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Transfer and einstellung effects of examples on devising computer algorithms Goddard, William 1976

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TRANSFER AND EINSTELLUNG EFFECTS OF EXAMPLES ON DEVISING COMPUTER ALGORITHMS by WILLIAM PHILIP GODDARD B. Ed., U n i v e r s i t y o f B r i t i s h C o lumbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics E d u c a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 1976 0 William Philip Goddard, 1 9 7 6 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS Date / 7 L i ABSTRACT T h i s s t u d y was m o t i v a t e d by the w r i t e r ' s o b s e r -v a t i o n t h a t the p r o v i s i o n o f s o l v e d examples t o s t u d e n t s l e a r n i n g t o d e v i s e computer a l g o r i t h m s d i d n o t a s s i s t and even seemed t o h i n d e r i n the development o f such s k i l l s . I t was s u r -mised t h a t t h i s might be due t o a number of f a c t o r s . The l e a r n e r might be d e l a y e d i n hiso.development o f the h e u r i s -t i c s n e c e s s a r y t o c r e a t e a l g o r i t h m s u s i n g s e l f - g e n e r a t e d m e d i a t o r s . He might be m i s l e d i n h i s e x p e c t a t i o n o f the d i f -f i c u l t y o f p e r f o r m i n g such t a s k s i n d e p e n d e n t l y . He might d i s p l a y r i g i d i t y (an E i n s t e l l u n g e f f e c t ) i n h i s l a t e r use o f the t e c h -n i q u e s demonstrated by p r e v i o u s l y p r o v i d e d examples. Grade n i n e s t u d e n t s were a s s i g n e d t o two groups a t random. Both groups were g i v e n a p r i n t e d i n t r o d u c t i o n t o computer program w r i t i n g i n the BASIC language and were asked t o s o l v e two problems, an easy problem and a h a r d e r c r i t e r i o n p roblem. B e f o r e the problems were a s s i g n e d one group was g i v e n a s o l v e d example w h i c h was v e r y s i m i l a r t o the easy problem. The second group was g i v e n a s h o r t h i s t o r y o f computers t o r e a d . A Chi-square- t e s t was used t o t e s t each o f t h e f o l l o w i n g h ypotheses: 1. The f i r s t p roblem was e a s i e r t h a n the second p r o b l e m f o r a l l s t u d e n t s . 2. The example h e l p e d t h e f i r s t g r o u p i n d o i n g t h e e a s y p r o b l e m c o m p a r i n g t h e p r o p o r t i o n o f c o r r e c t s o l u t i o n s t o t h e e a s y p r o b l e m i n e a c h g r o u p . 3 . The s e c o n d g r o u p had a h i g h e r p r o p o r t i o n o f c o r r e c t s o l u t i o n s f o r t h e " h a r d " p r o b l e m t h a n t h e f i r s t g r o u p . k. The s e c o n d g r o u p had a h i g h e r p r o p o r t i o n o f c o r r e c t s o l u t i o n s f o r t h e " h a r d " p r o b l e m t h a n t h e f i r s t g r o u p when o n l y t h o s e s t u d e n t s who c o r r e c t l y s o l v e d t h e f i r s t p r o b l e m were c o n s i d e r e d . The f i r s t , s e c o n d , and f o u r t h h y p o t h e s e s were f o u n d t o be s i g n i f i c a n t b eyond t h e .05 l e v e l . The c o n c l u s i o n was drawn t h a t t h e u s e o f examples t o t e a c h a l g o r i t h m d e v e l o p m e n t on t h e computer i s a t l e a s t s ometimes i n a d v i s a b l e i n t h a t examples may h i n d e r t r a n s f e r o f t r a i n i n g f r o m e a s y p r o b l e m s t o h a r d e r p r o b l e m s and do n o t i n c r e a s e t h e numbers who c a n i n d e p e n d e n t l y s o l v e a h a r d e r p r o b l e m . ( T h i s assumes t h a t t h e i n d e p e n d e n t s o l u t i o n o f h a r d e r p r o b l e m s i s t h e o n l y i n s t r u c t i o n a l g o a l . ) A t b e s t t h e p r o v i s i o n o f s u c h examples may be a waste o f t i m e , a t w o r s t i t may be a d i s t r a c t i o n . I t was f e l t t h a t f u r t h e r r e s e a r c h u s i n g a g r e a t e r number and v a r i e t y o f examples, c l a s s i f i e d i n some way, and u s i n g a v a r i e t y o f t e x t u a l m a t e r i a l i s b o t h w a r r a n t e d and d e s i r a b l e . It was also f e l t that a test instrument could he devised which would identify those students who would most benefit from a course in algorithm development on the computer. i v TABLE OF CONTENTS CHAPTER P a S e 1. THE PROBLEM 1 Background . 1 Purpose o f the Study 5 S i g n i f i c a n c e o f the Study 5 R e l a t e d S t u d i e s 8 T r a n s f e r 8 E i n s t e l l u n g 1° A l g o r i t h m Development 1^ Statement o f the Hypotheses 22 I I . THE DESIGN OF THE STUDY 23 I n t r o d u c t i o n 23 The I n s t r u c t i o n a l D e v i c e 23 The E x p e r i m e n t a l D e v i c e 23 The C o n t r o l D e v i c e 23 The T e s t I n s t r u m e n t 2i+ The P i l o t S t u d i e s 26 The S e l e c t i o n of C l a s s e s 26 I n s t r u c t i o n o f Teachers 2? F o r m a t i o n o f Groups 28 Proced u r e 28 Apparatus s 29 S t a t i s t i c a l A n a l y s i s 30 I n t r o d u c t i o n 30 V CHAPTER Page Data 30 N u l l Hypotheses 31 S t a t i s t i c a l Treatment 3 2 I I I . ANALYSIS OF THE. RESULTS 34 Summary of Data 34 Testing of the Hypotheses 34 Hypothesis One 34 Hypothesis Two 36 Hypothesis Three 37 Hypothesis Four 38 Conclusions 39 Discussion of Additional Data 39 IV. CONCLUSIONS AND IMPLICATIONS OF THE STUDY .. 40 Introduction 40 Discussion of the Conclusions 40 The Content of the Lesson 40 The Method of Instruction 41 Length of the Experiment 41 Group Makeup 41 Mechanics of R i g i d i t y 41 Limitations of the Study 42 Suggestions f o r Future Studies........... 43 FOOTNOTES 45 BIBLIOGRAPHY 48 v i CHAPTER Page APPENDIX A - INSTRUCTIONAL DEVICE 53 APPENDIX B - EXPERIMENTAL DEVICE 58 APPENDIX C - CONTROL DEVICE 59 APPENDIX D - TEST INSTRUMENT 60 APPENDIX E - COMPUTER PRINTOUT 6 l v i i LIST OF TABLES TABLE Page I . Summary o f Da t a 3^ I I . T e s t o f H y p o t h e s i s One 35 I I I . T e s t o f H y p o t h e s i s One (a) 35 IV. T e s t o f H y p o t h e s i s One(h) 35 V. T e s t o f H y p o t h e s i s Two 36 V I . T e s t o f H y p o t h e s i s Three 37 V I I . T e s t o f H y p o t h e s i s Four 38 v i i i LIST OF" FIGURES FIGURE P a S e 1. C o n c e p t u a l framework f o r L e a r n i n g t o D e v i s e A l g o r i t h m s 4 ACKNOWLEDGEMENT The a u t h o r wishes t o thank t h e t e a c h e r s and s t u d e n t s o f West Vancouver s c h o o l s f o r p a r t i c i p a t i o n , my committee f o r a n t i c i p a t i o n , my w i f e f o r r e h a b i l i t a t i o n , and P r o f e s s o r B. H i c k s ( U n i v e r s i t y o f I l l i n o i s ) f o r i n s p i r a t i o n . 1 CHAPTER I THE PROBLEM BACKGROUND I f s t u d e n t s are p r o v i d e d w i t h examples i n o r d e r t o t e a c h them t o d e v i s e t h e i r own algorithms,"'" t h e y may "become r i g i d or s t e r e o t y p e d i n t h e i r approach t o ' a l g o r i t h m d e v e l o p -ment. A l t h o u g h t e a c h e r - p r o v i d e d examples may s e r v e i n i t i a l l y 2 as m e d i a t o r s f o r the t r a n s f e r o f t r a i n i n g , those s t u d e n t s who become i n v o l v e d i n systems work-^ cannot c o n t i n u e t o be p r o v i d e d w i t h s u i t a b l e m e d i a t i n g examples. Systems work i s here d e f i n e d t o mean, " o r g a n i z i n g c o l l e c t i o n s o f men, machines and methods." Such s t u d e n t s must l e a r n e i t h e r t o g e n e r a t e the examples themselves o r t o use some h e u r i s t i c w h i c h i s not dependent on examples. Each such s t u d e n t w i l l use h i s s k i l l i n a d i f f e r e n t c o n t e x t f rom t h a t i n which i t was f i r s t l e a r n e d . P e r s o n n e l t e a c h i n g i n the a u t h o r ' s mathematics computer l a b o r a t o r i e s have remarked t h a t the p r o v i s i o n o f m e d i a t i n g examples i n the e a r l y s t a g e s o f t e a c h i n g s t u d e n t s t o d e v i s e a l g o r i t h m s seemed t o i n h i b i t the l a t e r independent work o f t h e s e s t u d e n t s . I t was s t a t e d t h a t those s t u d e n t s who were not p r o v i d e d w i t h sample methods p r o g r e s s e d more s l o w l y a t f i r s t b ut were more s u c c e s s f u l when l a t e r t h e y had t o t r a n s f e r t h e i r a l g o r i t h m development s k i l l s t o more d i f f i c u l t i n d i v i d u a l l y chosen problems f o r which no examples c o u l d be p r o v i d e d . 2 Of e i g h t e e n s t u d e n t s i n Computer S c i e n c e who were p o l l e d a t the end o f the c o u r s e w i t h r e s p e c t t o the v a l u e o f sample programs, s i x t e e n r e p o r t e d t h a t examples i n h i b i t e d p r o g r e s s o r were u s e l e s s . T h i s s t u d y i n v e s t i g a t e s such c l a i m s . I f these c l a i m s a r e v a l i d the common p r a c t i c e o f p r o v i d i n g f r e q u e n t examples i n s e n i o r mathematics and computer s c i e n c e c o u r s e s may be unsound. The e d u c a t i o n a l g o a l under c o n s i d e r a t i o n i n t h i s s t u d y was the a b i l i t y t o produce a p l a n o r a l g o r i t h m . T h i s g o a l i s c a l l e d " S y n t h e s i s 5 * 2 " i n the Taxonomy of E d u c a t i o n a l  O b j e c t i v e s . ^ There was no i n t e r e s t i n the p a r t i c u l a r a l g o r i t h m i t s e l f (a number sequence g e n e r a t o r ) , nor i n the p a r t i c u l a r type o f a l g o r i t h m . ( a computer program i n the BASIC l a n g u a g e ) , b u t f o r s i m p l i c i t y o f e x p e r i m e n t a t i o n p a r t i c u l a r c h o i c e s were made. The s p e c i f i c example i n t r o d u c e d p r i o r t o the c r i t e r i o n t a s k s had t o be c a p a b l e o f l o w e r o r d e r m e d i a t i o n o f t r a n s f e r o f t r a i n i n g i n a l g o r i t h m development w h i l e a t the same time i n h i b i t i n g o r a t l e a s t n o t a s s i s t i n g h i g h e r o r d e r t r a n s f e r by i n d u c i n g a r i g i d approach t o a l g o r i t h m development (an E i n s t e l l u n g e f f e c t ) . S e v e r a l p i l o t s t u d i e s were u n d e r t a k e n i n o r d e r t o d i s c o v e r s u i t a b l e s u b j e c t s and m a t e r i a l s . The a u t h o r ' s e x p e r i e n c e s w i t h the i d e a o f "computers i n the whole s c h o o l " ^ and subsequent seminars on the same t o p i c ( a t t h e U n i v e r s i t y o f I l l i n o i s , November 1971i J u l y 197*0 had demonstrated the d e s i r a b i l i t y o f d i s c o v e r i n g t h e b e s t means of l e a d i n g a s t u d e n t so t h a t , "His e f f o r t s h o u l d y i e l d a p r o d u c t . . . something t h a t can be o b s e r v e d t h r o u g h one or more o f the s e n s e s , and w h i c h i s c l e a r l y more 7 t h a n the m a t e r i a l s he began t o work w i t h . " I t i s d i f f i c u l t t o g e t s t u d e n t s t o produce a p l a n or a l g o r i t h m f o r the computer. Bloom n o t e s t h a t " . . . c u r r e n t programs overemphasize a c t i v i t i e s i n w hich the l e a r n e r f u n c t i o n s as a consumer and c r i t i c o f g i d e a s r a t h e r t h a n t h o s e i n w h i c h he f u n c t i o n s as a p r o d u c e r . " A l t h o u g h many s t u d e n t s expect t o l e a r n l a r g e l y by w o r k i n g examples t h r o u g h i m i t a t i o n o f a t e a c h e r or t e x t , i t i s d i f f i c u l t • t o see how any s e r i e s o f examples c o u l d l e a d a s t u d e n t t o p e r f o r m such s o p h i s t i c a t e d t a s k s as programming a computer t o p e r c e i v e sound and t h e r e b y o p e r a t e a r o b o t , or t o p l a y music. o These t a s k s are a c c o m p l i s h e d by the s t u d e n t s o f Seymour P a p e r t . Such s o p h i s t i c a t e d programming r e q u i r e s a g r e a t d e a l o f s y n -t h e s i s , "the c a t e g o r y o f the c o g n i t i v e domain wh i c h most c l e a r l y p r o v i d e s f o r c r e a t i v e b e h a v i o r on the p a r t o f the l e a r n e r . " ^ " 0 I n the " p r o d u c t i o n o f a p l a n or proposed s e t o f o p e r a t i o n s " " ^ t o p e r f o r m such t a s k s t h e r e c l e a r l y must be p r a c t i c e a t the l o w e r t a x o n o m e t r i c l e v e l s o f "comprehension", " a p p l i c a t i o n " , and " a n a l y s i s " . T h i s p r e s e n t experiment r e a l l y o p e r a t e s o n l y a t t h e s e l o w e r t a x o n o m e t r i c l e v e l s , but where s y n t h e s i s i s the g o a l o f i n s t r u c t i o n and i f examples can be shown t o be a h i n d r a n c e even i n low l e v e l p r e p a r a t i o n f o r t h a t g o a l , t h e n examples s h o u l d be a v o i d e d a t such l e v e l s o f i n s t r u c t i o n . 