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Students’ preconceptions of three vector quantities Aguirre, Jose M. 1981

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STUDENTS' PRECONCEPTIONS OF THREE VECTOR QUANTITIES  by JOSE M. AGUIRRE Teacher o f P h y s i c s , U n i v e r s i t y of C h i l e , C h i l e , 196 8 M.A., The U n i v e r s i t y o f B r i t i s h Columbia, 1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in  THE  FACULTY OF GRADUATE STUDIES  DEPARTMENT OF MATHEMATICS AND SCIENCE EDUCATION We accept t h i s t h e s i s as conforming to the r e q u i r e d  THE  standard  UNIVERSITY OF BRITISH COLUMBIA August, 1981 (c)  Jose M. A g u i r r e ,  1981  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t  the L i b r a r y s h a l l make  it  and  f r e e l y a v a i l a b l e for reference  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be  department or by h i s o r her understood t h a t  granted by  the head o f  representatives.  s h a l l not  be  allowed without my  permission.  /^fn . ^  I -in \  StMMftg'K.  SWCfCT'lb^  1  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date  It i s  copying or p u b l i c a t i o n of t h i s t h e s i s  f o r f i n a n c i a l gain  Department of  my  <  S  ;  Columbia  M\  written  ABSTRACT T h i s study was  d i r e c t e d towards the  identification  and a n a l y s i s of students' i n t u i t i v e ideas or about three kinematic v e c t o r q u a n t i t i e s — displacement,  and v e l o c i t y .  A list  preconceptions  vector position,  of ten i m p l i c i t v e c t o r  c h a r a c t e r i s t i c s d e r i v e d from a task a n a l y s i s of these three v e c t o r q u a n t i t i e s , served as the framework f o r the  develop-  ment of the i n t e r v i e w p r o t o c o l s used i n the study and  the  subsequent a n a l y s i s of the data. The  study proceeded i n two phases; i n the  c l i n i c a l - t y p e i n t e r v i e w s were conducted ten  students u s i n g two  first,  w i t h twenty grade  t a s k s which i n v o l v e d d e s c r i b i n g the  movement of a boat on a l a k e and a r i v e r ;  i n the second phas  a group i n t e r v i e w p r o t o c o l (G.I.P.) r e q u i r i n g a w r i t t e n response of ten  was  the f i r s t  developed  (based upon the tasks and the  results  phase) and administered to 8 c l a s s e s of grade  students, producing a t o t a l sample of 17>JS s u b j e c t s . A n a l y s i s of the r e s u l t s from the f i r s t phase y i e l d e d  a s m a l l but v a r i a b l e number (2 t o 5) number of i n f e r r e d r u l e s f o r each o f the ten i m p l i c i t v e c t o r c h a r a c t e r i s t i c s  d e a l t with i n the i n d i v i d u a l i n t e r v i e w s . r u l e s ' are hypothesized sets o f b e l i e f s or c o g n i t i v e  structures  These  (preconceptions)  used by the s u b j e c t s t o e x p l a i n  problem s i t u a t i o n posed by the i n t e r v i e w e r . gorization  yielded  p a r t upon c r i t e r i a  a  Further cate-  three broad types o f preconceptions  s c a l a r , t r a n s i t i o n a l , and v e c t o r i a l —  tive.  'inferred  —  which were based i n  d e r i v e d from the c u r r e n t p h y s i c s perspec-  The o v e r a l l r e s u l t s from the f i r s t phase  that most students possessed f a i r l y c o n s i s t e n t  indicated preconceptions  r e g a r d i n g the d i r e c t i o n a l and d i s t a n c e components i n v o l v e d i n the d e s c r i p t i o n  of the movement o f a boat on a l a k e .  For  example, a l l students r e c o g n i z e d the n e c e s s i t y o f u s i n g a r e f e r e n c e p o i n t and a frame o f r e f e r e n c e , however, they employed d i f f e r e n t references.  ' i n f e r r e d r u l e s ' i n the s e l e c t i o n o f these  Overall,  f o r f o u r v e c t o r c h a r a c t e r i s t i c s , most  of the s u b j e c t s used v e c t o r i a l type ' i n f e r r e d r u l e s ' they c o n s i d e r e d the q u a n t i t a t i v e  (where  aspects o f the v a r i a b l e s  i n v o l v e d , e.g., d i r e c t i o n and d i s t a n c e ) ;  f o r t h r e e o f the  v e c t o r c h a r a c t e r i s t i c s , a h i g h percentage o f s u b j e c t s made use  of scalar-type  f a i r l y primitive  ' i n f e r r e d r u l e s ' which corresponded t o  notions  (where they c o n s i d e r e d o n l y one  q u a l i t a t i v e or q u a n t i t a t i v e  variable);  f o r the remaining  three v e c t o r c h a r a c t e r i s t i c s , most o f the s u b j e c t s demonstrated t r a n s i t i o n a l - t y p e  'inferred rules'  (which con-  s i s t e d i n some combination o f the two p r i o r extreme categories).  The r e s u l t s in  general,  While  from t h e s e c o n d p h a s e o f t h e  corroborated  the f i n d i n g s  some new ' i n f e r r e d r u l e s '  percent of the i n f e r r e d rules discovered that  procedure,  the study  analytical  rules'  involving  task  a d i r e c t i o n a l as w e l l  terms cussed  i n a larger  i n terms  and c u r r i c u l u m  sets  components  v a l i d i t y of similar group o f  i m p l i c a t i o n s of both the r a t i o n a l  and t h e s u b s t a n t i v e  o f the v a r i o u s  the general  more h e t e r o g e n e o u s  which y i e l d e d the l i s t  characteristics,  f o r t h e way  as o t h e r  were e n h a n c e d b y f i n d i n g  The e d u c a t i o n a l  analysis,  of ' i n -  i n t u i t i v e l y approached a problem  t h e a p p r o a c h and r e s u l t s  subjects.  students  the a p p l i c a t i o n of  technique to generate sets  By d e v e l o p i n g a g r o u p q u e s t i o n n a i r e ,  response patterns  seem  of the i n t e r v i e w  w h i c h were h y p o t h e s i z e d t o a c c o u n t  grade ten s u b j e c t s  situation  Hence, i t would  illustrated  80  those  nor l i m i t e d t o a s m a l l group o f grade t e n  a particular  that  phase.  about  u s e d were common t o  were n e i t h e r an a r t i f a c t  I n summary,  ferred  from t h e f i r s t  were i d e n t i f i e d ,  i n the i n t e r v i e w sample.  the results  study,  of  of implicit results,  'inferred rules',  vector  expressed i n were  of t h e i r a p p l i c a b i l i t y to classroom developers.  disteachers  -iv-  TABLE OF CONTENTS Page ABSTRACT  i  L I S T OF TABLES.  xi  L I S T OF FIGURES  xiv  ACKNOWLEDGEMENTS  xv  CHAPTER ONE 1.0  Background  of  the  Study  1  1.1  D e f i n i t i o n of  1.2  Implicit vector Characteristics Physical Vector Quantities  1.3  General Statement of  1.4  Specific  Statements o f  the  1.5  Research  Questions  . . . .  14  1.6  Overview of  Study  15  1.7  Delimitation of  1.8  Justification  Terms  . .  the  Methods o f  of  the  of  . . . .  . . . .  Three  7  Problem  10  Problem  13  17  Study  the  4  Study  . . . .  19  CHAPTER TWO 2.0 •2.1  Introduction  . . . .  . . . .  . . . .  L i t e r a t u r e Review on S t u d e n t s ' Vector Quantities 2.1.1  Understanding ••  Students' D i f f i c u l t i e s i n Understanding Vector Quantities  21 22 22  Page  2.2  2.1.2  Viewing Students D i f f i c u l t i e s C r i t i c a l B a r r i e r Phenomena  2.1.3  P r e v i o u s Research About U n d e r s t a n d i n g on V e c t o r Quantities A.  Piagetian-Type  B.  S p e c i a l Experiments Vector Quantities  C.  Vector Quantities matics Education  Psychological  Context  of  as  Studies  the  to  31 Teach  Study  37  . . . .  37  Two A s p e c t s o f D e v e l o p m e n t a l P s y c h o l o g y : F o r m a l and F u n c t i o n a l  2.3  Educational 2.3.1 2.3.2  ...  38  ••  42  . . . .  43 46  Approach of  S i e g l e r ' s Approach C a s e ' s Approach . . . . Task  36  . . . .  2.2.2  2.2.4  36  i n Mathe-  Introduction  The N e o - P i a g e t i a n Development  28  Students'  2.2.1  2.2.3  a  47  Analysis Context  of  the  Study  T e a c h i n g - L e a r n i n g Problems Science Education . . . . C u r r i c u l u m Problems i n Education . . . .  . . . . . . . . . .  51  in 52  Science  . . . .  55  . . . .  58  CHAPTER.THREE 3.0  Introduction  3.1  The P i l o t S t u d y  3.2  R a t i o n a l Task A n a l y s i s Phenomena . . . .  -  . . . .  . . . .  58 of  Physical •• •  60  -vi-  3.2.1 3.2.2 3.3  3.4  Networks o f Quantities  the Three V e c t o r 6  68  The P h y s i c s A p p r o a c h  The Two T a s k s o f  the  T a s k One  3.3.2  Protocol of  3.3.3  T a s k Two  3.3.4  Protocol of  3.3.5  Further Discussion Protocol  Methods o f Interview  69  Study  3.3.1  3  70  • T a s k One  •  7  3  79 81  T a s k Two of  Data C o l l e c t i o n  :  Interview  86  Clinical  3.4.1  R e l i a b i l i t y and V a l i d i t y Interview  3.4.2  Interview  Format  3.4.3  Sample o f  Subjects  in  89  Clinical . . . .  92 ,  94 95  CHAPTER FOUR 4.0  Introduction  4.1  Data A n a l y s i s  4.3  Rules  . . . .  . . . .  99  . . . .  99  4.1.1  Inferred  4.1.2  An Example o f the D e r i v a t i o n o f Inferred Rules . . . .  4.1.3  4.2  98  C a t e g o r i z a t i o n of the I n f e r r e d R u l e s and C o n s t r u c t i o n o f t h e R u l e - M o d e l f o r RPS .  Format f o r P r e s e n t i n g R e s u l t s Data A n a l y s i s . . . . . . . . Results  of A n a l y s i s of  of  Interview  . . . .  101  105  Interview 110 Data  113  - V l l -  Page  4.4  4.3.1  I n f e r r e d Rules for Reference P o i n t f o r S t a t i o n a r y B o d i e s (RPS) . . . .  4.3.2  I n f e r r e d R u l e s f o r Frame o f R e f e r e n c e f o r a S t a t i o n a r y Body (FR)  .  113  119  4.3.3  I n f e r r e d Rules About Displacement o r Change o f L o c a t i o n (D) . . . .  4.3.4  I n f e r r e d Rules for A d d i t i o n D i s p l a c e m e n t s (AD)  4.3.5  of  130  .  139  I n f e r r e d Rules for S u b t r a c t i o n f o r V e c t o r P o s i t i o n s (SV) . . . .  .  146  4.3.6  I n f e r r e d Rules for Reference B o d i e s f o r M o v i n g O b j e c t s (RPM)  .  154  4.3.7  Inferred Rules for A n a l y s i s of Components (AC)  .  163  4.3.8  I n f e r r e d R u l e s f o r Components o f V e l o c i t i e s (CV)  .  170  4.3.9  I n f e r r e d Rules f o r Independence M a g n i t u d e s o f Components (IMC)  .  184  4.3.10  I n f e r r e d Rules f o r S i m u l t a n e i t y o f Component V e l o c i t i e s ( S C ) . . .  .  191  4.3.11  Subjects' Preconceptions Perspective  .  196  .  202  .  203  .  204  .  205  •  207  .  210  Two New V e c t o r  Characteristics  About . . . .  . . .  of L o c a t i o n  . . .  4.4.1  Independence P a t h (ILP)  from  4.4.2  Independence of D i r e c t i o n s o f Components (IDC) . . . . . . . . . .  4.5  Summary o f I n f e r r e d R u l e s f o r E a c h S u b j e c t i n t h e I n t e r v i e w Sample . .  4.6  Three Types o f 4.6.1  . . .  Preconceptions  Example o f the C a t e g o r i z a t i o n o f Inferred Rules . . .  of  -viii-  Pa'ge 4.6.2 4.6.3  4.6.4. 4.7  C a t e g o r i z a t i o n of Inferred i n t o the Three Types  Rules 211  C a t e g o r i z a t i o n of Inferred Rules of each S u b j e c t i n t o the Three Types  211  Discussion  215  Conclusion of  of  Table  4.25  P h a s e One  222  CHAPTER F I V E 5.0  Introduction  5.1  Meaning o f  225  Generalizability  in  this  Study  226  5.2  The G r o u p I n t e r v i e w  5.3  Construction of  Technique  the Group  226  Interview  Protocol 5.3.1  Tasks of  5.3.2  The G r o u p I n t e r v i e w (GIP)  5.3.3 5.4  . . . . the Group I n t e r v i e w  The P i l o t S t u d y  Administration of  231  Protocol 232  . . .  t h e GIP  233 . . . .  . . . .  Sample o f S u b j e c t s D e s c r i p t i o n of Administration Procedures . . . . . . . . Methodology of Data A n a l y s i s  . . . .  5.4.1 5.4.2  5.5  228  233 233  . . . .  235 238  CHAPTER SIX 6.0  Introduction  6.-1  Results  for  240 each V e c t o r C h a r a c t e r i s t i c  . . . .  240  -ix-  Page 6.1.1  Results for Reference for Stationary Bodies  6.1.2  244  Results for Displacement or ° f L o c a t i o n (D)  6.1.4  Results for Addition ments (AD)  6.1.5  Results for Subtraction of P o s i t i o n s (SVP)  6.1.6  Results for Reference M o v i n g O b j e c t s (RPM)  6.1.7 ' 6.1.8 6.1.9  6.3  242  R e s u l t s f o r Frame o f R e f e r e n c e (FR)  6.1.3  6.2  Point (RPS)  of  Change 246  Displace-  249  Vector  Bodies  251  for 253  Results for A n a l y s i s of (AC)..."  Components 255  Results for Composition of V e l o c i t i e s (CV)  2 5  8  R e s u l t s f o r Independence o f t u d e s o f Components (IMC)  Magni-  6.1.10  Results ponents  Com-  6.1.11  R e s u l t s f o r t h e Two New V e c t o r Characteristics  270  6.1.12  I n f e r r e d Rules f o r Independence o f L o c a t i o n s ' from P a t h 1ILP) • • •  ~7ri "i/u  6.1.13  I n f e r r e d Rules f o r Independence o f D i r e c t i o n s o f Components (IDC)  for Simultaneity of (SC)  Summary o f t h e I n f e r r e d S u b j e c t s i n t h e Sample General Discussion of T e c h n i q u e and R e s u l t s  Rules  266 269  . . . .  274  for A l l 279  Group I n t e r v e i w . . . .  279  -x-  Page  CHAPTER SEVEN the  282  7.0  Overview of  Study  7.1  Conclusions of  the Study  283  7.2  Educational Implications  287  7.3  Recommendations  for  F u r t h e r Research  290  . . . .  294  BIBLIOGRAPHY APPENDIX A  3  1  303  APPENDIX B APPENDIX C  0  -  3 2  2  APPENDIX D  •337  APPENDIX E  345  -xi-  LIST OF TABLES Page  1.1  List of  3.1 4.1  of  I m p l i c i t Vector  Three Vector  Characteristics  Quantities  (LIVC)  .  9  L i s t of Interview P r o t o c o l Questions f o r each V e c t o r C h a r a c t e r i s t i c List  of  Inferred  Point  for  4.2  Rules  of  4.3  List  of  Rules About  Stationary Rule-rModel  Inferred  Reference  for  Rules  (RPS)  115  RPS for  .  (FR)  122  4.5  List  4.6 4.7  (D) Rules of Rule-Model for D L i s t of I n f e r r e d Rules for A d d i t i o n D i s p l a c e m e n t s (AD) .  4.8  Rules of  4.9  List  of  Rule-Model Inferred  Vector  for  Rules  Rule-Model Inferred  118  Frame o f  Rules  of  88  Reference  4.4  of  of  Bodies  •  Displacement 132 138  of  141  for  (SVP)  Subtraction . . . .  . . . .  149  4.11  L i s t of I n f e r r e d Rules for Reference B o d i e s f o r M o v i n g O b j e c t s (RPM) R u l e s o f R u l e - M o d e l f o r RPM  156 162  4.13  L i s t of Inferred Components (AC)  165  4.14  Rules  4.15  List  of  Rule-Model Inferred  of V e l o c i t i e s 4.16  Table of  Rules  SVP  . . . .  Rules of  of  for  146  4.10  4.12  Rule-Model  for  129  f o r AD  Rules  Positions  FR  153  for Analysis of  f o r AC  Rules  170  for Composition . . . .  173  P a t h s Drawn by S u b j e c t s  182  -xii-  Page Table of  Paths  4.18  Rules  Rule-Model  4.19  L i s t of Inferred of Magnitudes of  4.20  Rules  4.21  L i s t of Inferred Rules o f Components (SC)  4.22  Rules"of  4.23  Table of I n f e r r e d Rules f o r Each S u b j e c t i n t h e Sample A b o u t E a c h V e c t o r Characteristic . . . .  206  C a t e g o r i z a t i o n of the Inferred Rules of E a c h . V e c t o r C h a r a c t e r i s t i c i n t o one o f the Type o f P r e c o n c e p t i o n s (Scalar, T r a n s i t i o n a l , or V e c t o r i a l )  212  4.25  C a t e g o r i z a t i o n of I n f e r r e d Rules S u b j e c t i n t o the T h r e e Groups  214  5.1  V e c t o r C h a r a c t e r i s t i c s and t h e Q u a n t i t y i n the G . I . P . for Each Vector Characteristic ,  230  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects Holding the I n f e r r e d Rules for RPS  242  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e S u b j e c t s H o l d i n g the I n f e r r e d Rules f o r FR .  244  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects H o l d i n g the I n f e r r e d Rules f o r D  246  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects H o l d i n g the I n f e r r e d Rules f o r AD . . . . . . .  249  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects H o l d i n g the I n f e r r e d Rules f o r SVP »  251  4.24  6.1  6.2  6.3  6.4  6.5  of  of  drawn b y S u b j e c t s  . . . .  183  4.17  183  f o r CV  Rules f o r Independence Components ( I M C ) . . . .  Rule-Model  Rule-Model  for  for  190  IMC for  186  Simultaneity  192 • 196  SC  of  Each  -xiii-  Page 6.6  6.7  6.8  6.9 6.10 6.11  6.12  6.13  6.14  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects H o l d i n g the I n f e r r e d Rules f o r RPM  253  . C o n t i n g e n c y T a b l e o f Female and Male Subjects H o l d i n g the I n f e r r e d Rules f o r AC  255  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e S u b j e c t s Holding the I n f e r r e d Rules f o r CV . .  258  Paths GIP  262  Drawn f o r  P a t h s Drawn f o r GIP. .  Question Question  21 o f 31 o f  the the 264  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e S u b j e c t s H o l d i n g the I n f e r r e d Rules f o r IMP ,  266  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects Holding the I n f e r r e d Rules f o r SC  269  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e S u b j e c t s H o l d i n g the I n f e r r e d Rules for- I L P  272  C o n t i n g e n c y T a b l e o f F e m a l e and M a l e Subjects Holding the I n f e r r e d Rules f o r IDC •  275  -xiv-  LIST OF FIGURES Page 2.1  A Particle  Moves i n a  Semicircular  Path  25  2.2  Reflection  i n a Plane Mirror  3.1  Network o f  a Kinematic Vector  29  Quantity  64  3.2  Network o f  3.3  Network o f V e c t o r D i s p l a c e m e n t  ••••  66  3.4  Network o f  V e c t o r Average  . . . .  67  3.5  Diagram o f  the Lake  . . . .  72  3.6 6.1  D i a g r a m o f t h e R i v e r f o r T a s k Two . . . . Common S t u d e n t s ' D r a w i n g s f o r I l l u s t r a t i n g the D i r e c t i o n i n which t h e M o t o r - B o a t was H e a d i n g S t u d e n t s ' Drawings I l l u s t r a t i n g the D i r e c t i o n i n which the Motor-Boat was H e a d i n g . . . .  6.2  Vector Position  for  . . . . . . . .  Velocity  Task  One  65  8  0  276 277  -xv-  ACKNOWLEDGEMENTS A number o f assisting  me t o c o m p l e t e  t h a n k them f o r  their  The most has  people  have been i n s t r u m e n t a l this  research.  i n f l u e n t i a l person  t o h i m ; he h a s  intellectual  I would l i k e  to  assistance. i n my g r a d u a t e  b e e n G a a l e n E r i c k s o n , my t h e s i s  indebted  in  always  stimulation,  advisor;  I  been a g r e a t  practical  program  am d e e p l y  source  h e l p and  of  emotional  support. I would l i k e of  my c o m m i t t e e  to  e x p r e s s my g r a t i t u d e  — Patricia Arlin,  Chandler,  Peter  Professor  Harry Cannon,  b u t was Also, and of  a continuous  I would l i k e  the examination  obtaining  who n o t o n l y  to  agreeing  Lastly,  subject  for  the  needed their must  support; patience  receive  David Bateson,  I must  Michael late  i n my c o m m i t t e e  and p e r s o n a l  support.  McDermott, B e t t y Howard, s e r v e as  t o Doug B l a c k f o r study.  support  Among them I w i s h t o  Anamuah-Mensah,  the  members  external  members  committee.  s t u d e n t s p r o v i d e d me c o n s t a n t criticism.  to  to  served  academic  thank L i l l i a n  gratitude  the  of  the  Walter Boldt,  and e s p e c i a l l y  source  David Whittaker for  Special in  Matthews,  to  and S h a r o n  and my d a u g h t e r s d u r i n g my s t u d i e s . for  Betty,  great  Numerous  help  graduate  constructive  especially  t h a n k my w i f e ,  my g r a t i t u d e  and  his  thank  Jophus  Haggerty. for  t h e much  Carmen and A l e j a n d r a , Carmen de S i l v a ,  her wonderful work.  my  for typist,  -1-  CHAPTER ONE  PROBLEM AND RESEARCH QUESTIONS OF THE STUDY  1.0  Background o f  It  is  educational in  the  Study  g e n e r a l l y acknowledged by b o t h t e a c h e r s researchers  that  students  encounter  understanding p h y s i c a l vector q u a n t i t i e s .  seems t o  be r e l a t e d  to  the  fact  that  (scalar and of  quantities)  direction direction  subtraction scalar  exist  of  subtraction,  findings at  Leboutet-Barrel, literature  (i.e.,  and r e s e a r c h  Kass,  1976;  regarding  t h o r o u g h l y r e v i e w e d i n C h a p t e r 2. researchers  have n o t been l o o k i n g  these  levels  Warren,  for  is  that  operations:  on).  Re-  quantities (Fisher,  1971).  these problems It  factor  operations  and so  problems w i t h  s c h o o l and c o l l e g e  1976;  magnitude  a d d i t i o n and  arithmetic  multiplication,  show t h a t  both high  (i.e.,  d i f f e r e n t ,from..±fae  deal with  with  magnitude  furthermore the  about o p e r a t i o n s  vectors)  quantities  addition, search  brings  do n o t  with only  and p h y s i c a l q u a n t i t i e s  (vector q u a n t i t i e s ) ;  difficulty  The problem  students  d i s t i n g u i s h between p h y s i c a l q u a n t i t i e s  and  1979;  The are  apparent  explanations  that with  respect  to  this  their•purpose  area of  than a n a l y z i n g  Why do most  structures  controlling vector that  quantities  t o be g r a s p e d by h i g h s c h o o l to  the  possess  Is  formal operations  too  students?  culties  related  factors  included i n these questions  it  certain  to  for  understand  characteristics?  are  instructional  have  quantities?  and p r o p o r t i o n a l i t y )  and t h e i r v e c t o r  physical vector  identifying  seem t o  Must s t u d e n t s  (e.g., the P i a g e t i a n  variables  quantities  students  physical vector  problem?  Instead,  it.  high school  i n grasping  a developmental mental  difficulties.  a p p e a r s to h a v e b e e n one o f  problem r a t h e r  difficulty  students'  abstract Or a r e  procedures?  Is  or  it  complex  the  diffi-  All  the  c o u l d be p a r t  of  the  problem. Some e d u c a t o r s Novak,  1977)  believe  and p s y c h o l o g i s t s that  teaching  a content-area  bring  to  the  about  it  that  learning  the most  is  (Ausubel,  important  factor  the knowledge t h a t  situation.  Ausubel f e l t  he i n t r o d u c e d a book  (1968)  1968;  with  in  subjects so  strongly  the  following  I f I had to reduce a l l the e d u c a t i o n a l p s y c h o l o g y t o j u s t one p r i n c i p l e , I w o u l d s a y t h i s : t h e most i m p o r t a n t s i n g l e factor i n f l u e n c i n g l e a r n i n g i s what t h e l e a r n e r a l r e a d y knows. A s c e r t a i n t h i s and t e a c h him a c c o r d i n g l y . It of  this  seems c l e a r ,  study,  conceptions  the questions  about  characteristics  then,  vector  prior  to  that to  in relation  ask  quantities formal  are:  to  the  Do s t u d e n t s  and t h e i r  instruction?  topic develop  vectorial Do t h e s e  conceptions  f o l l o w any p a t t e r n ?  rule-governed? have  It  is  the a u t h o r ' s  some i n t u i t i o n r e g a r d i n g  quantities be t h a t  the  instruction  the d i f f i c u l t i e s  is  that  These  intuitions  about  vector quantities  are  and t h e i r  about  has  the development o f  several  aspects of  forces  '  are  may  in  formal  interfering  to  these  problems.  pJie.conce.ptX.onA  characteristics.  quantities. conceptions  (force  vector  and i t  b e e n done t o u n c o v e r  vector  of  encounter  solutions  t e r m e d t h e Atudznt*  research  preconceptions  experiences;  these i n t u i t i v e notions of  students  characteristics  students  w i t h more f o r m a l p r e s e n t a t i o n s  Little  conceptions  contention that  b a s e d on t h e i r p r e v i o u s  one o f  •studied  Are these  is  students'  Piaget  (1973)  in children  a typical  regarding  vector  quantity),  particularly  t h e i r d i r e c t i o n a l a s p e c t and com-  position.  An e x t e n s i v e  analysis  presented  i n Chapter  Aguirre forces grade 8  (13  intuitive  to  students'  students of  to  to  sample of  find  associate vector  characteristics  several of  be  tasks  if  students  at  acquired  some  identified  on f o r c e s  those i n t u i t i o n s  components.  is  the It  with  through  i n an  inter-  that  some  direction  w o u l d be  14 y e a r o l d s u b j e c t s  characteristics  about  associated  were a b l e t o p r e d i c t  two f o r c e out  have a l r e a d y  T h e s e n o t i o n s were  An e x a m p l e  the r e s u l t a n t  children's beliefs  age)  of vector  force.  i n the  interesting  studied  14 y e a r s o f  responses  view s e t t i n g .  results will  e q u i l i b r i u m , and f o u n d t h a t  notions  the concept o f  Piaget's  2.  (1978)  in static  of  can  with other physical  also vector  -4quantities  s u c h as  While students' has  displacement  little  work h a s  preconceptions  been p a i d  to  knowledge p r i o r  the to  al.  model, (1976,  (1979) . for  of vector quantities,  problems  formal  of  1979).  subjects further  to  proposed  hypothetical methodology  problems  rules. is  is  to  presented it  is  when s t u d e n t s  problem s i t u a t i o n s This  are  is  based on the  i n t u i t i v e ideas which are  sented all  use  of  those  reality  b o t h found e v i d e n c e  fact  students that  properties.  properties  subjects  of  He  contention that  confronted with  (19 78)  Definition  by  rule-assessment  vector  and A g u i r r e of  Siegler used  students'  physical  as  i n which v e c t o r  belief  by  practical  2.  about  define  t h e most  content-areas.  his  the a u t h o r ' s  to what p h y s i c i s t s  attempts  and A r c h e n h o l d  the r u l e s  assess  i n Chapter  related  present.  identify  attention  these  (1978)  that  study  students'  described  A d e s c r i p t i o n of  some i n t u i t i v e n o t i o n s  illustrated  that  in specific  develop  is  judged  to  more  Some o f  and E a s l e y  a methodology to  I n summary,  1.1  study,  He a t t e m p t e d  solve  representing  instruction.  H o w e v e r , i t was  the present  velocity.  b e e n done s p e c i f i c a l l y  have been summarized i n D r i v e r et  and  i n 12 t o clearly  This  concrete  are  that  are  clearly  Piaget  (1973)  14 y e a r  old  vectorial.  Terms  Some o f  the  below.  It  terms is  being used  assumed t h a t  the p h y s i c a l q u a n t i t i e s  in this  the reader  i n c l u d e d i n the  study is  are  pre-  familiar  study,  such  with as  -5vector  position,  displacement,  the c a s e ,  the d e f i n i t i o n s  presented  i n Appendix A .  Definition  It  of  is  vector  the v e c t o r  plication  operation.  vector  that  ii)  y +  iii)  ( v + OJ)  =  this  requires  of  (zero)  are  and t h e vectors  and  direction  scalar  multi-  y_ and v_, t h e  property)  (y + v ) + w ( A s s o c i a t i v e  0  not  that:  (Commutative  a unique v e c t o r  is  p.l)  p o s s e s s e s magnitude  F o r each p a i r  y + v = v + y  If  quantities  ( J o r d a n , 1969,  addition operation  addition operation i)  those vector  quantity  a quantity  and o b e y s  of  and v e l o c i t y .  has  property)  the p r o p e r t y  that  v + 0 = v iv)  a unique v e c t o r v +  Definition  It explain  of  is  (-v)  of  discipline.  that  *  i n t u i t i v e n o t i o n used  phenomenon a n d w h i c h i s  experience.  and o r g a n i z a t i o n by t h e  of  a particular  subject's  the p r o p e r t y  =0.  preconception  a type  - v has  the  This  intuitive  formal  by a  subject  a product of  notion lacks  c o n c e p t i o n commonly  the  to the  scope  accepted  -6Definition  of  inferred  A rule use  is  rule  inferred  a consistent  to  exist  preconception  with a particular  question  if  a subject appears  i n a given  situation  o r problem posed  by the  to  to  deal  inter-  viewer.  Operational  d e f i n i t i o n of  The p r e s e n c e o f the  same b a s i c  of  regarding  reference  Definition  a rule  preconception  the questions choice  inferred  of  points  vector  A vector  the  is  rule  inferred  to deal  locate  In t h i s  direction  is  subject  uses  characteristic  a b o a t on a  of (e.g.,  lake).  characteristic  characteristic  describes  phenomena i n w h i c h t h e d i r e c t i o n p l a y s Note:  a  w i t h more t h a n h a l f  same v e c t o r to  if  study,  an a s p e c t o f  an i m p o r t a n t  physical  role.  p h y s i c a l phenomena i n w h i c h t h e  a decisive  factor  are  called vectorial  variable  physical  phenomena.  Definition  of  i m p l i c i t vector  An i m p l i c i t v e c t o r of  vectorial  elicited  characteristic  phenomena t h a t  i n the d e f i n i t i o n of  definitions study  physical  deals.  of  the  characteristic  three  is  a vector  vector  represents not  aspect  explicitly  quantity,  quantities  an  or  in  w i t h which  the this  -7-  1.2  Implicit Characteristics  of  Three P h y s i c a l V e c t o r  Quantities  The f o r m a l m a t h e m a t i c a l quantity  and i t s  section.  This  cular  and t h e o r e t i c a l those p r o p e r t i e s  aspect of  text-books  1980)  also  1980,  & Martindale,  usually  the  generally  1979;  Typical  school  of  Physics  some  by D . C .  quantities  quantities  istics  are  thought  t o be  important  i n terms  these p h y s i c a l q u a n t i t i e s .  when t e a c h e r s  use  granted  that  aspects  of  quantities  formal d e f i n i t i o n ,  the d e f i n i t i o n s  cover a l l  t h e p h y s i c a l phenomena t h a t represent.  the d i f f i c u l t i e s  by H e a t h , Giancoli,  through believes  Furthermore these  understanding  the  science  some  o f p h y s i c a l phenomena t h a t  represent.  of  It  e n c o u n t e r e d by s t u d e n t s  actually  may be  that for  important  these  T h i s may b e one o f  these  character-  they take  the  need  parti-  Physics  f o r m a l d e f i n i t i o n d o e s n o t make e x p l i c i t  vector  by  who do n o t  in describing  Fundamentals  previous  used  However, the a u t h o r  aspects or c h a r a c t e r i s t i c s  these  a.vector  i n the  physicists  introduce vector  such formal d e f i n i t i o n . that  presented  physical reality.  (P.S.S.C;  Macnaughton,  was  formal d e f i n i t i o n i s  mathematicians to concretize  properties  d e f i n i t i o n of  vector  the causes in  of  understanding  quantities. The d e f i n i t i o n s  quantities:  vector  of  the  position,  three  kinematic  displacement,  vector  and v e l o c i t y  -8-  (see  A p p e n d i x A)  vector in the  that  characterisits  the d e f i n i t i o n of three  reference However, include  that  are  a vector  a complete  the  d e f i n i t i o n of  considered  according  to  the  The v e c t o r  For  selection  example,  of  appropriate  description of  a vector  characteristics i n the u s u a l  author,  task analysis study.  the  considered  them.  quantity  does  that  not  not  requirement.  obviously  this  quantity.  points  for  includes  not e x p l i c i t l y  require  Those v e c t o r  to  study deals with  quantities  such a  rational  this  This  definitions,  were u n c o v e r e d b y e m p l o y i n g a  of  is  formal  are  the  three  fully  characteristics  quantities  explained  of  interest  of  interest  i n Chapter Three. are  listed  in  T a b l e 1.1. The l a s t in  Table  1.1  two i m p l i c i t  vector  characteristics  (ILP and IDC) were n o t a c t u a l l y  anticipated  by the  R a t i o n a l Task A n a l y s i s p e r f o r m e d upon t h e  vector  quantities.  These v e c t o r  characteristics  c o v e r e d by e m p i r i c a l l y a n a l y z i n g the in  the  first  in  the  study which are  to consider  phase o f  the  second phase of  the  study.  explained  two new v e c t o r the  study.  subjects'  three were u n -  responses  ( T h e r e w e r e two later).  It  listed  was  characteristics  in  phases decided the  -9Table  1.1 "  L i s t of Implicit Vector Characteristics of Three Vector Quantities : Vector P o s i t i o n , D i s p l a c e m e n t , and V e l o c i t y (LIVC)  1.  R e f e r e n c e P o i n t a s a means t o d e s c r i b e a s t a t i o n a r y body o r p o i n t (RPS).  the  location  2.  Frame o f R e f e r e n c e a s s o c i a t e d w i t h a r e f e r e n c e p o i n t as a means t o p r o v i d e q u a n t i t a t i v e l o c a t i o n o f a s t a t i o n a r y b o d y (FR) .  3.  D i s p l a c e m e n t , a q u a n t i t y w i t h m a g n i t u d e and d i r e c t i o n , w h i c h i s i n d e p e n d e n t from a l l p o s s i b l e p a t h s t o j o i n two l o c a t i o n s (D). I n t h i s s t u d y , change o f l o c a t i o n i s a synonym o f d i s p l a c e m e n t .  4.  A d d i t i o n o f d i s p l a c e m e n t s as an o p e r a t i o n t h a t r e l a t e s two c o n s e c u t i v e c h a n g e s o f l o c a t i o n s w i t h t h e t o t a l change o f l o c a t i o n (AD).  5.  S u b t r a c t i o n o f v e c t o r p o s i t i o n as an o p e r a t i o n t h a t r e l a t e s t h e v e c t o r p o s i t i o n a s s o c i a t e d t o two l o c a t i o n s o f a m o v i n g b o d y and t h e d i s p l a c e m e n t b e t w e e n t h e l o c a t i o n s (SVP).  6.  Reference b o d i e s ( s t a t i o n a r y or moving bodies) t o d e s c r i b e t h e m o t i o n o f an o b j e c t (RPM).  7.  A n a l y s i s o f t h e number o f combined form an a p p a r e n t  8.  C o m p o s i t i o n o f two s i m u l t a n e o u s a f f e c t i n g an o b j e c t ( C V ) .  9.  C o n s t a n c y o f e a c h one o f t h e m a g n i t u d e o f two v e l o c i t i e s e v e n when t h e y a r e i n t e r a c t i n g t o f o r m an a p p a r e n t s i n g l e m o t i o n (IMC).  component v e l o c i t i e s s i n g l e motion (AC). velocities  that  as  a  of  means  that are  10.  S i m u l t a n e i t y o f e f f e c t o f two component v e l o c i t i e s affecti n g a b o d y t o c r e a t e an a p p a r e n t s i n g l e m o t i o n ( S C ) .  11.  Independence o f a l o c a t i o n o f a body from the p a t h ( I L P ) . I t r e f e r s t o t h e f a c t t h a t i t i s n o t n e c e s s a r v t o know t h e p a t h f o l l o w e d b y a body when d e s c r i b i n g an i n s t a n t a n e o u s p o s i t i o n of i t . ..... • ._. _ M-  12.  C o n s t a n c y o f e a c h one o f t h e d i r e c t i o n s o f two v e l o c i t i e s , even when t h e y a r e i n t e r a c t i n g t o f o r m an a p p a r e n t single motion (IDC).  -10-  Th e i t e m s in  i n this  hierarchical order.  vector  not n e c e s s a r i l y  are  presented  presented  among a l l  these  through networks  in  Three. A detailed  istics  are  The r e l a t i o n s h i p s  characteristics  Chapter  list  description of  e a c h one o f  these  character-  and t h e r e a s o n s why t h e y were c o n s i d e r e d i n t h e  study are sections  being presented  i n Chapter Four.  In that  d e s c r i b i n g each v e c t o r c h a r a c t e r i s t i c  the a n a l y s i s t o do so  to  between the  of  data  c o l l e c t e d are  facilitate  i d e n t i f i e d preconceptions with  chapter,  together  developed.  the understanding of  present  It  the  was  with  decided  relationships  e a c h one o f  the  vector c h a r a c t e r i s t i c s . Furthermore, As the  was  the l i s t  mentioned e a r l i e r ,  items  included  o f T a b l e 1.1 i n the  was  aim o f  has  1.3  the  list  been shortened to  General Statement  of  of  to  is  regarding  Hereafter,  just  vector  analyze these  reference  implicit vector characteristics,  this  characteristics.  the Problem  K n o w l e d g e o f what t h e l e a r n e r b r i n g s situation  quantity.  i d e n t i f y and  students possess  of  not being e x p l i c i t l y  a vector  s t u d y was  vector characteristics.  made t o  of  exhaustive.  selection criterion  the c o n d i t i o n o f  this  the kinds o f preconceptions  is  the major  formal d e f i n i t i o n  The m a j o r  implicit  was n o t meant t o be  to  the  learning  a n i m p o r t a n t component i n p l a n n i n g a n e d u c a t i o n a l  T-1X-.  program.  Although there  among e d u c a t o r s t h e above search  what t h e  been a w i d e s p r e a d  and e d u c a t i o n a l  statement,  has  has  it  is  been d i r e c t e d  learner  already  researchers  agreement  with respect  indeed s u r p r i s i n g  to  how l i t t l e  towards  the problem o f  knows.  Shulman and T a m i r  re-  ascertaining suggest  that: T h e r e s h o u l d a l s o - b e an i n c r e a s e i n a g e n r e of b a s i c r e s e a r c h that i s science-specific r a t h e r than experiments to t e s t aspects o f p s y c h o l o g i c a l t h e o r i e s where s c i e n c e c o n t e n t h a p p e n s t o be a c o n v e n i e n t s u b j e c t - m a t t e r to use f o r r e s e a r c h m a t e r i a l . Such s c i e n c e r e l e v a n t b a s i c r e s e a r c h w o u l d be o n a t o p i c l i k e the c o g n i t i v e development o f s c i e n c e r e l e v a n t c o n c e p t s i n young c h i l d r e n , e.g., c a u s e and e f f e c t , s p a c e , t i m e , m a s s , momentum. T h i s s o r t o f r e s e a r c h i s c l e a r l y i n the t r a d i t i o n of Piaget. T.he puKpo&e would be. to Zde.nt.i6y i>ome Qene.n.a.1 normal e.xpectanci.e.6 ion. the. evolution o^ pan.ti.culan. concept* around w h i c h c u r r i c u l u m d e v e l o p e r s and program w r i t e r s could plan t h e i r c r e a t i v e endeavours. ( E m p h a s i s a d d e d ) ( S h u l m a n and T a m i r , 1 9 7 3 , p . 1 1 3 9 ) . The  shortage of  area of vector quantities culties  about  quantities  type o f  -— as  — may i n p a r t  i n an i n s t r u c t i o n a l  knowledge on the p a r t  the preconceptions  of  i n the  specific  of d i s t i n c t i v e for  some o f  in  i n s t r u c t i o n a l procedures.  be due t o t h e  setting.  This  is,  c u r r i c u l u m makers  physical the  these preconceptions  incapacity of  vectorial  concepts  students'  preconceptions.  students  to  grasp  the f a i l u r e  there  is  reflected t h e n , may  the to  a  teachers  character-  not  The d i f f i c u l t i e s ,  b u t may be due t o This  are  diffi-  vector-  and  h e l d by s t u d e n t s o n v e c t o r  and c o n s e q u e n t l y  not  a set  be r e s p o n s i b l e  istics the  research  e n c o u n t e r e d b y s t u d e n t s when c o n f r o n t e d w i t h  type problems lack of  this  consider  l a c k o f c o n s i d e r a t i o n may  -12c a u s e an  i n t e r f e r e n c e o r m i s m a t c h i n g b e t w e e n what  t e a c h e r w a n t s t o t e a c h and was  what s t u d e n t s  conjectured that this  vector  i s the case  a l r e a d y know.  i n the  broad  aim  methods o f d a t a a detailed  of t h i s  collection  examination  of  about v e c t o r q u a n t i t i e s . to  c a r r y out  of  the three kinematic  instruction  an  s t u d y was and  the  students'  The  first  concepts  yielded  d i f f e r e n t methods o f d a t a r e s u l t s of the The  study  first  interviews with  permit  preconceptions  t a s k r e q u i r e d , t h e n , was  of the v e c t o r i a l  dealt with  the L i s t  i n the  of I m p l i c i t  second  nature study.  Vector  phase u t i l i z e d students based  latter and  viding  a type  allowed  addition  i t has  instrument  techniques  developed  i n P h a s e One  by  the  t o be  summary, t h i s  field  of  t o be  classroom  for revealing  comparisons between the  to  data  was  classes of  cross-check  were a l s o two  situ-  vector characteristics.  the p o t e n t i a l  diagnostic  In  clinical-type,  students.  gathered  used data  of the  refined  The  students'  i n P h a s e Two s e t s c o u l d be  i n two  a much  thus  pro-  f o r use  so  as  a  analytical preconcepthat  made.  s t u d y makes a s i g n i f i c a n t  science education  on  interview data.  teachers. the  In  developed  more h e t e r o g e n e o u s " g r o u p o f s t u d e n t s , of v a l i d i t y  the  phases.  upon c o n c r e t e p r o b l e m  a d m i n i s t e r e d to i n t a c t  technique  larger  tions  these  individual,  Two  hence,  distinct  phase a group i n t e r v i e w instrument  w h i c h c o u l d be  In  c o l l e c t i o n were u s e d  a r e r e p o r t e d i n two  a t i o n s which i l l u s t r a t e d  This  of  appropriate  a n a l y s i s which would  in-depth analysis  task a n a l y s i s  to develop  C h a r a c t e r i s t i c s , w h i c h were d e s c r i b e d i n S e c t i o n 1.2.  the  It  quantities.  The  This  the  contribution  a r e a s : the d e s i g n  and  T!3^  application of in  the  study  conceptions  the various  and t h e about  methodological  substantive  vector  in two 1.4  this  a number o f more s p e c i f i c  terms  of  a number o f  of  students*  pre-  quantities.  The b r o a d p r o b l e m o f into  results  techniques' used  s t u d y was b r o k e n down  problems  research  and f u r t h e r  questions  i n the  described  following  sections.  Specific  Statements of  The g e n e r a l following 1)  specific  the Problem  p r o b l e m was  subdivided  into  the  statements:  To u n c o v e r , in  further  analyze,  students'  and l o o k f o r  preconceptions  about  the  patterns  the  12  vector  characteristics. 2)  To d e t e r m i n e w h e t h e r t h e r e was difference the  study  uncovered 3)  between regarding about  To c a t e g o r i z e  of  is,  and m a l e  the k i n d o f  the v e c t o r  significant subjects  of  preconceptions  characteristics.  the p r e c o n c e p t i o n s  characteristic That  female  any  for  each  vector  using a complexity c r i t e r i o n .  by a n a l y z i n g t h e  the v a r i a b l e s  number a n d t h e  (qualitative  a s s o c i a t e d w i t h each of  the  or  nature  quantitative)  identified  pre-  conceptions. 4)  To d e t e r m i n e the  students'  the degree of  correspondence  preconceptions  version  of vector  current  scientific  and t h e  characteristics community.  as  between  accepted h e l d by  the  5  Research  The attempted (1)  Questions  research questions which  What p r e c o n c e p t i o n s do s t u d e n t s h o l d about  Reference  1.2  Frame o f reference.  1.3  Displacement  or change o f  1.4  Addition of  displacements.  1.5  Subtraction of vector position.  1.6  Reference  1.7  A n a l y s i s o f component  1.8  Composition  How  point f o r stationary bodies.  bodies  of simultaneous  1.11  Independence  (3)  (4)  of  consistent are the preconceptions  listed  vector  some  path. interacting employed  t o solve problems  by  involving  characteristics? able t o be categoriz  fashion?  i s there a s i g n i f i c a n t d i f f e r e n c e between female male students regarding the identified  (5)  interacting  velocities.  of directions of  Are the identified preconceptions in  velocities.  o f l o c a t i o n s from  students when attempting the  velocities.  1.10.  Independence velocities.  location.  f o r objects i n motion.  Independence o f magnitudes velocities. S i m u l t a n e i t y o f component  1.12.  each o f  characteristics?  1.1  1. 9  study  t o answer were as f o l l o w s :  rfche f o l l o w i n g i m p l i c i t v e c t o r  (2)  the present  How g e n e r a l i z a b l e a r e t h e i d e n t i f i e d to other groups  of  students?  and  preconceptions? preconceptions  -15(6)  Which i d e n t i f i e d p r e c o n c e p t i o n s the  1.6  study  Overview of  and  Methods o f  Study  interviews with  in a series  of  students  the grade  transcription.  and a u d i o - t a p e d  Preconceptions with  characteristic  categorized terred  according to  rules  contextual  form,  rules  for  I n summary,  in  first  this  questions  i n t u i t i v e n o t i o n s about  listed  i n LIVC u s i n g a l a r g e r  decided  the  use  response  since  1 to  3 were  rules they The  was c a l l e d a  of  P h a s e Two was  the vector  set  rule-  addressed  sample  of  same c o n t e x t  Therefore,  interviews  format  to  (tasks)  subjects,  was  Because  and  the p r o t o c o l used for  it  one  from b o t h  h a d t o be u s e d ,  was a d a p t e d  facilitate  to un-  characteristics  a group i n t e r v i e w methodology.  similar protocol.  individual  These  situation.  t h e o b j e c t i v e s was t o compare t h e r e s u l t s  phases a  to  less  phase.  S i n c e the main o b j e c t i v e cover  the i n -  and  simply r u l e s .  characteristic  research  w h i c h were  finally,  task  vector  consistent  h y p o t h e t i c a l i n nature  each v e c t o r  model.  each  inferred rules,  i d e n t i f i e d i n one t y p e o f  level  The i n t e r -  to  i n a more g e n e r a l  and were c a l l e d  10  further  respect  their content;  were p r e s e n t e d  s h o u l d be c o n s i d e r e d as were o n l y  for  i n t h e L I V C were i d e n t i f i e d ;  p r e c o n c e p t i o n s were e x p r e s s e d as  of  at  semi-structured  whose a g e s r a n g e d b e t w e e n 15 and 17 y e a r s .  v i e w s were v i d e o - t a p e d  of  predominant i n  sample?  P h a s e one c o n s i s t e d individual  are  in  also the  use w i t h w r i t t e n  the a d m i n i s t r a t i o n of  the  1-16-  group the  interview.  interviewer  picted  understood carefully the  explained  i n the t a s k s  very c a r e f u l l y  in  When a d m i n i s t e r i n g  the nature  being  of  their  strument  constructed While  it  was  study,  it  nostic  test.  satisfied  that  all  students  to  serve  prepared  I n summary, second  addressed  i n both  phases.  In both phases, any k i n e m a t i c v e c t o r to  the  them t o  their  study  either  h e l p the  formal  i n s t r u c t i o n or  think  in this  of  as  second  (GIP).  phase  This  in-  standardized  the  for  use of  understanding  of  interfere  diag-  5. and 6 were  question  4 was  taught  preconceptions for  Here p r e c i s e l y this  present  had n o t y e t been  their  not  tests.  