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UBC Theses and Dissertations

An investigation of student ideas regarding two physical phenomena Axford, James Herbert 1986

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AN INVESTIGATION OF STUDENT IDEAS REGARDING TWO PHYSICAL PHENOMENA By JAMES HERBERT AXFORD B.Sc, University of B r i t i s h Columbia, 1972 B.Ed., University of Lethbridge, 1974 A THESIS SUBMITTED IN THE REQUIREMENTS MASTER PARTIAL FULFILLMENT OF FOR THE DEGREE OF OF ARTS 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 September, 198 6 ©James Herbert Axford, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department O f Mathematics and Science Education The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oct. i. igsfi /ai ABSTRACT The general problem investigated was the r e l a t i o n s h i p between the ideas that a student derives from a study of s c i e n t i f i c paradigms and those the student derives from other informal sources. There were s p e c i f i c facets of the general problem: to investigate i f students possess c h a r a c t e r i s t i c ideas, to investigate the e f f e c t of physics t r a i n i n g on student ideas, to investigate i f student ideas contain inconsistencies, and i f students with consistent ideas are successful i n applying s c i e n t i f i c paradigms. Student ideas of two physical phenomena, accelerated motion and simple harmonic motion, were investigated using three paper and pencil t e s t s . The boundaries between which ideas a student derived from informal sources and from the study of s c i e n t i f i c paradigms was not clear and therefore the case study methodology was used. Different facets of the investigation used d i f f e r e n t cases: varying from the classroom to the in d i v i d u a l student. It was found that the use of a variety of instruments allowed the ideas a student derives from informal sources and those the student derives from a study of the s c i e n t i f i c paradigms to be revealed. Students of d i f f e r e n t age/grade ranges were found to possess ideas that were c h a r a c t e r i s t i c . Physics i n s t r u c t i o n was found to have s i g n i f i c a n t e f f e c t s . However, i t was also found that some students retained ideas that were inconsistent with the s c i e n t i f i c paradigms even after physics i n s t r u c t i o n . This i l l u s t r a t e d that students could possess ideas that were inconsistent with each other. However, i t was shown that i t was possible for a student to resolve these inconsistencies. It was proposed that teaching strategies needed to be developed that would help students resolve inconsistent ideas. TABLE OF CONTENTS Abstract . . . . . . . . . i i L i s t of Tables . . . . . . . v i i Acknowledgements . . . . . . . v i i i Chapter I - INTRODUCTION 1. Background of the Problem 1 2. Purpose of t h i s Research . . . . . 6 3. Research Problem . . . . . . 7 4. Definitions . . . . . . . 8 4.1 S c i e n t i f i c Community. . . . . 8 4.2 Paradigm and Exemplar . . . . 8 4.3 Viewpoint . . . . . . . 1 0 4.4 Cha r a c t e r i s t i c Viewpoint . . . . 1 0 4.5 Paradigm completion questions . . . 1 1 4.6 Textbook problems . . . . . 1 1 5. Rationale for the Study . . . . . 1 1 6. Hypotheses . . . . . . . 1 3 7. Assumptions . . . . . . . 14 8. Delimitation of the Study . . . . 1 5 9. J u s t i f i c a t i o n of the Study . . . . 16 Chapter II - LITERATURE REVIEW 1. Introduction . . . . . . . 1 7 2. Acceleration: A Piagetian Task . . . 19 3. Experts and Novices . . . . . 2 1 4. Interview About Instances. . . . . 2 2 5. P r o j e c t i l e s and Pendula . . . . . 2 3 6. Hermeneutics . . . . . . . 2 6 i v Chapter III - METHODOLOGY 1. Recaptitulation of the Problem 2. Target Popultation 3. Sampling Plan . . . . . 4. Sample . . . . . . 5. Design . . . . . . 5.0 Introduction . . . . 5.1 Case Study Methodology 5.2 Physical Phenomena Under Study . 5.21 The Pendulum 5.22 Accelerated Motion 5.23 Free F a l l . . . . 5.24 P r o j e c t i l e Motion 5.25 Summary of Accelerated Motion 5.3 Data Gathering Instruments 5.30 Introduction 5.31 Paradigm completion questions 5.3 2 Comparison questions 5.3 3 Textbook questions 5.34 Summary of questions 5.4 Data C o l l e c t i o n . . . . 5.5 Analysis of Data 5.6 Interpretation of Results . 5.61 The F i r s t S p e c i f i c Problem 5.62 The Second S p e c i f i c Problem 5.63 The Third S p e c i f i c Problem 5.64 The Fourth S p e c i f i c Problem . . 62 Chapter IV - RESULTS AND CONCLUSIONS 1. Introduction . . . . . . . 64 2. The F i r s t S p e c i f i c Problem . . . . 65 2.1 Comments on the Textbook Problems . . 6 5 2.2 Comparison Questions . . . . 66 2.21 Harmonic Motion. . . . . 6 6 2.22 Accelerated Motion . . . . 69 2.3 Paradigm Completion Questions . . . 70 2.3lHarmonic and Accelerated Motion. . 71 2.4 Conclusions for the F i r s t S p e c i f i c Problem 72 3. The Second S p e c i f i c Problem . . . . 7 4 3.1 Textbook Problems . . . . . 75 3.2 Comparison Questions . . . . . 7 6 3.3 Textbook Problems . . . . . 77 3.4 Conclusions for the Second S p e c i f i c Problem 78 4. The Third S p e c i f i c Problm. . . . . 79 4.1 The F i r s t Case . . . . . . 79 4.2 The Second Case . . . . . . 80 4.3 Conclusions for the Third S p e c i f i c Problem 82 5. The Fourth S p e c i f i c Problem . . . . 83 6. General Conclusion . . . . . . 87 7. Implications . . . . . . . 88 Tables . . . . . . . . . 90 Bibliography . . . . . . . . 9 6 Appendix . . . . . . . . . 9 9 v i LIST OF TABLES Table Page 1 Harmonic Motion Comparison Question Results 91 2 Accelerated Motion Comparison Question Results 92 3 Paradigm Completion Questions 93 4 Paradigm Completion Questions 94 5 Combined Paradigm Completion Questions 94 6 Accelerated Motion Paradigm Completion Questions For Grade Twelve Students 95 v i i ACKNOWLEDGEMENTS A special acknowledgement i s extended to Dr. Walter Boldt, my thesis advisor, for both his advice and his patience. I would also thank the other member of my Committee Dr. C a r l i s l e . I wish to extend my sincere appreciation to my wife, Nancy, for her continued encouragement and support. In addition, I wish to thank my daughter, Mary Jane, for the summer holidays she f o r f e i t e d . v i i i 1 CHAPTER I - INTRODUCTION 1. B a c k g r o u n d o f t h e P r o b l e m . Kuhn i n h i s b o ok S t r u c t u r e o f S c i e n t i f i c R e v o l u t i o n s s t a t e s t h a t s c i e n c e d o e s n o t p r o g r e s s i n c r e m e n t a l l y ; b u t r a t h e r , h a s l o n g p e r i o d s o f ' n o r m a l s c i e n c e ' b r o k e n by b r i e f p e r i o d s o f c r i s i s d u r i n g w h i c h t h e f i e l d i s r e c o n s t r u c t e d . D u r i n g p e r i o d s o f n o r m a l s c i e n c e ' s c i e n t i f i c e n d e a v o u r s a r e b a s e d on " t h e a s s u m p t i o n t h a t t h e s c i e n t i f i c c o m m u n i t y knows what t h e w o r l d i s l i k e ... (and) assumes t h a t t h e y know t h e a n s w e r s and t h a t t h e y ( t h e a n s w e r s ) e x i s t ( 1 9 6 2 , pp. 5 . ) . " Kuhn u s e s t h e t e r m ' p a r a d i g m ' t o d e s c r i b e n o t o n l y t h e k n o w l e d g e s p e c i f i c t o t h e c o m m u n i t y b u t a l s o " t h e m e t h o d s , t h e p r o b l e m - f i e l d , a n d s t a n d a r d s o f s o l u t i o n a c c e p t e d ( p p . 1 0 2 ) . " T h i s i s why Kuhn a r g u e s t h a t s c i e n t i s t s a r e i n r e a l i t y j u s t ' p u z z l e s o l v e r s ' b e c a u s e t h e y h a v e b e e n g i v e n t h e " p i e c e s " and t h e " r u l e s " t o s o l v e t h e p u z z l e ( 1 9 6 2 , pp. 3 6 - 4 2 . ) . S t u d e n t s become s c i e n t i s t s by s t u d y i n g a n d p r a c t i c i n g t h e p a r a d i g m s o f t h e s c i e n t i f i c c o m m u n i t y t h a t a r e " r e v e a l e d i n i t s t e x t b o o k s , l e c t u r e s , and l a b o r a t o r y e x e r c i s e s " ( p . 4 3 S S R ) . Kuhn b e l i e v e s t h a t t h e a c c e p t e d p a r a d i g m s do n o t h a v e "an e x p l i c i t o r d i s c o v e r a b l e s e t o f r u l e s ... b u t may [ b e ] r e l a t e d b y r e s e m b l a n c e a n d by m o d e l i n g " and t h a t s c i e n t i s t s "work f r o m m o d e l s a c q u i r e d t h r o u g h e d u c a t i o n ... o f t e n w i t h o u t k n o w i n g o r n e e d i n g t o know what c h a r a c t e r i s t i c s h a v e g i v e n t h e s e m o d e l s t h e s t a t u s o f p a r a d i g m s " ( p p . 4 5 ) . M a s t e r m a n ( 1 9 7 0 ) s t a t e s t h a t Kuhn's c o n c e p t i o n o f 2 p a r a d i g m i s "a f u n d a m e n t a l i d e a a n d a new one i n t h e p h i l o s o p h y o f s c i e n c e " h o w e v e r , she c a u t i o n s t h a t Kuhn u s e s t h e t e r m p a r a d i g m " i n n o t l e s s t h a n t w e n t y - o n e d i f f e r e n t s e n s e s . " Kuhn a c k n o w l e d g e s t h a t h i s d e f i n i t i o n o f " p a r a d i g m " has two m a i n s e n s e s : one o f w h i c h i s " g l o b a l , e m b r a c i n g a l l t h e s h a r e d c o m m i t m e n t s o f a s c i e n t i f i c g r o u p ; " a nd t h e o t h e r w h i c h " i s o l a t e s a p a r t i c u l a r l y i m p o r t a n t s o r t o f commitment" ( K u h n , 1974, pp. 4 6 0 . ) . He i d e n t i f i e s t h r e e t y p e s c o m m i t m e n t s : 1. S y m b o l i c g e n e r a l i z a t i o n s a r e t h o s e e x p r e s s i o n s d e p l o y e d w i t h o u t q u e s t i o n b y t h e g r o u p , w h i c h c a n be c a s t i n some l o g i c a l f o r m ... t h e y a r e f o r m a l , o r t h e r e a d i l y f o r m a l i z a b l e , c o m p o n e n t s o f t h e d i s c i p l i n e . 2. M o d e l s a r e w h a t p r o v i d e t h e g r o u p w i t h p r e f e r r e d a n a l o g i e s o r , when d e e p l y h e l d , w i t h a n o n t o l o g y . A t one e x t r e m e t h e y a r e h e u r i s t i c ... a t t h e o t h e r , t h e y a r e o b j e c t s o f m e t a p h y s i c a l c o mmitment. 3. E x e m p l a r s a r e c o n c r e t e p r o b l e m s o l u t i o n s , a c c e p t e d by t h e g r o u p a s , i n q u i t e t h e u s u a l s e n s e , p a r a d i g m a t i c . T h i s s t u d y u s e d Kuhn's c o n c e p t o f s c i e n t i f i c p r o g r e s s a n d how s c i e n t i f i c c o m m u n i t i e s e d u c a t e s t u d e n t s t h r o u g h t h e u s e o f p a r a d i g m s . The g o a l o f e d u c a t i o n i s t o g i v e s t u d e n t s t h e g l o b a l c o m m i t m e n t s ; h o w e v e r , t h i s i s done p r i m a r i l y t h r o u g h t h e s c i e n t i f i c c o m m u n i t i e s commitment t o e x e m p l a r s . F r e d e r i c k Suppe s t a t e s t h a t Kuhn's e x e m p l a r s " a r e t h e a c c e p t e d a p p l i c a t i o n s o f t h e s y m b o l i c g e n e r a l i z a t i o n s t o v a r i o u s c o n c r e t e p r o b l e m s one f i n d s i n t h e e x a m p l e s a nd s o l u t i o n s t o t h e e x e r c i s e s i n s t a n d a r d t e x t b o o k s a n d 3 l a b o r a t o r y m a n u a l s . Suppe f u r t h e r s t a t e s t h a t t h e y h a v e t h e f o l l o w i n g c h a r a c t e r i s t i c s : 1. T h e y n o t o n l y i l l u s t r a t e how t o a t t a c h s y m b o l i c g e n e r a l i z a t i o n s t o n a t u r e , b u t a l s o i n d i c a t e w h i c h f o r m s o f t h e l a w s o r s y m b o l i c g e n e r a l i z a t i o n s a r e a p p l i c a b l e u n d e r v a r i o u s c i r c u m s t a n c e s . 2. S t u d e n t s may l e a r n f r o m t h e s t u d y o f e x e m p l a r s how t o ( m a t h e m a t i c a l l y ) m a n i p u l a t e t h e s y m b o l i c g e n e r a l i z a t i o n s t o o b t a i n s o l u t i o n s a s w e l l a s t o a t t a c h s y m b o l i c g e n e r a l i z a t i o n s t o n a t u r e . 3. I n t h e p r o c e s s o f t r a n s l a t i o n f r o m t h e i n f o r m a l d e s c r i p t i o n s o f t h e phenomena t o t h e i r c a n o n i c a l r e d e s c r i p t i o n s , v a r i o u s o t h e r s y m b o l l i e g e n e r a l i z a t i o n s may be e m p l o y e d , ( p . 4 84) Kuhn s u m m a r i z e s t h e i m p o r t a n c e o f e x e m p l a r s b y s t a t i n g t h a t a f t e r a p a r a d i g m has b e e n a c c e p t e d t h e a p p l i c a t i o n s o f t h e p a r a d i g m a r e p l a c e d " i n t o t e x t b o o k s f r o m w h i c h t h e f u t u r e p r a c t i t i o n e r w i l l l e a r n h i s t r a d e ... ( t h e ) p r o c e s s o f l e a r n i n g a t h e o r y d e p e n d s upon s t u d y o f a p p l i c a t i o n s i n c l u d i n g p r a c t i c e p r o b l e m s o l v i n g ... ( t h e s t u d e n t ) d i s c o v e r s m e a n i n g ... l e s s f r o m t h e i n c o m p l e t e t h o u g h s o m e t i m e s h e l p f u l d e f i n i t i o n s i n h i s t e x t t h a n b y o b s e r v i n g and p a r t i c i p a t i n g i n t h e a p p l i c a t i o n o f t h e s e c o n c e p t s t o p r o b l e m - s o l u t i o n ( s ) " ( 1 9 6 2 , pp. 4 6 . ) . E x e m p l a r s a r e a f o c a l p o i n t i n t h e e d u c a t i o n o f s t u d e n t s ; and e x e m p l a r s a r e a f o c a l p o i n t i n t h i s s t u d y . The s c i e n t i f i c c o m m u n i t y u s e s e x e m p l a r s t o e d u c a t e t h e s t u d e n t ; t h i s s t u d y u s e d e x e m p l a r s t o r e v e a l t h e i d e a s o f s t u d e n t s . E d u c a t i o n i s a p r o c e s s o f c h a n g e . The s c i e n t i f i c 4 c o m m u n i t y a l s o u n d e r g o e s c h a n g e . Kuhn a s s e r t s t h a t s c i e n c e c a n n o t r e j e c t a p a r a d i g m w i t h o u t a t t h e same t i m e , h a v i n g an a l t e r n a t e p a r a d i g m t o be a c c e p t e d . The t r a n s i t i o n b e t w e e n c o m p e t i n g p a r a d i g m s i s n o t done s l o w l y b u t " l i k e t h e G e s t a l t s w i t c h , i t m u s t o c c u r a l l a t o n c e ( t h o u g h n o t n e c e s s a r i l y i n an i n s t a n t ) . " ( 1 9 6 2 , pp. 149.) Kuhn e m p h a s i z e s t h a t t h e t r a n s i t i o n c a n n o t be a c h i e v e d by e x t e n d i n g o r r e f i n i n g t h e o l d p a r a d i g m b u t " r a t h e r i s a r e c o n s t r u c t i o n o f t h e f i e l d f r o m new f u n d a m e n t a l s , a r e c o n s t r u c t i o n t h a t c h a n g e s some o f t h e f i e l d s m o s t e l e m e n t a r y t h e o r e t i c a l g e n e r a l i z a t i o n s a s w e l l a s many o f i t s p a r a d i g m a t i c m e t h o d s and a p p l i c a t i o n s " ( 1 9 6 2 , p p . 8 4 . ) . Kuhn s t a t e s t h a t t h e s c i e n t i f i c c o m m u n i t y i s i n a s t a t e o f " c r i s i s " when i t u n d e r g o e s a t r a n s i t i o n f r o m one p a r a d i g m t o a n o t h e r . P i a g e t d e s c r i b e s a s i m i l a r p r o c e s s , " d i s e q u i l i b r i u m , " i n t e r m s o f t h e i n d i v i d u a l . P i a g e t d e s c r i b e s t h e p r o c e s s o f human i n t e l l e c t u a l d e v e l o p m e n t a s s u c c e s s i v e c h a n g e s i n s c h e m a t a . A p e r s o n i n t e r a c t s w i t h t h e w o r l d by t h e u s e o f s c h e m a t a . As a r e s u l t o f a p e r s o n ' s e x p e r i e n c e w i t h t h e w o r l d t h e s c h e m a t a must be m o d i f i e d t h r o u g h t h e p r o c e s s e s o f a s s i m i l a t i o n and a c c o m o d a t i o n . A s s i m i l a t i o n i s t h e a d d i t i o n o f i d e a s t h a t a r e c o n s i s t e n t w i t h t h o s e i d e a s a l r e a d y o r g a n i z e d . F o r i d e a s t o be c o n s i s t e n t t h e y must be c o n s i s t e n t w i t h e a c h o t h e r , w i t h o b s e r v e d i d e a s , a n d w i t h t h e p e r s o n ' s l o g i c a l and r e a s o n i n g s t r u c t u r e . A c c o m o d a t i o n i s a p r o c e s s t h a t m o d i f i e s t h e e x i s t i n g schema t o make i t c o n s i s t e n t w i t h new i d e a s . When a 5 p e r s o n c a n no l o n g e r u s e t h e p r o c e s s e s o f a s s i m i l a t i o n o r a c c o m o d a t i o n t o k e e p h i s / h e r p r e s e n t s c h e m a t a i n a g r e e m e n t w i t h t h e i r w o r l d , a p e r i o d o f d i s e q u i l i b r a t i o n h a s b e e n r e a c h e d . I t i s d u r i n g t h i s p e r i o d t h a t e s s e n t i a l l y new s c h e m a t a a r e f o r m e d . The p r o c e s s e s o f " c r i s i s " i n t h e s c i e n t i f i c c o m m u n i t y and " d i s e q u i l i b r i u m " i n t h e i n d i v i d u a l a r e t h e r e f o r e , a n a l o g o u s . T h i s s t u d y i d e n t i f i e d some o f t h e p a r a d i g m s o r s c h e m a t a o f s t u d e n t s . The s t u d y a l s o i d e n t i f i e d s t u d e n t s w i t h p a r a d i g m s o r s c h e m a t a t h a t w e r e i n c o n s i s t e n t w i t h e a c h o t h e r . S t u d e n t s w i t h i n c o n s i s t e n t p a r a d i g m s o r s c h e m a t a w e r e i n t e r p r e t e d t o be i n a s t a t e o f " c r i s i s " o r " d i s e q u i l i b r a t i o n . " The p r e s e n t r e s e a r c h p r o b l e m u s e d t h e c o n c e p t s o f p a r a d i g m s a n d s t u d e n t d e v e l o p m e n t t o i n v e s t i g a t e s t u d e n t a c q u i s i t i o n o f s c i e n t i f i c k n o w l e d g e . The e d u c a t i o n s y s t e m i s d e s i g n e d t h r o u g h g r a d e s and l e v e l s t o p r e s e n t s t u d e n t s w i t h e x p e r i e n c e s t h a t w i l l a l l o w them t o be i n c r e a s i n g l y a b l e t o i n t e r p r e t t h e w o r l d a r o u n d them i n t o more c o m p l e x p a r a d i g m s . S c i e n c e e d u c a t i o n t r i e s t o i m p a r t t o s t u d e n t s t h e s k i l l s a n d k n o w l e d g e t h a t w i l l a l l o w them t o i n t e r p r e t t h e i r p h y s i c a l w o r l d . The BC P h y s i c s C u r r i c u l u m G u i d e ( 1985) i d e n t i f i e s f o u r g o a l s ; one o f w h i c h i s " t o i n t r o d u c e t o s t u d e n t s t h e CONTENT o f p h y s i c s : t h e m a i n i d e a s , p r i n c i p l e s , a n d u n i f y i n g c o n c e p t s . " T h i s " c o n t e n t " i s w h a t Kuhn r e f e r s t o a s " g l o b a l p a r a d i g m s " ( 1 9 7 4 , pp. 4 6 0 . ) . I n a K u h n i a n s e n s e , t h e a i m o f s c i e n t i f i c e d u c a t i o n i s t o r e p l a c e t h e s t u d e n t s ' own p a r a d i g m s w i t h t h e s c i e n t i f i c c o m m u n i t i e s ' p a r a d i g m s . I n a 6 P i a g e t i a n s e n s e , t h e a i m o f e d u c a t i o n i s t o m o d i f y t h e s t u d e n t ' s p r e s e n t s c h e m a t a w i t h t o s c h e m a t a t h a t a r e a b l e t o d e a l i n c r e a s i n g l y w i t h more c o m p l e x k n o w l e d g e . R e c e n t w o r k i n v e s t i g a t i n g w hat has b e e n c a l l e d s t u d e n t " m i s c o n c e p t i o n s " , " p r e c o n c e p t i o n s " , " a l t e r n a t i v e f r a m e w o r k s " , " i n t u i t i v e r u l e s " , a nd t h e l i k e h a s shown t h a t s t u d e n t s do f o r m t h e i r own " v i e w p o i n t s " o f t h e p h y s i c a l w o r l d ( f o r r e v i e w s o f t h e l i t e r a t u r e s e e : D r i v e r and E a s l e y , 1978; M c D e r m o t t , 1984; B r a c e , 1 9 8 4 ) . T h i s s t u d y i n v e s t i g a t e d t h e " v i e w p o i n t " o f s t u d e n t s . The v i e w p o i n t o f a s t u d e n t d o e s n o t r e p r e s e n t a s i n g l e i d e a o r v i e w ; b u t r a t h e r a c o l l e c t i o n o f i d e a s t h a t may o r may n o t be c o n s i s t e n t w i t h e a c h o t h e r . G u n s t o n e a n d W a t t s (1985) s t a t e t h a t "most p e o p l e h a v e a s e l f - r e f e r e n c e s y s t e m t h a t n o t o n l y h e l p s them t o s o r t o u t t h e a n s w e r t o a p r o b l e m b u t h e l p s them s o r t o u t w h a t t h e p r o b l e m was i n t h e f i r s t p l a c e . " The " s e l f - r e f e r e n c e " s y s t e m may be i n t e r p r e t e d t o be w hat t h i s s t u d y r e f e r s t o a s " v i e w p o i n t . " G u n s t o n e and W a t t s i n d i c a t e t h a t t h e v i e w p o i n t s o f p e o p l e a s s i s t them t o a s k q u e s t i o n s . T h i s s t u d y u s e s t h e q u e s t i o n s p o s e d by s t u d e n t s t o i d e n t i f y t h e i r " v i e w p o i n t . " 2. P u r p o s e o f t h i s R e s e a r c h . S t u d e n t s i n s e c o n d a r y s c h o o l s ( g r a d e s e i g h t t o t w e l v e ) a r e i n a p e r i o d o f c o n s i d e r a b l e i n t e l l e c t u a l c h a n g e - a c h a n g e t h a t P i a g e t d e s c r i b e s a s g o i n g f r o m c o n c r e t e t o a b s t r a c t t h o u g h t . However C h a n d l e r (1975) c o n c l u d e s t h a t : ... t h e t a s k o f t h e a d o l e s c e n t i s n o t t o l e a r n t o p r o c e s s a l l e x p e r i e n c e t h r o u g h 7 h i g h l y f o r m a l a n d d e c e n t e r e d modes o f r e l a t i v i s t i c t h o u g h t . The p r o b l e m i s , r a t h e r , t o i d e n t i f y t h o s e e x p e r i e n c e s w h i c h a r e b e s t a p p r e c i a t e d f r o m m u l t i p l e p e r s p e c t i v e s , w h i l e p r e s e r v i n g a s a l e g i t i m a t e o p t i o n t h e r i g h t t o f e e l a p a r t i c u l a r way o r become c o m m i t t e d t o a p a r t i c u l a r v i e w o r g o a l o r c o u r s e o f a c t i o n i n t h e f a c e o f a v a i l a b l e a l t e r n a t i v e s . The s t u d e n t ' s v i e w p o i n t c o n t a i n s t h e i d e a s t h a t a r e h i s / h e r " a v a i l a b l e a l t e r n a t i v e s . " A d o l e s c e n t s t u d e n t s e n t e r s c i e n c e c l a s s e s w i t h a v i e w p o i n t a b o u t p a r t i c u l a r p h y s i c a l phenomena. S c i e n t i f i c p a r a d i g m s a b o u t t h e phenomena a r e i n t r o d u c e d i n s c i e n c e c l a s s e s t o t h e s t u d e n t . The r o l e o f e d u c a t i o n i s t o a s s i s t t h e s t u d e n t t o make t h e most a p p r o p r i a t e u s e o f t h e i d e a s t h a t c o m p r i s e t h e s t u d e n t ' s a v a i l a b l e a l t e r n a t i v e s . T h i s s t u d y i d e n t i f i e d some o f t h e s c i e n t i f i c i d e a s t h a t s t u d e n t s p o s s e s s a n d i n v e s t i g a t e d t h e i n c o n s i s t e n c i e s o f some s t u d e n t s ' i d e a s . An u n d e r s t a n d i n g o f t h e i d e a s s t u d e n t s p o s s e s s a n d t h e i n c o n s i s t e n c i e s t h a t may d e v e l o p i s o f b e n e f i t t o s c i e n c e e d u c a t o r s a n d may r e q u i r e t e a c h e r s t o c h a n g e t h e m e t h o d s u s e d t o i n t r o d u c e s c i e n c e c o n c e p t s t o s t u d e n t s . 