UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1981_A2 A38.pdf [ 15.77MB ]
Metadata
JSON: 831-1.0095383.json
JSON-LD: 831-1.0095383-ld.json
RDF/XML (Pretty): 831-1.0095383-rdf.xml
RDF/JSON: 831-1.0095383-rdf.json
Turtle: 831-1.0095383-turtle.txt
N-Triples: 831-1.0095383-rdf-ntriples.txt
Original Record: 831-1.0095383-source.json
Full Text
831-1.0095383-fulltext.txt
Citation
831-1.0095383.ris

Full Text

STUDENTS' PRECONCEPTIONS OF THREE VECTOR QUANTITIES by JOSE M. AGUIRRE Teacher of Physics, University of Chile, Chile, 196 8 M.A., The University of 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1981 (c) Jose M. Aguirre, 1981 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. It i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of /^fn 1. ^ SWCfCT'lb^ The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date StMMftg'K. <S ; M\ I -in \ ABSTRACT This study was directed towards the i d e n t i f i c a t i o n and analysis of students' i n t u i t i v e ideas or preconceptions about three kinematic vector quantities — vector position, displacement, and v e l o c i t y . A l i s t of ten i m p l i c i t vector c h a r a c t e r i s t i c s derived from a task analysis of these three vector quantities, served as the framework for the develop-ment of the interview protocols used i n the study and the subsequent analysis of the data. The study proceeded i n two phases; i n the f i r s t , c l i n i c a l - t y p e interviews were conducted with twenty grade ten students using two tasks which involved describing the movement of a boat on a lake and a r i v e r ; i n the second phas a group interview protocol (G.I.P.) requiring a written response was developed (based upon the tasks and the r e s u l t s of the f i r s t phase) and administered to 8 classes of grade ten students, producing a t o t a l sample of 17>JS subjects. Analysis of the results from the f i r s t phase yielded a small but variable number (2 to 5) number of inferred rules for each of the ten i m p l i c i t vector c h a r a c t e r i s t i c s dealt with i n the in d i v i d u a l interviews. These 'inferred rules' are hypothesized sets of b e l i e f s (preconceptions) or cognitive structures used by the subjects to explain a problem si t u a t i o n posed by the interviewer. Further cate-gorization yielded three broad types of preconceptions — scalar, t r a n s i t i o n a l , and v e c t o r i a l — which were based i n part upon c r i t e r i a derived from the current physics perspec-t i v e . The o v e r a l l r e s u l t s from the f i r s t phase indicated that most students possessed f a i r l y consistent preconceptions regarding the d i r e c t i o n a l and distance components involved i n the description of the movement of a boat on a lake. For example, a l l students recognized the necessity of using a reference point and a frame of reference, however, they em-ployed d i f f e r e n t 'inferred rules' i n the selection of these references. Overall, for four vector c h a r a c t e r i s t i c s , most of the subjects used v e c t o r i a l type 'inferred rules' (where they considered the quantitative aspects of the variables involved, e.g., d i r e c t i o n and distance); for three of the vector c h a r a c t e r i s t i c s , a high percentage of subjects made use of scalar-type 'inferred rules' which corresponded to f a i r l y primitive notions (where they considered only one qu a l i t a t i v e or quantitative v a r i a b l e ) ; for the remaining three vector c h a r a c t e r i s t i c s , most of the subjects de-monstrated transitional-type 'inferred r u l e s ' (which con-sisted i n some combination of the two p r i o r extreme categories). The r e s u l t s from the second phase o f the s t u d y , i n g e n e r a l , c o r r o b o r a t e d the f i n d i n g s from the f i r s t phase . W h i l e some new ' i n f e r r e d r u l e s ' were i d e n t i f i e d , about 80 p e r c e n t o f the i n f e r r e d r u l e s used were common to those d i s c o v e r e d i n the i n t e r v i e w sample. Hence, i t would seem t h a t the r e s u l t s were n e i t h e r an a r t i f a c t o f the i n t e r v i e w p r o c e d u r e , nor l i m i t e d t o a s m a l l group o f grade ten s tudent s In summary, the s tudy i l l u s t r a t e d the a p p l i c a t i o n o f a p a r t i c u l a r a n a l y t i c a l t e c h n i q u e to generate s e t s o f ' i n -f e r r e d r u l e s ' which were h y p o t h e s i z e d to account f o r the way t h a t grade t en s u b j e c t s i n t u i t i v e l y approached a problem s i t u a t i o n i n v o l v i n g a d i r e c t i o n a l as w e l l as o t h e r components By d e v e l o p i n g a group q u e s t i o n n a i r e , the g e n e r a l v a l i d i t y o f the approach and r e s u l t s were enhanced by f i n d i n g s i m i l a r response p a t t e r n s i n a l a r g e r more heterogeneous group o f s u b j e c t s . The e d u c a t i o n a l i m p l i c a t i o n s o f b o t h the r a t i o n a l t a sk a n a l y s i s , which y i e l d e d the l i s t o f 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 , and the s u b s t a n t i v e r e s u l t s , expres sed i n terms o f the v a r i o u s se t s o f ' i n f e r r e d r u l e s ' , were d i s -cus sed i n terms o f t h e i r a p p l i c a b i l i t y to c l a s s r o o m t e a c h e r s and c u r r i c u l u m d e v e l o p e r s . - i v -TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES. x i LIST OF FIGURES x i v ACKNOWLEDGEMENTS xv CHAPTER ONE 1.0 Background o f the Study 1 1.1 D e f i n i t i o n o f Terms . . . . . . . . . . 4 1.2 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 o f Three P h y s i c a l V e c t o r Q u a n t i t i e s 7 1.3 G e n e r a l S ta tement o f the Problem 10 1.4 S p e c i f i c S ta tements o f the Problem 13 1.5 Research Ques t ions . . . . 14 1.6 O v e r v i e w o f Methods o f S tudy 15 1.7 D e l i m i t a t i o n o f the S tudy 1.8 J u s t i f i c a t i o n o f the S tudy . . . . 17 19 CHAPTER TWO 2.0 I n t r o d u c t i o n . . . . . . . . . . . . 21 •2.1 L i t e r a t u r e Review on S t u d e n t s ' U n d e r s t a n d i n g V e c t o r Q u a n t i t i e s • • 22 2 . 1 . 1 S t u d e n t s ' D i f f i c u l t i e s i n U n d e r -s t a n d i n g V e c t o r Q u a n t i t i e s 22 Page 2 . 1 . 2 V i e w i n g S tudents D i f f i c u l t i e s as a C r i t i c a l B a r r i e r Phenomena B . S p e c i a l Exper iments t o Teach V e c t o r Q u a n t i t i e s C . V e c t o r Q u a n t i t i e s i n Mathe-m a t i c s E d u c a t i o n 2.2 P s y c h o l o g i c a l C o n t e x t o f the Study . . . . 2 . 2 . 1 I n t r o d u c t i o n . . . . 2 . 2 . 4 Task A n a l y s i s 3.1 The P i l o t S tudy 3.2 R a t i o n a l Task A n a l y s i s o f P h y s i c a l Phenomena . . . . • • • 28 2 . 1 . 3 P r e v i o u s Research About S t u d e n t s ' U n d e r s t a n d i n g on V e c t o r Q u a n t i t i e s A . P i a g e t i a n - T y p e S t u d i e s 31 36 36 37 37 2 . 2 . 2 Two A s p e c t s o f Deve lopmenta l P s y c h o l o g y : Formal and F u n c t i o n a l . . . 38 42 2 . 2 . 3 The N e o - P i a g e t i a n Approach o f Development • • S i e g l e r ' s Approach 43 C a s e ' s Approach . . . . . . . . 46 47 2 .3 E d u c a t i o n a l C o n t e x t o f the S tudy . . . . . . . . . . 51 2 . 3 . 1 T e a c h i n g - L e a r n i n g Problems i n S c i e n c e E d u c a t i o n . . . . 52 2 . 3 . 2 C u r r i c u l u m Problems i n S c i e n c e E d u c a t i o n . . . . . . . . 55 CHAPTER.THREE 3.0 I n t r o d u c t i o n - . . . . . . . . . . . . 58 58 60 - v i -3 . 2 . 1 Networks o f the Three V e c t o r Q u a n t i t i e s 6 3 3 . 2 . 2 The P h y s i c s Approach 3.3 The Two Tasks o f the Study 3 . 3 . 1 Task One • 3 . 3 . 3 Task Two 3 .3 .4 P r o t o c o l o f Task Two 3 . 3 . 5 F u r t h e r D i s c u s s i o n o f I n t e r v i e w P r o t o c o l 3.4 Methods o f Data C o l l e c t i o n : C l i n i c a l I n t e r v i e w 3 . 4 . 1 R e l i a b i l i t y and V a l i d i t y i n C l i n i c a l I n t e r v i e w . . . . 3 . 4 . 2 I n t e r v i e w Format , 3 . 4 . 3 Sample o f S u b j e c t s 68 69 70 3 . 3 . 2 P r o t o c o l o f Task One • 7 3 79 81 86 89 92 94 95 98 CHAPTER FOUR 4.0 I n t r o d u c t i o n 4.1 Data A n a l y s i s . . . . 99 4 . 1 . 1 I n f e r r e d Ru le s . . . . . . . . 99 4 . 1 . 2 An Example o f the D e r i v a t i o n o f I n f e r r e d R u l e s . . . . . . . . 101 4 . 1 . 3 C a t e g o r i z a t i o n o f 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 the R u l e - M o d e l f o r RPS . 105 4.2 Format f o r P r e s e n t i n g R e s u l t s o f I n t e r v i e w Data A n a l y s i s . . . . . . . . 110 4 .3 R e s u l t s o f A n a l y s i s o f I n t e r v i e w Data 113 - V l l -Page 4 . 3 . 1 I n f e r r e d Rules f o r 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 Rules f o r Frame o f Re fe rence f o r a S t a t i o n a r y Body (FR) 4 . 3 . 3 I n f e r r e d Rules About 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 (D) . . . . 4 . 3 . 4 I n f e r r e d Ru le s f o r A d d i t i o n o f D i s p l a c e m e n t s (AD) 4 . 3 . 5 I n f e r r e d Rules f o r 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) . . . . 4 . 3 . 6 I n f e r r e d Rules f o r Re ference B o d i e s f o r Moving O b j e c t s (RPM) 4 . 3 . 7 I n f e r r e d R u l e s f o r A n a l y s i s o f Components (AC) 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) 4 . 3 . 9 I n f e r r e d Rules f o r Independence o f Magni tudes o f Components (IMC) 4 . 3 . 1 0 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 ) . . . 4 . 3 . 1 1 S u b j e c t s ' P r e c o n c e p t i o n s About P e r s p e c t i v e . . . . 4 .4 Two New V e c t o r C h a r a c t e r i s t i c s . . . . . . 4 . 4 . 1 Independence o f L o c a t i o n from P a t h (ILP) 4 . 4 . 2 Independence o f 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 Rules f o r Each S u b j e c t i n the I n t e r v i e w Sample . . . . . 4.6 T h r e e Types o f P r e c o n c e p t i o n s 4 . 6 . 1 Example o f the C a t e g o r i z a t i o n o f I n f e r r e d Ru le s . . . . 113 119 130 . 139 . 146 . 154 . 163 . 170 . 184 . 191 . 196 . 202 . 203 . 204 . 205 • 207 . 210 - v i i i -Pa'ge 4 . 6 . 2 C a t e g o r i z a t i o n o f I n f e r r e d Ru le s i n t o the Three Types 211 4 . 6 . 3 C a t e g o r i z a t i o n o f I n f e r r e d R u l e s o f each S u b j e c t i n t o the Three Types 211 4 . 6 . 4 . D i s c u s s i o n o f T a b l e 4.25 215 4 .7 C o n c l u s i o n o f Phase One 222 CHAPTER FIVE 5 .0 I n t r o d u c t i o n 225 5.1 Meaning o f G e n e r a l i z a b i l i t y i n t h i s Study 226 5.2 The Group I n t e r v i e w Technique 226 5.3 C o n s t r u c t i o n o f the Group I n t e r v i e w P r o t o c o l . . . . 228 5 . 3 . 1 Ta sk s o f the Group I n t e r v i e w 231 5 . 3 . 2 The Group I n t e r v i e w P r o t o c o l (GIP) 232 5 . 3 . 3 The P i l o t S tudy . . . 233 5.4 A d m i n i s t r a t i o n o f the GIP . . . . . . . . . . . . 233 5 . 4 . 1 Sample o f S u b j e c t s 233 5 . 4 . 2 D e s c r i p t i o n o f A d m i n i s t r a t i o n P r o c e d u r e s . . . . . . . . 235 5 .5 Methodology o f Data A n a l y s i s . . . . 238 CHAPTER SIX 6.0 I n t r o d u c t i o n 240 6.-1 R e s u l t s f o r each V e c t o r C h a r a c t e r i s t i c . . . . 240 - i x -Page 6 . 1 . 1 R e s u l t s f o r Re fe rence 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) 242 6 . 1 . 2 R e s u l t s f o r Frame o f Re fe rence (FR) 244 6 . 1 . 3 R e s u l t s f o r D i s p l a c e m e n t o r Change ° f L o c a t i o n (D) 246 6 . 1 . 4 R e s u l t s f o r A d d i t i o n o f D i s p l a c e -ments (AD) 249 6 . 1 . 5 R e s u l t s f o r 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 s (SVP) 251 6 . 1 . 6 R e s u l t s f o r Re fe rence Bod ie s f o r Moving O b j e c t s (RPM) 253 6 . 1 . 7 R e s u l t s f o r A n a l y s i s o f Components ' ( A C ) . . . " 255 6 . 1 . 8 R e s u l t s f o r C o m p o s i t i o n o f V e l o c i t i e s (CV) 2 5 8 6 . 1 . 9 R e s u l t s f o r Independence o f M a g n i -tudes o f Components (IMC) 266 6 . 1 . 1 0 R e s u l t s f o r S i m u l t a n e i t y o f Com-ponents (SC) 269 6 . 1 . 1 1 R e s u l t s f o r the Two New V e c t o r C h a r a c t e r i s t i c s 270 6 . 1 . 1 2 I n f e r r e d Ru le s f o r Independence ~ 7 r i o f L o c a t i o n s ' from Pa th 1ILP) • • • " i / u 6 . 1 . 1 3 I n f e r r e d R u l e s f o r Independence o f D i r e c t i o n s o f Components (IDC) . . . . 274 6.2 Summary o f the I n f e r r e d Rules f o r A l l S u b j e c t s i n the Sample 279 6.3 G e n e r a l D i s c u s s i o n o f Group I n t e r v e i w T e c h n i q u e and R e s u l t s . . . . 279 -x-BIBLIOGRAPHY APPENDIX B APPENDIX D APPENDIX E Page 282 CHAPTER SEVEN 7.0 Overv iew o f the Study 7.1 C o n c l u s i o n s o f the S tudy 283 7.2 E d u c a t i o n a l I m p l i c a t i o n s 287 7 .3 Recommendations f o r F u r t h e r Research . . . . 290 294 APPENDIX A 3 0 1 303 APPENDIX C - 3 2 2 •337 345 - x i -LIST OF TABLES Page 1.1 L i s t o f 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 o f T h r e e V e c t o r Q u a n t i t i e s (LIVC) . 9 3.1 L i s t o f I n t e r v i e w P r o t o c o l Quest ions f o r each V e c t o r C h a r a c t e r i s t i c • 8 8 4.1 L i s t o f I n f e r r e d Rules About Re fe rence P o i n t f o r S t a t i o n a r y Bod ie s (RPS) 115 4.2 Ru le s o f Rule-rModel f o r RPS . 118 4 .3 L i s t o f I n f e r r e d Rules f o r Frame o f Re fe rence (FR) 122 4.4 Ru le s o f R u l e - M o d e l f o r FR 129 4 .5 L i s t o f I n f e r r e d Rules f o r D i s p l a c e m e n t (D) 132 4 .6 R u l e s o f R u l e - M o d e l f o r D 138 4 .7 L i s t o f I n f e r r e d Rules f o r A d d i t i o n o f D i s p l a c e m e n t s (AD) . 141 4 .8 Ru le s o f R u l e - M o d e l f o r AD 146 4 .9 L i s t o f I n f e r r e d Rules f o r 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 s (SVP) . . . . . . . . . . . . 149 4.10 R u l e s o f R u l e - M o d e l f o r SVP 153 4.11 L i s t o f I n f e r r e d Rules f o r Re ference B o d i e s f o r Moving O b j e c t s (RPM) 156 4.12 R u l e s o f R u l e - M o d e l f o r RPM 162 4.13 L i s t o f I n f e r r e d Rules f o r A n a l y s i s o f Components (AC) 165 4.14 Ru le s o f R u l e - M o d e l f o r AC 170 4.15 L i s t o f I n f e r r e d Rules f o r C o m p o s i t i o n o f V e l o c i t i e s . . . . 173 4.16 T a b l e o f Paths Drawn by S u b j e c t s 182 - x i i -4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 5.1 6.1 6.2 6.3 6.4 6.5 T a b l e o f Paths drawn by S u b j e c t s . . . . R u l e s o f R u l e - M o d e l f o r CV L i s t o f I n f e r r e d Ru le s f o r Independence o f Magni tudes o f Components ( I M C ) . . . . R u l e s o f R u l e - M o d e l f o r IMC L i s t o f I n f e r r e d Ru le s f o r S i m u l t a n e i t y o f Components (SC) R u l e s " o f R u l e - M o d e l f o r SC T a b l e o f I n f e r r e d Ru le s f o r Each S u b j e c t i n the Sample About Each V e c t o r C h a r a c t e r i s t i c . . . . C a t e g o r i z a t i o n o f the I n f e r r e d Rules o f Each .Vector 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 ( S c a l a r , T r a n s i t i o n a l , o r V e c t o r i a l ) C a t e g o r i z a t i o n o f I n f e r r e d Rules o f Each S u b j e c t i n t o the Three Groups V e c t o r C h a r a c t e r i s t i c s and the Q u a n t i t y i n the G . I . P . f o r Each V e c t o r C h a r a c t e r -i s t i c , C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d Ru le s f o r RPS C o n t i n g e n c y T a b l e o f Female and Male -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 . C o n t i n g e n c y T a b l e o f Female and Male 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 D C o n t i n g e n c y T a b l e o f Female and Male 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 AD . . . . . . . C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d Ru le s f o r SVP » Page 183 183 186 190 192 • 196 206 212 214 230 242 244 246 249 251 - x i i i -Page 6.6 C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d Ru le s f o r RPM 253 6.7 . C o n t i n g e n c y T a b l e o f Female and Male 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 AC 255 6.8 C o n t i n g e n c y T a b l e o f Female and Male 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 CV . . 258 6.9 Paths Drawn f o r Q u e s t i o n 21 o f the GIP 262 6.10 Paths Drawn f o r Ques t ion 31 o f the GIP. . 264 6.11 C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d R u l e s f o r IMP , 266 6.12 C o n t i n g e n c y T a b l e o f Female and Male 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 SC 269 6.13 C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d R u l e s for- ILP 272 6.14 C o n t i n g e n c y T a b l e o f Female and Male S u b j e c t s H o l d i n g the I n f e r r e d Ru le s f o r IDC • 275 -xiv-LIST OF FIGURES Page 2.1 A P a r t i c l e Moves i n a S e m i c i r c u l a r P a t h 25 2.2 R e f l e c t i o n i n a P l a n e M i r r o r 29 3.1 Network o f a K i n e m a t i c V e c t o r Q u a n t i t y 64 3.2 Network o f V e c t o r P o s i t i o n . . . . . . . . 65 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 V e l o c i t y . . . . 67 3.5 Diagram o f the Lake f o r Task One . . . . 7 2 3.6 Diagram o f the R i v e r f o r Task Two . . . . 8 0 6 .1 Common S t u d e n t s ' Drawings 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 w h i c h the M o t o r - B o a t was Heading 276 6.2 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 w h i c h the M o t o r - B o a t was Head ing . . . . 277 -xv-ACKNOWLEDGEMENTS A number o f peop le have been i n s t r u m e n t a l i n a s s i s t i n g me t o complete t h i s r e s e a r c h . I would l i k e t o thank them f o r t h e i r a s s i s t a n c e . The most i n f l u e n t i a l p e r s o n i n my graduate program has been Gaa len E r i c k s o n , my t h e s i s a d v i s o r ; I am d e e p l y i n d e b t e d t o h i m ; he has always been a g r e a t source o f i n t e l l e c t u a l s t i m u l a t i o n , p r a c t i c a l h e l p and e m o t i o n a l s u p p o r t . I would l i k e to expres s my g r a t i t u d e to the members o f my committee — P a t r i c i a A r l i n , W a l t e r B o l d t , M i c h a e l C h a n d l e r , P e t e r Matthews, and e s p e c i a l l y t o the l a t e P r o f e s s o r H a r r y Cannon, who not o n l y s e r v e d i n my committee bu t was a c o n t i n u o u s source o f academic and p e r s o n a l s u p p o r t . A l s o , I would l i k e t o thank L i l l i a n McDermott , B e t t y Howard, and D a v i d W h i t t a k e r f o r a g r e e i n g to se rve as e x t e r n a l members o f the e x a m i n a t i o n commit tee . S p e c i a l g r a t i t u d e t o Doug B l a c k f o r h i s g r e a t h e l p i n o b t a i n i n g the s u b j e c t f o r the s t u d y . Numerous g raduate 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 suppor t and c o n s t r u c t i v e c r i t i c i s m . Among them I w i s h to e s p e c i a l l y thank Jophus Anamuah-Mensah, D a v i d B a t e s o n , and Sharon H a g g e r t y . L a s t l y , I must thank my w i f e , B e t t y , f o r the much needed s u p p o r t ; and my daughter s Carmen and A l e j a n d r a , f o r t h e i r p a t i e n c e d u r i n g my s t u d i e s . Carmen de S i l v a , my t y p i s t , must r e c e i v e my g r a t i t u d e f o r h e r w o n d e r f u l work . - 1 -CHAPTER ONE PROBLEM AND RESEARCH QUESTIONS OF THE STUDY 1.0 Background o f the Study I t i s g e n e r a l l y acknowledged by b o t h t e a c h e r s and e d u c a t i o n a l r e s e a r c h e r s t h a t s t u d e n t s encounter d i f f i c u l t y i n u n d e r s t a n d i n g p h y s i c a l v e c t o r q u a n t i t i e s . The prob lem seems t o be r e l a t e d to the f a c t t h a t s t u d e n t s do no t 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 w i t h o n l y magnitude ( s c a l a r q u a n t i t i e s ) and p h y s i c a l q u a n t i t i e s w i t h magnitude and d i r e c t i o n ( v e c t o r q u a n t i t i e s ) ; f u r t h e r m o r e t h e f a c t o r o f d i r e c t i o n b r i n g s about o p e r a t i o n s ( i . e . , a d d i t i o n and s u b t r a c t i o n o f v e c t o r s ) d i f f e r e n t , from . . ± fae o p e r a t i o n s t h a t s c a l a r q u a n t i t i e s d e a l w i t h ( i . e . , a r i t h m e t i c o p e r a t i o n s : a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and so o n ) . Re-s e a r c h f i n d i n g s show t h a t problems w i t h the se q u a n t i t i e s e x i s t a t b o t h h i g h s c h o o l and c o l l e g e l e v e l s ( F i s h e r , 1979; L e b o u t e t - B a r r e l , 1976; Ka s s , 1976; W a r r e n , 1971) . The l i t e r a t u r e and r e s e a r c h r e g a r d i n g these problems a r e t h o r o u g h l y r ev i ewed i n C h a p t e r 2. I t i s a p p a r e n t t h a t r e s e a r c h e r s have no t been l o o k i n g f o r e x p l a n a t i o n s w i t h r e s p e c t t o t h i s a rea o f s t u d e n t s ' d i f f i c u l t i e s . I n s t e a d , t h e i r • p u r p o s e appears to have been one o f i d e n t i f y i n g the prob lem r a t h e r than a n a l y z i n g i t . Why do most h i g h s c h o o l s t u d e n t s seem t o have d i f f i c u l t y i n g r a s p i n g p h y s i c a l v e c t o r q u a n t i t i e s ? Is i t a d e v e l o p m e n t a l problem? Must s t u d e n t s pos ses s c e r t a i n menta l s t r u c t u r e s ( e . g . , the P i a g e t i a n f o r m a l o p e r a t i o n s f o r c o n t r o l l i n g v a r i a b l e s and p r o p o r t i o n a l i t y ) t o u n d e r s t a n d v e c t o r q u a n t i t i e s and t h e i r v e c t o r c h a r a c t e r i s t i c s ? I s i t t h a t p h y s i c a l v e c t o r q u a n t i t i e s a re too a b s t r a c t o r complex to be g ra sped by h i g h s c h o o l s t u d e n t s ? Or a r e the d i f f i -c u l t i e s r e l a t e d t o i n s t r u c t i o n a l p r o c e d u r e s ? A l l the f a c t o r s i n c l u d e d i n these q u e s t i o n s c o u l d be p a r t o f the p r o b l e m . Some e d u c a t o r s and p s y c h o l o g i s t s ( A u s u b e l , 1968; Novak, 1977) b e l i e v e t h a t the most i m p o r t a n t f a c t o r i n t e a c h i n g a c o n t e n t - a r e a i s the knowledge t h a t s u b j e c t s b r i n g t o the l e a r n i n g s i t u a t i o n . A u s u b e l f e l t so s t r o n g l y about i t t h a t he i n t r o d u c e d a book (1968) w i t h the f o l l o w i n g 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 would say t h i s : the most i m p o r t a n t s i n g l e f a c t o r i n f l u e n c i n g l e a r n i n g i s what the 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 . I t seems c l e a r , t h e n , t h a t i n r e l a t i o n to the t o p i c o f t h i s s t u d y , the q u e s t i o n s to ask a r e : Do s t u d e n t s d e v e l o p c o n c e p t i o n s about v e c t o r q u a n t i t i e s and t h e i r v e c t o r i a l c h a r a c t e r i s t i c s p r i o r t o f o rma l i n s t r u c t i o n ? Do these c o n c e p t i o n s f o l l o w any p a t t e r n ? A r e these c o n c e p t i o n s r u l e - g o v e r n e d ? I t i s the a u t h o r ' s c o n t e n t i o n t h a t s t u d e n t s have some i n t u i t i o n r e g a r d i n g the c h a r a c t e r i s t i c s o f v e c t o r q u a n t i t i e s based on t h e i r p r e v i o u s e x p e r i e n c e s ; and i t may be t h a t one o f the d i f f i c u l t i e s s t u d e n t s e n c o u n t e r i n f o r m a l i n s t r u c t i o n i s t h a t the se i n t u i t i v e n o t i o n s a r e i n t e r f e r i n g w i t h more f o r m a l p r e s e n t a t i o n s o f s o l u t i o n s t o these p r o b l e m s . These i n t u i t i o n s a re termed the Atudznt* ' pJie.conce.ptX.onA about v e c t o r q u a n t i t i e s and t h e i r c h a r a c t e r i s t i c s . L i t t l e r e s e a r c h has been done t o uncover s t u d e n t s ' p r e c o n c e p t i o n s about v e c t o r q u a n t i t i e s . P i a g e t (1973) • s t u d i e d the deve lopment o f c o n c e p t i o n s i n c h i l d r e n r e g a r d i n g s e v e r a l a s p e c t s o f f o r c e s ( f o r c e i s a t y p i c a l v e c t o r q u a n t i t y ) , p a r t i c u l a r l y t h e i r d i r e c t i o n a l a s p e c t and com-p o s i t i o n . An e x t e n s i v e a n a l y s i s o f P i a g e t ' s r e s u l t s w i l l be p r e s e n t e d i n C h a p t e r 2. A g u i r r e (1978) s t u d i e d c h i l d r e n ' s b e l i e f s about f o r c e s i n s t a t i c e q u i l i b r i u m , and found t h a t s t u d e n t s a t grade 8 (13 t o 14 y e a r s o f age) have a l r e a d y a c q u i r e d some i n t u i t i v e n o t i o n s o f v e c t o r c h a r a c t e r i s t i c s a s s o c i a t e d w i t h the c o n c e p t o f f o r c e . These n o t i o n s were i d e n t i f i e d t h r o u g h s t u d e n t s ' r e sponse s t o s e v e r a l t a s k s on f o r c e s i n an i n t e r -v iew s e t t i n g . An example o f those i n t u i t i o n s i s t h a t some s t u d e n t s i n the sample were a b l e t o p r e d i c t t h e d i r e c t i o n o f t h e r e s u l t a n t o f two f o r c e components . I t would be i n t e r e s t i n g t o f i n d out i f 14 y e a r o l d s u b j e c t s can a l s o a s s o c i a t e v e c t o r c h a r a c t e r i s t i c s w i t h o t h e r p h y s i c a l v e c t o r -4-q u a n t i t i e s such as d i s p l a c e m e n t and v e l o c i t y . W h i l e l i t t l e work has been done s p e c i f i c a l l y to s tudy s t u d e n t s ' p r e c o n c e p t i o n s o f v e c t o r q u a n t i t i e s , more a t t e n t i o n has been p a i d t o the problems o f r e p r e s e n t i n g s t u d e n t s ' knowledge p r i o r t o f o r m a l i n s t r u c t i o n . Some o f these a t tempts have been summarized i n D r i v e r and E a s l e y (1978) and A r c h e n h o l d e t a l . (1979) . However, i t was judged t h a t the most p r a c t i c a l m o d e l , f o r the p r e s e n t s t u d y , i s t h a t d e s c r i b e d by S i e g l e r (1976, 1979) . He a t tempted to i d e n t i f y the r u l e s used by s u b j e c t s t o s o l v e problems i n s p e c i f i c c o n t e n t - a r e a s . He f u r t h e r p roposed a methodology t o a s ses s s t u d e n t s ' use o f those h y p o t h e t i c a l r u l e s . A d e s c r i p t i o n o f h i s r u l e - a s s e s s m e n t methodology i s p r e s e n t e d i n Chapter 2. In summary, i t i s the a u t h o r ' s c o n t e n t i o n t h a t s t u d e n t s d e v e l o p some i n t u i t i v e n o t i o n s about p h y s i c a l r e a l i t y t h a t a re r e l a t e d to what p h y s i c i s t s d e f i n e as v e c t o r p r o p e r t i e s . T h i s i s i l l u s t r a t e d when s t u d e n t s are c o n f r o n t e d w i t h c o n c r e t e p r o b l e m s i t u a t i o n s i n which v e c t o r p r o p e r t i e s a r e c l e a r l y p r e s e n t . T h i s b e l i e f i s based on the f a c t t h a t P i a g e t (1973) and A g u i r r e (19 78) b o t h found e v i d e n c e i n 12 to 14 y e a r o l d s u b j e c t s o f i n t u i t i v e i d e a s which a re c l e a r l y v e c t o r i a l . 1.1 D e f i n i t i o n o f Terms Some o f the terms b e i n g used i n t h i s s tudy a r e p r e -s e n t e d b e l o w . I t i s assumed t h a t the r e a d e r i s f a m i l i a r w i t h a l l the p h y s i c a l q u a n t i t i e s i n c l u d e d i n the s t u d y , such as - 5 -v e c t o r 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 . I f t h i s i s n o t the c a s e , the d e f i n i t i o n s o f those v e c t o r q u a n t i t i e s a re p r e s e n t e d i n Appendix A . D e f i n i t i o n o f v e c t o r q u a n t i t y ( J o r d a n , 1969, p . l ) I t i s a q u a n t i t y t h a t p o s s e s s e s magnitude and d i r e c t i o n and obeys the v e c t o r a d d i t i o n o p e r a t i o n and the s c a l a r m u l t i -p l i c a t i o n o p e r a t i o n . F o r each p a i r o f v e c t o r s y_ and v_, the v e c t o r a d d i t i o n o p e r a t i o n r e q u i r e s t h a t : i ) y + v = v + y (Commutative p r o p e r t y ) i i ) y + (v + OJ) = (y + v ) + w ( A s s o c i a t i v e p r o p e r t y ) i i i ) a un ique v e c t o r 0 (zero) has the p r o p e r t y t h a t v + 0 = v i v ) a u n i q u e v e c t o r - v has the p r o p e r t y t h a t v + (-v) = 0 . D e f i n i t i o n o f p r e c o n c e p t i o n * I t i s a type o f i n t u i t i v e n o t i o n used by a s u b j e c t t o e x p l a i n a p a r t i c u l a r phenomenon and w h i c h i s a p r o d u c t o f the s u b j e c t ' s e x p e r i e n c e . T h i s i n t u i t i v e n o t i o n l a c k s the scope and o r g a n i z a t i o n o f the f o r m a l c o n c e p t i o n commonly a c c e p t e d by t h e d i s c i p l i n e . -6-D e f i n i t i o n o f i n f e r r e d r u l e A r u l e i s i n f e r r e d t o e x i s t i f a s u b j e c t appears t o use a c o n s i s t e n t p r e c o n c e p t i o n i n a g i v e n s i t u a t i o n t o d e a l w i t h a p a r t i c u l a r q u e s t i o n o r prob lem posed by the i n t e r -v i e w e r . O p e r a t i o n a l d e f i n i t i o n o f i n f e r r e d r u l e The p re sence o f a r u l e i s i n f e r r e d i f a s u b j e c t uses the same b a s i c p r e c o n c e p t i o n t o d e a l w i t h more than h a l f o f the q u e s t i o n s r e g a r d i n g the same v e c t o r c h a r a c t e r i s t i c ( e . g . , c h o i c e o f r e f e r e n c e p o i n t s t o l o c a t e a boa t on a l a k e ) . D e f i n i t i o n o f v e c t o r c h a r a c t e r i s t i c A v e c t o r c h a r a c t e r i s t i c d e s c r i b e s an a s p e c t o f p h y s i c a l phenomena i n which the d i r e c t i o n p l a y s an i m p o r t a n t r o l e . N o t e : I n t h i s s t u d y , p h y s i c a l phenomena i n w h i c h the v a r i a b l e d i r e c t i o n i s a d e c i s i v e f a c t o r a r e c a l l e d v e c t o r i a l p h y s i c a l phenomena. D e f i n i t i o n o f 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 An 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 r e p r e s e n t s an a s p e c t o f v e c t o r i a l p h y s i c a l phenomena t h a t i s n o t e x p l i c i t l y e l i c i t e d i n the d e f i n i t i o n o f a v e c t o r q u a n t i t y , o r i n the d e f i n i t i o n s o f the t h r e e v e c t o r q u a n t i t i e s w i t h w h i c h t h i s s t u d y d e a l s . -7-1.2 I m p l i c i t C h a r a c t e r i s t i c s o f Three P h y s i c a l V e c t o r  Q u a n t i t i e s The f o r m a l m a t h e m a t i c a l d e f i n i t i o n o f a . v e c t o r q u a n t i t y and i t s p r o p e r t i e s was p r e s e n t e d i n the p r e v i o u s s e c t i o n . T h i s f o r m a l d e f i n i t i o n i s g e n e r a l l y used by mathemat i c i ans and t h e o r e t i c a l p h y s i c i s t s who do n o t need t o c o n c r e t i z e those p r o p e r t i e s i n d e s c r i b i n g some p a r t i -c u l a r a s p e c t o f p h y s i c a l r e a l i t y . T y p i c a l s c h o o l s c i e n c e t e x t - b o o k s ( P . S . S . C ; 1980, Fundamentals o f P h y s i c s by H e a t h , Macnaughton, & M a r t i n d a l e , 1979; P h y s i c s by D . C . G i a n c o l i , 1980) a l s o u s u a l l y i n t r o d u c e v e c t o r q u a n t i t i e s t h r o u g h such f o r m a l d e f i n i t i o n . However, the a u t h o r b e l i e v e s t h a t the f o r m a l d e f i n i t i o n does no t make e x p l i c i t some a s p e c t s o r c h a r a c t e r i s t i c s o f p h y s i c a l phenomena t h a t t h e s e v e c t o r q u a n t i t i e s r e p r e s e n t . Fur thermore the se c h a r a c t e r -i s t i c s a re thought t o be i m p o r t a n t i n terms o f a c t u a l l y u n d e r s t a n d i n g t h e s e p h y s i c a l q u a n t i t i e s . I t may be t h a t when t e a c h e r s use the fo rma l d e f i n i t i o n , they take f o r g r a n t e d t h a t the d e f i n i t i o n s c o v e r a l l the i m p o r t a n t a s p e c t s o f the p h y s i c a l phenomena t h a t the se v e c t o r q u a n t i t i e s r e p r e s e n t . T h i s may be one o f the causes o f the d i f f i c u l t i e s e n c o u n t e r e d by s t u d e n t s i n u n d e r s t a n d i n g t h e s e q u a n t i t i e s . The d e f i n i t i o n s o f the t h r e e k i n e m a t i c v e c t o r q u a n t i t i e s : v e c t o r 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 - 8 -(see Appendix A) t h a t t h i s s tudy d e a l s w i t h i n c l u d e s v e c t o r c h a r a c t e r i s i t s t h a t a re no t e x p l i c i t l y c o n s i d e r e d i n the d e f i n i t i o n o f a v e c t o r q u a n t i t y . F o r example , the t h r e e q u a n t i t i e s r e q u i r e the s e l e c t i o n o f a p p r o p r i a t e r e f e r e n c e p o i n t s f o r a complete d e s c r i p t i o n o f them. However, the d e f i n i t i o n o f a v e c t o r q u a n t i t y does no t i n c l u d e such a r e q u i r e m e n t . Those 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 a re no t o b v i o u s l y c o n s i d e r e d i n the u s u a l f o rma l d e f i n i t i o n s , a c c o r d i n g t o the a u t h o r , were uncovered by employ ing a r a t i o n a l t a s k a n a l y s i s o f the t h r e e q u a n t i t i e s o f i n t e r e s t t o t h i s s t u d y . T h i s i s f u l l y e x p l a i n e d i n Chapter T h r e e . The v e c t o r c h a r a c t e r i s t i c s o f i n t e r e s t are l i s t e d i n T a b l e 1.1. The l a s t two 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 l i s t e d i n T a b l e 1.1 (ILP and IDC) were not a c t u a l l y a n t i c i p a t e d by the R a t i o n a l Task A n a l y s i s per formed upon the t h r e e v e c t o r q u a n t i t i e s . These v e c t o r c h a r a c t e r i s t i c s were u n -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 s u b j e c t s ' r e sponse s i n the f i r s t phase o f the s t u d y . (There were two phases i n the s tudy w h i c h a re e x p l a i n e d l a t e r ) . I t was d e c i d e d t o c o n s i d e r the two new v e c t o r c h a r a c t e r i s t i c s i n the second phase o f the s t u d y . - 9 -T a b l e 1.1 L i s t o f 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 " o f Three V e c t o r Q u a n t i t i e s : V e c t o r 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 . Re fe rence P o i n t as a means t o d e s c r i b e the l o c a t i o n o f a s t a t i o n a r y body o r p o i n t (RPS). 2. Frame o f Re ference 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 body (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 magnitude and d i r e c t i o n , which i s independent from a l l p o s s i b l e paths to j o i n two l o c a t i o n s (D) . In 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 changes o f l o c a t i o n s w i t h the 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 the v e c t o r p o s i t i o n a s s o c i a t e d to two l o c a t i o n s o f a moving body and the d i s p l a c e m e n t between the l o c a t i o n s (SVP) . 6. R e f e r e n c e b o d i e s ( s t a t i o n a r y o r moving b o d i e s ) as a means t o d e s c r i b e the mot ion o f an o b j e c t (RPM). 7 . A n a l y s i s o f the number o f component v e l o c i t i e s t h a t combined form an a p p a r e n t s i n g l e mot ion (AC) . 8. C o m p o s i t i o n o f two s imul t aneous v e l o c i t i e s t h a t a r e a f f e c t i n g an o b j e c t (CV) . 9. Cons t ancy o f each one o f the magnitude o f two v e l o c i t i e s even when t h e y a re i n t e r a c t i n g t o form an apparent s i n g l e m o t i o n (IMC). 1 0 . 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 a f f e c t -i n g a body t o c r e a t e an apparent s i n g l e mot ion (SC) . 1 1 . 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 the f a c t t h a t i t i s not n e c e s s a r v t o know the p a t h f o l l o w e d by 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 o f i t . ..... • ._. _ M-1 2 . Cons t ancy o f each one o f the 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 they a re i n t e r a c t i n g to form an apparent s i n g l e m o t i o n ( IDC) . -10-Th e i tems i n t h i s l i s t a re no t n e c e s s a r i l y p r e s e n t e d i n h i e r a r c h i c a l o r d e r . The r e l a t i o n s h i p s among a l l t h e s e v e c t o r c h a r a c t e r i s t i c s a re p r e s e n t e d through networks i n C h a p t e r T h r e e . A d e t a i l e d d e s c r i p t i o n o f each one o f these c h a r a c t e r -i s t i c s and the rea sons why t h e y were c o n s i d e r e d i n the p r e s e n t s tudy a re b e i n g p r e s e n t e d i n C h a p t e r F o u r . In t h a t c h a p t e r , s e c t i o n s 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 t o g e t h e r w i t h the a n a l y s i s o f da t a c o l l e c t e d a re d e v e l o p e d . I t was d e c i d e d t o do so t o f a c i l i t a t e the u n d e r s t a n d i n g o f the r e l a t i o n s h i p s between the i d e n t i f i e d p r e c o n c e p t i o n s w i t h 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 . F u r t h e r m o r e , the l i s t was no t meant t o be e x h a u s t i v e . As was ment ioned e a r l i e r , the major s e l e c t i o n c r i t e r i o n o f the i t ems o f T a b l e 1.1 was the c o n d i t i o n o f no t b e i n g e x p l i c i t l y i n c l u d e d i n the f o r m a l d e f i n i t i o n o f a v e c t o r q u a n t i t y . The major aim o f t h i s s tudy was t o i d e n t i f y and a n a l y z e the k i n d s o f p r e c o n c e p t i o n s s t u d e n t s pos se s s r e g a r d i n g these 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 . H e r e a f t e r , r e f e r e n c e i s made t o the l i s t o f 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 , t h i s has been s h o r t e n e d to j u s t v e c t o r c h a r a c t e r i s t i c s . 1.3 G e n e r a l S ta tement o f the Prob lem Knowledge o f what the l e a r n e r b r i n g s t o the l e a r n i n g s i t u a t i o n i s an i m p o r t a n t component i n p l a n n i n g an e d u c a t i o n a l T - 1X - . program. A l t h o u g h t h e r e has been a wide spread agreement among e d u c a t o r s and e d u c a t i o n a l r e s e a r c h e r s w i t h r e s p e c t to the above s t a t ement , i t i s indeed s u r p r i s i n g how l i t t l e r e -s e a r c h has been d i r e c t e d towards the problem o f a s c e r t a i n i n g what t h e l e a r n e r a l r e a d y knows. Shulman and Tamir sugges t t h a t : There s h o u l d a l s o - b e an i n c r e a s e i n a genre o f b a s i c r e s e a r c h t h a t i s s c i e n c e - s p e c i f i c r a t h e r than exper iments t o t e s t a s p e c t s 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 happens 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 t o 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 would be on a t o p i c l i k e t h e 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 . , cause and e f f e c t , space , t i m e , mass, 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 o f P i a g e t . 