teachers'  Research  i n t u i t i v e notions  since  i n the  questions  subjects  quantities;  was  such normative  a type  phase.  ideas  questions.  a kind of  research  problem s i t u a t i o n s . of  as  specifically  in this  portance  used  the purpose  may be c o n s i d e r e d  corresponded  asked  a Psychometric t e s t because i t  addressed  actual  questions  r e s p o n s e and t o w r i t e down t h e i r  c a n n o t be c o n s i d e r e d o r as  de-  the  a Group I n t e r v i e w P r o t o c o l  questionnaire  situations  read  space p r o v i d e d underneath the  called  interviews,  detail,  the q u e s t i o n s ,  The w r i t t e n i n s t r u m e n t was  experimental  in sufficient  and a f t e r  about  the  the group  dealing lies  with  the im-  i n t u i t i v e knowledge- c a n concepts with  them.  delivered This  in  author  -17believes tions  that  some o f  ("inferred  blocks"  (Hobbs,  and c o u l d w e l l vector  1.7  the  rules") 1977)  subjects'  c o u l d be c o n s i d e r e d as." s t u m b l i n g  or  "critical  interfere with  of  the  a b r o a d one w h i c h i s physics  quantities such a s :  selected  of  several  vector  of  implicit  Others,  cross-product meaning i s study,  of  was  position,  inner  limited  of  focused  preconceptions quantities.  of  study,  to the  to o n l y  those  analysis list  vectors their  three vector  and o n l y  to  These t h r e e  the  The r e a s o n s  the purpose  with respect  This  quantities,  of  this  for  study,  on u n c o v e r i n g the i n t u i t i v e n o t i o n s students  and  experience.  quantities. to  others  quantities.  product of  velocity)  are  i n the  physical vector  informal  physical  angular  described  or  to p h y s i c a l  vectorial quantities,  c o n s i d e r e d t o be more s i m p l e ,  first  there  were o m i t t e d b e c a u s e  r e s t r i c t i o n were r e l a t e d  usually  those  displacement, of  quantities,  the three  limited  or scalar  vectors,  characteristics  w h i c h was  duced  s t u d y was  properties;  the  vector  and l i n e a r and  remote from s t u d e n t s '  then,  (vector  the p r e s e n t  characteristics  s u c h as  study,  a s s o c i a t e d w i t h a number o f  force,  Likewise the  this  1980)  formal i n s t r u c t i o n of  In a d d i t i o n to  for  acceleration,  the  also  concepts.  moment'um.  basic  the  barriers"(Hawkins,  Study  The t o p i c c h o s e n f o r  other  preconcep-  quantities.  Delimitation  is  consistent  are  which  the ones  i n a formal i n s t r u c t i o n context.  These  are  introthree  -18-  quantities of  describe  motion, which  vector  t h e most  is  the a t t r i b u t e  quantities.  starting  Hence t h i s  point for  the  Furthermore tities,  s u c h as  acceleration, periences physics  o f most  courses Another  range This  of  ages  vectorial  seemed  felt  that  very abstract students,  to  like a  (.1978)  of  other vector  17)  or  and r e m o t e  and a r e school  only  angular  from the  taught  the  w h i c h was u s e d  i n the  studies,  quan-  ex-  in  advanced  curriculum. restricted  of  study.  Piaget*s  both of which r e p o r t e d  is  aware  to  It the  that  (1973) that  students  an i n t e r v i e w t e c h n i q u e  respect  may a l s o  respect  of  data.  To o v e r c o m e t h e s e p r o b l e m s  of  p h a s e one were c r o s s g e n e r a l i z a b i l i t y of  interview i t s e l f  validated  are  the  inferred rules  The p r o b l e m o f  was m i n i m i z e d b y u s i n g a  for  subjects  sample.  that  of  analysis  the  checked out  subjectivity  setting  and t h e  in part,  has  inferences  i n phase two.  i n p h a s e one were f u r t h e r  i n the  t o making  i n t r o d u c e problems  bias with  all  as  age.  samples.  second phase.  them  l i m i t a t i o n was  i n h e r e n t problems w i t h  generated  aspect  reasonable  p r e c o n c e p t i o n s were n o t found u n t i l  14 y e a r s  from s m a l l  the  defines  momentum, t o r q u e ,  important  The a u t h o r some  was  i n the high  (15  that  n a r r o w r a n g e was c h o s e n on t h e b a s i s  and A g u i r r e ' s  over  it  directional  study.  angular  are  relevant  findings That  is,  were in  the  i n an i n t e r v i e w  semi-structural  protocol  -19-  1.8  Justification  of  Physical topic (1)  of  vector quantities research  quantities  i n grade  students  appear  for  included  the reasons  for  is  these  barrier  see  informal  aspect of  (3) has  a wide range o f  (1978)  most s t u d e n t s  absent  The r e s u l t s notions  vector quantities,  in  has been  it  is  the nature of  an a r e a  of  of  research  study.  present  formulate  an  vector  study that  from t h e r e s e a r c h  an  is  literature.  into students'  intuitive  and t h e i r  specific  c a n be a p p l i e d t o p r o b l e m s o f i n s t r u c t i o n  two b r o a d w a y s :  instructional the  about  about p h y s i c s v e c t o r q u a n t i t i e s  characteristics  critical  h a v e gone t h r o u g h  of  conspiciously  them  i n which a d i r e c t i o n a l f a c t o r ,  ideas  of  physics  discussion of  intuitive  (5)  o u t some o f  restricted  t h e y h a v e had t h e c h a n c e t o  characteristics;  most  understanding vector  and t h e r e f o r e set  (2)  pro-  students'lack  (4)  experiences  underlying  i n most p h y s i c s  difficulties;  Hawkins  phenomena);  reasons:  find  of vector q u a n t i t i e s  (e.g.,  the  very important to  further understanding of  concepts  the f o l l o w i n g  to have d i f f i c u l t y  and i t  understanding  are  were c h o s e n a s  11 and 12 i n N o r t h A m e r i c a ;  quantities,  from  Study  the present  these  grams  the  Firstly,  packages  based  i n the development o f upon k n o w l e d g e o b t a i n e d  The m a i n p u r p o s e o f  these  packages  would  from be  -20-  t o make t h e  "right  c o n n e c t i o n " b e t w e e n what  already  know and what t h e  teach.  Secondly,  the  students  school curriculum is  study would p r o v i d e  science  w i t h a w e l l - d o c u m e n t e d and r i c h d e s c r i p t i o n o f tive use  notions  and c o n s i s t e n t  preconceptions  i n coping with problematic  quantities.  situations  trying  that  to  teachers  the  intui-  students  involving  vector  -21-  CHAPTER  TWO  LITERATURE REVIEW/ AND PSYCHOLOGICAL AND EDUCATIONAL CONTEXTS OF THE STUDY 2.0  Introduction  This chapter literature concepts; study;  related  deals with t o students*  (2) t h e g e n e r a l  first  extensive general  understanding  psychological context  topic  c o u l d be made u s e f u l .  the f o l l o w i n g aspects  literature  review  students;  ( i i ) the research  preconceptions  f o r the the f i n d -  With regard t o  a r e reviewed':  i s presented  lack of understanding  (1) t h e  of vectorial  (3) t h e e d u c a t i o n a l p e r s p e c t i v e i n w h i c h  ings of the research the  t h r e e main t o p i c s :  ( i ) an  t o show t h e  o f v e c t o r q u a n t i t i e s by  carried  about v e c t o r i a l  o u t so f a r on  concepts;  view o f e x p e r i m e n t s and i n s t r u c t i o n a l  students'  and ( i i i ) a r e -  u n i t s about  vectors,  which have been c r e a t e d w i t h  the s p e c i f i c  purpose o f  alleviating  the d i f f i c u l t i e s  u s u a l l y encountered by  In r e l a t i o n  to the psychological context,  students.  a discussion i s  -22-  presented  of  the  so-called  neo-Piagetian  approach,  and t h e m e t h o d o l o g i c a l  new v i e w o f d e v e l o p m e n t . a new t h e o r y o f section,  which  recommended i n  this  The p s y c h o l o g i c a l components  instruction is is  tools  psychological  also  b a s e d on t h i s  introduced i n  neo-Piagetian  of  this  view of  development. The f i n a l topic of  of  the  section  of  this  chapter  the E d u c a t i o n a l Context of  general  implications  of  this  t h e c u r r i c u l u m and t h e c o r r e s p o n d i n g are  discussed.  2.1  Literature Vector  Review on S t u d e n t s '  the type  studies  related  to  section.  discussed  2.1.1  research  for  strategies  of  the d i f f i c u l t i e s  are  about  development  Understanding  i n understanding  section.  of  teaching  c o u n t e r e d by s t u d e n t s  critical  study:  the  Quantities  Several  students'  deals with  in this  difficulties barrier  vector  A connection is  a n d what H a w k i n s  phenomena;" t h i s  Previous  research  quantities  Students'  vectorial  is  of  concepts made  has  reviewed  between  called  done i n a s p e c i a l  students'  also  Difficulties  is  (1978)  en-  sub-  preconceptions in this  i n Understanding  section.  Vector  Quantities  Research understanding  findings  vector  have  shown t h a t  quantities  and t h e i r  difficulties  in  characteristics  -23-  exist  at  both high  stance,  Fisher  physics  concepts  students. cepts  (1979)  asked  i n terms  She f o u n d  (5 o f  22)  quantities; In grade  s c h o o l and c o l l e g e  that  i n grade  teachers  of  levels. to r a t e  among t h e 11,  12,  difficulty,  their  two were r e l a t e d  to  con-  vector  o b l i q u e momentum  concepts  for  frame were among  students.  i d e n t i f i e d by t e a c h e r s ,  was  One s o u r c e  the v e c t o r  of  nature  concepts.  which of  and M u g h o l  the concepts  troublesome;  (1976)  taught  they found  that  five  T h e s e c o n c e p t s were p r e s e n t e d students  who were a s k e d  after  the concepts  grade  11 and 12  Kass  (1976)  the d i f f e r e n c e  circular  proving among  s c h o o l and  college  them on a s c a l e o f  felt  difficulty  A high percentage  that vector  of  concepts  difficult. asked  353 g r a d e  in difficulty  the p r i n c i p a l  motion,  establish  physical quantities  had been t a u g h t .  mechanics matched i n p a i r s . shows t h a t  to  were s t r i c t l y v e c t o r i a l .  to  to rate  (44 p e r c e n t )  were among t h e m o s t  tried  i n s c h o o l p h y s i c s were  20 c o n s i d e r e d m o s t d i f f i c u l t  city,  for  f i v e most d i f f i c u l t  two v e c t o r q u a n t i t i e s :  Johnstone  the  of  p r o j e c t i l e m o t i o n and t h r e e d i m e n s i o n a l v e c t o r s .  t h e most d i f f i c u l t  the  a set  perceived d i f f i c u l t y  i n t e r a c t i o n s .and c e n t r e o f mass r e f e r e n c e  of  For i n -  12 s t u d e n t s  to  b e t w e e n 20 c o n c e p t s T h e summary o f  clusters  of  her  difficulty  rate of  results are  c e n t r i f u g a l and c e n t r i p e t a l f o r c e s ,  p r o j e c t i l e motion,  moments, and c o m p o s i t i o n and  velo-  -24-  resolution are  of  strictly  forces.  sity  t h e s e p h y s i c a l phenomena  vectorial.  In another from t h e  A l l of  article  Warren  (1971)  f o l l o w i n g p r o b l e m w h i c h was  entrants  i n e n g i n e e r i n g and  reported  given to  results  148  univer-  science:  A motor v e h i c l e t r a v e l s w i t h u n i f o r m s p e e d on l e v e l g r o u n d , t u r n i n g t o t h e r i g h t i n a path of uniform curvature. T h e r e i s no w i n d . S k e t c h a p l a n showing (a) an a r r o w marked R r e p r e s e n t i n g t h e r e s u l t a n t o f a l l f o r c e s a c t i n g on t h e v e h i c l e i n the h o r i z o n t a l p l a n e . (b) An a r r o w marked F r e p r e s e n t i n g t h e r e s u l t a n t f o r c e o f f r i c t i o n e x e r t e d by the ground on t h e v e h i c l e . (c) S u i t a b l e l a b e l l e d a r r o w ( s ) r e p r e s e n t i n g any o t h e r f o r c e o r f o r c e s a c t i n g on the v e h i c l e i n t h i s p l a n e This all  the  is  a simple v e c t o r i a l problem.  students  unintelligible  attempted  projective  question a correctly, The n e x t y e a r  the  t h e number o f  correct  following:  23,  Warren adds force  i n or near This  drawings;  answers  for  and none f o r  were  answered  193 new  to  b,  c  c.  students;  e a c h c j u e s t i o n were  the  respectively.  c i r c u l a r motion i s  mechanics perspective)  (vectorial  motion considered  figures  gives  the  resultant  the d i r e c t i o n of motion.  r e a s o n may be t h a t  dimensions  nearly  a n d 13 r e s p o n d e d  t h e most p o p u l a r answer  concept of  (scalar  14  given to  a,  m y s t e r y and c o n f u s i o n on the p a r t  dimension  their  only  3 answered b ,  same p r o b l e m was  11,  that  the problem,  While  .  is  is  of  the  so o f t e n before  perspective.) .  s u r r o u n d e d by students. taught  extension  One  i n only to  two  I n one d i m e n s i o n  i n the d i r e c t i o n o f  the a p p l i e d  hence the o n e - d i m e n s i o n a l s o l u t i o n r e i n f o r c e s  the  one  the force,  students'  -25-  initial  i n t u i t i v e n o t i o n on f o r c e  stand, c i r c u l a r motion i t tion  of  simple  vector  is  and m o t i o n .  necessary  displacement,  to  have  velocity,  To u n d e r some and  appreciaaccelera-  tion. Usually, in  school  of  two p o i n t s  everyday speed;  is  i n terms  speech,  consequently  was  question  first  also given  is  students  clearly  is  vector quantities the d i f f e r e n c e  o f magnitude  velocity  t h e s e two t e r m s .  speed  the  displacement;it  distinguishing of  one o f  the This  often  of  introduced position  and d i r e c t i o n . used  as  a synonym o f  have much d i f f i c u l t y  stipulated  In  scientific  in meaning  c o n f u s i o n between v e l o c i t y  examined i n the  test  g i v e n by W a r r e n .  and The  was:  " A p a r t i c l e moves i n t h e p a t h shown ( F i g u r e 2.1), the speed i n c r e a s i n g u n i f o r m l y w i t h time i n the s e m i c i r c u l a r s e c t i o n , f r o m 10 m s ~ l t o 12 m s ~ i . For t h i s s e c t i o n of the path c a l c u l a t e the a v e r a g e s o f (a) t h e v e l o c i t y (b) t h e a c c e l e r a tion. (Assume T r = 2 2 / 7 ) " .  Figure  2.1  -26The s e m i c i r c u l a r of  11 ms  hence  The c h a n g e hence the was  of  it  path  takes  is  2/7  displacement  is  the  f r o m s p e e d when a s k e d  asked  for  students  the  left  i n Figure  speed that  148  them a s  This  Figure, answer  responding.  scalars  average  The c h a n g e  2.1,  path.  downward i n t h e  had d i s t i n g u i s h e d  t h e y gave the  the'average v o l o c i t y .  22 ms ^ t o  i n going along  among t h e  to c l a s s i f y  w i t h b u t one e x c e p t i o n  an a v e r a g e  7 ms ^ downwards.  g i v e n b y o n l y one s t u d e n t of  sec.  at  2 metres  average v e l o c i t y i s  though n i n e t e n t h s  taken  Al-  velocity  or  vectors,  s p e e d when of  hence the  velocity is  average  accele-  -2 ration  is  77 ms  by the r a t i o  to  of  the  left.  the magnitude  T h i s magnitude of  the change  •22 ms ^ , a n d t h e t i m e t a k e n b e t w e e n The c o r r e c t  answer was  g i v e n by o n l y  Warren c o n c l u d e d the difficulties aspects of  c a n be  the  seen  teaching  report  of  physics  of  the  about  team o f  university professors  their  vector  (1976)  studies  of  1973,  first  2/7  sec.  students.  by s a y i n g to  that  the  students'  c e r t a i n common  in pre-university  point to  identify  reported  t h a t more  understanding  the  courses, real  1975)  using  Two o t h e r  scalars,  in Paris several  (Malgrange,  students' understanding and b e t w e e n  of  force  and m o t i o n .  One  distinction  measurements.  team w e r e c o n c e r n e d  the r e l a t i o n s  a  Saltiel,  hundred s t u d e n t s .  and p h y s i c a l  s t u d i e s by t h i s  extensive  have been c o n d u c t e d by  p u b l i s h e d papers concerned the  between v e c t o r s ,  velocity,  velocity,  problem.  Leboutet-Barrel  and V i e n n o t  of  obtained  two p o i n t s ,  two  t o be r e l a t e d  b u t he d i d n o t go b e y o n d t h i s roots  the  is  between It  with  force  appears  and  that  -27two i n t u i t i v e t y p e s interaction forces,  forces  gravity,  associated velocity  of  forces  which act  tension of  and w h i c h becomes this  type of  the mass,  like  impulse or  The s e c o n d  ideas in the  about  motor  (it  relation speed as  its  are  was  assumed  the  that  frame  of  reference.  finishes  of  about  students'  students  perceived  communicated by  The r e s u l t s  of  reference. a_l  his  This  therefore, as  these  well studies  i n coordinatdifficulty  (197 3) .  article  is  concepts  as  force,  momentum, v e l o c i t y ,  students  at  e a c h age  saying  that  needed c o n c e r n i n g t h e growth o f  He adds  its  w i t h o u t any  the moving o b j e c t  et  3 00  translation  Motion and,  much more r e s e a r c h  level.  the  the  examined  e x p e r i e n c e d by s t u d e n t s of  force  a property  the body had a motor)  by P i a g e t  Leboutet-Barrel  speed  of  of  of  inertia.  showed t h a t  two f r a m e s  a l s o demonstrated  as  (which i n v o l v e d  the  field  same t i m e as  t h e movement o f  dimensions.  the d i f f i c u l t y  the  of  real  same d i r e c t i o n a s  researchers  i n t r i n s i c properties  ing motion w i t h i n was  team o f  (static  thought of  force  a m o v i n g b o d y as  mass and i t s  indicate  is  the composition of  of  to  zero at  Results  students:  and a k i n d  the  study reported  two d i m e n s i o n s . speed  a spring)  force  by t h e above  for  upon m a s s e s  w i t h m o t i o n w h i c h has  velocity;  students)  exist  speed,  energy  such in  that:  We n e e d t o know how s t u d e n t s i n t e g r a t e n e t w o r k s o f new c o n c e p t s and s t r u c t u r e t h e m . I t would be i n d i s p e n s a b l e to explore the area of preconceptions and i t w o u l d b e o f e x t r e m e i m p o r t a n c e t o make an i n v e n t o r y o f the e n t i r e range of erroneous c o n c e p t s e l a b o r a t e d s u b s e q u e n t l y by s t u d e n t s i n t h e c o u r s e o f l e a r n i n g and w h i c h d i s t o r t t h e c o n s t r u c t i o n o f c o h e r e n t system of knowledge. ( L e b o u t e t - B a r r e l , 1976, P . 4 6 5 )  -28-  2.1.2  Viewing  Students'  Barrier  Phenomena  Hawkins  (1980)  difficulties called  Difficulties  that  standing  fit  vectors  Hawkins'  most  class  It  is  of  which  learning he  very l i k e l y  students experience  in his  describes  Critical  and m a t h e m a t i c s  ban.n.-i2.fi phunomzna.  the d i f f i c u l t i e s  a  discussed a specific  i n school science  ch.ltlc.al  as  in  that  under-  definition.  c r i t i c a l barriers  as  follows:  (1) A r e c o n c e p t u a l o b s t a c l e s w h i c h c o n f i n e and i n h i b i t s c i e n t i f i c u n d e r s t a n d i n g , (2) are ' c r i t i c a l ' and so d i f f e r f r o m o t h e r c o n c e p t u a l d i f f i c u l t i e s i n t h a t t h e y : (a) i n v o l v e p r e c o n c e p t i o n s w h i c h the l e a r n e r r e t r i e v e s from p a s t experiences t h a t are i n c o m p a t i b l e w i t h s c i e n t i f i c . u n d e r s t a n d i n g , (b) a r e w i d e s p r e a d among a d u l t s as w e l l as c h i l d r e n , among t h e a c a d e m i c a l l y a b l e b u t s c i e n t i f i c a l l y n a i v e a s w e l l as t h o s e l e s s w e l l educated, (c) i n v o l v e n o t s i m p l y d i f f i c u l t y i n acquiring s c i e n t i f i c facts but i n a s s i m i l a t i n g c o n c e p t u a l f r a m e s f o r o r d e r i n g and r e t r i e v i n g i m p o r t a n t f a c t s , (d) a r e n o t n a r r o w i n t h e i r a p p l i c a t i o n b u t when o n c e s u r m o u n t e d p r o v i d e a key to the comprehension o f a wide r a n g e o f phenomena. To s u r m o u n t a c r i t i c a l b a r r i e r i s n o t m e r e l y t o overcome one o b s t a c l e b u t t o o p e n up s t i m u l a t i n g new p a t h w a y s t o s c i e n t i f i c understanding. ( H a w k i n s , 1980, p.3) Hawkins has people's  ideas  used  the  f o l l o w i n g example  to  investigate  about m i r r o r v i s i o n :  I have sometimes a s k e d s u b j e c t s t o i m a g i n e t h a t one w a l l o f t h e room we a r e s i t t i n g i n i s a l a r g e m i r r o r and t h e n t o d r a w , on a m a p . o f the room, the d i r e c t i o n i n which they would l o o k t o see a g i v e n o b j e c t ' i n t h e m i r r o r " . (Hawkins, 1978, p.l) He h a s b u t most  of  found a wide range of  them c l u s t e r  directions  a r o u n d t h e two e x t r e m e  as  answers  labelled  as  -29-  A and B i n F i g u r e Figure  2.2,  2.2  Subjects consisted children, in A  elementary  the p h i l o s o p h y of (the  correct  science.  response),  and the r e s t  selecting  direction B is  phenomena  This  clear  size  and r e p r e s e n t majority of  matter  is  are  barriers  high  easily to  t h e y do n o t  regard  this  The i d e a  of barrier  related  to o t h e r  physical  heat  observed  and  temperature,  i n many c o n -  learning for  at  least  students.  when t h e r e l a t e d  e v e n when s t u d e n t s  seem t o be a b l e  specific  incorrect  critical  s c h o o l and c o l l e g e  i n t r o d u c e d , but  (the  at  of  and s c a l e ,  generally present  conflict, to  are  B  example  c o n c e p t and o t h e r s  class  15 p e r c e n t c l u s t e r  i n between.  a clear  (Hawkins names  conceptions  About  scatter  school  and a g r a d u a t e  50 p e r c e n t a t  and a c t i o n a n d r e a c t i o n ) texts,  upper elementary  school teachers,  response),  phenomena.  of  p r o b l e m Hawkins has  These subject-  may see  to discuss said  a  the  it. that:  In  -30-  a teacher concerned to ' c o v e r the s u b j e c t ' — meaning, of course, to get through a textbook o r p r o m i s e d o u t l i n e — w i l l become e x a s p e r a t e d w i t h s t u d e n t s ' d i s a b i l i t i e s o r w i t h h i s own i n a b i l i t y t o make s u c h e l e m e n t a r y t h i n g s c l e a r . The f a c t t h a t p a t i e n t e x p l a n a t i o n i s no immediate cure i s a h a l l m a r k o f the c l a s s o f c r i t i c a l b a r r i e r phenomena. ( H a w k i n s , 1 9 7 8 , p . 5) The  f i e l d o f mechanics  examples o f  c r i t i c a l barrier  p o t e n t i a l l y may c o n t a i n many phenomena.  phenomena i n w h i c h d i r e c t i o n i s  Particularly,  a decisive  factor  in  in  their  comprehension. It that  seems r e a s o n a b l e ,  students'  vector  students  as  may be c o n s i d e r e d  For instance,  tend to  situations  ignore  Warren  i n which t h i s  vector  This belief further  factor  quantities  as  to  (1971)  is  decisive.  most o f as  phenomena c o u l d become  is  are  aimed a t  qualify of  as  beliefs  kind of  The q u e s t i o n s  uncovering students'  c r i t i c a l barriers. seems i m p e r a t i v e .  treat  inhibit about  "critical barriers"  It  addressed  The s e a r c h Hawkins has  for  the  mechanics  problem t h a t  beliefs  such  scalar.  These b e l i e f s  phenomena.  concerned.  tend to  and may  students on c u r r e n t views o f  is  in physical Findings  students  of  this  barrier  that  t h o u g h t h e y were  understanding.  with  physical  found  instruction  study  speculate  a critical  may be a c o n c e p t u a l o b s t a c l e  scientific  mechanics  point,  the d i r e c t i o n a l aspect  these would suggest t h a t  physical  this  p r e - i n s t r u c t i o n c o n c e p t i o n about  quantities  phenomenon.  at  the  in this  present research  t h a t may w e l l for said  these  kinds  that:  Though t h e e x i s t e n c e o f i n d i v i d u a l c r i t i c a l b a r r i e r s a p p e a r s t o have b e e n r e c o g n i z e d b y many t h o u g h t f u l and e x p e r i e n c e d t e a c h e r s , they appear n o t t o have been c o n s i d e r e d colZzctJ.ve.ly  as p r o v i d i n g i m p o r t a n t c l u e s t o t h e improvement o f the t e a c h i n g a r t , to c u r r i c u l u m making o r t o c o g n i t i v e s c i e n c e . (Hawkins, 1980, p . l ' l This  author  used  as  a  believes  starting  can h e l p to that  they  2.1.3  As  for  it  vectors.  concept of  About Students'  on s t u d e n t s ' of  found i n the  preconceptions  literature  Both of force;  these  studies  and b o t h u s e d  which  obstacles  sidered  scalar  as  (Raven,  reviewed below. to  students'  Other  1968,  (Piaget,  vector  only  A l l these not  aimed a t  quantities'  students'  are  for  vector  these q u a n t i t i e s  but  Aguirre,  approach to  two  conceptions  on t h e  related  investigations,  been  vector  1973;  students  1972).  conceptions,  to understand  about  a Piagetian  but  has  In f a c t ,  focused  Two o t h e r - s t u d i e s  have been r e p o r t e d  this  be  U n d e r s t a n d i n g on  physical quantities.  quantities  subjects  or  was m e n t i o n e d i n C h a p t e r O n e , t h e r e  data c o l l e c t i o n .  in  i n s t r u c t i o n a l procedures  w h i c h were m a i n l y c o n c e r n e d w i t h  related  can  Quantities  research  s t u d i e s were  are  preconceptions  c r i t i c a l barriers  Research  characteristics  of  students'  produce.  Vector  1978)  point  surmount the  Previous  little  that  were  con-  studies  directly helping  also  reviewed  section.  Piagetian-Type Piaget children  (4 t o  S t u d i es (1973)  created  12 y e a r s  of  nine tasks  age)  develop  to  find  notions  o u t how about  severe  -32-  aspects  of  forces  aspects  c o n s i d e r e d by him a r e  characteristics  —  of  force  force.  the c h i l d r e n ' s b e l i e f s stages, his  with  the v e c t o r  it  of  to  1973,  years  age)  either start  at of  at  their III  the  cannot  on t h e  1973,  Chapter 7),  role  of  that weights  appear  resultant  stage.  until of  Reviewing  understand  stage I I I . age.  acting  are  forces.  table,  upon a s m a l l  i n stage II  different  parallel,  (7 t o  results  At  (an e x a m p l e  and o t h e r  book c a l l e d  of  the  studies  task  as  summarizes about  Unde.su> tanding  these  forces  contradict  of  In the  They  force)  said  pull  and more when experiments  children's beliefs Cau&allty  stage  perceive  well.  directions  strict  (Piaget,  seem t o  a gravitational  in different Piaget  the e f f e c t  forces  the  t h e y use  seem t o  In another  10  are  H o w e v e r , when t h e  i n stage III  the d i r e c t i o n of  object  i n magnitude.  consider  force.  general  reviewed below.  which uses a c i r c u l a r  to  This  Some  p r o b l e m s where f o r c e s  subjects  t h e y come t o g e t h e r .  another  categorized  they r e t u r n to n o n - a d d i t i v i t y .  l e s s when t h e y p u l l  on f o r c e s  vector  c h i l d r e n do n o t  and t h e o b s e r v e d  subjects  The  developmental  each  12 y e a r s  when f o r c e s  predictions,  at  children  solve  a d d i t i v i t y of  angle  the  tasks,  or are  task,  the  three  force  are  Chapter 6),  an a n g l e  some  over  study  the  an a n g l e  arithmetic act  of  non-parallel forces  (Piaget, of  evident that  Piaget's  three  of  (A a n d B)  subjects  I n one o f with  i n terms  characteristics  corresponds results  is  precisely  quantity.  For each task P i a g e t  two l e v e l s  results,  being a vector  (1974) .  in In  this  -33book,  Piaget  claims  that:  When we come t o d i r e c t i o n s , we f i n d new p r o b l e m s , s i n c e we a r e no l o n g e r c o n c e r n e d w i t h s o many g e n e r a l c o o r d i n a t i o n s and r e l a t i o n s h i p s between the o p e r a t i o n a l forms and t h e e x p e r i m e n t a l c o n t e n t s , b u t r a t h e r concerned w i t h s p a t i a l o p e r a t i o n s , the p e c u l i a r c h a r a c t e r i s t i c o f which i s the c o n s t r u c t i o n o f forms o f w h i c h some c o r r e s p o n d , i n o t h e r r e s p e c t s , t o s i m i l a r forms e x i s t i n g i n the o b j e c t s thems e l v e s and s u b j e c t t o b e i n g r e a c h e d p e r c e p t i v e l y or'through physical experience. ( P i a g e t , 1974, p.35) This students' expected that  statement  preconceptions that  operations;  some f e a t u r e s  the d i r e c t i o n a l  of  factor  the It  is  include ideas  of  t h e way  it  is  position, possible  locations  concludes  of  not  displacement, that  characteristics,  spatial  Piaget  of  operations.  vector  but  vector  of  collected  importance  on s p a t i a l  formally define  and t h e i r v e c t o r  From t h e d a t a  the  these preconceptions  physicists  contain  indicates  t h e y may  particularly objects.  that:  In a l l cases s t u d i e d , the geometric s t r u c t u r i n g of d i r e c t i o n s i s l i n k e d to dynamics. That i s , what i s c a u s i n g t h e d i r e c t i o n o f m o t i o n . For i n s t a n c e , the s p a t i a l o p e r a t i o n of d i s p l a c e m e n t , d i r e c t i o n , e t c . , become e x p l a n a t o r y o n l y when c o n s o n a n t w i t h d y n a m i c s and a t t r i b u t e d t o o b j e c t s , since these determine, i n t h e i r causal c o n n e c t i o n s , the geometric r e l a t i o n s h i p s isomorphic w i t h those a p p r o p r i a t e to the o p e r a t i o n s . Needl e s s to say, the c a u s a l i n t e r p r e t a t i o n i s possible o n l y i f t h e s h a p e s and i n i t i a l p o s i t i o n o f t h e o b j e c t s a r e g i v e n , t h a t i s , by g r a n t i n g them' s p a t i a l as w e l l as k i n e t i c and d y n a m i c p r o p e r t i e s . ( P i a g e t , 1974, p.41) Efforts centrate  will  o n l y on the k i n e m a t i c a s p e c t s o f  problem s i t u a t i o n s cause of  b e made i n t h e p r e s e n t  depicted  the p h y s i c a l  i n the t a s k s ,  phenomena w i l l  research the that  n o t be  to  concrete is,  the  considered.  con-  -34If  the  subjects  the r e s e a r c h e r  refer will  to dynamic f a c t o r s  c a u t i o u s l y t r y t o move t h e  to the kinematic aspects In vector It  is  in their  of  the  summary, P i a g e t was  characteristics the purpose of  responses  discussion  task.  interested only  i n some  a b o u t one v e c t o r q u a n t i t y —  the present  study  t o u n c o v e r how  develop c o n c e p t i o n s about o t h e r v e c t o r q u a n t i t i e s formal  instruction.  To t h i s  end,  and v e l o c i t y t h a t  Piaget  found  out that  this  i t may be t h a t  students  However,  Aguirre about of  (1978) forces  this  that  in static  their  beliefs  of  The r e s e a r c h q u e s t i o n s  related  to  develop for  equilibrium.  forces  conceptions  the r e s e a r c h ,  12 y e a r s  that  the  The m a i n  of  age  looking  the concept of  s t u d y were m a i n l y forces,  two and t h r e e component  a c t i o n and r e a c t i o n , and c o m p o s i t i o n o f  about  Piaget  with.  on a c t i o n o f  i n systems w i t h  findings of  the  i n c h i l d r e n ' s minds,  aspects of  i n that  the c h i l d r e n ' s b e l i e f s  equilibrium  point  but to uncover the  study deals  about d i f f e r e n t  force.  Among t h e  as  r e s e a r c h was n o t d i r e c t e d t o w a r d  but  children  important to  that  c a r r i e d out a study about c h i l d r e n ' s  vectorial characteristic  forces,  is  students this  for  static  the c o n c e p t i o n s  s t u d y was n o t t o c a t e g o r i z e  preconceptions that  emphasis  it  study on f o r c e s ,  characteristics  beliefs  s i m i l a r to  displace-  preconceptions i n developmental stages  did with his  specific vector  forces.  the aim o f  subjects' (1973)  for  are  students  before  h o l d v e c t o r i a l c o n c e p t i o n s about v e c t o r p o s i t i o n , ment,  force.  forces.  the author found  have a l r e a d y  c l e a r l y have v e c t o r i a l  grasped  features.  that physical  -35-  T h e s e were r e l a t e d a g e d 14 were a b l e with different the  to  composition of solve  magnitude  problems  forces.  This  intuitions these  shows  and v a r i o u s  that  students  with v e c t o r i a l  angles.  intuitive vectorial  forces.  One O f t h e  whether  to  Raven children's  other  (1968,  beliefs  physical 1972)  about  and l i n e a r momentum. vectorial primary avoid  school  momentum as s t u d y was  of  postulated  acceleration.  is  is  tasks  was  Again, was  a common a p p r o a c h  determine intuitions  studies  at  all,  on  acceleration the since  He was  he  able  l i n e a r momentum b y that  is,  The p u r p o s e  of  he  used  to taking  considered  t h e momentum  sequence w i t h i n  behind his by u s i n g  treated  as  study  a  taken by r e s e a r c h e r s ;  i n which the d i r e c t i o n a l  This  produces  tion,  c o m p l e x phenomena i n t o  of  the  an u n i d i m e n s i o n a l  a scalar  being c o n t r o l l e d . transforming  to  to  b e i n g n e c e s s a r y for u n d e r s t a n d i n g  purpose  acceleration  up e x p e r i m e n t a l  of  situation;  as  of  that  related  quantities;  the developmental  concept  This  only  some v e c t o r i a l  two v e c t o r  quantity.  A similar  situation,  are  study  these concepts  momentum. of  composition  may be h o w e v e r ,  c a r r i e d out  properties  to determine  in  some p h y s i c a l  c h i l d r e n i n both s t u d i e s .  a scalar  items  Subjects  B u t he d i d n o t e m p h a s i z e  properties  the v e c t o r i a l  of  this  components  entities.  has  an u n i d i m e n s i o n a l p h y s i c a l  set  of  s t u d e n t s may a l s o d e v e l o p  relation  it  conceptions  intents  about  develop  features;  Children  containing  s a m p l e had n o t y e t b e e n i n s t r u c t e d  forces.  in  to  concept. they factor  a very s i m p l i f i e d simpler  set is  situaones,  -36-  i n which This  is  subjects  have  a totally  t o work w i t h  acceptable  a l l o w one t o u n c o v e r a l l that  B.  experiences  Special  students'  difficulties  created  properties These  special  of  .helping  students  to  to determine  A list  some o f  if  these v e c t o r i a l  illustrate  dimensions physics  (Greig, Vector  1957);  approach  to  and i n -  aim o f  no  of  successful.  follows:  1951);a  (Miller, vectors  a game  of  dramatic  1952); in  versatility  the concept  of  data  have been  navigation:  1971);  shown.  characteristics  and f o r c e  teaching  vectorial  the  is  the  quantities,  dramatically  (Dempster,  vector  aware o f  experiments  some p r o p e r t i e s  ( P e r r y and C a m p b e l l ,  screw n a i l  C.  (Baez,  the  prepared with  Does p r e s s u r e h a v e d i r e c t i o n ?  model t o  are  these tasks  between p r e s s u r e  conceptions  vector  Unfortunately there  available  distinction  being  i n which  grasp the vector  certain quantities.  of  teachers,  experiments  are  not  Quantities  i n understanding  tasks  rich  does  environment.  physical quantities  experimental  it  through formal  to Teach Vector  and p h y s i c s  variables.  but  the p o t e n t i a l l y  with their  Experiments Physicists  have  approach,  s t u d e n t s may h a v e d e v e l o p e d  formal  fewer  of  a  three for the  cross  product  1972). Quantities Researchers  interested usually  i n Mathematics i n mathematics  i n students'  treat  vectors  as  Education e d u c a t i o n have a l s o  understanding  has  been to  use  vectors.  a b s t r a c t mathematical  do n o t make c o n n e c t i o n s w i t h p h y s i c a l interest  of  vectors  as  reality.  a tool  to  been  But  they  entities, Their improve  and  main the  teaching this  of  other  task,  units  for  areas  mathematics use  these units  are  Mathematical  listed  (Troyer,  1970).  1969)  the understanding used  1966) ; V e c t o r s (Smith,  1963);  concrete  of  space,  vector  2.2.1  context  from p h y s i c a l  involving  formal  models  section  improve them h a v e  acceleration, for  the  however,  experimental  or  study  of  physics  situations  reality.  Study  d e a l s w i t h the broad  i n which t h i s  development the  none o f  point of view;  concrete  to  1971,  Introduction  This  issue  but  approach  P s y c h o l o g i c a l C o n t e x t .of t h e  2.2  i n Algebra  (Lamon,  objects  like velocity,  from a m a t h e m a t i c a l  directly  Solve  Vectors  studies  have used  look for  Approach to  1963);  the  vectors  obtained  1959);  1965) ; P l a n e G e o m e t r y b y V e c t o r  T h i s may be a n a c c e p t a b l e  should  some o f  : An A i d t o  Using Vectors to  force.  teachers  of  Complex Numbers and V e c t o r s  Some o f  a physical quantity  instructional  The t i t l e s  (Salzarulo,  (Glicksman,  (Scott,  Rosembloom,  (Pedley,  Equations  and Geometry Methods  below:  To a c c o m p l i s h  have c r e a t e d  situations.  Understanding  Geometry  Simultaneous  educators  i n school  i n H i g h S c h o o l Math  Vector  i n mathematics.  of  as  the  study  is  dichotomous  proposed  ^ functional development)  psychological  embedded. approach  First, to  the  by some p s y c h o l o g i s t s (or  competence  dichotomy.'  and  is  a  theoretical  study  of  discussed  —  performance  Reasons a r e  advanced  as  -38-  t o why t h e  performance  approach  for  through  the  Although,  logical  of  study  study.  Case's  to  assess  students'  is  Case.  devoted  current  This  this  to to  inthe  psycho-  Instruction.  Case  tradition'  in-  level  of  functioning  proposition  is  in  sympathy  the  methodo-  research.  c u r r e n t l y used  'Neo-Piagetian  explained  i n a three-step theory of  content-areas.  logy approach  T h i s model i s  Theory of  step  aim o f  preferred  and  subsection  second  the b a s i c  the  not d i r e c t l y r e l a t e d  (the  specific  c h o s e n as  Siegler,  is  a separate  struction)  with  work  component o f  proposes  in  the p r e s e n t  this  struction,  m o d e l was  Finally,  by some r e s e a r c h e r s  — rational  task  in  the  analysis  —  is  presented.  2.2.2  Two A s p e c t s  of  Developmental Psychology  : Formal  and F u n c t i o n a l  Some p s y c h o l o g i s t s Wollman, between  1978;  Case,  ( F l a v e l and W o h l w i l l ,  1978)  two i n t e r d e p e n d e n t  and f u n c t i o n a l . meaning o f  the  Flavell  et  two a s p e c t s  have made t h e aspects al.  distinction  of development:  (1969)  i n the  1969;  formal  have e x p l a i n e d  the  f o l l o w i n g way:  The f o r m a l a s p e c t has t o do w i t h t h e m o r p h o l o g y of the p r o c e s s : the sorts of c o g n i t i v e e n t i t i e s t h a t make up t h e s u c c e s s i v e o u t p u t s o f d e v e l o p ment and how t h e s e e n t i t i e s a r e c a u s a l l y , t e m p o r a l l y , and o t h e r w i s e i n t e r r e l a t e d . . . T h e o t h e r a s p e c t . . . has t o do w i t h f u n c t i o n and m e c h a n i s m , somewhat s p e c i f i e d i n r e l a t i o n t o e n v i r o n m e n t a l i n p u t s , b y w h i c h i t i n f a c t makes the c o g n i t i v e p r o g r e s s t h a t has been f o r m a l l y c h a r a c t e r i z e d . ( F l a v e l l and W o h l w i l l , 1 9 6 9 , p . 6 7 ) .  -39-  Th i s  dichotomy of  a useful  way t o  however,  it  development  organize  or  to  structure  s h o u l d be a p p l i e d c a u t i o u s l y  One p r o b l e m i s  that  structures  transituational,  free  —  Piaget not  are  a position (1972)  totally  has  above  it  that  that  research and  atemporal,  his  shown t o  to  be  questions;  reflectively.  the hypothesized  difficult  that  theoretical  dichotomy i s  automotan model o f  to  the  as  is  stated  or  aspect.  implies  been  and  mental  context-  defend.  Even  formal operations  were  context-free.  Another the  has  formal  aspect  the  model t h a t  is  competence  versus  development. and t h e  F l a v e l and W o h l w i l l  to  the  explained  related  to  performance  The f o r m e r  latter have  closely  is  related  functional this  position  follows: A competence m o d e l , which i s a f o r m a l , l o g i c a l , r e p r e s e n t a t i o n o f t h e s t r u c t u r e o f some d o m a i n . . a n a u t o m a t o n m o d e l (an e l a b o r a t e d v e r s i o n o f what Chomsky a p p a r e n t l y meant b y " p e r f o r m a n c e " ) , which r e p r e s e n t s the p s y c h o l o g i c a l p r o c e s s by which t h e i n f o r m a t i o n embodied i n competence a c t u a l l y g e t s a c c e s s e d and u t i l i z e d i n r e a l s i t u a t i o n s . The c o m p e t e n c e m o d e l g i v e s an a b s t r a c t , p u r e l y l o g i c a l r e p r e s e n t a t i o n o f what t h e o r g a n i s m knows o r c o u l d do i n a t i m e l e s s , i d e a l e n v i r o n m e n t , w h e r e a s t h e automaton model has t h e job o f d e s c r i b i n g a r e a l d e v i c e t h a t c o u l d p l a u s i b l y i n s t a n c e t h a t knowledge o r s k i l l , and i n s t a n c e i t w i t h i n t h e c o n s t r a i n t s (memory l i m i t a t i o n , r a p i d performance, etc.) u n d e r w h i c h human b e i n g s a c t u a l l y o p e r a t e . ( F l a v e l l and W o h l w i l l , 1 9 6 9 , p . 7 1 ) . Using  of  these d e s c r i p t i v e  development  formal  is  normally considered  t h e o r y o r competence  structures  that  as  Piaget's  an example  model o f development.  because P i a g e t ' s main i n t e r e s t mental  categories,  was  to  look for  were t r a n s i t u a t i o n a l  and  theory of This  a is  general context-free-  Considering or  Piaget  formalist,  t o be p r e d o m i n a n t l y a  rather  than a f u n c t i o n a l i s t  t o Wollman  (1978) , t h a t  content  human k n o w l e d g e  of  structure.  structuralist  he i s  For instance,  less  means,  according  concerned with  than with  its  the  organization  Wollman c l a i m s  or  that:  P i a g e t * s concern i s w i t h the fundamental u n d e r l y i n g o r g a n i z a t i o n o f performance on t a s k s s u c h as t h e seriation task. He f e e l s t h a t s u c h t a s k s r e v e a l an i d e n t i c a l organization or ' s t r u c t u r e ' , that i s , successf u l p e r f o r m a n c e on a l l s u c h t a s k s i n v o l v e s o n e b a s i c s t r u c t u r e o r scheme, t h e s e r i a t i o n scheme. ( W o l l m a n , 1978, p.8) Flavell that  Piaget's  a logical  that  d e s c r i p t i o n of  the  p r o g r a m has  systems o f  possess  not to  at  provide a  psychologists  (1978),  functional  with the  the  Siegler  s u c h as  (1976,  aspect of result  subject  and how t h a t  out  provide  intellectual points  in  psychological  knows a b o u t  specific  account,  understanding  the  knowledge  and C a s e  According  of  partly  a  a specific is  is  are  (1978) as  have  a  this  view-  encounter what  content-area and  organized.  development  related  content-areas  (1966),  function of  represented  more c l o s e l y  specific  to  a cognitive  functional aspect of  model) of  Pascual-Leone  been d e f i n e d  development.  or performance  already  the performance of  been to  different  1978),  the p h y s i c a l environment i s  In t h i s  pointed  t h e p r o c e s s e s by w h i c h t h e s e o p e r a t i o n s  b e e n more c o n c e r n e d w i t h w h a t h a s  point  have a l s o  and u t i l i z e d .  Other Wollman  of  children  development,  acquired  (1969)  primary research  description  operations their  and W o h l w i l l  to  and the  the  (or growth  possible  -41-  factors  that  fically  shown b y t h e work o f It  tomy  could influence  is  (formal  one  and  and f u n c t i o n a l  aspects,  however,  finds  it  useful  It  w o u l d be d i f f i c u l t , that  Siegler.  that  this  them as  dicho-  and  these  two  i n t e r a c t i o n must be  i n the view of  treated  spefi-  per-  unreflectively.  to d i s t i n g u i s h  t h e i r manner o f  is  or competence  s h o u l d n o t be a c c e p t e d  development,  of  Pascual-Leone again,  of  a position  This  worth s t r e s s i n g  formance models) Even i f  such growth.  this  author,  two t o t a l l y  aspects considered.  to  defend  independent  aspects  development. If  the  considered,  competence-performance  however,  it  does have  value,  a n d on t h e s e g r o u n d s  nature  of  then,  is  settle  the  research  not to  the  solve  issue but  was  the problems  vector  already  model o f  is  organized,  summary,  preconceptions, solving  which students content-areas.  it  the  i n the present  it  was  tasks,  develop  clear  o u t what  content-area  that  the  that  through t h e i r might shed  and  a  (i.e.,  specific performance  analysis  framework. of  students'  p e r f o r m a n c e when  l i g h t on t h e p r o c e s s  their beliefs that  the  to  research.  t h e more a p p r o p r i a t e  thought  T h i s means  The i n t e n t i o n ,  theoretical  study ;wex£~-tocfind  was  the  the dichotomy or  a specific  provides  obtained  specific  of  study.  and t o d e t e r m i n e how t h a t  development  In  this  knows a b o u t  quantities),  knowledge  of  heuristic  to c h a r a c t e r i z e  t o draw upon some o f  the broad goals  subject  used  use  cautiously  some u t i l i t y and  undertaken i n t h i s  m e t h o d o l o g i c a l knowledge f o r Since  dichotomy i s  with regard  the research  was  to  by  particular  very  contextual  oriented  students' for  i n nature.  preconceptions  instance,  intellectual  i n t o the  about  operations  of  theory.  crete  formal operations  understand students' to  find  N o r was  vector  vector  logical  Piaget's or  The a i m was  are  rather,  about  such p r e c o n c e p t i o n s  consistency  or  organization.  