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The development of a predictive test for mathematical success Biggs, Marjorie Dudley 1968

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THE DEVELOPMENT  OF A P R E D I C T I V E  T E S T FOR M A T H E M A T I C A L  M A R J O R I E DUDLEY B.Sc.  University  A THESIS  SUCCESS  BIGGS  of California  at Davis,  1963  S U B M I T T E D I N P A R T I A L F U L F I L L M E N T ' 'OF  THE R E Q U I R E M E N T S  FOR THE DEGREE, OF  MASTER  in  OF ARTS  the Faculty of Education  We  accept  required  this  thesis  as c o n f o r m i n g  to the  standard  THE U N I V E R S I T Y  OF B R I T I S H  November, 1968  COLUMBIA  In p r e s e n t i n g an  this  thesis  inpartial  advanced degree a t t h eU n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  agree that  permission  f u l f i l m e n t o f the requirements of British  available  for  for extensive  Columbia,  I agree  for  that  r e f e r e n c e and Study. copying of this  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by  his  of  this  written  representatives. thesis  for  financial  gain  shall  permission.  Department o f  ^f/Ccc.CLgS^^<^C.  The U n i v e r s i t y . o f B r i t i s h V a n c o u v e r 8, Canada  Date  I t i sunderstood  Columbia  Q e X L , ^ / f % y  that  copying o r p u b l i c a t i o n  n o t b e a l l o w e d w i t h o u t my  ABSTRACT It  was  literature available the  the  i n regards  undertaken  students;  courses  the  new  the  the  of  (2) to  t h a t have  analyze and  review  validity  tests;  courses  (3) to  of the  (1) to  predictive  for skills  content  study  counselling  of mathematics  by  mathematics  of t h i s  to the  mathematical  sequences  determine  purpose  the  analyze  and  would  content  concepts;  and  mathematical  of  be  these  (h) t o  counselling  test. All distinct were  v e r s i o n s of t h i s sections.  dependent  concepts section format.  and was The  administered the  test  of  just  the  two  test and  upon t h e  based test again  on  competence.  he  had  been  presented  whose of  both  0.70.  On  mathematical  exposed.  The  close to of  a correlation  correlation 0.50.  a student's  aspects  needed  for  second  in a teach-test  a s m a l l sample  s e c t i o n s had The  been  different  times.  two  solutions  administered, evaluated, revised  i s a good p r e d i c t o r measures  c o n s i s t e d of  comprehension  material  several  s e c t i o n s has  test  contained problems  to which  new  was  from  than  one  student'-s  procedures  scores less  Part  counselling  coefficient Each grade  of  students  coefficient between  section in  and  of  this  mathematics  mathematical  i i i TABLE  OF  CONTENTS  CHAPTER I.  PAGE THE I N V E S T I G A T I O N  •  1  The P r o b l e m  1  Statement  of the problem  1  of the study  2  Importance The  Teach-Test  3  . : i l . REVIEW OF THE L I T E R A T U R E  5  Literature First  on t h e P r e d i c t i o n  Year  Implications  III.  College  of Success i n  Mathematics  of Previous  5  Studies  9  THE D E S I G N . . : Selection  and P r e p a r a t i o n  10 of Test  Items  The T e s t s  IV.  10 11  S e c t i on o n e  11  Section  13  two  AN A N A L Y S I S OF THE DATA AND  CONCLUSIONS  l h  BIBLIOGRAPHY  l8  APPENDIX  20.  i v  LIST  OF  TABLE I.  TABLES  PAGE Analysis  of Results  15  CHAPTER THE Many u n i v e r s i t i e s entering program to  students. i s to  succeed  identify  and,  INVESTIGATION  have  The  I  administered  principal those  in turn,  to  a variety  purpose  areas  of  i n which  provide  a basis  such  of t e s t s a  testing  a student for  to  is  likely  effective  counselling. At  the  University  of  British  coefficient  determined  by  comparing  mathematical the  final  been  counselling test  scores  obtained  decreasing.  