3 . R e s e a r c h P r o b l e m . The g e n e r a l p r o b l e m was t o s t u d y t h e r e l a t i o n s h i p b e t w e e n t h e i d e a s t h a t a s t u d e n t d e r i v e s f r o m a s t u d y o f s c i e n t i f i c p a r a d i g m s a n d t h o s e t h e s t u d e n t d e r i v e s f r o m o t h e r i n f o r m a l s o u r c e s . The r a n g e o f p h y s i c a l phenomena i n v e s t i g a t e d was l i m i t e d t o a c c e l e r a t e d m o t i o n a nd h a r m o n i c m o t i o n . F o u r f a c e t s o f t h e g e n e r a l p r o b l e m w e r e a d d r e s s e d t h r o u g h s p e c i f i c p r o b l e m s . The s p e c i f i c p r o b l e m s w e r e : 8 3.1 Do t h e c l a s s e s t o be i n v e s t i g a t e d c o n s i s t o f s t u d e n t s t h a t p o s s e s s a ' c h a r a c t e r i s t i c v i e w p o i n t ? ' 3.2 What e f f e c t d o e s p h y s i c s t r a i n i n g h a v e o n t h e v i e w p o i n t s o f s t u d e n t s ? 3.3 Do s t u d e n t s p o s s e s s v i e w p o i n t s t h a t c o n t a i n i n c o n s i s t e n c i e s ? 3.4 Do s t u d e n t s p o s s e s s i n g a c o n s i s t e n t v i e w p o i n t , h a v e s u c c e s s i n a p p l y i n g s c i e n t i f i c p a r a d i g m s ? 4. D e f i n i t i o n s . 4.1 S c i e n t i f i c Community A s c i e n t i f i c c o m m u n i t y i s d e f i n e d b y Kuhn ( 1 9 7 4 , pp. 461.) a s f o l l o w s : ... t h e p r a c t i t i o n e r s o f a s c i e n t i f i c s p e c i a l i t y . Bound t o g e t h e r b y common e l e m e n t s i n t h e i r e d u c a t i o n a n d a p p r e n t i c e s h i p , t h e y s e e t h e m s e l v e s a n d a r e s e e n by o t h e r s a s t h e men r e s p o n s i b l e f o r t h e p u r s u i t o f a s e t o f s h a r e d g o a l s , i n c l u d i n g t h e t r a i n i n g o f t h e i r s u c c e s s o r s . S u c h c o m m u n i t i e s a r e c h a r a c t e r i z e d by t h e r e l a t i v e f u l l n e s s o f c o m m u n i c a t i o n w i t h i n t h e g r o u p a nd by t h e r e l a t i v e u n a n i m i t y o f t h e g r o u p ' s j u d g m e n t i n p r o f e s s i o n a l m a t t e r s . To a r e m a r k a b l e e x t e n t t h e members w i l l h a v e a b s o r b e d t h e same l i t e r a t u r e a n d d r a w n s i m i l a r l e s s o n s f r o m i t 4.2 P a r a d i g m a nd E x e m p l a r Kuhn " c o u l d n o t , when e x a m i n i n g t h e m e m b e r s h i p o f a s c i e n t i f i c c o m m u n i t y , r e t r i e v e e nough s h a r e d r u l e s t o a c c o u n t f o r t h e g r o u p ' s u n p r o b l e m a t i c c o n d u c t o f r e s e a r c h . " As a r e s u l t , Kuhn i n t r o d u c e d t h e t e r m " p a r a d i g m " t o r e p r e s e n t t h e " s h a r e d e x a m p l e s o f s u c c e s s f u l p r a c t i c e " t h a t a l l o w e d t h e s c i e n t i f i c c o m m u n i t y t o c o n t i n u e ( 1 9 7 4 , pp. 4 8 2 . ) . Kuhn 9 s t a t e s : "A p a r a d i g m i s w h a t t h e members o f a s c i e n t i f i c c o m m u n i t y , a n d t h e y a l o n e , s h a r e . " As s t a t e d e a r l i e r ( p p . 2) Kuhn h a s two m a i n s e n s e s o f p a r a d i g m : g l o b a l a nd s p e c i f i c . The g l o b a l p a r a d i g m i s composed o f "a s e t o f p a r a d i g m s . " The g l o b a l p a r a d i g m f o r m s a " d i s c i p l i n a r y m a t r i x " t h a t i s "composed o f o r d e r e d e l e m e n t s o f v a r i o u s s o r t s , e a c h r e q u i r i n g f u r t h e r s p e c f i c a t i o n ( p p . 4 6 2 ) . " Kuhn o u t l i n e s t h r e e t y p e s o f p a r a d i g m s : s y m b o l i c g e n e r a l i z a t i o n s , m o d e l s , and e x e m p l a r s . Kuhn s t a t e s t h a t p h i l o s o p h e r s g e n e r a l l y a s s e r t t h a t s c i e n t i s t s a t t a c h s y m b o l i c e x p r e s s i o n s t o n a t u r e t h r o u g h c o r r e s p o n d e n c e r u l e s . Kuhn a r g u e s t h a t he h a s n o t f o u n d a s c i e n t i f i c c o m m u n i t y w i t h a s e t o f r u l e s t h a t i s " n e a r l y s u f f i c i e n t i n number o r f o r c e t o a c c o u n t f o r t h e a c t u a l c o r r e l a t i o n s b e t w e e n f o r m a l i s m and e x p e r i m e n t made r e g u l a r l y and u n p r o b l e m a t i c a l l y b y members." I n a d d i t i o n , v e r y "few c o r r e s p o n d e n c e r u l e s a r e t o be f o u n d i n s c i e n c e t e x t s o r s c i e n c e t e a c h i n g ; " t h e r e f o r e , Kuhn c o n c l u d e s t h a t t h e r o l e o f c o r r e s p o n d e n c e r u l e s i s f u l f i l l e d by e x e m p l a r s . Kuhn d e s c r i b e s e x e m p l a r s a s f o l l o w s : S t u d e n t s o f p h y s i c s r e g u l a r l y r e p o r t t h a t t h e y h a v e r e a d t h r o u g h a c h a p t e r o f t h e i r t e x t , u n d e r s t o o d i t p e r f e c t l y , b u t n o n e t h e l e s s h a d d i f f i c u l t y s o l v i n g t h e p r o b l e m s a t t h e c h a p t e r ' s e n d . A l m o s t i n v a r i a b l y t h e i r d i f f i c u l t y i s i n s e t t i n g up t h e a p p r o p r i a t e e q u a t i o n s , i n r e l a t i n g t h e w o r d s a n d e x a m p l e s g i v e n i n t h e t e x t t o t h e p a r t i c u l a r p r o b l e m s t h e y a r e a s k e d t o s o l v e . O r d i n a r i l y , a l s o t h o s e d i f f i c u l t i e s d i s s o l v e i n t h e same way. The s t u d e n t d i s c o v e r s a way t o s e e h i s p r o b l e m a s l i k e a p r o b l e m he h a s a l r e a d y e n c o u n t e r e d . Once t h a t l i k e n e s s o r a n a l o g y h a s b e e n s e e n , o n l y m a n i p u l a t i v e d i f f i c u l t i e s r e m a i n . ... a n a c q u i r e d a b i l i t y t o s e e r e s e m b l a n c e s b e t w e e n a p p a r e n t l y 10 d i s p a r a t e problems plays i n the sciences a s i g n i f i c a n t p a r t of the r o l e u s u a l l y a t t r i b u t e d to correspondence r u l e s . ... These concrete probems with t h e i r s o l u t i o n s are what I p r e v i o u s l y r e f e r r e d to as exemplars, a community's standard examples ... A c q u i r i n g an arsenal of exemplars, j u s t as much as l e a r n i n g symbolic g e n e r a l i z a t i o n s , i s i n t e g r a l to the process by which a student gains access to the c o g n i t i v e achievements of h i s d i s c i p l a r y group (1974, pp. 470). 4.3 Viewpoint The viewpoint of a student i s assumed to be some form of mental r e p r e s e n t a t i o n of the p h y s i c a l world. The viewpoint would be s i m i l a r to th a t proposed by Johnson-Laird ( i n Gardner, 1985, pp. 368): Johnson-Laird p r e f e r s t o p o s i t at l e a s t three types of mental r e p r e s e n t a t i o n : p r o p o s i t i o n a l r e p r e s e n t a t i o n s that resemble n a t u r a l languages; mental models which are s t r u c t u r a l analogues of the world; and images t h a t are the perceptual c o r r e l a t e s of models from a p a r t i c u l a r p o i n t of view. We ought t o t h i n k of i n d i v i d u a l s as representing information at s e v e r a l d i f f e r e n t l e v e l s i f a b s t r a c t i o n : moreover, the form of r e p r e s e n t a t i o n at one l e v e l need not be the same as the form of re p r e s e n t a t i o n at another l e v e l . The viewpoint of the student i s s i m i l a r l y assumed t o c o n s i s t of ideas about f a m i l i a r p h y s i c a l phenomena. 4.4 C h a r a c t e r i s t i c viewpoint The c h a r a c t e r i s t i c viewpoint i s the hypothesized viewpoint, not of an i n d i v i d u a l , but of a group of students about a p a r t i c u l a r p h y s i c a l phenomena. As such, the c h a r a c t e r i s t i c viewpoint i s a c o l l e c t i o n of r e l a t e d ideas t h a t the group of students c h a r a c t e r i s t i c a l l y subscribe t o . 11 4.5 P a r a d i g m C o m p l e t i o n Q u e s t i o n s P a r a d i g m c o m p l e t i o n q u e s t i o n s a r e q u e s t i o n s i n w h i c h t h e s o l v e r i s s u p p l i e d o n l y w i t h s u f f i c i e n t d e t a i l s t o a l l o w h i m / h e r t o r e c o g n i z e t h e s c i e n t i f i c p a r a d i g m t h a t t h e q u e s t i o n e x e m p l i f i e s . 4.6 T e x t b o o k p r o b l e m s T e x t b o o k p r o b l e m s a s u s e d i n t h i s p a p e r , w i l l r e f e r t o t y p i c a l , w e l l - d e f i n e d p h y s i c s p r o b l e m s g i v e n i n p h y s i c s t e x t b o o k s f o r p e d a g o g i c a l u s e i n l e a r n i n g s c i e n t i f i c p a r a d i g m s . They h a v e c l e a r l y s t a t e d i n i t i a l s t a t e s and g o a l s t a t e s . The o p e r a t o r s f o r t h e s o l u t i o n a r e a l s o g i v e n and d e f i n e d i n a d v a n c e . 5. R a t i o n a l e f o r t h e S t u d y The c e n t r a l p r o b l e m o f t h i s r e s e a r c h i s t o s t u d y t h e r e l a t i o n s h i p b e t w e e n t h e i d e a s t h a t a s t u d e n t d e r i v e s f r o m a s t u d y o f s c i e n t i f i c p a r a d i g m s and t h o s e a s t u d e n t d e r i v e s f r o m o t h e r i n f o r m a l s o u r c e s . I n o r d e r t o i n v e s t a g a t e t h e i d e a s p o s s e s s e d by t h e s t u d e n t a f r a m e w o r k , o r some f o r m o f " k n o w l e d g e s t r u c t u r e " ( T h a g a r d , 1 9 8 4 ) , must be h y p o t h e s i z e d . T h e s e k n o w l e d g e s t r u c t u r e s a r e r e f e r r e d t o a s " s c h e m a t a , " " f r a m e s , " o r " s c r i p t s " a nd i n t h i s s t u d y a r e i d e n t i f i e d a s t h e s t u d e n t s ' v i e w p o i n t s . The s t u d e n t s ' v i e w p o i n t s c h a n g e a s t h e s t u d e n t s l e a r n . T h i s c h a n g e i s d e s c r i b e d by P i a g e t , t h r o u g h t h e p r o c e s s e s o f a s s i m i l a t i o n and a c c o m o d a t i o n t o s c h e m a t a ( v i e w p o i n t s ) . T h a g a r d s t a t e s , i n a g r e e m e n t w i t h K u hn, t h a t t h e d i f f i c u l t y a s c i e n t i s t w o u l d h a v e i n a b a n d o n i n g one p a r a d i g m ( v i e w p o i n t ) and a c c e p t i n g a new p a r a d i g m a s f o l l o w s : ... a d d i n g an e n t i r e f r a m e i s n o t t h e same as a c c e p t i n g a s e t o f p r o p o s i t i o n s . A new f r a m e m u s t be i n t e g r a t e d w i t h e x i s t i n g k n o w l e d g e b y e s t a b l i s h i n g i t s p l a c e i n t h e p r o c e s s i n g s y s t e m : new i n f o r m a t i o n i s u s e l e s s u n t i l p r o c e d u r a l c o n n e c t i o n s w i t h e x i s t i n g f r a m e s a r e i n p l a c e . Once a c o m p l e x f r a m e s y s t e m s u c h a s t h a t n e e d e d f o r s o l v i n g p h y s i c s p r o b l e m s i s f u n c t i o n i n g a s a w h o l e , p i e c e m e a l r e v i s i o n becomes p r o b l e m a t i c . ... [A p r i n c i p l e o f m e t h o d o l o g i c a l c o n s e r v a t i s m ] f o r f r a m e s y s t e m s m i g h t be s o m e t h i n g l i k e : I f y o u h a v e a f r a m e s y s t e m , i t i s u n r e a s o n a b l e t o g i v e i t up m e r e l y b e c a u s e t h e r e i s a n a v a i l a b l e a l t e r n a t i v e s y s t e m . T h i s p r i n c i p l e d e r i v e s i t s s u p p o r t f r o m two c o n s i d e r a t i o n s . F i r s t , b e c a u s e o u g h t i n p l i e s c a n , we c a n n o t be e n j o i n e d t o a b a n d o n a f r a m e s y s t e m i f i t i s n o t p o s s i b l e f o r us t o do s o . We c a n n o t r e p r o g r a m o u r s e l v e s : t h e c o n s t r u c t i o n a nd a l t e r a t i o n o f f r a m e s y s t e m s i s n o t w i t h i n o u r c o n s c i o u s c o n t r o l ... S e c o n d , e v e n i f i t w e r e p o s s i b l e , i t a p p e a r s t h a t t h e r e a r e e m p i r i c a l g r o u n d s f o r v i e w i n g t h e abandonment o f a f u n c t i o n i n g f r a m e s y s t e m a s u n d e r s i r a b l e ... The r a t i o n a l s t r a t e g y t o a d o p t when one l e a r n s o f t h e e x i s t e n c e o f a p l a u s i b l e a l t e r n a t i v e c o n c e p t u a l s y s t e m t o o n e ' s own i s t o a t t e m p t t o l e a r n t h e s y s t e m i n much t h e same way a s i t s p r o p o n e n t s h a v e d o n e . T h i s i s d i f f i c u l t , b e c a u s e o f i n t e r f e r e n c e f r o m t h e f r a m e s a l r e a d y i n p l a c e . However, i f one d o e s s u c c e e d i n d e v e l o p i n g t h e a l t e r n a t i v e f r a m e s y s t e m i n p a r a l l e l w i t h o n e ' s o r i g i n a l o n e , t h e n r a t i o n a l c h o i c e o f t h e s e c o n d o v e r t h e f i r s t i s p o s s i b l e . I n r e c e n t y e a r s t h e r e h a v e b e e n s t u d i e s w h i c h show t h a t p e o p l e c a n and do p o s s e s s a t t h e same t i m e , i d e a s t h a t may be i n c o n s i s t e n t w i t h e a c h o t h e r . M c D e r m o t t , i n h e r a r t i c l e , " R e s e a r c h o n C o n c e p t u a l U n d e r s t a n d i n g i n M e c h a n i c s " (1984) r e v i e w s r e s e a r c h o n t h e l e a r n i n g a nd t e a c h i n g o f b a s i c c o n c e p t s . A l t h o u g h t h e m o t i v a t i o n b e h i n d t h e d i f f e r e n t 13 research projects varied, i t was generally found that "students of d i f f e r e n t ages and educational backgrounds often begin t h e i r study of physics with very s i m i l a r ideas, many of which are i n c o n f l i c t with the concepts of physics. These ideas generally have an i n t u i t i v e basis and have proved to be highly r e s i s t a n t to i n s t r u c t i o n . " 6. Hypotheses It i s hypothesized that students' viewpoints are composed of a set of ideas; these viewpoints may be taken to be frames, schemata, s c r i p t s , paradigms, and so on. It i s further hypothesized that an in d i v i d u a l idea may be consistent with part and inconsistent with other parts of the viewpoint. The student i s hypothesized to acquire s c i e n t i f i c knowledge through two generalized sources: informal and formal. The hypotheses for the s p e c i f i c problems are: 6.1 As students mature t h e i r viewpoints w i l l undergo change through informal sources. The student also, acquires more complex levels of formal, or "logico-mathematical," reasoning s k i l l s . Therefore, i t i s hypothesized that students of a common age or grade l e v e l w i l l have view-points that are generally s i m i l a r , and that students of higher grades w i l l evidence a greater variety of ideas (although some of the ideas the students acquired e a r l i e r may be "hidden" by the acquired s c i e n t i f i c paradigms) and ideas that are more l i k e the s c i e n t i f i c paradigms. 6.2 Students who receive physics t r a i n i n g are exposed i n a formal manner to the s c i e n t i f i c paradigms. As a r e s u l t , 14 i t i s h y p o t h e s i z e d t h a t t h e v i e w p o i n t s o f s t u d e n t s w i l l show i n c r e a s e d e v i d e n c e o f i d e a s t h a t a r e b a s e d on t h e s c i e n t i f i c p a r a d i g m s w i t h i n c r e a s e d t r a i n i n g i n p h y s i c s . 6.3 I t i s h y p o t h e s i z e d t h a t o n c e a s t u d e n t p o s s e s s e s a n i d e a i t i s a l w a y s p r e s e n t . A s t u d e n t who h a s r e c e i v e d p h y s i c s t r a i n i n g w i l l , i n a d d i t i o n , h a v e t h e i d e a s g a i n e d d u r i n g s c i e n c e i n s t r u c t i o n . T h e s e a r e d e r i v e d f r o m w h a t has b e e n r e f e r r e d t o a s i n f o r m a l a n d f o r m a l s o u r c e s o f i d e a s . I t i s h y p o t h e s i z e d t h a t some o f t h e s e i d e a s w i l l be i n c o n s i s t e n t w i t h e a c h o t h e r . I t i s a d d i t i o n a l l y h y p o t h e s i z e d t h a t t h e s t u d e n t may r e v e r t t o i d e a s t h a t w e r e d e v e l o p e d b e f o r e p h y s i c s t r a i n i n g i n c a s e s s u c h a s t h e f o l l o w i n g : a n o v e l s i t u a t i o n ; o r one w here t h e s t u d e n t d o e s n o t r e c o g n i z e t h e a n a l o g y t o t h e " e x e m p l a r s " p r e v i o u s l y p r a c t i c e d w i t h d u r i n g t r a i n i n g ; o r t h e s t u d e n t may r e c o g n i z e p a r t o f e x e m p l a r and u s e e a r l i e r i d e a s t o " f i l l i n t h e b l a n k s . " 6.4 I t i s h y p o t h e s i z e d t h a t s t u d e n t s w i t h a c o n s i s t e n t s e t o f i d e a s w i l l be b e t t e r a b l e t o a p p l y t h e p a r a d i g m s o f s c i e n c e . 7. A s s u m p t i o n s 7.1 T h a t s t u d e n t s a t t h e s e c o n d a r y s c h o o l l e v e l h a v e v i e w p o i n t s a b o u t a w i d e r a n g e o f p h y s i c a l phenomenon and t h a t t h e s e v i e w s c a n be m a n i f e s t e d t h r o u g h v e r b a l and n o n - v e r b a l means. 7.2 T h a t s c i e n t i s t s h a v e v i e w p o i n t s on t h e same r a n g e o f phenomenon t h a t h a v e come a b o u t t h r o u g h t r a i n i n g a nd 15 consequently, are un l i k e l y to be commensurate with the viewpoints of untrained students. 7.3 It i s possible to i n i t i a t e students into the s c i e n t i f i c community through t r a i n i n g . The t r a i n i n g involves what Kuhn c a l l s "finger exercises" with the s c i e n t i f i c paradigm to induce G e s t a l t - l i k e s h i f t s i n the student. 8. Delimitation of the Study. The study w i l l be done i n the secondary school of a small town. The school i s a junior-senior secondary (grades 8-12) and has a student population of about 600. The population of the town i s r e l a t i v e l y stable and therefore most of the grade twelve students have attended the school for four or f i v e years. The students investigated ranged from grade 8 to grade 12. Special emphasis was placed on students who had taken physics courses. It should be noted that, Physics 11 may be taken either by grade 11 students or grade 12 students. The majority of students do not take Physics 11, and the minority that do tend to be the more academically able students. The author has taught at the school for the past four years, and i n that time has had contact with most of the students that participated i n the study. The physical phenomena studied was limited to accelerated motion and harmonic motion. Both of these phenomena are f a m i l i a r to a l l students and the students therefore had an experiential introduction to them p r i o r to taking physics courses. Not a l l students are expected to 16 h a v e h a d i d e n t i c a l e x p e r i e n c e s ; n o r t o h a v e drawn t h e same c o n c l u s i o n f r o m t h e i r e x p e r i e n c e s . I t i s t h o u g h t h o w e v e r , t h a t t h e r e s u l t s may be a p p l i e d t o o t h e r p h y s i c a l phenomena t h a t s t u d e n t s a r e f a m i l a r w i t h . 9. J u s t i f i c a t i o n o f t h e s t u d y . T h i s s t u d y w i l l h a v e two f u n d a m e n t a l j u s t i f i c a t i o n s o r a p r i o r i . The f i r s t , i s t h a t t h e t e a c h i n g o f t h e s c i e n t i f i c p a r a d i g m s t o s t u d e n t s h a s e d u c a t i o n a l v a l u e . The s e c o n d , i s t h a t t h e s t u d e n t s ' v i e w p o i n t s h a v e v a l u e . W h i l e t h e p a r a d i g m s o f s c i e n c e a n d o t h e r i d e a s i n t h e v i e w p o i n t o f t h e s t u d e n t may n o t be c o n s i s t e n t t h e y a l l h a v e v a l u e a n d a r e f u n c t i o n a l . F o r e x a m p l e , t h e s t u d e n t may h a v e b e e n a b l e t o c a t c h a b a s e b a l l l o n g b e f o r e b e i n g g i v e n a n y t r a i n i n g i n t h e s c i e n t i f i c p a r a d i g m s o f p r o j e c t i l e m o t i o n . F u r t h e r , i t i s u n l i k e l y t h a t a s t u d e n t a f t e r p h y s i c s i n s t r u c t i o n , w i l l a p p l y s c i e n t i f i c f o r m a l i s m t o t h e c a t c h i n g o f b a s e b a l l . The g o a l o f e d u c a t i o n i s t o t r a i n t h e s t u d e n t t o c h o o s e t h e a p p r o p r i a t e i d e a f o r a g i v e n s i t u a t i o n . I n o r d e r f o r e d u c a t i o n t o a c c o m p l i s h t h i s g o a l i t i s n e c e s s a r y f o r e d u c a t o r s t o u n d e r s t a n d t h e n a t u r e o f t h e d i f f e r e n c e s b e t w e e n t h e i d e a s t h a t f o r m t h e s t u d e n t s ' v i e w p o i n t s 17 CHAPTER I I - LITERATURE REVIEW 1. I n t r o d u c t i o n The g e n e r a l p r o b l e m was t o s t u d y t h e i d e a s t h a t a s t u d e n t d e r i v e s f r o m a s t u d y o f s c i e n t i f i c p a r a d i g m s and t h o s e t h e s t u d e n t d e r i v e s f r o m o t h e r i n f o r m a l s o u r c e s . Rene D e s c a r t e s a s s e r t e d a c l e a r d i s t i n c t i o n b e t w e e n r e a s o n and e i t h e r p e r c e p t i o n o r i m a g i n a t i o n a n d t h a t " n e i t h e r o f t h a t l a t t e r g i v e k n o w l e d g e , a l t h o u g h t h e y may h e l p o r h i n d e r i t " ( p p . x i v ) . D e s c a r t e s w i s h e d t o e s t a b l i s h a s y s t e m o f v a l i d k n o w l e d g e t h a t was b a s e d on c l e a r and d i s t i n c t r e a s o n . H i s p h i l o s o p h y was b a s e d on a n e x a m i n a t i o n o f h i s own m i n d , a n d o n l y a p p l i c a b l e t o o t h e r i n d i v i d u a l s b y e x t e n s i o n . D e s c a r t e s b e g a n h i s s e c o n d m e d i t a t i o n w i t h t h e s t a t e m e n t : " I w i l l ... p r o c e e d by c a s t i n g a s i d e a l l t h a t a d m i t s o f t h e s l i g h t e s t d o u b t , n o t l e s s t h a n i f I had d i s c o v e r e d i t t o be a b s o l u t e l y f a l s e ; a nd I w i l l c o n t i n u e o n t h i s t r a c k u n t i l I s h a l l f i n d s o m e t h i n g t h a t i s c e r t a i n , o r a t l e a s t u n t i l I s h a l l know w i t h c e r t a i n t y t h a t t h e r e i s n o t h i n g c e r t a i n . " T h i s a s s u m p t i o n l e a d s t o what C a m p b e l l c a l l s " p e r f e c t i o n - s e e k i n g " ; t h a t i s , t o r e j e c t a p o s s i b l e t r u t h t o a v o i d a c c e p t i n g s o m e t h i n g t h a t i s f a l s e . D e s c a r t e s c o n l u d e d h i s m e d i t a t i o n : . . . I f i n d I h a v e i n s e n s i b l y r e v e r t e d t o t h e p o i n t I d e s i r e d ; f o r , s i n c e i t i s now m a n i f e s t t o me t h a t b o d i e s t h e m s e l v e s a r e n o t p r o p e r l y p e r c e i v e d by t h e s e n s e s n o r by t h e f a c u l t y o f i m a g i n a t i o n , b u t b y t h e i n t e l l e c t a l o n e ; and s i n c e t h e y a r e n o t p e r c e i v e d b e c a u s e t h e y a r e s e e n and t o u c h e d , b u t o n l y b e c a u s e t h e y a r e u n d e r s t o o d [ o r r i g h t l y c o m p r e h e n d e d by t h o u g h t ] , I r e a d i l y d i s c o v e r t h a t t h e r e i s n o t h i n g more e a s i l y o r c l e a r l y a p p r e h e n d e d t h a n my own m i n d , ( p p . 