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The attitude and achievement of tenth grade general mathematics students as effected by the use of desk… Humphries, Leslie Rae 1972

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n  THE  A T T I T U D E AND ACHIEVEMENT OF TENTH GRADE  GENERAL MATHEMATICS  STUDENTS AS E F F E C T E D .  BY THE USE OF DESK CALCULATORS  by  L E S L I E RAE HUMPHRIES 3. A., U n i v e r s i t y  A  of British  Columbia,  1961  THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF MASTER OF ARTS  in  t h e Department of Education  We a c c e p t required  THE  this  thesis  as conforming  to the  standard  UNIVERSITY OF BRITISH NOVEMBER, 1972  COLUMBIA  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  further  for  scholarly  by h i s of  agree  this  written  thesis  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  it  that permission  p u r p o s e s may  representatives. thesis  for  available  be g r a n t e d  It  financial  is  by  the  gain  Columbia  shall  not  requirements  reference copying of  I  agree  and  copying or  be a l l o w e d  for  that  study.  this  thesis  Head o f my D e p a r t m e n t  understood that  of/2  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, C a n a d a  for  for extensive  permission.  Department  Date  freely  British  the  or  publication  without  my  ABSTRACT  The the  use  of  purpose of  to  see  to  determine whether  i n a grade ten  would produce a p o s i t i v e  toward mathematics,  quantitative and  s t u d y was  a desk c a l c u l a t o r  mathematics c l a s s attitude  the  thinking  to see  a  change i n a  if" achievement  (problem s o l v i n g )  i f t h e r e was  general  w o u l d be  pupil  in increased,  c o r r e l a t i o n b e t w e e n any  such  changes. Two  teachers, each having  g e n e r a l mathematics, class.  The  taught  to a r r i v e at  solutions  mathematics. manner, b u t  The did  use  the  g r o u p s were t a u g h t  and  grade,ten a  to  control i n the  in  in a  similar  desk c a l c u l a t o r s .  post-tested  use  calculators  t o a p r e a r r a n g e d program  have a c c e s s  were p r e - t e s t e d  of  were i n s t r u c t e d  encouraged to  control  not  classes  experimental and  experimental classes  o f desk c a l c u l a t o r s , and  classes  an  two  The  for attitude  and  achievement. A Mathematics A t t i t u d e yields  (1)  a  and  (.3)  a  composite  test  comprised sixteen  (A),  a calculation  (problem  constructed  "Preference for Calculation"  "Preference for Quantitative score,  T e s t was  solving)  Thinking"  score,  (Problem  (2)  situation,  situations  (B),  s i t u a t i o n , and  a  Solving)  "Mathematics A t t i t u d e "  three-part  which  score.  consisting  a quantitative  a distracter,  (C),  Th of  thinking a  iii non-inathematical a c t i v i t y . like  and a  The p u p i l s  were a s k e d t o s t a t e  d i s l i k e among t h e t h r e e o f f e r e d .  using b i s e r i a l  r , a l l b u t one  When u s i n g  On  item  of the items proved  analysis  satisfactory.  t h e Spearman-Brown p r o p h e c y f o r m u l a f o r  estimating  reliability,  sufficient  f o r d i f f e r e n t i a t i n g b e t w e e n means o f g r o u p s , b u t  not  the r e l i a b i l i t y  groups also  o f C o v a r i a n c e was  for a significant positive  change  f o r a s i g n i f i c a n t improvement  improvement  i n a c h i e v e m e n t was  At  in  t h e 0.05  attitude 0.08  and  positive  change The  calculator  conclusion  No  change  was  The a t t i t u d e  and  change  i n achievement.  a n d improvement that  At  improvement  significant correlation  drawn was  of a  i n achievement.  t h e use o f a in attitude,  desk but  achievement w i t h a c a r e f u l l y planned  correlation  and achievement  i n attitude  a significant  makes no s i g n i f i c a n t c h a n g e  perhaps might program.  i n attitude  and  The  no s i g n i f i c a n t  no s i g n i f i c a n t c h a n g e  no  in attitude,  change  l e v e l , t h e r e was  T h e r e was  the  checked f o r s i g n i f i c a n c e .  l e v e l , however, t h e r e was  achievement.  differences.  i n achievement.  of a suspected p o s i t i v e  the  f o u n d t o be  u s e d t o compare  correlation  in  was  sufficient for differentiation of individual Analysis  a  of a positive  change  in  attitude  found. test  c o n s t r u c t e d may,  be o f u s e t o o t h e r e x p e r i m e n t e r s .  with  modifications,  i V  TABLE OF CONTENTS  Chapter 1.  2.  Page INTRODUCTION  1  BACKGROUND  1  PURPOSE OF THE STUDY  3  DEFINITION OF TERMS  4  HYPOTHESES  5  REVIEW OF THE LITERATURE  6  REVIEW BY AIKEN  6  ATTITUDE  9  ATTITUDE AND ACHIEVEMENT  11  CALCULATORS, ATTITUDE AND ACHIEVEMENT 3.  . . .  DESIGN OF THE STUDY  12 16  PURPOSE  16  PROCEDURE  16  HYPOTHESES  IS  T E S T SELECTION  19  Arithmetic  19  Attitude  Quantitative  Thinking  Mathematical  Attitude  TEST CONSTRUCTION Definition  o f Terms  Achievement '.  . . . .  20 22 22 22  Item S e l e c t i o n  23  Scoring  24  Item A n a l y s i s  25  V Chapter  Page Reliability  25  STATISTICAL DESIGN 4.  5.  26  ANALYSIS OF THE RESULTS  28  FIRST HYPOTHESIS  28  SECOND HYPOTHESIS  29  THIRD HYPOTHESIS  29  SUMMARY AND CONCLUSIONS LIMITATIONS  31 :  IMPLICATIONS FOR FURTHER RESEARCH  32 35  APPENDICES FOOTNOTES  37  BIBLIOGRAPHY  42  APPENDIX A  SAMPLE PROBLEMS  45  APPENDIX B  DUTTON'S ATTITUDE SCALE  48  APPENDIX C  MATHEMATICS ATTITUDE TEST  50  APPENDIX D  IOWA TEST OF QUANTITATIVE THINKING.  55  APPENDIX E  ITEM ANALYSIS OF MATHEMATICS ATTITUDE TEST  64  Chapter  1  BACKGROUND  One with  the  the  chief  the  case.  of  low  the  achiever  determiner Aiken  mathematics  a  important  having  extreme  having  more  the  low  the  good  for  low  attitudes  of  a  That  low  on  of  the  toward  Studies negative  positive  attitude  is  necessarily  ability  mathematics  deal  attitudes  achievement  a  to  ability  i s not  research  attitudes."'''  has  i s how  "mathematical  g e n e r a l l y has  achiever  teacher  achievement  out  determiner  achiever  the  review  points  moderate  of  i n mathematics.  in his  toward less  problems  of  than have  may  be  students those  shown  attitude  that  while  toward  2 mathematics. Dutton the  negative  arithmetic  has  listed  attitudes  "takes  too  s e v e r a l areas  toward long"  that  contribute  a r i t h m e t i c some b e i n g  and  was  "boring  or  to  that  stale"  with  3 "too  much  memorization."  The diverse  field.  acquiring parental  acquiring of  of  a  Some  of  negative  attitude,  4  negative the  contributing  attitude  course  attitudes  are  content,  a  and  covers  factors  slow the  existing  ones.  toward  learning teacher's  approach. The c l a s s r o o m t e a c h e r s c a n n o t hope a l l t h e c o n t r i b u t i n g f a c t o r s b u t t h e y m i g h t be the  a the  rate,  . .  to e l i m i n a t e able to modify  2 The is  partly  pupil's  self  developed  mathematics. confidence According  I f he  may  be  Alpert generates  fails  "The  i n the  in relation  his experiences  reduced  to Aiken  experiences  by  concept  to  mathematics  i n working  with  i n a t a s k many t i m e s  then  and  increased.  hostility  teacher  may  be  must p r o v i d e  for  self  success  learning.""'  et a l l e a d  a negative  us  to conclude  attitude  and  low  a negative  achievement attitude  6 generates negative of  low  achievement.  attitude  improving  c h a n g e an  t o a more p o s i t i v e one  achievement.  a c h a n g e c a n be  To  brought  There are  about.  However, t h e a t t i t u d e  existing m i g h t be  indications  one  way  that  such  ' r e v e a l e d by  the p u p i l s  i s not  9  a unique a t t i t u d e but  a multidimensional  measure what a t t i t u d e s be  changed and  how  D u t t o n has  are present  they  can  g i v e n us  dimensions or f a c e t s . ^ arithmetic situation  b e c a u s e he requiring  transferred  Perhaps  the  connotations the  rewarding  c h a n g e d i s a complex  indications  i t s l o w and  arithmetic  calculations  negative  what a t t i t u d e s  o f the  I f a p u p i l were t o  finds  To can  problem.  attitude  dislike  tedious  then  c o u l d have a n e g a t i v e  any attitude  to i t .  P e r h a p s by tedious  be  and  attitude.  removal, o f  the  the  necessity  of  doing  s i t u a t i o n tiiat f o r m e r l y would  c o u l d have a p o s i t i v e  situation in itself  which p r e v i o u s l y  m i g h t be  c o u l d not  be  have  connotation.  interesting because of  and  the  3 negative a t t i t u d e  toward c a l c u l a t i o n s .  Several  such It  programs  have In  been  order  expressed  desk . - c a l c u l a t o r s  have b e e n  t h e i r optimism about have  u s e d i n home a n d  validate  t h e machines  c a r r i e d opinions  the r e s u l t s .  have  use i n s c h o o l .  f r o m o b s e r v a t i o n s b u t do  H e a d l i n e s such as  "Calculator  Takes S t i n g use  '  i n solving  The m a n u f a c t u r e r s o f desk c a l c u l a t o r s  Newspapers not  results."'  t o r e d u c e t h e work i n v o l v e d  q u e s t i o n s posed, industry.  attempted with varying  12 13  Out o f M a t h " have g e n e r a t e d q u e s t i o n s 14 o f c a l c u l a t o r s i n a mathematical program.  about t h e  PURPOSE OF THE STUDY  In  t h i s chapter various  w h i c h make c l a i m s program  on t h e a t t i t u d e  From t h e s e c l a i m s would  about  seem t h a t  to produce  about  the e f f e c t s  1. A p o s i t i v e  change  based  o f the low a c h i e v e r .  improved a t t i t u d e  outcomes  cited  of a calculator  and achievement  a calculator  several  s o u r c e s h a v e been  and achievement i t  b a s e d program  might b e e x p e c t e d  f o r t h e low a c h i e v e r : i n general  attitude  toward  mathematics. 2. A g e n e r a l  improvement  i n problem  solving  achievement. 3. A p o s i t i v e in  attitude  c o r r e l a t i o n between  a n d improvement  a positive  i n problem  change  solving  achievement. A proposal  f o r a s t u d y was p u t f o r t h w h i c h  compare t h e a t t i t u d e s  and achievement  o f four  would  classes  4 o f g r a d e 10 g e n e r a l m a t h e m a t i c s desk c a l c u l a t o r s  and two  c a l c u l a t o r s were to be each  having  classes  be  control  used.  control  w o u l d be  attitudes before  one  taught  students.  and  Two  c l a s s e s not u s i n g  Two  statistically  one e x p e r i m e n t a l  as s i m i l a r i l y  the experimental analysed  the  t e a c h e r s w o u l d be i n v o l v e d  as was  and a c h i e v e m e n t s w o u l d be measured  and a f t e r  classes using  period.  for significant  class.  The  feasible. by s e l e c t e d  The m e a s u r e s changes  The tests would  and  correlations.  DEFXNITION OF'TERMS  (a) Q u a n t i t a t i v e t h i n k i n g : supplied  data  operations question  together with  the a p p l i c a t i o n  of arithmetic to a r r i v e  to which the student  or procedures  a r e t o be  is  who  normally  that  a program  of Education  to a  given  to those  designed  by  the  t o be g i v e n  to  those It  whose a c a d e m i c a c h i e v e m e n t i s such, t o compete  successfully  with  more a b l e . (c) d e s k c a l c u l a t o r :  Underwood D i v i s i - S u m a  24 w h i c h  a machine  such  i s a t e n key  as the machine  *-  Chapter  steps  a r e n o t on an a c a d e m i c - t e c h n i c a l p r o g r a m .  t h e y w o u l d n o t be a b l e  others  at a s o l u t i o n  taken.  Columbia Department  students  of  of basic  has not b e e n t o l d what  (b) g e n e r a l m a t h e m a t i c s : British  the a s s i m i l a t i o n  A more d e t a i l e d 3 pages 2(5-2 7.  outline  i s t o be f o u n d i n  Olivetti  5 performing division  addition,  with  subtraction,  printed  multiplication,  and  output.  HYPOTHESES  Three  hypotheses  1. T h e r e attitude  toward  Mathematics 2. will  show  than Where  quantitative direction  using  The a c h i e v e m e n t greater  f o r those there  change  f o r the grade desk  there  f o r those  i n the achievement  change will  ten General  quantitative  not using  i s a  i n general  calculators.  i n doing  improvement  thinking  formulated.  be a p o s i t i v e  mathematics  students  calculators 3.  will  were  using  desk  desk  calculators.  