4 The c o n c e p t u a l framework f o r l e a r n i n g t o d e v e l o p a l g o r i t h m s was as f o l l o w s : I Task. J I \^Higher o r d e r s k i l l s , t a s k output./ 7 1 M e d i a t o r s , s e l f g e n e r a t e d \ Lower o r d e r s k i l l s . Task Output. M e d i a t o r s : i ) t e x t or t e a c h e r e x p l a n a t i o n i i ) s u b o r d i n a t e t a s k s i i i ) g r o u p i n t e r a c t i o n i v ) s o l v e d examples v ) s e l f - g e n e r a t e d h e u r i s t i c s A l g o r i t h and pro yelopment s o l v i n g Low o r d e r s k i l l s i n terms o f d i f f i c u l t y , c o m p l e x i t y , and remoteness from p a s t e x p e r i e n c e , 5 The background f o r t h i s framework was d e r i v e d l a r g e l y from the work o f R.M. Gagne and S.S. Lee. The background f o r the phenomenon o f E i n s t e l l u n g r i g i d i t y was d e r i v e d from A.S. and E.H. L u c h i n s and K.M. M i l l e r . The computer equipment and i n s t r u c t i o n t e c h n i q u e s , as w e l l as t h e problem i t s e l f , o r i g i n a t e d i n the a u t h o r ' s computer c e n t r e s a t S e n t i n e l and H i l l s i d e Secondary S c h o o l s i n West Vancouver, B.C. PURPOSE OF THE STUDY I t was the a u t h o r ' s purpose t o i n v e s t i g a t e whether the p r o v i s i o n of an example t o s t u d e n t s l e a r n i n g t o d e v i s e t h e i r own computer a l g o r i t h m s r e s u l t e d i n a r i g i d and t h e r e f o r e u n s u c c e s s f u l approach t o independent a l g o r i t h m development. SIGNIFICANCE OF THE STUDY 12 Gagne and Brown s t a t e d i n 1961 t h a t "guided d i s c o v e r y " was most e f f e c t i v e , " d i s c o v e r y " n e x t most e f f e c t i v e , and " r u l e and example" l e a s t e f f e c t i v e i n p r o d u c i n g t r a n s f e r o f c o n c e p t u a l l e a r n i n g . I t would seem a l o g i c a l e x t e n s i o n t o the Gagne and Brown model i f p r o v i s i o n were made f o r j u d g i n g the e f f e c t i v e n e s s o f examples t o mediate "guided d i s c o v e r y " or t o c o n t r i b u t e t o the r e l a t i v e i n e f f e c t i v e n e s s o f " r u l e and example". T h i s paper may a i d the s t u d y o f the pr o b l e m by d e t e r m i n i n g i f an example c o u l d i n h i b i t o r a t l e a s t f a i l t o  a s s i s t h i g h e r o r d e r t r a n s f e r o f a l g o r i t h m - b u i l d i n g l e a r n i n g 6 sets and yet be capable of lower order transfer. I f the guidance of discovery through teacher-provided mediating examples must be terminated at some point, the learner must either be able to i n t e n t i o n a l l y develop mediators for himself or he must learn to do without mediators altogether. For example, i f the learner becomes a bio.logist who investigates the rel a t i o n s h i p between the sizes of rabbit and fox populations, and i f he wishes to produce a computer algorithm f o r the guidance of game management personnel, he must be able to r e l a t e concepts i n ecology and s t a t i s t i c s to s k i l l s i n mathematics and computer programming. C a l l i n g upon his past experience i n these f i e l d s he must develop a new algorithm which may be used by those not possessing his competencies. He must be able to synthesize relevant techniques i n a f l e x i b l e rather than i n a stereotyped manner. If examples are provided for low order tasks the learner may be rendered a disservice i n several ways. He may be delayed i n his development of the h e u r i s t i c s necessary to create algorithms using self-generated mediators. He may be misled i n his expectation of task d i f f i c u l t y by early s u p e r f i c i a l successes and poor motivation might result- from un-accustomed large increments i n d i f f i c u l t y when mediation i s withdrawn i n advanced courses or at school leaving. He might display an E i n s t e l l u n g e f f e c t i n his handling of data and techniques. 7 There would seem t o he i m p l i c a t i o n s ' h e r e f o r w orkers i n programmed i n s t r u c t i o n , computer a s s i s t e d i n s t r u c t i o n , and t e x t - h o o k w r i t i n g as w e l l as f o r mathematics and s c i e n c e c l a s s r o o m t e a c h e r s . I t i s o f t e n assumed, i t appears, t h a t s p e c i f i c examples o f a p r o c e s s b e i n g t a u g h t w i l l a lways mediate t r a n s f e r t o h i g h e r o r d e r uses o f the p r o c e s s (see f o r example, J . W a l t h e r , Computer A s s i s t e d Mathematics Program, S c o t t Foresman and Co., 1 9 6 9 ) . Even i f i t i s c l a i m e d t h a t examples are used o n l y t o a i d i n the e x p l a n a t i o n o f terms and c o n c e p t s and not t o mediate t h e i r use, i t may be t h a t c a r e must be e x e r c i s e d i n the use o f examples so t h a t r i g i d i t y o f m e n t a l p r o c e s s e s i s n o t produced as ,an u n d e s i r a b l e s i d e e f f e c t . ' I t may be t h a t such t o p i c s as a l g o r i t h m development i n mathematics and computer programming s h o u l d be t a u g h t i n the same manner as the game o f c h e s s ; t h a t i s , the moves o f each p i e c e a r e e x p l a i n e d (the meaning o f symbols, commands) and t h e n t h e l e a r n e r i s s i m p l y a l l o w e d t o p l a y ( d e v i s e a l g o r i t h m s , w r i t e programs). A t a much l a t e r s t a g e , a f t e r the l e a r n e r has d e v e l o p e d h i s own methods, he might s a f e l y u n d e r t a k e the s t u d y o f s p e c i a l t e c h n i q u e s by comparing h i s methods w i t h those o f e x p e r t s . I t might be s a f e t o l e a r n by example a t t h i s s tage as t h e r e might be l e s s chance o f a d o p t i n g a r i g i d approach t o the t a s k . 8 RELATED STUDIES A foundation w i l l "be l a i d for the considerat ion of transfer of t r a i n i n g i n learning hierarchies "by a discuss ion of the work of R.M. Gagne. The work of A . S . and E . H . Luchins and K.M. M i l l e r w i l l introduce the phenomenon of E i n s t e l l u n g . M. Wertheimer and J . M . Scandura are c i t e d for t h e i r work on problem so lv ing and algorithms; G.M. Haslerud, P. Suppes, T .A. Romberg and S.S. Lee for studies on guided discovery, item s t ructure , cognit ive i n d i v i d u a l differences and chaining cues re spec t ive ly . Transfer. In 19*^9 Gagne J studied the problem of measuring t rans fer . He concluded that the best means were; raw score, percentage improvement due to the contr ibut ion of the trans-ferred task, percentage improvement during t r i a l s and presence or absence of t ransfer as measured by coe f f i c i ent s of c o r r e l a t i o n . He stated that other measures of the effect of t ransfer of t r a i n i n g suffered from lack of compat ib i l i ty of empiricism. , lk • In 1961 he invest igated the re la t ionsh ips between learning sets i n knowledge a c q u i s i t i o n and concluded that there was a po s i t i ve transfer to each new learning set from relevant subordinate past learning and that the c o r r e l a t i o n between the mastery of t h i s subordinate material , and achievement 9 was h i g h e s t a t the h i g h e s t l e v e l o f h i s l e a r n i n g h i e r a r c h y . 1 ^ I n 1961 J he a l s o i n v e s t i g a t e d the e f f e c t s o f v a r i a t i o n i n programming c o n c e p t u a l l e a r n i n g m a t e r i a l s on l e a r n i n g p roblem s o l v i n g as measured by the time r e q u i r e d , h i n t s r e q u i r e d and s c o r e o b t a i n e d . The problems used i n v o l v e d d e r i v i n g the f o r m u l a f o r the sum o f terms i n u n f a m i l i a r number s e r i e s . Such f o r m u l a w r i t i n g may be c o n s i d e r e d as ' low l e v e l a l g o r i t h m development. 1 £1 By 1962 Gagne was w o r k i n g w i t h a u t o - i n s t r u c t i o n a l d e v i c e s t o d e f i n e what he c a l l s " p r o d u c t i v e l e a r n i n g " . By t h i s phrase he meant the k i n d o f change i n human b e h a v i o r w h i c h p e r m i t s the i n d i v i d u a l t o p e r f o r m s u c c e s s f u l l y an e n t i r e c l a s s ( o r system) o f s p e c i f i c t a s k s r a t h e r t h a n one member of the c l a s s . At t h i s time he s t a t e d t h a t , " t h e r e a r e no i n s t a n c e s o f an i n d i v i d u a l who i s a b l e t o p e r f o r m what has been i d e n t i f i e d as a ' h i g h e r £Le.vel' l e a r n i n g s e t , and who t h e n shows h i m s e l f unable t o p e r f o r m a 'lower l e v e l ' l e a r n i n g s e t r e l a t e d t o i t . " He f u r t h e r s t a t e d t h a t the r a t e o f a t t a i n -ment o f l e a r n i n g s e t s i n an h i e r a r c h y comes t o depend t o an i n c r e a s i n g e x t e n t on the l e a r n i n g s e t s w h i c h have j u s t p r e v i o u s l y been a c q u i r e d and a c c o r d i n g l y t o a d e c r e a s i n g e x t e n t upon a b a s i c f a c t o r or a b i l i t y . 17 I n I 9 6 3 Gage p o i n t e d out Gagne*s q u e s t i o n i n g o f t h e a s s u m p t i o n t h a t the b e s t way t o l e a r n a performance i s t o p r a c t i c e t h a t performance. 10 "In conditioning, c l a s s i c a l or otherwise, one observes learning only a f t e r the animal has made the f i r s t response and one conceives of what i s learned as either a response or an association terminating i n a response, i n either case established by p r a c t i c i n g the response with reinforcement. Gagne challenged t h i s on the grounds that the responses required do not have to be learned at a l l - they are already i n the human's repertoire..." "In t r a i n i n g men to trouble shoot (Gagne 1962) complex equipment, there was no single task to be produced, rather i t was the learning of an elaborate set of rules pertaining to the flow of signals through a complex c i r c u i t -another cognitive structure (system) that proves e s s e n t i a l . Rather than response e l i c i t a t i o n and reinforcement, as i s implied by at l e a s t some interpretations of conditioning theory, the more important p r i n c i p l e s i n t r a i n i n g , i n Gagnes view, deal with task analysis, i n t r a task transfer, component task achievement, and sequencing." That i s , the r e a l i z a t i o n of the structure was more important than the response elements. E ins te Hung. L u c h i n s 1 0 pointed out possible deleterious effects of habituated behavior. His method was as follows. Several problems, a l l solvable by one somewhat complex procedure, were presented i n succession. The problems involved using three water jars of given capacity to produce a ce r t a i n volume of water. Then a s i m i l a r problem was given which could be solved by a more direct and simple method. This problem was c a l l e d an "extinction problem". He found that most subjects persisted i n attempting to use the complex method. Although the order of d i f f i c u l t y was reversed i n the present study, and although 11 o n l y two problems were p r e s e n t e d , the i d e a o f E i n s t e l l u n g r i g i d i t y i s s t i l l p r e s e n t i n t h a t the s t u d e n t has the oppor-t u n i t y and the tendency t o copy a t e c h n i q u e r a t h e r t h a n use a d i r e c t method. L u c h i n s found t h a t the tendency t o copy the p r e v i o u s method of s o l u t i o n i n subsequent problems was independent of e d u c a t i o n a l l e v e l , age, and I.Q. He f u r t h e r found t h a t attempts t o p r e v e n t t h i s m e n t a l r i g i d i t y or E i n s t e l l u n g e f f e c t were q u i t e i n e f f e c t i v e . He t r i e d t o p r e v e n t i t s occurence by t e l l i n g s u b j e c t s t o ' a v o i d b l i n d n e s s ' a f t e r d o i n g a c e r t a i n p r o b l e m and b e f o r e d o i n g the n e x t . He made the problems more c o n c r e t e , he p r o v i d e d f a c i l i t i e s f o r e x p e r i m e n t a t i o n , a l l t o no a v a i l . He observed t h a t a h a b i t o f p r o b lem s o l u t i o n , "ceases t o be a t o o l d i s c r i m i n a t e l y a p p l i e d , but becomes a p r o c r u s t e a n bed t o w h i c h the s i t u a t i o n must conform; when, i n a word, i n s t e a d o f the i n d i v i d u a l ^ m a s t e r i n g the h a b i t , the h a b i t masters the i n d i v i d u a l . " L u c h i n s recommended t h a t l e a r n e r s must become accustomed t o problems w i t h too much d a t a . They must be p r e v e n t e d from l e a r n i n g 'type e x e r c i s e s . ' He o b s e r v e d t h a t t h e u s u a l c l a s s r o o m p r a c t i c e o f t e a c h i n g a s p e c i f i c t o p i c and t h e n a s s i g n i n g problems o n l y on t h a t s p e c i f i c t o p i c c o n t r i b u t e s t o the f o r m a t i o n o f E i n s t e l l u n g e f f e c t s i n problem s o l v i n g . F o r example, a f t e r a l e