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8602964 Akdag, Fusun Sem iha THE EFFECTS OF COMPUTER PROGRAMMING ON YOUNG CHILDREN'S LEARNING The Ohio Slate University Ph.D. 1985 University Microfilms International 300 N Zeeb Road, Ann Arbor, Ml 48106 Copyright 1985 by Akdag, Fusun Semiha All Rights Reserved

THE EFFECTS OF COMPUTER PROGRAMMING ON YOUNG C H IL D R E N 'S LEARNING DISSERTATION P re s e n te d i n P a r t i a l F u l f i l l m e n t o f t h e R e q u ir e m e n ts f o r th e D e g re e D o c t o r o f P h ilo s o p h y i n t h e G ra d u a te S c h o o l o f The O h io S t a t e U n i v e r s i t y By F u s u n S e m ih a A k d a g, B. A., M. A. «i * t T he O h io S t a t e U n i v e r s i t y 1985 R e a d in g C o m m itte e : A p p ro v e d B y D r. S u za n n e K. D a m a rin D r. R ic h a r d D. H o w e ll D r. E l i s a L. K le in Lk r t'flju L M D r. L o r r e n L. S t u l l Adv i s o r D e p a rtm e n t o f E d u c a t io n a l T h e o ry a n d P r a c t ic e

C o p y r ig h t by F u sun S em iha Akdag 1985

To Those Who Gave Me S u p p o rt

ACKNOWLEDGEMENTS I e x p re s s s in c e r e a p p r e c ia t io n t o e v e ry o n e who h e lp e d me th ro u g h t h i s s tu d y and made i t p o s s ib le f o r me to c o m p le te my d o c t o r a l p ro g ra m. To D r. Suzanne K. D a m a rin, my a d v is o r, I am e s p e c ia lly g r a t e f u l f o r a l l h e r a d v ic e, g u id a n c e, e n th u s ia s m and s u p p o rt th ro u g h o u t t h i s s tu d y and a l l my d e g re e p ro g ra m. She has c o n t r ib u t e d t o b o th my p r o fe s s io n a l and p e rs o n a l d e v e lo p m e n t. I am s in c e r e ly t h a n k f u l to D r. L o r re n S t u l l, who a ls o a d v is e d me, 3 u p p ^i te d me and g u id e d me a l l th ro u g h my d e g re e p ro g ra m and my r e s e a r c h. S p e c ia l th a n k s t o D r. R ic h a rd H o w e ll f o r h is s u p p o rt, g u id a n c e and id e a s th ro u g h t h i s s tu d y. I am a p p r e c ia t iv e o f D r. E lis a K le in f o r h e r a d v ic e and a s s is ta n c e i n r e v ie w in g and c r i t i q u i n g t h i s docu m e n t. A p p r e c ia tio n i s e x te n d e d t o th e c o o p e r a tio n and a s s is ta n c e o f th e a d m in is t r a t io n and f a c u lt y o f C lin t o n E le m e n ta ry S chool and I n d ia n S p rin g s E le m e n ta ry S chool o f C olum bus, O h io. I am In d e b te d t o th e p r in c ip a l s o f b o th s c h o o ls M r. I r v i n and M rs. Balcom b who w e re v e ry h e lp f u l and s u p p o r t iv e th ro u g h o u t t h i s s tu d y. S p e c ia l th a n k s t o M rs, K ir k, M rs. C la r k, M rs. A rb o ro u g h and M rs. C n im ie le w s k i who a llo w e d me t o use t h e i r c la s s e s and who w e re e x tr e m e ly c o o p e r a tiv e and p a t ie n t th ro u g h o u t t h i s s tu d y. A warm th a n k yo u goes t o th e c h ild r e n who p a r t ic ip a t e d and gave me an I n s i g h t i n t h i 3 s tu d y. iii

W ith o u t them my e f f o r t s w o u ld n o t mean a n y th in g. S in c e re a p p r e c ia t io n goes to th e s t a f f o f E dgar D a le M ic ro C om puter L a b o r a to r y and 1RCC C om puter L a b o ra to ry in Ramseyer H a ll, The O hio s t a t e U n iv e r s it y. I am g r a t e f u l t o a l l o f them who p ro v id e d a s s is ta n c e. Thanks e x te n d e d t o F re d R u la n d o f th e O hio S ta te U n iv e r s it y f o r h is g r e a t h e lp i n s t a t i s t i c a l a n a ly s is o f th e d a ta and to k in d e r g a r te n te a c h e rs C a ro l P r ic e and D la n H o l l e t t f o r t h e i r tim e and s h a r in g t h e i r e x p e rie n c e s w it h me. To my f a m ily, I am g r a t e f u l f o r th e s u p p o rt and a p p r e c ia t io n o f my w o rk th e y a lw a y s com m unicated t o me a c ro s s th e m ile s.

VITA 1971-1 975 H a c e tte p e U n iv e r s it y, A n ka ra, T u rk e y. B. A. in P s y c h o lo g y, 1977-1979 B o g a z ic i U n iv e r s it y, I s t a n b u l, T u rk e y. M. A. i n P s y c h o lo g ic a l G uidance and C o u n s e lin g. G ra d u a te R esearch A s s o c ia te. 1979-1980 D ir e c t o r o f th e d a y - c a r e c e n te r a t E c z a c ib a s i. F a c to r y, I s t a n b u l, T u rk e y. 1981-1985 The O h io S ta te U n iv e r s it y, C olum bus, O hio, Ph.D. in E d u c a tio n a l T h e o ry and P r a c tic e. G ra d u a te R e se arch A s s is t a n t / G ra d u a te T e a c h in g A s s is t a n t. 1983 C u rric u lu m d e v e lo p e r and i n s t r u c t o r a t SUMMER TECH 1 983. 1984 C u rric u lu m d e v e lo p e r and i n s t r u c t o r a t SUMMER TECH 1 9 8 4. FIELDS OF STUDY M a jo r F ie ld : E a r ly C h ild h o o d E d u c a tio n M in o r F ie ld s : C h ild D e ve lo p m e n t E d u c a tio n a l T e c h n o lo g y V

TABLE OF CONTENTS Page D E D IC A TIO N...i i ACKNOWLEDGEMENTS... i i i V IT A... v L IS T OF L IS T OF TABLES...v i i i FIGURES...i x C h a p te r I. INTRODUCTION... 1 I n t r o d u c t i o n...,...1 I I. LITERATURE REVIEW... 6 P ro b le m S o l v i n g...-...6 S p a t ia l S k i l l...11 C o n c e p t F o r m a t io n...15 P ro b le m S t a t e m e n t...18 I I I. METHOD... 20 S u b je c t s 2 0 P r o c e d u r e...21 M e a s u re s... 23 P ro b le m S o l v in g...23 P i age t i an T a s k s... 23 C o n c e p t T e s t...24 Map T e s t...24 L e t t e r R e c o g n it io n...25 I n s t r u c t i o n a l T r e a tm e n ts... 26 Logo T r e a tm e n t... 27 C o m p u te r-a id e d I n s t r u c t i o n T r e a tm e n t...30 C o n t r o l T r e a tm e n t... 31

IV...RESULTS... 32 A n a ly s is o f P r e t e s t and P o s t t e s t D a ta...33 Peabody M ath R e a d in e s s T e s t... 37 P ia g e t ia n T e s t...41 C o n c e p t T e s t...44 Map T e s t... 48 Logo T e s t...51 V. DISCUSSION... 56 APPENDICES Peabody M ath R e a d in e s s T e s t... 58 P i age t i an T a s k s...5 9 C o n cept T e s t... 60 Map T e s t... 61 L im it a t io n s and S u g g e s tio n s f o r F u r t h e r R e s e a rc h...63 A. M e a s u re s... 66 REFERENCES... 86 v ii

