher than that of Task l e v e l s f i v e and s i x . The Guttman Scalogram Analysis supported the idea,of a l i n e a r hierarchy but re-arranged the items from the order hypothesized. (Table. 14). The f a c t that so few subjects f a i l e d any of the l e v e l s from three to seven makes i t very d i f f i c u l t to argue for the existence of a l i n e a r r e l a t i o n s h i p between these tasks. While i t i s possible that t h e l i n e a r p a t t e r n d o e s n o t a r i s e b e c a u s e o f t h e ' c e i l i n g ' e f f e c t t h a t o c c u r r e d i n t h e s e f i v e t e s t s , i t m a y w e l l b e t h a t t h e i t e m s o n t h e s e t e s t s d o n o t h a v e a l i n e a r r e l a t i o n s h i p t o o n e a n o t h e r a t a l l . A s e c o n d q u e s t i o n i s w h e t h e r o r n o t t a s k s a t a g i v e n l e v e l o f t h e h i e r a r c h y a r e o f s u f f i c i e n t l y e q u a l d i f f i c u l t y t h a t t h e y m a i n t a i n t h e s a m e r e l a t i o n s h i p t o o n e a n o t h e r w h e n t h e y a r e a n a l y z e d s t a t i s t i c a l l y . T a k e n i n d i v i d u a l l y , T a s k l e v e l s d o n o t a u t o m a t i c a l l y r e m a i n a t t h e s a m e p o i n t i n t h e h i e r a r c h y a c c o r d i n g t o t h e G u t t m a n S c a l o g r a m A n a l y s i s . T h e i t e m s o n t h e " S h a p e s " s u b - t e s t a p p e a r t o b e b y f a r t h e e a s i e s t w h e n a c u t - o f f o f t w o c o r r e c t r e s p o n s e s o u t o f t h r e e i s e m p l o y e d . T h e " C r a y o n " t a s k s , o n t h e o t h e r h a n d , a p p e a r e d t o b e h a r d e r t h a n b o t h t h e " B l o c k s " a n d t h e " S h a p e s " t a s k s . I n t e r m s o f l e v e l s , h o w e v e r , i f t h e d a t a a r e p o o l e d , t h e r e i s s o m e s u p p o r t f o r u s e o f t h e t a s k s i n t h e w a y t h e h y p o t h e s i z e d h i e r a r c h y i s c o n s t r u c t e d . T h e d i f f i c u l t y , a s w i t h t h e d i s c u s s i o n o f t h e s e v e n - s t a g e l i n e a r h i e r a r c h y , i s t h a t f a r t o o f e w s u b j e c t s f a i l e d i t e m s a t t h e S e m i - C o n c r e t e a n d C o n c r e t e l e v e l s m a k i n g ' c e i l i n g ' e f f e c t s a c o n f o u n d i n g f a c t o r i n t h e d i s c u s s i o n . C e r t a i n l y t h e r e i s n o c l e a r l i n e a r i t y i n t h e h i e r a r c h y , w i t h t h e e x c e p t i o n o f T a s k l e v e l s o n e a n d t w o . I n f a c t , i t c o u l d b e a r g u e d t h a t t h e r e i s e v i d e n c e t o s u p p o r t f u r t h e r b r a n c h i n g o f t h e h i e r a r c h y , w i t h T a s k l e v e l s t h r e e t o s e v e n c o n s i d e r e d o n e l e v e l . T h e s t r o n g e s t e v i d e n c e t o s u p p o r t t h e c o n c e p t o f a h i e r a r c h y a r o s e f r o m t h e G u t t m a n S c a l o g r a m A n a l y s i s o f t h e F o u r - l e v e l h i e r a r c h y u s i n g a c u t - o f f p o i n t o f 2 o u t o f 3 i t e m s c o r r e c t a s a b a s i s f o r p a s s i n g o r f a i l i n g a T a s k L e v e l ( T a b l e 1 5 ) . T h e r e w a s a v e r y c l e a r d i s t i n c t i o n b e t w e e n t h e A b s t r a c t a n d A b s t r a c t - C o n c r e t e , a n d b e t w e e n t h e A b s t r a c t -59 Concrete and the Semi-Concrete l e v e l s . However, the d i s t i n c t i o n was not as clear between the Semi-Concrete and Concrete l e v e l s , perhaps for one of the reasons proposed i n the discussion