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

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CONCEPTIONS  OF TEACHING MATHEMATICS:  THE IDEAS OF S I X P R E - S E R V I C E  SECONDARY MATHEMATICS  TEACHERS  by ROSALIND CHUI MEI LEE B.Sc, B.Ed., A THESIS  The The  U n i v e r s i t y of U n i v e r s i t y of  SUBMITTED  Toronto, Toronto,  1976 1977  I N PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE Department of  Curriculum  i n Mathematics  STUDIES Studies  Education  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 to the r e q u i r e d standard  •THE UNIVERSITY**OF June  BRITISH  COLUMBIA  1996  ( c ) R o s a l i n d Chui Mei Lee,  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)  Abstract  " I am a m a t h t e a c h e r . " images.  Elucidating pre-service  statement  and i d e a s  about  mathematics  investigation,  conclusion of  w i t h a mathematics  specialty;  a p p l i e d mathematics.  Results  different  of  content  of mathematics;  gave p r i o r i t y  teacher  teachers of  t h e i r ideas  t h e i r extended p r a c t i c u m .  conceptions  education.  Knowing t h e i r candidates  of  A l l had b a c h e l o r ' s  show a l l s i x t e a c h e r  teaching mathematics.  study i n pure  candidates  i n their teaching perspective  T h o s e who h a d s t u d i e d g r a d u a t e  expressed  a desire  t o make m a t h i n t e r e s t i n g a n d s t i m u l a t i n g f o r  formed and p a s t  experiences  are  the  influential  Some  development  skills.  that  level  or  had  Some e m p h a s i z e d  to student  at  degrees  emphasized mathematical p r o c e s s e s .  support the c o n s t r u c t i v i s t ideas  a  teaching mathematics  cognitive  Results  this  secondary-  t h r e e had pursued graduate  others  for  of  conceptions.  six pre-service  were i n t e r v i e w e d about  evokes a p l e t h o r a  p e r s o n a l meanings  teaching also provides  from which to consider a l t e r n a t i v e In t h i s  the  teachers'  adds t o t h e knowledge base f o r t e a c h e r  own b e l i e f s basis  This simple statement  and  mathematics  conceptions  are  i n s h a p i n g them.  students.  personally  Table of  Contents  Abstract Table of  i i Contents  i i i  Acknowledgements  v  CHAPTER 1:  1  Introduction  Stability  of  Conceptions  1  Influence  of  Conceptions  2  Teacher Education Research  2  Research Questions  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 Mathematics Education  6  The R o l e o f  9  Existing Beliefs  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  14  of Teaching Mathematics  A G e n e r a l Model f o r D e s c r i b i n g of  Conceptions  Teaching  The F r a m e w o r k f o r t h i s  16 Study  18  CHAPTER 3 : M e t h o d o l o g y  21  Development of the P r o t o c o l The P i l o t S t u d i e s The f i r s t p i l o t s t u d y The s e c o n d p i l o t s t u d y  21 22 22 23  The M a i n The The The The  24 24 26 27  Study program sample data c o l l e c t i o n d a t a p r o c e s s i n g and a n a l y s i s  - i i i -  CHAPTER 4 :  Results  30  Yvette  30  Shannon  35  Doug  43  Ray  50  Victoria  55  Mary  62  Summaries  67  CHAPTER 5 :  Conclusions,  Other  I s s u e s & Recommendations  71  Conclusions  71  Other  76  Issues  Recommendations  79  References  85  Appendix A: Views  of  teaching  and l e a r n i n g  90  Appendix B:  Sample q u e s t i o n s  92  A p p e n d i x C:  Sample  94  A p p e n d i x D: S a m p l e  field  notes  transcript  excerpt  - iv  96  -  Acknow1edgement s  F i r s t , t o my two p r i n c e s s e s , Dawn a n d D e n i s e , 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 .  whose  love,  trust  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 w e r e c o n c e r n e d f o r my s t u d i e s a s I was f o r t h e i r s .  and as  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 and mentor 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 .  to  F i n a l l y , t o t h e f r i e n d s a n d 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 the w r i t i n g of t h i s t h e s i s . T h e y 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 a n d 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 heartfelt  and humble  THANK  YOU  to a l l .  C h a p t e r One:  Pre-service what  it  Lortie  means  teachers  have  Introduction  already  t o t e a c h when t h e y e n t e r  (1975)  first  used the  constructed a teacher  how t h o s e  to  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  decided to enter. however,  i n that  narrow vantage  that is,  very  S t a b i l i t y of These relatively professional Tomchin, service  t h i s o c c u p a t i o n have  this  t h e y have o n l y seen the  full  teaching  actually  i s not a t r u e  responsibilities.  likely persist  for that  [p  spent  they  61] 12  have  apprenticeship,  show f r o m one  entire  kaleidoscope  He g o e s o n t o  formed d u r i n g t h i s p e r i o d ,  of  program.  the p r o f e s s i o n  r a t h e r than having viewed the  a teacher's  the v i s i o n of will  a student,  point,  which comprises  who c h o o s e  He p o i n t s o u t t h a t as  education  term " a p p r e n t i c e s h i p of o b s e r v a t i o n "  to describe 16 y e a r s  t h e i r own i d e a s  claim  skewed t h o u g h  it  individual.  Conceptions initial stable  for  program  1987;  conceptions  education  (Brown & B o r k o ,  Tabichnick  teachers,  of  teaching  have  students  throughout  1992;  and Z e i c h n e r ,  Goodman (1988)  been  found  to  their  be pre-  Goodman, 1988 ;. B o r k o , L a l i k & 1984) .  termed these  In  his  "intuitive  work  with  pre-  screens":  " . . . t h e i r pre-professional images formed an ' i n t u i t i v e screen through which they interpreted their professional e d u c a t i o n . . . . [ I n ] the [teacher education] program, these i n t u i t i v e s c r e e n s g a v e them a n 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 out o f the a c t i v i t i e s and i d e a s p r e s e n t e d t o t h e m . . . . W h e n exposed 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 tended t o a c t f i r s t on an intuitive r a t h e r t h a n an 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 an 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 1 3 0 - 1 3 1 ] 1  In o t h e r words,  Goodman f o u n d t h e t e a c h e r  education program s o l i d i f i e d  and/or c l a r i f i e d these p r e - p r o f e s s i o n a l  images  images  t h o s e who d i d  of  teaching.  He a l s o  c o n c e p t u a l change underwent their  incipient  Influence  of  found that  a good d e a l o f u n r e s t  a s t h e y w e r e g i v i n g up  Conceptions t h e i r ideas  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  (Thompson,  1984) .  These  ideas  i n turn  of  t e a c h i n g seem t o  as t h e e s s e n c e o f  desk,  (Dossey,  students  beliefs  about  1992;  its  very different  Thompson, 1 9 9 2 ) .  who l e a r n m a t h e m a t i c s  the  significance  its  teaching;  of  Teacher  and p u r p o s e s  with  ideas  in  the  their the  different  This  i t may h a v e  highlights  about mathematics  the b e l i e f s  and  that  i n the c l a s s r o o m and t o g a i n  for  subject  side of  for teaching  1991) .  be  insight  teaching.  Education Research  Borko, p r o g r a m s were rather  (Brophy,  t o a c q u i r e knowledge o f  potentially drive their actions into t h e i r goals  purposes  investigating teachers'  namely,  they give  From t h e o p p o s i t e  from teachers  n a t u r e and d i f f e r e n t  classroom experiences  the  influence their actions  c l a s s r o o m and f l a v o u r t h e messages about mathematics students  experience  ideas.  For mathematics teachers, strongly  r a t h e r t h a n i n t r o d u c e new  Lalik  a n d T o m c h i n (1987)  largely  than teacher  addressed identified  student  observed that  teacher  i n f o r m e d by r e s e a r c h on e x p e r i e n c e d  candidates.  teachers'  education teachers  T h e y " f o u n d o n l y a few s t u d i e s  conceptions of  t h e n e e d f o r more r e s e a r c h  teaching"  [p 78]  in exploring this  - 2 -  that  and  population's  conceptions Further, of  of  Lampert  research  161]  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  that  a n d most o f  than teacher  (1988) is  stated that  specific  that  thinking.  h i s teacher  and t y p e o f  For example,  range  his  (e.g.  candidates Lappan,  Civil,  mathematics  1990b)  1990).  (1991)  content  in t h e i r students'  s u c h as d i v i s i o n  to uncovering beliefs  & McDiarmid,  mathematics example,  Intervention studies  course ideas  elementary prospective  l e s s o n and  lack of,  (e.g.  on  prewith  The  mathematical  T i r o s h and  that  For example,  i n an a t t e m p t  to  of the nature of  1990) .  (1990b)  amount  knowing math  is  correct  elementary W i l c o x , Schram,  i n s t i t u t e d a " l e a r n i n g community" i n a  Little  t h o u g h some c o n t e n t  Ball's  during  content.  r u l e s and canons and p r o d u c i n g  R e s e a r c h on secondary p r e - s e r v i c e (Ball  before,  t h e s e have d e a l t  i n , and thus  have a l s o been c a r r i e d o u t .  & Lanier  1993)  have e x p l o r e d  and have c e n t r e d a r o u n d s p e c i f i c  d o i n g math a c c o r d i n g t o s e t answers  H o w e v e r , most o f  from exposing f a l l a c i e s  1990; B a l l ,  & Agostinelli,  w e r e made, more s t u d i e s  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 Graeber,  rather  progress.  conceptions.  elementary candidates studies  (Valli  c l a r i t y o f b o a r d w o r k a n d amount o f  statements  teachers'  Valli  behaviours  [p  e d u c a t i o n p r o g r a m and u s e d s u c h c r i t e r i a as  unit planning to evaluate  service  education i n mathematics"  through his teaching experiences  questioning,  Since those  to teacher  e x i s t e d o n l y a " s m a l l amount  was c o n c e r n e d w i t h o b s e r v a b l e  f o l l o w e d one i n d i v i d u a l and a f t e r  there  behaviour.  i n i t i a t e conceptual  mathematics.  teachers  i s known a b o u t  issues  i s e v e n more t h e i r views  limited of  have been i d e n t i f i e d .  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 t e a c h e r s and found a l a c k o f  - 3 -  change  teaching  For as  conceptual  understanding of d i v i s i o n  i n both groups.  Z a s l a v s k y and P e l e d  found 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 the commutative and p r o p e r t i e s of b i n a r y operations  (1996)  associative  i n the understandings of  20  prospective  teachers.  Research  Questions  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 s o c i e t a l m i l i e u of  backgrounds,  constant  p a r t i c u l a r l y i n the  change and u n c e r t a i n e c o n o m i c s .  a p r e - p r o f e s s i o n a l program v a r y i n age, have f u r t h e r academic q u a l i f i c a t i o n s . varying conceptions To h e l p f i l l this  of  what a r e  the  Some o f  come f r o m o t h e r p r o f e s s i o n s , It  is  likely  they w i l l  teachers of  i s known a b o u t  of  and  have  the kinds of  secondary  in this  study  is:  mathematics?  s t u d y were p u r s u i n g , o r had  w i t h graduate  qualitatively different  ideas  t e a c h i n g mathematics h e l d by a  work i n m a t h e m a t i c s .  f o r those  in  mathematics.  conceptions  subjects  graduate  subquestion: conceptions  the  Students  the primary research q u e s t i o n i n t h i s  individual  group of p r e - s e r v i c e  completed,  teaching  t h e gap i n what  population holds,  present  T h i s gave r i s e t o  studies  a  i n mathematics,  are  their  f r o m t h o s e who h o l d o n l y a  baccalaureate? This conceptions of the  study set of  out 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 e a c h i n g mathematics of  secondary mathematics. entire  landscape  a group of p r e - s e r v i c e  While the r e s u l t s  of p o s s i b l e conceptions,  comprehensive p i c t u r e of  these prospective  are not  the teachers  intended to  convey  t h e y do p r o v i d e a  teachers'  ideas,  thoughts,  attitudes,  and g o a l s  primary intent of  r e l a t e d to the t e a c h i n g of mathematics.  this research is  informs mathematics teacher the teacher d e s c r i b e d as  candidates  t o add t o the knowledge base  education.  w i t h what B r o w n ,  "encounters  The  In addition,  are not taught  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 653]  - 5 -  the study p r o v i d e d  Cooney and J o n e s  i n which teachers  that  t o be  (1990)  have  methodologies  important."  [p  Chapter  C o n s t r u c t i v i s m i n Mathematics The  influence  of  approximately  past  (Steffe  c o n s t r u c t i v i s m was relevance  Confrey, extent  1986;  that  the  Brophy,  1986b).  grounds  an e m e r g i n g  The b a s i c their  of  and t e a c h e r 1995a).  " p o l i t i c a l l y correct"  (Cobb,  1994a)!  knowledge  was p e r s o n a l  It  is  individuals  than r e c e i v i n g  it  from an e x t e r n a l  t a k e s a n d how i t  is  created  1990)  h i g h l i g h t e d the  r e a l i t y when he f o r m u l a t e d  (von G l a s e r s f e l d , was  as a (see  even  create source.  have been  issues  of  theorists. assertion  b y a n i n d i v i d u a l was u n i q u e a n d n o t  because i t  for  and as  that  (1989,  objective  1994),  constructivism is  rather  the  education  Simon,  1986a;  the b a s i s  1991)  1995;  premise of  research  Brophy,  i s b e i n g u s e d as (Confrey,  the  its  prominent to  teaching  created  constructivism"  (see  emerging models o f  Von G l a s e r s f e l d  an e x t e r n a l ,  paradigm  (Thompson,  as  over  perspective,  education  i t h a s become  perspective  & D'Ambrozio,  research  debate w i t h regard to  i n t e l l e c t u a l development  d e b a t e among c o n s t r u c t i v i s t  knowledge  new  learning  own k n o w l e d g e  What f o r m t h i s  intense  Now,  steadily  theory of  1995b; S t e f f e  being construed  of  education  As a r e l a t i v e l y  i n i n f o r m i n g mathematics  a constructivist  foundation for  1994).  increased  t r a d i t i o n a l process-product  theory  Simon,  30 y e a r s ago h a s  focus  a tentative for  Literature  Education  & Kieren,  and u s e f u l n e s s  compared t o the  The  c o n s t r u c t i v i s m i n mathematics  which began decade  2:  1990,  p 28).  f o r m e d as a r e s u l t  - 6 -  the  a replication  of  "radical  He a r g u e d of  that  an  that  knowledge  individual's  experiences Because  a n d how t h a t  p e r s o n made s e n s e o f  sense-making by i t s  argued there  nature  c a n be no p r e c i s e  is  individual  a n d i n t e r n a l , he  counters  which e x i s t s  else's  view which purports that  independently of  human e x p e r i e n c e  acquiring a personal understanding or  knowledge.  (Confrey,  there  and t h a t  "knowing" of  else  interpretative  rather than information t r a n s f e r  the r e a l i s t  also  o f what we t h i n k someone  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  u s i n g s h a r e d meanings This  experiences.  way o f k n o w i n g someone  We c a n o n l y h a v e o u r own u n d e r s t a n d i n g s knows.  those  is  1994) .  knowledge  learning  that  acts  is  external  knowledge. Another debate perspective  views  it  construction occurs ideas  constructs  existing  K o n o l d and J o h n s o n language  of  a r e many s i m i l a r  differences  t o accomodate  cognitive  that  then there  new c i r c u m s t a n c e s and i s  constructed. where  this  One  knowledge  called  is  ideas  "schema."  However,  between  an  when t h e r e "fit"  T h i s mental development  learning.  continual.  It  occurs  are with new to  - 7 -  account  ( E r i c k s o n , 1987,  or  construction  "increasingly viable descriptions with"  schema  consciously  L e a r n i n g o r knowledge  we come i n t o c o n t a c t  or  Recognition  r e - o r g a n i z a t i o n or c r e a t i o n of  constitutes  to  psychological  psychology,  t h e new s i t u a t i o n d o e s n o t  c a n be d e s c r i b e d a s d e v e l o p i n g phenomena t h a t  call  characteristics  t h e new c i r c u m s t a n c e s .  subconsciously,  individual  (1991)  schema a n d a new e n c o u n t e r .  schema,  is  t h r o u g h c o m p a r i n g a n d c o n t r a s t i n g new e x p e r i e n c e s  In the  when t h e r e  sufficient  the  are h e l d i n mental s t r u c t u r e s  individual's  for  as t h e a c t i v i t y o f  already held.  constructivism.  occurs  f o c u s s e s o n how t h e k n o w l e d g e  p 9) .  of  the  The o t h e r proponents result only  stance  follows  c l a i m knowledge  a sociocultural perspective.  formation f o r the  of p a r t i c i p a t i n g i n s o c i a l  i n the  context  of  these  individual  interactions.  interactions;  it  is  socially  Learning, or personal  construction,  the  observed  the process  of  on the e x t e r n a l p l a n e  making p e r s o n a l "learns"  sense of  a n d becomes  Cobb  (1994b)  complementary, specific  it.  debate  c o n s t r u c t i v i s m have education.  based on measurable  by the over  individual  has  shifted  quantities,  and i d e a s  the p r i n c i p a l s are to those  1990),  and person  are  of  so-called  examined,  understandings,  That  the  This approach allows  research  in  1986)  paradigm" to (Thompson,  to  mathematics whose  results  studies,"  where  from those  for  interactions.  adherents  "scientific  w i t h a "humanist p e r s p e c t i v e "  How a n i n d i v i d u a l study.  was  group.  from i n v e s t i g a t i o n s  from "a p r o c e s s - p r o d u c t  alternative  this  social  (Romberg a n d C a r p e n t e r ,  investigation into multiple  of  1991)  i n t e r n a l i z a t i o n , the  through s o c i a l  t h i n k i n g and d e c i s i o n - m a k i n g p r o c e s s e s " for  i n t e r n a l i z i n g what  i t s philosophical aspects,  studies"  Jones,  meaning  constructed  Wood, & Y a c k e l ,  being researched.  "field-based  perspective"  a  knowledge  t h e s e two p e r s p e c t i v e s  changed the course  The f o c u s  of  i n t o the  as  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  knowledge  Despite  Cobb,  Through t h i s  suggested  problems and i s s u e s  c r e a t i o n of  (e.g.  enculturated  has  individual  occurs  Knowledge has  through shared understandings. is  Those  the  w i t h an  to thoughts  "analytic  (Brown, Cooney "a focus 1992) .  are  on  and  teachers'  By a l l o w i n g  c o n s t r u c t i v i s m h a s o p e n e d t h e way  for  conceptions.  conceives  of  t e a c h i n g mathematics  is  the  import i s given to personal understandings  - 8 -  focus on  teaching There  is  a consequence  o f my b e l i e f  in a constructivist  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 .  to teach,  nor to conceive  of  teaching.  There  epistemology.  i s no one r i g h t  way  What a n i n d i v i d u a l b e l i e v e s  to  be t e a c h i n g m a t h e m a t i c s must h a v e 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 experiences  and  beliefs.  The R o l e o f  Existing Beliefs  The b e l i e f s articulated  of  teachers  (Thompson,  1984).  important  for researchers  Teachers'  thoughts  classroom  ( C l a r k and P e t e r s o n , is  results  studies.  other  Nevertheless,  a s w e l l as f o r t h e t e a c h e r s  1986).  t h i n k i n g and i s  be  k n o w i n g what t h e y a r e  The r e s e a r c h e r  is  themselves.  substantially influence their actions  what t h e t e a c h e r of  may n o t be i n t e g r a t e d a n d may n o t  in  gains  the insight  able to apply t h i s context  into  to  " . . . 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 the p s y c h o l o g i c a l c o n t e x t i n which the t e a c h e r p l a n s and 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 b e 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 about h i s o r h e r r o l e and about t h e dynamics 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 286-287] N o t o n l y do t e a c h e r s ' classroom,  they also  ten fifth-grade beliefs  affect  their actions  influence t h e i r students'  teachers  i n four schools,  Ford  beliefs. (1994)  i n the I n one s t u d y  beliefs;  computational (1992)  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  skills also  with t h e i r teachers' mathematics.was  applied to r e a l - l i f e s i t u a t i o n s .  found students' images  images o f m a t h e m a t i c s  of mathematics.  p e r c e i v e d as t h e c o n t e n t  - 9 -  their  applying  In another linked  The i m a g e s w e r e  i n a set  of  found t h e i r  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  students'  Brown  beliefs  of books,  study,  strongly  varied; as r u l e s  and  ideas  used to solve problems,  as e a s y a f t e r  being sorted out,  and  as  enjoyable. The t e a c h e r  also benefits  from t h i s type of  through increased self-awareness Nickson  (1988)  has  suggested  research  of personal b e l i e f s  that  teachers'  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  beliefs  activity  about  teaching.  may be  and development  but,  on the  other  hand, " 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 t h e m 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 images  of  themselves  candidates. alternative program  as t e a c h e r s  teaching perspectives  emphatically,  1992) .  a foundation to  (1995)  teacher  education"  teachers  Goodman (1988)  Growth" concepts  (p 1 2 3 ) .  w h i c h emerged as  philosophy, concepts the  authority,  " g u i d i n g images"  t e a c h i n g as a p r o b l e m o f  d i f f e r e d amongst  same t e r m s .  e v e n more  teachers'  the teacher  For example,  teaching:  (p 124)  -  teachers a  Children's three first  Interpretations of  candidates  and  " T e a c h i n g as  f o r the  even though t h e y  to nine of his subjects,  - 10  of  elementary  and autonomy were  control.  beliefs  Using interviews  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  Cooperation,  compare  [p 6 8 ] .  s t u d i e d 12 p r e - s e r v i c e  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 Control"  teacher  has y i e l d e d e v i d e n c e  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 .  