i n attitude  be a change  i n doing  thinking  toward  i n t h e same  quantitative  thinking.  Chapter  2  REVIEW OF THE LITERATURE  REVIEW BY AIKEN  In Aiken's attitudes  toward  (1969) r e v i e w o f t h e r e s e a r c h c o n c e r n i n g  m a t h e m a t i c s he d i s c u s s e s s e v e r a l  of measuring  attitudes  a r e no v a l i d  measures.^  superficially observation  even  though  some m a i n t a i n t h a t  A i k e n notes  that  "observationi s  t o be inadequate."- -^  A  1  second  method i s a q u e s t i o n a l r e where s t u d e n t s a r e a s k e d  to i n d i c a t e  there  t h e most o b j e c t i v e m e a s u r e b u t . . . t e a c h e r  ( i s found)  true or f a l s e  methods  to a statement  an o v e r a l l  to state  and t h e responses a r e t a b u l a t e d  attitude  or, i n another  example, a  17 preference  to a subject  The  field.  Thurstone and L i k e r t  t e c h n i q u e s were f o u n d  discriminating  Picture  resistance, heart beat  used  achievement  t o be  grade. o f a n x i e t y such  breathing characteristics,  as  electrical  blood pressure  r a t e were f o u n d b y A i k e n t o be u s e d b y a few 19  experimenters.  indicators  scalogram  A s t u d y by N e a l e i g h  p r e f e r e n c e was f o u n d  at the t h i r d  Physical  and  used.  p r e f e r e n c e s t o measure a t t i t u d e s and  proneness.^  skin  scaling  t o be p o p u l a r , b u t Guttman's  a n a l y s i s was i n f r e q u e n t l y picture  attitude  7  When t h e very  attitudes  definite attitudes  were m e a s u r e d i t was  toward a r i t h m e t i c  may  noted  be  "that  formed  as  20  early Junior  as  the  third  grade."  H i g h S c h o o l was  I t was  the  period  further  when a  noted  large  that  percentage 21  of prospective Aiken by  teachers summarizes  the  their  attitude.  section  on  measuring  attitudes  saying: . . . a t t i t u d e s a r e p r o b a b l y not v e r y s t a b l e i n t h e e a r l y grades. In a d d i t i o n , t h e p r e c i s e n e s s w i t h w h i c h p u p i l s can e x p r e s s t h e i r a t t i t u d e s v a r i e s w i t h l e v e l o f maturity. F i n a l l y i t i s c l e a r that a t t i t u d e s toward d i f f e r e n t a s p e c t s o f a r i t h m e t i c and m a t h e m a t i c s a r e m e a s u r e d by " g e n e r a l a t t i t u d e " i n s t r u m e n t s a d m i n i s t e r e d at d i f f e r e n t grade l e v e l s . A t t i t u d e toward m a t e r i a l s t o be l e a r n e d by r o t e , s u c h as t h e m u l t i p l i c a t i o n t a b l e , i s not t h e same v a r i a b l e a s a t t i t u d e t o w a r d w o r d p r o b l e m s and a l g e b r a i c symbols. Of more i m p o r t a n c e than the exact frequency of a t t i t u d e s at d i f f e r e n t grade l e v e l s , however, a r e t h e c a u s e s a n d e f f e c t s o f t h e s e attitudes. ^ I t was  analysis and  formed  noted  of the  Alpert  relationships  performance,  p e r f o r m a n c e as  that  of  al  (1963) made  among a t t i t u d e ,  they viewed the  a kind  et  self  level  of  perpetuating  an  expectation  expectation cycle  and  affecting  23 the  child's self  found higher  the  concept.  A n o t h e r s t u d y by  " c o r r e l a t i o n of a t t i t u d e  for arithmetic  and  Brown and  achievement  than f o r s p e l l i n g , reading  Abell  was  or  24 language."  D u r r a n c e f o u n d s i g n i f i c a n t c o r r e l a t i o n between  performance  i n m a t h e m a t i c s and  measures o f a t t i t u d e  and  25 anxiety.  I t was  positive  attitude A  study  by  n o t e d by  Cech  that  achievers  had  a  toward mathematics  than  more u n d e r a c h i e v e r s . 26  Jackson maintains  that  i t i s only  at  8 "the  extremes - h i g h l y p o s i t i v e  attitude  affects Aiken  achievement  or highly negative  i n any s i g n i f i c a n t  summarizes h i s r e v i e w  among a t t i t u d e ,  - that  way."^  o f the r e l a t i o n s h i p s  e x p e c t a t i o n , and p e r f o r m a n c e by s a y i n g :  C o l l e c t i v e l y , the findings o f studies relating p e r s o n a l i t y v a r i a b l e t o mathematics a t t i t u d e and a c h i e v e m e n t i n d i c a t e t h a t i n d i v i d u a l s w i t h more p o s i t i v e a t t i t u d e s and higher achievement tend t o have b e t t e r p e r s o n a l a n d s o c i a l a d j u s t m e n t t h a n t h o s e w i t h n e g a t i v e a t t i t u d e s a n d low a c h i e v e m e n t . These r e s u l t s must b e k e p t i n p e r s p e c t i v e , h o w e v e r . The c o r r e l a t i o n s a r e r e l a t i v e l y low, a n d i t i s a t r u i s m t h a t c o r r e l a t i o n does not imply c a u s a t i o n . Personals o c i a l adjustment, a t t i t u d e s , and achievement not o n l y i n t e r a c t w i t h each o t h e r , b u t they a r e t h e e f f e c t s o f o t h e r home, s c h o o l , a n d community v a r i a b l e s . In h i s review change a p u p i l ' s  Aiken  attitude  of himself i n r e l a t i o n  has a l s o  noted  that " i n order t o  toward mathematics h i s p e r c e p t i o n  t o m a t h e m a t i c s m a t e r i a l s must be  29 changed."  That  confidence,  p u p i l s who c o n s t a n t l y f a i l  develop  dislike  and h o s t i l i t y  loose  was n o t e d  self by L e r c h  30 and  others.  The p u p i l  w h i c h he c a n e x p e r i e n c e can  be b r o u g h t  text writer, At  about  must be p r o v i d e d w i t h success.  How s u c c e s s  i s up t o many p e o p l e :  situations i n experiences  the teacher, the  t h e home a n d o t h e r s .  least  one i n v e s t i g a t o r  (Tulock)  recommended  that  "games, c o n t e s t s a n d o t h e r a u d i o - v i s u a l a i d s be u s e d t o 31 heighten for  interest  the p u p i l  pleasant  3 2  While  Natkin  " t o a s s o c i a t e mathematics with  may a l t e r  subject."  i n mathematics."  his attitude  or anxiety  suggests  something  toward t h e  9 Aiken  expressed  general attitude with  saying  "the concept  t o w a r d m a t h e m a t i c s s h o u l d be  the a t t i t u d e s  mathematics,  h i m s e l f by  t o w a r d more s p e c i f i c  f o r example, problem  supplemented  aspects  solving  of a  and  of  routine  drill."  ATTITUDE  The  problem  o f the p u p i l s '  mathematics  i s not  has  not  been f o u n d ,  and  Norton  yet  formed. matics  has  helped  They f e l t i s largely  attitudes  a new  one. and  The  as  a cultural  to  a study  the  by  l a c k of  interest that  c o n d i t i o n e d by  toward  problem  Poffenberger  negative attitudes  phenomenon a n d  o f some c h i l d r e n a r e  They s t a t e  solution  such,  t o show how  the present  negative attitudes  are  i n mathethe  the f a m i l y .  that:  A t t i t u d e s a r e d e v e l o p e d i n t h e home i n some c a s e s b e f o r e the c h i l d begins s c h o o l . I n t h e f i r s t and s e c o n d g r a d e s he i s a f f e c t e d not o n l y by h i s t e a c h e r a n d h i s r e a d i n e s s t o d e a l w i t h numbers, b u t a l s o by t h e a t t i t u d e of h i s parents toward the s u b j e c t matter. He c a r r i e s i n t o h i s h i g h s c h o o l mathematics c l a s s e s a t t i t u d e s t h a t a r e l o n g i n b u i l d i n g and d i f f i c u l t t o c h a n g e . Certainly i t i s l o g i c a l t o e x n e c t t h a t t h e s t u d e n t who goes i n t o a c l a s s w i t h t h e t h o u g h t " H e r e i s a n o t h e r l o u s y math c l a s s " , i s s e v e r l y handicapped.34 A predetermined  attitude  is difficult  t o o v e r c o m e ; as  they  relate: S t u d e n t s w i t h an i n i t i a l l y n e g a t i v e a t t i t u d e t o w a r d m a t h e m a t i c s may go i n t o t h e c l a s s r o o m w i t h a m e n t a l a t t i t u d e s e t a g a i n s t t h e s u b j e c t w h i c h may be m a i n t a i n e d e v e n when p o s i t i v e i d e n t i f i c a t i o n w i t h t h e t e a c h e r i s made. It i s t h e r e f o r e of c o n s i d e r a b l e importance that p a r e n t s a n d t e a c h e r s i n t h e e a r l y g r a d e s make e v e r y e f f o r t to give p o s i t i v e experiences with a r i t h m e t i c . ^  In o r d e r toward  to determine  a r i t h m e t i c were, D u t t o n  scale.  Dutton's  scale  which a r e to b e checked is  an attempt  rupil  From  the a t t i t u d e s  has d e v e l o p e d  consists  agrees  an a t t i t u d e  with  them.  measure t h e a t t i t u d e s  the administration  has c o n c l u d e d t h a t  of pupils  o f twenty-two q u e s t i o n s  i f the person  to objectively  or teacher.  Dutton of  what  pupils  dislike  It  of a  of his scale  arithmetic  because  the f o l l o w i n g : 1. L a c k  problems without  o f understanding--confused practical  by  thought-  applications.  2. A r i t h m e t i c was hard--made p o o r  grades--irregular  attendance. 3.  Poor  teachers--punishment--frightening experiences.  4. Not s u r e o f s e l f - - i n s e c u r e - - m e n t a l b l o c k s — f e a r of  failure. 5.  Time f a c t o r s - - n o t  enough t i m e - - t a k e s  too long —  pressure. 6. A r i t h m e t i c was b o r i n g , s t a l e — s l o w l e a r n i n g , behind.  37 7. T o o much  drill —  I t seems f r o m that  For  the responses  i f some o f t h e r e a s o n s  removed o r r e d u c e d ,  memorization. r e c e i v e d by D u t t o n ,  for disliking  improvement  arithmetic  i n achievement  could  then,  were result.  again, Dutton s t a t e s : How p u p i l s f e e l t o w a r d a r i t h m e t i c i s i m p o r t a n t . L i k i n g a r i t h m e t i c has a p r o n o u n c e d e f f e c t upon t h e amount o f work a t t e m p t e d , t h e e f f o r t e x p e n d e d , a n d t h e l e a r n i n g that i s acquired.38  11 He a l s o c o n c l u d e s  that:  A p p a r e n t l y l a s t i n g a t t i t u d e s toward a r i t h m e t i c a r e d e v e l o p e d a t each g r a d e l e v e l . G r a d e s V a n d V I I were p r o n o u n c e d most c r u c i a l . Some c o n t r a d i c t i o n o f c o n c l u s i o n s by  S t r i g h t concludes  made b y g i r l s  i n general  edge i n a r i t h m e t i c , study It  that  found that  a consistently better  f o r a study score i s  s t u d i e s , w h i l e b o y s have a  geography, and s c i e n c e .  girls  made no a t t e m p t  occurs  liked  arithmetic  The S t r i g h t  better  to correlate arithmetic  slight  than  boys.  achievement and  40 attitude  toward  arithmetic. A T T I T U D E A-ID ACHIEVEMENT  It early of  has b e e n h y p o t h e s i z e d  and that  they p l a y 41  the student.  a role  a t t i t u d e s a r e formed  i n the g e n e r a l  achievement  A s t u d y by Fedon s e t o u t t o d e t e r m i n e  some o f t h e a t t i t u d e s h e l d used the s c a l e d  that  attitudes  toward a r i t h m e t i c .  Fedon  then  toward a m a t h e m a t i c a l program t o  evaluate  that program. I n h i s summary he s t a t e s : The s c a l e s d i s c u s s e d i n t h i s p a p e r a r e a n a t t e m p t t o c a s t l i g h t on some a s p e c t s o f t h e a r i t h m e t i c curriculum that a r e l i k e d and d i s l i k e d . A t t i t u d e s ^ l a y an i m p o r t a n t p a r t i n t h e s u c c e s s o f t h e a r i t h m e t i c program. I f we f e e l that they a r e a v a l i d c r i t e r i a f o r e v a l u a t i n g the e f f e c t i v e n e s s o f our program, then the a p p l i c a t i o n o f t h i s s c a l e w i l l provide b e t t e r o p p o r t u n i t i e s to study c h i l d r e n ' s r e a c t i o n s as they e x p e r i e n c e a r i t h m e t i c i n daily l i f e . 4  To histogram remedial  relate  2  a t t i t u d e and achievement, Stephens used a  t o compare class.  t h e a t t i t u d e s o f an a c c e l e r a t e d  The h i s t o g r a m  indicated that  and a  the accelerated  12 class  d i d h a v e a more p o s i t i v e  attitude  toward  arithmetic  43 than the remedial Another  class.  s t u d y by L y d a  and M o r s e r e a c h e s  the  conclusions: 1. When m e a n i n g f u l methods o f t e a c h i n g a r i t h m e t i c are used, changes i n a t t i t u d e s toward a r i t h m e t i c take place. N e g a t i v e a t t i t u d e s become p o s i t i v e , a n d t h e i n t e n s i t y o f p o s i t i v e a t t i t u d e s becomes e n h a n c e d . 