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1 INFORMATION TO USERS Tliis reproduction was made from a copy o f a docum ent sent to us for microfilming. While tin: m ost advanced technology has been used to photograph and reproduce this docum ent, the quality o f the reproduction is heavily dependent upon the quality o f the m aterial subm itted. The following explanation o f techniques is provided to help clarify markings or notations; w hich m ay appear on this reproduction. 1.T hc sign or target for pages apparently lacking from the docum ent photographed is Missing Pagc(s)". If it was possible to obtain the missing pagt (s) o r section, they arc spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure com plete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication o f either blurred copy because o f movement during exposure, duplicate copy, o r copyrighted materials that should not have been filmed. For blurred pages, a good image o f the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent fram e. 3. When a m ap, drawing or chart, etc., is part o f the material being photographed, a definite m ethod o f sectioning" the material has been followed. It is custom ary to begin filming at the upper left hand com er o f a large sheet and to con inuc from left to right in equal sections with small overlaps. If necessary, sect oning is continued again-beginning below the first row and continuing on until com plete. 4. F or illustrations th at cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints arc available upon request from the Dissertations Custom er Services Department. 5. Sou e pages in any docum ent may have indistinct print. In all cases the best avai able copy has been filmed. University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 43106

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3 Dziak, N ancy Jo PROGRAMMING COMPUTER GRAPHICS AND THE DEVELOPMENT OF CONCEPTS IN GEOMETRY The Ohio State University Ph.D. 188j> University Microfilms International 300 N. Zeeb Road, Ann Arbor, Ml Copyright 1985 by Dziak, Nancy Jo All Rights Reserved

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5 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this docum ent have been identified here with a check mark -J. 1. Glossy photographs or pag es 2. Colored Illustrations, paper or print 3. Photographs with dark background 4. Illustrations are poor copy 5. P ag es with black marks, not original copy 6. Print shows through as there is text on both sides of page 7. Indistinct, broken or small print on several pages ^ 8. Print ex ce ed s margin requirem ents 9. Tightly bound copy with print lost in spine 10. Com puter printout pages with indistinct print 11. P age(s) lacking when material received, and not available from school or author. 12. P agefs) seem to be missing in numbering only a s text follows. 13. Two pages num bered. Text follows. 14. Curling and wrinkled pag es 15. Other University Microfilms international

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7 PROGRAMMING COMPUTER GRAPHICS AND THE DEVELOPMENT OF CONCEPTS IN GEOMETRY DISSERTATION P re se n ted in P a r t i a l F u lfillm e n t o f th e Requirem ents f o r the Degree D octor of P hilosophy i n the G raduate School o f The Ohio S ta te U n iv e rsity By Nancy Jo D ziak, B.S., M.A. * * * * * The Ohio S ta te U n iv e rsity 1985 Reading Committee: Approved By Dr. Fred Damarin / I i \ I Dr. P h ilip C lark Dr. Suzanne Damarin Dr. G erald Winer A dvisor Department o f Psychology

8 C o p y rig h t by Nancy Jo Dziak 1985

9 DEDICATION To the stu d e n ts o f Mohawk M iddle School In Columbus, Ohio, and a l l those who care f o r them. ii

10 ACKNOWLEDGMENTS I wish to thank my a d v is o r, F r. Fred Damarin, fo r p ro v id in g me with many y e a rs o f p a tie n t and c a re fu l teaching* Damarin. I am a lso g r a te f u l to Dr* Suzanne Kidd Her e x p e rtis e in com puters and mathematics e d u catio n g re a tly helped t h i s d i s s e r t a t io n become p o s s ib le. Thanks to com m ittee members Dr. Gerald Winer and Dr. P h ilip C lark fo r t h e i r su g g e stio n s and t h e i r confidence in my a b i l i t i e s. I s h a ll always be In d eb te d to Dr. Delos Wickens and Dr. Carol Wickens fo r n u rtu rin g my e a rly i n te r e s t in psychology and re s e a rc h, and fo r o ffe rin g years o f generous personal su p p o rt. F in a lly, I would l ik e to thank my fam ily: Mrs. L il l ia n Dziak, B everly, R ich ard, and e sp e c ia lly my f a th e r, the l a t e P e te r D ziak, fo r t h e i r constant love and re sp e c t. iii

11 VITA November 18, 1953 Born - L akeside, Ohio 1976 B.S., Psychology, The Ohio S ta te U n iv e rs ity, Columbus, Ohio 1981 M.A., D evelopm ental P sychology, The Ohio S ta te U n iv e ris ty, Columbus, Ohio G raduate R esearch A s s o c ia te, U.S. Army G rant, I n s t r u c t i o n a l E f f e c ts on C la s s ic a l C o n d itio n in g, The Ohio S ta te U n iv e rs ity, Columbus, Ohio G raduate Teaching A s s o c ia te, In tro d u c to ry P sy ch o lo g y, The Ohio S ta te U n iv e rs ity, Columbus, Ohio G raduate R esearch A s s o c ia te, TABS-Math P r o je c t, The Ohio S ta te U n iv e rsity C o lle g e o f E d u o atio n, Columbus, Ohio Psychology C o n s u lta n t, B a tt e l l e Memorial I n s t i t u t e, Columbus, Ohio I n s t r u c t o r o f P sy ch o lo g y, U n iv e rs ity o f W isconsin C e n te r- M anitowoc, M anitowoc, W isconsin PAPER PRESENTED Damarin, P. and D ziak, N. In d iv id u a l d if f e r e n c e s in a t t r i b u t i o n a l b i a s. P ap er p re se n te d a t th e 2 4 th an n u al m eeting o f th e S o c ie ty o f M u ltiv a ria te E x p e rim e n ta l Psychology. D enver, C olorado, November, 1983 iv

12 TABLE OF CONTENTS DEDICATION ACKNOWLEDGMENTS VITA LIST OF TABLES LIST OF FIGURES Page i i i i i iv v i i i X INTRODUCTION 1 CHAPTER I. LITERATURE REVIEW The Development o f Concepts in Geometry P ia g e t - The C o n stru c tio n o f E u c lid ia n Space if if Overview if L o ca tin g a P o in t in Two Dimensions 6 C o n serv atio n o f Area 8 Measurement o f Area 9 U nderlying P r in c ip le s 10 V e r if ic a tio n 11 van H iele - L ev els o f Geometric Thought lif L evels lif Theory C urrent R esearch Programming G raphics - A pplied 21 Elem entary Geometry

13 An Example 21 U nderlying P rocesses 2k Computer Programming 26 G eneral Programming Programming and Improvement in 26 L o g ical and M athem atical A b i l i t i e s 28 Computers and Improvement in G eom etric and S p a tia l A b i l it i e s 32 Games 33 I I. HYPOTHESES AND SUMMERY OP UNDERLYING 37 PRINCIPLES T ra n sfe r o f T ra in in g 37 H ypothesis I 38 Sex D iffe re n c e s 39 H ypothesis I I 4-1 Age E f fe c ts 41 H ypothesis I I I 42 Grade E f f e c ts 42 H ypothesis IV 42 P re d ic tio n s from S ta n d ard ize d T e st S cores 43 H ypothesis V 43 I I I. METHODS 44 T e s ts 44 T est o f van H iele L evels o f G eom etric 44 Thought M astery T e st 48 S p a tia l O rie n ta tio n T est 48 S u b je c ts 48 vi

14 Procedure **9 A pparatus Design ^9 52 T reatm ent 52 S ta n d ard ize d T est Scores 56 IV. RESULTS 58 R e l i a b i l i t i e s 58 A n aly sis o f P re-t reatm ent C ontrol G roup/ 60 T reatm ent Group D iffere n ce s H ypotheses 61 H ypothesis H ypothesis I I I H ypothesis H ypothesis I I I IV H ypothesis V 6k V. DISCUSSION 82 FOOTNOTES 93 BIBLIOGRAPHY APPENDIXES A. Sample Item s from Burner (1981) 103 B. P r e t e s ts and P o s t - t e s t s 108 C. I n s t r u c t i o n a l M a te ria ls 137 vii

15 LIST OF TABLES Table Page 1. S u b je ct Breakdown by Grade, Sex, and 50 T reatm ent 2. S u b je c t Breakdown by Age Sex and T reatm ent R e l i a b i l i t i e s Geometry P r e t e s t - Age x Sex x T reatm ent ANOVA 68 5* Geometry P r e te s t - Sex x Grade x T reatm ent ANOVA Card R o ta tio n s P r e te s t - Age x Sex x T reatm ent ANOVA Card R o ta tio n s P r e te s t - Sex x Grade x T reatm ent ANOVA M astery T e st - P r e /P o s t- te s t D ifferen ce S cores i n Age x Sex x Treatm ent (TX) ANOVA M astery T e s t - P r e /P o s t- te s t D ifferen ce S cores i n Grade x Sex x Treatm ent (TX) ANOVA Geometry T est - P r e /P o s t- te s t D iffere n ce S cores i n Age x Sex x Treatm ent ANOVA 7k 11. Geometry T e st - P r e /P o s t- te s t D iffere n ce S cores i n Grade x Sex x Treatm ent (TX) ANOVA D iffe re n c e Score Means fo r the Four Groups Inv o lv ed i n th e S ig n if ic a n t Sex x T reatm ent I n t e r a c t io n (T able 11) 76 viii

16 13. Card R o ta tio n s T e st - P r e /P o s t- te s t D ifferen ce S co res in Age x Sex x T reatm ent ANOVA 77 lk» Card R o ta tio n s T e st - P r e /P o s t- te s t D iffere n ce S co res in Grade x Sex x T reatm ent ANOVA Stepw ise M u ltip le R eg ression A nalysis Using S ta n d a rd iz e d T est S cores to P re d ic t M astery T e st S cores S ig n if ic a n t C o r re la tio n s In v o lv in g S ta n d ard ize d T e s ts, Geometry T e s t, Card R o ta tio n s, and M astery T est fo r T reatm ent Group S u b je c ts Geometry T e st - P r e / P o s t - t e s t D iffere n ce S co res in an Age x Sex x T reatm ent ANOVA w ith 11 and 15 Year Old S u b je c ts Only 81 ix

17 LIST OF FIGURES Figure Page 1. Example o f i+0 x JfO g r id f o r low r e s o lu tio n Apple g ra p h ic s w ith a sample p ic tu r e. 2. Design o f the stu d y. 3* Geometry t e s t p r e te s t s c o re s fo r m ales. /+ Geometry t e s t p o s t - t e s t s c o re s f o r m ales. 3. Geometry t e s t p r e t e s t s c o re s fo r fem ales. 6. Geometry t e s t p o s t - t e s t s c o re s f o r fem ales. 7. Geometry t e s t d iffe re n c e s c o re s fo r m ales. 8. Geoemtry t e s t d iffe re n c e s c o re s fo r fem ales x

18 INTRODUCTION Computers have had an e s c a la tin g im pact on our s o c ie ty f o r the p a st th re e decades* The advent o f m icrocom puters i n 1975 brought com puter acc ess w ith in the f in a n c ia l reach o f in d iv id u a ls and many prim ary and secondary s c h o o ls. T his a r r i v a l o f the "com puter age" i s prom pting changes in school c u r r ic u la and fo rc in g re e v a lu a tio n o f what i s needed to t r a i n s tu d e n ts to be em ployable a d u lts (NCTM, 1980). Over one h a lf o f a l l the sc h o o ls i n the U nited S ta te s now have com puters. (G re e n fie ld, 198*+). Computer l i t e r a c y req u ire m e n ts a re b ein g proposed a c ro ss the co u n try, and i n s t r u c t i o n in com puter programming i s becoming common. In a d d itio n, th e re a p p ea rs to be a tre n d toward in tro d u c in g program ming to younger and younger c h ild re n. As p a r t i a l su p p o rt fo r th ese changes in th e schools, many c la im s have been made concerning th e c o g n itiv e b e n e f its o f le a rn in g to program. Most o f th e s e c la im s, however, ap pear to be g en erated more by e x p e c ta tio n and e x cite m en t th an by e m p iric a l evidence. 1

19 The push to teach com puter programming to c h ild re n i s g a in in g momentum, and we know very l i t t l e about the immediate or f a r re a c h in g e f f e c t s o f such in s tr u c tio n. M arketing p re s s u re from hardw are and softw are companies i s in c re a s in g ly a f f e c tin g d e c is io n s made by school a u th o r i ti e s. I t i s u n fo rtu n a te th a t many o f th ese d e c isio n s are bein g made w ith o u t the b e n e fit o f in p u t from research* C u rren t s tu d ie s concerning th e e f f e c ts o f computer programming i n s t r u c t i o n are th e re fo re p a rtic u la rly, tim e ly. I t i s hoped th a t r e s u l ts from such s tu d ie s w ill have a s i g n i f i c a n t im pact on the use o f com puters in the sc h o o ls and on th e development o f p ro to ty p ic c u r r ic u la. The p re s e n t study concerns in s tr u c tio n in programming g ra p h ic s. E lem entary g ra p h ic s a re norm ally in c lu d e d in in tro d u c to ry c o u rs e s, th e re fo re thousands o f s tu d e n ts a re exposed each y e a r to t h i s type o f tr a in in g, s u r p r i s in g ly, th e re has been very l i t t l e re s e a rc h done on t h i s to p ic (Pea and K urland, 198*+). As i s the case f o r computer programming i n g e n e ra l, we know l i t t l e about th e c o g n itiv e e f f e c t s o f le a r n in g to program g ra p h ic s. Does t h i s tr a in in g t r a n s f e r to o th e r task s? Are th e re sex, age, o r grade d if f e r e n c e s i n th e a c q u is itio n o r g e n e r a liz a b ility o f t h i s s k i l l?

20 The p re s e n t study I s designed to a d d ress th ese q u e stio n s. S p e c if ic a lly, t h i s re s e a rc h w ill examine some c o g n itiv e e f f e c ts o f le a r n in g to program low r e s o lu tio n g ra p h ic s, in BASIC, on th e Apple l i e computer L ines and p ic tu r e s are c re a te d on the g ra p h ic s sc re e n by programming th e com puter to p lo t sm all re c ta n g le s of c o lo r u sin g a s e r i e s o f h o riz o n ta l and v e r t i c a l coordi-. n a te s. As a r e s u l t, when s tu d e n ts le a r n to program g ra p h ic s they re c e iv e extended p r a c tic e in the use o f a c o o rd in a te system and i n v is u a liz in g t h e i r in te n d ed g ra p h ic s d is p la y. I t i s h y p o th esized th a t t h i s re p e a te d p r a c tic e in lo c a tin g p o in ts in two dim ensions, plannin g the d ir e c tio n and i n te r s e c t i o n o f l i n e s, and c o n s tru c t in g shapes on the sc re e n w ill improve th e s tu d e n ts ' a b i l i t i e s in s p a t i a l o r ie n ta tio n and elem entary geometry The follow ing review b eg in s w ith a d e s c rip tio n of th e two m ajor th e o rie s devoted to the development o f th e c h i l d 's conception o f geom etry. The subsequent d isc u s s io n w ill show how ex p erien ce w ith programming com puter g ra p h ic s may be u s e fu l in f a c i l i t a t i n g t h i s developm ent.

21 CHAPTER I LITERATURE REVIEW The Development o f Concepts in Geometry P ia g e t - The C o n stru ctio n o f E u c lid ia n Space Overview. The two m ajor works by P ia g e t concerning th e development o f geom etric and s p a t i a l th in k in g a re The C h ild 1s C onception of Space (P ia g e t and In h e ld e r, 1956) and The C h ild 's Conception o f Geometry (P ia g e t, In h e ld e r, and Szem inska, I9 6 0 ). These books a re c o n sid ere d to be a two volume s e r i e s on the same g e n e ra l to p ic ( F la v e ll, 1963)» The C h ild 's Conception o f Space d e s c rib e s th re e phases in the development o f geom etric thought; to p o lo g ic a l, p r o je c tiv e, and E u clid e a n space c o n stru c tio n s. T op o lo g ical space develops f i r s t and la y s th e fo u n d atio n f o r th e c o n s tru c tio n o f b oth p r o je c tiv e and E u c lid ia n sp ace. T o p o lo g ical r e la tio n s in c lu d e e n c lo s u re, c o n tin u ity, p ro x im ity, s e p a ra tio n, and o rd e r. The c h ild a t t h i s sta g e can rec o g n ize, fo r exam ple, t h a t two o b je c ts.

