UBC Theses and Dissertations

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UBC Theses and Dissertations

The relationship between volume conservation and a volume algorithm for a rectangular parallelepiped Feghali, Issa Nehme 1979

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THE RELATIONSHIP BETWEEN VOLUME CONSERV&TION AND ft VOLUHE ALGORITHM FOR A RECTANGULAR PARALLELEPIPED  by ISSA  NEHME FEGHALI  B.Sc. , C a l i f o r n i a S t a t e P o l y t e c h n i c U n i v e r s i t y , 1971 M.S., B a y l o r U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF . THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in THE FACULTY OF GRADUATE STUDIES (Mathematics Education)  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1979 @  Issa  N. F e g h a l i ,  1979 -  In p r e s e n t i n g  this  thesis  an a d v a n c e d d e g r e e a t the L i b r a r y I further for  agree  scholarly  by h i s of  shall  make  it  freely  that permission  for  It  fulfilment  of  of  Columbia,  gain  of  The U n i v e r s i t y  MathwnaMrs  of  British  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1WS  shall  D  e  c  24,  1979  Education  Columbia  the  requirements  reference copying of  I agree and this  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  financial  for  for extensive  permission.  Department  British  available  p u r p o s e s may be g r a n t e d  thesis  written  Date  the U n i v e r s i t y  representatives.  this  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT  C h a i r m a n : Dr.  D o u g l a s Owens  T h i s study between  the  degree t o of  a  which  present grade  at  =  level  the  programs  and  Piaget  that  volume in  by  grade  stage  4 are  our  theory  seems t o be  present  i t  as e a r l y  is  not  until how  formal  grade the  they  measures.  4,  find  children  operations.  In s u c h a  in  or  some  formal  can  V e r y few  curriculum  the  While  as  a need f o r r e s e a r c h  school  Height  concerning  i s introduced.  algorithm  to exhibit  X  between  c h i l d r e n understand  expected  Width  algorithm  apparent discrepancy  m u l t i p l y i n g the boundary  predicament there justify  that  volume  the  levels.  algorithm  the  operational  an  relationship  comprehension  Piaget*s  s c h o o l programs i n t r o d u c e claims  X  k n o w l e d g e and  a t which t h i s  (1960)  the  o f d i s p l a c e d volume and  Length  i s a consequence of  school  investigate  grade c h i l d r e n l e a r n the  "Volume f  to  of c o n s e r v a t i o n  sixth  x W x H) "  problem  designed  level  cuboid,  (V = L The  was  order to  to  suggest  modifications. Subjects were (N =  of t h r e e  classified 16)  and  suburban  as n o n c o n s e r v e r s  conservers  o f volume c o n s e r v a t i o n . experimental  schools  groups  (N = 32) The  and  (N = 5 7 ) ,  using  subjects  one  in  a  British partial  Columbia conservers  judgement-based  were t h e n  c o n t r o l g r o u p by  divided into randomizing  test two each  iii  conservation  group a c r o s s t h e  experimental using  an  school  approach programs  treatment of  groups  included  the  in  treatment respect  other  America.  by  a  using  on  was  method  different  items),  Volume  Achievement  the S t a n f o r d  Conservation, posttests  tests  A c h i e v e m e n t , and posttests fully  Treatment).  compensating  (27  was  tests  with  supplemented control  group  The  pretests and  were:  tests  were a n a l y z e d  two-way a n a l y s i s  of  Volume  Computation.  Achievement.  the  questions posttest per  cent  volume  algorithm  regardless and  retention  performance  of  Data  test,  the  using  a  covariance.  conservation  subjects of this  level.  The  Volume  from  separately  t o c o m p u t a t i o n and  their  (11  (45 i t e m s )  S u b j e c t s i n t h e volume t r e a t m e n t showed t h e y were a b l e apply  by  Multiplication  were: Volume C o n s e r v a t i o n ,  Multiplication  crossed  was  This  factors  items),  Achievement,  and r e t e n t i o n  algorithm  were u s e d : Volume C o n s e r v a t i o n  Achievement T e s t .  and r e t e n t i o n  volume  emphasized  i t e m s ) and t h e c o m p u t a t i o n s e c t i o n  Volume  the  this  on n u m e r a t i o n s y s t e m s .  achievement  (20  of  of  (N = 39)  that  o f t h e volume a l g o r i t h m . The  taught a unit  that  the  group  the  algorithm  Activities  (Multiplication  training  of  resembles  experimental  using  skills  discussion  = 30)  3 X 3  which  c u b e s and l a t e r  algorithm  included  Four  of  t a u g h t t h e volume  t o v a r i a t i o n s i n o t h e r f a c t o r s and  brief  (N  North  One  c o m p a r i s o n , o r d e r i n g , and f i n d i n g  The  multiplication  was  treatments.  (Volume T r e a t m e n t )  c u b o i d s by c o u n t i n g  taught  a  (N = 36)  used  "V = L x W x H."  three  to  comprehension  level.  On  the  g r o u p showed a 65  F o r the grade 6 s t u d e n t s i n the  study, c o n s e r v a t i o n l e v e l learning  the  comprehension On  in  volume  the  <  other  0.01)  (F =  groups  multiplication  10,33, p < 0.01)  on t h e M u l t i p l i c a t i o n  than those  involvement  was,  generally,  level  or  levels  those  achievement  Test.  i n the o t h e r groups  t r e a t m e n t s . The  between  the  pretest  d f = 2)  and  retention  independent  of the  (biserial In asked  of  r =  posttest  12.24,  on t h e  Volume  to teach  that  includes  improvement of  their  transition a)  each  test  achievement  a lower  to a higher  of  posttest  = 0.97,  2  achievement (biserial  the  the s t u d e n t s *  volume  independent of  ( X  from  of  objects.  treatments ( X  =  2  d f = 2)  and  s c o r e s between t h e  r = 0.13)  and  0.93, b)  pretest  retention  test  students  were  0. 09) . to  t o write reasons  explicit  an  found  and  volume  an addendum  e q u a l and  (F =  i n manipulating physical  regardless  of c o n s e r v a t i o n was  each  and  treatment  I t seems a p p r o p r i a t e , t h e r e f o r e ,  active  conservation scores  in  b e t t e r than  volume t r e a t m e n t d i d s i g n i f i c a n t l y  better  There  factor  the - c o m p u t a t i o n  volume a l g o r i t h m o f a c u b o i d u s i n g a method  students*  and  significant at  s u b j e c t s of the  Achievement P o s t t e s t . the  a  algorithm  significantly  Subjects of the p  not  levels.  the p o s t t e s t ,  performed  was  unequal  the  Conservation  for their  volumes.  Those  on t h e i t e m s o f u n e q u a l  Test  judgements i n i t e m s written  reasons  volumes than  involving were  of e q u a l  more  volumes.  V  TABLE OF CONTENTS  Page  ABSTRACT  .  L I S T OF TABLES  ,.  i i ix  ACKNOWLEDGEMENTS  xiii  CHAPTER I  ,  THE PROBLEM  Definition Statement  o f Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o f t h e Problem  Justification  Validation  of Piaget's  ....................... Cognitive  c f some o f P i a g e t ' s  Theory  Findings  7 7  .........  Need f o r R e s e a r c h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER I I  3 5  of t h e Problem  Implications  1  10 11  REVIEW OF RELATED LITERATURE . . . . . . . . . . . . . . . . . 13  Summary o f P i a g e t ' s Piaget's  Studies  Theory o f Volume  Reactions to Piaget's Training  13 ......................  T h e o r y and E x p e r i m e n t s ....... 23  Experiments ...............................  Discussion  of  Training  Experiments  other  cf  Conservation  Experiments  Involving  of L i t e r a t u r e  25  Volume  Tasks ...........................  Summary and I m p l i c a t i o n s  25  than  Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion  19  Reviewed  30  . . . . 34  vi  Age o f S u b j e c t s  .................................  N a t u r e o f Volume C o n s e r v a t i o n Choice  Tests  .............  35  o f Treatments ............................  39  Conclusion CHAPTER  III  34  43  PROCEDURES  45  Subjects  ..45  Description  o f the Treatments ......................  46  Volume T r e a t m e n t  46  Multiplication  47  Treatment  C o n t r o l Treatment ............................... Description  of Tests  Instruction  and T e s t i n g  Instructors  ...............................  49  P r o c e d u r e s .................  55  ............. ........................  55  S c h e d u l e o f I n s t r u c t i o n and T e s t i n g Design  48  .............  55  of t h e Study ................................  57  H y p o t h e s e s .........................................58 Statistical CHAPTER IV Tests  Analyses  60  RESULTS  62  Reliabilities  and Item  Volume A c h i e v e m e n t T e s t Volume C o n s e r v a t i o n Multiplication Preliminary  Analysis  63  ........................  63  .................  63  ...............................  64  ......................................  64  Assumptions o f A n a l y s i s Instructor  62  .........................  Achievement T e s t  Analysis  Covariates  Test  Analysis  Effect  of C o v a r i a n c e  ............  65  ...66  o f C o v a r i a n c e .............................  68  vii  Hypotheses  1-3 .,  68  Hypothesis  4  73  Study  74  Correlation  H y p o t h e s e s 5 and 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  74  H y p o t h e s e s 9 and 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  77  Hypothesis  78  10  T e s t s o f Independence  79  H y p o t h e s e s 6 and 8  79  Hypothesis  81  Post  11  Hoc Q u a l i t a t i v e  Transition Students'  Analyses  .......................  between C o n s e r v a t i o n Reasons  f o rtheir  L e v e l s ..........  86  Consistency Responses  between to  Conservation  Question  Test  Reasons  Levels  o f C o n s e r v a t i o n and 12  of  the  Volume  ............................  f o rtheir  R e s p o n s e s on  90  Question  12 CHAPTER V  82  R e s p o n s e s on Q u e s t i o n  11  Students'  82  92  SUMMARY, CONCLUSIONS AND RECOMMENDATIONS . . . . . . 97  Review  o f t h e Problem ..............................  F i n d i n g s and C o n c l u s i o n s Summary o f C o n c l u s i o n s  ...........................99 and D i s c u s s i o n  Importance o f C o n s e r v a t i o n Effect  97  Levels  102 ............102  o f T r e a t m e n t s .........................103  Transition  between C o n s e r v a t i o n  Levels  Effect  o f Mathematics Achievement  Effect  o f Sex  .......105  .............106 .106  viii  Limitations  o f the Study -  Implications  f o r Educational  .................107 P r a c t i c e ..............109  Eecommendations f o r F u t u r e R e s e a r c h  ...110  REFERENCES  ....114  APPENDIX  A  Description  of t h e I n s t r u c t i o n a l U n i t s  APPENDIX  B  Item  APPENDIX  C  Raw  APPENDIX  D  Unadjusted D e s c r i p t i v e  APPENDIX  E  Tests  Statistics  ......123 191  Data  ..195 Statistics  ...........199 .201  ix  LIST OF  TABLES  Table  Page  3.1  Experimental Design  4.1  F Values  4.2  Analysis of Covariance - Instructors E f f e c t  4.3  Analysis Scores  4.4  f o r E n t e r i n g of C o v a r i a t e s  Means,  of  of  Volume  Achievement  D e v i a t i o n s , and Group S i z e s  Pretest  Adjusted  Means, S t a n d a r d  Volume A c h i e v e m e n t  Adjusted  Scores  for  o f Volume  Treatments  by  L e v e l s ...................................  Conservation  of  Covariance  71  Standard  Conservation  4.7  D e v i a t i o n s , and Group  Posttest  S c o r e s f o r T r e a t m e n t s by  Levels  Means, S t a n d a r d  Volume  Achievement  Analysis  of  Covariance  72 D e v i a t i o n s , and Group Hetention  Test  Scores  Sizes for  of  Multiplication  Analysis Scores  72  Posttest  Scores... 4.9  71  Sizes  T r e a t m e n t s by C o n s e r v a t i o n L e v e l s . . . . . . . . . . . . . . . . . . . . . 4.8  68  Posttest  Scores  Achievement  of  .........  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71  Retention  4.6  57 65  o f C o v a r i a n c e o f Volume A c h i e v e m e n t  Analysis  4.5  .................................  73 of  Covariance  of M u l t i p l i c a t i o n  Retention  ................................................  74  X  4.10 A d j u s t e d of  Means, S t a n d a r d  Multiplication  Test 4.11  Achievement  Correlation Scores  Transition  Conservation 4.12 C e l l  a  Post  and B e t e n t i o n  Change  Significance  Between  Lower  Levels  Volume Test  Level  of  Contingency Table:  4.17 C o n t i n g e n c y  Coeffients  Between Volume A c h i e v e m e n t and t h e B e t e n t i o n  T r a n s i t i o n Op,  Test  Table:  or  Scores Test  . 78  . . . . . . 78 Staying  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80  T r a n s i t i o n Op V e r s u s T r e a t m e n t s on Test  . . . . . . . . . . . . . . . . . . . 80  T r a n s i t i o n Down V e r s u s  t h e P o s t t e s t and t h e B e t e n t i o n Tahle:  Down,  and  Between P r e t e s t - P o s t t e s t and  P o s t t e s t and t h e B e t e n t i o n  Contingency  Changed/  C o r r e l a t i o n C o e f f i e n t s and S i g n  Table:  Contingency  Levels  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77  Same V e r s u s T r e a t m e n t s  Pretest-Retention  . . . . . . . . . . . . . . . . . . 76  t h e P r e t e s t and e a c h o f t h e  SAT S c o r e s on t h e P o s t t e s t  4.16  Test  X T r e a t m e n t s on t h e  Moment C o r r e l a t i o n  Levels  Point-Biserial  4.19  or  Whose C o n s e r v a t i o n  Test  4.15  on  Higher  betwween  4.14 P e a r s o n ' s P r o d u c t  4.18  Coefficients  and B e t e n t i o n  Not  the  74  Sizes of Conservation  Did  the  P o s t t e s t and B e t e n t i o n  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76  Number o f S u b j e c t s  and  Sizes  on t h e P o s t t e s t and B e t e n t i o n  to  Pretest, Posttest 4.13  Achievement  and Group  Scores f o r Treatments  Biserial  and  Deviations,  Test  Treatments  . . . . . . . . . . . . . . . . 81  Pretest Conservation  Level  Versus  Sex 4.20  82 Contingency  Conservation  Table:  P r e t e s t - P o s t t e s t T r a n s i t i o n Among  L e v e l s by T r e a t m e n t s . . . . . . . . . . . . . . . . . . . . . 85  4.21 C o n t i n g e n c y  Table:  Among C o n s e r v a t i o n 4.22 C o n t i n g e n c y  Table:  Responses  Conservation 4.24  and 4.27  Retention  Table:  on  Reasons f o r the  Volume  11  of  88  Reasons f o r the  Test  to their  Volume  11  of  Reasons f o r the  Volume  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89  C o r r e c t o r I n c o r r e c t Response on Conservation  Pretest,  Conservation  with  Respect  Levels  to their  12  of  Posttest ......... 92  Reasons f o r the  Pretest  Responses  Volume .•  on  with  Respect  Question  12  to their of  95  Reasons f o r the  Volume  P o s t t e s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95  Responses  Conservation  of  to their  Respect  Question  Number o f S t u d e n t s  their  with  Test Versus  Responses  Conservation  11  Respect  Question  Number o f S t u d e n t s  their  4.29  on  Number o f S t u d e n t s  Conservation  with  Question  12 o f t h e Volume  their  4.28  on  Retention  Contingency  Item  to their  P o s t t e s t .................................89  Responses  Conservation  Respect  Question  Number o f S t u d e n t s  their  4.26  on  with  P r e t e s t ..................................  Responses  Conservation  Transition  L e v e l s by T r e a t m e n t s . . . . . . . . . . . . . . . 86  Number o f S t u d e n t s  their  4.25  Posttest-Retention Test  Number o f S t u d e n t s  their  Transition  L e v e l s by T r e a t m e n t s . . . . . . . . . . . . . . . 85  Among C o n s e r v a t i o n 4.23  P r e t e s t - R e t e n t i o n Test  on  with  Respect  Question  Retention  Test  12  to their of  Reasons f o r the  Volume  . . . . . . . . . . . . . . . . . . . . . . . . . . . 96  B1 Volume A c h i e v e m e n t P r e t e s t Item S t a t i s t i c s . . . . . . . . . . . . 191 B2 Volume C o n s e r v a t i o n  P r e t e s t Item S t a t i s t i c s  191  B3 Volume a c h i e v e m e n t B4 Volume C o n s e r v a t i o n B5 M u l t i p l i c a t i o n  B7 Volume C o n s e r v a t i o n  D1  Betention Betention  Betention  Volume a c h i e v e m e n t  Conservation  Levels  ,...192  Statistics  ..........192  Statistics T e s t Item T e s t Item  T e s t Item  193 Statistics Statistics  Statistics  U n a d j u s t e d Means, S t a n d a r d D e v i a t i o n s , of  D2  Statistics  P o s t t e s t Item  P o s t t e s t Item  B6 Volume a c h i e v e m e n t  B8 M u l t i p l i c a t i o n  P o s t t e s t Item  Volume  .,..194  .........194  and Group  Sizes  P o s t t e s t S c o r e s f o r T r e a t m e n t s by  ...................................  U n a d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and G r o u p of  193  achievement  T r e a t m e n t s by C o n s e r v a t i o n  Betention  Test  Scores  200  Sizes for  L e v e l s . . . . . . . . . . . . . . . . . . . . . 200  ACKNOWLEDGEMENTS  Sincere thesis  supervisor,  assistance extended Arlin,  for his  Thomas  Individually  their  time  this  study.  Bates,  Thanks  of J a r v i s  School  and  participation  colleagues,  Drive  Institute  Gratitude i s also  in this  study.  note  John  of  Taylor  gave g e n e r o u s l y o f  thanks  coding,  editing  is  Lynn  administrators  of  Richardson  Delta  School  also  Elementary for  expressed  to  Cannon, and t o my  most  of  t e a c h e r s , s e c r e t a r i e s and  and G a r r y  and  completion  assistance.  School,  and  Dr. P a t r i c i a  t o Dr. Todd R o d g e r s and  Elementary  Johnson, Lee H e r b e r t s  encouragement  the  Elementary  Cliff  special  And  they  statistical  and t h e p r i n c i p a l s ,  students  and  D r . Edward Hobns and D r . G a i l  i s also expressed  a l s o t o thank  District  testing,  guidance  and e x p e r t i s e i n t h e f o r m u l a t i o n a n d  I would l i k e  Beverley  study.  and c o l l e c t i v e l y ,  Thomas O'Shea f o r t h e i r  A  encouragement,  members o f my t h e s i s c o m m i t t e e :  Professor  School  t o E r . . D o u g l a s Owens, my  i n t h e development o f t h i s  to other  Spitler.  Dr.  a p p r e c i a t i o n i s expressed  their  my  friends  Roth  fortheir  help i n  of  a l l 'f o r  their  and s u g g e s t i o n s , .  finally, of B r i t i s h  I  am  grateful  Columbia  to c a r r y out the study.  t o the Educational  f o rproviding  financial  Research support  1  CHAPTER I  THE  The the  process  of selecting*  mathematics  curriculum  mathematics e d u c a t o r s . concern  and  textbooks  contain  many e d u c a t o r s until  hold  about  Elkind,  volume  12  theoretically  and  these  i n order  educators  suggest many  of  however, d e a l s algorithm  "Volume = L e n g t h Two introduce (Dilley  has  f o r example,  While  the  been  a  Such  widely  used  textbook  the  algorithm  The  1975-b;  the algorithm  or  can f o c u s on  present  study,  t h e i n t r o d u c t i o n o f t h e volume parallelepiped  i.e.,  (V = L x W x H) ". series  "V = L x W x H"  formally  to  positions of  in  British  Columbia  i n g r a d e 5 (age 10)  e t a l . , 1974 and E i c h o l z e t a l . , 1974). A n o t h e r  introduces  volume  curriculum  an e x a m i n a t i o n  rectangular  x Width x H e i g h t  the  of  three,  therefore,  the present  presentation. with  conserve  need,  volume  majority  Carpenter,  examine  to justify  volume  a  major  cognitive  as e a r l y a s g r a d e  1964;  in  o f many  f o r presenting  students.  experimentally  particularly  for  level  activities  i t s modification.  aspects  been,  (Ozgiris,  There  topics  the i n t e r e s t  t h a t most c h i l d r e n do n o t  age  1961-a).  timing  about t h e n e c e s s a r y  age  school  and  caught  has  analysts  appropriate  concepts to elementary  ordering has  There  among c u r r i c u l u m  abilities  PROBLEM  i n g r a d e 4 (age 9)  series  ( E l l i o t et  2  al,,  1974)  al.,  1975).  graders of  and  a  uses i t i n f o r m a l l y i n grade 3 This  last  series,  for  i n s i t u a t i o n s i n which t h e y rectangular  example,  are  parallelepiped  (age  8)  involves  t o compute  given  (Elliot  the  i t s length,  et  third volume  width  and  height. Most p r o p o n e n t s such  early  children  1960) IV  introduction  do  learning  not  it  [formal  edges"  (p. 4 0 8 ) .  notions 28),  or  6  volume  Piaget  that with  (age  likewise,  There position  discrepancy  "V  are  grade  levels  5  geometric  to  =  raises  curricula  that  be  the  profitably  placed  a  some  for  (Piaget et a l . , until  stage  they  multiplying  in  381).  can  boundary volume  school increasingly of  Osborn  premature  discrepancy  programs r e g a r d i n g  issue in  to i n the  should of  geometrical  (1976, pp.  stress  of  the  (1975) s u g g e s t e d  elementary  see  be  between  27-  volume  where  curriculum"  school  Piaget's  t h e : l e v e l at  which  introduced.  whether or n o t  presenting  Fabricant  formulas a t the studied  by  development  (p.  L x W x H"  justified  seriously  abilities  " i t i s not  learned  spontaneous  most s c h o o l  algorithm  that  most  a s e r i o u s l e a r n i n g problem;.  seems and  11).,Piaget  with  that  (1960) adds i n h i s d i s c u s s i o n of  warned  creates  claim  cognitive  simply  "knowledge the  and  disagree  c h i l d r e n u n d e r s t a n d how  a s c h i l d r e n grow o l d e r "  algorithms  the  necessary  grade  would  algorithm  f o r example, h o l d s  area  calculation  theory  the  operational] that  an  interferes  of  develop the  before  himself,  arrive at  of P i a g e t ' s  present  algorithm  This school at  the  that "teaching level  such f o r m u l a s (pp. .6-7).  has  to  would be It  was  of be  most as  a  3  result  of  the concerns  mentioned  above t h a t t h e  present  study  originated.  Definition It  i s necessary  which w i l l The  occur term  displaced  by  substance: with  of..Terms  to c l a r i f y  throughtout volume  the  the  refers  liquid  or  to  a three dimensional  a  hollow  1976,  p.  the  certain  measure  o b j e c t . The  gas.  c a p a c i t y which r e f e r s t o t h e  by  of  of  the  o b j e c t may  space  be  any  Volume i s n o t  t o be  confused  measure o f t h e  space  enclosed  o b j e c t . Even t h o u g h " i n t e r n a l  c o n t a i n e r ...  terms  study.  a three dimensional solid,  usage  volume  of  i s synonymous t o c a p a c i t y " ( K e r s l a k e ,  14),  the term  volume  usually  term  conservation refers  refers  to  non-hollow  objects. The attribute  of  an  changes o f other 1962,  pi  invariant nothing set of  153).  object  attributes  example, t h e  r e g a r d l e s s of i t s  i s added o r t a k e n  that a  (Wohlwill  and  volume o f a s u b s t a n c e  shape  and  position  away. L i k e w i s e , t h e  as  s e t as l o n g as n o t h i n g  The with 550)  term  cuboid  i s added o r t a k e n  rectangular, parallelepiped  (Webster's d i c t i o n a r y ,  is illustrated  however, have  been  by  edited  the shape of f i l l e d  confused  with  under Lowe,  remains long  as  n u m e r o u s n e s s of a  r e m a i n s unchanged d u r i n g changes i n the s p a c i a l the  certain  (or o b j e c t s ) r e m a i n s i n v a r i a n t  irrelevant For  to the concept  arrangement  away.  which i s synonymous by  Gove,  boxesi  rectangular  1971,  These prism  p.  terms, and  a  4  clarification right  needed.  r e c t a n g u l a r prism  (James used  is  and  James,  A  rectangular^parallelepiped i s a  r a t h e r than  just  1S76). T h r o u g h o u t  a  this  rectangular paper cuboid  prism will  be  i n t e r c h a n g a b l y with r e c t a n g u l a r p a r a l l e l e p i p e d . The  that  term  a l g o r i t h m r e f e r s to a procedure  guarantees  correctly 1978).. basic  and  a  i n the  Algorithms  division  have  of e g u a t i o n s The  width,  product  of these  of  computation includes, state the  of  in their  level  may  be  height  measures the  algorithm  and  comprehensioni  (V =  the  four  multiplication  and  with  dimensions  computing  refers at  level  of  are asked  cuboids,  dimensions, The  or  are  state  a  attachments volume  of  of the  total  cuboids cuboid  t r a n s f o r m a t i o n s on  with  given  known d i m e n s i o n s  resulting  a given  volume  cuboid.  from  a  level  where  the  of  computation  where s t u d e n t s  cuboids*  the  the l e v e l s  i n c l u d e s , f o r example, s i t u a t i o n s  to  the  to  comprehension asked  the  dimensions  and  partitioned  of  complex  applying  three  cuboid  systems  LxWxH)._  The  known  more  o b t a i n e d by  "V = L x W x H"  example, s i t u a t i o n s  of  The  volume a l g o r i t h m  the  cuboids  description  Papadimitriou,  of d i f f i c u l t y .  the  - of the  volume g i v e n d i a g r a m s o f  partitioned  and  steps  performed  a l g o r i t h m s , while s o l v i n g  measures o f  learning  for  (Lewis  are  of i n v e r s e matrices i s a  the  three  i f the steps  subtraction*  a cuboid  and  term  mastery  use  finding  length,  The  order  r a t h e r simple  by t h e  of  result  addition,  volume  algorithm  proper vary  operations,  one.  correct  of o r d e r e d  to non-  word of  students  diagram  of  or to s t a t e  the  proposed  dimensional  5  Statement o f t h e Problem The  purpose  of  relationship  between  the  to  degree  algorithm  by  Piaget  which  (Piaget,  objectives textbook  on and  regard  to  children.  the  order  first  of  a  cuboid  conservation  based  treatment factors  consisted  when t h e p r o d u c t  = 2 X 3 X 6,-  statements  such  supplemented cuboid  three  of  school  of  on  elementary with grade  and l e a r n i n g  36 = 4 X [  ] X [  programs.  be J.  treatment  algorithm  The  the task  able  method second  of varying given  to  This  d i s c u s s i o n o f t h e volume  study  implemented.  discovery  F o r example,  would  algorithm  t h e volume  learning  i s constant.  aims o f t h i s  based  sixth  were  using a guided  and c o n s i s t e d o f t e a c h i n g general  The  informative data  treatments  "V = L x W x H i " The t h i r d  treatment The  by a b r i e f  was  about l e a r n i n g t h e  the . student as  employed  f o r t h e volume o f a c u b o i d .  M  present  mainly  volume  1960).  of conservation,  consisted of teaching  on a p p r o a c h e s o f  of  used  level  and  showed t h e r e l a t i o n s h i p s among  levels  "V = I x W x H"  volume  level  widely  the  l e a r n t h e volume  algorithm  in  to gain information  treatment  The  a l s o provided  "V = L x W x H  of  Szeminska,  the  found  t h e volume o f a c u b o i d ,  The  36  volume  achievement,  of the algorithm  for  of  The s t u d y  determine  v a r i a t i o n s of t e s t s  and  Furthermore, the study  mathematics  In  using  usage  to  students  "V = L x W x H".  activities  series.  grade  Inhelder  the  was  of conservation  sixth  was d e t e r m i n e d  achievement  study  the l e v e l  f o r a cuboid  conservation  this  that  complete  task  algorithm  was of a  served- a s c o n t r o l  various numeration may b e ' l i s t e d  systems.  as f o l l o w s :  6  1. To  determine the v a r i o u s degrees to  partial  conservers,  volume a l g o r i t h m To  determine  the  two  teaching  for  a  the degree  H."  of e f f e c t i v e n e s s f o r each  methods on l e a r n i n g t h e volume  determine  algorithm  To  level  determine  levels  the  of a cuboid  conservation  5.  "V = L x W x  effect  on  of  learning  the t r a n s i t i o n  from  the  relationship  the  of  between  To  determine  achievement To  and  achievement algorithm Finally verifying according (not  and  for a  of  t o S i e g l e r and  unlike  investigating  this  one  volume  relationship  the  the  between  relationship  sex  degree  of  and  the  cuboid.  between  of c o n s e r v a t i o n  relationship  the  the  mathematics  of  volume.  between  mathematics  learning  the  volume  cuboid.  the r e s u l t s  aspects  volume  volume.  the l e v e l s  determine  the  sex and  d e g r e e o f l e a r n i n g t h e volume a l g o r i t h m f o r a  7.  algorithm  to another.  of conservation  To d e t e r m i n e  6.  of  cuboid. To  4.  conservers,  n o n - c o n s e r v e r s of volume l e a r n t h e  of a cuboid  2.  3.  and  which  one)  cognitive  of the  this  study  will  developmental theory  atlas have  (1976, become  development.  p.  360) a  be  useful  in  of P i a g e t  and  training  standard  studies  method f o r  7  Justification The  present  discrepancy  between  c o g n i t i v e theory introduce (age do  8) not  study  the  Problem  i s a consequence of a the  present  volume  algorithm  while  Piaget  and  the  of  a cuboid  of  volume b e f o r e  seems t o be  in  justify  school  its by  our  present  m o d i f i c a t i o n . DeVault expressed noting  the  a  need  a the  textbooks 3  most c h i l d r e n age  11  need f o r  curriculum  the  and  as e a r l y as g r a d e  a predicament t h e r e  to  programs  his followers claim that  conservation  about  some s c h o o l  6 ) . In such order  concern  school  of Piaget. S p e c i f i c a l l y ,  the  develop  of  (grade  research  or  suggest  f o r such  research  following:  Needed now i s t h e r e s e a r c h t h a t w i l l make t h e l i n k i n the continuum between the research of the behavioral s c i e n t i s t s and t h e work o f the mathematicians who have designed new programs for s c h o o l s ... The s t u d i e s most l i k e l y t o p r o d u c e u s e f u l r e s u l t s f o r c u r r i c u l u m work would be e x p e r i m e n t a l s t u d i e s ... (DeVault> 1966, pp. 637-639) Likewise, thinking  in  S t e f f e and  Hirstein  measurement  (1976) d i s c u s s e d  situations  and  children's  recommended  the  following: I n p l a n n i n g t h e m a t h e m a t i c s e x p e r i e n c e s .... the teacher should consider t h e s t a g e s o f c o g n i t i v e d e v e l o p m e n t . The proposed content and the methods of presenting that content should a l s o be c o n s i d e r e d . ( S t e f f e and H i r s t e i n , 1976, p. 35)  Implications Piaget observed  has  for several  children.  influential education  of Piaget's C o g n i t i v e  in  His  realizes  that  decades  theory  curriculum  Theory  has  planning Piaget  is  tested,  interviewed  and  become i n c r e a s i n g l y more because saying  "everybody something  in  that i s  8  relevant  to the  teaching  of c h i l d r e n "  (Duckworth,  1964-b,  p.  496) . A  central  attainment  theme i n t h e  of c e r t a i n c o n s e r v a t i o n  requirements  for  (Piaget,  p.  f o r the for  1941,  understanding  number  is  I t has a  problem  (I.Q.)  Engen,  (Van  .would  number],  seem the  e v e n been  predictor  The  solving  than  1971). Van  that  for  effect  these  5" and  the  relationship  importance  algorithm. necessity  between Volume  for  compare a c h o s e n evident  object to  be  that  of  that  is  a r e two is  names  necessary  conservation in  addition  Intelligence  children their  a t t e n t i o n on on  our  of and  Quotient  [nonconservers  r a t h e r than  be  in  taken  that of  activities traditional  in  addition  was  of  generalizing  to  measurement.  To  volume w i t h  the  volume c o n s e r v a t i o n  measured i s a r e q u i r e m e n t  designed  volume  seems  and  relevant to  o f number c o n s e r v a t i o n  conservation  volume  concepts  r e t e n t i o n of a d d i t i o n  success  T h i s study  conservation  unit  and  i s particularly  should  a p o s s i b l e analogy.  "8"  of number  conservation  solving  Even though c a r e  considered  48).  number  problem  volume c o n c e p t s ,  is  of  (p.  are  i s the  E n g e n , f u r t h e r , recommended  school should center  curriculum"  subtraction  suggest  +  of  Piaget  mathematical  reported  might enhance c o n s e r v a t i o n  arithmetic  It  generalization  better  subtraction  study.  "3  of  that  of  4 ) . F o r example  comprehension,  that  tasks  same number. T h i s c o n s e r v a t i o n  basic facts.  "it  c o g n i t i v e theory  be  measure  before,  the  volume apparent  volume i s t o  volume o f of the  a  an  to  seems t o  to f i n d  and  this  unit  an and  object. of  measurement  the can  9  be  meaningful. Piaget  position of  (Piaget  regarding  volume.  necessary  He  et  al.,  1960)  seems  to  c a l c u l a t i o n s i n measurement  considers  f o r any  the  meaningful  concept  hold  the  including  that  of c o n s e r v a t i o n  computation  same  in  both  to  area  be and  volume: ... C h i l d r e n a t t a i n a c e r t a i n k i n d o f c o n s e r v a t i o n o f a r e a (and volume), based on the p r i m i t i v e c o n c e p t i o n o f a r e a (and volume) as t h a t which i s bounded by l i n e s ( o r f a c e s ) . That understanding comes long before the ability to calculate areas and volumes by mathematical multiplication, i n v o l v i n g r e l a t i o n s between units of d i f f e r e n t powers i . . (p. 355)  Piaget's children level  do  and  volume o f  volume  not  conserve  thus  do  a cuboid  argues t h a t  e x p e r i m e n t s have a l s o r e v e a l e d volume b e f o r e  not  by  understand  simply  the  formal  how  they  that  most  operational  can  arrive  m u l t i p l y i n g i t s dimensions.  at  a  Piaget  :  The d e c o m p o s i t i o n and r e d e c o m p o s i t i o n of a continuum are o p e r a t i o n s which belong t o the l e v e l of f o r m a l o p e r a t i o n s . This explains why i t i s n o t u n t i l s t a g e IV t h a t c h i l d r e n u n d e r s t a n d how t h e y c a n a r r i v e a t an area or a volume simply by multiplying boundary edges.. ( P i a g e t e t a l . , 1960, p. 408) On  the  intellectual stimuli  other  hand,  development  of the  cognitive  words, discovery  Piaget and  favors  In  Piaget  development;  which i t i s b e n e f i c i a l  is  p r o c e e d s on  surroundings.  