LIST OF TABLES T able Page 1. S c h e d u le o f t e s t in g and t r e a t m e n t... 22 2. 2x3 f a c t o r i a l d e s ig n...26 3. P r e te s t tnean3 and s ta n d a rd d e v i a t io n s... 34 4. P ost t e s t means and s ta n d a rd d e v i a t io n s...34 5. A n a ly s is o f v a ria n c e f o r Peabody M ath R e a d in e s s p o s t t e s t.... 3 4 6. A n a ly s is o f v a ria n c e f o r Peabody M ath R e a d in e s s p r e t e s t... 37 7. A n a ly s is o f c o v a ria n c e f o r Peabody M ath R e a d in e s s p o s t t e s t..3 9 8. A n a ly s is o f v a ria n c e f o r P ia g e tia n p o s t t e s t... 41 9. A n a ly s is o f c o v a ria n c e f o r P ia g e tia n p o s t t e s t... 42 10. A n a ly s is o f v a ria n c e f o r C oncept p o s t t e s t...... 4 4 1 1. A n a ly s is o f v a ria n c e f o r C oncept p r e t e s t... 45 12. A n a ly s is o f c o v a ria n c e f o r C oncept p o s t t e s t...46 1 3. A n a ly s is o f v a ria n c e f o r Map p o s t t e a t...48 1 4. A n a ly s is o f c o v a ria n c e f o r Map p o s t t e s t...49 1 5. F re q u e n cy d i s t r i b u t i o n o f Logo p o s t t e s t s c o re s... 51 16. A n a ly s is o f d is k s...52 17. A n a ly s is o f c o v a ria n c e f o r Peabody M ath R e a d in e s s t e s t ( u s in g Logo t e s t s c o re s )...53 1 8. A n a ly s is o f c o v a ria n c e f o r P ia g e tia n ta s k s ( u s in g Logo t e s t s c o r e s )...53 19. A n a ly s is o f c o v a ria n c e f o r C oncept t e s t ( u s in g Logo t e s t s c o r e s )... 54 2 0. A n a ly s is o f c o v a ria n c e f o r Map t e s t ( u s in g Logo t e s t s c o r e s )... 55 v i i i

LIST OF FIGURES F ig u re Page 1. Sex by g ro u p i n t e r a c t i o n f o r Peabody p o a t t e s t...40 2. Sex by g ro u p i n t e r a c t i o n f o r P ia g e tia n p o s t t e 3 t...43 3- Sex by g ro u p i n t e r a c t i o n f o r C oncept p o a t t e s t... 47 4. Sex by g ro u p i n t e r a c t i o n f o r Map p o a t t e s t...50 ix

CHAPTER I INTRODUCTION I n t r o d u c t i o n The in t r o d u c t io n o f a m a ll and in e x p e n s iv e c o m p u te rs h a s th e p o t e n t ia l f o r a s u b s ta n tiv e im p a c t on e d u c a tio n a l th e o ry and p r a c t ic e. E x p e rts p r e d ic t t h a t m ic ro c o m p u te rs w i l l have a s i g n i f i c a n t in f lu e n c e on i n s t r u c t i o n d u r in g t h i s decade (G la s s, 1964; H a t f i e l d, 1 9 8 4 ). T h is i s s u p p o rte d by "S econd N a tio n a l S u rv e y o f I n s t r u c t io n a l Uses o f S chool C o m p u te rs " c o n d u c te d by John H o p k in s U n iv e r s it y w h ic h r e p o r t s t h a t one m i l l i o n c o m p u te rs a re now in A m e ric a n e le m e n ta ry and s e c o n d a ry s c h o o ls and f i f t e e n m i l l i o n s tu d e n ts and 5 0 0,DUG te a c h e r s, one f o u r t h o f th e n a t io n s s c h o o l te a c h e r s, used them l a s t y e a r (NCTM News B u l l e t i n, Novem ber, 1 9 8 5 ). The r e p o r t o f th e N a tio n a l C o u n c il o f T e a c h e rs o f M a th e m a tic s (NCTM, 1984) c o n fe re n c e on "The Im p a c t o f C o m p u tin g T e c h n o lo g y on S c h o o l M a th a n a tic s " p r o je c te d t h a t " a l l s tu d e n ts w i l l e x p e rie n c e a p p r o p r ia te a p p lic a t io n o f c o m p u te rs i n th e s tu d y o f each s c h o o l d i s c i p l i n e and a l l s tu d e n ts and te a c h e r s w i l l have a c c e s s to c a lc u la t o r s and c o m p u te rs f o r th e s tu d y o f m a th e m a tic s, in th e c la s s ro o m and a t heme" ( C o r b it t, 1984, p. 1 5 ). A lth o u g h th e r e i s g r e a t e n th u s ia s m f o r im p la n e n ta tio n o f m ic ro c o m p u te rs i n s c h o o ls and a lth o u g h i t has been

c la im e d t h a t th r o u g h I n t e r a c t i o n w it h c o m p u te rs c h ild r e n can p ro m o te t h e i r t h in k i n g s k i l l s su ch as p ro b le m s o lv in g and l o g i c a l t h in k i n g, t h e r e i s v e ry l i t t l e r e s e a r c h i n th e l i t e r a t u r e on th e c o g n it iv e e f f e c t s o f c o m p u te r u s e s, p a r t i c u l a r l y c o m p u te r p ro g ra m m in g by yo ung c h ild r e n. One o f th e e a r l i e s t and m o st le n g th y s t u d ie s i n t h i s a re a was s t a r t e d by Seymour P a p e rt and h i s c o lle a g u e s i n 1967- As a r e s u l t o f t h e i r s tu d y, th e c o m p u te r la n g u a g e L o g o, w h ic h ta k e s i t s name fro m th e G re e k w o rd lo g o s " m e a n in g " t h o u g h t ", was d e v e lo p e d t o c r e a te an e n v iro n m e n t i n w h ic h c h ild r e n o o u ld c o m m u n ic a te w it h c o m p u te rs. The re s e a r c h ana d e v e lo p m e n t w o rk f o r Logo by Seymour P a p e rt and h is c o lle a g u e s i s based o n tw o b e l i e f s : a ) C h ild r e n ca n l e a r n t o use c o m p u te rs i n a m a s t e r fu l w ay. b ) The way c h ild r e n le a r n t o use c o m p u te rs can e f f e c t th e way i n w h ic h th e y le a r n e v e r y t h in g e ls e ( P a p e r t, 1 9 8 0, p. 8 ). P a p e rt a s s e r t s t h a t a p p r o p r ia t e e a r ly and c o n tin u e d i n t e r a c t i o n w it h c o m p u te rs can a l t e r c o g n it iv e d e v e lo p m e n t ( P a p e r t, 1 9 8 0 ). H is w o rk d ra w s I m p lic a t io n s fro m P ia g e t 's s t u d ie s. L ik e P ia g e t, P a p e rt a s s e r t s t h a t c h i ld r e n a re b u i ld e r s o f t h e i r own i n t e l l e c t u a l s t r u c t u r e s and t h a t th e y need m a t e r ia ls t o b u ild w i t h. The Logo la n g u a g e was d e v e lo p e d s p e c i f i c a l l y f o r y o u n g c h ild r e n as a good e x a m p le o f a p p r o p r ia t e c o m p u te r-b a s e d m a t e r ia ls. F o r c h ild r e n, th e m o st s a l i e n t f e a t u r e o f Logo i s T u r t le g r a p h ic s, named o r i g i n a l l y f o r