The  Columbia the the  scores  in first  year  Education  C o l u m b i a was  invited  of B r i t i s h  procedures  for obtaining a counselling test  university This of  the  a student's  mathematics study  desired  study  ( l ) to  validity  c o n s i s t e d of the  (2)  to  analyze  courses  Department to  with  assist  at  in  t h a t would  final  score  in first  initial  steps  i n the  has  the  the provide year  development  test.  of the  review  of the  students  courses.  I.  S t at ement  the  mathematics  Mathematics  of  on  a d m i n i s t e r e d t o new  University  a better indication  correlation  problem.  the  PROBLEM  I t was  literature  available the  THE  purpose  i n regards  mathematical  sequences  the  of  to the  this predictive  counselling tests;  of mathematics  courses  that  have  been  and  content and  w o u l d be of these  ( h ) to  of  of the  Columbia years  mathematics time.  section  Files  recent  courses  content  study. assumed any  e s t i m a t i o n of the  course."'"  in  of. t h e  seem t o h a v e  reliable  British  the  (3)  students;  to  for skills  of the  new  was  University  and  mathematical  on  occasion  achievement students  that  or  an  than  grade  limitations  of  have been p o i n t e d  grades  out  a negative  test  of  California  at  mathematics  to  of  students  success  given  of  in  at the  same  3  and  .  scores  Coddington from  Aptitude Tests  mathematics  Los  Holland  between the  Scholastic  year  is a  standardized counselling tests  correlation  in first  the  University  given  predictor  Coddington  p o r t i o n of the  obtained  University  by  in a  at the  English test  a mathematics  on  aptitude test  final  a more r e l i a b l e  and  a.score  2  mathematics  the  concepts;  personnel  counselling center  show t h a t  courses  Similar  reported  analyze  test.  Import ance  mathematics  by  mathematics  determine  counselling  students  undertaken  Angeles.  courses  at  the  and  final  the  Holland wrote  a  "'"Boyce, R.W. a n d R.C. Paxson, " P r e d i c t i v e V a l i d i t y of E l e v e n T e s t s a t One S t a t e C o l l e g e , " E d u c a t i o n a l and P s y c h o l o g i c a l M e a s u r e m e n t , XXV ( W i n t e r 1 9 6 5 ) , 1143. 2  The  C o d d i n g t o n , E.A., "Scholastic Aptitude Tests A m e r i c a n M a t h e m a t i c a l M o n t h l y , LXX ( S e p t e m b e r ,  in Mathematics," 196-3), 750-55 -  H o l l a n d , J . L . , "The P r e d i c t i o n o f G r a d e s f r o m t h e C a l i f o r n i a P s y c h o l o g i c a l I n v e n t o r y and t h e S c h o l a s t i c A p t i t u d e T e s t s , " J o u r n a l of E d u c a t i o n a l Psychology, L (August, 1 9 5 9 ) , 135-42;  regression the  equation  Scholastic  Inventory  Aptitude Tests.  predictions  and  that  Aptitude Tests  from  of a single  Columbia  counselling accurately  Cooperative Aptitude  course test  a student's than such  School  to  search  which  learn  he c o u l d make b e t t e r  and Paxson  5  that  score  t h e mathematics as t h e A m e r i c a n  and C o l l e g e A b i l i t y  h i s population  of these  reports  , the University  i t s own  final  from t h e  of  mathematical i twould  more  i n a first  year  p o r t i o n o f an A m e r i c a n College Tests, the Tests,  or the Scholastic  Tests.  THE T E A C H - T E S T  f o ra test  learn' mathematics items  the scores  As a r e s u l t  t o develop  II. The  that  with the anticipation  predict  standardized  than  Personality-  d a t a by r e s t r i c t i n g  and Boyce  decided  test  mathematics  He f o u n d  university. h  several sections of  and t h e C a l i f o r n i a  the available  s t u d i e s by Heimer  British  from  to obtain a better predictor  Scholastic  to  using the scores  t o measure  a student's  l e dt o t h e development  would provide  mathematics  a measure  ability to  of a c o l l e c t i o n  o f an i n d i v i d u a l ' s  under t e s t - l i k e  conditions.  