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Relationships between classroom processes and student performance in mathematics : an analysis of cross-sectional… Taylor, Alan Richard 1987

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RELATIONSHIPS AND  BETWEEN  STUDENT  CLASSROOM  PERFORMANCE  IN  PROCESSES  MATHEMATICS  An A n a l y s i s o f C r o s s - s e c t i o n a l D a t a From  t h e 1985 P r o v i n c i a l  Assessment  o f Mathematics  By Alan B.Sc,  Richard  Taylor  The U n i v e r s i t y o f B r i t i s h  M.Ed., W e s t e r n  Washington  Columbia,  U n i v e r s i t y , 1976  A THESIS SUBMITTED I N PARTIAL F U L F I L L M E N T THE  REQUIREMENTS  FOR  DOCTOR O F  1967  THE DEGREE  OF  OF  EDUCATION  in THE (Department  We  FACULTY  o f Mathematics  accept to  THE  OF GRADUATE  this  and Science  thesis  the required  UNIVERSITY  OF  STUDIES  as conforming standard  BRITISH  Septenber,  Education)  COLUMBIA  1987  © Alan Richard Taylor,  1987  In  presenting  degree at  this  the  thesis  in  University of  partial  fulfilment  of  British Columbia, I agree  freely available for reference and study. I further copying  of  department  this or  publication of  thesis for by  his  or  the  representatives.  that the  It  is  granted  it  by the that  head of copying  my or  this thesis for financial gain shall not be allowed without my written  /*f/0  TlCS  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  £/9?~7 i  DE-6(3/81)  an advanced  Library shall make  understood  permission.  Department of  for  agree that permission for extensive  scholarly purposes may be her  requirements  &0#C>4 T/Qfl/  ii ABSTRACT  The the  purpose o f t h i s  use  of  schooling  survey  i n v e s t i g a t i o n was t o examine,  data,  relationships  between  and outcomes, as measured by s t u d e n t  The i n p u t s o f s c h o o l i n g were c o m p r i s e d  of  grouped  students  under  and t e a c h e r s '  1  each  inputs  o f a number  of the following categories:  backgrounds;  students  and  1  teachers'  p e r c e p t i o n s o f m a t h e m a t i c s ; c l a s s r o o m o r g a n i z a t i o n and solving  processes.  Outcome  measures  a c h i e v e m e n t on t e s t t o t a l , problem A  related  appropriateness  question of using  involved  exploration  cross-sectional  survey  output  question,  subsequent  address  longitudinal  this  study,  student  s o l v i n g and a p p l i c a t i o n s .  found  To  problem-  included  d e c i s i o n s b a s e d on t h e r e l a t i o n s h i p s variables.  of  achievement i n  mathematics. variables  through  which  of  data  the  t o make  among t h e i n p u t and results  utilized  from  the  i n s t r u m e n t s , were examined f i r s t w i t h p o s t - t e s t d a t a and  a  same second  w i t h t h e i n c l u s i o n o f p r e - t e s t d a t a as c o v a r i a t e s . Data c o l l e c t e d from t e a c h e r s and s t u d e n t s o f Grade 7 i n t h e 1985 in  B r i t i s h C o l u m b i a Assessment o f M a t h e m a t i c s were r e - a n a l y s e d order t o l i n k  responses  t o Teacher Q u e s t i o n n a i r e s w i t h t h e  students* r e s u l t s i n teachers' r e s p e c t i v e classrooms. were  received  from  students  i n 1816  classrooms  Responses across  the  p r o v i n c e and from 1073 t e a c h e r s o f Grade 7 m a t h e m a t i c s . The  d a t a underwent s e v e r a l s t a g e s o f a n a l y s i s .  Following  t h e n u m e r i c a l c o d i n g o f v a r i a b l e s and t h e a g g r e g a t i o n o f s t u d e n t data t o c l a s s calculated  level,  between  Pearson  pairs  product-moment c o r r e l a t i o n s  of variables.  Factor  analysis  were and  iii multiple  regression  stages of the A  of  the v a r i a b l e s  variety  significant  of  utilized  relationships  student behaviors,  were t e a c h e r s  were  found 1  and  were  student  subsequent  to  a t t i t u d e s toward problem s o l v i n g , and  methods  used  by  show t h a t  student  between Among  achievement  t h e number  teachers,  p e r c e p t i o n s o f mathematics, and socio-economic also  found  achievement.  t o be most s t r o n g l y r e l a t e d  approaches  Results  at  analysis.  number  t e a c h e r and  techniques  and  student  status.  background,  students'  and  t e a c h e r s ' p e r c e p t i o n s of mathematics, c l a s s r o o m o r g a n i z a t i o n and problem-solving in  student  however,  achievement. was  measuring lower solving thinking  processes  higher  The  for  f o r measurable  variances  amount o f v a r i a n c e accounted  achievement  cognitive levels  on  application  of behavior,  than  items which measured c o g n i t i v e b e h a v i o r  for, items,  on  problem-  at the  critical  level.  Through examination the c r o s s - s e c t i o n a l prediction  of  change  classroom  models.  a l l account  in  and  change  of the  standardized beta  longitudinal in  models,  achievement  process  variables  based was  D i f f e r e n c e s , however, were found  other c a t e g o r i e s .  weights  i t was on  found  from that  corresponding  similar  for  for variables  both  i n the  iv TABLE OF CONTENTS Page ABSTRACT  i i  REFERENCES  v i i  TABLES  Vii  FIGURES  ix  APPENDICES  ix  ACKNOWLEDGEMENTS  X  CHAPTER 1.  STATEMENT OF THE PROBLEM  1  1.1  BACKGROUND  1  1.2  PURPOSE OF THE STUDY  3  1.3  ASSUMPTIONS OF THE STUDY  3  1.4  SIGNIFICANCE OF THE STUDY  5  1.5  SUMMARY  6  REVIEW OF THE LITERATURE  8  RELATED LITERATURE ON THE EFFECTS OF SCHOOLING  8  CHAPTER 2. 2.1  2.2  2.3  2.4  Early Studies More R e c e n t F i n d i n g s F a c t o r s A f f e c t i n g S t u d e n t Outcomes  8 9 10  FACTORS AFFECTING THE LEARNING OF MATHEMATICS  12  Teacher Background and Behaviors Classroom Processes Student A t t i t u d e s  14 15 17  SIMILAR STUDIES OF A CROSS-SECTIONAL NATURE  20  The The  20  F i r s t I n t e r n a t i o n a l Mathematics Study N a t i o n a l Assessment o f E d u c a t i o n a l P r o g r e s s (NAEP) - A R e p l i c a t i o n S t u d y  SUMMARY  23 25  V  CHAPTER 3.  RESEARCH DESIGN AND METHODOLOGY  26  3.1  A MODEL OF INPUTS AND OUTCOMES OF SCHOOLING  26  3.2  POPULATION AND SAMPLING PLAN  29  The The  29 29  1985 M a t h e m a t i c s A s s e s s m e n t 1987 V a l i d a t i o n S t u d y  3.3  OVERVIEW  OF THE METHOD OF STUDY  3.4  INSTRUMENTATION Booklets f o r Students S t u d e n t s Background Items Students' Perceptions o f Mathematics Teacher Questionnaire 1  30 31 31 36 37 37  3.5  DESCRIPTION AND DEFINITION OF THE VARIABLES  38  3.6  DATA COLLECTION  41  1985 D a t a C o l l e c t i o n C a l c u l a t i o n s f o r Dependent V a r i a b l e s C a l c u l a t i o n s f o r Independent V a r i a b l e s 1986-87 D a t a C o l l e c t i o n  41 42 42 43  DATA ANALYSIS PROCEDURES  43  1985 A s s e s s m e n t D a t a Correlational Analysis Factor Analysis Multiple Regression 1987 V a l i d a t i o n S t u d y  43 44 44 44 46  SUMMARY  47  FINDINGS  48  4.1  PREPARATION OF THE DATA  48  4.2  DESCRIPTIVE ANALYSIS OF THE INDEPENDENT VARIABLES  50  Student Background V a r i a b l e s Teacher Background Classroom Organization Problem-Solving Processes Teachers' Perceptions o f Mathematics Students' Perceptions o f Mathematics  51 54 56 59 65 69  3.7  3.8  CHAPTER 4.  vi 4.3  CORRELATIONAL  ANALYSIS  73  S t u d e n t Background and Achievement T e a c h e r Background and S t u d e n t Achievement C l a s s r o o m O r g a n i z a t i o n and Achievement P r o b l e m - S o l v i n g P r o c e s s e s and S t u d e n t Achievement T e a c h e r s ' P e r c e p t i o n s o f Mathematics and Student Achievement S t u d e n t s ' P e r c e p t i o n s o f Mathematics and A c h i evement Interpretation of Correlation Coefficients 4.4  4.5  4.6  4.7  CHAPTER 5. 5.1  5.3  74 76 78 81 83 85 86  FACTOR ANALYSIS  87  Student Background F a c t o r s Teacher Background F a c t o r s Classroom O r g a n i z a t i o n F a c t o r s Problem-Solving Processes Teachers' Perceptions of Mathematical Students' Perceptions o f Mathematical Summary o f t h e F a c t o r A n a l y s e s  87 89 90 93 95 95 95  MULTIPLE REGRESSION  ANALYSIS  Topics Topics  96  Student Background Teacher Background Classroom O r g a n i z a t i o n Problem-Solving Process Teachers' P e r c e p t i o n s o f Mathematics Students' P e r c e p t i o n s o f Mathematics  97 98 99 100 101 102  THE PROVINCIAL MODELS  103  Problem S o l v i n g Test Total Applications  104 105 106  THE SURREY MODEL  108  1987 P o s t - T e s t M o d e l 1987 L o n g i t u d i n a l M o d e l  108 110  SUMMARY AND CONCLUSIONS  113  SIGNIFICANT FINDINGS AND  CONCLUSIONS  114  The F i n a l M o d e l s  129  IMPLICATIONS FOR DECISION MAKERS  134  vii 5.4  5.5  LIMITATIONS OF  THE  STUDY  13 6  E s t i m a t i o n o f C l a s s Means D e f i n i t i o n of V a r i a b l e s I m p a c t s o f I n d e p e n d e n t V a r i a b l e s on t h e Achievement of I n d i v i d u a l Students The C l a s s a s a U n i t o f A n a l y s i s  137 138  IMPLICATIONS FOR  139  138 138  FURTHER RESEARCH  REFERENCES  141 TABLES  1.  Domains and  Item Assignments  34  2.  Summary S t a t i s t i c s  3.  I n d e p e n d e n t V a r i a b l e s and  4.  Frequency  5.  M o t h e r Tongue  51  6.  Time Spent  52  7.  Educational Level of Parents  53  8.  Time Spent  58  9.  Problem-Solving  f o r Grade 7 T e s t B o o k l e t s  Distribution  36  T h e i r Sources  39  of Class Sizes  50  on Homework  on C l a s s r o o m  Activities  S t r a t e g i e s Taught  of Problem-Solving  60  10.  Sources  Exercises  61  11.  Activities  12.  Problem Types Assigned t o Students  63  13.  Classroom  64  14.  Number o f P r o b l e m - S o l v i n g  15.  Teachers'  16.  R e l i a b i l i t y Analyses  17.  D i s t r i b u t i o n s o f I n d e x Numbers f o r T e a c h e r s ' P e r c e p t i o n s of Mathematics  69  18.  Students'  70  19.  R e l i a b i l i t y Analyses  20.  D i s t r i b u t i o n s o f I n d e x Numbers f o r S t u d e n t s ' P e r c e p t i o n s of Mathematics  Used t o M o t i v a t e Students  62  F e a t u r e s t o Promote Problem S o l v i n g Activities  Perceptions of Mathematical of Teachers'  Sources  Used  Topics  Perception Scales  Perceptions of Mathematical of Students'  and  Topics  Perception Scales  64 66 68  72 72  Vlll  21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.  34.  C o r r e l a t i o n s Among S t u d e n t B a c k g r o u n d and A c h i e v e m e n t  Variables  C o r r e l a t i o n s Among T e a c h e r B a c k g r o u n d and S t u d e n t A c h i e v e m e n t  Variables  C o r r e l a t i o n s Among C l a s s r o o m V a r i a b l e s and Achievement  Process 82 and  C o r r e l a t i o n s Among S t u d e n t s ' P e r c e p t i o n s Achievement  and T h e i r  Principal Variables  Components o f S t u d e n t  Rotated Factor Variables Principal Variables  Matrix  o f Student  Background 88 Background 89  Matrix  o f Teacher  Background 90  Components o f C l a s s r o o m  Organization 91  Rotated Factor Variables Principal Variables  85  Background  Components o f T e a c h e r  Rotated Factor Variables Principal Variables  84  88  Matrix  o f Classroom  Organization 92  Components o f P r o b l e m - S o l v i n g  Rotated Factor Variables  Process 93  Matrix  of Problem-Solving  Student Background F a c t o r s Variables  and F a c t o r s  Classroom Organization Criterion Variables  96  R e g r e s s e d on  Factors  Criterion 98  R e g r e s s e d on 99  Problem-Solving Process Factors Criterion Variables Teachers' Perceptions Variables  Process 94  36.  39.  79  C o r r e l a t i o n s Among T e a c h e r s ' P e r c e p t i o n s Student Achievement  Numbers o f V a r i a b l e s  38.  77  Organization  C o r r e l a t i o n s Among P r o b l e m - S o l v i n g V a r i a b l e s and Achievement  35.  37.  75  R e g r e s s e d on 100  R e g r e s s e d on C r i t e r i o n 102  ix 40.  Students* Perceptions  R e g r e s s e d on C r i t e r i o n  Variables  103  41.  P r o v i n c i a l R e g r e s s i o n Model f o r Problem S o l v i n g  104  42.  P r o v i n c i a l R e g r e s s i o n Model f o r T e s t T o t a l  105  43.  P r o v i n c i a l R e g r e s s i o n Model f o r A p p l i c a t i o n s  107  44.  P o s t - T e s t R e g r e s s i o n Model  109  45.  L o n g i t u d i n a l R e g r e s s i o n Model f o r T e s t T o t a l  111  46.  C o r r e l a t i o n s Among P r o b l e m - S o l v i n g P r o c e s s Variables S i g n i f i c a n t R e l a t i o n s h i p s W i t h Achievement on Problem S o l v i n g and A p p l i c a t i o n s  47. 48.  Variances  i n Achievement Accounted F o r  119 125 129  FIGURES  FIGURE 1  F a c t o r s a f f e c t i n g outcomes o f s c h o o l i n g .  11  FIGURE 2  B l o c k i n g e f f e c t s on s c h o o l achievement.  23  FIGURE 3  A model o f i n p u t s and outcomes o f s c h o o l i n g .  27  FIGURE 4  E f f e c t s o f s c h o o l i n g on achievement i n problem s o l v i n g , t e s t t o t a l and a p p l i c a t i o n s .  131  E f f e c t s o f s c h o o l i n g on mathematics achievement based on c r o s s - s e c t i o n a l and l o n g i t u d i n a l data.  133  FIGURE 5  APPENDICES  A.  B r i t i s h Columbia Mathematics Assessment T e s t B o o k l e t s R, S, and T  153  B.  Teacher's Guide Q u e s t i o n n a i r e  232  C.  Coding o f V a r i a b l e s  251  X  ACKNOWLEDGEMENTS  I would l i k e t o thank my committee members who p l a y e d a key role  i n the  Robitaille, a  development  of  this  dissertation.  t o a l l aspects  of the task  s t a n d a r d s and w i l l i n g n e s s t o a l l o w scope objectives  Sherrill, the  David  as my r e s e a r c h s u p e r v i s o r , p r o v i d e d i n s p i r a t i o n and  commitment  pursue  Dr.  with  importance  of d e t a i l  H i s high  i n my work caused me t o  corresponding  with h i s meticulous reviews,  a t hand.  diligence.  Dr.  James  f o c u s s e d my a t t e n t i o n on  and a c c u r a c y  i n research.  Dr. John  Anderson p l a y e d an important r o l e i n r e s p o n d i n g t o my methods o f statistical  analyses  and Dr. Douglas Owens s e r v e d an  important  f u n c t i o n as a r e v i e w e r w i t h a p h i l o s o p h i c a l p e r s p e c t i v e . My this  family contributed s i g n i f i c a n t l y  dissertation  as  well.  They  t o t h e development o f provided  encouragement a t a l l s t e p s a l o n g t h e way. daughters  Charlene  publication  to  and Cindy  come  into  acknowledge some o f my and  support.  willing  to  computers.  Among discuss  f r i e n d s who  them was issues  I  which  i n helping this  would  also  also provided  Dr. Nand dealt  and  Brenda, my w i f e , and  share w i t h o t h e r s being.  support  Kishor with  like  to  encouragement  who  was  always  statistics  and  1  CHAPTER 1 STATEMENT OF THE PROBLEM  1.1  BACKGROUND  A major purpose o f c r o s s - s e c t i o n a l education  i s the collection  d e c i s i o n making and  learning  direction service  direction f o r  i n a number o f a r e a s r e l e v a n t t o t h e t e a c h i n g  f o r resource  maintenance  i n mathematics  of data t o provide  o f mathematics.  and  studies  These  allocation,  in-service  areas o f i n t e r e s t curriculum  training,  o f strengths  revision, pre-  further  and improvement  include  research,  and  o f weaknesses  as  d e m o n s t r a t e d by l e v e l s o f achievement by s t u d e n t s . During t h e l a s t  twenty years s e v e r a l  large,  quantitative  s t u d i e s o f m a t h e m a t i c s e d u c a t i o n have been c o n d u c t e d i n a number of j u r i s d i c t i o n s .  V a s t amounts o f d a t a r e l e v a n t  t o students'  a c h i e v e m e n t and a t t i t u d e s , t e a c h e r s ' b a c k g r o u n d s and a t t i t u d e s , and  c l a s s r o o m p r o c e s s e s were  collected.  were t h e F i r s t and Second I n t e r n a t i o n a l i n Mathematics Robitaille, Assessments (Carpenter, Lindquist,  Educational  Corbitt, Matthews  &  t h e 1977,  Mathematics  S t u d i e s o f Achievement  i n British  ( i n press)], the National  Progress  Kay, L i n d q u i s t  F o u n d a t i o n S u r v e y s (Fey, including  studies  [Husen, 1967; M c K n i g h t , T r a v e r s & Dossey, 1985;  1985; R o b i t a i l l e & Garden of  Among t h e s e  Silver,  i n the United  & Reys,  1983),  1980;  the National  States  Carpenter, Science  1979) and a number o f C a n a d i a n s t u d i e s  1981 and 1985 P r o v i n c i a l Columbia  (Robitaille  R o b i t a i l l e , 1981; R o b i t a i l l e & O'Shea, 1985).  Assessments o f  & S h e r r i l l , 1977;  2  Each direction  of  these  for  d e c i s i o n making  earlier.  Although  studies  data  reported in  were  information  most  of  collected  the on  providing  areas  inputs  e d u c a t i o n a l s y s t e m , as r e f l e c t e d by c l a s s r o o m p r o c e s s  noted of  the  variables,  t e a c h e r s ' a t t i t u d e s and t e a c h e r s ' b a c k g r o u n d s , and on o u t p u t s measured by s t u d e n t s ' a c h i e v e m e n t , a n a l y s e s o f t h e between them were n o t interest  to  the  reported.  investigators  each  interest  was,  "How  i n the  1985  Provincial  they  study.  R o b i t a i l l e and O'Shea (1985) r e p o r t e d t h a t one of  relationships  Nevertheless,  in  were  example,  of the  questions  Assessment  a t t i t u d e s and  of  For  of  Mathematics  are achievement l e v e l s r e l a t e d t o c e r t a i n aspects  students' backgrounds, t h e i r  as  o p i n i o n s , and  of  those  o f t h e i r t e a c h e r s ? " (p. 3 ) . An answer t o t h i s q u e s t i o n , however, r e q u i r e d f u r t h e r a n a l y s i s of the data. Among studies  the  have  reasons  why  these  and  other c r o s s - s e c t i o n a l  not  reported  relationships  outputs,  include  problems  associated  concern:  l e v e l s of data aggregation,  and c l a s s r o o m s , In inputs  the and  student  province  of  British  analysis  of  data  address could from  two  be  the  processes,  study,  and  areas  of  three  l i n k a g e s between  relationships  outcomes  for  Columbia from  design  the  aggregated who  to  1985  of  the  which  of  to  7  between  teachers  examined  provided  through  an  level.  in  the  further  Assessment  of  opportunity  to  stated earlier.  First,  Second,  questionnaires  mathematics,  educational  mathematics  Provincial  classroom  responded  perceptions  Grade  were  of the t h r e e concerns  teachers  with  inputs  and l a c k o f p r e - t e s t d a t a .  present  Mathematics,  between  and  on  data  results classroom  background  -  characteristics  could  be  students i n t h e i r classes. lack of a pre-test,  linked  t o achievement  3  results  I n order t o address the t h i r d  of  issue,  t h e 1985 Assessment was r e p l i c a t e d w i t h a  sample o f Grade 7 s t u d e n t s  and t e a c h e r s  from  a l a r g e suburban  s c h o o l d i s t r i c t d u r i n g t h e 1986-87 s c h o o l y e a r .  A p r e - t e s t and  a p o s t - t e s t were a d m i n i s t e r e d and p r e - t e s t s c o r e s were t r e a t e d as c o v a r i a t e s . using  A comparison  p o s t - t e s t data  only  was t h e n made between t h e r e s u l t s  and t h o s e  found  when p r e - t e s t d a t a  were a l s o i n c l u d e d i n t h e a n a l y s i s .  1.2  PURPOSE OF THE STUDY  The o b j e c t i v e s o f t h e p r e s e n t s t u d y were t h r e e f o l d : to  identify  those  through  classroom  behaviors  the analysis  processes,  which  are related  and t e a c h e r  survey  characteristics  i n a significant  a c h i e v e m e n t i n m a t h e m a t i c s ; second, i n which  of provincial  first, data, and  way t o s t u d e n t  t o t e s t a t h e o r e t i c a l model  r e l a t i o n s h i p s between i n p u t s and outcomes o f s c h o o l i n g  are hypothesized;  and t h i r d ,  t o use r e s u l t s  from  a subsequent  v a l i d a t i o n s t u d y t o compare f i n d i n g s from s u r v e y r e s e a r c h based on b o t h t h e p r e s e n c e and absence o f p r e - t e s t d a t a . also  expected  t o provide d i r e c t i o n  R e s u l t s were  f o r future research i n the  analysis o f cross-sectional data.  1.3  ASSUMPTIONS OF THE STUDY  Direction  f o r the determination  of  procedures,  data  c o l l e c t i o n and a n a l y s i s was p r o v i d e d b y two b a s i c a s s u m p t i o n s o f  4  the present study.  The f i r s t assumption  was  that what students  learn i n mathematics i s , i n part, a function of t h e i r a t t i t u d e s toward  i t and  the  teacher  course of i n s t r u c t i o n .  behaviors  Second, was  which  occur  the assumption  during  the  that student  learning i n the classroom can be measured by the aggregation of i n d i v i d u a l r e s u l t s to obtain c l a s s - l e v e l data.  In making these  assumptions  that  some evidence  existed  to  suggest  they  were  by  many  plausible. The  first  educators.  assumption  However, Willms  is  one  generally  and  Cuttance  held  (1985) reported that  studies p r i o r to the mid 1970s, f o r the most part, found l i t t l e evidence to support the notion that teachers or schools made a d i f f e r e n c e i n student learning. a  number  of  background learning, effect.  these  early  accounted and  that  Willms  and  for  They stated that evidence  studies suggested most  of  instructional Cuttance  the  variance  factors  proceeded  home and  to  had  no  from  student  in  student  significant  report that  those  findings were subsequently challenged by a number of more recent studies on e f f e c t i v e schooling, i n which i n s t r u c t i o n a l were p a r t i a l l e d  out from the others.  concluded  instructional  student than  that  learning.  others and  suggest  The  factors  do  more recent studies  make a d i f f e r e n c e i n  Some teachers appear to be since evidence  from  that i s the case, the f i r s t  study was  The  effects  more  effective  recent studies e x i s t s assumption  to  underlying the  proposed. second  assumption,  that data  aggregated  to  classroom  l e v e l can be used to t y p i f y student behaviors, has been adopted i n numerous other studies.  For example, the large quantitative  5 studies referenced e a r l i e r used aggregated data f o r a n a l y s i s and reporting.  Critics  of t h i s  practice  argue that w i t h i n  class  variances i n student achievement are not accounted  f o r when the  data  resulting i n  are aggregated  (Willms  & Cuttance,  1985),  lower magnitudes f o r c o r r e l a t i o n c o e f f i c i e n t s . however, was used  as a u n i t  of analysis  The classroom,  i n the present  study  since the b e n e f i t s i n t h i s  case outweighed the disadvantages.  For example, the classroom  i s a functional u n i t with which to  compare teacher behavior and second, student l e v e l r e s u l t s would not  be meaningful  sampling  i n this  study  due t o a m u l t i p l e  matrix-  design employed i n the 1985 P r o v i n c i a l Assessment of  Mathematics.  1.4  SIGNIFICANCE OF THE STUDY  A number of questions of s i g n i f i c a n c e t o d e c i s i o n makers were  addressed  direction process  i n the  current  f o r the a l t e r a t i o n and  activities,  study.  Answers  of various aspects  suggested  of classroom  f o r planning  both  pre-service  and  in-service  and i d e n t i f i e d  areas  f o r further  research.  The  questions were as follows: 1.  What r e l a t i o n s h i p s e x i s t among teacher background c h a r a c t e r i s t i c s and student background c h a r a c t e r i s t i c s ; and between these v a r i a b l e s and student achievement i n mathematics?  2.  What r e l a t i o n s h i p s e x i s t among types of classroom organizations and structures; and between these v a r i a b l e s and students achievement i n mathematics? 1  3.  What r e l a t i o n s h i p s e x i s t between d i f f e r e n t approaches to the teaching of problem s o l v i n g and students' achievements' i n mathematics?  6  1.5  4.  What r e l a t i o n s h i p s e x i s t among t e a c h e r s p e r c e p t i o n s o f mathematics and s t u d e n t s • p e r c e p t i o n s o f mathematics; and between t h e s e p e r c e p t i o n s and s t u d e n t s ' achievement i n mathematics?  5.  What d i f f e r e n c e s , i f any, e x i s t i n t h e s t r e n g t h s o f t h e r e l a t i o n s h i p s i n q u e s t i o n s 1 t o 4 when achievement i s measured a t d i f f e r e n t c o g n i t i v e b e h a v i o r l e v e l s ?  6.  How much v a r i a n c e i n s t u d e n t achievement i n mathematics i s accounted f o r by t h e e f f e c t s o f t e a c h e r and s t u d e n t background, c l a s s r o o m o r g a n i z a t i o n and p r o c e s s e s , and t e a c h e r s ' and s t u d e n t s ' p e r c e p t i o n s o f mathematics?  7.  What d i f f e r e n c e s o c c u r i n t h e r e s u l t s found through the a n a l y s i s of c r o s s - s e c t i o n a l data a f t e r l o n g i t u d i n a l data are included i n the a n a l y s i s ?  1  SUMMARY  The  importance o f t h i s  meaningful that  information  relationships  schooling  be  determined  i s based on a need t o c o l l e c t  f o r d e c i s i o n making.  between  established  for  study  the in  effective  inputs  order  I t i s essential  and  that  classroom  the  guidelines  organization,  a p p r o p r i a t e t e a c h i n g s t r a t e g i e s and t h e e f f i c i e n t resources. supported  A  need  for collection  by Randhawa and Fu  outcomes  of  (1973).  this  They  of  can  be  use  of  a l l o c a t i o n of  information  concluded,  was  after  a  survey o f l i t e r a t u r e on t h e e f f e c t s o f i n p u t v a r i a b l e s , t h a t t h e learning  environment  achievement. for  of  a  classroom  can  be  a  predictor of  This r e c o g n i t i o n brings with i t the r e s p o n s i b i l i t y  assessing  t h e environmental  v a r i a b l e s and examining  their  r e l a t i o n s h i p s w i t h achievement. The literature conceptual  next  chapter  deals  and t h e t h e o r e t i c a l framework,  based  with  a  review  perspectives on  the  of  pertinent  o f t h e study.  literature  review,  A is  developed  i n Chapter  affecting  student  instruments  and  3 i n which i n t e r r e l a t i o n s h i p s among f a c t o r s  outcomes procedures,  are  proposed.  definitions  Descriptions of  the  variables  methods o f a n a l y s i s are a l s o d e a l t w i t h i n Chapter 4  and  5  discuss  conclusions.  results  of  the  analyses  of  and  3.  the and  Chapters arrive  at  8  CHAPTER 2 REVIEW OP  Examinations of and  outcome  the  variables  are  LITERATURE  relationships  have  attempts t o determine the these v a r i a b l e s  THE  been  among a  conducted  e f f e c t s of  by  (Centra  cross-sectional point  design  i n time.  time was  identifies  a  student  learning  these studies  a  longitudinal  nature  used a  a  given  i n which  as a v a r i a b l e .  r e v i e w i s o r g a n i z e d on  among v a r i a b l e s  behaviors,  i n which d a t a were c o l l e c t e d a t  Others were o f  included  This  1980). Many o f  in  I n c l u d e d among  teacher c h a r a c t e r i s t i c s , teacher  & Potter,  input  researchers  schooling.  s t u d e n t c h a r a c t e r i s t i c s , s t u d e n t b e h a v i o r s and outcomes  number o f  the  which were examined  number  of  issues  basis  of the  relationships  i n these s t u d i e s .  dealt  with  in  the  It  also  analyses  of  data.  2.1  RELATED LITERATURE ON  Early  l i t e r a t u r e on  decades l e a d i n g up  t o the  conclusion  that teaching  learning.  For  IC 000 was  EFFECTS OF  SCHOOLING  Studies A r e v i e w o f the  two  THE  studies  of  in  (1960) r e p o r t e d  on t e a c h e r e f f e c t i v e n e s s , in  which  the  i t s inconsistencies  Educational  Hobson, M c P a r t l a n d , reported  e a r l y 1970s c o u l d  &  relationships  that  one  York,  the  to  the  student  literature  findings.  (Coleman, 1966)  between  in  i n more than  combined  across  Opportunity"  Mood, W e i n f e l d the  lead  does not make a d i f f e r e n c e i n  example, Gage  overwhelming  "Equality  teacher effectiveness  Campbell,  results  over  In  400  were input  variables was  and  achievement  concluded  that  school p o l i c i e s , and  per  achievement. also  variables  such  pupil  outcome measures were examined.  as  which  could  teacher s a l a r i e s ,  expenditures  had  be  the  difference.  conclusion  (Fey,  1969;  that  Jenks,  G e n t i s , Heyns & M i c h e l s o n ,  1972;  on  pupil  o f t h i s e a r l y work  teachers  Smith,  by  of curriculum,  effect  A number o f subsequent reviews  reached  manipulated  type  little  It  do  not  Ackland,  make  Bane,  a  Cohen,  Dunkin and B i d d l e , 1974.)  C r i t i c s o f t h e s e e a r l i e r s t u d i e s , however, c l a i m t h a t t h e i r f a i l u r e s t o uncover important number o f weaknesses was  contended  i n design  Levin,  linked  1968) , t h e  directly  to  those  use  the  several methodological data  and  t h a t the d i f f i c u l t y  t o s c h o o l r e s o u r c e s and and  i n p u t e f f e c t s were a t t r i b u t e d t o a analysis.  For  of s e p a r a t i n g the e f f e c t s  due  t o f a m i l y background  o f achievement curriculum  tests  (Bowles  which were  (Postlethwaite,  problems, d e a l i n g w i t h use  1975)  of  due  not and  aggregated  ( B u r s t e i n , 1980) , l e d t o i n c o n c l u s i v e f i n d i n g s . Other  found  researchers  argued  that  between t e a c h e r s ' b e h a v i o r s  relationships and  students'  could  not  be  achievement  in  e a r l y s t u d i e s because o f t h e type o f d a t a c o l l e c t e d . Biddle that  example, i t  (1974),  teachers  directed  at  f o r example, contended were  the  seldom  Coleman  Good, Grouws & Ebmeier  t h a t a b a s i c problem  observed.  et.  Dunkin and  The  same  a l . (1966) r e p o r t .  criticism For  was  example,  (1983) c l a i m e d t h a t i t d e a l t w i t h  and output v a r i a b l e s but not c l a s s r o o m  was  input  processes.  More Recent F i n d i n g s Willms critical  and  Cuttance  reviews  of  (1985) c l a i m e d t h a t as a r e s u l t o f the  earlier  studies,  "a  new  literature  is  10 emerging t h a t emphasizes w i t h i n s c h o o l p r o c e s s e s t h a t l i n k p u p i l inputs  to  studies  schooling  examined  classrooms  outcomes"  the  and  types  schools  (p.  of  290).  These  learning  (McLaughlin,  Evertson,  1980).  environments  within  Rutter, and  resources  s t y l e s (Brophy,  Further  w i t h i n the school  work done by  Anderson  (1979) a t t e m p t e d t o d e f i n e s c h o o l c l i m a t e and teacher  interactions.  Based  on  this  A  (1982) and  of  found t h a t the (SES) , t e a c h e r of  conclude  are  studies  of  input-output  e f f e c t s o f i n p u t s s u c h as  this  1980;  literature  Bridge,  that  schools  taught  are  and is  classroom  relationships  socio-economic  27) by  Judd do  related to  (Murnane, and  make  Mooch, a  academic  1981;  Clark,  1979;  status  For  and  outcomes. Lotto  Rutter,  difference.  and 1983)  example,  c o n t e n d s t h a t , " c h i l d r e n l e a r n more when talented,  highly  motivated  b e l i e v e t h a t t h e i r p u p i l s can l e a r n and who day  i t  c h a r a c t e r i s t i c s , t i m e s p e n t on s p e c i f i c t a s k s  Murnane (1981, p. they  other  p u p i l motivation  Reviewers of McCarthy,  and Moos  studied pupil literature  the  i s a f u n c t i o n o f what i s done d u r i n g c l a s s t i m e .  number  effects  and  (Bidwell  i n c r e a s i n g l y c l e a r t h a t what c h i l d r e n l e a r n from t h e i r experiences  Maughan,  A n d e r s o n , A n d e r s o n & Brophy, 1980)  a l l o c a t i o n of teaching Kasarda,  recent  1978;  M o r t i m o r e & Ouston, 1979); t e a c h e r b e h a v i o r s 1982a, b;  more  teachers  s t r u c t u r e the  who  school  so t h a t p u p i l s spend l a r g e amounts o f t i m e 'on t a s k ' w o r k i n g  at basic s k i l l  development."  F a c t o r s A f f e c t i n g Student Outcomes  I t can be c o n c l u d e d , b a s e d on t h e p r e c e d i n g  general  o f r e s e a r c h e v i d e n c e on f a c t o r s a f f e c t i n g s t u d e n t s ' that  variables  related  to  home  background  and  review  performance, students'  11 c h a r a c t e r i s t i c s have t h e most levels. that  However,  significant  findings of recent  impact on achievement  studies  s c h o o l - r e l a t e d f a c t o r s have some e f f e c t  have  also  as w e l l .  shown  A model  d e p i c t i n g t h e s e r e l a t i o n s h i p s i s shown i n F i g u r e l .  Home Background  s  Characteristics o f Students  School-Related Factors  \  F i g u r e 1.  The main categories  Student  Outcomes  F a c t o r s a f f e c t i n g outcomes o f s c h o o l i n g .  i n f l u e n c e s on s t u d e n t  i n t h e model  Characteristics  of  (shown i n F i g u r e  Students  I n t e r a c t i o n s among t h e s e  outcomes  and  1) :  fall  under  three  Home Background,  School-Related  Factors.  categories are also i l l u s t r a t e d  i n the  model. I n f l u e n c e s on academic performance have been t h e s u b j e c t o f extensive provide  research,  have on s t u d e n t s  motivation.  struggle  of the research  a comprehensive and a c c u r a t e  impacts they and  b u t none  with  appropriateness  with  The e f f e c t i v e the  complexity  o f instruments  methods o f a n a l y s i s .  account o f them o r o f t h e  different  schools of  has been a b l e t o  levels  research  the  t o measure  of a b i l i t y  continues t o  interactions, them  the  and a c c e p t a b l e  12 2.2  FACTORS AFFECTING THE  The  literature  learning  of  reported  earlier.  related  mathematics  Effectiveness  LEARNING OF MATHEMATICS  In  specifically  follows  a  pattern  r e p o r t i n g on  P r o j e c t , Good, Grouws  to  the  the  teaching  similar  Missouri  & Ebmeier  to  and that  Mathematics  (1983) made  the  following observation: When we began our program o f r e s e a r c h i n t h e e a r l y 1970s t h e r e was v e r y l i t t l e u s e f u l o r r e l i a b l e information available f o r d e s c r i b i n g t h e r e l a t i o n s h i p between c l a s s r o o m p r o c e s s e s (e.g., t e a c h e r b e h a v i o r ) and c l a s s r o o m p r o d u c t s (e.g., s t u d e n t achievement). What knowledge e x i s t e d i n 1970 about t h e e f f e c t s o f c l a s s r o o m p r o c e s s e s on s t u d e n t achievement was weak and c o n t r a d i c t o r y . A f t e r a decade o f e x t e n s i v e r e s e a r c h on c l a s s r o o m p r o c e s s e s (much o f t h i s r e s e a r c h s u p p o r t e d by t h e National I n s t i t u t e of Education), there i s now much p e r t i n e n t i n f o r m a t i o n about t h i s r e l a t i o n s h i p (p. 1 ) . This  observation  is  supported  by  evidence  from  l a r g e s c a l e , c o r r e l a t i o n a l s t u d i e s i n which t h e d a t a t h a t i t was produced  possible to identify  higher  Brophy and  achievement  E v e r t s o n , 1974;  in  some t e a c h e r s who students  Good and  than  several,  illustrated consistently  expected  Grouws, 1975).  (e.g.,  I t was  p o s s i b l e , through d e s i g n and a n a l y s i s , t o i d e n t i f y  instructional  p a t t e r n s t h a t d i f f e r e n t i a t e d t h e s e t e a c h e r s from  those who  less  definition  successful  effectiveness Evertson,  according  (e.g.,  1976;  Berliner  Rosenshine,  experimental  studies  instructional  behavior  (e.g.  Brophy, 1979;  to  an and  operational Tikunoff,  1979).  supported suggested  A  these by  1976;  number patterns  the  Good and Grouws, 1977;  also  of  were  Brophy  of and  field-based  of  correlational Stallings,  effective studies 1980).  13 In and  a  study  Grouws  effective found  of  fourth-grade  (1977)  identified  teachers  that  from  a  nine  sample  e f f e c t i v e n e s s was  environment,  environment, behavioral another  higher  a  problems  et. a l .  (1980),  over  one  the  class  They  a task-  relatively  as  a  the  learning  unit.  f o c u s i n g on mathematics,  few In  Evertson  found t h a t more e f f e c t i v e t e a c h e r s , i n c o n t r a s t  the f o l l o w i n g c h a r a c t e r i s t i c s : p r e s e n t a t i o n s and  of  demonstrated  t h e y spent more time  d i s c u s s i o n s w i t h l e s s time  expectations  management  less  with  relaxed  t o l e s s e f f e c t i v e ones a t the Grade 7 and 8 l e v e l s ,  higher  nine  associated  and  Good  hundred.  expectations,  teaching study  and  of i n s t r u c t i o n ,  non-evaluative  and  process-product  of  clarity  achievement  instruction,  effective  strongly  following behavioral clusters: focused  mathematics  students  and  on  content  on s e a t work, h e l d exhibited  stronger  skills.  In l i g h t difference  of these  i n the  f i n d i n g s , a case t h a t t e a c h e r s do make a  learning  example, Good, e t . a l . Mathematics  of  (1983),  mathematics  can  be  made.  For  i n t h e i r r e p o r t on t h e M i s s o u r i  Effectiveness Project  referenced  earlier,  confirm  t h i s p o s i t i o n w i t h t h e f o l l o w i n g statement: Our r e s e a r c h p r o v i d e s c o m p e l l i n g e v i d e n c e t h a t t e a c h e r s make a d i f f e r e n c e i n s t u d e n t l e a r n i n g and o f f e r s some u s e f u l i n f o r m a t i o n about how more o r l e s s e f f e c t i v e t e a c h e r s d i f f e r i n t h e i r b e h a v i o r and i n t h e i r e f f e c t s on s t u d e n t achievement (p. 13). The  size  considerably reported accounted  of  these  across  teacher  studies.  Good  effects,  however,  (1979, p.  54)  that  in  a  study  by  Inman,  for  26  percent  of  adjusted  varies  f o r example,  instructional  variables  variance  minority  in  14 students'  achievement  scores  but  adjusted variance f o r majority  for  only  12  students' their  between  background  attitudes,  output  a  number  and  variables  as  of  attitude,  classroom  of  the  students.  S i n c e t h e major purpose o f the p r e s e n t relationships  percent  input  by  i s t o examine  variables  background  processes  defined  study  and  of  such  as  teachers  and  organization;  and  achievement  in  student  mathematics, t h i s review proceeds t o examine f i n d i n g s from o t h e r s t u d i e s on t h e e f f e c t s o f these p a r t i c u l a r v a r i a b l e s . Teacher Background and  Behaviors  In g e n e r a l , s t u d i e s on t h e r e l a t i o n s h i p s between background characteristics  of  teachers  little  effects.  