2. A s s o c i a t e d w i t h m e a n i n g f u l methods o f t e a c h i n g a r i t h m e t i c and c h a n g e s i n a t t i t u d e a r e s i g n i f i c a n t g a i n s i n a r i t h m e t i c achievement, that i s , i n a r i t h m e t i c a l c o m p u t a t i o n a n d reasoning.'*'* This  study,  However,  then,  indicates  the e f f e c t  q u e s t i o n as A i k e n of  significant  prediction males."  attitude  and D r e g e r  contribution  o f achievement  that has  methods c h a n g e  attitudes.  on  i s open  achievement  conclude  that  "...the  hypothesis  o f mathematics a t t i t u d e s  i s borne  out  f o r females,  to  to but  not  4 5  CALCULATORS, ATTITUDE AND  A s t u d y by o p e r a t e d computing  F e h r , McMeen, a n d machines a t  ACHIEVEMENT  S o b e l , u s i n g hand  the grade  five  level  made t h e  hypothesis: P u p i l s who u s e c o m p u t i n g m a c h i n e s t o l e a r n a r i t h m e t i c w i l l g a i n s i g n i f i c a n t l y i n paper and p e n c i l commutations, and i n a r i t h m e t i c r e a s o n i n g o v e r t h o s e who do not use t h e computing machines. This was  h y p o t h e s i s was c o n c l u d e d by  s u b s t a n t i a t e d by  the experimenters  1. M a c h i n e - t a u g h t ability; 2. M a c h i n e - t a u g h t ability;  students  their  experiment.  It  that: g a i n more i n r e a s o n i n g  s t u d e n t s g a i n more i n c o m m u t a t i o n  13 3. M a c h i n e - t a u g h t s t u d e n t s l e a r n more a l s o , b e c a u s e t h e y u n d e r s t a n d m a c h i n e c o m p u t a t i o n a s w e l l as o r d i n a r y arithmetic; 4. The i n t e r e s t o f s t u d e n t s and t e a c h e r s i n a r i t h m e t i c i s h e i g h t e n e d by the use o f machines; 5. The m a c h i n e s f i t i n t o o u r p r e s e n t c u l t u r e . They g i v e a d d i t i o n a l learning.4*7 The was  not  c o n c l u s i o n reached  that  concludes  reached  that  by  Durrance  "...there  calculator will  by  Fehr,  Sobel  i n h i s experiment,  i s no p r o o f  significantly  McMeen, a n d  enable  for  he  t h a t the use  of  a student  achieve  to  the  48 in arithmetic." A with  s t u d y was  n i n t h grade  improvement p a p e r and  calculators  m e a s u r e d by  attitude and  their attitude  computational  test  this  developed  test  This test  by  does n o t ,  toward mathematics  the use  SMSG was  The  used  significant  and  (2)  skills.  The  showed no  of  test  results  PY011  Pro-  as  differences  however, d i f f e r e n t i a t e  t o measure t h e  the experimental d i d not  their  of  S t a n f o r d Diagnostic A r i t h m e t i c Test, Test used  in  (3)  for  in  among  in general.  B and C was  Stanford  to test  toward q u a n t i t a t i v e t h i n k i n g  P a r t s A,  skills  calculators  toward mathematics  skills  showed no  toward computation,  The  the  Cech u s i n g d e s k  to obtain computational r e s u l t s .  Math C o m p o s i t e  attitude.  by  g e n e r a l mathematics students  i n (1)  pencil  performed  test  computational  significant and  differences  the c o n t r o l  f o r improvement  more complex q u a n t i t a t i v e t h i n k i n g .  2,  groups.  in The  i n achievement  14 The e x p e r i m e n t d i d show s i g n i f i c a n t improvement the  e x p e r i m e n t a l g roup i n computing  using  calculators  of as  49 opposed  t o the group Various  effect  computing  without  g r a d e l e v e l s have b e e n u s e d t o t e s t  of c a l c u l a t o r s  o r s i m i l a r machines  of the p u p i l s .  The s t u d i e s  Cech  dissimilar results.  give  quite  attitude affects substantiates has  been  it  has n o t b e e n  by  on  I t seems t h a t  the  achievement  been  that  shown  Little  that  research  ten l e v e l  to see i f a t t i t u d e s  attitudes  are formed e a r l y ,  t h e y c a n be c h a n g e d  can  but  at  the  stage. of children  toward a r i t h m e t i c  formed e a r l y  i n t h e home.  Children's  be m o d e r a t e d  by a p o s i t i v e  i d e n t i f i c a t i o n with the  attitudes  and p o s i t i v e e x p e r i e n c e s w i t h a r i t h m e t i c . Dutton, learning  are  can  sometimes teacher  According  to  t h e amount o f work a t t e m p t e d , e f f o r t e x p e n d e d , acquired  arithmetic.  The  i s dependent  upon t h e a t t i t u d e  relationship of attitude  open  to question,  ship  has a p o s i t i v e c o r r e l a t i o n , white  not  and  I t i s supposed  b u t n o t much has  determined that  The a t t i t u d e s  found  the  Fehr et a l , Durrance,  f o r mathematics.  done a t t h e g r a d e  changed.  later  learning,  t h i s claim  be  them.  that  f o r Stephens  t h e r e l a t i o n s h i p was  indicated  and  toward  and achievement i s that  the r e l a t i o n -  A i k e n and  a predictor  Dreger  for girls  but  f o r boys. Studies  general  using  conclusions.  calculators  are not c o n s i s t e n t  Fehr et a l found that  t h e r e was  in their a  15 significant  improvement  i n arithmetic  reasoning, w h i l e Durrance achievement.  computation  f o u n d no s i g n i f i c a n t  and  change i n  Chapter  3  DESIGN O F THE STUDY PURPOSE  The p u r p o s e o f t h e s t u d y was t o d e t e r m i n e the  use o f a desk c a l c u l a t o r would  in a pupil's  attitude  ment i n q u a n t i t a t i v e if  produce a p o s i t i v e  toward mathematics, thinking  whether change  to see i f achieve-  would be i n c r e a s e d ,  t h e r e was a c o r r e l a t i o n b e t w e e n a n y s u c h  and t o see  changes.  PROCEDURE  Two t e a c h e r s , classes  o f grade t e n g e n e r a l  involved one  including  i n the experiment.  class  using  the experimenter, and four  mathematics Each  desk c a l c u l a t o r s ,  teacher  pupils,  were  h a d two  the other  classes,  u s i n g no  calculators. All  classes  administrators the  pupils'  available.  a t the start  programs,  were g r o u p e d by t h e s c h o o l of the school  the electives  year according t o  chosen and  No s t e p s were made t o r a n d o m i z e  Each c l a s s test,  involved  was pr e - t e s t e d  Iowa a c h i e v e m e n t  one  room  test.  the p u p i l s .  with the Dutton  t h e e x p e r i m e n t e r ' s own m a t h e m a t i c s  the  classes  attitude  The c a l c u l a t o r s  attitude t e s t and  were s i t u a t e d i n  t o be u s e d b y t h e e x p e r i m e n t a l g r o u p .  The c o n t r o l  17 g r o u p was due  to  of the  The  teachers  the  l e s s o n s t o be  similar  classes. use  in a different  the p r e s e n c e  discuss as  taught  as was While  the  errors  During taught  b e f o r e and  and  pr e s e n t e d .  different  the  being  the control  g r o u p was  taught  first  lessons the c o n t r o l  certain  i n columns and i n w h i c h one  then  third  quick addition  with  function  and  and  check.  r e a d and  The the  Groups and  add  pupils  numbers  were  copied,  and s u b t r a c t i o n .  one  machine.  groups  the p u p i l s  While  The  two  the  remaining  grouped  in  the  one was  add and  were e x p l a i n e d .  alternately  were  T h e y were i n s t r u c t e d  the o t h e r would m e n t a l l y  on  was  numbers  others  ways o f remembering a n d  operations  group  t o copy a n d  The  pupils  and  for accurracy.  of the r e s u l t s .  both  to f i n d  copying  checked  o p e r a t i o n o f the machine.  the machine,  to  lesson consisted of exercises for  In the experimental in pairs  taught  t h e p u p i l s were t o c o p y  l e s s o n p u p i l s were asked  The  kept and  number c o m b i n a t i o n s .  grouped i n p a i r s added.  teachers  g r o u p was  Some e x e r c i s e s were t i m e d  three digits  c l a s s was  experimental  o f numbers were s u p p l i e d w h i c h  with  Each  to  in calculating.  adding  In the second  contamination  during the experiment  some ways o f r e m e m b e r i n g numbers,  correctly  add.  t o pr e v e n t  calculators.  f e a s i b l e with  the c a l c u l a t o r s ,  eliminate  met  room  verify copying  using some numbers  l e s s o n s were s i m i l a r  o p e r a t i n g the machine.  with  The  machine were g r a d u a l l y broadened  in  scone.  18 The  lessons of the  g r o u p s were as lesson  dealt  s i m i l a r as was  with  employees.  The  computations  the weekly  control  while  the  The  o f q u e s t i o n was  questions.  The  were u t i l i z e d . computations, installment operations  was  standard The  earnings  expected not  questions  commission e a r n i n g s ,  buying. either  The  singly  in  fifty-seven  for  seven  pupils  i n the c o n t r o l  three tests  at the completion  twenty-  t o do  thirty  with  the  computations.  t h e number and  of  exercises  payroll  simple  interest  involved  and  the  i n the experimental groups.  weeks d u r i n g F e b r u a r y ,  All  of  basic  combination.  T h e r e were s i x t y p u p i l s and  one  group because o f  calculations or  t o do  of questions dealt  control  o f a number  varied,, only  types  the  F o r example,  expected  experimental  calculations  and  feasible.  g r o u p was  speedier type  experimental  grouns,  The e x p e r i m e n t  March a n d A p r i l  were a g a i n a d m i n i s t e r e d of the experimental  of  ran  1968.  to a l l  program.  HYPOTHESIS  Three hypotheses  were  1. T h e r e w i l l  a p o s i t i v e change i n g e n e r a l  attitude  be  formulated.  toward mathematics f o r the grade t e n  Mathematics 2.  students The  u s i n g desk  achievement  show g r e a t e r improvement  than  f o r those  *  For  using  further  calculators.  i n doing q u a n t i t a t i v e  will  not  General  f o r those  using  thinking  calculators  calculators.  details please  see  the  appendix.  3. Where t h e r e i s a c h a n g e i n quantitative direction  thinking  there w i l l  i n the achievement  be  attitude  toward  a change i n t h e  i n doing quantitative  same thinking.  T E S T SELECTION  Arithmetic A  Attitude test  to determine  a pupil's  attitude  toward  50 arithmetic  has b e e n c o n s t r u c t e d by  c o l l e g e students w r i t e statements and  dislikes  forty-five further  toward  and  twenty-two statements  The  statements  both  From  The  These  were  f e e l i n g s mentioned  of a person's  attitude  o f v a l u e f o r each  o f the  some  screened  degrees.  toward  to  refer  could  appear  elicit  t o be  The  statements  f o r an  arithmetic.  3y u s i n g methods s u g g e s t e d by T h u r s t o n e a scale  likes  retained.  c o u l d be c o n s i d e r e d as h a v i n g f a c e v a l i d i t y indication  were  e x a m i n a t i o n were f o u n d  or n e g a t i v e t o v a r y i n g  had  these statements,  o p e r a t i o n s or p r o c e d u r e s which  feelings.  positive  on  Dutton  concerning t h e i r  s t a t e m e n t s were r e t a i n e d .  to a r i t h m e t i c a l various  arithmetic.  Dutton.  and  Chave,  twenty-two s t a t e m e n t s  was  51 established. The they  These  s c a l e d v a l u e s range  p u p i l s were a s k e d  agreed w i t h .  They c o u l d c h e c k  they wished.  The  t a b u l a t e d and  totalled,  arrived  at.  That  to check  s c o r e s f o r each and  1 to  10.5.  the statements  a s many o r as  statement  an a v e r a g e  i s , i f five  from  checked  response  statements  few  that as  were  score  were checked  was whose  20 values  totalled  w o u l d be  response  reliability  score  average  achievement  test  using  a test-retest  score  the  average  i s reported  to  be  a test  administered  w h i c h c o u l d be  and  for this  scored  test  c h o s e n was 4:  Form Y-3S".  questions  arrive  at  The  t o use  course,  a  statistical  are  is  however, t o m e a s u r e t h e  twelve.  The  test  T h e r e was  was  students designed  a class  test  of  who  the  had  ability  principles The  ability  with  t h a t c o u l d be  of to  questions  situations.  There  t o do  abstract  mathematical  grades  version c o n s i s t i n g of  three multiple-choice questions  that  be  Thinking:  have s p e c i a l  f o r use  one  Educational  and  posed.  measurement  grade  which c o u l d  concerned with  f i n a n c e and  ability.  was  background.  