s s o n on t w o - d i g i t m u l t i p l i c a t i o n t h e t e a c h e r might a s s i g n as a f i r s t e x e r c i s e a q u e s t i o n i n s u b t r a c t i o n . He s u r m i s e d t h a t speeded t e s t s a l s o c o n t r i b u t e t o an E i n s t e l l u n g 12 e f f e c t i n that probably the most useful piece of information related to some problems i s the fact that they can be done i n so many minutes. The demand for speed causes blindness. Upon experimenting with changes i n these usual classroom procedures he was t o l d by the students, "I did what I was t o l d to do", 20 "you t r i c k e d us", "you taught us wrong." Luchins further stated that i f tests were interesting, i f substantial mark allowances were made f o r method of attack, and i f tests were presented as a method of helping students, there would be less mechanization of thinking fostered by them. 21 M i l l e r i n 1957 found r e s u l t s contradicting some of Luchins e a r l i e r tentative conclusions. He found that there i s a s i g n i f i c a n t (negative) r e l a t i o n between E i n s t e l l u n g r i g i d i t y or mechanization i n problem solving and i n t e l l i g e n c e . He also found that when sub-samples of technical and modern school boys were matched for i n t e l l i g e n c e and compared, the Ei n s t e l l u n g e f f e c t occurred s i g n i f i c a n t l y more often innthe modern school group (who were d r i l l e d ) than i n the technical school students (who were trained to search for alternative methods). This r e s u l t he discussed as a function of teaching methods and attitude to school. M i l l e r used a single school to regulate some factors not controlled i n the Luchin's study such as physical surroundings, teacher personality and socio-economic status of parents. The present writer has followed 13 M i l l e r ' s l e a d i n t h i s and assumed t h a t the d i s t r i b u t i o n o f i n t e l l i g a n c e and age i n t o sub-groups was a c h i e v e d by random-i z a t i o n . The M i l l e r method o f s c o r i n g ( c o u n t i n g the number o f c o n t r o l and e x t i n c t i o n problems s o l v e d ) was used and l i k e M i l l e r a p i l o t s t u d y was used t o determine s u i t a b i l i t y o f p roblem d i f f i c u l t y and S grade l e v e l f o r the r e s e a r c h a t hand. I n the d i s c u s s i o n o f h i s e x p e r i m e n t a l r e s u l t s M i l l e r remarked, "depending on what one c o n s i d e r s the aim o f e d u c a t i o n , i t c o u l d be argued t h a t i n s u i t i n g t e a c h i n g methods t o the l e s s a b l e modern s t r e a m p u p i l s , the more a b l e a r e p r e v e n t e d from d e v e l o p i n g as f l e x i b l y as t h e y o t h e r w i s e might have done. On the o t h e r hand i f i t i s c o n s i d e r e d d e s i r a b l e t o t r a i n p u p i l s t o approach the s o l u t i o n o f problems i n a mechana>gal way, the above f i n d i n g s need cause no c o n c e r n . " The p r e s e n t w r i t e r w i l l comment upon t h i s f u r t h e r i n C h a p t e r IV. I t i s s u f f i c i e n t a t t h i s p o i n t t o comment t h a t t h e s e l e c t i o n o f n o n - r i g i d s t u d e n t s f o r t r a i n i n g i n independent h i g h l e v e l a l g o r i t h m development on computers, might be more p r o d u c t i v e t h a n the s e l e c t i o n o f h i g h I.Q. s t u d e n t s . A l t h o u g h l a c k o f r i g i d i t y t ended t o accompany h i g h I.Q. the more s p e c i f i c and d i r e c t method o f s e l e c t i o n i s more a p p r o p r i a t e i n s c r e e n i n g i n d i v i d u a l t a l e n t s . M i l l e r ' s comments on s c h o o l morale and c l i m a t e are p a r t i c u l a r l y i m p o r t a n t t o t e a c h e r s c o n s i d e r i n g the i n s t a l l a t i o n o f computers and computer t r a i n i n g programs s i m i l a r t o t h o s e Ik o f the p r e s e n t w r i t e r . The p r e s e n t experiment was c a r r i e d out i n a s c h o o l w i t h a f r i e n d l y c l i m a t e but w i t h a ' s t r u c t u r e d ' academic approach t o e d u c a t i o n . The s c h o o l was i n an upper s o c i o - e c o n o m i c a r e a . A l g o r i t h m Development and Problem S o l v i n g . Wertheimer J d e s c r i b e d problem s o l v i n g as a s e a r c h f o r s t r u c t u r e which i s l i k e t he s e a r c h f o r the r e v e r s e d image i n an o p t i c a l i l l u s i o n . Perhaps some s t r u c t u r e s ( s u c h as new a l g o r i t h m s ) must be sought by t h e i n d i v i d u a l from the b e g i n n i n g w i t h o u t the a i d o f the m e d i a t o r s mentioned i n the c o n c e p t u a l framework ( f i g u r e 1). 2k Scandura r e l a t e d some e f f e c t s o f a l g o r i t h m l e a r n i n g and problem s o l v i n g . I t s h o u l d be noted t h a t t h i s i s n o t i d e n t i c a l w i t h the c e n t r a l i s s u e o f t h i s p a p e r , a l g o r i t h m development, however Scandura's r e s u l t s a re u s e f u l i n t h a t t h e y show what a l g o r i t h m s a r e , how,they can be g e n e r a l i z e d and how t h e y r e l a t e t o h i g h e r o r d e r t r a n s f e r i n problem s o l v i n g . Scandura d e s c r i b e d a l g o r i t h m s as f o l l o w s ; " A l g o r i t h m s e x i s t f o r s o l v i n g many t y p e s o f problems. The s t e p - b y - s t e p c o m p u t a t i o n a l p r o c e d u r e s used i n a r i t h -m e t i c perhaps p r o v i d e the most f a m i l i a r examples, but a l g o r i t h m s a re a l s o used i n d e a l i n g w i t h a l l s o r t s o f p r a c t i c a l and t h e o r e t i c a l problems - from ' t r o u b l e - s h o o t i n g ' t o m a t h e m a t i c a l p h y s i c s . A common f e a t u r e o f such p r o -cedures i s t h a t t h e y can be a p p l i e d m e c h a n i c a l l y w i t h o u t u n d e r s t a n d i n g . " 2 5 1 5 As a computer has no ' u n d e r s t a n d i n g ' and i s t r u l y ' m e c h a n i c a l ' i t o b v i o u s l y s o l v e s problems by means o f a l g o r i t h m s . The a l g o r i t h m s d e v e l o p e d f o r computer use u s u a l l y can be extended i n a p p l i c a t i o n t o human use and so Scandura's s t a t e m e n t would seem t o s u p p o r t the b e l i e f t h a t t h e r e e x i s t s a c o n t i n u i n g need f o r a l g o r i t h m d e v e l o p e r s and a l g o r i t h m u s e r s , who are not n e c e s s a r i l y d i s t i n c t p e r s o n s . C e r t a i n l y the p e r s o n w h o i o r i g i n a l l y d e v i s e d the a l g o r i t h m s h o u l d u n d e r s t a n d the t h e o r y o f i t s o p e r a t i o n c o m p l e t e l y . Scandura c o n t i n u e s ; "Many e d u c a t o r s and s u b j e c t m a t t e r s p e c i a l i s t s would m a i n t a i n t h a t a l g o r i t h m s a re l i m i t e d i n t h e i r u s e f u l n e s s t o s p e c i f i c p r o b lem s i t u a t i o n s and t h a t t r a n s f e r t o v a r i a n t s o f the o r i g i n a l problems r e q u i r e s an u n d e r s t a n d i n g o f u n d e r l y i n g p r i n c i p l e s . The f a c t t h a t u n d e r s t a n d i n g i s t y p i c a l l y d e f i n e d i n terms o f p e r -formance on some t r a n s f e r t a s k , however, poses a problem f o r r e s e a r c h i n g such c o n j e c t u r e s . To o p e r a t i o n a l l y d e f i n e u n d e r s t a n d i n g i n terms o f problem s o l v i n g t r a n s f e r would c l e a r l y l e a d t o c i r c u l a r i t y . One way t o overcome t h i s d i f f i c u l t y i s t o o p e r a t i o n a l l y d e f i n e ' u n d e r s t a n d i n g ' i n terms o f the amount o f i n f o r m a t i o n p r e s e n t e d . " 2 6 The p r e s e n t s t u d y , as w i l l be seen, v a r i e d the amount o f i n f o r m a t i o n t o determine the degree o f u n d e r s t a n d i n g . I n a s e r i e s o f t h r e e e x p e r i m e n t s ' ' ( ' * ~ y Scanduras demonstrated t h a t : a) S u c c e s s f u l p r o blem s o l v i n g does not n e c e s s a r i l y depend on an u n d e r s t a n d i n g o f the problem i n v o l v e d . ( T h i s l e n d s 16 s u p p o r t t o the modern n o t i o n o f 'systems* i n which symbol m a n i p u l a t i o n can o f t e n be s u b s t i t u t e d f o r u n d e r s t a n d i n g o f the p r o b l e m ) . b) T r a n s f e r does n o t depend on ' u n d e r s t a n d i n g * . That i s , i t i s p o s s i b l e t h a t s u b j e c t s d e t e c t r e l a t i o n s h i p s  and cues among a l g o r i t h m s and a r e a b l e t o m o d i f y them a c c o r d -i n g t o s y n t a c t i c c o n s t r a i n t s p r e s e n t between the a l g o r i t h m and the i n d i v i d u a l p r o b lem c h a r a c t e r i s t i c s . ( T h i s would seem t o be an e s s e n t i a l s k i l l f o r a systems programmer). c) I t i s p o s s i b l e t h a t , i n s t e a d o f a s k i n g , l i k e Gagne, 'What would S need t o know i n o r d e r t o do t h i s t a s k ? ' , one c o u l d be concerned w i t h the problem, 'Could i n f o r m a t i o n f a c i l i t a t e problem s o l v i n g ? ' . I n o t h e r words, the s t r u c t u r e d r e l a t i o n s h i p s c o u l d be o f prime i m p o r t a n c e . d) I t i s f e a s i b l e t o p r e d i c t s t u d e n t p r o blem s o l v i n g performance on a g i v e n t o p i c by s u b j e c t i v e l y a n a l y s i n g s t r u c t u r a l r e l a t i o n s h i p s between the performance c r i t e r i o n and t h e i n f o r m a t i o n p r e s e n t e d . e) Mere p r e s e n t a t i o n o f s u b o r d i n a t e m a t e r i a l i s n o t always s u f f i c i e n t t o ensure subsequent l e a r n i n g when the terms ( i e . symbols and words) d e n o t i n g the s u b o r d i n a t e n o t i o n s a r e used t o d e s c r i b e the h i g h e r o r d e r m a t e r i a l . That i s , Gagne was concerned w i t h s k i l l s w h i c h a r e more r e a d i l y t a u g h t and n o t 17 w i t h the p r e s e n t a t i o n of continuous discourse i n terms of sub-ordinate a s s o c i a t i o n s , concepts and p r i n c i p l e s . f ) There are at l e a s t some co n d i t i o n s under which p r e r e q u i s i t e p r a c t i c e has a greater e f f e c t on problem s o l v i n g performance immediately a f t e r l e a r n i n g than an equal amount of p r a c t i c e time at the c r i t e r i o n l e v e l . Scandura notes that the problem of g l o s s i n g over p r e l i m i n a r y t o p i c s so as to spend more time on 'main ideas *, p a r t i c u l a r l y when the p r e l i m i n a r y t o p i c s are r e l a t i v e l y u n f a m i l i a r to the students, i s a very r e a l problem i n many c o l l e g e classrooms, p a r t i c u l a r l y i n the t e c h n i c a l and s c i e n t i f i c areas. (This lends some support to the n o t i o n that the p r o v i s i o n of mediators i n p r e l i m i n a r y m a t e r i a l may be a means of g l o s s i n g over necessary p r a c t i c e i n a l g o r i t h m development). g) Problem s o l v i n g i s improved by the p r e -experimental a v a i l a b i l i t y of p r e r e q u i s i t e m a t e r i a l , even a f t e r repeated r e - i n t r o d u c t i o n of c r i t e r i o n l e v e l m a t e r i a l s , h i n t s and p r a c t i c e i n problem s o l v i n g , when the p r e r e q u i s i t e m a t e r i a l was presented p r i o r to the c r i t e r i o n m a t e r i a l , and by p r e r e q u i s i t e p r a c t i c e only when the p r e r e q u i s i t e m a t e r i a l came f i r s t . 