with  the  and i s ance  so-called  in this  2.2.3  of  the  subject's  advanced by s e v e r a l approach  is  development  theorists.  so named b e c a u s e as  its  starting  a c q u i r i n g new c o n c e p t i o n s .  factors  the  current  level  more e a s i l y factors  that  of  that  of  to  uncover  constrain  and  any d e g r e e  fits  of  well  Development,  functional or  perform-  Development  been  Wollman,  takes P i a g e t ' s  recently  1978).  The  account  of  to  suplement  a s s o c i a t e d with the  In other words, the process o f  development,  so  the  The r e a s o n  performance  for  process  Neo-  development that  the  this  is  might  that  would be b e t t e r  and  students'  functioning i n a given content-area  be a s s e s s e d .  con-  content-area  then,  of  to  a i m was  a p p r o a c h has  factors  considers  affecting  learn  p o i n t and a t t e m p t s  of  approach  the  systems  stages  Approach to  (c.f. it  w i t h an a c c o u n t  Piagetian  of  fit,  behaviour.  this  it  the  study,  The N e o - P i a g e t i a n A p p r o a c h o f  A s t h e name i m p l i e s ,  the  possess  "Neo-Piagetian"  p r i m a r i l y concerned with  account  to  a specific  if  taken  the  and f o r m a l  necessary  out  The a p p r o a c h  if  i n t e n d e d t o d e t e r m i n e what  quantities,  preconceptions  see  characteristics  d e s c r i p t i o n of  concrete it  not to  the  understood  -43and t h e t y p e o f related  to  assessment of  observed  performance  The g r o u n d w o r k f o r Pascual-Leone  (1969)  quent work.  and C a s e  (Wollman,  Pascual-Leone,  (19 7 8 ) .  s t u d y and i t Siegler's  T h e i r work i s  is  discussed  Approach:  two o t h e r  they are  closely  Siegler's  work  of  I n one o f aspects  of  formance, that  cern  earlier  development:  is  only  to  the  first  balance,  a n d u n a m b i g u o u s way;  subjects'  conceptions  the  focus  was  and b a s i c  relevant.  of  or  which they then apply  schemes  which appear (1976  strategies  children organize  t o be  are  similar  , p.Ill)  pre-  current  in  to  thinking.  upon  three  For the  Sigeler's  rules.  to  solve  assumes,that  specific  con-  i n an a c c u -  characterizes  This  entails  a series In other  students'  the r u l e s  present  main c o n -  knowledge i n terms  i n some w a y .  rule-governed with  he  per-  processes  and d i s t a n c e )  Specifically,  that  is  knowledge g o v e r n i n g t a s k  time speed,  i n terms  present  H i s aim i s  i n t h e s e two a r e a s .  assumption  Siegler  the  experience,  aspect i s  the  1978)  below.  t o d e t e r m i n e c h i l d r e n ' s knowledge about  students'  to  the  (1976,  (1976,1978,1979)  (1976)  specific  responsiveness  tent-areas (i.e., rate  studies  underlie differences  study,  Siegler  and e x p l a i n d e v e l o p m e n t a l d i f f e r e n c e s  his  subse-  emphasizing  functioning i n given content-areas.  characterize  and  Neo-Piagetian  related  i n more d e t a i l  dominantely concerned with d e s c r i b i n g level  done m a i n l y b y  importance o f  aspect o f development;  closely  1978).  doctoral dissertation  have r e c o g n i z e d the  functional  w o u l d be more  t h e movement was  in his  Besides  researchers  interest  of  the  of  rules  problems  words,  problem-solving  progressing  from  -44-  less  sophisticated  classic, first  Piegetian balance  rule  years)  t o more s o p h i s t i c a t e d  For  he p r o p o s e d 4 r u l e s :  the  the  t y p i c a l l y b e i n g u s e d by t h e y o u n g e r c h i l d r e n  and t h e f o u r t h r u l e by t h e o l d e s t B u t what d o e s  Siegler  task,  with age.  answers  by  it  mean t o  say  (over  someone i s  12  (5  years).  using a  rule?  saying:  I n one s e n s e , r u l e s c a n be t h o u g h t o f as a means o f s u m m a r i z i n g d a t a , as when a s e t o f r e s p o n s e s i s s y m b o l i z e d by a s i n g l e v e r b a l sentence, a mathematical equation, a production s y s t e m , o r some o t h e r f o r m a l i s m . Within this u s a g e i t makes l i t t l e s e n s e t o i n q u i r e i f r u l e governedness d e v e l o p s , because the r u l e s a r e m e r e l y representations (they are i n the e x p e r i m e n t e r ' s h e a d ) ; any f a i l u r e o f t h e r u l e s t o f i t t h e p a r t i c u l a r r e s p o n s e s t e l l s us o n l y t h a t t h e r e p r e s e n t a t i o n i s inadequate. An a l t e r n a t i v e a p p r o a c h , however, i s t h a t the r u l e statements not o n l y s u m m a r i z e d a t a b u t a l s o have some c o r r e s p o n d e n c e t o t h e way i n w h i c h t h e d a t a were g e n e r a t e d (they a r e i n t h e s u b j e c t ' s h e a d as w e l l as t h e e x p e r i menter's) ( S i e g l e r , 1978, p.19). For considered.  the present That i s ,  summarizing d a t a ,  Siegler  rules  but  some c o r r e s p o n d e n c e  it  to  (1977,  study,  He c i t e s  of  task,  shadows  s e c o n d a p p r o a c h was  were i n f e r r e d as  was a l s o  assumed  p . 394)'  says  that  rules  as  they  a  have  reports  that  i n each  case:  them.  other  representation (19 58)  (1966)  water  projection jar  a n d a p r o b a b i l i t y c o n c e p t t a s k d e s c r i b e d b y Chapman Siegler  of  generate  a number o f  I n h e l d e r and P i a g e t ' s  B r u n e r and K e n n e y ' s  a means  that  t h e way how s u b j e c t s  t a s k s have been s t u d i e d u s i n g approach.  the  task, (1975).  -45-  . The p r o p o s e d r u l e s have been found t o c h a r a c t e r i z e t h e p e r f o r m a n c e o f more t h a n 80% o f c h i l d r e n a g e d 5 t o 17 y e a r s a t a l e v e l o f p r o b a b i l i t y e x t r e m e l y u n l i k e l y t o be t h e p r o d u c t o f a random response process. ( S i e g l e r , 1 9 7 7 , p . 394) Even though the b r o a d , the present tion  and d a t a  1978,  1979)  rested  analysis  Methodology)  the  models o f  tasks.  rules  So  and  on t h e c o n c e p t s  he  has  in his  initial  studies of  time,  of  Siegler's  this  the  has  formulated  previous  speed,  the r u l e s  to  findings.  based  no s t u d y r e l a t e d  these concepts.  1978)  (Siegler,  to  upon w h i c h  areas. students'  was  found  to  Therefore,  formulate the  R u l e - A s s e s s m e n t methodology c o u l d n o t be u s e d  the present  study.  to provide data  One o f  t h e aims o f  such t h a t  mulate  the S i e g l e r - t y p e  tested  using  this  s t u d y was  i t w o u l d be p o s s i b l e  rule-model.  the c i t e d methodology.  T h e s e c a n be  1979)  on  i n these conceptual  t h e r e was no d a t a for  formula-  (Siegler,  and d i s t a n c e the r u l e s  (Rule-  these  research  about k i n e m a t i c v e c t o r g n a r r t i ± i e s so  collec-  decision  requires  on t h e b a l a n c e  investigations  literature  ..models o f  for  in  (Siegler,  s u b j e c t s might use  Siegler  was m e n t i o n e d e a r l i e r ,  preconceptions  fact  far,  steps  f o r m u l a t e d the models o f  Piaget"s  the  that  the  i n t h i s methodology  first  on t h e b a s i s  instance,  in  the  Siegler,  p r o p o s e d by him  The r e a s o n s  the r u l e s  For  as  methodologies  were n o t u s e d .  o f models o f  perform  But,  s i m i l i a r to that of  on the methodology i t s e l f ;  Assessment tion  study i s  t h e o r e t i c a l approach taken  to  in  in for-  subsequently  -46-  Case's  Approach:  believes  that  c a n be u s e d He a r g u e s by w h i c h  Case,  research  to  i n s t r u c t i o n a l problems  students  acquire  important questions the content of  Case  (Case,  types wants  of  conceptions.  t o answer the  "is  structures  He p r o p o s e d  in his  1978a).  process  and u t i l i z e o p e r a t i o n a l  i n s t r u c t i o n to  already have a v a i l a b l e " . three  approach  a f u n c t i o n a l a p p r o a c h may show t h e  and o t h e r more c o n t e x t u a l  cess of  psychologist,  b a s e d on t h e N e o - P i a g e t i a n  address  that  a developmental  structures One o f  how t o that  the  adopt  children  theory a  pro-  steps:  The f i r s t s t e p i s t o a n a l y z e t h e s t r u c t u r a l u n d e r p i n n i n g s o f the academic d i s c i p l i n e t h a t i s t o form t h e c o n t e n t o f i n s t r u c t i o n . The second step i s to assess c h i l d r e n ' s c u r r e n t l e v e l of functioning i n that d i s c i p l i n e . The t h i r d step i s to t a i l o r the i n s t r u c t i o n a l a c t i v i t i e s t h a t are presented to the l e v e l of functioning that is assessed, either with a view t o p r o m o t i n g a t r a n s i t i o n from t h i s l e v e l o f f u n c t i o n i n g t o a h i g h e r one ( i f t h a t i s t h e g o a l ) , or w i t h a view to adopting the m a t e r i a l presented to the c u r r e n t l e v e l of f u n c t i o n i n g ( i f t h a t i s the g o a l ) . ( C a s e , 1978a/ p . 2 0 4 ) Although  this  optimize  the development of  it  i n s t r u c t i o n a l p r o c e d u r e was d e v e l o p e d  c a n a l s o be u s e d  which conceptions acquired  one i s related it  is  to  students'  study  related  and u t i l i z e d b y  and make u s e  to  specific  three  strictly  instructional.  not s t r i c t l y  very useful  steps  the performance  for  related  of  structures,  the process  content-areas  by  are  students.  From t h e  to  operational  to  suggested by C a s e , The s e c o n d  aspect of to  step  only is  development;  instruction.  i n s t r u c t i o n a l purposes.  the  actually therefore,  However, i t  The f i r s t  last  step  is is  -47-  neither  directly related  velopment nor to analysis,  which  to  the  instruction.  functional aspect of But the  should i d e n t i f y  result  the major  given d i s c i p l i n a r y teaching unit)  and t h e  among t h e m , may h e l p t o  search  conceptions For are  about  specific  the p r e s e n t  relevant;  the  this  step,  academic  first  content,  analysis  the  of  Cin a  relationships for  that  students'  content  first  two to  area.  steps  instruction,  the p o s s i b l e  is  the  a structural  applications  list  further  of  in  a network o f  of  methodology used presented  approach a c t u a l l y  of  study;  is,  ing of  g r a d e 10  listed  i n Table  2.2.4  Task /Analysis  psychological  the  to  students  characteristics.  is  to  The  structural  section.  The s e c o n d the main  analysis step  function-  characteristics  1.1.  t h e more p r o m i s i n g l i n e s research  of  current  i n c o g n i t i v e development  of  objective  study the current l e v e l of the v e c t o r  re-  displayed  relationships.  type of  corresponds  about  1 --  i n C h a p t e r 3 where t h e  conceptual  below i n the next  that  of  characteristics  to perform t h i s  Case's  One o f  implict vector  expanded  among t h e s e v e c t o r  terms  analysis  b e e n p a r t i a l l y done i n C h a p t e r  lationships  the  concepts  which r e l a t e s  i n terms  step,  has  the development o f  is  only  this  work. Case's  This  the  aspects of  study,  third  c a n be c o n s i d e r e d o n l y of  focus  of  de-  utilizes  -48-  some f o r m o f Case,  Task A n a l y s i s  1978).  cation  is  The v a l u e  that  to  taken  i n the present  instruction.  The t y p e o f  task  analysis.  Task h i e r a r c h i e s  of  of  that  (1976)  refers argues  that  implicit that to  to  R.T.A.  is  its  Gagne's  describe  approach  eduappli-  was  is  i n the  previous  generally  called  abbreviated  as  hierarchical  (1970)  expert's  Gagne - t y p e of  work".  nature.  and  performance.  and  others.  is  not. in  Resnick  hierarchies  do n o t a r i s e  as  performances  on a v a r i e t y  of  are  "left  She adds t h a t  need  i n terms  a  R.T.A.).  and c e r t a i n l y n o t d e v e l o p m e n t a l ,  hierarchies R.T.A.  for  Gagne' s a n a l y s i s  p s y c h o l o g i c a l models  i n Gagne1s  of  b y Gagne  (.1978),  descriptions  is,  kind  sequence)  a kind of  that  analysis  1978;  t o be d i r e c t l y  a l l u d e d to  i n s t r u c t i o n a l purposes  psychological tasks;  appears  Wollman,  a c e r t a i n k i n d have been d e s c r i b e d  psychological  it  type of  (Thereafter  But a c c o r d i n g t o Wollman really  1978,  study.  analysis  One common f e a t u r e  for  this  Hence t h i s  (Step 1 i n C a s e ' s  rational  used  of  such r e s e a r c h  cable  section  CSiegler,  this  validation.  entirely is  the  reason  R e s n i c k goes on  of:  T y p i c a l l y a r a t i o n a l task a n a l y s i s i s d e r i v e d f r o m t h e s t r u c t u r e o f t h e s u b j e c t - m a t t e r and makes few e x p l i c i t a s s u m p t i o n s a b o u t t h e l i m i t a t i o n o f human memory c a p a c i t y o r p e r c e p t u a l e n c o d i n g p r o c e s s e s ( R e s n i c k , 1976,- p . 6 5 ) In c o n t r a s t of  conceptual  to  the r a t h e r  relationships  which c h a r a c t e r i z e s  R.T.A.,  formal,  logical  from a d i s c i p l i n a r y another  type o f  analysis perspective,  analysis  has  -49-  b e e n d e s c r i b e d as  an e m p i r i c a l t a s k  abbreviated  as  to  the question of  consider  seen by t h e students How a r e  student.  This  these  how t h a t  What schemes  questions familiar  schemes  cues  t o be e a s i l y  analysis  attempts  particular  task  are  o r p r e c o n c e p t i o n s do  significant  obviously requires  student  for  them?  i n f l u e n c e d by the cues?  with the  the  "knowledge"  and how t h i s  following  is  Are they  The answer  investigator  (preconceptions)  knowledge i s  What likely  to  to  the  task?  or preconceptions coordinated?  Resnick describes the  type of  (hereafter  p o s s e s s w h i c h a l l o w them t o u n d e r s t a n d t h e  irrelevant  the  E.T.A.I.  analysis  these  become  p o s s e s s e d by  coordinated.  an e m p i r i c a l t a s k  analysis  in  way:  I t i s b a s e d on i n t e r p r e t a t i o n o f t h e d a t a (errors, l a t e n c i e s , s e l f - r e p o r t s , e y e o r h a n d movements, e t c ) f r o m human p e r f o r m a n c e on a t a s k ; t h e a i m o f s u c h a n a l y s i s i s t o d e v e l o p a d e s c r i p t i o n (model) o f p r o c e s s e s t h a t would account f o r those d a t a " . ( R e s n i c k , 1976, p.65). Case types  of  task  research. culties  (1978)  analysis  that  tasks  students  related  that  carefully  assessed,  rationally  tasks  state.  both  in their  to uncover the  or content-areas  to vector quantities)  and t h a t  initial the  and p s y c h o l o g i c a l l y  the d e s i r e d  have used  ( R a t i o n a l and E m p i r i c a l )  the l e a r n e r ' s  to b r i n g  1979)  diffi-  show when e n c o u n t e r i n g s p e c i f i c  the  analyzed,  sequenced  (1976,  I n t h e i r work t h e y a t t e m p t  They s u g g e s t t h a t (e.g.,  and S i e g l e r  the  learner  T h i s means  t o be  taught  must be  carefully  k n o w l e d g e must  learning activities  (in the developmental from  his  t h a t Case  initial and  tasks.  be must  be  sense)  state  Siegler's  to  -50-  approaches  differ  educational  researchers,  need t o c l a s s i f y this  she  initial  more s u i t a b l e point task  of  as  approach state  spontaneously Wollman  i n that  learners  neo-Piagetian  learner's or  markedly from t h a t  (1978)  and f o r  analyses.  concrete advocates  to  of  t o be  t h e use  Wollman says  is or  Piagetian  no a t t e m p t formal.  the  or  Instead,  assessment of  the  s t r a t e g i e s which  the c r i t e r i o n  thinks that  than others  view,  there  i n terms  applies  o f most  task.  some c o n t e n t - a r e a s  treated of  from a  rational  he  are  functional  and e m p i r i c a l  that:  T a s k a n a l y s i s s h o u l d be a p p l i e d t o some s e v e r a l k e y s c i e n c e and s c i e n c e - r e l a t e d mathematics t o p i c s . To q u a l i f y as a k e y t o p i c , two c o n d i t i o n s s h o u l d be m e t : (1) t h e t o p i c should p l a y a c e n t r a l r o l e i n the s t r u c t u r e o f t h e s u b j e c t m a t t e r and (2) t h e t o p i c s h o u l d be p a r t i c u l a r l y r e c a l c i t r a n t t o current teaching attempts. ( W o l l m a n , 1978, p . 1 0 7 ) This and t h e i r  author believes  special  characteristics  s t a t e d by Wollman, Task A n a l y s i s , was  used In  was  fit  Rational  study  from t h e d i s c i p l i n e  students'  preconceptions  research  tasks.  two  conditions Therefore,  and  two k i n d s o f  a R a t i o n a l Task A n a l y s i s  Task A n a l y s i s  these  quantities  Empirical,  study.  in this  Empirical  the  physical vector  shown i n C h a p t e r 1.  two f o r m s :  i n the present  were u s e d : analysis  as  in its  summary,  that  that  that  expert's consisted  about  task  consisted  point of of  analyses  view,  analyzing  the c o n t e n t - a r e a  of  an and  an  the  included i n  -512.3  Educational Context of  In approach of  the  last  the  two d e c a d e s  i n science  teaching  1966;  (Shulman & T a m i r ,  Bruner,  Hawkins  1961;  1973;  Schwab,  the  more k n o w l e d g e ;  he c l a i m s  that  (p.20).  1973;  T h e y were r e a c t i n g which emphasizes instead  of  acquire  that  if  use  explaining In  only  of  n o t much w i l l  context,  understanding states  a  be  is  learned  Hawkins,  acquiring  1978).  knowledge  to  this  themselves  concept  phases,  is  expected  have  risen  to  possess  statements  that  a  and  approach  each  some : b a s i c » s c i e n t i f i c  and w i l l  scientific  gain  "Concept Approach"  1964;  i n successive  it  students w i l l literacy,  to  to  l i m i t e d curriculum approach,  According  organized  Wittrock,  according  t h e way t h e d i s c i p l i n e s  and i l l u s t r a t i n g  finished  scientific  knowledge  1962,  that  of  process  n o t much k n o w l e d g e  t h e method o f  knowledge.  this  scientific  Schwab  against  considering  a curriculum is  Hawkins  use  But,  O t h e r s i n t u r n have p r o p o s e d  (Shulman & T a m i r ,  is  for  upon the  Hawkins, 1978;  1962).  method i s  and a v a i l a b l e  focussed  the p r a c t i c e  (1978),  organized  the u n d e r l y i n g c u r r i c u l u m  has  i n q u i r y , which emphasizes  method.  Study  concept.  when t h e  a higher  course  level  foundation  of  for  explanations.  that:  T h i s v i e w , l i k e t h e method v i e w , r e p r e s e n t s c o n d i t i o n s necessary but not s u f f i c i e n t . It s u f f e r s f r o m t h e a m b i g u i t y o r c o n f u s i o n o f two v e r y d i f f e r e n t l e v e l s o f ' l e a r n ' and ' u n d e r stand' . V e r b a l t r a n s m i s s i o n and r e c e p t i o n f o c u s e d o n t h e u s e and i l l u s t r a t i o n o f s c i e n t i f i c  -52-  c o n c e p t s i s n o t t h e same as t e a c h i n g and l e a r n i n g which succeeds i n r e c o n s t r u c t i n g o l d e r t h o u g h t - h a b i t s . . . What h a s b e e n wrong w i t h t r a d i t i o n a l d i d a c t i c i n s t r u c t i o n has n o t been i t s emphasis upon the t r a n s m i s s i o n o f f a c t u a l knowledge b u t i t s n e g l e c t o f the d e v e l o p m e n t a l p r e c o n d i t i o n s f o r g e n u i n e and p r o d u c t i v e a s s i m i l a t i o n of t h i s knowledge. There i s a time f o r d i d a c t i c t e a c h i n g but i t i s a time which presupposes t h a t thoughth a b i t s w h i c h c a n a c c e p t , s t o r e , and r e t r i e v e f o r new u s e s a r e a l r e a d y d e v e l o p e d o r d e v e l o p i n g . . . ' O p t i m a l t e a c h i n g m u s t , t h e r e f o r e , be b o t h d i a g n o s t i c and n u r t u r a l . (Hawkins, 1978, p . 20-21) This similar  to  diagnostic sidering skills the  that  of  aspect  a view of  science  education  Hawkins, p a r t i c u l a r l y with of  teaching.  the knowledge,  that  students  the  It  is  argued  that  learning strategies, the  instructional  teaching-learning process  has  much t o  this  necessary.  examine  the  first  study  about tried,  students  possess  conceptions  the  situation, But  three  to  aspects  in particular,  a s p e c t by e x a m i n i n g the knowledge  that  p r i o r to  in a specific  the  con-  and  gain.  these  to  by  to  or  instruction.  done b y u n c o v e r i n g - a n d a n a l y z i n g s t u d e n t s '  instructional  2.3.1  research  The p r e s e n t  preconceptions T h i s was  approach,  very  regard  b r i n g to  practice is  a u t h o r has  pre-  content-area.  Teaching—Learning Problems i n Science E d u c a t i o n  The g e n e r a l teaching-learning do n o t e a s i l y  problem of problem.  the present  R e s e a r c h has  grasp v e c t o r i a l  quantities  research  is  shown t h a t that  are  a students  usually  -53included 1976) .  in physics  Where do t h e p r o b l e m s  methodology?  Is  these concepts? from s t u d e n t s ' t o be  the  Is  it  students  that  experience  the  and,  view.  author believes  do n o t c o n s i d e r  Obviously  this  In t h i s  siderations, account  Is are  it  1979;  the  not  therefore,  is  is  Kass,  instruction  "ready"  content-area  that  to  grasp  very  too  remote  complex  the  methodology view,  we s e e ,  students'  is  current  part  of  that  knowledge.  an o v e r a l l c u r r i c u l u m among o t h e r  content-area  there  con-  program m a i n l y t a k i n g  arranged  approach  to  or  is  into  discipline.  in hierarchical  an a p p r o a c h  n o t an e a s y  i n science  i n the  order; field.  no c o n s i d e r a t i o n  that  task.  programs.  students'  As  of  considers  First  beliefs  of  all,  i n many  For i n s t a n c e ,  preteaching  students' research topics  n o t much  conceptions  is  about  conceptions.  Another preconceptions alternate  is  to uncover students'  known a b o u t vectorial  methodologies  lie  views.  conceptions  included  Present  s u g g e s t e d by t h e e x p e r t  To c h a n g e  needed  of  usually  in this  the problems  c u r r i c u l u m makers,  structure  order being  some o f  students'  plan a content-area  The p r o g r a m i s  is  that  lie?  i n s t r u c t i o n methodology.  usually  that  it  (Fisher,  understood? This  in  G r a d e 11 and 1 2 .  important is  that  frameworks  reason  some o f  to  find  these  out the  students'  ( a l s o c a l l e d b y some  or misconceptions,  see  Driver  and  -54-  Easley,  1978,  p.62)  and i n t e r f e r e  with  important  reason  are  formal  and a l s o  conceptions  c a n be u s e d  Knowledge o f  to  in  in constructing  students which  indicates  conceptions"  related  instructional report  indicates  explored  (11-12 y e a r s o f  age)  the  being  studied  l a w o f moments,  between  force  struction  of that  did  of  not,  12-15  benefit  efforts after  to  related  "miswere  refute  Driver students  their thinking  about  centers and  inter-  the of  gravity,  relation After  topics,  in-  it  and c o n f l i c t i n g  a change  Their  students'  science  identified. these  an  s t u d e n t s were  systems,  to  in  (1977).  frameworks  the counter-examples encourage  factors  instruction.  teaching,  old  giving  Common  and Dawson  and r e a c t i o n , were  year  of  irrelevant  i n accommodating to  science.  and i m p l e m e n t a t i o n o f  (balancing action  in  has  and s i n k i n g b o d i e s  alternate  and m o t i o n )  itself,  frameworks  the d i f f i c u l t i e s  Prior  the u n i t s  apparent  floating  have  v i e w e d a n d a number o f  intuitive  flotation.  some p e r s i s t e d  t o new e x p e r i e n c e s .  topics  reject  despite  some o f  how t h e s e  the p o s s i b l e  i n the design  that  another alternate  a study with  sequence by R o w e l l  misconceptions, (1973)  report  to  is  these  alternate  the p r i n c i p l e o f  considered  This  learning tasks  the o p p o r t u n i t y to  understanding  extinction,  instruction.  students'  (1969)  also  instruction.  investigate  C o l e a n d Raven  students  to  t o u n c o v e r and a n a l y z e  frameworks,  been used  resistant  very  i n students'  was evidence thinking,  -55-  and a t  times  illustrated  produced o n l y c o n f u s i o n . the  communication d i f f i c u l t i e s  and s t u d e n t s when e a c h e n t e r interpretive  seeing  "a  n o t as  an a c t i v e  student  understanding  results as  (see  science  teaching  will  approach.  is  it  last  is  practice  thought  that  difficulties  apparent  Education  of  the.Educational  u n d e r l y i n g approach  claims;  to  1973;  shown t h a t  this  behind t h i s  the  the  than that  and s c i e n t i s t s  in particular.  the  student  scientist,  inquiry  regularities can  But research  of  Erickson  the  methodology  subtle  see  in  approach  method i s  H a w k i n s , 1978;  students  perspectives  This  the  used  c o u l d match  b y an i n q u i r i n g s t u d e n t .  have  in  was m e n t i o n e d t h a t  program,  and  persist.  two d e c a d e s .  very different  that:  different  1978)  by e m p h a s i z i n g  (Shulman & T a m i r ,  1978)  with  teacher  (Hawkins,  also  it  The r a t i o n a l e  which are  "discovered"  between  in  i n s t r u c t i o n that  a process-oriented  nature,  )  o f making s c i e n t i f i c  inquiry that  2.3  been the  i n the  a method o f  techniques  Kelly,  students'  topics  Section  a p p r o a c h has  for  also  we p e r s i s t  system"  introductory section  inquiry  of  a filing  if  C u r r i c u l u m Problem i n Science  Perspective  within  show t h a t  organism,  certain  In the  looks  into dialogue  study  frames.  The a b o v e  2.3.2  This  (1975)  and  Deadman and  the w o r l d adults  be  in has  from general, said  -56Thus perhaps the r e a l u t i l i t y o f the ' c h i l d as a s c i e n t i s t ' m e t h a p h o r i s n o t t h a t i t s u g g e s t s one c a n s i m p l i f y o r r e d u c e t h e complex body o f s c i e n t i f i c k n o w l e d g e and methods i n t o p e d a g o g i c a l l y d i g e s t i b l e p r o g r a m s , as h a s b e e n done t o date. Rather, i t i s to view both c h i l d and s c i e n t i s t as e n g a g e d i n t h e same b a s i c a c t i v i t y o f a t t e m p t i n g t o p e r c e i v e some sense of o r d e r i n t h e i r world — o n l y at d i f f e r e n t l e v e l s of a b s t r a c t i o n . (Erickson, 1975, p . 1 6 ) . The i m p l i c a t i o n s ment o f this  science  author.  ought  to  this  position  c u r r i c u l a seems c l e a r  The a p p r o a c h  proceed  from the  mena a s s o c i a t e d w i t h powerful  of  the  is  that  simple  for  develop-  t o E r i c k s o n and  to  c u r r i c u l u m development  and more c o n c r e t e  student's world  and s o p h i s t i c a t e d  the  conceptions  pheno-  toward the  of  the  more  expert's  world. C u r r i c u l u m development typically  i n science  for  teaching  a topic  and t h e n t r i e d o u t on s t u d e n t s . or otherwise,  referred to  the  This  or to  the  alternative as  the  ones a r e  sought.  curriculum-to-students  is  the  latter  investigated  building-up process  and t e a c h i n g  are  to  their  are  success,  then accepted This  could  mode i n  i n which  and t h e n ,  explored  or  be  contrast 1978).  students'  through a  i n which development,  c o m b i n e d , ways  first,  (Deadman & K e l l y ,  approach, first  materials,  established  According  s t u d e n t s - t o - c u r r i c u l u m mode  understanding  are  c u r r i c u l u m packages are  author prefers  gradual  has  f o l l o w e d a p r o c e d u r e whereby t h e c o n t e n t ,  and s t r a t e g i e s  modified,  subjects  research,  directly  with  -57-  th e students  b y w h i c h t h e i r u n d e r s t a n d i n g c a n be  Deadman and K e l l y to-students of  courses  ment o f  (19_781  mode i s but  it  suitable  is  individual  think  less  that:  for  wholly  why i t  suitable  quantities  ways  for  dealing with  special  the  in-depth develop-  least so  one o f  far  the  to  the  determine  topic of  characteristics.  why t h i s  c u r r i c u l u m mode t o  involved.  author  is  favouring  attempt  to  overcome  learning vectorial  concepts.  the  vector  The d e v e l o p -  employed have n o t p r o v i d e d s u f f i c i e n t  t h e t e a c h i n g and l e a r n i n g p r o b l e m s  the reason  in  or at  n o t been p o s s i b l e  and t h e i r  ment m e t h o d s of  has  for  aspects  (p. 8 )  T h i s may be t h e r e a s o n , reasons,  "The c u r r l c u l u m -  d e v e l o p i n g the broad  suitable  courses",  increased.  details  This  is  also  students-to-  some o f  the  problems  -58CHAPTER THREE  TASKS, INTERVIEW PROTOCOL, AND DATA COLLECTION OF PHASE ONE 3.0  Introduction This  chapter  the procedures first is  phase.  p e r t i n e n t to First,  presented,  which  to c a r r y out the rational  task  a network o f  broad  relationships  3.1  of  the  the  The P i l o t  the r e s e a r c h .  the  study  suitable The use  which  concepts.  These networks  i n the  these  study,  with  the  in  show  characteristics is  a d e s c r i p t i o n of  their respective  ends by d e s c r i b i n g the d a t a  format of  of  shows  the v e c t o r c h a r a c t e r i s t i c s  Following  The c h a p t e r  methodology, istics  used  the p i l o t  t a s k s were  among t h e v e c t o r  i n T a b l e 1.1  t h e two t a s k s  of  the  and  c o l l e c t i o n of  explained next,  presenting  of  tocols.  part is  instruments  report of  showed t h a t  first  terms  the  the data  a short  analysis  t h e method o f  listed  deals with  pro-  collection  i n t e r v i e w , and t h e  character-  s u b j e c t s who were i n t e r v i e w e d . Study  According  to Pines  et  al:  e v e r y r e s e a r c h program which i s to i n v o l v e the administration of c l i n i c a l interviews s h o u l d have a p i l o t s t u d y i n w h i c h t h e i n t e r v i e w c a n be d e v e l o p e d . The p i l o t i n t e r v i e w s h o u l d be c a r r i e d o u t i n a p o p u l a t i o n s i m i l a r i n a l l r e s p e c t s to the s u b j e c t s o f the research study. (Pines et al,1978,p.8). The determine are  main purposes if  suitable  the to  of  the p i l o t  interview tasks attain  the  goals  study were:  t o be u s e d of  the  i n the  study;  (ii)  (i)  to  study to  -59determine the  if  target  the  protocol questions  population  and i m p r o v e d u n t i l become  (if  not,  they are  familiar with  t h e y s h o u l d be  sufficiently  the r e l e v a n t  t a r g e t p o p u l a t i o n s u c h as  c a n be u n d e r s t o o d  by  paraphrased  clear);  (iii)  characteristics  of  to the  l a n g u a g e a b i l i t y and m a n i p u l a t i o n  dexterity. During eight were  t h e p r e l i m i n a r y work f o r  children  of  various  ages  (ranging  i n t e r v i e w e d which r e s u l t e d  t h a t were e v e n t u a l l y used  questions  about  .eventually  emerged  appealing  and c l e a r  task for for  8 mm f i l m - l o o p and l a r g e were u s e d . the  study  the  responded  to  the  problem because the models their  hands  to p o i n t to  imaginary or r e a l Initially in  understanding fact  lines the  vector  characteristics  form.  These q u e s t i o n s  question.  that  two o r  the  was  three.  cm)  i n terms  Even w i t h  sufficiently  of  a  objects  used,  geometry  students  experienced  Table  1.1)  were r e v i s e d  reduced,  which  c a r d b o a r d model  to  use  and drew  some  understood  this  were  presented  several  times  the  questions  after-pilot  difficulty  T h i s was  the o r i g i n a l questions  students  the  paper models were  the q u e s t i o n s .  (see  and  models.  L i k e w i s e t h e number o f  characteristic to  on the  a few o f  the  evident that  the v a r i o u s  i n both  p a r t i c u l a r l y when an  cm x 100  tasks  study  t o be  p e r m i t t e d the  subjects  due t o  was  seemed  P r e v i o u s l y , when s m a l l e r  subjects  17)  The two t a s k s  subjects;  (100  f r o m 13 t o  i n the  situations.  the  study,  i n numerous c h a n g e s  the tasks  the  the p i l o t  likely  just in  question  until  intent of on e a c h  testing,  reduction i n  the  it the  vector  from  five  questions,  -60-  however, i t s t i l l  seemed f e a s i b l e i n the p i l o t - s t u d y  to  determine i f the s u b j e c t h e l d c o n s i s t e n t p r e c o n c e p t i o n s f o r a p a r t i c u l a r v e c t o r c h a r a c t e r i s t i c by examining responses  to the two  or t h r e e r e l a t e d q u e s t i o n s .  their  Further-  more, the fewer q u e s t i o n s made i t p o s s i b l e to conduct  the  i n t e r v i e w i n 45 to 50 minutes. Three of the 8 Ss of the f i n a l phase o f the study were younger than 15 years i n grade 9 ) .  They gave l i t t l e  (2Ss were i n grade 8 and  thought  c h a r a c t e r i s t i c s i n c l u d e d i n task two, d i f f i c u l t y both i n understanding .and i n e x p r e s s i n g t h e i r i d e a s . i t was  to the v e c t o r demonstrated g r e a t  the q u e s t i o n of task  two  Based on these o b s e r v a t i o n s ,  Furthermore, the grade ten students have not  been f o r m a l l y taught v e c t o r s and  so i t enabled  focus on the k i n d s of p r e c o n c e p t i o n s  the study  students possess  v e c t o r q u a n t i t i e s b e f o r e they r e c e i v e any  instruction  to  about on  topic. In summary, the p i l o t study was  the t a s k s and the i n t e r v i e w p r o t o c o l . the t a s k s and  u s e f u l to improve I t a l s o showed t h a t  the use o f i n d i v i d u a l i n t e r v i e w s would  s u i t a b l e to address  3.2  IS  decided t o o n l y choose s u b j e c t s i n grade 10 f o r the  a c t u a l study.  the  pilot  the major q u e s t i o n s posed i n the  be study.  R a t i o n a l Task A n a l y s i s of P h y s i c a l Phenomena  The T a b l e 1.1  v e c t o r c h a r a c t e r i s t i c s t h a t were p r e s e n t e d i n i n Chapter  One  were o b t a i n e d by a n a l y z i n g v a r i o u s  p h y s i c a l s i t u a t i o n s , p a r t of which can be r e p r e s e n t e d  by  -61-  the  three vector quantities  study. of  After  the s i t u a t i o n s ,  properties for  that  that  that  some o f  reason  the most r e l e v a n t k i n e m a t i c  the formal  definitions generally  the uncovered c h a r a c t e r i s t i c s  t h e y were c a l l e d i m p l i c i t  it  intuitive  w o u l d be e a s i e r understanding of  which c h a r a c t e r i s t i c s students;  to  and i f  about v e c t o r  were n o t  vector  characteristics.  textbooks  Physics: Health, : Giancoli, of  b u t t h e s e b o o k s do n o t p r e s e n t characteristics  presented  review of  B.C. schools.  of  students  had any  characteristics; to had  physics  l S S O h - d e f j ne . f o r m a l l y t h e several  vector  c l e a r l y the  textbooks it  c a n be  among t h e m ,  vector quantities  p h y s i c a l phenomena.  P S S C , 1980;  M a c n a u g h t o n , and M a r t i n -  some v e c t o r q u a n t i t i e s  that  (e.g.,  i n Chapter One.  I n summary,  showing r e l a t i o n s h i p s  perties  character-  seemed t o be most t r o u b l e s o m e  main v e c t o r c h a r a c t e r i s t i c s  emphasizing  the  t h e r e were any o t h e r n o t i o n s s t u d e n t s  1979; P h y s i c s  them p r e s e n t  these  ex-  quantities.  Fundamentals of  an e x t e n s i v e  included  T h i s was  out i f  any o f  Many s c h o o l p h y s i c s  dale,  find  the  was d e t e r m i n e d  was h o p e d t h a t b y c l e a r l y s t i p u l a t i n g t h e s e  istics  in  It  i n t r o d u c e d i n the d e f i n i t i o n s .  that  aspects  t h e s e a s p e c t s were t h e n c o m p a r e d t o  particular vector quantity.  plicitly  It  finding  d e a l t w i t h - i n ;£he\ p r e s e n t  quantities,  essential This  author d i d  currently said  that  used all  of  i n i s o l a t i o n , without  and w i t h o u t s t r o n g l y represent  specific  pro-  The  task o f a n a l y z i n g  i n d e t a i l the v e c t o r i a l  . c h a r a c t e r i s t i c s o f the t h r e e p h y s i c a l q u a n t i t i e s was c a r r i e d out w i t h the use o f R a t i o n a l Task A n a l y s i s . Resnick says t h a t : R a t i o n a l task a n a l y s i s can be d e f i n e d as an attempt t o s p e c i f y p r o c e s s e s o r procedures t h a t would be used i n h i g h l y e f f i c i e n t performance o f some t a s k s . The r e s u l t i s a d e t a i l e d d e s c r i p t i o n o f an " i d e a l i z e d " performance - one t h a t s o l v e s the problem i n minimal moves, does l i t t l e ' b a c k - t r a c k i n g , makes few o r no e r r o r s . T y p i c a l l y a R a t i o n a l Task A n a l y s i s i s d e r i v e d from the s t r u c t u r e of the s u b j e c t matter and makes few e x p l i c i t assumptions about the l i m i t a t i o n s o f human memory c a p a c i t y o r p e r c e p t u a l encoding p r o c e s s . 1  (Resnick, 1 9 7 6 , p . 6 5 ) .  Usually,  the h i g h l y e f f i c i e n t performance and the  d e r i v a t i o n of properties  from the s t r u c t u r e of the s u b j e c t  matter a r e c a r r i e d out by an e x p e r t the procedure used i n the p r e s e n t  i n the f i e l d .  T h i s was  study.  A f t e r s e l e c t i n g t h r e e o f the e a r l i e s t q u a n t i t i e s g e n e r a l l y taught i n h i g h  vector  school__physics  (vector  p o s i t i o n , displacement, and v e l o c i t y ) , an a n a l y s i s was p e r formed t o determine the p h y s i c a l c h a r a c t e r i s t i c s t h a t they represent.  That i s , what a s p e c t s o f p h y s i c a l r e a l i t y do the  three q u a n t i t i e s deal with.  Both common and d i f f e r e n t  elements i n these q u a n t i t i e s were examined along w i t h mathematical a b s t r a c t i o n s .  their  A t the end o f t h i s a n a l y s i s , a  s e t o f e s s e n t i a l v e c t o r c h a r a c t e r i s t i c s o f the three p h y s i c a l q u a n t i t i e s was produced. three u n i v e r s i t y physics professors criticism.  T h i s s e t was shown t o f o r comments and  P r i v a t e d i s c u s s i o n s were h e l d w i t h each one  of the professors,  and notes were taken o f suggested changes  and i m p r o v e m e n t s . again was  with  the  reached  A revised  set  was p r o d u c e d ,  same p r o f e s s o r s .  and a f i n a l  set  A t the  of vector  and  end an  dicusssed  agreement  characteristics  was  produced. The w h o l e p r o c e s s Task A n a l y s i s . (see  Table  question  study, of  which i s  these  was  among t h e  different  three  was  decided  to  that  istics  as  four  depict  the v a r i o u s  properties. istics overall  of  networks  any v e c t o r  framework  works d e p i c t e a c h one o f  of  for  three  list  the  the  of  students'  implicit  and t h e i r which  properties. relationships  vector  the other  This  and  the g e n e r a l  their  character-  network p r o v i d e s  three.  characteristics  physical quantities  character-  diagramatically  among t h e c o n c e p t s  quantity.  vector  relationships  structural  newtwork shows  the vector  for  and t h e i r  concepts,  relations  The f i r s t  the  concepts  among t h e s e p h y s i c a l q u a n t i t i e s  research  Quantities  show c l e a r l y  present  Rational  characteristics  first  search  of  characteristics.  However,  felt  implicit the  the  the Three V e c t o r  does n o t  it  to  Networks o f  it  a procedure  of  implicit  characteristics  So,  list  directly relates  the  preconceptions  3.2.1  This derived  1.1)  of  constitutes  The o t h e r associated selected  for  the  three  net-  with this  study. The r a t i o n a l e primarily (1977)  on the  and t h e  for  semantic  active  preparing networks  structural  these networks of  is  based  L i n d s a y and Norman  networks  o f Norman and  Figure  3.1  Network o f a K i n e m a t i c Physical  kinematic s c a l a r quantity composition vectors  of  Vector  Quantity  Entity  kinematic vector quantity | an i n s t a n t a n e o u s l o c a t i o n i s independent of others l o c .  parallelogram method  direction  prop.  independent components  independence o f magnitudes  frame o f reference f  real number  units  r»P  reference point  Key  o f R e l a t i o n a l Terms:  sub = has a ( n o n - i n t e r s e c t i n g ) subclass" charac.= has as a c h a r a c t e r i s t i c o f o b e y i n g that operation  independence of d i r e c t i o n s  pr«p  vector subtraction  one o r more simultaneous components  p r o p . = has a s a p r o p e r t y cond. = needs c e r t a i n c o n d i t i o n s t o make q u a n t i t a t i v e  I CTl  I  Figure  3.2  Network o f V e c t o r P o s i t i o n  Physical  kinematic scalar quantity  (V.P.)  Entity  k i nematic vector quantity Sub  ck&rfrc ~~ v e c t o r subtraction  t r i a n g l e method  composition o f two V.P. an i n s t a n t a n e o u s location i s i n dependent o f o t h e r locations  r e a l number  vector position prop  prop  direction  magnitude  l  units  frame o f reference /Top reference point  Note  ;  Meaning o f r e l a t i o n a l terihd same as i n F i g u r e 3.1  r e a l number  units  I  Figure  3.3  Network o f V e c t o r Displacement  Physical  kinematic scalar quantity  (or Change o f L o c a t i o n )  Entity  kinematic vector quantity  tub displacement o f change o f l o c a t i o n  vector addition  composition of displacements  t r :Langle me1:hod  independent path  from  magnitude  direction  1  and  r e a l number  units  reference  Notet  prop  frame o reference  point  Meaning o f r e l a t i o n a l terms same as i n F i g u r e  3.1  real  number  Network o f V e c t o r Average V e l o c i t y  Physical  Entity  kinematic vector quantity  kinematic scalar quantity  velocity instantaneous velocity composition velocities S0& p a r a i l e logramj v e c t o r method addition  average  velocity  prop one o r more i n s t a n t a n e o u s components magnitude  units'  direction  independent components  sofc independence o f magnitudes  r e a l number  frame o f reference  prop  units  l (Ti  I  reference body stationary body  Note:  independence of d i r e c t i o n s  Meaning o f r e l a t i o n a l terms same as i n F i g u r e 3.1  moving r e f e r e n c e body  -68Rumelhart  (1975).  The t a s k s p r o p o s e d on a l l  the vector  depicted made t o tion  i n the  for  quantities  use  and t h e i r  f o u r networks o f  show t h e p r e c i s e  i n the4stud'y are  characteristics  concepts.  correspondence  c o n t a i n e d i n t h e networks and t h e  based  Efforts  were  between t h e protocols  of  informathe  tasks. The diagrams  and d i s c u s s i o n s  are  provided i n this  for  any k i n e m a t i c v e c t o r  all  vector  this  section.  The l i n e s  mathematical  characteristics. vector lated.  for  the v e c t o r  i n Table  the network to  position,  only  1.1  are  3.4  sub,  that  are  relationships (i.e.,  do n o t  indicate  3.2.2  The P h y s i c s  Approach  The p h y s i c s  approach represents  among  shown i n boxes,  define  the  are  vector two  explained  cliarac,  prop, the  the v e c t o r displacement,  average  network  i n some way  show r e s p e c t i v e l y  vector  reat  and  networks and  velocity.  c u r r e n t knowledge  (i.e.,  a p p l i e d mathematics, methodologies)  the  b e t w e e n t h e two  c o n n e c t e d by i t  Some more s p e c i f i c  3.2  depicts  characteristics,  The l i n e s  Figures  3.1  networks  The r e l a t i o n s h i p s  relationships  characteristics  the bottom o f cond.).  listed  four  connecting the d i f f e r e n t  which c o n t a i n the v e c t o r explicit  the  Figure  quantity.  characteristics  network.  of  used  theories,  research  the  physicists'  concepts,  methodologies,  to d e a l w i t h e x p e r i m e n t a l  definitions, problem s o l v i n g situations  -69-  similar •  to  The p h y s i c i s t s '  approach cepts  t h e ones d e p i c t e d  -  is  physics  3.1  approach the  formal  definitions  of  reader  are  3.4).  that  characteristics  The r e a s o n for  the the  istic.  concepts,  explanations  3 ^3  4.1.3  using  used  of  this  that  T h e Two T a s k s  of  of  vector  the  the  defi-  aspects  actually  implicit  out  re-  vector  for  approach  i n the  categorization  each v e c t o r  a vector  characterthe  characteristic  shown i n t h e  is  was  networks  presented  with  of Further  in  Study  for  the present  characteristics  position  (see  to  Four).  One i n c l u d e d t h e v e c t o r  displacement  quantities;  vector citaratrt^rlstic.  Two t a s k s were c r e a t e d  network o f  it  categorization  (Chapter  con-  the  by s i m p l y c o m p a r i n g  of  current view,  treating  of  important  important  list  identified  an i n f e r r e d r u l e  of  vector  the  physics  1.1).  of  a p p r o a c h was  the p h y s i c i s t ' s  Section  purpose  inferred rules This  is  The networks  form o f  study.  do n o t r e p r e s e n t  and l e a v e  Table  It  f o r d e f i n i n g the p h y s i c s  specific  content of  of  study.  the three  succint  (see  the  they represent  those networks  of  a structural  of  i n the networks  Then,  the concept being d e f i n e d .  present  of  to  usually  tasks  represented  i n the present  remind  nitions  the  c u r r e n t knowledge - t h e  partially  (Figures  as  Figure  (see  Figure  3.3).  