94) 18 Descartes came to believe that, as a r e s u l t of his r e f l e c t i o n s , "the mind exhibits powers of reasoning which i t imposes upon the world of sensory experience" rather than as he o r i g i n a l l y believed that " a l l experiences and thought aris e through the senses" (Gardener, 1985, pp. 52). Empiricists responded to Descartes' r a t i o n a l i s t views and the debate has continued through the centuries. Of importance to t h i s study i s the questioning of the role of sensory experience; and the use of "introspection" as a tool of inquiry. Jean Piaget attempted to use b i o l o g i c a l p r i n c i p l e s to create a "genetic epistemology." Piaget stated that "genetic epistemology attempts to explain knowledge, on the basis of i t s h i s tory, i t sociogenesis, and es p e s c i a l l y the psychological origin.of the notions and operations upon which i t i s based ... there i s a p a r a l l e l i s m between progress made in the l o g i c a l and r a t i o n a l organization of knowledge and the corresponding formative psychological processes" (cited i n M i l l e r , 1984). Piaget observed children doing informal tasks i n order to be able to characterize d i f f e r e n t stages of development: stages that had been hypothesized to be p a r a l l e l e d by mankind. The stages characterized by Piaget have proved "less robust than his s p e c i f i c experimental demonstrations" (Gardener, 1985). However, the experimental tasks developed by Piaget continue to be used and extended. Where Descartes used his own mind to determine what his ideas 19 were, Piaget observed children. Descartes wanted to only accept ideas that were proven truths; i n t h i s regard Descartes was a "perfection-seeker." Piaget wanted to know what ideas were used by a c h i l d regardless of the idea's truthfulness (for example a young ch i l d ' s not possessing the idea of conservation of volume when a l i q u i d changes shape), that i s , the c h i l d i s a "truth-seeker." A truth-seeker's goal i s the " t o t a l i t y " of knowledge and w i l l accept a greater r i s k of accepting fa l s e statements i n order to possess more of the " t o t a l i t y . " Kuhn developed a view of the history of science through the concept of "paradigm." The study used the paradigm concept to investigate students' ideas. In p a r t i c u l a r , the study used exemplars which are a p a r t i c u l a r form of paradigm. The works of Kuhn and Piaget formed a foundation for the study. 2. Acceleration: A Piagetian Task Trowbridge and McDermott (1981) extended Piaget's acceleration task (Piaget, 1970/1946.) to investigate university introductory physics students' understanding of the concept of acceleration. Understanding was defined as "the a b i l i t y to apply i t successfully i n interpreting simple motions of re a l objects." Student ideas were investigated s i m i l a r l y to Piaget's subjects, with individual interview demonstrations. Their r e s u l t s with university students d i f f e r e d from Piaget's results with children. Since a l l students were successful Trowbridge and McDermott concluded 20 that a "primitive i n t u i t i o n of acceleration as speeding up" was adequate for successful completion of the acceleration task and i t did not require consideration of a r a t i o (acceleration being the r a t i o of change i n v e l o c i t y to change i n time). In addition, they concluded that: f i r s t l y , students could be arranged into h i e r a r c h i c a l groups, from those with a nonkinematical approach to those with a qu a l i t a t i v e understanding; secondly, when the task was presented i n written form i t encouraged "almost automatic substitution i n memorized formulas;" and t h i r d l y , the d i f f i c u l t i e s students e x h i b i t i e d were very persistent. The present study used younger students and found s i m i l a r l y that students could be categorized and that ideas were persistent, even a f t e r i n s t r u c t i o n . However, th i s study's younger students demonstrated less written problem solving a b i l i t y (this was attributed i n large part to student's mathematical a b i l i t y ) . Trowbridge and McDermott used the individual interview demonstration strategy introduced by Piaget. The students were given s p e c i f i c guidance for the f i r s t tasks as follows: The interviewer explains that to make the comparison i t i s unnecessary to i d e n t i f y the cause of the acceleration ... i f a student does not notice ... the interviewer asks questions that serve to d i r e c t attention ... Thus the students are assisted i n making the observations necessary for comparing the accelerations. It remains for them to combine t h i s information i n a manner that permits successful resolution of the task. Piaget frequently used t h i s strategy to discover how children 21 reasoned. Piaget was i n v e s t i g a t i n g the c h i l d r e n ' s h y p o t h e t i c o - l o g i c o powers, t h a t i s how c h i l d r e n use observations to reason new conclusions. Since a student's ideas before and a f t e r the i n t e r v i e w may be d i f f e r e n t , what a student's i n i t i a l ideas were and what ideas were developed as a r e s u l t of the demonstration need t o be d i s t i n g u i s h e d . In p a r t i c u l a r , which ideas are t o be considered the student's "informal ideas?" This s t r a t e g y does, however, use " r e a l o b j e c t s " i n a f a m i l i a r s i t u a t i o n , whereas t h i s study used thought experiments or re p r e s e n t a t i o n s of objects i n f a m i l i a r s i t u a t i o n s . 3. Experts and Novices L a r k i n , McDermott, Simon, and Simon i n v e s t i g a t e d expert and novice physics problem s o l v i n g through " t h i n k i n g - a l o u d p r o t o c o l s " and computer s i m u l a t i o n s . The thin k - a l o u d p r o t o c o l s have been c r i t i c i s e d as a form of i n t r o s p e c t i o n that l a c k s o b j e c t i v i t y . E r i c s o n and Simon (1980) responded to the c r i t i c i c m s : "verbal reports are data ... v e r b a l i z i n g i n f o r m a t i o n ... a f f e c t ( s ) c o g n i t i v e processes only i f the i n s t r u c t i o n s r e q u i r e v e r b a l i z a t i o n s of in f o r m a t i o n t h a t would not otherwise (have been) attended t o . " The computer si m u l a t i o n s were based on the hypothesis t h a t human memory i s organized as a large set of productions ( c o n d i t i o n - a c t i o n p a i r s ) . The computer s i m u l a t i o n was done to show tha t t h e i r explanations were " o p e r a t i o n a l " and d i d not depend on "vague, m e n t a l i s t i c concepts." C h i , F e l t o v i c h , and Glaser i n v e s t i g a t e d expert and novice c a t e g o r i z a t i o n and 22 representatin of physics problems through sorting tasks followed by interviews. In both studies the experts were students at the doctoral l e v e l and the novices were taking university introductory physics courses. Both studies found fundamental differences between the c h a r a c t e r i s t i c s of novices and experts. Chi et a l . found that "experts tended to categorize problems into types that are defined by the major physics p r i n c i p l e s that w i l l be used i n solution, whereas novices tend to categorize them into types as defined by the e n t i t i e s contained i n the problem statement." Larkin et a l . found that experts "worked forward from the givens to the desired quantities" on the easy questions, whereas novices required "goals and subgoals to d i r e c t t h e i r search" for deciding what to do. Experts solved problems s i g n i f i c a n t l y faster but were slower to sort problems, which indicated that experts had stronger abstracted methods. This study used secondary school students. The grade twelve physics students investigated i n t h i s study could be i d e n t i f i e d as novices. The present study used paradigm completion questions i n which the subject proposed the problem and the investigator assigned the category to the problem. Chi et a l . did the reverse and associated "the categories of problems as representing internal schemata." 4 . Interview About Instances Osborne (1980) developed a method c a l l e d "Interview About Instances" to investibate "concept attainment." The method used a set of cards depicting instances or 23 n o n - i n s t a n c e s o f a c o n c e p t . The s t u d e n t was a s k e d t o c a t e g o r i z e e a c h c a r d a n d was t h e n a s k e d t o e x p l a i n t h e b a s i s o f t h e c a t e g o r i z a t i o n . O s b o r n e i d e n t i f i e d t h r e e a d v a n t a g e s : f i r s t l y , t h e method i s a p p l i c a b l e o v e r a w i d e r a n g e o f s t u d e n t s ; s e c o n d l y , t h e i n t e r v i e w c a n n o t be i g n o r e d by a s t u d e n t w h e r e a s a w r i t t e n a n s w e r may be o m i t t e d f o r a v a r i e t y o f r e a s o n s ; a n d t h i r d l y , i n a f o r m a l s i t u a t i o n t h e s t u d e n t may d e m o n s t r a t e a s u p e r f i c i a l u n d e r s t a n d i n g , f o r e x a m p l e , a b l i l i t y t o r e c a l l a f o r m u l a d o e s n o t g u a r a n t e e c o n c e p t u n d e r s t a n d i n g . T h i s s t u d y s i m i l a r l y f o u n d t h a t s t u d e n t s o f a l l g r a d e s w e r e a b l e t o r e l a t e t o t h e d i a g r a m s a nd o b s e r v e d s t u d e n t s h a v i n g some d i f f i c u l t i e s w i t h t h e w r i t t e n a s p e c t s . T h i s m e t h o d was a l s o u s e d b y W a t t s (1983) t o s t u d y s t u d e n t s ' i d e a s a b o u t t h e c o n c e p t o f f o r c e . W a t t s f o u n d e i g h t d i s t i n c i t v e " f r a m e w o r k s " t h a t p u p i l s e m l o y e d w i t h t h e c o n c p e t o f f o r c e . I n a d d i t i o n W a t t s commented: I n a n a l y s i n g t h e i n d i v i d u a l i n t e r v i e w t r a n s c r i p t s , an a t t e m p t was made t o c o n s t r u c t f r a m e w o r k s t h a t c a n a c c o u n t f o r s t a t e m e n t s by a p u p i l i n s u c h a way t h a t s t a t e m e n t s a r e c o m p a t i b l e w i t h e a c h o t h e r . The a s s u m p t i o n t h a t a l l o f a p e r s o n ' s s t a t e m e n t s a r e l o g i c a l l y c o n s i s t e n t t o a l i s t e n e r ( o r r e a d e r ) , i s d i f f i c u l t t o m a i n t a i n . However, i t i s one t h a t h a s t o be made a s a w o r k i n g h y p o t h e s i s , o t h e r w i s e i t i s t o o e a s y t o d i s c o u n t s e c t i o n s o f a s t u d e n t ' s d i s c o u r s e t h a t seem i n c o n s i s t e n t w i t h u n d e r s t a n d a b l e s e c t i o n s . I t was, h o w e v e r , t h e s e i n c o n s i s t e n c i e s t h a t w e r e a m a j o r f o c u s o f t h i s s t u d y . 5. P r o j e c t i l e s a n d P e n d u l a W a t t s a n d Z y l b e r s z t a j n (1981) a s s e s s e d t h e p o p u l a r i t y o f 24 some " a l t e r n a t i v e f r a m e w o r k s " o f f o r c e c o n c e p t s p o s s e s s e d by s t u d e n t s a n d t o e x a m i n e t e a c h e r a w a r n e s s o f t h e s e a l t e r n a t i v e f r a m e w o r k s . T h i s was i n v e s t i g a t e d u s i n g a p a p e r a n d p e n c i l q u e s t i o n a i r e b a s i c a l l y c o n s i s t i n g o f t w e l v e m u l t i p l e c h o i c e q u e s t i o n s . The f i r s t t h r e e c o n c e r n e d a s t o n e b e i n g t h r o w n v e r t i c a l l y u p w a r d and t h e s e c o n d t h r e e c o n c e r n e d t h e p r o j e c t i l e m o t i o n o f a c a n n o n b a l l . W a t t s and Z y l b e r s z t a j n f o u n d t h a t m o s t o f t h e p u p i l s a s s o c i a t e d f o r c e w i t h m o t i o n and "a c o n s i d e r a b l e number o f c h i l d r e n t h o u g h t t h a t a s t h e movement d e c r e a s e s , s o t h e f o r c e d e c r e a s e s . " T e a c h e r s w e r e f o u n d t o be a w a r e o f t h e a l t e r n a t i v e f r a m e w o r k s s t u d e n t s w e r e u s i n g . T h i s s t u d y , w h i l e n o t i n v e s t i g a t i n g f o r c e s , i n v e s t i g a t e d t h e m o t i o n o f f a l l i n g o b j e c t s . The m a j o r i t y o f s t u d e n t s i n v e s t i g a t e d , i n c l u d i n g s t u d e n t s w i t h g r a d e e l e v e n p h y s i c s t r a i n i n g , d i d n o t p o s s e s s t h e i d e a t h a t t h e a c c e l e r a t i o n o f g r a v i t y i s c o n s t a n t . The r e s u l t s w e r e m i x e d , b u t s t u d e n t s t e n d e d t o p o s s e s s t h e i d e a t h a t t h e a c c e l e r a t i o n was g r e a t e s t i n t h e m i d d l e o r a t t h e end o f t h e f a l l , i n a g r e e m e n t w i t h t h e r e s u l t s o f W a t t s and Z y l b e r s z t a j n . C a r a m a z z a , M c C l o s k e y , and G r e e n a s k e d u n i v e r s i t y s t u d e n t s t o s o l v e s i m p l e p r o b l e m s a b o u t t h e t r a j e c t o r i e s o f f a l l i n g o b j e c t s . They f o u n d t h a t " p e o p l e do a b s t a c t f r o m t h e i r e x p e r i e n c e w i t h t h e w o r l d g e n e r a l p r i n c i p l e s c o n c e r n i n g t h e m o t i o n o f o b j e c t s ... h o w e v e r , t h e s e p r i n c i p l e s a r e o f t e n s t r i k i n g l y a t v a r i a n c e w i t h t h e most f u n d a m e n t a l p h y s i c a l l a w s ... n o t o n l y by p e o p l e w i t h no f o r m a l i n s t r u c t i o n i n p h y s i c s , b u t a l s o by a l a r g e p r o p o r t i o n o f t h o s e who h a v e 25 completed high school or college physics courses." The students investigated were presented with a l i n e drawing of a b a l l and s t r i n g moving i n an arc as a pendulum. The students were then asked to indicate the path the b a l l would follow i f the s t r i n g were cut when the b a l l was i n the location indicated and moving i n the d i r e c t i o n indicated. The responses of the students divided into six categories. Caramazza et a l . conclude that: "although, the naive b e l i e f s would appear to have t h e i r o r i g i n i n experience ... It i s more l i k e l y that, as Piaget has argued, deduction plays a c r u c i a l role i n the development of models of the physical world." They also concluded that physics t r a i n i n g did necessarily provide a f u l l understanding of p r o j e c t i l e motion i t did have some e f f e c t . S p e c i f i c a l l y i n s t r u c t i o n provided some understanding of: (1) the importance of the b a l l ' s horizontal v e l o c i t y , and (2) the fact that the b a l l w i l l accelerate v e r t i c a l l y as i t f a l l s . The results of t h i s study were s i m i l a r , although, students with grade eleven physics t r a i n i n g s t i l l demonstrated an i n a b i l i t y to separate horizontal and v e r t i c a l motion. Clement (1982) investigated university students' "preconceptions" i n introductory mechanics. The f i r s t problem given the students involved a pendulum. Students were asked to draw and label arrows showing the d i r e c t i o n of each force acting on the pendulum. As with Watts and Zylbersztajn students possessed the "motion implies a force" misconception and included a force tangent to the s t r i n g . 26 This could indicate, as t h i s study found, that students possessed l i t t l e s c i e n t i f i c understanding of the motion of a pendulum. 6. Hermeneutics Heelan's book "Space-perception and the Philosophy of Science" was based on the thesis that: what we know i s not limited to the deliverances of a unique p r i v i l e g e d perceptual framework constituting an absolute trans-cultural empirical basis for a l l knowledge, and we can have access to a m u l t i p l i c i t y of possible perceptual horizons, both of Euclidean and of non-Euclidean structure, grounded both i n unaided perception and i n the use of special technologies ("readable" technologies) invented using s c i e n t i f i c theories. This approach i s what I c a l l "horizonal." With some important reservations as to the goal and function of philosophical analysis, the method used i n t h i s book as appropriate to th i s i s both phenomenological and hermeneutical. The empirical focus of t h i s book i s space perception. Its larger purpose, however, i s philosophical: i n p a r t i c u l a r i t i s about the philosophy of science. (pp. 3) It i s the larger purpose and the methodology that i s of interes t to t h i s study. The central concerns of phenomenology are: "(1) the a p o d i c t i c i t y of given objects (noemata), (2) how the subject receives these objects i n experience (noesis), and (3) the conditions of p o s s i b i l i t y of the noesis-noema structure i n the perceiving subject. Hermeneutics i s : "the science and art of interpreting texts and t e x t l i k e materials i n order to arr i v e at the "things themselves" about which the text speaks. The task of inter p r e t a t i o n i s led by fore-structures of understanding, 27 suggested by c u l t u r a l t r a d i t i o n s -- the "biases and prejucices" -- we share, and by clues i n the textual material i t s e l f . ... This search i s guided by appropriate fore-structures of understanding: these comprise (1) Vorsicht, or the resources of a common descriptive language; (29 V o r g r i f f , or a hypothesis about the sense of the materials being investigated; and (3) Vorhabe, or the c u l t u r a l l y acquired s k i l l and practices we need to understand, recognize, and name the objects i n our World, (pp. 220) Heelan uses the concepts of phenomenology and hermeneutics to develop his "horizonal" approach. This approach assumes that "plural v e r i d i c a l r e a l i s t i c perspectives are possible consonant with the p l u r a l i t y of d i f f e r e n t horizons of perception within a World: there i s not just one empirical basis for fact but a p l u r a l i t y of empirical bases" (pp. 177). This approach i s consistent with the thesis developed i n t h i s study that i t i s possible for students to possess inconsistent ideas within a consistent viewpoint. Heelan concludes that r e a l i t y i s composed of a "network" or horizons. As t h i s study divided ideas into those derived from informal sources and formal sources, Heelan proposes classes of horizons: One major d i v i s i o n between classes of horizons of perception i s that between "manifest images" of r e a l i t y and " s c i e n t i f i c images." The former represent objects as constituted i n th e i r essential form by a human s o c i a l or c u l t u r a l i n t e r e s t or by c r i t e r i a found only i n the mental l i f e of persons, and they are functions of persons; i n the words of W. S e l l a r s , "the 'manifest' image of man-in-the-world [ i s ] the framework in terms of which man encountered himself." A s c i e n t i f i c image, on the contrary, represents objects as constituted i n t h e i r 28 essential forms by systems of postulated (or theoretical) e n t i t i e s , related to one another and to some manifest World by s c i e n t i f i c theory, and encountered only through the mediation of instruments or technology, (pp. 177) The World of our place, time, and culture i s structured by horizons of perceptual p o s s i b i l i t i e s rooted i n the primordial i n t e n t i o n a l i t y of perception. Since access to these horizons i s learned together with that part of our natural language which provides descriptive terms for them, the creation of new horizons and the sharing of old i s a h i s t o r i c a l and c u l t u r a l process, both for the i n d i v i d u a l and for the community. ... Experimental s c i e n t i s t s -- l i k e most professional groups -- have access by v i r t u e of t h e i r profession to horizons that are c h a r a c t e r i s t i c a l l y t h e i r own ... (pp. 192) Although, what was of i n t e r e s t to t h i s study was the hermeneutical model, Heelan extends t h i s approach to outline a normative model for the "progressive aspect of s c i e n t i f i c knowledge." This model depends on the assumption that for older theories to be comparable with l a t e r theories: "(1) the older theory i n i t s own time was authentic (designating, that i s , through appropriate readable technologies, perceptual p r o f i l e s of s c i e n t i f i c essences), and (2) the older theory has a current and contemporary form. The science student, by analogy i n t h i s study, has been found to possess informal ideas that were both functional and persistent. 29 CHAPTER III - METHODOLOGY 1. Recapitulation of the Problem The general problem was to study the rel a t i o n s h i p between the ideas that a student derives from a study of s c i e n t i f i c paradigms and those the student derives from other informal sources. The range of physical phenomena investigated was limited to accelerated motion and harmonic motion. Four facets of the general problem were addressed through s p e c i f i c problems. The s p e c i f i c problems were: 1.1 Do the classes to be investigated consist of students that possess a 'char a c t e r i s t i c viewpoint?' 1.2 What e f f e c t does physics t r a i n i n g have on the viewpoints of students? 1.3 Do students possess viewpoints that contain inconsistencies? 1.4 Do students possessing a consistent viewpoint, have success i n applying s c i e n t i f i c paradigms? 2. Target Population The target population of t h i s study was grade twelve senior secondary students, although the study included junior secondary students (grades eight to ten). The students l i v e d i n and near a small coastal town. The town's secondary school had an enrollment of 5 50 students from grades eight to twelve. The town r e f l e c t e d a mixed socioeconomic status with, over the past few years, increased unemployment. The population was r e l a t i v e l y stable and the the grade twelve students generally had attended the school for f i v e years. 30 The grade twelve students that participated i n the study were of above average academic a b i l i t y . 