These  of ability items  H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n M a t h e m a t i c a l Monthly, LXXIII (October, 1966), 883-89. Tests  ^ B o y c e , R.W. a n d R.C. P a x s o n , " P r e d i c t i v e V a l i d i t y o f E l e v e n a t One S t a t e C o l l e g e " , E d u c a t i o n a l a n d P s y c h o l o g i c a l  M e a s u r e m e n t , XXV  (Winter  1965),  llk3-hT.  •k. are  similar  report to  t o t h e example  the term teach-test  these  make  i n form  items.  a significant contribution i n a 'mathematics  of  report  this  different  indicate  factors  failed  he u s e d  I t was a n t i c i p a t e d  success  have  will  i n the appendix.  needed  t o measure.  course. that  exclusively  that  such  the teach-test  f o r success  that  to refer  a measure  to the prediction The r e s u l t s  In t h i s  of a  given  measures  could person's  i n Chapter IV some  known p r e v i o u s  tests  CHAPTER REVIEW OF Although of  success  with  total  with  the  Because  most  success  studies  concerned with  of the  obsolete.  changes  curriculums the  This  of  a general  i n u n i v e r s i t y study,  p r e d i c t i o n of  populations,  success i n the and  a few  in college nature  i n the  nature  been the  college  characteristics  experience  at  the  deal  concerned  mathematics.  f i n d i n g s of p r e d i c t i v e s t u d i e s has  and  have been  level  of the  predictions  preparatory of u n i v e r s i t y  often  become  University  of  Columbia.  I.  LITERATURE IN  The  studies  inadequacy Tests  LITERATURE  i n u n i v e r s i t y have been  mathematics  British  of the  THE  II  for  of  the  samples  ON  THE  F I R S T YEAR by  Holland  1  PREDICTION COLLEGE and  mathematics that  d i d not  OF  SUCCESS  MATHEMATICS  Coddington  2  showed t h e  predictive  s e c t i o n of  the  Scholastic  Aptitude  represent  the  university population.  H o l l a n d , J . L . , "The P r e d i c t i o n o f C o l l e g e G r a d e s f r o m t h e C a l i f o r n i a P s y c h o l o g i c a l I n v e n t o r y and t h e S c h o l a s t i c A p t i t u d e T e s t s , " J o u r n a l of E d u c a t i o n a l Psychology, L (August,  1959), 135-1+2 .  The  C o d d i n g t o n , E.A., "Scholastic Aptitude Tests in Mathematics," American Mathematical M o n t h l y , LXX ( S e p t e m b e r , 1963) 750-55.  Wick success  i n h i s study  i n first  year  o f some  factors associated  c o l l e g e mathematics  arrived  with at four  conclusions. 1. For the p a r t i c i p a n t s i n this investigation there a p p e a r s t o be no s i g n i f i c a n t d i f f e r e n c e i n t h e q u a l i t y o f p r e p a r a t i o n f o r f i r s t - y e a r c o l l e g e m a t h e m a t i c s as b e t w e e n t h e e x p e r i m e n t a l (S.M.S.G.) a n d t h e t r a d i t i o n a l programs i n high s c h o o l mathematics. T h e r e i s some i n d i c a t i o n that t h e experimental mathematics m a t e r i a l s do h a v e a c u m u l a t i v e effect. 2. The h i g h s c h o o l m a t h e m a t i c s r e c o r d was c o n s i s t e n t l y the source o f t h e best p r e d i c t o r s o f success i n f i r s t - y e a r college mathematics. The m a t h e m a t i c s a v e r a g e o v e r g r a d e s t e n t o t w e l v e gave t h e h i g h e s t c o r r e l a t i o n w i t h success i n c o l l e g e a l g e b r a c o u r s e s , whereas t h e average i n grade 12 a p p e a r e d t o b e s l i g h t l y m o r e i n d i c a t i v e o f s u c c e s s i n b e g i n n i n g c a l c u l u s . A l l c o r r e l a t i o n s between t h e f a c t o r s a n a l y z e d and c o l l e g e mathematics grades a r e best d e s c r i b e d . .,as b e i n g l o w . 