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 which 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 c o u l d p l a n t h e i r c r e a t i v e endeavour s . (Emphasis added)(Shulman and T a m i r , 1973, p . 1 1 3 9 ) . The shor tage o f t h i s type o f r e s e a r c h i n the s p e c i f i c a r e a o f v e c t o r q u a n t i t i e s -— as a s e t o f d i s t i n c t i v e p h y s i c a l q u a n t i t i e s — may i n p a r t be r e s p o n s i b l e f o r some o f the d i f f i -c u l t i e s e n c o u n t e r e d by s t u d e n t s when c o n f r o n t e d w i t h v e c t o r -t y p e problems i n an i n s t r u c t i o n a l s e t t i n g . T h i s i s , t h e r e i s a l a c k o f knowledge on the p a r t o f c u r r i c u l u m makers and t e a c h e r s about t h e p r e c o n c e p t i o n s h e l d by s t u d e n t s on v e c t o r c h a r a c t e r -i s t i c s and c o n s e q u e n t l y these p r e c o n c e p t i o n s are not r e f l e c t e d i n the i n s t r u c t i o n a l p r o c e d u r e s . The d i f f i c u l t i e s , t h e n , may n o t be due t o t h e i n c a p a c i t y o f s t u d e n t s t o g r a s p the v e c t o r i a l c o n c e p t s b u t may be due to the f a i l u r e t o c o n s i d e r s t u d e n t s ' p r e c o n c e p t i o n s . T h i s l a c k o f c o n s i d e r a t i o n may -12-cause an i n t e r f e r e n c e or mismatching between what the t e a c h e r wants t o t e a c h and what students a l r e a d y know. I t was c o n j e c t u r e d t h a t t h i s i s the case i n the i n s t r u c t i o n o f v e c t o r q u a n t i t i e s . The broad aim o f t h i s study was t o develop a p p r o p r i a t e methods of data c o l l e c t i o n and a n a l y s i s which would p e r m i t a d e t a i l e d examination of the s t u d e n t s ' p r e c o n c e p t i o n s about v e c t o r q u a n t i t i e s . The f i r s t t ask r e q u i r e d , then, was to c a r r y out an i n - d e p t h a n a l y s i s of the v e c t o r i a l nature of the t h r e e k i n e m a t i c concepts d e a l t w i t h i n the study. T h i s task a n a l y s i s y i e l d e d the L i s t of 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 , which were d e s c r i b e d i n S e c t i o n 1.2. Two d i f f e r e n t methods of data c o l l e c t i o n were used hence, the r e s u l t s o f the study are r e p o r t e d i n two d i s t i n c t phases. The f i r s t phase u t i l i z e d i n d i v i d u a l , c l i n i c a l - t y p e , i n t e r v i e w s w i t h students based upon c o n c r e t e problem s i t u -a t i o n s which i l l u s t r a t e d these v e c t o r c h a r a c t e r i s t i c s . In the second phase a group i n t e r v i e w instrument was developed which c o u l d be a d m i n i s t e r e d to i n t a c t c l a s s e s o f s t u d e n t s . T h i s l a t t e r t e chnique allowed d a t a to be gathered on a much l a r g e r and more heterogeneous"group of s t u d e n t s , thus p r o -v i d i n g a type of v a l i d i t y c r o s s - c h e c k of the i n t e r v i e w d a t a . In a d d i t i o n i t has the p o t e n t i a l t o be r e f i n e d f o r use as a d i a g n o s t i c instrument by classroom t e a c h e r s . The a n a l y t i c a l t e chniques developed f o r r e v e a l i n g the s t u d e n t s ' preconcep-t i o n s i n Phase One were a l s o used i n Phase Two so t h a t comparisons between the two data s e t s c o u l d be made. In summary, t h i s study makes a s i g n i f i c a n t c o n t r i b u t i o n to the f i e l d o f s c i e n c e e d u c a t i o n i n two a r e a s : the d e s i g n and T ! 3 ^ a p p l i c a t i o n o f the v a r i o u s m e t h o d o l o g i c a l t e c h n i q u e s ' used i n the s tudy and the s u b s t a n t i v e r e s u l t s o f s t u d e n t s * p r e -c o n c e p t i o n s about v e c t o r q u a n t i t i e s . The b r o a d prob lem o f t h i s s tudy was b r o k e n down i n t o a number o f more s p e c i f i c problems and f u r t h e r d e s c r i b e d i n terms o f a number o f r e s e a r c h q u e s t i o n s i n the f o l l o w i n g two s e c t i o n s . 1.4 S p e c i f i c S ta tements o f the Problem The g e n e r a l prob lem was f u r t h e r s u b d i v i d e d i n t o the f o l l o w i n g s p e c i f i c s t a t e m e n t s : 1) To u n c o v e r , a n a l y z e , and look f o r the p a t t e r n s i n s t u d e n t s ' p r e c o n c e p t i o n s about th e 12 v e c t o r c h a r a c t e r i s t i c s . 2) To d e t e r m i n e whether t h e r e was any s i g n i f i c a n t d i f f e r e n c e between female and male s u b j e c t s o f the s tudy r e g a r d i n g the k i n d o f p r e c o n c e p t i o n s uncovered about the v e c t o r c h a r a c t e r i s t i c s . 3) To c a t e g o r i z e the p r e c o n c e p t i o n s f o r each v e c t o r c h a r a c t e r i s t i c u s i n g a c o m p l e x i t y c r i t e r i o n . T h a t i s , by a n a l y z i n g the number and the n a t u r e o f the v a r i a b l e s ( q u a l i t a t i v e o r q u a n t i t a t i v e ) a s s o c i a t e d w i t h each o f the i d e n t i f i e d p r e -c o n c e p t i o n s . 4) To d e t e r m i n e the degree o f co r re spondence between the s t u d e n t s ' p r e c o n c e p t i o n s and the a c c e p t e d v e r s i o n o f v e c t o r c h a r a c t e r i s t i c s as h e l d by the c u r r e n t s c i e n t i f i c community. 5 R e s e a r c h Q u e s t i o n s T h e r e s e a r c h q u e s t i o n s w h i c h t h e p r e s e n t s t u d y a t t e m p t e d t o a n s w e r w e r e a s f o l l o w s : (1) W h a t p r e c o n c e p t i o n s d o s t u d e n t s h o l d a b o u t e a c h o f rfche f o l l o w i n g 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 ? 1.1 R e f e r e n c e p o i n t f o r s t a t i o n a r y b o d i e s . 1.2 F r a m e o f r e f e r e n c e . 1.3 D i s p l a c e m e n t o r c h a n g e o f l o c a t i o n . 1.4 A d d i t i o n o f d i s p l a c e m e n t s . 1.5 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 . 1.6 R e f e r e n c e b o d i e s f o r o b j e c t s i n m o t i o n . 1.7 A n a l y s i s o f c o m p o n e n t v e l o c i t i e s . 1.8 C o m p o s i t i o n o f s i m u l t a n e o u s v e l o c i t i e s . 1. 9 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 i n t e r a c t i n g v e l o c i t i e s . 1.10. S i m u l t a n e i t y o f c o m p o n e n t v e l o c i t i e s . 1.11 I n d e p e n d e n c e o f l o c a t i o n s f r o m p a t h . 1.12. I n d e p e n d e n c e o f d i r e c t i o n s o f i n t e r a c t i n g v e l o c i t i e s . (2) H o w c o n s i s t e n t a r e t h e p r e c o n c e p t i o n s e m p l o y e d b y s t u d e n t s w h e n a t t e m p t i n g t o s o l v e p r o b l e m s i n v o l v i n g t h e l i s t e d v e c t o r c h a r a c t e r i s t i c s ? (3) A r e t h e i d e n t i f i e d p r e c o n c e p t i o n s a b l e t o b e c a t e g o r i z i n s o m e f a s h i o n ? (4) i s t h e r e a s i g n i f i c a n t d i f f e r e n c e b e t w e e n f e m a l e a n d m a l e s t u d e n t s r e g a r d i n g t h e i d e n t i f i e d p r e c o n c e p t i o n s ? (5) H o w 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 p r e c o n c e p t i o n s t o o t h e r g r o u p s o f s t u d e n t s ? -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 a re predominant i n the s tudy sample? 1.6 Overv iew o f Methods o f Study Phase one c o n s i s t e d i n a s e r i e s o f s e m i - s t r u c t u r e d i n d i v i d u a l i n t e r v i e w s w i t h s t u d e n t s a t the grade 10 l e v e l and whose ages ranged between 15 and 17 y e a r s . The i n t e r -v iews were v i d e o - t a p e d and a u d i o - t a p e d f o r f u r t h e r t r a n s c r i p t i o n . P r e c o n c e p t i o n s w i t h r e s p e c t t o each v e c t o r c h a r a c t e r i s t i c i n the LIVC were i d e n t i f i e 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 were expre s sed as i n f e r r e d r u l e s , w h i c h were c a t e g o r i z e d a c c o r d i n g t o t h e i r c o n t e n t ; f i n a l l y , the i n -t e r r e d r u l e s were p r e s e n t e d i n a more g e n e r a l and l e s s c o n t e x t u a l fo rm, and were c a l l e d s i m p l y r u l e s . These r u l e s s h o u l d be c o n s i d e r e d as h y p o t h e t i c a l i n n a t u r e s i n c e they were o n l y i d e n t i f i e d i n one t y p e o f t a sk s i t u a t i o n . The s e t o f r u l e s f o r each v e c t o r c h a r a c t e r i s t i c was c a l l e d a r u l e -m o d e l . In summary, r e s e a r c h q u e s t i o n s 1 to 3 were addre s sed i n t h i s f i r s t phase . S i n c e the main o b j e c t i v e o f Phase Two was t o u n -c o v e r i n t u i t i v e n o t i o n s about the v e c t o r c h a r a c t e r i s t i c s l i s t e d i n LIVC u s i n g a l a r g e r sample o f s u b j e c t s , i t was d e c i d e d to use a group i n t e r v i e w methodology . Because one o f the o b j e c t i v e s was t o compare the r e s u l t s from b o t h phases the same c o n t e x t ( tasks) had to be u s e d , and a l s o a s i m i l a r p r o t o c o l . T h e r e f o r e , the p r o t o c o l used i n the i n d i v i d u a l i n t e r v i e w s was adapted f o r use w i t h w r i t t e n re sponse format t o f a c i l i t a t e the a d m i n i s t r a t i o n o f the 1-16-group i n t e r v i e w . When a d m i n i s t e r i n g the group i n t e r v i e w s , the i n t e r v i e w e r e x p l a i n e d the e x p e r i m e n t a l s i t u a t i o n s d e -p i c t e d i n the t a s k s i n s u f f i c i e n t d e t a i l , r e a d the q u e s t i o n s v e r y c a r e f u l l y and a f t e r b e i n g s a t i s f i e d t h a t a l l s t u d e n t s u n d e r s t o o d the n a t u r e o f the q u e s t i o n s , asked them t o t h i n k c a r e f u l l y about t h e i r re sponse and to w r i t e down t h e i r i d e a s i n the space p r o v i d e d undernea th 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 used i n t h i s second phase was c a l l e d a Group I n t e r v i e w P r o t o c o l (GIP) . T h i s i n -s t rument cannot be c o n s i d e r e d as a k i n d o f s t a n d a r d i z e d q u e s t i o n n a i r e o r as a P s y c h o m e t r i c t e s t because i t was n o t c o n s t r u c t e d t o s e rve the purpose o f such n o r m a t i v e t e s t s . W h i l e i t was p r e p a r e d s p e c i f i c a l l y f o r use i n the p r e s e n t s t u d y , i t may be c o n s i d e r e d as a type o f t e a c h e r s ' d i a g -n o s t i c t e s t . In summary, r e s e a r c h q u e s t i o n s 5. and 6 were addre s sed i n t h i s second phase . Research q u e s t i o n 4 was addre s sed i n b o t h phase s . In b o t h p h a s e s , th e s u b j e c t s had n o t y e t been t a u g h t any k i n e m a t i c v e c t o r q u a n t i t i e s ; t h e i r p r e c o n c e p t i o n s c o r r e s p o n d e d to t h e i r i n t u i t i v e n o t i o n s f o r d e a l i n g w i t h a c t u a l p rob lem s i t u a t i o n s . Here p r e c i s e l y l i e s the i m -p o r t a n c e o f the s tudy s i n c e t h i s i n t u i t i v e knowledge- can e i t h e r h e l p the u n d e r s t a n d i n g o f concep t s d e l i v e r e d i n f o r m a l i n s t r u c t i o n o r i n t e r f e r e w i t h them. T h i s a u t h o r -17-b e l i e v e s t h a t some o f the s u b j e c t s ' c o n s i s t e n t p r e c o n c e p -t i o n s ( " i n f e r r e d r u l e s " ) 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 b l o c k s " (Hobbs, 1977) o r " c r i t i c a l b a r r i e r s " ( H a w k i n s , 1980) and c o u l d w e l l i n t e r f e r e w i t h the f o r m a l i n s t r u c t i o n o f v e c t o r q u a n t i t i e s . 1.7 D e l i m i t a t i o n o f the S tudy The t o p i c chosen f o r the s t u d y , v e c t o r q u a n t i t i e s , i s a b r o a d one which i s a l s o a s s o c i a t e d w i t h a number o f o t h e r p h y s i c s c o n c e p t s . In a d d i t i o n to the t h r e e p h y s i c a l q u a n t i t i e s s e l e c t e d f o r the p r e s e n t s t u d y , t h e r e a r e o t h e r s such a s : a c c e l e r a t i o n , f o r c e , and l i n e a r and a n g u l a r moment'um. L i k e w i s e the s tudy was l i m i t e d t o the a n a l y s i s o f s e v e r a l v e c t o r p r o p e r t i e s ; those d e s c r i b e d i n the l i s t o f i m p l i c i t c h a r a c t e r i s t i c s o f p h y s i c a l v e c t o r q u a n t i t i e s . O t h e r s , such as the i n n e r o r s c a l a r p r o d u c t o f v e c t o r s and c r o s s - p r o d u c t o f v e c t o r s , were o m i t t e d because t h e i r meaning i s remote from s t u d e n t s ' i n f o r m a l e x p e r i e n c e . T h i s s t u d y , t h e n , was l i m i t e d to o n l y t h r e e v e c t o r q u a n t i t i e s , ( v e c t o r p o s i t i o n , d i s p l a c e m e n t , v e l o c i t y ) and o n l y t o the b a s i c c h a r a c t e r i s t i c s o f those q u a n t i t i e s . The rea sons f o r t h i s r e s t r i c t i o n were r e l a t e d to the purpose o f t h i s s t u d y , w h i c h was f o c u s e d 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 o r p r e c o n c e p t i o n s o f s t u d e n t s w i t h r e s p e c t t o p h y s i c a l q u a n t i t i e s . These t h r e e v e c t o r i a l q u a n t i t i e s , which are u s u a l l y c o n s i d e r e d to be more s i m p l e , a re the ones i n t r o -duced f i r s t i n a f o r m a l i n s t r u c t i o n c o n t e x t . These t h r e e -18-q u a n t i t i e s d e s c r i b e the most r e l e v a n t d i r e c t i o n a l a s p e c t o f m o t i o n , which i s the a t t r i b u t e t h a t d e f i n e s them as v e c t o r q u a n t i t i e s . Hence t h i s seemed l i k e a r e a s o n a b l e s t a r t i n g p o i n t f o r the s t u d y . Fur thermore i t was f e l t t h a t o t h e r v e c t o r quan-t i t i e s , such as a n g u l a r momentum, t o r q u e , o r a n g u l a r a c c e l e r a t i o n , a re v e r y a b s t r a c t and remote from the ex-p e r i e n c e s o f most s t u d e n t s , and are o n l y t a u g h t i n advanced p h y s i c s c o u r s e s i n the h i g h s c h o o l c u r r i c u l u m . A n o t h e r i m p o r t a n t l i m i t a t i o n was the r e s t r i c t e d range o f ages (15 to 17) which was used i n the s t u d y . T h i s narrow range was chosen on the b a s i s o f P i a g e t * s (1973) and A g u i r r e ' s (.1978) s t u d i e s , b o t h o f which r e p o r t e d t h a t v e c t o r i a l p r e c o n c e p t i o n s were n o t found u n t i l s t u d e n t s a r e o v e r 14 y e a r s o f age . The a u t h o r i s aware t h a t an i n t e r v i e w t e c h n i q u e has some i n h e r e n t problems w i t h r e s p e c t t o making i n f e r e n c e s from s m a l l samples . I t may a l s o i n t r o d u c e problems o f b i a s w i t h r e s p e c t to the i n t e r v i e w i t s e l f and the a n a l y s i s o f d a t a . To overcome these problems i n p a r t , the f i n d i n g s o f phase one were c r o s s v a l i d a t e d i n phase two. T h a t i s , the g e n e r a l i z a b i l i t y o f the i n f e r r e d r u l e s t h a t were g e n e r a t e d i n phase one were f u r t h e r checked o u t i n the second phase . The prob lem o f s u b j e c t i v i t y i n an i n t e r v i e w s e t t i n g was m i n i m i z e d by u s i n g a s e m i - s t r u c t u r a l p r o t o c o l f o r a l l s u b j e c t s i n the sample . -19-1.8 J u s t i f i c a t i o n o f the S tudy P h y s i c a l v e c t o r q u a n t i t i e s were chosen as the t o p i c o f th e p r e s e n t r e s e a r c h f o r the f o l l o w i n g r e a s o n s : (1) the se q u a n t i t i e s are i n c l u d e d i n most p h y s i c s p r o -grams i n grade 11 and 12 i n N o r t h A m e r i c a ; (2) most s t u d e n t s appear to have d i f f i c u l t y u n d e r s t a n d i n g v e c t o r q u a n t i t i e s , and i t i s v e r y i m p o r t a n t t o f i n d o u t some o f the rea sons f o r these d i f f i c u l t i e s ; (3) s t u d e n t s ' l a c k o f u n d e r s t a n d i n g o f v e c t o r q u a n t i t i e s has r e s t r i c t e d them from f u r t h e r u n d e r s t a n d i n g o f a wide range o f p h y s i c s c o n c e p t s ( e . g . , see Hawkins (1978) d i s c u s s i o n o f c r i t i c a l b a r r i e r phenomena); (4) most s t u d e n t s have gone t h r o u g h i n f o r m a l e x p e r i e n c e s i n w h i c h a d i r e c t i o n a l f a c t o r , an u n d e r l y i n g a s p e c t o f v e c t o r q u a n t i t i e s , has been p r e s e n t and t h e r e f o r e they have had the chance to f o r m u l a t e an i n t u i t i v e s e t o f i d e a s about the n a t u r e o f v e c t o r c h a r a c t e r i s t i c s ; (5) i t i s an a r e a o f s t u d y t h a t i s c o n s p i c i o u s l y absent from the r e s e a r c h l i t e r a t u r e . The r e s u l t s o f r e s e a r c h i n t o s t u d e n t s ' i n t u i t i v e n o t i o n s about p h y s i c s v e c t o r q u a n t i t i e s and t h e i r s p e c i f i c c h a r a c t e r i s t i c s can be a p p l i e d to problems o f i n s t r u c t i o n i n two b r o a d ways : F i r s t l y , i n the development o f i n s t r u c t i o n a l packages based upon knowledge o b t a i n e d from the s t u d y . The main purpose o f these packages would be - 2 0 -t o make the " r i g h t c o n n e c t i o n " between what s t u d e n t s a l r e a d y know and what the s c h o o l c u r r i c u l u m i s t r y i n g t o t e a c h . S e c o n d l y , the s tudy would p r o v i d e s c i e n c e t e a c h e r s 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 the i n t u i -t i v e n o t i o n s and c o n s i s t e n t p r e c o n c e p t i o n s t h a t s t u d e n t s use i n c o p i n g w i t h p r o b l e m a t i c s i t u a t i o n s i n v o l v i n g v e c t o r q u a n t i t i e s . -21-CHAPTER TWO LITERATURE REVIEW/ AND PSYCHOLOGICAL AND EDUCATIONAL CONTEXTS OF THE STUDY 2.0 I n t r o d u c t i o n T h i s c h a p t e r d e a l s w i t h t h r e e main t o p i c s : (1) the l i t e r a t u r e r e l a t e d t o students* understanding o f v e c t o r i a l c oncepts; (2) the g e n e r a l p s y c h o l o g i c a l c o n t e x t f o r the study; (3) the e d u c a t i o n a l p e r s p e c t i v e i n which the f i n d -i n g s o f the r e s e a r c h c o u l d be made u s e f u l . With r e g a r d to the f i r s t t o p i c the f o l l o w i n g aspects are reviewed': ( i ) an e x t e n s i v e l i t e r a t u r e review i s presented t o show the g e n e r a l l a c k o f understanding o f v e c t o r q u a n t i t i e s by stu d e n t s ; ( i i ) the r e s e a r c h c a r r i e d out so f a r on st u d e n t s ' p r e c o n c e p t i o n s about v e c t o r i a l concepts; and ( i i i ) a r e -view o f experiments and i n s t r u c t i o n a l u n i t s about v e c t o r s , which have been c r e a t e d w i t h the s p e c i f i c purpose of a l l e v i a t i n g the d i f f i c u l t i e s u s u a l l y encountered by s t u d e n t s . In r e l a t i o n t o the p s y c h o l o g i c a l c o n t e x t , a d i s c u s s i o n i s -22-p r e s e n t e d o f the s o - c a l l e d n e o - P i a g e t i a n p s y c h o l o g i c a l a p p r o a c h , and the m e t h o d o l o g i c a l t o o l s recommended i n t h i s new v i e w o f deve lopment . The p s y c h o l o g i c a l components o f a new t h e o r y o f i n s t r u c t i o n i s a l s o i n t r o d u c e d i n t h i s s e c t i o n , which i s based on t h i s n e o - P i a g e t i a n view o f deve lopment . The f i n a l s e c t i o n o f t h i s c h a p t e r d e a l s w i t h the t o p i c o f the E d u c a t i o n a l C o n t e x t o f the s t u d y : development o f the g e n e r a l i m p l i c a t i o n s o f t h i s type o f r e s e a r c h f o r the c u r r i c u l u m and the c o r r e s p o n d i n g t e a c h i n g s t r a t e g i e s a r e d i s c u s s e d . 2 .1 L i t e r a t u r e Review on S t u d e n t s ' U n d e r s t a n d i n g o f  V e c t o r Q u a n t i t i e s S e v e r a l s t u d i e s r e l a t e d to the d i f f i c u l t i e s e n -c o u n t e r e d by s t u d e n t s i n u n d e r s t a n d i n g v e c t o r i a l c o n c e p t s a re d i s c u s s e d i n t h i s s e c t i o n . A c o n n e c t i o n i s made between s t u d e n t s ' d i f f i c u l t i e s and what Hawkins (1978) has c a l l e d c r i t i c a l b a r r i e r phenomena;" t h i s i s done i n a s p e c i a l sub-s e c t i o n . P r e v i o u s r e s e a r c h o f s t u d e n t s ' p r e c o n c e p t i o n s about v e c t o r q u a n t i t i e s i s a l s o rev iewed i n t h i s s e c t i o n . 2 . 1 . 1 S t u d e n t s ' D i f f i c u l t i e s i n U n d e r s t a n d i n g V e c t o r  Q u a n t i t i e s Research f i n d i n g s have shown t h a t d i f f i c u l t i e s i n u n d e r s t a n d i n g v e c t o r q u a n t i t i e s and t h e i r c h a r a c t e r i s t i c s -23-e x i s t a t b o t h h i g h s c h o o l and c o l l e g e l e v e l s . F o r i n -s t a n c e , F i s h e r (1979) asked t e a c h e r s to r a t e a s e t o f p h y s i c s c o n c e p t s i n terms o f p e r c e i v e d d i f f i c u l t y f o r t h e i r s t u d e n t s . She found t h a t among the f i v e most d i f f i c u l t c o n -c e p t s (5 o f 22) i n grade 11, two were r e l a t e d to v e c t o r q u a n t i t i e s ; 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 . In grade 12, two v e c t o r q u a n t i t i e s : o b l i q u e momentum 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 frame were among the most d i f f i c u l t c o n c e p t s f o r s t u d e n t s . One source o f d i f f i c u l t y , i d e n t i f i e d by t e a c h e r s , was the v e c t o r n a t u r e o f the c o n c e p t s . Johns tone and Mughol (1976) t r i e d to e s t a b l i s h w h i c h o f the c o n c e p t s t a u g h t i n s c h o o l p h y s i c s were p r o v i n g t r o u b l e s o m e ; they found t h a t f i v e p h y s i c a l q u a n t i t i e s among the 20 c o n s i d e r e d most d i f f i c u l t were s t r i c t l y v e c t o r i a l . These c o n c e p t s were p r e s e n t e d t o s c h o o l and c o l l e g e s t u d e n t s who were asked t o r a t e them on a s c a l e o f d i f f i c u l t y a f t e r th e c o n c e p t s had been t a u g h t . A h i g h p e r c e n t a g e o f grade 11 and 12 (44 p e r c e n t ) f e l t t h a t v e c t o r c o n c e p t s were among the most d i f f i c u l t . Kass (1976) asked 353 grade 12 s t u d e n t s to r a t e the d i f f e r e n c e i n d i f f i c u l t y between 20 c o n c e p t s o f mechanics matched i n p a i r s . The summary o f her r e s u l t s shows t h a t the p r i n c i p a l c l u s t e r s o f d i f f i c u l t y a r e c i r c u l a r m o t i o n , 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 , v e l o -c i t y , p r o j e c t i l e m o t i o n , moments, and c o m p o s i t i o n and -24-r e s o l u t i o n o f f o r c e s . A l l o f these p h y s i c a l phenomena a r e s t r i c t l y v e c t o r i a l . In ano ther a r t i c l e Warren (1971) r e p o r t e d r e s u l t s from the f o l l o w i n g prob lem which was g i v e n t o 148 u n i v e r -s i t y e n t r a n t s i n e n g i n e e r i n g and s c i e n c e : 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 speed on l e v e l g round , t u r n i n g t o the r i g h t i n a p a t h o f u n i f o r m c u r v a t u r e . 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 arrow marked R r e p r e s e n t i n g the 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 the 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 arrow marked F r e p r e s e n t i n g the 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 the v e h i c l e . (c) S u i t a b l e l a b e l l e d arrow(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 . T h i s i s a s i m p l e v e c t o r i a l p r o b l e m . W h i l e n e a r l y a l l the s t u d e n t s a t tempted the p r o b l e m , t h e i r f i g u r e s were u n i n t e l l i g i b l e p r o j e c t i v e d r a w i n g s ; o n l y 14 answered q u e s t i o n a c o r r e c t l y , 3 answered b , and 13 re sponded t o c . The nex t y e a r the same problem was g i v e n t o 193 new s t u d e n t s ; the number o f c o r r e c t answers f o r each c jues t ion were the f o l l o w i n g : 23, 11, and none f o r a , b , c r e s p e c t i v e l y . Warren adds t h a t the most p o p u l a r answer g i v e s the r e s u l t a n t f o r c e i n o r near the d i r e c t i o n o f m o t i o n . T h i s c o n c e p t o f c i r c u l a r mot ion i s su r rounded by mys te ry and c o n f u s i o n on the p a r t o f the s t u d e n t s . One r e a s o n may be t h a t mechanics i s so o f t e n t aught i n o n l y one d i m e n s i o n ( s c a l a r p e r s p e c t i v e ) b e f o r e e x t e n s i o n to two d i m e n s i o n s ( v e c t o r i a l perspect ive . ) . In one d i m e n s i o n the m o t i o n c o n s i d e r e d i s i n the d i r e c t i o n o f the a p p l i e d f o r c e , 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 s t u d e n t s ' -25-i n i t i a l i n t u i t i v e n o t i o n on f o r c e and m o t i o n . To u n d e r -s tand, c i r c u l a r m o t i o n i t i s n e c e s s a r y to have some a p p r e c i a -t i o n o f s i m p l e v e c t o r d i s p l a c e m e n t , v e l o c i t y , and a c c e l e r a -t i o n . U s u a l l y , one o f the f i r s t v e c t o r q u a n t i t i e s i n t r o d u c e d i n s c h o o l i s d i s p l a c e m e n t ; i t i s the d i f f e r e n c e o f p o s i t i o n o f two p o i n t s i n terms o f magnitude and d i r e c t i o n . In everyday speech , v e l o c i t y i s o f t e n used as a synonym o f speed ; c o n s e q u e n t l y s t u d e n t s have much d i f f i c u l t y i n d i s t i n g u i s h i n g c l e a r l y the s t i p u l a t e d s c i e n t i f i c meaning o f these two t e rms . T h i s c o n f u s i o n between v e l o c i t y and speed was a l s o examined i n the t e s t g i v e n by W a r r e n . The q u e s t i o n g i v e n was: " A p a r t i c l e moves i n the 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 t ime i n the s e m i c i r c u l a r s e c t i o n , from 10 ms~ l to 12 m s ~ i . F o r t h i s s e c t i o n o f the pa th c a l c u l a t e the averages o f (a) the v e l o c i t y (b) the a c c e l e r a -t i o n . (Assume T r =22/7 ) " . F i g u r e 2.1 -26-The s e m i c i r c u l a r p a t h i s t aken a t an average speed o f 11 ms hence i t t akes 2/7 s e c . i n g o i n g a l o n g t h a t p a t h . The change o f d i s p l a c e m e n t i s 2 metres downward i n the F i g u r e , hence the average v e l o c i t y i s 7 ms ^ downwards. T h i s answer was g i v e n by o n l y one s t u d e n t among the 148 r e s p o n d i n g . A l -though n i n e t e n t h s o f the s t u d e n t s had d i s t i n g u i s h e d v e l o c i t y from speed when asked t o c l a s s i f y them as s c a l a r s o r v e c t o r s , w i t h b u t one e x c e p t i o n they gave the average speed when asked f o r t h e ' a v e r a g e v o l o c i t y . The change o f v e l o c i t y i s 22 ms ^ t o the l e f t i n F i g u r e 2 . 1 , hence the average a c c e l e --2 r a t i o n i s 77 ms t o the l e f t . T h i s magnitude i s o b t a i n e d by the r a t i o o f the magnitude o f the change o f v e l o c i t y , •22 ms ^ , and the t ime t aken between the two p o i n t s , 2/7 s e c . The c o r r e c t answer was g i v e n by o n l y two s t u d e n t s . Warren c o n c l u d e d the r e p o r t by s a y i n g t h a t the s t u d e n t s ' d i f f i c u l t i e s can be seen to be r e l a t e d t o c e r t a i n common a s p e c t s o f the t e a c h i n g o f p h y s i c s i n p r e - u n i v e r s i t y c o u r s e s , b u t he d i d n o t go beyond t h i s p o i n t t o i d e n t i f y the r e a l r o o t s o f t h e p r o b l e m . L e b o u t e t - B a r r e l (1976) r e p o r t e d t h a t more e x t e n s i v e s t u d i e s about v e c t o r u n d e r s t a n d i n g have been c o n d u c t e d by a team o f u n i v e r s i t y p r o f e s s o r s i n P a r i s (Malgrange , S a l t i e l , and V i e n n o t 1973, 1975) u s i n g s e v e r a l hundred s t u d e n t s . One o f t h e i r f i r s t p u b l i s h e d papers c o n c e r n e d the d i s t i n c t i o n between v e c t o r s , s c a l a r s , and p h y s i c a l measurements . Two o t h e r s t u d i e s by t h i s team were c o n c e r n e d w i t h s tudent s ' u n d e r s t a n d i n g o f the r e l a t i o n s between f o r c e and v e l o c i t y , and between f o r c e and m o t i o n . I t appears t h a t -27-two i n t u i t i v e types o f f o r c e s e x i s t f o r s t u d e n t s : r e a l i n t e r a c t i o n f o r c e s which a c t upon masses ( s t a t i c f i e l d o f f o r c e s , g r a v i t y , t e n s i o n o f a s p r i n g ) and a k i n d o f f o r c e a s s o c i a t e d w i t h m o t i o n which has the same d i r e c t i o n as the v e l o c i t y and w h i c h becomes zero a t the same t ime as the v e l o c i t y ; t h i s type o f f o r c e i s thought o f as a p r o p e r t y o f the mass, l i k e impul se o r f o r c e o f i n e r t i a . The second s t u d y r e p o r t e d (which i n v o l v e d about 3 00 s tudent s ) by the above team o f r e s e a r c h e r s examined s t u d e n t s ' i d e a s about the c o m p o s i t i o n o f the movement o f t r a n s l a t i o n i n two d i m e n s i o n s . R e s u l t s showed t h a t s t u d e n t s p e r c e i v e d the speed o f a moving body as the speed communicated by i t s motor ( i t was assumed t h a t the body had a motor) w i t h o u t any r e l a t i o n t o the frame o f r e f e r e n c e . M o t i o n a n d , t h e r e f o r e , speed a r e i n t r i n s i c p r o p e r t i e s o f the moving o b j e c t as w e l l as i t s mass and i t s d i m e n s i o n s . The r e s u l t s o f the se s t u d i e s i n d i c a t e the d i f f i c u l t y e x p e r i e n c e d by s t u d e n t s i n c o o r d i n a t -i n g m o t i o n w i t h i n two frames o f r e f e r e n c e . T h i s d i f f i c u l t y was a l s o demonst ra ted by P i a g e t e t a_l (197 3) . L e b o u t e t - B a r r e l f i n i s h e s h i s a r t i c l e s a y i n g t h a t much more r e s e a r c h i s needed c o n c e r n i n g the growth o f such c o n c e p t s as f o r c e , momentum, v e l o c i t y , speed , energy i n s t u d e n t s a t each age l e v e l . He adds t h a t : We need t o know how s t u d e n t s i n t e g r a t e networks o f new c o n c e p t s and s t r u c t u r e them. I t would be i n d i s p e n s a b l e to e x p l o r e the a r e a o f p r e c o n c e p t i o n s and i t would be o f extreme impor tance to make an i n v e n t o r y o f the e n t i r e range o f e r roneous 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 the c o u r s e o f l e a r n i n g and which d i s t o r t the c o n -s t r u c t i o n o f c o h e r e n t system o f knowledge . ( L e b o u t e t - B a r r e l , 1976, P.465) -28-2 . 1 . 2 V i e w i n g S t u d e n t s ' D i f f i c u l t i e s as a C r i t i c a l B a r r i e r Phenomena Hawkins (1980) d i s c u s s e d a s p e c i f i c c l a s s o f l e a r n i n g d i f f i c u l t i e s i n s c h o o l s c i e n c e and mathemat ics which he c a l l e d ch.ltlc.al ban.n.-i2.fi phunomzna. I t i s v e r y l i k e l y t h a t the d i f f i c u l t i e s t h a t most s t u d e n t s e x p e r i e n c e i n u n d e r -s t a n d i n g v e c t o r s f i t i n h i s d e f i n i t i o n . Hawkins ' d e s c r i b e s c r i t i c a l b a r r i e r s as f o l l o w s : (1) A r e c o n c e p t u a l o b s t a c l e s which 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 from 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 e x p e r i e n c e s 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) are 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 the a c a d e m i c a l l y a b l e bu t s c i e n t i f i c a l l y n a i v e as w e l l as those l e s s w e l l e d u c a t e d , (c) i n v o l v e no t s i m p l y d i f f i c u l t y i n a c q u i r i n g s c i e n t i f i c f a c t s b u t i n a s s i m i l a t i n g c o n c e p t u a l frames 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 narrow i n t h e i r a p p l i c a t i o n b u t when once surmounted p r o v i d e a key to the comprehens ion o f a wide range o f phenomena. To surmount 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 to overcome one o b s t a c l e b u t t o open up s t i m u l a t i n g new pathways t o s c i e n t i f i c u n d e r s t a n d i n g . (Hawkins, 1980, p .3) Hawkins has used the f o l l o w i n g example to i n v e s t i g a t e p e o p l e ' s i d e a s about m i r r o r v i s i o n : I have sometimes asked s u b j e c t s to imagine t h a t one w a l l o f the 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 then to draw, on a map.of the room, the d i r e c t i o n i n w h i c h they would l o o k t o see a g i v e n o b j e c t ' i n the m i r r o r " . (Hawkins, 1978, p . l ) He has found a wide range o f d i r e c t i o n s as answers b u t most o f them c l u s t e r around the two extreme l a b e l l e d as -29-A and B i n F i g u r e 2 . 2 , F i g u r e 2.2 S u b j e c t s c o n s i s t e d o f upper e l ementa ry s c h o o l c h i l d r e n , e l ementa ry s c h o o l t e a c h e r s , and a g raduate c l a s s i n the p h i l o s o p h y o f s c i e n c e . About 15 p e r c e n t c l u s t e r a t A ( the c o r r e c t r e s p o n s e ) , 50 p e r c e n t a t B (the i n c o r r e c t r e s p o n s e ) , and the r e s t s c a t t e r i n between. The i d e a o f s e l e c t i n g d i r e c t i o n B i s a c l e a r example o f c r i t i c a l b a r r i e r phenomena. T h i s c o n c e p t and o t h e r s r e l a t e d to o t h e r p h y s i c a l phenomena (Hawkins names s i z e and s c a l e , hea t and t e m p e r a t u r e , and a c t i o n and r e a c t i o n ) a re e a s i l y ob se rved i n many c o n -t e x t s , and r e p r e s e n t b a r r i e r s t o l e a r n i n g f o r a t l e a s t a c l e a r m a j o r i t y o f h i g h s c h o o l and c o l l e g e s t u d e n t s . These c o n c e p t i o n s a re g e n e r a l l y p r e s e n t when the r e l a t e d s u b j e c t -mat te r i s i n t r o d u c e d , but even when s t u d e n t s may see the c o n f l i c t , they do no t seem t o be a b l e t o d i s c u s s i t . In r e g a r d t o t h i s s p e c i f i c prob lem Hawkins has s a i d t h a t : - 3 0 -a t e a c h e r c o n c e r n e d to ' c o v e r the s u b j e c t ' — meaning , o f c o u r s e , to ge t through a t ex tbook o r promised 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 to make such e l ementary 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 c u r e 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. (Hawkins, 1978, p . 5) The f i e l d o f mechanics p o t e n t i a l l y may c o n t a i n many examples o f c r i t i c a l b a r r i e r phenomena. P a r t i c u l a r l y , i n phenomena i n w h i c h d i r e c t i o n i s a d e c i s i v e f a c t o r i n t h e i r c o m p r e h e n s i o n . I t seems r e a s o n a b l e , a t t h i s p o i n t , t o s p e c u l a t e t h a t s t u d e n t s ' 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 p h y s i c a l v e c t o r q u a n t i t i e s may be c o n s i d e r e d as a c r i t i c a l b a r r i e r phenomenon. F o r i n s t a n c e , Warren (1971) found t h a t s t u d e n t s t e n d to i g n o r e the d i r e c t i o n a l a s p e c t i n p h y s i c a l s i t u a t i o n s i n which t h i s f a c t o r i s d e c i s i v e . F i n d i n g s such as these would sugges t t h a t most o f s t u d e n t s tend t o t r e a t p h y s i c a l v e c t o r q u a n t i t i e s as though t h e y were s c a l a r . T h i s b e l i e f may be a c o n c e p t u a l o b s t a c l e and may i n h i b i t f u r t h e r s c i e n t i f i c u n d e r s t a n d i n g . These b e l i e f s about mechanics phenomena c o u l d become " c r i t i c a l b a r r i e r s " f o r the i n s t r u c t i o n o f s t u d e n t s on c u r r e n t v iews o f mechanics phenomena. I t i s w i t h t h i s k i n d o f prob lem t h a t th e p r e s e n t s tudy i s c o n c e r n e d . The q u e s t i o n s addre s sed i n t h i s r e s e a r c h a re aimed a t u n c o v e r i n g s t u d e n t s ' b e l i e f s t h a t may w e l l q u a l i f y as c r i t i c a l b a r r i e r s . The s e a r c h f o r the se k i n d s o f b e l i e f s seems i m p e r a t i v e . Hawkins has s a i d t h a t : Though the 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 appears to have been r e c o g n i z e d by 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 no 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 to the improvement o f the t e a c h i n g a r t , t o 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 T h i s a u t h o r b e l i e v e s t h a t s t u d e n t s ' p r e c o n c e p t i o n s can be used as a s t a r t i n g p o i n t f o r i n s t r u c t i o n a l p r o c e d u r e s w h i c h can h e l p to surmount the c r i t i c a l b a r r i e r s o r o b s t a c l e s t h a t they p r o d u c e . 2 . 1 . 