22 5 or fe a tu re s a re c lo s e to g e th e r o r f a r a p a r t. He can u n d erstan d t h a t i n a s e r i e s o f o b je c ts A, B, C, o b je c t B i s between A and C, He can d i f f e r e n t i a t e between a toy being in s id e a box o r o u tsid e a box. There i s, however, no c o n s e rv a tio n o f d ista n c e o r a n g le s, and th e re i s no d i s t i n c t i o n between s t r a ig h t and curved l i n e s. There i s a ls o no u n d e rstan d in g th a t o b je c ts are r e l a te d in a common sp a c e. Each o b je c t in a sen se has i t s own sp a c e. The a re a betw een two o b je c ts o r fe a tu re s m ight be viewed a s b elo n g in g to the space o f one of the o b je c ts, o r i t may not be co n sid ered a t a l l (see review by Smock, ). T o p o lo g ical r e l a ti o n s c o n s titu te the c h i l d 's c o n c e p tio n o f space u n t i l about the age o f a t which tim e n o tio n s about p ro je c tiv e space begin s u rfa c in g. C o n stru c tio n o f p r o je c tiv e space in v o lv e s th e a b i l i t y to r e l a t e two o r more o b je c ts in the same space. An o b je c t may be seen from a " p o in t o f view" o r p e r s p e c tiv e, o r o b je c ts may be r e la te d to one a n o th e r, a 6 in the c o n s tr u c tio n o f a s t r a i g h t l i n e. The c h ild now can i d e n t i f y n o t only n e a r and f a r r e l a ti o n s, but th o se o f r i g h t and l e f t, in fro n t o f, behind, above, and below. C o n stru c tio n o f p r o je c tiv e space b eg in s w ith i d e n tif y in g s t r a i g h t l i n e s a t about age J*. The c h i l d 's p e rs p e c tiv e ta k in g th en g ra d u a lly im proves and becomes

23 l e s s e g o c e n tric w ith the a r r i v a l o f c o n c re te o p e ra tio n s. F in a lly, a t form al o p e ra tio n s th e c h ild p o ssesse s a co o rd in a te d system o f p e rs p e c tiv e s. He o r she u n d erstan d s p a r a l l e l and p e rp e n d ic u la r l i n e s and can conserve a n g le s. Much o f th e c o n s tru c tio n o f p ro je c tiv e space develops c o n c u rre n tly w ith concepts o f E u c lid ia n sp ace. E u c lid ia n space adds th e q u a n tif ic a tio n o f d ista n c e to concepts about o b je c t r e l a t i o n s. The c h ild now can t e l l, n o t only t h a t o b je c t A i s above o r below o b je c t B, but how fa r above o r below B o b je c t A i s lo c a te d. The C h i l d s C onception Geometry i s devoted e n tir e ly to d e s c rib in g th e c o n s tru c tio n o f E u c lid ia n sp a c e,b e g in n in g w ith spontaneous measurement and p ro g re s sin g th rough c o n se rv a tio n o f le n g th and a re a, measurement o f le n g th and a re a, and lo c a tin g a p o in t i n two dim ensions. The fo llo w in g d isc u s s io n w ill be r e s t r i c t e d to a. d e s c r ip tio n o f th e P ia g e tia n ta s k s f o r c o n se rv a tio n and measurement o f a re a and lo c a tin g a p o in t in two dim en sio n s. These s k i l l s a re d ir e c tly r e la te d to th e u n d e rsta n d in g and use o f a tw o-dim ensional c o o rd in a te system. L o ca tin g a P o in t in Two D im ensions. In t h is P ia g e tia n ta s k the s u b je c t i s p re se n te d w ith two i d e n t i c a l, w h ite, s l i g h t l y tr a n s lu c e n t p ie c e s o f p aper. One sh e e t has a re d dot in the upper r i g h t q u ad ran t.

24 7 The s u b je c t i s pro v id ed w ith a r u l e r, s t i c k s, and p ie c e s o f th re a d, and asked to p u t a red dot i n e x a c tly the same sp o t on th e second sh e e t o f p a p er. Since the paper i s tr a n s lu c e n t, th e c o r r e c t resp o n se can be e a s ily v e r if ie d by p la c in g s h e e t #2 on top o f th e o r ig in a l. T his i s a p a r t i c u l a r l y i l l u s t r a t i v e ta s k ; th e sev quence o f resp o n ses i s q u ite c le a r and in fo rm a tiv e. In the e a rly s ta g e s, up to age 6, s u b je c ts a ttem p t to p lace t h e i r re d dot by v is u a l in s p e c tio n. The m a te ria ls provided fo r m easuring a re e i t h e r n o t used a t a l l o r are n o t p ro p e rly u sed. A fte r age 6 one-dim ensional measurements a re o b se rv ed. A c h ild may m easure o b liq u e ly from the c o rn e r to th e d o t, c r make one h o riz o n ta l o r one v e r t i c a l m easurem ent. C h ild re n aged 6} to 7 dem onstrate t r a n s i t i o n to tw o-dim ensional measurement. They w ill s t i l l measure o b liq u e ly from th e c o rn e r but w ill try to m a in ta in th e slo p e w hile tr a n s f e r r in g the r u l e r. A t r i a l and e r r o r d isc o v ery o f tw o-dim ensional measurement o ccu rs betw een ag es 7 to 8. The s u b je c t, a f t e r much p r a c tic e, w i l l ta k e b o th h o riz o n ta l and v e r t i c a l m easurem ents. F in a lly, a f t e r about age 8}, s u b je c ts w ill e a s ily b e g in by m easuring i n two dim ensions, and they can r e a d ily e x p la in t h e i r rea so n s fo r doing t h i s.

25 C onservation o f A rea. U nderstanding the n o tio n o f a re a re q u ire s a co n cep tu al s te p beyond lo c a tin g a p o in t i n two dim ensions. The c h ild cannot sim ply count e.g. 3 u n its a c ro ss and 4 u n its up. He o r she must think, ift o f square u n its, which r e q u ire s a c o o rd in a tio n o f two dim ensions b efo re the ta s k has even begun. There are s e v e ra l ta s k s re p o rte d in P ia g e t (I960) f o r t e s tin g c o n serv a tio n o f a re a. One in v o lv e s a toy cow and toy houses on a p lay meadow. re a rra n g in g u n it sq u a res in a f ig u r e. A nother in v o lv e s A th ir d d e a ls w ith c u ttin g and r e lo c a tin g s e c tio n s o f a p ie c e o f p a p er. The fo llo w in g i s a d is c u s s io n o f the t h i r d ta s k. I n i t i a l l y, a square p ie c e o f p a p er i s shown to the c h ild. I t i s then c u t, and th e s e c tio n s a re reassem b led. For example, the square p ap er may be cu t d ia g o n a lly and the two s e c tio n s p u t to g e th e r to form a t r ia n g le The c h ild i s then asked i f t h i s new shape has the same amount o f "space" o r "room" as the o r i g in a l. As P ia g e t p o in ts o u t, th e term inology used in ask in g q u e stio n s i s more o f a d i f f i c u l t y in a re a ta s k s than in any of the. le n g th o r d is ta n c e ta s k s. With c o n se rv a tio n o f le n g th the c h ild can use i n t u i t i v e n o tio n s o f d ista n c e tra v e le d o r " s te p s ". In a re a ta s k s, however, the young c h ild probably has no in h e re n t

26 n o tio n s about "space" o r "roomu O therw ise, the child, would most lik e l y be ab le to conserve a re a. The c o n se rv a tio n o f a re a ta s k s e l i c i t a f a m ilia r s e r i e s o f resp o n ses o fte n seen i n o th e r ty p es o f conservatio n. Up to th e age o f 6, c h ild re n re g a rd any change o f shape as a change in the amount o f a re a. In te rm e d ia te resp o n ses occur between ages 6 and 7 and in v o lv e o c c a sio n a l c o n se rv a tio n. N on-conserving resp o n se s u s u a lly occur w ith th e l a r g e s t p e rc e p tu a l a l t e r a t i o n. For exam ple, i f a re c ta n g le i s c u t in to p ie c e s and reassem bled in to a n o th e r fo u r-sid e d fig u re, an in te rm e d ia te c h ild 'w o u ld say th a t th e a re a i s the same. I f the re c ta n g le i s cu t and reassem bled in to a pyram id, however, th e a re a i s s a id to be d i f f e r e n t. emerges a f t e r age 7* O p e ra tio n a l c o n se rv a tio n u su a lly At t h i s sta g e the c h ild can conserve through many d if f e r e n t ty p es o f tra n s fo rm a tio n s, and can v e rify th e response by re v e rs in g the tra n s fo rm a tio n o r by u sin g a m iddle term to p la c e on to p o f the c u t s e c tio n s. Measurement o f Area. The use o f a m iddle term to v e rify c o n se rv a tio n resp o n ses marks the b eg in n in g o f the a b i l i t y to measure a re a. In th e ta s k f o r measurement o f a re a, th e s u b je c t i 6 asked to compare v a rio u s f ig u re s : a r ig h t t r ia n g le, a r e c ta n g le, and an i r r e g u l a r f ig u r e. The c h ild i s given sm all c u t-o u ts o f sq u a re s and t r ia n g le s and i s askod, fo r exam ple, i f the r i g h t t r i a n g l e i s the same s iz e as the i r r e g u l a r fig u re.

27 10 A b ility to measure a re a s emerges around age? Younger c h ild re n may sim ply compare the two sh ap es v is u a lly and not use any o f th e sm all c u t- o u ts. I f prom pted, however, they may use j u s t a few c u t- o u ts and then make a judgm ent. A fte r about age 7, c h ild r e n w ill superim pose the c u t-o u ts on the fig u re and count the number o f s e c tio n s. Around age 8, c h ild re n become ab le to use u n it i n te r a t i o n o f j u s t one c u t-o u t to measure the v/hole fig u re. U nderlying P r in c ip le s. In a d d itio n to th e p re v io u s ly d e sc rib e d sequencing o f to p o lo g ic a l, p r o j e c ti v e, and E u c lid ia n space c o n s tru c tio n s, th e re a re two o th e r fundam ental p r in c ip le s p a r t i c u l a r l y im p o rtan t to the development o f geom etric th o u g h t. F i r s t, P i a g e t 's d e s c rip tio n s are m ainly concerned w ith th e c h i l d 's m ental r e p re s e n ta tio n s, not p e rc e p tio n s. He s e e s the c h ild developing m ental models o f space and s p a t i a l r e l a t i o n s, models t h a t allow the c h ild to m easure le n g th, d is ta n c e, a re a, and angles and e s t a b l is h "... a p ic tu r e o f space as a kind o f a ll-e n v e lo p in g c o n ta in e r made up o f a network of s i t e s o r su b sp aces. W ithin th e c o n ta in e r are o b je c ts, the th in g s c o n ta in e d, which move from s i t e to s i t e, now occupying o r f i l l i n g a g iv en s i t e, now le a v in g i t unoccupied and em pty." ( F l a v e l l, 1963, p. 335).

28 The c o n s tru c tio n of th is s p a t i a l model i s e s s e n t i a l ly th e r e s u l t o f the c h i l d 's a c tio n s, a c tio n s in v o lv in g 11 the m an ip u latio n o f o b je c ts in sp ace. T his i s the second u n d e rly in g p r in c ip le, not only o f the c o n s tr u c tio n o f E u c lid ia n sp ace, but a ls o o f th e development o f v i r t u a l l y a l l c o g n itiv e s t r u c tu r e s (P ia g e t, 1950)* P ia g e t w rite s, "O perations are nothing b u t i n t e r i o r i z e d a c tio n s whose e f f e r e n t im pulses do not develop In to e x te rn a l movements " (195*+ p. 1*+1). P i a g e t 's c o n v ic tio n th a t a c tio n s a re fundam ental to c o g n itiv e developm ent i s f u r t h e r e lu c id a te d in h is r e f l e c t io n s on e d u ca tio n (P ia g e t, 1970). He s tro n g ly encourages te a c h e rs to allo w and f o s t e r s tu d e n ts ' le a rn in g through the s t u d e n t 's own a c tio n s. V e r if ic a tio n. Since the t r a n s l a t i o n o f P i a g e t 's o r ig in a l works on c o n se rv a tio n, huge numbers o f s tu d ie s have appeared in the l i t e r a t u r e. Although only a r e l a t i v e l y sm all p o rtio n o f th ese in v o lv e c o n se rv a tio n o f le n g th and a re a, re s e a rc h on c o n se rv a tio n o f w eig h t, number, m ass, and volume provide i n s i g h t,in to s im ila r developm ental p ro c e sse s. In a review o f the conserve* tlo n s tu d ie s, C arp en ter s t a t e s, " P ia g e t's d e s c r ip tio n o f th e developm ent o f c o n se rv a tio n has g e n e ra lly been confirm ed u sin g a g re a t v a rie ty o f ex p erim en tal p ro c e d u re s, m a te r ia ls, and ty p es o f tra n sfo rm a tio n s" ( 1976, p. 5 2 ).

29 12 A popular focus of recent research has been the testing of the ages which Piaget claims mark the beginning of conservation. Studies of conservation of length using the standard Piagetian tasks generally find conservation occurring between the ageb of 6 and 8. Divers (1970) found that children aged 7 understand transitive relations involving length. Steffe and Carey (1972) also identified transitive relations of length, with implied conservation, at the age of 7* Swada and Nelson (1976) found the b eg inning o f le n g th c o n se rv a tio n in c h ild re n 9 and 6 y e a rs o ld. Music (1978) s tr e s s e d th e im portance of m otor a c t i v i t y i n c h i l d r e n 's judgm ents about d is ta n c e. F or exam ple, young c h ild r e n judge th e d is ta n c e a c ro ss a room a s f a r th e r i f they must c a rry a sack o f o b je c ts, then i f th ey are empty handed. A lso, th e d is ta n c e i s co nsidered l e s s i f th e c h ild ru n s r a t h e r th an w alks a c ro ss the room. Shantz and Smock (1966) g e n e ra lly confirm ed P i a g e t 's ages fo r c o n se rv a tio n o f d is ta n c e and use o f a c o o rd in a te system. Not surprisingly, Piaget's descriptions of the development of spatial and geometric thought has had (at least in theory) an impact on educational practice. There is not always a good match, however, between Piagetian theory and classroom applications. Kidder (1978, 1976) implies that the original 'Piagetian tasks

30 13 f o r c o n serv a tio n o f len g th may be too lim ite d to provide an e d u c a tio n a lly u se fu l t e s t of conserving o p e ra tio n s. He found th a t, although c h ild re n from 6 to 8 can conserve le n g th w ith the two s tic k s ta s k, most 9» H> and 13 y e a r o ld c h ild re n cannot conserve through E u c lid ia n tra n sfo rm a tio n s ( s lid e s, r o ta tio n s, and r e f l e c t i o n s ). Such tran sfo rm a tio n s re q u ire more complex m ental r e p re s e n ta tio n s. C u rricu la fo r teach in g geometry to c h ild re n o fte n use P ia g e tia n theory as a b ase, even though t h i s may not be e n tir e ly a p p ro p ria te. K idder (1976) o b serv es, "A ttem pts have been made, and are bein g made, to apply P ia g e tia n theory d ir e c tly in the classroom. T his i s so even though P iag et disco u rag es such a p ractice'.' ( p.225). As p rev io u sly m entioned, P ia g et o f f e r s some g e n e ra l g u id e lin e s to e d u c a to rs. S everal of th ese are r e f le c te d i n the fo llow ing recommendation o f UNESCO, 1956 "The Teaching o f M athem atics in Secondary Schools',' "20. I t i s im portant.(a) to guide the stu d e n t in to forming h is own id e a s and d isco v erin g m athem atical r e la tio n s and p ro p e rtie s h im se lf, r a th e r than im posing ready-made a d u lt thought upon him; (b) to make sure th a t he a cq u ires o p e ra tio n a l p ro cesses and id e a s before i n t r o ducing him to form alism ; (c ) not to e n tr u s t to autom atism any o p e ra tio n s th a t are not a lre ad y a s s im ila te d." (P ia g e t, 1970,.p. 48) In reg ard to a c tu a l classroom a p p lic a tio n s, however, th e re appears to be a gap between P ia g e t's d e s c rip tio n

