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The relation of Piaget's three stages in number conservation development to achievement in grade I arithmetic Dennis, Isobel Gertrude 1967

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THE RELATION OF PIAGET'S THREE STAGES IN NUMBER CONSERVATION DEVELOPMENT TO ACHIEVEMENT IN GRADE I ARITHMETIC by ISOBEL GERTRUDE DENNIS B.A., University of Western Ontario, 1 9 6 3 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of EDUCATION We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 6 7 In presenting this thesis in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library shal l make i t f ree ly avai lable for reference and Study. I further agree that permission for extensive copying of this thesis for scholar ly purposes may be granted by the Head of my Department or by h.fe representatives. It is understood that copying or publ ica t ion of th is thesis for f i n a n c i a l gain shal l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT J e a n P i a g e t d e s c r i b e s t h r e e s t a g e s i n t h e development of t h a t a s p e c t o f q u a n t i t a t i v e t h i n k i n g w h i c h he named c o n s e r v a t i o n of number. I n t h e f i r s t s t a g e , a c h i l d i s q u i t e unaware t h a t one-to-one p a i r i n g of two s e t s i m p l i e s e q u i v a -l e n c e of t h e s e t s . He i s u n a b l e t o make a c o r r e c t o n e - t o -one c o r r e s p o n d e n c e , and i f p r e s e n t e d w i t h two s e t s of o b j e c t s w h i c h have been matched u n i t f o r u n i t , b e l i e v e s t h a t one s e t has become g r e a t e r i f i t s u n i t s a r e s p r e a d o u t , o r s m a l l e r i f t h e y have been compressed. I n t h e second s t a g e , t h e c h i l d i s a b l e t o make a c o r r e c t c o r r e s p o n d e n c e , b u t does not b e l i e v e i n t h e c o n t i n u e d e q u i v a l e n c e of t h e s e t s when one i s s p a t i a l l y r e a r r a n g e d . I n t h e t h i r d s t a g e , the c h i l d main-t a i n s t h a t t h e matched s e t s r e m a i n e q u i v a l e n t even though the u n i t s of one s e t have been r e a r r a n g e d , t h a t i s , t h e c h i l d c o n s e r v e s number. P i a g e t p o s t u l a t e s t h a t c o n s e r v a t i o n i s a n e c e s s a r y c o n d i t i o n of m a t h e m a t i c a l u n d e r s t a n d i n g . I n t h i s s t u d y , I t was h y p o t h e s i z e d t h a t c h i l d r e n who a r e i n Stage 1 a t t h e b e g i n n i n g of t h e i r Grade 1 y e a r , and who a r e s t i l l i n Stage 1 a t t h e b e g i n n i n g of t h e second term show low achievement i n a r i t h m e t i c a t t h e end of t h e s c h o o l y e a r . I t was f u r t h e r h y p o t h e s i z e d t h a t each s t a g e i n c o n s e r -v a t i o n i s a s s o c i a t e d w i t h c o r r e s p o n d i n g l e v e l s i n t e r m i n a l achievement i n Grade 1 a r i t h m e t i c . One hundred f i f t y - s i x c h i l d r e n r e c e i v e d an i n d i v i d u a l i i i c o n s e r v a t i o n t e s t i n October of t h e i r f i r s t grade y e a r , and were t h e r e b y c l a s s i f i e d as b e i n g i n Stage 1, Stage 2, o r Stage 3 i n c o n s e r v a t i o n development. I n J a n u a r y , t h o s e c l a s s i f i e d as Stage 1 r e c e i v e d a second c o n s e r v a t i o n t e s t , and were a g a i n c l a s s i f i e d a c c o r d i n g t o t h e i r s t a g e i n c o n s e r v a t i o n development a t t h a t t i m e . I n May, t h e a r i t h m e -t i c s u b - t e s t of the S t a n f o r d Achievement T e s t , P r i m a r y I B a t t e r y was a d m i n i s t e r e d t o a l l g r o u p s . A s i g n i f i c a n t p r o p o r t i o n of the Stage 1 group s e l e c t e d by t h e January c o n s e r v a t i o n t e s t had achievement s c o r e s w h i c h f e l l below t h e median f o r a l l s u b j e c t s , w h i l e a s i g n i f i c a n t p r o p o r t i o n o f t h e Stage 3 group s e l e c t e d by t h e October t e s t had above-median achievement s c o r e s . Mean achievement s c o r e s f o r the two Stage 3 groups d i d not d i f f e r s i g n i f i c a n t l y from each o t h e r , but were h i g h e r t h a n mean achievement s c o r e s f o r the Stage 1 and Stage 2 g r o u p s . No s i g n i f i c a n t d i f f e r e n c e s were found among mean achievement s c o r e s of Stage 1 and Stage 2 g r o u p s . The r e s u l t s were i n t e r p r e t e d as b e i n g c o n s i s t e n t w i t h P i a g e t ' s t h e o r y . The s u p e r i o r i t y o f t h e mean t e r m i n a l achievement of e a r l y c o n s e r v e r s o ver t h a t of c h i l d r e n who had n o t d e v e l o p e d c o n s e r v a t i o n by Ja n u a r y appeared g r e a t enough t o be o f e d u c a t i o n a l i m p o r t a n c e . Some i n d i v i d u a l s c o r e s showed marked d e v i a t i o n from t h e p a t t e r n d e r i v e d from t h e group d a t a , however, and c a u t i o n i n use of the c o n s e r v a t i o n t e s t as a p r e d i c t i v e i n s t r u m e n t was recommended. It was proposed that the conservation test could be a useful diagnostic procedure for the teacher. V TABLE OF CONTENTS CHAPTER PAGE I. THE THEORETICAL BASIS OF THE PROBLEM 1 I I . STATEMENT OF THE PROBLEM 6 R e l a t i o n of the C o n s e r v a t i o n Concept t o E a r l y Mathematics E d u c a t i o n 6 Purposes of t h e Study 8 Statement of Hypotheses 9 III. SURVEY OF RELATED LITERATURE 12 V a l i d a t i o n of t h e Thr e e - s t a g e Theory of C o n s e r v a t i o n Development 12 E v i d e n c e of t h e S i g n i f i c a n c e of C o n s e r v a t i o n as a S t e p i n Number Concept Development 15 The R e l a t i o n of P i a g e t ' s Theory of Number Concept Development t o E d u c a t i o n a l P r a c t i c e . 19 The R e l a t i o n of C o n s e r v a t i o n t o Other V a r i a b l e s 20 TV. DESIGN AND PROCEDURE 22 The Sample . 22 The T e s t i n g Programme 23 Restatement of Hypotheses as E x p e c t a t i o n s f o r t h e E x p e r i m e n t a l Groups 23 D e s c r i p t i o n of t h e T e s t s 25 I n t e r v i e w s 33 V. SELECTED DATA AND STATISTICAL TREATMENT OF RESULTS 3^ v i CHAPTER PAGE I n f o r m a t i o n f rom Teacher I n t e r v i e w s 34 C o n s e r v a t i o n R e s u l t s 3^ Achievement T e s t R e s u l t s 35 S t a t i s t i c a l T e s t s of the N u l l Hypotheses D e r i v e d from t h e E x p e c t a t i o n s f o r t h e R e l a t i v e Achievement of t h e Groups as C l a s s i f i e d by t h e C o n s e r v a t i o n T e s t 35 A P o s t e r i o r i S t a t i s t i c a l T e s t s 39 Summary of R e s u l t s 40 V I . IMPLICATIONS OF RESULTS 42 C o n c l u s i o n s 42 The R e l a t i o n of t h e R e s u l t s t o P i a g e t ' s Theory . 43 E d u c a t i o n a l S i g n i f i c a n c e of R e s u l t s 45 The R e l a t i o n o f Rate of Development t o Achievement 50 G e n e r a l i t y of t h e Study 50 BIBLIOGRAPHY 52 APPENDIX A. C o n s e r v a t i o n T e s t Items 54 APPENDIX B. Supplementary Data 57 LIST OP TABLES v i i TABLE PAGE I . D i s t r i b u t i o n of S u b j e c t s i n t h e E x p e r i m e n t a l Groups R e s u l t i n g f r om C-Test I and C-Test I I . 34 I I . Summary of Achievement T e s t R e s u l t s 35 I I I . S i g n i f i c a n c e of P r o p o r t i o n s of 1^ and H i A c h i e v i n g Below Median i n A r i t h m e t i c J6 IV. Summary of A n a l y s e s of V a r i a n c e 37 V. S i g n i f i c a n c e of D i f f e r e n c e s Between Group Means i n A r i t h m e t i c Achievement 38 V I . F r e q u e n c i e s of S c o r e s on C o n s e r v a t i o n T e s t s . . . 58 V I I . T o t a l F r e q u e n c i e s and D i s t r i b u t i o n Among Groups of Raw S c o r e s on t h e A r i t h m e t i c Achievement T e s t 58 V I I I . P r o p o r t i o n s of E x p e r i m e n t a l Groups F a l l i n g Above and Below Median Achievement . 59 I X . C o m p o s i t i o n of E x p e r i m e n t a l Groups by S c h o o l C l a s s Membership . 59 X. Ranges and Mean Achievement S c o r e s of S c h o o l C l a s s e s 60 X I . Numbers of S u b j e c t s i n Each Group Who Counted f o r Two o r More C o n s e r v a t i o n Items 60 v i i i L IST OP FIGURES FIGURE PAGE 1. Schematic P r e s e n t a t i o n of t h e T e s t i n g Programme . . 23 2 . C o n f i g u r a t i o n s P r e s e n t e d i n C-Test I 55 3 . C o n f i g u r a t i o n s P r e s e n t e d I n C-Test I I 56 CHAPTER I THE THEORETICAL BASIS OP THE PROBLEM A c c o r d i n g t o Jean P i a g e t 1 s t h e o r y of i n t e l l e c t u a l development, t h e e s t a b l i s h m e n t of l o g i c a l s t r u c t u r e s w h i c h he c a l l s c o n s e r v a t i o n s marks t h e c h i l d ' s p r o g r e s s t h r o u g h t h e p e r i o d of p r e - o p e r a t i o n a l thought, ( a p p r o x i m a t e l y two t o seven y e a r s o f a g e ) , t o the p e r i o d of c o n c r e t e o p e r a t i o n s , ( a p p r o x i m a t e l y seven t o e l e v e n y e a r s ) . C o n s e r v a t i o n i s t h e u n d e r s t a n d i n g t h a t c e r t a i n p r o p e r t i e s , such as mass, w e i g h t , volume and number, r e m a i n i n v a r i a n t t h r o u g h o u t changes i n shape, s p a t i a l rearrangement, p a r t i t i o n i n g I n t o s u b s e t s o r p o r t i o n s , and t h e l i k e . C o n s e r v a t i o n , o r t h e permanence of wholes, I s a n e c e s s a r y c o n d i t i o n of a l l r a t i o n a l a c t i v i t y , though n o t a s u f f i c i e n t c o n d i t i o n o r e x p l a n a t i o n of i t . l One of the i m p o r t a n t c o n s e r v a t i o n s i s , i n P i a g e t ' s t e r m i n o l o g y , the c o n s e r v a t i o n of number, t h a t i s , t h e under-s t a n d i n g t h a t t h e c a r d i n a l i t y of a s e t , ( t h e e i g h t n e s s of e i g h t ) , remains i n v a r i a n t t h r o u g h o u t any and a l l t r a n s f o r m a -t i o n s o f t h e s e t . P i a g e t d i s t i n g u i s h e s t h r e e s t a g e s i n t h e development of t h e c o n s e r v a t i o n of number. T h i s t h e o r y was de v e l o p e d f r om a n a l y s i s of c h i l d r e n ' s r e s p o n s e s l n a v a r i e t y of •'•Jean P i a g e t , The C h i l d ' s C o n c e p t i o n of Number, Manchester U n i v e r s i t y P r e s s , 1953. P« 3» 2 problem s i t u a t i o n s , o f w h i c h t h e f o l l o w i n g i s an example: (a) M a t e r i a l s 1) e i g h t egg cups 11) t w e l v e eggs (b) P r o c e d u r e 1) E i g h t egg cups a r e p l a c e d i n a h o r i z o n t a l row l n f r o n t of t h e c h i l d . 11) The c h i l d i s a s k e d t o g e t one egg f o r each egg cup, as many eggs as cups. i l l ) The c h i l d i s p e r m i t t e d t o check h i s work by p l a c i n g t h e eggs i n the cups. E r r o r s I n s e l e c t i n g t h e number of eggs a r e c o r r e c t e d , so t h a t each cup c o n t a i n s an egg. Iv) The e i g h t eggs a r e removed from t h e cups and c l u s t e r e d t o g e t h e r n e a r t h e row of egg cups, v) The c h i l d i s a s k e d t h e c o n s e r v a t i o n q u e s t i o n : A r e t h e r e more eggs o r egg cups, o r a r e t h e r e t h e same number? (c) Responses i ) T y p i c a l of Stage 1: When as k e d t o g e t one egg f o r e v e r y egg cup, t h e c h i l d makes a row th e same l e n g t h as the row of egg cups, but c o n t a i n i n g an i n c o r r e c t number of eggs. When the eggs a r e removed from t h e cups and c l u s -t e r e d , he t h i n k s t h e r e a r e more cups t h a n eggs. 3 i i ) T y p i c a l of Stage 2: The c h i l d s e l e c t s t h e c o r r e c t number of eggs t o f i l l t h e cups. When t h e eggs a r e c l u s t e r e d i n f r o n t of t h e cups, he t h i n k s t h e r e a r e more cups t h a n eggs. i l l ) T y p i c a l o f Stage 3: The c h i l d s e l e c t s t h e c o r r e c t number of eggs t o f i l l t h e cups. When t h e eggs a r e c l u s t e r e d , he u n h e s i t a -t i n g l y a s s e r t s t h a t t h e number of eggs and egg cups i s t h e same. P i a g e t d e s c r i b e d t h e s e t h r e e s t a g e s i n number c o n s e r -v a t i o n development i n terms of t h e i n t e r r e l a t i o n s h i p s of t h e p e r c e p t u a l and l o g i c a l f a c t o r s w h i c h he deemed were t h e p s y c h o l o g i c a l bases o f t h e r e s p o n s e s . He I n f e r r e d t h a t t h e c h i l d I n t h e f i r s t s t a g e judges e q u a l i t y o r I n e q u a l i t y on t h e b a s i s o f q u a l i t a t i v e s i m i l a r i t i e s and d i f f e r e n c e s . That i s , the s e t w h i c h has t h e g e n e r a l l y b i g g e r appearance i s judged t o c o n t a i n t h e most el e m e n t s . The c h i l d i s i n c a p a b l e of l o g i c a l