3. The u s e o f m u l t i p l e r e g r e s s i o n t e c h n i q u e s f o r p r e d i c t i n g success r e p r e s e n t e d a c o n s i d e r a b l e improvement o v e r t h e u s e o f s i n g l e p r e d i c t o r v a r i a b l e s . The r e s u l t i n g m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s were h i g h enough t o w a r r a n t t h e u s e o f p r e d i c t i o n s b a s e d on t h e m u l t i p l e r e g r e s s i o n equations f o r s e l e c t i o n and placement o f s t u d e n t s , and f o r other g r o u p - c o u n s e l i n g procedures. h. The u s e o f p r o b a b i l i t y t a b l e s s u c h as d e s c r i b e d i n t h i s r e p o r t w o u l d a p p e a r t o b e an e f f e c t i v e way o f u s i n g p r e d i c t e d g r a d e s when c o u n s e l i n g w i t h i n d i v i d u a l s t u d e n t s . T h e l i m i t a t i o n s on t h e u s e o f m u l t i p l e r e g r e s s i o n e q u a t i o n s i n c o u n s e l i n g have l a r g e l y been removed by t h e advent o f c o m p u t e r f a c i l i t i e s a t most c o l l e g e s . 3 Boyce validity  and Paxson were  of eleven  tests  new  students  at this  had  obtained  scores  concerned with  used  at Troy  college during which, they  the predictive  State  the five  claimed,  College. previous  The years  averaged between t h e  W i c k , M.E., " S t u d y o f t h e F a c t o r s A s s o c i a t e d w i t h S u c c e s s i n F i r s t Y e a r C o l l e g e M a t h e m a t i c s , " The M a t h e m a t 1 c s T e a c h e r , L V I I ( N o v e m b e r , 1 9 6 5 ) , 6^7-^8.  7. twenty-sixth  and  thirty-eighth  percentile.  They  wrote,  The p r e d i c t i v e v a l i d i t y d a t a r e p o r t e d i n a t e s t m a n u a l a r e . o f t e n i n t e n d e d t o be t r u e o r r e p r e s e n t a t i v e o f a n a t i o n a l sample. Since the d i s c r e p a n c i e s i n p r e d i c t i v e v a l i d i t y c o e f f i c i e n t s a r e l i k e l y t o c h a n g e as a l o c a l c o l l e g e sample departs from a n a t i o n a l sample, v a l i d i t y s t u d i e s s h o u l d be p e r f o r m e d f o r e a c h c o l l e g e ; ^ Heimer wide of was  realized  variety  of p r i o r  understanding due  to  Another  and  regularity, considered  of v a r i a b i l i t y  and  value  between  was  presented  to  did  not  the  not  at  learn a l l . of  statistical  of  has  Anderson and  one  was  in a  he  written  type,  the  mathematical  learn manner  i t was  on  his hypothesis  potential  degrees  extent,  education.  material and  not  which  those  who  poorly  the  that the  He  could  presented  study  a  ability.  examinations  d i d not  a pilot  varied  or  predictive  i n a t e a c h - t e s t format.  supports  a high  due  to  received  variability  and  satisfactory  undertook  with  motivation  achievement who  had  In p a r t , t h i s  m a t e r i a l because  a test data  that  students  i n mathematics  of p r e v i o u s  students  them  Thus  mathematical  school  training  achievement.  i t obvious  distinguish  procedure  c o l l e g e bound  d i f f e r e n c e s i n student  source  validity  that  His teach-test  for obtaining a valid  measure  aptitude.'' and  Whittemore's  university  study  mathematics  related  o n l y one  department.  The  high high  B o y c e , R.W. a n d R.C. Paxson , ' " P r e d i c t i v e V a l i d i t y of E l e v e n T e s t s a t One S t a t e C o l l e g e , " E d u c a t i o n a l a n d P s y c h o l o g i c a l M e a s u r e m e n t , XXV ( W i n t e r 1 9 6 5 ) , 11^3. ^ H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n Mathematical Monthly, LXXIII (October, 1966), 833-89.  8 .  school to  from which  some c l a s s e s  Significant  t h e d a t a was o b t a i n e d  and'traditional  correlations  grades  and'the  score,  and between  grade  point  correlations  American  mathematics  Testing  mathematics  i n mathematics.  obtained  Program  grades  mathematics  mathematics  and t h e c o l l e g e  However no  involving  mathematics  t o others.  f o u n d between modern  College  modern  average were  were  t a u g h t modern  traditional  significant mathematics  6  grades. Wampler of  aptitude  college  eight  variables  of the  test.  