basic arithmetic ideas  or to i d e n t i f y  the  t o do Q u a n t i t a t i v e  i n v o l v e mainly  thinking  the  w i t h i n the  a  Iowa T e s t  s o l u t i o n s to the q u e s t i o n s  attempt,  test  as w e l l a s  "The  Ability  at  administered  easily,  Development T e s t  student  t o measure  mathematics course  s t a n d a r d i z a t i o n and The  test  m a t e r i a l a p p r o p r i a t e f o r the p u p i l s  the g e n e r a l  hour p e r i o d a l l o t e d  suitable  a  attained i n quantitative thinking, a test  i n v o l v e d with level,  Achievement  In s e l e c t i n g  sought which c o v e r e d  no  response  5 2  Selection.  the  the  of the  method on  Quantitative Thinking  10  then  six. The  0.94.  thirty,  nine  to  thirty-  administered  within  the  laborious analysis correct a  calculations of" t h e  the  by  question  arithmetical  student  in  s i x t y minute p e r i o d . the  The  students,  followed  by  operations.  know what  c h o i c e s were s u c h  is correct  b e f o r e he  a  The  have a g o o d a r i t h m e t i c a l  multiple  test  d i d not  but  did  require  s e l e c t i o n of test  did  s e n s e as that  could,  require  the  the  require  the  an  that  distracters  student  e x c e p t by  must  chance,  S3  select  the  correct  For  response.  e x a m p l e , one  of  In a s h o r t 4087 x  cut  198,  questions  method o f  one  What p r o d u c t this  the  first  multiplying  m u l t i p l i e s 4087 x  must t h e n be  r e s u l t i n order  was:  to  subtracted  obtain  the  200.  from correct  answer?  The  student  200  - 2 =  could  be  test  13 x 2 4087 x  3)  198  could  198  student has  but  not  test could place  his  by  the  0.91  for pupils  4100  the be  wrong  - 13 to  do  on  the  They  grade.  r e a d i l y by  It i s rated  reliability  same  them.  then  having  a mark s e n s e c a r d .  M e n t a l Measurements  of  with  and  response.  administered.  The  = 4087  machine s c o r e d  responses  Reliability.  4087 x 2 Not g i v e n  2  know what  to  been w i d e l y  excellent  x  4) 5)  13  know t h a t  distracted The  the  1) 2)  5 5  Yearbook. is said  as  good  T^e to  5 4  to  oe  about  22 Mathematical  Attitude  No t e s t  was f o u n d by t h e e x p e r i m e n t e r  measure t h e a t t i t u d e quantitative test  toward c a l c u l a t i n g  thinking.  that  and a l s o  I t was t h e n d e t e r m i n e d  would  toward  that  such a  s h o u l d be c o n s t r u c t e d .  TEST CONSTRUCTION  A  test  was s o u g h t w h i c h w o u l d  t h o s e who  like  simple  involving  planning,  as  well  as a t e s t  would p r e f e r were s e l e c t e d Definition  the l a t t e r  question or  but not the former. this  data  (A) :  questions  capacities,  those  who  Three  areas  type o f d i f f e r e n t i a t i n g .  the performance o f a  on a g i v e n  Thinking:  (B): the a s s i m i l a t i o n  to arrive  to which the student  specified  s e t o f numbers.  together with the a p p l i c a t i o n  of arithmetic  procedures  to a  t o l d what  a r e t o be t a k e n .  physical  ( C ) : an a c t i v i t y  or mental not i n v o l v i n g  of  of basic  at a s o l u t i o n  has n o t b e e n  Non-mathematical A c t i v i t y : either  from  o f Terms  or operations  operations  o r other mental  w h i c h might a l l o w  Quantitative supplied  evaluating  b u t do n o t l i k e  which would d i f f e r e n t i a t e  Calculation: operation  calculations  differentiate  mathematics.  stens  23 Item  Selection A grade  one  desirable  t e n g e o m e t r y c l a s s was  activity  mathematics and a l s o supplied  o u t s i d e mathematics.  were  one  that  liked  they  liked  the l e a s t .  1.  who  were a s k e d  t o put a check f o r the  t h e most a n d a c r o s s f o r t h e one  An e x a m p l e was  Solve  (Pi) , a n d  (C) and p r e s e n t e d a s a g r o u p o f  given as  (a) A d d up a c o l u m n o f 20 (b)  then  Some o f t h e s u g g e s t e d  (A), Quantitative Thinking  Non-mathematical A c t i v i t y to the p u p i l s ,  The a u t h o r  then p l a c e d i n the f o l l o w i n g  categories; Calculation  three  to think of  a n d one u n d e s i r a b l e a c t i v i t y i n  some o t h e r s o f h i s own.  or s u p p l i e d items  asked  they  follows:  three d i g i t  numbers 3  t h e e q u a t i o n 3x + 2 = 20  X  (c) Wash t h e b r e a k f a s t d i s h e s There situations, original classes  were i n t h e o r i g i n a l  each w i t h a statement  test  was  labelled  near and  as p o s s i b l e  thirty  f o r each c a t e g o r y .  The  students  9.  Since the students  were  t h e y were assumed t o b e a s  to the experimental  classes  i n attitude  achievement. The  responses  c l a s s e s were t h e n the t h i r t y certain of  some  Type I , a n d a d m i n i s t e r e d t o two  o f General Mathematics  General Mathematics  test  t a b u l a t e d f o r each  situations.  statements  the o r i g i n a l  o f t h e two G e n e r a l M a t h e m a t i c s 9 of the c a t e g o r i e s i n  From an i n s p e c t i o n  were r e v i s e d  statements  was  o f the  resnonses,  t o b e more s e l e c t i v e . "Watch  television".  One  This  24 statement The thus  was c h a n g e d t o "Watch Batman on  original  i t was r e v i s e d  negative  different  past  thirty  item  many s t a t e m e n t s  each  The f i n a l  Type I I , nine  from  this  or reassembled i n  consisted of sixteen f rom c a t e g o r y  r e v i s i o n was a d m i n i s t e r e d  A, B  t o a grade  nine  t h a t was h o m o g e n e o u s l y g r o u p e d a c c o r d i n g t o  were p l a c e d i n t h i s  Only class.  the teacher's  attitude  to indicate  and found  calculating  form  c o n t a i n i n g a statement  This f i n a l class  were r e j e c t e d  were c o m p a r e d w i t h  solving  was l a b e l l e d  two c l a s s e s o f g r a d e  marks i n m a t h e m a t i c s .  better  test  students.  combinations.  situations,  algebra  responses;  p o s i t i v e and  t a b u l a t i o n was made o f t h e r e s p o n s e s  tabulation,  C.  so t h a t i t e l i c i t e d b o t h  t o another  mathematics A  and  revised  administered  general  b u t no d i s l i k e  responses. The  and  h a d r e c e i v e d many l i k e  television".  followed closely  and a low p r e f e r e n c e  those  students with  The r e s u l t a n t  C+ o r  scores  expectations o f student  a strong preference f o r  b\ a p r e f e r e n c e  f o r problem  f o r non-mathematical  activities.  Scoring A s c o r e f o r e a c h c a t e g o r y was o b t a i n e d by a l l o t t i n g two  marks f o r e a c h p o s i t i v e  response  i n that  ( a c h e c k m a r k ) , one mark f o r e a c h b l a n k and  no marks f o r a n e g a t i v e r e s p o n s e  category.  Hence, t h e r e  category  i n that  (a cross)  i s a p o s s i b l e high  category, i n that  score o f  25 thirty-two,  and  categories,  A,  Scores preference and  a  low  score  B,  and  C,  were t h a n  toward c a l c u l a t i o n  three  scores. a t t i t u d e or  (A), q u a n t i t a t i v e t h i n k i n g  was  activities  (C).  d e r i v e d by a d d i n g  (B),  A  the A  and  scores.  Analysis An  item  a n a l y s i s was  methods s u g g e s t e d  by  twenty-seven percent biserial  ' r ' was  and  with  one  the  suggested the  found  i n the  or  a cross  on  this  scores  a l l items  minimum.  top  item  and  i n each  exception  solving category.  using bottom  were c o m p a r e d .  were f o u n d  The  test  A  category  t o be was  A  above  number  twelve  c o m p l e t e a n a l y s i s may  be  of A t t i t u d e Test for reliability The  using  s i x t e e n items  r e s p o n s e o f a c h e c k was scored  basis The  found  the  appendix.  performed. A  o f the  computed f o r e a c h  0.20  A test  basis.  p e r f o r m e d on  G a r r e t t " ^ whereby t h e  exception  problem  Reliability  was  Each o f the  a v a i l a b l e f o r an  t' e i n d i c a t e d n o n - m a t h e m a t i c a l  Item  of  ^ero.  received separate  mathematics a t t i t u d e s c o r e B  of  t o be  o d d - e v e n was  zero.  The  f o r each o f  correlation 0.4782.  The  0.5433, and  the  the s p l i t - h a l f  were s p l i t scored  one,  odd-even s c o r e s three  f o r the  an  and were  odd-even a  blank  tabulated  categories.  category  correlation  on  method  A  odd-even  f o r the category  f o r the category  C  odd-even  was B was  26 0.7001.  When u s i n g  the  Spearman-Brown n r o n h e c y  formula  S7  for  estimating  o f 0.64 70  f o r A,  obtained. 0.60  reliability  The  from  0.7043 f o r B,  reliability  which G a r r e t t ^  of  designated  toward Q u a n t i t a t i v e as  a distractor.  was  derived  from  Data f o r the scores A^,  B^  for  the  on  the  and  A.,  the  P  2  and  f o r the  A score Test  sum  The D^  scores  the  A taking  the  score  two and  Category  (C)  t h e s e two  the  subscores "Attitude was  toward  subscores  used  Mathematics" B).  (A +  Test  and  were  and  A ,  P>  Test  pre-test  scores and  obtained  Scale  and  and  2  2  the  designated A  B2  +  2  yielded  post-test,  were d e s i g n a t e d  scores while  respectively  post-test.  from t h e  Mathematics  toward Q u a n t i t a t i v e  d i f f e r e n c e between  of A t t i t u d e  the  (A)  "Attitude  pre-test  pre-test  was  for  yielded  Dutton A t t i t u d e  for  of  differentiating  i n d i v i d u a l s were o b t a i n e d from  f o r change i n a t t i t u d e  finding  for  of  + B-^ f o r t h e  Iowa A c h i e v e m e n t  P-^ and  Test  Mathematics A t t i t u d e  post-test.  d e s i g n a t e d D^  score  minimum  DESIGN  (B).  Thinking"  the  for  toward C a l c u l a t i o n "  The  were  differences.  Mathematics A t t i t u d e "Attitude  figures  above the  sufficient  STATISTICAL The  are  necessary  not  individual  halves,"  0.8236 f o r C  figures  b e t w e e n means o f g r o u p s , b u t differentiating  and  considers  5  snlit  the  post-test  toward Q u a n t i t a t i v e  Thinking  f o r change i n achievement  d i f f e r e n c e between the  and  was  post-test  Attitude  Thinking pre-test (B  2  found  and  by  -  B^).  by  pre-test  27 scores P  2  "  p  o f t h e Iowa A c h i e v e m e n t  o f C o v a r i a n c e was  from D u t t o n ' s T e s t  Attitude Test  a n d was  designated  l Analysis  scores  Test,  Test  used  to e v a l u a t e  the  (D-, and D2) , t h e M a t h e m a t i c s  (A^ + B-^ a n d A2 + 8 3 ) , a n d t h e Iowa  on Q u a n t i t a t i v e T h i n k i n g  (P-^ a n d P2) •  Achievement  The l e v e l s  of 59  s i g n i f i c a n c e were f o u n d f r o m a t a b l e by D i x o n a n d M a s s e y . The Achievement  correlation (P2 - P^)  Quantitative Thinking experimental  o f the scores  from Change i n  a n d Change i n A t t i t u d e  toward  (B2 - B-^) were c a l c u l a t e d f o r t h e  and the c o n t r o l groups.  were then c h e c k e d f o r s i g n i f i c a n c e number o f d e g r e e s o f f r e e d o m .  These c o r r e l a t i o n s  with  the respective  Chapter  4  ANALYSIS OF THE RESULTS  F I R S T HYPOTHESIS  The in  general  hypothesis  t h e r e w o u l d be a p o s i t i v e  a t t i t u d e toward mathematics f o r s t u d e n t s  calculators The for  that  was  number o f o b s e r v a t i o n s  t h e c o n t r o l group and s i x t y - o n e  made was  fifty-seven  f o r the  o f 118 o b s e r v a t i o n s .  Scale yielded results  experimental  Dutton's A t t i t u d e  as f o l l o w s :  CONTROL  EXPERIMENTAL  Mean  S. D.  Mean  S. D.  Pre-test  5.376  1.247  5.513  1.413  Post-test  5.326  1.127  5.562  1.303  F - r a t i o was f o u n d  freedom,  t o b e 0.89 w h i c h , w i t h  115 d e g r e e s o f  i s not s i g n i f i c a n t . The  the  using  tested using analysis o f covariance.  group f o r a t o t a l  The  change  subscore  results  o f the Mathematics A t t i t u d e Test  " A t t i t u d e toward Mathematics"  (A + B) were a s  follows: CONTROL  using  EXPERIMENTAL  Mean  S. D.  Mean  S. D.  Pre-test  33.70  6.902  32.20  6.623  Post-test  31.98  7.308  30.98  6.828  29 The  F-ratio  was f o u n d not  f o r t h e " A t t i t u d e toward Mathematics"  t o be 0.00 w h i c h w i t h  significant.  significant  subscore  115 d e g r e e s o f f r e e d o m i s  The c o n c l u s i o n must be t h a t  t h e r e was no  change i n a t t i t u d e . SECOND HYPOTHESIS  The quantitative those  hypothesis  that  t h i n k i n g would  u s i n g c a l c u l a t o r s than  was t e s t e d u s i n g analysis  the achievement show g r e a t e r those  improvement f o r  not using c a l c u l a t o r s  t h e Iowa A c h i e v e m e n t T e s t  of covariance.  