been a t t r i b u t e d t o him, for  Piaget  fact,  not i t s own and  considers he  only  ( M o d g i l l , 1974, education  saying  regardless  contrary education  questions pp.  that  r e j e c t s rote learning that  that  to  the  the has  a  tool  extent  126-127). In leads  of  what  t o be the  the  to  other  child  to  forces information  on  10  tie  student  who  i s not  ready  for i t :  T h i s i s a b i g d a n g e r o f s c h o o l - f a l s e accommodation which satisfies a c h i l d because i t agrees with a v e r b a l formula he has been g i v e n . T h i s i s a false equilibration which satisfies a c h i l d by accommodating t o words, t o a u t h o r i t y and n o t t o o b j e c t s as t h e y p r e s e n t t h e m s e l v e s t o him ... ( P i a g e t , 1964, p.4)  Validation  o f some o f P i a g e t ' s F i n d i n g s  Piaget's and the  computation world.  (1964),  (grade  Elkind  do  (1961-a) ,  not  until  found  develop  6). Likewise,  not  including  those  have b e e n examined  f o r example,  students  was  findings  that  at  about  conserved On  30%  of  other  "acquisition dependent  of  14 t h a t  on  least  Ogilvie  50%  75%  (1961)  of B r i t i s h  five  and  (1960)"  Piaget p.  experience p.  eight-year  hand,  specific  on  algorithm  grade  formal  dependent  (1971,  (1975-b) and  Uzgiris  of  American  Arlin  found  students  (1977)  age  that  11 i t  developed  reported  students i n B r i t i s h  that  Columbia  volume. the  (1972,  throughout  c o n s e r v a t i o n o f volume b e f o r e  volume c o n s e r v a t i o n . More r e c e n t l y , only  researchers  Carpenter  L o v e l l and  age  by  of volume c o n s e r v a t i o n  many  scientific  instructional  general maturation ( S i e g l e r and  223), t o be 179) olds  for  further  (grade 2 and  "V = L x W x H"  experiences  than  Atlas,  for  1976,  3)  i n order  p.  suggest can  far by  368).  considered  that  be f a r more  and  hypothesized  volume to  believe  r e a s o n i n g may  example,  necessary went  educators  less  Inhelder Graves  education  conservation.  Lovell  t h a t even s e v e n -  learn  how  to calculate  the  to  and  use  volume.  and the  11  Need f o r R e s e a r c h The  task  curriculum on  of  policies  the  researchers.  "are  instructional i s that  very  of  age  been n o t e d (Elkind,  ( L o v e l l and  t h a t the 1962;  verification one  Towler  should  Studies,  that  can  also  not  the  and  not "V  such expect  matter  " placing  conservation,  It necessary and  a  the  i n the  an  data volume  One  of  tested  in  reason  for  learning  been r e p o r t e d  not  13  1971;  conserve Graves,  considerably misleading  planning  as  636).  school  or  if  (Fogelman,  until  the all  a l l students  volume  activities,  used  without In  any  i n t r o d u c t i o n of students  proper  above, i n d i c a t e  algorithm  too  There  early  is  depend  presently  a  on  and the  volume  matter  of  exactness.  intention  of the  grade  the  conserve  to conserve.volume.  which  The  cultures  1970).  ones mentioned  14  also  1972).  with  is that  1961) . M e a n w h i l e , i t h a s  Wheatly,  x W x H"  never  volume b e f o r e  a d u l t s do  be  often  than  o f l e a r n i n g . I t seems, t h e r e f o r e , t h a t  rather than  was  conserve  d a n g e r of i n t r o d u c i n g t h e  process  preference  may  is  I t has  n e c e s s a r i l y delay  = L  harming the of  of  that  p.  school  makers r a t h e r  psychology 40).  Ogilvie,  in curriculum  volume.  is  not  and  1966,  the p.  policy  that  above have v a r i e d  volume a l g o r i t h m  one  do  majority  communities  case,  (DeVault,  t h e o r i e s to  found  theory  1967,  students  averages s t a t e d and  on  (Young,  on  DeVault  our awareness of  majority of  based  contexts"  limited  years  psychological  depends, u n f o r t u n a t e l y ,  educational  this  applying  of  the  investigator  r e l a t i o n s h i p between volume  a l g o r i t h m . . Such  data  was  used  to  provide  conservation for  making  12  recommendations r e l a t e d t o the volume  algorithm  p r i o r to grade  seems r e a s o n a b l e  ...  produce  results  useful  experimental  justification  studies"  to assert for  (DeVault,  6.  that  for  the  DeVault advocates t h a t " i t the  studies  curriculum 1966,  teaching  p.  639) .  most l i k e l y  work  would  to be  13  CHAPTER  REVIEW OF Even  though  interviewed  and  1950's, l i m i t e d  psychology  d e c a d e s , however, i t has foremost  of  reviewed summaries Ginsburg  theory  here. of and  his  and  to  opper,  which  Piaget's  individuals.  are  the  theory His  i n general  and  1969,  p.  may  more  find  in  In the l a s t  two  that  1964,  critical, p.  knowledge  5),.  Only  this  and  aspects study  are  comprehensive  (1963),  Maier  (1965),  Theory  seem t o be c o n s i s t e n t w i t h  h i s philosophy  is  (1957)..  of l e a r n i n g i n  c r e a t i v e , i n v e n t i v e and  Piaget  he  intellectual  distinguishes  discoverers  between  his  particular.  e d u c a t i o n a l goals c o n s i s t of c r e a t i n g i n d i v i d u a l s  active,  North was,  ix). to  tested,  i n general  of  relevant  Berlyne  of education  and  field  views i n F l a v e l l  (1969)  goals  influence  education  Summary o f P i a g e t ' s Piaget's  decades  become i n c r e a s i n g l y c l e a r  Interested readers  Opper  several  and  to a few  contributor  (Ginsburg  Piaget's  for  children,  until  development"  has  observed  educational  "the  RELATED LITERATURE  Piaget  American the  II  who  are  (Piaget,  development  of  l e a r n i n g . While development i s a spontaneous  and  14  genuine is  process  a process  a teacher  concerning  limited  or  an  Piaget  to only  b a s e d on  explains  which the  schemes a r e consist  of on  interpreting  a total  thus,  an  closed,  independent (in  the  transformation. operations  can  s t r u c t u r e s are case  of  rational,  the real  Piaget and  general  that  by  and  (Piaget,  as  that  illustrated  a given  p.  9).  sense), the  larger  ones t h a t i l l u s t r a t e  structure  the  but  his These which  mentally  way  or  them,  they  operation  are  he  tried  acquisition  not  " i t  is  i t i s always a p a r t ft.  structure  is,  which i s governed  as  in  a substructure  the  most  p.  of of  instance,  structures  and  laws  interdependence  ( P i a g e t , 1970,  describe  does  Rather,  i n many ways. F o r  of  attempt t o  structures  in"  under c e r t a i n  n a t u r a l number s y s t e m b e i n g  d i d not  are  modifying the  operations  Mathematically,  o r complex  and  21-22).  act,  by  a result  1964,  system o f  substructures  forming  "operations"  operations.  mathematical  be  by  of  8).  other  operations  structure"  pp..  understanding  from  by  8).  "experience  objects  p.  p.  caused  of behavior  1969,  actions  further, perceives  to other  and  Opper,  or  1964,  independently  linked  on"  manifested  and  (Piaget,  Piaget,  of  events them  constructed  exist  and  and  children,  patterns  "actions  interiorized  physically,  young  organized  nourished  problem  learning  e s s e n c e of t h e : d e v e l o p m e n t  in  and  knowledge,  ( P i a g e t , 1964,  the  even  (Ginsburg  of  a particular  that  are  child's  surroundings  totality  external stimulus  knowledge s t a r t s f i r s t , "schemes"  the  of  some the the  23). complicated  t o d i s c o v e r the  simplest  of knowledge. For  Piaget,  15  "...  the  central  (processes  of)  functioning  mental birth the  The f i r s t  and c o n t i n u e s  until  grows  surroundings; when t h e y  that  to  he  can  characterized  and  substages  stage,  a b o u t two y e a r s o f age*  longer  lasts by  be  between  seen.  12, t h e c h i l d  as  seriation.  the  child's  learns The  child  simultaneously,.  thinking  only  demonstrated to  reason  child  s t a r t s from  In t h i s  stage  and  his  second  use  of  reasoning.  which l a s t s  stage,  the  until  that the  he  rather  than only  In t h e t h i r d about  is  still  age 11  as  well  limited  e x i s t and a c t i o n s formal  systematically, perform  by l a n g u a g e  classification  and  operations directly  verbally  on r e a l i t y  is  c o n c e p t s and  symbolically. on  to  that are  operational,  the child's capability to define  also  I ti s  c a p a b l e a l s o o f c l a s s i f y i n g and  However,  stage,  symbols,  even  The  stated  (Ginsburg  and  1969).  The  noted  i s  objects  logically,  propositions  way  last by  can  Opper,  about  The  characterize  himself  The  t o group using  seriating  possible.  structures  u n t i l a b o u t t h e age o f s e v e n y e a r s .  the concrete .operational,  or  which  the sensorimotor,  distinguish  and  1964, p. 9) .  processes of t h e s e  d e v e l o p m e n t , and by growth i n i n t u i t i v e stage,  organization  a l s o d e v e l o p s t h e permanence o f o b j e c t s  no  preoperational,  (Piaget,  the  main s t a g e s  growth.  child  elaboration,  structures"  observed  i n t o four  o f development i s t o understand t h e  formation,  o f these  Piaget fall  problem  ages m e n t i o n e d  fixed  i n t h e above paragraph  o r u n i v e r s a l ; they  i n Piaget's  studies.  are not  in  are the approximate average  any ages  What i s f i x e d , however, i s t h e o r d e r  16  of  s u c c e s s i o n of the  every age  one  of  stages;  the stages  a normal  i n the  child  order they  a t which c h i l d r e n r e a c h a c e r t a i n  children*  mentioned.  s t a g e may  and  The  ( P i a g e t and  transition  i n t e r c h a n g e a b l e but preceeding prepares  one,  following  i t  i t  stage "Each  i t as  Inhelder,  example,  continues  is  not  from  the  structure,  and  results  1969,  to  p.  does n o t function  Inhelder,  1969,  next:  maturation,  growth, e s p e c i a l l y  available.  pp.  t h a t are r e s p o n s i b l e f o r the  possibilities The  for  or  later  153) i  The  cease in  in  the  developed  second  of t h e  factor, The  i f  the  a c t i o n s performed objects  on  Maturation  physical  themselves.  of  provides is  be  physical  or  experience  consists  of  physical  objects An  consists  experience  logical-mathematical results the  one  transmission  which  minimal  e x p e r i e n c e , can physical  recognizes  development from  n e r v o u s system  k n o w l e d g e a c q u i r e d by a b s t r a c t i n g t h e while  154-155)  experience, s o c i a l  development  logical-mathematicali  physical  (stage)  next  which i t i s sooner  most i m p o r t a n t l y , " e q u i l i b r a t i o n " .  objects  the  a subordinate  into  for  to  pp. „ 152-153) .  becomes i n t e g r a t e d i n i t s s t r u c t u r e s ( P i a g e t ,  ( P i a g e t and  t o the  organic  1969,  the  122).  four factors  and  and  stages but  Piaget  stage  integrating  perception,  and  p.  one  integrative.  (Piaget  sensorimotor  1973,  Inhelder,  f o r a s u b s e g u e n t one,  integrated"  thought  from  y e a r s and  amongst  r e m a i n s i n t h a t s t a g e depends on h i s d e g r e e o f i n t e l l i g e n c e milieu  months t o a few  vary,  The  one  social  a few  are  through  period  his  from  develops  example  properties  of  by a b s t r a c t i n g  rather of  than a  the  logical-  17  mathematical child and  who  experience*  was  p l a y i n g with  counted  ten,  a l s o got  ten.  counted  ten.  up  to ten  12,  then  Next, he The  Piaget,  1970,  The  third  the  the  the  process  of  development  pp.  9-11).  to  be  three  most i m p o r t a n t  the  other  (Piaget,  labels this  obviously  and  formal  but  not  by  education  and  "a  intellectual  1969,  p.  term  r e f e r s t o an  s t r u c t u r e s which i n t e r a c t s w i t h  and  itself  that  harmony  amongst  never  accordingly  (1975, p.  explain  behaviour.  by  the  Piaget  a t t a i n e d but  170)  human his  being  strives  continuously  and  the  of  adjustment  that t h i s  an  environment  Opper, 1969,  p.  (1957),  toward e q u i l i b r i u m  improved  The  which  the p a r t  Festinger  opinions,  however,  in  "equilibrium",  and  similarly  knowledge,  claims, is  and  (Ginsburg  is  Piaget  a c t i v e s t a t e of  open s y s t e m o f  172),. P i a g e t  child  f o r mental development.  1 5 7 ) . " The  connotation,  and  ( P u l a s k i , '1971,  s e r i e s o f a c t i v e c o m p e n s a t i o n s on  Inhelder,  modifies  the  f a c t o r f o r development i s " e q u i l i b r a t i o n "  of  as  themselves.  above a r e c o n s i d e r e d  sufficient  of  interaction,  verbal i n s t r u c t i o n s acquired  add p.  experience  pebbles  row  still  1964,  independent  of the  transmission  and  a  and  that h i s pebbles  configuration  and  direction  circle  subject i n response to e x t e r n a l disturbances  (Piaget in  discovered  factors described  necessary  consists  thus,  social  concerns  put h i s p e b b l e s i n a  pebbles i n a  which i s  by  Piaget*  the  16-17). P i a g e t  illustrated  The  He  p h y s i c a l experience  factor,  by  counted i n the  of t h e i r  logical-mathematical, to  put  pp.  given  pebbles. he  child,  regardless  superior  often  attitudes  and and  equilibrium i s (1975,  p.  23)  18  through  the  information attained  curiosity (1975, p.  a high  interacts  the  i_ a s s i m i l a t i o n ]  1969,  p.  filters and  on  fit  the  i t  other  pressure  in  modify  Assimilation of  construction of  "a c o g n i t i v e s y s t e m is  not  terms  at  in  (Ginsburg  act;  on  one  environment i n t o  his  of r e a l i t y  modifies  has  rest.  It  with  and  accomodation  hand he  which  line  and  every  new  of i t s s t r u c t u r e s  itself  |accomodation J"  i n p u t of the  the  the  equilibrium  can  demands  aspects  the  of  Thus,  environment  and  172).  inseparable  170).  degree  with  environmental  o f e x p l o r i n g and  Opper,  are  hand t h e own  two  learner  structures  h i s s t r u c t u r e s i n order  ( P i a g e t and  Inhelder,  1969,  pp.  to 4-  6). The  above  influencing does  the  thinks upon  description  development child  his  mental  postulated concepts  realization  effort  conservation  the  constructs  nearly  as  findings  that  revealed  notions the that  1969,  not  only  a d u l t , but  p.  8)  he  depending  In f a c t ,  understanding of  factors  Piaget  mathematical  specific  "conservation" of  an a t t r i b u t e  t h i s a t t r i b u t e remains i n v a r i a n t  tasks are  the  the  4). Conservation  irrelevant  successful in have  p.  and  that  capabilities.  attributes.  to determine the  various  stages  than  Opper,  attainment  1941,  under c h a n g e s o f o t h e r considerable  and  c a p a b i l i t i e s of  (Piaget,  r e f e r s to the  efficiently  s t r u c t u r e s and  depends upon t h e  attributes  mind  less  (Ginsburg  t h a t the  the  seem t o warn e d u c a t o r s  "think  differently"  of  Piaget  approximate  achieved  and  of  tasks.  these  latter  as  children  in  the  made a  ages a t  which  means by  which  He  the  i n the  has  has  not  former.,  been His  preoperational  19  stage  rely  h e a v i l y on  knowledge on  the  a t seven*  eleven  is  capabilities stimuli  not  of the of  the  not  increase  him;  understanding  experiments, 2),.  saying,  number a t  1969,  p.  however,  develop  surroundings.  He  that own  child  and  on  and  and  the  other  hand,  r e s p o n s e depend on  of  learning  doing  six,  volume  intellectual  regardless t h a t one  by  just  and  that  the  the  his readiness  he  does not  charges  of can  telling child  i n t e r p r e t s (Duckworth, 1964-a,  mental s t r u c t u r e s . P i a g e t , c o n s e q u e n t l y ,  unprofitably  the  i s only saying of the  that,  about age  n e c e s s i t a t e s c o n d i t i o n s i n which  holds,  acceleration  acquiring  99).  on t h e i r  understanding  manipulates,  Piaget  in  at e i g h t , w e i g h t a t n i n e  Inhelder,  child  the  interpretation his  perception  surroundings..He f u r t h e r reported  substance  ( P i a g e t and  Piaget  the  their  immediate  average, h i s s u b j e c t s conserved  length at  about  their  p.  child's  level  and  encourage  Americans  of  so:  T e l l an A m e r i c a n t h a t a c h i l d d e v e l o p s a certain way of t h i n k i n g at s e v e n , and he i m m e d i a t e l y s e t s about to t r y t o develop those same ways o f t h i n k i n g a t s i x o r even f i v e y e a r s o f age ... . Most o f t h e r e s e a r c h ... h a s n ' t worked because experimenters have not paid attention to the equilibrium t h e o r y ... . L e a r n i n g a f a c t by r e i n f o r c e m e n t d o e s n o t i n and of i t s e l f result in mental adaptation. ( P i a g e t , 1967, p. 343)  Piaget's  Studies  Piaget, thoroughly volumei b a s e 3 cm  Volume  Inhelder,  the  They  of  subject  and  Szeminska  (1960)  of  conservation  and  showed c h i l d r e n a s o l i d  x 3 cm  and  height  4 cm  and  studied  rather  measurement  nonpartitioned cuboid  they  told  them  that  of of the  20  block  was  a  condemned house  built  on an i s l a n d .  asked t h e c h i l d r e n t o use c e n t i m e t r e possible  houses  t o be b u i l t cm,  o r 3 cm x 4 cm  (pp^  et a l .  x 2 cm,  to  compare  another?  Is there  alternative cuboid  and  for  (i960)  pairs  a l s o used  this  method  by f u r t h e r q u e s t i o n i n g .  asked:  (a) whether  which h a d present space  more, house  as  the  out of a  the s u b j e c t t h e new  old  one  to  build  and  asked  as b i g as  one  357).  An  another  were  build  Then  cm  observed  continuously  volume  the r i s e  a new  looked  followed of  a  another of  the subject  was  t h e same room o r  one t h a t had  exactly  to  discover  like  the  i t .  i f  the  The child  o f t h e number o f c u b e s o r i f  and c o n s e r v e d i t ( p p . . 3 5 7 - 3 5 8 ) . checked t h e i r  i n t h e bottom  and  and  a s much  order  x  looked  They  in  the t o t a l  4  method.  in  technique.  x  1  (b) whether one c a n use t h e same c u b e s  e x p e r i m e n t e r s always  cm  x  cm  t h e cubes  a n d o l d h o u s e s had  water d i s p l a c e m e n t 3  1  (p.  above  to  watched.  on t h e c o n s e r v a t i o n  The  were  cuboids,  used  parallelepiped  relied  considered  two  dozen  methods d e s c r i b e d  tried  he  houses  (p. 3 5 7 ) .  experimenters totally  build  to give c h i l d r e n a c e r t a i n  The e x p e r i m e n t e r s  rectangular  base while  to  parallelepipeds  was  o r room  p r e l i m i n a r y two  different  an a u x i l i a r y  o f them: " a r e t h e s e  ask them t o f i n d *  partitioned  x 3 cm,  as much wood i n e a c h o f them?"  t h a t had t h e same s i z e The  2 cm  further  355-357).  showed c h i l d r e n s e v e r a l r e c t a n g u l a r them  order  t h a t have t h e same s p a c e ; t h e new  on i s l a n d s o f 2 cm  Piaget  cubes i n  They  They  in  built  results  using  a metal cuboid  of a container the  by  water  while  level.  a  o f 3 cm  the  child  They  then  21  asked  the  child  change i f t h e cm  x  18 cm  et a l . the  (age  They  also  showed  space  of  did  progress  2  the  measure  "nearly  those  of  by  they  could  volume  the  t e r m s of  space  recognized alter  or  the  with  object.  the  the i n t e r i o r  These  volume  boundary.  logical  (not  t h a t i s , when  enough.  the They  not and  the  one*  The  level  of  and  could  also  occupied  copy  calculate  number still  of  by  cubes  when  even  and  volume i t by asked  an o b j e c t t h e y necessary  d i d not conserve  displaced  to  expressing  s u c h a s " t w i c e as much"  h e i g h t . In f a c t ,  space  showed  They s t a r t e d  meaningfully  rearrangement  volume.  this  unit-cubes  They  o c c u p i e d or water  that  the  tall  of  previous  high."  about  surround  the  not  Children  using  t h e l e n g t h , width  t h e volume  of  though  the  multiplying  equated  interior  between d i m e n s i o n s ,  10):  t h r e e times as  the  358-385).  of  displaced.  (age 9 o r  but  (1960, pp.  contained inside  i n metrical relationships  correctly  three levels  show c o n s e r v a t i o n o f volume i n t h e s e n s e  correctly  measures  matter  original,  not  over  x 1  a house on a s m a l l e r b a s e t h e y c o n s t r u c t e d  o c c u p i e d o r water Level  m o d i f i e d t o 2 cm  experiments  understanding  to reconstruct  children  would  cm.  concept  relationships  than  was  o f t h e water  8 o r 9) : C h i l d r e n c o n s e r v e d  quantity  taller  x 9  level  d e s c r i b e d below:  the  it  the  in their  volume  i.e.,  mathematical)  x 2 cm  noted  are b r i e f l y  Level, 1  thought  arrangement of t h e cubes  understanding  asked  he  o r t o 2 cm  Piaget  levels  i f  to  volume i n  though  they  of the u n i t - c u b e s d i d not  22  Level  3  established  (age a  They  length,  width,  product  Children the  discovered  volume  terms  and  and  height;  of t h e i r  in  t h a t two  also conserved  s p a c e as  i n the and  space  and  displaced* notion  of i n t e r i o r (3)  et  volume  multiplication  grasp  decomposition  and  boundary  at  edges"  (2)  until  lengths the  an  area  or  (p. 408),..  IV  of  equal  i f  equal. to  Piaget  et  volume i s  of  that  occupied water  the  complete  measurement  involves  necessitates of  of a continuum operations*  the  simply  formal  space. are This  by  "The  operations  c h i l d r e n understand  volume  well  meanings i . e . , (1)  continuity  that a  and  and  of f o r m a l  stage  of  respect  of  conservation  (1960) c o n c l u d e d  level  product  of c o m p l e m e n t a r y volume o r  redecomposition  t o the  i t i s not  volume,  of  with  the  displacement.  i t s three  conservation  of three  operational  which belong  acquires  al.  of t h e  interpretations  conservation  Piaget  of  arrive  child  level  base and  d i m e n s i o n s were  (1960) seem t o s a y t h a t t h e c o n s e r v a t i o n when t h e  this  of t h e  volume  water  of  v o l u m e s were  respective linear  experiments  conservation  can  Children area  surrounding  developed  why  above): between t h e  of t h i s stage  The al.  and  relationship  volume.  the  12  explains how  they  multiplying  23  Reactions The  reactions  Piaget's from  t o P i a g e t ' s Theory of  p s y c h o l o g i s t s and  methods o f t e s t i n g and  criticism  to  Piaget's  and  stages  of French-Canadians.  conclusions,  (1964) l o o k e d  substages  and  added o t h e r s . The  data they  children  in  Quebec  that  attained  the  in  et  al.  he  mental development but  revealed  case  begin to  of  first  (Carpenter,  and  a p p e a r i n young  (1960) c o n c l u d e d " 1975-b)  (pp.  published  A s s e s s m e n t showed t h a t "on  Szeminska" Flavell  his the  facts  concluded  by t h e (pp.  the  later  measurement  of  700  students students  study  graders  "measurement than  same y e a r  National the  i n average s t u d i e s of  revealed  Piaget however,  results.  Assessment whole  He and  measurement  American  students  Piaget,  Inhelder  501-507) . reviewed  criticised described  and  of  somewhat  (1963)  work. He  from  than  contradictory  State  develop  results  own  3-13) i I n t h e  Michigan  and  later  children earlier  the  indicated  of  e l i m i n a t e d some  collected  second  that  than  existence  French-Canadian  declared  concepts  experiments  existence  the  they  (1975-a) r e p o r t e d t h a t h i s  the  concepts  varied  i n Geneva.  Carpenter that  f o r the  P i a g e t i a n s t a g e s a y e a r o r two  Piaget  have  validation  They g e n e r a l l y c o n f i r m e d  stages of  by  regarding  of mental development i n a p o p u l a t i o n  Piaget's  used  educators,  results.  Laurendeau  general  his  p r a i s e . Likewise the  have shown i n c o n s i s t e n t Pinard  and.Experiments  (2) t h e  Piaget in  use  Piaget's writing mainly  the of t h e  on  (1)  experiments qualitative  the and  and  questioned  gap  between  the t h e o r y  method  and  he the  24  role  o f the  language i n the i n t e r p r e t a t i o n  Piaget that  his  responded method  psychological complete. carry  He  on  and  and in  admitted  respecting  experiences  to  the  claimed those  accelerating suitable resolved.  the  a  but  tasks  process  that who  not  challenge  those do  of who not.  the  Piagetian  appear t o  the  to  be  side  with  However,  Piaget  learning  conservation  guidelines  abilities  cumulative  be  development  for  Piaget's l e v e l  to  controlled  consider  sufficient  a q u i s i t i o n of conservation  ages f o r p r e s e n t i n g  i s f a r from  t a s k s . Many s i d e w i t h and  than  educators  they  accelerating  guidelines  necessary  but  explained  rather  and  approve  development,  level  He  s t u d i e s under  to  the c o n s e r v a t i o n  conservation  through  outnumber  sound  the degree of  the be  of  Other educators  acguired  psychologists  seem  stages  criticisms.  that his research  most e d u c a t o r s  the  data,.  epistemological  ix).  regarding  consider  is  p.  particularly  (1971)  he  ( P i a g e t , 1963,  of  tasks.  study  even e n c o u r a g e d  Finally,  divided  of  Flavell's  further s t a t i s t i c a l l y  conditions  order  to  of  which  and are  l e a r n i n g . Shulman Piaget the  tasks  various a c t i v i t i e s  1  seem  question and  thus  i s by  no  to of the  means  25  Training,, E x p e r i m e n t s Educators  have  been  developmental theory related  to  seriously  studied Many  training;  others  sections  of  is  explain  experiments  p.  the  issues  of t r a i n i n g  have  using  (seminar,  equilibration  of  been  The  their  than "to  own  conservation  following these  volume, and  his  been l i t t l e  admitted  84)*  in their  some.of  than  out  drawn  i n volume.  Volume explain  evidence  theory  between  how  the  from  and  1962,  Lowe,  able  to  theory.  does not  suggest  this  University sufficiently  c o g n i t i v e development  transition  level  validating  Catholic  Specifically, little  seems t o  stages  be  non-conservation p. and  c o g n i t i v e d e v e l o p m e n t from  two  training  equilibration  the  in  have  a l o w e r l e v e l o f knowledge t o a  has  (Wohlwill  t h o u g h t t o a i d the to  summarize  set  higher"  there  laws  conservation  by  regarding  role  have  failure.  other  13)  transition  1971, the  tasks  p.  t h a t the  the  (Beilin*  will  and  success  reported  paper  t o be  himself  America)  about  claimed  made from  Unfortunately,  of  have  (1970,  i s judged  He  the  T r a i n i n g Experiments other  Piaget  theory;.  subject  have  this  of  transition  decades,  a stand  example,  this  in conservation  Discussion  that  for  for  tasks,. A number o f r e s e a r c h e r s , c o n s e q u e n t l y ,  conclusions.  studies  of P i a g e t t o take  education;  conservation  challenged,  153). test  known to  Training  various  ways  non-conservation  conservation* Even  investigate  though the  the  ultimate  learning  of  purpose a  volume  of  this  algorithm  study  i s to  it  seems  26  necessary volume  to  review  whenever  therefore,  they  include  weight, volume, training  regard  fact,  Rotbenberg  training 70).  differences varied  research  to the and  study  is  Orost  there  emphasis  Their  results  induce raised  extinction shown  others testing  test  subject  the  conditions  responses..  1964,  The  of  and  each  of  "for  four  1965;  of  et a l .  successful  Sigel,  1966).  procedures  could  studies  of these  aspects  have  acquisition are  explanation,  aspects  of  major s t u d i e s  of  A summary of  these  because  process  verbal  of (p.  because  be  In  every  same t y p e s  Nevertheless,  the  reversibility.  regarding  training.  and  to  in  Beilin,  most i m p o r t a n t  language  consistent  Mermelstein  what a p p e a r e d  conservation.  aspects  and  and  transfer-to-training"  conservation  Bruner,  substance  subtracting,  themselves  that  showed t h a t none o f t h e t r a i n i n g  conservation.  conservation  i n educational research  and  substance  important  substance,  been  reported  significant  (1967), f o r e x a m p l e , employed  1961;  number  have o c c u p i e d  (1969)  unusual  on  (Smedslund,  in  s t u d i e s have n o t  are  i n procedures  of  is  than  reviewed,  of  e f f e c t i v e n e s s of c o n s e r v a t i o n  not  procedures  It  studies  conservation  researchers  which r e p o r t no  This  for  The  other  success.  Unfortunately, with  relevant.  number.  most  many h a v e c l a i m e d  are  experiments i n areas  training  and  that  successful  training  of  adding-  screening,  what r e s e a r c h  has  will  be  discussed  study  the  effect  hereafter. Hohlwill  (1959)  "addition-subtraction"  was  the  treatment  first  to  on  acquisition  the  of  of  number  27  conservation.  The  matching a given representing changed the away  one  six,  Bothenberg training  As  who  and  that  instructing  when  the an  child  fully  explained  to the  results.  They  taking  asserted  was that  were "on  the  (1962)  and  Lowe  also reported  of language  success  in  has  i s concerned, i t does  contributes  been  used  to  o r i n demanding v e r b a l  Sonstroem  (1966)  saying  and  i s seeing"  in  experiments  Moreover,  the  effect  the of  and  most  l e a r n s not  to  (1965)  verbally  after  found  each  contradictory interfered  conservation  was  also  a l . . (1967), however,  training  when v e r b a l e x p l a n a t i o n  is  when he  language t r a i n i n g  substance  the  " i t  conservation et  of  explanations  Beilin  training  of  verbally  that  t h a t he  (p. 2 2 4 ) .  Mermelstein  observed that  facilitated  law  in  description  advocated  doing  conservation  c h i l d r e n the  above  verbal  not  intellectual  mainly  attribute,  rather than  insignificant  and  or  treatment  who  their  response. the  he  subjects,  what he  success  unsuccessful  Wohlwill  and  this factor  i s both  reported  replicated  Wohlwill  effect  responses.  the  believe  of  Language  in  but  most were t h o s e  Later,  experimenter  adding  that t h i s  pictures  (1969) u s e d s i m i l a r " a d d i t i o n - s u b t r a c t i o n "  c l e a r how  development.  three  conservation*  f a r as t h e  their  of  while  reported  b e n e f i t e d the  Orost  number  seem t o be  for  one  p l a y i n g a game o f  e i g h t e l e m e n t s . The  Wohlwill  conservation*  to  changes  or  of  i n i n d u c i n g number c o n s e r v a t i o n  of  inducing  seven  with  c o n f i g u r a t i o n of o b j e c t s  subjects  doorstep"  consisted  s e t of o b j e c t s  element.  successful the  treatment  processes.  procedures demanded  with  became  from  the  28  subjects  who  Wohlwill  and  The to  the  newly  Lowe, 1962;  effect  effect  aspects  acquired  are  of screening  of language*  put  non-perceived  action*  the  another that  level  and  that  further  claimed  enactive  ikonic success  yields  Bruner's  approach  including  Piaget  inducing  a pseudoconservation  identity*  He  identity  and  to  poured  seven-year-olds has  argued not  and  but  Piaget and  vice versa.  rejection induces The  of  effect  conflict  i n the  degree to  claimed  i s achieved.  conservation  He with  four-year-olds. other  accused  researchers Bruner  of  verbalization understanding advocated  development r a t h e r  and  235)  representations  the  shows  (p.  Langer  that screening  (1970)  of of  that than  reported  misleading  cues  conservation* of  extinction  research  d i d not  Smedslund resist  has  literature.  which c o n s e r v e r s  researcher*  conservers  a container  Moreover, Piaget  Bruner's hypothesis  substance  He  f o r c i n g the  it..Experimentally, Strauss  the  to  (1968)  that conservation  used  from  substance with  v e r b a l l y to a  to  conservation  not  related  seven-year-olds  been u n d e r a t t a c k by  himself.  1966;  misleading  183-207)  symbolic  mental s t r u c t u r e precedes language  follows  the  four-  and  by  respond  pp.  i n inducing  to  the  are l e d  (1966,  water t o be  his five-  the  who  watch t h e a t t a i n e d l e v e l . both  the  cues i s c l o s e l y  where t h e y  asking  of the  then  i t i s when  agree  Subjects  Bruner  in  (Sonstroem,  1970).  misleading  in a situation  screening technique predict  Roll,  conservation  resist  also  been  a  source  of  E x t i n c t i o n i s measured  by  deliberate  by  (1961)  e x t i n c t i o n as  reported  confusion that  did natural  trained  conservers.  29  Hall  and  natural Hall  Kingsley  conservers and  reported  (1968) resist  Kingsley  t h a t none  rejected  claim  that  e x t i n c t i o n more t h a n t r a i n e d o n e s d o .  replicated  of  Smedslund's  their  Smedslund's  17  natural  experiment  conservers  and  resisted  extinction. There  seems  conservation that  and  includes state  related seemed  mental  be  training.  by  trend  Piaget  in  inducing  (1968)  explains  reciprocity. of  Wallach  and  r e t u r n i n g t o the o r i g i n a l  Sprott  application  inversion-negation*  for  number  technique  dolls  by i n v e r s i o n -  Inversion-negation  c o n s i s t s of compensating  successful  of f i t t i n g  promising  operation  reciprocity  attributes* to  a  h a s two f o r m s , r e v e r s i b i l i t y  reversibility  the  while  be  by r e v e r s i b i l i t y  reversibility  negation  to  in  (1964) employed  what  reversibility,  via  of  conservation.  back i n t o t h e i r  variations  They  used t h e  beds a f t e r  they  have  been c l u s t e r e d o r s c a t t e r e d . The e x p e r i m e n t e r s d i d n o t  provide  training  already  for reversibility  knew i t i n r e a l l i f e conservation Anderson applied They  a  process..  (1967) c h e c k e d them  chance In  that  results  the  number  (1964) r e s u l t s  and  of  continuous  "reversibility  of Wallach  reversibility  in  W a l l a c h and S p r o t t * s  confirmed  Roll  later  i t  and  (p.  (1967).  apply  who  Wallach, Wall  p e r c e p t u a l cues  those  to  subjects  study,  conservation"  al*  a  to conservation  concluded  misleading  b u t gave t h o s e  441).  (1970)  would  as  seem  Roll's  asserted  i n an i n c r e a s e  well to  (1970)  and S p r o t t  liquid  be  not  using  necessary  findings,  (1964) that  as  quantity.  and  for  however,  Wallach  et  inversion-negation  o f number c o n s e r v a t i o n  i f  30  no  verbal explanation i s required.  Discussion  of  Elkind  (1961-a)  experiments. weight of  that  not  "the  conservation  the  age  similar  of  Piaget's  21%  11"  (p.  from  of  the  (1961-rb)  senior high Newton,  225). data  the  unit  extended  his  s c h o o l s t u d e n t s . The  data  of  showed t h a t 95%  to  of  reach  the  Moreover, scores  and  age  conservation girls  found were  and  conserved  a  (1961-b)  Levittown,  New  obtained  that,  of  study  on  by  Elkind  f o r the  of the  17-  a  pictured  100%  of high  weight  but  volume* On  Schoeppe  York. T h e i r " r e s u l t s  of  school  only  average,  conserve  volume.  grades, with  (1973)  strikingly  same mean age  68%  the  eighth-grade are  and from  I.Q. volume  h i g h e r number o f boys  28  the  6%  subjects  correlated  and  in  to junior  i n a l l secondary  significantly  (75%)  (1975-b)  469  students  positively  volume. N a d e l  Elkind's  those  conservation  of the secondary  Elkind  revealed  43%  replication  c o n s e r v a t i o n o f mass and  47%  sample  (NAEP). O n l y  of  and  cubes.  attain  however, o n l y  a  collected  volume  graduates them  from  Carpenter  1 3 - y e a r o l d s and  into  Massachusetts,  mass  Massachusetts,  the  answers t o  which i s p a r t i t i o n e d Elkind  for  d a t a he c o l l e c t e d  Assessment o f E d u c a t i o n a l P r o g r e s s  y e a r - o l d s gave c o r r e c t solid  with  o f volume d i d n o t i n most c a s e s  results  9-year-olds,  Tasks  some o f P i a g e t ' s c o n s e r v a t i o n  s t u d e n t s i n Newton,  appear b e f o r e the  National  replicated  f o r volume. The  elementary  reported  I n v o l v i n g Volume, C o n s e r v a t i o n  H i s r e s u l t s agreed  but  175  Experiments  than  replicated females  in  parallel  to  group"  (p.  309).  31  In of  volume, p a r t i c u l a r l y , their  subjects,  whose mean age was  reached the conception Elkind adults.  chose  Massachusetts results of  were  those  58%  study  was  Age  students  and  Ogilvie  of  (age  14)  only  as  I . Q.  Purdue  score.  and  in  were  significantly  a l l age  groups.  Elkind's  (1962)  last  University  and  they  (1961)  used  They f o u n d  of  the  can s p e e d  even  Piagetian  that  Piaget  asserted  that  " i t appears that  the j u n i o r school  assumed interior  researchers  9  a  not  i t  volume a s w e l l  possible as  developed  until  the  do 50 p e r c e n t  believed  is  e t a l . , 1960)  well  before fourth  of pupils  by a s i n g l e cube  container"  is  ( p . 124). On t h e  that  proper  volume to  occupied  to  various  on t h e a v e r a g e ,  up t h e a c q u i s i t i o n o f t h e that  methods  about  (Piaget  t h e amount o f w a t e r d i s p l a c e d  hand,  calculate  mass and w e i g h t b u t  replicated  independent o f the s i z e of the f u l l  They  His 92%  volume c a n n o t be a t t a i n e d ,  that  training  from  17 and 37 y e a r s .  students i n grades 6 t o  11 o r 12. I n f a c t ,  other  students  findings i n that  well  at  had  Elkind's results.  concepts.  realize  as  (1971)  c e r t a i n l y c o r r e c t when he  year  between  months,  t e s t i n g t o young  college  volume t h a n d i d g i r l s  191 B r i t i s h  concept age  varied  29%  5  his  with h i s p r e v i o u s  volume.  Wheatley  Lovell  volume  240  students conserved  on 71 c o l l e g e  question  of  that only  ( p i 309)  c o r r e l a t e d t o volume c o n s e r v a t i o n  and  confirmed  ages  consistent  more boys c o n s e r v e d Towler  extended  sample  whose  conserved  positively  finally, a  college  13 y e a r s  o f volume c o n s e r v a t i o n  (1962),  He  N a d e l and Shoeppe f o u n d  school  concepts.  " l e a r n " how volume,  to  before  32  the  concept  developed; on  of  volume  and t r u s t e d  t h a t such  c o n s e r v a t i o n o f both  (1971) learn  suggested  activities  interior  and  algorithm  the i n t e r i o r  Ozgiris  o f water d i s p l a c e m e n t )  (1964)  to the variation  The  collected  data  supported  Piaget's  conservation  20%  of Ozgiris'  the  above  collected  further  stated  the  that  of  181  30  levels  conservation  in  in  of  black  of  reported  Missouri.  Only 2  support  was  males,  were  white  varied  from  positively  of sex.  males,  30  significantly  classes  higher  white  and  significantly  78% o f a l l s u b j e c t a d u l t s c o n s e r v e d  on  The s u b j e c t s  one t o e i g h t . G r a v e s f o u n d scored  sex  and volume. She  and 30 b l a c k f e m a l e s .  i n adult basic education  Bat-haee,  o f r a c e and  o f mass, w e i g h t *  whom  b l a c k s and m a l e s s c o r e d  Moreover,  Illinois  volume.  performance  the effect  volume c o n s e r v a t i o n t a s k s , w h i t e s than  experiments.  a g e s he f o u n d i n  (1971) a l s o  pupils  conservation  were a l l e n r o l l e d grade  from  120 a d u l t s ; 30  females,  with  who had a mean age o f 12 y e a r s  Bat-Haee  (1972) i n v e s t i g a t e d  degree  tested  students  t o I*Q. s c o r e and age; i t was i n d e p e n d e n t  Graves  (p. 1 7 9 ) .  of the ordered  i n the  and w e i g h t b u t n o t  to  sequence o f P i a g e t i a n c o n s e r v a t i o n t a s k s i n t h e  d a t a he  related  order  and volume  and t h e a v e r a g e  graders,  volume*  weight*  elementary  sequence  sixth  findings  o f m a t e r i a l s used  120  of substance  months, c o n s e r v e d of  from  in  Lovell olds can  volume o f a c u b o i d  sequence o f c o n s e r v a t i o n o f s u b s t a n c e , respect  and e i g h t y e a r  Piaget's  is  attention  volume.  "V = L x W x H"  o r occupied tested  may f o c u s  occupied  t h a t e v e n some s e v e n  how t o use t h e  calculate  ( i n terms  than  their  that i n higher females.  mass, 67% w e i g h t  33  but  only  24% volume. G r a v e s  volume,  maturation  concluded  that  mathematics  conservation  are necessary  ... o f volume"  Price-Williams, of  experience  conservation  and tasks.  They  substance,  weight,  who  significantly  higher  significantly remaining that  volume.  of the students  subjects.  cognitive  had  The  in  study  g r o w t h may  suggested  "that  sciences  in  various  versions  of  liquid,  sample c o n s i s t e d o f a between  i n pottery-making  6 and  9  families.  pottery-making  scored  c o n s e r v a t i o n t a s k s than t h e  children the rest  in  number,  whose a g e s v a r i e d  i n substance  higher than  Spanish  Their  lived  the  (1969) s t u d i e d t h e r o l e  manipulation  experience  same  in  scored  higher  but  not  of the s u b j e c t s i n a l l four  conservation tasks..Price-Williams e t a l .  their  of  1972, p. 2 2 3 ) .  administered  o f 56 M e x i c a n s t u d e n t s half  case  an i n d i v i d u a l c a n a t t a i n  experiments  and  Those c h i l d r e n  other  (Graves,  particularly  conservation  years;  before  Gordon and Eaminez  Piagetian  total  the  a l o n e c a n n o t e x p l a i n d e v e l o p m e n t * " I t may  be t h a t e d u c a t i o n a l o r p r a c t i c a l e x p e r i e n c e s and  in  the  be a v e r y i m p o r t a n t  role  factor"  of  concluded skills  (p. 7 6 9 ) .  in  34  Summary and  ftqe o f  I m p l i c a t i o n s of L i t e r a t u r e  Subjects  Within c e r t a i n  limits,  process  of  example,  R o t h e n b e r g and  children  subject  t o be  important  involved. Inhelder that  preoperational 75%  in  studies;  successful cognitive (Kingsley  (1962) f a i l e d  is  the  and  others  have  children  s t r u c t u r e but and  (1959),  to  Hall, stage  have  are not  an  with  o l d e r ones.  1974,  of  pp.  12.5%  of  intermediate  s u b j e c t s advanced  finding  can  Beilin  be  that  Strauss  training  reached  to  standing  "on  the  the  across  is  " i n p o s s e s s i o n o f the yet  the level  seen  (1965),  the  125-126)  and most  proper  equilibration"  1 9 6 7 ) . T h e s e s u b j e c t s a r e s a i d t o be and  the  younger  level  experiments  noted  who  affect  with  cognitive  i n Modgil,  her  This particular  Wohlwill  with  transitional  Low  of  to  succeeded  the i n t e r m e d i a t e l e v e l  level.  (1970)  (1969)  (cited  one  seem  In number c o n s e r v a t i o n , f o r  subjects progressed  of  operational  Langer  does n o t  Orost  w h i l e W o h l w i l l and  reported  while  age  conservation training.  What a p p e a r s  many  Reviewed  at a  doorstep"  of  studies indicate  that  conservation. For  volume,  however,  the expected  percentage  (grade  7)  6 and  1961-a;  o f c o n s e r v e r s among  v a r i e s from  Carpenter,  this  study  20%  1975-b)i  conservers i s particularly of  the reviewed  t o 25%  (Uzgiris,  This expected  important  11 and  1964;  percentage  to t h i s  n e c e s s i t a t e s the i n c l u s i o n  12 y e a r  of  olds  Elkind, of  s t u d y . The  volume nature  volume c o n s e r v e r s  35  in  the  study  sample. because  Grade 6 seems t o be (1)  i t i s just  one  w h e r e i n most t e x t b o o k s e r i e s a  cuboid  and  conservers  Nature of The  (2)  f o r the  value  in  collaborators, experiments  this  have  students'  responses,  using  which  through  Mehler  nonverbal  a sufficient  developmental caused Some  and  should  (Inhelder  the  of  5 for  number  of  child's  some did  Epstein,  and  and  p r o c e d u r e s f o r such who  favour  main  Piagetian  to the  child's  Sinclair,  eminance  1969,  of language i n have  disagreed  verbal explanation  require  to  have  of  to  his  eliminate conducted  justification  have a t t e m p t e d  1968,  in  paid  researchers  pretrainning  (i.e.,  1963). I n o r d e r  not  in fact  stages  considerable  that  be  their  Piaget's  specifically  (Flavell,  they and  the  and,  v e r b a l responses  Researchers  grade  for  a  for  avoid  verbal  procedures  (Braine,  1959,  f o r example)*  Calhoun  (1971)  d i f f i c u l t y i n a s s e s s i n g c h i l d r e n ' s number  their  this  algorithm  justification  have  criticized  problem,  in  reported  answers  decisions  research  Bever,  to  attention  experiments  procedural  instructions  their  literature*  e m p h a s i s on  and  than  volume  f o r example, have e x p l a i n e d  Others  Piaget's  the  find  subjects'  seems  the  of h i s  type  higher  for  Tests  inferring  "special  justification  actions  to the  levels)  discussion  with  expect t o  choice  study.  given  while  conservation  Piagetian  can  grade  introduce  Volume C o n s e r v a t i o n  responses  p.5)."  one  a reasonable  recommended t h e  conservation  use  of  totally  assessment.  justification  seem t o  believe  that  36  without  such  mistakenly  inferring  Subjects  may  attributes might  justification  reported  correctly  the  error  by s i m p l y of  may  be c a u s e d by  by c o n c e n t r a t i n g  1963 a n d 1969).  o r , i n case  on  I  developmental stage  respond c o r r e c t l y  respond  focusing  a higher  (Smedslund,  alternative  type  chosing  volume  substance  or  which h a s shown t h a t  For  to  subjects.  on  irrelevant  example, the f i r s t  conservation  more s u b j e c t s a r e  (or l a s t )  classified  when  when i t was  ( B r a i n e r d , 1973, p. 1 7 4 ) . F o r e x a m p l e . R o l l  awareness that  using  conservation  argument further  conservation.  their  negligible  was n o t used  few o f h i s t r a i n e d number c o n s e r v e r s  of  the  nonverbal increase  when v e r b a l is  Wohlwill  that  training in  justifications  developmental  verbal  for  number  responses  became  students'  level  than  (1962) i n d i c a t e  were demanded.  of  as  (1970)  showed  procedures  conservation  justifications  reveal their  and Lowe  by  Research i s  conservers  reports that  criterion  child  testing,  weight a t t r i b u t e .  the j u s t i f i c a t i o n  a  and  Thus  the  responses  reduce  may  type  I  error. Researchers explanation argue  who  oppose  for inferring  that  type  the  I I e r r o r s may  the  requirement  subject's  are  t o o s t r i n g e n t . The argument o f t h o s e  be  based  on t h e t h e o r y  of Piaget  developmental  level,  researchers  himself. theory  Flavell  of  to  (1963), f o r  Piaget  i s here t r e a t e d as a dependent v a r i a b l e with  as  independent  variable  (p. 2 7 1 ) . " B r a i n e r d  that  has l o n g  maintained  that  which  seems  behavior  "Piaget  the  criteria  observed  explains  in  verbal  example,  the  that  a  be made by u s i n g  of  "language cognition  (1973)  also  ... c o g n i t i v e  37  structures further,  originate  considers  necessary 177).  rather  than  explanations  language"  sufficient  and  but  not  conditions f o r inference of cognitive structures  Brainerd  holds  criterion,  domain  which the  to  177).  Brainerd  Piaget's  theory  in  that  i f  t h e n one  one  the  choses  unduly  theoretical  concludes  "extraneous" type built  logic  adequate  explanation  (p.  in  restricts  construct  that  "from  type  I I e r r o r s but  of  error  as d o e s t h e  employ the  the  behavioral  (structure) the  judgement c r i t e r i o n  I and  source  to  (p.  applies  standpoint  risks  only  the  does  not  risk  explanation  of usual any  criterion  (p.178). Hobbs  (1975) s i d e d  necessary  and  with  sufficient  developmental l e v e l  and  conservation  testing.  Piagetian  type  testing  subjects  are  given  answers,  the  persistent  liable  cause  to  (Hobbs, 1975, Piaget's the or  subject's Hobbs'  nature with  the  easily ones,  applied He  discussing  a mistake i s not  taken  chances  to  subjectivity inference  position,  regarding  a c t i o n * does not  give  volume  though  in  value  and  at face  of t h e of  to  even  the  correct  experimenter  conservation  the  is  levels  verbal j u s t i f i c a t i o n  seem t o oppose (1963)  Brainerd's  explained  his epistemological studies necessitates  overlooked are  determining  conclusions that  repeated  for  the  272).  subjects  which  his  explained  (1975) p o s i t i o n . P i a g e t  of  while  subject's behaviour  incorrect  p.  Brainerd  as  i n order ...  and  to  "...  unearth  t o J. u s e ]  f r e e and  flexible  as  (1973)  that  the  interaction  what i s o r i g i n a l  methods,  of  including  and  verbal  p o s s i b l e * " Furthermore,  38  he  encouraged  conditions explained  educators  (Piaget,  to  1963,  p.  conduct s t u d i e s under c o n t r o l l e d i x ) . More r e c e n t l y , P i a g e t  (1973)  that  In f a c t i t i s a very general p s y c h o l o g i c a l law that the child can do something i n a c t i o n l o n g b e f o r e he r e a l l y becomes ' a w a r e of what i s i n v o l v e d 'awareness' occurs long after the action* In other words, the subject p o s s e s s e s f a r g r e a t e r i n t e l l e c t u a l powers t h a n he a c t u a l l y c o n s c i o u s l y u s e s . ( P i a g e t , 1973, p. 86) 1  Hobbs  (1975)  conservation o f the  test  occupy that  test  b a s e d on  subjects  space  the  developed  in  space  were  water  objects.  The  those  persistently  who  directly test of  second  related to  c o n s i s t e d of water  to  the  the  same  water, t r a n s f o r m i n g questioning the  other  the  p r o c e d u r e s used development of The  by  balls  of  of  of are  development  was  taken  to  of  the  reduce  The  part  I  of  and  of the  balls  ball  was  in  Test  and  the  glasses balls  turn  in and  useful in  used i n t h i s himself  in  Brainerd  Test  used  in this  caused  by  study. the  (1973)  justification  Test,  the  and  the  error  is  immersed i n i t . The  Volume C o n s e r v a t i o n type  detect  than s i z e ,  last  (1973)  for  to  of  a n t i c i p a t e d water l e v e l  (1963)  basis  and  size  designed  (1975) were p a r t i c u l a r l y  Piaget  part  objects  the  same s i z e  other  first  l e v e l to r i s e  i m m e r s i n g one  the  volume  that  with  was  of the  about the  Flavell the  the  weight, r a t h e r  j u d g e m e n t - b a s e d Volume C o n s e r v a t i o n the  test  Volume C o n s e r v a t i o n  position  theory  that  transformed  Hobbs  the  interpretations Piaget's  student  g l a s s i f the  directly  level,  each  In  water  space o c c u p i e d .  presenting  displaced  alone.  cause the  of the  judged  a  experimentally  varies  part  used  judgement  shown and  occupied  and  to  of  the  study.  In  further  care  students'  39  concentration  on  irrelevant  questions that allowed the  response  lines an  format  s u b j e c t s ' guessing  weight  nonconservers items  of  based  than  volume  for  the  test  but n o t volume were volume  who  Test  in  T e s t used of  balls,  no fact  broken  Further, on  s t u d e n t s as  developing  i n this  volume  study  the were  conservation  1 9 6 8 ) . The v a r i e t y  the  preparation  of  and d e t e c t i n g t h e c o n s e r v e r s o f w e i g h t  a d a p t a t i o n s o f Hobbs'  used  In  concentrated those  used  procedures  conservation  Conservation  asked.  t h e water rose*  classify  plasticene  example,  one o f f i v e  a l . , 1960, P i a g e t and I n h e l d e r ,  of t r a n s f o r m a t i o n s t o  his  and  t h e Volume C o n s e r v a t i o n  et  subjects  thought  o f volume. The c r i t e r i a  on P i a g e t ' s t e s t i n g  (Piaget  were  was made t o d e t e c t t h e s t u d e n t s  rather  For  c o n s i s t e d of darkening  t o whichever t h e student  effort  attributes.  testing.  i n this  (1975) p r o c e d u r e s  in  The p r o t o c o l o f t h e Volume  study  i s presented  i n detail  in  Chapter I I I .  Choice, o f Treatments The  concept  important  i s s u e because t h i s  relationship algorithm has  between  study  training  is  on  1977).  It  varying  factors  procedures designed  the  usage  and  compare  of  show  the  t h e volume A  concern  multiplication  abilities  o f t h e volume o f a c u b o i d  (Spitler,  i s c o n j e c t u r e d t h a t when s t u d e n t s of a f i x e d  i s a very  to  conservation.  about the necessary  i n the c a l c u l a t i o n  determine  treatment  "V = L x W x H" and volume  been e x p r e s s e d  involved  underlying  product  volumes  they  can  are p r o f i c i e n t i n rapidly  or dimensions  predict,  o f cuboids* For  40  example, i t i s probable would  successfully  subjects  are t o  condemned original. is  other  considered  to  be  This  the  C h a p t e r I I I and  details  (1958)  conjecture  p.  227).  The  intended in  to  opposite  that  changes  negation  the  seems  an  with  respect  most  of  concrete  conservations  (volume,  motion)  index  the  index  a r e an  first-order  justified  the in  within and  the  Piaget  "...  by is  a f f i r m a t i o n by  equal  a f f i r m a t i o n " (Brainerd,  1970,  previous  paragraph  of  is  o r more f a c t o r s factors.  via  inversion-  training  procedures.  inappropriate generalization  explained  t h a t the  weight, and  momentum,  require  first-order  and  that  only  area)  are  second-order  and  operational level*  conservations  for  Reciprocity  operational level  of f o r m a l  and  presumes r e v e r s i b i l i t y  substance,  density,  the  A.  Piaget. Inhelder  promising  (number, l e n g t h ,  a  manipulation  to v a r i a t i o n s i n other  He  for  An o u t l i n e i s g i v e n  be  i n the  however, a g a i n s t  which  from  basis  e f f e c t i v e n e s s of r e v e r s i b i l i t y  various conservations.  conservation  the  reciprocity.  proposed  in  calculation  s t u d e n t s i n c o m p e n s a t i n g one  the  Piaget cautioned, across  above may  in a related  procedure  to t r a i n  fact  by  factor  i n Appendix  conservation  or  problem  volume  study.  of  proficiency  replacement  c o m p e n s a t i n g c h a n g e s i n one  multiplication In  the  such  a base d i f f e r e n t  for  provided  mentioned  inversion-negation  and  are  on  a  provides  c o g n i t i v e theory  asserted  analogous  key  of  of  treatment  conjecture  treatment  cojitext of the  built  with island  height  be  multiplication  The  Piaget's  words i n t h i s  to  conservation.  solve  students  p r e d i c t the  building In  that  rectilinear He  added  that  successive  41  application  o f the two  negation  and  aspects  compensation),  n e c e s s i t a t e simultaneous and  Piaget,  Flamer,  1958,  1971)  aspects  p. 3 2 0 ) .  also  of r e v e r s i b i l i t y ,  while  to  She  further  maintains  conservation  i s not a c h i e v e d .  He  multiply  two  numbers t o g e t h e r  and  lengths  or  three  not  ..."  (Inhelder  of  and  various  one  other, could  Piaget  aspect  harm  expected  the learning  and  to  the  pp.  of the  volume  of the  i f  t h i n g to two  product  continuity  There  treatment  number  to multiply  the  p. 4 0 8 ) .  cognitive level  of  that t h e i r  involves  of  seems t o be  would l e a d  is  a  to  a  volume a l g o r i t h m and  of  effectiveness  three  criteria:  of  1 7 - 1 8 ) . The  to r e v e a l the e f f e c t  other  of  such  retention,  subjects  before  such  results  of t h i s  study  of m u l t i p l i c a t i o n  skills  in  volume a l g o r i t h m . experimental  treatment,  was  cuboid  = L x W x H"  "V  quite another  understand  learning  in  t h a t " i t i s one  would e x a m i n e t h e  ( P i a g e t , 1964,  The  holds  latter  respect  generalization,  proficiency  understanding  multiplication  temporary  with  training  to  l e n g t h s and  t h a t the  and  learning  that  ( P i a g e t , e t a l . , 1960,  conservation.  of  lead  a r e a o r a volume. The  possibility  are  ones  i n Green, Ford  separation  expense o f the  does  limited  aspects  held t h a t emphasizing  manipulation  space  second-order  (cited  the  (inversion-  learning.  1  Piaget  an  the  of both  Inhelder  objected  at the  reversibility  application  of r e v e r s i b i l i t y .  subjects  of  designed  s c h o o l p r o g r a m s used  treatment,  to t e a c h the using  an  i n North  volume  labeled algorithm  approach t h a t resembles America  and  volume for  a  those  particularly  in  42  British apply  Columbia.  the r e s u l t s  hand, t h i s  than  p.  normal  comparison,  (Elliot  the  example  boxes  building units  with  direct  may  not  t h e volume (s)  Dilley  et  used  i n this  algorithm,  to  this  treatment  their  ordering volume  similar  used  v. 4,  1971). involved of  as  some  closed w e l l as  of  whose  c o u n t i n g t h e number o f cubes and  (Eicholz  students  1974,  and Maggs,  et  a l . , 1974,  a l . , 1974, v. 6, p. 110; E l l i o t  to build  with  classroom.  (Thyer  and d i r e c t  respect  visible,  stating  blocks  of  guidebook,  cubes models o f p o l y h e d r a l s  be  145)..Later,  school  of a c t i v i t i e s  Teacher's  comparison  with  unit  building  e t h e r models f o r t h e t e a c h i n g o f volume i n  activities  (cuboids)  in  elementary  4) and o f measurement i n g e n e r a l  macroscopical  school  and r e q u i r e d more  o r d e r i n g , c o u n t i n g o f cubes,  et al.,  Preliminary  p.  in  for  the s e q u e n t i a l progress  c o n s i s t e n t with  particular  i n order to  t o s c h o o l p r o g r a m s . On t h e o t h e r  involvement,  is  Furthermore,  was  of the study  b e c a u s e i t was more c o m p r e h e n s i v e  students' active  treatment,  r e s e m b l a n c e was n e c e s s a r y  t r e a t m e n t ' was c o n s i d e r e d an improvement o v e r  approaches  cubes,  Such  nonstandard  cuboids  number o f b l o c k s and i n d i r e c t l y  v. 5,  p. .276;  e t a l . , 1974, v. 4, then  standard  unit  t o t h e ones g i v e n , c o u n t e d compared  then  ordered  the those  cuboids. . The  volume  simplification  a l g o r i t h m "7 = L x W x H" was i n t r o d u c e d  ( E i c h o l z e t a l . , 1974, v. 5, p. 276) o f c o u n t i n g  cubes i . e . , l e n g t h X width layer cubes  and  length  (Dilley  as a  X  yielded  width  t h e number o f c u b e s  X height  gave t h e t o t a l  e t a l . , 1974, v. 5, p. 134; E l l i o t  in  one  number o f  et a l . ,  1974,  43  v.  4,  p.  143).  algorithm  This  to various cases.  applied  to  p.  to p a r t i a l l y  111),  v. 6,  p.  (Eicholz  attachments  272)  and  of  For  covered to  this  ended  p.  a p p l y i n g the the  e t a l * , 1974,  (Eicholz  et  dimensional  is  provided  given  volume  algorithm  in  i n Appendix  al.,  was v.  6,  1974,  transformations  273)..An o u t l i n e  treatment  l e s s o n p l a n s are  (Dilley  cuboids  proposed 6,  by  example,  of cuboids  e t a l . , 1974;, v.  activities detailed  treatment  of  the  Chapter  main  III  and  A.  Conclusion T h e r e seems t o be a g r o w i n g b e l i e f -  among  educators  that  i.  intellectual can  development*  be a c c e l e r a t e d  (1969)  with  at least proper  for transitional subjects,  training.  Flavell  and  Hill  summarized:  The early Piagetian training studies had negative o u t c o m e s , b u t t h e p i c t u r e i s now c h a n g i n g . I f o u r reading of recent trends i s c o r r e c t , few on e i t h e r s i d e o f t h e Atlantic would now maintain that one cannot by any pedagogic means measurably %>ur, s o l i d i f y , o r o t h e r w i s e f u r t h e r t h e c h i l d * s c o n c r e t e - c " j f e r a t i o n a l p r o g r e s s , (p. 19)  Even t h o s e hold that  that learning i t  does  Fullerston, tool  who  not  are i n accord can  with  the  a c c e l e r a t e development  initiate  i t (Saedslund,  Piagetian but  they  1961;  maintain  Halford  1970). p i a g e t h i m s e l f c o n s i d e r s e d u c a t i o n  f o r stage  theory  to  be  and a  acceleration:  But i t r e m a i n s t o be d e c i d e d t o what e x t e n t i t [ e d u c a t i o n ] is b e n e f i c i a l . . . Consequently, i t i s h i g h l y probable that t h e r e i s an optimum r a t e o f d e v e l o p m e n t , t o e x c e e d o r f a l l  44  b e h i n d which would be e q u a l l y h a r m f u l . But we do n o t know its laws, and on this point a^s w e l l i t w i l l be up t o f u t u r e r e s e a r c h t o e n l i g h t e n us. ( P i a g e t , 1972, q u o t e d by f l o d g i l , 1974, pp. 126-127)  45  CHAPTER I I I  PROCEDURES This major the and  chapter  includes discussion  considerations: the choice of  and d e s c r i p t i o n  subjects,  t r e a t m e n t s , the p r e p a r a t i o n o f t e s t s t r e a t m e n t s were  of four  description  of  and t h e way t h e t e s t s  conducted.  Subjects The  study  was c o n d u c t e d  suburban  school  Columbia.  These  of  t h o s e used  family  of  the  in  America.  that  Canada,  grade  lcwer  students  mainland  a mathematics  In  1971,  the  appeared grade  in  North  1974).  America..  t o be r e p r e s e n t a t i v e  6 s t u d e n t s i n North The  classes  subjects  a  of B r i t i s h  program  typical  annual  average  metropolitan These  area  subjects  was  $10 664  were o f ages and  s o c i o e c o n o m i c s t a t u s s i m i l a r t o t h o s e o f most s u b u r b a n students  in  d i s t r i c t was $11 033 w h i l e t h e a v e r a g e  income i n t h e V a n c o u v e r  (Statistics  sixth  subjects followed  i n North  income  family  district  on  The of  sample the  chosen,  population  grade  6  therefore, of  suburban  A m e r i c a . .,  consisted  i n t h r e e s c h o o l s . Each  o f 171 s t u d e n t s o f s e v e n of  two  schools  had  grade 6  two  full  46  classes  of  of  6, one  class  of grade  o f grade  7 and  grade 6 combined.  grade  class test  grade  6, w h i l e t h e t h i r d  o r t r e a t m e n t day  sample  was  group.  The  two  treatments  control The  Each  two  6 combined,  S u b j e c t s who  which  were  both  of a c u b o i d i . e . , V  group  consisted  o f the t h r e e  algorithm  aim  groups  and  one  missed  The  one  two  at a r r i v i n g  = L x W x H .  reguired  and  underwent  aimed  of l e a r n i n g  any  final  to  be  about  of  control  different  at the  volume  treatment of the  v a r i o u s numeration about  systems. the  same  t h e same amount o f t i m e .  treatments i s described  of  this  f o r a c u b o i d "V  method b a s e d  in detail  treatment = L  was  x W x H"  to using  on a p p r o a c h e s o f p r e s e n t s c h o o l  below.  t h e volume o f c u b o i d s by c o u n t i n g c u b e s  outlined  concepts  below.  Complete  A.  for  In  such  finding  or  by  or  using  " V = L x W x H . "  main  Appendix  activities  volume  discovery  programs.  computing  The  include  the  a guided  volume  algorithm,  lessons  teach  programs  in  and  Treatment  The  the  class  from the study..The  e x p e r i m e n t a l groups  experimental  of d i f f i c u l t y  Volume  full  o f the, T r e a t m e n t s  t h r e e t r e a t m e n t s were b e l i e v e d  level  one  105 s t u d e n t s .  study i n c l u d e d  algorithm  grade  were e l i m i n a t e d  Description The  5 and  s c h o o l had  and and  activities detailed  of t h i s  lesson  treatment are  p l a n s are  provided  47  1.  Direct  comparison  of o b j e c t s .  2.  Direct  ordering of o b j e c t s .  3.  Indirect  comparison  of c l o s e d  4.  Standard  units:  dm  5.  Indirect  o r d e r i n g of c l o s e d  6.  Volume o f  7.  Volume o f p a r t i t i o n e d  8.  Algorithm  m, 3  and  3  boxes. cm . 3  boxes.  p o l y h e d r a l models b u i l t  f o r the  and  from  non-partitioned  volume o f a c u b o i d  9. A p p l i c a t i o n o f t h e diagrams of cuboids.  unit  volume  "V  algorithm  cubes.  cuboids.  = L X W  to  X.H."  cuboids  and  10. A p p l i c a t i o n o f the volume a l g o r i t h m t o t h e f o l l o w i n g cases: a. Word d e s c r i p t i o n o f c u b o i d s . b. C u b o i d s t o u c h i n g s i d e by s i d e . c . D i a g r a m s o f c u b o i d s w i t h some u n i t c u b e s a t t a c h e d o r removed. d. A t t a c h m e n t s o f h a l f c u b e s t o c u b o i d s . e. D i a g r a m s o f p a r t i a l l y c o v e r e d c u b o i d s . 11. A p p l i c a t i o n o f t h e volume a l g o r i t h m proposed dimensional t r a n s f o r m a t i o n s .  Multiplication This teaching emphasis  example, in  here  was  volume t r e a t m e n t ,  of a cuboid  developing  the  provided  that t h e i r  product  t h a t 24  statements  was  followed  "V  = L x W x H". A;  the  on  given  completing  Appendix  like  volume a l g o r i t h m  three factors  cuboids  with  Treatment  treatment, the  to  by  = 2 X 3 X 4, such  a brief The  a brief  of t h i s is.given  of  students  the  ].I {  are  and For  trained  ]. T h i s  volume o f  treatment below.  fixed. were  at The  v a r y i n g two  remained  = 6 X £  d i s c u s s i o n of  details outline  a s 24  aimed  "V = L x W x H".  skills  the  was  a  task cuboid  provided  in  48  il.  Review o f t h e c o m m u t a t i v e  and a s s o c i a t i v e  principles.  2. Prescribing t h e range f o r t h e m i s s i n g f a c t o r i n an i n e q u a l i t y i n v o l v i n g two f a c t o r s a t e a c h o f i t s s i d e s . 3. P r e s c r i b i n g t h e r a n g e f o r t h e inequality involving three factors  missing factor i n an a t each o f i t s s i d e s .  4. Effect on the product o f two f a c t o r s when t h e s e f a c t o r s a r e changed a d d i t i v e l y o r m u l t i p l i c a t i v e l y . (Note: " a d d i t i v e l y " and " m u l t i p l i c a t i v e l y " w i l l subsume decrease as w e l l a s i n c r e a s e . ) 5. Effect on t h e p r o d u c t o f t h r e e f a c t o r s when t h e s e f a c t o r s a r e changed a d d i t i v e l y or m u l t i p l i c a t i v e l y . (Note: " a d d i t i v e l y " and " m u l t i p l i c a t i v e l y " w i l l subsume decrease as w e l l a s i n c r e a s e . ) 6. E f f e c t on one o f two f a c t o r s c h a n g e d and t h e p r o d u c t i s f i x e d  when t h e o t h e r f a c t o r i s  7. E f f e c t on twc ( o r one) o f t h e t h r e e f a c t o r s when one (or two) o f t h e s e f a c t o r s i s (are) c h a n g e d and t h e p r o d u c t is fixed. 8.,Clarification  of the concept  o f volume.  9. . A l g o r i t h m f o r t h e volume o f a c u b o i d "V = L X W X H." (IQ. . A p p l i c a t i o n of partially partitioned  t h e volume a l g o r i t h m t o p a r t i t i o n e d , and p a r t i a l l y c o v e r e d c u b o i d s .  11..Application o f t h e volume a l g o r i t h m proposed d i m e n s i o n a l t r a n s f o r m a t i o n s .  Control  and  treatment  to  was  given  sensitization  about  unit  effects.  The  on n u m e r a t i o n  t h e same l e v e l  the c o n t r o l  treatment  group  with  f o r t h i s treatment  f o r the history  consisted  of  s y s t e m s and was b e l i e v e d  of d i f f i c u l t y  t h e two e x p e r i m e n t a l g r o u p s .  outline  to  o f c o n t r o l l i n g f o r any "Hawthorne", m a t u r a t i o n ,  instructional of  cuboids  Treatment  This purpose  to  The  (detailed  as the treatment following lesson  i s  a  an  t o be offered  general  p l a n s may be f o u n d  49  in  Appendix  A) .  1. .Review o f Base  10 p l a c e v a l u e  2. B u n d l i n g  i n fives  3. C o u n t i n g  i n Base 5  4.  Converting  and  numerals  i n s i x e s and  6.  Counting  i n Ease 6  7.  Converting  8. C o n v e r t i n g 9.  