McDill  and  Rigsby  of  records,  keeping  students'  For  researchers  did  and  achievement and  find  student  instance,  (1973)  achievement  and  Rutter  found  or that  et.  that  salary level  outcomes  had  no  t e a c h e r s w i t h more than  and  time,  the  relationship  with  However,  relationship  shown  (1979),  preparation  aspirations. the  al.,  have  the  between  a bachelor's  same  students* degree  was  s i g n i f i c a n t l y and p o s i t i v e l y c o r r e l a t e d . Brophy  (1982b),  or behaviors  i n a d i s c u s s i o n of teacher  associated with  student  characteristics  achievement g a i n s ,  listed  t h e f o l l o w i n g e i g h t c a t e g o r i e s based on r e s e a r c h f i n d i n g s o f the seventies:  teacher  opportunity  to  learn;  curriculum pacing; supportive identified student  expectations; classroom  following  achievement:  and  student  organization;  t e a c h i n g t o mastery; and  Rosenshine and  variables  clarity,  efficiency;  management  active teaching;  l e a r n i n g environment. the  teacher  as  strong  variability,  Furst  a  (1971),  correlates  enthusiasm,  of  task  15 orientation positive  and  opportunity  correlations  achievement Henderson  were  to  between  found  by  learn.  teacher  Kolb  Further  evidence  variability  (1977)  and  and  Cooney,  of  student  Davis  and  (1975).  In a major c r o s s - s e c t i o n a l study i n v o l v i n g more than 20 secondary  s t u d e n t s and  t h e i r t e a c h e r s , M c D i l l and R i g s b y  000  (1973)  found a p o s i t i v e r e l a t i o n s h i p between t h e e d u c a t i o n a l background of  teachers  and  student  achievement.  c o n f i r m e d i n a study by t h e New which  both  teacher  This  relationship  York S t a t e Department  education  and  experience  was  (1976) i n  showed  positive  c o r r e l a t i o n s w i t h s t u d e n t achievement. The  importance  Ward  (1979),  from  the  and  i n p r e s e n t i n g the  Schools  handicap  of t r a i n i n g  of  Council,  elementary  mathematics  in-service  results  i n which  he  teachers  education.  He  of  a  was  stressed  survey  undertaken  reported that  was  suggested  by  lack  of  that  because  the  training  main in  of  the  l i n e a r i t y o f t h e s u b j e c t , mathematics can s u f f e r most from  poor  teachers. Based teacher  on  results  background  learning. studies,  the  these  variables  Although variables  of  have  findings related  studies some  effect  are  not  the  behaviors  to  i t appears on  consistent and  that  student  across a l l professional  p r e p a r a t i o n o f t e a c h e r s have p o s i t i v e r e l a t i o n s h i p s w i t h s t u d e n t achievement. Classroom In  Processes a  comprehensive  instructional Stallings  practices  (1976)  study in  Grade  reported  that  of 1  classroom and out  Grade of  a  processes 3  and  classrooms,  possible  340  16 correlations processes,  between  108  were  mathematics  achievement  significantly  related  at  and  classroom  the  0.05  level.  The c l a s s r o o m p r o c e s s v a r i a b l e which c o r r e l a t e d t h e h i g h e s t w i t h achievement  was  mathematical  activities.  homework, and  the  amount  of  time  spent  by  students  Emphases p l a c e d on d r i l l ,  on  follow-up to  i n s t r u c t i o n w i t h s m a l l groups were among t h e o t h e r  v a r i a b l e s p o s i t i v e l y r e l a t e d t o s t u d e n t achievement. Further mathematical (Husen,  evidence activities  1967;  In  an  Assessment  the  has  and  of r e s u l t s  Educational  of  Progress  1974;  1976;  from  time  spent  on  i n numerous s t u d i e s .  Kaskowitz,  McDonald and E l i a s ,  analysis  of  importance  been p r o v i d e d  Stallings  H a r n i s c h f e g e r , 1974; 1979).  of  the in  Wiley  and  Ebmeier and Good, 1977-1978 N a t i o n a l  Mathematics,  Welch,  Anderson & H a r r i s (1982) r e p o r t e d t h a t t h e amount o f mathematics s t u d i e d accounted  f o r 34 p e r c e n t o f t h e v a r i a n c e i n achievement  f o r 17-year o l d s .  An important d i s t i n c t i o n among t y p e s o f t i m e -  related  activities  defined  three  academic  was  made  variables:  learning  time.  by  Berliner  allocated He  found  (1978)  time, that  in  which  engaged while  time,  a l l of  v a r i a b l e s were p o s i t i v e l y a s s o c i a t e d w i t h achievement, time accounted In  a  effective used  group  acceleration  of  students  need  extended  by  teachers  in by  and these  allocated  f o r l e s s v a r i a n c e than the o t h e r s .  study  small  he  Wiener  (1979)  of mathematics a t teaching  students of  to  and  remedial  r e s u l t s o f a study  Rosenberg, W i l s o n  i t was  & Bursuck  t h e Grade  present to  found  new  provide  that 2 and  skills  general  assistance.  This  the 3  most levels  for  the  review  for  finding  was  o f Grade 4 s t u d e n t s by S i n d e l a r ,  (1984)  i n which  i t was  found  work i n s m a l l groups promoted s t u d e n t s ' engaged time,  and  that that  engaged t i m e was  r e l a t e d to higher  by Moody, B a u s e l l & J e n k i n s ,  achievement l e v e l s .  (1974) and  Fisher, Berliner, Filby,  Marliave,  Cahen & Dishaw (1980), a l s o r e p o r t e d  s i z e s had  considerable  Results specific  from  i n f l u e n c e on  these  activities  studies  to  homework  students  were  engaged  also  existed  positive  to  Stallings,  1976;  subject.  in  student  on  suggest  working  and  time  in  with  small  (Moody  Fisher  et.  on  which  (Berliner,  Considerable  learning  1979;  time  activities  achievement.  student  Weiner,  1976)  mathematical  that  that  F o r example, t i m e s p e n t  (Stallings,  show  effect  learning.  i n mathematics i s p o s i t i v e l y c o r r e l a t e d  to  to  group  on  follow-up  related  that small  spent  student achievement i n the  1978)  17 Studies  evidence  groups  et.  had  al.,  a l . , 1980;  a  1974;  Sindelar  e t . a l . , 1984). Student A t t i t u d e s  In  addition  to  the  learning  of  principles,  facts  and  methods, i m p o r t a n t outcomes o f s c h o o l i n g a l s o i n c l u d e a t t i t u d e s , v a l u e s and  appreciation.  objectives  three  towards  methods have been u s e d mathematics.  These  i n t e r v i e w s , s e n t e n c e c o m p l e t i o n t e s t s and most p o p u l a r o f t h e s e methods i s t h e use as t h e T h u r s t o n e o r L i k e r t ( A i k e n , L i t t l e research that  outcomes r e l a t e  to  curriculum.  number o f  attitude  latter  from t h e a f f e c t i v e domain, w h i c h c o n s t i t u t e s a m a j o r  goal of the A  The  favorable  achievement.  measure  include  students'  observations,  attitude scales.  The  of a t t i t u d e s c a l e s such  1972).  evidence i s a v a i l a b l e t o support the b e l i e f  attitudes For  to  towards  example,  mathematics  Jackson  (1968),  lead in  a  to  higher  review  of  18 research  in  significant Caezza  this  found  relationships  (1970)  significant However,  area  and  Van  de  between Walle  relationships  low,  few  in  positive  studies  attitude  (1973),  studies  mathematics  (Torrance,  have  1966;  been  Wess,  however, a t t r i b u t e s  found  1970;  in  the  elementary  between  a t t i t u d e t o be d e f i n e d  a  no  level.  students'  of  1973).  findings  as a v a r i a b l e and  found  and  s t u d e n t s ' performance  number  Phillips,  inconclusive  achievement.  example,  t e a c h e r s ' a t t i t u d e s toward mathematics and in  and  reported  for  at  correlations  that  other  studies  Lester  (1980),  t o the  t o the  elusiveness  of  lack of r e l i a b l e  instruments. Several  specific  significant,  positive  performance  were  problem-solving  al.  (1980)  among j u n i o r in  a  and  10  that  between  (1973),  sixth  in  graders  in  a  study  involving  self-perception that  between  i s causally  between grades  5  10  to  the  2  of  5  s e l f - r a t i n g s of strength  of  the  good  achievement (1984),  students'  i n Grades  and  the  Evertson  Newman  analysis  i n mathematics Grades  related and  an  and  of  that  with  mathematics s t u d e n t s .  study  which  attitude  found  significant correlations  high school  and  of  follow  h i g h e r s e l f - e s t e e m than poor ones.  found  found  achievement  Robinson  behaviors  longitudinal  achievement  projects  correlations  found.  problem s o l v e r s had et.  research  2,  5  mathematics ability  and  relationship  diminishes. In  a  summary  achievement, H a r t correlation attitude  some  studies  (1977) commented t h a t  occurs,  affects  of  it  is  achievement  difficult or  vice  relating  attitude  to  even where s i g n i f i c a n t to  versa.  determine The  whether  direction  of  19 the  r e l a t i o n s h i p between t h e s e v a r i a b l e s was  Neal  (1979),  mathematics  in  reviewing  teachers,  found  that  a  a l s o found t h a t  affective variables  effect  than  background  status.  Further  attitudes  and  Frederickson  general  of  on  the  pupils'  such  as  of  achievement.  He  gender  relationships  achievement  was  found  stronger  or  marital  between  teacher  by  Edmonds  (1978) i n a study o f Grade 6 s t u d e n t s and  K y l e s and  attitudes  They  topics.  staff  toward  Sumner  (1977) i n v e s t i g a t e d  mathematics,  but  confirmed that  Similar  results  also  not  A l t h o u g h f i n d i n g s were not  found by  correlations  were  activities  reported  1984).  Of  1978;  further  between a t t i t u d e student (Kyles  consistent  between  and  in  the  perceptions  attitude  and  across these and  (Robinson, 1973;  Edmonds  the  d i r e c t i o n of  (Neal,  different topics  1969  primary  achievement  i n t e r e s t are  Sumner, 1977).  within  studies,  E v e r t s o n e t . a l . , 1980;  of  of  (1980).  Begle, 1979;  achievement  only  student perceptions d i f f e r e d  s e v e r a l o f the r e s e a r c h e r s  Frederickson,  in  details  survey conducted by the Assessment o f Performance U n i t  positive  and  schools.  work o f  subject.  of  enjoyment  o f t e a c h e r s have a  s t u d e n t s ' p e r c e p t i o n s o f d i f f e r e n t t o p i c s and  among  attitudes  teacher  e f f e c t on  variables  evidence  student  elementary The  positive  studies  mathematics had  the  by  (1969). Begle  812  also questioned  and  Newman,  relationships  ; Hart,  within  were  the  1977)  and  curriculum  20  2.3  SIMILAR STUDIES OF  In data,  much  of  the  researchers  A CROSS-SECTIONAL NATURE  earlier  work  reported  on  with  cross-sectional  cause-and-effeet  u s i n g measures o f a s s o c i a t i o n such as r e g r e s s i o n coefficients. that  would  another. (1985)  occur  in  one  Critics  of  this  approach such as  the  genuine  claim  family,  only  and  studied.  can  to  by  variable  student  other  While  used  non-school  longitudinal  to  i t is  measure  description of  a  similar  cross-sectional schooling The  correlation  measured  change  Willms and of  effect  Cuttance  school  can  the  the  be  data  can  may  be  that  strengths the  influence  of  and  revealed  of  They  ability,  cause-and-effeet necessary  in  to  be  report  cross-sectional association  data  between  e x t e n t t o which v a r i a n c e s i n  a t t r i b u t e d to those v a r i a b l e s . methods  nature  to  of  the  analysis  present  used  one,  in  which  two used  d a t a t o examine r e l a t i o n s h i p s between i n p u t s  s t u d e n t achievement,  of  follows.  F i r s t I n t e r n a t i o n a l Mathematics study One  of  and  of  for  assumed  the  s t u d e n t achievement can be  studies  a  relationship  factors  v a r i a b l e s under examination and  A  given  achievement  controlling  relationships,  be  and  o n l y through the a n a l y s i s o f l o n g i t u d i n a l d a t a .  that  casual  that  variables  accurately  relationships  These s t a t i s t i c s were used t o e s t i m a t e t h e  suggest  teacher  survey  o f the  schooling  f i r s t major i n t e r n a t i o n a l s t u d i e s on  the  learning  I n t e r n a t i o n a l Mathematics Study under  the  Evaluation  auspices of  of  the  Educational  of  mathematics  of the was  effects  the  First  (Husen, 1967), conducted i n  International Achievement.  Association A  total  for of  1964 the  twelve  countries the  participated  following  containing final  four  most  i n t h e study which  populations:  13-year  secondary  year;  olds;  and  involved  13-year  students i n  olds;  grade  level  mathematics  students  in  their  non-mathematics  students  in  their  f i n a l secondary y e a r . Husen  (1967) r e p o r t e d  that  t h e main o b s t a c l e  a n a l y s i s o f r e s u l t s was t h e l a c k o f c o n s i s t e n t number  of  independent  indices.  Different  variables  among  educational In  with  school  o l d group  variables,  to  often  different  feasible  applied  to  cultures  and  scores  were  reported  independent  the teacher  size  was  were t e a c h e r  opportunity  1  achievement  found  variables  which  and t h e s t u d e n t .  Among  t o be c o r r e l a t e d  a t the  The t e a c h e r v a r i a b l e s which c o r r e l a t e d t h e h i g h e s t  achievement  as  0.08  characteristics included  were  total  forty-five  the school,  0.12 l e v e l .  students  due  measurement o f a  operationally  interpretations  countries  t h e 13-year  characterized  with  with  i n the  systems.  correlated  the  variables  faced  learn.  and  which  fathers'  mathematics  to  training  These  0.19  were  ratings  correlations  respectively.  found  education  and t e a c h e r  to  (0.18),  of  were  Students'  correlate students'  positively interest  (0.30) and s t u d e n t s ' p l a n s and a s p i r a t i o n s  in  (0.18 t o  0.22). At  the  next  stage  dropped due t o o v e r l a p p i n g twenty-six four  independent  headings:  of  analysis  or unsuitable  variables  parental  next w i t h v a r i a b l e s  coding.  grouped  variables,  v a r i a b l e s and s t u d e n t v a r i a b l e s . was conducted  several  under  teacher  variables  were  There remained the  following  variables,  A stepwise r e g r e s s i o n under each h e a d i n g  school analysis  regressed  on t o t a l s c o r e .  The a v e r a g e amount o f v a r i a n c e a c r o s s c o u n t r i e s  i n t h e a c h i e v e m e n t o f 13-year o l d s a c c o u n t e d each s e p a r a t e h e a d i n g teacher variables,  follows:  1.3  f o r by v a r i a b l e s i n  p a r e n t a l v a r i a b l e s , 4.4  percent;  school variables,  1.3  percent; percent;  and s t u d e n t v a r i a b l e s , 7.5 p e r c e n t . A d i f f e r e n t a p p r o a c h t o t h e a n a l y s i s o f r e s u l t s from t h r e e subsequent I n t e r n a t i o n a l  Studies  i n science education,  reading  c o m p r e h e n s i o n and l i t e r a t u r e was r e p o r t e d by Coleman (1975). r e p o r t e d on a number o f m e t h o d o l o g i c a l  i s s u e s w h i c h were  in  studies.  the analysis of results  variable  was  achievement  from t h e s e in  each  Independent v a r i a b l e s on t h e o t h e r three Blocks: and  of  three  hand, were  He  faced  The dependent topic  areas.  clustered into  B l o c k 1- Home Background; B l o c k 2- Type o f S c h o o l  Program; and B l o c k  3- S c h o o l  Instruction.  The p u r p o s e o f  s e p a r a t i n g v a r i a b l e s i n t o t h e s e b l o c k s a p r i o r i was t o b r i n g some order i n t o the regression analyses. The e x p e c t a t i o n s , a c c o r d i n g t o Coleman, were t h a t B l o c k v a r i a b l e s were more i m p o r t a n t  than  Block  3 v a r i a b l e s and t h a t  s c h o o l v a r i a b l e s ( B l o c k 2) w o u l d show l i t t l e e f f e c t . r a t i o n a l e , v a r i a b l e s were e n t e r e d in  order  increments  of Block  number.  Using  this  into the regression analysis  Effects  were  then  identified  t o e x p l a i n e d v a r i a n c e s f o r each p r e c e d i n g B l o c k .  d i a g r a m shown i n F i g u r e 2, i n d i c a t e s t h e c a u s a l r e a s o n i n g t h e u s e o f t h i s sequence o f b l o c k s .  1  as The  behind  Block 1  F i g u r e 2.  The  National  Replication In school  Block 3  Block 2  a  Block 4  B l o c k i n g e f f e c t s on s c h o o l (Coleman, 1975, p.361).  Assessment  of  Educational  achievement  Progress  (NAEP)  -  A  study study  conducted  mathematics  Horn and Walberg  t o assess  achievement  t h e dependence  on a number  (1984) r e g r e s s e d  of input  of  high  factors,  t h e achievement and i n t e r e s t  s c o r e s o f a sample o f 17-year o l d s on each o t h e r and on f o u r t e e n other v a r i a b l e s . the  dependencies  The purposes o f t h e study of  students'  interest  were t o i n v e s t i g a t e and  achievement  in  mathematics on a l a r g e r s e t o f v a r i a b l e s through r e p l i c a t i o n o f the e a r l i e r NAEP  study.  Data from t h e 1977-78 NAEP were used i n t h e a n a l y s e s . and  Walberg  (1984) r e p o r t e d  t h a t a sample o f 1480 17-year  was drawn, u s i n g a s t r a t i f i e d , design: the  Horn  three-stage  national  olds  probability  t h e p r i m a r y sampling u n i t s were r e p r e s e n t a t i v e o f a l l  regions  and community  s i z e s i n the United  States.  At the  second  stage, schools  sampled third  from a l i s t stage  selected  a  within  primary sampling u n i t  o f a l l p u b l i c and p r i v a t e s c h o o l s .  random  within  each  each  sample  of  school.  age-eligible  Students  used a t the a n a l y s i s  At the  students  i n the  sample  a d m i n i s t e r e d t h e NAEP b o o k l e t e n t i t l e d "Number 1". p r o c e d u r e was  were  was were  A weighting  stage t o estimate n a t i o n a l  s t a t i s t i c s by c o m p e n s a t i n g f o r o v e r s a m p l i n g o f s e l e c t e d g r o u p s . Achievement was measured with a r e l i a b i l i t y The t e s t was levels  of  included  o f 0.92,  by a t e s t c o n s i s t i n g o f 55 i t e m s ,  u s i n g Cronbach's a l p h a c o e f f i c i e n t .  c o m p r i s e d o f f i v e c o n t e n t a r e a s and f o u r  behavior. the  The  following:  c o u r s e s , most advanced  independent instruction  variables (3  cognitive  under  levels),  number  c o u r s e , home e n v i r o n m e n t , TV,  SES, s e x and e t h n i c i t y .  study  homework,  These v a r i a b l e s were l i m i t e d i n number  by t h e i t e m s c o n t a i n e d i n t h e NAEP b o o k l e t and were s e l e c t e d the  basis  of  of  earlier  findings  of  relationships  on  between  b a c k g r o u n d , i n s t r u c t i o n and a c h i e v e m e n t . Univariate between the  analyses  achievement  most advanced  showed  high  positive  correlations  and b o t h t h e number o f c o u r s e s (0.62)  course taken (0.63).  r a n g i n g between 0.38  Moderate  correlations,  and 0.41 were f o u n d between a c h i e v e m e n t and  t r a d i t i o n a l i n s t r u c t i o n , home e n v i r o n m e n t and SES. of  0.23  frequency  and  0.21 of  were  reported  course-related  between  Multiple regression  for  achievement  Correlations  achievement  activities  and  r e s p e c t i v e l y , whereas t e l e v i s i o n e x p o s u r e c o r r e l a t e d  e f f e c t s of v a r i a b l e s  and  and  both  homework negatively.  t e c h n i q u e s were u s e d t o d e t e r m i n e t h e  on t h e outcome measures.  accounted  f o r 57  A r e d u c e d model  percent of the variance  and  showed t h a t each o f  12 v a r i a b l e s  was  statistically  significant  when t h e o t h e r s were c o n t r o l l e d . F i n d i n g s from t h e Horn and Walberg  (1984) study showed t h a t  mathematics achievement i s a f u n c t i o n o f t h e l e v e l and amount o f mathematics  coursework  influenced  by  instruction,  completed  student  interest  education  of  i n high in  parents  school.  It  mathematics, and  quality  is  also  traditional of  the  home  environment.  2.4  SUMMARY  This  chapter  researchers  has  discussed  i n examining  outcomes o f s c h o o l i n g . early  studies  and  the  the  the  relationships  number  relationships  more  significant  1980s.  issues  between limited ones  faced  by  inputs  and  findings  of  of  subsequent  R e s u l t s from numerous s t u d i e s  between s t u d e n t  r e l a t e d t o s t u d e n t and  of  I t r e p o r t e d on t h e  r e s e a r c h i n t h e 1970s and on  a  achievement  and  variables  t e a c h e r backgrounds, s t u d e n t and  teacher  a t t i t u d e s , and c l a s s r o o m p r o c e s s e s were then r e p o r t e d . In  the  literature of  next  review  instruments  chapter, just  and  a  conceptual  completed,  procedures,  methods o f a n a l y s i s a r e i n c l u d e d .  model,  i s presented. definition )  of  based A  on  the  description  variables  and  26  CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY This  chapter  contains  a  discussion  of a  model  of the  r e l a t i o n s h i p s between i n p u t s and outcomes o f s c h o o l i n g .  I t also  d e s c r i b e s t h e p o p u l a t i o n and t h e sampling  plan, i d e n t i f i e s the  r e l e v a n t v a r i a b l e s and t h e methods by which t h e y were measured, p r o v i d e s a d e s c r i p t i o n o f t h e i n s t r u m e n t s and t h e i r development, and d e s c r i b e s t h e d a t a c o l l e c t i o n and a n a l y t i c a l  3.1  procedures.  A MODEL OF INPUTS AND OUTCOMES OF SCHOOLING  In  the preceding  students'  achievement  reported that early background variance Fey,  accounted  i n student  1969; J e n c k s  However, that,  subsequent  after  chapter  t h e impact  i n mathematics  s t u d i e s found f o r almost achievement  students'  ability  a l l statistically (e.g., Coleman,  studies i n the l a t e f o r student  s c h o o l s d i d make a d i f f e r e n c e  schooling  was d i s c u s s e d .  e t . a l . , 1972; Dunkin  controlling  of  I t was  and f a m i l y significant  et. a l . ,  and B i d d l e ,  1970s and 1980s  ability  (e.g., Willms  on  1966;  1974). found  and background,  and C u t t a n c e , 1985;  Anderson, 1982; Murnane, 1981; Moos, 1979). A f u r t h e r search o f the l i t e r a t u r e ,  which f o c u s e d on those  f a c t o r s o f s c h o o l i n g t h a t have an e f f e c t on s t u d e n t s ' and  achievement i n mathematics, uncovered  related  t o these  organization, the  outcomes.  and t e a c h e r  Instructional background  attitudes  a number o f v a r i a b l e s processes,  and a t t i t u d e s  i n p u t s o f s c h o o l i n g shown t o have some such  classroom were  effect  among (e.g.,  Brophy, 1982a,b; Good, e t . a l . , 1983; E v e r t s o n , e t . a l . , 1980).  The  effects  interrelated. out  that,  of  input v a r i a b l e s ,  F o r example,  Neal  however,  (1969)  even where s i g n i f i c a n t  a r e complex  and H a r t  correlations  (1977)  occurred,  and  pointed i t was  d i f f i c u l t t o determine whether a g i v e n v a r i a b l e a f f e c t e d  student  a t t i t u d e and achievement o r v i c e v e r s a .  that a  model  for investigation  some o f t h e s e Based conceptual  on  should  provide  T h i s suggested f o r the  interaction  of  variables. the  model  preceding shown  in  review Figure  of 3  the is  literature,  the  presented  for  investigation.  Student Background Achievement Classroom Processes  ±  Student Perceptions of Mathematics  Legend.  Teacher Perceptions o f Mathematics  One way r e l a t i o n s h i p s o f i n t e r e s t Interactions of interest R e l a t i o n s h i p s not under study  Figure 3.  ^ <•*  A model o f i n p u t s and outcomes  —  of schooling.  A b a s i c assumption o f t h e model i s t h a t s t u d e n t s ' as  represented  students  1  by achievement  i n mathematics,  > a*  outcomes,  are f u n c t i o n s of  backgrounds and p e r c e p t i o n s , t e a c h e r s ' backgrounds and  p e r c e p t i o n s , and c l a s s r o o m  processes.  F o r t h e purposes o f t h i s  28 study, their  students'  and  teachers'  a t t i t u d e s and  factors,  however,  opinions  involve  perceptions are toward  complex  limited  mathematics  sets  of  t o what  are.  These  interactions.  For  example, i n e x p l o r i n g a number o f c a u s a l models f o r achievement, Parkerson,  Schiller,  Lomax  reciprocal  paths  causal  account  in  of  attempting  to  &  Walberg  (1984)  concluded  that  influence  should  be  into  obtain  a  better  taken  understanding  of  classroom l e a r n i n g . The  model  portrayed  in  Figure  3  classifies  student  and  t e a c h e r background v a r i a b l e s i n a t h e o r e t i c a l  s t r u c t u r e used  explain  As  variance  in  diagram, t h e impacts be  in  one  student  achievement.  direction.  inputs,  They  are  relatively  Two  determine which had on  students'  processes  and  correlations  such  as  of  classroom  these  the  teachers' not  or  between s t u d e n t s ' background and background  and  classroom  and  were  to  hence  students'  Neither  Hence,  c o n n e c t i n g t h e s e v a r i a b l e s a r e shown i n t h e model.  these  relationships  t h e i r perceptions or  processes.  classroom  interrelated, were  to  perceptions  Although  be  student  examined  teachers'  versa.  p e r c e p t i o n s may examined.  and  to interact with  relationships  vice  fixed  processes,  greater effect:  perceptions  were  the  t o have an e f f e c t on them.  and t e a c h e r s ' p e r c e p t i o n s , a r e expected achievement.  in  o f t h e s e p a r t i c u l a r f a c t o r s a r e expected  s t u d e n t s ' achievement i s not expected Other  shown  to  teachers  dotted  1  lines  29 3.2  The  POPULATION AND  1985  Mathematics Assessment  The  population  s t u d e n t s and funded  SAMPLING PLAN  the  present  study  consisted  of a l l  t e a c h e r s o f Grade 7 i n B r i t i s h Columbia p u b l i c  independent  provided  for  by  s c h o o l s as o f May,  the  Ministry  of  1985.  Education,  s t u d e n t s a t t h a t l e v e l as o f February, A total  o f 33 888  a r e t u r n r a t e o f 94  percent.  questionnaire, including  one  class  Grade  participated.  A  the  1985  was  statistics  population 35  of  890.  Grade 7 s t u d e n t s wrote t e s t b o o k l e t s f o r  one  of  Based on  and  S i n c e t e a c h e r s completed cases  where t h e y  7 mathematics, total  of  taught  fewer t e a c h e r s  1073  teacher  a t most  more  than  than  classes  questionnaires  were  returned. The  1987  Validation  The  sample f o r t h e  students  and  District,  a  teachers  i n v o l v e d 2146 of  1987 at  Grade  m i l l i o n people.  school  during  level  The May,  plan the  consisted of a l l i n Surrey  located i n a  classrooms.  district's  Assessment  study  7  district  s t u d e n t s from 104  the  Mathematics  validation  the  l a r g e suburban  a r e a o f about 1.5  part  study  1987  School  metropolitan  administration  S i n c e t h e study  to  replicate  198 6-87  school  was  the  1985  year,  all  s t u d e n t s and t e a c h e r s o f Grade 7 i n the d i s t r i c t were i n v o l v e d . Students consisting attitude During complete  were  of  achievement  scale the an  required  in  items,  September,  September attitude  to  complete background  1986  and  administration scale  which  test  questions  again  teachers  formed  booklets,  part  in were of  and  May, asked the  an  1987. to more  comprehensive  q u e s t i o n n a i r e from t h e 1985 Assessment.  were asked t o complete  the e n t i r e  Teachers  questionnaire during  the  May  administration.  3.3  OVERVIEW OP THE METHOD OP STUDY  The  independent  variables academic  which  experience,  shown  were  teachers' on  i n problem  total.  In  and dependent  from  t o be  included  those  related  in  the  included  perceptions the  other  solving,  examining  variables,  techniques of  the  of  factors  instruction,  were  mathematical relationships  and  mathematics.  hand,  to 1985  home backgrounds, t e a c h e r s • backgrounds  1  variables,  achievement test  which  classroom organization,  s t u d e n t s ' and  dependent  and  had  selected  The p o o l o f independent v a r i a b l e s  describing students  and  were  previous research  achievement,  Assessment.  variables  The  students'  applications among  and  independent  t h e u n i t s o f a n a l y s e s were t e a c h e r and  class. Responses Teacher tape  of  those  s t u d e n t s whose t e a c h e r s completed  Q u e s t i o n n a i r e were r e - s c o r e d  and  aggregated  variances  were  correlations  t o the  then  between  from  the  classroom l e v e l .  linked  to  independent  responses and  provincial Class  of  the data  means  teachers  dependent  and and  variables  determined. S i n c e a major g o a l o f t h e c u r r e n t s t u d y was model  to  explain  achievement, the  variance  regression  t h r e e dependent  i n s t u d e n t outcomes  t o determine a as  measures  of  e q u a t i o n s were determined w i t h each  of  v a r i a b l e s f o r each o f t h e i n p u t  categories.  31 F i n a l models were then determined  by r e g r e s s i n g a l l v a r i a b l e s on  each dependent v a r i a b l e . P r e - t e s t d a t a were not a v a i l a b l e from t h e p r o v i n c i a l and hence t h e r e was  no c o n t r o l a t t h a t s t a g e o f t h e a n a l y s i s f o r  the e n t r y - l e v e l  knowledge o f s t u d e n t s .  weights a r r i v e d  a t through  estimates  which may  extent  of  Consequently  regression  m u l t i p l e r e g r e s s i o n techniques  have c o n t a i n e d  differences in i n i t i a l the  study  students*  b i a s which  may  were  bias or e r r o r a t t r i b u t e d  characteristics.  have  occurred,  To  results  to  determine from  1986-7 V a l i d a t i o n Study were compared both w i t h and w i t h o u t  the pre-  t e s t d a t a as a c o v a r i a t e . F u r t h e r a n a l y s i s o f t h e 1986^87 d a t a , u s i n g a panel of  cross-lagged  c o r r e l a t i o n between t e a c h e r s • p e r c e p t i o n s and  students',  provided  an  indication  of  the  perceptions  extent  to  which  teachers' perceptions a f f e c t students' perceptions.  3.4  INSTRUMENTATION  Booklets f o r students I n t h e 1985  Assessment f o u r b o o k l e t s , e n t i t l e d Q, R,  T, were a d m i n i s t e r e d basis. was  The  Q  randomly t o Grade 7s' on a  booklet,  administered  which  contained  test  of the  matrix-sampling  design.  Each  of  booklets  R,  items,  population  whereas t h e o t h e r s were e v e n l y d i s t r i b u t e d t o t h e b a l a n c e , a  S,  using and  c o n s i s t e d o f 50 d i f f e r e n t m u l t i p l e - c h o i c e achievement items, of three a t t i t u d e scales, The  achievement  items  and  a common s e t o f background  measured  the  following  and  one-per-student  open-ended  t o a sample o f f o u r p e r c e n t  S,  seven  T one  items.  domains:  Number  and  Operation,  Problem S o l v i n g , Computers. the  non-curricular.  Test  five  curricular of  content  ranges  of  area  in  prescribed domains,  domains,  different  two  present new  all  (A c o p y A.)  not  of  each  to  of  the  present  items  and  statistics, in  from  the  two  analysis  items  were  of  equivalent  R,S  and  T  is  examined on  the  non-curricular  calculators  analysis.  the  booklets  study  and  from  four.  and achievement the  the  domain  across  allocate  Total in  first  booklets  schooling  test  included  deleted  made  Since  between i n p u t s  were  was  Test  contained  evenly  the  entitled  Applications  distributed  were  study,  The  items  the  two  domains  established.  sum o f  attempt  Appendix  achievement  options,  the  the  latter  in  Form  since  Q  they  and  results involved  items.  scan  Table in  and  whereas  were  and an  curriculum,  also  optical  items  probability  computers,  five  the  of  the  r o u t i n e s t o r y p r o b l e m s drawn from t h e  relationships  All  of  Topics,  and C a l c u l a t o r s and  whereas  deleted  Algebraic  domains measured c o n t e n t  purposes  were  difficulty.  contained  were  the  comprised  Achievement by  these  a n d A p p l i c a t i o n s were  was  consisted  of  curriculum,  For  domains  Total  domain  five  Columbia  two  Measurement,  P r o b a b i l i t y and S t a t i s t i c s ,  The f i r s t  British  latter  Geometry,  1  including  categories.  don't  know."  choice  Students  format  with  answered  on  sheets. lists  study.  Applications,  "I  multiple  the  items  As  reported  was c r e a t e d This  contained  earlier,  from items  d o m a i n was  not  in  each  Domain  contained  part  of  domain  the  7,  i n the  included entitled  first  original  four  table  of  specifications  and  therefore  no  attempt  had  d i s t r i b u t e items c o n t a i n e d i n i t e v e n l y a c r o s s The  attitude  follows:  scales  Mathematics  in  School,  in  the  Booklet  Only  from t h e Mathematics i n School  i n the  c u r r e n t study.  perceptions enjoyment  of  the  C a l c u l a t o r s and  In t h i s importance,  mathematics c u r r i c u l u m .  booklets R;  Computers,  difficulty ten  to  were  as  Gender  in  Booklet  T.  s c a l e were examined  s c a l e students  i n l e a r n i n g associated with  made  forms.  Booklet  Mathematics, results  S;  contained  been  were asked in  their  learning  major t o p i c s i n  and the  34 Table 1 Domains and Item Assignments Domain 1.  Item Numbers ( b o o k l e t and no.)  Number and O p e r a t i o n  R: S: T:  2.  Geometry  R: S: T:  1, 2, 3, 4, 37, 40, 41, 1, 2, 3, 4, 25, 26, 27, 1, 2, 3, 4, 16, 17, 39,  8, 9, 14, 15, 16 42, 43 11, 12, 13, 19, 48, 49, 50 9, 10, 11, 15, 47, 48, 49  12, 13, 19, 20, 26, 27, 31, 32, 38, 39, 44, 45 5, 6, 9, 10, 20, 21, 36, 37, 39, 40, 41, 42 12, 13, 14, 18, 19, 27, 28, 29, 40, 41, 42, 43  3 . Measurement  R: S: T:  21, 28, 48, 49, 50 14, 15, 16, 22, 38 35, 36, 37, 38, 44  4.  Algebraic Topics  R: S: T:  5, 29, 30, 33, 34, 35, 36, 46 23, 24, 30, 31, 32, 33, 43, 44 5, 6, 23, 24, 25, 26, 45, 46  5.  Problem  R: S: T:  17, 18, 25 34, 35, 47 8, 20, 50  6.  Test Total  R: S: T:  all all all  7.  Applications  R: S: T:  14, 30 2, 3, 4, 24, 39, 44 10, 26, 38, 39  Solving  items l i s t e d above items l i s t e d above items l i s t e d above  T e s t items from n o n - c u r r i c u l a r domains a r e n o t i n c l u d e d i n this table.  I n g e n e r a t i n g t h e 1985 achievement from  the University  selected  o t h e r s from  of  British  a  pre-determined  Columbia  i n s t r u m e n t s used  several different jurisdictions. table  items,  a contract  developed  i n testing  some  programs  team and from  Items were s e l e c t e d t o r e f l e c t  of specifications  and on t h e b a s i s o f  t h e i r psychometric p r o p e r t i e s . the  1977  and  National States, number other  Provincial  Assessment the  of  Second  surveys  Canadian  chosen of  1981  Sources f o r t h e s e items  of  Assessments  Educational  International conducted  t o measure a wide range  cognitive  behavior.  Problem-Solving students'  domain  ability  to  Study  Items  of  included  example,  consisted apply  of  prior  in  the  Zealand, in  and  the  pool  a l l questions which  knowledge  and and  different  items  the  United  Mathematics,  New  of a b i l i t y  For  Mathematics,  Progress  i n England,  provinces.  of  included  in  a in  were levels  in  the  measured unfamiliar  situations. To items  complete needed  jurisdictions, with  Grade  O'Shea  8  (1985,  following  to  the be  table  of  developed  specifications and  some,  required modification. students  i n the  p.14),  reported  criteria  Fall that  a  number  selected  of  from  new  other  These items were p i l o t e d of  1984.  items  Robitaille had  to  meet  b e f o r e they were c o n s i d e r e d f o r a d d i t i o n  the p o o l . Standard i t e m s t a t i s t i c s were computed f o r each o p t i o n o f each item. On t h e b a s i s o f t h e s e r e s u l t s , items which showed any o f t h e f o l l o w i n g c h a r a c t e r i s t i c s were e i t h e r eliminated or modified p r i o r to being c o n s i d e r e d f o r p o s s i b l e use i n t h e Assessment: • more than 95% o r fewer t h a n 10% o f t h e s t u d e n t s c o r r e c t l y answered t h e q u e s t i o n ; • not a l l d i s t r a c t o r s a t t r a c t e d  respondents;  • t h e b i s e r i a l c o r r e l a t i o n between t h e c o r r e c t answer and t o t a l t e s t s c o r e was l e s s than 0.20;  and the to  36 • t h e b i s e r i a l c o r r e l a t i o n between t h e c o r r e c t answer and t o t a l t e s t s c o r e was l e s s than t h e b i s e r i a l c o r r e l a t i o n between a d i s t r a c t o r and t h e t o t a l t e s t s c o r e . The t a b l e on f i n a l  of specifications  forms were reviewed  and t h e items chosen  f o r c o n t e n t v a l i d i t y by an A d v i s o r y  Committee, composed o f mathematics e d u c a t o r s . psychometric  properties of the s e t of t e s t  A summary o f t h e items  each form i s shown i n T a b l e 2 below (Anderson,  Summary S t a t i s t i c s  Table 2 f o r Grade 7 T e s t B o o k l e t s  Standard Dev.  R  55.6  9.0  0.87  S  51.0  8.3  0.83  T  55.2  8.3  0.84  These d a t a difficulty Students'  show t h a t  with  means  results  c o n t a i n e d on  1986).  Mean P e r c e n t Correct  Form  t o appear  Cronbach's A l p h a  t h e forms were o f s i m i l a r  ranging  between  were d i s t r i b u t e d  levels of  51.0 and 55.6 p e r c e n t .  i n a s i m i l a r way about  each  mean as r e f l e c t e d by s t a n d a r d d e v i a t i o n s r a n g i n g between 8.3 and 9.0.  V a l u e s o f Cronbach's c o e f f i c i e n t a l p h a i n d i c a t e  instrument  was  reliable.  Based  on t h e s e  results  that  each  i t can be  s t a t e d t h a t the.forms were r e l a t i v e l y p a r a l l e l i n n a t u r e . Students  1  The following  Background Items student areas:  background  items  age, gender,  gave  first  information language,  i n the  mathematics  program,  homework,  parents'  m e t r i c u n i t s o f measure. used  of  and  i t  was  study  assumed  that  students'  which  "Mathematics  measured  difficulty  and  operations  with  working  with  responses  basic facts; about  d o i n g geometry.  