th e t u r t l e r o b o t and m o re r e c e n t ly f o r t h e s c re e n r o b o t w h ic h ta k e s t h e fo rm o f an is o s c e le s t r i a n g l e. P a p e r t d e s c r ib e s th e t u r t l e a s an " o b j e c t - t o - t h i n k - w i t h " ( P a p e r t, 1 9 8 0, p. 1 1 ). Some im p o r t a n t f e a t u r e s o f th e t u r t l e a r e : 1) i t i s d y n a m ic, i n t h a t i t ca n move f o r w a r d and b a c k w a rd and i t can r o t a t e r i g h t a nd l e f t a b o u t i t s c e n t e r w i t h th e commands o f t h e u s e r ; 2 ) t h e t u r t l e i s u n d e r th e c o n t r o l o f th e c h i l d who d e c id e s how f a r th e t u r t l e s h o u ld m ove, w h ic h d i r e c t i o n i t s h o u ld f a c e, and w h a t f i g u r e i t s h o u ld d ra w ; 3) t h e t u r t l e i s c a p a b le o f m o v in g i n a p a r t i c u l a r d i r e c t i o n i n " t u r t l e s t e p s " and o f r o t a t i n g a b o u t i t s c e n t r a l p o i n t ; 4 ) a s t h e t u r t l e moves a c r o s s th e s c r e e n, i t c a n le a v e u t r a c k ( o r n o t, d e p e n d in g u p o n co m m a n ds). Thus t h e t u r t l e c a n dra w p i c t u r e s w it h d i f f e r e n t commands o f th e c h i l d. ( F o r a c o m p le te d e s c r i p t i o n o f L o g o la n g u a g e s e e A b e ls o n ( 1 9 8 1 ) ). T h e re a r e w id e ly d iv e r g e n t v ie w s on t h e c o g n it i v e e f f e c t s o f c o m p u te r p ro g ra m m in g i n g e n e r a l, and Logo i n p a r t i c u l a r. I n one r e c e n t s tu d y by Pea ( 1 9 8 3 ) * 8 t o 9 y e a r - o ld and 11 t o 12 y e a r - o ld s t u d e n t s who w e re t r a in e d i n L ogo f o r a o n e - y e a r p e r io d d i d n o t show a n y s i g n i f i c a n t g a in s i n p la n n in g s k i l l s o r d e c is io n m a k in g. I n a n o th e r s t u d y, C le m e n ts ( 1985) w o rk e d w it h tw o g ro u p s o f f i r s t g r a d e r s : t h e e jc p e r im e n ta l g ro u p le a r n e d L o g o p ro g ra m m in g a nd th e c o n t r o l g ro u p r e c e iv e d c o m p u te r - a id e d i n s t r u c t i o n (C A I) f o r a p e r io d o f tw e lv e w e e k s. P o s t - t e s t i n g show ed t h a t th e p ro g ra m m in g g ro u p s c o r e d s i g n i f i c a n t l y h ig h e r on tw o m e a s u re s o f c o g n it io n, r e f l e c t i v i t y and d iv e r g e n t t h i n k i n g, th a n th e o o n t r o l g ro u p. A lth o u g h t h e r e i s c o n t r o v e r s y as t o th e r e s u l t s o f m o s t s t u d ie s c i t e d, an

im p o r t a n t f a c t i s t h a t m oat o f them u sed L ogo b e ca u se th e r e s e a r c h e r s c o n s id e r e d Logo a u n iq u e p ro g ra m m in g la n g u a g e (H ow e, G shea & p la n e ( 1 9 7 9 ), G ourm an & B o u rn e ( 1 9 8 3 ), Pea ( 1 9 8 3 ), C le m e n ts ( 1 9 8 5 ) ). I t may be t h a t th e s e r e s e a r c h e r s c h o s e L ogo b e c a u s e i t b r in g s "a c h a lle n g e t o t r a d i t i o n a l fo rm s o f l e a r n i n g and i n s t r u c t i o n. And i t i s a new p o w e r fu l t o o l t h a t c a n f a c i l i t a t e a v e r y d i f f e r e n t ty p e o f l e a r n i n g th a n m o s t o f us e x p e r ie n c e d " ( C a r t e r, 1 9 8 3, p. 3 9 ). S e v e ra l s t u d ie s lo o k e d a t Logo i n th e d e v e lo p m e n t o f m a th e m a tic a l and g e o m e t r ic a l s k i l l s a nd c o n c e p ts. The E d in b u rg h Logo p r o j e c t (1 9 7 7 ) i s one o f th e e a r l i e s t r e s e a r c h s t u d ie s u s in g L o g o a s a t o o l f o r th e l e a r n i n g o f m a th e m a tic s. The r e s e a c h e r s i n t h i s p r o j e c t w o rk e d w it h 12 and 13 y e a r o ld b o y s i n a p r i v a t e s c h o o l f o r tw o y e a r s. S tu d e n ts w e re g iv e n s t a n d a r d iz e d m a th t e s t s b e fo r e and a f t e r th e p r o j e c t. The r e s u l t s o f th e s tu d y i n d i c a t e t h a t th e e x p e r im e n ta l (L o g o ) g ro u p im p ro v e d s l i g h t l y, b u t n o n - s i g n l f l c a n t l y o v e r th e c o n t r o l g ro u p o n a " b a s ic m a th s " t e s t. H o w e ver th e c o n t r o l g ro u p d id b e t t e r th a n t h e Logo g ro u p o n a t e s t o f "m a th a t t a in m e n t ". The B r o o k lin e P r o je c t ( 1 9 7 7 ), w h ic h was c o n d u c te d th r o u g h a c o l l a b o r a t i o n o f th e M IT Logo g ro u p a nd t h e B r o o k lin e P u b lic s c h o o ls in v e s t i g a t e d th e e f f e c t s o f Logo o n a g ro u p o f s i x t e e n s i x t h g ra d e s t u d e n t s i n th e a re a s o f p ro b le m s o lv in g and m a th e m a tic s, fle s e a r c h e r s r e p o r t e d t h a t o b j e c t iv e d a ta c o u ld n o t be g a th e r e d o n t h e a tu d y, and t h e d a ta o b ta in e d on t e s t s o f p ro b le m s o lv in g and m a th e m a tic s had in c o n c lu s iv e r e s u l t s ( W a tt, 1 9 8 2 ). F i n a l l y, i n a s tu d y done a t C a l i f o r n i a P o ly t e c h n ic I n s t i t u t e i n L o s A n g e le s (C r o n,