Problem of  state  " 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  Research with pre-service  and t h e i r  which 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  Raymond a n d S a n t o s  s h o u l d be a n i n t e g r a l p a r t o f  observations,  school,  i s equally important for  This metacognition provides  (Kagan,  different  regarding students,  cooperation  these used meant  pupils  f o l l o w e d the r u l e s  l a i d out by the t e a c h e r .  meant meant w o r k i n g t o g e t h e r desired result contrast  Similarly,  by s t a t u s :  In both instances,  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 .  i n philosophy is  institutional  harmoniously.  there  To t h e o t h e r s ,  However,  it  the the  obvious.  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 and p e r s o n a l a u t h o r i t y .  I am t h e t e a c h e r .  y o u r f r i e n d a n d we r e s p e c t  The f i r s t  authority: was  granted  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  each o t h e r .  Goodman (1988)  concludes,  " 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 a n d Buchmann (1989) role  existing beliefs  They g i v e issue  of  a stronger  e q u a l a c c e s s t o k n o w l e d g e where experience  Janice's  case,  that  was  her course  readings.  perspectives  of  to bias"  On t h e o t h e r h a n d , S a r a h ,  disadvantaged background,  [p 3 7 2 ] . her  t h e same a c a d e m i c  were a c t u a l l y r e i n f o r c e d b y h e r  of  the  r e l i e d on  went u n c h a l l e n g e d ,  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  the  teachers.  "both candidates  l i m i t e d and s u b j e c t  because her p e r s p e c t i v e s  as n o n - e t h n i c s t u d e n t s  s t a n d on  learning of pre-service  a s 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 '  personal  that  p l a y i n the  take  In  notions  achievement  i n t e r p r e t a t i o n of  who came f r o m a  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  provide  more f o r h e r p u p i l s . As Kagan  (1992)  reported i n a review of  the p r o f e s s i o n a l growth of teachers,  the  ideas  personal beliefs images  of  of  27 s t u d i e s  e l e m e n t a r y and s e c o n d a r y  dealing with  pre-service  t e a c h i n g t h e y had f o r m u l a t e d were  strong.  Their  o f what c o n s t i t u t e d g o o d 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  themselves  as a t e a c h e r  had been d e r i v e d from  - 11  -  "personal  biographies,"  i.e.  from the  remembered and t h e i r p a s t findings it  of  earlier  images  experiences  studies,  Kagan,  1992)  unique nature  of  conceptual  that  The s t a b i l i t y o f importance of programs teacher  had  as  with  and images  I n t h e one s t u d y where  exemplary teachers they students.  most n o t a b l y ,  was r e p o r t e d t h e s e b e l i e f s  the program.  of  Tabachnick  often  of  teacher  underscores  For teacher  and f o r p e r s o n a l p r o f e s s i o n a l  for reflection  cited  in  was a t t r i b u t e d t o  candidates  n e e d t o be made a w a r e o f  can then provide a b a s i s  it  1990,  during  the  course.  investigating their conceptions.  candidates  (1984) ,  remained unchanged  change d i d o c c u r ,  ideas  t o make a d i f f e r e n c e  & Zeichner  (Florio-Ruane & Lesmire,  p a r t i c u l a r methods the  In concert  their ideas.  the  education growth,  This  knowledge  from w h i c h change c a n be  effected.  The N a t u r e o f Because instructional at  various  Mathematics how m a t h e m a t i c s practice  i s p e r c e i v e d may s t r o n g l y  (Dossey,  characterizations  of  1992;  Ernest,  (1981)  this  of  Platonist,  f o r m a l i s t and  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 r e a l i t y w i t h immutable t r u t h s  set  apart  regarding  give  voice  human k n o w l e d g e  to these  t o be a n i d e a l ,  f r o m a n d a b o v e human  and m a t h e m a t i c i a n s  truths.  - 12  -  and  the  fallibilist.  Mathematical p r i n c i p l e s are d i s c o v e r e d r a t h e r than i n v e n t e d . independently of  looks  the d i s c i p l i n e , Davis  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  nature of mathematics:  section  mathematics.  F o l l o w i n g the h i s t o r i c a l development Hersh  1991),  affect  work t o  a  existence. They locate  exist and  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 axioms,  definitions  and theorems.  w h i c h have been c r e a t e d  These  and f o r m u l a s  predictions  of  r e a l - w o r l d phenomena,  primary purpose mathematical whose  the  models  meaning i s  agreed this  for  sense,  deduction  accurate  descriptions  these a p p l i c a t i o n s  cadre  mathematics  rules  of p r o f e s s i o n a l s  is  constructed.  i n which conclusions  a field  i s not  precepts  flux  as  rigid,  it  its  the b e l i e f  rules  not  the and  together and  known a s m a t h e m a t i c i a n s .  It  responds  are  d e v i s e d by  is  a field  can w i t h s t a n d r i g o r o u s  school holds  i n constant  Mathematics Its  This  invented.  and  strung  (axioms)  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 o n t h e n a t u r e fallibilist.  symbols  The f o r m u l a s  are merely symbols  d e r i v e d from s e t s of  upon by t h a t  via  of  which have been  c r e a t i o n of mathematics.  i n themselves  comprised  are manifested  and r u l e s  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  is  that  of  logical  proof.  of mathematics  mathematics  t o new q u e s t i o n s  and d e f i n i t i o n s  In  not  is  and  is  the  mutable, challenges.  "set  in  c a n a n d do c h a n g e 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  by  stone."  mathematicians. 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 work o f  mathematicians,  t e a c h e r s about  mathematics.  t o the use  of  consistent  with that  mathematics  they  are u s e f u l  i n d e s c r i b i n g the  t o be a f i x e d ,  students,  "presenting,"  knowledge  to  contrast,  a teacher  learners,  and s t r a t e g i e s  For example, non-changing  which c a r r i e s  if  entity  teaching  mathematics  - 13  -  i n the  could  to  transmitting  technique.  t o be a d y n a m i c  the  of lead  classroom  believes  t o be r e v e a l e d  a connotation of  c o u l d be a f a v o r e d  who b e l i e v e s  a teacher  from  thoughts  A t e a c h e r ' s view of mathematics  i n s t r u c t i o n a l techniques view.  have been d e r i v e d  By and  changing body of believes value  knowledge  mathematics  the  1986  " c a l l i n g on s e t  R e s e a r c h on C o n c e p t i o n s  pre-service  individual  c i t e d i n Dossey,  Research  aspect.  students  Someone  theoretical  1992,  who  of opinions  i n classroom  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  s t r u c t u r e d format (Hersh,  that  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  c o n t r i b u t i o n s of  Another teacher  c o u l d emphasize  discussions.  content  language and  could  in a  conceptions"  p 42).  of Teaching Mathematics  specific  to conceptions  teachers of  of  t e a c h i n g mathematics  secondary mathematics  held  has been d i f f i c u l t  by  to  locate. Wilson secondary  (1994)  teacher,  investigated Molly,  the e v o l v i n g b e l i e f s  i n the context  pedagogy and m a t h e m a t i c a l c o n t e n t .  of  a course  changed and the r e l a t i o n s h i p between  her views  of mathematics  this  v i e w was c o n s i s t e n t  w i t h her general  conceptual  view of mathematics  u n d e r s t a n d i n g broadened and deepened  course;  t o w a r d t h e e n d s h e was a b l e and use v a r i o u s  to  forms of  of  the  and  - that  it  answers.  during  identify real-world f u n c t i o n s under  at  functions  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  Molly's  functions  integrated  He f o u n d t h a t  M o l l y had a narrow view of  was a c o l l e c t i o n o f p r o c e d u r e s  of  which  prospective  t h i s u n d e r s t a n d i n g and  and t e a c h i n g m a t h e m a t i c s .  the coursework,  one  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  functions  beginning of  of  the  applications  different  circumstances. How d i d t h i s mathematics? However,  increased understanding affect  M o l l y b e l i e v e d her view had changed  Wilson  (1994)  her view of  teaching  significantly.  c l a i m e d s h e h a d s i m p l y f o u n d new ways t o  - 14  -  reduce  the monotony o f techniques, vibrant t o see  classroom lectures.  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  not f o r m u l a t e d a changed p e r s p e c t i v e  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 . alternative  mathematics  ways o f  to offer  Thus,  teaching mathematics,  and mathematics  t e a c h i n g were s t i l l  " a more  although  her views  [she  began]  of  r e l a t i v e l y narrow."  [p  367] Two o t h e r Brown  (1985)  studies  a n d Owens  dissertations.  (1987)  literature  (1985)  the  studies  sources. 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  teacher'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 a s he became s o c i a l i z e d i n t o  the  teaching profession.  the  Conceptions of mathematics,  a p p r o p r i a t e g o a l s and t a s k s "relative  f o r a math c l a s s r o o m ,  r e s p o n s i b i l i t i e s of  motivation,  teacher  his  ideas  and s t u d e n t s  d i s c i p l i n e , and e v a l u a t i o n "  comprised the a n a l y t i c a l framework. conceptions  of mathematics  of  teaching.  However,  p o i n t i n g once a g a i n t o t h e sought  and b e l i e f s  of mathematics  to describe  1992,  f o u n d was t h a t  and t e a c h i n g " s t a b i l i t y of  the c o n s t r u c t s  perspectives  o f m a t h e m a t i c s were l a r g e l y  experiences.  A d d i t i o n a l l y , the teacher of  p 225) pupils'  i n changing  college-level  - 15  -  most  (Brown & B o r k o , conceptions.  of  i n f l u e n c e d by candidates  the  concerning  four  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 .  mathematical perspective  about  "he d i d n o t c h a n g e some o f h i s  1992,  (1987)  about  t e a c h i n g had a s t r o n g i n f l u e n c e  conceptions  Owens  beliefs  (Brown & B o r k o ,  What she  basic general p 225),  -  - are both unpublished d o c t o r a l  T h i s h a s meant g l e a n i n g d e s c r i p t i o n s o f  through secondary Brown  which f i g u r e prominently i n the  He f o u n d  their  pre-college  found  math c o u r s e s  pre-service  the  incompatible with  their that  own.  These  pre-college  for prospective  findings confirm B a l l education figures  secondary  a n d M c D i a r m i d ' s (1990)  prominently i n content  teaching.  (1992)  has o f f e r e d  The e l e m e n t s  ideal.  used to describe  The f o u r t h ,  The f i r s t  context,  three  defines  Teaching  a framework f o r d e s c r i b i n g c o n c e p t i o n s  establish their d i s t i n c t i v e differences context,  understanding  teachers.  A General Model f o r D e s c r i b i n g Conceptions of Pratt  the conceptions are:  teacher,  learners,  a r e germane t o a n y t e a c h i n g  the o r g a n i z a t i o n a l s e t t i n g  of program,  factors  that  c o u l d i n f l u e n c e the t e a c h i n g and/or l e a r n i n g ,  education"  overriding goal,  class)  and i n c l u d e s any  " I d e a l " was u s e d t o d e s c r i b e  (p.205)  the  type  or e x p l i c i t ,  for the teacher.  such  of  contexts  classes,  250 t e a c h e r s  s u c h as p o s t - s e c o n d a r y  r e l i g i o u s education,  of  or  Pratt's derived  i n 5 c o u n t r i e s and a c r o s s  i n s t i t u t i o n s , government  and t r a i n i n g programs  as  "purposes  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 a n d t h e f r a m e w o r k was from i n t e r v i e w s of over  of  external  a n d c a n be c o n s t r u e d a s t h e c e n t r a l  implicit  content,  situation.  (e.g.  type  adult  type of  a  and  i n business  variety military and  industry. Pratt Engineering Being;  (1992)  delineated five  different  Facilitating Society.  conceptions  of  teaching:  - 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  Developmental - C u l t i v a t i n g the P e r s o n a l Agency,-  Intellect;  Nurturing  and S o c i a l Reform - S e e k i n g a  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  emphases i n t h e r e l a t i o n s h i p s between  - 16  the  -  arise  elements.  of  and hence  school,  impending examinations.  claim  of  Better  from d i f f e r i n g  The E n g i n e e r i n g c o n c e p t i o n h a s a s delivering is  content.  It  a primary focus.  are  concentrated  is  the  on i m p r o v i n g t e c h n i q u e s  tasks,  teacher  in this  case  exemplar of  skills  the values  embodies  the  "knowledge  like  delivery.  he i s a l s o  i m p l i c a t i o n that  teacher  s e e n t o be  the whole  a s much a s he h a s c o g n i t i o n o f  passed on t h r o u g h r o l e m o d e l l i n g "  it.  (p.  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  that  of knowledge.  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  the  The i d e a o f m o d e l l i n g , o r  Examples o f  by  acquired.  However,  constitutes corpus  Content  the Engineering conception,  expert,  the content.  Energies  and l e a r n i n g measured  as dominant e l e m e n t s .  conveys t h i s  content  [is]  of  for content  objectives  i s not o n l y the content  craftsman,  and i n f o r m a t i o n t r a n s m i s s i o n  gained and/or competencies  and c o n t e n t  a master  feature  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 .  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 , has  distinctive  teacher-centered  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 completion of  its  this  being  person  Hence,  212)  and l e a r n i n g  inherent  within  c o n c e p t i o n w e r e most  apprenticeships  an  often  and m e d i c a l  internships. The D e v e l o p m e n t a l desire of  to further  exercises  learners'  of  thought.  cognitive  and the r o l e of  ways o f  possess the  the  conception is  development  curriculum is  thinking since  capacity  c h a r a c t e r i z e d by the  to develop  produce d i s e q u i l i b r i u m , are The p e r s o n a l w o r t h o f Nurturing conception.  of her l e a r n e r s . t o s t i m u l a t e and  the c e n t r a l b e l i e f  whether  a p r e f e r r e d mode o f the  is  increasingly abstract  Inquiry-oriented exercises,  individual  -  the  The  focus  enhance  that  individuals  and complex  to arouse  forms  curiosity  or  instruction.  student  A s t r o n g bond between  - 17  teacher's  i s paramount  teacher  in  the  and l e a r n e r  is  evident. welfare  The t e a c h e r of  the  exhibits  student.  a h i g h degree of  The s t u d e n t ' s  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 control  is  knowledge identity  is  s e e n as b e i n g a v e h i c l e  c a n be  the  political  for  conducts  of  the teacher.  and b e l i e f s .  the  teacher, class  It  environmental  The F r a m e w o r k f o r T h i s (1992)  i n c l u d i n g but not  emphases,  learner's  but  a specific  equally strong,  to the  learners.  world."  Examples are  ideology religious, set  of  for  has  the  some f o r m s  "desirable  activities,  scope of  a  ideology, of  Study  own r o l e o f  [p 1 3 5 ] .  The t e a c h e r  goals of  teaching,  desirable  the  t e a c h i n g mathematics the  students'  role,  i n s t r u c t i o n a l approaches  Undergirding these i s  this  of pre-service  study,  and a c c e p t a b l e a philosophy of  my f r a m e w o r k  mathematics  - 18  to describe  teachers c o n s i s t e d of  -  as  mathematics  and  outcomes teaching  mathematics.  conceptions  centre  education.  l e g i t i m a t e mathematical procedures  For the  self-  c l e a r l y a r t i c u l a t e d and, b e i n g at  l e a r n e r and s o l i c i t s u p p o r t  limited to  teaching  instruction"  Content  i d e o l o g y may be  described a conception of  [a t e a c h e r ' s ]  appropriate  is  and r e l a t e s  and f e m i n i s t  Thompson  knowledge.  of  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  creating a "better  program,  This  or d e r i v e d from a d i f f e r e n t ,  the  development,  and i n t e r n a l l o c u s  through which the  c o n c e p t i o n l a b e l l e d S o c i a l Reform,  m i s s i o n to e n l i g h t e n the thereby  self-concept  and  personal  developed.  activities  convictions stage  self-knowledge  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  In the drives  concern f o r the  the the  of  conception of mathematics goal  the n a t u r e of mathematics and the c o n c e p t i o n o f  where t h e  l a t t e r encompasses an u n d e r l y i n g p h i l o s o p h y ,  f o r teaching mathematics,  and  a conception of  usefulness the  teacher  student  and  content.  The t e n t a t i v e to  work.  Pratt  (1988)  mathematics,  ideas  knowledge  The k n o w l e d g e  and p e d a g o g i c a l  (Shulman, 1986,  w o u l d be teacher Ball  p 6).  content  candidates.  In addition,  Ernest  with respect  (1989,  1991)  and  and about  feelings  about  knowledge,  i.e.  t h a t make i t  " t h e ways  cohort,  s i g n i f i c a n t and hence  to content  i n her  components  included  framework. i n h i s framework.  was s u b d i v i d e d i n t o k n o w l e d g e  category  matter,  and a t t i t u d e s  t e a c h i n g mathematics  of  the  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  teaching,  of  t e a c h i n g mathematics  - 19  -  His  math of:  (math p e d a g o g y  and math c u r r i c u l u m ) , context  to  elementary  teacher.  the  of  comprehensible  t h r o u g h h e r work w i t h t h i s  used s i m i l a r  other subject  oneself  c o m p o n e n t was c o m p r i s e d o f  beliefs  mathematics,  (1989,  i n f o r m e d my  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 , The k n o w l e d g e  in  the  and E r n e s t  prominent i n her study of  found apprehension of mathematics feelings  of  I drew  An example o f an i d e a o f math t e a c h i n g  " t e a c h i n g as t e l l i n g , "  personal  (1988)  o f math t e a c h i n g and l e a r n i n g ,  r e p r e s e n t i n g and f o r m u l a t i n g the s u b j e c t others"  and c o n t e n t ,  knowledge  teacher  relationships  of mathematics teachers a l s o  outlined 3 strands:  a  t h i s model l i e s  the elemental  the teacher  the  framework,  The a p p e a l o f  frameworks o f f e r e d by B a l l  i n r e l a t i o n to mathematics. content  (1992).  and s t u d e n t ,  study conceptions  Ball  on the r o l e o f  In developing the  i n b r i n g i n g to the fore  between  1991)  a perspective  students.  h e a v i l y on t h e work o f its  teaching  f o r math  ( s c h o o l and s t u d e n t s ) ,  and  education  (educational psychology,  education).  The c a t e g o r y  conception of mathematics, attitudes attitude  educational  of b e l i e f s  models o f  education.  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 toward teaching  By i n q u i r i n g perspectives, teaching to  this  mathematics  was s u b d i v i d e d i n t o b e l i e f s  the nature of mathematics, and p r i n c i p l e s of  aims,  Finally,  t e a c h i n g and the  category  toward mathematics  of: learning of  and an  mathematics.  into perspectives  of  content  s t u d y goes beyond l o o k i n g a t  i n v e s t i g a t i n g p a r t i c u l a r views  - 20 -  of  as w e l l a s general  teaching  instructional  views  of  mathematics.  Chapter 3:  Noddings position"  (1990)  1994)  h i s own v i e w s  researcher  that  "post-epistemological  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 .  paradigm necessitates  (Denzin & L i n c o l n , represent  has dubbed c o n s t r u c t i v i s m a  which y i e l d s  constructivist  Methodology  for  it  a qualitative research  i s o n l y by a l l o w i n g the  methodology  subject  u s i n g h i s own w o r d s a n d n o t t h o s e  i n s i g h t into his understandings of  The  to  imposed by  a concept  the  c a n be  gained. S i n c e the aim of t h i s their personal perspectives the general  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  express  on the t e a c h i n g of mathematics,  selected  i n t e r v i e w guide technique  s t r u c t u r e d approach ensured that  the  (Patton,  This semi-  same t o p i c s d e l i n e a t e d  framework were c o v e r e d w i t h each p e r s o n y e t t o expand upon c e r t a i n t o p i c s  1990).  I  in  the  i t gave room f o r f l e x i b i l i t y  or explore others.  It  also  allowed  for  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 . Development of  the  An i n i t i a l  Protocol  set  of questions  a d i s c i p l i n e , mathematics mathematics,  the r o l e of  u n d e r l y i n g goals of Contrast  and p o t e n t i a l p r o b e s  as a s c h o o l s u b j e c t , the teacher,  view of  t e a c h i n g m a t h e m a t i c s were  c a n be e f f e c t i v e  so t h a t  was  view of  t e a c h i n g changed over  for  juxtaposed  the year?" (Ball  message  the student,  and  composed.  An example  -  is  ideas,  "Has y o u r  N o t i n g that mathematics  & McDiarmid,  - 21  as  about  i l l u m i n a t i n g and c l a r i f y i n g  i n c o r p o r a t e d i n t o the p r o t o c o l .  E n g l i s h are often  on mathematics  1990)  and  t h e q u e s t i o n "How  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 , English?"  was a l s o  included.  Whether u n d e r l y i n g p h i l o s o p h i e s are e x p l i c i t o r i m p l i c i t , (1988)  p o i n t s out  [p 1 2 4 ] .  "the roots of  With t h i s  ( A p p e n d i x A)  i n mind,  potential  probe d u r i n g the  The P i l o t  Studies  Two p i l o t p r o j e c t s  validate  the  their perspectives  I selected  "Views of  [are]  The f i r s t p i l o t  from the  Columbia. for  These listening. lengths  of  were c a r r i e d o u t p r i o r  to the main s t u d y . as w e l l  as  study. volunteer  the U n i v e r s i t y of  w e r e i n t e r v i e w e d i n my o f f i c e  at  the  British  university  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 .  