of  fractions;  scale  the  equations;  the present items  c o n s i s t e d o f t e n items of  the  topics:  operations  about  importance,  with  responded  basic  with  decimals;  estimation;  memorizing  s o l v i n g word problems;  system; working  The s c a l e was adapted  different  of several  following  basic  learning  Students  t o each  perceptions  enjoyment  solving  the metric  scale.  students'  As a r e s u l t  toward  "Mathematics i n S c h o o l . "  i n School"  percents;  attitudes  o f t h e i r a t t i t u d e s toward  c o n t a i n e d i n S c a l e R, e n t i t l e d The  (1977) and Whitaker  students'  i n t h e mathematics c u r r i c u l u m . examined  had been  Mathematics  mathematics were composites topics  items  with  assessments.  Based on f i n d i n g s o f K y l e s and Sumner (1982),  familiarity  The m a j o r i t y o f t h e s e  i n previous p r o v i n c i a l  Students' Perceptions  education,  perimeter  learning  and a r e a ; and  on a f i v e - p o i n t L i k e r t - t y p e  from one developed  f o r use i n the  Second I n t e r n a t i o n a l Study o f Mathematics. Teacher Q u e s t i o n n a i r e The  Teacher  Questionnaire  s e c t i o n s , which a r e l i s t e d Mathematics Computers. information listed.  i n School,  was  comprised  as f o l l o w s :  Problem  F o r t h e purpose examined  was  Background  Solving,  four  major  Information,  and C a l c u l a t o r s and  of the current  limited  of  t o the f i r s t  study,  however,  three  sections  A copy o f t h e q u e s t i o n n a i r e i s i n c l u d e d i n Appendix B.  38  Items teachers'  in  background  preparation  development periods,  the  and  for  activities,  section  teaching  frequency  included  mathematics, and  a number o f s p e c i f i c  questions  length  activities  on  professional  of  mathematics  i n the  classroom.  Items i n t h e "Mathematics i n S c h o o l " s e c t i o n d e a l t w i t h t h e same t o p i c s as those  i n the student booklet.  section  questions  problem  solving,  were  included  strategies  on  taught  In t h e  teachers• to  problem-solving  a t t i t u d e s toward  students  and  activities  used t o f a c i l i t a t e t h e l e a r n i n g o f problem s o l v i n g .  3.5  DESCRIPTION AND  Dependent mathematical Problem  DEFINITION OF THE  variables achievement  Solving,  were in  VARIABLES  selected  three  A p p l i c a t i o n s and  S o l v i n g v a r i a b l e measured s t u d e n t  as  indicators  categories Test  Total.  achievement on  to r e q u i r e higher order t h i n k i n g s k i l l s  of  of  interest:  The  Problem-  items  designed  whereas t h e A p p l i c a t i o n  v a r i a b l e a s s e s s e d s t u d e n t achievement on r o u t i n e s t o r y problems. Overall Total  achievement  in  was  measured  by  the  Test  variable. Categories  of  independent  i n c l u d e d the f o l l o w i n g : Students'  Perceptions  Mathematics, Processes.  A  number  major  topics  in  scales—importance,  Student  v a r i a b l e s i n the  of  Teachers'  Organization items  present  study  Background, Teacher Background,  of Mathematics,  Classroom  w i t h i n each c a t e g o r y . of  mathematics  were used  Perceptions  of  and  Problem-Solving  to  measure v a r i a b l e s  S i n c e t e a c h e r s ' and s t u d e n t s ' p e r c e p t i o n s the  curriculum  difficulty  and  involved  three  enjoyment—the  sum  different of  each  s c a l e was used as a v a r i a b l e f o r f u r t h e r a n a l y s i s a f t e r a review of  t h e i r psychometric  properties.  A l i s t i n g o f each  independent  v a r i a b l e and i t s source, i s shown i n T a b l e 3.  Table 3 Independent V a r i a b l e s and T h e i r Sources Source  Variable A.  Student  Background  1. 2. 3. 4. 5.  Language f i r s t spoken Language spoken a t home now Time spent on mathematics homework Father's l e v e l o f education Mother's l e v e l o f e d u c a t i o n  B.  Teacher  1. 2. 3. 4. 5. 6. 7.  Years e x p e r i e n c e P r e f e r e n c e t o t e a c h mathematics Proportion of teaching load Attendance a t c o n f e r e n c e s Attendance a t workshops Mathematics c o u r s e s completed Mathematics e d u c a t i o n c o u r s e s completed  C.  Student P e r c e p t i o n s o f Mathematics  Background  and Item Number  Background I n f o r m a t i o n Student B o o k l e t 1 2 7 9 10  Background I n f o r m a t i o n Teacher Q u e s t i o n n a i r e 1 2 3 5 6 8  Scale R Student B o o k l e t  P e r c e p t i o n s o f importance, d i f f i c u l t y and enjoyment o f the f o l l o w i n g t o p i c s : 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  Adding, s u b t r a c t i n g , m u l t i p l y i n g , and d i v i d i n g f r a c t i o n s Adding, s u b t r a c t i n g , m u l t i p l y i n g , and d i v i d i n g d e c i m a l s Working w i t h p e r c e n t s L e a r n i n g about e s t i m a t i o n Memorizing b a s i c f a c t s S o l v i n g equations S o l v i n g word problems L e a r n i n g about t h e m e t r i c system Working w i t h p e r i m e t e r and a r e a Doing geometry  2 3 4 5 6 7 8 9 10  40 D.  Teacher P e r c e p t i o n s o f Mathematics  Scale R Teacher Questionnaire  P e r c e p t i o n s o f importance, d i f f i c u l t y t o t e a c h and enjoyment i n t e a c h i n g t o p i c s l i s t e d under "Student P e r c e p t i o n s o f Mathematics" E. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. G. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.  Classroom  Organization  Background I n f o r m a t i o n Teacher Questionnaire  Type o f c o u r s e Frequency o f t e s t i n g Number o f c l a s s e s p e r week Length o f p e r i o d Time on homework a c t i v i t i e s Questioning Seatwork Working i n s m a l l groups Working a t s t a t i o n s Time on c o m p u t a t i o n a l d r i l l Giving lecture style instruction Problem-Solving  Processes  10 11 12 13 14 15 16 17 18 19 20 ' Scale S Teacher Q u e s t i o n n a i r e  P e r c e p t i o n o f s t u d e n t enjoyment E x p e c t a t i o n o f performance S a t i s f a c t i o n t o teach Ease o f t e a c h i n g Uses o f d i f f e r e n t s t r a t e g i e s I n s e r v i c e involvement Uses o f m a t e r i a l s r e s o u r c e s A c t i v i t i e s f o r motivation Frequency o f t e a c h i n g Problem t y p e s used O r g a n i z a t i o n o f room  In  order  to  u n d e r l y i n g items,  examine  common  1 2 5 7 8 9 10 11 12 14 15  dimensions  3 (with t h e e x c e p t i o n  and Teacher P e r c e p t i o n s ) , was s u b s e q u e n t l y t h e 1985 Assessment.  into related clusters. variables  from  constructs  a f a c t o r a n a l y s i s o f v a r i a b l e s w i t h i n each o f  t h e c a t e g o r i e s shown i n T a b l e  from  or  the  After  o f Student  undertaken u s i n g data  a n a l y s i s they  were  combined  To d i s t i n g u i s h t h e s e new combinations o f original  ones  they  are  referred  to  as  factors,  r a t h e r than v a r i a b l e s .  ten-item  scales  under  S i n c e t h e sums o f each o f t h e  Students'  and  Teachers'  Perceptions  produced o n l y 3 v a r i a b l e s f o r each o f t h e s e c a t e g o r i e s , a f a c t o r analysis  was  n o t conducted.  d e s c r i p t i o n , these  However,  f o r purposes  o f ease o f  sums a r e r e f e r r e d t o as f a c t o r s when r e s u l t s  are d i s c u s s e d .  3.6  DATA COLLECTION  A d e s c r i p t i o n of the procedures administration  and  collection  questionnaires i s presented  of  in this  used f o r t h e d i s t r i b u t i o n , test  booklets  section.  and  teacher  D i s c u s s i o n of the  c r i t e r i a used f o r t h e a g g r e g a t i o n o f d a t a i s a l s o i n c l u d e d . 1985 Data C o l l e c t i o n I n t h e 1985 P r o v i n c i a l Assessment o f Mathematics,  booklets  f o r s t u d e n t s were packaged by s c h o o l , a t a c e n t r a l  location, i n  numbers  10  Since  which a  exceeded  matrix  reported  sampling  i n t e r l e a v e d a t t h e packaging  enrollments  by  used,  percent.  design  was  booklets  were  stage.  Administration instructions  t o t e a c h e r s d i r e c t e d them t o d i s t r i b u t e t h e b o o k l e t s t o s t u d e n t s i n t h e o r d e r they appeared i n each package. A l e t t e r t o p r i n c i p a l s i n s t r u c t e d them t o a s s i g n one c l a s s code number more than  t o each t e a c h e r .  one c l a s s  I n cases  where a t e a c h e r  o f Grade 7 mathematics,  t e a c h e r was a s s i g n e d t o t h e f i r s t  t h e code  table cycle.  In r e s p o n d i n g  the  was  t o answer c l a s s - s p e c i f i c  instructed  r e l a t i v e t o the assigned  class.  f o r that  c l a s s met d u r i n g t h e week o r  i n t h e time teacher  taught  to the questionnaire, questions  Both optical  students  scan  and t e a c h e r s  sheet.  Teachers  responded  and t h e i r  t o questions  students  on an  used  t h e same  c l a s s code numbers t o a l l o w f o r l i n k a g e s between t h e i r  responses  t o be made. Calculations Means aggregated were  and v a r i a n c e s t o the class  power  number know"  f o r Dependent V a r i a b l e s  tests,  o f items response  f o r each level.  these  was  reported scores.  Since  statistics  responded  achievement  included  t h e assessment  were  t o by each  variable  calculated  student.  booklets using the  The " I don't  i n the determination  I n c l u s i o n o f t h a t response  were  of  these  i s c o n s i s t e n t with  p a s t p r a c t i c e f o r p r o v i n c i a l assessments i n B r i t i s h Columbia and with  the National  United  States.  Assessment  Since  students  of Educational responded  Progress  t o only  i n the  one i n t h r e e  items, c l a s s s t a t i s t i c s a r e r e p o r t e d as e s t i m a t e s . Calculations  f o r Independent V a r i a b l e s  Options  f o r a l l non-achievement  items  i n s t r u m e n t s were l a b e l l e d a l p h a b e t i c a l l y . values  f o r responses,  The w e i g h t i n g s  the options  were  contained  i n the  In order t o c a l c u l a t e re-coded  numerically.  a s s i g n e d t o each o p t i o n a r e r e p o r t e d i n Appendix  C. Due students As  a  to  the matrix-sampling  that  was  d i d n o t have an o p p o r t u n i t y t o respond result,  calculations.  missing A  class  summing t h e w e i g h t i n g s number  design  of  respondents.  responses mean was  were  not  calculated  of options selected For teachers,  used, a l l  t o each used  f o r each  and d i v i d i n g options  item.  i n the item  by  by t h e  f o r single-  response  items  were  However,  options  given  the  f o r multiple-response  Teacher Q u e s t i o n n a i r e ,  were a s s i g n e d  o r a "0" i f t h e y were n o t . response  weighting  assigned  items,  included  a "1" i f t h e y  Teachers'  scores  to i t . i n the  were  chosen  f o r each m u l t i p l e  item was a r r i v e d a t by summing t h e " l " s .  1986-87 Data C o l l e c t i o n Similar were  procedures  followed  t o those  used  f o r t h e packaging  used i n t h e 1986-87 V a l i d a t i o n administered  during  arrangements  were  September made  and d i s t r i b u t i o n Study. and  to  used t h e same c l a s s  assigned  centrally.  Two  scale.  i t consisted In  May,  of  teachers  again  were  the  following  May,  all  booklets  and  To ensure t h a t t e a c h e r s and  Questionnaire  only  the  were  asked  to  were  were used.  "Mathematics  q u e s t i o n n a i r e used i n t h e 1985 p r o v i n c i a l  3.7  of materials  codes f o r each s e s s i o n , t h e y  v e r s i o n s o f t h e Teacher  September  assessment  Since the booklets  collect  q u e s t i o n n a i r e s a f t e r each s i t t i n g . students  i n t h e 1985  in  complete  In  School"  the  full  assessment.  DATA ANALYSIS PROCEDURES  1985 Assessment Data C l a s s means and v a r i a n c e s were computed f o r each relating by  t o students'  domain.  classroom Appendix  Teacher processes  C.  respondents  background, variables were  t o each  item.  on background,  coded  A l l calculations  perceptions,  numerically were  based  and achievement perceptions as  on  variable  reported  the  number  and in of  44 Correlational  Analysis  At t h e second stage o f a n a l y s i s , association  among  determined.  t h e independent  Pearson  correlation  measure o f t h i s a s s o c i a t i o n . provided  measures  an index o f t h e degree o f  and dependent  variables  c o e f f i c i e n t s were  used  was as a  The r e s u l t i n g c o r r e l a t i o n m a t r i c e s  of the s t a t i s t i c a l  interrelationships  among  variables. Factor Analysis At  t h e next  stage,  patterns  of  whether  variables.  information  on were  analysis  relationships  independent  variables  factor  the  Results  common,  located.  was  existed of  this  underlying  As  to identify  within  groups  procedure  of  provided  dimensions  a by-product  number o f v a r i a b l e s t o be i n v e s t i g a t e d  used  on  which  of t h i s process the  i n the following  stage of  a n a l y s i s was reduced. The Student and  variables Background,  Problem-Solving  loadings  calculated  combinations discussed variables been  Multiple  Teacher  in  to  of  the  category  Background,  Processes  under T e a c h e r s by  each  were  the  variables  according  reduced  subscales,  of  under  were  Classroom  clustered,  factor  referred  and S t u d e n t s  calculating  based  to  Since 1  as  on  factors  the  f o r each  factor  Resulting  number  P e r c e p t i o n s had  the t o t a l  of  Organization  analysis.  composition. 1  headings  and of  already  of the three  t h e y were n o t f a c t o r a n a l y z e d . Regression  Subsequently  the s e t of factors  within  each  category  r e g r e s s e d on each o f t h e t h r e e dependent o r c r i t e r i o n  was  variables.  This resulted  i n production of a family of regression equations  for  each  dependent  was  conducted d u r i n g t h i s s t a g e t o determine which  the best p r e d i c t o r s were t h e n  variable.  A  step-wise r e g r e s s i o n  o f s u c c e s s i n each  r e g r e s s e d a g a i n on  each  f a c t o r s were  equation.  dependent  reference to category, to a r r i v e at the f i n a l  analysis  A l l factors  variable,  without  models.  The f u n c t i o n a l r e l a t i o n s h i p between s t u d e n t achievement  and  t h e independent v a r i a b l e s c o u l d be d e s c r i b e d as f o l l o w s : Student Achievement where: SB TB SP TP CO PS  = = = = = =  = f (SB, TB, SP, TP, CO,PS)  Student Background Teacher Background Student P e r c e p t i o n s Teacher P e r c e p t i o n s Classroom O r g a n i z a t i o n Problem-Solving Processes  A g e n e r a l l i n e a r m u l t i p l e r e g r e s s i o n model d e s c r i b i n g r e l a t i o n s h i p i s shown below:  where: observed s c o r e o f t h e i t h c l a s s on Y 1, M classrooms 1,..., 6 independent v a r i a b l e s J c o n s t a n t term /3j — u n s t a n d a r d i z e d r e g r e s s i o n coefficient X --j = v a l u e o f t h e j independent  4  t  n  T  •  £ J  .th  v a r i a b l e on t h e 1 " t r i a l = r e s i d u a l o r e r r o r term  this  46 1987  V a l i d a t i o n Study Analysis  involved used  on  of  two  provincial  from The  data,  with  It by  their arrival  1987  Validation  employed  the  same method  post-test class  The  second  achievement  i n v o l v e d an  controlled  students  for  the  variance  l e a r n i n g which  1  i n these c l a s s e s .  took  in  i n the r e g r e s s i o n  extent  to  place  In t h i s p r o c e s s ,  correlation  which  c o e f f i c i e n t s provided  teachers'  unanswered  Provincial  i n the  Assessment.  Study  employed  panel  correlation  prior  to  class pre-test covariates  equations.  and  students'  s i m i l a r , t h e q u e s t i o n o f which had remained  analysis  achievement  means f o r achievement i n mathematics were t r e a t e d as  Although  Study  which combined r e g r e s s i o n a n a l y s i s w i t h a n a l y s i s  variance.  contributed  the  first  dependent v a r i a b l e s .  of covariance, of  data  approaches.  the  means as  the  time  To  as  were  a g r e a t e r impact on t h e  other  address  (Campbell  &  the  perceptions  a n a l y s i s of  a third  measures o f  this  results  the  1985  issue, the V a l i d a t i o n  variable, using Stanley,  from  1963).  a  cross-lagged To  determine  which v a r i a b l e had t h e g r e a t e r e f f e c t on t h e o t h e r , c o r r e l a t i o n s between  teachers  perceptions students'  1  a t Time 2;  perceptions  cross-lagged  perceptions and  at  1  teachers • perceptions  a t Time 1 were compared.  c o r r e l a t i o n s was  significantly  the o t h e r , t h i s would p r o v i d e evidence g r e a t e r e f f e c t on t h e  Time  other.  and  students'  a t Time 2  and  I f one  the  of  more p o s i t i v e  o f which v a r i a b l e had  than the  3.8  SUMMARY  This plan,  the  chapter  has  instruments  a n a l y s i s procedures. the a n a l y s i s  and  described and The  the  variables, next  i n Chapter 5,  population and  data  and  collection  chapter presents the the  sampling  results  r e l a t i o n s h i p s between  r e s u l t s and t h e q u e s t i o n s under study a r e d i s c u s s e d .  and of  these  48  CHAPTER 4 FINDINGS  Analyses preliminary  of  the  data  analyses,  involved  correlational  and m u l t i p l e r e g r e s s i o n a n a l y s e s . t h e d a t a were p r e p a r e d  PREPARATION OF THE DATA  The  p r e l i m i n a r y a n a l y s i s stage  the  data  into  f o r f u r t h e r study.  of students' numerical  aggregated deviations  the  were  phases:  factor  analyses  analyses,  A t each s t a g e o f t h e p r o c e s s ,  involved i n i t i a l The  responses  form.  to  distinct  f o r t h e subsequent s t e p .  4.1  coding  four  step  t o background  Following class  first  level  calculated  this  for  involved the r e -  and a t t i t u d e  step,  where  preparation of  responses  means  each  and  items were  standard  variable  under  investigation. At Teacher data  t h e next  stage,  Questionnaire  were  teacher, response required  then  were  stored  containing item  teachers'  and  re-coded  i n the  numerical means  responses  t o numerical  form  values  of  one  A total  to  f o r those o f 1073  from t h e  form.  record  assigned  calculated  m u l t i p l e responses.  t o items  These  f o r each  each  single  items  which  teacher  records  were produced i n t h i s manner. C l a s s and t e a c h e r r e c o r d s were matched a t t h e next stage o f analysis. class  sizes  I n some cases were  found  no match was  t o be u n u s u a l l y  found,  whereas i n o t h e r s  small or large.  Since  t e a c h e r s answered t h e i r q u e s t i o n n a i r e s w i t h r e s p e c t t o a s i n g l e  class,  no more t h a n  one  match was  expected  taught  Grade 7 mathematics t o more t h a n  i n c a s e s where they  one  class.  s m a l l o r l a r g e c l a s s e s were deemed t o be due missing could  class have  principals  Unusually  t o students  code numbers o r p r o v i d i n g i n c o r r e c t  resulted o r by  from  non-assignment  t e a c h e r s , o r by  of  omission  ones.  class  of the  either This  codes  by  assigned  code  i n c a s e s where  they  number by one o r more s t u d e n t s i n each c l a s s . The were  non-assignment  assigned,  the  of c l a s s  failure  of  codes o r , teachers  to  ensure  s t u d e n t s r e c o r d e d them on t h e i r answer s h e e t s , may to p o l i t i c a l were  reasons.  instructed  non-achievement taken,  by  F o r example, t e a c h e r s  their  items  superintendents  in this  the  first  asked  to  assessment complete  r e s u l t s from t h e i r Because o f and  g r e a t e r than  number was would  have  booklet  Q  a  some o f  the  This  position  was  which  where  teachers  could  be  such  considerations, class  40 were dropped  from  sizes  was were  linked  less  the a n a l y s i s .  w r i t t e n each was  of  administered  of  40  13  the  booklets  randomly  R,  to  S 4  ranges  was  selected  f o r matched  study i s shown i n T a b l e  and  than  The  to  13  lower  since  few,  4.  records  which  T.  percent  s t u d e n t s were r e q u i r e d from  g r e a t e r than t h i s number were known t o e x i s t . size  this  s e l e c t e d so t h a t a t l e a s t f o u r s t u d e n t s i n each c l a s s  upper bound  class  d e c i s i o n s taken  In a d d i t i o n ,  Columbia  questionnaire  districts  classes.  population, at l e a s t An  in British  due  omit  i n p a r t , because o f a number o f unpopular  by t h e p r o v i n c i a l M i n i s t r y o f E d u c a t i o n .  have been  i n some  to  assessment.  that a l l  each  i f any,  Since of  the  class. classes  A d i s t r i b u t i o n of remained  in  the  50 Table 4 Frequency D i s t r i b u t i o n o f C l a s s S i z e s  Class Size 13-18  145  20  20  19-24  255  35  55  25-30  292  40  95  37  5  100  31-40  A total an  average  percent  o f 729 class  from  the  were r e c e i v e d .  c l a s s e s remained i n t h e p r e s e n t  size  code numbers  on  As  4.2  23.  This  number o f  Teacher  a  study loss  of  32  q u e s t i o n n a i r e s which  this  l o s s was from  likely  missing  Q u e s t i o n n a i r e s , non-assignment  o r non-completion  with  class  of  o f code numbers on  due  code  answer  students.  this  section  discussed.  major  teacher  I t c o u l d have r e s u l t e d  DESCRIPTIVE ANALYSIS OF THE  In  represented  mentioned e a r l i e r ,  numbers t o c l a s s e s , s h e e t s by  of  total  to s e v e r a l reasons.  and  Cumulative Percent  Percent  Frequency  distributions  student  organization,  perceptions mathematics.  of  of  Results are presented  categories:  classroom  INDEPENDENT VARIABLES  responses  background,  and  presented  i n each o f t h e f o l l o w i n g  problem-solving  mathematics  are  teacher  background,  processes,  student  teacher  perceptions  of  51 Student Background V a r i a b l e s A  total  of five  background  examined:  language f i r s t  home, time  spent  variables  f o r students  spoken, language c u r r e n t l y  on t h e l a s t  mathematics homework  were  spoken a t  assignment,  l e v e l o f f a t h e r s ' e d u c a t i o n , and l e v e l o f mothers' e d u c a t i o n . Students tongue.  responded  t o two q u e s t i o n s r e l a t e d t o t h e i r mother  R e s u l t s a r e shown i n T a b l e 5.  Table 5 Mother Tongue (Percent) English Language f i r s t  spoken  89  C u r r e n t home language  94  Eighty-nine percent their  language  first  spoke  a  These r e s u l t s language  other  6  identified  E n g l i s h as  The p r o p o r t i o n o f s t u d e n t s f o r  whom E n g l i s h was t h e language percent.  11  o f t h e students  spoken-  Non-English  currently  spoken a t home was 94  show t h a t o f those than  English,  students  45  percent  who  first  did  not  c u r r e n t l y speak t h e i r mother tongue a t home. For level,  t h e purpose  student  selected  o f aggregating  responses  were  assigned  and a "1" i f a n o n - E n g l i s h  index number was then determined equal  results a  t o t h e classroom  "2" i f E n g l i s h was  response  was chosen.  f o r each c l a s s .  I n d i c e s were  t o 1 + x, where x was e q u i v a l e n t t o t h e p e r c e n t a g e  students  f o r whom  distribution  English  was  t h e language  o f index numbers f o r language f i r s t  An  spoken.  of The  spoken o f a l l  52 classes  i n the analysis  d e v i a t i o n o f 0.13.  had  a  mean  of  1.89  and  a  standard  F o r language c u r r e n t l y spoken, t h e mean was  1.94 and t h e s t a n d a r d d e v i a t i o n was 0.09.  These r e s u l t s  relate  t o t h e d a t a shown i n T a b l e 5, where 89 p e r c e n t spoke E n g l i s h as their  first  language and 94 p e r c e n t  currently  speak E n g l i s h a t  home. S t u d e n t s were asked which one, o f f i v e t i m e i n t e r v a l s , most c l o s e l y approximated t h e amount o f t i m e t h e y s p e n t on t h e i r mathematics homework assignment.  last  Table 6 reports the r e s u l t s .  Table 6 Time Spent on Homework Amount o f Time  P e r c e n t o f Respondents  None  4  I - 10 minutes  29  I I - 30 minutes  52  31-60 minutes  12  More than 60 minutes  3  The d a t a r e p o r t e d i n T a b l e 6 show t h a t t h e v a s t m a j o r i t y o f students  (85 p e r c e n t ) spent 30 o r fewer minutes  mathematics homework assignment. to  this  students  item  is  spending  relatively 11^-30  The d i s t r i b u t i o n  symmetric,  minutes  on  with their  assignment and s i m i l a r numbers spending e i t h e r more t h a n one hour.  on t h e i r  52  last  o f responses percent  last  of  homework  no t i m e o r e l s e  53 Weightings to  5  a s s i g n e d t o each o p t i o n ranged  f o r "more  than  60 minutes."  from 1 f o r "none"  The d i s t r i b u t i o n  of  index  numbers f o r c l a s s e s had a mean o f 2.82 and a s t a n d a r d d e v i a t i o n of  0.33.  These  results  show  t h a t t h e average  amount o f  time  spent on t h e l a s t mathematics homework assignment by s t u d e n t s i n c l a s s e s was c l o s e s t t o t h e 11-30 minute time The response  items  on  rates.  parents'  educational  These r e s u l t s  may  interval.  levels  indicate  received  that  not know t h e e d u c a t i o n a l l e v e l s o f t h e i r p a r e n t s . 50 p e r c e n t d i d n o t s e l e c t an e d u c a t i o n a l l e v e l o r male g u a r d i a n  and 46 p e r c e n t  mother o r female  guardian.  items  have  could also  s i n g l e parent f a m i l i e s . who  selected  one  failed  to select  i n part,  students d i d F o r example,  f o r their father  The l a r g e o m i s s i o n  resulted,  from  one f o r t h e i r r a t e f o r these t h e number o f  T a b l e 7 shows r e s u l t s o f t h o s e  of the educational  levels  low  students  for either  their  parents or guardians. Table 7 Educational Level o f Parents (Adjusted Percent) Level of Education Attended  P e r c e n t o f Respondents Mother Father  L i t t l e o r None  2  3  Elementary  4  6  Secondary  45  40  Post Secondary  49  51  54 The  largest  secondary" of  f o r both mother and f a t h e r .  respondents  had  attended  secondary percent  p r o p o r t i o n o f responses  indicated  a college,  training. for their  that  university  This fathers  level  mother o r female  to a  guardian  form o f p o s t -  response  rate  guardians.  Ninety-four  s c h o o l a t e i t h e r a secondary  t o 91 p e r c e n t  o f 51  school Although  t h e s e r e s u l t s suggest t h a t p a r e n t s were w e l l educated,  i t should  noted  compared  male  "post  of fathers.  be  higher  or  was  F o r example, 49 p e r c e n t  o r some o t h e r  compared  p e r c e n t o f mothers a t t e n d e d or  their  by c a t e g o r y  that the question  completion Columbia  referred  a t each r e s p e c t i v e l e v e l . i s compulsory  until  age  t o attendance  level.  16,  Weightings lowest,  i t i s expected  s i m i l a r r e s u l t s were  f o r levels  of education  t o 4 f o r the highest.  that  school a t l e a s t t o the  I n t h e 1987 v a l i d a t i o n study, conducted  Surrey School D i s t r i c t ,  than  Since schooling i n B r i t i s h  v i r t u a l l y a l l p a r e n t s would have a t t e n d e d secondary  rather  i n the  found.  ranged  from  The d i s t r i b u t i o n  1 f o r the  of class-index  numbers f o r mothers' l e v e l o f e d u c a t i o n had a mean o f 3.37 and a standard of  d e v i a t i o n o f 0.36.  education  had a mean  On t h e o t h e r hand,  o f 3.39 and a  standard  fathers' level deviation of  0.29. Teacher Background Teacher background v a r i a b l e s c o n s i s t e d o f seven related  to  experience,  preference  to  teach  mathematics,  p r o p o r t i o n o f t e a c h i n g l o a d , p r o f e s s i o n a l development and p r o f e s s i o n a l t r a i n i n g .  55 p e r c e n t  activities  A discussion of results follows.  Responses t o t h e q u e s t i o n on t e a c h i n g e x p e r i e n c e that  questions  of teachers  had taught  indicated  11 o r more y e a r s , i n  5  contrast  t o only  6 percent  second y e a r o f t e a c h i n g . fewer t h a n in  two y e a r s  education  i n either  their  first  The s m a l l p r o p o r t i o n o f t e a c h e r s  o f experience  because  government.  who were  may have r e f l e c t e d  of restraint  F o r example,  policies  class  sizes  5  or with  cutbacks  of the p r o v i n c i a l  increased  in  British  Columbia d u r i n g r e c e n t y e a r s w h i l e s t u d e n t e n r o l l m e n t s d e c l i n e d . The  v a s t m a j o r i t y , 95 p e r c e n t ,  of teachers  indicated that,  g i v e n a c h o i c e , t h e y would n o t a v o i d t e a c h i n g mathematics. 3 percent and  r e p o r t e d t h a t they  2 p e r c e n t were  up  avoid  teaching  t o twenty  of teachers  percent  i n d i c a t e d t h a t mathematics  of t h e i r  teaching  p e r c e n t r e p o r t e d t h a t more than  f o r t y percent  spent  These  teaching  teachers level.  mathematics.  specialize This  the subject  undecided.  S i x t y three percent took  would  Only  i n teaching  result  could  be  Only  8  o f t h e i r time was  data  suggest  mathematics  due t o a  load.  at  limited  that  few  t h e Grade practice  7 of  p l a t o o n i n g f o r t h e t e a c h i n g o f mathematics and t h e s m a l l numbers of  Grade 7 c l a s s e s i n many elementary Teachers  development teacher  responded t o two q u e s t i o n s r e l a t e d t o p r o f e s s i o n a l activities.  attended  in  a workshop  The f i r s t  a mathematics  previous three years. at  schools.  asked  The second q u e s t i o n d e a l t w i t h  during  p e r c e n t had a t t e n d e d  or not the  session a t a conference  ( o t h e r than a t a conference)  mathematics  whether  the previous  a conference  three  i n the  attendance  o r an i n - s e r v i c e day years.  Fifty-one  and 59 p e r c e n t had a t t e n d e d a  mathematics workshop o r i n - s e r v i c e day w i t h i n t h a t time p e r i o d . Two q u e s t i o n s d e a l t w i t h p r o f e s s i o n a l t r a i n i n g . question  asked  how many p o s t - s e c o n d a r y  courses  The f i r s t  i n mathematics  56 had been s u c c e s s f u l l y completed, w h i l e t h e second asked number  of  successfully  education. section  completed  courses  in  mathematics  Courses were d e f i n e d as the e q u i v a l e n t o f a 1.5  at  the  University  of  British  Columbia.  post-secondary  courses  completed  or  six  i n mathematics  more.  The  S i n c e Grade 7 i s p a r t o f t h e results  for  the  completed  12 of  percent  had  teachers,  22  s c h o o l program i n  question  on  mathematics  I t i s common p r a c t i c e a t  l e v e l f o r t e a c h e r s t o t e a c h s e v e r a l s u b j e c t s , and  t h e r e f o r e s p e c i a l i s t s w i t h mathematics majors a r e more l i k e l y teach  at  no  i n mathematics e d u c a t i o n .  elementary  c o u r s e s completed were not s u r p r i s i n g . t h e elementary  only  same p r o p o r t i o n  p e r c e n t , had not completed any c o u r s e s  Columbia,  and  unit  Twenty-two  p e r c e n t o f t h e t e a c h e r s o f Grade 7 mathematics had  British  f o r the  the  secondary  level.  It  was  surprising  d i s a p p o i n t i n g , however, t o f i n d t h a t 22 p e r c e n t  of the  to and  teachers  had completed no mathematics methods c o u r s e s . Classroom  Organization  Teachers clustered  were asked  under  the  t e a c h e r s were asked and,  with  related  some  to  the  a total  category  of  t o respond  questions, type  of  period  spent  Organization.  The  with reference to a s i n g l e  class  the  Classroom  most  course,  a l l o c a t e d t o mathematics, and mathematics  o f e l e v e n q u e s t i o n s which were  recent  frequency  one. of  Questions  testing,  time  the p r o p o r t i o n o f t h e most r e c e n t on  several  different  classroom  activities. The indicate slower  q u e s t i o n which d e a l t w i t h c o u r s e t y p e asked whether  students),  the  program  r e g u l a r , or  they  o f f e r e d was  enriched.  teachers to  modified  Ninety-three  (for  percent  57 indicated that their Each  of  the  other  responses.  The  that  o r no  little  r e f e r e n c e c l a s s was two  high  categories  response  streaming  percent  mathematics  almost  once a week and percent  of  p e r week and teachers and  suggests level  Columbia.  or  quizzes gave  o f weeks.  quizzes  in them  Only  either  1  only  all.  of the teachers o f f e r  five  mathematics  E l e v e n p e r c e n t o f f e r fewer than f i v e c l a s s e s  t h a t the  Ninety-seven  length  order  to  determine  mathematics each week, a new products  results  indicated study,  ranged between 190  p e r c e n t o f the  o f p e r i o d s were between  31  amount  in  v a r i a b l e was  and  the  item  asked  percent  percent  of the spent  265 minutes p e r week. approximately how  time  c r e a t e d by  amount o f time  teachers  on t o answer q u e s t i o n s  of  spent  calculating  l e n g t h s o f p e r i o d s each week.  t h a t f o r 89  time spent on mathematics was One  the  between t h e numbers and  remained i n t h e  one  the  60 minutes. In  The  or  of  class  percent  couple  gave t e s t s  18 p e r c e n t o f f e r more.  indicated  tests  Forty-seven  once every r e p o r t i n g p e r i o d o r not a t  c l a s s e s each week.  r e g u l a r program  gave  once e v e r y  t h a t they  Seventy-one p e r c e n t  percent  schools i n B r i t i s h  day.  percent  4  i n mathematics a t t h e  teachers  every  42  indicated  the  a r e g u l a r program.  received  f o r the  o c c u r s a t Grade 7 i n elementary Ten  on  i n mathematics  The 223  many s t u d e n t s  c l a s s e s which  average weekly  minutes. they  had  d u r i n g t h e most r e c e n t p e r i o d .  i n d i c a t e d t h a t they  had  called Twenty-  c a l l e d upon l e s s than  q u a r t e r o f the c l a s s and 45 p e r c e n t c a l l e d upon more than  one-  half.  58 Six period  questions  spent  on  dealt  with  proportions  a number o f d i f f e r e n t  reported i n Table  of  the  most  activities.  recent  Results  8.  Table 8 Time Spent on Classroom (Percent)  Activity  None  Activities  P e r c e n t o f C l a s s Time 1-25 26-50 51-100  Homework  8  83  7  2  Seatwork  2  30  43  25  S m a l l Groups  59  35  4  Work S t a t i o n s  93  7  -  2  -  53  44  2  1  10  63  24  3  Computational  Drill  Explaining Topics  I n 92  percent  on homework-related t h e amount o f time centers,  where  of the c l a s s e s at l e a s t activities. students  in  93  allocated for this. drill.  some time  spent  percent  a t work s t a t i o n s  of  topics  to  the  activity  no  L i t t l e time a l s o was  spent on  computational  time  p e r c e n t who  was  spent  showed t h a t the  (37 p e r c e n t o f a l l c l a s s e s ) spent  o f the p e r i o d on t h i s a c t i v i t y .  entire  or  with  classes  majority i n the category  c l a s s e s teachers  spent  the  between 1 and 25 p e r c e n t o f the p e r i o d on d r i l l ,  the  was  T h i s c o n t r a s t e d markedly  F o r example, a c l o s e r l o o k a t t h e 44  than one-tenth  are  spent class.  part  of  This  the  In 90 p e r c e n t of  period explaining  result,  less  when combined  new with  59 others  showing  groups  or  at  little  or  activity  appear t o o r g a n i z e  no  time  centers,  their  allocated  f o r work  indicates that  in  small  most  teachers  t e a c h i n g o f mathematics i n a  lecture-  s t y l e manner. Problem-Solving Teachers Solving  Processes responded  Processes  responses. solving,  The  and  eleven  category,  questions  in-service  solving  to  five  of  which  dealt with  activities,  varieties  questions  of  in  the  Problem-  required  multiple  a t t i t u d e s toward  frequency  approaches  of  and  problem  teaching  problem  resources  used  by  teachers. Teacher a t t i t u d e s toward problem s o l v i n g i n v o l v e d to  four  questions  enjoyment  of  satisfaction to teach. enjoyed their  and  Only  students of  easy  27  percent  solving would  teachers  to  questions, this  their  achievement  problem s o l v i n g and it  to  in  i n teaching the t o p i c ;  problem  percent  related  perform were  Based  appears  to  student  solving;  easy  their  found i t  t h a t most o f t h e i r  students  expected  on  satisfied  on  of  they  well  be  how  percent  an even lower  teach.  there  28  problem and  thought  and  perceptions  responses  that with  that  most  of  Only  45  teaching  of  topic. their  p r o p o r t i o n , 19 p e r c e n t , results  a  need  for  the  f o r more  found  latter  two  in-service  on  topic. Responses t o t h e  support  to  the  question  suggestion  o p p o r t u n i t i e s i n t h i s area.  on that  in-service there  is  involvement need  lends  for  more  F o r example, 70 p e r c e n t o f t e a c h e r s  i n d i c a t e d they had not a t t e n d e d any workshops on problem  solving  i n t h e p a s t y e a r and o n l y 9 p e r c e n t had a t t e n d e d more than  one.  T h i r t y - n i n e p e r c e n t o f t e a c h e r s answered t h a t problem class.  s o l v i n g e v e r y day, In c o n t r a s t ,  25  they  taught  as a r e g u l a r p a r t o f t h e mathematics  percent  indicated  that  they  taught i t  o n l y as a u n i t from time t o t i m e . F i v e items  from t h i s  category involved m u l t i p l e  Teachers were asked t o s e l e c t which t h e y s u b s c r i b e d . strategies  taught,  responses.  from t h e o p t i o n s l i s t e d ,  those to  The q u e s t i o n s d e a l t w i t h p r o b l e m - s o l v i n g  sources  of  exercises  a c t i v i t i e s used t o m o t i v a t e s t u d e n t s , problem  used,  different  t y p e s a s s i g n e d and  f e a t u r e s used i n the c l a s s r o o m . There  were  five  which t o choose.  different  problem-solving s t r a t e g i e s  R e s u l t s a r e shown i n T a b l e Table 9 Problem-Solving S t r a t e g i e s  Strategy  9.  