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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 P H I L I P GODDARD B. E d . , U n i v e r s i t y  of B r i t i s h Columbia,  1963  A THESIS SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics Education  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA SEPTEMBER,  0  1976  William Philip Goddard, 1 9 7 6  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  fulfilment of  the requirements f o r  the U n i v e r s i t y of B r i t i s h Columbia,  make i t  freely available  that permission  for  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  that  study. thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of  this  written  representatives. thesis  It  is understood that copying or p u b l i c a t i o n  f o r f i n a n c i a l gain s h a l l  permission.  Department of The U n i v e r s i t y o f B r i t i s h  Columbia  2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1WS  Date  / 7  L  not be allowed without my  i  ABSTRACT T h i s s t u d y was m o t i v a t e d b y t h e w r i t e r ' s v a t i o n t h a t t h e p r o v i s i o n o f s o l v e d examples l e a r n i n g t o d e v i s e computer  obser-  to students  a l g o r i t h m s d i d n o t a s s i s t and  seemed t o h i n d e r i n t h e d e v e l o p m e n t  o f such s k i l l s .  even  I t was  m i s e d t h a t t h i s m i g h t be due t o a number o f f a c t o r s .  sur-  The  l e a r n e r m i g h t be d e l a y e d i n h i s o . d e v e l o p m e n t o f t h e h e u r i s tics  necessary to create algorithms using  mediators. ficulty  self-generated  He m i g h t 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 t h e d i f -  of p e r f o r m i n g such tasks independently.  rigidity  He m i g h t  display  (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 t h e 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. G r a d e n i n e s t u d e n t s were a s s i g n e d t o two g r o u p s a t random.  B o t h g r o u p s were g i v e n a p r i n t e d  computer  p r o g r a m w r i t i n g i n t h e BASIC l a n g u a g e a n d w e r e  to  introduction to  s o l v e two p r o b l e m s , a n e a s y p r o b l e m and a h a r d e r  problem.  asked  criterion  B e f o r e t h e p r o b l e m s w e r e a s s i g n e d one g r o u p 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 t h e e a s y  problem.  The s e c o n d g r o u p was g i v e n a s h o r t h i s t o r y o f c o m p u t e r s  to read.  A C h i - s q u a r e - t e s t was u s e d t o t e s t e a c h o f t h e following  hypotheses: 1.  The f i r s t  p r o b l e m was e a s i e r t h a n t h e s e c o n d  problem f o r a l l students.  2. the to  The example  easy problem  correct  solutions  correct when  only  problem  those  were  s t u d e n t s who  training  from  increase  t h e numbers  it  may  problems  using  who  the f i r s t  group. of  than the f i r s t the  group  first  drawn  hypotheses  that  the use of  on t h e computer examples  may  i s at  examples  the independent instructional  may  be a waste  examples least  hinder transfer  t o harder problems  that  were  level.  of  a n d do n o t  can independently solve  i s the only  was  and v a r i e t y a variety  desirable.  of  a harder  solution  goal.) of time,  of  At best the a t worst  distraction.  It number  was  i n that  assumes  o f such  t h e .05  beyond  easy problems  (This  be a  than  solved  and f o u r t h  development  inadvisable  provision  solutions  had a h i g h e r p r o p o r t i o n  correctly  second,  conclusion  teach algorithm  harder  group  f o r the "hard" problem  first,  The  problem.  of correct  had a h i g h e r p r o p o r t i o n  f o r the "hard" problem  t o be s i g n i f i c a n t  sometimes  i n doing  considered.  The found  group  group.  group  The s e c o n d  solutions  the proportion  i n each  The s e c o n d  k.  to  comparing  the easy problem 3.  helped the f i r s t  felt  that  further  o f examples,  of textual  research  classified  material  using  a  greater  i n some way,  i s both warranted  and  and  It was also f e l t that a test instrument could he devised which would i d e n t i f y those students who would most benefit from a course i n algorithm development on the computer.  iv TABLE OF CONTENTS CHAPTER 1.  P  THE PROBLEM .  o f the Study  Significance  o f the Study  Related Studies Transfer  e  1 5 5 8 8  Einstellung  1°  A l g o r i t h m Development  1^  Statement II.  S  1  Background Purpose  a  o f t h e Hypotheses  THE DESIGN OF THE STUDY  22 23 23  Introduction 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  Studies  26  The S e l e c t i o n o f 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  Procedure  28  Apparatus s  29  Statistical  Analysis  Introduction  30 30  V  CHAPTER  III.  Page Data  30  N u l l Hypotheses  31  S t a t i s t i c a l Treatment  3  ANALYSIS OF THE. RESULTS  34  Summary of Data  34  T e s t i n g o f the Hypotheses  34  Hypothesis One  34  Hypothesis Two  36  Hypothesis Three  37  Hypothesis Four  38 39  Conclusions Discussion IV.  2  o f A d d i t i o n a l Data  CONCLUSIONS AND IMPLICATIONS OF THE STUDY ..  39 40  Introduction  40  Discussion  40  o f the C o n c l u s i o n s  The Content of the Lesson  40  The Method of I n s t r u c t i o n  41  Length of the Experiment  41  Group Makeup  41  Mechanics o f R i g i d i t y  41  Limitations  of the Study  Suggestions f o r F u t u r e S t u d i e s . . . . . . . . . . .  42 43  FOOTNOTES  45  BIBLIOGRAPHY  48  vi 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  6l  vii L I S T OF  TABLES  TABLE I. II. III. IV. V. VI. VII.  Page Summary o f D a t a  3^  T e s t o f H y p o t h e s i s One  35  T e s t o f H y p o t h e s i s One ( a )  35  Test of Hypothesis  35  One(h)  T e s t o f H y p o t h e s i s Two  36  T e s t o f H y p o t h e s i s Three  37  Test of Hypothesis Four  38  viii L I S T OF" FIGURES FIGURE 1.  P  C o n c e p t u a l framework f o r L e a r n i n g to Devise Algorithms  a  S  e  4  ACKNOWLEDGEMENT The students  author wishes t o thank t h e t e a c h e r s and  o f West V a n c o u v e r  schools f o r p a r t i c i p a t i o n ,  my c o m m i t t e e f o r a n t i c i p a t i o n , and P r o f e s s o r B. H i c k s inspiration.  my w i f e f o r r e h a b i l i t a t i o n ,  (University of I l l i n o i s ) f o r  1  CHAPTER I THE  PROBLEM  BACKGROUND If  students  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 a l g o r i t h m s , " ' " t h e y may "become rigid ment.  or stereotyped i n t h e i r approach to 'algorithm Although  t e a c h e r - p r o v i d e d e x a m p l e s may s e r v e  developinitially  2 as m e d i a t o r s  f o r the t r a n s f e r of t r a i n i n g ,  those  students  who become i n v o l v e d i n s y s t e m s work-^ c a n n o t c o n t i n u e provided with suitable mediating  examples.  Systems work i s  h e r e 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 and  methods."  t o be  o f men, m a c h i n e s  S u c h 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 u s e some h e u r i s t i c w h i c h i s n o t  d e p e n d e n t on e x a m p l e s .  Each such  in  student w i l l  use h i s s k i l l  a d i f f e r e n t c o n t e x t f r o m t h a t i n w h i c h i t was f i r s t Personnel  teaching i n the author's  learned.  mathematics  computer l a b o r a t o r i e s have remarked t h a t t h e p r o v i s i o n o f mediating  examples i n the e a r l y s t a g e s  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 of these  students.  of teaching students t o  the l a t e r  I t was s t a t e d t h a t t h o s e  n o t p r o v i d e d w i t h s a m p l e methods p r o g r e s s e d first  i n d e p e n d e n t work s t u d e n t s who were  more s l o w l y a t  b u t were more s u c c e s s f u l when l a t e r t h e y h a d 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  individually  c h o s e n p r o b l e m s f o r w h i c h no e x a m p l e s c o u l d be p r o v i d e d .  2 Of e i g h t e e n  students  p o l l e d a t t h e end o f t h e c o u r s e  i n Computer  o r were This  who  were  w i t h respect to the value  sample programs, s i x t e e n r e p o r t e d progress  Science  t h a t examples  of  inhibited  useless. study  i n v e s t i g a t e s such c l a i m s .  I f these  claims  a r e v a l i d t h e 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 m a t h e m a t i c s and computer s c i e n c e  be  courses  may  unsound.  The e d u c a t i o n a l g o a l u n d e r 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  is called  "Synthesis  Objectives.^ itself type but  to produce a p l a n or a l g o r i t h m . 5*2"  T h e r e was no i n t e r e s t i n t h e p a r t i c u l a r  ( a number s e q u e n c e g e n e r a t o r ) ,  f o r s i m p l i c i t y of experimentation The  specific  of lower  language),  p a r t i c u l a r choices  order mediation  of 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 i n h i b i t i n g or at l e a s t not a s s i s t i n g higher inducing a r i g i d  algorithm  nor i n the p a r t i c u l a r  example i n t r o d u c e d p r i o r t o t h e  t a s k s h a d t o be c a p a b l e  goal  i n t h e Taxonomy o f E d u c a t i o n a l  o f a l g o r i t h m . ( a c o m p u t e r p r o g r a m i n t h e BASIC  made.  This  were  criterion  of t r a n s f e r  a t t h e same t i m e o r d e r t r a n s f e r by  approach to a l g o r i t h m development (an E i n s t e l l u n g  effect).Several  pilot  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  to  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  experiences  w i t h the i d e a of  " c o m p u t e r s i n t h e w h o l e s c h o o l " ^ a n d s u b s e q u e n t s e m i n a r s 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 d e m o n s t r a t e d  the d e s i r a b i l i t y  t h e b e s t means o f l e a d i n g a s t u d e n t so t h a t , should y i e l d a product... something t h r o u g h one  o r more o f t h e s e n s e s , and w h i c h 7  get s t u d e n t s to produce  discovering  "His  t h a t c a n be  t h a n t h e m a t e r i a l s he b e g a n t o w o r k w i t h . " to  of  effort  observed  i s c l e a r l y more It is  difficult  a p l a n or a l g o r i t h m f o r the  Bloom notes t h a t " . . . c u r r e n t programs overemphasize i n w h i c h t h e l e a r n e r f u n c t i o n s as a c o n s u m e r and  computer.  