3 P r e v i o u s Research About S t u d e n t s ' U n d e r s t a n d i n g on  V e c t o r Q u a n t i t i e s As i t was ment ioned i n Chapter One, t h e r e has been l i t t l e r e s e a r c h on s t u d e n t s ' p r e c o n c e p t i o n s about v e c t o r c h a r a c t e r i s t i c s o f p h y s i c a l q u a n t i t i e s . In f a c t , o n l y two s t u d i e s were found i n the l i t e r a t u r e ( P i a g e t , 1973; A g u i r r e , 1978) which were m a i n l y c o n c e r n e d w i t h s t u d e n t s c o n c e p t i o n s o f v e c t o r s . B o t h o f t h e s e s t u d i e s f o c u s e d on the s t u d e n t s ' c o n c e p t o f f o r c e ; and b o t h used a P i a g e t i a n approach f o r d a t a c o l l e c t i o n . Two o t h e r - s tud ie s r e l a t e d t o v e c t o r q u a n t i t i e s have been r e p o r t e d bu t these q u a n t i t i e s were c o n -s i d e r e d as s c a l a r (Raven, 1968, 1972) . A l l t h e s e s t u d i e s a re r ev i ewed be low. Other i n v e s t i g a t i o n s , not d i r e c t l y r e l a t e d t o s t u d e n t s ' c o n c e p t i o n s , but aimed a t h e l p i n g s u b j e c t s t o u n d e r s t a n d v e c t o r q u a n t i t i e s ' a re a l s o r ev i ewed i n t h i s s e c t i o n . P i a g e t i a n - T y p e S tud i es P i a g e t (1973) c r e a t e d n i n e t a s k s t o f i n d out how c h i l d r e n (4 t o 12 y e a r s o f age) d e v e l o p n o t i o n s about severe -32-a s p e c t s o f f o r c e s — f o r c e b e i n g a v e c t o r q u a n t i t y . The a s p e c t s c o n s i d e r e d by him are p r e c i s e l y the v e c t o r c h a r a c t e r i s t i c s o f f o r c e . F o r each t a sk P i a g e t c a t e g o r i z e d the c h i l d r e n ' s b e l i e f s i n terms o f t h r e e d e v e l o p m e n t a l s t a g e s , w i t h two l e v e l s (A and B) a t each s t a g e . Rev iewing h i s r e s u l t s , i t i s e v i d e n t t h a t c h i l d r e n do no t u n d e r s t a n d the v e c t o r c h a r a c t e r i s t i c s o f f o r c e u n t i l s tage I I I . T h i s c o r r e s p o n d s to s u b j e c t s o v e r 12 y e a r s o f age . Some g e n e r a l r e s u l t s o f P i a g e t ' s s tudy a r e rev iewed be low. In one o f the t a s k s , which uses a c i r c u l a r t a b l e , w i t h t h r e e n o n - p a r a l l e l f o r c e s a c t i n g upon a s m a l l o b j e c t ( P i a g e t , 1973, C h a p t e r 6 ) , c h i l d r e n i n s tage I I (7 t o 10 y e a r s o f age) cannot s o l v e problems where f o r c e s a r e e i t h e r a t an a n g l e o r a r e d i f f e r e n t i n magn i tude . A t the s t a r t o f the t a s k , when f o r c e s a re p a r a l l e l , they use s t r i c t a r i t h m e t i c a d d i t i v i t y o f f o r c e s . However, when the f o r c e s a c t a t an a n g l e and the obse rved r e s u l t s seem t o c o n t r a d i c t t h e i r p r e d i c t i o n s , they r e t u r n t o n o n - a d d i t i v i t y . I n s t age I I I some s u b j e c t s appear t o c o n s i d e r the e f f e c t o f the a n g l e on the r e s u l t a n t f o r c e . In another t a sk ( P i a g e t , 1973, C h a p t e r 7 ) , s u b j e c t s i n s tage I I I seem t o p e r c e i v e the r o l e o f the d i r e c t i o n o f the f o r c e s as w e l l . They s a i d t h a t we ight s (an example o f a g r a v i t a t i o n a l f o r c e ) p u l l l e s s when they p u l l i n d i f f e r e n t d i r e c t i o n s and more when t h e y come t o g e t h e r . P i a g e t summarizes the se exper iments on f o r c e s and o t h e r s t u d i e s about c h i l d r e n ' s b e l i e f s i n ano ther book c a l l e d Unde.su> tanding Cau&allty (1974) . In t h i s -33-book , P i a g e t c l a i m s t h a t : 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 are no l o n g e r concerned w i t h so 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 the 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 c o n c e r n e d 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 th e c o n s t r u c t i o n o f forms o f which 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 them-s e l v e s and s u b j e c t to b e i n g reached p e r c e p t i v e l y o r ' t h r o u g h p h y s i c a l e x p e r i e n c e . ( P i a g e t , 1974, p.35) T h i s s ta tement i n d i c a t e s the impor tance o f the s t u d e n t s ' p r e c o n c e p t i o n s on s p a t i a l o p e r a t i o n s . I t i s no t e x p e c t e d t h a t the se p r e c o n c e p t i o n s i n c l u d e i d e a s o f the way t h a t p h y s i c i s t s f o r m a l l y d e f i n e v e c t o r p o s i t i o n , d i s p l a c e m e n t , and t h e i r v e c t o r o p e r a t i o n s ; bu t i t i s p o s s i b l e t h a t they may c o n t a i n some f e a t u r e s o f v e c t o r c h a r a c t e r i s t i c s , p a r t i c u l a r l y the d i r e c t i o n a l f a c t o r o f s p a t i a l l o c a t i o n s o f o b j e c t s . From t h e d a t a c o l l e c t e d P i a g e t c o n c l u d e s t h a t : I n a l l ca ses s t u d i e d , the g e o m e t r i c s t r u c t u r i n g o f d i r e c t i o n s i s l i n k e d to dynamic s . T h a t i s , what i s c a u s i n g the d i r e c t i o n o f m o t i o n . F o r 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 o f 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 consonant w i t h dynamics and a t t r i b u t e d t o o b j e c t s , s i n c e these d e t e r m i n e , i n t h e i r c a u s a l c o n n e c t i o n s , the geometr i c r e l a t i o n s h i p s i s o m o r p h i c 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 . Need-l e s s t o s ay , the c a u s a l i n t e r p r e t a t i o n i s p o s s i b l e o n l y i f the shapes and i n i t i a l p o s i t i o n o f the o b j e c t s a re 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 dynamic p r o p e r t i e s . ( P i a g e t , 1974, p .41) E f f o r t s w i l l be made i n the p r e s e n t r e s e a r c h t o c o n -c e n t r a t e o n l y on the k i n e m a t i c a s p e c t s o f the c o n c r e t e prob lem s i t u a t i o n s d e p i c t e d i n the t a s k s , t h a t i s , the cause o f the p h y s i c a l phenomena w i l l no t be c o n s i d e r e d . -34-I f the s u b j e c t s r e f e r to dynamic f a c t o r s i n t h e i r r e sponse s the r e s e a r c h e r w i l l c a u t i o u s l y t r y t o move the d i s c u s s i o n t o the k i n e m a t i c a s p e c t s o f the t a s k . In summary, P i a g e t was i n t e r e s t e d o n l y i n some v e c t o r c h a r a c t e r i s t i c s about one v e c t o r q u a n t i t y — f o r c e . I t i s the purpose o f the p r e s e n t s tudy to uncover how s t u d e n t s d e v e l o p 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 b e f o r e f o r m a l i n s t r u c t i o n . To t h i s end , i t may be t h a t s t u d e n t s 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 , d i s p l a c e -ment , and v e l o c i t y t h a t are s i m i l a r to the c o n c e p t i o n s t h a t P i a g e t found f o r f o r c e s . However, i t i s i m p o r t a n t t o p o i n t o u t t h a t the aim o f t h i s s tudy was no t t o c a t e g o r i z e the s u b j e c t s ' p r e c o n c e p t i o n s i n d e v e l o p m e n t a l s tages as P i a g e t (1973) d i d w i t h h i s s t u d y on f o r c e s , b u t t o u n c o v e r the s p e c i f i c p r e c o n c e p t i o n s t h a t s t u d e n t s d e v e l o p f o r the 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 t h i s s tudy d e a l s w i t h . A g u i r r e (1978) c a r r i e d out a s tudy about c h i l d r e n ' s b e l i e f s about f o r c e s i n s t a t i c e q u i l i b r i u m . The main emphasis o f t h i s r e s e a r c h was no t d i r e c t e d toward l o o k i n g f o r v e c t o r i a l c h a r a c t e r i s t i c o f f o r c e s i n c h i l d r e n ' s m i n d s , b u t t h e i r b e l i e f s about d i f f e r e n t a s p e c t s o f the c o n c e p t o f f o r c e . The r e s e a r c h q u e s t i o n s i n t h a t s tudy were m a i n l y r e l a t e d t o the c h i l d r e n ' s b e l i e f s on a c t i o n o f f o r c e s , s t a t i c e q u i l i b r i u m i n systems w i t h two and t h r e e component f o r c e s , 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 f o r c e s . Among the f i n d i n g s o f the r e s e a r c h , the a u t h o r found t h a t c h i l d r e n about 12 y e a r s o f age have a l r e a d y gra sped p h y s i c a l c o n c e p t i o n s t h a t c l e a r l y have v e c t o r i a l f e a t u r e s . - 3 5 -These were r e l a t e d t o c o m p o s i t i o n o f f o r c e s . C h i l d r e n aged 14 were a b l e t o s o l v e problems c o n t a i n i n g components w i t h d i f f e r e n t magnitude and v a r i o u s a n g l e s . S u b j e c t s i n the sample had n o t y e t been i n s t r u c t e d about c o m p o s i t i o n o f f o r c e s . T h i s shows t h a t s t u d e n t s d e v e l o p some p h y s i c a l i n t u i t i o n s w i t h v e c t o r i a l f e a t u r e s ; i t may be however , t h a t these i n t u i t i v e v e c t o r i a l c o n c e p t i o n s a re o n l y r e l a t e d to f o r c e s . One Of the i n t e n t s o f t h i s s tudy i s to de te rmine whether s t u d e n t s may a l s o d e v e l o p some v e c t o r i a l i n t u i t i o n s i n r e l a t i o n to o t h e r p h y s i c a l e n t i t i e s . Raven (1968, 1972) has c a r r i e d out s t u d i e s on c h i l d r e n ' s b e l i e f s about two v e c t o r q u a n t i t i e s ; a c c e l e r a t i o n and l i n e a r momentum. But he d i d no t emphasize the v e c t o r i a l p r o p e r t i e s o f these c o n c e p t s a t a l l , s i n c e he used p r i m a r y s c h o o l c h i l d r e n i n b o t h s t u d i e s . He was a b l e t o a v o i d the v e c t o r i a l p r o p e r t i e s o f l i n e a r momentum by t a k i n g an u n i d i m e n s i o n a l p h y s i c a l s i t u a t i o n ; t h a t i s , he c o n s i d e r e d momentum as a s c a l a r q u a n t i t y . The purpose o f the momentum s t u d y was t o d e t e r m i n e the d e v e l o p m e n t a l sequence w i t h i n a s e t o f i t ems p o s t u l a t e d as 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 momentum. A s i m i l a r purpose was b e h i n d h i s s tudy o f the c o n c e p t o f a c c e l e r a t i o n . A g a i n , by u s i n g an u n i d i m e n s i o n a l s i t u a t i o n , a c c e l e r a t i o n was t r e a t e d as a s c a l a r c o n c e p t . T h i s i s a common approach t aken by r e s e a r c h e r s ; they s e t up e x p e r i m e n t a l t a s k s i n which the d i r e c t i o n a l f a c t o r i s b e i n g c o n t r o l l e d . T h i s produces a v e r y s i m p l i f i e d s i t u a -t i o n , t r a n s f o r m i n g complex phenomena i n t o s i m p l e r o n e s , - 3 6 -i n w h i c h s u b j e c t s have to work w i t h fewer v a r i a b l e s . T h i s i s a t o t a l l y a c c e p t a b l e a p p r o a c h , but i t does no t a l l o w one t o uncover a l l the p o t e n t i a l l y r i c h c o n c e p t i o n s t h a t s t u d e n t s may have d e v e l o p e d through f o r m a l and i n -f o r m a l e x p e r i e n c e s w i t h t h e i r e n v i r o n m e n t . B . S p e c i a l E x p e r i m e n t s to Teach V e c t o r Q u a n t i t i e s P h y s i c i s t s and p h y s i c s t e a c h e r s , b e i n g aware o f the s t u d e n t s ' d i f f i c u l t i e s i n u n d e r s t a n d i n g v e c t o r q u a n t i t i e s , have c r e a t e d s p e c i a l exper iments i n which the v e c t o r i a l p r o p e r t i e s o f p h y s i c a l q u a n t i t i e s a re d r a m a t i c a l l y shown. These e x p e r i m e n t a l t a s k s a r e p r e p a r e d w i t h the aim o f . h e l p i n g s t u d e n t s to g ra sp the v e c t o r c h a r a c t e r i s t i c s o f c e r t a i n q u a n t i t i e s . U n f o r t u n a t e l y t h e r e i s no d a t a a v a i l a b l e t o d e t e r m i n e i f these t a s k s have been s u c c e s s f u l . A l i s t o f some o f the se v e c t o r i a l exper iments f o l l o w s : Does p r e s s u r e have d i r e c t i o n ? (Dempster, 1951) ; a d r a m a t i c d i s t i n c t i o n between p r e s s u r e and f o r c e ( M i l l e r , 1952) ; a model t o i l l u s t r a t e some p r o p e r t i e s o f v e c t o r s i n t h r e e d imens ions (Baez, 1957) ; v e c t o r n a v i g a t i o n : a game f o r p h y s i c s (Per ry and C a m p b e l l , 1971) ; v e r s a t i l i t y o f the screw n a i l approach t o t e a c h i n g the concep t o f c r o s s p r o d u c t ( G r e i g , 1972) . C . V e c t o r Q u a n t i t i e s i n Mathemat ics E d u c a t i o n R e s e a r c h e r s i n mathemat ics e d u c a t i o n have a l s o been i n t e r e s t e d i n s t u d e n t s ' u n d e r s t a n d i n g o f v e c t o r s . But they u s u a l l y t r e a t v e c t o r s as a b s t r a c t m a t h e m a t i c a l e n t i t i e s , and do not make c o n n e c t i o n s w i t h p h y s i c a l r e a l i t y . T h e i r main i n t e r e s t has been t o use v e c t o r s as a t o o l t o improve th e t e a c h i n g o f o t h e r a reas i n mathemat i c s . To a c c o m p l i s h t h i s t a s k , mathematics e d u c a t o r s have c r e a t e d i n s t r u c t i o n a l u n i t s f o r use i n s c h o o l s i t u a t i o n s . The t i t l e s o f some o f the se u n i t s a r e l i s t e d b e l o w : Complex Numbers and V e c t o r s i n H i g h S c h o o l Math ( P e d l e y , 1966) ; V e c t o r s : An A i d t o M a t h e m a t i c a l U n d e r s t a n d i n g (Smith , 1959) ; A p p r o a c h t o V e c t o r Geometry ( T r o y e r , 1963) ; U s i n g V e c t o r s t o S o l v e S imul taneous E q u a t i o n s ( S a l z a r u l o , 1963) ; V e c t o r s i n A l g e b r a and Geometry ( G l i c k s m a n , 1965) ; P l a n e Geometry by V e c t o r Methods ( S c o t t , 1970) . Some o f the s t u d i e s (Lamon, 1971, Rosembloom, 1969) have used c o n c r e t e o b j e c t s to improve the u n d e r s t a n d i n g o f v e c t o r space , bu t none o f them have used a p h y s i c a l q u a n t i t y l i k e v e l o c i t y , a c c e l e r a t i o n , o r f o r c e . T h i s may be an a c c e p t a b l e approach f o r the s t u d y o f v e c t o r s from a m a t h e m a t i c a l p o i n t o f v i e w ; however , p h y s i c s t e a c h e r s s h o u l d l o o k f o r c o n c r e t e e x p e r i m e n t a l s i t u a t i o n s o b t a i n e d d i r e c t l y from p h y s i c a l r e a l i t y . 2.2 P s y c h o l o g i c a l C o n t e x t .of the Study 2.2 .1 I n t r o d u c t i o n T h i s s e c t i o n d e a l s w i t h the broad p s y c h o l o g i c a l c o n t e x t i n w h i c h t h i s s tudy i s embedded. F i r s t , a t h e o r e t i c a l i s s u e i n v o l v i n g the dichotomous approach t o the s t u d y o f development as proposed by some p s y c h o l o g i s t s i s d i s c u s s e d — the f o r m a l ^ f u n c t i o n a l (or competence and per formance models o f development) d i c h o t o m y . ' Reasons a r e advanced as -38-to why the per formance model was chosen as the p r e f e r r e d approach f o r the p r e s e n t s t u d y . T h i s model i s e x p l a i n e d t h r o u g h the work o f S i e g l e r , and C a s e . A l t h o u g h , t h i s s tudy i s no t d i r e c t l y r e l a t e d t o i n -s t r u c t i o n , a s e p a r a t e s u b s e c t i o n i s devoted t o th e p s y c h o -l o g i c a l component o f C a s e ' s Theory o f I n s t r u c t i o n . Case proposes (the second s t ep i n a t h r e e - s t e p t h e o r y o f i n -s t r u c t i o n ) t o a s ses s s t u d e n t s ' c u r r e n t l e v e l o f f u n c t i o n i n g i n s p e c i f i c c o n t e n t - a r e a s . T h i s p r o p o s i t i o n i s i n sympathy w i t h the b a s i c aim o f t h i s r e s e a r c h . F i n a l l y , the methodo-l o g y approach c u r r e n t l y used by some r e s e a r c h e r s i n the ' N e o - P i a g e t i a n t r a d i t i o n ' — r a t i o n a l t a sk a n a l y s i s — i s p r e s e n t e d . 2 . 2 . 2 Two A s p e c t s o f Deve lopmenta 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 Some p s y c h o l o g i s t s ( F l a v e l and W o h l w i l l , 1969; Wol lman, 1978; Ca se , 1978) have made the d i s t i n c t i o n between two i n t e r d e p e n d e n t a s p e c t s o f deve lopment : f o r m a l and f u n c t i o n a l . F l a v e l l e t a l . (1969) have e x p l a i n e d the meaning o f the two a s p e c t s i n 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 to do w i t h the morphology o f the p r o c e s s : the s o r t s o f c o g n i t i v e e n t i t i e s t h a t make up the 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 these e n t i t i e s are 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 . . . The 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 mechanism, 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 , by which 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 d ichotomy o f development has been shown t o be a u s e f u l way t o o r g a n i z e o r to s t r u c t u r e r e s e a r c h q u e s t i o n s ; however , i t s h o u l d be a p p l i e d c a u t i o u s l y and r e f l e c t i v e l y . One prob lem i s t h a t i t i m p l i e s t h a t the h y p o t h e s i z e d m e n t a l s t r u c t u r e s are t r a n s i t u a t i o n a l , a t e m p o r a l , and c o n t e x t -f r e e — a p o s i t i o n t h a t i s d i f f i c u l t t o d e f e n d . Even P i a g e t (1972) has s t a t e d t h a t h i s f o r m a l o p e r a t i o n s were no t t o t a l l y c o n t e x t - f r e e . A n o t h e r t h e o r e t i c a l model t h a t i s c l o s e l y r e l a t e d to the above d ichotomy i s the competence v e r s u s per formance o r automotan model o f deve lopment . The former i s r e l a t e d t o the f o r m a l a s p e c t and the l a t t e r to the f u n c t i o n a l a s p e c t . F l a v e l and W o h l w i l l have e x p l a i n e d t h i s p o s i t i o n as f o l l o w s : A competence m o d e l , w h i c h 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 the s t r u c t u r e o f some domain . . a n automaton model (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 by " p e r f o r m a n c e " ) , w h i c h 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 ge 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 competence model 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 th e organ i sm 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 , whereas the automaton model has the 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 the 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 pe r fo rmance , e t c . ) under which 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 , 1969, p . 7 1 ) . U s i n g the se d e s c r i p t i v e c a t e g o r i e s , P i a g e t ' s t h e o r y o f development i s n o r m a l l y c o n s i d e r e d as an example o f a f o r m a l t h e o r y o r competence model o f deve lopment . T h i s i s because P i a g e t ' s main i n t e r e s t was to l ook f o r g e n e r a l menta l s t r u c t u r e s t h a t were t r a n s i t u a t i o n a l and c o n t e x t - f r e e -C o n s i d e r i n g P i a g e t to be p r e d o m i n a n t l y a s t r u c t u r a l i s t o r f o r m a l i s t , r a t h e r t h a n a f u n c t i o n a l i s t means, a c c o r d i n g t o Wollman (1978) , t h a t he i s l e s s concerned w i t h the c o n t e n t o f human knowledge than w i t h i t s o r g a n i z a t i o n o r s t r u c t u r e . F o r i n s t a n c e , Wollman c l a i m s t h a t : P i a g e t * s c o n c e r n 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 per formance on t a s k s such as the s e r i a t i o n t a s k . He f e e l s t h a t such t a sk s r e v e a l an i d e n t i c a l o r g a n i z a t i o n o r ' s t r u c t u r e ' , t h a t i s , s u c c e s s -f u l per formance on a l l such t a s k s i n v o l v e s one b a s i c s t r u c t u r e o r scheme, the s e r i a t i o n scheme. (Wollman, 1978, p .8) F l a v e l l and W o h l w i l l (1969) have a l s o p o i n t e d out t h a t P i a g e t ' s p r i m a r y r e s e a r c h program has been to p r o v i d e a l o g i c a l d e s c r i p t i o n o f the systems o f i n t e l l e c t u a l o p e r a t i o n s t h a t c h i l d r e n pos se s s a t d i f f e r e n t p o i n t s i n t h e i r deve lopment , not t o p r o v i d e a p s y c h o l o g i c a l d e s c r i p t i o n o f the p r o c e s s e s by which these o p e r a t i o n s a r e a c q u i r e d and u t i l i z e d . O t h e r p s y c h o l o g i s t s such as P a s c u a l - L e o n e (1966) , Wollman (1978) , S i e g l e r (1976, 1978) , and Case (1978) have been more c o n c e r n e d w i t h what has been d e f i n e d as a f u n c t i o n a l a s p e c t o f deve lopment . A c c o r d i n g to t h i s v i e w -p o i n t the r e s u l t or per formance o f a c o g n i t i v e encounter w i t h the p h y s i c a l env i ronment i s p a r t l y a f u n c t i o n o f what the s u b j e c t a l r e a d y knows about a s p e c i f i c c o n t e n t - a r e a and how t h a t s p e c i f i c knowledge i s r e p r e s e n t e d and o r g a n i z e d . In t h i s a c c o u n t , the f u n c t i o n a l a s p e c t o f development (or the per formance model) i s more c l o s e l y r e l a t e d t o the growth o f u n d e r s t a n d i n g o f s p e c i f i c c o n t e n t - a r e a s and the p o s s i b l e -41-f a c t o r s t h a t c o u l d i n f l u e n c e such g rowth . T h i s i s s p e f i -f i c a l l y shown by the work o f P a s c u a l - L e o n e and S i e g l e r . I t i s wor th s t r e s s i n g a g a i n , however, t h a t t h i s d i c h o -tomy ( formal and f u n c t i o n a l a s p e c t s , or competence and p e r -formance models) s h o u l d no t be a c c e p t e d u n r e f l e c t i v e l y . Even i f one f i n d s i t u s e f u l to d i s t i n g u i s h these two a s p e c t s o f deve lopment , t h e i r manner o f i n t e r a c t i o n must be c o n s i d e r e d . I t would be d i f f i c u l t , i n the v iew o f t h i s a u t h o r , to de fend a p o s i t i o n t h a t t r e a t e d them as two t o t a l l y independent a s p e c t s o f deve lopment . I f the competence-per formance d ichotomy i s c a u t i o u s l y c o n s i d e r e d , however , i t does have some u t i l i t y and h e u r i s t i c v a l u e , and on these grounds was used to c h a r a c t e r i z e the n a t u r e o f the r e s e a r c h under t aken i n t h i s s t u d y . The i n t e n t i o n , t h e n , i s n o t t o s o l v e the problems o f the d ichotomy o r to s e t t l e the i s s u e bu t t o draw upon some o f the t h e o r e t i c a l and m e t h o d o l o g i c a l knowledge f o r use i n the p r e s e n t r e s e a r c h . S i n c e the b r o a d g o a l s o f t h i s s t u d y ; w e x £ ~ - t o c f i n d o u t what a s u b j e c t a l r e a d y knows about a s p e c i f i c c o n t e n t - a r e a ( i . e . , v e c t o r q u a n t i t i e s ) , and t o de te rmine how t h a t s p e c i f i c knowledge i s o r g a n i z e d , i t was c l e a r t h a t the per formance model o f development p r o v i d e s the more a p p r o p r i a t e framework. In summary, i t was thought t h a t the a n a l y s i s o f s t u d e n t s ' p r e c o n c e p t i o n s , o b t a i n e d t h r o u g h t h e i r per formance when s o l v i n g s p e c i f i c t a s k s , might shed l i g h t on the p r o c e s s by w h i c h s t u d e n t s d e v e l o p t h e i r b e l i e f s w i t h r e g a r d to p a r t i c u l a r c o n t e n t - a r e a s . T h i s means t h a t the r e s e a r c h was v e r y c o n t e x t u a l o r i e n t e d i n n a t u r e . The aim was no t t o see i f s t u d e n t s ' p r e c o n c e p t i o n s about v e c t o r c h a r a c t e r i s t i c s f i t , f o r i n s t a n c e , i n t o the l o g i c a l d e s c r i p t i o n o f the systems o f i n t e l l e c t u a l o p e r a t i o n s o f c o n c r e t e and f o r m a l s t ages o f P i a g e t ' s t h e o r y . Nor was i t i n t e n d e d t o de te rmine what c o n -c r e t e o r f o r m a l o p e r a t i o n s a r e n e c e s s a r y to l e a r n to u n d e r s t a n d v e c t o r q u a n t i t i e s , r a t h e r , the aim was t o uncover s t u d e n t s ' p r e c o n c e p t i o n s about a s p e c i f i c c o n t e n t - a r e a and t o f i n d out i f such p r e c o n c e p t i o n s pos ses s any degree o f c o n s i s t e n c y o r o r g a n i z a t i o n . The approach t aken i n t h i s s t u d y , t h e n , f i t s w e l l w i t h the s o - c a l l e d " N e o - P i a g e t i a n " Approach t o Development , and i s p r i m a r i l y concerned w i t h the f u n c t i o n a l o r p e r f o r m -ance account o f the s u b j e c t ' s b e h a v i o u r . 2 . 2 . 3 The N e o - P i a g e t i a n Approach o f Development As the name i m p l i e s , t h i s approach has been r e c e n t l y advanced by s e v e r a l t h e o r i s t s . ( c . f . Wol lman, 1978) . The approach i s so named because i t t akes P i a g e t ' s account o f development as i t s s t a r t i n g p o i n t and a t tempts to suplement i t w i t h an a c c o u n t o f the f a c t o r s a s s o c i a t e d w i t h the p r o c e s s o f a c q u i r i n g new c o n c e p t i o n s . In o t h e r words , the Neo-P i a g e t i a n approach c o n s i d e r s the p r o c e s s o f development and the f a c t o r s a f f e c t i n g t h a t deve lopment , so t h a t the s t u d e n t s ' c u r r e n t l e v e l o f f u n c t i o n i n g i n a g i v e n c o n t e n t - a r e a might more e a s i l y be a s s e s s e d . The rea son f o r t h i s i s t h a t the f a c t o r s t h a t c o n s t r a i n per formance would be b e t t e r u n d e r s t o o d -43-and th e t y p e o f assessment o f i n t e r e s t would be more c l o s e l y r e l a t e d t o obse rved performance (Wollman, 1978) . The groundwork f o r the movement was done m a i n l y by P a s c u a l - L e o n e (1969) i n h i s d o c t o r a l d i s s e r t a t i o n and subse-quent work . B e s i d e s P a s c u a l - L e o n e , two o t h e r N e o - P i a g e t i a n r e s e a r c h e r s have r e c o g n i z e d the importance o f emphas i z ing the f u n c t i o n a l a s p e c t o f deve lopment ; they a re S i e g l e r (1976, 1978) and Case (19 7 8 ) . T h e i r work i s c l o s e l y r e l a t e d t o the p r e s e n t s t u d y and i t i s d i s c u s s e d i n more d e t a i l be low. S i e g l e r ' s A p p r o a c h : S i e g l e r ' s work (1976,1978,1979) i s p r e -d o m i n a n t e l y c o n c e r n e d w i t h d e s c r i b i n g the s u b j e c t s ' c u r r e n t l e v e l o f f u n c t i o n i n g i n g i v e n c o n t e n t - a r e a s . H i s aim i s t o c h a r a c t e r i z e 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 i n t h i n k i n g . I n one o f h i s e a r l i e r s t u d i e s (1976) the focus was upon t h r e e a s p e c t s o f deve lopment : s p e c i f i c knowledge g o v e r n i n g t a sk p e r -formance , r e s p o n s i v e n e s s t o e x p e r i e n c e , and b a s i c p r o c e s s e s t h a t u n d e r l i e d i f f e r e n c e s i n these two a r e a s . F o r the p r e s e n t s t u d y , o n l y the f i r s t a s p e c t i s r e l e v a n t . S i g e l e r ' s main c o n -c e r n i s t o d e t e r m i n e c h i l d r e n ' s knowledge about s p e c i f i c c o n -t e n t - a r e a s ( i . e . , b a l a n c e , t ime speed , and d i s t a n c e ) i n an accu-r a t e and unambiguous way; S p e c i f i c a l l y , he c h a r a c t e r i z e s s t u d e n t s ' c o n c e p t i o n s i n terms o f r u l e s . T h i s e n t a i l s the a s sumpt ion t h a t c h i l d r e n o r g a n i z e knowledge i n terms o f r u l e s o r schemes w h i c h they t h e n a p p l y to s o l v e a s e r i e s o f problems w h i c h appear to be s i m i l a r i n some way. In o t h e r words , S i e g l e r (1976 , p . I l l ) a s s u m e s , t h a t s t u d e n t s ' p r o b l e m - s o l v i n g s t r a t e g i e s a r e r u l e - g o v e r n e d w i t h the r u l e s p r o g r e s s i n g from -44-l e s s s o p h i s t i c a t e d t o more s o p h i s t i c a t e d w i t h age . F o r the c l a s s i c , P i e g e t i a n b a l a n c e t a s k , he proposed 4 r u l e s : the f i r s t r u l e t y p i c a l l y b e i n g used by the younger c h i l d r e n (5 year s ) and the f o u r t h r u l e by the o l d e s t (over 12 y e a r s ) . But what does i t mean t o say someone i s u s i n g a r u l e ? S i e g l e r answers by s a y i n g : In one sense , r u l e s can be thought o f as a means o f summariz ing d a t a , as when a s e t o f r e sponses 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 s e n t e n c e , a m a t h e m a t i c a l e q u a t i o n , a p r o d u c t i o n sys tem, o r some o t h e r f o r m a l i s m . W i t h i n t h i s usage i t makes l i t t l e sense to 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 r e p r e s e n t a t i o n s ( they a r e 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 the r u l e s t o f i t the p a r t i -c u l a r r e sponse s t e l l s us o n l y t h a t the r e p r e s e n t a -t i o n i s i n a d e q u a t e . 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 s tatements not o n l y summarize d a t a bu t a l s o have some c o r r e s p o n d e n c e t o the way i n w h i c h the da t a were genera ted (they a r e i n the s u b j e c t ' s head as w e l l as the e x p e r i -m e n t e r ' s ) ( S i e g l e r , 1978, p . 1 9 ) . F o r the p r e s e n t s t u d y , the second approach was c o n s i d e r e d . T h a t i s , r u l e s were i n f e r r e d as a means o f summariz ing d a t a , bu t i t was a l s o assumed t h a t t h e y have some c o r r e s p o n d e n c e to the way how s u b j e c t s genera te them. S i e g l e r (1977, p . 394)' says t h a t a number o f o t h e r t a s k s have been s t u d i e d u s i n g r u l e s as a r e p r e s e n t a t i o n a p p r o a c h . He c i t e s I n h e l d e r and P i a g e t ' s (19 58) p r o j e c t i o n o f shadows t a s k , B r u n e r and K e n n e y ' s (1966) water j a r t a s k , and a p r o b a b i l i t y c o n c e p t t a sk d e s c r i b e d by Chapman (1975) . S i e g l e r r e p o r t s t h a t i n each c a s e : -45-. The proposed r u l e s have been found to c h a r a c t e r -i z e the per formance o f more than 80% o f c h i l d r e n aged 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 the p r o d u c t o f a random response p r o c e s s . ( S i e g l e r , 1977, p . 394) Even though the b r o a d , t h e o r e t i c a l approach taken i n the p r e s e n t s t u d y i s s i m i l i a r t o t h a t o f S i e g l e r , the c o l l e c -t i o n and d a t a a n a l y s i s m e t h o d o l o g i e s proposed by him ( S i e g l e r , 1978, 1979) were not u s e d . The rea sons f o r t h i s d e c i s i o n r e s t e d on the methodology i t s e l f ; i n t h i s methodology ( R u l e -Assessment Methodology) the f i r s t s t eps r e q u i r e s the f o r m u l a -t i o n o f models o f the r u l e s t h a t s u b j e c t s might use to p e r f o r m the t a s k s . So f a r , S i e g l e r has f o r m u l a t e d the se models o f r u l e s on the b a s i s o f p r e v i o u s r e s e a r c h f i n d i n g s . F o r i n s t a n c e , i n h i s s t u d i e s on the b a l a n c e ( S i e g l e r , 1978) and on the c o n c e p t s o f t i m e , speed , and d i s t a n c e ( S i e g l e r , 1979) he has f o r m u l a t e d the models o f the r u l e s based on P i a g e t " s i n i t i a l i n v e s t i g a t i o n s i n these c o n c e p t u a l a r e a s . B u t , as was ment ioned e a r l i e r , no s tudy r e l a t e d t o s t u d e n t s ' p r e c o n c e p t i o n s 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 was found i n the l i t e r a t u r e so t h e r e was no d a t a upon which to f o r m u l a t e ..models o f the r u l e s f o r these c o n c e p t s . T h e r e f o r e , the S i e g l e r ' s Ru le -As se s sment methodology c o u l d n o t be used i n the p r e s e n t s t u d y . One o f the aims o f t h i s s tudy was i n f a c t t o p r o v i d e d a t a such t h a t i t would be p o s s i b l e t o f o r -mulate the S i e g l e r - t y p e r u l e - m o d e l . These can be s u b s e q u e n t l y t e s t e d u s i n g the c i t e d methodology . -46-C a s e ' s A p p r o a c h : Ca se , a deve lopmenta l p s y c h o l o g i s t , b e l i e v e s t h a t r e s e a r c h based on the N e o - P i a g e t i a n approach can be used t o addres s i n s t r u c t i o n a l problems (Case , 1978a) . He argues t h a t a f u n c t i o n a l approach may show the p r o c e s s by which s t u d e n t s a c q u i r e and u t i l i z e o p e r a t i o n a l s t r u c t u r e s and o t h e r more c o n t e x t u a l types o f c o n c e p t i o n s . One o f the i m p o r t a n t q u e s t i o n s Case wants to answer " i s how t o adopt the c o n t e n t o f i n s t r u c t i o n to the s t r u c t u r e s t h a t c h i l d r e n already have a v a i l a b l e " . He proposed i n h i s t h e o r y a p r o -ce s s o f t h r e e s t e p s : The f i r s t s t ep i s t o a n a l y z e the 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 to form the c o n t e n t o f i n s t r u c t i o n . The second s t e p i s t o a s se s s c h i l d r e n ' s c u r r e n t l e v e l o f f u n c t i o n i n g i n t h a t d i s c i p l i n e . The t h i r d s t ep i s t o 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 p r e s e n t e d to the l e v e l o f f u n c t i o n i n g t h a t i s a s s e s s e d , e i t h e r w i t h a v iew to p romot ing 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 the g o a l ) , o r w i t h a v iew to a d o p t i n g the m a t e r i a l p r e s e n t e d to the c u r r e n t l e v e l o f f u n c t i o n i n g ( i f t h a t i s the g o a l ) . (Case, 1978a/ p.204) A l t h o u g h t h i s i n s t r u c t i o n a l p rocedure was d e v e l o p e d t o o p t i m i z e the development o f s t u d e n t s ' o p e r a t i o n a l s t r u c t u r e s , i t c an a l s o be used t o s tudy and make use o f the p r o c e s s by w h i c h c o n c e p t i o n s r e l a t e d t o s p e c i f i c c o n t e n t - a r e a s a r e a c q u i r e d and u t i l i z e d by s t u d e n t s . From the t h r e e s t eps suggested by C a s e , o n l y the l a s t one i s s t r i c t l y i n s t r u c t i o n a l . The second s t ep i s a c t u a l l y r e l a t e d to the per formance a s p e c t o f deve lopment ; t h e r e f o r e , i t i s n o t s t r i c t l y r e l a t e d t o i n s t r u c t i o n . However, i t i s v e r y u s e f u l f o r i n s t r u c t i o n a l p u r p o s e s . The f i r s t s t e p i s -47-n e i t h e r d i r e c t l y r e l a t e d to the f u n c t i o n a l a s p e c t o f d e -ve lopment nor t o i n s t r u c t i o n . But the r e s u l t o f t h i s a n a l y s i s , which s h o u l d i d e n t i f y the major c o n c e p t s Cin a g i v e n d i s c i p l i n a r y t e a c h i n g u n i t ) and the r e l a t i o n s h i p s among them, may h e l p t o focus the s e a r c h f o r s t u d e n t s ' c o n c e p t i o n s about s p e c i f i c a s p e c t s o f t h a t c o n t e n t a r e a . F o r the p r e s e n t s t u d y , o n l y the f i r s t two s t eps a re r e l e v a n t ; the t h i r d s t e p , which r e l a t e s t o i n s t r u c t i o n , can be c o n s i d e r e d o n l y i n terms o f the p o s s i b l e a p p l i c a t i o n s o f t h i s work. C a s e ' s f i r s t s t e p , a s t r u c t u r a l a n a l y s i s o f the academic c o n t e n t , has been p a r t i a l l y done i n C h a p t e r 1 - -the development o f the l i s t o f i m p l i c t v e c t o r c h a r a c t e r i s t i c s . T h i s a n a l y s i s i s f u r t h e r expanded i n C h a p t e r 3 where the r e -l a t i o n s h i p s among these v e c t o r c h a r a c t e r i s t i c s i s d i s p l a y e d i n terms o f a network o f c o n c e p t u a l r e l a t i o n s h i p s . The methodology used t o p e r f o r m t h i s type o f s t r u c t u r a l a n a l y s i s i s p r e s e n t e d below i n the next s e c t i o n . The second s t e p o f C a s e ' s approach a c t u a l l y c o r r e s p o n d s to the main o b j e c t i v e o f t h e s t u d y ; t h a t i s , t o s tudy the c u r r e n t l e v e l o f f u n c t i o n -i n g o f grade 10 s t u d e n t s about the v e c t o r c h a r a c t e r i s t i c s l i s t e d i n T a b l e 1 .1 . 2 . 2 . 4 Task /Ana lys i s One o f the more p r o m i s i n g l i n e s o f c u r r e n t p s y c h o l o g i c a l r e s e a r c h i n c o g n i t i v e development u t i l i z e s -48-some form of Task A n a l y s i s C S i e g l e r , 1978, Wol lman, 1978; C a s e , 1978) . The v a l u e o f t h i s type o f a n a l y s i s f o r e d u -c a t i o n i s t h a t such r e s e a r c h appears t o be d i r e c t l y a p p l i -c a b l e t o i n s t r u c t i o n . Hence t h i s k i n d o f approach was t aken i n the p r e s e n t s t u d y . The type o f a n a l y s i s a l l u d e d to i n the p r e v i o u s s e c t i o n (Step 1 i n C a s e ' s sequence) i s g e n e r a l l y c a l l e d a r a t i o n a l t a sk a n a l y s i s . ( T h e r e a f t e r a b b r e v i a t e d as R . T . A . ) . One common f e a t u r e o f R . T . A . i s i t s h i e r a r c h i c a l n a t u r e . Task h i e r a r c h i e s o f a c e r t a i n k i n d have been d e s c r i b e d and used f o r i n s t r u c t i o n a l purposes by Gagne (1970) and o t h e r s . But a c c o r d i n g t o Wollman (.1978), Gagne' s a n a l y s i s i s n o t . r e a l l y p s y c h o l o g i c a l and c e r t a i n l y no t d e v e l o p m e n t a l , i n t h a t i t r e f e r s t o a k i n d o f e x p e r t ' s pe r fo rmance . R e s n i c k (1976) argues t h a t Gagne - t y p e h i e r a r c h i e s do n o t a r i s e as p s y c h o l o g i c a l d e s c r i p t i o n s o f per formances on a v a r i e t y o f t a s k s ; t h a t i s , p s y c h o l o g i c a l models are " l e f t e n t i r e l y i m p l i c i t i n Gagne 1 s w o r k " . She adds t h a t t h i s i s the r e a s o n t h a t Gagne ' s h i e r a r c h i e s need v a l i d a t i o n . R e s n i c k goes on t o d e s c r i b e R . T . A . i n terms o f : T y p i c a l l y a r a t i o n a l t a sk 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 o f the 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 sumpt ions about the 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 .65) In c o n t r a s t t o the r a t h e r f o r m a l , l o g i c a l a n a l y s i s o f c o n c e p t u a l r e l a t i o n s h i p s from a d i s c i p l i n a r y p e r s p e c t i v e , w h i c h c h a r a c t e r i z e s R . T . A . , another type o f a n a l y s i s has -49-been d e s c r i b e d as an e m p i r i c a l t a sk a n a l y s i s ( h e r e a f t e r a b b r e v i a t e d as E . T . A . I . T h i s type o f a n a l y s i s a t tempts t o c o n s i d e r the q u e s t i o n o f how t h a t p a r t i c u l a r t a sk i s seen by the s t u d e n t . What schemes o r p r e c o n c e p t i o n s do the s t u d e n t s pos se s s which a l l o w them t o u n d e r s t a n d the t a sk? How a r e these schemes o r p r e c o n c e p t i o n s c o o r d i n a t e d ? What i r r e l e v a n t cues a re s i g n i f i c a n t f o r them? A r e they l i k e l y t o be e a s i l y i n f l u e n c e d by the cues? The answer to the se q u e s t i o n s o b v i o u s l y r e q u i r e s the i n v e s t i g a t o r t o become f a m i l i a r w i t h the "knowledge" ( p r e c o n c e p t i o n s ) po s se s sed by the s t u d e n t and how t h i s knowledge i s c o o r d i n a t e d . R e s n i c k d e s c r i b e s an e m p i r i c a l t a sk a n a l y s i s i n the f o l l o w i n g way: I t i s based on i n t e r p r e t a t i o n o f the d a t a ( e r r o r s , l a t e n c i e s , s e l f - r e p o r t s , eye o r hand movements, e tc ) from human per formance on a t a s k ; the aim o f such 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 . 