l y m a n i p u l a t i n g t h e r e l a t i o n s h i p s between t h e s p a t i a l c h a r a c t e r i s t i c s o f the matched s e t s , c o - o r d i n a t i o n of whi c h would y i e l d t h e i d e a of t h e u n i t ( M o n e n e s s M ) . S i n c e he has l i t t l e n o t i o n o f t h e u n i t , he has no h e s i t a t i o n i n a s s e r t i n g t h e i n e q u a l i t y of t h e groups w h i c h he has j u s t matched one-f o r - o n e , as i n t h e case of t h e eggs and egg cups. H i s men t a l p r o c e s s e s a r e c h a r a c t e r i z e d by i r r e v e r s i b i l i t y : he i s unable t o r e p r e s e n t t o h i m s e l f s uch l o g i c a l c o n s i d e r a t i o n s as " I c o u l d put a l l t h e eggs back i n t h e cups"; " I f t h e eggs were s p r e a d out a g a i n t h e r e would be one egg f o r each cup." P e r c e p t i o n i s i r r e v e r s i b l e , and t h e Stage 1 c h i l d i s p e r c e p t i o n - b o u n d . I n t h e second s t a g e t h e c h i l d i s a b l e t o a n a l y z e the c o n f i g u r a t i o n i n an i n t u i t i v e ( p e r c e p t i o n - d e p e n d e n t ) way. Thus when t h e elements of two matched s e t s a r e o p p o s i t e one a n o t h e r , he sees t h e n u m e r i c a l e q u i v a l e n c e . He can r e l a t e d e n s i t y t o number ( t h e more c l o s e l y packed t h e more elements t h e r e a r e ) , o r he can r e l a t e e x t e n s i o n t o number ( t h e more space t h e elements t a k e up, t h e more t h e r e a r e ) , but has not l o g i c a l l y c o - o r d i n a t e d t h e s e r e l a t i o n s h i p s t o g r a s p the n a t u r e of t h e u n i t . The correspondence he makes i s n o t t r u l y a q u a n t i f y i n g o p e r a t i o n , and when t h e one-to-one c o r r e s p o n -dence of two matched s e t s i s not v i s u a l l y e v i d e n t , he does not b e l i e v e i n t h e i r e q u a l i t y . I f t h e p o s s i b i l i t y i s p o i n t e d o u t , t h e Stage 2 c h i l d f o r e s e e s t h e r e t u r n of t h e elements t o t h e i r matched p o s i t i o n , but t h i s e m p i r i c a l c o n s i d e r a t i o n does not a c t as a l o g i c a l p r e m i s e . I t does not i m p l y , f o r example, t h a t t h e eggs w i l l a l ways f i t a g a i n i n t o t h e cups, o r t h a t t h e number of eggs and egg cups was t h e same d u r i n g t h e i n t e r v a l when correspondence was not v i s u a l l y e v i d e n t . I n P i a g e t * s terms, the c h i l d ' s thought has not y e t de v e l o p e d l o g i c a l r e v e r s i b i l i t y . I n t h e t h i r d s t a g e , t h e I n t u i t i v e l y g r a s p e d r e l a t i o n s of d e n s i t y and e x t e n s i o n t o number become l o g i c a l l y 5 m a n i p u l a b l e . D i f f e r e n c e s i n d e n s i t y and/or e x t e n s i o n between two equated s e t s a r e u n d e r s t o o d t o compensate f o r each o t h e r , hence be e q u a t a b l e . The u n i t i s thus c o m p l e t e l y d e f i n e d , and u n i t f o r u n i t m a t c h i n g becomes a q u a n t i f y i n g o p e r a t i o n , c h a r a c t e r i z e d by r e v e r s i b i l i t y . I t i s t h e n u n d e r s t o o d t h a t s p a t i a l arrangements of matched s e t s a r e i m m a t e r i a l t o t h e n u m e r i c a l v a l u e of t h e s e t s , s i n c e any and a l l arrangements can be r e a r r a n g e d t o demonstrate one-to-one c o r r e s p o n d e n c e . I n Stage 3» t h e l o g i c a l o p e r a t i o n has triumphed over p e r c e p -t u a l i n t u i t i o n . 2 I n i t s t h i r d s t a g e of development, t h e n , c o n s e r v a t i o n of number becomes a s t a b l e , f u n c t i o n a l , c o g n i t i v e s t r u c t u r e . P i a g e t s t a t e s t h a t c o n s e r v a t i o n o f some s o r t i s the p o s t u l a t e of any m a t h e m a t i c a l u n d e r s t a n d i n g whatsoever.3 That t h e con-s e r v a t i o n o f number i s a n e c e s s a r y c o n d i t i o n o f u n d e r s t a n d i n g t h e most e l e m e n t a r y a r i t h m e t i c a l o p e r a t i o n s f o l l o w s as a r e a s o n a b l e h y p o t h e s i s from t h e t h e o r y . 2 T h i s e x p o s i t i o n of t h e p s y c h o l o g y of the t h r e e s t a g e s i n c o n s e r v a t i o n of number i s drawn f r e e l y f r om P i a g e t , The  C h i l d ' s C o n c e p t i o n of Number, pp. 41-85. 3lb l d . , pp. 3, 4. CHAPTER I I STATEMENT OF THE PROBLEM I . RELATION OF THE CONSERVATION CONCEPT TO EARLY MATHEMATICS EDUCATION I n most e d u c a t i o n a l systems, f o r m a l i n s t r u c t i o n i n mathematics i s begun i n Grade 1, when most p u p i l s a r e f i v e and o n e - h a l f t o seven y e a r s o l d . A t t h i s age, c h i l d r e n a r e a t v a r i o u s s t a g e s i n a c q u i r i n g t h e c o n s e r v a t i o n of number, a c c o r d i n g t o t h e f i n d i n g s of P i a g e t and t h o s e who f o l l o w e d h i s l e a d . The c h i e f c o n t e n t a r e a s of t h e Grade 1 a r i t h m e t i c p r o -gramme a r e (1) t h e a c q u i s i t i o n of c e r t a i n number c o n c e p t s 1 ; (2) t h e a c q u i s i t i o n of s k i l l l n r e a d i n g , w r i t i n g , and under-s t a n d i n g i n terms of p o s i t i o n a l p l a c e v a l u e , t h e numerals t o 100; (3) d i s c o v e r y and mastery of a d d i t i o n and s u b t r a c t i o n f a c t s t o e i g h t . C u r r e n t l y t h e r e i s s t r o n g emphasis on d e v e l o p i n g I n s i g h t I n t o u n d e r l y i n g m a t h e m a t i c a l p r i n c i p l e s , and methods a r e adapted t o t h i s end. T h i s o b j e c t i v e i s e x p l i c i t l y s t a t e d i n the i n t r o d u c t o r y pages of a r e c e n t l y p u b l i s h e d Grade 1 a r i t h m e t i c textbook. 2 These a u t h o r s f e e l ^-Maurice L. H artung, e t . a l . . C h a r t i n g the Course i n A r i t h m e t i c , Chicago: S c o t t , Foresman Company, I960, pp. 7-18. ^ M a u r i c e L. Hartung, e t . a l . , S e e i n g Through  A r i t h m e t i c 1, T e a c h e r s ' E d i t i o n , T o r o n t o : W. J . Gage, 1965, pp. 4 and 5' 7 t h a t c o u n t i n g a l o n e i s n o t an adequate p r e l i m i n a r y t r a i n i n g f o r u n d e r s t a n d i n g numbers and n u m e r i c a l o p e r a t i o n s , and t h e i r t e x t i n c l u d e s a good d e a l of p r e p a r a t o r y work b e f o r e l a u n -c h i n g i n t o a d d i t i o n and s u b t r a c t i o n , and s y m b o l i c e x p r e s s i o n of t h e s e o p e r a t i o n s . I f t h e c o n s e r v a t i o n of number i s c e n t r a l i n t h e development of number u n d e r s t a n d i n g , c h i l d r e n who have n o t a c q u i r e d t h i s concept p r i o r t o t h e i r e n t r y i n t o Grade 1 o r d u r i n g the p r e p a r a t o r y s t a g e s of the Grade 1 programme c o u l d be e x p e c t e d t o show low achievement i n a r i t h m e t i c . The f a i l u r e t o a c h i e v e c o n s e r v a t i o n c o u l d be e x p e c t e d t o be e s p e c i a l l y c l e a r l y r e f l e c t e d when the p r o -gramme emphasizes u n d e r s t a n d i n g of m a t h e m a t i c a l p r i n c i p l e s , as opposed t o mere a c q u i s i t i o n o f m e c h a n i c a l s k i l l s . I f a c h i l d ' s development i n c o n s e r v a t i o n i s an a c c u r a t e i n d e x of h i s growth I n t h e a b i l i t y t o t h i n k q u a n t i -t a t i v e l y , t h e t h r e e s t a g e s i n c o n s e r v a t i o n development c o u l d be e x p e c t e d t o be a s s o c i a t e d w i t h c o r r e s p o n d i n g l e v e l s i n achievement i n a r i t h m e t i c . However, c h i l d r e n a r e e x p e c t e d t o know i n c r e a s i n g amounts of a r i t h m e t i c as t h e i n s t r u c t i o n a l programme p r o -g r e s s e s , t h a t i s , t h e s t a n d a r d of achievement s h i f t s w i t h the passage of t i m e . The most c r u c i a l of t h e s e s t a n d a r d s i s , i n s c h o o l systems where g r a d i n g i s s t i l l u sed, th e t e r m i n a l one on the b a s i s of w h i c h p a s s i n g o r f a l l i n g grades a r e a s s i g n e d , and d e c i s i o n s as t o f u t u r e grade placement made. Can a t e s t o f number c o n s e r v a t i o n a d m i n i s t e r e d n e a r the 8 b e g i n n i n g of t h e y e a r be e x p e c t e d t o p r e d i c t t h e l e v e l of t e r m i n a l achievement i n Grade 1 a r i t h m e t i c ? Would th e c h i l d r e n who a r e c l a s s i f i e d as Stage 1 a t t h e b e g i n n i n g of t h e y e a r c o n t i n u e t o be d e f i c i e n t i n t h e i r g r a s p of a r i t h m e -t i c t h r o u g h o u t t h e y e a r , so t h a t t h e i r t e r m i n a l achievement s c o r e s were s i g n i f i c a n t l y below t h o s e of c h i l d r e n c l a s s i f i e d as Stage 2 o r Stage 3» or would t h e p r o g r e s s o f Stage 1 c h i l d r e n i n t o Stage 2 o r Stage 3 e n a b l e them t o " c a t c h up" t o t h e o t h e r s i n u n d e r s t a n d i n g the a r i t h m e t i c i n s t r u c t i o n , so t h a t no d i f f e r e n t i a l achievement would be e v i d e n t a t t h e end of t h e y e a r ? The same problem can be s t a t e d w i t h r e f e r e n c e t o c h i l d r e n c l a s s i f i e d as b e i n g i n Stage 2 a t t h e b e g i n n i n g of t h e y e a r . The q u e s t i o n i n a more g e n e r a l i z e d way i s t h i s : Does t h e r a t e of development o f number c o n s e r v a t i o n a f f e c t t h e amount of l e a r n i n g of i n s t r u c t i o n a l m a t e r i a l ? T h i s i s a complex problem, and t h e d e s i g n of t h e p r e s e n t s t u d y p e r m i t s o n l y t h a t i t i n d i c a t e whether o r not t h e r e i s a r e l a t i o n s h i p between t h e r a t e of c o n s e r v a t i o n development and a r i t h m e t i c achievement w h i c h w a r r a n t s more tho r o u g h I n v e s t i g a t i o n . I I . PURPOSES OF THE STUDY T h i s s t u d y a t t e m p t s t o f i n d e v i d e n c e f o r t h e e x i s t e n c e of the f o l l o w i n g r e l a t i o n s h i p s w h i c h a r e h y p o t h e s i z e d On t h e b a s i s of P i a g e t * s t h e o r y of the development of t h e concept of number: 9 1. A r e l a t i o n s h i p between f a i l u r e t o a c h i e v e c o n s e r -v a t i o n a t t h e b e g i n n i n g , o r d u r i n g t h e f i r s t term of f o r m a l a r i t h m e t i c I n s t r u c t i o n , and low t e r m i n a l achievement l n Grade 1 a r i t h m e t i c . 2. A r e l a t i o n s h i p between P i a g e t ' s t h r e e s t a g e s of number c o n s e r v a t i o n and achievement l e v e l s l n Grade 1 a r i t h m e t i c . The s t u d y as a whole p r o v i d e s a t e s t of t h e t h e o r y t h a t number c o n s e r v a t i o n d e v e l o p s over time t h r o u g h t h r e e d i s t i n c t s t a g e s . I I I . STATEMENT OF HYPOTHESES A. Wi t h r e s p e c t t o f a i l u r e t o conserve a t the b e g i n -n i n g o r d u r i n g the f i r s t term of t h e i n s t r u c t i o n a l programme i n Grade 1 a r i t h m e t i c , i t was h y p o t h e s i z e d t h a t : 1. C h i l d r e n i n the f i r s t s t a g e of c o n s e r v a t i o n development p r i o r t o the i n i t i a t i o n of f o r m a l i n s t r u c t i o n I n Grade 1 a r i t h m e t i c show low t e r m i n a l achievement i n Grade 1 a r i t h m e t i c . 2. C h i l d r e n who remai n i n the f i r s t s t a g e of c o n s e r v a t i o n development th r o u g h o u t t h e f i r s t term i n Grade 1 show low t e r m i n a l achievement i n Grade 1 a r i t h m e t i c . B. With r e g a r d t o the r e l a t i o n of s t a g e of c o n s e r v a -t i o n t o t e r m i n a l achievement, i t was h y p o t h e s i z e d t h a t : 1. The t h r e e s t a g e s of number c o n s e r v a t i o n a r e a s s o c i a t e d w i t h c o r r e s p o n d i n g l e v e l s o f t e r m i -n a l achievement i n Grade 1 a r i t h m e t i c , when the s t a g e c l a s s i f i c a t i o n i s made nea r t h e time of t h e i n i t i a t i o n of f o r m a l i n s t r u c t i o n i n a r i t h m e t i c . 