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Perceptivist mathematics education Copeland, Brian Dwight 1989

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PERCEPTIVIST  MATHEMATICS  EDUCATION BY B.Ed, 1982,  B R I A N DWIGHT C O P E L A N D M.A., T h e U n i v e r s i t y o f B r i t i s h 1985.  Columbia,  T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF THE R E Q U I R E M E N T S FOR THE D E G R E E OF M A S T E R OF A R T S .  IN THE F A C U L T Y OF G R A D U A T E S T U D I E S (DEPARTMENT OF  We  accept  EDUCATION)  t h i s t h e s i s as conforming required standards  THE U N I V E R S I T Y OF B R I T I S H A u g u s t 198 9 © Brian  Dwight  Copeland,  to the  COLUMBIA  1989  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  University  of  British  Columbia,  available for  copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and study. scholarly  or for  her  purposes  representatives.  financial gain  of  Mathematics  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  October  2 ,  1 9 8 9  the  that  agree  may  be  It  is  shall not  Education  requirements  I agree  I further  permission.  Department  of  that  the  Library  an  advanced  shall make it  permission for  granted  by  understood be  for  allowed  the that  without  extensive  head  of  my  copying  or  my  written  i i  ABSTRACT The  aim  of  perceptivist  this  paper  i s to  philosophy  constructivitist  theory  of  of  knowledge  i n secondary mathematics  summary  of perceptivism  a program  the  context The  and  main  thesis  of  i n mathematics  lead  to  actual  with  this  that  the  i s  to  that  deal  makes  a  now,  act  as  transformer" real,  lessons  with.  where  constructivism  The  i t was  an to  to  a  provided within  ideas. the  way  to  ideas  into  activities  that  traditional  problem abilities  are  for  too  much  mathematics  age  The  processing past  more  previously, i s  computer.  students  most  innovation  education  possible  information boost  a  working  i s  i s that  information  not  and  computational  of the personal  computer "step  computation  up to  mathematics.  practical aimed  the  reality  utilitarian  perceptual The  paper  The  a  are discussed  i s t o emphasize  often  with  development  can  this  perceptions.  to deal  possible  A  constructivism  perceptivist-constructivist  practice  students  education.  lesson materials  apply  acquisition  of p e r c e p t i v i s t - c o n s t r u c t i v i s t  translate  needed  and  and  education  problem  and  outline  part  at a p a r t i a l i n the  of  the  paper  realization  classroom.  The  of  consists  of  perceptivism-  lessons  concern  concepts  and  mathematics elementary concludes in  the  skills  from  curriculum function  with  secondary  traditional  areas;  theory  a report  the  on  classrooms  arithmetic,  and field of  calculus. t e s t s of  the  author.  the  secondary algebra, The  paper  materials  iv  TABLE OF CONTENTS Page ABSTRACT  i  L I S T OF T A B L E S  v i  L I S T OF F I G U R E S CHAPTER I : A PROBLEM I N MATHEMATICS EDUCATION The n e e d f o r an e m p h a s i s on a p p l i c a t i o n s M a t h e m a t i c s must become more m e a n i n g f u l The c o m p u t e r - a r e s o u r c e t o b e u t i l i z e d P h i l o s o p h y c a n h e l p show t h e way  i  v i i . . . .  . . . .  . . . .  1 6 8 10 13  CHAPTER I I : A BRIEF D E S C R I P T I O N OF A P A R T OF PERCEPTIVISM Orientation t o the world Orientation t o being educated Education rooted i n perception Ways o f s e e i n g P e r c e p t i v i s m and mathematics education . . . .  15 15 20 26 29 34  CHAPTER I I I : A BRIEF DESCRIPTION . CONSTRUCTIVISM Knowledge i s a c t i v e l y c o n s t r u c t e d A l l experience i s theory laden The r o l e o f a l t e r n a t e c o n c e p t i o n s  37 37 40 42  OF  C H A P T E R I V : A N I N T E R P R E T A T I O N OF P E R C E P T I V I S M . . . The s u b s t a n t i v e r o l e o f t h e c o m p u t e r Trajectories using t h e computer t o do mathematics Extension of the concept the t r a j e c t o r y envelope Perceptivist-constructivist modelling . . . . Computer assisted exploration of the exponential function F u n c t i o n s as models o f r e a l i t y  46 47 50 53 60 66 72  C H A P T E R V: A N I N T E R P R E T A T I O N OF C O N S T R U C T I V I S M . . 93 The a c t i v e c o n s t r u c t i o n o f m e a n i n g 93 A p p e a l i n g t o e x i s t i n g knowledge 100 Alternate conceptions and conceptual conflict 103 Summary 105  V  CHAPTER V I : P E R C E P T I V I S T - C O N S T R U C T I V I S T CALCULUS ON THE COMPUTER Rationale Numerical d i f f e r e n t i a t i o n Numerical integration  108 108 112 119  CHAPTER V I I : REPORT ON A FIELD CALCULUS UNIT Purpose o f t h e f i e l d t e s t I n s t r u c t i o n a l sequence F i e l d notes Questionnaire results  132 132 134 136 140  TEST  OF  C H A P T E R V I I I : C O N C L U S I O N AND RECOMMENDATIONS Conclusions Suggestions f o rf u r t h e r research Recommendations BIBLIOGRAPHY APPENDIX  . . .  THE  . . . 148. 149 153 156 161 171  vi  LIST OF TABLES Page Table  1.  B r a u n e r ' s o u t l i n e o f t h e ( n i n e ) modes o f perception. (Brauner, 1986, p.13) . . .  25 86  Table  2.  Decay  Table  3.  World population figures i n h i b i t e d growth model  Table Table Table Table Table  4. 5. 6. 7. 8.  o f carbon-14  Disease spread i n h i b i t e d growth Student Calculus  i n a model  of  Analysis students  of  mean  the 88  city  Questionnaire Unit  Analysis question  for using  the 90  -  Computer 143  response  to  each 144  responses  Computer Math: activities  by  individual .147  Possible  program  of 160  vii  LIST OF FIGURES Page Figure  1.  Orientation to the world  19  Figure  2.  Hofstadter's depiction of Complement a r i t y : s e e i n g i n d i f f e r e n t ways . . .  32  Figure  3.  Computer  used  to  amplify  power  of  concepts  49  Figure  4.  Trajectories  Figure  5.  Lob v s . C l o t h e s l i n e T r a j e c t o r y  Figure  6.  The  Figure  7.  Hits  Figure  8.  Kigure  9.  Hitting Specific Locations Inside the Envelope ; Perceptivist-Constructivist modelling; d i s t a n c e f a l l e n as a function of time elapsed since dropped  61  "Periodic" Properties  Variation  63  11. " P e r i o d i c "  Variation  Figure Figure  10.  envelope Inside  .  51  of trajectories  . . . .  53  . . . . .  54  the Envelope  55  of of  Chemical Cultural  Development Figure  12.  Figure  13.  Figure  Figure  Figure  57  64  "Periodic" V a r i a t i o n i n Heartbeat  . . .  65  "Periodic" Variation in Sunspot Activity 14. E x p o n e n t i a l Function and Population Growth (Smaller growth r a t e corresponds to lower curve)  68  15. E x p o n e n t i a l Function and Inflation (smaller i n f l a t i o n rate corresponds to lower curve)  70  16. E x p o n e n t i a l Interest "  71  Function  and  65  Compound  v i i i Figure  17.  Bouncing b a l l ,  rapid  decay  Figure  18.  Bouncing b a l l ,  slow decay  Figure  19.  Water  Figure  20. A v e r a g e  depth v a r i a t i o n value  of  a  21. Average  Figure  22.  Figure  75  in tidal  cycles  function  computer Figure  75  on  77  the  • tidal  .  77  depth  78  Terminal Velocity in Motion Resistance . . . 23. P o s i t i o n as Function of Time  with with  80  Terminal Velocity  81  Figure  24.  Damped O s c i l l a t o r y M o t i o n  83  Figure  25.  World population  84  Figure  26.  014  Figure  27.  Figure  28.  World population, model Inhibited growth spread  Figure Figure Figure Figure Figure  Figure  29. 30. 31. 32. 33.  34.  decay  projections  .  Percentage buying growth Computer equations  solution  Computer equations  solution  87 inhibited  growth 88  model  of  disease 90  a  product,  inhibited 90  of •  degree  of  degree  one 97 two 97  Computer equations  solution  Computer equations  solution  Computer equations  solution  of  degree  three 98  of  degree  four 98  of  degree  five 99  ix Figure  35. Computer  solution  equations  of  degree  six  . . . .  99  Figure  36. G r a p h i n g  Figure  37. S l o p e by s e c a n t  Figure  38. Time v s . p a t i e n t t e m p e r a t u r e  Figure  39. Time v s . c i t y ' s  Figure  4 0 . D o l l a r s s p e n t o n a d v e r t i s i n g v s . T.V. sets sold 41. Time v s . f r a c t i o n o f d o c t o r s a c c e p t i n g a medicine  Figure Figure  42. W i d t h  a face  of  rectangle  103 approximation  113 . . . . . .  population  rectangle  115 115  vs.  area  formed  116 117  of 118  Figure  43. Time o f y e a r  vs. sales  119  Figure  44. y = x  Figure  45. Month v s . s a l e s  Figure  46. Time v s . v e l o c i t y  Figure  47. Time v s . R a t e  of Energy  Use  (4 H r s )  Figure  48. Time v s . R a t e  of Energy  Use  (24 h r s ) . . 124  Figure  49. A r e a between two c u r v e s  12 6  Figure  50. A v e r a g e  127  Figure  51. Volume o f r e v o l u t i o n  120  2  value  i n that  month  122 123  of a function  . . 124  128  1  CHAPTER I A PROBLEM IN MATHEMATICS EDUCATION Mathematics trouble.  Compared  societies  no  theory  we  establish  that  available major  this  —  purpose  ways  industrial  i n North  America i s  Since  i n particular.  i s to  a guiding philosophy  i s to establish that  by  the philosophy  i n which  way  and provide  rationale  computer  unit  This  be  used  thesis  mathematics calculus unit.  to  lessons A field  used  to  i s now second  o f argument  and  suggestions might  be  to  generate  f o r ways  i n which  improve  presents  or  mathematics  various and  test  a  computer computer  of the calculus  i s r e p o r t e d and d i s c u s s e d . This  centered and  can  A  education  suggestions  integrated  arguments  provides  mathematics  a  whole  of education  i t c a n be  integrated  a  One o f t h e m a i n  provide  In particular  education.  as  Without a guiding  improved.  the  Dewey t h e r e h a s  t o do b e t t e r .  thesis  i s i n  leading  Perceptivism-constructivism.  demonstration about  America  i n education  are not l i k e l y of  other  achievement  theory  education  purposes  North  (McKnight, 1987).  guiding  mathematics  i n  to  mathematics  •lagging behind been  education  lack  chapter around  i s concerned with declining  of effort  i n using  a group  achievement t h e computer  i n  of problems mathematics  as a  resource  2 to  remedy t h e p r o b l e m .  mathematics beyond  mere  without  a  The  problem  performance  kindergarten  computation,  i s especially  guiding  exploited  increase  meaningful mathematics. i s developed  in five  poor  performance  students  i s discussed.  one  suggestion  performance. is  to  emphasize  details.  (d)  development relieving (e) in  the  I t i s then  of  argued  that  be  achievement  and  suggestion  i s  the  The  this  details  secondary  to  i n the of  to  literature technical assist  the  meaningfulness  burden  of  philosophy  can  exploited  by  computation, suggest  from the  to  i s  improve  the  able  and  in  applications  the computer gives  can  the  literature  computer  Second,  by  on  12,  that  (a)  meaningfulness  student  computation  mathematics  stress  applications  which the freedom  of  (b) A  grade  argument  stages.  the  the  improve The  First,  poor.  mathematics  second  The  of  in  in  (c) A  to  to  i t i s unlikely  can  of  be  philosophy  computer  chapter  i s twofold.  help  ways  burden improve  performance.  Second r a t e s k i l l l e v e l s High with  school  the  technological  graduates  mathematical society.  are  not  equipped  complexities  Henry  Pollak  to  of  (1987),  deal modern  a  noted  3 industrial industry include the  mathematician,  expects the  ability  ability  techniques  mathematical ability  the  ability  solve  to  an  attitude  work  complex  formulated  form.  calculation  does  or  problem,  The not  present  provide such  way  adequately prepare  for  further  Usiskin  (1985)  education.  abstract  situations,  and  Students of  a belief  render  to  problem  concepts  toleration  techniques to  mathematical  world  on  situations.  and  of  ability  real  These  mathematically,  repertoire the  skills  t o have.  problem  cooperatively  formulated problems  i n any  up  problems,  disposed to  mathematical  place  set  t o see m a t h e m a t i c a l  common a n d  or  to  what  employees  f o r m u l a t i o n s from  in  well  level  to apply a f l u i d to  the  entry  summarized  solving, principles  also  open  needed  ended,  not  i n the u t i l i t y  problems regimen skills  into  of  a  low  and  well level  abilities  students f o r the Commenting  of  on  remarks:  The biggest problem in secondary school mathematics today is recognized by a l l , regardless of feelings toward new math or back t o b a s i c s . I t i s t h a t a l a r g e number perhaps a m a j o r i t y - of high school graduates lack the mathematical know-how to cope e f f e c t i v e l y i n society, q u a l i f y f o r the jobs they would l i k e , or q u a l i f y f o r the t r a i n i n g programs ( i n c l u d i n g those i n college) l e a d i n g to the jobs they would l i k e . ( U s i s k i n , 1985, p.8)  work this  4 College level  of  Leitzel  and  university  college  and  Osborne  officials  note  entering  mathematical  summarize  i n these  the  poor  competence.  terms.  A b o u t 45% of the graduating s e n i o r s i n the United States enter some form of post• secondary education, and a b o u t 45% of this group lack the mathematical skills and understandings needed f o r success i n postsecondary school mathematics. ( L e i t z e l and O s b o r n e , 1985, p.150) A  survey  of  undergraduate  computer  science  enrolment  in  colleges the  is  enrollment  presumably Anderson  is  have  been  national  showed t h a t  historically  of  students apply  (Carpenter, Swafford, industrial  do  courses  low  level  current two  Another  courses  i n high  year 37%  that  of  should  school  (Albers,  the  National  in  in  mathematics  computational  levels  skills  but  basic  skills  that  a  concepts  problem-solving  Kouba,  Linquist,  Internationally/  nations,  of  public  Progress  understand  1987) .  at  (U.S.A.),  not  Brown,  15%  and  1987).  level  reasonable  mathematics  mathematics.  Educational  at  that  completed  a  of  in  for precalculus  Loftsgaarden,  Assessment  they  remedial  and  On  clearly  documented  mathematics for  programs  particularly  majority nor  Silver  in  could  situations  compared those  were  with the  and other  Orient,  5  average  North  (McKnight, and  American  Crosswhite,  Cooney,  students  Dossey, K i f e r ,  1987).  "underachieving  are not  Swafford,  McKnight  curriculum"  competitive  notes  expects  Travers  how  less  of  our our  students, has them spend l e s s time s t u d y i n g mathematics and has fewer students e n r o l l e d i n advanced mathematics than  students  Assessment lack  of  i n other  countries.  o f Mathematics mathematical  The I n t e r n a t i o n a l  and Science  ability  found  compared  a  to  similar leading  jurisdictions. F o r t y percent o f Korea's 13-year-old students understand measurement and geometry concepts and are. s u c c e s s f u l at s o l v i n g even more complex problems. Less than 10 percent of those from O n t a r i o (French) and the U n i t e d S t a t e s have the same l e v e l s k i l l s (Lapointe, Mead and P h i l l i p s , 1989, p.10). There students  we  mathematics Skills students  is  reason  "pass"  to  that  of  a l l the  i n s c h o o l , many do not get a good  education.  Placement  believe  The New  Test  e n t e r i n g New  private institutions.  is  Jersey  given  Jersey p u b l i c  to  College  Basic  virtually a l l  c o l l e g e s and some  A f t e r a n a l y z i n g the 1984 r e s u l t s  the C o u n c i l r e p o r t e d : Of those students who completed a t r a d i t i o n a l c o l l e g e p r e p a r a t o r y program i n mathematics (Algebra 1, Geometry and A l g e b r a 2) , l e s s  6 t h a n 4 p e r c e n t were p r o f i c i e n t i n e l e m e n t a r y algebra. E v e n more s t a r t l i n g o n l y 36% of these students were proficient in computation. A m o n g s t u d e n t s who completed o n l y one y e a r o f a l g e b r a (N=2,030), o n l y one s t u d e n t was proficient. ( M o r a n t e , F a s k o w a n d M e n d i t t o 1 9 8 4 , p . 30) Voelker  (1982),  science,  in a  stated  that  graduates  fail"  literacy.  Passing  understanding North (1989)  the  "90  to  courses  high  of  a l l high  school  not  school  for seem  last  place  in  nations  of Education  (1989).  similar  deficiencies  among  performance  has  already  be  entirely  not  scientific to  equate  s e n i o r s have  industrialized  students.  in  school  criteria does  and  chemistry  may  secondary  Department  indicate  that  of  to  on.  12th  among  study  percent  meet  what went  fallen  U.S.  to  American  achievement  similar  North fallen  coincidental  year  mathematical according Other  to  studies  biology  American behind  this  and  industrial  Japan's, w i t h our  a  fact  problems  education.  The  need f o r an emphasis on The  need  more r e l e v a n t , long  time.  to  move  applied  Carson,  mathematics format  i n 1913,  Among the many education during  applications  has  education  toward  been r e c o g n i z e d f o r a  wrote:  changes the last  a  in mathematical twenty years  7 one e l e m e n t a t l e a s t a p p e a r s t h r o u g h o u t ; a desire to relate the subject to r e a l i t y , to e x h i b i t i t as a l i v i n g body o f t h o u g h t w h i c h can and does influence human life at a multitude of points. (Carson, 1913, p.35) The  198 9  (NCTM)  National  Standards  education attempt  need  Council  of  Document not  to refine  get  Teachers  emphasized bogged  of  Mathematics  that  mathematics  down  in  an  endless  computation.  It i s a common a s s u m p t i o n t h a t mathematics computations are necessary before one can study algebra or geometry or investigate applied problems. This assumption i s not warranted. Too many s t u d e n t s a r e r e f u s e d a n opportunity to learn the mathematics that would make i t possible f o r them to be p r o d u c t i v e members o f s o c i e t y b e c a u s e they are not p r o f i c i e n t at s k i l l s which are now done b e s t on a c a l c u l a t o r o r c o m p u t e r . (NCTM, 1 9 8 9 , p.30) Similarly,  Ralston  spend  refining  are  time  not  shifted  required. to  integrating  making and  argues  that  there  i s no  computational s k i l l s Rather,  time  and  mathematics  expanding  an  reason  i f the  effort  to  skills  should  be  intellectually  activity.  The focus of a l l mathematics teaching must soon become t e a c h i n g s t u d e n t s t o u n d e r s t a n d mathematics rather than teaching them to m a n i p u l a t e symbols; t h a t ' s what t h e computers are f o r . ( R a l s t o n , 1985, p.39)  Fey  a n d Good  higher  up  focussing skills,  comment i n a s i m i l a r v e i n on  cognitive  predominantly  leaving  technicalities evolve  the  scale.  low  the intellectual t o chance,  toward  communicating  on  becoming  a  level  means  we m u s t a i m Rather  than  computational  meaningfulness  mathematics  about t h e r e a l  that  ofthe  education  must  of understanding  and  world.  It should be p o s s i b l e f o r every teacher t o p l a c e g r e a t e r s t r e s s on s i t u a t i o n s i n w h i c h mathematics i s used t o model t h e s t r u c t u r e o f real-life situations. There i s a growing supply o f resource material f o r t h i s purpose and i t suse w i l l force attention to important themes i n t h e c u r r i c u l u m of the future. (Fey a n d Good, 1 9 8 5 , p.52)  Mathematics must become more meaningful Preoccupation the  broader  students it  students A  natural.  context  the material  o f mathematics, whether  they  forces  understand  I t d o e s n o t seem u n r e a s o n a b l e t o t h i n k  would  cynical,  c a l c u l a t i o n , a t t h e expense o f  intellectual  t o cover  or not.  with  quickly pick  mechanical  Nigel  Ford  u p o n how t o p l a y  approach  to learning  that  t h e game. would  claims:  There i s i n c r e a s i n g support f o r t h e idea t h a t t h e way s t u d e n t s t h i n k , b e l i e v e a n d v a l u e i s by no means synonymous w i t h e x p o s u r e t o , a n d performance of, exercises i n the comprehension, recall and manipulation of  seem  9 information and ideas. Most course-based activities certainly do not preclude the possibility that i n terms of a student's personal acceptance and valuing of, and commitment t o i n f o r m a t i o n and i d e a s , he i s doing no more than "going through the motions" w i t h shallow and short-term, i f any, effects. (Ford, 1979, p.215) Roy  Forbes  notes  Educational "back  to  that  Progress basics"  the  (NAEP)  Assessment  1979 r e s u l t s  indicate  mathematics  preceeding  1979  expense  understanding.  of  National  maintained  emphasis  low  level  i n  the  skills  of that  years  at  the  D u r i n g a p e r i o d when t h e p u b l i c h a s p l a c e d great e m p h a s i s on t h e " b a s i c s , " assessment data show t h a t m a t h e m a t i c s a c h i e v e m e n t h a s declined, especially i n problem-solving and understanding of concepts. (NAEP, 1 9 7 9 , p.7) Present mathematics emphasis  relating  education on  meaningful Standards  emphasis  on back  extensive  mathematics Document  lists  reform from  includes  a curriculum  computation for  four  driven  toward  students.  at least  bringing by  more  The  NCTM  important  goals  t o b r i n g i n g mathematics back t o t h e  student.  • Mathematics should be studied as an integrated whole so that students u n d e r s t a n d i t as a dynamic d i s c i p l i n e and an i n t e g r a l p a r t o f o u r c u l t u r e .  • Mathematics should a b i l i t i e s t o reason • Mathematics context.  help build logically.  should  be  taught  students'  i n a  natural  • Students s h o u l d be encouraged t o c r e a t e , invent, and p a r t i c i p a t e . (NCTM, 1 9 8 9 , p . 3 5 ) Only  by m e a n i n g f u l l y  possibility motions" this  of  involving  discouraging  just  and o f encouraging  way  can  "underachieving  a  the student, "going  actual  "nation  at  i s there through  learning.  risk"  curriculum", especially  the  Only  revitalize  a  i n  i t s  i n mathematics.  The computer - a r e s o u r c e t o be u t i l i z e d The in  computer  mathematics  claim  that  seems  education.  i s greater  including  books  piece  realized. potential will  of  than  that  and  Educational but has  not develop  of  and w r i t i n g . "  technology  had  an  enormous  Walker  "the potential  education  a  t o have  (1984)  computers  makes  for  o f any p r i o r  i t spotential held  effect.  i t s own u s e a s d e C e c c o  The  the  improving  invention,  But t h e computer  television  little  potential  may  i s just not  be  tremendous technology  remarks:  A l l these f a c i l i t i e s and equipment [computer, C.A.I.], I must r e m i n d y o u , a r e much more s o p h i s t i c a t e d t h a n a n y t h e o r y o f t e a c h i n g we presently have. The temptation i n a technological society i s to allow our f a n t a s t i c machines t o determine our research problems and our e d u c a t i o n a l p r a c t i c e . Iti s f a r m o r e i m p o r t a n t t h a t we s u b o r d i n a t e these  11 machines to the theoretical and practical instructional problems which, undoubtedly, the machines can h e l p us s o l v e . (de C e c c o , 1 9 6 8 , p . 4 1 8 ) By of  relieving  the learner  computational  burden  the  potential  f o runderstanding.  to  discovered  have  exception Standards  of  the  Document  o f a c o n s i d e r a b l e amount  this  computer  The r e s t fact,  education  expands  of society  with  the  community.  the seems  apparent The  NCTM  says:  Most current mathematics programs fail to reflect t h e impact of the technological revolution affecting our society. The availability of low-cost calculators, computers and r e l a t e d new t e c h n o l o g y have already dramatically changed t h e nature o f b u s i n e s s , i n d u s t r y , government, s c i e n c e s and social sciences. Unfortunately, most students are not educated t o p a r t i c i p a t e i n t h i s new s o c i e t y . (NCTM, 1 9 8 9 , p . 2 5 ) Released  from  t h e burden  would be a b l e t o p u r s u e personal  relevance,  intellectual literacy. examples  of  computation,  student  t h e meaning o f mathematics, i t s applicability  u s e f u l n e s s , as t h e medium Usiskin  the  claims  o f how t e c h n o l o g y  that  and of  history  and techniques  general  quantitative  provides liberate.  