from Base  5 with  Description.of  posttests  different and  retention  Volume A c h i e v e m e n t portion  of  the  Achievement  piloted seventh  Test  using  suitability study  of  without  Test  The  and  the  and t o i m p r o v e  grade  a sixth  results  the  level testing  pretests,  Test  Multiplication the  Volume  Test  class  were  and  were u s e d t o r e v i s e to  a  Achievement  Achievement grade  and  The p o s t t e s t s  Test,  scheme o f c o n s e r v a t i o n l e v e l s , the  as  Volume  Conservation  pilot  renaming  (SAT).  the  and t h e M u l t i p l i c a t i o n  grade c l a s s .  renaming  Conservation  Test  grade c l a s s ,  10  The p r e t e s t s c o n s i s t e d o f t h e  Achievement  Volume  6  administered  Conservation  a fifth  classification  tests.  6  Tests  of  T e s t . The  Achievement  were  5 t o Base  and  tests consisted  Volume  5  10 t o Base  without  T e s t , t h e Volume  the S t a n f o r d  and t h e r e t e n t i o n Test,  tests  5  e x p r e s s i n g numbers i n Base  5 w i t h and  S u b t r a c t i o n i n Base  Three  10 t o Base  n u m e r a l s from Base  A d d i t i o n i n Base  1Q.  e x p r e s s i n g numbers i n Base  from Base  5. B u n d l i n g  numerals  concepts  confirm  a the the  ( s i x t h ) chosen f o r t h e major instruments.  Each  of  the  50  revised except The  tests the  be d e s c r i b e d  Volume C o n s e r v a t i o n  Volume C o n s e r v a t i o n  the  answer s h e e t s The  by in  will  their  (1960),  test  the  an  effort  students, student line  to  avoided  and  as  was  of the with  used  nonconservation, last  part  test  third  as  (pages  11  for  classification  intended  those  part  partial  first  one  of  conservation and  12)  the  judgement; t h i s i n the  i s a d e s c r i p t i o n of  (1968) and  the  a  the  and  (1975)  levels.  and  During  this  interaction  among  experimenter. by  Each  darkening  as "more" o r " l e s s " part  (pages 2 and  to The  give  the  second  part  7-11)  was  the  three  and  part  (pages  conservation  the  4-  volume to  categories,  were a s k e d t o  of  see  students  designed  validity  a  were  3,  conservation. In  students  In  procedures  group.  the  provided  third  E.  used  Hobbs  subjects that associate  (pages  in  below  procedures  answer s h e e t  procedures.  subjects  for their  following  was  on  conservation  and  separate  test  reasons the  a  i n Appendix  described  minimal the  students  to i d e n t i f y  the  class  tests  E.  explained  p o s s i b l e . The  the  w e i g h t . The  classify  Inhelder  made t o keep  on  included  based  judgement. Terms s u c h  much a s  E)  was  volume  t a s k s t o the  between  show t h e  familiarity  with  Test  experimenter  was  are  i n Appendix  P i a g e t and  responded  Appendix  6)  the  Test  c o p i e s of the  i s completely  detection tests for  demonstrated test  given  Volume C o n s e r v a t i o n  Piaget  this  are  Test  i n t u r n and  the give  information test.  The  test.  1, The experimenter displayed, side by side, three i d e n t i c a l t e s t tubes p a r t i a l l y f i l l e d to the same level with coloured water. The l e v e l s were marked a r o u n d t h e t u b e s . The e x p e r i m e n t e r t o l d t h e group that the levels were the same. The experimenter then displayed two i d e n t i c a l b a l l s and a l a r g e r b a l l o f p l a s t i c i n e close to  51  the tubes. He t o l d t h e group t h a t two o f t h e b a l l s were t h e same and t h e t h i r d was l a r g e r . He a s k e d a student to come forward and confirm that the water l e v e l s i n the t u b e s were t h e same, t h a t two o f t h e b a l l s were the same and that t h e t h i r d was l a r g e r . I f t h e s t u d e n t d i s a g r e e d , t h e e x p e r i m e n t e r a s k e d him t o a d j u s t the amount of water or plasticine by adding or d e l e t i n g . E a c h s t u d e n t was g i v e n a p e n c i l and an answer b o o k l e t which c o n s i s t e d o f 12 d i f f e r e n t c o l o u r e d answer s h e e t s . 2. The e x p e r i m e n t e r a s k e d , "What w i l l happen i f I p u t t h i s b a l l ( r i g h t ) i n t o t h i s t u b e ( r i g h t ) ? Where w i l l t h e water level be?" The experimenter p u t t h e b a l l i n one o f t h e t u b e s , t h e water r o s e and s t u d e n t s were i n s t r u c t e d t o t u r n t o page 2 and o b s e r v e t h e drawn r e s u l t . T h i s q u e s t i o n was suggestive by nature and was intended to f a m i l i a r i z e s t u d e n t s w i t h t h e q u e s t i o n i n g and a n s w e r i n g p r o c e s s . 3. The e x p e r i m e n t e r a s k e d , "What w i l l happen i f I put t h i s o t h e r b a l l (middle) i n t o t h i s t u b e ( m i d d l e ) ? Where will the water level be a f t e r I put t h e b a l l i n ? D a r k e n t h e l i n e n e a r e s t t o where the water l e v e l w i l l be a f t e r I put the ball i n . " The s u b j e c t s d a r k e n e d a l i n e showing t h e i r j u d g e m e n t and t u r n e d t o page 3. Then the e x p e r i m e n t e r put the ball in the tube and p o i n t e d o u t t h e l e v e l t o t h e students. 4. S t e p 3 was r e p e a t e d u s i n g t h e t h i r d t u b e and t h e p l a s t i c i n e b a l l t h a n t h e ones in the first two s t u d e n t s used page 3 f o r t h e i r r e s p o n s e s .  larger tubes;  5. A l l tubes and balls were removed. The e x p e r i m e n t e r d i s p l a y e d two new t u b e s p a r t i a l l y filled with the same amount of water, a p l a s t i c i n e b a l l , and a s t e e l b a l l o f t h e same s i z e a s t h e plasticine ball. The experimenter pointed out that the balls were of t h e same s i z e . A d i f f e r e n t s t u d e n t was called on for verification. The student was a l s o a s k e d t o compare t h e w e i g h t o f t h e s t e e l b a l l and t h e p l a s t i c i n e b a l l u s i n g a double-pan balance scale. The e x p e r i m e n t e r p u t t h e p l a s t i c i n e b a l l i n one o f t h e t u b e s , t h e water r o s e . S i m i l a r g u e s t i o n s t o the ones in section 3 above were asked using the s t e e l b a l l . S t u d e n t s r e s p o n d e d on page 4, t h e n the experimenter put the s t e e l b a l l i n the tube. 6. Step 5 was repeated using a s t e e l b a l l which was s m a l l e r but h e a v i e r than a b a l l of plasticine. Students r e s p o n d e d on page 5, t u r n e d t o page 6 and t h e e x p e r i m e n t e r put the s t e e l b a l l i n the tube. 7. A l l tubes and b a l l s were removed. Two new t e s t t u b e s partially filled with water were displayed. The experimenter presented two c u b e s ; one made o f g l a s s , t h e o t h e r o f aluminum..Both c u b e s had t h e same size but one  52  was h e a v i e r than the o t h e r . A s t u d e n t came f o r w a r d and c o n f i r m e d t h e s e f a c t s . The e x p e r i m e n t e r p u t t h e g l a s s cube i n one o f the t u b e s and t h e water r o s e t o a c e r t a i n l e v e l . The e x p e r i m e n t e r s a i d , "From now on I w i l l not show you the answers. If I put t h e aluminum cube i n t o t h i s o t h e r t u b e , where w i l l t h e water level be? Darken the line nearest to where t h e w a t e r l e v e l w i l l b e . " S t u d e n t s were i n s t r u c t e d to respond and then turn to page 7. The e x p e r i m e n t e r d i d n o t put t h e aluminum cube i n t h e t u b e . Subjects 6  who  demonstrated  These  failed  two  evidence  of  o f t h e g u e s t i o n s on associating  pages 4,  volume w i t h  part  of the t e s t  and  weight.  s u b j e c t s were c l a s s i f i e d as n o n c o n s e r v e r s o f volume.  second  5  The  immediately f o l l o w e d .  8. The experimenter presented two t e s t t u b e s , a n d , f i v e p l a s t i c i n e b a l l s o f the same s i z e . He p u t one o f t h e b a l l s i n one o f t h e t u b e s and t h e water r o s e . He t h e n r o l l e d one of the b a l l s i n t o a sausage shape i n f r o n t of the group. The experimenter then said, " I f I put t h e s a u s a g e i n t o t h i s o t h e r t u b e , where w i l l t h e water l e v e l b e ? D a r k e n t h e l i n e n e a r e s t to where the water level will be." The children responded on page 7, t u r n e d t o page 8 b u t t h e e x p e r i m e n t e r d i d n o t put t h e s a u s a g e i n t h e t u b e . 9. Step 8 was r e p e a t e d by t r a n s f o r m i n g one of the balls into nine or ten s m a l l p i e c e s , one of t h e b a l l s i n t o a s m a l l p i e c e and a l a r g e p i e c e , and f i n a l l y t h e last ball into three similar b u t f l a t t e n e d p i e c e s . I n each o f t h e above mentioned t r a n s f o r m a t i o n s the students used a separate answer sheet (pages 8,9,10) t o darken a l i n e i n d i c a t i n g t h e i r judgement. 1 0 . . A l l t u b e s and b a l l s were removed. Two b e a k e r s w i t h t h e same amount o f water and two p l a s t i c i n e b a l l s o f t h e same size were p r e s e n t e d . A s t u d e n t came f o r w a r d and c o n f i r m e d t h e s e f a c t s . The e x p e r i m e n t e r p u t one o f the balls into one of t h e b e a k e r s . He t r a n s f o r m e d t h e o t h e r b a l l i n t o a " p a n c a k e " s h a p e i n f r o n t o f t h e g r o u p . He t h e n s a i d , " I f I put t h e 'pancake' i n t o t h i s o t h e r b e a k e r , where will the water l e v e l be? Darken t h e l i n e n e a r e s t t o where t h e water level will be." The children responded (page 1 1 ) . The e x p e r i m e n t e r d i d n o t put t h e 'pancake' i n t h e beaker but he said, " I f you i n d i c a t e d t h a t t h e l e v e l o f t h e water w i l l be h i g h e r t h a n t h e l e v e l i n t h e o t h e r b e a k e r , e x p l a i n why i t w i l l be h i g h e r . I f you indicated that the level will be lower, explain why i t w i l l be s o . And i f you i n d i c a t e d t h a t t h e water l e v e l w i l l be the same as the other beaker, explain why i t will be t h e same." The s t u d e n t s r e s p o n d e d and t u r n e d t o page 12.  53  11. The b e a k e r c o n t a i n i n g t h e ball was replaced by an i d e n t i c a l b e a k e r w i t h w a t e r a t t h e same l e v e l . S t e p 10 was repeated using two identical b o x e s made o f m a r b l e . The e x p e r i m e n t e r put the first box and i t s detached top s i m u l t a n e o u s l y i n t o one o f t h e b e a k e r s . They sank. He t h e n p l a c e d t h e t o p o f t h e o t h e r box on t h e box and s a i d , " I f I seal the t o p t o t h e box, p u t t h e box i n t h e o t h e r b e a k e r and i t s i n k s , where w i l l t h e water l e v e l be? Darken the line nearest to where the water level w i l l be." The e x p e r i m e n t e r asked f o r r e a s o n s as i n s t e p 10. Success i n the r e s p o n s e s on  sheets  was  7 through  11.  correct  i f  succeeded  i n a l l 5 responses  Those  the  above t e s t  who  succeeded  nonconservers. partial  correct  The  rest  comments used  students*  judgement  considered guestion  was  darkened.  classified  considered  Students as  who  conservers.  one  or  none  were  classified  as  of  the  students  were  classified  as  two  Piagetian believed  and  The  reasoning.  conservation  students  than  and  computation  pages  However, due  the  two  p r o t o c o l . I t was are a b l e  and  12  between  the  difficulties  comments were  not  Furthermore,  the  m a r b l e b o x e s was  classification was  to  11  scheme. T h i s  not  a  guestion  not c o n s i s t e n t with  the  i n c l u d e d because i t  was  more e a s i l y  to g i v e reasons  for  for equality.  purpose  any  on  classification.  u n e q u a l g u a n t i t i e s and  that  students  degree of c o n s i s t e n c y  concerning  guestioning  inequality  the  v e r b a l communications these  page 12  the  involved  by  i n the conservation  on  of  scores  item  five  in  to r e v e a l the  interpreting  reveal  was  were  written  were o n l y  part  Each  s c o r i n g the  conservers.  The  in  line  measured by  of  the  Mathematics Achievement T e s t  possible  correlation  between  general  mathematics  achievement.  s e c t i o n of the  Stanford  volume The  Achievement T e s t  was  to  achievement arithmetic was  used  54  (Madden  e t a l . , 1 9 7 3 ) . Two r e l i a b i l i t y  half estimate formula,  and t h e  are  coefficient  stated  coefficients,  based  on  the  split  Kuder-Eichardson  i n t h e manual t o be 0.90 w i t h  a standard  P error  of 2.9. T h i s Mathematics  multiple  choice  items  Achievement  and was  administered,  a maximum o f 35 m i n u t e s . The s c o r e was d e t e r m i n e d The  by t h e number  items  questions  found  i n the three  the  student  attachment  of  Measurements without number  units.  Scores  choice  students  to  predict  products  series.  and  a  drawing  asked  to  were test  given  responses.  Heath  were g i v e n  and  Project  of a cuboid, find  the  of  o r an  volume.  i n the form  Achievement T e s t  of  number by  missing  consisted  the  was  factor.  changes, such  not  and  items  the  a X b = c X d,  were  students  as i n c r e a s e , decrease  were d e t e r m i n e d  11  a X b X c > d X e X f ,  given  In other  of  I n some o f  a s s e r t i o n s o f t h e form  a X b X c = d X e X f ,  scores  Elementary  were a l s o d e t e r m i n e d  when f a c t o r s c h a n g e d a d d i t i v e l y  Students'  are v a r i a t i o n s of  and 9 s h o r t answer i t e m s .  where one o f t h e f a c t o r s this  responses.  responses.  items  a X b > c X d ,  predict  textbook  given  on t h i s  Multiplication  multiple items  was  dimensions  of c o r r e c t  The  of c o r r e c t  test  was composed o f 27 q u e s t i o n s f o r e a c h  cuboids, of  45  o f e a c h s u b j e c t on t h i s  I n v e s t i g a t i n g School, Mathematics,  M a t h e m a t i c s . The t e s t  contains  a s recommended, i n  o f t h e Volume A c h i e v e m e n t T e s t  Mathematics,  which  Test  or  by t h e number  asked were  to asked  or doubling, i n multiplicatively. of their correct  55  Instruction  and  Testing^Procedures  Instructors Three were u s e d same sex  male to carry  sex  each  not teach  o f them t a u g h t  a l l treatments.  morning  in  to  order  and p r o v i d e d  instruction.  group every  He a l s o  review  lessons,  treatments; The  investigator  retention assisted and  they  test  of  trained  printed  Volume  the t e a c h e r such  the  guides  a  handle  way  instructors and m a t e r i a l s  met w i t h t h e i n s t r u c t o r s treatment  problems  as a  period  and  insure  The i n s t r u c t o r s o n l y c a r r i e d  out the  d i d n o t a d m i n i s t e r any o f t h e t e s t s . administered the  volume  pretest,  conservation.  i n a d m i n i s t e r i n g the Mathematics  the  in  the  The i n v e s t i g a t o r d i d  and a f t e r n o o n d u r i n g t h e  uniformity of instruction. 4-day  He  teachers  The c h o i c e o f  were r a n d o m i z e d  any c f t h e t r e a t m e n t s .  daily  were a l l c e r t i f i e d  was made i n o r d e r t o e x c l u d e  The i n s t r u c t o r s  before the treatments for  who  out the t h r e e treatments.  instructors  variable.  that  instructors  Achievement  Pretest,  Two  posttest  and  female  teachers  Achievement  Pretest  posttest  and r e t e n t i o n  test.  Schedule  of I n s t r u c t i o n  and T e s t i n g  In the b e g i n n i n g of the experiment that  the  more a b o u t were  also  reason  for including  t h e way  grade  informed  6  that  them i n t h e s t u d y  students the  the students  learn  were  told  was t o l e a r n  mathematics.  outcome o f t h e s t u d y  They  would n o t  56  affect  their  grades a t school  d i a g n o s i s or  e v a l u a t i o n . The  not  any  to  to teach  d i s c u s s the The  nor  would i t s e r v e  classroom  mathematics d u r i n g  treatment  experiment  the  began by  giving  Conservation  conservers, the  pupils  of  within  order  all  classes  Boys and  for  sex.,  s u b j e c t s of each treatment days  at  pretests The lasted  the  four  of b o t h  groups f o r the  to  classify  nonconservers.  at  each  school  level.  Then  girls  were r a n d o m i z e d  This  procedure  schools.  began a f t e r consecutive  the  three  of  of  across  separately  Three  s c h o o l days.  c l a s s e s were t a u g h t  same c l a s s  pretesting  the  school for  i n the t h r e e  p e r i o d . On  each  of  the  of  after  days  recess  after  Volume Test  days were r e s e r v e d  class and  the  one  treatments  Achievement were  i n the  Test  administered.  f o r these  and the  predetermined  the  period  one  days  At e a c h s c h o o l  school  Achievement  names  were r e s e r v e d  gave a t e a c h  the  as  listed  determined  i n such a  Test,  were the  schools  Two  names  group were:randomized  experiment  The  students  The  moved t c a l l t h r e e  one  nor  conservation.  instructors  recess,  treatment  p r e t e s t s : mathematics  a t each o f the  of the  requested  randomization.  treatments for  students  beginning  and  and  conservation  treatments.  the  used  were  students.  volume  to conservation  each  to balance  was  conservers  according  the t h r e e in  partial  subjects  initially  Test  individual  p e r i o d of  t o p i c s with t h e i r  a c h i e v e m e n t , volume a c h i e v e m e n t and Volume  teachers  for  four  way  days  that  instruction  they  before  afternoon. the and  Volume the  Conservation  Multiplication  Three c o n s e c u t i v e  p o s t t e s t s . The  school  retention tests  of  57  volume  conservation,  achievement these  were g i v e n s e v e n  seven  classroom  volume  weeks  the  b a s i c concern  main f a c t o r s .  treatments.  Test,  the  multiplication  posttests.  teachers  One  of the  resumed  During regular  of t h i s factor  Study  study  was  necessitated  made up  conservation, while the other  Volume  and  i n s t r u c t i o n i n mathematics.  The  volume  weeks a f t e r  classroom  Design  two  achievement  There  were  achievement the  also  four  P o s t t e s t , the  Multiplication  were used  Pretest  and  as c o v a r i a t e s .  design i s given i n Table  was  three l e v e l s  composed o f t h e  dependent  Achievement  Posttest  Test. Scores  Mathematics Achievement A  schematic  the  Retention and  on  of  three  variables,  Volume a c h i e v e m e n t  M u l t i p l i c a t i o n ! Achievement R e t e n t i o n Achievement  of t h e  considering  the  the Volume  Pretest  representation  (SAT)  of  the  3-1.  T a b l e 3.1 Experimental Design Conservation l e v e l s and treatments  Multiplication  Volume  Control  Nonconservers Partial  conservers  Conservers A Campbell  Pretest-Posttest and  Stanley  C o n t r o l Group d e s i g n  (1963, p.  13)  was  used  which  with  appears  blocking  in on  58  the  conservation  factor..  Schematically  the  design  is  as  follows:  Randomized  G1  Pretest  Treatment 1  Posttest  Retention  Test  assignment  G2  Pretest  Treatment  2  Posttest  Retention  Test  to  G3  Pretest  Treatment  3  Posttest  Retention  Test  groups  Hypotheses While certain of  searching  f o r answers  statistical  the hypotheses  described  to  the  hypotheses i n n u l l  listed  below  were  above w h i l e t h e o t h e r s  aims  form  deduced  were based  of  the  study  were t e s t e d . Some from  the  design  on t h e aims  of the  study.  H 1.  There  achievement  are  no  scores,  among c o n s e r v a t i o n  H 2.  There  achievement  are  H 3. T h e r e achievement  are  on  differences  in  t h e p o s t t e s t and r e t e n t i o n  volume test,  groups.  no  scores,  among t r e a t m e n t  significant  significant on  differences  in  t h e p o s t t e s t and r e t e n t i o n  volume test,  groups.  no  scores,  between c o n s e r v a t i o n  significant on  interactions  in  t h e p o s t t e s t and r e t e n t i o n  and t r e a t m e n t .  volume test,  59  H 4.  There  are  no  significant  multiplication  achievement  retention  among t r e a t m e n t  H 5. The the  test,  transition  pretest  posttest  to  and  from  a  achievement  scores,  scores  and  a lower l e v e l of c o n s e r v a t i o n  on  groups.  l e v e l of conservation,  test,  on  on  in  the p o s t t e s t  higher  retention  differences  is  the  independent  posttest  and  on  of  the  volume  retention  test  respectively.  H 6.  The  transition  the  pretest  to  a  posttest  and  H 7.  transition  the  The  pretest  posttest  to a  a lower l e v e l of c o n s e r v a t i o n  higher  retention  and  achievement  from  test,  from  lower  on  level  on  the  of t r e a t m e n t s .  l e v e l of c o n s e r v a t i o n  of  test,  the  of c o n s e r v a t i o n ,  i s independent  a higher  retention  scores  level  on  conservation,  is  independent  posttest  and  on  on of  the  volume  retention  test  respectively.  H 8.  The  transition  the  pretest  to  a  from  a higher  lower  posttest  and r e t e n t i o n  H 9.  volume a c h i e v e m e n t  The  retention scores.  test,  test,  l e v e l of conservation,  level  of conservation,  i s independent  scores,  on  the  on  on the  of treatments.  posttest  are not r e l a t e d t o mathematics  and  achievement  60  H10.  The  initial  level  mathematics achievement  H11. The i n i t i a l  level  of c o n s e r v a t i o n  i s independent of  scores..  of conservation  i s not  related  to  sex.  H12. The volume a c h i e v e m e n t s c o r e s related  to sex.  Statistical In  order  posttest  to test  score  and  Analyses  hypotheses  retention  1-4, e a c h d e p e n d e n t v a r i a b l e  test  separately  by  covariance.  The a n a l y s i s was c a r r i e d  program found hoc  on t h e p o s t t e s t a r e n o t  BMDP2V. across  using a 3 X 3 f u l l y  In t h e c a s e s treatments  score,  crossed out  c o m p a r i s o n s were made t o d e t e r m i n e  analyzed  two-way a n a l y s i s o f using  where s i g n i f i c a n t  and c o n s e r v a t i o n  was  the  d i f f e r e n c e s were  levels,  which  computer  Scheffe  groups  post  differed  significantly. Hypotheses involved  a f o r c e d dichotomy while  interval  scale.  recommended testing  the  A  biserial  and S t a n l e y  two v a r i a b l e s , one o f w h i c h  the other  was measured on  correlation (1970, p . 168)  coefficient was  used  of t r a n s i t i o n a l cases  was e x p e c t e d  statistic..  However,  amonq t h e t h r e e  an as for  h y p o t h e s e s . H y p o t h e s e s 6 and 8 were t e s t e d  t a b l e s and t h e C h i S q u a r e  number  levels  by G l a s s  these  frequency  5 and 7 d e a l t with  using since  conservation  t o be s m a l l i t was n o t a p p r o p r i a t e  to  use  61  the  number  Instead for  of  t r a n s i t i o n a l cases  t r a n s i t i o n s were c l a s s i f i e d  each treatment  continuity  t o be  Hypothesis  11  variable.  The  recommended this  the  (Glass  on  and  variables tested  truly It  (1977, p.  Stanley,  and  ordinal  1970,  and  tested  using  Hypothesis and  the  Stanley,  p.  ordinal  scores  for testing  correlation  analysis  is  coefficient included  c o r r e l a t i o n c o e f f i c i e n t (Glass  and  a variable  on  an  of two  hypothesis  12  as  with  109-113). H y p o t h e s i s 10 This  for 332).  an  dealt  pp..176-178). F i n a l l y , h y p o t h e s i s  using  and  recommended  interval scales.  K e n d a l l ' s Tau  9  change"  correction  tied  used  levels.  "no  1970,  variable  McSweeny was  scales  pp.  Yates'  Stanley,  nominal  267).  1970,  " c h a n g e " vs  allow f o r  P e a r s o n ' s product-moment  cn  and  a  M a r a s c u i l o and  dichotomous v a r i a b l e  was  (Glass  with  interval  using  Stanley,  (Glass  as  Wilcoxon two-sample t e s t by  usage o f  applied dealt  hypothesis  variables  i n order to  between e a c h p a i r o f  involved  interval  was and a  scale.  a point-biserial correlation coefficient  1970,  pp.  163-164).  62  CHAPTER IV  RESULT S This chapter are  sections  covariance, hoc  on t e s t  analyses,  correlation  qualitative  contains  c o n t a i n s the r e s u l t s of the  study,  analyses.  study.  preliminary study,  tests  The  Included  a n a l y s i s of  f o r i n d e p e n d e n c e and  section  post  of p r e l i m i n a r y  two p a r t s , t h e v a r i a b l e c o v a r i a t e s and t h e  study  instructor  effect. .  Tests R e l i a b i l i t i e s Three posttests Volume  different  tests  were  and r e t e n t i o n t e s t s .  Achievement  and Item  Analysis  administered  The p r e t e s t s  Volume  T e s t , t h e Volume C o n s e r v a t i o n  of  Test  Achievement  Achievement  of  Achievement  statistics  Achievement  item Test,  multiplication  T e s t , t h e Volume C o n s e r v a t i o n  analysis the  The t e s t  will  Volume  achievement  c a n be f o u n d  Test.  be  Conservation  test.  i n Appendix  A B.  of  complete  Test the  T e s t and t h e  reliabilities  reported  the  and t h e  The p o s t t e s t s and t h e r e t e n t i o n t e s t s c o n s i s t e d  Multiplication summary  pretests,  consisted  a r i t h m e t i c computation s e c t i o n of the Stanford (SAT).  as  and  a  f o r t h e Volume Test list  and  the  of  item  63  Volume A c h i e v e m e n t The in  Hoyt  e s t i m a t e s of r e l i a b i l i t i e s  the p r e t e s t ,  0.94  Test  posttest  respectively.  percent  of  and r e t e n t i o n  The d i f f i c u l t y  subjects  80.7 i n t h e p r e t e s t ,  pretests,  from  0.78  level,  were as  test.  ranged  The  0.94,  0.95 and  defined  by  ranged from  from  item 0.35  t o 0.79 i n t h e p o s t t e s t  i n the r e t e n t i o n  consistency)  23.4 t o 78.4 i n t h e p o s t t e s t  coefficients  f r o m 0.35  test  responding c o r r e c t l y ,  24.6 t o 87.1 i n t h e r e t e n t i o n correlation  (internal  the  8.2 t o  and from  point-biserial to  and  0.74 from  i n the 0.39  to  test.  Volume C o n s e r v a t i o n T e s t The posttest  Hoyt and  retention  respectively.. in  The  the p r e t e s t ,  68.4%  to  were  difficulty  level  from  from  0.46  to  Hoyt  retention  to  93.0%  correlation  0.75  in  were 0.86  the  0.85  pretest, and  0.82  r a n g e d f r o m 35.7% t o 76.6% posttest  test.  The i t e m  from  0.46  and  from  point-biserial  to  i n the p o s t t e s t  0.78  in  and from  the  0.33 t o  Test  estimate of r e l i a b i l i t i e s  from  the  test.  Achievement  test  ranged  ranged  in  0.78,  59.6% t o 86.0% i n t h e  coefficients  Multiplication  level  "reliabilities  test  0.77 i n t h e r e t e n t i o n  The  of  91.2% i n t h e r e t e n t i o n  correlation pretest,  estimate  and 0.79  in  the  respectively..The  8.2% t o 85.4% i n t h e p o s t t e s t retention  coefficients  test.  ranged  posttest  from  The  and  item  0.34  to  and  difficulty from  7.6%  point-biserial 0.67  in  the  64  posttest  and  f r o m 0.34  t o 0.55  i n the r e t e n t i o n  Preliminary  test.  Analysis  Covariates The  Stanford  Achievement  Achievement P r e t e s t the  four  were c o n s i d e r e d  dependent  Volume A c h i e v e m e n t Posttest  and  regression  variables,  Betention  covariates  analysis  was  was  computer  program  BMD02B  Pretest  entered  Achievement  Posttest  Achievement  Betention  Multiplication  df=1,103)  (F=11.33,  df=2,102).  Volume  carried  two  Volume  t h e SAT  Pretest  these r e s u l t s .  df= 1,103)  however, had two the  using  removes  the any  low. only  t h e Volume  f o r both the  (F=58.39,'  Similarly,  covariates  Achievement  summarizes  and  To  step-wise  out  that  a covariate  Test  Test.  a 5% i n c l u s i o n and a 558  automatically  (F=78.63,  Posttest,  (F=35. 31,  as  Achievement  multiple  r e s u l t s o f the a n a l y s i s r e v e a l e d  Achievement  had  which  of  for  Posttest,  Betention  when i t s s i g n i f i c a n c e l e v e l becomes t o o  The  Test  sequence  Volume  covariates  Achievement  Achievement  analysis  The  the  Multiplication  using  levels.  and  as p o s s i b l e  conducted  deletion  variable  a  (SAT)  Volume  Test,  Multiplication  determine the  Test  and  Volume  the  Volume  df=1,103).. covariates,  Achievement  the  The SAT  Pretest  the M u l t i p l i c a t i o n Betention (F=43.35,  (F=21.81,  df=1,103)  df=2,102).  and  the  Table  4.1  65  T a b l e 4.1 F Values f o r E n t e r i n g of Dependent  V o l Ach P r e SAT V o l Ach P r e V o l Ach P r e SAT V o l Ach P r e  V o l Ach P o s t Mul Ach P o s t V o l Ach B e t Mul Ach Ret  F Value  df  Covariate  Variable  Covariates  1,103 1,103 2,102 1,103 1,103 2, 102  78.63 35.31 11.33 85. 39 43.35 21.81  V o l Ach P r e : Volume A c h i e v e m e n t P r e t e s t V o l Ach P o s t : Volume A c h i e v e m e n t P o s t t e s t V o l Ach R e t : Volume A c h i e v e m e n t R e t e n t i o n t e s t Mul Ach P o s t : M u l t i p l i c a t i o n A c h i e v e m e n t P o s t t e s t Mul Ach R e t : M u l t i p l i c a t i o n A c h i e v e m e n t R e t e n t i o n t e s t SAT: S t a n f o r d A c h i e v e m t T e s t  Assumptions o f A n a l y s i s Analysis pp.  752-753)  blocking  of Covariance  of covariance was  as  prefered  to  recommended analysis  was f e a s i b l e on c o n s e r v a t i o n  by of  level  Winer variance  only  be r e l a t e d t o f u t u r e  covariance  adjustment  achievement  intended  to  (i.e., reduce  treatment e f f e c t Further are  related  (Elashoff,  measured  believed  of  dependent  variables  SAT and p r e t e s t bias  and  volume  f o r the r e g r e s s i o n  volume  increase  (i.e.,  achievement)  the  ( C o c h r a n , 1957, p. 2 6 2 ) .  j u s t i f i c a t i o n s f o r using to  were  randomly at  was  accuracy o f the  the  fulfillment  analysis of  of  certain  covariance assumptions  1969, p. 3 8 5 ) : 1. R a n d o m i z a t i o n was f u l f i l l e d  individuals assigned  past  volume a c h i e v e m e n t . I n t h e a n a l y s i s o f  and m u l t i p l i c a t i o n a c h i e v e m e n t )  on c o v a r i a t e s  since  while  a c h i e v e m e n t i n c o m p u t a t i o n and volume c a l c u l a t i o n were to  (1971,  assigned to  randomly  treatments.  t o g r o u p s and g r o u p s  2. C o v a r i a t e s ,  having  since were been  t h e b e g i n n i n g o f t h e e x p e r i m e n t , were i n d e p e n d e n t  66  of  t r e a t m e n t s . 3. C o v a r i a t e s a r e measured a c c u r a t e l y  standardized 0.90)  and  SAT the  Volume  Achievement  =  0.94).  4. The  reliability variables in to  (Kuder-Eichardson  on  t h e c o v a r i a t e s was  the p r e l i m i n a r y determine  analysis,  reliability Test  believed  a linear  the r e l a t i o n s h i p significant  relationship  assumption  of  linearity.  5.  h o m o g e n e i t y o f c o v a r i a n c e and  no  was  scatter  variables 1969, to  p.  v e r s u s c o v a r i a t e s f o r each 392).  indicate  respect  Winer  that the  to  homogeneity  coefficients  ( p . 7 7 2 ) . " 6.  of  variables  dependent  conservation of  level  of  The  within  was  s u b j e c t s i n some c e l l s .  fulfillment  of t h i s  "statistical  convenience"  Elashoff  assumption only  and the  dependent (Elashoff,  "there evidence is  on  of  interaction  of the  covariance  each  not t e s t e d  variables  treatment group  assumption  used  of the assumption  plots  assumptions  fact  confirmed  (1971) f u r t h e r c l a i m s t h a t analysis  of  In  model was  treatment-slope  =  dependent  linear.  found  Fulfillment  comparing  the  between d e p e n d e n t  The  through  t o be  estimate  regression  covariates.  done  of  the  coefficient  (Hoyt  regression  using  robust  with  ...  regression  of normal  distribution  treatment  group  at  each  because  o f t h e s m a l l sumber  (1969)  explains  that  the  i . e . , normality i s required f o r (p.  386).  Instructor.Effect The  proposed  experimental design  (Treatments  X  problem  treatment  in a  necessary  the  Conservation  inclusion  group of  Levels). in  one  in  chapter However, of  instructors  the as  3 a  was  behaviour  schools a  3X3  factor.  made The  67  experimental  design  became  Levels X Instructors). determine  the  Achievement  Analysis  effect  Posttest  of and  Posttest, . Since the i n t e n t that  any i n s t r u c t o r  level The  be  interaction,  X T r e a t m e n t s on summary effect  of  the  in  and t h e 3-way i n t e r a c t i o n s (p  <  0.10).. The (p < 0. (10)  Multiplication of  covariance  2-way i n t e r a c t i o n  to necessitate  2-way  Instructors  Posttest.  A  f o r the i n s t r u c t o r s  reduce  Levels)  the  was  design  as was s u g g e s t e d  not  involving  thought  to  was be  t h e o r i g i n a l 2-way  possible to  significant  instructors  was n o t  restructuring  d e s i g n o f t h e s t u d y . I t was t h e r e f o r e  Conservation  were  i n T a b l e 4.2.  the e f f e c t of i n s t r u c t o r s  and  only  was  Achievement  a single  instructors  Volume  be i d e n t i f i e d , t h e s i g n i f i c a n c e  just  enough  the  was t o i n s u r e  t h e main e f f e c t o f i n s t r u c t o r s  strong  both  o f t h e 3-way a n a l y s i s  Since  significant,  were c o n d u c t e d t o  o f t h e n u l l h y p o t h e s i s was s e t a t 0.10.  analysis  i s presented  Conservation  Achievement  t o be s i g n i f i c a n t  the  X  Multiplication  nonsignificant  found  of covariance  the  main e f f e c t o f i n s t r u c t o r s to  (Treatments  instructors  effect will  f o rthe rejection  found  and  3X3X3  to  3X3  i n chapter  pool  across  (Treatments 3.  X  68  Analysis  T a b l e 4.2 of Covariance - I n s t r u c t o r s E f f e c t df  Source o f V a r i a t i o n V o l Ach P o s t Main E f f e c t Instr Tr Cons 2- way I n t e r a c t i o n s Instr X Tr I n s t r X Cons Cons X Tr 3- way I n t e r a c t i o n s Error Mul Ach P o s t Main E f f e c t Instr Tr Cons 2- way I n t e r a c t i o n s Instr X Tr I n s t r X Cons Cons X Tr 3- way I n t e r a c t i o n s Error  significance  MS  2 2 2  39.54 517.02 93. 