2. The t h r e e s t a g e s of number c o n s e r v a t i o n development a r e a s s o c i a t e d w i t h c o r r e s p o n d i n g l e v e l s i n t e r m i n a l achievement i n Grade 1 a r i t h m e t i c , when t h e c o n s e r v a t i o n c l a s s i f i c a -t i o n i s made a t t h e b e g i n n i n g of t h e second term. 3. C h i l d r e n who have a c h i e v e d Stage 2 o r Stage 3 development i n number c o n s e r v a t i o n a t t h e time of t h e i n i t i a t i o n of f o r m a l i n s t r u c t i o n i n a r i t h m e t i c show s u p e r i o r t e r m i n a l achievement i n Grade 1 a r i t h m e t i c t o th o s e who a c h i e v e t h e s e s t a g e s d u r i n g the f i r s t t e r m. C. The h y p o t h e s i s r e l a t i n g t o t h r e e - s t a g e development of c o n s e r v a t i o n o v e r t i m e p o s t u l a t e s t h a t : 1. C h i l d r e n i n t h e f i r s t s t a g e of c o n s e r v a t i o n development show the tendency t o r e a c h h i g h e r s t a g e s of development over a four-month p e r i o d . A t e n t a t i v e answer t o the q u e s t i o n of t h e r e l a t i o n of r a t e of c o n s e r v a t i o n development t o a r i t h m e t i c achievement can be f o r m u l a t e d f rom t h e i n f o r m a t i o n p r o v i d e d by t h e acceptance or rejection of some or a l l of the above hypotheses. CHAPTER I I I SURVEY OF RELATED LITERATURE I . VALIDATION OF THE THREE-STAGE THEORY OF CONSERVATION DEVELOPMENT C r o s s - s e c t i o n a l s t u d i e s by D o d w e l l , Feigenbaum and E l k l n d of c h i l d r e n ' s performance on v a r i o u s P i a g e t i a n number t a s k s a t d i f f e r e n t age l e v e l s l e n d u n e q u i v o c a l s u p p o r t t o t h e r a t h e r modest p o s t u l a t i o n t h a t t h e r e i s improvement i n p e r -formance w i t h i n c r e a s e i n age. I n a s t u d y of 250 Canadian c h i l d r e n i n Grades 1 t o 3, D o d w e l l found a c o r r e l a t i o n of .52 between age and s c o r e on a b a t t e r y of P i a g e t t a s k s r e l a t e d t o t h e development of the concept of number.! Fiegenbaum found t h e above-median-age subgroup o f h i s f o u r - t o - s e v e n - y e a r - o l d s u b j e c t s was c o n s i s -t e n t l y s u p e r i o r t o t h e below-median-age subgroup i n correspondence and c o n s e r v a t i o n t a s k s , u s i n g two l e v e l s of t a s k c o m p l e x i t y . ^ D o d w e l l f o u n d , i n a d d i t i o n , t h a t a c h i l d c o u l d be a s s i g n e d t o one of t h e t h r e e s t a g e s on t h e b a s i s of h i s v e r -b a l r e s p o n s e s t o each number t a s k , though t h e r e was some i p . C. Do d w e l l , " C h i l d r e n ' s U n d e r s t a n d i n g of Number and R e l a t e d Concepts," Canadian J o u r n a l of P s y c h o l o g y , 14, I960, pp. 191-205. 2Kenneth D. Feigenbaum, "Task C o m p l e x i t y and IQ as V a r i a b l e s i n P i a g e t ' s Problem of C o n s e r v a t i o n , " C h i l d  Development, 34, 1963, pp. 423-432. 13 d i f f i c u l t y i n d i s t i n g u i s h i n g a Stage 2 from a Stage 3 response. 3 However, a c h i l d would g i v e v a r y i n g t y p e s o f r e s ponse t o t h e f i v e t a s k s w h i c h c o n s t i t u t e d the t e s t , and c o u l d n o t be d e f i n i t e l y a s s i g n e d t o one s t a g e on t h e t e s t as a whole. But t h e " A - s c o r e " a measure of t h e " g l o b a l n e s s , " o r "Stage 1-ness" of t h e r e s p o n s e s had a h i g h n e g a t i v e c o r r e l a -t i o n w i t h t h e t o t a l t e s t s c o r e . That i s , t h e more p e r c e p t i o n - b o u n d a c h i l d was, t h e l o w e r h i s s c o r e i n number concept development. E l k l n d c l a s s i f i e d h i s t a s k s , r a t h e r t h a n h i s s u b j e c t s , c h o o s i n g c o n s e r v a t i o n t a s k s r e q u i r i n g t h r e e l e v e l s of l o g i c a l s o p h i s t i c a t i o n f o r t h e i r s o l u t i o n . ^ I n h i s d i s c u s s i o n , he e x p l i c i t l y r e l a t e s t h e s e l e v e l s t o t h e t h r e e s t a g e s of development p o s t u l a t e d by P i a g e t . He found a s y s t e m a t i c s u p e r i o r i t y w i t h age on t h e t a s k s r e q u i r i n g Stage 2 and Stage 3 t y p e s of judgment f o r t h e i r c o r r e c t s o l u t i o n . The s t u d i e s c i t e d above demonstrate t h a t t h e r e i s p r o -g r e s s i o n w i t h age i n t h e a b i l i t y of c h i l d r e n t o s o l v e Piaget*s c o n s e r v a t i o n o f number problems, and s t r o n g l y s u p p o r t t h e c o n t e n t i o n t h a t p r o g r e s s i o n l i e s a l o n g t h e d i m e n s i o n of i n c r e a s i n g dominance of l o g i c a l over p e r c e p t u a l modes of s o l u t i o n . The s t u d i e s do n o t c l a r i f y t h i s q u e s t i o n : A r e 3Dodwell, op_. c i t . , pp. 191-205. ^ D a v i d E l k i n d , "The Development of Q u a n t i t a t i v e T h i n k i n g : A S y s t e m a t i c R e p l i c a t i o n of P i a g e t ' s S t u d i e s , " J o u r n a l of G e n e t i c P s y c h o l o g y , 98, 1961, pp. 219-2?. 14 P i a g e t ' s s t a g e s d i s t i n c t and q u a l i t a t i v e l y d i s c e r n i b l e i n t h e i n d i v i d u a l c h i l d ' s development? T h i s q u e s t i o n i s o f some imp o r t a n c e i n t h e v a l i d a t i o n of P i a g e t ' s t h e o r y , s i n c e he proposes t h a t i t i s t h e i n d i v i d u a l ' s i n t e r a c t i o n w i t h t h e environment w h i c h p r o v i d e s t h e m a t e r i a l f o r t h e b u i l d i n g o f c o g n i t i v e s t r u c t u r e s . Hence b o t h i n n a t e and e n v i r o n m e n t a l d i f f e r e n c e s a f f e c t the r a t e of development. P i a g e t ' s s t a g e s a r e t h e r e f o r e not d e f i n e d by age b a r r i e r s w h i c h c o u l d be d i s c o v e r e d t h r o u g h c r o s s - s e c t i o n a l i n v e s t i g a t i o n . A s t a g e t h e o r y of development p o s t u l a t e s n o t m e r e l y change w i t h age, b u t i n v a r i a n t o r d e r i n t h e p r o g r e s s i o n of s t a g e s . A s t u d y by W o h l w i l l g i v e s t e n t a t i v e s u p p o r t f o r t h e c o n s i s t e n t o r d e r i n g of t h e development of number u n d e r s t a n -d i n g i n t h r e e s t a g e s c h a r a c t e r i z e d by i n c r e a s i n g use of l o g i c a l as opposed t o p e r c e p t u a l f u n c t i o n s . 5 By means of a sc a l o g r a m a n a l y s i s , a s e r i e s of t e s t s , w h i c h were n o n - v e r b a l a d a p t a t i o n s of P i a g e t ' s correspondence, c o n s e r v a t i o n , and o r d i n a l - c a r d i n a l correspondence t a s k s , was o r d e r e d t o form a s c a l e of number development. The seventy-two k i n d e r g a r t e n and n u r s e r y s c h o o l p u p i l s t e s t e d showed t h r e e main t y p e s of response p a t t e r n s . F o u r t e e n s u b j e c t s f a i l e d a l l t e s t s ; e i g h t s u b j e c t s passed t h o s e t e s t s i n w h i c h some p e r c e p t u a l s u p p o r t f o r t h e c o r r e c t s o l u t i o n was a v a i l a b l e ; t h e r e m a i n i n g 5Joachim F. W o h l w i l l , "A Study of t h e Development of Number Concept by Scalogram A n a l y s i s , 1 * J o u r n a l of G e n e t i c  P s y c h o l o g y , 97. I 9 6 0 , pp. 345-77. 15 s u b j e c t s passed a l l i t e m s , i n c l u d i n g items r e q u i r i n g under-s t a n d i n g of r e l a t i o n s h i p s between numbers, w i t h o u t p e r c e p t u a l s u p p o r t f o r t h e c o r r e c t answer. I t s h o u l d be s t a t e d t h a t t h e p a t t e r n s were somewhat b l u r r e d by e x c e p t i o n s , and t h e items d i d n ot a l l s c a l e w e l l , but t h e e v i d e n c e i s a t l e a s t sugges-t i v e t h a t t h e r e a r e t h r e e l e v e l s of n u m e r i c a l u n d e r s t a n d i n g a t t a i n e d i n i n v a r i a n t o r d e r . I I . EVIDENCE OF THE SIGNIFICANCE OF CONSERVATION AS A STEP IN NUMBER CONCEPT DEVELOPMENT I n W o h l w i l l ' s s t u d y , t h e c o n s e r v a t i o n t e s t was w e l l -d e f i n e d i n i t s p l a c e i n t h e s c a l e , w i t h a s i g n i f i c a n t number of s u b j e c t s p a s s i n g the i t e m below but f a i l i n g i t , and a s i g -n i f i c a n t number p a s s i n g i t but f a l l i n g t h e s u c c e e d i n g one. T h i s s u g g e s t s t h a t c o n s e r v a t i o n i s an i m p o r t a n t s t e p i n t h e d e v e l o p m e n t a l sequence r e p r e s e n t e d by W o h l w i l l * s t a s k s . F u r t h e r s u p p o r t of t h e c o n t e n t i o n t h a t c o n s e r v a t i o n i s a n i m p o r t a n t c o g n i t i v e achievement, and n o t j u s t a n o t h e r l i t t l e knack t h a t a c h i l d can l e a r n , can be d e r i v e d f rom t h e r e s u l t s o f e x p e r i m e n t a l a t t e m p t s t o i n d u c e c o n s e r v a t i o n i n n o n - c o n s e r v i n g c h i l d r e n . W i t h one n o t a b l e e x c e p t i o n , t h e s e a t t e m p t s have been l a r g e l y u n s u c c e s s f u l . W o h l w i l l and Lowe used t h r e e n o n - v e r b a l t r a i n i n g p r o c e d u r e s on groups on non-c o n s e r v i n g k i n d e r g a r t e n c h i l d r e n (mean age 70 months.)^ ^ J . F. W o h l w i l l and Rowland C. Lowe, " E x p e r i m e n t a l A n a l y s i s of t h e Development of the C o n s e r v a t i o n of Number," C h i l d Development, 3 3 : 1 . 1962, pp. 5 3 - 6 7 . A l t h o u g h t h e t r a i n i n g groups made a s i g n i f i c a n t improvement from p r e - t e s t t o p o s t - t e s t , t h e i r improvement was e q u a l l e d by t h a t of t h e c o n t r o l group. Feigenbaum and S u l k i n compared the e f f e c t s of two v e r y b r i e f t r a i n i n g p r o c e d u r e s on non-c o n s e r v i n g k i n d e r g a r t e n p u p i l s r a n g i n g i n age f r o m 6 l t o 77 months.7 Though one t r e a t m e n t was a l i t t l e more e f f e c t i v e t h a n t h e o t h e r , the o v e r - a l l e f f e c t i v e n e s s of e i t h e r t r e a t -ment was not i m p r e s s i v e , as w e l l over h a l f of b o t h groups s t i l l f a i l e d t h e p o s t - t e s t a f t e r i n s t r u c t i o n . Lack of a con-t r o l group l n t h i s s t u d y makes i t i m p o s s i b l e t o e v a l u a t e t h e s i g n i f i c a n c e o f t h e t r a i n i n g p r o c e d u r e s . W a l l a c h and S p r o t t , on t h e o t h e r hand, were a l m o s t c o m p l e t e l y s u c c e s s f u l i n i n d u c i n g number c o n s e r v a t i o n by means of a t e c h n i q u e w h i c h r e p e a t e d l y demonstrated th e r e v e r -s i b l e n a t u r e o f s p a t i a l r earrangements of a s e t . S F o u r t e e n of t h e i r f i f t e e n n o n - c o n s e r v i n g s u b j e c t s p a s s e d t h e p o s t - t e s t a f t e r t h i s i n s t r u c t i o n , w h i l e none of t h e f i f t e e n c o n t r o l s u b j e c t s showed improvement. T h i s s u c c e s s i s r e m a r k a b l e i n v i e w of t h e c o m p a r a t i v e f a i l u r e of o t h e r e x p e r i m e n t a l a t t e m p t s t o t e a c h c o n s e r v a t i o n of number. I n e x p l a i n i n g why t h e y succeeded where o t h e r s f a i l e d , W a l l a c h and S p r o t t .D. Feigenbaum and Howard S u l k i n , " P i a g e t ' s Problem of C o n s e r v a t i o n of D i s c o n t i n u o u s Q u a n t i t i e s : A T e a c h i n g E x p e r i e n c e , " J o u r n a l of G e n e t i c P s y c h o l o g y , 105, 1964, pp. 91-97. ^ L i s e W a l l a c h and R i c h a r d L. S p r o t t , " I n d u c i n g Number C o n s e r v a t i o n i n C h i l d r e n , " C h i l d Development, 35, 1964, pp. 1057-71. 