An i n i t i a l  calculus  to predict  aptitude  The b e s t  several:'.measures  performance i n  selection  of eleven  factors  was t h e s c o r e made  subjects  with  of using  of eleven predictor  was c h o s e n .  twenty-two  be u s e d  different  variable  achievement  the feasibility  could  made up a p o o l  criterion  the  which  mathematics.  measuring scores  studied  was made, a n d t h e s e variables.  coefficient  geometry.  s c o r e s on t h e L o c a t i o n s  Tests,  The f i v e Division  calculus  prediction was 0 . 9 5 f o r  who h a d c o m p l e t e d t e n s e m e s t e r  analytic  The  on a s t a n d a r d i z e d  combination of five  The m u l t i p l e  tests  hours  v a r i a b l e s , were  Test, Inference 7  Test,  Wide Range  V o c a b u l a r y . T e s t , and.Cube  Comparison  Test.  H e i m e r , R.T., "A T e a c h - T e s t P r o c e d u r e F o r O b t a i n i n g M e a s u r e s o f M a t h e m a t i c a l A p t i t u d e , " The A m e r i c a n Mathemat i c a l Monthly, LXXIII, (October, 1966), 833-89. Wampler, J o e F., " P r e d i c t i o n o f A c h i e v e m e n t i n C o l l e g e M a t h e m a t i c s , " The M a t h e m a t i c s T e a c h e r , L I X ( A p r i l , 1 9 6 6 ) , 3 6 4 - 9 .  From a that  the  grade and  largest  in  the  study  an  advanced  and  the  students  c o e f f i c i e n t of  predictor  courses  1,2hk  of  algebra  variables  multiple  course were  percentile  Kinzer  score  was  and  found  c o r r e l a t i o n when used  as  grades  i n the  on  Ohio  the  Kinzer  the six  the  criterion prerequisite  State  Psychological g  Examination.  This  II.  Since score  used  the to  IMPLICATIONS  predictive predict  many v a r i a b l e s , is  enrolled,  British of  the  the  including  the  Counselling to  c o e f f i c i e n t obtained  Columbia.  I t was  important  w o u l d be  for  used  students  i n the  f o r the  value  PREVIOUS  have  0-.75.  STUDIES  demonstrated  i n mathematics university Department  of  the  the  from  that  a test  entering that  multiple  prediction  of  the this  is  that  the  dependent  at- w h i c h  investigate  anticipated  factor  OF  the  studies  success  Columbia decided  specifically  an  c o e f f i c i e n t has  the  student  University  predictive w o u l d be  University  of  upon  of  validity written  British  c o e f f i c i e n t would  be  regression  equation  success  mathematics.  in  K i n z e r , J.R. a n d L.G. K i n z e r , " P r e d i c t i n g Grades Advanced Mathematics," J o u r n a l of.Applied Psychology, XXXVII ( J u n e , 1953), 182-8U.  that  in  CHAPTER I I I THE  All  versions  distinct were  sections.  dependent  concepts items  of t h i s  and  that  Part  upon t h e  DESIGN  counselling one  contained  precedures to which  were t o d e t e r m i n e  he  i f the  of the t e a c h - t e s t  referred  t o as t h e p r e - t e a c h - t e s t  based  The  test  was  students and to  on new  at the enroll  test  material  administered  i n Mathematics University i n the  would  be  These  items.  of the  The  12,  Mathematics  of mathematical  an  and  Columbia  four  under-  second  and  two  solutions  items w i l l  secondary  be section  format.  schools  to  Mathematics  to students  planning  and  validity  items used  have  of the  counselling  c o n t e n t were been  made  included  in  this'  paper. I. At entrance few of  S E L E C T I O N AND  the undergraduate to the U n i v e r s i t y  y e a r s c a n be three  college  l.P-3 .  1968.  decreased i f i t s precise  none  whose  exposed  s t u d e n t had  in several  of  the r e l i a b i l i t y  been  of  presented i n a teach-test  11,  Since  available,  problems  had  content.  of B r i t i s h  autumn  consisted  student:'s comprehension  standing  was  test  P R E P A R A T I O N OF level  nearly  of B r i t i s h  TEST  ITEMS  a l lpersons Columbia  categorized  by  h a v i n g been  preparatory  mathematics  seeking,  i n the  enrolled  programs.  