3. Sampling Plan The sampling plan included the following classes; one grade eight science c l a s s , one grade nine science c l a s s , one grade ten science c l a s s , and one grade twelve algebra c l a s s . Since the secondary school was the only one i n the town, the sample represents the accesible population i n the school. The school i t s e l f , was a somewhat biased sample of the extended target population of schools i n B r i t i s h Comlumbia. Nevertheless, i t was hoped that generalizations arrived at i n the study apply since what was at issue was how students' viewpoints are modified by s c i e n t i f i c t r a i n i n g . This was expected to be more a function of developmental and educational level than s p e c i f i c environmental and s o c i a l factors. 4. Sample The junior secondary classess selected were the author's grade eight science c l a s s , the author's grade nine science c l a s s , and another teacher's grade ten c l a s s . The author had established an easy and relaxed rapport with the students of both classes. The grade eight class consisted of twenty-two students. The students of the grade eight class were generally of average academic a b i l i t y and no students were repeating the course. The grade nine class consisted of fourteen students of low academic a b i l i t y . The grade ten science class consisted of sixteen students. The grade ten 31 c l a s s , while taught by another teacher, was taught i n the author's classroom. The author had presented demonstrations and lectures to the grade ten class and therefore a rapport had been established. The grade twelve algebra class was not taught by the author. The algebra class was composed of thirteen students; f i v e with no physics background, seven having successfully completed grade eleven physics, and one having successfully completed grade twelve physics. The students who had physics t r a i n i n g received i t from the author i n previous semesters. The grade twelve physics class consisted of four above average a b i l i t y students. The classroom atmosphere was relaxed and informal. At the time of the study the students had e s s e n t i a l l y completed the grade twelve course. 5. Design  5.0 Introduction The primary problem to be researched was to study the re l a t i o n s h i p between the ideas that students derived from s c i e n t i f i c paradigms during t r a i n i n g and those that students derive from other informal sources. It had been hypothesized that these ideas may be inconsistent with each other. This s i t u a t i o n was analogous to that faced by physicists at the turn of the century. Light had been thought to be a perfect wave process for at least a century. However i n 1905 E i n s t e i n showed that l i g h t behaved l i k e p a r t i c l e s . Other experiments followed which confirmed that l i g h t behaved l i k e p a r t i c l e s . These experiments did not however, show that 32 l i g h t did not have a wave nature. This wave-particle duality was interpreted by Bohr with his "complementarity p r i n c i p l e . " The complementarity p r i n c i p l e states that: ...no s i t u a t i o n can a r i s e i n which both of the complementary aspects of the phenomenon show up simultaneously and rigorously. The contradiction arises from the fact that one has to measure the properties. The i n t e r a c t i o n between the object of observation and the measuring apparatus cannot be neglected ... (Mehra, pp.36.) The analogy was relevant for two reasons. F i r s t l y , the student had been hypothesized to hold ideas that may be inconsistent, or be complementary, with each other. These ideas, i t had been further hypothesized, may be integrated to d i f f e r e n t l e v e l s within the viewpoints of d i f f e r e n t students. As such, i t i s l i k e l y , following the complementarity p r i n c i p l e that a measuring instrument may reveal one idea while at the same time concealing another idea. The second relevance of the analogy was the i n t e r a c t i o n between the object and the observer. McDermott (1984) states: "Because res u l t s and methods are so c l o s e l y intertwined i n research, i t i s important i n interpreting the findings to bear i n mind the procedures used." McDermott summarizes several c h a r a c t e r i s t i c s including: "The investigator's perception of student thinking may d i f f e r i f only one question i s asked about a concept rather than many, or i f only one context i s used rather than several." This was p a r t i c u l a r l y s i g n i f i c a n t i n the problem of revealing student ideas. The conclusion was that the methodology must include a variety of 33 instruments; each of which may reveal d i f f e r e n t ideas. The case study methodology was best suited to meet t h i s c r i t e r i a as noted below. 5.1 Case Study Methodology The methodology used was that of the case study. Yin states that a case study i s an empirical inquiry that: -investigates a contemporary phenomenon within i t s r e a l - l i f e context; when -the boundaries between phenomenon and context are not c l e a r l y evident; and -multiple sources of evidence are used, (pp. 2 3.) The f i r s t point was e a s i l y met but the second and t h i r d point had p a r t i c u l a r s i g n i f i c a n c e . For t h i s research problem i t had been hypothesized that the student may possess ideas that are derived from d i f f e r e n t sources. These d i f f e r e n t kinds of ideas were i d e n t i f i e d with how the ideas were acquired; either through informal sources (interaction with the environment) or through more formal sources (science classes, textbooks, formal experimentation, magazines, e t c . ) . Piaget characterized two sim i l a r c l a s s i f i c a t i o n s of knowledge. Piaget hypothesized that: Our knowledge stems neither from sensation nor from perception alone but from the entire action, of which perception merely constitutes the function of s i g n a l i z a t i o n . The c h a r a c t e r i s t i c of i n t e l l i g e n c e i s not to contemplate but to "transform" and i t s mechanism i s e s s e n t i a l l y operatory. Operations consist of i n t e r i o r i z e d and coordinated actions i n group structures ... there are two ways of modifying i t s pos i t i o n , i t s movements, or i t s c h a r a c t e r i s t i c s i n order to explore i t s nature: t h i s i s action known as "physical." The other consists i n enriching the object 34 with c h a r a c t e r i s t i c s or new relationships which r e t a i n i t s c h a r a c t e r i s t i c s or previous relationships, yet completing them by systems of c l a s s i f i c a t i o n , numerical order, measure, and so f o r t h : these actions are known as "logico-mathematical." (1971, pp. 67) . Thus Piaget has divided the a c q u i s i t i o n of knowledge, which he considers to be a process that i s "operatory," into two methods: the "physical" and the "logico-mathematical." These are analogous to what had been i d e n t i f i e d i n t h i s study, as "informal" and "formal." Both of these, i n turn, r e l a t e to what Yin has c a l l e d the "phenomenon" and the "context." In t h i s case, the context was the student's environment from which the student had informal (or physical) sources of knowledge, and the phenomenon the formal (or logico-mathematical) sources. Piaget recognized that the a c q u i s i t i o n of knowledge must always be primarily "active i n the elementary stages of knowledge formation." However, Piaget continued that even physical or experimental knowledge cannot be detached from a l l logico-mathematical organization. Piaget used physics to develop t h i s theme: ... physics, as the most developed science of experience, i s a perpetual assimilation of experimental f a c t with logico-mathematical structures, since the very refinement of the experience serves as logico-mathematical instruments used as necessary intermediaries between the subject and the objects to be reached. ... I f , appearing to be based on sensation, physical knowledge i s constantly withdrawn more and more, the reason i s that i t never proceeds from sensation nor even from pure perception but, at the very outset, i t implies a 35 * logico-mathematical schematization of perceptions as well as actions exercised on the objects. Beginning by such schematization, i t i s natural therefore that these logico-mathematical additions become more and more important with the development of physical knowledge and that, consequently, physical knowledge i s constantly withdrawn more and more from perception as such. (1971a, pp. 73) * (footnote given: In the sense of an organization of f a c t , thanks to the intervention of sensorimotor "schemes.") Thus Piaget showed that the boundaries are not clear and d i s t i n c t between whether knowledge has been derived from physical (or informal) sources and logico-mathematical (or formal) sources. The problem investigated thus, met Yin's second c r i t e r i a . Yin's t h i r d c r i t e r i a that "multiple sources of evidence are used." The introduction to t h i s section (section 5.0) showed that student ideas could be interpreted by an analogy to the complementarity p r i n c i p l e . It was concluded that t h i s study would have to use a variety of instruments. This was Yin's t h i r d and f i n a l condition. The research problem to be investigated therefore had met the conditions that d i s t i n g u i s h t h i s problem as being the case study. This problem was investigated with the study of more than a single case and therefore used a multiple-case design. The advantage to using a multiple-case design was not merely to make more attempts at finding an "appropriate" case but rather to use " r e p l i c a t i o n l o g i c . " Yin (pp. 48) states r e p l i c a t i o n l o g i c i s to "select each case so that i t either (a) predicts s i m i l a r r e s u l t s (a l i t e r a l r e p l i c a t i o n ) or (b) 36 produces contrary results but for predictable reasons (a the o r e t i c a l r e p l i c a t i o n ) . " Multiple cases are not used to esta b l i s h the incidence of a phenomenon - sampling l o g i c i s needed to es t a b l i s h those measures. The analysis also "must follow cross-experiment rather than within-experiment design and l o g i c . " The i d e n t i f i c a t i o n and description of the cases i s included with the presentation of the method of the analysis of data (section 5 . 5 ) . 5 .2 Physical Phenomena Under Study This study investigated two s p e c i f i c areas of physics: harmonic motion (the pendulum) and accelerated motion (free f a l l and p r o j e c t i l e motion). 5 . 2 1 . The Pendulum a) Possible Variables: period frequency length of the s t r i n g mass of the bob height of the dropping point (=amplitude) force of the push used to s t a r t the o s c i l l a t i o n v e l o c i t y of the bob energy (kinetic and potential) of the bob forces - tension i n the s t r i n g - gravity b) Common Relevant Formulae: T = 2 *3 .14( L/g ) 1 / 2 f = 1/T 37 2 E R + E p = k or 0.5mv + mgh= k c) Theory and Common Problem Types: The motion of the pendulum re s u l t s from the vector imbalance of the tension i n the s t r i n g p u l l i n g the bob toward the center and the force of gravity p u l l i n g the bob downward. The imbalance of the forces i s c a l l e d the restoring force (that which t r i e s to return the bob to i t s equilibrium point) and i s given by: F = -mg sin(A) where A i s the angle the s t r i n g makes with the v e r t i c a l . Simple harmonic motion i s defined as motion where the restoring force i s proportional to the displacement. For a pendulum t h i s i s not true ( i t i s proportional to sin( A ) ) . However, for angles less than 15 degrees the difference i s less than one percent and i s used as a good approximation. Therefore, although a student may consider the use of forces, i t was considered u n l i k e l y he/she w i l l be aware that the f i r s t formula i s an approximation. The f i r s t formula leads to three problems. The f i r s t two are either to f i n d the period or the length of a pendulum when given the other two variables ('g' i s usually assumed to be for Earth and not given e x p l i c i t l y ) . The t h i r d one i s to use a pendulum to fi n d the acceleration of gravity. While a l l three problems are a l g e b r a i c a l l y i d e n t i c a l , the f i r s t two are usually associated with "simple harmonic problems" at the simplest l e v e l . The t h i r d i s fundamentally d i f f e r e n t , i n that the variables for length and period are seen as d i r e c t 38 properties of the pendulum i t s e l f whereas the gr a v i t a t i o n a l acceleration i s a variable of the pendulum's "environment." It therefore i s usually used to calculate the acceleration of gravity on Earth, or some other planet. The second formula simply r e l a t e s period and frequency and could therefore be used to introduce a second step into the f i r s t three problem types. The t h i r d formula represents conservation of energy and i s shown i n two forms. The forms are generally interpreted as the f i r s t being more q u a l i t a t i v e and the second more quantitative. Qualitative problems would deal with a f t e r the bob i s released i t s potential i s transformed into k i n e t i c energy which i s then converted back into potential energy and so on. Quantitative problems could be related to how fa s t the bob i s t r a v e l i n g a f t e r being released from a given height or the reverse. A t h i r d group of problems i s possible that uses ranking: for example, "Where does the bob tr a v e l with the greatest speed?" 5.22 Accelerated Motion An object undergoes accelerated motion when the object i s acted on by an external unbalanced force. The acceleration i s i n the same d i r e c t i o n as the force. For t h i s study the object w i l l be a rock and the force w i l l be gravity. Gravity i s the force that an object, such as a rock, experiences as a r e s u l t of being i n the Earth's g r v i t a t i o n a l f i e l d (the rock also has a gr a v i t a t i o n a l f i e l d , but i t s e f f e c t on the Earth i s rather i n s i g n i f i c a n t ) . In a 39 gr a v i t a t i o n a l f i e l d the object experiences a force that i s proportional to i t s mass and inversely proportional with the distance squared. The distance i s taken from the Earth's center and for the surface of the Earth, may be considered a constant. Thereore, the force of gravity on the Earth's surface i s d i r e c t l y proportional to the mass. The acceleration a body undergoes i s inversely proportional to i t s mass. The r e s u l t of t h i s i s that the acceleration of gravity on the Earth's surface i s a constant and i s independent of the objects mass. It must be remembered that i f the distance from the Earth's center (altitude) varies s i g n i f i c a n t l y i n a given s i t u a t i o n the force of gravity w i l l not be a constant. The motion of an object w i l l be the vector r e s u l t of i t s i n i t i a l state of motion and the e f f e c t s of gravity. Any horizontal motion w i l l be unaffected by gravity; only v e r t i c a l motion i s affected. The i n i t i a l conditions of the s i t u a t i o n used i n t h i s study was a person holding a rock at the top of a c l i f f that overlooked a large body of water that had a sailboat on i t . If a rock i n that s i t u a t i o n i s released the only force acting on the rock i s the force of gravity and the rock undergoes accelerated motion i n a downward d i r e c t i o n . Accelerated motion was divided into two categories: "free f a l l " and " p r o j e c t i l e motion." An object was i n "free f a l l " when there was no horizontal motion and i n " p r o j e c t i l e motion" when there was horizontal motion. 5.23 Free F a l l 40 a) Possible Variables: height of c l i f f , distance of f a l l mass of rock i n i t i a l v e r t i c a l speed of the rock (up, down, none) time of f a l l energy (kinetic and potential) of the rock v e l o c i t y of rock at d i f f e r e n t heights or times acceleration of gravity b) Common Relevant Formulae: d = v Q t + 0.5gt 2 v = v Q + gt d = (v 2 - v Q 2 ) / ( 2 g ) 2 + Ep = k or 0.5mv + mgh = k c) Common Problem Types: The simplest problems involved no i n i t i a l v e l o c i t y (VQ = 0) and some examples are: How long does a rock take to reach the water i f the c l i f f height i s 50 meters? What i s the ve l o c i t y of the rock that f a l l s o ff a 50 meter c l i f f , when i t enters the water? If a rock i s dropped from a c l i f f and takes 3.19 seconds to land at the base, how high i s the c l i f f ? The second level of problems involved an i n i t i a l v e r t i c a l v e l o c i t y . The acceleration i s always downwards, but the i n i t i a l v e l o c i t y may be either upwards or downwards. Therefore, when the problems are solved the vector nature of variables must be used. The problems remain b a s i c a l l y the same as the ones previously described. 41 As was previously stated, the motion does not have any dependence on i t s mass; i t was expected, however, that some students would have an idea that suggests "the heavier one w i l l get there f i r s t " (Gunstone and Watts). Therefore, questions involving the use of mass were expected to be posed by students. 5.24 P r o j e c t i l e Motion P r o j e c t i l e motion i s the same as free f a l l with the addition of some independent horizontal motion. The previous discussion on free f a l l w i l l apply as well to p r o j e c t i l e motion. Therefore, only the possible variables, formulae, and problems that are needed i n addition to those for free f a l l w i l l be mentioned. a) Possible Varibles: horizontal distance angle of i n i t i a l motion b) Common Relevant Formulae: d = v h t c) Common Problem Types: The simplest case i s when there i s no i n i t i a l angle of motion r e l a t i v e to the horizontal. This then results i n problem that are a combination of accelerated motion and simple motion. The accelerated motion i s the v e r t i c a l component and the simple motion i s the horizontal component. An example i s : A car driv i n g at 50 km/h accidentaly drives off a 50 meter c l i f f , how far from the base of the c l i f f does i t land? 42 Problems, i n which there i s an i n i t i a l angle of motion r e l a t i v e the horizontal, must be solved with the use of vectors. The i n i t i a l motion must be resolved into horizontal and v e r t i c a l components. The problem i s then solved as a combination of free f a l l with i n i t i a l motion and horizontal motion. 5.25 Summary of Accelerated Motion Problems; The problems have been shown i n the preceeding two sections to be i n four basic groups: 1. Problems with no i n i t a l motion. 2. Problems with i n i t i a l v e r t i c a l motion. 3. Problems with i n i t i a l horizontal motion but no i n i t i a l v e r t i c a l motion. 4. Problems with i n i t i a l v e r t i c a l and horizontal motion. The problems generally range i n d i f f i c u l t y from the simpler to the more d i f f i c u l t . However, within each type there w i l l be variations that may be more s i g n i f i c a n t . 5.3 Data Gathering Instruments  5.30 Introduction The re l a t i o n s h i p between the ideas that students derived from s c i e n t i f i c paradigms and from other informal sources was studied through the use of: a) School records of the subjects past achievement i n selected science and mathematics courses. b) Paradigm-completion questions. c) Comparison questions. d) Textbook questions. The school records of the students was reviewed to make an assessment of the general academic a b i l i t y of the classes 43 studied. Past student a b i l i t y i n physics was infer r e d from school records of physics grades. The l a s t three types of questions were d i f f e r e n t forms of paper and pencil t e s t s . The rationales for the development and examples of each of the three types of questions are described i n the following subsections. 5.31 Paradigm-completion questions The idea underlying these questions came from Kuhn's writing on the development of science i n The Structure of  S c i e n t i f i c Revolutions (1962) and his a r t i c l e A Function for  Thought Experiments (1964). But, f i r s t a look at a l a t e r work by Kuhn, Logic of Discovery or Psychology of Research. In t h i s work Kuhn was asked to compare his own views of s c i e n t i f i c development with those of S i r Karl Popper. He re p l i e d that they were both concerned with the dynamic process by which s c i e n t i f i c knowledge i s acquired and that they both often turn to history for data and concluded that "from t h i s pool of shared data, we draw many of the same conclusions." However, he continued on to state that "What demands attention i s not so much the peripheral area i n which our occasional secondary disagreements are to be i s o l a t e d but the central region i n which we appear to agree ... our intentions are often quite d i f f e r e n t when we say the same things." What was of importance to t h i s study was not how Kuhn and Popper d i f f e r e d , but rather the method Kuhn chose to "explore the separation." Kuhn states: I am at once perplexed and intrigued about how best to explore the separation. How am 44 I to persuade S i r K a r l , who knows everything I know about s c i e n t i f i c development and who has somewhere or other said i t , that what he c a l l s a duck can be seen as a rabbit? How am I to show him what i t would be l i k e to wear my spectacles when he has already learned to look at everything I can point to through his own? In t h i s s i t u a t i o n a change i n strategy i s c a l l e d f o r , and the following suggests i t s e l f . Reading over once more a number of S i r Karl's p r i n c i p a l books and essays, I encounter again a series of recurrent phrases which, although I understand them and do not quite disagree, are locutions that I could never have used i n the same places. ... these metaphors, which s t r i k e me as patently inappropriate, may prove more useful than straightforward descriptions. They may that i s , be symptomatic of contextual differnces that a careful l i t e r a l expression hides. If that i s so, then these locutions may function ... as the rabbit-ear, the shawl, or the ribbon-at-the-throat which one i s o l a t e s when teaching a f r i e n d to transform his way of seeing a gestalt diagram, (pg. 3, 1970) The student had been hypothesized to possess his/her own informal ideas and may possess the formal ideas or s c i e n t i f i c paradigms. Both types of ideas are based on physical phenomena that are r e a d i l y available to both students and s c i e n t i s t s . However i t has been shown that s c i e n t i s t s view physical phenomena through what Piaget refers to as "logico-mathematical organization." In addition, s c i e n t i s t s have been given the exemplars and models that accompany the paradigm. The "paradigm-completion" questions functioned i n an analagous r o l e to the phrases which Kuhn used to illuminate the differences between his paradigm and Popper's. The questions were designed so that the view through the students "spectacles" was revealed. 