Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Conceptions of teaching mathematics: the ideas of six pre-service secondary mathematics teachers Lee, Rosalind Chui Mei 1996

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

Item Metadata

Download

Media
831-ubc_1996-0392.pdf [ 4.01MB ]
Metadata
JSON: 831-1.0054656.json
JSON-LD: 831-1.0054656-ld.json
RDF/XML (Pretty): 831-1.0054656-rdf.xml
RDF/JSON: 831-1.0054656-rdf.json
Turtle: 831-1.0054656-turtle.txt
N-Triples: 831-1.0054656-rdf-ntriples.txt
Original Record: 831-1.0054656-source.json
Full Text
831-1.0054656-fulltext.txt
Citation
831-1.0054656.ris

Full Text

CONCEPTIONS OF TEACHING MATHEMATICS: THE IDEAS OF S I X P R E - S E R V I C E SECONDARY MATHEMATICS TEACHERS b y ROSALIND CHUI MEI LEE B . S c , The U n i v e r s i t y o f T o r o n t o , 1976 B . E d . , The U n i v e r s i t y o f T o r o n t o , 1977 A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f C u r r i c u l u m S t u d i e s i n M a t h e m a t i c s E d u c a t i o n We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d •THE UNIVERSITY**OF B R I T I S H COLUMBIA J u n e 1996 ( c ) R o s a l i n d C h u i M e i L e e , 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) A b s t r a c t " I am a math t e a c h e r . " T h i s s i m p l e s t a t e m e n t e v o k e s a p l e t h o r a o f i m a g e s . E l u c i d a t i n g p r e - s e r v i c e t e a c h e r s ' p e r s o n a l m e a n i n g s f o r t h i s s t a t e m e n t adds t o t h e k n o w l e d g e b a s e f o r t e a c h e r e d u c a t i o n . K n o w i n g t h e i r own b e l i e f s a n d i d e a s a b o u t t e a c h i n g a l s o p r o v i d e s t e a c h e r c a n d i d a t e s a b a s i s f r o m w h i c h t o c o n s i d e r a l t e r n a t i v e c o n c e p t i o n s . I n t h i s i n v e s t i g a t i o n , s i x p r e - s e r v i c e t e a c h e r s o f secondary-m a t h e m a t i c s were i n t e r v i e w e d a b o u t t h e i r i d e a s o f t e a c h i n g m a t h e m a t i c s a t t h e c o n c l u s i o n o f t h e i r e x t e n d e d p r a c t i c u m . A l l h a d b a c h e l o r ' s d e g r e e s w i t h a m a t h e m a t i c s s p e c i a l t y ; t h r e e h a d p u r s u e d g r a d u a t e s t u d y i n p u r e o r a p p l i e d m a t h e m a t i c s . R e s u l t s show a l l s i x t e a c h e r c a n d i d a t e s h a d d i f f e r e n t c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s . Some e m p h a s i z e d t h e c o n t e n t o f m a t h e m a t i c s ; o t h e r s e m p h a s i z e d m a t h e m a t i c a l p r o c e s s e s . Some gave p r i o r i t y i n t h e i r t e a c h i n g p e r s p e c t i v e t o s t u d e n t d e v e l o p m e n t a n d c o g n i t i v e s k i l l s . Those who h a d s t u d i e d g r a d u a t e l e v e l m a t h e m a t i c s e x p r e s s e d a d e s i r e t o make math i n t e r e s t i n g and s t i m u l a t i n g f o r s t u d e n t s . R e s u l t s s u p p o r t t h e c o n s t r u c t i v i s t i d e a s t h a t c o n c e p t i o n s a r e p e r s o n a l l y f o r m e d a n d p a s t e x p e r i e n c e s a r e i n f l u e n t i a l i n s h a p i n g t h e m . T a b l e o f C o n t e n t s A b s t r a c t i i T a b l e o f C o n t e n t s i i i A c k n o w l e d g e m e n t s v CHAPTER 1: I n t r o d u c t i o n 1 S t a b i l i t y o f C o n c e p t i o n s 1 I n f l u e n c e o f C o n c e p t i o n s 2 T e a c h e r E d u c a t i o n R e s e a r c h 2 R e s e a r c h Q u e s t i o n s 4 CHAPTER 2 : The L i t e r a t u r e 6 C o n s t r u c t i v i s m i n M a t h e m a t i c s E d u c a t i o n 6 The R o l e o f E x i s t i n g B e l i e f s 9 The N a t u r e o f M a t h e m a t i c s 12 R e s e a r c h on C o n c e p t i o n s o f T e a c h i n g M a t h e m a t i c s 14 A G e n e r a l M o d e l f o r D e s c r i b i n g C o n c e p t i o n s o f T e a c h i n g 16 The Framework f o r t h i s S t u d y 18 CHAPTER 3 : M e t h o d o l o g y 21 D e v e l o p m e n t o f t h e P r o t o c o l 21 The P i l o t S t u d i e s 22 The f i r s t p i l o t s t u d y 22 The s e c o n d p i l o t s t u d y 23 The M a i n S t u d y The p r o g r a m 24 The s a m p l e 24 The d a t a c o l l e c t i o n 26 The d a t a p r o c e s s i n g a n d a n a l y s i s 27 - i i i -CHAPTER 4 : R e s u l t s 30 Y v e t t e 3 0 S h a n n o n 3 5 Doug 4 3 R a y 5 0 V i c t o r i a 55 M a r y 62 Summaries 67 CHAPTER 5 : C o n c l u s i o n s , O t h e r I s s u e s & Recommendat ions 71 C o n c l u s i o n s 71 O t h e r I s s u e s 76 R e c o m m e n d a t i o n s 7 9 R e f e r e n c e s 85 A p p e n d i x A : V i e w s o f t e a c h i n g and l e a r n i n g 90 A p p e n d i x B : Sample q u e s t i o n s 92 A p p e n d i x C: Sample f i e l d n o t e s 94 A p p e n d i x D: Sample t r a n s c r i p t e x c e r p t 96 - i v -Acknow1edgement s F i r s t , t o my two p r i n c e s s e s , Dawn and D e n i s e , whose l o v e , t r u s t a n d c o o p e r a t i o n made "Mom's b o o k " a r e a l i t y . S e c o n d , t o t h e s u b j e c t s o f my r e s e a r c h , a g r e a t g r o u p who were as c o n c e r n e d f o r my s t u d i e s as I was f o r t h e i r s . T h i r d , t o t h e p a t i e n t members o f my c o m m i t t e e who f o c u s s e d my t h i n k i n g a n d s h a r p e n e d my v o c a b u l a r y : Tony C l a r k e , a f r i e n d a n d m e n t o r t o a l l g r a d u a t e s t u d e n t s ; G a a l e n E r i c k s o n , e v e r u n d e r s t a n d i n g ; a n d , most i m p o r t a n t l y , A n n A n d e r s o n , who showed me a w h o l e new d i m e n s i o n t o t h e c o n c e p t o f support w i t h h e r c o n s i s t e n t e n c o u r a g e m e n t . F i n a l l y , t o t h e f r i e n d s and c o l l e a g u e s who " h e l d my h a n d " t h r o u g h t h e w r i t i n g o f t h i s t h e s i s . They p u s h e d , d r a g g e d , c a j o l e d , a n d b r i b e d me a n d r e g a l e d me w i t h t h e i r own w r i t i n g a d v e n t u r e s and s u c c e s s s t o r i e s t o make s u r e I k e p t p l u g g i n ' away. A h e a r t f e l t and humble THANK YOU t o a l l . C h a p t e r One: I n t r o d u c t i o n P r e - s e r v i c e t e a c h e r s have a l r e a d y c o n s t r u c t e d t h e i r own i d e a s o f what i t means t o t e a c h when t h e y e n t e r a t e a c h e r e d u c a t i o n p r o g r a m . L o r t i e (1975) f i r s t u s e d t h e t e r m " a p p r e n t i c e s h i p o f o b s e r v a t i o n " [p 61] t o d e s c r i b e how t h o s e who c h o o s e t h i s o c c u p a t i o n have a c t u a l l y s p e n t 12 t o 16 y e a r s o r more w a t c h i n g p r a c t i t i o n e r s o f t h e p r o f e s s i o n t h e y h a v e d e c i d e d t o e n t e r . He p o i n t s o u t t h a t t h i s i s n o t a t r u e a p p r e n t i c e s h i p , h o w e v e r , i n t h a t as a s t u d e n t , t h e y have o n l y s e e n t h e show f r o m one n a r r o w v a n t a g e p o i n t , r a t h e r t h a n h a v i n g v i e w e d t h e e n t i r e k a l e i d o s c o p e w h i c h c o m p r i s e s a t e a c h e r ' s f u l l r e s p o n s i b i l i t i e s . He goes o n t o c l a i m t h a t t h e v i s i o n o f t e a c h i n g f o r m e d d u r i n g t h i s p e r i o d , skewed t h o u g h i t i s , w i l l v e r y l i k e l y p e r s i s t f o r t h a t i n d i v i d u a l . S t a b i l i t y o f C o n c e p t i o n s T h e s e i n i t i a l c o n c e p t i o n s o f t e a c h i n g have b e e n f o u n d t o be r e l a t i v e l y s t a b l e f o r e d u c a t i o n s t u d e n t s t h r o u g h o u t t h e i r p r e -p r o f e s s i o n a l p r o g r a m (Brown & B o r k o , 1992; Goodman, 1988 ;. B o r k o , L a l i k & T o m c h i n , 1 9 8 7 ; T a b i c h n i c k a n d Z e i c h n e r , 1984) . I n h i s work w i t h p r e -s e r v i c e t e a c h e r s , Goodman (1988) t e r m e d t h e s e " i n t u i t i v e s c r e e n s " : " . . . t h e i r p r e - p r o f e s s i o n a l images f o r m e d a n ' i n t u i t i v e s c r e e n 1 t h r o u g h w h i c h t h e y i n t e r p r e t e d t h e i r p r o f e s s i o n a l e d u c a t i o n . . . . [ I n ] t h e [ t e a c h e r e d u c a t i o n ] p r o g r a m , t h e s e i n t u i t i v e s c r e e n s gave them an o r i e n t a t i o n p o i n t f r o m w h i c h t h e y made s e n s e o u t o f t h e a c t i v i t i e s a n d i d e a s p r e s e n t e d t o t h e m . . . . W h e n e x p o s e d t o new i d e a s o r e x p e r i e n c e s , s t u d e n t s t e n d e d t o a c t f i r s t o n a n i n t u i t i v e r a t h e r t h a n a n i n t e l l e c t u a l l e v e l . No m a t t e r how l o g i c a l a n i d e a seemed, i f i t d i r e c t l y c o n t r a d i c t e d a s t u d e n t ' s i n t u i t i v e s c r e e n , i t was u s u a l l y r e j e c t e d . . . . M o s t s t u d e n t s t e n d e d t o be i n f l u e n c e d b y t h o s e p e o p l e o r e x p e r i e n c e s t h a t l e g i t i m a t e d t h e i r e x i s t i n g ' i n t u i t i v e s c r e e n . ' " [pp 130-131] I n o t h e r w o r d s , Goodman f o u n d t h e t e a c h e r e d u c a t i o n p r o g r a m s o l i d i f i e d a n d / o r c l a r i f i e d t h e s e p r e - p r o f e s s i o n a l images r a t h e r t h a n i n t r o d u c e new i m a g e s o f t e a c h i n g . He a l s o f o u n d t h a t t h o s e who d i d e x p e r i e n c e c o n c e p t u a l c h a n g e u n d e r w e n t a good d e a l o f u n r e s t as t h e y were g i v i n g up t h e i r i n c i p i e n t i d e a s . I n f l u e n c e o f C o n c e p t i o n s F o r m a t h e m a t i c s t e a c h e r s , t h e i r i d e a s o f t e a c h i n g seem t o be s t r o n g l y i n f l u e n c e d b y what t h e y p e r c e i v e as t h e e s s e n c e o f t h e s u b j e c t (Thompson, 1984) . T h e s e i d e a s i n t u r n i n f l u e n c e t h e i r a c t i o n s i n t h e c l a s s r o o m a n d f l a v o u r t h e messages a b o u t m a t h e m a t i c s t h e y g i v e t h e i r s t u d e n t s ( D o s s e y , 1 9 9 2 ; Thompson, 1 9 9 2 ) . From t h e o p p o s i t e s i d e o f t h e d e s k , s t u d e n t s who l e a r n m a t h e m a t i c s f r o m t e a c h e r s w i t h d i f f e r e n t b e l i e f s a b o u t i t s n a t u r e and d i f f e r e n t p u r p o s e s f o r t e a c h i n g i t may h a v e v e r y d i f f e r e n t c l a s s r o o m e x p e r i e n c e s ( B r o p h y , 1991) . T h i s h i g h l i g h t s t h e s i g n i f i c a n c e o f i n v e s t i g a t i n g t e a c h e r s ' i d e a s a b o u t m a t h e m a t i c s a n d i t s t e a c h i n g ; n a m e l y , t o a c q u i r e k n o w l e d g e o f t h e b e l i e f s t h a t p o t e n t i a l l y d r i v e t h e i r a c t i o n s i n t h e c l a s s r o o m a n d t o g a i n i n s i g h t i n t o t h e i r g o a l s and p u r p o s e s f o r t e a c h i n g . T e a c h e r E d u c a t i o n R e s e a r c h B o r k o , L a l i k and T o m c h i n (1987) o b s e r v e d t h a t t e a c h e r e d u c a t i o n p r o g r a m s were l a r g e l y i n f o r m e d b y r e s e a r c h o n e x p e r i e n c e d t e a c h e r s r a t h e r t h a n t e a c h e r c a n d i d a t e s . T h e y " f o u n d o n l y a few s t u d i e s t h a t a d d r e s s e d s t u d e n t t e a c h e r s ' c o n c e p t i o n s o f t e a c h i n g " [p 78] a n d i d e n t i f i e d t h e n e e d f o r more r e s e a r c h i n e x p l o r i n g t h i s p o p u l a t i o n ' s - 2 -c o n c e p t i o n s o f t e a c h i n g and r e l a t i n g t h e s e t o t e a c h i n g b e h a v i o u r . F u r t h e r , L a m p e r t (1988) s t a t e d t h a t t h e r e e x i s t e d o n l y a " s m a l l amount o f r e s e a r c h t h a t i s s p e c i f i c t o t e a c h e r e d u c a t i o n i n m a t h e m a t i c s " [p 161] a n d most o f t h a t was c o n c e r n e d w i t h o b s e r v a b l e b e h a v i o u r s r a t h e r t h a n t e a c h e r t h i n k i n g . F o r e x a m p l e , V a l l i ( V a l l i & A g o s t i n e l l i , 1993) f o l l o w e d one i n d i v i d u a l t h r o u g h h i s t e a c h i n g e x p e r i e n c e s b e f o r e , d u r i n g a n d a f t e r h i s t e a c h e r e d u c a t i o n p r o g r a m a n d u s e d s u c h c r i t e r i a as amount a n d t y p e o f q u e s t i o n i n g , c l a r i t y o f b o a r d w o r k and amount o f l e s s o n a n d u n i t p l a n n i n g t o e v a l u a t e h i s p r o g r e s s . S i n c e t h o s e s t a t e m e n t s were made, more s t u d i e s have e x p l o r e d p r e -s e r v i c e t e a c h e r s ' c o n c e p t i o n s . H o w e v e r , most o f t h e s e have d e a l t w i t h e l e m e n t a r y c a n d i d a t e s and have c e n t r e d a r o u n d s p e c i f i c c o n t e n t . The s t u d i e s r a n g e f r o m e x p o s i n g f a l l a c i e s i n , a n d t h u s l a c k o f , m a t h e m a t i c a l u n d e r s t a n d i n g on c e r t a i n t o p i c s s u c h as d i v i s i o n ( e . g . T i r o s h a n d G r a e b e r , 1 9 9 0 ; B a l l , 1990b) t o u n c o v e r i n g b e l i e f s t h a t k n o w i n g m a t h i s d o i n g math a c c o r d i n g t o s e t r u l e s and c a n o n s and p r o d u c i n g c o r r e c t a n s w e r s ( e . g . C i v i l , 1 9 9 0 ) . I n t e r v e n t i o n s t u d i e s o n e l e m e n t a r y c a n d i d a t e s h a v e a l s o b e e n c a r r i e d o u t . F o r e x a m p l e , W i l c o x , S c h r a m , L a p p a n , & L a n i e r (1991) i n s t i t u t e d a " l e a r n i n g c o m m u n i t y " i n a m a t h e m a t i c s c o n t e n t c o u r s e i n an a t t e m p t t o i n i t i a t e c o n c e p t u a l c h a n g e i n t h e i r s t u d e n t s ' i d e a s o f t h e n a t u r e o f m a t h e m a t i c s . R e s e a r c h o n s e c o n d a r y p r e - s e r v i c e t e a c h e r s i s e v e n more l i m i t e d ( B a l l & M c D i a r m i d , 1990) . L i t t l e i s known a b o u t t h e i r v i e w s o f t e a c h i n g m a t h e m a t i c s t h o u g h some c o n t e n t i s s u e s have b e e n i d e n t i f i e d . F o r e x a m p l e , B a l l ' s (1990b) s t u d y i n v e s t i g a t e d s e c o n d a r y as w e l l as e l e m e n t a r y p r o s p e c t i v e t e a c h e r s and f o u n d a l a c k o f c o n c e p t u a l - 3 -u n d e r s t a n d i n g o f d i v i s i o n i n b o t h g r o u p s . Z a s l a v s k y a n d P e l e d (1996) f o u n d c o n c e p t u a l d i f f i c u l t i e s w i t h t h e c o m m u t a t i v e a n d a s s o c i a t i v e p r o p e r t i e s o f b i n a r y o p e r a t i o n s i n t h e u n d e r s t a n d i n g s o f 20 p r o s p e c t i v e t e a c h e r s . R e s e a r c h Q u e s t i o n s T h o s e e n t e r i n g t h e t e a c h i n g p r o f e s s i o n c a n n o t be c o n s i d e r e d a c o h o r t w i t h homogeneous b a c k g r o u n d s , p a r t i c u l a r l y i n t h e p r e s e n t s o c i e t a l m i l i e u o f c o n s t a n t change and u n c e r t a i n e c o n o m i c s . S t u d e n t s i n a p r e - p r o f e s s i o n a l p r o g r a m v a r y i n a g e , come f r o m o t h e r p r o f e s s i o n s , a n d have f u r t h e r a c a d e m i c q u a l i f i c a t i o n s . I t i s l i k e l y t h e y w i l l have v a r y i n g c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s . To h e l p f i l l t h e gap i n what i s known a b o u t t h e k i n d s o f i d e a s t h i s p o p u l a t i o n h o l d s , t h e p r i m a r y r e s e a r c h q u e s t i o n i n t h i s s t u d y i s : what a r e t h e i n d i v i d u a l c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s h e l d b y a g r o u p o f p r e - s e r v i c e t e a c h e r s o f s e c o n d a r y m a t h e m a t i c s ? Some o f t h e s u b j e c t s i n t h i s s t u d y were p u r s u i n g , o r h a d c o m p l e t e d , g r a d u a t e work i n m a t h e m a t i c s . T h i s gave r i s e t o a s u b q u e s t i o n : f o r t h o s e w i t h g r a d u a t e s t u d i e s i n m a t h e m a t i c s , a r e t h e i r c o n c e p t i o n s q u a l i t a t i v e l y d i f f e r e n t f r o m t h o s e who h o l d o n l y a b a c c a l a u r e a t e ? T h i s s t u d y s e t o u t t o e x p l o r e and c o n s e q u e n t l y d e s c r i b e t h e c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s o f a g r o u p o f p r e - s e r v i c e t e a c h e r s o f s e c o n d a r y m a t h e m a t i c s . W h i l e t h e r e s u l t s a r e n o t i n t e n d e d t o c o n v e y t h e e n t i r e l a n d s c a p e o f p o s s i b l e c o n c e p t i o n s , t h e y do p r o v i d e a c o m p r e h e n s i v e p i c t u r e o f t h e s e p r o s p e c t i v e t e a c h e r s ' i d e a s , t h o u g h t s , a t t i t u d e s , a n d g o a l s r e l a t e d t o t h e t e a c h i n g o f m a t h e m a t i c s . The p r i m a r y i n t e n t o f t h i s r e s e a r c h i s t o a d d t o t h e k n o w l e d g e b a s e t h a t i n f o r m s m a t h e m a t i c s t e a c h e r e d u c a t i o n . I n a d d i t i o n , t h e s t u d y p r o v i d e d t h e t e a c h e r c a n d i d a t e s w i t h what B r o w n , Cooney and J o n e s (1990) h a v e d e s c r i b e d as " e n c o u n t e r s i n w h i c h t e a c h e r s a r e n o t t a u g h t m e t h o d o l o g i e s b u t a r e e n c o u r a g e d t o f i n d o u t what t h e y b e l i e v e t o be i m p o r t a n t . " [p 653] - 5 -C h a p t e r 2 : The L i t e r a t u r e C o n s t r u c t i v i s m i n M a t h e m a t i c s E d u c a t i o n The i n f l u e n c e o f c o n s t r u c t i v i s m i n m a t h e m a t i c s e d u c a t i o n r e s e a r c h w h i c h b e g a n a p p r o x i m a t e l y 30 y e a r s ago has i n c r e a s e d s t e a d i l y o v e r t h e p a s t d e c a d e ( S t e f f e & K i e r e n , 1 9 9 4 ) . As a r e l a t i v e l y new p e r s p e c t i v e , c o n s t r u c t i v i s m was t h e f o c u s o f i n t e n s e d e b a t e w i t h r e g a r d t o i t s r e l e v a n c e a n d u s e f u l n e s s i n i n f o r m i n g m a t h e m a t i c s e d u c a t i o n r e s e a r c h c o m p a r e d t o t h e t r a d i t i o n a l p r o c e s s - p r o d u c t p a r a d i g m (see B r o p h y , 1 9 8 6 a ; C o n f r e y , 1 9 8 6 ; B r o p h y , 1 9 8 6 b ) . Now, i t has become p r o m i n e n t t o t h e e x t e n t t h a t a c o n s t r u c t i v i s t p e r s p e c t i v e i s b e i n g u s e d as t h e b a s i s f o r a t e n t a t i v e t h e o r y o f i n t e l l e c t u a l d e v e l o p m e n t ( C o n f r e y , 1 9 9 4 ) , as g r o u n d s f o r an e m e r g i n g t h e o r y o f l e a r n i n g (Thompson, 1991) a n d as a f o u n d a t i o n f o r e m e r g i n g m o d e l s o f t e a c h i n g a n d t e a c h e r e d u c a t i o n (see S i m o n , 1995b; S t e f f e & D ' A m b r o z i o , 1995; S i m o n , 1 9 9 5 a ) . I t i s e v e n b e i n g c o n s t r u e d as " p o l i t i c a l l y c o r r e c t " (Cobb, 1 9 9 4 a ) ! The b a s i c p r e m i s e o f c o n s t r u c t i v i s m i s t h a t i n d i v i d u a l s c r e a t e t h e i r own k n o w l e d g e r a t h e r t h a n r e c e i v i n g i t f r o m a n e x t e r n a l s o u r c e . What f o r m t h i s k n o w l e d g e t a k e s and how i t i s c r e a t e d have b e e n i s s u e s o f d e b a t e among c o n s t r u c t i v i s t t h e o r i s t s . V o n G l a s e r s f e l d (1989, 1990) h i g h l i g h t e d t h e a s s e r t i o n t h a t t h e k n o w l e d g e c r e a t e d b y an i n d i v i d u a l was u n i q u e and n o t a r e p l i c a t i o n o f an e x t e r n a l , o b j e c t i v e r e a l i t y when he f o r m u l a t e d " r a d i c a l c o n s t r u c t i v i s m " (von G l a s e r s f e l d , 1990, p 2 8 ) . He a r g u e d t h a t k n o w l e d g e was p e r s o n a l b e c a u s e i t was f o r m e d as a r e s u l t o f a n i n d i v i d u a l ' s - 6 -e x p e r i e n c e s a n d how t h a t p e r s o n made s e n s e o f t h o s e e x p e r i e n c e s . B e c a u s e s e n s e - m a k i n g b y i t s n a t u r e i s i n d i v i d u a l and i n t e r n a l , he a l s o a r g u e d t h e r e c a n be no p r e c i s e way o f k n o w i n g someone e l s e ' s k n o w l e d g e . We c a n o n l y have o u r own u n d e r s t a n d i n g s o f what we t h i n k someone e l s e k n o w s . C o m m u n i c a t i o n among i n d i v i d u a l s t h e n becomes i n t e r p r e t a t i v e a c t s u s i n g s h a r e d m e a n i n g s r a t h e r t h a n i n f o r m a t i o n t r a n s f e r ( C o n f r e y , 1994) . T h i s c o u n t e r s t h e r e a l i s t v i e w w h i c h p u r p o r t s t h a t t h e r e i s k n o w l e d g e w h i c h e x i s t s i n d e p e n d e n t l y o f human e x p e r i e n c e a n d t h a t l e a r n i n g i s a c q u i r i n g a p e r s o n a l u n d e r s t a n d i n g o r " k n o w i n g " o f t h a t e x t e r n a l k n o w l e d g e . A n o t h e r d e b a t e f o c u s s e s on how t h e k n o w l e d g e i s c o n s t r u c t e d . One p e r s p e c t i v e v i e w s i t as t h e a c t i v i t y o f t h e i n d i v i d u a l where k n o w l e d g e c o n s t r u c t i o n o c c u r s t h r o u g h c o m p a r i n g and c o n t r a s t i n g new e x p e r i e n c e s t o i d e a s a l r e a d y h e l d . K o n o l d and J o h n s o n (1991) c a l l t h i s p s y c h o l o g i c a l c o n s t r u c t i v i s m . I n t h e l a n g u a g e o f c o g n i t i v e p s y c h o l o g y , i d e a s o r c o n s t r u c t s a r e h e l d i n m e n t a l s t r u c t u r e s c a l l e d " s c h e m a . " R e c o g n i t i o n o c c u r s when t h e r e a r e many s i m i l a r c h a r a c t e r i s t i c s b e t w e e n a n i n d i v i d u a l ' s schema and a new e n c o u n t e r . H o w e v e r , when t h e r e a r e s u f f i c i e n t d i f f e r e n c e s t h a t t h e new s i t u a t i o n does n o t " f i t " w i t h e x i s t i n g s c h e m a , t h e n t h e r e i s r e - o r g a n i z a t i o n o r c r e a t i o n o f new schema t o a c c o m o d a t e t h e new c i r c u m s t a n c e s . T h i s m e n t a l d e v e l o p m e n t t o a c c o u n t f o r new c i r c u m s t a n c e s c o n s t i t u t e s l e a r n i n g . I t o c c u r s c o n s c i o u s l y o r s u b c o n s c i o u s l y , a n d i s c o n t i n u a l . L e a r n i n g o r k n o w l e d g e c o n s t r u c t i o n c a n be d e s c r i b e d as d e v e l o p i n g " i n c r e a s i n g l y v i a b l e d e s c r i p t i o n s o f t h e phenomena t h a t we come i n t o c o n t a c t w i t h " ( E r i c k s o n , 1987, p 9) . - 7 -The o t h e r s t a n c e f o l l o w s a s o c i o c u l t u r a l p e r s p e c t i v e . T h o s e p r o p o n e n t s c l a i m k n o w l e d g e f o r m a t i o n f o r t h e i n d i v i d u a l o c c u r s as a r e s u l t o f p a r t i c i p a t i n g i n s o c i a l i n t e r a c t i o n s . Knowledge has m e a n i n g o n l y i n t h e c o n t e x t o f t h e s e i n t e r a c t i o n s ; i t i s s o c i a l l y c o n s t r u c t e d t h r o u g h s h a r e d u n d e r s t a n d i n g s . L e a r n i n g , o r p e r s o n a l k n o w l e d g e c o n s t r u c t i o n , i s t h e p r o c e s s o f t h e i n d i v i d u a l i n t e r n a l i z i n g what was o b s e r v e d o n t h e e x t e r n a l p l a n e ( e . g . Cobb, Wood, & Y a c k e l , 1991) a n d m a k i n g p e r s o n a l s e n s e o f i t . T h r o u g h t h i s i n t e r n a l i z a t i o n , t h e p e r s o n " l e a r n s " a n d becomes e n c u l t u r a t e d i n t o t h e s o c i a l g r o u p . Cobb (1994b) has s u g g e s t e d t h e s e two p e r s p e c t i v e s a r e c o m p l e m e n t a r y , one b e c o m i n g t h e b a c k g r o u n d t o t h e o t h e r d e p e n d i n g o n t h e s p e c i f i c p r o b l e m s a n d i s s u e s b e i n g r e s e a r c h e d . T h i s a p p r o a c h a l l o w s f o r c r e a t i o n o f k n o w l e d g e b y t h e i n d i v i d u a l t h r o u g h s o c i a l i n t e r a c t i o n s . D e s p i t e d e b a t e o v e r i t s p h i l o s o p h i c a l a s p e c t s , a d h e r e n t s t o c o n s t r u c t i v i s m have c h a n g e d t h e c o u r s e o f r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n . The f o c u s has s h i f t e d f r o m i n v e s t i g a t i o n s whose r e s u l t s a r e b a s e d o n m e a s u r a b l e q u a n t i t i e s , s o - c a l l e d " s c i e n t i f i c s t u d i e s , " t o " f i e l d - b a s e d s t u d i e s " (Romberg a n d C a r p e n t e r , 1986) where t h e t h o u g h t s a n d i d e a s o f t h e p r i n c i p a l s a r e e x a m i n e d , f r o m t h o s e w i t h an " a n a l y t i c p e r s p e c t i v e " t o t h o s e w i t h a " h u m a n i s t p e r s p e c t i v e " ( B r o w n , Cooney a n d J o n e s , 1 9 9 0 ) , f r o m " a p r o c e s s - p r o d u c t p a r a d i g m " t o " a f o c u s o n t e a c h e r s ' t h i n k i n g a n d d e c i s i o n - m a k i n g p r o c e s s e s " (Thompson, 1992) . By a l l o w i n g f o r a l t e r n a t i v e u n d e r s t a n d i n g s , c o n s t r u c t i v i s m has o p e n e d t h e way f o r i n v e s t i g a t i o n i n t o m u l t i p l e c o n c e p t i o n s . How a n i n d i v i d u a l c o n c e i v e s o f t e a c h i n g m a t h e m a t i c s i s t h e f o c u s o f t h i s s t u d y . T h a t i m p o r t i s g i v e n t o p e r s o n a l u n d e r s t a n d i n g s o n - 8 -t e a c h i n g i s a c o n s e q u e n c e o f my b e l i e f i n a c o n s t r u c t i v i s t e p i s t e m o l o g y . T h e r e i s no o n t o l o g i c a l r e a l i t y t o t e a c h i n g . T h e r e i s no one r i g h t way t o t e a c h , n o r t o c o n c e i v e o f t e a c h i n g . What an i n d i v i d u a l b e l i e v e s t o be t e a c h i n g m a t h e m a t i c s must have b e e n c o n s t r u c t e d f r o m h i s / h e r e x p e r i e n c e s a n d b e l i e f s . The R o l e o f E x i s t i n g B e l i e f s The b e l i e f s o f t e a c h e r s may n o t be i n t e g r a t e d and may n o t be a r t i c u l a t e d (Thompson, 1 9 8 4 ) . N e v e r t h e l e s s , k n o w i n g what t h e y a r e i s i m p o r t a n t f o r r e s e a r c h e r s as w e l l as f o r t h e t e a c h e r s t h e m s e l v e s . T e a c h e r s ' t h o u g h t s s u b s t a n t i a l l y i n f l u e n c e t h e i r a c t i o n s i n t h e c l a s s r o o m ( C l a r k a n d P e t e r s o n , 1 9 8 6 ) . The r e s e a r c h e r g a i n s i n s i g h t i n t o what t h e t e a c h e r i s t h i n k i n g a n d i s a b l e t o a p p l y t h i s c o n t e x t t o r e s u l t s o f o t h e r s t u d i e s . " . . . w e c a n make s e n s e o f [ r e s e a r c h ] f i n d i n g s o n l y i n r e l a t i o n t o t h e p s y c h o l o g i c a l c o n t e x t i n w h i c h t h e t e a c h e r p l a n s a n d d e c i d e s . . . . [where] t h e p s y c h o l o g i c a l c o n t e x t i s t h o u g h t t o be composed o f a m i x t u r e o f o n l y p a r t i a l l y a r t i c u l a t e d t h e o r i e s , b e l i e f s and v a l u e s a b o u t h i s o r h e r r o l e a n d a b o u t t h e d y n a m i c s o f t e a c h i n g and l e a r n i n g . " [ C l a r k and P e t e r s o n , 1986, pp 2 8 6 - 2 8 7 ] N o t o n l y do t e a c h e r s ' b e l i e f s a f f e c t t h e i r a c t i o n s i n t h e c l a s s r o o m , t h e y a l s o i n f l u e n c e t h e i r s t u d e n t s ' b e l i e f s . I n one s t u d y o f t e n f i f t h - g r a d e t e a c h e r s i n f o u r s c h o o l s , F o r d (1994) f o u n d t h e i r b e l i e f s a b o u t p r o b l e m s o l v i n g i n m a t h e m a t i c s was p a r a l l e l e d b y t h e i r s t u d e n t s ' b e l i e f s ; b o t h g r o u p s d e s c r i b e d p r o b l e m s o l v i n g as a p p l y i n g c o m p u t a t i o n a l s k i l l s a p p l i e d t o r e a l - l i f e s i t u a t i o n s . I n a n o t h e r s t u d y , B r o w n (1992) a l s o f o u n d s t u d e n t s ' images o f m a t h e m a t i c s l i n k e d s t r o n g l y w i t h t h e i r t e a c h e r s ' images o f m a t h e m a t i c s . The images were v a r i e d ; m a t h e m a t i c s . w a s p e r c e i v e d as t h e c o n t e n t i n a s e t o f b o o k s , as r u l e s a n d - 9 -i d e a s u s e d t o s o l v e p r o b l e m s , as e a s y a f t e r b e i n g s o r t e d o u t , a n d as e n j o y a b l e . The t e a c h e r a l s o b e n e f i t s f r o m t h i s t y p e o f r e s e a r c h a c t i v i t y t h r o u g h i n c r e a s e d s e l f - a w a r e n e s s o f p e r s o n a l b e l i e f s a b o u t t e a c h i n g . N i c k s o n (1988) has s u g g e s t e d t h a t t e a c h e r s ' b e l i e f s may be c o u n t e r p r o d u c t i v e t o c u r r i c u l a r change and d e v e l o p m e n t b u t , o n t h e o t h e r h a n d , " t h e y may come t o a p p r e c i a t e t h e e f f e c t t h e s e [ b e l i e f s ] h a v e o n t h e i r p e r f o r m a n c e as t e a c h e r s o f m a t h e m a t i c s a n d , i f n e c e s s a r y , c a l l them i n t o q u e s t i o n . " [p 246] K n o w i n g t h e i r own b e l i e f s r e g a r d i n g s t u d e n t s , s c h o o l , a n d t h e i r images o f t h e m s e l v e s as t e a c h e r s i s e q u a l l y i m p o r t a n t f o r t e a c h e r c a n d i d a t e s . T h i s m e t a c o g n i t i o n p r o v i d e s a f o u n d a t i o n t o compare a l t e r n a t i v e t e a c h i n g p e r s p e c t i v e s w h i c h a r i s e d u r i n g t h e i r p r e p a r a t i o n p r o g r a m ( K a g a n , 1992) . Raymond a n d S a n t o s (1995) s t a t e e v e n more e m p h a t i c a l l y , " C o n f r o n t i n g and c h a l l e n g i n g p r e s e r v i c e t e a c h e r s ' b e l i e f s s h o u l d be a n i n t e g r a l p a r t o f t e a c h e r e d u c a t i o n " [p 6 8 ] . R e s e a r c h w i t h p r e - s e r v i c e t e a c h e r s has y i e l d e d e v i d e n c e o f d i f f e r e n t i n t e r p r e t a t i o n s f o r t h e same c o n c e p t . U s i n g i n t e r v i e w s a n d o b s e r v a t i o n s , Goodman (1988) s t u d i e d 12 p r e - s e r v i c e e l e m e n t a r y t e a c h e r s a n d f o u n d two b r o a d p r a c t i c a l p h i l o s o p h i e s o f t e a c h i n g : " T e a c h i n g as a P r o b l e m o f C o n t r o l " and " T e a c h i n g as t h e F a c i l i t a t i o n o f C h i l d r e n ' s G r o w t h " (p 1 2 3 ) . C o o p e r a t i o n , a u t h o r i t y , and autonomy were t h r e e c o n c e p t s w h i c h emerged as " g u i d i n g i m a g e s " (p 124) f o r t h e f i r s t p h i l o s o p h y , t e a c h i n g as a p r o b l e m o f c o n t r o l . I n t e r p r e t a t i o n s o f t h e s e c o n c e p t s d i f f e r e d amongst t h e t e a c h e r c a n d i d a t e s e v e n t h o u g h t h e y u s e d t h e same t e r m s . F o r e x a m p l e , t o n i n e o f h i s s u b j e c t s , c o o p e r a t i o n meant - 10 -p u p i l s f o l l o w e d t h e r u l e s l a i d o u t b y t h e t e a c h e r . To t h e o t h e r s , i t meant meant w o r k i n g t o g e t h e r h a r m o n i o u s l y . I n b o t h i n s t a n c e s , t h e d e s i r e d r e s u l t was a s m o o t h l y o p e r a t i n g c l a s s r o o m . H o w e v e r , t h e c o n t r a s t i n p h i l o s o p h y i s o b v i o u s . S i m i l a r l y , t h e r e were two d i s t i n c t i n t e r p r e t a t i o n s o f a u t h o r i t y : i n s t i t u t i o n a l a u t h o r i t y and p e r s o n a l a u t h o r i t y . The f i r s t was g r a n t e d b y s t a t u s : I am t h e t e a c h e r . The s e c o n d was g r a n t e d b y t h e p u p i l s : I am y o u r f r i e n d a n d we r e s p e c t e a c h o t h e r . Goodman (1988) c o n c l u d e s , " W h i l e o n t h e s u r f a c e t h e s e s t u d e n t s [ t e a c h e r s ] e x p r e s s e d a common p h i l o s o p h y , t h e y u n i q u e l y i n t e r p r e t e d t h e images w i t h i n e a c h p e r s p e c t i v e , and t h u s t h e i r p r a c t i c a l p h i l o s o p h y o f t e a c h i n g d i f f e r e d s i g n i f i c a n t l y . " [p 129] F e i m a n - N e m s e r and Buchmann (1989) t a k e a s t r o n g e r s t a n d o n t h e r o l e e x i s t i n g b e l i e f s p l a y i n t h e l e a r n i n g o f p r e - s e r v i c e t e a c h e r s . T h e y g i v e as a n e x a m p l e two e l e m e n t a r y c a n d i d a t e s ' p e r s p e c t i v e s o f t h e i s s u e o f e q u a l a c c e s s t o k n o w l e d g e where " b o t h c a n d i d a t e s r e l i e d o n p e r s o n a l e x p e r i e n c e t h a t was l i m i t e d and s u b j e c t t o b i a s " [p 3 7 2 ] . I n J a n i c e ' s c a s e , b e c a u s e h e r p e r s p e c t i v e s went u n c h a l l e n g e d , h e r n o t i o n s t h a t e t h n i c m i n o r i t i e s were n o t c a p a b l e o f