i n doing  The r e s u l t s  results  of the test  with  were a s  follows: CONTROL  EXPERIMENTAL  Mean  S. D.  Mean  S. D.  Pre-test  11.82  4.445  13.08  3.625  Post-test  12.74  4.249  14.51  3.404  A  F-ratio  o f 3.33 w i t h  significant at  a t t h e 0.05 l e v e l .  t h e 0.08 l e v e l .  significant thinking  115 d e g r e e s  o f freedom  was n o t  I t i s , however,  T h e c o n c l u s i o n must be t h a t  change i n t h e a c h i e v e m e n t  significant there  i s no  i n doing q u a n t i t a t i v e  by u s i n g a c a l c u l a t o r - b a s e d p r o g r a m . THIRD HYPOTHESIS  The  raw s c o r e s  asked t o e s t a b l i s h  were h a n d l e d  the necessary  by a c o m p u t e r a n d i t was  calculations.  c o m p u t e r was p r e - p r o g r a m m e d w i t h o u t  prior  Since the  knowledge o f any  30 results, first  the t h i r d  hypothesis  attitude  towards  that  i n the achievement  tested  by  their  attitude  quantitative  and  was  t o be  f o r the c o n t r o l  conclusion change  Test  thinking  found  correlation  in  along  with  the  i n attitude  achievement  that  toward  i n doing  as  by  0.2105.  there  be  quantitative  of  t h e 0.05  i s no  was  change by  the  doing  Achievement  groups  Neither  quantitative  i n  -the I o w a  a  thinking  measured  the achievement  f o r the experimental  be  will  change  c o r r e l a t i o n between  i s s i g n i f i c a n t at  then  there  quantitative  thinking  measured  groups,  figures must  as  and  i s a positive  thinking  The  quantitative  Attitude  there  i n doing  correlation.  toward  Mathematics  Test  tested  where  quantitative  change  in  was  two. The  in  hypothesis  -0.0492, the  level.  correlation  thinking thinking.  and  The of  change  Chapter  SUMMARY AND  Research there  i n mathematics. are  formed  CONCLUSIONS  into attitude  i s a positive  5  formation  correlation  of attitude  Some o f t h e a t t i t u d e s  in junior  high  school.  of  the reasons f o r d i s l i k i n g  if  a desk c a l c u l a t o r  indicates  that  and achievement  toward  mathematics  I t was t h o u g h t t h a t  mathematics  some  might b e d e c r e a s e d  c o u l d be u s e d t o work t h e t e d i o u s  calculations. The p u r p o s e o f t h e s t u d y was t o d e t e r m i n e the  use o f a desk c a l c u l a t o r would change a p u p i l ' s  toward mathematics, thinking  t o see i f achievement  w o u l d be i n c r e a s e d ,  c o r r e l a t i o n between any such Four c l a s s e s were u s e d w i t h control The  whether  experimenter's For tests  a n d t o s e e i f t h e r e was a changes.  two t e a c h e r s .  a n d one e x p e r i m e n t a l  scale,  Each t e a c h e r class  thinking  using  t h e Iowa A c h i e v e m e n t  own M a t h e m a t i c s A t t i t u d e  t h e program  increases  desk  Test  i n achievement  and c o r r e l a t i o n  calculators.  using  Dutton's  and t h e  Test.  used, the grade l e v e l changes  pupils  t a u g h t one  and p o s t - t e s t e d  u s e d t o measure p o s s i b l e  mathematics,  i n quantitative  o f g r a d e 10 G e n e r a l M a t h e m a t i c s  c l a s s e s were p r e - t e s t e d  attitude  attitude  tested,  in attitude  and the  toward  i n quantitative  between a n y s u c h c h a n g e s ,  there  3 2  is  no e v i d e n c e  attitude doing  fora positive  correlation  and achievement w h i l e  quantitative  significant  o f change i n  the i n c r e a s e d achievement i n  t h i n k i n g u s i n g desk c a l c u l a t o r s  a t t h e 0.05 l e v e l  i t i s significant  i s not  a t t h e 0.08  level.  LIMITATIONS  The students as had  s t u d y was c a r r i e d  out w i t h g e n e r a l  at the grade t e n l e v e l .  mathematics  No r a n d o m i z a t i o n  was  t h e g r o u p s were n o t n r e - s e l e c t e d f o r t h e p r o g r a m  done  as they  been p l a c e d by t h e a d m i n i s t r a t i o n i n t h e i r r e s p e c t i v e  classes.  Often  some s p e c i f i c  t h e placement  program b e i n g  i s done i n c o n j u n c t i o n w i t h  predominant  p r o g r a m p r e d o m i n a n c e was n o t t a k e n a s s u m e d t h a t some e q u i v a l e n c e could  i n a class.  i n t o account.  of attitude  be accomodated f o r i n t h e t e s t  The  I t was  and achievement  results  by a n a l y s i s o f  covariance. Poffenberger s 1  pupils  study  are established early.  positive  mathematical  suggested  I t i s only a f t e r s e v e r a l  experiences  have b e e n e n c o u n t e r e d  a c h a n g e o f any d e g r e e c o u l d o c c u r . formed e a r l y and s e v e r a l p o s i t i v e encountered experience strength general  that the a t t i t u d e s o f  that  Because a t t i t u d e s a r e  experiences  must be  b e f o r e a c h a n g e i s made c o u l d mean t h a t a n y w o u l d have t o be l o n g - l a s t i n g  to shift  attitude  and o f c o n s i d e r a b l e  at the grade t e n l e v e l .  mathematics s t u d e n t s  have n o t met w i t h  The  a great  deal  33 of  success  attitudes  i n m a t h e m a t i c s a n d h e n c e may have d i f f e r e n t from  The the  students  two g r o u o s were t a u g h t  environment  attitude.  using  conventional i n an o l d e r  chairs.  Going  outside,  frequently  desks.  study.  t a b l e s and  part  Pupils  i n the experimental  frequently They  do n o t l i k e  like  Other f a c t o r s include  when o p e r a t i n g .  i n order  to conduct  changes  in their  their  a n d rooms may a g a i n  in a  feeling of have h a d a  on a t t i t u d e s a n d a c h i e v e m e n t .  that  may h a v e a f f e c t e d t h e r e s u l t s o f  the noise The n o i s e  factors  a p a r t i c u l a r seat  Changes i n s e a t s influence  trip  group.  security.  negative  a  e f f e c t on t h e a t t i t u d e o f t h e  row, a n d any c h a n g e s t h r e a t e n  study  relatively  The p h y s i c a l  particular  the  in a  b u i l d i n g using  i n the rain.  established patterns.  slight  their  The e x p e r i m e n t a l  Some room c h a n g e s were n e c e s s a r y the  on  to the older b u i l d i n g necessitated  have had a n e g a t i v e taking  classes.  i n d i f f e r e n t rooms, a n d  may have h a d some i n f l u e n c e  g r o u p s were t a u g h t  pupils  mathematics  The c o n t r o l g r o u p s were t a u g h t  new c l a s s r o o m  could  i n regular  g e n e r a t e d by t h e c a l c u l a t o r s may have b e e n d i s t r a c t i n g t o t h e  pupils.  The p r o g r a m  possible  f o r any i n t e r a c t i o n b e t w e e n p u p i l , t e a c h e r ,  calculator.  s e l e c t e d may n o t have b e e n t h e b e s t  The d u r a t i o n  o f t h e s t u d y was s e v e n weeks,  w h i c h may n o t have b e e n l o n g  enough t o remove a p o s s i b l e  H a w t h o r n e e f f e c t o r t o overcome a n y deep s e a t e d feelings  toward  and  mathematics.  negative  34 It  has b e e n a r g u e d  that  while  was l e a r n i n g t o o p e r a t e t h e m a c h i n e s , could  have been d o i n g  thought  i n that  h a v e a n edge o v e r and  copying  thought  teacher  An  Reading,  there  teacher  fair  attitude test  remembering, were  influence.  may have b e e n a b i a s f o r I t was h o p e d b y t h e t o a minimum.  i s open t o s u s p i c i o n  i n that  i t is  known i f t h e p u p i l c a n r e l a t e what h i s f e e l i n g s a r e He may p u t down what he t h i n k s  may n o t h a v e m e a s u r e d what we  selection  o f the c o r r e c t  should  s u c h a s he w o u l d f i n d The quantitative  concepts  measured.  responses f o r  Hence t h e Iowa  what c a n be done on t h a t  do on a n o p e n - e n d e d t e s t i n a real  involved  life  The o r g a n i z a t i o n  test,  question  problem s i t u a t i o n .  i n problem  t h i n k i n g a r e not c l e a r l y  what was m e a s u r e d was n o t t h a t changed.  choice  i s open-ended.  t e s t shows o n l y  what a p u p i l m i g h t  have  feels.  r e s p o n s e may b e q u i t e d i f f e r e n t  r e s u l t s f r o m one w h i c h  achievement  he i s  t h a n what he a c t u a l l y  A t e s t which uses m u l t i p l e  not  i n reading,  b i a s was k e p t  e x p e c t e d t o p u t down, r a t h e r  in  I t was  remembering, a n d c o p y i n g  fora particular class.  toward a s u b j e c t .  We  group  t h e e x p e r i m e n t a l groups would not  two t e a c h e r s  that  group  t h e p r o c e d u r e u s e d was  t o be s e p a r a t e f r o m m a c h i n e o p e r a t i o n  experimenter  not  that  the c o n t r o l groups  numbers.  With one  the c o n t r o l  some i r r e l e v a n t e x e r c i s e s .  by t h e e x p e r i m e n t e r  to both groups  the experimental  solving or i n  understood.  Perhaps  w h i c h was i n f l u e n c e d o r  of a pupil's  thinking  was n o t  35  tested, aptly  and  i t i s this  s h a p e d by  the c a l c u l a t o r ,  A s t u d y was pupils  organization,  rather  a t a s c h o o l whose s o c i o - e c o n o m i c l e v e l stable,  home s i t u a t i o n s .  European  cultural  background.  economic  influences  to other  grades, types of c l a s s e s ,  economic  situations.  present  The  The pupils  s t u d y was  short  of sufficient  firmly  length  o f the d e e p - s e a t e d a t t i t u d e s .  younger  pupils  f o r a longer period  may  machines  n o t have used.  What k i n d program higher  of  I t has  strong  social  and  to g e n e r a l i z e social  or  mathematics  entrenched.  The  such, i t  t o overcome the A study  involving  o f time might  t h e market  A study using  yield  results  have been  newer, q u i e t e r noise  electronic  level  machines  of the  calculators  might  worth. of particular  of attitude  level  on  the d i s t r a c t i n g  prove o f s i g n i f i c a n t The  i s one  changes.  T h e r e a r e now which  mathematics  d u r a t i o n and, a s  inertia  significant  tested.  FURTHER RESEARCH  are usually  of a r e l a t i v e l y  n o t have b e e n  cultural,  with grade ten general  whose a t t i t u d e s  s t u d y was  cultural,  make i t d i f f i c u l t  IMPLICATIONS FOR  i s more  t h a n what was  done w i t h g r a d e t e n g e n e r a l  lower middle c l a s s ,  may  perhaps, which  s t u d e n t s were n o t  or achievement  t h e same f o r p u p i l s  o f achievement, but poor  i s changed? w i t h an  analysed.  Would  initially  attitude?  the  36  During had  the operation of the c a l c u l a t o r ,  t o know what  However,  d i d n o t work,  then  they  i t m i g h t be a d v i s a b l e t o t e a c h  o f t h e d a t a b y means o f a f l o w  o p e r a t i o n s were t o be p e r f o r m e d  calculations procedure  were done b y t h e m a c h i n e .  might  lead  used.  method i n s o l v i n g  chart.  That i s  s a y , t h a t s t u d e n t s w o u l d be r e q u i r e d t o show what  a n d what  a  Perhaps  o p e r a t i o n s were t o b e  and e r r o r  If multiplication  try division.  organization to  a n d what  some w o u l d u s e a t r i a l  the q u e s t i o n . would  numbers  the p u p i l s  numbers  before the The f l o w  to a better insight  into  chart  how  to s o l v e  problem. The  pupil  role  of the c a l c u l a t o r  in facilitating  t o a c h i e v e b e t t e r i s not c l e a r l y  the a d d i t i o n a l calculator  experience  required o r g a n i z a t i o n o f the s i t u a t i o n a r e performed pupil  understood.  i n working with  t o do more work i n a g i v e n  o r w h e t h e r some o t h e r  to g r e a t e r achievement  the Whether  the a i d o f the  time,  whether the  before  factors  are questions  the c a l c u l a t i o n s  are enabling the y e t t o be  investigated. The attitude  teacher  i n order  to conduct  mathematics a t t i t u d e be a v a l i d the  test  needs i n f o r m a t i o n about  test  and r e l i a b l e  c o u l d be u s e f u l  remedial  a pupil's  procedures.  c o n s t r u c t e d f o r the study  test.  With  to others.  The seems t o  some m o d i f i c a t i o n s ,  FOOTNOTES  38  "•"Lewis R. A i k e n J r . , " A t t i t u d e s Toward Mathematics," S t u d i e s i n Mathematics V o l . XIX, ed. James W. W i l s o n and L. Ray C a r r y , SMSG, S t a n f o r d U n i v . , 196-9, p.12. T  P . D. C r i s t a n t i e l l o , " A t t i t u d e Toward Mathematics and the P r e d i c t i v e V a l i d i t y o f a Measure o f Q u a n t i t a t i v e A p t i t u d e , " J o u r n a l o f E d u c a t i o n a l Research. LV (DecemberJanuary, 1962) pp.184-186. 