1? e l i m i n a t e many a l t e r n a t i v e s t o c o n c l u d e t h a t t h e i r s u c c e s s i s due t o t h e i r p a r t i c u l a r i n s t r u c t i o n a l p r o c e d u r e . However, t h e y n e g l e c t t o c o n s i d e r i m p o r t a n t d i f f e r -ences between t h e i r e x p e r i m e n t a l group and t h e e x p e r i m e n t a l groups of comparable s t u d i e s . The mean age o f t h e i r group was 83 months, a y e a r o r more o l d e r t h a n t h e s u b j e c t s used by W o h l w i l l and Lowe and by Feigenbaum and S u l k i n . F u r t h e r m o r e , t h e i r p r e - t e s t p r o c e d u r e e l i m i n a t e d c h i l d r e n from t h e e x p e r i -ment who were a l m o s t a s s u r e d l y i n t h e f i r s t s t a g e of c o n s e r v a t i o n development. The p r e - t e s t c o n s i s t e d of a p a i r of t w o - s t e p items such t h a t i f a c h i l d f a i l e d t h e f i r s t s t e p , ( t h a t of a s s e r t i n g t h e e q u a l i t y of a s e t of s i x c h e c k e r s p l a c e d on a s e t of s i x c a r d s ) , t h e second s t e p ( t h a t of r e a r r a n g i n g the c h e c k e r s and p o s i n g t h e c o n s e r v a t i o n q u e s t i o n ) c o u l d n o t be c a r r i e d o u t . The e i g h t c h i l d r e n who f a i l e d t he f i r s t s t e p , t h a t i s , who d e n i e d t h e n u m e r i c a l e q u a l i t y of two s e t s of o b j e c t s p l a c e d i n d i r e c t one-to-one c o r r e s p o n d e n c e , were not i n c l u d e d i n the s t u d y . A c c o r d i n g t o P i a g e t , c h i l d r e n i n Stage 1 judge on t h e b a s i s o f g l o b a l p e r -c e p t i o n ( t h u s s u p p o s i n g t h a t the c a r d s , b e i n g b i g g e r , a l s o a r e more), and do not u n d e r s t a n d one-to-one correspondence as th e b a s i s of n u m e r i c a l e q u a l i t y . I n t h e o t h e r t e a c h i n g s t u d i e s c i t e d t h e r e was no a c c i d e n t a l e l i m i n a t i o n of Stage 1 s u b j e c t s from t h e e x p e r i m e n t a l g r o u p s . The s i g n i f i c a n t r e s u l t of t h e s e t h r e e t e a c h i n g e x p e r i -ments i s t h i s : t h a t W a l l a c h and S p r o t t ' s r a t h e r " o l d " Stage 2 c h i l d r e n were r e c e p t i v e t o a s h o r t - t e r m "one-shot" i n s t r u c -t i o n a l p r o c e d u r e , whereas the younger c h i l d r e n , many of whom were u n d o u b t e d l y l n Stage 1, used by W o h l w i l l and Lowe and by Feigenbaum and S u l k i n , were t o a l a r g e e x t e n t u n a f f e c t e d by such p r o c e d u r e s . s-A s t u d y by C o x f o r d , 9 though i t d e a l s w i t h i n s t r u c t i o n i n P i a g e t * s s e r l a t i o n problems r a t h e r t h a n w i t h c o n s e r v a t i o n of number, i s r e l e v a n t here.10 C o x f o r d c l a s s i f i e d h i s s u b j e c t s , who ranged l n age from t h r e e y e a r s , s i x months, t o seven y e a r s , f i v e months, as t o t h e i r d e v e l o p m e n t a l s t a g e on u n d e r s t a n d i n g of o r d i n a l s e r l a t i o n . He s e l e c t e d t w e l v e Stage 1 c h i l d r e n and t w e l v e Stage 2 c h i l d r e n f o r i n s t r u c t i o n . Con-t r o l groups were c a r e f u l l y matched f o r s t a g e placement, age, and IQ. A f t e r I n s t r u c t i o n l n s e r l a t i o n p r o c e d u r e , t h e Stage 1 c h i l d r e n showed no s i g n i f i c a n t g a i n s over t h e i r c o n t r o l group. The Stage 2 c h i l d r e n , on t h e o t h e r hand, p r o f i t t e d s u f f i c i e n t l y f r o m i n s t r u c t i o n t o s i g n i f i c a n t l y o u t p e r f o r m t h e i r c o n t r o l group on t h e p o s t - t e s t . I n g e n e r a l , t h e r e s u l t s of t h e t e a c h i n g e x p e r i m e n t s c i t e d h e r e s u p p o r t t h e c o n t e n t i o n t h a t a c h i l d ' s r e c e p t i v i t y t o i n s t r u c t i o n d e s i g n e d t o i n c r e a s e h i s u n d e r s t a n d i n g o f 9A. F. C o x f o r d J r . , " E f f e c t s of I n s t r u c t i o n on t h e Stage Placement of C h i l d r e n i n P i a g e t ' s S e r i a t i o n E x p e r i -ments, " A r i t h m e t i c Teacher, 11, 64, pp. 4-9. ^ A c c o r d i n g t o P i a g e t , c o n s e r v a t i o n of c a r d i n a l number and development of o r d i n a l number u n d e r s t a n d i n g a r e c l o s e l y r e l a t e d , b o t h d e v e l o p i n g i n t h r e e synchronous s t a g e s . The  C h i l d ' s C o n c e p t i o n of Number, pp. 147-149. ! 9 number may be dependent upon h i s h a v i n g d e v e l o p e d t h e c o g n i -t i v e s t r u c t u r e s w h i c h P i a g e t p o s t u l a t e s i n h i s s t a g e t h e o r y of number concept development. I I I . 'THE RELATION OP PIAGET'S THEORY OP NUMBER CONCEPT DEVELOPMENT TO EDUCATIONAL PRACTICE A number of r e v i e w e r s of the l i t e r a t u r e , n o t a b l y L u n z e r . H I s a a c s , ^ 2 and F l a v e l l , 1 3 have e x p r e s s e d t h e o p i n i o n t h a t P i a g e t ' s t h e o r y of number concept development has i m p l i -c a t i o n s f o r e d u c a t i o n a l p r a c t i c e . The l a t t e r two propose t h a t P i a g e t * s r e s e a r c h i n s t r u m e n t s can be used as t e s t s of r e a d i -ness f o r i n s t r u c t i o n i n number work. I s a a c s a s s e r t s s t r o n g l y t h a t such t e s t s s h o u l d be used t o a s s u r e t h a t a c h i l d has a f u l l , f u n c t i o n a l g r a s p of t h e i d e a of number b e f o r e any f o r -mal t e a c h i n g o r l e a r n i n g i s imposed on him. However, t o d a t e o n l y one i n v e s t i g a t i o n has examined t h e r e l a t i o n of p e r -formance on P i a g e t i a n number t a s k s w i t h achievement i n s c h o o l a r i t h m e t i c . D o d w e l l o b t a i n e d a c o r r e l a t i o n of .59 between the s c o r e s of k i n d e r g a r t e n c h i l d r e n on a s e t of f i v e of l ^ E . A. L u n z e r , Recent S t u d i e s i n B r i t a i n Based on t h e  Work of Je a n P i a g e t . (London: N a t i o n a l F o u n d a t i o n f o r Edu-c a t i o n a l R e s e a r c h i n England and Wales, i960.) 1 2 N a t h a n I s a a c s , New L i g h t on C h i l d r e n ' s Idea of Number. (London: Ward Lock E d u c a t i o n a l Company, L i m i t e d , 196*377 P. 37. 13John H. F l a v e l l , The Developmental P s y c h o l o g y of  Jean P i a g e t . ( T o r o n t o : D. Van N o s t r a n d Company, 1963), pp. 365-69. 20 Piaget*s number tasks (three of which were conservation pro-blems), and t h e i r scores on a test of arithmetic progress administered at the end of the f i r s t term i n Grade 1 arithme-t i c . 1 ^ He also constructed a group t e s t , using p i c t o r i a l representations of the o r i g i n a l problems, which he f e l t had value as an arithmetic readiness t e s t . Brace and Nelson, adopting a Piagetian d e f i n i t i o n of the "true concept of number,** evaluated the number concept development of 124 children ranging i n age from f i v e years, four months to s i x years f i v e m o n t h s . T h e f i f t y - f i v e Items of the test included four conservation items. They found that f o u r - f i f t h s of the group f a i l e d the conservation items, and concluded from t h i s that teachers need to provide experi-ence which develops cardinal number understanding before using numerals. They make other recommendations f o r the order of presentation of number work on the basis of t h e i r findings. IV. THE RELATION OF CONSERVATION TO OTHER VARIABLES The co r r e l a t i o n of i n t e l l i g e n c e quotients and scores on batteries of Piagetian number problems has been found by ^ P . C. Dodwell, "Children's Understanding of Number Concepts: Characteristics of an Individual and of a Group Test,** Canadian Journal of Psychology, 14, I960, pp. 191-205. 1 5 A l e c Brace and L. Doyal Nelson, "The Pre-school Child's Concept of Number," The Arithmetic Teacher, 12, (February, 1965), pp. 126-33. 21 D o d w e l l ^ and by Elkind-*-''7 t o be c o n s i s t e n t l y low but p o s i -t i v e . On t e s t s of c o n s e r v a t i o n a l o n e , Felgenbaum found t h a t t h e above-average IQ group made s i g n i f i c a n t l y b e t t e r s c o r e s t h a n t h e average and below average group.18 I n t h e s t u d y j u s t c i t e d , D o d w e l l f o u n d s t a t i s t i c a l l y i n s i g n i f i c a n t but c o n s i s t e n t d i f f e r e n c e s i n the number concept s c o r e s of c h i l d r e n c l a s s i f i e d i n t h r e e s o c i o - e c o n o m i c groups a c c o r d i n g t o t h e i r f a t h e r ' s o c c u p a t i o n s . These d i f -f e r e n c e s f a v o u r e d t h e h i g h e r s o c i o - e c o n o m i c g r o u p s . N e i t h e r D odwell^? n o r Feigenbaum^O have found s i g n i -f i c a n t o r c o n s i s t e n t sex d i f f e r e n c e s i n t h e a b i l i t y of boys and g i r l s t o do P i a g e t * s number t a s k s . l 6 D o d w e l l , op_. c i t . , i960. ^ E l k i n d , op_. c i t . , pp. 219-2?. ^ F e i g e n b a u m , o£. c i t . , pp. 423-32. i^p. c. D o d w e l l , " R e l a t i o n s Between th e U n d e r s t a n d i n g of t h e L o g i c of C l a s s e s and of C a r d i n a l Number i n C h i l d r e n , " Canadian J o u r n a l of P s y c h o l o g y . 16 (2), 1962, pp. 152-60. 2 0 F e l g e n b a u m , op_. c i t . , pp. 423-32. CHAPTER IV DESIGN AND PROCEDURE I . THE SAMPLE The 215 Grade 1 p u p i l s a t t e n d i n g t h r e e s c h o o l s i n t h e N o r t h Vancouver S c h o o l D i s t r i c t made up t h e o r i g i n a l s u b j e c t p o o l . These p u p i l s were members of seven c l a s s e s , one of w hich was a combined Grade 1 and 2. The s u p e r i n t e n d e n t and a s s i s t a n t s u p e r i n t e n d e n t o f t h e s c h o o l d i s t r i c t judged t h a t t h e t h r e e s c h o o l s chosen s e r v e d low, m i d d l e and h i g h Income g r o u p s . A v a r i e t y of e t h n i c groups, i n c l u d i n g I t a l i a n , E a s t I n d i a n , S c a n d i n a v i a n , Japanese, N o r t h A m e r i c a n I n d i a n , as w e l l as a m a j o r i t y of w h i t e c h i l d r e n o f undetermined o r i g i n made up t h e sample. Whi l e not f o r m a l l y a r e p r e s e n t a t i v e sample, t h e r e seems t o be l i t t l e r e a s o n t o suppose t h a t t h e group does not p r o v i d e a c r o s s - s e c t i o n of t h e u r b a n , p u b l i c s c h o o l p o p u l a t i o n . T h i r t y - s i x c h i l d r e n randomly chosen from t h e c l a s s e s i n numbers p r o p o r t i o n a l t o the c l a s s s i z e c o n s t i t u t e d a p i l o t group w h i c h r e c e i v e d a p r e l i m i n a r y form of t h e c o n s e r v a t i o n t e s t . These were e l i m i n a t e d f r om the s t u d y p r o p e r , as were a number of p u p i l s who were r e p e a t i n g Grade 1. These e l i m i n a -t i o n s p l u s t h e l o s s of p u p i l s who moved d u r i n g t h e y e a r , o r who were 111 on t e s t i n g days, l e f t a group of 156 s u b j e c t s . W i t h t h e e x c e p t i o n of one 7-7 y e a r - o l d , t h e i r ages as of 23 January 1967 ranged from 6-0 t o 7-1 y e a r s . There were ?6 boys and 80 g i r l s . Of t h e 156 s u b j e c t s , 118 had a t t e n d e d k i n d e r g a r t e n o r o t h e r p r e - s c h o o l c l a s s e s . I I . THE TESTING PROGRAMME D u r i n g t h e f i r s t two weeks of October, t h e e n t i r e s u b j e c t group (SG) r e c e i v e d a s i x - i t e m t e s t of number c o n s e r -v a t i o n ( C - t e s t I ) . On the b a s i s of t h e i r s c o r e s , t h e s u b j e c t s were c l a s s i f i e d as Stage 1, Stage 2 and Stage 3. and the r e s u l t i n g groups d e s i g n a t e d 1^, I2 and I3, r e s p e c t i v e l y . D u r i n g the second week of t h e f o l l o w i n g J a n u a r y , 1^ r e c e i v e d a s i m i l a r form of t h e c o n s e r v a t i o n t e s t ( C - t e s t I I ) and was regrouped a c c o r d i n g t o t h e i r s t a g e c l a s s i f i c a t i o n a t t h a t t i m e . These groups were d e s i g n a t e d 11^, H 2 a n d 113- D u r i n g the l a s t week i n May, a l l s u b j e c t s r e c e i v e d t h e a r i t h m e t i c s u b - t e s t of t h e S t a n f o r d Achievement T e s t , P r i m a r y I B a t t e r y , Form W. F i g u r e 1 g i v e s a s c h e m a t i c r e p r e s e n t a t i o n of t h i s programme. I I I . RESTATEMENT OF HYPOTHESES AS EXPECTATIONS FOR THE EXPERIMENTAL GROUPS The hypotheses g e n e r a t e d by the t h e o r y of number con-s e r v a t i o n development, as l i s t e d on pp. 9-11. a r e r e l a t e d t o the e x p e c t a t i o n s w i t h r e s p e c t t o the a r i t h m e t i c achievement of the groups I ] _ , 1 3 . I3. H i . IT-2 a n d IT-3 i n t h e f o l l o w i n g manner: 23 October J a n u a r y May Read from l e f t t o r i g h t , t h e f i g u r e i n d i c a t e s t h e o r d e r i n which t e s t s were a d m i n i s t e r e d , the sub-groups which were d e r i v e d by means of the t e s t s , and which sub-groups r e c e i v e d f u r t h e r t e s t i n g . F i g u r e 1. Schematic p r e s e n t a t i o n of the t e s t i n g programme. 24 A. Hypotheses r e l a t e d t o f a i l u r e t o c o n s e r v e a t the b e g i n n i n g , o r d u r i n g t h e f i r s t term o f , t h e i n s t r u c t i o n a l programme i n a r i t h m e t i c : 1. The p r o p o r t i o n of 1^ o b t a i n i n g achievement s c o r e s w h i c h f a l l below t h e median f o r t h e e n t i r e group w i l l exceed the p r o p o r t i o n of 1^ o b t a i n i n g above-median s c o r e s . 2. The p r o p o r t i o n of 11^ o b t a i n i n g s c o r e s w h i c h f a l l below t h e group median achievement s c o r e w i l l exceed t h e p r o p o r t i o n of 11^ o b t a i n i n g above-median s c o r e s . B. Hypotheses r e g a r d i n g t h e r e l a t i o n of s t a g e c l a s s i -f i c a t i o n t o t e r m i n a l achievement i n a r i t h m e t i c : 1. The mean achievement s c o r e of w i l l exceed t h a t of I 2 w h i c h w i l l i n t u r n exceed t h a t of 2. The mean achievement s c o r e of II j w i l l exceed t h a t of I I 2 , w h i c h w i l l i n t u r n exceed t h a t of I I I . 