t h e same a c a d e m i c a c h i e v e m e n t as n o n - e t h n i c s t u d e n t s were a c t u a l l y r e i n f o r c e d b y h e r i n t e r p r e t a t i o n o f h e r c o u r s e r e a d i n g s . On t h e o t h e r h a n d , S a r a h , who came f r o m a d i s a d v a n t a g e d b a c k g r o u n d , was f r u s t r a t e d a t n o t b e i n g a b l e t o p r o v i d e more f o r h e r p u p i l s . As Kagan (1992) r e p o r t e d i n a r e v i e w o f 27 s t u d i e s d e a l i n g w i t h t h e p r o f e s s i o n a l g r o w t h o f e l e m e n t a r y and s e c o n d a r y p r e - s e r v i c e t e a c h e r s , t h e i d e a s o f t e a c h i n g t h e y h a d f o r m u l a t e d were s t r o n g . T h e i r p e r s o n a l b e l i e f s o f what c o n s t i t u t e d good t e a c h i n g a n d t h e i r p a r t i c u l a r images o f t h e m s e l v e s as a t e a c h e r h a d b e e n d e r i v e d f r o m " p e r s o n a l - 11 -b i o g r a p h i e s , " i . e . f r o m t h e images o f e x e m p l a r y t e a c h e r s t h e y h a d remembered a n d t h e i r p a s t e x p e r i e n c e s as s t u d e n t s . I n c o n c e r t w i t h f i n d i n g s o f e a r l i e r s t u d i e s , most n o t a b l y , T a b a c h n i c k & Z e i c h n e r (1984) , i t was r e p o r t e d t h e s e b e l i e f s and images o f t e n r e m a i n e d u n c h a n g e d d u r i n g t h e p r o g r a m . I n t h e one s t u d y ( F l o r i o - R u a n e & L e s m i r e , 1 9 9 0 , c i t e d i n K a g a n , 1992) where c o n c e p t u a l change d i d o c c u r , i t was a t t r i b u t e d t o t h e u n i q u e n a t u r e o f t h a t p a r t i c u l a r methods c o u r s e . The s t a b i l i t y o f t h e i d e a s o f t e a c h e r c a n d i d a t e s u n d e r s c o r e s t h e i m p o r t a n c e o f i n v e s t i g a t i n g t h e i r c o n c e p t i o n s . F o r t e a c h e r e d u c a t i o n p r o g r a m s t o make a d i f f e r e n c e and f o r p e r s o n a l p r o f e s s i o n a l g r o w t h , t e a c h e r c a n d i d a t e s n e e d t o be made aware o f t h e i r i d e a s . T h i s k n o w l e d g e c a n t h e n p r o v i d e a b a s i s f o r r e f l e c t i o n f r o m w h i c h change c a n be e f f e c t e d . The N a t u r e o f M a t h e m a t i c s B e c a u s e how m a t h e m a t i c s i s p e r c e i v e d may s t r o n g l y a f f e c t i n s t r u c t i o n a l p r a c t i c e ( D o s s e y , 1992; E r n e s t , 1 9 9 1 ) , t h i s s e c t i o n l o o k s a t v a r i o u s c h a r a c t e r i z a t i o n s o f m a t h e m a t i c s . F o l l o w i n g t h e h i s t o r i c a l d e v e l o p m e n t o f t h e d i s c i p l i n e , D a v i s a n d H e r s h (1981) have d e l i n e a t e d t h r e e b r o a d p h i l o s o p h i e s r e g a r d i n g t h e n a t u r e o f m a t h e m a t i c s : P l a t o n i s t , f o r m a l i s t a n d f a l l i b i l i s t . The P l a t o n i s t p h i l o s o p h y h o l d s m a t h e m a t i c s t o be a n i d e a l , a r e a l i t y w i t h i m m u t a b l e t r u t h s s e t a p a r t f r o m and above human e x i s t e n c e . M a t h e m a t i c a l p r i n c i p l e s a r e d i s c o v e r e d r a t h e r t h a n i n v e n t e d . T h e y e x i s t i n d e p e n d e n t l y o f human k n o w l e d g e and m a t h e m a t i c i a n s work t o l o c a t e a n d g i v e v o i c e t o t h e s e t r u t h s . - 12 -The f o r m a l i s t p h i l o s o p h y s t a t e s m a t h e m a t i c s i s c o m p r i s e d o f a x i o m s , d e f i n i t i o n s a n d t h e o r e m s . T h e s e a r e m a n i f e s t e d v i a s y m b o l s w h i c h have b e e n c r e a t e d and f o r m u l a s and r u l e s w h i c h have b e e n i n v e n t e d . E v e n t h o u g h t h e s e may be u s e d t o g i v e a c c u r a t e d e s c r i p t i o n s a n d p r e d i c t i o n s o f r e a l - w o r l d phenomena, t h e s e a p p l i c a t i o n s a r e n o t t h e p r i m a r y p u r p o s e f o r t h e c r e a t i o n o f m a t h e m a t i c s . The f o r m u l a s a n d m a t h e m a t i c a l m o d e l s i n t h e m s e l v e s a r e m e r e l y s y m b o l s s t r u n g t o g e t h e r whose m e a n i n g i s d e r i v e d f r o m s e t s o f r u l e s (ax ioms) d e v i s e d b y a n d a g r e e d u p o n b y t h a t c a d r e o f p r o f e s s i o n a l s known as m a t h e m a t i c i a n s . I n t h i s s e n s e , m a t h e m a t i c s i s c o n s t r u c t e d . I t i s a f i e l d o f l o g i c a l d e d u c t i o n i n w h i c h c o n c l u s i o n s c a n w i t h s t a n d r i g o r o u s p r o o f . The t h i r d p h i l o s o p h i c a l s c h o o l on t h e n a t u r e o f m a t h e m a t i c s i s t h e f a l l i b i l i s t . T h i s s c h o o l h o l d s t h e b e l i e f t h a t m a t h e m a t i c s i s m u t a b l e , a f i e l d i n c o n s t a n t f l u x as i t r e s p o n d s t o new q u e s t i o n s and c h a l l e n g e s . M a t h e m a t i c s i s n o t r i g i d , i t s r u l e s and d e f i n i t i o n s n o t " s e t i n s t o n e . " I t s p r e c e p t s c a n and do change t h r o u g h d e b a t e a n d d i s c u s s i o n b y m a t h e m a t i c i a n s . Though t h e s e p h i l o s o p h i c a l p e r s p e c t i v e s have b e e n d e r i v e d f r o m t h e work o f m a t h e m a t i c i a n s , t h e y a r e u s e f u l i n d e s c r i b i n g t h e t h o u g h t s o f t e a c h e r s a b o u t m a t h e m a t i c s . A t e a c h e r ' s v i e w o f m a t h e m a t i c s c o u l d l e a d t o t h e u s e o f i n s t r u c t i o n a l t e c h n i q u e s a n d s t r a t e g i e s i n t h e c l a s s r o o m c o n s i s t e n t w i t h t h a t v i e w . F o r e x a m p l e , i f a t e a c h e r b e l i e v e s m a t h e m a t i c s t o be a f i x e d , n o n - c h a n g i n g e n t i t y t o be r e v e a l e d t o s t u d e n t s , " p r e s e n t i n g , " w h i c h c a r r i e s a c o n n o t a t i o n o f t r a n s m i t t i n g k n o w l e d g e t o l e a r n e r s , c o u l d be a f a v o r e d t e a c h i n g t e c h n i q u e . By c o n t r a s t , a t e a c h e r who b e l i e v e s m a t h e m a t i c s t o be a d y n a m i c a n d - 13 -c h a n g i n g b o d y o f k n o w l e d g e c o u l d e m p h a s i z e t h a t a s p e c t . Someone who b e l i e v e s m a t h e m a t i c s t o be d e r i v e d t h r o u g h a c o n s e n s u s o f o p i n i o n s c o u l d v a l u e t h e c o n t r i b u t i o n s o f i n d i v i d u a l s t u d e n t s i n c l a s s r o o m d i s c u s s i o n s . A n o t h e r t e a c h e r w i t h a f o r m a l i s t p h i l o s o p h y may p r e s e n t c o n t e n t i n a s t r u c t u r e d f o r m a t " c a l l i n g on s e t t h e o r e t i c a l l a n g u a g e and c o n c e p t i o n s " ( H e r s h , 1986 c i t e d i n D o s s e y , 1992, p 4 2 ) . R e s e a r c h o n C o n c e p t i o n s o f T e a c h i n g M a t h e m a t i c s R e s e a r c h s p e c i f i c t o c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s h e l d b y p r e - s e r v i c e t e a c h e r s o f s e c o n d a r y m a t h e m a t i c s has b e e n d i f f i c u l t t o l o c a t e . W i l s o n (1994) i n v e s t i g a t e d t h e e v o l v i n g b e l i e f s o f one p r o s p e c t i v e s e c o n d a r y t e a c h e r , M o l l y , i n t h e c o n t e x t o f a c o u r s e w h i c h i n t e g r a t e d p e d a g o g y a n d m a t h e m a t i c a l c o n t e n t . He e x a m i n e d how h e r u n d e r s t a n d i n g o f f u n c t i o n s c h a n g e d a n d t h e r e l a t i o n s h i p b e t w e e n t h i s u n d e r s t a n d i n g a n d h e r v i e w s o f m a t h e m a t i c s and t e a c h i n g m a t h e m a t i c s . He f o u n d t h a t a t t h e b e g i n n i n g o f t h e c o u r s e w o r k , M o l l y h a d a n a r r o w v i e w o f f u n c t i o n s a n d t h i s v i e w was c o n s i s t e n t w i t h h e r g e n e r a l v i e w o f m a t h e m a t i c s - t h a t i t was a c o l l e c t i o n o f p r o c e d u r e s t o be a p p l i e d t o o b t a i n c o r r e c t a n s w e r s . M o l l y ' s c o n c e p t u a l u n d e r s t a n d i n g b r o a d e n e d a n d d e e p e n e d d u r i n g t h e c o u r s e ; t o w a r d t h e e n d she was a b l e t o i d e n t i f y r e a l - w o r l d a p p l i c a t i o n s o f f u n c t i o n s a n d u s e v a r i o u s f o r m s o f f u n c t i o n s u n d e r d i f f e r e n t c i r c u m s t a n c e s . How d i d t h i s i n c r e a s e d u n d e r s t a n d i n g a f f e c t h e r v i e w o f t e a c h i n g m a t h e m a t i c s ? M o l l y b e l i e v e d h e r v i e w h a d c h a n g e d s i g n i f i c a n t l y . H o w e v e r , W i l s o n (1994) c l a i m e d she h a d s i m p l y f o u n d new ways t o r e d u c e - 14 -t h e m o n o t o n y o f c l a s s r o o m l e c t u r e s . She h a d l e a r n e d new i n s t r u c t i o n a l t e c h n i q u e s , n o t f o r m u l a t e d a c h a n g e d p e r s p e c t i v e t o o f f e r " a more v i b r a n t a n d m e a n i n g f u l image o f m a t h e m a t i c s . T h u s , a l t h o u g h [she began] t o s e e a l t e r n a t i v e ways o f t e a c h i n g m a t h e m a t i c s , h e r v i e w s o f m a t h e m a t i c s and m a t h e m a t i c s t e a c h i n g were s t i l l r e l a t i v e l y n a r r o w . " [p 367] Two o t h e r s t u d i e s w h i c h f i g u r e p r o m i n e n t l y i n t h e l i t e r a t u r e -B r o w n (1985) a n d Owens (1987) - a r e b o t h u n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n s . T h i s has meant g l e a n i n g d e s c r i p t i o n s o f t h e s t u d i e s t h r o u g h s e c o n d a r y s o u r c e s . B r o w n (1985) i n v e s t i g a t e d t h e c h a n g e s i n one p r e - s e r v i c e t e a c h e r ' s c o n c e p t i o n o f m a t h e m a t i c s t e a c h i n g as he became s o c i a l i z e d i n t o t h e t e a c h i n g p r o f e s s i o n . C o n c e p t i o n s o f m a t h e m a t i c s , b e l i e f s a b o u t t h e a p p r o p r i a t e g o a l s and t a s k s f o r a math c l a s s r o o m , and b e l i e f s a b o u t t h e " r e l a t i v e r e s p o n s i b i l i t i e s o f t e a c h e r a n d s t u d e n t s c o n c e r n i n g m o t i v a t i o n , d i s c i p l i n e , a n d e v a l u a t i o n " (Brown & B o r k o , 1992, p 225) c o m p r i s e d t h e a n a l y t i c a l f r a m e w o r k . What she f o u n d was t h a t p u p i l s ' c o n c e p t i o n s o f m a t h e m a t i c s t e a c h i n g h a d a s t r o n g i n f l u e n c e i n c h a n g i n g h i s i d e a s o f t e a c h i n g . However , "he d i d n o t change some o f h i s most b a s i c g e n e r a l c o n c e p t i o n s o f m a t h e m a t i c s and t e a c h i n g " (Brown & B o r k o , 1 9 9 2 , p 2 2 5 ) , p o i n t i n g once a g a i n t o t h e s t a b i l i t y o f c o n c e p t i o n s . Owens (1987) s o u g h t t o d e s c r i b e t h e c o n s t r u c t s o f f o u r p r e - s e r v i c e t e a c h e r s r e g a r d i n g m a t h e m a t i c s and t e a c h i n g m a t h e m a t i c s . He f o u n d t h e i r p e r s p e c t i v e s o f m a t h e m a t i c s were l a r g e l y i n f l u e n c e d b y p r e - c o l l e g e e x p e r i e n c e s . A d d i t i o n a l l y , t h e t e a c h e r c a n d i d a t e s f o u n d t h e m a t h e m a t i c a l p e r s p e c t i v e o f c o l l e g e - l e v e l math c o u r s e s i n c o m p a t i b l e w i t h - 15 -t h e i r own. T h e s e f i n d i n g s c o n f i r m B a l l and M c D i a r m i d ' s (1990) c l a i m t h a t p r e - c o l l e g e e d u c a t i o n f i g u r e s p r o m i n e n t l y i n c o n t e n t u n d e r s t a n d i n g f o r p r o s p e c t i v e s e c o n d a r y t e a c h e r s . A G e n e r a l M o d e l f o r D e s c r i b i n g C o n c e p t i o n s o f T e a c h i n g P r a t t (1992) has o f f e r e d a f ramework f o r d e s c r i b i n g c o n c e p t i o n s o f t e a c h i n g . The e l e m e n t s u s e d t o d e s c r i b e t h e c o n c e p t i o n s a n d h e n c e e s t a b l i s h t h e i r d i s t i n c t i v e d i f f e r e n c e s a r e : t e a c h e r , l e a r n e r s , c o n t e n t , c o n t e x t , i d e a l . The f i r s t t h r e e a r e germane t o any t e a c h i n g s i t u a t i o n . The f o u r t h , c o n t e x t , d e f i n e s t h e o r g a n i z a t i o n a l s e t t i n g ( e . g . t y p e o f s c h o o l , t y p e o f p r o g r a m , t y p e o f c l a s s ) and i n c l u d e s a n y e x t e r n a l f a c t o r s t h a t c o u l d i n f l u e n c e t h e t e a c h i n g a n d / o r l e a r n i n g , s u c h as i m p e n d i n g e x a m i n a t i o n s . " I d e a l " was u s e d t o d e s c r i b e t h e " p u r p o s e s o f a d u l t e d u c a t i o n " ( p . 2 0 5 ) and c a n be c o n s t r u e d as t h e c e n t r a l o r o v e r r i d i n g g o a l , i m p l i c i t o r e x p l i c i t , f o r t h e t e a c h e r . P r a t t ' s r e s e a r c h was a s t u d y i n a d u l t e d u c a t i o n and t h e f r a m e w o r k was d e r i v e d f r o m i n t e r v i e w s o f o v e r 250 t e a c h e r s i n 5 c o u n t r i e s a n d a c r o s s a v a r i e t y o f c o n t e x t s s u c h as p o s t - s e c o n d a r y i n s t i t u t i o n s , g o v e r n m e n t a n d m i l i t a r y c l a s s e s , r e l i g i o u s e d u c a t i o n , and t r a i n i n g p r o g r a m s i n b u s i n e s s a n d i n d u s t r y . P r a t t (1992) d e l i n e a t e d f i v e d i f f e r e n t c o n c e p t i o n s o f t e a c h i n g : E n g i n e e r i n g - D e l i v e r i n g C o n t e n t ; A p p r e n t i c e s h i p - M o d e l i n g Ways o f B e i n g ; D e v e l o p m e n t a l - C u l t i v a t i n g t h e I n t e l l e c t ; N u r t u r i n g -F a c i l i t a t i n g P e r s o n a l Agency,- and S o c i a l R e f o r m - S e e k i n g a B e t t e r S o c i e t y . The d i s t i n c t i o n s among t h e c o n c e p t i o n s a r i s e f r o m d i f f e r i n g emphases i n t h e r e l a t i o n s h i p s b e t w e e n t h e e l e m e n t s . - 16 -The E n g i n e e r i n g c o n c e p t i o n has as i t s d i s t i n c t i v e f e a t u r e d e l i v e r i n g c o n t e n t . I t i s t e a c h e r - c e n t e r e d a n d i n f o r m a t i o n t r a n s m i s s i o n i s a p r i m a r y f o c u s . L e a r n e r s a r e v i e w e d as n o n - p r o b l e m a t i c . E n e r g i e s a r e c o n c e n t r a t e d on i m p r o v i n g t e c h n i q u e s f o r c o n t e n t d e l i v e r y . C o n t e n t i s o f t e n b r o k e n down i n t o d i s c r e t e o b j e c t i v e s and l e a r n i n g m e a s u r e d b y c o m p l e t i o n o f t a s k s , s k i l l s g a i n e d a n d / o r c o m p e t e n c i e s a c q u i r e d . The A p p r e n t i c e s h i p c o n c e p t i o n , l i k e t h e E n g i n e e r i n g c o n c e p t i o n , h a s t h e t e a c h e r a n d c o n t e n t as d o m i n a n t e l e m e n t s . H o w e v e r , t h e t e a c h e r i n t h i s c a s e i s n o t o n l y t h e c o n t e n t e x p e r t , he i s a l s o s e e n t o be a n e x e m p l a r o f t h e v a l u e s o f t h e c o n t e n t . The i d e a o f m o d e l l i n g , o r b e i n g a m a s t e r c r a f t s m a n , c o n v e y s t h i s i m p l i c a t i o n t h a t t h e w h o l e p e r s o n e m b o d i e s t h e c o n t e n t as much as he has c o g n i t i o n o f i t . H e n c e , " k n o w l e d g e [ i s ] p a s s e d o n t h r o u g h r o l e m o d e l l i n g " ( p . 212) a n d l e a r n i n g c o n s t i t u t e s a c q u i r i n g b o t h i n f o r m a t i o n and t h e v a l u e s i n h e r e n t w i t h i n t h a t c o r p u s o f k n o w l e d g e . E x a m p l e s o f t h i s c o n c e p t i o n were most o f t e n f o u n d i n s u c h a r e a s as v o c a t i o n a l a p p r e n t i c e s h i p s a n d m e d i c a l i n t e r n s h i p s . The D e v e l o p m e n t a l c o n c e p t i o n i s c h a r a c t e r i z e d b y t h e t e a c h e r ' s d e s i r e t o f u r t h e r t h e c o g n i t i v e d e v e l o p m e n t o f h e r l e a r n e r s . The f o c u s o f e x e r c i s e s a n d t h e r o l e o f c u r r i c u l u m i s t o s t i m u l a t e a n d e n h a n c e l e a r n e r s ' ways o f t h i n k i n g s i n c e t h e c e n t r a l b e l i e f i s t h a t i n d i v i d u a l s p o s s e s s t h e c a p a c i t y t o d e v e l o p i n c r e a s i n g l y a b s t r a c t a n d c o m p l e x f o r m s o f t h o u g h t . I n q u i r y - o r i e n t e d e x e r c i s e s , w h e t h e r t o a r o u s e c u r i o s i t y o r p r o d u c e d i s e q u i l i b r i u m , a r e a p r e f e r r e d mode o f i n s t r u c t i o n . The p e r s o n a l w o r t h o f t h e i n d i v i d u a l s t u d e n t i s p a r a m o u n t i n t h e N u r t u r i n g c o n c e p t i o n . A s t r o n g b o n d b e t w e e n t h e t e a c h e r a n d l e a r n e r i s - 17 -e v i d e n t . The t e a c h e r e x h i b i t s a h i g h d e g r e e o f c o n c e r n f o r t h e p e r s o n a l w e l f a r e o f t h e s t u d e n t . The s t u d e n t ' s s e l f - k n o w l e d g e and d e v e l o p m e n t , i n d i c a t e d b y an i n c r e a s i n g l y p o s i t i v e s e l f - c o n c e p t a n d i n t e r n a l l o c u s o f c o n t r o l i s c o n s i d e r e d more i m p o r t a n t t h a n c o n t e n t k n o w l e d g e . C o n t e n t k n o w l e d g e i s s e e n as b e i n g a v e h i c l e t h r o u g h w h i c h t h e l e a r n e r ' s s e l f -i d e n t i t y c a n be d e v e l o p e d . I n t h e c o n c e p t i o n l a b e l l e d S o c i a l R e f o r m , a s p e c i f i c i d e o l o g y d r i v e s t h e a c t i v i t i e s o f t h e t e a c h e r . T h i s i d e o l o g y may be r e l i g i o u s , p o l i t i c a l o r d e r i v e d f r o m a d i f f e r e n t , b u t e q u a l l y s t r o n g , s e t o f c o n v i c t i o n s a n d b e l i e f s . I t i s c l e a r l y a r t i c u l a t e d a n d , b e i n g a t c e n t r e s t a g e f o r t h e t e a c h e r , becomes t h e g u i d i n g p r i n c i p l e s b y w h i c h he c o n d u c t s t h e c l a s s and r e l a t e s t o t h e l e a r n e r s . The t e a c h e r has a m i s s i o n t o e n l i g h t e n t h e l e a r n e r and s o l i c i t s u p p o r t f o r t h e i d e o l o g y , t h e r e b y c r e a t i n g a " b e t t e r w o r l d . " E x a m p l e s a r e some f o r m s o f e n v i r o n m e n t a l a n d f e m i n i s t e d u c a t i o n . The Framework f o r T h i s S t u d y Thompson (1992) d e s c r i b e d a c o n c e p t i o n o f t e a c h i n g m a t h e m a t i c s as i n c l u d i n g b u t n o t l i m i t e d t o " d e s i r a b l e g o a l s o f t h e m a t h e m a t i c s p r o g r a m , [a t e a c h e r ' s ] own r o l e o f t e a c h i n g , t h e s t u d e n t s ' r o l e , a p p r o p r i a t e t e a c h i n g a c t i v i t i e s , d e s i r a b l e i n s t r u c t i o n a l a p p r o a c h e s a n d e m p h a s e s , l e g i t i m a t e m a t h e m a t i c a l p r o c e d u r e s a n d a c c e p t a b l e o u t c o m e s o f i n s t r u c t i o n " [p 1 3 5 ] . U n d e r g i r d i n g t h e s e i s a p h i l o s o p h y o f t e a c h i n g m a t h e m a t i c s . F o r t h e s c o p e o f t h i s s t u d y , my f r a m e w o r k t o d e s c r i b e t h e c o n c e p t i o n s o f p r e - s e r v i c e m a t h e m a t i c s t e a c h e r s c o n s i s t e d o f t h e - 18 -c o n c e p t i o n o f t h e n a t u r e o f m a t h e m a t i c s a n d t h e c o n c e p t i o n o f t e a c h i n g m a t h e m a t i c s where t h e l a t t e r encompasses a n u n d e r l y i n g p h i l o s o p h y , a g o a l f o r t e a c h i n g m a t h e m a t i c s , a p e r s p e c t i v e o n t h e r o l e o f t h e t e a c h e r a n d a c o n c e p t i o n o f s t u d e n t s . I n d e v e l o p i n g t h e f r a m e w o r k , I drew h e a v i l y o n t h e work o f P r a t t ( 1 9 9 2 ) . The a p p e a l o f t h i s m o d e l l i e s i n i t s u s e f u l n e s s i n b r i n g i n g t o t h e f o r e t h e e l e m e n t a l r e l a t i o n s h i p s b e t w e e n t h e t e a c h e r a n d s t u d e n t , t h e t e a c h e r and c o n t e n t , a n d t h e s t u d e n t a n d c o n t e n t . The t e n t a t i v e f r a m e w o r k s o f f e r e d b y B a l l (1988) a n d E r n e s t ( 1 9 8 9 , 1991) t o s t u d y c o n c e p t i o n s o f m a t h e m a t i c s t e a c h e r s a l s o i n f o r m e d my w o r k . B a l l (1988) o u t l i n e d 3 s t r a n d s : k n o w l e d g e o f a n d a b o u t m a t h e m a t i c s , i d e a s o f math t e a c h i n g and l e a r n i n g , f e e l i n g s a b o u t o n e s e l f i n r e l a t i o n t o m a t h e m a t i c s . The k n o w l e d g e component was c o m p r i s e d o f c o n t e n t k n o w l e d g e a n d p e d a g o g i c a l c o n t e n t k n o w l e d g e , i . e . " t h e ways o f r e p r e s e n t i n g a n d f o r m u l a t i n g t h e s u b j e c t t h a t make i t c o m p r e h e n s i b l e t o o t h e r s " ( S h u l m a n , 1986, p 6 ) . An e x a m p l e o f a n i d e a o f math t e a c h i n g w o u l d be " t e a c h i n g as t e l l i n g , " p r o m i n e n t i n h e r s t u d y o f e l e m e n t a r y t e a c h e r c a n d i d a t e s . I n a d d i t i o n , t h r o u g h h e r work w i t h t h i s c o h o r t , B a l l f o u n d a p p r e h e n s i o n o f m a t h e m a t i c s s i g n i f i c a n t and h e n c e i n c l u d e d p e r s o n a l f e e l i n g s w i t h r e s p e c t t o c o n t e n t i n h e r f r a m e w o r k . E r n e s t (1989, 1991) u s e d s i m i l a r components i n h i s f r a m e w o r k . H i s b r o a d c a t e g o r i e s were k n o w l e d g e , b e l i e f s and a t t i t u d e s o f t h e math t e a c h e r . The k n o w l e d g e c a t e g o r y was s u b d i v i d e d i n t o k n o w l e d g e o f : m a t h e m a t i c s , o t h e r s u b j e c t m a t t e r , t e a c h i n g m a t h e m a t i c s (math p e d a g o g y a n d m a t h c u r r i c u l u m ) , c l a s s r o o m o r g a n i z a t i o n a n d management f o r math t e a c h i n g , t h e c o n t e x t o f t e a c h i n g m a t h e m a t i c s ( s c h o o l a n d s t u d e n t s ) , a n d - 19 -e d u c a t i o n ( e d u c a t i o n a l p s y c h o l o g y , e d u c a t i o n a l a i m s , m a t h e m a t i c s e d u c a t i o n ) . The c a t e g o r y o f b e l i e f s was s u b d i v i d e d i n t o b e l i e f s o f : c o n c e p t i o n o f t h e n a t u r e o f m a t h e m a t i c s , m o d e l s o f t e a c h i n g a n d l e a r n i n g m a t h e m a t i c s , a n d p r i n c i p l e s o f e d u c a t i o n . F i n a l l y , t h e c a t e g o r y o f a t t i t u d e s was s u b d i v i d e d i n t o a n a t t i t u d e t o w a r d m a t h e m a t i c s a n d a n a t t i t u d e t o w a r d t e a c h i n g m a t h e m a t i c s . By i n q u i r i n g i n t o p e r s p e c t i v e s o f c o n t e n t as w e l l as i n s t r u c t i o n a l p e r s p e c t i v e s , t h i s s t u d y goes b e y o n d l o o k i n g a t g e n e r a l v i e w s o f t e a c h i n g t o i n v e s t i g a t i n g p a r t i c u l a r v i e w s o f t e a c h i n g m a t h e m a t i c s . - 20 -C h a p t e r 3 : M e t h o d o l o g y N o d d i n g s (1990) has dubbed c o n s t r u c t i v i s m a " p o s t - e p i s t e m o l o g i c a l p o s i t i o n " w h i c h y i e l d s an i m p o r t a n t m e t h o d o l o g i c a l p e r s p e c t i v e . The c o n s t r u c t i v i s t p a r a d i g m n e c e s s i t a t e s a q u a l i t a t i v e r e s e a r c h m e t h o d o l o g y ( D e n z i n & L i n c o l n , 1994) f o r i t i s o n l y b y a l l o w i n g t h e s u b j e c t t o r e p r e s e n t h i s own v i e w s u s i n g h i s own words and n o t t h o s e i m p o s e d b y t h e r e s e a r c h e r t h a t i n s i g h t i n t o h i s u n d e r s t a n d i n g s o f a c o n c e p t c a n be g a i n e d . S i n c e t h e a i m o f t h i s s t u d y was t o a l l o w i n d i v i d u a l s t o e x p r e s s t h e i r p e r s o n a l p e r s p e c t i v e s o n t h e t e a c h i n g o f m a t h e m a t i c s , I s e l e c t e d t h e g e n e r a l i n t e r v i e w g u i d e t e c h n i q u e ( P a t t o n , 1 9 9 0 ) . T h i s s e m i -s t r u c t u r e d a p p r o a c h e n s u r e d t h a t t h e same t o p i c s d e l i n e a t e d i n t h e f r a m e w o r k were c o v e r e d w i t h e a c h p e r s o n y e t i t g a v e room f o r f l e x i b i l i t y t o e x p a n d u p o n c e r t a i n t o p i c s o r e x p l o r e o t h e r s . I t a l s o a l l o w e d f o r i n t e r a c t i o n f o r p r o b i n g and c l a r i f i c a t i o n . D e v e l o p m e n t o f t h e P r o t o c o l A n i n i t i a l s e t o f q u e s t i o n s and p o t e n t i a l p r o b e s o n m a t h e m a t i c s as a d i s c i p l i n e , m a t h e m a t i c s as a s c h o o l s u b j e c t , message a b o u t m a t h e m a t i c s , t h e r o l e o f t h e t e a c h e r , v i e w o f t h e s t u d e n t , a n d u n d e r l y i n g g o a l s o f t e a c h i n g m a t h e m a t i c s were c o m p o s e d . C o n t r a s t c a n be e f f e c t i v e f o r i l l u m i n a t i n g a n d c l a r i f y i n g i d e a s , so t h a t was i n c o r p o r a t e d i n t o t h e p r o t o c o l . A n e x a m p l e i s "Has y o u r v i e w o f t e a c h i n g c h a n g e d o v e r t h e y e a r ? " N o t i n g t h a t m a t h e m a t i c s a n d E n g l i s h a r e o f t e n j u x t a p o s e d ( B a l l & M c D i a r m i d , 1990) t h e q u e s t i o n "How - 21 -w o u l d y o u compare t e a c h i n g m a t h e m a t i c s t o t e a c h i n g o t h e r s u b j e c t s , l i k e E n g l i s h ? " was a l s o i n c l u d e d . W h e t h e r u n d e r l y i n g p h i l o s o p h i e s a r e e x p l i c i t o r i m p l i c i t , Goodman (1988) p o i n t s o u t " t h e r o o t s o f t h e i r p e r s p e c t i v e s [are] v i s u a l " [ s i c ] [p 1 2 4 ] . W i t h t h i s i n m i n d , I s e l e c t e d " V i e w s o f t e a c h i n g a n d l e a r n i n g " ( A p p e n d i x A) as r e p r e s e n t i n g some common n o t i o n s o f t e a c h i n g t o u s e as a p o t e n t i a l p r o b e d u r i n g t h e i n t e r v i e w s . The P i l o t S t u d i e s Two p i l o t p r o j e c t s were c a r r i e d o u t p r i o r t o t h e m a i n s t u d y . T h e i r g e n e r a l p u r p o s e was t o r e f i n e my i n t e r v i e w t e c h n i q u e as w e l l as v a l i d a t e t h e i n t e r v i e w q u e s t i o n s . The f i r s t p i l o t s t u d y . A p r e l i m i n a r y p i l o t s t u d y was c o n d u c t e d w i t h two v o l u n t e e r s t u d e n t s f r o m t h e e l e m e n t a r y B . E d , p r o g r a m a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . B o t h s t u d e n t s were i n t e r v i e w e d i n my o f f i c e a t t h e u n i v e r s i t y f o r 3 5 - 4 5 m i n u t e s . E a c h i n t e r v i e w was a u d i o - t a p e d a n d t r a n s c r i b e d . T h e s e i n t e r v i e w s a l l o w e d me t o p r a c t i c e my q u e s t i o n i n g a n d my l i s t e n i n g . W h i l e t r a n s c r i b i n g t h e t a p e s , I n o t e d t h e c o m p a r a t i v e l e n g t h s o f q u e s t i o n s a n d a n s w e r s , a n d any d i s t r a c t i n g comments I made. T h i s p r e p a r e d me t o l i s t e n more c l o s e l y a n d make f e w e r i n c o n s e q u e n t i a l comments f o r t h e n e x t s e t o f i n t e r v i e w s . I a l s o r e v i e w e d t h e d a t a c u r s o r i l y w h i c h gave me p r e l i m i n a r y e x p e r i e n c e i n l o o k i n g f o r c a t e g o r i e s o f i d e a s . The c o n t e x t o f t h e s e d i s c u s s i o n s was t e a c h i n g m a t h e m a t i c s t o e l e m e n t a r y a g e d c h i l d r e n . W h i l e t h e s e were e l e m e n t a r y t e a c h e r c a n d i d a t e s a n d h e n c e r e p r e s e n t e d a d i f f e r e n t g r o u p f r o m t h o s e i n t h e - 22 -m a i n s t u d y , t h e d a t a showed an i n t e r e s t i n g c o n t r a s t i n i d e a s o f t e a c h i n g m a t h e m a t i c s . One saw m a t h e m a t i c s as b e i n g - a n i n e x t r i c a b l e p a r t o f o t h e r c o n t e n t a r e a s . She d e s c r i b e d u s i n g i n t e r d i s c i p l i n a r y p r o j e c t s t o t e a c h m a t h e m a t i c s t o h e r y o u n g p u p i l s t h r o u g h i n t e g r a t i o n w i t h o t h e r s u b j e c t s . The o t h e r p e r s o n b e l i e v e d m a t h e m a t i c s c o u l d o n l y be t a u g h t i n i s o l a t i o n f r o m o t h e r s u b j e c t s u s i n g a s t a n d a r d d r i l l a n d p r a c t i c e f o r m a t . The s e c o n d p i l o t s t u d y . F o r t h e s e c o n d p i l o t i n v e s t i g a t i o n , f o u r v o l u n t e e r s f r o m t h e B . E d , s e c o n d a r y math p r o g r a m were c h o s e n . T h e s e i n t e r v i e w s , c o n d u c t e d i n d i f f e r e n t l o c a t i o n s , l a s t e d 3 5 - 4 5 m i n u t e s . They were a u d i o - t a p e d a n d t r a n s c r i b e d . W i t h t h i s s e t o f i n t e r v i e w s , I l e a r n e d t o p r o b e a n d c l a r i f y more e f f e c t i v e l y . F o r e x a m p l e , d u r i n g t h e i n t e r v i e w I l i s t e n e d f o r t h e words " i t " o r " t h a t " u s e d as t h e s u b j e c t o f a s e n t e n c e a n d s o u g h t t o c l a r i f y t h e c o n c e p t b e i n g r e f e r r e d t o . T h r o u g h r e p e t i t i o n a n d i n c r e a s i n g f a m i l i a r i t y w i t h t h e p r o t o c o l , I r e l i e d l e s s o n t h e w r i t t e n q u e s t i o n s b e f o r e me a n d f o c u s s e d more on what t h e i n t e r v i e w e e was s a y i n g . T h i s r e s u l t e d i n a more c o n v e r s a t i o n a l i n t e r v i e w s t y l e . I d e v e l o p e d a g r a p h i c t e m p l a t e o f t h e q u e s t i o n s I w o u l d u s e i n t h e m a i n s t u d y ( A p p e n d i x B ) . I t s n o n - l i n e a r f o r m a t was u s e f u l f o r m a k i n g n o t e s d u r i n g t h e i n t e r v i e w . I a l s o d e c i d e d t o b e g i n a l l i n t e r v i e w s o f t h e m a i n s t u d y w i t h "What does i t mean t o y o u t o t e a c h m a t h ? " D u r i n g t h e s e c o n d p i l o t , t h i s q u e s t i o n g e n e r a t e d t h e most i l l u s t r a t i v e a n s w e r s f r o m w h i c h I c o u l d f o l l o w - u p w i t h f u r t h e r q u e s t i o n i n g . - 23 -The r e s u l t s o f t h i s p i l o t s t u d y showed s u b s t a n t i a l d i f f e r e n c e s i n i d e a s o f t e a c h i n g m a t h e m a t i c s and s u g g e s t e d t h a t t h e r e a l s o w o u l d be d i f f e r e n c e s i n t h e i d e a s o f t h e p r e - s e r v i c e t e a c h e r s i n t h e m a i n s t u d y . The M a i n S t u d y The p r o g r a m . The B a c h e l o r o f E d u c a t i o n p r o g r a m a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a (UBC) f o r s e c o n d a r y m a t h e m a t i c s i s a t w e l v e - m o n t h p r o g r a m c o n s i s t i n g o f c o u r s e work and two p r a c t i c e t e a c h i n g e x p e r i e n c e s . The f i r s t p r a c t i c u m o c c u r s d u r i n g t h e f a l l s e m e s t e r f o r two weeks a n d t h e s e c o n d o c c u r s d u r i n g t h e s p r i n g s e m e s t e r f o r n i n e t e e n w e e k s . S t u d e n t s a r e a s s i g n e d t o t h e same s c h o o l f o r b o t h p r a c t i c a . The s a m p l e . S e v e n s t u d e n t s f r o m t h e s e c o n d a r y math B . E d , p r o g r a m a t UBC were s e l e c t e d . T h r e e were m a l e , f o u r were f e m a l e . A l l h a d a f o u r - y e a r B . S c . s p e c i a l i z i n g i n m a t h e m a t i c s , a r e q u i r e m e n t f o r e n t r a n c e . Of t h e s e s e v e n , one h a d a M a s t e r s d e g r e e i n s t a t i s t i c s and two were w o r k i n g t o w a r d s M a s t e r s d e g r e e s i n p u r e m a t h e m a t i c s . The f o u r t h h a d a c o m b i n e d E n g l i s h a n d m a t h e m a t i c s s p e c i a l t y a n d t h e f i f t h h a d a c o m b i n e d S p a n i s h a n d m a t h e m a t i c s s p e c i a l t y . The f i n a l two h a d a m a j o r i n m a t h e m a t i c s o n l y . The s e v e n who c o m p r i s e d t h e s a m p l e was t h e g r o u p I s u p e r v i s e d as f a c u l t y a d v i s o r . F o r t h a t c l a s s o f 41 math s p e c i a l i s t s , I was one o f t h r e e f a c u l t y a d v i s o r s . To ease t r a v e l l i n g d u r i n g t h e p r a c t i c a , I was a s s i g n e d t h o s e whose t e a c h i n g s c h o o l s c l u s t e r e d a b o u t a m a i n h i g h w a y ; s o , t h e d e c i s i o n was g e o g r a p h