31 IJf of the development of geometric thinking and instruction in geometric concepts. A theory is needed that combines both formal instruction in geometry and Piagetian principles of development. Two Dutch mathematics educators, P.Mi van Hiele and D. van Hiele-Qeldof, have formulated and attempted to implement such a theory. v a n Hiele - Levels of Geometric Thought van H iele and van H iele-q eld o f (1958) s tu d ie d th e ro le o f i n t u i t i o n i n geom etry and th e development o f geometry le a rn in g i n th e c h ild. U n til re c e n tly, (W irszup, 1976)i t h e i r work was la r g e ly ig n o red in th e U nited S t a te s. F ren d en th al (1973) helped p o p u la riz e t h e i r w ritin g s i n Europe. M athem atics e d u c a to rs i n th e S o v ie t Union, however, have been I n te r e s te d i n t h e i r work sin c e I In th e e a rly 's, re s e a rc h e rs a t th e S o v ie t Academy o f P edagogical S cien ces conducted s tu d ie s t h a t l e n t v a l i d i t y to the van H iele th eo ry and developm ental l e v e ls ( S to ly a r, 1965). L e v e ls. Because o f t r a n s l a t i o n d i f f i c u l t i e s, d e s c r ip tio n s o f th e van H iele le v e ls d i f f e r somewhat depending upon th e so u rc e. D e sc rip tio n s in v a rio u s amounts o f d e t a i l can be found in Mayberry (1983), Geddes (1982), Burger (1981), and Wirszup (1976). The fo llo w in g i s tak en from B urger (1981). I t i s f e l t

32 t h a t t h i s d e s c r ip tio n I s th e c le a r e s t and most concise 15 o f th o se a v a ila b le i n E n g lish. There are 5 le v e ls ; the co n te x t o f each and th e form o f reaso n in g re q u ire d are b r i e f l y d e s c rib e d. ( In B urger (1981) the le v e ls are numbered 0 to k* In W irszup (1976) they are numbered 1 to To avoid c o n fu sio n, the Burger d e s c rip tio n s a re renum bered here from 1 to 5.) L evel 1 ( v is u a liz a tio n ) C ontext: b a sic geom etric shapes ( t r i a n g l e s, sq u a re s, r e c ta n g le s, p a ra lle lo g ra m s, rhom bi, o th e r polygons, c i r c l e s, e t c. ). Form o f Reasoning: v is u a l i d e n t i f i c a t i o n and com parison o f the shapes as a whole. At t h i s le v e l the stu d e n t le a r n s to reco g n ize and name sim ple sh apes. L evel 2 ( d e s c r ip tio n ) C ontext: p r o p e r tie s of g eo m etrical sh apes. Form o f Reasoning: in fo rm al a n a ly s is o f th e component p a r ts of shapes ( s id e s, a n g le s, d ia g o n a ls, lin e s o f symmetry, e t c. ), and a com parison o f d if f e r e n t shapes acco rd in g to t h e i r p r o p e r tie s. At t h i s l e v e l, the stu d e n t e s ta b lis h e s some n ecessary p ro p e rtie s o f g e o m e tric al sh apes.

33 1 6 l e v e l 3 ( a b s t r a c t i o n ) C o n te x t: r e l a t i o n s h i p s among the p r o p e r tie s o f shapes* Form o f R easoning: l o g ic a l p a r t i a l o rd e r in g o f th e p r o p e r tie s o f shapes le a d in g to the fo rm u la tio n o f a b s tr a c t d e f i n i ti o n s and o la s s in c lu s io n s (eg. p a ra lle lo g ra m s r e c ta n g le s s q u a re s ). At t h i s le v e l th e s u f f i c i e n c y o f p r o p e r tie s to determ ine a shape i s le a r n e d. L ev el 4- (d e d u o tio n ) C o n tex t: a g e o m e tric a l system oom plete w ith u n d e fin e d te rm s, p o s tu la te s, an u n d e rly in g l o g i c a l system, d e fin e d te rm s, and th eo rem s. Form of R easoning: d e d u c tio n o f sta te m e n ts w ith in th e c o n s t r a i n ts o f the m athem a tic a l system* At t h i s l e v e l, th e s tu d e n t l e a r n s o f th e n e c e s s ity f o r eaoh component in th e m ath em atical system and s tu d ie s th e c l a s s i c a l r e s u l t s in E u c lid ia n geom etry.

34 17 Level 5 ( r ig o r ) C ontext: v a rio u s geom etries Form o f R easoning: rig o ro u s m athem atical study w ith in s e v e ra l g eo m etries, a stu dy o f the p r o p e r tie s o f the system s o f p o s tu la te s, ( e. g. 't h e incom pleteness o f E u c lid 's p o s tu la te s, and the com pletion Theory. by H i l b e r t. ) The van H ie le s saw the development o f \ geom etric th o u g h t as a p ro g re s sio n o f d is c o n tin u itie s : "The d i s c o n ti n u i ti e s a r e... jumps in the le a rn in g curve, and th ese jumps re v e a l th e presence o f le v e ls. The le a r n in g p ro c e ss h as sto p p ed ; l a t e r on i t w ill s t a r t i t s e l f once a g a in. In th e meantime, th e p u p il seems to have 'm a tu re d ' The te a c h e r does n o t succeed i n f u r th e r e x p la n a tio n o f th e s u b je c t. He and th e o th e r s tu d e n ts who have rea ch e d th e new l e v e l seem to speak a language which cannot be u n d ersto o d by th e p u p ils who have n o t y e t reached th e new level'.' (van H ie le, 1958, p. 75)*. In re g a rd to t h e i r d is c u s s io n o f d i s c o n tin u itie s, th e van H ie le s resem ble P ia g e t. U nlike P ia g e t, however, th e van H ie le s e x p la in developm ent a s m ainly the r e s u l t o f i n s t r u c t i o n o r a " p ro c e ss o f apprenticeship'^ r a t h e r th an a consequent o f m a tu ra tio n.

35 18 The " a p p re n tic e s h ip " which le a d s to p r o g r e s s from one le v e l to th e next c o n s is ts o f fiv e p h ases. The f i r s t phase i s c a lle d " in fo rm a tio n ". The s tu d e n t le a r n s to g e n e ra lly reco g n ize th e f i e l d o f in v e s tig a tio n by exam ining the i n s t r u c t i o n a l m a te r ia ls p re se n te d to him. N ext, the s tu d e n t engages in " d ir e c te d o r ie n ta tio n ". He o r she e x p lo re s the m a te r ia ls, and the s u b je c t m atter i s p re se n te d in a manner th a t allow s the c h a r a c te r is tic s tr u c tu r e s to p ro g re s s iv e ly appear to the s tu d e n t. L a te r, in the "e x p la n a tio n " p h ase, the a c q u ire d e x p erien c es a re lin k e d to l i n g u i s t i c sym bols, and the s tu d e n t can tak e p a rt in classroom d is c u s s io n s about the to p ic in q u e stio n. During th e n ex t p hase, c a lle d " f r e e o rie n ta tio n ',' th e stu d e n t knows many o f th e methods and m a te ria ls th a t have been p re s e n te d, b u t th e se co n cep ts are not e f f e c tiv e ly o rg an iz ed. F in a lly, i n th e " in te g r a tio n " phase, the stu d e n t a c q u ire s a c o o rd in a te d overview o f th e methods a t h is o r h e r d is p o s a l (W irszup, 1976, p.83)* A c e n tr a l p r in c ip le u n d e rly in g th e c h i l d 's p ro g re ssio n through th e se phases i s t h a t i t i s b e s t fo r the c h ild to le a rn geom etry. i n a manner a k in to guided d isc o v ery. The s tr u c tu r e o f the s u b je c t m a tte r i s supposed to " p ro g re s s iv e ly appear" to th e c h ild, and not be sim ply

36 19 imposed by the teacher* T h is recom mendation o f discovery type le a rn in g i s s im ila r to P i a g e t 's g u id e lin e s fo r the e sta b lish m e n t o f te a c h in g tec h n iq u e s th a t f o s te r a c tiv e le a rn in g. C urrent R esearch. Two re c e n t NSF p r o je c ts, one a t Oregon S ta te (B u rg er, 1981) and a n o th e r a t Brooklyn College (Qeddes, 1982), have employed van H ie le s ' theory and le v e ls to a s s e s s the geom etric th in k in g o f some prim ary and secondary sch o o l s tu d e n ts here in th e U nited S ta te s. P re lim in ary fin d in g s in d ic a te th a t the theory i s u se fu l fo r such an a sse ssm e n t, and th a t in g e n e ra l, stu d e n t grade le v e l i s r e l a te d to t h e i r e stim a te d van H iele le v e l. In both p r o je c ts, s tu d e n ts were in te rv ie w e d and asked to make judgm ents about shapes based on th e sh ap es' geom etric p ro p e rtie s (see Appendix A f o r sample item s from B urger, (1981). R eports o f th e se in te rv ie w s in d ic a te th a t s tu d e n ts i n the prim ary g rad e s ten d to c a te g o riz e geom etric shapes m ainly in term s o f v is u a l c h a r a c t e r i s t i c s. Many stu d e n ts as o ld as 13 and 1J+ have d i f f i c u l t y w ith deductive rea so n in g and w ith making a p p ro p ria te c la s s in c lu s io n s in v o lv in g geom etric sh a p es. Secondary school stu d e n ts who have s tu d ie d geom etry a re b e t t e r a t d e d u ctio n, but are o fte n confused about th e r o l e s o f p o s tu la te s and theorem s. In a d d itio n, secondary sch o o l s tu d e n ts r a r e ly understand the id e a o f a m ath em atical system.

37 C r itic s o f the p re s e n t U.S. geometry curriculum suggest t h a t van H ie le 's th eo ry may be a u s e fu l to o l fo r re v is in g c u rre n t i n s t r u c t i o n (C oxford, 1978, W irszup, 1976). They com plain th a t i n most American sch o o ls today geometry i s b a re ly m entioned d u rin g the elem entary y e a rs. Then, in th e te n th g ra d e, the s tu d e n ts are p resen te d w ith a form al geom etry under the assum ption th a t they have a lre a d y p ro g re sse d through s e v e ra l low er le v e ls o f geom etric th in k in g. The goal o f a van H iele type curriculum would be to f o s t e r th e g rad u al development o f an i n i u i t i v e sen se o f geom enty, and to provide in s t r u c t i o n in r e la te d m athem atical p ro ced u res t h a t would g e n e ra liz e to o th e r a re a s o f lo g ic and m easurement. One technique th a t may h e lp f o s t e r a s tu d e n t's i n t u i t i v e sense o f geom etry would be to encourage the stu d e n t to use geometry in some a p p lie d c o n te x t. I f t h i s were a ls o a m o tiv a tin g c o n te x t, the e d u c a tio n a l g a in s could be s i g n i f i c a n t. Programming com puter g ra p h ic s i s an a p p lic a tio n o f g eom etric concepts in v o lv in g a two- dimemsional co o rd in a te system and some p r o p e r tie s o f sim ple shapes and l i n e s. Programming i s a ls o a m o tiv a tin g ex p erien ce fo r many s tu d e n ts (G re e n fie ld, 198**). In a d d itio n, the a c t iv i t y o f programming g ra p h ic s in c o rp o r a t e s many o f the im p o rtan t developm ental p r in c ip le s

38 2 1 p rev io u sly d isc u sse d. The fo llo w in g s e c tio n w ill e la b o ra te on th ese e d u c a tio n a lly and developm entally u sefu l fe a tu re s o f g ra p h ic s programming. Programming G raphics - A pplied E lem entary Geometry An Example. The Apple BASIC g ra p h ic s sc re e n, in low re s o lu tio n, i s a rra n g ed i n a 1*0 by re c ta n g le s (in fo rm a lly c a lle d s q u a r e s ). g r id of The sq u a res are numbered as in d ic a te d in F ig u re 1. P lo ttin g p ic tu r e s on the sc ree n in v o lv e s d e s ig n a tin g a p o in t o r s e t o f p o in ts to be c o lo re d. F or exam ple, "PLOT 5*10" means c o lo r in th e square th a t i s fiv e sq u a res a c ro ss and te n sq uares down. H o riz o n ta l and. v e r t i c a l lin e s can be p lo tte d p o in t by p o in t, o r more e a s i l y, by using the coms^nds HLIN and VLIN. For exam ple, "HLIN 0,39 a t 7" w m a h o r iz o n ta l lin e from square 0 to square 39 a t 7 square^ijow n on th e v e r t i c a l a x is. Now c o n sid e r some o f th e mtjytal p ro c e sse s in v o lv ed in u sin g t h i s g ra p h ic s system. Suppose a c h ild wanted to c o n s tru c t the p ic tu r e o f a house shown in F ig u re 1. He o r she would obviously need to u n d erstan d how to lo c a te a p o in t in two dim ensions. T h is r e q u ir e s the o p e ra tio n s o f c o n se rv a tio n and m easurement o f le n g th, combining th e sq u ares to make th e door and window a p a r t i c u l a r s iz e in v o lv e s c o n s e rv a tio n and measurement

39 2 2 F ig u re 1. Example o f 40 x 40 g r id f o r low r e s o l u ti o n Apple g ra p h io s w ith a sam ple p i c t u r e.

40 23 of a re a. L ocating the door In th e m iddle o f th e house re q u ire s some knowledge o f symmetry. In o rd e r to draw the square form ing the body o f th e house, the c h ild must have some n o tio n s about th e p r o p e r tie s o f a sq u are: equal le n g th s id e s th a t meet to form fo u r r ig h t a n g le s. The most d i f f i c u l t ta s k i n c o n s tru c tin g t h i s house, however, i s draw ing the ro o f. There a re th re e main fe a tu re s th a t the c h ild may d isc o v e r. F i r s t s th e ro o f w ill not be p o in te d u n le s s i t c o n ta in s an odd number o f sq u a re s. The h e ig h t o f the ro o f i s r e l a te d to the w idth o f the house. As the w idth o f th e house in c re a s e s the h e ig h t o f the ro o f in c r e a s e s. To make the ro o f, one cannot sim ply s t a r t connecting sq u a re s to form a d iag o n al on one slo p e then a r b i t r a r i l y s to p and begin draw ing th e o th e r slo p e. U nless the c h ild i s very f o r tu n a te, the ro o f w ill not be sym m etrical, o r i t may n o t e x a c tly meet both s id e s o f the house. The c h ild needs to c a lc u la te the w idth of th e house, fin d th e m id -p o in t, extend up a v e r t i c a l l i n e, and draw th e d iag o n a l u n t i l i t i n t e r s e c t s th a t l i n e. I t i s q u ite p o s s ib le, o f c o u rse, to draw th e ro o f using t r i a l and e r r o r m ethods, and t h i s i s fre q u e n tly done. A djustm ents made through t r i a l and e r r o r, however, can s t i l l be q u ite i n s t r u c t i v e. The s tu d e n t must supply

41 c o o rd in a te s fo r each square o r lin e th a t i s drawn. When a lin e i s re lo c a te d the c o o rd in a te s change, o fte n in a sy stem atic manner. For example, a h o riz o n ta l move o f one w all o f the house re q u ire s a change in a l l th e X c o o rd in a te s, but the Y c o o rd in a te s rem ain the same. Programming these k in d of changes fo rc e s the stu d e n t to examine the r e la tio n s among the s e t s of numbers used as c o o rd in a te s. U nderlying P ro cesses. There are s e v e ra l o th e r b a sic c h a r a c te r is tic s o f th is programming a c t i v i t y th a t support i t s ed u catio n al p o te n tia l. F i r s t, i t i s a c tiv e. The c h ild i s a c tiv e ly c o n stru c tin g p ic tu r e s on the screen and making and c o rre c tin g h is o r h e r own m istak es. The im portance of a c tio n to the c h i l d 's le a rn in g and development has? already been d isc u sse d. Second, i t i s re p r e s e n ta tio n a l. The c h ild can imagine in a p e rc e p tu a l sense a p ic tu re to be drawn, b u t in o rd er to program i t the stu d e n t must o v erlay t h i s network o f s p a t i a l lo c a tio n s. T hird, programming i s m o tiv a tin g. S tudents enjoy most kinds o f programming. Perhaps t h i s i s because in many ways a computer i s l i k e a te le v is io n s e t, only b e tte r because i t i s i n te r a c t iv e (G re e n fie ld, 1981*). C hildren seem to p a r tic u la r ly enjoy programming g rap h ic s. G raphics are r e l a t i v e l y easy to program, and they a re, l i t e r a l l y, c o lo r f u l.