Taught  Percent of Teachers  Look f o r a p a t t e r n  84  Guess and  42  Make a  check  list  70  Make a s i m p l e r problem  72  Work backwards  46  The solving  results strategy  from  i n T a b l e 9 show t h a t t h e most p o p u l a r t a u g h t was  "look f o r a pattern."  p e r c e n t o f t e a c h e r s r e p o r t e d they taught t h i s t o o n l y 42 p e r c e n t who  t a u g h t "guess and  Eighty-four  strategy  check."  problem-  compared  Another t h e y used Table  question  asked  teachers  f o r problem-solving  to  identify  exercises.  sources  Results are  shown i n  10.  T a b l e 10 of Problem-Solving E x e r c i s e s  Sources Source  Percent of  Textbook  93  Mathematics c o n t e s t s  37  Problem-solving  67  booklets  Professional journals  19  Book o f p u z z l e s  59  Results popular  indicate  source  for  that  the  textbook  problem-solving  is  percent  exercises.  who  used  T h i s low  professional response  r e f l e c t s the  low  in  each  the  most  Ninety-three  as  a  rate to professional  o f t h e N a t i o n a l C o u n c i l o f Teachers section  far  i t compared t o o n l y  journals  u n f o r t u n a t e g i v e n The A r i t h m e t i c Teacher,  problem-solving  by  Teachers  exercises.  p e r c e n t o f t h e t e a c h e r s i n d i c a t e d t h e y used 19  the  source  for  journals i s  t h e elementary  journal  o f Mathematics (NCTM), has  issue.  The  result,  a  however,  r a t e o f membership i n t h e NCTM among t e a c h e r s  o f Grade 7. R e s u l t s showing percentages types  of  activities  a r e shown i n T a b l e  to 11.  motivate  o f t e a c h e r s who students  used  f o r problem  different solving,  62 T a b l e 11 A c t i v i t i e s Used t o M o t i v a t e Type o f A c t i v i t y  Students  P e r c e n t o f Teachers  C o m p e t i t i v e games  41  Problem o f t h e day  41  Puzzles or brain teasers  81  L i b r a r y f i l e o f problems  31  Contests  32  " P u z z l e s o r b r a i n t e a s e r s " was t h e most p o p u l a r  selection.  E i g h t y - o n e p e r c e n t o f t h e t e a c h e r s i n d i c a t e d t h a t t h e y used activity. example,  Other the  selections  next  most  were  popular  markedly  less  selections  this  popular.  were  For  "competitive  games" and "problem o f t h e day," each o f which were s e l e c t e d by 41 p e r c e n t and  of the teachers.  "library  percent,  T a b l e 12.  of  problems"  p o p u l a r were " c o n t e s t s "  selected  by  only  32  and  31  types  of  respectively.  Teachers problems  file  The l e a s t  they  were  also  assigned  asked  to  t o students.  indicate  certain  Results  are reported i n  63 T a b l e 12 Problem Types A s s i g n e d t o s t u d e n t s Type o f Problem More than one  Percent of 35  answer  I n f o r m a t i o n t o be  collected  59  Can be s o l v e d i n s e v e r a l ways  89  Can be s o l v e d c o l l e c t i v e l y  44  Too much o r l i t t l e i n f o r m a t i o n  64  Based on  the data  contained  type  o f problem a s s i g n e d was  than  one  way.  and  35  percent  one  i n Table  one  12,  the most  which c o u l d be  Problems which c o u l d be  t h o s e w i t h more than  Teachers  popular  s o l v e d i n more  solved c o l l e c t i v e l y  answer were t h e  of teachers, respectively,  least  and  popular with  indicating  that  44  they  assigned these types. In o r d e r t o determine promote  problem  solving,  d i f f e r e n t options.  what c l a s s r o o m teachers  f e a t u r e s were used  selected  R e s u l t s a r e shown i n T a b l e  from 13.  among  to  five  64 T a b l e 13 Classroom F e a t u r e s t o Promote Problem S o l v i n g P e r c e n t o f Teachers  Type o f F e a t u r e Problem-solving  14  centre  B u l l e t i n board d i s p l a y e d  15  Problem o f t h e week  36  Contests with the c l a s s  52  Students make up problems  52  Few  teachers  indicated  that  they  f e a t u r e s t o promote problem s o l v i n g . 15  percent  of teachers  respectively,  used  these  F o r example, used  classroom  o n l y 14 and  either  a  problem-  s o l v i n g c e n t r e o r a b u l l e t i n board d i s p l a y f o r problems. The  numbers  of d i f f e r e n t  and c l a s s r o o m a c t i v i t i e s used a r e r e p o r t e d i n T a b l e 14.  approaches,  sources  of material  i n t h e t e a c h i n g o f problem  I t shows t h e p e r c e n t a g e s  solving  o f teachers  who use up t o a maximum o f f i v e d i f f e r e n t s o u r c e s o r a c t i v i t i e s . T a b l e 14 Number o f P r o b l e m - S o l v i n g A c t i v i t i e s and Sources (Percent)  Activity  Used  Number o f Responses S e l e c t e d 1 2 3 4 5  S t r a t e g i e s Taught  16  16  30  16  Sources  17  23  35  18  7  Variety of A c t i v i t i e s  26  37  25  9  3  Problem Types  14  25  31  18  Classroom  52  35  10  of Exercises  Features  2  22  12 1  65  Table  14  between one response  and  percentages  of the  For  instruct  solving  the  five  items.  teachers  all  lists  example,  students  strategies.  options  however,  used  percent  that  percent  indicated  of  problem-  they  taught  taught only  one.  f o r problem-solving  a l l five  exercises.  sources.  The  most  Only  7  frequent  t h r e e , chosen by 35  percent  respondents.  Most t e a c h e r s , two  68  i n d i c a t e d t h a t t h e y used t h r e e  number o f d i f f e r e n t sources used was of the  selected  f o r each o f t h e m u l t i p l e -  f i v e s t r a t e g i e s , compared t o 16 p e r c e n t who  percent,  who  i n t h r e e o r more d i f f e r e n t  Twenty-two  o r more d i f f e r e n t s o u r c e s  activities  four a c t i v i t i e s all  teachers  i t shows  S i x t y p e r c e n t of respondents  or  of  63 to  and  percent, motivate  indicated  they  students.  Only  used e i t h e r 9  one  percent  used  an even s m a l l e r p r o p o r t i o n , 3 p e r c e n t ,  used  five. Based on t h e r e s u l t s , t h r e e o r more d i f f e r e n t problem t y p e s  were a s s i g n e d by  61 p e r c e n t  a s s i g n e d o n l y one t y p e and In  responding  to  of the teachers.  Fourteen  percent  12 p e r c e n t a s s i g n e d a l l f i v e .  the  question  f e a t u r e s used i n the c l a s s r o o m  on  the  number o f  different  t o promote problem s o l v i n g ,  only  13 p e r c e n t i n d i c a t e d t h a t they had t h r e e o r more o f t h e f e a t u r e s listed.  Most, 52 p e r c e n t , used o n l y one  f e a t u r e whereas o n l y 1  p e r c e n t employed a l l f i v e . Teachers  1  P e r c e p t i o n s o f Mathematics  In S c a l e R, curriculum  t e a c h e r s r a t e d each o f t e n major t o p i c s i n the  according  to  their  perceptions  of  its  importance,  66 e a s i n e s s t o t e a c h , and to  respond  to  positive.  selected Table  on  a  enjoyment t o t e a c h .  five-point  Likert  scale,  Teachers ranging  were  from  asked  negative  R e s u l t s , showing t h e p r o p o r t i o n s o f t e a c h e r s  t h e two  positive  who  o p t i o n s f o r each i t e m a r e r e p o r t e d i n  15.  T a b l e 15 P e r c e p t i o n s o f Mathematical (Percent*)  Teachers'  Topic  Importance  Topics  Easy t o Teach  Enjoyable t o Teach  Fractions  81  51  85  Decimals  99  76  94  Percents  97  48  92  Estimation  86  41  67  Basic Facts  90  58  52  Equations  94  34  85  Word Problems  96  15  75  M e t r i c System  93  43  68  Perimeter  85  64  86  71  54  83  & Area  Geometry  * P e r c e n t o f t e a c h e r s s e l e c t i n g t h e two Operations important,  with decimals  with  important.  99  Other  percent  r a t e d by t e a c h e r s as t h e most  rating  important  and  lowest r a t i n g s  geometry.  The  by  more  very  and  fractions  important  or  p e r c e n t o f t e a c h e r s were p e r c e n t s , e q u a t i o n s , word problems,  to  as  as  90  The  rated  it  than  t h e m e t r i c system.  topics  was  p o s i t i v e options.  f o r importance were g i v e n  low  rating  for  fractions,  67 r e l a t i v e t o o t h e r s , c o u l d be p a r t l y because o f Canada's of  the m e t r i c  rather  than  relatively  system, common  low  i n which g r e a t e r use  fractions.  importance  i s given  Although  rating,  the  Geometry  Provincial  adoption  to  decimal  received  Mathematics  R e v i s i o n Committee has g i v e n i t g r e a t e r prominence i n t h e r e v i s i o n of the  reported  This  the  15  percent  contrasted  percent.  decimals  the  most d i f f i c u l t .  r a t e d word problems as  sharply  with  its  whereas  importance  easy t o  For teach.  rating  of  96  A l a r g e p o r t i o n o f t e a c h e r s d i d not r e p o r t t h a t any  topics,  with  a r e a , were easy t o  the  They l e a s t enjoyed  exceptions  of  decimals  and  perimeter  of and  teach.  T e a c h e r s enjoyed  still  the e a s i e s t t o p i c t o teach  t h a t word problems was  example, o n l y  recent  curriculum.  T e a c h e r s found they  a  teaching  decimals  and  percents  the  most.  t e a c h i n g t h e memorization o f b a s i c f a c t s ,  but  52 p e r c e n t r a t e d i t as e n j o y a b l e t o t e a c h . Overall,  important. enjoyed  teachers They  found  perceived several  that  most  topics d i f f i c u l t  topics to  were  teach  but  t e a c h i n g almost a l l o f them.  In o r d e r  to  gain  a measure o f the  the three s c a l e s , a r e l i a b i l i t y a r e shown i n T a b l e  16.  reliability  a n a l y s i s was  o f each  conducted.  of  Results  68 T a b l e 16 of Teachers'  R e l i a b i l i t y Analyses  Perception  Inter-Item C o r r e l a t i o n Mean  Scale  Scales Cronbach s Alpha 1  Importance  0.17  0.68  D i f f i c u l t y t o Teach  0.31  0.82  Enjoyment t o Teach  0.25  0.77  Inter-item average  degree  Cronbach's each  correlation  means  provided  an  o f a s s o c i a t i o n among t h e items  alpha,  scale's  on t h e o t h e r  internal  hand,  provided  consistency  based  index  of the  i n each  scale.  an e s t i m a t e  of  on  the  item  an  inter-item  correlations. The  Difficulty  to  Teach  scale,  with  c o r r e l a t i o n mean o f 0.31 and a Cronbach's C o e f f i c i e n t 0.82 was t h e most that  each  reliable  of the three  s c a l e consisted of only  coefficients  scales.  10 items,  Considering  their  reliability  o f 0.68, 0.82 and 0.77 were r e l a t i v e l y  the b a s i s o f these  data,  i t was d e c i d e d  Alpha o f  high.  On  t o sum t h e s c o r e s f o r  each o f t h e t h r e e s c a l e s . As score  a result  f o r each  o f t h e summing p r o c e s s ,  of the three  numbers a t t h e next  stage  s c a l e s which  of analysis.  teachers were  received a  used  as  Characteristics  d i s t r i b u t i o n s o f t h e s e i n d i c e s a r e r e p o r t e d i n T a b l e 17.  index of the  69 T a b l e 17 D i s t r i b u t i o n o f Index Numbers T e a c h e r s ' P e r c e p t i o n s o f Mathematics  for  Weighting Range  Variable Importance Difficulty  t o Teach  10-50  42.53  4.17  10-50  31.36  6.19  10-50  38.79  4.89  Enjoyment t o Teach  These r e s u l t s importance to  Standard Deviation  Mean  show t h a t , on average,  teachers' ratings of  were h i g h e r than t h e i r o t h e r two r a t i n g s .  t e a c h r a t i n g s , which were t h e lowest, w i t h  Difficulty  a mean o f 31.36,  were a l s o t h e most d i v e r s e , w i t h a s t a n d a r d d e v i a t i o n o f 6.19. Students•  Perceptions  Students  o f Mathematics  responded  t h e i r teachers.  t o t h e same t o p i c s  From t h e i r p e r s p e c t i v e s they  in  terms o f i t s importance,  to  learn.  Table  i n scale  difficulty  18 r e p o r t s  R as d i d  r a t e d each  to learn,  the proportions  s e l e c t e d t h e two p o s i t i v e o p t i o n s f o r each item.  topic  and enjoyment  o f students  who  70 T a b l e 18 students' Perceptions of Mathematical (Percent*)  Topics  Importance  Easy t o Learn  Fractions  87  64  37  Decimals  66  44  86  Percents  58  83  41  Estimation  88  57  42  Basic Facts  70  41  72  Equations  63  84  51  Word Problems  86  67  46  M e t r i c System  55  59  69  P e r i m e t e r & Area  57  60  51  Geometry  53  43  52  Topic  Enjoyable t o Learn  * P e r c e n t o f s t u d e n t s s e l e c t i n g t h e two p o s i t i v e o p t i o n s .  Students the  most  students least  rated estimation, fractions,  important selecting  important  topics.  Geometry, w i t h  t h e two p o s i t i v e  topic.  and word problems as  Overall,  53 p e r c e n t  of the  o p t i o n s , was ranked  as t h e  the  importance  s t u d e n t s were lower than those o f t h e i r t e a c h e r s . difference  i n ratings  between  students  ratings  The g r e a t e s t  and t e a c h e r s  d e c i m a l s , p e r c e n t s , e q u a t i o n s and t h e m e t r i c system. r a t i n g s of students percentage  f o r each o f t h e s e t o p i c s ,  of  were f o r Importance  were a t l e a s t 30  p o i n t s below t h o s e o f t h e i r t e a c h e r s .  Equations respectively,  and were  percents, reported  with  ratings  as e a s i e s t  of  to learn  84 by  and  83  students.  Memorizing  basic  facts,  geometry  hand, were found h a r d e s t .  and d e c i m a l s ,  Students  on  the  other  reported that s i x of the ten  t o p i c s were e a s i e r t o l e a r n than t e a c h e r s had r e p o r t e d t h e y were to  teach.  F o r example, d i f f e r e n c e s f o r p e r c e n t s , e q u a t i o n s , and  word problems were  found  o f 35, 50 and 52 p e r c e n t a g e  when s t u d e n t s '  ratings  points  were compared  respectively  with  those of  t h e i r teachers.  In c o n t r a s t , o n l y 44 p e r c e n t o f s t u d e n t s r a t e d  decimals  t o l e a r n whereas 76 p e r c e n t  it  as easy  easy t o t e a c h . Operations  students  as  a  with topic  decimals they  was  found  rated  by  enjoyable  c o n t r a s t e d w i t h a r a t i n g o f o n l y 37 p e r c e n t fractions. learn  Students  than  example, and  of teachers rated  their  ratings  area,  percentage  and  to  percent  learn.  of This  f o r operations with  found e i g h t o f t h e t o p i c s l e s s e n j o y a b l e t o teachers  found  enjoyable  f o r fractions,  percents,  geometry  differences  points.  86  had  to  teach.  equations,  For  perimeter  greater  than  30  There was a p o s i t i v e d i f f e r e n c e , however, i n  r a t i n g s between s t u d e n t s and t e a c h e r s on memorizing b a s i c f a c t s . F o r example,  72 p e r c e n t  of students  found  this  topic  enjoyable  compared,to o n l y 52 p e r c e n t o f t h e i r t e a c h e r s . The perception  reliability  coefficients  scales are reported  f o r each  i n Table  19.  of  the  Both  c o r r e l a t i o n means and Cronbach's a l p h a s a r e l i s t e d .  student-  inter-item  72 T a b l e 19 A n a l y s e s o f Student P e r c e p t i o n S c a l e s  Reliability  Inter-Item C o r r e l a t i o n Mean  Scale  Cronbach's Alpha  Importance  0.22  0.74  D i f f i c u l t y t o Learn  0.19  0.70  Enjoyment t o L e a r n  0.25  0.77  The with  a  most r e l i a b l e o f t h e s c a l e s reliability  correlation retained  mean  for  coefficients.  of  further  coefficient 0.25.  of  was "Enjoyment t o Learn", 0.77  A l l three  analysis  based  A sum was c a l c u l a t e d  and  scales, on  Distributions for  o f index numbers,  the perception scales  inter-item  however,  their  f o r each  c l a s s means o f t h e sums were used i n f u r t h e r  an  were  reliability  o f t h e s c a l e s and analysis.  comprised  o f c l a s s means,  a r e r e p o r t e d i n T a b l e 20.  The grand  mean and s t a n d a r d d e v i a t i o n a r e r e p o r t e d f o r each o f t h e s c a l e s . T a b l e 20 D i s t r i b u t i o n o f Index Numbers f o r S t u d e n t s ' P e r c e p t i o n s o f Mathematics Weighting Range  Mean  Standard Deviation  Importance  10-50  37.02  2.17  D i f f i c u l t y t o Learn  10-50  35.59  2.47  Enjoyment t o L e a r n  10-50  34.13  2.67  Variable  Based on  these  data,  students  h i g h e r than t h e i r r a t i n g s  1  ratings  of  importance  o f d i f f i c u l t y o r enjoyment.  mean f o r t h i s r a t i n g , however, was  5.51  were  The  grand  percentage p o i n t s  lower  than t h a t o f t h e i r t e a c h e r s . I n examining t h e r e l a t i o n s h i p s perceptions examined. because  of  mathematics,  Ratings they  only  between s t u d e n t  and  the  rating  importance  teacher was  of d i f f i c u l t y and enjoyment were not compared  were  from  two  perspectives—the  learning  of  mathematics and t h e t e a c h i n g o f mathematics. A c o r r e l a t i o n o f 0.15 perceptions of the I t was  low,  determine panel  a  the  importance o f mathematics i n t h e  direction  correlation from t h e  third  perceptions  of  (Campbell 1987  variable,  of  in  teachers  students  variables This  was  result  and  Stanley,  which  The  that  importance o f mathematics may  1963)  was  conducted,  Time was  introduced  students  between  the  a t Time 2 were  of teachers  l a t t e r two  teachers'  study.  cross-lagged  c o r r e l a t i o n between t h e  whereas between t h e  suggests  1985  a  correlations  of perceptions  student  I n an attempt t o  relationship,  a t Time 1 and  a t Time 1. 0.16  this  level.  v a l i d a t i o n study.  compared w i t h c o r r e l a t i o n s and  found between t e a c h e r and  but s i g n i f i c a n t a t t h e 0.05  using data as  was  a t Time 2 first  i t was  perceptions  two 0.10.  of  have a g r e a t e r e f f e c t on those  the of  t h e i r s t u d e n t s than v i c e v e r s a .  4.3  CORRELATIONAL ANALYSIS  The  second  phase  i n the  examination  of  the  data  involved  c o r r e l a t i o n a l a n a l y s e s t o t e s t f o r concomitant v a r i a t i o n between  specific  variables.  These  i d e n t i f i e d i n Chapter  analyses  examined  relationships  1 and p r o v i d e d i n p u t s t a t i s t i c s f o r f a c t o r  analysis. In  order  to  obtain  additivity  applied  to  p.82).  the  Pearson  achieve of  normality  effects,  distributions  of  error  square-root  for  each  effects  transforms  variable  (Kirk,  product-moment c o r r e l a t i o n s were then  among a l l p a i r s o f v a r i a b l e s  of i n t e r e s t .  and  to were  1982,  calculated  Results are presented  by c a t e g o r i e s o f v a r i a b l e s . Student  Background and Achievement  Correlations variables  and  achievement  shown i n T a b l e 21. denoted  w i t h an  between c l a s s on  indices  the  Statistically  asterisk.  three  f o r student criterion  significant  background domains  are  relationships  are  75 T a b l e 21 Among Student Background V a r i a b l e s  Correlations Variable  SB1  SB1 Lang. 1 s t Spoken  SB2  100  SB3  81* 100  SB2 Lang, a t Home SB3 Time on Homework SB4 F a t h e r s  Problem  A2 T e s t  SB5  Al  A2  A3  -4  14*  20*  2  -1  2  -4  12*  15*  5  3  6  -1  5  5  4  20*  27*  29*  20*  21*  25*  72*  61*  100  2  Education  1  SB5 Mothers* Al  SB4  and Achievement  100  66*  Education  100  Solving  100  Total  100  83*  A3 A p p l i c a t i o n s Note.  100  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above a r e based on t h e computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100. *p<0.05.  Fifteen among  statistically  the variables.  between  language  educational  parents'  level  solving,  spoken,  and  particularly  level  total between between  strong  and  and s t u d e n t  educational  a t home; l e v e l ; and  achievement on problem  spoken  parents'  found  ones a r e  spoken  applications.  language  with  language  and f a t h e r s '  and  were  o f t h e more i n t e r e s t i n g  spoken  of education  test  relationships  Some  first  mothers'  significant correlations  Among  first  educational  correlations  of  and  these, currently  levels 0.81  and  were 0.66  respectively. Non-significant  correlations  achievement may be due, i n p a r t ,  between  language  spoken and  t o l i t t l e v a r i a n c e among c l a s s  indices class  f o r language indices  on  spoken.  language  F o r example, first  spoken  85 p e r c e n t o f t h e and  95  percent  on  language spoken a t home were 1.8 o r g r e a t e r o u t o f a maximum o f 2.  These  results reflect  E n g l i s h as a f i r s t Correlations 0.61, test  the high proportion  o f students with  language who remained i n t h e sample. between s u b t e s t s were h i g h .  They  ranged  from  between problem s o l v i n g and a p p l i c a t i o n s , t o 0.83, between total  latter  and a p p l i c a t i o n s .  two  subtests  was  A h i g h e r c o r r e l a t i o n between t h e  expected  since  a l l of  the  c o n t a i n e d i n a p p l i c a t i o n s were a l s o c o n t a i n e d i n t h e t e s t However,  no  test  items  were  common  t o problem  items total.  solving  and  applications. Teacher Background Table  and Student  22 l i s t s  Achievement  the correlations  among t e a c h e r  background  v a r i a b l e s and between t h e s e v a r i a b l e s and s t u d e n t achievement i n problem  solving,  test  total  and a p p l i c a t i o n s .  An a s t e r i s k i s  used t o denote s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s .  77 T a b l e 22 C o r r e l a t i o n s Among Teacher Background and Student Achievement Variable  TBI TB2  TBI  Experience  100  TB2  Preference  TB3 TB4  27*  TB4  Conferences  TB5  Workshops  7* 10*  5  9*  0  100  11* 100  Al  Problem  A2  A3  0  5  1  -13* -2  0  -5  TB7  4  3  32* 19* 100  16* 100  Courses  Al  7*  9* 16*  TB6 Math Courses TB7 Math Ed.  TB6  4  100  TB3 P r o p o r t i o n  TB5  Variables  3  -2  -7* -7*  22*  4  20*  6  41*  3  5  2  g*  13*  100  72*  61*  100  Solving  A2 T e s t T o t a l  4  12* 13*  100  7*  83*  A3 A p p l i c a t i o n s Note.  8*  100  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above a r e based on t h e computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100. *p<0.05.  Fifteen  statistically  significant correlations  out  of  a  t o t a l o f 21 were found between p a i r - w i s e r e l a t i o n s h i p s among the independent found  variables.  between  attendance completed attendance  at  Coefficients  experience  and  conferences  and at  mathematics workshops  of  0.20  preference to  and  workshops,  education and  or  teach  mathematics,  mathematics  courses  mathematics  g r e a t e r were  courses  completed,  education  and  courses  completed. A  negative  and  statistically  significant correlation  is  shown between p r e f e r e n c e t o t e a c h mathematics and t h e number of  78 mathematics e d u c a t i o n c o u r s e s taken. p a r t , t o responses  from t e a c h e r s who  but  attended  who  may  have  mathematics readily who  education  available.  were  required  a  post  courses  I t may to  a  be due,  in  p r e f e r t o t e a c h mathematics secondary  were  also  take  T h i s r e s u l t may  not  reflect  institution  required  or  results  from  mathematics  methods  where  else  not  teachers  course  but  p r e f e r not t o t e a c h mathematics. Eight  out  of  a  background v a r i a b l e s  total and  of  student  statistically significant. correlations attendance and  were  a t workshops  applications,  education  low,  courses  and  21  correlations achievement  Although the  and  completed  student  and  were  teacher  found  to  be  the magnitudes o f a l l t h e s e  strongest  between  between  ones  were  achievement  the  number  achievement  on of  on  between  test  total  mathematics the  same  two  organization variables  and  domains. Classroom  O r g a n i z a t i o n and Achievement  Relationships between Table  these  23.  A  among  variables  classroom and  student  number o f s t a t i s t i c a l l y  c o r r e l a t i o n s , are included i n the  achievement significant,  results.  are but  shown  in  negative  79 T a b l e 23 Among Classroom O r g a n i z a t i o n and Achievement  Correlations  Var.  Cl  Cl  C2  C3  -6  4  100  2  100  C2 C3  C4  C5  C7  C8  2 - 7 -15* -11* -3 0  -2  100 -40*  C4  C6  100  C5  C8  1  -2  15* 19* 21*  -4  -1  16*  -6  0  1  1  2  0  -4  4  -3  -4  1  2  17* -11* -4  1  4  0  5  2  3  10*  0  2  -2  -26*  1  1  -2  -11* -2  0  0  -7  3  13*  9*  -2  2  100  -1  7* 13* -8*  3  22* -6 100  CIO  12* 100  Cll Al  Problem  A2  Test  A3  Application  Note.  Solving  Total  A3  2  100  C9  A2  1  100  C7  -2  C l l Al  5  5  100  C6  9*  C9 CIO  Variables  4  8* 10*  4 8*  -5  0  2  -1  -6  -3  -3  -4  100  -3  -6  0  100  72  61  100  83 100  1.  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above a r e based on t h e computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100.  2.  Independent v a r i a b l e s were as f o l l o w s : C l = course t y p e ; C2 = frequency o f t e s t i n g ; C3 = number o f c l a s s e s ; C4 = l e n g t h o f c l a s s e s ; C5 = time on homework a c t i v i t i e s ; C6 = number o f s t u d e n t s q u e s t i o n e d ; C7 = time on seatwork; C8 = time i n s m a l l groups; C9 = time a t a c t i v i t y c e n t e r s , C10 = time on c o m p u t a t i o n a l d r i l l ; C l l = time i n t r o d u c i n g new topics.  *p<0.05.  80 N e g a t i v e and -0.15  and  number  -0.11  of  Although  statistically were  students  these  found  significant  between  questioned  relationships  and  are  not  e v i d e n c e t o suggest t h a t t e a c h e r s may more  with  modified  classes  r e l a t i o n s h i p between time in  i n t r o d u c i n g new  (-0.26). new  spent on  topics also  T h i s r e s u l t may  material  than  using  a  correlations  type  of  course  time  spent  and  on  of both  seatwork.  s t r o n g , t h e y p r o v i d e some emphasize t h e s e  with  enriched  individual  showed a  activities  ones.  The  seatwork and  time  negative c o r r e l a t i o n  i n d i c a t e that teachers tend t o present  lecture-style  rather  than  a  discovery  approach. Positive  and  statistically  significant  correlations  found between 10 p a i r s o f c l a s s r o o m o r g a n i z a t i o n v a r i a b l e s . example, period  the  number of  correlated  significantly  frequency of t e s t i n g , related drill,  activities, and  students  working  likely  with  the  i n t h e most following  For  recent  variables:  l e n g t h o f p e r i o d , time spent on homework-  i n t r o d u c i n g new  activities  questioned  were  at  activity  topics.  involve  centers,  computational  S i n c e each o f t h e p r e c e d i n g  question-and-answer  exchanges,  s i g n i f i c a n t c o r r e l a t i o n s w i t h h i g h e r magnitudes were expected. The  only  classroom  organization variables  which  showed  s t a t i s t i c a l l y s i g n i f i c a n t and p o s i t i v e c o r r e l a t i o n s w i t h s t u d e n t achievement were type o f c o u r s e and number o f c l a s s e s p e r week. Type  of  course  (0.15), t e s t of  classes  significant, domains.  correlated  total per but  significantly  (0.19) and  week lower,  also  with  applications  showed  positive  problem  (0.21). and  The  solving number  statistically  c o r r e l a t i o n s w i t h the t h r e e achievement  81 Problem-Solving Table variables, activities.  24  P r o c e s s e s and Student Achievement lists  related  correlations to  teacher  among p r o b l e m - s o l v i n g behaviors  and  R e l a t i o n s h i p s between t h e s e v a r i a b l e s  achievement a r e a l s o r e p o r t e d .  process  classroom and  student  82 T a b l e 24 C o r r e l a t i o n s Among P r o b l e m - S o l v i n g P r o c e s s V a r i a b l e s and Achievement Var.  PS1  PS1  100  PS2  PS3  52* 100  PS 2  PS5  PS6  PS7  PS8  PS9  PS10  PS11  A2  A3  16*  15*  16*  19*  14*  18*  17*  16*  19*  18*  36* 26*  16*  4  16*  19*  14*  22*  14*  22*  30*  26*  9  13*  12*  19*  18*  10*  13*  9*  15*  14*  3  12*  2  9*  8*  5  10*  4  2  2  15*  40*  34*  18*  43*  24*  10*  8*  11*  13*  12*  8*  15*  23*  11*  8*  9*  52*  21*  40*  40*  10*  11*  10*  24*  37*  46*  6  10*  11*  16*  22*  6  8*  8*  37*  8*  8*  9*  7*  5  2  72  61  24* 100  PS4 PS 5  100  PS 6  100  PS 7  100  PS8  100  PS 9  100  PS10  100  PS11  100  Al  Problem  A2  Test  A3  Application  Note.  Al  31* 26*  100  PS 3  PS4  Solving  100  Total  100  83 100  1.  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above are based on the computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100.  2.  Independent v a r i a b l e s were as f o l l o w s : PS1 = p e r c e p t i o n o f s t u d e n t enjoyment; PS2 = p e r c e p t i o n of s t u d e n t achievement; PS3 = s a t i s f a c t i o n w i t h t e a c h i n g ; PS4 = ease o f t e a c h i n g ; PS5 = number o f s t r a t e g i e s ; PS6 = a t t e n d a n c e o f i n - s e r v i c e ; PS7 = s o u r c e s o f e x e r c i s e s ; PS8 = number o f m o t i v a t i o n a l a c t i v i t i e s ; PS9 = frequency o f t e a c h i n g ; PS10 = problem t y p e s ; PS11 = c l a s s r o o m f e a t u r e s .  *p<0.05. Positive shown f o r 50  and o f the  statistically 55  significant  correlations  r e l a t i o n s h i p s between p a i r s  of  are  problem-  83 s o l v i n g process v a r i a b l e s .  The two  s t r o n g e s t r e l a t i o n s h i p s were  between t e a c h e r s ' p e r c e p t i o n s o f s t u d e n t enjoyment o f and perceptions  of  between  number  the  student of  achievement different  of  variables  showed  problem  sources  number o f d i f f e r e n t a c t i v i t i e s used pairs  in  of  d i f f e r e n t problem-solving  correlations  sources  of  (0.43);  the  exercises  strategies  (0.40)  and  number o f d i f f e r e n t  number o f problem t y p e s a s s i g n e d features  (0.40); and  of  and the  students. 0.52.  Both Other  between t h e number o f  taught  and b o t h numbers of  different sources  solving,  e x e r c i s e s and  to motivate  r e l a t i v e l y s t r o n g c o r r e l a t i o n s were found  their  problem  of  (0.40) and  types  used  e x e r c i s e s and  both  number o f  classroom  number o f m o t i v a t i o n a l a c t i v i t i e s  used  and  number o f c l a s s r o o m f e a t u r e s (0.46). The were  relationships  stronger  variables. found  than  among p r o b l e m - s o l v i n g  those  found  among  classroom  F o r example, 12 c o r r e l a t i o n s  among t h e  former  variables  process  variables  organization  g r e a t e r than  compared  to only  0.3 0 were  1 among  the  latter. Twenty-six  out  problem-solving  of  process  a  total  of  variables  33  and  relationships  student  p o s i t i v e and s t a t i s t i c a l l y s i g n i f i c a n t . low  magnitude,  relationships  were  achievement and test total Teachers  1  with  less  between  student  than  achievement  The  expectations  performance on  were  Most o f t h e s e were o f a 0.10.  teachers'  (0.30) and on a p p l i c a t i o n s  problem  strongest of  student  solving  (0.22),  (0.26).  P e r c e p t i o n s o f Mathematics and s t u d e n t Achievement  Relationships importance  10  between  of,  the  between  /  difficulty  teachers' to  teach  perceptions and  the  of  enjoyment  the in  84 teaching of  mathematics a r e r e p o r t e d  interest  between  are a l s o d i s c u s s e d .  these  i n this section.  perceptions  Correlations  and s t u d e n t  achievement  These r e l a t i o n s h i p s a r e shown i n T a b l e 25.  T a b l e 25 C o r r e l a t i o n s Among T e a c h e r s ' P e r c e p t i o n s and s t u d e n t Achievement Variable  TP1  TP1  Importance  TP2  Difficulty  TP3  Enjoyment  Al  Problem  A2  Test  A3  TP2  100  20* 100  TP3  Al  58*  4  10*  6  37*  9*  16*  15*  5  3  71*  61*  100  A2  -1  Solving  100  Total  100  Applications  Note:  A3  83* 100  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above a r e based on t h e computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100. *p<0.05.  All ratings and  three  of the pair-wise  o f importance,  statistically  importance  difficulty  significant.  and enjoyment  was  relationships  among  and enjoyment  were  positive  relationship  between  The  the strongest,  however,  teacher  with  a  c o r r e l a t i o n c o e f f i c i e n t o f 0.58. Of  the three teacher-perception variables,  to-teach r a t i n g correlated the highest Correlations  between  this  rating  the d i f f i c u l t y -  w i t h s t u d e n t achievement.  and s t u d e n t  achievement  on  problem s o l v i n g , t e s t t o t a l and a p p l i c a t i o n s were 0.09, 0.16 and  85 0.15  respectively.  they  were a l l s i g n i f i c a n t a t t h e 0.05 l e v e l .  statistically variables  Although  these c o r r e l a t i o n s  The o n l y  other  s i g n i f i c a n t c o r r e l a t i o n between t e a c h e r - p e r c e p t i o n  and s t u d e n t  achievement  was between  r a t i n g and s t u d e n t performance on t e s t Students'  were n o t h i g h ,  t h e importance  total.  P e r c e p t i o n s o f Mathematics and Achievement  T h i s s e c t i o n examines r e l a t i o n s h i p s between s t u d e n t  ratings  of t h e importance, d i f f i c u l t y t o l e a r n and enjoyment i n l e a r n i n g mathematics. perceptions solving,  I t also  looks  o f mathematics  test  total  at relationships  and t h e i r  achievement  and a p p l i c a t i o n s .  r e l a t i o n s h i p s i s reported  between  A  on  summary  their problem  of  these  i n T a b l e 26.  T a b l e 26 C o r r e l a t i o n s Among S t u d e n t s ' P e r c e p t i o n s and T h e i r Achievement Variable  SP1  SP1  Importance  100  SP2  Difficulty  SP3  Enjoyment  Al  Problem  A2  Test  A3  Applications  Note:  Solving  SP2 57* 100  SP3  Al  A3  53*  9*  14*  12*  59*  7*  15*  14*  17*  27*  19*  71*  61*  100  100  Total  The c o r r e l a t i o n c o e f f i c i e n t s reported on t h e computed c o r r e l a t i o n s rounded p l a c e s and m u l t i p l i e d by 100.  *p<0.05.  A2  100  83* 100  above a r e based t o two d e c i m a l  86 S t r o n g r e l a t i o n s h i p s are shown among s t u d e n t r a t i n g s o f importance,  difficulty  and  enjoyment  of  mathematics.  C o r r e l a t i o n c o e f f i c i e n t s between t h e s e p a i r s of v a r i a b l e s between  0.53  and  0.59.  student  perceptions  Although  and  variables  were  strongest  relative  r a t i n g and  achievement.  their  positive  and  were  number  student  significant,  between  the  the  enjoyment  a p p l i c a t i o n s were 0.17,  0.27  and and  of  variables  achievement  correlations, aggregation points  out  impact not  to  A  in in  each  of  problem  restriction  however, the  may  class  a number o f on  be  at  the  v a r i a b l e s were found.  solving, the  due  to  categories  test  low  the  of  effects  (1980)  class variances,  this  magnitude  level the of  of  and  total  magnitude  Burstein  v a r i a n c e s were examined i n each o f in  input  for  and these  of  data  example,  i n m u l t i l e v e l data a n a l y s i s  Within  for  the  on  level.  issues  results.  accounted  increases  i n t h i s c h a p t e r show  s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s between  applications.  no  criterion  of C o r r e l a t i o n C o e f f i c i e n t s  but p o s i t i v e and  were  between  c o r r e l a t i o n s between enjoyment  C o r r e l a t i o n c o e f f i c i e n t s reported  can  the  ranged  respectively.  Interpretation  a  on  statistically  problem s o l v i n g , t e s t t o t a l and 0.19  correlations  achievement  relationships The  all  the  in this  analysis.  which case, Class  achievement domains correlations  with  but  input  87 4.4  FACTOR ANALYSIS  A f a c t o r a n a l y s i s o f the v a r i a b l e s w i t h i n each o f the categories process on  was  was  which  conducted  employed  the  at  the  next  stage  of  analysis.  t o determine common u n d e r l y i n g  criterion  variables  were  located  identify  t h o s e v a r i a b l e s which were most l i k e l y  included  i n a regression  analyzed  within  the  equation.  categories  The  of  and  student  background,  the  theoretical  model.  reduced  3  to  perception  in  e f f e c t s of  Since  each  of  c a t e g o r i e s , no  the the  inputs  number student  to  v a r i a b l e s were f a c t o r -  in  examine  hence  t o be u s e f u l l y  o r g a n i z a t i o n and p r o b l e m - s o l v i n g  to  This  dimensions  background, c l a s s r o o m order  input  of  teacher processes  identified variables  perception  f a c t o r a n a l y s i s was  in  had  and  the been  teacher  conducted on  these  variables. A  criterion  eigenvalue  of  commonly used  1.0  or  greater  for factor identification (Nie,  1975,  p.479).  is  Using  an  this  c r i t e r i o n , v a r i a b l e s were grouped i n t o f a c t o r s w i t h i n each i n p u t category.  The  was  considered  was  greater  loading of a v a r i a b l e i n t o i t s r e s p e c t i v e f a c t o r significant  than 0.30  (Spencer & Bowers, 1976,  where a v a r i a b l e loaded level,  i t was  highest category  assigned  level.  A  i f i t s c o r r e l a t i o n with  i n t o more t h a n one  the  p.10).  f a c t o r at  In  of  these  f a c t o r s by  cases  the  t o the f a c t o r i n t o which i t loaded  discussion  factor  major  0.30  at  the  input  follows.  Student Background F a c t o r s The background  first  factor  variables:  analysis language  included first  the  spoken,  five  student  language  spoken  88 c u r r e n t l y i n t h e home, time spent on homework, f a t h e r ' s l e v e l o f e d u c a t i o n , and mother's l e v e l  of education.  R e s u l t s a r e shown  i n T a b l e 27.  T a b l e 27 P r i n c i p a l Components o f s t u d e n t Background Eigenvalue  Percent Variance  1  2.0786  41.6  41.6  2  1.4053  28.1  69.7  3  0.9820  19.6  89.3  4  0.3382  6.8  96.1  5  0.1959  3.9  100.0  Factor  An student  Variables  analysis  o f the data  background  variables  i n Table would  c u m u l a t i v e v a r i a n c e o f 69.7 p e r c e n t . the v a r i a b l e s  (after  Cumulated Percent  27 shows t h a t t h e f i v e  yield  two  with  3 i t e r a t i o n s ) as shown i n T a b l e 28.  Factor 1  a  A Varimax r o t a t i o n grouped  T a b l e 28 R o t a t e d F a c t o r M a t r i x o f Student Background Variable  factors  Variables Factor 2  Home Language 1st Language Homework  0.9318 0.9258 -0.2143  0.1455 0.1735 0.0677  Father Education Mother E d u c a t i o n  -0.0312 0.0900  0.9127 0.9027  89 Inspection of the loadings into factor  (Factor  variables. not  show  using  Both  fathers  Teacher Background second  with  the  language  of  attendance  at  mathematics  the  i t was  and mothers'  1  criterion  (Spencer  dropped  education,  from  the  on t h e o t h e r  i n t o F a c t o r 2.  