i c a l l y b a s e d . S i x were a s s i g n e d t o s e c o n d a r y s c h o o l s (Grades 8 - 1 2 ) ; one was a s s i g n e d t o a j u n i o r s e c o n d a r y - 24 -s c h o o l ( G r a d e s 7 - 9 ) . I n s e l e c t i n g t h e s e p r o s p e c t i v e t e a c h e r s f o r t h e s t u d y I c o n s i d e r e d them " t y p i c a l " c a s e s ( M i l e s & Huberman, 1994) who were c o n v e n i e n t t o a c c e s s . As f a c u l t y a d v i s o r , i t was my r e s p o n s i b i l i t y t o a c t as l i a i s o n b e t w e e n them a n d t h e i r r e c e i v i n g i n s t i t u t i o n , t o e n s u r e a s m o o t h w o r k i n g r e l a t i o n s h i p b e t w e e n them and t h e i r s c h o o l a d v i s o r , t o l e n d a d v i c e a n d s u p p o r t f o r t h i s p h a s e o f t h e i r p r o f e s s i o n a l t r a i n i n g , a n d t o e v a l u a t e t h e i r i n t r o d u c t o r y t e a c h i n g a d v e n t u r e s f o r m a t i v e l y and s u m m a t i v e l y . T h i s r o l e p r o v i d e d a d v a n t a g e s a n d d i s a d v a n t a g e s t o t h e r e s e a r c h . The o b v i o u s d i s a d v a n t a g e was t h e power c o n n o t a t i o n i n h e r e n t t h r o u g h my r o l e as f a c u l t y a d v i s o r . R a t h e r t h a n s p e a k i n g f r e e l y , t h e t e a c h e r c a n d i d a t e s c o u l d have f e l t i n t i m i d a t e d t o g i v e " r i g h t " a n s w e r s , i . e . a n s w e r s I m i g h t be l o o k i n g f o r . To m i t i g a t e t h i s e f f e c t , I h a d i n f o r m e d them o f my i n t e r e s t i n u s i n g them f o r t h i s s t u d y when we f i r s t met a n d I h a d e m p h a s i z e d t h a t I t o o was a s t u d e n t , as t h e y w e r e . I t o l d them what I was l e a r n i n g was how t o i n t e r v i e w a n d how t o a n a l y z e i n t e r v i e w d a t a , a n d t h e t o p i c t h a t i n t e r e s t e d me was i d e a s a b o u t t e a c h i n g m a t h e m a t i c s . T h r o u g h t h i s " e q u a l f o o t i n g " e x p l a n a t i o n , I h o p e d t o e s t a b l i s h r a p p o r t w i t h t h e g r o u p . I a l s o i n f o r m e d them t h a t t h e y c o u l d " o p t o u t " o f t h e i n t e r v i e w w i t h me a t a n y t i m e w i t h no c o n s e q u e n c e t o t h e i r p r a c t i c u m e v a l u a t i o n . (The f a c t t h a t I am s t i l l i n t o u c h w i t h most o f t h e i n d i v i d u a l s o n a p e r s o n a l b a s i s a t t e s t s , i n my v i e w , t o t h e m u t u a l c o m f o r t l e v e l we a t t a i n e d . ) I c o n d u c t e d c l a s s r o o m v i s i t s w i t h e a c h t e a c h e r c a n d i d a t e a b o u t e v e r y two w e e k s . We a l w a y s d e b r i e f e d a f t e r t h e c l a s s r o o m o b s e r v a t i o n s , a n d t h e s u m m a r i e s o f o u r d i s c u s s i o n s were w r i t t e n i n t h e m o n t h l y - 25 -f o r m a t i v e e v a l u a t i o n s . When t h e f i n a l , summat ive e v a l u a t i o n s were composed a t t h e c o n c l u s i o n o f t h e p r a c t i c u m , t h e y c o n t a i n e d no s u r p r i s e s t o t h e t e a c h e r c a n d i d a t e s as t h e y were s i m p l y s u m m a r i e s o f t h e c o n v e r s a t i o n s we h a d b e e n h a v i n g . I n t h e P a s s / F a i l g r a d i n g scheme f o r t h e p r a c t i c u m e x p e r i e n c e , a l l r e c e i v e d a " P a s s . " By w o r k i n g w i t h them d u r i n g b o t h p r a c t i c a , I was a b l e t o e s t a b l i s h a c o m f o r t a b l e r e l a t i o n s h i p w i t h e a c h p e r s o n . Our e a s e w i t h e a c h o t h e r was an a d v a n t a g e t o t h e r e s e a r c h . A p p e n d i x D, an e x c e r p t f r o m one o f t h e t r a n s c r i p t s , p r o v i d e s a s e n s e o f t h e f l a v o u r o f t h e i n t e r v i e w s . Our f a m i l i a r i t y meant t h a t d u r i n g t h e i n t e r v i e w , e i t h e r o f us c o u l d e v o k e s p e c i f i c i n c i d e n c e s w h i c h t h e o t h e r p e r s o n a l s o knew o f t o h i g h l i g h t o r p r o v i d e a n e x a m p l e o f t h e t o p i c we were p u r s u i n g . The d a t a c o l l e c t i o n . E a c h p e r s o n was i n t e r v i e w e d o n c e . The i n t e r v i e w s were c o n d u c t e d d u r i n g t h e f i n a l week o f t h e n i n e t e e n - w e e k p r a c t i c u m a n d t h e f o l l o w i n g week, a f t e r e a c h p e r s o n h a d b e e n g i v e n h i s / h e r f i n a l e v a l u a t i o n . The f i r s t two i n t e r v i e w s were c o n d u c t e d i n t h e p r a c t i c u m s c h o o l s . One o c c u r r e d a t t h e e n d o f t h e s c h o o l d a y and t h e o t h e r d u r i n g a s p a r e p e r i o d . A l t h o u g h b o t h were s u c c e s s f u l i n t e r v i e w s , t h e s i t u a t i o n s seemed t o c o n s t r a i n t h e s u b j e c t s ' r e s p o n s e s i n t h a t t h e y f o c u s s e d o n t h e i m m e d i a c y o f t h e p r e v i o u s c l a s s w h i c h t e n d e d t o o v e r s h a d o w t h e b r o a d e r t h o u g h t s a n d f e e l i n g s r e g a r d i n g t e a c h i n g i n g e n e r a l . I n one c a s e , we became i n c r e a s i n g l y d i s t r a c t e d t o w a r d t h e end o f t h e i n t e r v i e w b y a n t i c i p a t i n g t h e b e l l r i n g i n g t o s i g n a l t h e b e g i n n i n g o f t h e n e x t p e r i o d . - 26 -T h e r e a f t e r , i n t e r v i e w s were c o n d u c t e d i n c a f e s . T h e s e v e n u e s were more c o n d u c i v e t o b r i n g i n g o u t t h e p e n s i v e n e s s f r u i t f u l i n i n v e s t i g a t i n g c o n c e p t i o n s . P r o b l e m s w i t h b a c k g r o u n d n o i s e were m i n o r a n d d i d n o t hamper t h e i n t e r v i e w p r o c e s s . E a c h i n t e r v i e w b e g a n w i t h t h e q u e s t i o n "What does i t mean t o y o u t o t e a c h m a t h ? " From t h e r e s p o n s e , t h e i n t e r v i e w t r a v e l l e d i t s own p a t h t o c o v e r a r e a s p e r t i n e n t t o t h e i n v e s t i g a t i o n . S u b j e c t s were p r o b e d f o r e x a m p l e s , c l a r i f i c a t i o n , and f u r t h e r t h o u g h t s on t h e t o p i c . T h o s e who d i d n o t have a c l e a r p e r s o n a l c o n c e p t i o n o f t e a c h i n g i n t h e f o r m o f a n i m a g e , m e t a p h o r o r m o t t o were shown t h e d i a g r a m s " V i e w s o f t e a c h i n g a n d l e a r n i n g " ( A p p e n d i x A ) . The i n t e r v i e w s were a u d i o - t a p e d . E a c h l a s t e d f r o m 25 t o 40 m i n u t e s . I t o o k f i e l d n o t e s u s i n g t h e s c h e m a t i c d e v e l o p e d f r o m t h e s e c o n d p i l o t s t u d y and made a d d i t i o n a l n o t e s a f t e r e a c h i n t e r v i e w . A p p e n d i x C shows how t h e s c h e m a t i c was u s e d . The d a t a p r o c e s s i n g and a n a l y s i s . I t r a n s c r i b e d t h e t a p e o f e a c h i n t e r v i e w . Pseudonyms were g i v e n t o e a c h t e a c h e r c a n d i d a t e . One t a p e was n o t u s a b l e as t h e s u b j e c t h a d i n a d v e r t e n t l y p u l l e d t h e m i c r o p h o n e o u t o f t h e r e c o r d e r . T h i s o c c u r r e d w h i l e he was a d j u s t i n g h i s p o s i t i o n . The m i c r o p h o n e a p p a r e n t l y h a d moved j u s t enough t o d i s c o n n e c t b u t n o t enough t o f a l l o u t , so i t s d i s l o c a t i o n was n o t n o t i c e d u n t i l s e v e r a l weeks l a t e r , d u r i n g t r a n s c r i p t i o n . S i n c e t h e p u r p o s e o f t h e s t u d y was t o l o o k a t s n a p s h o t s o f i n d i v i d u a l s ' c o n c e p t i o n s t a k e n a t t h e same p o i n t d u r i n g t h e i r y e a r -l o n g p r o g r a m , I d e c i d e d n o t t o a r r a n g e a s u b s t i t u t e i n t e r v i e w . T h i s - 27 -l e f t s i x c a s e s f o r a n a l y s i s . A p p e n d i x D p r o v i d e s a s a m p l e e x c e r p t f r o m one o f t h e i n t e r v i e w s . The t r a n s c r i p t s were a n a l y z e d u s i n g t h e t e c h n i q u e o f p a t t e r n c o d i n g ( M i l e s a n d Huberman, 1 9 9 4 ) . I r e a d e a c h t w i c e . Then c o d e s c o r r e s p o n d i n g t o c a t e g o r i e s d e a l i n g w i t h m a t h e m a t i c s , i t s t e a c h i n g a n d / o r i t s l e a r n i n g were w r i t t e n b e s i d e t h e a p p r o p r i a t e " c h u n k " ( M i l e s & H u b e r m a n , 1994, p 56) i n t h e e n t i r e i n t e r v i e w . As I g e n e r a t e d t h e s e , I a l s o w r o t e memos; t h e s e c o r r e s p o n d e d t o code n o t e s a n d t h e o r e t i c a l n o t e s ( S t r a u s s a n d C o r b i n , 1 9 9 0 ) . E a c h c o d e d e n t r y on t h e t r a n s c r i p t was t h e n t r a n s f e r r e d t o a s e p a r a t e i n d e x c a r d w i t h a l l t h e i n f o r m a t i o n n e c e s s a r y t o r e - l o c a t e i t i n t h e raw d a t a . E a c h t r a n s c r i p t g e n e r a t e d a p p r o x i m a t e l y 40 t o 60 c a r d s . These c a r d s were t h e n s o r t e d b y c a t e g o r y . F o r e a c h p e r s o n , I w r o t e a summary o f t h e i d e a s c o n t a i n e d o n t h e c a r d s f o r e a c h c a t e g o r y . To i n c r e a s e t h e t r u s t w o r t h i n e s s o f t h e d a t a , I b e g a n t h e p r o c e s s o f c o n d u c t i n g member c h e c k s ( L i n c o l n & Guba, 1985) o f t h e s e s u m m a r i e s . I was a b l e t o l o c a t e f o u r o f t h e s i x s u b j e c t s a n d s e n t e a c h a c o p y o f t h e f i r s t d r a f t o f my summary o f h i s / h e r i n t e r v i e w . E a c h p e r s o n was a s k e d t o r e a d i t and t e l l me w h e t h e r I h a d c a p t u r e d h i s / h e r t h o u g h t s f a i r l y a n d a c c u r a t e l y . T h r e e were r e t u r n e d . T h e i r w r i t t e n comments i n d i c a t e d no c h a n g e s were n e c e s s a r y . Two s u g g e s t e d i n c l u d i n g one more p o i n t ; t h e s e were i n c o r p o r a t e d . A d e s c r i p t i o n o f e a c h p e r s o n ' s c o n c e p t i o n s was w r i t t e n w i t h s u p p o r t i n g e v i d e n c e f r o m t h e d a t a . T h e s e d e s c r i p t i o n s c o n s t i t u t e c h a p t e r 4 . A n a l y z i n g components w i t h i n e a c h c a s e gave r i s e t o - 28 -i n t e r e s t i n g c o m p a r i s o n s a c r o s s t h e c a s e s , i n c l u d e d i n c h a p t e r 5 . T h e s e c o m p a r i s o n s a r e - 29 -C h a p t e r 4 : R e s u l t s I n t h i s c h a p t e r , I d e s c r i b e e a c h s u b j e c t ' s a c a d e m i c b a c k g r o u n d , c o n c e p t i o n o f m a t h e m a t i c s , and v i e w o f t e a c h i n g m a t h e m a t i c s . B e c a u s e o f t h e s e m i - s t r u c t u r e d a p p r o a c h o f t h e i n t e r v i e w s , t h e d e s c r i p t i o n s f o r e a c h c a s e v a r y s l i g h t l y , a l l o w i n g f o r t h e d i f f e r e n t f o c i o f e a c h t e a c h e r c a n d i d a t e t o come t o t h e f o r e as t h e y d i d d u r 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 s . Y v e t t e A c a d e m i c B a c k g r o u n d Y v e t t e h a d a 4 - y e a r B . S c . w i t h a m a j o r i n m a t h e m a t i c s . She i s a " t r a d i t i o n a l " s t u d e n t , i . e . one who went d i r e c t l y i n t o a B . S c . p r o g r a m a f t e r c o m p l e t i n g h i g h s c h o o l , a n d s u b s e q u e n t l y o n t o a B . E d , p r o g r a m . C o n c e p t i o n o f M a t h e m a t i c s Y v e t t e saw m a t h e m a t i c s as a n i n f l e x i b l e c o r e o f c o n t e n t a n d i t s p r o b l e m s as h a v i n g one answer w h i c h c o u l d be a r r i v e d a t t h r o u g h a v a r i e t y o f m e t h o d s . S c i e n c e s a l s o f i t t h i s c a t e g o r y a n d c o n t r a s t e d w i t h a r t s s u b j e c t s . " . . . m a t h i s p o s s i b l y , n o t a s . . . compared t o an a r t s t y p e c o u r s e , i t ' s n o t as f l e x i b l e i n some r e s p e c t s . I mean y o u c a n be f l e x i b l e i n t h e way y o u a p p r o a c h a p r o b l e m . L o t s o f k i d s m i g h t s o l v e p r o b l e m s d i f f e r e n t l y , b u t , t h e r e ' s u s u a l l y one a n s w e r . . . . B u t f o r E n g l i s h , s a y y o u ' r e w r i t i n g an e s s a y , y o u c a n a l w a y s , y o u c a n a p p r o a c h t h i n g s d i f f e r e n t l y , and t h e r e ' s u s u a l l y n o t an a n s w e r . . . " [p 9 ] " . . . I p r e f e r um, t h e s c i e n c e s f o r e x a m p l e , b e c a u s e t h e r e seems t o be a n a n s w e r [ s m a l l l a u g h ] , y o u when y o u ' r e a s k e d a q u e s t i o n , y o u ' r e f i n d i n g a n a n s w e r . . . . I do l i k e t h e i d e a t h a t , h e r e ' s a q u e s t i o n , t h e r e ' s a n a n s w e r . T r y and f i n d i t . " [p 5-6] - 30 -Y v e t t e h a d a p o s i t i v e d i s p o s i t i o n t o w a r d s m a t h e m a t i c s . She e n j o y e d i t a g r e a t d e a l d u r i n g h i g h s c h o o l and a t t r i b u t e s t h a t t o one p a r t i c u l a r t e a c h e r . She a l s o e n j o y e d t h e m a t h e m a t i c s c o u r s e s she t o o k i n u n i v e r s i t y , " . . . b u t I s t a r t e d t o n o t see t h e r e l e v a n c e i n them a n y m o r e , so I g u e s s t h a t ' s where I k i n d o f g o t l o s t . " [p 5] H e r e n j o y m e n t o f t h e s u b j e c t came f r o m t h e c h a l l e n g e she e x p e r i e n c e d i n s o l v i n g p r o b l e m s a n d s a t i s f a c t i o n she f o u n d i n g e t t i n g t h e " r i g h t a n s w e r . " " B u t I a l s o l i k e d , math was c h a l l e n g i n g , y e t I was a b l e t o s u c c e e d a t i t a n d I f o u n d t h a t t h a t was s a t i s f y i n g f o r me, t o know t h a t t h i s was, um, y o u ' r e g i v e n a p r o b l e m and i t ' s h a r d , a n d I , I was c h a l l e n g e d a n d I WANTED t o f i g u r e i t o u t . . . . f o r me, I e n j o y e d t h e c h a l l e n g e , o f math and um, I f e l t a s a t i s f a c t i o n t o g e t a n a n s w e r a t t h e e n d . " [p 5] V i e w o f T e a c h i n g M a t h e m a t i c s To Y v e t t e , t e a c h i n g m a t h e m a t i c s meant p r e s e n t i n g c o n t e n t u s i n g a s e t o f e f f e c t i v e p r e s e n t a t i o n s k i l l s . T h e s e s k i l l s n e e d e d t o accommodate t h e d i f f e r e n t s e n s o r y m o d a l i t i e s t h a t d e f i n e d i f f e r e n t l e a r n i n g s t y l e s . " . . . [ t e a c h i n g ] b e g i n s w i t h i n t r o d u c i n g new c o n c e p t s t o s t u d e n t s , um, t e a c h i n g s o m e t h i n g o r i n t r o d u c i n g s o m e t h i n g t h a t ' s d i f f e r e n t t o t h e m , t h a t t h e y ' v e n e v e r e x p e r i e n c e d b e f o r e , t h a t maybe t h e y p o s s i b l y h a v e , b u t t e a c h i n g i t i n a d i f f e r e n t w a y . " [p 1] " I t h i n k t h a t y o u s h o u l d t r y and have a v a r i e t y o f , o f , um, m e t h o d s , s o , t o l e a r n math b e s t . I t h i n k t h a t some s t u d e n t s u n d e r s t a n d b y b e i n g shown how t o do s o m e t h i n g , um, b o t h , so v i s u a l , some s o r t o f v i s u a l p r e s e n t a t i o n , um, o r a l l y , e x p l a i n , t r y i n g t o e x p l a i n t h i n g s i n d i f f e r e n t w a y s . . . . I t h i n k t h a t t r y i n g t o have a v a r i e t y b e c a u s e n o t a l l t h e s t u d e n t s a r e gonna l e a r n t h e same w a y . . . " [p 2] She b e l i e v e d t h e s t u d e n t was r e s p o n s i b l e f o r h i s own l e a r n i n g . "My r o l e w o u l d be t o , um, i n t r o d u c e t h e new m a t e r i a l , b u t t h e n t o um, k i n d o f l i k e a , um, a manager o r a f a c i l i t a t o r , someone who f a c i l i t a t e s t h e i r l e a r n i n g , so a l t h o u g h I may n o t be r e s p o n s i b l e f o r e v e r y i n d i v i d u a l ' s l e a r n i n g . . . I c a n , um, t r y t o s p a r k s o m e t h i n g i n them t o have them l e a r n . Um, so my r o l e w o u l d be a f a c i l i t a t o r . " [p 2] - 31 -H e r t e a c h i n g p e r s p e c t i v e c a r r i e d o v e r t o a l l s u b j e c t s . She d e s c r i b e d s i m i l a r i t i e s i n t e a c h i n g m a t h e m a t i c s a n d E n g l i s h . " I t h i n k t h a t t h e r e ' s a l o t o f s i m i l a r i t i e s , j u s t i n t e a c h i n g a n y t h i n g b e c a u s e y o u , um, i t ' s u s u a l l y y o u ' r e t e a c h i n g s o m e t h i n g t h a t ' s new t o t h e m , and y o u want them t o p r a c t i c e i t , h a v e some g u i d e d p r a c t i c e maybe, a n d , um, work on i t i n d i v i d u a l l y , work o n i t maybe i n g r o u p s . I ' m , e v e n i n E n g l i s h , s a y E n g l i s h a n d m a t h , y o u m i g h t t e a c h them s o m e t h i n g new, o r r e a d a new b o o k , b u t t h e n work o n i t t o g e t h e r and d i s c u s s i t , j u s t t h e same way y o u w o u l d d i s c u s s a math p r o b l e m o r work on i t t o g e t h e r . . . . I t h i n k t h a t j u s t b y what I remember, o f b e i n g i n t h o s e k i n d o f [ E n g l i s h ] c l a s s e s a l o t o f t h i n g s ARE [ s i c ] t h e same, y o u l i s t e n t o t h e t e a c h e r d e m o n s t r a t e s o m e t h i n g new, y o u work on i t w i t h o t h e r s t u d e n t s , y o u d i s c u s s i t . . . . T h e same b a s i c s . " [p 9] S h o w i n g t h e r e l e v a n c e o f a t o p i c t o s t u d e n t s was i m p o r t a n t t o Y v e t t e . I f a t o p i c c o u l d be made t o seem r e l e v a n t , t h e n s t u d e n t s w o u l d be more m o t i v a t e d t o l e a r n and some o f t h e d i s c i p l i n e p r o b l e m s w o u l d d i s a p p e a r . Y v e t t e c i t e d t h e i m p o r t a n c e o f r e l e v a n c e t o u n d e r s t a n d i n g i n h e r own m a t h h i s t o r y : " T h e r e were some math c o u r s e s i n u n i v e r s i t y t h a t . . . I s t a r t e d t o n o t see t h e r e l e v a n c e i n them any m o r e , so I g u e s s t h a t ' s where I k i n d o f g o t l o s t . " [p 5] She m e a s u r e d h e r s e l f u s i n g t h i s s t a n d a r d . " I t h i n k t h a t t h e r e a r e t i m e s where I c o u l d have more o f t e n showed t h e r e l e v a n c e , o r showed, um, um, j u s t b e e n more e n c o u r a g i n g p e r h a p s , b u t I have e s p e c i a l l y l a t e l y t r i e d r e a l l y h a r d t o , t o show when t h i n g s a r e u s e f u l . " [p 4] Y v e t t e h a d n o t d e v e l o p e d a m o t t o o r m e t a p h o r f o r t e a c h i n g a n d t h u s I showed h e r t h e " V i e w s o f t e a c h i n g and l e a r n i n g . " She c h o s e #2, t h e g u i d e a n d t h e t r a v e l l e r , as most r e p r e s e n t a t i v e o f h e r c o n c e p t i o n o f t e a c h i n g . " . . . I t h i n k 2 ' s a good e x a m p l e ' c a u s e i t ' s t h e g u i d e a n d t h e t r a v e l l e r a n d t h e g u i d e i s , um, o b v i o u s l y g u i d i n g t h e t r a v e l l e r [ l a u g h t e r ] a l o n g a n d s h o w i n g them d i f f e r e n t t h i n g s . . . . t h e p e r s o n s t a n d i n g was t h e g u i d e and t h e , t h e f e l l o w l y i n g down was t h e t r a v e l l e r . Um, j u s t ' c a u s e i t k i n d o f l o o k s l i k e t h e y ' r e i n f r o n t , t h e y ' r e p o i n t i n g e v e r y t h i n g o u t , b u t t h e t r a v e l l e r ' s n o t j u s t a n i n n o c e n t b y s t a n d e r , so t o s p e a k . The t r a v e l l e r i s - 32 -a c t i v e l y i n v o l v e d . . . . a n d I t h i n k t h a t r e l a t e s t o t e a c h i n g b e c a u s e as a t e a c h e r y o u c a n g u i d e t h e s t u d e n t s and t h a t ' s what y o u ' r e t h e r e t o d o , g u i d e them and show them d i f f e r e n t t h i n g s , b u t t h e y a r e a c t i v e l y a p a r t o f i t i n t h i s e x p l o r i n g i d e a . . . i n t h e c l a s s r o o m , I s u p p o s e , y o u ' r e e x p l o r i n g m a t h e m a t i c s . " [p 12] C h a r a c t e r i s t i c s o f a good math t e a c h e r . Y v e t t e t o u c h e d u p o n f o u r q u a l i t i e s she t h o u g h t were i m p o r t a n t f o r a g o o d t e a c h e r o f m a t h e m a t i c s . T h e s e were a d e s i r e t o t e a c h m a t h , t h e a b i l i t y t o h a v e a g o o d r a p p o r t w i t h s t u d e n t s , t h e k n o w l e d g e t o e x p l a i n c o n c e p t s , a n d good p r e s e n t a t i o n s k i l l s . " I t h i n k y o u have t o want t o t e a c h m a t h . . . " [p 6] " [ A g o o d math t e a c h e r i s s]6meone who c a n , I g u e s s , m o t i v a t e t h e s e s t u d e n t s t o l e a r n , b u t , um, a t t h e same t i m e i t has t o be someone, um as p a r t o f t h a t , t h e p e r s o n has t o be a b l e t o r e l a t e w e l l t o t h e s t u d e n t s , have a good r a p p o r t , um, so t h a t t h e s t u d e n t s want t o l i s t e n a n d a r e , a r e a c t i v e l y i n v o l v e d . U m . . . j u s t t h i n g s l i k e i f y o u mumble o r i f y o u d o n ' t e x p l a i n t h i n g s c a r e f u l l y , um, t h a t w o u l d make f o r um, a d i f f i c u l t t e a c h e r t o u n d e r s t a n d . . . " [p 7] " . . . Y o u n e e d t o know how t o e x p l a i n t h i n g s , e l s e , um, e l s e I d o n ' t know how y o u c o u l d t e a c h i t . . . . I f y o u d o n ' t know what y o u ' r e t e a c h i n g , t h e n when k i d s come up w i t h t r i c k y q u e s t i o n s l i k e t h e y d o , y o u ' r e n o t gonna be a b l e t o e x p l a i n i t , a n d y o u r , y o u r e x p l a n a t i o n s f o r t h i n g s a r e n ' t gonna make t o o much s e n s e t o t h e m ! " [p 7] When a s k e d t o e l a b o r a t e on t h e s e a s p e c t s , she f o c u s s e d a g a i n o n what she b e l i e v e s t e a c h i n g t o be a l l a b o u t - c l e a r p o r t r a y a l o f c o n c e p t s . " . . . i f I g r a d u a t e d f r o m h i g h s c h o o l a n d I d i d w e l l i n m a t h , a n d I know, y o u know, a l l t h e math G r a d e 8 t h r o u g h 12, a n d , i t ' s p o s s i b l e t h a t I c o u l d be a good math t e a c h e r i f i t ' s j u s t n a t u r a l i n s t i n c t t o , um, y o u have t h e c o m m u n i c a t i o n s k i l l s m a y b e . . . . [To be] a g o o d t e a c h e r . Um, good c o m m u n i c a t i o n s k i l l s , um, someone y o u c a n u n d e r s t a n d , s p e a k s c l e a r l y , um, t r i e s h a r d t o h a v e t h e s t u d e n t s l e a r n , t r i e s t o d e m o n s t r a t e t h i n g s i n d i f f e r e n t w a y s , n o t d o i n g i t o n t h e b o a r d b u t a l s o s a y i n g s o m e t h i n g , d o i n g s o m e t h i n g a c t i v e l y . " [p 8] She was a b l e t o e l a b o r a t e o n what c l e a r p r e s e n t a t i o n m e a n t , e . g . s p e a k i n g c l e a r l y , b u t was u n a b l e t o g i v e e x a m p l e s o f how a t e a c h e r " t r i e s h a r d t o have s t u d e n t s l e a r n . " T h i s u n d e r s c o r e s Y v e t t e ' s s t r o n g b e l i e f i n t e a c h i n g as a s e t o f good d e l i v e r y s k i l l s . - 33 -V i e w o f s t u d e n t s . Y v e t t e v i e w e d s t u d e n t s as h a v i n g d i f f e r e n c e s i n a b i l i t i e s a n d s h a p e d h e r g o a l o f t e a c h i n g a r o u n d t h i s . She w o u l d h e l p e a c h s t u d e n t t o a c h i e v e what h e / s h e c o u l d a c h i e v e . "My g o a l , I g u e s s , i s t o , t h a t t h e s t u d e n t s w i l l a c h i e v e , some s o r t o f um, l e v e l h i g h e r t h a n where t h e y ' r e a t . . . . S o , my g o a l c a n ' t be t h e same f o r e v e r y s t u d e n t I d o n ' t t h i n k , b u t , um, d i f f e r e n t s t u d e n t s a c h i e v e a t a d i f f e r e n t l e v e l . " [p 1] She a l s o saw s t u d e n t d i f f e r e n c e s i n d i s p o s i t i o n t o w a r d s m a t h e m a t i c s . Though p r o b l e m - s o l v i n g was a s t i m u l u s f o r h e r , Y v e t t e a c k n o w l e d g e d t h a t n o t a l l s t u d e n t s e n j o y e d t h i s t y p e o f c h a l l e n g e . " . . . I was c h a l l e n g e d and I WANTED t o f i g u r e i t o u t . Now I ' m s u r e a l l s t u d e n t s a r e n ' t l i k e t h a t , b u t f o r me, I e n j o y e d t h e c h a l l e n g e . . . t h e r e a r e d e f i n i t e l y i s some t h a t I c a n t e l l t h i n k t h e same way, t h a t t h e y e n j o y a c h a l l e n g e a n d t h e y , t h e y k i n d o f l i k e t h e i d e a t h a t , um, t h i s i s a , y o u know, a t r i c k y q u e s t i o n , l e t ' s t r y a n d f i n d t h e a n s w e r . " [p 5-6] D u r i n g h e r p r a c t i c u m , she became aware o f o t h e r f a c t o r s a n d p e o p l e i n f l u e n c i n g t h e l i v e s o f t h e t e e n a g e r s . T h i s i n c r e a s e d h e r s y m p a t h y t o w a r d s t h e s t u d e n t s , b u t d i d n o t a f f e c t t h e way she t a u g h t . " . . . y o u r e a l i z e t h a t t h e s t u d e n t has so many o u t s i d e f a c t o r s t h a t y o u d o n ' t know a b o u t , o r y o u may o r may n o t know a b o u t , a n d t h a t r e a l l y a f f e c t s a s t u d e n t ' s l i f e . . . . I g u e s s i t d o e s n ' t r e a l l y c h a n g e t h e way I t e a c h . I t j u s t c h a n g e s t h e way I , I t r y t o u n d e r s t a n d t h e m . . . S o , b u t I d o n ' t know i f my t e a c h i n g , my t e a c h i n g s t y l e , I d o n ' t t h i n k , has b e e n a f f e c t e d b y t h a t . " [p 13] C o n c e p t i o n o f l e a r n i n g m a t h . When a s k e d how math i s b e s t l e a r n t , Y v e t t e f o c u s s e d a g a i n o n d i f f e r e n t l e a r n i n g s t y l e s . She m e n t i o n e d t h e n e e d t o accommodate d i f f e r e n t s e n s o r y m o d a l i t i e s t o h e l p i n d i v i d u a l s t u d e n t s l e a r n , a n d how s t u d e n t p r a c t i c e o f math q u e s t i o n s was i m p o r t a n t . She c l a i m e d " t h e r i g h t m i n d s e t " [p 9] was i m p o r t a n t f o r s u c c e s s i n m a t h . She h a d d i f f i c u l t y e l a b o r a t i n g o n t h i s c o n c e p t i t ' s more - 34 -s c i e n t i f i c , I d o n ' t know! I t ' s more m a t h e m a t i c a l ! " [p 9] L a t e r , she d e s c r i b e d i t as a c t i v e i n v o l v e m e n t i n the p r o b l e m . " . . . t h e y ge t down t o work, t h e y ' r e , t h e y ' r e t h i n k i n g about a p r o b l e m , t h e y r e a d i t , and t h e y , t h e y ask q u e s t i o n s t h a t show t h a t t h e y ' r e a c t i v e l y , um, i n v o l v e d i n what t h e y ' r e d o i n g , whereas someone e l s e p r o b a b l y w o u l d n ' t even ask a q u e s t i o n i f t h e y ' r e n o t a c t i v e l y i n v o l v e d i n what t h e y ' r e d o i n g . So , a mind s e t would be an a c t i v e m i n d . Someone t h a t ' s r e a d y t o t h i n k . " [p 10] I t i s i n t e r e s t i n g t o no te t h a t w h i l e Y v e t t e c l a i m e d t h a t t h e t e a c h i n g o f math was s i m i l a r t o the t e a c h i n g o f o t h e r s u b j e c t s , l e a r n i n g math r e q u i r e d a " r i g h t , " m a t h e m a t i c a l mind s e t . Shannon A c a d e m i c and P e r s o n a l B a c k g r o u n d Shannon had an Honours B . S c . ( 4 - y e a r ) i n m a t h e m a t i c s . B e f o r e e n t e r i n g t h e B . E d , p r o g r a m she t u t o r e d p r i v a t e l y . She had a l s o worked f u l l - t i m e i n a r e s t a u r a n t and s u b s e q u e n t l y became t h e manager . A f t e r c o m p l e t i n g h e r t e a c h e r p r e p a r a t i o n p r o g r a m , Shannon' s p l a n s were t o r e t u r n t o h e r home p r o v i n c e and f i n d a t e a c h i n g j o b o r h e l p h e r h u s b a n d manage a r e s t a u r a n t . C o n c e p t i o n o f M a t h e m a t i c s Shannon d e s c r i b e d mathemat ic s as h a v i n g b o t h t h e o r e t i c a l and p r a c t i c a l components . She c l e a r l y f a v o u r e d the p r a c t i c a l a s p e c t . "In u n i v e r s i t y , I h a t e d t h e o r y . . . . I l i k e a p p l i e d math . I d o n ' t l i k e t h e o r e t i c a l m a t h . " [p 5] To d i s t i n g u i s h between t h e two, Shannon c h a r a c t e r i z e d them: " T h e o r y t o me i s m e m o r i z a t i o n . When I had an exam i n my, i n my c o u r s e s where i t was t h e o r y , I had t o memorize a l l t h e p r o o f s , and t o me when y o u a p p l y , you l e a r n how t o USE [ s i c ] t h o s e p r o o f s . . . . m y f o u r t h y e a r c o u r s e i n math, I r e a l l y e n j o y e d . I t was, a whole t h i n g was a b o u t , u h , s i m u l a t i o n ? And g o i n g t o t h e GM [ G e n e r a l M o t o r s ] p l a n t ? And w a t c h i n g how the r o b o t s . . . m a k e c a r s . . . t r y i n g t o f i n d a sys tem t h a t would put more o u t p u t o f - 35 -c a r s . . . . I t h i n k t h a t ' s e x c e l l e n t . . . . I t h i n k t h a t ' s more i m p o r t a n t . " [p 5-6] F o r S h a n n o n , t h e r e was a b l a c k a n d w h i t e n e s s t o m a t h e m a t i c s . " I l i k e s e e i n g a r i g h t o r wrong a n s w e r . A n d y o u know math i s one o r t h e o t h e r . " [p 7] S h a n n o n saw a d i s t i n c t d i f f e r e n c e b e t w e e n t h e m a t h e m a t i c s i n u n i v e r s i t y a n d t h a t t a u g h t i n h i g h s c h o o l . " [ W ] h a t y o u l e a r n i n u n i v e r s i t y i s t o t a l l y d i f f e r e n t t h a n what y o u l e a r n i n h i g h s c h o o l . " [p 10] V i e w o f T e a c h i n g M a t h e m a t i c s The s t u d e n t - t e a c h e r r e l a t i o n s h i p was p r i m a r y f o r S h a n n o n . H e r g o a l was t o make a d i f f e r e n c e t o e a c h s t u d e n t , p a r t i c u l a r l y l o w e r a c h i e v e r s . " . . . I wanna make a d i f f e r e n c e t o e a c h s t u d e n t a n d e s p e c i a l l y t h e ones t h a t a r e n ' t d o i n g w e l l . " [p 20] " I want t o make a d i f f e r e n c e t o e a c h o n e . . . t h e ones t h a t a r e p r o b l e m c h i l d r e n a r e t h e ones I ' m , more c o n c e r n e d a b o u t ? " [p 21] F o r h e r , s u c c e s s f u l t e a c h i n g meant b e i n g a b l e t o " c o v e r e v e r y s t u d e n t . " [p 11] She w o r r i e d t h a t when she h a d h e r own c l a s s r o o m , she w o u l d n ' t f i n d enough t i m e t o a p p r o a c h e a c h o n e , a n d some w o u l d n o t g e t a t t e n t i o n f r o m h e r . She t r i e d e s t a b l i s h i n g i n d i v i d u a l r e l a t i o n s h i p s w i t h e a c h s t u d e n t d u r i n g h e r p r a c t i c u m b u t f o u n d t h i s t a s k d a u n t i n g . S H : [ I n t h e b e g i n n i n g o f t h e p r a c t i c u m ] I t h o u g h t . . . I h a d t o do e v e r y t h i n g . I r e a l i z e d y o u c a n ' t do e v e r y t h i n g . You c a n ' t e x p e c t y o u r s e l f t o do e v e r y t h i n g . I : What w o u l d be e x a m p l e s o f t h e e v e r y t h i n g ? S H : Of e v e r y t h i n g ? M e a n i n g h e l p e v e r y s t u d e n t i n t h a t c l a s s , go t o e v e r y i n d i v i d u a l s t u d e n t . [p 11] When p r o b e d f o r what she w a n t e d t o a c c o m p l i s h w i t h e a c h s t u d e n t , h e r r e s p o n s e was " I want them t o p a s s . . . " [ p 19] E l a b o r a t i n g o n one p a r t i c u l a r s t u d e n t who h a d p o o r a t t e n d a n c e and d i s c i p l i n e p r o b l e m s , she s a i d , " I want t o see h i m p a s s . I want t o see h i m p a s s t h e c o u r s e , ' c a u s e I w o u l d f e e l l i k e I ' m t h e f a i l u r e , w h i c h maybe i s w r o n g . " [p 19] - 36 -She a l s o gave two e x a m p l e s o f s t u d e n t s who were " p r o b l e m k i d s " b u t whom she saw as h a v i n g p o t e n t i a l t o s u c c e e d i n s c h o o l . She a t t e m p t e d t o s t r i k e up a r e l a t i o n s h i p w i t h e a c h , [p 19-20] F o r S h a n n o n , n o t o n l y was more math l e a r n t b y t h e s t u d e n t when she w o r k e d w i t h them i n d i v i d u a l l y , b u t a f i r m e r r e l a t i o n s h i p c o u l d be b u i l t i n t h i s m a n n e r . " I g e t a l o t more done w i t h o n e - t o - o n e . A n d t h e y u n d e r s t a n d i t more t h a n i f I do i t i n f r o n t o f t h e w h o l e c l a s s . A n d i t ' s i m p o s s i b l e t o do t h a t , u n f o r t u n a t e l y . . . I w i s h I c o u l d do i t t o e v e r y s t u d e n t . . . I t ' s j u s t t h e way t h e y e v e n a c t a n d , l i k e e v e n when I h e l p s t u d e n t s h e r e a t t h e i r s e a t s t h e y ' r e s t - , t h e y ' r e h a p p y when t h e y u n d e r s t a n d i t . . . . y o u d o n ' t have t i m e t o go a r o u n d , [p 9] M o d e l f o r t e a c h i n g . . A n e x e m p l a r f o r Shannon was t h e p r o t a g o n i s t i n S t a n d a n d D e l i v e r , a m o v i e she v i e w e d i n one o f h e r t e a c h e r e d u c a t i o n c o u r s e s . He w o r k e d w i t h h i s s t u d e n t s t h e way she w a n t e d t o work w i t h h e r s a n d he r e p r e s e n t e d a m o d e l f o r h e r t o f o l l o w . She saw a p a r a l l e l b e t w e e n h e r s t u d e n t s a n d h i s i n t h e m o v i e . "So t h i s t e a c h e r [ i n t h e m o v i e ] t o o k i t u p o n h i m s e l f a n d he made math so e n j o y a b l e t h a t t h e k i d s e v e n came i n o n W E E K E N D S ! . . . A n d I u s e d t h a t m o t t o , I l o v e d t h a t m o v i e , and I l o v e d how he c o u l d r e l a t e t o t h o s e k i d s . My G r a d e 1 0 ' s a r e i d e n t i c a l t o t h o s e k i d s . . . . A n d t h a t ' s why I ' m b u d d y - b u d d y w i t h them b e c a u s e HE was a n d he g o t t h r o u g h w i t h them t h a t way. " [p 13] S h a n n o n r e l a t e d e v e n more w i t h t h e m o v i e when t h e r e was a t e a c h e r s ' s t r i k e d u r i n g h e r p r a c t i c u m . T h i s meant c l a s s e s were c a n c e l l e d a n d c o n s e q u e n t l y , some s t u d e n t s w a n t e d e x t r a h e l p o u t s i d e r e g u l a r h o u r s . She saw t h i s a l l i a n c e as e p i t o m i z i n g h e r g o a l o f a i d i n g s t u d e n t s t h r o u g h e s t a b l i s h i n g a p e r s o n a l r e l a t i o n s h i p . " . . . m y G r a d e 1 0 ' s r e m i n d me o f them b e c a u s e one o f t h e s t u d e n t s when we were a l l o w e d t o h e l p [ d u r i n g t h e s t r i k e ] he was h e r e w i t h me ' t i l 6 o ' c l o c k . . . . N o w how many s t u d e n t s w i l l do t h a t ? A n d t h i s k i d i s a 50% a v e r a g e . A n d I f e l t g r e a t . " [p 14] - 37 -S h a n n o n b e l i e v e d a n o n - a u t h o r i t a r i a n r e l a t i o n s h i p e n h a n c e d s t u d e n t l e a r n i n g . I n b e i n g f r i e n d l y , h o w e v e r , she e x p e r i e n c e d p r o b l e m s w i t h c l a s s r o o m c o n t r o l . " I c a n ' t be b u d d y - b u d d y w i t h them, b e c a u s e t h e n t h e y t a k e a d v a n t a g e o f me. So I d o n ' t know where my l i m i t s a r e . . . . B e c a u s e I f i n d when I ' m t h e i r f r i e n d h e l p i n g t h e m , t h e y l e a r n m o r e . A n d t h e n , and t h e n t h e y t a k e a d v a n t a g e somet imes - " [p 14] When she was shown t h e d i a g r a m s i n " V i e w s o f T e a c h i n g a n d L e a r n i n g , " she d e c i d e d none o f them c o n v e y e d h e r c o n c e p t i o n o f t e a c h i n g as c l e a r l y as t h e m o v i e . C o n c e p t i o n o f t e a c h e r . S h a n n o n ' s c o n c e p t i o n o f a t e a c h e r was a n o m n i s c i e n t p e r s o n who was e x p e c t e d t o be so i n f r o n t o f s t u d e n t s and p e e r s . " I s o m e t i m e s w o n d e r , a n d t h a t s h o u l d n ' t come o u t o f me" [p 3] a n d " I d o n ' t l i k e h e s i t a t i n g ' c a u s e I want them t o l e a r n m a t h " [p 6] was how she d e s c r i b e d what h e r b e h a v i o u r i n f r o n t o f s t u d e n t s s h o u l d b e . S h a n n o n b e l i e v e d t h o s e i n p o s i t i o n s o f a u t h o r i t y s h o u l d h a v e a l l t h e a n s w e r s a n d make them a v a i l a b l e t o o t h e r s . She e l a b o r a t e d on t h i s when d i s c u s s i n g what t o t e l l s t u d e n t s who a s k e d why E u c l i d e a n p r o o f s were s t u d i e d : " . . . w h a t ' s t h e r e a s o n t h e y n e e d g e o m e t r y , I d o n ' t know a n a n s w e r . T h i s i s n o t t h e r i g h t a n s w e r , b u t I t e l l them w e l l i n o r d e r t o g r a d u a t e , t o go t o u n i v e r s i t y , y o u n e e d y o u r g r a d e 1 2 . So y o u n e e d g e o m e t r y . T h a t ' s n o t a g o o d enough a n s w e r . I w i s h t h e y a l s o t e a c h us how t o t e l l t h e s t u d e n t s t h e r e a s o n f o r e a c h [ t o p i c ] , w i t h o u t me m a k i n g up s o m e t h i n g ? . . . . I u s e d t o have a t e a c h e r s a y i t ' s b e c a u s e i t makes y o u t h i n k . I s t h a t t h e a n s w e r ? I d o n ' t k n o w . " [p 3] A n d l a t e r i n t h e i n t e r v i e w , S H : B e c a u s e l i k e I s a i d I d o n ' t know t h e r e a s o n why we l e a r n g e o m e t r y . I w i s h I d i d . A n d I w i s h t h e y w o u l d t e l l us m o r e , o f how t o . . . ' c a u s e I know i t ' s my own o p i n i o n a n d I d o n ' t know what t h a t i s f o r g e o m e t r y . . . I : Ok, where do y o u t h i n k y o u c a n f i n d t h a t o u t ? S H : H o p e f u l l y , f r o m u n i - , f r o m p r o f s w i l l t e l l me! [p 6] - 38 -I n t h i s e x c e r p t , Shannon l a i d o u t a h i e r a r c h i c a l p a t h w a y f o r r e l a y i n g i n f o r m a t i o n . The p r o f e s s o r s w o u l d t e l l h e r why g e o m e t r y was t a u g h t so she i n t u r n c o u l d t e l l t h e s t u d e n t s . The i d e a o f t h e t e a c h e r b e i n g a l l - k n o w i n g p u t g r e a t p r e s s u r e o n h e r . She was c o n c e r n e d h e r i g n o r a n c e o f c o n t e n t o r c o n t e n t - r e l a t e d i s s u e s w o u l d c a u s e h e r t o l o s e f a c e w i t h o t h e r s t a f f o n c e she became a c e r t i f i e d t e a c h e r . " Y o u know I , I know I was l u c k y I c o u l d go t o [ s c h o o l a d v i s o r ] . . . a n d s a y I d o n ' t u n d e r s t a n d t h i s q u e s t i o n . . . . B u t when I become my own t e a c h e r c a n I go t o o t h e r t e a c h e r s ? A n d t h e y s a i d y e s . I w o u l d f e e l more w e i r d . I know I ' m a s t u d e n t [ t e a c h e r ] , so I c a n do t h a t , b u t as a t e a c h e r , c a n I do t h a t ? To me, I ' l l l o o k s t u p i d . " [p 8] " . . . I was t e r r i f i e d ' c a u s e I f o r g o t e v e r y t h i n g . . . . T o me I d o n ' t remember t h i s s t u f f t h a t I l e a r n e d i n h i g h s c h o o l and I f e e l s t u p i d . " [p 10] The t e a c h e r was n o t a l l o w e d t o w o n d e r , g u e s s , o r n o t know. T h i s p u t a g r e a t s t r a i n o n h e r as she t r i e d t o a c c o m p l i s h t h i s i d e a l . " [ T h e s t u d e n t s a r e ] c o m i n g h e r e so y o u c a n t a l k t o t h e m , l i s t e n t o t h e m , t e a c h t h e m . . . . Y o u have t o make y o u r s e l f . . . e v e n t h o u g h y o u d o n ' t f e e l w e l l , y o u h a v e t o g e t a c r o s s t h a t t h i s i s i m p o r t a n t , e v e n t h o u g h y o u m i g h t n o t t h i n k i t i s , p e r s o n a l l y , b u t y o u g o t t a g e t t o , t o them t h a t i t i s i m p o r t a n t . " [p 18] C o n s i s t e n t w i t h t h e n o t i o n o f o m n i s c i e n c e , she h e l d i n d i s d a i n t h o s e i n t h e p r o f e s s i o n she p e r c e i v e d t o be n o t a l l - k n o w i n g . "When I h a d an exam i n my, i n my [ u n i v e r s i t y m a t h e m a t i c s ] c o u r s e s where i t was t h e o r y , I h a d t o m e m o r i z e a l l t h e f o r m u l a s , . . . a l l t h e p r o o f s . . . . b u t n o t e v e n t h e p r o f knows t h a t b y h e a r t , a n d y e t t h e y e x p e c t t h e s t u d e n t s t o . I d o n ' t a g r e e w i t h t h a t . " [p 5] R e q u i r e m e n t s t o t e a c h m a t h e m a t i c s . " P a t i e n c e . A l o t o f p a t i e n c e . " [p 9] S h a n n o n ' s r e p l i e s t o t h i s s e c t i o n o f t h e i n t e r v i e w seemed t o be i n f l u e n c e d b y t h e r a m b u n c t i o u s n a t u r e o f h e r s t u d e n t s , a n d b y t h e d i f f i c u l t y she h a d m a k i n g t h e t r a n s i t i o n f r o m s u c c e s s f u l p r i v a t e - 39 -t u t o r i n g t o t e a c h i n g a c l a s s . She d e s c r i b e d t h e j o b : " [ B ] e i n g a t e a c h e r y o u a l w a y s h a v e t o have y o u r w i t s t o y o u r s e l f . You c a n ' t h a v e a d a y where y o u d o n ' t f e e l w e l l . You have t o make y o u r s e l f [ t e a c h ] . " [p 17] T e a c h i n g m e t h o d s . S h a n n o n v o i c e d s t r o n g o p p o s i t i o n a g a i n s t t e a c h i n g f o r u n d e r s t a n d i n g as a d v o c a t e d b y t h e t e a c h e r e d u c a t i o n p r o g r a m . D u r i n g h e r p r a c t i c u m , she n o t i c e d how many s t u d e n t s c o u l d n o t do s i m p l e a r i t h m e t i c w i t h o u t t h e i r c a l c u l a t o r s . She was adamant t h a t t h e y s h o u l d be d r i l l e d o n number f a c t s , h e r r a t i o n a l e b e i n g t h a t was t h e way she h a d l e a r n t them a n d t h e number f a c t s h a d s t u c k i n h e r memory. " . . . t h e s t u d e n t s t h e y c a n ' t e v e n a d d , and m u l t i p l y , a n d s u b t r a c t ; i t ' s d i s g u s t i n g . . . . w h a t was i t when I l e a r n e d how t o a d d a n d s u b t r a c t ? Why d i d i t s t a y w i t h me compared t o t h e s e s t u d e n t s ? . . . . A n d a b o u t m a k i n g c a l c u l a t o r s - t h e y c a n ' t u s e them a n y m o r e . Make them t h i n k a g a i n . Go b a c k i n t i m e . . . . [ f o r ] a d d i n g a n d s u b t r a c t i n g - d r i l l i n g . I ' m s o r r y , I t h i n k t h a t ' s t h e b e s t way. I was d r i l l e d m u l t i p l y i n g . A n d I l e a r n e d i t . " [p 1-2] T e a c h i n g , f o r S h a n n o n , meant d i r e c t i n s t r u c t i o n , t e l l i n g . C o o p e r a t i v e m e t h o d s , as she u n d e r s t o o d f r o m h e r s c h o o l a d v i s o r s , were c o n s i d e r e d " e x t r a " a n d a w a s t e o f t i m e . " . . . y o u ' r e b e i n g t o l d t h a t y o u c a n ' t do c o o p e r a t i v e l e a r n i n g b e c a u s e t h i s c u r r i c u l u m has t o be c o v e r e d b y t h i s t i m e . . . . A n d i t ' s l i k e , I ' m s o r r y y o u o n l y have t h i s amount o f t i m e t o c o v e r t h i s m a t h . I f y o u do c o o p e r a t i v e l e a r n i n g , y o u w o n ' t have t i m e t o c o v e r i t . T h a t ' s what I ' m b e i n g t o l d b y t h e s e g u y s . A n d i f t h e y t e l l y o u t o do i t , b u t t h e n when I a t t e m p t i t , o h y o u ' v e w a s t e d a c l a s s . T h a t w a s n ' t a s . . . a s p r o d u c t i v e as i f y o u t a u g h t a l e s s o n a n d l e t them do t h e w o r k . . . . e v e n t h o u g h t h i s way I know, w i t h c o o p e r a t i v e l e a r n i n g y o u c a n see t h e y r e a l l y u n d e r s t a n d i t . " [p 16] I n c o m p a r i s o n t o t e a c h i n g E n g l i s h , Shannon a s s e r t e d t h a t t e a c h i n g m a t h e m a t i c s was more c o n s t r a i n i n g , b e c a u s e i n E n g l i s h " y o u c a n g e t away w i t h so much m o r e . " [p 15] To e l a b o r a t e , she c i t e d w a t c h i n g a s t u d e n t t e a c h e r o f E n g l i s h show m o v i e s i n c l a s s . Shannon f e l t t r a p p e d b y t h e c u r r i c u l u m w h i c h h a d t o be c o v e r e d , and f e l t t h e o n l y way t o do i t was - 40 -t h r o u g h d i r e c t i n s t r u c t i o n . O t h e r , more s t u d e n t - c e n t r e d t e c h n i q u e s she w i s h e d t o u s e were c o n s i d e r e d t o be e x t r a c u r r i c u l a r a n d s u p e r f l u o u s a c c o r d i n g t o h e r i n t e r p r e t a t i o n o f what h e r s c h o o l a d v i s o r s w a n t e d . H e r g o a l was t o show s t u d e n t s a p p l i c a t i o n s o f m a t h e m a t i c s , h e r f a v o u r i t e a s p e c t o f t h e d i s c i p l i n e . She d e s c r i b e d a n u n d e r g r a d u a t e m a t h c o u r s e she t h o r o u g h l y e n j o y e d when she went t o a l o c a l a u t o m o t i v e p l a n t a n d s t u d i e d t h e o u t p u t o f t h e r o b o t i c s a s s e m b l y l i n e . The t e r m p r o j e c t f o r t h e c o u r s e was t o d e v e l o p a s y s t e m t o i n c r e a s e t h e e f f i c i e n c y o f t h e l i n e . " A n d I t h o u g h t , t h a t i s g r e a t e v e n f o r h i g h s c h o o l . To t a k e t h e k i d s t o t h e G e n e r a l M o t o r s p l a n t t o see why c e r t a i n math i s a p p l i e d , a n d t o u n d e r s t a n d t e c h n o l o g y t h a t ' s c o m i n g o u t . " [p 5] O t h e r I s s u e s D i l e m m a w i t h c o n t e n t . S h a n n o n f e l t v e r y u n c o m f o r t a b l e t h a t she was a u n i v e r s i t y m a t h m a j o r , b u t c o u l d n o t remember t h e j u n i o r h i g h s c h o o l l e v e l m a t h e m a t i c s she was t o t e a c h . " I f e e l s t u p i d . A n d I d o n ' t u n d e r s t a n d t h a t . " [p 10] She b l a m e d t h e t e a c h e r e d u c a t i o n p r o g r a m a t UBC f o r n o t p r e p a r i n g h e r , i . e . n o t r e v i e w i n g more c o n t e n t w i t h t h e t e a c h e r c a n d i d a t e s . She was adamant t h a t c o n t e n t r e v i e w be a component o f t e a c h e r p r e p a r a t i o n . " . . . I t h i n k t h e F a c u l t y o f E d u c a t i o n s h o u l d be t e a c h i n g us math t e a c h e r s , r e v i e w i n g e v e r y t h i n g we have t o t e a c h i n h i g h s c h o o l . . . . I h a d t o do a l l t h e homework, b e c a u s e I d i d n ' t u n d e r s t a n d i t . I n s t e a d o f t e a c h i n g us - , a l o t o f t h e s t u f f we l e a r n e d i n o u r c o n c e n t r a t i o n c o u r s e s was u s e l e s s . " [p 8] D i l e m m a w i t h s t u d e n t r e l a t i o n s h i p s . - 41 -S h a n n o n w a n t e d t o be a f r i e n d t o t h e s t u d e n t s , b u t was t o l d b y h e r s c h o o l a d v i s o r s t h a t as a t e a c h e r , she h a d t o be " m e a n . " S i n c e h e r v i e w o f t h e t e a c h e r - s t u d e n t r e l a t i o n s h i p l e a n e d h e a v i l y t o w a r d s t u d e n t s , she a l l o w e d them t o have a g r e a t d e a l o f c o n t r o l o f t h e f l o w o f t h e l e s s o n a n d t h e c l a s s . T h e n , b e i n g y o u n g a d o l e s c e n t s , t h e y w o u l d t a k e a d v a n t a g e o f t h i s , a n d c l a s s e s w o u l d become u n m a n a g e a b l e . Shannon d i d h a v e p r o b l e m s w i t h c l a s s management, " . . . a n d I d o n ' t know where my m i d - p o i n t i s . " [p 14] She was t r o u b l e d b y n o t f i n d i n g a w o r k a b l e b a l a n c e b e t w e e n b e i n g a f r i e n d t o t h e s t u d e n t s and b e i n g t h e t e a c h e r , t h e a u t h o r i t y f i g u r e . P e r s p e c t i v e o n a u t h o r i t y . S h a n n o n h a d a v e r y s t r o n g b e l i e f t h a t t h o s e i n a u t h o r i t y s h o u l d know a l l t h e a n s w e r s . F o r S h a n n o n , t h e r e were r i g h t a n s w e r s a n d w r o n g a n s w e r s , a n d " t h e y " who knew t h e r i g h t a n s w e r s s h o u l d t e l l " u s " ( t h e o t h e r s ) . T h i s a t t i t u d e r e s o n a t e s w i t h P e r r y ' s (1970) d e s c r i p t i o n o f d u a l i t y i n h i s scheme o f i n t e l l e c t u a l a n d e t h i c a l d e v e l o p m e n t . T h i s p l a y e d o u t f o r Shannon o n s e v e r a l f r o n t s . F i r s t , t h e p r o f e s s o r s a t UBC were t h e a u t h o r i t i e s i n t h e t e a c h e r p r e p a r a t i o n p r o g r a m a n d t h e y were t h e o n e s who s h o u l d t e l l t h e p r e - s e r v i c e t e a c h e r s a l l t h e y n e e d e d t o know. S e c o n d , when t h e p r e - s e r v i c e t e a c h e r s were i n t h e c l a s s r o o m i n f r o n t o f s t u d e n t s , t h e y i n t u r n became t h e a u t h o r i t i e s f o r t h e s e y o u n g e r o n e s . T h e r e f o r e t h e y n e e d e d t o know a l l t h e a n s w e r s t o s t u d e n t s ' q u e s t i o n s . T h i s c a u s e d Shannon g r e a t w o r r y . E v e n h e r c o n v i c t i o n t o h e r g o a l i n t e a c h i n g , " . . . t o make a d i f f e r e n c e t o e a c h o n e " [p 2 1 ] , c r e a t e d d i s s o n a n c e w i t h t h i s f u n d a m e n t a l p e r s p e c t i v e . " A n d I , I d o n ' t know i f t h a t ' s r i g h t o r w r o n g . I d o n ' t - 42 -k n o w . . . I d o n ' t know i f t h a t ' s c o r r e c t t o be t h i n k i n g t h a t way o r n o t . " [p 21] U n d e r l y i n g h e r t e a c h i n g p h i l o s o p h y , she h a d an e x t r e m e l y s t r o n g b e l i e f i n a r i g h t v e r s u s wrong way o f t h i n k i n g and was s e a r c h i n g f o r t h e c o r r e c t o n e . Doug A c a d e m i c B a c k g r o u n d Doug was t h e o n l y s u b j e c t w i t h a l i b e r a l a r t s d e g r e e ( B . A . ) . H i s m a t h s p e c i a l t y q u a l i f i e d h i m f o r t h e B . E d , p r o g r a m i n s e c o n d a r y m a t h e m a t i c s . C o n c e p t i o n o f M a t h e m a t i c s Doug b e l i e v e d m a t h e m a t i c a l k n o w l e d g e was c r e a t e d t h r o u g h n e g o t i a t i o n a n d a g r e e m e n t . C o n v e n i e n t and u s e f u l d e f i n i t i o n s were f o r m e d b y s c h o l a r s i n t h e f i e l d a n d e v o l v e d c o n t i n u a l l y as n e e d a r o s e . He c i t e d t h e f u n c t i o n as an e x a m p l e . " I mean math i s b a s e d on d e f i n i t i o n , and I t h i n k a h , a n i m p o r t a n t p a r t i s t o show [ s t u d e n t s ] t h a t , t h a t we c r e a t e t h e s e d e f i n i t i o n s as p e o p l e . We make up t h i s , h e r e ' s a d e f i n i t i o n , r i g h t ? A n d a h , we c o u l d j u s t as e a s i l y s a y t h e d e f i n i t i o n i s t h i s . . . . a f u n c t i o n u s e d t o have more t h a n one p o s s i b i l i t y . F o r i n s t a n c e t h e s q u a r e r o o t . . . w a s a t one t i m e c o n s i d e r e d a f u n c t i o n , r i g h t ? . . . n o w i n o r d e r t o make i t a f u n c t i o n we s a y t h e p o s i t i v e s q u a r e r o o t . S o . . . i t ' s , c h a n g e d o v e r t i m e b e c a u s e i t became a more c o n v e n i e n t d e f i n i t i o n , r i g h t ? So we d e f i n e t h e s e t h i n g s . . . . I t ' s c o n v e n i e n t t o l o c a t e t h e c e n t r e o f t h e c i r c l e a t t h e c e n t r e p o s i t i o n a t (0,0) b u t i f , I mean i t c o u l d j u s t as e a s i l y have b e e n (1,1) o r s o m e t h i n g . We j u s t c h o o s e w h a t e v e r i t s u i t s us a t t h e t i m e . " [p 3] Doug e n j o y e d t h e s t r u c t u r e and o r d e r l i n e s s o f m a t h e m a t i c a l c o n t e n t . " [ T ] h e r e ' s a c e r t a i n c l e a n l i n e s s t o m a t h , o r p u r i t y t o m a t h , t o t h e , t o t h e o r d e r a n d s t r u c t u r e and e v e r y t h i n g , and how i t ' s l a i d o u t ? You know, i t ' s v e r y c l e a n t h a t way? A n d I l i k e t h a t ? V e r y l o g i c a l ? I t h i n k t h a t , y o u know l i k e , t h a t math i s b e a u t i f u l i d e a , o f t h e r a t i o n a l i t y , and s i m p l i c i t y I t h i n k I r e a l l y l i k e . " [p 9] - 43 -The common p e r c e p t i o n o f m a t h e m a t i c s as a t o o l o f s c i e n c e a n d e n g i n e e r i n g t r o u b l e d h i m . He b e l i e v e d i t masked p u r e m a t h e m a t i c s as a d i s c i p l i n e u n t o i t s e l f . I n a b r o a d e r c o n t e x t , he f e a r e d s o c i e t y as a w h o l e was s l o w l y l o s i n g i t s a e s t h e t i c s i d e . . . . [ M ] a t h e m a t i c s i s a , i s o f t e n s e e n n o t as a d i s c i p l i n e , n e c e s s a r i l y b y i t s e l f . Or as a , j u s t a way o f t h i n k i n g , b u t i t ' s s e e n as a p a r t o f s c i e n c e , a n d , and e n g i n e e r i n g . . . . a l l o u r ways o f l o o k i n g a t t h e w o r l d , o u r , o u r p h i l o s o p h y , o u r s o c i a l s t u d i e s , o u r e c o n o m i c s , p s y c h o l o g y , t h e y ' v e a l l , t h e y ' v e a l l s t a r t e d t o c a l l t h e m s e l v e s s c i e n c e s , l i k e s o f t s c i e n c e s , r i g h t ? . . . I t h i n k t h a t may be c h e a p e n i n g e a c h o f t h o s e o t h e r a r e a s , y o u k n o w ? . . . I t h i n k we l o s e s i g h t o f o t h e r ways o f l o o k i n g a t p s y c h o l o g y . . . . I d o n ' t n e c e s s a r i l y b e l i e v e t h a t i t i s a s c i e n c e . Or e c o n o m i c s , as a s c i e n c e , r i g h t ? . . . T h i n g s become v a l i d when t h e y ' r e s c i e n t i f i c , o r t h e y become v a l i d when we c a n b a c k them up w i t h numbers a n d so o n . I d o n ' t n e c e s s a r i l y a g r e e w i t h a n y o f t h a t , a n d y e t , t h e r e ' s a d a n g e r i n t e a c h i n g math i s t h a t i t ' s s e e n as a b i g p a r t o f t h a t s i d e o f t h e w o r l d . " [p 13-15] V i e w o f T e a c h i n g M a t h e m a t i c s To Doug, t e a c h i n g m a t h e m a t i c s meant t e a c h i n g r e a s o n i n g , t h i n k i n g a n d c o m m u n i c a t i o n s k i l l s t h r o u g h m a t h e m a t i c a l c o n t e n t . He saw t h e o r d e r i n m a t h e m a t i c s as c o n d u c i v e t o t e a c h i n g t h i n k i n g a n d f o s t e r i n g t h e a b i l i t y t o c o m m u n i c a t e w e l l t h o u g h t - o u t i d e a s . C o m m u n i c a t i o n e n t a i l e d n o t o n l y i n t e r p e r s o n a l c o m m u n i c a t i o n , i . e . t h e e x c h a n g e of . i d e a s w i t h o t h e r s , b u t a l s o c o m m u n i c a t i o n w i t h o n e s e l f . " B e c a u s e I t h i n k t h a t math i s a b o u t . . . p u t t i n g t h i n g s i n an o r d e r e d f a s h i o n . . . t h i n k i n g i n a s t r u c t u r e d o r d e r e d m a n n e r . . . a n d so t h a t l e n d s i t s e l f t o b o t h t h i n k i n g and t o c o m m u n i c a t i o n . " [p 1] " . . . y o u ' r e t e a c h i n g p e o p l e t h e t h i n k i n g and r e a s o n i n g s k i l l s c o n c e p t u a l l y , a n d p a t t e r n s , and p u t t i n g t h i n g s t o g e t h e r a n d so o n ? " [p 4] " I t h i n k t h e t h i n g t h a t w e ' r e t r y i n g t o do i s t o t e a c h them t o t h i n k a n d t o c o m m u n i c a t e . . . . A n d I a l s o t h i n k t h a t one o f t h e most i m p o r t a n t t h i n g s a b o u t l e a r n i n g how t o c o m m u n i c a t e i s l e a r n i n g how t o c o m m u n i c a t e w i t h y o u r s e l f . . . t h i n k i n g w i t h i n y o u r s e l f , p u t t i n g y o u r own f e e l i n g s o r n o t i o n s o r w h a t e v e r i n t o b e i n g a b l e t o p l a y w i t h them as a t h o u g h t . . . and t h a t ' s a p a r t o f c o m m u n i c a t i o n as much as me t e l l i n g y o u o r us d i s c u s s i n g s o m e t h i n g . " [p 1] - 44 -He p l a c e d t e a c h i n g i n a b r o a d e r p e r s p e c t i v e , one t h a t e m b o d i e d t h e p u r p o s e o f e d u c a t i o n . " . . . w e ' r e t e a c h i n g v a l u e s . . . . I t h i n k t h a t t e a c h i n g o f v a l u e s - I t h i n k t h e c o n s t r u c t i o n o f v a l u e s i s s o m e t h i n g t h a t w e ' r e a l l i n v o l v e d i n , o r may be i n v o l v e d i n t o , f o r o u r own v a l u e s . A n d w e ' r e s - , and i f t h e r e ' s d e m o c r a t i c e d u c a t i o n , o r a n y o f a number o f t h i n g s w e ' r e t e a c h i n g p e o p l e t o q u e s t i o n , t o t h i n k f o r t h e m s e l v e s and t o s t a r t t o c r e a t e a t some p o i n t , t o g i v e them t h e t o o l s n e c e s s a r y t o c r e a t e v a l u e s , t o c r e a t e t h e i r own v a l u e s , r a t h e r t h a n a c c e p t . . . w h a t e v e r t h a t d o e s n ' t a l l o w them t o q u e s t i o n , t o c r e a t e . " [p 2] T h i s o v e r - r i d i n g g o a l o f e d u c a t i o n was l i n k e d t o h i s v i e w o f what i t meant t o be a p e r s o n , and t h o u g h Doug n e v e r e l a b o r a t e d o n h i s p e r s p e c t i v e o f human d e v e l o p m e n t , he c l e a r l y saw i t as a n e l e m e n t a l component o f l i f e , b o t h h i s own a n d h i s s t u d e n t s ' . " B u t . . . t h e p r i m a r y , p r i m a r y g o a l t h o u g h i s , a h t o h e l p p e o p l e d e v e l o p as human b e i n g s , y o u know? o r t o grow as human b e i n g s a n d t h a t ' s t r u e f o r me t o o . " [p 10] C o n c e p t i o n o f t e a c h e r . Doug b e l i e v e d a math t e a c h e r was a r o l e m o d e l f o r a d u l t h o o d . As s u c h , t h e t e a c h e r w o u l d model t h e q u a l i t i e s he b e l i e v e d i m p o r t a n t t o f o s t e r i n s t u d e n t s : t h e a b i l i t y t o r e a s o n , t h e w i l l i n g n e s s t o l e a r n , t h e w i l l i n g n e s s t o make m i s t a k e s . " . . . l e t ' s s a y a t t h i s p a r t i c u l a r age l e v e l , I mean t h e s e a r e p e o p l e t h a t a r e c o m i n g o u t o f c h i l d h o o d l e a r n i n g t o be a d u l t s , r i g h t ? So w e ' r e , so e v e n i f i t ' s j u s t as a r o l e m o d e l , y o u know, a n d maybe as a p o o r o n e , b u t as t h e o n l y one t h a t t h e y s e e i n a , o u t o f a g r o u p o f 3 0 p e o p l e , um, y o u know, w e ' r e t e a c h i n g t h i n g s l i k e what i t means t o be an a d u l t , o r what i t means t o f u n c t i o n s o c i a l l y . . . a n d t h i n g s l i k e t h a t . " [p 2] " W e l l , I t h i n k t h a t ' s a w o r t h w h i l e r o l e m o d e l . Someone who q u e s t i o n s . . . . I f I g e t p r e s e n t e d w i t h a q u e s t i o n , a n d I t r y a n d p r e t e n d t h a t t h e r e ' s one way t o answer i t , o r t h a t t h e r e ' s one r i g h t a n s w e r t o e v e r y m a t h e m a t i c a l q u e s t i o n , t h e n I ' v e c r e a t e d a c e r t a i n r o l e m o d e l , b u t I w o u l d c a l l i t a n e g a t i v e o n e . I f I show t h a t . . . I c a n make m i s t a k e s and l e a r n f r o m t h a t , o r i f I show t h a t I c a n t h i n k t h r o u g h t h i n g s and t r y o r . . . I see d i f f e r e n t p e o p l e c o m i n g up w i t h d i f f e r e n t a n s w e r s and I a c c e p t d i f f e r e n t a n s w e r s , t h e n I ' m s h o w i n g a v e r y d i f f e r e n t p e r s o n , a v e r y d i f f e r e n t b e l i e f i n what i s i d e a l a n d r i g h t , w h e t h e r t h e r e ' s a r i g h t a n d a w r o n g , o r w h e t h e r t h e r e ' s many d i f f e r e n t r i g h t s o r many d i f f e r e n t - 4 5 -p o s s i b i l i t i e s . . . . So r a t h e r t h a n t e a c h i n g them what i s r i g h t a n d m o d e l i n g t h a t , I ' m t e a c h i n g them t h e a b i l i t y t o q u e s t i o n . " [p 3] Doug w a n t e d t o show s t u d e n t s t h a t m a k i n g d e c i s i o n s i n v o l v e d a m b i g u i t y a n d t h a t a n s w e r s were n o t a l w a y s c l e a r - c u t . He w a n t e d them' t o r e a l i z e t h a t " t h e " a n s w e r was n o t a l w a y s a v a i l a b l e and was i n f a c t s o m e t i m e s t h e r e s u l t o f m a k i n g many c h a n g e s , o u t r i g h t m i s t a k e s a n d g o i n g up b l i n d a l l e y s . He a l s o w a n t e d them t o see t h a t d e c i s i o n s c o u l d be made t h r o u g h c o n s e n s u s a f t e r r e a s o n a b l e p r o s a n d c o n s o f a n a r g u m e n t were w e i g h e d . T h i s f o r m o f c r i t i c a l t h i n k i n g was what he h o p e d t o e n g e n d e r t h r o u g h h i s t e a c h i n g , a n d f o r h i m i t d i s t i n g u i s h e d a g o o d t e a c h e r f r o m a b a d o n e . " A g o o d t e a c h e r i s a good s t u d e n t , someone who shows a n a b i l i t y t o l e a r n a n d a w i l l i n g n e s s t o l e a r n , d e m o n s t r a t e s t h e q u a l i t i e s t h a t y o u want t o see and t h a t , t h a t y o u ' r e e n c o u r a g i n g i n a s t u d e n t , i n a p o s i t i v e a p p r o a c h t o l i f e m a y b e . . . . s o someone w h o ' s r i g i d a n d t h i n k s k n o w l e d g e i s a c e r t a i n t h i n g . . . m a k e s a b a d t e a c h e r . " [p 8] " . . . I t h i n k y o u have t o be w i l l i n g l i k e t o , t o make m i s t a k e s , o r , o r h a v e , show t h a t s o m e t h i n g a l i t t l e i s h a r d and y o u h a v e t o t h i n k a b o u t i t . " [p 5] H i s work w i t h and i n t e r e s t i n s t u d e n t s f r o m a l t e r n a t i v e p r o g r a m s c a u s e d h i m t o c o n s i d e r h i s i n t e r a c t i o n and i n f l u e n c e o n them as a t e a c h e r e v e n more d e e p l y . " L i k e , l i k e I q u e s t i o n e d why I was t e a c h i n g m a t h e m a t i c s when, y o u know, I g o t p e o p l e c o m i n g i n t o t h e room whose m o t h e r j u s t k i c k e d them o u t o f t h e home. A n d a l l t h e s e v a r i o u s t h i n g s g o i n g o n , a n d I ' m s a y i n g , I ' m t h i n k i n g l i k e , I ' m t e a c h i n g , y o u know, q u a d r i l a t e r a l s , a n d what t h e h e l l does t h a t have t o do w i t h t h i s p e r s o n ' s l i f e ? B u t , i n a c t u a l f a c t , I t h i n k i t d i d make a d i f f e r e n c e a t t h a t p o i n t i n t i m e , t o do t h a t , y o u know?" [p 10] D o u g ' s p r i m a r y f o c u s was o n t e a c h i n g , r a t h e r t h a n t e a c h i n g m a t h e m a t i c s ; t h e c o n t e n t o f h i s c l a s s e s was s e c o n d a r y . " N o , I d o n ' t t h i n k i t ' s t h a t i m p o r t a n t t h a t t h e y e n d up a t t h e e n d o f g r a d e 12 w i t h a huge k n o w l e d g e o f m a t h . I t h i n k i t ' s i m p o r t a n t t h a t t h e y end up w i t h a n a b i l i t y t o r e a s o n , a n d a n a b i l i t y t o c o m m u n i c a t e . B u t I t h i n k , a h , I t h i n k f o r t h e most p a r t , i t ' s l i k e l y t h a t t h e two w i l l be l i n k e d . You know l i k e , n o t t h a t i f t h e y c a n ' t do f r a c t i o n s t h e y c a n ' t [ t h i n k ] , b u t as t h e y g a i n t h e a b i l i t y t o do f r a c t i o n s , t h e y g a i n t h e a b i l i t y t o t h i n k . " [p 6] - 46 -" . . . I d o n ' t t h i n k y o u t e a c h r e a s o n i n g s k i l l s s p e c i f i c a l l y , y o u t e a c h a c o u r s e o f s o m e t h i n g and t h r o u g h s t u d y i n g t h a t c o u r s e t h e y l e a r n r e a s o n i n g s k i l l s . . . " [p 7] He a d v o c a t e d more l o g i c i n t h e math c u r r i c u l u m , b e l i e v i n g t h a t t h e s k i l l s a c q u i r e d c o u l d be u s e d b y s t u d e n t s t o d e l i b e r a t e p r e s s i n g a n d c o n t r o v e r s i a l i s s u e s o f t h e d a y . " . . . I ' d l o v e t o be t e a c h i n g a t some p o i n t a n d s a y , t e a c h a l i t t l e b i t o f b a s i s f o r some k i n d o f a h , l o g i c a l s k i l l s ? o f s a y a way o f b r e a k i n g up an argument o r s o , y o u know t h e s o r t o f l o g i c t h a t y o u do i n a , i n a p h i l o s o p h y c o u r s e ? J u s t a v e r y b a s i c l e v e l a n d them g i v e them a , a , an argument on s o m e t h i n g t h a t ' s c u r r e n t , l i k e l o g g i n g , y o u know i n some s m a l l town i n B . C . o r w h a t e v e r , a n d h a v e them b r e a k up t h e a r g u m e n t , y o u know? A n d , a n d t h e r e f o r e a p p l y some o f t h o s e t h i n k i n g and r e a s o n i n g s k i l l s d i r e c t l y t o s o m e t h i n g y o u ' r e i n v o l v e d i n . " [p 6] T h u s , m a t h e m a t i c s was a v e h i c l e t o h e l p s t u d e n t s d e v e l o p t h e i r t h i n k i n g . Doug made i t c l e a r t h a t m a t h e m a t i c a l c o n t e n t t o o k a b a c k s e a t t o t h e l a r g e r g o a l o f d e v e l o p i n g t h e a b i l i t y o f c r i t i c a l a n a l y s i s . "So c l e a r l y , I mean i f t h e y d i d n ' t l e a r n m a t h , b u t t h e y , t h e y were a b l e t o s i t down a n d , and r e a s o n w i t h me o v e r s o m e t h i n g a n d a r g u e c o h e s i v e l y a b o u t s o m e t h i n g , t h a t w o u l d be much, much b e t t e r t h a n b e i n g a b l e t o s i t and m e m o r i z e t h e s t e p s t o m u l t i p l y f r a c t i o n s a n d y e t n o t be a b l e t o i n t h e l o n g r u n , y o u know, c r i t i c a l l y a n a l y z e somebody e l s e ' s s t a t e m e n t , y o u know, o r t o q u e s t i o n what we see i n t h e m e d i a o r s o m e t h i n g l i k e t h a t , b a s e d o n , y o u know, r a t i o n a l r e