42 25 There appear to be good reaso n s to expect th a t i n s t r u c t i o n i n programming g ra p h ic s may have some b e n e f ic ia l e f f e c t s in a d d itio n to the sim ple tra n s m iss io n o f a te c h n ic a l s k i l l. O ccasional re se a rc h on the e d u c a tio n a l advantages o f com puter programming i n s t r u c t i o n has been conducted o ver the p a st 13 y e a rs. There i s some in d ic a tio n th a t th e re a re c o g n itiv e b e n e f its o f le a r n in g to program, p a r t i c u l a r l y in the a re a o f m athem atics.

43 2 6 Computer Programming General Programming Most o f the p rev io u s re se a rc h on computer programming has not concerned the c o g n itiv e b e n e f its o f le a rn in g to program, but has focused on programming r u le s, memory o rg a n iz a tio n, and debugging s t r a t e g i e s. Brooks, (1977) g e n e ra te d a model th a t i d e n t i f i e s 101* r u le s n e ce ssa ry fo r a p a r t i c u l a r type o f programming a c t i v i t y. Green and Barstow (1978) d e sc rib e over a hundred r u le s th a t most programmers posses fo r s o r tin g and se a rc h in g a lg o rith m s. S e v e ra l s tu d ie s concern memory o rg a n iz a tio n a l d iff e re n c e s between e x p e rt and novice programmers (A delson,.1981; Reltm an, 1980; Schneiderm an, 1977). McKeithen (1981) found th a t e x p e rts used d if f e r e n t mnemonic s t r a t e g i e s f o r remembering chunks o f programming commands. E xperienced programmers organized the term s acc o rd in g to meaning o r fu n c tio n, whereas novices grouped th e term s acco rd in g to su rfa c e c h a r a c t e r i s t i c s such as word le n g th o r s p e llin g. M ental re p r e s e n ta tio n s o f program s have been examined by DuBonlay, O 'Shea and Monk (1981) and Hoc (1977). E x p ert/n o v ic e d iffe re n c e s in debugging s t r a t e g i e s have been s tu d ie d by Gould ( ), Gould and Drongowski (197*+), and Youngs (197**). Gould (1975) found th a t e x p e rts ten d to read programs fo r flow

44 27 of control, while novices read programs line by line. Richard Mayer has attempted to analyze processes involved in learning BASIC and. has offered some specific recommendations for teaching BASIC to novices (Mayer, 1979)* He has also experimented with different techniques for instruction in programming and has determined that learning and transfer, to tests of programming concepts, can be increased by providing concrete models and encouraging students to reformulate technical information in their own words (Mayer, 1975» 1976, 1981) His work on concrete models and the use of advance organizers is built on information processing models similar to those of Qreeno (1973)» Much of what is claimed about the basic cognitive processes involved in computer programming appears to be based on intuitive analyses of what programmers do. It seems as though such abilities as general Intelligence, mathematics ability, analogical reasoning, and knowing the principles of conditional logic would be important to competent programming. There is, however, a remarkable lack of research in this area. Although instruments such as the Programmer's Aptitude Test have been available since 1956* a reliable relation between these scores and performance in the workplace has not been established (Bell, 1976). The need for research

45 28 on b a sic a b i l i t i e s in v o lv ed in, and a ffe c te d by, computer programming i s evidenced by re c e n t NIE funding o f over 1 /2 m illio n d o lla r s in g ra n ts fo r the study o f "The Demands and C o g n itiv e Consequences o f Computer L earning". Programming and. Improvement in L o g ical and M athem atical A b i l i t i e s As e a r ly as 1969* F eurzeig e t a l. advocated th a t c h ild r e n be ta u g h t programming in m athem atics c la s s e s. He f e l t th a t le a r n in g to program would help the developm ent o f m athem atical concepts in fo u r b a sic ways: 1. Programming p ro v id es j u s t i f i c a t i o n fo r and i l l u s t r a t i o n o f form al m athem atical r ig o u r. 2. Programming encourages the study o f m athem atics th rough e x p lo ra tio n. 3. C h ild re n achieve in s ig h t in to c e r ta in m athem atical concepts through programming. Programming p ro v id es a language i n which c h ild re n may d e sc rib e t h e i r own problem so lv in g p ro c e s s e s. There a p p e a rs to be modest evidence f o r a l l fo u r o f th e se c la im s. Ross and Howe, in t h e i r review (1981), s t a t e t h a t, " In,te rm s o f s t a t i s t i c a l r e s u l t s the re s e a rc h o f the l a s t decade in to "m athem atics through programming" has been more encouraging than d isc o u ra g in g, but only m ild ly s o,"

46 There i s some i n d ir e c t evidence th a t le a rn in g to program im proves u n d erstan d o f m athem atical rig o u r. Howe, O 'Shea, and P lane (1980) re p o rt a sm all study of 11 y e a r o ld boys who were ta u g h t m athem atics using LOGO. A fte r two y e a rs, th e te a c h e r r a te d the stu d e n ts in the study as b e t t e r a b le to e x p re ss m athem atical id e a s and t h e i r own u n d e rsta n d in g d i f f i c u l t i e s. DuBonlay (1978) tau g h t LOGO programming to S c o ttis h te a c h e r tr a in e e s and found th a t, a f t e r the i n s t r u c t i o n, s e v e ra l o f them had in c re a s e d i n s i g h t in to the im portance o f c le a r, p re c is e e x p la n a tio n s. P a p e rt (1980) r e p o r ts th a t many c h ild re n le a rn in g LOGO adopt a sy s te m a tic, te s tin g 'a p p ro a c h to w ritin g program s. The e x p lo r a tio n o f m athem atics through programming has re c e iv e d some a t t e n t i o n. P a p e rt, in h is popular book W indstorm s ( ), c iteb numerous examples o f c h ild r e n s ' spontaneous e x p lo ra tio n s while w hile working w ith LOGO and P a p e r t 's T u rtle Geometry. Dwyer (1975) has s tu d ie d c u r r ic u la th a t s u c c e s s fu lly used BASIC as a to o l f o r th e encouragem ent o f ex p lo rin g a wide v a r ie ty o f unique problem s, such as o p e ra tin g a m echanical band-o rg an. Howe e t a l. (1979) d e sc rib e s e v e ra l exam ples i n which a c h ild w ill take advantage of m ista k e s in programming and produce a d i f f e r e n t and b e t t e r p ro d u ct th an th a t which was o r ig in a lly in te n d e d.

47 30 S ev eral s tu d ie s have dem onstrated th a t programming h elps provide i n s i g h t in to m athem atical con cep ts. F eu rzeig e t a l. (1969) re p o rt th a t a group o f seven to nine y e a r old c h ild r e n a cq u ired a "m eaningful u n d erstanding o f concepts l i k e v a ria b le and procedure a f t e r le a rn in g to debug program s. M iln er (1973) found in c re a s e d comprehension o f the n o tio n of a v a ria b le in 11 y ear o ld c h ild re n a f t e r th ey le a rn e d to w rite re c u rsiv e program s. Howe e t a l. (1979) a ls o found improved un d erstanding o f v a ria b le s i n 11 y e a r o ld boys follow ing I n s tr u c tio n in LOGO. Clem ent, Lochhead, and Soloway (1982) re p o rt t h a t c o lle g e s tu d e n ts could more e a s ily solve a lg e b ra word problem s in th e c o n te x t o f w ritin g a computer program : th a n by sim ply w ritin g an a lg e b ra ic e q u a tio n. They note t h a t, "Computer programming a p p a re n tly encourages an a c tiv e, p ro ce d u ral view of e q u a tio n s th a t many s tu d e n ts f a i l to use in the co n te x t o f a lg e b ra ". T here ap p ears to be some su p p o rt f o r the F eu rzeig claim t h a t programming p ro v id es a language f o r the d e s c r ip tio n o f problem so lv in g p ro c e ss e s. S ta tz (1973) found improvement in problem so lv in g on ta s k s re q u irin g re c u rs io n and anagram s, a f t e r s u b je c ts had re c e iv e d in s t r u c t i o n in LOGO. Seidman (1981) r e p o r ts t h a t le a rn in g c o n d itio n a l branch sta te m e n ts in LOGO a f f e c ts

48 31 s u b je c ts ' u n d e rsta n d in g o f c o n d itio n a l lo g ic. P apert and G o ld ste in (1972) encourage the te a c h in g o f the term inology o f programming i n o rd e r to provide the s tu d e n t w ith c o n cep ts about a lg o rith m s and procedures t h a t are u s e fu l in problem so lv in g a c t i v i t i e s. Pea and K urland (198^) have c r i t i c i z e d s e v e ra l o f th ese s tu d ie s which a re o fte n c ite d a s su p p o rt o f t r a n s f e r of. t r a i n in g from programming. They argue t h a t, i n the Howe e t a l. (1979) stu d y, te a c h e r e v a lu a tio n s may have been u n r e lia b le, sin c e the r a t e r s were aware o f which s tu d e n ts had re c e iv e d the LOGO t r a i n in g. The Dwyer (1975) and Howe e t a l. (1979) re s e a rc h showed an in c re a s e i n math e x p lo ra tio n u sin g b o th BASIC and LOGO, but Pea and K urland p o in t out t h a t t h i s e x p lo ra tio n was only w ith in the com puter environm ent. They f u r th e r n o te th a t p h ra se s l i k e "m eaningful u n d erstan d in g " a re o f te n not o p e ra tio n a liz e d, and th a t many o f th e re s e a rc h r e p o r ts a re m erely a n e c d o ta l. These c r i t ic i s m s a re w ell tak e n, b u t th e p rev io u s re s e a rc h fin d in g s do n o t w arran t d ism is sa l. In s te a d, th e p a s t re s e a rc h sh o u ld be viewed as p o s s ib ly in d ic a tin g a tre n d, and im proved and more form al re s e a rc h tech n iq u es sh o u ld.b e employed i n the f u tu r e. Pea and Kurland recommend s e v e ra l re s e a rc h m o d ific a tio n s, th ey w rite,

49 32 "... these studies suffer in not linking level of programming skill to specific outcomes expected, and the critical studies of 'low level' transfer expected from level I and II Celementary] programming Bkills remain to be carried out." Computers and' Improvement in Geometric and Spatial Abilities For over a decade, Seymour Papert has been using LOGO and his Turtle Geometry to teach children mathematical concepts and the "powerful ideas" encountered in programming. Papert (1980) claims that the use of Turtle graphics teaches a new kind of geometry. (The "turtle", seen as a small triangle, leaves a line trail as it moves around the computer screen.) Distance is measured in "turtle steps", and angles are made by turning the turtle right or left from 1. to 360. The child learns experimentally that, e.g. to draw an equilateral triangle each angle must be made with a turn of 120. The manner in which the turtle draws, and the ease with which right and left turns are made, makes LOGO graphics particularly useful for promoting an understanding of angles and arcs. Papert's reports are encouraging, but his research is based mainly on observations made of a relatively small number of selected children in a specialized laboratory.

50 33 Except fo r P a p e r t 's and h is a s s o c ia te s ' work w ith LOGO, th e re i s l i t t l e o th e r re s e a rc h on programming g ra p h ic s and g e o m e tric a l thin k in g * There i s, however, some in fo rm a tio n concerning th e e f f e c t s o f c e r ta in computer games and videogam es on s p a t i a l and m athem atical a b i l i t i e s * Games. The com puter g ra p h ic s sc re e n, being designed as a system o f X and Y c o o rd in a te s, c o n v en ien tly le n d s i t s e l f to games t h a t r e q u ir e the u se r to sp e c ify c o o rd in a te s. Two such games a re Harpoon and Sonar* The follow ing d e s c r ip tio n s o f both games are pro v id ed by t h e i r d eveloper, Jam es L evin (1981)*^ "Harpoon i s a com puter game ( w r itte n i n P a sc al f o r an Apple I I Computer) t h a t p re s e n ts the p la y e rs w ith a drawing o f a s h a r k 's f i n on th e computer sc re e n, w ith two p e rp e n d ic u la r l i n e s i n te r s e c t i n g over th e sh a rk. Each l i n e has i t s e n d p o in ts la b e le d w ith numbers. The program asks th e p la y e r s to s p e c ify the p o s itio n o f th e sh ark l e f t and r i g h t and th en i t s p o s itio n up and down* A fter th ey e n te r th e two num bers, a 'h a rp o o n ' f l i e s a c ro ss th e sc re e n to th e p o s itio n they have s p e c if ie d. I f th a t s p o t i s c lo se enough to th e sh a rk, th en th e harpoon h i t s th e sh a rk, and th e sh ark s in k s o u t of view. I f the harpoon m isse s, th en a 'sp la sh * occurs on the sc re e n to mark the s p o t, and th e p la y e rs can tr y again u sin g th e s p la s h mark as feedback* Sonar i s a n o th e r game program (a ls o w ritte n i n P a sc a l fo r th e Apple I I Computer) th a t te a c h e s math s k i l l s w ith in th e framework o f a game. T his game i s s im ila r to Harpoon, a s the p la y e rs have the g o al o f h i t t i n g a sh ark w ith a harpoon. But i n Sonar, the shark doesn't i n i t i a l l y appear on th e sc re e n, but i s hidden underw ater. The p l a y e r 's 'sonar* readout t e l l s where th e shark i s h id in g, g iv in g X and Y c o o rd in a te

51 numbers. The players try to move the 'crosshairs' to that spot on the screen. Then the harpoon flies to that spot and if it is close enough to the shark's position, the shark surfaces and then is harpooned. Otherwise, the harpoon splashes into the water, and the coordinate numbers of their guess are displayed as feedback," Levin (1981) r e p o r ts t h a t b oth games have been te s te d w ith 10 y e a r o ld s tu d e n ts, and th a t th e se c h ild re n found th e games ^ c h a lle n g in g and m o tiv a tin g ". The s tu d e n ts ' s k i l l s a t id e n tif y in g p o s itio n s w ith in 3*+ the co o rd in ate system developed q u ic k ly. When working in only one dimemsion, c h ild r e n reach ed p ro fic ie n c y ( a d e v ia tio n o f 1 3# on th e f i r s t guess) w ith in ;10 games. Levin su g g e sts t h a t s tu d e n ts le a r n to have an " i n t u i t i v e f e e l" fo r numbers by f r e e ly c o n v ertin g between numbers and l in e le n g th s ; In a d d itio n to th e se e d u c a tio n a l com puter games, some videogames In v o lv e s im ila r s p a t i a l s k i l l s. Sm all and Small (1982) d e sc rib e s t r a t e g i e s f o r the videogame "B attlezo n e" a s r e q u irin g m ental r o t a t io n and v is u a liz a tio n o f images in th re e dim erasions. The once p o p u lar videogame "Space In v ad ers" i s d e sc rib e d as re q u ir in g the sim ultaneous c o o rd in a tio n o f h o r iz o n ta l and v e r t i c a l axes and the a b i l i t y tp p r e d ic t th e in te r s e c tio n o f im aginary lin e s (Lowery and K nirk, 1982). Almost no form al re s e a rc h has been conducted on tr a n s f e r o f videogame s k i l l s to s p a t i a l and geom etric s k i l l s. There i s one unpublished study conducted a t

52 th e H arvard G raduate School o f E ducation th a t has examined 35 t h i s t r a n s f e r Iss u e (Gagnon, 198^). S u b je c ts In th e stu d y, 58 u n dergraduate and g ra d u a te s tu d e n ts, were exposed to fiv e h o u rs o f spaced p r a c tic e a t videogames* Two games were used: T arg, a tw o-dim ensional^m aze game and B a ttle z o n e, a th re e -d im e n s io n a l war game* S u b jects were p r e te s te d and p o s t - t e s te d u sin g th re e s p a t ia l a b i l i t y t e s t s : th e Guilford-Zim m erm an S p a tia l O rie n ta tio n T est ( i n which judgm ents a re made about p o s itio n s o f a b o at bow r e l a ti v e to the t e s t ta k e r ), th e G u ilfo rd - Zimmerman S p a tia l V is u a liz a tio n T e st ( in which judgments a re made about th e r o t a t i o n o f an alarm c lo c k ), and th e Employee A p titu d e Survey V isu a l P u rs u it T est ( in which s u b je c ts tr a c e l i n e s through a c i r c u i t board m aze). Gagnon found s i g n i f i c a n t c o r r e la tio n s between videogame sc o re s and s p a t i a l t e s t scores* Male s u b je c ts scored h ig h er than fem ale s u b je c ts on the s p a t ia l t e s t s & on the b a se lin e m easure o f one o f the videogam es. There was no o v e ra ll e f f e c t o f th e videogame p r a c tic e, but th e re was a tre a tm e n t by se x in te r a c tio n * Female s u b je c ts in the tre a tm e n t group improved on the s p a t ia l v is u a liz a tio n t e s t s i g n i f i c a n t l y more than d id female s u b je c ts in the c o n tro l group. There was no s ig n if ic a n t improvement fo r male s u b je c ts ^.