factor  analysis  the  seven  teacher  years of experience, preference t o teach,  teaching workshops,  education  involved  load,  attendance  mathematics  courses  at  courses  completed.  conferences, completed  Results  of  p r i n c i p a l components a n a l y s i s a r e shown i n T a b l e 29. T a b l e 29 P r i n c i p a l Components o f Teacher Background  Variables  Eigenvalue  Percent Variance  1  1.8828  26.9  26.9  2  1.0976  15.7  42.6  3  0.9853  14.1  56.7  4  0.9564  13.6  70.3  5  0.8738  12.5  82.8  6  0.6667  9.5  92.3  7  0.5374  7.7  100.0  Factor  &  Factors  background v a r i a b l e s : proportion  as  Consequently,  hand, l o a d e d s i g n i f i c a n t l y  The  correlations  +/-0.30  1976, p.10).  factor.  high  The c o r r e l a t i o n o f -0.2143 f o r homework, however, i s  significant  Bowers,  1)  t h e language and homework  Cumulated Percent  and the  90 The with  principal  eigenvalues  cumulative rotated  components a n a l y s i s greater  than  1.0,  v a r i a n c e o f 42.6 p e r c e n t .  factor  matrix  grouped  extracted  which  two  factors  accounted  for a  A f t e r three i t e r a t i o n s the  the  variables  into  the  two  c l u s t e r s shown i n T a b l e 30. T a b l e 30 F a c t o r M a t r i x o f Teacher Background V a r i a b l e s  Rotated  Factor 1  Variable Conferences Workshops Math Courses Math Ed. Courses T e a c h i n g Load Experience Preference  Factor  Factor 2  0.6584 0.6562 0.5917 0.5720 0.4041  -0.0034 -0.0724 0.3595 0.4316 -0.1039  0.0367 -0.1083  0.6892 0.6754  1 shows s i g n i f i c a n t  loadings  f o r variables related  t o p r o f e s s i o n a l p r e p a r a t i o n and p r o p o r t i o n o f t e a c h i n g l o a d . is  notable,  however,  that  mathematics methods c o u r s e s but  not  as  highly,  significantly preference  with  into  the  numbers  of  mathematics  completed a l s o loaded Factor  2.  v a r i a b l e s measuring  t o t e a c h mathematics.  Factor teaching  It and  significantly, 2  correlated  experience  and  T h i s suggests t h a t t h e r e i s an  u n d e r l y i n g a t t r i b u t e common t o t h e v a r i a b l e s i n each o f t h e two factors. Classroom O r g a n i z a t i o n A  new  mathematics classes  variable was  p e r week  Factors consisting  created and  from  length  of  the  total  the product of  period.  of The  time  spent  on  the  number  of  new  variable  replaced  t h e s e two v a r i a b l e s a t t h i s s t a g e o f t h e a n a l y s i s .  r e s u l t i n g t e n classroom organization  The  v a r i a b l e s were t h e n f a c t o r -  a n a l y z e d and t h e p r i n c i p a l components a n a l y s i s i s shown i n T a b l e 31.  Principal  T a b l e 31 components o f Classroom O r g a n i z a t i o n Eigenvalue  Percent Variance  1  1.4730  14.7  14.7  2  1.2625  12.7  27.4  3  1.2041  12.0  39.4  4  1.0685  10.7  50.1  5  1.0025  10.0  60.1  6  0.9615  9.6  69.7  7  0.8629  8.6  78.3  8  0.8105  8.1  86.5  9  0.7047  7.0  93.5  10  0.6499  6.5  100.0  Factor  The  Variables  principal  components  with eigenvalues greater  analysis  than one.  60 p e r c e n t o f t h e v a r i a n c e .  extracted  five  factors  These f a c t o r s accounted f o r  A f t e r seven i t e r a t i o n s t h e r o t a t e d  f a c t o r m a t r i x grouped t h e v a r i a b l e s i n T a b l e 32.  Cumulated Percent  i n t o f i v e c l u s t e r s as shown  92 T a b l e 32 o f Classroom O r g a n i z a t i o n  Rotated Factor Matrix Variable  Factor  Seatwork Lecture Questioning  Factor 2  Factor 3  Factor 4  Factor 5  -0. 7455 0. 6998 0. 4998  0. 0939 -0. 0462 0. 3432  -0. 1494 -0. 3075 0. 1120  -0. 1194 -0. 2744 0. 2324  -0. 0343 0. 0520 -0. 1554  Comp. D r i l l Test Freq.  -0. 0121 -0. 0263  0. 7683 0. 6633  0. 0131 -0. 0260  0. 1335 -0. 1367  0. 1604 -0. 2440  S m a l l Group Work S t a t i o n  -0. 0997 0. 1666  -0. 1497 0. 1898  0. 7605 0. 6957  -0. 0750 -0. 0350  -0. 1768 0. 2698  Homework T o t a l Time  0. 2331 -0. 1649  -0. 1081 0. 1268  0. 0047 -0. 1164  0. 7852 0. 5313  -0. 1537 0. 2053  Course Type  -0. 0003  -0. 0731  0. 0220  0. 0167  0. 8760  All  1  Variables  v a r i a b l e s w i t h i n each c l u s t e r loaded  each o f t h e i r f i v e r e s p e c t i v e f a c t o r s .  significantly  Factor  1 was a composite  o f time spent on seatwork, time spent on i n t r o d u c i n g new and  t h e number o f s t u d e n t s q u e s t i o n e d i n t h e most r e c e n t  It  i s notable,  negatively  however,  on t h e f i r s t  time  spent  f a c t o r and t h a t significantly,  on  topics period.  seatwork  loaded  t h e number o f s t u d e n t s  questioned  also  Factor  The second f a c t o r was comprised o f v a r i a b l e s r e l a t e d  2.  loaded  that  into  t o time spent on c o m p u t a t i o n a l Factor  3 combined v a r i a b l e s  groups  and  Time spent on  time  spent  on time  a t work  on homework-related  mathematics  clustered  drill  into  but  highly,  and f r e q u e n c y spent  stations  into  of t e s t i n g .  i n working  i n small  or a c t i v i t y  centers.  activities the  less  fourth  and t o t a l factor.  time  spent  Factor  c o n s i s t e d o f a s i n g l e v a r i a b l e r e l a t e d t o the type o f course.  5  93 Problem-Solving Processes The  next  variables  factor analysis  specifically  solving.  involved eleven classroom  related  to  the teaching  of  process problem  The p r i n c i p a l components a n a l y s i s i s r e p o r t e d i n T a b l e  33. T a b l e 33 P r i n c i p a l Components o f P r o b l e m - S o l v i n g P r o c e s s V a r i a b l e s Eigenvalue  Percent Variance  1  3.0491  27.7  27.7  2  1.6281  14.8  42.5  3  1.0750  9.8  52.3  4  0.9359  8.5  60.8  5  0.8105  7.4  68.2  6  0.7509  6.8  75.0  7  0.6702  6.1  81.1  8  0.6427  5.8  86.9  9  0.5308  4.9  91.8  10  0.4701  4.2  96.0  11  0.4366  4.0  Factor  Based suggested. variables  on  these  results,  a  Cumulated Percent  100.0  three-factor  solution  is  I t accounted f o r 52.3 p e r c e n t o f t h e v a r i a n c e among i n this  category.  After  five  iterations  the rotated  f a c t o r m a t r i x , shown i n T a b l e 34, l i s t s t h e f a c t o r l o a d i n g s .  94 T a b l e 34 Rotated F a c t o r Matrix o f Problem-Solving Process V a r i a b l e s Variable  Factor 1  Factor 3  Exercises Activities Problem t y p e s Strategies Class Feature Test Freq.  0.7697 0.7460 0.6754 0.6437 0.6265 0.3588  0.0325 0.1151 0.1080 0.0385 0.0776 0.1559  0.0587 0.0495 -0.0366 -0.1151 0.3735 0.1978  Expected Ach. Expected E n j . Satisfaction Ease t o Teach  0.1724 0.1508 0.1062 -0.0927  0.8000 0.7238 0.6650 0.5403  -0.2026 0.0126 0.1878 0.3872  0.0843  0.0387  0.8656  In-service  The of  Factor 2  variables  different  approaches  teachers.  They  different  sources  strategies problem  which l o a d e d  and  were of  and  into  F a c t o r 1 i n v o l v e d numbers  sources  comprised  of  exercises  and  problem  types  of  the  materials following  activities;  taught;  frequency  used  by  variables: variety of  of  testing  s o l v i n g ; and number o f d i f f e r e n t c l a s s r o o m f e a t u r e s used  t o encourage problem display  board  teacher  solving,  or contests.  attitude  toward  such as a p r o b l e m - s o l v i n g c e n t e r , Variables i n Factor 2 related  problem  solving.  They  to  included  t e a c h e r s ' e x p e c t a t i o n s o f s t u d e n t s ' achievement i n and enjoyment o f problem and  solving,  t h e ease  which  loaded  involvement. respective  they into  their found  s a t i s f a c t i o n with i n teaching i t .  Factor  3  A l l variables  factors.  related loaded  to  teaching the topic The s i n g l e  teachers'  significantly  variable  in-service with  their  95 Teachers'  P e r c e p t i o n s o f Mathematical  Teachers'  perceptions  Topics  o f mathematics  were  comprised  t h e i r r a t i n g s o f t e n major t o p i c s i n t h e c u r r i c u l u m . were  fractions,  decimals,  percent,  estimation,  of  The t o p i c s  basic  facts,  e q u a t i o n s , word problems, m e t r i c system, p e r i m e t e r and a r e a , and geometry.  Each o f t h e s e t o p i c s was r a t e d a c c o r d i n g t o t e a c h e r s '  perceptions  of  their  enjoyment t o t e a c h .  importance,  reduced  to  teach  R e s u l t s f o r each r a t i n g were summed  topics t o obtain a single score. obtained  difficulty  f o r each o f t h e t h r e e  I n t h i s way a t o t a l ratings.  Since  and  across  s c o r e was  this  procedure  t h e number o f v a r i a b l e s from 30 t o 3, no f a c t o r a n a l y s i s  was conducted referred  i n this  category.  t o as f a c t o r s  The r e m a i n i n g  f o r r e p o r t i n g purposes  3 v a r i a b l e s are i n t h e remaining  analysis. S t u d e n t s ' P e r c e p t i o n s o f Mathematical Three students' and  rating  scores  perceptions  Topics  were determined  o f t h e importance,  i n a similar difficulty  way f o r  to learn,  enjoyment i n l e a r n i n g f o r t h e same t o p i c s as were r a t e d by  teachers.  Since only  was conducted.  3 v a r i a b l e s r e s u l t e d , no f a c t o r a n a l y s i s  F o r r e p o r t i n g purposes t h e s e  variables will  be  r e f e r r e d t o as f a c t o r s . Summary o f the F a c t o r  Analyses  The v a r i a b l e s i n each c a t e g o r y were reduced result  of the factor analyses.  i n numbers which r e s u l t e d .  Table  i n number as a  35 shows t h e d i f f e r e n c e s  96 T a b l e 35 Numbers o f V a r i a b l e s and F a c t o r s Number of V a r i a b l e s  Category  Number o f F a c t o r s  Student  Background  5  2  Teacher  Background  7  2  Classroom O r g a n i z a t i o n  10  5  Prob. S o l v i n g P r o c e s s e s  11  3  Teachers  Perceptions  30  3  Students• Perceptions  30  3  Total  93  18  As  1  shown i n T a b l e  under e x a m i n a t i o n factors.  35, t h e number o f independent  i n t h e c u r r e n t study was reduced  from 93 t o 18  The number o f v a r i a b l e s under Teachers*  and Students*  P e r c e p t i o n s was reduced total  rating  topics.  variables  scores,  from 30 t o 3 i n each case by d e t e r m i n i n g  calculated  by  summing  raw  scores  across  F a c t o r s c o r e s , on t h e o t h e r hand, which were c a l c u l a t e d  i n t h e r e m a i n i n g c a t e g o r i e s , were determined  by f i r s t c o n v e r t i n g  variable  z-scores  scores t o z-scores.  The o r i g i n a l  c o n v e r t e d t o f a c t o r s c o r e s i n s t a n d a r d s c o r e form  were  then  ( K e r l i n g e r and  Pedhazur, 1973, p. 365).  4.5  MULTIPLE REGRESSION ANALYSIS  At were  this  used  to  stage  of a n a l y s i s  analyze  m u l t i p l e r e g r e s s i o n techniques  the r e l a t i o n s h i p s  c r i t e r i o n v a r i a b l e s and s e t s o f p r e d i c t o r  between each  of the  factors.  purpose  The  was  t o determine  control  the best  linear  f o r o t h e r confounding  contributions  made  by  a  prediction  factors number  equation  i n order  of  and t o  t o evaluate the  variables  on  student  achievement i n mathematics. The in  first  stage  which  factors  regressed  i n turn  case  a  which  process, the  each  regression  were  continued  with  to  significant. determine  each  be  input  was  categories  domains. used  first  was  to  t h e one which  variable.  until  the  determine  stage  of variance  In  was  this  correlated  Successive  F-ratio  were  I n each  p r e d i c t o r s of success.  The purpose a t t h i s  accounted  the  achievement  criterion  entered  of multiple regressions  procedure  the best  t h e amount  be  of  on t h e t h r e e  the f a c t o r entered  highest  could  from  step-wise factors  involved a series  factors  no  longer  o f t h e a n a l y s i s was t o  i n student  achievement  f o r by t h e f a c t o r s  i n each  separate  stage  factors  from  which input  category. At  t h e second  categories  were  entered  of analysis, using  the  step-wise  purpose o f t h i s p r o c e s s was t o develop the  total  variance  determined.  accounted  The  f o r by  a l l factors  could  U s i n g t h i s p r o c e s s any v a r i a n c e s which were into  be  shared  account.  Background  The  two  the step-wise Factor  procedure.  a g e n e r a l model i n which  by f a c t o r s from d i f f e r e n t c a t e g o r i e s were t a k e n Student  a l l input  2,  fathers'  student  background  factors  were  regressed,  method, on each o f t h e t h r e e c r i t e r i o n  comprised educational  o f two v a r i a b l e s r e l a t e d backgrounds,  was  using  variables.  t o mothers'  selected  first  case based on h i g h e r c o r r e l a t i o n s w i t h t h e c r i t e r i o n  and  i n each  variables.  98 Factor  2 remained  i n the prediction  equation  whereas F a c t o r 1 d i d n o t meet t h e c r i t e r i a  f o r each  f o r entry.  domain Results  a r e shown i n T a b l e 36.  T a b l e 36 Student Background F a c t o r s Regressed on C r i t e r i o n Variable  Step  Factor  Mult R  R  2  Variables  F  p  Problem S o l v i n g  1  2  0.2193  0.0481  36.72  <0.001  Test Total  1  2  0.2601  0.0677  52.75  <0.001  Applications  1  2  0.3005  0.0903  72.14  <0.001  These 9.0  data  percent  problem  of  significant accounted  The  2 accounts  i n student  total  effect  of  level  and this  f o r 4.8,  6.8 and  achievement  on t h e  application factor  i n each  is  case.  variables  statistically The v a r i a n c e s of parents  f o r a s t a t i s t i c a l l y s i g n i f i c a n t portion of the variance  thinking,  achievement i t accounts  score  comprised  and,  on  questions  which  test  higher  f o r somewhat more v a r i a n c e  in  particular,  on  order  on t h e t e s t  application  questions  o f r o u t i n e s t o r y problems. Background  Using  t h e step-wise  factors  were  Neither  factor  solving  test  a t t h e 0.001  student  Teacher  Factor  f o r show t h a t w h i l e t h e e d u c a t i o n a l l e v e l  accounts  total  that  the variance  solving,  respectively.  in  show  or  regressed remained  test  total.  method, on  each  t h e two of  teacher  background  the c r i t e r i o n  variables.  i n the predictor However,  equation  Factor  1,  f o r problem comprised  of  99 variables  involving  professional  equation f o r the a p p l i c a t i o n  preparation,  remained  i n the  variable.  Teacher background accounted f o r a p p r o x i m a t e l y 1 p e r c e n t o f the As  variance shown  i n student  i n prior  measured,  achievement  research,  appears  t o have  teacher  no  The  questions.  as  effect  currently  on  problem  e f f e c t on a p p l i c a t i o n s .  Organization f i v e classroom  the  c r i t e r i o n variables  5,  comprised  remained  as  variables.  of a  a  organization  factors  were r e g r e s s e d  a t t h e next s t e p o f t h e p r o c e s s .  single  predictor  variable factor  on  the  type  f o r a l l of  F a c t o r 4, which r e l a t e d  mathematics and t h e time  of  on  Factor course,  the c r i t e r i o n  t o t h e t o t a l time spent on  spent on homework r e l a t e d  remained i n t h e r e g r e s s i o n only.  background  significant  s o l v i n g o r t e s t t o t a l and l i t t l e Classroom  on a p p l i c a t i o n  activities,  equation f o r the t e s t t o t a l  variable  A summary o f r e s u l t s i s shown i n T a b l e 37.  T a b l e 37 Classroom O r g a n i z a t i o n F a c t o r s Regressed on C r i t e r i o n V a r i a b l e s Variable Problem Test  Solving  Total  Applications  Step  Factor  Mult R  R  2  F  P  1  5  0.1687  0 .0285  19.54  <0 .001  1 2  5 4  0.2169 0.2338  0 .0471 0 .0547  32.93 19.26  <0 .001 <0 .001  1  5  0.2155  0.0464  32.47  <0.001  100 Classroom of  O r g a n i z a t i o n F a c t o r 5 accounted  t h e v a r i a n c e on problem  and  4.6 p e r c e n t  the  prediction  additional  solving,  on a p p l i c a t i o n s . . equation  0.8 p e r c e n t  f o r 2.8 p e r c e n t  4.7 p e r c e n t  on t e s t  F a c t o r 2, which  f o r test  total,  remained i n  accounted  o f t h e v a r i a n c e on t h a t  total  f o r an  variable.  All  f a c t o r s which remained i n t h e e q u a t i o n s were s i g n i f i c a n t a t t h e 0.001  level.  Problem-Solving Process A  total  of three  Problem-Solving  Process  r e g r e s s e d on each o f t h e c r i t e r i o n v a r i a b l e s .  factors  were  F a c t o r 2, which  was a measure o f t e a c h e r a t t i t u d e toward t h e t e a c h i n g o f problem solving,  remained  i n a l l three  equations.  Factor  1,  r e l a t e d t o v a r i e t i e s o f approaches used t o t e a c h problem a l s o remained i n t h e r e g r e s s i o n application  variables.  equation  solving  e q u a t i o n f o r t h e t e s t t o t a l and  The t h i r d  teacher p a r t i c i p a t i o n i n in-service prediction  which  factor,  which  activities,  related  to  remained i n t h e  f o r the problem-solving c r i t e r i o n v a r i a b l e .  R e s u l t s a r e summarized i n T a b l e 38.  T a b l e 38 P r o b l e m - S o l v i n g P r o c e s s F a c t o r s Regressed on C r i t e r i o n V a r i a b l e s Variable Problem Test  Solving  Total  Applications  Factor  Mult R  R^  F  p  1 2  2 3  0..1833 0..2010  0..0336 0..0404  24,.89 15..35  <0..001 <0..001  1 2  2 1  0..2377 0.,2513  0..0565 0..0631  42..88 24..09  <0.,001 <0..001  1 2  2 1  0..2212 0..2418  0..0489 0..0584  36,.84 22..19  <0..001 <0..001  Step  101 Results percent  i n Table  38 show t h a t  2 accounted  and F a c t o r 3 accounted f o r an a d d i t i o n a l  the v a r i a n c e i n student test  Factor  total  variance  variable,  1 added  2 was  percent.  4.9  percent.  an a d d i t i o n a l  A l l values  Factor  On t h e  f o r 5.6 p e r c e n t 0.7  v a r i a n c e i n s t u d e n t achievement on a p p l i c a t i o n s Factor  0.7 p e r c e n t o f  achievement i n problem s o l v i n g .  F a c t o r 2 accounted  and F a c t o r  f o r 3.4  1 added  of the F s t a t i s t i c  of the  percent.  The  accounted f o r by  an a d d i t i o n a l  reported  1.0  i n the table  were s i g n i f i c a n t a t t h e 0.001 l e v e l . Based solving,  on  which  expectations  was  in  the  attitude  of variables  performance  i n teaching  variance  to  teacher  comprised  criterion variables. slightly  data,  o f student  effectiveness measurable  these  problem student  toward  related  and t h e i r  solving  to  teacher  perceptions  accounted  achievement  problem  on  of  f o r some a l l three  Involvement i n i n - s e r v i c e a c t i v i t i e s added  variance  accounted  f o r i n problem  solving  whereas t h e v a r i e t y o f d i f f e r e n t approaches used i n t h e t e a c h i n g of  problem  solving  accounted  f o r part  of  achievement on t h e t e s t t o t a l and a p p l i c a t i o n Teachers* At  the  variance  in  variables.  P e r c e p t i o n s o f Mathematics the  importance,  next  stage  difficulty  were r e g r e s s e d  of  analysis  i n teaching  and  on each o f t h e problem  application variables.  the  three  enjoyment  solving,  scores  for  i n teaching  test  total  R e s u l t s a r e r e p o r t e d i n T a b l e 39.  and  102 Table  39  T e a c h e r s ' P e r c e p t i o n s Regressed on C r i t e r i o n Variable  Step  Problem S o l v i n g Test T o t a l  the  Mult R  2 2 1 2  0.0944 0.1587 0.1810 0.1468  1 1 2 1  Applications  Factor  Factor  2, which was comprised  difficulty  R  0.0089 0.0252 0.0328 0.0216  Teachers* for test  <0.05 <0.001 <0.001 <0.001  t o t e a c h mathematics, remained in. t h e r e g r e s s i o n  solving, ratings  total.  the v a r i a n c e  6.64 19.06 12.48 16.26  of teachers' perceptions of  0.9, 2.5 and 2.2 p e r c e n t o f t h e v a r i a n c e problem  p  F  2  e q u a t i o n f o r each o f t h e dependent v a r i a b l e s .  on  Variables  test  total  f o r importance I t accounted  I t accounted f o r  i n student  and a p p l i c a t i o n s also  remained  achievement  respectively.  i n the equation  f o r an a d d i t i o n a l  0.8 p e r c e n t o f  i n s t u d e n t achievement on t h a t domain.  S t u d e n t s ' P e r c e p t i o n s o f Mathematics Students* difficulty on  ratings  i n learning  the c r i t e r i o n  Results  o f mathematics  t o importance,  and enjoyment i n l e a r n i n g were  variables  are reported  i n regard  a t t h e next  i n T a b l e 40.  step  regressed  i n the analysis.  103 T a b l e 40 S t u d e n t s ' P e r c e p t i o n s Regressed on C r i t e r i o n Variable Problem Test  Step  Solving  Total  Applications  Students' (Factor  Factor  Mult R  1  3  0.1671  0.0279  20.88  <0.001  1  3  0.2714  0.0736  57.79  <0.001  1 2  3 2  0.1960 0.2105  0.0384 0.0443  29.04 16.84  <0.001 <0.001  ratings  3) remained  criterion variance  I t accounted  i n learning  percent  comprised  students*  equation  on  difficulty  f o r applications  percent o f the variance  applications.  percent  of the  7.4 p e r c e n t on t e s t Factor  ratings,  and accounted  mathematics  f o r each o f t h e  f o r 2.8  i n achievement i n problem s o l v i n g ,  t o t a l and 3.8  4.6  f o r enjoyment  i n the p r e d i c t i o n equation  variables.  of  Variables  2,  which  remained  i n the  f o r an a d d i t i o n a l  i n achievement on t h a t  was  0.6  domain.  THE PROVINCIAL MODELS  General  models  f o r each  of the c r i t e r i o n  variables  were  determined through r e s u l t s from t h e second s t a g e i n t h e m u l t i p l e regression  analysis.  categories  were  At  regressed  this  stage  on each  factors  a l l input  o f t h e dependent  using  the step-wise procedure.  By u s i n g  input  categories,  s h a r e d among  any v a r i a n c e  from  f o r b e f o r e t h e t o t a l amount o f v a r i a n c e  variables  f a c t o r s from a l l o f t h e them was  was determined.  accounted  104 Problem S o l v i n g Five out of the t o t a l regression  equation  o f 18 f a c t o r s  f o r problem  remained  solving.  i n the f i n a l  Results  of  the  r e g r e s s i o n a r e shown i n T a b l e 41.  T a b l e 41 R e g r e s s i o n Model f o r Problem S o l v i n g  Provincial Input Category  Factor Number  Step  Mult R  R  R Ch  2  FCh  2  P  S. Back.  2  1  0.1979  0. 0392  0.0392  25.24 <0 .001  C l a s s Org.  5  2  0.2574  0. 0663  0.0271  17.92 <0 .001  S. P e r c e p t .  3  3  0.2974  0. 0884  0.0221  15.01  Prob.  Solv.  2  4  0.3173  0. 1007  0.0122  8.38 <0 .05  Prob.  Solv.  3  5  0.3265  0. 1066  0.0060  4.10 <0 .05  Note.  <0 .001  R^Ch = Incremental change i n R^ FCh = F r a t i o f o r t h e i n c r e m e n t a l change i n R . 2  Based on t h e d a t a r e p o r t e d i n T a b l e 41, t h e e f f e c t f a c t o r s accounted  f o r a t o t a l o f 10.7 p e r c e n t o f t h e v a r i a n c e i n  s t u d e n t achievement i n problem s o l v i n g . accounted comprised  parents'  educational  percent of the variance. accounted  f o r t h e second  comprised  of course  percent.  The o t h e r  the  additional  follows:  The g r e a t e s t e f f e c t i s  f o r by F a c t o r 2 from s t u d e n t background. of  of a l l  accounted  f o r 3.9  F a c t o r 5, from c l a s s r o o m o r g a n i z a t i o n , largest  type  and  amount o f v a r i a n c e .  i t explained  f a c t o r s which remained  amount  student  levels,  This factor,  of  ratings  variance  they  o f enjoyment,  an  I t was  additional  2.7  i n t h e e q u a t i o n and explained  2.2 p e r c e n t ;  were  as  teachers'  105 a t t i t u d e s toward problem s o l v i n g ,  1.2 p e r c e n t ;  and t e a c h e r i n -  s e r v i c e i n problem s o l v i n g , 0.6 p e r c e n t . The  i n p u t s o f s c h o o l i n g d i d n o t e x p l a i n much v a r i a n c e i n  s t u d e n t performance i n problem s o l v i n g . o n l y because o f l i m i t a t i o n s  T h i s r e s u l t may be not  due t o t h e d e f i n i t i o n  of variables  and t h e a g g r e g a t i o n o f d a t a b u t a l s o t o many o t h e r f a c t o r s which could  have  aptitude  effect.  are  among  problem s o l v i n g ,  F o r example,  student  characteristics  but are l i k e l y  ability  which  either  are  and  spatial  important  for  inherent i n students or  d i f f i c u l t t o teach. Test  Total Out  o f t h e o r i g i n a l 18 f a c t o r s 5 remained i n t h e r e g r e s s i o n  equation total.  explaining variance  i n student  achievement  on  test  R e s u l t s a r e shown i n T a b l e 42.  Provincial Input Category  Factor Number  T a b l e 42 R e g r e s s i o n Model f o r T e s t T o t a l  Step  Mult R  R  2  R Ch  Fch  z  P  S. P e r c e p t .  3  1  0.2808  0.0789  0.0789  52.99  <0.001  S. Back.  2  2  0.3576  0.1279  0.0490  34.75  <0.001  C l a s s Org.  5  3  0.4058  0.1647  0.0368  27.17  <0.001  Prob.  Solv.  2  4  0.4300  0.1849  0.0202  15.29  <0.001  T. P e r c e p t .  1  5  0.4414  0.1949  0.0100  Note.  7.61  R^Ch = Incremental change i n R FCh = F r a t i o f o r t h e i n c r e m e n t a l change i n R . z  2  <0.05  106 The the  effects  variance  of the factors  i n student  accounted  achievement  f o r 19.5 p e r c e n t  on t e s t  total.  of  Students  1  p e r c e p t i o n s o f t h e i r enjoyment o f l e a r n i n g mathematics  accounted  for  This  was  f o r an a d d i t i o n a l  4.9  t h e g r e a t e s t amount o f v a r i a n c e  f o l l o w e d by p a r e n t s '  a t 7.9 p e r c e n t .  l e v e l s of education  p e r c e n t , c o u r s e type f o r 3.7 p e r c e n t , t e a c h e r s ' a t t i t u d e s toward problem s o l v i n g f o r 2.0 p e r c e n t and t e a c h e r s ' p e r c e p t i o n s o f t h e importance o f mathematics f o r 1.0 p e r c e n t . Students' accounted  p e r c e p t i o n s o f enjoyment  i n learning  mathematics  f o r c o n s i d e r a b l y more v a r i a n c e on t e s t t o t a l  problem s o l v i n g .  I t accounted  f o r 7.9 p e r c e n t  on t h e former and o n l y 2.2 p e r c e n t on t h e l a t t e r . f a c t o r s common t o both domains a l s o accounted  than  on  of the variance The o t h e r two  f o r more v a r i a n c e  on t e s t t o t a l t h a n on problem s o l v i n g . Applications Five  factors  applications.  also  remained  i n the regression equation f o r  T a b l e 43 shows t h e r e s u l t s .  107 T a b l e 43 P r o v i n c i a l R e g r e s s i o n Model f o r A p p l i c a t i o n s Input Category  Factor Number  Step  Mult R  R  2  R Ch  Fch  2  P  S. Back.  2  1  0.2884  0.0832  0.0832  56.17  <0. 001  S.  3  2  0.3498  0.1223  0.0391  27.56  <0. 001  C l a s s Org.  5  3  0.3974  0.1579  0.0356  26.10  <0. 001  Prob.  Solv.  1  4  0.4184  0. 1751  0.0172  12.81  <0. 001  Prob.  Solv.  2  5  0.4282  0.1833  0.0082  Percept.  Note.  6.21  <0. 001  R^Ch = Incremental change i n R^ FCh = F r a t i o f o r t h e i n c r e m e n t a l change i n R . 2  The  five  accounted  for  achievement explained followed  f a c t o r s which remained i n t h e r e g r e s s i o n e q u a t i o n  on  18.3  percent  applications.  the greatest by  of  student  the  variance  Parents'  amount  of  perceptions  level  variance of  in of  student education  at  8.3  percent,  enjoyment  in  learning  mathematics a t 3.9 p e r c e n t , c o u r s e type a t 3.6 p e r c e n t , of problem-solving  activities  variety  and m a t e r i a l s a t 1.7 p e r c e n t and  t e a c h e r s ' a t t i t u d e s toward problem s o l v i n g a t 0.8 p e r c e n t . The the  r e g r e s s i o n equation  5 factors  i n common  criterion variables. explained  with  f o r applications the equations  However, t h e l e v e l  shared  f o r t h e o t h e r two  of parents'  c o n s i d e r a b l y more v a r i a n c e i n s t u d e n t  different  problem-solving  activities  t e a c h e r s e x p l a i n e d an a d d i t i o n a l  education  performance on  t h i s domain than on problem s o l v i n g o r t e s t t o t a l . of  4 out of  The v a r i e t y  and m a t e r i a l s used  by  1.7 p e r c e n t o f t h e v a r i a n c e i n  performance on t h e a p p l i c a t i o n domain b u t d i d n o t e n t e r i n t o t h e  108 final This  e q u a t i o n f o r e i t h e r o f t h e o t h e r two dependent v a r i a b l e s . may  indicate  measurable,  that  these  activities  have  ability  to solve  e f f e c t on s t u d e n t s '  problems b u t l i t t l e  a  l i m i t e d , but routine  story  o r no e f f e c t on t h e i r performance on items  r e q u i r i n g h i g h e r c o g n i t i v e s k i l l s o r on o v e r a l l t e s t r e s u l t s .  4.7  THE SURREY MODEL  Since results,  pre-test  data  the current  district  to  for,  The  Provincial  1985  whether,  i t would  applying  provincial pre-test of  data.  t h e models  existed.  the f i n a l  Assessment  was  i n the F a l l  were model.  replicated  o f 1986 and a  district.  t h e same  scores  alter  scores  urban  o f 1987 t o a l l s t u d e n t s and t e a c h e r s o f  A model was determined and  for provincial  pre-test  significantly  of a pre-test  i n the Spring  Grade 7 i n t h a t  available  when  Mathematics  through a d m i n i s t r a t i o n post-test  not  study examined r e s u l t s from a l a r g e  determine  controlled  were  A  first  using  procedures  second  only  as  reported  model was then  were used as c o v a r i a t e s . were  then  compared  post-test  f o r t h e 1985  developed  Results  scores,  i n which  based on each  t o see i f any  differences  Based on t h e s e d i f f e r e n c e s , a judgment was made on t h e  appropriateness pre-test  of using  information,  the analysis  o f survey  f o r decision-making  at  data,  without  the p r o v i n c i a l  level.  1987  P o s t - T e s t Model The  same method o f a n a l y s i s  Assessment  was  applied  to  used w i t h  post-test  data  results  from t h e 1985 from  t h e 1987  109 validation  study.  S i n c e t h e purpose o f t h i s  p a r t o f t h e study  was t o compare f i n d i n g s based  on t h e a n a l y s e s o f c r o s s - s e c t i o n a l  data  only  and l o n g i t u d i n a l  criterion  variable.  classrooms  completed  returned  data, As test  questionnaires  reported  test  earlier,  booklets. and  total  A  matches  was  used  students  total  were  from  o f 100  found  as t h e 104  teachers  between  97  t e a c h e r s and c l a s s r o o m s . The factors  final from  p o s t - t e s t model  was  determined  a l l of the input categories, using  method, on c l a s s means f o r t e s t t o t a l . 21  factors  by  remained  i n the f i n a l  Four  regressing  t h e step-wise  out of a t o t a l of  regression equation.  Results  a r e r e p o r t e d i n T a b l e 44.  T a b l e 44 P o s t - T e s t R e g r e s s i o n Model Input Category  Factor Number  Step  Mult R  R  S. P e r c e p t .  3  1  0.4194  S. Back.  1  2  Prob.  3 4  Solv.  T. Back. Note.  R Ch  Fch  P  0 .1759  0.1759  17.07  <0. 000  0.5050  0 .2550  0.0791  8.391 <0. 001  3  0.5414  0 .2931  0.0381  4.200 <0. 044  4  0.5798  0 .3361  0.0431  4.994 <0. 028  2  2  R^Ch = Incremental change i n r FCh = F r a t i o f o r t h e i n c r e m e n t a l change i n R . 2  These d a t a show t h a t a l l f a c t o r s accounted  f o r 33.6 p e r c e n t  o f t h e v a r i a n c e i n s t u d e n t achievement on t e s t t o t a l . from  student  p e r c e p t i o n , comprised  Factor 3  of students' perceptions of  t h e i r enjoyment i n l e a r n i n g mathematics, e x p l a i n e d 17.6 p e r c e n t  110 o f t h e v a r i a n c e i n s t u d e n t achievement on t e s t t o t a l . from  student  background,  language f i r s t spoken and accounted  for  an  which  comprised  of  Factor 1  variables  language c u r r e n t l y spoken i n t h e home,  additional  7.9  percent  of  the  variance  s t u d e n t achievement on the same c r i t e r i o n v a r i a b l e . v a r i a n c e i n achievement o f 3.8 from  problem-solving  p e r c e n t was  processes.  on t e a c h e r s a t i s f a c t i o n  on  I t was  in  Additional  e x p l a i n e d by F a c t o r 3 comprised  of  i n t e a c h i n g problem s o l v i n g ,  variables the  number  o f d i f f e r e n t p r o b l e m - s o l v i n g s t r a t e g i e s t a u g h t and t h e number o f workshops  on  background  problem  F a c t o r 4,  solving on  the  attended other  by  teachers.  hand, was  a  p r o p o r t i o n o f t e a c h i n g l o a d i n mathematics and a d d i t i o n a l 4.3 1987  Teacher  measure o f accounted  the  f o r an  p e r c e n t o f t h e v a r i a n c e i n s t u d e n t achievement.  L o n g i t u d i n a l Model A t t h i s s t a g e o f a n a l y s i s , achievement means f o r t e s t  total  from t h e p r e - t e s t r e s u l t s were i n t r o d u c e d as c o v a r i a t e s i n t o the final  regression  equation.  Using  this  procedure,  student  knowledge and b e h a v i o r s a t t h e b e g i n n i n g o f t h e s c h o o l y e a r were controlled. were which  As  explained occurred  a  result,  the  by  factors  related  during  the  year  variances more  under  i n achievement closely  to  examination.  showing v a r i a n c e s e x p l a i n e d , a r e r e p o r t e d i n T a b l e  45.  which  processes Results,  Ill T a b l e 45 L o n g i t u d i n a l R e g r e s s i o n Model f o r T e s t T o t a l Factor Number  Input Category Pretest  Step  Mult R  1  0.6361  R  R Ch  FCh  P  0 .4046  0.4046  54.36  <0 .000  2  2  S. P e r c e p t .  1  2  0.6906  0 .4770  0.0724  10.94  <0 .001  C l a s s Org.  2  3  0.7275  0 .5293  0.0523  8.67  <0 .004  T. Back.  4  4  0.7469  0 .5579  0.0286  4.98  <0 .029  S. Back.  1  5  0.7636  0 .5831  0.0252  4.60  <0 .035  Note.  R^Ch = Incremental change i n R^. FCh = F r a t i o f o r t h e i n c r e m e n t a l change i n R . 2  The  final  longitudinal  f a c t o r s out o f a t o t a l variance  i n student  model  for test  total  retained 5  o f 22 and e x p l a i n e d 58.3 p e r c e n t  achievement  for test  total.  of the  The p r e - t e s t  e x p l a i n e d 40.5 p e r c e n t o f t h e v a r i a n c e , w i t h t h e b a l a n c e o f 17.8 percent  accounted  retained. variance  f o r by  the r e s t  of the factors  which  were  I n c l u d e d among t h e o t h e r f a c t o r s which accounted f o r i n student  t h e importance  achievement  were s t u d e n t s '  o f mathematics, c l a s s r o o m  perceptions of  o r g a n i z a t i o n F a c t o r 2,  t e a c h e r background F a c t o r 4 and s t u d e n t background F a c t o r 1. An a d d i t i o n a l  7.2 p e r c e n t o f t h e v a r i a n c e was e x p l a i n e d by  students'  perceptions  Classroom  organization  variables  on  the  of  the  Factor  number  of  importance 2,  which  students  of was  mathematics. comprised  questioned  and  of the  p r o p o r t i o n o f c l a s s time spent working i n s m a l l groups e x p l a i n e d an  additional  Teacher  5.2  percent  background  Factor  of 4,  the which  variance was  a  in  achievement.  measure  of the  112 proportion  of teaching  additional  2.9 p e r c e n t  and  student  i n mathematics,  of the variance  background  language f i r s t  load  Factor  1,  accounted  i n student  comprised  f o r an  achievement  of variables  on  spoken and language c u r r e n t l y spoken i n t h e home,  e x p l a i n e d an a d d i t i o n a l 2.5 p e r c e n t . A comparison differences. 33.6  F o r example,  percent  percent.  t h e c r o s s - s e c t i o n a l model  model,  of  after  students  were  the  in  Students'  each  model,  entry  level  controlled,  perceptions  mathematics, their  in  of  f o r example, perceptions  some  their  of  the  remained i n t h e r e g r e s s i o n e q u a t i o n  knowledge  explained  only  cases  were  i n the  of  learning  post-test  importance  of  behaviors  Mother tongue and p r o p o r t i o n o f t e a c h i n g  remained  models.  effects  o f classroom  model whereas  o r g a n i z a t i o n remained  a factor  f o r measurable v a r i a n c e model.  one  on p r o b l e m - s o l v i n g i n student  factor  model  mathematics  a f t e r entry l e v e l  However,  17.8  different.  were c o n t r o l l e d . i n both  and  which e x p l a i n e d t h e  enjoyment  remained  explained  achievement whereas t h e  The f a c t o r s and t h e i r c o m p o s i t i o n ,  variances  whereas  between t h e two models shows some  of the variance i n student  longitudinal behaviors  of results  load  measuring t h e  i n the longitudinal processes  accounted  achievement i n t h e p o s t - t e s t  113  CHAPTER 5 SUMMARY AND CONCLUSIONS  The purpose o f t h i s of  survey data,  outcomes,  study was t o examine,  relationships  as measured  by  between  inputs  t h r o u g h t h e use o f s c h o o l i n g and  s t u d e n t achievement  i n mathematics.  The i n p u t s o f s c h o o l i n g were comprised o f a number o f v a r i a b l e s grouped under each o f t h e f o l l o w i n g teachers'  backgrounds,  mathematics,  students'  classroom  categories:  s t u d e n t s ' and  and t e a c h e r s '  perceptions of  organization  and  problem-solving  processes. Outcome total,  measures  problem  variable  solving  provided  mathematics variables distinct  included  a  and  students' applications.  measure  whereas  the  achievement  of  The  overall  problem  solving  of cognitive  behavior.  t e s t e d s t u d e n t s ' achievement on t e s t  measure  critical  thinking.  on t h e o t h e r which  Test  hand,  were  items  and  total  were judged by committees  P r o v i n c i a l Assessment  in  application a t two  solving, f o r  items i n t e n d e d t o  i n the  comprised  test  performance  Problem  example,  problems  test  were d e s i g n e d t o measure s t u d e n t achievement levels  variable,  on  application  of routine  involved  story  i n t h e 1985  o f Mathematics t o be o f a lower c o g n i t i v e  level. A  related  appropriateness large-scale relationships address  this  problem of using  assessment  involved  cross-sectional to  make  found among t h e i n p u t question,  exploration  results  from  survey  decisions  of  data  based  from on  and output v a r i a b l e s . a subsequent  the a  the To  longitudinal  114 study  which  with  utilized  p o s t - t e s t data  data.  