3» a) The mean achievement s c o r e of I 2 w i l l exceed t h a t o f I l 2 « b) The mean achievement s c o r e of I3 w i l l exceed t h a t of II3. C. The h y p o t h e s i s r e l a t i n g t o t h e t h r e e - s t a g e development of c o n s e r v a t i o n o v er t i m e : 1. I i w i l l show an i n c r e a s e i n mean c o n s e r v a t i o n s c o r e when r e t e s t e d a f t e r a four-month i n t e r -v a l . IV. DESCRIPTION OF THE TESTS The C o n s e r v a t i o n T e s t M a t e r i a l s . A t w e n t y - f o u r by t h i r t y - s i x Inch f l a n n e l -b o a rd eovered w i t h p a l e y e l l o w f a b r i c s e r v e d as a background on w h i c h r e d , b r i g h t y e l l o w , and b l a c k f e l t c u t - o u t s were d i s p l a y e d . I n c o n s p i c u o u s k n o t s of w h i t e t h r e a d g u i d e d t h e examiner i n p l a c i n g the c u t - o u t s so t h a t t h e s t i m u l u s p a t -t e r n s were u n i f o r m f o r a l l s u b j e c t s . E i g h t s e t s of c u t - o u t s were used, each s e t c o n s i s t i n g on seven u n i t s i d e n t i c a l i n s i z e , shape, and c o l o u r , as f o l l o w s : (a) C i r c l e s i ) one s e t of each c o l o u r , one and o n e - h a l f i n c h e s i n d i a m e t e r , 11) one s e t b l a c k , t h r e e i n c h e s l n d i a m e t e r . (b) Squares i ) one s e t of each c o l o u r , one and o n e - h a l f i n c h e s i n w i d t h , i l ) one s e t r e d , t h r e e i n c h e s i n w i d t h . I n a d d i t i o n , two p i n k f e l t s t a r s were used t o demonstrate t o the c h i l d how t h e f l a n n e l b o a r d works. T e s t p r o c e d u r e . The examiner (E) f i r s t demonstrated how a f e l t s t a r c l i n g s t o t h e f l a n n e l s u r f a c e , and i n v i t e d 26 t h e s u b j e c t (S) t o p l a c e a s t a r below E's s t a r . When S showed t h a t he c o u l d f o l l o w t h i s i n s t r u c t i o n , E proceeded w i t h t h e t e s t . The f o l l o w i n g s t e p s c o n s t i t u t e d t h e proce d u r e f o r t h e f i r s t t h r e e i t e m s : (a) E p r e s e n t e d two s e t s of c u t - o u t s , each s e t i n a p i l e , s e l e c t i n g one f o r h e r s e l f and g i v i n g t h e o t h e r t o S. (b) S was t o l d , "You and I w i l l t a k e t u r n s p u t t i n g t h e c i r c l e s ( o r s q u a r e s , where a p p r o p r i a t e ) on t h e f l a n n e l b o a r d , so t h a t you put on j u s t as many as I do." ( T h i s statement preceded t h e f i r s t i t e m o n l y . ) (c) E p l a c e d h e r c u t - o u t s one a t a ti m e i n a h o r i z o n -t a l row, p e r m i t t i n g S t o p l a c e h i s c u t - o u t below t h a t of E each t i m e . (d) As t h e l a s t i t e m was p l a c e d , E remarked, " T h i s i s a l l I have; i s t h a t y o ur l a s t one, t o o ? " (e) E t h e n moved one s e t t o form a new c o n f i g u r a t i o n . ( f ) The c o n s e r v a t i o n q u e s t i o n was posed, e.g., "Are t h e r e more r e d ones o r more b l a c k ones, o r a r e t h e r e t h e same number?" T h i s q u e s t i o n was accom-p a n i e d by a p p r o p r i a t e g e s t u r e s , so t h a t c h i l d r e n who were unsure of c o l o u r names would u n d e r s t a n d t h e q u e s t i o n . (g) S was a s k e d t o remove h i s s e t of c u t - o u t s from t h e b o a r d . A t t h i s t ime E r e c o r d e d h i s response t o t h e c o n s e r v a t i o n q u e s t i o n . P r e c e d i n g t h e f o u r t h i t e m , S was t o l d t o p l a c e h i s l i t t l e c u t - o u t i n t h e m i d d l e of E's b i g one. F o r t h e f o u r t h and f o l l o w i n g i t e m s , E p l a c e d h e r c u t - o u t s i n v a r i o u s p a t -t e r n s , r a t h e r t h a n i n a h o r i z o n t a l row. I n o t h e r r e s p e c t s t h e l a s t t h r e e i t e m s proceeded i n t h e same manner as t h e f i r s t t h r e e . Diagrams of t h e c o n f i g u r a t i o n s p r e s e n t e d by-each i t e m a r e g i v e n l n Appendix A. C o n s t r u c t i o n of t h e Items. The s i x items of t h e t e s t c o n s t i t u t e d a t h r e e - s t e p Gutman-type s c a l e . T h i s was accom-p l i s h e d by u s i n g items a t two l e v e l s of d i f f i c u l t y d e s i g n e d i n s u ch a way t h a t Stage 1 c h i l d r e n would f a i l b o t h t y p e s , Stage 2 c h i l d r e n would pass o n l y t h e e a s i e r i t e m s , and Stage 3 c h i l d r e n would pass b o t h t h e e a s i e r and t h e more d i f f i c u l t i t e m s . The t h r e e d i f f i c u l t items were s t a n d a r d c o n s e r v a t i o n i t e m s , d i r e c t l y a n a l a g o u s t o P i a g e t 1 s egg-cup problem as d e s c r i b e d i n C hapter I . I n t h e s e i t e m s , t h e two s e t s of f e l t shapes a r e p l a c e d i n one-to-one correspondence l n h o r i z o n t a l rows. When one row I s s p r e a d o u t , condensed o r c l u s t e r e d , g l o b a l p e r c e p t i o n y i e l d s m i s l e a d i n g i n f o r m a t i o n about th e n u m e r i c a l e q u a l i t y of t h e s e t s , and t h e i n t u i t i v e p e r c e p t u a l a n a l y s i s of the Stage 2 c h i l d g i v e s no h i n t of e q u a l i t y . Thus, o n l y t h o s e c h i l d r e n who have a t h o r o u g h g r a s p of t h e c o n s e r v a t i o n p r i n c i p l e were ex p e c t e d t o pass t h e s e Items. I n t h e t h r e e e a s i e r i t e m s , S p l a c e s one s m a l l c u t - o u t 28 on each of E's l a r g e ones. E moves t h e s m a l l c u t - o u t s t o form a s i m i l a r c o n f i g u r a t i o n b e s i d e t h e s e t of l a r g e c u t - o u t s , i n t h i s way k e e p i n g one-to-one p a t t e r n i n g a v a i l a b l e f o r t h e i n t u i t i v e p e r c e p t u a l a n a l y s i s of t h e Stage 2 c h i l d , w h i l e t h e Stage 1 c h i l d , u s i n g g l o b a l p e r c e p t i o n , i s d e c e i v e d by t h e " b i g n e s s " of the s e t of l a r g e c u t - o u t s . T h e r e f o r e , Stage 2 but n ot Stage 1 c h i l d r e n were e x p e c t e d t o pass t h e s e i t e m s . The t h r e e e a s i e r items were s e l e c t e d from a p o o l of s i x w h i c h were a d m i n i s t e r e d i n c o n j u n c t i o n w i t h the t h r e e s t a n d a r d items t o the p i l o t group i n a p r e l i m i n a r y t e s t . W i t h two e x c e p t i o n s , the 36 s u b j e c t s showed s c a l e - t y p e r esponse p a t t e r n s , t h a t i s , i f t h e y passed any of the d i f f i -c u l t i t e m s , t h e y passed a l l the easy ones as w e l l . The t h r e e easy items w h i c h showed t h e g r e a t e s t power of d i s c r i m i n a t i o n between Stage 1 and Stage 2 were chosen f o r i n c l u s i o n i n t h e c o n s e r v a t i o n t e s t . S c o r i n g . Such answers as " I t h i n k t h e y a r e the same;" "There a r e the same number;" "Same number;" or even m e r e l y "Same," were c r e d i t e d as c o r r e c t , and one p o i n t a c c o r d i n g l y awarded. The answers "Yes;" "These a r e more;" "These a r e b i g g e r ; " and " I don't know," were counted i n c o r r e c t and no p o i n t g i v e n . The t e s t s c o r e s were c o n v e r t e d t o s t a g e c l a s s i -f i c a t i o n s i n the f o l l o w i n g manner: Score Stage 0,1 1 2-4 2 5.6 3 29 The second form of t h e c o n s e r v a t i o n t e s t . F o r t h e Jan u a r y r e - t e s t o f t h e Items i n v o l v e d i d e n t i c a l p r o c e -dures and employed c o n f i g u r a t i o n s s i m i l a r t o t h o s e used i n the i n i t i a l t e s t . However, wherever squares had been used i n the i n i t i a l t e s t , c i r c l e s o f t h e same d i m e n s i o n were s u b s t i t u -t e d and, s i m i l a r l y , squares were s u b s t i t u t e d f o r c i r c l e s . V a l i d i t y of the c o n s e r v a t i o n t e s t . S i n c e the t h r e e s t a n d a r d items a r e such d i r e c t t r a n s f o r m a t i o n s of P i a g e t ' s provoked correspondence t a s k s , ^ t h e r e i s l i t t l e r e a s o n t o doubt t h a t c h i l d r e n who responded p r o m p t l y and c o r r e c t l y on a l l t h r e e , as w e l l as t o the t h r e e e a s i e r i t e m s , s a t i s f y P i a g e t ' s c r i t e r i o n f o r Stage 3 development, namely, t h e y demonstrate t h e i r u n d e r s t a n d i n g t h a t u n i t f o r u n i t m a t c h i n g i m p l i e s e q u a l i t y of t h e matched s e t s . T h i s c o n f i d e n c e i s f u r t h e r j u s t i f i e d by the f a c t t h a t t h e t e s t p r o c e d u r e p r e s e n -t e d t he c o n s e r v a t i o n problem i n i t s s t r o n g form, t h a t i s , t h e e q u a l i t y of t h e s e t s a f t e r t h e correspondence was made was no t made e x p l i c i t b e f o r e the s e t s were r e a r r a n g e d . Thus t h e c h i l d had t o n o t i c e t h a t t h e " t a k i n g t u r n s " p r o c e d u r e had, i n f a c t , produced "equal s e t s , and use t h i s knowledge i n c o n c l u -d i n g t h a t t h e s e t s were s t i l l e q u a l a f t e r r e arrangement. The case i s not so c l e a r w i t h r e s p e c t t o the t h i r t e e n Stage 3 s u b j e c t s who answered o n l y two of t h e t h r e e s t a n d a r d items c o r r e c t l y , n o r t h e seven who counted f o r more t h a n one i t e m . However, t h o s e who f a i l e d o n l y one i t e m a l m o s t i T h e C h i l d ' s C o n c e p t i o n of Number, pp. 41 -63. 30 i n v a r i a b l y m i s s e d t h e f i r s t one. S i n c e t h e t e s t d i d not i n c l u d e any p r a c t i c e i t e m s , m i s s i n g t h e f i r s t might be con-s t r u e d as a " s l i p " t h e c h i l d makes b e f o r e g r a s p i n g t h e n a t u r e of t h e problem a t hand. I t might seem t h a t t h e use o f coun-t i n g s h o u l d d i s q u a l i f y a c h i l d from Stage 3 c l a s s i f i c a t i o n , s i n c e he has o b v i o u s l y n o t reasoned d i r e c t l y t h a t one-to-one p a i r i n g i m p l i e s e q u i v a l e n c e . I t was n o t always p o s s i b l e t o t e l l , however, what f u n c t i o n c o u n t i n g had i n t h e c h i l d ' s a r r i v i n g a t h i s answer. Only t h r e e c h i l d r e n m e t h o d i c a l l y c ounted b o t h s e t s b e f o r e r e p o r t i n g t h e i r answers. Others p o i n t e d o r bobbed t h e i r heads as the second s e t was b e i n g moved by t h e examiner, perhaps i n o r d e r t o a r r i v e a t t h e i r d e c i s i o n , o r perhaps t o check t h e d e c i s i o n t h e y had a l r e a d y made. ( A f t e r t h e t h i r d o r f o u r t h i t e m , most c h i l d r e n a n t i c i -p a t e d t h e c o n s e r v a t i o n q u e s t i o n , sometimes a n s w e r i n g b e f o r e i t was asked.) A s m a l l number of c h i l d r e n were perhaps m i s c l a s s i f l e d as Stage 3 by t h e s c o r i n g a s c o r r e c t o f items on w h i c h t h e y used c o u n t i n g . G e n e r a l l y , however, the t e s t showed s a t i s f a c t o r y power t o d i s t i n g u i s h Stage 3 c h i l d r e n i n a c c o r d a n c e w i t h P i a g e t ' s s p e c i f i c a t i o n . The d i s t i n c t i o n made by t h e t e s t between Stage 2 and Stage 1 cannot be r e g a r d e d w i t h the same degree of c o n f i d e n c e , s i n c e i t depended e n t i r e l y upon a s i n g l e c r i t e r i o n , namely, g l o b a l n e s s of p e r c e p t i o n . P i a g e t , however, d e s c r i b e s Stage 1 c h i l d r e n as e x h i b i t i n g , as w e l l as g l o b a l n e s s of p e r c e p t i o n , t h e i n a b i l i t y t o produce a one-to-one c o r r e s p o n d e n c e . S i n c e 31 the test procedure forced the c h i l d to make a correct corres-pondence of his seven cut-outs with the examiner's seven cut-outs, i t provided no way of distinguishing those who could do so from those who might have f a i l e d . Also, Stage 1 thinking, as Piaget describes i t , i s c h a r a c t e r i s t i c of children one to three years younger than the present sample. It i s probable, therefore, that the thinking of many of the children who were c l a s s i f i e d as Stage 1 had a c t u a l l y acquired some of the attributes of Stage 2 thinking, although i n t u i -t i v e perceptual analysis was s t i l l absent. The shortcomings of a r i g i d l y worded and scored te s t i n g procedure were demonstrated also by the case of one boy who, having responded i n c o r r e c t l y to a l l s i x items, remarked, "I should be saying same number, because they're the same. Like i f there's ten of these, there's ten there." One may suppose that t h i s c h i l d was so thoroughly at home with the nature of number that i t never occurred to him that the examiner would ask such a f o o l i s h question as, "Are there the same number?" He assumed the question must r e f e r to the r e l a t i v e apparent size of the sets, and answered accordingly. A test with greater f l e x i b i l i t y i n the procedures and per-mitting attention to q u a l i t a t i v e as well as quantitative aspects of the response might have yielded an accurate evaluation of thi s child's grasp of the conservation p r i n c i p l e . 