24  il.54 20.15 3.63  0. 22 0.00 0.03  4 4 4 8 76  12.77 52. 10 25.58 21.02 25.65  0.50 2.03 1.00 0.82  0.74 0. 10 0.41 0. 59  2 2 2  11.40 71.80 25.10  1.29 8.10 2.83  0.28 0.01 0.07  4 4 4 8 76  20.55 4.58 16.09 13.00 8.86  2.32 0.52 1.82 1.47  0.07* 0.72 0. 14 0. 18  * p < 0.10 ( I n s t r u c t o r e f f e c t o n l y ) I n s t r : I n s t r u c t o r ; T r : Treatment; Cons: C o n s e r v a t i o n  Analysis  Level  of Covariance  1-3 The  test  significant  of (p  hypothesis <  Achievement P o s t t e s t ANCOVA in  Table  0.05) scores  1  difference among  f o r t h e Volume A c h i e v e m e n t 4.3.  Post  hoc  (p. 58)  analysis  revealed  was  found  conservation Posttest using  scores Sheffe's  that in  Volume  groups. c a n be  a  The found  method  of  69  multiple  comparisons  superiority  showed  of the conservers  g r o u p . No s i g n i f i c a n t conservation difference  a  group over  difference  g r o u p s a t t h e 0.05 found  in  significant  Volume  was  g r o u p s a t t h e 0.05  Volume A c h i e v e m e n t  Eetention  Test  among  T h e r e was  Achievement  among c o n s e r v a t i o n  <  0.05)  the p a r t i a l - c o n s e r v e r s  found  level.  (p  any  no  other  significant  Eetention Test  level.  scores  The ANCOVA f o r t h e  s c o r e s c a n be f o u n d  in  Table  4.4. The  means,  treatments Pretest  by c o n s e r v a t i o n  can  standard  standard  be  found  deviations  conservation  levels  Volume A c h i e v e m e n t The  adjusted  treatments Posttest Tables  levels  in  group  for  the  Eetention  means,  for  Table  and  and  the  4.5.  Volume  Test  of  of  Achievement means,  treatments  Achievement  c a n be f o u n d  for  sizes  Volume  deviations  levels  group  The u n a d j u s t e d  sizes  standard  by c o n s e r v a t i o n  by  P o s t t e s t and  in  Appendix  D.  and g r o u p s i z e s o f  the  Volume  Achievement  and Volume A c h i e v e m e n t E e t e n t i o n T e s t c a n be f o u n d  4.6  marginal  deviations  and 4.7 r e s p e c t i v e l y . I n a l l values  determined  for  the  by c a l c u l a t i n g  means  the  and  of  these  standard  weighted  in  tables  the  deviations are  average  of  the  cell  values. The  test  of  significant  (p  Achievement  Posttest  4.3).  Post  hoc  comparisons treatment  over  <  hypothesis 0.001)  showed  a  (p. 58)  difference  scores  analysis  2  was  found  among t r e a t m e n t  using  Sheffe's  significant  multiplication  revealed in  groups  method  <  0.01)  a  Volume  (see T a b l e  of  superiority  treatment.(p  that  multiple of and  volume over  70  control found  treatment  between  t h e 0.05 was  0.01).  multiplication  Sheffe's  groups  (see  method o f  superiority  of  0.01)  treatment  control  significant  4.4).  of  over  control was  hypothesis"  interaction  was  Achievement  conservation The 13.06  conservers,  15.18  treatment  group  the  adjusted  were  14. 52  for  means  on  Retention  in  volume  (37%) the  10.82  (40%)  (46%)  that  was  and  10.89  at  found  between  level.  f o r the (65%)  were  partial f o r the  multiplication  group.  Likewise  Retention Test (42%)  for  f o r t h e c o n s e r v e r s , 17.49 group,  no  Achievement  scores  Volume A c h i e v e m e n t 11. 23  No  treatments  f o r the  f o r the c o n t r o l  (55%)  group  0.01) .  multiplication  and  Test  treatment  Volume A c h i e v e m e n t P o s t t e s t  12.30  treatment  treatment  <  Volume  a t t h e 0.05  f o r the n o n c o n s e r v e r s ,  c o n s e r v e r s , 14.87  the  (p  f o r t h e c o n s e r v e r s , 17.42  9.87  significant  revealed  treatments  group,  and  (54%)  multiplication group.  (56%)  a  3  f o r the nonconservers,  treatment  partial  and  using  significant interaction  a d j u s t e d means on t h e  (48%)  volume  levels  analysis  level.  level. Similarly,  Volume  hoc  a t t h e 0.05  the in  among  between  s c o r e s between c o n s e r v a t i o n l e v e l s no  scores  treatment  Posttest 0.05  difference  multiplication  found  at  Test  showed  (p. 58)  was  treatment  (p < 0.001)  Post  found  treatment  difference  control  Retention  Table  over  difference  test  and  a significant  volume t r e a t m e n t  significant  . The  significant  multiple comparisons  and  and  No  treatment  i n Volume A c h i e v e m e n t  treatment  <  <  level. Similarly,  found  (p  (p  13.51  (50%)  (40%)  f o r the  the (65%)  for  the  control  71  T a b l e 4 .3 A n a l y s i s o f C o v a r i a n c e o f Volume A c h i e v e m e n t df  Source Conservation Treatment Conservation Error  Level Level  * p < 0.05  Analysis  Posttest Scores  ==—————= ——=== =--------==-----:==== = =  — =—— —— —— — — = — 3  X Treatment  2 2 4 95  MS  88. 88 339.56 31. 50 27.74  F  3. 20* 12. 24** 1. 14  ** p < 0.001  T a b l e 4.4 o f C o v a r i a n c e o f Volume A c h i e v e m e n t E e t e n t i o n S c o r e s df  Source Level  Conservation Treatment Conservation Error  Level  X Treatment  2 2 4 95  MS 74. 16 217.01 23.35 27. 79  F 2.67 7.81* 0.84  * p < 0.001  T a b l e 4.5 Means, S t a n d a r d D e v i a t i o n s , and Group S i z e s o f Volume A c h i e v e m e n t P r e t e s t S c o r e s f o r T r e a t m e n t s by C o n s e r v a t i o n L e v e l s (Maximum S c o r e = 27) Conservation Level  Volume  •••Treatments • „, Multiplication Control  Total  Non-conservers  7.21 (5. 23) 19  7 .61 (4 .88) 18  4.45 (3.98) 20  6. 37 (4.68) 57  Partial-conservers  10.17 (8. 23) 6  8 .67 (7 -71) 6  5.75 (4.50) 4  8. 50 (7. 10) 16  Conservers  9.18 (6.85) 11  10 .60 (6 .17) 15  14.00 (8.85) 6  10. 75 (6.9D 32  8.31 (6. 23) 36  8.92 (5 .81) 39  6.53 (5.02) 30  8. 03 (5.73) 105  •  Total  72  T a b l e 4.6 A d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and Group S i z e s o f Volume A c h i e v e m e n t P o s t t e s t S c o r e s f o r T r e a t m e n t s by C o n s e r v a t i o n L e v e l s (Maximum S c o r e = 27) Conservation Level  • Volume  Treatments Multiplication  Control  Total  Non-conservers  18.33 (5.63) 19  11 .60 (7 .27) 18  9.35 (6.87) 20  13.06 (6.58) 57  Partial-conservers  15.49 (7.41) 6  7 .83 (10.17) 6  8.29 (7.05) 4  10.82 (8.36) 16  Conservers  16.91 (4.29) 11  14 .92 (7 .54) 15  12*65 (8.59) 6  15. 18 (6. 62) 32  17.42 (5.52) 36  12 .30 (7 .82) 39  9.87 (7.24) 30  13.36 (6.86) 105  ~  Total  T a b l e 4.7 A d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and Group S i z e s o f Volume A c h i e v e m e n t R e t e n t i o n T e s t S c o r e s f o r T r e a t m e n t s by C o n s e r v a t i o n L e v e l s (Maximum S c o r e = 27) Conservation Level  Volume  ..Treatments Multiplication  Control  Total  Non-conservers  19.00 (6.51) 19  13 .91 (6 .34) •18  10 .81 (6 .90) 20  14.52 (6.59) 57  Partial-conservers  14.04 (8. 13) 6  8 .96 (10.00) 6  10 .41 (8 .66) 4  11. 23 (8.96) 16  Conservers  16.76 (5.08) 11  14 .84 (7 .96) 15  11 .48 (9 .05) 6  14. 87 (7.17) 32  Total  17.49 (6.34) 36  13 .51 (7 .53) 39  10 .89 (7 .56) 30  14. 12 (7.13) 105  73  Hypothesis The  4 test  of  hypothesis  significant  difference  Achievement  Posttest  significant  difference  Retention  scores was  found  Multiplication  Table  f o r the r e t e n t i o n  and  analysis  posttest  using  level,  multiplication over c o n t r o l The  showed  treatment  treatment  means,  over  groups.  level.  The  found  in  be  s c o r e s i n T a b l e 4.9.  multiple  comparisons,  significant volume t r e a t m e n t  standard  Achievement  for  can  deviations  Posttest  and  be f o u n d i n T a b l e  A n a l y s i s of C o v a r i a n c e  the  superiority  of  (p <  0.01)  and  and  group  Retention  sizes  test  4.10.  df  MS  Scores F  2  23.46  2.46  Treatment  2  98.54  10.33*  4  19.16  2.01  94  9.54  *  p < Q. 001  L e v e l X Treatment  of  scores  Conservation Level  Error  Post  at  T a b l e 4.8 of M u l t i p l i c a t i o n P o s t t e s t  Source  Conservation  No  Achievement  a t t h e 0.05  s c o r e s can  test  a  Multiplication  treatment  groups  that  (p < 0 . 0 1 ) .  Multiplication treatments  a  in  in Multiplication  Posttest  Sheffe's  revealed  found  among  s c o r e s among t r e a t m e n t  4.8  (p* 59)  (p < 0.001) was  ANCOVA f o r t h e  hoc  4  74  Table Analysis of Covariance  4.9  of M u l t i p l i c a t i o n  Source  Retention  df  1  w~—  Conservation  Level  Treatment Conservation  MS —•  L e v e l X Treatment  Error  ,  Scores F  ,  . ,  ,  2  4.62  0.68  2  7.81  1.15  4  3.42  0.51  94  6.76  T a b l e 4. 10 A d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and G r o u p S i z e s o f M u l t i p l i c a t i o n A c h i e v e m e n t P o s t t e s t and R e t e n t i o n T e s t S c o r e s f o r T r e a t m e n t s (Maximum S c o r e = 20)  Test  Volume  Treatments Multiplication  Control  Total  Posttest  9.67 (3.69) 36  12.59 (3.01) 39  10.29 (4.08) 30  10.9 3 (3. 55) 105  10.65 (2.99) 36  11.34 (3.53) 39  10.76 (3.14) 30  10.95 (3.23) 105  Retention  Test  Correlation  Hypotheses  5 and, 7  Biserial testing revealed  Study  correlation  coefficients  o f h y p o t h e s e s 5 and 7 t h a t the t r a n s i t i o n  were  (p. 5 9 ) . The t e s t from  a lower  calculated  for  of hypothesis  to a higher  level  5 of  75  conservation  between  the  pretest  independent  o f volume a c h i e v e m e n t  Similarly,  the  conservation  transition  between  found  independent  level.  Table  between lower test  4.11  X  treatments  Table  4. 13 levels  the  0.05  retention  achievement  scores  correlation  and t r a n s i t i o n  level.. l e v e l of  test  was  at the  0.05  coefficients to a higher  or  a t t h e p o s t t e s t and t h e r e t e n t i o n  shows t h e c e l l on  the  shows  the  went  the  found  sizes  of  conservation  p r e t e s t , p o s t t e s t and r e t e n t i o n number  of  students  whose  up, down o r r e m a i n e d t h e same between  p r e t e s t and e a c h o f t h e p o s t t e s t and t h e r e t e n t i o n t e s t . Likewise,  transition found  the t e s t  from  a  independent  pretest retention  and test  another a general the  was  hypothesis  a t t h e 0.05  revealed  achievement and  level  the t r a n s i t i o n  independent  7  scores  nonconservers, 4. 1 2 ) .  13  the  between  the  (see T a b l e from  between " t h e  p r e t e s t and t h e  4.11).  one c o n s e r v a t i o n  level  to  o f volume a c h i e v e m e n t s c o r e s  there  was  levels;  In  improvement i n t h e s u b j e c t s ' c o n s e r v a t i o n  32 c o n s e r v e r s  that  t o a lower l e v e l of conservation i s  volume  posttest  p r e t e s t there  Table  of  higher  of  the  Even though  and  4.12  at  was  a lower to a higher  shows t h e b i s e r i a l  Table  conservation the  volume  of conservation  levels.  test.  of  the p o s t e s t  scores  t h e p r e t e s t and  volume a c h i e v e m e n t s c o r e s level  levels  from  and  were  57 n o n c o n s e r v e r s ,  16 p a r t i a l  while  i n the r e t e n t i o n  test  partial  conservers  there  conservers were  and 56 c o n s e r v e r s  36 (see  76  T a b l e 4.11 B i s e r i a l C o r r e l a t i o n C o e f f i c i e n t s Between Volume A c h i e v e m e n t S c o r e s on t h e P o s t t e s t , R e t e n t i o n T e s t and T r a n s i t i o n t o a H i g h e r o r Lower L e v e l o f C o n s e r v a t i o n Transition  Higher  Posttest-pretest  0. 13  Retention  0. 09 (0.34)  Test-Pretest  i Number i n (  Cell  Transition  (0.18)1  ) i n d i c a t e s the s i g n i f i c a n c e  Lower  0.03  (0.79)  0.03  (0.77)  level  T a b l e 4.12 S i z e s o f C o n s e r v a t i o n L e v e l s X T r e a t m e n t s on t h e Retention Test P r e t e s t , P o s t t e s t and  Volume  Treatments Multiplication  Control  Total  19* 19 13  18 10 11  20 17 12  57 46 36  6 3 6  6 8 5  4 4 2  16 15 13  Conservers  11 14 17  15 21 23  6 9 16  32 44 56  Total  36 36 36  39 39 39  30 30 30  105 105 105  Conservation Level Non-conservers  Partial-conservers  _  * The f i r s t number r e f e r s t o t h e p r e s t e s t The s e c o n d number r e f e r s t o t h e p o s t t e s t The t h i r d number r e f e r s t o t h e r e t e n t i o n  test  77  T a b l e 4.13 Number o f S u b j e c t s Whose C o n s e r v a t i o n L e v e l s C h a n g e d / D i d Not Change Between t h e P r e t e s t and e a c h o f t h e P o s t and E e t e n t i o n T e s t Level Posttest Betention  Op  L e v e l Down  23 32  Test  Same L e v e l  11 8  Total  73 65  105 105  H y p o t h e s e s 9 and 12 Pearson's calculated revealed  product-moment  for  testing  pretest  pretest  Volume  SAT  Point  test  and  were n o t  found  to  12  be  the p o i n t - t i s e r i a l  significant  levels  0.001)  Test scores  to by  were  found the SAT. found  levels  the  correlation  between t h e volume  scores.  correlation  volume a c h i e v e m e n t s c o r e s  The t e s t  were  measured  summarizes  significant  hypothesis  <  was  ( r = 0.37, p < 0.001) t o t h e  4.14  and t h e SAT  biserial of  Table  the  scores  testing  correlated  summarizes  sex.  (p. 5 9 ) .  scores  Achievement B e t e n t i o n  scores.  achievement  the  achievement  significantly  coefficients  for  9  c o r r e l a t e d ( r = 0.35, p  mathematics  Similarly, be  hypothesis  coefficient  t h a t t h e Volume A c h i e v e m e n t P o s t t e s t s c o r e s  t o be s i g n i f i c a n t l y  to  of  correlation  coefficients  were  (p. .60). The t e s t  calculated  revealed  that  on t h e p o s t t e s t and t h e r e t e n t i o n correlated correlation  between t h e volume  to  sex.  Table  coefficients  achievement  4.13  and t h e  scores  and  78  T a b l e '4. 14 P e a r s o n ' s P r o d u c t Moment C o r r e l a t i o n C o e f f i e n t s and S i g n i f i c a n c e L e v e l s Between Volume A c h i e v e m e n t S c o r e s and SAT S c o r e s on t h e P o s t t e s t and t h e R e t e n t i o n T e s t SAT V o1u me~Ach i eve m e n t ~ P o s t t e s t Volume A c h i e v e m e n t R e t e n t i o n  Significance  0. 35  5700026 0.00011  0.37  Test  T a b l e 4.15 P o i n t - B i s e r i a l C o r r e l a t i o n C o e f f i e n t s and S i g n i f i c a n c e L e v e l s Between Volume A c h i e v e m e n t S c o r e s and Sex on t h e P o s t t e s t and t h e R e t e n t i o n T e s t Sex Volume A c h i e v e m e n t P o s t t e s t Volume A c h i e v e m e n t R e t e n t i o n  Hypothesis  initial  Tau  correlation  of h y p o t h e s i s level  of  used  measured  for testing  subjects their  who  did  took  were  10  (p. 6 0 ) .  p = 0.08)  The was  of  this  test found  of  by t h e c o m p u t a t i o n  was  calculated for  showed t o be  section  hypothesis  that  achievement  o f SAT. The  consisted  of  sample  a l l 146  Pretest regardless of  i n other p a r t s o f the experiment.  A l l other  t e s t e d u s i n g t h e d a t a o f t h e 105 s u b j e c t s  n o t m i s s any t e s t o r t r e a t m e n t  day.  the  independent  mathematics  t h e Volume C o n s e r v a t i o n  participation  hypotheses  0.(|3 0.23  coefficient  conservation  ( K e n d a l l ' s Tau = 0.09, scores  0.14 0.12  Test  10  Kendall's testing  Significance  who  79  T e s t s o f Independence  H y p o t h e s e s 6 and 8 Frequency calculated of  for testing  hypothesis  higher  t a b l e s were made and C h i S q u a r e s t a t i s t i c s  of  was f o u n d  conservation independent  the C h i Square f o r t r a n s i t i o n treatments a and  lower  from  the  of treatments up o r  (p. 5 9 ) . The  no  a lower  pretest  to  a  and t h e  a t t h e 0.05  transition  test  up  level; versus  the transition  from  t o a h i g h e r l e v e l o f c o n s e r v a t i o n between t h e p r e t e s t  0.05  level  test  was f o u n d  with  Chi  frequencies of t r a n s i t i o n pretest  between  was 0.93 w i t h d f = 2. S i m i l a r l y ,  the r e t e n t i o n  the  6 and 8  6 revealed that the t r a n s i t i o n  level  posttest  of hypotheses  were  and  represented  each  of  i n Table  independent  square  up  of  versus  of  0.97  treatments  and  treatments  at  d f = 2. The between  the  t h e p o s t t e s t and t h e r e t e n t i o n t e s t a r e  4.16. The f r e q u e n c i e s of t r a n s i t i o n  no  transition  up v e r s u s  treatments  of  t h e p o s t t e s t and t h e r e t e n t i o n  up  or  between t h e p r e t e s t and e a c h test  are represented i n Table  4.17. Likewise, transition between  from  the  treatments  the  a higher  pretest  of to  and  Likewise,  conservation  hypothesis a  lower  posttest  a t t h e 0.05 l e v e l ;  down o r no t r a n s i t i o n 2.  test  the  down v e r s u s  the t r a n s i t i o n  from  8  revealed that the  level was  Chi  of  found  square  treatments  conservation  independent for  transition  was 0.91 w i t h  a h i g h e r t o a lower  between t h e p r e t e s t and  the  retention  of  df =  l e v e l of test  was  80  found  independent  square  o f 3.48  of  treatments  at  t h e 0.05  and d f = 2. The f r e q u e n c i e s o f  o r no t r a n s i t i o n  down v e r s u s  treatments  each  o f t h e p o s t t e s t and t h e r e t e n t i o n  Table  4.18.  level  with C h i  transition  down  between t h e p r e t e s t a n d test  are.represented i n  T a b l e 4. 16 C o n t i n g e n c y T a b l e : T r a n s i t i o n Up, Down, o r S t a y i n g t h e Same V e r s u s T r e a t m e n t s Between P r e t e s t - P o s t t e s t and Pretest-Retention Test Treatment  Volume Multiplication Control  Transition Up Post Retention Post Retention Post Retention  Transition Down  No Transition  25 21 25 25 21 19  5 4 4 4 2 0  6 11 10 10 7 11  T a b l e 4. 17 C o n t i n g e n c y T a b l e : T r a n s i t i o n Up V e r s u s T r e a t m e n t s on t h e P o s t t e s t a n d t h e R e t e n t i o n T e s t Transition  Treatment Volume Multiplication Control  Post Retention Post Retention Post Retention  Up  No T r a n s i t i o n  30 25 29 29 23 19  6 11 10 10 7 11  P o s t t e s t : "chi s g u a r e = 0.93, d f = 2, p = 0.63 E e t e n t i o n t e s t : C h i s g u a r e = 0.97, d f = 2, p =  0.61  Up  81  T a b l e 4.18 C o n t i n g e n c y T a b l e : T r a n s i t i o n Down V e r s u s T r e a t m e n t s on t h e P o s t t e s t a n d t h e R e t e n t i o n T e s t Treatment  Transition  Volume Multiplication Control Posttest: Retention  Post Retention Post Retention Post Retention  the  hypotheses  5 4 4 4 2 0  31 32 35 35 28 30  in this  s e c t i o n are r e l a t e d  of the study.  about t h e t r a n s i t i o n in  No T r a n s i t i o n  C h i s q u a r e = 0.91, d f = 2, p = 0.63 t e s t : C h i s q u a r e = 3.48, d f = 2, p = 0. 18  The r e s u l t s r e p o r t e d to  Down  However, a d d i t i o n a l f i n d i n q s  between c o n s e r v a t i o n l e v e l s  t h e s e c t i o n of Post  directly  Hoc Q u a l i t a t i v e  are  included  Analyses.  Hypotheses 11 The W i l c o x o n two-sample t e s t for the  testing initial  of level  significantly  (p  hypothesis of  using t i e d  11 ( p . 6 0 ) . The t e s t  conservation  of  < 0.05) b e t t e r t h a n  frequencies o f the p r e t e s t conservation represented hypothesis  scores  the  was a p p l i e d  revealed  males'  was  that found  t h a t o f t h e f e m a l e s . The levels  versus  sex i s  i n T a b l e 4.19. The sample used f o r t e s t i n g o f t h i s consisted  o f t h e 150 s u b j e c t s  t o whom t h e p r e t e s t  82  T a b l e 4.19 Table: Pretest Conservation  Contingency Conservation  Level  Males  Nonconservers P a r t i a l Conservers Conservers Total  was a d m i n i s t e r e d . nonconservers conservers 30  were  were  males and 20  33  chapter However,  Transition The  posttest, test  directly  in  to  additional  tests  contingency  females.  females,  the  The  partial  and t h e c o n s e r v e r s  were  Analyses  the  previous sections o f t h i s  hypotheses  findings  of  hypotheses  which  of  this  seem  study.  to  be  of  section.  6  and  conservation  pretest to retention  following,  74  Conservatipn^Leyels  between  was  and  45  the  are r e p o r t e d i n t h i s  between  transition  and  Hoc p u a l i t a t i v e  reported  some  significance  males  males  78 22 50 150  females.  results relate  76  Sex  Total  45 9 20 74  13 males and 9 f e m a l e s  Post The  Females  33 13 30 76  T h e r e were  L e v e l Versus  independent however,  are  tables  of  of  test  levels  observations  These t a b l e s are i n c l u d e d as T a b l e s  revealed from  and p o s t t e s t  treatments  transition  8  at based  that pretest  to  to  retention  t h e 0.05 l e v e l . on  the  the  The  detailed  among c o n s e r v a t i o n 4.20, 4.21 and 4.22.  levels.  83  1. very  The  r e g r e s s i o n of c o n s e r v e r s  rarely.  Only  two  o f 32  n o n c o n s e r v e r s between t h e 32  conservers  and  retention  regressed  2.  19  .3.  of  of  Even  to  of the  any  partial  and  progressed other  there  levels.  regressed as  back t o  multiplication  but  two  the  44  to the  pretest  conservers  posttest  and  the  throughout  the  a  their i n the  the  those as  of 30  original  (29%)  the  partial  conservation. improvement  groups,  the  control  progress  with  respect  subjects  in  all  s u b j e c t s i n the (6.66%)  pretest level  of  groups  test.  In  two  i n the c o n t r o l and  i n the  other  not  posttest,  retention between  the  were f o u r i n e a c h  test.  volume'  the and  who  regressed  of  conservers  test.  of the  experiment  14  c o n t r o l group r e g r e s s e d  there  higher  p r e t e s t and  general  subjects the  4 of  to  i n the c o n t r o l group d i d  the  retention  treatments  stability  is,  difference in  conservers  of  treatment  between  pretest to retention high  was  That  much  regressed  and  level  t o have u n d e r g o n e a s t e a d y  subjects  posttest  to a higher  among a l l  g r o u p s . The  who  The  the  observable  n o n c o n s e r v e r s and  though  regress  treatment group  between  seem t o be  (33%)  levels  seemed  progressed  from  57  conservation  seem  between  none  n o n c o n s e r v e r s and  progressed  conservation  Two  nonconservers  regressed  p o s t t e s t . None o f  o f c o n s e r v a t i o n . F o r example between t h e  conservers  to  the  to nonconservers  T h e r e does n o t  posttest  group  p r e t e s t and  occurred  test.  progress  levels  (6.25%) c o n s e r v e r s  test. . Similarly,  to  retention  the  regressed  to nonconservers  is  conservation not  level  surprising..  Some  research  84  reported  i n chapter  stability  of  misleading  seem  to  explained  as  show  by  on  conservers  most  in  experience  in  volume  conservers  acquired  i n the  who  were  of  so  and  activities t h a t they  t r e a t m e n t s to  in the  understanding  volume  of  of the of  who  did  s u b j e c t s i n the other  two  Tables  in  have  not  could volume  4.21,  were  and  partial  groups.  The  disturbed  the  applied  volume c o n s e r v a t i o n  the  other  groups  4.20,  regressed  incorrectly  conservation.  resisted  group  experience  could  showed  t o note t h a t  multiplication  c o n t r o l group c o u l d volume  even  control  conservers.  those  and  curious  level  influence  of  the  partial  was  instability  partial  level  i n the  much as t h e  the.  the  that  i t  of subjects  g r o u p s . The  activities 4.22  conservation  However,  level  regress  treatment be  their  cues.  conservation  2 i n d i c a t e d that natural conservers  knowledge  tasks.  have used t h e i r  Those  intuitive  85  T a b l e 4.20 Contingency Table: P r e t e s t - P o s t t e s t T r a n s i t i o n Among C o n s e r v a t i o n L e v e l s by T r e a t m e n t s Pretest  Nonconservers  Volume T r e a t m e n t Nonconservers P a r t i a l conservers Conservers M u l t i p l i c a t i o n Treatment Nonconservers P a r t i a l conservers Conservers C o n t r o l Treatment Nonconservers P a r t i a l conservers Conservers  15 3 1  Post Partial  Conservers  1 1  3 2 9  2 4 2  8 0 13  3  2 2 5  1  8 2 0 15 1 1  Test Conservers  1 0  T a b l e 4.21 Contingency Table: Pretest-Retention Test T r a n s i t i o n Among C o n s e r v a t i o n L e v e l s by T r e a t m e n t s Pretest  Nonconservers  Volume T r e a t m e n t Nonconservers P a r t i a l conservers Conservers l u l t i p l i c a t i o n Treatment Nonconservers P a r t i a l conservers Conservers C o n t r o l Treatment Nonconservers P a r t i a l conservers Conservers  Retention Test P a r t i a l Conservers  Conservers  11 2 0  3 1 2  5 3 9  8 3 0  1 3 1  9 0 14  12 0 0  1 1 0  7 3 6  86  T a b l e 4.22 Contingency Table: P o s t t e s t - R e t e n t i o n Test T r a n s i t i o n ftmong C o n s e r v a t i o n L e v e l s by T r e a t m e n t s t  Posttest  Nonconservers  Volume T r e a t m e n t Non-cqnservers Partial-conservers Conservers  4 2 1  3 1 13  8 2 0  1 4 1  1 2 20  10 2 0  2 0 0  5 2 9  Seasons f o r t h e i r  Students'  Question the  students  were a s k e d  reasons  were  classification response  classification  Question  intended  to provide  scheme. The r e a s o n  or  11 was  first  unclassifiable.  Test  was a p a r t  for  their  validity  student  classified reason  as  coded  unclassifiable  reason The  consistent,  A reason  i f i t was n o t p o s s i b l e t o  as was  understand  g i v e n by t h e s t u d e n t .  c o n s i s t e n t and  classified  according  1.  general  Same:  f o r the  was c l a s s i f i e d  i f i t d i d not c o n t r a d i c t the response.  the  The  information f o r the  g i v e n by e a c h  A  of  guestion the  responses.  consistent as  11  scheme. I n t h i s  to give reasons  on q u e s t i o n  inconsistent,  R e s p o n s e s on  11 o f t h e Volume C o n s e r v a t i o n  conservation  . Conservers  12 0 0  M u l t i p l i c a t i o n Treatment Non-conservers Partial-conservers Conservers C o n t r o l Treatment Non-conservers Partial-conservers Conservers  Betention Test P a r t i a l Conservers  inconsistent  responses  to the f o l l o w i n g nine  use o f the term  were  further  attributes:  "same" w i t h  reference to the  87  2.  two  b a l l s without s p e c i f y i n g  two  b a l l s are s t i l l  Size:  g e n e r a l use  the  t h e a t t r i b u t e . Example:  same."  of the term  "size".  3. Volume: s p e c i f i c  use o f t h e term  "volume".  4.  Amount: s p e c i f i c  use o f t h e t e r m  "amount".  5.  Boom: s p e c i f i c  use o f t h e t e r m  "room".  6. Mass: s p e c i f i c  use o f t h e term  "mass".  7.  Weigbt: would  8.  specific be  use  of  t h e same b e c a u s e  Shape: s p e c i f i c  t h e term  "weight".  i t ' s g o t t h e same  use o f t h e term  the  ball  There none  in  "Eecause was  was the  pretest. be the  posttest  posttest  104  look the  and  none  was  same  based  105  i n the  an  t e s t . A l l of these  support the c o r r e c t  response  the  4. 24  responses i n the  retention  incorrect  from  on T a b l e s 4.23,  responses i n the p r e t e s t ,  of  length".  i n the r e t e n t i o n  eliminated  classifiable  i n the retention  consisted  11  above. "Because  u n c l a s s i f i a b l e response i n the  response  and  inconsistent two  one  Observations  made about  weight.  heated."  only  unclassifiable  they  Example: " I t  "shape".  9. O t h e r : r e f e r e n c e t o a r e a s o n o t h e r t h a n t h e Examples:  "The  pretest,  test.  This  data  of  and 4.25  pretest,  t e s t , . There  can  105 i n  were  ten  ten i n the posttest responses  r e s p o n s e and  except  a r e a s o n which  of " e q u i v a l e n c e "  or  "same  and two  might water  level. " 1.  In  frequent size.  the  pretest,  posttest  reason given f o r a c o r r e c t  The  second  and  third  and  retention  response  was  t e s t the related  most to  most f r e q u e n t r e a s o n s g i v e n f o r a  88  correct  response  reasons the  for  were  a correct  conservation level 2. The r e a s o n s  weight  and  amount  respectively.  r e s p o n s e d i d n o t seem  The  t o be a f f e c t e d  by  of students.  given  by  conservers  t o be more e v e n l y  for  distributed  their  responses  seem  attributes  t h a n t h e r e a s o n s g i v e n by n o n c o n s e r v e r s  correct  among t h e v a r i o u s and  partial  conservers. 3. responses to  The  reason  given  i n the pretest,  s h a p e , room and  most  posttest  frequently  for  and r e t e n t i o n  test  incorrect related  weight.  T a b l e 4.23 Number o f S t u d e n t s w i t h R e s p e c t t o t h e i r Reason f o r t h e i r R e s p o n s e on Q u e s t i o n 11 o f t h e Volume C o n s e r v a t i o n P r e t e s t Reasons 5 6  Response  1  2  3  4  Q u e s t i o n 11 C o r r e c t Nonconservers P a r t i a l conservers Conservers  2 1 0  10 2 12  0 0 4  7 1 5  0 0 0  Q u e s t i o n 11 I n c o r r e c t Nonconservers P a r t i a l conservers Conservers  0 22 0  0 2 0  1i 0 0  4 4 0  42 11  0  7  8  9  0 0 1  7 2 9  1 0 0  0 0 1  0 0 0  5i 0 0  8i 1 0  72 0 0  Size 3. Volume 2» Code f o r t a b l e s 4.21, 4.22 , 4.23:: 1. Same 9. O t h e r 7. Weight 8. .Shape 4. amount 5. Room 6. , Mass i means 1 i n c o n s i s t e n t r e s p o n s e i n c l u d e d means 2 i n c o n s i s t e n t r e s p o n s e s ; i n c l u d e d . 2  89  T a b l e 4.24 Number o f S t u d e n t s w i t h R e s p e c t t o t h e i r R e a s o n f o r t h e i r R e s p o n s e on Q u e s t i o n 11 o f t h e Volume C o n s e r v a t i o n P o s t t e s t Reasons Response  1~ ~2  Q u e s t i o n 11 C o r r e c t Nonconservers P a r t i a l conservers Conservers  2 1 4  Q u e s t i o n 11 I n c o r r e c t Nonconservers P a r t i a l conservers Conservers  0 1* 0  3  T  * means 1 i n c o n s i s t e n t * means 4 i n c o n s i s t e n t  15 2 17 1 * 11  0  1 0 4  4  5  6  1 0 0 0 0 0 3 10 2 0  1* 0 0  0 0 0  0 0 0  1 1 0  7  8 ~~9  5 5  1 2* 0  2* 0 2  7* 1 0  3 1 0  6 2 0  response i n c l u d e d . responses i n c l u d e d .  T a b l e 4.25 Number o f S t u d e n t s w i t h R e s p e c t t o t h e i r R e a s o n f o r t h e i r Response on Q u e s t i o n 11 o f t h e Volume C o n s e r v a t i o n Retention Test 1  Response  2  3  4  Reasons 6 5  7  8  9  6 3 9  0 0 0  1 1 2  4 0 0  5i 0 0  6 3 0  4i 4 0  Q u e s t i o n 11 C o r r e c t Nonconservers P a r t i a l conservers Conservers  1 0 5  8 1 24  0 0 5  ,_..... — f 1 0 0 0 0 0 8 1 2  Q u e s t i o n 11 I n c o r r e c t Nonconservers P a r t i a l conservers Conservers  0 0 0  0 1 0  0 0 0  0 0 0  i  means 1 i n c o n s i s t e n t Most  reason  of  related  subjects,  to  who  size.  including  retention reason  those  test,  conservers  nine  who  related are only  response  included.  answered q u e s t i o n However,  a  conservers  answered to  0 0 0  weight.  considerable  gave a  number  of  i n e a c h o f t h e p r e t e s t and  question If  11 c o r r e c t l y  those  weight c o n s e r v e r s  11  correctly nine  gave  classified  and not volume  a as  conservers  90  there in  this  test only as  i s doubt about study.  (items  the  However, an 4,  5 and  6)  weight c o n s e r v e r s . nonconservers  prevented  some  of of  to  information  Question  of, t h e  12  Question marble  quantities  and  was  responses  easily  reports responses  conservation. are  about the  The  not  This  than  be  reported  in  the  reason  for  by  their  sufficient of  the  Test.  to  concerning  the  the  conservation two of  unequal  the  usual  i n c l u d e d because i t to  g i v e reasons  equality.  12  given  Test  typical  about  guestion  reasons  their  Responses  involved  able  for  observations tc  have  validity  of  guestion  might  were  Test  part  therefore,  who  could  Volume C o n s e r v a t i o n  a  used  were c l a s s i f i e d  seem t o p r o v i d e  p r o t o c o l . I t was  students  those  students  of C o n s e r v a t i o n ^ a n d  not,  that more  i n the  by  test  conservation  factors  Volume C o n s e r v a t i o n  scheme.  believed  between  not  do  was  questioning  paragraph  11  the  Piagetian  inequality  given  Volume C o n s e r v a t i o n  boxes  classification  to d e t e c t  Language  reasons  Levels  12 o f  the  s u b j e c t s from e x p r e s s i n g  scheme used  between  of  weight c o n s e r v e r s  f o r conclusive evidence  Consistency  conservation  part  designed  These  guestion  classification  two  was  those  of the  earlier  volume.  more a p p r o p i a t e l y . The responses  validity  the  the  levels  students  following  for  following  consistency  and the  The  was  section  found of  for  their  of  this  chapter. The or  numbers o f s t u d e n t s  incorrectly  are  who  represented  answered g u e s t i o n i n Table  4.26  12  correctly  f o r a l l students  91  who took there  that p a r t i c u l a r was  a  conservation For  the  that  positive  Chi  C h i Square  there  between  was  the  responses  On  pretest  Square  not  a  responses  was 7.82  significantly  of  conservation  question  12  in  and  between  the  on  posttest level  question  of 12.  (p < 0.05) and f o r t h e  was 15.59 (p < 0 . 0 0 1 ) . . I t  level  on  the  relationship  o f s u b j e c t s and t h e i r  pretest  posttest  test.  was  positive of  surprising relationship  subjects  and  the r e t e n t i o n t e s t  their  a t t h e 0.05  level. There  was a g e n e r a l  question and  12  among a l l c o n s e r v a t i o n  (64%) p a r t i a l  answered  guestion  nonconservers, (73%)  (15%  followed number  conservers  conservers  and 35  of  In the r e t e n t i o n  answered  i t correctly.  a l l conservation  followed  nonconservers  test.  conservers  (58%) p a r t i a l  who  The  test,  conservers  who answered  groups  by p a r t i a l c o n s e r v e r s  answered  percents  the  other  question  i n the pretest, of  i t correctly  48  was in  seem t o have i m p r o v e d t h e  ( 5 % ) . On  seems t o have i n c r e a s e d s t e a d i l y retention  (5 8%)  I n t h e p o s t t e s t 32 o f 77 (42%)  12. The c o n s e r v e r s  improvement)  on  nonconservers,  14 o f 24  among  by t h e n o n c o n s e r v e r s of  answers  between t h e p r e t e s t and p o s t t e s t t h e r e  improvement  answering q u e s t i o n  (37%)  and 29 o f 50  i t correctly.  (65%) c o n s e r v e r s  correct  g r o u p s between t h e p r e t e s t  (76%) p a r t i a l  answered  appears t h a t  general  most  12 c o r r e c t l y .  (51%) n o n c o n s e r v e r s ,  60 o f 93 It  conservers  16 o f 21  conservers  o f 53  and  a  of  t h e p o s t t e s t . . In t h e p r e t e s t 29 o f 78  14 o f 22  27  improvement  conservers  seem t o have  (12%)  hand  the  12 c o r r e c t l y posttest and  and  partial  increased  in  92  the  posttest  and  then  decreased  increased  percents of c o r r e c t  followed  by  the tendency  decreased of  i n the r e t e n t i o n  responses  of  these  percents could possibly  regression  toward  the  be  mean  test.  two  groups  attributed in  The  to  successive  observations.  T a b l e 4.26 C o n t i n g e n c y T a b l e : C o r r e c t o r I n c o r r e c t Response on Item 12 o f t h e Volume C o n s e r v a t i o n P r e t e s t , P o s t t e s t and R e t e n t i o n T e s t V e r s u s C o n s e r v a t i o n L e v e l s Incorrect  Correct  Test Pretest Nonconservers P a r t i a l conservers Conservers  29 14 29  49 8 21  Posttest Nonconservers P a r t i a l conservers Conservers  32 16 35  45 5 13  Retention Test Nonconservers P a r t i a l conservers Conservers  27 14 60  26 10 33  P r e t e s t : N = 150, C h i s q u a r e = 7.82, d f = 2, p = 0.0201 P o s t t e s t : N = 146, C h i s q u a r e = 15.59, d f = 2, p = 0.0004 R e t e n t i o n t e s t : N = 170, C h i s q u a r e = 2.59, d f = 2, p = 0.2735  S t u d e n t s ' R e a s o n s f o r t h e i r R e s p o n s e s on All on  the  question  r e a s o n s g i v e n by 12  inconsistent,  were  first  unclassifiable,  few  c a s e s o f no  in  the  two  unciassifiable  posttest  response. and  one  the  There i n the  responses  Question  students f o r t h e i r classified  o r no  as  response.  were none i n t h e retention in  12  the  test.  pretest,  judgement  consistent,  There  were v e r y  pretest, There none  one  were o n l y in  the  93  posttest 4.27,  and  4.28  three  and  4.29  excluding  the  no  4.27,  and  4.29  4.28  pretest,  104  retention  test.  all  reasons  response  and  unclassifiable  contain  103  responses of the  and  given  classifiable  posttest  and  inconsistent  to the f o l l o w i n g  ten  Weight:  specific  "Because  i t i s t h e same  use  of  the  term  the  of  the  use  o f t h e term  "size".  4.  Boom: s p e c i f i c  use  o f the term  "room".  5.  Shape: s p e c i f i c  use  o f t h e term  "shape".  6.  Space:  use  of t h e term  "space".  specific  Same:  use o f t h e t e r m  general  specifying  any  reference ether  8. C l o s e d c o n t a i n e r : closed  lower  because  Open c o n t a i n e r : one  n o t go  a l i d and  "the  term  in it. water  reference to the fact  o f the boxes because  Air inside:  to  "same"  without  attribute.  will  i t has  Example:  "amount".  r e f e r e n c e t o the f a c t  water  classified  weight."  specific  the  responses  "weight".  Size:  10.  of  attributes:  3.  in  students  cases,. T a b l e s  r e a s o n s were  Amount: s p e c i f i c  9.  by  responses  101  2.  is  data of T a b l e s  summarize  consistent  according  7..  The  test.  The  1.  i n the r e t e n t i o n  i t was  reference to the f a c t  that  when  the  box  Example: " I t w i l l  be  won't go i n . " that  the  water  went  open.  t h a t the c l o s e d  box  keeps  a i r inside i t .  0. O t h e r :  r e f e r e n c e to a reason other than the  above.  Examples:  "Because  i t i s b i g g e r " . "Because more p r e s s u r e  will  the  than the  be on  water  open  box."  94  There  were f o u r  five  in  these  responses  and  a  the  posttest  reason  inequality  except  two  which  might  pretest,  those  who  that  water can  The  not  the c l o s e d  weight,  the  support  the  retention an  pretest,  test, i l l  incorrect  the  and  correct  retention  go i n t o t h e c l o s e d  of  response  response  given reason  was  box,  that  test  most  r e l a t e d to the  there i s a i r i n s i d e  pretest,  g a v e an amount  most f r e g u e n t l y 3.  of  in  of  that  the  the  facts  water  closed  water c a n n o t  of  go  went box. into  box.  In  who  consisted  posttest  or that  most f r e q u e n t l y  those  s i x i n the  answered c o r r e c t l y gave a r e a s o n  t h e open box,  2.  and  responses  o r d i f f e r e n t water l e v e l s .  1. I n t h e  in  inconsistent  There  posttest  i n c o r r e c t response or  s i z e . The  reason  and  retention  gave a r e a s o n  t e s t most o f related  r e l a t e d t o weight  to  was  the  difference  in  given.  does n o t  the  frequency  for  their correct  seem t o be  of reasons  g i v e n by  or i n c o r r e c t  any  observable  the t h r e e c o n s e r v a t i o n groups  responses.  95  T a b l e 4.27 Number o f S t u d e n t s w i t h B e s p e c t t o t h e i r S e a s o n f o r t h e i r Response on Q u e s t i o n 12 o f t h e Volume C o n s e r v a t i o n P r e t e s t Reasons 6 7  1  2  Q u e s t i o n 12 C o r r e c t Nonconservers P a r t i a l conservers Conservers  5 1 1  0 0 0  0 0 0  0 0 1  0 0 0  1 0 1  0 0 0  0 0 0  2 1 2  ' 1 0 0  0 0 0  0 0 0  4 1 0  Q u e s t i o n 12 I n c o r r e c t 142 Nonconservers 1 P a r t i a l conservers Conservers 7  3  : 4  Response  5  8  10  ,„,,,• 0  1 3 3  3 1 3  1 0 1  1 1 0  0 0 1  2 0 0  9 11 6 9 91 2 2i  Code f o r t a b l e s 4.24, 4.25, 4.26: 1. Weight 2. Amount 3. S i z e 4._Room 5. Shape 6. S p a c e 7. Same 8. Water d o e s n o t go i n , box c l o s e d , e t c . .9. Water went i n , box o p e n , e t c . .10. A i r i n s i d e 0. O t h e r * means 1 i n c o n s i s t e n t r e s p o n s e included. means 2 i n c o n s i s t e n t r e s p o n s e s i n c l u d e d . 2  T a b l e 4.28 Number o f S t u d e n t s w i t h R e s p e c t t o t h e i r Reason f o r t h e i r Response on Q u e s t i o n 12 o f t h e Volume C o n s e r v a t i o n P o s t t e s t Reasons 7 6  Response  1  2  3  4  5  Q u e s t i o n 12 C o r r e c t Nonconservers P a r t i a l conservers Conservers  0 1 1  0 0 0  0 0 0  0 0 5  0 0 0  0 1 3  0 0 0  1 0 2  2 2 6  0 0 0  0 0 1  0 0 0  2 1 2  Q u e s t i o n 12 i n c o r r e c t 62 Nonconservers P a r t i a l conservers 2 1 Conservers 1 means 1 i n c o n s i s t e n t 2 means 2 i n c o n s i s t e n t  response included. responses included.  8  9  10  0  17 5 12  5 1 4  2 1 3  2 0 0  2i 0 0  0 0 0  5i 0 2  1 1i 2  96  T a b l e 4.29 Number o f S t u d e n t s w i t h R e s p e c t t o t h e i r R e a s o n f o r t h e i r R e s p o n s e on Q u e s t i o n 12 o f t h e Volume C o n s e r v a t i o n R e t e n t i o n Test Eesponse  1  2  Q u e s t i o n 12 C o r r e c t Nonconservers P a r t i a l conservers Conservers  0 0  2 0 0  Q u e s t i o n 12 I n c o r r e c t Nonconservers 7 P a r t i a l conservers 0 Conservers 3  0 0 3  3  4  0 0  0 0  . Reasons ~5 6 7  0 0  0  1  8  0 1  7 0  0  1  0  2  0  1 0 5  0 0 0  0 1 1  0 0 1  1» 1 5i  5 10 5 0 1  9  10  0  6 0 10*  2 2 4  0 1 6*  1 0 0  0 0 1  2 2 0  2  1 means 1 i n c o n s i s t e n t r e s p o n s e i n c l u d e d . means 2 i n c o n s i s t e n t r e s p o n s e s i n c l u d e d .  2  Most s t u d e n t s who give of  an  explicit  reason  t h e c o n t a i n e r s was  go i n s i d e  it.  answered  correctly  of 10;  63  (79%)  i n the  In the  retention  test  previously  noted t h a t  appears  correctly  gave an  that to  reason  i t give  volumes i n q u e s t i o n  correctly  was  related  ( c l o s e d ) and 40  gave r e a s o n  gave  It  open  s u b j e c t s who  11  which  pretest  correctly  question  responded  of 8,  answered 46 8,  9 or  52  9 or  (77%) 10.  most o f t h o s e  On  who  more  easier  for  explicit  12 t h a n a b o u t  would  (77%)  could  that  one  (would  not)  subjects  who  10; i n t h e p o s t t e s t gave r e a s o n  s u b j e c t s who  8,  responded was  answered  reasons about  equality  correctly  related to  students  who  was to  size.  responded  inequality  i n question  50  9 or  t h e o t h e r hand, i t  imprecise reason that was  t o the f a c t  water  correctly  o f 66  t o q u e s t i o n 12  11.  of  97  CHAPTEfi V  SUMMARY, CONCLUSIONS AND This chapter findings  and  implications future  contains  the  a brief  review  conclusions,  f o reducational  RECOMMENDATIONS of t h e  limitations  practice,  problem,  of  the  the  study,  and r e c o m m e n d a t i o n s  for  research.  Review o f t h e P r o b l e m The between  purpose o f t h i s the  which s i x t h cuboid apparent  level  study  of c o n s e r v a t i o n  grade students  "V = L x W x H". discrepancy  i s t o determine t h e r e l a t i o n s h i p  learn The  o f volume and t h e d e g r e e t o  the  volume  problem  i s  a c o n s e g u e n c e o f an  the time t o i n t r o d u c e  of Piaget concerning  the  volume a l g o r i t h m  of a  used  introduce the algorithm (Dilley  a  school  c o g n i t i v e theory  widely  of  between t h e p r e s e n t  the  Two  algorithm  programs - and  cuboid.  textbook  series  in  "V = L x W x H" i n  British  grade  5  Columbia (age 10)  e t a l . , 1974 and E i c h o l z e t a l . , 1974). A n o t h e r  series  i n t r o d u c e s t h e a l g o r i t h m f o r m a l l y i n g r a d e 4 (age- 9) ( E l l i o t e t al.,  1974) and u s e s i t i n f o r m a l l y i n g r a d e 3 (age 8 ) . Most p r o p o n e n t s o f P i a g e t ' s  theory  would  disagree  with  98  such  early  introduction  children  do n o t  learning  i t before  example,  holds  operational) area or  develop  necessary  a l g o r i t h m and  the necessary  grade 6 that  (age  " i t  that children  volume s i m p l y  Piaget  of the  by  Piaget  not  until  understand  how  any  abilities  for  (1960) h i m s e l f , f o r stage  they  IV  can  meaningful  (formal  arrive  m u l t i p l y i n g boundary edges"  (1960) c o n s i d e r s t h e c o n c e p t for  most  cognitive  11).  is  claim that  (p.  of c o n s e r v a t i o n  computation  in  both  at  an  408). to  be  area  and  volume: ... C h i l d r e n a t t a i n a c e r t a i n k i n d o f c o n s e r v a t i o n o f a r e a £ a n d v o l u m e ] , b a s e d on t h e p r i m i t i v e conception of area (and volume) as t h a t which i s bounded by l i n e s (or f a c e s ) . That understanding comes long before the ability to calculate areas and volumes by mathematical multiplication, involving relations between units of d i f f e r e n t powers ... ( P i a g e t , 1960, p. 355) On  the other  formal  hand, many e d u c a t o r s  scientific  may  be  specific  i n s t r u c t i o n a l experiences  and  general  maturation  (1960)" 223),  (Siegler for  necessary further and order  3) to  reasoning  believe that "acquisition  than  and  1976,  considered  p.  to  suggest  can  learn  how  c a l c u l a t e the  T h e r e seems t o be programs and  the t h e o r y  t o use  and  the  dependent dependent  Inhelder  368).  Lovell  t h a t even s e v e n  less  by  education  f o r volume c o n s e r v a t i o n .  more  far  hypothesized  Atlas,  example,  far  Graves and  e i g h t year  algorithm  and  p.  on  (1972,  179)  olds  on  Piaget  experience  (197 1,  of  p.  to  be  went  (grade  "V = L x W x H"  2 in  volume. a discrepancy of P i a g e t .  i n t r o d u c e the  volume a l g o r i t h m  This p o s i t i o n  seems t o be  of  between t h e  Some o f t h e a cuboid  b a c k e d by  some  as  present  present early  educators  school  programs  as g r a d e who  3.  claim  99  that  scientific  instruction argue  than  that  calculation  necessarily  have  such  of the  all  that  the  1962;  to  justify  as DeVault assert  training  and  h i s p r o p o n e n t s seem  to  volume i s a p r e r e q u i s i t e f o r  any-  volume.  our  .who  that  has  advocates the  p.  volume..  conserve  majority  of  1971;  been acknowledged " i t  seems  s t u d i e s most l i k e l y be  Studies conserve  Graves,  1972),.  a need f o r r e s e a r c h  school curriculum  that  algorithm  a d u l t s do n o t  Wheatley,  not  to  by  or suggest such  experimental  its  educators  reasonable produce  in  ...  to  useful results  studies  (DeVault,  639)."  The  aims o f t h i s  restated  related  ought  students  present  T h i s need  one volume  T o w l e r and  c u r r i c u l u m work would  in  this  t o each of the  Aim  1..  conservers, learn  However,  i n t r o d u c t i o n o f the  F i n d i n g s and  be  d e p e n d e n t on  a p r e d i c a m e n t t h e r e seems t o be  modification.  1966,  acre  P i a g e t and  of  until  (Elkind,  order  for  delay  indicated  volume In  to  = L x W x H"  is  maturation.  conservation  meaningful  "V  reasoning  the  To  aims w i l l  results  superiority  the  conservers,  volume a l g o r i t h m The  which  were s t a t e d i n C h a p t e r  1  will  s e c t i o n . A l s o a summary o f t h e f i n d i n g s  determine  partial  Findings. (p < 0.05)  study  Conclusions  be  reported.  various  degrees  and  nonconservers  of a c u b o i d of the  o f the  "V  = L x W x  p o s t t e s t showed a  conservers  group o v e r  to of  which volume  H". significant the  partial  100  conservers  group  in  the  significant  difference  was  groups  t h e 0.05  at  found i n  Volume  conservation Aim of for  2.  t h e two  Volume  l e v e l . There  Achievement  was  no  significant  teaching  Retention  Test  scores  between  level.  methods on  learning  the  f o r each  volume  algorithm  a cuboid. The  r e s u l t s showed  a significant superiority  volume t r e a t m e n t g r o u p o v e r m u l t i p l i c a t i o n 0.01)  and  over c o n t r o l  Achievement between group  t r e a t m e n t group  Posttest.  No  multiplication at  the  retention  test  treatment  group  over  control  0.05  was  a  treatment  f o u n d between  of  conservation  a  (p < 0.01);  multiplication  a t t h e 0.05  on  was  found  treatment of  the  of volume (p <  no  0.01)  significant  t r e a t m e n t group  and  level.  effect  of  learning  the  from  one  volume  the  transition  The  transition  from  a lower to a higher l e v e l of  between t h e p r e t e s t  and  i n d e p e n d e n t o f volume a c h i e v e m e n t  Similarly,  Volume  results  superiority  (p <  volume  l e v e l t e another.  Findings. conservation  cuboid  the  of  the  control  treatment group  group  3. To d e t e r m i n e t h e  algorithm  and  Similarly, significant  on  difference  group  over m u l t i p l i c a t i o n  t r e a t m e n t group  Aim  treatment  t r e a t m e n t group  (p < 0.01)  significant  level.  showed  control  difference  be  No  difference  determine the degree o f e f f e c t i v e n e s s  Findings.  and  Posttest.  f o u n d between any o t h e r c o n s e r v a t i o n  g r o u p s a t t h e 0.05 To  Achievement  the postest  was  found  s c o r e s a t t h e 0.05  t h e t r a n s i t i o n from a l o w e r t o  a  higher  to  level.  level  of  101  conservation  between  the  pretest  and t h e r e t e n t i o n t e s t  found  t o be i n d e p e n d e n t  c f volume  0.05  level.  the t r a n s i t i o n  level  of c o n s e r v a t i o n  achievement  Likewise,  scores  was  found  between  achievement  to  from a h i g h e r  be  the  another  was  found  scores  there  was  conservation ftim,4. levels  To  initial  a total  level  were  were  improvement  of  conservation b e t t e r than  To  Findings. be  Similarly, not  found aim  level  to  achievement  the  subjects*  33  revealed  that  o f m a l e s was f o u n d  t o be  females..  Out  76 were males and 74 were f e m a l e s .  The  males  and  45  females,  the  partial were  females. determine t h e r e l a t i o n s h i p  The Volume  (p  achievement  significantly  between sex and t h e  for a  cuboid.  achievement P o s t t e s t s c o r e s  significantly  To  Pretest  13 m a l e s and 9 f e m a l e s and t h e c o n s e r v e r s  t h e Volume  6.  in  that of the  d e g r e e o f l e a r n i n g t h e volume a l g o r i t h m  to  o f volume  level.  o f volume.  (p < 0.01)  30 m a l e s and 20  found  a t t h e 0.05  f r o m one c o n s e r v a t i o n  The Volume C o n s e r v a t i o n  nonconservers  Rim,5.  volume  d e t e r m i n e t h e r e l a t i o n s h i p between sex and t h e  150 s t u d e n t s ,  conservers  of  p r e t e s t and t h e p o s t t e s t and  independent  general  of conservation  significantly of  a  be  the  levels.  Findings. the  to  at  t o a lower  independent  between t h e p r e t e s t and t h e r e t e n t i o n t e s t Even t h o u g h t h e t r a n s i t i o n  scores  was  <  0.05)  Eetention  c o r r e l a t e d with  were  correlated Test  determine the r e l a t i o n s h i p  t o sex.  scores  s e x a t t h e 0.05  not  were  level.  between m a t h e m a t i c s  1  102  a c h i e v e m e n t and t h e l e v e l s Findings. be  The i n i t i a l  independent  pretest  of  a s measured  Aim  of c o n s e r v a t i o n  the  level  o f volume.  of conservation  was  mathematics achievement  by t h e c o m p u t a t i o n  scores  to  of the  s e c t i o n o f SAT.  7. To d e t e r m i n e t h e r e l a t i o n s h i p  between  a c h i e v e m e n t and t h e d e g r e e o f l e a r n i n g t h e volume a  found  mathematics algorithm f o r  cuboid. Findings.  found the  Volume  t o be s i g n i f i c a n t l y mathematics  Similarly, to  The  be  (r  Importance to  result than The  of Conservation  learn  i n this  conservers  nonconservers not  the  on  to  measured by SAT.  scores  the  volume  connection  were  degree  found  Volume  an  to  which  sixth  only  grade  significant  scored  higher  on t h e Volume A c h i e v e m e n t P o s t t e s t .  score  significantly  was  between t h e l e v e l o f  a l g o r i t h m . ; The  significantly  on t h e p o s t t e s t ; p a r t i a l  the  study  was t h a t t h e c o n s e r v e r s  conservers  d i d not  This  relationship  lower, than  p o s t t e s t . On t h e r e t e n t i o n t e s t , groups  correlated  p < 0.001) c o r r e l a t e d t o t h e  Levels.  o f volume and t h e  the p a r t i a l  though  =0.37,  Test  were  and D i s c u s s i o n  determine  conservation students  scores  scores  scores.  Summary o f C o n c l u s i o n s  attempt  pretest  Volume A c h i e v e m e n t R e t e n t i o n  o f SAT  Posttest  (r = 0.35, p < 0.001)  achievement  significantly  pretest  Achievement  higher  conservers  than  scored  the  lower,  t h e n o n c o n s e r v e r s on t h e  the scores of the conservation  Achievement  Test  did  not  differ  103  significantly. group s c o r e d  Furthermore*  65%  on  Volume A c h i e v e m e n t be  able  to  apply and  conservation  level.  So are  far  this  factor  who  levels,  f o r a cuboid, regardless  have r e a c h e d  learning 5.  the  If this the  unreasonable.  so,  then  algorithm  So  as t h e  long  the  present  there  may  by  students  in  also i n f l u e n c e the  those  learnability  will  Effect  be  discussed  of Treatments. did  of  the  later  and  Volume  Achievement  subjects  who  were i n e a c h o f t h e  level  not  an  no  data  in to be  grades  4  programs w h i c h  do  not  be  reasonability  is  support should of  volume the  may  the i d e a  not  who  place  volume. However, which  would The  time t o i n t r o d u c e  the  chapter. were  the  Retention other  take  that  algorithm.,  in this  b e t t e r , on  Posttest  their  likewise  volume c o n s e r v a t i o n  Subjects  significantly  has  grades  for  does not  for further research, regarding  treatment  school  volume a l g o r i t h m  f a c t o r s ether than  the  as d e f i n e d  i n , say,  l e a r n e r s have become c o n s e r v e r s  be  algorithm,  study  of  i s  to  at  grade  level  might  criterion  study  and  that influences successful  present  volume  i n t r o d u c t i o n of the the  the  this  level  factor  volume a l g o r i t h m be  learnability,  before  as a  6th  volume a l g o r i t h m  conservation  unimportant  of the  present  i n l e a r n i n g the  that  treatment  C h i l d r e n i n g r a d e 6 seem  I t i s p o s s i b l e , although  i t ,  relatively  need  Test.  volume a l g o r i t h m  students  volume  Volume A c h i e v e m e n t P o s t t e s t  comprehension  as  study.  support  the  the  of the  concerned, i t appears, t h a t c o n s e r v a t i o n  important  and  each of the Retention  computation  students  in  the  volume  Volume  Achievement  Test,  than  treatments;  the  those  subjects  104  in  the  other  two  treatments d i d not  volume p e r f o r m a n c e . The treatment  did  Achievement treatment volume did  differ  posttest  which was In the  those  who  significantly.  level  i s , the  better,  than ' those  and  the  on  who  were  This  in  volume  were  indicated  in  that was  the  treatment at  the  successful.  multiplication  material  taught.  short,  subjects  i t may who  be c o n c l u d e d  were i n t h e  that,  at the p o s t t e s t  multiplication  m a t e r i a l b u t t h e s u b j e c t s who  volume  did significantly  treatment  the m u l t i p l i c a t i o n  treatment,  were  b e t t e r than  on  the  level,  treatment l e a r n e d  the m u l t i p l i c a t i o n  in  the  control  treatment  l e a r n e d the  Multiplication  who  were i n the  in  multiplication  the  those  multiplication  s t u d e n t s had  significantly  were i n t h e  the c o n t r o l treatment;  treatment  not  That  significantly  Posttest,  and  s u b j e c t s who  differ  those  Volume  in who  the were  Achievement  Posttest. The  conclusions  volume t r e a t m e n t teaching  sixth  cuboids  of cubes; for  this  factors  on  included  building  multiplication  activities  them w i t h  method l a t e r  computing  treatment  is  by  i s better than the  g r a d e r s t h e volume a l g o r i t h m o f  volume t r e a t m e n t of  m e n t i o n e d above seem t o s u g g e s t  the  consisted their  used  volume mainly  of of  p r o d u c t and  constant; t h i s  t a s k was  cuboid.  c o u n t i n g the  the algorithm cuboid.  studying the varying factors  s u p p l e m e n t e d by  t o t h e volume a l g o r i t h m "V = L x W x  treatment  f o r d e t e r m i n i n g the  c u b e s and  a  a  that  H.  H  The  "V  the in The  volume number  = L x W x  H"  multiplication  effect  of  when t h e i r a brief  varying product  application  105  The  r e s u l t s of  conjecture in  made  varying  the  is  one  to  thing  to  or  multiply  not,  therefore,  area  or  of  three  lengths  a volume ...  predict  cuboids. claim  numbers t o g e t h e r  the  are p r o f i c i e n t  rapidly  support Piaget's  or  support  s t u d e n t s who  dimensions  two  lengths  i s an  and  and  (Piaget  and  On  the  that  " i t  quite  another  understand  that  et a l . ,  1960,  408)." Transition  study revealed students*  lower to  between C o n s e r v a t i o n that  there  conservation  achievement  scores  a higher  each of the of  volumes  two  product  do  a f i x e d p r o d u c t can  r e s u l t s seem t o  multiply  their p.  the  study  i n Chapter 2 t h a t  f a c t o r s of  determine contrary,  this  was,  or t h e i r  posttest  and  appears  conservation treatments could  have  level and  was  been  (sensitization)  the  during  were "on from  the  one  doorstep" stage  Uncontrollable Volume  which s u b j e c t s  students'  Conservation  by  Test  found  of  scores.  a  and  independent  other  Possible test  lasted  than  factors  influence  could  (peer  about  have to  who  developed the  influence) classroom  to two  e s p e c i a l l y those  development  the  subjects'  Growth i s s u s p e c t e d  i n g r a d e 6,  outside  volume  pretest  the  some f a c t o r ( s )  experiment  discussions  their  the  t r a n s i t i o n from  influence,  cognitive  the  treatments.  of c o n s e r v a t i o n ,  of  The  improvement  peer  the  of  between t h e  'Hawthorne e f f e c t ' .  have been a f a c t o r b e c a u s e months  of  of  improvement o f  t e s t was  achievement  growth, and  and  influenced  volume  an  treatments.  retention  that  Results  regardless  of c o n s e r v a t i o n  volume a c h i e v e m e n t s c o r e s It  generally,  levels  level  Levels.  next. of  the  could  have  106  influenced  the r e s u l t s  since  these  itself  could  about  conservation  later  tests have  tests.  were  tasks  level  to  to the f a c t  (Hawthorne  effect)  and  the  test  mathematics of  The  in  own  errors in  o f the s t u d e n t s from  could  have  been  one  partially  t h a t t h e y were c h o s e n f o r t h e e x p e r i m e n t  were  correlated  Re s t i l t s  of  -of  to  feeling  the  mathematics  o f volume  was  achievement s c o r e s  study and  the  achievement  by t h e c o m p u t a t i o n s e c t i o n o f SAT.  The  initial  f o u n d t p be i n d e p e n d e n t o f  measured  by t h e  computation  SAT. above-mentioned  competency  their  test  seriously  a c h i e v e m e n t s c o r e s on t h e p o s t t e s t  of conservation  section  next  think  and c o n s e q u e n t l y t o t h e i n f l u e n c e  volume  s c o r e s measured level  to correct  of_Hathematics- Achievement.  that  retention  the  .test  t h e p r e t e s t . The  some s u b j e c t s t o and  retention  worthy.  Effect revealed  and t h e to  the development  attributed  special  identical  influenced  Finally,  conservation  of the p o s t t e s t  results  seem  to  i n m a t h e m a t i c s c o m p u t a t i o n may  volume  achievement  mathematics  or  achievement  vice  score  indicate  versa. and  suggest  volume  a  that  a  competency  Furthermore,  the  conservation  level  seem t o be i n d e p e n d e n t . Effect the  volume  o f Sex. algorithm  conservation volume test hand,  the  was  of  o f se^c on t h e d e g r e e o f  o f a c u b o i d and on t h e  o f volume  algorithm  levels  The e f f e c t  was a  examined. cuboid,  The  level  of  degree o f l e a r n i n g  the  a t the p o s t t e s t  n o t f o u n d t o be r e l a t e d  males  initial  were f o u n d t o have  learning  t o sex.  and On  a significantly  retention the (p <  other 0.01)  107  higher  initial  The  level  of c o n s e r v a t i o n t h a n  above-mentioned  volume c o n s e r v a t i o n has such  as  (1971).  Graves The  initial active  Grave,  (1972),  of  a l s o r e p o r t e d by  Elkind  of the  of  manipulative  females.  males o v e r  males  to  males  skills.  in  females  other  (1961-b and  1962)  the  of c o n s e r v a t i o n c o u l d be  participation  involving  been  superiority  level  superiority  the  researchers and  Wheatley  females  attributed practical  in  the  t o the  more  experiences  ( P r i c e - W i l l i a m s e t a l . , 1969  There  of the  Study  are  several limitations  of the  limitations  are  related  of s u b j e c t s chosen f o r  experiment  while  other  procedures  of the  study.  The  area.  the  this  consisted  missed  any  was  study  test  32,  to  study On  grade  limited  the  other  6  of day  105  and  upon t h e to t h i s  who  171  these the  conseguences of  grade  students slightly  students  a  higher  area.  The those  were e l i m i n a t e d , t h e  final  only  but  of  the  when  students.  hand, t h e  students  Some o f  greater metropolitan  conservers  is  study.  f a m i l y income was  or t r e a t m e n t  reduced  are  were s i x t h  originally  g e n e r a l i z a t i o n based  students. were  limitations  T h e i r average  nonconservers, Any  type  a v e r a g e income o f t h e  subjects  sample  t o the  s u b j e c t s of t h i s  suburban  who  and  1972).  Limitations  than  in  There  16 p a r t i a l  results  and  followed  a  57  conservers.  conclusions  or to a s i m i l a r s u b j e c t s used  were  in  population this  of of  study  mathematics program  108  typical  of those  socioeconomic  used  i n North  status  jsimilar  s t u d e n t s i n North America. be  r e p r e s e n t a t i v e of the  grade  6 s t u d e n t s and  America, to  They were o f  those  of  suburban  T h e s e s u b j e c t s seem,  could  be  and  grade  therefore,  p o p u l a t i o n o f suburban  findings  ages  North  6 to  American  generalized  to  this  larger population. The  limitations  constraints the  due  tests.  treatment, active  of  the  from  usual skills  "V = L x W x H." Volume  involved  Furthermore,  Achievement  Multiplication  The  conservation  levels  conservation  test.  by  Test,  approaches  in  learning  o n l y one  relating  more  multiplication  version  Volume  classification was  required  i n the  involve  t r e a t m e n t s , the  i n elementary  A c h i e v e m e n t T e s t was  the experiment.  limited  school  the  procedures  constraints  and  than i s normal  other experimental treatment,  multiplication  the  experimental  more c o m p r e h e n s i v e  involvement  different  to  t o t h e t r e a t m e n t s and  One was  relating  of  students' The  treatment,  was  i t s emphasis  on  o f the a l g o r i t h m o f each  at d i f f e r e n t subjects  of  the  Test  and  stages  into  a c h i e v e d u s i n g a judgement-based  The  the t r e a t m e n t s  generalizations and  test  of  instruments  volume  classrooms.  Conservation  used  to  this used.  of  volume volume  study  are  109  Implications for Educational Implication adjusted  mean s c o r e  Posttest  and  indicate  that  volume  1. The of  students  algorithm  their  levels  Since  conservation the  "V  The  This  are  the  indicate  successful.  cubes;  computing  activity  volume  future  2.  by  volume  to teach  oriented  volume  of the  method.  a  The be  the  volume  affected  6  and  "V  are  school  volume a l g o r i t h m  cuboid school  of  Piaget.  ways  for  of t h i s  study  treatment  was  counting  seems, of  delay  programs.  determining  It  proven  a  theory  = L x W x H"  cuboid,.  not  possible  volume  an  using  suggest the  conclusions  c u b e s and  be  prevalent  to the  b a s e d on  to  programs t h a t  for  that the  of the  seem  algorithm,  therefore,  justification  algorithm  learning  school  grade  activity-oriented  the  not  outlines  b u i l d i n g them w i t h  later  did  respect  results  T h i s seems t o  seem t o  algorithm  research  The  the  with  not  present  t o say  T h i s t r e a t m e n t was  the  appropriate  the  justified  do  to  does n o t ,  of  an  volume.  the  prior  Test.  had  Achievement  = L x W x H."  the  study  m a t t e r of  that  of c u b o i d s  Volume  capable  "V  of  learning  T h i s i s not  Implication  of  in  algorithm  s e c t i o n on  pursuing  volume t r e a t m e n t the  level  a criterion,  x W x H."  practices  cuboid  conservation  introducing = L  a  of c o n s e r v a t i o n  as  unreasonable. of  both  on  o f such s t u d e n t s  factor  introduce  65%  i n grade 6 are  for  the  important  of the  Volume A c h i e v e m e n t E e t e n t i o n  achievement s c o r e s by  students  Practice  the the  was  volume number  used  for  therefore,  a cuboid  using  an  110  Implication were  The  mathematics computation s c o r e s  f o u n d t o be p o s i t i v e l y  scores.  This  correlated  seems t o s u g g e s t  computation vice  3,  may  indicate  that  SAT  w i t h volume a c h i e v e m e n t  a competency  a competency  of  in  i n volume  mathematics  achievenent or  versa. Implication  learning  the  4. F e m a l e s seem t o be as c a p a b l e a s  volume  algorithm  However, t h e s u p e r i o r i t y initial active  level  of  manipulative  the t e a c h i n g acquisition  of the males to  of c o n s e r v a t i o n c o u l d  participation  involving  of  volume  o f volume  males  purpose  concepts  by  the  t o which s t u d e n t s l e a r n  "V = L x W x H."  conclusions  and i m p l i c a t i o n s  needed on t h i s It larger students  On  the  experiences  be  programs i n  beneficial  to  the  determine  the  females.  was  to  of c o n s e r v a t i o n  the  of  t h e volume  basis  volume  algorithm  of  the  and for a  findings,  of the study, f u r t h e r research i s  topic.  i s recommended t h a t t h e e x p e r i m e n t be s a m p l e . , The only  in  practical  study  between  cuboid  females  t o t h e more  may  relationship deqree  H."  f o r Future Besearch  this  the l e v e l  the  = L x W x  Activity-oriented  conservation  of  "V  in  be a t t r i b u t e d  in  skills.  Recommendations The  f o r a cuboid  males  sample  16 o f which were  of  this partial  sample m i g h t i n f l u e n c e t h e r e s u l t s  found  study  replicated consisted  conservers. in this  A  study.  on of  a 105  larger  111  It on  is  also  subjects  could or  level by  the  and  high  than  volume  have developed l e a r n i n g  conservation choice  and  of  a  more A  six.  grade  The  habits  that  "on  and  be  replicated  independence could  have  of been  chosen. Students i n grade  the  became c o n s e r v e r s e a r l y  lower  were v e r y e f f e c t i v e , doorstep"  i n the  of  volume  experiment.  The  consequently a lower l e v e l  importance  6  of  volume  of  conservation  clearly.  further  recommendation  volume l e a r n i n g comprehension  i s to  at higher c o g n i t i v e  i s investigated  conduct  a study,  in  which  l e v e l s than computation  with  respect  to  a  and  correlation  volume c o n s e r v a t i o n . . It  and  experiment  achievement  have been  development might r e v e a l the level  the  grade l e v e l  those students could  with  that  i n a grade lower  conservation influenced  recommended  i s also  recommended t h a t  a volume c o n s e r v a t i o n  parallel  forms  of,  a  volume c o n s e r v a t i o n  retention but  t e s t be  not  developed  identical  to,  conservation  p r e t e s t , . The  i d e n t i c a l conservation  the  posttest  retention  pretest,  greater  peer  were n o t It which  influence  were  investigated  recommended  made  the  are  volume  t e s t s given  have  allowed  in a  s e n s i t i z a t i o n e f f e c t than i f they  in  that  the  Post  noted t h a t  there  the Hoc  following Qualitative  observations Analyses  be  further.  