1984) Logo was used w it h t h i r t y s tu d e n ts i n K in d e r g a r te n th r o u g h s e v e n th g ra d e s. The p r e t e s t s and p o s t t e s t s in c lu d e d e s t im a t io n o f a n g le s, e s t im a t io n o f le n g t h s o f l i n e s and c u r v e s, s e g m e n t-to -w h o le r e l a t i o n s h i p s and f o l lo w in g d i r e c t i o n s on a g ra p h. R e s u lts in d ic a t e d t h a t a l l c h ild r e n u s in g Logo r e g a r d le s s o f age and g ra d e, had s i g n i f i c a n t g a in s i n le a r n in g on th e s k i l l s b e in g m e a sure d. The c u r r e n t s tu d y e xa m in e d some o f th e I m p lic a t io n s o f Logo p ro g ra m m in g on y o u n g c h i l d r e n 's c o g n it io n. A g ro u p o f k in d e r g a r t e n s tu d e n ts w e re i n s t r u c t e d on th e use o f th e Logo la n g u a g e i n tw o s e s s io n s t o t a l l i n g one h o u r e ach w e e k, o v e r a f i v e week p e r io d. The Logo commands F o rw a rd, B a ck, R ig h t, L e f t, C le a r s c r e e n, D raw, P e n c o lo r, B a c k g ro u n d c o lo r and R e p e a t w e re I n tr o d u c e d as w e l l as th e p ro c e d u re s f o r r i g h t a n g le s and a r c s i n o r d e r t o make i t e a s ie r f o r th e c h ild r e n t o draw s q u a re s and c u rv e s. A n o th e r g ro u p o f k in d e r g a r t e n s tu d e n ts was in t r o d u c e d t o some e d u c a tio n a l s o ftw a r e and a t h i r d g ro u p s e rv e d as a c o n t r o l g ro u p f o r th e s tu d y. A l l t h r e e g ro u p s w e re p r e - and p o s t t e s te d on th e s p e c i f i c c o n c e p ts : r i g h t / l e f t, r i g h t a n g le, o p e n /c lo s e d c u rv e, s t r a ig h t / c u r v e d l i n e as w e l l as on g e n e r a l m ath and sp a t lo - g e cue t r i e s k i l l s t o f i n d o u t w h e th e r th e Logo g ro u p im p ro v e d s i g n i f i c a n t l y o v e r th e o t h e r tw o g ro u p s. R e s u lts w e re a n a ly z e d u s in g a n a ly s is o f v a r ia n c e and c o v a r ia n c e and th e im p lic a t io n s o f th e re s e a r c h a r e r e p o r te d.

CHAPTER I I LITERATURE REVIEW I t has been a rg u e d t h a t Logo may e f f e c t th e d e v e lo p m e n t o f c o g n it iv e s k i l l s such as p ro b le m s o lv in g o r p la n n in g s k i l l s and m a th e m a tic s kn o w le d g e i n c h ild r e n (R o b e r ts, 1 9 8 4 ). F o r th e p u rp o s e s o f t h i s s tu d y th r e e s e ts o f c o g n it iv e s k i l l s t h a t a re im p o r ta n t in m a th e m a tic s e d u c a tio n a r e re v ie w e d ; th e s e a r e (1 ) p ro b le m s o lv in g, ( 2 ) s p a t i a l s k i l l s, and ( 3 ) c o n o e p t fo r m a tio n. P ro b le m S o lv in g : I n An Agenda f o r A c t io n (NCTM, 198C) th e g ro u p o f m ath e d u c a to rs p ro p o s e d t h a t th e d e v e lo p m e n t o f p ro b le m s o lv in g s k i l l s s h o u ld be th e fo c u s o f m a th e m a tic s e d u c a tio n t h i s d e ca d e. A re v ie w o f th e l i t e r a t u r e on m a th e m a tic a l p ro b le m s o lv in g shews t h a t t h i s te rm i s used w it h a t le a s t tw o d i f f e r e n t m e a n in g s. The m ost common m e a nin g r e f e r s t o th e s o lu t i o n o f w o rd o r v e r b a l p ro b le m s. U nder t h i s i n t e r p r e t a t i o n, a q u e s tio n l i k e " M ary had f o u r o o o k ie s ; she gave one t o J im. How many d id sh e have l e f t? " i s c o n s id e re d t o be an e xam ple o f a m a th e m a tic a l p ro b le m s o lv in g a c t i v i t y f o r c h ild r e n. The second m e a n in g o f p ro b le m s o lv in g i s f a r m ore g e n e ra l and in c lu d e s m ore th a n th e s o lu t i o n o f w ord p ro b le m s. P o ly a d e fin e s 6

p ro b le m s o lv in g a s " t h e c o n s c io u s s e a rc h f o r some a c t io n a p p r o p r ia t e t o a t t a i n some c l e a r l y c o n c e iv e d, b u t n o t im m e d ia te ly a t t a in a b le a im " (C r u is h a n k, F it z g e r a ld & J e n s e n, 1 980, p. 2 3 0 ). T h is v ie w may be a m ore u s e fu l g u id e i n th e p r e p a r a t io n o f c h ild r e n t o be e f f e c t iv e p ro b le m s o lv e r s i n e v e ry d a y l i f e. As N e ls o n and K i r k p a t r i c k (1 9 7 5 ) p o in t o u t th e m a in p u rp o s e o f "a n y i n s t r u c t i o n i n m a th e m a tic s a t th e e a r ly c h ild h o o d le v e l i s t o h e lp th e c h ild see o r d e r and m eaning i n th e s it u a t io n s and e v e n ts t h a t o c c u r i n h is d a y - to - d a y a c t i v i t i e s. Once th e c h ild le a r n s how t o r e l a t e a s i t u a t i o n t o i t s m a th e m a tic a l r e p r e s e n t a t io n, he can see new m eaning i n i t and b e g in t o e x e r o is e some o o n t r o l o v e r i t " ( p. 7 0 ). A c o o rd ln g t o c o n te m p o ra ry P ia g e tia n t h e o r i s t s ( M e lr in g, 1 9 8 0 ), p ro b le m s o lv in g i n c h ild r e n i s e f f e c t e d by tw o f a c t o r s : 1. E g o c e n tr ic t h in k i n g i n w h ic h th e c h ild * s syste m o f th o u g h t depends o n w h a t i s d i r e c t l y s e e n, h e a rd, o r o th e r w is e e x p e rie n c e d. 2. S o d a l f a c t o r s in f lu e n c in g th e change fro m s u b je c t iv e th o u g h t t o o b je c t iv e r e a l i t y. The t h in k in g a b i l i t y o f yo ung c h ild r e n i s a f f e c t e d by e g o c e n tris m. T h a t i s, c h ild r e n a re u n a b le t o ta k e i n t o a c c o u n t a n o th e r p e r s o n 's p o in t o f v ie w. They i n t e r p r e t and use w o rd s i n te rm s o f t h e i r own e x p e rie n c e s, u na w a re o f th e f a c t t h a t o th e r c h ild r e n o r a d u lt s may have had d i f f e r e n t e x p e rie n c e s and may have d i f f e r e n t p o in t s o f v ie w. C h ild r e n 's t h i n k i n g i s s e lf - c e n t e r e d ; th e y t h in k t h a t e v e ry o n e t h in k s th e same way th e y do and t h a t th e y a re " r i g h t " i n e v e r y th in g th e y