6 5 ) . Case (1978) and S i e g l e r (1976, 1979) have used b o t h types o f t a sk a n a l y s i s ( R a t i o n a l and E m p i r i c a l ) i n t h e i r r e s e a r c h . In t h e i r work they a t tempt t o uncover the d i f f i -c u l t i e s t h a t s t u d e n t s show when e n c o u n t e r i n g s p e c i f i c t a s k s . They sugges t t h a t the t a s k s o r c o n t e n t - a r e a s t o be t a u g h t ( e . g . , t a s k s r e l a t e d t o v e c t o r q u a n t i t i e s ) must be c a r e f u l l y a n a l y z e d , t h a t the l e a r n e r ' s i n i t i a l knowledge must be c a r e f u l l y a s s e s s e d , and t h a t the l e a r n i n g a c t i v i t i e s must be r a t i o n a l l y and p s y c h o l o g i c a l l y ( i n the d e v e l o p m e n t a l sense) sequenced t o b r i n g the l e a r n e r from h i s i n i t i a l s t a t e to the d e s i r e d s t a t e . T h i s means t h a t Case and S i e g l e r ' s -50-approaches d i f f e r marked ly from t h a t o f most P i a g e t i a n e d u c a t i o n a l r e s e a r c h e r s , i n t h a t t h e r e i s no a t tempt o r need t o c l a s s i f y l e a r n e r s as c o n c r e t e o r f o r m a l . I n s t e a d , t h i s n e o - P i a g e t i a n approach advoca te s assessment o f the l e a r n e r ' s i n i t i a l s t a t e i n terms o f the s t r a t e g i e s w h i c h he o r she s p o n t a n e o u s l y a p p l i e s to the c r i t e r i o n t a s k . Wollman (1978) t h i n k s t h a t some c o n t e n t - a r e a s a r e more s u i t a b l e than o t h e r s t o be t r e a t e d from a f u n c t i o n a l p o i n t o f v i e w , and f o r th e use o f r a t i o n a l and e m p i r i c a l t a sk a n a l y s e s . Wollman says t h a t : Task 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 key s c i e n c e and s c i e n c e - r e l a t e d mathemat ics t o p i c s . To q u a l i f y as a key t o p i c , two c o n d i t i o n s s h o u l d be met : (1) the t o p i c s h o u l d 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 the s u b j e c t mat te r and (2) the 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 c u r r e n t t e a c h i n g a t t e m p t s . (Wollman, 1978, p.107) T h i s a u t h o r b e l i e v e s t h a t p h y s i c a l v e c t o r q u a n t i t i e s and t h e i r s p e c i a l c h a r a c t e r i s t i c s f i t these two c o n d i t i o n s s t a t e d by Wol lman, as was shown i n C h a p t e r 1. T h e r e f o r e , Task A n a l y s i s , i n i t s two forms : R a t i o n a l and E m p i r i c a l , was used i n the p r e s e n t s t u d y . I n summary, i n t h i s s tudy two k i n d s o f t a sk a n a l y s e s were u s e d : a R a t i o n a l Task A n a l y s i s t h a t c o n s i s t e d o f an a n a l y s i s from t h e d i s c i p l i n e e x p e r t ' s p o i n t o f v i e w , and an E m p i r i c a l Task A n a l y s i s t h a t c o n s i s t e d o f a n a l y z i n g the s t u d e n t s ' p r e c o n c e p t i o n s about the c o n t e n t - a r e a i n c l u d e d i n the r e s e a r c h t a s k s . -51-2.3 E d u c a t i o n a l C o n t e x t o f the Study In the l a s t two decades the u n d e r l y i n g c u r r i c u l u m approach i n s c i e n c e t e a c h i n g has f o c u s s e d upon the p r o c e s s o f i n q u i r y , w h i c h emphasizes the p r a c t i c e o f s c i e n t i f i c method. (Shulman & T a m i r , 1973; Hawkins , 1978; W i t t r o c k , 1966; B r u n e r , 1961; Schwab, 1962) . B u t , a c c o r d i n g t o Hawkins (1978) , method i s the use o f knowledge t o g a i n more knowledge ; he c l a i m s t h a t i f not much knowledge i s o r g a n i z e d and a v a i l a b l e f o r use not much w i l l be l e a r n e d ( p . 2 0 ) . O t h e r s i n t u r n have proposed a "Concep t A p p r o a c h " (Shulman & T a m i r , 1973; Schwab 1962, 1964; Hawkins , 1978) . They were r e a c t i n g a g a i n s t t h a t l i m i t e d c u r r i c u l u m a p p r o a c h , which emphasizes o n l y the method o f a c q u i r i n g knowledge i n s t e a d o f c o n s i d e r i n g the way the d i s c i p l i n e s themse lves a c q u i r e t h a t knowledge . A c c o r d i n g t o t h i s c o n c e p t approach a c u r r i c u l u m i s o r g a n i z e d i n s u c c e s s i v e phase s , each e x p l a i n i n g and i l l u s t r a t i n g some : b a s i c » s c i e n t i f i c c o n c e p t . In t h i s c o n t e x t , i t i s expected t h a t when the c o u r s e i s f i n i s h e d s t u d e n t s w i l l have r i s e n t o a h i g h e r l e v e l o f s c i e n t i f i c l i t e r a c y , and w i l l possess a f o u n d a t i o n f o r u n d e r s t a n d i n g s c i e n t i f i c s ta tements and e x p l a n a t i o n s . Hawkins s t a t e s t h a t : T h i s v i e w , l i k e the 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 n e c e s s a r y bu t no t s u f f i c i e n t . I t s u f f e r s from the 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 -s t a n d ' . 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 focused on the use 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 no t the same as t e a c h i n g and l e a r n i n g w h i c h 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 has been 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 no 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 bu 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 genuine and p r o d u c t i v e a s s i m i l a t i o n o f t h i s knowledge . There i s a t ime 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 t ime which presupposes t h a t t h o u g h t -h a b i t s which can a c c e p t , s t o r e , and r e t r i e v e f o r new uses a re 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 must , 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) T h i s a u t h o r has a v iew o f s c i e n c e e d u c a t i o n v e r y s i m i l a r to t h a t o f Hawkins , p a r t i c u l a r l y w i t h r e g a r d to the d i a g n o s t i c a s p e c t o f t e a c h i n g . I t i s argued t h a t by c o n -s i d e r i n g the knowledge , the l e a r n i n g s t r a t e g i e s , and the s k i l l s t h a t s t u d e n t s b r i n g t o the i n s t r u c t i o n a l s i t u a t i o n , the t e a c h i n g - l e a r n i n g p r o c e s s has much t o g a i n . But t o p r a c t i c e t h i s a p p r o a c h , r e s e a r c h about these t h r e e a s p e c t s i s n e c e s s a r y . The p r e s e n t s tudy t r i e d , i n p a r t i c u l a r , t o examine the f i r s t a s p e c t by examining the knowledge o r p r e c o n c e p t i o n s t h a t s t u d e n t s pos ses s p r i o r t o i n s t r u c t i o n . T h i s was done by 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 ' p r e -i n s t r u c t i o n a l c o n c e p t i o n s i n a s p e c i f i c c o n t e n t - a r e a . 2 . 3 . 1 Teaching—Learning Problems i n S c i e n c e E d u c a t i o n The g e n e r a l problem o f the p r e s e n t r e s e a r c h i s a t e a c h i n g - l e a r n i n g p r o b l e m . Research has shown t h a t s t u d e n t s do n o t e a s i l y g r a s p v e c t o r i a l q u a n t i t i e s t h a t a r e u s u a l l y - 5 3 -i n c l u d e d i n p h y s i c s Grade 11 and 12. ( F i s h e r , 1979; K a s s , 1976) . Where do the problems l i e ? Is i t the i n s t r u c t i o n methodology? I s i t t h a t s t u d e n t s a r e no t " r e a d y " to g r a s p the se c o n c e p t s ? I s i t t h a t th e c o n t e n t - a r e a i s v e r y remote from s t u d e n t s ' e x p e r i e n c e a n d , t h e r e f o r e , i s too complex t o be unders tood? T h i s a u t h o r b e l i e v e s t h a t some o f the problems l i e i n the i n s t r u c t i o n methodology . P r e s e n t m e t h o d o l o g i e s u s u a l l y do n o t c o n s i d e r s t u d e n t s ' c u r r e n t knowledge . O b v i o u s l y t h i s methodology i s p a r t o f an o v e r a l l c u r r i c u l u m v i e w . In t h i s v i e w , c u r r i c u l u m makers , among o t h e r c o n -s i d e r a t i o n s , p l a n a c o n t e n t - a r e a program m a i n l y t a k i n g i n t o account the s t r u c t u r e o f t h a t c o n t e n t - a r e a o r d i s c i p l i n e . The program i s u s u a l l y a r r anged i n h i e r a r c h i c a l o r d e r ; t h a t o r d e r b e i n g sugges ted by the e x p e r t i n the f i e l d . As we s ee , i n t h i s approach t h e r e i s no c o n s i d e r a t i o n o f s t u d e n t s ' v i e w s . To change to an approach t h a t c o n s i d e r s s t u d e n t s ' c o n c e p t i o n s i s no t an easy t a s k . F i r s t o f a l l , r e s e a r c h i s needed t o uncover s t u d e n t s ' b e l i e f s i n many t o p i c s i n c l u d e d i n s c i e n c e programs . For i n s t a n c e , no t much i s known about s t u d e n t s ' p r e t e a c h i n g c o n c e p t i o n s about v e c t o r i a l c o n c e p t i o n s . A n o t h e r i m p o r t a n t r e a s o n t o f i n d out the s t u d e n t s ' p r e c o n c e p t i o n s i s t h a t some o f these ( a l so c a l l e d by some a l t e r n a t e frameworks o r m i s c o n c e p t i o n s , see D r i v e r and -54-E a s l e y , 1978, p.62) are very r e s i s t a n t t o e x t i n c t i o n , and i n t e r f e r e w i t h fo rma l i n s t r u c t i o n . T h i s i s a n o t h e r i m p o r t a n t r e a s o n t o uncover and a n a l y z e these a l t e r n a t e frameworks, and a l s o to i n v e s t i g a t e how these i n t u i t i v e c o n c e p t i o n s can be used i n i n s t r u c t i o n . Knowledge o f s t u d e n t s ' a l t e r n a t e frameworks has been used i n c o n s t r u c t i n g l e a r n i n g t a s k s i n s c i e n c e . C o l e and Raven (1969) r e p o r t a s tudy w i t h 12-15 yea r o l d s t u d e n t s which i n d i c a t e s the p o s s i b l e b e n e f i t o f g i v i n g s t u d e n t s the o p p o r t u n i t y t o r e j e c t i r r e l e v a n t f a c t o r s i n u n d e r s t a n d i n g the p r i n c i p l e o f f l o t a t i o n . Common " m i s -c o n c e p t i o n s " r e l a t e d t o f l o a t i n g and s i n k i n g b o d i e s were a l s o c o n s i d e r e d i n the d e s i g n and i m p l e m e n t a t i o n o f an i n s t r u c t i o n a l sequence by R o w e l l and Dawson (1977) . T h e i r r e p o r t i n d i c a t e s t h a t d e s p i t e e f f o r t s t o r e f u t e s t u d e n t s ' m i s c o n c e p t i o n s , some p e r s i s t e d a f t e r i n s t r u c t i o n . D r i v e r (1973) e x p l o r e d some o f the d i f f i c u l t i e s s c i e n c e s t u d e n t s (11-12 y e a r s o f age) have i n accommodating t h e i r t h i n k i n g t o new e x p e r i e n c e s . P r i o r t o t e a c h i n g , s t u d e n t s were i n t e r -viewed and a number o f a l t e r n a t e frameworks about the t o p i c s b e i n g s t u d i e d ( b a l a n c i n g sys tems , c e n t e r s o f g r a v i t y , the law o f moments, a c t i o n and r e a c t i o n , and r e l a t i o n between f o r c e and motion) were i d e n t i f i e d . A f t e r i n -s t r u c t i o n o f the u n i t s r e l a t e d t o these t o p i c s , i t was apparent t h a t the c o u n t e r - e x a m p l e s and c o n f l i c t i n g e v i d e n c e d i d n o t , o f i t s e l f , encourage a change i n s t u d e n t s ' t h i n k i n g , -55-and a t t imes produced o n l y c o n f u s i o n . T h i s s tudy a l s o i l l u s t r a t e d the communicat ion d i f f i c u l t i e s between t e a c h e r and s t u d e n t s when each e n t e r i n t o d i a l o g u e w i t h d i f f e r e n t i n t e r p r e t i v e f rames . The above r e s u l t s show t h a t i f we p e r s i s t i n s e e i n g "a s t u d e n t as a f i l i n g sys tem" (Hawkins, 1978) and no t as an a c t i v e o r g a n i s m , s t u d e n t s ' d i f f i c u l t i e s i n u n d e r s t a n d i n g c e r t a i n t o p i c s w i l l a l s o p e r s i s t . 2 . 3 . 2 C u r r i c u l u m Problem i n S c i e n c e E d u c a t i o n In the i n t r o d u c t o r y s e c t i o n o f 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 (see S e c t i o n 2.3 ) i t was ment ioned t h a t the i n q u i r y approach has been the u n d e r l y i n g approach used i n s c i e n c e t e a c h i n g i n the l a s t two decades . T h i s approach l o o k s f o r a method o f i n s t r u c t i o n t h a t c o u l d match the t e c h n i q u e s o f making s c i e n t i f i c c l a i m s ; t h i s method i s the i n q u i r y a p p r o a c h . The r a t i o n a l e b e h i n d t h i s methodology i s t h a t i t i s thought t h a t by emphas i z ing s t u d e n t i n q u i r y w i t h i n a p r o c e s s - o r i e n t e d program, the s u b t l e r e g u l a r i t i e s o f n a t u r e , w h i c h a r e apparent to the s c i e n t i s t , can be " d i s c o v e r e d " by an i n q u i r i n g s t u d e n t . But r e s e a r c h and p r a c t i c e (Shulman & T a m i r , 1973; Hawkins , 1978; Deadman and K e l l y , 1978) have shown t h a t s t u d e n t s see the w o r l d from v e r y d i f f e r e n t p e r s p e c t i v e s than t h a t o f a d u l t s i n g e n e r a l , and s c i e n t i s t s i n p a r t i c u l a r . E r i c k s o n (1975) has s a i d t h a t : -56-Thus 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 ' methaphor i s n o t t h a t i t sugges t s one can s i m p l i f y o r reduce the complex body o f s c i e n t i f i c knowledge 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 programs , as has been done t o d a t e . R a t h e r , i t i s to v i e w b o t h c h i l d and s c i e n t i s t as engaged i n the 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 o f o r d e r i n t h e i r w o r l d — o n l y a t d i f f e r e n t l e v e l s o f a b s t r a c t i o n . ( E r i c k s o n , 1975, p . 1 6 ) . The i m p l i c a t i o n s o f t h i s p o s i t i o n f o r the d e v e l o p -ment o f s c i e n c e c u r r i c u l a seems c l e a r to E r i c k s o n and t o t h i s a u t h o r . The approach i s t h a t c u r r i c u l u m development ought to p r o c e e d from the s i m p l e and more c o n c r e t e pheno-mena a s s o c i a t e d w i t h the s t u d e n t ' s w o r l d toward the more p o w e r f u l and s o p h i s t i c a t e d c o n c e p t i o n s o f the e x p e r t ' s w o r l d . C u r r i c u l u m development i n s c i e n c e s u b j e c t s has t y p i c a l l y f o l l o w e d a p r o c e d u r e whereby the c o n t e n t , m a t e r i a l s , and s t r a t e g i e s f o r t e a c h i n g a t o p i c a re e s t a b l i s h e d f i r s t , and then t r i e d o u t on s t u d e n t s . A c c o r d i n g to t h e i r s u c c e s s , o r o t h e r w i s e , the c u r r i c u l u m packages a re then a c c e p t e d o r m o d i f i e d , o r a l t e r n a t i v e ones a re sought . T h i s c o u l d be r e f e r r e d to as the c u r r i c u l u m - t o - s t u d e n t s mode i n c o n t r a s t t o the s t u d e n t s - t o - c u r r i c u l u m mode (Deadman & K e l l y , 1978) . T h i s a u t h o r p r e f e r s the l a t t e r a p p r o a c h , i n which s t u d e n t s ' u n d e r s t a n d i n g i s i n v e s t i g a t e d f i r s t and t h e n , through a g r a d u a l b u i l d i n g - u p p r o c e s s i n which deve lopment , r e s e a r c h , and t e a c h i n g are combined , ways a r e e x p l o r e d d i r e c t l y w i t h - 5 7 -t h e s t u d e n t s by which t h e i r u n d e r s t a n d i n g can be i n c r e a s e d . Deadman and K e l l y ( 1 9 _ 7 8 1 t h i n k t h a t : "The c u r r l c u l u m -t o - s t u d e n t s mode i s s u i t a b l e f o r d e v e l o p i n g the b r o a d a s p e c t s o f c o u r s e s bu t i t i s l e s s s u i t a b l e f o r the i n - d e p t h d e v e l o p -ment o f i n d i v i d u a l c o u r s e s " , (p. 8 ) T h i s may be the r e a s o n , o r a t l e a s t one o f the r e a s o n s , why i t has n o t been p o s s i b l e so f a r t o de te rmine w h o l l y s u i t a b l e ways f o r d e a l i n g w i t h the t o p i c o f v e c t o r q u a n t i t i e s and t h e i r s p e c i a l c h a r a c t e r i s t i c s . The d e v e l o p -ment methods employed have not p r o v i d e d s u f f i c i e n t d e t a i l s o f the t e a c h i n g and l e a r n i n g problems i n v o l v e d . T h i s i s a l s o the r e a s o n why t h i s a u t h o r i s f a v o u r i n g the s t u d e n t s - t o -c u r r i c u l u m mode t o a t tempt t o overcome some o f the problems i n l e a r n i n g v e c t o r i a l c o n c e p t s . -58-CHAPTER THREE TASKS, INTERVIEW PROTOCOL, AND DATA COLLECTION OF PHASE ONE 3.0 I n t r o d u c t i o n T h i s c h a p t e r d e a l s w i t h the i n s t r u m e n t s and the p r o c e d u r e s p e r t i n e n t t o the da t a c o l l e c t i o n o f the f i r s t phase . F i r s t , a s h o r t r e p o r t o f the p i l o t s tudy i s p r e s e n t e d , w h i c h showed t h a t the t a s k s were s u i t a b l e t o c a r r y o u t the f i r s t p a r t o f the r e s e a r c h . The use o f r a t i o n a l t a sk a n a l y s i s i s e x p l a i n e d n e x t , which shows the method o f p r e s e n t i n g the v e c t o r c h a r a c t e r i s t i c s i n terms o f a network o f c o n c e p t s . These networks show broad r e l a t i o n s h i p s among the v e c t o r c h a r a c t e r i s t i c s l i s t e d i n T a b l e 1.1 F o l l o w i n g these i s a d e s c r i p t i o n o f the two t a s k s used i n the s t u d y , w i t h t h e i r r e s p e c t i v e p r o -t o c o l s . The c h a p t e r ends by d e s c r i b i n g the d a t a c o l l e c t i o n methodo logy , the format o f the i n t e r v i e w , and the c h a r a c t e r -i s t i c s o f the s u b j e c t s who were i n t e r v i e w e d . 3.1 The P i l o t S tudy A c c o r d i n g t o P i n e s e t a l : e v e r y r e s e a r c h program which i s t o i n v o l v e t h e a d m i n i s t r a t i o n o f c l i n i c a l i n t e r v i e w s s h o u l d have a p i l o t s tudy i n w h i c h the i n t e r -v iew can 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 out 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 t o the s u b j e c t s o f the r e s e a r c h s t u d y . (P ines e t a l , 1 9 7 8 , p . 8 ) . The main purposes o f the p i l o t s tudy were : ( i ) t o de te rmine i f the i n t e r v i e w t a s k s t o be used i n the s t u d y a re s u i t a b l e t o a t t a i n the g o a l s o f the s t u d y ; ( i i ) t o - 5 9 -determine i f the p r o t o c o l q u e s t i o n s can be u n d e r s t o o d by the t a r g e t p o p u l a t i o n ( i f n o t , they s h o u l d be p a r a p h r a s e d and improved u n t i l t h e y a re s u f f i c i e n t l y c l e a r ) ; ( i i i ) t o become f a m i l i a r w i t h the r e l e v a n t c h a r a c t e r i s t i c s o f the t a r g e t p o p u l a t i o n such as language a b i l i t y and m a n i p u l a t i o n d e x t e r i t y . D u r i n g the p r e l i m i n a r y work f o r the p i l o t s t u d y , e i g h t c h i l d r e n o f v a r i o u s ages ( r ang ing from 13 t o 17) were i n t e r v i e w e d which r e s u l t e d i n numerous changes i n b o t h the t a s k s t h a t were e v e n t u a l l y used i n the s tudy and the q u e s t i o n s about the t a sk s i t u a t i o n s . The two t a s k s w h i c h . e v e n t u a l l y emerged f o r the s tudy seemed t o be s u f f i c i e n t l y a p p e a l i n g and c l e a r f o r the s u b j e c t s ; p a r t i c u l a r l y when an 8 mm f i l m - l o o p and l a r g e (100 cm x 100 cm) c a r d b o a r d model were u s e d . P r e v i o u s l y , when s m a l l e r paper models were u s e d , the s u b j e c t s re sponded t o the t a s k s i n terms o f a geometry prob lem because the models p e r m i t t e d the s t u d e n t s to use t h e i r hands t o p o i n t to the v a r i o u s o b j e c t s and drew i m a g i n a r y o r r e a l l i n e s on the mode l s . I n i t i a l l y the s u b j e c t s e x p e r i e n c e d some d i f f i c u l t y i n u n d e r s t a n d i n g a few o f the q u e s t i o n s . T h i s was l i k e l y due t o the f a c t t h a t the o r i g i n a l q u e s t i o n s were j u s t the v e c t o r c h a r a c t e r i s t i c s (see T a b l e 1.1) p r e s e n t e d i n q u e s t i o n fo rm. These q u e s t i o n s were r e v i s e d s e v e r a l t imes u n t i l i t was e v i d e n t t h a t the s t u d e n t s u n d e r s t o o d the i n t e n t o f the q u e s t i o n . L i k e w i s e the number o f q u e s t i o n s on each v e c t o r c h a r a c t e r i s t i c was r e d u c e d , a f t e r - p i l o t t e s t i n g , from f i v e t o two o r t h r e e . Even w i t h t h i s r e d u c t i o n i n q u e s t i o n s , -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 subject held consistent preconceptions for a p a r t i c u l a r vector c h a r a c t e r i s t i c by examining t h e i r responses to the two or three re l a t e d questions. Further-more, the fewer questions made i t possible to conduct the interview i n 45 to 50 minutes. Three of the 8 Ss of the f i n a l phase of the p i l o t study were younger than 15 years (2Ss were i n grade 8 and IS i n grade 9). They gave l i t t l e thought to the vector c h a r a c t e r i s t i c s included i n task two, demonstrated great d i f f i c u l t y both i n understanding the question of task two .and i n expressing t h e i r ideas. Based on these observations, i t was decided to only choose subjects i n grade 10 for the actual study. Furthermore, the grade ten students have not been formally taught vectors and so i t enabled the study to focus on the kinds of preconceptions students possess about vector q uantities before they receive any i n s t r u c t i o n on the t o p i c . In summary, the p i l o t study was useful to improve the tasks and the interview protocol. I t also showed that the tasks and the use of i n d i v i d u a l interviews would be suitable to address the major questions posed i n the study. 3.2 Rational Task Analysis of Physical Phenomena The vector c h a r a c t e r i s t i c s that were presented i n Table 1.1 i n Chapter One were obtained by analyzing various physical s i t u a t i o n s , part of which can be represented by -61-the t h r e e v e c t o r q u a n t i t i e s d e a l t w i t h - in ;£he\ p r e s e n t s t u d y . A f t e r f i n d i n g the most r e l e v a n t k i n e m a t i c a s p e c t s o f the s i t u a t i o n s , these a s p e c t s were then compared to the p r o p e r t i e s t h a t th e f o r m a l d e f i n i t i o n s g e n e r a l l y i n c l u d e d f o r t h a t p a r t i c u l a r v e c t o r q u a n t i t y . I t was d e t e r m i n e d t h a t some o f th e uncovered c h a r a c t e r i s t i c s were no t ex-p l i c i t l y i n t r o d u c e d i n the d e f i n i t i o n s . T h i s was the rea son t h a t they were c a l l e d 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 . I t was hoped t h a t by c l e a r l y s t i p u l a t i n g these c h a r a c t e r -i s t i c s i t would be e a s i e r t o f i n d ou t i f s t u d e n t s had any i n t u i t i v e u n d e r s t a n d i n g o f any o f these c h a r a c t e r i s t i c s ; which c h a r a c t e r i s t i c s seemed to be most t roub le some t o s t u d e n t s ; and i f 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 had about v e c t o r q u a n t i t i e s . Many s c h o o l p h y s i c s t ex tbooks ( e . g . , PSSC, 1980; Fundamentals o f P h y s i c s : H e a l t h , Macnaughton, and M a r t i n -d a l e , 1979; P h y s i c s : G i a n c o l i , l SSOh-def j ne . f o r m a l l y the main v e c t o r c h a r a c t e r i s t i c s o f s e v e r a l v e c t o r q u a n t i t i e s , bu t the se books do no t p r e s e n t c l e a r l y the e s s e n t i a l c h a r a c t e r i s t i c s p r e s e n t e d i n C h a p t e r One. T h i s a u t h o r d i d an e x t e n s i v e r e v i e w o f p h y s i c s t ex tbooks c u r r e n t l y used i n B . C . s c h o o l s . In summary, i t can be s a i d t h a t a l l o f them p r e s e n t some v e c t o r q u a n t i t i e s i n i s o l a t i o n , w i t h o u t showing r e l a t i o n s h i p s among them, and w i t h o u t s t r o n g l y emphas i z ing t h a t v e c t o r q u a n t i t i e s r e p r e s e n t s p e c i f i c p r o -p e r t i e s o f p h y s i c a l phenomena. The task of analyzing 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 of the three physical quantities was c a r r i e d out with the use of Rational Task Analysis. Resnick says that: Rational task analysis can be defined as an attempt to specify processes or procedures that would be used i n highly e f f i c i e n t performance of some tasks. The r e s u l t i s a de t a i l e d d e s c r i p t i o n of an "i d e a l i z e d " performance - one that solves the problem i n minimal moves, does l i t t l e 'back-tracking 1, makes few or no errors. T y p i c a l l y a Rational Task Analysis i s derived from the structure of the subject 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 of human memory capacity or perceptual encoding process. (Resnick, 1 9 7 6 , p . 6 5 ) . Usually, the highly e f f i c i e n t performance and the der i v a t i o n of properties from the structure of the subject matter are c a r r i e d out by an expert i n the f i e l d . This was the procedure used i n the present study. A f t e r s e l e c t i n g three of the e a r l i e s t vector quantities generally taught i n high school__physics (vector p o s i t i o n , displacement, and v e l o c i t y ) , an analysis was per-formed to determine the physical c h a r a c t e r i s t i c s that they represent. That i s , what aspects of physical r e a l i t y do the three quantities deal with. Both common and d i f f e r e n t elements i n these quantities were examined along with t h e i r mathematical abstractions. At the end of t h i s a n a l y s i s , a set of e s s e n t i a l vector c h a r a c t e r i s t i c s of the three physical quantities was produced. This set was shown to three u n i v e r s i t y physics professors for comments and c r i t i c i s m . Private discussions were held with each one of the professors, and notes were taken of suggested changes and improvements . A r e v i s e d s e t was p r o d u c e d , and d i c u s s s e d a g a i n w i t h the same p r o f e s s o r s . A t the end an agreement was r eached and a f i n a l s e t o f v e c t o r c h a r a c t e r i s t i c s was p r o d u c e d . The whole p r o c e s s c o n s t i t u t e s a p r o c e d u r e o f R a t i o n a l Task A n a l y s i s . T h i s d e r i v e d l i s t o f i m p l i c i t c h a r a c t e r i s t i c s (see T a b l e 1.1) d i r e c t l y r e l a t e s t o the f i r s t r e s e a r c h q u e s t i o n o f the s t u d y , which i s the s e a r c h f o r s t u d e n t s ' p r e c o n c e p t i o n s o f these i m p l i c i t c h a r a c t e r i s t i c s . 3 . 2 . 1 Networks o f the Three V e c t o r Q u a n t i t i e s However, i t was f e l t t h a t the l i s t o f 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 does no t show c l e a r l y the r e l a t i o n s h i p s among the t h r e e d i f f e r e n t c o n c e p t s and t h e i r p r o p e r t i e s . So , i t was d e c i d e d t o p r e s e n t the s t r u c t u r a l r e l a t i o n s h i p s among these p h y s i c a l q u a n t i t i e s and t h e i r v e c t o r c h a r a c t e r -i s t i c s as f o u r networks o f c o n c e p t s , which d i a g r a m a t i c a l l y d e p i c t the v a r i o u s r e l a t i o n s among the concept s and t h e i r p r o p e r t i e s . The f i r s t newtwork shows the g e n e r a l c h a r a c t e r -i s t i c s o f any v e c t o r q u a n t i t y . T h i s network p r o v i d e s the o v e r a l l framework f o r the o t h e r t h r e e . The o t h e r t h r e e n e t -works d e p i c t the v e c t o r c h a r a c t e r i s t i c s a s s o c i a t e d w i t h each one o f t h r e e p h y s i c a l q u a n t i t i e s s e l e c t e d f o r t h i s s t u d y . The r a t i o n a l e f o r p r e p a r i n g these networks i s based p r i m a r i l y on the semant ic networks o f L i n d s a y and Norman (1977) and the a c t i v e s t r u c t u r a l networks o f Norman and Figure 3.1 Network of a Kinematic Vector Quantity P h y s i c a l E n t i t y kinematic kinematic s c a l a r quantity vector quantity composition of vectors | an instantaneous l o c a t i o n i s independent of others l o c . parallelogram method vector subtraction one or more simultaneous components d i r e c t i o n frame of reference r e a l number fr»P reference p o i n t pr«p u n i t s prop. independent components independence of magnitudes independence of d i r e c t i o n s I CTl I Key of R e l a t i o n a l Terms: sub = has a (non-intersecting) subclass" prop. = has as a property charac.= has as a c h a r a c t e r i s t i c of obeying cond. = needs c e r t a i n conditions to make q u a n t i t a t i v e that operation Figure 3.2 Network of Vector P o s i t i o n (V.P.) kinematic scalar quantity ~~ vector subtraction ck&rfrc t r i a n g l e method composition of two V.P. an instantaneous l o c a t i o n i s i n -dependent of other locations Physical E n t i t y k i vector nematic quantity Sub vector p o s i t i o n prop magnitude r e a l number prop d i r e c t i o n units frame of reference r e a l number u n i t s /Top reference point l I Note ; Meaning of r e l a t i o n a l terihd same as i n Figure 3.1 Figure 3.3 Network of Vector Displacement (or Change of Location) Physical E n t i t y kinematic sc a l a r quantity kinematic vector quantity tub displacement of change of l o c a t i o n vector a d d i t i o n t r : me1 Langle :hod composition of displacements independent from path  magnitude r e a l number units 1 and d i r e c t i o n frame o reference prop r e a l number reference point Notet Meaning of r e l a t i o n a l terms same as i n Figure 3.1 Network of Vector Average V e l o c i t y kinematic s c a l a r quantity Physical E n t i t y S0& parai l e logramj method vector addition kinematic vector quantity v e l o c i t y composition v e l o c i t i e s average v e l o c i t y instantaneous v e l o c i t y prop one or more instantaneous components prop independent components magnitude units' d i r e c t i o n frame of reference sofc independence of magnitudes independence of d i r e c t i o n s r e a l number un i t s l (Ti I reference body stationary body moving reference body Note: Meaning of r e l a t i o n a l terms same as i n Figure 3.1 -68-Rumelhart (1975). The t a s k s proposed f o r use i n t h e 4 s t u d ' y a r e based on a l l the v e c t o r q u a n t i t i e s and t h e i r c h a r a c t e r i s t i c s d e p i c t e d i n the f o u r networks o f c o n c e p t s . E f f o r t s were made to show the p r e c i s e cor re spondence between the i n f o r m a -t i o n c o n t a i n e d i n the networks and the p r o t o c o l s o f the t a s k s . The diagrams and d i s c u s s i o n s o f the f o u r networks a re p r o v i d e d i n t h i s s e c t i o n . F i g u r e 3.1 d e p i c t s the network f o r any k i n e m a t i c v e c t o r q u a n t i t y . The r e l a t i o n s h i p s among a l l v e c t o r c h a r a c t e r i s t i c s l i s t e d i n T a b l e 1.1 a re shown i n t h i s ne twork . The l i n e s c o n n e c t i n g the d i f f e r e n t boxes , which c o n t a i n the v e c t o r c h a r a c t e r i s t i c s , do no t d e f i n e e x p l i c i t m a t h e m a t i c a l r e l a t i o n s h i p s between the two v e c t o r c h a r a c t e r i s t i c s . The l i n e s o n l y i n d i c a t e t h a t the two v e c t o r c h a r a c t e r i s t i c s connec ted by i t a re i n some way r e -l a t e d . Some more s p e c i f i c r e l a t i o n s h i p s are e x p l a i n e d a t the bottom o f the network ( i . e . , sub , c l i a r a c , p r o p , and c o n d . ) . F i g u r e s 3.2 t o 3.4 show r e s p e c t i v e l y the networks f o r the v e c t o r p o s i t i o n , the v e c t o r d i s p l a c e m e n t , and v e c t o r average v e l o c i t y . 3 . 2 . 2 The P h y s i c s Approach The p h y s i c s approach r e p r e s e n t s the p h y s i c i s t s ' c u r r e n t knowledge ( i . e . , t h e o r i e s , c o n c e p t s , d e f i n i t i o n s , a p p l i e d mathemat i c s , r e s e a r c h m e t h o d o l o g i e s , prob lem s o l v i n g methodo log ie s ) used to d e a l w i t h e x p e r i m e n t a l s i t u a t i o n s -69-s i m i l a r t o the ones d e p i c t e d as the t a s k s o f the s t u d y . • The p h y s i c i s t s ' c u r r e n t knowledge - the p h y s i c s approach - i s p a r t i a l l y r e p r e s e n t e d i n th e networks o f c o n -c e p t s ( F i g u r e s 3.1 t o 3 . 4 ) . T h e n , they r e p r e s e n t the p h y s i c s approach i n the p r e s e n t s t u d y . I t i s i m p o r t a n t t o remind the r e a d e r t h a t those networks do no t r e p r e s e n t the f o r m a l d e f i n i t i o n s o f the t h r e e v e c t o r q u a n t i t i e s ; d e f i -n i t i o n s a re u s u a l l y s u c c i n t and l e a v e i m p o r t a n t a s p e c t s ou t o f the c o n c e p t b e i n g d e f i n e d . The networks a c t u a l l y r e -p r e s e n t a s t r u c t u r a l form o f the l i s t o f 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 (see T a b l e 1 . 1 ) . The r e a s o n f o r d e f i n i n g the p h y s i c s approach was f o r the s p e c i f i c purpose o f u s i n g i t i n the c a t e g o r i z a t i o n o f the i n f e r r e d r u l e s i d e n t i f i e d f o r each v e c t o r c h a r a c t e r -i s t i c . T h i s approach was used by s i m p l y compar ing the c o n t e n t o f an i n f e r r e d r u l e o f a v e c t o r c h a r a c t e r i s t i c w i t h the p h y s i c i s t ' s c u r r e n t v i e w , shown i n the networks o f c o n c e p t s , o f t r e a t i n g t h a t v e c t o r c i t a r a t r t ^ r l s t i c . F u r t h e r e x p l a n a t i o n s o f t h i s c a t e g o r i z a t i o n i s p r e s e n t e d i n S e c t i o n 4 . 1 . 3 (Chapter F o u r ) . 3 ^3 The Two Tasks o f the Study Two t a s k s were c r e a t e d f o r the p r e s e n t s t u d y . Task One i n c l u d e d the v e c t o r c h a r a c t e r i s t i c s o u t l i n e d i n the network o f v e c t o r p o s i t i o n (see F i g u r e 3.2) and the network o f d i s p l a c e m e n t (see F i g u r e 3 . 3 ) . These c h a r a c t e r i s t i c s - 7 0 -c o r r e s p o n d to i tems one t h r o u g h f i v e i n T a b l e 1 .1 . Task Two i n c l u d e d the v e c t o r c h a r a c t e r i s t i c s o u t l i n e d i n network o f average v e l o c i t y (See F i g u r e 3 . 4 ) , which a l s o c o r r e s p o n d e d t o i tems s i x through t e n i n T a b l e 1 .1 . (Items 11 and 12 were o n l y t r e a t e d i n the second p h a s e ) . I t was thought t h a t the t a s k s s h o u l d be as r e l a t e d as p o s s i b l e t o some common e x p e r i e n c e s o f the s u b j e c t s s i n c e a major purpose i n th e s tudy was the s e a r c h f o r i n t u i t i v e n o t i o n s which may be pos se s sed by the s t u d e n t s . S i n c e b o a t i n g on l a k e s and r i v e r s i s v e r y p r e v a l e n t i n B r i t i s h Co lumbia one t a s k r e f e r r e d t o a l a k e s i t u a t i o n and the o t h e r one t o a r i v e r s i t u a t i o n . In summary the c r i t e r i a used f o r t a sk s e l e c t i o n were : 1) t h a t the t a sk s e t t i n g s h o u l d be f a m i l i a r to most o f the s u b j e c t s , 2) t h a t the t a s k s must i l l u s t r a t e d r a m a t i c a l l y the v e c t o r c h a r a c t e r i s t i c s , and f i n a l l y 3) t h a t the t a s k s s h o u l d be o f some i n t e r e s t o r a p p e a l t o the s t u d e n t . 3 . 3 . 1 Task One B r i e f l y , the s i t u a t i o n s i m u l a t e d i n t a sk one was as f o l l o w s : a l a k e i n which t h e r e a re a c o u p l e o f y o u n g s t e r s f i s h i n g i n a row-boat . The sub jec t s b e i n g i n t e r v i e w e d i s asked t o imagine h e r s e l f / h i m s e l f s t a n d i n g on a dock f o r the -71-purpose o f k e e p i n g t r a c k o f the good f i s h i n g spo t s found by the c o u p l e . T h i s p e r s o n i s w a i t i n g f o r some f r i e n d s who are coming t o f i s h l a t e r and he / she has to t e l l them where to l o c a t e the good f i s h i n g s p o t s . The f o l l o w i n g m a t e r i a l s were used i n the i n t e r v i e w s : - a 8 mm f i l m l o o p ( approx imate ly 4 m i n . long) showing the round t r i p o f a row-boat on a l a k e . The f i l m was produced e s p e c i a l l y f o r the s t u d y ; the l o c a t i o n was Deer l a k e , Burnaby , B r i t i s h C o l u m b i a . - a c a r d b o a r d model o f the l a k e (lm by l m ) , - a t o y boa t and t h r e e toy human f i g u r e s , - shee t s o f paper (27 cm by 21 cm) d e p i c t i n g a s chemat ic d iagram o f the l a k e . The c o n c r e t e prob lem s i t u a t i o n was p r e s e n t e d t o th e i n t e r v i e w e e u s i n g b o t h the f i l m l o o p and the c a r d b o a r d model t o accompany an o r a l d e s c r i p t i o n . A schemat ic d i agram o f the l a k e i s shown i n F i g u r e 3 . 5 . T h i s d i agram shows the s t a r t i n g l o c a t i o n o f the row-boat (the w h a r f ) , the permanent l o c a t i o n o f the i n t e r v i e w e e ( r e p r e s e n t e d by S on the d o c k ) , and s e v e r a l good f i s h i n g spo t s ( P ^ , P 2 , P 2 » and P ^ ) . The t a sk c o n s i s t e d i n q u e s t i o n i n g the subject- on how h e / s h e would d e s c r i b e the t r i p o f the row-boat and the l o c a t i o n s o f the f i s h i n g s p o t s . T h i s was judged t o be a s u i t a b l e ta sk t o f i n d o u t s u b j e c t s ' p r e 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 , whose c h a r a c t e r i s t i c s a re i n c l u d e d i n the network o f F i g u r e 3 . 2 . - 7 2 -F i g u r e 3 . 5 Diagram o f the l a k e , i n c l u d i n g the dock (where the s u b j e c t S s t a n d s ) , the w h a r f , the row-boa t , the f i s h i n g spo 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 f o l l o w i n g a s p e c t s o f t h i s network were l o o k e d f o r : ( i ) the need f o r a r e f e r e n c e p o i n t to l o c a t e an o b j e c t ; ( i i ) the need f o r a frame o f r e f e r e n c e to p r o v i d e a p r e c i s e l o c a t i o n o f an o b j e c t , ( t h i s i s shown through the k i n d o f c o o r d i n a t e s used by the s u b j e c t which can be q u a l i -t a t i v e o r q u a n t i t a t i v e ) ; ( i i i ) the need to p r o v i d e a-d i r e c t i o n o f the l o c a t i o n o f an o b j e c t , t h i s can be c o n -s i d e r e d p a r t o f a s p e c t ( i i ) . Task one was a l s o used t o f i n d out p r e c o n c e p t i o n s about some a s p e c t s o f v e c t o r d i s p l a c e m e n t , which were i n -c l u d e d i n the network o f F i g u r e 3 . 