32 The .Stanford Achievement T e s t S u i t a b i l i t y of t h e t e s t . The a r i t h m e t i c s u b - t e s t of t h e S t a n f o r d Achievement T e s t , P r i m a r y I B a t t e r y , Form W, s e r v e d as a measure of a r i t h m e t i c achievement. T h i s t e s t , p u b l i s h e d i n 1964, samples p u p i l s 1 u n d e r s t a n d i n g of measures, p r o b l e m - s o l v i n g , and number c o n c e p t s . These a r e t h e main a r e a s of s t u d y c o v e r e d by t h e S e e i n g Through A r i t h m e t i c I t e x t . The items t e n d t o r e q u i r e f l e x i b i l i t y i n t h e a p p l i c a -t i o n of t e c h n i q u e s and u n d e r s t a n d i n g s a c q u i r e d from a r i t h m e t i c i n s t r u c t i o n , r a t h e r t h a n performance of r o u t i n i z e d p r oblems. F o r i n s t a n c e , i n s t e a d of m e r e l y m a r k i n g the c l o c k t h a t shows seven o ' c l o c k , t h e c h i l d must mark the c l o c k t h a t shows the t ime when most c h i l d r e n g e t up i n t h e morning. I t i s f e l t , t h e r e f o r e , t h a t the S t a n f o r d t e s t c o n s t i t u t e d a s u i t a b l e t e s t f o r p u p i l s who had r e c e i v e d t h e S e e i n g Through  A r i t h m e t i c programme, and a good measure of m a t h e m a t i c a l u n d e r s t a n d i n g as w e l l . P r o c e d u r e . The i n v e s t i g a t o r a d m i n i s t e r e d the t e s t a c c o r d i n g t o the p r i n t e d i n s t r u c t i o n s t o the p u p i l s i n t h e i r r e g u l a r c l a s s r o o m s . I n 43 of t h e i t e m s , p u p i l s mark t h e c o r r e c t r e s p o n s e t o a problem r e a d by t h e examiner. The r e m a i n i n g 20 items a r e a d d i t i o n and s u b t r a c t i o n examples. S i n c e the p u p i l s were u n f a m i l i a r w i t h a d d i t i o n and s u b t r a c -t i o n w r i t t e n i n v e r t i c a l form, t h e examiner supplemented the p r i n t e d i n s t r u c t i o n s (which i n c o r p o r a t e d a b r i e f l e s s o n i n t h e I n t e r p r e t a t i o n of examples i n h o r i z o n t a l form) w i t h 33 I n s t r u c t i o n i n r e a d i n g examples i n t h i s f o rm. V. INTERVIEWS A t t h e time of the f i r s t t e s t i n g , t h e I n v e s t i g a t o r h e l d an i n f o r m a l i n t e r v i e w w i t h each of the t e a c h e r s of t h e c l a s s e s i n v o l v e d \n t h e s t u d y l n o r d e r t o a s c e r t a i n what t o p i c s had a l r e a d y been i n t r d d u c e d i n t h e i r a r i t h m e t i c i n s t r u c t i o n . The t e a c h e r s were r e q u e s t e d t o p r o c e e d w i t h t h e i r u s u a l a r i t h m e t i c c o u r s e . They were not i n f o r m e d of t h e r e s u l t s o f e i t h e r o f t h e c o n s e r v a t i o n t e s t s . CHAPTER V SELECTED DATA AND STATISTICAL TREATMENT OP RESULTS I . INFORMATION FROM TEACHER INTERVIEWS A l l seven t e a c h e r s r e p o r t e d t h a t t h e y were u s i n g t h e t e x t , S e e i n g Through A r i t h m e t i c I , and t h e accompanying work-s h e e t s i n t h e i r c l a s s e s . By t h e time t h e f i r s t c o n s e r v a t i o n t e s t was a d m i n i s t e r e d , a l l had p r e s e n t e d t h e m a t e r i a l i n geometry w h i c h b e g i n s t h i s t e x t . A l l had d e a l t w i t h o n e - t o -one m a t c h i n g and c a r d i n a l r e c o g n i t i o n of groups as g r e a t as f o u r o r f i v e . Two t e a c h e r s had t a u g h t r a t i o n a l c o u n t i n g t o t e n , and r e c o g n i t i o n of the numerals t o t e n . One had t a u g h t r o t e c o u n t i n g t o 100. I I . CONSERVATION TEST RESULTS T a b l e I shows the d i s t r i b u t i o n of s u b j e c t s i n t o groups 1^, I 2 . I3, IIi» I I 2 a ^ d II3, on t h e b a s i s of t h e i r s c o r e s on C - t e s t I and C - t e s t I I . TABLE I DISTRIBUTION OF SUBJECTS IN THE EXPERIMENTAL GROUPS RESULTING FROM C-TEST I AND C-TEST I I Te s t Stage 3 Stage 2 Stage 1 Stage 2 Stage 3 C - t e s t I 42 (I3) 70 ( I 2 ) 44 ( I i ) C - t e s t I I 20 ( H i ) 1 7 ( 1 1 2 ) 7 (H3) I I I . ACHIEVEMENT TEST RESULTS 35 Raw scores on the arithmetic sub-test of the Stanford Achievement Test ranged from 8 to 62, 63 being the maximum possible score. The mean f o r the entire group f e l l at 35.83. the median at 36. Table II shows the achievement test r e s u l t s f o r the experimental groups. TABLE II SUMMARY OF ACHIEVEMENT TEST RESULTS Group N Range Mean S.D. *3 42 21-61 40.14 10.44 12 70 8-62 34.86 12.67 II 44 10-54 32.50 11.26 ' » 3 7 32-54 42.57 6.97 Sub-groups 17 10-45 30.53 10.04 Of I], 30.65 I H i 20 12-54 11.59 IV. STATISTICAL TESTS OF NULL HYPOTHESES DERIVED FROM THE EXPECTATIONS FOR THE RELATIVE ACHIEVEMENT OF GROUPS AS CLASSIFIED BY THE CONSERVATION TEST The Expected Low Achievement of 1^ and The re s u l t s of s t a t i s t i c a l tests of the n u l l hypo-theses that proportions of I j and 11^ obtaining below-median scores i n arithmetic achievement do not s i g n i f i c a n t l y exceed .50 appear i n Table I I . A z-value of 1.64 was the c r i t e r i o n 36 adopted f o r s i g n i f i c a n c e a t the .05 l e v e l , u s i n g a o n e - t a i l e d t e s t . The n u l l h y p o t h e s i s was a c c e p t e d i n t h e case of I ^ , but i n t h e case of I I l t i t was c o n c l u d e d t h a t t h e p r o p o r t i o n of t h e group o b t a i n i n g below-median s c o r e s was s i g n i f i c a n t l y g r e a t e r t h a n .50. TABLE I I I SIGNIFICANCE OF PROPORTIONS OF I i AND H i ACHIEVING BELOW MEDIAN IN ARITHMETIC Group N P r o p o r t i o n P r o p o r t i o n z EQ Below Median Expected under HO I l 44 .59 .50 1.20 a c c e p t e d I I I 20 .70 .50 1.79 r e j e c t e d (p<.05) The E x p e c t e d R e l a t i v e Achievement of t h e E x p e r i m e n t a l Groups Three one-way a n a l y s e s of v a r i a n c e were a p p l i e d t o t e s t t h e n u l l h y p o t h e s i s t h a t t h e r e i s no r e l a t i o n s h i p between c o n s e r v a t i o n s t a g e c l a s s i f i c a t i o n and t e r m i n a l achievement i n a r i t h m e t i c . The groups i n v o l v e d i n each t e s t were (1) Groups 11, I2 and I3; (2) Groups I I l t II2 and I I 3 ; (3) Groups I3, 12, I I I , 1*2 and I I3. The f i r s t of t h e s e a n a l y s e s y i e l d e d an F - r a t i o s i g n i f i c a n t a t t h e .01 l e v e l , and t h e second a n F - r a t i o s i g n i f i c a n t a t t h e .05 l e v e l . The i n c l u s i o n I n t h e t h i r d a n a l y s i s of Groups I2 and II3, whose numbers, 70 and 7 r e s p e c t i v e l y , a r e w i d e l y d i v e r g e n t , r e n d e r s t h e o b t a i n e d 37 s i g n i f i c a n c e a t t h e .01 l e v e l s u b j e c t t o r e s e r v a t i o n . I n h i s d i s c u s s i o n of p r o c e d u r e s i n cases w h i c h show such a d e p a r t u r e from t h e assumptions o f th e a n a l y s i s o f v a r i a n c e model, L i n d q u i s t s u g g e s t s t h a t a n o b t a i n e d F - r a t i o s i g n i f i c a n t a t the .01 l e v e l s h o u l d be r e g a r d e d as i n d i c a t i n g an a c t u a l s i g -n i f i c a n c e a t t h e .02 l e v e l . 1 W i t h t h i s a l l o w a n c e , t h e r e s u l t of t h e t h i r d a n a l y s i s can be r e g a r d e d w i t h c o n f i d e n c e i n i t s s t a t i s t i c a l s i g n i f i c a n c e a t b e t t e r t h a n t h e .05 l e v e l . The n u l l h y p o t h e s i s was a c c o r d i n g l y r e j e c t e d , and t h e c o n c l u s i o n adopted t h a t t h e r e i s a s i g n i f i c a n t r e l a t i o n s h i p between con-s e r v a t i o n s t a g e c l a s s i f i c a t i o n and a r i t h m e t i c achievement. T a b l e IV summarizes the r e s u l t s o f th e a n a l y s e s . TABLE IV SUMMARY OF ANALYSES OF VARIANCE N u l l H y p o t h e s i s Sum o f Squares Between W i t h i n T o t a l d f F " l a - " I 3 1328.22 21402.71 22730.93 2, 153 4.75** *II 1 - < " l l 2 - < < ' l l 3 844.50 4738.50 5583.00 2, 41 3.65* ^ I I 2 = ^ 1 1 3 2115.56 20615.36 22730.92 4, 151 3 .87** **p<.01 *p<.05 I n s p e c t i o n o f t h e group means i n d i c a t e d t h a t t h e exp e c t e d s u p e r i o r i t y of 113 over I I ] _ , and of I3 o v e r I I 3 had l E . F . L i n d q u i s t , D e s i g n and A n a l y s i s of Experiments I n P s y c h o l o g y and E d u c a t i o n , B o s t o n : H o u g h t o n - M i f f l i n Company, 1953. pp. 8 3 - 8 6 . 38 not m a t e r i a l i z e d , t h e d i f f e r e n c e s b e i n g i n t h e wrong d i r e c -t i o n i n b o t h c a s e s . Where d i f f e r e n c e s between group achievement means o c c u r r e d i n t h e e x p e c t e d d i r e c t i o n , one-t a i l e d t - t e s t s of t h e s i g n i f i c a n c e of t h e s e d i f f e r e n c e s were a p p l i e d . T a b l e V p r e s e n t s the r e s u l t s of t h e s e t e s t s . TABLE V SIGNIFICANCE OF DIFFERENCES BETWEEN GROUP MEANS IN ARITHMETIC ACHIEVEMENT H y p o t h e s i s Observed D i f -f e r e n c e d f t tabled (p <-o5) H0: M I 3 = ^ l 2 v s . H i : ^ I 3 ^ I 2 5.28 153 2 . 2 9 1 . 9 8 t Ho: M~LZ= ^ I i v s . H i : M I2>M1\ 2 . 3 6 153 1 . 0 3 1 .98* Ho: ^13= ^ I i v s . H i : M I f M I i 7.64 153 2 . 9 9 1 .98* Ho: ^ I l 3 = ^ I I l v s . H i : MTlfMTl\ 1 1 . 9 2 41 2 . 4 9 1 .68 H 0: ^ I 2 = ^ I l 2 v s . H i : ^ I 2 > ^ I I 2 4 . 3 3 151 1 .37 1 .65 t i n o r d e r t o m a i n t a i n a n o v e r a l l s i g n i f i c a n c e l e v e l of . 0 5 . t h e t a b l e d v a l u e f o r p<.025 when df= 153 was adopted, s i n c e t h e s e t h r e e t e s t s were not independent. The H y p o t h e s i z e d R i s e i n C o n s e r v a t i o n T e s t S c o r e s The mean c o n s e r v a t i o n s c o r e f o r I t r o s e from .19 a t t h e f i r s t t e s t i n g t o 2 . 1 9 a t t h e Ja n u a r y r e - t e s t i n g . T h i s i n c r e a s e was found t o be s i g n i f i c a n t a t t h e p< . 0 0 5 l e v e l (df= 4 3 ) , u s i n g a o n e - t a i l e d t e s t f o r c o r r e l a t e d samples. 39 V. A POSTERIORI STATISTICAL TESTS Comparison of I^ and I I 3 Since, contrary to expectation, the mean achievement score for I I 3 exceeded that of I 3 , i t was decided to apply the Scheffe s t a t i s t i c for a posteriori determination of the significance of mean differences. The obtained F, .60, f e l l far below the F^ value, namely 9*72.2 Accordingly, the d i f -ference in mean achievement between I 3 and I I 3 was judged to be non-slgnlfleant. Proportions of I 3 and I I 3 With Above-Medlan Scores Since the data indicated a more marked superiority of I 3 and I I 3 than was expected, a test of the significance of the proportion of these groups scoring above the median was of interest. Having only seven subjects, II3 was too small for application of the z-test. However, six of these did obtain above-median scores. The proportion of I3 obtaining above-median scores was .69, which was significantly d i f -ferent at the .01 level from .50, the proportion expected under the null hypothesis. (z= 2.47) Proportions of 1 3 and I I 2 obtaining below-median scores were .46 and .65. respectively. These proportions do not di f f e r significantly from .50. 2George A. Ferguson, St a t i s t i c a l Analysis in Psycho- logy and Education, New York: McGraw-Hill, Incorporated, I 9 6 6 , p. 296^ R e l a t i o n s h i p Between C o n s e r v a t i o n Stage and K i n d e r g a r t e n  A t t e n d a n c e A c h l - s q u a r e t e s t of the r e l a t i o n s h i p between k i n d e r -g a r t e n a t t e n d a n c e and c o n s e r v a t i o n s t a g e c l a s s i f i c a t i o n r e v e a l e d no s i g n i f i c a n t r e l a t i o n s h i p . (X= 6.433, df =2) R e l a t i o n s h i p of Age and Stage C l a s s i f i c a t i o n The mean ages i n months f o r I ] _ , I 2 and 13 were 65, 66 and 65 r e s p e c t i v e l y , as of January 1967. T h e r e f o r e , no r e l a -t i o n s h i p between age and c o n s e r v a t i o n s t a g e was e v i d e n t . Mean D i f f e r e n c e I n Achievement Between Counters and Non- c o u n t e r s I n s p e c t i o n of achievement s c o r e s of the 21 c h i l d r e n who counted f o r two o r more items on t h e c o n s e r v a t i o n t e s t r e v e a l e d a l a r g e number of h i g h s c o r e s . T h e i r mean s c o r e was found t o be 41.5. A t w o - t a i l e d t - t e s t of t h e d i f f e r e n c e between t h i s mean and 3^.7, the mean s c o r e f o r the remainder of the s u b j e c t s y i e l d e d t= 2.44, which i s s i g n i f i c a n t a t the .05 l e v e l (df= 154). V I . SUMMARY OF RESULTS 1. The mean s c o r e o b t a i n e d by 1^ on C-Test I I s i g n i -f i c a n t l y exceeded t h e i r mean s c o r e on C-Test I . 