The h i s t o r y o f m a t h e m a t i c s i s f i l l e d w i t h t h e development o f techniques t h a t take d i f f i c u l t p r o b l e m s and. make t h e m a u t o m a t i c . Usually these automatic procedures provide a benefit; with tedious calculations o u t o f t h e way,  many  understanding the mathematics i t s e l f  The could  freedom  be  were  in  the  needed  pre-information  i n which t o be  curriculum  the  will  by  of  computer  mathematics. degree  culture, a  low  they  before  reworking  importance of Use  the  the  to the  age  force  reassessed. on  given  applications of  are not  Technology  curriculum have  explore  skills  the  computers.  will  from c a l c u l a t i o n  used.to  Computational  i d e a s and purposes o f becomes e a s i e r . ( U s i s k i n , 1985, p.16)  of  level  computers  the  skills  may  turn  i t s head.  For nearly every function of i n t e r e s t , the computer utilities make a l l ... questions a c c e s s i b l e i n some i n t e l l e c t u a l l y h o n e s t a n d mathematically powerful form to students who have not f o l l o w e d t h e c o n v e n t i o n a l regimen o f s k i l l development. They open a f a s t t r a c k t o the polynomial, trigonometric, exponential and algebraic functions that model interesting phenomena in the physical, biological, economic and social worlds. C o m p u t i n g o f f e r s an o p p o r t u n i t y t o t u r n t h e secondary school mathematics curriculum on i t s head. I n s t e a d o f m e e t i n g a p p l i c a t i o n s as a reward f o r years of preparation, students can now begin with t h e most natural and motivating aspect of mathematics — i t s applications. (Fey and Good, 1985, p.48) What i s n e e d e d , given  by  the  that  shows  t o i n d i c a t e w h a t t o do computer,  where  to  i s  a  look  with  philosophy for  the  the  of  freedom  education  meaning  of  13 mathematics.  In education,  this  i s a t hand  i n t h e form  of Perceptivism-Constructivism.  P h i l o s o p h y can h e l p show the way Because for next  education  so l o n g  we  without  describes  have  has  been  any rudder  been  without  buffeted  from  t o steady  the situation  i n these  a  philosophy  one  fad to the  the course.  Brauner  terms:  Ever since the p u b l i c school stopped trying to implement t h e program d e r i v e d from t h e P r a g m a t i s m o f J o h n Dewey . . . t h e y h a v e b e e n without a working philosophy of education. That t h i r t y - y e a r i n t e l l e c t u a l drought has l e d to such a shortage o f working t h e o r i e s i n t h e area of teaching practises that policy considerations provide what little basis there i s for justifying the current curriculum and i t s methods. Indeed, t h e grounds f o r p u b l i c s c h o o l p r a c t i s e have been so parched f o r so long that a whole generation of teachers has grown to r e t i r e m e n t a g e w i t h o u t e v e r k n o w i n g j u s t how i t i s that a working philosophy of education can i n f o r m c l a s s r o o m p r a c t i s e s .. . (Brauner, 1986, p . l ) This  not to  blueprint. logically  say that  a  philosophy  I t i s n o t as i f t h e r e deduce  specific  of  education  i s some f o r m a l  educational  epistemological or axiological  What  considered  of  conditions  practitioner  for  needs  here  rational  good  reasons  way t o  practises  metaphysical, i s being  i s a  premises.  i s a looser notion, action.  A  from  that  reflective  f o r what he d o e s .  This  14 looser also  notion  consistent  practise.  foundation Meaningful,  new p r o g r a m s .  of  might  B u t what  many  i s needed i s a  systematic  to build  innovation i n  a p p l i e d mathematics,  very  may  provide  would  be  facilitated  a part  o f most  f o r and s t r u c t u r e o f t h e  different  depending  I I , the writer will that  computer  to  bear  of secondary  the theory  concern  an  o r i e n t a t i o n t o what  theoretical  foundation  suggestions  as  calculation  that  to  describe  on t h e p r o b l e m  improve  of  gives.  views  on  the  t h e program.  chapter  applicability  on w h i c h  f o r example,  be  Perceptivism  the  to  But t h e grounds  reasons behind In  theoretical  c e r t a i n p r a c t i s e s and that  philosophy  t h e computer,  program  other  p r a c t i s e s may b e s o u n d .  theoretical  by  that  be c o n s i s t e n t w i t h  existing a  means  the  two  o f how  parts t o use  meaningfulness  mathematics.  The t w o  an o r i e n t a t i o n t o t h e w o r l d  what  i t means will to  t o be  then do  be  with  the mathematical  educated.  used the  to  parts and This  generate  freedom  use of the  and  from  computer  CHAPTER I I A BRIEF DESCRIPTION OF A PART OF PERCEPTIVISM As  a  major  theory  parts;  orientation theory.  an  t o being  as they  Perceptivism  orientation  Chapter  insofar the  of education,  to  the  educated. and a  II  might  describes bear  has world,  moral  the  three  and  first  social  two  on t h e m a t h e m a t i c a l  an  parts  uses  of  computer.  O r i e n t a t i o n t o the world Perceptivism of  British  philosophy According off  Columbia  t o Brauner  impinge  signals.  (1985)  These  on  Any  impressions. others  and Some  anything  impressions  packaging  can  Brauner,  give  ways o f " s e e i n g impressions,  be  they  unnoticed, packaging  to  notions,  Such  formulae  ever  c a n become c o n c e p t s .  as  generate  subjective  things".  or  impulses,  received  left  rise  i f  gives  information  received  are  Private  exists  of these  can  impressions  Brauner's  t o the world.  that  contain  a l l signals  widespread p u b l i c usage for  orientation  organism,  to.  at the University  C.J. Brauner.  impulses  an  are attended  individualistic for  Dr.  C o n s t e l l a t i o n s or packages  that  of  by  developed  r e s t s on a c e r t a i n  impulses.  data.  i s being  achieve A  concept  16 i s a p u b l i c f o r m u l a f o r f o c u s i n g a t t e n t i o n on a l i m i t e d number o f c h a r a c t e r i s t i c s of the e n t i t y under c o n s i d e r a t i o n . ( B r a u n e r , 1986, p.17) By  using publicly  users In  available  can communicate  fact  the concepts  packaging  formulae  and "see t h i n g s "  language  i n similar  ways.  shape t h e p e r c e p t i o n .  The c o n c e p t s o f o r d i n a r y language are the i n s t r u m e n t s by which aspects of experience both r e a l and imagined a r e s i n g l e d out and imbued with significance. In a mature language setting concepts generate perception. ( B r a u n e r , 1 9 8 6 , p . 16) In of  a  Brauner's person  received. usually  Concepts, tested,  the world  obtained  Brauner's  being  agreement to  process impulses  available  the  chaotic. that  the  beneficial  Without be  to  publicly  are usually  would  regard are similar  perception i s a  attending  impressions.  concepts  this  human  selectively  widely  packaging  view,  ways  stability  This  author  the latter's  and of of has  views i n  Cassirer's.  The construction of our perceptive world begins w i t h such acts of dividing up t h e ever-flowing series o f sensuous phenomena. In t h e m i d s t o f t h i s s t e a d y f l u x o f phenomena there are retained certain determinate ( p e r c e p t i v e ) u n i t s w h i c h , f r o m now o n , s e r v e as fixed centers of orientation. The particular phenomenon could not have any  17 characteristic meaning except i f thus referred t o those centers. A l l further progress of objective knowledge, a l l clarification and determination of our perceptive world depends upon this ever p r o g r e s s i n g development. ( C a s s i r e r , 1923, p.165) The  fixed  centers  conceptual  systems.  perception  possible.  of  orientation These  Brauner  are  concepts  fixed  claims  centers  and make  that  Concepts, notions and d i s p o s i t i o n s are the prime instruments f o r s e n s i t i z i n g people t o p a r t i c u l a r constellations of impressions i n their physical, social, mental or moral surroundings. (Brauner, 1986, p.11) It best can  i s only  conceptual be  through  repertoire  meaningfully  schooled). individual understands.  t h e development  The  available  called  theoretical  determine Brauner  what  and use o f t h e  that  educated  (as  structures he  sees  an  individual opposed  used and  to  by  the  how  he  i s i n agreement w i t h C a s s i r e r .  T h e r e i s n o f a c t u a l i t y ... a s a n a b s o l u t e ... i m m u t a b l e d a t u m ; b u t w h a t we c a l l a f a c t i s always t h e o r e t i c a l l y o r i e n t e d i n some way, seen i n regard t o some ... context and implicitly determined thereby. Theoretical e l e m e n t s do n o t somehow b e c o m e a d d e d t o a "merely factual," but they enter into the definition of the factual itself. ( C a s s i r e r , 1923, p.475)  18 The  world  becomes  of  sensory  and  are  literally  because Brauner J.M.  they seems  "knowable"  conceptual blind  lack  to  only through equipment.  certain basic  be  substantial  in  modes  mediation  The  features  certain  to  the  uneducated  of  of  the  world  perception.  agreement  with  Jauch.  When we t r y t o u n d e r s t a n d n a t u r e , we should look at the phenomena as i f they were m e s s a g e s t o be u n d e r s t o o d . Except t h a t each message appears to be random until we e s t a b l i s h a code t o r e a d i t . T h i s code t a k e s the form of an abstraction, that i s , we c h o o s e t o i g n o r e c e r t a i n t h i n g s as i r r e l e v a n t a n d we t h u s p a r t i a l l y s e l e c t the content of the message by a free choice. These irrelevant signals form the "background noise" which w i l l l i m i t the accuracy of our message. But s i n c e the code i s not a b s o l u t e there may be s e v e r a l messages i n the same raw m a t e r i a l o f t h e d a t a , so c h a n g i n g the code will result i n a message of equally deep significance i n something that was merely n o i s e b e f o r e a n d c o n v e r s e l y : i n a new code a f o r m e r m e s s a g e may b e d e v o i d o f m e a n i n g . T h u s a c o d e p r e s u p p o s e s a f r e e c h o i c e among different, complementary aspects, each of w h i c h has equal claim to r e a l i t y , i f I may use t h i s dubious word. Some o f t h e s e aspects may be completely unknown to us now but they may reveal themselves t o an o b s e r v e r with a different system of a b s t r a c t i o n s . ( J a u c h , 1973, p.237)  In  Figure  Perceptivism's view  1,  the  writer  orientation  of mathematics,  to  offers the  a visual world.  summary  of  The  writer's  a v e r s i o n of mathematical  realism,  X energy I input  HETR PROCESSOR (wisdom).  SENSOR(S)  g sensing and relayin process I  •  i  PROCESSOR (knowing)  1  rt O  rt  photons, chemicals pressures, heat, sound etc. interface with organism  O  n  I—'  a  l-There is a tendency to perceive (a) What has been perceived in the past (b) What fits into a subject-field pattern j (c)What is in agreement with others , (d)What is important to the subject '-Errors seem to be unavoidable in the process .-Only parts of reality are relayed by "g"  ft  perception of reality depends on the interaction subject, object  gi  selected physical and logical elements of the input are coded and relayed on.  hi  p«rttf*'»«  r  energy **J  1: understanding takes place, input compared to stored patterns and models-response signal possible jt  analysis, evaluation, speculation, synthesis-reflection on accumulated thought and experience  20 implicit In  in  Figure  the  is  substance  1,  received  figure  as  elaborated  Y  gives  signals.  Signals  impressions.  The  dispositions  (impressions,  formulae  central  (concepts,  together  to  form  a  off  in  Appendix.  impulses  can  octagon  be  which  theories).  are  packaged  consists  notions)  conceptual  the  and  as  of  private  of  public  Concepts  can  build  system.  O r i e n t a t i o n t o b e i n g educated A by we  conceptual  setting  out  select  generated  of in  dispositions to  have  these  free,  content  of  just  formed of  signals.  notions  of  the  recognizable  culturally modes  as  grouping the  message. noise".  the  system  the  the  choice  perception.  so  perceptions  ways  part  and  and  of  the  so  come  i s .  If  publicly  individual a  among  Perceivers  of  develop  and  world  presupposes  delimited, of  way  repertoire  Perception  perception  Individuals  different  are  of  "background  in  conceptual  complementary  and  interaction  concepts.  but  mode  choice  are  the  a  perceive  private  verifiable certain  events  and  to  notions  form  the  interpretation  of  the  signals  Impressions  generates  conditions  partially  Irrelevant  system  has  relatively different have  the  21 capacity  to  different In be  see things  i n a  number  of  fundamentally  ways.  order  equipped  culture.  t o be c o n s i d e r e d with  educated,  the conceptual  Brauner  a person  world-builders  must  of the  claims:  Perceptivism i s built around t h e idea that t h e e d u c a t e d p e r s o n h a s a g r e a t e r command o f more worthwhile ways of perceiving things t h a n s o m e o n e who h a s n o t h a d t h e o p p o r t u n i t y t o g e t an e d u c a t i o n . Hence t h e p u r p o s e o f education i s t o equip the learner with the capacity to perceive things i n the different ways t h a t t h e most w o r t h w h i l e approaches t o understanding provide. (Brauner, 1988, p.2) Brauner (a)  has i d e n t i f i e d  The  first  perception. justified language  By  standard  on  a  mode  depicted  example,  Wayne  (c) of  The t h i r d  widely  i s  standard  accepted  and  and t h e o r i e s o f ordinary  i s generated,  f o r example,  seeing  politeness  i n  others.  i s mythic  perception.  By  drawing  and mythology,  generate  Gretzky symbol,  mode  modes o f p e r c e p t i o n .  perception  principles  folklore  sex  of  perception  elements  adolescent  basic  t h e most  table,  The s e c o n d religion,  mode  using  concepts,  recognizing (b)  basic  nine  mythic  as  hero,  Rambo  as  i s theoretic  the physical sciences  such  t h e images perception, Mick  Jagger  and for as  law and  order  man.  perception.  The  areas  as p h y s i c s ,  chemistry  and  biology  generate  understand example, energy  them  water  obscure  is  thematic  history  boiling  your  humanities,  develop  The themes  not t o t a l l y  serve the  of high  i s the Earth  Marxist  studies, their  as  often  points  surplus  of  view  value  Veblen's  theme  Hawaiian  (e)  The  fifth  Disciplines law,  suntans  such  political  referred  partially  through verifiable  the  of  equilibrium  use  uses  makes  of  shows  ostentation, perception.  education,  as t h e s o c i a l  theses.  repressed  as  which  sociology,  economics,  of the unconscious  expression economic  science,  makes i t  o f t h e worker,  thesistic  as psychology,  to collectively  perception  theory  i s  winter  then  f o r example,  consumption  i n B.C.'s mode  that  which  t o see b u s i n e s s as e x p l o i t a t i o n  up  major  unfalsifiable but  possible  of conspicuous  mode  as t h e  through  f o rthematic perception, of  around  philosophy,  grouped  accounts  are unverifiable,  theory  kinetic  rotating  such  who For  (d) T h e f o u r t h  Fields  arbitrary  as a b a s i s  those  i s t h e escape  o f t h e Sun.  classical  that  perception.  sunset  view  power  theoretic  perception.  and  themes.  o f such  achieve  particles,  to  yet  theories  geography, sometimes  sciences,  shape  quasi-quantitative,  For  example,  dreams  thoughts. t h e analogy  Freud's  viewable  Keynes  as t h e  theory  of  o f an e n g i n e  to  be  tuned,  R.P.M.  to explain  abstract  absurdum  variation,  primary  perception.  literature,  poetry  the  most  for  example,  retreat  Paul  recognizing  a  and  sculpture  arts  provide  perception  of  operational concepts how  i t becomes t o make  new  which  appearance,  perception.  arts  as ad  mode  i s  such  as  to  create  experience,  of imperialism i n d e p i c t i o n o f New to  i s primal  as dance,  Gogh s t a r r y s k y , a P i c a s s o  seventh  Melville's  t h e forms  change  o f human  i n i t srelation  such  as  machinery  u s e words  conception  mode  such  reductio  discursive  portraits  (h) T h e e i g h t h  nondiscursive  The  theatre,  Scott's  Puritanism  Dick.  work,  (g)  sixth  and t h e p e r c e p t i o n seeing  i n The R a j Q u a r t e t ,  England  Disciplines  example,  The  f a r reaching  ( f ) The  the conceptual  i n conversation.  the  t h e R.P.M., i n  cycle.  understanding f o r  t o increase  t o decrease  provide  formal  form,  functional  fuel  perception.  and mathematics  needed t o a l l o w  primal  less  more  the business  i s relational  logic  of  giving  and sometimes  order mode  sometimes  life  i n Moby  perception.  The  music,  p a i n t i n g and  allow  t h e renewed,  f o r example,  a  Van  face.  ( i ) T h e n i n t h mode i s  Through  the use of t e c h n i c a l  possible ones,  to  how  see  how  mechanisms  t o f i x o l d ones, f o r  24 example, flat  how  to f i xa  1 i s Brauner's  perception. types  F o r each of concepts  generate  a  includes  an  Mathematics relational axioms,  perception  obscure  of the nine  are given.  type  of  of  each  a  might  concepts Brauner  of  perception.  perception.  The  refers are the  of the various  branches  geometry,  An example  of  of  analysis, relational  be t o see a m a t h e m a t i c a l f u n c t i o n i n a raw  data  or  expressed  t h e m a t h e m a t i c a l mode  a  the sources  These  type  algebra,  and s t a t i s t i c s ) .  modes o f  perception.  relational  and proofs  (arithmetic,  the learner  promotes  most  to repair  of perception  used  of  i n a problem  Acquiring for  i s part  collected,  equation  mode  example  theorems  probability  of  certain  summary  p r o p o s i t i o n s t o which Brauner  mathematics  set  a n d how  tire. Table  and  carburetor  a  genuine  discipline.  new  way  of  to  an  algebraic  i n ordinary  language.  of perception seeing  understanding Morris  see  of  Kline's  opens  the world a  up and  traditionally  remarks  here  relevant. The meaning and purpose o f almost a l lof mathematics do n o t l i e i n t h e s e r i e s o f l o g i c a l l y r e l a t e d c o l l e c t i o n s o f symbols b u t in what t h e s e c o l l e c t i o n s have t o t e l l us about our world. ( K l i n e , 1953, p.15)  are  25  Table 1 Brauner's (Brauner,  f'1  til  B  outline of the 1986, p.13)  (nine)  modes  of  perception.  :sa_ iiii  1  II  ;|J_  til  ill i l l  ii  «B C  si  IJtlJl  i 3 -I  .S  ii UJ  3c  ill  i l si  11  fUli  ^ i  11  s 1  a  III  i-a.f  i  illlll  S 5 u .  n .1 S  :  8-2._  Jill!  iii  _ _ Z O C O  LU X  <3  Q. - J U l CO  26  Education rooted i n perception The c e n t r a l i d e a of concepts g e n e r a t i n g p e r c e p t i o n c r e a t e s an image of the l e a r n e r not as a blank sheet t o be  trained  rational,  in  stimulus  emerging  response  perceiver,  managing v a r i o u s c o n c e p t u a l c o g n i t i v e t a s k s undertaken conceptual  fruit just  a  systems.  matter and  alternatives. rationality,  of  on  making  the the  selecting  The  correct  from  a preparedness  respond  level  i n informed of  education  the world.  aims  t o use  ways t o then  bear  i s not  response  but  of  conceptual  at  developing  t o n o v e l t y with  The world i s always n o v e l .  new  i s not  f a c t s or concepts but f a c i l i t y understand  in  Learning  possible  t o respond  student must be prepared  goal  higher  s u r v i v e and  vine.  Perceptivism  open, but not empty, minds.  to  engaged  a  A c o n c e p t u a l s u r v i v a l of the  which p e r c e p t i o n s  which wither  balancing  The  The  as  ecosystem with v a r i o u s p o s s i b l e p e r c e p t i o n s  determines and  actively  but  i n e d u c a t i o n take p l a c e i n a  competing f o r acceptance. fittest  patterns  existing problems simple  knowledge of  life.  mastery  of  i n a p p l y i n g knowledge t o  Brauner s t a t e s :  Instead of teaching it [curriculum] for concept mastery alone and thereby l i m i t i n g i t to the top twenty percent in academic ability, an important change has been  27 introduced. Perceptivism initiates the search f o r content on the basis of the p e r c e p t i o n t h a t i s d e s i r e d and then insists that the concepts needed t o produce i t a r e taught i n such a way that the learner a c t u a l l y achieves t h e sought f o rp e r c e p t i o n . (Brauner, 1987, p.27) Brauner  claims  students  find  mathematics  best  by  rote,  concept  mastery  of  secondary  too  school  abstract.  then,  the  meaningfulness  of the learning  will  be m i n i m a l  the students without  curriculum can  the majority  especially  applicability the  that  and  c a n do i s t o p e r f o r m  really  instruction  be a t t e n u a t e d .  understanding. on  perception  In and  because  the rituals By  basing  the  problem  --  —  The way t o make t e a c h i n g l e s s a b s t r a c t , w h i l e preserving the integrity of the concepts involved, i s t o make s u r e t h a t t h e s t u d e n t has t h e a c t u a l p e r c e p t i o n t h a t t h e c o n c e p t s fosters. Indeed having the relevant p e r c e p t i o n i s so important i f t h e range and e f f e c t i v e n e s s o f academic t e a c h i n g a r e t o be extended to the entire student body, Perceptivism would reconstruct the entire curriculum around p r e f e r r e d perceptions. (Brauner, 1987, pp.12-13) Perceptivism individual with  tries  t o balance  interests  t h e need  and understandings  t h e need t o expand understanding  publicly  available  spontaneous  and a b i l i t i e s  of the  student  by developing t h e  modes o f p e r c e p t i o n .  interests  t o respect the  R e l y i n g on t h e  of the student  i s  28 blind,  i t  leaves  the  student mastery  Focusing  on  concept  interests  and  abilities  the  concepts  unrelated  to  previously  wither the  balance  education  involvement  in  The  never  Perceptivism One to  of  provide  toward  an  education  (i)  of  that  results  knowledge  bag  the  unused,  and  What  has  to  achieve  assumed  i n  lengthy  with  computational  balance  for  the  many jungle.  advocated  writing this  current  academic in  a  is  by  thesis  educational  fragmented in  is  trend  specialization.  helpful  organization, interact  shape a c t i o n  bodies  grab  the  because  approach  addressing  In to the  issues:  perception  (ii)  to  pedantic,  information  to  motivations  Perceptivism  following  difficult  the  attainable.  antidote  this  knowledge.  so  the  narrow,  learner.  calculation,  makes  the  the  was  from  more  when  so  emerging  computer  i s empty  vine, of  technical  unempowered.  respecting  learner  the  experience  made t h e  without  the  on  mathematics  students  of  uneducated,  of of  information  (Table  synthesis,  knowledge, isolated  that  to  discussing  interpret  modes  experience,  of and  1). by  discussing  that  facts  informs  by  knowledge but  life.  a  relationships be  relevant  not  just  system  a of  29 (iii)  application  organization This  brings  use  of  knowledge,  o f knowledge knowledge  by  stressing  the  f o r use i . e . f o r perception.  out of passive  r e t e n t i o n and  into  (Figure 1).  (iv)  using  rational based  knowledge  view  of self  n o t on  aspect  bring  about  and world.  This  intensive, exclusive  o f human  repertoire  to  of  an  informed,  rationality  absorption  knowledge  b u t an e x t e n s i v e ,  perceptual  possibilities  with  i s one  inclusive and  their  interaction.  Ways o f s e e i n g According by  (nine)  possible  basic  conceptual  different  schemes  form  analogous Higher  to Perceptivism,  a  i n  a  characterized  schemes. of  conceptual  interactions  learning  place  ways  of  The  seeing  the  ecosystem a  perceptual  conceptual  i s mediated schemes  conflict,  make  world. with  biological  and t h e c o n s t r u c t i o n dynamic  by  perception  The  a l lt h e ecosystem.  o f meaning  take  environment, competition  and  completion. What  does  different  ways?  it the  i s .  The  i t mean Is this  to  perceive  possible?  f o l l o w i n g examples  suggestion-  plausible.  the  world  Perceptivism  are offered  to  i n  claims render  What  i s a  Sun  sunset?  drops  i n the  horizon. the  with  i s the  ways,  the latter moon,  use  the  of  and  a  t h e naked  Galileo's  perceptions  can  to  a r t depicted  detail.  Da  attention  t o such  looked  level  Vinci  at only  view  and be  the  Giotto's  different  perception. smooth  body?  moon  more  a  The  to  be  perfect  romantically  The  controversy  observations social  man  without  using  turmoil  new  physiological  and Michelangelo detail.  