53 To d a te, re s e a rc h on th e p o s s ib le t r a n s f e r o f programming s k i l l s to geom etric and s p a t i a l a b i l i t i e s 36 has been lim ite d and la r g e ly an ecdotal* D espite the w idespread in c lu s io n o f g ra p h ic s programming in to school c u r r ic u la, t h i s p a r t i c u l a r type o f programming has re c e iv e d v i r t u a l l y no re s e a rc h a tt e n t i o n. Most o f the work on t r a n s f e r o f t r a i n in g concerns th e language LOGO, y e t BASIC i s th e r e s id e n t language in e s s e n t ia ll y a l l m icrocom puters p u rch ased by s c h o o ls. There i s no a v a ila b le re s e a rc h in fo rm a tio n on th e c o g n itiv e req u irem en ts fo r, o r p o ss ib le b e n e f its o f, le a rn in g to program BASIC g ra p h ic s. I t i s t h i s d e a rth o f in fo rm a tio n th a t m o tiv a te s the p re se n t stu d y.

54 CHAPTER II HYPOTHESES AND SUMMARY OF UNDERLYING PRINCIPLES T ra n s fe r o f T ra in in g The lim ite d evidence fo r t r a n s f e r of programming s k i l l s to o th e r c o g n itiv e a b i l i t i e s may appear more s u b s ta n tia l when c o n sid e re d in l i g h t o f the d i f f i c u l t i e s encountered i n d e m o n stratin g any type o f problem so lv in g tra n s fe r» even in a d u lts. Although i t has lo n g been reco g n ized t h a t t r a n s f e r o f tr a in in g i s c e n tr a l to le a rn in g and e d u c a tio n (Ferguson, 1956, 195^; M essick, 198*f), re s e a rc h in d ic a te s th a t s u b je c ts g e n e ra lly have g re a t d i f f i c u l t y n o tin g s i m i l a r i t i e s between "problem isom orphs" (problem s w ith i d e n t i c a l u n d e rly in g lo g ic but s l i g h t l y d if f e r e n t a p p earan ces) (se e B ryant, Brown, and Campione, 1983; Gick and Holyoak, 1982; Simon and Hays, 1976). Problem isom orphs sure exam ples o f n e ar t r a n s f e r. Many of the programming s tu d ie s have attem p ted to fin d t r a n s f e r to ta s k s re q u irin g r e l a te d b u t d i f f e r e n t u n d erly in g s t r a te g ie s ( f a r t r a n s f e r ). These s tu d ie s focused on t r a n s f e r to g e n eral problem so lv in g o r to a broad range o f s p a t ia l a b i l i t i e s. The p re s e n t stu d y w ill in c lu d e t e s t s to a sse ss th re e d if f e r e n t l e v e ls o f t r a n s f e r. A t e s t o f g ra p h ic s 37

55 38 programming competence w i l l a s s e s s n ear t r a n s f e r. I t i s a p ap er and p e n c il v e rs io n of a c t i v i t i e s perform ed a t the com puter. A t e s t o f van H iele l e v e ls o f geom etric thought w ill be used to measure medium t r a n s f e r. The item s on th is t e s t in c lu d e examples o f th e k in d s o f sh a p e s, l i n e s, and lo g ic re q u ire d fo r g ra p h ic s programming. The t h i r d t e s t i s a sta n d a rd iz e d t e s t o f s p a t i a l o r ie n ta tio n ( f a r tr a n s fe r ). T his t e s t re q u ir e s the s u b je c t to make judgm ents about the o r ie n ta tio n o f v a rio u s sh ap es i n two dim ensions. As Pea and K urland (198^) recommend, - th is stu d y w ill examine low le v e l t r a n s f e r from elem en tary programming s k i l l s, and the t e s t s w ill a s s e s s ta s k s and p ro c e sse s c lo s e ly r e la te d to a c tu a l programming a c t i v i t i e s. H ypothesis I T ra in in g in programming BASIC com puter g ra p h ic s w ill t r a n s f e r to : a. a paper and p e n c il m astery t e s t b. a t e s t o f van H iele l e v e ls o f geom etric thought c. a t e s t o f s p a t i a l o r ie n ta tio n (Card R o ta tio n s). There w ill be a main e f f e c t f o r tre a tm e n t; improvement in t e s t sc o re s o f the tre a tm e n t group w ill be s i g n i f ic a n t l y g r e a te r than improvement i n t e s t s c o re s o f th e c o n tro l group d. There w ill be a d i f f e r e n t amount o f tr a n s f e r fo r each t e s t. The most t r a n s f e r w ill be to th e m astery t e s t, the l e a s t t r a n s f e r to th e s p a t i a l o r ie n ta tio n t e s t.

56 39 Sex Differences R esearch on sex d iff e re n c e s i n s p a t i a l a b i l i t i e s r a th e r c o n s is te n tly dem o n strates an advantage fo r m ales (see Maccoby and J a c k lin, 1974; Nyborg, 1983 fo r rev iew s). T his advantage g e n e ra lly b eg in s a t p u b erty and in c re a s e s u n t il about age An i n te r n a t i o n a l study o f m athem atics a b i l i t i e s i n s tu d e n ts in 19 c o u n trie s found th a t a t age 10, boys o u tsc o red g i r l s by 1 /4 o f a sta n d a rd d e v ia tio n. At age 14, th e advantage fo r boys In c re a s e d to 1 /2 o f a sta n d a rd d e v ia tio n, and by th e l a s t y e a r o f secondary sc h o o l, boys o u tsc o red g i r l s by 1 f u l l sta n d a rd d e v ia tio n (Comber and Keeves, 1973)* D eclines in m athem atical a b i l i t i e s i n g i r l s between the ages o f 11 and 15 y e a rs have been re p o rte d by Ross and Simpson, (1971)* S ta f f o rd (1972) found th a t boys began to o u tsc o re g i r l s on t e s t s o f q u a n tita tiv e rea so n in g a f t e r age 12. P o s t-p u b e rta l advantages f o r m ales on t e s t s o f s p a t i a l o r i e n t a t i o n and v is u a liz a tio n have been re p o rte d by Emmett (19t9)«Maccoby, (1966), S l a t e r (1971). V arious th e o r ie s have been fo rm u lated i n an attem p t to account f o r th e se d if f e r e n c e s. G enetic in flu e n c e s, b ra in l a t e r a l i z a t i o n, and en v ironm ental p re s s u re s may a l l p lay a r o l e. T h e o rie s r e l a te d to hormonal changes, however, appear to have th e most s u b s ta n t ia l e m p iric a l su p p o rt (Nyborg, 1985)*

57 In th e c o n te x t of th e p re s e n t stu d y, s o c ia liz a tio n 40 in flu e n c e s may have an im p o rta n t e f f e c t. O ften, videogames are a c h i l d 's f i r s t c o n ta c t w ith any kind of com puter. The v io le n t themes of m ost videogam es a t t r a c t m ales and seem to d isc o u rag e fem ales (G re e n fie ld, 1981* ) Malone (1981) dem onstrated th a t the ap p eal o f an e d u c a tio n a l computer game co u ld be a l t e r e d by v a ry in g the amount of a g g re s s io n -lik e re in fo rc e m e n ts. When the rein fo rce m en ts were made more a g g re s s iv e, th e gam e's p o p u la rity in c re a se d fo r boys and d ecreased f o r g i r l s. I f boys a re highly m otivated to play videogam es, t h i s ex citem en t may g e n e ra liz e to o th e r ty p es of com puters. Computers are a sso c ia te d w ith m athem atics and, i n d i r e c t l y, w ith male dominated o c cu p a tio n s r e q u ir in g advanced m athem atics. F in a lly, t e l e v i s i o n shows and com m ercials fo r computers u su a lly c a s t m ales in th e r o l e s of p e rso n s who re c e iv e, program, and purchase com puters. A ll o f th ese s o c ia l in flu e n c e s may combine to pro v id e e x tr a m o tiv atio n fo r males to l e a r n programming and r e l a t e programming to o th e r a c t i v i t i e s. Although the Gagnon C1964) stu d y d id fin d a treatm ent e f f e c t f o r fem ales and not fo r m ales, the larg e body of re s e a rc h on sex d if fe re n c e s in s p a t i a l a b i l i t i e s and the m o tiv a tio n a l f a c to r s j u s t d isc u sse d su p p o rt the follow ing h y p o th eses.

58 H ypothesis II There will be sex differences favoring males, a. Test scores of males will be significantly higher than scores for females on all three instruments. There will be a sex by treatment interaction. b. W ithin the tre a tm e n t gro u p, male s u b je c ts. w ill improve s i g n i f i c a n t l y more than fem ale s u b je c ts. Age Effects D e sc rip tio n s o f th e developm ent o f th e c h i l d 's conceptions o f space and geom etry dem onstrate t h a t s p a t i a l a b i l i t i e s Improve w ith age (L aurendeau and P in a rd, 1970; P ia g e t, I9 6 0 ). Although m ost P ia g e tia n ta s k s In v o lv in g geometry are accom plished by age n in e, c o n se rv a tio n and measurement o f volume may n o t be ach iev ed u n t i l age 11, C onservation through tra n s fo rm a tio n s may n o t be achieved u n t il a f t e r age 13 (K id d er, 1976, 1978). The c h i l d 's a b i l i t y to program com puter g ra p h ic s would be expected to in c re a se w ith improved l o g i c a l re a so n in g a b i l i t i e s. V arious kinds o f lo g ic a l re a s o n in g c a p a c itie s co n tin u e to develop up to age 13 ( F l a v e l l, 1963). S tu d ie s in v o lv in g van H iele l e v e ls found in c r e a s e s i n le v e ls o f geom etric th in k in g w ith a g e. They a ls o n o ted th a t o ld e r c h ild re n are b e t t e r a b le to b e n e f it from c lu e s, h in ts, and in s tr u c tio n s c o n ce rn in g s o lu tio n s t r a t e g i e s (B urger, 1981; Geddes, 1982).

59 kz H ypothesis I I I T est sco res and th e amount o f t r a s f e r o f t r a in in g w ill in c re a se w ith age. a. Scores on a l l th re e t e s t s w ill s i g n i f ic a n t l y in c re a s e as th e s u b j e c t s ' ages in c r e a s e. b. There w ill be a tre a tm e n t by age i n te r a c t io n ; the amount o f improvement in th e tre a tm e n t group w ill s i g n i f i c a n t l y In c re a s e as the s u b je c ts ' ages in c r e a s e. Grade E f fe c ts Grade and age e f f e c t s a re confounded w ith one a n o th e r. The van H iele th e o ry, however, would su p p o rt a p re d ic tio n o f improved t e s t sc o re s on th e b a s is of grade alo n e. They propose t h a t th e le v e l o f geom etric thought i s g re a tly in flu e n c e d by form al in s t r u c t i o n in geom etry. The re s e a rc h on c h ild re n in. d i f f e r e n t grades shows, i n g e n e ra l, an in c re a s e in van H iele le v e l w ith in c re a s e s i n the s u b j e c t 's g ra d e.(b u rg e r, 1981; Geddes, 1982). H ypothesis-iv T est sco/.-s and th e amount o f t r a n s f e r o f tr a in in g w ill in c re a se w ith g rad e. a. Scores on a l l th re e t e s t s w ill s ig n i f ic a n t l y in c re a se as th e s u b je c t '. grade in c r e a s e s.

60 *f3 b. There w ill be a tre a tm e n t by grade i n te r a c t io n ; th e amount of improvement in the tre a tm e n t group w ill s i g n i f ic a n t l y in c re a s e as the s u b j e c t 's grade in c r e a s e s. P re d ic tio n s from S ta n d a rd iz e d T e sts There has been l i t t l e re s e a rc h concerning th e c o g n itiv e a b i l i t i e s re q u ire d f o r programming, alth o u g h th e re i s c le a r ly a need f o r such in fo rm a tio n (P ea and K urland, 1981f). T h is ty p e o f d a ta would be u s e fu l in id e n tify in g th e optim al grade o r age le v e l f o r programming i n s t r u c t i o n. Such in fo rm a tio n m ay.also h elp id e n tif y s tu d e n ts w ith p a r t i c u l a r l y s tro n g programming p o t e n t i a l. S ince programming g ra p h ic s i s e s s e n tia lly an e x c e ris e i n a p p lie d m athem atics, i t i s expected t h a t, out o f a b a tte r y o f e ig h t sta n d a rd iz e d t e s t s c a le s, a t e s t o f a p p lie d m athem atics.m ill be the b e st p r e d ic to r o f g ra p h ic s programming s k i l l ( c f. Cronbach, 1984). H ypothesis V Out o f a b a tte r y o f e ig h t s ta n d a rd iz e d t e s t s, the b e st s in g le p r e d ic to r o f programming s k i l l w i l l be a. t e s t of a p p lie d m athem atics. T his w ill be dem onstrated by th e a p p lie d m athem atics t e s t score being i d e n t i f i e d a s the f i r s t p r e d ic to r in a stepw ise m u ltip le re g re s s io n.

61 CHAPTER I I I METHODS T e sts T est o f van H iele L evels o f Geom etric Thought P a s t attem p ts to t e s t c h ild re n f o r l e v e ls o f g eom etric th in k in g, acco rd in g to th e van H iele th e o ry, have focused on the c h i l d 's v e rb a l r e p o r ts concerning p r o p e r tie s o f shapes (B urger, 1981; Geddes, 1982). Because o f th e a n tic ip a te d la rg e numbers o f s u b je c ts i n th e p re s e n t stu d y, i t was c o n sid e re d p r e fe r a b le to a d m in iste r a paper and p e n c il t e s t r a t h e r than conduct in d iv id u a l in te rv ie w s. In a d d itio n, a focus on th e p r o p e r tie s o f l i n e s r a t h e r th an shapes seemed more a p p ro p ria te. When a c h ild program s the com puter to draw p ic tu r e s, he o r she g e n e ra lly program s a s e r i e s o f l i n e s, add l o c a te s them a t th e a p p ro p ria te c o o rd in a te s to form sh ap es. R ecall t h a t th e se l i n e s can be c o n s tru c te d one square a t a tim e ( e.g. PLOT 6,5 ; PLOT 7»5; PLOT 8,5)» o r can be drawn u sin g HLIN and VLIN ( e.g. HLIN 6,8 a t 5)* Since the stu d e n ts re c e iv e extended p r a c tic e working w ith l i n e s r a th e r th an sh ap es, i t was tho u g h t th a t kk

62 any improvement in geom etric th in k in g m ight be b e t t e r tapped by t e s t item s th a t r e f e r r e d to l i n e s. As w ith sh ap es, l i n e s can be reg ard ed as p o ss e s sin g v a rio u s p r o p e r tie s. One lin e drawn on the com puter sc re e n has le n g th, and by v ir tu e o f th e manner i n which i t i s programmed, i t has d ir e c tio n ( e.g. h o r iz o n ta l l i n e from 6 to 8 a t 5)» I f a com parison i s made o f two l i n e s on the sc re e n, they can be p a r a l l e l, p e rp e n d ic u la r, in te r s e c t i n g (o r heading fo r an i n t e r s e c t i o n ), and n o n - in te r s e c tin g ( o r s e p a ra tin g from one a n o th e r). These main p r o p e r tie s o f l i n e s, i n the c o n te x t o f programming com puter g ra p h ic s, were used to develop item s fo r a s s e s s in g th e s u b je c ts ' van H iele le v e l o f geom etric th o u g h t. In a d d itio n, four t e s t item s were taken d ir e c tly from r e p o r ts o f p re v io u s re s e a rc h (B urger, 1981). The fo llo w in g i s an ite m by item d e s c rip tio n o f the geom etry le v e l t e s t used in th e p re s e n t s tu d y.(s e e Appendix B f o r p a r a l l e l forms o f a l l t e s t s ). L evel I -.V is u a liz a tio n. Item 1 - v is u a l com parison of d i f f e r e n t ty p es o f l i n e s. Item 2 - i d e n t i f i c a t i o n and naming o f sim ple l i n e s. Item 3 - s im ila r to B urger (1981), com parison o f sim ple shapes*

63 4 6 Item s 4, 5» and 6 - taken from B urger, t h i B s e t o f items could measure levelb 1, 2, or 3 depending on the student's answers to item 6. Level 2 - D e sc rip tio n. Item 7 - in fo rm a l a n a ly s is o f sh ap es, l in e s o f symmetry. Item 8 - com parison o f the p r o p e r tie s o f both l in e s i n o rd e r to p r e d ic t the lo c a tio n o f t h e i r i n te r s e c t i o n. Item 9 - com parison o f th e p r o p e r tie s o f l i n e s, p r e d ic tio n o f t h e i r in te r s e c t i o n, s u b tle in tro d u c tio n o f the co n cep ts o f p a r a l l e l and p e rp e n d ic u la r. Level 3 - A b s tra c tio n. Item 10 - sim ple t e s t o f in c lu s io n, the shaded a re a could in c lu d e any number o f v e r t i c a l o r s l i g h t l y angled l in e s th a t f a l l w ith in th e m iddle t h i r d o f the sc re e n. Which o f the examples to th e r i g h t are in c lu d e d in t h i s s e t? Item 11 - the s tu d e n t must a s s e s s w hether o r not the combined p r o p e r tie s o f a l l th re e l i n e s a re s u f f i c i e n t to cause j o i n t in te r s e c tio n. Item 12 - the stu d e n t must a s s e s s w hether o r not the p r o p e r tie s o f each l i n e, when combined, a re s u f f i c i e n t to form a sq u a re.