30 31 In 1967 Scandura added these concl u s i o n s ; h) Knowing how to use an al g o r i t h m i s d i f f e r e n t 18 from knowing when t o use i t . There i s a tendency t o t r y t o s o l v e a group o f problems a l l i n the same way. ( c f . E i n s t e l l u n g ) . i ) The more g e n e r a l the i n t r o d u c t o r y m a t e r i a l and examples are the b e t t e r use w i l l be made of them. The p r e s e n t a u t h o r would comment t h a t s o l v e d examples a r e p r o b a b l y the l e a s t g e n e r a l form o f i n s t r u c t i o n . To remem-b e r a r u l e and an example may be s i m p l y t a k i n g the p a t h o f l e a s t r e s i s t a n c e . I n e x a m i n i n g a s o l v e d example a t a p r e r e q u i s i t e l e v e l w h i c h i s i n t e n d e d t o t r a n s f e r g e n e r a l i z e d s k i l l s t o h i g h e r o r d e r t a s k s t h e l e v e l o f t r a n s f e r may be too low - t h a t i s , i t may be too s p e c i f i c . The S_ might s i m p l y see an exemplar o f a problem type r a t h e r t h a n o f a co n c e p t . The e f f e c t o f cues l e a r n e d may t h e n have i n d u c e d an E i n s t e l l u n g e f f e c t . Scandura n o t e d t h a t the r e s u l t s o f h i s ex p e r i m e n t , however, were not u n e q u i v o c a b l e as the e f f e c t may depend upon the m a t e r i a l (groups and t o p o l o g y ) used. The same may be s a i d f o r the r e s u l t s of the p r e s e n t e x p e r i m e n t . Scandura used an example wh i c h a s s i s t e d Ss i n s o l v i n g the f i r s t p r o b lem p r e s e n t e d i n much the same way as the p r e s e n t E. Scandura, however, p r o v i d e d the h e l p f u l example t o a l l Ss and so was unable t o d i s c r i m i n a t e between the e f f e c t o f p r o v i d i n g or not p r o v i d i n g the example. The p r e s e n t s t u d y l o o k e d a t t h i s q u e s t i o n and t h e r e b y r e i n f o r c e d and expanded Scandura's s t a t e m e n t s c o n c e r n i n g the d i f f e r e n c e s i n achievement 19 on e x t i n c t i o n problems between those who a r e and t h o s e who a r e not " r u l e - u s e r s " . Scandura a l s o s e t a p r e c e d e n t f o r the use o f h i s t o r i c a l m a t e r i a l as a f i l l e r f o r the c o n t r o l group and he c o n f i r m e d e x p e r i m e n t a l l y t h a t t h i s s h o u l d not confound r e s e a r c h i n t o t r a n s f e r o f t r a i n i n g . Haslerud-^ 2 i n r e v i e w i n g the work o f K a t o n a (who m a i n t a i n e d t h a t t e a c h i n g by examples was b e s t ) and C r a i g (who found t h a t the more guidance p r o v i d e d the more e f f i c i e n t t h e d i s c o v e r y ) c o u n t e r e d t h e i r work w i t h an e x p e r i m e n t d e a l i n g w i t h c o d i n g problems which showed t h a t u n l e s s a p r i n c i p l e was d e r i v e d by the l e a r n e r ( r a t h e r t h a n s t a t e d and demonstrated t h r o u g h examples) t h e r e was no t r a n s f e r o f the p r i n c i p l e t o o t h e r problems. One would e x p e c t t o f i n d s i m i l a r r e s u l t s from e x p e r i m e n t s c o n c e r n i n g problem s o l v i n g i n computer programming. 33 S u p p e s ^ s t a t e d t h a t the l e a r n i n g o f s i m p l e m a t h e m a t i c a l c o n c e p t s i s more c l o s e l y a s s o c i a t e d w i t h an ' a l l or none' a s s u m p t i o n t h a n by a s i m p l e i n c r e m e n t a l assump-t i o n : That i s , r e s e a r c h w h i c h attempted t o show the p r o c e s s o f such l e a r n i n g may s i m p l y have shown the p r o b a b i l i t y p t h a t the s u b j e c t had r e a c h e d the c o n d i t i o n e d s t a t e (when p = l ) . T h i s may throw doubt upon the r e l e v a n c e o f some s t a t e m e n t s c o n c e r n i n g ' t r a n s f e r o f t r a i n i n g ' but i t h e l p e d t o e x p l a i n t h o s e s t u d i e s w h i c h showed 'lower o r d e r t r a n s f e r ' w i t h no 20 accompanying ' h i g h e r o r d e r t r a n s f e r ' . Suppes-^, l i k e Gagne, worked on the c r e a t i o n o f a model wh i c h can p r e d i c t from an item's s t r u c t u r e the p r o c e s s a s u b j e c t must go t h r o u g h i n f i n d i n g a c o r r e c t r e s p o n s e . I t s h o u l d be n o t e d t h a t i n t h i s paper a ' c o r r e c t r e s p o n s e ' was not equated w i t h ' l e a r n i n g a c o n c e p t ' . Romberg-^, j _ n c o n t r a s t t o Suppes, s t a t e d t h a t , " . . . i t seems p l a u s i b l e t h a t many i n s t r u c t i o n a l p r o c e d u r e s i n mathematics c o u l d be g u i d e d by a p p r o p r i a t e u t i l i z a t i o n o f i n f o r m a t i o n on c o g n i t i v e i n d i v i d u a l d i f -f e r e n c e s , but t h i s i s not the case. As i n d i v i d u a l i z a t i o n of i n s t r u c t i o n and computer management become a r e a l i t y , a p t i t u d e and a b i l i t y d a t a s h o u l d become e x t r e m e l y u s e f u l . Perhaps i n the n e x t decade c r a c k s i n the i r o n c u r t a i n w i l l appear." Among the a p t i t u d e s t o which Romberg r e f e r s s h o u l d be i n c l u d e d t h e a b i l i t y t o r e s i s t E i n s t e l l u n g r i g i d i t y . L ee-^'-^ c o n t i n u e s t o i n v e s t i g a t e the e f f e c t o f cha-ining cues ( m e d i a t o r s ) on l e a r n i n g 'complex c o n c e p t u a l r u l e s * such as l a b e l l i n g o b j e c t s o f v a r i o u s s i z e s c o l o u r s and shapes. He and Gagne seem t o b e l i e v e i n the e x i s t e n c e o f unique b o d i e s o f component l e a r n i n g w h i c h are n e c e s s a r y f o r the t r a n s f e r from l o w e r t o h i g h e r l e v e l c o n c e p t . However c h a i n i n g i s o n l y p o s s i b l e when a l l the l i n k s i n the c h a i n c a n be found. I n t o d a y ' s i n t e r d i s c i p l i n a r y systems, d e s i g n cues may be h a r d t o f i n d and the i d e n t i f i c a t i o n o f component l e a r n i n g f o r tomorrow's t a s k i m p o s s i b l e . T r a i n i n g i n ' l o o k i n g 21 outside the problem1 and 'avoiding blindness' may be what is required for at least some members of the population. 22 STATEMENT OF THE HYPOTHESIS I t i s hypothesized, t h a t a group o f s t u d e n t s who succeed on a low o r d e r computer a l g o r i t h m development t a s k a f t e r the p r o v i s i o n o f a s p e c i f i c example w i l l have p r o p o r t i o n a l l y fewer s u c c e s s e s on a h i g h e r o r d e r t a s k t h a n w i l l a group o f s t u d e n t s who succeeded on the low o r d e r t a s k w i t h o u t the use o f the s p e c i f i c example. I n p a r t i c u l a r : 1. The f i r s t p roblem i s e a s i e r t h a n the second problem f o r a l l s t u d e n t s . 2. The example h e l p s the f i r s t group i n d o i n g the "easy" problem comparing the p r o p o r t i o n o f c o r r e c t s o l u t i o n s t o the easy p r o b l e m i n b o t h groups. 3 . The second group have a h i g h e r p r o p o r t i o n o f c o r r e c t s o l u t i o n s f o r the "hard" problem t h a n t h e f i r s t group. k. The second group have a h i g h e r p r o p o r t i o n o f c o r r e c t s o l u t i o n s f o r the "hard" problem t h a n t h e f i r s t group when o n l y those s t u d e n t s who c o r r e c t l y s o l v e the f i r s t problem are c o n s i d e r e d . 2 3 CHAPTER I I THE DESIGN OF THE STUDY INTRODUCTION The f o l l o w i n g f o u r s e t s o f m a t e r i a l were f i r s t p r e p a r e d . The I n s t r u c t i o n a l D e v i c e . The i n s t r u c t i o n a l d e v i c e c o n s i s t e d o f a t h r e e page d u p l i c a t e d b o o k l e t w h i c h c o n t a i n e d , " D i r e c t i o n s t o Teachers and S t u d e n t s " , and "The L e s s o n " . ( c f . Appendix A) "The L e s s o n " p r o v i d e d i n s t r u c t i o n i n the use o f the BASIC computer language. P r i n t e d m a t e r i a l s were used i n o r d e r t o c o n t r o l t h e t e a c h e r v a r i a b l e . The E x p e r i m e n t a l D e v i c e . The e x p e r i m e n t a l d e v i c e c o n s i s t e d o f a s i n g l e d u p l i c a t e d page showing a s o l v e d example o f a program t o p r i n t the odd numbers. ( c f . Appendix B) The i n t e n t o f t h i s example was t o make the 'easy' t a s k v e r y easy f o r the e x p e r i m e n t a l group. The C o n t r o l D e v i c e . The c o n t r o l d e v i c e c o n s i s t e d o f a s i n g l e d u p l i c a t e d page g i v i n g a b r i e f h i s t o r y o f computers. ( c f . Appendix C) T h i s was t o be used by the c o n t r o l group w h i l e the e x p e r i m e n t a l 24 group used the e x p e r i m e n t a l d e v i c e . The T e s t I n s t r u m e n t . The t e s t i n s t r u m e n t c o n s i s t e d o f a s i n g l e d u p l i c a t e d page w h i c h i d e n t i f i e d the s t u d e n t , h i s group (A f o r e x p e r i m e n t a l , B f o r c o n t r o l ) , and a s s i g n e d two t a s k s . ( c f . Appendix D) Task one ( the 'easy' t a s k ) was t o program the computer t o p r i n t the even numbers. Task two (the ' d i f f i c u l t * t a s k ) was t o program the computer t o p r i n t the F i b o n a c c i numbers. The r e a s o n s f o r h a v i n g the a l g o r i t h m development s k i l l s d e v e l o p e d on a computer were: 1. Computer language c o u l d be used t o d e v e l o p a l g o r i t h m s . 2. I t would be p o s s i b l e t o f i n d Ss h a v i n g no p r e v i o u s e x p e r i e n c e w i t h computer languages t h e r e b y h e l p i n g t o c o n t r o l the e f f e c t o f p a s t l e a r n i n g e x p e r i e n c e s . 3 . As f o u r new terms (computer commands) were t o be i n t r o d u c e d the c o n f o u n d i n g e f f e c t o f p r e v i o u s l y l e a r n e d v o c a b u l a r y was re d u c e d . A t the same time complete a l g o r i t h m s c o u l d be programmed u s i n g o n l y these f o u r new commands (LET, PRINT, END, GOTO). 4. As the l e a r n e r s would never have been i n s t r u c t e d i n computer languages by any o t h e r i n s t r u c t i o n a l mode the c o n f o u n d i n g e f f e c t o f p r e v i o u s l e a r n i n g methods was reduc e d . 2 5 5. The e v a l u a t i o n of success could be accomplished q u i c k l y by an 'unbiased* computer w i t h l i t t l e of any k i n d of teacher or experimenter e f f e c t . 6. The whole sub jec t o f e d u c a t i o n a l uses of computers was of p a r t i c u l a r i n t e r e s t to the exper imenter . 26 THE PILOT STUDIES The m a t e r i a l s d e s c r i b e d above were used i n two p i l o t s t u d i e s t o determine t h e i r s u i t a b i l i t y as t o grade l e v e l , v o c a b u l a r y l e v e l , time l i m i t s and problem d i f f i c u l t y . The p i l o t s t u d i e s were e x e c u t e d i n the same manner as the main experiment w h i c h i s d e s c r i b e d l a t e r . I t was found i n the f i r s t p i l o t s t u d y w i t h grade s i x s t u d e n t s o f s e v e r a l West Vancouver e l e m e n t a r y s c h o o l s t h a t the grade s i x l e v e l was too low t o produce a s u f f i c i e n t number o f c o r r e c t programs. I n the second p i l o t s t u d y which used grade n i n e s t u d e n t s from a s i n g l e West Vancouver s e c o n d a r y y s c h o o l , i t was found t h a t r e v i s i o n i n the v o c a b u l a r y l e v e l o f b o t h the i n s t r u c t i o n a l d e v i c e and the t e s t , i n s t r u m e n t were r e q u i r e d . These needed r e v i s i o n s were u n d e r t a k e n and the f i n a l v e r s i o n s w h i c h appear i n the app e n d i c e s were p r o -duced. Time l i m i t s as d e s c r i b e d l a t e r were s e t on the b a s i s o f the second p i l o t s t u d y so t h a t no s t u d e n t r e q u i r e d more t i m e . THE SELECTION OF CLASSES C l a s s e s w h i c h were t o be used f o r the experiment were those c l a s s e s a t a se c o n d a r y s c h o o l i n West Vancouver, B.C. whi c h were made a v a i l a b l e t o the E by t h e i r mathematics t e a c h e r s . These t e a c h e r s agreed t o a t t e n d a meeti n g t o l e a r n about the experiment and t h e n t o a d m i n i s t e r the m a t e r i a l s under the guidance o f the E. A m u t u a l l y a g r e e a b l e hour was found so t h a t 27 a l l c l a s s e s c o u l d use the m a t e r i a l s a t the same time ( w i t h t h e i r own mathematics t e a c h e r i n a t t e n d a n c e ) t o ensure t h a t t h e r e would be no i n t e r a c t i o n between c l a s s e s . INSTRUCTION OF TEACHERS The t e a c h e r s whose c l a s s e s were t o be used f o r the experiment a t t e n d e d a meeting one week p r i o r t o the e x p e r i m e n t a l s e s s i o n a t which the E e x p l a i n e d the purpose o f the ex p e r i m e n t . The E a l s o e x p l a i n e d t h a t t h o s e s t u d e n t s who a d m i t t e d t o p r e v i o u s e x p e r i e n c e w i t h computers were n o t t o be used i n the exp e r i m e n t . Each t e a c h e r was p r o v i d e d w i t h a t a b l e o f random numbers and shown how t o a s s i g n s t u d e n t s t o group A or group B. Time l i m i t s were e x p l a i n e d t o t h e t e a c h e r s and the n e c e s s i t y o f p r e v e n t i n g any t a l k i n g or c o p y i n g by the s t u d e n t s was emphasized. I t was e x p l a i n e d t h a t the E would go from room t o room w h i l e the e x p e r i m e n t a l s e s s i o n was i n p r o g r e s s t o ensure t h a t groups were formed and t r e a t e d i n a u n i f o r m manner (the f o u r c l a s s r o o m s t o be used were i n the same wi n g o f the b u i l d i n g ) . The d i r e c t i o n s i n c l u d e d i n the i n s t r u c t i o n a l d e v i c e were d i s c u s s e d and r e a s o n s g i v e n f o r these d i r e c t i o n s . The t e a c h e r s were t o l d t h a t t h e y were t o a l l o w t e n minutes f o r t h e f o r m a t i o n o f groups and d i s t r i b u t i o n ( f a c e downward) o f "The L e s s o n " and the Group A or Group B