Whether  on t h e s u r f a c e  i n the  Lamentation  Michelangelo's  rotating  i n  not  eye.  i s  o f t h e Sun.  the  human over  Creation  body  or also  as  Christ  o f Man.  painted t h e human  o f " t h e bone beneath t h e s k i n "  differences  sunset  initiate.  Medieval  is  the  a  rock-like  initial  attest  below  spherical  shows  i t might  the  attached,  perfect  cratered  as  telescope  that  perceived  or a planetary  using  surrounding the  i t  globe  perceived  us  a nonstandard  telescope  mountainous heavenly  event  i t that  disappears  tells  being  i s  globe  and  has  the observer's  same  heavenly  sense  the observer  to. o b s c u r e  This  The  sky  But science  Earth,  around  Common  body  at the  generates perceived as  with  opposed  the i n to  31 4.  What  i s a  table?  I t appears  stable,  impenetrable  object.  surface  of the table  and  i t i s  spread  mostly  out- at  microscope  with  empty  regular  makes  5.  Rather  than  night,  look  reflecting, next  looking at  perceivable.  This  commonly o b s e r v e d one  to  package using  five  electron  microscope  intervals. the  of  electron of  a  The  how  sky.  See i t  See t h e d a r k n e s s  v a n Gogh  to  i n the sky at  i n the  "starry  sky" i s  perception  of a  Each of t h e examples pay  attention  s i g n a l s and impressions  cornerstone  The  atoms  achievement  diffusing.  This  some  of the table.  phenomena.  concepts.  with  i s a different  show  a  i n on t h e  space  light  blurring,  t o be  zoom  at the objects  the  to the light.  But  an  possible  nonstandard perception  initially  to  and  i n different  theoretical  Perceptivism's  ways  principle view  i s a  of  being  analogy.  This  educated. Let analogy diagram directly, can  only  contain  the writer i s  with  (Figure  reference 2) .  unmediated perceive only  summarize  We by  to  with  an  the  cannot  selection  Hofstadter perceive  information.  reality  and p r o c e s s i n g .  projections of reality,  partial  (1979)  We  each of which  I t i s as  i f we  can  32  Figure  2.  Hofstadter's depiction of Complementarity: s e e i n g i n d i f f e r e n t ways (Hofstadter, 1979, p.l)  33 only  see  reality the  the walls  remains  possible Herbert  Only  contained  to  Hofstadter's  the directly  projections.  information  on  get  an  unperceivable  by paying  adequate  generator  of  projections, i s i t  conception  should  while  attention to a l l the  i n the various  F e i g l ' s comment  photograph,  be t a k e n  of  reality.  seriously.  There a r e n o t two d i f f e r e n t s o r t s o f r e a l i t y , but there a r e two ways of providing a conceptual framework f o r i t s d e s c r i p t i o n . I n f a c t , a t l e a s t s o i t s e e m s t o me, t h e r e a r e a great many "perspectives" o r frames - the extremes b e i n g t h e p u r e l y e g o c e n t r i c as t h e "lower limit" and t h e completely physical account as t h e "upper l i m i t " . In between a r e t h e many h a l f w a y ( o r p a r t way) h o u s e s o f t h e possible manifest images. ( F e i g l , 1 9 6 7 , p. 145)  nine on  There  i s nothing  modes  of perception.  t h e development  magical  about  Progress  o f more  the latter  He  has added  the  primal  wrote  explicit  and  about  art  music.  operational  The  such  discussion  i n January  perception,  important  Brauner  as  The  painting, was  1988. always,  or  i n part, Indeed  i n September  of perception.  o f two a d d i t i o n a l modes,  the operational.  nondiscursive  schemes.  s e v e n modes  mention  eight  depends,  conceptual  when t h e w r i t e r b e g a n w o r k i n g w i t h 1987,  seven,  first  primal  concerns  sculpture brought  I t reflects  a  up mode  and i n of  but e s p e c i a l l y since the  34  Industrial  Revolution,  mechanisms by which operational working  what  important modes  from  a clock  the  of the  I t depends on an  experience,  works,  f o r when y o u r  message  but  knowledge  of  the  of a piece of technology, social  ( e . g . how  t o look  a  t e c h n o l o g y works.  abstraction,  principles  physical  reflecting  how t o f i x a  c a r doesn't  i n a l l of this  principle  of  understanding behind the  work) .  or  bicycle, But t h e  i s n o t t h e number o f  the plasticity  of  human  specifics.  P e r c e p t i v i s m and mathematics e d u c a t i o n In  a  summarize  compact  Perceptivism  possible.  Students  perception  which  way  an  mathematics  up  the  best can  has  i n  an  them  now  with  does.  lessens  this  more  written form  t h e modes  to as of  t o see t h e world t h e The of this  computational  of  was  applicable  the realization  possibility  trouble, program  complexity.  problem  i n The  and so opens  relevant,  meaningful  f o r more s t u d e n t s .  i s only  through  meaningfully  schooled) .  II  equipped  person  t h e development  conceptual repertoire be  as  allow  been  computer  mathematics It  will  with  Chapter  c a n be  educated  traditionally,  personal  form,  The  possible  called  theoretical  and use o f t h e  that  educated  an  (as  structures  individual opposed  used  by  to the  35 individual A  determine  mathematics  concept  is  preoccupied  within  doomed  to  the  that  perceptivist  applied,  type  education  and to  how at  reach A  that  the  needs  difficulties  towards  the  A  inherent  implies.  It  to  locate time  goals  of  i n the  required  approach is  framework  Association  of  a  overcome  only that  America  to  within we  challenge.  Mathematical  is  to  The paradox of our times is that as mathematics becomes increasingly powerful, only the powerful seem t o b e n e f i t f r o m i t . The a b i l i t y to think mathematically—broadly interpreted—is absolutely crucial to advancement in virtually every career .... Confidence i n dealing with data, skepticism in analyzing arguments, persistence in penetrating complex p r o b l e m s , and literacy, i n communicating about t e c h n i c a l matters — these are the enabling a r t s o f f e r e d by the new m a t h e m a t i c a l s c i e n c e s . Whatever e l s e the MAA* may do i n the remaining years of this century, i t must work t o e n s u r e t h a t these  "The  of  mathematics  the  computer  or  education  i t s  interdisciplinary  age  level  fails of  seen.  fact  mathematics  ecology  strive  i t is mere  irrelevance.  information  Steen's  not  liturgy  Perceptivism of  respond to  does  and  stops  conceptual  conceptual,  mathematics  seen  that  i t s own  wants  computational  is  schooling.  isolation  education  this  and  just  with  itself  the  education  mastery  perception  what  can  36 new liberating, mathematical arts available to a l l students. ( S t e e n , 1987, Mathematics education  is  must  no  longer  technological, mathematical the  luxury  literacy of  social  conditions  technological especially both  overall ship will,  of  of  a  the  the  solve  increasingly quantitative,  essential.  is  and  of  at  our  of  most  thinking  education.  \  The  i n the  form  is  providing without  drift the  in  the  years.  But  just  problems  to  of  solutions.  somehow  ten  Modern  large,  will  And  gone.  computers  we  of  for twelve  in society  rudder  an  retention  mathematics  philosophy  mathematical  abilities  population  hints  itself,  becoming  6)  a  world,  to  calculators and  In  made  p.  because  luxury.  fast  demand  progress  working  without  is  the  school  of  problems  a  catering  percent  in  better  competitive  twenty  population  do  are  like  an a  computer  mathematics  37  CHAPTER I I I A BRIEF DESCRIPTION. OF CONSTRUCTIVISM Constructivism acquisition. philosophy  i s  I t would of  Perceptivism.  provide  a  chapter  i s concerned  all  coherent  theory  developed Both  of  framework with  i s  three  passively  received.  The  experience  i s theory  laden  notions,  i n Brauner's  meaningful  learning.  conceptions  of things An  knowledge  so t h a t  integral  is  s o r t i n g out the tangle  part  (1988)  perspectives thesis.  claim  past  This o f most i s that  rather  claim  sense,  than  i s that a l l experience  are important  claim  a r e endemic  meaning.  these  first  second  The t h i r d  of  Brauner  constructed  (1988)  with the  central claims  The  actively  by  for this  of Constructivism.  knowledge  of  appear t o be c o m p a t i b l e  education  called  versions  a  i s that  and to  alternate  i n the construction  of higher  of conceptions  learning that  then  compete  for attention.  Knowledge i s a c t i v e l y It  may  be  tempting  something  of a blank  a  presentation  clear  practise Somehow  constructed to  sheet of  think  ready new  of  the  learner  t o be w r i t t e n on. ideas  i s  given,  a n d r e w a r d t o f o l l o w , l e a r n i n g seems the rational  activity  of the learner  as If. with  probable. i s  left  38 out.  Simplified  stimulus-response empirical Strike  behaviorism model  conditions  works  learning  seems  to  and that  imply  given  that  a  the right  i s inevitable.  Kenneth  argues:  Behaviourists have interpreted traditional e m p i r i c i s m i n s u c h a way t h a t e p i s t e m o l o g y i s seen as u n r e l a t e d t o l e a r n i n g . Learning i n t u r n i s n o t seen as a r a t i o n a l a c t i v i t y . I t is something that happens t o people under proper empirical conditions. ( S t r i k e , 1982, p.51) But  the working  critical principle  i n  out of understanding Constructivism.  i s implicit  i n Dewey f o r  by t h e learner i s  The  constructivist  example.  No t h o u g h t , n o i d e a , c a n p o s s i b l y b e c o n v e y e d as a n i d e a f r o m one p e r s o n t o a n o t h e r . When it i s told, i t i s t o b e o n e t o whom i t i s t o l d , a n o t h e r f a c t , n o t a n i d e a . ... O n l y b y w r e s t l i n g with t h e conditions of t h e problem at f i r s t hand, s e e k i n g a n d f i n d i n g h i s own way o u t , d o e s h e t h i n k . (Dewey, 1 9 7 4 , p . 9 8 ) Piaget  was a l s o  contemporary  teaching  inappropriate transmission of  a constructivist.  insofar  o f mathematics as  o f knowledge  intellectual  He t h o u g h t t h a t  i t rested rather  than  most  and science on  the  was  simple  t h e development  independence.  To u n d e r s t a n d i s t o d i s c o v e r ... t h e g o a l o f i n t e l l e c t u a l education i s n o t t o k n o w how t o  39 repeat or r e t a i n ready-made t r u t h s . It is in l e a r n i n g t o m a s t e r t h e t r u t h by o n e s e l f at t h e r i s k o f l o s i n g a l o t o f t i m e and o f g o i n g through a l l the roundabout ways that are inherent i n real activity. ( P i a g e t , 1973, p.218) This learning in for  type  of  a  seems  to  be  mathematics example,  simplified implicit  classrooms.  claims  The  transmission i n much  of  account  what  NCTM S t a n d a r d s  goes  of on  Document  that  In most classrooms, the conception of learning is that students are passive absorbers of information, storing i t in e a s i l y r e t r i e v a b l e f r a g m e n t s as a r e s u l t of repeated practise and reinforcement. Research f i n d i n g s from psychology indicate that l e a r n i n g does not occur by passive absorption (Resnick, 1986) . Instead, i n d i v i d u a l s a p p r o a c h e a c h new t a s k w i t h p r i o r knowledge, a s s i m i l a t e new information and c o n s t r u c t t h e i r own new meanings. (NCTM, 1 9 8 9 , p.28) Similarly,  Howson a r g u e s t h a t many s e c o n d a r y  operate  the  on  basis  of  simple  classrooms  transmission.  One i s l i k e l y to f i n d a stereotyped form of teaching i n the bulk of secondary schools classrooms, a form which r e l i e s h e a v i l y on the textbook and t h e t r a d i t i o n a l p a t t e r n o f exposition-examples-exercises. Apparatus i s r a r e l y used, and c l a s s t e a c h i n g i s s t i l l the norm. (Howson, 1985, p.75)  40 As  a r e s u l t much o f what  because  i t i s not believed,  regurgitated. the  This  students  course  i s learned  - we  design  cannot  must  but be  accept  simply  regarded this  the  learner  abilities  specific  as  constructor  t o handle,  construct  forgotten  memorized as  a  and  fault  of  as a b a s i c  flaw i n  i s a central  concern  and d e l i v e r y .  What c o n s t r u c t i v i s t s a d v o c a t e with  i s simply  meaning  process  are  more  performances.  of  meaning.  Basic  and apply  information  important  than  Lochhead  to  isolated  writes:  W h a t I s e e a s c r i t i c a l t o t h e new c o g n i t i v e science i s t h e r e c o g n i t i o n t h a t knowledge i s not an e n t i t y which can-be s i m p l y t r a n s f e r r e d f r o m t h o s e who h a v e t o t h o s e who d o n ' t ... Knowledge i s something which each i n d i v i d u a l l e a r n e r must c o n s t r u c t f o r and by himself. This view o f knowledge as an individual construction ... i s u s u a l l y r e f e r r e d t o a s constructivism. (Lochhead, 1985, p.14)  A l l experience i s theory laden All students and  experience  come t o c l a s s w i t h  concepts.  subjective, sense. reality describe,  i s theory  Many  For  them  function  as  understand  these  notions  their  may i n  subjective  viable and  so t h a t  a repertoire of  of  undeveloped  laden  models. adapt  to  secondary experiences  be  private,  Brauner's  (1988)  constructions The the  learner world  of can only  through  these  capable  of producing  learners  with  descriptive  the  shares  and  then  becomes  are in  obvious.  used  of constraints.  limits  of  Third, objective  a renegotiation meaning,  The p a r a l l e l s Education  with  becomes  t h e young an a p p r e c i a t i o n  i s  which there  intuitive  web  there  of public are  reality.  o f t h e terms  within  i n  First,  personal,  i s t h e common  language  knowledge  framework  of  the  unlimited  and  New  and  subjective  or  of culture  structures  routines,  But  i n these  a  are  regularities  arbitrary  within  there  they  reality.  others.  language.  constructing  constraints.  with  systems  Second,  constraining  model  bonds  arises  the subjective  culture  for  and  are three  knowledge.  f o r new  and models  of the unifying  constituted  are  to  Because  r e l a t i o n s and  are not t o t a l l y  learner  there  search  which  means  constructions  conceptions.  concepts,  can a c t i v e l y  meanings  because  existing  the  Learning  of  the limits  reference of  these  Brauner's  Perceptivism  a process  of  creating  of the fact  that many worlds are possible, that meaning and r e a l i t y are created and n o t discovered, that negotiation i s the a r t of constructing new meanings by which individuals can regulate t h e i r r e l a t i o n s with each other. It will not, I think, be an i m a g e o f human d e v e l o p m e n t t h a t l o c a t e s a l l of the sources of change inside the individual, the sole child. F o r i f we h a v e  42 learned anything from the dark passage in h i s t o r y t h r o u g h w h i c h we a r e now m o v i n g i t i s t h a t man, s u r e l y , i s not i n island, entire of i t s e l f b u t a p a r t o f t h e c u l t u r e t h a t he i n h e r i t s and t h e n r e c r e a t e s . The power t o recreate reality, to reinvent culture, we w i l l come t o r e c o g n i z e , i s w h e r e a t h e o r y o f development must begin this discussion of mind. ( B r u n e r , 1986, p.149) x  The  role  of a l t e r n a t e  Within  the  be  viewed  as  to  become  more  domain. world  the  are  the  of  an  paradigm  designed  expert  must  signals  and  the  to  in  a  of  and  transform  situation  can  what  to  understandings relevant  experience  pay  that  concepts of  are  Because  to  may  the can  are  mathematics  at and  the  to  packaged swarm is  often select scheme a  and  perceptions  science.  of  in  to, to of in and the  problematic  Simple,  core  the  noise  attended  expert  fumble  attention to. from  novice  the  competing  understand  novice  result  be  can  knowledge  out  t o be  conceptual  often The  sort  a  a  orientation  impressions  effective  quickly.  to  t h e o r i e s he  education  particular  A v a i l a b l e are  reliable  and  able  signals  concepts.  most  situation  be  Which  meaning?  possession  by  process  message.  notions  knowing  a  learner  construct  use  constructivist  R e f e r r i n g back t o Brauner's  from the how  conceptions  an In  fail,  not  informal generated expert's contrast,  43 novices  are likely  t o use rote  guidance  of the perceptions  concepts  and t h e o r i e s .  with  simple, be  called  likely  t o be q u i c k ,  procedures.  expert  irrelevant  conceptions,  no  way  processing  because  procedures  i s chaotic.  of  reliable  this  knowledge  formal  are a l l that's  to  from  remedial  powerful,  formal  efforts  must  focus  ways  of looking  c a n do t h i s  only  of perceptions, based  hold.  processing, must  response  have worth  according  t o what  on t h e concepts  This  would  from  a  has been  They  must  matters.  and able t o a  repertoire i s sensible  and notions  represent  sense  o f ownership  to life  are willing  choosing  of  as o u t l i n e d i n  at the world.  i fthey  and a c t independently,  reasonable  a t t e n t i o n on t h e r o l e  They must have a sense  They  currently  i s  called  of significant  an educated  and  be  Learners  a sense t h a t  modes  there  out the relevant  could  focusses  have  of  this  to relevant  Education's  paragraph.  on t o them.  various  think  because  access  t o sort  i n creating expert  previous  passed  powerful  processing.  education  that  the  understandings  processing  access  Constructivism  the  b y more  When r a n d o m r o t e p r o c e d u r e s with  expert  intuitive  intuitive  available,  on  generated  without  When i n f o r m a t i o n c a n b e m a n a g e d  informal,  could  algorithms  a  they  rational  44 response  with  accustomed of  faith  will  reasoning an  to  an  open  rote  allowed  power and  so  often  holds  alternate  experience conceptions Hewson  &  to  the  compete  and  struggle  understanding  needs that  Gertzog  accommodating and  that and  that  be  ways  (1982)  of  learning current  is  at  things. rational  learner's a  new,  juggling  ongoing the  only  active  process  of  the  a c t i v e p a r t i c i p a t i o n of  to  take to  survival.  write  construct  this  and  looking  over  notions.  involves  an  assimilating to  of  needs for  of  interacting with  This  requires  capable  learner's  Learning  expansion  repertoire.  learner,  the  involves  and  accommodation t h a t the  will  be  inquiry  of  ideas.  educated  reorganization  that  background  plausible  conceptual  memory,to t a k e p r i o r i t y  some p r i v a t e  ideas  child  procedures  on  a  A  and  response based  Current  Becoming  rules  best  incompatible,  various  of  mind.  at  against  concepts.  empty  would  Constructivism occur  not  acceptance  have  uneducated  but  about rival  account  of  force  alternate  Posner, the  past  Strike,  processes  views  meaningful,  of  of  things  applicable  involves.  A c c o m m o d a t i o n may ... h a v e t o w a i t u n t i l s o m e unfruitful attempts at assimilation are worked through. It rarely seems c h a r a c t e r i z e d by e i t h e r a f l a s h o f i n s i g h t , i n w h i c h o l d i d e a s f a l l away t o be replaced  45 by new visions, o r as a steady logical progression f r o m one commitment t o another. R a t h e r , i t i n v o l v e s much f u m b l i n g a b o u t , many false starts and mistakes, and frequent reversals of direction. (Posner, S t r i k e , Hewson & G e r t z o g , 1982, p.223) In give  chapters a  I I and I I I t h e w r i t e r has attempted t o  brief  Construction  description  as t h e o r i e s .  later  to  shed  paper  —  can  secondary  light the  on  the  Chapters  to  Perceptivism  of the descriptions  make  them  educational mathematics  role  central be  of  education.  will  used  of  this  improve  the  problem  used  to  to  and  curriculum  the  the  i n such thesis  computer  a  and attempt  Constructivism,  given,  and  be  IV and V t o f o l l o w w i l l  basis  relevant  Perceptivism  theories  mathematics  instruction? interpret  These  computer  school  of  on t h e  way  problem, i n  as t o the  improving  46  CHAPTER IV AN INTERPRETATION OP PERCEPTIVISM The a  specific  certain  theoretical  particular  problem  theoretical  a  III.  large  an  to  schooled  society  just  yet are poorly  p p . 10-35.).  problems  calculus  education  been  this  bear  on  unit  Templeton  designed  constructivist Secondary  by  getting are  technological (IAEP,  t h e enormous  wasted  because  utilized.  i n Chapter  IV  and C o n s t r u c t i v i s m  as  VI  the  will  detail  author,  principles,  i n Vancouver.  I.  These  Chapter  Chapter by  students  students  t o be  effectively  practise.  particular,  applications  background  Perceptivism  i s getting  Too many  advanced  seems  detailed  interpret  In  i s that  the motions"  understood  i s not being  have  perceptivistat  i n Chapters I I  Realistic  o f t h e computer  and V w i l l may  described  i n an  Against  The  Perceptivism-  through  ever  a  called  i s lacking.  "going  to  education.  education.  but not educated.  computer  they  mathematics  relate  i n education  i n mathematics  mathematics  be  potential the  i s  I t has been  important than  1988,  perspective  percentage of the student population  meaningful  more  chapter i s t o  i n\mathematics  The p r o b l e m  inadequate  seem  of this  perspective  Constructivism. and  purpose  and  a  using delivered  Chapter V I I  will  47  report  on  field  interpret  the  conclusions  tests,  of  results.  a n d make  the  calculus  Chapter  VIII  unit  will  and  report  recommendations.  The s u b s t a n t i v e r o l e o f t h e computer The in  computer has so f a r been  mathematics  guiding  philosophy  to  inform  to  think  education the  education  the by  because  or theory  i t s effective  information  there  There  will  I t provides  analogy,  to free  so  may  they  taking  over  students  be  i s  understanding, major  students  become  of mathematics  t o construct  human p e r c e p t i o n  tasks,  actively  concepts  and  education  i n t h e computer  t o use an  from digging  application  mathematical  temptation  Perceptivism-Constructivism  the manipulative to  no  i n  the opportunity,  freed  been  problems  age p h i l o s o p h y  solve  anomaly  acquisition  i s the  needed t o advance mathematics education age.  has  o f knowledge  use.  technology  itself.  t r e a t e d as an  industrial  ditches  by  hand  sky scrapers.  By  t h e computer engaged  i n  frees the  interpretation of the  that  i n the information  shape  quantitative  age.  F e y a n d Good  argue:  Computing o f f e r s an o p p o r t u n i t y t o t u r n t h e secondary school curriculum on i t s head. I n s t e a d o f m e e t i n g a p p l i c a t i o n s as a r e w a r d for years of preparation, students can begin  48 w i t h t h e most of mathematics  Years not  n a t u r a l and m o t i v a t i n g aspect — i t sapplications. (Fey & Good, 1985, p.49)  of preoccupation  be needed.  this.  with  low l e v e l  The c o m p u t e r  Ralston  calculation  can take  care  may  o f much o f  claims:  No s o u n d a r g u m e n t c a n b e a d d u c e d t o s u p p o r t a thesis that claims that high school students must be v e r y s k i l l f u l a t p o l y n o m i a l algebra, trigonometric identities, the solution of linear o r q u a d r a t i c systems o f equations o r any o f t h e m y r i a d m a n i p u l a t i v e t a s k s t h a t a r e part o f t h e current high school mathematics curriculum. ( R a l s t o n , 1985, p.37) What  I  am  computer"  into  Brauner's  model  been see  able,  computer  range  available way  we  literacy  can  3, t o c h a n g e  can detect  processing of  useful  t o a l l through see  the  world.  c a n become  an  and 3 ) .  i n part,  extend  we  "plug  processing  1  i n Figure  because, we  that  conceptual  electron microscopes).  information The  i s  (see Figures  of impulses  radar, the  the  as shown  the world  range  suggesting  unit  has  t h e way  we  i t has extended t h e (e.g.v i a t e l e s c o p e s ,  the  manner  possibilities concept  mathematical  made  c a n change t h e  Quantitative, mode  for  utilization.  procedures  t h e computer  active  i n  Science  I n an analogous  through  the  mathematical  of perception  as  Figure  3.  Computer used  t o amplify, power  of  concepts.  50 previously  laborious,'  unmanageable  calculation  is  automated. With  t h e computer,  seem more r e a l i z a b l e .  