64 L evel U - D eduction. w? Item 13 - sim ple m athem atical d ed u ctio n In c lu d in g one v a r ia b le, t h i s d ed u ctio n ta k e s p lac e w ith in the c o n s tr a in ts o f th e law s o f a rith m e tic. Item 14 - ded u ctio n in v o lv in g t r a n s i t i v e r e l a t i o n s. Item s 15 and 16 - p a r a l l e l and p e rp e n d ic u la r l i n e s are d e fin e d. The s tu d e n t must make d ed u ctio n s w ith in t h i s d e sc rib e d system. The f i n a l item i s an a p p lie d example o f lo c a tin g a p o in t i n two dim ensions. T h is was n o t in te n d e d to t e s t van H iele l e v e l, however, i t could be c o n stru e d as a more d i f f i c u l t v e rs io n o f item 11. There a re no le v e l 5 ite m s. The m athem atical R igor le v e l i s f a r beyond th e a n tic ip a te d c a p a c itie s o f any p o te n tia l s u b je c ts. The item s j u s t d e sc rib e d, and t h e i r p a r a l l e l form s, are th e end r e s u l t o f th re e i t e r a t i o n s o f t e s ti n g. Three p i l o t s tu d ie s were conducted and a p p ro p ria te re v is io n s were made fo llo w in g each stu d y. The p i l o t re s e a rc h in c lu d e d a t o t a l o f 97 s u b je c ts w ith an age range o f 3 to 16 y e a rs. Follow ing each stu d y, th e item s th a t were r e ta in e d were those th a t d isc rim in a te d among s u b je c ts and were c o rr e la te d w ith s u b je c t age.

65 48 M astery T est T his i s a p e n c il and p a p er t e s t t h a t m easures g ra p h ic s programming com petence. The s u b je c ts are given examples o f g ra p h ic s o u tp u t and asked to w rite programs th a t would produce such o u tp u t. They a re then given example g ra p h ic s program s and asked to draw the expected o u tp u t. R e su lts from p i l o t t e s t i n g su g g ested only m inor re v is io n s in w ording. S p a tia l O rie n ta tio n T est Card R o ta tio n s i s a m arker t e s t f o r th e s p a t ia l o r ie n ta tio n f a c to r i n the K it o f F a c to r R eferenced C ognitive T e s ts (E.T.S., Ekstrom, French, Harman, and Dermen, 1976). I t re q u ir e s s u b je c ts to make judgments about shapes t h a t have been r o ta te d i n two dim ensions. Although th e t e s t manual su g g e sts t h a t t h i s t e s t i s s u ita b le fo r grad es 8-16, p i l o t t e s t i n g in d ic a te d th a t the t e s t was unable w ith c h ild r e n in g rad es s i x, seven, and e ig h t. R e l i a b i l i t y o f t h i s t e s t av erag es.8 5. I t i s composed o f two p a r a l l e l forms each tim ed fo r th re e m inutes. S u b je c ts S u b je c ts were 95 m ales and fem ales who p a rtic ip a te d in th e stu d y by com pleting both th e p r e t e s t s and p o s t - t e s t a The s u b je c ts were s i x t h, se v e n th, and e ig h th grade s tu d e n ts

66 k9 a t a midwest urban m iddle sc h o o l. R a c ia l com position o f the school was ap proxim ately 40$ in n e r - c it y b la c k s, and 60$ poor and working c la s s w h ite s. The w hite s tu d e n ts were bussed in from a d is ta n c e o f about s ix m ile s. Two c la s s e s a t each grade l e v e l were in c lu d e d in th e stu d y. The c la s s e s were s e l e c te d acco rd in g to mathemat i c s a b i l i t y and sc h e d u lin g c o n s t r a i n ts. A ll c la s s e s used in th e stu d y were d e s c rib e d by t h e i r m athem atics te a c h e rs as having average a b i l i t y f o r t h e i r grade in t h a t p a r t i c u l a r sc h o o l. A ll s i x c la s s e s were scheduled to p a r t i c i p a te in th e stu d y on th e same e ig h t days. There were 22 s ix th g rad e, 39 se v e n th g rad e, and 34 e ig h th grade s tu d e n ts. O v e ra ll th e re were 43 males and 32 fem ales in c lu d e d i n the s tu d y. S u bjects* ages were 11, 12, 13» 14, and 13* The number o f. s u b je c ts a t each age were 7, 30, 31> 21, and 6 r e s p e c tiv e ly. A d d itio n a l breakdowns by age, se x, and tre a tm e n t group a re in c lu d e d i n T ab les 1 and 2. A pparatus P rocedure E ig h t Apple l i e com puters w ith b la c k and w hite m o n ito rs, and one d isk d riv e a tta c h e d to each, were used in the tre a tm e n t. On the f i f t h and s ix th days,

67 50 Table 1 Subject Broakdown_by Grade._5ex. and Treatnent Group VARIABLE CODE N FOR ENTIRE POPULATION 95 CRADE SEX TX 0. e TX 1. 3 SEX 1. 9 TX 0. 3 TX i. 1 1 ' 1,1 6 CRADE SEX TX 1. 7 SEX TX TX l. 13 CRADE B. 16 SEX TX 0. 3 TX SEX 1. TX 0. B Codeat TX - t r a a t a e n t - 0» c o n tr o l g ro u p, 1 - t r a a t a a n t g ro u p SEX - 0 * a a l e a, 1 f a a a la a

68 5 1 T able 2 S a b te c t Breakdown by Age. Sex, and Treatment Group VARIABLE CODE N FOR ENTIRE POPULATION 9 5 ACE SEX 0, 3 TX 0. 2 ' """ TX 1. 1 SEX 1. 4 TX 0. 1 TX 1. J ACE SEX TX 0. a TX 1. 7 SEX 1- TX 0. a TX 1. 7 ACC SEX TX 0. 3 TX SEX TX 0. a TX 1- a ACE SEX 0. 7 TX 0 3 TX 1. * SEX TX 0. TX 1. 9 ACC IS. 6 SEX 0. 3 TX 0. I TX 1. 2 SEX 1-3 Tx 0. 2 TX 1. 1 Codes: TX treatm ent con trol group, 1» treatm ent group SEX ** 0 m ales, 1 - fem ales

69 th re e a d d itio n a l com puters w ith c o lo r m o n ito rs were added. 52 C la ss s iz e s ranged from 10 to lk s tu d e n ts. S h arin g o f com puters was r o ta te d among s u b je c ts. The com puter programming in s t r u c t i o n was giv en in a com puter la b o ra to ry lo c a te d i n th e basem ent o f the sc h o o l. The room was la r g e, approxim ately 30 x 30 f e e t, w ith com puter ta b le s along th e w a lls, a ch alk b o ard, and one la r g e work a re a lo c a te d i n th e m iddle. The room was com fortably warm, w ell l i t, and i s o l a t e d from th e r e s t o f th e b u ild in g. Design The study was conducted u sin g a sta n d a rd p r e t e s t / p o s t - t e s t c o n tro l group design (Campbell and S ta n le y, 1963)* The d esig n i s summarized in F ig u re 2. For the c o n tro l group, the " a lte r n a te a c t i v i t y w ith te a c h e r" c o n s is te d m ainly in review o f re c e n t le s s o n s and a d d itio n a l e x e rc is e s, and some i n depth stu d y o f to p ic s t h a t were o f p a r t i c u l a r i n t e r e s t to th e B tu d en ts. The te a c h e rs ag reed n o t to te a c h any to p ic s r e l a te d to geom etry to th e c o n tro l group d u rin g the stu d y. T reatm ent S u b je c ts were randomly d iv id e d i n to tre a tm e n t and c o n tro l groups. Form I o f Card R o ta tio n s, M astery, and the van H iele geometry le v e l t e s t was given to a l l

70 P r e te s ts Treatm ent P o s t- te s ts S u b je cts - ( P a r a lle l forms) Treatm ent 1 2 geometry t e s t I n s tr u c tio n geometry t e s t Group 3 Card R o tatio n s in BASIC Card R o tatio n s C ontrol Group m astery t e s t G raphics m astery t e s t geometry t e s t A lte rn a te geometry t e s t Card R o tatio n s A c tiv ity w ith Card R o tatio n s IT m astery t e s t Teacher m astery t e s t Figure 2. Design o f the stu d y. VJi Vj4

71 s u b je c ts as a p r e t e s t (se e Appendix B fo r v e rb a l t e s t 54 in s tr u c tio n s ) * A fte r com pleting the p r e t e s t s, the s u b je c ts were th e n t o ld which o f them would be in c lu d e d i n th e com puter c la s s ( th e tre a tm e n t g ro u p ), and which o f them would rem ain i n the c la s s w ith t h e i r r e g u la r te a c h e r (th e c o n tr o l group)* The n a tu re o f th e stu d y was e x p la in e d to a l l s u b je c ts i n g e n e ra l term s. The im portance o f the c o n tro l group was emphasized* S u b je c ts were to ld th a t they had been random ly a ssig n e d to g ro u p s, and th a t the c o n tro l group s tu d e n ts would e v e n tu a lly be allow ed to ;.le a rn about th e com puters from t h e i r r e g u la r m athem atics and sc ie n c e tea ch e rs* I t was f u r th e r e x p la in e d th a t th e re were only e ig h t com puters i n th e la b o ra to ry, and th e r e f o re i t was n o t f e a s ib le to take the e n tir e c la s s down to th e la b o r a to r y a t one time* The stu d y was conducted on Tuesdays and Thursdays over a p e rio d o f fo u r c o n se c u tiv e weeks. P r e te s tin g was done on th e f i r s t Tuesday* The fo llo w in g s ix Tuesdays and T hursdays were tre a tm e n t days, and p o s t - t e s ti n g was done on th e l a s t Thursday* The tre a tm e n t c o n siste d i n s ix c la s s p e rio d s o f 50 m inutes each, conducted over a p e rio d o f 20 days. The c o n te n t o f each s e s s io n was as fo llo w s:

72 Day 1 - o r i e n t a t i o n to com puters and c la s s p ro c e d u re s, FOR...NEXT command. Day 2 - b eg in n in g g ra p h ic s programming, d e s c rip tio n o f the g ra p h ic s sc re e n as c o o rd in a te system, p r a c tic e w ith p re w ritte n program s. Day 3 - more g ra p h ic s, p l o tt i n g shapes p o in t by p o in t, in tr o d u c tio n o f th e commands HLIN and VLIN.. Day k - p l o t t i n g h o r iz o n ta l and v e r t i c a l l i n e s, making r e c ta n g le s, making th e s tu d e n t's i n i t i a l s i n b lo ck form. Day 3 - s u b je c ts work o u t a p ic tu r e o f t h e i r own on th e g ra p h ic s w orksheet, and beg in to program t h a t p ic tu r e as a p r o je c t. Day 6 - s u b je c ts f i n i s h p r o je c ts and run them on c o lo r m o n ito rs. The w orksheet and i n s t r u c t i o n a l m a te ria ls were p re v io u sly developed by the Technology and B asic S k i l l s in M athem atics p r o je c t conducted a t The Ohio S ta te U n iv e rsity C ollege o f E d u ca tio n. E ig h t in s t r u c t i o n a l b o o k le ts had been developed, each fo cu sin g on a. d if f e r e n t BASIC programming command. The f i r s t b o o k let in th e s e r i e s o f e ig h t i s c a lle d FOR...NEXT. I t was used here on day 1 because i t in c lu d e s an o r ie n ta tio n

73 5 6 to th e com puter and because t h i s command i s sometimes used i n g ra p h ic s programming. The fo llo w in g fiv e days o f tre a tm e n t used th e b o o k let e n t i t l e d G ra p h ic s.(s e e Appendix C fo r th e s e m a te r ia ls ). S ta n d ard ize d T est S cores The fo llo w in g in fo rm a tio n was c o lle c te d from school re c o rd s o f both tre a tm e n t and c o n tro l group s u b je c ts : b ir th d a te sex a p titu d e t e s t sc o re s language non-language t o t a l d a te o f t e s t i n g achievem ent t e s t s re a d in g v o c a b u la rly com prehension date o f t e s ti n g a rith m e tic com putation co n cep ts a p p lic a tio n s d ate o f t e s ti n g The sta n d a rd iz e d a p titu d e t e s t t h a t had p re v io u s ly been given to the s tu d e n ts wa6 th e S h o rt Form T est o f Academic A ptitude (SFTAA). T h is i s an in stru m e n t d e riv e d from th e C a lif o r n ia T e st o f M ental M atu rity (CTMM), C a lif o r n ia T e s tin g Bureau, The achievem ent t e s t sc o re s were from the Comprehensive T est o f B asic S k i l l s (CTBS), C a lif o rn ia T e s tin g Bureau. S tudent

74 57 s c o re s were re p o rte d in s ta n in e s (10-90)» w ith a mean o f 50 and a sta n d a rd d e v ia tio n o f 20. In general} o v e r a ll s tu d e n t sc o re s were about 1 /2 sta n d a rd d e v ia tio n below th e mean.