of the seven-stage l i n e a r hierarchy. The C o e f f i c i e n t of Reproducibility of 0.95 i s marginally high enough to provide s t a t i s t i c a l v a l i d a t i o n . A review of the p i l o t t e s t led to support for the ideas of both a l i n e a r progression from Cpncrete to Abstract and the p o s s i b i l i t y of a f o u r - l e v e l hierarchy. Data from the main study supports, to some extent, both of these ideas. It i s important to note that i t i s not possible to generalize from the r e s u l t s of the p i l o t test to the main study because only seven children were involved i n the p i l o t t e s t . However, some valuable ideas arose which were worth considering. One of the major differences between the r e s u l t s of the p i l o t test and the main study was the f a c t that so many ch i l d r e n answered most of the questions on the main study on items at Task Levels 3 to 7. On the p i l o t t e s t no subject received more than 11 out of 21 items correct on the "Blocks" sub-test. For a l l the other sub-tests the mean scores were well below 50 percent of the t o t a l possible. As the seven subjects chosen for the P i l o t Test were from the research pool of 49 children who q u a l i f i e d f o r the study, i t seems un l i k e l y that the c h a r a c t e r i s t i c s of t h i s group would be d i f f e r e n t to those chosen for the main study. The major difference i n the two s i t u a t i o n s i s that the subjects i n the main study were taught, while those i n the p i l o t t e s t were not. The percentage of subjects who passed at each l e v e l of the hierarchy, comparing the Main Study and the P i l o t Test, was noteworthy: main study from task l e v e l seven to task l e v e l one: 96%, 96%, 93%, 89%, 89%, 71%, 36%; p i l o t t e s t from task l e v e l seven to task l e v e l one: 38%, 52.4%, 28.6%, 28.6%, 23,8%, 0%, 0%. 60 It appears that teaching was e f f e c t i v e at a l l l e v e l s of the hierarchy, p a r t i c u l a r l y at task l e v e l two " T a l l i e s " , where the change was from 0% to 11%. The most important point i s that the more abstract the task, the greater i s the percentage increase i n the main study r e s u l t s . At task l e v e l s s i x and seven the percentage change i s approximately 100%, at task l e v e l s three, four and f i v e the change i s over 200%; and, at task l e v e l s one and two the change i s i n f i n i t e because the p i l o t t e s t baselines were 0% i n each case. One factor other than teaching, deserves mention: i t i s d i f f i c u l t to assess the extent to which the subjects understood the i n s t r u c t i o n s on the-.-pilot t e s t . The p i l o t t e s t was, i n part, used as a vehicle for standardizing the i n s t r u c t i o n s for the main study. If comprehension of the i n s t r u c t i o n s on the p i l o t t e s t was poor, t h i s may have depressed the scores. Another question emerging from the p i l o t t e s t was whether or not the tasks at the Abstract and Abstract-Concrete Levels were teachable. Results indicated that the subjects were most l i k e l y ready for the type of symbolic teaching involved i n these tasks. Limitations of the Study From the discussion thus f a r , i t i s apparent that the high rate of success on the more concrete tasks, which resulted i n many subjects fi n d i n g the items at several of the Task Levels too easy, i s a s i g n i f i -cant factor i n t h i s study. The factors of teaching, and the lack of actual difference i n the tasks