a s o n i n g . " [p 10] H o w e v e r , he v o i c e d a c a v e a t i n b e i n g a t e a c h e r o f m a t h e m a t i c s . One c o u l d be s e e n as a n a d v o c a t e o f t h e " s c i e n t i f i c " way o f l o o k i n g a t t h i n g s ; t h i s i n f a c t c o n t r a d i c t e d h i s p e r s o n a l w o r l d v i e w . " B u t I , I f i n d t h a t a d a n g e r i n t e a c h i n g m a t h e m a t i c s i s t h a t y o u e n d up e n c o u r a g i n g t h i s s c i e n t i f i c v i e w o f t h e w o r l d . . . . T h i s h a s g o t t o do w i t h t h e f a c t t h a t m a t h e m a t i c s i s a , i s o f t e n s e e n n o t as a d i s c i p l i n e , n e c e s s a r i l y b y i t s e l f . Or as a , j u s t a way o f t h i n k i n g , b u t i t ' s s e e n as a p a r t o f s c i e n c e , a n d , a n d e n g i n e e r i n g a n d d a d a da da d a . S o , b y e n c o u r a g i n g p e o p l e i n math t o be s u c c e s s f u l m a t h e m a t i c a l l y . . . i t c a n be i n t e r p r e t e d as e n c o u r a g i n g them t o go on i n t o t h e s c i e n c e w h i c h a l o n e i s n ' t w r o n g , b u t may a l s o e n c o u r a g e t h i s v i e w o f t h e w o r l d , y o u know, w h i c h i s t h a t t h e w o r l d o p e r a t e s o n s c i e n t i f i c p r i n c i p l e s , o r , a n d so o n . " [p 14] H e r e , Doug c o n t r a s t e d h i s g o a l f o r t e a c h i n g w i t h o t h e r g o a l s . Whereas h i s p r i m a r y f o c u s was t o f o s t e r a n d e n c o u r a g e t h o u g h t a n d d e v e l o p m e n t i n - 47 -a d o l e s c e n t s , he f e a r e d t h a t , due t o t h e p o p u l a r l y p e r c e i v e d n a t u r e o f m a t h e m a t i c s , h i s t e a c h i n g c o u l d be s e e n as e n c o u r a g i n g s t u d e n t s t o a c c e p t t h e i n c r e a s i n g l y " s c i e n t i f i c " v i e w o f t h e w o r l d . T e a c h i n g m e t h o d s . Though Doug was a s t r o n g a d v o c a t e o f t e a c h i n g f o r c o n c e p t u a l u n d e r s t a n d i n g , he b e l i e v e d t h e r e was a l s o a p l a c e f o r t e a c h i n g p r o c e d u r e s . He f e l t as s t u d e n t s w o r k e d t h r o u g h b o t h a s p e c t s , t h e i n t e r p l a y w o u l d g e n e r a t e more c o n c e p t u a l u n d e r s t a n d i n g a n d h e l p s t u d e n t s d e v e l o p t h e s k i l l s o f a n a l y s i s and s y n t h e s i s . " y o u n e e d t o t e a c h p e o p l e t h e p r o c e d u r e s and t h i s a n d t h a t , a n d y o u t e a c h them t h e c o n c e p t i o n s a n d e v e r y t h i n g e l s e , b u t I t h i n k o v e r t h e l o n g r u n o f p r a c t i c e a n d , and g o i n g i n b e t w e e n t h o s e two d i f f e r e n t s i d e s o f i t , o r p r a c t i s i n g t h e p r o c e d u r a l t e r m s , t h e y , t h e y p i c k up a g r e a t e r u n d e r s t a n d i n g o f b o t h . " [p 4] Doug r e g a r d e d t h e s e c o n d a r y math c u r r i c u l u m o v e r l y g e a r e d t o w a r d p r o v i n c i a l exams, l e a v i n g l i t t l e t i m e f o r c r e a t i v i t y i n t h e c l a s s r o o m . N o t o n l y was t h e c u r r i c u l u m s e t and e a c h u n i t n e a t l y s u p p l i e d b y a t e x t b o o k c h a p t e r , b u t t h e t y p e s o f q u e s t i o n s g e n e r a t e d were a l l p r o c e d u r a l . Doug c o n t r a s t e d t h i s i n f l e x i b i l i t y w i t h S o c i a l S t u d i e s , E n g l i s h , a n d G r a p h i c A r t s , where he saw t h e t e a c h e r w i t h more f r e e d o m t o d e v e l o p h e r own u n i t s and d e s i g n a v a r i e t y o f s t u d e n t a c t i v i t i e s . "some s u b j e c t s a f f o r d f a r more o p p o r t u n i t y . . . . i n S o c i a l s y o u a l m o s t c r e a t e y o u r own u n i t e v e r y t i m e i t seems t o m e . . . . [whereas i n math] h e r e ' s a u n i t , and h e r ' s t h e t e x t b o o k . A n d t h e t e x t b o o k c h a p t e r i s t h e u n i t , y o u know? A n d t h e r e ' s v e r y l i t t l e f r e e d o m . . . " [p 11] O t h e r I s s u e s A d o l e s c e n t D e v e l o p m e n t . Doug h a d a s p e c i a l i n t e r e s t i n t e a c h i n g " a l t e r n a t i v e " a d o l e s c e n t s , i . e . t h o s e whose e m o t i o n a l p r o b l e m s p r e c l u d e d them f r o m j o i n i n g a - 48 -r e g u l a r c l a s s r o o m . He b e l i e v e d math c l a s s c o u l d be u s e d as one p o i n t o f s t r u c t u r e i n a n o t h e r w i s e c h a o t i c l i f e . " . . . t h a t p e r s o n m i g h t n e e d a , a s e n s e o f o r d e r and s t r u c t u r e , a n d t h i n g s , a n d so o n , w h i c h a r e g u l a r i t y i n t h e c l a s s r o o m g i v e s t h e m , t h e d i s c i p l i n e i n t h e c l a s s r o o m g i v e s t h e m . . . . a n d i t g i v e s s o m e t h i n g t o t h i n k a b o u t o u t s i d e o f t h e m s e l v e s . " [p 10] S t u d e n t t e a c h i n g . Doug f e l t c o n s t r a i n e d i n c l a s s b e c a u s e o f h i s s t a t u s as a p r e -s e r v i c e t e a c h e r . He f e l t t h e e v a l u a t i v e a s p e c t o f t h e p r a c t i c u m f o r c e d h i m t o be t h e o p p o s i t e t y p e o f t e a c h e r f r o m what he w a n t e d t o d e p i c t . " W e l l , i f y o u ' r e a s t u d e n t t e a c h e r l i k e me, a n d , y o u ' r e j u s t t r y i n g t o p l a y i n t o what someone e l s e e x p e c t s . . . . I t h i n k we c a n [ t e a c h t h i n k i n g s k i l l s ] j u s t as s i m p l y b y , as s h o w i n g a w i l l i n g n e s s t o make m i s t a k e s i n f r o n t o f t h e m , y o u know. A n d s h o w i n g a w i l l i n g n e s s t o n o t have t h e a n s w e r , l i k e - , a n d I t h i n k t h a t ' s a d a n g e r o f s t u d e n t t e a c h i n g i s t h a t y o u f e e l l i k e y o u h a v e t o be p r e p a r e d and y o u have t o w a l k i n t h e r e a n d y o u c a n ' t make m i s t a k e s a n d a l l t h e r e s t o f i t . " [p 5] W o r l d v i e w . T h r o u g h o u t h i s d e s c r i p t i o n s o f t e a c h i n g a n d o f t h e t e a c h e r , Doug was c o n s i s t e n t i n h i s p h i l o s o p h y o f h e l p i n g s t u d e n t s d e v e l o p t h e i r t h i n k i n g a b i l i t i e s so t h a t t h e y c o u l d a n a l y z e a n d s u b s e q u e n t l y c h o o s e v a l u e s f o r t h e m s e l v e s . He c o n t r a s t e d two v i e w s : " K n o w l e d g e i s a t o p - d o w n t h i n g , t h e r e IS a n a n s w e r , w e ' r e l o o k i n g f o r v a r i o u s t r u t h s , o r t h e r e a r e t r u t h s , y o u know w h e t h e r t h e y ' r e B i b l i c a l , o r m y t h o l o g i c a l , o r s c i e n t i f i c , o r a h , o r w h e t h e r y o u know, we j u s t c r e a t e o u r own p e r c e p t i o n s o f t h e w o r l d as we want t o , a n d a h , y o u know, we u s e o u r a b i l i t i e s t h e way we c h o o s e t o . " [p 8] Doug f i t h i s p h i l o s o p h y o f t e a c h i n g m a t h e m a t i c s i n w i t h h i s l a r g e r p i c t u r e o f l i f e . D: You know what t h e d a n g e r i s i n t e a c h i n g m a t h , I t h i n k ? F o r me, t h e s c i e n t i f i c v i e w o f l o o k i n g a t t h e w o r l d i s m y t h o l o g y ? A n d , i t ' s y o u know, i t ' s as v a l i d ? I : M y t h o l o g y . You mean c o n s t r u c t e d ? D: Y e a h . I t ' s as v a l i d , and i t ' s a v a l i d way o f l o o k i n g a t t h e w o r l d , b u t i t ' s as v a l i d as t h e G r e e k m y t h o l o g y , r i g h t ? A n d we l o o k a t t h e G r e e k m y t h o l o g y a n d t h i n k , what a b u n c h o f s t o r i e s t h a t w e r e n ' t t r u e , b u t i n t h e same way w e ' r e j u s t - 49 -m a k i n g up o u r own m o d e l s f o r t h e w o r l d u s i n g s c i e n t i f i c l a n g u a g e , b u t we r e a l l y b u y i n t o i t . . . a d a n g e r o f t e a c h i n g math i s t h a t y o u t e a c h , o r y o u l e a d p e o p l e i n t o b e l i e v i n g t h a t , t h a t much more? So i f y o u t e a c h p e o p l e t h i n k i n g s k i l l s y o u c a n t e a c h them t o q u e s t i o n a n d a n s w e r t h a t f o r t h e m s e l v e s , and i f t h e y w a n t , t o t h e e x t e n t t h a t t h e y want t o b u y i n t o i t , o r want t o u s e s c i e n t i f i c m y t h o l o g y , t h a t ' s f i n e " [p 14] Ray A c a d e m i c a n d P e r s o n a l B a c k g r o u n d R a y h a d a B . S c . i n A p p l i e d M a t h e m a t i c s . He h a d a l s o c o m p l e t e d 2 y e a r s o f a n E n g i n e e r i n g d e g r e e and some g r a d u a t e l e v e l m a t h e m a t i c s c o u r s e s . P r i o r t o e n t e r i n g t h e B . E d , p r o g r a m he t u t o r e d p r i v a t e l y a n d w o r k e d as a n a c t u a r i a l a s s i s t a n t i n an i n s u r a n c e f i r m . C o n c e p t i o n o f M a t h e m a t i c s R a y ' s c o n c e p t i o n o f m a t h e m a t i c s e m p h a s i z e d i t s u t i l i t a r i a n n a t u r e . He saw m a t h e m a t i c s as a t o o l f o r o t h e r f i e l d s . " . . . w e u s e d math i n p h y s i c s , we u s e math i n b i o l o g y , u s e math i n e v e r y o t h e r [ s c i e n c e ] " [p 4] Ray b e l i e v e d m a t h e m a t i c s was a c c e s s i b l e t o a l l a n d c o u l d be f u n f o r a l l , e s p e c i a l l y i f t h e t o p i c s were r e l e v a n t t o common e v e n t s . " I t c a n be a c c e s s i b l e t o e v e r y s i n g l e k i d o u t t h e r e . " [p 10] V i e w o f T e a c h i n g M a t h e m a t i c s " I t h i n k t e a c h i n g i s 20% t e a c h i n g a n d . . . 8 0 % d e a l i n g w i t h s t u d e n t s i n d i f f e r e n t ways and t h e i r p r o b l e m s . . . . y o u d e a l w i t h k i d s o n a n i n d i v i d u a l b a s i s . . . . 8 0 % o f y o u r t i m e i s i n t e r a c t i n g w i t h t h e k i d s , a n d , a h , g e t t i n g t o know t h e k i d s , b e i n g t h e i r f r i e n d , t r y t o be a r o l e m o d e l a n d s t e e r i n t h e r i g h t d i r e c t i o n . " [p 1-2] F o r R a y , t h e t e a c h e r t o o k on m u l t i p l e r o l e s o f c o u n s e l l o r , g u i d e , f r i e n d a n d p a r e n t d e p e n d i n g on t h e t y p e o f c l a s s as w e l l as i n d i v i d u a l s t u d e n t s . " R o l e as a t e a c h e r . . . depends on d i f f e r e n t g r a d e l e v e l s , b u t I t h i n k y o u ' r e t h e r e t o g u i d e them i n t h e ' r i g h t ' d i r e c t i o n , n o t as - 50 -a p a r e n t , n o t e v e n as a t e a c h e r . I g u e s s y o u t a k e d i f f e r e n t r o l e s as a t e a c h e r . " [p 2] " A t one p o i n t y o u a r e t h e t e a c h e r , y o u have d i s c i p l i n e r u l e s , a t t h e n e x t p o i n t , y o u ' r e a p a r e n t , a t t h e n e x t p o i n t y o u ' r e a f r i e n d , y o u ' r e a c o u n s e l l o r . A h . . . i t ' s j u s t a m a z i n g how t h e d i f f e r e n t r o l e s come u p o n y o u . " [p 13] The most p r o m i n e n t o f t h e s e f o r Ray was c o u n s e l l o r , f o r he f o u n d h i m s e l f b o t h b e f o r e a n d a f t e r c l a s s d e a l i n g w i t h a v a r i e t y o f s t u d e n t s who were g o i n g t h r o u g h d i v e r s e p r o b l e m s . "So y o u know y o u t a k e t h e t i m e t o s i t down w i t h t h e s e k i d s a n d move them a s i d e . . . . I g u e s s [you] g e t t o know t h e k i d s . " [p 2] V i e w o f S t u d e n t s . R a y ' s i d e a s o f t e a c h i n g were l a r g e l y i n f l u e n c e d b y h i s 9A/10A c l a s s , o f whom he s p o k e a g r e a t d e a l . 9A/10A was a t e r m i n a l m a t h c l a s s composed o f g r a d e s 9 and 10 s t u d e n t s , a c l a s s t h e d e p a r t m e n t h e a d h a d f o u g h t t o e s t a b l i s h . Ray and t h e d e p a r t m e n t h e a d t e a m - t a u g h t t h i s c l a s s . He d e s c r i b e d them as "22 s t u d e n t s w i t h d i f f e r e n t b e h a v i o u r a l p r o b l e m s " [p 1] and t h o u g h he s p o k e o f t h e c h a l l e n g e s t h e y p r e s e n t e d he n e v e r c l a i m e d t h e y d r a i n e d h i s e n t h u s i a s m o r e n e r g y . I n s t e a d , he s p o k e r e p e a t e d l y o f t h e p e r s o n a l p r o b l e m s e a c h s t u d e n t was h a v i n g a n d how t h o s e i n h i b i t e d t h e i r l e a r n i n g . He saw them as i n d i v i d u a l s w i t h e m o t i o n a l a n d b e h a v i o u r a l p r o b l e m s , k i d s who came f r o m d y s f u n c t i o n a l f a m i l i e s a n d a c t e d o u t t h e i r f r u s t r a t i o n s i n s c h o o l . " . . . a n d I r e a l i z e d t h a t , e v e n a f t e r y o u l e c t u r e d them y o u j u s t s a t t h e r e and h e l p e d o u t i n d i v i d u a l s t u d e n t s a n d most o f them w o u l d r e a c t a f t e r 5 , 10 m i n u t e s , t h e y ' d g e t u p , t h r o w s o m e t h i n g , o r t h e y ' d have a t e m p e r t a n t r u m . S o , o k , f o r t h a t c l a s s y o u a c t u a l l y t a u g h t 20% o f t h e t i m e . 80% o f t h e t i m e y o u w o u l d a c t u a l l y d e a l w i t h them r e g a r d i n g t h e i r e m o t i o n a l b e h a v i o u r s . . . . Y o u a r e d e a l i n g , u h , w i t h a v a r i e t y o f p r o b l e m s . " [p 1] - 51 -(When Ray r e t u r n e d my d r a f t summary o f h i s comments, he n o t e d t h a t f o r " r e g u l a r " s t u d e n t s , t h e r a t i o o f t e a c h i n g t i m e t o t i m e s p e n t d e a l i n g w i t h b e h a v i o u r a l p r o b l e m s was p r o b a b l y 5 0 : 5 0 . ) Ray s p o k e o f o t h e r s t u d e n t s as m e c h a n i s t i c a l l y a t t e n d i n g s c h o o l . He g a v e t h i s as t h e r e a s o n most s t u d e n t s were d i s i n t e r e s t e d i n m a t h e m a t i c s . " I t h i n k most o f them f e e l t h e y have t o come t o s c h o o l , do t h e a s s i g n e d homework, g o , come b a c k t h e n e x t d a y . T h e y f i n d t h a t t h a t ' s t h e i r j o b . . . a n d I t h i n k t h a t ' s why t h e y have a d i s l i k e f o r . . . a n y s u b j e c t . " [p 5] I n a d d i t i o n , he b e l i e v e d s t u d e n t s saw math as " . . . s o m e t h i n g q u i t e h a r d , a n d a t e r r o r , a n d a w a s t e o f t i m e , has no u s e . " [p 3] H o w e v e r , he c l a i m e d s t u d e n t s h a d an "unknown c u r i o s i t y a b o u t m a t h e m a t i c s " [p 5] a n d t h a t o n c e t h e c o n n e c t i o n o f math t o d a i l y l i f e was made, t h e y w o u l d be i n t e r e s t e d i n t h e s u b j e c t . A c c o r d i n g t o R a y , s t u d e n t s n e e d e d t o s e e t h e u s e f u l n e s s o f m a t h e m a t i c s t o b r e a k down t h e i r e m o t i v e b a r r i e r s . Though he c l a i m e d t h e r e was b e a u t y i n m a t h , he s t r e s s e d t h a t " k i d s h a v e t o a p p r e c i a t e t h e b e a u t y o f a l g o r i t h m s a t a l a t e r s t a g e . " [p 10] R e l a t i n g h i s own e x p e r i e n c e , he s t a t e d he d i d n ' t see " t h e b e a u t y o f a l g o r i t h m s " [p 10] u n t i l 3 r d y e a r u n i v e r s i t y , when he c o u l d "make t h e c o n n e c t i o n . . . . B u t i n h i g h s c h o o l , t h e y have t h e o p p o r t u n i t y [ t o make t h e c o n n e c t i o n ] b e c a u s e y o u a r e t h e t e a c h e r , o r as a c o u n s e l l o r , o r as a f r i e n d w h o ' s t h e r e t o h e l p them make t h e c o n n e c t i o n . " [p 10] G o a l s o f a math t e a c h e r . A m a t h t e a c h e r n e e d e d t o "hook k i d s o n t o s c h o o l " [p 4] b y " h o o k i n g them o n t o m a t h . " [p 14] Ray b e l i e v e d t h a t b y s h o w i n g s t u d e n t s t h e u s e f u l n e s s a n d r e l e v a n c e o f m a t h e m a t i c s t h e y w o u l d become i n t e r e s t e d i n t h e s u b j e c t m a t t e r . - 52 -" . . . k i d s [need] t o u n d e r s t a n d t h a t math i s u s e d i n e v e r y a s p e c t o f o u r l i f e , i n e v e r y s u b j e c t o f l i f e , w h e t h e r i t ' s m u s i c f o r n o t e s . . . f o r p h y s i c s . . . o u t on t h e f o o t b a l l f i e l d , a v e r a g e i n c l a s s , k e e p i n g s c o r e s , a h , f i n d i n g p e r c e n t a g e s . I t ' s j u s t t h e d a i l y k n o w l e d g e , a n d most k i d s d o n ' t have t h a t . . . . T h e y have t o have a r e a s o n a n d I t h i n k t h a t ' s what a l o t o f k i d s l a c k . A r e a s o n . Why a r e t h e y l e a r n i n g t h i s . " [p 9] " A g a i n , i f i t d o e s n ' t p e r t a i n t o them I t h i n k t h e y s e e math a n d s c i e n c e , s c i e n c e s as a t o o l t o go on t o h i g h e r e d u c a t i o n a n d u n i v e r s i t y , so i f t h e y ' r e n o t g o i n g t o g o , why b o t h e r l e a r n i n g i t ? T h a t has t o be c h a n g e d . T h e y have t o be t a u g h t t h e u s e f u l n e s s i n math e v e n i f y o u d o n ' t go o n t o u n i v e r s i t y , i n y o u r d a i l y l i f e , i n w h a t e v e r o c c u p a t i o n y o u ' r e i n , d e a l i n g w i t h t h e b a n k , b u y i n g a c a r , g o i n g t o t h e g r o c e r y s t o r e . . . " [p 7] The t e a c h e r a l s o n e e d e d t o combat p s y c h o l o g i c a l b a r r i e r s t h e s t u d e n t s m i g h t have r e g a r d i n g m a t h e m a t i c s b y m a k i n g c l a s s e s e n j o y a b l e . " B u t t h e y have t o t a k e t h i s [math] c l a s s w h i c h was b e s t o w e d u p o n them b y t h e s c h o o l . W h i c h i s s a d , so y o u t r y a n d make t h e c l a s s f u n f o r t h e m . " [p 3] " . . . y o u c a n u s e math t o b o o s t t h e i r c o n f i d e n c e l e v e l , f o r k i d s who h a v e no way o f p a s s i n g , who f a i l e d b e f o r e . You s h a r e , y o u show them t h e u s e f u l n e s s o f math d u r i n g c l a s s t i m e . " [p 3] Q u a l i t i e s o f a math t e a c h e r . "So I t h i n k t h e a t t r i b u t e s o f a good math t e a c h e r number one i s t o make t h e k i d s l o v e m a t h . . . u n d e r s t a n d why math i s t h e r e , what u s e m a t h i s . " [p 5] To R a y , a t e a c h e r o f m a t h e m a t i c s n e e d e d t o be a p e r s o n o f m a t h e m a t i c s . " . . . i f y o u j u s t k e e p somebody who i s n o t a math t e a c h e r t o t e a c h m a t h , t h e y d o n ' t u n d e r s t a n d t h e b e a u t y and t h e a p p r e c i a t i o n o f t h e d a i l y l i f e , t h e y c a n ' t c o n v e y t h a t message a c r o s s . . . y o u have t o be a b l e t o g i v e o u t t h a t message t o t h e k i d s t h a t y o u a r e h a v i n g a w o n d e r f u l t i m e d o i n g t h i s . . . y o u have t o l o v e d o i n g m a t h e m a t i c s t o o . " [p 11] He g a v e h i m s e l f as a n e x a m p l e o f someone who was a l w a y s l o o k i n g f o r a p p l i c a t i o n s a n d o p p o r t u n i t i e s o f u s i n g m a t h e m a t i c s . " . . . s o m e b o d y who e n j o y s i t a l w a y s l o o k s f o r e x a m p l e s . F o r e x a m p l e , i f y o u ' r e d r i v i n g on a l o n g t r i p , u h , what I t e n d t o do s o m e t i m e s i s u n c o n s c i o u s l y l o o k a t t h e s i g n p o s t e d as y o u go a l o n g o n t h e h i g h w a y t h a t s a y s one m i l e , two m i l e s , t h r e e m i l e s . A n d l o o k a t my s p e e d o m e t e r and t r y and do a mess o f c a l c u l a t i o n s . A n d y o u do i t u n c o n s c i o u s l y b e c a u s e y o u e n j o y d o i n g i t . " [p 12] T e a c h i n g m e t h o d s . - 53 -R a y a d v o c a t e d methods w h i c h w o u l d g i v e s t u d e n t s t h a t " c o n n e c t i o n " t o t h e r e a l w o r l d . He s u g g e s t e d g u e s t s p e a k e r s . He e n v i s i o n e d i n v i t i n g s a n i t a t i o n w o r k e r s , s e c r e t a r i e s and c a s h i e r s t o t e l l t h e c l a s s how t h e y u s e d math i n t h e i r w o r k . D u r i n g h i s p r a c t i c u m he h a d p l a n n e d a f i e l d t r i p t o a b a s e b a l l game where t h e s t u d e n t s h a d an a s s i g n m e n t t o c o m p l e t e . ( U n f o r t u n a t e l y , t h e game was r a i n e d o u t . ) He a l s o s u g g e s t e d h a v i n g s t u d e n t s d e v i s e t h e i r own math p r o b l e m s b a s e d o n t h e i r e x p e r i e n c e s . He c a l l e d t h e s e " p o s i t i v e p r o b l e m s " [p 8 ] . P l a y i n g games a n d u s i n g p u z z l e s , r i d d l e s , and p r o b l e m s o f i n t r i g u e were o t h e r methods o f " h o o k i n g them o n t o m a t h " [p 1 4 ] . He s t a t e d a t e a c h e r w o u l d a c q u i r e a bank o f t h e s e t h r o u g h e x p e r i e n c e , a n d c l a i m e d t o h a v e a l r e a d y c o l l e c t e d q u i t e a few w h i c h he u s e d . He t r i e d t o s t a y away f r o m l e c t u r i n g . He f e l t t h a t h a v i n g l i f e e x p e r i e n c e a n d " t r i c k s " t o s h a r e w i t h t h e s t u d e n t s w o u l d a l l o w f o r more v a r i e t y i n c l a s s a n d move away f r o m a t r a d i t i o n a l l e c t u r e s t y l e . M e t a p h o r . Ray h a d d e v e l o p e d h i s own m e t a p h o r f o r t e a c h i n g . I t was o f t h e t e a c h e r as a g a r d e n e r a n d s t u d e n t s as p l a n t s u n d e r h i s s u p e r v i s i o n . T h i s m e t a p h o r h i g h l i g h t e d s t u d e n t i n d i v i d u a l i t y . The c a r e a n d c o n c e r n he f e l t f o r them i s e v i d e n t i n h i s d e s c r i p t i o n . " I t h i n k a g a r d e n e r w o u l d be a g r e a t m e t a p h o r . B e c a u s e y o u know i f y o u have a n u r s e r y , y o u g e t p l a n t s and y o u n u r t u r e t h e m , b e c a u s e when t h e y ' r e a t a y o u n g age t h a t ' s when t h e y ' r e most s u s c e p t i b l e f o r d i s e a s e o r t o f a l l o f f and y o u know as a g a r d e n e r y o u , y o u t e n d t o them i n d i f f e r e n t f o r m s . I n d i f f e r e n t f l o w e r b e d s , y o u f e r t i l i z e them d i f f e r e n t l y . Y e a h , I t h i n k t h a t ' s a v e r y good a n a l o g y o f i t . . . . S o I t h i n k y o u ' r e c o n s t a n t l y t e n d i n g f o r t h e i r b e s t u s e . . . . E v e r y p l a n t i s t o t a l l y d i f f e r e n t . . . . A s a g a r d e n e r y o u ' r e c o n s t a n t l y w a t c h i n g o u t f o r , a n d I t h i n k what I meant b y d i s e a s e was y o u know, k i d s f a l l i n g o f f t h e wagon. T h e y l o s e i n t e r e s t and y o u know y o u t r y a n d change y o u r methods o f p l a n t i n g o r I mean y o u ' r e a l w a y s a d a p t i n g . . . . y o u c a r e f o r t h e m , - 54 -y o u n u r t u r e t h e m . . . . A n d I t h i n k t h a t , t h a t p e r t a i n s t o b e i n g a t e a c h e r t o o i f y o u t h i n k a b o u t i t . " [p 14-15] O t h e r I s s u e s C o n c e p t u a l c h a n g e . R a y ' s v i e w o f t e a c h i n g m a t h e m a t i c s h a d c h a n g e d w i t h h i s p r a c t i c u m e x p e r i e n c e f r o m one w h i c h was c o n t e n t - o r i e n t e d t o one w h i c h was s t u d e n t -o r i e n t e d . " L e t ' s s e e , y e a h , my v i e w s have c h a n g e d . A h , d r a s t i c a l l y , I t h i n k . . . . m y p r e v i o u s a s s u m p t i o n was y o u j u s t g o , y o u t e a c h , y o u l e a v e , y o u mark p a p e r s . B u t no t h a t ' s n o t i t . T e a c h i n g i s more t h a n t h a t . T e a c h i n g i s , i t ' s a m u l t i t u d e o f j o b s . " [p 13] V i c t o r i a A c a d e m i c a n d P e r s o n a l B a c k g r o u n d V i c t o r i a h a d a 3 - y e a r B . A . w i t h D i s t i n c t i o n i n E c o n o m i c s , a n d a 4 -y e a r B . S c . i n M a t h e m a t i c s a c h i e v e d one y e a r a f t e r h e r B . A . A n M . S c . i n m a t h e m a t i c s was i n p r o g r e s s . She c a l l e d h e r s e l f a " p r o f e s s i o n a l s t u d e n t . " [p 6] V i c t o r i a was a w h i z k i d i n m a t h e m a t i c s f r o m p u b l i c s c h o o l t h r o u g h u n i v e r s i t y a n d g r a d u a t e s c h o o l . She s k i p p e d g r a d e s a n d was a b l e t o l e a r n m a t h e f f o r t l e s s l y , a b s o r b i n g i t . " . . . y o u ' v e a l w a y s b e e n a b l e t o p u t me somewhere w i t h t h e t e x t b o o k and I c a n l e a r n t h e c o u r s e f a s t e r t h a n y o u c a n t e a c h i t t o m e . " [p 13] As a g r a d u a t e s t u d e n t i n U B C ' s m a t h e m a t i c s d e p a r t m e n t she h a d t a u g h t an u n d e r g r a d u a t e c o u r s e f o r t h o s e w i t h no math b a c k g r o u n d who were p l a n n i n g t o e n r o l l i n a B . E d , p r o g r a m . C o n c e p t i o n o f M a t h e m a t i c s V i c t o r i a d i f f e r e n t i a t e d l o w e r l e v e l m a t h e m a t i c s f r o m g r a d u a t e l e v e l m a t h e m a t i c s . She s t a t e d a t t h e l o w e r l e v e l i t was a l l known a n d f o r m u l a i c ; i t e x e m p l i f i e d t h e s c i e n t i f i c n a t u r e o f m a t h . G r a d u a t e l e v e l - 55 -m a t h , h o w e v e r , i n v o l v e d " f u z z y , g r e y a r e a s " [p 7] w h i c h r e q u i r e d i n t u i t i o n a n d c r e a t i v i t y t o a c h i e v e r e s u l t s ; t h i s was t h e a r t i s t i c s i d e o f m a t h . " . . . w h e r e [ t h e p u b l i c s c h o o l i s ] a t i n math i s a known a r e a a n d i t ' s n o t c h a n g i n g a l l t h a t m u c h . . . . y o u ' r e n o t g o i n g t o s u d d e n l y h a v e e q u a t i o n s o f l i n e s c h a n g i n g d e s p e r a t e l y . I mean i t ' s known what a l i n e i s , and how t o s o l v e i t . [p 7] " . . . as a n u n d e r g r a d y o u have t h e r i g h t o r w r o n g a n d I mean y o u h a v e , y o u ' l l h a v e , y o u ' r e d o i n g a p r o o f ? A n d t h e r e ' s d i f f e r e n t ways o f d o i n g t h e p r o o f , b u t y o u know y o u ' r e t r y i n g t o p r o v e t h e same t h i n g i n t h e e n d . . . . W h e r e a s when y o u g e t i n t o g r a d m a t h , y o u s t a r t g e t t i n g i n t o . . . t h e f u z z y g r e y a r e a s t h e r e , y o u ' r e g e t t i n g i n t o t h e edges o f math r a t h e r t h a n , r a t h e r t h a n t h i s known a r e a . . . . y o u ' r e j u s t s t i l l d i s c o v e r i n g , y o u ' r e s t i l l c r e a t i n g i t , ok? I t ' s s t i l l b e i n g c r e a t e d . ' [p 6-7] " I t h i n k [ m a t h e m a t i c s ] i s a l i t t l e o f e a c h [ a r t a n d s c i e n c e ] , a n d I t h i n k t h e shame o f i t i s we t e a c h i t t o o much as a s c i e n c e . . . [ M ] o s t p e o p l e as an u n d e r g r a d have o n l y s e e n i t as a . s c i e n c e . A n d i t ' s c r e a t i v e , y o u have t o be c r e a t i v e t o be a g o o d m a t h e m a t i c i a n . " [p 19] V i c t o r i a c l a i m e d s t u d e n t s g e n e r a l l y saw math as r i g i d , " t h i s c o l d h a r d o b j e c t " [p 1 ] , t o t a l l y e x t e r n a l i z e d , a n d d i f f i c u l t t o c o m p r e h e n d . I n c o n t r a s t she saw m a t h e m a t i c s , i n p a r t i c u l a r h e r s u b - s p e c i a l t y o f d i s c r e t e m a t h e m a t i c s , as f u n a n d e x p l i c a b l e t o t h e l a y - p e r s o n as " g l o r i f i e d c o n n e c t - t h e - d o t s . " [p 6] She went f u r t h e r t o c l a i m t h a t e l e m e n t a r y t e a c h i n g u s u a l l y r e w a r d s m e m o r i z a t i o n i n m a t h e m a t i c s , a t r a i t n o t d i r e c t l y l e a d i n g t o b e c o m i n g a g o o d m a t h e m a t i c i a n . I n h i g h e r y e a r s where u n d e r s t a n d i n g r a t h e r t h a n m e m o r i z a t i o n was r e q u i r e d , t h e t r a n s i t i o n o f t e n p r o v e d d i f f i c u l t f o r t h e s t u d e n t who was f o r m e r l y s u c c e s s f u l d o i n g memory w o r k , [p 9] V i c t o r i a c l a i m e d m a t h e m a t i c s ' a e s t h e t i c q u a l i t y was o f t e n o v e r s h a d o w e d b y i t s u t i l i t a r i a n n a t u r e f o r p r a g m a t i c p u r p o s e s . " . . . w e ' r e t h e b u i l d i n g b l o c k s o f o t h e r s c i e n c e s . . . . w e ' r e c r e a t i n g t o o l s , b a s i c a l l y . . . . f o r me I f i n d math a j o y a n d b e a u t y a n d e v e r y t h i n g e l s e i n m a t h , b u t n o b o d y ' s g o i n g t o f u n d , no c o m p a n y ' s g o i n g t o go o u t t h e r e a n d , um, f u n d m a t h e m a t i c i a n s o r s a y t h a t - 56 -m a t h e m a t i c i a n ' s a r e a t a l l u s e f u l u n l e s s t h e y a c t u a l l y c r e a t e s o m e t h i n g , o k ? . . . c r e a t i n g a t o o l t h a t w i l l be u s e f u l i n t h e r e a l w o r l d . " [p 8] C o n c e p t i o n o f M a t h e m a t i c i a n V i c t o r i a f o r m u l a t e d h e r i d e a s o f a m a t h e m a t i c i a n t h r o u g h a l g e b r a i s t s she h a d met as a g r a d u a t e s t u d e n t a t Q u e e n ' s U n i v e r s i t y . A l l were c r e a t i v e , a r t i s t i c and m u s i c a l , " t h i n g s t h a t y o u d o n ' t t h i n k a m a t h e m a t i c i a n s h o u l d b e . " [p 20] " T h e y a l l p l a y a n i n s t r u m e n t o f some s o r t . Um, t h e y a l l a r e i n v o l v e d , go t o t h e t h e a t r e s , t h e y a l l do t h i s , v e r y a r t s y . . . s t u f f . " [p 20] She d e s c r i b e d t h e m a t h e m a t i c i a n ' s work as c h e c k i n g f o r m e a n i n g a n d u n d e r s t a n d i n g ; when d e a l i n g w i t h a p r o b l e m i t must make s e n s e " l o g i c a l l y a n d i n t u i t i v e l y . " [p 10] V i e w o f T e a c h i n g M a t h e m a t i c s V i c t o r i a h a d two g o a l s i n t e a c h i n g m a t h e m a t i c s . The f i r s t was t o d i s p e l t h e n o t i o n o f a m a t h e m a t i c i a n as a "geek" o r " n e r d , " a n d t h e s e c o n d was t o change t h e commonly p e r c e i v e d n a t u r e o f m a t h e m a t i c s as b e i n g r i g i d a n d u n a t t a i n a b l e f o r t h e a v e r a g e p e r s o n . " . . . t e a c h i n g m a t h e m a t i c s i s . . . m a k i n g math human r a t h e r t h a n h a v i n g i t t h i s c o l d h a r d o b j e c t t h a t t h e y [ s t u d e n t s ] have t o g e t o v e r a n d d e a l w i t h a n d s u r v i v e r a t h e r t h a n s o m e t h i n g t h e y e n j o y . " [p 1] " . . . j u s t m a k i n g i t [math] v e r y , a h , t o u c h a b l e ? I f t h a t makes s e n s e ? M a k i n g , m a k i n g math as I s a i d , v e r y h u m a n . . . " [p 2] " . . . w h a t I w o u l d l i k e t o do i s I ' d l i k e t o make math f u n . . . " [p 18] " A n d I t h i n k y o u ' r e s o r t o f s a y i n g t h a t h e y , I ' m human a n d I ' m d o i n g math a n d I e n j o y m a t h . T h e r e f o r e , y o u a r e human a n d y o u , y o u c a n e n j o y math t o o . y o u d o n ' t have t o see t h i s p e r s o n t h a t w i t h t h e , v e r y s t e r e o t y p i c a l , l i k e t h e g l a s s e s w i t h t h e t a p e on i t , t h e g e e k , t h e geek p a c k w i t h t h e p e n l e a k i n g t h r o u g h a n d y o u know, t h e p o c k e t p r o t e c t o r t y p e t h i n g . . . . B r e a k i n g t h a t s t e r e o t y p e . " [p 13] She saw t h e f e a r o f math i n t h e u n i v e r s i t y s t u d e n t s she h a d t a u g h t a n d became c o n c e r n e d t h a t t h i s f e a r w o u l d t r a n s f e r t o t h e c h i l d r e n t h e y - 57 -w o u l d e v e n t u a l l y t e a c h . A s c e n a r i o she d e p i c t e d f o r t h i s was a f o r c e d h a l f - h o u r o f i n s t r u c t i o n i n math v e r s u s a o n e - a n d - a - h a l f h o u r o f m u s i c o r a r t i n s t r u c t i o n i n t h e e l e m e n t a r y c l a s s r o o m . V i c t o r i a w a n t e d t o h e l p them a n d o t h e r s overcome t h e i r f e a r a n d p a r a n o i a o f m a t h e m a t i c s a n d h a v e them e n j o y t h e s u b j e c t . She a d v o c a t e d g e t t i n g t h e message t o s t u d e n t s t h a t " m a t h c a n be e n j o y e d b y humans" [p 13] and d i s p e l t h e s t e r e o t y p e o f t h e " m a t h n e r d " , p a s s i n g o n i n s t e a d a p o s i t i v e a t t i t u d e a n d f o s t e r i n g a w i l l i n g n e s s t o e x p l o r e m a t h . M e t a p h o r f o r t e a c h i n g . V i c t o r i a h a d b e g u n t o d e v e l o p a m e t a p h o r f o r t e a c h i n g . T e a c h e r as P a i n t e r , a n d S t u d e n t as P a i n t i n g was how she c o n c e p t u a l i z e d i t . The s t u d e n t was a p