two p r i n c i p l e s First,  instruction  t o be a i m e d  Mathematics  should  and  how  application can  now  need  be  student  we  not s t a r t  the  to  rather  than  learn  about  i n part the  interconnections  of  itself  the  Second, the  i t s  interactions  by  computers.  applicability, various  ways  but  of the  computer,  mathematics  synthesis  of  specialization,  Students  u s i n g t h e dynamic  managed  With  involve  and i s o l a t i o n .  systems  world  i t .  n o t be d e l a y e d .  knowledge  world  and end w i t h  and use.  t o our understanding  perceive  structured  fragmentation  i t s h o u l d be p o s s i b l e f o r  at actual perception  must c o n t r i b u t e i m m e d i a t e l y world  of Perceptivism  of  can  understand  of  conceptual  Students  can  limits  and  perceiving the  (refer to Figure 3 ) .  T r a j e c t o r i e s - u s i n g the computer t o do mathematics Consider principles  i n action  for students an  the following  phenomena.  computer  (refer to Figure 4).  opportunity  trigonometric  in a  illustration  to  display  functions Actual  cases  learning  of  modelling  could  two  activity  Trajectories provide  t h e use in  of these  be  quadratic real  and world  demonstrated  or  51  Figure  4. T r a j e c t o r i e s  appealed  to;  shot  puts,  case  the  below,  cannon  streams  balls, of  water.  function which  where  velocity  of  (Kraushaar, y  g  i s  the 1982, =  the  the  The  functions  Figure  4.  the  the  and  9  rocks, two  arrows,  dimensional  trajectory  gravitational  projectile  is  constant,  the  projection  given V  the angle  pp.117-120).  x tan  students  In  models  (6) - g x  to  listed  plot  (1 + t a n  2  2V Get  footballs,  (0) )  2  some  below  2  trajectories,  yield  the  as  below.  trajectories  in  52 Functions: y = x * . 2 7 - ( ( 9 . 8 x ) /2000) (1+.073) y = x * . 5 8 - ( ( 9 . 8 x ) / 2 0 0 0 ) (1 + . 3 4 0 ) y = x * l . 0- ( ( 9 . 8 x ) / 2 0 0 0 ) (1 + 1 . 00) y = x * 1 . 7 - ( ( 9 . 8 x ) /2000) (1+3.00) y = x * l l - ( (9. 8 x ) / 2 0 0 0 ) (1+130.) y'= ( 1 0 0 0 0 - x ) / 2 0 0 2  2  2  2  2  2  The  following  questions  encouraged t o r e f l e c t i)  Which angle  Which  angle  maximum iii)  v)  what  9.8  m/sec  0 m/sec  Explain 9  and  2  with  the  angles  happens 2  i f g  on E a r t h  i n outer  t o go  half  the  ... 0 =  for  friends  changes, m/sec  happens  2  on t h e Moon  with  as v  examples  example,  increases,  from  playing  everyday football,  a hose.  5 how p o i n t s  two t r a j e c t o r i e s .  on t h e x - a x i s  Get students  f o r t h e two t r a j e c t o r i e s a theory  f o r example,  space.  Give  i n Figure  develop  furthest?  i f 0 = 90?  t o 1.6  constant.  squirting  them  the projectile  what  experience,  hit  t o go  ... Why?  and document g  Notice  activity.  explain  Explain  to  c a n be  distance?  ...  from  students  computer  the projectile  distance? causes  how  What h a p p e n s t o t h e p r o j e c t i l e 0?  iv)  on t h e i r  causes  ... s h o r t e s t ii)  illustrate  of the "lob".  can be  t o explore  are related. A s k them  to  how Have  apply  53  Figure  the  5.  Lob  theory  throwing opposed while  vs. C l o t h e s l i n e Trajectory  to  ordinary  passes to  he's  zone  in  experience,  football  defense,  against  squirting  standing behind  for  a wall.  a  example,  man-to-man  friend  as  with  water  F i g u r e 5 uses  these  functions: y y y y  = = = =  x * 4 . 7 0 - ( ( 9 . 8 x ) / 2 0 0 0 ) (1+22.11) x * 0 . 2 1 - ( ( 9 . 8 x ) / 2 0 0 0 ) (1+00.05) x * 1 . 9 6 - ( (9.8x ) /2000) (1+03.85) x * 0 . 5 1 - ( (9.8x ) /2000) (1+00.26) 2  2  2  2  E x t e n s i o n o f t h e concept By and  appealing  t a r g e t s out  develop  and  to  of  explore  the  range the  - the t r a j e c t o r y idea  of  for a idea  of  targets within  certain a  envelope  v,  range  students  trajectory  can  envelope.  For  given  v  y  V (-^r ) 2  =  and  0  g  - x  The  Figure  target i)  6  shows g=10.  space  into  points the  envelope  Is;  /200  envelope  and  2  o  the  (^A  2  V =1000  of  z  Function: y = (10000-x )  6.  equation  2  2  2  Figure  the  v,  the  three the  trajectories  envelope  The  outside  given  of  for a  envelope  trajectory  curve  divides  with the  regions; curve  which  cannot  be  hit  with  55 ii)  points  on  the  curve  which  can  be  hit  which  can  at  only  one  h i t with  two  discuss  the  angle, iii)  points  inside  the  curve  be  angles. Let  students,  explore,  possibilities.  The  document computer  computational  chores  interpretation,  analysis  student.  Computer  concept  development  The space, This  business with  can  Figure  7.  be  a  managed (refer  of  given  Hits  Inside  to  hitting V,  done u s i n g  takes  leaving and  the  care  the  up  calculation Figure a  Envelope  the of  to  the  facilitates  3).  target finding  formula,  of  tasks  synthesis  involves the  and  in  the  the  target  angle  0.  where  l  =  m  (Vo ) / g ,  the  2  range  factor  for  the  projectile. Figure  7  show  a  trajectories students  h i t  some  there  one  h i t with  be  some  points  with  They w i l l  find,  gradually  no  angle  the discriminant solution  discriminant solutions  will  be  ( i . e . two  shows two p o i n t s  different  be  zero and  positive  so  angles).  chosen  of course, be  so  there  i s  be Figure  i n s i d e the t r a j e c t o r y envelope  w h i c h c a n be h i tw i t h  two d i f f e r e n t  just  (x,y) t h e  will  F o r example,  so  f o r some  f o r some there  that  negative  solution),  angle)  Have  teacher  will  ( i . e . no will  ( i . e . one  two  1982, pp.117-120).  (x,y) t h e d i s c r i m i n a n t  will  (x,y)  being  (see Kraushaar,  coordinates. for  point  two 8  each o f  trajectories.  Functions: y = x * l . 0 0 - ( ( 9 . 8 x ) / 2 0 0 0 ) (1 + 0 1 . 00) y = x * l . 4 3 - ( (9. 8x ) /2000) (1+02.04) y = x * 3 . 2 7 - ( ( 9 . 8 x ) / 2 0 0 0 ) (1 + 1 0 . 6 9 ) y = (10000-x ) /200 2  2  2  2  For to  another be h i t .  angles.  example, Since  assume  that  the point  i t i s i n s i d e the envelope,  S u b s t i t u t i n g them,  (55,30) i s expect  two  57  X-AXIS: Y-AXIS: Figure  9...±18 STEP 8...68 STEP  8. H i t t i n g Envelooe  9 = tan  - 1  5  5  Specific  100  100  Locations  (55)  2  2  - 2  Inside  the  (100) (30)  55  e e If  a  68°13' 50°24' point  obtained.  on  the x-axis  Consider  6 = tan'  1  i s chosen  two  00)  2  - 50  2  - 2 (100) (0)  55 75°20'  9 =  14°40'  If  (47,40)  discriminant  becomes  are  (50,0),  T 100  9 =  angles  on  the  zero  envelope  so o n l y  i s  one angle  ]  chosen,  the  i s obtained  58  e  =  tan" T  100  1  ±  (100)  -  2  (47)  -  2  2(100)40  47 9 If  =  65°4'  (60,60)  obtained  outside  because  The  computer  computer  and can  points in  of  actual  the  views  about  objects  what  language  squirting  with  concepts  and  functions,  from  and  is  physics  Second,  the  student  can  such  as  to  gravity,  angle  these  in  turn  be  of  encouraged. passes,  be  compared  velocities,  launch,  related  functions,  Third,  football  shots  the  photographs  is  of  due  trigonometric  so  the  taking  happening.  systems  cannon  can  by  motion  compared  descriptions  water  of  calculation  stop  be  conceptual  three  use  First,  is  zero.  trajectories.  moving,  believe  Ordinary  height  perceptivist  of  angle  illustrates  extensive  can  of  acceleration  and  no  i s less than  education.  simulations  actually  interaction  the  perception  of  chosen,  exercise  elaborate allows  is  discriminant  mathematics  computer  with  envelope  trajectory  important  care  the  to  range,  quadratic  coordinates,  curves  59 from  mathematics.  discrimination rather die  than  from  This  and help  leaving  promote  assimilation  ideas  lack of use).  would  and  i n isolation  Brauner  conceptual  accommodation (to wither  and  argues:  Whether i t i s i n the sciences, the humanities, the social sciences or the l o g i c a l sciences, secondary a b s t r a c t i o n s aim to generate a n u n c o m m o n mode o f p e r c e p t i o n that i s sometimes a t odds with ordinary perception. As a r e s u l t . . . e d u c a t i o n has a strong responsibility to provide the kind of h e l p t h a t m a k e s t h e u n i q u e mode o f p e r c e p t i o n aimed a t a genuine and l i k e l y outcome. (Brauner, 1988, p.24) The  trajectory  learn  new  project  ideas  offers  i n the context  such  an  opportunity  o f a ".genuine a n d  to  likely  outcome." The  author  examples perception.  wishes  of  a  to provide  mathematics  What  the  i s  the  perception,  the  computer  and  modes  of  fashion.  should  learning  to facilitate  t h e mathematics  education  reader  throughout  system  the reader  resulting  plugged  information operating  perception  (Table  see  into  across 1)  i n  some an  more  rooted  i n  operating i n  the  processing  with  actual  conceptual (Figure  of the  3)  nine  integrating  60  Perceptivist-constructivist The the  acquisition  ability  to  use  of  mathematics.  modelling  because  at  table  of  five  of  and  mathematics quadratic  a  i s to  main  the  gap  to  number,  with  pairs,  as  as  a  graph,  as  rule  a  because  in English. and  working  the  The  concept  a main  twelve  and  a  goal  secondary  knowledge  exponential  a  real  of  linear,  logarithmic  impulses  the  use  of  concepts  world.  organized constructed  These Data  can  in tables and  often  modelling  world  and  that  and a  In  be  graphs. rule  the  our  from A  of as  function, perception experiences  model  devised.  case such  curve,  shape  9).  selectively  those  equation,  collected  data, ' a  Figure  be  concepts.  concepts be  (See  may  would  involves  problem,  predictions  operation, variable,  of  be  reality  ordered  and  these  integral.  and  between  world  of  build  off  slope, the  real  in  set  a  eleven  model  mathematics  of  tasks  as  components;  gives  attended  central  and  functions.  mathematical Reality  function,  basis  Perceptionist-constructivist four  a  this  sentence  polynomial,  the  i s the  (data),  grade  trigonometric  of  of  does  levels;  as  present  It  It  values  (equation)  concept  i t bridges  experience.  exists  the  i t , i s one  secondary  and  modelling  The  can  then  (model)  61  T i  function:  Figure  9.  ordered pairs t a b l e of values graph rule English sentence, perception.  leading  to  Perceptivist-constructivist modelling; distance fallen as a function of time elapsed since dropped  62 rule  will  in turn  Predictions  can  against  data.  model puts  real is  i n f l u e n c e the be  made The  consistently  the  way  we  variation  of  elements  (Figure  10),  (Figure The  12)  concept  occurring  and  (d)  Traditional trapped three  i n the  realms  applications  possible past,  is  personal  take now, the  mathematics of the  sterile  constructivist  to  is critical  so  that  shape  in  whereas  computers,  as  the  power  Fey  and  chemical fall  from  heartbeat  to  of  (Figure seeing  13).  what's  often  The  Good  other  a  picture  What pipe  our  is  quickly  accurate  of  the  an  Perceptivist-  proceed  just  gets  What r e s u l t s  student.  i t was  computing  to  full,  the  and  forgetting  mathematics.  the  concepts  of  education  is  data,  phenomena.  model,  alternative  and  example,  sunspots  and.their interaction.  intellectually  begin  of  various diverse  realm  and  For  impulses  occurrence  of p e r i o d i c i t y  i n these  (c)  reality  models  rise  tested  through  properties  Ob)  world.  and  between  world:  (a)  the  model  Our  the  11)  see  Reality,  model.  perceive  (Figure  the  there.  periodic  civilizations  with  we  interaction  c o n s t r a i n t s i n our  shape  way  makes  can this  in  the  information  age  argue.  dream  to  63  Kins AjfHiq.iv i  Figure  10.  "Periodic" V a r i a t i o n of Chemical ( C o n s i d i n e , 1976, p.1731)  Properties  gure  11.  "Periodic" Development  Variation  of  Cultural  65  Figure  12.  "Periodic" (Considine,  Variation 1976, p.1242)  Figure  13.  "Periodic" Variation i n ( C o n s i d i n e , 1976, p.2123)  i n  Sunspot  Heartbeat  Activity  66 For 1. 2. 3.  a given function f ( x ) , f i n d — f ( x ) f o r x = a; x so t h a t f ( x ) = a; x so t h a t maximum o r minimum v a l u e s of f(x) occur; 4. t h e r a t e o f c h a n g e i n f n e a r x = a; 5. the average value of f over the i n t e r v a l ( a , b) For nearly every function of interest, computer u t i l i t i e s make a l l f i v e questions a c c e s s i b l e i n some i n t e l l e c t u a l l y h o n e s t and mathematically powerful form to students who have not f o l l o w e d the c o n v e n t i o n a l regimen of s k i l l development. (Fey  Computer  assisted  & Good,  exploration  of  the  function,  and  i t s  1985,  p.  48)  exponential  function The  exponential  logarithmic various  f u n c t i o n , can  discussed  applications i n business,  science.  The  compound  interest,  absorption, inflation.  functions  forgetting,  Graphs, are  student's  task  concepts  shifts  that  to the  weld  model.  The  becomes  approachable  at  perceptivist  crunching  to  the  reasoning  to  real  as  phenomena  as  r a d i o a c t i v e decay,  the  push  mastery the of  mathematics  informed world  such  values  notion  of and  and  solutions  a  button.  educated  shifts  from  to The  application  experience the  drug  growth  and  application  problems.  to and  of  together  regard  medicine  population  tables  available  with  the  ecology,  model  probability,  equations  the  be  inverse  and  of the  person number  of q u a n t i t a t i v e  67 Following  are three  exponential  function  to  to  be  (a)  able  population  of  years  of  Earth and  graphs  as  growth,  (b)  population  4 percent  capacity Useful around 1.  be d e t e r m i n e d ?  2.  What h a p p e n s  How  the  The  effect at  first  of  our  100  present 1,  growth  rates.  Carrying  as  billion.  introduced  mean  i n human  3.  What  would  rate  could  30 be  2,  3  developed  much w o r r y i n g  about  i n this  might  regard?  look  like?  What  would  terms?  speaking there  How  the carrying capacity of  t h e graph  a  successful  problem"  you t h i n k  f i g u r e mean?  i s Canada doing  i f we e x c e e d  What w o u l d  this  Do  (c)  and p r o j e c t i n g  exercises  the growth  it  Graphically  lives:  such as:  does  "population  likely  their and  the  starting  billion  and  to  of the  are  investments.  1 4 ) , shows  arbitrarily  issues  t h e use  inflation,  term  growth  of 5  discussions  Earth?  relevant  annual population  i s  What  the  of long  (Figure  population  of  i n s i t u a t i o n s students  perceive  compounding growth set  examples  what  look  solution like  nothing?  the  graphically?  i s the "population  i s a problem  to  problem"?  or i s i t a l l just  so  68  Funct i o n s : y«5*(l+.02>Tx y=5*(l+.03>tx y=5*d + .0i:)tx y=30  /  X-AXIS: Y-AXIS: Figure  O...iOO 8,..100  14.  Get  and t o being beyond  perception to  do  mere  by  way  i t .  fact  Exploit by  information  of  educated,  of t h e world.  construction the  5 5  E x p o n e n t i a l Function and P o p u l a t i o n Growth (Smaller growth r a t e corresponds t o lower curve)  Perceptivism, world  STEP STEP  or  i t sorientation  i s suggesting  concept  the  computer  the learner. processing-  The  what  t o do.  to  actual  mastery  Constructivism to  i s showing allow  computer  practicable.  to the  how  active  i s making  The  student  69 uses  the  calculating  power  beyond the  technical details  mastery,  understanding  of  the  computer  i n order  through  to  to  attain  use  get  concept  and  actual  perception. The effect 100 A  second  of  inflation  years  at  future  over  be  and  algebra  mathematics and  50  and  and  percent  of  of  are and  to  the  act  a  useful to  relevant  considerably  (their  income  strategies  majority  could lot  senior  devices  for  simply  to  students.  opportunity  students  of  scientists  of  mathematics  more  A in  serve  an  the  living  filter  offers  on  is  of  future  but  vast  make  available  as  focus  problems  for  next  burger  students.  useful  a s s i s t e d mathematics and  by  word  institutions  alienate  lives  planning  attempted  the  could  the  the  inflation.  standard  their  shows  over  annual for  student's  Computer a  $50  questions  contrived  classes  and  15  and on  even  (Figure 15),  $2.50 h a m b u r g e r  Financial  education  mystify  a  years  mathematicians  higher  graphs,  point  years).  discussed  the  10  inflation  next  producing  5,  Discussions of  the  3,  of  on  reference  included. effects  group  to  education than  at  present. The compound  third amount  set of  of an  graphs,  (Figure  investment.  16),  illustrates  Emergency,  short  70  Tunct i ons: y=2. 50* < 1 + . 05 > : fx y = 2 . 5 0 * C l + .03:>Tx y = 2 . 5 0 * ( 1 + . 10.) t x y = 2 . 5 0 * < l + . 15:>tx y=50  X-AXIS: Y-AXIS:  0...±00 O...1O0  STEP STEP  5 5  E x p o n e n t i a l F u n c t i o n and Inflation (smaller inflation rate corresponds lower curve)  term  and long  term  savings  could  be  discussed.  Functions: y=5*C1+.05)tx  y=5*(i+.io>tx y=5*(l+.i5>tx y=50  /  4-  f  5 5 Figure  16.  E x p o n e n t i a l F u n c t i o n and  Interpolation investment in  ability  t h e computer  actual  quantitative world. a  indicates  can  make  them  that  with  the  mechanism  we  the  experience  students ideas  o f t h eburden  of here  of  development  an informed,  oftheir  the i n  fail  educated,  experiences  By p e r m i t t i n g mathematics t of u n c t i o n  filter  of  a n ds o i t becomes p o s s i b l e  t o develop  understanding  Interest  value  even  Conceptual occur  the  The w r i t e r ' s  progress  relieves  c a nthen  t h e student  t o find  times.  computations.  mathematics for  be used  at various  t h e classroom  limited if  could  Compound  t o educate  o f the  simply as by  simple  72 default.  In  potential are  of  failing  the  to  computer  in  l e a v i n g uneducated the  70%  F u n c t i o n s as models o f If of  mathematics  intellectual  philosophy  of  some e s o t e r i c student and  now  indispensable difficulties  education  tautology.  I t can  i t can  The  tool  needed to  from  the  respect looking  to at  the  to  in  world  so  The  computer.  is  they  more  than  showing  the new  seems  to  be  an  computational mathematical  (see F i g u r e place  3).  for  the  must  quickly  constructing  meaning  by  new  integrating that  perceptivist  in  students  orienting  measure  world  no  explaining  essential The  we  the  phenomena  of  things  see  by  applying  there  a  i t is  overcome the  responsibility  and  a  this  computer  But  old  achieve  show do  information.  knowledge,  perceptions. is  of  say  information,  previous  us  encountered  not  the  help  ways.  to  education  out.  within  i t must  This  given  i s to  education  t o model r e a l  transmission  filtered  educational  reality  functions is  the  mathematics  respectability  that  useful  be  exploit  can  i t  knowledge  competing achieve  facilitator  NCTM s t a n d a r d s  relating  to  with  ways  of  the  desired  in this  process  documents  argues:  73 Computer technology provides tools, especially spreadsheets and graphing utilities, t h a t make t h e s t u d y of function concepts and t h e i r a p p l i c a t i o n s a c c e s s i b l e t o • a l l s t u d e n t s i n g r a d e s 9-12. This technology makes i t p o s s i b l e f o r s t u d e n t s t o o b s e r v e t h e behaviour of many types of functions, including direct and inverse variation, general polynomial, radical, step, exponential, logarithmic, and sinusoidal. A l l students should use a graphing u t i l i t y t o i n v e s t i g a t e how t h e g r a p h o f y = a f ( b x + c ) + d i s r e l a t e d t o t h e graph o f y = f ( x )f o r v a r i o u s c h a n g e s o f t h e p a r a m e t e r s a, b, c, and d. (NCTM, 1 9 8 9 , p . 1 5 5 ) The  Bouncing  Ball  "Bouncing b a l l " of  bouncing  ball  allowed  to  absolute  value  the  damped  well. for  an and  released  bounce.  motions  the ball,  of  different  ball  compared  t h e development from  the  the of  functions  ground  exponential,  ball  and  by students.  model  elasticity  variable  damping  The p e r f o r m a n c e  could  the performance  to  reasonably  for different  gravity  and  then  of a  be  of  modelled  different  ball  tennis). The  bouncing  ball  needs  three  functions  it: 1.  o f a model  i s found  bouncing  (e.g. lacrosse)  with  above  combination  to allow  c a n be e x p l o r e d  actual  (e.g.  A  and cosine  Opportunities  effect  details  the sine  or cosine  t o g e t up a n d down,  to  model  74 2.  absolute  3.  e x p o n e n t i a l t o g e t t h e damping After  those in  balls,  decay  would bounce each  to  bounce,  hopefully  different  be  behavior  of  frequencies,  about  set.  encourage  higher  Preformal student  as  would  be  specific  comparisons  but bounced  such  amplitudes  and t h e p o s s i b i l i t y  decay  could  will  the  Questions  didn't  exercises  17 a n d 1 8 ) , s t u d e n t s  o n t h e moon  that  values,  effect.  exploratory  model  with  rates.  happen  with  (Figures  position  bouncing  t o do away w i t h n e g a t i v e  playing  following  a  and  value  with  what  of a  ball  and  higher  experimentation  independence  and  confidence.  Tide Equation "Tide tides. to  for y,  i n this five  a given slope  interval  sine  modelled existing  the  ability  case,  how  a graphing  to  model  I t i s useful  program  can help  of the basic features of a function: value value  o f x, v a l u e  at a point,  minimum  and average value  Water depth the  explores  The s i n e f u n c t i o n seems a d e q u a t e .  note,  provide  equation"  i n a tidal  function. and  o r maximum  on an  i n a  values  given on  c a n be m o d e l l e d  and way  The a v e r a g e  a  an  interval.  region  Amplitude  discussed  knowledge.  of x t o produce  frequency  that  value  can  connects of a  with be  with  continuous  75 Funct i on: y«70*<2.71tc-.05*x)J+abs<cosC.25*x))  i i i i i  i i ', i i i.  A \  I  II  <. A i. / \ i.  I/  u  II'  !  '{  V  r  X-flXIS: Y-AXIS: Figure  V ^ N ----  O...29Q STEP 9...S3 STEP  17.  Bouncing b a l l ,  5  rapid  ±0 decay  Function: y=70*C2. 71T c-. 0 2 * x ) ) #abs <: s i n <. SO*x))  il  II H .  I1 )\ l) i! 11 i tsI I 1 i U II II M l ||  i  I "Ml II If I IP*  J'i .  i I  ! I I  ' 11, ."i ,i  It  I  .'I  ii AA  r t A A *.. 1 ! ! f '/ '/ '1 ii ii i if i.« i n i ' it) n A * i f f [ H  ! ! ! M V I 2 V T I  :  Y-rtXIS: Figure  18.  H  H  V VVV\AAAAA.-  • • 2QO STEP 8...58 STEP  2 •  Bouncing b a l l ,  ,  slow  i.6 5 decay  .  function is,  c a n be  average  average  J  water  value  depth  investigated  over  a  tidal  here,  that  cycle.  This  c a n be c a l c u l a t e d as f o l l o w s ;  dx  f ( X )  A  meaningfully  (-b-a) the  definite  integral  Variations changing changes will in  on  the  i n tidal  A  exercise  account  discussion  accelerating  on  the  velocity  and  velocity,  average  a  seasonal  of actual  so compactly  to  lead  represents  while  i s of value computer  drag  a  tides stored  full  formal  behave  while  experiencing  a i r  objects  earth  here  given  distance.  c a n be  gives  falling  towards  possible  used  Resistance"  how  What  between  and  model.  with  resistance.  be  include  Resistance  of  gravity  might  deeper water  u n p a c k much o f t h e i n f o r m a t i o n  "Motion  This  basic  depth.  the mathematical  can  done on t h e m a c h i n e .  t h e model t o r e f l e c t  Motion with  to  being  The  i s the exploration  the  expressions  concepts  coefficient  of  terminal  and a c c e l e r a t i o n due  investigated numerically. explain  ball the  and kind  the a  difference  feather of  for  i n  falling  informed,  The  to  data  behavior Earth.  quantitative  77 Fu.icti on: y » 3 * s i n <0.5*x)+20  XrftXIS: Y-AXIS:  Figure  19.  9 0  J -«•;£« . . J.QO S T E P ..38 STEP  Water  depth  1  5  variation  i nt i d a l  cycles  Funct i ons: y = 3 * s i n ( 0 . 5 * x >+20 y = » 3 * s i n <0. 5*r.) +20 y » 3 * s i n(0.5*x)+20  fi&saik. • mm ft • / •xisgjinw-' sssniiii suramin bh&siiiiii  t  -Kami «in KBSIIIIII  nuu. S~%Slfi Y-AXIS: z  Figure  20.  g - . . ± e o STEP fl.»,38 STEP Average  value  5 i  o f a f u n c t i o n on t h e computer  78 discussion  o f aspects  competency should  o f t h e world  that  mathematical  enable.  Funct i o n : v = 3 * s i n (. 0. 5*x > +20  ii\V\'t. FI 1111 >.  Ii,  •i I I Pi .<ll  CIIMI llrwll II I m i n i m i iiin i MINI Mill Mil! I mill mil inn i  IIN  u 11 r\ ,k ill I jjjV.vi 111 IK-iltl III lllll III m I1 1 m  m Ml  Mill I I lllll film Ill Mill !! IIIi l ..tin u m m viim u m m i l lm  nil im  1  m i n inn m i n i m i inn  .