activities  critic  of 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 A l t h o u g h many s t u d e n t s e x p e c t t o l e a r n l a r g e l y b y examples through i m i t a t i o n o f a t e a c h e r or t e x t , to  s e e how  any s e r i e s  producer." working  i t is difficult •  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 s o u n d and t h e r e b y o p e r a t e a r o b o t , o r t o p l a y m u s i c . o  T h e s e t a s k s a r e a c c o m p l i s h e d b y t h e 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 of s y n thesis,  " t h e c a t e g o r y o f t h e c o g n i t i v e d o m a i n w h i c h most  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 t h e p a r t o f t h e  clearly  learner."^"  In  the " p r o d u c t i o n of a p l a n or proposed  to  p e r f o r m s u c h 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 t h e  t a x o n o m e t r i c l e v e l s o f "comprehension", "analysis".  T h i s present experiment  these lower taxonometric l e v e l s , g o a l o f i n s t r u c t i o n and  0  set of operations""^  "application",  and  r e a l l y operates only at  b u t where s y n t h e s i s i s t h e  i f e x a m p l e s c a n be  shown t o be  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 , e x a m p l e s s h o u l d be a v o i d e d a t s u c h l e v e l s  of  a then  instruction.  lower  4 The a l g o r i t h m s was  c o n c e p t u a l framework f o r l e a r n i n g as  to  develop  follows: I  J  Task.  I  \^Higher order s k i l l s ,  1  Mediators,  self  Lower o r d e r  task  output./  7  generated\  skills.  Task Output. Mediators: i)text  or t e a c h e r  ii)subordinate iii)group iv)solved  tasks  interaction examples  v)self-generated Algorith and p r o  explanation  heuristics  yelopment solving  Low  order s k i l l s  difficulty,  i n terms  complexity,  remoteness from past  of  and  experience,  5 The b a c k g r o u n d f o r t h i s f r a m e w o r k was  derived  l a r g e l y f r o m t h e w o r k o f R.M. Gagne a n d S.S. L e e .  The b a c k g r o u n d  f o r t h e 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 A.S.  a n d E.H. L u c h i n s a n d K.M. M i l l e r .  The c o m p u t e r  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 p r o b l e m originated Hillside  i n t h e a u t h o r ' s computer  from equipment  itself,  c e n t r e s a t S e n t i n e l and  S e c o n d a r y S c h o o l s i n West V a n c o u v e r , B.C. PURPOSE OF THE STUDY I t was t h e a u t h o r ' s p u r p o s e t o i n v e s t i g a t e  the  provision  their  o f a n example  own c o m p u t e r  unsuccessful  whether  to students learning t o devise  algorithms resulted  i na rigid  approach t o independent a l g o r i t h m  and t h e r e f o r e  development.  S I G N I F I C A N C E OF THE STUDY  12 Gagne a n d B r o w n s t a t e d discovery" and  was most e f f e c t i v e , " d i s c o v e r y "  " r u l e and example"  the  Gagne a n d B r o w n m o d e l  the  effectiveness  or t o c o n t r i b u t e  that  "guided  n e x t most e f f e c t i v e ,  least e f f e c t i v e i n producing  of conceptual l e a r n i n g .  example".  i n 1961  transfer  I t w o u l d seem a l o g i c a l e x t e n s i o n t o i f p r o v i s i o n w e r e made f o r j u d g i n g  o f examples  t o mediate  "guided  to the r e l a t i v e ineffectiveness  discovery" o f " r u l e and  T h i s p a p e r may a i d t h e s t u d y o f t h e p r o b l e m b y  d e t e r m i n i n g i f an example  could  a s s i s t higher order t r a n s f e r  inhibit  or at least f a i l to  of algorithm-building  learning  6 s e t s and yet be capable  of lower order t r a n s f e r .  I f the guidance of d i s c o v e r y through p r o v i d e d mediating  examples must be terminated  teacherat some p o i n t ,  the l e a r n e r must e i t h e r be able to i n t e n t i o n a l l y  develop  mediators f o r h i m s e l f or he must l e a r n t o do without altogether.  mediators  For example, i f the l e a r n e r becomes a bio.logist  who  i n v e s t i g a t e s the r e l a t i o n s h i p between the s i z e s o f r a b b i t  and  fox p o p u l a t i o n s , and  i f he wishes to produce a computer  a l g o r i t h m f o r the guidance of game management p e r s o n n e l , must be able to r e l a t e concepts to s k i l l s  statistics  i n mathematics and computer programming.  upon h i s past experience new  i n ecology and  a l g o r i t h m which may  competencies. techniques  Calling  i n these f i e l d s he must develop  be delayed  a  be used by those not p o s s e s s i n g h i s  He must be able to s y n t h e s i z e r e l e v a n t  i n a f l e x i b l e r a t h e r than i n a s t e r e o t y p e d manner. I f examples are p r o v i d e d f o r low order t a s k s  l e a r n e r may  he  be rendered  a d i s s e r v i c e i n s e v e r a l ways.  the He  i n h i s development of the h e u r i s t i c s necessary  c r e a t e algorithms u s i n g s e l f - g e n e r a t e d mediators.  He  may to  may  be m i s l e d i n h i s e x p e c t a t i o n of task d i f f i c u l t y by e a r l y s u p e r f i c i a l successes  and poor m o t i v a t i o n might r e s u l t - from  accustomed l a r g e increments  i n d i f f i c u l t y when mediation i s  withdrawn i n advanced courses  or at s c h o o l l e a v i n g .  d i s p l a y an E i n s t e l l u n g e f f e c t i n h i s h a n d l i n g of data techniques.  un-  He  might  and  7 T h e r e w o u l d seem t o he  implications'here for  w o r k e r s i n programmed i n s t r u c t i o n , and  text-hook  classroom  computer a s s i s t e d i n s t r u c t i o n ,  w r i t i n g as w e l l as f o r m a t h e m a t i c s and  teachers.  I t i s o f t e n assumed, i t a p p e a r s ,  s p e c i f i c examples o f a process t r a n s f e r to higher  science  being taught  that  w i l l always mediate  o r d e r uses of 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 M a t h e m a t i c s Program, S c o t t Foresman and  Co.,  1969).  Even i f i t i s claimed  t h a t examples are  o n l y t o a i d i n t h e e x p l a n a t i o n o f t e r m s and to  mediate t h e i r use,  t h e use  i t may  be  p r o d u c e d as ,an u n d e s i r a b l e ' I t may  be  side  c o n c e p t s and  t h a t c a r e must be  o f e x a m p l e s so t h a t r i g i d i t y  t h a t s u c h t o p i c s as a l g o r i t h m  the  t h e game o f c h e s s ;  and  are e x p l a i n e d  exercised i n i s not  effect.  c o m p u t e r p r o g r a m m i n g s h o u l d be  each piece  not  of mental processes  i n m a t h e m a t i c s and same manner as  used  development taught  in  t h a t i s , t h e moves o f  ( t h e m e a n i n g o f s y m b o l s , commands)  then the l e a r n e r i s simply allowed  to play  w r i t e programs).  A t a much l a t e r s t a g e ,  d e v e l o p e d h i s own  m e t h o d s , he  (devise  a f t e r the  algorithms,  learner  might s a f e l y undertake the  study  of s p e c i a l techniques  by c o m p a r i n g h i s methods w i t h t h o s e  experts.  s a f e t o l e a r n by  as the  I t m i g h t be  t h e r e m i g h t be task.  example a t t h i s  l e s s chance o f a d o p t i n g  a rigid  has  of  stage  approach  to  8  RELATED STUDIES A foundation w i l l "be l a i d f o r the c o n s i d e r a t i o n of t r a n s f e r of t r a i n i n g i n l e a r n i n g h i e r a r c h i e s "by a d i s c u s s i o n 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 f o r t h e i r work on problem s o l v i n g and algorithms; G.M. Haslerud, P. Suppes, T.A. Romberg and S.S.  Lee f o r studies on guided discovery,  item s t r u c t u r e , c o g n i t i v e i n d i v i d u a l differences  and chaining  cues r e s p e c t i v e l y . Transfer. In 19*^9 Gagne transfer.  J  studied the problem of measuring  He concluded that the best means were; raw score,  percentage improvement due to the c o n t r i b u t i o n of the t r a n s f e r r e d task, percentage improvement during t r i a l s and presence or absence of t r a n s f e r as measured by c o e f f i c i e n t s  of  correlation.  effect  He stated that other measures of the  of t r a n s f e r of t r a i n i n g suffered from l a c k of c o m p a t i b i l i t y of empiricism. ,  In 1961 he  lk  •  i n v e s t i g a t e d the r e l a t i o n s h i p s  between l e a r n i n g sets i n knowledge a c q u i s i t i o n and concluded that there was a p o s i t i v e t r a n s f e r to each new l e a r n i n g set from r e l e v a n t subordinate past l e a r n i n g and that the c o r r e l a t i o n between the mastery of t h i s subordinate m a t e r i a l , and achievement  9 was  highest  a t the  highest  l e v e l of h i s l e a r n i n g  hierarchy.  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 of 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 p r o b l e m s o l v i n g as m e a s u r e d by required  and  score obtained.  d e r i v i n g the  f o r m u l a f o r the  number s e r i e s . low  1962  devices to define t h i s p h r a s e he  sum  instances  At  was  working with  w h a t he  calls  "productive  kind  as  t i m e he  tasks  'higher  i s able  a  shows h i m s e l f  unable to perform a He  auto-instructional learning".  t h a n one  "there  'lower l e v e l '  i n an h i e r a r c h y  who  and  f a c t o r or  no  rate  of  to a  set  attain-  comes t o d e p e n d t o  accordingly  been  then  learning  l e a r n i n g s e t s w h i c h have  p r e v i o u s l y been a c q u i r e d  entire  member o f  are  £Le.vel' l e a r n i n g s e t , and  the  By  t o p e r f o r m w h a t has  f u r t h e r s t a t e d t h a t the  ment o f l e a r n i n g s e t s i n c r e a s i n g e x t e n t on  rather  stated that,  i n d i v i d u a l who  e x t e n t upon a b a s i c  '  o f change i n human b e h a v i o r  i d e n t i f i e d as  r e l a t e d to i t . "  considered  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  this  o f an  be  £1  (or system) of s p e c i f i c  class.  involved  of terms i n u n f a m i l i a r  Gagne  meant t h e  w h i c h p e r m i t s the  the  problems used  hints  development. 1  class  The  learning  time r e q u i r e d ,  S u c h f o r m u l a w r i t i n g may  level algorithm By  the  on  an  just  decreasing  ability.  17 I n I 9 6 3 Gage the  assumption that  to p r a c t i c e that  the  pointed b e s t way  performance.  out  Gagne*s q u e s t i o n i n g  to l e a r n a performance i s  of  10 "In c o n d i t i o n i n g , c l a s s i c a l or otherwise, one observes l e a r n i n g only a f t e r the animal has made the f i r s t response and one conceives of what i s l e a r n e d as e i t h e r a response or an a s s o c i a t i o n t e r m i n a t i n g i n a response, i n e i t h e r case e s t a b l i s h e d by p r a c t i c i n g the response with r e i n f o r c e m e n t . Gagne challenged t h i s on the grounds that the responses r e q u i r e d do not have to be l e a r n e d at a l l - they are a l r e a d y i n the human's repertoire..." "In t r a i n i n g men to t r o u b l e shoot (Gagne 1962) complex equipment, there was no s i n g l e task to be produced, r a t h e r i t was the l e a r n i n g of an e l a b o r a t e s e t of r u l e s p e r t a i n i n g to the flow of s i g n a l s through a complex c i r c u i t another c o g n i t i v e s t r u c t u r e (system) t h a t 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 i m p l i e d by at l e a s t some i n t e r p r e t a t i o n s of c o n d i t i o n i n g 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, d e a l w i t h t a s k a n a l y s i s , i n t r a t a s k t r a n s f e r , 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 i n s te H u n g . Luchins of h a b i t u a t e d  1 0  pointed  behavior.  