75 CHAPTER IV RESULTS R e l i a b i l i t i e s T e s ts f o r s p a t i a l o r i e n t a t i o n (C ard R o ta tio n s ), van H iele l e v e ls o f geom etric th o u g h t, and course m astery were each a d m in iste re d as p r e t e s t s and p o s t - t e s t s u sin g p a r a l l e l form s. R e l i a b i l i t i e s could be c a lc u la te d u sin g the c o r r e la tio n between p a r a l l e l forms o r C ronbach's Alpha, a measure o f i n t e r n a l c o n siste n c y. These f ig u r e s a re re p o rte d i n T able 3«The r e l i a b i l i t y fo r Card R o ta tio n s and th e geom etry t e s t a re somewhat low. The pre and p o s t - t e s t s were a d m in iste re d 2k days a p a r t, however, and the s u b je c ts were in sch o o l ta k in g c la s s e s t h a t m ight have a ffe c te d t h e i r m athem atics and v e rb a l a b i l i t i e s d u rin g t h i s tim e. There i s a n o tab ly la rg e d if fe r e n c e between the tre a tm e n t and c o n tro l groups on th e c o r r e la tio n between p a r a l l e l forms o f th e geom etry t e s t. A p o s s ib le e x p la n a tio n fo r t h i s concerns m o tiv a tio n a l d iff e re n c e s between the two g ro u p s. During th e 56

76 Table 3 R e li a b i l i ti e s Card R otations T o tal su b je c ts Treatm ent group C ontrol group C o rre la tio n s between p a r a l le l forms.69 ;70.67 Alpha c o e ffic ie n t P r e te s t Post t e s t Geometry t e s t T otal s u b je c ts Treatm ent group C ontrol group Mastery t e s t Treatm ent group Vji VO

77 6 0 p r e t e s t in g a l l s u b je c ts appeared s u f f c ie n t ly m otivated to perform w ell on th e t e s t s. During the p o s t - t e s ti n g, however, some fem ale c o n tro l s u b je c ts seemed u n in te re s te d, and a few were i r r i t a t e d a t having to tak e th e t e s t s ag ain and re c e iv e n o th in g in r e tu r n. The m astery t e s t was q u ite r e l i a b l e. C o rre la tio n s between p a r a l l e l forms o f th e m astery t e s t were not c a lc u la te d because v i r t u a l l y a l l s u b je c ts scored zero on the p r e t e s t. A nalyses o f P re -tre a tm e n t C o n tro l G roup/t reatm ent Group D iffe re n c e s The purpose o f the a n a ly se s c o n ta in e d in T ab les it through 7 i s to i d e n t i f y any d if f e r e n c e s between the tre a tm e n t and c o n tro l groups th a t m ight b ia s the r e s u l t s o r confound l a t e r a n a ly s e s. There i s only one in s ta n c e o f any s ig n i f ic a n t e f f e c t in c lu d in g tre a tm e n t group. T hat i s a 3-way tre a tm e n t x sex x grade i n te r a c t io n in th e e r r o r term o f the a n a ly s is o f th e geom etry p r e t e s t (T able 3 ). As can be seen in F ig u re s 3 and 5, the reaso n fo r th is i n te r a c t io n i s a grade r e l a te d d iff e re n c e i n the s u p e r io r ity o f th e tre a tm e n t o r c o n tro l groups dependent on se x. For the s ix th grade m ales, th e c o n tro l group has a h ig h e r mean on the p r e t e s t. For

78 6 1 the s ix th grade fem ales, the tre a tm e n t group has a h ig h e r mean sc o re. ' J u s t the re v e rs e i s tru e fo r the e ig h th grade. For th e m ales, the tre a tm e n t group has the h ig h e r sc o re. mean i s h ig h e r. For the fem ales, the c o n tro l g ro u p 's Since s u b je c ts were randomly assig n ed to tre a tm e n t groups, th e re i s no ap parent reason why t h is d iffe re n c e should o c cu r. I f i t i s found th a t th e re i s a s ig n if ic a n t grade e f f e c t in the p r e / p o s t- t e s t a n a ly s is, however, t h i s in te r a c t io n would pose a problem. E ight treatm en t by sex ANOVA's were run using each o f th e sta n d a rd iz e d t e s t sc o re s as the dependent v a r ia b le. There were no s ig n i f ic a n t p re -tre a tm e n t d iffe re n c e s between treatm en t and c o n tro l groups in any o f th ese a n a ly se s. Hypotheses H ypothesis I T his h y p o th esis p re d ic te d a main e f f e c t fo r trea tm e n t on a l l th re e t e s t s. There i s a la rg e main e f f e c t fo r treatm en t on the m astery t e s t (T ables 8 and 9)* T his i s i n both th e age x sex x tre a tm e n t, and the grade x sex x treatm en t ANOVA's. The c o n tro l s u b je c ts

79 6 2 scored zero on both the m astery p ro te s t and p o s t - t e s ts. The mean m astery t e s t score fo r the treatm ent group p o s t - t e s t was out of a p o ssib le 18 p o in ts. There a re no main e f f e c ts fo r treatm en t on the geometry t e s t (T ables 10 and 11) o r the Card d o ta tio n s t e s t (T ables 13 and lif). H ypothesis II H ypothesis I l a p re d ic ts s ig n if ic a n tly h ig h er t e s t sc o re s fo r males than fo r fem ales. T his i s the case f o r the Card R o tatio n s p r e te s t (Table?) T his i s the only s i g n if ic a n t main e f f e c t fo r sex. H ypothesis l i b p r e d ic ts more improvement fo r males than fem alesf a tre a tm e n t by sex in te r a c tio n. There was a treatm en t by sex i n te r a c tio n, but i t in d ic a te d more improvement fo r fem ales r a th e r than males (T able 1 1 ). T his I n te r a c tio n o ccurred in the treatm en t x sex x grade ANOVA fo r the geometry t e s t. T his in te r a c tio n i s graphed in F ig u re s 3 through 8. As can be noted from the p lo ts, the fe m a le s.in the treatm en t group improved more than did the fem ales in the c o n tro l group. For th e m ales, however, the c o n tro l group improved more than the trea tm e n t group. The means and p ost hoc com parisons fo r t h is in te r a c tio n are co n tain ed in Table 12.

80 6 3 H ypothesis I I I H ypothesis I l i a p re d ic te d a s i g n i f i c a n t in c re a s e in t e s t s c o re s w ith in c r e a s e s i n age. There were no s i g n i f i c a n t e f f e c t s fo r age (T ab les 6, 10, 1 3 ). The h ig h e s t mean sc o re on the m astery t e s t was 12, fo r the 13 y e a r o ld s u b je c ts. Twelve y e a r o ld s u b je c ts sco red th e h ig h e s t on both Card P o ta tio n s and th e geom etry te s t', *v?.9 and 2J+.3 r e s p e c tiv e ly. H ypothesis I l l b p re d ic te d in c re a s in g improvement w ith age, a tre a tm e n t by age i n t e r a c t i o n. There were no tre a tm e n t by age i n te r a c t io n s (T ab les 8, 10, 13)* H ypothesis IV H ypothesis IVa p re d ic te d an in c re a s e in t e s t sc o re s w ith g ra d e. The m astery t e s t in c r e a s e s w ith grade: 8.7» 1 0.2, 1 1.8, f o r s i x t h, s e v e n th, and e ig h th g rad es r e s p e c tiv e ly, but th e s e d if f e r e n c e s a re not s i g n i f i c a n t. The geom etry t e s t a ls o in c re a s e s w ith g rad e: 22.3* 23*ki and 23*7 f o r s ix th, se v e n th, and e ig h th g rad e s, but t h i s too i s n o t s i g n i f i c a n t. On th e Card R o ta tio n s t e s t, th e se v en th grade o u tsc o re d both th e s ix th and e ig h th g ra d e s. H ypothesis IVb p re d ic te d in c re a s in g improvement w ith g ra d e, tre a tm e n t by grade i n te r a c t io n s. There a re no tre a tm e n t by grade i n te r a c t io n s (T ables 9, 11, Xlf>.

81 6*4 H y p o th e s is V T his h y p o th esis p re d ic te d th a t the a p p lie d m athem atics sta n d a rd iz e d t e s t sc o re would be the b e st p r e d ic to r o f sc o re s on th e programming m astery t e s t. T his t e s t was not even s i g n i f ic a n t in th e re g re s s io n a n a ly s is. The b e st p r e d ic to r, w ith a m u ltip le R o f,*+1, was th e non-language t e s t (T able 1 5 ) T able 16 d is p la y s s i g n i f i c a n t c o r r e la tio n s in v o lv in g sta n d a rd iz e d t e s t sc o re s and th e m astery, geom etry, and card r o ta tio n s t e s t s. N on-language sta n d a rd iz e d t e s t sc o re does r e f l e c t th e h ig h e s t c o r r e la tio n ( o f th e sta n d a rd iz e d t e s t s u b t e s t s ). The h ig h e s t c o r r e la tio n w ith m astery, c o n sid e rin g a l l t e s t s, i s the geom etry p o s t - t e s t. T his in d ic a te s th a t s i m il a r s k i l l s may be re q u ire d f o r both t e s t s, and t h a t some t r a n s f e r o f tr a in in g could be e x p ected.

82 65 0 C ontrol Croup -* T roatount Croup N»»h Tout S to ru a O n ti F igure 3 > Geometry t e s t p r e t e s t so o res f o r m ales, 0 0 C ontrol Croup H - X T ru alau n t Group H.an Tout Seorua 21 W, 7 Oradu F igure 4. Geometry t e s t p o s t - t e s t sc o re s f o r m ales.

83 fab 0 *0 itonlrul «ruu ' % j! T realst'ri'. ui'iw i' Naan T aat Scoraa 21 7 S Maan T aat Scoraa Figure 6. Geometry test post-test scores for females.

84 control group X X treatment group difference scores * Grade Figure 7. Gooaetry. t e s t d ifferen ce scores for males control group X treatment group difference scores Grade Figure 8, Geometry te e t difference scorea for fem ales.

85 Table 4 Geometry Pretest - Age x Sex x Treatment ANOVA SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS AGE SEX TX WAY INTERACTIONS AGE SEX AGE TX SEX TX WAY INTERACTIONS AGE SEX TX EXPLAINED RESIDUAL TOTAL

86 Table 5 Geometry P r e te s t - Sex x Grade x Treatm ent ANOVA SOURCE OF VARIATION SUM OF SQUARES DF MEAN SQUARE F SIGNIF OF F MAIN EFFECTS SEX GRADE TX WAY INTERACTIONS SEX GRADE SEX TX GRADE TX WAY INTERACTIONS SEX GRADE TX EXPLAINED RESIDUAL TOTAL

87 Table 6 Card R o tatio n s P r e te s t - Age x Sex x T reatm ent ANOVA SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF- SQUARE F OF F MAIN EFFECTS AGE SEX TX WAY INTERACTIONS AGE SEX AGE TX SEX TX WAY INTERACTIONS AGE SEX TX EXPLAINED RESIDUAL TOTAL

88 Table 7 Card Rotations Pretest ~ Sex x Grade x Treatment ANOVA SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS SEX GRADE TX WAY INTERACTIONS SEX GRADE SEX TX GRADE TX WAY INTERACTIONS SEX GRADE TX EXPLAINED RESIDUAL 27512^ TOTAL

89 Table 8 rastery rtc/ruen.- UCOW L/i.1igi a. ii cauiutsii w Source of Variation Sum Of squares DP Mean Square F Signif of F Main Effects Age Sex TX Way Interactions Age Sex Age Tx Sex Tx Way Interactions Explained Residual TOTAL j PO

90 Table 9 Mastery Test - Pre/Post-test Difference Scores in Grade x Sex x Treatment (TX) ANOVA Sum of Mean Signif Source of Variation Squares DF Square F of F Main Effects Sex Grade TX Way Interactions Sex Grade Sex TX l.* Grade TX Way Interactions ti Sex Grade TX Explained Residual Total

91 Table 10 Geometry Tesfr ~ Pre/Post-test Difference Scores in Age x Sex x Treatment ANOVA SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS AGE SEX TX WAY INTERACTIONS AGE SEX AGE TX SEX TX WAY INTERACTIONS AGE SEX TX EXPLAINED RESIDUAL TOTAL

92 Table.11 Geometry Test - Pre/Post-teet Difference Scores in Grade x Sex x Treatment (TX) ANOVA ucuiue ti.y iuou no/ rwov- Source of Variation Sum Squares DF Mean Square F Signif of F Main effects Sex Grade TX Way Interactions Sex Grade Sex TX Grade TX way Interactions Sex Grade TX Explained Residual Total a Ul

93 7 6 Table 12 Difference Score Means for the Four_Groupe IhvolTed.in the S ig n ifica n t Sex x Treatment In te r assign.c.tfll^gju 1 Mean d lffirtn o t Croup K poetprat*at Halt Control Mala Treatment female Control female treatm ent Poat H op Oomparlaona Leaat S ig n ifica n t D ifference T ea t.(s teel and lo r r le. 1980, p ). lad t >0,j\^E rror Mean Square (1/H^ 1/N2) d f i_r ldual_source^ df - 83 Kean Square - 15*252 t a ta tia tlo required fo r algnificanoe a t *05 le v e l with 83 df Comparlaon Croupa. d iffere n c e acorea Male Control - Male Treatment lad> V l U /19 1/24) needed fur.05 a lg n ifica n o e - M.S. female Treatment - Female Control «1.970 ia d.o j \/l U /2 4 1/29) needed for.05 algnificanoe - *

94 Table 13 Card Rotations Test - Pre/Post-test Difference Scores in Age x Sex x Treatment ANOVA SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS AGE SEX TX WAY INTERACTIONS AGE SEX AGE TX SEX TX WAY INTERACTIONS AGE SEX TX EXPLAINED RESIDUAL ' TOTAL j ^3

95 Table 14 Card R otations T est - P r e /P o s t- te s t D ifference Scores In Grade x Sex x Treatm ent ANOVA SOURCE OF VARIATION SUM OF SQUARES DF MEAN SQUARE F SIGNIF OF F MAIN EFFECTS SEX GRADE TX WAY INTERACTIONS SEX GRADE SEX TX GRADE TX WAY INTERACTIONS SEX GRADE TX EXPLAINED RESIDUAL TOTAL

96 Table 15 M astery T eet S cores VARIABLE SUMMARY TABLE MULTIPLE R R SQUARE RSQ CHANGE SIMPLE R B BETA NLG CCT CPU COM LG (CONSTANT) NLG non-language CCT * math concepts CPU «math computation LG * language COM reading comprehension vo

97 80 Table 16 S ig n if ic a n t C o rre la tio n s In v o lv in g S tan d ard ized T e s ts. Geometry T e s t. Card R o ta tio n s ^ and M astery T est fo r T reatm ent Group S u b je c ts Correlations with Mastery Test correlation significance level N on-language *^1.003 T o ta l (language & non-language Math concepts *31 *02 Card R o tatio n s p r e t e s t Card R o ta tio n s p o s t - t e s t NS Geometry p r e t e s t *3*f.009 Geometry p o s t - t e s t.^ Math A p p lic a tio n s NS Correlations with Card Rotations pretest Geometry p r e t e s t :.ifl.001 Geometry p o s t - t e s t Correlations with Card Rotations post-test Geometry p r e t e s t.^ Geometry p o s t - t e s t *2k.03

98 Table 17 Geometry Test - Pre/Post-test Difference Scores in an Age x Sex x Treatment ANOVA with 11 and 15 Year Old Subjects Only SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS AGE SEX TX WAY INTERACTIONS AGE SEX AGE TX SEX TX WAY INTERACTIONS AGE SEX TX EXPLAINED RESIDUAL TOTAL

99 CHAPTER V DISCUSSION A m ajor s tr e n g th o f the p re s e n t stu d y i s i t s e c o lo g ic a l re le v a n c e. The re s e a rc h was conducted under circu m stan ces t h a t id e a lly fo s te re d " a convergence of both th e n a t u r a l i s t i c and th e ex p erim e n tal approaches (B ro n fen b ren n er, 1977 p. 514)* The s e t t i n g was a n a tu r a lly o c c u rrin g s i t u a t i o n i n which a la r g e number o f com puter naive s u b je c ts o f v a rio u s ages were lo c a te d i n a school th a t had j u s t i n s t a l l e d a com puter la b o ra to ry. There were no r e g u la r com puter c la s s e s being tau g h t a t the tim e, and the sch o o l a d m in is tra tio n was e a g e r to g e t the com puters i n to im m ediate u se. The la b o ra to ry was too sm all to accommodate an e n ti r e c la s s a t once, so alm ost any use o f th e com puters would re q u ire th e d iv is io n o f c la s s e s. There were only th re e m ild ex p erim en tal i n te r v e n tio n s. Pre and p o s t - t e s ti n g was conducted, u sin g p e n c il and p ap er sc h o o l-ty p e t e s t s. The s u b je c ts were tr a in e d in g ra p h ic s programming, u sin g i n s t r u c t i o n a l m a te ria ls s im ila r to those th a t m ight have e v e n tu a lly been 82