2. A s t a t i s t i c a l l y s i g n i f i c a n t p r o p o r t i o n of H i , but not of 1^, o b t a i n e d achievement s c o r e s below the group median. A s i g n i f i c a n t p r o p o r t i o n of 13 o b t a i n e d above-median achievement s c o r e s , w h i l e p r o p o r t i o n s of H i and I I 2 d i d n o t s i g n i f i c a n t l y d i f f e r from .50. 41 3. The comparisons o f I2 and I3 y i e l d e d no s i g n i -d i f f e r e n c e i n mean achievement between 1^ and 13, but showed the e x p e c t e d s u p e r i o r i t y of 13 over t h e o t h e r two gro u p s . S i m i l a r l y , t h e comparisons of H i , I I 2 and I I3 showed no s i g -n i f i c a n t d i f f e r e n c e i n mean achievement between 11^ and Il2» but y i e l d e d t h e e x p e c t e d s u p e r i o r i t y of I I 3 over t h e o t h e r two g r o u p s . Comparisons of I 3 and I2 w i t h I I 3 and I I 2 , r e s p e c t i v e l y , f a i l e d t o r e v e a l t h e ex p e c t e d s u p e r i o r i t y of t h e f o r m e r groups i n a r i t h m e t i c achievement. 4. No s i g n i f i c a n t r e l a t i o n s h i p s were e v i d e n t between k i n d e r g a r t e n a t t e n d a n c e o r age and c o n s e r v a t i o n s t a g e . 5. A s i g n i f i c a n t d i f f e r e n c e i n mean achievement was found between t h e group of s u b j e c t s who counted f o r two o r more c o n s e r v a t i o n items and the remainder of t h e s u b j e c t group. The d i f f e r e n c e was i n f a v o u r o f t h e s u b j e c t s who used c o u n t i n g . CHAPTER V I IMPLICATIONS OF RESULTS I . CONCLUSIONS I t was t h e purpose o f t h i s s t u d y t o p r o v i d e e v i d e n c e as t o t h e v a l i d i t y of P i a g e t * s t h e o r y of t h r e e - s t a g e d e v e l o p -ment i n number c o n s e r v a t i o n , and t o i n v e s t i g a t e t h e r e l a t i o n s h i p between Grade 1 c h i l d r e n ' s development i n c o n s e r v a t i o n of number and t h e i r t e r m i n a l achievement i n a r i t h m e t i c . W i t h r e s p e c t t o t h e sample I n v e s t i g a t e d , t h e s e c o n c l u s i o n s f o l l o w from t h e r e s u l t s : 1. S u b s t a n t i a l numbers of c h i l d r e n i n Stage 1 of con-s e r v a t i o n development demonstrate t h e more advanced s t a g e s of development a f t e r a time l a p s e of f o u r months. 2. F a i l u r e t o d e v e l o p beyond Stage 1 i n c o n s e r v a t i o n by t h e time of t h e onset of f o r m a l i n s t r u c t i o n i n a r i t h m e t i c i s not s i g n i f i c a n t l y a s s o c i a t e d w i t h low a r i t h m e t i c achievement a t t h e end of t h e Grade 1 y e a r . However, low t e r m i n a l achievement i s c h a r a c t e r i s t i c o f p u p i l s who a r e s t i l l i n Stage 1 a t the b e g i n n i n g of t h e second term. 3. W h i l e t h e r e a r e no s i g n i f i c a n t d i f f e r e n c e s i n achievement among Stage 1 and Stage 2 groups as s e l e c t e d by e i t h e r t h e f i r s t o r second c o n s e r v a t i o n t e s t s , c h i l d r e n who have r e a c h e d Stage 3 by t h e b e g i n n i n g o f t h e i r Grade 1 y e a r o r by t h e b e g i n n i n g of the second term show a c h i e v e -ment s u p e r i o r t o t h a t of Stage 1 o r Stage 2 groups. 4 . E a r l y Stage 2 c o n s e r v a t i o n development i s not a s s o c i a t e d w i t h any s i g n i f i c a n t advantage i n achievement over l a t e Stage 2 development. E a r l y Stage 3 development i s n o t a s s o c i a t e d w i t h g r e a t e r s u p e r i o r i t y of achievement t h a n i s l a t e r Stage 3 development. 5. N e i t h e r age n o r p r e - s c h o o l e x p e r i e n c e a r e s y s t e m a t i c a l l y a s s o c i a t e d w i t h s t a g e i n c o n s e r v a -t i o n development. 6. The use of c o u n t i n g on t h e c o n s e r v a t i o n t e s t i s a s s o c i a t e d w i t h h i g h a r i t h m e t i c a chievement. I I . THE RELATION OF THE RESULTS TO PIAGET'S THEORY C o n s e r v a t i o n as a Thr e e - s t a g e Developmental P r o c e s s The s t u d y does n ot p r o v i d e a r i g o r o u s t e s t of t h e t h e o r y t h a t c o n s e r v a t i o n d e v e l o p s t h r o u g h t h r e e d i s t i n c t s t a g e s i n i n v a r i a n t o r d e r . However, t h e f i n d i n g s a r e c o n s i s -t e n t w i t h t h a t h y p o t h e s i s . The mean c o n s e r v a t i o n s c o r e f o r 1^ r o s e as e x p e c t e d between t h e October and January t e s t i n g s . The h i g h e r mean achievement s c o r e of I 2 o v e r 1^, I I 2 and 11^ (though n ot s t a t i s t i c a l l y s i g n i f i c a n t ) , c o u p l e d w i t h t h e f a c t t h a t a s l i g h t l y g r e a t e r p r o p o r t i o n of t h e fo r m e r group 44 o b t a i n e d above-median s c o r e s s u g g e s t s t h a t t h e r e i s a r e a l d i f f e r e n c e between Stage 1 and Stage 2 even as d i s t i n g u i s h e d by t h e p r e s e n t t e s t , and t h a t Stage 2 c h i l d r e n a r e c l o s e r t o a c q u i r i n g c o n s e r v a t i o n t h a n a r e Stage 1 c h i l d r e n . ! The s i g -n i f i c a n t l y s u p e r i o r achievement of both Stage 3 groups i s f i r m e v i d e n c e of a r e a l d i s c o n t i n u i t y between Stage 2- and Stage 3 t h i n k i n g i n the c h i l d . Repeated t e s t i n g of t h e Stage 1 group a t f r e q u e n t i n t e r v a l s would be r e q u i r e d t o show t h a t a l l who r e a c h e d Stage 3 had i n f a c t passed t h r o u g h t h e second s t a g e . C o n s e r v a t i o n as t h e N e c e s s a r y C o n d i t i o n of M a t h e m a t i c a l  U n d e r s t a n d i n g The f i n d i n g t h a t a s i g n i f i c a n t p r o p o r t i o n of t h e c h i l -d r e n who were s t i l l c l a s s i f i e d as Stage 1 i n J a n u a r y f e l l below t h e group median i n a r i t h m e t i c achievement p r o v i d e s s u p p o r t f o r t h e t h e o r y t h a t c o n s e r v a t i o n i s a n e c e s s a r y condi-t i o n of m a t h e m a t i c a l u n d e r s t a n d i n g . A d d i t i o n a l s u p p o r t d e r i v e s f rom the f i n d i n g t h a t b o t h t h o s e c h i l d r e n who were i n Stage 3 i n October and t h o s e who had a t t a i n e d Stage 3 by J a n u a r y showed s i g n i f i c a n t l y g r e a t e r mean achievement t h a n t h e comparable Stage 2 and Stage 1 c h i l d r e n . Age and Stage Placement The l a c k of s y s t e m a t i c r e l a t i o n s h i p i n t h i s s t u d y between age and c o n s e r v a t i o n s t a g e i s e a s i l y a c c o u n t e d f o r by t h e f a c t t h a t t h e age-range of t h e sample encompassed o n l y a i T a b l e V I I I , Appendix B p r e s e n t s p r o p o r t i o n s of a l l groups f a l l i n g above and below t h e median. t h i r t e e n month span ( e x c l u d i n g the 7-7 y e a r - o l d ) . In the s t u d i e s of D o d w e l l 2 and Fiegenbaum3 which demonst rated the s u p e r i o r per formance of o l d e r c h i l d r e n on P i a g e t i a n t a s k s , the samples r e p r e s e n t e d age ranges of t h r e e and f o u r y e a r s , r e s p e c t i v e l y . With such samples , s y s t e m a t i c d i f f e r e n c e s between o l d e r and younger c h i l d r e n were e v i d e n t . P r e - s c h o o l E x p e r i e n c e and Stage Placement S i n c e the c h i l d r e n i n t h i s sample had a t t e n d e d p r i v a t e k i n d e r g a r t e n s and p r e - s c h o o l c l a s s e s of v a r i o u s k i n d s , the degree of u n i f o r m i t y i n the amount and k i n d of a r i t h m e t i c I n s t r u c t i o n they had r e c e i v e d i s i m p o s s i b l e t o e s t i m a t e . I t i s l i k e l y t h a t a l l c h i l d r e n who had a t t e n d e d p r e - s c h o o l c l a s -ses had been exposed t o p lanned and/or i n c i d e n t a l number e x p e r i e n c e s . The absence of r e l a t i o n s h i p between p r e - s c h o o l a t t e n d a n c e and c o n s e r v a t i o n s tage suggests t h a t such e x p e r i -ence i s not more e f f e c t i v e t h a n e x p e r i e n c e s t h a t a r e a v a i l a b l e t o the young c h i l d a t home i n f a c i l i t a t i n g number c o n s e r v a t i o n deve lopment . I I I . EDUCATIONAL SIGNIFICANCE OF RESULTS The C r i t e r i o n of Low Achievement The group median on the a r i t h m e t i c achievement t e s t f e l l a t 361 which co r responds t o a grade s c o r e of 1.8. T h i s 2 D o d w e l l , I960. 3peigenbaum, 1963« 46 i s two p o i n t s below t h e t e s t norm f o r c h i l d r e n i n t h e n i n t h month of Grade 1 . Below-median s c o r e s t h e r e f o r e f e l l below t h e grade norm, and may, on t h a t ground, be c o n s i d e r e d as r e p r e s e n t i n g i n a d e q u a t e achievement i n a r i t h m e t i c . E d u c a t i o n a l S i g n i f i c a n c e of Mean D i f f e r e n c e s The mean s c o r e s of I ^ , l£, I i , I I 3 . I I 2 a**d 1 1 ^ c o r -responded t o grade norms 1 . 9 . 1 . 8 , 1 . 7 . 2 . 1 , 1 . 6 and 1 . 6 , r e s p e c t i v e l y . The d i f f e r e n c e between t h e grade s c o r e s f o r c o n s e r v e r s ( I 3 and I I 3 ) , and n o n - c o n s e r v e r s as of J a n u a r y ( 1 1 2 and H i ) I s g r e a t enough t o w a r r a n t c o n s i d e r a t i o n from an e d u c a t i o n a l p o i n t of v i e w . The c o n s e r v e r s show average o r b e t t e r achievement, but c h i l d r e n who a r e s t i l l n o n - c o n s e r v e r s by t h e second term a r e somewhat r e t a r d e d i n a r i t h m e t i c a c hievement. N e v e r t h e l e s s , I n s p e c t i o n of i n d i v i d u a l s c o r e s i n d i -c a t e s t h a t c a u t i o n s h o u l d be used i n a p p l y i n g t h i s g e n e r a l i z a t i o n t o i n d i v i d u a l p u p i l s . The v a r i a b i l i t y i n a l l groups was l a r g e , w i t h some c o n s e r v e r s showing low a c h i e v e -ment s c o r e s , and a few n o n - c o n s e r v e r s (as of January) showing average t o h i g h achievement. There i s no r e a s o n t o suppose, of c o u r s e , t h a t c o n s e r v e r s a r e i n v a r i a b l y endowed w i t h t h e a t t e n t i v e n e s s t o i n s t r u c t i o n , good work h a b i t s and o t h e r a t t r i b u t e s w h i c h c o n t r i b u t e t o s a t i s f a c t o r y achievement. I n t h e case of h i g h - a c h i e v i n g n o n - c o n s e r v e r s , i t i s q u i t e pos-s i b l e t h a t t h e s e c h i l d r e n a c t u a l l y a c q u i r e d t h e a b i l i t y t o c o n s e r v e number w e l l b e f o r e t h e May achievement t e s t . The 47 fact remains, however, that two conservation testings failed to predict their high achievement. The predictive inadequacy of one administration of the conservation test is further demonstrated by the finding that a group ( I I 3 ) selected from those for whom low achievement would be predicted on the basis of the f i r s t test (1-^) show achievement equal to that of the group for whom high achievement was predicted. The use of a conservation test to derive firm expectations for achievement, or to select r i g i d instructional groupings is clearly not justified by the results. Conservation and Counting The place of counting in early arithmetic instruction is currently a matter of debate. Piaget found no connection between the a b i l i t y to count and the belief in the Invariant equivalence of two matched sets. Children who demonstrated f i r s t and second stage thinking in conservation problems con-tinued to do so after counting one of the sets. He concludes that the developmental process whereby one-to-one matching becomes a quantifying operation for the child is not begun by numerals as such.^ Brace and Nelson found that counting is not a reliable index of the extent to which a child has developed the "true concept of number." While they do not explicitly suggest that the order of teaching should follow the order of development, they do say, "It would appear that ^The Child's Conception of Number, p. 64. a thorough understanding of c a r d i n a l number i s necessary before the c h i l d can have f a c i l i t y w i t h o r d i n a l number and before he can r e a l l y a p preciate the s i g n i f i c a n c e of the coun-t i n g process.5 This suggests that understanding of counting i s a product of previous number understanding, r a t h e r than a primary s k i l l which c o n t r i b u t e s to subsequent mathematical understanding. The authors of Seeing Through A r i t h m e t i c p r e s c r i b e an i n s t r u c t i o n a l programme i n which the study of c a r d i n a l values of sets up to and i n c l u d i n g those w i t h t e n members, and the ordering of n a t u r a l numbers t o ten precedes the i n t r o d u c t i o n of r a t i o n a l counting. They recommend t h i s on the ground that counting cannot be a meaningful process u n t i l c a r d i n a l i t y and order are understood.6" On the other hand, Williams recommends that r o t e coun-t i n g be taught i n kindergarten, i n view of h i s f i n d i n g that mathematical achievement was c o r r e l a t e d w i t h r o t e counting a b i l i t y (r.