45 A form of thought experiment was used to explore the viewpoints of the students. Thought experiments, as applied to t h i s problem, were not used i n t h e i r usual forms: s c i e n t i s t s have used them to perform "impossible" experiments, Piaget used them i n his development studies, and teachers have used them as learning devices. Thought experiments were instead, used to provide a setting or context, that allowed the students to use t h e i r "metaphors" i n a manner that revealed the ideas of t h e i r viewpoints. Several thought experiments are reviewed below i n order to show how they usually function and demonstrate t h e i r c h a r a c t e r i s t i c s . Kuhn, i n his a r t i c l e A Function For Thought Experiments, gave two examples of thought experiments. 1. Piaget dealt with children, exposing them to an actual laboratory s i t u a t i o n and then asked them questions about i t . In s l i g h t l y more mature subjects, however, the same ef f e c t might have been produced by questions alone i n the absence of any physical exhibit. If those same questions had been self-generated, we would be confronted with the pure thought experimental s i t u a t i o n ... (pg. 310, 1964) 2. (Galileo) asks his two interlocutors to imagine two planes, CB v e r t i c a l and CA i n c l i n e d , erected the same v e r t i c a l distance over a horizontal plane, AB ... His object i s to make them r e a l i z e that, using the concept of speed then current, they can be forced to admit that motion along the perpendicular i s simultaneously faster then, equal i n speed to, and slower than the motion along the i n c l i n e . His further object i s , by the impact of t h i s paradox, to make his interlocutors and readers r e a l i z e that speed ought not to be attributed to the whole of a motion, but rather i t s parts. In short, the thought experiment i s , as G a l i l e o 46 himself points out, a propaedeutic to the f u l l discussion of uniform and accelerated motion ... Galieleo's thought experiment brought the d i f f i c u l t y to the fore by confronting readers with the paradox i m p l i c i t i n t h e i r mode of thought. As a r e s u l t , i t helped them to modify t h e i r conceptual apparatus, (pg. 316, 1964) The most famous thought experiments are those devised by E i n s t e i n . Einstein's philosphy included Gedanken or thought experiments. His theory of science and knowledge emphasized an experiential foundation and i s summarized by M i l l e r (1984): ... the axiomatic structure (A) of a theory i s b u i l t psychologically on the experiences (E) of the world of perceptions. Inductive lo g i c cannot lead from the (E) to the (A). The (E) need not be r e s t r i c t e d to experimental data, nor to perceptions; rather, the (E) may include the data of Gedanken experiments. ... (pg. 44) Ein s t e i n made extensive use of thought experiments, not only in explaining phenomenon to others but also for his own formulations of concepts. M i l l e r (1984) outlines three of Einstein's thought experiments. 1. The thought experiment of 1895 that led to the special r e l a t i v i t y theory of 1905 concerned the experiences of a moving observer who t r i e s to catch up with a point on a l i g h t wave whose source i s at rest. What i s involved i n t h i s thought experiment i s the picture of a moving observer, the i n t u i t i o n of catching up with something that at f i r s t was moving faster than you, and the Anschauung or customary i n t u i t i o n of l i g h t as a wave phenomenon. By the customary i n t u i t i o n of l i g h t I mean depicting l i g h t as an e n t i t y that has properties that are abstracted from cert a i n properties of water waves. Throughout the next decade Ein s t e i n pondered t h i s thought experiment. ... (pg 243) 47 2. The thought experiment of 1907 that led to the 1915 general r e l a t i v i t y theory concerned an observer i n free f a l l and the consequences of there being "for him during t h i s f a l l no g r a v i t a t i o n a l f i e l d - at least i n his immediate v i c i n i t y " ... Ei n s t e i n "saw" objects that dropped with him, and thus f e l l at r e l a t i v e r e s t , i n the context of Newton's law of motion applied to the f a l l i n g observer and to an observer on the ground. Although s c i e n t i s t s have many times "seen" objects f a l l i n g side by side, Einstein "saw" what cognitive s c i e n t i s t r e f e r to as the "deep structure" i n t h i s scene; that i s , he "saw" the r e l a t i o n between gravity and acceleration so long as he also assumed the exact equality of g r a v i t a t i o n a l and i n e r t i a l masses, (pg. 244) 3. Straightway i n the r e l a t i v i t y paper, Einstein emphasized that the basic problems confronting physical theory concerned not the constitution of matter but understanding the equivalence of viewpoints between moving observers. For t h i s purpose, he began with the simplest thought experiment i l l u s t r a t i n g the problems of electromagnetic induction: a magnet and conducting loop i n r e l a t i v e i n e r t i a l motion. No mathematics was necessary to demonstrate that Maxwell's electrodynamics led observers on the wire loop and the magnet to d i f f e r e n t interpretations for the ph y s i c a l l y measurable e f f e c t , the current induced i n the conductor, (pg. 117) These f i v e examples of thought experiments were used, with the help of Kuhn (1964), to examine how thought experiments function and what purpose they serve. Kuhn posited an answer, which he thought was incomplete: "the new understanding produced by thought experiments i s not an understanding of nature but rather of the s c i e n t i s t ' s conceptual apparatus ... the thought experiment's function i s to a s s i s t i n the elimination of p r i o r confusion by forcing the s c i e n t i s t to recognise contradictions that had been 48 inherent i n his way of thinking from the s t a r t . " Kuhn i d e n t i f i e d four c h a r a c t e r i s t i c s of thought experiments; and M i l l e r i d e n t i f i e d a f i f t h c h a r a c t e r i s t i c from Einstein's thought experiments: 1. I t must allow those who perform i t or study i t to employ concepts i n the same ways they have been employed before. Only i f that condition i s met can the thought experiment confront i t s audience with unanticipated consequences of t h e i r normal conceptual operations, (pg 3 20) 2. [thought experiments function by] confronting the s c i e n t i s t with a contradiction or c o n f l i c t i m p l i c i t i n his mode of thought. Recognizing the contradiction then [appears] propaedeutic to i t s elimination ... (pg. 329) 3. Though the imagined s i t u a t i o n need not be even p o t e n t i a l l y r e a l i z a b l e i n nature, the c o n f l i c t deduced from i t must be one that nature i t s e l f could present, (pg 333) 4. The c o n f l i c t which confronts the s c i e n t i s t in the experimental s i t u a t i o n must be one that, however unclearly seen, has confronted him before. Unless he has already had that much experience, he i s not yet prepared to learn from thought experiments alone, (pg. 333) 5. These experiments concerned a mixture of Anschauungen [ i n t u i t i o n ] with mental imagery based on a strong image-perception l i n k , (pg. 243) These f i v e c h a r a c t e r i s t i c s , together with Kuhn's strategy formed the basis for the paradigm-completion questions. To summarize the development for the paradigm completion questions to t h i s point: a student was presented with a thought experiment that revealed the ideas of the student's viewpoint through the "metaphores" the student used. A 49 general c h a r a c t e r i s t i c of thought experiments i s t h e i r apparent s i m p l i c i t y , t h i s too, was present i n the paradigm completion questions. The questions are usually of situations and objects f a m i l i a r to the student. The questions are usually given i n a v i s u a l , diagramatic or sketch form. The problem however, must be given i n such a form that the student, i f not f a m i l i a r with the s c i e n t i f i c paradigm, w i l l be able to see his/her viewpoint. Two questions were developed using t h i s r a tionale. The f i r s t problem was i n the area of accelerated motion. The student was presented with a picture showing a boy standing at the top of a c l i f f holding two rocks. At the foot of the c l i f f was a lake or ocean with a sailboat some distance out from the foot of the c l i f f . The second problem was a picture of a pendulum. Both of these are events that are found i n the student's environment and are easy to v i s u a l i z e . The student was asked to provide " t y p i c a l " problems to accompany the s i t u a t i o n . The ideas of a student's viewpoint were revealed not only by the problems posed by the student, but also by the "locutions" or "metaphores" that the student used to state the problem. It must be stressed that t h i s use of thought experiments was fundamentally d i f f e r e n t from the usual application. Usually the student i s given both the s i t u a t i o n and the question. In t h i s instance the student was only given the s i t u a t i o n and expected to generate the question(s). It was the selection of the question, and the way i n which the 50 question was stated, that revealed the ideas of the student's viewpoint. A f i n a l c h a r a c t e r i s t i c of thought experiments that has not been mentioned, i s that they are invariably q u a l i t a t i v e and not quantitative. In t h i s a p p l i c a t i on of thought experiments the students were allowed to give quantitive, as well as quanlitative questions. 5.32 Comparison questions The purpose of these questions was to investigate whether students had acquired the s c i e n t i f i c paradigms. Unlike the paradigm completion questions, which were extremely open ended, these questions had a limited set of possible answers for the student to select. In addition, these questions only explored a limited set of possible paradigms, whereas the paradigm completion questions allowed the student complete freedom. These questions are c a l l e d comparison questions because students were be asked to compare the outcome of two given s i t u a t i o n s . Students were asked to compare two situations i n order to explore t h e i r acceptance of s c i e n t i f i c paradigms i n a manner that was s t r i c t l y non-mathematical. In addition, each question was given as a multiple choice question. An example i s : A B C D 1. A l f r e d and Betty each have a pendulum consisting of a b a l l attached to a s t r i n g . Each pendulum i s 1 meter long but Alfred's b a l l i s 250 grams whereas Betty's i s 500 grams. C i r c l e (A) i f Alfred's pendulum has a shorter period, (B) i f Betty's has a shorter period, (C) i f they 51 have a common period, and (D) i f the question cannot be answered. This example investigated the student's idea regarding the e f f e c t of mass on a pendulum's period. The r e s u l t s of t h i s may have been d i f f e r e n t from the paradigm completion question, i n that without d i r e c t i o n the student may not have thought of the r e l a t i o n s h i p as even e x i s t i n g . 5.33 Textbook Questions These are t y p i c a l problems found i n most introductory textbook of physics, and hence paradigmatic i n the usual sense of the word. As defined e a r l i e r these are well defined problems that have "clear stated i n i t i a l states and goal states." They are numerical problems given i n a real physical s i t u a t i o n . In addition, the 'operators for the solution are also given and defined i n advance. ' To help i n t h i s regard students were provided with the formulae, i f necessary, to solve the problems. A diagram of the problem was provided and space to solve the problem. An example i s (without the diagram): The diagram below shows Fred standing at the top of a 50 meter high c l i f f . Fred i s holding a red rock. How long does i t take for the red rock to reach the water after Fred drops i t ? This problem t y p i f i e s a textbook problem, but at the same time i s outside the d e f i n i t i o n . To q u a l i f y under the d e f i n i t i o n i t needs to have a c l e a r l y defined i n i t i a l state. In order to solve t h i s problem the acceleration of gravity i s 52 needed and not provided. Kuhn i n Function of Measurement in Modern Physical Science (In ISIS, 1961) argues that measurement has a "special e f f i c a c y " that has been given to science through textbooks. In t h i s case, since i t i s assumed to be on the surface of the Earth, the acceleration of 2 gravity i s 9.8 m/s . Textbooks inform the student, by examples, that: f i r s t of a l l , gravity can be assumed to be constant anywhere on the surface of the earth (regardless of a l t i t u d e , subsurface density, e t c . ) ; and secondly that he/she 2 2 may use 9.8 m/s instead of 9.80665 m/s (or some other more or less accurate f i g u r e ) . The problem also f a i l s for a more general reason. Kuhn describes the role of textbooks i n science experimentation (ISIS, p. 165): the application of a physical theory involves some approximation (in f a c t , the plane i s not " f r i c t i o n l e s s , " the vacuum i s not "perfect," the atoms are not "unaffected" by c o l l i s i o n s ) , and the theory i s not therefore expected to y i e l d quite precise r e s u l t s ... That no experiment gives quite the expected numerical r e s u l t i s sometimes c a l l e d "The F i f t h Law of Thermodynamics." ... what s c i e n t i s t s seek in numerical tables i s not usually "agreement" at a l l , but what they often c a l l "reasonable agreement." ... "Reasonable agreement" varies from one part of science to another, and within any part of science i t varies with time ... That, I think, i s why the tables are there: they define "reasonable agreement." By studying them, the reader learns what can be expected of the theory. ... Without the tables, the theory would be e s s e n t i a l l y incomplete. Students of physics, having done t h e i r own experimentation and the practice problems, which aft e r a l l represent r e a l l i f e (or experimental) s i t u a t i o n s , have been informed what i s 53 important and what may be ignored. In t h i s case, the problem should be treated as i f i t had been done i n a vacuum. Textbook problems were therefore ones whose " i n i t i a l states" and "goal states" are c l e a r l y defined only within the s c i e n t i f i c paradigm. 5.34 Summary of Questions The conclusion of the design introduction subsection (5.0) was that there must be "a variety of instruments; each of which may reveal d i f f e r e n t ideas." Three instruments were developed. The paradigm completion problems were completely open ended and placed only two basic conditions on the student. F i r s t l y , that the student was f a m i l i a r with the s i t u a t i o n . Secondly, that the student understood how to state a question - not the substance of the question, but how to form and state a question. The comparison problems required no mathematical or problem solving a b i l i t y , but were designed to investigate whether the students had "acquired the s c i e n t i f i c paradigms." The textbook problems required that the student possessed the global s c i e n t i f i c paradigm and was also, able to apply the s p e c i f i c paradigms i n exemplary puzzles. The paradigm completion questions were characterized as investigating the more informal ideas, whereas the textbook questions, the more formal ideas. The comparison questions allowed for both types of ideas. While the question types were given these characterizations they a l l had the p o s s i b i l i t y of revealing any ideas of the student's 54 viewpoint, i n p a r t i c u l a r when the viewpoint contained ideas that were inconsistent. 5.4 Data C o l l e c t i o n The data was coll e c t e d by giving paper and pencil exercises to the f i v e classes of students i d e n t i f i e d i n the section 4. Sample. The testing was done during the c l a s s ' regularly scheduled science (or algebra) c l a s s . The students were informed that the exercise was not for c r e d i t toward t h e i r coursework and would not form any part of the evaluation process for the class grades. The students were informed that the author was interested i n t h e i r responses for his own use. As expected, student response to the exercise was p o s i t i v e . This was expected for two reasons: f i r s t l y , that students i n general wish to do the best that they can i n any si t u a t i o n which i s reasonable; and secondly, that a good rapport had been established by the author with the students. The students were each given the paradigm completion questions f i r s t ; followed by the comparison questions; and l a s t l y the textbook problems. The paradigm completion questions for accelerated motion and harmonic motion were presented to the students i n the following manner. 1. The students were given a sheet of foolscap. 2 . The students were t o l d to draw a picture of a c l i f f overlooking the ocean, that had a boat out at sea. They were then t o l d to draw a boy at the top of the c l i f f holding one or two rocks. A sample diagram was simultaneously done for them on an overhead projector. 3 . The students were then t o l d to make up science questions that they think might be asked about the picture; and to write them below the 55 picture (additional paper was available i f needed). Science questions were explained to be those that might be asked by s c i e n t i s t s , by physics textbooks, given to grade eleven or twelve physics students on an exam. The students were asked to give f i v e to ten questions that were d i f f e r e n t . 4. The students were t o l d they could give whatever dimensions for the height of the c l i f f , mass of the rocks, and so on i f they wished. 5. The students were t o l d when they drew the boat that i t was there i n case they want an observer or something on the water. They were not t o l d they can throw the rock at i t , although that was the intention. 6 . The students were t o l d that they do not have to be able to solve the problem, although i t should be possible to solve i t , perhaps by someone older or with more physics t r a i n i n g . This was to allow students to present problems they thought were reasonable but s t i l l have been problems they could not solve themselves. 7 . After a short while they were given the pendulum paradigm completion" question. This question was provided on a sheet of paper as shown i n the appendix. It was explained that the dotted pendula were to allow them to describe the pendulaa i n d i f f e r e n t places. The general instructions for accelerated motion were repeated. The grade eight and nine classes required that a pendulum be shown to them. After the students fi n i s h e d the paradigm completion questions they were coll e c t e d and the students were given the comparison questions. The instructions were read to the students and p a r t i c u l a r emphasis was given to ensure that a clear d e f i n i t i o n of the period of a pendulum was given. When the students were fini s h e d they were given the textbook problems as well as a sheet of formulae i f they requested i t . They were instructed to t ry the problems without the use of the formulae sheet i f possible. 5.5 Analysis of Data 56 The i n i t i a l analysis of the data involved: transcribing the paradigm completion questions and tabulating the re s u l t s of the comparison questions. This set of data provided the most basic l e v e l of description for each student. This set of data formed a system that Cavallo (1979) refers to as an "object system." An object system requires three concepts: 1. attributes which constitute the methodological object (basic a t t r i b u t e s ) ; 2. attributes which constitute the f i e l d against which the basic attributes are observed or measured (supporting a t t r i b u t e s ) ; 3. a set of appearances for each basic and supporting a t t r i b u t e , (p. 52) In t h i s case, each student had a set of a t t r i b u t e s : t h e i r grade l e v e l , the amount of physics t r a i n i n g , the questions asked i n the paradigm completion exercise, the selections made of the comparison questions, and answers to the textbook questions. The supporting attributes are: the other possible grades, other possible levels of physics t r a i n i n g , and a l l the other possible answers to each of the three exercises. For the comparsion questions the supporting attributes consist of simply the other choices. For the textbook questions there were two sets: the 'r i g h t ' answer and the 'wrong' answers. For the paradigm completion exercises the supporting attributes were simply a l l possible questions. The set of a l l possible questions i s an open set, as i s the set of wrong answers. It i s impossible to meet Cavallo's t h i r d c r i t e r i a with an open set and so some l i m i t s were placed on them. The general research problem was considered by four 57 s p e c i f i c problems. The problems themselves provided the basis for l i m i t i n g the data. This was what Cavallo refers to as transforming an object system into a general system; and i s done through two steps: 1. Depending on the purposes of the investigation ... not a l l possible appearances of attributes w i l l be recognized as d i s t i n c t . Generally, a p a r t i t i o n i s defined on each set of appearances ... This step constitutes a determination of a resolution l e v e l for the investigation. 2. assignment: to each attribute of an interpretation-free symbol or name (and) to each of the lumped classes (partitions) a symbol which i s also interpretation-free (p. 53) The data was analyzed by investigating the s p e c i f i c problems i n the order presented at the s t a r t of t h i s chapter. The need for investigating the problems i n that order was that the r e s u l t s of each level of investigation was used to provide a background of data for the next l e v e l . This was necessary, because the background provided was used to close the set of possible answers. The f i r s t s p e c i f i c problem was investigated using entire classes as the case l e v e l . The second s p e c i f i c problem was investigated using only the grade 12 students divided into three cases; those with grade 12 physics, grade 11 physics and no physics. The f i n a l two s p e c i f i c problems were investigated using the student as the case l e v e l . Therefore i n terms of resolution, the four s p e c i f i c problems began with the coarsest resolution and proceeded to the f i n e s t resolution. 