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 While t r a n s c r i b i n g the tapes, questions  and answers,  I noted the  set  of  comparative  a n d a n y d i s t r a c t i n g comments  l i s t e n more c l o s e l y a n d make f e w e r  comments f o r t h e n e x t  interviews.  I made.  inconsequential  I a l s o reviewed the  c u r s o r i l y w h i c h g a v e me p r e l i m i n a r y e x p e r i e n c e  in looking for  data categories  ideas. The c o n t e x t  of  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  e l e m e n t a r y aged c h i l d r e n . candidates  a  questions.  Both students  T h i s p r e p a r e d me t o  of  as  interviews.  elementary B . E d , program at  35-45 m i n u t e s .  [sic]  t e a c h i n g t o use  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 students  visual"  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 interview  Goodman  t e a c h i n g and l e a r n i n g "  a s r e p r e s e n t i n g some common n o t i o n s o f  Their general  like  and hence  W h i l e t h e s e were e l e m e n t a r y  represented a different  - 22 -  to  teacher  group from those  in  the  main study,  the data  mathematics. content  One saw m a t h e m a t i c s  areas.  mathematics  showed a n i n t e r e s t i n g c o n t r a s t as b e i n g - a n  of  teaching  inextricable part  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  of to  t o her young p u p i l s through i n t e g r a t i o n w i t h o t h e r  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 from o t h e r  i n ideas  subjects  using a standard d r i l l  The s e c o n d p i l o t For the  c o u l d o n l y be t a u g h t and p r a c t i c e  other teach  subjects.  in isolation  format.  study.  second p i l o t  investigation,  secondary  math p r o g r a m were c h o s e n .  different  locations,  lasted  four volunteers  These  interviews,  35-45 m i n u t e s .  from the B . E d ,  conducted  in  They were a u d i o - t a p e d  and  transcribed. With t h i s effectively. "it" the  or  familiarity before  of  u s e d as  the  being referred  main study  subject  a s e n t e n c e and sought  to.  notes d u r i n g the  of  Through r e p e t i t i o n and  interview  a graphic template Its  interview.  interviewee  of  to  second p i l o t ,  clarify  was  questions  saying.  This  I would use  in  style.  the questions  n o n - l i n e a r format  was u s e f u l  for  t h i s question generated  from which 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  - 23 -  t h e most  illustrative  questioning.  the  making  I also decided to begin a l l interviews  t h e m a i n s t u d y w i t h "What d o e s i t mean t o y o u t o t e a c h m a t h ? " the  words  increasing  I r e l i e d l e s s on the w r i t t e n  more o n what t h e  (Appendix B ) .  more  interview I l i s t e n e d for the  i n a more c o n v e r s a t i o n a l  I developed  I l e a r n e d t o probe and c l a r i f y  d u r i n g the  w i t h the p r o t o c o l ,  me a n d f o c u s s e d  resulted  interviews,  For example,  "that"  concept  set  of  During answers  The r e s u l t s ideas  of  of  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  t e a c h i n g mathematics  differences The M a i n  i n the  ideas  of  and s u g g e s t e d t h a t  the p r e - s e r v i c e  there  teachers  in  a l s o w o u l d be  i n the main study.  Study  The p r o g r a m . The B a c h e l o r o f Columbia  (UBC) f o r s e c o n d a r y m a t h e m a t i c s  c o n s i s t i n g of first  p r a c t i c u m occurs d u r i n g the f a l l  assigned The  d u r i n g the to the  teaching experiences.  The  f o r two weeks a n d  f o r n i n e t e e n weeks.  the  Students  same s c h o o l f o r b o t h p r a c t i c a .  from the  T h r e e were m a l e ,  specializing seven,  a twelve-month program  semester  s p r i n g semester  British  sample.  Seven s t u d e n t s selected.  the U n i v e r s i t y of  is  c o u r s e w o r k a n d two p r a c t i c e  second occurs are  E d u c a t i o n program at  s e c o n d a r y m a t h B . E d , p r o g r a m a t UBC w e r e  f o u r were f e m a l e .  i n mathematics,  a requirement for entrance.  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  towards Masters degrees  i n pure mathematics.  E n g l i s h and mathematics  s p e c i a l t y and the  and mathematics  specialty.  A l l had a f o u r - y e a r  The f i n a l  Of  B.Sc.  these  a n d two w e r e w o r k i n g The f o u r t h h a d a c o m b i n e d  fifth  had a combined S p a n i s h  two h a d a m a j o r  i n mathematics  only. The s e v e n who c o m p r i s e d t h e faculty three  advisor.  faculty  assigned so,  those  For that  advisors. whose  class  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 of  41 m a t h s p e c i a l i s t s ,  To e a s e 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 ,  teaching schools  c l u s t e r e d about  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 .  secondary  schools  I was one  (Grades 8 - 1 2 ) ;  of was  a main highway;  S i x were a s s i g n e d  one was a s s i g n e d t o a j u n i o r  - 24 -  I  as  to secondary  school study  (Grades  7-9).  In selecting  I c o n s i d e r e d them " t y p i c a l " c a s e s  were c o n v e n i e n t As f a c u l t y between  to  support  advisor,  was my r e s p o n s i b i l i t y t o a c t i n s t i t u t i o n , to ensure  of  their professional  i n t r o d u c t o r y teaching adventures  as  answers  disadvantage  c o u l d have f e l t  intimidated to give  looking for.  To m i t i g a t e  had emphasized t h a t  I t o o was a s t u d e n t ,  l e a r n i n g was how t o topic that  Through t h i s  i n t e r v i e w w i t h me a t evaluation. individuals comfort  footing"  I also  we  two w e e k s .  and t h e  ideas  this  and  evaluate  summatively. research.  that  I am s t i l l  teacher  answers,  effect,  they were.  about  interview  teaching  data,  mathematics.  they c o u l d "opt to t h e i r  i n my v i e w ,  met a n d  to the  rapport  out"  of  the  practicum the mutual  attained.)  We a l w a y s  w i t h each teacher  debriefed  summaries o f o u r d i s c u s s i o n s  after  candidate  the classroom  were w r i t t e n i n t h e  - 25 -  I  I t o l d t h e m what  i n t o u c h w i t h most o f  attests,  i.e.  I had informed  I hoped t o e s t a b l i s h  i n f o r m e d them t h a t  I conducted classroom v i s i t s every  lend advice  s t u d y when we f i r s t  as  explanation,  on a p e r s o n a l b a s i s  level  "right"  a n y t i m e w i t h no c o n s e q u e n c e  (The f a c t  a smooth w o r k i n g  the  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 e s t e d me was  "equal  w i t h the group.  who  liaison  to the  Rather than speaking f r e e l y ,  i n u s i n g them f o r t h i s  and t h e  the  was t h e p o w e r 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  advisor.  I m i g h t be  as  t r a i n i n g , and t o  f o r m a t i v e l y and  t h e m o f my i n t e r e s t  I was  to  r o l e p r o v i d e d advantages and d i s a d v a n t a g e s  faculty  candidates  ( M i l e s & H u b e r m a n , 1994)  them and t h e i r s c h o o l a d v i s o r ,  f o r t h i s phase  The o b v i o u s role  it  them and t h e i r r e c e i v i n g  This  teachers for  access.  r e l a t i o n s h i p between  their  these prospective  about  observations, monthly  formative  evaluations.  composed a t to the  the  teacher  conversations  When t h e  c o n c l u s i o n of candidates  final,  summative e v a l u a t i o n s  the p r a c t i c u m ,  a s t h e y were  we h a d b e e n h a v i n g .  the p r a c t i c u m experience,  t h e y c o n t a i n e d no  s i m p l y summaries o f  In the  a l l received  a  an a d v a n t a g e  to the research.  the  transcripts,  provides  familiarity specific  meant  that  incidences  p r o v i d e an example The d a t a  week,  of  One o c c u r r e d a t  to c o n s t r a i n the  thoughts became  interview,  t o p i c we were  week o f  the  other  f r o m one  of  interviews.  e i t h e r o f us c o u l d  Our  evoke  to highlight  The i n t e r v i e w s  the nineteen-week  were  or  conducted  p r a c t i c u m and the  following  evaluation.  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  t h e end o f  s c h o o l day and t h e o t h e r d u r i n g a  subjects'  the previous  and f e e l i n g s  the  successful responses  class  interviews,  i n that  the b e l l  they  the  regarding teaching  focussed  i n general. end o f  the  -  26  -  on the  interview the  spare seemed  the broader  I n one c a s e ,  r i n g i n g to s i g n a l the beginning of  period.  schools.  situations  which tended t o overshadow  i n c r e a s i n g l y d i s t r a c t e d toward the  anticipating  establish  pursuing.  i n t e r v i e w e d once.  A l t h o u g h b o t h were  immediacy of  flavour of  each p e r s o n had been g i v e n h i s / h e r f i n a l  The f i r s t  period.  the  for  collection.  final  after  d u r i n g the  the  to  Our e a s e w i t h e a c h  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  E a c h p e r s o n was d u r i n g the  I was a b l e  A p p e n d i x D, a n e x c e r p t  a sense of  the  "Pass."  r e l a t i o n s h i p w i t h each p e r s o n .  was  surprises  P a s s / F a i l g r a d i n g scheme  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 , a comfortable  were  by  next  we  Thereafter, more c o n d u c i v e conceptions. hamper t h e  interviews  were  t o b r i n g i n g out  conducted  i n cafes.  the pensiveness  Problems w i t h background n o i s e  interview  to cover  image,  From t h e r e s p o n s e , to the  interview travelled its  investigation.  I took  second p i l o t  field  notes  were p r o b e d  topic.  Those  teaching  i n the  form of  "Views of  Each l a s t e d  u s i n g the  schematic  s t u d y a n d made a d d i t i o n a l n o t e s schematic  The d a t a p r o c e s s i n g  and  I t r a n s c r i b e d the tape to each teacher  candidate.  was  for  who an  teaching  and  of  each  One t a p e  enough t o d i s c o n n e c t  long program,  after  each  interview.  from  the  interview.  Pseudonyms w e r e  was n o t u s a b l e the  as  taken at  the  This  s t u d y was t o  out,  so  a substitute  -  had  occurred had its  during look at  snapshots  t h e same p o i n t d u r i n g t h e i r  I decided not to arrange  - 27  weeks l a t e r ,  given  subject  The m i c r o p h o n e a p p a r e n t l y  several  Since the purpose of  the  recorder.  b u t not enough t o f a l l  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  i n d i v i d u a l s ' conceptions  developed  40  analysis.  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 .  transcription.  f r o m 25 t o  used.  i n a d v e r t e n t l y p u l l e d the microphone out of  of  own p a t h  on the  shown t h e d i a g r a m s  were a u d i o - t a p e d .  A p p e n d i x C shows how t h e  moved j u s t  Subjects  you  (Appendix A ) .  The i n t e r v i e w s minutes.  the  a c l e a r personal conception of  m e t a p h o r o r m o t t o were  learning"  investigating  "What d o e s i t mean t o  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  d i d not have  in  process.  areas p e r t i n e n t  examples,  fruitful  were  were m i n o r a n d d i d n o t  Each i n t e r v i e w began w i t h the q u e s t i o n t o t e a c h math?"  These venues  interview.  yearThis  left  s i x cases  one o f  the  for  analysis.  a sample  were a n a l y z e d u s i n g t h e  ( M i l e s and Huberman, 1 9 9 4 ) .  technique  I read each t w i c e .  corresponding to c a t e g o r i e s d e a l i n g w i t h mathematics, and/or  its  Huberman, also  l e a r n i n g were w r i t t e n b e s i d e 1994,  wrote  (Strauss  p 56)  memos;  it  approximately  i n the  40 t o  For each p e r s o n ,  first  to  60 c a r d s .  the  locate  draft  a summary o f  the  its  codes  teaching  "chunk"  (Miles & these,  n o t e s and t h e o r e t i c a l transcript  information  ideas  four of  and t e l l  t h e s e were  notes  was  then  generated category.  c o n t a i n e d on the  cards  4.  the data,  the  six  his/her  me w h e t h e r  Three  subjects  1985)  I began of  these  and sent  interview.  the  summaries.  each a copy  Each person  I had c a p t u r e d h i s / h e r  were r e t u r n e d .  process  Their written  each p e r s o n ' s  from the d a t a .  A n a l y z i n g components  These  was w r i t t e n  descriptions  - 28 -  with  constitute  w i t h i n each case gave r i s e  was  comments  Two s u g g e s t e d i n c l u d i n g one  conceptions  of  thoughts  incorporated.  A d e s c r i p t i o n of supporting evidence  of  I  necessary  were t h e n s o r t e d by  ( L i n c o l n & Guba,  i n d i c a t e d no c h a n g e s w e r e n e c e s s a r y .  chapter  cards  pattern  As I g e n e r a t e d  Each t r a n s c r i p t  trustworthiness  and a c c u r a t e l y .  point;  t o code  These  o f my summary o f  to read i t  fairly  interview.  of  Then  appropriate  index card w i t h a l l the  c o n d u c t i n g member c h e c k s  asked  from  category.  I was a b l e the  the  Each coded e n t r y on the  raw d a t a .  I wrote  To i n c r e a s e of  entire  1990).  to a separate  to r e - l o c a t e  each  i n the  these corresponded  and C o r b i n ,  transferred  for  excerpt  interviews.  The t r a n s c r i p t s coding  Appendix D provides  to  more  i n t e r e s t i n g comparisons across included i n chapter  the  cases,  5.  - 29 -  These  comparisons  are  Chapter  In t h i s  chapter,  I describe  conception of mathematics, the  4:  Results  each s u b j e c t ' s  and v i e w o f  s e m i - s t r u c t u r e d approach of the  academic  teaching mathematics.  interviews,  Because  the d e s c r i p t i o n s  each case v a r y s l i g h t l y ,  a l l o w i n g f o r the d i f f e r e n t  candidate  fore  t o come t o t h e  background,  f o c i of  as t h e y d i d d u r i n g t h e  each  of  for teacher  individual  interviews.  Yvette Academic Background Yvette  had a 4 - y e a r  "traditional" after  student,  B.Sc.  i.e.  one who went d i r e c t l y i n t o a B . S c .  completing high school,  Conception of Yvette  w i t h a major i n mathematics.  a  program  and s u b s e q u e n t l y on t o a B . E d , p r o g r a m .  Mathematics saw m a t h e m a t i c s a s a n i n f l e x i b l e c o r e o f  content  p r o b l e m s a s h a v i n g one a n s w e r w h i c h c o u l d be a r r i v e d a t v a r i e t y of methods. arts  She i s  Sciences  also  fit  t h i s category  and  its  through a  and c o n t r a s t e d  with  subjects. " . . . 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 . Lots of k i d s might solve 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 b e 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 an answer. T r y a n d f i n d i t . " [p 5 - 6 ]  - 30 -  to  Yvette enjoyed  it  had a p o s i t i v e d i s p o s i t i o n towards m a t h e m a t i c s .  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  particular  teacher.  in university, more, of  She a l s o  "...but  so I guess t h a t ' s  the  subject  problems  enjoyed  I started where  of  and s a t i s f a c t i o n  Teaching  effective  accommodate learning  see  came f r o m t h e c h a l l e n g e  To Y v e t t e , set  to not  I k i n d o f got  she  that  courses  to she  one took  the relevance  i n them any  lost."  Her  [p 5]  she e x p e r i e n c e d  found i n g e t t i n g  " B u t I a l s o l i k e d , m a t h was at i t and I found t h a t t h a t t h i s w a s , um, y o u ' r e g i v e n c h a l l e n g e d a n d I WANTED t o c h a l l e n g e , o f m a t h a n d um, at the end." [p 5] View of  the mathematics  She  the  in  enjoyment  solving  "right  answer."  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 was s a t i s f y i n g f o r me, t o know t h a t a p r o b l e m a n d i t ' s h a r d , a n d I , I was 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 I f e l t a s a t i s f a c t i o n t o get an answer  Mathematics t e a c h i n g mathematics presentation  the d i f f e r e n t  skills.  meant p r e s e n t i n g c o n t e n t These  skills  sensory modalities that  needed  define  using a  to  different  styles.  " . . . [ 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 different 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 have, but t e a c h i n g i t i n a d i f f e r e n t way." [p 1] " I t h i n k t h a t y o u s h o u l d t r y a n d h a v e a v a r i e t y o f , o f , um, methods, 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 , s o 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 to e x p l a i n things i n d i f f e r e n t w a y s . . . . I think that t r y i n g t o have a v a r i e t y because not a l l the 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  the  student  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 , s o 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 h a v e them l e a r n . Um, s o 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  -  Her 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 over  to a l l subjects.  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 mathematics and  She  English.  "I t h i n k that there's a l o t of s i m i l a r i t i e s , just i n teaching 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 , a n d 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, w o r k o n i t i n d i v i d u a l l y , w o r k o n i t maybe i n g r o u p s . I ' m , even i n E n g l i s h , say E n g l i s h and math, y o u m i g h t t e a c h t h e m 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 w o r k o n i t t o g e t h e r a n d 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 r e m e m b e r , 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 w o r k 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 . " [p 9] Showing the r e l e v a n c e Yvette.  If  disappear.  Yvette  l e a r n a n d some o f  c i t e d the  h e r own m a t h h i s t o r y :  that's  I k i n d of got  "I think that  showed t h e r e l e v a n c e , perhaps, things  but  and the  the relevance lost."  there  [p 5]  are  o r showed,  important  then students  to would  relevance  to understanding i n in university  i n them a n y m o r e ,  so I  She m e a s u r e d h e r s e l f  using  t i m e s w h e r e I c o u l d h a v e more  um, um, j u s t  lately tried  b e e n more  guess this  often  encouraging  r e a l l y hard to,  t o show when  [p 4]  had not d e v e l o p e d a motto o r metaphor f o r t e a c h i n g and thus  showed h e r t h e  guide  importance of  I have e s p e c i a l l y  are u s e f u l . " Yvette  I  was  the d i s c i p l i n e problems would  " T h e r e w e r e some m a t h c o u r s e s  s t a r t e d t o not see  where  standard.  a t o p i c to students  a t o p i c c o u l d be made t o seem r e l e v a n t ,  be more m o t i v a t e d t o  that...I  of  "Views of  traveller,  t e a c h i n g and l e a r n i n g . "  a s most r e p r e s e n t a t i v e  She c h o s e #2,  of her conception  the of  teaching. " . . . I t h i n k 2 ' s a good example ' c a u s e i t ' s t h e g u i d e and 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 and 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 a n d t h e , t h e f e l l o w l y i n g down was t h e traveller. 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 out, but the t r a v e l l e r ' s not j u s t an 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 . . . . and 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 because a s 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 a n d 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 a n d show them d i f f e r e n t t h i n g s , b u t t h e y are a c t i v e l y a part of i t i n t h i s e x p l o r i n g i d e a . . . i n the c l a s s r o o m , I suppose, y o u ' r e e x p l o r i n g mathematics." [p 12] Characteristics Yvette  t o have  concepts, "I  a good math  teacher.  touched upon f o u r q u a l i t i e s  a good t e a c h e r ability  of  of mathematics.  These  she t h o u g h t  were a d e s i r e  a good r a p p o r t w i t h s t u d e n t s ,  and good p r e s e n t a t i o n t h i n k you have  were  important  to teach math,  the knowledge  to  for the  explain  skills.  t o want t o t e a c h m a t h . . . "  [p 6]  " [ A g o o d m a t h 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 h a s t o be s o m e o n e , um a s p a r t o f t h a t , t h e p e r s o n h a s 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 , h a v e a g o o d r a p p o r t , um, s o t h a t t h e s t u d e n t s want t o l i s t e n and a r e , are a c t i v e l y i n v o l v e d . Um...just things 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 g o n n a 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 g o n n a 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 what  she b e l i e v e s  on t h e s e a s p e c t s ,  t e a c h i n g t o be a l l a b o u t  she  - clear  focussed portrayal  again  on  of  concepts. " . . . i f I g r a d u a t e d from h i g h s c h o o l and I d i d w e l l i n math, and I know, y o u know, a l l t h e math Grade 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 g o o d m a t h 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 h a v e 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, g o o d 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 demonstrate t h i n g s i n d i f f e r e n t ways, not d o i n g i t on the b o a r d but a l s o s a y i n g something, d o i n g something a c t i v e l y . " [p 8] She was a b l e speaking c l e a r l y ,  to elaborate  o n what  b u t was u n a b l e  "tries  h a r d t o have  belief  i n teaching  students as  a set  to give  learn." of  c l e a r p r e s e n t a t i o n meant, examples  o f how a  This underscores  good d e l i v e r y  - 33 -  skills.  e.g.  teacher  Yvette's  strong  View of Yvette  students. viewed students  shaped her g o a l of achieve  as h a v i n g d i f f e r e n c e s  teaching around t h i s .  what h e / s h e  could  i n a b i l i t i e s and  She w o u l d h e l p e a c h s t u d e n t  to  achieve.  "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 mathematics. acknowledged  saw s t u d e n t  differences  i n d i s p o s i t i o n towards  T h o u g h 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 , that  not a l l students  enjoyed  this  type of  Yvette  challenge.  " . . . 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 . 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 and f i n d the 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 , s h e became influencing  the  towards  students,  the  l i v e s of  aware o f o t h e r f a c t o r s  the teenagers.  This increased her  but d i d not a f f e c t  t h e way s h e  and  people  sympathy  taught.  " . . . y o u r e a l i z e t h a t t h e s t u d e n t h a s s o 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 guess 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 , h a s b e e n a f f e c t e d b y t h a t . " [p 13]  Conception of  l e a r n i n g math.  When a s k e d how m a t h i s b e s t  learnt,  Yvette  different  learning styles.  different  sensory modalities to help individual  student  practice  She m e n t i o n e d t h e n e e d t o  o f math q u e s t i o n s  She c l a i m e d " t h e  focussed  was  r i g h t mind s e t "  a g a i n on  accommodate  students  l e a r n , a n d how  important. [p 9] was  important for  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  - 34 -  success more  scientific, described  I don't it  as  know!  active  It's  more m a t h e m a t i c a l ! "  involvement  i n the  [p 9]  Later,  she  problem.  " . . . t h e y g e t down t o w o r k , t h e y ' r e , t h e y ' r e t h i n k i n g a b o u t a p r o b l e m , t h e y r e a d i t , a n d t h e y , t h e y a s k 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 , w h e r e a s someone e l s e p r o b a b l y w o u l d n ' t e v e n a s k 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] It  is  interesting  teaching  of  m a t h was  math r e q u i r e d a  to  note  that  s i m i l a r to  the  while  Yvette  teaching  " r i g h t , " mathematical  mind  of  claimed that  other  the  subjects,  learning  set.  Shannon Academic  and P e r s o n a l Background  Shannon had an Honours B . S c . entering  the  full-time  in a restaurant  completing return  to  manage  a  B . E d , p r o g r a m she  her  teacher  i n mathematics.  tutored privately.  and s u b s e q u e n t l y  became  p r e p a r a t i o n program,  h e r home p r o v i n c e  and f i n d  She h a d a l s o the  Shannon's  a teaching  Before worked  manager. plans  job or help  After  were her  to husband  restaurant.  Conception of  Mathematics  Shannon d e s c r i b e d mathematics practical  (4-year)  components.  She  clearly  as  having both t h e o r e t i c a l  favoured the  "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 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 the  two,  like  practical  aspect.  a p p l i e d math.  Shannon c h a r a c t e r i z e d  and  I  don't  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 h a d a n 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 h a d t o m e m o r i z e a l l t h e p r o o f s , a n d t o me when y o u a p p l y , y o u l e a r n how t o USE [ s i c ] those p r o o f s . . . . m y f o u r t h y e a r course i n math, I r e a l l y e n j o y e d . It was, a w h o l e t h i n g was a b o u t , u h , s i m u l a t i o n ? A n d 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 ? A n d w a t c h i n g how t h e 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 s y s t e m t h a t w o u l d p u t more o u t p u t o f  - 35 -  c a r s . . . . I think that's i m p o r t a n t . " [p 5 - 6 ]  e x c e l l e n t . . . . I think  that's  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 " I l i k e s e e i n g a r i g h t o r wrong answer. 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 u n i v e r s i t y and t h a t  taught  more  to mathematics.  A n d y o u know m a t h i s  between the mathematics  in  i n high school.  "[W]hat you 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 l e a r n i n h i g h s c h o o l . " [p 10] View of  one  t h a n what  you  Teaching Mathematics  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 .  g o a l was t o make a d i f f e r e n c e  to each s t u d e n t ,  Her  p a r t i c u l a r l y lower  achievers. " . . . I wanna make a d i f f e r e n c e ones t h a t a r e n ' t d o i n g w e l l . "  t o each s t u d e n t and e s p e c i a l l y [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 o n e s 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 o n e s I ' m , more c o n c e r n e d a b o u t ? " [p For her, student." wouldn't  successful  [p 11] find  t e a c h i n g meant b e i n g a b l e t o  "cover  a t t e n t i o n from her.  a n d some w o u l d n o t  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  21]  every  She w o r r i e d t h a t when s h e h a d h e r own c l a s s r o o m ,  enough t i m e t o a p p r o a c h each one,  the  she get  relationships  w i t h each student d u r i n g her p r a c t i c u m but found t h i s task d a u n t i n g . SH: [ 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 everything. 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 ? SH: 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 to every i n d i v i d u a l student. [p 11] When p r o b e d f o r what s h e 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 her response particular said, 'cause  was  " I want them t o p a s s . . . " [ p  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  " I want t o s e e I would f e e l  him pass. like  19]  student,  E l a b o r a t i n g o n one  and d i s c i p l i n e p r o b l e m s ,  I want t o s e e  him pass the  course,  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 . "  - 36 -  she  [p  19]  She a l s o she  g a v e two e x a m p l e s  of  students  who were  saw a s h a v i n g p o t e n t i a l t o s u c c e e d  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 r o b l e m k i d s " b u t whom  i n school.  [p  She a t t e m p t e d  to  19-20]  F o r S h a n n o n , n o t o n l y was more m a t h l e a r n t b y t h e  student  when  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 in this  she  built  manner.  " I g e t a l o t more done w i t h o n e - t o - o n e . And 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 . And 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 - , they're 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 h a v e t i m e t o go a r o u n d , [p 9] Model f o r  teaching..  An exemplar  f o r S h a n n o n was t h e p r o t a g o n i s t  a movie she v i e w e d with his  students  represented students  i n one o f h e r t e a c h e r  education courses.  t h e way s h e w a n t e d t o w o r k w i t h h e r s  a model f o r her t o f o l l o w .  and h i s  i n S t a n d and D e l i v e r ,  i n the  He w o r k e d  a n d he  She saw a p a r a l l e l b e t w e e n  her  movie.  "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 m a t h s o 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 , a n d I l o v e d how he c o u l d r e l a t e to those 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 t h e m 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 teachers'  s t r i k e d u r i n g her p r a c t i c u m .  cancelled  and c o n s e q u e n t l y ,  regular students  hours.  was  T h i s meant c l a s s e s  some s t u d e n t s  wanted e x t r a h e l p  were outside  She saw t h i s a l l i a n c e a s e p i t o m i z i n g h e r g o a l  through e s t a b l i s h i n g a personal  a  of  aiding  relationship.  " . . . 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 w e r e 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 ? And t h i s k i d i s a 50% a v e r a g e . And I f e l t g r e a t . " [p 14]  - 37  -  Shannon b e l i e v e d learning.  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 enhanced  In being f r i e n d l y ,  however,  she e x p e r i e n c e d p r o b l e m s  student with  classroom 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 t h e m , 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 w h e r e my l i m i t s a r e . . . . B e c a u s e 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 . And t h e n , a n d t h e n t h e y t a k e a d v a n t a g e s o m e t i m e s - " [p 14] When s h e was shown t h e d i a g r a m s Learning," as  s h e d e c i d e d none o f  c l e a r l y as  the  them c o n v e y e d h e r c o n c e p t i o n o f  conception of  a teacher  t o be s o i n f r o n t o f  and t h a t  s h o u l d n ' t come o u t o f me"  I want  behaviour  them t o  i n front  answers  of  students those  to t e l l  was a n o m n i s c i e n t p e r s o n who and p e e r s .  [p 3]  and  " I sometimes  "I don't  like  was  wonder,  hesitating  [p 6] was how s h e d e s c r i b e d what  her  should be.  i n p o s i t i o n s of  a n d make them a v a i l a b l e  when d i s c u s s i n g what were  students  l e a r n math"  Shannon b e l i e v e d the  teaching  teacher.  expected  'cause  and  movie.  Conception of Shannon's  i n "Views of T e a c h i n g  a u t h o r i t y s h o u l d have  to others.  students  She e l a b o r a t e d  on  all this  who a s k e d why E u c l i d e a n p r o o f s  studied: " . . . 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 answer, 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 need geometry. T h a t ' s not a good 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 u s 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 h a v e 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 . Is t h a t the answer? I don't know." [p 3]  And l a t e r SH:  I: SH:  i n the  I  interview,  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 geometry. I wish I d i d . And I w i s h t h e y would 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 . . . Ok, w h e r e 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 ? 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 -  In t h i s  excerpt,  information. she  Shannon l a i d out  The p r o f e s s o r s  i n turn could t e l l The i d e a o f  her. issues  the  h e r why g e o m e t r y  for  relaying  was t a u g h t  so  students.  She was c o n c e r n e d h e r  certified  would t e l l  the teacher  would cause her to  a h i e r a r c h i c a l pathway  b e i n g a l l - k n o w i n g put great p r e s s u r e  ignorance  lose  of  content  face with other  or  staff  on  content-related once  s h e became  a  teacher.  " 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 ? And t h e y s a i d yes. 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 ] , s o 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 a n d I f e e l stupid." [p 10] The t e a c h e r great  was n o t a l l o w e d t o w o n d e r ,  s t r a i n o n h e r as  guess,  o r n o t know.  she t r i e d t o a c c o m p l i s h t h i s  This put  a  ideal.  "[The s t u d e n t s are] coming here so you can t a l k t o them, l i s t e n t o t h e m , t e a c h t h e m . . . . Y o u h a v e 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 , you have t o get 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 , even though you might not t h i n k i t i s , p e r s o n a l l y , but you 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] Consistent those  w i t h the n o t i o n of omniscience,  i n the p r o f e s s i o n  she p e r c e i v e d  she h e l d i n d i s d a i n  t o be n o t a l l - k n o w i n g .  "When I h a d a n 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 ] courses w h e r e 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 expect the students t o . I d o n ' t a g r e e w i t h t h a t . " [p 5] Requirements "Patience. Shannon's  to teach  mathematics.  A l o t of p a t i e n c e . " replies  to this  [p 9]  s e c t i o n of  the  i n t e r v i e w seemed t o  i n f l u e n c e d by the rambunctious nature of her s t u d e n t s , difficulty  she had m a k i n g t h e  and by  t r a n s i t i o n from s u c c e s s f u l  - 39 -  the  private  be  tutoring to teaching a class. you always  have  t o have y o u r w i t s  where y o u d o n ' t Teaching  She d e s c r i b e d t h e  feel  well.  to y o u r s e l f .  You h a v e  have  teacher  a  [teach]."  day [p  17]  methods.  u n d e r s t a n d i n g as  advocated  by the t e a c h e r  s h e n o t i c e d how many s t u d e n t s  without t h e i r calculators. o n number f a c t s ,  "[B]eing a  You c a n ' t  t o make y o u r s e l f  Shannon 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  practicum,  job:  c o u l d n o t do s i m p l e  She was adamant  had s t u c k  for  education program.  her r a t i o n a l e being that  t h e m a n d t h e number f a c t s  teaching  i n her  that  During her arithmetic  t h e y s h o u l d be  was t h e way s h e h a d  drilled  learnt  memory.  " . . . t h e s t u d e n t s t h e y c a n ' t even add, and m u l t i p l y , and 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 subtract? Why d i d i t s t a y w i t h me c o m p a r e d t o t h e s e s t u d e n t s ? . . . . A n d about 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 use them anymore. Make t h e m 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 and 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 the best 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] Teaching, Cooperative considered  f o r S h a n n o n , meant d i r e c t  methods, "extra"  as  instruction, telling.  she u n d e r s t o o d from h e r s c h o o l a d v i s o r s ,  and a waste of  were  time.  " . . . 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 h a s 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 h a v e t h i s amount o f t i m e t o c o v e r t h i s math. 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 h a v e t i m e t o cover 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 . And 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 class. 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 t h e m 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 s e e t h e y r e a l l y u n d e r s t a n d i t . " [p 16] In comparison to teaching mathematics  of  that  teaching  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  w i t h s o much m o r e . " teacher  E n g l i s h , Shannon a s s e r t e d  [p 15]  To e l a b o r a t e ,  E n g l i s h show m o v i e s  in class.  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 ,  -  she c i t e d w a t c h i n g a Shannon f e l t  and f e l t  40  -  away  student  trapped by  t h e o n l y way t o do i t  the was  through d i r e c t  instruction.  Other,  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  w i s h e d t o u s e w e r e 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  superfluous  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 H e r g o a l was t o show s t u d e n t s favourite course  aspect of  she t h o r o u g h l y e n j o y e d  and s t u d i e d the o u t p u t o f for  the  line. kids  course  "And I thought,  that  when s h e went t o a l o c a l a u t o m o t i v e  a system to  to understand technology that's  the  plant  project  efficiency  of  To t a k e  the the  why c e r t a i n m a t h i s a p p l i e d ,  coming o u t . "  Dilemma w i t h  content.  Shannon f e l t  very uncomfortable that  and  [p 5]  s h e was a u n i v e r s i t y m a t h  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  s h e was t o t e a c h .  "I feel  She b l a m e d t h e t e a c h e r  stupid.  content  t h i n k the  teachers,  w i t h the teacher  r e v i e w be a component o f  F a c u l t y of  t o do a l l t h e homework,  courses  And I d o n ' t u n d e r s t a n d t h a t . "  -  [p 10]  was  E d u c a t i o n s h o u l d be t e a c h i n g u s m a t h  the  stuff  i n h i g h s c h o o l . . . . I had Instead  of  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  [p 8]  Dilemma w i t h s t u d e n t  She  preparation.  because I d i d n ' t understand i t .  , a l o t of  was u s e l e s s . "  candidates.  teacher  r e v i e w i n g e v e r y t h i n g we h a v e t o t e a c h  t e a c h i n g us  mathematics  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 ,  n o t r e v i e w i n g more c o n t e n t  adamant t h a t "...I  increase  The t e r m  math  Issues  major,  i.e.  line.  i s g r e a t even f o r h i g h s c h o o l .  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  Other  her  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  the r o b o t i c s assembly  was t o d e v e l o p  wanted.  a p p l i c a t i o n s of mathematics,  the d i s c i p l i n e .  she  relationships.  - 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 school advisors of  the  that  as a t e a c h e r ,  teacher-student  a l l o w e d them t o have and the of  class.  this,  s h e h a d t o be  "mean."  a great deal of  c o n t r o l of  the  Since her  management,  flow of  the  "...and I don't  Shannon d i d  view  students  she  advantage  have  know w h e r e my m i d - p o i n t  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  being a f r i e n d to the  her  lesson  they would take  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 .  [p 14]  b u t was t o l d b y  r e l a t i o n s h i p leaned h e a v i l y toward students,  Then, b e i n g young a d o l e s c e n t s ,  problems w i t h c l a s s is."  students,  and b e i n g t h e t e a c h e r ,  the  between  authority  figure. Perspective  on a u t h o r i t y .  Shannon had a v e r y know a l l t h e answers,  and " t h e y "  others). duality  in his  the  t h e ones know. front ones.  of  scheme o f  authorities  resonates  questions.  they  fronts.  the p r e - s e r v i c e  (1970)  "And I ,  and wrong  "us"  i n t u r n became  [p 2 1 ] ,  I don't  (the  d e s c r i p t i o n of This  the p r o f e s s o r s  teachers  a t UBC  were  i n the  the a u t h o r i t i e s  to  classroom  for these  in  younger  to  students'  "...to  make a  worry. i n teaching,  created  know i f  were  a l l t h e y needed  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 each one"  answers  should t e l l  First,  teachers  T h i s c a u s e d Shannon g r e a t  perspective.  right  should  p r e p a r a t i o n program and t h e y  Even her c o n v i c t i o n t o her g o a l difference  in authority  i n t e l l e c t u a l and e t h i c a l development.  when t h e p r e - s e r v i c e  Therefore  were  with Perry's  i n the teacher  who s h o u l d t e l l  students,  those  who knew t h e r i g h t a n s w e r s  f o r Shannon on s e v e r a l  Second,  that  For Shannon, t h e r e  This attitude  p l a y e d out were  answers.  strong b e l i e f  dissonance  that's  - 42 -  with this  r i g h t or wrong.  fundamental I  don't  know...I [p 21]  don't  know i f  that's  correct  U n d e r l y i n g her teaching  belief  i n a r i g h t versus  correct  t o be t h i n k i n g t h a t  philosophy,  w r o n g way o f  way o r  she h a d a n e x t r e m e l y  t h i n k i n g a n d was  searching  not." strong for  the  one.  Doug Academic  Background  Doug was t h e math s p e c i a l t y  only subject  q u a l i f i e d him f o r  with a l i b e r a l arts  degree  the B . E d , program i n  (B.A.).  His  secondary  mathematics. Conception of  Mathematics  Doug b e l i e v e d negotiation  and agreement.  formed by s c h o l a r s He c i t e d t h e  mathematical  i n the  f u n c t i o n as  knowledge  Convenient  and u s e f u l  f i e l d and e v o l v e d an  was c r e a t e d  through  definitions  were  c o n t i n u a l l y as n e e d  arose.  example.  " I mean m a t h i s b a s e d o n d e f i n i t i o n , a n d 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 ? And ah, we c o u l d j u s t a s 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 h a v e more t h a n one p o s s i b i l i t y . For instance the square 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 definition, right? So we d e f i n e t h e s e t h i n g s . . . . I t ' s convenient 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 h a v e b e e n (1,1) o r something. 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 u s a t t h e t i m e . " [p 3] Doug e n j o y e d  the  structure  and o r d e r l i n e s s  of  mathematical  content. " [ 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 to math, o r p u r i t y t o math, 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 a n d e v e r y t h i n g , a n d how i t ' s l a i d out? Y o u k n o w , 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 ? Very logical? I t h i n k t h a t , y o u know l i k e , t h a t m a t h 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 engineering discipline w h o l e was  troubled him. unto i t s e l f .  slowly  He b e l i e v e d In a broader  losing its  as a t o o l o f  science  and  i t masked p u r e m a t h e m a t i c s context,  aesthetic  he f e a r e d  society  as as  a a  side.  . . . [ M ] a t h e m a t i c s i s a , i s o f t e n seen 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 by i t s e l f . Or a s 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 a s 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 l l o u r ways o f l o o k i n g at the w o r l d , our, our p h i l o s o p h y , our s o c i a l s t u d i e s , our economics, psychology, they've a l l , they've a l l s t a r t e d to 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 b e 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 necessarily believe that i t i s a science. Or e c o n o m i c s , a s 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 t h e m up w i t h n u m b e r s a n d s o o n . I d o n ' t n e c e s s a r i l y agree w i t h any o f t h a t , and 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 m a t h 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 side of the w o r l d . " [p 1 3 - 1 5 ] View of Teaching To D o u g ,  Mathematics  t e a c h i n g mathematics  meant t e a c h i n g r e a s o n i n g ,  and 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 . i n mathematics ability  as c o n d u c i v e  not o n l y i n t e r p e r s o n a l communication, others,  but  also  He saw t h e  t o t e a c h i n g t h i n k i n g and f o s t e r i n g  t o communicate w e l l t h o u g h t - o u t  communication w i t h  ideas.  i.e.  thinking order  the  Communication e n t a i l e d  t h e exchange of. i d e a s  with  oneself.  "Because 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 , and 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 and 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 t h e m 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 m o s t 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 communicate 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 t h e m as a t h o u g h t . . . a n d 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 a s much a s me t e l l i n g y o u o r u s 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 , purpose of  one t h a t  embodied  the  education.  " . . . w e ' r e teaching v a l u e s . . . . I t h i n k that teaching of values - I t h i n k the c o n s t r u c t i o n of values i s something that we're 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 - , a n d 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 of things we're teaching people to question, to t h i n k for t h e m s e l v e s a n d 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 t h e m 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 . . . whatever 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 , to create." [p 2] This o v e r - r i d i n g goal of meant t o be a p e r s o n , perspective component 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  a n d 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  o f human d e v e l o p m e n t , life,  it  he c l e a r l y saw i t  b o t h h i s own a n d h i s  as an  elemental  students'.  " B u t . . . t h e p r i m a r y , p r i m a r y g o a l though i s , ah 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 g r o w a s human b e i n g s t h a t ' s t r u e f o r me t o o . " [p 10] Conception of  teacher.  Doug b e l i e v e d a m a t h t e a c h e r such, foster  the teacher  willingness  was a r o l e m o d e l f o r a d u l t h o o d .  w o u l d m o d e l t h e q u a l i t i e s he b e l i e v e d  i n students:  and  the a b i l i t y to reason,  important  the w i l l i n g n e s s to  As to  learn,  the  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 , right? So w e ' r e , s o e v e n i f i t ' s j u s t a s a r o l e m o d e l , y o u k n o w , a n d maybe as a p o o r o n e , b u t a s 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 a n 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 things l i k e that." [p 2] "Well, I t h i n k t h a t ' s a worthwhile r o l e model. Someone who q u e s t i o n s . . . . I f I get p r e s e n t e d w i t h a q u e s t i o n , and I t r y and p r e t e n d t h a t t h e r e ' s one way t o a n s w e r i t , o r t h a t t h e r e ' s one r i g h t answer 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 model, but I would c a l l i t a negative one. I f I show t h a t . . . I c a n make m i s t a k e s a n d 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 a n d I a c c e p t d i f f e r e n t a n s w e r s , then I'm showing a very d i f f e r e n t person, 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  -  45  -  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 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 . " Doug w a n t e d t o show s t u d e n t s  that making d e c i s i o n s  a m b i g u i t y and t h a t  answers  realize  a n s w e r was n o t a l w a y s  that  sometimes up b l i n d  "the"  the r e s u l t alleys.  engender teacher  available  He a l s o w a n t e d them t o s e e after  T h i s form of  reasonable critical  through his teaching, from a bad  clear-cut.  o f m a k i n g many c h a n g e s ,  made t h r o u g h c o n s e n s u s were w e i g h e d .  were n o t a l w a y s  involved  He w a n t e d them' t o  a n d was i n  fact  outright mistakes that  and [p 3]  decisions  p r o s and cons o f  and g o i n g  c o u l d be  an argument  t h i n k i n g was what he h o p e d  to  and f o r him i t d i s t i n g u i s h e d a good  one.  " A g o o d t e a c h e r i s a g o o d 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 and a w i l l i n g n e s s t o l e a r n , demonstrates the q u a l i t i e s t h a t y o u want t o s e e a n d 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 knowledge i s a c e r t a i n t h i n g . . . m a k e s a bad t e a c h e r . " [p 8] " . . . I t h i n k y o u h a v e 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 a n d y o u h a v e t o t h i n k about i t . " [p 5] H i s work w i t h and i n t e r e s t caused him to consider h i s teacher  e v e n more  i n students  from a l t e r n a t i v e  programs  i n t e r a c t i o n and i n f l u e n c e on them as  a  deeply.  " 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 w h e n , y o u k n o w , 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 t h e m 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 d o e s t h a t h a v e t o do w i t h t h i s person'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 k n o w ? " [p 10] Doug's p r i m a r y focus mathematics;  the content  was o n t e a c h i n g ,  o f h i s c l a s s e s was  rather than  teaching  secondary.  " 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 think i t ' s important t h a t t h e y e n d 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 communicate. 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 . Y o u 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 you teach 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 , you 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 learn reasoning s k i l l s . . . " [p 7] He a d v o c a t e d more l o g i c skills  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  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  controversial  issues  of  the  the  t o d e l i b e r a t e p r e s s i n g and  day.  " . . . 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 a n a r g u m e n t 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 and 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 t o w n i n B . C . o r w h a t e v e r , a n d h a v e t h e m 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 a n d 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 you're involved i n . " [p 6] Thus,  m a t h e m a t i c s was a v e h i c l e  Doug made i t  clear  l a r g e r goal of  to help students  that mathematical content  develop  their  thinking.  took a back seat t o  d e v e l o p i n g the a b i l i t y of c r i t i c a l  the  analysis.  "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 w e r e a b l e t o s i t down a n d , a n d 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 m u c h , 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 memorize 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 and 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 s e e 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] However,  he v o i c e d a c a v e a t  i n being a teacher  One c o u l d be s e e n a s a n a d v o c a t e o f t h e things;  this  i n fact  of  mathematics.  " s c i e n t i f i c " way o f  l o o k i n g at  contradicted his personal world view.  "But I , I f i n d t h a t a danger i n t e a c h i n g mathematics i s t h a t you 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 . O r a s 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 and da da 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 m a t h 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 a s e n c o u r a g i n g t h e m t o go o n 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 s o o n . " [p 14] Here,  Doug c o n t r a s t e d h i s g o a l  h i s primary focus  was t o f o s t e r  for teaching with other goals. and encourage  - 47  -  thought  Whereas  and development  in  adolescents,  he f e a r e d  mathematics,  his  accept  the  that,  t e a c h i n g c o u l d be s e e n as  increasingly  Teaching  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  " s c i e n t i f i c " view of  procedures.  he b e l i e v e d  He f e l t  as  i n t e r p l a y would generate develop  the  skills  "you need you teach over the different they pick  of  exams,  world.  there  of  was a l s o  students  teaching a place  teaching the  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  analysis  and  students  synthesis.  the  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  leaving l i t t l e  time f o r c r e a t i v i t y  textbook  of  develop  conceptual  worked t h r o u g h b o t h a s p e c t s ,  and each u n i t n e a t l y  English,  for  for  N o t o n l y was t h e c u r r i c u l u m s e t chapter,  procedural.  to  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 and t h a t , and them t h e c o n c e p t i o n s and e v e r y t h i n g e l s e , b u t I t h i n k 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 between t h o s e two sides of i t , or p r a c t i s i n g the p r o c e d u r a l terms, they, 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 provincial  the  students  methods.  T h o u g h Doug was a s t r o n g a d v o c a t e understanding,  encouraging  of  but  the types  Doug c o n t r a s t e d  and G r a p h i c A r t s ,  this  questions  were  with Social  w h e r e he saw t h e t e a c h e r  classroom.  s u p p l i e d by a  generated  inflexibility  h e r own u n i t s a n d d e s i g n a v a r i e t y o f  i n the  toward  all Studies,  w i t h more f r e e d o m  student  to  activities.  "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 . . . . [ w h e r e a s i n math] h e r e ' s a u n i t , a n d h e r ' s t h e t e x t b o o k . And the 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? And 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] Other  Issues Adolescent  Development.  Doug h a d a s p e c i a l i.e.  those  whose  interest  i n teaching  "alternative"  adolescents,  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 -  regular classroom. structure  He b e l i e v e d m a t h c l a s s  i n an o t h e r w i s e  chaotic  c o u l d be u s e d a s one p o i n t  of  life.  " . . . t h a t p e r s o n might need a, a sense o f o r d e r and s t r u c t u r e , and t h i n g s , and so o n , w h i c h a r e g u l a r i t y i n the c l a s s r o o m g i v e s them, the d i s c i p l i n e i n the classroom gives t h e m . . . . a n d i t gives 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] Student  teaching.  Doug f e l t service  constrained i n class  teacher.  He f e l t  h i m t o be t h e o p p o s i t e  because of h i s  the e v a l u a t i v e  type of  teacher  aspect  of  status  as  a pre-  the p r a c t i c u m forced  f r o m what he w a n t e d t o  depict.  " 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 , a s 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 k n o w . And 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 answer, l i k e - , and I t h i n k t h a t ' s a danger of s t u d e n t t e a c h i n g i s t h a t you f e e l l i k e you have t o be p r e p a r e d a n d y o u h a v e 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] World  view.  Throughout h i s d e s c r i p t i o n s of was c o n s i s t e n t  i n his philosophy of helping students  t h i n k i n g a b i l i t i e s so t h a t values  t e a c h i n g and o f  for themselves.  they could analyze  He c o n t r a s t e d two  the  teacher,  develop  Doug  their  and s u b s e q u e n t l y  choose  views:  "Knowledge i s a t o p - d o w n t h i n g , t h e r e IS an 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 , or m y t h o l o g i c a l , or s c i e n t i f i c , or ah, or whether you k n o w , 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 picture D: I: D:  of  h i s p h i l o s o p h y of t e a c h i n g mathematics  in with his  larger  life. Y o u 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 ? For 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 a s v a l i d ? Mythology. Y o u mean c o n s t r u c t e d ? Yeah. I t ' s a s v a l i d , a n d 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 a s 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 ? And 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 you t e a c h , o r you 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 , a n d 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 , that's f i n e " [p 14]  Ray Academic and P e r s o n a l Background Ray had a B . S c . years of  i n Applied Mathematics.  He h a d a l s o  a n E n g i n e e r i n g d e g r e e a n d some g r a d u a t e  courses.  Prior  w o r k e d as  an a c t u a r i a l a s s i s t a n t  Conception of Ray's  mathematics  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 i n an i n s u r a n c e  firm.  Mathematics  c o n c e p t i o n of mathematics  He saw m a t h e m a t i c s physics,  level  completed 2  as  emphasized i t s u t i l i t a r i a n  a t o o l for other f i e l d s .  we u s e m a t h i n b i o l o g y ,  nature.  " . . . w e u s e d math i n  use math i n e v e r y o t h e r  [science]"  [p  4] R a y 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 for  a l l , especially  c a n be a c c e s s i b l e View of  if  the t o p i c s  to every  t o a l l a n d c o u l d be  were r e l e v a n t  t o common e v e n t s .  s i n g l e k i d out t h e r e . "  fun "It  [p 10]  Teaching Mathematics  " 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 a n d 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 % of your time i s i n t e r a c t i n g w i t h the 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] For Ray, friend  the  and p a r e n t  teacher  t o o k on m u l t i p l e r o l e s  d e p e n d i n g on the t y p e  of  class  of  counsellor,  a s w e l l as  guide,  individual  students. " 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  - 50 -  as  a p a r e n t , n o t e v e n as a t e a c h e r . as a t e a c h e r . " [p 2]  I guess you take  different  roles  " 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 h a v e d i s c i p l i n e r u l e s , the next p o i n t , y o u ' r e a p a r e n t , at the next p o i n t y o u ' r e a friend, you're a counsellor. 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 m o s t p r o m i n e n t o f  t h e s e f o r R a y was c o u n s e l l o r ,  both before  class  and a f t e r  going through diverse  dealing with a variety  problems.  the k i d s . "  Ray's  grades  [p 1]  t e a c h i n g were  largely  a great deal.  9 a n d 10 s t u d e n t s ,  [you]  who  were  time to get  to  sit  know  of  a class  Ray and t h e d e p a r t m e n t "22 s t u d e n t s  the department  with different  the challenges  He saw them as  e m o t i o n a l and b e h a v i o u r a l p r o b l e m s ,  class  head had  head team-taught  the p e r s o n a l problems each student learning.  9A/10A  9 A / 1 0 A was a t e r m i n a l m a t h  a n d t h o u g h he s p o k e o f  inhibited their  families  the  i n f l u e n c e d by h i s  claimed they drained h i s enthusiasm or energy.  repeatedly those  of  He d e s c r i b e d them as  problems" never  ideas  to e s t a b l i s h .  class.  guess  students  Students.  o f whom he s p o k e  composed o f fought  of  [p 2]  View of  class,  f o r he f o u n d h i m s e l f  "So y o u know y o u t a k e  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  at  this  behavioural  they presented Instead,  he  he  spoke  was h a v i n g a n d how individuals with  k i d s who came f r o m d y s f u n c t i o n a l  and a c t e d out t h e i r f r u s t r a t i o n s  i n school.  " . . . a n d I r e a l i z e d t h a t , even 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 a n d 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 t h e m 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 temper t a n t r u m . So, ok, f o r t h a t c l a s s you 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 R a y r e t u r n e d my d r a f t for  "regular"  students,  the  summary o f  r a t i o of  He g a v e t h i s  as  students  comments,  he n o t e d  t e a c h i n g time to time spent  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 Ray spoke o f o t h e r  his  that  dealing  50:50.)  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  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  school.  in  mathematics. " I t h i n k most o f them f e e l t h e y h a v e 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 . They 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 h a v e a d i s l i k e for...any subject." [p 5] I n a d d i t i o n , he b e l i e v e d hard,  and a t e r r o r ,  claimed students that  once  the  and a waste of  subject.  usefulness  of mathematics  A c c o r d i n g t o Ray, to break  T h o u g h he c l a i m e d t h e r e to appreciate  R e l a t i n g h i s own e x p e r i e n c e , [p 10]  connection....But connection] friend  until  i n high school,  there  Goals of  A math t e a c h e r  usefulness the  subject  students  university, t h e y have  and  t h e y w o u l d be  needed  t o see  i n m a t h , he s t r e s s e d  a l g o r i t h m s at  the teacher,  [p 5]  he  the  barriers.  a later  that  stage."  [p  "the beauty  the o p p o r t u n i t y  o r as a c o u n s e l l o r , [p  "kids 10]  of  when he c o u l d "make  t o h e l p t h e m make t h e c o n n e c t i o n . "  a math  them o n t o m a t h . "  was made,  quite  However,  mathematics"  he s t a t e d he d i d n ' t s e e  3rd year  because you are  who's  of  [p 3]  down t h e i r e m o t i v e  was b e a u t y  the beauty  "...something  h a s no u s 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  i n the  algorithms"  time,  saw m a t h a s  h a d a n "unknown c u r i o s i t y a b o u t  interested  have  students  the  [ t o make o r as  the  a  10]  teacher. needed t o  [p 14]  and r e l e v a n c e  "hook k i d s o n t o s c h o o l "  Ray b e l i e v e d  [p 4] b y  t h a t by showing s t u d e n t s  of mathematics  matter.  - 52 -  t h e y w o u l d become  "hooking the  interested  in  " . . . k i d s [need] t o u n d e r s t a n d t h a t m a t h i s u s e d i n e v e r y a s p e c t o f our l i f e , i n every s u b j e c t of l i f e , whether i t ' s music f o r n o t e s . . . f o r p h y s i c s . . . o u t on the 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 , keeping scores, ah, f i n d i n g percentages. I t ' s j u s t the 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 h a v e t h a t . . . . T h e y h a v e t o h a v e 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 m a t h 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 o n 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 , s o 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 h a s t o be c h a n g e d . T h e y h a v e t o be t a u g h t t h e u s e f u l n e s s i n m a t h 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 whatever 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 the bank, 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  s t u d e n t s might have r e g a r d i n g mathematics by making c l a s s e s " B u t t h e y h a v e t o t a k e t h i s [math] them b y t h e s c h o o l . Which i s s a d , f u n f o r t h e m . " [p 3]  the  enjoyable.  c l a s s w h i c h was b e s t o w e d u p o n s o y o u t r y a n d make t h e c l a s s  " . . . y o u c a n u s e m a t h 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 t h e m t h e u s e f u l n e s s o f m a t h d u r i n g c l a s s t i m e . " [p 3] Qualities  of  a math  teacher.  "So I t h i n k t h e 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 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 m a t h i s t h e r e , what u s e m a t h i s . " [p 5] To R a y ,  a teacher  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  mathematics. " . . . i f y o u j u s t k e e p somebody who i s n o t a m a t h t e a c h e r t o t e a c h math, 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 m e s s a g e a c r o s s . . . y o u h a v e 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 a s a n e x a m p l e o f applications  someone who was a l w a y s  looking for  and 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 . For e x a m p l e , i f y o u ' r e d r i v i n g o n 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 . And l o o k a t my s p e e d o m e t e r a n d t r y a n d 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] Teaching methods.  -  53  -  Ray a d v o c a t e d to the  real world.  s a n i t a t i o n workers,  methods w h i c h w o u l d g i v e He s u g g e s t e d secretaries  u s e d math i n t h e i r work. trip  to a baseball  complete.  experiences.  guest speakers. and c a s h i e r s  game where t h e s t u d e n t s  devise  teacher have  would acquire  already  w i t h the  the c l a s s  had an a s s i g n m e n t  " p o s i t i v e problems" riddles,  " h o o k i n g them o n t o m a t h " a bank o f  collected  from l e c t u r i n g . share  of  to t e l l  how  they  field  to  He a l s o  suggested  t h e i r own m a t h p r o b l e m s b a s e d o n t h e i r  He c a l l e d t h e s e  o t h e r methods  "connection"  He e n v i s i o n e d i n v i t i n g  t h e game was r a i n e d o u t . )  P l a y i n g games a n d u s i n g p u z z l e s , were  that  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  (Unfortunately,  having students  students  and p r o b l e m s o f [p 1 4 ] .  students  that  having l i f e  a  and c l a i m e d  He t r i e d t o s t a y  experience  w o u l d a l l o w f o r more v a r i e t y  away f r o m a t r a d i t i o n a l l e c t u r e  intrigue  He s t a t e d  these through experience,  q u i t e a few w h i c h he u s e d .  He f e l t  [p 8 ] .  and " t r i c k s " i n class  to  away  to  a n d move  style.  Metaphor. Ray had d e v e l o p e d teacher  as  a gardener  h i s own m e t a p h o r f o r  and s t u d e n t s  T h i s metaphor h i g h l i g h t e d student he f e l t  for  them i s  evident  teaching.  as p l a n t s u n d e r h i s individuality.  in his  It  was o f  the  supervision.  The c a r e  and  concern  description.  " 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 , you get p l a n t s and y o u n u r t u r e them, 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 a n d y o u know a s 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 . In 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 . Yeah, 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 for t h e i r best u s e . . . . E v e r y plant is 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 w a g o n . They l o s e i n t e r e s t a n d y o u know y o u t r y a n d c h a n g e y o u r m e t h o d s 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  -  you n u r t u r e t h e m . . . . A n d I t h i n k t h a t , that p e r t a i n s 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 1 4 - 1 5 ]  Other  to being a  Issues Conceptual  change.  