= . 5 1 ) . i n a sample of 595 kindergarten entrants. 7 Recent a r t i c l e s by Ashlock^ and by Mastaln and Noss o f9 5Brace and Nelson, op_. c i t . , p. 132. 6Hartung, _e_t,. al., 1965, pp. 223-24. 7A. H. Wi l l i a m s , "Mathematical Concepts, S k i l l s and A b i l i t i e s of Kindergarten Entrants," The A r i t h m e t i c Teacher, 12:4, 1965, pp. 261-8. 8Robert B. Ashlock, "Planning Mathematics I n s t r u c t i o n f o r Pour and Five-Year-Olds," The A r i t h m e t i c Teacher, 1 3 : 5 . 1966, pp. 397-99 . 9Rlchard K. Mastain and Bernice C. Nossof, "Mathema-t i c s i n the Kindergarten," The A r i t h m e t i c Teacher, 1 3 : 1 . 1966, pp. 32-36 . 49 include rational counting in the l i s t of number activ i t i e s suitable for kindergarten pupils. While i t would be unwise to place much faith in the gratuitous finding that the twenty-one children who counted for two or more items of the conservation test showed superior achievement, i t suggests that complete understanding of cardinality (conservation) need not precede understanding of counting. Eighteen of these subjects were classified as non-conservers on the f i r s t t e s t . 1 ^ Either they did not count accurately, or did not think of counting early enough in the test to get more than four items correct. But the important fact may be that they did recognize the problem as one in which counting was an appropriate method for arriving at the answer. This would suggest that, although their counting s k i l l was not necessarily reliable, counting was a meaningful operation for them. In these cases, meaningful counting appears to precede the thorough grasp of the conser-vation of number, and to be an index of adequate to superior achievement in Grade 1 arithmetic. For some children, at least, i t does not seem necessary or perhaps desirable to delay instruction in counting until cardinal number is thoroughly understood, nor on the other hand to postpone experiences designed to develop cardinal number understanding un t i l counting Is mastered. . l^Numbers of subjects in each group who counted on the conservation test are given in Table XL, Appendix B. 50 IV. THE RELATION OF RATE OF DEVELOPMENT TO ACHIEVEMENT I f i t i s assumed t h a t a l l t h o s e who t e s t e d as Stage 1 i n O ctober were a t t h e same p o i n t i n t h e i r development of q u a n t i t a t i v e t h i n k i n g , t h e n t h e s e p u p i l s , whose c h r o n o l o g i c a l age d i d not d i f f e r s i g n i f i c a n t l y from t h o s e who were i n Stage 3 a t t h a t t i m e , had d e v e l o p e d somewhat more s l o w l y t h a n t h e i r Stage 3 c l a s s m a t e s . But some of them a c h i e v e d Stage 3 by J a n u a r y , l e a v i n g b e h i n d t h o s e who had been c l a s s i f i e d w i t h them. T h i s may be i n t e r p r e t e d as i n d i c a t i n g an a c c e l e r a t i o n i n c o g n i t i v e development r e l a t i v e t o t h e c h i l d ' s own p r e v i o u s r a t e o f development. P u p i l s showing t h i s a c c e l e r a t i o n have a t e r m i n a l achievement i n a r i t h m e t i c e q u a l t o t h a t of p u p i l s who were ahead l n development a t the b e g i n n i n g of t h e y e a r . These r e s u l t s s u ggest t h a t t h e r a t e of development i n q u a n t i -t a t i v e t h i n k i n g i s v a r i a b l e , and t h a t p u p i l s who undergo t h e " s p u r t " i n development e a r l y i n t h e y e a r a r e a b l e t o r e o r g a n i z e vague o r c o n f u s e d p r e v i o u s l e a r n i n g s and become e q u a l l y p r o f i c i e n t w i t h t h o s e who were o r i g i n a l l y ahead. F u r t h e r i n v e s t i g a t i o n would be r e q u i r e d t o see whether t h i s a c c e l e r a t i o n i s a r e l i a b l e phenomenon, and i f so, whether i t s l a t e o c c u r r e n c e i s a s s o c i a t e d w i t h permanent d i s a b i l i t y i n l e a r n i n g a r i t h m e t i c . V. GENERALITY OF THE STUDY The e f f e c t , i n t h i s s t u d y , of v a r y i n g e f f i c i e n c y 51 and methods among the t e a c h e r s i s unknown. However, n e a r l y a l l groups c o n s i s t e d of some p u p i l s from each of the t e a c h i n g c l a s s e s . Achievement s c o r e ranges f o r t h e c l a s s e s i n d i c a t e t h a t t h e r e was a wide v a r i a t i o n i n achievement w i t h i n each c l a s s . H Two o f t h e c l a s s e s had a d i f f e r e n t t e a c h e r d u r i n g the second term, so t h a t t h e sample r e c e i v e d i n s t r u c t i o n from a t o t a l of n i n e t e a c h e r s . I t may be assumed, t h e r e f o r e , t h a t d i f f e r e n c e s i n t e a c h i n g a b i l i t y d i d not c o n t r i b u t e s y s t e m a t i -c a l l y t o t h e f i n d i n g s w i t h r e s p e c t t o c o n s e r v a t i o n and a r i t h m e t i c achievement i n Grade 1. The g e n e r a l i t y o f t h e r e l a t i o n s h i p between c o n s e r v a -t i o n and Grade 1 a r i t h m e t i c achievement i s dependent upon t h e c o m p a r a b i l i t y o f o t h e r i n s t r u c t i o n a l programmes i n a r i t h m e t i c w i t h t h a t c o n t a i n e d i n S e e i n g Through A r i t h m e t i c JE. The r e l a t i o n s h i p might not be e v i d e n t , f o r i n s t a n c e , i n c l a s s e s r e c e i v i n g programmes wh i c h emphasize s e t t h e o r y o r geometry. However, where u n d e r s t a n d i n g of the d e c i m a l system of numera-t i o n and the b a s i c o p e r a t i o n s of a r i t h m e t i c form t h e c o r e of the mathematics programme, t h e r e l a t i o n s h i p s h o u l d h o l d , w i t h c h i l d r e n o f comparable age and c u l t u r a l background i n r e a -s o n a b l y good m e n t a l and p h y s i c a l h e a l t h . l l T a b l e s I X and X i n Appendix B show th e d i v i s i o n of the t e a c h i n g c l a s s e s a c r o s s t h e e x p e r i m e n t a l g roups, and t h e means and ranges i n a r i t h m e t i c achievement f o r t h e t e a c h i n g c l a s s e s . BIBLIOGRAPHY BIBLIOGRAPHY A s h l o c k , R o b e r t B. " P l a n n i n g Mathematics I n s t r u c t i o n f o r F o u r and F i v e - Y e a r - O l d s , " The A r i t h m e t i c Teacher, 13 (May, 1966), pp. 397-99. B r a c e , A l e c and L. D o y a l N e l s o n . "The P r e - s c h o o l C h i l d ' s Concept of Number," The A r i t h m e t i c Teacher, 12 ( F e b r u a r y , 1965), pp. 126-33. C o x f o r d , A. F. J r . " E f f e c t s of I n s t r u c t i o n on the Stage Placement of C h i l d r e n i n P i a g e t ' s S e r i a t i o n E x p e r i -ments," The A r i t h m e t i c Teacher, 11 ( J a n u a r y , 1964), PP. 4-9. D o d w e l l , P. C. " C h i l d r e n ' s U n d e r s t a n d i n g of Number and R e l a -t e d C oncepts," Canadian J o u r n a l of P s y c h o l o g y , 14, I960, pp. 191-205. . " C h i l d r e n ' s U n d e r s t a n d i n g o f Number Concepts: C h a r a c t e r i s t i c s o f an I n d i v i d u a l and of a Group T e s t , " Canadian J o u r n a l of P s y c h o l o g y , 15, 1961, pp. 29-36. . " R e l a t i o n s Between t h e U n d e r s t a n d i n g of the L o g i c of C l a s s e s and of C a r d i n a l Number i n C h i l d r e n , " Canadian  J o u r n a l of P s y c h o l o g y . 16, 1962, pp. 152-60. E l k i n d , D a v i d . "The Development of Q u a n t i t a t i v e T h i n k i n g : A S y s t e m a t i c R e p l i c a t i o n of P i a g e t ' s S t u d i e s , " J o u r n a l of  G e n e t i c P s y c h o l o g y , 98, 1961, pp. 219-2?. Feigenbaum, Kenneth D. "Task C o m p l e x i t y and IQ as V a r i a b l e s i n P i a g e t ' s Problem of C o n s e r v a t i o n , " C h i l d Development. 34, 1963, pp. 423-32. , and Howard S u l k i n . " P i a g e t ' s Problem of Conserva-t i o n o f D i s c o n t i n u o u s Q u a n t i t i e s : A Tea c h i n g E x p e r i e n c e , " J o u r n a l of G e n e t i c P s y c h o l o g y , 105, 1964, pp. 91-97. F e r g u s o n , George A. S t a t i s t i c a l A n a l y s i s I n P s y c h o l o g y and  E d u c a t i o n . T o r o n t o : M c G r a w - H i l l Book Company, 1966. F l a v e l l , John H. The Deve1opmental P s y c h o l o g y of Jean  P i a g e t . T o r o n t o : D. Van N o s t r a n d Company, 1963. Gruen, G e r a l d E. " E x p e r i e n c e s A f f e c t i n g t h e Development of Number C o n s e r v a t i o n I n C h i l d r e n , " C h i l d Development, 36:2, 1965, PP. 963-79. 53 Hartung, Maurice L., et. a l . Seeing Through Arithmetic 1: Teachers1 Edition. Toronto: W. J. Gage, 1965. . et. a l . Charting the Course in Arithmetic. Chicago: Scott, Foresman Company, I960. Hunt, J. MeV. Intelligence and Experience. New York: Ronald Press Company, 1961. Isaacs, Nathan. Some Aspects of Piaget's Work. London: National Froebel Foundation, 1961. . New Light on Children's Idea of Number. London: Ward Lock Educational Company, I965. Lindquist, E. F. Design and Analysis of Experiments In  Psychology and Education. Boston: Houghton-Mifflin Company, 1953. Lunzer, E. A. Recent Studies ln Britain Based on the Work of  Jean Piaget. London: National Foundation for Educa-tional Research in England and Wales, No. 4 , i 9 6 0 . Mastain, Richard K. and Berniee C. Nossof. "Mathematics in the Kindergarten," The Arithmetic Teacher, 13 (January, 1 9 6 6 ) , pp. 32-37. Piaget, Jean. Logic and Psychology. Manchester University Press, 1953. . The Child's Conception of Number. New York: W. W. Norton and Company, 1965. Wallach, Lise and Richard L. Sprott. "Inducing Number Conservation in Children," Child Development, 35, 1964, pp. 1057-71. Williams, A. H. "Mathematical Concepts, S k i l l s , and Abi l i t i e s of Kindergarten Entrants," The Arithmetic  Teacher, 12 (April, I 9 6 5 ) . pp. 2 6 1 - 6 8 . Wohwill, Joachim F. "A Study of the Development of Number Concept by Scalogram Analysis," Journal of Genetic  Psychology, 97. I 9 6 0 , pp. 3^5-77. , and Rowland C. Lowe. "Experimental Analysis of the Development of Conservation of Number," Child Develop- ment, 33:1, 1962, pp. 153-67. APPENDIX A CONSERVATION TEST ITEMS Figure 2. C-test I. Configurations produced by (a) one-to-one matching and'(b) rearrangement of one of the matched sets. 56 (a) 1. (b) 0 • • • • • • • • • • • • • 2. i i i i i i ^ n 3. • • • • • • • •LXE 4. mm m m m 5. WW a O Q D O O 6.' L e g e n d : ® — b l a c k H — r e d ll 1 — y e l l o w F i g u r e d . C - t e s t H . C o n f i g u r a t i o n s p r o d u c e d b y ( a ) o n e - t o - o n e m a t c h i n g a n d ( b ) r e a r r a n g e m e n t o f o n e o f t h e m a t c h e d s e t s . APPENDIX B SUPPLEMENTARY DATA 58 TABLE V I FREQUENCIES OF SCORES ON CONSERVATION TESTS Sc o r e Frequency C-Test I C-Test I I T o t a l 6 41 5 5 11 2 4 20 5 3 45 11 2 16 4 1 10 3 0 41 19 184 49 TABLE V I I TOTAL FREQUENCIES AND DISTRIBUTION AMONG GROUPS OF RAW SCORES ON THE ARITHMETIC ACHIEVEMENT TEST Sc o r e *3 12 I I I I 3 112 H i T o t a l 55-63 5 2 7 46-54 10 14 (4)* 2 2 28 37-4-5 10 20 (12) 4 5 3 42 28-36 13 11 (16) 1 7 8 40 19-27 4 15 (5) 3 2 24 10-18 7 (7) 2 5 14 0-9 1 1 T o t a l 42 70 (44) 7 17 20 156 * S l n c e I i i s composed of I I 3 , n 2 and I I I , f i g u r e s l n b r a c k e t s a r e n o t i n c l u d e d i n summing t o t h e r i g h t . 59 TABLE V I I I PROPORTIONS OF EXPERIMENTAL GROUPS FALLING ABOVE AND BELOW MEDIAN ACHIEVEMENT Group N P r o p o r t i o n Above Median P r o p o r t i o n Below Median 42 .69 • 31 12 70 . 5 ^ .46 II 44 .41 • 59 Sub- I I 3 7 6 out of 7 (.86) 1 out of 7 (.14) groups Il£ 17 • 35 .65 o f I i H i 20 .30 .70 TABLE IX COMPOSITION OF EXPERIMENTAL GROUPS BY SCHOOL CLASS MEMBERSHIP S c h o o l C l a s s 13 12 H 113 112 H i T o t a l A 1 4 9 (2) 2 15 2 9 (10) 1 7 2 19 B 3 4 13 (8) 1 2 5 25 4 8 8 (10) 2 5 3 26 C 5 10 9 (5) 1 1 3 24 6 5 10 (3) 1 1 1 18 7 11 12 (6) 1 1 4 29 T o t a l 42 70 (44) 7 17 20 156 TABLE X RANGES AND MEAN ACHIEVEMENT SCORES OF SCHOOL CLASSES 60 S c h o o l C l a s s N Range Mean A 1 15 2k - 53 40.30 2 19 Ik - 51 31.58 B 3 25 10 - 5k 31.16 4 26 10 - 5k 32.92 C 5 24 26 - 54 38.54 6 18 8 - 62 33.78 7 29 12 - 61 40.79 NUMBERS OF SUBJECTS TABLE X I IN EACH GROUP WHO COUNTED FOR TWO OR MORE CONSERVATION ITEMS I3 I 2 I i I I 3 I I 2 H i Counters 3 12 6 3 1 0 Non-counters 39 58 38 4 16 20 

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