page t o each s t u d e n t . They were t o l d t o a l l o w f i f t e e n minutes f o r the r e a d i n g o f t h i s m a t e r i a l , f i v e minutes f o r d i s t r i b u t i n g ( f a c e downward) the t e s t i n s t r u m e n t , 28 ten minutes for working on task one, and f i f t e e n minutes for working on task two (the teacher ensured, v i s u a l l y , that no student worked on task one when to l d to go on to task two). I t was explained that after the b e l l sounded s i g n a l l i n g the end of the test period the teacher was to c o l l e c t a l l materials and ask students to report whether they would have l i k e d more time or verbal explanations, and whether they would l i k e to meet with the E to discuss the materials. The teachers were asked to report the answers to these questions to the E. FORMATION OF THE GROUPS The population was formed as described above from members of four grade nine classes which were made available to the E by their mathematics teachers. Students who had previous computer experience were rejected from the sample. Each student was randomly assigned to group A or group B using a table of random numbers. Ninety-two subjects were used; forty-nine i n group A and forty-three i n group B. PROCEDURE On the day of the experiment the E was an observer i n the four experimental classrooms while the teachers handled the materials as described above. E ensured that the table of random numbers was used c o r r e c t l y and that the correct materials were distributed, the correct times kept, and that there was no int e r a c t i o n between Ss. 29 A l l t e a c h e r s r e p o r t e d t h a t t h e i r s t u d e n t s found the t ime l i m i t s adequate and t h a t no s t u d e n t e x p r e s s e d a d e s i r e f o r v e r b a l e x p l a n a t i o n . Three of the f o u r t e a c h e r s r e p o r t e d t h a t t h e i r s t u d e n t s e x p r e s s e d a d e s i r e t o meet w i t h the E and d i s c u s s the m a t e r i a l s . The E v i s i t e d t h e s e c l a s s e s d u r i n g t h e i r r e g u l a r mathematics c l a s s e s d u r i n g t h e week f o l l o w i n g t h e e x p e r i m e n t a l s e s s i o n and answered q u e s t i o n s d i r e c t e d t o him by t h e s t u d e n t s . The s t u d e n t s ' t a s k performances were judged b o t h t h r o u g h computer o p e r a t i o n upon t h e i r programs and by o b j e c t i v e o b s e r v a t i o n s o f s t u d e n t programming e r r o r s . The c r i t e r i o n f o r t a s k s u c c e s s was the a c t u a l p r o d u c t i o n by the computer, u s i n g S_s programs, of the r e q u i r e d number sequences. I n t h i s way the marker v a r i a b l e was c o n t r o l l e d . APPARATUS The a p p a r a t u s c o n s i s t e d i n i t i a l l y o f a H e w l e t t -P a c k a r d 2 0 0 7 system computer w i t h c a r d r e a d e r and BASIC s o f t w a r e . The f i n a l v e r s i o n o f the e xperiment used a H e w l e t t -P a c k a r d 9 8 3 0 computer w i t h h i g h speed p r i n t e r , o p t i c a l mark c a r d r e a d e r and BASIC s o f t w a r e . 30 STATISTICAL ANALYSIS I n t r o d u c t i o n . A C h i - s q u a r e d t e s t was used t o i n v e s t i g a t e the d i f f e r e n c e i n ; 1. The p r o p o r t i o n o f s u c c e s s e s on the easy t a s k as compared t o the second t a s k f o r a l l Ss and w i t h i n each group. 2. The p r o p o r t i o n o f s u c c e s s e s on the easy t a s k i n group A as compared t o group B. 3. The p r o p o r t i o n o f s u c c e s s e s on the d i f f i c u l t t a s k i n group A as compared t o group B. k. The p r o p o r t i o n o f s u c c e s s e s on the d i f f i c u l t t a s k i n group A as compared t o those i n group B who were s u c c e s s f u l on the easy t a s k . D a t a . F o r each s t u d e n t the two t a s k s were marked as s u c c e s s e s o r r f a i l u r e s on the b a s i s o f whether t h e y produced the c o r r e c t computer p r i n t - o u t . I n a d d i t i o n a s u b j e c t i v e d e c i s i o n was made on whether each group A s u b j e c t attempted t o use the method o f the p r o v i d e d example and o f t h e f i r s t t a s k i n o r d e r t o do the second t a s k . A s u b j e c t i v e d e c i s i o n was made on whether each group B s u b j e c t attempted t o use the method o f t a s k one i n o r d e r t o do the second t a s k . 31 N u l l Hypotheses. H 1. There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o -p o r t i o n o f Ss who succeeded on the f i r s t t a s k compared t o the p r o p o r t i o n of those who succeeded on the second t a s k . H l a . There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o -p o r t i o n o f Ss i n group A who succeeded on t h e f i r s t t a s k compared t o the p r o p o r t i o n o f those who succeeded on the second t a s k . H l b . There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o -p o r t i o n o f Ss i n group B who succeeded on the f i r s t t a s k as compared t o the p r o p o r t i o n o f t h o s e who succeeded on the second t a s k . H 2. There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f Ss i n group A who succeeded on the f i r s t t a s k compared t o the p r o p o r t i o n o f Ss i n group B who succeeded on the f i r s t t a s k . H 3« There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o -p o r t i o n o f Ss i n group A who succeeded on the second t a s k compared t o the p r o p o r t i o n o f Ss i n group B who succeeded on the second t a s k . H 4 . There i s no s i g n i f i c a n t d i f f e r e n c e i n the p r o -p o r t i o n o f S_s i n group A who succeeded on the second t a s k compared t o the p r o p o r t i o n o f Ss i n group B who succeeded on the second t a s k i f o n l y those s u b j e c t s who were s u c c e s s f u l on the f i r s t t a s k were examined. 32 S t a t i s t i c a l Treatment. The f o l l o w i n g c h i - s q u a r e d s t a t i s t i c s were c a l c u l a t e d w i t h one degree o f freedom and were t e s t e d a t the .05 l e v e l o f s i g n i f i c a n c e . A l i n d i c a t e d t h e number o f s u c c e s s e s o f Ss i n group A on t a s k 1. A2, B l , B2 were i n t e r p r e t e d s i m i l a r l y . A*2 and B*2 i n d i c a t e d the number o f s u c c e s s e s i n groups A and B on t a s k 2 o f those Ss who succeeded on t a s k 1. 'N' r e f e r r e d t o the t o t a l number o f Ss i n v o l v e d i n the e x p e r i m e n t . '#Correct* r e f e r r e d t o the number o f s u c c e s s e s i n each c a t e g o r y . 33 HI # C o r r e c t # I n c o r r e c t A l + B l E F A2+B2 G H H l a # C o r r e c t # I n c o r r e c t A l E F A2 G H H l b # C o r r e c t # I n c o r r e c t B l E F B2 G H H2 # C o r r e c t # I n c o r r e c t A l E F B l G H H3 # C o r r e c t # I n c o r r e c t A2 E F B2 G H H4 # C o r r e c t # I n c o r r e c t A*2 E F B*2 G H I n each case the computer c a l c u l a t e d the c h i - s q u a r e d 39 s t a t i s t i c a c c o r d i n g t o the formula-^ 7 CHI SQUARED = N (EH — F G ) 2 (E+F) (G+H) (E+G) (F+H) ( c f . Appendix E f o r the computer s t a t i s t i c a l t r e a t m e n t ) 34 CHAPTER I I I ANALYSIS OF THE RESULTS The f o l l o w i n g t a b l e summarizes the d a t a o b t a i n e d . TABLE I SUMMARY OF DATA Number o f Ss who: Group A Group B Had n e i t h e r t a s k c o r r e c t . 6 20 Task 1 c o r r e c t o n l y . 31 9 Task 2 c o r r e c t o n l y . 1 0 Tasks 1&2 c o r r e c t . 11 14 One or b o t h t a s k s c o r r e c t . 43 23 Task 1 c o r r e c t . 42 23 Task 2 c o r r e c t . 12 14 Showed E i n s t e l l u n g e f f e c t . 14 1 Number 49 43 TESTING OF THE HYPOTHESES H y p o t h e s i s one. H y p o t h e s i s one was t h a t t h e r e would be no s i g n i -f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f s u b j e c t s who succeeded on the f i r s t t a s k as compared t o the p r o p o r t i o n o f those who succeeded on the second t a s k . Hypotheses l a and l b d e a l t w i t h each group o f Ss s e p a r a t e l y . The f o l l o w i n g t a b l e s summarize the r e s u l t s o b t a i n e d : 35 TABLE I I TEST OF HYPOTHESIS ONE HI # C o r r e c t # I n c o r r e c t A l + B l 65 (71%) 27 (29$) A2+B2 26 (28%) 66 (72%) Degrees o f freedom = 1, C h i - s q u a r e = 33-069 TABLE I I I TEST OF HYPOTHESIS ONE(a) HIa # C o r r e c t # I n c o r r e c t A l 42 (86%) 7 (14%) A2 12 (2i+%) 37 (76%) Degrees o f "freedom = 1, C h i - s q u a r e = 37.121 TABLE IV TEST OF HYPOTHESIS ONE(b) H l b # C o r r e c t # I n c o r r e c t B l 23 (53%) 20 (47%) B2 r 4 (33%) 29 (67%) Degrees o f freedom = 1, C h i - s q u a r e = 3*842 The c r i t i c a l v a l u e o f C h i squared a t the .05 l e v e l o f s i g n i f i c a n c e ^ 0 w i t h one degree o f freedom i s 3«84l S i n c e a l l t h r e e v a l u e s o f C h i - s q u a r e d o b t a i n e d exceeded the c r i t i c a l v a l u e , the N u l l h y p o t h e s i s was r e j e c t e d and i t was co n c l u d e d t h a t t h e r e was a s i g n i f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f s u b j e c t s who succeeded on th e f i r s t t a s k 36 as compared t o the second t a s k . T h i s r e s u l t was o b t a i n e d even w i t h group B who had no example t o a s s i s t them i n d o i n g the f i r s t t a s k . I t may t h e r e f o r e be c o n c l u d e d t h a t the f i r s t t a s k was found t o be e a s i e r and i s t h e r e f o r e a 'lower o r d e r t a s k * i n computer a l g o r i t h m development. H y p o t h e s i s two. H y p o t h e s i s two was t h a t t h e r e would be no s i g n i -f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f S_s i n group A who succeeded on the f i r s t t a s k compared t o the p r o p o r t i o n o f Ss i n group B who succeeded on the f i r s t t a s k . The t a b l e below summarizes the r e s u l t s o b t a i n e d : TABLE V TEST OF HYPOTHESIS TWO H2 # C o r r e c t # I n c o r r e c t A l 42 (86%) 7 (14%) B l 23 (53%) 20 (47%) Degrees o f freedom = 1, Chi-squa^e = 11.471 S i n c e the c h i - s q u a r e d v a l u e o b t a i n e d exceeded the c r i t i c a l v a l u e the n u l l h y p o t h e s i s was r e j e c t e d and i t was c o n c l u d e d t h a t t h e r e was a s i g n i f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f Ss i n each group who succeeded on the f i r s t t a s k . I t may t h e r e f o r e ^ be c o n c l u d e d t h a t the s o l v e d example p r o v i d e d t o group A a s s i s t e d them i n p e r f o r m i n g the f i r s t t a s k 37 by a c t i n g as a m e d i a t o r between the i n s t r u c t i o n a l m a t e r i a l and the f i r s t t a s k . H y p o t h e s i s t h r e e . H y p o t h e s i s t h r e e was t h a t t h e r e would be no s i g n i -f i c a n t d i f f e r e n c e i n the p r o p o r t i o n o f Ss i n group A who s u c -ceeded on the second t a s k compared t o the p r o p o r t i o n o f Ss i n group B who succeeded on the second t a s k . The t a b l e below summarizes the r e s u l t s o b t a i n e d : TABLEAVI TEST OF HYPOTHESIS THREE H3 # C o r r e c t # I n c o r r e c t A2 12 (24%) 37 (76%) B2 14 (33%) 29 (67%) Degrees o f freedom = 1, C h i - s q u a r e = .735 S i n c e the c h i - s q u a r e d v a l u e o b t a i n e d d i d not exceed the c r i t i c a l v a l u e i t was c o n c l u d e d t h a t t h e r e was no s t a t i s -t i c a l l y s i g n i f i c a n t d i f f e r e n c e between the groups i n r e g a r d t o the p r o p o r t i o n who succeeded on the second t a s k . I t cannot t h e r e f o r e be c o n c l u d e d t h a t Ss i n group A as a group were h i n d e r e d by the example i n p e r f o r m i n g the second t a s k . E v i d e n t l y the example was n e i t h e r a h i n d r a n c e n or a h e l p t o the whole o f group A on the second t a s k . 