Ray's view of experience  t e a c h i n g mathematics had changed w i t h h i s p r a c t i c u m  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  student-  oriented. " L e t ' s s e e , y e a h , my v i e w s h a v e t h i n k . . . . m y previous assumption l e a v e , y o u mark p a p e r s . B u t no than that. Teaching i s , i t ' s a  changed. Ah, d r a s t i c a l l y , I was y o u j u s t g o , y o u t e a c h , y o u t h a t ' s not i t . T e a c h i n g i s more m u l t i t u d e o f j o b s . " [p 13]  Victoria Academic and P e r s o n a l Background Victoria year  B.Sc.  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  mathematics student."  had 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 , and a 4-  was i n p r o g r e s s .  after  She c a l l e d h e r s e l f  her B.A. a  in  "professional  [p 6]  Victoria university  was a w h i z k i d i n m a t h e m a t i c s  and g r a d u a t e  school.  l e a r n math e f f o r t l e s s l y , p u t me somewhere  mathematics  it  absorbing i t .  department  t o me."  from p u b l i c s c h o o l  "...you've  always  [p 13]  she h a d t a u g h t  As a g r a d u a t e  Victoria  to  faster  i n UBC's  course  for  those  i n a B.Ed, program.  Mathematics differentiated  l e v e l mathematics. it  student  an u n d e r g r a d u a t e  to  been a b l e  and I c a n l e a r n t h e c o u r s e  w i t h no m a t h b a c k g r o u n d who w e r e p l a n n i n g t o e n r o l l Conception of  through  She s k i p p e d g r a d e s a n d was a b l e  w i t h the textbook  than you can teach  formulaic;  An M . S c .  She s t a t e d  e x e m p l i f i e d the  lower l e v e l mathematics at  the  lower l e v e l  it  from  was a l l known a n d  s c i e n t i f i c nature of math.  - 55  -  graduate  Graduate  level  math,  however,  intuition of  involved "fuzzy,  grey  and c r e a t i v i t y t o a c h i e v e  areas"  [p 7]  results;  which r e q u i r e d  t h i s was t h e  artistic  side  math. " . . . 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 m a t h i s a known a r e a a n d i t ' s not changing a l l that m u c h . . . . y o u ' r e not going to suddenly have e q u a t i o n s of l i n e s changing 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 . [p 7] " . . . a s a n u n d e r g r a d y o u h a v e t h e r i g h t o r w r o n g a n d I mean y o u have, y o u ' l l have, y o u ' r e doing a proof? And t h e r e ' s different 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 fuzzy grey areas there, you're g e t t i n g i n t o t h e e d g e s o f m a t h 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 area....you're just s t i l l discovering, you're s t i l l creating i t , ok? I t ' s s t i l l being created.' [p 6-7] " I t h i n k [mathematics] i s a l i t t l e o f each [ a r t and s c i e n c e ] , and 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 a s 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 . science. A n d i t ' s c r e a t i v e , y o u h a v e 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]  Victoria hard object" In contrast discrete  claimed students  [p 1 ] , she  generally  totally externalized,  saw m a t h e m a t i c s ,  mathematics,  as  saw m a t h a s  rigid,  and d i f f i c u l t  to  "this  comprehend.  i n p a r t i c u l a r her s u b - s p e c i a l t y  fun and e x p l i c a b l e [p 6]  to the  She went  lay-person  further  connect-the-dots."  elementary  t e a c h i n g u s u a l l y rewards memorization i n mathematics,  to claim  successful  d o i n g memory w o r k ,  Victoria overshadowed  for  the  student  that a  In higher  w h e r e 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 r a n s i t i o n often proved d i f f i c u l t  of  as  "glorified  not d i r e c t l y l e a d i n g t o becoming a good m a t h e m a t i c i a n .  who was  cold  trait years  the formerly  [p 9]  claimed mathematics'  aesthetic  q u a l i t y was  by i t s u t i l i t a r i a n nature f o r pragmatic  often  purposes.  " . . . w e ' r e the b u i l d i n g blocks of other s c i e n c e s . . . . we're 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 m a t h 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  -  mathematician's are at a l l u s e f u l unless they a c t u a l l y create 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] Conception of Mathematician Victoria algebraists All  were  formulated her  ideas  of  s h e h a d met a s a g r a d u a t e  creative,  a mathematician student  a r t i s t i c and m u s i c a l ,  mathematician should be."  at  "things  through  Queen's  University.  that you d o n ' t  [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 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 artsy...stuff." [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 understanding;  View of  Teaching  Victoria dispel  "logically  [p 10] Mathematics  h a d two g o a l s  the n o t i o n of  s e c o n d was t o c h a n g e being r i g i d  are  work a s c h e c k i n g f o r m e a n i n g a n d  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  and i n t u i t i v e l y . "  think a  i n teaching mathematics.  a m a t h e m a t i c i a n as the  a "geek" o r  The f i r s t  " n e r d , " and  was the  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  and u n a t t a i n a b l e  f o r the  average  to  as  person.  " . . . 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 m a t h 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 get o v e r and d e a l w i t h and s u r v i v e r a t h e r t h a n something 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 sense? M a k i n g , m a k i n g m a t h as I s a i d , v e r y h u m a n . . . " [p 2] "...what 18]  I w o u l d l i k e t o do i s  I ' d l i k e t o make m a t h f u n . . . "  [p  " 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 and I e n j o y math. 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 geek, t h e geek pack w i t h t h e pen l e a k i n g t h r o u g h and 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 stereotype." [p 13] She saw t h e a n d became  fear  o f math i n the u n i v e r s i t y s t u d e n t s  concerned that  this  fear  -  would t r a n s f e r  57  -  to the  she had children  taught they  would e v e n t u a l l y h a l f - h o u r of or art  i n s t r u c t i o n i n the  them e n j o y  the  A scenario  the  overcome subject.  "math n e r d " ,  Metaphor f o r  Painter,  as  oeuvre,  and not a s c u l p t u r e d u r i n g the  Her metaphor  left  individual  getting [p 13]  and d i s p e l t h e  a metaphor  the  had l e f t  for  teacher/painter  where  the master  students  stereotype  of  and f o s t e r i n g  teaching.  emphasized g r e a t l y  the  composition of  friends,  impressions influence  t h e home,  on the of  every  t h e i r mark o n t h e  he o r s h e on t h e  and l i f e  canvas. event  it.  w o u l d be  work.  entered your classroom,  Teacher  a  the  student.  i n general.  student's  student's  canvas,  the  These  The p a i n t i n g was i n the  the  chipping  embodied  canvas.  as  The  was a d d i n g o n t o  artist  c a r r y i n g t h e work o f o t h e r s  the  helping to create  attitude  help  and have  t h e message t o  the  first  their  accumulation of  music  V i c t o r i a wanted to  c r e a t i o n of  were t h e p a r e n t s ,  had a l l  hour of  P a i n t i n g was how s h e c o n c e p t u a l i z e d  a p a i n t i n g where  prepped p a i n t i n g ,  forced  math.  had begun t o d e v e l o p  student  was a  and p a r a n o i a of mathematics  b y humans"  was  others  classroom.  She a d v o c a t e d  student  When t h e  this  teaching.  and Student  away s u b s t a n c e  fear  for  a one-and-a-half  p a s s i n g on i n s t e a d a p o s i t i v e  to explore  Victoria  elementary their  " m a t h c a n be e n j o y e d  willingness  she d e p i c t e d  i n s t r u c t i o n i n math v e r s u s  them and o t h e r s  that  teach.  These an  past;  each person  each was  it.  "I l i k e the idea of a p a i n t e r ? . . . the student i s the p a i n t i n g y o u ' r e quote unquote c r e a t i n g . Mind you 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 , their parents, their friends, a l l t h i s , a l l this is 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 what's happening i s you, y o u ' r e given a p a i n t i n g t h a t ' s prepped and s t a r t e d , and 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 . And h o p e f u l l y p o s i t i v e l y , ok? And the reason I l i k e p a i n t i n g i s because you are  -  58  -  adding on. You're not, taking away..." [p 15]  it's  not  like  s c u l p t i n g where  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 t r e p i d a t i o n she  felt  i n the r o l e of  she had t h e o p p o r t u n i t y t o d e s i g n , The i n f l u e n c e inherent  c o u l d be n e g a t i v e  i n that  was  teacher. create,  as  It  shown b y  you're  the  i n t i m i d a t e d her  that  and draw on t h e p a i n t i n g .  a s w e l l as p o s i t i v e a n d t h e  power  frightening.  " . . . t h a t ' s the scary p a r t of t e a c h i n g , i s your a b i l i t y to a f f e c t the l i v e s i s not 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 . And 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 m a t h t e a c h e r disciplinarian. members o f  the  She r e m e m b e r e d t h e p h y s i c a l f o r c e class  and l i t t l e  how s h e p e r c e i v e d t h e r o l e o f  The a b i l i t y t o f o r e s e e was  else.  the  Requirements f o r a teacher  explanations  who was a  strict  he u s e d w i t h v a r i o u s  T h i s may h a v e h a d a b e a r i n g o n  teacher. of  mathematics.  student  d i f f i c u l t i e s and  important for a teacher  of  generate  mathematics.  " I t h i n k t h e r e a l g o a l a n d t h e r e a l i m p o r t a n c e i s n ' t s o much a c t u a l l y p h y s i c a l l y knowing 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 be, l i e ? and t h e y c a n ' t e x p l a i n t h e problems. 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 s e e 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 , o k ? 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] Confidence  i n the  courses  took to achieve  it  subject  person becoming a teacher  m a t t e r was k e y . t h i s was n o t  The number o f  important.  o f m a t h h a d no f e a r  need t o e n j o y math and not f e a r  it  of  [because]...it  university  The f a c t  the  that  subject,  transmits."  the  was. [p  "You 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 a s w e l l : a s e n s e o f h u m o u r , knowledge that  those  and use o f  alternative  approaches/methods.  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 h a v e  - 59 -  She a l s o  commented  empathy f o r  the  slower  learners.  herself  as  This  extremely  is notable  quick to  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  learn  mathematics.  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 [the  students  another  are]  going through at  i m p o r t a n t component  of  this  development  "[w]ith  stage i n t h e i r  a good math  life"  [p 3]  c o n s i d e r e d a model t e a c h e r .  b e c a u s e he was a s u b s t i t u t e , go a b s o l u t e l y  ape o n . "  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  She d e s c r i b e d h i m a s h a v i n g  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 the p o i n t that  he was n o t o v e r l y s t r i c t  t h e m f o r who t h e y w e r e . " She c l a i m e d t h a t 17]  and r i g i d ,  [p 18]  nor demanding,  T h a t was what  she h e r s e l f  was  "still  and would l i k e t o change t o  was h e r o t h e r  interests:  t e a c h e r s must h a v e q u o t e u n q u o t e reinforces  her goal  Teaching  that  it  meant t o c o n v e y  W h i l e she p r e f e r r e d n o t t o  times necessary  t o do s o .  [p 3]  that  made  emulate.  and w h i t e "  students  What s h e  music. [p 20]  [p  saw  in  "...math This  notion  "touchable."  "teach," f o r not  she c l a i m e d i t liking  w a r m i n g up t h e c l a s s  - 60 -  as  this  were with  was  "cold,  cooperative review  to  at  break.  she c o n s i d e r e d e f f e c t i v e  pairing, paired quizzes,  was  t h i s method  image o f m a t h e m a t i c s  s h e was t r y i n g t o  A l t e r n a t i v e methods ones:  ok?"  She  information through l e c t u r i n g  Her reasons  a l i e n a t e d and promoted t h e  hard t h i n g "  dance,  a personality,  presence,"  "accepting  accept...the [p 17]  kids  methods.  The w o r d " t e a c h " Victoria.  just  very black  gardening,  f o r m a k i n g m a t h human a n d  "a  she wanted t o  "just  the  to the k i d s .  w i t h o u t w a n t i n g t o c h a n g e them t o o d e s p e r a t e l y . " her favour  someone  " [ a ] n d y o u know w i t h s u b s t i t u t e s  [p 18]  was  teacher.  D u r i n g h e r p r a c t i c u m she had t h e o p p o r t u n i t y t o w i t n e s s she  what  questions. solve  Another strategy  she  the b e l i e f  that  i n the group t h a t  by the  student. the  students.  Student  development  it  [p 4]  if  e x p l a i n t h e i r answer.  Victoria  the  student  used i t  an e r r o r  students  overcome  their  growth i n  Other  Issues Students' Victoria  [p 4]  students  the  t h e same p o i n t t h e y ' r e  So,  t h e i r fear  to  t h i s was  student.  going,  i n a d d i t i o n to her goal of  o f math, she a l s o  saw v a l u e  more  in  done  However, It  "is  h e y I c a n do helping fostering  self-esteem.  view of  mathematics.  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  c u r r i c u l u m w i t h the  students'  lives.  undulations with identity crises divorcing parents, assimilate  " W a l k me  to the c l a s s ,  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  i s good."  is  She i l l u s t r a t e d  would e x p l a i n h i s / h e r steps to the o t h e r s .  occasionally painful but...at this  she u s e d t o e n c o u r a g e  On t h e s u r f a c e  advocated  occured.  was a c o n c e r n o f V i c t o r i a ' s .  was a p h r a s e  With  the a l g o r i t h m used  when s h e 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 .  so t h a t  this,  concretizes  c a n more e a s i l y d i s c e r n where  View of  through i t " fully  The p r o c e s s  The s t e p s become n a i l e d down s o t h a t  student  and  h i s / h e r a n s w e r was c o r r e c t .  v e r b a l i z a t i o n aided understanding, V i c t o r i a  group work f o r h e r c l a s s e s .  this  person  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  convince the others  made,  l i k e d t o u s e was t o h a v e e a c h  it  of  In the throes  and deep f a m i l i a l  the of  secondary  emotional  problems such  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  s o m e t h i n g so c o n c r e t e .  and  M a t h was c o m p l e t e l y d i f f e r e n t  Life.  - 61  -  as  from  " . . . i t makes m a t h t h i s t h i n g w h e r e 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 s o 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 ] separations, 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 are g r e y a r e a s i n t h e i r l i f e , and a l o t o f t h i n g s are subject t o change. And 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 , a n d s a y i n g l e a r n t h e s e , do t h i s And i t ' s , i t ' s such c o n t r a s t t o the 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 m a k e s , 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 ]  Mary Academic  Background  Mary had a B . S c . had taught  i n mathematics  introductory statistics  teaching assistant. d u r i n g the  Her secondary  Conception of  graduate  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 , and  so  t h e c u r r i c u l u m she  Mathematics  content.  graduate  as a  She  taught.  To M a r y , m a t h was n o t  problems,  to medical students  i n t e r v i e w s h e made c o m p a r i s o n s b e t w e e n  t o o k a n d t h e one s h e  involve  and an M . S c . i n S t a t i s t i c s .  Rather,  it  " c r u n c h i n g numbers" nor d i d i t was p r o c e s s :  d e d u c i n g and r e a s o n i n g , work,  logic.  really  logical thinking, At a higher  solving  level,  i.e.  i t became p h i l o s o p h y .  " [ T ] o me, m a t h i s n o t s o 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 , a n d how t o d e d u c e t h i n g s a n d r e a s o n . It's, it's logic. I f y o u ' r e t a l k i n g a b o u t m a t h a t a much more advanced l e v e l , then i t ' s almost 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 numbers, 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 math." [p 1] Mary's solving  for  likened  it  challenge problems.  a t t r a c t i o n to mathematics she e n j o y e d  was t h e  doing puzzles,  and e n j o y e d  came w i t h e x p l o r i n g d i f f e r e n t  To h e r ,  that  was t h e  of  problem-  and " f i g u r i n g t h i n g s  to doing crossword puzzles, that  element  the  out."  She  s t i m u l a t i o n and  ways t o s o l v e  puzzles  and  fun i n math.  " I e n j o y e d m a t h e m a t i c s n o t s o 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 a s much a s  -  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. It's like a puzzle. I love doing p u z z l e s , I love s o l v i n g p r o b l e m s . . . . I l i k e homework...where y o u h a v e t o go home a n d 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 a n d 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] View of Teaching  Mathematics  Teaching students  to  l e a r n how t o t h i n k a n d how t o  major g o a l s of M a r y ' s t e a c h i n g . coping w i t h problems  These  l e a r n were  s k i l l s would prepare  they would encounter  i n their adult  the  students  for  life.  " . . . 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 , s o 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 and 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 t h e m 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 k i d s a r e a b l e t o t h i n k a n d t a c k l e t h e p r o b l e m . " [p 9]  that  " I d o n ' t t h i n k i t ' s a s i m p o r t a n t t o p u s h as much c o n t e n t t o t e a c h k i d s how t o t h i n k " [p 10] In fact, students'  the philosophy of  t h i n k i n g s k i l l s c a r r i e d over  She saw no d i f f e r e n c e  between  B o t h meant h e l p i n g s t u d e n t s s o l v i n g problems. persuasive geometry,  t e a c h i n g mathematics  The l i n e a r ,  e s s a y was t h e  s k i l l s of  l o g i c a l process  is  in  general.  and t e a c h i n g E n g l i s h .  analysis,  strategizing  used to w r i t e  same one u s e d t o f o r m u l a t e  it  developing  f o r Mary to t e a c h i n g  t e a c h i n g mathematics  develop  as  as  the  a proof  and  a i n Euclidean  according to Mary.  "[Teaching] r e a l l y i s n ' t about mathematics. That 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 s o o n s o f o r t h . I t ' s n o t s o much a b o u t t h e c o n t e n t a s much as a b o u t , y o u k n o w , how t o , how t o s o l v e a problem. I t ' s l i k e a n y 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 a n e s s a y . You s t i l l have 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 a n 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 m a t h h a s n o t h i n g t o do w i t h m a t h . I t h a s 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 . It's all 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 -  Another goal of  t e a c h i n g mathematics  f o r M a r y was t o c h a n g e  the  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 h a v e 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 hating arithmetic, or - . Just dry. J u s t s i t t i n g i n c l a s s , and y o u know, k i l l i n g an h o u r , and g e t t i n g out o f t h e r e , c r u n c h i n g numbers, 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 teaching,  she d e s c r i b e d a t e a c h e r  that...gets  as  t h e i r c u r i o s i t y up a n d ,  solving a problem..."  the purpose  "somebody who p o s e s i s able  to guide  energy,  Mary s t a t e d  [p 7]  several  times that  p o s s i b l y m i r r o r i n g the energy  practicum. knowledge  Enthusiasm, confidence, to e x p l a i n the concept  Being able  t o have f u n w i t h the  p a r t i c i p a t i n g were a l s o  a teacher  she f e l t  students,  listed.  her b e l i e f s ,  Teaching  g e t t i n g them i n t e r e s t e d  time f o r proper lesson p r e p a r a t i o n .  The  good  it."  unsuccessful  a n d how  [p 4]  ( A p p e n d i x A) r e p r e s e n t i n g  However, that  teaching the concept  none o f  possible  them seemed t o c a p t u r e  the  t e a c h i n g m a t h was t e a c h i n g p r o b l e m - s o l v i n g .  methods.  Mary began her p r a c t i c u m b e l i e v i n g t h a t most e f f i c i e n t  and good  M a r y was shown t h e d i a g r a m s  essence of  characteristics.  The  k i d s t o t h i n k about  teaching.  and  element.  would " t h i n k v e r y c a r e f u l l y about  of  of  she needed d u r i n g h e r  T i m e was a c r i t i c a l  teacher  conceptions  s h o u l d have a l o t  t h o r o u g h l y were o t h e r  r e q u i r e d a l o t of  gonna get  f o r a mathematics  the a b i l i t y to handle questions  teacher  you're  questions  them t h r o u g h  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 teacher,  of  way o f  teaching.  she c h a n g e d h e r  facts  was  When she n o t i c e d how s t u d e n t s  i n s o l v i n g the d i v e r s i t y of  G r a d e 10 g e o m e t r y  drilling  ideas.  - 64 -  the  were  E u c l i d e a n proof problems  in  " 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 s e e 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 stumped r i g h t away. You know. They 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 had not t a u g h t the geometry p a r t I would not have r e a l i z e d t h a t , I d o n ' t t h i n k . " [p 1 0 - 1 1 ] Now s h e d i s a g r e e d Instead,  l a i d out,  students  t o t h i n k about  believed  an e f f e c t i v e  is  which had l o g i c  l e s s o n p l a n would l e a d students  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 .  come o u t v e r y n a t u r a l l y ,  well."  and f l o w ,  the t o p i c by u s i n g p e r t i n e n t  students.  