3 . P r e c o n c e p t i o n s about -73-the f o l l o w i n g a s p e c t s o f t h i s network were l o o k e d f o r : ( i ) the need f o r a way t o d e s c r i b e changes o f l o c a t i o n o f a moving o b j e c t (which can be g i v e n by the l e n g t h o f a s t r a i g h t l i n e j o i n i n g the two c o n s e c u t i v e l o c a t i o n s — i t s magnitude — and the a n g l e t h a t the s t r a i g h t l i n e forms w i t h a s e l e c t e d r e f e r e n c e l i n e — i t s d i r e c t i o n ) ; ( i i ) the need f o r a frame o f r e f e r e n c e and r e f e r e n c e p o i n t ; ( i i i ) the need f o r an o p e r a t i o n t o r e l a t e c o n s e c u t i v e changes o f l o c a t i o n ( a d d i t i o n o f v e c t o r d i s p l a c e m e n t s ) ; ( iv) the need t o r e a l i z e t h a t a change o f l o c a t i o n i s independent o f the pa th f o l l o w e d by the o b j e c t between two l o c a t i o n s ; and (v) the need f o r an o p e r a t i o n t o r e l a t e two d i f f e r e n t l o c a t i o n s o f an o b j e c t r e l a t i v e t o a chosen r e f e r e n c e p o i n t w i t h the r e s u l t a n t change o f l o c a t i o n ( 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 to p r o -duce the r e s u l t a n t d i s p l a c e m e n t ) . T h i s a s p e c t i s a l s o p a r t . o f the network o f v e c t o r p o s i t i o n , see F i g u r e 3 . 2 . Task one was a l s o s u i t a b l e to f i n d out the u n d e r -s t a n d i n g o f a d o l e s c e n t s u b j e c t s about p e r s p e c t i v e t a k e n . T h i s p e r s p e c t i v e a s p e c t was not c o n s i d e r e d 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 p r e s e n t s t u d y . I t was i n c l u d e d t o f i n d ou t i f s u b j e c t s a t these ages s t i l l have an e g o c e n t r i c c o n -c e p t i o n o f p e r s p e c t i v e . 3 . 3 . 2 P r o t o c o l o f Task One There were an average o f two q u e s t i o n s f o r each v e c t o r c h a r a c t e r i s t i c . Sentences i n p a r e n t h e s i s r e f e r r e d - 7 4 -to what the i n t e r v i e w e r (I) was d o i n g d u r i n g the i n t e r v i e w . The f i l m - l o o p and the c a r d b o a r d model were used i n the a p p r o p r i a t e t imes i n the i n t r o d u c t i o n ; fu r thermore the s u b j e c t s were encouraged t o make use o f the shee t s w i t h the s chemat ic d iagram o f the l a k e to e x p l a i n t h e i r v i e w s . The i n t r o d u c t i o n o f the p r o t o c o l , g i v e n be low, was r e p e a t e d a lmos t v e r b a t i m to each o f the s u b j e c t s i n t e r v i e w e d . I n t r o d u c t i o n o f P r o t o c o l : " Imagine t h a t you a re s t a n d i n g on a dock o f a ca lm l a k e and you suddenly see a c o u p l e o f y o u n g s t e r s g e t t i n g ready t o go f i s h i n g i n a row-boa t . S i n c e you are w a i t i n g t h e r e f o r a c o u p l e o f f r i e n d s who are coming l a t e r t o f i s h , you d e c i d e to c a r e f u l l y observe the t r i p o f the row-boa t . You a r e p a r t i c u l a r l y i n t e r e s t e d i n remembering the p l a c e s where they seem t o c a t c h more f i s h . You w i l l see a speeded up v e r s i o n o f t h i s f i s h i n g t r i p i n the ' f i l m ; the camera s h o o t i n g the f i l m was put on the same dock where you a r e supposed t o be s t a n d i n g so t h a t what you w i l l see i n the f i l m c o r r e s p o n d s to what you would see i f you a c t u a l l y were on the d o c k . Here i s a c a r d b o a r d model o f the l a k e which r e p r e s e n t s what you w i l l see i n the f i l m . I s e v e r y t h i n g c l e a r ? . . . Do you have any q u e s t i o n s ? I f n o t we can now see the f i l m " . The e n t i r e f i l m was shown, w h i c h c o n t a i n s s e v e r a l l o c a t i o n s o f the row-boat i n i t s round t r i p . The t r i p s t a r t e d and ended a t t h e whar f , the l o c a t i o n s o f the f i s h i n g spot s d e p i c t e d i n F i g u r e 3 . 5 were shown i n the f i l m . W h i l e the f i l m was b e i n g shown, the i n t e r v i e w e r was d e s c r i b i n g the t r i p and was p o i n t i n g o u t the l o c a t i o n s o f the f i s h i n g s p o t s . Immediate ly a f t e r the f i l m ended , the i n t e r v i e w e r made use o f the c a r d b o a r d model t o b e g i n the i n t e r v i e w . The q u e s t i o n s r e l a t e d to 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 d i s c u s s e d i n ta sk one are g i v e n be low. Ques t ions About Re ference 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)  P e r s p e c t i v e T a k e n , and Frame o f Reference (FR)  Ques t ion 1; (Items 1 and 2 o f LIVC) You a re h e r e on the d o c k , and as you saw i n the f i l m the f i r s t f i s h i n g s p o t found by the y o u n g s t e r s was more o r l e s s here (I l o c a t e d the t o y boa t i n the c o r r e s p o n d i n g spot on the c a r d b o a r d mode l ; P^ on F i g u r e 3 . 5 ) . How would you d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g s p o t t o your f r i e n d s ? Ques t ion 2 ; ( P e r s p e c t i v e taken) a) ( I f S uses a p a r t i c u l a r body o r p l a c e as a r e f e r e n c e p o i n t ) C o u l d you d e s c r i b e the l o c a t i o n o f t h a t f i s h i n g spot from o t h e r p l a c e s as w e l l ? What p l a c e s ? b) ( I f S d e s c r i b e s the l o c a t i o n from o t h e r p l a c e s ) I s the d e s c r i p t i o n o f t h i s l o c a t i o n the same o r d i f f e r e n t when i t i s done from o t h e r p l a c e s ? Q u e s t i o n 3: (Item 2 o f LIVC) C I f S has used no q u a n t i t y o r o n l y one , e i t h e r d i s t a n c e o r d i r e c t i o n , when answer ing q u e s t i o n 1) You have j u s t d e s c r i b e d the l o c a t i o n o f the f i r s t f i s h i n g s p o t . How c o u l d you make the d e s c r i p t i o n -76-o f t h a t l o c a t i o n ( f i r s t f i s h i n g spot) more p r e c i s e f o r your f r i e n d s ? Ques t ion 4: (Items 1 and 2 o f LIVC) The boa t c o n t i n u e d moving , i t went t o t h i s second f i s h i n g spo t (I l o c a t e d the t o y boat i n t h a t c o r r e s p o n d i n g s p o t ; P^ o n F i g u r e 3.5) How would you d e s c r i b e the l o c a t i o n o f t h i s second f i s h i n g spot t o your f r i e n d s ? Q u e s t i o n 5 : (Items 1 and 2 o f LIVC) A n o t h e r f i s h i n g spot was about here (I l o c a t e d the t o y boa t a t a p o i n t , P^ , about the same d i s t a n c e from S as P^ w a s ) . (a) How would you d e s c r i b e the 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 spo t t o your f r i e n d s ? (b) ( I f S uses o n l y d i s t a n c e i n h e r / h i s d e s c r i p t i o n ) The d i s t a n c e from the l a s t spo t t o you and from t h e f i r s t spot t o you are a lmos t the same. How c o u l d you d i s t i n g u i s h these two l o c a t i o n s ? Ques t ions About P e r s p e c t i v e s Q u e s t i o n 6; ( P e r s p e c t i v e t aken and Item 2 o f L I V C ) . a) From where you are (on the d o c k ) , c o u l d you a p p r o x i m a t e l y d e s c r i b e the p a t h f o l l o w e d by the boa t ? b) P l e a s e draw t h a t p a t h i n t h i s map o f the l a k e ( A f t e r S has drawn the path) Do you t h i n k t h i s i s the a c t u a l path? c) What do you need t o know i n o r d e r to have a b e t t e r a p p r o x i m a t i o n o f the p a t h t aken by the boat? - 7 7 -d) A f r i e n d o f yours i s by the t r e e and draws the path he sees f o r the boat, you compare both drawings, yours and h i s . Do you expect t o f i n d them the same o r d i f f e r e n t ? Q uestions About Displacements Q u e s t i o n 7 : (Item 3 of LIVC) The i n i t i a l l o c a t i o n o r s t a r t i n g p o i n t o f the boat was here b e s i d e the wharf ( P ^ ) . We can say t h a t the boat has had a change of l o c a t i o n when i t moved from the s t a r t i n g p o i n t (PQ ) to the f i r s t f i s h i n g spot ( P ^ ) . (a) How c o u l d you d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g spot w i t h r e s p e c t t o the s t a r t i n g spot? ( I f S uses d i s t a n c e i n h e r / h i s d e s c r i p t i o n ) . I s t h a t the d i s t a n c e a l o n g the path f o l l o w e d by the boat between the two spots? (b) Then, the boat moved from the f i r s t f i s h i n g spot t o the second spot (I moves the boat from P ^ to P ^ ) How c o u l d you d e s c r i b e the l o c a t i o n o f the second f i s h i n g spot w i t h r e s p e c t t o the f i r s t one? Q u e s t i o n 8: (Item 3 o f LIVC) ( I f S uses the l e n g t h o f the s t r a i g h t l i n e between two l o c a t i o n s ) . Does your d e s c r i p t i o n f o r a change of l o c a t i o n say anything about the path f o l l o w e d by the boat between the two l o c a t i o n s ? Q uestions About A d d i t i o n o f Displacements  Q u e s t i o n 9; (Item 4 of LIVC) Suppose you have e s t i m a t e d the d i s t a n c e s from the - 7 8 -s t a r t i n g p o i n t (PQ) to the f i r s t f i s h i n g spot (Pj_) t o be 100m, and from the f i r s t spot to the second one to be 150 m. I f your f r e i n d s were a t the w h a r f , how c o u l d you t e l l them t o go d i r e c t l y t o the second spot? Ques t ion 10: (Item 4 o f LIVC) Suppose you a l s o e s t i m a t e d the d i s t a n c e from the second spot t.o the t h i r d one t o be l p o m. How c o u l d you t e l l your f r i e n d s t o go d i r e c t l y from the s t a r t i n g p o i n t t o the t h i r d spot ? Ques t ions About 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  Q u e s t i o n 11 : 1 ( I t e m 5 o f LIVC) Suppose you have e s t i m a t e d the d i s t a n c e from where you a r e s t a n d i n g on the f i r s t f i s h i n g spot t o be 200 m, and a l s o from the dock to the second spo t to be 300 m. I f your f r i e n d s were a t the f i r s t s p o t . How c o u l d you t e l l them t o t o go d i r e c t l y t o the second spot? Q u e s t i o n 12 : (Item 5 o f LIVC) Suppose you a l s o e s t i m a t e d the d i s t a n c e from where you a re s t a n d i n g on the dock t o the t h i r d f i s h i n g spo t t o be 300 m. I f your f r i e n d s were a t the f i r s t s p o t . How c o u l d you t e l l them t o go d i r e c t l y t o the t h i r d spot? Immediate ly a f t e r f i n i s h i n g w i t h t a s k one the i n t e r v i e w , c o n t i n u e d w i t h t a sk two. The i n t e r v i e w e r t r i e d t o make a smooth t r a n s i t i o n between the two t a s k s by s a y i n g t h a t the same two f r i e n d s were now u s i n g a motor -boa t on the r i v e r . -79-3 ., 3 - 3 Task Two The s i t u a t i o n s i m u l a t e d i n t a sk two was the f o l l o w i n g : a r i v e r on which t h e r e i s a motor -boa t w i t h two p e o p l e , a p e r s o n on a b r i d g e , ( aga in the s u b j e c t i s asked to imagine t h a t h e / s h e i s the p e r s o n who i s o b s e r v i n g the movement o f the m o t o r - b o a t , which moves a g a i n s t and w i t h the c u r r e n t , and a c r o s s the r i v e r . D u r i n g the i n t e r v i e w the s u b j e c t was a sked t o d e s c r i b e and e x p l a i n the motor -b o a t ' s movements. The c o n c r e t e prob lem s i t u a t i o n was p r e s e n t e d t o the s u b j e c t s u s i n g the f o l l o w i n g m a t e r i a l s : — - a 8 mm f i l m l o o p ( a p p r o x i m a t e l y 4 m i n . long) showing the d i f f e r e n t movements o f the motor -boa t on t h e r i v e r . T h i s i s a _ f i l m produced f o r the H a r v a r d P r o j e c t P h y s i c s c o u r s e , - a c a r b b o a r d model o f the r i v e r and the b r i d g e (1 by 1 m) , w h i c h a l s o i n c l u d e s one dock on each s i d e o f the r i v e r , - a: t o y b o a t and t h r e e t o y human f i g u r e s , - shee t s o f paper (27 cm by 21 cm) d e p i c t i n g a s chemat i c d i agram o f the r i v e r (see F i g u r e 3 . 6 ) . An i n t r o d u c t i o n e x p l a i n i n g the t a sk was p r e s e n t e d t o t h e s u b j e c t , a f t e r w h i c h a s e m i - s t r u c t u r e d p r o t o c o l was u s e d . As b e f o r e , the s u b j e c t s were q u e s t i o n e d on how he / she would d e s c r i b e the movements o f the b o a t . T h i s was judged t o be a s u i t a b l e t a s k , to d e t e r m i n e the s t u d e n t s ' p reconcept ions about F i g u r e 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 the s u b j e c t s t a n d s ) , the two d o c k s , and the m o t o r - b o a t the v e c t o r c h a r a c t e r i s t i c s s i x t o t e n l i s t e d i n L I V C , a l l o f w h i c h a re i n c l u d e d i n the network o f average v e l o c i t y i n F i g u r e s 3 . 4 . P r e c o n c e p t i o n s about the f o l l o w i n g a s p e c t s o f t h i s network were examined: ( i ) the need f o r d e s c r i b i n g the m o t i o n o f an o b j e c t w i t h r e s p e c t t o any r e f e r e n c e body , (be i t s t a t i o n a r y o r i n m o t i o n ) ; and the need to d i s t i n g u i s h t h a t a s i n g l e m o t i o n can be d e s c r i b e d d i f f e r e n t l y i f v iewed - 8 1 -from two d i f f e r e n t r e f e r e n c e b o d i e s , ( p a r t i c u l a r l y i f one o f r e f e r e n c e body i s i n r e l a t i v e m o t i o n w i t h r e s p e c t t o the o t h e r ) ; ( i i ) the need f o r r e a l i z i n g t h a t an apparent s i n g l e mot ion can be formed by two o r more component movements; ( i i i ) the need f o r r e a l i z i n g t h a t a moving o b j e c t may be the r e s u l t o f the i n t e r a c t i o n o f two s imul t aneous movements . ( i . e . , the c o m p o s i t i o n o f v e c t o r q u a n t i t i e s ) ; ( iv ) the need f o r r e a l i z i n g t h a t even when two movements i n t e r a c t t o p r o -duce a s i n g l e r e s u l t a n t movement, each one o f them m a i n t a i n s i t 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 ( e . g . , magnitude and d i r e c t i o n ) d u r i n g the i n t e r a c t i o n ; and f i n a l l y (v) the need f o r r e a l i z i n g t h a t two o r more movements c o u l d a f f e c t a body a t the same t i m e . 3.3.4 P r o t o c o l o f Task Two The g e n e r a l format o f t h i s i n t e r v i e w p r o t o c o l i s s i m i l a r to t h a t o f t a sk o n e . I n t r o d u c t i o n o f P r o t o c o l ; " Imagine now t h a t you a r e s t a n d i n g on a b r i d g e over a wide and ca lm r i v e r . You a re h e r e on the b r i d g e (I show the s u b j e c t the p o i n t S on the c a r d b o a r d m o d e l , diagrammed i n F i g u r e 3.6) o b s e r v i n g your f r i e n d s , who are r i d i n g i n a m o t o r - b o a t . The speed o f the motor cannot be changed so the speed o f the boa t due t o the motor remains t h e same. Your f r i e n d s on t h e i r r i d e w i l l go a g a i n s t and w i t h the s t r e a m , and a l s o a c r o s s the r i v e r . As b e f o r e , you d e c i d e i t would be i n t e r e s t i n g t o observe and d e s c r i b e a l l -82-the moves t h a t the boa t w i l l make. T h i s f i l m w i l l show you how the boa t might move; the camera to take t h i s f i l m was p l a c e d on the b r i d g e where you are t o imagine y o u r s e l f t o be s t a n d i n g . So what you w i l l see i n the f i l m c o r r e s p o n d s to what you would see i f you a c t u a l l y were on the b r i d g e . We w i l l see p a r t o f the f i l m now and then we can d i s c u s s what we have seen u s i n g t h i s c a r d b o a r d m o d e l . A g a i n , I am i n t e r e s t e d i n your i d e a s and e x p l a n a t i o n s o f what you w i l l see i n the f i l m . I s e v e r y t h i n g c l e a r ? . . . Do you have any q u e s t i o n s ? . . . I f not we can see the f i r s t p a r t o f the f i l m " . In t h i s t a sk the f i l m was shown i n two p a r t s ; d i s c u s s i o n and q u e s t i o n s were asked a f t e r each p a r t . The c a r d b o a r d model was used as a c o n c r e t e r e f e r e n t when d i s c u s s i n g v a r i o u s q u e s t i o n s i r i the i n t e r v i e w . The q u e s t i n s r e l a t e d to 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 c o r r e s p o n d i n g t o t a sk two are p r e s e n t e d i n the f o l l o w i n g s e c t i o n s . Ques t ions About Reference Body f o r Moving O b j e c t s (RPM) Opening Q u e s t i o n : Have you had any e x p e r i e n c e w i t h motor -boa t s on l a k e s o r r i v e r s ? ( T h i s q u e s t i o n was a sked to o n l y f i n d ou t i f the s u b j e c t had any p r e v i o u s e x p e r i e n c e w i t h m o t o r - b o a t s ) . ' The f i r s t p a r t o f the f i l m was viewed showing, the water m o v i n g , the boa t g o i n g a g a i n s t the s tream and two . s t a t i o n a r y buoys b e s i d e the b r i d g e . -83-Ques t ion 1 3 * : (Item 6 o f LIVC) In the f i l m you saw the motor -boa t moving , (a) W i t h r e s p e c t t o what was the b o a t moving? ( I f S does no t u n d e r s t a n d t h i s q u e s t i o n , i t was pa raphra sed as f o l l o w s : How c o u l d you t e l l t h a t the boa t was moving? ( I f S ment ions t h a t the boa t was moving r e l a t i v e t o the b r i d g e o r s h o r e ) . (b) Was the boa t moving w i t h r e s p e c t t o the water? Why do you t h i n k so? ( I f S says t h a t the boa t was a l s o moving r e l a t i v e t o the moving .water) (c) Was the movement o f the boa t w i t h r e s p e c t t o the b r i d g e (or whatever s t a t i o n a r y body the s u b j e c t mentions) and the movement o f i t w i t h r e s p e c t t o the water the same o r d i f f e r e n t ? Why do you t h i n k so? Ques t ions About A n a l y s i s o f Components (AC) and C o m p o s i t i o n  o f V e l o c i t i e s (CV) Ques t ion 14: (Items 7 and 8 o f LIVC) a) Imagine t h a t the motor -boa t i s moving ups t ream a g a i n s t the c u r r e n t . (I moves the t o y boa t i n t h a t d i r e c t i o n , i n the c a r d b o a r d m o d e l ) . How would you d e s c r i b e i t s mot ion from the b r i d g e ? b) What d i f f e r e n t movements a re a f f e c t i n g the boa t when i t goes upstream? What a re t h e y ? *Numbering of que s t i o n s f o r task two continued on from task one. -84-c) Imagine now t h a t the motor -boa t i s moving downstream w i t h the c u r r e n t (I moves the toy-boa t i n the a p p r o p r i a t e d i r e c t i o n ) . How would you d e s c r i b e i t s m o t i o n from the b r i d g e ? d) What d i f f e r e n t movements are a f f e c t i n g the boa t when i t goes downstream? "What are they? Ques t ion 15; (Item 7 and 8 o f LIVC) y Imagine t h i s t ime t h a t your f r i e n d s i n the m o t o r -boa t want to c r o s s the r i v e r . They l e a v e from t h i s dock (dock 1, see F i g u r e 3.6) head ing t h a t way (I pu t s the boa t b e s i d e the dock 1 and e x p l a i n s the way how the boa t w i l l s t a r t h e a d i n g ) . a) How would you d e s c r i b e i t s mot ion from the b r i d g e ? P l e a s e , take the toy boa t and show me the k i n d o f movement you w i l l see u s i n g the * c a r d b o a r d m o d e l . b) Now c o u l d you draw t h a t pa th on t h i s d iagram o f the r i v e r ? (I hands S a shee t o f paper w i t h the s chemat i c d iagram on i t ) . Why do you t h i n k i t f o l l o w s the p a t h you have drawn? * A l m o s t a t t h e end o f t h e d a t a 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 a b o u t t h e i n d e p e n d e n c e 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 as 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 a c r o s s t h e r i v e r , w i l l t h e b o a t r e a c h t h e o t h e r s i d e h e a d i n g t h i s w a y 3 o r t h i s way^jk ?The 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 and t h e c a r d b o a r d model t o show t h e two h e a d i n g ways. T h i s 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 15a, and was a s k e d o f o n l y 5 s u b j e c t s . -85-c) What d i f f e r e n t movements are a f f e c t i n g the motor -boa t when i t goes a c r o s s .the r i v e r ? What are they? The p a r t o f the f i l m w i t h the motor -boa t c r o s s i n g the r i v e r i s shown. Ques t ion 16: a) You j u s t saw the motor -boa t moving a c r o s s the r i v e r i n the f i l m . Was the movement o f the boa t due t o the motor or due t o the c u r r e n t ? b) Suppose the speed o f the c u r r e n t i s 5 km/hr and the speed produced by the motor on the boa t i s 10 k m / h r . C o u l d you guess o r e s t i m a t e the speed o f the boat when i t was g o i n g a c r o s s the r i v e r as you saw i t i n the f i l m ? Ques t ions About S i m u l t a n e i t y o f Components  Ques t ion 17 : (Item 10 o f LIVC) ( I f S ment ions t h a t the boat , i s b e i n g a f f e c t e d by the speed o f the c u r r e n t and the speed "due t o the motor ; o t h e r w i s e the I sugges t s the e x i s t e n c e o f the two com-ponents) . A r e these movements, the movement o f the s t ream and the one due t o the motor , a f f e c t i n g the m o t i o n o f the boa t t o g e t h e r o r one a f t e r the o t h e r ? How c o u l d you ex-p l a i n your re sponses ? Ques t ions About Independence o f Magni tudes o f Components Ques t ion 18: (Item 9 o f LIVC) ( I f S ment ions t h a t the boa t i s b e i n g a f f e c t e d -86-s i m u l t a n e o u s l y by the speed o f the c u r r e n t and the speed due t o the m o t o r ) . When the boa t was j u s t c r o s s i n g the r i v e r , you s a i d t h a t b o t h the speed o f the c u r r e n t and the speed due to the motor were a f f e c t i n g the b o a t . Do you t h i n k t h a t the speed o f the boa t due t o the motor i s a f f e c t e d o r changed because o f the speed o f the c u r r e n t ? Why do you t h i n k so? Ques t ions About C o m p o s i t i o n o f V e l o c i t i e s  Q u e s t i o n 19 : (Item 8 o f LIVC) F i n a l l y , suppose your f r i e n d s want to c r o s s the r i v e r w i t h the m o t o r - b o a t and r e a c h dock 2. T h a t i s , l e a v e from dock 1 and end up a t dock 2. I s t h e r e any way they c o u l d do i t ? How? C o u l d you p l e a s e draw the pa th on t h i s shee t o f paper t h a t you might see i f t h e y f o l l o w e d your s u g g e s t i o n . 3 . 3 . 5 F u r t h e r D i s c u s s i o n o f I n t e r v i e w P r o t o c o l Three r e l e v a n t i s s u e s r e l a t e d t o the i n t e r v i e w p r o t o c o l a re d i s c u s s e d be low. These i s s u e s a r e : (1) the use o f d i f f e r e n t number o f q u e s t i o n s f o r some o f the v e c t o r c h a r a c t e r i s t i c s , (2) o c c a s i o n a l l y , use o f one q u e s t i o n to i d e n t i f y i n f e r r e d r u l e s f o r more than one v e c t o r c h a r a c t e r i s t i c , and (3) a summary t a b l e o f the i n t e r v i e w p r o t o c o l q u e s t i o n s a s s o c i a t e d w i t h each o f the v e c t o r c h a r a c t e r i s t i c s . - 8 7 -1. Each v e c t o r c h a r a c t e r i s t i c i n the i n t e r v i e w p r o t o c o l was addres sed w i t h a v a r y i n g 'number1 o f q u e s t i o n s (see T a b l e 3 . 1 ) . The main r e a s o n f o r t h i s i s due to the n a t u r e o f the v e c t o r c h a r a c t e r i s t i c i t s e l f and on the t a s k s chosen f o r the s t u d y . F o r example , the n a t u r e o f a d d i t i o n o f d i s p l a c e m e n t s i n t a sk one a l l o w e d f o r a t l e a s t two q u e s t i o n s f o r t h i s v e c t o r c h a r a c t e r i s t i c — each one t r y i n g t o uncover the s u b j e c t s ' i d e a s w i t h r e s p e c t to a s i t u a t i o n t h a t r e q u i r e s add ing two g i v e n d i s p l a c e m e n t s . To de te rmine whether a sub jec t possessed a c o n s i s t e n t v iew about a d d i t i o n o f d i s p l a c e m e n t s s e v e r a l o t h e r q u e s t i o n s , each t ime u s i n g a d i s t i n c t p a i r o f d i s p l a c e m e n t s , c o u l d have been u s e d . However, due to the t ime l i m i t a t i o n s 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 two q u e s t i o n s were a s k e d . On the o t h e r hand , c o n s i d e r i n g the n a t u r e o f the s i m u l t a n e i t y c h a r a c t e r i s t i c o f two v e l o c i t y components , e i t h e r the s u b j e c t sees the two v e l o c i t i e s a f f e c t i n g the boa t ( i n t a sk two) a t the same/time t3r.-Tme.-after the o t h e r . There a re o n l y the se two p o s s i b i l i t i e s , and b o t h a re c o v e r e d i n t h e same q u e s t i o n . In the t a s k s used i n the s tudy t h e r e was no o t h e r way t o check f o r t h i s v e c t o r c h a r a c t e r i s t i c . 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 the t a s k s chosen f o r the s tudy i n some i n s t a n c e s a l l o w e d f o r a l a r g e r number o f q u e s t i o n s t o be asked about a s p e c i f i c v e c t o r c h a r a c t e r i s t i c . 2. A t l e a s t f o u r q u e s t i o n s were used to uncover the s u b j e c t ' s p r e c o n c e p t i o n o f two v e c t o r c h a r a c t e r i s t i c s . -88-These q u e s t i o n s were broad i n scope ( i . e . , Q s . l , 4:, 5, 13d) such t h a t the re sponses t o them p e r m i t t e d i n f e r r e d r u l e s t o be a s s i g n e d f o r two v e c t o r c h a r a c t e r i s t i c s . T h i s was the case f o r the i n t r o d u c t o r y q u e s t i o n s f o r each one o f t a s k , q u e s t i o n 1, 4, and 5 i n t a sk one ; and q u e s t i o n 13d i n t a sk two. T h i s i s a l e g i t i m a t e approach bu t i t i s recommended t h a t the p r e c o n c e p t i o n s o b t a i n e d through these broad q u e s t i o n s be checked by u s i n g p r o v i n g q u e s t i o n s , w h i c h s h o u l d be more s p e c i f i c i n s cope . F o r i n s t a n c e , p r e c o n c e p t i o n s about .frame o f r e f e r e n c e were o b t a i n e d from q u e s t i o n s 1, 4, and 5; these q u e s t i o n s were a l s o s u i t a b l e f o r a s c e r t a i n i n g p o s s i b l e p r e c o n c e p t i o n s about r e f e r e n c e p o i n t . These p r e c o n c e p t i o n s were then c r o s s - c h e c k e d by a n a l y z i n g the s u b j e c t ' s re sponse to q u e s t i o n 3. 3. The f o l l o w i n g t a b l e p r e s e n t s a l i s t o f q u e s t i o n s from the i n t e r v i e w p r o t o c o l which was used t o o b t a i n the i n f e r r e d r u l e s f o r each o f the v e c t o r c h a r a c t e r -i s t i c s . T a b l e 3.1 L i s t o f I n t e r v i e w P r o t o c o l Ques t ions f o r each V e c t o r C h a r a c t e r i s t i c V e c t o r C h a r a c t e r i s t i c Number o f Quest ions i n P r o t o c o l Task One: 1. R e f e r e n c e P o i n t f o r 1 ,4 ,5 S t a t i o n a r y B o d i e s (RPS) 2. Frame o f Re fe rence (FR) 1 , 3 , 4 , 5 a , 5 b , 6 c 3. D i s p l a c e m e n t (D) 7a ,7b ,8 4. A d d i t i o n o f D i s p l a c e m e n t s 9,10 (AD) -89-T a b l e 3.1 - c o n t i n u e d V e c t o r C h a r a c t e r i s t i c Number o f Ques t ions i n p r o t o c o l 5. S u b t r a c t i o n o f V e c t o r s 11,' 12 P o s i t i o n (SVP) ( P e r s p e c t i v e Taken) 2a, 2b, 6d Task Two: 6. Re fe rennce P o i n t f o r 13a, 13b, 13c Moving O b j e c t s (RPM) 7. A n a l y s i s o f Components 14b, 14d, 15c (AC) 15a, 15b, 16a, 8. C o m p o s i t i o n o f V e l o c i t i e s 14a, 14c , (CV) 16b, 19 9. Independence o f M a g n i - 18 tudes o f Components (IMC) . 10. S i m u l t a n i e t y o f Components 17 (SC) 3.4 Methods o f Data C o l l e c t i o n Methodology : C l i n i c a l I n t e r v i e w Due t o the f a c t t h a t one o f the main g o a l s o f t h i s s tudy was to uncover s t u d e n t s ' p r e c o n c e p t i o n s o f s e l e c t e d p h y s i c a l q u a n t i t i e s , i t was thought t h a t a method t h a t w i l l " d i g " d e e p l y i n s u b j e c t s ' t h i n k i n g would be more adequate . A k i n d o f P i a g e t i a n c l i n i c a l i n t e r v i e w would a c c o m p l i s h t h a t a i m . The inadequacy o f t r a d i t i o n a l t e s t i n g p r o c e d u r e s ( i . e . , w r i t t e n q u e s t i o n n a i r e s o r t e s t s ) has been noted by s e v e r a l r e s e a r c h e r s i n t e r e s t e d i n c o n c e p t u a l deve lopment . F o r i n -s t a n c e , E a s l e y reminds us t h a t : -90-P i a g e t p o i n t e d out i n 1929 t h a t t e s t s cannot p r o v i d e enough i n f o r m a t i o n to d e c i d e what s t r u c t u r e s a r e i n v o l v e d 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 o f menta l a c t i v i t y w i t h o u t g r a s p i n g the p s y c h o l o g i c a l o p e r a t i o n s i n t h e m s e l v e s ' (p.281) . . . . The o v e r r i d i n g ques t f o r r e l i a b i l i t y w h i c h appears to be the dominant c o n c e r n i n the 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 o f 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 pa rad igm, i s n e c e s s a r i l y a slow and nonmechanica l p r o c e d u r e . I t b e g i n s i n s u b j e c t i v e , bu t h o p e f u l l y e d u c a t e d , judgements and moves toward o b j e c t i v i t y as i t a t t a i n s completeness i n a c c o u n t i n g f o r the t o t a l p r o t o c o l . ( E a s l e y , 1974, p.289) P i n e s (1980) comments f u r t h e r t h a t t h e r e i s s t i l l some d i s s a t i s f a c t i o n w i t h s o p h i s t i c a t e d t e s t i n g and a s s o c i a t i o n a l i s t i c methods f o r e v a l u a t i n g and d e s c r i b i n g c o g n i t i v e s t r u c t u r e s . He c i t e s as an example S tewar t (1979) , and says t h a t t h e r e i s s u f f i c i e n t e v i d e n c e o f a move towards c a p i t a l i z i n g upon p e r s o n a l , f l e x i b l e i n t e r v i e w t e c h n i q u e s ( e . g . , P i n e s , 1979; P i n e s e t ajL, 1978) . P i n e s goes on t o say t h a t "The t r a n s i t i o n , a l t h o u g h slow and d i f f i c u l t , w i l l , I b e l i e v e , u l t i m a t e l y r e s u l t i n a d i s t i n c t advancement f o r the f i e l d o f s c i e n c e e d u c a t i o n " . ( P i n e s , 1981, p.361) One o f the problems o f the c l i n i c a l method ment ioned by P i n e s (1980) i s t h a t o f non-response by the s u b j e c t . He c l a i m s t h i s prob lem i s g e n e r a l l y more acu te w i t h younger c h i l d r e n , b u t no d i f f i c u l t i e s o f t h i s n a t u r e were e n -c o u n t e r e d w i t h the a d o l e s c e n t s u b j e c t s i n t h i s s t u d y . S i n c e no n o n - r e s p o n d e r s were p r e s e n t i n the sample, the i n t e r n a l -91-and e x t e r n a l v a l i d i t y o f the s tudy i s i n c r e a s e d ( P i n e s , 1980, p . 362) . A c l i n i c a l i n t e r v i e w i s l i t e r a l l y a f a c e - t o - f a c e m e e t i n g . I t has been d e f i n e d as "a c o n v e r s a t i o n d i r e c t e d to a d e f i n i t e purpose o t h e r than s a t i s f a c t i o n i n the c o n v e r s a -t i o n i t s e l f " , (Bingham, e t ajL, 1959, p . 3) o r what c o u l d be c a l l e d a " p r o f e s s i o n a l c o n v e r s a t i o n " ( G a r r e t , 1972, p . 5 ) . I t s c h i e f g o a l i s to a s c e r t a i n the n a t u r e and e x t e n t o f an i n d i v i d u a l ' s knowledge about a p a r t i c u l a r domain by i d e n t i -f y i n g the r e l e v a n t c o n c e p t i o n s the s u b j e c t h o l d s and the p e r c e i v e d r e l a t i o n s h i p s among those c o n c e p t i o n s . The c l i n i c a l i n t e r v i e w has two i m p o r t a n t f e a t u r e s . One i s u s i n g the i n t e r v i e w t o e l i c i t and c o l l e c t s p e c i f i c k i n d s o f i n f o r m a t i o n . The second i s a l l o w i n g the s u b j e c t t o t ake the l e a d , i n d u c i n g h e r / h i m to t a l k more and more f r e e l y . The o b j e c t i o n s t o the c l i n i c a l method as b e i n g u n s t r u c t u r e d , s u b j e c t i v e , and prone t o e x p e r i m e n t a l b i a s can 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 i n t e r v i e w i n a d -v a n c e . The a n a l y s i s o f the v e c t o r 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 s e l e c t e d f o r t h i s s t u d y , u s i n g R a t i o n a l Task A n a l y s i s , p r o v i d e d enough i n f o r m a t i o n to p r e p a r e r e l e v a n t and i n dep th q u e s t i o n s w i t h r e s p e c t t o the s e l e c t e d c o n t e n t - a r e a . I t was a l s o expec ted t h a t t h i s r e s e a r c h e r ' s y e a r s o f e x p e r i e n c e u s i n g the i n d i v i d u a l i n t e r -view as a da t a c o l l e c t i o n methodology would r e s u l t i n an o b j e c t i v e and i n s i g h t f u l i n t e r p r e t a t i o n o f the da t a g e n e r a t e d -92-by the i n t e r v i e w s . F i n a l l y , the use o f v i d e o t a p e f o r r e c o r d i n g the i n t e r v i e w s a l l o w e d f o r a c c u r a t e t r a n s c r i p t i o n and added a degree o f o b j e c t i v i t y to the d a t a . 3 .4 .1 R e l i a b i l i t y and V a l i d i t y i n C l i n i c a l I n t e r v i e w Any r e s e a r c h p r o j e c t must t ake i n t o c o n s i d e r a t i o n the problems c o n c e r n i n g the e x t e r n a l and i n t e r n a l v a l i d i t y o f t h e da t a c o l l e c t i o n methodology . I n t e r n a l and e x t e r n a l v a l i d i t y , a re o b v i o u s l y r e l a t e d s i n c e i f i n t e r n a l v a l i d i t y i s l a c k i n g , e x t e r n a l v a l i d i t y i s no l o n g e r an i m p o r t a n t c o n s i d e r a t i o n . The r e l i a b i l i t y o f a measurement p r o -cedure i s a l s o c o n c e r n e d w i t h a c c u r a c y and so i t i s a l s o r e l a t e d to the i s s u e o f i n t e r n a l v a l i d i t y . In o t h e r words , i f a measurement i s not a c c u r a t e o r cannot be r e p r o d u c e d , i t does no t make much sense t o de te rmine how w e l l i t i s measur ing what i t i s supposed to measure. However, an i n -s trument can measure something v e r y " x e T i a b ' r y , b u t t h i s does no t i m p l y t h a t i t i s measur ing what i t i s supposed to measure i . e . , i t may s t i l l have low v a l i d i t y . One can f i n d t e s t s w i t h h i g h r e l i a b i l i t y but the c o n s t r u c t b e i n g measured i s e i t h e r unknown o r v e r y t r i v i a l . R e f e r r i n g to 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 c l i n i c a l i n t e r v i e w methodo logy , P i n e s e t al^ (1978) have s t a t e d : " t h a t the c l i n i c a l i n t e r v i e w does no t produce i d e n t i c a l r e s u l t s from one i n t e r v i e w to the next w i t h the same s u b j e c t , even w i t h th e same i n t e r v i e w e r " . However, i t -93-i s t h e i r c o n t e n t i o n t h a t : " t h e c l i n i c a l i n t e r v i e w i s r e -l i a b l e i n t h a t u n i f o r m i t y o f knowledge assessment i s p o s s i b l e and t r a i n e d i n t e r v i e w e r s w i l l show [ i n t e r r a t e r r e l i a b i l i t y ] c o r r e l a t i o n s i n the assessment o f r = .9 o r b e t t e r " An i m p o r t a n t a s p e c t o f r e l i a b i l i t y assessment w i t h c l i n i c a l i n t e r v i e w d a t a , then a c t u a l l y r e l a t e s to the i n -t e r v i e w e r judgments about the d a t a ; t h a t i s , what i n f o r m a -t i o n o r segment o f the s u b j e c t ' s a c t i v i t y i s c o n s i d e r e d as e v i d e n c e f o r the u n d e r s t a n d i n g o f a c e r t a i n c o n t e n t - a r e a . The prob lem i s to o b t a i n the e x a c t word ing o f the s u b j e c t ' s r e s p o n s e s ; from which the p a r t s . t h a t e l i c i t the s u b j e c t ' s u n d e r s t a n d i n g (or h i s / h e r c o n c e p t i o n ) about the p r o b l e m a t i c s i t u a t i o n under d i s c u s s i o n can be s e l e c t e d . T h i s i s s u e can be s o l v e d , i n p a r t , by v i d e o t a p i n g or a u d i o - t a p i n g the i n t e r v i e w , so the a n a l y s t can i d e n t i f y a t what p o i n t the s u b j e c t seemed to have used the language o r e x h i b i t e d the b e h a v i o u r t h a t r e p r e s e n t s a givenMCDnEsptiDn. However, the s e l e c t i o n and i n t e r p r e t a t i o n o f i n t e r v i e w d a t a s t i l l remains v e r y much a m a t t e r o f good judgment on the p a r t o f the a n a l y s t . R e l i a b i l i t y on these judgments o f da t a can be e s t i m a t e d , a c c o r d i n g t o P i n e s e t a l . (1978) , by a s k i n g o t h e r e x p e r i e n c e d i n t e r v i e w e r s t o go through the same i n t e r v i e w d a t a — i n w r i t t e n o r taped form — making the same t y p e s o f judgments . Agreement o r d i sagreement o f judgments p r o v i d e s e s t i m a t e s o f r e l i a b i l i t y . In the p r e s e n t s t u d y , an independent a n a l y s t went through the i n t e r v i e w d a t a o f 6 ' r andomly s e l e c t e d s u b j e c t s (6 o f 20) f o r 3 randomly s e l e c t e d v e c t o r c h a r a c t e r i s t i c s ( 3 o f 10) . The i n v e s t i g a t o r e x p l a i n e d the a n a l y t i c p r o c e d u r e to the a n a l y s t . The t r a n s -c r i b e d i n t e r v i e w p r o t o c o l s f o r one s u b j e c t was used t o d e -mons t ra te the a n a l y t i c . p r o c e d u r e . R e s u l t s o f the independent a n a l y s t showed an agreement o f 9 4.4 p e r c e n t r e g a r d i n g the c l a s s i f i c a t i o n o f the c o n s i s t e n t p r e c o n c e p t i o n s . V a l i d i t y judgments i n c l i n i c a l i n t e r v i e w d a t a a t tempt t o ensure t h a t the a n a l y s t ' s c l a i m s a r e a r e a s o n a b l e i n t e r p r e t a t i o n o f the s u b j e c t ' s c o g n i t i v e s t r u c t u r e . A g a i n , one way o f c h e c k i n g the i n t e r p r e t a t i o n o f i n t e r v i e w d a t a i s t h r o u g h the use o f e x t e r n a l a n a l y s t s . A n o t h e r r o u t e , employed i n t h i s s t u d y , was to ask the s u b j e c t s e v e r a l q u e s t i o n s about the same a s p e c t o f the c o n t e n t - a r e a ( i . e . , same v e c t o r c h a r a c t e r i s t i c ) . In t h i s way the a n a l y s t i s a b l e t o c a r r y out a , t y p e o f c o n s i s t e n c y check f o r p a r t i c u l a r c o n c e p t i o n s h e l d by a s u b j e c t . . 3 . 4 . 