a i n t i n g where t h e t e a c h e r / p a i n t e r was a d d i n g o n t o t h e o e u v r e , a n d n o t a s c u l p t u r e where t h e m a s t e r a r t i s t w o u l d be c h i p p i n g away s u b s t a n c e d u r i n g t h e c r e a t i o n o f t h e w o r k . H e r m e t a p h o r e m p h a s i z e d g r e a t l y t h e c o m p o s i t i o n o f t h e s t u d e n t . When t h e s t u d e n t f i r s t e n t e r e d y o u r c l a s s r o o m , he o r she e m b o d i e d t h e p r e p p e d p a i n t i n g , c a r r y i n g t h e work o f o t h e r s on t h e c a n v a s . T h e s e o t h e r s were t h e p a r e n t s , f r i e n d s , t h e home, a n d l i f e i n g e n e r a l . T h e s e h a d a l l l e f t t h e i r i m p r e s s i o n s on t h e c a n v a s . The p a i n t i n g was a n a c c u m u l a t i o n o f t h e i n f l u e n c e o f e v e r y e v e n t i n t h e s t u d e n t ' s p a s t ; e a c h i n d i v i d u a l h a d l e f t t h e i r mark o n t h e s t u d e n t ' s c a n v a s , e a c h p e r s o n was h e l p i n g t o c r e a t e i t . " I l i k e t h e i d e a o f a p a i n t e r ? . . . t h e s t u d e n t i s t h e p a i n t i n g y o u ' r e q u o t e u n q u o t e c r e a t i n g . M i n d y o u i t ' s a p a i n t i n g t h a t comes t o y o u p r e p p e d a n d s t a r t e d , o k ? . . . S o t h e y have, t h e i r home l i f e , t h e i r p a r e n t s , t h e i r f r i e n d s , a l l t h i s , a l l t h i s i s a l l t u r n i n g i n t o t h e p a i n t i n g and a d d i n g t o t h e p a i n t i n g , ok? So w h a t ' s h a p p e n i n g i s y o u , y o u ' r e g i v e n a p a i n t i n g t h a t ' s p r e p p e d a n d s t a r t e d , a n d y o u , y o u and t h e s t u d e n t a r e h e l p i n g work c r e a t e more o f i t . L i k e y o u ' r e a d d i n g on t o i t . A n d h o p e f u l l y p o s i t i v e l y , ok? A n d t h e r e a s o n I l i k e p a i n t i n g i s b e c a u s e y o u a r e - 58 -a d d i n g o n . Y o u ' r e n o t , i t ' s n o t l i k e s c u l p t i n g where y o u ' r e t a k i n g a w a y . . . " [p 15] T h i s was a s t r o n g c o n c e p t i o n f o r V i c t o r i a as shown b y t h e t r e p i d a t i o n she f e l t i n t h e r o l e o f t e a c h e r . I t i n t i m i d a t e d h e r t h a t she h a d t h e o p p o r t u n i t y t o d e s i g n , c r e a t e , a n d draw o n t h e p a i n t i n g . The i n f l u e n c e c o u l d be n e g a t i v e as w e l l as p o s i t i v e a n d t h e power i n h e r e n t i n t h a t was f r i g h t e n i n g . " . . . t h a t ' s t h e s c a r y p a r t o f t e a c h i n g , i s y o u r a b i l i t y t o a f f e c t t h e l i v e s i s n o t j u s t p o s i t i v e . I t c a n be n e g a t i v e . A n d t h a t , t h a t ' s s o m e t h i n g t h a t , t o some e x t e n t , s c a r e s m e . " [p 17] She r e c a l l e d a h i g h s c h o o l math t e a c h e r who was a s t r i c t d i s c i p l i n a r i a n . She remembered t h e p h y s i c a l f o r c e he u s e d w i t h v a r i o u s members o f t h e c l a s s a n d l i t t l e e l s e . T h i s may have h a d a b e a r i n g o n how she p e r c e i v e d t h e r o l e o f t h e t e a c h e r . R e q u i r e m e n t s f o r a t e a c h e r o f m a t h e m a t i c s . The a b i l i t y t o f o r e s e e s t u d e n t d i f f i c u l t i e s a n d g e n e r a t e e x p l a n a t i o n s was i m p o r t a n t f o r a t e a c h e r o f m a t h e m a t i c s . " I t h i n k t h e r e a l g o a l and t h e r e a l i m p o r t a n c e i s n ' t so much a c t u a l l y p h y s i c a l l y k n o w i n g t h e m a t h ? . . . S o y o u ' l l have p e o p l e . . . w h o u n d e r s t a n d t h e s t u f f so w e l l t h a t t h e y c a n ' t see where t h e p r o b l e m s c o u l d b e , l i e ? and t h e y c a n ' t e x p l a i n t h e p r o b l e m s . A n d I t h i n k t h a t ' s what i s i m p o r t a n t f o r a g o o d m a t h t e a c h e r , i s t o see where t h e d i f f i c u l t y i s g o i n g t o come up a n d what i s t h e r e a l l y i m p o r t a n t CONCEPT t o g e t a c r o s s , ok? What do t h e y [ t h e s t u d e n t s ] r e a l l y n e e d t o u n d e r s t a n d a b o u t t h i s ? " [p 12] C o n f i d e n c e i n t h e s u b j e c t m a t t e r was k e y . The number o f u n i v e r s i t y c o u r s e s i t t o o k t o a c h i e v e t h i s was n o t i m p o r t a n t . The f a c t t h a t t h e p e r s o n b e c o m i n g a t e a c h e r o f math h a d no f e a r o f t h e s u b j e c t , w a s . " Y o u n e e d t o e n j o y math a n d n o t f e a r i t [ b e c a u s e ] . . . i t t r a n s m i t s . " [p 14] She e n u m e r a t e d o t h e r r e q u i r e m e n t s as w e l l : a s e n s e o f humour , k n o w l e d g e a n d u s e o f a l t e r n a t i v e a p p r o a c h e s / m e t h o d s . She a l s o commented t h a t t h o s e who do w e l l i n m a t h e m a t i c s w o u l d n o t have empathy f o r t h e - 59 -s l o w e r l e a r n e r s . T h i s i s n o t a b l e when we r e c a l l t h a t V i c t o r i a d e s c r i b e d h e r s e l f as e x t r e m e l y q u i c k t o l e a r n m a t h e m a t i c s . To V i c t o r i a , u n d e r s t a n d i n g a d o l e s c e n t d e v e l o p m e n t " [ w ] i t h what [ t h e s t u d e n t s a r e ] g o i n g t h r o u g h a t t h i s s t a g e i n t h e i r l i f e " [p 3] was a n o t h e r i m p o r t a n t component o f a good math t e a c h e r . D u r i n g h e r p r a c t i c u m she h a d t h e o p p o r t u n i t y t o w i t n e s s someone she c o n s i d e r e d a m o d e l t e a c h e r . She was p a r t i c u l a r l y i m p r e s s e d b y h i m b e c a u s e he was a s u b s t i t u t e , " [ a ] n d y o u know w i t h s u b s t i t u t e s t h e k i d s go a b s o l u t e l y ape o n . " [p 18] She d e s c r i b e d h i m as h a v i n g " a p r e s e n c e , " a c a l m a n d c o n t r o l l e d manner w h i c h meant b u s i n e s s t o t h e k i d s . She made t h e p o i n t t h a t he was n o t o v e r l y s t r i c t n o r d e m a n d i n g , j u s t " a c c e p t i n g them f o r who t h e y w e r e . " [p 18] T h a t was what she w a n t e d t o e m u l a t e . She c l a i m e d t h a t she h e r s e l f was " s t i l l v e r y b l a c k a n d w h i t e " [p 17] a n d r i g i d , a n d w o u l d l i k e t o change t o " j u s t a c c e p t . . . t h e s t u d e n t s w i t h o u t w a n t i n g t o change them t o o d e s p e r a t e l y . " [p 17] What she saw i n h e r f a v o u r was h e r o t h e r i n t e r e s t s : g a r d e n i n g , d a n c e , m u s i c . " . . . m a t h t e a c h e r s must have q u o t e u n q u o t e a p e r s o n a l i t y , o k ? " [p 20] T h i s n o t i o n r e i n f o r c e s h e r g o a l f o r m a k i n g math human and " t o u c h a b l e . " T e a c h i n g m e t h o d s . The w o r d " t e a c h " meant t o c o n v e y i n f o r m a t i o n t h r o u g h l e c t u r i n g t o V i c t o r i a . W h i l e she p r e f e r r e d n o t t o " t e a c h , " she c l a i m e d i t was a t t i m e s n e c e s s a r y t o do s o . H e r r e a s o n s f o r n o t l i k i n g t h i s m e t h o d was t h a t i t a l i e n a t e d and p r o m o t e d t h e image o f m a t h e m a t i c s as t h i s " c o l d , h a r d t h i n g " [p 3] t h a t she was t r y i n g t o b r e a k . A l t e r n a t i v e methods she c o n s i d e r e d e f f e c t i v e were c o o p e r a t i v e o n e s : p a i r i n g , p a i r e d q u i z z e s , w a r m i n g up t h e c l a s s w i t h r e v i e w - 60 -q u e s t i o n s . A n o t h e r s t r a t e g y she l i k e d t o u s e was t o h a v e e a c h p e r s o n s o l v e a q u e s t i o n o n h i s / h e r own, t h e n f o r m a g r o u p w i t h o t h e r s a n d c o n v i n c e t h e o t h e r s i n t h e g r o u p t h a t h i s / h e r answer was c o r r e c t . W i t h t h e b e l i e f t h a t v e r b a l i z a t i o n a i d e d u n d e r s t a n d i n g , V i c t o r i a a d v o c a t e d g r o u p work f o r h e r c l a s s e s . The p r o c e s s c o n c r e t i z e s t h e a l g o r i t h m u s e d b y t h e s t u d e n t . The s t e p s become n a i l e d down so t h a t i f a n e r r o r i s made, t h e s t u d e n t c a n more e a s i l y d i s c e r n where i t o c c u r e d . V i e w o f s t u d e n t s . S t u d e n t d e v e l o p m e n t was a c o n c e r n o f V i c t o r i a ' s . She i l l u s t r a t e d t h i s when she d e s c r i b e d one o f h e r t e a c h i n g t e c h n i q u e s . "Walk me t h r o u g h i t " [p 4] was a p h r a s e she u s e d t o e n c o u r a g e s t u d e n t s t o more f u l l y e x p l a i n t h e i r a n s w e r . On t h e s u r f a c e t o t h e c l a s s , t h i s was done so t h a t t h e s t u d e n t w o u l d e x p l a i n h i s / h e r s t e p s t o t h e o t h e r s . H o w e v e r , V i c t o r i a u s e d i t as a t o o l t o f a c i l i t a t e g r o w t h o f t h e s t u d e n t . I t " i s o c c a s i o n a l l y p a i n f u l b u t . . . a t t h e same p o i n t t h e y ' r e g o i n g , h e y I c a n do t h i s , t h i s i s g o o d . " [p 4] S o , i n a d d i t i o n t o h e r g o a l o f h e l p i n g s t u d e n t s o v e r c o m e t h e i r f e a r o f m a t h , she a l s o saw v a l u e i n f o s t e r i n g t h e i r g r o w t h i n s e l f - e s t e e m . O t h e r I s s u e s S t u d e n t s ' v i e w o f m a t h e m a t i c s . V i c t o r i a c o n t r a s t e d t h e b l a c k and w h i t e n e s s o f t h e s e c o n d a r y c u r r i c u l u m w i t h t h e s t u d e n t s ' l i v e s . I n t h e t h r o e s o f e m o t i o n a l u n d u l a t i o n s w i t h i d e n t i t y c r i s e s and deep f a m i l i a l p r o b l e m s s u c h as d i v o r c i n g p a r e n t s , i t w o u l d be d i f f i c u l t f o r them t o a c c e p t a n d a s s i m i l a t e s o m e t h i n g so c o n c r e t e . M a t h was c o m p l e t e l y d i f f e r e n t f r o m L i f e . - 61 -" . . . i t makes math t h i s t h i n g where t h e y c a n ' t u n d e r s t a n d b e c a u s e so much o f t h e i r l i f e , I mean w i t h t h e i r [ p a r e n t a l ] s e p a r a t i o n s , w i t h what t h e y ' r e g o i n g t h r o u g h a t t h i s s t a g e i n t h e i r . l i f e . . . . t h e r e a r e g r e y a r e a s i n t h e i r l i f e , a n d a l o t o f t h i n g s a r e s u b j e c t t o c h a n g e . A n d i n math w e ' r e g i v i n g them a s e t , a c a r v e d i n s t o n e r u l e s b a s i c a l l y , and s a y i n g l e a r n t h e s e , do t h i s A n d i t ' s , i t ' s s u c h c o n t r a s t t o t h e r e s t o f t h e i r l i v e s t h a t i t s o m e t i m e s makes , i t a l i e n a t e s them f r o m i t , I t h i n k . " [p 3 - 4 ] M a r y A c a d e m i c B a c k g r o u n d M a r y h a d a B . S c . i n m a t h e m a t i c s and an M . S c . i n S t a t i s t i c s . She h a d t a u g h t i n t r o d u c t o r y s t a t i s t i c s t o m e d i c a l s t u d e n t s as a g r a d u a t e t e a c h i n g a s s i s t a n t . H e r s e c o n d a r y s c h o o l i n g o c c u r r e d i n V i e t n a m , a n d so d u r i n g t h e i n t e r v i e w she made c o m p a r i s o n s b e t w e e n t h e c u r r i c u l u m she t o o k a n d t h e one she t a u g h t . C o n c e p t i o n o f M a t h e m a t i c s To M a r y , math was n o t " c r u n c h i n g n u m b e r s " n o r d i d i t r e a l l y i n v o l v e c o n t e n t . R a t h e r , i t was p r o c e s s : l o g i c a l t h i n k i n g , s o l v i n g p r o b l e m s , d e d u c i n g a n d r e a s o n i n g , l o g i c . A t a h i g h e r l e v e l , i . e . g r a d u a t e w o r k , i t became p h i l o s o p h y . " [ T ] o me, math i s n o t so much what y o u know, um, i t ' s , i t ' s how t o t h i n k , how t o s o l v e p r o b l e m s , and how t o d e d u c e t h i n g s a n d r e a s o n . I t ' s , i t ' s l o g i c . I f y o u ' r e t a l k i n g a b o u t math a t a much more a d v a n c e d l e v e l , t h e n i t ' s a l m o s t p h i l o s o p h y . . . . J u s t s i t t i n g i n c l a s s , a n d y o u know, k i l l i n g a n h o u r , a n d g e t t i n g o u t o f t h e r e , c r u n c h i n g n u m b e r s , c o m p l e t e l y m e a n i n g l e s s . To me t h a t ' s n o t m a t h . " [p 1] M a r y ' s a t t r a c t i o n t o m a t h e m a t i c s was t h e e l e m e n t o f p r o b l e m -s o l v i n g f o r she e n j o y e d d o i n g p u z z l e s , a n d " f i g u r i n g t h i n g s o u t . " She l i k e n e d i t t o d o i n g c r o s s w o r d p u z z l e s , and e n j o y e d t h e s t i m u l a t i o n a n d c h a l l e n g e t h a t came w i t h e x p l o r i n g d i f f e r e n t ways t o s o l v e p u z z l e s a n d p r o b l e m s . To h e r , t h a t was t h e f u n i n m a t h . " I e n j o y e d m a t h e m a t i c s n o t so much b e c a u s e o f . . . t h e a c t u a l s u b j e c t s o r a r e a s o f m a t h e m a t i c s t h a t I h a d come a c r o s s as much as - 62 -i t i s a p r o b l e m - s o l v i n g t h i n g f o r me. I t ' s l i k e a p u z z l e . I l o v e d o i n g p u z z l e s , I l o v e s o l v i n g p r o b l e m s . . . . I l i k e h o m e w o r k . . . w h e r e y o u have t o go home and c r a c k y o u r h e a d o v e r some p r o b l e m s . [The u n i v e r s i t y p r o f e s s o r and I] w e ' d e x c h a n g e i d e a s o n how many d i f f e r e n t ways y o u c a n s o l v e a p r o b l e m . T h a t was c h a l l e n g i n g f o r me. T h a t was f u n . " [p 2] V i e w o f T e a c h i n g M a t h e m a t i c s T e a c h i n g s t u d e n t s t o l e a r n how t o t h i n k a n d how t o l e a r n were t h e m a j o r g o a l s o f M a r y ' s t e a c h i n g . These s k i l l s w o u l d p r e p a r e s t u d e n t s f o r c o p i n g w i t h p r o b l e m s t h e y w o u l d e n c o u n t e r i n t h e i r a d u l t l i f e . " . . . t o me i t ' s v e r y i m p o r t a n t f o r p e o p l e t o l e a r n how t o t h i n k l o g i c a l l y . . . T o me, [math] i s n o t l i k e t e a c h i n g a k i d how t o a d d , so much as t e a c h i n g h i m how t o t h i n k a b o u t s o l v i n g a p r o b l e m . . . . a l l o w i n g them to. do t h e i r own t h i n k i n g a n d c o m i n g up w i t h s o l u t i o n s a n d s o l v i n g t h i n g s . . . a n d t h a t s h o u l d t h e y e n c o u n t e r o t h e r p r o b l e m s i n l i f e , n o t n e c e s s a r i l y t o do w i t h m a t h e m a t i c s , t h a t t h e y a r e a b l e t o s o l v e them y o u k n o w . " [p 1] " . . . t h e most i m p o r t a n t p a r t o f l e a r n i n g m a t h e m a t i c s i s t h a t t h e k i d s a r e a b l e t o t h i n k and t a c k l e t h e p r o b l e m . " [p 9] " I d o n ' t t h i n k i t ' s as i m p o r t a n t t o p u s h as much c o n t e n t as i t i s t o t e a c h k i d s how t o t h i n k " [p 10] I n f a c t , t h e p h i l o s o p h y o f t e a c h i n g m a t h e m a t i c s as d e v e l o p i n g s t u d e n t s ' t h i n k i n g s k i l l s c a r r i e d o v e r f o r M a r y t o t e a c h i n g i n g e n e r a l . She saw no d i f f e r e n c e b e t w e e n t e a c h i n g m a t h e m a t i c s a n d t e a c h i n g E n g l i s h . B o t h meant h e l p i n g s t u d e n t s d e v e l o p s k i l l s o f a n a l y s i s , s t r a t e g i z i n g a n d s o l v i n g p r o b l e m s . The l i n e a r , l o g i c a l p r o c e s s u s e d t o w r i t e a p e r s u a s i v e e s s a y was t h e same one u s e d t o f o r m u l a t e a p r o o f i n E u c l i d e a n g e o m e t r y , a c c o r d i n g t o M a r y . " [ T e a c h i n g ] r e a l l y i s n ' t a b o u t m a t h e m a t i c s . T h a t i t i s t h a t t h e y ' r e l e a r n i n g how t o t h i n k . How t o p u t t h i n g s i n o r d e r . . . a n d make s t r a t e g i e s f o r t h i n g s , a n d so on so f o r t h . I t ' s n o t so much a b o u t t h e c o n t e n t as much as a b o u t , y o u know, how t o , how t o s o l v e a p r o b l e m . I t ' s l i k e any o t h e r c o u r s e . . . T o me, t h e y ' r e a l l t h e s a m e . . . y o u know f o r e x a m p l e , t o w r i t e an e s s a y . You s t i l l h a v e t o come up w i t h a t o p i c t h a t y o u want t o w r i t e a b o u t , o r an a r g u m e n t , s e t up how y o u ' r e g o i n g t o a r g u e y o u r p o i n t . . . . I t ' s j u s t a d i f f e r e n t , um, medium, o f s o l v i n g a p r o b l e m . " [p 13] " . . . t h e more I t e a c h t h e more I r e a l i z e t h a t t e a c h i n g math has n o t h i n g t o do w i t h m a t h . I t has t o do w i t h t e a c h i n g . . . . Y o u go f r o m E n g l i s h t o b i o l o g y t o math t o p h y s i c s t o , u h , S p a n i s h . I t ' s a l l d e v e l o p i n g a way o f l e a r n i n g , s h o w i n g k i d s how t o l e a r n . " [p 14] - 63 -A n o t h e r g o a l o f t e a c h i n g m a t h e m a t i c s f o r M a r y was t o c h a n g e t h e p o p u l a r p e r c e p t i o n o f m a t h e m a t i c s as b o r i n g , and n u m b e r - c r u n c h i n g . " I have s e e n a l o t o f k i d s g r o w i n g up h a t i n g m a t h e m a t i c s a n d h a t i n g a r i t h m e t i c , o r - . J u s t d r y . J u s t s i t t i n g i n c l a s s , a n d y o u know, k i l l i n g a n h o u r , a n d g e t t i n g o u t o f t h e r e , c r u n c h i n g n u m b e r s , c o m p l e t e l y m e a n i n g l e s s . To me, t h a t ' s n o t m a t h . " [p 1-2] C o n s i s t e n t w i t h t h e p h i l o s o p h y o f p r o b l e m - s o l v i n g as t h e p u r p o s e o f t e a c h i n g , she d e s c r i b e d a t e a c h e r as "somebody who p o s e s q u e s t i o n s t h a t . . . g e t s t h e i r c u r i o s i t y up a n d , i s a b l e t o g u i d e them t h r o u g h s o l v i n g a p r o b l e m . . . " [p 7] I n e n u m e r a t i n g q u a l i t i e s she b e l i e v e d n e c e s s a r y f o r a m a t h e m a t i c s t e a c h e r , M a r y s t a t e d s e v e r a l t i m e s t h a t a t e a c h e r s h o u l d have a l o t o f e n e r g y , p o s s i b l y m i r r o r i n g t h e e n e r g y she f e l t she n e e d e d d u r i n g h e r p r a c t i c u m . E n t h u s i a s m , c o n f i d e n c e , t h e a b i l i t y t o h a n d l e q u e s t i o n s a n d k n o w l e d g e t o e x p l a i n t h e c o n c e p t t h o r o u g h l y were o t h e r c h a r a c t e r i s t i c s . B e i n g a b l e t o have f u n w i t h t h e s t u d e n t s , g e t t i n g them i n t e r e s t e d a n d p a r t i c i p a t i n g were a l s o l i s t e d . Time was a c r i t i c a l e l e m e n t . The g o o d t e a c h e r r e q u i r e d a l o t o f t i m e f o r p r o p e r l e s s o n p r e p a r a t i o n . The g o o d t e a c h e r w o u l d " t h i n k v e r y c a r e f u l l y a b o u t t e a c h i n g t h e c o n c e p t a n d how y o u ' r e g o n n a g e t k i d s t o t h i n k a b o u t i t . " [p 4] M a r y was shown t h e d i a g r a m s ( A p p e n d i x A) r e p r e s e n t i n g p o s s i b l e c o n c e p t i o n s o f t e a c h i n g . H o w e v e r , none o f them seemed t o c a p t u r e t h e e s s e n c e o f h e r b e l i e f s , t h a t t e a c h i n g math was t e a c h i n g p r o b l e m - s o l v i n g . T e a c h i n g m e t h o d s . M a r y b e g a n h e r p r a c t i c u m b e l i e v i n g t h a t d r i l l i n g f a c t s was t h e most e f f i c i e n t way o f t e a c h i n g . When she n o t i c e d how s t u d e n t s were u n s u c c e s s f u l i n s o l v i n g t h e d i v e r s i t y o f E u c l i d e a n p r o o f p r o b l e m s i n G r a d e 10 g e o m e t r y she c h a n g e d h e r i d e a s . - 64 -" T h a t comes f r o m G r a d e 10 g e o m e t r y . ' C a u s e e v e r y p r o b l e m ' s d i f f e r e n t , r i g h t ? . . . . a n d y o u see t h a t g o i n g f r o m one p r o b l e m t o a n o t h e r , t h e y j u s t , t h e y , t h e y ' r e s tumped r i g h t away. You know. T h e y d o n ' t have t h e a b i l i t y t o t h i n k on t h e i r o w n . . . . L i k e i f I h a d n o t t a u g h t t h e g e o m e t r y p a r t I w o u l d n o t have r e a l i z e d t h a t , I d o n ' t t h i n k . " [p 10-11] Now she d i s a g r e e d w i t h s i m p l y p r e s e n t i n g a c o n c e p t t o s t u d e n t s . I n s t e a d , she b e l i e v e d g o o d t e a c h i n g e n t a i l e d a g o o d l e s s o n p l a n , one w h i c h was c a r e f u l l y l a i d o u t , w h i c h h a d l o g i c and f l o w , a n d w h i c h l e a d s t u d e n t s t o t h i n k a b o u t t h e t o p i c b y u s i n g p e r t i n e n t e x a m p l e s . M a r y b e l i e v e d a n e f f e c t i v e l e s s o n p l a n w o u l d l e a d s t u d e n t s t o u n d e r s t a n d i n g , a n d i t was w o r t h s p e n d i n g h o u r s t o d e v e l o p . " Y o u r most p r o d u c t i v e h o u r i s t h e h o u r i n w h i c h y o u do y o u r l e s s o n p l a n " [p 14] a n d a g o o d one " w i l l come o u t v e r y n a t u r a l l y , f l o w , and t h i n g s w i l l work o u t v e r y w e l l . " [p 8] "Um, how w o u l d I r e c o g n i z e [a good l e s s o n p l a n ] ? The f l o w o f t h e l e s s o n , t h e l o g i c . . . y o u c a n t e l l when a l e s s o n p l a n has b e e n c a r e f u l l y t h o u g h t o u t , t h a t t h e r e was e x a m p l e s l e a d i n g up t o a c e r t a i n i d e a , and t h e n g e t k i d s t o t h i n k a b o u t i t a n d t h e n , a n d f i n a l l y c l u i n g i n on t h e i r own, what i s g o i n g o n , why t h a t i s , c e r t a i n t h i n g s h a p p e n , as o p p o s e d t o j u s t g i v i n g t h e c o n c e p t t o t h e k i d s . " [p 5] V i e w o f s t u d e n t s . M a r y e n j o y e d t h e s t u d e n t s . She e n j o y e d t h a t t h e G r a d e 9 s t u d e n t s were s t i l l " f u l l o f p e r s o n a l i t y " [p 16] a n d f o u n d them " s o c u t e . " [p 11] The f a c t t h a t t h e y c o u l d be q u i t e o u t s p o k e n i n c l a s s d i d n o t p e r t u r b h e r . She a t t r i b u t e d t h i s t o t h e i r i n t r i n s i c c u r i o s i t y . She c o n s i d e r e d d i s c i p l i n i n g a w a s t e o f t i m e e v e n t h o u g h i t was a t t i m e s n e c e s s a r y . Some o f t h e d i s c i p l i n e p r o b l e m s , i n M a r y ' s o p i n i o n , came f r o m t h e s t u d e n t s b e i n g " s p o o n f e d , " [p 7] r a t h e r t h a n b e i n g p u s h e d t o t h i n k i n d e p e n d e n t l y . M a r y b e l i e v e d t h a t s t u d e n t s h a d an i n h e r e n t a b i l i t y t o s t r a t e g i z e ( t h i n k ) a n d i f t h e y were g i v e n t h e o p p o r t u n i t y t o d e v e l o p - 65 -t h i s , c o n t e n t u n d e r s t a n d i n g w o u l d n a t u r a l l y f o l l o w . To h e r , s t u d e n t s h a d l o s t t h e i r a b i l i t y t o t h i n k b e y o n d a g i v e n e x a m p l e b e c a u s e t h e y were a c c u s t o m e d t o d o i n g m a t h e m a t i c s v i a r o t e p r o c e d u r e s . " . . . t h e y d o n ' t e v e n . . . t h i n k t h e y ' r e c a p a b l e o f s o l v i n g s o m e t h i n g u n l e s s t h e y h a v e s e e n i t b e f o r e . . . A n d t h a t ' s a s h a m e . " [p 9] " I d o n ' t c a r e a b o u t l i n e a r e q u a t i o n s . I c a r e i f t h e k i d s c a n r e c o g n i z e i t r i g h t away, a n d a r e t h e y h a v i n g f u n , a r e t h e y , y o u know, f o r g e t t i n g t h e y a r e l e a r n i n g a b o u t m a t h , a n d t h e i r n a t u r a l [ c u r i o u s ] s e l v e s come o u t . " [p 15] " I mean i f y o u a l l o w them t i m e t o t h i n k , y o u ' d be a m a z e d . . . " [p 7] O t h e r I s s u e s C o n c e p t u a l C h a n g e . B e c a u s e M a r y h a d e m i g r a t e d t o B r i t i s h C o l u m b i a a f t e r c o m p l e t i n g h i g h s c h o o l i n V i e t n a m , she was a b l e t o compare t h e two m a t h e m a t i c s c u r r i c u l a . H e r o b s e r v a t i o n s were a f a c t o r i n c h a n g i n g h e r i d e a s o f t e a c h i n g f r o m a d r i l l and p r a c t i c e , l e a r n - t h e - f a c t s a p p r o a c h t o one t h a t s o u g h t t o d e v e l o p t h i n k i n g a b i l i t i e s . " . . . I t h i n k k i d s h e r e compared t o o t h e r c o u n t r i e s a r e q u i t e weak i n m a t h e m a t i c s , um, and t h e y ' r e n o t p u s h e d h a r d e n o u g h . T h e y ' r e n o t a l l o w e d t o t h i n k . A n d t h e y a r e b e i n g j u s t , t h e y ' r e j u s t s p o o n f e d a l l t h e t i m e . . . . I h a d j u s t f i n i s h e d h i g h s c h o o l i n my c o u n t r y , I came h e r e , um - d o n ' t l a u g h , i t was r e a l l y e a s y . I h a d t o do a G r a d e 12 e q u i v a l e n c y a l l o v e r a g a i n . I t was a j o k e ! You know, I c o u l d do [ t h a t ] when I was i n G r a d e 9 i n my own c o u n t r y . " [p 6-7] " I ' m v e r y u s e d t o b e i n g d r i l l e d . . . . a n d h a v i n g t a u g h t d u r i n g my p r a c t i c u m I r e a l i z e d t h a t t h e r e , t h e r e i s a n o t h e r way o f t e a c h i n g . " [p 9-10] A n o t h e r f a c t o r i n e f f e c t i n g a change i n h e r i d e a s o f t e a c h i n g was t e a c h i n g g e o m e t r y , as e x p l a i n e d a b o v e . T e a c h i n g p r e p a r a t i o n and d u t i e s . M a r y s p e n t much t i m e p r e p a r i n g h e r l e s s o n s b e c a u s e she b e l i e v e d t h i s was t h e most i m p o r t a n t a s p e c t o f h e r j o b . I n f a c t , she was so - 66 -c o n c e r n e d t h a t s t u d e n t s u n d e r s t o o d t h e c o n c e p t b e i n g t a u g h t she d i d n ' t seem t o b a l a n c e h e r t i m e . " Y o u know [ s c h o o l a d v i s o r ] t o l d me I s p e n t way t o o much t i m e t r y i n g t o p e r f e c t t h e l e s s o n , t r y i n g t o make i t f u n f o r t h e k i d s a n d he s a i d I c a n ' t a l w a y s do t h a t b e c a u s e I ' l l d r a i n m y s e l f r i g h t o u t o f i t . " [p 6] M a r y r e s i s t e d c a r r y i n g o u t o t h e r d u t i e s n o r m a l l y a s s o c i a t e d w i t h t e a c h i n g : a d m i n i s t r a t i v e d e t a i l s , m a r k i n g a t t e n d a n c e , f o l l o w i n g up o n n o n - a t t e n d a n c e , d e a l i n g w i t h p a r e n t s and c o u n s e l l o r s . She c o n s i d e r e d i t a w a s t e o f h e r p r e p a r a t i o n t i m e and h e r t e a c h i n g t i m e . Summaries To e n c a p s u l a t e t h e i r i d e a s and f a c i l i t a t e c o m p a r i s o n , t h e c o n c e p t i o n s o f t h e s e s i x p r o s p e c t i v e t e a c h e r s a r e s u m m a r i z e d . Y v e t t e : C o n t e n t P r e s e n t a t i o n " I t e a c h m a t h . " Y v e t t e ' s n o t i o n o f t e a c h i n g m a t h e m a t i c s c e n t r e d a b o u t c o n t e n t p r e s e n t a t i o n . She f e l t s t u d e n t s were r e s p o n s i b l e f o r t h e i r own l e a r n i n g , a n d h e r r o l e was t o p r e s e n t t h e m a t e r i a l e f f e c t i v e l y . T h i s meant u s i n g a v a r i e t y o f t e c h n i q u e s t o f a v o u r d i f f e r e n t l e a r n i n g s t y l e s . S h a n n o n : A Buddy S h a n n o n ' s c o n c e p t i o n o f t e a c h i n g m a t h e m a t i c s f o c u s e d o n b u i l d i n g a o n e - t o - o n e r e l a t i o n s h i p w i t h h e r s t u d e n t s . T h i s was e v i d e n c e d b y h e r d e s i r e t o f o r m a r e l a t i o n s h i p w i t h e a c h , h e r f r u s t r a t i o n a t n o t b e i n g a b l e t o a c c o m p l i s h t h i s , h e r c l a i m t h a t h e l p i n g t h o s e who d i d p o o r l y was a h i g h p r i o r i t y , and h e r c h o i c e o f a r o l e m o d e l - someone p o r t r a y e d as v a l u i n g h i s s t u d e n t s o v e r a l l . - 67 -S h a n n o n h a d a s t r o n g a n d p e r v a s i v e s e n s e o f d u a l i t y , i . e . t h e r e e x i s t e d a r i g h t way v e r s u s a wrong way. She b e l i e v e d t h o s e i n a u t h o r i t y s h o u l d h a v e c o r r e c t a n s w e r s f o r a l l q u e s t i o n s . A t UBC, t h o s e were t h e p r o f e s s o r s ; i n t h e c l a s s r o o m , i t was h e r s e l f . S h a n n o n f a v o u r e d t h e u t i l i t y o f m a t h e m a t i c s a n d h e r g o a l i n t e a c h i n g was t o i m p a r t t h a t a p p l i c a b i l i t y . She a d v o c a t e d d r i l l i n g as t h e most e f f e c t i v e way t o t e a c h number f a c t s . D o u g : A R o l e M o d e l D o u g ' s c o n c e p t i o n was d r i v e n b y h i s b e l i e f t h a t s t u d e n t s were i n a d e v e l o p m e n t a l p h a s e b e f o r e a d u l t h o o d . B e i n g a t e a c h e r meant b e i n g a r o l e m o d e l f o r them a n d d i s p l a y i n g t h e p r i n c i p l e s he v a l u e d . F o r e x a m p l e , b y s h o w i n g them he somet imes s t u m b l e d when s o l v i n g a m a t h e m a t i c s p r o b l e m he b e l i e v e d he was d e m o n s t r a t i n g t h a t m a k i n g m i s t a k e s a n d l e a r n i n g f r o m them was p a r t o f e v e r y o n e ' s l i f e . Doug b e l i e v e d m a t h e m a t i c a l k n o w l e d g e was c r e a t e d a n d c o n s t a n t l y c h a n g i n g . He w a n t e d t o s h a r e t h i s p e r s p e c t i v e w i t h s t u d e n t s . He a l s o b e l i e v e d m a t h e m a t i c s h a d a n o t h e r p u r p o s e f o r t r o u b l e d t e e n a g e r s . He c l a i m e d t h e r i g i d i t y and r o u t i n e n e s s i n t h e m a t h e m a t i c s t h e y were l e a r n i n g p r o v i d e d them a f o r m o f s t a b i l i t y . R a v : M u l t i p l e R o l e s , A G a r d e n e r Ray v i e w e d t e a c h i n g m a t h e m a t i c s as i n v o l v i n g s e v e r a l d i f f e r e n t r o l e s i n c l u d i n g t h a t o f c o u n s e l l o r , p a r e n t a n d f r i e n d as w e l l as t e a c h e r . H i s g o a l was f o r k i d s t o l o v e l e a r n i n g math a n d he b e l i e v e d t h i s c o u l d be a c h i e v e d i f s t u d e n t s f o u n d m a t h e m a t i c s u s e f u l , f u n , a n d r e l e v a n t . On a b r o a d e r s c a l e , he b e l i e v e d t h e t e a c h e r ' s r o l e was t o - 68 -"hook them t o s c h o o l . " R a y ' s m e t a p h o r f o r t e a c h i n g was a g a r d e n e r t a k i n g c a r e o f p l a n t s . V i c t o r i a : C h a n g i n g Images , A P a i n t e r V i c t o r i a ' s g o a l was t o d i s p e l t h e n o t i o n t h a t m a t h e m a t i c s was a " c o l d , h a r d o b j e c t " p r a c t i c e d o n l y b y g e e k s and n e r d s . She w a n t e d t o i n s t i l i n h e r s t u d e n t s t h a t d o i n g math r e q u i r e d c r e a t i v i t y a n d i n t u i t i o n as w e l l as a s e n s e o f l o g i c . She f e l t l o w e r l e v e l math was t o o p r o c e s s -o r i e n t e d t o a d d r e s s t h e c r e a t i o n o f m a t h e m a t i c s . T h u s , t h e t r u e n a t u r e o f m a t h e m a t i c s c o u l d n o t be r e a l i z e d . S c h o o l math was d i f f e r e n t f r o m r e a l m a t h . V i c t o r i a b e l i e v e d t h e r i g i d i t y and c e r t a i n t y i n s c h o o l m a t h e m a t i c s a l i e n a t e d s t u d e n t s b e c a u s e i t c o n t r a s t e d w i t h u n c e r t a i n t i e s i n o t h e r a r e a s o f t h e i r l i f e . She c a r r i e d . t h i s c o m p l e x i t y o f l i f e i n t o h e r m e t a p h o r f o r t e a c h i n g . The t e a c h e r was a p a i n t e r ; t h e s t u d e n t t h e t a p e s t r y b e i n g w o r k e d o n . O t h e r s h a d w o r k e d o n i t e a r l i e r - f o r m e r t e a c h e r s , p a r e n t s , f r i e n d s , f a m i l y - a n d now, as one o f t h e s t u d e n t ' s t e a c h e r s , i t was h e r t u r n t o c o n t r i b u t e t o t h e t a p e s t r y . M a r y : P r o b l e m - S o l v i n g M a r y ' s c o n c e p t i o n o f t e a c h i n g math was c o n s i s t e n t w i t h h e r i d e a s o f m a t h e m a t i c s as a d i s c i p l i n e , h e r b e l i e f i n t h e p u r p o s e o f t e a c h i n g m a t h e m a t i c s a n d h e r v i s i o n o f a math t e a c h e r . E s s e n t i a l l y , " i t r e a l l y i s n ' t a b o u t m a t h " b u t i t was a b o u t c r i t i c a l t h i n k i n g a n d p r o b l e m s o l v i n g i n p r e p a r a t i o n f o r l i f e . C o n s e q u e n t l y , she i n f u s e d h e r t e a c h i n g w i t h p r o b l e m s r e l e v a n t t o t h e s t u d e n t s and e n c o u r a g e d s t u d e n t s t o s t r e t c h t h e i r t h i n k i n g b e y o n d t h e r o t e and l o o k f o r s t r a t e g i e s b e y o n d t e x t e