100 83 s e le c te d fo r use by the te a c h e rs. The c la s s e s were randomly d iv id ed In to tre a tm e n t and c o n tro l groups, as opposed to the te a c h e r s ' s e le c tio n o f stu d e n ts to go to the la b o ra to ry. The stu d e n ts a ls o had an u n fa m ilia r te a c h e r fo r the computer c la s s e s. In many sc h o o ls, however, a d if f e r e n t te a c h e r i s h ire d to run computer programming c la s s e s, so t h i s was not a d e p a rtu re from what m ight norm ally occur. I t i s w idely recognized th a t re s e a rc h w ith e c o lo g ic a l rele v an c e i s o fte n d i f f i c u l t to conduct (G ibbs, 1979; B ronfenbrenner, 1977)* T hat being the c a se, t h i s study was run under u n u su ally fo rtu n a te circu m stan ces. I t seems c le a r, however, th a t alm ost anytim e a new tea ch in g technique o r technology i s in tro d u c ed in to th e classroom a " n a tu ra l experim ent" i s provided. H ypothesis I p re d ic te d tr a n s f e r o f tr a in in g to the m astery t e s t, the geometry t e s t, and to the t e s t o f s p a t ia l o r ie n ta tio n. I t a ls o p re d ic te d th a t the amount o f t r a n s f e r would be ranked in the o rd er j u s t l i s t e d. The ANOVA's fo r the m astery t e s t, as expected, showed a la rg e tre a tm e n t e ff e c t (T ables 8 and 9 ). For the geometry t e s t, th e re was a treatm en t e f f e c t fo r fem ales (Table 1 1 ), and fo r Card R o ta tio n s, th e re were no trea tm e n t e f f e c t s (T ables 13 &!*+) The p re d ic tio n of

101 84 th e d i f f e r e n t i a l amount o f t r a n s f e r was th u s confirm ed. The c o r r e la tio n s betw een th e m astery t e s t and th e geom etry t e s t, and between th e m astery t e s t and th e Card R o ta tio n s t e s t (T able 16) in d ic a te th a t the s k i l l s re q u ire d f o r th e geom etry t e s t may be more c lo s e ly r e l a te d to com puter programming s k i l l s than are s p a t i a l o r i e n t a t i o n a b i l i t i e s. The la c k o f t r a n s f e r to Card R o ta tio n s and the low c o r r e la tio n between t h i s t e s t and th e m astery t e s t su p p o rt Pea and K u rla n d 's (198*+) su g g e stio n th a t t r a n s f e r from com puter programming to broad c o g n itiv e a b i l i t i e s i s u n lik e ly. More su c ce ss w ith t r a n s f e r i s to be ex p ected when u sin g t e s t s th a t m easure a b i l i t i e s c lo s e ly r e l a te d to programming s k i l l s. The Card R o ta tio n s t e s t o f s p a t i a l o r i e n t a t i o n was in c lu d e d p a r tly because th e p a s t re s e a rc h on g e n e r a liz a tio n o f com puter programming tr a in in g was lim ite d and not y e t co n v in cin g. The p re s e n t stu d y, however, adds f u r th e r evidence to the lik e lih o o d t h a t no rem arkable wide ran g in g t r a n s f e r o f t r a in in g w ill r e s u l t from s u b je c ts ' le a r n in g to program com puters. The main tre a tm e n t e f f e c t o f i n t e r e s t i n t h i s stu d y i s th e tre a tm e n t by sex i n te r a c t io n re p o rte d in Table 11 and graphed in F ig u res 3-8* The f d iffe re n c e between tre a tm e n t and c o n tro l group

102 8 5 improvement i s only s i g n i f i c a n t f o r th e fem ales. T his i s an o v e r a ll improvement o f 2.2 p o in ts on a t e s t w ith a mean fo r a l l s u b je c ts o f and a sta n d a rd d e v ia tio n o f 4.6* The h ig h e s t p o s s ib le sco re on t h i s t e s t i s 38. Though s i g n i f i c a n t, t h i s i s a sm all e f f e c t. Sex by tre a tm e n t ANOVA's were run on a l l 17 item s to determ ine i f t h i s improvement fo r fem ales was g e n e ra liz e d o r only r e f l e c t e d improvement on s p e c if ic ite m s. The r e s u l t s in d ic a te d t h a t the advantage fo r fem ales wa6 not r e s t r i c t e d to any p a r t i c u l a r s e t o f ite m s. T h is tre e -ment by sex i n t e r a c t i o n i s very s im ila r to the r e s u l t s re p o rte d by Gagnon ( That stu d y in v o lv e d t r a n s f e r from videogames to s p a t i a l v i s u a li z a t io n. R e c a ll t h a t h ere too th e re was a p r e t e s t / p o s t - t e s t improvement fo r fem ales, and no improvement f o r m ales. These two s tu d ie s co n sid ered to g e th e r may in d ic a te some s p e c ia l r e l a t i o n between fem ales and com puter g ra p h ic s t h a t cannot be accounted fo r by re fe re n c e to sta n d a rd sex d iffe re n c e s in s p a t i a l a b i l i t i e s. There are several possible explanations for this treatment by sex interaction. The first relates to a distinction between spatial abilities and training in spatial abilities. Studies involving assessment of

103 spatial abilities following puberty consistently report 8 6 an advantage for males. Research concerning instruction in spatial abilities, however, is not quite as definitive. Most of the studies find no sex differences in improvement following training. There are only a small number of studies in which greater improvement for males is reported (McGee, 1979 review; Smith and Litman, 1979)* Another p o s s ib le rea so n fo r the in a c c u ra te h y p o th esis concerns the p u b e rta l s t a t u s of the s u b je c ts. Reviews o f sex d iffe re n c e s i n s p a t i a l a b i l i t i e s u s u a lly in d ic a te th a t the g r e a te r advantage f o r m ales occurs a f t e r pu b erty (Nyborg, 1983)* In the p re s e n t stu d y, 8635 o f th e s u b je c ts were aged 12 to 1l+t r i g h t a t p u b e rty. I f hormones a re an im p o rtan t f a c to r in e le v a te d s p a t i a l a b i l i t i e s in m ales, th en p r e d ic tio n s based on hormones" would be l e s s a c c u ra te i n s i t u a t i o n s l ik e th e p re s e n t stu d y where th e re i s an u n c e rta in m ixture o f pre and p o s t-p u b e rta l hormonal l e v e ls fo r both sex es. C arefu l r e f l e c t i o n on th e b e h av io r o f the s u b je c ts ' d u rin g t r a in in g and t e s t i n g su g g e sts, however, th a t the most p la u s ib le e x p la n a tio n o f th e sex d iffe re n c e i n tre a tm e n t e f f e c t r e s t s on v a rio u s s o c ia l and m o tiv a tio n a l f a c to r s. Although d i f f i c u l t to u n ra v e l, s i t u a t i o n a l and " in tr a p e r s o n a l" c o n te x ts roust be co n sid ere d whenever e d u c a tio n a l measurements a re taken to e v a lu a te le a rn in g and t r a n s f e r (M essick, 1981+)

104 87 The in te r p e r s o n a l b eh av io rs o f th e m ales in the s u b je c t pool su g g e st th a t th e re was an i n te r a c t io n o c c u rrin g between th e p e er group and th e.in s tr u c tio n a l s e t t i n g. The p r e t e s t and p o s t - t e s t graphs fo r m ales c le a r ly dem onstrate t h a t, w hile th e tre a tm e n t group improved only s l i g h t l y, the c o n tro l group m ales improved s i g n i f i c a n t l y. (T h is was not th e case fo r fe m a le s.) What would cause th e male c o n tro l group to perform so much b e t t e r on th e p o s t - t e s t? Inform al o b se rv a tio n s in d ic a te d t h a t th e m ales i n t h i s school were very c o m p e titiv e, and t h a t most o f them r e a l ly wanted to be chosen to go down to th e com puter la b o ra to ry. I o c c a s io n a lly n o tic e d m ales in th e tre a tm e n t group bragging to c o n tro l group m ales about t h e i r programming su c c e sse s. I t i s my h y p o th e sis t h a t, a t th e tim e o f p o s t - t e s ti n g, many o f th e c o n tro l group m ales b e lie v e d th a t t h e i r fu tu re a c c e ss to th e com puters depended on t h e i r perform ance on the t e s t s. As a r e s u l t, they were very m o tiv ated. The fem ale c o n tro l group, on the o th e r hand, did not seem very i n t e r e s t e d in ta k in g the com puter c la s s, and t h e i r p re and p o s t - t e s t sc o re s are s im ila r,. The o th e r m ajor f a c to r c o n tr ib u tin g to th e la c k o f a s i g n i f ic a n t improvement fo r m ales was the behavior

105 o f the tre a tm e n t group. The same c o m p e titio n t h a t in s p ir e d the c o n tro l group worked to d i s t r a c t th e tre a tm e n t group m ales from t h e i r programming ta s k s. Many o f th ese s u b je c ts were e a s ily e x c ite d and e a s ily f r u s t r a t e d. There were o c c a sio n a l d is c ip lin e problem s, and s e v e ra l m ales o fte n seemed more i n t e r e s t e d i n th e a c t i v i t i e s o f o th e r s tu d e n ts th en in t h e i r own programming. As a r e s u l t, th e m ales sp e n t c o n sid e ra b ly l e s s tim e on ta s k th an did th e fem ales. I t i s b e lie v e d th a t t h i s com bination o f an u n u su a lly m o tiv ated c o n tro l group and a d i s tr a c te d tre a tm e n t group- caused th e sc o re s o f the m ales i n the tre a tm e n t group to be g e n e ra lly low er than ex p ected, and the c o n tro l group s c o re s to be h ig h e r than ex p ected, th u s e lim in a tin g a s i g n i f i c a n t trea tm e n t e f f e c t fo r m ales. H ypothesis I I I p re d ic te d an in c re a s e in t e s t s c o re s, and an in c re a s e in improvement w ith age. There were no s i g n i f ic a n t r e s u l t s f o r h y p o th e sis I I I. T his may be because the age range in th e p re s e n t study i s f a irly, lim ite d, and age i s confounded w ith g rad e. As a r e s u l t, th e re a re s u b je c ts o f th e same age w ith d if f e r e n t amounts o f e d u c a tio n a l e x p e rie n c e, and s u b je c ts o f d if f e r e n t ages w ith the same amount o f e d u c a tio n a l ex p erien c e.

106 There i s one a n a ly s is, however, t h a t i s somewhat s u r p r is in g and i s r e la te d to age. T able 17 i s an age x sex x tre a tm e n t ANOVA o f sc o re s on the geom etry t e s t fo r 11 and 15 y e a r o ld s u b je c ts o n ly, (th e extrem e g ro u p s). Cronbach and Snow (1977) recommend u sin g extrem e groups i n such a c a se, "M easuring an a p titu d e, dropping c ases from the m iddle o f the d i s t r i b u t i o n, and randomly d iv id in g c ase s i n each o f the t a i l s to form tre a tm e n t groups produces a com paratively pow erful d e sig n. " (p. 59) In t h i s a n a ly s is, th e re i s a s i g n i f ic a n t main e f f e c t f o r tre a tm e n t, p ^. O l. T h is a n a ly s is must be c o n sid ere d e x p lo ra to ry, and th e e f f e c t i n te r p r e te d c a u tio u s ly, however, because th e re a re only 13 s u b je c ts in c lu d e d. I f i t can be assumed t h a t t h is r e s u l t may be m eaningful, th e re a re probably two d i f f e r e n t p ro c e ss e s a t work h e re. The 11 y e a r o ld s u b je c ts could have * b e n e fite d from th e t r a in in g because they may be more i n t e l l i g e n t th an t h e i r 12 y e a r old c la ssm a te s. They have managed to keep up w ith o ld e r c h ild re n and n o t be r e ta in e d in a g rad e. The 15 y e ar o ld s u b je c ts, a lth o u g h, they may have been h eld back a g rad e, probably b e n e f it by being o ld e r.

107 9 0 H ypothesis IV p re d ic te d t e s t score in c re a s e s w ith grade and in c re a s e s in improvement w ith grade. Although th e re was a h in t o f score in c re a s e s by grade on the m astery and geometry t e s t s, th e re were no s ig n if ic a n t r e s u l t s fo r h y p o th esis IV. I t i s im p o rtan t to keep in mind th a t the m astery t e s t, geometry t e s t, and s p a t ia l o rie n ta tio n t e s t were a l l a s se ss in g a b i l i t i e s th a t had not been form ally tau g h t to the s u b je c ts. The m athem atics te a c h e rs a l l in d ic a te d th a t they had not covered any o f the concepts r e f le c te d in the t e s t ite m s. T h erefo re, even though i t m ight be expected th a t s u b je c ts in h ig h er grades would have b e t t e r g e n e ra l le a rn in g and a n a ly s is s k i l l s, in t h i s c ase, an in c re a s e in grade le v e l does not imply in c re a se d exposure to the re le v a n t geom etric con cep ts. H ypothesis V p re d ic te d th a t the a p p lied m athem atics sc a le o f the C a lifo rn ia T est o f Basic S k i l l s would be the b e st p r e d ic to r o f sc o re s on the m astery t e s t. T his was not th e c ase. The b e st p re d ic to r was the non-language s c a le. T his sc a le was a lso th e b est p re d ic to r in re g re s s io n a n aly ses fo r the geometry le v e l t e s t and fo r the Card R o tatio n s t e s t. S u p e r f ic ia lly, t h i s may not seem lik e an unusual r e s u l t. Non-language s c a le s, however, are g e n e ra lly c ate g o riz e d w ith g e n eral in te llig e n c e and a p titu d e

108 9 1 t e s t s (C ronbach, 198**). M easures o f in te llig e n c e and a p titu d e do not norm ally c o r r e la te as w ell w ith t e s t s o f s p e c i f i c a b i l i t i e s as do a p p lie d t e s t s in a re a s r e l a te d to the s p e c if ic a b i l i t y. I t may be th a t g ra p h ic s programming, s p a t i a l o r ie n ta tio n, and van H ie le l e v e ls o f geom etric thought are a l l r e la te d more to g e n e ra l i n te l li g e n c e th an to s p e c i f i c a ll y tr a in e d a b i l i t i e s. In r e tr o s p e c t, i t i s f e l t th a t the o r ig in a l d esig n o f t h i s stu d y should have in c lu d e d more c o n sid e ra tio n o f the I n tr a p e rs o n a l and s i t u a t i o n a l c o n te x ts t h a t p ro b ab ly e x p la in th e most n o ta b le r e s u l t, the sex by tre a tm e n t i n te r a c t io n. I t would have been b e n e f ic ia l i f some o f th ese f a c to r s could have been m easured and s e p a ra te d o u t from the v a rian ce due to tre a tm e n t e f f e c t s. In conducting a s im ila r study in th e f u tu r e, i t may be a d v is a b le to o b ta in eg. r a tin g s from te a c h e rs o f d is c ip lin e problem s w ith in d iv id u a l s tu d e n ts. A q u e s tio n n a ire about the s u b je c ts ' a t t i t u d e s tow ard com puters m ight.also be g iv en. T h is'm ig h t in c lu d e a m easure o f f r u s t r a t i o n le v e l fo llo w in g a programming c o u rse, a m easure o f m o tiv a tio n d u rin g the c la s s, o r some item s co n cern in g p e rc e iv e d p e er p re ssu re to perform.

109 9 2 Such d a ta could then be analyzed u sin g an a n a ly s is o f c o v a ria n c e. The im portance o f the v a rio u s c o n te x tu a l f a c to r s could then be more c lo s e ly examined.. T h is k ind o f a n a ly s is o f co v ariance technique i s com patible w ith an e c o lo g ic a l approach to re s e a rc h. S itu a tio n a l v a r ia b le s, such as peer p re s s u re, a re taken i n to acco u n t, and th e re i s minimal d is ru p tio n induced by the r e s e a rc h e r. I t i s common th a t re s e a rc h in th e f i e l d, in t h i s case i n the s c h o o ls, o fte n produces d a ta th a t i s the r e s u l t o f many s u b tle in f lu e n c e s, and main e f f e c t s a re fre q u e n tly d i f f i c u l t to o b ta in. The r e s u l t s o f th e p re s e n t stu d y seem to su p p o rt B ro n fen b ren n er' s (1977) r e la te d comment - "To c o rru p t, only s l i g h t l y, th e term inology o f ex p erim en tal d esig n : 'I n e c o lo g ic a l re s e a rc h, the p r in c ip a l main e f f e c t s are l ik e l y to be i n te r a c t io n s ". (p» 516)

110 FOOTNOTES 1. Reproduced w ith p erm issio n o f D. R eidel P u b lish in g Co, B oston, MA., D e sc rip tio n s from E d u catio n al S tu d ie s in M athem atics. 12?1981) J+21-**3*f* 2. Exact c o r r e la tio n s and ANOVA s t a t i s t i c s were not a v a ila b le. For more in fo rm a tio n see re fe re n c e fo r D. Gagnon. 3. A n tic ip a te d ages o f s u b je c ts to be used in th e p re s e n t study a re 11 to 15 y e a rs. 93

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