38 Hypothesis f o u r . Hypothesis four was that there would he no s i g n i -f i c a n t d i f f e r e n c e i n the p r o p o r t i o n of Ss of group A who succeeded on the second task compared to the p r o p o r t i o n of Ss i n group B who succeeded on the second task i f only those Ss who were s u c c e s s f u l on the f i r s t task were examined. The t a b l e below summarizes the r e s u l t s obtained: TABLE VII TEST OF HYPOTHESIS FOUR H4 # Correct # I n c o r r e c t A*2 11 (26%) 31 (74%) B*2 14 (61%) 9 (39%) Degrees of freedom = 1, Chi-squared = 7 .551 Since the chi-squared value obtained exceeded the c r i t i c a l value the n u l l hypothesis was r e j e c t e d and i t was concluded that there was a s i g n i f i c a n t d i f f e r e n c e i n the p r o p o r t i o n of Ss i n the two groups who succeeded on the second task i f only those who were s u c c e s s f u l on the f i r s t task were examined. I t may ther e f o r e be concluded t h a t the solved example hindered any t r a n s f e r which might have occurred from the f i r s t task to the second task f o r those who s u c c e s s f u l l y completed the f i r s t task. 39 CONCLUSIONS C o n s i d e r i n g a l l f o u r hypotheses t o g e t h e r i t may he c o n c l u d e d t h a t the p r o v i s i o n o f an example w h i c h mediated t r a n s f e r o f s k i l l s t o an easy t a s k made no d i f f e r e n c e t o the number o f c o r r e c t s o l u t i o n s on the h a r d e r , c r i t e r i o n t a s k , but the s o l u t i o n o f the e a s i e r p r o b l e m (when a s o l v e d example was p r o v i d e d ) d i d h i n d e r the f u r t h e r u t i l i z a t i o n ( t r a n s f e r ) o f s k i l l s a c q u i r e d i n p e r f o r m i n g the f i r s t t a s k . DISCUSSION OF ADDITIONAL DATA The above r e s u l t s a r e c o n f i r m e d by the s u b j e c t i v e s c r u t i n y o f u n s u c c e s s f u l S_ programs. F o u r t e e n S_s i n group A att e m p t e d t o use the method o f s o l u t i o n demonstrated i n the example and a p p r o p r i a t e t o the f i r s t t a s k . Only one S i n group B attempted such a method. 40 CHAPTER IV CONCLUSIONS AND IMPLICATIONS OF THE STUDY INTRODUCTION The c e n t r a l h y p o t h e s i s o f t h i s s t u d y was t h a t the use o f a s p e c i f i c example w h i c h would mediate the t r a n s f e r o f s k i l l s from i n s t r u c t i o n t o the s o l u t i o n o f an easy problem would a l s o i n h i b i t t h e t r a n s f e r o f tho s e same s k i l l s from an easy problem t o a more d i f f i c u l t problem. That i s , the s t u d e n t was not u n s u c c e s s f u l because o f exposure t o the easy t a s k , b u t because o f h i s s o l u t i o n o f the t a s k u s i n g a s o l v e d example. As s i g n i f i c a n t r e s u l t s were o b t a i n e d i t was n e c e s s a r y t o c o n s i -der p o s s i b l e r e a s o n s . DISCUSSION OF THE CONCLUSIONS The Content o f the L e s s o n . I n c h o o s i n g the m a t e r i a l f o r i n s t r u c t i o n the e x p e r i m e n t e r used p i l o t s t u d i e s t o a s s i s t him t o choose a s u i t a b l e r e a d i n g and grade l e v e l . I t may have been t h a t the p a r t i c u l a r c h o i c e s o f w r i t t e n m a t e r i a l and grade l e v e l as w e l l as the i n t e r e s t v a l u e o f the s u b j e c t m a t t e r i n f l u e n c e d the r e s u l t s . However i t was the purpose o f the s t u d y o n l y t o f i n d i f r i g i d i t y c o u l d be i n d u c e d by the use o f a s p e c i f i c example. Of course o t h e r examples were used i n the i n s t r u c t i o n common t o bo t h c r e x p e r i m e n t a l groups and the e f f e c t s o f the s e examples s h o u l d be c o n s i d e r e d i n f u r t h e r r e s e a r c h . 41 The Method o f I n s t r u c t i o n . F o r purposes o f e x p e r i m e n t a t i o n w r i t t e n d i r e c t i o n s , l e s s o n s , examples, and t a s k s e l e c t i o n were used. I t may be t h a t the r e s u l t s o f t h i s s t u d y r e f l e c t S_s r e a c t i o n t o w r i t t e n m a t e r i a l . Such t e c h n i q u e s as o r a l d i s c u s s i o n and computer d e m o n s t r a t i o n might have produced d i f f e r e n t r e s u l t s due t o d i f f e r e n c e s i n e d u c a t i o n a l c l i m a t e , m o t i v a t i o n , and e x p e c t a -t i o n o f s u c c e s s . L e n g t h o f the E x p e r i m e n t . A l t h o u g h p i l o t s t u d i e s and o b s e r v a t i o n s d u r i n g e x p e r i m e n t a t i o n c o n f i r m e d t h a t no S_ wanted more t i m e , i t i s • p o s s i b l e t h a t the one hour l o n g e x p e r i e n c e i n d u c e d f a t i g u e i n some Ss and t h e r e b y i n f l u e n c e d the r e s u l t s . Group Makeup. A l t h o u g h random assignment t o groups was n e c e s s a r y i n o r d e r t o have e q u i v a l e n t groups and a l t h o u g h i t was made obviou s t h a t assignment t o groups was random, t h e r e may have been a c u r i o s i t y about the m a t e r i a l s o t h e r s were r e c e i v i n g w h i c h d i s t r a c t e d some Ss. Mechanics o f R i g i d i t y . No c o n c l u s i o n s have been formed as t o t h e p o s s i b l e causes o f r i g i d i t y d i s c u s s e d i n the i n t r o d u c t i o n : Whether an 42 o r i g i n a l p r e d i s p o s i t i o n t o r i g i d i t y i s i n n a t e o r l e a r n e d , whether the l e a r n e r i s d e l a y e d "by examples i n h i s development o f n e c e s s a r y h e u r i s t i c s , m i s l e d i n h i s e x p e c t a t i o n o f t a s k d i f f i c u l t y or h i n d e r e d i n some o t h e r way. These q u e s t i o n s a re beyond the scope o f t h i s paper. I t has been demonstrated, however, t h a t r i g i d i t y can be i n d u c e d i n a t l e a s t some Ss by the use o f an example. LIMITATIONS OF THE STUDY Some l i m i t a t i o n s o f the s t u d y have a l r e a d y been i m p l i e d i n the above d i s c u s s i o n s o f the c o n c l u s i o n s w i t h r e s p e c t t o the l e s s o n c o n t e n t , method o f i n s t r u c t i o n , l e n g t h o f the experiment and group makeup. There are a l s o o f course the u s u a l l i m i t a t i o n s o f the as s u m p t i o n o f r a n d o m i z a t i o n o f i n t e l l i g e n c e . A f u r t h e r l i m i t a t i o n i s t h a t a l l the Ss i n the f i n a l e xperiment came from the same s c h o o l . T h i s s c h o o l i s i n an upper s o c i o - e c o n o m i c d i s t r i c t and has a h i s t o r y o f e d u c a t i o n a l i n n o v a t i o n . F o r t h e purposes o f the q u e s t i o n a t hand, whether r i g i d i t y can be i n d u c e d , the c h o i c e o f a s i n g l e s c h o o l ( w i t h i t s p a r t i c u l a r s o c i o - e c o n o m i c s t a t u s and I.Q. range) was i d e a l i n t h a t i t c o n t r o l l e d a number o f v a r i a b l e s . F o r the purpose o f g e n e r a l i z i n g the r e s u l t s o f t h i s s t u d y f u r t h e r , e s p e c i a l l y f o r t e x t - b o o k w r i t e r s and t e a c h e r e d u c a t o r s , the l i m i t a t i o n s are g r e a t . F u r t h e r s t u d i e s f o r g e n e r a l i z a t i o n s h o u l d be conducted i n as wide a v a r i e t y o f s c h o o l s as p o s s i b l e . 43 SUGGESTIONS FOR FUTURE STUDIES Scandura demonstrated the f e a s i b i l i t y o f p r e d i c t i n g p r o b lem s o l v i n g performance by s u b j e c t i v e l y a n a l y s i n g s t r u c t u r a l r e l a t i o n s h i p s between the c r i t e r i o n and the i n f o r m a t i o n p r e -s e n t e d . I n the p r e s e n t s t u d y a s t r u c t u r a l r e l a t i o n s h i p i n v o l -v i n g c o p y i n g was demonstrated i f a l l p r i n t e d m a t e r i a l s up t o t h e c r i t e r i o n l e v e l t a s k were c o n s i d e r e d as ' i n f o r m a t i o n p r e s e n t e d ' . T h i s s t r u c t u r e was i n a p p r o p r i a t e a t the f i n a l c r i t e r i o n l e v e l , y e t i t would seem t o the a u t h o r t h a t p r e c i s e l y t h i s s t r u c t u r e i s p r e s e n t e d t o s t u d e n t s u s i n g many t e x t books i n h i g h s c h o o l mathematics and computer s c i e n c e . T y p i c a l l y the i n t r o d u c t i o n t o a t o p i c i s f o l l o w e d by t h r e e or f o u r s p e c i -f i c s o l v e d examples and t h e n by a s e t o f e x e r c i s e s . The s t u d e n t who uses t h e s e examples as g e n e r a l i z a t i o n s o b t a i n s m e d i a t i o n o f t r a n s f e r from what he has r e a d t o the e x e r c i s e s w h i c h he must now dp_. The s t u d e n t who a t t e m p t s t o a p p l y the methods o f the examples too r i g i d l y t o the e x e r c i s e s f a i l s on the more d i f f i c u l t e x e r c i s e s w h i c h , i n h i g h e r l e v e l c o u r s e s , may o f t e n be c o n s i d e r e d the t r u e c r i t e r i o n l e v e l . Indeed, the f i r s t few problems i n the e x e r c i s e are f r e q u e n t l y i n t e n d e d as no more t h a n d r i l l and p r a c t i c e i n the use o f v o c a b u l a r y , d e f i n i t i o n s , and n o t a t i o n . L u c h i n s and M i l l e r have a l r e a d y demonstrated t h a t some s t u d e n t s have a p r e d i s p o s i t i o n t o t h i s s o r t o f E i n s t e l l u n g r i g i d i t y and the p r e s e n t s t u d y shows t h a t the example may r e i n f o r c e t h i s p r e d i s p o s i t i o n ( i f E i n s t e l l u n g 4 4 r i g i d i t y i s thought o f as a l e a r n e d b e h a v i o r ) . T h i s w i l l h i n d e r what Gagne c a l l s p r o d u c t i v e l e a r n i n g . I t seems t h e r e -f o r e t h a t f u r t h e r r e s e a r c h s h o u l d be u n d e r t a k e n w h i c h i n v e s t i -g a t e s r i g i d i t y u s i n g a g r e a t e r v a r i e t y o f examples, c l a s s i f i e d i n some way, and a g r e a t e r v a r i e t y o f t e x t u a l m a t e r i a l w h i c h c o n t a i n s e f f e c t i v e w arnings t o a v o i d b l i n d n e s s . Such r e s e a r c h s h o u l d use l a r g e r b l o c k s o f i n s t r u c t i o n , u s i n g a whole t e x t book t o p i c and t a k i n g c o g n i z a n c e o f Scandura's c a v e a t t h a t Ss w i l l c o n t i n u e t o respond i n a r i g i d way u n l e s s c o n f r o n t e d w i t h feedback w h i c h i n d i c a t e s t h a t 'the r u l e has changed'. I f i t i s f e l t t h a t E i n s t e l l u n g r i g i d i t y i s somehow a f u n c t i o n o f p e r s o n a l i t y , t h e n i t would be a p p r o p r i a t e t o c o n s t r u c t i n s t r u m e n t s which would i d e n t i f y w hich s t u d e n t s would most b e n e f i t from a c o u r s e i n such d i s c i p l i n e s as a l g o r i t h m development on the computer. I t c o u l d perhaps be shown t h a t a l t h o u g h t h e r e i s a n e g a t i v e c o r r e l a t i o n between i n t e l l i g e n c e and E i n s t e l l u n g r i g i d i t y i t i s p o s s i b l e t o c o n s t r u c t s p e c i f i c t e s t s f o r r i g i d i t y w h i c h are b e t t e r f o r the purpose t h a n i n t e l l i g e n c e t e s t s . 45 FOOTNOTES k5a 1 A l g o r i t h m - "A p r e s c r i b e d s e t o f w e l l d e f i n e d r u l e s o f p r o c e s s e s f o r the s o l u t i o n o f a problem i n a f i n i t e number o f s t e p s , e.g. a f u l l s t a t e m e n t o f an a r i t h m e t i c p r o -cedure f o r e v a l u a t i n g s i n x t o a s t a t e d p r e c i s i o n . " A m e rican  S t a n d a r d V o c a b u l a r y f o r I n f o r m a t i o n P r o c e s s i n g (American S t a n d a r d s A s s o c i a t i o n , 1966) p.9« The term "program" was n o t used as i t might i m p l y a d i r e c t t r a n s l a t i o n i n t o computer language o r s i m p l y the r e v i s i o n o f e x i s t i n g programs. 2 M e d i a t o r - a b r i d g e f o r the t r a n s f e r o f t r a i n i n g whereby r e l e v a n t m a t e r i a l and s k i l l s a r e r e c a l l e d . ^Systems work - " O r g a n i z i n g c o l l e c t i o n s o f men, machines and methods." American S t a n d a r d V o c a b u l a r y , o p . c i t . , p.2 6 . S t u d e n t s i n the a u t h o r ' s grade e l e v e n computer s c i e n c e c o u r s e u n d e r t a k e the p r o d u c t i o n o f packages f o r such t a s k s a s ; s c h o o l s c h e d u l i n g , r e p o r t i n g , a t t e n d a n c e , a c c o u n t i n g , computer a s s i s t e d i n s t r u c t i o n , c h e m i s t r y l a b o r a t o r y r e p o r t marking, a n a l y s i s o f f u n c t i o n s , g r a p h i n g o f c o n i e s , s u n r i s e p r e d i c t i o n , and g e n e t i c a n a l y s i s . 4 / Benjamin S. Bloom ( e d . ) , Taxonomy o f E d u c a t i o n a l O b j e c t i v e s , Handbook I : The C o g n i t i v e Domain (New York: D a v i d McKay Company I n c . , 1 9 5 6 ) , p.170. -'For the use o f the terms " m e d i a t o r " , " t r a n s f e r " , l o w e r l e v e l " , " h i g h e r l e v e l " , c f . R o b e r t M. Gagne, "The . A c q u i s i t i o n o f Knowledge, " P s y c h o l o g i c a l Review, 59 35-65• 1962. F o r the use o f " E i n s t e l l u n g " c f . A.S. and E.H. L u c h i n s , " M e c h a n i z a t i o n i n Problem S o l v i n g , the E f f e c t o f Einstellung»." P s y c h o l o g i c a l Monograph, 54, No.6, p . l , 1942. 6W. P. Goddard, "Computers and The Whole School','' J o u r n a l o f E d u c a t i o n a l Data P r o c e s s i n g , 6:108-120, 1969 . n Benjamin S. Bloom ( e d . ) , Taxonomy o f E d u c a t i o n a l  O b j e c t i v e s , Handbook I : The C o g n i t i v e Domain, (New York: D a v i d McKay Company I n c . , 1956) p.162 , ( S y n t h e s i s ) . ^ I b i d . , p.166. o Seymour P a p e r t , "Teaching C h i l d r e n Thinking-',' (memo LOGO L a b o r a t o r y , 545 Technology Square, Cambridge Mass.). "^Bloom, op. c i t . , p. 1 6 2 . 