out p o s s i b l e d e l e t e r i o u s e f f e c t s  His method was  as f o l l o w s .  Several  problems, a l l s o l v a b l e by one  somewhat complex procedure, were  presented i n s u c c e s s i o n .  problems i n v o l v e d u s i n g  The  three  water j a r s of given c a p a c i t y to produce a c e r t a i n volume of water.  Then a s i m i l a r problem was  by a more d i r e c t and an  simple method.  " e x t i n c t i o n problem".  i n attempting to use of d i f f i c u l t y was  He  g i v e n which c o u l d be T h i s problem was  reversed  called  found that most s u b j e c t s  the complex method. i n the present  persisted  Although the study, and  solved  order  although  11 o n l y two  problems were p r e s e n t e d , the  idea of E i n s t e l l u n g  rigidity  is still  s t u d e n t has  t u n i t y and  the  the  L u c h i n s found t h a t the  level,  age,  and  I.Q.  ineffective.  o c c u r e n c e by t e l l i n g s u b j e c t s  d o i n g a c e r t a i n p r o b l e m and  before  the  he  p r o b l e m s more c o n c r e t e ,  experimentation,  a l l t o no  to  He  further  d o i n g the  He  or  tried  to  'avoid b l i n d n e s s '  provided  avail.  He  next.  facilities  the  was  found that attempts to prevent t h i s mental r i g i d i t y E i n s t e l l u n g e f f e c t were q u i t e  use  tendency t o copy  method o f s o l u t i o n i n s u b s e q u e n t p r o b l e m s  independent of e d u c a t i o n a l  its  oppor-  tendency t o copy a t e c h n i q u e r a t h e r t h a n  a d i r e c t method. previous  p r e s e n t i n t h a t the  He  prevent after made  for  observed that a h a b i t  of  problem s o l u t i o n , " c e a s e s 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 , b u t becomes a p r o c r u s t e a n bed t o w h i c h t h e s i t u a t i o n must c o n f o r m ; when, i n a w o r d , i n s t e a d o f t h e 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 prevented from l e a r n i n g the u s u a l  For  classroom p r a c t i c e of teaching  He  formation  They must He  observed  a specific  on t h a t s p e c i f i c t o p i c  be that  topic  and  contributes  of 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 .  e x a m p l e , a f t e r a l e s s o n on  teacher  much d a t a .  'type e x e r c i s e s . '  then a s s i g n i n g problems only t o the  too  m i g h t a s s i g n as a f i r s t  t w o - d i g i t m u l t i p l i c a t i o n the exercise  a question  surmised t h a t speeded t e s t s a l s o c o n t r i b u t e  i n subtraction.  t o an E i n s t e l l u n g  12 e f f e c t i n t h a t p r o b a b l y the most u s e f u l p i e c e  of  information  r e l a t e d t o some problems i s the f a c t t h a t they can be done i n so many minutes.  The demand f o r speed causes b l i n d n e s s .  Upon  experimenting w i t h changes i n these u s u a l classroom procedures he was t o l d by the students, "I d i d what I was t o l d t o do", "you  t r i c k e d us", "you taught us wrong." Luchins f u r t h e r s t a t e d t h a t  20  i f t e s t s were i n t e r e s t i n g ,  i f s u b s t a n t i a l mark allowances were made f o r method o f a t t a c k , and  i f t e s t s were presented as a method o f h e l p i n g  students,  there would be l e s s mechanization of t h i n k i n g f o s t e r e d by them. 21  Miller  i n 1957 found r e s u l t s c o n t r a d i c t i n g some  o f Luchins e a r l i e r t e n t a t i v e c o n c l u s i o n s .  He found t h a t  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 s o l v i n g and i n t e l l i g e n c e . He a l s o found t h a t when sub-samples of t e c h n i c a l and modern s c h o o l boys were matched f o r i n t e l l i g e n c e and compared, the E i 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 o f t e n modern s c h o o l school  innthe  group (who were d r i l l e d ) than i n the t e c h n i c a l  students (who were t r a i n e d t o search f o r a l t e r n a t i v e  methods).  T h i s r e s u l t he d i s c u s s e d  methods and a t t i t u d e t o s c h o o l . to r e g u l a t e  as a f u n c t i o n o f t e a c h i n g  M i l l e r used a s i n g l e  school  some f a c t o r s not c o n t r o l l e d i n the Luchin's study  such as p h y s i c a l surroundings, teacher p e r s o n a l i t y economic s t a t u s  of p a r e n t s .  The present w r i t e r has  and s o c i o followed  13 M i l l e r ' s l e a d i n t h i s and intelligance ization.  and  The  age  assumed t h a t t h e d i s t r i b u t i o n  i n t o s u b - g r o u p s was  achieved  M i l l e r method o f s c o r i n g ( c o u n t i n g t h e  o f c o n t r o l and  e x t i n c t i o n p r o b l e m s s o l v e d ) was  Miller a pilot  s t u d y was  p r o b l e m d i f f i c u l t y and  randomnumber  u s e d and  like  used t o determine s u i t a b i l i t y  of  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 of h i s experimental Miller  by  of  results  remarked,  " d e p e n d i n g on what one c o n s i d e r s t h e a i m o f e d u c a t i o n , i t c o u l d be a r g u e d t h a t i n s u i t i n g t e a c h i n g methods t o t h e l e s s a b l e m o d e r n s t r e a m p u p i l s , t h e more a b l e a r e p r e v e n t e d f r o m 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 t h e o t h e r h a n d 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 p r o b l e m s i n a mechana>gal way, t h e a b o v e f i n d i n g s n e e d c a u s e no c o n c e r n . " The  present  i n C h a p t e r IV.  writer w i l l  comment u p o n t h i s f u r t h e r  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  the s e l e c t i o n of n o n - r i g i d students  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 d e v e l o p m e n t on c o m p u t e r s , m i g h t be p r o d u c t i v e t h a n t h e s e l e c t i o n o f h i g h I.Q.  students.  l a c k o f r i g i d i t y t e n d e d t o a c c o m p a n y h i g h I.Q. and  Although  t h e more  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  individual  more  specific  i n screening  talents. Miller's  comments on s c h o o l m o r a l e and  p a r t i c u l a r l y important o f c o m p u t e r s and  to teachers  c o n s i d e r i n g the  climate  are  installation  computer t r a i n i n g programs s i m i l a r t o  those  Ik of the present w r i t e r .  The p r e s e n t  experiment  was c a r r i e d o u t  i n a school with a f r i e n d l y climate but with a 'structured' academic approach t o e d u c a t i o n . socio-economic  The s c h o o l was i n a n u p p e r  area.  A l g o r i t h m D e v e l o p m e n t and P r o b l e m S o l v i n g . Wertheimer  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  J  f o r s t r u c t u r e which i s l i k e i n an o p t i c a l i l l u s i o n .  the search f o r the reversed  image  P e r h a p s some s t r u c t u r e s ( s u c h a s new  a l g o r i t h m s ) must be s o u g h t b y t h e i n d i v i d u a l f r o m t h e b e g i n n i n g without  the a i d of the mediators  mentioned i n the conceptual  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  l e a r n i n g and problem s o l v i n g .  of algorithm  I t s h o u l d be n o t e d  that  this  i s not i d e n t i c a l w i t h the c e n t r a l issue of t h i s paper, a l g o r i t h m development , however S c a n d u r a ' s r e s u l t s a r e u s e f u l i n t h a t  they  show what a l g o r i t h m s a r e , h o w , t h e y c a n be g e n e r a l i z e d a n d how they r e l a t e t o higher order t r a n s f e r i n problem  solving.  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 p r o b l e m s . 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 u s e d i n a r i t h m e t i c p e r h a p s p r o v i d e t h e most f a m i l i a r e x a m p l e s , b u t a l g o r i t h m s a r e 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 s u c h p r o c e d u r e s i s t h a t t h e y c a n 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 understand i n g . " 5 2  15  As 'mechanical'  a c o m p u t e r h a s no  'understanding'  and i s t r u l y  i t o b v i o u s l y s o l v e s p r o b l e m s b y 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 c o m p u t e r u s e u s u a l l y c a n be  e x t e n d e d i n a p p l i c a t i o n t o human u s e a n d s o S c a n d u r a ' s s t a t e m e n t w o u l d seem t o s u p p o r t  the b e l i e f that there  exists  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  who a r e n o t 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 person  w h o i o r i g i n a l l y devised of i t s operation  the a l g o r i t h m should  users,  understand the theory  completely.  Scandura  continues;  "Many e d u c a t o r s a n d 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 are 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 problem 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 of 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 understanding 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 f o r m a n c e on some t r a n s f e r t a s k , h o w e v e r , p o s e s a p r o b l e m 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 understanding 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 this d i f f i c u l t y i s to operationally define 'understanding' i n t e r m s o f t h e 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  study,  a s w i l l be s e e n , v a r i e d t h e  amount o f i n f o r m a t i o n t o d e t e r m i n e t h e d e g r e e o f In a s e r i e s of three  experiments'' '*~ (  understanding. y  Scanduras  demonstrated t h a t : a)  S u c c e s s f u l p r o b l e m s o l v i n g does n o t n e c e s s a r i l y  d e p e n d on a n u n d e r s t a n d i n g  of the problem i n v o l v e d .  (This  lends  16 support  t o the modern n o t i o n o f  manipulation the  c a n o f t e n be  'systems* i n w h i c h  symbol  s u b s t i t u t e d f o r understanding  of  problem). b)  T r a n s f e r does n o t  depend on  'understanding*.  