next in  Most  one of  11. the  new  students w i l l  offered will  in British  have  British  Columbia  British  With  of  involved  basic  the  skills  skill  fifty-seven in  individuals  had a d i f f e r e n t  on t r i g o m e t r i c  that  could  above  that  section,  have  been  mathematics  are needed  one.  This  obtained  of B r i t i s h  THE  series  test  was  11  a secondary school  case.  and f o r t y - f i v e in British  the test  acquired  programs.  year  by Many  mathematics  after  Forty-five  analyzing administration,  students  enrolled  students i n Mathematics  Columbia wrote  calculations  administering  For the f i r s t  All  of the students'  For  e a c h p r o b l e m t w o c o m p a r i s o n s w e r e made.  the  number  o f s t u d e n t s i n each  the items  Columbia.  d e v e l o p e d by  and m o d i f y i n g i n each  concepts  TESTS  i t e m s were p r e p a r e d .  Mathematics  coming  for readily  the concepts presented i n f i r s t  of the test  results  have  items i n the f i r s t  at the University  Section  ' Those  of items based  II.  versions  program.  i n any o f t h e t h r e e  the items tested  courses  Some p e r s o n s  courses.  the exception  comprehending  currently-  courses i n the previous  Columbia w i l l  and t h e p r e - t e a c h - t e s t  students  courses  secondary schools.  Columbia mathematics  mathematics  chosen  taken mathematics  completed mathematics  from beyond of  have  and answers  c o u r s e who  12 a t  this examination. were,  collected.  One was  obtained  a  between correct  12 .  solution. and  The o t h e r  was "between t h e t o p t w e n t y - o n e  the bottom twenty-one  wrote  the test.  The s t u d e n t s '  most  of the alternatives  form  of these All  second 'not  and t h i r d  the  in  given'.  were  presented  space  version.  whose  scores  For both  were  i n the upper of the test  mathematical  items.  instructions  as m e n t i o n e d  alternatives Choices  necessary,  answers  Columbia.  of these  were p r o v i d e d  If  item.  m u l t i p l e choice  alternatives  choices.  each  The  selection  items  were  given  were never  a new  27%  questions  groups were There  question  F o r most  inserted  analyzed. were  five  'not g i v e n ' w i t h  t h e same  of t h e items the p r e v i o u s l y used  s e l e c t e d by t h e s t u d e n t s  by t h e s t u d e n t s .  at a  who w r o t e t h e  constructed.  including  was  included  students  s e l e c t e d from t h e seven  choice  was  a d m i n i s t r a t i o n s the papers  F o r each  above.  instructions  Thirty-two  or lower was  For the  t h e c o r r e c t answer i n  when h i s m u l t i p l e c h o i c e  in British  A f o u r t h form  five  choice  including  t o the one-hundred-seventeen'pupils  third  forty-two  with  choices  v e r s i o n w r i t t e n by e i g h t y - o n e  school  given  who  used t o provide  were m u l t i p l e c h o i c e .  was t o p r o v i d e  Forty-six  t h e second  secondary  12  selected f o r the multiple  formats  the student  designated  'not  answers were  a d m i n i s t r a t i o n , seven  were  stated' that  i n Mathematics  questions.  subsequent  given'  students  students  from  were e l i m i n a t e d .  among t h e w r o n g  13.  During who An  t h e week  currently  were  appropriate  computing  the  this  The  two.  administered  some k n o w l e d g e  articles  i n part  from  The  the second  group  scores  was  one w h i c h  obtained  on  this  computed.  dependent  were  of  questions  upon t h e The  indicated  student's  students  t h a t they had  included i n the teach-test  further analysis.  sequence  obtained  score  and t h i r d  correlation  correlation test  Columbia.  A l l of the students'  collected.  