The  intent  the  same i n s t r u m e n t s  and  was  then,  to  with  see  the  i f the  were examined  inclusion  first  of pre-test  same g e n e r a l  conclusions  c o u l d be made.  5.1  SIGNIFICANT FINDINGS AND  Analyses  of p r o v i n c i a l  CONCLUSIONS  data  from  the  1985  used t o address r e s e a r c h q u e s t i o n s 1 t o 6. q u e s t i o n 7, r e s u l t s from t h e 1987 in  two  ways.  analysis  First,  to  explain  covariates  subsequently  in  the  p o s t - t e s t data  the  amount  of  were  r e s u l t s t o see what, i f any, In under  were  used  variance  in  in  the  student  f o r by t h e v a r i a b l e s w i t h i n  Second, p r e - t e s t d a t a were i n c l u d e d  regression  explained  address  v a l i d a t i o n study were examined  achievement i n mathematics accounted  as  In o r d e r t o  only  each c a t e g o r y under study.  Assessment were  then  equations. compared  differences  with  The  variances  the  preceding  existed.  t h e f o l l o w i n g d i s c u s s i o n each o f t h e r e s e a r c h q u e s t i o n s examination  is  stated  first.  Related  findings  follow  d i r e c t l y a f t e r each r e s p e c t i v e q u e s t i o n . 1.  What r e l a t i o n s h i p s e x i s t among t e a c h e r background c h a r a c t e r i s t i c s and s t u d e n t background c h a r a c t e r i s t i c s ; and between t h e s e v a r i a b l e s and s t u d e n t s ' achievement i n mathematics?  Strong student first  relationships  background  spoken and  fathers'  and  were  found  variables.  They  between were  two  pairs  students'  of  language  t h e language they c u r r e n t l y speak a t home, and  mothers'  coefficient  between  between t h e  second  the pair  levels first i t was  of  education.  pair 0.66.  of  The  variables The  strong  correlation was  0.81  and  relationship  115 found  between  language  unexpected due  between  the  educational  as s t r o n g as  the  a  number  completed  the  and  the  parents  on  the and  background  of  v a r i a b l e s were  not  background v a r i a b l e s j u s t statistically  significant  and  workshops  mathematics  (0.32),  education  and  mathematics  courses  completed  three  years  may  have  indicated  attended that  many  a r e a c t i v e i n t h e a r e a o f p r o f e s s i o n a l development  attend  a t t e n d one  both  types  of  s e s s i o n s whereas t h o s e  who  do  form o f i n - s e r v i c e p r o b a b l y do not a t t e n d t h e  relatively  mathematics  strong  courses  relationship  and  the  not wish t o t a k e take  post-secondary  methods  between  number  c o u r s e s s u c c e s s f u l l y completed may  to  both  correlation  mathematics s e s s i o n s a t c o n f e r e n c e s  previous  t e a c h e r s who  not  high  study  r e l a t i o n s h i p found between t h e number o f mathematics  workshops and  The  of  i n the  not  Examples i n c l u d e d c o r r e l a t i o n s between  at conferences  (0.41). The  likely  The  was  t o t h e h i g h p r o p o r t i o n o f mothers  examples o f s t u d e n t  however,  attendance  during  levels  among t e a c h e r  c o r r e l a t i o n s were found.  courses  spoken  attended school a t the high school l e v e l or higher.  Relationships  cited;  currently  t h e i r mother tongue.  o t h e r hand, i s l i k e l y due f a t h e r s who  and  t o the l a r g e p r o p o r t i o n of students  f o r whom E n g l i s h was found  first  courses  of  the  not  other.  number  mathematics  education  i n d i c a t e t h a t t e a c h e r s who  mathematics c o u r s e s  either,  provided  may  that  of  do  choose  option  is  a v a i l a b l e i n t h e i r t e a c h e r e d u c a t i o n program. Statistically  significant  r e l a t i o n s h i p s were found  t h e e d u c a t i o n a l l e v e l s o f p a r e n t s ' and all  three  variables  criterion ranged  variables.  from  0.20  The  between  between  s t u d e n t s ' achievement correlations  the  among  educational  on  these  levels  of  116 b o t h p a r e n t s and fathers'  educational  Since parents* of  achievement on  l e v e l s of  socio-economic  positive  in  Murnane, 1981;  students* the  and  the  found  numerous  between  of  were  (e.g.  between  the  education  number  of  student  Husen,  1967;  most  strongly  with  courses  workshops  completed.  attended  and  a p p l i c a t i o n s were 0.12  and  number o f mathematics methods c o u r s e s completed  0.09  and  0.13  a p p l i c a t i o n s , on t h e  respectively.  These  correlations,  low  found  in  by  1979;  and  Ward,  2.  What r e l a t i o n s h i p s e x i s t among t y p e s o f c l a s s r o o m o r g a n i z a t i o n s and s t r u c t u r e s ; and between t h e s e v a r i a b l e s and s t u d e n t s ' achievement i n mathematics?  studies  the  several  were  i n small  T h i s r e s u l t may  types  Rigsby,  1973;  Rutter  et. a l . ,  found  pairs to  of  be  classroom  statistically  For example, the c o r r e l a t i o n between p r o p o r t i o n  a c t i v i t y c e n t e r s are to  and  between  variables  time spent working 0.22.  McDill  S i m i l a r r e s u l t s were  1979.  Relationships  significant.  t o draw c o n c l u s i o n s .  and  other  however, were too  organization  and  S t a t i s t i c a l l y s i g n i f i c a n t c o r r e l a t i o n s found  s t u d e n t achievement on t e s t t o t a l and hand,  and  the  number o f workshops a t t e n d e d  mathematics  respectively.  between t h e  measures  confirmed  factor  which c o r r e l a t e d  s t u d e n t achievement on t e s t t o t a l and 0.13  as  Horn & Walberg, 1984).  achievement were t h e  Correlations  studies  between  applications.  viewed  study  this  0.29  on  be  current  other  teacher variables  number  achievement  education could  status,  relationships  achievement,  The  level  problem s o l v i n g t o  of  groups and  at  activity  i n d i c a t e t h a t s t u d e n t s who probably i n small  activities  in  which  centers  was  are working a t  group c o n f i g u r a t i o n s they  of  would  likely  due be  117 engaged.  Time  numbers  of  students  correlation classroom  spent  on  called  coefficient  practice  homework-related  of  upon 0.17.  to  activities  answer  This  and  questions  finding  may  the  had  a  reflect  a  i n which homework-related a c t i v i t i e s  include  s t u d e n t q u e s t i o n i n g w h i l e d i s c u s s i n g t h e answers. Several  significant  organization these  were  variables the  relationships  correlated  following:  students questioned  type  (-0.15), and  and l e n g t h o f p e r i o d (-0.40). c o u r s e t y p e and teachers classes  tend than  between  negatively. of  course  Included  and  the  ask  more  i n enriched  among  number  of  t h e number o f c l a s s e s p e r week  The  n e g a t i v e r e l a t i o n s h i p between  number o f s t u d e n t s q u e s t i o n e d may to  classroom  questions  ones.  of  indicate that  students  Activities  such  in as  modified drill  and  p r a c t i c e and d i s c u s s i o n o f homework, which i n v o l v e q u e s t i o n - a n d answer  techniques,  are  likely  c l a s s e s than i n e n r i c h e d ones. number o f It  emphasized  that  as  numbers  in  modified  The n e g a t i v e c o r r e l a t i o n between  c l a s s e s p e r week and  indicates  more  l e n g t h o f p e r i o d was of  classes  in a  expected.  week i n c r e a s e ,  t h e i r d u r a t i o n decreases. The o n l y r e l a t i o n s h i p between c l a s s r o o m o r g a n i z a t i o n v a r i a b l e s and s t u d e n t achievement o f i n t e r e s t which c o r r e l a t e d s i g n i f i c a n t l y was  t h e type o f c o u r s e .  C o r r e l a t i o n s between  c o u r s e t y p e and s t u d e n t achievement i n problem s o l v i n g , t o t a l and a p p l i c a t i o n s were 0.15,  0.19  and  0.21  test  respectively.  These r e s u l t s l i k e l y i n d i c a t e t h a t s t u d e n t s i n e n r i c h e d c l a s s e s perform b e t t e r than those i n r e g u l a r o r m o d i f i e d ones.  This  f i n d i n g i s not s u r p r i s i n g s i n c e one would expect t h a t s t u d e n t s i n e n r i c h e d c l a s s e s a r e h i g h e r a c h i e v e r s on average  than  those  118 i n r e g u l a r o r m o d i f i e d ones.  The magnitude o f t h e  however, i s lower than expected.  correlation,  T h i s r e s u l t i s l i k e l y due  t h e s m a l l numbers o f c l a s s e s (7 p e r c e n t )  to  i n t h e study which were  r e p o r t e d as m o d i f i e d o r e n r i c h e d . 3.  What r e l a t i o n s h i p s e x i s t between d i f f e r e n t approaches t o the t e a c h i n g o f problem s o l v i n g and s t u d e n t s ' achievement i n mathematics?  A number o f r e l a t i v e l y s t r o n g r e l a t i o n s h i p s were  found  among p r o b l e m - s o l v i n g p r o c e s s v a r i a b l e s and between t h e s e s t u d e n t achievement.  R e l a t i o n s h i p s among t h e  independent  v a r i a b l e s w i t h c o r r e l a t i o n c o e f f i c i e n t s g r e a t e r than 0.20 shown i n T a b l e  46.  and  are  119 T a b l e 46 C o r r e l a t i o n s Among P r o b l e m - S o l v i n g P r o c e s s Variable PS1  Enjoyment  PS2  Achievement  PS3  Satisfaction  PS4  Easiness  PS1  PS 2  PS 3  PS4  100  52  31  26  100  36  26  100  24 100  PS 5 PS 5  Strategies  PS 6  In-service  PS 7  Exercises  PS8  Activities  PS 9  Freq./Teach  PS9  Problem Types  PS11  Features  Note.  Variables*  PS 6  PS 7  PS8  40  34  100  PS9  PS10  PS11  43  24  100  -  23 100  52  21  40  40  100  24  37  46  100  22 100  37 100  The c o r r e l a t i o n c o e f f i c i e n t s r e p o r t e d above a r e based on the computed c o r r e l a t i o n s rounded t o two d e c i m a l p l a c e s and m u l t i p l i e d by 100. * Only c o r r e l a t i o n s >0.20 a r e r e p o r t e d .  Results  i n d i c a t e that a strong  r e l a t i o n s h i p e x i s t e d between  t e a c h e r s ' e x p e c t a t i o n s o f s t u d e n t s ' enjoyment o f problem s o l v i n g and  their  topic.  expectations  In addition,  problem-solving student variables  of students'  their  correlated  enjoyment  and  achievement  s a t i s f a c t i o n with strongly  achievement.  ranged from 0.31 t o 0.52.  with  their  on  t h e same  the teaching  of  perceptions of  Correlations  among  These r e s u l t s suggest  these that  120 a  common  evidence  construct  these  into  These  among  variables  these  Further  g a i n e d t h r o u g h t h e f a c t o r a n a l y s e s i n which t h e y a l l l o a d e d factor.  relationship  variables.  was  same  common  underlie  variables  the  of a  may  may  a l l be  a  measure o f  found  among t h e  t e a c h e r a t t i t u d e toward problem s o l v i n g . Other numbers  of  materials number  relationships different  different  approaches  sources  activities correlated  variables suggest  interest  or resources u t i l i z e d  of d i f f e r e n t  solving  of  used  that  and  sources  by t e a c h e r s .  of exercises  t o motivate  a t 0.52.  correlated  were  between  problem  solving  of  different  sources  common c o n s t r u c t  of exercises may a l s o  likely  These  also  use  a  be  teaching  viewed  as measures  results  of  activities.  A  these v a r i a b l e s  of teacher  t o the  number  a l s o l o a d e d i n t o one f a c t o r a t t h e f a c t o r a n a l y s i s could  the other  o f approaches  and m o t i v a t i o n a l  underlie  f o r problem  among  and 0.46.  who u s e a v a r i e t y  teaching  and t h e number o f  students  teachers  different  F o r example, t h e  Relationships 0.24  of  since  stage.  flexibility  they They  i n the  o f problem s o l v i n g .  Several between  statistically  s i g n i f i c a n t r e l a t i o n s h i p s were  problem-solving  achievement magnitude  process  i n mathematics. than  variables,  were  several  variables  Although  correlations  are  worthy  of  they  and were  among comment.  students'  of a  the  found  lesser  independent  For  example,  t e a c h e r s ' e x p e c t a t i o n s o f s t u d e n t s ' enjoyment o f and achievement in  problem  solving  showed  the strongest  achievement.  These v a r i a b l e s  and  problem  0.22 w i t h  solving,  correlated  relationships respectively  0.19 and 0.30 w i t h  test  with  a t 0.16 total,  121 and  0.18  and  0.26  with  applications.  These  findings  s i m i l a r ones i n s t u d i e s by Good and Grouws (1977), al.  (1980) and Brophy  confirm  Evertson et.  (1982b).  The number o f d i f f e r e n t s t r a t e g i e s t a u g h t , p a r t i c i p a t i o n i n in-service activities, and number o f problem with  all  three  number o f d i f f e r e n t s o u r c e s o f e x e r c i s e s , types taught a l s o c o r r e l a t e d  of  the  achievement  r e l a t i o n s h i p s between d i f f e r e n t in  mathematics  Furst  (1971),  were  also  Cooney,  Davis  domains.  approaches  found  in  used  Henderson  Positive  and  s t u d i e s by  and  significantly  achievement  Rosenshine (1975)  and  (1977). Based on t h e s e r e s u l t s , s t u d e n t achievement was be a s s o c i a t e d w i t h t e a c h e r e x p e c t a t i o n s o f s t u d e n t s , in in-service activities t e a c h i n g o f problem 4.  and  significantly  and w i t h f l e x i b i l i t y o f approach  t o the  solving.  p e r c e p t i o n s o f the enjoyment related  difficulty  to  (0.20),  importance  and  strongest  relationship  ratings.  importance  in  teaching  one  another.  enjoyment  enjoyment (0.58). was  Relationships  mathematics and strongest  found t o  involvement  of, d i f f i c u l t y mathematics The  of  and  difficulty  between  importance teacher  s t u d e n t achievement were l e s s these  correlations,  which  (0.37),  and  and the  enjoyment  perceptions pronounced.  were  were  importance  These r e s u l t s showed t h a t  between  in  correlation  c o e f f i c i e n t s among t h e s e v a r i a b l e s were as f o l l o w s : and  Kolb  What r e l a t i o n s h i p s e x i s t among t e a c h e r s • p e r c e p t i o n s o f mathematics and s t u d e n t s ' p e r c e p t i o n s o f mathematics; and between t h e s e p e r c e p t i o n s and s t u d e n t s • achievement i n mathematics?  Teachers' teaching  and  of The  statistically  122 significant,  were  between  the  difficulty  to  teach  rating  and  achievement. Students'  perceptions  of  mathematics  were  more  r e l a t e d than those o f t h e i r t e a c h e r s .  Correlation  among t h e i r  importance  (0.53),  importance  enjoyment have All  r a t i n g s were as  (0.59).  difficulty  These  common p e r c e p t i o n s  data  of  (0.57),  indicate  the  three  and  that  ratings  coefficients and  0.27  student  with  earlier  the  studies  attitude  and  Robinson,  1973;  These  enjoyment three  found  ratings  criterion  similar  achievement  students  Evertson e t . a l . ,  results  mathematics r e l a t e basis of t h i s  indicate  that  significantly  finding,  between  variables.  1980;  tend  to  achievement.  correlated  Wess,  and  f o r mathematics.  A  relationships  (e.g.  enjoyment  difficulty  of t h e i r ratings correlated s i g n i f i c a n t l y with  F o r example, and  and  follows:  strongly  number  between  1970;  0.17 of  student  Phillips,  1973;  and Newman, 1984). students'  to their  perceptions  achievement.  curriculum developers  and  On  teachers  t o c o n s i d e r s t u d e n t s * p e r c e p t i o n s of mathematics as an  of the need  important  f a c t o r i n success i n l e a r n i n g the subject. 5.  What d i f f e r e n c e s , i f any, e x i s t i n t h e s t r e n g t h s o f the r e l a t i o n s h i p s i n q u e s t i o n s 1 t o 4 when achievement i s measured a t d i f f e r e n t c o g n i t i v e behavior l e v e l s ?  Peterson part,  and  researchers  processes  that  Fennema have  (1985)  not  facilitate  investigated students*  l e v e l t a s k s a r e t h e same as t h o s e for  such  Rosenshine in  the  investigation (1979).  present  was  An attempt  study  by  claimed  whether  learning  of  i n high l e v e l  pointed  out  t o address  examining  that,  the  by  for  the  the  classroom  low  cognitive  tasks.  Good  A need  (1983)  t h i s concern was  effects  of  most  and made  independent  123 variables as w e l l  upon s t u d e n t performance on r o u t i n e  as on h i g h e r o r d e r p r o b l e m - s o l v i n g  application  or  items  critical-thinking  questions. As r e p o r t e d e a r l i e r , t h r e e c r i t e r i o n v a r i a b l e s were examined i n t h e c u r r e n t  study.  The v a r i a b l e s were t e s t  problem s o l v i n g and a p p l i c a t i o n s . was  comprised  The p r o b l e m - s o l v i n g  v a r i a b l e , on t h e o t h e r hand, c o n s i s t e d of c o g n i t i v e behavior.  application  of discussion  between  on  show  although  that  significantly  the  two  with  parents* both,  example, c o r r e l a t i o n s problem  student  criterion  v a r i a b l e were h i g h e r than  No  be  ,  achievement  fathers*  level  at t h i s  o n l y r e s u l t s i n v o l v i n g t h e s e two v a r i a b l e s w i l l  Relationships  variables  levels  with  the  were  between achievement on 0.29  and  correlations  0.20  and  teacher  correlate application  were  variables  showed  For  applications  respectively  0.20  and  examination  they were w i t h problem s o l v i n g .  l e v e l o f e d u c a t i o n and 0.25 significant  variables  under  educational  correlations  found  solving  background  with  w i t h mothers'.  found  v a r i a b l e s and achievement i n problem s o l v i n g . 7  The  o f items a t a lower  F o r t h e purposes  compared.  and  variable  o f t e s t items d e s i g n e d t o measure s t u d e n t  achievement a t a h i g h c o g n i t i v e b e h a v i o r l e v e l .  point,  total,  between  teacher  However, 5 o f the  statistically  significant  c o r r e l a t i o n s with the a p p l i c a t i o n v a r i a b l e . A s i m i l a r pattern classroom domains. with  organization For  example,  achievement  on  was  found between t h e r e l a t i o n s h i p s  variables the  type  problem  and of  the  course  solving,  two  achievement  correlated and  at  with  at  0.21  0.15 with  .124 achievement  on  applications.  process v a r i a b l e s with  correlated  applications  exceptions teaching promote  were  of  than  of  problem  correlations  between  solving  solving.  stronger r e l a t i o n s h i p  and  Both  with  the  problem-solving  a t e i t h e r t h e same l e v e l  with  problem  problem  Most  solving. both  classroom of  these  achievement on  or higher  Two  notable  in-service  on  the  features  used  to  variables problem  showed  a  solving  than  mathematics  were  on the a p p l i c a t i o n domain. Although  a l l students  significantly variables, problem  related  they  difficulty however,  to  The  higher  significantly  also  on  on  both  criterion  applications  only teachers' perception of  i n teaching. were  of  achievement  correlated  solving.  which c o r r e l a t e d  perceptions  1  The  higher  than  mathematics  with the c r i t e r i o n v a r i a b l e s magnitudes with  of  the  applications  on  was  correlations, than  problem  solving. Based independent  on  variables  applications, to  this  examination with  i t i s apparent  performance  on  achievement  the on  relationships problem  solving  t h a t t h e y a r e more s t r o n g l y  applications.  statistically significant  of  A  summary  relationships  of  the  and  related  number  i s shown i n T a b l e  of  47.  of  125 Significant  Input  Category  Stud.  Bckgrd.  T a b l e 47 R e l a t i o n s h i p s With Achievement S o l v i n g and A p p l i c a t i o n s  on Problem  Applications Relationships*  Number o f Variables  Problem-Solving Relationships*  5  2  2  7  0  5  C l a s s Org.  11  2  2  P. S o l v . P r o c e s s  11  8  9  3  1  1  3  3  3  40  16  Teach.  Bckgrd.  Teach.  Percept.  Stud. P e r c e p t . Total  Number o f c o r r e l a t i o n s s i g n i f i c a n t a t t h e 0.05 l e v e l .  * Note.  As  shown i n T a b l e  correlated This  22  47, 16 o u t o f a t o t a l  significantly  compared  independent  with  22  with  achievement  significant  on problem  correlations  solving.  between  v a r i a b l e s and achievement on a p p l i c a t i o n s .  relationships,  only  two  correlated  s o l v i n g than w i t h a p p l i c a t i o n s . number o f r e a s o n s . application-type Therefore  o f 40 v a r i a b l e s  more  highly  Of t h e s e  with  problem  These r e s u l t s may be due t o a  F o r example, more time i n s c h o o l i s spent on items  stronger  than  on  relationships  critical could  thinking develop  questions. between  i n p u t s o f s c h o o l i n g and achievement on t h o s e t y p e s o f items. 6.  the  How much v a r i a n c e i n s t u d e n t achievement i n mathematics i s accounted f o r by t h e e f f e c t s o f t e a c h e r and s t u d e n t background, c l a s s r o o m o r g a n i z a t i o n and p r o c e s s e s , and t e a c h e r s ' and s t u d e n t s ' p e r c e p t i o n s o f mathematics.  the  126 As r e p o r t e d the  variables  earlier,  a factor  contained  within  analysis  each  was conducted w i t h  input  category'  Factor  s c o r e s were determined and t h e f a c t o r s w i t h i n each c a t e g o r y were then  regressed,  criterion One  using  t h e s t e p - w i s e method,  student  background  remained  factor,  i n the regression  criterion variables.  I t explained  parents'  of the  percent  socio-economic  on  as  of  5 percent of the variance i n  applications.  status,  levels  e q u a t i o n f o r each o f t h e  s t u d e n t achievement on problem s o l v i n g , 9  each  variables.  education,  and  on  7 p e r c e n t on t e s t  These  reflected  results  by  total  indicate  parents'  levels  that of  e d u c a t i o n , has an e f f e c t on achievement i n mathematics. Teacher student  background  achievement.  measuring  factors  Only  professional  results  i t appears t h a t  amount  teacher  achievement respectively.  explained on  3,  problem An  variance  in  comprised o f v a r i a b l e s  remained  i n the regression  of variance. background,  Based  factors on t h e s e  as d e f i n e d  i n the  e f f e c t on s t u d e n t achievement.  Classroom o r g a n i z a t i o n program,  factor,  little  No o t h e r t e a c h e r background  any a p p r e c i a b l e  c u r r e n t study, has l i t t l e  of  one  preparation,  equation f o r a p p l i c a t i o n s . explained  explained  f a c t o r number 5,  5 and 5 p e r c e n t solving,  additional  1  test  of the variances i n  total  percent  comprised o f type  of  and  applications  the  variance  achievement on t e s t t o t a l was accounted f o r by a second  in  factor,  comprised o f v a r i a b l e s measuring t o t a l time spent on mathematics and time spent on homework-related  activities.  These  findings  p r o v i d e some e v i d e n c e t h a t a d d i t i o n a l time spent i n mathematics  127 and  additional  proportions  of class  r e l a t e d a c t i v i t i e s can a f f e c t s t u d e n t Two p r o b l e m - s o l v i n g  process  three  r e g r e s s i o n equations.  which  measured  performance ease found  teacher  Factor  solving,  2, comprised o f student  and t h e i r  I t accounted  i n achievement  f o r 3,  6  on problem  respectively.  participation  i n in-service activities,  percent  whereas  of the variance a  approaches  of  third and  factor,  sources  applications.  importance explaining some  variables  enjoyment  and  s a t i s f a c t i o n o f and  of  A  and  5  percent  solving,  second  test  factor,  of  Results  explained  teacher  involving  the  uses  materials  accounted  variance  indication  attitudes  of  i n student  f o r the  that  problem  achievement.  teacher  solving, different  test  regressions  toward  same total  show t h e  solving  They a l s o  involvement  to  an a d d i t i o n a l  i n problem  these  and  related  i n achievement  from  of the  total  a d d i t i o n a l amount o f v a r i a n c e on achievement i n both and  homework-  achievement.  applications  1  on  i n t e a c h i n g t h e t o p i c , e x p l a i n e d t h e most v a r i a n c e i n  achievement. variances  spent  f a c t o r s remained i n each o f t h e  expectations  on problem  time  in  in  provide  in-service  a c t i v i t i e s and t h e i r use o f a f l e x i b l e approach i n t e a c h i n g may a f f e c t student  l e a r n i n g o f mathematics.  Teachers'  perceptions  mathematics  explained  achievement  on  respectively. achievement  on  solving,  additional  test  the  difficulty  1, 3 and 2 p e r c e n t  problem An  of  total  1  was  test  and  of  accounted  t h e enjoyment  they  experienced  applications  the variance f o r by  p e r c e p t i o n s o f t h e importance o f mathematics. of  teaching  of the variances i n  total  percent  in  i n teaching  in  teachers'  Their perceptions mathematics d i d  128 not  account  for  any  substantial  amount  of  variance  in  experienced  in  achievement. Perceptions  of  the  enjoyment  l e a r n i n g mathematics accounted achievement percent  on  achievement perceptions  on problem  on of  mathematics. that might  the  These  results  be b e n e f i c i a l .  of  which  category  achievement  on  explained  v a r i a n c e s accounted  they  total  percent by  experienced some  should  in  of  learning  to confirm mathematics  be c o g n i z a n t  e f f o r t s to provide  of the students'  evidence  perceptions  and 4  o f student  students  with  i n mathematics.  remained  each  1  explained  provide  Teachers  on t e s t  additional  positive  and make c o n c e r t e d  Factors  7 percent  was  difficulty  enjoyable experiences  input  An  applications  t h e development  attitudes  f o r 3 percent of the variance i n  solving,  applications.  students  i n r e g r e s s i o n equations  different  criterion  amounts  variable.  f o r i s shown i n T a b l e 48.  of A  f o r each  variances  summary  in  o f the  129 T a b l e 48 V a r i a n c e s i n Achievement Accounted (Percent) Input Category  Prob. S o l v i n g  For  Test Total  Applications  Student  Bckgrd.  5  7  9  Teacher  Bckgrd.  0  0  1  Class Organization  3  5  5  P. S o l v i n g P r o c e s s  4  6  6  Student  Percept.  3  7  4  Teacher  Percept.  1  3  2  The ranged  v a r i a n c e s e x p l a i n e d f o r achievement i n problem  from  none by  s t u d e n t background  solving  t e a c h e r background f a c t o r s t o 5 p e r c e n t factors.  T h i s compared w i t h  a range by  by the  same f a c t o r s r e s p e c t i v e l y o f 0 t o 7 p e r c e n t f o r t e s t t o t a l and to 9 percent f o r a p p l i c a t i o n s . each  category  achievement  on  explained test  total  I t i s apparent  considerably and  on  that factors  more  applications  variance than  on  1  from in  problem  solving. The F i n a l Models One in  o f t h e g o a l s o f t h i s study was  achievement  c a t e g o r y and section.  accounted  for  i n developing  each o f t h e c r i t e r i o n v a r i a b l e s , any  factors  from  factors  within  each  t h e s e r e s u l t s have been d i s c u s s e d i n t h e  However,  account  by  t o examine t h e v a r i a n c e s  a  final  i t was  common v a r i a n c e s which may different  categories.  To  input  preceding  r e g r e s s i o n model f o r  important have been address  t o take shared  this  into among  issue a l l  130 f a c t o r s were r e g r e s s e d , u s i n g t h e s t e p - w i s e t h e t h r e e achievement v a r i a b l e s . weights,  from  the  1985  drawn  weights  from  the  are  final  Provincial  Assessment  standardized models  and  regression  discussed  beta  that  weights  independent  have on The  provides  the  one  variables  measure  earlier,  of  the  the  associated with  in  of  change  that  could  A comparison effects  input  category  (Pedhauzer,  be  beta  standard d e v i a t i o n u n i t . weights  classroom  are  additive,  organization  and  predicted  in  The  conceptual comprised  latter  model  those  weights  problem-solving  category,  (Figure  3)  the  changes  S i n c e the e f f e c t s a t t r i b u t e d t o associated process  originally  introduced  of those v a r i a b l e s contained  in  with  variables  were summed t o produce a b e t a weight f o r t h e c l a s s r o o m category.  1982). standard  dependent v a r i a b l e when t h e v a l u e o f t h e i n p u t c a t e g o r y by one  1987  which a l l  relative  each  dependent achievement v a r i a b l e s  units  beta  coefficients,  v a l u e o f each b e t a weight i n d i c a t e s t h e number o f  deviation  of  5.  f a c t o r s were r e g r e s s e d on the c r i t e r i o n v a r i a b l e s . of  each  R e s u l t s , summarized w i t h  v a l i d a t i o n study a r e shown i n F i g u r e s 4 and Beta  method, on  processes  shown Chapter  in 3,  i n each o f t h e two  the is sub-  categories. The  r e l a t i v e e f f e c t s on s t u d e n t achievement i n mathematics  p r e d i c t e d by v a r i a b l e s a s s o c i a t e d w i t h each i n p u t c a t e g o r y t h e 1985 are  Provincial  shown  for  each  a p p l i c a t i o n domains.  Assessment a r e shown i n F i g u r e 4. of  the  problem  solving,  test  from  Results total  and  131 0.50  0.40 Beta Weights  0.30  0.20  0.10  Student Bckgrd. Legend:  Student Percept.  Classroom Processes  Achievement i n Problem S o l v i n g Achievement i n T e s t T o t a l Achievement i n A p p l i c a t i o n s  F i g u r e 4.  »•  > -t •  •>  E f f e c t s o f s c h o o l i n g on achievement i n problem s o l v i n g , t e s t t o t a l and a p p l i c a t i o n s .  Teacher background  variables  d i d not show any  e f f e c t on s t u d e n t achievement i n t h e f i n a l model. is  also  attributed  limited  Teacher Percept.  change  corresponding  to teachers' perceptions. in  achievement  change  in  them,  on  and  test  no  appreciable  Little  They  effect  predicted  a  total,  given  a  substantial  change  in  achievement on e i t h e r problem s o l v i n g o r a p p l i c a t i o n s . Classroom p r o c e s s e s , w i t h a b e t a weight o f 0.32, twice  the  variables  effect  on  associated  perceptions. weights o f 0.17  The  achievement with  latter  and 0.13  i n problem  student two  background  categories  respectively.  solving  were  and  predicted than d i d students•  assigned  beta  132 Achievement change  on  applications  i n variables  associated with  b e t a weight f o r t h i s c a t e g o r y was 0.26  and  0.17  could  I t was  background  might  achievement  in  affected  classroom  0.38,  f o r s t u d e n t background  respectively.  be  most  by  processes.  The  compared w i t h weights and  students*  of  perceptions  o f i n t e r e s t t o note t h a t change i n s t u d e n t  have  a  considerably  applications  than  on  greater  problem  effect  solving  or  on test  total. T e s t t o t a l r e s u l t s might be a f f e c t e d l e s s t h a n t h o s e on t h e other  two  criterion  variables.  On  would l i k e l y o t h e r two. change  the  variables  by  change  i n classroom  o t h e r hand, change i n s t u d e n t s *  process  perceptions  a f f e c t achievement i n t h i s domain more t h a n The  test  i n teachers•  total  domain was  perceptions  may  t h e o n l y one result  i n an  i n which  a  appreciable  change i n s t u d e n t  achievement.  o n l y 0.10,  be c o n s i d e r a b l y l e s s t h a n a s i m i l a r change i n  i t may  However, w i t h  i n the  a b e t a weight  of  the other input c a t e g o r i e s . 7.  from  What d i f f e r e n c e s o c c u r i n t h e r e s u l t s found through the a n a l y s i s o f c r o s s - s e c t i o n a l d a t a a f t e r l o n g i t u d i n a l data are i n c l u d e d i n the analysis?  Results,  showing t h e  the  validation  sectional  1987  results,  based  relative study, on  effects  of  input categories  a r e shown i n F i g u r e 5.  p o s t - t e s t data  only  are  Crosscompared  w i t h t h o s e from t h e l o n g i t u d i n a l a n a l y s i s i n which p r e - t e s t d a t a were  included.  test  total.  The  only  criterion  variable  reported  upon  is  0.60  0. 50  0.40 Beta Weights  0.30  0.20  0.10 Pretest Legend:  Student Perc.  Student Bckgrnd.  Classroom Processes  Cross-sectional results Longitudinal results  F i g u r e 5.  Based students'  E f f e c t s o f s c h o o l i n g on mathematics achievement based on c r o s s - s e c t i o n a l and l o n g i t u d i n a l d a t a .  on  data  shown  perceptions  i n Figure  show  5,  student  background  considerably stronger  s t u d e n t e n t r y - l e v e l b e h a v i o r s a r e not c o n t r o l l e d are.  When t h e s e b e h a v i o r s were not c o n t r o l l e d ,  f o r s t u d e n t background was  0.26  compared t o 0.16  Students'  the  other  weights The  Teacher Bckgrnd.  p e r c e p t i o n s , on  o f 0.27  w i t h t h e c o n t r o l and 0.42  differences  expected  since  between  the  hand,  they  in are  beginning  student not of  likely a  school  when  t h a n when they the beta  weight  when t h e y were.  were  assigned  beta  variables  was  without.  background to  effects  and  change year  a and  great  deal  the  end.  134 Consequently,  r e s u l t s from t h e p r e - t e s t would have accounted f o r  a substantial portion of t h e i r effect Likewise,  part  mathematics However, based in  of  would  effect be  weight  of  students'  accounted  o f 0.27  achievement.  perceptions  f o r by  caution  can s t i l l  f o r students'  perceptions,  should  Findings,  based  mathematics, classroom The  on  should  These  taken  results  when  student not  interpreting  background  be  indicate  and  attributed  on t h e i r that  survey  some data.  perceptions  solely  to  of  current  practices. relative  were  found  data  yielded  longitudinal of  be  experiences  have an a p p r e c i a b l e e f f e c t  o f mathematics.  of  the pre-test.  on l o n g i t u d i n a l r e s u l t s , suggests t h a t s t u d e n t  perceptions  two  e f f e c t s of classroom processes  t o be s i m i l a r a beta data  data  background. these  also  the beta  a given year  sets  the  on s t u d e n t s '  weight  i n this  were  i n both  also  cases.  o f 0.23  category. found  The  compared Similar  f o r the  on achievement cross-sectional  t o 0.25  f o r the  r e s u l t s with  effects  of  both  teacher  Based on t h e s e f i n d i n g s , s i m i l a r e f f e c t s r e l a t e d t o input  categories  could  be  expected  from  survey  results.  5.3  IMPLICATIONS FOR DECISION MAKERS  F i n d i n g s from  t h e c u r r e n t study p r o v i d e some d i r e c t i o n f o r  d e c i s i o n makers who e i t h e r s e t p o l i c y o r e l s e a r e p r a c t i t i o n e r s i n t h e e d u c a t i o n a l system. to identify  Although  causal relationships,  t h e f i n d i n g s do n o t c l a i m  they a r e based  t h e s t r e n g t h s o f r e l a t i o n s h i p s between v a r i a b l e s .  on measures o f Implications  135 based on t h e f i n d i n g s a r e d i r e c t e d a t t h e M i n i s t r y o f teacher educators Among  the  collection  of  allocation direction  resources, future  the  the  assessments  development  research.  are  the  i n d e c i s i o n making f o r the of  (Learning  curriculum,  Assessment  from c r o s s - s e c t i o n a l s u r v e y s .  1987  interpreting classroom  provincial  and  Branch,  i n f o r m a t i o n c o l l e c t e d , however, i s u s u a l l y l i m i t e d t o  p o s t - t e s t data from  of  information to a s s i s t  for  1984). The  and t e a c h e r s o f mathematics. purposes  of  Education,  validation  study,  results.  processes  on  For  care  example,  student  background  differences and  found  students'  should  be  although  achievement  a n a l y s i n g b o t h c r o s s - s e c t i o n a l and considerable  Based on f i n d i n g s  were  taken  when  effects similar  of  after  l o n g i t u d i n a l d a t a , t h e r e were in  the  effects  perceptions  of  student  of  mathematics  on  students'  perceptions  of  achievement. Relationships mathematics  and  implication  found their  between achievement  for curriculum  in  developers.  the  subject  These  results  t h e importance o f t h e a f f e c t i v e domain as an important achievement. developers  Based  should  on  the  stress  d e s i g n o f c u r r i c u l u m and  findings i n this  the  importance  i n the  of  area,  this  identification  have  an  confirm  factor in curriculum  domain  in  of resources  the to  support i t . A number o f i m p l i c a t i o n s a r e d i r e c t e d t o t e a c h e r and  to  between  teachers. student  flexibility  of  F o r example, achievement approach  and  significant and the  relationships  in-service use  educators  of  some  found  involvement, classroom  136 organizational in-service The  practices  provide  implications  were  based  findings,  on  however,  Chapter 1 and  5.4  the  r e f e r r e d t o i n t h i s s e c t i o n were d i r e c t e d  Willms s i m i l a r to  (in the  selection bias study  l e v e l s o f the  findings are  from  subject  to  THE  and are  r e f e r r e d to i n the  present  system.  study.  assumptions  These  reported  in  follow.  in  one,  listing  limitations  c i t e d problems o f  q u a l i t y of data. also  educational  STUDY  press),  present  the the  l i m i t a t i o n s which  LIMITATIONS OF  current  and  planning.  a t d e c i s i o n makers a t s e v e r a l They  direction for pre-service  subject  to  The  of  studies  aggregated  data contained  some o f  the  data, in  the  r e s t r i c t i o n s he  f o l l o w i n g comment:  . . . e f f e c t i v e n e s s s t u d i e s w i l l c o n t i n u e t o be based on d a t a t h a t a r e l e s s than w h o l l y adequate. N a t i o n a l d a t a on s c h o o l i n g outcomes and p u p i l s ' c h a r a c t e r i s t i c s a r e u s u a l l y d e r i v e d from m u l t i - p u r p o s e s u r v e y s t h a t have a number o f competing r e s e a r c h g o a l s and p r i o r i t i e s . Researchers u s u a l l y need t o make compromises t h a t determine the l e n g t h and c o n t e n t o f t e s t s and q u e s t i o n n a i r e s , the method o f d a t a c o l l e c t i o n , and the sample d e s i g n . Along w i t h t h i s problem, t h e r e i s o f t e n r e s i s t a n c e from p u p i l s , p a r e n t s , t e a c h e r s and a d m i n i s t r a t i o n who view the c o l l e c t i o n as an i n c u r s i o n upon t h e i r r i g h t t o c o n f i d e n t i a l i t y , o r view i t w i t h s u s p i c i o n , not s h a r i n g the g o a l s o f t h o s e c o l l e c t i n g the d a t a , and f e a r i n g t h e d a t a w i l l be used t o h o l d them a c c o u n t a b l e (p. 3 ) . Some e v i d e n c e  of  resistance  the number o f c l a s s e s which had This  was  demonstrated  in a  from t e a c h e r s was  apparent  t o be dropped from the  number  of  ways; two  of  by  analysis.  