themselves, have already been mentioned; i t i s pertinent to consider a few a d d i t i o n a l points. The i n i t i a l subject pool was 125 subjects, eventually reduced to 49, from which the subjects for both the p i l o t t e s t and the main study were chosen., Thus 60% of the population were eliminated from the study., The s e l e c t i o n of the 49 subjects was arri v e d at - with the exception of s i x who could already answer four or f i v e questions at the most abstract l e v e l of the h i e r a r -chy - by giving a pretest which has many developmental c h a r a c t e r i s t i c s to i t . Many 5 year old c h i l d r e n , f o r instance, are unable to conserve d i s c r e t e e n t i t i e s which they w i l l have no d i f f i c u l t y conserving one year l a t e r . It i s conceivable that the 49 subjects who formed the research pool had c h a r a c t e r i s t i c s which would d i s t i n g u i s h them' from the 76 not chosen, other than the a b i l i t y to s u c c e s s f u l l y complete the tasks presented on the pretest. Two variables which could be considered would be socio-economic status and i n t e l l i g e n c e . It i s possible that the c h i l d r e n who formed the research pool were from a higher socio-economic cla s s than those who f a i l e d the pretest; i t i s also possible that they were more i n t e l l i g e n t than those who f a i l e d the pretest. The factor of i n t e l l i g e n c e i s p a r t i c u l a r l y relevant to the issue of ' c e i l i n g out' on the tasks designed. I t may be, for instance, that brighter children do not require tasks presented i n a very concrete form such as "Blocks" or "Crayons" and that any form of concrete presentation either two-dimensional, or three-dimensional, w i l l s u f f i c e . Many child r e n who experience d i f f i c u l t y with c a r d i n a l numbers resort to the use of concrete aids such as f i n g e r s . These ch i l d r e n were eliminated from t h i s study by the pretest. As mentioned i n the discussion of previous research, i t i s not possible to discuss s k i l l h ierarchies for children without considering developmental c h a r a c t e r i s t i c s . This i s p a r t i c u l a r l y true f o r chi l d r e n at 5 years of age where cognitive think-ing and str a t e g i e s are r a p i d l y changing. Recommendations for Future Research The Guttman Scalogram Analysis validated the four l e v e l hierarchy using two out of three items as the p a s s - f a i l c u t - o f f point. The c o e f f i c i e n t of r e p r o d u c i b i l i t y was 0.95 which was s t a t i s t i c a l l y s i g n i f i c a n t . The question that a r i s e s i s the extent to which t h i s f i n d i n g has any relevance to education. The formulation of an opera-t i o n a l d e f i n i t i o n of numbers, as outlined by Resnick (1970) suggests that addition equations are at the top of the hierarchy - that i s , at the highest l e v e l of d i f f i c u l t y . The r e s u l t s of the present study present the p o s s i b i l i t y of taking addition equations and presenting them at three d i f f e r e n t l e v e l s of d i f f i c u l t y before a c t u a l l y presenting them a b s t r a c t l y . Results for the hierarchy as presently constituted would need r e p l i c a t i n g before i t could j u s t i f i a b l y be used i n an educational s e t t i n g . A second issue which a f f e c t s the arrangement of items i n the form of a hierarchy i s item d i f f i c u l t y . Results showed some support for the c r i