I t was  conservation cases of  or  t e s t could  which  identical. i s also  1.  and  posttest  levels  regression,  of the  was  subjects partial  a general in  progress i n  the  a l l g r o u p s . However, i n  conservers  who  were  given  112  experiences  in  most.,Further subjects,  needed  investigation  the  may  Students  were  not  for  gave  a  Further of  volume studies between  Such  on  the  i n the  response  be  Volume  responses.  Those  the  a  It  to  is  undertaken  where s t u d e n t s I t was  noted who  were  while  A and  other  related  validate.  to the  recommended  that  determine  the  to  i n t e r r o g a t i o n methods o f  were asked  t h a t the  answered  decreased who  steadily  the  used.  volume.  p o s t t e s t and  On  test  reason  needed  and  validity  i n c o r r e c t response  and  is  nonverbal  conservers,  test. 12  of the  12 of t h e Volume C o n s e r v a t i o n  responses. and  g a v e an  11  sufficient  about  conservation.  seemed t o have i m p r o v e d  question  of  particularly  the c o r r e c t response  number o f n o n c o n s e r v e r s  retention  provide  investigation  Question  conservers  The  support  c o n s e r v a t i o n of  their  to  judgement  correct  unegual volumes  increased  seem  final  which might  relationship  asked,  number c f s t u d e n t s  methodological  two  subjects,  for their  considerable  3.  of  Test, to write reasons  f o r the  assessing  level  question  information  assessing  o f volume a c t i v i t i e s  in  did  students  interviewing  where r e g r e s s i o n o c c u r s .  r e v e a l the e f f e c t  responses  weight.  cases  includes  the  conservers.  Conservation  a reason  for  seem t o h a v e r e g r e s s e d  which  of the conservation  partial 2.  activities  investigation,  is  stability  volume  Test to  i n the  of 12  reasons partial  correctly,  retention  answered q u e s t i o n i n the  write  number  question  concerned  pretest,  12  test.  correctly  posttest  o t h e r h a n d , most c o r r e c t r e s p o n s e s  supported  by  explicit  reasons.  and to  Further  113  research  is  conservation  needed f o r q u e s t i o n  in this  conservation  and  the  elementary  investigate  the  degree  of  research  algorithms  f o r the area  In  further  summary, school  a cuboid  be j u s t i f i e d  be c o n d u c t e d  between  that i s , area equals  for  of  may  is  times  skills in  be d e s i g n e d  conservation  to and  region,  width.  needed  before  p r a c t i c e o f i n t r o d u c i n g t h e volume  (V = L x W x H) i n g r a d e s e a r l i e r  is  and t h e d e g r e e  of a r e c t a n g u l a r  length  research  It  to determine the  tasks  area  volume  algorithm  investigated.  involving multiplication  "A = L x W",  prevalent  was  conservation  relationship algorithm  between  l e a r n i n g a volume  s c h o o l . F o r example, r e s e a r c h  the  the  relationship  skills  between v a r i o u s  l e a r n i n g other  learning  the  that similar  relationship of  study  multiplication  recommended  and c a s e s  o f i n e q u a l i t y o f volume i n g e n e r a l .  Finally,  involving  12 i n p a r t i c u l a r  the  algorithm  than grade 6 can  on t h e b a s i s o f t h e c o g n i t i v e t h e o r y  of Piaget.  114  REFERENCES  Arlin, P. K. The a p p l i c a t i o n o f P i a g e t i a n t h e o r y t o instructional decisions (Eeport No. 78:1, E d u c a t i o n a l R e s e a r c h I n s t i t u t e o f B r i t i s h C o l u m b i a ) . B. C. , 1977.  B a t - H a e e , M. A . C o n s e r v a t i o a o f mass, weight, and volume i n i n t e r m e d i a t e g r a d e s . P s y c h o l o g i c a l R e p o r t s , 1971, 28, 163168. Beilin, H. Learning and o p e r a t i o n a l convergence i n l o g i c a l thought development. 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I n d u c i n g number conservation c h i l d r e n . C h i l d D e v e l o p m e n t . 1964, 35, 1057-1071.  in  Wallach, L., W a l l , A. J . , & A n d e r s o n , L. Number c o n s e r v a t i o n : The role of reversibility, addition-subtraction, and m i s l e a d i n g cues,. Child,, D e v e l o p m e n e n t , 1967, 38, 425-442.  122  Winer, B. J . S t a t i s t i c a l p r i n c i p l e s i n e x p e r i m e n t a l d e s i g n . Y o r k : M c G r a w - H i l l Book Company, 1971.  New  Wohlwill, J . F. On e s s a i d a p p r e n t i s s a g e dans l e domaine de l a c o n s e r v a t i o n du nombre. Etudes d ,'epistomologie Genetigue, 1959, 9, 125-135. 1  Wohlwill, J . F., & Lowe R. C. E x p e r i m e n t a l a n a l y s i s development of the conservation of number.. D e v e l o p m e n t . 1962, 33, 153-167.  of the Child  appendix a  DESCRIPTION  OF THE INSTRUCTIONAL  UNITS  124  T r e a t m e n t ft Lesson Behavioral  1  Objectives:  1. Given two models of cuboids ( c l o s e d boxes o r s o l i d b l o c k s ) t h e volume o f w h i c h differ macroscopically, the students will be a b l e t o s t a t e which one has t h e g r e a t e r volume. 2. G i v e n f i v e models of c u b o i d s (closed boxes or solid blocks) the volume of any two of which differ m a c r o s c o p i c a l l y , t h e s t u d e n t s w i l l be a b l e to order the f i v e m o d e l s by volume. 3. Given two closed b o x e s , t h e volumes o f which do n o t n e c e s s a r i l y d i f f e r m a c r o s c o p i c a l l y , and given a set of decimetre c u b e s t o be u s e d a s u n i t s , t h e s t u d e n t s w i l l be a b l e t o b u i l d models c o n g r u e n t t o t h e closed boxes, and t h e r e b y t o s t a t e t h e volume o f e a c h box. 4. G i v e n f i v e c l o s e d b o x e s , t h e volume o f any two o f which do n o t n e c e s s a r i l y d i f f e r m a c r o s c o p i c a l l y , and g i v e n a s e t of d e c i m e t r e c u b e s t o be used a s u n i t s , t h e s t u d e n t s w i l l be a b l e t o b u i l d models c o n g r u e n t t o the c l o s e d b o x e s , and t h e r e b y t o o r d e r t h e f i v e b o x e s by volume. 5. G i v e n a p i c t u r e o f p o l y h e d r a l model built from unit cubes, some o f which may n o t be v i s i b l e , and g i v e n a s e t o f u n i t cubes, the s t u d e n t s w i l l be able to build the p i c t u r e d model and s t a t e i t s volume.  Outline: 1.  D i r e c t comparison  2.  Direct ordering  3.  I n d i r e c t comparison of  of  of  objects.  objects. closed  boxes.  3.1. Need for units; volume i s t h e cubes. 3.2. - N o n - s t a n d a r d u n i t s : D i s c u s s i o n .  number o f  unit  125  4.  Standard  units:  m,  5.  Indirect  o r d e r i n g of c l o s e d  6.  Volume o f p o l y h e d r a l m o d e l s b u i l t  7.  Worksheet.  3  dm  3  and  cm . 3  boxes. from  unit  cubes.  Materials: 1. C a r d b o a r d boxes. 2. 1 i n c h - c u b e s . . 3. Some d e c i m e t r e c u b e s and c e n t i m e t r e c u b e s . 4. A p o s t e r o f p o l y h e d r a l m o d e l s b u i l t f r o m u n i t 5. A cm r u l e r .  cubes.  Activities: G i v e e a c h s t u d e n t 25 i n c h - c u b e s ( r e f e r t o t h e s e cubes as simply "cubes" and n o t as " i n c h - c u b e s " ) . Ask t h e s t u d e n t s t o l a y t h e b l o c k s a s i d e b e c a u s e t h e y w i l l be used later in the period. 1.  (2 min)  Direct  comparison  of  objects:  Display the two c l o s e d c a r d b o a r d boxes A and B o f s i z e s 6 cm X 4 cm X 2 cm and 40 cm X 20 cm X 10 cm r e s p e c t i v e l y . Ask the students to guess which i s b i g g e r , which o c c u p i e s more s p a c e and w h i c h h a s t h e g r e a t e r volume. C o n c l u d e t h a t box B i s bigger than box A and t h a t any one o f t h e f o l l o w i n g s e n t e n c e s describes t h i s fact. a . Box b. Eox c . Box d. Eox 2.  B B B B  i s b i g g e r t h a n box A t a k e s up more room t h a n box A o c c u p i e s more s p a c e t h a n box A h a s a l a r g e r volume t h a n box A.  (3 min)  Direct  o r d e r i n g of  objects:  D i s p l a y , i n t h i s o r d e r , t h e f i v e c l o s e d b o x e s F, G, H, I, and J of sizes 2 dm , 4 dm , 16 dm , 1 dm and 10 dm r e s p e c t i v e l y . Ask t h e s t u d e n t s t o h e l p t o o r d e r t h e b o x e s from largest to smallest. Allow time f o r responses then o r d e r the boxes by c o m p a r i n g any two b o x e s and t h e n p u t t i n g a third in i t s p r o p e r p o s i t i o n between t h e f i r s t two and so on. 3  3  3  3  3  126  3.  (5 min)  Indirect  comparison  of c l o s e d  boxes:  3. 1 • Need f o r , u n i t s : Display a closed cardboard box i n e a c h o f two d i s t a n t l o c a t i o n s of t h e c l a s s r o o m (use box P and box Q) and ask the students to compare the v o l u m e s o f t h e b o x e s w i t h o u t moving them. L e a d t h e d i s c u s s i o n i n o r d e r t o c o n c l u d e t h a t perception may be d e c e i v i n g ; d e t e r m i n e and compare the volumes u s i n g t h e following a c t i v i t i e s : a . Use u n i t s ( s m a l l e r b o x e s p r o v i d e d ) to build next to each box a c u b o i d w i t h t h e same shape and volume a s t h a t o f the box. b . Count t h e number o f u n i t s and w r i t e t h e v o l u m e s o f the boxes on the board. (Volume o f P = 7 u n i t s , Volume of Q = 8 units). c . Compare t h e volumes o f the b o x e s u s i n g t h e numbers of u n i t s f o u n d i n b. could  Stress that i n t h i s p r o c e s s , a_vy u n i t s o f t h e same be u s e d , but t h e same u n i t s must be used t h r o u g h o u t . 3-2.  Non-standard  u n i t s : -Discussion,:  be  Encourage i n d i v i d u a l used as u n i t s .  of  D i s c u s s with the s t u d e n t s the - f e a s i b i l i t y some o f t h e s u g g e s t e d u n i t s . 4.  (5 min)  size  s t u d e n t s to suggest  Standard  units:  dm  3  and  i t e m s which and  could  convenience  cm : 3  Ask the students about the common u n i t s f o r m e a s u r i n g l e n g t h ( d e s i r e d answer: m, dm, cm, . . . ) . I f n e c e s s a r y use a cm ruler to measure length. Lead the discussion i n order to conclude that m , dm , cm are c o n s i s t e n t with the units we u s u a l l y use t o measure l e n g t h . Show c u b e s o f volume 1 dm and 1 cm . Emphasize that t h e s e volumes a r e the u s u a l m e t r i c u n i t s used i n i n d u s t r y and commerce. Use d e c i m e t r e c u b e s t o build a congruent shape and to measure the volume of box B (4 dm X 2 dm X 1 dm) and use c e n t i m e t r e cubes to build a congruent shape and to measure the volume of box A (6 cm X 4 cm X 2 cm). W r i t e on t h e board statements such as "Volume o f box B = 8 dm " and "Volume o f box A = 48 cm ." 3  3  3  3  3  3  5.  (5 min)  Indirect  3  ordering  of c l o s e d  boxes:  Display t h r e e c l o s e d b o x e s ( I , F, K) and a s k t h e s t u d e n t s t o s u g g e s t how t o o r d e r t h e boxes from largest to smallest using t h e 1 dm b l o c k s and t h e method d e s c r i b e d i n s e c t i o n 3. 1 above. D e t e r m i n e t h e volume o f e a c h box and order the boxes accordingly. 3  127  6. cubes:  (5  min)  Volume o f p o l y h e d r a l  models b u i l t  from  unit  D i s p l a y a p o s t e r (#1.1) o f two a r r a n g e m e n t s o f c u b e s . The first o f t h e a r r a n g e m e n t s i n c l u d e s one b l o c k " a b s e n t " a l t h o u g h i t may a p p e a r t o be " p r e s e n t . " The s e c o n d o f the arrangements includes one block "present" a l t h o u g h i t may a p p e a r t o be " a b s e n t . " Ask t h e s t u d e n t s t o use the cubes (inch-cubes) i n order to build c u b o i d s e x a c t l y l i k e t h e ones p i c t u r e d i n t h e p o s t e r and t o s t a t e t h e volumes i n t e r m s of t h e c u b e s . 7.  (10 min)  Worksheet:  D i s p l a y boxes X, Y, W and Z i n one a r e a o f the classroom (station #1). Display also boxes #1 and #2 as w e l l as 15 d e c i m e t r e cub-es i n a n o t h e r a r e a ( s t a t i o n #2). G i v e e a c h o f the students a copy of t h e a t t a c h e d w o r k s h e e t and e x p l a i n i t t o them. D i v i d e t h e p u p i l s i n t o 3 g r o u p s . L e t two of the groups each be of a b o u t 1/4 t h e c l a s s and t h e t h i r d o f a b o u t 1/2 o f t h e c l a s s . Ask e a c h o f t h e s m a l l e r g r o u p s t o s t a r t on station #1 o r #2 and t h e l a r g e r group on t h e s e a t work. Instruct the students i n the s m a l l e r groups t o f i n i s h the work a t t h e i r own s t a t i o n , t h e n , w i t h y o u r a p p r o v a l , t o move t o t h e o t h e r s t a t i o n and f i n i s h t h e work t h e r e , and t h e n t o r e t u r n to t h e i r seats f o r the seat work. Similarly instruct the students i n the l a r g e r g r o u p t o f i n i s h t h e i r s e a t work f i r s t t h e n w a i t f o r y o u r a p p r o v a l i n moving to the stations. Send students f r o m t h i s l a r g e r g r o u p t o t h e s t a t i o n s as t h e y f i n i s h t h e i r s e a t work and as s p a c e and o r d e r a t t h e stations allow. At t h e end o f t h e p e r i o d c o l l e c t t h e w o r k s h e e t s .  128  12S  Hasc:  First:  last:  lesson  (Station  Al -  Worksheet  #1)  1. Examine  t h e two  fccxes less  lettered  ihich  occupies  which  has t h e s i a l l e r  1 and i .  s p a c e , ¥ o r X? volume  I or  X?  2. Examine t h e f o u r boxes l e t t e r e d W, X, I , and Z; t h e n o r d e r t h e a f r c a l a r g e s t t c s . a l l e s t and r e c o r d year a n s w e r s b e l o w .  (largest)  ,  ,  (smallest)  ( S t a t i o n 12) 3. B u i l d s t a c k s c f c u b e s s i m i l a r t o t h e s e b c r e s and following: The  volume  c f fccx 11 i s :  cubes.  The  vclune  c f fccx #2 i s :  cubes.  ( S e a t work) 4. f o r each of tie a o d e l . C c u n t the nuaber  Figure Vcluae = 5.1ist  cubes  Voluae =  a  Figure C  B cubes  Voluse =  1, E , and C i n o r d e r  (Largest)  the  figures below use the c u b e s t o b u i l d c f c u b e s and r e c o r d t h e v c l u a e .  Figure  A  answer  (Smallest)  cubes  130  Treatment A Lesson  2  Behavioral Objectives: 1. G i v e n a c u b o i d o r a diagram of a cuboid, which is completely p a r t i t i o n e d or p a r t i a l l y p a r t i t i o n e d i n t o u n i t c u b e s , t h e s t u d e n t s w i l l be a b l e to build a layer and determine the volume o f t h e l a y e r , t h e number o f l a y e r s and the t o t a l volume. 2. G i v e n a diagram of a non-partitioned cuboid, the dimensions shown e i t h e r by n u m e r a l s o r by t h e f a c t o f i t s e d g e s b e i n g marked i n u n i t s e g m e n t s , the s t u d e n t s w i l l be able to determine the volume o f a l a y e r , t h e number o f l a y e r s and t h e t o t a l volume o f the c u b o i d .  Outline: 1. F a l l o w  up  o f the  worksheet  2.  Volume o f p a r t i t i o n e d  3.  Algorithm  f o r the  4. A p p l i c a t i o n o f t h e diagrams of c u b o i d s .  and  from  non^partitioned cuboids.  volume o f volume  the p r e v i o u s l e s s o n .  a cuboid algorithm  "V  = L X W X to  cuboids  5. . Worksheet.  Materials: 1. T h r e e c a r d b o a r d b o x e s . 2. Some d e c i m e t r e cubes. 3. A p o s t e r o f : a p a r t i t i o n e d c u b o i d , a p a r t i a l l y p a r t i t i o n e d c u b o i d and a n o n - p a r t i t i o n e d cuboid,.  H." and  131  activities: 1. jesson: Give previous can c o u n t  (3  min)  Fellow  up o f t h e , w o r k s h e e t  from  the  previous  each student his corrected worksheet from lesson. E x p l a i n t h a t i n o r d e r t o compare volumes t h e c u b e s and compare t h e numbers o b t a i n e d .  F o r i l l u s t r a t i o n c o u n t t h e c u b e s o f the s h a p e s in each student f o l l o w s on h i s own w o r k s h e e t , r e p o r t t h e o f t h e s h a p e s and s t a t e t h e i r o r d e r . 2. cuboids:  (7  min)  Volume o f p a r t i t i o n e d  Give each student of a p a r t i t i o n e d c u b o i d students t o b u i l d with pictured i n the poster, s t a t e t h e volume o f the  and  the one  #4 as volumes  non-partitjoned  25 i n c h - c u b e s . D i s p l a y a p o s t e r (#2.1) of dimensions 2, 2 and 5. Ask the t h e c u b e s a c u b o i d e x a c t l y l i k e t h e one t o c o u n t t h e number o f blocks and to cuboid.  Draw on t h e b o a r d a d i a g r a m o f a n o n - p a r t i t i o n e d c u b o i d o f d i m e n s i o n s L = 4, W = 3 and H = 2. W r i t e t h e s e d i m e n s i o n s along the edges. Ask the students to guess t h e number o f c u b e s n e c e s s a r y t o b u i l d the c u b o i d . P a r t i t i o n the top l a y e r then the r e s t o f t h e c u b o i d . Ask t h e s t u d e n t s t o d e t e r m i n e t h e volume by u s i n g t h e b l o c k s t o b u i l d a model then count the number of b l o c k s used. 3. _I  = _ 1  (12 _ J  min)  Algorithm  f o r the  volume o f a  cuboid  ___  D i s p l a y a c l o s e d c a r d b o a r d box (H) whose dimensions are L = 4 dm, W = 2 dm and H = 2 dm. Display also about 20 decimetre cubes. Discuss with the students how one can d e t e r m i n e t h e volume of t h e box u s i n g t h e a v a i l a b l e c u b e s . L e a d the a c t i v i t i e s u s i n g the cubes i n order t o determine: a. The number o f d e c i m e t r e c u b e s a l o n g t h e l e n g t h o f t h e b o t t o m l a y e r . W r i t e on t h e b o a r d " L ( l e n g t h ) = 4 dm" i.e./ 4 c u b e s f i t a l o n g t h e l e n g t h , o f t h e bottom l a y e r . b. The number o f decimetre cubes a l o n g the width bottom l a y e r . W r i t e oh t h e b o a r d "W (width) = 2 dm" c. B u i l d t h a t can  of  the  a l a y e r and c o n c l u d e t h a t t h e number of blocks f i t i n t h e b o t t o m l a y e r i s g i v e n by L X W. /  d. The number o f d e c i m e t r e c u b e s a l o n g t h e box. W r i t e on t h e b o a r d "H ( h e i g h t ) = 2 dm"  height of  the  e. B u i l d a s h a p e c o n g r u e n t t o t h e box and conclude that the total volume i s the volume cf one' l a y e r (L X W)  132  multiplied V = L 2 W X  by E.  the  number  of  layers  (H)  i.e.,  Display another cardboard box (R) whose d i m e n s i o n s a r e 3 dm, 3 dm and 2 dm. D i s p l a y a l s o about 10 decimetre cubes. Discuss with t h e s t u d e n t s how one c a n d e t e r m i n e t h e volume o f t h e box u s i n g t h e a v a i l a b l e c u b e s , knowing t h a t t h e r e are not enough cubes to build a shape c o n g r u e n t t o t h e box o r even b u i l d ' a shape congruent t o a l a y e r . Develop a g a i n the a l g o r i t h m "V = L X W X H" by f o l l o w i n g s t e p s a to e of the previous activity but w i t h o u t a c t u a l l y b u i l d i n g a shape s i m i l a r t o t h e box. Test the algorithm "V = L X W X H" c a r d b o a r d box (#1) o f d i m e n s i o n s 4 dm, 3 dm f. Find the d e c i m e t r e s and  of the  length, width m u l t i p l y them.  and  g. B u i l d a s h a p e s i m i l a r t o t h e and c o u n t t h e number o f c u b e s .  box  h.  g. •  Compare t h e  4. (6 min) cuboids:  r e s u l t s i n f* and  Application  of  using and 1  height with  a dm: of  different  the  decimetre  the, volume a l g o r i t h m t o  Present a p o s t e r (#2.2) of d i a g r a m s o f c u b o i d s a l g o r i t h m as d i r e c t e d .  box  in  cubes  diagrams and  apply  a. R e f e r t o t h e p a r t i t i o n e d c u b o i d on t h e p o s t e r . Without actually using the blocks develop a g a i n the a l g o r i t h m "V = L X W X H" by v e r b a l l y f o l l o w i n g s t e p s a t o e o f the previous a c t i v i t y . b. Refer to the partially partitioned and the nonp a r t i t i o n e d c u b o i d s . I n e a c h c a s e d e t e r m i n e t h e volume of e a c h l a y e r , t h e number o f l a y e r s and t h e t o t a l volume..Use the algorithm "V = L X W X H" to compute the volume. Compare t h e two answers. 5. , (7 min)  Worksheet:  Give each of the s t u d e n t s a copy of the attached worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e p e r i o d .  133  Posters of Lesson  No. 2.2  A2  131  First;  Dane: l a s t :  l e s s o n K2 - Worksheet  1, F i n a the v c l u n e o f each o f t h e f i g u r e s below, the cubes t o b u i l d mcdels of t h e f i g u r e s ) .  (you  Figure F  Figure C Volane o f a l a y e r = ,  ?oluae of a l a y e r =  Huafcer o f l a y e r s = _  Hunber of l a y e r s =  Voluae =  foluae =  cubes  2. vclome o f t h e fcettes l a y e r Number c f l a y e r s = Yolune =  3. I = 1  Bay  -  H = V =  4. V c l n a e o f t h e box =  •  cubes  use  135  Treatment , A  Lesson  Behavioral  3  objectives:  1. G i v e n e i t h e r a d i a g r a m o f a n o n - p a r t i t i o n e d c u b o i d w i t h its dimensions marked, or a word description of the dimensions of a cuboid without a diagram, the students will be able t o use t h e volume a l g o r i t h m i n o r d e r t o d e t e r m i n e t h e volume o f t h e c u b o i d . 2. G i v e n a d i a g r a m o f c u b o i d s t o u c h i n g s i d e by s i d e and all o f t h e r e q u i r e d d i m e n s i o n s , t h e s t u d e n t s w i l l be a b l e t o use t h e volume a l g o r i t h m i n " order t o determine the t o t a l volume o f t h e c u b o i d s . 3. G i v e n a d i a g r a m o f a c u b o i d t o which t h e r e a r e a t t a c h e d half cubes (rectangular parallelepipeds or triangular p r i s m s ) t h e s t u d e n t s w i l l be a b l e t o d e t e r m i n e the total volume. 4. Given partially the t o t a l  a diagram of a partitioned c u b o i d which i s c o v e r e d , t h e s t u d e n t s w i l l be a b l e t o determine volume o f t h e c u b o i d .  Outline: 1. F o l l o w  up o f t h e w o r k s h e e t  from  the p r e v i o u s l e s s o n .  2. Application o f t h e volume a l g o r i t h m t o t h e f o l l o w i n g cases: a. Word d e s c r i p t i o n o f c u b o i d s . b. C u b o i d s t o u c h i n g s i d e by s i d e . C. D i a g r a m s o f c u b o i d s w i t h some u n i t c u b e s a t t a c h e d or removed. d. A t t a c h m e n t s o f h a l f c u b e s t o c u b o i d s . e. D i a g r a m s o f p a r t i a l l y c o v e r e d c u b o i d s . 3.  Worksheet.  136  1. P o s t e r s o f c u b o i d s some units attached cubes to c u b o i d s .  t o u c h i n g s i d e by or removed and  s i d e , cubo,ids attachments of  with half  Activities,: 1. lessen:  (7  min)  Follow  up o f t h e  worksheet from  the  previous  Give each student his corrected worksheet from the p r e v i o u s l e s s o n and e x p l a i n t h a t "V = L X W X H" , h e l p s us to compute the volume (the number of c u b e s needed t o b u i l d a s i m i l a r shape) o f e a c h o f t h e s h a p e s drawn on t h e w o r k s h e e t . F o r i l l u s t r a t i o n e x p l a i n , as e a c h s t u d e n t f o l l o w s on his own w o r k s h e e t , t h a t i n #2 t h e volume o f t h e t o p l a y e r (of L = 4 and W = 2) i s L X W = 4 X 2 = 8 , t h a t t h e r e a r e 6 l a y e r s and t h a t t h e volume i s t h e number of cubes in one layer (8) multiplied by the number of layers (6) i.e., V = 4 X 2 X 6 = 48. 2. (18 min) following cases: a.  A p p l i c a t i o n of the  Word d e s c r i p t i o n  of  volume a l g o r i t h m  Cuboids touching  s i d e by  dimensions Substitute W X H" and illustrate l a y e r and  side:  P r e s e n t t h e p o s t e r (#3.1) o f two c u b o i d s t o u c h i n g s i d e s i d e . Ask s t u d e n t s t o compute t h e volume o f e a c h c u b o i d and t o d e t e r m i n e t h e t o t a l volume. c. removed:  the  cuboid:  Write on t h e b o a r d a v e r b a l d e s c r i p t i o n of t h e o f a r e c t a n g u l a r box (L = 6 dm, W = 2 dm, H = 5 dm). the given numbers for L, W, and H i n "V - L X compute t h e volume. Draw a d i a g r a m on t h e b o a r d and that "L X W" gives the number of blocks i n one L X W X H i s t h e t o t a l volume. b.  to  D i a g r a m s of c u b o i d s  with  some u n i t  cubes a t t a c h e d  by add or  P r e s e n t t h e p o s t e r (#3.2) o f c u b o i d s w i t h some u n i t cubes a t t a c h e d o r removed. Lead the s t u d e n t s t o compute t h e volume o f the cuboid as i f there were nothing a t t a c h e d o r removed. Determine the volume of the unit blocks to be added or s u b t r a c t e d . Add o r s u b t r a c t t o d e t e r m i n e t h e volume. d.  Attachments  of h a l f  cubes t o  cuboids:  137  Present the poster (#3.3) of the diagrams of the a t t a c h m e n t s o f h a l f cubes t o c u b o i d s . In each case ask the students to d e t e r m i n e t h e volume o f t h e c u b o i d , t h e volume o f t h e a t t a c h e d h a l f c u b e s and add t o d e t e r m i n e t h e t o t a l volume. €  »  Diagrams of  partially  covered  cuboids:  P r e s e n t t h e p o s t e r (#3.4) o f a partially covered cuboid (4X3X6). Point o u t t h a t t h e b l o c k s o f t h e t o p l a y e r a r e shown and t h e r e i s a t o t a l o f 6 l a y e r s . D e t e r m i n e t h e length, width and h e i g h t ' of t h e c u b o i d and use t h e a l g o r i t h m t o compute t h e volume. C o u n t t h e number o f b l o c k s i n t h e t o p layer (12) and c o n c l u d e t h a t e a c h o f t h e 6 l a y e r s has 12 b l o c k s . D e t e r m i n e t h e t o t a l volume (6x12). Compare t h e two a n s w e r s . 3.  (10  min)  Wo r k s h e e t :  G i v e each o f the s t u d e n t s a copy o f the a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t the end o f t h e p e r i o d .  138  P e s t e r s of Lessen ft3  No- 3-1  Ho. 3.2  139  First:  Ha&e: L a s t :  Lesson_A3_;,Worksheet 1. A box has a l e n g t h o f 14, a width of 11 and a h e i g h t o f 3. What i s t h e volume of the box?  _  _  What i s t h e volume 5.  A p i l e of cubes and h a l f cubes What i s t h e volume  6 What i s t h e volume  wmmi k a e t a l b l o c k and a h a l f o f a d i f f e r e n t b l o c k on t o p  A p i l e o f cubes p a r t i a l l y covered  Bhat i s t h e volume  t h a t i s t h e volume  140  Treatment A Lesson  4  Behavioral Objective: 1. G i v e n a d i a g r a m o r a word d e s c r i p t i o n o f a c u b o i d o f known dimensions and g i v e n a proposed additive or multiplicative dimensional transformation, the students w i l l be a b l e t o s t a t e t h e volume o f t h e c u b o i d t h a t would r e s u l t a f t e r the transformation.  Outline: 1. F o l l o w  up o f t h e w o r k s h e e t from  2. Application of proposed d i m e n s i o n a l  the p r e v i o u s l e s s o n .  t h e volume a l g o r i t h m t o c u b o i d s transformations.  3.  Summary o f g e n e r a l i z a t i o n s .  4.  Worksheet.  Materials: 1. & p o s t e r o f a p a r t i t i o n e d  cuboid.  with  141  Activities: 1. lesson:  (5  min)  Fellow  up o f t h e worksheet  from  the previous  1  Give each student h i s corrected worksheet from t h e p r e v i o u s l e s s o n and e x p l a i n t h a t i n #2 f o r e x a m p l e , t h e r e a r e 2 h a l f - c u b e s a t t a c h e d t o a r e c t a n g u l a r p i l e of cubes o f L = 3, W = 3 a n d H = 2. The volume o f t h e 2 h a l f - c u b e s i s 1, t h e volume o f t h e p i l e i s 3 X 3 X 2 = 18 ( u s i n g V = L X W X H) and t h e t o t a l volume i s t h e r e f o r e 18 + 1 = 19. Similarly f o r #6, e x p l a i n t h a t t h e volume o f t h e b l o c k i s 5 X 3 X 4 = 60 ( u s i n g V = L X W x H ) , t h e volume o f the h a l f block on t o p i s (3 X 2 X 1)/2 = 3 and t h e t o t a l volume i s t h e r e f o r e 60 + 3 = 63. with  2. (18 min) A p p l i c a t i o n o f t h e volume a l g o r i t h m proposed dimensional t r a n s f o r m a t i o n s :  W r i t e on t h e b o a r d (#4.1) of a partitioned r e p l a c e L , W, and H by t h e volume (240) and r e p l a c e V  to cuboids  "V = L X W X H", d i s p l a y a poster c u b o i d o f d i m e n s i o n s 6, 4 and 10 and numbers 6, 4 and 10. Compute t h e by 240.  Apply t h e f o l l o w i n g c h a n g e s t o t h e f a c t o r s ( L , W, H) and observe the changes i n the product (volume). Encourage t h e students t o s t a t e and t e s t c o n j e c t u r e s a b o u t t h e e f f e c t on t h e volume when t h e d i m e n s i o n s a r e c h a n g e d . 2.1. A d d i t i v e ( o r m u l t i p l i c a t i v e ) i n c r e a s e - i n one, two o r t h r e e o f t h e d i m e n s i o n s produces a d d i t i v e (or m u l t i p l i c a t i v e ) i n c r e a s e i n t h e volume. 2 . 1 . 1 . A d d i t i v e i n c r e a s e i n o n e , two o r t h r e e o f t h e dimensions: Ask t h e s t u d e n t s t o use t h e a l g o r i t h m "V = L X W X H" t o c a l c u l a t e t h e volume o f t h e c u b o i d i f i t s l e n g t h i n c r e a s e s t o 7 u n i t s . . W r i t e "7 X 4 X 10 = 280" u n d e r n e a t h "6 X 4 X 10 =240." C o n t i n u e by a s k i n g t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o the volume (240) i f t h e width increases o r the height i n c r e a s e s . A l l o w t i m e f o r r e s p o n s e s , w r i t e on t h e b o a r d t h e two sentences "6 X 5 X 10 = " and " 6 X 4 X 1 1 = _", d i v i d e t h e s t u d e n t s i n t o two g r o u p s and a s k e a c h g r o u p t o c o m p l e t e one of t h e statements. S o l i c i t t h e a n s w e r s (300,264) and c o m p l e t e t h e s t a t e m e n t s w r i t t e n on t h e b o a r d . Ask the students t o p r e d i c t what h a p p e n s t o t h e volume (240) i f any one o f t h e d i m e n s i o n s i n c r e a s e s . When conjectures are made test them u s i n g e x a m p l e s s u c h a s "8 X 4 X 10 = 320" and "6 X 4 X 12 = 288." Lead the discussion i n order to c o n c l u d e t h a t t h e volume i n c r e a s e s i f any one o f t h e d i m e n s i o n s  142  increases. S i m i l a r l y , ask t h e s t u d e n t s t o p r e d i c t what happens t o t h e volume (240) i f two o f t h e d i m e n s i o n s i n c r e a s e . A l l o w t i m e f o r responses and test them using examples such as "7 X 5 X 10 = 350" and "6 X 6 X 10 = 360." L e a d t h e d i s c u s s i o n i n o r d e r t o c o n c l u d e t h a t t h e volume i n c r e a s e s i f two of the dimensions increase. Similarly, lead the d i s c u s s i o n i n order t o conclude the volume increases i f a l l dimensions increase. "7 X 5 X 11 = 385" f o r i l l u s t r a t i o n . 2.1.2. dimensions:  Multiplicative-increase  that Ose  i n one o r two o f t h e  Ask t h e s t u d e n t s t o p r e d i c t what happens to the volume (240) i f any one of the dimensions i s m u l t i p l i e d by "2" ( d o u b l e d ) . A l l o w t i m e f o r r e s p o n s e s and ask three groups to t e s t them u s i n g "12 X 4 X 10 = .__", "6 X 8 X 10 = _ _ _ " and "6 X 4 X 20 = " (answer: 480),. C o n c l u d e t h a t t h e volume i s multiplied by 2 ( d o u b l e d ) . C o n t i n u e by a s k i n g t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e volume (240) i f any one of the dimensions i s multiplied by 3 (tripled) or 4. Test the conjectures using: 6 X (3 X 4) X 10 = 720 = 6 X 4 X (4 X 10) = 960 =  (3 X 240) and (4 X 2 4 0 ) .  Lead t h e d i s c u s s i o n i n o r d e r t o c o n c l u d e that the volume is multiplied by 2 ( d o u b l e d ) , 3 ( t r i p l e d ) , ... i f any one o f t h e d i m e n s i o n s i s m u l t i p l i e d by 2 ( d o u b l e d ) , 3 ( t r i p l e d ) , ... L i k e w i s e , a s k t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e volume (240) i f e a c h o f two d i m e n s i o n s i s m u l t i p l i e d by 2; t h e n if one i s m u l t i p i e d by 2 and a n o t h e r by 3, e t c . Ose e x a m p l e s such as t h e ones written below to e s t a b l i s h that i f one d i m e n s i o n i s m u l t i p l i e d by a whole number and a n o t h e r d i m e n s i o n is also multiplied by a whole number, t h e volume w i l l be m u l t i p l i e d by t h e p r o d u c t o f t h e two numbers. (5 X 6) X (2 X 4) X 10 = 30 X 8 X 10 = 2400  (5 X 2) X  240  6 X (3 X 4) X 6 X 12 X 20 =  (3 X 2) X  240  (2 X 10) = 1440  2.2» A d d i t i v e (or m u l t i p l i c a t i v e ) d e c r e a s e i n one, two o r t h r e e o f t h e d i m e n s i o n s , p r o d u c e s a d d i t i v e (or m u l t i p l i c a t i v e ) d e c r e a s e i n t h e , volume,. :  Model the d i s c u s s i o n of t h i s s e c t i o n a f t e r the d i s c u s s i o n of t h e p r e v i o u s s e c t i o n (2.1). F o r each o f t h e generalizations  143  below (lettered a, b, ...) ask t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e volume (240) i f t h e proposed changes in the dimensions are applied, t e s t the students* p r e d i c t i o n s using t h e g i v e n e x a m p l e s and c o n c l u d e t h e g e n e r a l i z a t i o n . 2.2.1, A d d i t i v e , d e c r e a s e , i n one,, two o r t h r e e o f t h e d i m e n s i o n s : (move q u i c k l y t h r o u g h t h i s s e c t i o n ) a. The volume d e c r e a s e s i f any, one of the decreases. E x a m p l e s : 5 X 4 X 10 = 200; 6 X 3 X 10 = 180.  dimensions  b. The volume decreases decrease. E x a m p l e s : 5 X 3 X 10 = 150;  dimensions  i f  any  two  5 X 4 X 3 =  of the 60  c . The volume d e c r e a s e s i f a l l d i m e n s i o n s Example: 4 X 3 X 8 = 96 2.2.2. dimensions:  Multiplicative  a . The volume w i l l be one o f t h e d i m e n s i o n s  decrease.  d e c r e a s e i n one,or  d i v i d e d by i s divided  2 ( h a l v e d ) , 3, by 2, 3, . . .  two  of  the  ...  i f any  Examples: 6 X (4/2) X 10 = 240/2 6 1 2 X 10 = 120 6 6  X 4 X (10/5) = 240/5 X 4 X 2 = 48  b. If one and a n o t h e r volume w i l l  o f t h e d i m e n s i o n s i s d i v i d e d by a whole number d i m e n s i o n i s d i v i d e d by a whole number, the be d i v i d e d by t h e p r o d u c t o f t h e two numbers.  Examples: (6/2) X (4/2) X 10 = 240/(2 X 2) = 3 X 2 X 10 = 60  240/4  2.3. A d d i t i v e i n c r e a s e i n one.pr tuo of the dimensions a d d i t i v e d e c r e a s e i n one, o r two c a n p r o d u c e any one o f t h e following: a. b. c.  