study.  outlined in 3.2)  These  and t h e  Task the network  characteristics  -70-  correspond Task  to  items  one t h r o u g h f i v e  Two i n c l u d e d t h e v e c t o r  network o f  average v e l o c i t y  corresponded  to  items  11 and 12 were o n l y It as  was  possible  since  intuitive  through ten  that  i n the the  which  i n Table  1.1.  second  tasks  purpose  i n the  outlined  3.4),  s t u d y was  on l a k e s  and r i v e r s  one t o  In  a river  is  referred  in  also (Items  phase).  s h o u l d be as  some common e x p e r i e n c e s  C o l u m b i a one t a s k  the other  Figure  1.1.  of  the  the  related  subjects  search  n o t i o n s w h i c h may be p o s s e s s e d by t h e  Since boating British  characteristics (See  treated  thought  to  a major  six  i n Table  for  students.  very prevalent  to a lake  in  situation  and  situation.  summary t h e c r i t e r i a  used  for  task  selection  were: 1)  2)  that  the  most  of  that  t h e t a s k s must  vector 3)  that  setting  s h o u l d be  to  subjects, illustrate  characteristics,  to  familiar  tasks the  and  s h o u l d be o f  dramatically  the  finally some i n t e r e s t  or  student.  T a s k One  Briefly, follows: fishing asked  the  the  appeal  3.3.1  task  to  a  lake  the  situation  i n which there  i n a row-boat. imagine  simulated are  i n t a s k one was  a couple of  The s u b j e c t s b e i n g  herself/himself  standing  as  youngsters  interviewed on a d o c k  is for  the  -71-  purpose by the  of  keeping  couple.  who a r e  This  coming t o  where t o  track  locate  the  person  fish  the  of  is  later  good  a  8 mm f i l m  showing the  the  spots  some  has  to  found  friends tell  them  spots.  were u s e d  i n the  interviews:  (approximately  4 min.  long)  round t r i p  The f i l m was  for  and h e / s h e  fishing  loop  fishing  waiting  The f o l l o w i n g m a t e r i a l s -  good  of  a row-boat  produced e s p e c i a l l y  l o c a t i o n was D e e r l a k e ,  on a  for  the  Burnaby,  lake. study;  British  Columbia. -  a c a r d b o a r d model o f  the  -  a toy boat  t o y human  -  sheets of  paper  schematic  diagram  The c o n c r e t e interviewee model  to  the  the  the  lake  starting  is  location of  S on the d o c k ) ,  the  and s e v e r a l  a  was p r e s e n t e d  l o o p and t h e  shown i n F i g u r e  l o c a t i o n of  depicting  lake.  problem s i t u a t i o n film  lm),  figures,  cm b y 21 cm)  of  using both the  the permanent  and  (27  (lm b y  accompany an o r a l d e s c r i p t i o n .  diagram of shows  and t h r e e  lake  to  the  cardboard  A schematic 3.5.  This  the row-boat interviewee  good f i s h i n g  (the  diagram wharf),  (represented spots  by  (P^,P2,P2»  P^). The t a s k  how h e / s h e locations  would d e s c r i b e of  the  T h i s was subjects'  consisted  fishing judged  are  the  t r i p of  the  subject- on  the row-boat  and  the  spots.  t o be a s u i t a b l e  preconceptions  characteristics  i n questioning  about  vector  task  to  position,  i n c l u d e d i n the network o f  find  out  whose Figure  3.2.  -72Figure 3 . 5 D i a g r a m o f t h e l a k e , i n c l u d i n g t h e d o c k (where t h e s u b j e c t S s t a n d s ) , the w h a r f , the row-boat, the f i s h i n g s p o t s (P , P „ , P , , and P ) , and a t r e e  P r e c o n c e p t i o n s about  the  were l o o k e d f o r :  the need f o r  an o b j e c t ; a precise kind of tative  (ii)  (i)  the need f o r  l o c a t i o n of  coordinates  sidered  of  part  the of  about cluded  (iii)  aspect  of  (this  this  p o i n t to  reference is  network  to  provide  shown t h r o u g h  s u b j e c t w h i c h c a n be the need to p r o v i d e  an o b j e c t ,  locate  this  the  quali-  a-  c a n be  con-  (ii).  also  some a s p e c t s o f  of  a reference  frame  used by the  l o c a t i o n of  T a s k one was  a  an o b j e c t ,  or q u a n t i t a t i v e ) ;  direction  following aspects  used  to  find  out  vector displacement,  i n the network of  Figure  3.3.  preconceptions w h i c h were i n -  Preconceptions  about  -73-  the  following aspects of  (i)  the need f o r  a moving o b j e c t straight  line  magnitude a  a way t o  reference  for  a frame  for  an o p e r a t i o n t o  (addition of  by the o b j e c t  of  to  change o f duce the of  out  if  ception  3.3.2  subjects of  suitable  subjects  a s p e c t was  at  t h e s e ages  and  the  (iii)  This see to  (v)  of  the  the  study. still  with  need need  location realize followed  need  of  an  for object  resultant  aspect Figure  find  its  the  the path  locations  about  is  also  propart .  3.2.  out the  under-  perspective  taken.  It  as  was  another  vector  included to  h a v e an e g o c e n t r i c  Protocol of  T a s k One  characteristic.  Sentences  two q u e s t i o n s  for  i n parenthesis  find  con-  perspective.  T h e r e were an a v e r a g e o f vector  (ii)  the need t o  not considered  i n the present  —  v e c t o r p o s i t i o n to  vector position,  adolescent  perspective  of  a  l i n e forms  changes  point with  displacement).  also  locations  independent of  two d i f f e r e n t  l o c a t i o n of  point;  (iv)  for:  length of  direction);  displacements);  (subtraction  resultant  characteristic  of  straight  consecutive  a chosen r e f e r e n c e  T a s k one was  This  relate  relate  location  of  the  and r e f e r e n c e  location is  the network o f  standing  that  b e t w e e n two l o c a t i o n s ;  an o p e r a t i o n t o relative  changes  two c o n s e c u t i v e  line — its  reference  vector  a change  looked  ( w h i c h c a n be g i v e n b y t h e  j o i n i n g the  of  n e t w o r k were  describe  — and t h e a n g l e  selected  that  this  each  referred  -74t o what t h e  interviewer  (I)  was d o i n g d u r i n g t h e  The f i l m - l o o p and t h e c a r d b o a r d m o d e l were u s e d  interview. in  the  i n t r o d u c t i o n ; furthermore  the  s u b j e c t s were e n c o u r a g e d  t o make u s e  of  with  the  the  explain  appropriate  times  schematic  i n the  diagram of  lake  to  the  sheets their  The i n t r o d u c t i o n o f  t h e p r o t o c o l , g i v e n b e l o w , was  almost  each o f  verbatim to  Introduction of on a d o c k o f youngsters  Protocol:  a calm lake  getting  S i n c e you are  coming l a t e r  the  t r i p of  the  to  that  you are  go f i s h i n g  fish,  for  in a  a couple of  you d e c i d e You a r e  see  trip  i n the' f i l m ;  a  speeded  the camera  same d o c k where y o u a r e  t h a t what y o u w i l l would  see  if  see  friends  to c a r e f u l l y  film.  questions?  If  of  the Is  the  f i l m was row-boat  started  and e n d e d a t  fishing  spots  film.  While  describing  the  the  i n Figure  f i l m was  trip  in its  so  Here i s  a  what y o u  . . . Do y o u h a v e  will any  film".  round t r i p . the l o c a t i o n s  several  The of  trip the  3 . 5 were shown i n  b e i n g shown, t h e  a n d was  p u t on  t o what y o u  shown, w h i c h c o n t a i n s  the wharf,  depicted  the  fishing  standing  corresponds  everything clear?  observe  interested  this  lake which represents  n o t we c a n now see  The e n t i r e locations  film  who  f i l m was  t o be  of  c a t c h more  y o u a c t u a l l y w e r e on t h e d o c k .  c a r d b o a r d model o f i n the  up v e r s i o n o f shooting the  i n the  a couple  particularly  supposed  standing  row-boat.  r e m e m b e r i n g t h e p l a c e s where t h e y seem t o You w i l l  see  to  repeated  interviewed.  and y o u s u d d e n l y see  row-boat.  fish.  the  ready  subjects  "Imagine  waiting there  are  in  the  views.  the  interviewer  p o i n t i n g out the  locations  was of  the  fishing  interviewer  spots.  Immediately  made u s e  of  after  the  film  ended,  the c a r d b o a r d model to  begin  the the  interview. The q u e s t i o n s characteristics  Questions  discussed  Question  (Items  You a r e  less  (I  s p o t on t h e  1 and 2 o f  spot  located  spot to Question a)  below.  Bodies  the  (RPS)  (FR)  and as  y o u saw i n t h e  i n the  was more o r  corresponding  P^ o n F i g u r e  l o c a t i o n of  film  3.5).  the  first  fishing  friends?  2; (If  given  found by the y o u n g s t e r s  cardboard model;  your  vector  LIVC)  the toy boat  How w o u l d y o u d e s c r i b e  the  Stationary  Reference  h e r e on the d o c k ,  fishing  here  Point for  T a k e n , and Frame o f  1;  first  t o e a c h one o f  i n t a s k one a r e  About Reference  Perspective  the  related  (Perspective S uses  taken)  a particular  body o r p l a c e  as  a  reference  point) C o u l d you d e s c r i b e spot b)  (If Is  Question  from o t h e r S describes  the  places the  different  when i t  3:  2 of  C If  S has  is  as  well?  location  the d e s c r i p t i o n of  (Item  l o c a t i o n of  this  that  What  the  done f r o m o t h e r  used  no q u a n t i t y  same o r  places?  or only one,  o r d i r e c t i o n , when a n s w e r i n g  You h a v e  just spot.  places)  LIVC)  distance  fishing  places?  from o t h e r  location  fishing  described  the  question  l o c a t i o n of  How c o u l d y o u make t h e  either  the  1)  first  description  -76-  of  that  precise Question  4:  location  (first  for  friends?  (Items  The b o a t fishing  your  1 and 2 o f  spot  (I  you d e s c r i b e  Question  5:  at  location  this  first  Question a)  About 6;  distance  spot  t o you are  (Perspective  c)  (I  located  t h e same  the  distance  to your  in her/his  spot  two  friends?  description)  t o you and  almost  the  the  from  same.  How  locations?  taken and Item 2 o f (on t h e d o c k ) ,  LIVC).  c o u l d you  the path followed  by  boat?  Please  draw t h a t  (After  S has  is  fishing  Perspectives  approximately describe  b)  How w o u l d  second  f i s h i n g spot  from t h e l a s t  From where y o u a r e  the  that  How w o u l d y o u d e s c r i b e  c o u l d you d i s t i n g u i s h these  Questions  second  LIVC)  (a)  last  S uses o n l y  The d i s t a n c e the  this  a p o i n t , P^, about  P^ w a s ) . of  in  F i g u r e 3.5)  n  f i s h i n g s p o t was a b o u t h e r e  from S as  (If  o  this  friends?  (Items 1 and 2 o f  toy boat  (b)  P^  the l o c a t i o n of  to your  Another  i t went t o  l o c a t e d the toy boat  spot;  more  LIVC)  continued moving,  corresponding  spot  f i s h i n g spot)  the a c t u a l  p a t h i n t h i s map o f  drawn t h e p a t h )  the  Do y o u t h i n k  this  path?  What do y o u n e e d t o know i n o r d e r t o h a v e better  lake  approximation of  a  t h e p a t h t a k e n by t h e  boat?  -77d)  A friend  o f yours  p a t h he s e e s drawings,  i s by t h e t r e e  and draws t h e  f o r t h e b o a t , y o u compare  yours and h i s .  both  Do y o u e x p e c t t o f i n d  them t h e same o r d i f f e r e n t ? Q u e s t i o n s About Question  7:  Displacements  (Item  3 o f LIVC)  The i n i t i a l  location or starting  was h e r e b e s i d e t h e w h a r f has  h a d a c h a n g e o f l o c a t i o n when  point you  (PQ) t o t h e f i r s t  describe  respect  fishing  t o the s t a r t i n g  spot?  i t moved f r o m t h e s t a r t i n g  spot  (P^).  (I moves t h e b o a t  respect to the f i r s t 8:  (If  spot with  ( I f S uses d i s t a n c e  i n her/his  fishing  from P ^ t o P ^ )  o f t h e second  fishing  spot t o the  How  c o u l d you  spot with  one?  (Item 3 o f LIVC) S uses  two l o c a t i o n s ) . location  could  t h e two s p o t s ?  describe the location  Question  (a) How  fishing  (b) T h e n , t h e b o a t moved f r o m t h e f i r s t spot  the boat  Is that the distance along the path followed  t h e b o a t between  second  of the boat  can say t h a t  the l o c a t i o n o f the f i r s t  description). by  ( P ^ ) . We  point  the length o f the s t r a i g h t Does y o u r d e s c r i p t i o n  say a n y t h i n g about  b e t w e e n t h e two  the path  line  between  f o r a change o f  f o l l o w e d by t h e b o a t  locations?  Q u e s t i o n s About A d d i t i o n o f Displacements Question  9;  ( I t e m 4 o f LIVC)  Suppose you have e s t i m a t e d t h e d i s t a n c e s from t h e  -78starting 100m,  point  (P Q )  to  and f r o m t h e  If  your f r e i n d s  to  go d i r e c t l y t o  Question  10:  the  first  spot  were a t the  (Item  tell the  spot  t.o t h e  your f r i e n d s  second  4 of  third  Questions Question  to  the  also  to  the d i s t a n c e l p o m.  one t o be  11:  1  (Item  5 of  of Vector  them  from  the  How c o u l d y o u starting  point  to  were a t  on the  the  12:  (Item  first  the  first  the  standing  300 m.  c o u l d you t e l l  If  the d i s t a n c e  fishing  second spot.  second  5 of  s p o t t o be  continued with  task  two.  200  300 m.  the d i s t a n c e  your  them t o  the  third  were a t  the  f r o m where  fishing first  the t h i r d  spot  s p o t . How spot?  f i n i s h i n g w i t h t a s k one t h e The i n t e r v i e w e r t r i e d  were now u s i n g  a motor-boat  interview,  t o make  smooth t r a n s i t i o n b e t w e e n t h e two t a s k s b y s a y i n g same two f r i e n d s  m, and If  How c o u l d y o u t e l l  go d i r e c t l y t o  after  t o be  LIVC)  your f r i e n d s  Immediately  spot  f r o m where  spot?  on t h e dock t o  them t o  Position  LIVC)  Suppose you a l s o e s t i m a t e d  t o be  150 m.  spot?  estimated  About S u b t r a c t i o n  go d i r e c t l y t o  you are  be  s e c o n d one t o be  go d i r e c t l y f r o m t h e  from t h e dock t o  Question  to  spot?  standing  friends  (Pj_)  how c o u l d y o u t e l l  Suppose you have e s t i m a t e d you a r e  spot  LIVC)  third to  fishing  the wharf,  Suppose you a l s o second  first  that  on t h e  a the  river.  -79-  3 ., 3 - 3  T a s k Two  The s i t u a t i o n following: people, to  a river  simulated i n task  on w h i c h t h e r e  a p e r s o n on a b r i d g e ,  imagine t h a t he/she  movement o f  the  subject  boat's  and a c r o s s was a s k e d  (again  the  a motor-boat with the  subject  i s the p e r s o n who i s  the motor-boat,  the c u r r e n t ,  is  two was  is  the r i v e r .  to describe  asked  observing  w h i c h moves a g a i n s t During the  interview  and e x p l a i n t h e m o t o r -  movements.  subjects -  u s i n g the  a 8 mm f i l m  following  loop  on t h e r i v e r . T h i s Harvard -  Project  of  the  -  a: t o y b o a t  -  sheets  of  schematic  4 min.  movements o f  Physics  the  used.  subject,  would d e s c r i b e a  suitable  and t h r e e  bridge  i n c l u d e s one d o c k o n e a c h  paper  (27  diagram of  t o y human  figures,  cm b y 21 cm) the r i v e r  depicting a  (see  the  which a semi-structured  Figure  3.6).  presented p r o t o c o l was  s u b j e c t s were q u e s t i o n e d o n how h e / s h e  t h e movements o f  task,  the  river,  after  As b e f o r e ,  motor-boat  and t h e  An i n t r o d u c t i o n e x p l a i n i n g t h e t a s k was the  long)  course,  the r i v e r  (1 b y 1 m ) , w h i c h a l s o side  —  i s a _ f i l m produced f o r  a c a r b b o a r d model o f  to  materials:  (approximately  showing the d i f f e r e n t  to  the  and w i t h  The c o n c r e t e p r o b l e m s i t u a t i o n was p r e s e n t e d the  two  the boat.  to determine the  T h i s was  j u d g e d t o be  s t u d e n t s ' preconceptions about  Figure  3.6  Diagram o f the r i v e r , i n c l u d i n g the b r i d g e (where t h e s u b j e c t s t a n d s ) , t h e two d o c k s , and t h e m o t o r - b o a t  the v e c t o r which are Figures this  Preconceptions  n e t w o r k were  a  to  ten  examined:  about (i)  an o b j e c t w i t h r e s p e c t  stationary  that  six  i n c l u d e d i n the network o f  3.4.  motion of it  characteristics  single  or  i n motion);  the to  listed  i n LIVC,  average v e l o c i t y  the  for  describing  any r e f e r e n c e  and t h e n e e d t o  of  in  following aspects  need  m o t i o n c a n be d e s c r i b e d  all  body,  of the (be  distinguish  differently  if  viewed  -81-  f r o m two d i f f e r e n t of  reference  other);  body i s  (ii)  the  (i.e., for  r e a l i z i n g that  i n t e r a c t i o n of  the c o m p o s i t i o n of  resultant  d u r i n g the  3.3.4  the  vector  quantities);  movement,  format  that  of  Introduction of  bridge  (I  task  show t h e  the with  so  same. the  decide  the  need  to  pro-  interact  them m a i n t a i n s  magnitude  and f i n a l l y  this  and  (v)  the  interview protocol  need a  the  "Imagine  subject  speed  Your stream,  now t h a t  you are  You are  standing  h e r e on  the  t h e p o i n t S on the c a r d b o a r d m o d e l ,  3.6) o b s e r v i n g y o u r f r i e n d s , The speed  of  the boat  due t o  friends  on t h e i r  ride will  and a l s o a c r o s s interesting  who  are  the motor cannot  of  i t w o u l d be  is  one.  o v e r a w i d e and c a l m r i v e r .  i n a motor-boat.  changed  of  Protocol;  diagrammed i n F i g u r e riding  (e.g.,  movements .  same t i m e .  The g e n e r a l  on a b r i d g e  the  single  two o r more movements c o u l d a f f e c t  T a s k Two  to  to  may be  (iv)  e a c h one o f  interaction;  Protocol of  similar  a moving o b j e c t  e v e n when two movements  r e a l i z i n g that  body a t  one  an a p p a r e n t  two s i m u l t a n e o u s  own p h y s i c a l c h a r a c t e r i s t i c s  direction) for  the  a single  if  motion with respect  r e a l i z i n g that  r e a l i z i n g that  duce its  of  for  (particularly  f o r m e d b y two o r more component movements;  the need f o r  result  bodies,  in relative  the need  m o t i o n c a n be (iii)  reference  the motor  to observe  remains  go a g a i n s t  the r i v e r .  be  and  As b e f o r e ,  and d e s c r i b e  you all  -82-  t h e moves  that  how t h e b o a t  the boat w i l l  m i g h t move;  make.  This  t h e camera  to  film w i l l  take  this  p l a c e d on t h e b r i d g e where y o u a r e  to  be  i n the f i l m  standing.  So what y o u w i l l  t o what y o u w o u l d We w i l l  see  part  what we h a v e  see of  seen  if  see  now and t h e n we c a n  using  this  cardboard model.  see  film.  Is  . . .  n o t we c a n s e e  If  In t h i s discussion  task  and e x p l a n a t i o n s  everything clear?  the  f i l m was  and q u e s t i o n s  various  The q u e s t i n s characteristics the  following  Questions  only  after  part  out  of  any  the  film".  each p a r t .  The  when  iri the i n t e r v i e w .  related  to  e a c h one o f task  the  two a r e  vector presented  in  sections.  if  (RPM)  Have y o u h a d a n y e x p e r i e n c e w i t h m o t o r -  or rivers?  The f i r s t water moving,  I am  parts;  referent  ( T h i s q u e s t i o n was a s k e d  the subject  h a d any p r e v i o u s  with motor-boats).  .stationary  Again,  A b o u t R e f e r e n c e Body f o r M o v i n g O b j e c t s  on l a k e s find  questions  discuss  . . . Do y o u h a v e  first  a concrete  corresponding to  Opening Q u e s t i o n : boats  as  bridge.  o f what y o u w i l l  shown i n two  were a s k e d  c a r d b o a r d m o d e l was u s e d discussing  the  to  corresponds  film  ideas  was  imagine y o u r s e l f  the  i n your  questions?  film  y o u a c t u a l l y were o n t h e  interested i n the  show y o u  part  to  experience '  of  the boat  buoys b e s i d e  the  f i l m was v i e w e d s h o w i n g , t h e  going against the  bridge.  the  s t r e a m and two  -83-  Question  13*:  In  (Item  the f i l m  (a)  With respect  not  understand t h i s  follows: (If  y o u saw t h e m o t o r - b o a t  shore).  the water?  b o a t was a l s o  question,  (If  was p a r a p h r a s e d  that  (b)  S  t h e b o a t was m o v i n g ? to  Was t h e b o a t m o v i n g w i t h so?  (If  stationary  respect  body t h e s u b j e c t  i t with  respect  different?  Why do y o u t h i n k  to  S says  that  the  the  (c)  the b r i d g e  mentions)  to the water  the  respect  m o v i n g r e l a t i v e t o the m o v i n g .water) the boat with  does  as  t h e b o a t was m o v i n g r e l a t i v e  movement o f  of  it  Why do y o u t h i n k  t h e movement o f  Questions  moving,  t o what was t h e b o a t m o v i n g ?  S mentions t h a t  whatever  LIVC)  How c o u l d y o u t e l l  bridge or to  6 of  and  Was (or  the  same o r  so?  A b o u t A n a l y s i s o f Components  (AC) and C o m p o s i t i o n  V e l o c i t i e s (CV)  Question a)  14:  (Items  7 and 8 o f  Imagine  that  against  the c u r r e n t .  that  the motor-boat i s (I  moving  upstream  moves t h e t o y b o a t  d i r e c t i o n , i n the cardboard model).  would you d e s c r i b e b)  LIVC)  What d i f f e r e n t b o a t when i t  its  motion  movements a r e  goes upstream?  *Numbering o f q u e s t i o n s task one.  f o r task  from t h e affecting What a r e  two c o n t i n u e d  in  How bridge? the  they?  on from  -84-  c)  I m a g i n e now t h a t downstream w i t h boat you  d)  describe  15;  its  motion  b o a t want t o c r o s s 1,  see  moves t h e toy-  from t h e  g o e s downstream?  time t h a t your the r i v e r .  F i g u r e 3.6)  bridge? the  "What a r e  they?  y  friends  They l e a v e  h e a d i n g t h a t way  t h e d o c k 1 and e x p l a i n s  How w o u l d  affecting  ( I t e m 7 and 8 o f LIVC)  Imagine t h i s  start  (I  What d i f f e r e n t movements a r e  Question  beside  the c u r r e n t  moving  i n the a p p r o p r i a t e d i r e c t i o n ) .  b o a t when i t  (dock  the motor-boat i s  i n the motorfrom t h i s (I  puts  dock  the  t h e way how t h e b o a t  boat  will  heading). a)  How w o u l d y o u d e s c r i b e bridge? me  Please,  the kind  cardboard b)  take  its  motion  the toy boat  o f movement y o u w i l l  model.  from  and show  see  river?  (I  hands S a s h e e t  s c h e m a t i c d i a g r a m on i t ) . follows  using  the  *  Now c o u l d y o u draw t h a t p a t h o n t h i s the  the  diagram of  o f paper w i t h  Why do y o u t h i n k  the it  t h e p a t h y o u h a v e drawn?  * Almost a t t h e end o f t h e data c o l l e c t i o n i n phase ono, a q u e s t i o n t h a t seemed a p p r o p r i a t e t o f i n d o u t s t u d e n t s ' c o n c e p t i o n s about t h e independence o f d i r e c t i o n o f components was s u g g e s t e d t h r o u g h t h e r e s p o n s e o f one s u b j e c t . The q u e s t i o n was p h r a s e d a s f o l l o w s : F o r t h e c a s e o f t h e b o a t g o i n g across the r i v e r , w i l l the boat reach t h e other side heading t h i s w a y 3 o r t h i s way^jk ? T h e i n t e r v i e w e r u s e d t h e t o y - b o a t a n d t h e c a r d b o a r d m o d e l t o show t h e t w o h e a d i n g w a y s . This q u e s t i o n was i n c l u d e d i m m e d i a t e l y , a f t e r q u e s t i o n 1 5 a , a n d was a s k e d of only 5 subjects.  -85c)  What d i f f e r e n t movements a r e m o t o r - b o a t when i t What a r e  Question a)  is  goes a c r o s s  the f i l m with  the motor-boat  saw t h e m o t o r - b o a t m o v i n g a c r o s s  i n the f i l m .  Was t h e movement o f  due t o t h e m o t o r o r due t o Suppose the  the speed o f  the  the c u r r e n t i s  as  y o u saw i t  i n the  17:  otherwise  (Item 10 o f  river  film?  the boat, i s  t h e c u r r e n t and t h e suggests  Are these  being affected  s p e e d "due t o  the e x i s t e n c e  of  t o g e t h e r o r one a f t e r your  the other?  by  the motor;  t h e two c o m -  movements, t h e movement o f  and t h e one due t o t h e m o t o r , a f f e c t i n g  (If  the  speed  LIVC)  S mentions t h a t  the I  ponents) .  Question  the  is  A b o u t S i m u l t a n e i t y o f Components  the speed o f  Questions  boat  5 k m / h r and  C o u l d you guess o r e s t i m a t e  t h e b o a t when i t was g o i n g a c r o s s  (If  plain  the  current?  of  Questions  the  speed p r o d u c e d by the motor on the b o a t  10 k m / h r .  boat  crossing  16:  river  Question  .the r i v e r ?  shown.  You j u s t  b)  the  they?  The p a r t o f the r i v e r  affecting  the  stream  the motion of  the  How c o u l d y o u e x -  responses?  A b o u t I n d e p e n d e n c e o f M a g n i t u d e s o f Components 18:  (Item 9 o f  S mentions that  LIVC) the boat  is  being  affected  -86-  simultaneously due t o you  the motor).  said  due t o the  by the  that  think  c u r r e n t and t h e  When t h e b o a t was  both the  speed o f  t h e b o a t due  changed because o f  the  to  just  speed  crossing  the  t h e c u r r e n t and t h e the b o a t .  t h e motor  speed o f  is  river,  speed  Do y o u t h i n k affected  the c u r r e n t ?  that  or  Why do y o u  so?  Questions Question  About Composition of 19:  (Item  Finally, with  8 of  suppose your f r i e n d s  the motor-boat  c o u l d do i t ?  3.3.5  that  and r e a c h d o c k  vector  you might  draw t h e  see  if  Is  there  related  issues  discussed  different  below.  number o f  characteristics, to  the  That i s ,  leave  any way t h e y  p a t h on t h i s  Three r e l e v a n t are  2.  identify  These  questions  (2)  and  characteristics.  rules  (3)  of  suggestion.  Protocol  to  the  issues for  occasionally,  inferred  characteristic,  sheet  they followed your  interview protocol questions vector  2.  Interview  question vector  dock  Further Discussion of  protocol of  want t o c r o s s  How?  C o u l d you p l e a s e paper  Velocities  LIVC)  f r o m d o c k 1 and end up a t  use  the  t h e m o t o r were a f f e c t i n g  speed o f  river  speed of  interview are:  some o f use  of  (1) the one  f o r more t h a n  one  a summary t a b l e o f  the  associated with  the  each of  the  -871. protocol (see  Each v e c t o r c h a r a c t e r i s t i c was a d d r e s s e d  Table 3.1).  nature tasks  of  chosen for  addition of  the  for  study.  this  to uncover the  situation  that  for  characteristic  displacements  two q u e s t i o n s trying  w i t h a v a r y i n g 'number 1 o f  The m a i n r e a s o n  the v e c t o r  i n the i n t e r v i e w  this  itself  For example,  is  due t o  the nature  a d d i t i o n of  subjects'  ideas with respect  a d d i n g two g i v e n  displacements  several  to  asked.  On t h e o t h e r h a n d ,  simultaneity characteristic  other  either boat  the subject  (in task  There are in  two)  sees at  questions,  were  t h e two v e l o c i t i e s  affecting  and b o t h a r e  In the t a s k s used  was no o t h e r way t o c h e c k f o r  this  vector  i n the  study  for  larger  number o f  vector  characteristic.  questions  At least  the s u b j e c t ' s  i n some i n s t a n c e s t o be a s k e d  four questions  preconception of  the other. covered there  characteristic.  The n a t u r e o f o t h e r v e c t o r c h a r a c t e r i s t i c s and t h e chosen  the  two v e l o c i t y c o m p o n e n t s ,  t h e same/time t3r.-Tme.-after t h e  t h e same q u e s t i o n .  2.  could  limitations  two q u e s t i o n s  o n l y t h e s e two p o s s i b i l i t i e s ,  the study  view  c o n s i d e r i n g the nature o f of  a  displacements.  H o w e v e r , due t o t h e t i m e  a s s o c i a t e d w i t h an i n t e r v i e w o n l y  least  — e a c h one  each time u s i n g a d i s t i n c t p a i r of d i s p l a c e m e n t s , have been u s e d .  of  i n t a s k one a l l o w e d f o r a t  To d e t e r m i n e w h e t h e r a s u b j e c t p o s s e s s e d a c o n s i s t e n t about  the  and on t h e  vector characteristic  requires  questions  allowed for about  a  tasks a  specific  were u s e d t o u n c o v e r  two v e c t o r  characteristics.  -88-  These q u e s t i o n s such t h a t t o be  task, in  the responses  assigned  the case  for  for  the  question  task  were b r o a d  two.  them p e r m i t t e d  two v e c t o r  4,  This  is  a legitimate  the preconceptions  questions  which  s h o u l d be more s p e c i f i c  questions for  1,  for  a b o u t .frame  4,  a n d 5;  of  the  3.  subject's  i n scope.  For  reference  e a c h one  questions  from the  the  it  is  presents  each o f  these  questions, instance,  about  from  suitable  reference by  3.  a list  of  i n t e r v i e w p r o t o c o l w h i c h was u s e d for  13d  through  were a l s o  response to question  inferred rules  of  were o b t a i n e d  preconceptions  The f o l l o w i n g t a b l e  was  and q u e s t i o n  obtained  these questions  possible  This  approach but  13d)  rules  T h e s e p r e c o n c e p t i o n s were t h e n c r o s s - c h e c k e d  analyzing  obtain  inferred  be c h e c k e d by u s i n g p r o v i n g  ascertaining  point.  Q s . l , 4:, 5 ,  characteristics.  and 5 i n t a s k o n e ;  broad  preconceptions  (i.e.,  introductory questions  1,  recommended t h a t  to  i n scope  the v e c t o r  to  character-  istics.  Table List  Vector  of  3.1  Interview P r o t o c o l Questions Vector Characteristic  Characteristic  Task One: 1. Reference Point for S t a t i o n a r y B o d i e s (RPS) 2. Frame o f R e f e r e n c e (FR) 3. D i s p l a c e m e n t (D) 4. A d d i t i o n of Displacements (AD)  for  each  Number o f Protocol  Questions  1,4,5 1,3,4,5a,5b,6c 7a,7b,8 9,10  in  -89-  Table  3.1  -  continued  Number  Vector Characteristic  of Questions  in  protocol 5.  Subtraction of Vectors P o s i t i o n (SVP) ( P e r s p e c t i v e Taken)  11,' 12 2a,  T a s k Two: Referennce P o i n t for 6. M o v i n g O b j e c t s (RPM) 7. A n a l y s i s o f Components (AC) 8. Composition of V e l o c i t i e s (CV) I ndependence of M a g n i 9. t u d e s o f Components (IMC) . 10. S i m u l t a n i e t y o f Components (SC)  3.4  Methods o f  6d  2b,  13a,  13b,  13c  14b,  14d,  15c  14a, 16b, 18  14c, 19  15a,  15b,  16a,  17  Data C o l l e c t i o n Methodology :  Clinical  Interview  Due t o study  was  physical "dig"  fact  to uncover quantities,  deeply  A kind of aim.  the  Piagetian  stance,  it  was  of  interested  Easley  the main g o a l s o f  preconceptions  thought  this  selected  a method t h a t  or tests)  has  i n conceptual that:  testing  procedures  been n o t e d by development.  will  adequate.  i n t e r v i e w would a c c o m p l i s h  traditional  reminds us  that  of  t h i n k i n g w o u l d be more  clinical  questionnaires  researchers  one o f  students'  in subjects'  The i n a d e q u a c y  written  that  that (i.e.,  several For i n -  -90-  P i a g e t p o i n t e d o u t i n 1929 t h a t t e s t s c a n n o t p r o v i d e enough i n f o r m a t i o n t o d e c i d e what structures are involved i n a c h i l d ' s t h i n k i n g , " and P i a g e t and I n h e l d e r l a t e r c h a r a c t e r i z e d t e s t s as g i v i n g o n l y the ' r e s u l t s o f e f f i c i e n c y of mental a c t i v i t y without grasping the psychological operations i n themselves' (p.281) .... The o v e r r i d i n g q u e s t f o r r e l i a b i l i t y w h i c h a p p e a r s t o be t h e d o m i n a n t c o n c e r n i n t h e t e s t p a r a d i g m , i s doomed . t o . g e n e r a t e many e r r o r s i n the i d e n t i f i c a t i o n of c o g n i t i v e s t r u c t u r e s . . . A n a l y s i s o f p r o t o c o l , i n the s t r u c t u r a l i s t p a r a d i g m , i s n e c e s s a r i l y a s l o w and n o n m e c h a n i c a l procedure. I t begins i n s u b j e c t i v e , but h o p e f u l l y e d u c a t e d , j u d g e m e n t s and moves t o w a r d o b j e c t i v i t y as i t a t t a i n s c o m p l e t e n e s s i n accounting for the t o t a l p r o t o c o l . (Easley, 1974, p.289) Pines  (1980)  comments f u r t h e r  some d i s s a t i s f a c t i o n associationalistic cognitive  capitalizing  say I  field  is  1979;  as  sufficient  still  and  an example S t e w a r t evidence of  flexible  Pines et  testing  is  e v a l u a t i n g and d e s c r i b i n g  He c i t e s  upon p e r s o n a l ,  ultimately result of  science  One o f by P i n e s  for  there  ajL,  (1979),  a move t o w a r d s  interview techniques  1978) .  P i n e s goes on t o  " T h e t r a n s i t i o n , a l t h o u g h s l o w and d i f f i c u l t ,  believe,  the  there  Pines,  that  sophisticated  methods  structures.  and s a y s t h a t  (e.g.,  with  that  i n a d i s t i n c t advancement  education".  the problems of  (1980)  is  that  of  the c l i n i c a l  1981,  this  children,  b u t no d i f f i c u l t i e s o f  for  p.361)  method m e n t i o n e d  n o n - r e s p o n s e by t h e  He c l a i m s  countered with  problem i s  (Pines,  will,  subject.  g e n e r a l l y more a c u t e w i t h y o u n g e r  the adolescent  no n o n - r e s p o n d e r s were p r e s e n t  this  subjects i n the  n a t u r e were e n in this sample,  study. the  Since  internal  -91and e x t e r n a l 1980,  p.  v a l i d i t y of  meeting.  It  a definite  Its  has  a  been d e f i n e d other  (Bingham,  "professional  chief  goal  individual's fying  interview is  purpose  itself",  called  is  to  perceived  to  increased  of  et  (Pines,  the  take the  ajL,  ascertain  p . 3)  (Garret,  1972,  among t h o s e  domain by  of  an  identi-  and  the  conceptions.  two i m p o r t a n t  elicit  The second  be  p.5).  and e x t e n t  subject holds  to  conversa-  o r what c o u l d  the nature  the  directed  i n the  a particular  interview to  lead,  1959,  i n t e r v i e w has  information.  face-to-face  "a c o n v e r s a t i o n  conversation"  relationships  using  literally a as  conceptions  The c l i n i c a l  kinds  is  than s a t i s f a c t i o n  knowledge about  the r e l e v a n t  One i s  study  362).  A clinical  tion  the  is  features.  and c o l l e c t  specific  a l l o w i n g the  i n d u c i n g her/him to  talk  subject  more and more  freely. The o b j e c t i o n s unstructured,  to  the  subjective,  clinical  method as  and p r o n e t o  experimental  c a n be d i m i n i s h e d by p l a n n i n g a p r o t o c o l vance. three  The a n a l y s i s  the v e c t o r  physical quantities  Rational prepare the  of  Task A n a l y s i s , relevant  selected  researcher's v i e w as  selected  interview i n  It  was  experience  for  using  this  study,  also  using  and i n s i g h t f u l  interpretation  the  to  with respect expected the  of  to  that  this  individual  a data c o l l e c t i o n methodology would r e s u l t  objective  ad-  of  p r o v i d e d enough i n f o r m a t i o n  content-area. of  bias  characteristics  and i n d e p t h q u e s t i o n s  years  being  the data  inter-  i n an generated  -92-  by t h e  interviews.  Finally,  recording  the  and a d d e d  a degree o f  3.4.1  interviews  the problems the data  validity, is  objectivity  is  related  must  the  issue of  a measurement  it  d o e s n o t make much s e n s e t o  strument  is  what i t  is  supposed  consideration  and i n t e r n a l  validity  Internal  and  external  internal  validity  if  no l o n g e r  an  important  a measurement and so  it  is  also  In other  o r c a n n o t be  determine  pro-  reproduced,  how w e l l  to measure.  words,  it  is  H o w e v e r , an i n -  can measure something v e r y " x e T i a b ' r y , b u t t h i s  not imply that  it  is  find  with high r e l i a b i l i t y but  measured  is  either  Referring clinical  may s t i l l  what i t  i.e.,  tests  it  measuring  measure  stated:  transcription  Interview  into  accuracy  not accurate  for  data.  internal validity.  if  measuring  is  the  since  The r e l i a b i l i t y o f concerned with  videotape  accurate  take  external  validity  of  in Clinical  obviously related  external  also  to  to  c o l l e c t i o n methodology.  consideration. cedure  project  c o n c e r n i n g the  are  lacking,  use  allowed for  R e l i a b i l i t y and V a l i d i t y  Any r e s e a r c h  of  the  have  unknown o r v e r y  identical  the c l i n i c a l  results  same s u b j e c t ,  supposed  low v a l i d i t y . the  to One c a n  construct  being  trivial.  t o v a l i d i t y and r e l i a b i l i t y when u s i n g  i n t e r v i e w methodology, "that  is  does  f r o m one  even w i t h the  Pines  et  al^ (1978)  i n t e r v i e w does not i n t e r v i e w to  the  have  produce  next with  same i n t e r v i e w e r " .  the  However,  it  -93-  is  their  liable  contention that:  i n that  possible  uniformity of  and t r a i n e d  reliability]  "the  clinical  knowledge a s s e s s m e n t  interviewers  correlations  interview is  i n the  will  show  re-  is  [interrater  assessment of  r  =  .9  or  better" An i m p o r t a n t clinical  interview data,  terviewer tion or  judgments  segment o f  evidence The  for  understanding  subject  seemed  the to  represents  the  selection  and i n t e r p r e t a t i o n  experienced data of  —  an  good  according to  i n w r i t t e n or  taped  of  analyst  in-  informa-  the  subject's  the  subject's  about  the  problematic  This  issue  audio-taping  what p o i n t  the  judgment  et  al.  of  the  can  be other  same i n t e r v i e w same  types  judgments  In the p r e s e n t  went t h r o u g h t h e  the  remains  by a s k i n g  form — making the of  of  data  (1978),  go t h r o u g h t h e  reliability.  still  on the p a r t  judgments  the  However,  interview data  can  the  language or e x h i b i t e d  of  as  elicit  Agreement o r disagreement  estimates  independent  to  the  content-area.  givenMCDnEsptiDn.  Pines  interviewers  judgments.  provides  a  R e l i a b i l i t y on t h e s e  estimated,  or  with  considered  exact wording of  can i d e n t i f y at  have u s e d  of  is  a certain  by v i d e o t a p i n g  that  analyst.  of  to  what  c a n be s e l e c t e d .  behaviour  v e r y much a m a t t e r  is,  activity  conception)  analyst  relates  that  the p a r t s . t h a t  discussion  in part, so  the  (or h i s / h e r  s i t u a t i o n under  interview,  the d a t a ;  subject's  to o b t a i n  from w h i c h  solved,  then a c t u a l l y  about the  r e l i a b i l i t y assessment  the understanding  problem i s  responses;  be  aspect of  interview  study, data  of  6'randomly selected  selected  vector  explained cribed  the  subjects  (6 o f  20) f o r  characteristics  ( 3 of  10).  analytic  interview protocols  monstrate analyst  showed an a g r e e m e n t of  Validity attempt  to  Again,  judgments that  of  one way o f  is  route,  (i.e.,  questions same v e c t o r  able  Interview  Before menter  was  tried  t o have  the  comfortable with  i n general  age,  analysts. t o ask  reasonable  interview  Another  the  subject  the  In t h i s  content-area  way t h e  check  analyst  for  subject.  and g e n e r a l  nature  he/she  was  that  assured  and i n t e r e s t s .  terms  search  the  a  structure.  an i n f o r m a l c h a t w i t h  seemed  that  the  data  are  interview i t s e l f ,  name,  explained  de-  Format  starting  feel  claims  consistency  h e l d by a  asking her/his to  interview  same a s p e c t o f  characteristic).  conceptions  to  independent  i n t e r p r e t a t i o n of  to c a r r y out a , t y p e of  particular  . 3.4.2  the  the  trans-  regarding  cognitive  external  study,  about  The  preconceptions.  in clinical  of  of  9 4.4 p e r c e n t  checking the  employed i n t h i s  several  of  Results  subject's  t h r o u g h the use  investigator  the a n a l y s t .  the a n a l y s t ' s  the  The  one s u b j e c t was u s e d  the c o n s i s t e n t  ensure  interpretation  is  for  to  the a n a l y t i c . p r o c e d u r e .  classification  data  procedure  3 randomly  the  the  the  the After  situation,  the purpose of  the  of  experi-  student, the the  subject author  the present  interview.  (For  example,  i n t e r v i e w was n o t a t e s t  i n t e r v i e w e r was v e r y i n t e r e s t e d  in their  re-  but  answers  ^95^ and t h e way t h a t that  problems  we c o u l d d e v e l o p more i n t e r e s t i n g  materials  for  After was  they thought about  use  i n the  this  chat,  shown a n d t h e  When i t  was  situation, with  the  that  the use  of  film  first  the  this  t a s k was  the subject understood  projection  Again,  p r o j e c t i o n were  to  semi-structured  task  the  required  to  subject  complete  interview,  the  Sample  Piaget standing  of  conceptions age  12.  ceptions that  it  by the  the e n t i r e  the room.  forces,  Aguirre until  in-  film to  under-  in a  designated The t i m e  i n t e r v i e w ranged set  from 4 5 to  up p r i o r t o  The s u b j e c t was  asked  if  A t the end o f  thanked f o r h e r / h i s  the he/she  the  cooperation.  Subjects  (1973),  that  showing  task  seemed  subjects.  interview taped.  s u b j e c t was of  the  and t h e  subject  V i d e o t a p e e q u i p m e n t was  entering  together  situation.  minded h a v i n g the  3.4.3  the problem  protocol  introductory questions  school attended  50 m i n u t e s .  presented.  two was made s m o o t h l y ,  The i n t e r v i e w s were c a r r i e d o u t room o f  one  began.  i n i t i a t e d when t h e  the problem  so  science  task  c a r d b o a r d m o d e l and t h e n p r e s e n t i n g  troduction.  stand  and h e l p f u l  the c a r d b o a r d model o f  the  The t r a n s i t i o n  this  schools).  introduction of  clear  like  in his  studies of  found t h a t  his  students'  subjects  under-  do n o t  develop  c o u l d be d e f i n e d as  vectorial until  after  (1978)  clear  con-  age  14 o r  d i d not later.  may be n o n p r o d u c t i v e t o  find  vectorial  These r e s u l t s select  subjects  suggested younger  than  13 y e a r s  of  age  physical  concepts  in  11,  grade  was  grade  20  subjects  this  included  the t a r g e t  10  For  for  One o f  in this  17 the  — e l e v e n males  and  t h e y were a l l  study  i n the  study.  students'  for  the  their  Science  parents.  s e l e c t i o n of  sample were  teachers  subjects:  their classes,  were g i v e n t h e p a r e n t  the concepts  selected  at  Junior  volunteers  for  a  consent  letters.  Among t h e  by t h e this  phase o f of  a  sufficient  is  for  about  infer  all  that  the  search  a particular  phase o f students  of  20 were  teachers.  c o n c e p t development the use since  interest  subjects  a n d 50 p e r c e n t g i r l s ,  for  schools  t h e s u b j e e l s ;:who d e m o n s t r a t e d  50 p e r c e n t b o y s  Actually,  from  researh  sampling  randomly s e l e c t e d  a  Secondary  c o n s i d e r i n g the c o n s t r a i n t  this  in-  Frederick Banting  who o b t a i n e d p e r m i s s i o n ,  the aim o f  (1)  well-adjusted,  i n two j u n i o r h i g h  (Sir  e x p l a i n e d the need o f  conceptions  of  Two i m p o r t a n t  t h e y had o b t a i n e d c o n s e n t  S e c o n d a r y S c h o o l and Mary H i l l  to  study  consisted  and n i n e f e m a l e s .  Coquitlam School D i s t r i c t  study  this  teachers.  and a f t e r  School)  for  taught  T h e s e two c r i t e r i a were c h e c k e d w i t h  voluntary basis  Junior  the  usually  the sample  r e l a t i v e l y h e a l t h y and  The s u b j e c t s  of  are  (2) t h e y h a d n o t b e e n f o r m a l l y t a u g h t  cluded the  study  since  years).  a s p e c t s were c o n s i d e r e d i n t h e that  Also,  population selected  (ages 15 t o  Phase  study.  the  a research  program c n  t y p i c a l sample  is  individuals'  content-area.  s t u d y was n o t t o  In other words, statistically  i n some d e f i n e d p o p u l a t i o n o f g r a d e  10  -97have as  those  the of  similar preconceptions p o s s e s s e d by t h e  a i m was a  set  of  ahout v e c t o r age  level.  further  to o b t a i n data tentative  about  subjects  characteristics  i n .the,«?sample.  which would p e r m i t the  hypotheses  characteristics These hypotheses  cross-checking  vector  regarding  the  h e l d by s t u d e n t s  Two- o f - t h e  formulation  preconceptions at  were t h e n s u b m i t t e d  i n Phase  Rather,  study.  this to  -98-  CHAPTER FOUR  RESULTS OF PHASE ONE OF THE STUDY 4.0  Introduction The major  chapter  sections. 1)  divided  i n t o the  The s e c t i o n s  the d e s c r i p t i o n of sis  2)  is  and  examples  the d e s c r i p t i o n of  following  refers  to:  t h e methods of  those  the  seven  of  data  analy-  methods;  format of  results  presentation; 3)  4)  the  findings  in  a list  for  all  of  of  p h a s e one w h i c h a r e  inferred rules  vector  and  presented  rule-models  characteristics;  the i n f e r r e d r u l e s  for  each s u b j e c t  in  the  sample; 5)  6)  the d e s c r i p t i o n of  an a n a l y t i c a l  step  further  the  data;  the d e s c r i p t i o n of this  7)  categorize  type of  interview  results  obtained  to  using  categorization;  the conclusions of according  to  addressed  by t h i s  p h a s e one a r e  the r e s e a r c h phase.  presented  questions  being  -99-  4.1  Data Analysis  The d a t a The f i r s t  third  for  steps  sistent  corresponded  these  the  terms  looking for of  to  or  consistent  each s u b j e c t  to  all  the process of  inferred rules  of  The second  and  expressing  the  inferred rules a c c o r d i n g to  to produce a  in less  contemporary  the task  contextual  'rule-model'  for  con-  and  a  The f o u r t h s t e p e n t a i l e d  inferred rules  steps.  patterns  characteristic.  i n terms  perspective.  