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 detect r e l a t i o n s h i p s and  c u e s among a l g o r i t h m s  and  are able  i n g to s y n t a c t i c c o n s t r a i n t s present and  the  t o be  one  between the  i n d i v i d u a l problem c h a r a c t e r i s t i c s .  an e s s e n t i a l s k i l l c)  Gagne,  t o m o d i f y them  facilitate  algorithm  ( T h i s w o u l d seem  f o r a systems programmer).  I t i s p o s s i b l e t h a t , i n s t e a d of asking,  'What w o u l d S n e e d t o know i n o r d e r  c o u l d be  accord-  concerned w i t h the problem,  problem s o l v i n g ? ' .  r e l a t i o n s h i p s c o u l d be d)  'Could  this  task?',  information  I n other words, the  of prime  It i s feasible  t o do  like  structured  importance.  to p r e d i c t student  s o l v i n g p e r f o r m a n c e on a g i v e n t o p i c by  problem  subjectively analysing  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 the  information e)  not  presented. 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  material i s  a l w a y s s u f f i c i e n t t o e n s u r e s u b s e q u e n t l e a r n i n g when t h e  t e r m s ( i e . s y m b o l s and are used to d e s c r i b e was  and  words) d e n o t i n g  the h i g h e r  the  subordinate  order m a t e r i a l .  notions  T h a t i s , Gagne  c o n c e r n e d 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  not  17 with the presentation ordinate  of continuous discourse  i n terms o f sub-  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 a r e a t l e a s t some c o n d i t i o n s  p r e r e q u i s i t e p r a c t i c e has a g r e a t e r  under w h i c h  e f f e c t on problem s o l v i n g  performance i m m e d i a t e l y a f t e r l e a r n i n g t h a n an e q u a l amount o f p r a c t i c e time a t the c r i t e r i o n l e v e l . problem o f g l o s s i n g over p r e l i m i n a r y  Scandura n o t e s t h a t t h e t o p i c s so as t o spend  more time on 'main i d e a s *, p a r t i c u l a r l y when t h e 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 c l a s s r o o m s , 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 a r e a s .  ( T h i s l e n d s some s u p p o r t t o  the n o t i o n t h a t t h e p r o v i s i o n o f m e d i a t o r s 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 o f g l o s s i n g over n e c e s s a r y p r a c t i c e i n algorithm g)  development). Problem s o l v i n g i s improved by t h e p r e -  experimental a v a i l a b i l i t y of prerequisite material, 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  even  materials,  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 p r e s e n t e d p r i o r t o 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 o n l y when the p r e r e q u i s i t e  material  came f i r s t . 30 31  I n 1967 h)  Scandura added these  Knowing how t o use an a l g o r i t h m  conclusions; i s different  18 f r o m k n o w i n g when t o u s e i t .  There i s a tendency  s o l v e a g r o u p o f p r o b l e m s a l l i n t h e same way. i)  to t r y to  (cf. Einstellung).  The more g e n e r a l t h e i n t r o d u c t o r y m a t e r i a l a n d  examples a r e the b e t t e r use w i l l The  be made o f them.  p r e s e n t a u t h o r w o u l d comment t h a t s o l v e d e x a m p l e s  are 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 a n d a n e x a m p l e may be s i m p l y t a k i n g t h e p a t h o f l e a s t resistance.  I n examining  l e v e l which  a s o l v e d example a t a p r e r e q u i s i t e  i s intended to transfer generalized s k i l l s to  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 t o o l o w - t h a t is, of  i t may be t o o s p e c i f i c . a problem  The S_ m i g h t s i m p l y s e e a n  type r a t h e r than o f a concept.  The e f f e c t o f  c u e s l e a r n e d may t h e n have i n d u c e d a n E i n s t e l l u n g Scandura  noted  that the r e s u l t s  were n o t u n e q u i v o c a b l e material  of h i s experiment,  of the present Scandura  effect. however,  a s t h e e f f e c t may depend u p o n t h e  (groups and t o p o l o g y ) used.  the r e s u l t s  exemplar  used  The same may be s a i d f o r  experiment.  an example w h 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 problem  p r e s e n t e d i n much t h e same way a s  t h e p r e s e n t E.  however, p r o v i d e d t h e h e l p f u l  Scandura,  to  a l l Ss a n d s o was u n a b l e  of  p r o v i d i n g o r n o t p r o v i d i n g t h e example.  example  t o d i s c r i m i n a t e between the e f f e c t The p r e s e n t  study  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 a n d e x p a n d e d Scandura's  statements  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 p r o b l e m s b e t w e e n t h o s e who are  not  are  and  those  "rule-users". Scandura a l s o s e t a p r e c e d e n t f o r the use  h i s t o r i c a l m a t e r i a l as he  a filler  confirmed experimentally  research  i n t o t r a n s f e r of Haslerud-^  the  f o r the  that this  c o n t r o l group  should  with  coding  derived  by  by  e x a m p l e s was  learner  problems.  One  no  efficient  experiment  dealing  a principle  ( r a t h e r t h a n s t a t e d and was  (who  Craig  t h e more  p r o b l e m s w h i c h showed t h a t u n l e s s the  confound  b e s t ) and  c o u n t e r e d t h e i r w o r k w i t h an  through examples) there other  not  and  the work of K a t o n a  f o u n d t h a t t h e more g u i d a n c e p r o v i d e d discovery)  of  training.  i n reviewing  2  maintained that teaching (who  who  was  demonstrated  t r a n s f e r of the p r i n c i p l e t o  would expect to 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 of  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 'all  o r n o n e ' a s s u m p t i o n t h a n by  tion:  That i s , r e s e a r c h  o f s u c h l e a r n i n g may the  subject  T h i s may  had  a simple  with  simply  r e a c h e d the  h a v e shown t h e conditioned  ' t r a n s f e r of t r a i n i n g ' but  t h o s e s t u d i e s w h i c h showed  assumpprocess  probability  state  (when  p  p=l).  o f some s t a t e m e n t s  i t helped to  'lower order  an  incremental  w h i c h a t t e m p t e d t o show t h e  throw doubt upon the r e l e v a n c e  concerning  simple  explain  transfer' with  no  that  20  accompanying  'higher order  Suppes-^, l i k e  transfer'. Gagne, w o r k e d o n t h e c r e a t i o n o f a  model which can p r e d i c t from an item's s t r u c t u r e the process 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 It  s h o u l d be n o t e d t h a t i n t h i s p a p e r  was n o t e q u a t e d w i t h  response.  a 'correct  response'  'learning a concept'.  Romberg-^, j _  n c o n  trast  t o Suppes, s t a t e d  that,  " . . . 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 m a t h e m a t i c s c o u l d be g u i d e d b y 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 , b u t t h i s i s n o t t h e c a s e . As i n d i v i d u a l i z a t i o n o f i n s t r u c t i o n a n d c o m p u t e r management become a r e a l i t y , a p t i t u d e a n d 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 t h e n e x t decade c r a c k s i n t h e i r o n c u r t a i n w i l l appear." Among t h e a p t i t u d e s t o w h i c h Romberg 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 Lee-^'-^ cha-ining cues rules* and of for  refers  to resist Einstellung  rigidity.  continues t o i n v e s t i g a t e the e f f e c t of  ( 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  s u c h 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  shapes. unique  He a n d Gagne seem t o b e l i e v e  i n the existence  b o d i e s o f component l e a r n i n g w h i c h a r e n e c e s s a r y  the t r a n s f e r from lower t o h i g h e r l e v e l concept.  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 t h e l i n k s c a n be f o u n d .  However  i n the chain  I n today's i n t e r d i s c i p l i n a r y systems,  design  c u e s may be h a r d t o f i n d a n d t h e 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 .  Training i n 'looking  21 outside the problem  1  and 'avoiding blindness' may be what i s  required for at least some members of the population.  22  STATEMENT OF THE HYPOTHESIS It  i s hypothesized, t h a t a group o f s t u d e n t s  who s u c c e e d on a l o w o r d e r c o m p u t e r  algorithm  task after the p r o v i s i o n of a s p e c i f i c  example  development will  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 will  a g r o u p o f s t u d e n t s who s u c c e e d e d on t h e l o w 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 In  particular:  1.  The f i r s t  problem f o r a l l 2. the  example.  problem i s e a s i e r than the second  students.  The e x a m p l e  helps the f i r s t  group i n d o i n g  "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 problem i n both groups. 3.  The s e c o n d g r o u p have a h i g h e r p r o p o r t i o n o f  correct solutions  f o r the "hard" problem than the f i r s t  group. k.  The s e c o n d g r o u p 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 than the 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 t h e first  problem are considered.  23  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  first  prepared. The  Instructional The  Device.  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 booklet which contained, Students",  a n d "The L e s s o n " .  " D i r e c t i o n s t o Teachers and  ( c f . A p p e n d i x 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 t h e u s e o f t h e BASIC c o m p u t e r  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 variable. The  Experimental The  Device.  experimental device consisted of a single  d u p l i c a t e d page s h o w i n g a s o l v e d e x a m p l e o f a p r o g r a m t o p r i n t the  odd n u m b e r s .  ( c f . A p p e n d i x B)  was t o make t h e ' e a s y ' The  The i n t e n t o f t h i s  example  t a s k very easy f o r the experimental  group.  C o n t r o l Device. The  c o n t r o l device  consisted of a single duplicated  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 . A p p e n d i x C)  T h i s was t o be u s e d b y t h e c o n t r o l g r o u p w h i l e t h e e x p e r i m e n t a l  24 group used t h e e x p e r i m e n t a l The  Test  device.  Instrument. The  t e s t instrument  page w h i c h i d e n t i f i e d  consisted of a single duplicated  t h e s t u d e n t , h i s g r o u p (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 ) , a n d a s s i g n e d two t a s k s . one  ( t h e 'easy'  e v e n numbers.  skills  Task  t a s k ) was t o p r o g r a m t h e c o m p u t e r t o p r i n t t h e T a s k two ( t h e ' d i f f i c u l t *  the computer t o p r i n t The  ( c f . A p p e n d i x D)  t h e F i b o n a c c i numbers.  