The o n l y  between  i n which  alteration  the third  the articles  and f o u r t h appeared  changed. For  113  pupils i n  f o l l o w e d by a s e r i e s  on t h e t e a c h - t e s t p o r t i o n o c c u r r e d  was  was  of the subject matter  and answers  administration.  mathematics  The t e a c h - t e s t p o r t i o n c o n s i s t e d o f f o u r  omitted  calculations  Columbia  d i d measure  in British  of the w r i t t e n explanations.  whose r e s p o n s e s  were  test  the score  s p r i n g grade  test.  analysis.  to nineteen  between  students  of B r i t i s h  that this  school  the correct solution  understanding  items  indication  coefficient  explanatory  which  f o r an i t e m  113 a t a s e c o n d a r y  Section  for  was  and t h e s t u d e n t s '  brief  University  hundred  12 t o o k t h i s  of success i n a college l e v e l  correlation  test  were used  test  Mathematics  i n Mathematics  and'the  an i n i t i a l  likelihood  course  enrolled  program  center  To g i v e  of spring vacation three  a d m i n i s t r a t i o n s and'the  coefficients  were  on s e c t i o n o n e a n d t w o .  coefficient a l s o was  between  obtained.  Mathematics  computed between For the later  t h e s p r i n g grade  the  group  a  and t h e t e a c h -  CHAPTER I V AN A N A L Y S I S For  the first  numerical action ship  between  students full  written  Table  I  of the item  The c o r r e l a t i o n  spring  i s 0.819.  provides a analysis  and  grade  .68.  was  analysis  of B r i t i s h  forty  on t h e m a t h e m a t i c a l c o u n s e l l i n g  for  answers.  correct  In the future  answer  For  section.  between  sections  rations  was  .52  a n d .56  correlation  was  .48.  grades  and part  test.  will  The c o r r e l a t i o n  respectively.  F o r about offered  f o r which the  c o e f f i c i e n t.  and t h i r d  For Mathematics  .The c o r r e l a t i o n b e t w e e n .67.  include  be d e c r e a s e d .  one a n d t w o f o r t h e s e c o n d  t w o was  to  i n the choices  t h e number o f i t e m s  i s 'not g i v e n '  the second  and by t h e  i n the decision  o f t h e q u e s t i o n s a c h a n g e was made  113  o f the. t e s t s  Columbia  113 students r e s u l t e d  half  subsequent  The KR20 o f t h e  Mathematics items  complete  by t h e Mathematics  The computer  at the University  CONCLUSIONS  coefficient f o rthe r e l a t i o n -  the scores obtained  and t h e i r  test  section  description  taken.  OF THE DATA AND  the  administ113 t h e  spring  T A B L E I - A N A L Y S I S OF  1  2  4  3  Math o =tfc H H UN  J- -p pq II cd S3  S  6  5  o o 00  „ Vlcvi  -=t -P II cd  S  •  H  H  O  II  =te  a  H  H . i H OJ II CO A| || O II O fl  fl  8  10  9  11  12  13  Run #2 • R u n #3 27#=21.8J • 27% = 3 1 . 59  12  o =«= oj H  0  7  15.  RESULTS  =te  o • cd o H CU PL, CQ  frOJ  OJ  fH  u  (U H is OJ O Ii  J  CM  Q) H ft CM ft  II  S-i  CU H is ro  ;0  II  IH  CU o • cd H U PH CO  ft  D  1  -.8  37  5  2  20  1  12  20  2  7  27  10  1  16  2  9  19  3  32  . 34  11  3  20  10  11  18  4  hi  29  11  2  16  4  18  18.  5  21  31  !3  0  18  6  6  6  28  2 9,  11  0  18  8  . 7  21  7  2 . 13  5  1  7  M7  13  18  8  9  42  19  2  21  M3  2 . 14  M5  3  15  17 •  ft ft  0  & '  a u o o  H ft CO  (3  <D  t— OJ CD  II  a  -  n H +J cd CQ P3 -H fl -H !H 0) S O 0) -p . PS ft to H  -=f  pq  2  .346  3  .372  -  10  . 310  -  D  11  30  8  28  6  • 459  14  31  8  .448  7  .189  4'  .458  -  5  .378 .460  -  14  30  9  6  22  5  3  14  9  16  21  -  9  10  20  22  8  3  11  11  11  18  -  11  . 343  11  0  0  0  0  0  15  0  0  0  0  15  -. 019  12  0  0  0  0  0  20  0  0  0  0  20  .079  13  0  2  0  0  2  34  0  0  0  0  31  • 277  14  0  0  0  0  0  39  0  0  0  '0  36  .242  15  1  3  1  1  1  D  D = item  Dropped  M  item  Modified  C  Corre ct  n  number  of  subjects  14  D  16.  TABLE I ( C o n t i n u e d )  1  2  3  16  19  29  k  5  6  11  3  15  7  8  9  23.  