t e r i a of d i f f i c u l t y of addition items as out-l i n e d i n Chapter Three. It would be possible, then, to conceive of a 4 l e v e l hierarchy with 3 l e v e l s of item d i f f i c u l t y at each l e v e l . Further research i s indicated given the information a v a i l a b l e from t h i s study. Replication of the r e s u l t s using a larger population of kindergarten c h i l d r e n would be valuable, to determine i f the Semi-Concrete and Concrete l e v e l s could be distinguished simply by increasing the number of subjects. A larger population of subjects would also allow the variable of age to be studied i n more d e t a i l . It i s possible, for instance, given the research a v a i l a b l e about developmental variables that younger kindergarten c h i l d r e n may perform d i f f e r e n t l y than older kindergarten c h i l d r e n . Future research might also consider the r o l e socio-economic status and i n t e l l i g e n c e . Other possible areas of i n v e s t i g a t i o n include studies of grade one c h i l d r e n p r i o r to exposure to formal teaching, comparisons of early grade ones with kindergarten and comparisons of r e s u l t s from these areas with other research that provides data relevant to the h i e r a r c h i c a l and developmental under-standing of the concept of numbers. REFERENCES Bateman, B. D. Essentials of teaching San Rafael, C a l i f o r n i a : Dimensions, 1971. Ber e i t e r , C , & Engelmann, S. Teaching disadvantaged c h i l d r e n i n preschool. Englewood C l i f f s , N.J.: P r e n t i c e - H a l l , 1966. Brainserd, C. J . Inducing o r d i n a l and cardi n a l representations of the f i r s t f i v e natural numbers. Journal of Experimental C h i l d Psychology 1974, 18, 520-534. Bruner, J . S., Olver, R. R., & Greenfield, P. M. Studies i n Cognitive Growth. In J . S. Bruner (Ed.), New York: John Wiley and Sons, 1967. B u e l l , R. R. Piagetian theory i n t o inquiry action. In J . I. Athey & D. C. Rubadeau (Eds.), Educational implications of Piaget's theory. Toronto, Ontario: Xerox, 1970. Edwards, A. L. Experimental design i n psychological research. New York: Holt, Rinehart & Winston, 1968. Engelmann, S. Relationship between psychological theories and the act of teaching. Journal of School Psychology, 1967. Engelmann, S. Conceptual learning. San Rafael, C a l i f o r n i a : Dimensions 1969a. Engelmann, S. Preventing f a i l u r e i n the primary grades. Chicago: Science Research Associates 1969b. Gagne', R. M. & Paradise, N. E. A b i l i t i e s and learning sets i n know-ledge a c q u i s i t i o n . Psychological Monographs, 1961a, 75,-(Whole No. 518). 65 Gagne'', R. M., & Brown, L. T. Some factors i n the programming of conceptual learning. Journal of Experimental Psychology, 1961b, 62, 313-321. Gagne/, R. M. The a c q u i s i t i o n of knowledge, Psychological Review, 1962a, 69, 355-365. Gagne*, R. M., Mager, J . R., Garstens, H. L., & Paradise, N. E. Factors i n acquiring knowledge of a mathematical task. Psychological Monographs, 1962, 76, 7_ (Whole No. 526). Gagne'', R. M. The learning requirements for enquiry. Journal Research i n science teaching, 1963, 1, 144-153. Gagne*, R. M. Problem s o l v i n g . In A. W. Melbon (Ed.), Categories of human learning. New York: Academic Press, 1964. Gagne', R. M. Some factors i n learning non-metric geometry. Child Development, 1965, 30, 42-49. Gagne*, R. M. Contributions of learning to human development. Psychological Review, 1968, 75, 177-191. Gagne'', R. M. The conditions of