2  3 w i l b u r H. Dutton, "Measuring A t t i t u d e s Toward A r i t h m e t i c , " Elementary S c h o o l J o u r n a l September, 1954, T  p29.  "'"Thomas P o f f e n b e r g e r and Donald Norton, " F a c t o r s i n the Formation o f A t t i t u d e s Toward Mathematics," Journal of E d u c a t i o n a l Research. V o l . 52, January 1959, pp.171-176. 5  Lewis R. A i k e n J r . , op. c i t . , p. 38. Lewis R. A i k e n J r . , op. c i t . , c i t i n g R. A l p e r t , G. Slettwagen, and D. Becker, " P s y c h o l o g i c a l F a c t o r s i n Mathematics E d u c a t i o n , " Report summary i n N e w s l e t t e r No. l"> SMSG, S t a n f o r d U n i v e r s i t y , 1963.  f  ^Wesley J . Lyda and E v e l y n C l a y t o n Morse. " A t t i t u d e , Teaching Methods and A r i t h m e t i c Achievement," The A r i t h m e t i c T e a c h e r March 1963, pp.136-138. T  o  Joseph P h i l i p Cech, "The E f f e c t o f the use o f Desk C a l c u l a t o r s has on A t t i t u d e and Achievement i n Ninth-Grade G e n e r a l Mathematics C l a s s e s / ' ' D i s s e r t a t i o n A b s t r a c t s , December 1970, 31A, p.2784-A. ^Lewis R. A i k e n J r . , op. c i t . ,  p.4.  W i l b u r H. Dutton, " A t t i t u d e s o f J u n i o r H i g h S c h o o l P u p i l s Toward A r i t h m e t i c , " The S c h o o l Review 1956, pp.18-22. 1 0  T  """""Victor Rodney Durrance, "The E f f e c t o f the R o t a r y C a l c u l a t o r on A r i t h m e t i c Achievement i n Grades S i x , Seven, and E i g h t , " D i s s e r t a t i o n A b s t r a c t s . 25, p.6307.  12  Joseph P. Cech, op. c i t .  ^ H o w a r d F. Fehr, George McMeen, Max S o b e l . "Using Hand-Operated Computing Machines i n L e a r n i n g A r i t h m e t i c , " The A r i t h m e t i c Teacher October 1956, pp.145-150. T  1965.  "^Hobart 15  (Okla.) Democrat-Chief, Wednesday, March  Lewis R. A i k e n J r . , op. c i t . ,  p.2.  31,  39  - ^ I b i d , p.2, c i t i n g Morrisett, L. M. and Vinsonhaler, J . , (Eds.). "Mathematical Learning," Monographs of the Society for Research In Child Development, XXX, No. 1 (1965). I b i d , p.3, c i t i n g Brown, K. E., and A b e l l , T. L. "Research i n the Teaching of Elementary School Mathematics," Arithmetic Teacher. XII (November, 1965), pp. 5^7-9+9* 1 7  • ^ I b i d , p.5, c i t i n g Nealeigh, T. R. Development and V a l i d a t i o n of a Non-verbal Attitude and Achievement Index f o r Mathematics, unpublished doctoral d i s s e r t a t i o n , Ohio State U n i v e r s i t y , 1967. (Dissertation Abstracts, XXVIII-A, p.3567). 1 9  Ibid,  p.5.  I b i d , p.6, c i t i n g Fedon, J . P. T h e Role o f Attitude i n Learning Arithmetic," A r i t h m e t i c Teaf>hpr V (December, 1958), pp.304-310. 2 0  7  I b i d , p.6, c i t i n g Reys, R. E., and Delon, F. G. "Attitudes of Prospective Elementary School Teachers Towards Arithmetic," Arjtftmetlc Teacher, XV ( A p r i l , 1968), pp.363-366. 2 1  2 2  Ibld,  p.8.  ^ I b l d , p.8, c i t i n g Alpert, R., Stellwagon, G., and Becker, D. "Psychological Factors i n Mathematics Education," Report summary i n Newsletter No. 15, School Mathematics Study Group, Stanford U n i v e r s i t y , 1963. 2  2 l f  2  I b i d , p.9, c i t i n g Brown and A b e l l .  5 l b i d , p.10, c i t i n g Alpert et a l .  *°Ibid, p.10, c i t i n g Degnan, J . A. "General Anxiety and Attitudes Toward Mathematics i n Achievers and Underachievers i n Mathematics," Graduate Research i n Education and Related D i s c i p l i n e s , I I I (1967), pp.49-62. 27 'Ibid, p.12, c i t i n g Jackson, P. W. L i f e i n Classrooms (New York: Holt, Rlnehart & Winston, I968). I b i d , p.22. 2 8  I b i d , p.37, c i t i n g Bassham, H., Murphy, M., and Murphy, Katherine. "Attitude and Achievement i n Arithmetic," Arithmetic Teacher XI (February, 1964), pp.66-72. 2 q  T  3°Ibid, p.37, c i t i n g Lerch, H. H., "Arithmetic I n s t r u c t i o n Changes Pupils' Attitudes Toward Arithmetic," Arithmetic Teacher, VIII (March, 1961), pp.117-119. 31lbid, p.37, c i t i n g Tulock, Mary K. "Emotional Blocks i n Mathematics," Mathematics Teacher, L (December, 1957), pp.572-576.  40  ^ I b i d , p . 3 8 , c i t i n g N a t k i n , G. L. The Treatment o f Mathematical A n x i e t y Through Mediated T r a n s f e r o f A t t i t u d e Toward Mathematics, u n p u b l i s h e d d o c t o r a l d i s s e r t a t i o n , Indiana U n i v e r s i t y , 1966. ( D i s s e r t a t i o n A b s t r a c t s , XXVTI-A, p.4137). 2  3 3  I b i d , p.40.  ^Thomas P o f f e n b e r g e r and Donald Norton, " F a c t o r s l n the Formation o f A t t i t u d e toward Mathematics," J o u r n a l o f E d u c a t i o n a l R e s e a r c h V o l . 5 2 , No. 5 , January, 1 9 5 9 , p.175* T  3  ^Ibid,  p.174.  i l b u r H. Dutton, "Measuring A t t i t u d e s Toward A r i t h m e t i c , " Elementary S c h o o l J o u r n a l September, 1 9 5 4 , pp.24-31. T  3 7  I b i d , p.29.  W i l b u r H. Dutton, " A t t i t u d e s o f J u n i o r H i g h S c h o o l P u p i l s Toward A r i t h m e t i c , " The S c h o o l Review 1 9 5 6 , p p . 1 8 - 2 2 . 3 8  T  3 9  I b i d , p.22.  ^ V i r g i n i a M. S t r i g h t , "A Study o f the A t t i t u d e s Toward A r i t h m e t i c o f Students and Teachers i n the T h i r d , F o u r t h , and S i x t h Grades," The A r l ^ h m o t i n T^rbav, October, i 9 6 0 , pp.2P0-286. "''-'-Aiken, p . 5 , c i t i n g L. M. M o r r i s e t t and J . V i n s o n h a l e r Eds. "Mathematical L e a r n i n g , " Monographs o f the S o c i e t y f o r Research i n C h i l d Development, XXX, No. 1 , 1965, p.132. 42 J . P e t e r Fedon, "The Role o f A t t i t u d e i n L e a r n i n g A r i t h m e t i c , " The A r i t h m e t i c Teacher, December, 1 9 5 8 , p . 3 1 0 . ^ L o i s Stephens, "Comparison o f A t t i t u d e and Achievement Among J u n i o r High S c h o o l Mathematics C l a s s e s , " The A r i t h m e t i c Teacher, November, i 9 6 0 , p p . 3 5 1 - 3 5 6 . " • L e s l e y J . Lyda and E v e l y n C. Morse, " A t t i t u d e s , Teaching Methods and A r i t h m e t i c Achievement," The A r i t h m e t i c Teacher, March, 1 9 6 3 , pp.136-138. ^ L e w i s R. A i k e n and R. M. Dreger, "The E f f e c t o f A t t i t u d e s on Performance i n Mathematics," J o u r n a l o f E d u c a t i o n a l Psychology, V o l . 5 2 , 1 9 6 1 , p . 2 6 . ^^Howard F . Fehr, George McMeen and Max S o b e l , "Using Hand-Operated Computing Machines i n L e a r n i n g A r i t h m e t i c , " The A r i t h m e t i c Teacher, October, 1956, pp. 1 4 5 - 1 5 0 . ^Ibid,  p.148.  41  ^ V i c t o r R. Durr-ance, "The E f f e c t o f the R o t a r y C a l c u l a t o r on A r i t h m e t i c Achievement i n Grades S i x , Seven and E i g h t , " D i s s e r t a t i o n A b s t r a c t s . 2 5 , p. -6307. ^ J o s e p h P. Cech, "The E f f e c t the use o f Desk C a l c u l a t o r s has on A t t i t u d e and Achievement i n Ninth-Grade General Mathematics C l a s s e s , " D i s s e r t a t i o n A b s t r a c t s , 31A, December 1970, p.2784A. 5°Wilbur H. Dutton, "Measuring A t t i t u d e s A r i t h m e t i c , " Elementary S c h o o l J o u r n a l . September, pp.24-31.  Toward 1954,  L. Thurstone and E. J . Chave, The Measurement o f A t t i t u d e , Chicago: U n i v e r s i t y o f Chicago, 1948. 5 D u t t o n , "Measuring A t t i t u d e s , " p.26. 2  53E. F. L i n d q u i s t , The Iowa T e s t s o f E d u c a t i o n a l Development G e n e r a l Manual. Chicago: Science Research A s s o c i a t e s , March, 1969, P - 7 . ^*P. A. Lappan J r . , "Review o f 'The Iowa T e s t s o f E d u c a t i o n a l Development ," Mental Measurements Yearbook, 1  V o l , VI pps. 8 7 2 - 8 7 3 .  ^ S h e l d o n S. Meyers, "Annotated B i b l i o g r a p h y o f Mathematics T e s t s , " E v a l u a t i o n i n Mathematics, Washington: NCTM 2 6 t h Yearbook, 1 9 6 I , p . 2 0 1 . Education,  ^ H e n r y E . G a r r e t t , S t a t i s t i c s i n Psychology and David McKay Co. Inc., New York, 1958, p . 3 3 9 . *?Ibid,  P.339.  5 lbid,  p.351.  8  Dixon and F. J . Massey J r . , I n t r o d u c t i o n t o S t a t i s t i c a l A n a l y s i s 2d ed., McGraw-Hill Book Company, New York, 1957, p p . 3 0 6 - 3 1 0 .  BIBLIOGRAPHY  43  A i k e n , Lewis R. J r . , " A t t i t u d e s toward Mathematics," S t u d i e s i n Mathematics, V o l . XIX, ed. James W i l s o n and L. Ray C a r r y , SMSG, S t a n f o r d U n i v e r s i t y , 1969. Cech, Joseph P., " The E f f e c t the use o f Desk C a l c u l a t o r s has on A t t i t u d e and Achievement i n Ninth-Grade G e n e r a l Mathematics C l a s s e s , " D i s s e r t a t i o n A b s t r a c t s , 31A December 1 9 7 0 . C r i s t a n t i e l l o , P. D., " A t t i t u d e Toward Mathematics and the P r e d i c t i v e V a l i d i t y o f a Measure o f Q u a n t i t a t i v e A p t i t u d e , " J o u r n a l o f E d u c a t i o n a l Research, LV (December-January  1962).  Dutton, Wilbur H., " A t t i t u d e s o f J u n i o r High S c h o o l Toward A r i t h m e t i c , " The School Review, 1956.  Pupils  , "Measuring A t t i t u d e s Toward A r i t h m e t i c , " Elementary School J o u r n a l , September, 1954. Dixon, W. J . and Massey, F. J . J r . , I n t r o d u c t i o n t o S t a t i s t i c a l A n a l y s i s 2nd ed., McGraw-Hill Book Company, New York, 1957. Durrance, V i c t o r R., "The E f f e c t o f the R o t a r y C a l c u l a t o r on A r i t h m e t i c Achievement i n Grades S i x , Seven and E i g h t , " Dissertation Abstracts, 25. Fedon, J . P e t e r , "The Role o f A t t i t u d e i n L e a r n i n g The A r i t h m e t i c Teacher. December, 1958.  Arithmetic,"  Fehr, Howard F., George McMeen and Max S o b e l , "Using HandOpera ted Computing Machines i n L e a r n i n g A r i t h m e t i c , " The A r i t h m e t i c Teacher, October, 1956. G a r r e t t , Henry E., S t a t i s t i c s i n Psychology and David McKay Co. Inc., New York, 1958.  Education,  Hobart, (Oklahoma) Democrat-Chief, Wednesday, March 31*  1965«  Lappan, P. A. J r . , "Review o f 'The Iowa Tests o f E d u c a t i o n a l Development ," Mental Measurements Yearbook, V o l . VI. 1  L i n d q u i s t , E. F., The Iowa T e s t s o f E d u c a t i o n a l Development General Manual, Chicago: S c i e n c e Research A s s o c i a t e s , March, 1959. Lyda, Wesley J . , and E v e l y n C. Morse, " A t t i t u d e s , Teaching Methods and A r i t h m e t i c Achievement," The A r i t h m e t i c Teacher, March, 1963. Meyers, Sheldon S., "Annotated B i b l i o g r a p h y o f Mathematics T e s t s , " E v a l u a t i o n i n Mathematics, Washington: NCTM 2 6 t h Yearbook, 1961.  PofrenDerger, Thomas and Donald Norton, " F a c t o r s I n the Formation o f A t t i t u d e s toward Mathematics," J o u r n a l o f Educational, Research, V o l . 5 2 , No. 5 January, 1 9 5 9 . 5  Stephens, L o i s , "Comparison o f A t t i t u d e s and Achievement Among J u n i o r High S c h o o l Mathematics C l a s s e s , " The A r i t h m e t i c Teacher, November, i 9 6 0 , N  S t r i g h t , V i r g i n i a M., "A Study o f the A t t i t u d e s Toward A r i t h m e t i c o f Students and Teachers i n the T h i r d , F o u r t h , and S i x t h Grades," The A r i t h m e t i c T e a c h e r October, i 9 6 0 . T  Thurstone, L. L. and E . J . Chave, The Measurement o f A t t i t u d e Chicago: U n i v e r s i t y o f Chicago, 1 9 4 8 .  T  APPENDIX A  SAMPLE PROBLEMS  Some o f the q u e s t i o n s asked were s i m i l a r t o the f o l l o w i n g . A.  P a y r o l l Computation Hours  Name a b c d e f g TOTALS B.  45 42 40 40 43 36 32  Gross Pay  Rate per hour 3-25 3.00 2.75 2.75 2.75 2.50 2.50  Income Tax  Pension  Net Pay  Commission 1.  C a l c u l a t e the commission r e c e i v e d  by the agent i n each  o f the f o l l o w i n g : Sales a) 1 5 6 0 . 0 0 b) 2.  Rate o f Commission 15#  750.00  22%  C a l c u l a t e the r a t e o f commission i n each o f the f o l l o w i n g :  Sales a) 4000.00  Commission 340.00  b) 7 0 , 0 0 0 . 0 0 3.  Rate  4200.00  C a l c u l a t e the amount o f s a l e s i n each o f the f o l l o w i n g :  Commission a) 320.00  Rate h%  b) 4 6 0 . 0 0 C.  Commission  Sales  15%  Simple I n t e r e s t 1.  C a l c u l a t e the i n t e r e s t i n each o f the f o l l o w i n g :  Principal a) 4 5 0 . 0 0  Rate 5%  b) 1 2 5 0 . 0 0  3%%  2.  3.  Interest  3 i yr.  