2 I n t e r v i e w Format B e f o r e s t a r t i n g the i n t e r v i e w i t s e l f , the e x p e r i -menter t r i e d to have an i n f o r m a l c h a t w i t h the s t u d e n t , a s k i n g h e r / h i s name, age , and i n t e r e s t s . A f t e r the s u b j e c t seemed t o f e e l c o m f o r t a b l e w i t h the s i t u a t i o n , the a u t h o r e x p l a i n e d i n g e n e r a l terms the purpose o f the p r e s e n t r e -s e a r c h and g e n e r a l n a t u r e o f the i n t e r v i e w . (For example , he / she was a s s u r e d t h a t the i n t e r v i e w was no t a t e s t b u t t h a t the 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 i n t h e i r answers ^95^ and the way t h a t they thought about problems l i k e t h i s so t h a t 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 and h e l p f u l s c i e n c e m a t e r i a l s f o r use i n the s c h o o l s ) . A f t e r t h i s c h a t , the c a r d b o a r d model o f t a sk one was shown and the i n t r o d u c t i o n o f t h i s t a sk was p r e s e n t e d . When i t was c l e a r t h a t the s u b j e c t u n d e r s t o o d the prob lem s i t u a t i o n , the use o f the s e m i - s t r u c t u r e d p r o t o c o l t o g e t h e r w i t h the f i l m p r o j e c t i o n began . The t r a n s i t i o n to t a sk two was made smooth ly , showing f i r s t the c a r d b o a r d model and then p r e s e n t i n g the ta sk i n -t r o d u c t i o n . A g a i n , i n t r o d u c t o r y q u e s t i o n s and the f i l m p r o j e c t i o n were i n i t i a t e d when the s u b j e c t seemed t o u n d e r -s t and the prob lem s i t u a t i o n . The i n t e r v i e w s were c a r r i e d out i n a d e s i g n a t e d room o f the s c h o o l a t t e n d e d by the s u b j e c t s . The t ime r e q u i r e d to complete the e n t i r e i n t e r v i e w ranged from 4 5 to 50 m i n u t e s . V i d e o t a p e equipment was s e t up p r i o r to the s u b j e c t e n t e r i n g the room. The s u b j e c t was asked i f h e / s h e minded h a v i n g the i n t e r v i e w t a p e d . A t the end o f the i n t e r v i e w , the s u b j e c t was thanked f o r h e r / h i s c o o p e r a t i o n . 3 .4 .3 Sample o f S u b j e c t s P i a g e t (1973) , i n h i s s t u d i e s o f s t u d e n t s ' u n d e r -s t a n d i n g o f f o r c e s , found t h a t h i s s u b j e c t s do no t d e v e l o p c o n c e p t i o n s t h a t c o u l d be d e f i n e d as v e c t o r i a l u n t i l a f t e r age 12. A g u i r r e (1978) d i d no t f i n d c l e a r v e c t o r i a l c o n -c e p t i o n s u n t i l age 14 o r l a t e r . These r e s u l t s sugges ted t h a t i t may be n o n p r o d u c t i v e to s e l e c t s u b j e c t s younger than 13 y e a r s o f age f o r t h i s s t u d y . A l s o , s i n c e the p h y s i c a l c o n c e p t s i n c l u d e d i n t h i s s tudy are u s u a l l y t aught i n grade 11, the t a r g e t p o p u l a t i o n s e l e c t e d f o r t h i s s tudy was grade 10 (ages 15 t o 17 y e a r s ) . F o r Phase One o f the s tudy the sample c o n s i s t e d o f 20 s u b j e c t s — e l e v e n males and n i n e f ema le s . Two i m p o r t a n t a s p e c t s were c o n s i d e r e d i n the s e l e c t i o n o f s u b j e c t s : (1) t h a t they were a l l r e l a t i v e l y h e a l t h y and w e l l - a d j u s t e d , and (2) they had not been f o r m a l l y t aught the c o n c e p t s i n -c l u d e d i n the s t u d y . These two c r i t e r i a were checked w i t h the s t u d e n t s ' t e a c h e r s . The s u b j e c t s f o r the sample were s e l e c t e d a t a v o l u n t a r y b a s i s and a f t e r they had o b t a i n e d c o n s e n t from t h e i r p a r e n t s . S c i e n c e t e a c h e r s i n two j u n i o r h i g h s c h o o l s o f C o q u i t l a m S c h o o l D i s t r i c t ( S i r F r e d e r i c k B a n t i n g J u n i o r Secondary S c h o o l and Mary H i l l J u n i o r Secondary School ) e x p l a i n e d the need o f v o l u n t e e r s f o r a r e s e a r h s tudy t o t h e i r c l a s s e s , the sub j e e l s ;:who d e m o n s t r a t e d i n t e r e s t were g i v e n the p a r e n t c o n s e n t l e t t e r s . Among the s u b j e c t s who o b t a i n e d p e r m i s s i o n , c o n s i d e r i n g the c o n s t r a i n t o f s ampl ing 50 p e r c e n t boys and 50 p e r c e n t g i r l s , 20 were randomly s e l e c t e d by the t e a c h e r s . A c t u a l l y , f o r t h i s phase o f a r e s e a r c h program c n c o n c e p t development the use o f a t y p i c a l sample i s s u f f i c i e n t s i n c e the s e a r c h i s f o r i n d i v i d u a l s ' c o n c e p t i o n s about a p a r t i c u l a r c o n t e n t - a r e a . In o t h e r words , the aim o f t h i s phase o f the s tudy was no t t o s t a t i s t i c a l l y i n f e r t h a t a l l s t u d e n t s i n some d e f i n e d p o p u l a t i o n o f grade 10 - 9 7 -have s i m i l a r p r e c o n c e p t i o n s about v e c t o r c h a r a c t e r i s t i c s as those po s se s sed by the s u b j e c t s i n . t h e , « ? s a m p l e . R a t h e r , the aim was t o o b t a i n da t a which would p e r m i t the f o r m u l a t i o n o f a s e t o f t e n t a t i v e hypotheses r e g a r d i n g the p r e c o n c e p t i o n s ahout v e c t o r c h a r a c t e r i s t i c s h e l d by s t u d e n t s a t t h i s age l e v e l . These hypotheses were then s u b m i t t e d to f u r t h e r c r o s s - c h e c k i n g i n Phase Two- o f - t h e s t u d y . - 9 8 -CHAPTER FOUR RESULTS OF PHASE ONE OF THE STUDY 4.0 I n t r o d u c t i o n The c h a p t e r i s d i v i d e d i n t o the f o l l o w i n g seven major s e c t i o n s . The s e c t i o n s r e f e r s t o : 1) the d e s c r i p t i o n o f the methods o f d a t a a n a l y -s i s and examples o f those methods; 2) the d e s c r i p t i o n o f the format o f r e s u l t s p r e s e n t a t i o n ; 3) the f i n d i n g s o f phase one w h i c h a re p r e s e n t e d i n a l i s t o f i n f e r r e d r u l e s and r u l e - m o d e l s f o r a l l v e c t o r c h a r a c t e r i s t i c s ; 4) 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 i n the sample ; 5) the d e s c r i p t i o n o f an a n a l y t i c a l s t ep to f u r t h e r c a t e g o r i z e the i n t e r v i e w d a t a ; 6) the d e s c r i p t i o n o f r e s u l t s o b t a i n e d u s i n g t h i s type o f c a t e g o r i z a t i o n ; 7) the c o n c l u s i o n s o f phase one a r e p r e s e n t e d a c c o r d i n g to the r e s e a r c h q u e s t i o n s b e i n g addre s sed by t h i s phase . - 9 9 -4.1 D a t a A n a l y s i s The d a t a a n a l y s i s o f phase one was done i n f o u r s t e p s . The f i r s t one c o n s i s t e d o f l o o k i n g f o r p a t t e r n s o r c o n s i s t e n t p r e c o n c e p t i o n s i n the re sponses o f each s u b j e c t to a l l q u e s t i o n s f o r each v e c t o r c h a r a c t e r i s t i c . The second and t h i r d s teps c o r r e s p o n d e d t o the p r o c e s s o f e x p r e s s i n g the c o n -s i s t e n t p r e c o n c e p t i o n s i n terms o f i n f e r r e d r u l e s and c a t e g o r i z i n g these i n f e r r e d r u l e s a c c o r d i n g to a contemporary P h y s i c s p e r s p e c t i v e . The f o u r t h s t e p e n t a i l e d the t a sk o f e x p r e s s i n g the i n f e r r e d r u l e s i n l e s s c o n t e x t u a l and more g e n e r a l terms t o produce a ' r u l e - m o d e l ' f o r each v e c t o r c h a r a c t e r i s t i c . 4 .1 .1 I n f e r r e d R u l e s : The s e a r c h f o r c o n s i s t e n t p r e c o n c e p -t i o n s i n the s u b j e c t s ' re sponses t o the q u e s t i o n s on 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 was done by means o f e m p i r i c a l t a sk a n a l y s i s . T h i s was b a s i c a l l y an a n a l y s i s and i n t e r p r e -t a t i o n o f a s t u d e n t ' s per formance on a p a r t i c u l a r a s p e c t o f th e t a s k ; the g o a l was to uncover the s u b j e c t ' s p r e c o n c e p t i o n o f d e a l i n g w i t h t h a t a s p e c t . I f a s u b j e c t appeared t o be u s i n g a common " s t r a t e g y " t o cope w i t h more than h a l f o f the q u e s t i o n s r e l a t e d t o 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 , i t was assumed t h a t the s u b j e c t h e l d a c o n s i s t e n t p r e c o n c e p t i o n about t h a t p a r t i c u l a r a s p e c t . The pre sence o f a c o n s i s t e n t p a t t e r n o f r e s p o n s e s c o u l d mean t h a t s u b j e c t s h o l d p a r t i c u l a r -100-k i n d s o f i d e a s which become e x p l i c i t when they a re f aced w i t h c e r t a i n t y p e s o f c o g n i t i v e t a s k s . I f t h i s were the c a s e , i t c o u l d be h y p o t h e s i z e d t h a t a s u b j e c t pos se s se s some type o f r u l e t o d e a l w i t h these s i t u a t i o n s . T h i s i s a c t u a l l y an i n f e r e n t i a l s t ep because t h e r e i s no s t r a i g h t f o r w a r d way t o know i f a s u b j e c t i s r e a l l y u s i n g these r u l e s ; hence they a r e r e f e r r e d t o as ' i n f e r r e d r u l e s ' i n t h i s s t u d y . In summary, a c o n s i s t e n t p r e c o n c e p t i o n used by a s u b j e c t to d e a l w i t h a p a r t i c u l a r a s p e c t ( i . e . , 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 ) i n one -of the t a s k s was expre s sed as an i n -f e r r e d r u l e . I t c annot be c l a i m e d t h a t a s u b j e c t w i l l use an i n -f e r r e d r u l e f o r a s p e c i f i c v e c t o r c h a r a c t e r i s t i c ( e . g . , r e f e r e n c e p o i n t ) f o r an e x p e r i m e n t a l s i t u a t i o n d i f f e r e n t from the one d e p i c t e d i n the t a s k , even when the q u e s t i o n s be r e -l a t e d t o the same v e c t o r c h a r a c t e r i s t i c ( e . g . , r e f e r e n c e p o i n t ) . To make t h i s c l a i m i t would be n e c e s s a r y t o c r e a t e s e v e r a l t a s k s s u i t a b l e f o r t h a t v e c t o r c h a r a c t e r i s t i c , and d e t e r m i n e whether the s u b j e c t uses the s a m e " i n f e r r e d r u l e " t o cope w i t h the t a s k s . In: t h i s s ense , the p r e s e n t r e s e a r c h i s v e r y c o n t e x t u a l ; the i n f e r r e d r u l e s are o n l y a p p l i c a b l e to the c o n t e x t o f the t a s k s ( i . e . , boa t s on l a k e s and r i v e r s ) . The d e f i n i t i o n and the o p e r a t i o n a l d e f i n i t i o n o f i n f e r r e d r u l e s are r e p e a t e d here (see s e c t i o n o f d e f i n i t i o n o f terms i n C h a p t e r One) t o f a c i l i t a t e the comprehens ion o f the d e v e l o p e d examples i n the f o l l o w i n g s e c t i o n . -101-D e f i n i t i o n o f I n f e r r e d R u l e : An i n f e r r e d r u l e i s a c o n s i s t e n t p r e c o n c e p t i o n t h a t a s u b j e c t appears t o be u s i n g i n a g i v e n s i t u a t i o n t o d e a l w i t h a p a r t i c u l a r q u e s t i o n o r p rob lem posed by the i n t e r v i e w e r . (An example o f the use o f t h i s d e f i n i t i o n : i n ta sk one , the e x p e r i m e n t a l s i t u a t i o n was a row-boat moving on a l a k e ; a g i v e n s i t u a t i o n was the l o c a t i o n o f s e v e r a l f i s h i n g s p o t s ; and a c o n s i s t e n t p r e -c o n c e p t i o n h e l d by some s u b j e c t s was the use o f s e v e r a l b o d i e s as r e f e r e n c e t o l o c a t e the s p o t s ) . O p e r a t i o n a l D e f i n i t i o n o f I n f e r r e d R u l e : A r u l e i s i n f e r r e d i f a s u b j e c t uses the same b a s i c p r e c o n c e p t i o n to d e a l w i t h more t h a n h a l f o f the q u e s t i o n s r e g a r d i n g the same v e c t o r c h a r a c t e r i s t i c . 4.1.2 An example o f the D e r i v a t i o n o f I n f e r r e d Rules f o r the 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 : Re ference P o i n t  f o r S t a t i o n a r y B o d i e s Three c o n s i s t e n t p r e c o n c e p t i o n s and t h e r e f o r e t h r e e r u l e s were i n f e r r e d from the i n t e r v i e w d a t a f o r R e f e r e n c e 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) . Examples o f how each one o f the i n f e r r e d r u l e s was i d e n t i f i e d a re now p r e s e n t e d . The q u e s t i o n s about Re fe rence P o i n t used i n the i n t e r v i e w p r o t o c o l were as f o l l o w s : Ques t ion 1 : How would you d e s c r i b e the 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 spo t t o y o u r f r i e n d s ? Q u e s t i o n 4 : How would you d e s c r i b e the l o c a t i o n o f t h i s second f i s h i n g spot t o your f r i e n d s ? -102-Ques t ion 5 : How c o u l d you d e s c r i b e the 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 spo t t o your f r i e n d s ? Case One: A s u b j e c t u s i n g more than one r e f e r e n c e p o i n t t o l o c a t e each one o f the f i s h i n g s p o t s . The f o l l o w i n g re sponse s a re from T r a c y (15 y e a r s ) : (Q . l ) " I w i l l p r o b a b l y p o i n t them and say so many f e e t t h a t way . . . I can say 40 f e e t from the dock . . . i t w i l l be about 20 f e e t from the wharf t o my r i g h t . . . R e s e a r c h e r ' s Comments: S u b j e c t was u s i n g two b o d i e s as r e f e r e n c e : the dock and the whar f . (Q.4) " . . . I c o u l d say 20 f e e t pass the r e d one (buoy o r r e d thumb tack p l a c e d a t the f i r s t f i s h i n g spo t on the model) . . . from h e r e (the dock) about 100 f e e t o r something l i k e t h a t " . R e s e a r c h e r ' s Comments: S u b j e c t a g a i n was u s i n g two b o d i e s as r e f e r e n c e : the buoy and the dock . (Q.5). " I w i l l say i t ' s near t h a t bay t h e r e . . . v e r y n e a r the shore l i n e t h e r e . . . " R e s e a r c h e r ' s Comments: S u b j e c t k e p t u s i n g two b o d i e s o r p l a c e s as r e f e r e n c e : bay and shore l i n e . D e r i v a t i o n o f I n f e r r e d R u l e : . The re sponses o f T r a c y t o the t h r e e q u e s t i o n s r e g a r d i n g s e l e c t i o n o f r e f e r e n c e p o i n t show a c o n s i s t e n t p a t t e r n when asked to l o c a t e o b j e c t s . . . t h a t o f u s i n g more t h a n one body o r p l a c e as r e f e r e n c e . Based on t h i s c o n s i s t e n t p r e c o n c e p t i o n and u s i n g the o p e r a t i o n a l d e -f i n i t i o n the f o l l o w i n g r u l e can be i n f e r r e d f o r R e f e r e n c e P o i n t f o r S t a t i o n a y B o d i e s : " E a c h one o f the l o c a t i o n s o f the 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 s e v e r a l r e f e r e n c e b o d i e s o r p l a c e s " . -103-T h i s i n f e r r e d r u l e was c a l l e d RPS-1 . Case Two: A s u b j e c t u s i n g one body o r p l a c e as a r e f e r e n c e but a d i s t i n c t one when l o c a t i n g each one o f the f i s h i n g s p o t s . The f o l l o w i n g re sponse s a r e from Hans (17 y e a r s ) : (Q . l ) " . . . h o w f a r i t ' s from the shore . . . ( I : what shore?) . . . t h a t one by the f o r e s t ( shore i n f r o n t o f the w h a r f ) . " R e s e a r c h e r ' s Comments: S u b j e c t was u s i n g one p l a c e as a r e f e r e n c e : a shore i n f r o n t o f the whar f . (Q.4) " . . . how f a r i t ' s from the o t h e r f i s h i n g spo t (the f i r s t one) . . . " R e s e a r c h e r ' s Comments: S u b j e c t was a g a i n u s i n g one p l a c e as r e f e r e n c e but a d i f f e r e n t one than w h e n . l o c a t i n g the f i r s t f i s h i n g s p o t , and i t was p r e c i s e l y t h a t s p o t . (Q.5) " . . . f rom the shore . . . how f a r ou t i t ' s from the s h o r e " (the shore c l o s e r to the l a s t f i s h i n g s p o t ) . R e s e a r c h e r ' s Comment: S u b j e c t kept u s i n g o n l y one p l a c e as a r e f e r e n c e but a d i f f e r e n t r e f e r e n c e p o i n t was used on each o c c a s s i o n . D e r i v a t i o n o f I n f e r r e d R u l e : The c o n s i s t e n t v iew o f Hans to l o c a t e the d i f f e r e n t f i s h i n g spo t s i s e x p r e s s e d i n h i s r e sponse s t o the q u e s t i o n s . Each re sponse c o n t a i n s o n l y one r e f e r e n c e body o r p l a c e bu t a d i s t i n c t one f o r each d e s c r i p t i o n . Based on t h i s c o n s i s t e n t p r e c o n c e p t i o n , the f o l l o w i n g r u l e can be i n f e r r e d f o r 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 : -104-" E a c h o f the l o c a t i o n s o f the f i s h i n g spo 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 body . b u t a d i s t i n c t one f o r each l o c a t i o n " . T h i s i n f e r r e d r u l e was c a l l e d RPS-2 Case T h r e e : A s u b j e c t u s i n g one body o r p l a c e as a r e f e r e n c e and t h e same one when l o c a t i n g each one o f the f i s h i n g s p o t s . The f o l l o w i n g re sponse s are from P r e s t o n (16 y e a r s ) : (Q . l ) " . . . from here (the dock) I w i l l p r o b a b l y use degrees . . . maybe 40 degrees n o r t h from where I am s t a n d i n g (the d o c k ) " R e s e a r c h e r ' s Comments: S u b j e c t was u s i n g one body as a r e f e r e n c e : the dock . (Q.4) " from here (the dock) . . . i t w i l l be about 15 degrees t o the n o r t h . . . o r u s i n g the wharf i t i s s t r a i g h t a c r o s s " . R e s e a r c h e r ' s Comments: S u b j e c t was a g a i n u s i n g one body as a r e f e r e n c e , t h e d o c k . He a l s o ment ioned the p o s s i b i l i t y o f u s i n g the w h a r f , but he expre s sed i n the sense t h a t he w i l l use e i t h e r one and n o t b o t h . (Q.5) " . . . i t i s i n the i n d e n t a t i o n o f l a n d ( the c l o s e r shore t o the: .'last. W i s h i n g spot) . . . i t w i l l be i n the sorrth . . . o r i t would be about 180 degrees toward t h e south from here (the d o c k ) " . • R e s e a r c h e r ' s Comments: S u b j e c t kep t u s i n g o n l y one body as a r e f e r e n c e , even though he ment ioned two p l a c e s he w i l l use e i t h e r one , one o f which i s the d o c k . D e r i v a t i o n o f I n f e r r e d R u l e : From P r e s t o n ' s r e s p o n s e s , i t i s c l e a r t h a t he was c o n s i s t e n t l y u s i n g the same body as a r e f e r e n c e t o l o c a t e each o f the f i s h i n g s p o t s . Based on t h i s c o n s i s t e n t p r e c o n c e p t i o n the f o l l o w i n g i n f e r r e d r u l e can be p r o d u c e d . -105-"Each one o f the l o c a t i o n s o f the f i s h i n g spo 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 body and the same one f o r each l o c a t i o n " . T h i s i n f e r r e d r u l e was c a l l e d RPS-3 . In the above example , the i n f e r r e d r u l e s have been p r e s e n t e d i n a p a r t i c u l a r o r d e r . T h i s p r o c e s s o f o r d e r i n g the i n f e r r e d r u l e s was done by a n a l y z i n g t h e l e v e l o f d i f f e r e n t i a t i o n o f each i n f e r r e d r u l e . That i s , by a n a l y z i n g how t h e c o n t e n t - o f t h e i n f e r r e d r u l e c o u l d be d i f f e r e n t i a t e d from t h a t o f a p h y s i c i s t ' s p e r s p e c t i v e . T h i s v iew i s c o n -t a i n e d i n the s o - c a l l e d P h y s i c s Approach i n S e c t i o n 3 . 1 . 2 . A d e t a i l e d e x p l a n a t i o n o f how t h i s c a t e g o r i z a t i o n was done i s p r e s e n t e d i n the next s e c t i o n , which a l s o d e s c r i b e s the c o n s t r u c t i o n o f a r u l e - m o d e l . The methodology d e s c r i b e d above was used t o produce 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 i n the sample and f o r each 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 . A t the end o f t h i s p r o c e s s , each s u b j e c t was a s s i g n e d a maximum o f 10 i n f e r r e d r u l e s , one f o r each v e c t o r c h a r a c t e r i s t i c . I f a s u b j e c t d i d not pos se s s a c o n s i s t e n t p r e c o n c e p t i o n f o r a v e c t o r c h a r a c t e r i s t i c , t h e n , no r u l e was a s s i g n e d t o h e r / h i m f o r t h a t v e c t o r c h a r a c t e r i s t i c . 4 . 1 . 3 C a t e g o r i z a t i o n o f t h e I n f e r r e d Rules 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 Two a s p e c t s were c o n s i d e r e d i n t h e c a t e g o r i z a t i o n o f the i n f e r r e d r u l e s p r e s e n t e d i n S e c t i o n 4 . 1 . 2 . F i r s t , the -106-c o n t e n t o f each i n f e r r e d r u l e was compared w i t h th e c o r r e s -ponding v e c t o r c h a r a c t e r i s t i c - r e f e r e n c e p o i n t - i n the network o f v e c t o r p o s i t i o n (see F i g u r e 3.2) t o d e t e r m i n e how d i f f e r e n t i a t e d i t was from th e one i n the ne twork . C o n t e n t , i n t h i s c o n t e x t , i n c l u d e s b o t h the number o f v a r i a b l e s p r e -sent and the n a t u r e o f t h e s e v a r i a b l e s ( tha t i s , q u a l i t a t i v e o r q u a n t i t a t i v e ) . Second , the i n f e r r e d r u l e s were o r d e r e d a c c o r d i n g t o the P h y s i c s A p p r o a c h ; the i n f e r r e d r u l e s f o r RPS whose c o n t e n t most d i f f e r e d from the p h y s i c i s t ' s p e r s p e c t i v e was c o n s i d e r e d as the l e a s t d i f f e r e n t i a t e d among the i n f e r r e d r u l e s , and i t w a s . c a t e g o r i z e d a t the lowes t l e v e l , (the f i r s t i n f e r r e d r u l e i n the example d e v e l o p e d i n S e c t i o n 4 . 1 . 2 ) . The i n f e r r e d r u l e f o r RPS whose c o n t e n t was most s i m i l a r t o the c o r r e s p o n d i n g one i n th e network — one un ique r e f e r e n c e p o i n t — was c o n s i d e r e d as the most d i f f e r -e n t i a t e d p r e c o n c e p t i o n and i t was c l a s s i f i e d a t th e h i g h e s t l e v e l (the t h i r d i n f e r r e d r u l e i n S e c t i o n 4 . 1 . 2 ) . The i n -f e r r e d r u l e whose c o n t e n t was c o n s i d e r e d i n between th e two extreme l e v e l s o f d i f f e r e n t i a t i o n was c l a s s i f i e d i n the i n t e r m e d i a t e l e v e l ( the second i n f e r r e d r u l e i n S e c t i o n 4 . 1 . 2 Other v e c t o r c h a r a c t e r i s t i c s may have more than one i n t e r -media te i n f e r r e d r u l e s . A f t e r c a t e g o r i z i n g the i n f e r r e d r u l e s , t h e y were e x p r e s s e d i n more g e n e r a l and l e s s c o n t e x t u a l t e r m s . The r e s u l t s o f t h i s l a s t p r o c e d u r e y i e l d e d th e r u l e - m o d e l f o r the f i r s t v e c t o r c h a r a c t e r i s t i c , r e f e r e n c e p o i n t f o r s t a t i o n a r y b o d i e s . The g e n e r a l r u l e s i n t h i s r u l e - m o d e l -107-s h o u l d s t i l l be c o n s i d e r e d as b e i n g v e r y t e n t a t i v e s i n c e 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 c o n t e x t and s t i l l l a c k e m p i r i c a l s u p p o r t i n o t h e r r e l a t e d c o n t e x t s . The g e n e r a l r u l e s i n the h y p o t h e t i c a l r u l e - m o d e l a re c l o s e r i n form t o those used by S i e g l e r (1978, 1979) w i t h h i s r u l e - a s s e s s m e n t methodo logy . Re fe rence P o i n t i n the P h y s i c s A p p r o a c h : A s h o r t d e s c r i p t i o n o f the approach ' used by p h y s i c i s t s t o choose r e f e r e n c e p o i n t s i s p r e s e n t e d be low. The network o f v e c t o r p o s i t i o n (see F i g u r e 3.2 and d e f i n i t i o n i n Appendix A ) , which a c t u a l l y r e p r e s e n t s the P h y s i c s A p p r o a c h , show t h a t t h i s q u a n t i t y ( v e c t o r p o s i t i o n ) needs magnitude and d i r e c t i o n t o be c o m p l e t e l y d e f i n e d . Both magnitude and d i r e c t i o n need t o be measured w i t h r e s p e c t t o a s p e c i f i c frame o f r e f e r e n c e which i s r e l a t e d t o a s p e c i f i c r e f e r e n c e p o i n t ( the s o - c a l l e d o r i g i n o f the frame o f r e f e r e n c e o r c o o r d i n a t e s s y s t e m ) . As i s shown i n F i g u r e 3 . 2 , p h y s i c i s t s u s u a l l y use the same frame o f r e f e r e n c e and f o r e x t e n s i o n the same r e f e r e n c e p o i n t t o d e s c r i b e (magnitude and d i r e c t i o n ) t h e l o c a t i o n s o f d i f f e r e n t b o d i e s o r the i n s t a n -taneous l o c a t i o n s o f a moving body. T h i s i s the c u r r e n t approach used by p h y s i c i s t s : and p r o v i d e s the norm t o which t h e i n f e r r e d c o u l d be compared. C a t e g o r i z a t i o n o f the F i r s t I n f e r r e d R u l e : Based on the approach d e s c r i b e d above , the c o n t e n t o f the c o n s i s t e n t p r e -c o n c e p t i o n t h a t most d i f f e r e d from i t was i n c l u d e d i n the f o l l o w i n g i n f e r r e d r u l e : -108-" E a c h o f the l o c a t i o n s o f the f i s h i n g spo t s a re d e s c r i b e d u s i n g s e v e r a l r e f e r e n c e b o d i e s o r p l a c e s " ( I n f e r r e d r u l e RPS-1) T h i s i n f e r r e d r u l e shows the use o f more than one r e f e r e n c e t o l o c a t e b o d i e s . S u b j e c t s h o l d i n g t h i s c o n c e p t i o n d i d n o t p e r c e i v e the impor tance o f s e l e c t i n g o n l y one body as a r e f e r e n c e to l o c a t e a l l s p o t s . I n s t e a d , they used s e v e r a l d i f f e r e n t ones to d e s c r i b e the l o c a t i o n o f each - f i s h i n g s p o t . The i n f e r r e d f u l e RPS-1 was r e p h r a s e d i n the f o l l o w i n g g e n e r a l form and i t c o r r e s p o n d e d to the f i r s t r u l e i n the r u l e - m o d e l f o r r e f e r e n c e p o i n t : "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 p l a c e s ) t o l o c a t e a s t a t i o n a r y body" (Rule R P S - I ) . C a t e g o r i z a t i o n o f the Second I n f e r r e d R u l e : The c o n s i s t e n t p r e c o n c e p t i o n t h a t was judged t o be next c l o s e s t to the p h y s i c i s t s p e r s p e c t i v e was: " E a c h one o f the l o c a t i o n s o f the f i s h i n g spo 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 body bu t a d i s t i n c t one f o r each l o c a t i o n " ( I n f e r r e d r u l e R P S - 2 ) . The i n f e r r e d r u l e shows the use o f o n l y one r e f e r e n c e body as a p h y s i c i s t would do i t bu t t h a t r e f e r e n c e body i s no t a lways the same. T h i s a s p e c t makes t h i s c o n c e p t i o n s t i l l d i f f e r e n t from a p h y s i c i s t ' s p e r s p e c t i v e , even though i t c o n t a i n s an i m p o r t a n t f e a t u r e o f i t . The i n f e r r e d r u l e RPS-2 was r e p h r a s e d i n the f o l l o w i n g g e n e r a l form and i t c o r r e s p o n d e d t o the second r u l e o f the r u l e - m o d e l : -109-"Use o f one unique r e f e r e n c e p o i n t (body or place) to l o c a t e a s t a t i o n a r y body but 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 are used to l o c a t e o t h e r b o d i e s " . (Rule RPS-II) C a t e g o r i z a t i o n o f the T h i r d I n f e r r e d Rule: The c o n s i s t e n t p r e c o n c e p t i o n whose content was c l o s e s t to the p h y s i c i s t ' s p e r s p e c t i v e was: "Each one of the l o c a t i o n s o f 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 the 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) T h i s i n f e r r e d r u l e shows the use o f o n l y one r e f e r e n c e body f o r each one of the l o c a t i o n s . T h i s a c t u a l l y corresponds to the procedure used by p h y s i c i s t s . The i n f e r r e d r u l e RPS-3 was rephrased i n the f o l l o w i n g g e n e r a l form and i t corresponded to the t h i r d r u l e i n the ru l e - m o d e l : "Use o f one unique r e f e r e n c e p o i n t (body or pla c e ) t o l o c a t e a l l r e q u i r e d s t a t i o n a r y b o d i e s " . (Rule RPS-III) T h i s s e t of t h r e e r u l e s , summarized below, thus c o n s t i t u t e s the rule-model f o r the v e c t o r c h a r a c t e r i s t i c , r e f e r e n c e p o i n t f o r s t a t i o n a r y b o d i e s . Rule Model f o r RPS: Rule RPS-I: 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 (bodies o r p l a c e s ) t o l o c a t e a s t a t i o n a r y body. Rule RPS-II: Use o f one unique r e f e r e n c e p o i n t (body or place) t o l o c a t e a s t a t i o n a r y body but 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 re used t o l o c a t e o t h e r b o d i e s . Rule RPS-III: Use o f one unique r e f e r e n c e (body or pla c e ) t o l o c a t e a l l r e q u i r e d s t a t i o n a r y b o d i e s . -110-T h i s methodology was used to produce the r u l e - m o d e l s f o r each one o f the 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 . A t the end o f t h i s p r o c e s s t h e r e were 10. r u l e - m o d e l s , one f o r each v e c t o r c h a r a c t e r i s t i c . 4.2 Format f o r P r e s e n t i n g R e s u l t s o f I n t e r v i e w Data A n a l y s i s The r e s u l t s o f the a n a l y s i s o f the i n t e r v i e w d a t a f o r each v e c t o r c h a r a c t e r i s t i c i n terms o f i n f e r r e d r u l e s w i l l be p r e s e n t e d i n a format w h i c h c o n t a i n f i v e s u b - s e c t i o n s Each o f the se s u b - s e c t i o n s a r e f u l l y e x p l a i n e d be low b e f o r e r e p o r t i n g the a c t u a l r e s u l t s . 1. I n t r o d u c t i o n o f V e c t o r C h a r a c t e r i s t i c : T h i s sub-s e c t i o n c o n s i s t s o f a s h o r t d e s c r i p t i o n o f the p h y s i c s background r e l a t e d to the v e c t o r c h a r a c t e r i s t i c , rea sons to c o n s i d e r i t as an i m p l i c i t c h a r a c t e r i s t i c , and reasons to i n c l u d e i t i n the p r e s e n t .study 2. Ques t ions i n the I n t e r v i e w P r o t o c o l R e l a t e d t o  the V e c t o r C h a r a c t e r i s t i c : T h i s s u b - s e c t i o n c o n s i s t s o f a l i s t o f a l l q u e s t i o n s from the i n t e r v i e w p r o t o c o l o f the t a s k s r e l a t e d to t h a t s p e c i f i c v e c t o r c h a r a c t e r i s t i c . 3. L i s t o f I n f e r r e d R u l e s : T h i s s u b - s e c t i o n c o n t a i n s a l l the i n f e r r e d r u l e s i d e n t i f i e d from the i n t e r v i e w d a t a f o r the s p e c i f i c v e c t o r c h a r a c t e r i s t i c . The i n f e r r e d r u l e s were o b t a i n e d -111-by f o l l o w i n g the same procedure i l l u s t r a t e d i n S e c t i o n 4 . 1 . 2 , and a r e p r e s e n t e d i n the o r d e r o b t a i n e d from the l e v e l o f d i f f e r e n t i a -t i o n a n a l y s i s — which was e x p l a i n e d i n S e c t i o n 4 . 1 . 3 . The i n f e r r e d r u l e s a r e i d e n t i f i e d by u s i n g the a b b r e v i a t i o n f o r the v e c t o r c h a r a c t e r i s t i c p l u s a d i g i t . These d i g i t s a r e used to d i f f e r e n t i a t e among i n f e r r e d r u l e s o f 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 ( i . e . , the f i r s t i n f e r r e d r u l e f o r r e f e r e n c e p o i n t f o r a s t a t i o n a r y o b j e c t would be R P S - 1 ) . The number o f female and male s u b j e c t s a s s i g n e d to each i n f e r r e d r u l e i n the l i s t i s p r o v i d e d i n p a r e n t h e s i s ( e . g . , 6F , 7M f o r RPS-1) a t the end o f the i n f e r r e d r u l e s t a t ement . A t the bottom o f the l i s t o f i n f e r r e d r u l e s the c a l c u l a t e d v a l u e o f the C h i - s q u a r e t e s t s t a t i s t i c i s p r o v i d e d . '" The "two v a r i a b l e s used i n the s t a t i s t i c a l t e s t was i n f e r r e d r u l e c l a s s i f i c a t i o n and sex . I t was d e c i d e d to s tudy i f t h e r e was s i g n i -f i c a n t d i f f e r e n c e , r e g a r d i n g p r e c o n c e p t i o n s about v e c t o r c h a r a c t e r i s t i c s , between female and male s u b j e c t s because i t i s u s u a l l y found i n the l i t e r a t u r e t h a t boys do b e t t e r than g i r l s i n s c i e n c e (Hobbs, e t a l , 1979; N a t i o n a l Assessment o f E d u c a t i o n a l P r o g r e s s , (1978); E r i c k s o n , e t a l , -112-1980) . Research f i n d i n g s r e p o r t e d u s u a l l y r e l a t e t o s c i e n c e ach ievement , t h i s means a f t e r i n s t r u c t i o n . I t was thought i t would be i n t e r e s t i n g t o f i n d out i f t h e r e was s i g n i f i c a n t sex d i f f e r e n c e r e g a r d i n g p r e -i n s t r u c t i o n knowledge about v e c t o r c h a r a c t e r -i s t i c s . Even t h o u g h , a s m a l l sample was used (11 boys and 9 g i r l s ) , the r e s u l t s o f the C h i - s q u a r e t e s t s t a t i s t i c i s r e p o r t e d . The c a l c u l a t e d C h i - s q u a r e v a l u e s have i n d i c a t e d t h a t t h e r e was no s i g n i f i c a n t d i f f e r e n c e between the i n f e r r e d r u l e s o f each v e c t o r c h a r a c t e r i s t i c a c c o r d i n g to sex . D i s c u s s i o n o f R e s u l t s : Under t h i s s u b - s e c t i o n each o f the i n f e r r e d r u l e s p r e s e n t e d i n sub-s e c t i o n 3 i s d i s c u s s e d and a n a l y z e d i n d e t a i l . The a n a l y s i s here i s suppor ted by p r o v i d i n g s e l e c t i v e i n t e r v i e w e x c e r p t s f rom a few s u b j e c t s . P r e s e n t a t i o n o f e x c e r p t s w i l l have the f o l l o w i n g f o r m a t : name o f s u b j e c t , age , number o f q u e s t i o n i n p a r e n t h e s i s ( i . e . , Q. 4 ) . W i t h i n an e x c e r p t , any e x p l a n a t i o n o f terms o r f u r t h e r i n t e r v i e w e r ' s q u e s t i o n s (I) a re denoted i n p a r e n t h e s i s . Ru le M o d e l : In the f i n a l s u b - s e c t i o n the i n -f e r r e d r u l e s were r e p h r a s e d i n a more g e n e r a l and l e s s c o n t e x t u a l form thus y i e l d i n g the -113-g e n e r a l r u l e s . These more g e n e r a l r u l e s w i l l be i d e n t i f i e d u s i n g the a b b r e v i a t i o n f o r the g i v e n v e c t o r c h a r a c t e r i s t i c and a Roman n u m e r a l . The Roman numeral r u l e c o r r e s p o n d s t o the A r a b i c numeral i n f e r r e d r u l e ( e . g . , RPS-1 c o r r e s p o n d s to R P S - I ) . 4 .3 R e s u l t s o f A n a l y s i s o f I n t e r v i e w Data T h i s s e c t i o n c o n t a i n s the r e s u l t s o f u s i n g the i n t e r v i e w a n a l y s i s methodology , which was d e l i n e a t e d i n S e c t i o n s 4 . 1 . 2 and 4 . 1 . 3 , f o r each o f the f i r s t t e n v e c t o r c h a r a c t e r i s t i c s l i s t e d i n S e c t i o n 1 .2 . Some t r a n s c r i p t i o n s o f i n t e r v i e w d a t a a re p r e s e n t e d i n Appendix B . T h i s c o r r e s p o n d s to re sponses to the q u e s t i o n s on .two v e c t o r c h a r a c t e r i s t i c s of a l l s u b j e c t s i n the sample . R e s u l t s are p r e s e n t e d s e p a r a t e l y f o r each v e c t o r c h a r a c t e r i s t i c f o l l o w i n g the format d e s c r i b e d i n the p r e -v i o u s s e c t i o n . 4 . 3 . 1 . I n f e r r e d R u l e s f o r Re fe rence 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) A t the b e g i n n i n g o f the s tudy i t was c o n j e c t u r e d t h a t a r e f e r e n c e p o i n t was one unique v e c t o r c h a r a c t e r i s t i c -114-w h i c h was used when d e s c r i b i n g the l o c a t i o n s o f b o d i e s . The da t a c o l l e c t e d i n t h i s s tudy showed- t h a t the d e s c r i p -t i o n s used by s t u d e n t s t o l o c a t e a s t a t i o n a r y body o r p l a c e ( e . g . , a f i s h i n g spot i n t a sk 1 ) and those used to l o c a t e a moving body ( e . g . , a moving boa t on the r i v e r i n t a sk 2) were two d i f f e r e n t t h i n g s . T h u s , i t became e v i d e n t t h a t i t was n o t p o s s i b l e to i d e n t i f y common c r i t e r i a f o r b o t h c a s e s . Because o f these f i n d i n g s i t was d e c i d e d t o t r e a t r e f e r e n c e p o i n t as two d i s t i n c t v e c t o r c h a r a c t e r i s t i c s . The f i r s t , which i s t r e a t e d i n t h i s s e c t i o n , r e f e r s to the c h o i c e o f r e f e r e n c e b o d i e s o r p o i n t s when d e s c r i b i n g s t a t i o n a r y o b j e c t s o r p l a c e s . The second type r e f e r s t o the c h o i c e o f s t a t i o n a r y r e f e r e n c e b o d i e s a n d / o r moving r e f e r e n c e b o d i e s ( e . g . , water o f the r i v e r i n t a sk 2) when d e s c r i b i n g o b j e c t s i n m o t i o n . T h i s became v e c t o r c h a r a c t e r i s t i c number & i n LIVC (see C h a p t e r O n e ) . T h i s s e c t i o n d e a l s o n l y w i t h the v e c t o r c h a r a c t e r -i s t i c r e f e r e n c e p o i n t f o r s t a t i o n a r y i c b D i i i e s (RPS) and the aim was to f i n d o u t how s t u d e n t s d e a l t w i t h the prob lem o f d e s c r i b i n g the l o c a t i o n o f s t a t i o n a r y b o d i e s o r i n -s t an taneous l o c a t i o n s o f a moving o b j e c t . D i d they see the need f o r a r e f e r e n c e p o i n t t o d e s c r i b e a l o c a t i o n ? D i d they d e s c r i b e these l o c a t i o n s w i t h r e s p e c t t o a s p e c i f i c r e f e r e n c e p o i n t ? What d i d they choose as r e f e r e n c e p o i n t s ? Ques t ions o f I n t e r v i e w P r o t o c o l f o r RPS Q u e s t i o n s : 1, 4, and 5 i n t a sk one . Q u e s t i o n 1: How would you d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g spo t t o your f r i e n d s ? Ques t ion 4 : How would you d e s c r i b e the l o c a t i o n o f the second f i s h i n g spo t t o your f r i e n d s ? Ques t ion 5; How would you d e s c r i b e the 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 spo t t o your f r i e n d s ? T a b l e 4.1 L i s t o f I n f e r r e d Ru le s About Re ference  P o i n t f o r S t a t i o n a r y Bod ie s (RPS) Three i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , which have the f o l l o w i n g common open ing s e n t e n c e : " E a c h one o f the l o c a t i o n s o f the f i s h i n g spo t s i s d e s c r i b e d u s i n g . . . I n f e r r e d Rule RPS-1 : S e v e r a l re ference bod ie s o r p l a c e s " . (6F, 7M). I n f e r r e d Ru le RPS-2 : Only one r e f e r e n c e body b u t a d i s t i n c t one f o r each l o c a t i o n " ( 2 F , 2 M ) . I n f e r r e d Rule RPS-3 : One unique r e f e r e n c e body f o r a l l l o c a t i o n s " ( I F , , ; 2 M ) „ (X 2 = . 