learning. Holt, Rinehart & Winston, 1970. Gronlund, H. E. Constructing an achievement t e s t . Englewood C l i f f s , New Jersey: P r e n t i c e - H a l l , 1970. Karplus, R. The science curriculum: Improvement study. In I. J . Ashley & D. C. Rubadeau (Eds.), Educational implications of Piaget's theory. Toronto, Ontario: Serox, 1970. K i l p a t r i c k , J . Cognitive theory and the school mathematics study group programme. In I. J . Ashley & D. C. Rubadeau (Edsi), Educational implications of Piaget's theory. Toronto, Ontario: Xerox, 1970. Mager, R. F., & Clark, C. Explorations i n student-controlled i n s t r u c t i o n . Psychological Reports, 1963, 13, 71-76. M i l l e r , S. A. Non-verbal assessment of conservation of number. Child  Development, 1976, 47, 722-728. Resnick, L. B. Design of an early learning curriculum. Learning Research and Development Centre, University of Pittsburgh, 1967. Resnick, L. B. Behaviour analysis i n curriculum design: A h i e r a r c h i c a l l y sequenced introductory mathematics curriculum. Monograph 2, Learning Research and Development Centre, University of Pittsburgh, 1970. Saxe, G. B. A developmental analysis of notational counting. Child Development, 1977, 48, 1512-1520. Si e g e l , L. S. The Sequence of development of c e r t a i n number concepts i n preschool c h i l d r e n . Developmental Psychology 1971, 5_, 357-361. Uprichard, A. E. & P h i l l i p s , E. R. An intraconcept analysis of r a t i o n a l number addition: A v a l i d a t i o n study. Journal for Research i n Mathematics Education, 1977, Q_, 7-16. Wong, M. C , Resnick, L. B., & Boozer, R. F. The sequence of development of some early mathematics behaviours. C h i l d Development, 1971, 42, 1767-1778. APPENDIX A MEASURING INSTRUMENT Bas i c a d d i t i o n f a c t s . ( u s i n g the numbers one to nine i n c l u s i v e l y ) sums are no l e s s than s i x and no grea te r than n i n e . (1) 1 + 5 (5) 5 + 1 (2) 1 + 6 (6) 6 + 1 (3) 1 + 7 (7) 7 + 1 (4) 1 + 8 (8) 8 + 1 (9) 2 + 4 (13) 4 + 2 (10) 2 + 5 (14) 5 + 2 (11) 2 + 6 (15) 6 + 2 (12) 2 + 7 (16) 7 + 2 (17) 3 + 3 (21) 4 + 3 (18) 3 + 4 (22) 5 + 3 (19) 3 + 5 (23) 6 + 3 (20) 3 + 6 (24) 4 + 4 (26) 5 •+ 4 (25) 4 + 5 T o t a l : 26 Fac ts APPENDIX B INSTRUCTIONS FOR THE POSTTEST Task Level 7. (Three items were given). E to S: Here are two sets of crayons (E points). Join these two sets of crayons to make one set. How many crayons are there now? S responds: This w i l l be repeated f o r items 2 and 3. Task Level 6. (Three items w i l l be given). Same as Task Level 7. Substitute blocks for crayons. Task Level 5. (Three items). E to S: Here are two sets of f i s h (E points). I f you joined these two sets of f i s h to make one new set, how many f i s h w i l l there be? S responds: Task Levels 4 and 3 w i l l be the same. Substitute shapes and then matchsticks f o r f i s h . (Three items at each task l e v e l ) . Task Level 2. (Three items). E to S: Here are two numbers which I want you to add together (E p o i n t s ) . I f you wish you may use the t a l l y marks to help you. When you add these numbers together, how many are there now? S responds: 69 Task Level 1. (Three items). E to S: Here are two numbers that I want you to add together. How many do they make when you add them? (If the subject begins to use his f i n g e r s , the experimenter should i n d i c a t e that t h i s i s not allowed). S responds: A P P E N D I X C P R E T E S T E N T E R I N G B E H A V I O U R S E a c h c h i l d w a s p r e s e n t e d w i t h t h e n u m b e r s o n e t o