t h in k (W a d sw o rth, 1 978; B ie h le r, 1 9 7 9 ). M o re o v e r e g o c e n tr ic c h ild r e n c a n n o t d e c e n te r, m eaning th e y c a n n o t d i r e c t t h e i r a t t e n t i o n t o m ore th a n one a s p e c t o f a s i t u a t i o n a t a tim e and th e y c a n n o t re v e rs e o p e r a tio n s m e n ta lly. The e x is te n c e o f e g o c e n tris m im p lie s t h a t w it h p re s c h o o l c h ild r e n th e p ro b le m s o lv in g e m p h a s is s h o u ld be b a s ic a lly n o n v e r b a l; th e s e c h ild r e n le a r n b e s t w it h le s s a b s t r a c t and le s s fo r m a l h a n d s -o n m a t e r ia ls ( M a f f e l & B u c k le y, 1 9 8 0 ). S o c ia liz e d sp eech and th o u g h t, th e a b i l i t y t o ta k e th e v ie w s o f o th e r s i n t o a c c o u n t, d o e s n o t d e v e lo p u n t i l th e age o f seven o r e ig h t y e a r s. O nly th e n a re th e c h ild r e n a b le t o d e c e n te r and s o lv e p ro b le m s by m e n ta lly r e v e r s in g a c t io n s. The p ro b le m s yo u n g c h ild r e n a re exposed t o s h o u ld c a r r y th e f o llo w in g c h a r a c t e r is t ic s i n o r d e r to a s s u re th e c h ild r e n have e x p e rie n c e w it h th e b a s ic p ro b le m s o lv in g s k i l l s. F i r s t, th e p ro b le m s s h o u ld h ave v e ry few a b s t r a c t co m p onents. S econd, th e p ro b le m s h o u ld r e q u ir e s e v e r a l s te p s t o re a c h th e s o lu t io n. T h ir d, th e id e a l p ro b le m s s h o u ld have m u lt ip le c o r r e c t a n s w e rs. F i n a l l y, th e p ro b le m s s h o u ld r e q u ir e a n a ly s is and s y n th e s is o f in f o r m a tio n. A p p ly in g known a lg o r ith m s t o s im p le e x e r c is e s w it h o n e -s te p s o lu t io n s d oes n o t p re p a re th e c h i ld f o r co m p le x e v e ry day p r o b le m - s o lv in g s i t u a t i o n s (C ru is h a n k e t a l., 1 9 8 0 ). A o o o rd in g t o G o f f in 4 T u l l (1 9 8 5 ) th e r e a re e ig h t q u e s tio n s t h a t i l l u c l d a t e th e p rim a ry c h a r a c t e r i s t ic s o f good p ro b le m s f o r yo u n g c h ild r e n :

9 - I s th e p ro b le m i n t e r e s t i n g and m e a n in g fu l to y o u n g c h ild r e n? Does i t a llo w c h ild r e n t o a p p ly t h e i r k n o w le d g e i n new s it u a t io n s? - Can th e p ro b le m be s o lv e d a t v a r io u s l e v e l s o f c o m p le x ity? - I s th e p ro b le m e a s i l y u n d e rs to o d? - A re c h ild r e n r e q u ir e d t o make d e c is io n s? - Can c h ild r e n o b s e rv e fe e d b a c k t o t h e i r a c tio n s? - Can c h ild r e n e v a lu a t e t h e i r s o lu t io n? - Can c h ild r e n use c o n c r e te a c t io n s t o g a t h e r in f o r m a t io n and s o lv e th e p ro b le m? - A re t h e r e o p p o r t u n it ie s t o c o o p e ra te and become a w a re o f d i f f e r e n t p o in t s o f v ie w? G iv e n th e c o n d it io n s s u g g e s te d by C ru is h a n k e t a l. ( 1 9 8 0 ), Logo w o u ld a p p e a r t o be a good p r o b le m - s o lv in g t o o l. I t h a s a lm o s t no a b s t r a c t c o m p o n e n ts ; t h e r e i s l i t t l e o r no v e r b a l e m p h a s is. The p ro b le m s t r u c t u r e ( f i g u r e, d ra w in g e t c. ) i s c a r r ie d o u t a s a command o r command w h ic h ca n be e x e c u te d by p r e s s in g k e y s w h ic h i n t u r n c a u s e s th e t u r t l e to dra w a f i g u r e on th e s c re e n. S e c o n d ly, t h e r e i s m ore th a n one p o s s ib le way t o re a c h th e s o lu t i o n t o a Logo p ro b le m. G e n e r a lly s p e a k in g, th e c h ild m u st d iv id e th e p ro b le m i n t o s m a lle r s te p s w h ic h i s a good te c h n iq u e i n p ro b le m s o lv in g (P e rlm a n, 1 9 7 6 1. The p ro c e d u re s o r p ro g ra m s a r e w r i t t e n s t e j^ - b y - s t e p and e ach s te p can be seen o n th e s c re e n a s s o o n as i t i s e n te r e d. T h u s, th e c h ild can t h i n k, e s tim a te, t r y, make c o r r e c t io n s a n d, f i n a l l y, re a c h a

10 s o lu t i o n. T h i r d l y, th e r e i s no s in g le " r i g h t " s o lu t i o n t o a p ro b le m. A f i g u r e can be d ra w n i n a num ber o f w a y s ; f o r e xa m p le i f th e p ro b le m i s t o draw a s q u a re th e f o l l o w in g p r o v id e s e v e r a l s o lu t i o n s : FD 30 RT 90 FD 30 RT 90 FD 30 RT 90 FD 30 RT 90 o r REPEAT 4 [FD 30 RT 9 0 ] o r REPEAT 4 [R T 90 FD 3 0 ] F i n a l l y, t h e r e i s n o t a s ta n d a r d, know n fo r m u la t o a p p ly t o a g iv e n p ro b le m i n L o g o. A lth o u g h some o f th e b a s ic p ro c e d u re s may be used r e p e a t e d ly, an o v e r a l l p ro b le m, w h ic h can be f a r m ore c o m p le x, r e q u ir e s c l a s s i f i c a t i o n, t r i a l - a n d - e r r o r, d e d u c tio n and e s t im a t io n w h ic h a re d e fin e d as b a s ic th o u g h t p ro c e s s e s i n p ro b le m s o lv in g (S c h w e ig e r & W h e a tle y, 1 9 7 5 ). Logo a ls o f i t s th e p ro b le m s o lv in g c r i t e r i a p ro p o s e d by G o f f in & T u l l ( 1 9 8 5 ). The c h i l d d e c id e s w h a t t o c r e a t e w it h Logo and can s o lv e th e p ro b le m a t v a r io u s l e v e l s o f c o m p le x ity : he o r sh e ca n use t r i a l - a n d - e r r o r o r d e d u c tiv e r e a s o n in g ; ca n use c o n c r e te a c t io n s l i k e u s in g h i s o r h e r own body i n space t o g a th e r in f o r m a t io n i n o r d e r t o s o lv e th e p ro b le m. S in c e th e fe e d b a c k i s im m e d ia te and o b s e r v a b le, th e c h ild c a n e v a lu a t e th e s o lu t i o n. M o re o v e r, m ic ro c o m p u te rs can be u sed w it h g ro u p s o r p a i r s o f c h ild r e n as w e l l as w it h i n d i v i d u a l s and t h i s a llo w s th e c h i l d t o engage i n c o o p e r a tiv e e f f o r t s and t o v ie w