1+  5  10  11  -  12  13  D  17  k  3  0  0  3  1*5  0  0  2  18  k2  .1*62  18  1  0  0  0  0  30 .  0  k  0  13  27  . 332  19  3  6  1  0  5  D  20  0  2  0  0  2  D  21  0  0  0  0  0  D  22  2  0  0  0  0  M22 .  1  13  5  10 .  22  .175  23  1  0  0  0  0  Mkk  0  0  2  15  in  . 1*07  2k  5  13  2  0  11  17  •lU  16  12  28  17  .391  25  0  22  6  2  Ik  18  5  15  8  29  18  .1*16  26  0  6  1  0  5  M31  ' 0  7  1  2k  28  .376  M37  0  3  18  26  3k  .380  3  11  2k  .511*  27  0  3  0  0  3  M27.  28.  0  0  0  0  0  D  29  0  8  2  0  6  M2 9.  1  16  30  7  . 5  0  1  k  MU3  0  5  31  lU  29  10  1  18  16  13  17  32  1  3  1  1  1  32  0  2  33  2  8  0  2.  6  13  8  16  3k  1  3  0  1  2  MlU  0  8  10  13  0  2  11  19  1  11  35  D = item  Dropped  M = item  Modified  C = Correct n = number o f s u b j e c t s  .  -  D 5. 10  20  26  .3.1*8  12  1*0  . 110  16  . 212  3  ik  29.  . 1*17  3;  •17  13  .1*1*0  lU  .H-97  19  . 3 52  8  29  TABLE 1  2  3  4  5  6  36  1  5  0  1  37  0  3  0  38  2  8  39  37  4o  I  17  (Continued)  7  8  9  10 11  12  13  •4  33  0  10  4  25  30  .382  0  3  D  0  0  8  38  2  12  13  31  35  . 298  25  4  2  19  12  18  21  19  29  12  .158  8  14  3  1  10  26  2  15  22  23  .388  4l  0  3  1  1  1  2 4.  0  6  42  21  33  13'  1  21  21  5  16  12  30  M21*  . 209  8  D  h3  3  15  0  1 . 14  46  0  5  11  23  • 43  .314  44  0  15  2  0  11  4i  0.  6  6  22  38  .403  45  9  37  16  3  18  D  46  1  22  1  2  22  28  M25*  .258  47  2  2  0  0  .2  D  48  3  36  15  3  18  D  49  0 . 10  0  0  10  36  33  .367  50 .  3  11  5  0  6  D  51  2  19  0  "3  16  25  52  0  26  2  12  D  53  '0  0  0  0  •54  0  0  0  55  0  - 4  56  0  3  57  0  19  19  0 . •4  5  21  1  6  0 . M35  0  5  7  26  32  .340  0  0  M4O  0.  1  0  .13  •37  . 329  0  0  '4  D  0  0  3  D  0  0  0.  3  1  11  39  ,447  12  D  item  Dropped  M  item  Modified  *  accidentally  C  Correct  n  number  of  8  19  modified  subjects  • 42  D  18.  BIBLIOGRAPHY Anderson,  James  and  "Predictive Freshman  Utility  R.W.  Mathematics  Tests  and at  R.C.  One  State  The  E.A.  Criteria  for  Advanced  Mathematics  Teacher,  619-20.  "Predictive  College,"  Validity  Educat i o n a l  (Winter 196-5),  "Scholastic  American  Whlttmore.  C o u r s e s , " The  Paxson.  M e a s u r e m e n t , XXV  Coddington,  G.  of Certain  ( O c t o b e r , 196-7),  LX  Boyce,  Robert  of Eleven  and  Psychological  lll*3-l*7.  Aptitude  Mathematical Monthly;  Tests i n Mathematics," LXX  (September,  1963),  750-55.  King,  W.H. to  "Differences  Types  i n Mathematical Achievement  of Secondary  Locations,"  S c h o o l s and  E d u c a t i o n a l .Research,  Related  Their Geographical VIII  (November,  19-6-5),  7^-80.  Kinzer,  J.R.  College XXXVII  Wampler, Joe  and  L.G.  Mathematics," (June, 1953),  F.  The  "Predicting  Journal  Grades  of A p p l i e d  in  Advanced  Psychology,  182-81*.  "Prediction  Mathematics," 36U-9.  Kinzer.  of Achievement  Mathematics  Teacher,  in  College  LIX  (April,  19 6-6),  APPENDIX  20. A TEACH-TEST  Radicals when  o f t h e same  order which  expressed i n the simplest  Hence- 7 / 3 The  and - 2 / 3  radical. written  are like  sum o f t w o l i k e  ITEM  form  h a v e t h e same  are c a l l e d  radicals.  radicals.  radicals  c a n be w r i t t e n  T h e sum o f t w o . u n l i k e r a d i c a l s , i n i t s simplest  like  radicand  form,  cannot  each  with  a  single  of which i s  be e x p r e s s e d w i t h  one  r a d i ca l . Example s: 5/2  (i)  +  7/27  (ii)  7/2  -  3/3  =  (5 + 7 ) / 2  =  12/2  = 7*3/3  +• U-/75  =  (iii)  7/3  +  (iv)  2/3  +2/5"  Simplify-  1) 2)  the following  / 2 + 3/2 - 7/3  5/2  +  2/12  = 7/3 = 2(/3  -  21/3  3/3  3/1  =  (21-3+20)/3  =  38/3  +  + I+.5/3 + 20/3  5> 2 /  +  /5)  .  statements:  3)  2 / 2 8 " - 5/"i+T + 5/63"  

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