reasons  developed  t a s k ) was t o p r o g r a m  f o r h a v i n g t h e a l g o r i t h m development  on a computer were:  1.  C o m p u t e r l a n g u a g e c o u l d be u s e d t o d e v e l o p  2.  I t w o u l d 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  algorithms.  experience  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 past l e a r n i n g experiences. 3. be  As f o u r new t e r m s ( c o m p u t e r commands) w e r e t o  introduced the confounding  v o c a b u l a r y was r e d u c e d .  effect of previously learned  A t t h e same t i m e  c o u l d be programmed u s i n g o n l y t h e s e  complete  algorithms  f o u r new commands ( L E T ,  PRINT, END, GOTO). 4. in  As t h e l e a r n e r s w o u l d n e v e r have b e e n  instructed  c o m p u t e r l a n g u a g e s b y a n y o t h e r i n s t r u c t i o n a l mode t h e  confounding  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  reduced.  25  5.  The e v a l u a t i o n o f s u c c e s s c o u l d be a c c o m p l i s h e d  q u i c k l y by an ' u n b i a s e d * t e a c h e r or e x p e r i m e n t e r 6.  computer w i t h l i t t l e  o f any k i n d o f  effect.  The whole s u b j e c 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  experimenter.  26 THE PILOT  STUDIES  The m a t e r i a l s d e s c r i b e d a b o v e w e r e u s e d i n two p i l o t s t u d i e s to determine t h e i r  s u i t a b i l i t y as t o grade  level,  v o c a b u l a r y l e v e l , time l i m i t s and p r o b l e m d i f f i c u l t y . pilot  s t u d i e s were e x e c u t e d i n t h e same manner a s t h e m a i n  experiment which i s described l a t e r . pilot  The  I t was f o u n d i n t h e f i r s t  s t u d y w i t h g r a d e 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 t h e g r a d e s i x l e v e l was t o o l o w t o produce a s u f f i c i e n t pilot  number o f c o r r e c t p r o g r a m s .  I n the second  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  Vancouver  s e c o n d a r y y s c h o o l , i t was f o u n d t h a t r e v i s i o n  vocabulary l e v e l of both the i n s t r u c t i o n a l i n s t r u m e n t were r e q u i r e d .  West  i n the  d e v i c e and t h e test,  These needed r e v i s i o n s were u n d e r t a k e n  a n d t h e f i n a l v e r s i o n s w h i c h a p p e a r i n t h e a p p e n d i c e s were p r o duced. the  Time l i m i t s a s d e s c r i b e d l a t e r w e r e s e t o n t h e b a s i s o f  s e c o n d 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 w e r e t o be u s e d f o r t h e e x p e r i m e n t w e r e t h o s e c l a s s e s a t a s e c o n d a r y s c h o o l i n West V a n c o u v e r , w h i c h were made a v a i l a b l e t o t h e E b y t h e i r  mathematics  B.C.  teachers.  These t e a c h e r s a g r e e d t o a t t e n d a m e e t i n g t o l e a r n about t h e e x p e r i m e n t and t h e n t o a d m i n i s t e r t h e m a t e r i a l s under the g u i d a n c e o f t h e E.  A m u t u a l l y a g r e e a b l e h o u r was f o u n d s o t h a t  27 all  c l a s s e s c o u l d u s e t h e m a t e r i a l s a t t h e same t i m e  own m a t h e m a t i c s t e a c h e r be no i n t e r a c t i o n  i n attendance)  (with t h e i r  t o ensure t h a t there  would  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 w e r e t o be u s e d f o r t h e experiment  attended  a m e e t i n g one week p r i o r  to the experimental  s e s s i o n a t which the E e x p l a i n e d the purpose of the The E a l s o e x p l a i n e d t h a t t h o s e previous  experiment.  s t u d e n t s who a d m i t t e d t o  e x p e r i e n c e w i t h c o m p u t e r s were n o t t o be u s e d i n t h e  experiment.  E a c h 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 r a n d o m  numbers a n d shown how t o a s s i g n s t u d e n t s  t o g r o u p A o r g r o u p 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 t h e 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 emphasized.  or c o p y i n g by the s t u d e n t s  was  I t was e x p l a i n e d t h a t t h e E w o u l d go f r o m room t o  room w h i l e t h e 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 g r o u p s were f o r m e d and t r e a t e d i n a u n i f o r m manner classrooms  t o be u s e d were i n t h e same w i n g o f t h e  building).  The d i r e c t i o n s i n c l u d e d i n t h e i n s t r u c t i o n a l d e v i c e 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  directions.  (the four  were The  teachers  were t o l d t h a t t h e y were t o a l l o w t e n m i n u t e s 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 "  t h e G r o u p A o r G r o u p B page t o e a c h s t u d e n t .  and  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 of 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) t h e t e s t  instrument,  28 t e n minutes f o r working on t a s k one, and f i f t e e n minutes f o r working on task two (the teacher ensured,  visually,  t h a t no  student worked on t a s k one when t o l d t o go on to t a s k two). I t was e x p l a i n e d t h a t a f t e r the b e l l sounded s i g n a l l i n g the end o f the t e s t p e r i o d the teacher was to c o l l e c t a l l m a t e r i a l s and ask students to r e p o r t whether they would have l i k e d more time or v e r b a l e x p l a n a t i o n s , and whether they would l i k e to meet w i t h the E to d i s c u s s the m a t e r i a l s . asked  The teachers were  t o r e p o r t the answers to these questions t o the E. FORMATION OF THE GROUPS The p o p u l a t i o n was formed as d e s c r i b e d above from  members of f o u r grade nine c l a s s e s which were made a v a i l a b l e to the E by t h e i r mathematics t e a c h e r s .  Students who had  p r e v i o u s computer experience were r e j e c t e d from the sample. Each student was randomly a s s i g n e d t o group A or group B u s i n g a t a b l e o f random numbers.  Ninety-two s u b j e c t s were used;  f o r t y - n i n e i n group A and f o r t y - t h r e e i n group B. PROCEDURE On the day o f the experiment the E was an observer i n the f o u r experimental  classrooms  the m a t e r i a l s as d e s c r i b e d above.  while the teachers E ensured  handled  t h a t the t a b l e o f  random numbers was used c o r r e c t l y and t h a t the c o r r e c t m a t e r i a l s were d i s t r i b u t e d , the c o r r e c t times kept, and t h a t there was no i n t e r a c t i o n between Ss.  29  A l l teachers  reported  that their  t i m e l i m i t s a d e q u a t e and  t h a t no  for verbal explanation.  Three of the  that their  students  expressed  d i s c u s s the m a t e r i a l s .  The  student  students  expressed  found  a desire  four teachers  reported  a d e s i r e t o meet w i t h t h e E  E visited  these  the  classes  and  during  t h e i r r e g u l a r m a t h e m a t i c s 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 the experimental him  by t h e  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 to  students. The  students'  t a s k p e r f o r m a n c e s were judged b o t h  t h r o u g h c o m p u t e r o p e r a t i o n u p o n t h e i r p r o g r a m s and observations  of student  f o r task success  was  programming e r r o r s .  The  t h e a c t u a l p r o d u c t i o n by  m a r k e r v a r i a b l e was  objective  criterion  the  S_s p r o g r a m s , o f t h e r e q u i r e d number s e q u e n c e s .  by  computer,  I n t h i s way  using the  controlled. APPARATUS  The Packard  of a  2007 system computer w i t h c a r d r e a d e r  software. Packard  apparatus consisted i n i t i a l l y  The  f i n a l v e r s i o n of the  and  BASIC  software.  BASIC  experiment used a  9830 computer w i t h h i g h speed p r i n t e r ,  card reader  and  Hewlett-  Hewlett-  o p t i c a l mark  30 S T A T I S T I C A L ANALYSIS Introduction. A C h i - s q u a r e d t e s t was u s e d t o i n v e s t i g a t e t h e difference 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 o n t h e e a s y  task  a s c o m p a r e d t o t h e s e c o n d t a s k f o r a l l Ss a n d w i t h i n e a c h 2.  group.  The p r o p o r t i o n o f s u c c e s s e s o n t h e e a s y t a s k i n  g r o u p A a s c o m p a r e d t o g r o u p B. 3.  The p r o p o r t i o n o f s u c c e s s e s o n t h e d i f f i c u l t  t a s k i n g r o u p A a s c o m p a r e d t o g r o u p B. k.  The p r o p o r t i o n o f s u c c e s s e s o n t h e d i f f i c u l t  t a s k i n g r o u p A a s c o m p a r e d t o t h o s e i n g r o u p B who were s u c c e s s f u l on t h e easy  task.  Data. For  e a c h s t u d e n t t h e two t a s k s were m a r k e d a s  successes o r r f a i l u r e s the  c o r r e c t computer  on t h e b a s i s o f whether t h e y p r o d u c e d print-out.  d e c i s i o n was made o n w h e t h e r to  In addition a subjective  each group A s u b j e c t  u s e t h e method o f t h e p r o v i d e d  example  t a s k i n o r d e r t o do t h e s e c o n d t a s k . was  made o n w h e t h e r  attempted  and o f t h e f i r s t  A subjective decision  each group B s u b j e c t attempted t o use  t h e method o f t a s k one i n o r d e r t o do t h e s e c o n d  task.  31 Null  Hypotheses. H 1.  p o r t i o n o f Ss who  T h e r e 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 t h e s u c c e e d e d on t h e f i r s t  p r o p o r t i o n o f t h o s e who  pro-  t a s k compared t o t h e  s u c c e e d e d on t h e s e c o n d  task.  H l a . T h e r e 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 t h e p o r t i o n o f Ss i n g r o u p A who to  s u c c e e d e d on t h e f i r s t  t h e p r o p o r t i o n o f t h o s e who H lb.  There  task  s u c c e e d e d on t h e s e c o n d  pro-  compared  task.  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 t h e  p o r t i o n o f Ss i n g r o u p B who  s u c c e e d e d on t h e f i r s t  c o m p a r e d t o t h e p r o p o r t i o n o f t h o s e who  task  succeeded on the  pro-  as second  task. 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 t h e  p r o p o r t i o n o f Ss i n g r o u p A who  s u c c e e d e d on t h e f i r s t  c o m p a r e d t o t h e p r o p o r t i o n o f Ss i n g r o u p B who the  first  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 t h e  p o r t i o n o f Ss i n g r o u p A who  s u c c e e d e d on t h e s e c o n d  c o m p a r e d t o t h e p r o p o r t i o n o f Ss i n g r o u p B who second  succeeded  on  task. H 3«  the  task  pro-  task  succeeded  on  task. 