1 1 I b i d . , S y n t h e s i s 5.2, p . 1 7 0 . 12 R o b e r t M. Gagne and L a r r y T. Brown,."Some F a c t o r s i n the Programming o f C o n c e p t u a l L e a r n i n g , " J o u r n a l o f  E x p e r i m e n t a l P s y c h o l o g y , 12:p.313, I 9 6 I . 46 3R.M. Gagne, H. F o s t e r , and M.H. Crowley, "The Measurement o f T r a n s f e r o f T r a i n i n g , " P s y c h o l o g i c a l B u l l e t i n , 45: 97-130, 1948. 1 4 R.M. Gagne and N.E. P a r a d i s e , " A b i l i t i e s and L e a r n i n g S e t s i n Knowledge A c q u i s i t i o n , " P s y c h o l o g i c a l  Monograph, 75:1-23, 1961. 1 ^Robert M. Gagne and L a r r y T. Brown, "Some F a c t o r s i n the Programming o f C o n c e p t u a l L e a r n i n g , " J o u r n a l o f E x p e r i -m e n t a l P s y c h o l o g y , 62:313-321, 1 9 6 1 . -1 ^ R.M. Gagne, "The a c q u i s i t i o n o f Knowledge," P s y c h o l o g i c a l Review, 69:p.360, 1962 . 1 7 N.L. Gage, "Paradigms f o r R e s e a r c h on T e a c h i n g , " Handbook o f R e s e a r c h on T e a c h i n g , (Rand McNally,1963)1 p.135. 18 A.S. and E.H. L u c h i n s , "New E x p e r i m e n t a l Attempts a t P r e v e n t i n g M e c h a n i z a t i o n i n Problem S o l v i n g , " J o u r n a l o f  G e n e r a l Psychology." 42:p.297, 1950. 1 9 yA.S. L u c h i n s , " M e c h a n i z a t i o n i n P r o b l e m S o l v i n g , the E f f e c t o f E i n s t e l l u n g , " P s y c h o l o g i c a l Monograph, 84:p.93,1942. 2 0 I b i d . . p.91. 21 Kenneth M. M i l l e r , " E i n s t e l l u n g R i g i d i t y , I n t e l l i g e n c e , and T e a c h i n g Methods," B r i t i s h J o u r n a l o f  E d u c a t i o n a l P s y c h o l o g y , 27:127-134, 1957. op I M d l , p. 134. 2 3 ^M. Wertheimer, P r o d u c t i v e T h i n k i n g , (Harper and Row, 1945). 24 J.M. Scandura, " A l g o r i t h m L e a r n i n g and Problem S o l v i n g , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 3 4 : p . l , I966A. 2 5 I b i d . , p . l . 26 . . o p . c i t . 2 7 I b i d . , p . l . 2 8 J.M. Scandura, "Problem S o l v i n g and P r i o r L e a r n i n g , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 34:4-7, 1966B. 2 9 . . . yJ.M. Scandura, " P r i o r L e a r n i n g , P r e s e n t a t i o n Order, and P r e r e q u i s i t e P r a c t i c e i n Problem S o l v i n g , " The J o u r n a l o f E x p e r i m e n t a l E d u c a t i o n , 34:p.l2, I966C. 47 30 J J.M. Scandura, E. Woodward, and F. Lee, "Rule G e n e r a l i t y and C o n s i s t e n c y i n Mathematics L e a r n i n g , " A m e rican E d u c a t i o n a l R e s e a r c h J o u r n a l , 4 : p . 3 0 3 » I 9 6 7 A . 31 J.M. Scandura and J.N. W e l l s , "Advance O r g a n i z e r s i n L e a r n i n g A b s t r a c t M a t h e m a t i c s , " American E d u c a t i o n a l  R e s e a r c h J o u r n a l , 4 : p . 2 9 5 , I 9 6 7 B . ^2G.M. H a s l e r u d and S. Megers, "The T r a n s f e r V a l u e o f G i v e n and I n d i v i d u a l l y D e r i v e d P r i n c i p l e s , " J o u r n a l o f -E d u c a t i o n a l P s y c h o l o g y . 4 9 : 2 9 3 - 2 9 8 , 1 9 5 8 . 33 -^P. Suppes, " M a t h e m a t i c a l Concept F o r m a t i o n m C h i l d r e n , " American P s y c h o l o g i s t , 2 1 : p . l 3 9 , 1 9 6 6 . 34 P. Suppes and M. J e r n a n , "Computer A s s i s t e d I n s t r u c t i o n , " The B u l l e t i n o f the N a t i o n a l A s s o c i a t i o n o f  Secondary S c h o o l P r i n c i p a l s , p . l , Feb. 1 9 7 0 . 3< ^T.A. Romberg, "Research i n Mathematics E d u c a t i o n , " Review of E d u c a t i o n a l R e s e a r c h , 7 : p . 4 8 l , 1 9 6 9 . ^ 6 S . S . Lee and R.M. Gagne, " E f f e c t s o f C h a i n i n g Cues on the A c q u i s i t i o n o f a Complex C o n c e p t u a l R u l e , " J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y , 8 0:p.468, I 9 6 9 . 37 S.S. Lee, " T r a n s f e r from Lower L e v e l t o H i g h e r L e v e l Concept," J o u r n a l o f V e r b a l L e a r n i n g and V e r b a l  B e h a v i o r , 7 : p - 9 3 0 , 1 9 6 8 . ^ 8 S . S . Lee and R.M. Gagne, "The E f f e c t s o f the Degree o f Component L e a r n i n g on the A c q u i s i t i o n o f a Complex C o n c e p t u a l R u l e , " J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y , 8 3 : 1 3 - 1 8 , 1 9 7 0 . 39 -^Deobold B. VanDalen, U n d e r s t a n d i n g E d u c a t i o n a l  R e s e a r c h , ( M c G r a w - H i l l S e r i e s i n E d u c a t i o n , I 9 6 6 ) , p . 4 l 3 . 4 0 Roger E. K i r k , E x p e r i m e n t a l D e s i g n P r o c e d u r e s f o r the B e h a v i o r a l S c i e n c e s , (Belmont:Brooks C o l e P u b l i s h i n g Company, 1968), p . 5 3 0 . BIBLIOGRAPHY 48a A l l e n , F. "Teaching f o r G e n e r a l i z a t i o n i n Geometry," The  Mathematics Teacher, 43 :245-251, 1950 . Alspaugh, C a r o l Ann. " I d e n t i f i c a t i o n of some Components of Computer Programming," Journa l f o r Research i n  Mathematics Education, 3 : 2 ; 8 9 - 9 8 , 1972 . American Standard Vocabulary f o r Information Processing, American Standards A s s o c i a t i o n , 1966 . A u s t i n , C A . "The Laboratory Method i n the Teaching of Geometry, "The Mathematics Teacher, 20:286-294, 1927 . Ausubel, D.P. Learning by Discovery-Rationale and Mystique, Urbana Bureau of E d u c a t i o n a l Research, U n i v e r s i t y of I l l i n o i s , I 9 6 I . B e l l a r d i n e l l i , Mario. C r i t e r i a and E v a l u a t i o n Tools i n the  T r a i n i n g of Programmers, I n t e r n a t i o n a l Conference on T r a i n i n g f o r Information, Rome, 1971• Bloom, Benjamin S. Taxonomy of E d u c a t i o n a l O b j e c t i v e s: Handbook I , The Cog n i t i v e Domain, New York: David McKay C o l L t d . , 1956 . B r e s k i n , Stephen et a l . "Non Verbal R i g i d i t y .& P e r s e v e r a t i o n , " J o u r n a l of Psychology, 75:239-42, 1970 . B r e s k i n , Stephen and R i c h , Kenneth. " C o r r e l a t i o n of Non V e r b a l R i g i d i t y and I n t e l l i g e n c e i n Elementary School C h i l d r e n , " Perceptual and Motor S k i l l s , 3 2 : 1 9 4 , 1 9 7 1 . Bry, Peter M. and Nawas, M. " R i g i d i t y : A Function of R e i n f o r c e -ment," Per c e p t u a l and Motor S k i l l s , 29:1;118, 1969 . C a t t e l l , R.B. and T i n e r , G. "The V a r i e t i e s of S t r u c t u r a l R i g i d i t y , " J o u r n a l of P e r s o n a l i t y , 1 7 : 3 2 1 - 3 2 4 , 1949 . C a t t e l l , R.B. and Winder, A.E. " S t r u c t u r a l R i g i d i t y i n R e l a t i o n to Learning Theory and C l i n i c a l Psychology," P s y c h o l o g i c a l Review, 59:23-28, 1952 . Cunningham, John D. " R i g i d i t y i n Ch i l d r e n ' s Problem S o l v i n g , " School Science and Mathematics, 6 6 : l l 4,No.5 8 2 whole, I 9 6 6 . Danver J . "The Computer as a Study Tool," B u l l e t i n of the  N a t i o n a l A s s o c i a t i o n of Secondary School P r i n c i p a l s , 54:18-26, 1970 . Durward, M. Lynn. "An E v a l u a t i o n of Computer Education i n Vancouver Secondary Schools," Vancouver Board of School Trustees, Department of Planning and E v a l u a t i o n , 1 9 7 3 . 49 Durward, M. Lynn and E l l i s , E.N. "An Evaluation of Computer Science 11E," Vancouver Board of School Trustees, Department of Planning and Evaluation, 1973-Gage, N.L. "Paradigms for Research on Teaching," Handbook of  Research on Teaching, New York: Rand McNally, p.89-141, 1963: Gagne, R.M. " M i l i t a r y Training and P r i n c i p l e s of Learning," American Psychologist. 17:83-91, 1962. Gagne, R.M. "The A c q u i s i t i o n of Knowledge," Psychological  Review, 59*355-365, 1962. Gagne, R.M. "Learning and P r o f i c i e n c y y i n Mathematics," The Mathematics Teacher, 56:620-626, 1963. Gagne, R.M. . The Conditions of Learning. New York: Holt Rinehart, 1965. Gagne, R.M. Psychological P r i n c i p l e s i n Systems Development. New York: Holt Rinehart and Winston, 1962. Gagne R.M. and Brown, L.T. "Some Factors i n the Programming of Conceptual Learning," The Journal of Experimental  Psychology, 63:313-321, 1961. Gagne, R.M., Foster, H., Crowley, M.E. "The Measurement of Transfer of Training," Psychological B u l l e t i n , 45:97-130, 1948. Gagne, R.M., Mayor, J.R., Gastons, H.C., and Paradise, N.E. "Factors i n Acquiring Knowledge of a Mathematical Task," Psychological Monograph, 76 : 1 -21 , 1962. Gagne, R.M. and Paradise, N.E. " A b i l i t i e s and Learning Sets i n Knowledge Acq u i s i t i o n , " Psychological Monograph, 75:1-23, 1961. Gagne, R.M. and Smith,E.C. "A Study of the E f f e c t s of Verbal-i z a t i o n on Problem Solving," The Journal of Experimental  Psychology, 63 :12-18, 1962. Goddard, W.P. "Computers for the Whole School," Journal of  Educational Data Processing, 6:108-120, I969. Gorman, B. and Breskin S. "Non Verbal R i g i d i t y , C r e a t i v i t y , and Problem Solving," Perceptual and Motor S k i l l s , 29:3 ;715-718, 1969. 50 H a s l e r u d , G.M. and Meyers, S. "The T r a n s f e r Value o f G i v e n and I n d i v i d u a l l y D e r i v e d P r i n c i p l e s , " J o u r n a l o f Educa- t i o n a l P s y c h o l o g y , 49 :293-298, 1958 . Heimer, R a l p h T. " C o n d i t i o n s o f L e a r n i n g i n Mathematics Sequence Theory Development," Review o f E d u c a t i o n a l P s y c h o l o g y . 3 9 : 4 9 3 - 5 0 8 , 1969. . Henderson, K. " S t r a t e g i e s f o r Te a c h i n g by the D i s c o v e r y Method," U p d a t i n g Math, S e c t i o n IV whole, 1958 . Henderson, K. "Research on T e a c h i n g Secondary S c h o o l M a t h e m a t i c s , " Handbook o f R e s e a r c h on T e a c h i n g . New York: Rand M c N a l l y , p . l 0 0 7 - 1 0 3 0 , 1963 . H e n d r i x , G. "The V a r i a b l e Paradox," S c h o o l S c i e n c e and Math, 59:461-464, 1959-H u l l , T.E. I n t r o d u c t i o n t o Computing. Englewood C l i f f s N.J.: P r e n t i c e H a l l , 1966 . Jenson, R.S. "The Development and T e s t i n g o f a- Computer A s s i s t e d I n s t r u c t i o n U n i t Designed t o Teach D e d u c t i v e R e a s o n i n g , " U n p u b l i s h e d D o c t o r a l D i s s e r t a t i o n , F l o r i d a S t a t e U n i v e r s i t y , 1966 . Katona, G. O r g a n i z i n g and M e m o r i z i n g S t u d i e s i n t h e P s y c h o l -ogy o f L e a r n i n g and T e a c h i n g . Columbia U n i v e r s i t y P r e s s , 1940 . K i r k , Roger. E x p e r i m e n t a l D e s i g n : P r o c e d u r e s f o r t h e B e h a v i o r a l S c i e n c e s . Belmont, C a l i f o r n i a : Brooks C o l e P u b l i s h i n g Co., p.530 , 1968 . Lee, S.S. " T r a n s f e r from Lower L e v e l t o H i g h e r L e v e l Concept," J o u r n a l o f V e r b a l L e a r n i n g and V e r b a l B e h a v i o r . 7 :930-937 , 1968 . Lee, S.S. and Gagne, R.M. " E f f e c t s o f C h a i n i n g Cues on the A c q u i s i t i o n o f a Complex C o n c e p t u a l R u l e , " J o u r n a l o f  E x p e r i m e n t a l P s y c h o l o g y , 80:468-474, 1969 . Lee, S.S. and Gagne, R.M. "The E f f e c t s o f the Degree o f Component L e a r n i n g on the A c q u i s i t i o n o f a Complex C o n c e p t u a l R u l e , " J o u r n a l o f E x p e r i m e n t a l P s y c h o l o g y , 83 :13-18, 1970 . L e i c h l i t e r , V.P. "Businessmenaand Mathematics T e a c h e r s , " The Mathematics Teacher. 60:448-451, 1967 . 51 L e v i t t , E. "The Water J a r E i n s t e l l u n g T e s t as a Measure o f R i g i d i t y , " P s y c h o l o g i c a l B u l l e t i n , 5 3 : 3 4 7 - 3 7 0 , 1956 . L e v i t t , E. and Z e h l e r , S.L. "The V a l i d i t y o f the E i n s t e l l u n g T e s t as a Measure of R i g i d i t y , " J o u r n a l o f A b n o r m a l l and S o c i a l P s y c h o l o g y , 48 : 5 7 3 - 5 8 0 , 1953-Loupe, M i c h a e l J . T r a i n i n g and T r a n s f e r o f Problem S o l v i n g . M i n n e a p o l i s : M i n n e s o t a U n i v e r s i t y , 1972 . L u c h i n s , A.S. "On Recent Usage of E i n s t e l l u n g & i t s E f f e c t as a T e s t o f R i g i d i t y , " J o u r n a l o f C o n s u l t i n g P s y c h o l o g y , 1 5 : 8 9 - 9 4 , 1 9 5 1 . L u c h i n s , A.S. " M e c h a n i z a t i o n i n Problem S o l v i n g - The E f f e c t o f E i n s t e l l u n g , " P s y c h o l o g i c a l Monographs, 54<£No. 61-95» 1942 . L u c h i n s , A.S. and E.H. "New E x p e r i m e n t a l Attempts a t P r e -v e n t i n g M e c h a n i z a t i o n i n Problem S o l v i n g , " J o u r n a l o f  G e n e r a l P s y c h o l o g y , 42 : 2 7 9 - 2 9 7 , 1950 . McDermott, C.W. I n d u s t r i a l A p p l i c a t i o n s o f M a t h e m a t i c a l  Models and A b s t r a c t M a t h e m a t i c a l Systems f o r use i n  E l e c t i v e Graduate Mathematics C o u r s e s . U n p u b l i s h e d D o c t o r a l D i s s e r t a t i o n , Auborn U n i v e r s i t y , 1 9 6 7 . Medin, Douglas L. L e a r n i n g t o L e a r n i n C h i l d r e n . Washington D.C.: O f f i c e o f E d u c a t i o n , 1972 . M i l l e r , K.M. " E i n s t e l l u n g R i g i d i t y , I n t e l l i g e n c e - , and T e a c h i n g Methods," B r i t i s h J o u r n a l o f E d u c a t i o n a l P s y c h o l o g y , 2 7 : 1 2 7 - 1 3 4 , 1957 . P a p e r t , Seymour. T e a c h i n g C h i l d r e n T h i n k i n g , Logo L a b o r a t o r y , 1967 . P r i m a v e r a , L o u i s H. and Simon, W i l l i a m E. "Non V e r b a l R i g i d i t y and Academic Achievement i n H i g h S c h o o l and C o l l e g e , " P e r c e p t u a l and Motor S k i l l s , 33:2PT.2 ;1178 , 1 9 7 1 . R i e d e s e l , C.A. and Suydem, M.N. "Problem S o l v i n g , Some S u g g e s t i o n s from R e s e a r c h , " The A r i t h m e t i c Teacher, 1 6 : 4 6 7 - 4 7 8 , 1 9 6 9 . Rokeach, M. " G e n e r a l i z e d M e n t a l R i g i d i t y as a F a c t o r i n E t h n o c e n t r i c i s m , " J o u r n a l o f Abnormal and S o c i a l  P s y c h o l o g y , 4 3 : 2 5 9 - 2 7 8 , 1948 . Romberg, T.A. "Research i n Mathematics E d u c a t i o n , " Review of  E d u c a t i o n a l R e s e a r c h . 