•-rr..  Al l l l \  IK  II I II I I I 4 I . II t I 1 lllll I  II  m in  Earth,  an o b j e c t  .fx  .•I'ITN  I fr '  I III..  1 1  11 f'. iiif'.  i  II fv^lll  miln  II  III  "Ill  1  5  K I\ I iV-i  i mn III  n  III  III  ii il III i m lllll i mn II i mi . um in II! i i u m i u m n m i mi i t mil i n mil Li JI in IIIJILIUIU ill JJIl J l U U I I U i l tmil  ... II.  - .  u  .  depth  i s dropped from a height  a i r resistance  distance  .il IIV. /mi ;i nit v.-i'i i n n um Mil  Illlllll HIM" t lllll  O . ..166 STEP 9...38 STEP  F i g u r e 21. Average t i d a l If  lill  II  II  i  01JJJL jji II JJJU.LJ J U U i I UJJ I LUll U.UilU J  X-AXIS : Y-AXIS:  1 11 IS. II lllf. I IIIIK  ignored,  h above t h e  the v e l o c i t y  (v)  and  (d) are given by  v = at = 32t d - .5at  2  = 16t , where a i s a c c e l e r a t i o n due t o 2  gravity Velocity and  increases  indefinitely  supposedly a l l o b j e c t s  there  i s no mention o f mass.  situation i s different Some  fall  devices  even  with  respect  t o time  at t h e same r a t e  since  With a i r r e s i s t a n c e t h e  (see equations 16 and 17 below).  depend  on  the e f f e c t s  of a i r  ,1  The of  computer motion v  facilitates  under  =  (v  t  i s terminal  x  p v  Also  =  height  y  y  - yt  Q  =  y»  Notice  +  I/P  *  - g/p  using  but  what  will  =  (v  + v)  G  equation  are  from  and  assume v  *  that  the  velocity  (g/p  a  and  objects  cannon and  *  by;  0.  (l-exp(-pt))  should  does  not  free  fall  Obviously balls.  proceed  to  student  mathematical (Figure  can  be  feathers  Below,  increase is  then  velocity  (17)  velocity  corresponding  in  the  i s given  =  0  terminal  terminal  17.  plotted  get  ground.  g/p  rather  of  to  a-exp(-pt))  *  T  i/p  influence  g/p  gravity,  +  physical  height  differently falls  * t  =  t  above ground  above  t  to  v  (16)  object  v  e*  use  due  Experimentation  determine  The  0 and  is  - g/p  i s height  indefinitely  factors  =  the  below  attained.  0  e x p (x)  of  s i m p l i c i t y use y  which  velocity,  and  * exp(-pt)  where y For  i s time,  the =  in  ,  t  is acceleration  g/p  resistance,  type  velocity,  is original  0  s i m p l i c i t y assume v  v  of  this  i s drag c o e f f i c i e n t ,  t  g  i n v e s t i g a t i o n of  constraint  + v ) *exp(-pt)-v  0  where v  For  the  the  22) . studied behave  different  can  explore  free the  Functions:  °  u  i) ya(32/0.5)*<:2.71t<-0.S*x5)-32/0.5 Wi y«<32/1.5)*(2.71t(-1.5*x) )-32/1.5 ml y=<32/.20)*C2.71tO.20*x> )-32/.20 W1y»C32/. 19>*<2.71*<-. 19*x) )-32/. 19 v) y=C32/. 1S)*<2.71T<-. 16*x) )-32/. IS  (U)  on)  X-AXIS: Y-AXIS: F i g u r e 22.  falls  are plotted  different 1.  = 21  Person v  3.  T  T  x  t h e student  with parachute  can  p = 1.5  ft/sec. with  no p a r a c h u t e  p = .15  = 213 f t / s e c .  Person v  and  possibilities.  Person v  2.  Q...56 STEP 5 -208...280 STEP 50 Terminal Velocity i n Motion with  unbuttoned  = 64 f t / s e c .  o v e r c o a t p=.5  ( F i g u r e 23)  Resistance  explore  the  81 F u n c t i ons:  )) y « 5 0 0 0 - 2 1 . 3 * x + l / 1 . 5 * C 2 1 . 3 > * < l - 2 . 7 1 T O l . S * x > ) in"y=5000-213.*x + l / . 15*C212. ) *< 1-2. 7IT C-. I5*x> >' iSiy=5000-64. 0*x+1 / 0 . 5* C S4. 0) * < 1 - 2 . 71T ( - . 50*x) )  ( in;  "v.  O . . . -40 S T E P ! Q . , . 5Q6Qi STEP  X-AXIS:  y-f»xis:  Position Velocity  F i g u r e 23  Damped O s c i l l a t o r y "Damped gradually or  amplitude  of  spring.  the physical  attempts  andforth  motion  function  and adjustments  function  decrease  Motion"  A sine  and period.  exponential gradually  Oscillatory  behavior  o f Time  with  Terminal  Motion  reducing back  vibrating  periodic  as Function  over system.  then  time,  will  capture t h e made f o r  by an appropriate  cause  t h eamplitude t o  duplicating  Actual  o f a pendulum  c a n be  Multiplying  will  t o model t h e  t h e behavior  pendulum  or  spring  82  systems  c a n be  studied  constants.  Once  discussion  of a real  constructing  and  an e x p o n e n t i a l  accomplished  t h e form  A*  the  pendulum  reverse  decaying  a  and attempt  They  damping  destructive  periodic  function  t  The e q u a t i o n i s  Students t o model  24.  should  of  this also  and suggest  vibration,  amplitudes  can study  an a c t u a l  the  spring  by a d j u s t i n g parameters i n  students  should  fading  behavior.  decreasing  trigonometric  See F i g u r e  applications  variation.  quantitative  gradually  function exp(-bx).  development  research  of  using  function.  appropriate  i s the vehicle f o r  and pendulums  i n oscillation,  general  their  situation  t (X)*exp(-bx).  examples p r o v i d e d or  informed,  p e r i o d i c phenomena w i t h  be  using  understanding.  examples  can  of  world  springs  are both  Modelling  again,  meaningful  Vibrating out  and modelled  At this also  type be  be  of  able  point i n able  to  functional to  generate  a p p l i c a t i o n s , , f o r example  amplification  and sound  theory.  U n i n h i b i t e d Growth Model "Growth different growth a  models  (c)  the rate  provides  of growth:  (and decay),  limit,  where  Models"  for  three  (a) u n i n h i b i t e d e x p o n e n t i a l  (b) i n h i b i t e d  inhibited  outlines  exponential  exponential  i s proportional  only  growth to the  growth t o to  limit  remaining  83  F u n c t i ons: i) y=2*sinC2*x-3.1 4 5 ",h y=4*sin<2*x-3. 1 4) iii) y=S*sinC4*x-3. 14) ',*) y='5>*sin < l * x - 3 . i 4)  *<2.71t(-x/5:i ) *<2.71T<-x/3> ) *<2.7l1 Ox/3> ) *<2. 71TC-X/3) ) k  (ill)  I  P.  X - A X I S : " - A X I S :  Figure  24.  9-. . . ± 5 S T E P ± . . 1 0 STEP  Damped O s c i l l a t o r y  room  f o r growth  rate  also  Application population psychology the  diverse  (unlike  depends  give  Motion  t h e previous  on t h e p o p u l a t i o n  exercises theory,  i  i n business,  economics,  t h e student  case size  where t h e remaining).  ecology,  physics,  medicine,  sociology and  a chance t o s e eand e x p l o r e  applications o f the three  models.  84  If  a  population  proportional  to i t s size,  then t h e p o p u l a t i o n P(t)  reproduces  = P  0  a normal  itself sort  at  a  rate  of situation,  P i s g i v e n by  exp(kt)  where  k  is  the  growth  p e r year  percent  as a  of  decimal  fraction. Figure  25  population  below under  i s an  investigation of future  the dual  growth and p e r c e n t a g e  assumptions  increases  ranging  world  of uninhibited from  .5% -  3%.  Functions: '") y=5*2.71t(.010*x:> ii) y=5*2.71T<;.015*x:> iii*) y=5*2. 71T <.. 0 2 0 * x ) w> y=5*2.71TC.030*x) vi) y=5*2. 71T (.'. 005*x') v.) y=5*2.7i-r<:. ooo*.-<:> -  "(vi)  X-AXIS: V-AXIS: Figure  25.  -10O...100 STEP Q...SO STEP S World p o p u l a t i o n  ±6  projections  85  E x p o n e n t i a l Decay If the  a population  present P(t)  declines  population  = P  where  at a rate  proportional to  then,  exp(-kt)  0  k  i s the percent  of  yearly  decline  as  a  decimal. The  constant  of  values  to  show,  is  something  k f o r carbon-14  i n Table  2 and graph  f o r example,  object's  like  C-14  that  5750  remains  i s .00012.  i n Figure  the half  years  life  and that  the object  The  table  26 can be  used  of the isotope  i fonly  i s about  70% o f an 3000  years  old.  I n h i b i t e d Growth Model If to  a  population  growth  the  s u c h dP = k P ( L - P ) dt  growth P(t)  rate  Table L  D  reaches  The  = 30  certain  i s given  limits and k  by  P„*L  inflection  change  by  where L i s t h e l i m i t  then the population  = P  The  i s constrained  graph  3 depict billion.  + exp(-LKt) * (L-PJ point  of the curve  a maximum a n d t h e n i n Figure world  27  population  and  where  declines table  the rate  of  i s P = .5L. of  values  i n  under t h e c o n s t r a i n t o f  86 Table 2 Decay  of carbon-14  Age  of object  P e r c e n t C-14  x  100*(2.7?(-.00012*x))  0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. 9000. 10000. 11000. 12000. 13000. 14000. 15000. 16000. 17000. 18000. 19000. 20000. If a  100.00 88.72 78.72 " 69.84 6 9 . 84 54.98 48 . 78 43.28 38.40 34.07 30.23 2 6 . 82 2 3 . 80 21.11 18.73 1 6 . 62 14.75 13.08 1 1 . 61 10.30 9.14  the issue  could  be  quantitative basis.  an  analogous  growth  Table  of  o f 2,000 p e o p l e .  the  mathematical  students.  on  infect  similarity  using  t h e model  4 and F i g u r e  situation  spreads  town  to  carried  gradually a  remaining  the The could  in  28  which  entire  as  depict disease  population  physical basis for be  discussed  with  87  Functions: y = 1 0 0 * C 2 . 7 1 T < - . 0 0 0 1 2 * x !>) y=70 y=50  •  ***.  X -AXIS: V-AXIS: Figure  9 .. .29908 STEP STEP 5 8 ., .±09  26.  C-14  1Q90  decay  I n h i b i t e d Growth (L-P) A third  model o f growth  a s s u m e s dP = k ( L - P )  where  dt L-P  i s the available  The p o p u l a t i o n P(t)  =  room  f o r growth  a t any time i s g i v e n  L(1-exp(-kt))  a n d P=0  by  when t = 0 .  88 Functions: <5 * 3 0 ) /  »*»  y=  if)  y=cs*30)  (5+2.  JM) y = C 5 * 3 0 5 / ( 5 + 2 .  i») y= ( 5 * 3 0  5  7IT  / C5+2.  X-AXIS: V-AX IS : Figure  711  < - 3 0 * . 0 1 *x5  27.  ( - 3 0 * .  71TC-30*.  0 2 0 *  x5*  5  ( 3 0 - 5 5  5 5  World  inhibited  population,  from present  0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5  0 0 5 * x ) * < 3 0 - 5 5  8 ... 59 STEP 9, , , -40 S T E P  Table 3 World population figures (2% a n n u a l g r o w t h r a t e ) Years  * C 3 0 - 5 5 :>  / <s+2.7i-r c - 3 0 * . o o i * x 5*00-55:» 5  growth  f o rt h e i n h i b i t e d  model  growth  model  World population i n b i l l i o n s (5*30)/(5+2.7t(-30*.020*x)*(30-5) 5.00 14.15 2 3 . 98 28.40 2 9 . 63 2 9 . 93 2 9 . 98 30.00  89  Table 4 Disease model Time  spread  i n a  city  using  growth  i n weeks Number o f p e o p l e i n f e c t e d (10*2000)/(10+2.7T(-2000*.003*x)*(2000-10) 0. 00 0.10 0.20 0.30 0.40 0.50 0. 60 0.70 0. 80 0. 90 1.00 1.10 1.20 1.30 1.40 1.50 1. 60 1.70 1. 80 1. 90 2 . 00  Assume  thelimit  1 0 . 00 18.11 32.70 5 8 . 69 104.25 181.25 307.81 497.19 751.35 1045.08 1331.22 1567.13 1736.31 1845.87 1912.21 1950.76 1972.62 1984.85 1991.64 1995.40 1997.47  L, p e r c e n t a g e  will  buy a product  been  determined  rate  k =  is  the inhibited  i s 100% i . e . L=l.  that  .07 s o t h a t  of t h epopulation  t h e percent  Suppose  acceptance  the population buying  i t  that has  increase  the  product  given by P (t) = 1 ( l - e x p ( - . 0 7 t ) )  The  graph  advertise  i n Figure  29  shows  t h e number  t o g e tany given degree  o f times t o  o f acceptance.  90  F u n c t i on: y«<10*2000  X-AXIS: V-AXIS: Figure  28.  / (. 10+2. 71 T< - 2 0 0 0 * . 003*x * < 2 0 0 0 - 1 0 )  0...2 STEP .1 O...2009 STEP Inhibited  190  growth model  of disease  spread  Funct i ons: y » 1 * (. 1 - 2 . 71 f (. -. 0 7 * x ) .)  I / :  V.  /  .. . .  X-AXIS: V-AXIS: Figure  29.  8...60 STEP 0...1 STEP Percentage growth  5 .±  buying  a  product,  inhibited  As make  t h e examples  the  reality  program  meaningful  f o r more  evidence  i n place to  remains  a  Computer  before  of  problem  describing  concrete abstract  relations.  may b e c r i t i c a l The  can explore  and  relations  computer  facilitates  Perceptivist-Constructivist computer, makes  because  the  of i t s information  Perceptivist-Constructivist the technological  learning  mathematics.  computer  of  can help  than  application The  g e t us beyond  f o r many  education.  just  an  theory basis  and  thus  informed  of The  power,  mathematics  fantasy.  concept  study  formulating  processing  possibility  development,  to  realization  Perceptivist-Constructivist  t h e computer  made  literacy.  realistic  makes  need  mathematics  rather  be  of learning  perception  basic  failure  must  before  Computer a i d e d  the  education.  domains  Learners  to attaining  This  programs  no  of the  master  i n mathematics  them.  properties  to  a The  have  o f t h e powers  literacy.  mathematics  learners  before.  educators  students  can  mathematics  ever  mathematics  quantitative  chronic  that  than  enable  t h e computer  of  t o take advantage  integrated  available  application  that  help  fundamentals  illustrate,  students  suggests  computer  above  a  academic i n  turn  f o r concept meaningful use  of the  c o m p u t a t i o n f o r t h e few  92 and  moving  examples have  toward meaningful mathematics  provided  included  across examples  population  ecology,  medicine.  A l l the  ordinary are  language  guided  by  Perceptivism, build  a  B r a u n e r ' s modes  examples  t h e range  curriculum  rooted  all  because  i t i s mathematics  we  can  mathematics access,  using  our ordinary  c a n move  —  be  the  business,  language.  c a n be expanded  to  In this  way  mathematics f o r  something we  share.  to  which  we  language.  I f we as  world  something  i n  such  toward about  and  discussed  i n perception.  education  perception  philosophy  o f examples  mathematics  a l lrelate  can  age  The  economics  as t e c h n i c a l  information  of  physics,  meteorology,  as w e l l  an  from  fora l l .  t o which I t i s a l l have  93  CHAPTER V AN INTERPRETATION OF CONSTRUCTIVISM The three  present  basic  Chapter  principles  I I I .  instruction  computation  on  second and  the  indicated  existing  of  tedious,  conceptions belief, the just  the  of things  rather  than  teacher.  With  that  simply  and  should  draw  way  that  allow  f o r the  provisional  or  taking  explore  competing  care many  for  his  on t h e a u t h o r i t y o f  f o r u s e means  repeating,  this.  principle  can be  of  third  t h e computer  relying  Understanding  recalling  of  might  i nthe  the burden  i n a  should  student  i n that  encourage  The  assimilation  conceptions.  to  experience  instruction and  the learner  instruction  knowledge.  alternate the  i s able  interpret  indicated  relieving  indicated  existing  accommodation  By  computer  that  involve  to  as o u t l i n e d  principle  actively  to  attempt  of Constructivism first  principle  to  i s an  o f meaning.  relate  relates  The  should  construction  The  chapter  more  i t involves  than  actually  believing.  The a c t i v e c o n s t r u c t i o n o f meaning Constructivists information instruction are  a l l that  and  make  knowledge.  i s to  transmit  i s required.  a  distinction When  the  information, Knowledge,  between  purpose  of  explanations  the gaining  of  94 expertise  i n information  cannot  simply  be  with  the  i s  construct  by  active  reconstruction  because  can  example.  with to  i s usually  There  polished, teaching  often  student.  The  theory  product means. that  make  Eric  i t i s a  "pointlessness and t h a t of  a  curriculum  finished  strand  of premature  for a  came"  starting proceeding  developing  theory;  product  and  The  remote  (McPherson,  of  finished  to  geometry  dominated  by t h e  formalism"  was o f t e n  future  the  i s not the  and m i s d i r e c t e d  curriculum  on  compactness  regard  long  a  mathematics  formalism  with  The  abstract,  than  and f i n a l l y  notes  an  familiarity,  but premature  the traditional  never  of  Rather  abstraction  McPherson  preparation  students  the  i n part,  the trajectory  i t incomprehensible.  i s the goal  or  difficulties.  t h e problem  axiomatized  extreme  must  reconstruction  at least  witness  and b a s i c  dumps  and  re-invention,  difficult,  and discovery  refined,  can  active  t o mathematics. ideas  exploration  the  or  this;  i s also  approach  intuitive  with  learner.  necessary.  re-invention  help  constructed  the  learner  Discovery, i s  be  of  of technical,' computational  computer  formal  I t must  the  himself.  mathematics  and a p p l i c a t i o n ,  participation  something  Discovery, in  transmitted.  active  Knowledge  processing  that  1985, p. 6 7 ) .  " t o o much f o r many  95  Howson  echos  abstract  a  similar  curricula  possibility  concern  which  of meaningful  seem  with to  mathematics  an  elitist,  eliminate  the  f o r s o many.  Each stage i s seen as a p r e p a r a t i o n f o rt h e next: primary prepares f o r lower secondary, lower secondary f o r upper secondary, which i n its turn prepares for tertiary education. Yet by t h i s stage only a minute p r o p o r t i o n o f t h e a g e c o h o r t may r e m a i n w i t h i n t h e f o r m a l education system; has a l l t h i s " p r e p a r a t i o n " been j u s t a waste o f time and e f f o r t f o r t h e g r e a t m a s s o f t h e s t u d e n t s who a r e n o l o n g e r there? (Howson, 1 9 8 6 , p. 29) The  computer  alternative students. to  to  twelve  Vancouver.  having  associated  a  Different should  of  algebra  concept  using  line  of  with  real  can  only  be asked  by t h e  a  from  a n  the x  axis  c a n be  software variables. the  however  and  the  of i t s easily  suitably  F o r degree x  i n  polynomial  graph  t o discover these  author  graphing  the  exist  on  reconstruct  degree  cross  real  Secondary  where  graphing  possibilities  to  a  product  Templeton i s  solutions  equations  straight  at  function crosses  investigated  finished  exercise, given  exercise  The  reconstruction  the  students  theorem  perspective. equation  this  The  fundamental  make  dumping  Consider  grade  selected  can  axis  and  (Figure 30).  one  once.  students Second  96 degree 31) . are  equations One  are represented  possibility  i s that  other,  possibilities?  backwards  making  up e q u a t i o n s  -  0  solution  (x-3)  (x+5)  =>  possibilities  equations, complex  including  roots?  (Figure  and  s i x  (Figure  must  Discovery, student's  re-invention job.  t h e computer.  wide to  range their  +  -  there  33), degree 35)  are  or  I t i s easy  of p o s s i b i l i t i e s conclusions.  The  =  theorem What  for  degree  two  of  multiple  and  three  five  (Figure  (Figure  suggested  solution  and  and s a t i s f y  the  i s  the  where i t  i s the computational f o r students  34)  possibilities.  p o s s i b l e now,  NCTM  work  0) .  reconstruction  What makes t h i s i n the past,  15  f o r degree  discover the other  was i m p r a c t i c a l of  the factor  2x  What  can  using  are  Possibilities  four  student  2  (Figure  twice.  Students  the possibility  32) , d e g r e e degree  x  parabolas  i t crosses  the  (e.g.  by  power  to explore themselves  Standards  a as  document  claims: Calculators and computers with appropriate software transform t h e mathematics classroom i n t o a l a b o r a t o r y much l i k e t h e e n v i r o n m e n t i n many s c i e n c e c l a s s e s , w h e r e s t u d e n t s u s e technology to investigate, conjecture, and verify their findings. In this setting, the teacher encourages experimentation and provides . opportunities for students to summarize ideas and e s t a b l i s h connections with previously studied topics. (NCTM, 1 9 8 8 , p . 1 2 8 )  97 Function: y = 3 * < x - 1 5 + 5 * < x +4 . " >  V = 3 * * < X - ± > + 5 * * < X + -*> X-AXIS: -28...20 STEP 5 Y-AXIS: -500...580 STEP Figure  30.  Computer s o l u t i o n  58  o f degree  one  equations  two  equations  Funct i on: y=xT2-x-12  Y-AXIS'Figure  31.  - 5 8 8 . . .588  STLf  Computer s o l u t i o n  ^-  o f degree  98  Function: y=xT3-3*xf2-10*x+24  •SP  i  V=xt3-3*xt2-ie*X+24 X - A X I S : - 2 0 . . . 2 0 STEP 5 V - A X I S : -50O...500 STEP 58 Figure  32.  Computer s o l u t i o n  o f degree three equations  Funct i on: y=xt4-4*xT3-7*x-t'2+34*x-24  V = X t - 4 - 4 ^ X t 3 - ? * X t 2 + 34"*X-24 X-AXIS: -20...20 STEP 5 Y - A X I S : - 5 6 9 , . . 5 9 9 STEP 50 Figure  33.  Computer s o l u t i o n  o f degree  four equations  99 F unction: y=xT5-3*XT4-11*xT3+27*xt2+10*x-24  V=Xt5-3*Xt4-ii*Xt3+27*Xt2+iO*X-24 X-AXIS: -28...20 STEP 5 Y-AXIS: -580...506 STEP 56 Figure  34.  Computer  solution  o f degree  five  equations  Function: y=x1 6-xT5-l7*x-r4+5*xT3+S4*x-r2-4*x-48 k  X-AXIS: Y-AXIS"Figure  35.  -20...20 STEP 5 -SO©...50O STEP Computer  solution  58  o f degree  s i x equations  100  A p p e a l i n g t o e x i s t i n g knowledge Kilpatrick  (1987)  must t a k e  account  Teaching,  using  understanding, transmission at  argues  and knowledge.  procedures  that  at  sharply  of  student's  become  behaviour. for  guiding  transferring students change teacher's their  by  interviews and modify  computer  shifting  information The with  burden  become  t o understand.  presentations  structures  the  can  of as  might  be the overt  process  than  simply  account  be  what  encouraged t o  for  from  the  understanding  emphasis more  to  and  a  deviations  means  attempts  aim  than  into  Less  that  inside  becomes  information  i s put  on  emphasis  on  infer  cognitive  them. facilitate  focus  transmission of construction  the help  they  Student  expectations  teaching  The  a n d how  critical.  efforts  teacher  now  as  rather  Taking  the  must  interesting  learning  knowledge.  think i s  inferred  communication  student  from  assimilation  conceptions  more  Linguistic  procedures  that  competing  Processes  generating  distinguished  i n  facilitated.  aim  using  behaviour,  head  constructivist  experience  of information,  accommodation  the  of existing  becomes  repetitive  that  of to  this  classroom  so t h a t  of  process  activity  knowledge  i s placed  o f t h e computer,  type  with  from  construction. the  student,  the teacher  then  101 becomes for  a construction consultant.  the teacher  suggesting computer  c a n be  watching,  learning strategies. monitor  between  spent  encourages  students  Much  The such  and  teacher  can  relate  of class  time  diagnosing  and  p u b l i c nature consultation  and  between  of both  students  themselves. The  computer  knowledge examples  by  t o which  trajectory damped which  the  the student  simulation,  oscillatory  study  within  Brauner's  sense.  and  to help  to  existing  Brauner's standard assuring  with  This  t h e new  that  perception knowledge  mastery.  seems  ordinary  familiar  and  to the  language, a  in  handle"  understanding to  deliver  i s  and pivotal  of perception, of  The  problems  language,  generates,  consists  Brauner  of  mathematical  at the level  that  relate.  to "provide  This  i t  practical  resistance,  ordinary  i s likely  knowledge.  claim  of  existing  of  a r e a l l examples  context  relate  knowledge  concept  the  to  can already  motion  motion  ideas  c a n be . d i s c u s s e d and a r e p r o b a b l y  student  just  facilitating  new  factual  on the i n  and not  recall  argues:  The k e y t o t h e c o n v e r s i o n of imagery back i n t o l i n g u i s t i c form f o r use i n t h i n k i n g and c o m m u n i c a t i o n i s o r d i n a r y l a n g u a g e .... To learn effectively they [students] need two things that are often not provided:  or  102 1.  T o b e s h o w n w h e r e new c o n c e p t s s t a n d i n relation to familiar concepts, and categories of experience. To be encouraged t o form trustworthy ordinary language accounts of t h e key concepts and t h e c a t e g o r i e s o f experience to which they belong. ( B r a u n e r , 1 9 8 6 , p . 19)  2.  The help  computational  to  existing simple  make  the  algebraic  representation of  of  Problems parabolic  face.  moving,  likely  by  of  computer  etc.  relate  to  esoteric.  A  or  relation  Mathematical  familiar  expanding  as making  f a c e o r an e l l i p i c a l  to increase  NCTM  and  also  properties of the  i n a more  stretching  networking  ears  can  f u n c t i o n s and  the function  eyes,  c o u l d be s e t such  facilitated  i t less  Various  c a n be examined  The  The  a  be m i r r o r e d i n noses,  that  imposing  i s t h e use of quadratic  t o make  variations  and  a n d s o make  relations  can  o f t h e computer  unfamiliar  experience  example  power  context,  ( f i g . 36) .  a circular  face, a  face.  ideas  and  assisted  the meaningfulness  experiences,  applications, of  i s  mathematics.  claims:  Instruction that focuses on networks of m a t h e m a t i c a l i d e a s r a t h e r t h a n s o l e l y on t h e nodes o f t h e networks i n i s o l a t i o n w i l l serve to instill i n s t u d e n t s an u n d e r s t a n d i n g o f , and appreciation f o r , both t h e power and beauty of mathematics. Developing mathematics as an integrated whole also serves to increase the potential for r e t e n t i o n and t r a n s f e r of mathematical ideas.  103 Connecting mathematics with and with daily affairs u t i l i t y of the subject.  other disciplines underscores the (NCTM,  1989, p.  149)  Functions: y=.25*xT2-7 y=sqr ( 1G-XT2:> y=-sqr Cl£-xt2:> y=-sqr<4-<x-l)t2)+7 y=-sqr(2-<x-4)T2)+8 y = s q r ( 2 - C x - 4 > T2)+8 y=sq r ( 2 - < x +4.'» T 2) +8 y=-sqr C 2 - C x + 4 > T 2 ) + 8  \  /  i . . . » . . . > . . . » . . . > .  ^-AXIS : V-AXIS: Figure  \ . . . . . . . . . . . . . ,  -i.O . . .± Q -10...±8  36.  Graphing  a  S  T  E  P  S  T  E  P  1 1  face  A l t e r n a t e conceptions and c o n c e p t u a l Posner, explained conceptual  Strike,  how  for  change.  accommodation  with  Hewson the  and  conflict  Gertzog  constructivist,  Through  active  (1982) learning  construction  p r e - e x i s t i n g knowledge,  the  have i s and  learner  104 changes the he  or  she  stimuli be  way  separate  "errors"  belief.  the  new  "errors"  doing  for  the  student's  that they  the  will  says  obligation  the  teacher's  obligation  the  project.  Constructivist  trusting to  relationship.  solve problems  trust the  teachers often  interest this  and  the  to  Because  to  A  that  respect  social  dwells  helpful  must  demands  c r e a t e and  of  honest of  the  will  so p o s e  is  and  dwells  meaning  and  consultant  on  depends  trust  on  efforts.  academic  a constant  a  students  Students  be  not  students  that  spontaneous  often  to  contract  on  teaching then  tendencies  compete  change  construct  their  to  i f  constructively  to  teacher  even  It  r a t h e r one  a  should  better.  but  be  keeping  believes  c o n s t r u c t meaning.  relationship  constructivist  rapidly  to  opposing  trusting  situation  and  work  way  i f he  conceptual  one  than  develop  learn  react  the  conflict  constructivist  not  so  conceptions  they  to  The  to  only  reward  teacher  student's  cognitive  because  is  Rather  activities  teachers  classroom what  world.  student  rote learning.  the  the  allows  The  and  c o n c e p t u a l i z e s and  simple,  conceptions  important  on  and  she  Allowing  occur  for  in  or  perceives  created.  just  he  must Given  student  disciplines a  difficult  challenge  to  teachers. the  students  computer can  be  can  make  encouraged  calculations to  so  investigate  105 whatever p o s s i b i l i t i e s the  program  Students  don't  example is  a  of  they  find  conceptual need  change  to rely  i n  themselves believe  and  i t .  to  simply  to  aim  Students  decide  information.  the  construction  change.  This  (Figure  3) .  The  the student  meaningful  possible.  of algebra  The  problem  investigate for  believe  and  why  f o r the teacher There  of  makes  authority.  understanding  reason through  the benefit  the information  to  tot r y  i s every  i s precisely  into  This  more  can  to  transmit  t h e computer"  empowers  what  i s no r e a s o n  conceptual  system  theorem  There  at  "plugging  point.  much  on t e a c h e r  of the fundamental  case  plausible.  of  processing  boost  i n processing  ability  t o assume  responsibility  f o r the  construction of  understanding.  Summary The  i n t e g r a t e d use o f t h e computer  education literacy age and  can help than  i s presently  understanding concept  g i v e more s t u d e n t s  we  need  mastery  to  more q u a n t i t a t i v e  t h e case. to  move  actual  The  information  beyond  computation  perception  combined P e r c e p t i v i s t - C o n s t r u c t i v i s t of  the  theory  addressed principles. actual  i n  which this  thesis  First,  perception  are  relevant center  instruction  and  use.  i n mathematics  theory. to on  should  Mathematics  i s The  the  parts  problem  five be  the  basic  aimed should  at not  106 start  and  end  immediately  to  we  i t .  perceive  use  need  not  involve  its in  our  itself  but  understanding  With  be  the  student's  with  the  knowledge  and  an  way  Knowledge  can and  ideas  forgotten.  involve  the  learner  in  Knowledge  i s something  and  themselves.  Discovery,  makes such  The  this as  of  a  on  and  realistic  or  less  goal.  is  knowledge  applied  problems  of  student  can  be  i t  instruction development  should of  no  active  Only belief.  believed  of  meaning.  construct  for  alternative.  of  an  is  actively  reconstruction  power in  than  the  is  computer  abstract  field  s c h o l a r l y dream and  more  Fourth,  i n s t r u c t i o n should  draw  i n a way  understanding. relate to  studied  can  must  the  a  and  which  not  is  relate to e x i s t i n g experience  existing  because  There  a l l  for  and  should  isolation.  compete  learners  how  rather  construction  involvement,  mathematics,  and  to  computational  active  of  basis,  that  the  and  application  instruction should  that  re-invention  necessary.  forced  Third,  world  instruction  ongoing  understanding  quickly  by  be  contribute  interaction  s p e c i a l i z a t i o n , fragmentation this  the  Second,  of  on  of  computer t h i s  delayed.  synthesis  must  immediately  "crunch allow  alternate  the  and  that  taps  Meaningful,  experience with  the  the  numbers".  for  the  of  computer  evolution  provisional  the  Fifth, and  conceptions.  107 With  the  student might  can be  relying for It  computer explore  competing  involves  perceptual These  unit. unit.  many  care  of  the  conceptions  for his belief,  of  more  than  just  recalling  actually believing,  tedious,  the  things  that  rather  on t h e a u t h o r i t y o f t h e t e a c h e r .  u s e means  Chapter  taking  than  simply  Understanding and  repeating.  on  a non-inferential,  be  operationalized i n  basis. five  VI,  principles will  a  Chapter  Perceptivist-Constructivist VII will  Chapter  recommendations.  VIII  report draws  on  field  conclusions  calculus  tests and  of the makes  108  CHAPTER VI PERCEPTIVIST-CONSTRUCTIVIST CALCULUS ON THE COMPUTER The  computer  makes  numerical  approach  do  i t " becomes  with  theoretical  The  line,  c a n be t i e d  a  emphasis  firm  than  of  central  a curve,  rate  c a n be l e f t  ideas  away.  The  and associated  f o rc o l l e g e ( i . e .  of calculus—limit,  o f change,  and slope  t o an e x i s t i n g  grasp  "What y o u c a n  right  algorithms,  o f t h e program  conceptual  possible.  answerable  techniques,  under  The  to calculus  derivation  manipulative later).  t h e e x p l o r a t i o n and use o f a  notion  of a of  area  tangent  function.  i sproviding students  o f t h e uses  of calculus  with the associated manipulative  with  rather  techniques.  Rationale Mathgrapher, by  Human  is  have  a curve  calculated  under  a  psychology, be  integral  derivative, (definite using  curve  approximation.  can  by Steven  R e l a t i o n s Media,  packages, under  written  and other slope  t o have  or derivatives  at a point,  and area  options.  The  approximation.  solutions  The  from  from  s o c i o l o g y , economics,  shown  graphing  calculated  Problems  and published  computer  integral)  a secant i s  Losse  that  at a point.  physics,  and ecology  involve In this  area  trapezoidal  business, medicine  slope  definite  w a y we c a n  109 attempt and  to  so  build  understanding  approach  the  NCTM  rather  just  standard  rote  recall  addressed  to  calculus.  T h i s s t a n d a r d does not advocate t h e formal study of calculus i n high school for a l l students or even f o r a l l c o l l e g e - i n t e n d i n g students. Rather, i t c a l l s f o r o p p o r t u n i t i e s for students to systematically, but i n f o r m a l l y , i n v e s t i g a t e the c e n t r a l ideas of c a l c u l u s — l i m i t , area under a curve, r a t e of change, and slope of a tangent line—that contribute to deepening of their understanding of function and i t s utility in representing and answering questions about r e a l - w o r l d phenomena. The instructional approach to these concepts f o r a l l students, i n c l u d i n g c o l l e g e intending students, should be highly exploratory and based on geometric and numerical experiences that capitalize on computer and calculator-based graphing utilities. The instructional activities s h o u l d be aimed at p r o v i d i n g students with firm conceptual underpinnings of calculus, rather than with the a s s o c i a t e d manipulative t e c h n i q u e s . [emphasis added] (NCTM, 1 9 8 9 , p.180) M.  K.  Heid  manipulation hope o f  remarks programs  in at  a  similar the  c a l c u l u s b e y o n d mere  vein  college  how  level  symbolic offer  computation.  If students were allowed free use of symbolic . manipulation programs such as muMath, c a l c u l u s t e a c h e r s could concentrate on d e v e l o p m e n t o f c o n c e p t s and as a result construct exams which test more than technique. The entire calculus curriculum w o u l d n e e d t o be rethought. ( H e i d , 1983, p.218)  the  110 Numerical  integration  trapezoidal  approximation.  introduced  by  trapezoids  and  (i.e.  gets  of  denoted is  follow  definite If  the  o f an o b j e c t ,  travelled  by  from and  as  approaches  some  function  a t o b.  v(t) =  even  5t  from  integral  how  are changing  time  perceptions  computer  familiarity  with  meaningful.  needed,  i s  formalization.•  a  distance  to  with  What  to  Rather  graphs  justification,  and  problems ecology definite as t h e  comes f i r s t  the come  the  theorem  on i n t e r v a l s ,  make will  b  sociology,  What  of  total  applications,  symbolism  computer  the  and  a to b  fundamental  mentioned.  value.  from  i s  F o r example, i f  then  2  give values of quantities  functions actual  show  (F(x))  i s the derivative  business, physics, psychology, medicine  limit  a curve.  The  i s not  a  of  meaning  velocity,  calculus  number  out the geometric  which  the object,  can be  value).  as area under of  by  specific  amount o f F f r o m  i s given  of  the  t h e area under t h e curve  distance,  changing  processes  t o some  point  derivative  F'(x) then  velocity  i f the area  that  Mathgrapher  increasing  and c l o s e r  integral  the total  the  seeing  on  Infinite  gradually  closer  Exercises  proceeds  i sthe  with  enough  work  of the  later,  when  verification  and  111 Notice  how  development  of  the  theory  Initial  insights,  refined,  some  give  shape  above i n  constructivist-  approach  the opportunity,  toward  a  constructionist "reinventing"  the  ideas  product.  would  to  to  that  would  gives  work  more  than giving  polished,  product outright  t h e m t o make  sense of  Numerical by  secant  formed  process packages to  depict  could  (i.e.  gets  and  asking  be used,  by  as  on w h i c h t h e s e c a n t  does  and c l o s e r n o t have  so f a r t o some  a  differentiation.  o r some  with  gradually  o f t h e segment  could  t h e g e o m e t r i c meaning derivative")  Mathgrapher  processes,  closer  Mathgrapher numerical  on  introduced  i fthe slope  to  ("definite  c a n be  and seeing  value).  display  Infinite  of the interval  a limit  to  the student the  proceeds  the size  approaches specific  differentiation  integration,  decreasing  likely  it.  approximation.  numerical  is  finished  gradually  a process of  produce meaningful l e a r n i n g refined,  the  Perceptivist-  such be  gradually  Perceptivist-  The  be  knowledge  explored,  mathematics  likewise,  the  science.  are  The  product.  claim  and  generalized  to  student  finished  and  some  finished  recapitulates  mathematics  hunches  abandoned,  to a  approach  be done by  of slope  as t a n g e n t .  at a  visual Other hand, point  112 The  meaning  developed  by  using  disciplines concept. simply  By  by  and  previous  the  a  a  retain  memorize and  then  and  wide  0  at  of  integration  then  useful  values  on  Finding  an the  points.  experience  in  connected  to  student  can  organization more  ideas  a  of  investigated  the  and  again  range  critical  easily  experience,  the  is  points.  calculus  apply  once  be  table  slope  are  a  can  activities,  They  is  change  theory  of  and  meaningful  Numerical  from  of  initial  plausible  ideas.  likely  of to  rather than  just  f o r g e t them.  differentiation  Calculus and  is  concerned  slopes  of  calculus,  the  Mathgrapher approximation. side,  x  slope to f  rate  idea  the  understand,  the  Problems  generating  knowledge  pivotal  each  mathematics  looking for critical to  intuitively  integral  how  graphing,  rooting  curves  i t .  show  leads  achieve  the  Minimum-maximum  interval slope  of  Ax  and  The  a x  point -  Ax.  - f (x -  Ax)  at x  a  is  under called  calculus.  point  and  The  areas  former  differential  slope  be;  (x + A x ) 2  latter  finds  Ax  finding  curves.  Choose +  with  create  computer  by a  secant gap  on  estimates  Narrow  the  curve  y  Figure  i*  gap  =  x  2  to  get  find  a better  the  slope  approximation.  at  different  For  points  the (see  37).  une 1 1  o n  :  tv  V = X + 2 X-AXIS : V  ft  —  X "I S  Figure  Is  a  gets based  -2<Q . . .-20 S T E P 5 -T>f-»s-!i . . . p f m S T F P  :  37.  Slope  steady  answer  s m a l l e r and on 1.  2.  3  by  secant  approximation  approached  smaller?  as  the  Numerical  secant  x  spread  differentiation  is  ideas  The  slope of the  the  secant  The  TAB a r e  smaller  secant  x  the  spread,  tangent  at P  and  approximately value the  chosen  closer  the  the  slope  of  equal. for  Ax,  slope  of  the the  secant  3.  AB  at  P.  If  the  will  will  Ax  be  is  =  2  to  the  small  close  conventional d(x )  be  enough,  enough  to  notation  slope  slope  of  the  the  the  slope  slope  f o r the  tangent  at  P.  problem  (some n u m b e r a t  x =  of  AB The  above  is  5) .  dx This  is  the If  the  "definite a  at  slope  at  expression  any  x  derivative")  general  curve d  called  point  (x ) , i s c a l l e d 2  =  and  for  5  (in a  is  the  x  i s desired,  the  derivative  a  way  number.  slope  of  the  at  any  then and  dx is  a  point is Doing of  x  temperature  =  days  -.8°/day,  as  Calculus  can  population 100,000  3  =  at  =  gives  matter  of  calculus  Consider  Then  P  a  differential  y t  (as  that  the fact  slope the  derivative  2x).  change.  patient's  function  is  starts +  an as  - . l x  2  example  from  a  function  of  +  1.2x  .6°/day,  from  2000*t . 2  used  + 7  at  illustrated be  involves  100,000 At  time  rates  medicine.  time  i s given  A by  98.6. days,  -.2°/day,  i n Figure in  finding  t  grows =  5  10  38.  geography. and  at  to  years  A  city's  an  amount  dP/dt  is  115 20,000  people/year  people/year.  X - S X I S : V - X IS :  Figure  This  V—A X IS :  39.  Here estimates  10  9 .  . .  ±  Time v s . c i t y ' s  an  i tw i l l  example sell  i t i s  i n Figure  40,000  39.  I  temperature  f *:  5 S T E P  - - - S « m ifil iFS ;g  i s  years  i s illustrated  Time v s . p a t i e n t  v = j.« w y t« ts + ii «%j  Figure  at  Q . . . .12 S T E P .1 3 & . . .i i O S T E P  38.  X - r t X I S :  and  ±  STEP  5! ifli 1H1  ft  population  from  n T.V.  sets  business. after  A  spending  firm x on  116 advertising  x  x = 100 d n / d x dN/dx  i s -  T.V.'s money  thousands  100  T.V.'s/lOOO  i s  100  T.V.'s/lOOO,  / 1000.  A t some p o i n t  at  X - A X I S : —  A  dollars.  advert,  x  =  300  advertising  at  At x  dN/dx  =  200  i s  -300  i s a waste  of  £  — x -t i: + oiaa*x -*• w  v  of  (see F i g u r e 40).  i- u n c t l o n : V=- v f 2 + 3 O O -*• v. +  V  i n  V T c;  Figure  '8 . . .  „qi  :  40.  O  :,2 f5 Ft A  Dollars sold  Consider  S T E P  «i  spent  another  R T £  on  i> 3 P  ±  « 8 *3  advertising  example  from  v s . T.V.  medicine.  percentage  ( e x p r e s s e d as a d e c i m a l f r a c t i o n )  who  a new  accept  months  dP/dx  medicine  i s 6%  at  12  months  The  of doctors  i s g i v e n i n F i g u r e 41.  p e r month,  sets  i t i s  At  6  1.8%  117 per of  month  a n d a t 18 m o n t h s  change slows  i t .55% p e r month.  as t h e l i m i t i n g  value  The  rate  i s approached.  Funct i on:  y= i -s*<p <•'. -0. 2*'<)  v'asi.'-fc.XV' <. — it . X - A X I S : V - A X I S :  Figure  >  O . . . 2 - 4 S T E P 8 .' . . ± S T E P  41.  Time v s . f r a c t i o n medicine  Geometry p r o v i d e s long  piece  width  x  of string  and  length  Notice  t h e maximum  slope  there  introduced  i s  X  t h  month  20  x  found  -  area  (sin(x)  that  that  Optimization  a 4 0 cm.  a rectangle area  at x  20x =  -  10.  theory  a  could  of x . 2  The be  (see Figure 42). business  finds that  year +  so  accepting  Consider  t o form  i s attained  0.  the  doctors  example. used  example, from  of  of  being  up a n d down company  y = 40,000 be  this  and explored  Another An  1 , ±  i s  cos(x)).  i s the following. i t ssales  given Using  by  during the  the  function  t h e graph  i tcan  118  X - r i X I S : Y - f i X I S :  Figure  B*. . . 2 9 S T E P 8 < . . ± 2 0 S T E  42.  S'  Width of formed  ±  5  P  rectangle  vs  a t x = 2 mo  i s -$52930/mo  at  i s $3870/mo  x = 7 mo  at  x = 8 mo  at  x = 12 mo  The  significance  can  be  with  of  rectangle  i s -$45318/mo i s $55125  of positive,  discussed  area  /mo  negative  reference  and zero  to the calculus  slope (see  Figure 43). The  computer  calculus  come  creates  alive  the  immediately  applications  t o which the student  is  a mystery  the  then  less  world.  possibility with  can r e l a t e .  a n d more a u s e f u l  way  of  having  meaningful Calculus of  seeing  119  i'  Y = . 4 © y Q < S I H < X ) + C C S < X > > X - A X I S : 8 . . . 2 - 4 S T E P ±  y-flxis:  Figure  -seaeo,..seeee  43.  Numerical  Time o f y e a r  t h e curve  the integral  find  trapezoidal  the  chart  Figure  sales  integration  Consider Use  versus  S T E J  option  44 g e n e r a t e d  2 5 10 15 20 25 30 100  2  and  on t h e i n t e r v a l  on t h e Mathgrapher  approximations  below  No.  y = x  compare  by t h e computer.  Trapezoids  program t o  t o the area. i t with  Area  0<x<10.  F i l l i n  versions  of  120  I I  i"-K.  I I I !  i i r i iXr w X - A X I S v —A X I S :  44 .  Is  steady  trapezoids on  •  - 2 P . . . 2 0 S T E P -iQ . . . 5 8 6 S T E P  Figure  a  ?  i i i ! i i .i. . r. > ^ _ _•—j _ _i _ _ , - T- -r i i . . . . • • • • • * • « •  •  1  y = x  5  *  •  5 8  2  answer  increases?  approached Numerical  as  the  number  integration  of  i s based  3 ideas; 1.  The  sum  trapezoids  of  the  areas  of  i s approximately  approximating  equal  to the  area  under t h e curve. 2.  3.  The  finer  get  to the actual  I f we  we  make t h e d i v i s i o n s  we  area.  use a s u f f i c i e n t l y close  the closer  get  as  the  f o r the application  fine  to the actual  division  area  a t hand.  we  will  as needed  to  121 The ,  x  0  conventional n o t a t i o n f o r the dx.  2  0  to  a  general x  10.  dx)  2  This  is called  I t i s a number, expression  this that  a matter  of  is a  fact  the  f o r an  is called  function  the  the  definite  area  integral  +  3  is  from  curve.  If  i s desired, (i.e. integral  expression  i t is Y = x  above  integral  under the  indefinite  general  problem  f o r the  and  is  area  a  (as  C)  3 The with  under  the  graph  y  t r a p e z o i d a l approximation.  used to l i m i t  area  improve  -  the  =  =  is  2  found  below  More t r a p e z o i d s can  approximation  C'dx  x  by  forcing  i t to  be  a  693.2..  -12  Business northern of  skis  can  use  climate in  x  ski  month  t h  ideas company is  where  S i s i n thousands  total  sales f o r the  J  7* ( 1 - c o s  o  Consider in  physics.  keeps v(x)  going =  3x  2  an A  + 2x.  of  year  example particle and The  integral  determines  given  (;c/6*x) )  faster  from  by  dollars  i s given = of  the  moves  faster  i t s sales function  See  such  according  distance the  (S)  The  by, Figure  application in  A  below,  f o r t h a t month.  $84, 000. the  calculus.  a  of c a l c u l u s way  to the  particle  45.  goes  that  i t  equation  122 v=" * ( 1 - c e s ( 3 . T  14*x/6) )  '/I  I v I%  I I  I !  I!  X - A X I S : Y - A X I S : Figure  between For  Q. -2  45.  i  Pi.  ;  i  1  example  and  i  r  V..  STEP STEP  15  Month vs.  time  i  sales  i  I  i n that  month  time  2  i s  the  definite  from  time! =  1  to  time  + 2*x  dx  2  =  5  the  integral. particle  goes,  3*x  This computation are  2  procedure of  calculation—for force  148.  change  Figure  power as  by  the  distance  46  allows  determined through  example,  changes,  See  generalized  quantities  undergoing  work as  =  as  for  the  variables  that  course velocity  current  of  the  changes,  changes.  123  Fun<:t i e n s : y=2*vcT2+2*v;  ' l l l l l  X - A X I S : V — A X I S :  Figure  O . . . ? S T E P ft, , . 2 0 9 S T E P  46.  Time v e r s u s  Another  example  . 5  i  0  velocity  of  calculation  flux  concerns the rate  of use  uses  electrical  according  where the  k  energy  four  hours  the  10*x*exp(-x)  For  a whole  day  the  dx  9.1  family  10*x*exp(-x)  dx  =  A  family  x  i s time.  During  uses;  kwh.  See  Figure  47.  uses; 9.98  in  t o the e q u a t i o n below,  and  family =  quantities  of e l e c t r i c i t y .  i s i n kilowatt-hours,  first  of  kwh.  See  Figure  48.  124 (  — -  .nin  n II ir.  in i ni.....  rJlllllli.. J! 11 f i: 11 n. •-All IIIIII iJUMI I il! lr>\. Jll III I I ill*. illllll I IIIIIK  .  IIIIHIY l l l l l l l l IIIIIII IKliltHII IIII illIIIV.. alilini I l i l l i l l l l l ' (111 I il 11!!(l!!lilfl UJtlJLiJ  iJUJi.lJl.lJ.  X - A X I S V —AX I S Figure  8 . . . 2-4 S T E P 8 . . .5 STEP  47.  Time v e r s u s  Rate  ± . 5 o f Energy  Use: 4 hours  Functions: y=10*x*e. ;p ( — x !> ,-  ' mm rsiprgista • • • • .«..-! t«381STv . . ;jEll!!.?l^!il.-iS?!i!Si . ^i!I^PI£3pi!>f|L  .  .  saiiiii^S'iteMtiiav-  •  •  ^fejiKiii^ldaL  3B8 r;*ca! aa sat saa FW*.  a* 113 i H £fil IRS -"3 r.i I It ITr*u_  X - A X I S : V - A X I S :  Figure  48.  8 . . . 2-4 S T E P 8 , . . 5 STEP  ± .5  Time v e r s u s  o f Energy  Rate  U s e : 24 h o u r s  125 A  whole  the  web  of  questions  calculation the  to  1.  why  use  2.  what  3.  i s the model  geometry,  distribution  area  between  the  a r e a b e t w e e n t h e two to x =  3 i s given  2  The  procedure  the  computation  =  2  of  the  the  company?  integral  curves.  With  The  from  regard  curves,  curves below.  to  consider  area  from  x  by;  15  i s g e n e r a l and (see f i g u r e  average  connect  i t is?  i d e a o f a r e a b e t w e e n two  (x +3)-x  The  as  application  geometric  -2  to  are t h e r e f o r the power  the  =  addressed  realistic?  an  the  be  reality.  implications  Consider  can  value A  the  computer takes  care  of  49).  of  a  continuous  function  f is  0 given  by  \ ,  f (x) d x  .  This corresponds  to  some  (b^aj value  of  A* ( b - a )  the i s the  example, field  the  area  by  milligrams  at time the per  such  under  following  of medicine.  bloodstream given  function  The x  that  the  the  curve  situation amount A  after  function millilitre.  A  from  could  of drug  a dosage = The  rectangular  3e"  -86x  a  to  b.  occur in a  area For  in  the  patient's  i s administered i s ,  amount  where of  A  is  in  drug  in  the  126  F u n c t i ons: y=x1*2+3 v= (x t2+3) - (. x T2 . ' >  X - f t X I S : V - A X I S :  Figure body The  - 5 . . . 5 Q . . , -4Q  49.  Area  i s going average  interval  S T E P S T E P  i 5  between two curves  down  as t h e body  metabolizes  t h e drug.  amount  o f t h e drug  i n t h e body  over t h e  [a,b] i s g i v e n by  Over t h e f i r s t  two hours  t h e average i s  3 * e x p (-.86*x) d x = 2.87/2  = 1.435 m l  ( F i g u r e 50)  2 Consider figure formed. by  another  i srotated  example  about  The volume  f o r geometry.  the x axis  of the solid  i n space,  I f a plane a solid i s  of revolution  i s given  127  X-flXIS: V - A X J S : F i g u r e 50.  7C*  where rotated. =  1  Q...2  D : = : 3 S T E P Average value  [ f (x) ] a,  2  b  x = 2  generated  has  = ± of a function  dx, are  F o r example  and  . _L  STEP  the  7t*[exp(x)]  2  dx  of  i f the curve  i s rotated volume,  bounds  given =  about  the  plane  y = exp(x)  the  x  axis  by  86.1.  See  Figure  51.  region  between the  x  shape  128 F u n c t i ons: y=exp \. %.') VT * [ e x p ( x ) l 2 A  dx =  86.1  y = 3 . 1 4 * C exp <: x ) ) 1 2  i  j  n  A I .1 I ! /! I I • A\ I I i  IMM  X-AXIS: V A X I =  - 5 ... 5 S T E P O , . .5QQ STEP  —  Figure  51.  Volume o f  J.  s j ^ - L 11 J . I . t . - :  5 0  revolution  Summary The use  examples  i n this  of t h e computer  chapter  have  shown  represents the p o s s i b i l i t y  that of  1.  t u r n i n g t h e m a t h e m a t i c s c u r r i c u l u m on i t s h e a d .  2.  providing powerful students  (not  complete Algebra 3.  techniques  just  the  f o r t h e use  10%  who  the  of  a l l  meaningfully  12 n o w ) .  