F i n d the r a t e o f i n t e r e s t i n each o f the f o l l o w i n g :  Principal a) $ 5 0 0 . 0 0 b)  Time - 2 yr.  1200.00  Interest 30.00 90.00  Rate  Time 1 yr. l iyr.  C must be i n v e s t e d a t 6% t o y ia el lc du l a$t6e0 . 0the 0 i pn rt ei rn ec si tp ail nt h8a tmonths.  4.  C a l c u l a t e the time i n each o f the f o l l o w i n g : Principal  a) $750.00 b) 3 5 0 . 0 0 D.  Installment 1.  Interest  40.00 19.25  Rate  47  Time  h% 5\%  Buying  Tom Barrow borrowed  $100.  He promised to repay the l o a n  i n 10 i n s t a l l m e n t s o f $10 each a t l~k% monthly on the unpaid balance.  Make a t a b l e  showing:  Payments on p r i n c i p a l Balance due I n t e r e s t charges Each monthly payment T o t a l i n t e r e s t charges T o t a l o f repayments and i n t e r e s t  APPENDIX  B  DUTTON'S A T T I T U D E SCALE  49  1.  I think about arithmetic problems outside of school and l i k e to work them out.  2.  I don't f e e l sure of myself i n arithmetic.  3.  I enjoy seeing how r a p i d l y and accurately I can work arithmetic problems.  h,  I l i k e arithmetic, but I l i k e other subjects just as w e l l .  5.  I l i k e arithmetic because i t i s p r a c t i c a l .  6.  I don't think arithmetic i s fun, but I want to do well ln i t .  7.  I am not enthusiastic about arithmetic, but I have no r e a l d i s l i k e for i t e i t h e r .  8.  Arithmetic i s as important as any other subject.  9.  Arithmetic i s something you have to do even though i t i s not enjoyable.  10.  Sometimes I enjoy the challenge presented by an arithmetic problem.  11.  I have always been a f r a i d of arithmetic.  12.  I would l i k e to spend more time i n school working arithmetic.  13.  I detest arithmetic and avoid using i t at a l l times.  Ih.  I enjoy doing problems when I know how  15.  I avoid arithmetic because I am not very good with f i g u r e s .  16.  Arithmetic t h r i l l s me, other subject.  17.  I never get t i r e d of working with numbers.  18.  I am a f r a i d of doing work problems.  19.  Arithmetic i s very i n t e r e s t i n g .  20.  I have never l i k e d arithmetic.  21.  I think arithmetic i s the most enjoyable taken.  22.  I can't see much value i n arithmetic.  to work them w e l l .  and I l i k e i t better than any  subject I have  APPENDIX C  MATHEMATICS ATTITUDE TEST  51  NAME BLOCK DATE ATTITUDE TEST FOR  MATHEMATICS  You w i l l be p r e s e n t e d w i t h 16 s i t u a t i o n s  i n which you  have t h r e e c h o i c e s ( A ) , ( B ) , ( C ) . From these t h r e e you a r e asked  choices  t o s e l e c t the one you l i k e the l e a s t and the  one you l i k e the most.  Put a check ( N / ) i n the square  f o r the one you l i k e the b e s t .  Put a c r o s s ( X ) i n the square  f o r the one you l i k e the l e a s t .  EXAMPLE 1 . a . Add up a column o f 2 0 t h r e e d i g i t b. Solve the e q u a t i o n 3 x + 2 = 2 0 . c. Wash the b r e a k f a s t d i s h e s .  SCORES  ft  X  3  C  numbers. A B C  V  X  Add  up a s e r i e s o f f r a c t i o n s .  F i n d t h e number when f o u r more t h a n t h e number i s t h r e e l e s s t h a n t h r e e t i m e s t h e number. Watch Batman on TV. Add  up a c o l u m n o f 15 numbers.  D e t e r m i n e t h e number o f n i n e i n c h needed t o c o v e r t h i s f l o o r . Learn  to  Multiply  tiles  waltz. a series of fractions.  F i n d t h e l e n g t h o f a r e c t a n g l e when t h e p e r i m e t e r a n d w i d t h a r e known. Pai.nt a p i c t u r e o f some c o l o r f u l Multiply  five,  two d i g i t  stones.  numbers.  Two men t o g e t h e r c a n do a p i e c e o f work i n s i x days. One man w o r k i n g a l o n e c a n do t h e j o b i n t e n d a y s . How l o n g w o u l d i t t a k e t h e s e c o n d man h i m s e l f ? Type out t h i s Find  page.  t h e s q u a r e o f 3.2.  You p a i d $60 f o r a r a d i o a t a s a l e w h i c h a d v e r t i s e d t h a t a l l p r i c e s were r e d u c e d by one t h i r d . What was t h e o r i g i n a l p r i c e ? Dust  the l i v i n g  room.  Work some s u b t r a c t i o n decimals.  questions  involving  Mary i s t e n a n d h e r a u n t i s f o r t y . When was M a r y o n e - s e v e n t h a s o l d a s h e r a u n t ? G i v e an o r a l r e p o r t  i n Social  Studies.  53  7.  F i n d the  b.  T h e r e a r e 120 a p p l e and p e a c h t r e e s i n an orchard. T h e r e a r e 2/3 as many p e a c h t r e e s as a p p l e t r e e s . How many t r e e s o f e a c h a r e there?  c . Run 8.  root of  l a p s d u r i n g P.  169.  E.  a.  F i n d out another.  b.  The h e a d o f a f i s h i s 10 i n c h e s l o n g , t h e t a i l i s as l o n g a s t h e head p l u s one h a l f o f t h e body; t h e body i s a s l o n g a s t h e head and the t a i l t o g e t h e r . How l o n g i s the f i s h ?  c . Rake up 9.  square  ft 8  a.  a.  what p e r c e n t  the  F i n d out another.  leaves  on  one  the  what p e r c e n t a g e  number i s o f  lawn. one  number i s o f  b . A b a s e b a l l team won 25 more games t h a n i t lost. I f i t won 3/5 o f i t s games, how many did i t play? c . Make a d r a w i n g o f a c o f f e e t a b l e t h a t a r e g o i n g t o have b u i l t . 10.  a.  Round o f f s e v e n t y hundred.  numbers t o t h e  you  nearest  b . A t o y t r a i n c o n s i s t s o f an e n g i n e , a d i n i n g c a r and a c a b o o s e . The w h o l e t r a i n i s 26 inches long. The r e l a t i o n o f t h e l e n g t h s o f t h e d i n i n g c a r a n d t h e e n g i n e i s (3,2) and the r e l a t i o n o f the l e n g t h s o f the caboose and the d i n i n g c a r i s ( 1 , 2 ) . What i s t h e l e n g t h o f each s e c t i o n ? c . Mow 11.  the  lawn.  a . Work o u t f i v e d i v i s i o n q u e s t i o n s numbers o f s e v e n d i g i t s .  involving  b. A policeman chases a t h i e f . The policeman t a k e s two s t e p s w h i l e t h e t h i e f t a k e s t h r e e . But e a c h s t e p t h e p o l i c e m a n t a k e s c o v e r s as much d i s t a n c e a s two o f t h e t h i e f ' s s t e p s . How many s t e p s w i l l t h e p o l i c e m a n t a k e b e f o r e he c a t c h e s t h e t h i e f ? c.  Design  a d e s k f o r y o u r homework.  C  54  12.  a . D e t e r m i n e t h e w e i g h t o f 990 s e v e n t y - f i v e pounds e a c h . b. Determine c.  13.  the cost  o f making  L i s t e n t o an o r a l r e p o r t a b o u t weeds.  a. Determine  bags  by  weighing  your  own  tent.  a classmate  t h e volume o f a f i s h  tank.  b . On a r e m o t e i s l a n d t h e r e a r e two t r i b e s , The T e e ' s a n d t h e L ' s . The T e e ' s a l w a y s t e l l the t r u t h . The L ' s a l w a y s l i e . A s t r a n g e r t o t h e i s l a n d comes t o a g r o u p o f t h r e e n a t i v e s who a l l l o o k a l i k e . He a s k s t h e f i r s t what t r i b e he b e l o n g s t o but c a n ' t make o u t t h e answer s o he a s k s t h e s e c o n d n a t i v e what he s a i d a n d g e t s the r e p l y "He s a i d he was a T e e . " The t h i r d s a y s t h e s e c o n d was a l i e r . To what t r i b e d i d t h e t h i r d man b e l o n g ? c. 14.  See  a stage  play.  a . C a l c u l a t e t h e p r i c e o f an a r t i c l e i s t o be r e d u c e d by l / 3 .  i fi t  b . You have p l a n n e d t o t a k e a b o a t t r i p f o r 20 p e o p l e , two were u n a b l e t o g o . You e a c h had t o p a y an a d d i t i o n a l t h r e e dollars. How much was t h e b o a t r e n t a l ? c. 15.  Write a l e t t e r gift.  a . Compute t h e w e i g h t at  c.  Go  algebraic  for a birthday  o f 1000  t e n pounds f o r e a c h  b . S o l v e an  16.  o f thanks  yards of wire  1000  feet.  equation.  f o r a car r i d e with your  grandparents.  a . F i n d t h e common d i v i s o r o f f o u r b . Graph an a l g e b r a i c e q u a t i o n . c.  Sing  i n a mixed  choir.  numbers.  fi  B G  APPENDIX D  IOWA TEST OF QUANTITATIVE THINKING  56  ABILITY TO  DO  QUANTITATIVE THINKING DIRECTIONS: In working these problems, pay no a t t e n t i o n to the suggested answers u n t i l a f t e r you have got your own answer. I f your answer agrees w i t h one o f the suggested answers, mark the c o r r e s p o n d i n g box on the answer s h e e t . I f your answer does not agree w i t h any o f the answers g i v e n , mark the l a s t box i n the c o r r e s p o n d i n g row on the answer s h e e t . There a r e many problems t o which the c o r r e c t answer i s not g i v e n . Do not rework a problem simply because your answer i s not among those suggested. I n s t e a d mark the f i f t h box ("Not g i v e n " or "None o f these") and go on to the next problem. The t h r e e sample problems have been marked c o r r e c t l y on the answer s h e e t . Notice how they a r e marked, and mark the remaining problems s i m i l a r l y . Samples: 0. What i s the c o s t o f 3 pounds o f b u t t e r a t 500 per pound? 1) $1.15 4) $ 1 . 6 5 2) $ 1 . 3 5 5) Not g i v e n .  3)  00.  Mrs. Smith, I n paying f o r 600 worth of g r o c e r i e s , gave the clerk a dollar b i l l . How much change should she r e c e i v e ?  1)  2)  3) 000.  4) 5)  50^  3JC  NOTE:  600  Not  given.  ]fc }|C jjc 3JC J^C 3^C jjc 3^C  Do your f i g u r i n g on s c r a t c h paper. these pagesJ  Make no marks on  A three-pound beef r o a s t c o n t a i n s f i v e ounces o f bone. The weight o f the bone i s what p a r t o f the t o t a l weight? 1) 3/80 h) 15/16 2) 5 A 8 5) Not g i v e n .  3) 2.  25£  300  I f one g a l l o n o f p a i n t w i l l cover 200 square f e e t , approximately how many g a l l o n s of p a i n t w i l l be needed t o cover 380 square f e e t ? 1) 1 g a l l o n h) k g a l l o n s 2) 2 g a l l o n s 5) 5 g a l l o n s . 3) 3 g a l l o n s  J^C Sjt  1.  $1.50  5/19  The i n g r e d i e n t s f o r making 2k c h o c o l a t e marshmallow b a l l s , as g i v e n i n a p a r t i c u l a r r e c i p e , a r e : 2 squares (2 oz.) unsweetened c h o c o l a t e 1 cup evaporated m i l k 1/2 cup g r a n u l a t e d sugar 12 marshmallows, h a l v e d 1 cup f i n e l y chopped walnuts  How many ounces of chocolate would be needed i n order t o make 3 dozen marshmallow b a l l s ? 1) 2 1 A  4)  2 ) 2 1/2 3)  3.  3  1/3  In a short-cut method of multiplying 4 0 8 7 x 198, one f i r s t m u l t i p l i e s 4087 x 2 0 0 . What product must then be subtracted from t h i s r e s u l t i n order to obtain the correct answer? 1)  13 x 2  4)  How  many pencils s e l l i n g at 2 for 5£ can be bought for 600?  1) 3 0  3)  4)  7.  6  5) Not given.  12  The map of a summer camp was drawn according to the scale 1 inch = 1/8 mile. What i s the length of the camp ground i n miles i f the corresponding distance on the map measures 1 0 inches? 1 ) . 8 Mile 4 ) 8 0 miles 2 ) 1 . 2 5 miles 5) Not given. 3 ) 8 . 2 miles  NOTE:  6.  4087 x 2  5) Not given.  2) 15 5.  1/2  5) Not given.  2 ) 4087 x 13 3 ) 198 x 2 4.  3  5 7  I f you do not recognize quickly how a problem should be worked, skip i t and go on to the next. You may come back to the harder problems l a t e r i f time permits.  What i s the next term i n the series 2 0 , 6 . 1 . 8 , 1)  .0162  2)  .0486  3)  .0540  4)  .0810  5)  Not  The following quotation i s from an automobile pamphlet.  .54,  .162?  given.  club  " I f you are i n an accident while your car i s t r a v e l i n g under 40 miles an hour, there i s one change i n 4 4 that someone w i l l be k i l l e d . I f an accident occurs while your car i s t r a v e l i n g over 40 miles an hour, there i s one chance i n 19 that someone w i l l be k i l l e d . " Which of the following statements best interprets the meaning of t h i s quotation? 1 ) A car t r a v e l i n g f a s t e r than 40 miles an hour w i l l k i l l 2 5 more people than w i l l a car t r a v e l i n g under 40 miles an hour. 2 ) I f a l l people drove faster than 40 miles an hour, one i n 19 would be k i l l e d . 3 ) I f a l l people drove at 3 0 miles an hour, nobody would be killed. 4 ) I f people drive more than 40 miles an hour, the danger of f a t a l accidents increases. 5) Out of 1,000 people t r a v e l i n g by car, 19 w i l l be k i l l e d by f a s t d r i v e r s , and 4 4 by slow d r i v e r s .  8.  E x a c t l y how many hundreds a r e c o n t a i n e d i n the number  63,517?  1) 63.517 9.  2) 517  3) 635.17  To make bookshelves, cut i n t o three equal 1) 2 f e e t , 6 inches 3) 3 f e e t , 5 inches  4) 6 3 , 5 0 0  58  5) Not g i v e n .  