expressing general  of  each v e c t o r  categorizing  p h a s e one was done i n f o u r  i n the responses  preconceptions  Physics  of  one c o n s i s t e d  preconceptions questions  analysis  of  and more  each  vector  characteristic.  4.1.1  Inferred  Rules:  The s e a r c h  tions  i n the  subjects'  cular  vector  characteristic  task  analysis.  tation the of  of  task;  using  was  t h e g o a l was  related  assumed  about  that  to  that  pattern of  that  consistent  to  If  a subject  cope w i t h  aspect  be  more t h a n h a l f  of  subject held  a consistent  characteristic,  The p r e s e n c e  of  subjects  of  preconception to  the  c o u l d mean t h a t  interpre-  appeared  vector  responses  parti-  empirical  and  subject's  a particular  aspect.  on a  on a p a r t i c u l a r  to  particular  of  an a n a l y s i s  uncover the  aspect.  preconcep-  the questions  was done b y means  performance  a common " s t r a t e g y "  questions  to  T h i s was b a s i c a l l y  a student's  dealing with  responses  for  the it  preconception a  consistent  hold  particular  -100-  kinds  of  i d e a s w h i c h become e x p l i c i t when t h e y a r e  with certain case, type  it of  types  rule  t o know i f  to deal with  summary,  t o as  these  If  this  a subject  situations. is  r e a l l y using  'inferred rules'  a consistent  characteristics)  were  the  possesses This  is  actually  no s t r a i g h t f o r w a r d  way  these r u l e s ;  they  in this  hence  study.  In  p r e c o n c e p t i o n u s e d by a s u b j e c t aspect  (i.e.,  i n one -of t h e  some  one o f  t a s k s was  the  to  vector  expressed  as  an i n -  rule. It  ferred  c a n n o t be c l a i m e d t h a t  rule  reference  for  a specific  point)  t h e one d e p i c t e d lated  is  with a particular  ferred  tasks.  step because there  a subject  referred  deal  cognitive  c o u l d be h y p o t h e s i z e d t h a t  an i n f e r e n t i a l  are  of  faced  to  point).  the  for  i n the  to  cope w i t h  the  is  very contextual; of  that  vector  In: t h i s  the  situation  (e.g.,  the  (i.e.,  sense,  boats  rules  of  i n C h a p t e r One)  terms  are  repeated  the d e v e l o p e d examples  to  here  (see  to  the present  are  only  on l a k e s  following  re-  create and rule" research  applicable and  rivers).  d e f i n i t i o n of  section  facilitate  i n the  be  from  reference  same"inferred  The d e f i n i t i o n and t h e o p e r a t i o n a l inferred  different  characteristic,  inferred rules  tasks  an i n -  (e.g.,  w o u l d be n e c e s s a r y  s u b j e c t uses  tasks.  the  characteristic  characteristic  for  use  e v e n when t h e q u e s t i o n s  claim i t  suitable  determine whether the  the c o n t e x t  task,  same v e c t o r  tasks  vector  an e x p e r i m e n t a l  T o make t h i s  several  a subject w i l l  of  definition  the comprehension o f section.  to  -101Definition  of  consistent  preconception that  using  Inferred  Rule:  i n a given situation  or  p r o b l e m posed by t h e  of  this  was  definition:  a subject  to  several  conception held  interviewer.  i n task one,  bodies  as  reference  Operational  to  Definition  a subject  uses  more t h a n h a l f  of  of  to  (An e x a m p l e  be  the  same b a s i c  the  questions  the  the use  of  use  situation  and a c o n s i s t e n t  the  pre-  several  spots).  Inferred  the  of  question  a g i v e n s i t u a t i o n was  s u b j e c t s was locate  appears  a  the e x p e r i m e n t a l  f i s h i n g spots;  by some  is  deal with a p a r t i c u l a r  a r o w - b o a t moving on a l a k e ;  location of  if  An i n f e r r e d r u l e  Rule:  A rule  is  inferred  p r e c o n c e p t i o n to d e a l regarding  the  with  same v e c t o r  characteristic.  4.1.2 A n e x a m p l e  of  the D e r i v a t i o n of  the  Implicit Vector  for  Stationary  were  Point  for  one o f  the  Characteristic  :  preconceptions  Bodies  inferred rules  The q u e s t i o n s  for  Reference  Point  and t h e r e f o r e  three  i n f e r r e d from the i n t e r v i e w d a t a Stationary  Rules  Bodies  Three c o n s i s t e n t rules  Inferred  about  i n t e r v i e w p r o t o c o l were as  (RPS).  was  Examples o f  i d e n t i f i e d are  Reference  for  Reference how e a c h  now p r e s e n t e d .  P o i n t used  in  the  follows:  Question  1:  How w o u l d y o u d e s c r i b e t h e l o c a t i o n o f t h i s f i r s t f i s h i n g spot to your friends?  Question  4:  How w o u l d y o u d e s c r i b e t h e l o c a t i o n o f t h i s second f i s h i n g spot to your f r i e n d s ?  -102Question Case  One:  5:  A subject locate  The  e a c h one o f  reference:  Researcher's reference: (Q.5). Researcher's as  " . . . I (buoy first here like  fishing  from T r a c y  of  Subject  Point  years):  the  two b o d i e s  Subject  a g a i n was  u s i n g two  Comments:  Subject  kept  bay and s h o r e  Inferred  to  of  as  Stationay  of  or  Tracy to  reference  locate  p r e c o n c e p t i o n and u s i n g  following rule  . . .  line.  selection  when a s k e d  bodies  u s i n g two b o d i e s  R u l e : . The r e s p o n s e s  regarding  pattern  as  dock.  " I w i l l say i t ' s n e a r t h a t bay t h e r e very near the shore l i n e there . . . "  consistent  for  (15  was u s i n g  u s i n g more t h a n one b o d y o r p l a c e  finition  spots.  wharf.  t h e buoy and t h e  questions  a consistent  this  to  c o u l d s a y 20 f e e t p a s s t h e r e d one o r r e d thumb t a c k p l a c e d a t t h e f i s h i n g s p o t on t h e model) . . . f r o m ( t h e dock) a b o u t 100 f e e t o r s o m e t h i n g that".  Comments:  reference:  Derivation three  Comments:  the dock and the  (Q.4)  places  are  the  point  " I w i l l p r o b a b l y p o i n t them and s a y so many f e e t t h a t way . . . I c a n s a y 40 f e e t f r o m t h e d o c k . . . i t w i l l b e a b o u t 20 f e e t f r o m t h e w h a r f t o my r i g h t . . .  Researcher's  of  u s i n g more t h a n o n e r e f e r e n c e  following responses (Q.l)  as  How c o u l d y o u d e s c r i b e t h e l o c a t i o n o f t h i s l a s t f i s h i n g spot to your friends?  point  objects  reference.  . . .  show that  Based  the o p e r a t i o n a l  can be i n f e r r e d f o r  the  on  de-  Reference  Bodies:  " E a c h one o f t h e l o c a t i o n s o f t h e f i s h i n g s p o t s is described using several reference bodies or places".  -103This  inferred  C a s e Two:  r u l e was  A subject but  called  u s i n g one b o d y o r  a distinct  fishing  Comments:  a shore  (Q.4)  a  each  f r o m Hans  (17  reference one o f  as  but  reference fishing  S u b j e c t was of  the  S u b j e c t was  a different  spot,  (Q.5)  and i t  one  was  reference  Comment: but  as  a  wharf.  again  fishing  u s i n g one  place  than w h e n . l o c a t i n g  precisely  S u b j e c t kept  a different  ( I : what (shore  u s i n g one p l a c e  that  using  reference  the  spot.  " . . . f r o m t h e s h o r e . . . how f a r o u t from the s h o r e " (the shore c l o s e r last fishing spot).  Researcher's  the  years):  " . . . how f a r i t ' s f r o m t h e o t h e r s p o t ( t h e f i r s t one) . . . " Comments:  a  are  in front  Researcher's  first  one when l o c a t i n g  as  " . . . h o w f a r i t ' s from the shore . . . s h o r e ? ) . . . t h a t one by t h e f o r e s t i n f r o n t of the w h a r f ) . "  Researcher's reference:  place  spots.  The f o l l o w i n g r e s p o n s e s (Q.l)  RPS-1.  it's to the  o n l y one p l a c e  p o i n t was  used  on  as each  occassion. Derivation locate  of  Inferred  the d i f f e r e n t the  Rule:  fishing  responses  to  reference  body o r p l a c e  description. following Stationary  rule  questions.  The c o n s i s t e n t spots  is  view o f  expressed  in  Each response contains  only  a distinct  one  B a s e d on t h i s  consistent  preconception,  Bodies:  inferred  for  Reference  each  Point  to  his  but  c a n be  for  Hans  for  the  one  -104-  "Each of the l o c a t i o n s of the f i s h i n g spots i s d e s c r i b e d u s i n g o n l y one r e f e r e n c e b o d y . b u t a d i s t i n c t one f o r e a c h l o c a t i o n " . This  inferred  Case T h r e e : and t h e The  r u l e was A subject  called using  RPS-2 one b o d y o r p l a c e  same one when l o c a t i n g  following responses (Q.l)  are  from P r e s t o n  the  (16  a  reference  fishing  spots.  years):  " . . . f r o m h e r e ( t h e dock) I w i l l p r o b a b l y u s e d e g r e e s . . . maybe 40 d e g r e e s n o r t h f r o m where I am s t a n d i n g ( t h e d o c k ) "  R e s e a r c h e r ' s Comments: reference:  e a c h one o f  as  the  (Q.4)  S u b j e c t was  using  one b o d y as  a  dock. "  f r o m h e r e ( t h e dock) . . . i t w i l l 15 d e g r e e s t o t h e n o r t h . . . o r u s i n g i t is straight across".  Researcher's  Comments:  S u b j e c t was  a reference,  the dock.  He a l s o m e n t i o n e d t h e p o s s i b i l i t y  using use  the wharf,  either  is  clear  consistent produced.  one o f of  to  sense t h a t  Subject kept  using  he  as of  will  o n l y one body  e v e n t h o u g h he m e n t i o n e d two p l a c e s  that  reference  one b o d y  both.  Comments:  a reference, one,  i n the  using  " . . . i t i s i n the i n d e n t a t i o n of land ( t h e c l o s e r s h o r e t o the: .'last. W i s h i n g s p o t ) . . . i t w i l l be i n t h e sorrth . . . o r i t w o u l d be a b o u t 180 d e g r e e s t o w a r d t h e s o u t h f r o m here (the d o c k ) " .  •Researcher's  Derivation  he e x p r e s s e d  one and n o t  (Q.5)  either  but  again  be a b o u t the wharf  which  Inferred he was locate  is  Rule:  the  preconception  of  From P r e s t o n ' s  the  the  use  dock.  consistently each  he w i l l  as  using  fishing  the  responses,  it  same b o d y as  a  spots.  following inferred  B a s e d on  this  rule  be  can  -105-  " E a c h one o f t h e l o c a t i o n s o f t h e f i s h i n g s p o t s i s d e s c r i b e d u s i n g o n l y one r e f e r e n c e b o d y and t h e same one f o r e a c h l o c a t i o n " . This  i n f e r r e d r u l e was In the  presented the  above  differentiation  was  of  the  the  inferred rule  so-called  presented  i n the next  section,  for  not possess  vector  4.1.3  then,  subject  con3.1.2.  categorization  which a l s o  i n the  was  describes  At the  no r u l e was  for  assigned  If a  produce and  end o f  a maximum o f  preconception  to  sample  characteristic.  10  for  this inferred  a subject  did  vector  to her/him  for  characteristic.  of  the  Inferred  the Rule-Model  for  RPS  Two a s p e c t s were c o n s i d e r e d the  by a n a l y z i n g  differentiated  a b o v e was u s e d  assigned  each v e c t o r  Categorization of  of  T h i s view i s  how t h i s  characteristic.  a consistent  characteristic, that  each  s u b j e c t was  one f o r  level  a rule-model.  each i m p l i c i t v e c t o r each  been  ordering  Physics Approach i n S e c t i o n of  process,  of  That i s ,  perspective.  explanation  inferred rules  have  process  c o u l d be  A detailed  construction of  rules,  This  inferred rule.  The m e t h o d o l o g y d e s c r i b e d the  inferred rules  order.  a physicist's  i n the  done i s the  of  RPS-3.  done by a n a l y z i n g t h e  each  how t h e c o n t e n t - o f  tained  example,  in a particular  inferred rules  from t h a t  called  inferred rules  presented  Rules  and C o n s t r u c t i o n  i n the c a t e g o r i z a t i o n  in Section  4.1.2.  First,  of  the  -106content  of  each  ponding v e c t o r  inferred  r u l e was  characteristic  network o f v e c t o r  position  differentiated  was  in  this  sent or  it  context,  and t h e  nature  quantitative).  according  to  the  of  the  was  level,  (the  Section most  4.1.2).  similar  level  extreme  and i t  as  point  third  — was  level  Other v e c t o r  characteristics  mediate  inferred After  expressed results the  was  of  first  stationary  (the  i n more g e n e r a l  vector  for  the  among  lowest  developed  in  network — t h e most  considered  at  the  4.1.2).  inferred rule  one  differhighest  The i n -  i n between  classified  was  the  in  two  the  i n Section  may h a v e more t h a n one  4.1.2  inter-  rules.  categorizing  this  as  i n Section  second  ordered  RPS whose c o n t e n t  classified  was  pre-  qualitative  were  at  one i n t h e  differentiation  intermediate  is,  i n t h e example  inferred rule  of  variables  differentiated  considered  r u l e whose c o n t e n t was levels  least  for  Content,  physicist's  was.categorized  inferred rule  the  inferred rules  from the  the  p r e c o n c e p t i o n and i t  (the  ferred  differed  in  network.  (that  the  -  corres-  t o d e t e r m i n e how  inferred rules  the corresponding  unique reference entiated  the  The i n f e r r e d r u l e  to  3.2)  one i n t h e  Approach;  considered  first  Figure  these variables  Physics  inferred rules,  (see  the  point  b o t h t h e number o f  Second,  RPS whose c o n t e n t most perspective  - reference  from the  includes  compared w i t h  last  the  inferred rules,  and l e s s  procedure  The g e n e r a l  were  terms.  y i e l d e d the rule-model  characteristic,  bodies.  contextual  they  reference rules  point  in this  The for  for  rule-model  -107-  should  still  be c o n s i d e r e d as  being very tentative  t h e y were i d e n t i f i e d i n a v e r y s p e c i f i c lack  e m p i r i c a l support The g e n e r a l  are  closer  with his  i n other related  rules  i n form t o  and  still  contexts.  i n the h y p o t h e t i c a l rule-model  those  u s e d by S i e g l e r  (1978,  1979)  rule-assessment methodology.  Reference  Point  i n the  of  the approach' used  is  presented  Physics Approach:  by p h y s i c i s t s  definition  A short  t o choose  description  reference  (see  i n Appendix A ) , which a c t u a l l y  Physics Approach,  show t h a t  this  m a g n i t u d e and d i r e c t i o n t o  F i g u r e 3.2 represents  quantity  (vector  and the  position)  be c o m p l e t e l y d e f i n e d .  B o t h m a g n i t u d e and d i r e c t i o n n e e d t o be m e a s u r e d w i t h to  a specific  specific of  3.2, for  frame  reference  reference  of  reference  point  (the  or coordinates  physicists extension  and d i r e c t i o n ) taneous  u s u a l l y use  the  so-called  same r e f e r e n c e  of  related  origin  As i s  t h e same frame  the l o c a t i o n s  locations  which i s  system).  of  to  of  the  different This  and p r o v i d e s  respect a frame  shown i n F i g u r e of  reference  point to describe  a moving body.  a p p r o a c h u s e d by p h y s i c i s t s : the  points  below.  The n e t w o r k o f v e c t o r p o s i t i o n  needs  context  since  (magnitude  bodies  or the  is  current  the  and  t h e norm t o  instan-  which  i n f e r r e d c o u l d be c o m p a r e d .  Categorization  of  approach d e s c r i b e d conception following  the F i r s t above,  Inferred Rule:  the content of  t h a t most d i f f e r e d inferred  rule:  from  it  was  Based  on t h e  the c o n s i s t e n t included  in  pre-  the  -108-  "Each of the l o c a t i o n s of the f i s h i n g spots are described using s e v e r a l reference bodies or places" ( I n f e r r e d r u l e RPS-1) This to  inferred rule  locate  shows  bodies.  perceive  the  reference  to  different  ones  the use  Subjects  importance of locate  all  holding this selecting  spots.  to d e s c r i b e  o f more t h a n o n e  conception did  o n l y one b o d y as  Instead,  the  reference  they used  l o c a t i o n of  not  a  several  each - f i s h i n g  spot. The i n f e r r e d general  f o r m and i t  rule-model  for  fule  RPS-1 was  corresponded  reference  rephrased  to  the  first  preconception physicists  the Second I n f e r r e d  that  was  perspective  rule  following in  the  point:  "Use of m u l t i p l e r e f e r e n c e p o i n t s or places) to locate a s t a t i o n a r y (Rule R P S - I ) . Categorization of  i n the  judged  Rule:  t o be n e x t  (bodies body" The  closest  consistent to  the  was:  " E a c h one o f t h e l o c a t i o n s o f t h e f i s h i n g s p o t s i s d e s c r i b e d u s i n g o n l y one r e f e r e n c e b o d y b u t a d i s t i n c t one f o r e a c h l o c a t i o n " (Inferred r u l e RPS-2). The i n f e r r e d r u l e as  shows  the use  a p h y s i c i s t w o u l d do i t  always  the  different contains  same.  This  of  but that  from a p h y s i c i s t ' s an i m p o r t a n t f e a t u r e  form and i t  rule-model:  reference  a s p e c t makes  this  of  corresponded  body  body i s  conception  perspective,  not still  even though  it  it.  T h e i n f e r r e d r u l e RPS-2 was general  o n l y one r e f e r e n c e  to  rephrased the  i n the  following  second r u l e of  the  -109-  "Use o f one u n i q u e r e f e r e n c e p o i n t (body o r p l a c e ) t o l o c a t e a s t a t i o n a r y body b u t d i f f e r e n t unique r e f e r e n c e p o i n t s a r e used to locate other bodies". (Rule RPS-II) Categorization preconception perspective  of the Third whose c o n t e n t  Inferred Rule: was  The c o n s i s t e n t  c l o s e s t to the p h y s i c i s t ' s  was:  " E a c h one o f t h e l o c a t i o n s o f t h e f i s h i n g s p o t s i s d e s c r i b e d u s i n g t h e same r e f e r e n c e body". ( I n f e r r e d r u l e RPS-3) This  i n f e r r e d r u l e shows t h e u s e o f o n l y  for  e a c h one o f t h e l o c a t i o n s .  the  p r o c e d u r e u s e d by p h y s i c i s t s . The  general  This  one r e f e r e n c e  a c t u a l l y corresponds to  i n f e r r e d r u l e RPS-3 was r e p h r a s e d  form and i t c o r r e s p o n d e d  body  to the t h i r d  i n the following r u l e i n the  rule-model: "Use o f one u n i q u e r e f e r e n c e p o i n t (body o r place) to locate a l l required s t a t i o n a r y bodies". (Rule RPS-III) This constitutes reference  s e t of three  r u l e s , summarized b e l o w ,  the rule-model  point  f o r the vector  for stationary  thus  characteristic,  bodies.  R u l e M o d e l f o r RPS: Rule RPS-I:  Use o f m u l t i p l e r e f e r e n c e p o i n t s (bodies o r places) t o l o c a t e a s t a t i o n a r y body.  R u l e R P S - I I : Use o f one u n i q u e r e f e r e n c e p o i n t (body o r p l a c e ) t o l o c a t e a s t a t i o n a r y body b u t d i f f e r e n t u n i q u e r e f e r e n c e p o i n t s a r e used t o l o c a t e other bodies. Rule R P S - I I I :  Use o f one u n i q u e r e f e r e n c e (body o r place) t o locate a l l required s t a t i o n a r y bodies.  -110-  T h i s m e t h o d o l o g y was for  e a c h one o f  end o f vector  4.2  this  the  will  to produce  vector  the  rule-models  characteristics.  t h e r e w e r e 10. r u l e - m o d e l s ,  one  At for  the each  characteristic.  Format f o r  each v e c t o r  Presenting  of  Results  of  I n t e r v i e w Data A n a l y s i s  the a n a l y s i s  of  the i n t e r v i e w  characteristic  be p r e s e n t e d  Each of  implicit  process  The r e s u l t s for  used  1.  the  actual  are  rules  sub-sections  e x p l a i n e d below  of  Characteristic:  section  consists  physics  background r e l a t e d  implicit  a short  reasons  before  This  d e s c r i p t i o n of to  the  sub-  the  vector  to consider  characteristic,  i n the present  and r e a s o n s  it  as  to  an  include  .study  Questions i n the Interview P r o t o c o l Related to the Vector C h a r a c t e r i s t i c : This sub-section consists  of  all  questions  interview protocol of  the  tasks  specific 3.  inferred  results.  characteristic,  2.  fully  Introduction of Vector  it  of  i n a format which c o n t a i n f i v e  these sub-sections  reporting  i n terms  data  List  of  contains the  of  a  list  vector Inferred all  the  related  the to  that  characteristic. Rules:  This  inferred rules  interview data  characteristic.  from  for  the  sub-section identified  specific  The i n f e r r e d r u l e s  from  vector were  obtained  -111by f o l l o w i n g t h e in  same p r o c e d u r e  Section 4.1.2,  order tion  obtained analysis  Section  and a r e  from the  l e v e l of  by u s i n g  the v e c t o r  characteristic  digits  used  rules  of  (i.e.,  for  first  vector  a stationary  each  female  to  provided  i n parenthesis  At  the c a l c u l a t e d statistic in  the  is  value  ficant  statistical  literature science of  and  difference,  subjects  w o u l d be  and male  RPS-1).  subjects  i n the 6F,  list  list  7M f o r  of  study  if  regarding  boys et  rules  test used  rule  sex.  because  (Hobbs,  RPS-1)  inferred  inferred  it  is  1979;  Educational Progress,  signi-  between female  usually  do b e t t e r  al,  t h e r e was  preconceptions  characteristics,  that  is  statement.  the C h i - s q u a r e  t e s t was  was d e c i d e d t o  about v e c t o r male  of  reference  p r o v i d e d . '" T h e "two v a r i a b l e s  classification It  for  (e.g.,  the  These  among i n f e r r e d  inferred rule  the bottom of  for  characteristic  inferred rule  the  are  a digit.  object  assigned  t h e end o f  plus  the  in  abbreviation  inferred rule  The number o f  at  explained  to d i f f e r e n t i a t e  a particular  the  point  the  in  differentia-  The i n f e r r e d r u l e s  identified  are  presented  — w h i c h was  4.1.3.  illustrated  found i n  than g i r l s  National  (1978);  and the  in  Assessment  Erickson,  et  al,  -112-  1980).  Research  relate after be  to  findings  science  achievement,  instruction.  interesting  to  reported  was  thought  find  out  if  sex  instruction  knowledge about  difference  vector  (11 b o y s  and 9 g i r l s ) ,  The c a l c u l a t e d indicated  that  difference vector  the  section  3 is  format: in  according to  discussed  of  is  used  the  have  Under  this  and a n a l y z e d  supported  of  each  sex. sub-section in  sub-  in  detail.  by p r o v i d i n g  from a  few  excerpts  will  have  name o f  subject,  age,  number o f  question  W i t h i n an  excerpt,  (i.e.,  any e x p l a n a t i o n questions  (I)  of  are  Q. 4 ) .  terms o r denoted  in  Rule Model:  In the  ferred  were r e p h r a s e d  rules  contextual  final  further  the  subjects.  of  parenthesis  and l e s s  was  significant  interview excerpts  Presentation  character-  inferred rules  Results:  here  pre-  reported.  inferred rulespresented  The a n a l y s i s selective  is  t h e r e was no  characteristic  each o f  was  Chi-square values  between the  Discussion of  would  sample  the r e s u l t s  statistic  it  regarding  Even though, a s m a l l  test  means  there  istics.  Chi-square  this  It  significant  usually  following  interviewer's  parenthesis.  sub-section i n a more  form t h u s  yielding  the  in-  general the  -113general  rules.  will  i d e n t i f i e d using  be  for  T h e s e more g e n e r a l  the g i v e n v e c t o r  Roman n u m e r a l .  4.3  The Roman n u m e r a l  rule  RPS-1 c o r r e s p o n d s  (e.g.,  of  the A r a b i c  Analysis of  section  analysis 4.1.2  in  the  This  inferred RPS-I).  of  using  the  of  of  the  first  ten  vector  1.2.  i n t e r v i e w data are  corresponds  in  to  responses  characteristics  of  presented  to  all  the subjects  sample.  characteristic  4.3.1.  each  in Section  .two v e c t o r  Results  vious  rule  Data  the r e s u l t s  for  Some t r a n s c r i p t i o n s  on  to  and a  m e t h o d o l o g y , w h i c h was d e l i n e a t e d  listed  i n Appendix B.  numeral  Interview  contains  and 4 . 1 . 3 ,  characteristics  questions  characteristic  to  This  Sections  abbreviation  corresponds  Results  interview  the  rules  are  presented  following  the  separately f o r format  each  described  vector  i n the  pre-  section.  Inferred Bodies  Rules  for  Point  the  it  for  Stationary  (RPS)  At the beginning of a reference  Reference  p o i n t was  study  one u n i q u e v e c t o r  was  conjectured  characteristic  that  -114w h i c h was u s e d when d e s c r i b i n g t h e The d a t a  collected in this  tions  used by s t u d e n t s  place  (e.g.,  locate task  were two d i f f e r e n t  evident  that  criteria decided  it  for both cases. to  treat  reference  characteristics. section,  refers  to  to  in  task  vector  2)  characteristic  of  to  find  the need f o r  a reference  they d e s c r i b e  reference  point?  Questions  of  Questions: Question of  the  1,  1:  first  of  these  common  these  is  findings  treated  in  reference water  of  (see  stationaryicbDiiies  bodies  the  with  or i n -  p o i n t to describe  a  to  a  reference  one.  to your  location  friends?  see  location?  f o r RPS  the  the  the problem  Did they  How w o u l d y o u d e s c r i b e  became  and  a moving o b j e c t .  as  river  character-  (RPS)  stationary bodies  What d i d t h e y c h o o s e  points  Chapter One).  the v e c t o r  dealt  or  This  locations with respect  f i s h i n g spot  was  The s e c o n d  i n motion.  only with  and 5 i n t a s k  it  this  bodies  or places.  (e.g.,  Interview Protocol 4,  in  reference  o u t how s t u d e n t s  locations  on t h e r i v e r  identify  stationary  d e s c r i b i n g the l o c a t i o n o f  stantaneous  Did  point for  to  became  number & i n L I V C  section deals  reference  a i m was  which  bodies  used  two d i s t i n c t v e c t o r  objects  the c h o i c e of  body o r  it  of  when d e s c r i b i n g o b j e c t s  This istic  to  the c h o i c e of  a n d / o r moving r e f e r e n c e  Thus,  p o i n t as  The f i r s t ,  when d e s c r i b i n g s t a t i o n a r y type r e f e r s  things.  Because  descrip-  1 ) and t h o s e  a moving boat  was n o t p o s s i b l e  bodies.  the  a stationary  i n task  (e.g.,  of  showed- t h a t  locate  a f i s h i n g spot  a moving body  2)  to  study  locations  specific points?  Question of  4:  the second  Question this  5;  last  How w o u l d y o u d e s c r i b e fishing  spot to your  friends?  How w o u l d y o u d e s c r i b e  fishing  spot t o your  Table  the l o c a t i o n  the l o c a t i o n of  friends?  4.1  L i s t o f I n f e r r e d Rules About Reference P o i n t f o r S t a t i o n a r y B o d i e s (RPS)  Three i n f e r r e d characteristic,  rules  w h i c h have  were o b t a i n e d  for this  vector  t h e f o l l o w i n g common o p e n i n g  sentence: " E a c h one o f t h e l o c a t i o n s i s described using . . . Inferred places".  Rule RPS-1: (6F, 7M).  of the f i s h i n g  spots  Several reference bodies o r  I n f e r r e d Rule RPS-2: O n l y one r e f e r e n c e b o d y b u t a d i s t i n c t one f o r e a c h l o c a t i o n " ( 2 F , 2 M ) . Inferred  Rule RPS-3:  for  locations" (IF,,;2M) „  (X2  Discussion of Inferred  all  = .21, df = 2  Rule RPS-1: reference  or places  a r e used  This  p>.05)  c a n be r e f e r r e d  t o as t h e  p o i n t c o n c e p t i o n " because s e v e r a l to locate  s a m p l e o f 2 0 , 13 u s e d that  body  Results  "multiple  believed  One u n i q u e r e f e r e n c e  this  a stationary  conception.  t h e l o c a t i o n o f an o b j e c t  object.  These  bodies  From t h e  subjects  c a n b e s t be  described  -116b y "using s e v e r a l  different  from i n t e r v i e w d a t a  will  reference  illustrate  bodies.  Some  more c i e a r l y  excerpts  this  con-  ception.  S u z z a n e , 1 7 : (Q.4) " I t ( t a s k 1, s e c o n d f i s h i n g s p o t ) i s s o r t of i n the middle but i t ' s c l o s e r to the w h a r f . . . t h e r e a r e two b u o y s on t h e w a t e r . . . t e l l them (her f r i e n d s who a r e c o m i n g t o f i s h ) where t h e f i r s t one ( f i r s t f i s h i n g s p o t ) i s a n d t e l l them t h a t t h e s e c o n d one i s f a r t h e r b u t n o t much and i t ' s c l o s e r t o t h e t r e e " . Dale,  15 (Q.4) " I t ( t a s k 1, s e c o n d f i s h i n g s p o t ) is j u s t p a s t t h e f i r s t f i s h i n g s p o t , i n t h e same a n g l e , a c o u p l e o f h u n d r e d o f y a r d s f r o m me and i t i s r i g h t i n front of this corner".  These using  two e x c e r p t s reference  points.  the middle o f  the  the wharf;  fishing  and a t r e e ;  first  several  show t h a t  fishing  t o be a so  as  spot,  "try  to  Inferred  lake,  spot, to  of  three:  It  appears  type  body i s  rule  p o i n t c o n c e p t i o n " , because a  used  to  locate  spot)  the of  location  trip  as  (task  (the w h a r f ) ,  the boat.  reference  used  each  points  fishing  the d o c k ) ,  and t h e  These p o i n t s selected  to  as  the  distinct  spot.  a reference 1,  the  strategy  c a n be r e f e r r e d  This  five:  first  Rule RPS-2:  point for  among t h e  the  chance•vrxf..suecess.  were:the observer's  (fishing  and a c o r n e r .  of  t r y and m a x i m i z e t h e i r  T h r e e p l a c e s were more o f t e n  ing  the buoys,  and D a l e made use  himself,  were  S u z z a n e made u s e  i n c l u d e e v e r y t h i n g you c a n "  "changing reference reference  both subjects  last  point, the  they start-  stop  were a l s o  found  by s u b j e c t s h o l d i n g  -117-  RPS-I.  A common p r a c t i c e  or  place  to  Ss  holding  describe  this  the  the wharf fishing that  the  spot  spot),  shore cerpts  rule,  three  closest  closest  from t h e  select  followed  the  first  object  they used and f o r  (the  to  the  l o c a t i o n being described.  l o c a t i o n of  (the  was  the  to  tree  the  last  place  to  the  fishing  interview data  last  four  procedure. spot  spot),  the  object  they used  fishing  To  they  for  closest spot  object  Among t h e  fishing  (the  the  this  closest  used second to the  spot).  clearly illustrated  Exthis  practice.  Julie,  15:  (Q.l)  " I ' d t e l l them ( h e r f r i e n d s ) t o go f r o m t h i s p o i n t (she showed t h e w h a r f ) o u t s t r a i g h t i n angle . . . a l i t t l e b i t i n angle".  (Q.4)  " . . . t u r n t h e b o a t i n a n g l e and h e a d f o r the t r e e s o r whatever landmark from the shore ( c l o s e s t shore to the second fishing spot).  (Q.5)  " . . . go t o where t h e c u r v e i s (closest s h o r e t o t h e l a s t f i s h i n g s p o t ) a n d go a b o u t 15 f e e t a n d t h e r e i t i s (she meant t o c a r r y t h e b o a t t o t h a t s h o r e and s t a r t , r o w i n g f r o m t h e r e ) .  These e x c e r p t s reference selected  point to the  show t h a t  describe  closest  object  the  subject  e a c h one o f to  the  the  changed  the  locations  l o c a t i o n as  a  and  reference  point. Inferred reference  R u l e RPS-3:  This  rule  c a n be c a l l e d t h e  p o i n t c o n c e p t i o n " because  same o b j e c t  as  a reference  p o i n t to  each s u b j e c t locate  the  "constant  used  the  different  -118-  fishing  spots.  observer's point  for  location the t r i p  conception, as  spots;  the s t a r t i n g  (the dock i n t a s k (the w h a r f ) .  two c o n s i s t e n t l y  a reference  fishing  The most common p l a c e s  1)  or the  F o r the  used  are  3Ss  holding  their position  the other  made c o n s i s t e n t . u s e  point  (the wharf)  from i n t e r v i e w d a t a  Preston,  16:  as  locations  a reference  show t h i s  this  (the  the  subject  the  starting  point to describe  Excerpts  of  dock)  three of  point.  consistent  practice:  (Q.l)  from h e r e (the dock) I w i l l p r o b a b l y u s e d e g r e e s . . . maybe 4 0 ° n o r t h f r o m where I am s t a n d i n g h e r e . . . "  (Q.4)  " . . . from h e r e ( t h e dock) i t about 1 5 ° to the n o r t h . . .  (Q.5)  " . . . i t i s i n the i n d e n t a t i o n of the l a n d . . . o r i t ' d be a b o u t 1 8 0 ° t o w a r d t h e s o u t h from h e r e ( t h e d o c k ) " .  These e x c e r p t s sistently  used  different  fishing  the  clearly  same p l a c e  for  show t h a t  this  to d e s c r i b e  the  will  subject  be  con-  location  of  spots.  Table Rule-Model  selected  Reference  4.2 Point  for  Stationary  Bodies  Rule RPS-I:  "Use o f m u l t i p l e r e f e r e n c e p o i n t s (bodies or places) to l o c a t e a s t a t i o n a r y b o d y " .  Rule RPS-II:  " U s e o f one r e f e r e n c e p o i n t (body o r p l a c e ) to l o c a t e a s t a t i o n a r y body b u t d i s t i n c t ones t o l o c a t e o t h e r s t a t i o n a r y bodies".  RuleRPS-III:  "Use o f one u n i q u e r e f e r e n c e p o i n t (body o r p l a c e ) t o l o c a t e a number o f s t a t i o n ary b o d i e s " .  -119-  Rule RPS-III view of  selecting  that  occupies  It  it  was  actually  corresponds  a reference  point.  the h i g h e s t  n o t an e a s y t a s k  to  This  position  the is  physicists'  the  reason  i n the r u l e - m o d e l .  to decide which  inferred  rule  b e t w e e n RPS-1 a n d RPS-2 s h o u l d f o l l o w i n c o m p l e x i t y . was  taken because  point,  even though i t  the d i f f e r e n t  4.3.2  this  Inferred  was  not the  for  Frame o f  Rules  Stationary  Body  The r e a s o n  for  was  to  students  have about  tions, of  locations  characteristic  is  of  notion of  a reference  to  the need f o r  to  relate  is  including  point  it  for  a  frame  of  bodies  reference  conceptions  quantitative  descrip-  or p l a c e s .  This  to  the previous  one,  essentially  is  an e x t e n s i o n  of  point.-  required.  Reference  e x p r e s s e d b y an a n g l e )  to  that  a body o r p l a c e  After  a reference  to  the r e f e r e n c e  t e n d e d t o become a frame  of  quantitative  body  (or  location with  p o i n t c a n be m e a s u r e d .  point)  (i.e., respect  When t h e s e  p o i n t has  reference.  only  from w h i c h  and d i r e c t i o n o f  a specific  the  point refers  a n o t h e r b o d y b u t no  the d i s t a n c e  expressed,  for  related  location of  reference  Reference  the  stationary  as  are  same r e f e r e n c e  u n c o v e r what k i n d o f  the c h o i c e of  been c h o s e n ,  ties  reference  (F.R.)  closely  point —  description  o n l y one  m a k i n g more p r e c i s e  reference  the  considers  spots.  characteristic  has  also  RPS-2  then been  quantiex-  -120Specifically, vector with of  characteristic  the problem of  different  flat  expressed  including  this  t o d e t e r m i n e how s u b j e c t s  deal  (i.e.,  (i.e.,  distance  for  in their  d i d t h e y use  to  spots  the need o f expressed  i n order  D i d t h e y use  and d i r e c t i o n )  reason  fishing  D i d t h e y see  i n degrees)  spots?  points  was  the  describing quantitatively  points  surface.  description  ing  then,  to  the  i n task using  locate  1) o n a  a  in units  quantitative and/or  relate  angle  the v a r i o u s  more t h a n one v a r i a b l e descriptions  locations  and w h i c h  these v a r i a b l e s  fish-  (distance reference  to  the  various  locations? Originally vector after  it  characteristic a n a l y z i n g the  the preconceptions to v e c t o r frame  was  of  — that  first  of  reference.  similar  T h i s was  frame  of  was d e c i d e d t o c o n s i d e r and t h e  Frame o f position intuitive  to  it  was  those  aspect of only  responses  t o v e c t o r p o s i t i o n were u s e d preconceptions  position —  obtained  the  and i n f e r r e d r u l e s  for  frame  because i t i n nature  was  selected  thought that  than the  latter,  of  the q u e s t i o n s for  was  for  locations. reference related  identification  rather  the  vector  position,  frame  data  reference  as  for  that  related  describing the  but  realized  reference^and"vector  Thus,  of  another  the case because both  on the q u a n t i t a t i v e  characteristic,  vector  interviews  focussed it  include  uncovered through the questions  p o s i t i o n were  characteristics,  intended to  of than  former  and b a s i c  is  reference. vector more  notions  -121of  it  w o u l d more  to  formal  l i k e l y be  evident  of  Interview  for  FR  Questions  1,  3,  4,  and  6c  Question  1:  the  first  Question either  that  for  your  Question this  Question of  5a,  fishing 3:  (If  spot  to your  S has  1).  the  location  location  u s e d no q u a n t i t y  (first  of  friends?  when  fishing  or  only  one,  answering  How c o u l d y o u make t h e  description  spot)  more  precise  friends? 4;  How w o u l d y o u d e s c r i b e  second  fishing  5 : (a) :  fishing  fishing  spot  is  first  spot to  the  your  location  friends?  How w o u l d y o u d e s c r i b e  last  the  One.  How w o u l d y o u d e s c r i b e  this  than  i n Task  distance or d i r e c t i o n ,  question  of  prior  instruction.  Questions  ' of  i n subjects  at  spot to  the  your  spot,  but  location  friends?  same d i s t a n c e  fishing  the  (This  from the  in a  dock  different  direction). (b): same d i s t a n c e  Question  6: (c) :  followed  by the  in  to  path  taken  two f i s h i n g  from you  you d i s t i n g u i s h  order  These  (S  on d o c k ) .  spots  are at  How c o u l d  them? (After boat)  S has  What do y o u n e e d  have a b e t t e r by t h e  drawn t h e  boat?  approximation  path to of  know the  the  -122-  Table List  These tative  used  Inferred  inferred rules  and q u a l i t a t i v e  meters, being  of  4.3  degrees)  interpreted Five  were e x p r e s s e d  terms.  The u s e  i n nature.  and o t h e r d e s c r i p t i o n s to  qualitative  inferred rules  characteristic,  of  s u c h as  w h i c h have  in  i n both q u a n t i units  (e.g.,  interpreted  Where u n i t s  t h e r i g h t were u s e d , as  (FR)  i n t h e d e s c r i p t i o n was  quantitative  farther,  Rules  north,  were  as  not  south,  closer,  the d e s c r i p t i o n s  were  nature.  were o b t a i n e d the  for  this  vector  f o l l o w i n g common o p e n i n g  sentence: " P r e c i s e d e s c r i p t i o n o f e a c h one o f t h e l o c a t i o n s o f t h e f i s h i n g s p o t s i s done by u s i n g . . . Inferred  Rule F R - 1 :  a given reference  Only  a quantitative  p o i n t and t h e  distance  location)  as  a  (between  coordinate"  (1F,2M) . Inferred pressed  as  Inferred tative are  Rule FR-2:  Rule FR-3:  d i r e c t i o n as  used  — one f o r  direction" Inferred  are  i n degrees)  as  a coordinate."  Both q u a l i t a t i v e coordinates the distance  direction  distance  (OF, 3 M ) .  and q u a l i -  b u t two r e f e r e n c e and a n o t h e r  (ex-  one  points for  the  (1F,1M)  Rule FR-4:  qualitative points  an a n g l e  Only a q u a n t i t a t i v e  Both quantitative  d i r e c t i o n as used  — one  coordinates  for  distance  but  the d i s t a n c e  two  and  reference  and a n o t h e r  one  for  -123-  the d i r e c t i o n " Inferred  (5F,3M)  Rule FR-5:  qualitative  Both q u a n t i t a t i v e  d i r e c t i o n as  coordinates  erence p o i n t for both distance Cx  2  =3.67,  df  not assigned  Discussion Inferred the  inferred  This  asking  distance question  a fishing  in  a p o s i t i o n where t h e d i s t a n c e s  of  the  spot  was d i s t a n c e .  s p o t s were i d e n t i c a l ,  spots. to  recognize to  17:  5b,  (1F,1M)  (1F,1M)  coordinate the o n l y  were  Question to  the  distance  this  difficulty  of  this  interview data  subjects  and  last  a l o n e was  two  using  concep-  location  5b p u t first  as  coordinate  p o i n t and t h e  thus  the d i s t a n c e  20 s u b j e c t s made u s e  Hans,  as  However, t h e s u b j e c t s  adherance  from t h e i r  ref-  interpreted  i n d i s t i n g u i s h i n g between the  not appear their  2 Ss  r u l e c o u l d be  used between a r e f e r e n c e  sufficient  ;  same  rule).  of  fishing  the  and d i r e c t i o n "  = S p > 05  one q u a n t i t a t i v e  Even a f t e r  subjects  with  and  Results:  Rule F R - 1 :  "only  tion".  of  distance  not  locations  this  rule did  and m a i n t a i n e d  criterion.  Only  inferred rule;  same  3 out  of  excerpts  follow.  (Q.l)  " . . . how f a r away the s h o r e . . . "  (Q.3)  " . . . how f a r  it  [they]  is  are  away f r o m  from me..."  (Q.5a)  " . . . from t h e s h o r e from t h e s h o r e "  . . . how f a r  out  is  (Q.5b)  "What do y o u mean? (Question i s rephrased) . . . I d o n ' t know . . . maybe how many m e t r e s a c r o s s a r e (S r e f e r r e d t o t h e d i s t a n c e b e t w e e n t h e two f i s h i n g s p o t s ) " .  -124-  (_Q.6c)  " . . . f r o m where I am i t ' s  Kelley,15:(Q.l)  d o n ' t know any o t h e r  "I  (Q.4)  " . . . f r o m t h e t r e e so f a r . . . i t 60 y a r d s b u t . . . and t h e n so f a r where I am . . . "  i s about o u t f r o m . ..  (Q.5a)  "How f a r  am"  (Q.5b)  " . . . I d o n ' t know . . . f r o m t h e a n g l e t h a t t h e y a r e (S r e f e r r e d t o t h e a n g l e f o r m e d b y t h e two s t r a i g h t l i n e s j o i n i n g t h e two s p o t s w i t h t h e dock) . . . you c o u l d say 200 m i n one d i r e c t i o n (S r e f e r r e d t o t h e d i s t a n c e b e t w e e n t h e two s p o t s ) . . . o r y o u make an i s o s c e l e s t r i a n g l e and make t h e s e s i d e s 200 m e t r e s , t h e n t h e y w i l l be equal".  (Q.6c)  " . . . the d i s t a n c e s , from the p o i n t s t h r o u g h t h e y w e n t and t h e a n g l e s o f how much t h e y t u r n e d (S r e f e r r e d t o t h e  two e x c e r p t s  only distance  as  as  a  w h i c h was observe  which 3 of  show t h a t  t h e s e Ss  There i s  I  judged  This  rule  d i r e c t i o n as  situation  by use  e x p r e s s e d by a n g l e s  made u s e  of  t o be  insufficient  of  summarized  coordinate".  i n degrees.  direction  It  is  only,  interesting  T h e r e were  conception;  as  two  seems t o be more p r i m a r y t h a n d i r e c t i o n . holding this  distance  in a  hold-  coordinate,  subjects  t h e y d i d n o t use  Subjects  locations  quantitative  as  a  20  that  displacements)"  some i n d i c a t i o n o f  it  c o u l d be  preconception differentiated  dimensional  f r o m where I  (FR-3 R u l e ) .  "only quantitative  to  is  way".  formed by c o n s e c u t i v e  here but  Rule FR-2:  this  away i t  a coordinate.  appearing  to q u a l i f y  ing  that is guess the  (Q.3)  These  Inferred  difficult".  " . . . f i n d a p o i n t i n that shore i n l i n e with the spot . . . then, distance ( I : What d i s t a n c e ? ) . , .  angles  direction  very  excerpts  from  -125-  their  interview data  Preston,  16:  (Q.l)  " . . . f r o m where . . . I ' l l p r o b a b l y u s e d e g r e e s . . . maybe i t ' s 4 0 ° n o r t h f r o m where I am s t a n d i n g " .  (Q.3)  (Not a s k e d  (Q.4)  " f r o m h e r e . . . i t w i l l be a b o u t 1 5 ° to the n o r t h " " i t i s i n the i n d e n t a t i o n of the land o r a t 9 o ' c l o c k u s i n g t h a t d i r e c t i o n as 12 o ' c l o c k , i t w i l l be i n t h e s o u t h . . . " a b o u t 1 8 0 ° toward t h e s o u t h from h e r e " .  (Q.5a)  Steve,  15:  follows.  to  this  subject).  (Q.5b)  " i t ' d be 200 m e t r e s t o t h e n o r t h w e s t and 200 m t o t h e s o u t h - e a s t (S u s e d d i s t a n c e expressed i n metres because I i n t r o d u c e d them i n t h e q u e s t i o n 5 b . What d i d y o u mean b e f o r e b y d e g r e e s ? ) . . . o h , by d e g r e e s I mean . . . t h i s i s my z e r o l i n e (a l i n e p a s s i n g t h r o u g h t h e d o c k and w h a r f ) a n d my f r a c t i o n i s g o i n g up l i k e t h i s (S showed s e v e r a l angles; 2 0 ° , 3 0 ° , 9 0 ° , using his locat i o n as a r e f e r e n c e p o i n t )  (Q.6c)  " . . . lake wharf from (2nd  (Q.l)  ". . . I w o u l d - s a y : ' S"tart from the w h a r f t h e r e and go o u t a b o u t 70 d e g r e e s a n g l e t o a l m o s t be p a r a l l e l t o t h e t r e e (I: a n y t h i n g e l s e t h a t you c o u l d t e l l to your friends?) . . . straight in front o f t h i s c u r v e (a b a y on t h e l a k e ) is"  (Q.3)  " . . .  (Q.4)  "when y o u a r e i n t h e f i r s t f i s h i n g s p o t t u r n t o w a r d . t h e s h o r e . . . f r o m where I am s t a n d i n g , i t w i l l be r i g h t b e h i n d the f i r s t f i s h i n g s p o t " .  (Q.5a)  " f i r s t end up i n t h e bank a n d p a r a l l e l t o t h e f i r s t one ( f i r s t f i s h i n g s p o t ) I guess . . . i n the f i r s t curve i n t h a t bank ( I : What e l s e c o u l d be i m p o r t a n t t o mention to your f r i e n d s ? ) . . . (pause)  p r o b a b l y the d i s t a n c e s a c r o s s the ( I : What d i s t a n c e s ? ) . . . f r o m t h e to here ( f i r s t f i s h i n g s p o t ) , here ( f i r s t f i s h i n g spot) to here fishing spot)-.."  ( l o n g pause)  . . .  -126(Q.5b)  " t h a t f i s h i n g s p o t i s 200 m e t r e s (the v a r i a b l e d i s t a n c e was i n t r o d u c e d by i n t e r v i e w e r i n q u e s t i o n 5b) t o t h e r i g h t a b o u t 7 0 d e g r e e s a n g l e . . . and t h e o t h e r one i s 200 m t o t h e l e f t a b o u t 70 d e g r e e s a n g l e from t h i s p o i n t o f view (the d o c k ) "  (Q.6c)  " . . . distances from t h e w h a r f and so o n " .  Both the  subjects  interviewer  even a f t e r angle  with  respect  introduced i t  to  the  Rule F R - 3 :  First  all  variables  was  and d i s t a n c e ,  Inferred of  d i d not mention the d i s t a n c e  the d i s t a n c e  both,  it  is  (distance  in  as  This  a pair  rule  a type of  is  reference  Secondly, two  "qualitative  This  coordinates  conception".  