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 t h e  p o r t i o n o f S_s i n g r o u p A who  s u c c e e d e d on t h e s e c o n d  c o m p a r e d t o t h e p r o p o r t i o n o f Ss i n g r o u p B who the  s e c o n d t a s k i f o n l y t h o s e s u b j e c t s who  the  first  t a s k were  examined.  pro-  task  succeeded  on  were s u c c e s s f u l on  32 Statistical  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  w i t h one d e g r e e o f f r e e d o m a n d w e r e t e s t e d significance.  a t t h e .05 l e v e l  A2, B l , B2 w e r e i n t e r p r e t e d  and B*2 i n d i c a t e d  of  similarly.  t h e number o f s u c c e s s e s i n g r o u p s A a n d  B on t a s k 2 o f t h o s e Ss who s u c c e e d e d o n t a s k 1. t o t h e t o t a l number o f Ss i n v o l v e d referred  calculated  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  g r o u p A on t a s k 1. A*2  were  'N' r e f e r r e d  i n the experiment.  t o t h e number o f s u c c e s s e s i n e a c h  category.  '#Correct*  33 HI  #Correct Al+Bl A2+B2  Hla Al A2  Hlb Bl B2 H2  Al Bl H3 A2 B2  H4 A*2 B*2  In statistic  F H  E G #Correct  #Incorrect  E G  F H  #Correct  #Incorrect  E G  F H  #Correct  #Incorrect  E G  F H  #Correct  #Incorrect  E G  F H  #Correct  #Incorrect  E G  F H  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  a c c o r d i n g t o the formula-^ CHI  SQUARED =  7  N (EH — FG) (G+H) (E+G) (F+H) 2  (E+F)  (cf.  #Incorrect  Appendix E f o r the computer s t a t i s t i c a l  treatment)  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  obtained.  TABLE I SUMMARY OF DATA Group A  Group B  Had n e i t h e r t a s k c o r r e c t . Task 1 c o r r e c t o n l y . Task 2 c o r r e c t only. T a s k s 1&2 c o r r e c t . One o r b o t h t a s k s c o r r e c t . Task 1 c o r r e c t . Task 2 c o r r e c t . Showed E i n s t e l l u n g e f f e c t .  6 31 1 11 43 42 12 14  20 9 0 14 23 23 14 1  Number  49  43  Number o f Ss who:  TESTING OF THE Hypothesis  HYPOTHESES  one. Hypothesis  one was t h a t t h e r e w o u l d be no  signi-  f i c a n t d i f f e r e n c e i n t h e p r o p o r t i o n o f s u b j e c t s who s u c c e e d e d on t h e f i r s t  t a s k as compared t o t h e p r o p o r t i o n o f those  succeeded on t h e second t a s k . e a c h g r o u p o f Ss s e p a r a t e l y . the r e s u l t s  obtained:  who  Hypotheses l a and l b d e a l t w i t h The f o l l o w i n g t a b l e s  summarize  35 TABLE I I TEST OF HYPOTHESIS ONE HI  # Correct  Al+Bl 65 ( 7 1 % ) A2+B2 26 ( 2 8 % ) D e g r e e s o f f r e e d o m = 1, C h i - s q u a r e  # Incorrect  =  27 ( 2 9 $ ) 66 (72%)  33-069  TABLE I I I TEST OF HYPOTHESIS ONE(a) HIa  # Correct Al  # Incorrect  42 ( 8 6 % )  A2  12 (2i+%)  D e g r e e s o f "freedom = 1, C h i - s q u a r e  7 (14%)  37 (76%)  = 37.121  TABLE I V TEST OF HYPOTHESIS ONE(b) Hlb  # Correct  Bl 23 (53%) B2 r 4 (33%) D e g r e e s o f f r e e d o m = 1, C h i - s q u a r e The  critical  level of significance^ Since  in  i t was c o n c l u d e d  20 ( 4 7 % )  29 (67%)  = 3*842  o f C h i s q u a r e d a t t h e .05  w i t h one d e g r e e o f f r e e d o m i s 3«84l  a l l three values  exceeded t h e c r i t i c a l and  0  value  # Incorrect  value,  o f Chi-squared  theN u l l hypothesis  obtained was r e j e c t 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  t h e p r o p o r t i o n o f s u b j e c t s who s u c c e e d e d o n t h e f i r s t  task  36 a s compared  to the second t a s k .  e v e n w i t h g r o u p B who the  first  t a s k was task*  task.  T h i s r e s u l t was  had no e x a m p l e t o a s s i s t t h e m i n d o i n g  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 t h e  f o u n d t o be e a s i e r a n d i s t h e r e f o r e a  i n computer a l g o r i t h m  Hypothesis  order  two. t h a t t h e r e w o u l d be no  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 s u c c e e d e d on t h e f i r s t g r o u p B who  summarizes  'lower  first  development.  H y p o t h e s i s two was  in  obtained  o f S_s i n g r o u p A  t a s k compared  who  t o the p r o p o r t i o n  s u c c e e d e d on t h e f i r s t  the r e s u l t s  signi-  task.  The  table  o f Ss below  obtained: TABLE V  TEST OF HYPOTHESIS H2  # Correct  TWO #  Incorrect  Al 42 ( 8 6 % ) 7 (14%) Bl 23 ( 5 3 % ) 20 ( 4 7 % ) D e g r e e s o f f r e e d o m = 1, C h i - s q u a ^ e = 11.471 Since c r i t i c a l value  the chi-squared  value  the n u l l hypothesis  concluded that there  was  provided  exceeded the  r e j e c t e d and i t 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 e a c h g r o u p who I t may  was  obtained  s u c c e e d e d on t h e f i r s t  t h e r e f o r e ^ be c o n c l u d e d t h a t t h e s o l v 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  task. example task  37 by a c t i n g as a m e d i a t o r between and t h e f i r s t Hypothesis  the i n s t r u c t i o n a l  material  task.  three. Hypothesis  t h r e e was t h a t t h e r e w o u l d be no  signi-  f i c a n t d i f f e r e n c e i n t h e p r o p o r t i o n o f Ss i n g r o u p A who c e e d e d on t h e s e c o n d t a s k compared  t o t h e p r o p o r t i o n o f Ss i n  g r o u p B who s u c c e e d e d on t h e s e c o n d t a s k . summarizes  the r e s u l t s  suc-  The t a b l e  below  obtained: TABLEAVI  TEST OF HYPOTHESIS H3  THREE  # Correct  # Incorrect  A2 12 ( 2 4 % ) B2 14 (33%) D e g r e e s o f f r e e d o m = 1, C h i - s q u a r e Since the  critical  tically  the chi-squared  value  value  i t was c o n c l u d e d  =  (76%) (67%)  37 29  .735  obtained  t h a t t h e r e was no  s i g n i f i c a n t d i f f e r e n c e between  hindered  the whole  I t cannot  t h a t Ss i n g r o u p A a s a g r o u p w e r e  by t h e example  E v i d e n t l y t h e example  statis-  the groups i n r e g a r d t o  t h e p r o p o r t i o n who s u c c e e d e d on t h e s e c o n d t a s k . t h e r e f o r e be c o n c l u d e d  d i d not exceed  i n performing  the second  was n e i t h e r a h i n d r a n c e  o f g r o u p A on t h e s e c o n d  task.  task.  nor a help to  38  Hypothesis  four. Hypothesis  f o u r was t h a t t h e r e 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 t h e p r o p o r t i o n o f Ss o f 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 t h e second t a s k i f o n l y those Ss 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. below summarizes the r e s u l t s  The t a b l e  obtained:  TABLE V I I TEST OF HYPOTHESIS FOUR H4  # Correct 11 (26%) 14 (61%)  A*2 B*2  # Incorrect  Degrees o f freedom = 1, C h i - s q u a r e d  31 (74%) 9 (39%)  = 7.551  Since 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 t h e 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 concluded  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 t h e  p r o p o r t i o n o f Ss i n t h e two groups who succeeded on the second t a s k i f o n l y those 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.  I t may t h e r e f o r e be concluded  that the solved  example h i n d e r e d any t r a n s f e r w h i c h might have o c c u r r e d from the f i r s t t a s k t o the second t a s k f o r those who s u c c e s s f u l l y completed t h e f i r s t  task.  39 CONCLUSIONS Considering he  a l l four hypotheses together  c o n c l u d e d t h a t t h e p r o v i s i o n o f an example w h i c h mediated  transfer of s k i l l s  t o a n e a s y t a s k made no d i f f e r e n c e t o t h e  number o f c o r r e c t s o l u t i o n s on t h e h a r d e r , the  i t may  c r i t e r i o n task, but  s o l u t i o n o f t h e e a s i e r p r o b l e m (when a s o l v e d  provided) d i dhinder skills  acquired  the further u t i l i z a t i o n  i n performing  the f i r s t  e x a m p l e was  (transfer) of  task.  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  scrutiny of unsuccessful  S_ p r o g r a m s .  by t h e s u b j e c t i v e  F o u r t e e n S_s i n g r o u p A  a t t e m p t e d t o u s e t h e method o f s o l u t i o n d e m o n s t r a t e d i n t h e example and a p p r o p r i a t e  t o the f i r s t  B a t t e m p t e d such a method.  task.  O n l y one S i n g r o u p  40 CHAPTER I V CONCLUSIONS AND IMPLICATIONS OF THE STUDY INTRODUCTION The  central hypothesis of t h i s  s t u d y was t h a t t h e  use o f a s p e c i f i c example w h i c h would m e d i a t e t h e t r a n s f e r of  skills  from i n s t r u c t i o n t o t h e s o l u t i o n o f an easy problem  w o u l d a l s o i n h i b i t t h e t r a n s f e r o f t h o s e same s k i l l s e a s y p r o b l e m t o a more d i f f i c u l t was n o t u n s u c c e s s f u l b e c a u s e because  problem.  That i s ,  from an the student  o f exposure t o the easy t a s k , b u t  of h i s s o l u t i o n of the task using a solved  example.  As s i g n i f i c a n t r e s u l t s w e r e 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  possible reasons. DISCUSSION OF THE CONCLUSIONS  The  Content o f the Lesson. In  choosing the m a t e r i a l f o r i n s t r u c t i o n the  experimenter 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  s u i t a b l e r e a d i n g and grade l e v e l .  a  I t may have b e e n t h a t t h e  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 t h e i n t e r e s t v a l u e o f t h e s u b j e c t m a t t e r the  results.  However i t was t h e p u r p o s e  influenced  of the study only to  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 b y t h e u s e o f a s p e c i f i c example.  Of c o u r s e o t h e r e x a m p l e s w e r e u s e d i n t h e i n s t r u c t i o n  common t o b o t h c r e x p e r i m e n t a l g r o u p s a n d t h e e f f e c t s e x a m p l e s 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  research.  of these  41 The  Method o f I n s t r u c t i o n . For purposes of experimentation 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 u s e d .  I t may be  t h a t t h e 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 material.  Such t e c h n i q u e s  demonstration  as o r a l d i s c u s s i o n and computer  m i g h t have p r o d u c e d 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 of success. Length  o f the Experiment. Although  pilot  experimentation confirmed  s t u d i e s and o b s e r v a t i o n s  t h a t no S_ w a n t e d more t i m e , i t i s •  p o s s i b l e t h a t t h e one h o u r l o n g e x p e r i e n c e i n some Ss a n d t h e r e b y  during  induced  fatigue  influenced the r e s u l t s .  