5.6 Interpretation of the Results 58 The next subsections outline how the analysis was implemented for each of the subproblems. The general strategy was to use the pattern-matching method of case-study of analysis as presented by Yin: For case-study analysis, one of the most desirable strategies i s the use of a pattern-mathcing l o g i c . Such a l o g i c compares an empirically based pattern with a predicted one (or with several alternative p r e d i c t i o n s ) . If the patterns coincide, the res u l t s can help a case study to strengthen i t s i n t e r n a l v a l i d i t y , (pp. 103) The following subsections outline the patterns for each s p e c i f i c problem that was predicted by the associated hypothesis. 5.61 The F i r s t S p e c i f i c Problem: Problem: Do the classes to be investigated consist of students that possess a 'characteristic viewpoint?' Hypothesis: As students mature t h e i r viewpoints w i l l undergo change through informal sources. In addition, the student acquires increasingly more complex levels of formal, or "logico-mathematical," reasoning s k i l l s . Therefore, i t i s hypothesized that students of a common age or grade l e v e l w i l l have viewpoints that are generally s i m i l a r , and that students of higher grades w i l l evidence a greater variety of ideas (although some of the ideas the students acquired e a r l i e r may be "hidden" by the l a t e r acquired s c i e n t i f i c 5 9 paradigms) and ideas that are more l i k e the s c i e n t i f i c paradigms. The hypothesis led to two tasks. The f i r s t task was to demonstrate that students of a p a r t i c u l a r grade had a c h a r a c t e r i s t i c viewpoint. The second task was to show that the c h a r a c t e r i s t i c viewpoints were d i f f e r e n t . The textbook questions had a heavy reliance on the "logico-mathematical" organization. Therefore, i t was expected that students without e x p l i c i t t r a i n i n g to solve physics problems would i n general, not be successful. The success i n solving problems was a d i r e c t measure of t h i s a b i l i t y . The comparison questions were expected to show that students of higher grade leve l s would tend to display the s c i e n t i f i c paradigm. This was done by comparing the proportion of each class that answered the question i n accordance with the s c i e n t i f i c paradigm. The grade twelve class contained students that have had e x p l i c i t physics t r a i n i n g and those with no t r a i n i n g . As a r e s u l t , only items on which there was no s i g n i f i c a n t difference between those with and without physics t r a i n i n g were used. The paradigm completion questions that were asked by the students were sorted into groups. The groups were established by using the nature of the question, not whether i t was a question that could be answered. So, for example, the following two questions were grouped together: (1) How long would i t take for a rock to f a l l from a f i f t y meter 60 c l i f f to the ground? and (2) How long would i t take the rock to f a l l ? The two questions are sorted together on the 'form' of the question not on i t s "mathematical completeness." For t h i s s p e c i f i c problem these groups of questions were further grouped into four c l a s s i f i c a t i o n s : questions that were not able to be c l a s s i f i e d i n the f i r s t sorting; questions that are largely of a descriptive nature, or those that r e l a t e more to the problem setting than to the physics of the problem; questions that do not conform to the s c i e n t i f i c paradigms; and questions that do conform to the s c i e n t i f i c paradigm, with an emphasis on the form rather than the s p e c i f i c d e t a i l s . From the hypothesis i t was expected that senior grades would show an increasing tendency toward questions that conformed to the s c i e n t i f i c paradigm. 5.62 The Second S p e c i f i c Problem: Problem: What e f f e c t does physics t r a i n i n g have on the viewpoints of students? Hypothesis: Students who receive physics t r a i n i n g are exposed i n a formal manner to the s c i e n t i f i c paradigms. As a r e s u l t , i t i s hypothesized that the viewpoints of students w i l l show increased evidence of ideas that are based on the s c i e n t i f i c paradigms with increased t r a i n i n g i n i physics. The only students with physics t r a i n i n g were the grade 12 students, and the other students were not considered. There was expected to be a noticeable difference i n student 61 a b i l i t y to solve textbook questions. The "within group differences" had been i d e n t i f i e d i n the F i r s t S p e c i f i c Problem for the comparison questions. The paradigm completion problems were expected to show a s i g n i f i c a n t difference not only i n the 'form' of the questions but also i n the content and structure of the questions. Therefore t h i s s p e c i f i c problem required that the students' questions that were based on the s c i e n t i f i c paradigms be assessed. 5.63 The Third S p e c i f i c Problem; Problem: Do students possess viewpoints that contain inconsistencies? Hypothesis: It was hypothesized that once a student possesses an idea i t i s always present. A student who has received physics t r a i n i n g w i l l , i n addition, have the ideas gained during science i n s t r u c t i o n . These are derived from what has been referred to as informal and formal sources of ideas. It i s hypothesized that some of these ideas w i l l be inconsistent with each other. It i s a d d i t i o n a l l y hypothesized that the student may revert to ideas that were developed before physics t r a i n i n g i n cases such as the following: a novel s i t u a t i o n ; or one where the student does not recognize the analogy to the "exemplars" previously practiced with during t r a i n i n g ; or the student may recognize part of exemplar and use e a r l i e r ideas to " f i l l i n the blanks." 62 This required i d e n t i f y i n g students with inconsistent ideas. This i d e n t i f i c a t i o n was done by observing a student using d i f f e r e n t ideas for what were e s s e n t i a l l y the same physical phenomena i n d i f f e r e n t parts of the inve s t i g a t i o n . 5.64 The Fourth S p e c i f i c Problem: Problem: Do students possessing a consistent viewpoint, have success i n applying s c i e n t i f i c paradigms? Hypothesis: I t was hypothesized that students with a consistent set of ideas w i l l be better able to apply the paradigms of science. It would appear to be a truism that students with a consistent viewpoint must by d e f i n i t i o n be successful; since, one of the paradigms i s exemplars. However, i t had also been hypothesized that a student must possess ideas that were derived before exposure to s c i e n t i f i c paradigms. These ideas, or some of these ideas, are bound to be inconsistent with the s c i e n t i f i c paradigms. Since the student cannot "erase" these ideas; the student must possess an inconsistent viewpoint. This i s a similar conclusion to that reached by Kuhn when he looked for the "rules" of the s c i e n t i f i c community and found neither a s u f f i c i e n c y of r u l e s . This problem appears to lead to a paradox: either to accept that the student can "erase" ideas or that a consistent viewpoint i s not necessary to apply paradigms. This issue was discussed by Campbell (1981) i n his a r t i c l e , "Can Inconsistency by Reasonble?" Campbell begins: We cannot know something [in the propositional sense] unless i t i s true. The 6 3 things that we know, therefore, must be l o g i c a l l y consistent. Moreover, we cannot know something unless we are j u s t i f i e d i n believing i t . But i t does not obviously follow that the things that we are j u s t i f i e d i n believing must be consistent with each other. For we can be j u s t i f i e d i n believing something that turns out to be f a l s e . Knowledge e n t a i l s truth and hence consistency. Rationally j u s t i f i e d b e l i e f does not e n t a i l truth and i t may not e n t a i l consistency. Campbell states that the ideal for someone seeking knowledge i s "to accept as many true statements and as few f a l s e statements" as possible. This may be accomplished by either "truth-seeking" or "perfection-seeking." The truth-seeker's goal i s the " t o t a l i t y i t s e l f " and w i l l accept some r i s k of f a l s e statements to gain true statements. The perfection-seeker's goal i s to accept no f a l s e statements. Using these d e f i n i t i o n s "truth-seeking i s compatible with t o l e r a t i n g inconsistency and incompatible with perfection-seeking." Campbell concludes that i t i s wiser to "retain t e n t a t i v e l y the inconsistency i n our b e l i e f s , rather than to r e j e c t one or more of them b l i n d l y or to invent ad hoc explanations or to suspend b e l i e f e n t i r e l y . " The student i s therefore hypothesized to be more of a truth-seeker and w i l l i n g to accept some inconsistencies. The problem was therefore interpreted to be: can a student have a consistent viewpoint that contains inconsistent ideas? 64 CHAPTER IV - RESULTS AND CONCLUSIONS 1. Introduction. The general research problem under investigation was to study the relat i o n s h i p between the ideas that a student derives from a study of s c i e n t i f i c paradigms and those the student derives from other informal sources. Four facets of t h i s general problem were investigated by four s p e c i f i c problems. The s p e c i f i c problems were investigated through tes t i n g the students of f i v e secondary school classes. The physical phenomena investigated was limited to accelerated motion and harmonic motion and the tes t i n g consisted of three paper and pencil t e s t s . As outlined i n Chapter I I I , the s p e c i f i c problems were arranged sequentially such that the res u l t s from the f i r s t were used for the second, and so on. This chapter i s organized i n a sim i l a r sequence. For each s p e c i f i c problem the re s u l t s and interpretations are given separately for the textbook problems, the comparison questions, and paradigm completion questions. Followed by a general discussion and conclusion(s) to the s p e c i f i c problem. After each s p e c i f i c problem has been discussed t h i s chapter ends with general conclusions and directions for further study. The students of the grade eight and nine classes were given the B r i t i s h Columbia Classroom Achievement Tests for Grade 7/8 of S c i e n t i f i c Processes and S c i e n t i f i c Literacy/Knowledge. The student's scores on these two tests were t o t a l l e d . The grade eight's mean score (x=51.5; 65 sx=11.4) was lower than the grade nine's mean score (x=5 9.3; s =14.8). The grade nine c l a s s ' mean score was s i g n i f i c a n t l y lower than the grade eight c l a s s ' at the alpha=0.10 l e v e l . [For the t - t e s t t = -1.92, t = 1.70] As a re s u l t the grade nine scores must be interpreted with caution. 2. The F i r s t S p e c i f i c Problem. The problem i s : Do the classes to be investigated consist of students that possess a c h a r a c t e r i s t i c viewpoint? 2.1 Comments on the Textbook Problems Students i n grades eight, nine, and ten were generally had no ideas on how to solve the textbook problems. This was e s s e n t i a l l y true even i f the students were given sample problems i n addition to the formulae. Eight students were able to i d e n t i f y that one example was i d e n t i c a l to the f i r s t accelerated motion problem and were therefore able to successfully solve the f i r s t problem. The f i r s t i n t e r p r e t a t i o n i s that students without grade twelve algebra (grade eleven algebra may s u f f i c e but was not investigated) or physics t r a i n i n g were generally not able to solve the accelerated motion or harmonic motion textbook problems. This lack of success was the r e s u l t of three possible factors. F i r s t l y , the students had not been given any p r i o r i n s t r u c t i o n or practice i n using the formulae of accelerated motion, or harmonic motion. Secondly, i t can safely be assumed that many of the students did not possess s u f f i c i e n t mathematical s k i l l to attempt solving the problems. Thirdly, given that the problems were novel to the students and the 6 6 students possessed limited mathematical s k i l l s , the student were not motivated to attempt the problems. 2 . 2 Comparison Questions  2 . 2 1 Harmonic Motion The re s u l t s of the comparison questions were as shown i n Table 1; and are given as percentages for comparison. The percentages are shown for two student groupings: within a l l the students by grade; and within the grade twelve students by l e v e l of physics t r a i n i n g . It must be remembered that within the grade twelve students the number of students within each category was small ( f i v e with physics twelve, seven with physics eleven, and 5 with no physics) and therefore care needs to be taken with t h e i r interpretation. The s i x t h and seventh items showed strong t r a i n i n g e f f ects i n addition to grade l e v e l and w i l l be discussed i n both the f i r s t and second s p e c i f i c problem sections. The f i r s t and eighth questions were e s s e n t i a l l y answered co r r e c t l y by a l l grade twelve students while the other classes displayed a general increase with grade l e v e l . The f i r s t question concerned the rel a t i o n s h i p between the length of a pendulum and i t s period. The eighth question concerned the r e l a t i o n s h i p between the pendulum's period and the d i r e c t i o n of swing. The second to f i f t h questions concerned the r e l a t i o n s h i p between the period of a pendulum and respectively: mass, i n i t i a l angle, "push," and a l t i t u d e above sea l e v e l . Less than half the students i n each of the grade levels selected the correct answer. Due to the lack of 67 student success i n questions one and two i t was not possible to i n t e r p r e t the responses of students to question nine. As a r e s u l t there were three interpretations for harmonic motion, without considering questions s i x , seven or nine for the stated reasons. The second interpretation was that students tended to possess the idea that factors, such as the mass of the pendulum, influenced the pendulum's period; when i n f a c t , the factors do not influence the period. This i n t e r p r e t a t i o n was i n agreement with Inhelder and Piaget (1958, pp. 6 7-7 9.) who considered the pendulum to exemplify a physical phenomena that required students to employ "operations of exclusion" and made sim i l a r observations of children. Questions two, three, and four e s s e n t i a l l y related the period of a pendulum to some idea of "force" or "impetus." Students with or without t r a i n i n g and of a l l grade lev e l s showed a strong reluctance to eliminate the idea that these factors were i n f l u e n t i a l . Question six showed a t r a i n i n g e f f e c t as stated e a r l i e r , for the grade twelve students; but for the remaining students t h e i r responses showed a reluctance to eliminate the material of the pendulum " s t r i n g " as being related to i t s period. Questions seven and eight related the period of a pendulum to i t s location (hemisphere of pendulum and d i r e c t i o n of swing). Grade eight and nine students s t i l l showed a reluctance to eliminate the idea that these variables were i n f l u e n t i a l , whereas the grade ten students had generally successfully eliminated these 68 variables. Question f i v e was to investigate student understanding of the relat i o n s h i p between the effects of gravity and period of the pendulum (the acceleration of gravity being reduced with a l t i t u d e ) . The res u l t s through a l l grades were b a s i c a l l y mixed, with perhaps, the exception of the grade twelve physics students. It was believed that the majority of students did not interpret the question i n the intended manner - i t may have been better to have placed the pendulum on the moon, for instance. The t h i r d i nterpretation regards the rel a t i o n s h i p between the length and period of a pendulum. As stated i n the f i r s t i n terpretation, grade eight students were unable to eliminate other variables; and t h i s was also the r e s u l t for question one which related period to length. However as i s seen i n Table 1, grade eight students were equally divided between choosing whether the pendulum's period would be longer or shorter. This implied that grade eight students were guessing and do not possess the s c i e n t i f i c paradigm of the r e l a t i o n s h i p beteen length and period. Grade ten students did generally better (69%); and grade twelve students possessed the s c i e n t i f i c paradigm. It i s int e r e s t i n g to note that when the grade twelve algebra students were given the te s t , many of the students were observed to swing an imaginary penudulum while answering these questions. This could imply that the grade twelve students were either better able to v i s u a l i z e t h i s phenomena 69 or more i n c l i n e d to perform a thought experiment of t h e i r own (or perhaps both). The t h i r d i nterpretation was then: that students of grade eight do not posses the s c i e n t i f i c paradigm of the r e l a t i o n s h i p between the period of a pendulum and i t s length, the majority of grade ten students, and most grade twelve students possess the s c i e n t i f i c paradigm. 2.22 Accelerated Motion The majority of questions on accelerated motion, as shown i n Table 2, demonstrated clear evidence of the e f f e c t s of physics t r a i n i n g , or were eliminated as being related to the students problem solving a b i l i t i e s . Questions one and nine showed no t r a i n i n g e f f e c t but a d i s t i n c t grade e f f e c t . Question one asked students what would happen i f two i d e n t i c a l rocks were dropped at the same time. It was s i g n i f i c a n t to note that although no students believed a s p e c i f i c rock would land f i r s t , some students believed i t would not be possible to predict which would land f i r s t . The significance was interpreted to be that students were not guessing and did i n f a c t believe that i t was impossible, for whatever reasons, to predict which rock would land f i r s t . This r e s u l t was contrary to the general s c i e n t i f i c p r i n c i p l e s of Newtonian mechanics which are being developed i n students of that age; the most fundamental of which i s "determinism." The d e f i n i t i o n of "determinism" taken from M i l l e r (1984, pp. 175.) i s : "a system w i l l occupy every point on i t s possible tr a j e c t o r y and/or that a system w i l l develop on the tra j e c t o r y that i s required by the i n i t i a l previous 70 conditions because every successive state i s entailed by the one previous." The fourth interpretation was that the idea of "determinism" was possessed by the grade twelve students; but not by a l l the grade ten or eight students. Question nine asked students i f i t was possible to determine the time taken for the rock to f a l l to the bottom of the water. This question i s , i n addition to being impossible to solve as stated, an inappropriate question within the Newtonian paradigm (this i s because i n water the f l u i d e f f e c t s could not be ignored). This f i f t h i n t e r p r e t a t i o n i s then that: students of grade twelve were better able to recognize a problem as being "paradigmatic" than were grade tens; and s i m i l a r l y grade tens were better than grade eights. 2.3 Paradigm Completion Questions The were 165 student paradigm completion responses for harmonic motion and 313 responses for accelerated motion. Both the harmonic motion and accelerated motion questions were divided into four categories as follows: appropriate questions, inappropriate questions, descriptive questions, and those that were unable to be c l a s s i f i e d . Whether a question was appropriate or unappropriate was determined, as outlined i n Chapter III (section 5.61) by the "form" of the question matching the s c i e n t i f i c paradigm. The re s u l t s of t h i s categorization are shown i n Table 3. The following section discusses the results and interpretations for harmonic and accelerated motion. 71 2.31 Harmonic and Accelerated Motion It was observed that students were able to produce more questions about accelerated motion (313 vs. 165) and that there was fewer u n c l a s s i f i a b l e accelerated motion questions (9% vs. 20%). This was interpreted to be the r e s u l t of three factors: f i r s t l y , students had more previous experiential i n t e r a c t i o n with accelerated motion; secondly, the accelerated motion problems offered the students more "props" for t h e i r problems; and t h i r d l y , as a r e s u l t of the f i r s t two factors, accelerated motion was more e a s i l y v i s u a l i z e d by students. The si x t h interpretation was then: student viewpoints of accelerated motion were generally more robust than t h e i r viewpoints of harmonic motion. The seventh interpretation was that students asked the same form of question regardless of the phenomena being investigated. This was based on the observation that students tended to ask the same type of questions, with the exception of u n c l a s s i f i e d questions, for both accelerated and harmonic motion. The support for t h i s statement i s shown i n Table 4, which i s the same as Table 3 with the following modifications: u n c l a s s i f i e d questions were omitted, the percentage of each of the other three question types was replaced with i t s rank for each grade. It i s seen that 0.75 of the c e l l s agree with each other. The eighth int e r p r e t a t i o n was that as students of grade increased the questions changed from being more descriptive to being more appropriate. Support for t h i s i s shown i n 72 Table 5, which shows the percentage of a l l questions answered by grade l e v e l . It was evident from the table that acceptable questions increased from 33% for grade eights to 70% for grade twelves, meanwhile descriptive questions dropped from 42% to 6%. The implication i s that as students grade l e v e l increased the student's focus i n the question changed from the story of the problem to the physics of the problem. This r e s u l t i s si m i l a r to that found by investigation of problem r e c a l l by students ( S i l v e r , 1981; K r u t e t s k i i , 1976) who found that good problem solvers tend to r e c a l l the "structural features of the problem" whereas poor problem solvers tend to r e c a l l the " s p e c i f i c d e t a i l s of a problem statement." 