flow,  and w h i c h  examples. to  one lead  Mary  understanding,  " Y o u r most p r o d u c t i v e  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 "  "will  to  she b e l i e v e d good t e a c h i n g e n t a i l e d a good l e s s o n p l a n ,  w h i c h was c a r e f u l l y  and i t  w i t h simply p r e s e n t i n g a concept  [p 14]  and t h i n g s  will  and a good work o u t  hour  one  very  [p 8] "Um, how w o u l d I r e c o g n i z e [a g o o d 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 h a s 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 get k i d s t o t h i n k about i t and t h e n , and f i n a l l y c l u i n g i n o n 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 opposed t o j u s t g i v i n g t h e c o n c e p t t o the k i d s . " [p 5] View of  students.  Mary enjoyed were s t i l l The f a c t her.  "full  that  the  students.  of p e r s o n a l i t y "  they  a waste of  Some o f students  being  independently. strategize  [p 16]  to t h e i r  the Grade 9  time even though i t  "spoonfed,"  [p 7]  Mary b e l i e v e d and i f  d i d not  was a t  times  She  11]  perturb considered  i n M a r y ' s o p i n i o n , came f r o m  students  the  think  had an i n h e r e n t a b i l i t y  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  - 65 -  [p  necessary.  r a t h e r than b e i n g pushed t o  that  students  and f o u n d them "so c u t e . "  intrinsic curiosity.  the d i s c i p l i n e problems,  (think)  that  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  She a t t r i b u t e d t h i s  disciplining  She e n j o y e d  develop  to  this,  content  had l o s t  To h e r ,  students  t h e i r a b i l i t y t o t h i n k beyond a g i v e n example because t h e y  accustomed  to doing mathematics  even...think it  understanding would n a t u r a l l y f o l l o w .  they're  before...And  capable  that's  of  v i a rote procedures.  "...they  s o l v i n g something unless  a shame."  were  don't  t h e y have  seen  [p 9]  " I d o n ' t care about l i n e a r e q u a t i o n s . I c a r e i f the k i d s can 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 about math, and 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 t h e m t i m e t o t h i n k , y o u ' d be a m a z e d . . . " Other  [p 7]  Issues Conceptual Because  Change.  Mary had e m i g r a t e d t o B r i t i s h  high school  i n Vietnam,  curricula.  Her o b s e r v a t i o n s  teaching sought  from a d r i l l  to develop  she was a b l e  t o compare t h e  were a f a c t o r  and p r a c t i c e ,  Columbia a f t e r two  completing  mathematics  i n changing her ideas  learn-the-facts  of  a p p r o a c h t o one  that  thinking abilities.  " . . . I t h i n k k i d s h e r e c o m p a r e d 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, a n d 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 . They're not allowed to t h i n k . And t h e y are 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 had 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 k n o w , 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] Another factor  in effecting  t e a c h i n g geometry,  a change  as e x p l a i n e d  T e a c h i n g p r e p a r a t i o n and Mary spent this  was t h e most  i n her ideas  of  teaching  was  above. duties.  much t i m e p r e p a r i n g h e r l e s s o n s important aspect  of her job.  - 66 -  b e c a u s e she In fact,  believed  s h e was  so  concerned that  students  seem t o b a l a n c e  her  understood the concept  being taught  she  didn't  time.  " 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] Mary r e s i s t e d teaching:  c a r r y i n g out o t h e r d u t i e s  administrative details,  non-attendance,  normally associated  marking attendance,  dealing with parents  f o l l o w i n g up o n  and c o u n s e l l o r s .  a waste 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  with  She c o n s i d e r e d  it  time.  Summaries To e n c a p s u l a t e conceptions Yvette:  of  t h e i r ideas  these  Content  six prospective  about  content  their  own l e a r n i n g ,  different  comparison,  teachers are  the  summarized.  Presentation  "I teach math."  effectively.  and f a c i l i t a t e  Yvette's  presentation.  notion of  She f e l t  t e a c h i n g mathematics  students  a n d h e r r o l e was t o p r e s e n t  T h i s meant u s i n g a v a r i e t y o f  learning  centred  were r e s p o n s i b l e the  for  material  techniques  to  favour  styles.  Shannon: A Buddy Shannon's one-to-one desire able  to  conception of  t e a c h i n g mathematics  r e l a t i o n s h i p with her students. form a r e l a t i o n s h i p w i t h each,  to accomplish t h i s ,  her c l a i m that  a h i g h p r i o r i t y , and h e r c h o i c e valuing his  students  over  of  focused  on b u i l d i n g a  T h i s was e v i d e n c e d b y  her f r u s t r a t i o n at h e l p i n g those  not  her  being  who d i d p o o r l y  a r o l e m o d e l - someone p o r t r a y e d  all.  - 67 -  was as  Shannon had a s t r o n g and p e r v a s i v e existed  a right  way v e r s u s  s h o u l d have c o r r e c t professors;  i n the  answers  was  the u t i l i t y  impart that  t h e most e f f e c t i v e  it  duality,  She b e l i e v e d  for a l l questions.  classroom,  Shannon f a v o u r e d t e a c h i n g was t o  a w r o n g way.  sense of  i.e.  those  in  A t UBC, t h o s e  there authority  were  the  herself.  of mathematics  applicability.  way t o t e a c h number  and h e r g o a l  She a d v o c a t e d  in  drilling  as  facts.  Doug: A R o l e Model Doug'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  developmental r o l e model f o r example,  before  adulthood.  Being a teacher  a n d l e a r n i n g f r o m them was p a r t o f  had another purpose  claimed the r i g i d i t y  and r o u t i n e n e s s  l e a r n i n g p r o v i d e d them a f o r m o f Rav: M u l t i p l e  Roles,  teacher. this  His goal  a  For  for  making  life. and  constantly  with students.  He  troubled teenagers. they  also He  were  stability.  A Gardener  of  counsellor,  was f o r k i d s t o  c o u l d be a c h i e v e d  relevant.  was c r e a t e d  i n the mathematics  Ray v i e w e d t e a c h i n g mathematics including that  everyone's  this perspective  b e l i e v e d mathematics  roles  meant b e i n g  p r o b l e m h e 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  He w a n t e d t o s h a r e  were i n a  s t u m b l e d when s o l v i n g a  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 changing.  students  t h e m 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 .  b y s h o w i n g t h e m he s o m e t i m e s  mathematics mistakes  phase  that  On a b r o a d e r  if  students  scale,  as  involving several  parent love  and f r i e n d as w e l l  l e a r n i n g m a t h a n d he  found mathematics  he b e l i e v e d  - 68 -  different  useful,  as believed fun,  t h e t e a c h e r ' s r o l e was  and to  "hook them t o s c h o o l . " taking care Victoria:  of  C h a n g i n g Images,  hard object"  instil  i n her students  o r i e n t e d to address  real  alienated  that  She f e l t  lower  She w a n t e d  c o u l d n o t be r e a l i z e d .  Thus,  the  a to  intuition  l e v e l m a t h was t o o  the c r e a t i o n of mathematics.  believed  process-  true  S c h o o l m a t h was d i f f e r e n t  students  t h e r i g i d i t y and c e r t a i n t y  because i t  their  metaphor  for  tapestry  b e i n g worked on.  life.  parents,  teachers,  it  i n school  contrasted with uncertainties  She c a r r i e d . t h i s c o m p l e x i t y o f  teaching.  teachers,  nature from  The t e a c h e r  was a p a i n t e r ;  Others had worked on i t  friends,  f a m i l y - a n d now,  life  the  in  other  into  her  student  the  earlier  as one o f  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  mathematics  -  the  former student's  tapestry.  Problem-Solving Mary's  conception of  mathematics  as  t e a c h i n g m a t h was c o n s i s t e n t  a d i s c i p l i n e , her b e l i e f  mathematics  and h e r v i s i o n o f  isn't  math" but  about  in preparation for problems their  was  d o i n g math r e q u i r e d c r e a t i v i t y and  logic.  areas of  of  mathematics  math. Victoria  Mary:  gardener  A Painter  p r a c t i c e d o n l y by geeks and n e r d s .  a sense of  mathematics  t e a c h i n g was a  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  "cold,  as w e l l as  metaphor f o r  plants.  Victoria's  of  Ray's  relevant  it  life. to the  t h i n k i n g beyond the  examples.  She h e r s e l f  students  Essentially,  ideas  teaching "it  really  c r i t i c a l t h i n k i n g and p r o b l e m s o l v i n g  Consequently,  rote  i n the purpose of  a math t e a c h e r .  was a b o u t  with her  she  infused her teaching  and encouraged  and l o o k f o r  students  strategies  to  beyond  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  - 69 -  with  stretch text the  most  enjoyable this  as  elements  fun too.  o f mathematics and wanted h e r s t u d e n t s Teaching mathematics  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 -  is  to  experience  t e a c h i n g t h i n k i n g and  teaching  Chapter 5:  C o n c l u s i o n s , Other I s s u e s & Recommendations  Conclusions The f i r s t conceptions  of  research question for this  the  show c l e a r d i f f e r e n c e s  f o r each p e r s o n .  individual  student  p l a c e d on c o g n i t i v e emphasized content it  lives,  and t h a t  i n the conceptions  Perspectives  skills  of  differed  Those  i n whether  significance  or p s y c h o l o g i c a l development.  it  was  thinking,  that  Those  i t was r e l e v a n t  to  students'  conception of  teaching  mathematics  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  single  is nevertheless  dominant f e a t u r e s conceptions  of  t h e i r view.  is helpful in this  Yvette Her o b j e c t i v e  Pratt's  t o be a b l e (1992)  to describe  model f o r  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 way.  in t h e i r students,  into a the  describing  regard. mathematics.  of her subject  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  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  students  useful  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  t h e most e f f e c t i v e  who  mathematics  encompasses s e v e r a l it  was  fun.  Though each p e r s o n ' s  category,  on  pre-service  wanted t o demonstrate v a r i o u s a s p e c t s o f  involved c r i t i c a l  teaching  ranged from an emphasis  to h i g h l i g h t i n g content.  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  - that  to  mathematics?  The r e s u l t s  the  what, a r e  t e a c h i n g mathematics h e l d by i n d i v i d u a l s p r e p a r i n g  teach secondary  mathematics  s t u d y was:  self-esteem  and c u l t i v a t e  t y p i f i e d a Nurturing conception.  t o mature and g a i n s e l f - c o n f i d e n c e  - 71  -  in  development  For them,  was a n i m p o r t a n t  in  helping objective.  Shannon 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 the  students  i n that  she  wanted  t o h e l p t h e m p a s s t h e c o u r s e w h e r e a s Doug was c l e a r he was p r e p a r i n g h i s students ability  for adulthood. also  Doug's  a t t e n t i o n to his students'  illustrated Pratt's  "cultivating  the  intellect"  (1992)  (p 213)  Developmental conception  dominates.  typified  the Developmental conception.  students  for  reasoning where  Mary's perception  also  She f o c u s s e d o n p r e p a r i n g  t h e i r f u t u r e when s h e t a u g h t ;  her emphasis  lay  in  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 problems by t h i n k i n g c r i t i c a l l y through mathematical questions. showed a s p e c t s o f believed that this.  for  subject  the  students  content,  I n t h i s way,  everywhere  i n his students.  least  h a l f the  job of  He  a n d he w a n t e d s t u d e n t s  he h o p e d t o e n g e n d e r  a deeper  a t t e n t i o n he f e l t  to  appreciation  He a l s o d e m o n s t r a t e d c o n c e r n  b y g i v i n g them i n d i v i d u a l  claimed at  mathematics  the A p p r e n t i c e s h i p and N u r t u r i n g c o n c e p t i o n s .  m a t h was p r e s e n t  realize  Ray's view of teaching  for  they needed.  He  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  but e n t a i l e d "being there"  i n other capacities  for  the  teenagers. Though P r a t t ' s of  teaching  i n general,  included are focus  when t h e y t e a c h .  content-focussed; mathematics  it  the d i f f e r e n t  displayed aspects of  fun  (1992)  model i s u s e f u l lacks  subject  for describing  specificity.  conceptions  What n e e d s t o  a s p e c t s of mathematics on which t e a c h e r s  V i c t o r i a ' s idea of the teacher  a Nurturing conception,  as a p a i n t e r  but her p r i m a r y g o a l  she wanted t o d i s p e l n e g a t i v e  and m a t h e m a t i c i a n s .  stereotypes  subject  was  of  R a y b e l i e v e d he c o u l d show s t u d e n t s  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 a n d t h i s w o u l d  them o n t o t h e  be  and o n t o s c h o o l .  - 72  Mary a l s o  -  sought  to  "hook"  show  the  students  that  m a t h e m a t i c s was r e l e v a n t  p e r t a i n e d to t h e i r everyday  by f o r m u l a t i n g problems which  activities.  The s e c o n d r e s e a r c h q u e s t i o n w a s : i n mathematics mathematics  degree  hold a q u a l i t a t i v e l y different  from those  Mary,  Ray,  with only a  and V i c t o r i a  i n mathematics.  Yvette  to the  individual.  Victoria's share  a l l s i x teacher However,  d e s c r i p t i o n s of  t h e i r enjoyment All  three  mathematics.  of  candidates  situations  that  the subject  accomplish t h i s . her students  for students  as o b j e c t i v e  could relate  and  too,  t o have f u n d o i n g  and d i f f i c u l t  like  to school,  to think c r i t i c a l l y .  both stated that  a c o u n s e l l o r and a  forming mathematical  similar  a b i l i t y of  to Doug's goal  teaching students  perceptions  the  fun would  for  to reason  was  However, Mary  also  o f math c l a s s  and  She w a n t e d them t o f i n d m a t h i n t e r e s t i n g a n d e x c i t i n g  - 73 -  that  he w a n t e d t o d r a w t h e m  learn content.  wanted t o s p e c i f i c a l l y change s t u d e n t s '  the  to  Ray b e l i e v e d  t o and t h e y would f i n d  This i s  to  students.  f o r t e a c h i n g was t o f o s t e r  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  mathematics.  unique  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  a n d he b e l i e v e d t h a t  Mary's goal  t e a c h i n g and i n f a c t ,  specialty.  had v a r y i n g c o n c e p t i o n s  with their  i n drawing students  students  graduate  science  what d i s t i n g u i s h e d M a r y ' s , R a y ' s  meant b e i n g o t h e r t h i n g s  as w e l l ,  a  degree w i t h a mathematics  She s t a t e d w a n t i n g t o make m a t h " h u m a n . "  to mathematics  teaching  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  V i c t o r i a ' s goal  However,  studies  baccalaureate?  arts  indicated a desire  being a teacher friend.  conception of  and Shannon had b a c h e l o r o f  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 comprehend.  w i t h graduate  had completed a l l o r p a r t o f  d e g r e e s a n d Doug h a d a l i b e r a l Specifically,  do t h o s e  for  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  boring.  M a r y , Ray and V i c t o r i a b e l i e v e d  a n d c o u l d be e n j o y a b l e Yvette  there  o r i e n t e d toward g e t t i n g  empathetic  facilitator, friend,  Doug a l s o his  teaching  to  instil  students  a love  having different  how s h e  mathematics,  enjoyed  a love  views  of  teaching,  an  f o r math as p a r t  the teacher  F o r some,  t h e i r view of mathematics  memorization.  h e l d P l a t o n i s t views  there  of  candidates  appeared  and t h e i r  to  expressed  was one a n s w e r .  came t o  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 focus  doing  mathematics  as b e i n g a b o d y o f  as h e l p i n g s t u d e n t s  on u s i n g v a r i o u s methods  Shannon's p r o p e n s i t y to d r i l l i n g  are c o n s i s t e n t  math.  - 74 -  Yvette  Shannon c l a i m e d math had a  and even d e s c r i b e d t h e o r e t i c a l  T h e y b o t h saw m a t h e m a t i c s  Yvette's  of mathematics.  f i n d i n g t h e a n s w e r when i t  implying there  o r wrong answer,  content.  and t h r o u g h b e i n g  teaching.  Shannon and Y v e t t e  right  an  Shannon.  d i d not express g e n e r a t i n g  be a s t r o n g l i n k b e t w e e n  stated  subject  f o r m a t h , t h e y w e r e more  through the m a t e r i a l by b e i n g  i n the case of Yvette,  i n the case of  which  " t u r n i n g k i d s on" to the  a l s o v a r i e d i n t h e i r view of mathematics.  of  such  goal.  Besides  views  accessible  students.  was no r e f e r e n c e  Rather than t r y i n g to  effective  was  While they both d e s c r i b e d aspects of mathematics  enjoyed,  itself.  mathematics  and Shannon, on the o t h e r hand, d i d not e x p r e s s  inclinations. they  for a l l  that  t o convey  as  content learn  and  that  topics  and  with a P l a t o n i c view  of  Doug h e l d a f a l l i b i l i s t  view.  c r e a t e d t h r o u g h human e n d e a v o u r  He c o n s i d e r e d m a t h e m a t i c s  and s u b j e c t  m a t h e m a t i c s was d y n a m i c a n d f a l l i b l e . this  belief  example,  to change.  As  He was v e r y c l e a r  to  such,  i n connecting  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 .  he f e l t  would l e a r n that  t h a t by showing s t u d e n t s life  i n general,  he t o o made m i s t a k e s ,  and mathematics  be  For they  i n p a r t i c u l a r , was  not  predetermined. Mary and V i c t o r i a  a l s o v i e w e d math as a dynamic s u b j e c t .  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 level  mathematics  as p h i l o s o p h y .  w h i c h she e n c o u r a g e d processes  she b e l i e v e d  doing mathematics stated  students  rather than content  Her c o n c e p t i o n o f  and  in  t o use the t h i n k i n g and r e a s o n i n g  inherent  i n the s u b j e c t .  s h e w a n t e d t o show s t u d e n t s  Victoria  described She e x p l i c i t l y  t h i s aspect of mathematics.  t h i s v i e w o f mathematics and the  she d e s c r i b e d i s n o t e v i d e n t .  graduate  t e a c h i n g was one  as r e q u i r i n g c r e a t i v i t y and i n t u i t i o n .  a r e l a t i o n s h i p between  Mary  This lack of  an apparent  However,  teaching  methods  r e l a t i o n s h i p can  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 techniques known)  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  and n o n - s c h o o l math  something that t o promote  (still  being created):  s c h o o l math  how c o u l d y o u  meant  creativity?  Correspondingly, h i s view of  mathematics.  - 75 -  academic  teaching included  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 of  teach  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 m e t h o d s  R a y 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 background.  (all  this  aspect  These  links  and t h e i r v i e w s  between  of  the  interviewees'  t e a c h i n g mathematics  I n p a r t i c u l a r , Thompson ( 1 9 8 4 ) , found  "the  and i t s  teachers'  teaching,  beliefs,  regardless  who s t u d i e d t h r e e  views,  124)  She s u g g e s t e d  actual  of whether  career,  mathematics  they are c o n s c i o u s l y albeit  of  subtle,  or  role i n shaping  instructional behaviour."  (p  t h e c o n n e c t i o n between t h e i r v i e w o f math and t h e i r various  and 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 .  the p r o s p e c t i v e  findings. teachers,  about  t e a c h i n g b e h a v i o u r may h a v e b e e n d i m i n i s h e d b y  institutional  mathematics  inservice  and p r e f e r e n c e s  characteristic patterns  of  support the research  unconsciously held, play a significant, the teachers'  perspectives  teachers  t h e y may f i n d  in this  study,  the exigencies  not a l l o w them t o t e a c h  i n ways t h a t  Similarly,  for  when t h e y embark o n t h e i r  o f a new,  full-time position  are  compatible with t h e i r  fully  will  beliefs.  Other  Issues  In a d d i t i o n to t h e i r perspectives seven areas of  on t e a c h i n g m a t h e m a t i c s ,  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  I  noted  candidates.  Adolescence All  the teacher  candidates  during their practicum. scripted  i n t o the  t h e y were g r e a t l y became  became more c o g n i z a n t  Q u e s t i o n s r e g a r d i n g t h i s age  interviews, affected  yet  of  adolescence  group were  e v e r y p e r s o n made comments i n d i c a t i n g  by t h e i r teenaged  students.  Adolescence  a r e a l i t y embodied 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 ,  pressure,  identity crises  t h e y were d e a l i n g w i t h ,  and u n s t a b l e  not j u s t  not  families.  a p a r t i c u l a r age,  - 76 -  peer  T h e y became but w i t h  had  aware  the  that  psychological d e e p l y about  tasks  of  that  Victoria commented t h a t  and Mary,  expressed  the mathematics  at  procedures class  junior  in  Lives saw m a t h e m a t i c a l  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  from the r o i l i n g  by c o n t r a s t ,  s c h o o l math w o u l d a l i e n a t e mathematical procedures  adults  same  what  i n her  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  and s e p a r a t e  This would give  other  university.  were a s o u r c e o f  Victoria,  as  someone who h a d m a j o r e d  gave them an o p p o r t u n i t y t o f o c u s  different  In  Shannon  s h e was t o t e a c h  who w o r k e d w i t h t r o u b l e d y o u t h ,  therapeutic.  studies,  i n v o l v e the  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  School Mathematics i n Students'  as  felt  " r e a l " math.  the school c u r r i c u l u m d i d not  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 ,  Doug,  to  m a t h e m a t i c i a n s worked w i t h and d e v e l o p e d .  