a board 8 f e e t 9 i n c h e s l o n g was s e c t i o n s . How l o n g was each s e c t i o n ? 2) 2 f e e t , 11 inches 4) 3 f e e t , 10 inches 5) Not g i v e n .  10.  An agent r e c e i v e d commissions a t the r a t e o f 2 . 5 $ on a s a l e of ^2,500, 3% on a s a l e o f $3,000, 3.2$ on a s a l e o f $3}200, 3«5# on a s a l e o f $3>500, and so on. According to t h i s p a t t e r n , how much money would h i s commission amount to on a s a l e o f $4,000? 1) $16 2) $40 3) $160 4) $400 5) Not g i v e n .  11.  A f a m i l y bought a two-family house, p l a n n i n g to r e n t one apartment and l i v e r e n t - f r e e i n the o t h e r . They were not i n t e r e s t e d i n making any f u r t h e r p r o f i t . Y e a r l y expenses on the house were estimated a t $140 f o r t a x e s ; $340 f o r i n s u r a n c e , upkeep and d e p r e c i a t i o n ; and $200 f o r other expenses. What monthly r e n t should the f a m i l y charge the tenant? 1) $29 2) $55 3) $57 4) $64 5) $68.  Problems 12 and 13 a r e based on the c i r c l e graph below, r e p r e s e n t i n g the budget p l a n o f a f a m i l y .  12.  Which o f these c o n c l u s i o n s i s supported by t h i s graph? 1) The f a m i l y spends too much f o r f o o d . 2) The f a m i l y i s wealthy. 3) The f a m i l y has a v e r y low income. 4) The f a m i l y i s e x t r a v a g a n t . 5) None o f the c o n c l u s i o n s g i v e n above can be drawn.  13.  On the b a s i s of t h i s budget p l a n , how much would be a l l o t e d f o r r e n t i f the f a m i l y ' s income were $ 4 , 0 0 0 ? 1) $400 2) $480 3) $ 1 , 0 0 0 4) $ 1 , 6 0 0 5) Not g i v e n .  14.  Height i n Inches No. o f Students  52  2  The t a b l e above the students i n the 40 students 1 student i s 53  53  54  55  56  57  58  59  60  1  0  2  3  1  5  12  8  Over  60  6  g i v e s , to the n e a r e s t i n c h , the h e i g h t s o f a certain class. The t a b l e shows t h a t o f i n the c l a s s , 2 students are 52 i n c h e s t a l l , inches t a l l , e t c .  E x a c t l y one-eighth o f the members o f the c l a s s a r e the same h e i g h t (to the n e a r e s t i n c h ) . How t a l l a r e the students i n t h a t group? 1) 55 i n c h e s 2) 56 i n c h e s 3) 58 i n c h e s 4) 60 i n c h e s 5) Not g i v e n . 15.  I t has been recommended t h a t a person watching t e l e v i s i o n s i t a t l e a s t 6 f e e t away from the s e t i f the s c r e e n measures 10 i n c h e s a c r o s s , and 7 i n c h e s f a r t h e r away f o r each a d d i t i o n a l i n c h o f s c r e e n . What i s the minimum d i s t a n c e s a t i s f y i n g t h i s requirement f o r a 20-inch screen? 1) 11 f e e t , 8 i n c h e s 2) 11 f e e t , 10 inches 3) 12 f e e t 4) 17 f e e t , 8 i n c h e s 5) Not g i v e n .  16.  A housewife purchased l i n e n s from a w h o l e s a l e r f o r r e s a l e from her home. She i n v e s t e d $104.00 i n merchandise, and made s a l e s amounting to $ 7 9 * 5 0 . Her i n v e n t o r y then showed t h a t she s t i l l had merchandise on hand f o r which she had p a i d $42.00. What was her gross p r o f i t , to date? 1) $ 1 7 . 5 0 2) $24.50 3) $ 3 7 . 5 0 4) $ 6 6 . 5 0 5) Not g i v e n .  17.  Many f i r e i n s u r a n c e p o l i c i e s operate under an 80$ c l a u s e . T h i s c l a u s e p r o v i d e s t h a t the owner of a house i n s u r e d f o r 80$ o f i t s v a l u e w i l l r e c e i v e the complete c o s t o f damages (up to the f a c e v a l u e o f the p o l i c y ) i n case o f f i r e , w h i l e the owner o f p r o p e r t y i n s u r e d f o r l e s s than 80$ o f the v a l u e w i l l r e c e i v e o n l y 60/80 o f damage c o s t s r e s u l t i n g from the f i r e . A house v a l u e d a t $15,000 was i n s u r e d f o r $7,500. Damages caused by f i r e c o s t $2,000. How much should the company pay the owner? 1) $1,000 2) $1,500 3) $ 1 , 6 0 0 4) $2,000 5) Not g i v e n .  Problems 1 8 t o 2 0 a r e based on the f o l l o w i n g  graph.  60  FEDERAL BUDGET: RECEIPTS AND EXPENDITURES (In B i l l i o n s o f D o l l a r s ) I 00  L  \  io 10  1  /  io 5°  Vo  3<? ko io o mo  I  /  /  /  / iS.  F/c  \  It? two f/ft  I  •-jf--.,.  IV  l/S  /  -  \  /.-.  cei 'HI  'Vi  "/3  'TO  'ft esr  18.  What i s the t o t a l o f the e s t i m a t e d r e c e i p t s f o r the years 1952 and 1953 combined? 1) 32 b i l l i o n s 2) 70 b i l l i o n s 3 ) 132 b i l l i o n s 4) 155 b i l l i o n s 5) 287 b i l l i o n s .  19.  F o r which one o f the f o l l o w i n g years were the r e c e i p t s approximately the same as the e x p e n d i t u r e s ? 1) 19^1 2) 1944 3) 1949 4) 1952 5) None o f t h e s e .  20.  Assuming the p o p u l a t i o n o f the U n i t e d S t a t e s t o be 150 m i l l i o n , what i s the e s t i m a t e d per c a p i t a t a x f o r 1953?  1) $470  2 ) $570  3 ) $700  4) $ 1 , 0 5 0  5) S I , 3 3 0 .  21.  People w i t h c h e c k i n g accounts i n Bank A pay the bank 100 f o r every check i s s u e d but n o t h i n g f o r d e p o s i t s made, w h i l e those d e a l i n g w i t h Bank T> pay 50 f o r each check i s s u e d and 50 f o r each d e p o s i t made. What k i n d o f d e p o s i t o r w i l l save money oy d e a l i n g w i t h Bank B? 1) A l l d e p o s i t o r s 2) Those who i s s u e checks more f r e q u e n t l y than they make deposits. 3) Those who make d e p o s i t s more f r e q u e n t l y than they i s s u e checks. 4) Those who make d e p o s i t s j u s t as f r e q u e n t l y as they i s s u e checks. 5) There i s no d i f f e r e n c e , s i n c e i n the l o n g r u n each person d e p o s i t s as much as he withdraws.  22.  I n p l a y i n g a game, John makes a score o f - 3 5 and Henry makes a score o f +65. What i s the d i f f e r e n c e i n t h e i r scoras? 1) 30 2 ) 35 3) 90 4) 100 5) Not g i v e n .  23.  During World War I I , about 9i% o f g i r l h i g h s c h o o l graduates were r e c r u i t e d f o r the Army Cadet Nurse Corps. T h i s i s e q u i v a l e n t to s a y i n g t h a t the Nurse Corps r e c r u i t e d which of the f o l l o w i n g ? 1) 9 out o f 15 g i r l g r a d u a t e s . 2) 9 out o f 50 g i r l g r a d u a t e s . 3) 10 out o f 95 g i r l g r a d u a t e s . 4) 19 out o f 100 g i r l g r a d u a t e s . 5) 19 out of 200 g i r l graduates.  24.  Mr. N o l l accepted a temporary job f o r a week. At the end o f the week he was p a i d $ 6 0 . He was t o l d t h a t he had completed 5/6 o f the work, and c o u l d r e t u r n the next week to complete the job a t the same r a t e o f pay. How much would he be p a i d to complete the job? 1) $6 2) $10 3) $12 4) $50 5) Not g i v e n .  25.  The annual premium f o r an i n s u r a n c e p o l i c y i s f i g u r e d a t the r a t e o f 54# per $100. For premiums p a i d f o r 3-year p e r i o d s i n advance, the 3-year premium i s s e t a t 2 1/2 times the 1-year premium. What would be the average y e a r l y r a t e on a 3-year premium p a i d i n advance? 1) 36<z> per $100 2) 45^ per $100 3) 5 V per $100 4) 64^ per $100 5) I t would depend upon the f a c e v a l u e o f the p o l i c y .  26.  The hundreds d i g i t o f a t h r e e - d i g i t number i s Jh, the tens d i g i t i s t , and the u n i t s d i g i t i s u.. Which of 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 h i s number? 1) h x t x u 2) h + t + u 3) 100 h + l O t + u 4) ( 1 0 0 h ) ( 1 0 t ) ( u ) 5) None of t h e s e .  6 1  27.  / 5 yrf».  To determine the d i s t a n c e between p o i n t s A and B on o p p o s i t e s i d e s o f a pond, l i n e BC i s l a i d o f f p e r p e n d i c u l a r to the l i n e o f s i g h t between A and B, as shown i n the diagram above. The d i s t a n c e s BC and CA are then measured. What i s the d i s t a n c e between A and B i n yards? ( Do not t r y to estimate the d i s t a n c e from the f i g u r e , s i n c e i t i s not drawn to s c a l e . ) 1) 18 2) 20 3) 21 4) 23 5) Not g i v e n .  28.  62  One commonly used method o f a l l o w i n g f o r a n n u a l d e p r e c i a t i o n i n v a l u e c o n s i s t s o f d e d u c t i n g t h e same per c e n t each y e a r o f the v a l u e a t the b e g i n n i n g o f t h a t y e a r . A c c o r d i n g t o t h i s method, what i s the v a l u e a t the end o f two y e a r s o f a c a r c o s t i n g $ 2 , 0 0 0 new i f t h e r a t e o f d e p r e c i a t i o n i s 25% per year? 1)  $1,000  2)  $1,125  3)  $1,500  40  $1,250  5)  Not  given.  29.  The energy r e q u i r e m e n t o f a d u l t s 2 0 - 6 0 y e a r s o l d f o r 1 hour o f s l i g h t e x e r c i s e i s .4- c a l o r i e per pound of body w e i g h t . The energy r e q u i r e m e n t f o r a d u l t s 6 0 - 7 0 y e a r s o l d i s 10% l e s s . What i s t h e c o r r e s p o n d i n g c a l o r i e r e q u i r e m e n t per pound o f body w e i g h t f o r t h e s e o l d e r persons? 1) . 0 9 6 2 ) . 0 6 3 ) . 0 3 40 . 3 6 5) Not g i v e n .  30.  An a u t o m o b i l e d e a l e r d e t e r m i n e s t h e t r a d e - i n v a l u e o f a c e r t a i n make and model o f c a r by means o f the f o l l o w i n g table. Y = Number o f Y e a r s Since Purchased T = T r a d e - i n Value in Dollars  2  1 2,000  3  1,700  1,4-00  1,100  ^  Which o f the f o l l o w i n g f o r m u l a s e x p r e s s e s the r e l a t i o n s h i p between t r a d e - i n v a l u e (T) and the age o f the c a r ( Y ) ? 1) 3)  T = 2 , 0 0 0 - 150Y T = 1,100 + 600Y  5) None o f t h e s e .  2)  40  T = 2,500 T = 2,300 -  500Y  300Y  31.  What i s t h e a r e a i n square f e e t o f the c i t y l o t shown i n t h e diagram above? 1)  4-,800  2)  7,600  3)  8,000  40  8,125  5)  Not  given.  Use the f o l l o w i n g i n f o r m a t i o n  l n s o l v i n g problems ^2 and  ^3.  Let A = the cash p r i c e o f an a r t i c l e o f merchandise, I = the i n s t a l l m e n t p r i c e , i . e . , the t o t a l c o s t when p u r c h a s i n g on the i n s t a l l m e n t p l a n . D = the down payment, n = the number o f monthly payments. Then r , the approximate r a t e o f i n t e r e s t p a i d when p u r c h a s i n g on the i n s t a l l m e n t p l a n , i s g i v e n by the formula: 24(1 - A) r = (A - D)(n + 1) 32.  Which 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 the amount o f each payment? I - D A - D A I 1) n 2) n 3)n 4) n 5) None o f these.  33.  A t e l e v i s i o n s e t s e l l s f o r $300 cash, or $50 down and $40 per month f o r 7 months. What i s the approximate r a t e of i n t e r e s t p a i d by an i n d i v i d u a l p u r c h a s i n g t h i s s e t on the i n s t a l l m e n t plan?  1) 7%  2) 10%  3) 17%  4) 26%  5) 3 6 $ .  APPENDIX E  ITEM ANALYSIS OF MATHEMATICS ATTITUDE TEST  65  A - Attitude  Toward  Calculation  Top 27% Question Number  Number o f Responses  Bottom  Percent o f Responses  r  bis  Percent o f Responses  27% Number o f Responses  1  22  69  49  22  7  2  24  75  48  25  8  3  14  44  53  6  2  4  18  56  34  22  5  8  25  50  -  7 0  6  23  72  37  37  12  7  15  47  45  10  3  8  26  81  71  10  3  9  15  47  45  10  3  10  20  62  37  28  9  11  10  32  30  10  3  12  20  62  33  28  13  12  37  61  -  9  14  20  62  64  6  2  15  21  65  29  37  12  16  28  87  63  22  7  0  66  B - A t t i t u d e Toward Q u a n t i t a t i v e T h i n k i n g  Top 27% Question Number  Number o f Responses  Bottom 27%  Percent o f Responses  r  bis  Percent o f Responses  Number o f Responses  1  15  h6  66+  -  0  2  15  h6  4-5  9  3  3  15  he  36  16  5  4  16  50  68  -  0  5  28  88  63  25  8  6  20  63  68  3  1  7  22  69  26  44  8  15  4-6  66  -  14  9  12  37  50  3  10  Ih  44  65  -  11  18  56  60  6  2  12  12  37.5  31  10  13  15  4-6  64  3  1  14  17  53  4-9  13  4  15  8  25  35  6  2  16  11  34  34-  9  3  + the value  i s g r e a t e r than  * the value  i s t o o low t o b e  +  09*  the i n d i c a t e d acceptable.  number.  0  1 0  67  C - A t t i t u d e Toward Non Mathematical  Top 27% Question Number  Number o f Responses  Activity-  Bottom 27%  Percent o f Responses  r  bis  Percent o f Responses  Number o f Responses  1  20  77  63  15  4  2  18  69  72  4  1  3  22  85  72  12  3  4-  18  69  65  8  2  5  21  81  82  -  0  6  5  19  34  4  1  7  12  46  66  -  0  8  19  73  79+  9  22  85  10  17  11  +  +  -  0  50  38  10  65  62  8  2  23  88  68  19  5  12  6  23  39  4  1  13  26  100  77  +  31  8  14  21  81  82  +  -  0  15  21  81  79+  4  1  16  19  73  79+  -  0  + the value  i s g r e a t e r than  t h e i n d i c a t e d number.  

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