which  were  137 non-completion o f t h e Teacher Q u e s t i o n n a i r e o r t h e absence o f a class  code number  either  on s t u d e n t  test  q u e s t i o n n a i r e s which were r e t u r n e d . was s u f f i c i e n t  t o exclude  results  of a class  file  in  i n the province  found. found  This  district  was  approximately  the present  study  those  from t h e a n a l y s i s  c o u l d n o t be found.  f o r several other smaller Since  o r on  E i t h e r o f these c o n d i t i o n s  s i n c e a matching t e a c h e r the largest  booklets  half  only  F o r example,  24 matches were  o f t h e number  which  were  re-analysis of  data  districts. involved  a  c o l l e c t e d i n t h e 1985 P r o v i n c i a l Assessment o f Mathematics and a replication  o f i t , f o r purposes o f examining e f f e c t s when p r e -  test  are  data  introduced  into  the  l i m i t a t i o n s were due t o pre-determined design  and  instrumentation  used  Subsequent methods o f a n a l y s i s used f u r t h e r t o these l i m i t a t i o n s .  analysis,  a  number  of  parameters based on t h e  in  the  1985  Assessment.  i n t h e c u r r e n t study  added  A l i s t of limitations follows:  E s t i m a t i o n o f C l a s s Means S i n c e t h e 1985 study used a m u l t i p l e m a t r i x sampling  design  i n which s e v e r a l t e s t b o o k l e t s were used, c l a s s achievement and perception students  means were  wrote  estimates.  a given  achievement  perception s c a l e contained e s t i m a t e was reduced  Approximately item  i n B o o k l e t R.  one  i n three  and responded The s t a n d a r d  somewhat by t h e r e l a t i v e l y  t o the error of  l a r g e number o f  items i n each achievement domain. The  same l i m i t a t i o n a p p l i e d t o c l a s s means f o r achievement  i n t h e v a l i d a t i o n sample. on  results  from  P e r c e p t i o n means, however, were based  a l l students  since a l l booklets  perception scale i n that administration.  contained the  138  D e f i n i t i o n of As  Variables  indicated  earlier,  independent  variables  i n the  1985  Mathematics Assessment were chosen on t h e b a s i s o f what p r e v i o u s r e s e a r c h had t o say about t h e i r e f f e c t s on s t u d e n t achievement. However,  their  operational  definitions  were  based  on  the  m u l t i p l e c h o i c e items used t o measure them i n t h a t s t u d y .  This  l i m i t a t i o n i s common t o o t h e r s i m i l a r s t u d i e s where independent variables  were  defined  in  however, makes comparisons different  attributes  may  a  similar  across have  way.  This  practice,  s t u d i e s more d i f f i c u l t  been  measured  to  since  represent  the  same v a r i a b l e . Impacts o f  Independent V a r i a b l e s on t h e Achievement  of  I n d i v i d u a l Students The differ an  effects  from one  attempt  to  independent v a r i a b l e s  (Luecke & McGinn,  control  some  by  examined  effects  However,  student  performance of  analysis.  this  level  the  level  are  variables  necessary  m a t r i x sampling was  on to  the c l a s s  t o t h e m a t r i x sample  P r o v i n c i a l Assessment the student r a t h e r  In  grouped and  each  group.  utilize  employed  then  this i n the  results.  Analysis  p r e s e n t s t u d y used Due  may  1975).  have  o r some o t h e r c r i t e r i a  independent data  studies  not f e a s i b l e t o r e p o r t s t u d e n t l e v e l  The C l a s s as a U n i t o f The  for  Since multiple  study i t was  achievement  s t u d e n t t o another  students  strategy.  have on  o f Mathematics  than t h e c l a s s  and  plan  t e a c h e r as u n i t s employed  i t was  for this  i n the  of 1985  not f e a s i b l e t o use purpose.  Burstein  139 (1980) I d e n t i f i e d a number o f i s s u e s c r e a t e d i n a g g r e g a t i n g d a t a to  the  class  variance likely  level.  was  had  not  For  example,  accounted  a moderating  in this  f o r and  effect  on  the  study  absence  within of  the magnitude o f  class  this  data  correlation  coefficients. These analyses  limitations  and  the  established  shaped  reporting  parameters  the  of  methods  results.  l i m i t i n g , the  employed  However,  accuracy  of  in  the  they  also  conclusions  drawn i n t h e study.  5.5  IMPLICATIONS FOR  Several for  FURTHER RESEARCH  findings  further  from  research.  relationships  between  the  c u r r e n t study  These  include  classroom  a  process  achievement o f s t u d e n t s o f d i f f e r e n t  have  implications  need  to  examine  variables  and  l e v e l s of a b i l i t y ,  d e t e r m i n a t i o n o f e f f e c t s o f s c h o o l - r e l a t e d v a r i a b l e s on achievement.  In  addition,  further  e f f e c t s o f a number o f independent correlate  significantly  with  r e s e a r c h on  means  criterion  achievement may  for  variables  student  i n the  achievement  how  they  First, affected  were q u e s t i o n s  of  relationships the  student  nature  and  provide  three  direction  environment.  study  were  domains.  t o the c l a s s  class These two  between t e a c h e r s ' b e h a v i o r s  and  interest  t h e s e q u e s t i o n s t h e t e a c h e r and  the  for  achievement  primary  current  on  achievement r e s u l t s were aggregated reasons.  and  v a r i a b l e s which were found t o  t o enhance t h e e f f e c t i v e n e s s o f t h e e d u c a t i o n a l The  the  the  of  their  to this  level  respective classes study.  To  answer  c l a s s were t h e l o g i c a l u n i t s  of  140 analysis. the  1985  Second, t h e m u l t i p l e m a t r i x Provincial  Assessment  booklet.  Consequently  meaningful.  I t would be  sampling  utilized  more  student-level of  interest,  d e s i g n used than  one  test  were  not  examine  the  results  however,  to  e f f e c t s o f t e a c h e r b e h a v i o r s on s t u d e n t s o f d i f f e r e n t ability.  Further  research  could  address  this  in  levels  question  of by  u t i l i z i n g a single t e s t booklet. Willms variables  and  Cuttance  such  s t r a t e g i e s and  as  (1985)  class  reported  size,  school expenditure  school do  not  that, size,  instructional  appear t o have s t r o n g  d i r e c t e f f e c t s on c o g n i t i v e achievement, t h e y may e f f e c t s by the o v e r a l l  facilitating  effective  t e a c h i n g and  f u n c t i o n of t h e s c h o o l " (p. 290).  i n t h i s a r e a which w i l l t h e c l a s s r o o m may  "...although  have  indirect  contributing  to  Further research  control f o r school e f f e c t s  from  outside  provide a d d i t i o n a l i n s i g h t i n t o the e f f e c t s of  c l a s s r o o m p r o c e s s e s on s t u d e n t achievement. 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T e l l how much you l i k e the t o p i c .  I f you a r e not sure what a t o p i c means, l e a v e i t s t h r e e answers blank.  1.  Adding, s u b t r a c t i n g , m u l t i p l y i n g and d i v i d i n g  fractions  A.  B.  not a t a l l important  very d i f f i c u l t  dislike  not  difficult  dislike  undecided  undecided  undecided  important  ea sy  like  very  very  important  important  easy  Adding, s u b t r a c t i n g , m u l t i p l y i n g and d i v i d i n g  a lot  like a l o t  decimals  A.  B.  not a t a l l i m p o r t a n t  very d i f f i c u l t  dislike  not  difficult  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  important  easy  c. a lot  like a l o t  Working w i t h p e r c e n t s A.  B.  not a t a l l important  very d i f f i c u l t  dislike  not  difficult  dislike  undecided  undecided  undecided  important  ea sy  like  very  very  important  important  easy  a lot  like a l o t  156  L e a r n i n g about  estimation  A.  C.  not a t a l l important  very d i f f i c u l t  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  v e r y easy  like a lot  Memorizing b a s i c  dislike  a lot  facts  A.  B.  not a t a l l important  very d i f f i c u l t  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  v e r y easy  like a lot  Solving  C. dislike  a lot  equations A.  B.  C.  not a t a l l important  very d i f f i c u l t  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  v e r y important  v e r y easy  like a lot  dislike  a lot  o  S o l v i n g word problems £L  B^ not a t a l l important  very d i f f i c u l t  dislike  a lot  not i m p o r t a n t  difficult  dislike  undecided  undecided  undecided  important  easy  like  very i m p o r t a n t  v e r y easy  like a lot  157 -  8.  Learning  about  not  at  not  important  the  metric  a l l important  3  system  very  d i f f i c u l t  dislike  a  d i f f i c u l t  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  Working  with  perimeter  and  not  at  a l l important  not  important  easy  like  a  l o t  B^ very  d i f f i c u l t  C_j_  dislike  a  d i f f i c u l t  dislike  undecided  undecided  undecided  important  easy  like  very  very  Doing  important  l o t  area  A;_  10.  -  easy  like  a  lot  l o t  geometry  A. not  at  a l l important  not  important  B. very  d i f f i c u l t  C. dislike  a  d i f f i c u l t  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  easy  like  a  l o t  l o t  158  - 4 -  STUDENT BACKGROUND  1.  2.  What  What  language A.  English  B.  French  C.  Another  language A. B. C.  3.  4.  A.  Regular Program Early  C.  Late French  D.  "Programme-Cadre  help A.  class  French  with  home  now?  i n English  Immersion de  Fran^ais"  i s taught i n  go t o a L e a r n i n g A s s i s t a n c e  Centre  i n your  school  mathematics?  There  i s no L e a r n i n g A s s i s t a n c e this  B.  Yes, I do.  C.  No,  your  i n your  English French  I  Do y o u s o m e t i m e s in  most o f t e n  Immersion  Mathematics  in  6.  speak?  a r e you i n ?  Do y o u s o m e t i m e s for  to  language  B.  In this  learn  language  do you speak  English French Another  What p r o g r a m  A. B.  5.  d i d you f i r s t  INFORMATION  Centre  f o r mathematics  school.  don't.  go  t o an ESL c l a s s  (English  as a Second  school?  A.  There  B. C.  Yes, I do. No, I d o n ' t .  i s no E S L c l a s s  i n this  school.  Language)  159  - 5-  How long d i d i t take you t o do your assignment? A. B. C. D. E.  8.  E.  I d i d n ' t have any homework t o do y e s t e r d a y . L e s s than 30 minutes Between 30 minutes and 1 hour From 1 t o 2 hours More than 2 hours  Very l i t t l e o r no s c h o o l i n g a t a l l Elementary s c h o o l Secondary s c h o o l C o l l e g e , u n i v e r s i t y o r some o t h e r form o f post-secondary t r a i n i n g I don't know.  What was the h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by your mother o r female g u a r d i a n ? A. B. C. D. E.  11.  class.  What was the h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by your f a t h e r o r male g u a r d i a n ? A. B. C. D.  10.  We never have mathematics homework i n t h i s Between 1 and 10 minutes Between 11 and 30 minutes Between 31 and 60 minutes More than an hour  About how much time d i d you spend d o i n g homework i n a l l s u b j e c t s yesterday? A. B. C. D. E.  9.  l a s t mathematics homework  Very l i t t l e o r no s c h o o l i n g a t a l l Elementary s c h o o l Secondary s c h o o l C o l l e g e , u n i v e r s i t y o r some o t h e r form o f post-secondary t r a i n i n g I don't know.  Here i s a l i s t o f reasons f o r s t u d y i n g mathematics. b e l i e v e i s most important? A. B. C. D. E.  Which do you  To p r e p a r e f o r the next y e a r ' s mathematics c o u r s e To l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y To l e a r n how t o use mathematics t o s o l v e problems i n the r e a l world To l e a r n t o t h i n k l o g i c a l l y To l e a r n what mathematics i s  160  - 6 -  12.  Which of these reasons f o r s t u d y i n g mathematics do you b e l i e v e t o be l e a s t important? A. B. C. D. E.  To p r e p a r e f o r the next y e a r ' s mathematics c o u r s e To l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y To l e a r n how to use mathematics t o s o l v e problems i n the r e a l world To l e a r n t o t h i n k l o g i c a l l y To l e a r n what mathematics i s  Both answers g i v e n f o r q u e s t i o n s 13-16 a r e c o r r e c t . I f you were asked each q u e s t i o n , which one o f the two answers comes t o mind f i r s t ?  13.  How  much does a b i c y c l e A. B.  14.  How  How  About 70 degrees About 20 degrees  f a r i s i t from P r i n c e George t o P r i n c e Rupert? A. B.  16.  About 15 k i l o g r a m s About 35 pounds  What i s the temperature i n t h i s room? A. B.  15.  weigh?  About 700 k i l o m e t r e s About 450 m i l e s  much g a s o l i n e can the gas tank i n a l a r g e c a r hold? A. B.  About 20 g a l l o n s About 90 l i t r e s  161  1.  2.  The statement " t h i r t y i s l e s s  than f o r t y - f i v e "  A.  30 >'45  B.  30 < 45  C.  45 < 30  D.  45 ^ 30  E.  I don't know.  Which i s the c o r r e c t name f o r t h e m i s s i n g 3 x 26 = (3 x  3.  4.  [3  A.  2  B.  6  C.  20  D.  26  E.  I d o n ' t know.  J o e packs tomatoes 4 t o a box. box i s he now p a c k i n g ?  Multiply:  A.  the f o u r t h  B.  the f i f t h  C.  the s i x t h  D.  the e i g h t e e n t h  E.  I d o n ' t know.  403 x 59 A.  24 337  B.  5 642  C.  23 777  D.  3 627  E.  I don't know.  i s shown by  number?  + (3 x 6)  I f he has packed 18 tomatoes,  which  KINDS OF LUNCHES STUDENTS EAT  D.  300  E.  I don't know.  Mike f l i p s 2 dimes. l a n d heads?  D. E.  Which s t a t i s t i c  What i s the p r o b a b i l i t y t h a t t h e y " w i l l b o t h  2  3 I don't know.  t e l l s you which event happened the most  A.  mode  B.  mean  C.  median  D.  ra ng e  E.  I don't know.  frequently?  163  Which p a i r o f l i n e segments shown below have l e n g t h s the r a t i o o f 1 t o 4?  PI  1  1  Ql  1  +•  R  H  Sl-  9.  which a r e i n  H  A.  P and Q  B.  R and S  C.  P and S  D.  R and T  E.  I don't know.  1  H  H  I-  The f o l l o w i n g diagram o f a p l a y g r o u n d i s drawn t o a s c a l e o f 1 cm = 2 m. What i s the a c t u a l l e n g t h o f the l o n g e s t s i d e o f t h e playground?  5 cm  10 cm A.  2 m  B.  10 m  C.  20 m  D.  100 m  E.  I don't know.  - 10 -  The  c h a r t shows the p o p u l a t i o n  o f the e a r t h a t d i f f e r e n t  times.  Year  1650  1700  1750  1800  1850  1900  1950  Population in Billions  0.60  0.62  0.80  0.95  1.20  1.70  2.55  Which 50 year p e r i o d  showed the l a r g e s t g a i n  A.  1700-1750  B.  1800-1850  C.  1850-1900  D.  1900-1950  E.  I don't know.  Which one o f the f o l l o w i n g keys would you push t o g e t back the answer t o a c a l c u l a t i o n which you had s t o r e d i n the  E.  I d o n ' t know.  i n population?  165 - 11 -  Which one o f the f o l l o w i n g  shapes i s a  the  cylinder?  A.  inside  figure.  B.  o u t s i d e the  C.  on the boundary o f the  D.  n e i t h e r i n s i d e , o u t s i d e nor on the boundary o f the f i g u r e .  E.  I don't know.  figure. figure.  166  - 12 -  14.  15.  3 Mrs. Smith baked 48 c o o k i e s . B i l l y a t e — o f the c o o k i e s and B e t t y 1 a t e — o f the c o o k i e s . In a l l , how many c o o k i e s were eaten? 8  Subtract:  A.  16  B.  18  C.  20  D.  24  E.  I don't know.  51.2 - 4.35 A.  46.95  B.  46.85  C.  17.7  D.  7.7  E.  16.  Which number  I don't know.  i s largest? A.  f  •>• I E.  I don't know.  167  - 13 17.  18.  M a r b l e s a r e a r r a n g e d i n the shape o f a t r i a n g l e on t h e f l o o r . How many marbles a r e t h e r e i n a t r i a n g l e w i t h 7 marbles i n the base?  A.  12  B.  28  C.  42  D.  49  E.  I don't know.  The s t i c k m a n below i s Mr. B i g . He i s 9 paper c l i p s t a l l o r 6 buttons t a l l . There i s a n o t h e r s t i c k m a n , Mr. S h o r t , who i s 6 paper c l i p s t a l l . How many b u t t o n s t a l l would he be?  C.  4  D.  5  E.  I don't know.  - 14 How many p a i r s o f p a r a l l e l p l a n e s a r e t h e r e i n t h e f o l l o w i n g  A.  2  B.  3  C.  4  D.  6  E.  I don't  know.  In which t r i a n g l e i s a n g l e X an obtuse  250  g i s how many k i l o g r a m s ? A.  25  B.  250  C.  0.25  D.  2.5  E.  I don't  know.  angle?  169  - 15 -  22.  23.  You wish t o c a l c u l a t e 6% o f 85 on t h e calculator. Which one o f t h e f o l l o w i n g sequences o f k e y s t r o k e s w i l l l i k e l y g i v e the c o r r e c t answer?  A. B. c D.  BESSEB ESSE EEEEEE EEEEE  E.  I don't know.  An i m a g i n a r y computer c a n draw p i c t u r e s When g i v e n t h e i n s t r u c t i o n MOVE i t w i l l draw a p i c t u r e  MC- |MRl ; M - | M  E  m al  9!i~  uEEB  on a t e l e v i s i o n s c r e e n .  this  MOVE R MOVE  When g i v e n t h e i n s t r u c t i o n i t w i l l draw t h i s  Which one o f t h e f o l l o w i n g  like  0.  picture  sets  of instructions  A.  R R R R  B.  MOVE R R R  C.  MOVE MOVE MOVE MOVE  D.  MOVE R MOVE R MOVE R MOVE  E.  I don't know.  would draw a square?  - 16 -  An i m a g i n a r y computer w i l l i n p u t two numbers and p r i n t these i n s t r u c t i o n s a r e g i v e n t o i t :  t h e i r sum  TELL A TELL B WRITE A + P  What would  the same computer  print  i f g i v e n these  instructions?  TELL A TELL B WRITE A * B  A.  the sum o f the numbers  B.  the p r o d u c t o f the  C.  the q u o t i e n t o f the  D.  the  E.  I don't know.  d i f f e r e n c e of  numbers numbers  the numbers  How many d i f f e r e n t r o u t e s a r e t h e r e from A t o B? o n l y up and t o t h e r i g h t . B  A* A.  8  B.  10  C.  15  D.  18  E.  I  You may  travel  Keeping the top up, i n how many d i f f e r e n t ways can the cube on l e f t be p l a c e d i n the square h o l e i n the f i g u r e on the r i g h t ?  O Which one o f the f o l l o w i n g diagrams shows f l i p image o f the man shown to the r i g h t ?  the  1 7 2  - 18 -  29.  30.  Of t h e f o l l o w i n g e x p r e s s i o n s , i n c r e a s e d b y 5? A.  5 -  n  B.  n +  5  C.  5 <  n  D.  5 n  E.  I don't  which  If  x  and y  were would  given  represents  a  number  n  know.  "Mike p a i d x d o l l a r s f o r y metres How m u c h d i d o n e m e t r e c o s t ? "  operations  one  numerical  rope.  values,  you use t o f i n d  A.  addition  B.  subtraction  C.  multiplication  D.  division  E.  I don't  know.  of  which  one o f t h e f o l l o w i n g  t h e p r i c e o f one metre  of  rope?  - 19 -  12m  The mat and the f l o o r shown on the r i g h t a t e s i m i l a r shapes. How many mats would be needed t o c o v e r the f l o o r ?  A.  4  B.  6  C.  9  D.  10  E.  I don * t know.  6m  4f0  - 20 Ax  The  (V + •) i s e q u a l t o A.  A x • + V  B.  A x V + •  C.  (A x V) + (A x •)  D.  (A + V) x (A + •)  E.  I don't  know.  s o l u t i o n o f 2n + 8 = 20 i s : A.  12  B.  14  C.  6  D.  10  E.  I don't  know.  I f n = 5, then In + 4 = A.  14  B.  18  C.  20  D.  11  E.  I don't  know.  Which one o f the f o l l o w i n g e x p r e s s i o n s r e p r e s e n t s t w i c e a number l e s s 5? A.  2x + 10  B.  2x - 10  C.  2x - 5  D.  2x + 5  E.  I don't  know.  17 5 - 21 -  37.  Written  Which  as a percent,  —  A.  5%  B.  0.5%  =  C.  20%  D.  50%  E.  I don't  one o f t h e f o l l o w i n g  know.  i s a  quadrilateral?  D.  39.  Which  E.  one o f the f o l l o w i n g  i s a measure  A.  diameter  B.  radius  C.  area  D.  circumference  E.  I don't  know.  I don't  know.  o f distance around  a  circle?  176  - 22 -  40.  41.  42.  43.  Divide:  45 ) 1232 A.  25 remainder 7  B.  27 remainder 17  C.  29 remainder 27  D.  207 remainder 17  E.  I don't know.  3.008 w r i t t e n  i n words i s  A.  three  hundred  B.  three  thousand  C.  t h r e e and e i g h t  hundredths.  D.  three  thousandths.  E.  I don't know.  and e i g h t  Which one o f the f o l l o w i n g  Multiply:  eight.  numbers i s l a r g e s t ?  A.  0.694  B.  0.07  C.  0.76  D.  0.0816  E.  I don't know.  0.01 x 2300 A.  23  B.  230  C.  2 300  D.  23 000  E.  I don't  eight.  know.  177  - 23 -  44.  Which one o f t h e f o l l o w i n g f i g u r e s i s c o n g r u e n t t o the f i g u r e shown t o the r i g h t ?  B.  E.  45.  I don't know.  E s t i m a t e t h e number o f degrees i n a n g l e Y o f t h i s t r i a n g l e .  Y  A.  60°  B.  90°  C.  30°  D.  120°  E.  I don ' t know.  178  46.  A t what time d i d the h i g h e s t temperature  k  47.  Ii Will •  A.  3 am  B.  2 pm  C.  4 pm  D.  midnight  E.  I don't know.  reading occur?  k \ m : A ::  Four s p i n n e r s a r e shown below. Suppose you LOSE the game i f the p o i n t e r l a n d s on 1. Which s p i n n e r would you choose?  - 25 -  48.  I n t h e m e t r i c s y s t e m , what does t h e p r e f i x " c e n t i " mean? A. B.  49.  j—  o f t h e u n i t o f measure o f t h e u n i t o f measure  C.  10 t i m e s t h e u n i t o f measure  D.  100 t i m e s t h e u n i t o f measure  E.  I d o n ' t know.  5 m e t r e s i s t h e same l e n g t h a s : A.  50 c e n t i m e t r e s  B.  500 c e n t i m e t r e s  C.  50 m i l l i m e t r e s  D.  500 m i l l i m e t r e s  E.  I d o n ' t know.  179  180  181  1 -  SCALE  For  each  1.  Men A.  2.  5.  6.  7.  items  better  strongly Disagree  have  Q  >  D  i  g  e  J  A.  Strongly . Disagree  My  father  A.  Strongly _. ^ Disagree  My  mother  . A.  Strongly . ^ •* Disagree  My  father  . A.  2  . _ B.  „ B.  _. Disagree  <  U  „ C.  just  doing  n  c  e  c  doing  Strongly 1  Disagree  d  e  d  D  .  „ Undecided  than  you  feel.  women. A  g  r  e  e  than  ^ D.  E  .  Strongly Agree  boys.  „ Agree  '„ E.  Strongly Agree  » Agree  ,-. E.  Strongly Agree  3  J  girls.  „ j -^ J Undecided  a s much  how  as a  ^ D.  man  2  does. ^ , D. A g r e e  ~ o o C. U n d e c i d e d  3  r. E.  Strongly Agree 3  2  ^ C.  r, J •J J Undecided  „ D.  » Agree  E.  Strongly Agree 3  mathematics.  „ ^. B. D i s a g r e e  i s usually able  him t o  i  than  describes  mathematics.  „ ~B. D i s a g r e e  enjoys  ~ C.  ^. Disagree  „ ~B. D i s a g r e e  enjoys 2  3  c  best  natural a b i l i t y i n mathematics  A woman n e e d s a c a r e e r  ask  which  and engineers  t o know m o r e m a t h e m a t i c s  Strongly . Disagree  3  MATHEMATICS  t h e answer  scientists  more  3  need  choose  G E N D E R AND  3  Strongly Disagree  Boys A.  4.  make  Girls A.  3.  of these  S:  „ „ - , • - , ^ C. U n d e c i d e d  to help  me  with  my  ~ D.  » Agree  mathematics  E.  Strongly _ Agree 3  homework  2  i f I  help. „ ~. B. D i s a g r e e 3  O  C.  M  J  -  J  J  Undecided  n  D.  *  Agree  c  E.  Strongly Agree  182  - 2 My mother i s u s u a l l y a b l e ask her t o h e l p . A.  Strongly Disagree 1  t o h e l p me w i t h my mathematics homework i f I  „ . B. D i s a g r e e  , . C. Undecided  D. Agree 3  My mother t h i n k s t h a t l e a r n i n g mathematics i s i m p o r t a n t  10.  11.  A.  Strongly Disagree  My  f a t h e r t h i n k s t h a t l e a r n i n g mathematics i s important  A.  Strongly Disagree  My  f a t h e r wants me  3  1  Strongly A. _. Disagree 3  12.  1  Strongly . Disagree 2  1  15.  '  o „ J ^ C. Undecided  - „ ^ . C. Undecided  ~ « D. Agree 3  Stronqly * Agree y  f o r me. ~ E.  Strongly Agree  f o r me.  ^ D. Agree  „ E.  Strongly Agree  „ „ D. Agree  „ E.  Strongly Agree  „ » D. Agree  ^ E.  Strongly " Agree  ^ * D. Agree  P E.  Strongly Agree  3  3  J  t o do w e l l i n mathematics.  „ „. B. D i s a g r e e  C. Undecided  3  J  t o do w e l l i n mathematics.  „ B. D i s a g r e e  C. Undecided  1  G i r l s can do b e t t e r than boys i n mathematics. . Strongly A. ^. ^ Disagree  14.  „ „. B. D i s a g r e e  My mother wants me A.  13.  -  _ _. B. D i s a g r e e  E.  ' B. D i s a g r e e 3  „ „ ^ C. Undecided  My mother i s u s u a l l y v e r y i n t e r e s t e d i n h e l p i n g me w i t h mathematics. A.  Stronqly . * Disagree  My  f a t h e r i s u s u a l l y v e r y i n t e r e s t e d i n h e l p i n g me  A  *  y  Strongly r,^ „ ~ Disagree  B. D i s a g r e e 3  B  „ »  Disagree 3  C. Undecided  „ „ * ^ C. Undecided  D. Agree  E.  Strongly . J; Agree  w i t h mathematics.  ^ * D. Agree  „ E.  Strongly * Agree J  183  - 3 -  STUDENT BACKGROUND  1.  2.  3.  4.  5.  What  What  language  d i d you f i r s t  A.  English  B.  French  C.  Another  language A.  English French  C.  Another  What program A.  Regular Program Early  C.  Late French  D.  "Programme-Cadre  class  French  English  B.  French  A.  with  home  now?  i n English  Immersion de  Francais"  i s taught i n  go t o a L e a r n i n g A s s i s t a n c e  Centre  i n your  school  mathematics?  There in  6.  i n your  Immersion  Mathematics  A.  help  most o f t e n  a r e you i n ?  Do y o u s o m e t i m e s for  speak?  language  B.  In this  to  language  do you speak  B.  learn  INFORMATION  i s no L e a r n i n g A s s i s t a n c e this  B.  Yes, I do.  C.  No, I  Do y o u s o m e t i m e s i n your school?  Centre  f o r mathematics  school.  don't.  go t o an ESL c l a s s  A.  There  B. C.  Yes, I do. No, I d o n ' t .  i s no ESL c l a s s  (English  i n this  a s a Second  school.  Language)  -  7.  8.  9.  How l o n g d i d i t t a k e y o u assignment? We  Between  1 and  C.  Between  11  and  30  minutes  D.  Between  31  and  60  minutes  E.  More  t h a n an  mathematics 10  homework  time d i d you  spend  B e t w e e n 30 m i n u t e s From 1 t o 2 hours More than 2 hours  C.  Secondary  D.  College,  E.  post-secondary I d o n ' t know.  of  1  yesterday.  or  college  a t t e n d e d by  your  ata l l  highest  level  or  some o t h e r  form  of  training  of  school  or  college  a t t e n d e d by  your  female guardian? Very  Elementary  l i t t l e  C.  Secondary  D.  College,  or  no  schooling  school university  or  some o t h e r  form  of  training  I don't  know.  l i s t  reasons f o r studying  of  ata l l  school  post-secondary  i s a  i n a l l subjects  school  A.  E.  class.  hour  school  university  B.  believe  and  t o do  V e r y l i t t l e o r no s c h o o l i n g Elementary school  the  i n this  d o i n g homework  C. D. E.  or  homework  minutes  I d i d n ' t have any homework L e s s t h a n 30 m i n u t e s  What was  mathematics  hour  What was t h e h i g h e s t l e v e l f a t h e r or male g u a r d i a n ?  Here  last  A. B.  mother  11.  your  B.  A b o u t how m u c h yesterday?  have  184  -  A.  A. B.  10.  never  t o do  4  mathematics.  Which  i s most important?  A.  To  prepare f o r the  B.  To  learn  how  to perform calculations  C.  To  learn  how  t o use  in  the  real  next year's mathematics mathematics  world  D.  To  learn  to  E.  To  learn  what mathematics  think  logically i s  to  course  accurately  solve  problems  do  you  185  5 -  12.  Which o f t h e s e reasons f o r s t u d y i n g mathematics do you b e l i e v e t o be l e a s t important? A. B. C. D. E.  To p r e p a r e f o r the next y e a r ' s mathematics c o u r s e To l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y To l e a r n how t o use mathematics t o s o l v e problems .in the r e a l world To l e a r n t o t h i n k l o g i c a l l y To l e a r n what mathematics i s  B o t h answers g i v e n f o r q u e s t i o n s 13-16 a r e c o r r e c t . I f you were asked each q u e s t i o n , which one o f the two answers comes t o mind f i r s t ?  13.  How  much does a b i c y c l e A. B.  14.  How  How  About 70 d e g r e e s About 20 d e g r e e s  f a r i s i t from P r i n c e George to P r i n c e Rupert? A. B.  16.  About 15 k i l o g r a m s About 35 pounds  What i s the temperature i n t h i s room? A. B.  15.  weigh?  About 700 About 450  kilometres miles  much g a s o l i n e c a n the gas tank i n a l a r g e c a r hold? A. B.  About 20 g a l l o n s About 90 l i t r e s  - 6ACHIEVEMENT SURVEY  1.  2.  3.  4.  By r o u n d i n g o f f t o t h e n e a r e s t t e n , an e s t i m a t e o f 91 x 29 would be A.  270  B.  279  C.  2 700  D.  27 000  E.  I don't know.  I f John had 2300 marbles, how many bags o f 10 marbles each c o u l d he make? A.  23  B.  230  C.  2 300  D.  23 000  E.  I don't know.  Sue has 58£. I f a p p l e s c o s t 11£ each, what i s the g r e a t e s t number of whole a p p l e s t h a t Sue c a n buy? A.  4  B.  5  C.  6  D.  47  E.  I don't know.  As o f June 1, 1976, the p o p u l a t i o n o f Canada was 22 589 416. Round o f f 22 589 416 t o the n e a r e s t t e n thousand. A.  22 580 000  B.  23 000 000  C.  22 600 000  D.  22 590 000  E.  I don't know.  186  187  - 7 -  The heavy l i n e shows one edge o f cube have?  A.  6  B.  5  C.  9  D.  12  E.  I d o n ' t know.  the c u b e .  How many edges does  the  188 - 8 -  7.  What i s t h e n u m e r i c a l v a l u e o f t h e computer language e x p r e s s i o n shown i n the box below?  2 + 3 - 2  8-  A.  0  B.  2  C.  12  D.  18  E.  I don't  know.  An i m a g i n a r y computer has commands c a l l e d PRAX and ADDO PRAX does t h e f o l l o w i n g : P r i n t o u t the v a l u e o f X Do t h e command c a l l e d ADDO ADDO does t h e f o l l o w i n g : Add 2 t o the c u r r e n t v a l u e o f X Do t h e command c a l l e d PRAX  What w i l l  the following  s e t o f i n s t r u c t i o n s do? X = 2 PRAX  A.  p r i n t o u t t h e odd numbers g r e a t e r than 0  B.  p r i n t o u t a l l i n t e g e r s g r e a t e r than 2  C.  p r i n t o u t t h e even numbers g r e a t e r t h a n 0  D.  p r i n t o u t a l l i n t e g e r s l e s s than 2  E.  I don't know.  -  189  9 L  9.  Which one o f t h e f o l l o w i n g appears  lines  p e r p e n d i c u l a r t o LM?  M  B.  D.  10.  *  E.  Which one o f t h e f o l l o w i n g i s INCORRECT?  statements  about  I don't  the diagram  A.  Plane  X  i s parallel  B.  Plane  X  i s oblique t o plane  C.  Plane  W  i s parallel  D.  Plane  Z i s perpendicular t o plane  E.  I don't  know.  know.  to plane  to plane  Z. Y. Y. Y.  shown  below  190  - 10 -  11.  About how much w i l l  t h i s grocery  bill  total?  •ocono• groceries  # # # # #  12.  13.  $0.43 $1.67 $0.17 $0.93 $2.89  A.  between $3 and $4  B.  between $6 and $7  C.  between $9 and $10  D.  between $12 and $15  E.  I don't know.  John had 12 b a s e b a l l c a r d s . does John have l e f t ?  Divide:  He gave  A.  4  B.  6  C.  8  D.  9  E.  I don't know.  .12 ).036 A.  3  B.  0.003  C.  0.3  D.  0.03  E.  I d o n ' t know.  o f them t o J i m .  How many  - 11 -  Which one o f t h e f o l l o w i n g s t a t e m e n t s i s t r u e ? A.  100° C i s the b o i l i n g  p o i n t o f water.  B.  212° C i s the b o i l i n g  p o i n t o f water.  C.  32° C i s the f r e e z i n g p o i n t o f water'.  D.  10° C i s the f r e e z i n g p o i n t o f water.  E.  A ten-year-old  I don't know.  boy i s l i k e l y  t o weigh:  A.  35 grams  B.  75 grams  C.  35 k i l o g r a m s  D.  75 k i l o g r a m s  E.  I don't know.  The a r e a o f each o f the 6 SMALL s q u a r e s shown below i s 4.  What i s the p e r i m e t e r o f the LARGE r e c t a n g l e ?  A.  10  B.  12  C.  20  D.  24  E.  I don't know.  192 12  17.  -  I f on the r o l l o f a d i e t h e p r o b a b i l i t y j, then the p r o b a b i l i t y  t h a t a f i v e w i l l appear i s  t h a t a f i v e o_r a t h r e e w i l l appear i s :  A.  B.  _1_ 36 1 3  D. E.  12  I d o n ' t know.  2, 3, 4, 4, 5, 6, 8, 8, 9, 10 18.  For a p a r t y game each number shown above was p a i n t e d on a d i f f e r e n t Ping Pong b a l l , and the b a l l s were t h o r o u g h l y mixed up i n a bowl. I f a b a l l i s p i c k e d from t h e bowl by a b l i n d f o l d e d p e r s o n , what i s the p r o b a b i l i t y t h a t t h e b a l l w i l l have a 4 on i t ?  A.  1 2 1 4  C.  1 5  D.  1 10  E.  I d o n ' t know.  193 - 13 -  19.  What i s 24% o f $150.00?  100 B.  $ 24.00  C.  $ 36.00  D.  $174.00  E.  I don't know.  I don't know.  21.  L i n e s t h a t a r e i n the same p l a n e and do not i n t e r s e c t a r e c a l l e d A.  parallel  lines.  B.  perpendicular  C.  skew  D.  oblique  E.  I don't know.  lines. lines.  lines.  - 14 -  A.  30  B.  40  C.  240  D.  19  E.  23.  24.  194  I don't know.  Which one o f the f o l l o w i n g i s the same a s "18 more than a number e q u a l s . 44"? A.  18n = 44  B.  i f -  C.  n + 18 = 44  0.  n = 18 + 44  E.  I don't  "How f a r w i l l J a n walk 10 m i n u t e s ? "  44  know.  i f she walks a t the r a t e o f 1 k i l o m e t r e i n  What a d d i t i o n a l i n f o r m a t i o n i s needed  to solve t h i s  A.  where she was going  B.  how f a s t  C.  how long she walked  D.  how much she was c a r r y i n g  E.  I don't know.  she was w a l k i n g  problem?  195  - 15 -  25.  Simplify;  A.  0  B.  26.  Subtract:  infinity  C.  6  D.  cannot be done  E.  I don't know.  7 - -7 6  7  t  C.  - 4 E.  27.  ESTIMATE the sum:  I don't know.  347.0 + 738.0 + 1.327  A.  1 000  B.  2 000  C.  10 000  D.  100 000  E.  I don't know.  196 - 16 -  28.  Which one o f the f o l l o w i n g keys would you push t o s t o r e the answer t o a c a l c u l a t i o n so you c o u l d use i t l a t e r on? [Cj|cEJ  ! »»/o»  ! M C ! |MRI [ M - j JM^J  EBEB EEEB  M+ MR  EBEB BBSS  CE  + I don't know.  29.  The f o l l o w i n g sequence o f k e y s t r o k e s i s c a r r i e d o u t on the c a l c u l a t o r :  n  u. ; iCE.  I C  What w i l l l i k e l y be shown i n the d i s p l a y ?  : M C |MR; :  iM-  jx ; [v~"j jj=j I :  6.  69.  76.  636.  E.  I don't know.  EEEE BEEB EEEB fT  1  +/_! ! =  197  - 17 30.  31.  32.  To s o l v e f o r b i n the e q u a t i o n 2b + 3 = 10, the f i r s t s t e p s h o u l d be t o A.  d i v i d e both s i d e s by 3.  B.  m u l t i p l y both s i d e s by 2.  C.  subtract  D.  add 3 t o both s i d e s .  E.  I don't know.  Which one o f the f o l l o w i n g s t a t e m e n t s i s NOT  I f y = 15 -  A.  2a + a = 3a  B.  3a - a = 2a  C.  a + a = a  D.  2a - a = 2  E.  I don't know.  true?  x  what happens t o y a s x  53.  3 from both s i d e s .  increases?  A.  y decreases  B.  y  increases  C.  y  remains the same  D.  c a n n o t t e l l what happens t o y  E.  I d o n ' t know.  If x and y a r e odd numbers, what i s t r u e about x + y? A.  I t i s odd.  B.  I t i s even.  C. .  I t may be e i t h e r even o r odd  D.  depending on what x and y a r e . You cannot t e l l a t a l l .  E.  I don't know.  198 - IS -  34.  What i s the minimum number o f t i l e s they a r e a l l f a c i n g the same way?  t h a t must be t u r n e d so t h a t  •*  •  •i •  •  35.  A.  4  B.  5  C.  6  D.  11  E.  I don't know.  •  •  Three t e n n i s p l a y e r s named P a t , Wendy and L e s l i e a r e w a l k i n g t o the courts. P a t , the b e s t p l a y e r o f the t h r e e , always t e l l s the t r u t h . Wendy sometimes t e l l s the t r u t h , w h i l e L e s l i e never t e l l s the t r u t h . Who i s Pat?  A  B  A.  A  B.  B  C.  C  D.  either A or B  E.  . I don't  know.  199  36.  In which one o f the f o l l o w i n g t r a n s l a t i o n o f the f i r s t ?  D.  37.  diagrams  E.  i  In which one o f the f o l l o w i n g r o t a t i o n o f the f i r s t ?  A.  diagrams  I don't know.  i s the second f i g u r e a  B.  D.  38.  i s the second f i g u r e a  C.  E.  I don't know.  Which u n i t s h o u l d be used t o measure how much l i q u i d A.  kilolitre  B.  millimetre  C.  metre  D.  millilitre  E.  I don't know.  a glass  holds?  200  20  39.  P and Q a r e t h e c e n t r e s o f the 2 squares shown below. d i s t a n c e i n c e n t i m e t r e s from P t o Q?  What i s the  2 cm  2 A.  1  B.  2  cm  2cm  C.  40.  D.  V2  + 2  E.  I.don't know.  The diagram shown below i l l u s t r a t e s a ruler-and-compass method o f c o p y i n g an a n g l e . I f the c o n s t r u c t i o n l i n e s a r e l a b e l l e d as shown i n the f i g u r e , what o r d e r c o u l d be f o l l o w e d when c o m p l e t i n g the construction?  A.  1, 5, 2, 3 and 4  B.  1, 5, 3, 4 and 2  C.  1, 5, 3, 2 and 4  D.  1, 5, 4, 3 and 2  E.  I don't know.  - 21 -  41.  An incomplete  figure  is  shown to the  right.  Which one of the f o l l o w i n g shows the completed f i g u r e , g i v e n t h a t m i s a l i n e of symmetry?  m  o o  42.  E.  The f i g u r e to the r i g h t shows a cube w i t h one c o r n e r c u t o f f and shaded. Which one o f the f o l l o w i n g d r a w i n g s shows how the cube would look when viewed d i r e c t l y from above?  I don'  202  43.  The graph below shows what a boy d i d d u r i n g a p e r i o d o f 24 h o u r s .  Which i s the BEST e s t i m a t e f o r the number o f hours he spent watching TV?  44.  A.  3  B.  6  C.  9  0.  12  E.  I don't know.  Pat was t e s t i n g h i s model p l a n e . H i s f r i e n d s guessed how long i t would s t a y i n the a i r . The plane s t a y e d up f o r 17 minutes. Who guessed c l o s e s t t o the c o r r e c t time? F. •• •'•  Bob  Carol  yiiii Ijill  [ 1  |  14  16  |Jv:o>:-i:v:v:v.J:->:-:-.v:  ;  Steven  < ,-  i  ;  i  , it  2  4  ,,  ..I  m  6  m 8  10  12  Time (In Minutes)  A.  Susan  B.  Bob  C.  Carol  D.  Steven  E.  I don't know.  18  20  22  In f o u r months, the v o l l e y b a l l team spent t h e f o l l o w i n g amounts t r a v e l l i n g t o games: 1st 2nd 3rd 4th  month month month month  -  $17.9 5 $22.40 $ 8.25 $15.80  What was the average amount spent on t r a v e l l i n g  A.  $10.10  B.  $64.40  C.  $32.20  D.  $16.10  E.  I don't know.  The median t e s t mark was 37 o u t o f 50. B i l l y How many c h i l d r e n s c o r e d h i g h e r than B i l l y ? A.  more than h a l f  B.  l e s s than h a l f  C.  \ exactly  each month?  s c o r e d 30 o u t o f 50.  half  D.  none  E.  I don't know.  I am t h i n k i n g o f two numbers. When you add them you g e t 36. When you s u b t r a c t them you g e t 8. To f i n d both numbers the most u s e f u l p r o b l e m - s o l v i n g t e c h n i q u e would be t o A.  guess and check.  B.  draw a p i c t u r e o r diagram.  C.  solve a simpler  D.  work  E.  I don't know.  backwards.  problem.  204 24 -  48.  There a r e 13 boys and 15 g i r l s group i s boys?  *  28  B -  • 15  C.  M 13  A  B  i n a group.  