3 9 : 4 7 3 - 4 9 1 , 1 9 6 9 . 5 2 Saarni, Carolyn. Piagetian Operations and F i e l d Independence  as Factors i n Children's Problem Solving Performance. Berkley: University of C a l i f o r n i a , 1 9 7 1 • Scandura, J.M. "Algorithm Learning and Problem Solving," Journal of Experimental Education, 3 4 : 1 - 6 , I966A. Scandura, J.M. "Problem Solving and P r i o r Learning," Journal  of Experimental Education, 3 4 : 7 - 1 1 , 1966B. Scandura, J.M. "Prior Learning, Presentation Order, and Prerequisite Practice i n Problem Solving," Journal of Experimental Education, 3 4 : 1 2-14, I 9 6 6 C . Scandura, J.M. and Wells, J.N. "Advance Organizers i n Learning Abstract Mathematics," American Educational Research  Journal. 4 f 2 9 5 - 3 0 1 , 1 9 6 7 . Scandura, J.M., Woodward, E., and Lee, F. "Rule Generality and Consistency i n Mathematics Learning," American Educational Research Journal, 4 : 3 0 3 - 3 1 9 , 1 9 6 7 . Suppes, P. "Mathematical Concept Formation i n Children," American Psychologist, 2 1 : 1 3 9 - 1 5 0 , I 9 6 6 . Suppes, P. "Some Theoretical Models for Mathematics Learning," Journal for Research and Development i n Education, l : l ; 5 - 2 2 , I 9 6 7 . Suppes, P., Hyman, Lester, and Jerman, M. Linear St r u c t u r a l Models for Response and Latency Performance i n Arithmetic  on Computer Controlled Terminals. Stanford: Psychology Series, I n s t i t u t e for Mathematical Studies i n the S o c i a l Sciences, Technical Report No. 9 0 , Stanford University, 1 9 6 6 . Suppes, P. and Jernan, M. "Computer Assisted Instruction," The B u l l e t i n of the National Association of Secondary  School P r i n c i p a l s , 54:27-40. 1970. Van Dalen, Deobold B. Understanding Educational Research. McGraw H i l l Series i n Education, p .4 1 3 , 1 9 6 6 . Walther, J. Computer Assisted Mathematics Program. Scott Foresman and Co., I 9 6 9 . Wertheimer, M. Productive Thinking. Harper and Row Co., 1 9 5 9 -Woodward, E.L. Comparitive Study of Teaching Strategies Involving Advance Organizers and Post Organizers and  Discovery and Non-Discovery Techniques Where Instruction  i s Mediated by Computer. Tallahassee: Doctoral Thesis, F l o r i d a State University, 1 9 6 6 . .APPENDICES 53a APPENDIX A THE INSTRUCTIONAL DEVICE 53b DIRECTIONS TO TEACHERS AND STUDENTS Hand out "THE LESSON" ( t h e s e t h r e e p a g e s ) * t o e v e r y s t u d e n t . 2 . Hand out "GROUP A" page t o h a l f the s t u d e n t s ( a p p r o x i m a t e l y ) and "GROUP B" page t o the o t h e r h a l f u s i n g a RANDOM p r o c e s s of d i s t r i b u t i o n . 3. S t u d e n t s have 15 minutes t o r e a d t h e s e t h r e e pages p l u s t h e i r e x t r a GROUP A or B page. 4. S t u d e n t s have 10 minutes t o work on Problem 1. 5. S t u d e n t s have the r e s t o f the p e r i o d t o work on Problem 2 . NOTE: P l e a s e f e e l f r e e t o t u r n back t o the L e s s o n a t any time but p l e a s e do n o t work on Problem 1 when asked t o work on Problem 2. THE LESSON Groups A & B A program i s a l i s t o f i n s t r u c t i o n s f o r a computer. You are now g o i n g t o l e a r n how t o program a computer. The l i s t o f i n s t r u c t i o n s t h a t makes up a computer program must be w r i t t e n i n a s p e c i a l language. We w i l l use the s p e c i a l computer language c a l l e d BASIC. A f t e r you have s t u d i e d the language c a l l e d BASIC you w i l l w r i t e two programs o f your own. I w i l l t h e n p u t y o u r programs i n t o the computer t o see i f the computer 'understands* you. THE BASIC LANGUAGE The computer must have a name f o r e v e r y number. T h i s name must be a l e t t e r o f the a l p h a b e t . A statement which g i v e s a name t o a number must b e g i n w i t h the word LET. F o r example, i f we w i s h t o g i v e the name X t o the number 9, we w i l l w r i t e t h i s s t a tement ^Appendix n o t e : O r i g i n a l was on t h r e e 8^ * 14 pages. 54 f o r the computer: LET X = 9 I f we w i s h t o g i v e t h e name B t o the number 4, we w i l l w r i t e the s t a t e m e n t : LET B = 4 Now, i f we w i s h t o have the computer add the number c a l l e d X t o t h e number c a l l e d B, we must g i v e a name t o t h e answer. L e t us choose A ( a l t h o u g h any l e t t e r e x c e p t X and B would do j u s t as w e l l ) . We w r i t e the s t a t e m e n t : LET A = X + B N o t i c e t h a t the name or l e t t e r w h i c h i s g e t t i n g a new v a l u e must be w r i t t e n on the l e f t o f the e q u a l s i g n . The computer must a l r e a d y know the numbers b e l o n g i n g t o any l e t t e r name on the r i g h t o f an e q u a l s i g n . The computer obeys your program i n s t r u c t i o n s o r s t a t e m e n t s one a t a t i m e , s t a r t i n g w i t h the f i r s t s t a t e m e n t . A statement which t e l l s the computer t o p r i n t an answer must b e g i n w i t h the word, PRINT. F o r example, i f we w i s h t o p r i n t the v a l u e o f A, c a l c u l a t e d b e f o r e , we would g i v e the s t a t e m e n t : PRINT A The l a s t statement i n a program must be the s i n g l e word END. Each statement must be w r i t t e n on a s i n g l e l i n e o f your program. The computer w i l l obey your s t a t e m e n t s i n the same o r d e r you w r i t e them. We now know enough computer statements t o w r i t e a computer program. 5 5 Here i s our program: LET X = 9 LET B = 4 LET A = X + B PRINT A END When I put t h i s program i n t o the computer i t w i l l p r i n t o n l y what we t o l d i t t o PRINT. T h i s i s what i t w i l l p r i n t on the page ' 1 3 We have a s p e c i a l r e a s o n f o r numbering each statement o f our program. You w i l l see the r e a s o n l a t e r . Our program now l o o k s l i k e t h i s : 1. LET X = .9 2 . LET B = 4 3. LET A = X + B 4 . PRINT A 5 • END The r e a s o n we number each statement i s t h a t we sometimes w i s h the computer t o go back t o a p r e v i o u s s t a t e m e n t . F o r t h i s we use the statement b e g i n n i n g w i t h the words GO TO. I f we w i s h the computer t o r e t u r n t o statement number 1 we w r i t e : GO TO 1 F o r example, i f we w i s h the computer t o do the p r e v i o u s program a g a i n and a g a i n , we would w r i t e t h i s program: 1. LET X = 9 2 . LET B = 4 3. LET A = X + B 4 . PRINT A 5 . GO TO 1 6. END 5 6 T h i s i s what the computer would p r i n t : _ 1 3 1 3 The computer w i l l n e ver s t o p , o f c o u r s e , u n t i l I push a HALT b u t t o n . I t w i l l do statement 1 , t h e n s t a t e m e n t s 2 , 3 , 4 t h e n a t statement 5 i"t w i l l r e t u r n t o do s t a t e m e n t s 1 , 2 , 3 , 4 and so on a l l over a g a i n . As you might guess, the computer w i l l n e v e r ge t t o s t e p 6 i n our example, b u t t h i s s t e p must be i n the program anyway. Of course we do not u s u a l l y want the computer t o p r i n t the same number a g a i n and a g a i n . Suppose we wanted the computer t o use a v a l u e o f 9 f o r X, but we wanted B t o s t a r t w i t h a v a l u e o f 4 and t h e n t o grow 6 g r e a t e r each time the computer d i d a c a l c u l a -t i o n . We c o u l d t h e n w r i t e the f o l l o w i n g program: 1 . LET X = 9 2 . LET B = 4 3 . LET A = X + B 4 . PRINT A 5 . LET B = B + 6 6 . GO TO 3 7. END Statement 5 means, l e t the new v a l u e o f B e q u a l 6 more t h a n the o l d v a l u e o f B. T h i s i s what the computer would p r i n t . 57 I f we had wanted the computer t o add 9 t o the v a l u e o f B, f o r each c a l c u l a t i o n , i n s t e a d o f 6, statement 5 c o u l d have heen: 5. LET B = B + X T h i s i s what the computer would have p r i n t e d : 1 3 2 2 31 40 I f we had wanted the f i r s t v a l u e o f B t o be p r i n t e d b e f o r e t h e o t h e r number, so t h a t the computer would p r i n t : 1 3 2 2 31 40 our program would have l o o k e d l i k e t h i s : 1. LET X = 9 2 . LET B = 4 3 . PRINT B 4. LET A = X + B 5. PRINT A 6. LET B = B + X 7. GO TO 4 8. END APPENDIX B THE EXPERIMENTAL DEVICE 5 8 a GROUP A A FINAL EXAMPLE OF A PROGRAM. Here is a program that w i l l print the odd numbers like this without ending: 1. LET X = 1 2 . PRINT X 3 . LET X = X + 2 4 . PRINT X 5 . GO TO 3 6. END APPENDIX C THE CONTROL DEVICE GROUP B A SHORT HISTORY OF COMPUTERS. Here i s a l i s t o f names and da t e s w h i c h have t o do w i t h computers. 1. M u l l e r 1786 2. Babbage 1820 3. H o l l e r i t h 1889 4. A i t k e n 1937 5. B e l l Telephone 1938 6. E c k e r t 19^9 1. Proposed the i d e a o f a c a l c u l a t i n g machine. 2. D e s i g n e d a c a l c u l a t i n g machine. 3. Developed a d a t a p r o c e s s i n g machine. 4. B u i l t t he Mark I computer. 5. B u i l t e l e c t r o m a g n e t i c c a l c u l a t o r s . 6. Used e l e c t r o n i c t e c h n i q u e s f o r c o m p u t a t i o n . 60:: APPENDIX D THE TEST INSTRUMENT 60a IMPORTANTi GROUP.. Make sure you p u t your Group i n the b l a n k below. You w i l l f i n d whether you are group A or group B by l o o k i n g a t the t o p o f the o t h e r s i n g l e page you were g i v e n . NAME, SCHOOL GROUP A & B USE SCRAP PAPER FOR ROUGH WORK, THEN COPY YOUR FINAL ANSWERS ON THIS PAGE Problem 1: W r i t e a computer program which w i l l make the computer p r i n t the even numbers, l i k e t h i s , w i t h o u t e n d i n g . ~2 6 8 10 Problem 2: W r i t e a computer program which w i l l make the computer p r i n t the f o l l o w i n g sequence o f numbers, w i t h o u t e n d i n g . 1 2 3 5 8 13 21 Note t h a t i f you l e t X = 0 and Y = 1 t h e n the f i r s t number i n the sequence i s F = X + I f you now l e t X be the same as Y was, and i f you l e t Y be the same as F was, you get the second number i n the sequence F = X + Y and so on. APPENDIX E THE COMPUTER PRINTOUT PROGRAM 10 REM; CHI SQUARE FOR TWO BY TWO ARRAYS 20 PRINT "CORRECT","INCORRECT" 3 0 PRINT " 4 0 READ R,K 5 0 I F R#2 AND K#2 THEN 2?0 60 MAT A = ZER 7 0 REM;MATRIX OF DATA IS CLEARED 80 LET N=C=0 9 0 REM;N=GRAND TOTAL, C= CHI SQUARED 100 FOR E=l TO R 110 REM;E IS CURRENT ROW 1 2 0 FOR D=l TO K 1 3 0 REM;D IS CURRENT COLUMN 140 READ A(E,D) 1 5 0 REM;READ DATA FOR ROW E AND COLUMN D 160 PRINT A(E,D), 170 LET N=N+A(E,D) 180 REM5 ADD EACH TOTAL TO GRAND TOTAL 190 LET A(E,K+1)=A(E,K+1)+A(E,D) 200 LET A(R+1,D)=A(R+1,D)+A(E,D) 210 REM;STEPS 1 9 0,200 ADD ROW & COLUMN TOTALS 220 NEXT D 2 3 0 PRINT 240 NEXT E 2 5 0 PRINT "DEGREES OF FREEDOM = " ( R - l ) * ( K - l ) , 2 6 0 GOTO 2 9 0 2 7 0 PRINT "SWITCH TO PROGRAM FOR R BY K CHI SQUARE." 2 80 GOTO 410 2 9 0 LET T = N * ( ( A B S ( A ( l , l ) * A ( 2 f 2 ) - A ( 2 , l ) * A ( 1 . 2 ) ) ) T 2 ) 3 0 0 LET C=T/(A(l , 3)*A(2 , 3)*A ( 3 i l)*A ( 3»2)) 3 1 0 PRINT "CHI SQUARE = "C 3 2 0 PRINT "================================================ 3 3 0 REM;THIS IS CHI SQUARE ACCORDING TO VAN DALEN PAGE 413. 3 4 0 GOTO 40 3 5 0 DATA 2,2,65 , 2 7,26,66 3 6 0 DATA 2,2,42,7,12,37 3 7 0 DATA 2 , 2 , 2 3 , 2 0 , 1 4 , 2 9 3 8 0 DATA 2,2,42,7 , 2 3,20 3 9 0 DATA 2,2,12,37,14 , 2 9 400 DATA 2,2,11 , 3 1,14,9 410 END PRINTOUT 62 CORRECT INCORRECT 6 5 2 6 DEGREES OF 2 7 6 6 FREEDOM = 1 CHI SQUARE = 3 3 . 0 6 9 1 2 4 4 2 4 2 1 2 DEGREES OF 7 3 7 FREEDOM = 1 CHI SQUARE = 3 7 . 1 2 1 2 1 2 1 2 2 3 14 DEGREES OF 2 0 2 9 FREEDOM = 1 CHI SQUARE = 3 . 8 4 2 2 5 0 4 1 4 42 2 3 DEGREES OF 7 2 0 FREEDOM = 1 CHI SQUARE = 1 1 . 4 7 0 5 8 9 0 1 1 2 14 DEGREES OF 3 7 2 9 FREEDOM = 1 CHI SQUARE = 0 . 7 3 5 3 6 6 5 1 6 11 14 DEGREES OF 31 9 FREEDOM = 1 CHI SQUARE = 7 . 5 5 1 3 7 1 6 3 6 

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