2.4 Conclusions for the F i r s t S p e c i f i c Problem During the preceeding discussion eight interpretations were i d e n t i f i e d ; they were as follows: 1. Students without grade twelve algebra (grade eleven algebra may s u f f i c e but was not investigated) or physics t r a i n i n g were generally not able to solve the accelerated motion or harmonic motion textbook problems. 2. Students tended to believe that factors, such as the mass of the pendulum, influenced the pendulum's period; when i n f a c t , the factors do not influence the period. 3. Students of grade eight do not posses the s c i e n t i f i c paradigm of the re l a t i o n s h i p 73 between the period of a pendulum and i t s length, the majority of grade ten students, and most grade twelve students possess the s c i e n t i f i c paradigm. 4. The idea of "determinism" was acquired by the grade twelve students; but not by a l l grade the ten or eight students. 5 . Students of grade twelve were better able to recognize a problem as being "paradigmatic" than were grade ten students; and s i m i l a r l y grade ten students were better than grade eight students. 6. Student viewpoints of accelerated motion were generally more robust than t h e i r viewpoints of harmonic motion. 7. Students asked the same form of question regardless of the phenomena being investigated. 8. As a student's grade level increased the questions changed from being more descriptive to being more appropriate. Two tasks were i d e n t i f i e d i n Chapter III with respect to the f i r s t s p e c i f i c problem: f i r s t l y , to show students of a class a possessed a c h a r a c t e r i s t i c viewpoint; and secondly, that the viewpoints were d i f f e r e n t . It i s clear from the breadth and depth of the eight interpretations that students of d i f f e r e n t classes, did indeed possess c h a r a c t e r i s t i c 74 viewpoints. Some ideas were common to a l l the students investigated, such as the preference for accelerated motion; whereas, some ideas were more common i n one class than others, such as the rel a t i o n s h i p between a pendulum's period and length. This second idea i s an example also of the second task - to show classes are d i f f e r e n t . I t i s also evident that i n a l l cases where the classes d i f f e r , those of higher grade levels showed better reasoning s k i l l s (problem solving for example) or evidence of the s c i e n t i f i c paradigm (for example the idea of "determinism".) 3. The Second S p e c i f i c Problem The s i x t h i n t e r p r e t a t i o n from the f i r s t s p e c i f i c problem stated that: students were better able to relate to accelerated motion than to harmonic motion. This was interpreted to be p a r t i a l l y , a r e s u l t of students having more informal, experiential actions with accelerated motion. However, as was seen i n the comparison questions acceleration showed strong effects of t r a i n i n g . Whereas, harmonic motion showed a persistence of " n o n - s c i e n t i f i c " b e l i e f s even after t r a i n i n g . This was demonstrated by the res u l t s of questions two, three, and seven which showed that students through grade twelve did not believe: from question two, that the time to reach the bottom of the c l i f f i s independent of the rock's mass; from question three, that a dropped rock and a rock thrown h o r i z o n t a l l y take the same time to f a l l ; and from question seven, that the rate of acceleration i s a constant. These three ideas are fundamental to accelerated motion of 75 p r o j e c t i l e s ; but these ideas did not seem to be learned from informal experiences. However, while students seemed to have less experience with harmonic motion, students seem to acquire the idea of how the period of a pendulum i s related to i t s length even without i n s t r u c t i o n . The conclusion i s that informal, experiential actions are i n s u f f i c i e n t to develop the ideas of s c i e n t i f i c paradigms. The second s p e c i f i c problem i s : what e f f e c t does physics t r a i n i n g have on the viewpoints of students? This w i l l be investigated using only the phenomena of accelerated motion. 3.1 Paradigm Completion Questions The summary of the questions asked by the grade twelve students i s shown i n Table 6. It i s evident from the table that students with grade twelve physics posed considerably more and considerably better questions; whereas students without t r a i n i n g were not able to i d e n t i f y the s c i e n t i f i c paradigm. Additional evidence of the students without t r a i n i n g not possessing the s c i n t i f i c paradigm was that none of them posed a question that asked how long the rock would take to f a l l . This i s clear evidence that physics t r a i n i n g a l t e r s students viewpoints to include more s c i e n t i f i c paradigms - at least i n the form of producing exemplars. The high number of u n c l a s s i f i a b l e questions posed by grade eleven physics students showed that they possessed part, but not a l l , of the s c i e n t i f i c paradigm. This was demonstrated by the following two questions: A boy stands atop a 40 m c l i f f and throws a 90g stone into the water below. How hard 76 does he throw the stone? A man drops a rock weighing 5 g down a c l i f f at a height of 15 m. If the boat i s 6 1/2 meters from the c l i f f how far i s the man from the boat? These problems were the f i r s t questions given by two d i f f e r e n t students; and as a r e s u l t could be expected to be the students most paradigmatic problem. The f i r s t sentence of each problem i s within the s c i e n t i f i c paradigm (although the f i r s t problem may have needed to provide more data on the "throw"); the mass of the rock i s not needed, and the second student noted above the question that the "5 g" was extraneous information. The second sentence does not follow the paradigm. 3.2 Comparison Questions Questions two through seven show clear evidence of the e f f e c t of t r a i n i n g . The students with grade twelve physics possessed the s c i e n t i f i c paradigms those with no physics t r a i n i n g demonstrated the majority did not. Students with physics eleven showed, as i n the previous section, that some of the students, but not a l l , possessed the s c i e n t i f i c paradigm. This was demonstrated by the re s u l t s of three of the questions where less than 60% of the students with grade eleven physics demonstrated they possessed the s c i e n t i f i c paradigm. Question three asked i f a rock throw h o r i z o n t a l l y or dropped would land f i r s t . This concept i s fundamental to p r o j e c t i l e motion; that i s , the horizontal and v e r t i c a l motions are independent. However, i n t u i t i o n seems to t e l l 77 students i n c o r r e c t l y , that the object that t r a v e l l s the "longer distance" must take more time. The f i f t h question asked what angle would make the p r o j e c t i l e t r a v e l the furthest out from the c l i f f . The students demonstrated a lack of using ideas of symmetry; i t may be infe r r e d that some other idea may have been i n t e r f e r r i n g but would require further in v e s t i g a t i o n . Question seven asked students about the rate of acceleration. This r e s u l t i s i n p a r t i a l agreement with Trowbridge and McDermott (1981) who concluded that "although most students could define acceleration i n an apparently acceptable manner af t e r i n s t r u c t i o n , they could not use t h i s d e f i n i t i o n to determine a procedure for comparing the accelerations of two moving objects." It would appear that i n t h i s case rather than accepting the s c i e n t i f i c paradigm; students may have t r i e d to v i s u a l i z e the rock at d i f f e r e n t times of i t s descent and arrived at the incorrect conclusion. 3.3 Textbook Problems The grade twelve physics students were e s s e n t i a l l y able to answer a l l f i v e questions. One physics twelve student made two algebraic mistakes, and the other was not able to solve questions four and f i v e c o r r e c t l y ( i t must be noted that t h i s student had fi n i s h e d grade twelve physics during the previous school year). A l l the grade eleven physics students requested to be given the formulae and were provided with them. One student didn't hand i n the problems; one student did not attempt any 78 problems; two students only attempted the f i r s t problem, one of them did the problem c o r r e c t l y ; the remaining three students attempted at least the f i r s t three, however only one student c o r r e c t l y answered even the f i r s t question. A l l the physics eleven students had been able to solve si m i l a r problems when they were taking physics eleven; and at the present time a l l the students were passing algebra eleven; i n fa c t four of the students recieved B l e t t e r grades i n both physics eleven (the kinematics section) and algebra twelve. The students with no physics t r a i n i n g were given the formalae, but were not able to solve the problems. The students were then given example problems. Three students used these examples and a l l were able to solve the f i r s t two problems; two students solved the t h i r d problem; and one student solved the fourth. For a student to solve the textbook problems the student must not only possess the s c i e n t i f i c paradigms, but must also be able to apply the paradigms. It was evident i n the f i r s t two sections that the grade eleven physics students possessed part, but not a l l , of the s c i e n t i f i c paradigms. However, as was shown by the student with no physics; i t i s possible to solve textbook problems with the use of exemplars. Thus i t i s possible for students to be successful i n taking a physics course, but not have adopted the s c i e n t i f i c paradigms. 3.4 Conclusions for the Second S p e c i f i c Problem The second s p e c i f i c problem was: "what e f f e c t does physics t r a i n i n g have on the viewpoints of students." It was hypothesized that the viewpoints of students with increased t r a i n i n g would show increased evidence of possessing the s c i e n t i f i c paradigms. This was indeed, found to be true. However, i t was observed that the grade eleven students had d i f f i c u l t y applying the paradigms. 4. The Third S p e c i f i c Problem. The problem was: Do students possess viewpoints that contain inconsistencies? The problem was not to determine the frequency of students with inconsistent viewpoints; but rather to i d e n t i f y some students that possess inconsistent viewpoints. Two cases of students with inconsistent viewpoints were i d e n t i f i e d . 4.1 The F i r s t Case The ninth comparison question asked students i f i t was possible to determine the time taken for the rock to f a l l to the bottom of the water. Two grade eight students responded c o r r e c t l y to queston nine; however the students had e a r l i e r posed the following paradigm completion questions: How long does i t take for the rock to get to the bottom of the ocean? The boy throws a rock into the water the distance between the top of the water and the c l i f f i s 25 meters, i f the weight of the rock i s 0.72 kg and you know what water density i s . The top of the water to the ocean f l o o r i s 32 meters. How long w i l l i t take for the rock to reach the bottom of the ocean. It i s s i g n i f i c a n t that the students asked the question on th e i r own and must have seen i t as a reasonable question. No 80 other students asked t h i s question, athough eleven students asked questions either about "how far the dropped object would sink" or "how fast the dropped object would sink." The only possible conclusion i s that when la t e r asked the same question the students used d i f f e r e n t ideas. It i s possible to argue that t h i s i s not an idea that the students "possessed" but rather an idea that was formed at the time. This i s no solution however, since i n both s i t u a t i o n s , i f the students did not "possess" the idea then they must have reasoned the "ideas" from other ideas. To have reached d i f f e r e n t "ideas" then, the students must hold other ideas which are inconsistent (or possibly a worse s i t u a t i o n i n which the students reasoning i s inconsistent). Therefore i t must be concluded that i n d i f f e r e n t situations students may reveal ideas that are are inconsistent. 4.2 The Second Case The responses of a student with grade eleven physics to the paradigm completion question and the comparison questions w i l l be outlined. The students responses to the textbook problems w i l l then be analyzed. The student posed the following two problems i n response to the paradigm completion task: If the man threw the a rock at 5 km/hr up i n the a i r , how long would i t take to land i n the water? If the man threw a rock h o r i z o n t a l l y at a speed of 10 km/hr and i t took 2 0 sees to land how far away did the rock land from the c l i f f ? 81 The diagram showed the c l i f f ' s height t o be 50 meters. The f i r s t question i s paradigmatic. The second would be paradigmatic, except f o r the mention of the time taken. The time i n a d d i t i o n t o being i n c o r r e c t f o r the s i t u a t i o n , showed t h a t the student has not separated h o r i z o n t a l and v e r t i c a l motion. The student answered a l l the comparison question c o r r e c t l y ; with the exception of questions three and seven. The student's response to question three i n d i c a t e d t h a t a dropped rock would land before a h o r i z o n t a l l y thrown rock; and t h i s i s i n agreement w i t h the second paradigm completion question posed. However, the student responded c o r r e c t l y t o question s i x , t h a t the h o r i z o n t a l d i s t a n c e the rock goes v a r i e s d i r e c t l y w i t h i t s i n i t i a l speed. An i n c o n s i s t e n c y t h a t could p o s s i b l y be explained by the student using symmetry (the question has the word "twice" and so does the answer) to s e l e c t the answer (although the m a j o r i t y of students d i d n o t ) . The student's response to question seven i n d i c a t e d t h a t the a c c e l e r a t i o n of g r a v i t y i s g r e a t e s t at the s t a r t . Based on the f i r s t two tasks the sudent would appear to have p a r t i a l l y adopted the s c i e n t i f i c paradigms of a c c e l e r a t e d motion. The student attempted a l l of textbook problems. For f i r s t question the student r e v e r t e d t o unaccelerated motion and used the equation "d = v t . " The student i n doing t h i s , demonstrated the most b a s i c i n c o n s i s t e n y p o s s i b l e - t h a t of simply i d e n t i f y i n g the motion 82 as accelerated. In response to question three the student, who had posed a very si m i l a r question i n response to the paradigm completion, again reverted to unaccelerated motion. The student attempted question four. A diagram showed a c l i f f and a p r o j e c t i l e following a path that i s horizontal for roughly one-third of i t s length and then approximately parabolic (with some tendency to straighten out). The diagram showed that the student believed the motion to be accelerated (although the straight portion could possibly be interpreted as an idea si m i l a r to "impetus"). However, the acceleration i s c l e a r l y not the greatest at the s t a r t as was chosen e a r l i e r . Therefore, the student was incosistent about where a f a l l i n g body's acceleration i s greatest. The student had also i d e n t i f i e d the s i t u a t i o n as accelerated motion. In addition the student, although not treating horizontal and v e r t i c a l motion as being independent, uses the equation "d = 2 2 (v -VQ )/2g". The formula was i n c o r r e c t l y applied, the i n i t i a l horizontal speed was used for VQ and 9.8 was used for v (possibly because there were no other numbers available, i t was however an i n t e r e s t i n g choice). It was evident that the student possessed very inconsistent ideas about i d e n t i f y i n g a f a l l i n g object as being a form of accelerated motion. 4.3 Conclusions It i s evident from the two cases that students can possess inconsistent viewpoints. The inconsistencies can be is o l a t e d regarding s p e c i f i c ideas, such as the f i r s t case; however, the inconsistencies can be f a i r l y fundamental and 83 pervasive, such as the idea of a f a l l i n g object being a form of accelerated motion. 5. The Fourth S p e c i f i c Problem The fourth problem was: do students possessing a consistent viewpoint, have success i n applying s c i e n t i f i c paradigms? The viewpoints of two grade twelve physics students were investigated. The pseudonyms Chris and Pat were chosen. Both students by graduation had taken physics, mathematics and chemistry to the grade twelve l e v e l achieving f i n a l l e t t e r grades of A or B i n these courses. The two students were i n competition with each other. It was acknowledged by both students i n t h e i r "post mortems" of exam and test r e s u l t s that: Pat had better "natural" a b i l i t y but was more careless. Support for both of these assessments were observed i n cl a s s . During labs, Chris preferred not to work with Pat, since Pat did things too quickly. Chris tended to be a hard worker and as a r e s u l t , a good natured r i v a l r y persisted. Both students responded c o r r e c t l y to a l l the comparison questions and solved a l l problems c o r r e c t l y (Pat did make a careless mistake with a vector, but otherwise the question was co r r e c t ) . However, the students'- responses to the paradigm completion questions were very d i f f e r e n t . Their responses are l i s t e d below: Pat had eleven questions: 1. What would the period be i f the length of rope was 295 cm (on earth)? 84 2. What would the gravity (in m/s ) i f the period was 8 sec and the length of rope was 780 cm long? 3. Would the period be d i f f e r e n t for a pendulum i f you change the height you started the pendulum from? 4. What would the length of the pendulum be i f the period was 12 sec on earth? -i 5. Would i t be possible to have two pendulum with the same period but d i f f e r e n t lengths on the same planet? 6. What would the length of the pendulum be i f the period was 12 sec on the moon? (force of gravity 1/6 that of earth) 7. If (assume the length of the rope for the pendulum i s constant and gravity i s constant) you double the mass of the pendulum how much w i l l i t e f f e c t the period of the pendulum? 8. If the length of the rope for the pendulum i s increased by 4 times how would t h i s e f f e c t the period i f gravity i s constant? 9. If the length of the rope for the pendulum i s increased 4 times how much would gravity have to be changed for the period to stay the same? 10. What would be the period of a pendulum that was hung from a t a l l b uilding with a height of 900 m? 11. How much would the period change i f gravity was cut i n half and the length was increased 8 times. Chris had four questions. 1. What i s the period of A pendulum that i s 2 m long with a mass of 200 kg. 2. An average human w i l l pass out when he/she experiences a ce r t a i n number of g forces. What speed w i l l the pendulum have to be at point "E" i n order to make the human inside to pass out. 85 3. If a pendulum was attached to an e l a s t i c with a constant of .15. If the pendulum was l e t go at point "A" what w i l l be the stretch of the e l a s t i c at point "C." Length of„elastic o r i g i n a l l y = 10 cm g=9.8 m/s Length of pendulum = 1 m mass - 10 kg 4. With the force of the earth's rotation a pendulum w i l l spin when i t i s l e t go with a 2 m s t r i n g on a 25 kg b a l l how long w i l l i t take to make 1 rev? [A diagram accompanied th i s problem showing a pendulum from on top with three swings showing horizontal rotation] The instructions given to the students were to propose f i v e to ten d i f f e r e n t problems about pendula. Pat's viewpoint, as demonstrated by the proposed problems, showed a f l e x i b i l i t y and richness of ideas about harmonic motion. Pat's problems dealt simply and e f f e c t i v e l y with most of the possible variations that were i d e n t i f i e d i n Chapter III (Section 5.21) Pat did not propose a question about the e f f e c t on the period of a pendulum of i n i t i a l "push"; however, Pat did include some int e r e s t i n g variations comparing pendula (see question five) and combined e f f e c t questions (see question nine). Pat displayed inconsistent ideas about the e f f e c t of the mass of a pendulum on i t s period. In the comparison questions Pat responded c o r r e c t l y i n d i c a t i n g no dependence. However, Pat's seventh question asks what e f f e c t doubling the mass would have on a pendulum's period. Pat's viewpoint must have contained an idea that said i t was a good question to ask. This was an inconsistency for a "perfection-seeker" who must rej e c t 86 anything that turns out to be f a l s e . A "truth-seeker" would tol e r a t e t h i s idea as being part of the " t o t a l i t y " of harmonic motion. Ch r i s ' viewpoint, by comparison, seemed to be impoverished and meagre. Chris' f i r s t problem was paradigmatic but the focus of the remaining questions involved other phenomena: c i r c u l a r motion, e l a s t i c i t y , and Foucault's pendulum. Chris' problems did not focus on harmonic motion; but rather, used the pendulum as a "setting" for the problem's statement. Chris, at the time, thought he was proposing very "clever" problems and was proud of them. However, Chris did not recognize the purpose of the paradigm completion task. Chris revealed the most paradigmatic exemplar and then moved outside the paradigm. The possible interpretations are that: Chris only possessed the one idea; or that Chris did not recognize other variations of the idea as being s i g n i f i c a n t . The f i r s t was not possible due Chris' success on the other tasks. The interpetation that Chris did not r e a l i z e that the exemplar or formula (note that the exemplar uses the formula without algebraic manipulation) provide a locus for a family of ideas. Therefore C h r i s ' viewpoint consists of ideas that are poorly integrated. An advantage would be that any ideas that are inconsistent would not be i n c o n f l i c t . In conclusion, i t was found that there were two ways a student could have a consistent viewpoint, and s t i l l have ideas that are themselves inconsistent. F i r s t l y , a 87 " t r u t h - s e e k i n g " s t u d e n t c o u l d p o s s e s s i n c o n s i s t e n c i e s t h a t a r e a c c e p t a b l e . S e c o n d l y , i t was s e e n t h a t i n c o n s i s t e n c i e s t h a t w e r e p o o r l y i n t e g r a t e d c o u l d a l s o a p p e a a r t o be c o n s i s t e n t t o t h e s t u d e n t , i n t h a t t h e y w o u l d n o t be u s e d a t t h e same t i m e . S t u d e n t s w i t h c o n s i s t e n t , i n c l u d i n g t h e b r o a d e n e d d e f i n i t i o n o f c o n s i s t e n t , c a n be s u c c e s s f u l i n a p p l y i n g t h e p a r a d i g m s o f s c i e n c e . 