o th e r s 'a p p r o a c h e s a n d /o r s o l u t i o n s t o p ro b le m s. S n a t la l S k i l l : One o f th e m oat Im p o r ta n t g o a ls o f m a th e m a tic s e d u c a tio n i s " t h e d e v e lo p m e n t o f s t u d e n t s ' a b i l i t i e s t o t h in k and re a s o n i n n u m e r ic a l and s p a t i a l s i t u a t i o n s '' ( H a t f i e l d, 1 9 8 5, p. 2 7 ). S p a t ia l s k i l l has been d e fin e d i n a num ber o f w a ys. A c c o r d in g t o G u ilf o r d (1 9 6 7 ) i t i s th e a b i l i t y " t o m ove, t u r n, t w i s t o r r o t a t e an o b je c t o r o b je c t s and t o r e c o g n iz e a new a p p e a ra n c e o r p o s i t i o n a f t e r th e p r e s c r ib e d m a n ip u la t io n h a s been p e r fo r m e d. T h u r s to n s (1 9 5 0 ) d e f in e s s p a t i a l s k i l l as th e a b i l i t y " t o p e r c e iv e s p a t i a l p a t t e r n s a c c u r a t e ly and t o com pare them w it h each o t h e r " ( H a r r is, 1978* p. 4 0 5 ). S p a t ia l c o n c e p ts a r e c o n s tr u c te d th r o u g h i n t e r a c t i o n w it h th e p e r c e iv e d e n v iro n m e n t and a re o n ly one a s p e c t o f th e d e v e lo p m e n t o f g e n e ra l c o g n it iv e s t r u c t u r e s a c c o r d in g t o P ia g e t. Thus th e a c q u i s i t i o n o f s p a t i a l c o n c e p ts i s a p ro d u c t o f g e n e r a l i n t e l l e c t u a l d e v e lo p m e n t (S m ock, 1 9 7 6 ). S p a t ia l a b i l i t i e s, t h e i r c o n c e p t u a liz a t io n and d e v e lo p m e n t, and th e v a r io u s in f lu e n c e s o n them h a v e been s t u d ie d fro m a v a r i e t y o f p e r s p e c tiv e s (MoGee, 197 9; L ib e n, 1 9 8 1 ). I t i s g e n e r a lly a g re e d t h a t s p a t i a l a b i l i t i e s a r e im p o r ta n t b eca u se o u r k n o w le d g e a b o u t th e e n v iro n m e n t and th e w o r ld i s in f lu e n c e d by o u r p e r c e p t io n s. S e c o n d ly, many le s s o n s i n a r it h m e t ic and i n o t h e r s u b je c t s u t i l i z e p h y s ic a l m a t e r ia ls and p ic t u r e s t h a t m u st be in t e r p r e t e d and u sed i n s p e c i f i c w ays (Y o u n g, 1 9 8 2 ).

T h e re i s a d ir e c t r e l a t i o n betw een g e o m e try and s p a c e. A c c o rd in g t o W e ln zw e ig ( 1 9 7 8 ), g e o m e try i s " t h e s t r u c t u r e we im pose on space as a fra m e w o rk a b o u t w h ic h to o rg a n iz e o u r e x p e rie n c e s so t h a t we may a c c o u n t f o r them, e x p la in them and p r e d ic t w h a t w i l l happen i f (p. 1 7 3 )* C o p ela n d (197 9) d e f in e s g e o m e try as m a th e m a tic s c o n c e rn e d w it h " p o s i t i o n o r l o c a t i o n i n s p a c e " ( p. 2 5 4 ). T u r t le g e o m e try f i t s t h i s d e f i n i t i o n n ic e ly. A s p e c ia l f u n c t io n o f th e t u r t l e i s to s e rv e as a f i r s t r e p r e s e n t a t io n o f fo rm a l m a th e m a tic s f o r a c h i ld. C h ild r e n can i d e n t i f y w it h th e t u r t l e and a re th u s a b le to b r in g t h e i r kn o w le d g e a b o u t t h e i r own b o d ie s and how th e y move i n t o th e w o rk o f le a r n in g fo r m a l g e o m e try. The g o a l o f c h ild r e n 's f i r s t e x p e rie n c e s u s in g th e t u r t l e i n a le a r n in g e n v iro n m e n t i s n o t t o le a r n fo r m a l r u le s b u t t o d e v e lo p i n s ig h t s i n t o movement w i t h i n tw o -d im e n s io n a l s p a c e. I n th e Logo e n v iro n m e n t, c h ild r e n a re e n c o u ra g e d t o " p la y t u r t l e ", w h ic h in v o lv e s m o vin g t h e i r b o d ie s as th e t u r t l e on th e s c re e n m ust move i n o r d e r t o make a d e s ir e d p a t t e r n ( P a p e r t, 1 9 8 0 ). Movement t h e o r ie s e m p h a size th e id e a o f m o to r a c t i v i t i e s f o r th e a c q u is i t i o n o f s p a t ia l c o n o e p ts. H aw kins (1 9 6 4 ); R u s s e l ( 196 5 ) ; B a rs c h ( 1967) (1 9 6 8 ); D im o r d s te in (1 9 7 0 ) and B a r l i n A B a r l i n (1 9 7 1 ) a l l b e lie v e i n th e u s e fu ln e s s o f movement t o a id th e yo u n g c h ild in d e v e lo p in g s p a t ia l p e r c e p tio n and s p a t ia l c o n o e p ts (M u s ic k, 1 9 7 8 ). T u r t le g e o m e try p ro v id e s th e u s e r w it h th e c o n c r e te means t o e x p lo r e s p a t ia l g e o m e try c o rre s p o n d in g r o u g h ly t o a c h i l d 's own movement th ro u g h space (Howe, c it e d i n U p ir i s, 1981) and m o to r a c t i v i t y i s " o f

13 enorm ous im p o rta n c e f o r th e u n d e rs ta n d in g o f s p a t ia l t h in k in g " (C o p e la n d, 1 9 7 9, p. 2 7 8 ). As P ia g e t A I n h e ld e r (1 9 6 7 ) e x p la in, th e c h ild can r e c o n s t r u c t a p e r s p e c tiv e p o in t o f v ie w i f he o r she has o c c u p ie d th e p o s i t i o n t o w h ic h i t c o rre s p o n d s. F u rth e rm o re, ta s k s in v o lv in g s p a t ia l c o n c e p ts r e q u ir e c h ild r e n t o make Judgem ents u s in g r e la t io n s h ip s w h ic h m ust be c o o r d in a te d g r a d u a lly. So, " t h e p ro b le m o f le a r n in g i s e s s e n t ia lly bow t o f i n d a k in d o f 'b e s t f i t * betw een th e s t r u c t u r e o f a ta s k and th e s t r u c t u r e o f p e r s o n 's t h in k in g " (D ie n e s, 1 971, p. 2 5 ). T u r t le g e o m e try i s a p e r f e c t t o o l i n t h i s sense because t u r t l e p la y can f a c i l i t a t e a c h i l d 's a c q u i s it io n o f b a s ic c o n c e p ts su ch as r i g h t, l e f t, fo r w a r d, and b a ck. P ia g e t A I n h e ld e r (196 7) d e s c r ib e d th r e e ty p e s o f s p a c e : t o p o lo g ic a l space c o n c e rn s th e i n t e r n a l r e l a t i o n s w i t h i n and among g ro u p s o f is o la t e d o b je c t s, p r o je c t iv e space c o n c e rn s r e l a t i o n s o f o b je c t s t o th e s u b je c t and E u c lid e a n space c o n c e rn s r e la t io n s o f o b je c t s t o o b je o t s. Logo a p p e a rs t o be a good t o o l f o r d e v e lo p in g t o p o lo g ic a l o o n c e p ts suoh as n e a r, b e s id e, f a r fro m, in s id e, o u ts id e, b e tw e e n ; p r o je c t iv e c o n o e p ts suoh as a b o v e, b e lo w, t o th e l e f t o f, t o th e r i g h t o f and E u c lid e a n o o n c e p ts w h ic h a re q u a n t i t a t i v e e le m e n ts su ch as how f a r, a b o v e, b e lo w, t o th e r i g h t o f, t o th e l e f t o f and a n g le m easure. The m ovem ents o f th e t u r t l e on th e s c re e n w it h " t u r t l e s te p s " d e n o te th e s e c o n c e p ts i n a g a m e - lik e, n o n - th r e a te n in g le a r n in g s i t u a t i o n. S p a t ia l a b i l i t y has been o f p a r t i c u l a r I n t e r e s t t o p s y c h o lo g is t s because th e r e a r e r e p o r te d p e r s is t e n t se x d if f e r e n c e s