x a m p l e s . She h e r s e l f f o u n d p r o b l e m s o l v i n g and d o i n g p u z z l e s t h e most - 69 -e n j o y a b l e e l e m e n t s o f m a t h e m a t i c s a n d w a n t e d h e r s t u d e n t s t o e x p e r i e n c e t h i s as f u n t o o . T e a c h i n g m a t h e m a t i c s i s t e a c h i n g t h i n k i n g a n d t e a c h i n g p r o b l e m s o l v i n g w o u l d be M a r y ' s a x i o m . - 70 -C h a p t e r 5 : C o n c l u s i o n s , O t h e r I s s u e s & R e c o m m e n d a t i o n s C o n c l u s i o n s The f i r s t r e s e a r c h q u e s t i o n f o r t h i s s t u d y was: what, a r e t h e c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s h e l d b y i n d i v i d u a l s p r e p a r i n g t o t e a c h s e c o n d a r y m a t h e m a t i c s ? The r e s u l t s show c l e a r d i f f e r e n c e s i n t h e c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s f o r e a c h p e r s o n . P e r s p e c t i v e s r a n g e d f r o m a n e m p h a s i s o n t h e i n d i v i d u a l s t u d e n t t o h i g h l i g h t i n g c o n t e n t . Those p r e - s e r v i c e t e a c h e r s who e m p h a s i z e d t h e s t u d e n t d i f f e r e d i n w h e t h e r s i g n i f i c a n c e was p l a c e d o n c o g n i t i v e s k i l l s o r p s y c h o l o g i c a l d e v e l o p m e n t . Those who e m p h a s i z e d c o n t e n t w a n t e d t o d e m o n s t r a t e v a r i o u s a s p e c t s o f m a t h e m a t i c s - t h a t i t i n v o l v e d c r i t i c a l t h i n k i n g , t h a t i t was r e l e v a n t t o s t u d e n t s ' l i v e s , a n d t h a t i t was f u n . T h o u g h e a c h p e r s o n ' s c o n c e p t i o n o f t e a c h i n g m a t h e m a t i c s e n c o m p a s s e s s e v e r a l a s p e c t s w h i c h c a n n o t be s l o t t e d e x c l u s i v e l y i n t o a s i n g l e c a t e g o r y , i t i s n e v e r t h e l e s s u s e f u l t o be a b l e t o d e s c r i b e t h e d o m i n a n t f e a t u r e s o f t h e i r v i e w . P r a t t ' s (1992) model f o r d e s c r i b i n g c o n c e p t i o n s i s h e l p f u l i n t h i s r e g a r d . Y v e t t e e x h i b i t e d an E n g i n e e r i n g a p p r o a c h t o t e a c h i n g m a t h e m a t i c s . H e r o b j e c t i v e i n t e a c h i n g was t o d e l i v e r t h e c o n t e n t o f h e r s u b j e c t i n t h e most e f f e c t i v e way. Doug a n d S h a n n o n , whose p r i m a r y e m p h a s i s i n t e a c h i n g m a t h e m a t i c s was t o f o s t e r s e l f - e s t e e m and c u l t i v a t e d e v e l o p m e n t i n t h e i r s t u d e n t s , t y p i f i e d a N u r t u r i n g c o n c e p t i o n . F o r t h e m , h e l p i n g s t u d e n t s t o m a t u r e and g a i n s e l f - c o n f i d e n c e was a n i m p o r t a n t o b j e c t i v e . - 71 -S h a n n o n i d e n t i f i e d a s h o r t - t e r m g o a l f o r t h e s t u d e n t s i n t h a t she w a n t e d t o h e l p them p a s s t h e c o u r s e whereas Doug was c l e a r he was p r e p a r i n g h i s s t u d e n t s f o r a d u l t h o o d . D o u g ' s a t t e n t i o n t o h i s s t u d e n t s ' r e a s o n i n g a b i l i t y a l s o i l l u s t r a t e d P r a t t ' s (1992) D e v e l o p m e n t a l c o n c e p t i o n where " c u l t i v a t i n g t h e i n t e l l e c t " (p 213) d o m i n a t e s . M a r y ' s p e r c e p t i o n a l s o t y p i f i e d t h e D e v e l o p m e n t a l c o n c e p t i o n . She f o c u s s e d o n p r e p a r i n g s t u d e n t s f o r t h e i r f u t u r e when she t a u g h t ; h e r e m p h a s i s l a y i n d e v e l o p i n g t h e i r a b i l i t y t o s o l v e p r o b l e m s b y t h i n k i n g c r i t i c a l l y t h r o u g h m a t h e m a t i c a l q u e s t i o n s . R a y ' s v i e w o f t e a c h i n g m a t h e m a t i c s showed a s p e c t s o f t h e A p p r e n t i c e s h i p a n d N u r t u r i n g c o n c e p t i o n s . He b e l i e v e d t h a t math was p r e s e n t e v e r y w h e r e a n d he w a n t e d s t u d e n t s t o r e a l i z e t h i s . I n t h i s way, he hoped t o e n g e n d e r a d e e p e r a p p r e c i a t i o n f o r t h e s u b j e c t i n h i s s t u d e n t s . He a l s o d e m o n s t r a t e d c o n c e r n f o r s t u d e n t s b y g i v i n g them i n d i v i d u a l a t t e n t i o n he f e l t t h e y n e e d e d . He c l a i m e d a t l e a s t h a l f t h e j o b o f t e a c h i n g h a d n o t h i n g t o do w i t h c o n t e n t , b u t e n t a i l e d " b e i n g t h e r e " i n o t h e r c a p a c i t i e s f o r t h e t e e n a g e r s . T h o u g h P r a t t ' s (1992) model i s u s e f u l f o r d e s c r i b i n g c o n c e p t i o n s o f t e a c h i n g i n g e n e r a l , i t l a c k s s u b j e c t s p e c i f i c i t y . What n e e d s t o be i n c l u d e d a r e t h e d i f f e r e n t a s p e c t s o f m a t h e m a t i c s o n w h i c h t e a c h e r s f o c u s when t h e y t e a c h . V i c t o r i a ' s i d e a o f t h e t e a c h e r as a p a i n t e r d i s p l a y e d a s p e c t s o f a N u r t u r i n g c o n c e p t i o n , b u t h e r p r i m a r y g o a l was c o n t e n t - f o c u s s e d ; she w a n t e d t o d i s p e l n e g a t i v e s t e r e o t y p e s o f m a t h e m a t i c s a n d m a t h e m a t i c i a n s . Ray b e l i e v e d he c o u l d show s t u d e n t s t h e f u n i n m a t h e m a t i c s a n d how i t was commonly u s e d and t h i s w o u l d " h o o k " them o n t o t h e s u b j e c t and o n t o s c h o o l . M a r y a l s o s o u g h t t o show - 72 -s t u d e n t s t h a t m a t h e m a t i c s was r e l e v a n t b y f o r m u l a t i n g p r o b l e m s w h i c h p e r t a i n e d t o t h e i r e v e r y d a y a c t i v i t i e s . The s e c o n d r e s e a r c h q u e s t i o n was: do t h o s e w i t h g r a d u a t e s t u d i e s i n m a t h e m a t i c s h o l d a q u a l i t a t i v e l y d i f f e r e n t c o n c e p t i o n o f t e a c h i n g m a t h e m a t i c s f r o m t h o s e w i t h o n l y a b a c c a l a u r e a t e ? M a r y , R a y , a n d V i c t o r i a h a d c o m p l e t e d a l l o r p a r t o f a g r a d u a t e d e g r e e i n m a t h e m a t i c s . Y v e t t e and Shannon h a d b a c h e l o r o f s c i e n c e d e g r e e s a n d Doug h a d a l i b e r a l a r t s d e g r e e w i t h a m a t h e m a t i c s s p e c i a l t y . S p e c i f i c a l l y , a l l s i x t e a c h e r c a n d i d a t e s h a d v a r y i n g c o n c e p t i o n s u n i q u e t o t h e i n d i v i d u a l . H o w e v e r , what d i s t i n g u i s h e d M a r y ' s , R a y ' s a n d V i c t o r i a ' s d e s c r i p t i o n s o f t e a c h i n g m a t h e m a t i c s was how t h e y w a n t e d t o s h a r e t h e i r e n j o y m e n t o f t h e s u b j e c t w i t h t h e i r s t u d e n t s . A l l t h r e e i n d i c a t e d a d e s i r e f o r s t u d e n t s t o have f u n d o i n g m a t h e m a t i c s . V i c t o r i a ' s g o a l i n t e a c h i n g m a t h e m a t i c s was t o d i s p e l t h e c o m m o n l y - h e l d image o f m a t h e m a t i c s as o b j e c t i v e and d i f f i c u l t t o c o m p r e h e n d . She s t a t e d w a n t i n g t o make math " h u m a n . " Ray b e l i e v e d t h a t b e i n g a t e a c h e r meant b e i n g o t h e r t h i n g s t o o , l i k e a c o u n s e l l o r a n d a f r i e n d . H o w e v e r , i n d r a w i n g s t u d e n t s t o s c h o o l , he w a n t e d t o draw them t o m a t h e m a t i c s as w e l l , and he b e l i e v e d t h a t f o r m i n g m a t h e m a t i c a l s i t u a t i o n s t h a t s t u d e n t s c o u l d r e l a t e t o a n d t h e y w o u l d f i n d f u n w o u l d a c c o m p l i s h t h i s . M a r y ' s g o a l f o r t e a c h i n g was t o f o s t e r t h e a b i l i t y o f h e r s t u d e n t s t o t h i n k c r i t i c a l l y . T h i s i s s i m i l a r t o D o u g ' s g o a l f o r t e a c h i n g a n d i n f a c t , b o t h s t a t e d t h a t t e a c h i n g s t u d e n t s t o r e a s o n was more i m p o r t a n t t h a n h a v i n g s t u d e n t s l e a r n c o n t e n t . H o w e v e r , M a r y a l s o w a n t e d t o s p e c i f i c a l l y change s t u d e n t s ' p e r c e p t i o n s o f math c l a s s a n d m a t h e m a t i c s . She w a n t e d them t o f i n d math i n t e r e s t i n g a n d e x c i t i n g f o r - 73 -how i t c o u l d be a p p l i e d r a t h e r t h a n b e i n g o n l y n u m b e r - c r u n c h i n g a n d b o r i n g . M a r y , Ray and V i c t o r i a b e l i e v e d t h a t m a t h e m a t i c s was a c c e s s i b l e a n d c o u l d be e n j o y a b l e f o r a l l s t u d e n t s . Y v e t t e a n d S h a n n o n , o n t h e o t h e r h a n d , d i d n o t e x p r e s s s u c h i n c l i n a t i o n s . W h i l e t h e y b o t h d e s c r i b e d a s p e c t s o f m a t h e m a t i c s w h i c h t h e y e n j o y e d , t h e r e was no r e f e r e n c e t o " t u r n i n g k i d s o n " t o t h e s u b j e c t i t s e l f . R a t h e r t h a n t r y i n g t o i n s t i l a l o v e f o r m a t h , t h e y were more o r i e n t e d t o w a r d g e t t i n g s t u d e n t s t h r o u g h t h e m a t e r i a l b y b e i n g a n e f f e c t i v e f a c i l i t a t o r , i n t h e c a s e o f Y v e t t e , a n d t h r o u g h b e i n g a n e m p a t h e t i c f r i e n d , i n t h e c a s e o f S h a n n o n . Doug a l s o d i d n o t e x p r e s s g e n e r a t i n g a l o v e f o r math as p a r t o f h i s t e a c h i n g g o a l . B e s i d e s h a v i n g d i f f e r e n t v i e w s o f t e a c h i n g , t h e t e a c h e r c a n d i d a t e s a l s o v a r i e d i n t h e i r v i e w o f m a t h e m a t i c s . F o r some, t h e r e a p p e a r e d t o be a s t r o n g l i n k b e t w e e n t h e i r v i e w o f m a t h e m a t i c s a n d t h e i r e x p r e s s e d v i e w s o f t e a c h i n g . S h a n n o n a n d Y v e t t e h e l d P l a t o n i s t v i e w s o f m a t h e m a t i c s . Y v e t t e s t a t e d how she e n j o y e d f i n d i n g t h e answer when i t came t o d o i n g m a t h e m a t i c s , i m p l y i n g t h e r e was one a n s w e r . Shannon c l a i m e d m a t h h a d a r i g h t o r w r o n g a n s w e r , a n d e v e n d e s c r i b e d t h e o r e t i c a l m a t h e m a t i c s as m e m o r i z a t i o n . T h e y b o t h saw m a t h e m a t i c s as b e i n g a b o d y o f c o n t e n t a n d t h e y u n d e r s t o o d t h e i r j o b as a t e a c h e r as h e l p i n g s t u d e n t s l e a r n t h a t c o n t e n t . Y v e t t e ' s f o c u s on u s i n g v a r i o u s methods t o c o n v e y t o p i c s a n d S h a n n o n ' s p r o p e n s i t y t o d r i l l i n g a r e c o n s i s t e n t w i t h a P l a t o n i c v i e w o f m a t h . - 74 -Doug h e l d a f a l l i b i l i s t v i e w . He c o n s i d e r e d m a t h e m a t i c s t o be c r e a t e d t h r o u g h human e n d e a v o u r and s u b j e c t t o c h a n g e . As s u c h , m a t h e m a t i c s was d y n a m i c and f a l l i b l e . He was v e r y c l e a r i n c o n n e c t i n g t h i s b e l i e f w i t h what he w a n t e d t o c o n v e y t h r o u g h h i s t e a c h i n g . F o r e x a m p l e , he f e l t t h a t b y s h o w i n g s t u d e n t s he t o o made m i s t a k e s , t h e y w o u l d l e a r n t h a t l i f e i n g e n e r a l , a n d m a t h e m a t i c s i n p a r t i c u l a r , was n o t p r e d e t e r m i n e d . M a r y a n d V i c t o r i a a l s o v i e w e d math as a d y n a m i c s u b j e c t . M a r y c h a r a c t e r i z e d m a t h e m a t i c s as p r o c e s s r a t h e r t h a n c o n t e n t and g r a d u a t e l e v e l m a t h e m a t i c s as p h i l o s o p h y . H e r c o n c e p t i o n o f t e a c h i n g was one i n w h i c h she e n c o u r a g e d s t u d e n t s t o u s e t h e t h i n k i n g a n d r e a s o n i n g p r o c e s s e s she b e l i e v e d i n h e r e n t i n t h e s u b j e c t . V i c t o r i a d e s c r i b e d d o i n g m a t h e m a t i c s as r e q u i r i n g c r e a t i v i t y a n d i n t u i t i o n . She e x p l i c i t l y s t a t e d she w a n t e d t o show s t u d e n t s t h i s a s p e c t o f m a t h e m a t i c s . H o w e v e r , a r e l a t i o n s h i p b e t w e e n t h i s v i e w o f m a t h e m a t i c s a n d t h e t e a c h i n g methods she d e s c r i b e d i s n o t e v i d e n t . T h i s l a c k o f a n a p p a r e n t r e l a t i o n s h i p c a n p o s s i b l y be a t t r i b u t e d t o V i c t o r i a ' s u n d e r s t a n d i n g s o f i n s t r u c t i o n a l t e c h n i q u e s a n d t h e d i f f e r e n t i a t i o n she made b e t w e e n s c h o o l math ( a l l known) a n d n o n - s c h o o l math ( s t i l l b e i n g c r e a t e d ) : how c o u l d y o u t e a c h s o m e t h i n g t h a t i s c o m p l e t e l y known b y u s i n g i n s t r u c t i o n a l methods meant t o p r o m o t e c r e a t i v i t y ? Ray h a d a u t i l i t a r i a n v i e w w h i c h was s u p p o r t e d b y h i s a c a d e m i c b a c k g r o u n d . C o r r e s p o n d i n g l y , h i s v i e w o f t e a c h i n g i n c l u d e d d e m o n s t r a t i n g a n d u s i n g many a p p l i c a t i o n s t o show s t u d e n t s t h i s a s p e c t o f m a t h e m a t i c s . - 75 -T h e s e l i n k s b e t w e e n t h e i n t e r v i e w e e s ' p e r s p e c t i v e s o f m a t h e m a t i c s a n d t h e i r v i e w s o f t e a c h i n g m a t h e m a t i c s s u p p o r t t h e r e s e a r c h f i n d i n g s . I n p a r t i c u l a r , Thompson ( 1 9 8 4 ) , who s t u d i e d t h r e e i n s e r v i c e t e a c h e r s , f o u n d " t h e t e a c h e r s ' b e l i e f s , v i e w s , and p r e f e r e n c e s a b o u t m a t h e m a t i c s a n d i t s t e a c h i n g , r e g a r d l e s s o f w h e t h e r t h e y a r e c o n s c i o u s l y o r u n c o n s c i o u s l y h e l d , p l a y a s i g n i f i c a n t , a l b e i t s u b t l e , r o l e i n s h a p i n g t h e t e a c h e r s ' c h a r a c t e r i s t i c p a t t e r n s o f i n s t r u c t i o n a l b e h a v i o u r . " (p 124) She s u g g e s t e d t h e c o n n e c t i o n b e t w e e n t h e i r v i e w o f math a n d t h e i r a c t u a l t e a c h i n g b e h a v i o u r may have b e e n d i m i n i s h e d b y v a r i o u s i n s t i t u t i o n a l a n d o t h e r c o n s t r a i n t s on t h e i r p r a c t i c e . S i m i l a r l y , f o r t h e p r o s p e c t i v e t e a c h e r s i n t h i s s t u d y , when t h e y embark o n t h e i r c a r e e r , t h e y may f i n d t h e e x i g e n c i e s o f a new, f u l l - t i m e p o s i t i o n w i l l n o t a l l o w them t o t e a c h i n ways t h a t a r e f u l l y c o m p a t i b l e w i t h t h e i r b e l i e f s . O t h e r I s s u e s I n a d d i t i o n t o t h e i r p e r s p e c t i v e s on t e a c h i n g m a t h e m a t i c s , I n o t e d s e v e n a r e a s o f i n t e r e s t i n g c o m p a r i s o n among t h e t e a c h e r c a n d i d a t e s . A d o l e s c e n c e A l l t h e t e a c h e r c a n d i d a t e s became more c o g n i z a n t o f a d o l e s c e n c e d u r i n g t h e i r p r a c t i c u m . Q u e s t i o n s r e g a r d i n g t h i s age g r o u p were n o t s c r i p t e d i n t o t h e i n t e r v i e w s , y e t e v e r y p e r s o n made comments i n d i c a t i n g t h e y were g r e a t l y a f f e c t e d b y t h e i r t e e n a g e d s t u d e n t s . A d o l e s c e n c e h a d became a r e a l i t y e m b o d i e d i n k i d s w i t h b e h a v i o u r a l p r o b l e m s , p e e r p r e s s u r e , i d e n t i t y c r i s e s and u n s t a b l e f a m i l i e s . T h e y became aware t h a t t h e y were d e a l i n g w i t h , n o t j u s t a p a r t i c u l a r a g e , b u t w i t h t h e - 76 -p s y c h o l o g i c a l t a s k s o f t h a t age g r o u p . The t e a c h e r c a n d i d a t e s f e l t d e e p l y a b o u t t h e p u p i l s t h e y were w o r k i n g w i t h . S c h o o l M a t h Compared t o " R e a l " M a t h V i c t o r i a a n d M a r y , b o t h o f whom h a d u n d e r t a k e n g r a d u a t e s t u d i e s , commented t h a t s c h o o l math was n o t e q u i v a l e n t t o " r e a l " m a t h . I n o t h e r w o r d s , t h e m a t h e m a t i c s o f t h e s c h o o l c u r r i c u l u m d i d n o t i n v o l v e t h e same t e c h n i q u e s a n d e n g e n d e r t h e same t y p e s o f u n d e r s t a n d i n g s as what p r o f e s s i o n a l m a t h e m a t i c i a n s w o r k e d w i t h a n d d e v e l o p e d . S h a n n o n e x p r e s s e d d i s m a y t h a t t h e m a t h e m a t i c s she was t o t e a c h i n h e r j u n i o r h i g h c l a s s e s was u n f a m i l i a r t o h e r , someone who h a d m a j o r e d i n m a t h e m a t i c s a t u n i v e r s i t y . S c h o o l M a t h e m a t i c s i n S t u d e n t s ' L i v e s D o u g , who w o r k e d w i t h t r o u b l e d y o u t h , saw m a t h e m a t i c a l a l g o r i t h m s as t h e r a p e u t i c . He b e l i e v e d t h e r e p e t i t i o n a n d r o u t i n e n e s s o f t h e p r o c e d u r e s were a s o u r c e o f s t a b i l i t y and c e r t a i n t y f o r t h e t e e n s . M a t h c l a s s g a v e them an o p p o r t u n i t y t o f o c u s on s o m e t h i n g c o m p l e t e l y d i f f e r e n t a n d s e p a r a t e f r o m t h e r o i l i n g t u r m o i l o f t h e i r p e r s o n a l l i v e s . V i c t o r i a , b y c o n t r a s t , w o r r i e d t h a t t h e b l a c k - a n d - w h i t e n e s s o f s c h o o l math w o u l d a l i e n a t e s t u d e n t s . She r e a s o n e d t h a t t h e r i g i d i t y o f m a t h e m a t i c a l p r o c e d u r e s was o p p o s i t e t o t h e t u m u l t o f t h e i r a d o l e s c e n c e . T h i s w o u l d g i v e them e v e n g r e a t e r c a u s e t o d i s t a n c e t h e m s e l v e s f r o m a d u l t s a n d a u t h o r i t y f i g u r e s . T e a c h i n g M a t h e m a t i c s Compared t o O t h e r S u b j e c t s " I t ' s n o t so much a b o u t t h e c o n t e n t as a b o u t y o u know, how t o s o l v e a p r o b l e m . I t ' s l i k e any o t h e r c o u r s e . . . . T o me, t h e y ' r e a l l t h e s a m e . " [ M a r y , p 13-14] - 77 -" . . . a l o t o f t h i n g s ARE t h e same, y o u l i s t e n t o t h e t e a c h e r d e m o n s t r a t e s o m e t h i n g new, y o u work o n i t w i t h o t h e r s t u d e n t s , y o u d i s c u s s i t . . . t h e same b a s i c s . " [ Y v e t t e , p 9 ] " . . . I t h i n k t h e t h i n g w e ' r e t r y i n g t o do i s t e a c h them t o t h i n k a n d t o c o m m u n i c a t e . E n g l i s h i s o b v i o u s f o r t h a t , b u t I t h i n k m a t h has one o f t h e g r e a t e s t p o t e n t i a l s f o r t h a t . " [Doug, p 1 -2 , p a r a p h r a s e d ] T h e s e e x c e r p t s i n d i c a t e t h e s e t e a c h e r c a n d i d a t e s h a d i d e a s o f t e a c h i n g w h i c h were n o t l i n k e d t o c o n t e n t . T h e y f e l t t h e i r p e r s o n a l p h i l o s o p h y o f t e a c h i n g h e l d t r u e f o r a l l s u b j e c t s . T e a c h e r ' s R o l e i n t h e M a t h e m a t i c s C l a s s r o o m Doug a n d Shannon h a d o p p o s i n g v i e w s o f what i m p r e s s i o n t h e t e a c h e r o f m a t h e m a t i c s s h o u l d g i v e s t u d e n t s . Doug b e l i e v e d i n s h o w i n g s t u d e n t s i t was a c c e p t a b l e t o make m i s t a k e s b y h a v i n g them see a t e a c h e r make m i s t a k e s d u r i n g c l a s s . I n c o n t r a s t , Shannon b e l i e v e d t h e m a t h t e a c h e r s h o u l d p r e s e n t h e r s e l f as k n o w i n g a l l t h e a n s w e r s and n o t e v e n h e s i t a t i n g t o t h i n k a b o u t t h e m . C o n c e p t u a l Change M a r y a n d Ray s p o k e o f how t h e i r c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s h a d c h a n g e d d u r i n g t h e p r a c t i c u m . M a r y i d e n t i f i e d t h e t e a c h i n g o f E u c l i d e a n p r o o f s as t h e c a u s e f o r h e r c h a n g e . F o r R a y , i t was h i s c l o s e c o n t a c t w i t h t h e s t u d e n t s i n h i s t e r m i n a l math c l a s s . The r e s u l t s o f t h i s s t u d y s u p p o r t t h e u n d e r l y i n g b e l i e f t h a t i d e a s o f t e a c h i n g a r e p e r s o n a l a n d have b e e n c o n s t r u c t e d t h r o u g h p a s t e x p e r i e n c e s a n d a t t i t u d e s . F o r e x a m p l e . Shannon b e l i e v e d t h e d r i l l a p p r o a c h was t h e b e s t way t o l e a r n number f a c t s b e c a u s e t h a t was how she was t a u g h t t h e m . H e r p r e v i o u s work i n m a n a g i n g a r e s t a u r a n t may a l s o h a v e i n f l u e n c e d h e r b e l i e f t h a t b e i n g a b l e t o make m e n t a l c a l c u l a t i o n s - 78 -q u i c k l y was v e r y i m p o r t a n t . R a y , who s t u d i e d a p p l i e d m a t h e m a t i c s i n a c t u a r i a l a n d e n g i n e e r i n g c o u r s e s , w a n t e d t o e m p h a s i z e a p p l i c a t i o n s i n h i s t e a c h i n g . Doug h a d a b a c k g r o u n d i n l i b e r a l a r t s w h i c h may h a v e c o n t r i b u t e d t o h i s e p i s t e m o l o g i c a l s t a n c e ; h i s b e l i e f i n k n o w l e d g e c o n s t r u c t i o n c a r r i e d o v e r t o h i s p e r s p e c t i v e i n t e a c h i n g m a t h e m a t i c s . The r e s u l t s a l s o s u p p o r t i n p a r t t h e l i t e r a t u r e w h i c h i n d i c a t e s t h e b e l i e f s o f t e a c h e r c a n d i d a t e s a r e s t a b l e . The v i e w s o f t h e s e p r o s p e c t i v e t e a c h e r s were e x p r e s s e d a t t h e e n d o f t h e i r p r a c t i c a , n e a r t h e c o n c l u s i o n o f t h e i r p r o g r a m . They h a d gone t h r o u g h t h e same t e a c h e r p r e p a r a t i o n p r o g r a m , y e t t h e i r i d e a s o f t e a c h i n g m a t h e m a t i c s were d i f f e r e n t f r o m e a c h o t h e r a n d t h e i n f l u e n c e f r o m e a c h o f t h e i r b a c k g r o u n d s i s c l e a r . H o w e v e r , t h e i r i d e a s a r e n o t f i x e d and t h i s s t u d y shows t h a t t h e p r a c t i c u m does h a v e a n i n f l u e n c e i n t h e c o n s t r u c t i o n o f t h e i r c o n c e p t i o n s . M o s t n o t a b l y , M a r y a n d Ray a c k n o w l e d g e d a change i n t h e i r c o n c e p t i o n o f t e a c h i n g r e s u l t i n g f r o m t h e i r p r a c t i c a e x p e r i e n c e s . I n a d d i t i o n , a l l t h e t e a c h e r c a n d i d a t e s e x p r e s s e d a g r e a t e r a w a r e n e s s o f a d o l e s c e n c e b e c a u s e o f t e a c h i n g . Whether o r n o t t h i s a w a r e n e s s d i r e c t l y a f f e c t e d t h e i r a c t i o n s i n t h e c l a s s r o o m ( Y v e t t e c l a i m e d i t d i d n o t ) , i t c o n t r i b u t e s t o t h e i r k n o w l e d g e and e x p e r i e n c e and may i n f l u e n c e t h e i r f u t u r e t e a c h i n g . Recommendat i o n s B a s e d o n t h e f i n d i n g s o f t h i s s t u d y , f o u r r e c o m m e n d a t i o n s a r e o f f e r e d . - 79 -1. Pre-service teachers be given a range of experiences during their practica to expose them to different types of students as well as different types of topics. R e s e a r c h o n t h e i n f l u e n c e o f p r a c t i c a on p r o s p e c t i v e t e a c h e r s ' c o n c e p t i o n s show v a r y i n g r e s u l t s (Brown & B o r k o , 1 9 9 2 ) , a n d t h e r e i s s t r o n g e v i d e n c e t h a t c o n c e p t i o n s r e m a i n u n c h a n g e d (Kagan, 1992) . H o w e v e r , two s u b j e c t s i n t h i s s t u d y c l e a r l y i d e n t i f i e d a c h a n g e i n t h e i r t e a c h i n g p e r s p e c t i v e as a r e s u l t o f t h e i r p r a c t i c a e x p e r i e n c e s . W h i l e i t i s i m p o s s i b l e f o r t e a c h e r c a n d i d a t e s t o have i n f i n i t e e x p e r i e n c e s d u r i n g a f i n i t e p r a c t i c u m , i n c r e a s i n g t h e b r e a d t h o f t h e i r e x p e r i e n c e t o i n c l u d e d i f f e r e n t t y p e s o f c l a s s e s ( e . g . a c a d e m i c a n d n o n - a c a d e m i c ) a n d v a r y i n g m a t h e m a t i c a l t o p i c s ( e . g . g e o m e t r y a n d a l g e b r a ) w i l l e x p o s e them t o d i f f e r e n t c o n t e x t s u n d e r w h i c h t h e y may more c l o s e l y e x a m i n e p e r s o n a l i d e a s o f t e a c h i n g . W i l s o n (1994) s t a t e s : " b y g i v i n g [ p r e - s e r v i c e t e a c h e r s ] o p p o r t u n i t i e s t o r e f l e c t o n t h e i r own c o n c e p t i o n s w h i l e l e a r n i n g ( o r r e - l e a r n i n g ) m a t h e m a t i c s t h a t t h e y w i l l have t o t e a c h t h e m s e l v e s [ s ] u c h e x p e r i e n c e s w i l l a l l o w them t o make a c c o m m o d a t i o n s i n t h e i r b e l i e f s y s t e m s t o f i r s t a c k n o w l e d g e a l t e r n a t i v e p e r s p e c t i v e s o f m a t h e m a t i c s a n d m a t h e m a t i c s t e a c h i n g , and t h e n ( p e r h a p s ) embrace t h e m . " [p 368] 2. More studies on conceptions of prospective secondary mathematics teachers be carried out. W i t h r e s e a r c h t h a t p o i n t s t o t h e e x i s t e n c e o f t e a c h e r c a n d i d a t e s ' b e l i e f s o n t e a c h i n g p r i o r t o t h e i r e n t e r i n g a p r o f e s s i o n a l p r o g r a m ( e . g . F e i m a n - N e m s e r & Buchmann, 1 9 8 9 ) , t h e s t a b i l i t y o f t h e s e p e r s p e c t i v e s d u r i n g t h e p r o g r a m ( e . g . T a b i c h n i c k & Z e i c h n e r , 1 9 8 5 ) , a n d t h e i r i n f l u e n c e o n a c t i o n s i n t h e c l a s s r o o m a f t e r t h e p r o g r a m ( e . g . Thompson, 1 9 8 4 ) , i t becomes i m p o r t a n t t o know t h e s e b e l i e f s a n d i d e a s . - 80 -I n t h e s m a l l g r o u p w h i c h c o m p r i s e d t h e s a m p l e f o r t h i s s t u d y , m a j o r d i f f e r e n c e s i n p e r s p e c t i v e s o f t e a c h i n g m a t h e m a t i c s were f o u n d . The i n t e r v i e w e e s h a d gone t h r o u g h t h e i d e n t i c a l t e a c h e r p r e p a r a t i o n p r o g r a m y e t t h e i r i d e a s were d i f f e r e n t . T h i s s t r o n g l y s u g g e s t s a d d i t i o n a l p e r s p e c t i v e s w o u l d have b e e n l a i d o p e n i f t h e s a m p l e h a d c o n s i s t e d o f d i f f e r e n t i n d i v i d u a l s . F u r t h e r , u s i n g a l a r g e r s a m p l e c o u l d h a v e r e s u l t e d i n some c o m m o n a l i t i e s i n c o n c e p t i o n s . I d e a s a b o u t t e a c h i n g math c a n be i n v e s t i g a t e d u s i n g a v a r i e t y o f t e c h n i q u e s i n a d d i t i o n t o t h e t y p e o f i n t e r v i e w u s e d i n t h i s s t u d y . F o r e x a m p l e , t e a c h e r c a n d i d a t e s c o u l d be a s k e d t o : p r e p a r e m i n i - l e s s o n s o n s e l e c t e d t o p i c s ; r e s p o n d t o e p i s o d e s d e p i c t i n g v a r i o u s t y p e s o f s t u d e n t e n c o u n t e r s a n d / o r d i f f e r e n t t e a c h i n g s t y l e s ; o b s e r v e a v i d e o t a p i n g o f t h e i r own t e a c h i n g . I n e a c h o f t h e s e , t h e y c o u l d be p r o b e d f o r t h e r e a s o n s b e h i n d t h e i r r e s p o n s e s - what t h e y l i k e d and d i d n ' t l i k e a b o u t o t h e r s ' t e a c h i n g , what l e d them t o make t h e d e c i s i o n s t h e y made, what t h o u g h t s g u i d e d t h e i r a c t i o n s . I n t h i s way, t e a c h e r c a n d i d a t e s c a n e x p l o r e a n d a r t i c u l a t e t h e i r own i d e a s w h i l e t h e y a b i d e w i t h i n t h e s u p p o r t e d e n v i r o n m e n t o f a p r e p a r a t i o n p r o g r a m . T e a c h e r e d u c a t o r s c a n a l s o u s e t h e s e s t r u c t u r e d o p p o r t u n i t i e s t o b u i l d o n e x i s t i n g c o n c e p t i o n s , c h a l l e n g i n g t h e m , i f n e c e s s a r y , b u t u l t i m a t e l y , h e l p i n g t h e p r e - s e r v i c e t e a c h e r s t o b u i l d i n t e g r a t e d a n d a p p r o p r i a t e c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s . 3. Longitudinal studies be carried out for individual pre-service and novice teachers. - 81 -B r o w n a n d B o r k o (1992) have n o t e d t h a t most s t u d i e s o n t e a c h e r d e v e l o p m e n t have u s e d s e v e r a l t e a c h e r s r a t h e r t h a n a s i n g l e t e a c h e r . H o w e v e r , l o n g i t u d i n a l i n v e s t i g a t i o n s d u r i n g t h e t e a c h e r p r e p a r a t i o n p r o g r a m s u c h as W i l s o n ' s (1994) i n q u i r y w o u l d h e l p i d e n t i f y e v o l v i n g t h o u g h t s a n d b e l i e f s . " S n a p s h o t s " o f c o n c e p t i o n s c o u l d be t a k e n t h r o u g h a n a l y s i s o f i n t e r v i e w s a n d j o u r n a l w r i t i n g s o f t e a c h e r c a n d i d a t e s , as w e l l as u s i n g methods m e n t i o n e d a b o v e , a t v a r i o u s t i m e s d u r i n g t h e p r o g r a m . I n a d d i t i o n , f o l l o w i n g t h e same p e r s o n f r o m a p r e p a r a t i o n p r o g r a m t h r o u g h t o when she has a c c u m u l a t e d some s u b s t a n t i v e t e a c h i n g e x p e r i e n c e , c o n c e p t u a l d e v e l o p m e n t , f a c t o r s i n f l u e n c i n g i t , a n d how c o n c e p t i o n s r e l a t e t o h e r a c t i o n s i n t h e c l a s s r o o m c a n be e x p l o r e d . One e x a m p l e o f s u c h a s t u d y i s t h e c a s e o f F r e d , a n o v i c e t e a c h e r s t u d i e d b y Cooney (1985) a n d Brown (1985) who d e s c r i b e h i s d i l e m m a when h i s v i e w o f t e a c h i n g m a t h e m a t i c s c o n f l i c t s w i t h t h a t o f h i s s t u d e n t s . L o n g i t u d i n a l s t u d i e s o f t e a c h e r d e v e l o p m e n t s h o u l d i n c l u d e t h e o r i e s o f a d u l t d e v e l o p m e n t as p a r t o f t h e i r f r a m e w o r k . T h e s e w o u l d p r o v i d e a more h o l i s t i c c o n t e x t f o r u n d e r s t a n d i n g c h a n g e s i n i d e a s . Owens ( 1 9 8 7 ) , f o r e x a m p l e , u s e d P e r r y ' s (1970) scheme o f i n t e l l e c t u a l d e v e l o p m e n t a n d t h e c o n s t r u c t o f a u t h o r i t y t o i n v e s t i g a t e t h e b e l i e f s o f p r o s p e c t i v e t e a c h e r s . H o w e v e r , he t o o a p p l i e d i t t o a s m a l l s a m p l e o f p r o s p e c t i v e t e a c h e r s r a t h e r t h a n u s i n g i t i n a l o n g i t u d i n a l c o n t e x t . 4. Investigations of conceptions of teaching mathematics be carried out with other populations of teachers and learners. K n o w i n g s t u d e n t s ' c o n c e p t i o n s o f t e a c h i n g m a t h e m a t i c s c o u l d p r o v i d e i n s i g h t i n t o c l a s s r o o m i n t e r a c t i o n s . Thompson (1992) has - 82 -c l a i m e d t h a t none o f t h i s t y p e o f r e s e a r c h e x i s t s . H o w e v e r , i n B r o w n ' s (1985) s t u d y , s t u d e n t s ' i d e a s were i n v e s t i g a t e d as a p o t e n t i a l f a c t o r i n t h e s o c i a l i z a t i o n o f a n o v i c e t e a c h e r . Those i d e a s were f o u n d t o be i n f l u e n t i a l i n c h a n g i n g , a t l e a s t s u p e r f i c i a l l y , t h e b e l i e f s a n d a c t i o n s o f t h e t e a c h e r . My p a r t i c u l a r i n t e r e s t l i e s i n A d u l t B a s i c E d u c a t i o n ( A B E ) . T h e s e i n s t r u c t o r s a r e c h a r g e d w i t h t e a c h i n g p r e - c o l l e g e l e v e l c o n t e n t t o a d u l t s . T y p i c a l l y , t h e i r b a c k g r o u n d e n t a i l s a B . E d , d e g r e e a n d s e v e r a l y e a r s ' t e a c h i n g e x p e r i e n c e . I n s t r u c t o r s new t o t h e f i e l d w i l l u s u a l l y have t e a c h i n g e x p e r i e n c e o n l y a t t h e s e c o n d a r y l e v e l . K n o w i n g how t h e y c o n c e i v e o f t e a c h i n g m a t h e m a t i c s t o a d u l t s and how t h e y p e r c e i v e t h e i r r o l e w o u l d be i m p o r t a n t i n f o r m a t i o n . ABE s t u d e n t s b e g i n t h e i r p r o g r a m w i t h e x p e r i e n c e s o f s c h o o l , m a t h e m a t i c s , and t e a c h e r s remembered f r o m h i g h s c h o o l . E l u c i d a t i n g t h e i r i d e a s o f m a t h e m a t i c s a n d i t s t e a c h i n g w o u l d p r o v i d e k n o w l e d g e t h r o u g h w h i c h a n i n s t r u c t o r c o u l d i m p r o v e h e r i n s t r u c t i o n a l a p p r o a c h . I n t e r n a t i o n a l s t u d e n t s b r i n g d i f f e r e n t c u l t u r a l n o t i o n s o f t e a c h i n g a n d t h e t e a c h e r t o a C a n a d i a n c l a s s r o o m . E x p l o r i n g t h e s e c o u l d u n c o v e r p o t e n t i a l s o u r c e s f o r c o n f l i c t o r m i s u n d e r s t a n d i n g s i n t h e c l a s s r o o m . R e s e a r c h i n g t e a c h e r c a n d i d a t e s ' c o n c e p t i o n s o f t e a c h i n g math y i e l d s k n o w l e d g e f o r m a t h e m a t i c s t e a c h e r e d u c a t i o n p r o g r a m s ; i t a l s o s e r v e s p e r s o n a l t e a c h e r d e v e l o p m e n t . Those who have b e e n i n v o l v e d t h i n k a b o u t t h e i r i d e a s , t h e i r m i n d s p i q u e d b y t h e r e s e a r c h e r ' s q u e s t i o n s . One c a n d i d a t e e x p l i c i t l y s t a t e d a f t e r h e r i n t e r v i e w t h a t she h a d n e v e r c o n s i d e r e d h e r own i d e a s b e f o r e and t h e i n t e r v i e w h e l p e d h e r t h i n k i n - 83 -d i f f e r e n t ways a b o u t t e a c h i n g . I t i s h o p e d t h a t t h i s s t u d y p r o v o k e d t h e m i n d s o f t h e o t h e r s as w e l l . - 84 -REFERENCES B a l l , D . L . (1988) . U n l e a r n i n g t o t e a c h m a t h e m a t i c s . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 8., 4 0 - 4 8 . B a l l , D . L . ( 1 9 9 0 a ) . B r e a k i n g w i t h e x p e r i e n c e i n l e a r n i n g t o t e a c h m a t h e m a t i c s : The r o l e o f a p r e s e r v i c e methods c o u r s e . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 10 ( 2 ) , 1 0 - 1 6 . B a l l , D . L . ( 1 9 9 0 b ) . P r o s p e c t i v e e l e m e n t a r y and s e c o n d a r y t e a c h e r s ' u n d e r s t a n d i n g o f d i v i s i o n . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 2 1 , 1 3 2 - 1 4 4 . B a l l , D . L . & M c D i a r m i d , G.W. ( 1 9 9 0 ) . The s u b j e c t - m a t t e r p r e p a r a t i o n o f t e a c h e r s . I n W . R . H o u s t o n ( E d . ) , Handbook o f r e s e a r c h o n t e a c h e r e d u c a t i o n : A p r o j e c t o f t h e A s s o c i a t i o n o f T e a c h e r E d u c a t o r s (pp 4 3 7 - 4 4 9 ) . New Y o r k : M a c m i l l a n . B o r k o , H . , L a l i k , R . , & T o m c h i n , E . ( 1 9 8 7 ) . S t u d e n t t e a c h e r s ' u n d e r s t a n d i n g s o f s u c c e s s f u l a n d u n s u c c e s s f u l t e a c h i n g . T e a c h i n g a n d T e a c h e r E d u c a t i o n , 3 ( 2 ) , 7 7 - 9 0 . B r o p h y , J . ( 1 9 8 6 a ) . T e a c h i n g and l e a r n i n g m a t h e m a t i c s : Where r e s e a r c h s h o u l d be g o i n g . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 17, 3 2 3 - 3 4 6 . B r o p h y , J . (1986b) . Where a r e t h e d a t a ? A r e p l y t o C o n f r e y . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 17, 3 6 1 - 3 6 8 . B r o p h y , J . ( E d . ) . ( 1 9 9 1 ) . A d v a n c e s i n r e s e a r c h o n t e a c h i n g : T e a c h e r s ' k n o w l e d g e o f s u b j e c t m a t t e r as i t r e l a t e s t o t h e i r t e a c h i n g p r a c t i c e ( V o l . 2 ) . G r e e n w i c h , CT: J A I P r e s s . B r o w n , C A . ( 1 9 8 5 ) . A s t u d y o f t h e s o c i a l i z a t i o n t o t e a c h i n g o f a b e g i n n i n g s e c o n d a r y m a t h e m a t i c s t e a c h e r . U n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f G e o r g i a , A t h e n s . B r o w n , C A . & B o r k o , H . ( 1 9 9 2 ) . B e c o m i n g a m a t h e m a t i c s t e a c h e r . I n D . A . Grouws ( E d . ) , Handbook o f r e s e a r c h o n m a t h e m a t i c s t e a c h i n g a n d l e a r n i n g : A p r o j e c t o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s (pp 2 0 9 - 2 3 9 ) . New Y o r k : M a c m i l l a n . B r o w n , L . ( 1 9 9 2 ) . The i n f l u e n c e o f t e a c h e r s on c h i l d r e n ' s image o f m a t h e m a t i c s . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 12 ( 2 ) , 2 9 - 3 3 . B r o w n , S . I . , C o o n e y , T . J . , & J o n e s , D. ( 1 9 9 0 ) . M a t h e m a t i c s t e a c h e r e d u c a t i o n . I n W.R. H o u s t o n ( E d . ) , Handbook o f r e s e a r c h o n t e a c h e r e d u c a t i o n : A p r o j e c t o f t h e A s s o c i a t i o n o f T e a c h e r E d u c a t o r s (pp 6 3 9 - 6 5 6 ) . New Y o r k : M a c m i l l a n . C i v i l , M . ( 1 9 9 0 ) . "You o n l y do math i n m a t h " : A l o o k a t f o u r p r o s p e c t i v e t e a c h e r s ' v i e w s a b o u t m a t h e a m t i c s . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 10 ( 1 ) , 7 - 9 . C l a r k , C M . & P e t e r s o n , P . L . ( 1 9 8 6 ) . T e a c h e r s ' t h o u g h t p r o c e s s e s . I n M . C W i t t r o c k ( E d . ) , Handbook o f r e s e a r c h o n t e a c h i n g (pp 2 2 5 -2 9 6 ) . New Y o r k : M a c m i l l a n . - 85 -C o b b , P . Wood, T . & Y a c k e l , E . ( 1 9 9 1 ) . A n a l o g i e s f r o m t h e p h i l o s o p h y a n d s o c i o l o g y o f s c i e n c e f o r u n d e r s t a n d i n g c l a s s r o o m l i f e . S c i e n c e E d u c a t i o n , 75 ( 1 ) , 2 3 - 4 4 . C o b b , P . ( 1 9 9 4 a ) . An e x c h a n g e : C o n s t r u c t i v i s m i n m a t h e m a t i c s a n d s c i e n c e e d u c a t i o n . I n E d u c a t i o n a l R e s e a r c h e r , 2 3 , 4 . C o b b , P . ( 1 9 9 4 b ) . Where i s t h e mind? C o n s t r u c t i v i s t a n d s o c i o c u l t u r a l p e r s p e c t i v e s on m a t h e m a t i c a l d e v e l o p m e n t . I n E d u c a t i o n a l R e s e a r c h e r , 23., 1 3 - 2 0 . C o n f r e y , J . (1986) . A c r i t i q u e o f t e a c h e r e f f e c t i v e n e s s r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 17, 3 4 7 - 3 6 0 . C o n f r e y , J . (1994) . A t h e o r y o f i n t e l l e c t u a l d e v e l o p m e n t ( P a r t 1) . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 14 (3.), 2 - 8 . C o o n e y , T . J . (1985) . A b e g i n n i n g t e a c h e r ' s v i e w o f p r o b l e m - s o l v i n g . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 16 ( 5 ) , 3 2 4 - 3 3 6 . D a v i s , P . J . & H e r s h , R. (1981) . The M a t h e m a t i c a l E x p e r i e n c e . L o n d o n : P e n g u i n B o o k s . D e n z i n , N . K . & L i n c o l n , Y . S . ( E d s . ) . ( 1 9 9 4 ) . Handbook o f Q u a l i t a t i v e R e s e a r c h . T h o u s a n d O a k s , CA: S a g e . D o s s e y , J . A . ( 1 9 9 2 ) . The n a t u r e o f m a t h e m a t i c s : I t s r o l e and i n f l u e n c e . I n D . A . Grouws ( E d . ) , Handbook o f R e s e a r c h o n M a t h e m a t i c s T e a c h i n g a n d L e a r n i n g : A p r o j e c t o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s (pp 39-48) . New Y o r k : M a c m i l l a n . E r i c k s o n , G . L . (1987, A p r i l ) . C o n s t r u c t i v i s t e p i s t e m o l o g y a n d t h e p r o f e s s i o n a l d e v e l o p m e n t o f t e a c h e r s . P a p e r p r e s e n t e d a t t h e m e e t i n g o f t h e A m e r i c a n E d u c a t i o n a l R e s e a r c h A s s o c i a t i o n , W a s h i n g t o n , D . C . E r n e s t , P . (1989) . The k n o w l e d g e , b e l i e f s a n d a t t i t u d e s o f t h e m a t h e m a t i c s t e a c h e r : A m o d e l . J o u r n a l o f E d u c a t i o n f o r T e a c h i n g , 15 ( 1 ) , 13-33 . E r n e s t , P . ( 1 9 9 1 ) . The p h i l o s o p h y o f m a t h e m a t i c s e d u c a t i o n . L o n d o n : The F a l m e r P r e s s . F e i m a n - N e m s e r , S . & Buchmann, M . (1989) . D e s c r i b i n g t e a c h e r e d u c a t i o n : A f r a m e w o r k a n d i l l u s t r a t i v e f i n d i n g s f r o m a l o n g i t u d i n a l s t u d y o f s i x s t u d e n t s . The E l e m e n t a r y S c h o o l J o u r n a l , 89 ( 3 ) , 3 6 6 - 3 7 7 . F o r d , M . I . ( 1 9 9 4 ) . T e a c h e r s ' b e l i e f s a b o u t m a t h e m a t i c a l p r o b l e m s o l v i n g i n t h e e l e m e n t a r y s c h o o l . S c h o o l S c i e n c e a n d M a t h e m a t i c s , 94 ( 6 ) , 3 1 4 - 3 2 2 . Goodman, J . (1988) . C o n s t r u c t i n g a p r a c t i c a l p h i l o s p o h y o f t e a c h i n g : A s t u d y o f p r e s e r v i c e t e a c h e r s ' p r o f e s s i o n a l p e r s p e c t i v e s . T e a c h i n g a n d T e a c h e r E d u c a t i o n , 4 ( 2 ) , 1 2 1 - 1 3 7 . K a g a n , D. ( 1 9 9 2 ) . P r o f e s s i o n a l g r o w t h among p r e s e r v i c e a n d b e g i n n i n g t e a c h e r s . R e v i e w o f E d u c a t i o n a l R e s e a r c h , 62 ( 2 ) , 1 2 9 - 1 6 9 . - 86 -K o n o l d , C . & J o h n s o n , D . K . (1991) . P h i l o s o p h i c a l and p s y c h o l o g i c a l a s p e c t s o f c o n s t r u c t i v i s m . I n L . P . S t e f f e ( E d . ) , E p i s t e m o l o q i c a l f o u n d a t i o n s o f m a t h e m a t i c a l e x p e r i e n c e (pp 1 - 1 3 ) . New Y o r k : S p r i n g e r - V e r l a g . L a m p e r t , M . ( 1 9 8 8 ) . What c a n r e s e a r c h on t e a c h e r e d u c a t i o n t e l l us a b o u t i m p r o v i n g q u a l i t y i n m a t h e m a t i c s e d u c a t i o n ? T e a c h i n g a n d T e a c h e r E d u c a t i o n , 4 ( 2 ) , 1 5 7 - 1 7 0 . L i n c o l n , Y . S . & G u b a , E . G . ( 1 9 8 5 ) . N a t u r a l i s t i c I n q u i r y . Newbury P a r k , CA: Sage P u b l i c a t i o n s . L o r t i e , D . C . ( 1 9 7 5 ) . S c h o o l t e a c h e r : A s o c i o l o g i c a l s t u d y . C h i c a g o : U n i v e r s i t y o f C h i c a g o P r e s s . M i l e s , M . B . & Huberman, A . M . ( 1 9 9 4 ) . Q u a l i t a t i v e d a t a a n a l y s i s : A n e x p a n d e d s o u r c e b o o k , 2nd e d . T h o u s a n d O a k s , CA: S a g e . N i c k s o n , M . (1988) . P e r v a s i v e themes and some d e p a r t u r e p o i n t s f o r r e s e a r c h i n t o e f f e c t i v e m a t h e m a t i c s t e a c h i n g . I n D . A . G r o u w s , T . J . Cooney & D. J o n e s ( E d s . ) , P e r s p e c t i v e s on r e s e a r c h o n e f f e c t i v e m a t h e m a t i c s t e a c h i n g ( p p . 2 4 5 - 2 5 2 ) . R e s t o n , V A : N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s & L a w r e n c e E r l b a u m A s s o c i a t e s . N o d d i n g s , N . ( 1 9 9 0 ) . C o n s t r u c t i v i s m i n m a t h e m a t i c s e d u c a t i o n . I n R . B . D a v i s , C A . M a h e r & N . N o d d i n g s ( E d s . ) , C o n s t r u c t i v i s t v i e w s o n t h e t e a c h i n g a n d l e a r n i n g o f m a t h e m a t i c s (pp 7-18) . R e s t o n , V A : N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s . Owens, J . ( 1 9 8 7 ) . A s t u d y o f 4 p r e s e r v i c e s e c o n d a r y m a t h e m a t i c s t e a c h e r s ' c o n s t r u c t s o f m a t h e m a t i c s a n d m a t h e m a t i c s t e a c h i n g . ( D o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f G e o r g i a ) . D i s s e r t a t i o n A b s t r a c t s I n t e r n a t i o n a l , 48A, 5 8 8 . P a t t o n , M . Q . (1990) . Q u a l i t a t i v e e v a l u a t i o n a n d r e s e a r c h m e t h o d s , 2nd e d . Newbury P a r k , CA: S a g e . P e r r y , W . G . , J r . ( 1 9 7 0 ) . Forms o f i n t e l l e c t u a l a n d e t h i c a l d e v e l o p m e n t i n t h e c o l l e g e y e a r s : A scheme. New Y o r k : H o l t , R i n e h a r t & W i n s t o n . P r a t t , D . D . (1992) . C o n c e p t i o n s o f t e a c h i n g . A d u l t E d u c a t i o n Q u a r t e r l y , 4 2 , 2 0 3 - 2 2 0 . Raymond, M . A . & S a n t o s , V . (1995) . P r e s e r v i c e e l e m e n t a r y t e a c h e r s a n d s e l f - r e f l e c t i o n : How i n n o v a t i o n i n m a t h e m a t i c s t e a c h e r p r e p a r a t i o n c h a l l e n g e s m a t h e m a t i c s b e l i e f s . J o u r n a l o f T e a c h e r E d u c a t i o n , 46 ( 1 ) , 5 8 - 7 0 . R o m b e r g , T . A . & C a r p e n t e r , T . P . ( 1 9 8 6 ) . R e s e a r c h on t e a c h i n g a n d l e a r n i n g m a t h e m a t i c s : Two d i s c i p l i n e s o f s c i e n t i f i c i n q u i r y . I n M . C W i t t r o c k ( E d . ) , Handbook o f r e s e a r c h on t e a c h i n g ( 3 r d e d . , pp 8 5 0 - 8 7 3 ) . New Y o r k : M a c m i l l a n . S h u l m a n , L . (1986) . Those who u n d e r s t a n d : Knowledge g r o w t h i n t e a c h i n g . I n E d u c a t i o n a l R e s e a r c h e r , 15, 4 - 1 4 . - 87 -S i m o n , M . A . ( 1 9 9 5 a ) . E l a b o r a t i n g m o d e l s o f m a t h e m a t i c s t e a c h i n g : A r e s p o n s e t o S t e f f e a n d D ' A m b r o s i o . J o u r n a l f o r r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n , 2 6 , 1 6 0 - 1 6 2 . S i m o n , M . A . ( 1 9 9 5 b ) . R e c o n s t r u c t i n g m a t h e m a t i c s p e d a g o g y f r o m a c o n s t r u c t i v i s t p e r s p e c t i v e . J o u r n a l f o r r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n , 2 6 , 1 1 4 - 1 4 5 . S t e f f e , L . P . & K i e r e n , T . ( 1 9 9 4 ) . R a d i c a l c o n s t r u c t i v i s m a n d m a t h e m a t i c s e d u c a t i o n . J o u r n a l f o r r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n , 25., 7 1 1 - 7 3 3 . S t e f f e , L . P . & D ' A m b r o s i o , B . S . ( 1 9 9 5 ) . T o w a r d a w o r k i n g m o d e l o f c o n s t r u c t i v i s t t e a c h i n g : A r e a c t i o n t o S i m o n . J o u r n a l f o r r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n , 26., 1 4 6 - 1 5 9 . S t r a u s s , A . & C o r b i n , J . (1990) . B a s i c s o f Q u a l i t a t i v e R e s e a r c h : G r o u n d e d T h e o r y P r o c e d u r e s a n d T e c h n i q u e s . Newbury P a r k , CA: S a g e . T a b i c h n i c k , B . R . & Z e i c h n e r , K . M . ( 1 9 8 4 ) . The i m p a c t o f t h e s t u d e n t t e a c h i n g e x p e r i e n c e on t h e d e v e l o p m e n t o f t e a c h e r p e r s p e c t i v e s . J o u r n a l o f T e a c h e r E d u c a t i o n , 35 . (6) , 2 8 - 3 6 . Thompson, A . G . ( 1 9 8 4 ) . The r e l a t i o n s h i p o f t e a c h e r s ' c o n c e p t i o n s o f m a t h e m a t i c s a n d m a t h e m a t i c s t e a c h i n g t o i n s t r u c t i o n a l p r a c t i c e . E d u c a t i o n a l S t u d i e s i n M a t h e m a t i c s , 15, 1 0 5 - 1 2 7 . Thompson, A . G . ( 1 9 9 2 ) . T e a c h e r s ' b e l i e f s and c o n c e p t i o n s : A s y n t h e s i s o f t h e r e s e a r c h . I n D . A . Grouws ( E d . ) , Handbook o f R e s e a r c h o n M a t h e m a t i c s T e a c h i n g and L e a r n i n g : A P r o j e c t o f t h e N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s (pp. 1 2 7 - 1 4 6 ) . New Y o r k : M a c m i l l a n . Thompson, P .W. ( 1 9 9 1 ) . To e x p e r i e n c e i s t o c o n c e p t u a l i z e : A d i s c u s s i o n o f e p i s t e m o l o g y a n d m a t h e m a t i c a l e x p e r i e n c e . I n L . P . S t e f f e ( E d . ) , E p i s t e m o l o g i c a l f o u n d a t i o n s o f m a t h e m a t i c a l e x p e r i e n c e (pp 2 6 0 - 2 8 1 ) . New Y o r k : S p r i n g e r - V e r l a g . T i r o s h , D. & G r a e b e r , A . O . ( 1 9 9 0 ) . E v o k i n g c o g n i t i v e c o n f l i c t t o e x p l o r e p r e s e r v i c e t e a c h e r s ' t h i n k i n g a b o u t d i v i s i o n . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 2 1 , 9 8 - 1 0 8 . V a l l i , L . & A g o s t i n e l l i , A . (1993) . T e a c h i n g b e f o r e a n d a f t e r p r o f e s s i o n a l p r e p a r a t i o n : The s t o r y o f a h i g h s c h o o l m a t h e m a t i c s t e a c h e r . J o u r n a l o f T e a c h e r E d u c a t i o n , 44., 1 0 7 - 1 1 8 . V o n G l a s e r s f e l d , E . (1989) . C o n s t r u c t i v i s m i n e d u c a t i o n . I n T . H u s e n & N . P o s t l e t h w a i t e ( E d s . ) , I n t e r n a t i o n a l e n c y c l o p e d i a o f e d u c a t i o n : S u p p l e m e n t a r y V o l . 1 (pp 162-163) . O x f o r d : P e r g a m o n . V o n G l a s e r s f e l d , E . ( 1 9 9 0 ) . A n e x p o s i t i o n o f c o n s t r u c t i v i s m : Why some l i k e i t r a d i c a l . I n R . B . D a v i s , C A . M a h e r & N . N o d d i n g s ( E d s . ) , C o n s t r u c t i v i s t v i e w s on t h e t e a c h i n g a n d l e a r n i n g o f m a t h e m a t i c s (pp 19-29) . R e s t o n , VA: N a t i o n a l C o u n c i l o f T e a c h e r s o f M a t h e m a t i c s . W i l c o x , S . K . , S c h r a m , P . , L a p p a n , G. & L a n i e r , P . ( 1 9 9 1 ) . The r o l e o f a l e a r n i n g c o m m u n i t y i n c h a n g i n g p r e s e r v i c e t e a c h e r s ' k n o w l e d g e a n d b e l i e f s a b o u t m a t h e m a t i c s e d u c a t i o n . F o r t h e L e a r n i n g o f M a t h e m a t i c s , 11 ( 3 ) , 3 1 - 3 9 . - 88 -W i l s o n , M . R . ( 1 9 9 4 ) . One p r e s e r v i c e s e c o n d a r y t e a c h e r ' s u n d e r s t a n d i n g o f f u n c t i o n : The i m p a c t o f a c o u r s e i n t e g r a t i n g m a t h e m a t i c a l c o n t e n t a n d p e d a g o g y . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 2 5 , 3 4 6 - 3 7 0 . Z a s l a v s k y , 0 . & P e l e d , I . ( 1 9 9 6 ) . I n h i b i t i n g f a c t o r s i n g e n e r a t i n g e x a m p l e s b y m a t h e m a t i c s t e a c h e r s a n d s t u d e n t t e a c h e r s : The c a s e o f b i n a r y o p e r a t i o n . J o u r n a l f o r R e s e a r c h i n M a t h e m a t i c s E d u c a t i o n , 2 7 , 7 9 - 9 5 . - 89 -APPENDIX A "Views of teaching and learning" - 90 -1 VIEWS OF TEACHING AND LEARNING: A piece of clay 13 being moulded by the potter. LU Many people work on the building site. They are InvoK In clearing, carrying, building, planning and supervlsln s A child Is throwing stones Into a pond, and watching ripples spread outwards. A guide and a traveller are moving through hilly terrain. There are a lot of hills, and one or two are very tall Indeed. The view of the landscape changes as they ascend the higher ground. 1.6 L7J A petrol pump attendant Is putting petrol Into a car. The driver, who sometimes uses self-service petrol stations, will 3 0 0 0 be able to drive away. 1.7 A gardener, surrounded by a range of garden equlprr 13 tending some of the different types of plant In garden. He prefers the garden as It Is, but realises there are many types of garden. © Chi ld ren's Learning in Science Project -- 91 -APPENDIX B Sample q u e s t i o n s - 92 -What does i t mean t o y o u t o t e a c h math? TEACHER How w o u l d y o u d e s c r i b e y o u r r o l e as a m a t h t e a c h e r ? A t t r i b u t e s o f a g o o d math t e a c h e r MATHEMATICS How w o u l d y o u c h a r a c t e r i z e / d e s c r i b e math? What message a b o u t m a t h do y o u want t o g i v e ? STUDENTS What i s a s t u d e n t ' s j o b ? How do t h e y l e a r n math? PEDEGOGICAL CONT. KNOWLEDGE Can anyone t e a c h math? What (k, s , a) do t h e y need? What a r e e f f e c t i v e t e a c h i n g s t r a t e g i e s ? Cf o t h e r s u b j e c t s PHILOSOPHY m o t t o , m e t a p h o r d i a g r a m s ( p h i l o s o p h y ) Have y o u r v i e w s o f t e a c h i n g math c h a n g e d f r o m t h e b e g i n n i n g o f t h e y e a r ? - 93 -APPENDIX C Sample notes - 94 -0 W h a t d o e s i t m e a n t o y o u t o t e a c h m a t h ? T E A C H E R H o w w o u l d y o u d e s c r i b e y o u r r o l e a s s m a t h t e a c h e r ? A t t r i b u t e s o f a g o o d m a t h t e a c h e r T2) S T U D E N T S W h a t i s a s t u d e n t ' s H e w d o t h e y l e a r . - . m M A T H E M A T I C S H o w w o u l d y o u c h a r a c t e r i z e / d e s c r i b e m a t h ? W h a t m e s s a g e a b o u t m a d o y o u w a n t t o g i v e ? 0 P H I L O S O P H Y m o t t o , m e t a p h o r P E D E G O G I C A L C O N T . K N O W L E D G E C c . - . o n y o r i E t e a c h m a t h ? W h a t ( k , s , a ) d o t h e y n e e d ? W h a t a r e e f f e c t i v e t e a c h i n g s t r a t e g i e s ? C f o t h e r s u b j e c t s d i a g r a m s — ^ ^ • <y>Aj-N***)^  ( p h i l o s o p h y ) V> H a v e y o u r v i e w s o f t e a c h i n g m a t h c h a n g e d f r o m t h e b e g i n n i n g o f t h e y e a r ? 'a-9 5 Appendix D Sample Transcript Excerpt T h i s e x c e r p t i s t a k e n f r o m p a g e s 5 t o 9 o f t h e i n t e r v i e w w i t h V i c t o r i a . She a n d t h e i n t e r v i e w e r a r e i d e n t i f i e d b y V and I r e s p e c t i v e l y . I : Ok, o k . Now, y o u have a d e g r e e i n m a t h e m a t i c s ? V : Um, my s e c o n d d e g r e e , y e p . I : Ok, w h a t , what k i n d o f , w h a t ' s y o u r a c a d e m i c b a c k g r o u n d ? V : Um, my f i r s t d e g r e e i s a B a c h e l o r s , a B a c h e l o r s o f A r t s w i t h D i s t i n c t i o n i n E c o n o m i c s . My s e c o n d d e g r e e i s a B a c h e l o r s o f S c i e n c e H o n o u r s i n M a t h e m a t i c s . My t h i r d d e g r e e t h a t I ' v e b e e n w o r k i n g o n i s my M a s t e r s o f S c i e n c e i n , um, d i s c r e t e m a t h e m a t i c s , a n d my f o u r t h d e g r e e , w h i c h I ' m d o i n g r i g h t now, i s my B a c h e l o r o f E d u c a t i o n , i n m a t h e m a t i c s . I : Ok, so y o u r B . A . i n E c o n o m i c s i s a 3 o r 4 - y e a r ? V : 3 - y e a r . I : 3 - y e a r . A n d y o u r B . S c . H o n o u r s ? V : 4 - y e a r . I : 4 - y e a r ? Ok. A n d what was y o u r m a j o r i n t h a t ? V : Um, m a t h e m a t i c s . Um, o t h e r o n e ' s , e c o n o m i c s . I : Ok, s o , r e a l l y y o u c o u l d d e s c r i b e y o u r s e l f as a m a t h e m a t i c i a n . V : Y e a h . I : Ok. How w o u l d y o u c h a r a c t e r i z e [V: o r p r o f e s s i o n a l s t u d e n t ( l a u g h t e r ) ] N o . . . h o w w o u l d y o u c h a r a c t e r i z e math? V : Ok, y o u ' r e g o i n g t o have t o do a b i t more o f . . . . What do y o u mean b y c h a r a c t e r i z e math? I : What a d j e c t i v e s w o u l d y o u u s e t o d e s c r i b e m a t h e m a t i c s ? V : Ok, I ' m g o i n g t o s t a r t a n 1 u s e , I ' m gonna t a k e a l o o k a t my a r e a i n d i s c r e t e m a t h e m a t i c s a n d , I a c t u a l l y t h i n k , I t h i n k i t ' s f u n ? um, a n d I t h i n k a way t o do i t w o u l d b e , I mean I d e s c r i b e , I c a n d e s c r i b e d i s c r e t e m a t h e m a t i c s t o a n y b o d y w a l k i n g o n t h e s t r e e t a n d t h e y c a n u n d e r s t a n d i t b e c a u s e i t ' s g l o r i f i e d , what I do i s g l o r i f i e d c o n n e c t -t h e - d o t s . E v e r y o n e knows how t o c o n n e c t t h e d o t s ! Ok. W e ' r e j u s t g o n n a do i t a h e c k o f a l o t more a n d a l i t t l e b i t t o u g h e r , b a s i c a l l y . I do h y p e r g r a p h s w h i c h i s m u l t i p l e l i n e s , and t h i s t y p e o f t h i n g . U m . . . s o , I g u e s s i t ' s , i t ' s d i f f e r e n t b e c a u s e when y o u ' r e a n , as a n u n d e r g r a d y o u have t h e r i g h t o r wrong and I mean y o u h a v e , y o u ' l l h a v e , y o u ' r e d o i n g a p r o o f ? A n d t h e r e ' s d i f f e r e n t ways o f d o i n g t h e p r o o f , b u t y o u know y o u ' r e t r y i n g t o p r o v e t h e same t h i n g i n t h e e n d . I mean y o u ' r e g o i n g f r o m , um, i f e p s i l o n d e l t a t y p e t h i n g , ok p r o v e t h i s i s c o n t i n u o u s , p r o v e i t ' s d i s c o n t i n u o u s . You c a n ' t have s o m e t h i n g t h a t ' s w e l l , i t ' s s o r t o f c o n t i n u o u s i n t h i s a r e a . I t ' s e i t h e r c o n t i n u o u s o r n o t . Whereas when y o u g e t i n t o g r a d m a t h , y o u s t a r t g e t t i n g i n t o t h i n g s , um, s o r t o f t h e p r o o f s o f - w e l l , we d o n ' t know i f i t ' s t r u e o r n o t y e t , ok? W e ' r e t r y i n g t o p r o v e i t , and i t ' s u p , i t ' s up t o y o u t o make s u r e y o u ' r e m a k i n g t h e r i g h t c h o i c e s and so i t m i g h t mean i n t u i t i v e l y , y o u s o r t o f g o , w e l l i t SHOULD h a p p e n . . . W e l l , i f i t s h o u l d - 97 -h a p p e n t h e n p r o v e i t . Um, and t h e y d o n ' t know. A n d I mean y o u ' r e i n t h e f u z z y g r e y a r e a s t h e r e , y o u ' r e g e t t i n g i n t o edges o f m a t h , [ I : o h r e a l l y ? ] r a t h e r t h a n , r a t h e r t h a n t h i s known a r e a . ' C a u s e y o u ' r e s t a r t i n g o f f , I mean c a l c u l u s i s p r e t t y much known, ok? You move o u t f u r t h e r , a n d f u r t h e r , a n d my, I do h y p e r g r a p h s o r , um, s e t t h e o r y a n d , I ' m , I was w o r k i n g u n d e r one o f t h e t o p guys i n t h e f i e l d . Um, a n d i t ' s s o r t o f , a v e r y , i t ' s , i t ' s , i t ' s , y o u ' r e j u s t s t i l l d i s c o v e r i n g , y o u ' r e s t i l l c r e a t i n g i t , ok? I t ' s s t i l l b e i n g c r e a t e d . A n d l i k e t h a t t o me i s n e a t b e c a u s e I know t h e math t h a t we t e a c h i n h i g h s c h o o l s o r l i k e go o u t , i t ' s b e e n h e r e f o r e v e r . I mean o u r p a r e n t s d i d t h i s , o u r g r a n d p a r e n t s d i d t h i s . N o t h i n g ' s c h a n g e d . Whereas k i n d o f , n o , i t ' s n o t t h a t , i t ' s c h a n g i n g a l l t h e t i m e . I : I s i t c h a n g i n g a t t h e l e v e l t h a t t h e s t u d e n t s a r e l e a r n i n g a b o u t math o r , o r c a n y o u c o n - , c o n v e y t h a t t o t h e s t u d e n t s ? I s i t r e a s o n a b l e t o c o n v e y t h a t t o them? V : Um, hmm. I t h i n k w h a t ' s more r e a s o n a b l e t o c o n v e y t o them i s how t h e a d v e n t o f t e c h n o l o g y i s c h a n g i n g how we do m a t h . I mean as f a r a s , w e l l i t ' s n i c e t o know how m u l t i p l i c a t i o n a n d a l l t h a t w o r k s , b u t I d o n ' t e x p e c t y o u t o s i t , t o have t o s i t down a n d l o n g h a n d do 158 m i l l i o n d o t e t c . e t c . e t c . I mean a h o r r i b l e number t i m e s w h a t e v e r d i v i d e d i n t o s o m e t h i n g b e c a u s e y o u p u n c h i t i n t o y o u r c a l c u l a t o r . So i t ' s n o t r e a l l y w o r t h y o u r t i m e a n d e f f o r t b e c a u s e I know y o u c a n do i t , so t h a t , so w e ' l l j u s t u s e t h e c a l c u l a t o r t o do i t , ok? Whereas we u s e d t o have t o u s e s l i d e r u l e s , e t c , e t c . t o p r o v e o r t o go t h r o u g h t h a t . W e l l now, we d o n ' t h a v e t o , now w e ' r e n o t a s k i n g p u r e "do i t " t y p e q u e s t i o n s , w e ' r e now a s k i n g q u e s t i o n s t h a t a c t u a l l y r e q u i r e more t h o u g h t . I : Ok, we. Who's we? V : Um, we b e i n g m a t h e m - , b e i n g t e a c h e r s i n g e n e r a l o f m a t h . [ I : o k . . . ] W e ' r e n o t , w e ' r e n o t a s k i n g , w e ' r e n o t , w e ' r e n o t gonna g i v e a g r a d e 12 exam a n d t e l l s t u d e n t s t o m u l t i p l y t h i s b y t h i s o r do l o n g d i v i s i o n o n t h i s . L o n g d i v i s i o n , b y t h e way was s o m e t h i n g t h a t a t one p o i n t i n t i m e o n l y 5 p e o p l e i n t h e w o r l d c o u l d d o , ok? A n d i t ' s t h e same t h i n g as w i t h n e g a t i v e n u m b e r s . N e g a t i v e numbers a r e s o m e t h i n g , um, t h a t many m a t h e m a t i c i a n s w r o t e t h e s i s e s [ t h e s e s ] o n a n d w r o t e p a p e r s o n why n e g a t i v e numbers c a n ' t e x i s t . So t h a t , I t h o u g h t t h a t was k i n d o f n e a t . B u t , um, I t h i n k i t ' s h a r d t o b e c a u s e where t h e y ' r e a t i n math i s a known a r e a and i t ' s n o t c h a n g i n g a l l t h a t much. I t h i n k y o u c a n b r i n g some o f t h e new s t u f f i n p o s s i b l y , b u t i t , i t , what t h e y ' r e d e a l i n g w i t h , i s n o t , I mean, y o u ' r e n o t g o i n g t o s u d d e n l y h a v e e q u a t i o n s o f l i n e s c h a n g i n g d e s p e r a t e l y . I mean i t ' s known what a l i n e i s , a n d how t o s o l v e i t . W e ' r e n o t h a v i n g t h a t much change h a p p e n i n g t h e r e . The c h a n g e s i n how math i s b e i n g t a u g h t a r e more t e c h n o l o g y -b a s e d . I : Ok. T h a t ' s r e a l l y i n t r i g u i n g . I ' m r e a l l y g l a d y o u c a n a d d t h a t k i n d o f p e r s p e c t i v e o n i t . B e c a u s e a l o t o f - [ l a u g h t e r ] n o , t r u l y , b e c a u s e a l o t o f , um, s t u d e n t t e a c h e r s , w e l l , p e o p l e who a r e g e t t i n g t h e i r B . E d . ' s d o n ' t have a n i n - d e p t h m a t h e m a t i c a l b a c k g r o u n d as y o u d o . [ l a u g h t e r ] Can we, o k , w e ' l l go b a c k t o t h a t [V: ok] b e c a u s e I want t o c o n t i n u e , o f c o u r s e , t h a t t e a c h i n g a s p e c t . A n d I ' d l i k e t o go b a c k t o t h e f i e l d o f m a t h e m a t i c s . V : Ok, I ' v e s i d e t r a c k e d on t h a t . - 98 -I : N o , n o , t h a t ' s f i n e . T h a t ' s p e r f e c t . Um, y o u made one s t a t e m e n t a n d y o u s a i d , w e l l math i s c r e a t e d o r , o r d i s c o v e r e d , a n d i s i t one o r b o t h , o r what? V : Um, I t h i n k t h e new t h e o r i e s a r e p r o p o s e d , I g u e s s , I mean, y o u ' r e , y o u ' r e m a k i n g h y p o t h e s i s , um, and t h e r e f o r e t h a t ' s a d i s c o v e r y , i t ' s a d i s c o v e r y o f t h e h y p o t h e s i s . Then y o u do t h e p r o o f , a n d t h a t ' s c r e a t i n g . I mean t h e two a r e t i e d t o g e t h e r , I d o n ' t know i f y o u c a n r e a l l y d i s t i n - , d i s t i n g u i s h b e t w e e n t h e m . B e c a u s e y o u ' r e n o t , what w e ' v e d o n e , i s w e ' v e s e t , w e ' v e s e t i n s t o n e t o some e x t e n t t h e b a s i s o n w h i c h math i s b u i l t . A n d now o t h e r s c i e n t i s t s a r e u s i n g t h i s m a t h , ok? So w e ' r e , w e ' r e t h e b u i l d i n g b l o c k s o f o t h e r s c i e n c e s . I : F o r e x a m p l e , a l g e b r a , [V: um -] some f i n i t e math a t t h e b a s i c l e v e l . . . V : A t t h e b a s i c l e v e l . A n d I mean, I mean c h e m i s t r y u s e - , u s e s u s a l o t , b i o l o g y u s e s u s , p h y s i c s u s e s u s , a n d I mean a l o t o f o u r m a t h , t h e h i g h e r l e v e l , w e ' r e t r y i n g t o c l o s e l y a p p r o x i m a t e what t h e y ' r e d o i n g i n p h y s i c s , a n d what t h e y ' r e d o i n g i n c h e m i s t r y . W e ' r e t r y i n g , w e ' r e t r y i n g t o g e t t h e s t u f f w r i t t e n b e f o r e t h e y n e e d t o u s e i t , so when t h e y n e e d t o u s e i t t h e y c a n j u s t p u l l t h i s t h i n g o u t a n d g o , g r e a t t h i s i s a t o o l I n e e d t o u s e . W e ' r e c r e a t i n g t o o l s , b a s i c a l l y . I : I s t h a t t h e p u r p o s e o f , o f d e v e l o p i n g m a t h e m a t i c s ? V : Um, y e a h , b a s i c - , I mean, m a t h , f o r me I f i n d math a j o y a n d b e a u t y a n d e v e r y t h i n g e l s e i n m a t h , b u t , n o b o d y ' s g o i n g t o f u n d , no c o m p a n y ' s g o i n g t o go o u t t h e r e a n d , um, f u n d m a t h e m a t i c i a n s o r s a y t h a t m a t h e m a t i c i a n s a r e a t a l l u s e f u l u n l e s s t h e y a c t u a l l y c r e a t e s o m e t h i n g , ok? I ' m n o t s a y i n g t h a t e v e r y m a t h e m a t i c i a n ' s o u t t h e r e i s g o i n g t o c r e a t e s o m e t h i n g w o n d e r f u l o r a n y t h i n g , b u t y o u n e e d t o have somebody o c c a s i o n a l l y c r e a t i n g a t o o l t h a t w i l l be u s e f u l i n t h e r e a l w o r l d . O t h e r w i s e , math w i l l n e v e r g e t f u n d i n g a n d p e o p l e w i l l k i n d o f g o , w e l l , my c o m p u t e r , I w i l l p u t t h i s i n t o my c o m p u t e r . I d o n ' t n e e d t o know how t o do t h i s , a n d t h a t ' s , t h a t ' s d a n g e r o u s . I : L e t ' s t a l k a b o u t humans and m a t h . V : [ l a u g h t e r ] Ok! I : What d o e s i t t a k e f o r someone t o be g o o d a t math? A n d i f y o u t h i n k i t ' s d i f f e r e n t a t d i f f e r e n t l e v e l s , p l e a s e s a y s o . V : Oh, d e f i n i t e l y . Um, d e f i n i t e l y v e r y d i f f e r e n t g o i n g f r o m , I ' m j u s t g o i n g t o l o o k a t i t , um, i n e l e m e n t a r y s c h o o l , i t ' s p u r e m e m o r i z a t i o n . - 99 -

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0054656/manifest

Comment

Related Items