6 . G e n e r a l C o n c l u s i o n s The g e n e r a l p r o b l e m was t o s t u d y t h e r e l a t i o n s h i p b e t w e e n t h e i d e a s t h a t a s t u d e n t d e r i v e s f r o m a s t u d y o f s c i e n t i f i c p a r a d i g m s a n d t h o s e t h e s t u d e n t d e r i v e s f r o m o t h e r i n f o r m a l s o u r c e s . The f i r s t t w o s p e c i f i c p r o b l e m s showed t h a t s t u d e n t s p o s s e s s e d a c h a r a c t e r i s t i c v i e w p o i n t a n d t h a t a v a r i e t y o f i n s t r u m e n t s w e r e n e e d e d t o r e v e a l s t u d e n t i d e a s . The r e s u l t s o f t h e t h i r d a n d f o u r t h s p e c i f i c p r o b l e m n e e d t o be c o m p a r e d . I n t h e f i r s t c a s e , s t u d e n t s w e r e f o u n d t o p o s s e s s i n c o n s i s t e n t i d e a s . The i n c o n s i s t e n t i d e a s i d e n t i f i e d i n t h e t h i r d p r o b l e m l e a d t o an i n a b i l i t y t o a p p l y t h e s c i e n t i f i c p a r a d i g m s . However, i n t h e i n c o n s i s t e n t i d e a s i d e n t i f i e d i n t h e f o u r t h s p e c i f i c p r o b l e m d i d n o t l e a d t o d i f f i c u l t y . T h e s e t h r e e c a s e s c a n be i n t e r p r e t e d t o be p a r t s o f a c o n t i n u u m : t h e f i r s t b e i n g a n i n c o n s i s t e n c y t h a t was u n r e s o l v a b l e ; t h e C h r i s ' b e i n g a n i n c o n s i s t e n c y t h a t was a v o i d e d ( o r p a r t l y a c c o m o d a t e d ) ; a n d t h e P a t ' s , w h e r e t h e i n c o n s i s t e n t i d e a s h a v e b e e n r e s o l v e d . The r e l a t i o n s h i p b e t w e e n i n f o r m a l l y and f o r m a l l y d e r i v e d i d e a s r a n g e d f r o m 88 unresolved to resolved. The a b i l i t y to apply s c i e n t i f i c paradigms depended, i n part, on the resolution of the ideas. The general conclusion was that as the r e l a t i o n s h i p between the ideas a student derives from informal sources and from s c i e n t i f i c paradigms becomes more resolved the students are better able to apply the paradigms. 7. Implications The goal of science education i s to impart to students the s c i e n t i f i c paradigms so that the students w i l l be better able to interpret increasingly complex range of physical phenomena. Students must be helped to e s t a b l i s h a close r e l a t i o n s h i p between t h e i r informal ideas and the s c i e n t i f i c paradigms. Educators must focus therefore, not only on the s c i e n t i f i c paradigms but also on the students informal ideas. The need to e s t a b l i s h a close r e l a t i o n s h i p suggests that before s c i e n t i f i c paradigms are introduced i t could be b e n e f i c i a l to s t a r t with the student's informal ideas. The student may not be aware that he/she possesses these informal ideas. This could be achieved by class discussion with the goal of the class discovering i t s c h a r a c t e r i s t i c viewpoint. The students could also perform thought experiments or observe Piagetian type tasks i n order that the student's informal ideas create predictions that may be i n c o n f l i c t with actual observations. Once the student's ideas have been established, the s c i e n t i f i c paradigms could be integrated more e a s i l y by the student. This study was limited to students at the secondary 89 l e v e l learning accelerated motion. As a student proceeds to extend the learning new s c i e n t i f i c paradigms of motion, there i s a need to maintain the rela t i o n s h i p s . The relationships do not seem as obvious to students as to possessors of the paradigms. It i s the relationships however, that w i l l give the student an e f f e c t i v e viewpoint. 90 TABLES 91 HARMONIC MOTION COMPARISON QUESTION RESULTS quest, students by grade students by physics t r a i n i n g VIII IX X XII none p h y l l phyl2 1. A* 44 36 69 94 80 100 100 B 44 50 19 0 0 0 20 C 11 7 6 0 0 0 0 D 0 7 6 0 0 0 0 2. A 50 21 25 29 80 14 0 B 44 43 56 24 0 29 40 c* 6 21 19 41 60 29 60 D 39 14 0 12 40 0 0 3. A 22 57 38 6 0 0 20 B 22 7 56 35 0 71 20 c* 17 21 6 47 60 29 60 D 39 14 0 12 40 0 0 4. A 28 7 25 24 20 43 0 B 28 36 56 35 40 29 40 C* 28 21 13 35 40 14 60 D 17 36 6 6 0 14 0 5 . A 22 57 31 18 0 29 20 B* 33 7 19 41 40 29 60 C 28 14 50 18 40 14 0 D 17 21 0 18 20 29 20 6 . A 33 21 6 6 0 14 0 B 28 29 56 23 60 14 0 c* 17 36 31 70 40 71 100 D 22 14 6 0 0 0 0 7. A 33 21 0 12 20 14 0 B 11 14 13 0 0 0 0 C* 22 14 75 77 40 86 100 D 33 50 13 12 40 0 0 8 . A 22 14 13 0 0 0 0 B 28 29 0 0 0 0 0 C* 39 21 88 94 80 100 100 D 11 36 0 6 0 0 0 9. A 0 29 0 6 0 14 0 B* 28 29 31 59 60 43 80 C 50 21 44 24 20 29 20 D 22 21 25 6 20 29 20 Table 1. Student responses shown as percents by grade on the l e f t and by amount of physics t r a i n i n g on the r i g h t . 92 ACCELERATED MOTION COMPARISON QUESTION RESULTS quest, students by grade students by physics t r a i n i n g VIII IX X XII none phyll phyl2 1. A 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 C* 61 64 81 100 100 100 100 D 0 0 0 0 0 0 0 2. A 56 71 81 30 60 29 0 B 11 14 6 0 0 0 0 C* 33 14 13 65 20 71 100 D 0 0 0 6 20 0 0 3. A 61 71 88 35 60 43 0 B 0 7 6 0 0 0 0 C* 11 0 0 65 40 57 100 D 28 21 6 0 0 0 0 4. A 6 7 0 0 0 0 0 B 67 71 88 23 40 14 80 c* 17 21 13 77 60 86 80 D 11 0 0 0 0 0 0 5. A 11 7 6 30 60 29 0 B* 33 50 75 59 40 43 100 c 6 0 6 12 0 29 0 D 50 43 13 0 0 0 0 6. A 0 0 6 6 0 14 0 B 45 21 19 18 40 0 20 c* 22 64 44 65 40 71 80 D 6 7 19 0 0 0 0 E 28 7 13 12 20 14 0 7. A 17 0 6 18 0 43 0 B 11 29 19 12 40 0 0 C 22 21 31 30 60 29 0 D* 28 43 38 42 0 29 100 9. P 33 29 19 0 0 0 0 NP* 67 71 81 100 100 100 100 Table 2. Student responses shown as percents by grade on the l e f t and by amount of physics t r a i n i n g on the r i g h t . The aste r i s k indicates the correct response. 93 PARADIGM COMPLETION QUESTIONS HARMONIC MOTION c l a s s i f i c a t i o n grade 8 grade 9 grade 10 grade u n c l a s s i f i e d 20 34 20 10 descriptive 28 38 11 7 inappropriate 20 7 33 24 appropriate 32 21 36 59 raw t o t a l number 50 29 45 41 ACCELERATED MOTION c l a s s i f i c a t i o n grade 8 grade 9 grade 10 grade 12 a l l un c l a s s i f i e d 9 4 6 16 9 descriptive 43 39 27 4 28 inappropriate 23 2 4 21 19 22 appropriate 26 33 46 61 42 raw t o t a l number 70 93 67 83 313 Table 3. Student responses for both harmonic motion and accelerated motion to the paradigm completion questions are shown as percents for each c l a s s i f i c a t i o n . The raw t o t a l score indicates how many questions were posed by each grade. 94 PARADIGM COMPLETION QUESTIONS HARMONIC MOTION c l a s s i f i c a t i o n grade 8 grade 9 grade 10 grade 12 descriptive 2 1 3 3 inappropriate 3 3 2 2 appropriate 1 2 1 1 ACCELERATED MOTION c l a s s i f i c a t i o n grade 8 grade 9 grade 10 grade 12 descriptive 1 1 2 3 inappropriate 3 3 3 2 appropriate 2 2 1 1 Table 4. Student responses for both harmonic motion and accelerated motion to the paradigm completion questions are shown as a ranking for each c l a s s i f i c a t i o n within each grade. COMBINED PARADIGM COMPLETION QUESTIONS c l a s s i f i c a t i o n grade 8 grade 9 grade 10 grade 12 descriptive 42 44 23 6 inappropriate 25 22 29 24 appropriate 33 34 47 70 Table 5. Total student paradigm completion responses, excluding those that were u n c l a s s i f i a b l e , shown as a percent for each grade l e v e l . ACCELERATED MOTION PARADIGM COMPLETION QUESTIONS FOR GRADE TWELVE STUDENTS c l a s s i f i c a t i o n no physics physics 1 1 physics 1 2 u n c l a s s i f i e d 3 8 2 descriptive 2 1 0 inappropriate 3 4 9 appropriate 1 1 25 4 5 Table 6. Paradigm completion responses for grade twelve students with and without physics t r a i n i n g . The numbers indicate the actual number of questions posed. 96 BIBLIOGRAPHY Brace, G. Students' notions of force and motion: a review  of the l i t e r a t u r e . An unpublished paper, 1984, Dept of Mathemtics and Science Education, University of B r i t i s h Columbia. Caramazza, A., McCloskey, M., & Green, B. Naive b e l i e f s i n "sophisticated" subjects: misconceptions about t r a j e c t o r i e s of objects. Cognition, 1981, 9, 117-123 Campbell, R. Can inconsistencies be reasonable. Canadian  Journal of Philosophy, 1981, 11(2), 245-270. Cavallo, R. E. The Role of Systems Methodology i n Social  Science Research. Boston: Martinus Nijhoff Pub., 1979. Chandler, M. J. Relativism and the problem of epistemological loneliness. Human Deve1opment, 1975, 18, 171-180. Chi, M. T. H., Feltovich, P. J. & Glaser, R. Categorization and representation of physics problems by experts and novices. Cognitive Science, 1981, 5, 121-152 Clement, J. Students' preconceptions i n introductory mechanics. American Journal of Physics, 1982, 50(1), 66-71. Descartes, R. [A Discourse on Method] (J. Veitch, trans) New York: E. P. Dutton & Co. Inc., 1957 Driver, R. & Easley, J. Pupils and paradigms: a review of l i t e r a t u r e related concept development i n adolescent science students. Studies i n Science Education, 1978, 5, 61-84. Ericsson, K. A. & Simon, H. A. Verbal reports as data. Psychological Review, 1980, 87(3), 215-251. Gardener, H. The Mind's New Science. New York: Basic Books, Inc., 1985. Gunstone, R. & Watts, M. Force and Motion. In R. Driver, E. Guesne, & A. Tiberghien (Eds.) Children's Ideas In  Science. Philadelphia: Open University Press, 1985. Heelan, P. A. Space-perception and the Philosophy of Science. Los Angeles: Unniversity of C a l i f o r n i a Press, 1983. 97 Inhelder, B. & Piaget, J. The Growth of Logical Thinking (A. Parsons & S. Milgram trans.). New York: Basic Books, Inc., Pub., 1958. Jacoby, L. L. On interpreting the e f f e c t s of r e p e t i t i o n : solving a problem versus remembering a solution. Journal  of Verbal Learning and Verbal Behavior, 1978, 17(6), 649-667. Johnson, P. E., Cox, D. L., & Curran, T. E. Psychological r e a l i t y of physical concepts. Psychometric Science, 1970, 19(4), 245-246. K r u t e t s k i i , V. A. The Psychology of Mathematical A b i l i t i e s  i n Schoolchildren. J. K i l p a t r i c k & I. Wirszup (Eds.). Chicago: University of Chicago Press, 1976. Kuhn, T. S. The function of measurement i n modern physical science. ISIS, Quarterly Journal of Science Society, 1961, 52, 161-193. Kuhn, T. S. The Structure of S c i e n t i f i c Revolutions. Chicago: University of Chicago Press, 1962. Kuhn, T. S. A function for thought experiments. In Melanges  Alexandre Koyre. Paris: Herman, 1964. Kuhn, T. S. Logic of discovery or pyschology of research. In Lakatos, I. & Musgrave, A. (Eds.) C r i t i c i s m and the  Growth of Knowledge. Cambridge: University Press, 1970. Kuhn, T. S. Second thoughts on paradigms. In Suppe, F. The  Structure of S c i e n t i f i c Theories. Chicago: University of I l l i n o i s Press, 1974. Larkin, J . , McDermott, J . , Simon, D. P. & Simon, H. A. Expert and novice performance i n solving physics problems. Science. June 1980, 208(20), 1335-1342. McCloskey, M., Caramazza, A., & Green, B. Curvilinear motion i n the absense of external forces: naive b e l i e f s about motion of objects. Science, 1980, 210(5), 1139-1141. McDermott, L. Research on conceptual understanding i n mechanics. Physics Today, 1984, 24-32. Masterman, M. The nature of a paradigm. In Lakatos, I. & Musgrave, A. C r i t i c i s m and the Growth of Knowledge. Cambridge: Cambridge University Press, 1970. Mehra, J. The Quantum P r i n c i p l e : Its Interpretation and Epistemoloqy. Boston: D. Reidel Pub. Co., 1974. 9 8 M i l l e r , A. I. Imagery i n S c i e n t i f i c Thought Creating  20th-century Physics. Boston: Birkhauser, 1984. Osborne, R. J. A method for investigating concept understanding i n science. European Journal of Science  Education, 1980, 2(3), 311-321. Osborne, R.J. & G i l b e r t , J.K. A technique for exploring students' views of the world. Physics Education, 1980, 15,376-379. Piaget, J. [The Child's Conception of Movement and Speed] (G.E.T. Holloway & M.J. MacKenzie trans.). London: Routledge and Kegan Paul Ltd., 1970. ( O r i g i n a l l y published, 1946.) Piaget, J. [Psychologie and Epistemology] (A. Rosin, trans.). New York: Grossman Publishers, 1971. ( O r i g i n a l l y published, 1970.) Province of B r i t i s h Columbia, Ministry of Education, Curriculum Development Branch. Physics 11 Curriculum  Guide. V i c t o r i a , B.C.: Queens Printer, 1986. S i l v e r , E. A. Student perceptions of relatedness among mathematical verbal problems. Journal for Research i n  Mathematics Education, 1979, 10, 195-210. S i l v e r , E. A. Recall of mathematical problem information: solving related problems. Journal for Research i n  Mathematics Education, 1981, 12, 54-64. Suppe, F. Structure of S c i e n t i f i c Theories. Chicago: University of I l l i n o i s Press, 1974. Thagard, P. Frames, knowledge, and inference. Synthese, 1984, 61, 233-259. Trowbridge, D. E. & McDermott, L. C. Investigation of student understanding of the concept of acceleration i n one dimension. American Journal of Physics, 49(3), 242-253. Watts, D.M. A study of schoolchildren's a l t e r n a t i v e frameworks of the concept of force. European Journal of  Science Education, 1983, 5(2), 217-232. Watts, D.M. & Zylbersztajn, A. A survey of some children's ideas about force. Physics Education, 1981, 16, 360-365. Yin, R.K. Case Study Research Design and Methods. Beverly H i l l s : Sage Publications, 1985. 99 APPENDIX 100 Use the following diagram to answer questions 1 - 15. The diagram below shows Fred standing at the top of a 50 meter high c l i f f . Fred i s holding a red rock and a blue rock. The boat i s 200 meters from the base of the c l i f f and i s not moving. The scene is2on Earth where the acceleration of gravity i s 9.8 m/s . There i s no wind blowing and a i r resisrance may be ignored. INSTRUCTIONS: For each question, se l e c t the BEST answer and c i r c l e the l e t t e r of your choice. 1. If both rocks weigh 2 kilograms and Fred drops them straight down toward the water at the same time: A. The red rock w i l l h i t the water f i r s t . B. The blue rock w i l l h i t the water f i r s t . C. They w i l l h i t the water at the same time. D. It i s not possible to predict which w i l l land f i r s t . 2. If the red rock weighs 4 kilograms and the blue rock weighs 2 kilograms and Fred drops them straight down toward the water at the same time: A. The red rock w i l l h i t the water f i r s t . B. The blue rock w i l l h i t the water f i r s t . C. They w i l l h i t the water at the same time. D. It i s not possible to predict which w i l l land f i r s t . t MULTIPLE CHOICE QUESTIONS 101 3. If both rocks weigh 2 kilograms and Fred drops the red rock straight down toward the water but throws the blue rock h o r i z o n t a l l y outward: A. The red rock w i l l h i t the water f i r s t . B. The blue rock w i l l h i t the water f i r s t . C. They w i l l h i t the water at the same time. D. It i s not possible to predict which w i l l land f i r s t . 4. If the red rock weighs 2 kilograms and the blue rock weighs 4 kilograms and Fred drops them straight down toward the water at the same time. A. The red rock w i l l have a higher speed than the blue rock when they enter the water. B. The blue rock w i l l have a higher speed than the red rock when they enter the water. C. They w i l l have the same speed when they enter the water. D. It i s not possible to predict what t h e i r speed w i l l be. 5. If Fred wishes to throw the red rock out from the c i f f as far as possible, what angle should he throw the rock at? A. He should throw the rock h o r i z o n t a l l y . B. He should throw the rock at 4 5 degrees above the horizontal. C. He should throw the rock at 6 0 degrees above the horizontal. D. It doesn't matter what angle he throws i t at. 6. Fred throws both rocks h o r i z o n t a l l y toward the boat from the c l i f f . If the red rock i s thrown twice as fast as the blue rock then the red rock w i l l land i n the water: A. In the same place as the blue rock. B. A l i t t l e farther out than the blue rock. C. Twice as far out as the blue rock. D. Much farther out i n the water than the blue rock. E. I t i s impossible to predict where they w i l l land. 7. If Fred drops the red rock straight down, i t w i l l accelerate at the fastest rate: A. At the s t a r t of i t s f a l l . B. During the middle of i t s f a l l . C. At the end of i t s f a l l . D. It w i l l always have the same rate of acceleration. E. The rate of acceleration i s unknown. 102 INSTRUCTIONS: Some of the following problems are possible to solve. Some of the following problems are NOT possible to solve. You do not have to solve the following problems just c i r c l e whether they are possible (P) or not possible (NP) to solve. NP 8. The red rock weighs 4.3 kilograms. How long does i t take to f a l l to the water a f t e r Fred drops i t ? NP 9. The red rock weighs 3.5 kilograms. How long does i t take to reach the bottom of the water? NP 10. Fred throws the red rock h o r i z o n t a l l y out from the c l i f f . If the rock weighs 3 kilograms, how fa r from the base of the c l i f f w i l l i t land? NP 11. If Fred throws the blue rock at an angle of 30 degrees to the horizontal with a speed of 8 m/sec, w i l l he h i t the boat? NP 12. If Fred throws the red rock upward with a speed of 5 m/sec, with what speed w i l l i t enter the water. NP 13. If Fred drops the red rock, which weighs 5 kilograms, what w i l l the speed of the rock be when i t has f a l l e n halfway to the water? NP 14. If Fred throws the blue b a l l toward the boat at an angle of 6 0 degrees from the horizontal, how high w i l l the rock r i s e above the water? NP 15. What i s the k i n e t i c energy of the blue rock just before i t enters the water? 103 INSTRUCTIONS: Solve the following questions i n the space provided. If you think you can solve a problem but cannot remember the formulas you may ask for them. I f , aft e r being given the formulas, you s t i l l cannot solve a problem you may ask for a sample problem. Use the following diagram to answer questions 1 - 5 . The diagram below shows Fred standing at the top of a 50 meter high c l i f f . Fred i s holding a red rock and a blue rock. The boat i s 200 meters from the base of the c l i f f and i s not moving. The scene is2on Earth where the acceleration of gravity i s 9.8 m/s . There i s no wind blowing and a i r resistance may be ignored. t 1. How long does i t take for the red rock to reach the water after Fred drops i t ? 2. With what speed does the blue rock enter the water a f t e r Fred drops i t ? 104 If Fred throws the red rock straight up at 7 m/s, how high w i l l the rock r i s e above the surface of the water? If Fred throws the red rock h o r i z o n t a l l y towards the boat at 6 m/s, how far w i l l i t land from the base of the Fred throws the red rock upwards with a speed of 10 m/s. If the rock enters the water with a2speed of 2 5 m/s, was the acceleration of gravity 9.8 m/s ? 105 INSTRUCTIONS: Al f r e d and Betty each have a pendulum consisting of a b a l l attached to a s t r i n g . In the following questions you are to indicate whether (A)lfred's pendulum or (B)etty's pendulum has a shorter period. The period of a pendulum i s how long i t takes to make one complete swing. A pendulum that takes 10 seconds for 10 complete swings has a period of 1 second per swing. In each of the following problems c i r c l e (A) i f Alfred's pendulum has a shorter period; (B) i f Betty's pendulum has a shorter period; (C) i f they have a common period, that i s , neither one i s shorter; and (D) i f you don't know or the question cannot be answered. A B C D 1. Each b a l l weighs 500 grams but Alfred's st r i n g i s 1 meter whereas Betty's i s 2 meters long. A B C D 2. Each pendulum i s 1 meter long but Alfred's b a l l i s 250 grams whereas Betty's i s 500 grams. A B C D 3. Both pendulum are 1 meter long and weigh 500 grams but A l f r e d s t a r t s his pendulum out further than Betty does. A B C D 4. The pendulum are i d e n t i c a l however when they are started Betty pushes her pendulum downward to help i t get started. 106 A B C D 5. The pendulum are i d e n t i c a l however A l f r e d takes his pendulum to the top of a very t a l l mountain (Mt. Everest) while Betty remains i n Ladysmith. A B C D 6. The pendulum are i d e n t i c a l however Betty changes the o r i g i n a l s t r i n g which i s cotton for a piece of stainless steel wire. A B C D 7. The pendulum are i d e n t i c a l however Betty takes her pendulum to New Zealand while A l f r e d remains i n Ladysmith. A B C D 8. The pendulum are i d e n t i c a l but A l f r e d swings his i n a north-south l i n e whereas Betty swings hers i n an east-west l i n e . A B C D 9. Alfred's pendulum i s twice as long as Betty's; but, Betty's pendulum weighs twice as much as Alfreds pendulum. INSTRUCTIONS: Answer the following questions i n the space provided. If you do not know or remember the formulas for pendulum these w i l l be given to you. 1. What i s the period of a 2.5 meter pendulum that has a 500 gram bob? 2. A 500 gram bob pendulum has a period of 3.3 seconds. What i s i t s length? 3. Describe the how the energy (both k i n e t i c and potential) of a pendulum varies throughout i t s swing. 

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