I n s p a t ia l a c t i v i t y. I n a number o f s tu d ie s a s sum m arized by V i t t i g 4 P e te rs e n C19T9) * o n ly 20> t o 25% o f fe m a le s s u rp a s s th e a v e ra g e s c o re s o f m a le s ; t h i s has been docum ented I n num erous s t u d ie s o f A r o s t o s i, 1958; M c F a rla n e 4 S m ith, 1964; T y le r 1965; M acooby, 1966; G a ra l 4 S c h e in fe ld, 1968; Macooby 4 J a c k lln, 1 9 7 4. The p a r t i c u l a r ta s k s i n w h ic h th e sex d if f e r e n c e s have a p p e a re d a re th e embedded f i g u r e t e s t, m e n ta l r o t a t io n s, g e o m e tric s k i l l s, map r e a d in g and P ia g e tia n ta s k s. I n a l l o f th e s e ta s k s yo u n g c h ild r e n do n o t e x h i b i t d if f e r e n c e s t o any la r g e e x te n t and th e a d v a n ta g e o f th e m a le s does n o t o c c u r u n t i l m id o h lld h o o d o r l a t e r ( H a r r is, 1 9 7 6 ). The d a ta s u g g e s t t h a t m e a sures o f s p a t ia l a c t i v i t y do n o t show d if f e r e n c e s betw een boys and g i r l s i n p re s c h o o l y e a rs, b u t by th e age o f e ig h t h ig h e r p e rfo rm a n c e o f m a le s b e g in s t o emerge and by a d o le s c e n c e th e s e d if f e r e n c e s become c o n s is te n t (W ille r m a n, 1 9 7 9 ). R e c e n tly, w it h th e a d v e n t o f m ic ro c o m p u te rs, th e is s u e o f g e n d e r d if f e r e n c e s has been b ro a d e n e d t o In c lu d e th e a re a o f c o m p u te rs and se x d if f e r e n c e s. R esearch has been i n i t i a t e d t o d e te rm in e i f a c h i l d ' s age and s e x I s r e la t e d t o th e c h o ic e o f th e c o m p u te r as an a c t i v i t y and th e ty p e s o f o o m p u te r a c t i v i t i e s p r e f e r r e d. I n a s tu d y done I n th e G r a n ite s c h o o l syste m ( 1 9 8 1 ), se x d if f e r e n c e s i n c o m p u te r s e le c t io n w e re r e p o r te d a t a l l g ra d e le v e ls. F in d in g s o f th e re s e a rc h done a t B a l l S ta te U n iv e r s it y ( 1983) and th o s e o f a n o th e r s tu d y by S w ig g e r, C a m p b e ll 4 S w ig g e r (1 9 8 3 ) s u g g e s t t h a t p re s c h o o l c h ild r e n have n o t le a rn e d y e t t h a t c o m p u te rs, a lo n g w it h m a th e m a tic s, a re c h a r a c te r iz e d as a m a le dom ain by some

(Williams, 1384). C oncept F fipnu^t-ioqj The te rm " c o n c e p t" i s d e fin e d i n a v a r ie t y o f w a ys. S im p ly d e fin e d, a c o n c e p t i s " t h ^ la b e l o f a s e t o f th in g s t h a t have s o m e th in g i n oommon" ( A r c h e r, 1 9 6 6, p. 3 7 ). C o n cepts a re a ls o r e f e r r e d t o as th e "means by w h ic h t h in k in g i s c a r r ie d o n " ( H a r r e, 1 9 6 6, p. 3 ) and as " th e b a s is o f fo r m a l e d u o a tio n " (H e n d e rs o n, 1 9 7 0, p. 1 6 6 ). I n th e re s e a r c h l i t e r a t u r e, t h e o r ie s c o n c e rn in g c o n c e p t le a r n in g a re v a r ie d, b u t e v e ry t h e o r i s t a c k n o w le d g e s o o n c e p t d e v e lo p m e n t as a c r i t i c a l p a r t o f th e le a r n in g p ro c e s s ; much o f th e w o rk i n w h ic h s tu d e n ts s u b s e q u e n tly engage depends upon a c le a r u n d e rs ta n d in g o f b a s ic c o n o e p ts ( B i l l i n g s, 1 9 8 3 ). A lth o u g h th e r e i s no a rg u m e n t a s t o th e im p o rta n c e o f o o n c e p t le a r n in g (H e n d e rs o n, 197 0 ; M a rk le A Tiem ann, 1970; S ow der, 1 980) d e f i n i t i o n s v a ry f o r t h r e e re a s o n s : a ) th e r e a re r e a l d if f e r e n c e s in th e n a tu r e o f o o n c e p ts w i t h i n and a c ro s s d i s c i p lin e s b ) th e num ber o f e n t i t l e s c a lle d c o n c e p ts i s v e ry la r g e and c ) an i n d i v i d u a l 's o o n c e p t o f th e same t h in g o r c la s s e s o f th in g s changes re m a rk a b ly w ith in c r e a s in g le a r n in g and m a tu r a tio n (K la u s m le r, G h a ta la A P ra y e r, 1 9 7 4 ). B ru n e r, Goodnow A A u s tin (1 9 5 6 ) I n d ic a t e t h a t c o n c e p tu a liz in g h e lp s th e i n d i v i d u a l in th e f o l lo w in g w a ys: 1. r e d u c in g th e c o m p le x ity o f th e e n v iro n m e n t 2. i d e n t i f y i n g th e o b je c t s o f th e r e a l w o r ld a b o u t them 3. r e d u c in g th e n e c e s s ity o f c o n s ta n t le a r n in g

16 4. p r o v id in g th e d i r e c t i o n f o r in s t r u m e n t a l a c t i v i t y 5. o r d e r in g and r e l a t i n g c la s s e s o f e v e n ts Gagne (1 9 7 7 ) o u t lin e d a le a r n in g h ie r a r c h y t o d e s c r ib e w h a t th e i n d iv id u a l a t any p a r t i c u l a r p o in t i n tim e i s a b le t o le a r n. The h ie r a r c h y shows th e r e l a t i o n b etw e e n o o n o e p ts and p ro b le m s o lv in g. I n t h i s h ie r a r c h y th e c a p a b i l i t i e s a r e c u m u la tiv e and i n t e r n a l. They a re c u m u la tiv e i n th e sense t h a t c o n o e p ts a re le a r n e d, rem em bered and th e n used f o r s u b s e q u e n t h ig h e r - o r d e r le a r n in g. The a b i l i t y t o le a r n any p a r t i c u l a r c o n c e p t i s d e te rm in e d by a n o th e r p r e r e q u is it e c a p a b i l i t y, t h a t o f le a r n in g d is c r im in a t io n s. F o r e xam p le suppose t h a t th e o o n c e p t t o be le a rn e d i s " s t r a i g h t l i n e. " The f i r s t s ta g e o f le a r n in g b e g in s w it h th e d is c r im in a t io n o f a s t r a i g h t l i n e fro m a c u rv e d l i n e ; th e le a r n e r f i r s t re s p o n d s a f f i n n a t i v e l y t o a p ic t u r e o f a s t r a i g h t l i n e and n e g a t iv e ly t o p ic t u r e o f a c u rv e d l i n e. I n th e second s ta g e th e same s t r a i g h t l i n e i s in tr o d u c e d w it h v a r io u s c u rv e s and th e le a r n e r in d ic a t e s w h ic h i s s t r a i g h t. P r a c tic e c o n tin u e s u n t i l th e le a r n e r c o n s is t e n t ly d is c r im in a t e s s t r a i g h t fro m n o t s t r a i g h t. T hus, th e r e i s a g e n e r a liz a t io n o f th e o r i g i n a l d is c r im in a t io n le a r n in g t o o th e r s e t t in g s. I n th e t h i r d s ta g e, th e r e i s a v a r i a t i o n i n th e i r r e l e v a n t d im e n s io n s o f s t r a ig h t and n o r ^ s t r a ig h t l i n e s. A f t e r t h i s s ta g e has been m a s te re d, th e r e i s p r o o f t h a t th e le a r n e r has a c q u ir e d th e c o n c e p t o f l i n e s t r a ig h t n e s s. A c c o rd in g t o Gagne & W h ite (197 6) "a m a th e m a tic a l c o n c e p t i s an a b s t r a c t id e a t h a t e n a b le s a s tu d e n t t o c la s s i f y o b je c t s o r e v e n ts