What f r a c t i o n o f t h e  ±1  13 28 I don't know.  49.  50.  How many s q u a r e s must be shaded t o show 35% o f the s t r i p ?  A.  0.035  B.  0.35  C.  3.5  D.  35  E.  I don't know.  If 4 volleyballs  c o s t $96.00, how much w i l l  A.  $960.00  B.  $240.00  C.  $ 24.00  D.  $384.00  E.  I don't know.  10 v o l l e y b a l l s  cost?  205  206 -  SCALE T;  1.  i n school?  Never R a r e l y (about once a week) Sometimes (a c o u p l e o f times a week) F r e q u e n t l y (almost e v e r y day)  Choose t h e one answer which b e s t d e s c r i b e s how your c l a s s used c a l c u l a t o r s on mathematics t e s t s t h i s y e a r . A. B. C. D.  5.  Never R a r e l y (about once a week) Sometimes (a c o u p l e o f times a week) F r e q u e n t l y (almost every day)  How o f t e n do you use a c a l c u l a t o r A. B. C. D.  4.  Yes No  How o f t e n do you use a c a l c u l a t o r o u t s i d e s c h o o l ? , A. B. C. D.  3.  CALCULATORS AND COMPUTERS  Do you own a c a l c u l a t o r ? A. B.  2.  1 -  Not a t a l l ; we weren't a l l o w e d t o use them on t e s t s . We were a l l o w e d t o use them on some t e s t s , i f we wanted t o . We were a l l o w e d t o use them a l l t e s t s , i f we wanted t o . We were r e q u i r e d t o use them on some t e s t s .  I n what ways do you use a c a l c u l a t o r (Mark a l l t h a t a p p l y . ) A. B. C. D.  t o do mathematics i n t h i s  Not a t a l l ; we're n o t a l l o w e d t o use c a l c u l a t o r s To c a l c u l a t e answer t o problems To check answers F o r games and f u n  class?  i n this class.  207 - 2 -  6.  Some people say t h a t Grade 7 s t u d e n t s s h o u l d NOT be a l l o w e d t o calculators in school. How do you f e e l about t h i s ? A. B. C. D. E.  use  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Some people say t h a t i f s t u d e n t s a r e a l l o w e d t o use c a l c u l a t o r s , then i t s h o u l d NOT be n e c e s s a r y f o r them t o l e a r n how t o a d d , s u b t r a c t , m u l t i p l y , o r d i v i d e by hand. How do you f e e l about t h i s ? A. B. C. D. E.  8.  Do you have a computer home? A. B. C.  9.  Yes No I don't  (one t h a t w i l l do more than p l a y games)  know.  What have you used a computer t o do? A. B. C. D. E.  10.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  N o t h i n g , I ' v e never used a computer. To p l a y games To w r i t e s t o r i e s o r l e t t e r s To w r i t e my own programs To l e a r n about mathematics o r o t h e r s u b j e c t s  Where d i d you g e t A. B. C.  (Mark a l l t h a t a p p l y . )  most o f your e x p e r i e n c e w i t h computers?  I h a v e n ' t had any e x p e r i e n c e w i t h c o m p u t e r s . A t home In c o u r s e s taken a t s c h o o l or elsewhere  at  - 3 -  FOR ITEMS 11-20, CHOOSE THE ANSWER THAT BEST DESCRIBES YOUR OPINION.  11.  I would A. B. C. D. E.  12.  I feel A. B. C. D. E.  13.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  h e l p l e s s around computers. Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Computers c a n be used t o t e a c h mathematics. A. B. C. D. E.  15.  t o l e a r n more about computers.  E v e r y s t u d e n t s h o u l d be t a u g h t , i n s c h o o l , how t o use a computer. A. B. C. D. E.  14.  like  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Computers c a n be used t o t e a c h s u b j e c t s o t h e r t h a n mathematics. A. B. C. D. E.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  208  209 -  16.  computers.  computers.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Computers a r e g a i n i n g too much c o n t r o l over p e o p l e ' s A. B. C. D. E.  20.  being a b l e t o use  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  I enjoy u s i n g A. B. C. D. E.  19.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  I f e e l c o n f i d e n t about A. B. C. D. E.  18.  -  Using computers i s more s u i t a b l e f o r boys than f o r g i r l s . A. B. C. D. E.  17.  4  lives.  S t r o n g l y Disagree Disagree Undecided Agree S t r o n g l y Agree  I am a b l e t o work w i t h computers as w e l l as most o t h e r s my A. B. C. D. E.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  age.  210  - 5 STUDENT BACKGROUND  1.  2.  3.  4.  What  What  language A.  English  B.  French  C.  Another  language  English French  C.  Another  What program  Regular Program  C.  Late French  D.  "Programme-Cadre de  class  French  with  i n English  Immersion Francais"  i s taught i n  go t o a L e a r n i n g A s s i s t a n c e  Centre  i n your  school  mathematics?  There  i s no L e a r n i n g A s s i s t a n c e this  B.  Yes, I do.  C.  No,  I  Do y o u s o m e t i m e s your  now?  English French  Do y o u s o m e t i m e s  in  home  Immersion  Mathematics  in  6.  i n your  are-you i n ?  Early  A.  speak?  language  A.  help  to  most o f t e n  B.  In this  for  learn  language  do you speak  A. B.  A. B.  5.  d i d you f i r s t  INFORMATION  Centre  don't.  go t o an ESL c l a s s  (English  a s a Second  school?  A. B.  T h e r e i s no E S L c l a s s Yes, I do.  C.  No,  I  f o r mathematics  school.  don't.  i n this  school.  Language)  211  - 6 7.  How l o n g d i d i t take you t o do your assignment? A. B. C. D. E.  8.  E.  I d i d n ' t have any homework t o do y e s t e r d a y . L e s s than 30 minutes Between 30 minutes and 1 hour From 1 t o 2 hours More than 2 hours  Very l i t t l e o r no s c h o o l i n g a t a l l Elementary s c h o o l Secondary s c h o o l C o l l e g e , u n i v e r s i t y o r some o t h e r form o f post-secondary t r a i n i n g I don't know.  What was the h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by your mother o r female g u a r d i a n ? A. B. C. D. E.  11.  class.  What was the h i g h e s t l e v e l o f s c h o o l o r c o l l e g e a t t e n d e d by your f a t h e r o r male g u a r d i a n ? A. B. C. D.  10.  We never have mathematics homework i n t h i s Between 1 and 10 minutes Between 11 and 30 minutes Between 31 and 60 minutes More than an hour  About how much time d i d you spend d o i n g homework i n a l l s u b j e c t s yesterday? A. B. C. D. E.  9.  l a s t mathematics homework  Very l i t t l e o r no s c h o o l i n g a t a l l Elementary s c h o o l Secondary s c h o o l C o l l e g e , u n i v e r s i t y o r some o t h e r form o f post-secondary t r a i n i n g I don't know.  Here i s a l i s t o f r e a s o n s f o r s t u d y i n g mathematics. b e l i e v e i s most i m p o r t a n t ? A. B. C. D. E.  Which do you  To p r e p a r e f o r the next y e a r ' s mathematics c o u r s e To l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y To l e a r n how t o use mathematics t o s o l v e problems i n t h e r e a l world To l e a r n t o t h i n k l o g i c a l l y To l e a r n what mathematics i s  212  -  12.  7  -  Which o f these reasons f o r s t u d y i n g mathematics do you b e l i e v e t o be l e a s t important? A. B. C. D. E.  To p r e p a r e f o r the n e x t y e a r ' s mathematics c o u r s e To l e a r n how t o p e r f o r m c a l c u l a t i o n s a c c u r a t e l y To l e a r n how t o use mathematics t o s o l v e problems i n the r e a l world To l e a r n t o t h i n k l o g i c a l l y To l e a r n what mathematics i s  B o t h answers g i v e n f o r q u e s t i o n s 13-16 a r e c o r r e c t . I f you were asked each q u e s t i o n , which one o f the two answers comes t o mind f i r s t ?  13.  How  much does a b i c y c l e A. B.  14.  How  How  room?  About 70 d e g r e e s About 20 d e g r e e s  f a r i s i t from P r i n c e George t o P r i n c e Rupert? A. B.  16.  About 15 k i l o g r a m s About 35 pounds  What i s the temperature i n t h i s A. B.  15.  weigh?  About 700 k i l o m e t r e s About 450 m i l e s  much g a s o l i n e can the gas tank i n a l a r g e c a r h o l d ? A. B.  About 20 g a l l o n s About 90 l i t r e s  ACHIEVEMENT  The  The  value  value  o f 572 +  005 +  A.  18  650  B.  96  410  C.  148  205  D.  186  410  E.  I don't  of 3 + 4 ( 5 + 2 )  Subtract:  A.  25  B.  26  C.  31  D.  49  E.  I don't  73 i s  know.  i s  "  know.  2008 -  The  18  SURVEY  greatest  189  A.  819  B.  1181  C.  1819  D.  2181  E.  I  don't  common f a c t o r A.  2  B.  6  C.  120  D.  60  E.  I don't  know.  o f 24 a n d  know.  30  -  5.  When  the input  i s x the output i s :  Input 3  A.  19  B.  2x -  1  C  2x +  1  D.  x  E.  I don't  Output 7  4  9  5  11  6 7  13 15 17  8 x  6.  214  9 -  know.  F o r how many m o n t h s w a s t h e r a i n f a l l  more  than  5  cm?  25  E 20  u  \  ^ 15_J  2 10' £  5  Jan  Feb  Mar  Apr  A.  3  B.  4  C.  6  D.  9  E.  I don't  May  know.  Jun Jul MONTHS  Aug  Sep  Oct  Nov  Dec  - 10 Test Which  marks:  3 , 4,  5 , 4,  one o f t h e f o l l o w i n g  mark  cuts,  shown  t h e most  I don't  a cake  To f i n d  frequency 1 2 1 3 3 6  fora  B.  solve a simpler  C.  guess  D.  work  E.  I don't  pieces  backwards. know.  0 1 2 3 4 5  there  technique  pattern.  and check.  0 1 2 3 4 5  i s c u t s o t h a t a l l c u t s a r e made  problem-solving  look  frequency  know.  how many  A.  5  data?  0 1 2 3 4 5  D.  below,  useful  this  mark  1 1 0 3 4 5  the center.  5 , 0, 4,  mark  B.  E.  the diagram  represents  frequency  5  In  tables  1 1 0 3 4 6  0 1 2 3 4  through  5 , 5 , 3 , 3 , 1 , 4,  frequency  0 1 2 3 4 5  mark  5,  problem.  will  would  be a f t e r  be t o  10  216  - 11 -  9.  Add:  \ + \  A.  2 5  1 5  1 6 _5 6  E.  10.  11.  I d o n ' t know.  I f t h e r e a r e 300 c a l o r i e s i n 900 g o f a c e r t a i n f o o d , how many c a l o r i e s a r e t h e r e i n a 300 g p o r t i o n o f t h a t same food? A.  27  B.  33  C.  100  D.  270  E.  I d o n ' t know.  Which one o f the f o l l o w i n g shows a d i s c o u n t A.  30jz! o f f $3  B.  35jzf o f f $3  C.  40jzf o f f $3  D.  45s* o f f $3  E.  I d o n ' t know.  o f 10%?  217  - 12 -  13.  A.  AD and DC  B.  CA and DB  C.  CB and AD  D.  AE and EB  E.  I don't know.  Which one o f t h e f o l l o w i n g  A.  i s NOT a p a r a l l e l o g r a m ?  B.  E.  14.  I f two l i n e  I don't know.  segments a r e e q u a l i n l e n g t h , they a r e A.  horizontal.  B.  congruent.  C.  parallel.  D.  perpendicular.  E.  I don't know.  -  15.  16.  13  I f 3 c a k e s a r e each c u t i n t o t h i r d s , A.  1  B.  3  C.  6  D.  9  E.  I don't know.  218  -  how many p i e c e s a r e t h e r e ?  The w i d t h o f t h i s r e c t a n g l e i s how much l e s s 18.3 cm  13.6 cm  17.  A.  4.3 cm  B.  4.7 cm  C.  5.3 cm  D.  5.7 cm  E.  I don't know.  W r i t t e n as a decimal,  o  =  A.  0.12  B.  0.8  C.  0.125  D.  0.18  E.  I don't know.  than the l e n g t h ?  219 -  18.  14  -  o Which one o f the f o l l o w i n g diagrams shows t h e s l i d e image o f the man shown t o the r i g h t ?  9  o  O B.  c.  o E.  19.  I d o n ' t know.  In the diagrams shown below, how many p i e c e s t h e same s i z e a s A a r e needed t o c o v e r B?  A.  2  B.  3  C.  4  D.  6  E.  I d o n ' t know.  220 15  20.  21.  -  I am a number between 25 and 40. I have a remainder o f two when d i v i d e d by both 6 and by 9. Who am I ? A.  26  B.  29  C.  32  D.  38  E.  I  Think o f a c a l c u l a t o r a s a p e r s o n . A p e r s o n ' s e a r s and eyes a r e l i k e t h e calculator's A.  chip.  B.  keys.  C.  battery.  D.  memory.  E.  I don't know.  CE IMC  MR  M-  M+  •as mass QDE10B  22.  You w i s h t o c a l c u l a t e the square r o o t o f 64 on t h e c a l c u l a t o r . Which one o f t h e f o l l o w i n g sequences o f k e y s t r o k e s w i l l l i k e l y g i v e t h e c o r r e c t answer? A. B. C. D. E.  EBB SHE HEEHHE SEE I d o n ' t know.  C  ICE  JMC  MR  M-|[M+]  0000 SCHEIE  msmB QDESB  221  - 16  23.  The t a b l e shows t h e numbers o f v a r i o u s c o i n s found i n a box,  Coin  Number found  $1 (silver dollar) 50jzf ( f i f t y - c e n t p i e c e ) 250 ( q u a r t e r ) lOjzf (dime) 50 ( n i c k e l ) 10 (penny)  Which one o f  OS LL1  f o l l o w i n g graphs shows  :  1  i  It  Si  1  this?  tr  1  co z  the  2 6 1 3 8 3  •±i  CD  1  l  a  10t 25t 50t S1.  Z  2  B. It  5i 10t 25C 50<t 51.  COIN  COIN  8T 6 UJ  UJ  m  a 4  5 3 Z  5 Z  2 D.  It  5C  It  104 25$ 50$ S1. COIN  5$  104 25$ 50$ S1. COIN  E.  I don't  know.  222 - 17 -  24.  25.  I f the measure o f the s i d e o f each square i s d u n i t s , how the r e c t a n g l e ?  4 + d  units  B.  8 + d  units  C.  4 x d  units  D.  8 x  units  E.  I don't know.  In the formula z~ if  26.  A.  long i s  = R,  I = 250, P = 1000, and T = 2, then R i s  A.  1 8  B.  1 2  C.  1  D.  50  E.  I don't know.  Tom has y m a r b l e s and Mary has x m a r b l e s . Mary has more marbles than Tom. Which s e n t e n c e shows t h i s r e l a t i o n ? A.  x = y  B.  x < y  C.  x > y  D.  x > 2y  E.  I don't know.  223 - 18 -  Which one o f t h e f o l l o w i n g  Which one o f t h e f o l l o w i n g b i s e c t i n g an angle?  29.  i s most l i k e a r i g h t t r i a n g l e ?  illustrates  t h e c o r r e c t procedure f o r  What i s t h e d i a m e t e r o f a c i r c l e w i t h a r a d i u s o f 4? A.  8  B.  6  C.  4  D.  2  E.  I don't  know.  224  - 19 -  Sparky Spencer spun a s p i n n e r 100 times and made a r e c o r d o f results.  Outcome  A  Number of times  5 5  Which s p i n n e r d i d he most l i k e l y  B  C  30  15  use?  I f the p r o b a b i l i t y t h a t i t w i l l r a i n on a g i v e n day i s the p r o b a b i l i t y t h a t i t w i l l NOT r a i n i s : A.  0.36  B.  0.64  C.  99.64  D.  99.36  E.  I don't  6 years  B.  9 yea r s  C.  7 years  D.  5 years  E.  I don't  0.36,  then  know.  The average age of 4 c h i l d r e n i s 6 y e a r s . c h i l d r e n a r e 4 y e a r s , 8 y e a r s and 3 y e a r s , fourth child? A.  his  know.  I f the ages o f 3 of the what i s the age of the  225  - 20 -  A list  34.  o f i n s t r u c t i o n s f o r a computer A.  program.  B.  disk.  C.  terminal.  D.  memory.  E.  I don't know.  A set of i n s t r u c t i o n s  f o r an i m a g i n a r y  i s called a  computer  i s as f o l l o w s :  1.  A r r a n g e the t h r e e names Sandy, Dale, and Pat i n a l p h a b e t i c a l order  2.  Remove the l a s t  3.  I f o n l y one name i s to s t e p 4  4.  P r i n t o u t the names i n r e v e r s e  5.  Go back t o s t e p 2  What w i l l  the computer  name from the l i s t left,  print?  A.  Pat  B.  Dale, P a t  C.  Dale, P a t , Sandy  D.  Pat,  E.  I don't know.  Dale  stop, otherwise  order  go on  - 21 -  Which u n i t would  The  The  usually  be used f o r t h e mass o f sugar?  A.  kL  B.  km  C.  kg  D.  km  E.  I don't know.  2  temperature on a sunny  summer day would  A.  5° C e l s i u s  B.  25° C e l s i u s  C.  55° C e l s i u s  D.  85° C e l s i u s  E.  I don't know.  thickness  most l i k e l y be:  of a dime i s about:  A.  1 cm  B.  1 dm  C.  1m  D.  1 mm  E.  I don't know.  Mr. Jones p u t a f e n c e around h i s r e c t a n g u l a r garden. The garden 10 m long and 6 m wide. How many metres o f f e n c i n g d i d he use? A.  16 m  B.  30 m  C.  32 m  D.  60 m  E.  I don't know.  227  - 22 39.  A map o f B.C. i s t o be drawn so t h a t 1 m i l l i m e t r e r e p r e s e n t s 5 k i l o m e t r e s . I f the a c t u a l d i s t a n c e between Vernon and P e n t i c t o n i s 125 k i l o m e t r e s , how many m i l l i m e t r e s a p a r t s h o u l d these two p o i n t s be on t h e map? A.  125  B.  625  c.  120  D.  25  E. .  40.  I don't know.  Which one o f the f o l l o w i n g f i g u r e s shows an a c u t e a n g l e ?  A.  D.  41.  I don't know.  Which one o f t h e f o l l o w i n g i s a diagram o f a l i n e ?  A.  M  B.  M  N N N -«  M  -«8—o—  M  I don't know.  N  228  42.  Which one o f the f o l l o w i n g  patterns  c a n be made i n t o a pyramid?  J  43.  Along which l i n e can the t r a p e z o i d and c o r n e r t o c o r n e r ?  A.  q  B.  r  C.  s  D.  t  E.  I don't know.  be f o l d e d e x a c t l y  edge t o edge  229 - 24 What i s the a r e a o f the shaded p o r t i o n o f t h i s  figure?  T  A.  54  B.  96  C.  120  D.  60  E.  45.  46.  I don't know.  3n i s e q u a l t o A.  "h + 3  B.  W. - 3  C.  7k x 3  D.  'D.  E.  I don't  -T-  3  know.  Which l i s t c o n t a i n s a l l o f the whole numbers which make t h i s a t r u e statement?  A.  5  B.  7  C.  0,1,2,3,4,5,6  D.  0,1,2,3,4,5  E.  I don't know.  230  25 -  Simplify:  48.  Multiply:  A.  36  B.  64  C.  12  D.  32  £.  I don't  12 x  2~  A.  14-r  know.  4 B.  30  C.  33  D.  24-7 4 I don't  49.  How  many  shakes  know.  c a n I buy w i t h 'A  A.  2  B.  3  C.  4  D.  5  E.  I don't  know.  $4.20  ?  231  26  50.  Joyce  has 50^.  Which of the f o l l o w i n g can she buy?  8C 3<t  71  10C  A.  3 a p p l e s and 3 i c e cream cones  B.  5 a p p l e s and 3 b a l l o o n s  C.  4 i c e cream cones and a c h o c o l a t e bar  D.  3 c h o c o l a t e bars and a p e n c i l  E.  I don 1 1  know.  APPENDIX B Teacher's Guide Questionnaire  GRADE 7 FEACHER'S GUIDE QUESTIONNAIRE  234 -  1  M A T H E M A T I C S  INSTRUCTIONS FOR  -  A S S E S S M E N T  THE  TEACHER QUESTIONNAIRE  You a r e r e q u e s t e d t o use t h e Answer S h e e t a t t a c h e d t o t h i s q u e s t i o n n a i r e i n responding to these q u e s t i o n s . T h i s copy o f the Answer Sheet has a school facility code bubbled i n , w h e r e a s c o p i e s t o be u s e d by s t u d e n t s i n y o u r c l a s s do n o t . Their responses will be tracked through use of the Class Header Sheet i n c l u d e d i n t h i s package. The A n s w e r S h e e t h a s b e e n d e s i g n e d t o accommodate b o t h s t u d e n t and teacher responses. Four s t e p s to f o l l o w i n c o m p l e t i n g y o u r Answer S h e e t f o l l o w : 1.  In the s e c t i o n l a b e l l e d t h e word " T e a c h e r " f i l l your  Grade, i n the  and u n d e r bubble f o r Grade  2.  Indicate  3.  DO NOT c o m p l e t e t h e s e c t i o n you a r e a n s w e r i n g .  indicating  4.  Your  of  7.  Gender.  questionnaire  consists  five  which  form  sections.  S c a l e R - Mathematics i n School S c a l e S - Problem S o l v i n g S c a l e T - C a l c u l a t o r s and Computers Background I n f o r m a t i o n C l a s s s i z e and T e x t b o o k s Used Please complete the e n t i r e q u e s t i o n n a i r e on t h e d e s i g n a t e d Answer Sheet. Note that bubbles for the Background I n f o r m a t i o n a n d t h e C l a s s S i z e and T e x t b o o k s s e c t i o n s a r e on the r e v e r s e s i d e of the s h e e t . If you teach Mathematics t o more t h a n one Grade 7 class, q u e s t i o n s w i t h r e f e r e n c e t o one c l a s s o n l y s h o u l d be r e s p o n d e d t o f o r t h e f i r s t G r a d e 7 Math c l a s s w h i c h o c c u r s i n t h e week or i n your t i m e t a b l e c y c l e . After you have completed your responses, place this q u e s t i o n n a i r e and y o u r Answer S h e e t i n t h e same e n v e l o p e a s t h e g r e e n c l a s s H e a d e r S h e e t and t h e S t u d e n t A n s w e r S h e e t s f o r your c l a s s . A f t e r a d m i n i s t r a t i o n the envelope, t o g e t h e r w i t h all student test booklets, should be returned to your principal.  -  SCALE R:  z -  235  MATHEMATICS  IN SCHOOL  Please record your responses ON THE ANSWER SHEET i n the s e c t i o n l a b e l l e d "FORM R".  For e a c h o f t h e i t e m s i n t h i s s c a l e , t h r e e r e s p o n s e s a r e r e q u i r e d . C o n s i d e r o n l y t h e c l a s s d e s i g n a t e d by y o u r p r i n c i p a l f o r t h i s q u e s t i o n naire. A. B. C.  1.  T e l l how i m p o r t a n t you t h i n k t h e t o p i c i s f o r t h i s c l a s s . T e l l how e a s y i t i s t o t e a c h t h e t o p i c t o t h i s c l a s s . T e l l how much y o u l i k e t e a c h i n g t h e t o p i c t o t h i s c l a s s .  Adding,  subtracting, multiplying  and d i v i d i n g  fractions  C.  B.  2.  not  a t a l l important  very d i f f i c u l t  dislike  not  important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  Adding,  easy  s u b t r a c t i n g , m u l t i p l y i n g and d i v i d i n g  a lot  like a l o t  decimals  not  a t a l l important  very d i f f i c u l t  dislike  not  important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  easy  a lot  like a l o t  Working w i t h p e r c e n t s B. not  a t a l l important  very d i f f i c u l t  dislike  not  important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very  very  important  easy  a lot  like a l o t  - 3 4.  236  Learning about estimation B.  5.  not a t a l l important  very  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a lot  B.  not a t a l l important  very  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a lot  difficult  dislike a lot  Solving equations A.  7.  dislike a lot  Memorizing b a s i c f a c t s A.  6.  difficult  C.  B. difficult  dislike a lot  not a t a l l important  very  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a lot  Solving word problems A.  B. difficult  dislike a lot  not a t a l l important  very  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a lot  237 -  8.  4  -  Learning about the metric system A.  B.  not a t a l l important  very d i f f i c u l t  dislike a lot  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a l o t  Working with perimeter and area  10.  A.  B.  not a t a l l important  very d i f f i c u l t  dislike a lot  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a lot  Doing geometry  A.  B.  not at a l l important  very d i f f i c u l t  dislike a lot  not important  difficult  dislike  undecided  undecided  undecided  important  easy  like  very important  very easy  like a l o t  -  SCALE S: Please  5 -  238  PROBLEM SOLVING  r e c o r d y o u r r e s p o n s e s ON THE ANSWER SHEET i n t h e s e c t i o n l a b e l l e d "FORM S".  F o r Items 1-7, mark t h e r e s p o n s e w h i c h b e s t d e s c r i b e s y o u r o p i n i o n a b o u t e a c h s t a t e m e n t w i t h r e s p e c t t o t h e c l a s s d e s i g n a t e d by y o u r p r i n c i p a l f o r this questionnaire. 1.  Most o f my A. B. C. D. E.  2.  solving.  perform w e l l i n problem  solving.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  adequate i n s t r u c t i o n  f o r developing  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  My d i s t r i c t p r o v i d e s a d e q u a t e a s s i s t a n c e and r e s o u r c e s teaching o f problem s o l v i n g . A. B. C. D. E.  5.  problem  The t e x t b o o k s I use p r o v i d e problem-solving skills. A. B. C. D. E.  4.  enjoy  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Most o f ray s t u d e n t s A. B. C. D. E.  3.  students  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  I am s a t i s f i e d  w i t h my  teaching  A. Strongly Disagree B. Disagree C. Undecided D. .Agree E. S t r o n g l y Agree  o f problem  solving.  f o r the  239  - 66.  A l l m a t h e m a t i c s t e a c h e r s s h o u l d a t t e n d a t l e a s t one workshop o n problem s o l v i n g each y e a r . A. B. C. E.  It  i s easy A. B. C. D. E.  8.  S t r o n g l y Disagree Disagree Undecided Agree S t r o n g l y Agree  t o teach problem  solving.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Which o f t h e s e p r o b l e m - s o l v i n g s t r a t e g i e s do y o u t e a c h i n y o u r classes? (Mark a l l t h a t a p p l y . ) A. B. C. D. E.  Look f o r a p a t t e r n Guess and c h e c k Make a s y s t e m a t i c l i s t o r t a b l e S o l v e a s i m p l e r problem Work backwards  How many workshops on p r o b l e m s o l v i n g year? A. B. C. D. E.  10.  have y o u a t t e n d e d  None  1 2 3  More than 3  What s o u r c e s do you use t o p r o v i d e s t u d e n t s w i t h exercises? (Mark a l l t h a t a p p l y . ) A. B. C. D. E.  i n the p a s t  Textbook Mathematical contests Problem-solving booklets Professional journals Books o f p u z z l e s  problem-solving  240 - 7 -  11.  Which of the following do you use to motivate your students to p a r t i c i p a t e i n problem-solving a c t i v i t i e s ? (Mark a l l that apply.) A. Competitive games B. Problem of the day or week * C. Puzzles or brain teasers 0. Library or f i l e of interesting problems E. Contests  12.  Which of the following best characterizes your teaching of problem solving i n mathematics? I teach problem solving: A. As a unit from time to time B. Almost every day, as a regular part of the mathematics c l a s s C. One period every 2 or 3 weeks D. At the end of a major topic or chapter E. One period a week  13.  When you grade students' work i n problem solving, f o r which of the following do you give marks? (Mark a l l that apply.) A. B. C. D. E.  14.  Which of the following types of problems do you assign to your students? (Mark a l l that apply.) A. B. C. D. E.  15.  I don't give p a r t i a l c r e d i t . I t ' s a l l or nothing. For the appropriate diagram or equation For the procedures used (computation, etc.). For the f i n a l answer For checking the answer  Problems Problems Problems Problems Problems  with more than one correct answer which require students to c o l l e c t information which can be solved more than one way which students work on c o l l e c t i v e l y i n groups with either too much or too l i t t l e information  Which of the following a c t i v i t i e s do you have i n your class? (Mark a l l that apply.) A. A problem-solving center B. A b u l l e t i n board display on problem solving C. Problem of the week D. Problem-solving contests within the c l a s s E. Students make up problems for others to solve  -a SCALE T;  241  CALCULATORS AND COMPUTERS  P l e a s e r e c o r d y o u r r e s p o n s e s ON THE ANSWER SHEET i n t h e s e c t i o n l a b e l l e d "FORM T".  When r e s p o n d i n g ,  1.  s t u d e n t s use c a l c u l a t o r s  calculators?  I do n o t a l l o w c a l c u l a t o r s i n my c l a s s . Each s t u d e n t may b r i n g h i s o r h e r own. Each s t u d e n t must b r i n g h i s o r h e r own. C a l c u l a t o r s a r e provided f o r the students.  I n what ways do y o u use a c a l c u l a t o r work? (Mark a l l t h a t a p p l y . ) A. B. C. D.  i n mathematics?  Not a t a l l F o r d r i l l and p r a c t i c e t o enhance c o m p u t a t i o n a l s k i l l s To work on p r o b l e m s F o r o t h e r t o p i c s such a s e s t i m a t i n g , f i n d i n g p a t t e r n s , and s o on  How a r e s t u d e n t s i n y o u r c l a s s p r o v i d e d w i t h A. B. C. D.  5.  Never R a r e l y (perhaps l e s s t h a n once a week) Sometimes (a c o u p l e o f tiroes a week) F r e q u e n t l y ( a l m o s t e v e r y day)  I n what ways do y o u have y o u r (Mark a l l t h a t a p p l y . ) A. B. C D.  4.  Yes No  How o f t e n do y o u use a c a l c u l a t o r o u t s i d e s c h o o l ? A. B. C. D.  3.  class.  Do you own a c a l c u l a t o r ? A. B.  2.  consider only the designated  for non-instructional school  None To c a l c u l a t e s t u d e n t s ' marks, g r a d e s , and s o on To c h e c k s t u d e n t s ' answers on a s s i g n m e n t s To p r e p a r e w o r k s h e e t s o r t e s t s  242 - 9 -  6.  Some people say that Grade 7 students should not be allowed to use c a l c u l a t o r s i n s c h o o l . How do you f e e l about t h i s ? A. * B. C. D. E.  7.  Some people say that i f students are allowed to use c a l c u l a t o r s , then i t i s not necessary f o r them to l e a r n how t o add, s u b t r a c t , m u l t i p l y , or d i v i d e by hand. How do you f e e l about t h i s ? A. B. C. D. E.  8.  Yes No  What a p p l i c a t i o n s of computers have you had experience with? (Mark a l l that apply.) A. B. C. D. E.  10.  S t r o n g l y Disagree Disagree Undecided Agree . S t r o n g l y Agree  Do you have a computer (one that w i l l do more than p l a y games) a t home? A. B.  9.  S t r o n g l y Disagree Disagree Undecided Agree S t r o n g l y Agree  None Games Word processing Computer-assisted Programming  instruction  Where d i d you get most of your experience with computers? A. B. C.  I haven't had any experience with computers. On my own Through s p e c i a l courses  243 - l O -  ll.  Which o f the f o l l o w i n g i s c l o s e s t i n your s c h o o l ? A. B. C. D. E.  12.  13.  14.  i n your  mathematics c l a s s ?  Not used I use i t a s a t e a c h i n g t o o l t o d e m o n s t r a t e S t u d e n t s l e a r n computer programming. S t u d e n t s use s o f t w a r e p a c k a g e s . I use the computer f o r r e c o r d - k e e p i n g .  what k i n d s o f computer s o f t w a r e do y o u r mathematics? (Mark a l l t h a t a p p l y . ) A. B. C. D. E.  t y p e o f computer o r g a n i z a t i o n  T h e r e a r e no computers a t a l l i n t h i s s c h o o l . T h e r e a r e computers i n some o r a l l c l a s s r o o m s . Computers a r e p r o v i d e d i n one o r more p l a c e s f o r use by t e a c h e r s w i t h t h e i r s t u d e n t s . Computers a r e p r o v i d e d i n one o r more p l a c e s w h i c h a r e s t a f f e d by s p e c i a l i s t s o r r e s o u r c e p e r s o n s . Computers a r e used f o r a d m i n i s t r a t i v e p u r p o s e s o n l y .  How i s a computer used apply.) A. B. C. D. E.  to the  (Mark a l l t h a t  concepts.  s t u d e n t s use  None D r i l l and p r a c t i c e E d u c a t i o n a l games t o r e i n f o r c e s k i l l s o r T u t o r i a l (to teach a s k i l l or concept) S i m u l a t i o n (to p r o v i d e a>model o f a r e a l  for  learning  concepts situation)  Which g r o u p o f s t u d e n t s i n your m a t h e m a t i c s c l a s s makes the most o f computers i n s c h o o l ? A. B. C. D. E.  None, my s t u d e n t s do not use computers i n s c h o o l . The low a b i l i t y s t u d e n t s S t u d e n t s o f average a b i l i t y High a b i l i t y students They a l l make e q u a l use o f c o m p u t e r s .  use  FOR ITEMS 15-20, CHOOSE THE OPTION WHICH BEST DESCRIBES YOUR OPINION.  15.  I f we a l l o w computers t o be u:>ed i n s c h o o l , t h e y may t a k e o v e r some o f t h e major f u n c t i o n s o f t e a c h e r s . A. B. C. D. E.  16.  The computer s o f t w a r e t h a t i s a v a i l a b l e a p p r o p r i a t e and w e l l - d e s i g n e d . A. B. C. D. E.  17.  f o r t e a c h i n g mathematics i s  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  S t u d e n t s have an o p p o r t u n i t y t o be c r e a t i v e when they a r e t a u g h t m a t h e m a t i c s by computer. A. B. C. D. E.  18.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  More boys t h a n g i r l s A. B. C. D. E.  seem t o use computers f o r d o i n g m a t h e m a t i c s .  S t r o n g l y Disagree Disagree Undecided Agree S t r o n g l y Agree  245 -  19.  I t i s e s s e n t i a l t h a t computers t e a c h e r s o f mathematics. A. B. C. D. E.  20.  12 -  become an i n s t r u c t i o n a l  tool for a l l  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  To be s u c c e s s f u l i n modern s o c i e t y , n e a r l y e v e r y o n e w i l l computer s k i l l s . A. B. C. D. E.  Strongly Disagree Disagree Undecided Agree S t r o n g l y Agree  need  246  - 13 TEACHER BACKGROUND INFORMATION  P l e a s e r e c o r d y o u r r e s p o n s e s ON THE ANSWER SHEET i n t h e s e c t i o n l a b e l l e d "BACKGROUND INFORMATION".  1.  F o r how many y e a r s J u n e , 1985? A. B. C. D. E.  2.  4.  o f your c u r r e n t teaching  altogether?  load  i s mathematics?  0-20% 21-40% 41-60% 61-80% 81-100%  B.C. A s s o c i a t i o n o f M a t h e m a t i c s T e a c h e r s P r o v i n c i a l Intermediate Teachers A s s o c i a t i o n B.C. P r i m a r y T e a c h e r s A s s o c i a t i o n N a t i o n a l C o u n c i l o f Teachers o f Mathematics L o c a l M a t h e m a t i c s PSA None o f t h e above  Have you a t t e n d e d three years? A. B.  6.  mathematics  To which o f t h e f o l l o w i n g a s s o c i a t i o n s do y o u b e l o n g ? (Mark a l l t h a t a p p l y . ) A. B. C. D. E. F.  5.  teaching  Yes No Undecided  What p e r c e n t A. B. C. D. E.  mathematics as o f  1-2 y e a r s 3-5 y e a r s 6-10 y e a r s 11-15 y e a r s More t h a n 15 y e a r s  I f y o u had a c h o i c e , would y o u a v o i d A. B. C.  3.  w i l l y o u have been t e a c h i n g  a mathematics s e s s i o n a t a c o n f e r e n c e  i n the l a s t  Yes No  Have y o u a t t e n d e d a workshop ( o t h e r t h a n a t a c o n f e r e n c e ) o r an i n s e r v i c e day i n m a t h e m a t i c s i n t h e L a s t t h r e e y e a r s ? A. B.  Yes No  247  - 14 -  7.  A t what l e v e l s h o u l d s t u d e n t s f i r s t be t a u g h t m a t h e m a t i c s by someone who s p e c i a l i z e s i n t h e t e a c h i n g o f m a t h e m a t i c s ? A. B. C. D. E.  8.  How many p o s t - s e c o n d a r y c o u r s e s i n m a t h e m a t i c s have you s u c c e s s f u l l y completed? ( e . g . , F o r UBC 3 u n i t s = 2 c o u r s e s ) A. B. C. D. E.  9.  A t no l e v e l Primary Intermediate J u n i o r Secondary S e n i o r Secondary  0 1 or 2 3-5 6-9 10 o r more  How many p o s t - s e c o n d a r y c o u r s e s i n m a t h e m a t i c s e d u c a t i o n have you s u c c e s s f u l l y completed? ( e . g . , F o r UBC 3 u n i t s = 2 c o u r s e s ) A. B. C. D. E.  0 1 or 2 3-5 6-9 10 o r more  QUESTIONS 10-20 REFER TO THE SPECIFIC MATHEMATICS CLASS DESIGNATED BY YOUR PRINCIPAL FOR THIS QUESTIONNAIRE.  10.  Which o f the f o l l o w i n g students? A. B. C. D. E. F. G.  11.  to these  F u l l - y e a r c o u r s e , r e g u l a r program F u l l - y e a r c o u r s e , m o d i f i e d (slower students) Full-year course, enriched Semester c o u r s e , r e g u l a r program Semester c o u r s e , m o d i f i e d ( s l o w e r s t u d e n t s ) Semester c o u r s e , e n r i c h e d Other  On t h e a v e r a g e , mathematics? A. B. C. D. E.  b e s t d e s c r i b e s the course o f f e r e d  how o f t e n do you g i v e t h i s c l a s s  tests or quizzes i n  Almost e v e r y day Once a week Once e v e r y c o u p l e o f weeks Once e v e r y r e p o r t i n g p e r i o d I a l m o s t never g i v e t e s t s o r q u i z z e s i n m a t h e m a t i c s .  - 15 -  12.  How  many mathematics p e r i o d s does A. B. C. D. • E.  13.  How  long A. B. C. D. E.  248  t h i s c l a s s have e a c h c a l e n d a r week?  3 4 5 6 More than 6  i s each mathematics  period?  30 m i n u t e s o r l e s s 31-45 m i n u t e s 46-60 m i n u t e s 61-75 m i n u t e s More t h a n 75 m i n u t e s  QUESTIONS 14-20 REFER TO THE LAST MATHEMATICS PERIOD DURING WHICH YOU TAUGHT THIS CLASS.  14.  What p e r c e n t o f t h a t m a t h e m a t i c s p e r i o d was s p e n t on a c t i v i t i e s r e l a t e d t o homework from t h e p r e v i o u s day ( e . g . , d i s c u s s i n g , c o r r e c t i n g ) ? A. B.  C. D. E. F.  15.  How  1-10% 11-25% 26-50% 51-75% 76-100%  many s t u d e n t s d i d you c a l l on t o answer q u e s t i o n s ? A. B. C. D. E. F.  16.  None  None One or two L e s s than o n e - q u a r t e r o f t h e c l a s s About h a l f the c l a s s Between a h a l f and t h r e e - f o u r t h s o f the c l a s s Almost e v e r y s t u d e n t  What p e r c e n t o f t h a t mathematics p e r i o d d i d y o u r s t u d e n t s spend w o r k i n g i n d i v i d u a l l y on s e a t w o r k ? A. B.  C. D. E. F.  None  1-10% 11-25% 26-50% 51-75% 76-100%  249 -  17.  What p e r c e n t o f t h a t m a t h e m a t i c s p e r i o d d i d your s t u d e n t s working i n s m a l l g r o u p s ? A. ' B. C. D. E. F.  18.  None 1-10% 11-25% 26-50% 51-75% 76-100%  spend  _ .  What p e r c e n t o f t h a t m a t h e m a t i c s p e r i o d d i d y o u r s t u d e n t s spend on computational d r i l l ? A. B. C. D. E. F.  20.  spend  None 1-10% 11-25% 26-50% 51-75% 76-100%  What p e r c e n t o f t h a t mathematics p e r i o d d i d your s t u d e n t s working a t s t a t i o n s o r a c t i v i t y c e n t r e s ? A. B. C. D. E. F.  19.  16 -  None 1-10% 11-25% 26-50% 51-75% 76-100%  What p e r c e n t o f t h a t mathematics p e r i o d t o p i c s t o the e n t i r e c l a s s ? A. B. C. D. E. F.  None 1-10% 11-25% 26-50% 51-75% 76-100%  d i d you spend e x p l a i n i n g  new  250  -  17 -  CLASS S I Z E AND  P l e a s e r e c o r d y o u r answer 1/ 2 a n d 3 o f t h e s e c t i o n  1.  TEXTBOOKS  DSED  t o t h e n e x t t h r e e q u e s t i o n s on i t e m s l a b e l l e d Achievement Survey.  Which one o f t h e f o l l o w i n g i n d i c a t e s t h e s i z e o f the c l a s s f o r which you a r e r e s p o n d i n g t o t h i s questionnaire?  A.  1 - 1 5  B. 16 - 20 C. 21 - 25 D. E.  2.  3.  Which basic  26-30 31 o r l a r g e r  one o f t h e f o l l o w i n g t e x t b o o k s i s u s e d a s t h e text i n your classroom? A.  Essentials  of Mathematics  B.  Mathematics  C.  School  D.  Contemporary  E.  Other  1  1  Mathematics  1  Mathematics  Which o f t h e f o l l o w i n g t e x t b o o k s a r e u s e d a s s u p p l e m e n t a r y t e x t s ( n o t t h e b a s i c one) i n y o u r classroom? Check a l l t h a t a p p l y . A.  Essentials  of Mathematics  B.  Mathematics  C.  School  D.  Contemporary  E.  Other  1.  1  Mathematics  1  Mathematics  END OF  QUESTIONNAIRE  Thank you f o r y o u r  co-operation  APPENDIX C Coding o f  Variables  252  O p t i o n Codes f o r Independent V a r i a b l e s Variable  Item Number  Number Codes by Opt A  B  C  D  E  1 2 7 9 10  2 2 1 1 1  1 1 2 2 2  1 1 3 3 3  4 4 4  5  —  1 2 3 5 6 8 9  1 1 1 2 2 1 1  2 3 2 1 1 2 2  3 2 3  4  5  4  5  3 3  4 4  5 5  1 2 3 4 5 6 7 8 9 10  1 1 1 1 1 1 1 1 1 1  2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3 3  4 4 4 4 4 4 4 4 4 4  5 5 5 5 5 5 5 5 5 5  1 2 3 4 5 6 7 8 9 10  1 1 1 1 1 1 1 1 1 1  2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3 3  4 4 4 4 4 4 4 4 4 4  5 5 5 5 5 5 5 5 5 5  Student Background SB1 SB2 SB3 SB4 SB5 B.  -  Teacher Background TBI TB2 TB3 TB4 TB5 TB6 TB7 Student P e r c e p t i o n SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8 SP9 SP10  D.  Teacher Perception TP1 TP2 TP3 TP4 TP5 TP6 TP7 TP8 TP9 TP10  253  E.  Classroom O r g a n i z a t i o n C01 C02 C03 C04 C05 C06 C07 C08 C09 C010 COll  G.  10 11 12 13 14 15 16 17 18 19 20  2 5 3 15 1 1 1 1 1 1 1  1 2 5 7 8 9 10 11 12 14 15  1 1 1 1 1,0 1 1,0 1,0 1 1,0 1,0  1 4 4 38 2 2 2 2 2 2 2  3 3 5 53 3 3 3 3 3 3 3  2 6 68 4 4 4 4 4 4 4  -  1 7 83 5 5 5 5 5 5 5  Problem S o l v i n g P r o c e s s e s PS1 PS 2 PS 3 PS 4 *PS5 PS 6 *PS7 *PS8 PS 9 *PS10 *PS11  * M u l t i p l e response  items  2 2 2 2 1,0 2 1,0 1,0 5 1,0 1,0  3 3 3 3 1,0 3 1,0 1,0 3 1,0 1,0  4 4 4 4 1,0 4 1,0 1,0 2 1,0 1,0  5 5 5 5 1,0 5 1,0 1,0 4 1,0 1,0  6 6 6 6 6 6 6  

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