2 1 , d f = 2 p>.05) D i s c u s s i o n o f R e s u l t s I n f e r r e d Rule RPS-1 : T h i s can be r e f e r r e d t o as the " m u l t i p l e r e f e r e n c e p o i n t c o n c e p t i o n " because s e v e r a l b o d i e s o r p l a c e s a re used to l o c a t e a s t a t i o n a r y o b j e c t . From the sample o f 20, 13 used t h i s c o n c e p t i o n . These s u b j e c t s b e l i e v e d t h a t the l o c a t i o n o f an o b j e c t can b e s t be d e s c r i b e d -116-by "using s e v e r a l d i f f e r e n t r e f e r e n c e b o d i e s . Some e x c e r p t s from i n t e r v i e w d a t a w i l l i l l u s t r a t e more c i e a r l y t h i s c o n -c e p t i o n . Suzzane , 17 : (Q.4) " I t ( task 1, second f i s h i n g spot) i s s o r t o f i n the m i d d l e but i t ' s c l o s e r to the wharf . . . t h e r e are two buoys on the water . . . t e l l them (her f r i e n d s who are coming t o f i s h ) where the f i r s t one ( f i r s t f i s h i n g spot) i s and t e l l them t h a t the second one i s f a r t h e r but no t much and i t ' s c l o s e r t o the t r e e " . D a l e , 15 (Q.4) " I t ( task 1, second f i s h i n g spot) i s j u s t p a s t the f i r s t f i s h i n g s p o t , i n the same a n g l e , a c o u p l e o f hundred o f y a r d s from me and i t i s r i g h t i n f r o n t o f t h i s c o r n e r " . These two e x c e r p t s show t h a t b o t h s u b j e c t s were u s i n g s e v e r a l r e f e r e n c e p o i n t s . Suzzane made use o f f i v e : the m i d d l e o f the l a k e , the w h a r f ; the b u o y s , the f i r s t f i s h i n g s p o t , and a t r e e ; and Dale made use o f t h r e e : the f i r s t f i s h i n g s p o t , h i m s e l f , and a c o r n e r . I t appears t o be a " t r y t o i n c l u d e e v e r y t h i n g you c a n " type s t r a t e g y so as t o t r y and maximize t h e i r chance•vrxf. .suecess. I n f e r r e d Rule RPS-2 : T h i s r u l e can be r e f e r r e d t o as the " c h a n g i n g r e f e r e n c e p o i n t c o n c e p t i o n " , because a d i s t i n c t r e f e r e n c e body i s used t o l o c a t e each f i s h i n g s p o t . Three p l a c e s were more o f t e n used as a r e f e r e n c e p o i n t , they w e r e : t h e o b s e r v e r ' s l o c a t i o n (task 1, the d o c k ) , the s t a r t -i n g p o i n t f o r the t r i p ( the w h a r f ) , and the l a s t s t o p ( f i s h i n g spot) o f the b o a t . These p o i n t s were a l s o found among the r e f e r e n c e p o i n t s s e l e c t e d by s u b j e c t s h o l d i n g -117-R P S - I . A common p r a c t i c e was t o s e l e c t the c l o s e s t o b j e c t o r p l a c e to the l o c a t i o n b e i n g d e s c r i b e d . Among the f o u r Ss h o l d i n g t h i s r u l e , t h r e e f o l l o w e d t h i s p r o c e d u r e . To d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g spot they used the wharf (the c l o s e s t o b j e c t t o the s p o t ) , f o r the second f i s h i n g spot they used the t r e e (the c l o s e s t o b j e c t t o t h a t s p o t ) , and f o r the l a s t f i s h i n g spot they used the shore (the c l o s e s t p l a c e to the l a s t f i s h i n g s p o t ) . E x -c e r p t s from the i n t e r v i e w da ta c l e a r l y i l l u s t r a t e d t h i s p r a c t i c e . J u l i e , 15 : (Q . l ) " I ' d t e l l them (her f r i e n d s ) t o go from t h i s p o i n t (she showed the wharf) ou t s t r a i g h t i n ang le . . . a l i t t l e b i t i n a n g l e " . (Q.4) " . . . t u r n the boa t i n ang le and head 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 f i s h i n g s p o t ) . (Q.5) " . . . go to where the c u r v e i s ( c l o s e s t shore to the l a s t f i s h i n g spot) and go about 15 f e e t and t h e r e i t i s (she meant t o c a r r y the boa t to t h a t shore and s t a r t , rowing from there ) . These e x c e r p t s show t h a t the s u b j e c t changed the r e f e r e n c e p o i n t t o d e s c r i b e each one o f the l o c a t i o n s and s e l e c t e d the c l o s e s t o b j e c t t o the l o c a t i o n as a r e f e r e n c e p o i n t . I n f e r r e d Rule RPS-3: T h i s r u l e can be c a l l e d the " c o n s t a n t r e f e r e n c e p o i n t c o n c e p t i o n " because each s u b j e c t used the same o b j e c t as a r e f e r e n c e p o i n t t o l o c a t e the d i f f e r e n t -118-f i s h i n g s p o t s . The most common p l a c e s s e l e c t e d a re the o b s e r v e r ' s l o c a t i o n (the dock i n t a sk 1) o r the s t a r t i n g p o i n t f o r the t r i p (the w h a r f ) . F o r the 3Ss h o l d i n g t h i s c o n c e p t i o n , two c o n s i s t e n t l y used t h e i r p o s i t i o n (the dock) as a r e f e r e n c e p o i n t t o d e s c r i b e the l o c a t i o n s o f t h r e e f i s h i n g s p o t s ; the o t h e r s u b j e c t made c o n s i s t e n t . u s e o f the s t a r t i n g p o i n t (the wharf) as a r e f e r e n c e p o i n t . E x c e r p t s from i n t e r v i e w da ta show t h i s c o n s i s t e n t p r a c t i c e : P r e s t o n , 16: (Q. l ) from here (the dock) I w i l l p r o b a b l y use degrees . . . maybe 4 0 ° n o r t h from where I am s t a n d i n g here . . . " (Q.4) " . . . from here (the dock) i t w i l l be about 1 5 ° t o 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 o f the l a n d . . . o r i t ' d be about 1 8 0 ° toward the south from here (the d o c k ) " . These e x c e r p t s c l e a r l y show t h a t t h i s s u b j e c t c o n -s i s t e n t l y used the same p l a c e to d e s c r i b e the l o c a t i o n o f d i f f e r e n t f i s h i n g s p o t s . T a b l e 4.2 R u l e - M o d e l f o r Re ference P o i n t f o r S t a t i o n a r y B o d i e s Rule R P S - 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 p l a c 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 " . Ru le R P S - I I : "Use 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 bu t d i s t i n c t ones to l o c a t e o t h e r s t a t i o n a r y b o d i e s " . Ru leRPS- I I I : "Use o f one unique 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 -a r y b o d i e s " . -119-Rule RPS- I I I a c t u a l l y c o r r e s p o n d s to the p h y s i c i s t s ' v iew o f s e l e c t i n g a r e f e r e n c e p o i n t . T h i s i s the r e a s o n t h a t i t o c c u p i e s the h i g h e s t p o s i t i o n i n the r u l e - m o d e l . I t was no t an easy t a sk t o d e c i d e which i n f e r r e d r u l e between RPS-1 and 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 . RPS-2 was t a k e n because t h i s a l s o c o n s i d e r s o n l y one r e f e r e n c e p o i n t , even though i t was n o t the same r e f e r e n c e p o i n t f o r the d i f f e r e n t s p o t s . 4 . 3 . 2 I n f e r r e d Ru le s f o r Frame o f Re fe rence f o r a  S t a t i o n a r y Body ( F . R . ) The r e a s o n f o r i n c l u d i n g the frame o f r e f e r e n c e c h a r a c t e r i s t i c was to uncover what k i n d o f c o n c e p t i o n s s t u d e n t s have about making more p r e c i s e q u a n t i t a t i v e d e s c r i p -t i o n s , o f l o c a t i o n s o f s t a t i o n a r y b o d i e s or p l a c e s . T h i s c h a r a c t e r i s t i c i s c l o s e l y r e l a t e d to the p r e v i o u s one , r e f e r e n c e p o i n t — i t e s s e n t i a l l y i s an e x t e n s i o n o f the n o t i o n o f a r e f e r e n c e p o i n t . - Re fe rence p o i n t r e f e r s o n l y t o the need f o r the c h o i c e o f a body o r p l a c e from which t o r e l a t e the l o c a t i o n o f a n o t h e r body bu t no q u a n t i t a t i v e d e s c r i p t i o n i s r e q u i r e d . A f t e r a r e f e r e n c e body (or p o i n t ) has been c h o s e n , the d i s t a n c e to and d i r e c t i o n o f ( i . e . , as e x p r e s s e d by an angle) a s p e c i f i c l o c a t i o n w i t h r e s p e c t t o t h a t r e f e r e n c e p o i n t can be measured. When these q u a n t i -t i e s a re e x p r e s s e d , the r e f e r e n c e p o i n t has then been ex-tended t o become a frame o f r e f e r e n c e . -120-S p e c i f i c a l l y , t h e n , the rea son f o r i n c l u d i n g t h i s v e c t o r c h a r a c t e r i s t i c was t o de termine how s u b j e c t s d e a l w i t h the prob lem o f d e s c r i b i n g q u a n t i t a t i v e l y the l o c a t i o n s o f d i f f e r e n t p o i n t s ( i . e . , f i s h i n g spo t s i n t a s k 1) on a f l a t s u r f a c e . D i d they see the need o f u s i n g a q u a n t i t a t i v e d e s c r i p t i o n ( i . e . , d i s t a n c e expre s sed i n u n i t s a n d / o r a n g l e e x p r e s s e d i n degrees) i n o r d e r to l o c a t e the v a r i o u s f i s h -i n g spo t s ? D i d they use more than one v a r i a b l e ( d i s t a n c e and d i r e c t i o n ) i n t h e i r d e s c r i p t i o n s and which r e f e r e n c e p o i n t s d i d they use t o r e l a t e these v a r i a b l e s to the v a r i o u s l o c a t i o n s ? O r i g i n a l l y i t was i n t e n d e d t o i n c l u d e a n 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 — t h a t o f v e c t o r p o s i t i o n — but a f t e r a n a l y z i n g the f i r s t i n t e r v i e w s i t was r e a l i z e d t h a t the p r e c o n c e p t i o n s uncovered through the q u e s t i o n s r e l a t e d to v e c t o r p o s i t i o n were s i m i l a r t o those o b t a i n e d f o r frame o f r e f e r e n c e . T h i s was the case because b o t h v e c t o r c h a r a c t e r i s t i c s , frame o f r e f e r e n c e ^ a n d " v e c t o r p o s i t i o n , f o c u s s e d on the q u a n t i t a t i v e a s p e c t o f d e s c r i b i n g l o c a t i o n s . T h u s , i t was d e c i d e d to c o n s i d e r o n l y the frame o f r e f e r e n c e c h a r a c t e r i s t i c , and the re sponses f o r the q u e s t i o n s r e l a t e d t o v e c t o r p o s i t i o n were used as d a t a f o r the i d e n t i f i c a t i o n o f p r e c o n c e p t i o n s and 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 . Frame o f r e f e r e n c e was s e l e c t e d r a t h e r than v e c t o r p o s i t i o n because i t was thought t h a t the former i s more i n t u i t i v e i n n a t u r e than the l a t t e r , and b a s i c n o t i o n s -121-o f i t would more l i k e l y be e v i d e n t i n s u b j e c t s p r i o r to f o r m a l i n s t r u c t i o n . Ques t ions o f I n t e r v i e w f o r FR Ques t ions 1, 3, 4, 5a , and 6c i n Task One. Ques t ion 1: How would you d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g spot t o y o u r f r i e n d s ? Ques t ion 3: ( I f S has used no q u a n t i t y or o n l y one , e i t h e r d i s t a n c e o r d i r e c t i o n , when answer ing q u e s t i o n 1 ) . How c o u l d you make the d e s c r i p t i o n ' o f t h a t l o c a t i o n ( f i r s t f i s h i n g spot) more p r e c i s e f o r your f r i e n d s ? Ques t ion 4; How would you d e s c r i b e the l o c a t i o n o f t h i s second f i s h i n g spo t t o your f r i e n d s ? Ques t ion 5 : (a) : How would you d e s c r i b e the 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 spo t t o your f r i e n d s ? (Th i s f i s h i n g spo t i s a t the same d i s t a n c e from the dock than the f i r s t f i s h i n g s p o t , bu t i n a d i f f e r e n t d i r e c t i o n ) . ( b ) : These two f i s h i n g spot s a re a t the same d i s t a n c e from you (S on d o c k ) . How c o u l d you d i s t i n g u i s h them? Ques t ion 6: (c) : ( A f t e r S has drawn the p a t h f o l l o w e d by the boat) What do you need t o know i n o r d e r t o have a b e t t e r a p p r o x i m a t i o n o f the p a t h taken by the boat? -122-T a b l e 4.3 L i s t o f I n f e r r e d Ru le s (FR) These i n f e r r e d r u l e s were e x p r e s s e d i n b o t h q u a n t i -t a t i v e and q u a l i t a t i v e t e rms . The use o f u n i t s ( e . g . , m e t e r s , degrees) i n the d e s c r i p t i o n was i n t e r p r e t e d as b e i n g q u a n t i t a t i v e i n n a t u r e . Where u n i t s were not used and o t h e r d e s c r i p t i o n s such as n o r t h , s o u t h , c l o s e r , f a r t h e r , to the r i g h t were u s e d , the d e s c r i p t i o n s were i n t e r p r e t e d as q u a l i t a t i v e i n n a t u r e . F i v e i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , which have the f o l l o w i n g common open ing s e n t e n c e : " P r e c i s e d e s c r i p t i o n o f each one o f the l o c a -t i o n s o f the f i s h i n g spo t s i s done by u s i n g . . . I n f e r r e d Rule F R - 1 : O n l y a q u a n t i t a t i v e d i s t a n c e (between a g i v e n r e f e r e n c e p o i n t and the l o c a t i o n ) as a c o o r d i n a t e " (1F,2M) . I n f e r r e d Ru le F R - 2 : O n l y a q u a n t i t a t i v e d i r e c t i o n (ex-p r e s s e d as an a n g l e i n degrees) as a c o o r d i n a t e . " (OF, 3M). I n f e r r e d Ru le F R - 3 : Both q u a l 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 as c o o r d i n a t e s bu t two r e f e r e n c e p o i n t s a r e used — one f o r the d i s t a n c e and ano ther one f o r the d i r e c t i o n " (1F,1M) I n f e r r e d R u l e F R - 4 : 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 as c o o r d i n a t e s but two r e f e r e n c e p o i n t s a r e used — one f o r the d i s t a n c e and ano ther one f o r -123-the d i r e c t i o n " (5F,3M) I n f e r r e d R u l e F R - 5 : Both 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 as c o o r d i n a t e s w i t h the same r e f -e rence p o i n t f o r b o t h d i s t a n c e and d i r e c t i o n " (1F,1M) Cx2 = 3 . 6 7 , d f = S p > 05 ; 2 Ss (1F,1M) were no t a s s i g n e d i n f e r r e d r u l e ) . D i s c u s s i o n o f R e s u l t s : I n f e r r e d Rule F R - 1 : T h i s r u l e c o u l d be i n t e r p r e t e d as the " 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 c o o r d i n a t e c o n c e p -t i o n " . Even a f t e r a s k i n g q u e s t i o n 5b, the o n l y c o o r d i n a t e s u b j e c t s used between a r e f e r e n c e p o i n t and the l o c a t i o n o f a f i s h i n g spot was d i s t a n c e . Ques t ion 5b put s u b j e c t s i n a p o s i t i o n where the d i s t a n c e s to the f i r s t and l a s t f i s h i n g spo t s were i d e n t i c a l , thus d i s t a n c e a l o n e was not s u f f i c i e n t i n d i s t i n g u i s h i n g between the two l o c a t i o n s o f the s p o t s . However, the s u b j e c t s u s i n g t h i s r u l e d i d n o t appear t o r e c o g n i z e t h i s d i f f i c u l t y and m a i n t a i n e d t h e i r adherance t o the d i s t a n c e c r i t e r i o n . O n l y 3 ou t o f 20 s u b j e c t s made use o f t h i s i n f e r r e d r u l e ; same e x c e r p t s from t h e i r i n t e r v i e w da ta f o l l o w . Hans , 17 : (Q. l ) " . . . how f a r away [they] a r e from the s h o r e . . . " (Q.3) " . . . how f a r i t i s away from m e . . . " (Q.5a) " . . . from the shore . . . how f a r out i s from the s h o r e " (Q.5b) "What do you mean? (Quest ion i s rephrased) . . . I d o n ' t know . . . maybe how many metres a c r o s s are (S r e f e r r e d t o the d i s t a n c e between the two f i s h i n g s p o t s ) " . -124-(_Q.6c) " . . . from where I am i t ' s v e r y d i f f i c u l t " . K e l l e y , 1 5 : ( Q . l ) " . . . f i n d a p o i n t i n t h a t shore t h a t i s i n l i n e w i t h the spot . . . t h e n , guess the d i s t a n c e ( I : What d i s t a n c e ? ) . , . (Q.3) " I d o n ' t know any o t h e r way" . (Q.4) " . . . from the t r e e so f a r . . . i t i s about 60 y a r d s b u t . . . and then so f a r ou t f rom. .. where I am . . . " (Q.5a) "How f a r away i t i s from where I am" (Q.5b) " . . . I d o n ' t know . . . from the a n g l e t h a t they a re (S r e f e r r e d to the a n g l e formed by the two s t r a i g h t l i n e s j o i n i n g the two spot s w i t h the 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 the d i s t a n c e between the two spots ) . . . o r you make an i s o s c e l e s t r i a n g l e and make these s i d e s 200 m e t r e s , then they w i l l be e q u a l " . (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 went and the a n g l e s o f how much they t u r n e d (S r e f e r r e d to the a n g l e s formed by c o n s e c u t i v e d i s p l a c e m e n t s ) " These two e x c e r p t s show t h a t these Ss made use o f o n l y d i s t a n c e as a c o o r d i n a t e . There i s some i n d i c a t i o n o f d i r e c t i o n a p p e a r i n g here but I judged i t to be i n s u f f i c i e n t t o q u a l i f y as a (FR-3 R u l e ) . I n f e r r e d R u l e F R - 2 : T h i s r u l e c o u l d be summarized as " o n l y q u a n t i t a t i v e d i r e c t i o n as c o o r d i n a t e " . S u b j e c t s h o l d -i n g t h i s p r e c o n c e p t i o n d i f f e r e n t i a t e d l o c a t i o n s i n a two d i m e n s i o n a l s i t u a t i o n by use o f q u a n t i t a t i v e d i r e c t i o n o n l y , w h i c h was expre s sed by a n g l e s i n d e g r e e s . I t i s i n t e r e s t i n g t o observe t h a t t h e y d i d no t use d i s t a n c e as a c o o r d i n a t e , which seems to be more p r i m a r y than d i r e c t i o n . There were 3 o f 20 s u b j e c t s h o l d i n g t h i s c o n c e p t i o n ; e x c e r p t s from -125-t h e i r i n t e r v i e w da ta f o l l o w s . P r e s t o n , 16: (Q. l ) " . . . from where . . . I ' l l p r o b a b l y use degrees . . . maybe i t ' s 4 0 ° n o r t h from where I am s t a n d i n g " . (Q.3) (Not asked to t h i s s u b j e c t ) . (Q.4) " f rom here . . . i t w i l l be about 1 5 ° to the n o r t h " (Q.5a) " i t i s i n the i n d e n t a t i o n o f the l a n d 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 the south . . . " about 1 8 0 ° toward the south from h e r e " . (Q.5b) " i t ' d be 200 metres t o the n o r t h west and 200 m t o the s o u t h - e a s t (S used d i s t a n c e expre s sed i n metres because I i n t r o d u c e d them i n the q u e s t i o n 5b. What d i d you mean b e f o r e by degrees? ) . . . o h , by degrees I mean . . . t h i s i s my zero l i n e (a l i n e p a s s i n g t h r o u g h the dock and wharf) and 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 a n g l e s ; 2 0 ° , 3 0 ° , 9 0 ° , u s i n g h i s l o c a -t i o n as a r e f e r e n c e p o i n t ) (Q.6c) " . . . 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 l a k e ( I : What d i s t a n c e s ? ) . . . from the wharf t o here ( f i r s t f i s h i n g s p o t ) , from here ( f i r s t f i s h i n g spot) t o here (2nd f i s h i n g s p o t ) - . . " S t e v e , 15 : (Q. l ) " . . . I w o u l d - s a y : ' S " tart from the wharf t h e r e and go out about 70 degrees a n g l e t o a lmost be p a r a l l e l t o the 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 t o your f r i e n d s ? ) . . . s t r a i g h t i n f r o n t o f t h i s c u r v e (a bay on the l ake ) i s " (Q.3) " . . . ( long pause) . . . (Q.4) "when you a re i n the f i r s t f i s h i n g spo t t u r n t o w a r d . t h e shore . . . from 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 the bank and 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 spot) I guess . . . i n the f i r s t c u r v e 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 m e n t i o n to your f r i e n d s ? ) . . . (pause) -126-(Q.5b) " t h a t f i s h i n g spot i s 200 metres (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 the r i g h t about 7 0 degrees a n g l e . . . and the o t h e r one i s 200 m t o the l e f t about 70 degrees a n g l e from t h i s p o i n t o f v iew (the d o c k ) " (Q.6c) " . . . d i s t a n c e s . . . from p o i n t to p o i n t . . . from the wharf t o t h i s ( f i r s t f i s h i n g spot) and so o n " . Both s u b j e c t s d i d no t ment ion the d i s t a n c e u n t i l th e i n t e r v i e w e r i n t r o d u c e d i t as p a r t o f q u e s t i o n 5b. B u t , even a f t e r the d i s t a n c e was i n t r o d u c e d , the Ss d i d not use b o t h , ang le and d i s t a n c e , as a p a i r o f c o o r d i n a t e s measured w i t h r e s p e c t t o the same r e f e r e n c e p o i n t . I n f e r r e d Rule F R - 3 : T h i s r u l e i s made up o f s e v e r a l a s p e c t s . F i r s t o f a l l i t i s a type o f a q u a l i t a t i v e ' r u l e s i n c e the two v a r i a b l e s ( d i s t a n c e and d i r e c t i o n ) ment ioned are n o t a s s o c i a t e d w i t h u n i t s . S e c o n d l y , s u b j e c t s h o l d i n g t h i s c o n -c e p t i o n saw- the need to use two q u a l i t a t i v e c o o r d i n a t e s i n a two d i m e n s i o n a l s i t u a t i o n . And t h i r d l y , a l t h o u g h these s u b j e c t s used two v a r i a b l e s . t h e y r e l a t e d them t o d i f f e r e n t r e f e r e n c e p o i n t s . T h i s r u l e c o u l d be summarized a s : " q u a l i t a t i v e c o o r d i n a t e s w i t h d i f f e r e n t r e f e r e n c e p o i n t s c o n c e p t i o n " . O n l y two o f the 20 s u b j e c t s h e l d t h i s i n -f e r r e d r u l e . E x c e r p t s from t h e i r i n t e r v i e w d a t a a r e p r e -sen ted be low. Suzzane , 17 (Q. l ) " . . . 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 to say the degrees . . . o r a map (response to q u e s t i o n 3) " . . . j u s t t o t e l l them to go s t r a i g h t o r go w i t h them . . . " . -127-CQ.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 t h e m " . (Q.4) " i t ' s s o r t o f on the m i d d l e b u t i t ' s c l o s e r to the wharf . . . t h e r e a re two buoys on the w a t e r , t e l l them where the f i r s t one i s and t e l l them t h a t the second 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 to the t r e e " . (Q.5a) " I t ' s r i g h t o f f the s i d e , to my l e f t . . . i t i s p a s t the green b u o y " . (Q.5b) "One i s a t ang le ( I : What do you mean?) . . . w e l l one i s t o your r i g h t and the 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 ang le . . . the o t h e r one i s a b i t i n an a n g l e " (Q.6c) (Quest ion no t asked o f t h i s s u b j e c t ) . I n f e r r e d Rule F R - 4 : The o n l y d i f f e r e n c e between t h i s i n -f e r r e d r u l e and the p r e v i o u s one i s t h a t i n t h i s case the s u b j e c t s used q u a n t i t a t i v e d i s t a n c e , a l o n g w i t h a q u a l i t a t i v e d i r e c t i o n c o o r d i n a t e . . L i k e the p r e v i o u s i n f e r r e d r u l e these s u b j e c t s used d i f f e r e n t r e f e r e n c e p o i n t s f o r d i s t a n c e and d i r e c t i o n . T h i s p r e c o n c e p t i o n was the most common one found i n the sample o f s u b j e c t s , s i n c e 8 o f 20 h e l d i t . E x c e r p t s from t h e i r i n t e r v i e w d a t a f o l l o w s : " . . . j u s t about h a l f way to the m i d d l e , a-Ad h a l f way t o the r i g h t s i d e . . . i t ' s about 50 f e e t from me" (not asked to t h i s s u b j e c t ) " . . . you e x p l a i n to them a g a i n by d i s t a n c e . . . about 100 f e e t . . . maybe you c o u l d say a l o n g the shore j u s t pass the t r e e . . . " B e c k y , 15 : (Q. l ) (Q.3) (Q.4) (Q.5a) " i t i s near the shore on the l e f t s i d e o f the l a k e . . . i t ' s no t 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-(Q.5b) " . . . you c o u l d say 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 r e s p e c t t o a spo t on r i g h t s i d e o f s u b j e c t ) . . . you 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 about the o t h e r spot?) I t ' s about 7 5 ° a n g l e . . . " (Q.6c) " . . . the waves i t produces when i t moves i n o r d e r t o see i t . . . " L a r r y , 1 6 : ( Q . l ) " I c o u l d g i v e them the d i s t a n c e from t h i s (the wharf) i n f r o n t o f i t , and t h a t t h e r e i s a t r e e t o the r i g h t " . (Q.3) " . . . I t i s e x a c t l y i n the m i d d l e o f the l a k e . I f I am i n t h i s dock and you look t h a t t r e e the boat i s i n the m i d d l e o f them" . (Q.4) " I w i l l t e l l them: f i n d the 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 coming from t h a t t r e e to here (second f i s h i n g spot) and back t o the d o c k , and t h e r e i s the spo t about 30 f e e t from the t r e e " . (Q.5a) "From the s m a l l dock up to the l e f t s i d e (Q.5b) " . . . they a re b o t h 200 ya rds and one i s on the l e f t s i d e o f the c o a s t and the o t h e r i n the c e n t e r o f the l a k e . . . " (Q.6c) " . . . I need a compass ( I : How c o u l d you use i t ? ) . . . I d o n ' t know . . . we need markers l i k e the t r e e , c o r n e r s . . . " I n f e r r e d Rule F R - 5 : The o n l y d i f f e r e n c e between t h i s r u l e and r u l e FR-4 i s t h a t these s u b j e c t s used the same r e f e r e n c e p o i n t t o expre s s 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 . T h i s r u l e i s most s i m i l a r to what p h y s i c i s t s d e f i n e as a p o l a r frame o f r e f e r e n c e , the o n l y i n g r e d i e n t m i s s i n g i s the q u a n t i t a t i v e d i r e c t i o n , which can be ex-p r e s s e d as the a n g l e i n degrees t h a t the v e c t o r p o s i t i o n forms w i t h one r e f e r e n c e l i n e p a s s i n g t h r o u g h the r e f e r e n c e -129 p o i n t . T h i s c o n c e p t i o n was h e l d by o n l y 2 o f 20 s u b j e c t s which i n d i c a t e s t h a t i t i s n o t v e r y common among grade 10 s t u d e n t s . E x c e r p t s from t h e i r i n t e r v i e w da ta f o l l o w : I a n , 15 : (Q . l ) "You can t e l l how f a r o f f from the dock i t i s , i n what d i r e c t i o n . . . you can see from here i n the d i r e c t i o n o f the t r e e . . . " (Q.3) (Not asked t o t h i s s u b j e c t ) . (Q.4) " I t ' d be d i r e c t l y to the n o r t h - e a s t about 5 ° t o the N . E . s t r a i g h t out 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) " . . . you can do i t o f how f a r i s from here (the dock) . . . from here (the dock) i s 150 y a r d s o f f . . . (Q.5b) " . . . t h i s one i s r e a l l y c l o s e t o the s h o r e , the o t h e r one i s o u t j u s t i n the m i d d l e o f the l a k e . . . o r one i s on the l e f t (of o b s e r v e r ' s l o c a t i o n ) and the o t h e r on the r i g h t . . . " (Q.6c) " t o have a rough i d e a where i t went , how many t imes i t s t o p p e d , what d i r e c t i o n went i n and came back . . . " A l t h o u g h , Ian made use o f q u a n t i t a t i v e d i r e c t i o n i n some p o i n t , i t was n o t a c o n s i s t e n t c r i t e r i o n . He was more i n c l i n e d t o ment ion q u a l i t a t i v e d i r e c t i o n combined w i t h q u a n t i t a t i v e d i s t a n c e . T a b l e 4.4 . R u l e - M o d e l f o r Frame o f Re fe rence 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 to 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 " . Ru le F R - I I : "Use 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 degrees) 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 : "Use 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 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 ; bu t two r e f e r e n c e p o i n t s are u s e d , one f o r each c o o r d i n a t e " . Rule F R - I V : "Use 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 as c o o r d i n a t e s to 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 ; bu t two r e f e r e n c e p o i n t s are u s e d , one f o r each c o o r d i n a t e " . Ru le FR-V : "Use 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 the same r e f e r e n c e p o i n t f o r b o t h c o o r d i n a t e s " . 4 . 3 . 3 I n f e r r e d R u l e s about 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 (D) A l t h o u g h t h i s i s a w e l l d e f i n e d k i n e m a t i c v e c t o r q u a n t i t y , i t was c o n s i d e r e d i m p o r t a n t to i n c l u d e i t as an 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 f o r the f o l l o w i n g r e a s o n s : (1) I t can 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 i n com-p a r i s o n t o the v e c t o r p o s i t i o n c o n c e p t , which i s more " a r t i f i c i a l " o r remote from e x p e r i e n c e , (2) I t i s a s u i t a b l e concep t t o c l e a r l y d i f f e r e n t i a t e between a s c a l a r q u a n t i t y ( e . g . , the l e n g t h o f the path) and a v e c t o r i a l q u a n t i t y ( e . g . , d i s p l a c e m e n t ) . The f i r s t i s s c a l a r because i t o n l y needs a magn i tude , the l e n g t h o f the p a t h between two l o c a t i o n s , t o be c o m p l e t e l y d e s c r i b e d ; and the second i s v e c t o r i a l because i n a d d i t i o n t o magnitude i t a l s o needs d i r e c t i o n to be c o m p l e t e l y d e s c r i b e d , and f i n a l l y , (3) I t c o r r e s p o n d s , i n p a r t , to a v e r y common and i n t u i t i v e n o t i o n -131-t h a t t h e c l o s e s t d i s t a n c e between two l o c a t i o n s i s the l e n g t h o f the s t r a i g h t l i n e t h a t j o i n s them. From the p i l o t s tudy a t r e n d was obse rved f o r the s u b j e c t s t o choose the l a s t s t o p p i n g p o i n t o f a moving o b j e c t as a r e f e r e n c e p o i n t ( e . g . , use o f the f i r s t f i s h i n g spo t t o d e s c r i b e l o c a t i o n o f the second f i s h i n g s p o t ) . I f t h i s i s an i n t u i t i v e c o n c e p t i o n o f s e l e c t i n g r e f e r e n c e p o i n t , i t may h e l p s t u d e n t s t o under s t and the c o n c e p t o f d i s p l a c e m e n t . P a r t i c u l a r l y , i t s h o u l d f a c i l i t a t e the u n d e r -s t a n d i n g o f p a r t i a l c o n s e c u t i v e d i s p l a c e m e n t s , s i n c e p a r t i a l d i s p l a c e m e n t s a s s o c i a t e two c o n s e c u t i v e l o c a t i o n s o f a moving body . Ques t ions o f I n t e r v i e w P r o t o c o l Q u e s t i o n s : 6, 7a, 8a i n t a s k o n e . Ques t ion 6 : From where you are (on the dock) c o u l d you a p p r o x i m a t e l y d e s c r i b e the p a t h f o l l o w e d by the boa t ? P l e a s e , draw t h a t p a t h i n t h i s map o f the l a k e . Q u e s t i o n 7a: We can say t h a t the boa t has had a change o f l o c a t i o n when i t movers from the s t a r t i n g p o i n t ( S . P . ) t o the f i r s t f i s h i n g spo t ( P ^ ) . How c o u l d you d e s c r i b e the l o c a t i o n o f the f i r s t f i s h i n g s p o t w i t h r e s p e c t t o t h e s t a r t i n g p o i n t ? ( I f S i n c l u d e s the d i s t a n c e i n h e r / h i s d e s c r i p t i o n ) Is t h a t d i s t a n c e a l o n g the p a t h f o l l o w e d by the boa t ? Quest ion 7b: T h e n , the boa t moved from the f i r s t f i s h i n g spot t o the second one a t here (P2) Then how c o u l d you d e s c r i b e the l o c a t i o n o f t h i s f i s h i n g s p o t w i t h r e s p e c t t o the f i r s t one? -132-Ques t ion 8a : ( I f S i n c l u d e s the d i s t a n c e i n h e r / h i s d e s c r i p t i o n ] . You ment ioned the d i s t a n c e i n your d e s c r i p t i o n . Does a change o f l o c a t i o n t e l l a n y t h i n g about the p a t h f o l l o w e d by the boa t between the two l o c a t i o n s ? T a b l e 4 .5 L i s t o f I n f e r r e d Ru le s f o r D i s p l a c e m e n t (D) o r  Change o f L o c a t i o n In these i n s t a n c e s the p h y s i c a l phenomena b e i n g d e s c r i b e d was the boa t moving from one f i s h i n g spot t o a n o t h e r ; t h i s means, a change o f l o c a t i o n , which was used as a synonym o f d i s p l a c e m e n t . F i v e i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , w h i c h have the f o l l o w i n g common open ing s e n t e n c e : \ " A change o f l o c a t i o n o f the boa t moving from one f i s h i n g spo t t o ano ther i n d i c a t e s . . . I n f e r r e d Ru le D - l : The s t a r t i n g and f i n a l l o c a t i o n s " . ( I F , 1 M ) I n f e r r e d Rule D - 2 : The l e n g t h o f the s t r a i g h t l i n e j o i n i n g the two l o c a t i o n s and t h a t l e n g t h i s independent from the p a t h " . (0F,3M) I n f e r r e d Rule D - 3 : Both the d i s t a n c e a l o n g the p a t h and the 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 l o c a t i o n I n f e r r e d Rule D - 4 : Both the l e n g t h o f the s t r a i g h t l i n e j o i n i n g the two l o c a t i o n s w h i c h i s independent from p a t h and q u a l i t a t i v e d i r e c t i o n " . (7F,3M) -133-T n f e r r e d Rule D -5 : Both the l e n g t h o f the s t r a i g h t l i n e j o i n i n g the two l o c a t i o n s which i s independent from pa th and q u a n t i t a t i v e d i r e c t i o n " . (0F,1M) Cx2 = 6.80 d f = 5 p > . 0 5 ; IS(F) was no t a s s i g n e d i n f e r r e d r u l e ) D i s c u s s i o n o f R e s u l t s I n f e r r e d Rule D - l T h i s r u l e s t a t e s t h a t * a change o f l o c a t i o n i n d i c a t e s the aimed l o c a t i o n o f a t r i p and s t a r t i n g l o c a t i o n . There was no ment ion o f the p o s s i b l e ways o r p a t h i n r e a c h i n g t h a t f i n a l l o c a t i o n . No q u a l i t a t i v e o r q u a n t i t a t i v e v a r i a b l e s such as d i s t a n c e to the f i n a l l o c a t i o n o r d i r e c t i o n o f mot ion was p r o v i d e d . In summary, t h i s c o n c e p t i o n r e l a t e s o n l y the s t a r t i n g and the e n d i n g p o i n t s o f a moving o b j e c t but no t what happens i n between. T h i s was no 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 (15 t o 17 y e a r s o f age) s i n c e o n l y 2 among 20 h e l d t h i s c o n c e p t i o n . E x c e r p t s from i n t e r v i e w d a t a b e s t i l l u s t r a t e t h i s c o n c e p t i o n . Suzzane , 17 : (Q.7a) " J u s t i t j o e s s t r a i g h t . . . ( I : What c o u l d you t e l l t o 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 d o n ' t know . . . " (Q.8a) " Y e h , i t i s . . . i t t e l l s where the boa t went . . . " D a l e , 15 : (Q.7a) " W e l l i t s t a r t e d here (wharf) and went t o t h a t p o i n t ( f i r s t f i s h i n g spot) . . . they p a d d l e d s t r a i g h t o f f from the w h a r f " . CQ.7b) " . . . yeh . . . from here (wharf) t o here ( f i r s t f i s h i n g spot) . . . i t was s t r a i g h t o f f . Both s u b j e c t s s o l e l y emphasized the s t a r t i n g and e n d i n g p o i n t s i n t h e i r r e s p o n s e s . -134-I i i f e r r e d Rule D - 2 : S u b j e c t s h o l d i n g t h i s c o n c e p t i o n thought t h a t the d i s t a n c e between the two l o c a t i o n s was an i m p o r t a n t f a c t o r t o c o n s i d e r when d e s c r i b i n g a change o f l o c a t i o n . N o t i c e t h a t they were not c o n s i d e r i n g the d i s t a n c e a l o n g the p a t h ; these s u b j e c t s were d i f f e r e n t i a t i n g between the two f i s h i n g s p o t s . T h i s c o n c e p t i o n d i d no t i n -c l u d e the 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 o f l o c a t i o n . Among the 20 . s u b j e c t s , o n l y 3 h e l d t h i s c o n c e p t i o n , which i s shown t h r o u g h the f o l l o w i n g e x c e r p t s . Hans , 17 : (Q.7a) "How f a r out 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 f i s h i n g s p o t . (He meant the s t r a i g h t d i s t a n c e ) . (Q.7b) "Same t h i n g as b e f o r e . . . how f a r i t i s " (Q.8a) " I d o n ' t u n d e r s t a n d what you mean ( I : q u e s t i o n i s repeated) . . . no ( I : Why?) You can'if^-feell the a c t u a l p a t h , you c o u l d go around i n d i f f e r e n t ways " . T h i s s u b j e c t as the o t h e r two i n c l u d e d o n l y the s t r a i g h t d i s t a n c e i n h i s r e s p o n s e s ; t h e r e was no ment ion o f d i r e c t i o n a t a l l . I n f e r r e d Rule D - 3 : The o n l y s u b j e c t h o l d i n g t h i s c o n c e p t i o n c o n s i d e r e d the d i s t a n c e a l o n g the p a t h . T h i s was c a l c u l a t e d by the p r o d u c t o f the speed (assumed to be c o n s t a n t ) t imes the i n t e r v a l o f t ime taken i n moving from the wharf t o the f i r s t f i s h i n g s p o t . The d i r e c t i o n was expre s sed by the rl35-. angle formed by the two s t r a i g h t l i n e s p a s s i n g through the s t a r t i n g and ending p o i n t s from the o b s e r v e r ' s l o c a t i o n , A c t u a l l y , t h i s was not a d i r e c t i o n a s s o c i a t e d w i t h the change of l o c a t i o n but r a t h e r w i t h the d i r e c t i o n o f the v e c t o r p o s i t i o n a s s o c i a t e d w i t h the f i r s t f i s h i n g s p o t . Only 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 which i s shown i n the e x c e r p t s from i n t e r v i e w d a t a . Wayne, 15: (Q.7a) "You c o u l d look a t the f i r s t f i s h i n g spot and make an imaginary l i n e (from o b s e r v e r ' s l o c a t i o n on the dock); then look a t the wharf ( s t a r t i n g p o i n t ) , you n o t i c e i t w i l l be an angle between the l i n e s . You can note how f a r over i t i s ( s i z e of the a n g l e ) . You can a l s o t e l l how f a r a p a r t they are (the two l o c a t i o n s ) . I f they t r a v e l a t the same speed a l l the time, you c o u l d time how long i t takes t o go from one spot t o the 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 important 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 ? ) ... the speed, the d i r e c t i o n ... (Q.8a) "No ... ( I : Why not?) ... because he went from the s t a r t i n g p o i n t t o the f i s h i n g spot ... he went z i g z a g g i n g ... I n f e r r e d Rule D-4: T h i s c o n c e p t i o n o f change of l o c a t i o n was ve r y c l o s e t o the formal d e f i n i t i o n of d i s p l a c e m e n t s . The o n l y . i n g r e d i e n t m i s s i n g was the q u a n t i t a t i v e d i r e c t i o n ; s u b j e c t s h o l d i n g t h i s c o n c e p t i o n i n c l u d e d a q u a l i t a t i v e d i r e c t i o n . These students c o u l d r e a d i l y grasp the formal concept o f displacement i f they were i n s t r u c t e d how to express the q u a l i t a t i v e d i r e c t i o n i n a q u a n t i t a t i v e form. T h i s was a common c o n c e p t i o n among grade 10 students s i n c e 10 of . 