1? and t o s p e c if y w h e th e r th e s p e c i f i c o b je c t s o r e v e n ts a r e e x a m p le s o r n o n e x a m p ie s o f th e a b s t r a c t id e a * ( B i l l i n g s, 1 9 8 3, p. 1 8 ). F e h r ( 1966) s t a t e s t h a t th e a c q u i s i t i o n o f b a s ic c o n c e p ts i s a n e c e s s a ry and d e s ir e d outcom e o f m a th e m a tic a l i n s t r u c t i o n b u t t h e r e a r e th r e e f u r t h e r o u tc o m e s t h a t s h o u ld r e s u l t fro m th e le a r n in g o f b a s ic c o n c e p ts : 1. S k i l l i n m a n ip u la t io n, m e a n in g s k i l l i n n u m e r ic a l c a l c u l a t i o n and i n r e c o g n it io n o f m a th e m a tic a l r e l a t i o n s. 2. A p p lic a t io n o f th e o o n c e p ts t o p ro b le m s o lv in g. 3. E s t h e t ic s a t i s f a c t i o n t h a t d e r iv e s fro m th e k n o w le d g e and use o f m a th e m a tic a l c o n o e p ts. Logo a p p e a rs t o be a u s e f u l t o o l t h a t may c o n t r i b u t e t o th e s e o u tco m e s i n yo ung c h ild r e n. The c h i l d ca n p r a c t ic e w it h n um bers, c o o r d in a te s and g e o m e trio r e l a t i o n s h i p s, b e co m in g a w a re o f th e r e l a t i o n s w i t h i n a n o n - t h r e a t e n in g e n v iro n m e n t. T h is e n v iro n m e n t a llo w s a l o t o f t im e, p r a c t i c e and l e i s u r e t o th e c h i l d f o r fo r m in g r e l a t i o n s b e tw e e n h im s e lf and th e t u r t l e. The c h i l d i s l e f t t o be f r e e t o e x p lo r e a t h i s own pace w it h o u t b e in g " r i g h t * o r "w ro n g * o r t o le a r n i n one p r e - d e te r m ln e d w a y. As s e e n i n th e l i t e r a t u r e, p ro b le m s o lv in g, s p a t i a l s k i l l s and c o n c e p t le a r n in g a re t h r e e im p o r t a n t s k i l l s i n m a th e m a tic s e d u c a tio n and t h e r e may be g e n d e r d if f e r e n c e s r e la t e d t o th e s e Is s u e s.

P ro b le m S ta te m e n t M ic ro c o m p u te rs a re becom ing I n c r e a s in g ly p o p u la r and a c c e s s ib le i n e d u c a tio n ; th e y s h o u ld be in t e g r a t e d I n t o th e c u r r ic u lu m t o enhance th e d e v e lo p m e n t o f o o g n it iv e s k i l l s, e s p e c ia lly in young c h ild r e n. I n b o th s c h o la r ly and p r o fe s s io n a l l i t e r a t u r e, th e need f o r re s e a rc h i n t h i s a re a has been em phasized r e p e a te d ly. The p u rp o s e o f th e c u r r e n t s tu d y i s t o exam ine th e e f f e c t s o f i n s t r u c t i o n on o o m p u te r p ro g ra m m in g (L o g o ) o n tw o a s p e c ts o f yo u n g c h ild r e n s m a th e m a tic a l g ro w th : s p a t ia l c o n c e p ts and p ro b le m s o lv in g. I n a d d it io n, p a r t i c u l a r a t t e n t i o n i s p a id t o s e x d if f e r e n c e s i n th e s e a re a s. The f o llo w in g h y p o th e s e s fo rm th e b a s is o f th e s tu d y : H y p o th e s is 1: The s tu d e n ts w i l l m a s te r th e f o l lo w in g Logo c o n c e p ts and com m ands: FORWARD, BACK, RIGHT, LEFT, BACKGROUND COLOR, PENCOLOR, DRAW, CLEARSCREBI and REPEAT; m a s te ry w i l l be d e m o n s tra te d b o th i n th e s c o re s on th e Logo p o s t t e s t and i n a n a ly s e s o f c h ild r e n s 1 d is k s. H y p o th e s is 2 : P rogram m ing i n th e Logo la n g u a g e i s m ore e f f e c t i v e th a n c o m p u te r-b a s e d i n s t r u c t i o n i n th e d e v e lo p m e n t o f s p e c i f i c g e o m e tric c o n c e p ts : r i g h t a n g le, l e f t / r i g h t, o p e n /c lo s e d c u rv e, s t r a ig h t / c u r v e d

l i n e and p ro b le m s o lv in g s k i l l s. The r e l a t i v e e f f e c t iv e n e s s w i l l be d e m o n s tra te d th ro u g h a n a ly s is o f In c re a s e i n s c o re s fro m p r e t e s t t o p o s t t e s t. h y p o th e s is 3; T h e re w i l l be no s i g n i f i c a n t d if f e r e n c e s betw een se xes w it h re s p e c t t o t h e i r s c o re s on th e p r e - t e s t o r th e p o s t - t e s t.

CHAPTER H I METHOD The s tu d y used a p r o - and p o s t t e s t e x p e r im e n ta l d e s ig n t o d e m o n s tra te th e e ff e c t iv e n e s s o f Logo tr e a tm e n t on k in d e r g a r t e n c h ild r e n. S u b je c ts The s u b je c t s w e re e i g h t y - f o u r k in d e r g a r t e n s tu d e n ts fro m tw o s c h o o ls i n an in t e g r a t e d u rb a n p u b lic s c h o o l s yste m w it h mean age o f 6.3 y e a r s. The e x p e rim e n ta l g ro u p had f i f t y - s i x k in d e r g a r t e n s tu d e n ts fro m m o rn in g and a fte r n o o n s e c tio n s o f one s c h o o l. T w e n ty - e ig h t boya and t w e n t y - e ig h t g i r l s fro m b o th s e c tio n s w e re ra n d o m ly a s s ig n e d t o each o f th e tw o tre a tm e n t g ro u p s. The c o n t r o l g ro u p s u b je c t s w e re f i f t e e n b oys and f i f t e e n g i r l s fro m a n o th e r p u b lic s c h o o l m atched w it h th e e x p e rim e n ta l s c h o o l on a o c io -e c o n o m ic and o th e r r e le v a n t v a r ia b le s. S in c e th e p u b lic s c h o o l syste m doe3 n o t r e q u ir e e n tra n c e o r IQ t e s t s f o r a d m is s io n, th e r e w e re n o t any s ta n d a r d iz e d m easures t o use i n c o m p a rin g th e c h ild r e n. R e s u lts o f th e in t e r v ie w s w it h c la s s ro o m te a c h e r s and s tu d e n ts in d ic a t e d t h a t 20