n i n e i n c l u s i v e l y . T h e s e n u m b e r s w e r e p r e s e n t e d i n a r a n d o m o r d e r 3 , 8 , 2 , 7 , 9 , 1 , 6 , 4 , 5 E a c h c h i l d w a s s h o w n o n e n u m b e r a t a t i m e a n d a s k e d t o i d e n t i f y i t v e r b a l l y . To p a s s t h e t e s t a c h i l d h a d t o i d e n t i f y a l l n i n e n u m b e r s c o r r e c t l y . E a c h c h i l d h a d t o o r a l l y r e c i t e t h e n u m b e r s o n e t o n i n e i n c l u s i v e l y T h e e x p e r i m e n t e r s a i d t o t h e s u b j e c t : I w o u l d l i k e y o u t o c o u n t f o r m e . P l e a s e c o u n t u p t o t h e n u m b e r n i n e . I f t h e s u b j e c t h e s i t a t e d , t h e e x p e r i m e n t e r c o u l d h e l p t h e s u b j e c t b y g i v i n g e i t h e r t h e f i r s t , s e c o n d o r b o t h o f t h e f i r s t n u m b e r s i n t h e s e r i e s . I f t h e e x p e r i m e n t e r w a s u n s u r e w h e t h e r t h e s u b j e c t c o u l d c o m p l e t e t h e t a s k h e a s k e d t h e s u b j e c t t o r e p e a t t h e s e r i e s . E : W o u l d y o u c o u n t u p t o t h e n u m b e r n i n e a g a i n p l e a s e . ( N o h e l p i s p e r m i s s i b l e a t t h i s t i m e ) N o e r r o r w a s p e r m i t t e d i n o r d e r t o p a s s . T h e e x p e r i m e n t e r t h e n p l a c e d s i x b l o c k s b e f o r e t h e s u b j e c t . E . P l e a s e c o u n t t h e s e b l o c k s . C o u n t o u t l o u d s o t h a t I c a n h e a r y o u . S u b j e c t c o u n t s b l o c k s . 71 E. How many are there? Subject responds. The examiner then spread the blocks out so that they covered a greater surface area. Then the examiner asked the question again. E. How many are there now? Subject responds. E. Are there more blocks now? Subject responds. (The task was repeated i f error.occurred. The next time, however, four blocks were used. If the subject got the correct answer, one more item was given with seven blocks to make sure the second response was not a guess.) APPENDIX D INSTRUCTIONS FOR TEACHING Task Level 7 (real objects) On the table were two sets of crayons i n front of each c h i l d (item one was 4 + 4 ) . There were four pencils i n one set and four i n the other. The experimenter said to the subjects: E: Here are two sets of pencils (E p o i n t s ) . This i s one set of pencils (points) and t h i s i s another set of pencils (points). Watch me while I count t h i s set of p e n c i l s . (E counts demonstration set on his righthand side.) Now you count t h i s set (E points to the set on the subjects' l e f t side).' Subjects responded. E: Now watch while I count the other set. Now you count your other set. (E points to each subject's righthand set i n turn).. E: Now I am going to take t h i s set (E's lefthand set because i t was upside-down to the subjects) and j o i n i t to t h i s set. (E points to each and then joins them-.).! E: Now you take t h i s set (subjects' righthand set) and j o i n i t to t h i s set. Subjects responded. E: Now, how many crayons are there altogether? E i obtained a verbal response from each subject. E. then counted his pencils and gave the correct answer to the subjects. (Praise was used when applicable,)). 