Only  ferred  rule.  Excerpts  sented  below.  Suzzane,  17  rule  (Q.l)  two o f  d i d not  use  measured  several  subjects  And t h i r d l y ,  not  although  summarized reference held  two  con-  coordinates  them t o  20 s u b j e c t s  the  holding this  qualitative  related  aspects.  since  mentioned are  c o u l d be  from t h e i r  But,  point.  with different the  5b.  coordinates  made up o f  and d i r e c t i o n )  two v a r i a b l e s . t h e y  points.  of  until  question  a qualitative' rule  a two d i m e n s i o n a l s i t u a t i o n . used  of  i n t r o d u c e d , t h e Ss  as  saw- t h e n e e d t o u s e  subjects  part  same r e f e r e n c e  associated with units. ception  . . . from p o i n t to p o i n t . . . to t h i s ( f i r s t f i s h i n g spot)  these  different as: points  this  in-  interview data are p r e -  " . . . s t r a i g h t i n f r o n t o f the wharf . . . you p r o b a b l y have t o say t h e d e g r e e s . . . o r a map ( r e s p o n s e t o q u e s t i o n 3) " . . . j u s t t o t e l l them t o go s t r a i g h t o r go w i t h them . . . " .  -127CQ.3)  " . . . j u s t t o t e l l them t o o r go w i t h t h e m " .  (Q.4)  " i t ' s s o r t o f on t h e m i d d l e b u t i t ' s c l o s e r t o t h e w h a r f . . . t h e r e a r e two b u o y s on t h e w a t e r , t e l l them where t h e f i r s t one i s and t e l l them t h a t t h e s e c o n d one i s f a r t h e r b u t n o t much a n d i t ' s c l o s e r to the t r e e " .  (Q.5a)  "It's i t is  (Q.5b)  "One i s a t a n g l e ( I : What do y o u mean?) . . . w e l l one i s t o y o u r r i g h t and t h e o t h e r t o the l e f t . . . on the r i g h t s i d e i s a t an a n g l e . . . t h e o t h e r one i s a b i t i n an a n g l e "  (Q.6c)  Inferred ferred  rule  subjects  quantitative  coordinate. .  subjects  used d i f f e r e n t  in  the  sample  Excerpts  Becky,  This  (Q.l)  this  that  distance,  Like  points  since  interview data  between t h i s  in-  in this  the  case a  qualitative  inferred rule for  t h e most  8 of  . . .  subject).  along with  the p r e v i o u s  reference  subjects,  from t h e i r  15:  one i s  p r e c o n c e p t i o n was  of  of  The o n l y d i f f e r e n c e  direction  direction.  not asked  and t h e p r e v i o u s  used  straight  r i g h t o f f t h e s i d e , t o my l e f t past the green buoy".  (Question  Rule FR-4:  go  distance common one  20 h e l d  these  and found  it.  follows:  " . . . j u s t a b o u t h a l f way t o t h e m i d d l e , a-Ad h a l f way t o t h e r i g h t s i d e . . . i t ' s a b o u t 50 f e e t f r o m me"  (Q.3)  (not asked  to t h i s  subject)  (Q.4)  " . . . y o u e x p l a i n t o them a g a i n b y d i s t a n c e . . . a b o u t 100 f e e t . . . maybe y o u c o u l d s a y a l o n g the shore j u s t pass the t r e e . . . "  (Q.5a)  " i t i s near the shore on the l e f t s i d e o f the lake . . . i t ' s not f a r o u t , j u s t about 5 f e e t from t h e e d g e " .  -128-  Larry,  (Q.5b)  " . . . y o u c o u l d s a y i n an a n g l e and 50 f e e t ( d i s t a n c e o f P4 respect to a spot on r i g h t s i d e o f s u b j e c t ) . . . y o u c o u l d say about 1 2 0 ° a n g l e i f you s t a r t from here . . . (I: What a b o u t t h e o t h e r s p o t ? ) I t ' s about 7 5 ° a n g l e . . . "  (Q.6c)  " . . . t h e waves i t p r o d u c e s i n o r d e r t o see i t . . . "  16:(Q.l)  " . . . I t i s e x a c t l y i n the middle of the lake. I f I am i n t h i s d o c k and y o u l o o k t h a t t r e e the boat i s i n the middle of them".  (Q.4)  " I w i l l t e l l t h e m : f i n d t h e t r e e and t h i s dock (the w h a r f ) , the s t r a i g h t l i n e c o m i n g from t h a t t r e e t o h e r e ( s e c o n d f i s h i n g s p o t ) and b a c k t o t h e d o c k , and t h e r e i s t h e s p o t a b o u t 30 f e e t f r o m t h e tree".  (Q.5a)  "From the  (Q.5b)  " . . . t h e y a r e b o t h 200 y a r d s and one i s on t h e l e f t s i d e o f t h e c o a s t and t h e o t h e r i n the c e n t e r o f the l a k e . . . "  (Q.6c)  " . . . I n e e d a compass ( I : How c o u l d y o u use i t ? ) . . . I d o n ' t know . . . we n e e d markers l i k e the t r e e , c o r n e r s . . . "  s m a l l d o c k up t o  and r u l e  FR-4 i s  point  express q u a n t i t a t i v e  define  as  this there  (Q.3)  Rule F R - 5 :  direction.  moves  " I c o u l d g i v e them t h e d i s t a n c e from ( t h e w h a r f ) i n f r o n t o f i t , and t h a t i s a t r e e to the r i g h t " .  Inferred  to  when i t  This a polar  that  rule  The o n l y these  is  frame  difference  subjects  most s i m i l a r of  missing  is  the q u a n t i t a t i v e  pressed  as  the angle  reference,  and  the  same  that  l i n e passing  side  rule  reference  qualitative  t o what  physicists  the o n l y  ingredient  d i r e c t i o n , w h i c h c a n be  i n degrees  f o r m s w i t h one r e f e r e n c e  left  between t h i s  used  distance  the  the v e c t o r  ex-  position  through the  reference  -129 point. which  This  indicates  students. Ian,  c o n c e p t i o n was  15:  that i t is  Excerpts (Q.l)  h e l d by o n l y  not  very  from t h e i r  "You can t e l l  2 of  20  subjects  common among g r a d e  i n t e r v i e w data how f a r  off  10  follow:  from the  dock  i t i s , i n what d i r e c t i o n . . . y o u c a n see from h e r e i n t h e d i r e c t i o n o f t h e t r e e . . . " (Q.3)  (Not a s k e d  (Q.4)  " I t ' d be d i r e c t l y t o t h e n o r t h - e a s t a b o u t 5 ° t o the N . E . s t r a i g h t o u t from the o t h e r one ( f i r s t f i s h i n g s p o t ) .  (Q.5a)  " . . . y o u c a n do i t o f how f a r h e r e (the dock) . . . from h e r e i s 150 y a r d s o f f . . .  (Q.5b)  " . . the the (of the  (Q.6c)  " t o h a v e a r o u g h i d e a where i t w e n t , how many t i m e s i t s t o p p e d , what d i r e c t i o n went i n and came b a c k . . . "  Although, some p o i n t ,  it  more i n c l i n e d  to  subject).  of  quantitative  not a consistent  to mention q u a l i t a t i v e  with quantitative  i s from (the dock)  . t h i s one i s r e a l l y c l o s e t o t h e s h o r e , o t h e r one i s o u t j u s t i n t h e m i d d l e o f l a k e . . . o r one i s on t h e l e f t o b s e r v e r ' s l o c a t i o n ) and t h e o t h e r on right . . . "  I a n made u s e  was  this  criterion.  direction in He was  d i r e c t i o n combined  distance.  Table Rule-Model  for  4.4  Frame o f  . Reference  Rule F R - I :  "Use o f o n l y one q u a n t i t a t i v e d i s t a n c e as a c o o r d i n a t e t o l o c a t e a s t a t i o n a r y b o d y i n a two d i m e n s i o n a l s i t u a t i o n " .  Rule F R - I I :  " U s e o f o n l y one q u a n t i t a t i v e d i r e c t i o n (as an a n g l e e x p r e s s e d i n d e g r e e s ) as a c o o r d i n a t e t o l o c a t e a s t a t i o n a r y body i n a two d i m e n s i o n a l s i t u a t i o n " .  -130-  Rule F R - I I I :  " U s e o f b o t h q u a l i t a t i v e d i r e c t i o n and q u a l i t a t i v e d i s t a n c e as c o o r d i n a t e s t o l o c a t e a s t a t i o n a r y b o d y i n a two d i m e n s i o n a l s i t u a t i o n ; b u t two r e f e r e n c e p o i n t s a r e u s e d , one f o r e a c h c o o r d i n a t e " .  Rule FR-IV  :  "Use o f b o t h q u a n t i t a t i v e q u a l i t a t i v e d i r e c t i o n as l o c a t e a s t a t i o n a r y body mensional s i t u a t i o n ; but p o i n t s a r e u s e d , one f o r  Rule FR-V  :  " U s e o f b o t h q u a n t i t a t i v e d i s t a n c e and q u a l i t a t i v e d i r e c t i o n i n a two d i m e n s i o n a l s i t u a t i o n ; w i t h t h e same r e f e r e n c e p o i n t for both c o o r d i n a t e s " .  4.3.3  Inferred  Rules  Location  (D)  Although  this  quantity, implicit (1)  It  parison  it  was  vector  about  is  D i s p l a c e m e n t o r Change o f  a well  considered  d i s t a n c e and c o o r d i n a t e s to i n a two d i two r e f e r e n c e each c o o r d i n a t e " .  defined kinematic  important to  characteristic  for  the  vector  include i t  the v e c t o r  "artificial"  or  p o s i t i o n concept,  remote  from e x p e r i e n c e ,  concept to  clearly differentiate  (e.g.,  the  length of  (e.g.,  displacement).  needs a magnitude, locations,  t o be  the path)  the  which (2)  and a v e c t o r i a l scalar  length of  the p a t h between  direction  t o be c o m p l e t e l y d e s c r i b e d , to  because  and the it  a  suitable  quantity  quantity  is  because i n a d d i t i o n to magnitude  in part,  more  is  The f i r s t  vectorial  corresponds,  It  is  i n com-  between a s c a l a r  completely described;  an  following reasons:  c a n be c o n s i d e r e d a more i n t u i t i v e c o n c e p t to  as  it two  second also  and f i n a l l y ,  only  is  needs (3)  It  a v e r y common and i n t u i t i v e n o t i o n  -131that  the c l o s e s t  length of  the  distance  straight  line  From t h e p i l o t subjects object  to  as  choose  spot to  describe  If  is  this  point,  it  last point  that  two l o c a t i o n s joins  a t r e n d was stopping (e.g.,  l o c a t i o n of  the  may h e l p s t u d e n t s  of  to  Particularly,  partial  displacements  point of  use  of  the  for  first  fishing  fishing  spot).  selecting  reference  the concept  should f a c i l i t a t e displacements,  two c o n s e c u t i v e  the  a moving  the  understand  consecutive  associate  observed  second  it  is  them.  an i n t u i t i v e c o n c e p t i o n o f  displacement. standing  study  the  a reference  between  of  the  under-  since  locations  partial  of  a  moving body. Questions  of  Questions:  Interview Protocol  6, 7 a , 8a i n t a s k  Question  6:  one.  From where y o u a r e  you approximately d e s c r i b e boat?  Please,  Q u e s t i o n 7a:  draw t h a t  We c a n s a y  of  l o c a t i o n when i t  to  the  first  describe respect distance along  the to  the path  spot  to  respect  to  the  the  the  had a  lake. change  point  S includes  Is  that  (S.P.)  with the  distance  boat?  moved f r o m t h e  location of one?  the  f i s h i n g spot  (If  s e c o n d one a t  first  the  How c o u l d y o u  description)  the  has  starting  first  point?  the boat  map o f  the boat  (P^).  could  f o l l o w e d by  in this  f o l l o w e d by the  c o u l d you d e s c r i b e with  that  spot  starting  Q u e s t i o n 7b: T h e n , fishing  path  location of  in her/his  the path  movers f r o m t h e  fishing  the  (on t h e d o c k )  here this  (P2)  first T h e n how  fishing  spot  -132-  Question  8a:  (If  S includes  the d i s t a n c e  description]. in  in  her/his  You m e n t i o n e d t h e  your d e s c r i p t i o n .  Does  distance  a change  of  location  tell  a n y t h i n g about  the  path  followed  by the b o a t between  the  two  locations?  Table List  In described another; as  of  I n f e r r e d Rules for Displacement Change o f L o c a t i o n  these was  instances  the  physical  means,  a synonym o f  of  location,  r u l e s were o b t a i n e d  which have  the  Rule D - l :  The s t a r t i n g  Inferred  Rule D-2:  The l e n g t h o f  two l o c a t i o n s  joining  w h i c h was  used  for  this  vector  and t h a t  and f i n a l  length  the is  locations".(IF,1M)  straight  line  independent  joining  from  the  (0F,3M) Rule D-3:  quantitative Inferred  to  l o c a t i o n o f the b o a t moving from spot to another i n d i c a t e s . . .  Inferred  Inferred  spot  \ "A change o f one f i s h i n g  path".  being  f o l l o w i n g common o p e n i n g  sentence:  the  fishing  or  displacement.  inferred  characteristic,  a change  (D)  phenomena  t h e b o a t m o v i n g f r o m one  this  Five  4.5  along  the  path  and  d i r e c t i o n m e a s u r e d from the o b s e r v e r ' s l o c a t i o n  Rule D-4: the  Both the d i s t a n c e  Both the  two l o c a t i o n s  and q u a l i t a t i v e  length of  which i s  direction".  the  straight  independent  (7F,3M)  from  line path  the  -133-  Tnferred line  Rule D-5:  j o i n i n g the  Both the  two l o c a t i o n s  p a t h and q u a n t i t a t i v e Cx  2  = 6.80  inferred Discussion  of  df  = 5  location  indicates  the  or  T h e r e was  was  independent  p > .05;  IS(F)  rule  states  was  that*a  aimed l o c a t i o n o f  final  variables  relates  a moving o b j e c t  age)  location.  s u c h as  only but  the  distance  starting  since  only  a t r i p and  to the  Suzzane,  17:(Q.7a)  15:  Both  best  starting  ways  final  I n summary,  or  this  illustrate  location  points  i n between. (15  or  this  and t h e e n d i n g  2 among 20 h e l d  from i n t e r v i e w d a t a  of  No q u a l i t a t i v e  n o t what h a p p e n s  Excerpts  Dale,  assigned  change  n o t a common c o n c e p t i o n among a d o l e s c e n t s of  not  the p o s s i b l e  d i r e c t i o n o f m o t i o n was p r o v i d e d .  years  from  (0F,1M)  no m e n t i o n o f  i n reaching that  conception of  is  straight  Results This  quantitative  the  rule)  Rule D - l  path  which  direction".  Inferred  location.  length of  This  to  17  conception. this  conception.  "Just i t joes s t r a i g h t . . . (I: What c o u l d you t e l l to your f r i e n d s about t h i s change o f l o c a t i o n ? ) . . . I don't know . . . "  (Q.8a)  "Yeh, i t is . . . b o a t went . . . "  (Q.7a)  " W e l l i t s t a r t e d h e r e (wharf) and went to t h a t p o i n t ( f i r s t f i s h i n g spot) . . . t h e y p a d d l e d s t r a i g h t o f f from t h e wharf".  CQ.7b)  " . . . y e h . . . f r o m h e r e (wharf) t o h e r e ( f i r s t f i s h i n g s p o t ) . . . i t was s t r a i g h t off.  subjects  ending points  solely  in their  it  emphasized  responses.  tells  the  where  starting  the  and  -134-  Iiiferred  Rule D-2:  Subjects  thought t h a t  the d i s t a n c e  an i m p o r t a n t  factor  of  location.  distance between clude  the  to consider  the path;  Hans,  when d e s c r i b i n g a  17:  straight  This  conception d i d not  3 held this  following  differentiating  of  in-  location.  conception,  which  excerpts.  (Q.7b)  "Same t h i n g a s  (Q.8a)  " I d o n ' t u n d e r s t a n d what y o u mean ( I : q u e s t i o n i s r e p e a t e d ) . . . no ( I : Why?) Y o u can'if^-feell t h e a c t u a l p a t h , y o u c o u l d go a r o u n d i n d i f f e r e n t w a y s " .  subject  as  the other  in his  Rule D-3:  fishing  . . . how f a r  it  two i n c l u d e d o n l y  responses;  The o n l y  the d i s t a n c e  i n t e r v a l of  before  there  was  is"  the  no m e n t i o n  all.  by t h e p r o d u c t o f  first  only  the  "How f a r o u t i t i s . . . ( I : I s t h a t d i s t a n c e a l o n g the path?) . . . from t h e r e (wharf) t o t h e r e ( f i r s t fishing s p o t . (He meant t h e s t r a i g h t distance).  distance  considered  change  (Q.7a)  d i r e c t i o n at  Inferred  was  t h e y were n o t c o n s i d e r i n g  spots.  20 . s u b j e c t s ,  This  the  two l o c a t i o n s  t h e s e s u b j e c t s were  two f i s h i n g  shown t h r o u g h t h e  of  the  conception  t h e d i r e c t i o n a s s o c i a t e d w i t h a change  Among t h e is  between  Notice that  along  holding this  the  time  spot.  subject holding this  along  speed taken  the p a t h .  T h i s was  conception calculated  (assumed t o b e c o n s t a n t )  times  i n moving from the wharf t o  T h e d i r e c t i o n was  e x p r e s s e d by  the  the  rl35-.  angle the  f o r m e d b y t h e two s t r a i g h t  starting  Actually,  and e n d i n g  vector position  but rather with  associated with  shown i n t h e e x c e r p t s Wayne, 1 5 :  from  the d i r e c t i o n  the f i r s t  interview  of the  fishing  spot.  which i s  data.  (Q.7a)  "You c o u l d l o o k a t t h e f i r s t f i s h i n g s p o t a n d make a n i m a g i n a r y l i n e ( f r o m o b s e r v e r ' s l o c a t i o n on t h e d o c k ) ; then l o o k a t t h e w h a r f ( s t a r t i n g p o i n t ) , y o u n o t i c e i t w i l l be a n a n g l e b e t w e e n the l i n e s . You c a n n o t e how f a r o v e r i t i s (size of the angle). You c a n a l s o t e l l how f a r a p a r t t h e y a r e ( t h e two l o c a t i o n s ) . I f they t r a v e l a t t h e same s p e e d a l l t h e t i m e , y o u c o u l d t i m e how l o n g i t t a k e s t o go f r o m one spot t o t h e o t h e r , t h i s w i l l g i v e you the d i s t a n c e . ( I : What i s i m p o r t a n t t o c o n s i d e r f o r a change o f l o c a t i o n ? ) ... t h e s p e e d , t h e d i r e c t i o n ...  (Q.8a)  "No ... ( I : Why n o t ? ) ... b e c a u s e he went f r o m t h e s t a r t i n g p o i n t t o t h e f i s h i n g s p o t ... he went z i g z a g g i n g ...  Inferred  Rule  was v e r y  c l o s e t o the formal  D-4:  This conception  o f change o f  d e f i n i t i o n of  location  displacements.  o n l y . i n g r e d i e n t m i s s i n g was t h e q u a n t i t a t i v e d i r e c t i o n ;  subjects holding this direction  .  conception  These students  included a  could readily  qualitative  grasp  concept  of displacement  i f t h e y were i n s t r u c t e d  express  the q u a l i t a t i v e  direction  T h i s was a common c o n c e p t i o n since  10  o f . 20  mention here that  location,  associated with the  one s u b j e c t among 20 h e l d t h i s c o n c e p t i o n  Only  through  p o i n t s from t h e o b s e r v e r ' s  t h i s was n o t a d i r e c t i o n  change o f l o c a t i o n  The  lines passing  that these  how t o  i n a q u a n t i t a t i v e form.  among g r a d e 10  subjects held i t .  the formal  students  I t i s important  s t u d e n t s were e x p l i c i t l y  t h e d i s t a n c e c o n s i d e r e d was n o t a l o n g  to aware  the path;  -136-  some o f  them s a i d  that  t h e r e were many ways  of. m o v i n g from one l o c a t i o n t o the so  conceptions far,  it  Considering  characteristics the  students  way.  T h i s was  and f o r  f o l l o w i n g are  Lisa,  15:  the  in a quantitative  case  for  it  way,  was  in a  the v e c t o r  reference  dis-  characteristic.  from i n t e r v i e w  data:  " T h e s t a r t i n g p o i n t i s r i g h t on t h e end o f t h e w h a r f and t h e f i r s t s p o t i s a b i t to the r i g h t i n a s l i g h t angle . . . and e s t i m a t e s t h e amount o f m i l e s b e t w e e n t h e two . . . a l o n g t h e way f o l l o w e d by the b o a t " .  (Q.7b)  . "From t h i s ( f i r s t f i s h i n g spot) to t h i s ( s e c o n d f i s h i n g s p o t ) i s a l m o s t t h e same d i s t a n c e (as t h e d i s t a n c e f r o m w h a r f t o f i r s t f i s h i n g s p o t ) . . . maybe 50 o r 75 m e t r e s more . . . and t h e y h a v e t o t u r n some d e g r e e s " .  "along  the d i s t a n c e questions  Lori-Ann,  of  to  (Q.7a)  Although  distance  frame  excerpts  (Q.8a)  said:  the  analyzed  seemed  when t h e y c o n s i d e r e d a s e c o n d v a r i a b l e  placement,  two  vector  on o n l y one v a r i a b l e  qualitative  The  all  seems t h a t most o f  concentrate and  of  the o t h e r .  Cor p a t h s )  " . . . no . . . b e c a u s e t h e b o a t c a n a r o u n d i n d i f f e r e n t ways . . . " i n the response to  the  first  t h e way f o l l o w e d b y t h e b o a t " , along  the p a t h ;  show t h a t  between  the  15:(Q.9a)  the responses  she was r e f e r r i n g  question this  for to  go  meant  the  the  Lisa  other  straight  locations.  "How f a r i t i s , u s i n g f e e t o r m i l e s (I: Is t h a t d i s t a n c e a l o n g the path or the s t r a i g h t distance?) . . . along the p a t h , how f a r t h e b o a t w i l l go t o reach that p o i n t . (The f i r s t f i s h i n g spot. I: Suppose the p a t h i s t h i s one. I drew a c u r v e p a t h ) . . . o h ,  -137-  I see .... I am t a l k i n g a b o u t t h e s t r a i g h t distance ( I : anything e l s e to c o n s i d e r f o r t h i s change o f l o c a t i o n ? ) ... i f t h e b o a t were g o i n g b a c k and f o r t h i t w i l l be a much g r e a t e r d i s t a n c e •.. a l s o i f y o u want t o d e s c r i b e t h e e x a c t p a t h y o u c o u l d use s o u t h , n o r t h , e a s t , o r w e s t , and go so much t h i s way and t h e n t u r n a r o u n d t h e o t h e r way". (Q.8a)  Rule D-5:  Inferred formal  definition  holding they  " i t t e l l s t h e end o f t h e p a t h b u t y o u c a n ' t t e l l t h e p a t h b e c a u s e i t c o u l d go s t r a i g h t o r l i k e t h a t ... o r l i k e t h a t (she s k e t c h e d two d i f f e r e n t c u r v e p a t h s ) "  this  l o c a t i o n but  referring  location  as  realized  about  of  the  a p o t s and Since  using these  these  direction),  of  the  change  that these  of  vector position  the  two  the  observer's  subjects quantitative  instructional the  l o c a t i o n to another; i t students  would e a s i l y  knowledge of measuring a n g l e s  trans-  t o a new  but  inferred  r.ule  situation. conception,  included  i n this  also  considers  the  d i s t a n c e between the  (The  length  the  straight  and  subjects  show them t h e ^ i m p o r t a n c e o f k n o w i n g  expected  The  these  the  the  of q u a n t i t a t i v e d i r e c t i o n ,  the need o f u s i n g  to  c l o s e s t to  Although,  direction  point.  o f m o v i n g f r o m one  their  similar  to the  fishing  ( d i s t a n c e and  w o u l d be fer  the  a reference  t a s k w o u l d be  conception  made u s e  to the d i r e c t i o n  associated with  direction  the  of displacement.  conception  were n o t  variables  T h i s was  of  line  joining  s u b j e c t s were d i f f e r e n t i a t i n g  the d i s t a n c e  along  the path.  two  locations.  them)  thus  between t h i s  Three  students  distance among  -138-  20 h e l d  this  excerpts Becky,  c o n c e p t i o n , which  from t h e  15  interview  shown i n t h e  following  data.  (Q.7a)  " . . . the d i s t a n c e of the lake?  (Q.7b)  " go a b o u t 110 d e g r e e s ( l i n e j o i n i n g d o c k as a r e f e r e n c e p o i n t ) a n g l e b y u s i n g the p r o t r a c t o r toward t h a t c o r n e r on t h e r i g h t s i d e o f t h e l a k e and a b o u t 50 f e e t ( I : Is t h i s d i s t a n c e a l o n g the path or the s t r a i g h t distance b e t w e e n t h e two l o c a t i o n ? ) . . . the s t r a i g h t d i s t a n c e . . . ( I : What do y o u t h i n k i s i m p o r t a n t when y o u r e f e r t o a p a r t i c u l a r change o f l o c a t i o n ? ) . . . where i t s t a r t s and where i t g o e s , t h e d i s t a n c e and t h e a n g l e " .  (Q.8a) Steve,  is  15:(Q.7a)  "...  it  and i n what  does not t e l l  side  you the path  . . . "  (Q n o t u n d e r s t o o d when f i r s t a s k e d . I: What do y o u t h i n k i s i m p o r t a n t t o c o n s i d e r f o r a change o f l o c a t i o n ? . . . d i r e c t i o n and d i s t a n c e . . . (I: d i s t a n c e along the path or s t r a i g h t ? ) . . . the s t r a i g h t distance..."  (Q.7b)  " . . . i n t h e same way and d i s t a n c e . . . "  (Q.8a)  "No . . . t h e y - c a n go anywhere a r o u n d whole l a k e to a r r i v e to t h a t p o i n t , t h e y c a n go s t r a i g h t t o o " .  Table Rule-Model  for  ..  the  degrees the  4.6 Displacement  Rule D - I :  " A d i s p l a c e m e n t o r change o f l o c a t i o n i n d i c a t e s t h e s t a r t i n g and f i n a l locations".  Rule D - I I ;  " A d i s p l a c e m e n t o r change o f l o c a t i o n i n d i c a t e s the l e n g t h of the s t r a i g h t l i n e j o i n i n g two l o c a t i o n s " .  R u l e D - I I I : " A d i s p l a c e m e n t o r change o f l o c a t i o n i n d i c a t e s b o t h the l e n g t h o f the p a t h and a q u a n t i t a t i v e d i r e c t i o n measured from the o b s e r v e r ' s location".  -139-  Rule D-IV:  " A d i s p l a c e m e n t o r change o f l o c a t i o n i n d i c a t e s both the l e n g t h of the s t r a i g h t l i n e j o i n i n g two l o c a t i o n s a n d a q u a l i t a t i v e d i r e c t i o n o f the f i n a l l o c a t i o n with r e s p e c t to the i n i t i a l one".  Rule D-V :  " A d i s p l a c e m e n t o r change o f l o c a t i o n i n d i c a t e s both the l e n g t h of the s t r a i g h t l i n e j o i n i n g two l o c a t i o n s and a q u a n t i t a t i v e d i r e c t i o n g i v e n by t h e a n g l e o f t h e s t r a i g h t l i n e j o i n i n g t h e two l o c a t i o n s with another reference l i n e " .  4.3.4  Inferred  Rules  for Addition of  The a d d i t i o n o f operation,  which i s  s i t i o n of  any v e c t o r  this  o p e r a t i o n as  based on t h i s mathematic students,  displacements  applied  to  quantity.  an i m p l i c i t  author's  belief  Displacement  is  The r e a s o n vector that  for  the concept.  taught  for  understood (or  Usually,  generally  done w i t h o u t c o n n e c t i n g t h e v e c t o r i a l  ponds  to  by " a r r o w s "  the magnitude  of  knowledge i s  students  teachers  develop  (of  to)  this is entities  the arrow)  know v e r y w e l l  not e a s i l y This  shows that  are corres-  the this  sort  transferred  to  author also  believes  some p r e c o n c e p t i o n s  to  or  by  These v e c t o r i a l e n t i t i e s  kinematic v e c t o r i a l problems. that  and i t  and the arrow i t s e l f  Physics  abstract  i n mathematics  whose l e n g t h  direction.  is  related  is  represented  including  a formal concept  operation  with physical quantities.  compo-  characteristic  t h e c o n c e p t must be b a s e d o n  first  vectorial  the combination or  o p e r a t i o n t o be more e a s i l y  some i n t u i t i v e n o t i o n o f  a defined  (AD)  cope  solve  with  -140-  situations (or  i n which t h e y have  distances)  conceptions  with different  c o u l d be p a r t  would enhance This  the  ties.  as  this  conceptions  combining p a r t i a l of  :  9  distance  from the  have a l s o  the  formal  the  study  s t u d e n t s have  (P^)  estimated  it  spot  are  at  to  them t o  concept.  combination of  of  vector  is  to  and  quanti-  uncover  developed  about  one.  starting  spot  friends  that  displacements.  Suppose you have  fishing  pre-  P r o t o c o l f o r AD  Question  fishing  These  which possess magnitude  of  9 and 10 i n t a s k  tell  the  and c o n s e c u t i v e  Interview  Questions;:  second  of  considers  part  displacements  i m p l i c i t aspects  an i m p l i c i t c h a r a c t e r i s t i c  what k i n d s o f  first  the  displacements,  The aim o f  Questions  of  then,  combine  directions.  understanding  section,  two c o n s e c u t i v e direction,  to  the  as  established  point  (S.P.)  about  100 m, a n d y o u  from t h i s  (P2) as starting  the  spot  about  to  (P-^)  150 m.  p o i n t how a r e  go d i r e c t l y t o  the  the  second  to If  the your  you  going  fishing  spot? Note:  This question introducing expected  was  the  that  used  a d d i t i o n of  all  o r most o f  use- t h e  arithmetic  percent  of  strategy major  to  purpose  of  the  actually correct  the purpose  displacement. the  a d d i t i o n as  subjects find  only with  It  subjects  it  response.  these questions  was  to  was  will  a solution,  d i d use  of  as  and  100  a  Because  the  familiarize  -141-  the  subjects with  responses the  analysis  of  the  10:  distance  from t h i s  (P3) and i t  tell  your f r i e n d s this  The p u r p o s e the It  to  this  uncover  the  that  further  students' was  sufficient  to  illustrate  about  vector  These  of  to  vector  junction Note:  of  the  situation.  would d i s c o v e r  by  a d d i t i o n , but  it  i n s t r u c t i o n a l use  to vector  one q u e s t i o n w o u l d  subjects'  conceptions  4.7  Inferred Rules for A d d i t i o n D i s p l a c e m e n t s (AD)  and s e c o n d ,  the d i r e c t i o n  Both aspects  of  c o n t a i n two a s p e c t s .  the c r i t e r i o n to o b t a i n the magnitude o f  quantitative).  o u t how  characteristic.  inferred rules  displacement;  find kind  students  thought t h a t  Table List  starting  i n t u i t i v e ideas about  It  fishing  spot?  solving this  addition.  this  third  the  How c o u l d y o u  q u e s t i o n was  the o p e r a t i o n o f for  this  o  estimate  go d i r e c t l y f r o m t h e  go a b o u t  important  t  100 m.  fishing  in  rules.  (P2)  about  not expected  themselves is  spot  is  the  9 were n o t c o n s i d e r e d  inferred  third  of  subjects was  situation  Suppose you c o u l d a l s o  spot  to  kind of  question  Question  point Note:  for  this  are  the  First,  resultant  (qualitative  s e p a r a t e d by the  or  con-  and.  Despite  the  fact  that  t h e r e was  no s e a r c h  for  a  be  -142consistency for  this  of  responses,  vector  ceptions  were  parallelism  characteristic,  classified  i n the d a t a  Three i n f e r r e d characteristic, sentence Inferred  Rule AD-2:  addition  of  Inferred  Rule A D - 3 :  addition  of  (X 2  .14  Discussion of  df  two c o n s e c u t i v e  arithmetic did that  A magnitude  equal  con-  maintain  for  this  to  the  smaller  arithmetic  than  and no d i r e c t i o n  A magnitude  smaller  and d i r e c t i o n  = 3 p >  vector  consecutive  and no d i r e c t i o n i s  A magnitude  This  .05;  2Ss  cited".(IF,1M) the  is  cited"(5F,7M).  than the is  arithmetic  arithmetic  cited".(2F,2M)  (1F,1M)  were  not  rule).  rule  to o b t a i n  c o u l d a l s o be of  displacements  c o n c e p t i o n made u s e the magnitude  was  from b e i n g p a r a l l e l  they concentrated Excerpts  only  all;  of  The a n g l e about to  "the  it  of  the  resultant  formed by  the  under  out  use  this  may be due  on f i g u r i n g  displace-  1 2 0 ° and h e n c e  justify  The two s u b j e c t s  not mention d i r e c t i o n at  called  composing v e c t o r  displacements.  addition.  the d i s t a n c e .  to  two n o n - p a r a l l e l t h e b o a t has . . .  holding this  addition  far  subjects'  f o l l o w i n g common o p e n i n g  addition conception"  two n o n - p a r a l l e l  much t o o  question  Results:  Subjects  arithmetic  the  of of  inferred  Rule A D - 1 :  arithmetic  rules  were o b t a i n e d  the magnitudes  assigned  of  rules  the magnitudes  =  o n l y one  different  inferred  the magnitudes  Inferred  the  was  presentation.  "The r e s u l t a n t displacements  a d d i t i o n of  ments.  as  w h i c h have  Rule A D - l i  Inferred  because there  to  how t o  from i n t e r v i e w d a t a w i l l  the was  of rule the  fact  find illustrate  -143-  this  conception.  Note:  The e s t i m a t e d  distances  were d i f f e r e n t whether Suzzane,  17  for  S.P.  each  (SP.P1  P-^P^  P^,  a  subject depending  t h e y had e a r l i e r  (Q.10)  to  offered  an  n  16:  (Q.10)  yards,  2P3  estimation.  = 50 m, P 1 P 2 = 60m,  (SP.P^^ = 200  P  upon  P2P3 =  . . . I f I need a map"'.'., about ( I : Why do y o u t h i n k so?) ... or something . . . " Larry,  d  P ^  100)  200m about  = 200,  220m  P2P3=200)  " . . . 600 y a r d s . . . ( I : B u t y o u a s k e d y o u r f r i e n d s t o go d i r e c t l y from t h e w h a r f t o t h i s s p o t , P3?) • • • w e l l , i t i s 600 yards . . . " Both addition,  appear  particularly  distances. she  subjects  Suzzane  t o be u s i n g  Larry:  hesitated  roughly, estimated  he  just  between  strict  added 200  arithmetic  the  and  t h e d i s t a n c e by a d d i n g  three  220: the  it  seems  partial  distances. Inferred  Rule AD-2:  realized  that  Subjects  t h e y c o u l d n o t use  t h e d i s t a n c e s were n o t observed metic  the  addition of  different It  that  was  kind of  way a p h y s i c i s t  most  i n the  result  strategies that  of  to  they  smaller  common s t r a t e g i e s  obtain  addition  it  this  is  than the They  the  because  and a l s o  arith-  result.  problem  worth examining  sort  of  problem.  were what c o u l d be c a l l e d  they  tried  a numerical  would s o l v e  but  tackling  conception  same d i r e c t i o n ,  s h o u l d be  w o u l d do i t  methods  arithmetic  t h e component d i s t a n c e s .  not expected  students'  holding this  the  the the  The  -144-  *distance  comparison s t r a t e g y '  strategy'.  The f i r s t  one c o n s i s t e d  length of  the  line  distance,  for  instance with  strategy  adequate  resultant.  and a r b i t r a r y u n i t  many t i m e s  that  estimated.  unit  This  proportionality. assuming  fits  requires  t h o u g h t h e y were n o t )  the  'scale  This  (draw a t  20,  15:(Q.10)  Of c o u r s e ,  classified  used  these  to  to  be  direct  consisted  triangle  in  (even  Theorem strategy  triangle.  of  from  ="200,  find  i t would be a v a l i d  r u l e d i d not mention  the subjects  strategies,  excerpts  (SP.P^  This  and s e e i n g how  strategy'  under t h i s  More t h a n h a l f  shown i n t h e f o l l o w i n g Kevin,  scale)  and u s i n g t h e P y t h a g o r a s '  t h e t r i a n g l e were a r i g h t a n g l e  12 o f  P^.  i n c h o o s i n g an  the understanding of  if  sample,  a known  approach'  t o be r i g h t a n g l e  the r e s u l t a n t .  all.  the  i n t o the d e s i r e d d i s t a n c e  find  at  SP t o  consists  to  direction  in estimating  the d i s t a n c e  The ' P y t h a g o r a s '  the t r i a n g l e s  Subjects'  'Pythagoras'  S P . P ^ by c o m p a r i n g b y e y e w i t h  v e r y much r e s e m b l e s  the magnitude o f  and t h e  P2P3  in  which w i l l  the be  interview data. =  300)  " . . . I h a v e t o f i g u r e o u t t h e d i s t a n c e from h e r e (SP) t o h e r e (P3) . . . 300 s q u a r e d p l u s 300 s q u a r e d . . . 900 p l u s 900 . . . 1800 . . . and t h e s q u a r e d r o o t o f t h a t w i l l be t h e d i s t a n c e " . Russ,  15:  (Q.10)  ( S P ^  = 400m, P ^ . = 300m P 2 P 3  =  300)  " . . . t a k e from t h e math . . . a b o u t 8 00m . . . we have t o e s t i m a t e because i t ' s .not g o i n g t h r o u g h t h e c u r v e l i n e (he meant p a s s i n g t h r o u g h a l l s p o t s ) . . . ( I : c o u l d i t be 1000m?) . . . no because i t goes i n s t r a i g h t l i n e " . Tracy,  1 5 : ( Q . 1 0 ) ( S P . P ] L = 200 "...  going i n t h i s  feet,. path  P ^  = 150,  (passing  =  through a l l  200)  -145spots) or j u s t s t r a i g h t ? . . . C_I: n o , d i r e c t l y from h e r e , S P , t o h e r e , P3) . . . i t w i l l be a b o u t 475 f e e t b e c a u s e t h e y w i l l be l o s i n g some f e e t . . i f y o u go t h i s way ( p a s s i n g t h r o u g h a l l s p o t s , zigzagging p a t h ) i s l o n g e r i f y o u go s t r a i g h t . . t h i s way i s s h o r t e r " . Sandy,  1 5 : (Q. 10) ( S P . P ^ = 100  yards,  P ^  = 120 P 2 P 3  =  200)  (Question not c l e a r f o r h e r . I : How f a r w i l l y o u t e l l them t o go?) . . . 300 y a r d s . . . ( I : Why do y o u t h i n k s o ? ) . . . s a y t h i s s p o t ( P 2 ) i s h e r e (she assumed t r a n s l a t i n g P2 t o be i n l i n e w i t h SP. and P 3 , i t s o r t s o f g e o m e t r i c a l p r o j e c t i o n o f P2 o v e r t h e l i n e SP.P3) . . . i t i s a b o u t 200 y a r d s . . . b e c a u s e f r o m t h i s (new l o c a t i o n o f P~) t o t h i s ( P 3 ) i s 200 . . . t h e n , i t may be 400 . . . Inferred  Rule AD-3:  one w i t h  regard  of  resultant  under  this  This  to the  rule  is  strategies  displacement;  but  similar  to  used  find  to  i n a d d i t i o n the  r u l e mentioned the d i r e c t i o n  ingredient.  Excerpts of  subjects  holding  Preston,  16:  this  i n t e r v i e w data  conception  (Q.10)(SP.P1  the  previous  the magnitud subjects  as  an i m p o r t a n t  of  some o f  the 4  follow.  = 200m, P P  = 100 P 2 P 3  =  200)  " . . . go t o a w e s t e r l y d i r e c t i o n w i t h a few d e g r e e s o f f . . . l i k e a b o u t 400m f r o m t h e s t a r t i n g p o i n t t o t h i s ( P 3 ) . . . ( I : Why do y o u t h i n k i s 400?) . . . b e c a u s e t h e a n g l e w i l l b e c u t o f f u n l e s s y o u go i n a z i g z a g g i n g m o t i o n , o r f r o m h e r e (P2) t o h e r e ( P 3 ) , b u t i f y o u go s t r a i g h t t h e a n g l e w i l l be s h o r t e r so i t w i l l be a s t r a i g h t l i n e s h o r t e r i n a p p r o x i m a t e l y 100m". Becky,  1 5 : ( Q . l ) ( S P . P 1 = 60 f e e t ,  P ^  = 40,  P2P3  -  40).  " . . . y o u c o u l d e s t i m a t e a b o u t 120 f e e t (I: Why i s 120?) . . . b e c a u s e f r o m h e r e (SP) t o h e r e (P2) i s 1 0 0 , t h e n y o u a d d a b o u t 2 0 . From t h i s (P2) t o h e r e (P3) i s 4 0 b u t y o u a r e g o i n g t h a t way, y o u a r e g o i n g s t r a i g h t p a t h . So i t ' s a 120 f e e t and a b o u t 85 d e g r e e s a n g l e a n d a b o u t 50 f e e t away f r o m t h e l e f t side".  -146Both direction the of  subjects  in addition  resultant the  e x p l i c i t l y mentioned a  has  to  t o be  the b e l i e f smaller  that  quantitative  the magnitude  than the  arithmetic  of  addition  components.  Table Rule-Model  Rule A D - I :  Rule A D - I I :  for  4.8  Addition  of  Displacements  " T h e r e s u l t a n t o f t h e a d d i t i o n o f two n o n p a r a l l e l d i s p l a c e m e n t s has a m a g n i t u d e e q u a l to the a r i t h m e t i c a d d i t i o n of the magnitudes and d i r e c t i o n i s n o t c o n s i d e r e d " . " T h e r e s u l t a n t o f t h e a d d i t i o n o f two n o n p a r a l l e l d i s p l a c e m e n t s has a magnitude smaller than the a r i t h m e t i c a d d i t i o n of the magnitudes .and d i r e c t i o n i s n o t c o n s i d e r e d " .  R u l e A D - I I I : " T h e r e s u l t a n t o f t h e a d d i t i o n o f two n o n p a r a l l e l d i s p l a c e m e n t s has a magnitude s m a l l e r than the a r i t h m e t i c a d d i t i o n of the magnitudes and d i r e c t i o n i s c o n s i d e r e d " .  4.3.5  Inferred  Rules  for  Subtraction  for  Vector  Position  7"^  . (S.V.P.) Similar displacements, explicit for  the  and w e l l  including this  characteristic the  to  addition of  vector  are  the  case of  the v e c t o r i a l  subtraction defined  to  displacements,  characteristic  vector  vectorial  operation similar  of  as  addition  quantities  operation.  an i m p l i c i t  needs o t h e r  the kinds  an  The r e a s o n s  including  i n c l u s i o n of of  is  vector  the ones g i v e n f o r but  of  this  a r g u m e n t s as  well.  -147First  it  vectors it  is  c a n be a r g u e d is  more a b s t r a c t  more r e m o t e  be c o n s i d e r e d as compared t o Also  there  before cepts of  that  previous  use  different  of  points,  (magnitude  association  of  locations  or  vector  it  handle a l l  these concepts  purpose  vector  intuitive  Did  including  with  this  notion of  strategies  locations that  this  was  of  understood These  describe  for  of  points,  formal  and  fact  that  the  the  stationary  be a b l e t o  instruction,  situation.  relate  of  a moving body.  o p e r a t i o n as  to  con-  locations  association  subjects w i l l  how t o d e a l w i t h  can  "natural"  an  but  implicit  t o u n c o v e r how s u b j e c t s  sort  the  more  vector.  p o i n t to  before  d i d t h e y use  they consider  is  t o be  reference,  not expected  characteristic  confronted  of  of  Hence i t  p o s i t i o n to p a r t i c u l a r  instantaneous is  of  of  and d i r e c t i o n )  Obviously,  the  subtraction  frame  coordinates  which  concepts  one r e f e r e n c e  becaus  c o n c e p t i o n when  displacement  formally grasping are:  experience.  a more " a r t i f i c i a l "  several  subtraction o f  than the a d d i t i o n o p e r a t i o n  from s t u d e n t s '  a d d i t i o n of are  the o p e r a t i o n of  c o p e when  D i d t h e y have  the problem? the  What  two g i v e n  two d i s t a n c e s  any kind  distances were  not  parallel? It vectors in  may b e t h a t  contains  the o p e r a t i o n of  more i m p l i c i t  the r a t i o n a l tasks a n a l y s i s .  unknown i m p l i c i t students'  aspects  preconceptions,  dealing with  aspects If  that  subtraction  t h a t w^fe n o t - d e t e c t e d is  c o u l d be r e v e a l e d  the case, through  w h i c h may become m a n i f e s t  the p r o b l e m a t i c  situation.  of  these  the when  -148-  Questions  of  Interview Protocol  Questions:  11 and 12 i n t a s k  Question  11:  yourself  (on t h e d o c k )  at  P^)  for  is  different  and ~92 a spot to  Suppose you have  and i t  distance  Note:  e  o  n  (P^).  this  to  about  subjects)  How a r e  spot This  to  this  (different  is  estimated  spot P 2  (the  Your  friends  are  tell  placed  distances  other  line) .  q u e s t i o n was  make u s e  of  solution.  the  As w i t h served  used  them t o  only with  subjects with  t h a t most o f  strict  dock, at  go  w e r e us  the P^,  this  directly  agreed d i d use  a  12:  from y o u r s e l f  with  arithmetic  situation  would to  expectation:  find  a  18 .  subtraction.  because t h i s their  i n the a n a l y s i s  of  question responses the  rules.  Suppose you have a l s o  estimated  to  Figure  this  spot  understanding of  situation)  Your  friends  again  tell  them t o go d i r e c t l y t o  are  of  of  subjects  f a m i l i a r i z a t i o n procedure  not being considered  inferred  sort  subtraction  the a d d i t i o n o p e r a t i o n , as  the purpose this  the  arithmetic  Results  among 20 s u b j e c t s  Question  (thumb t a c k  from  (P2)?  was e x p e c t e d  are  spot  you g o i n g t o  familiarizing It  this  the d i s t a n c e  and y o u h a v e a l s o  straight  a  estimated  200 m e t r e s  from y o u r s e l f r  one.  (P3* see and i t  in this  spot  this  is  about  (P^).  other  spot  the  3.5 f o r 300  distance best  metres.  How c o u l d y o u (P^)?  -149-  Note:  It  was  thought  characteristic about  that  one q u e s t i o n  would  subtraction  of  Table List  of  inferred rules  refers  to  the magnitude  vector  positions  second,  of  (magnitude  the d i r e c t i o n  Both aspects  are  Again, although  vector  subjects'  as  conceptions  4.9 of  c o n t a i n two a s p e c t s .  The  first  the  resultant  of  subtracting  two  of  resultant  displacement);  and  (qualitative  or  quantitative).  s e p a r a t e d by the c o n j u n c t i o n a n d . i n the case of  t h e r e was  one q u e s t i o n  addition of for  thedifferent  subjects'  ferred  rules  to maintain p a r a l l e l i s m  Three  inferred rules  characteristic,  vector  quantities.  Inferred Rules for S u b t r a c t i o n V e c t o r P o s i t i o n (SVP)  These  the  show t h e  on t h i s  this  displacements,  vector  characteristic  c o n c e p t i o n s were c l a s s i f i e d i n the data  were o b t a i n e d  w h i c h have t h e  for  as  in-  presentation.  this  vector  f o l l o w i n g common o p e n i n g  sentence: "The r e s u l t a n t o f the s u b t r a c t i o n non-paxalLel vector p o s i t i o n s for s p o t s has . . . Inferred  Rule SVP-1:  subtraction  of  A magnitude  the magnitudes  equal  to  o f two two f i s h i n g v the  and d i r e c t i o n i s  arithmetic not  cited."  (3F,4M) Inferred  Rule SVP-2:  arithmetic not c i t e d . "  A magnitude d i f f e r e n t  subtraction of (1F,3M)  the magnitudes  from  the  and d i r e c t i o n  is  -150-  Inferred  Rule SVP-3;  arithmetic cited".  subtraction  = 2.46  assigned Discussion of Inferred the  df  'arithmetic  This  the  and d i r e c t i o n  this  strategy,  Earlier  be r a r e  find  by  the  0°  to  or  parallel  for  one.  expected.  vectorial  is  (1M) was  not  rule).  it  that  two v e c t o r say  (the  Excerpts  c o u l d a l s o be r e f e r r e d conception'  result  that  students develop  it  the  by t h e m s e l v e s .  very  would  The a n g l e  the  straight  lines  were  angle  was  actually  about  40  This  was likely operation formed  different  that  from i n t e r v i e w d a t a w i l l  sample,  different  abstract  sufficiently  obtain  sample.  c o n f i r m e d what  suggested  p o s i t i o n was  to  In the  the is  to  subtracting  positions  of  a n o t i o n which  was  of  displacement.  35 p e r c e n t  This  subtraction  1 8 0 ° to  rule  the r e s u l t a n t  accepted  of  IS  two n o n - p a r a l l e l v e c t o r  a high percentage  to  p > .05;  subtraction  of of  20 u s e d  from t h e  = 3  an i n f e r r e d  Rule SVP-1;  the magnitude  was  the magnitudes  from  Results  the magnitudes  7 of  of  different  (5F,3M) (X 2  as  A magnitude  in  from  line  degrees).  show t h e  subj^ect's  conceptions. (S:  stands  Dale,  15:  for  subject's  (Q.12).  ( S ^  location,  on t h e  = 100 m e t r e s ,  " . . . i t w i l l be 150 m e t r e s up t o h a v e 2 5 0 " . Hans,  17:  (Q.17) ( S ^  = 100m,  S.P3  dock)  S.P3 ...  = 250m) you j u s t  add  = 400m)  " . . . a b o u t 300m . . . ( I : Why do y o u t h i n k i t is 300?) . . . b e c a u s e t h i s d i s t a n c e (S.P,) i s 100 a n d t h e o t h e r one i s 400 "  them  -151-  Kelly,  1 5 : (Q.12) (S.Pj^ = 200m, "... tell and t h a t joinging and t e l l . . . well then i t ' s  These three subtracting  S.P3  =  600m)  t h e a n g l e between the f i r s t s p o t s p o t ( t h e a n g l e f o r m e d by t h e l i n e s each spot w i t h the s u b j e c t ' s l o c a t i o n ) them t o go 400m . . . ( I : Why 400m?) t h e r e a r e 200 h e r e and 600 t h e r e . . . a b o u t 400m"  s u b j e c t s were s t r i c t l y a r i t h m e t i c a l l y  the magnitudes  of  the  two n o n - p a r a l l e l v e c t o r  positions. Inferred  Rule SVP-2:  were aware line  of  the  Subjects  fact  that  holding.this preconception  the  two d i s t a n c e s  w h i c h w o u l d n o t a l l o w them t o u s e  traction. that  the  greater to  length of  important aspect the d i s t a n c e  than the r e s u l t  the  find  Another  two d i s t a n c e s .  