i n t r o d u c i n g meaningful  applications  of mathematics  immediately. The  computer  described  aided,  i n this  numerical  chapter  approach  i s aimed  at the  to  calculus  following:  Enabling  students  mathematical useful  eyes  cognitive  and  ideas  daily  to  with  through  mathematics  the  ideas  i n the  combat  isolation  from  world  perceive  ideas  repertoire  knowledge  the  as  training.  with  fragmentation,  see to  mathematical  disciplines  student's  the  and  working  from  other active  tendencies  of  alienation  beliefs  of  the  of  learner  life.  Actively of  and  intellectual  Engaging  in  to  involving  knowledge  reducing  the  by  the  learner i n the c o n s t r u c t i o n  offering  emphasis  i n c r e a s i n g  meaningful  on  brute  student  examples,  calculation  independence  and and  experimentation. Networking through rather  new  applications with than  Allowing  concrete knowledge just  elaborated than  to a  past  view  f u r t h e r fragmenting  the  knowledge  than  knowledge  construction  to  take  experience gradually  place and  initially,  by the  the  as  integrating,  knowledge. acceptance through  of  believable,  i t i s used,  learner.  knowledge  can  New rather  tested  Finally, be  new  numerical,  experimentation.  becomes  memorizable, upon  and  to  experience  and  rather  formalized.  130 The  perceptivist-constructivist  integrated start the  mathematics  working kind  standards  of  provides  systematically calculus  document,  theory  which  the insight  toward  program  of  needed  to  the realization  of  suggested  i n part  computer  by  the  NCTM  suggests:  Computing technology makes t h e fundamental concepts and applications of calculus accessible to a l l students. The a r e a u n d e r a f i n i t e p o r t i o n o f a curve, f o r example, can be approximated geometrically by p a r t i t i o n i n g ... A l l students could use a graphing u t i l i t y t o i n v e s t i g a t e and s o l v e o p t i m i z a t i o n problems, including the maximum-minimum problems traditionally associated with the first college-level course i n calculus, without c o m p u t i n g a d e r i v a t i v e . ... Using interactive graphing utilities, college-intending students could examine other characteristics of the graphs of functions, including continuity, asymptotes, end b e h a v i o r ( i . e . b e h a v i o r a s x -> <») a n d c o n c a v i t y ... Computing technology also permits the foreshadowing of analytic ideas f o r college intending students. From a computer-graphics perspective, f o r example, a differentiable f u n c t i o n c a n be v i e w e d as a f u n c t i o n h a v i n g the property that a small portion of i t s g r a p h , when h i g h l y m a g n i f i e d , a p p r o x i m a t e s a l i n e segment.... Instead of devoting large blocks of time to developing a mastery of paper-and-pencil manipulative s k i l l s , more t i m e a n d e f f o r t should be s p e n t on d e v e l o p i n g a c o n c e p t u a l understanding of k e y i d e a s and t h e i r a p p l i c a t i o n s . A l l students s h o u l d have t h e b e n e f i t o f a computer enhanced  131 i n t r o d u c t i o n t o some o f t h e t y p e s o f p r o b l e m s f o r w h i c h t h e c a l c u l u s was d e v e l o p e d . (NCTM, 1 9 8 9 , p p . 1 8 2 - 1 8 3 ) The the  realization  extent  that  constructivism can  show  how  manageable. through seeing  real the  represents education.  of this  perceptivism  c a n show to The life  world more  program  make  how  student  can  t o do  the  becomes  terms  than  show  of  what  to  i t and t h e  computational  applications, i n  i s facilitated  the  the  computer  involved, process  calculus.  schooling—it  do,  complexities  actively i n  to  of This  represents  132  CHAPTER V I I REPORT ON A FIELD TEST OF THE CALCULUS UNIT The tests  aim o f t h i s  twelve  i n Chapter VI. students  Vancouver,  field  i s t o report  of the Perceptivist-constructivist  outlined  This  chapter  B.C.,  chapter  teacher  notes  which  then  be  were r e a l i z e d  major  goal  schools.  concept  mastery  results  this, will  o f t h e extent  be to  i n t h e classroom.  paper  has been  principles  o f t h e computer  o f mathematics  must  t othe  Perceptivist-constructivist  of this  Perceptivism  instruction  of the  test  utilization  meaningfulness  given  i n  1989.  Following  i n terms  Perceptivist-constructivist effective  sequence  questionnaire  suggest  School  t h e purpose  outlined.  and discussed  Purpose o f t h e f i e l d The  discuss  and student  they  principles  first  unit  t o 41 g r a d e  1988 t o A p r i l  The i n s t r u c t i o n a l  will  summarized  Secondary  September  field  calculus  was g i v e n  Templeton  from  will  tests.  students  at  The u n i t  on  aim beyond t o actual suggests  can inform the t o improve t h e  education  suggests  that  mere  philosophy  also  that  proficient  i n the discriminate  i n secondary curriculum  factual  perception  t o s h o w how  and  recall  and  and use.  The  students  must  become  selection  and  skilled  133 use  of  the  Constructivism actively they  meaningful  challenge  new  way  concepts  of  the  field  test  teacher  principles  mathematics  classroom.  The  author  calculus, adapted  from  by  given see  a  in  determine  valid  treatment turn,  author  could  which  set the  experimental noncomputer  an  on  M.  made  a  type  comparison  in  in  must for The extent  see  the  the  computer  unit  in  exercises Notes  were  questionnaire and  an  students  was  could  Perceptivist-constructivist  in  the  classroom.  for  a  This  computer  study  Heid of  This,  experimental  compared  Kathleen study  later  of  be  group.  comparison.  could  (1982).  teacher  stage  this  the  instructional  student  performance  control  determine  Morel  a-  could  that  students  applied calculus  and  shape  group  to  realized  and  meaning,  competition  being  i f the  the  survive  was  experimental  taking  that  students  part  become  experience  and  Bittinger  the  to  in  must  e s t a b l i s h e d concepts.  designed  based  present  to  older,  the  perception.  students  and  with  theoretical  kept  and  use  which  of  c o n s t r u c t i o n of  and  purpose  modes  that  integrate past  ongoing,  to  suggests  involved i n the  must  belief  different  with  that  of  short  has  performance  work  assisted  stops  (1988)  in  in  using  a of  fact the  computer no  as  mention  calculus.  program meets most o f t h e  five  effect  applied  principles.  of  computer  concept  Although  of P e r c e p t i v i s t - c o n s t r u c t i v i s tp r i n c i p l e s i s  made, h e r the  a t o o l to teach  Heid  following  summarizes  motivated  development  on  criteria her  resequencing  student  set  out  study of  of  in the  skill  and  in  the  performance  manner:  During the first 12 weeks of an applied calculus course, two classes of college students (n=39) studied calculus concepts using graphical and symbol-manipulation computer programs to perform routine manipulations. Only t h e l a s t 3 weeks were spent on skill development. Class t r a n s c r i p t s , student interviews, f i e l d notes, -and t e s t r e s u l t s w e r e a n a l y z e d f o r p a t t e r n s of understanding. Students showed better understanding of course concepts and performed a l m o s t a s w e l l on a f i n a l exam o f r o u t i n e s k i l l s a s a c l a s s o f 100 s t u d e n t s who had p r a c t i c e d the s k i l l s f o r the e n t i r e 15 weeks. ( H e i d , 1988, p.3)  Instructional The integral worked area value  sequence  usual  topics  calculus on  under of  a  antiderivative traditional,  applied  were p r e s e n t e d  slopes, a  in  curve,  rates area  function,  of  to the  change,  students.  and They  minimum-maximum,  between  two  curves,  volume  of  revolution  and  major  difference  from  applications. noncomputer  differential  A  courses  was  the  average  "concepts  135 first"  approach  teaching they  of  up,  calculation  a  student  computer.  Set  only and  up,  sequence of  and  could  more  solving This  was  Computers  facilitated  graphing,  differentiation  Computers  easily  place  of  routine  as  priority.  were  the were  to  important done  analysis shown  used  by  were how  Hand-in  the  done  to  by  handle  assignments  for evaluation.  problem  the  well  calculations  hand.  i n action.  of  disciplines  f u n c t i o n s by  calculus  that  as  The  possibilities  teaching  later  in  3.  the  were  student  2.  instruction.  perceptual  r e p o r t i n g and  exams w e r e u s e d Instruction  1.  the  range  Students  polynomial  and  understanding  Graphing  student.  in  wide  concepts.  the  and  preceded  skills  from  develop  curriculum  concepts,  open  Problems  to  immediately and  the  computation,  of  the  ideas  of  because:  and  provided  development  seeing  involve  especially  integration.  data  ideas  for  discussion  d i d n ' t need  to  so  take  i n the abstract.  Computers through  realistic  problems  information processing  changing, solving  made  and  generating curve  tables  fitting.  of  such  manageable as  values,  parameter equation  136 Formal,  abstract  some i n t u i t i v e actively they  did  not  on  need  be  calculations  derive  i s  of magic  and  in  best  here  factual  was  to  The  be  Field  first  delivered notes  the  the  were  because  teacher  to  developed,  computer  manages  back  computer students  that  the  with  a  Indeed  i t might  be  a l l possible  worlds,  to  formulae.  principles, this due  as  least  approach  is  i t s complexity. as  mastery,  to  insincere  to  or at  directly  concept  possible, to  proceed  promise  i t s shortcomings. toward  do  by  relevance,  with formalism, later.  as  functions  generated  temptation and  students  the  bad  first  and  demonstrated  perception  as  proceed  recall  shifted  leave  with beginners  later  be  start  were  having the  But  formalism  has  on  giving  of  from  premature  to  to  mystery  the  perception.  must  then  likely  counterproductive  past  rely  until  Students  the  concepts  that  regression.  goal  from  w i t h o l d as  and  a l l functions  using  The  New  manipulation of  preferable,  delayed  i s present.  simply  argued  to operate  the b l i n d  were  calculations.  might  which  sense  to  assimilated  technical It  limits  i n exploration  knowledge.  merged and  like  understanding  involved  transmit  the  ideas  f o r those  or The  actual  using  relevance balance  meaning who  a  and  need i t ,  137 The  author  walking  around  students. here.  The  offered  the  got  teacher's  time  individual usually  time  than  in  the  formative  the  calculations,  Most  of the  spent  t e n minutes  i n  class  observing  and  Formal  the  learning  Freed  will  engaged  this  i n  learning  evaluation  into  observed  was  process  an o n g o i n g  principles practise  discussing  from  the  times.  spend  the  activity.  lessons, teaching  of particular  B e c a u s e t h e t e a c h e r was  with  required  the teacher  t o the overcoming  actual  translating  time  at  were  c a n become  process  help  helping  lessons  i n length.  "something-for-everybody"  Perceptivist  were  observers.  formally.  that  learning.  may  instructing  suggests  to  What i s  teacher  Observation  directed  the  The  specific  than  as  problems.  offering  was  neutral  of lesson presentation the teacher  of  summarized  interpreted  did  individual,  Rather  are  helping  solving  was  involved  majority  be  students w i t h problems.  less  duty  to  notes  and  while  s e t of observations, which  computer  right  notes  generating study.  o r n o t by f u t u r e ,  little  more  should  field  observing  of these  o f an h y p o t h e s i s  Because  the  results  i s a preliminary  students  collected  classroom  results  confirmed  spent  the  The  substance  be  regularly  actively as  i t  obstacles involved occurred,  process.  were  observable  i n the classroom. temperatures,  as  Students  populations,  138 profits,  e t c . , as  mathematics. to  relate  Another onto  One  to  example,  because  the  concept  were the  observed  nature  demanded  a  relevant  mathematics  perception  demanded  problem,  example,  sparked  streams  of water,  their  behavior  etc.  The  of  was  as  computer felt  was  e t c . and as  guilty.  to rely  on  verify One  could on  set  before  the  The  problems solve  science, for  footballs, to  explain  distance,  their  of  be  about  their  arc,  cooperatively ideas to  hunches.  a student remarked,  the computers  rate  trajectory,  became  out  two  ideas to  path,  in  difference  that  ideas needed  then  students t r i e d would  The  velocity,  helped  language,  discussions  situation  like  i t s  different  the  areas  and  to bear.  etc.  catch the  the  going  ordinary  numerous  such  city  to  occasion,  about  brought  medicine."  especially  problems  what  from  learning  too easy  the  economics,  experimental  a  for  visualize  other  one  "easy  things,  of  could  On  s t u d e n t s engage ideas  mathematics,  it  of  of  relevant  much e a s i e r  to  talking  population  actual  we  process."  the  c o u r s e as  dosage  " I t was when  to  the  of  business, population,  The  students  use  applications  change.  the  related  memorization,  understanding  between  if  the  initially  The  students  a  of  student explained,  economics,  also  ideas  student described  divorces,  problem.  the  these  see Some  "I  found  f o r many q u e s t i o n s  139 when  I  could  easily  have  done  the  calculations  in  my  head." Constructivist emerging active was  in  practise.  giving  student  concern.  remarked,  i t  encouraged For  roots  set  theorem  equation.  the  to  The  work  playing  "The  with  initiated  the  actually  learning  learning  of  applied faces. whether and A  Ideas they  concept  was  can work.  a I  did  curves tried  Past  assimilation  the  make  out;  was  (see  student  reflected commented,  on  promoted through  "The  computer students  then  aid.  By  I  was made  situation Figure  computer were  the  student  that  same  c o o p e r a t i v e c o n s t r u c t i o n o f m e a n i n g was  students  to  computer  the  experiences  rational  One  know  (The  to  was  computer  teacher  The  for  problem  roots  the  drawn  36)  shows upon  dialogue.  observable  calculations. can  a  concerning  the  the  not  easier.)  be  think  exercise  great  not  beneficial  to  of  calculus."  graphing  was  time  solving  was  investigation.  computer,  calculus  with  "It  from  through  little  formalism  power  observed  problem  being—use  backward  computer  that  algebra,  hint  calculating  empowered s t u d e n t explained,  only  in  with the of  be  learning  students  example, theorem  with  also  Because  student  fundamental  simply  information.  premature  themselves." the  promoted  directed, A  because  could  Independent  c o n s t r u c t i o n was  spent  was  principles  draw t h e  graphs  as One on  140 the  screen  think  easily.  about  (sic),  what  Then the  i n s t e a d of  function  waste  simulated  could on  observed  be  'the  questionnaire  the  degree  the  basic  five  Constructivist pp.105-107)  17  of  The  to  5,  principle active  Questions  the  to  graphs  is  graph."  In  example,  past  conflicting  views  were  frequently  situations.  5.  and  11,  which  no  Thirty  aim  at  aware  of  Perceptivist-  in practise,  (see  knowledge  Question  1,  2,  of 3  principle  1.  concerned  synthesis  of  Question  and  19 4,  6,  8,  9  knowledge, concerned  and  assimilation  student  the  perception,  principle  concept  are  explicit  however.  to  questions  the  realized  18  10  given  students of  had  2.  12,  was The  c o n s t r u c t i o n of  experiences, 20  5)  unit.  actual  knowledge,  and  the  themselves,  4,  concerned  (Table  students  Questions  A)  and  principles  concerned  principle  on  program being  principles  16  for  the  time  results  determining  15,  exercises,  of problem  end  past  draw  discussing details  the  3.  to  drawn  at  and  time  more  that  Students  The  the  spend  computer.  Questionnaire  students  can  information  modelling  experience  we  replies  7  principle  relation  questions and  and  13,  to 14,  accommodation,  were r e c e i v e d .  A n a l y s i s o f mean response t o each q u e s t i o n The  together  mean with  response the  to  each  standard  question deviation  was and  calculated t-score  141 (critical  value  Table  Results  that  5.  student  be  some  one, i n  acceptable  item  the  rejected  items 6).  H  6.  see  The a l t e r n a t i v e  2 a n d 3. actual  f o r each  of  i s accepted f o r  f o r t h e group.. realized  seems  one,  rejected  was  seems t o  hypothesis  principle  was  0  i n  partially  to principles  0  H  students  being  null  for  the students  principle  appear  results  two, s y n t h e s i s  f o r each  students.  of t h e items  that  This  this  Thus  i t  the principle  concerning  particular,  i tprovides  facilitated calculus  problems  encouraged For  Q  was  and  three,  was  H  i s of  that  0  was  Thus, i t  realized  together  the  by t h e  with  the  perceptivist  classroom.  In  t h a t t h e computer  solving  promoted  accepted  hypothesis  one  i n  interdisciplinary  H  was  some e v i d e n c e  setting which  5 and 18.  evidence,  realized  principle  knowledge, null  the  4,  principle  were  o f knowledge,  principle  provides  principles  the  The  as  given  There  that  five  and consequently that  i n Table  suggest and  3 and 17.  are  perception.  For  would  two  (see Table  1, 2,  actual  to  however w i t h r e g a r d  perception  accepted  questions  a n e u t r a l 3.  practise.  Consider  each  was  evidence  realized  The  a r e summarized  response  principles  items  2.16).  of  actual  meaningful  p e r c e p t i o n and  thinking. active f o r items  rejected  construction 6,  f o r item  8 and 9 7.  of  while  Students  142 did in  not r e a l i z e the face  "black some  box" aspect  actually  work  doing  might  need  was  D  and  positive  done  four,  12.  The  was  bothersome changed  This  suggests  i n reaching and manual  10  19  but accepted f o r  here  i s mixed.  t o t h e work t h e n e g a t i v e  items  suggest  For  principle  accommodation, for  items  evidence special to  14,  H  0  five,  16  and  don't  concept  was a c c e p t e d  15,  20.  the principles  importance  i s the evidence  that  the  involved  with  ideas  authority  relying  so  extent., suggest  partially  realized  that students  without  a  to  much  three  constructivist  of Of  seemed become on  f o rprinciple  f o rprinciples  i n t h e computer  rejected  realized.  them  with evidence  and  preponderance  were  on  why.  13 b u t  allowed  together lesser  The  Evidence  students  assimilation  computer  of the teacher.  The  results  realize  f o r item  indicates  feel  compromise  10 a n d 19 d o s u g g e s t  they  further  experience,  were a b l e t o r e l a t e 1 1 a n d 12 may  (see  calculation.  t o past  and  evidence  a  to  by n o t  f o r themselves  relation  on i t e m s  Apparently, the  or short  calculation  f o r items  results  unit.  f o r example).  principle  11  guilty  calculation  rejected  items  of this  t o be  computer  For  felt  the  p.138,  T h i s was d i s a p p o i n t i n g  o f t h e computer  who  note  between  H  principle.  of the goal  students  field  this  the five,  and f o u r t o  principles  classroom.  were  143  Table 5 Student  Questionnaire  - Computer C a l c u l u s  Unit  Directions 1) 2) 3)  Please c i r c l e the number of the response which most c l o s e l y matches your agreement or disagreement with the statements as you understand them. There i s definitely no r i g h t or wrong, favourable or unfavourable response - j u s t your honest opinion counts. Please t h i n k about and review t o y o u r s e l f what happened i n the computer classroom. Base your responses on what you r e c a l l a c t u a l l y happened i n the computer classroom.  Questions 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)  There were many r e a l world examples used t o show the uses of calculus. The questions used made me t h i n k about how s i t u a t i o n s would a c t u a l l y look i f I were to experience them myself. The computer took care of most of the c a l c u l a t i o n s f o r me. The computer questions made me use knowledge from d i f f e r e n t sources e.g. physics, business, common sense. When I could get problems that r e q u i r e d knowledge from d i f f e r e n t sources i t helped me understand the mathematics. I t was e a s i e r f o r me t o get i n v o l v e d i n l e a r n i n g c a l c u l u s with the help o f the computer. I found the c a l c u l u s u n i t b o r i n g . I couldn't r e l a t e t o i t . When the computer d i d the c a l c u l a t i o n s I could concentrate on what the r e s u l t s meant. Not knowing the d e t a i l s of how the computer made i t s c a l c u l a t i o n s l e f t me with an incomplete understanding. I d i d n ' t enjoy the c a l c u l u s u n i t because i t was too abstract and incomprehensible. I had to draw on knowledge I already had t o answer many of the c a l c u l u s problems. Mathematics i s no more nor l e s s comprehensible with the computer than without i t . The teacher spent a l o t l e s s time, compared t o a r e g u l a r classroom, making c l a s s p r e s e n t a t i o n s , l e c t u r e s e t c . I worked more on my own or i n student groups i n the computer classroom i n s t e a d of r e l y i n g on the teacher f o r i n s t r u c t i o n . The computer encouraged me t o d i s c u s s things more with other students to t r y t o f i g u r e out what was going on. The use of the computer i n s o l v i n g d i f f e r e n t problems helped me understand and b e l i e v e i n c a l c u l u s rather than accept i t on the teacher's a u t h o r i t y . C a l c u l u s does not have any uses. C a l c u l u s i s not r e l a t e d to any subject areas outside of mathematics. C a l c u l u s i s not u s e f u l i n r e l a t i n g things i n my l i f e past, present or f u t u r e . By i n t r o d u c i n g d i f f e r e n t subject areas i n t o c a l c u l u s problems I got confused. I t d i d n ' t help me understand.  Responses Strong Agree  Agree  Neutral  1)  5  4  3  20)  5  4  3  -  Disagree 2  Strong Disagree 1  2  1  144  Table 6 A n a l y s i s o f mean r e s p o n s e ( c r i t i c a l t=2.16)  t o each  question  o CM n  -  cn  <M  „  N  e\i  _ in  ^  «?  co  £  ^  CO  CO CO  *~.  CO  CO CO CO  CO  i-  CM  cn  CO  ~~  -  °.  CM  co  m  CO  in  CM CD*  m  •>a-  cn evi  ^  "* cvj  CM  —  CM  oo  m  to  «  «9  ^  CO  T-  CM  -  n  co  in co  .  ^1  CO  CO  CO  o>  CD CM CO  o CO  CO CD  " i s i i  3  145  B)  A n a l y s i s o f responses by i n d i v i d u a l Positive  5,  6,  8,  9,  positively of  responses 11,  stated  principles  18,  7,  questions,  null  also  and  H  H  x  students  on  some  students  tenuous  scored  that  constructivist the  computer There  the  suggest  0  some  on b o t h  H  x  Q  group  H , the 0  i s  a  neutral. and  on  positive  on  positive  of  fourteen  Perceptivist-  realized  i n  practise.  of q u e s t i o n s .  i s some see  of  this  positive  had H  principles  7.  i s not  had  group  on  response  seeing  students  As  evidence  to  Perceptivist-  being  realized i n  classroom.  of  some  tentative  a  evidence  f o rthe  perceptivist-constructivist  computer-mathematics that  of  are, there  theoretical  stated  both  latter  being  i s , i n summary,  realization in  H  as t h e r e s u l t s  suggest  on  x  indication  principles  negatively  i n Table  students  This  t o 17,  analyzed  S i x students Eight  1),  responses  response  H  (group  4,  the realization  student's  student  negative.  gave  were  a  were  questions.  Constructivist Six  i s that  3,  f o r the realization  a r e summarized  on n e g a t i v e .  0  2),  1, 2,  16  Negative  scores  scores  15,  indicate  evidence  i s that  x  student's  H  14,  (group  offer  hypothesis,  negative but  20  Students  3.  13,  questions,  the results  neutral Ten  12,  19,  principles. basis,  (4-5) on q u e s t i o n s  i n practise.  10,  students  teacher  should  classroom. be  able  Field t o observe  goals notes the  146 realization  of  classroom. students a  some  The  i n total  majority  of  constructivist  of  the  five  questionnaire  results  s e e some p r i n c i p l e s  individual  students  principles  partially  computer-mathematics  classroom.  principles  indicate  realized see  i n the that  and t h a t  perceptivist-  realized  i n the  147 Table 7 Analysis o f responses by i n d i v i d u a l ( c r i t i c a l t=2.57) Score i  i  1  H /H,  S.D.  1  3.8  .7  4.28  H  t  2  . 