G r o u p Makeup. Although  r a n d o m a s s i g n m e n t t o g r o u p s was  i n o r d e r t o have e q u i v a l e n t g r o u p s a n d a l t h o u g h obvious  necessary  i t was made  t h a t a s s i g n m e n t t o g r o u p s was r a n d o m , t h e r e may have  b e e n a c u r i o s i t y a b o u t t h e m a t e r i a l s o t h e r s were  receiving  w h i c h d i s t r a c t e d some S s . Mechanics o f R i g i d i t y . No c o n c l u s i o n s have b e e n f o r m e d a s 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 original predisposition to r i g i d i t y whether the l e a r n e r i s delayed of necessary difficulty  "by e x a m p l e s i n h i s d e v e l o p m e n t  h e u r i s t i c s , misled i n h i s expectation of task  or hindered  i n some o t h e r way.  beyond t h e scope o f t h i s p a p e r . however, t h a t r i g i d i t y by  i s innate or learned,  These q u e s t i o n s a r e  I t has been d e m o n s t r a t e d ,  c a n be i n d u c e d  i n a t l e a s t some Ss  the use o f an example. LIMITATIONS OF THE Some l i m i t a t i o n s  STUDY  o f t h e s t u d y have a l r e a d y b e e n  i m p l i e d i n t h e above d i s c u s s i o n s o f t h e c o n c l u s i o n s respect to the lesson content,  method o f i n s t r u c t i o n ,  o f t h e e x p e r i m e n t a n d g r o u p makeup. the u s u a l l i m i t a t i o n s intelligence.  length  There a r e a l s o o f course  of the assumption of randomization  of  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 t h e Ss i n t h e  f i n a l e x p e r i m e n t came f r o m t h e same s c h o o l . an upper s o c i o - e c o n o m i c  district  educational innovation. hand, whether r i g i d i t y school  with  This school i s i n  and has a h i s t o r y o f  F o r the purposes of the question a t c a n be i n d u c e d ,  (with i t s p a r t i c u l a r  the choice  socio-economic  of a single  s t a t u s a n d I.Q.  r a n g e ) 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 . For  the purpose o f g e n e r a l i z i n g the r e s u l t s  e s p e c i a l l y f o r text-book limitations  are great.  w r i t e r s and t e a c h e r  of this  study f u r t h e r ,  educators, the  Further studies f o r generalization  s h o u l d be c o n d u c t e d i n a s w i d e 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 problem  the f e a s i b i l i t y of  s o l v i n g p e r f o r m a n c e 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 b e t w e e n t h e c r i t e r i o n and sented.  predicting  the i n f o r m a t i o n p r e -  I n the p r e s e n t study a s t r u c t u r a l r e l a t i o n s h i p  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 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 presented'.  invol-  T h i s s t r u c t u r e was  to  'information  i n a p p r o p r i a t e a t the  final  c r i t e r i o n l e v e l , y e t i t w o u l d seem t o t h e a u t h o r t h a t p r e c i s e l y this  structure i s presented  t o s t u d e n t s u s i n g many t e x t b o o k s  i n h i g h s c h o o l m a t h e m a t i c s and the i n t r o d u c t i o n to a t o p i c fic  s o l v e d e x a m p l e s and  s t u d e n t who  uses these  computer s c i e n c e .  Typically  i s f o l l o w e d by t h r e e or f o u r  t h e n by a s e t of e x e r c i s e s .  speci-  The  e x a m p l e s 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 f r o m w h a t he has r e a d t o t h e e x e r c i s e s which  he must now  dp_.  The  s t u d e n t who  attempts  to apply  the  methods o f t h e e x a m p l e s t o o r i g i d l y t o t h e e x e r c i s e s f a i l s t h e more d i f f i c u l t may  o f t e n be  first  e x e r c i s e s which,  i n higher l e v e l  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 .  on  courses, Indeed,  the  few p r o b l e m s i n t h e e x e r c i s e a r e f r e q u e n t l y i n t e n d e d  no more t h a n d r i l l  and p r a c t i c e  i n t h e use  of vocabulary,  definitions,  and n o t a t i o n .  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  L u c h i n s and M i l l e r have a l r e a d y  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 t h e e x a m p l e may  as  this  t h e p r e s e n t s t u d y shows t h a t  reinforce this predisposition  ( i f Einstellung  44  rigidity  i s t h o u g h t o f as a l e a r n e d b e h a v i o r ) .  h i n d e r w h a t Gagne c a l l s  productive  fore that further research  learning.  This  will  I t seems t h e r e -  s h o u l d be u n d e r t a k e n w h i c h  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, i n some way, contains should  investiclassified  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  e f f e c t i v e warnings to avoid b l i n d n e s s .  use l a r g e r b l o c k s  Such  research  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 S c a n d u r a ' s  c a v e a t t h a t Ss  continue  confronted  t o r e s p o n d i n a r i g i d way  feedback which i n d i c a t e s t h a t  unless  'the r u l e h a s  with  changed'.  I f i t i s f e l t that 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 w o u l d be a p p r o p r i a t e construct instruments  which would  although  on t h e c o m p u t e r .  there  i s a negative  and E i n s t e l l u n g r i g i d i t y  tests.  would  algorithm  I t c o u l d p e r h a p s be shown t h a t c o r r e l a t i o n between  intelligence  i t i s possible to construct  t e s t s f o r r i g i d i t y which are b e t t e r f o r the purpose intelligence  to  i d e n t i f y which students  m o s t b e n e f i t f r o m a c o u r s e i n s u c h d i s c i p l i n e s as development  will  specific than  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 processes f o r the s o l u t i o n of 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 a n 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 . " American Standard Vocabulary f o r Information 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 , 1 9 6 6 ) p.9« The t e r m " p r o g r a m " 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 simply t h e r e v i s i o n o f e x i s t i n g programs. 2 Mediator - a bridge 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 . ^ S y s t e m s w o r k - " 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." A m e r i c a n S t a n d a r d V o c a b u l a r y , op.cit., p.26. Students i n t h e author's grade e l e v e n computer s c i e n c e course undertake the p r o d u c t i o n o f packages f o r such tasks 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 , attendance, 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 , chemistry l a b o r a t o r y r e p o r t marking, analysis o f functions, graphing of conies, sunrise 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 / B e n j a m i n S. B l o o m ( 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 D o m a i n (New Y o r k : D a v i d McKay Company I n c . , 1 9 5 6 ) , p . 1 7 0 . -'For t h e u s e o f t h e t e r m s " 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 K n o w l e d g e , " P s y c h o l o g i c a l R e v i e w , 59 35-65• 1962. F o r t h e u s e o f " E i n s t e l l u n g " c f . A.S. a n d E.H. 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 , t h e E f f e c t o f Einstellung»." P s y c h o l o g i c a l M o n o g r a p h , 5 4 , No.6, p . l , 1 9 4 2 . W. P. G o d d a r d , "Computers a n d The Whole School','' J o u r n a l o f E d u c a t i o n a l D a t a P r o c e s s i n g , 6:108-120, 1 9 6 9 . 6  n  B e n j a m i n S. B l o o m ( 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 D o m a i n , (New Y o r k : D a v i d McKay Company I n c . , 1 9 5 6 ) p . 1 6 2 , ( S y n t h e s i s ) . ^Ibid.,  p.166.  o  Seymour P a p e r t , " T e a c h i n g C h i l d r e n Thinking-',' (memo LOGO L a b o r a t o r y , 545 T e c h n o l o g y S q u a r e , C a m b r i d g e M a s s . ) . " ^ B l o o m , op. c i t . , p. 1 6 2 . 1 1  Ibid.,  Synthesis  5.2, p . 1 7 0 .  12 R o b e r t M. Gagne a n d L a r r y T. Brown,."Some F a c t o r s i n the Programming o f Conceptual 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 R.M. Gagne, H. F o s t e r , and M.H. C r o w l e y , "The Measurement o f T r a n s f e r of 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 , 3  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 M o n o g r a p h , 75:1-23, 1961. 1  ^ 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 of Conceptual L e a r n i n g , " J o u r n a l of 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 K n o w l e d g e , " P s y c h o l o g i c a l R e v i e w , 69:p.360, 1 9 6 2 . 17  N.L. Gage, " P a r a d i g m s 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 M c N a l l y , 1 9 6 3 ) 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 A t t e m p t s at Preventing Mechanization i n Problem S o l v i n g , " J o u r n a l of G e n e r a l P s y c h o l o g y . " 42:p.297, 1950. 19  A.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 , t h e 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 M o n o g r a p h , 84:p.93,1942. Ibid.. p.91. y  2 0  21  K e n n e t h 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 M e t h o d s , " 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  IMdl,  23  ^M.  Row,  p. 134.  Wertheimer, P r o d u c t i v e T h i n k i n g ,  (Harper  and  1945).  24  J.M. S c a n d u r a , " A l g o r i t h m L e a r n i n g and P r o b l e m 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. Ibid., p.l. 2 5  26 27  . . op.cit. Ibid., p . l .  28 J.M. S c a n d u r a , " P r o b l e m S o l v i n g and P r i o r 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 E d u c a t i o n , 34:4-7, 1966B. 29 . . . J.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 P r o b l e m 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 2 , I966C. The  y  47 30  J.M. S c a n d u r a , E. Woodward, and F. L e e , " R u l e G e n e r a l i t y and C o n s i s t e n c y i n M a t h e m a t i c s L e a r n i n g , " American E d u c a t i o n a l Research J o u r n a l , 4:p.303» I967A. J  31  J.M. S c a n d u r a and J.N. 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"The V a l i d i t y o f t h e E i n s t e l l u n g T e s t as a M e a s u r e o f 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 , 4 8 : 5 7 3 - 5 8 0 , 1953L o u p e , 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 P r o b l e m 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.  Solving.  L u c h i n s , A.S. "On R e c e n t U s a g e o f 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 , 15:89-94, 1951. L u c h i n s , A.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 M o n o g r a p h s , 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 A t t e m p t s a t P r e v e n t i n g Mechanization i n Problem S o l v i n g , " J o u r n a l of G e n e r a l Psychology, 42:279-297, 1950. 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"Non V e r b a l R i g i d i t y and A c a d e m i c A c h i e v e m e n t 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 M o t o r S k i l l s , 3 3 : 2 P T . 