a n u m e r i c a l answer  of  were n o t  arithmetic  that  and P^ must  applying arithmetic  tried  i n t h e i r endeavour  several  distance  strategy',  strategies.  strategy',  displacements.  show t h e u s e .20  i n estimating  of  w h i c h was a l s o  Excerpts  found f o r  from i n t e r v i e w d a t a  t h e s e and o t h e r  strategies  to  Among 'length a  b y c o m p a r i n g b y e y e w i t h a known l e n g t h , -afid  'Pythagoras of  which c o n s i s t s  be  subtraction  t h e more common were a g a i n what c a n be c a l l e d t h e comparison  sub-  s u b j e c t s o b s e r v e d was  between  Subjects  in  the  addition will  by f o u r o f  better the  subjects.  Wayne,  15:  (Q.12) (S.P-j^ = 200m,  S.P3  = 600m)  " . . . y o u c o u l d e s t i m a t e a b o u t 400m . . . (I: Why 4 0 0 m ? ) . . . i t ' s n o t p e r f e c t c o n t i n u o u s s t r a i g h t l i n e . . . ( I : W i l l i t be more o r l e s s  -152-  t h a n 400?) . . . a l i t t l e c o u l d you get the e x a c t two s p o t s ? ) . . . m e a s u r e l i n e s g o i n g from me t o t r i g n o m e t r y or the angle b e t w e e n t h e two s p o t s " Lisa,  15:(Q.12)(S.P1  = 200m,  S.P3  b i t less ( I : How d i s t a n c e between t h e the a n g l e between the t h e two s p o t s , u s i n g to c a l c u l a t e the d i s t a n c e = 450m)  " I s u b t r a c t t h e d i s t a n c e f r o m me t o t h e f i r s t s p o t f r o m t h e d i s t a n c e f r o m me t o t h e s e c o n d s p o t . . . and s u b t r a c t a l s o t h e d i s t a n c e f r o m t h e f i r s t s p o t t o here (the p r o j e c t i o n p o i n t o f on t h e s t r a i g h t l i n e c o n n e c t i n g t h e s u b j e c t w i t h P3) ... I h a v e t o s u b t r a c t t h a t b e c a u s e I am g o i n g i n s t r a i g h t l i n e , t h e o t h e r way i s f a r t h e r o u t " .  Although both subjects it  is  clear  that  t h e y d i d so  used  because  two d i s t a n c e s were n o t p a r a l l e l operations  c o u l d n o t be  Inferred  Rule SVP-3:  one w i t h  the d i f f e r e n c e  ception cited as  subjects,  20,  c o u l d mean t h a t  these  understand  is  formally presented  to  they r e a l i z e d  and c o n v e n t i o n a l  is  similar  subjects  that  the  arithmetic  to hold  the  previous  or  precon-  quantitative)  A high percentage this  s u b j e c t s w o u l d be of  to  holding this  (qualitative  consider.  the o p e r a t i o n  from i n t e r v i e w d a t a  15:  that  appeared  to  Becky,  rule  the d i r e c t i o n  8 of  strategies,  used.  This  an i m p o r t a n t a s p e c t  two d i f f e r e n t  vector  conception. in a better  subtraction  i n an i n s t r u c t i o n a l  s^e^tting.  of This  position  when  it  Excerpts  follow.  (Q.12) ( S ^  = 50m  S.P3  = 200m)  " i t ' s a b o u t 150 away ( I : Why?) well, t h a t i s 200 and t h i s 5 0 , and i t i s n o t v e r y much i n a n a n g l e . . . t h e n i t i s 150 . . . maybe a l i t t l e b i t more ( I : Why a l i t t l e b i t more?) . . . a b i t more o r l e s s and i t i s 85 d e g r e e s a n g l e " .  -153-  Tracy,  15:(Q.12)(S.P1  = 200m,  S.P3  = 600m)  " I t e l l them t o go . . . (pause) . . . a b o u t 400 o r 500 . . . b e c a u s e t h e y have t o go t h i s way . . . t h e y h a v e t o go f a r t h e r . . . I t e l l t o go t o t h e l e f t a b o u t 450m . . . o r 500m . . . ( I : Why 500?) . . . b e c a u s e i s q u i c k e r f o r me t o go . . . l i k e from me t o t h e r e (a p o i n t w h i c h c o r r e s p o n d s t o t h e p r o j e c t i o n o f P^ on t h e s t r a i g h t l i n e j o i n i n g s u b j e c t ' s l o c a t i o n and P3) i s a b o u t 200m and from me t o h e r (P3) i s 600 . . . t h e n i t w i l l be 400 b u t he i s h e r e ( P i ) t h e n he h a s t o t a k e a l o n g e r way . . . a b o u t 4 5 0 " .  Both of  subjects  obtaining  provided d i r e c t i o n i n a d d i t i o n to  the magnitude  Table Rule-Model  of  the  ways  resultant.  4.10 f o r SVP  Rule  SVP-I:  The r e s u l t a n t o f t h e s u b t r a c t i o n o f two n o n p a r a l l e l v e c t o r p o s i t i o n s has a magnitude equal to the a r i t h m e t i c s u b t r a c t i o n of the m a g n i t u d e s and d i r e c t i o n i s n o t c o n s i d e r e d " .  Rule  SVP-II:  " T h e r e s u l t a n t o f t h e s u b t r a c t i o n o f two n o n - p a r a l l e l v e c t o r p o s i t i o n s has a magnitude d i f f e r e n t from the a r i t h m e t i c s u b t r a c t i o n o f t h e m a g n i t u d e s and d i r e c t i o n i s n o t c o n s i d e r e d " .  Rule  SVP-III:  " T h e r e s u l t a n t o f t h e s u b t r a c t i o n o f two n o n - p a r a l l e l v e c t o r p o s i t i o n s has a magnitude d i f f e r e n t from the a r i t h m e t i c s u b t r a c t i o n o f t h e m a g n i t u d e s and d i r e c t i o n i s c o n s i d e r e d " .  -1544.3.6  I n f e r r e d Rules  for  Moving O b j e c t s  (RPM)  The. a i m o f  this  second p a r t  t o u n c o v e r how s t u d e n t s bodies  affected  situation  (task  b o a t w h i c h was motor,  by d i f f e r e n t 2)  was  to d e s c r i b e  affected  a motion?  D i d they r e a l i z e  this  specific  a motor-  to address that  that  the  the boat point to  t h e b o a t had  to d i f f e r e n t  reference  " r e f e r e n c e b o d y " was u s e d  frame  its  following is  i n motion  describe different  reference  points?  points?  particular vector characteristic,  a second r e f e r e n c e  might  describing  river.  instead  of  the  "reference  T h i s was done b e c a u s e t h e m o v i n g b o d y o f w a t e r as  p o i n t was  The  the motion of  the  tell  of motions with respect  For  the problem of  the need f o r a r e f e r e n c e  What d i d t h e y c h o o s e a s  reference  b o t h by the m o t i o n c r e a t e d by  How d i d s t u d e n t s  t h e y see  phrase  with  about  for  kinds of motions.  s e c t i o n attempted  questions.  kinds  dealt  and a l s o b y t h e c u r r e n t o f This  Did  Reference Bodies  point  c o u l d be  to which the motion o f  the  used boat  refer.  Questions Questions:  of  Interview Protocol  13b,  Question  13c,  13b:  film with  the  and 13d i n t a s k (After  subject  two.  has  seen p a r t  t h e b o a t moving on the r i v e r ) .  t o what i s Question  f o r RPM  With  the respect  t h e b o a t moving?  13c:  water?  of  Was t h e b o a t m o v i n g w i t h r e s p e c t  to  -155Question  13d;  (If  question  13c):  Is  to  you  with  (or b r i d g e  respect  Note:  It  to  is  s u b j e c t has the or  shore)  the  understanding  the phrase:  of  tell  that  To u n d e r s t a n d to consider  question  an i m p o r t a n t material. to  take  "with that  the  that  only  asked,  is  closer  factor  to  consider  to",  appears  that but  n o t t o be  questions. in  to  (this  The f o l l o w i n g  facilitate  the  "How c o u l d y o u  moving?" of  t h e p r o b l e m , we  20 s u b j e c t s  and o n l y  to a  problem,  when p r e p a r i n g  of  case.  question it  is  curriculum  teachers  u n d e r s t a n d -ttle  findings  understood  6 understood  semantic  and s c i e n c e  the  to  situation:  students  the  these  in  had d i f f i c u l t i e s  used  10 o u t o f  this  granted  respect  the  boat  encountered  with respect  seriousness  Textbook w r i t e r s  for  this  of  was  the  the  13b and 1 3 c ) .  the boat i s  13b when f i r s t  Although  subjects  questions  understanding  13c.  some s u b j e c t s  respect  different?  to mention here  In p a r t i c u l a r  part  for  speed o f  same o r  the meaning o f  equivalent  question  the  answer  the boat w i t h  and t h e  very important that  "yes"  understanding  is  have  speed of  the water  difficulties  a  often  seem  phrase  the present  study  show  -156-  Table List  of  4.11  Inferred Rules : for f o r Moving O b j e c t s  Reference (RPM)  Each i n f e r r e d r u l e covers relates  to  stationary  the b r i d g e ,  the b r i d g e ,  aspect refers river)  as  inferred  reference  to  the  and moving) Four  are  body.  both types  two a s p e c t s . bodies  shore), of  The  (i.e.,  observer  and t h e  the moving water  reference  bodies  on  second  In the d e s c r i p t i o n of of  first  (the the  (stationary  s e p a r a t e d by t h e c o n j u n c t i o n a n d .  inferred rules  characteristic,  the  selection  a reference rules,  or  Bodies  were o b t a i n e d  w h i c h have  the  for  this  vector  f o l l o w i n g common o p e n i n g  sentence: " D e s c r i p t i o n of the motion o f the motor-boat on t h e r i v e r b y a n o b s e r v e r o n t h e b r i d g e . . . Inferred lative  to  R u l e RPM-1:  Does n o t d i s t i n g u i s h movements  a stationary  a moving body  (e.g.,  body  (e.g.,  the water)"  bridge)  (3F,0M).  re-  and r e l a t i v e  to  -157I n f e r r e d Rule RPM-2: D i s t i n g u i s h e s movement r e l a t i v e t o an i m p l i c i t reference point to  a moving body  (i.e.,  the observer)  (e.g., the w a t e r ) " .  I n f e r r e d Rule RPM-3:.  and r e l a t i v e  (1F,2M)  D i s t i n g u i s h e s movements r e l a t i v e t o  s e v e r a l s t a t i o n a r y bodies  (e.g., the b r i d g e , shore) and  r e l a t i v e t o a moving body  (e.g., the w a t e r ) " .  I n f e r r e d Rule RPM-4: one  s t a t i o n a r y body  moving body  (X  2  (4F,6M)  D i s t i n g u i s h e s movements r e l a t i v e t o (e.g., the bridge)  (e.g., t h e w a t e r ) " .  = 2.90 d f = 4  was n o t a s s i g n e d  and r e l a t i v e t o a  (0F,3M).  p > .05; 1S(1M) an i n f e r r e d  rule).  Discussion of Results I n f e r r e d Rule RPM-1:  T h i s r u l e can be i n t e r p r e t e d t o mean  that only " s t a t i o n a r y bodies" bodies.  can be used as r e f e r e n c e  Subjects holding t h i s conception  d i d not r e l a t e  the motion o f t h e boat "with r e s p e c t t o " the moving water. They s a i d t h a t i t was moving "with" be  t h e water.  This could  i n t e r p r e t e d as meaning t h a t they d i d n o t see the i n d e -  pendent motion o f t h e boat w i t h r e s p e c t t o the water, compared t o the motion w i t h r e s p e c t t o the b r i d g e .  This  i n d i c a t e s t h a t t h e moving water i t s e l f was n o t used as a r e f e r e n c e body t o d e s c r i b e the motion o f t h e boat. example, i f the motor o f t h e boat i s not r u n n i n g ,  For then  the v e l o c i t y o f t h e boat w i t h r e s p e c t t o the water would  -158be  "zero".  Since only  conception ception  best  Julie,15  the  respect  to  Excerpts  from  inter-  conception.  (Q.13d)  " . . . . i t i s t h e same . . . ( I : Why?) . . . w e l l I am n o t m o v i n g so i f t h e b o a t i s m o v i n g and t h e r i v e r i s m o v i n g c a r r y i n g i t so i t ' l l b e t h e same s p e e d " .  . . .  it  wasn't  " y e h , b u t t h e w a t e r was a l s o  (Q.13d)  " I think-it will do y o u t h i n k so?) speed u p " .  it  They r e a l i z e d t h a t c a r r i e d the boat  i t was n o t c l e a r  the boat  the  show t h a t  the independent motion of  the water.  water.  with respect  moving".  be t h e same ( I : Why . . . because i t c a n ' t  responses of both subjects  of  moving  " . . . ( a f t e r s e v e r a l a t t e m p t s t o make t h e q u e s t i o n u n d e r s t a n d a b l e , I : How c o u l d y o u t e l l when an o b j e c t i s m o v i n g ? ) . . . w e l l , i t g o e s p a s t y o u and I g u e s s y o u c o u l d see t h e s c e n e r y . . . t h a t i s , m o v i n g away f r o m i t . . . i t ' s n o t a t ^ t h e same stationary scenery".  (Q.13c)  t h e i r answers speed  this  n o t a common c o n -  " . . . a little bit f a s t but a b i t  own spaexl, and t h a t in  10 s u b j e c t s .  is  this  (Q.13c)  15:(Q.13b)  to  it  hold  " . . . i t was m o v i n g s l o w l y b e c a u s e i t was a g a i n s t t h e c u r r e n t . . . ( I : How c o u l d y o u t e l l t h a t t h e b o a t was m o v i n g ? ) . . . when i t moves b e c a u s e t h e e n g i n e and t h e c u r r e n t ( I : Was i t m o v i n g w i t h r e s p e c t t o you?) . . . i t c o u l d i t be . . . ( I : was i t moving w i t h r e s p e c t t o something e l s e ? ) .. not r e a l l y .  differentiating  respect  19 s u b j e c t s  indicate that  illustrates  (Q.13b)  The not  would  among t h e g r a d e  view data  Becky,  this  3 out of  if  t h e y were  the boat  the water  along with  they  it  with  had  its  but  differentiated  to themselves,  or  with  -159Inferred  R u l e RPM-2:  explicitly describe  that  need  for  other of  the motion, i t with these  respect  to  b o a t moved w i t h fact  that  the b o a t ,  Suzanne,  15:  this  which  it  c a n be to  are  respect  to  the water. respect  to  17:(Q,13b)  body  It  could differentiate  That i s , the water  t h e y were  of  a motion.  themselves  to  cannot  c o n s c i o u s l y aware  c a r r i e d the boat.  be  the On t h e  the motion  and t h e m o t i o n  t h e y saw t h a t  the  independent of  the  Three subjects  among  d e s c r i b i n g the motion of  shown t h r o u g h t h e  " . . . it moving Was i t ... it moving (pause)  r u l e d i d not  inferred that  themselves.  conception of is  this  reference  body t o d e s c r i b e  students  the water  twenty h e l d  Lori,  subjects  hand, these with  but  respect  a reference  the boat  with  holding  m e n t i o n any s t a t i o n a r y  describing said  Students  following  excerpts:  l o o k e d l i k e t h e b o a t was . . . I saw a l i t t l e wave ( I : m o v i n g w i t h r e s p e c t t o what?) was m o v i n g s l o w l y . . . (was i t w i t h r e s p e c t t o yo^r?) . . . y e h . . . . . .  (Q,13d)  " . . . i t is different rushes i t more"  (Q.13b)  "By l o o k i n g at i t you can t e l l it's m o v i n g and f a s t e r t h a n t h e w a t e r ( I : Is the boat moving w i t h r e s p e c t to the shore?) . . . y e h . . . (pause) ( I s i t m o v i n g w i t h r e s p e c t t o you?) yeh, i t ' s getting c l o s e r " .  (Q.13c)  "Yeh, . . . me a n d so  (Q.13d)  "I  . . .  the  water  the water i s coming i s the boat"  d o n ' t know  . . .  towards  I d o n ' t understand  (I r e p h r a s e d t h e q u e s t i o n ) ... be d i f f e r e n t b e c a u s e t h e w a t e r boat they are both moving".  . . .  . . .  it will and t h e  -160Both reference using  it  point,  themselves  looked at  like  •respect Inferred  to  expressed  point. to  that  body.  t h e y were Suzzane  and L o r i  moving".  said  Both excerpts  They a l s o  themselves  implicitly  said: "by  looking  show  differentiated  and t h e  speed  "it  the the  with  water. The s u b j e c t s  in this  reference  holding  inferred rule  the  the motor-boat.  Ten o f  preconcep-  u s e d more t h a n one  b o d y and a m o v i n g b o d y t o  c o n c e p t i o n , which i s  excerpts  clear  was m o v i n g "  it's  respect the  was  a reference  the boat  the motion of  Dale,  it  Rule RPM-3:  stationary  this  as  reference  speeds w i t h  d i d n o t e x p l i c i t l y m e n t i o n any  but  you can t e l l  implicit  tion  subjects  describe  20 s u b j e c t s  shown t h r o u g h t h e  held  following  :  15:(Q.13b)  ( I : How c o u l d y o u t e l l t h a t t h e b o a t was m o v i n g ? ) . . . b e c a u s e t h e / e n g i n e . . . i t was g o i n g w i t h t h e c u r r e n t . . . i t p a s s e d t h i s ( t h e d o c k s ) and i t w i l l be g o i n g u n d e r t h e b r i d g e ( I : Was i t moving w i t h r e s p e c t t o you?) . . . yeh, i t was c o m i n g t o w a r d s m e " .  (Q.13c)  " p r o b a b l y i t was g o i n g f a s t e r t h a n w a t e r . . . t h e w a t e r was h e l p i n g i t move" . . .  the to  (Q.13d)  " t h e e n g i n e was r u n n i n g , t h e n , i t w i l l be d i f f e r e n t . . . b e c a u s e . . . t h e r e i s d i f f e r e n t s p e e d b e t w e e n t h e w a t e r 'and the boat. I f t h e e n g i n e i s o f f maybe i t w i l l go t h e same s p e e d a s t h e w a t e r and i f t h e e n g i n e i s on i t w i l l p r o b a b l y be f a s t e r t h a n the w a t e r . I f the water g o e s 10 m i l e s / h r , t h e b o a t w i l l go maybe 12 m i l e s / h r . "  -161-  Hans,  17:  Steve,  (Q.13b)  (How c o u l d y o u t e l l t h a t t h e b o a t was moving?) . . . "by t h o s e f i x e d buoys . . . (pause) . . . ( I : Was i t m o v i n g w i t h r e s p e c t t o y o u ? ) . . . y e h . . . ( I : Was i t moving w i t h r e s p e c t t o a n y t h i n g e l s e ? ) . . . y e h , the shore . . .  (Q.13c)  "Was t h e m o t o r on? c o u l d go f a s t e r  (Q.13d)  " D i f f e r e n t speeds . . . because w i t h m o t o r i t w i l l go f a s t e r "  15:(Q.13b)  (bridge,  buoy,  or  shore)  the  different  reference  bodies.  Inferred  R u l e RPM-4:  speed  to  also  as of  a reference the boat  Subjects holding  have e x p l i c i t l y chosen a body  with respect  (the b r i d g e )  differentiated  the water,  this  and were aware  the bridge  a  that  to  and  respect  the  the  boat  water  the boat  was  and a l s o w i t h r e s p e c t  T h i s does n o t n e c e s s a r i l y  to  reference.  the motion of  moving w i t h r e s p e c t  boat.  the  conception as  Thus t h e y b e l i e v e d t h a t  ceived  it  body  point,  with  was a m o v i n g b o d y .  the water.  . . .  c h o s e more t h a n one s t a t i o n a r y  they d i f f e r e n t i a t e d  These s t u d e n t s  yes)  " I t w i l l be d i f f e r e n t ( I : Why do y o u t h i n k so?) . . . on t h e w a t e r i t ' l l go o n l y 20 m i l e s / h r (He assumed b e f o r e t h a t s p e e d o f b o a t due t o t h e e n g i n e i s 20) b y l o o k i n g f r o m my p o i n t o f v i e w i t ' l l seem t o move f a s t e r b e c a u s e t h e current i s pushing i t . . . "  subjects  dock,  (I:  " . . . r e s p e c t t o t h e w a t e r ( I : Was i t moving w i t h r e s p e c t t o a n y t n i n g e l s e ? ) . . . w i t h r e s p e c t t o y o u and where I am and b o t h s i d e s o f t h e r i v e r " .  (Q.13d)  All  ...  mean t h a t  they  a r e l a t i o n s h i p b e t w e e n t h e s e two m o t i o n s o f  to  con-  the  -162-  Excerpts  from i n t e r v i e w d a t a  best  illustrated  this  conception:  Kevin,  Larry,  15:  16:  (Q.13b)  " . . . t o t h e w a t e r . . . ( I : Was i t moving w i t h r e s p e c t t o a n y t h i n g e l s e ? ) . . . t h e b r i d g e . . . (pause) . . .  (Q.13d)  " I t h i n k i t w i l l be d i f f e r e n t b e c a u s e t h e w a t e r i s m o v i n g and I am j u s t s t a n d i n g t h e r e , t h e b o a t w i l l be g o i n g f a s t e r compared t o me b u t l e s s f a s t compared t o t h e w a t e r "  (Q.13b)  " i t was m o v i n g w i t h r e s p e c t t o t h e r i v e r ( I : Was i t m o v i n g w i t h r e s p e c t t o something else?) . . . to the b r i d g e " .  (Q.13c)  " . . . i t w i l l be d i f f e r e n t b e c a u s e I am i n a f i x e d p o s i t i o n and t h e r i v e r i s m o v i n g so i t w i l l be d i f f e r e n t .  T h e s e two s u b j e c t s specific at  the  respect  stationary  body  have e x p l i c i t l y c h o s e n (the  bridge)  as  same t i m e t h e y h a v e d i f f e r e n t i a t e d to  one s t a t i o n a r y  one  a reference, the  speeds  b o d y and w i t h r e s p e c t  to  and with  a moving  body.  Table Rule-Model  Rule  RPM-I:  Rule RPM-II:  4.12 f o r RPM  " D e s c r i p t i o n s o f t h e m o t i o n o f an o b j e c t r e l a t i v e t o a s t a t i o n a r y r e f e r e n c e b o d y and r e l a t i v e to a moving r e f e r e n c e body are equivalent". " D e s c r i p t i o n o f t h e m o t i o n o f an o b j e c t r e l a t i v e t o an i m p l i c i t s t a t i o n a r y reference p o i n t and r e l a t i v e t o a m o v i n g r e f e r e n c e body a r e u n e q u i v a l e n t " .  -163-  Rule RPM-III:  " D e s c r i p t i o n s o f t h e m o t i o n o f an o b j e c t r e l a t i v e to s e v e r a l s t a t i o n a r y reference b o d i e s and r e l a t i v e t o a m o v i n g r e f e r e n c e body a r e u n e q u i v a l e n t " .  R u l e RPM-IV;  " D e s c r i p t i o n s o f t h e m o t i o n o f an o b j e c t r e l a t i v e t o one s t a t i o n a r y r e f e r e n c e b o d y and r e l a t i v e t o a m o v i n g r e f e r e n c e b o d y are u n e q u i v a l e n t " .  4.3.7  Inferred  Rules  The n o t i o n o f component m o t i o n s is  a decisive  the  aspect  vector a  aspect  is  to  consider  vectors  is  example  of  characteristic important ties  yet  it  is  textbooks. easily the  to  of  gravitational  with constant  is  the  in  of  Usually,  teaching  not considered  aspect represents  that  an i m p l i c i t i n that,  it  is  case of  field.  single  motions.  projectile  A few s t u d e n t s of  constant  at  i n most  an  least  not  motion can  be  A common t e x t -  motion i n  realize  two m o t i o n s ,  Some s t u d e n t s ,  the  that a horizontal  v e l o c i t y and a v e r t i c a l  acceleration.  quanti-  physics  some s t u d e n t s do  an a p p a r e n t  as a  c o n s i d e r i n comprehending v e c t o r  that  the r e s u l t  component w i t h  it  study,  combining several is  a moving body  t o be a c c o m p l i s h e d .  This  the  an u n d e r s t a n d i n g  been c a l l e d  The p r o b l e m i s  book e x a m p l e  motion i s  what h a s  (A.C.)  number o f  affect  not e x p l i c i t l y treated  recognize  result  if  o r even worse  i n the present  aspect  the  very s u p e r f i c i a l l y  s e p a r a t e a s p e c t by i t s e l f .  clear  Components  simultaneously  treated  quantities,  Analysis of  differentiating  that  composition of  this  for  component  however,  have  -164the b e l i e f  that  if  the path of  line,  the body i s  it  a curved path,  is  a moving body i s  only being affected the body i s  a  straight  b y one m o t i o n ;  being affected  if  b y more  t h a n one m o t i o n . An i m p o r t a n t a s p e c t quantity  is  particular position  to  see  how many c o m p o n e n t s a r e  situation,  of.vector  how t h e s e  quantities),  be a n a l y z e d when o n l y For  the  last is  purpose  of  this  kind of  conceptions  part-of  single  Questions:  Question different  14b,  14b:  a  how t h e y  can  them i s  of  t h e number o f  For t h i s to  about  Do s t u d e n t s  com-  apparent.  reason,  the main  u n c o v e r what analysis  realize  of  that  a motion  c o m b i n i n g a number o f m o t i o n s ? all  or  some o f  the  Do  component  body?  Question  14d:  (or  (Boat  motions they?  14d,  (Boat  motions they?  What a r e  of  (i.e.,  of  s t u d y was have  in a  Interview Protocol  What a r e  different  the  motion.  motions a f f e c t i n g  of  finding  students  differentiate  Questions  the  vector  present  combined  and a l s o  a c r u c i a l aspect.  c a n be t h e r e s u l t students  are  the r e s u l t a n t  two s t e p s ,  components  apparent  i n w o r k i n g w i t h any  (or  15c  i n task  two.  moving a g a i n s t speeds)  are  moving w i t h speeds)  are  the current)  affecting  the  the c u r r e n t ) affecting  the  What boat?  What boat?  -165Question  15c:  (Boat m o v i n g a c r o s s  What d i f f e r e n t are  motions  are  the  affecting  river)  the boat?  What  they?  Table List  of  4.13  Inferred Rules for A n a l y s i s o f Components ( A . C . )  Two i n f e r r e d r u l e s characteristic,  were o b t a i n e d  which have  the  for  this  vector  f o l l o w i n g common o p e n i n g  sentence: " A n a l y s i s o f components f o r t h e m o t i o n o f the m o t o r - b o a t on the r i v e r p r o d u c e s . . . Inferred being  Rule A C - 1 :  either  due t o  t h e m o t i o n due  the motor".  Inferred current  One component a f f e c t i n g  Two c o m p o n e n t s  and one. due t o  the motion  : t h e one due  the engine".  = 1.80 d f  not assigned  Discussion of  to the c u r r e n t o r  = 2  p >  (9F,  .05;  an i n f e r r e d  IS  rule).  Results The p r e c o n c e p t i o n e x p r e s s e d  inferred  rule  to the  this  refers  the motion of  subject  inferred  fact  the boat,  same component i n e a c h  the o n l y  the  (1M) was  Rule A C - 1 :  the  to  9M)  Inferred  affects  this  (0F,1M)  Rule A C - 2 :  (X2  it,  it  was  situation.  holding this  rule.  that  only not  one  in  this  component  necessarily  The r e s p o n s e s  conception best  of  illustrate  rl66Hans,  17:  (Q.14b)(Question not c l e a r for him, which i s repeated) s t i l l I d o n ' t understand . . . C I : when t h e b o a t i s g o i n g a g a i n s t t h e c u r r e n t , what i s a f f e c t i n g i t s m o t i o n ? ) ... the c u r r e n t . . . (I: a n y t h i n g e l s e ? ) ... ( l o n g pause) . . . ( I : R e c a l l t h a t t h e e n g i n e i s on) . . . t h e n , y o u h a v e t h e c u r r e n t and t h e e n g i n e . . . " (Q.14d)  " . . . j u s t one m o t i o n . . . j u s t the motor".  (Q.15c)  " . . .  Considering  just  two,  and 1 4 d ,  it  (I:  the c u r r e n t  the responses  prompted response)  ...  to  is  14b  clear  Which one?)  and t h e  (excluding that  this  motor"  the subject  saw o n l y one component i n a c t i o n upon t h e b o a t ,  although  seems t o  It  cult  shift  from t h e c u r r e n t t o  t o d e t e r m i n e what e f f e c t  o c c u r r e d when t h e  introduced  t h e component due t o  after  hint  that  affecting ject  the  the boat  seemed  pretty  the motor.  the motor  14d.  On t h e o t h e r  sure i n h i s  response  m e n t i o n e d two components.  It  clearly  two c o m p o n e n t s  are  differentiate  not acting  t h e components fused  in a parallel  these reasons  the  inferred  rule  related  Inferred  Rule A C - 2 :  That i s , the  the  speed o f  the  see  the  how t h e  s u b j e c t was t o one  hand,  the  t o Q. 1 5 c ,  sub-  he  thi-s—subject  can  o n l y when t h e y soon  s u b j e c t became  con-  components as  interact. holding  as  For the  component.  Subjects holding this acting  t h e b o a t due t o  t h e b o a t due t o  even  as  classified  two components  speed o f  but  component  way upon t h e b o a t ,  become p a r a l l e l ,  and he d i d n o t  differentiated  the  seems t h a t  diffi-  interviewer  i n 14b,  s u b j e c t m e n t i o n e d o n l y one i n Q.  is  it  conception  on the  boat.  the c u r r e n t ,  the motor.  A high  and  percent  -167-  of  the  sample,  ception. motions are  90 p e r c e n t  This  result  affecting  quite  reason.  This  tasks  18 o f  20,  held  m i g h t be e x p e c t e d  the boat  obvious. Other  or  i n that  i n the e x p e r i m e n t a l  t a s k was  s u c h as  chosen  projectile  for  suitable  students'  Some s u b j e c t s  i n t u i t i v e notions.  boat  s u c h as  considered other t h e w i n d and t h e  students  the  eddys.  Under  assume t h a t  these circumstances  also  was  the  no d i s c u s s i o n Excerpts  what h a s  Kevin,  the  taken  the boat  independently of  that  specific much  holding  affecting  this the the  easier  for  the  t h e r e was no w i n d  t h e r e were o n l y  two  boat. for  granted  acquires speed  the  that  speed  subjects  of  the  with regard  from i n t e r v i e w d a t a  will  to  would  current  p r o d u c e d by the e n g i n e . .  or questions  been e x p r e s s e d  15:  situation  f l o w i n g s m o o t h l y w i t h no w h i r l p o o l s o r  components a f f e c t i n g was  two  uncovering  situation  i n t e r v i e w e r t o l d them t h a t  r i v e r was  the  " i n t e r n a l current of  To make t h e e x p e r i m e n t a l  It  for  components  river".  and t h e  con-  motion are  more c o m p l e x and m i g h t n o t be  preconception  this  this  show more  There issue. clearly  above.  (Q.14b)  " . . . t h e m o t o r i s p u s h i n g t h a t way ( a g a i n s t t h e c u r r e n t ) and the r i v e r t h i s way . . . T h e r e a r e two m o t i o n s a f f e c t i n g the b o a t . . . "  (Q.14d)  (Q. n o t a s k e d  (Q.15c)  of  this  subject)  " T w o , f o r w a r d and s i d e w a y . . . t h e m o t o r p u s h i n g f o r w a r d and t h e r i v e r t h i s way" ( a l o n g t h e d i r e c t i o n o f t h e current).  -168-  Wayne,  15:  Randy,  15  (Q.14b)  " . . . I d o n ' t u n d e r s t a n d . . . y o u mean the f o r c e s . . . (I: W e l l , you c o u l d s a y t h a t ) . . . t h e r e a r e two t h e m o t o r and t h e s t r e a m " .  (Q.14d)  " . . . two . . . t h e m o t o r and t h e s t r e a m p u s h i n g and s p e e d i n g up t h e b o a t " .  (A.15c)  "two  (Q.14b)  " . . . the e n g i n e , the c u r r e n t going against i t . . . (I: anything e l s e , s u p p o s e t h a t t h e r e i s no w i n d ) . . . . not, that's a l l " .  (Q.15c)  "...  • For a l l  of  . . .  subjects,  two c o m p o n e n t s were a f f e c t i n g that  depicted  i n the task  they  for  some  introduced the  engine  and t h e  t h e c u r r e n t and t h e  these  observed  the  subjects  is  the  internal currents,  this  Obviously,  if  the n a t u r a l  conditions of  considered  i n the  task;  wind,  current  i n the middle than i n the  a much more c o m p l e x and d i f f i c u l t An i n t e r e s t i n g characteristic resultant city  that  v e l o c i t y of  due t o  another  was  finding  whirl  shore;  t h e number o f  regarding  the  affecting  wind,  bridge.  a river  were  stronger  would  become  this  vector  considered  the  two component v e l o c i t i e s  the r i v e r ;  components  shown when  pools, it  t h e m o t o r and v e l o c i t y d u e . t o  moved a c r o s s  situation  task.  i n d e p e n d e n t component a f f e c t i n g  when t h i s  also  underneath the  a few s u b j e c t s  the  was  least  the waves,  currents  i.e.,  was  at  experimental  a complex one,  present  It  and t h e d i f f e r e n t all  motor".  the a c t i o n of  the boat.  river".  this  the  the  stream)  as  motor-boat  brought to  the boat.  (velo-  three  These  -169same s u b j e c t s  responded  when t h e b o a t  moved a l o n g t h e  stream) .  This  is  quite  that  t h e r e were two c o m p o n e n t s stream  (upstream  understandable  for  have n o t y e t been f o r m a l l y i n t r o d u c e d to of  vector quantities.  that  some s u b j e c t s  three  components;  asked  if  It  i n the  is  the c o m p o s i t i o n surprising  differentiated  u n f o r t u n a t e l y these  down-  s u b j e c t s who  actually quite  sample  or  these  s u b j e c t s were  t h e y saw any r e l a t i o n s h i p f o r  these  three  not com-  ponents . The ception to  'three  discussed  a similar  component c o n c e p t i o n * and a  u n d e r i n f e r r e d r u l e A C - 1 seem t o  factor,  that  the d i r e c t i o n s or  f o r m e d b y t h e c o m p o n e n t s make a d i f f e r e n c e . possibility easily  that  subjects  when t h e y a r e  view data  of  con-  differentiate  not p a r a l l e l .  the three  ponent c o n c e p t i o n ' w i l l  best  illustrate  angle  There i s  the  c o m p o n e n t s more  Excerpts  subjects holding  the  point  from  inter-  the  'three  com-  it.  These  sub-  jects  i d e n t i f i e d t h e two c o m p o n e n t s i n Q. 14b and Q . 1 4 d .  Dale,  15:  Ian,  15:  (Q.15c)  " T h r e e . . . because the water goes t h i s w a y , t h e b o a t g o e s t h a t way (S showed t h e d i r e c t i o n i n w h i c h t h e b o a t was p o i n t i n g - t h e m o t i o n due t o t h e m o t o r ) , and t h e n i t g o e s l i k e t h a t (S showed t h e m o t i o n a l o n g t h e a c t u a l p a t h f o l l o w e d by the b o a t along the d i a g o n a l ) . . . "  (Q.15c)  " . . . s t i l l three . . . the p r o p e l l e r , t h e c u r r e n t and t h e b o a t b e i n g s i d e ways makes i t a l o t h a r d e r " .  -170-  Becky,  15  (Q.15c)  "two o r t h r e e . . . i t ' s g o i n g s i d e w a y s because i t ' s a g a i n s t the c u r r e n t , the c u r r e n t i s p u s h i n g down and t h e m o t o r i s t r y i n g to push i t a c r o s s . (I: b e f o r e y o u m e n t i o n e d o n l y two m o t i o n s , t h e c u r r e n t and t h e e n g i n e , how t h i s t h i r d one i s b e i n g p r o d u c e d ? ) . . . I t h i n k i t i s because the boat i s going sideways o f the c u r r e n t i n s t e a d o f going down".  The r e s p o n s e o f show t h e  "third  the  subjects  t o Q.15c  clearly  component".  Table Rule-Model  three  for  4.14  Analysis of  Components  Rule A C - I :  " T h e a n a l y s i s o f component v e l o c i t i e s f o r a moving b o d y , which i s a c t u a l l y b e i n g a f f e c t e d b y two s i m u l t a n e o u s and independent v e l o c i t i e s , produces only one c o m p o n e n t " .  Rule A C - I I :  " T h e a n a l y s i s o f component v e l o c i t i e s f o r a moving body, which i s a c t u a l l y b e i n g a f f e c t e d b y two simultaneous—-and i n d e p e n d e n t v e l o c i t i e s , p r o d u c e s t h e two components".  4.3.8  Inferred  Rules  This vector  for  Composition of V e l o c i t i e s  characteristic  c a n be c o n s i d e r e d  the r e c i p r o c a l aspect o f  the p r e v i o u s  of  a composition or  components.  simultaneous  Since  if  velocities  needs  components  or p a r t i a l  respect  composition of  to  one,  the  vector  as  analysis  combination of  t o be p e r f o r m e d ,  velocities  (C.V.)  must b e known. quantities  two  the With aspects  -171must be c o n s i d e r e d : will  be c o m b i n e d  how t h e m a g n i t u d e s  (this  will  be c a l l e d  magnitudes)  and how t h e d i r e c t i o n o f  be c o m b i n e d  (this  The  will  i n t e r a c t i o n of  d u c e an a p p a r e n t It  was  conceptions  single  expected  about  affecting  expected of  that  part  the  intuitive  cities?  that  t h e components  directions).  would have d e f i n i t e  component v e l o c i t i e s ,  notions  However, i t  including  students'  velocities.  the  f o r m a l method calcuof  What k i n d  How do t h e y c o m b i n e t h e m a g n i t u d e s  for  this  preconcepof  i n t e r a c t i o n of motions  p a t h do t h e y a n t i c i p a t e  com-  not  precise  How do t h e y c o m b i n e t h e d i r e c t i o n s ?  resultant  was  The p u r p o s e  to uncover the  about  even  t h e e x a c t number o f  and d i r e c t i o n .  composition of  pro-  ,  students  quantities,  will  and d i r e c t i o n s w i l l  differentiate  s t u d y was  t h e y have?  interaction of  s t u d e n t s ' would have d e v e l o p e d t h e  o f magnitude  about  components  motion.  a moving body.  lations  tions  the  the  i n t e r a c t i o n of  combination of  composing v e c t o r  of  the  both magnitudes  when t h e y c o u l d n o t ponents  be  of  of  do velo-  What k i n d  of  g i v e n component  motions? It  was  realized  that  if  considered  Particularly,  it  is  this  reference  point or reference  the water  on the  facilitated cities bridge.  is  if  river.  the  composition of  that  object  the d i s c u s s i o n  subjects  n o t an e a s y  is  one  about  velocities,  possible  situation*is  composition of  reference  already  task.  a moving body —  The e x p e r i m e n t a l  done f r o m a f i x e d If  is  body,  possess  velo-  — i.e.,  the  the n o t i o n of  t h e y may a t t e m p t  to  use  it  the when  -172trying  to  from t h e  e x p l a i n the  i n t e r a c t i o n of  velocities  observed  bridge.  Questions  of  Questions: Question  Interview Protocol 14a,  14a:  current)  14c,  15a,  f o r CV  15b,  16a,  14c:u  current).  and  19.  (Motor-boat moving upstream a g a i n s t  How w o u l d y o u d e s c r i b e  Question  16b,  the motion  from t h e  ( M o t o r - b o a t moving downstream w i t h  How w o u l d y o u d e s c r i b e  the  the motion  from  bridge?  the the  bridge? Note:  T h e s e two q u e s t i o n s to  f a m i l i a r i z e the  was e x p e c t e d predict  the a c t u a l  later  i n the  these  questions  of Question  bridge?  15a:  o r most o f  resultant loop.  the  task.  m o t i o n as  Hence,  for  this  the boat  It  subjects  would  was shown  the response  were n o t c o n s i d e r e d i n \ h e  the motion  take  introduction  to  analysis  characteristic.  across  the r i v e r ) .  t h a t you w i l l  see  from  How the  and show me t h e p a t h y o u  see.  draw t h a t Question the  film  all  (Motor-boat moving  Please  Question  in  film  an  s u b j e c t s with the  the i n f e r r e d r u l e s  would you d e s c r i b e  will  that  were used as  15b:  In t h i s  path? 16a:  film)  diagram of  the r i v e r ,  Could you e x p l a i n that (After  Was t h e  seeing speed o f  the boat the boat  due t o t h e w a t e r o r due t o  the  c o u l d you  prediction?  crossing that  motor?  the  river  y o u saw i n  the  -173Question  1(5 b :  Suppose t h e  5 k m / h r and t h e  speed o f  what do y o u t h i n k i s  the  speed o f  the c u r r e n t the motor  is  b o a t due  to  is  10 k m / h r ,  speed o f  the boat with r e s p e c t  to  you? Question the  19:  river  just  Finally,  from t h i s  in front  could  suppose your f r i e n d s  d o c k on t h i s  on the o t h e r  do i t ?  follow i n this  Table List  of  refers  ones.  to  the  of  resultant which  direction that  magnitudes  of  i n t e r a c t i o n of T h i s was  e a c h one o f  The second of  the  the magnitudes  the  is  any way  they  resultant  path  different  two a s p e c t s .  of  the  taken by c o n s i d e r i n g if  t h e y drew a  the components,  it  the  aspect refers  the  of  components;  v e l o c i t y was  different  aspect  from was  the path the  assumed  interaction  to  the  the  two com-  the  T h i s was '  resultant if  the  of  interaction  two component v e l o c i t i e s .  of  that  from  The f i r s t  directions  done b y c o m p a r i n g t h e m a g n i t u d e with  dock  for Composition (CV)  s u b j e c t s were c o n s i d e r i n g  directions. of  other  a d i r e c t i o n of motion d i f f e r e n t  of  these  there  cross  diagram?  p a t h drawn b y s u b j e c t s ;  shows  the  and draw t h e  these rules  They have  ponent v e l o c i t i e s .  Is  to  to  4.15  I n f e r r e d Rules of V e l o c i t i e s  The f o r m a t previous  side.  C o u l d you e x p l a i n  the boat w i l l  side  want  velocity  magnitude  from t h e  magnitudes  -174-  of  the components;  magnitudes.  it  was  assumed t o be an i n t e r a c t i o n  The two a s p e c t s  are  s e p a r a t e d by t h e  of  conjunction  and. Three  inferred rules  characteristic,  were o b t a i n e d  w h i c h have  the  for  this  vector  f o l l o w i n g common o p e n i n g  sentence: " T h e r e a r e two n o n - p a r a l l e l v e l o c i t i e s a f f e c t i n g -simultaneously the motor-boat, t h e p r e d i c t e d r e s u l t a n t v e l o c i t y shows that . . .  Inferred  Rule C V - 1 :  Its  d i r e c t i o n corresponds  direction  of  with  no i n t e r a c t i o n o f  this  corresponds magnitude,  to  the  t h e component w i t h g r e a t e r m a g n i t u d e ,  to  directions;  the magnitude  showing w i t h  this  of  showing  and bh^e m a g n i t u d e  t h e component w i t h  no i n t e r a c t i o n o f  greater  magnitude".  (1F,0M). Inferred  Rule C V - 2 :  between o f  those of  interaction to  of  of  this  Rule C V - 3 :  between  those  an i n t e r a c t i o n o f  and i t s  t o one  showing w i t h magnitude  this  t h e component w i t h g r e a t e r  Its  directions;  showing w i t h  and i t s  an  magnitude,  d i r e c t i o n corresponds  the components,  in  corresponds  no i n t e r a c t i o n o f m a g n i t u d e s " .  Inferred  of  d i r e c t i o n corresponds  the components,  directions;  the magnitude  showing w i t h  Its  magnitude  C3F,5M) to  one  this  in  -175corresponds  to one*  showing w i t h  (X  this  Discussion of Inferred this  df  Rule C V - 1 :  Suzzanne,  related her  17:  - -  = 2  p >  O n l y one  She d i d n o t  between d i r e c t i o n s  demonstrate  the  components (5F,6M)  .05).  Rules:  conception.  questions  from those o f  an i n t e r a c t i o n o f m a g n i t u d e s " .  = 1.40,  2  different  subject see  or magnitudes. to  the v e c t o r  among t h e  any k i n d o f  20 h e l d  interactions  Her r e s p o n s e s  characteristic  to  will  the best  conception.  (Q.14a)  " . . . n o t v e r y f a s t ( I : Why?) because i t ' s going a g a i n s t i t and t h e w a t e r i t i s p u l l i n g b a c k " .  (Q.14c)  " . . . f a s t e r because the pushes i t a b i t t o o "  (Q.15a)  " . . . t h a t the water i s coming toward me, t o go a c r o s s i t wiJ.1 be a l i t t l e h a r d e r b e c a u s e Tshe w a t e r pushes i t from a s i d e but i t c o u l d make i t a c r o s s (she m e a n t : t o w a r d the o t h e r d o c k ) .  (Q.15b)  /  stream  K V  water  yeh they w i l l r e a c h the other s i d e "  ' ' J — « — I KV Note:  Questions  16 and 19 were n o t a s k e d  response to question  because of  her  15  * S e v e r a l k i n d s o f v a l u e s were g i v e n by t h e m a g n i t u d e o f t h e r e s u l t a n t : v a l u e s i n between t h e two m a g n i t u d e s ( e . g . , 7.5 k m / h r ) , v a l u e s i n between t h e l a r g e r m a g n i t u d e and t h e a r i t h m e t i c a d d i t i o n o f t h e m a g n i t u d e s ( e . g . , 12.5 k m / h r ) , and v a l u e s e q u a l t o t h e a r i t h m e t i c a d d i t i o n of the magnitudes.  -176Her r e s p o n s e s interaction affecting speed o f totally  between  showed t h a t  the d i r e c t i o n s  the b o a t .  It  t h e b o a t due cancel  the  to  tude  is  to  the  n o t much two m o t i o n s  she b e l i e v e d  t h e m o t o r was  speed o f  does not n e c e s s a r i l y  was  of the  seems t h a t  This  equal  there  large  t h e b o a t due t o  mean t h a t  speed o f  the  that  enough the  to  to  current.  resultant  t h e b o a t due  the  magni-  the  engine;  V  that  is  not c l e a r  seems t h a t action,  she was  she was  Inferred  only  not  show a c l e a r  the  students.  the of  that These  the  the  ference  but  w i t h one o f the w i n n e r .  a  holding this  responses  that  it  is  the  directions.  20 s u b j e c t s  the  to  them as  "mixed v a l u e " . fighting"  a winner,  Excerpts  of  the  t h e m o t o r and t h e  the this  the  interference  the  the  greater  from i n t e r v i e w d a t a  two  speed  to  resultant  T h e y saw t h i s  between  view,  10  s p e e d due  that  did  magnitude  a common v i e w among g r a d e seen  consider-  answers  possessed  they d i d not consider  of  conception  Their  between  s u b j e c t s m i g h t have  "sort  inter-  t h e y were  v e l o c i t y and the magnitudes  two c a n be a as  the  it  under i n f e r r e d r u l e C V - 1 .  differentiation  s p e e d due  current,  But because  her view of  The s u b j e c t s  Eight of  shows  between  classified  interaction of  resultant  components. which  sure about  through t h e i r  ing  the  so  Rule C V - 2 :  made c l e a r  of  from h e r r e s p o n s e s .  inter-  speeds being  follow.  -177(Q.14a)  " i t w i l l move s l o w e r . . . ( I : Why do y o u t h i n k so?) . . . because i t f i g h t i n g the c u r r e n t . . . "  (Q.14c)  " . . . it  (Q.15a)  " i t w i l l s t a r t going the c u r r e n t goes t h i s  CQ.15b)  (First,  will  be g o i n g  he drew t h i s  faster  is  . . . "  sideways way"  because  path  I : Remember t h a t t h e y o n l y want t o c r o s s t h e r i v e r ? ) . . . o h , then i t i s a s t r a i g h t line: (Q.16a)  (Not a s k e d t o  (Q.16b)  " . . . i t w i l l be 10 k m / h r b e c a u s e how f a s t c a n g o "  (Q.'19)  yeh, j u s t go down f i r s t , 4  this  subject) that's  cross over . . . i t w i l l t h e n t h e y t u r n up t o t h e  &  )  (Q.14a)  " . . . s l o w e r . . . ( I : Why do y o u t h i n k so?) . . . because the current i s pushing V i t back"  (Q.14b)  "... with  (Q.15a, 15b)  " . . . t h e b o a t m o v i n g . . . ( I : What w i l l be t h e p a t h ? ) . . . s t r a i g h t a c r o s s . . . t h e c u r r e n t w i l l p u s h i t down a b i t I : remember (He drew t h i s p a t h . ^ j that they only want t o c r o s s the r i v e r ? ) . . . o h t h e y w i l l go t o t h i s p o i n t (a p o i n t c e r t a i n d i s t a n c e down the dock) . . . because the c u r r e n t a g a i n s t t h e s i d e , i t ' l l p r o b a b l y come down (He drew t h i s p a t h : /t\ I:Is i t a straight line?) . . . i t ' s probably a curve a l i t t l e b i t " .  f a s t e r . . . because the boat the stream . . . "  4  (Q.16a)  "...  due  to  the  motor"  is  going  -178-  Kevin,  (Q.16b)  " . . . 10 k m / h r b e c a u s e t h e s p e e d o f t h e b o a t due t o t h e e n g i n e i s n o t b e i n g a f f e c t e d much"  CQ.19)  t h e y go up a b i t P u s h them down, ^ ^ do t h e y h a v e t o / not too f a r  15:  (Q.14a)  " . . .  (Q.14b)(not  slower asked  so t h e c u r r e n t y ( I : How much j go up?) /  . . . "  to  this  subject)  (.Q.15a)  " . . . the way  (Q.15b)  "  (Q.16a)  " . . . t h a t speed (observed from b r i d g e ) i s due t o t h e e n g i n e "  (Q.16b) ( n o t (Q.19)  river will  asked to  this  push  them down  the  the  subject)  " t h e y have v Ys  t o s o r t o f t u r n i n t h i s way so t h e c u r r e n t p u s h i n g them k e e p s them i n s t r a i g h t l i n e . I t looks l i k e they are going t h a t way A <& but the c u r r e n t wouldn't c]^. l e t them t o go t h a t way so t h e y w i l l be f i n i s h i n g going i n s t r a i g h t l i n e . They aim the b o a t t h a t way ( J ) but the c u r r e n t keeps the boat going s t r a i g h t a c r o s s .  fr  For directions cularly .other  these was  clearly  through  hand,  three  the  the  subjects, observed  answers  responses  for  for  the  interaction  in their Q.15b  Q.16  of  responses,  and Q . 1 9 .  parti-  On t h e  show no i n t e r a c t i o n  magnitudes. Inferred  Rule C V - 3 :  inferred  rule  directions  Subjects  classified  expressed both kinds  and m a g n i t u d e s .  of  under  this  interactions:  The r e s u l t a n t  velocity  has  of  -179-  direction  and m a g n i t u d e d i f f e r e n t  components. 20,  held  More t h a n 5 0 p e r c e n t o f  this  preconception.  s u b j e c t s made e x p l i c i t u s e the a c t u a l resultant present  an  velocity,  interesting  believed will  percentage  that  the  of  the boat  result  of  the  sample,  11 o f  methods  these  to  magnitude o f  find the  u n d e r s t a n d i n g seems t o  crossing  that  some  the  river,  subjects  the i n t e r a c t i o n of  magnitudes  p r o d u c e a m a g n i t u d e whose n u m e r i c a l v a l u e w i l l  10 k m / h r a r e  magnitude o f  resultant).  the magnitude o f larger  15:  of  Other  the r e s u l t a n t  magnitude of Excerpts  Dale,  o f b o t h components  magnitudes  be  students.  just  v i e w o b s e r v e d was  between the magnitudes and  of precise  the b a s i c  the case o f  the  of  E v e n when none o f  d i r e c t i o n and t h e a c t u a l  i n a large In  from t h o s e  (i.e.,  components, subjects will  the components.  from i n t e r v i e w d a t a  be  in  5 km/hr  7 km/hr  believed  be g r e a t e r  that  than  the  ^ follows.  (Q.14a)  " . . . i f t h e e n g i n e i s r u n n i n g and i t has a speed o f 5 m i l e s / h r and the r i v e r a l s o has 5 m i l e s / h r , the b o a t w o n ' t move i t w i l l s t a y . . . i f t h e b o a t c o u l d go 10 m i l e s / h r , t h e s p e e d ( o f t h e b o a t w i t h r e s p e c t t o b r i d g e ) w i l l be 5 miles/hr."  (Q.14c)  " y o u see t h e b o a t goes a l o t f a s t e r . . . t h e w a t e r w i l l be m o v i n g a n d i t w i l l be m o v i n g t h r o u g h t h e w a t e r . . . ( I : Why w i l l i t move f a s t e r ? ) . . . b e c a u s e i t ' s moving w i t h the w a t e r , w i t h o u t the e n g i n e i t i s a l r e a d y g o i n g . . . i t has the speed o f the water p l u s the speed of the boat".  (Q.15a)  " i t ' l l go i n s t r a i g h t l i n e and t h e w a t e r w i l l be p u s h i n g i t l i k e t h a t (He t o o k  -180-  t h e t o y b o a t and showed t h e d r i f t i n g to the current) . . . "  due  (Q.15b) Note:  Question  16b was  asked  before  Q.  16a.  CQ.16b)  " . . . i t g o e s 7 o r 8 m i l e s / h r . . . ( I : Why d o n ' t y o u add them?) . . . b e c a u s e t h e w a t e r g o e s t h i s way (4-) and t h e b o a t t h a t way (->•) . . . I t h i n k i t ' l l go 15 b u t i n t h a t a n g l e (\) . . . I am n o t t o o s u r e a b o u t t h a t "  (Q.19)  " . . . y e h , t h e y c a n h e a d t h e b o a t up t h i s way a b i t and t h e c u r r e n t w i l l p u s h them down a bit. The p a t h i s a s l i g h t a r c : ^ . . . i t c o u l d be a s t r a i g h t l i n e / but I doubt i t . . . the boat w i l l be h e a d i n g t h a t way and t h e r i v e r w i l l b e p u s h i n g t h a t way (4-) s o i t becomes e v e n "  Tracy,15:(Q.14a)  " . . . i t ' s moving slower a g a i n s t the c u r r e n t "  because  it's  moving  (Q.14b)  " . . . same s p e e d as because i t depends  t h e c u r r e n t o r more o f t h e power o f t h e m o t o r ,  (Q.15a)  " . . . i t ' l l go s t r a i g h t b u t down a b i t . . . i t ' l l go t h i s way 0 » ) w i t h t h e s. c u r r e n t p u s h i n g down and h e r e (P i n d r a w i n g : ^ they b a c k up and go v e r y s l o w t o ^ | ^ — the dock".  (Q.15b)  ( I : p l e a s e , draw t h e p a t h i f to c r o s s the r i v e r )  (Q.16a)  " I t ' s b o t h , i t ' s a k i n d of c o m b i n a t i o n o f the c u r r e n t and t h e e n g i n e . . . t h e e n g i n e p u s h t o go f o r w a r d a n d t h e c u r r e n t p u s h t o go down . . . a c t u a l l y t h e s p e e d (of b o a t ) w i l l be s l o w e r t h a n i f i t were g o i n g w i t h t h e c u r r e n t the speed i s combined p u s h i n g i t i n a g e n e r a l d i r e c t i o n w i l l be w i t h t h e m o t o r and t h e current".  (Q.16b)  " . . . i t ' l l go a b o u t down a b i t " .  (Q.19)  (She p r o p o s e d too methods) " . . . t h e s l o w e r and l o g i c a l way and t h e q u i c k e r way  they only  7 miles/hr.,  it'll  want  slow  -181-  These s u b j e c t s and d i r e c t i o n  of  felt  the r e s u l t a n t  the correspondent v a l u e s subjects smaller said to  believed that than the  that  the  the boat  the  speed  of  the c u r r e n t .  subjects  reveal  the d i r e c t i o n of  their  out  estimate  if  it  give  interpretative regarding It  the  that  At least,  it  the r e s u l t a n t  is  to  does  ask  about  i n the sample.  kinds of  strategies  The t a b l e a n d t h e number o f  are  the  Because  of  detect  not  explicitly it  can  further questions  to  t o more p r e c i s e l y  However,  students'  presented of  most  the r i v e r  (question  consists  has  the  paths  preconceptions  motions.  just crossing  the o t h e r dock  They  i n f o r m a t i o n t o p r o v i d e some  i n t e r a c t i o n of  reaching  to  o r ways by w h i c h  the r e s u l t a n t .  sufficient  conjectures  the problems o f  was  the r e s u l t s  was d e c i d e d t o r e p o r t t h e d i f f e r e n t  by s u b j e c t s  these  drawn b y  possible It  from  the boat  the paths  have p r e c o n c e i v e d i d e a s  the magnitude o f  drawn s t i l l  down f o r  c o n c e p t i o n s about  seems n e c e s s a r y  students  Both of  the r e s u l t a n t  the r e s u l t a n t motion.  show t h e m a g n i t u d e o f be e s t i m a t e d ;  slows  c a n be s a i d  of motions.  magnitude  t h e b o a t due t o t h e m o t o r .  of  it  the  the components.  the magnitude of  speed o f  summary,  that  v e l o c i t y were d i f f e r e n t  speed  interaction  find  of  sure  that  "fight" In  pretty  19)  paths  subjects  drawn  tackled  ( q u e s t i o n 15)  and  i n d i f f e r e n t ways;  separately.  drawing of  s u b j e c t s who u s e d  that  the type of path.  path  both  -182-  Table (Question  15b  4.16  : Motor-Boat Moving A c r o s s  the  River)  Frequency  Path  15  3 Total Note:  When q u e s t i o n  2 cross  15b  20 first After  people the  river,  asked  3 Ss  mentioning  drew t h i s to  t^iem  that  on t h e b o a t o n l y w a n t e d  t h e y drew t h i s  A y\  *  1/  path  to  -183Table  4.17  (Question 19: Motor-Boat T r y i n g t o Reach Dock 2) Could you ... Draw the Path t h a t the Boat W i l l f o l l o w i n t h i s Diagram?  Frequency  Path  No method 20  Table  4.18  Rule-Model f o r Composition  Rule  CV-I  Rule CV-II  of V e l o c i t i e s  "The composition of two n o n - p a r a l l e l v e l o c i t i e s , which are s i m u l t a n e o u s l y a f f e c t i n g a moving body, produces a r e s u l t a n t - v e l o c i t y whose magnitude and d i r e c t i o n are those o f the component v e l o c i t y w i t h g r e a t e r magnitude". "The c o m p o s i t i o n of two n o n - p a r a l l e l v e l o c i t i e s , which are s i m u l t a n e o u s l y a f f e c t i n g a moving body, produces a r e s u l t a n t - v e l o c i t y whose magnitude  -184-  i s t h a t o f t h e component v e l o c i t y w i t h g r e a t e r m a g n i t u d e and whose d i r e c t i o n i s i n b e t w e e n t h e d i r e c t i o n s o f t h e two component v e l o c i t i e s " . Rule C V - I I I :  4.3.9  " T h e c o m p o s i t i o n o f two n o n - p a r a l l e l v e l o c i t i e s , which are simultaneously affecting a m o v i n g b o d y , p r o d u c e s a r e s u l t a n t whose magnitude* i s d i f f e r e n t from t h o s e o f the c o m p o n e n t s a n d whose d i r e c t i o n i s i n b e t w e e n t h e d i r e c t i o n s o f t h e component velocities".  Inferred  Rules  Components  istics In  Independence  vector  that  best  characteristic  fits  with  magnitudes magnitude  section of  is  b o a t due t o  to  to  the  t h e motor and measured  to  after  and  to  quantities.  inferred  of  the  case,  one  the v e l o c i t y o f  t h e one  current, of  to  the  the  corresponding (which  the v e l o c i t y  w i t h r e s p e c t .to t h e are  condition  with respect  the  implicit.  science  independence  the magnitude  two v e l o c i t i e s  note  is  t h e b o a t due t o  and measured  These  See  vector  one c o r r e s p o n d i n g  corresponds  the current)  of  character-  being  In the present  and t h e o t h e r m a g n i t u d e  actually  the  independence  the  the components.  the v e l o c i t y of  bridge.  refers  one o f  textbooks,  when i n t r o d u c i n g c o m p o s i t i o n o f  water;  is  v e r y r a r e l y mention the  This  Magnitudes  the c o n d i t i o n of  curriculum developers,  teachers  of  (I.M.C.)  This  fact,  for  shore  simultaneously  rule C V - 3 .  of  or  affecting  -185th e motor-boat, of  the boat  and t h e i r  which  is  combination produces  different  when o b s e r v e d ,  for  portant  to consider  cities  aspect is  that  instance,  respect  their  That i s ,  values.  magnitude  is  tude.  to  of  In o t h e r words,  resultant  are  to  this  of magnitudes,  sufficient  to  magnitude  of  of  boat  due t o  subjects  only  one o f are  the  the  a  "new"  i n t e r a c t i o n of  p o i n t conserves  means  that  its  with  magni-  the magnitudes and f r o m  for  vector characteristic, i t was  realized  a question  the components.  boat  the  it  of the  due t o  in  t h e m o t o r and v e l o c i t y  T h e v e l o c i t y due t o  the  It  if  was assumed  that  that  vector  characteristic.  the  effect  of  of  i n c l u d e d i n the p i l o t  be  affecting  this  speed  would  Recall that  from t h e o t h e r component,  about  in-  independence  independence o r dependence  t h e m o t o r upon t h e  c u r r e n t was  thought that  the constancy or  current).  show t h e i r v i e w a b o u t  due t o  of  that  maintain  e a c h component m e a s u r e d  t h e one c o n s i d e r e d .  t h e components  more,  fact  velocities  bodies  two i n d e p e n d e n t c o m p o n e n t s  (velocity of  m o t o r was  of  ask  two t h e r e  the boat  the  velo-  magnitude.  dependence  task  reference  reference this  The i m -  t h e component  a result  velocities  i n t e r a c t i o n of  i n d e p e n d e n t among t h e m s e l v e s  With regard  of  of  their  components,  a specific  components  in this  despite  p r o d u c e d as  the magnitudes respect  to  f r o m t h e component  from t h e b r i d g e .  the magnitudes  expressed with  a velocity  the  speed  the boat study;  all  of  one  would  of  due t o eight  Furtherthe the  boat  -186subjects was  h e l d the  always  by the  the  river  view t h a t  the  same b e c a u s e  and t h e  s p e e d due  t h e b o a t was  speed of  the  to  the  being  r i v e r was  current  carried  always  the  same. In  summary,  characteristic  was  the purpose  of  including this  to  if  subjects  find  out  whether o r n o t would r e a l i z e  the  tude  v e l o c i t y due  of  one component  from t h e  other  Questions  of  Questions: Question river  to  of  magni-  the  motor)  component.  Interview  18:  : Is  In the  the  Protocol  case of  speed o f  to water)  t h e b o a t due  to  the  the boat  t h e b o a t due affected  or  current?  Table List  independence  sample  18  with respect of  (i.e.,  i n the  vector  just to  the motor  changed by the  inferred  characteristic,  rules  w h i c h have  the (or  speed  How?  4.19  o f I n f e r r e d Rules f o r Independence M a g n i t u d e s o f Components ( I . M . C . )  Four  crossing  were o b t a i n e d the  of  for  this  vector  f o l l o w i n g common o p e n i n g  sentence: " T h e r e a r e two component v e l o c i t i e s affecting s i m u l t a n e o u s l y the motor-boat, the magnitude o f t h e component due t o t h e m o t o r i s . . . Inferred  Rule IMC-1:  with  component due  the  changed because of to  the  current".  the  interaction  (5F,4M)  -187-  Inferred  Rule IMC-2:  with  component due  the  decreased because of to  the  Inferred  Rule IMC-3:  with the  component due t o  Inferred  Rule IMC-4:  current".  increased the  = 1.73  2  Discussion Inferred  of  the  of  was  affected,  was  decreased or  Q.18), more,  of  19)  Excerpts Sandy,  (1F,3M)  lS(lM),was  (speed i s of  used  not  increased. view,  9 of  that  IMC-1.  t h e b o a t due  as  to  a synonym  the  it  19 was  (one  s u b j e c t was  a common v i e w .  (considering IMC-2,  w o u l d be  from t h e )  motor the  A high proportion of  asked  Further-  dependence in in-  one p a c k a g e )  almost  i n the dependence  interview data  to  80  speed  sub-  not  preconceptions  and IMC-3 as  among t h e m a g n i t u d e s ,  15:(Q.18  conception  here  s u b j e c t s were d i v i d e d accoxfli-ng  rules  inter-  they d i d not e l a b o r a t e whether  w h i c h shows  independence (15  .05;  the  current".  Subjects holding this  speed  among t h e m a g n i t u d e s ferred  p >  velocity)  yet  held this  if  (2F,1M)  question)  Rule IMC-1:  magnitude  jects  = 3  the  interaction  Results  mentioned that of  the  n o t changed because o f  df  asked the  because of  to  interaction  (1F,2M)  current".  a c t i o n w i t h t h e component due (X  the  or  percent  category.  follow  " . . . y e h , because the r i v e r wants t o p u s h t h e b o a t w i t h i t and go i n t h e same way . . . a n d i f t h e b o a t i s f i g h t i n g a g a i n s t i t . . . then i t has to . . . "  -188Wayne,  15:  Inferred result  Rule  of  b o a t due Only for  (Q.18)  3 of  them  15:  17:  Inferred result  of  b o a t due 3 of  (Q.18  the to  believed  velocities,  the  d e c r e a s e d o r wa,s  view;  excerpts  of  that  as  a  speed o f  slowed  the  down.  interview  data  " y e s , i t w i l l s l o w down b e c a u s e the c u r r e n t i s a k i n d of p u l l i n g down t h i s way (^** ^ ) and t h e m o t o r i s f i g h t i n g w i t h i t t o go to the shore, t h e n , i t i s s l o w i n g down". " y e s . . . i t ' s s l o w i n g down b e c a u s e the c u r r e n t i s pushing a s i d e " .  (Q.18. )  them  15:  )  .  Rule  of  subjects  follow.  IMC-3:  These  interaction  t h e m o t o r was  19 h e l d t h i s  two o f  Dale,  interaction  19 h e l d t h i s  ••" Hans,  These  t h e m o t o r was  two o f  Tracy,  IMC-2:  the to  " i t i s changed because the r i v e r i s pushing against i t . I f i t were calm water, there wasn't anything to s t o p , but the c u r r e n t i s pushing i t , i t w i l l s l o w down o r s p e e d u p " .  view;  of  subjects  believed  velocities,  increased  excerpts  of  or  the  that  as  speed o f  speeded up.  interview data  a the  Only for  follow.  (Q.18.)  " y e s , a f f e c t e d . . . i t makes t h e b o a t goes f a s t e r . . . i t (the c u r r e n t ) w i l l a f f e c t the boat but not the r i v e r (he meant t h e s p e e d o f t h e b o a t due t o the motor does n o t a f f e c t the speed of the r i v e r ) .  -189-  Julie,  15:  (Q.18)  " y e h , a l i t t l e b i t ( I : Why?) . . . i f i t i s h e a d i n g t h i s way the c u r r e n t i s going to push i t down t h e n i t s g o i n g t o h e l p the speed".  Inferred  Rule IMC-4:  inferred  rule  due  to  the  motor.  These  subjects  velocities  interact  subjects view;  result  velocities  i n the  that  sample  (4 o f was  Excerpts  from i n t e r v i e w d a t a (Q.18  )  s p e e d due  where  physics  to  the of  these  A low p e r c e n t a g e  held or  seem t o  boat  independence  situation  19)  this  the  understood  a low p r o p o r t i o n ,  some s t u d e n t s  a sophisticated  15:  the  among t h e m s e l v e s .  even though i t  shows  i n the  under  speed o f  the  have g r a s p e d  grasp of  Lisa,  the  c u r r e n t does n o t a f f e c t  of  this  classified  h e l d the view t h a t  magnitudes  of  Subjects  h a v e an  this intuitive  concept.  follow.  " . . . y o u mean t h e t o t a l s p e e d ? ( I : No, the speed o f the b o a t due t o t h e e n g i n e ) . . . oh, i t stays t h e same . . . j u s t t h e w a t e r p u s h i n g i t . . . ( I : What do y o u mean b y the t o t a l speed?) . . . the speed o f the w a t e r and t h e s p e e d o f t h e b o a t  -190Kevin,  15:  (.Q.18  )  " N o , I d o n ' t t h i n k so i f it's g o i n g t h i s way ( c r o s s i n g t h e r i v e r ) . . . o n l y the d i r e c t i o n of the boat (be meant d i r e c t i o n o f m o t i o n o f boat) i s being affected.  Table  4.20  Rule-Model  Rule  IMC-I:  f o r IMC  " T h e r e a r e two s i m u l t a n e o u s component v e l o c i t i e s a f f e c t i n g a body; the magnitude o f t h e component w i t h g r e a t e r m a g n i t u d e c h a n g e s due t o t h e i n t e r a c t i o n w i t h t h e o t h e r o n e " .  Rule IMC-II:  " T h e r e a r e two s i m u l t a n e o u s component v e l o c i t i e s a f f e c t i n g a body; the magnitude o f t h e component w i t h g r e a t e r m a g n i t u d e d e c r e a s e s due t o t h e i n t e r a c t i o n w i t h t h e other one".  Rule I M C - n i :  " T h e r e a r e two s i m u l t a n e o u s component v e l o c i t i e s a f f e c t i n g a body; the magnitude o f t h e component w i t h g r e a t e r magnitude i n c r e a s e s due t o t h e i n t e r a c t i o n w i t h t h e o t h e r one" .  Rule IMC-IV:  " T h e r e a r e two s i m u l t a n e o u s component v e l o c i t i e s a f f e c t i n g a body; the magnitude o f t h e component w i t h g r e a t e r m a g n i t u d e r e m a i n s cons-tairt d u r i n g j t i h e i n t e r a c t i o n " .  Rules it  cannot  be  texts.  Rules  cases:  either  magnitudes observe  it.  II  and I I I  said  for  I  that  are  actually  these w i l l  and IV r e a l l y  subjects  observe  interacting  be  very  found i n o t h e r  represent the  contextual;  the  two  independence  velocities  or  t h e y do  con-  general of not  -191-  4.3.10  Inferred  Rules  for  Simultaneity  Component V e l o c i t i e s  The  notion of  of  (S.C.)  simultaneity of  components  is  an  i m p o r t a n t one i n u n d e r s t a n d i n g p h y s i c a l phenomena. During  the p i l o t data  subjects  had the b e l i e f  affects with  the boat  greater  greater then,  the  vector  that  one o f  the o t h e r ,  magnitude.  In task  corresponds  to  the  components  this  composition of  composition of  before  finding,  aspect  one  the v e l o c i t y  as  it  with  the  the  motor;  velocity  was  one o f  components  velocities,  or  (i.e.,  forces)  is  but g e n e r a l l y without mentioning the  aspect;  and t h e  same o p e r a t i o n o f  to o b t a i n the r e s u l t a n t  Usually, taught  the  some  decided  the  implicit  characteristics. The  used  was  t h e one due t o  current.  simultaneity  found t h a t  two,  the boat  Due t o  was  and i t  one w o u l d a f f e c t  to m a i n t a i n the  11,  before  magnitude  this  due t o  collection it  the  first;  simultaneous  addition of later  the  of  It  that,  the d i f f e r e n c e  cases  s h o u l d be made e x p l i c i t l y .  i n grade  simultaneity  consecutive  addition  is  displacements.  displacements  same o p e r a t i o n  displacements*.  taught  vectorial  consecutive  vectorial  is  is  used  to  seems t o  this  author  between c o n s e c u t i v e  and  compare  simultaneous  However, p r e v i o u s  to  * They a r e o b t a i n e d t h r o u g h t h e use o f t h e e q u a t i o n v . A t , ( v e l o c i t y times i n t e r v a l o f time) which i s used a c c o r d i n g t o t h e number o f c o m p o n e n t s .  -192-  doing  that  before  it  may be b e s t  formal  between  the  instruction,  two  of  the  realize (in  that  task  of  (i.e.,  17  to  mentions engine  ask  (Task  the motor-boat the  the  the  for  students'  decided  was  the  students  that  motor-boat  was  the boat  other?  Why do y o u t h i n k  Table  the  v e l o c i t y of  are  together  and  about  affect  the  boat  effective  directly. affecting  river?  (If  b o a t due t o  the c u r r e n t )  S the  Were t h e s e  o r one a f t e r  the  so?  4.21  Simultaneity of  d i d not  beliefs  the q u e s t i o n  crossing  to  indirect  t h e most  How many m o t i o n s  affecting  of  simultaneity  time?  techniques  two v e l o c i t i e s :  simultaneity  to  Did students  it  Two):  The a u t h o r  about  two v e l o c i t i e s  for  s t u d y was  Protocol  motions  about  difference  the  do t h e  when i t  Rules  of  affecting  same  and t h e v e l o c i t y due  Inferred  part  f i n d i n g out  same t i m e ? ) ;  Question  students,  the  upon a b o d y .  trying several ways  a p p r o a c h was  this  two v e l o c i t i e s  Interview  simultaneity the  the  of  alternative  if  can d i s c e r n  of  velocities  occur at  After  out  i n t u i t i v e notions  two)  Questions  at  purpose  students'  interacting  find  cases.  Thus t h e uncover  to  find  Components  a large v a r i a t i o n of  components  i n the  sample;, o f  views  subjects.  -193I;n f a c t ,  it  seemed,  at  10  Cages  grade  from t h e d a t a  collected,  f r o m 15 t o 17) h a v e  already  c o n c e p t i o n o f s i m u l t a n e i t y o f components all  subects,  last  15 o f 17  q u e s t i o n because  i t was e v i d e n t  held  affecting  the motor-boat  which  firmly  sample  of surprising  of motions.  8 and 9  each o f grades grades  data  (ages  that  are  o f some  ideas  about  explanation  f r o m 13 t o 15)  I t m i g h t be t h a t  data  sequential  is  as w e l l  This  that  the  from each as  two f r o m  students  in  the s i m u l t a n i e t y  because  t h e sample  used  this  i n the p i l o t  small.  n o t much v a r i a n c e the s u b j e c t s ,  Only  i n the r e s u l t s ,  after  the view t h a t  two s u b j e c t s before  is  that  of  most o f  t h e two v a r i a b l e s ,  affect  held  i t a t t h e same  o f 17 t h o u g h t t h a t  one v e l o c i t y  the other.  Two i n f e r r e d r u l e s characteristic,  of the f i n d i n g  the f a c t  differentiating  t h e two v e l o c i t i e s  the boat  sentence:  ex-  or simultaneously.  W h a t e v e r may be t h e e x p l a n a t i o n  affects  t h e y were  this  o f c e r t a i n p h y s i c a l phenomena; h o w e v e r ,  c a n n o t be g e n e r a l i z e d  time.  the  practically  both v e l o c i t i e s  8 and 9 h a v e n o t y e t g r a s p e d  s t u d y was  since  i n c l u d e d two s t u d e n t s  10 and 1 1 .  characteristic  grasped  i n view o f the p i l o t  One p o s s i b l e  of the p i l o t  grades  that  together  showed t h e e x i s t e n c e  action  of  was s o r t  students  Cthree s u b j e c t s were n o t a s k e d  haustive)  finding  quite  that  which have  were t h e n o b t a i n e d  for this  vector  t h e f o l l o w i n g common o p e n i n g  -194-  "The s i n g l e a p p a r e n t movement o f t h e m o t o r b o a t i s t h e r e s u l t o f two v e l o c i t i e s t h a t are a f f e c t i n g i t . . . Inferred  Rule  SC-1:  one  Inferred  Rule  SC-2:  At  (X  = 2.4  df = 1  asked  the  2  not  after  the o t h e r " .  t h e same t i m e " p  > .05  (0F,2M)  (9F,6M)  ; 3 Ss  (3M)  were  question). *  Discussion  of  Results:  Inferred  Rule  SC-1:  the  that  t h e two  view  However, t h e i r and when one the  was  interviewer  first  time)  excerpt  Ian,  views  he  Only  two  s u b j e c t s o f 17  velocities  were e x p r e s s e d  asked  not a c t s i m u l t a n e o u s l y . in tentative  the q u e s t i o n a second  a c t e d a s he  hold  d i d n o t h e a r him  language  time  (because  very well  c h a n g e d h i s m i n d a s i s shown i n t h e  the  second  below.  15:  (Q.17)  Hans, 1 7 : ( Q . 1 7 ) :  "... p r o b a b l y t h e r i v e r , t h e r i v e r i s t r y i n g t o p u s h i t down a t t h e same t i m e t h a n t h e p r o p e l l e r i s p u s h i n g f o r w a r d ... the r i v e r a f f e c t s i t f i r s t . "... one a f t e r t h e o t h e r ... t h e y a r e not g o i n g w i t h each o t h e r . I f they w i l l be g o i n g w i t h e a c h o t h e r t h e y w i l l go t h i s way ( i n the d i r e c t i o n o f the c u r r e n t ) T h e y a r e two d i f f e r e n t m o t i o n s ...  B e c a u s e h i s r e s p o n s e was interviewer answer  do  seemed t o  t h e q u e s t i o n was  not  clearly  repeated.  His  h e a r d by "second  the  thought  was: "... p r o b a b l y a t t h e same t i m e ( I : What do y o u mean?) ... t h e m o t o r i s p u s h i n g t h e b o a t and t h e c u r r e n t i s p u s h i n g a t t h e s i d e ... t h e y h a p p e n a t t h e same t i m e ... ( I : B e f o r e y o u were s a y i n g s o m e t h i n g d i f f e r e n t , what do y o u a c t u a l l y t h i n k ? ) ... a t t h e same t i m e " .  -195-  Notice  that  these  two s u b j e c t s  probably.  B o t h were n o t s u r e  view about  the p r o b l e m a t i c s i t u a t i o n .  Inferred  Rule SC-2:  from i n t e r v i e w d a t a  of  o r d i d n o t have a  The f o l l o w i n g of  made u s e  students  are  some  holding  the  term  definite  excerpts  the s i m u l t a n e i t y  conception.  Lisa,  15:  (Q.17)  " . . . when i t ' s g o i n g down t h e s t r e a m ( b o a t m o v i n g downstream) i t ' s t o g e t h e r . . . b e c a u s e t h e b o a t and r i v e r go i n t h e same way, t h e n , y o u add t h e m . When i t ' s g o i n g up t h e r i v e r . . . y o u ' r e t a k i n g i t o f f (he meant s u b t r a c t s s p e e d o f r i v e r from s p e e d o f b o a t due t o e n g i n e ) ( I : and when i t i s crossing?) . . . the r i v e r i s a f f e c t i n g i t . . . the r i v e r i s pushing i t down . . . t o g e t h e r . . . I t h i n k " .  P r e s t o n , 1 6 : ( Q . 1 7 ) " more l i k e l y t o g e t h e r . . . t h e e n g i n e p u s h i n g f o r w a r d a t t h e same t i m e t h a n t h e c u r r e n t p u s h i n g down so i t ' l l be a v e r y s l i g h t a n g l e , o t h e r w i s e i t ' l l go i n steps ( ^ ) i f i t were one on a time. $y-^-^V Kevin,  15:(Q.17)  " A b o u t t h e same t i m e . . . . b e c a u s e t h e m o t o r k e e p s g o i n g a n d t h e r i v e r k e e p s g o i n g so t h e y h a v e t o a f f e c t t h e b o a t a t t h e same time"  Becky,  15:(Q.17)  " . . . t o g e t h e r I t h i n k ( I : Why do y o u t h i n k i t ' s together?) . . . b e c a u s e one ( t h e e n g i n e ? ) i s p u l l i n g i t and one ( t h e c u r r e n t i s p u s h i n g i t t h e o t h e r way . . . i t ' s together".  For affecting  all  these  the boat  when some s u b j e c t s about the  together. made u s e  same t i m e ,  views not very  subjects  This of  I think,  stable.  t h e two v e l o c i t i e s were c l a i m c a n b e made e v e n  terms  such as:  more  w h i c h c o u l d make t h e  However,  their  explanations  likely, subjects' show  -196a  steady  conviction.  Table Rule-Model  of  4.22  S i m u l t a n e i t y of  Component V e l o c i t i e s  Rule S C - I :  " T h e s i n g l e a p p a r e n t movement o f a b o d y i s t h e r e s u l t o f two component v e l o c i t i e s that are a f f e c t i n g i t s e q u e n t i a l l y " .  Rule S C - I I :  " T h e s i n g l e a p p a r e n t movement o f a b o d y t h e r e s u l t o f two component v e l o c i t i e s are a f f e c t i n g i t s i m u l t a n e o u s l y " .  4.3.11  Subjects'  In t h i s need o f frames  study  referring of  present  coordinate  have  system  the  the  reference  study  reference  P r e c o n c e p t i o n s About  locations  has  It  reference  positions  of  or  was  an e g o c e n t r i c  other  individual  to p o i n t s Findings of  subjects  a particular  points.  find  and the a  when c h o o s i n g  o u t how t h e s e  l o c a t i o n from  Do t h e y h a v e a is  use  the  described  several  different from  different  angles?  found t h a t  they are  objects  decided to  something t h a t  With regard has  understanding  o r i e n t e d around themselves  points.  perspective  of  Perspective  been emphasized.  shown t h a t  s u b j e c t s would d e s c r i b e different  importance of  is that  t h e above  young c h i l d r e n view of  observing  subjects  to  seeing  an o b j e c t  located  question,  Piaget  ( u n t i l 7 years things.  of  (19 56) age)  They b e l i e v e  from a p a r t i c u l a r  i n o t h e r p l a c e s must  see  have  that  if  location, it  in  the  -197same way t h a t - t h e y until  see  c h i l d r e n are  more t h a t  they develop  perspectives  different  a. c h i l d w i l l is  able  to  coordinate  Although 12 y e a r s  of  that  used  Child's  the  age,  conceptions  of  it  still  and p o i n t s  be one o f  the  matic  vector It  pective istic; in  was its  Questions  (first  as  Space,  view  Piaget  of  sources  of  are  has  are to  was  of  well  check  he/she view.  over their-  different  from  i n The  reasoned  view c o o r d i n a t i o n s ;  able  that  when  points  i n understanding  difficulty  also  said  relations  study  It  or  accept  stage are  a situation  1956).  not  age  they  ('The t h r e e m o u n t a i n t a s k '  that  if  perspective  then t h i s  i n understanding  could kine-  quantities.  i n c l u s i o n was  of  of  important  using  is  discrimination  possible  this  thought  not considered  Questions: Question  of  s h o u l d be e m p h a s i z e d  subjects  points  was  this  perspective  have d i f f i c u l t y  relations  at  a number o f  perspective  by,Piaget  points  it  years of  That i s ,  viewpoints.  subjects  Conception of  subjects  Children  simple  Piaget,  "progressive  from o t h e r  different  to  10-12  perspectives".  seen  master  range o f  a more  from t h e i r s .  to c o o r d i n a t e  According  i n the  and c o o r d i n a t i o n o f that  it.  older  as  that  to check  2a;  2b,  (If  6d i n t a s k  S mentions  response to  fishing  spot)  Q.l)  pers-  character-  Piaget's  findings  sample.  Interview Protocol for 2a,  aspect of  an i m p l i c i t v e c t o r  rather  than h i s  this  Perspectives  one.  one o r  two p l a c e s  C o u l d you d e s c r i b e  from o t h e r  places  as  well?  as  reference  that  location  What p l a c e s ?  -198-  Question  2b:  reference of  this  (If  S mentions  points).  fishing  Will  several  p l a c e s or bodies  the d e s c r i p t i o n s  s p o t done  of  from o t h e r p l a c e s  the  as  locations  be t h e  same o r  different? Question  6d:  boat).  If  (After  a friend of  p a t h he s e e s yours or  for  and h i s ,  the  of  later.  an i m p l i c i t  vector  find  the  drawings,  them t h e  same  describing  other  l o c a t i o n of was  of  inferred  rules  number o f  here  subjects  from i n t e r v i e w  data  ages,  not appear  but  18 d i d n o t h a v e much p r o b l e m the  first  shown i n t h e i r  T h i s was  ^question  that  not a it  was  fishing  responses  spot to  from  question  a d e s c r i p t i o n c a n be  surprising  result  a  to  2b and 6d was  surprise asked  to understand question  d i d mention that  because the  part  not considered  i n terms  and e x c e r p t s  them e v e n s a i d  subjects'  subjects  was  of  this  characteristic.  20 s u b j e c t s ,  this  from anywhere.  subjects  i n terms  perspective  views  the  places; Some o f  regarding  views.  Among t h e  did  the  t r e e a n d draws  to  The f i n d i n g s  were r e p o r t e d  different  showing those  the  by the  f o l l o w e d by  c a n y o u compare b o t h  s t u d y were n o t a n a l y z e d  holding  2a.  is  Do y o u e x p e c t  Results:  Results  in  yours  the boat  because the n o t i o n of as  drawn t h e p a t h  different?  Discussion of  S has  these 2b.  the d e s c r i p t i o n s  s p o t was b e i n g o b s e r v e d  done  considering  find  5 of  14  14 s u b j e c t s  The o t h e r were  nine  different  from d i f f e r e n t  points  )  -199-  of  view.  15 t o and  T h i s would  17,  are  do n o t a p p e a r  able  observed  suggest  that  subjects  from d i f f e r e n t will  these  t o h a v e an e g o c e n t r i c  to discriminate perspectives  interview data  at  reference  illustrate  t h e above  viewpoint,  of  points.  ages,  an  object  Excerpts  from  points.  Larry,  16:  (2b)  " . . . they w i l l d i f f e r e n t " . . . i f you are s t a n d i n g i n d i f f e r e n t s p o t s y o u w i l l be s e e i n g t h e same p o i n t b u t t h e d e s c r i p t i o n s w i l l be d i f f e r e n t "  Kevin,  15:  (2b)  " i t w i l l be d i f f e r e n t , you a r e i t from a d i f f e r e n t a n g l e  looking  at  L o r i - A n n , 1 5 : (2b)  " i t w i l l be d i f f e r e n t b e c a u s e i t i s f r o m d i f f e r e n t p l a c e s , i t i s t h e same p o i n t ( f i s h i n g s p o t ) ' b u t i t w o u l d h a v e t o be d i f f e r e n t b e c a u s e from t h i s p l a c e (dock) w x l l be f a r t h e r t h a n from t h e r e (wharf)"  Randy,  " p r o b a b l y t h e y w i l l b e d i f f e r e n t ( I : Why do y o u t h i n k so?) . . . you p r o b a b l y have t o go e i t h e r f a r t h e r from the w h a r f , o r from t h e t r e e , o r y o u m i g h t go i n d i f f e r e n t a n g l e from the shore t h a n i t would from t h e wharf"  15:  (2b)  From t h e s e e x c e r p t s did seen  not possess  ized  an e g o c e n t r i c  from d i f f e r e n t While that  understand seemed  to  reference  a n a l y z i n g the  a possible the  the  view o f  transcript  Q.2b,  the o b j e c t .  fact  of  the these  subjects  perspectives  of  a body  points.  the q ue s t i o n  locating  should consider  clear  data  r e a s o n why 5 o f t h e  i n t e n t i o n of  interpret  a method f o r  i t was  was as  the author  14 s u b j e c t s  that  these  a request That i s ,  using distances  to  real-  did  not  subjects describe  t h e method or angles, .  -200-  they b e l i e v e d  that  from d i f f e r e n t both.  if  places,  Wayne,  17:  15:  (2b)  jects  (2b)  the  was  that  same.  expected  respond with  result that  was  to  about  explain  the path observed  (responses the path  for  seen  angle  this  question  that  a larger  the c o r r e c t  that  and f i n a l l y  of  percentage  answer  where t h e o b s e r v e r find  from 6d),  from d i f f e r e n t  or  point.  of  that is.  two s u b j e c t s  if  different 8 of  sub-  should  subjects  subjects is,  14  places  ages o f  a few s u b j e c t s ,  t h e p a t h w o u l d be d i f f e r e n t  places;  distance,  will  C o n s i d e r i n g the range  same no m a t t e r  location  " . . . i t w i l l be t h e same ( I : Why do y o u t h i n k so?) . . . i t doesn't r e a l l y m a t t e r where y o u a r e , y o u c o u l d be anywhere i t w o u l d n ' t m a t t e r much d i f f e r e n c e (he showed t h e d i s t a n c e s f r o m d i f f e r e n t p l a c e s to the s p o t ) " .  points  said  a  "they (descriptions) are s i m i l a r i n a way, t h e y ( d i s t a n c e s ) a r e a l l s t r a i g h t i n f r o n t o f t h e w h a r f . . . l i k e t h e buoy (buoy l o c a t e d i t t h e f i r s t f i s h i n g spot) i s i n f r o n t of the wharf  When a s k e d reference  to d e s c r i b e  t h e y must u s e  A g a i n , e x c e r p t s from d a t a  Suzzane,  be  t h e y have  would  the path  is  from t h e  4 of  observed  14,  expressed  from  different  d i d not understand  views of  the group o f  p a t h was  independent of  interview data  will  s u b j e c t s who u n d e r s t o o d the  observer.  the  The u n e x p e c t e d  the  question. Excerpts  it  show that  the the  -201-  St eve,  15:  (6d)  " i t ' s a b o u t t h e same . . . b e c a u s e he ( t h e o b s e r v e r b y t h e t r e e ) w i l l be a b l e t o j u d g e t h e d i s t a n c e s o f t h e s p o t s n e a r him b e t t e r t h a n me (on t h e dock) . . . t h e n , i t w i l l be o n l y a l i t t l e b i t different".  Kevin,  15:  (6d)  " G e n e r a l l y t h e same b e c a u s e t h e y (friends i n t h e b o a t ) a r e g o i n g t o t h e same s p o t . . . she ( t h e o b s e r v e r by t h e t r e e ) w i l l be a b l e t o see more e x a c t l y t h e s p o t s c l o s e r t o h e r , n e a r me I ' l l be a b l e t o s e e i t more a c c u r a t e " .  Tracy,  15:  (6d)  " i t w i l l n o t be e x a c t l y b u t t h e y w i l l be s i m i l a r . . . f i r s t o f a l l , he ( t h e o b s e r v e r by t h e t r e e ) i s l o o k i n g a t from a d i f f e r e n t p o i n t o f v i e w and i t w i l l be d i f f i c u l t j u s t f r o m memory t o remember and o b t a i n e x a c t l y t h e same . . . b u t essentially they s h o u l d be t h e s a m e " .  Essentially that  subjects  t h e p a t h must be t h e  places;  the  m i g h t be  excerpts  drawings  slightly  observers  look  these  to are  Becky,  15:  same when o b s e r v e d  the  different  locate  paths due t o  instantaneous  15:  (6d)  from d i f f e r e n t  (6d)  the  conception  from  (made by t h e  different  students)  the d i f f i c u l t i e s locations.  f r o m s u b j e c t s who b e l i e v e d  different  Lori-Ann,  of  possessed  viewing  that  of  The f o l l o w i n g the  paths  positions.  " T h e y p r o b a b l y w i l l be d i f f e r e n t b e c a u s e she ( t h e o b s e r v e r by t h e t r e e ) i s l o o k i n g a t f r o m d i f f e r e n t a n g l e . . . when t h e boat i s l e a v i n g h e r e , the boat i s going away f r o m me b u t i t ' l l be c o m i n g t o h e r . . . she w i l l g e t t h i s a n g l e w h i c h I d o n ' t g e t . . . p a t h s w i l l p r o b a b l y be d i f f e r e n t b e c a u s e two d i f f e r e n t p e o p l e see d i f f e r e n t t h i n g s and a l s o d i f f e r e n t paths" " t h e y ( t h e p a t h s ) w i l l be d i f f e r e n t b e c a u s e . . . i t ' s n o t g o i n g t o be e x a c t l y t h e same b e c a u s e e v e r y b o d y s e e s i t d i f f e r e n t . . . because they are standing' in  different  position  . . . "  -202-  It their in  may be t h a t  responses  for  these  several  their explanations.  were a l s o p r e s e n t  subjects;-were-.mot of  While  them u s e d  some e l e m e n t s  of  uncertainty  subjects,  former group of  stated  c l e a r l y t h a t paths  must r e a l l y be t h e  latter  subjects b a s i c a l l y  stated  that  about  the term " p r o b a b l y "  i n the p r e v i o u s group o f  i m p o r t a n t d i f f e r e n c e was t h e  sure  the  subjects same,  the paths  while  would  the  be  different.  4.4.  "Two New V e c t o r  Characteristics  (Items 11 and 12 o f  While analyzing the interview data this  author that  two o t h e r i m p l i c i t v e c t o r  c o u l d be c o n s i d e r e d . all  One o f  three vector quantities. 1.2  is  characteristics  that  the  w h i c h means  T h e two e x t r a  in  vector that  not e x p l i c i t l y present  in  other the  vector  were a c t u a l l y u n c o v e r e d by e m p i r i c a l l y  subjects'  responses  to questions  the following  vector characteristics:  displacement,  composition of v e l o c i t i e s ,  of  of  magnitudes  at  p e r f o r m e d on t h e  A s was m e n t i o n e d e a r l i e r  c a n be i d e n t i f i e d .  characteristics analyzing  analysis  not exhaustive, are  to  characteristics  (Chapter One), the l i s ± o f .rrmp'ilicit  characteristics  definitions  appeared  t h o s e was n o t a n t i c i p a t e d  through the r a t i o n a l task  Section  it  LIVC)  components.  frame o f  related  to  reference,  and i n d e p e n d e n c e  -203The two new v e c t o r Independence of  of  Locations  D i r e c t i o n s of  identified  4.4.1  is  discussed  Independence  showed t h a t  while  it  that  some  was  that  a particular  vector  phase o f in this  seemed  finding,  characteristic  to  explained  later  i t was i n the  Since,  subjects  thought second  thought  the  Later,  phase,  it  on t h e  pilot  i t was  not  however,  was  through t h e i r  evident  responses path.  to r e c o n s i d e r  this  phase.  understanding  a path  points  passes.  or  the  The l o c a t i o n o f as  a set  determined  of  reference  frame  is  this  One i m p o r t a n t a s p e c t  is  of  independent  that  that  e a c h one o f  (i.e., the  of  is  an i n -  through which a  coordinates  these  a coordinate each  the path t h a t  the q u a n t i t a t i v e  particle points  with respect  l o c a t i o n of  c a n be d e t e r m i n e d i n d e p e n d e n t l y , o f T h i s means  sequence  locations  c a n be d e s c r i b e d  to.  originally  study.  first  imply  (ILP)  below.  number o f  ( o r body)  to  the  was  subsections.  was  the  these  comprehend c e r t a i n v e c t o r q u a n t i t i e s  For p h y s i c i s t s finite  two  l o c a t i o n was d e p e n d e n t  The i m p o r t a n c e o f aspect  to  called:  How e a c h o f  characteristic. obvious  first  subjects  (IDC).  characteristic  vector  i n the  on t h i s  and I n d e p e n d e n c e .  (ILP)  L o c a t i o n s , . f r o m .Path  a n a l y z i n g the data  Based  from P a t h  i n the next  of  this  t o be a n i m p l i c i t  considered  were  Components  Actually,  data  characteristics  to  a  system). point it  description of  belongs 1 a  -204location is  (set  of  coordinates  independent of  fact  that  boats  instantaneous  stops  dependent understand  the path  at  of  fishing  the path  kinematic vector  textbooks  and s c i e n c e aspect.  to  include this  of  the  4.4.2  18. the  or  stop points  (by i t )  is  quantities  and v e l o c i t y .  For a l l  physics  do n o t e m p h a s i z e it  i n the  to  vector  Usually,  these reasons,  row-  in-  important  s u c h as  The  (e.g.,  a m o v i n g body a r e  vector characteristic  this  was  in-  decided  second  phase  study. Independence  was  of  D i r e c t i o n s of  Subjects speed  of  classified the boat  affected.  questions  the  speed  Will  showed t h e  cases with the boat this  ponents.  the  question  IMC-4  said  that  t h e m o t o r d o e s n o t c h a n g e when  of  the c u r r e n t ,  the boat A  w a y : ^>  but  its  view w i t h  reach the other  4-  1/  or t h i s  direction further  side A  way: £ *  of  «r  1/  JA'>C  two  illustrated  this  The r e s -  whether a s u b j e c t  independence of d i r e c t i o n s  T h e r e was v a r i a n c e  j e c t s who were a s k e d  character-  answering  t o y on t h e c a r d b o a r d m o d e l ) .  question  o r had n o t g r a s p e d  (IDC)  vector  under i n f e r r e d r u l e  due t o  the r i v e r heading t h i s  ponses to  subjects  The i n t e r v i e w e r pursued, t h i s  such as:  (Interviewer  Components  in identifying this  g i v e n b y a number o f  interacting with is  of  teachers  The m a i n f a c t o r istic  locations  followed  displacement,  direction)  f o l l o w e d by t h e p a r t i c l e .  spots)  position,  dependence  o r magnitde and  of  between t h e r e s p o n s e s o f  question.  It  was  had comsub-  then decided  to  -205-  i n c l u d e the independence of d i r e c t i o n s as another v e c t o r c h a r a c t e r i s t i c i n the second phase of the The (IDC)  importance o f c o n s i d e r i n g t h i s  of v e c t o r q u a n t i t i e s as e x p l a i n e d below.  of  composition  This vector  c h a r a c t e r i s t i c i n a sense complements the one  i s t i c s completely  study.  characteristic  r e l a t e s to the complete understanding  dependence of magnitudes of components.  implicit  The  about i n two  character-  d e s c r i b e the independence of components.  T h i s means t h a t to a c t u a l l y understand the  independence  p r o p e r t y of v e c t o r q u a n t i t i e s , s u b j e c t s must understand both the independence o f magnitudes and d i r e c t i o n s o f i n t e r a c t i n g components.  T h i s and  the s u b j e c t s '  suggestion  g i v e n through responses to q u e s t i o n 18 were the main reasons i n d e c i d i n g t o i n c l u d e t h i s v e c t o r (IDC)  4.5  i n the second phase o f the  characteristic  study.  Summary o f I n f e r r e d Rules f o r Each S u b j e c t Interview  Table phase one.  i n the  Sample  4.23  i s a summary t a b l e of data c o l l e c t e d i n  I t shows the i n f e r r e d r u l e s f o r each s u b j e c t  about each one  o f the v e c t o r c h a r a c t e r i s t i c s .  corresponds to one  s u b j e c t and  Each  row  shows the p a r t i c u l a r i n -  f e r r e d r u l e f o r each v e c t o r c h a r a c t e r i s t i c .  In a d d i t i o n  to the name o f the s u b j e c t , the f i r s t column a l s o i n c l u d e s the f o l l o w i n g i n f o r m a t i o n : sex  (F or M),  age  i n years,  and  -206-  Table 4.23 Table o f I n f e r r e d  Subjects  1 2 3 NR  Dale M,15,YY  •  Suzanne F,17,YY  1 2 3 4 5 NR  M,16,--  •  Russ M,15,YY  •  «  •  •  *  a  •  •  Julie F,15,YY  Hans M,17,YY  •  •  •  •  «  •  •  Kevin M,15,-- •  *  •  •  t  •  a  •  •  •  •  *  .f  Lori F,15,YN  •  *  a  L o r i Ann F,15,YN  •  «  •  *  t  «  Randy M,15,YY  •  *