73 T h e d e m o n s t r a t i o n s w e r e o m i t t e d f r o m i t e m s t w o a n d t h r e e . T h e p r o c e d u r e w a s a s f o l l o w s : E : H e r e a r e t w o s e t s o f c r a y o n s ( E p o i n t s ) . T h i s i s o n e s e t o f c r a y o n s ( p o i n t s ) a n d t h i s i s a n o t h e r s e t o f c r a y o n s ( E p o i n t s ) . Now h o w m a n y c r a y o n s a r e t h e r e ? ( E p o i n t s t o s e t o n s u b j e c t s ' l e f t ) ) Now h o w m a n y i n t h i s s e t ? ( E p o i n t s t o s e t o n s u b j e c t s ' , r i g h t . ) ) . E : Now y o u j o i n t h i s s e t ( o n s u b j e c t s ' r i g h t ) t o t h i s s e t ( o n s u b j e c t s ' l e f t ) : S u b j e c t s r e s p o n d e d . E : How m a n y c r a y o n s a r e t h e r e a l t o g e t h e r ? S u b j e c t s r e s p o n d e d . E : T h e r e a r e s e v e n c r a y o n s a l t o g e t h e r . ( P r a i s e d i f a p p r o p r i a t e ) . T a s k L e v e l 6 T h i s w a s i d e n t i c a l t o t a s k l e v e l 7 e x c e p t c r a y o n s w e r e r e p l a c e d b y b l o c k s . T a s k L e v e l 5 T h e s u b j e c t s w e r e s h o w n t h e c a r d w i t h t h e i t e m o n i t . E : H e r e a r e t w o s e t s o f f i s h ( E p o i n t s ) . T h i s i s o n e s e t o f f i s h ( E p o i n t s t o g r o u p o n l e f t o f c a r d ) a n d t h i s i s a n o t h e r s e t o f f i s h ( E p o i n t s t o t h e g r o u p o n t h e r i g h t o f t h e c a r d ) . E : F i r s t , c o u n t t h i s s e t o f f i s h . S u b j e c t s r e s p o n d e d . E : N o w , c o u n t t h i s s e t o f f i s h . S u b j e c t s r e s p o n d e d . E : Now i f t h i s s e t i s a d d e d t o t h i s s e t ( E p o i n t s t o e a c h s e t ) , h o w m a n y f i s h a r e t h e r e a l t o g e t h e r ? S u b j e c t s r e s p o n d e d . E . c o u n t s t h e m a n d s u p p l i e s c o r r e c t a n s w e r . ( R e p e a t f o r i t e m s t w o a n d t h r e e ) . T a s k L e v e l s 3 a n d A w e r e t h e s a m e a s 5 e x c e p t ' f i s h ' w e r e r e p l a c e d b y ' s h a p e s ' a n d ' m a t c h s t i c k s ' . T a s k L e v e l 2 E . p l a c e d t h e c a r d b e f o r e t h e s u b j e c t s . E : We a r e g o i n g t o a d d . t h e s e t w o n u m b e r s t o g e t h e r ( E p o i n t s t o e a c h n u m b e r o n t h e c a r d , f i r s t t h e l e f t h a n d n u m b e r a n d t h e n t h e r i g h t -h a n d n u m b e r ) . To h e l p u s a d d t h e s e t w o n u m b e r s t o g e t h e r , we c a n c o u n t t h e t a l l y m a r k s u n d e r t h e n u m b e r s . ( E d e m o n s t r a t e s w i t h t h e f i r s t n u m b e r ) . E : Now I w a n t y o u t o a d d t h e s e t w o n u m b e r s t o g e t h e r . I w a n t y o u t o u s e t h e t a l l y m a r k s t o h e l p y o u g e t a n a n s w e r t o t h e q u e s t i o n . S u b j e c t s s o l v e t h e i t e m s a n d g i v e t h e i r a n s w e r s . E- a d d s t h e n u m b e r s u s i n g t h e t a l l y m a r k s a n d g i v e s t h e c o r r e c t a n s w e r . ( R e p e a t e d f o r i t e m s t w o a n d t h r e e ) . T a s k L e v e l 1 E . t o S u b j e c t s : Do y o u r e m e m b e r t h a t l a s t t i m e we a d d e d n u m b e r s t o g e t h e r we u s e d t a l l y m a r k s t o h e l p u s ? T h i s t i m e I w a n t y o u t o a d d t o g e t h e r t h e s e t w o n u m b e r s a n d t e l l me t h e a n s w e r . F i r s t , y o u l o o k a t t h i s n u m b e r ( E p o i n t s t o t h e l e f t h a n d n u m b e r ) a n d t h e n y o u a d d i t t o t h i s n u m b e r ( E p o i n t s t o t h e o t h e r n u m b e r ) . How 75 many do they make altogether? Subjects responded. E-adds the numbers out loud by counting on from the f i r s t number to the second. (e.g. 5 + 3 - 5-6,7,8)- answer 8) (Repeated for items two and three).