dismay that  mathematics  candidates  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 c h o o l m a t h was n o t e q u i v a l e n t  and engender  professional  The t e a c h e r  " R e a l " Math  the mathematics of  techniques  group.  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  words,  age  turmoil  worried that  students.  of  of  the  teens.  their personal  the black-and-whiteness  She r e a s o n e d t h a t  cause to d i s t a n c e  Math  completely  was o p p o s i t e t o t h e t u m u l t o f  them even g r e a t e r  and a u t h o r i t y  on s o m e t h i n g  algorithms  lives. of  the r i g i d i t y  their  of  adolescence.  themselves  from  figures.  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  Subjects  " I t ' s n o t s o 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 k n o w , how t o s o l v e a p r o b l e m . I t ' s l i k e a n y o t h e r c o u r s e . . . . T o me, t h e y ' r e a l l the same." [Mary, 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 w o r k o n i t w i t h o t h e r s t u d e n t s , 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]  you  " . . . 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 t h e m t o t h i n k and t o communicate. E n g l i s h i s o b v i o u s f o r t h a t , but I t h i n k math h a s 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 , paraphrased] These  excerpts  i n d i c a t e these teacher  t e a c h i n g w h i c h were n o t philosophy of Teacher's  linked to content.  teaching held true for a l l  They f e l t  their  mathematics  it  was a c c e p t a b l e  should give  herself  In contrast,  personal  impression the  Doug b e l i e v e d  i n showing a teacher  and not  teacher  even  Change  mathematics  had changed d u r i n g the p r a c t i c u m .  teaching of  Euclidean proofs  was h i s c l o s e  contact  The r e s u l t s  of  teaching  Mary i d e n t i f i e d the  as t h e c a u s e f o r h e r c h a n g e .  w i t h the  this  of  students  and a t t i t u d e s .  study support the u n d e r l y i n g b e l i e f  For example.  because that  Her p r e v i o u s work i n managing a r e s t a u r a n t  i n f l u e n c e d her 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  -  78  -  it  class.  that  ideas  past  Shannon b e l i e v e d t h e  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 them.  For Ray,  i n h i s t e r m i n a l math  t e a c h i n g a r e p e r s o n a l and have been c o n s t r u c t e d t h r o u g h  was t a u g h t  make  them.  M a r y a n d R a y s p o k e o f how t h e i r c o n c e p t i o n s  experiences  teacher  students  Shannon b e l i e v e d t h e math  as k n o w i n g a l l t h e answers  h e s i t a t i n g t o t h i n k about Conceptual  students.  o f what  t o make m i s t a k e s b y h a v i n g them s e e  during class.  should present  have  of  Role i n the Mathematics Classroom  of  of  had i d e a s  subjects.  Doug a n d S h a n n o n h a d o p p o s i n g v i e w s  mistakes  candidates  drill was how may  she  also  calculations  q u i c k l y was v e r y actuarial his  important.  Ray,  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  and e n g i n e e r i n g c o u r s e s ,  teaching.  wanted t o emphasize  Doug h a d a b a c k g r o u n d i n l i b e r a l  contributed to his epistemological c o n s t r u c t i o n c a r r i e d over The r e s u l t s  also  the b e l i e f s  of  prospective  teachers  the  teacher  conclusion of  different  i n part  candidates  were  are  expressed  t h e i r program.  is  However,  the  t h e i r ideas  in  have  mathematics. indicates  of  these  their practica,  through the  from each of  were  their  their  ideas  are not  an i n f l u e n c e  f i x e d and t h i s  s t u d y shows  i n the c o n s t r u c t i o n of  the  in  their  their  Most n o t a b l y ,  conception of  t e a c h i n g r e s u l t i n g from t h e i r p r a c t i c a e x p e r i e n c e s .  a l l the  affected  because of  their actions  contributes future  teacher  Mary and Ray a c k n o w l e d g e d  that  conceptions.  adolescence  teacher  clear.  p r a c t i c u m does have  addition,  near  same  t e a c h i n g mathematics  influence  in  knowledge  The v i e w s  t h e end o f  of  w h i c h may  l i t e r a t u r e which  stable.  at  applications  i n teaching  T h e y h a d gone  from each o t h e r and the  backgrounds  his belief  to his perspective  support  p r e p a r a t i o n program, yet  stance;  arts  in  candidates  teaching.  expressed  t o t h e i r knowledge  a greater  Whether o r not t h i s  i n the classroom  (Yvette  and e x p e r i e n c e  a change  In  awareness  awareness  claimed i t  of  directly  did not),  a n d may i n f l u e n c e  it  their  teaching.  Recommendat i o n s  Based on the  findings of  this  study,  f o u r recommendations  - 79 -  are  offered.  1.  Pre-service  practica  teachers be given a range of experiences  to expose them to different  different  types of  during  types of students as well as  topics.  Research on t h e influence o f p r a c t i c a on prospective conceptions  show v a r y i n g r e s u l t s  strong evidence However,  that  conceptions  two s u b j e c t s  teaching perspective it  their  i n this  (Brown & B o r k o , 1 9 9 2 ) , remain unchanged  teachers'  and there i s  (Kagan,  1992) .  s t u d y c l e a r l y i d e n t i f i e d a change  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 .  i s impossible f o r teacher  candidates  t o have  infinite  i n their While  experiences  during a f i n i t e practicum, increasing thebreadth o f t h e i r experience t o include different  types  of classes  varying mathematical topics to different ideas  contexts  of teaching.  ( e . g . academic and non-academic) and  ( e . g . geometry  and algebra)  will  u n d e r w h i c h t h e y may more c l o s e l y  Wilson  (1994)  expose  examine  them  personal  states:  "by g i v i n g [ p r e - s e r v i c e teachers] 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 t h e m 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 acknowledge 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 mathematics and m a t h e m a t i c s t e a c h i n g , a n d t h e n ( p e r h a p s ) e m b r a c e t h e m . " [p 3 6 8 ]  2.  More studies  on conceptions of prospective  teachers be carried  secondary mathematics  out.  With research that points t o the existence beliefs  on teaching p r i o r  influence 1984),  i t becomes  t h e s t a b i l i t y o f these  (e.g. Tabichnick & Zeichner,  on actions  candidates'  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 program ( e . g .  Feiman-Nemser & Buchmann, 1 9 8 9 ) , d u r i n g t h e program  of teacher  i n the classroom after  1985),  t h e program  i m p o r t a n t t o know t h e s e b e l i e f s  - 80 -  perspectives and t h e i r ( e . g . Thompson,  and ideas.  In the  s m a l l group which comprised the  major d i f f e r e n c e s The i n t e r v i e w e e s program yet additional  their  i n perspectives h a d gone  perspectives  would have been  resulted  candidates  selected  topics;  respond to episodes  and/or d i f f e r e n t  own t e a c h i n g .  teaching  reasons  behind t h e i r responses  others'  teaching,  explore  a l s o use  pre-service teaching  3. novice  depicting various  these,  observe  teachers  to b u i l d  had  sample  - what t h e y  liked  I n t h i s way,  types  of  t h e y made,  Teacher  integrated  of  the  like  candidates  on  student  a videotaping  while they abide w i t h i n  i f necessary,  For  mini-lessons  and d i d n ' t  teacher  of  study.  t h e y c o u l d be p r o b e d f o r  a p r e p a r a t i o n program.  c h a l l e n g i n g them,  sample  in this  prepare  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 on  conceptions,  suggests  using a variety  i n t e r v i e w used  t h e i r own i d e a s  supported environment of  preparation  the  l e d them t o make t h e d e c i s i o n s  guided t h e i r actions. and a r t i c u l a t e  investigated  styles;  found.  conceptions.  c o u l d be a s k e d t o :  In each of  what  in  study,  were  Further, using a larger  type of  teacher  thoughts  l a i d open i f  t e a c h i n g m a t h c a n be  example,  their  This strongly  i n some c o m m o n a l i t i e s  i n a d d i t i o n to the  encounters  i d e n t i c a l teacher  were d i f f e r e n t .  Ideas about techniques  through the  individuals.  this  t e a c h i n g mathematics  ideas  consisted of d i f f e r e n t c o u l d have  of  sample f o r  about what can  the  educators  can  existing  but u l t i m a t e l y ,  helping  and a p p r o p r i a t e c o n c e p t i o n s  the of  mathematics.  Longitudinal  studies  be carried  out for individual  teachers.  - 81  -  pre-service  and  Brown a n d B o r k o development However,  (1992)  h a v e n o t e d t h a t most s t u d i e s  have used s e v e r a l  teachers rather than a s i n g l e  l o n g i t u d i n a l investigations during the teacher  program s u c h as W i l s o n ' s  (1994)  andb e l i e f s .  analysis  o f interviews and j o u r n a l w r i t i n g s of teacher  "Snapshots"  of conceptions  as u s i n g methods m e n t i o n e d above,  program.  teacher.  preparation  i n q u i r y would h e l p i d e n t i f y  thoughts  well  on teacher  evolving  c o u l d be t a k e n  candidates,  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  experience,  conceptual development,  conceptions  relate  to her actions  factors  (1985)  a n d B r o w n (1985)  t e a c h i n g mathematics Longitudinal  provide Owens  i n f l u e n c i n g i t , a n d how  a novice teacher  studies  of teacher  f o r example,  development  should include  as p a r t o f t h e i r framework.  a more h o l i s t i c c o n t e x t  These  used P e r r y ' s  (1970)  scheme o f i n t e l l e c t u a l  prospective  teachers.  prospective  teachers rather than using i t i n a l o n g i t u d i n a l  of authority to investigate  the beliefs of  However, he t o o a p p l i e d i t t o a s m a l l sample o f context.  of conceptions of teaching mathematics be  out with other populations Knowing s t u d e n t s ' provide  would  f o r u n d e r s t a n d i n g changes i n i d e a s .  and the construct  Investigations  studied by  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  development  4.  One  c o n f l i c t s with that of h i s students.  o f a d u l t development  (1987),  teaching  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 .  example o f such a s t u d y i s t h e case o f F r e d ,  theories  as  at various times during the  p r o g r a m t h r o u g h t o when s h e h a s a c c u m u l a t e d some s u b s t a n t i v e  Cooney  through  of teachers and  learners.  conceptions o f teaching mathematics  insight into classroom i n t e r a c t i o n s .  - 82 -  carried  could  Thompson (1992) h a s  claimed that (1985) the  study,  the  t h i s type of  students'  s o c i a l i z a t i o n of  influential of  none o f  ideas  i n changing,  at  adults.  are  Typically,  Those  ideas  superficially,  of  were  found to  the b e l i e f s  and  in  be actions  level  o n l y at  content  to  a B . E d , degree and  the secondary  field  level.  will  These  several usually  K n o w i n g how  t o a d u l t s a n d how t h e y p e r c e i v e  they  their  important information.  ABE s t u d e n t s mathematics,  i n A d u l t B a s i c E d u c a t i o n (ABE).  I n s t r u c t o r s new t o t h e  t e a c h i n g mathematics  w o u l d be  ideas  lies  t h e i r background e n t a i l s  teaching experience.  conceive  their  i n Brown's  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  charged w i t h teaching p r e - c o l l e g e  teaching experience  role  least  However,  teacher.  instructors  have  were  a novice teacher.  My p a r t i c u l a r i n t e r e s t  years'  research exists.  b e g i n t h e i r program w i t h experiences  and t e a c h e r s  remembered f r o m h i g h s c h o o l .  of mathematics and i t s  of  school,  Elucidating  t e a c h i n g would p r o v i d e  knowledge  t h r o u g h w h i c h an i n s t r u c t o r c o u l d improve h e r i n s t r u c t i o n a l a p p r o a c h . International the  teacher  potential  students  bring different  to a Canadian classroom.  sources  f o r mathematics  serves personal teacher about  candidates'  t h e i r ideas,  conceptions of  teacher  development.  c o n s i d e r e d h e r own i d e a s  before  and  classroom.  t e a c h i n g math  education programs;  it  also  T h o s e who h a v e b e e n i n v o l v e d t h i n k  t h e i r minds p i q u e d by the r e s e a r c h e r ' 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  teaching  E x p l o r i n g these could uncover  f o r c o n f l i c t or misunderstandings i n the  Researching teacher y i e l d s knowledge  c u l t u r a l notions of  her i n t e r v i e w that  and t h e  - 83 -  questions. she had  never  i n t e r v i e w helped her t h i n k i n  different minds of  ways a b o u t the others  teaching. as  It  i s hoped t h a t  well.  -  84  -  this  study provoked  the  REFERENCES Ball,  D . L . (1988) . 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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 and pedagogy. J o u r n a l for Research i n Mathematics E d u c a t i o n , 25, 346-370. Z a s l a v s k y , 0. & P e l e d , I . (1996). Inhibiting factors i n generating 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 binary operation. J o u r n a l for Research i n Mathematics Education, 27, 79-95.  - 89 -  APPENDIX A "Views of t e a c h i n g and l e a r n i n g "  - 90 -  1  VIEWS OF TEACHING AND LEARNING:  Many people work on the building site. They are InvoK In clearing, carrying, building, planning and supervlsln  s  A piece of clay 13 being moulded by the potter.  LU  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 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.  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  ©  C h i l d r e n ' s Learning i n Science Project - 91 -  APPENDIX B Sample  -  questions  92  -  What d o e s 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 a s a math t e a c h e r ? A t t r i b u t e s o f a good math t e a c h e r  MATHEMATICS How w o u l d y o u characterize/describe math? What m e s s a g e 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 a n y o n e t e a c h math? What ( k , s , a) do t h e y n e e d ? 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 motto, metaphor diagrams (philosophy) Have y o u r v i e w s o f t e a c h i n g m a t h changed from the b e g i n n i n g of the  year?  - 93 -  APPENDIX C Sample notes  -  94  -  0  What  does  i t  mean  t o you  t o teach  math?  TEACHER How s  would  math  y o u  describe  your  role  a s  teacher?  Attributes  o f a good  math  teacher  MATHEMATICS How  would 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 do  T2)  message  you  want  about ma t o  give?  STUDENTS What Hew  i s a  student's  do they  lear.-. m  PEDEGOGICAL Cc. .  onyoriE  -  What What Cf  CONT. teach  subjects  PHILOSOPHY motto,  metaphor  d i a g r a m s — ^ ^ • <y>Aj-N***)^ (philosophy) Have  your  changed  V>  views  from  o f teaching  t h ebeginning  KNOWLEDGE math?  (k, s , a) do they  a r ee f f e c t i v e  other  0  math o f t h e year?  95  teaching  need? s t r a t e g i e s ?  'a-  Appendix D Sample T r a n s c r i p t  Excerpt  T h i s e x c e r p t i s t a k e n from pages 5 t o 9 o f the 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 a n d I r e s p e c t i v e l y . I:  Ok, o k .  Now, y o u h a v e a d e g r e e  V:  Um, my s e c o n d d e g r e e ,  I:  Ok, w h a t ,  what k i n d o f ,  i n mathematics?  yep. what's  your academic  background?  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 Honours 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 .  V:  3-year.  I:  3-year.  V:  4-year.  I:  4-year?  V:  Um, m a t h e m a t i c s .  I:  Ok,  V:  Yeah.  so,  i n Economics i s a 3 o r  And your B . S c .  Ok.  4-year?  Honours?  A n d what was y o u r m a j o r i n Um, o t h e r o n e ' s ,  that?  economics.  r e a l l y you 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 .  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 (laughter)] 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 h a v e t o do a b i t more o f . . . . 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  would you use t o d e s c r i b e  student  What do y o u mean  mathematics?  V: Ok, I ' m g o i n g t o s t a r t a n u s e , I ' m g o n n a 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 mathematics t o anybody w a l k i n g on the s t r e e t and t h e y can 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 the-dots. 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. We're 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 , a n d 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 , a s a n u n d e r g r a d y o u h a v e 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 , you're doing a proof? 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 continuous, prove i t ' s d i s c o n t i n u o u s . You c a n ' t h a v e s o m e t h i n g t h a t ' s w e l l , i t ' s sort of continuous i n t h i s area. I t ' s either continuous or not. 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 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 not y e t , ok? W e ' r e t r y i n g t o p r o v e i t , a n d 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 a n d s o 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 1  -  97  -  happen t h e n prove i t . Um, a n d t h e y d o n ' t k n o w . 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 math, [ I : oh really?] 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 . 'Cause 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 k n o w n , o k ? Y o u 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 g u y s i n t h e f i e l d . Um, a n d i t ' s sort of, a very, i t ' s , i t ' s , i t ' s , you're just s t i l l discovering, you're s t i l l c r e a t i n g i t , ok? I t ' s s t i l l being created. 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 m a t h 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 been here 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 grandparents d i d t h i s . N o t h i n g ' s changed. Whereas k i n d o f , n o , i t ' s not t h a t , i t ' s changing a l l the time. I: Is i t changing at the l e v e l that the students are math o r , o r c a n y o u c o n - , convey t h a t t o t h e s t u d e n t s ? t o c o n v e y t h a t t o them?  l e a r n i n g about Is i t reasonable  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 t h e m 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 a s 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 h a v e 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 something because you punch i t i n t o your 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 , s o t h a t , s o 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 , o k ? W h e r e a s we u s e d t o h a v e 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 " d o 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: ok...] W e ' r e n o t , w e ' r e not a s k i n g , w e ' r e n o t , w e ' r e not 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 on 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 o n e 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 do, ok? And i t ' s the same t h i n g a s 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 n u m b e r s 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 w h e r e t h e y ' r e a t i n m a t h 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 . I t h i n k you can 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 equations of l i n e s changing desperately. 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 c h a n g e h a p p e n i n g there. The c h a n g e s i n how m a t h 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 based. 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 you can add t h a t k i n d o f p e r s p e c t i v e on i t . Because a l o t of - [ l a u g h t e r ] no, 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 an 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 . [laughter] C a n 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 to continue, of course, that teaching aspect. A n d I ' d l i k e t o go b a c k to the f i e l d of mathematics. V:  Ok,  I've  s i d e t r a c k e d on  that.  - 98 -  I: No, no, 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 m a t h 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, a n d 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 of the hypothesis. T h e n y o u do t h e p r o o f , a n d t h a t ' s creating. 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 between them. 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 , o k ? 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: For example, level...  algebra,  [V: um -]  some f i n i t e m a t h a t  the  basic  V: At the 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 . We're t r y i n g , we're 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 , s o when t h e y need t o use i t t h e y can j u s t p u l l t h i s t h i n g out and go, g r e a t t h i s i s a t o o l I need t o use. We're c r e a t i n g t o o l s , b a s i c a l l y . I:  Is  that  the purpose of,  of developing  mathematics?  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 m a t h 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 mathematicians are at a l l u s e f u l unless they a c t u a l l y create something, ok? I ' m not s a y i n g t h a t every m a t h e m a t i c i a n ' s out 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 h a v e 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 get f u n d i n g and p e o p l e w i l l k i n d o f go, 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:  Let's  t a l k a b o u t humans a n d m a t h .  V:  [laughter]  Ok!  I: What d o e s i t t a k e t h i n k i t ' s d i f f e r e n t at V: Oh, d e f i n i t e l y . j u s t g o i n g to look at memorization.  f o r someone t o be g o o d a t math? And i f d i f f e r e n t l e v e l s , please say so.  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 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  -  99  -  you I'm  

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