2 ; 1 1 7 8 , 1 9 7 1 . R i e d e s e l , C.A. and Suydem, M.N. " P r o b l e m S o l v i n g , Some S u g g e s t i o n s f r o m R e s e a r c h , " The A r i t h m e t i c T e a c h e r , 16:467-478, 1969. R o k e a c h , 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 A b n o r m a l and S o c i a l Psychology, 43:259-278, 1948. Romberg, T.A. "Research i n Mathematics E d u c a t i o n a l Research. 39:473-491,  E d u c a t i o n , " Review of 1969.  52  Saarni, Carolyn. P i a g e t i a n Operations and F i e l d Independence as Factors i n Children's Problem S o l v i n g Performance. Berkley: U n i v e r s i t y of C a l i f o r n i a , 1 9 7 1 • Scandura, J.M. 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"Some T h e o r e t i c a l Models f o r Mathematics Learning," J o u r n a l f o r Research and Development i n Education, l:l;5-22,  I967.  Suppes, P., Hyman, L e s t e r , and Jerman, M. L i n e a r S t r u c t u r a l Models f o r Response and Latency Performance i n Arithmetic on Computer C o n t r o l l e d Terminals. Stanford: Psychology S e r i e s , I n s t i t u t e f o r Mathematical Studies i n the S o c i a l Sciences, T e c h n i c a l Report No. 9 0 , Stanford U n i v e r s i t y , 1 9 6 6 . Suppes, P. and Jernan, M. "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 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 , 54:27-40. 1970. Van Dalen, Deobold B. Understanding Educational Research. McGraw H i l l S e r i e s i n Education, p . 4 1 3 , 1 9 6 6 . Walther, J . Computer A s s i s t e d 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 S t r a t e g i e s I n v o l v i n g Advance Organizers and Post Organizers and Discovery and Non-Discovery Techniques Where I n s t r u c t i o n i s Mediated by Computer. Tallahassee: D o c t o r a l Thesis, F l o r i d a State U n i v e r s i t y , 1 9 6 6 .  .APPENDICES  53a  APPENDIX A  THE INSTRUCTIONAL DEVICE  53b DIRECTIONS TO TEACHERS AND STUDENTS Hand o u t "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 o u t "GROUP A" page t o h a l f t h e 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 t h e 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 m i n u t e s t o r e a d t h e s e t h r e e p a g e s p l u s t h e i r e x t r a GROUP A o r B p a g e .  4.  S t u d e n t s have 10 m i n u t e s  5.  S t u d e n t s have t h e r e s t o f t h e p e r i o d t o w o r k on P r o b l e m  NOTE:  t o work on P r o b l e m  1. 2.  P l e a s e f e e l f r e e t o t u r n back t o t h e L e s s o n a t any time b u t p l e a s e do n o t w o r k o n P r o b l e m 1 when a s k e d t o w o r k on P r o b l e m 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.  now g o i n g t o l e a r n how t o p r o g r a m a c o m p u t e r .  You a r e  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 c o m p u t e r p r o g r a m must be w r i t t e n i n a s p e c i a l language. c a l l e d BASIC. you w i l l  We w i l l u s e t h e s p e c i a l c o m p u t e r  A f t e r y o u have s t u d i e d t h e language  w r i t e two p r o g r a m s o f y o u r own.  I will  programs i n t o t h e computer t o see i f t h e computer  called  language BASIC  then put your 'understands*  you. THE BASIC LANGUAGE The  c o m p u t e r must h a v e a name f o r e v e r y number.  be a l e t t e r  of the alphabet.  A s t a t e m e n t w h i c h g i v e s a name t o  a number must b e g i n w i t h t h e w o r d L E T . to  T h i s name must  F o r e x a m p l e , i f we w i s h  g i v e t h e name X t o t h e number 9, we w i l l w r i t e t h i s  ^Appendix  note:  O r i g i n a l was on t h r e e 8^ * 14 p a g e s .  statement  54 for  t h e computer: LET X = 9  I f we w i s h t o g i v e t h e name B t o t h e number 4, we w i l l the  write  statement: LET B = 4  Now, i f we w i s h t o h a v e t h e c o m p u t e r a d d t h e number c a l l e d X to Let  t h e number c a l l e d B, we m u s t g i v e a name t o t h e a n s w e r . u s c h o o s e A ( a l t h o u g h a n y l e t t e r e x c e p t X a n d B w o u l d do  j u s t as w e l l ) .  We w r i t e t h e s t a t e m e n t : LET A = X + B  N o t i c e t h a t t h e name o r 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 o n t h e l e f t  of the equal sign.  The c o m p u t e r  must a l r e a d y know t h e numbers b e l o n g i n g t o a n y 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 c o m p u t e r o b e y s y o u r p r o g r a m  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 t h e f i r s t statement. A statement which t e l l s  t h e c o m p u t e r t o p r i n t a n a n s w e r must  b e g i n w i t h t h e w o r d , PRINT. the  F o r e x a m p l e , i f we w i s h t o p r i n t  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 w o u l d g i v e t h e s t a t e m e n t : PRINT A  The  l a s t s t a t e m e n t i n a p r o g r a m must be t h e s i n g l e w o r d END.  E a c h s t a t e m e n t must be w r i t t e n o n a s i n g l e The c o m p u t e r w i l l w r i t e them.  line  o f your program.  obey y o u r s t a t e m e n t s i n t h e same o r d e r y o u  We now know e n o u g h c o m p u t e r s t a t e m e n t s t o w r i t e  a computer program.  55  Here i s our program: LET X =  9  LET B =  4  LET A = X + B PRINT A END When I p u t t h i s p r o g r a m i n t o t h e c o m p u t e r i t w i l l w h a t we  told  i t t o PRINT.  T h i s i s what i t w i l l  print  only  p r i n t on t h e  page '  We  13  have a s p e c i a l r e a s o n f o r n u m b e r i n g e a c h s t a t e m e n t o f o u r  program. looks l i k e  You w i l l  see t h e r e a s o n l a t e r .  Our p r o g r a m  this:  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 e a c h s t a t e m e n t i s t h a t we the  sometimes  c o m p u t e r t o go b a c k t o a p r e v i o u s s t a t e m e n t .  c o m p u t e r t o r e t u r n t o s t a t e m e n t number 1 we GO TO  For  I f we  we  wish  write:  1  e x a m p l e , i f we w i s h t h e c o m p u t e r t o do t h e p r e v i o u s  a g a i n a n d a g a i n , we w o u l d 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  6.  wish  For this  u s e t h e s t a t e m e n t b e g i n n i n g w i t h t h e w o r d s GO TO. the  now  END  1  program  56  _  T h i s i s what the computer w o u l d  print:  13 13  The c o m p u t e r w i l l button. at  It will  statement  5  n e v e r s t o p , o f c o u r s e , u n t i l I p u s h a HALT do s t a t e m e n t 1 ,  i"t w i l l  on a l l o v e r a g a i n . get  then statements 2 , 3 , 4 then  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  As y o u m i g h t g u e s s , t h e c o m p u t e r w i l l  t o s t e p 6 i n our example, b u t t h i s  never  s t e p must be i n t h e  p r o g r a m anyway. Of c o u r s e we  do n o t u s u a l l y w a n t t h e c o m p u t e r t o p r i n t  number a g a i n and a g a i n .  t h e same  Suppose we w a n t e d t h e c o m p u t e r t o u s e  a v a l u e o f 9 f o r X, b u t we w a n t e d B t o s t a r t w i t h a v a l u e o f 4 a n d t h e n t o grow 6 g r e a t e r e a c h t i m e t h e c o m p u t e r d i d a tion.  We  could then w r i t e the f o l l o w i n g LET X =  9  2.  LET B =  4  3.  LET A = X + B  4.  PRINT A  5.  LET B = B +  6.  GO TO  S t a t e m e n t 5 means, l e t t h e new o l d v a l u e o f B.  program:  1.  7.  calcula-  6  3  END v a l u e o f B e q u a l 6 more t h a n t h e  T h i s i s what t h e computer w o u l d  print.  57 I f we h a d w a n t e d t h e c o m p u t e r t o add 9 t o t h e v a l u e each c a l c u l a t i o n ,  i n s t e a d o f 6,  5.  o f B, f o r  s t a t e m e n t 5 c o u l d have  heen:  LET B = B + X  T h i s i s what t h e c o m p u t e r w o u l d have p r i n t e d : 13 22 31  40 I f we h a d w a n t e d t h e 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 t h e c o m p u t e r w o u l d  13 22 31  40 o u r p r o g r a m w o u l d have l o o k e d l i k e  this:  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  8.  END  4  print:  APPENDIX B  THE EXPERIMENTAL DEVICE  58a GROUP A A F I N A L EXAMPLE  OF A PROGRAM.  Here i s a program that w i l l p r i n t the odd numbers l i k e t h i s 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 Here  i sa list  COMPUTERS.  o f names a n d d a 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 1 8 2 0  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 idea o f a c a l c u l a t i n g  2.  Designed a c a l c u l a t i n g  3.  Developed a data p r o c e s s i n g  4.  B u i l t t h e Mark I  5.  B u i l t electromagnetic  6.  Used  electronic  machine.  machine. machine.  computer. calculators.  techniques f o r computation.  60::  APPENDIX D THE TEST INSTRUMENT  60a IMPORTANTi  Make s u r e y o u p u t y o u r Group i n t h e b l a n k b e l o w . You w i l l f i n d w h e t h e r y o u a r e g r o u p A o r g r o u p B b y l o o k i n g a t t h e t o p o f t h e o t h e r s i n g l e page y o u were g i v e n .  GROUP..  NAME, SCHOOL GROUP A & B  USE SCRAP PAPER FOR ROUGH WORK, THEN COPY YOUR F I N A L ANSWERS ON THIS PAGE P r o b l e m 1:  W r i t e a c o m p u t e r p r o g r a m w h i c h w i l l make t h e c o m p u t e r p r i n t t h e e v e n numbers, l i k e t h i s , without ending. ~2  6 8 10  P r o b l e m 2:  W r i t e a c o m p u t e r p r o g r a m w h i c h w i l l make t h e computer p r i n t t h e 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 t h e s e q u e n c e i s F = X + I f y o u now l e t X be t h e same a s Y was, and i f y o u l e t Y be t h e same a s F was, y o u g e t the s e c o n d number i n t h e s e q u e n c e F = X + Y and s o o n .  APPENDIX  T H E COMPUTER  E  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 I S CLEARED 80 LET N=C=0 9 0 REM;N=GRAND TOTAL, C= C H I SQUARED 100 FOR E = l TO R 110 REM;E I S CURRENT ROW 1 2 0 FOR D = l TO K 1 3 0 REM;D I S 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 , 2 0 0 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 4 1 0 2 9 0 LET T = N * ( ( A B S ( A ( l , l ) * A ( 2 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 I S CHI SQUARE ACCORDING TO VAN DALEN PAGE 4 1 3 . 3 4 0 GOTO 40 3 5 0 DATA 2 , 2 , 6 5 , 2 7 , 2 6 , 6 6 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 , 4 2 , 7 , 2 3 , 2 0 3 9 0 DATA 2 , 2 , 1 2 , 3 7 , 1 4 , 2 9 400 DATA 2 , 2 , 1 1 , 3 1 , 1 4 , 9 410 END f  62 PRINTOUT  CORRECT  INCORRECT 27 66  65 26  DEGREES  OF FREEDOM  42 12  DEGREES  DEGREES  OF FREEDOM  33.06912442  = 1  CHI  SQUARE  =  37.12121212  = 1  CHI  SQUARE  =  3.842250414  CHI  SQUARE  =  11.47058901  CHI  SQUARE =  0.735366516  CHI  SQUARE  7 20  OF FREEDOM  12  = 1 37 29  14 OF FREEDOM  11  = 1 31 9  14 DEGREES  =  29  OF FREEDOM  23  DEGREES  SQUARE  20  42 DEGREES  CHI  7 37  23  14  = 1  OF FREEDOM  = 1  =  7.551371636  

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