20 s u b j e c t s h e l d i t . I t i s important t o mention here t h a t these students were e x p l i c i t l y aware t h a t the d i s t a n c e c o n s i d e r e d was not along the path; -136-some o f them s a i d t h a t t h e r e were many ways Cor paths) of. moving from one l o c a t i o n t o the o t h e r . C o n s i d e r i n g the c o n c e p t i o n s o f a l l v e c t o r c h a r a c t e r i s t i c s a n a l y z e d so f a r , i t seems t h a t most o f the s t u d e n t s seemed to c o n c e n t r a t e on o n l y one v a r i a b l e i n a q u a n t i t a t i v e way, and when t h e y c o n s i d e r e d a second v a r i a b l e i t was i n a q u a l i t a t i v e way. T h i s was the case f o r the v e c t o r d i s -p l a c e m e n t , and f o r the frame o f r e f e r e n c e c h a r a c t e r i s t i c . The f o l l o w i n g are e x c e r p t s from i n t e r v i e w d a t a : L i s a , 15: (Q.7a) "The s t a r t i n g p o i n t i s r i g h t on the end o f the wharf and the f i r s t spot i s a b i t t o the r i g h t i n a s l i g h t ang le . . . and e s t i m a t e s the amount o f m i l e s between the two . . . a l o n g the 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) t o t h i s (second f i s h i n g spot) i s a lmos t the same d i s t a n c e (as the d i s t a n c e from wharf t o f i r s t f i s h i n g spot) . . . maybe 50 o r 75 metres more . . . and they have to t u r n some d e g r e e s " . (Q.8a) " . . . no . . . because the boa t can go around i n d i f f e r e n t ways . . . " A l t h o u g h i n the re sponse to the f i r s t q u e s t i o n L i s a s a i d : " a l o n g the way f o l l o w e d by the b o a t " , t h i s meant the d i s t a n c e a l o n g the p a t h ; the re sponses f o r the o t h e r two q u e s t i o n s show t h a t she was r e f e r r i n g t o the s t r a i g h t d i s t a n c e between the l o c a t i o n s . L o r i - A n n , 15 : (Q.9a ) "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 : I s t h a t d i s t a n c e a l o n g the p a t h o r the s t r a i g h t d i s t a n c e ? ) . . . a l o n g the p a t h , how f a r the boa t w i l l go t o r e a c h t h a t p o i n t . (The f i r s t f i s h i n g s p o t . I : Suppose the p a t h i s t h i s one . I drew a c u r v e path) . . . o h , - 1 3 7 -I see .... I am t a l k i n g about the s t r a i g h t d i s t a n c e ( I : a n y t h i n g 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 the boat were going back 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 you want to d e s c r i b e the exact path you c o u l d use south, n o r t h , e a s t , or west, and go so much t h i s way and then t u r n around the o t h e r way". (Q.8a) " i t t e l l s the end o f the path but you c a n ' t t e l l the path because i t c o u l d go s t r a i g h t or l i k e t h a t ... or l i k e t h a t (she sketched two d i f f e r e n t curve p a t h s ) " I n f e r r e d Rule D - 5 : T h i s was the c o n c e p t i o n c l o s e s t to the f o r m a l d e f i n i t i o n o f d i s p l a c e m e n t . Although, the s u b j e c t s h o l d i n g t h i s c o n c e p t i o n made use of q u a n t i t a t i v e d i r e c t i o n , they were not r e f e r r i n g t o the d i r e c t i o n of the change of l o c a t i o n but t o the d i r e c t i o n o f the v e c t o r p o s i t i o n a s s o c i a t e d w i t h the f i s h i n g apots and u s i n g the o b s e r v e r ' s l o c a t i o n as a r e f e r e n c e p o i n t . S i n c e these s u b j e c t s r e a l i z e d about the need o f u s i n g these two q u a n t i t a t i v e v a r i a b l e s ( d i s t a n c e and d i r e c t i o n ) , the i n s t r u c t i o n a l t a s k would be t o show them the ^importance of knowing the d i r e c t i o n o f moving from one l o c a t i o n to another; i t would be expected t h a t these students would e a s i l y t r a n s -f e r t h e i r knowledge of measuring angles to a new but s i m i l a r s i t u a t i o n . The c o n c e p t i o n , i n c l u d e d i n t h i s i n f e r r e d r.ule a l s o c o n s i d e r s the d i s t a n c e between the two l o c a t i o n s . (The l e n g t h o f the s t r a i g h t l i n e j o i n i n g them) thus these s u b j e c t s were d i f f e r e n t i a t i n g between t h i s d i s t a n c e and the d i s t a n c e along the p a t h . Three students among -138-20 h e l d t h i s c o n c e p t i o n , which i s shown i n the f o l l o w i n g e x c e r p t s from the i n t e r v i e w d a t a . Becky , 15 (Q.7a) " . . . the d i s t a n c e and i n what s i d e o f the l a k e ? (Q.7b) (Q.8a) S t e v e , 15 : (Q.7a) (Q.7b) (Q.8a) " go about 110 degrees ( l i n e j o i n i n g dock as a r e f e r e n c e p o i n t ) a n g l e by 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 the r i g h t s i d e o f the l a k e and about 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 p a t h o r the s t r a i g h t d i s t a n c e between the 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 you t h i n k i s i m p o r t a n t when you 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 goes , the d i s t a n c e and the a n g l e " . " . . . i t does no t t e l l you the p a t h . . . " (Q no 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 you 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 a l o n g the p a t h o r s t r a i g h t ? ) . . . the s t r a i g h t d i s t a n c e . . . " " . . . i n the same way and d i s t a n c e . . . " . . the degrees "No . . . t h e y - c a n go anywhere around the whole l a k e to a r r i v e to t h a t p o i n t , they can go s t r a i g h t t o o " . T a b l e 4 .6 R u l e - M o d e l f o r D i s p l a c e m e n t 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 the s t a r t i n g and f i n a l l o c a t i o n s " . R u l e 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 o f 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 " . Ru le D - I I I : " A d i s p l a c e m e n t or 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 l o c a t i o n " . -139-Rule D - I V : " A d i s p l a c e m e n t or 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 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 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 w i t h r e s p e c t to the i n i t i a l o n e " . Rule D-V : " A d i s p l a c e m e n t or 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 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 the a n g l e o f the s t r a i g h t l i n e j o i n i n g the two l o c a t i o n s w i t h a n o t h e r r e f e r e n c e l i n e " . 4 . 3 . 4 I n f e r r e d Ru le s f o r A d d i t i o n o f D i s p l a c e m e n t (AD) The a d d i t i o n o f d i s p l a c e m e n t s i s a d e f i n e d v e c t o r i a l o p e r a t i o n , which i s a p p l i e d to the c o m b i n a t i o n o r compo-s i t i o n o f any v e c t o r q u a n t i t y . The r e a s o n f o r i n c l u d i n g t h i s o p e r a t i o n as an 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 i s based on t h i s a u t h o r ' s b e l i e f t h a t f o r a f o r m a l c o n c e p t o r mathematic o p e r a t i o n t o be more e a s i l y u n d e r s t o o d by s t u d e n t s , the c o n c e p t must be based on (or r e l a t e d to) some i n t u i t i v e n o t i o n o f the c o n c e p t . U s u a l l y , t h i s o p e r a t i o n i s f i r s t t a u g h t i n mathemat ics and i t i s g e n e r a l l y done w i t h o u t c o n n e c t i n g the v e c t o r i a l e n t i t i e s w i t h p h y s i c a l q u a n t i t i e s . These v e c t o r i a l e n t i t i e s a r e r e p r e s e n t e d by " a r r o w s " whose l e n g t h (of the arrow) c o r r e s -ponds t o the magnitude and the arrow i t s e l f shows the d i r e c t i o n . P h y s i c s t e a c h e r s know v e r y w e l l t h a t t h i s s o r t o f a b s t r a c t knowledge i s no t e a s i l y t r a n s f e r r e d t o s o l v e k i n e m a t i c v e c t o r i a l p r o b l e m s . T h i s a u t h o r a l s o b e l i e v e s t h a t s t u d e n t s d e v e l o p some p r e c o n c e p t i o n s to cope w i t h -140-s i t u a t i o n s i n which t h e y have to combine d i s p l a c e m e n t s (or d i s t a n c e s ) w i t h d i f f e r e n t d i r e c t i o n s . These p r e -c o n c e p t i o n s c o u l d be p a r t o f the i m p l i c i t a s p e c t s t h a t would enhance the u n d e r s t a n d i n g o f the f o r m a l c o n c e p t . T h i s s e c t i o n , t h e n , c o n s i d e r s the c o m b i n a t i o n o f two c o n s e c u t i v e d i s p l a c e m e n t s , w h i c h pos se s s magnitude and d i r e c t i o n , as an i m p l i c i t c h a r a c t e r i s t i c o f v e c t o r q u a n t i -t i e s . The aim o f t h i s p a r t o f the s tudy i s to uncover what k i n d s o f c o n c e p t i o n s s t u d e n t s have d e v e l o p e d about combin ing p a r t i a l and c o n s e c u t i v e d i s p l a c e m e n t s . Ques t ions o f I n t e r v i e w P r o t o c o l f o r AD  Q u e s t i o n s ; : 9 and 10 i n t a sk o n e . Ques t ion 9 : Suppose you have e s t a b l i s h e d the d i s t a n c e from the s t a r t i n g p o i n t ( S . P . ) t o the f i r s t f i s h i n g spo t (P^) as about 100 m, and you have a l s o e s t i m a t e d i t from t h i s spo t (P- )^ t o the second f i s h i n g spo t (P2) as about 150 m. I f your f r i e n d s a r e a t the s t a r t i n g p o i n t how are you g o i n g t o t e l l them t o go d i r e c t l y t o the second f i s h i n g spot? N o t e : T h i s q u e s t i o n was used o n l y w i t h the purpose o f i n t r o d u c i n g the a d d i t i o n o f d i s p l a c e m e n t . I t was e x p e c t e d t h a t a l l o r most o f th e s u b j e c t s w i l l use- the a r i t h m e t i c a d d i t i o n as a s o l u t i o n , and 100 p e r c e n t o f s u b j e c t s a c t u a l l y d i d use i t as a s t r a t e g y t o f i n d the c o r r e c t r e s p o n s e . Because the major purpose o f these q u e s t i o n s was t o f a m i l i a r i z e -141-the s u b j e c t s w i t h t h i s k i n d o f s i t u a t i o n the re sponse s f o r q u e s t i o n 9 were n o t c o n s i d e r e d i n the a n a l y s i s o f the i n f e r r e d r u l e s . Q u e s t i o n 10: Suppose you c o u l d a l s o e s t i m a t e the d i s t a n c e from t h i s spo t (P2) t o t h i s t h i r d f i s h i n g spo t (P3) and i t i s about 100 m. How c o u l d you t e l l your f r i e n d s t o go d i r e c t l y from the s t a r t i n g p o i n t t o t h i s t h i r d f i s h i n g spot? N o t e : The purpose o f t h i s q u e s t i o n was to f i n d out how the s u b j e c t s go about s o l v i n g t h i s k i n d o f s i t u a t i o n . I t was n o t e x p e c t e d t h a t s t u d e n t s would d i s c o v e r by themse lves the o p e r a t i o n o f v e c t o r a d d i t i o n , b u t i t i s i m p o r t a n t f o r f u r t h e r i n s t r u c t i o n a l use t o uncover the s t u d e n t s ' i n t u i t i v e i d e a s about v e c t o r a d d i t i o n . I t was thought t h a t one q u e s t i o n would be s u f f i c i e n t t o i l l u s t r a t e the s u b j e c t s ' c o n c e p t i o n s about t h i s v e c t o r c h a r a c t e r i s t i c . T a b l e 4.7 L i s t o f I n f e r r e d Ru le s f o r A d d i t i o n o f  D i s p l a c e m e n t s (AD) These i n f e r r e d r u l e s c o n t a i n two a s p e c t s . F i r s t , t h e c r i t e r i o n t o o b t a i n the magnitude o f the r e s u l t a n t d i s p l a c e m e n t ; and s e c o n d , the d i r e c t i o n ( q u a l i t a t i v e o r q u a n t i t a t i v e ) . Both a s p e c t s a r e 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 . N o t e : D e s p i t e the f a c t t h a t t h e r e was no s e a r c h f o r a -142-c o n s i s t e n c y o f r e s p o n s e s , because t h e r e was o n l y one q u e s t i o n f o r t h i s v e c t o r c h a r a c t e r i s t i c , the d i f f e r e n t s u b j e c t s ' c o n -c e p t i o n s were c l a s s i f i e d as i n f e r r e d r u l e s t o m a i n t a i n p a r a l l e l i s m i n the d a t a p r e s e n t a t i o n . Three i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , which have the f o l l o w i n g common open ing sentence "The r e s u l t a n t o f two n o n - p a r a l l e l c o n s e c u t i v e d i s p l a c e m e n t s o f the boat has . . . I n f e r r e d Rule A D - l i A magnitude e q u a l t o the a r i t h m e t i c a d d i t i o n o f the magnitudes and no d i r e c t i o n i s c i t e d " . ( I F , 1 M ) I n f e r r e d Rule A D - 2 : 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 o f the magnitudes and no d i r e c t i o n i s c i t e d " ( 5 F , 7 M ) . I n f e r r e d Rule A D - 3 : A magnitude s m a l l e r t h a n the a r i t h m e t i c a d d i t i o n o f the magnitudes and d i r e c t i o n i s c i t e d " . ( 2 F , 2 M ) (X 2 = .14 d f = 3 p > . 0 5 ; 2Ss (1F,1M) were not a s s i g n e d i n f e r r e d r u l e ) . D i s c u s s i o n o f R e s u l t s : I n f e r r e d Rule A D - 1 : T h i s r u l e c o u l d a l s o be c a l l e d " t h e a r i t h m e t i c a d d i t i o n c o n c e p t i o n " o f composing v e c t o r d i s p l a c e -ments . S u b j e c t s h o l d i n g t h i s c o n c e p t i o n made use o f a r i t h m e t i c a d d i t i o n t o o b t a i n the magnitude o f the r e s u l t a n t 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 . The a n g l e formed by the two c o n s e c u t i v e d i s p l a c e m e n t s was about 1 2 0 ° and hence was much too f a r from b e i n g p a r a l l e l t o j u s t i f y the use o f a r i t h m e t i c a d d i t i o n . The two s u b j e c t s under t h i s r u l e d i d no t ment ion d i r e c t i o n a t a l l ; i t may be due to the f a c t t h a t they c o n c e n t r a t e d o n l y on f i g u r i n g out how t o f i n d the d i s t a n c e . E x c e r p t s from i n t e r v i e w d a t a w i l l i l l u s t r a t e -143-t h i s c o n c e p t i o n . N o t e : The e s t i m a t e d d i s t a n c e s S . P . to P^, P-^P^ a n d P 2 P 3 were d i f f e r e n t f o r each s u b j e c t depending upon whether they had e a r l i e r o f f e r e d an e s t i m a t i o n . Suzzane , 17 (Q.10) ( S P . P 1 = 50 m, P 1 P 2 = 60m, P 2 P 3 = 100) . . . I f I need a map"'. ' . , about 200m ( I : Why do you t h i n k so?) . . . about 220m o r something . . . " L a r r y , 16: (Q.10) (SP.P^^ = 200 y a r d s , P ^ = 200, P 2 P 3 =200) " . . . 600 y a r d s . . . ( I : But you asked your f r i e n d s to go d i r e c t l y from the wharf t o t h i s s p o t , P3?) • • • w e l l , i t i s 600 y a r d s . . . " Both s u b j e c t s appear t o be u s i n g s t r i c t a r i t h m e t i c a d d i t i o n , p a r t i c u l a r l y L a r r y : he j u s t added the t h r e e d i s t a n c e s . Suzzane h e s i t a t e d between 200 and 220: i t seems she roughly , e s t i m a t e d the d i s t a n c e by a d d i n g the p a r t i a l d i s t a n c e s . I n f e r r e d Rule A D - 2 : S u b j e c t s h o l d i n g t h i s c o n c e p t i o n r e a l i z e d t h a t they c o u l d n o t use a r i t h m e t i c a d d i t i o n because the d i s t a n c e s were no t i n the same d i r e c t i o n , and a l s o t h e y obse rved t h a t the r e s u l t s h o u l d be 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 o f the component d i s t a n c e s . They t r i e d d i f f e r e n t k i n d o f s t r a t e g i e s to o b t a i n a n u m e r i c a l r e s u l t . I t was not expec ted t h a t they would s o l v e the prob lem the way a p h y s i c i s t would do i t b u t i t i s wor th examining the s t u d e n t s ' methods o f t a c k l i n g t h i s s o r t o f p r o b l e m . The most common s t r a t e g i e s were what c o u l d be c a l l e d the -144-* d i s t a n c e compar i son s t r a t e g y ' and the ' P y t h a g o r a s ' s t r a t e g y ' . The f i r s t one c o n s i s t e d i n e s t i m a t i n g the l e n g t h o f the l i n e SP .P^ by comparing by eye w i t h a known d i s t a n c e , f o r i n s t a n c e w i t h the d i s t a n c e SP t o P^ . T h i s s t r a t e g y v e r y much re sembles the ' s c a l e a p p r o a c h ' t o f i n d the magnitude o f r e s u l t a n t . T h i s c o n s i s t s i n c h o o s i n g an adequate and a r b i t r a r y u n i t (draw a t s c a l e ) and s e e i n g how many t imes t h a t u n i t f i t s i n t o the d e s i r e d d i s t a n c e t o be e s t i m a t e d . T h i s r e q u i r e s the u n d e r s t a n d i n g o f d i r e c t p r o p o r t i o n a l i t y . The ' P y t h a g o r a s ' s t r a t e g y ' c o n s i s t e d i n assuming the t r i a n g l e s to be r i g h t ang le t r i a n g l e (even though t h e y were not) and u s i n g the P y t h a g o r a s ' Theorem t o f i n d the r e s u l t a n t . Of c o u r s e , i t would be a v a l i d s t r a t e g y i f the t r i a n g l e were a r i g h t ang le t r i a n g l e . S u b j e c t s ' c l a s s i f i e d under t h i s r u l e d i d n o t ment ion d i r e c t i o n a t a l l . More than h a l f o f the s u b j e c t s i n the sample , 12 o f 20, used these s t r a t e g i e s , which w i l l be shown i n the f o l l o w i n g e x c e r p t s from i n t e r v i e w d a t a . K e v i n , 15 : (Q.10) ( S P . P ^ ="200, P 2 P 3 = 300) " . . . I have t o f i g u r e out the d i s t a n c e from here (SP) to here (P3) . . . 300 squared p l u s 300 squared . . . 900 p l u s 900 . . . 1800 . . . and the squared r o o t o f t h a t w i l l be the d i s t a n c e " . Russ , 15 : (Q.10) ( S P ^ = 400m, P ^ . = 300m P 2 P 3 = 300) " . . . t ake from the math . . . about 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 the 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 spots) . . . ( 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 " . T r a c y , 15 : (Q.10) (SP.P ] L = 200 f ee t , . P ^ = 150, = 200) " . . . g o i n g i n t h i s p a t h (pa s s ing t h r o u g h a l l -145-spots ) 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 , SP, to h e r e , P3) . . . i t w i l l be about 475 f e e t because they w i l l be l o s i n g some f e e t . . i f you go t h i s way (pa s s ing through a l l s p o t s , z i g z a g g i n g path) i s l o n g e r i f you go s t r a i g h t . . t h i s way i s s h o r t e r " . Sandy, 15: (Q. 10) (SP.P^ = 100 y a r d s , P ^ = 120 P 2 P 3 = 200) (Quest ion n o t c l e a r f o r h e r . I : How f a r w i l l you t e l l them t o go?) . . . 300 y a r d s . . . ( I : Why do you t h i n k s o ? ) . . . say t h i s spot (P 2 ) i s here (she assumed t r a n s l a t i n g P2 to be i n l i n e w i t h SP. and P3, 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 the l i n e SP.P3) . . . i t i s about 200 y a r d s . . . because from 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 . . . I n f e r r e d Rule A D - 3 : T h i s r u l e i s s i m i l a r to the p r e v i o u s one w i t h r e g a r d t o the s t r a t e g i e s used to f i n d the magni tud o f r e s u l t a n t d i s p l a c e m e n t ; but i n a d d i t i o n the s u b j e c t s under t h i s r u l e ment ioned the d i r e c t i o n as an i m p o r t a n t i n g r e d i e n t . E x c e r p t s o f i n t e r v i e w da ta o f some o f t h e 4 s u b j e c t s h o l d i n g t h i s c o n c e p t i o n f o l l o w . P r e s t o n , 16: ( Q . 1 0 ) ( S P . P 1 = 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 degrees o f f . . . l i k e about 400m from the s t a r t i n g p o i n t t o t h i s (P 3 ) . . . ( I : Why do you t h i n k i s 400?) . . . because the a n g l e w i l l be c u t o f f u n l e s s you go i n a z i g z a g g i n g m o t i o n , o r from here (P2) t o here ( P 3 ) , b u t i f you go s t r a i g h t the ang le 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". B e c k y , 15 : (Q. l ) ( S P . P 1 = 60 f e e t , P ^ = 40, P 2 P 3 - 4 0 ) . " . . . you c o u l d e s t i m a t e about 120 f e e t ( I : Why i s 120?) . . . because from here (SP) t o h e r e (P2) i s 100, then you add about 20. From t h i s (P2) to here (P3) i s 4 0 bu t you a re g o i n g t h a t way, you are 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 about 85 degrees a n g l e and about 50 f e e t away from the l e f t s i d e " . -146-Both s u b j e c t s e x p l i c i t l y ment ioned a q u a n t i t a t i v e d i r e c t i o n i n a d d i t i o n to the b e l i e f t h a t the magnitude o f the r e s u l t a n t has t o be 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 o f the components . T a b l e 4.8 R u l e - M o d e l f o r A d d i t i o n o f D i s p l a c e m e n t s Rule A D - I : "The r e s u l t a n t o f the 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 e q u a l t o the a r i t h m e t i c a d d i t i o n o f the magnitudes and d i r e c t i o n i s not c o n s i d e r e d " . Rule A D - I I : "The r e s u l t a n t o f the 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 o f the magni tudes .and d i r e c t i o n i s not c o n s i d e r e d " . Rule A D - I I I : "The r e s u l t a n t o f the 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 t h a n the a r i t h m e t i c a d d i t i o n o f 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 I n f e r r e d Rules f o r 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 . 7"^  ( S . V . P . ) S i m i l a r t o the ca se o f the v e c t o r i a l a d d i t i o n o f d i s p l a c e m e n t s , the s u b t r a c t i o n o f v e c t o r q u a n t i t i e s i s an e x p l i c i t and w e l l d e f i n e d v e c t o r i a l o p e r a t i o n . The rea sons f o r i n c l u d i n g t h i s o p e r a t i o n as an 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 a r e s i m i l a r t o the ones g i v e n f o r i n c l u d i n g the a d d i t i o n o f d i s p l a c e m e n t s , but the i n c l u s i o n o f t h i s v e c t o r c h a r a c t e r i s t i c needs o t h e r k i n d s o f arguments as w e l l . -147-F i r s t i t can be argued t h a t the o p e r a t i o n o f s u b t r a c t i o n o f v e c t o r s i s more a b s t r a c t than the a d d i t i o n o p e r a t i o n becaus i t i s more remote from s t u d e n t s ' e x p e r i e n c e . Hence i t can be c o n s i d e r e d as a more " a r t i f i c i a l " c o n c e p t i o n when compared t o a d d i t i o n o f d i s p l a c e m e n t which i s more " n a t u r a l " A l s o t h e r e are s e v e r a l p r e v i o u s c o n c e p t s to be u n d e r s t o o d b e f o r e f o r m a l l y g r a s p i n g s u b t r a c t i o n o f v e c t o r . These c o n -c e p t s a r e : use o f one r e f e r e n c e p o i n t t o d e s c r i b e l o c a t i o n s o f d i f f e r e n t p o i n t s , frame o f r e f e r e n c e , a s s o c i a t i o n o f c o o r d i n a t e s (magnitude and d i r e c t i o n ) f o r p o i n t s , and the a s s o c i a t i o n o f v e c t o r p o s i t i o n to p a r t i c u l a r s t a t i o n a r y l o c a t i o n s o r i n s t a n t a n e o u s l o c a t i o n s o f a moving body . O b v i o u s l y , i t i s n o t expec ted t h a t s u b j e c t s w i l l be a b l e t o h a n d l e a l l the se c o n c e p t s b e f o r e fo rma l i n s t r u c t i o n , bu t the purpose o f i n c l u d i n g t h i s o p e r a t i o n as an 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 was t o uncover how s u b j e c t s cope when c o n f r o n t e d w i t h t h i s s o r t o f s i t u a t i o n . D i d they have any i n t u i t i v e n o t i o n o f how t o d e a l w i t h the problem? What k i n d o f s t r a t e g i e s d i d they use t o r e l a t e the two g i v e n d i s t a n c e s D i d they c o n s i d e r the f a c t t h a t the two d i s t a n c e s were not p a r a l l e l ? I t may be t h a t the o p e r a t i o n o f s u b t r a c t i o n o f v e c t o r s c o n t a i n s more i m p l i c i t a s p e c t s t h a t w^fe not -detected i n the r a t i o n a l t a s k s a n a l y s i s . I f t h a t i s the c a s e , these unknown i m p l i c i t a s p e c t s c o u l d be r e v e a l e d through the s t u d e n t s ' p r e c o n c e p t i o n s , w h i c h may become m a n i f e s t when d e a l i n g w i t h the p r o b l e m a t i c s i t u a t i o n . -148-Quest ions o f I n t e r v i e w P r o t o c o l  Q u e s t i o n s : 11 and 12 i n t a s k one . Ques t ion 11 : Suppose you have e s t i m a t e d the d i s t a n c e from y o u r s e l f (on the dock) to t h i s spot (thumb tack i s p l a c e d a t P^) and i t i s about 200 metres ( d i f f e r e n t d i s t a n c e s were us f o r d i f f e r e n t s u b j e c t s ) and you have a l s o e s t i m a t e d the d i s t a n c e from y o u r s e l f t o t h i s o t h e r spot P 2 (the d o c k , P^ , and ~92 a r e o n a s t r a i g h t l i n e ) . Your f r i e n d s a re a t t h i s spot (P^) . How are you g o i n g t o t e l l them t o go d i r e c t l y t o t h i s spot ( P 2 ) ? N o t e : T h i s q u e s t i o n was used o n l y w i t h the purpose o f f a m i l i a r i z i n g the s u b j e c t s w i t h t h i s s o r t o f s i t u a t i o n I t was expec ted t h a t most o f the s u b j e c t s would make use o f s t r i c t a r i t h m e t i c s u b t r a c t i o n t o f i n d a s o l u t i o n . R e s u l t s agreed w i t h e x p e c t a t i o n : 18 . among 20 s u b j e c t s d i d use a r i t h m e t i c s u b t r a c t i o n . As w i t h the a d d i t i o n o p e r a t i o n , because t h i s q u e s t i o n se rved as a f a m i l i a r i z a t i o n p r o c e d u r e t h e i r r e sponse s a re n o t b e i n g c o n s i d e r e d i n the a n a l y s i s o f the i n f e r r e d r u l e s . Ques t ion 12: Suppose you have a l s o e s t i m a t e d the d i s t a n c e from y o u r s e l f t o t h i s spot (P3* see F i g u r e 3.5 f o r b e s t u n d e r s t a n d i n g o f s i t u a t i o n ) and i t i s about 300 m e t r e s . Your f r i e n d s a re a g a i n i n t h i s spot (P^ ) . How c o u l d you t e l l them t o go d i r e c t l y to t h i s o t h e r spo t (P^)? -149-N o t e : I t was thought t h a t one q u e s t i o n on t h i s v e c t o r c h a r a c t e r i s t i c would show the s u b j e c t s ' c o n c e p t i o n s about s u b t r a c t i o n o f v e c t o r q u a n t i t i e s . T a b l e 4.9 L i s t o f I n f e r r e d Ru le s f o r 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 (SVP) These i n f e r r e d r u l e s c o n t a i n two a s p e c t s . The f i r s t r e f e r s to the magnitude o f the r e s u l t a n t o f s u b t r a c t i n g two v e c t o r p o s i t i o n s (magnitude o f r e s u l t a n t d i s p l a c e m e n t ) ; and the s econd , the d i r e c t i o n ( q u a l i t a t i v e o r q u a n t i t a t i v e ) . Both a s p e c t s a re s e p a r a t e d by the c o n j u n c t i o n and . A g a i n , as i n the case o f a d d i t i o n o f d i s p l a c e m e n t s , a l t h o u g h t h e r e was one q u e s t i o n f o r t h i s v e c t o r c h a r a c t e r i s t i c t h e d i f f e r e n t s u b j e c t s ' c o n c e p t i o n s were c l a s s i f i e d as i n -f e r r e d r u l e s t o m a i n t a i n p a r a l l e l i s m i n the da t a p r e s e n t a t i o n . Three i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , which have the f o l l o w i n g common open ing s e n t e n c e : "The r e s u l t a n t o f the s u b t r a c t i o n o f two n o n - p a x a l L e l v e c t o r p o s i t i o n s f o r two f i sh ingv spo t s has . . . I n f e r r e d Rule S V P - 1 : A magnitude e q u a l to the a r i t h m e t i c s u b t r a c t i o n o f the magnitudes and d i r e c t i o n i s n o t c i t e d . " (3F,4M) I n f e r r e d Ru le S V P - 2 : 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 the magnitudes and d i r e c t i o n i s no t c i t e d . " (1F,3M) -150-I n f e r r e d Rule S V P - 3 ; 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 the magnitudes and d i r e c t i o n i s c i t e d " . (5F,3M) (X 2 = 2.46 d f = 3 p > . 0 5 ; IS (1M) was no t a s s i g n e d an i n f e r r e d r u l e ) . D i s c u s s i o n o f R e s u l t s I n f e r r e d Ru le S V P - 1 ; T h i s r u l e c o u l d a l s o be r e f e r r e d t o as the ' a r i t h m e t i c s u b t r a c t i o n c o n c e p t i o n ' o f s u b t r a c t i n g the magnitudes 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 t o o b t a i n the magnitude o f the r e s u l t a n t d i s p l a c e m e n t . In the sample , 7 o f 20 used t h i s s t r a t e g y , 35 p e r c e n t o f the sample . T h i s was a h i g h p e r c e n t a g e f o r a n o t i o n which i s v e r y d i f f e r e n t from the a c c e p t e d o n e . T h i s r e s u l t c o n f i r m e d what was e x p e c t e d . E a r l i e r i t was sugges ted t h a t i t would l i k e l y be r a r e t o f i n d t h a t s t u d e n t s d e v e l o p the a b s t r a c t o p e r a t i o n o f v e c t o r i a l s u b t r a c t i o n by t h e m s e l v e s . The a n g l e formed by the two v e c t o r p o s i t i o n was s u f f i c i e n t l y d i f f e r e n t from 0 ° t o 1 8 0 ° t o say t h a t t h e s t r a i g h t l i n e s were i n l i n e o r p a r a l l e l (the a n g l e was a c t u a l l y about 40 d e g r e e s ) . E x c e r p t s from i n t e r v i e w d a t a w i l l show the subj^ect ' s c o n c e p t i o n s . (S: s t ands f o r s u b j e c t ' s l o c a t i o n , on the dock) D a l e , 15: (Q.12). ( S ^ = 100 m e t r e s , S . P 3 = 250m) " . . . i t w i l l be 150 metres . . . you j u s t add them up t o have 250" . Hans , 17 : (Q.17) ( S ^ = 100m, S . P 3 = 400m) " . . . about 300m . . . ( I : Why do you t h i n k i t i s 300?) . . . because t h i s d i s t a n c e ( S . P , ) i s 100 and the o t h e r one i s 400 " -151-K e l l y , 15: (Q.12) (S.Pj^ = 200m, S . P 3 = 600m) " . . . t e l l the a n g l e between the f i r s t spo t and t h a t spot (the ang le formed by the l i n e s j o i n g i n g each spot w i t h the s u b j e c t ' s l o c a t i o n ) and t e l l them t o go 400m . . . ( I : Why 400m?) . . . w e l l t h e r e a r e 200 here and 600 t h e r e . . . then i t ' s about 400m" These t h r e e 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 s u b t r a c t i n g the magnitudes o f the 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 . I n f e r r e d Rule S V P - 2 : S u b j e c t s h o l d i n g . t h i s p r e c o n c e p t i o n were aware o f the f a c t t h a t the two d i s t a n c e s were not i n l i n e which would n o t a l l o w them t o use a r i t h m e t i c sub-t r a c t i o n . A n o t h e r i m p o r t a n t a s p e c t t h a t s u b j e c t s ob se rved was t h a t the l e n g t h o f the d i s t a n c e between and P^ must be g r e a t e r than the r e s u l t o f a p p l y i n g a r i t h m e t i c s u b t r a c t i o n to the two d i s t a n c e s . S u b j e c t s i n t h e i r endeavour to f i n d a n u m e r i c a l answer t r i e d s e v e r a l s t r a t e g i e s . Among the more common were a g a i n what can be c a l l e d the ' l e n g t h compar i son s t r a t e g y ' , which c o n s i s t s i n e s t i m a t i n g a d i s t a n c e by compar ing by eye w i t h a known l e n g t h , -afid the ' P y t h a g o r a s s t r a t e g y ' , which was a l s o found f o r a d d i t i o n o f d i s p l a c e m e n t s . E x c e r p t s from i n t e r v i e w d a t a w i l l b e t t e r show the use o f these and o t h e r s t r a t e g i e s by f o u r o f the .20 s u b j e c t s . Wayne, 15: (Q.12) (S.P-j^ = 200m, S . P 3 = 600m) " . . . you c o u l d e s t i m a t e about 400m . . . ( I : Why 4 0 0 m ? ) . . . i t ' s not 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-than 400?) . . . a l i t t l e b i t l e s s ( I : How c o u l d you ge t the e x a c t d i s t a n c e between the two spots? ) . . . measure the a n g l e between the l i n e s g o i n g from me t o the two s p o t s , u s i n g t r i g n o m e t r y o r the ang le t o c a l c u l a t e the d i s t a n c e between the two s p o t s " L i s a , 1 5 : ( Q . 1 2 ) ( S . P 1 = 200m, S . P 3 = 450m) " I s u b t r a c t the d i s t a n c e from me to the f i r s t s p o t from the d i s t a n c e from me to th e second s p o t . . . and s u b t r a c t a l s o the d i s t a n c e from the f i r s t spot t o here (the p r o j e c t i o n p o i n t o f on the s t r a i g h t l i n e c o n n e c t i n g the s u b j e c t w i t h P3) ... I have t o s u b t r a c t t h a t because I am g o i n g i n s t r a i g h t l i n e , the o t h e r way i s f a r t h e r o u t " . A l t h o u g h b o t h s u b j e c t s used two d i f f e r e n t s t r a t e g i e s , i t i s c l e a r t h a t they d i d so because they r e a l i z e d t h a t the two d i s t a n c e s were no t p a r a l l e l and c o n v e n t i o n a l a r i t h m e t i c o p e r a t i o n s c o u l d no t be u s e d . I n f e r r e d Rule S V P - 3 : T h i s r u l e i s s i m i l a r to the p r e v i o u s one w i t h the d i f f e r e n c e t h a t s u b j e c t s h o l d i n g t h i s p r e c o n -c e p t i o n c i t e d the d i r e c t i o n ( q u a l i t a t i v e o r q u a n t i t a t i v e ) as an i m p o r t a n t a s p e c t t o c o n s i d e r . A h i g h p e r c e n t a g e o f s u b j e c t s , 8 o f 20, appeared t o h o l d t h i s c o n c e p t i o n . T h i s c o u l d mean t h a t these s u b j e c t s would be i n a b e t t e r p o s i t i o n t o u n d e r s t a n d the o p e r a t i o n o f v e c t o r s u b t r a c t i o n when i t i s f o r m a l l y p r e s e n t e d i n an i n s t r u c t i o n a l s^e^tting. E x c e r p t s from i n t e r v i e w d a t a f o l l o w . B e c k y , 15 : (Q.12) ( S ^ = 50m S . P 3 = 200m) " i t ' s about 150 away ( I : Why?) w e l l , t h a t i s 200 and t h i s 50, and i t i s no t v e r y much i n an a n g l e . . . then 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 degrees a n g l e " . -153-T r a c y , 1 5 : ( Q . 1 2 ) ( S . P 1 = 200m, S . P 3 = 600m) " I t e l l them t o go . . . (pause) . . . about 400 o r 500 . . . because they have to go t h i s way . . . they have t o go f a r t h e r . . . I t e l l to go t o the l e f t about 450m . . . o r 500m . . . ( I : Why 500?) . . . because i s q u i c k e r f o r me to 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 to the p r o j e c t i o n o f P^ on the 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 about 200m and from me t o h e r (P3) i s 600 . . . then i t w i l l be 400 but he i s here (Pi) then he has to take a l o n g e r way . . . about 450" . Both s u b j e c t s p r o v i d e d d i r e c t i o n i n a d d i t i o n to ways o f o b t a i n i n g the magnitude o f the r e s u l t a n t . T a b l e 4.10 R u l e - M o d e l f o r SVP The r e s u l t a n t o f the 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 e q u a l t o the a r i t h m e t i c s u b t r a c t i o n o f the magnitudes and d i r e c t i o n i s no t c o n s i d e r e d " . "The r e s u l t a n t o f the 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 the magnitudes and d i r e c t i o n i s not c o n s i d e r e d " . "The r e s u l t a n t o f the 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 the magnitudes and d i r e c t i o n i s c o n s i d e r e d " . Rule S V P - I : Ru le S V P - I I : Ru le S V P - I I I : - 1 5 4 -4 . 3 . 6 I n f e r r e d Rules f o r Re ference B o d i e s f o r  Moving O b j e c t s (RPM) The. aim o f t h i s second p a r t about r e f e r e n c e p o i n t was t o uncover how s t u d e n t s d e a l t w i t h the prob lem o f d e s c r i b i n g b o d i e s a f f e c t e d by d i f f e r e n t k i n d s o f m o t i o n s . The s p e c i f i c s i t u a t i o n ( task 2) was to d e s c r i b e the mot ion o f a motor -boa t which was a f f e c t e d b o t h by the mot ion c r e a t e d by i t s motor , and a l s o by the c u r r e n t o f the r i v e r . T h i s s e c t i o n a t tempted to addres s the f o l l o w i n g q u e s t i o n s . How d i d s t u d e n t s t e l l t h a t the boa t i s i n m o t i o n D i d they see the need f o r a r e f e r e n c e p o i n t t o d e s c r i b e a mot ion? D i d they r e a l i z e t h a t the b o a t had d i f f e r e n t k i n d s o f mot ions w i t h r e s p e c t t o d i f f e r e n t r e f e r e n c e p o i n t s ? What d i d they choose as r e f e r e n c e p o i n t s ? F o r t h i s 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 , the phra se " r e f e r e n c e body" was used i n s t e a d o f " r e f e r e n c e p o i n t T h i s was done because t h e moving body o f water c o u l d be used as a second r e f e r e n c e frame to which the m o t i o n o f the boa t might r e f e r . Ques t ions o f I n t e r v i e w P r o t o c o l f o r RPM Q u e s t i o n s : 13b, 13c , and 13d i n t a sk two. Ques t ion 13b: ( A f t e r s u b j e c t has seen p a r t o f the f i l m w i t h the boa t moving on the r i v e r ) . W i t h r e s p e c t t o what i s the b o a t moving? Ques t ion 13c : Was the boa t moving w i t h r e s p e c t t o t h e water? -155-Ques t ion 13d; ( I f s u b j e c t has a " y e s " answer f o r q u e s t i o n 1 3 c ) : I s the speed o f the boa t w i t h r e s p e c t t o you (or b r i d g e o r shore) and the speed o f the boa t w i t h r e s p e c t t o the water the same o r d i f f e r e n t ? N o t e : I t i s v e r y i m p o r t a n t t o ment ion here the d i f f i c u l t i e s t h a t some s u b j e c t s e n c o u n t e r e d i n u n d e r s t a n d i n g the meaning o f these q u e s t i o n s . In p a r t i c u l a r the s u b j e c t s had d i f f i c u l t i e s i n u n d e r s t a n d i n g the p h r a s e : w i t h r e s p e c t t o ( t h i s i s p a r t o f q u e s t i o n s 13b and 1 3 c ) . The f o l l o w i n g e q u i v a l e n t q u e s t i o n was used to f a c i l i t a t e the u n d e r s t a n d i n g o f the s i t u a t i o n : "How c o u l d you t e l l t h a t the boa t i s moving?" To u n d e r s t a n d the s e r i o u s n e s s o f the p r o b l e m , we have t o c o n s i d e r t h a t o n l y 10 o u t o f 20 s u b j e c t s u n d e r s t o o d q u e s t i o n 13b when f i r s t a s k e d , and o n l y 6 u n d e r s t o o d q u e s t i o n 13c . A l t h o u g h t h i s i s c l o s e r t o a semant ic p r o b l e m , i t i s an i m p o r t a n t f a c t o r to c o n s i d e r when p r e p a r i n g c u r r i c u l u m m a t e r i a l . Textbook w r i t e r s and s c i e n c e t e a c h e r s o f t e n seem t o take f o r g r a n t e d t h a t s t u d e n t s u n d e r s t a n d -ttle phra se " w i t h r e s p e c t t o " , bu t the f i n d i n g s o f the p r e s e n t s tudy show t h a t t h i s appears no t t o be the c a s e . -156-T a b l e 4.11 L i s t o f I n f e r r e d Ru le s : f o r Re ference B o d i e s  f o r Moving O b j e c t s (RPM) Each i n f e r r e d r u l e c o v e r s two a s p e c t s . The f i r s t r e l a t e s t o s t a t i o n a r y r e f e r e n c e b o d i e s ( i . e . , o b s e r v e r on the b r i d g e , the b r i d g e , o r the s h o r e ) , and the second a s p e c t r e f e r s to the s e l e c t i o n o f the moving water (the r i v e r ) as a r e f e r e n c e body . In the d e s c r i p t i o n o f the i n f e r r e d r u l e s , b o t h types o f r e f e r e n c e b o d i e s ( s t a t i o n a r y and moving) are s e p a r a t e d by the c o n j u n c t i o n and . Four i n f e r r e d r u l e s were o b t a i n e d f o r t h i s v e c t o r c h a r a c t e r i s t i c , which have the f o l l o w i n g common open ing s e n t e n c e : " D e s c r i p t i o n o f the mot ion o f the motor -boa t on the r i v e r by an o b s e r v e r on the b r i d g e . . . I n f e r r e d Rule RPM-1: Does no t d i s t i n g u i s h movements r e -l a t i v e to a s t a t i o n a r y body ( e . g . , b r i d g e ) and r e l a t i v e t o a moving body ( e . g . , the w a t e r ) " ( 3 F , 0 M ) . -157-Inferred Rule RPM-2: Distinguishes movement r e l a t i v e to an i m p l i c i t reference point ( i . e . , the observer) and r e l a t i v e to a moving body (e.g., the water)". (1F,2M) Inferred Rule RPM-3:. Distinguishes movements r e l a t i v e to several stationary bodies (e.g., the bridge, shore) and r e l a t i v e to a moving body (e.g., the water)". (4F,6M) Inferred Rule RPM-4: Distinguishes movements r e l a t i v e to one stationary body (e.g., the bridge) and r e l a t i v e to a moving body (e.g., the water)". (0F,3M). (X 2 = 2.90 df = 4 p > .05; 1S(1M) was not assigned an inf e r r e d r u l e ) . Discussion of Results Inferred Rule RPM-1: This