6  2  4.0  .8  4.76  Hj  3  .8  3  4.0  1  3.7  H,  1.6  .7  -4.83  H,  4  3.2  1.2  .63  H„  2.6  .5  -2  H.  5  3.3  .9  1.25  H  1.5  .8  -1.52  Ho  6  4.0  .9  4.16  1.5  .5  -7.5  H>  7  4.0  .9  4.16  H  t  1.5  .5  -7.5  H,  8  4.0  1  3.7  H,  1.1  .4  -11.88  H»  9  3.3  .9  1.25  H„  1.8  . 7  -4.13  10  3.7  .8  3.33  H,  3.0  . 6  11  3.2  1.1  .69  H.  1.6  .7  -4.8  H,  12  3.4  .7  2.1  H.  3.8  . 4  5.0  H,  -3.1  H,  0  0  Mean  Group 2  Mean  Student  t  students  t  . D.  Ho/H;  -4.0 0  H> H„  0  Hi H.  (contradict) 13  3.5  .8  2.38  H,  2.1  .7  14  3.5  .8  2.4  H,  3.0  .6  15  4.2  .6  7.5  H,  2.1  . 9  16  4.2  .6  7.5  H,  2.5  .1  17  3.2  1.1  .69  H„  1.5  .5  -2.5  Hi  18  3.5  .7  2.63  H,  2.1  .4  -5.62  Hi  H„  2.0  0  -2.50  Hi  0  H.  -2.43 "  -1.1  Hi H.  19  3.3  .7  1.57  20  3.6  .6  3.75  H,  2.6  .7  -1.37  H. H.  21  3.5  .7  2.63  H  t  2.5  .8  -1.52  22  3.5  .5  3.84  H,  2.1  ,4  -5.  23  2.7  1.1  -1.03  1.6  .7  -4.82  H,  24  3.2  1  . 7  -2.41  H.  25  3.9  1.1  0  -5.0  Hi  26  3.7  1.3  .8  -1.5  H.  27  3.4  .9  .8  -1.5  Ho  28  3.2  1  .5  +3.00  Ho  H.  62  H,  (Contradict) .74 3.10  H„ H,  2.3 1  2.00  H„  1.5  1.67  H.  2.5  .74  H„  3.6  .6  .62  • H.  3.5  . 4  3.12  Ho  .9  1.66  H„  2.3  . 9  -1.89  Ho  (contradict) 29  3.1  (contradict) 30  3.4  148  CHAPTER V I I I CONCLUSION AND RECOMMENDATIONS The  main  thesis  m a j o r way t o i m p r o v e to  translate  into  of this secondary  that  principles  problems  with  have  information  transmission,  necessary  overwhelming. innovation  from  mitigates  software.  Theoretical  report  support  o f these  In and  on a  this  basic  extent  chapter  translating  age  were  technological  with  been  o f these  appropriate  practical  have  was  computational  of reality  conclusions  . The c o n c l u s i o n s of  integrated to  the  of  examples  provided  i n  claims.  principles  computer  amount  t o student,  computer  test  The  pragmatic,  large  teacher  arguments,  field  final  discussed.  a  of  t h ee f f e c t s of both  i s t h e personal  a  that  information  problems  and  type  that  actively  perception.  i n perceptions  The  that  been  and  inherent  this  that  o f knowledge and  t o actual  agenda  difficulties  curriculum  as c o n s t r u c t o r  utilitarian  assumed  a  Perceptivist-Constructivist  the learner  traditional  that  education i s  the learner  leads  has been  mathematics  p r a c t i s e by emphasizing  involves  paper  which into  a  focus  will  be  stated  on a ) t h e f i v e  Perceptivist-Constructivist  mathematics  the principles  program were  p r a c t i s e by a teacher  a n d b) t h e observed  as  and students i n  149 a  grade  for  12, c o m p u t e r  further  informed  calculus field  research  computer  will  usage  mathematics,  many  selection  t e n such  of  Finally,  some  Professional with  new  (b)  must  theory,  mathematics.  Finally,  boards  along  to  and  the  term,  offered.  ministries terms  The  (a)  to  deal  competent  with  philosophy,  computer  technology,  be  developed i n  ..to  secondary  f u n " usage  wastes  t o do much g o o d . of  of  actual  These  concern.  computer  "friday  A  discussed.  teachers  deal  should  or  i n  and  be  of  equip  and i s u n l i k e l y and  constructivism,  redone  able  mathematics  curricula  the long  be  Random  resource  be  areas  secondary  themselves.  will  will  organized,  i n  complexities.  exploit  valuable  In  suggest  must  programs  systematically  Once  questions,  classroom  Suggestions  practical  are i n three  Organized  rethink  i s  questions  development  professional learning  follow.  recommendations  recommendations  test.  education  a  (c)  need  to  active  construction,  perception,  perceptivism.  the entire  curriculum  Perceptivist-Constructivist  needs  to  be  how  a  lines.  Conclusions The  writer  philosophy knowledge  of  of  education,  acquisition,  application  to  this  the  paper  has  perceptivism,  shown a  theory  c o n s t r u c t i v i s m and t h e i r problem  of  computer  usage  of  joint i n  150 secondary  mathematics,  secondary how  the theory  have in 1.  mathematics  been  suggest  education.  might  given.  can  actually  Five  Detailed translate  basic  to  improve  examples  into  principles are  of  practice developed  the paper. Aim at a c t u a l perception In  so  doing  beyond  process  goes  applied. the  of real  learner  recall  with,  utilitarian  of  the  factual  perception  and  Figure  use  beyond  The c o m p u t e r  synthesis  domains.  By  perception,  process  encouraged.  fragmented  utilitarian by  choosing  collection  of  well  This  schooling  this  to  becomes  completion  mathematics  interaction  than  The and  an  that  by  Refer to  having to  modes  of them  repertoire  a specialized, knowledge  eclectic, manages  applying  knowledge  symbolize  conceptual  learner  tested  of  different  approach  toward  from  to  complexities.  dynamic  Rather  one.  mastery  I , f o r example.  academic  works  get  knowledge  in  and t h e languages  is  to  concepts.  academic  forcing  interact, a fluid  learner  concept  facilitates  and  to  situations.  encouraged  i n that  educational  Promote  world  of,  mere  education  1 i n Chapter  i s  and  managing t h e computational  2.  ways  the  conceptual  the  integrated, information best  of  systems,  a as  the  situation  this  process  by  capability. Chapter  I I f o r an  student  Active is  that  With  teacher  1  and  processing 3  i n  c o n s t r u c t i o n o f knowledge  by  than  Figure  i t s passive  graphing  reception.  understanding,  can  and  a face  relate  Computer less  dependent  for  i n Chapter  example,  into  the  the teacher  knowledge  domains.  This  competence,  promotes  empowers  learning  encourages  from  concepts  handling  dependent  on  on  their  own  to  Figure  36  knowledge  and  V,  the incorporation of past  experience  the  a t once  reconstruction, re-  students  more  to  i s  practical  a r e manageable  begin  c a n make  i t  relevant  rediscovery.  Refer,  Facilitate  so  for  student  and  resources.  and  Table  t h e computer,  and/or  algorithms  doing  information  of mathematics  the  invention  on  active  immediately  the  facilitates  example.  optional  applications  of  computer  reconstruction, re-invention or rediscovery  not  more  to  rather  necessary.  so  The  boosting  Refer  Encourage'the the  dictates.  situation. the student  ordinary  language  process  exploits  the integration students  to  of  use  By to or  so draw  other  existing experience a l l  their  152 abilities. project 5.  Allow help  f o r example,  i n Chapter  clarify  explore taking  conflict  ideas.  away,  the  computer  ideas  so  example,  of  support  limitations  instructional  into  The  help  theoretically  classroom.  does that  practise.  with  this  computer  concepts them.  theorem  to  of  indicate  and  As  an  algebra  field  need  and t h a t  give  here  motivated  i s reason  suggestions  been  tentative  preliminary can  actually  lessons and  developed  i s the idea  there  has  i s  some  the  practise.  test  the theory  t o be  that  into  The p r o g r a m ' s  has promoted  practise  out  The  evidence  but  techniques  the writer  to carry  translated  calculus  to the hypothesis  translated  that  be  computer  i s  V. supplied  and discussed.  with  can  been  actually a  understanding  and reuse  fundamental  has  given  What  the  to  can v i e f o r b e l i e f .  things  i n , retain  Evidence  Details  be  t o do  i n Chapter  can  ideas  students  exercise.  exercise  theory  and  see  enables  i s e x p l o i t e d here  students  to believe  into applications  as t h e i r  Competing  doing-is-believing empowers  and accommodation t o  By p l u n g i n g computer  different shape.  to the trajectory  IV.  f o r conceptual  right  The  Refer,  that  further. theory  to believe work  i n the  153  Suggestions f o r Many  further research  questions  possibility  of  mathematics  classroom  Certain  suggest  extensive  theoretically need  empirical  more  field  work  i s needed  1.  Can  the  the  without for  computer toward  lowering  example,  calculus  on needs  benefits  of  use  of  shown  be  competition mathematics  from  involved or  overload? differentiate  do It  been  made  example,  evidence  relevant  Can  away this Heid  first  year  almost students  as on  done use  to and  from  be  done  (1988) , computer  well basic  as skills  understanding.  interfering,  benefit  have  Further  determine the  the  trade-offs  necessitate.  applications  students  possibility.  levels?  how  do  computer  introducing  the  For  application?  to  in  mathematics  conceptual  research  By  shift  instructed  c o m p u t e r u s a g e may 2.  provide  the  questions.  students  better  real  claims  performance  has  traditionally and  to  as  usage  verification.  following  computation  a  motivated  would  answering  computer  becomes  that  to  themselves  they may  competing  clarify  or  conceptual  confuse? conflict  in  applied  simply  suffer  well  curriculum  be and  ideas,  does Do and  computer information  necessary  to  instruction.  Different levels Can  ability  levels  of conceptual  teachers  mathematics teachers  tolerate  develop  prepared  instructional  theory,  understanding  of  and  teach  Under what to  handle  applications  of  training  and  present  applied  c o n d i t i o n s can philosophy,  proficiency needed  to  and teach  mathematics?  teacher,  in-service  an  the  computer  Perceptivist-constructivist adequacy  different  conflict.  curriculum?  be  may  The  present  programs  teacher  needs  to  be  analyzed. Whenever  results  constructivist  computer  do  they  represent  or  just  Hawthorne  over  the  computer results Is  long i s just  because  there  computer  favourable  more  as  to  student  whether  approach lecture  engage  student  task.  i s required or  time-on-task  m i g h t be t h a t  and,  improvement  not  the  produces  of i t s novelty.  expectation  knowledge  Evidence  obtained,  a technological fad that  teacher  the  are  substantive  effect?  exploration  traditional  Perceptivist-  mathematics  genuine  term  to  as  But perhaps  as a  active  result,  the  with  the  approach?  t h e computer  an  simple  than  with  yield  The  can help t o  constructor more  of  time-on-  transmission i s superior  because  of direct  environment. and Do  Student  students  draw  using  computer  Are  chaotic  more  public  student Is  Does  than  with  resolved  with  lecture  by  from  the burden  the  student  of to  approaches?  cooperative with  nature  learning  traditional of  the  with  computer  methods?  computer  Does  encourage  cooperation?  interdisciplinary  longer  based  easily  than  I s t h e freedom  than  more  more  help  different  mathematics  experience  presentations?  exploited  investigate there  activities  conceptions  computer  computation  the  on a n d i n t e g r a t e p a s t  lecture  presentations?  than  computer  learning  with  interdisciplinary  increasing  already  cognitive Or  i s  methods?  learning, help  the  incorporate  r e p e r t o i r e and  the  retained  traditional  computer  t o the mathematics,  active  perceived  learning  r e a l i s m and r e l e v a n c e ,  relate  longer.  an  be  Perceptivist-Constructivist  alternate  with  his  i n v e s t i g a t i o n may  mathematics  traditional  to  control of the learning  o f f task.  while  Is  teacher  "realism  so  and  subject  impossible?  student i t into  retain i t relevance"  as i n t r o d u c i n g more c o m p l e x i t i e s , difficult  by  making  156 10.  Is  interdisciplinary  and  computer  used  more  than  teaching?  Ford  (1979)  of  students  believing the  student  hopefully  learning  "going  little. to  ownership.  wrote  from  about  through The  way  the  world  see  motivate  learning  the  Whether  the  motions"  in a to  is  prevalence  computer  student  this  traditional  the  the  accepted  new  so  empowers way  feel  a  needs  but  would  sense  of  empirical  investigation. Recommendations On wish  the theoretical  to  use  Perceptivism  the and  understanding  what  plans  available.  software, of  Pilot  comparisons integrated, classes  need  teachers  sequences  c a n be  a  will  be  to  lessons programs  between  in  and  some  to  do  with  for the  training  area.  using of  other  basis  teacher  computer  should  be  need  t o be  traditional  to  the  integration  developed and  mobilize computer.  developed  Perceptivist-constructivist made.  background  developed  into  programs  who  a  trying  be  teachers  or  provide  opportunities i n this  mathematics  and  they  to  Professional in-service  must p r o v i d e  Hardware,  need  Constructivism,  theories,  Action  mathematics  computer  educational  computer.  level,  and  made  so  that  computer mathematics  157 Only  when  the  computer  is  systematically  knowingly  exploited,  using  principles,  can  mathematics  empirically  computer  compared  outlined  i n Table  computer  lesson  packages.  in  an  configuration grades  form,  an  attempt  initial  attempt  suggested,  restructure  perceptions  students  computer  would  be  but  an  curriculum Figure  development  not  that  needed  required an part  might  one  be  offered  in  outline  Brauner  (1987)  i n terms  to  of  attain.  add-on of  to  computer  constitute,  just  tested  suggests  as  as  Perceptivist-  8  curriculum  integral  8  would  of  a  mathematics  mathematics  and  capabilities, w o u l d be 12,  are  are  achieve  to,  the  classroom  be  to  the The  such  i t .  Refer  useful  in  a to  3.  Table  of  to  would  such  various  sequences  This  legitimately  programs  Table  of lesson  theoretical  Programs  are  the  goals.  8-12.  as  The  exploit  Constructivist  in  to traditional.  8 are needed  systematically capabilities  sound  and  coordinated  plan  education the  listed  be  f o r the  involving  computer.  The  horizontally,  philosophy,  basic  across  curriculum.  actual  The  perception.  curriculum Curriculum  the  improvement  computer  the  table,  s y s t e m a t i c a l l y e x p l o i t e d , throughout grades  to deliver  achieving  hopefully  would  aim  topics,  8at and  158 the  actual  gradually reached  perceptions  f i l l  they  the table  concerning  the  as  would  be  a  central be  number  to  enable  ( e . g . mean, 12,  function  various  computer  symbolic  manipulation  be  the  level  importance specific  be  a  at  which  details and  philosophy,  perception  cooperation  information  table to  and  but  constructivism  The  indicate the  concept  that  computer  the  IV,  and and could  quickly  call  for  t h e demands  a  program. to  realize  i s not confined by  of  t o produce  need  to  Perceptivism-  information  involves  The  not i n the  the  t h e computer  provided  In  graphing  i n  a  could  graph).  lies  age m a t h e m a t i c s  insights  be  i s possible.  reconcile  a  involving  processing  and m i n i s t r i e s of education  mathematics.  of  as  here  t h e p o t e n t i a l o f t h e computer  promise  bar  central  information  the  mathematics  contemporary, Boards  i n  might  o u t l i n e d i n Chapter  student  on  capabilities  such  the  integrated  number  repeater,  o f t h e recommendation  consensus  that  as  of  i s  completed  perceptions  capabilities  used t o boost  to  8,  actual  might  The  computer  computer  decimal  consensus  specification  grade  concept and v a r i o u s  exploited  grade  In  would  implications  principles.  design  program.  produce,  educational  Perceptivist-constructivist, mathematics  to  practical  Perceptivist-constructivist table  aim  the  age  growing  interdependence information  of different  management  knowledge  function  interdependence.  Meaningful  meaningful  as a whole.  the  modes  actual computer  education  of perception  perceptions assisted,  be  information  and t h e  of technology  i n  mathematics  at students  the  long  this  requires  A curriculum  and aimed  should  domains  based  on  achieving  term  age, e d u c a t i o n a l  goal  of  reform.  3^ 1»  o  tn i> ca <  o 1—•  <  CO  Mathgrapher  Triangles  of  Problem-solving booklet  Spreadsheet Linear regression  Graph lo solve any equation Interest  Houghton Mifflin (Games) Software (Math)  Enrichmeni  program  Functions booklet  World population  Mathgrapher  Equation booklet  Quad, equation  P& S package  Ohm's Law  Single machines  Saga as equation solver  Saga  Simulation & problem solving  10B disk Alg. practice Class spreadsheet  < -c ~ £i > a o 2^ n.  Quad, equation Herons Sim. equation solver  •<£ S 3 2 Herediiy Dog  Angles, lines, triangles Polygons Circles Co-ord. geo  j  Bar graph  Prob. and statistics  Possible  Measurement Saga linear program functions Proofchecker Logo Brochmann Logo  Square root Pythagoras Equation solving Operations on integers  Trajectory angle Decimal repealer Math 8 disk Dynalist  Angles Lines Triangles  Evaluate expressions Mean Fraction—decimal Table values Geometry quiz  Graphing  Symbolic manipulation  Table 8 Computer Math:  Formula evaluation  Grade  Applications sofitvare  Programming Logo  Programming Basic  PROGRAM ASPECT  160  Activities.  J  rsl  1  161  BIBLIOGRAPHY Albers, D. J . , Anderson, R. D. & Loftsgaarden P. (1987) . U n d e r g r a d u a t e p r o g r a m s i n t h e m a t h e m a t i c a l and computer sciences t h e 1985-1986 survey. Washington, D.C: Mathematical Association of America. A y e r s , T., D a v i s , G., D u b i n s k y , E . , & L e w i n , P. ( 1 9 8 8 ) . 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I n s t i t u t e f o r Research in Educational Finance and Governance Policy N o t e s , 5 ( 3 ) , 3-4.  170  A  P  P  E  N  D  I  X  The r o l e o f mathematical r e a l i s m i n p e r c e p t i v i s m  171  page  In  order  to  19  simplifies  importance. implicit to  be  in  the  by  aspects In  i t s use  in  extent  realism  that  is  to  i t  perception  is  appears  aspects  of  applications  demonstrate  revealed  to  and  on  considerable  mathematical  meant  1  the  that  student  description  of  reality.  Perceptivism realism  as  insists  that  with  the  September is  not  the  the  writer  finds  with  mathematical  not.  Brauner  realism  logical  as  things  having  in  have  says  that  nothing  to  implied.  Brauner  about  Brauner  reality. compatible  himself  content  sees  mathematics  Whether  world do  of  with  that  mathematical  be  system. the  of  Perceptivism  to  (personal  p o i n t e d .out  nothing  whereas  seems  self  has  handmaiden  Brauner's  position  for  1989)  might  conventionalist consistent  w r i t e r , Brauner  15,  Perceptivism  writer  viewed  of  mathematical  are  the  discussions  accounts  the  mathematics  communication,  This  to  point  Figure  mathematical  perceive  writer  of  of  of  computerized  the  meaning  through  to  explanation,  theoretical  diagram  The  supplied  a  version  possible  reality.  the  A  facilitate  does  with  that  as  a a  system  experience  is  mathematics  as  such. In  Table  "examples",  1  on  page  25,  Brauner provides  along examples  side of  the  heading  perceptions  in  172  the  different  between  modes.  seeing  standard,  identifying  and  relational  two  i n  primal,  and  cases  primary  perception  i s  or  this  view  possible  i n the other  thematic  perception.  direct. '  that  At  observable.  only  are  used  at least  probable  just  concepts  o f knowledge.  the various  accepted  perception-enabling,  on  par with some  respect  dubious  world  seeing,  inferred  knowledge and  reality  i n would about  choice  i s  concepts  the  public  hold  that  aspects  theories  from  of  other  i s a v a i l a b l e , from  conceptual  and  theoretical,  b u i l d e r s but they  arbitrary  i s the  i s never  mathematical  to epistemological  or  cases"  or  realism  A free  i n Brauner's  but not i n relational  confirmed  mathematical  as  conventional  i s  that  and  the  perception  an  the extent  only  i s possible.  But perception,  they  reality,  some  latter "actual  best  uninferred"  cases,  forum,  give  at  of  thematic  that  objectionable  "direct,  best  To  theories  finds  cognitive  a  hold  but  to  In the  o f t h e two " e x c e p t i o n a l  implicit  among  to  perception  regard  respectively.  recognition  writer  characterization  domains  with  possible  theoretic,  operational  i n pre-existing perceptions  What  and  and  wants  not  to distinguish  thesistic,  or recognizing  identification pattern  i s careful  mythic,  Brauner  perception"  He  a r e a l l on  immediacy  grounds.  None  except can  be  173 assumed than it  to give  a  more  "direct"  any o t h e r .  For this  reason  i s reasonable  to claim  that  least  probable  knowledge  discussion  of  available  i n Copeland  (1985).  extension  Perceptivism, is  based  on  to  the  of  explain  ordinary  mediate  happens. view  that  relationships argues  A  to  the  i n the world.  pedagogic  of  a  to  i n  realism of  as  stands  up  standard such  as  realism perceived  Quine,  way  mathematics  as  to  nicely  perception "table"  In mathematics  relations  realism,  on  What m a t h e m a t i c a l the  basic  center  language  perception.  i s  considerations  uses  Even  and  at  more  realism  gives  philosophical  directness  competition.  concepts  thing  writer  that  can give  of  P h i l o s o p h i c a l l y , the view  the  "horse"  and  reality  of reality.  issue  The p e d a g o g i c  and  students.  holds  mathematics  this  the  ' both  simplicity  motivate  the writer  i n the d i r e c t i o n of mathematical  considerations. the  of  of aspects  detailed  The  perception  the  or same  interprets i s are  actual  f o r example,  that  From among the various conceptual schemes best suited t o these various pursuits, one— the phenomenalistic—claims epistemological priority. Viewed from within the phenomenalistic conceptual scheme, the ontologies of physical objects and mathematical o b j e c t s a r e myths. The q u a l i t y of myth, however, i s r e l a t i v e ; r e l a t i v e , i n  174 this case, t o t h e e p i s t e m o l o g i c a l point of view. This point of view i s one among various, corresponding to one among our various i n t e r e s t s and purposes. (Quine, 1953, p.19) There  i s no  theoretical have other  fact,  point  a s much form  version  Brauner  of view.  o r as l i t t l e  o f knowledge.  of mathematical  perception,  which  scientific  perception  agrees,  apart  Mathematical immediacy Beth,  knowledge  to reality  i n arguing  realism,  Brauner  from  compares  does  i n physics,  not  some can  as any  for a  new  mathematical accept,  w h i c h he  with  does.  Deductive theories cannot, i n general, provide an adequate description of" m a t h e m a t i c a l s t r u c t u r e s ; t h e r e f o r e , i t seems l i k e l y t h a t our knowledge o f such s t r u c t u r e s has, at least partly, an intuitive, an immediate character ... I t does n o t f o l l o w of course that a l lmathematics c o n s i s t s of such knowledge, or that the deductive theories o f modern mathematics should be judged e x c l u s i v e l y by t h e i r c o n f o r m i t y t o o u r mathematical i n t u i t i o n . I t seems w i s e r t o suppose, with Bernays, that mathematics r e s u l t s from our r a t i o n a l l y remodelling those fundamental i n s i g h t s which o r i g i n a t e from our mathematical intuition. In the respect m a t h e m a t i c s might be r e a s o n a b l y compared t o physics. (Beth, 1959, p.643) What  this  interpretation pedagogic thinking  writer  of Perceptivism  possibility about  i s claiming  that  mathematics,  i s that  i s a  provides  a  realistic  philosophical a  mathematics  useful  way  education  and of and  175 the  mathematical uses  Perceptivism would debate  be  may  neither  between  formalism  also  of the be  defensible  surprising realism,  is s t i l l  an  open  computer.  nor  Other  and  useful.  upsetting  idealism, question.  forms  of This  since  the  intuitionism  and  176  Appendix References Beth,  E. W. ( 1 9 5 9 ) . The f o u n d a t i o n s o f m a t h e m a t i c s . York: H a r p e r and Row.  Brauner, 15,  C. T. 1989.  1989. (M.A.  P e r s o n a l communication, Thesis defense).  New  September  Copeland, B. (1985). A proposal concerning the a p p l i c a t i o n of the p r i n c i p l e of complementarity i n philosophy. M.A. thesis, University of British Columbia. Unpublished. Quine, W. V. (1953) . From a l o g i c a l point Cambridge: Harvard U n i v e r s i t y P r e s s .  of  view.  


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