Additive increase i n the A d d i t i v e decrease i n the No change i n t h e volume  and  volume volume  Ask t h e s t u d e n t s whether t h e y can p r e d i c t what h a p p e n s to the volume i f one of the d i m e n s i o n s i n c r e a s e s and a n o t h e r d e c r e a s e s . A l l o w t i m e f o r p r e d i c t i o n s and ask t h r e e groups to t e s t them u s i n g t h e f o l l o w i n g e x a m p l e s :  144  6 X 5 X 9 = 270 ( t h e volume i n c r e a s e s ) 7 X 4 X 3 = 84 ( t h e volume d e c r e a s e s ) 8 X 3 X 10 = 240 ( t h e volume d o e s n o t change) Conclude that t h e volume may i n c r e a s e , d e c r e a s e o r s t a y t h e same i f one d i m e n s i o n i n c r e a s e s and a n o t h e r d e c r e a s e s ; e a c h example h a s t o be examined i n d i v i d u a l l y . 3.  (2 min) Summary  of generalizations;  Summarize t h e g e n e r a l i z a t i o n s asking this series of guestions answer t h e m :  made in this and encourage  lesson by students to  i . What h a p p e n s t o t h e volume i f we i n c r e a s e o n e , three of t h e dimensions?  two  or  i i . What h a p p e n s t o t h e volume i f we d e c r e a s e o n e , two o r t h r e e o f t h e dimensions? i i i . What h a p p e n s t o t h e volume i f we i n c r e a s e one o f d i m e n s i o n s and d e c r e a s e a n o t h e r ?  the  i v . a. What happens t o t h e volume i f we d o u b l e o n l y one dimension? b. What happens t o t h e volume i f we double two dimensions? c. What h a p p e n s t o t h e volume i f we d o u b l e a l l t h r e e dimensions? v. What h a p p e n s t o t h e volume and h a l v e a n o t h e r d i m e n s i o n ? 4.  (10 min)  i f we d o u b l e  one  dimension  worksheet,:  G i v e each o f t h e s t u d e n t s a copy o f the a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e p e r i o d .  Poster  of  Lesson  146  First;  Kane:Last;  fresscn  a4-MorKsheet  C o s p l e t e the f o l l o w i n g : 1. A p l a s t i c  hex. I f ve i n c r e a s e d tfce width of t h i s box by t h r e e u n i t s and the l e n g t h and t h e h e i g h t s t a y e d t k e sane, then i n the new bcx what would be t h e length?  iiath?  Height? Volune? 2. II a e t a l bcx. I f we decreased the h e i g h t by 3 u n i t s and t h e t i e l e n g t h by 2 u n i t s but t h e w i d t h s t a y e d tfce s a n e , then i n the new bcx what would be t h e length? lidth? Height? Ycluae? 3- 1 wooden b c x . I f we decreased t h e l e n g t h by 1 u n i t and we i n c r e a s e d t h e h e i g h t by 2 u n i t s but t h e w i d t h s t a y e d t h e sa«e, t h e n i n the new bcx what would be t h e length? lidth?  _  Height? _ • Vcluae?  4. A r e c t a n g u l a r p i l e o f c u b e s . I f we t r i p l e d t h e h e i g h t o f t h i s p i l e and t h e l e n g t h and the w i d t h s t a y e d t h e same, t h e n i n t h e new p i l e what would be the Length? width? Height? Volume? 5. A r e c t a n g u l a r I f we d o u b l e d the width but same, t h e n i n would be t h e  p i l e o f cubes. t h e l e n g t h and h a l v e d t h e h e i g h t s t a y e d the t h e new p i l e what  Length? Width? Height? Volume? 6-  A c a r d b o a r d box f u l l of c u b e s . I f we m u l t i p l i e d t h e h e i g h t by 3 and d i v i d e d t h e l e n g t h b y 6 and t h e w i d t h by 2, t h e n i n t h e new box what would be t h e Length? Width? Height? Volume?  148  Treatment B  Lesson 1 Behavioral  Objectives:  a, b a n d n d e n o t e n a t u r a l  numbers.  1. Given a multiplication e q u a t i o n w i t h t h e same t h r e e f a c t o r s on e a c h s i d e o f t h e e q u a l sign, such that the factors on b o t h s i d e s a r e n o t i n t h e same o r d e r and t h a t one o r two f a c t o r s o f one s i d e a r e m i s s i n g , the students w i l l be a b l e t o s t a t e t h e m i s s i n g f a c t o r s . 2. Given an assertion of the form a X b > a X c, a X b < a X c , c X a > b X a o r c X a < b X a such t h a t a is known, b u t o n l y one o f b o r c i s known, t h e s t u d e n t s w i l l be a b l e t o a s c r i b e c o r r e c t l i m i t s t o the range of v a l u e s f o r t h e unknown number. 3. G i v e n an a s s e r t i o n o f t h e f o r m a X b X n > a X c X n o r n X c X a > n X b X a s u c h t h a t n and a a r e known and o n l y one o f b o r c i s known, t h e s t u d e n t s w i l l a b l e t o a s c r i b e c o r r e c t l i m i t s t o the range of values f o r the unknown number t o make.the a s s e r t i o n t r u e .  Outline: 1. Review o f t h e c o m m u t a t i v e  and a s s o c i a t i v e  principles.  2. Prescribing the range f o r t h e m i s s i n g f a c t o r i n an i n e q u a l i t y i n v o l v i n g two f a c t o r s a t e a c h o f i t s s i d e s . 3. P r e s c r i b i n g t h e r a n g e f o r t h e m i s s i n g factor i n an i n e q u a l i t y i n v o l v i n g three f a c t o r s a t each o f i t s s i d e s 4.  Worksheet.  Materials: 1. ft p o s t e r o f 3 X 5 g r i d .  149  Activities: 1.a.  (5 min)  Review o f th__.Commutative  principle:  Present the poster (#1.1) of 3 X 5 grid and ask t h e s t u d e n t s t o t e l l t h e number o f rows (3) and the number of squares p e r row ( 5 ) . Ask the s t u d e n t s t o g i v e a m u l t i p l i c a t i o n s e n t e n c e t o d e s c r i b e t h e t o t a l number o f s q u a r e s (3 X 5 = 1 5 ) . W r i t e " 3 X 5 = 15" on t h e b o a r d . D e c l a r e t h a t t h e numbers 3 and 5 a r e c a l l e d t h e f a c t o r s w h i l e 15 i s c a l l e d t h e p r o d u c t . Turn the poster 90 d e g r e e s and ask s i m i l a r q u e s t i o n s t o the ones i n t h e p r e v i o u s p a r a g r a p h . C o n c l u d e t h a t t h e sentence describing the total number o f s q u a r e s i s now " 5 X 3 = 15". W r i t e " 5 X 3 = 15" on t h e b o a r d u n d e r n e a t h " 3 X 5 = 15".. L e a d the discussion to illustrate that b o t h "3 X 5" and "5 X 3" describe the same number of squares and are therefore e q u i v a l e n t . W r i t e t h e s t a t e m e n t "3 X 5 = 5 X 3" on t h e b o a r d . Ask t h e s t u d e n t s t o make a g e n e r a l i z a t i o n a b o u t t h e o r d e r of t h e f a c t o r s b a s e d on t h e statement "3 X 5 = 5 X 3". . A l l o w time f o r responses and emphasize t h e commutative p r i n c i p l e ( w i t h o u t m e n t i o n i n g t h e term "commutative") i . e . , t h e o r d e r of multiplying t h e f a c t o r s does n o t a f f e c t t h e p r o d u c t . T e s t t h i s p r i n c i p l e using 3 X 4 and 4 X 3 by d e t e r m i n i n g the answer to 3X4 and 4 X 3., C o n f i r m t h e c o m m u t a t i v e p r i n c i p l e by a s k i n g the s t u d e n t s t o compute t h e answer (180) t o e a c h side of the s e n t e n c e 12 X 15 = 15 X 12. 1. b.  (5 min)  Review o f t h e a s s o c i a t i v e  principle:  Write " 2 X 3 X 4 " on t h e b o a r d and ask t h e s t u d e n t s i f i t makes any d i f f e r e n c e i n c h o o s i n g t h e numbers t o be multiplied first, 2 X 3 or 3X4. Then write (2X3) X 4 = _____ and 2 X ( 3 X 4 ) = on t h e b o a r d . Ask t h e s t u d e n t s to help in the steps f o r finding t h e a n s w e r s and c o n c l u d e t h a t one may m u l t i p l y t h e f i r s t two f a c t o r s f i r s t o r the last two first. Write on t h e b o a r d (2 X 3) X 4 = 2 X (3 X 4 ) . . A s k t h e s t u d e n t s t o make a g e n e r a l i z a t i o n a b o u t t h e order of multiplying the factors b a s e d on t h e s e n t e n c e w r i t t e n on t h e b o a r d . A l l o w t i m e f o r r e s p o n s e s and e m p h a s i z e t h e a s s o c i a t i v e p r i n c i p l e (without m e n t i o n i n g t h e t e r m " a s s o c i a t i v e " ) f o r t h e p r o d u c t o f any t h r e e factors i . e . , the grouping o f the f a c t o r s does not a f f e c t the p r o d u c t . . T e s t and c o n f i r m t h i s p r i n c i p l e u s i n g "5 X 3 X 8". W r i t e t h e f o l l o w i n g s t a t e m e n t s on t h e b o a r d and ask the s t q d e n t s t o f i n d t h e numbers which c o m p l e t e t h e s t a t e m e n t s . X8X  in  an  1 1 = 8 X 6 X 1 1 ;  7 X 1 2 X 4  = _____  X 7 X  2. (10 min) P r e s c r i b i n g . . t h e r a n g e f o r t h e m i s s i n g f a c t o r i n e q u a l i t y i n v o l v i n g two f a c t o r s a t e a c h o f i t s s i d e s . W r i t e on  the  board  "3 X [  J  >  3X5"  and  follow  the  150  discussion below whole numbers t h a t  i n order to lead the students c a n f i t i n t h e £ ].  to prescribe  T: I f we r e p l a c e £ J by 10, w i l l i t be t r u e ? ( R e s p o n s e : " Y e s " . W r i t e 10 on t h e b o a r d u n d e r n e a t h [ J) T: I s t h e r e a number g r e a t e r t h a n 10 t h a t would make i t t r u e ? I f s o , t e l l me s u c h a number? (Write p u p i l s * r e s p o n s e s i n a s c e n d i n g o r d e r . Leave s p a c e s for contingencies) T: What i f we r e p l a c e £ ] by 1, w i l l ( R e s p o n s e : "No",. Ask how we t e l l )  i t be t r u e ?  T: Are t h e r e any numbers l e s s t h a n 10 t h a t t r u e ? I f s o , which numbers? ( W r i t e them w i t h t h e s e q u e n c e i n o r d e r )  would  make i t  T: What i s t h e l e a s t number we have b e e n a b l e t o r e p l a c e £ ] by, so f a r ? Does t h a t mean t h a t 6 i s t h e l e a s t whole number we c a n u s e ? Can anyone name a s m a l l e r whole number t h a n 6 t o make i t t r u e ? Why n o t ? T: ( P o i n t o u t t h e "gap" i n t h e s e q u e n c e on t h e b o a r d probably looks l i k e t h i s : Numbers t h a t  make i t t r u e : 6,  8,  10,  64,  which  100)  Are t h e r e any numbers between t h e s e two ( p o i n t t o 6 a n d 8) t h a t make i t t r u e ? How a b o u t h e r e ? ( p o i n t t o 10 and 64) etc. T: Could t r u e ? Why  we p o s s i b l y l i s t not?  a l l o f t h e numbers t h a t make i t  T: So w i t h o u t a c t u a l l y l i s t i n g them a l l , what could we write t h a t would c l e a r l y i n d i c a t e a l l o f t h e numbers t h a t make i t t r u e ? C o u l d t h i s do? (write on t h e b o a r d "Any number g r e a t e r t h a n 5" o r "n > 5") Similarly l e t the following statements.  students  prescribe  answers  to  the  3 X 8 < [ J X 8 12 an  X 15  >  i  ] X 12  3. (5 min) P r e s c r i b i n g , the..range f o r t h e m i s s i n g f a c t o r i n i n e q u a l i t y i n v o l v i n g three f a c t o r s a t each o f i t s s i d e s :  Write the statement " 2 X 3 X 6 < 2 X [ ] X 6" on t h e board a n d f o l l o w t h e s t e p s below i n o r d e r t o l e a d t h e s t u d e n t s t o p r e s c r i b e numbers t h a t can f i t i n t h e £ ], (Answer: "Any number g r e a t e r t h a n 3" o r "n > 3 " ) .  151  3.1. Determine true. . 3.2. D e t e r m i n e false. 3.3. than  a number, a number,  s a y 10, t h a t s a y 2, t h a t  makes t h e s t a t e m e n t  makes  than  3.5. D e t e r m i n e two o r t h r e e numbers make t h e s t a t e m e n t t r u e .  10,  say  50,  less which  between 10 and 50 t h a t  3.6. Establish that i t i s impossible numbers t h a t make t h e s t a t e m e n t t r u e . 3.7- W r i t e a s e n t e n c e t h a t p r e s c r i b e s make t h e s t a t e m e n t t r u e . ("Any number "n > 3") . Similarly l e t the students prescribe the I ] in order to make " 6 X 2 X 1 ] < 6 X 2 X 20" t r u e .  and it.  statement  Establish a l o w e r bound; F i n d a l l t h e numbers 10 which make t h e s t a t e m e n t t r u e .  3.4. D e t e r m i n e a l a r g e r number makes t h e s t a t e m e n t t r u e .  4.  the  to l i s t  a l l the  a l l t h e numbers t h a t g r e a t e r than 3" o r numbers t h a t f i t i n the statement  (10 min) W o r k s h e e t :  G i v e each o f t h e s t u d e n t s a copy o f t h e a t t a c h e d worksheet explain i t t o them. Move a r o u n d and h e l p them t o c o m p l e t e At t h e end o f t h e p e r i o d c o l l e c t t h e w o r k s h e e t s .  152  Poster  No- 1-1  of  Lesson  Bl  First:  Name: L a s t :  Lesson  B1 -  Worksheet  Complete t h e f o l l o w i n g :  1. 5 X 7 X 12 = 12 X 5 X 2. 9 X 3.  (14 X 10) = 10 X  (23 X 7) X 13 =  (  ( X  X  )  13) X  4. 7 X 9 > 7 X [ ] What number o r numbers c o u l d  go i n t h e _ ]?  5. 53 X 16 > 16 X [ ] What number o r numbers c o u l d  go i n t h e £ ]?  6.  22 X 25 X 26 < 22 X £ ] X 26 What number o r numbers c o u l d go i n t h e £ ]?  7.  213 X 19 X [ ] < 213 X 19 X 6 What number o r numbers c o u l d go i n t h e £ ]?  in  each of the f o l l o w i n g w r i t e :  <  #  = o r > i n the  8. 910. 11.  4 X(2 X 3)  2) X 3  12.  9 X  X 13) X 9  (8 X  13)  154  Treatment  lesson Behavioral  B  2  O b j e c t i v e s : a, b, c and P d e n o t e  natural  numbers.  1. G i v e n an a s s e r t i o n o f t h e form a X b = P where a , b and P a r e known, t h e s t u d e n t s w i l l be a b l e to describe the effect on P when any one o f t h e f o l l o w i n g c h a n g e s i s a p p l i e d a d d i t i v e l y or m u l t i p l i c a t i v e l y : i . I n c r e a s e i n a, o r b , o r b o t h , i i . D e c r e a s e i n a, o r b , o r b o t h . ' i i i . I n c r e a s e s i n a t o g e t h e r with d e c r e a s e i n b (or v i c e versa) . 2. G i v e n an assertion o f t h e form a X b X c = P the s t u d e n t s w i l l be a b l e t o d e s c r i b e t h e e f f e c t on P when any one of the f o l l o w i n g changes i s applied a d d i t i v e l y or m u l t i p l i c a t i v e l y to the f a c t o r s . i . I n c r e a s e i n a, b o r c i i . D e c r e a s e i n a, b o r c i i i . I n c r e a s e i n one o r two o f t h e f a c t o r s a, b o r c arid d e c r e a s e i n two o r one o f t h e f a c t o r s a, b o r c  1. F o l l o w  up o f t h e w o r k s h e e t  from  the previous lesson.  2. E f f e c t on t h e p r o d u c t o f two f a c t o r s when t h e s e f a c t o r s are changed additively or multiplicatively. (Note: " a d d i t i v e l y " and " m u l t i p l i c a t i v e l y " w i l l subsume decrease as w e l l as i n c r e a s e . ) 3Effect on the product o f t h r e e f a c t o r s when t h e s e f a c t o r s are changed a d d i t i v e l y or m u l t i p l i c a t i v e l y . . ( N o t e : " a d d i t i v e l y " and " m u l t i p l i c a t i v e l y " w i l l subsume decrease as w e l l a s i n c r e a s e . ) 4.  Worksheet  155 i  Materials: (The activities of this lesson a r e mainly number m a n i p u l a t i o n s and do n o t r e q u i r e p h y s i c a l m a t e r i a l s )  Activities: 1. lesson:  (4  min)  F o l l o w up o f t h e w o r k s h e e t  from  the previous  Give each student h i s corrected worksheet from the p r e v i o u s l e s s o n and e x p l a i n t h a t i n #4, any number l e s s t h a n 9 can qo i n t h e £ ] and make t h e s t a t e m e n t t r u e . Any number q r e a t e r than or e q u a l t o 9 w i l l make the statement false. S i m i l a r l y , i n #7 any number l e s s t h a n 6 w i l l make t h e s t a t e m e n t true and any number g r e a t e r t h a n o r e q u a l t o 6 w i l l make t h e statement f a l s e .  these  2. (20 factors  min) E f f e c t , o n . t h e p r o d u c t o f two f a c t o r s a r e changed a d d i t i v e l y o r , m u l t i p l i c a t i v e l y . ,  Write on t h e b o a r d t h e s t a t e m e n t " 6 X 4 = 24". A p p l y f o l l o w i n g c h a n g e s t o t h e f a c t o r s (4 and 6) and o b s e r v e changes i n t h e product (24). Encourage t h e s t u d e n t s t o and t e s t c o n j e c t u r e s about t h e e f f e c t on t h e p r o d u c t when f a c t o r s a r e changed.  when the the state the  2.1. A d d i t i v e (or m u l t i p l i c a t i v e ) i n c r e a s e i n one o r two of t h e f a c t o r s , produces a d d i t i v e (or m u l t i p l i c a t i v e ) i n c r e a s e i n the product. 2.1.1. A d d i t i v e  increase  i n one o r two o f t h e f a c t o r s :  Ask t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e p r o d u c t i f the f a c t o r 6 i s replaced by a larger number, say 7 o r 8 (desirable r e s p o n s e : " I t i n c r e a s e s " ) . Allow time f o r r e s p o n s e s and w r i t e on t h e b o a r d " 7 X 4 = 2 8 " u n d e r n e a t h " 6 X 4 = 24." C o n t i n u e by a s k i n g t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e product (24) i f any one o f t h e f a c t o r s increases. When conjectures are made t e s t them using examples such as " 8 X 4 = 32" and " 6 X 7 = 42." L e a d t h e d i s c u s s i o n i n o r d e r t o conclude that t h e p r o d u c t i n c r e a s e s i f any one o f t h e f a c t o r s i n c r e a s e s . W r i t e on t h e b o a r d 6  1  increases  X  4  i  =  24  I  stays increases t h e same  S i m i l a r l y , ask t h e s t u d e n t s t o p r e d i c t what happens t o t h e p r o d u c t (24) i f b o t h o f t h e f a c t o r s i n c r e a s e . Allow time f o r  156  r e s p o n s e s and t e s t them u s i n g e x a m p l e s s u c h as " 7 X 5 = 35" and "10 X 6 = 60." Lead the d i s c u s s i o n i n order to conclude t h a t t h e p r o d u c t i n c r e a s e s i f b o t h o f t h e f a c t o r s i n c r e a s e . W r i t e on the board 6 X 4  i  increases  =  i  increases  24  1  increases  2.1.2. M u l t i p l i c a t i v e i n c r e a s e factors:  i n one  or  two  of  the  Ask t h e s t u d e n t s t o p r e d i c t what h a p p e n s to the product (24) if any one o f t h e f a c t o r s i s m u l t i p l i e d by 2 ( d o u b l e d ) . A l l o w t i m e f o r r e s p o n s e s and ask two g r o u p s t o t e s t them using "12 X 4 = (answer: 4 8 ) " and " 6 X 8 = (answer: 4 8 ) . " Conclude that the product is multiplied by 2 (doubled). Continue by a s k i n g the s t u d e n t s t o p r e d i c t what h a p p e n s t o t h e p r o d u c t (24) i f any one o f the factors is multiplied by 3 ( t r i p l e d ) o r 4. T e s t t h e c o n j e c t u r e s u s i n g : 6 X 4 = 24 ( o r i g i n a l s t a t e m e n t ) 6 X 12 = 72 = 3 X 24 and 24 X 4 = S6 = 4 X 24. Lead the discussion i n order to conclude t h a t the product i s m u l t i p l i e d by 2 ( d o u b l e d ) , 3 ( t r i p l e d ) , ... i f any one of the f a c t o r s i s m u l t i p l i e d by 2 ( d o u b l e d ) , 3 ( t r i p l e d ) , ... W r i t e on the board 6  1  X  4  i  =  24  1  stays is is t h e same m u l t i p l i e d multiplied by 3 by 3 Likewise, ask the s t u d e n t s t o p r e d i c t what happens t o t h e p r o d u c t (24) i f one of the two f a c t o r s i s m u l t i p l i e d by 2 and t h e o t h e r by 3, e t c . Use e x a m p l e s s u c h as t h e one w r i t t e n below t o e s t a b l i s h t h a t i f one f a c t o r i s m u l t i p l i e d by a whole number and another by a whole number, t h e p r o d u c t w i l l be m u l t i p l i e d by t h e p r o d u c t o f t h e two numbers. 6 X 4 = 24 ( o r i g i n a l s t a t e m e n t ) 30 X 8 = 240 = 10 X 24 of in  2.2. Additive (or m u l t i p l i c a t i v e ) d e c r e a s e i n one o r two the f a c t o r s produces a d d i t i v e (or m u l t i p l i c a t i v e ) „ d e c r e a s e t h e p r o d u c t . (Move q u i c k l y t h r o u g h t h i s s e c t i o n )  Model the d i s c u s s i o n of t h i s s e c t i o n a f t e r the discussion of the p r e v i o u s s e c t i o n (2.1). For each o f the generalizations below (lettered a, b, «..) ask the s t u d e n t s t o p r e d i c t what  157  h a p p e n s t o t h e p r o d u c t (24) i f t h e p r o p o s e d changes in factors are a p p l i e d , t e s t t h e students' p r e d i c t i o n s using g i v e n e x a m p l e s and c o n c l u d e t h e g e n e r a l i z a t i o n . 2.2.1. (move q u i c k l y  A d d i t i v e d e c r e a s e i n one o r two o f t h e f a c t o r s : through t h i s section)  a. The product decreases. Examples: Write  the the  decreases  i f  5 X 4 = 20; 6 X 3 =  any  one  of  the f a c t o r s  18.  on t h e b o a r d 6 X 4  1  =  24  i  i  stays decreases decreases t h e same b . The p r o d u c t d e c r e a s e s i f b o t h o f t h e f a c t o r s E x a m p l e : 5 X 3 = 15. W r i t e on t h e b o a r d 6 X 4  i decreases  i  =  decrease.  24  I  decreases  decreases  2.2.2. M u l t i p l i c a t i v e . d e c r e a s e f a c t o r s : (move g u i c k l y t h r o u g h t h i s  i n one, two o r t h r e e o f t h e section)  a . The p r o d u c t w i l l be d i v i d e d by 2 (halved), any one o f t h e f a c t o r s i s d i v i d e d by 2, 3, ...  3,  ... i f  Example: 6 X 2 =  12 =  24/2  W r i t e on t h e b o a r d '6  J  X  stays t h e s a ine  4  i  is divided by 2  24  I  is divided by 2  B. The p r o d u c t (24) w i l l be- d i v i d e d by t h e p r o d u c t o f two numbers i f one o f t h e f a c t o r s i s d i v i d e d by one o f t h e numbers and a n o t h e r f a c t o r i s d i v i d e d by t h e o t h e r number. Examples: 3 X 2 = 6 = 2 X 1 = 2 = write  24/4 24/12  on t h e b o a r d  158  6  . 4  3  is  24  4  4  is divided by 4  4  is divided by 12  divided by 3 2.3- A d d i t i v e (or, m u l t i p l i c a t i v e ) i n c r e a s e i n one o f t h e f a c t o r s and a d d i t i v e ( o r m u l t i p l i c a t i v e ) , d e c r e a s e i n t h e o t h e r f a c t o r c a n p r o d u c e any one o f t h e f o l l o w i n g : t  a- A d d i t i v e ( o r m u l t i p l i c a t i v e ) b. A d d i t i v e ( o r m u l t i p l i c a t i v e ) c . No c h a n g e i n the,, p r o d u c t 2.3.1. a d d i t i v e decrease  increase i n the product decrease i n the product  A d d i t i v e i n c r e a s e i n one o f t h e f a c t o r s and i n the other f a c t o r :  Ask t h e s t u d e n t s whether t h e y c a n p r e d i c t what h a p p e n s t o the product i f one o f t h e f a c t o r s i n c r e a s e s and t h e o t h e r d e c r e a s e s . A l l o w t i m e f o r p r e d i c t i o n s and ask t h r e e groups t o t e s t them u s i n g t h e f o l l o w i n g e x a m p l e s : 6 X 4 = 24 5 X 9 = 7 X 3 = _____ 8 X 3 = ,  (original statement) (45, t h e p r o d u c t i n c r e a s e s ) (21, t h e p r o d u c t d e c r e a s e s ) (24, t h e p r o d u c t d o e s n o t change)  Conclude that t h e p r o d u c t may i n c r e a s e , d e c r e a s e o r s t a y t h e same i f one f a c t o r i n c r e a s e s and t h e o t h e r d e c r e a s e s ; each example has t o be examined i n d i v i d u a l l y . Write  on t h e b o a r d 6 X 4  increases  =  decreases  24 i n c r e a s e s , decreases o r s t a y s t h e same  3. (4 min) E f f e c t on,, t h e p r o d u c t o f t h r e e f a c t o r s when t h e s e f a c t o r s a r e changed a d d i t i v e l y o r m u l t i p l i c a t i v e l y : (move q u i c k l y through t h i s section) Write of the board the statement "4 X 6 X 10 = 240" and ask the students whether t h e y c a n p r e d i c t t h e change i n t h e p r o d u c t (240) o f t h r e e f a c t o r s (4, 6, 10) when these factors vary. Ask t h i s following series o f q u e s t i o n s and e n c o u r a g e students t o answer them by making generalizations. When n e c e s s a r y t e s t t h e g e n e r a l i z a t i o n s u s i n g n u m e r i c a l examples. i. What h a p p e n s t o t h e p r o d u c t i f we i n c r e a s e one, two o r t h r e e o f t h e f a c t o r s ? (answer: i t i n c r e a s e s ) i i . .What h a p p e n s t o t h e p r o d u c t i f we d e c r e a s e t h r e e of t h e f a c t o r s ? (answer: i t d e c r e a s e s )  o n e , two o r  159  i i i . What h a p p e n s t o t h e p r o d u c t i f we i n c r e a s e one o f t h e f a c t o r s and d e c r e a s e t h e o t h e r ? (answer: we c a n ' t t e l l ; i t may i n c r e a s e , d e c r e a s e o r s t a y t h e same. Each example has t o be examined i n d i v i d u a l l y ) i v . a. What happens t o t h e p r o d u c t i f we d o u b l e o n l y one factor? b. What h a p p e n s t o the product i f we double two factors? c. What h a p p e n s t o t h e p r o d u c t i f we d o u b l e a l l t h r e e factors? (answers:  i t w i l l be m u l t i p l i e d  by 2, 4 o r 8 ) .  v. What h a p p e n s t o t h e p r o d u c t i f we m u l t i p l y by 6 and d i v i d e another by 2? (answer: m u l t i p l i e d by 6 and d i v i d e d by 2. 4.  one factor i t w i l l be  (7 min) W o r k s h e e t :  Give each o f and e x p l a i n i t t o have enough t i m e a-s t h e y c a n . Move end o f t h e p e r i o d  t h e s t u d e n t s a copy o f t h e a t t a c h e d worksheet them. Warn t h e s t u d e n t s that they may n o t t o f i n i s h t h e work and a s k them t o do a s much a r o u n d and h e l p them t o c o m p l e t e i t . At t h e c o l l e c t the worksheets.  160  Name:Last:  , First:.  ;  L e s s o n B2-Worksheet  Complete t h e f o l l o w i n g :  1.  16  X  1  24  =  1  decreases  384 1  stays t h e same  ?  If 16 were r e p l a c e d by a s m a l l e r number and 24 s t a y e d same, what s h o u l d happen t o t h e p r o d u c t ?  2-  24  I  increases  X  15  . I  =  increases  360  i ?  I f 24 were replaced by a larger number replaced by a larger number, what s h o u l d product?  3.  35  *  X  28  1  =  the  and 15 were happen t o t h e  980  4  becomes becomes 3 ? twice t i m e s as as b i g big I f 35 were r e p l a c e d by a number replaced by a number three happen t o t h e p r o d u c t ?  t w i c e as b i g and 28 were times a s b i g , what s h o u l d  161  4.  17  X  4  9  X  14  1  2142  I  1  stays decreases decreases ? t h e same I f 9 were r e p l a c e d by a s m a l l e r number, 14 were replaced by a smaller number and 17 s t a y e d t h e same, what s h o u l d happen t o t h e p r o d u c t ?  5.  3  i  become twice as b i g  X  36  I  X  becomes one-third as b i g  13  I  becomes 3 times as b i g  1404  i  ?  I f 3 were r e p l a c e d by a number twice as b i g , 13 were replaced by a number three times as b i g and 36 were r e p l a c e d by a number o n e - t h i r d as b i g , what s h o u l d happen to the product?  162  Treatment B Lesson 3  Behavioral  Objectives:  a , b, c and P d e n o t e n a t u r a l  numbers.  1. G i v e n an a s s e r t i o n o f t h e form a X b = P where a and b a r e known and g i v e n a c o n d i t i o n a l s t a t e m e n t t h a t P r e m a i n s f i x e d w h i l e one f a c t o r i s r e p l a c e d by one o f i t s m u l t i p l e s o r d i v i s o r s , t h e s t u d e n t s w i l l be a b l e to anticipate a s u i t a b l e r e p l a c e m e n t f o r b. 2., G i v e n an a s s e r t i o n o f t h e form a X b X c = P s u c h t h a t a l l v a r i a b l e s a r e known and g i v e n a c o n d i t i o n a l statement that P r e m a i n s f i x e d w h i l e one o r two o f t h e f a c t o r s a r e r e p l a c e d by t h e i r m u l t i p l e s o r d i v i s o r s , t h e s t u d e n t s w i l l be able to anticipate suitable replacement for the remaining f a c t o r or f a c t o r s .  Outline: 1.  Follow  up o f t h e w o r k s h e e t  from t h e p r e v i o u s  2. Effect on one o f two f a c t o r s c h a n g e d and t h e p r o d u c t i s f i x e d  lesson.  when t h e o t h e r f a c t o r i s  3. E f f e c t on two ( o r one) o f t h e t h r e e factors when one (or two) o f t h e s e f a c t o r s i s (are) c h a n g e d and t h e p r o d u c t i s fixed. 4.  Worksheet  Materials: (The activities of this lesson are mainly number m a n i p u l a t i o n s and do n o t r e q u i r e p h y s i c a l materials)  163  Activities; 1. lesson:  (4  min)  Fellow  up o f t h e  Give each student his p r e v i o u s l e s s o n and e x p l a i n how answers t o #2 and #5.  worksheet f r o m  corrected one can  the^previous  worksheet from the obtain the correct  F o r #2 w r i t e "24 X 15 = 360" on t h e b o a r d and e x p l a i n t h a t if 24 and 15 were r e p l a c e d by l a r g e r numbers t h e p r o d u c t would increase. Illustrate by writing "30 X 20 = 600" underneath "24 X 15 = 360." Similarly for #5, write 3 X 36 X 13 = 1404 then 6 X 12 X 39 = ••„-.• E x p l a i n t h a t the product will be multiplied by 2 X 3 and d i v i d e d by 3. T h e r e f o r e t h e number t o f i l l i n t h e b l a n k w i l l be 2 X 1404 = 2808. 2. (8 min) f a c t o r i s changed  E f f e c t on one and t h e p r o d u c t  o f two f a c t o r s i s .fixed;  when t h e  other  Write on the board the statement " 9 X 6 = 54" and u n d e r n e a t h i t w r i t e "18 X I ] = 54" and t e l l t h e s t u d e n t s that in the examples on t h e b o a r d , t h e p r o d u c t 54 was f i x e d w h i l e t h e f a c t o r 9 was d o u b l e d t o 18. Ask t h e s t u d e n t s t o suggest a numbers that can go i n the i_ ] t o make t h e s t a t e m e n t t r u e (answer: 3 ) . C o n t i n u e by telling the students that 9 was m u l t i p l i e d by 2 t o g e t 18 and ask them t o p r e d i c t what happened t o 6 t o g e t 3. H e l p t h e s t u d e n t s c o n c l u d e t h a t 6 was d i v i d e d by 2 s i n c e 9 was m u l t i p l i e d by 2. Similarly write on t h e b o a r d " 5 X 8 = 40" and underneath i t w r i t e "10 X ( ] = 40". Ask t h e s t u d e n t s t o g u e s s how 10 was obtained from 5 (multiplied by 2 o r d o u b l e d ) and t o p r e d i c t what would happen t o 8 i n o r d e r t o t o g e t t h e same product 40 (divide i t by 2 or h a l v e i t ) . A l l o w t i m e f o r r e s p o n s e s and r e p l a c e "£ J " by "4". W r i t e on t h e b o a r d " 4 X 6 = 24" and ask the students to predict what h a p p e n s t o one o f t h e f a c t o r s i f t h e p r o d u c t (24) i s f i x e d and t h e o t h e r f a c t o r i s m u l t i p l i e d by 2. Test the s t u d e n t s * p r e d i c t i o n s u s i n g "8 X J. ] = 24" and "J, ] X 12 = 24". Lead t h e s t u d e n t s t o c o n c l u d e t h a t i f t h e p r o d u c t i s f i x e d and one o f t h e f a c t o r s i s m u l t i p l i e d by 2 (doubled) the other s h o u l d be d i v i d e d by 2 ( h a l v e d ) . Similarly, r e f e r t o "3 X 12 = 36" and a s k t h e s t u d e n t s t o p r e d i c t what happens t o one o f t h e f a c t o r s i f t h e product is fixed and one of the factors i s divided by 3. T e s t t h e s t u d e n t s ' p r e d i c t i o n s u s i n g "1 X £ ] = 36" and "£ ] X 4 = 36". Lead t h e s t u d e n t s t o c o n c l u d e t h a t i f t h e p r o d u c t i s f i x e d and one o f t h e f a c t o r s i s d i v i d e d by 3 t h e o t h e r f a c t o r should be m u l t i p l i e d by 3.  164  Write on the board " 9 X 8 = 72" and ask t h e s t u d e n t s t o p r e d i c t what s h o u l d happen t o one o f t h e f a c t o r s i f t h e p r o d u c t i s f i x e d and t h e o t h e r f a c t o r i s m u l t i p l i e d or divided by a number. A l l o w time f o r r e s p o n s e s and t e s t them u s i n g : 3 X £ ] 18 X [ J £ ] X 72 £ ] X 2  = = = =  72 72 72 72  (answer:3 X [ 2 4 ] = 72) (answer: 18 X £ 4 ] = 72) (answer: £ (I J X 72 = 72) (answer: £ 3 6 ] X 2 = 72)  Help the students generalize that i f the product i s f i x e d and one o f t h e f a c t o r s i s m u l t i p l i e d ( d i v i d e d ) by a number the o t h e r f a c t o r s h o u l d be d i v i d e d ( m u l t i p l i e d ) by t h e same number. 3. (13 min) E f f e c t on..two ( o r one) o f t h e t h r e e when one (or two) o f t h e s e f a c t o r s i s ( a r e ) c h a n g e d m u l t i p l i c a t i v e l y and t h e p r o d u c t i s f i x e d .  factors  3.1. M u l t i p l i c a t i v e i n c r e a s e , i n , one o f t h e ^ f a c t o r s and m u l t i p l i c a t i v e d e c r e a s e i n a n o t h e r and v i c e v e r s a (move q u i c k l y through t h i s s e c t i o n ) . Write o f t h e b o a r d t h e s t a t e m e n t "4 X 6 X 10 = 240" and underneath i t w r i t e "8 X £ ] X 10 = 240". T e l l the students t h a t t h e p r o d u c t 240 i s f i x e d and one o f t h e f a c t o r s 10 i s a l s o fixed while a n o t h e r f a c t o r 4 was m u l t i p l i e d by 2 t o become 8. Ask t h e s t u d e n t s t o p r e d i c t what s h o u l d happen t o 6 i n o r d e r t o f i n d t h e number (3) t h a t should go i n £ ] and make t h e statement t r u e . T e s t t h e s t u d e n t s ' p r e d i c t i o n by r e p l a c i n g t h e s u g q e s t e d number i n £ ] and c o m p l e t e the statement. Conclude t h a t 6 was d i v i d e d by 2. Refer t o the statement "4 X 6 X 10 = 240" and ask t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o one o f t h e f a c t o r s i f the product and one o f the f a c t o r s are f i x e d while a f a c t o r i s multiplied by a number. Ask t h e s t u d e n t s to test their predictions using: 12 X £ ] X 10 = 240 £ ] X 6 X 20 = 240 4 X 1 X £ ] = 240  (answer: 12 X £ 2 ] X 10 = 240) (answer: £ 2 ] X 6 X 2 0 = 240) (answer: 4 X 1 X £ 6 0 ] - 240)  Help the students t o c o n c l u d e t h a t i n t h i s c a s e i f one f a c t o r i s m u l t i p l i e d by a number, t h e o t h e r s h o u l d be divided by t h e same number. S i m i l a r l y , ask t h e s t u d e n t s t o p r e d i c t what h a p p e n s t o one o f t h e f a c t o r s i f a n o t h e r f a c t o r i s d i v i d e d by a number while t h e t h i r d f a c t o r and t h e p r o d u c t a r e f i x e d . Ask the students t o t e s t t h e i r p r e d i c t i o n s using 2 X £ ] X 10 = 240 £ ] X 2 X 10 = 240 4 X 1 X £ ] = 240  ( a n s w e r : 2 X £ 1 2 ] X 10 = 240) ( a n s w e r : £ 1 2 ] X 2 X 10 = 240) (answer: 4 X 1 X £ 6 0 ] = 240)  165  3. 2 I n c r e a s e versa  i n two o f t h e f a c t o r s and d e c r e a s e i n one a n d v i c e  B e f e r t o "4 X 6 X 10 = 2 4 0 " and t e l l t h e s t u d e n t s t h a t t h e p r o d u c t 240 i s f i x e d w h i l e 6 i s m u l t i p l i e d by 2 and 10 i s multiplied by 2. , W r i t e "£ J X 12 X 20 = 240" u n d e r n e a t h "4 X 6 X 10 = 240" and a s k t h e s t u d e n t s t o p r e d i c t what should happen t o t h e f a c t o r 4. T e s t t h e s t u d e n t s ' p r e d i c t i o n and l e a d the discussion t o conclude that since two f a c t o r s were multiplied by 2 e a c h and t h e p r o d u c t i s f i x e d t h e t h i r d f a c t o r s h o u l d be d i v i d e d by 2 X 2 o r 4. S i m i l a r l y , a s k t h e s t u d e n t s t o p r e d i c t what s h o u l d happen to the factor 6 i f 10 was d i v i d e d by 5 and 4 by 2. T e s t t h e p r e d i c t i o n s and h e l p t h e s t u d e n t s t o c o n c l u d e t h a t t h e f a c t o r 6 s h o u l d b e m u l t i p l i e d by t h e p r o d u c t 5 X 2 ( o r 1 0 ) . T e s t this conclusion using 2 X [ ] X 2 = 240  (answer:2 X [ 6 0 ] X 2 = 240)  Write "4 X 12 X 8 = 384" a n d t e l l t h e s t u d e n t s t h a t t h e p r o d u c t 384 i s f i x e d and t h e f a c t o r 4 i s m u l t i p l i e d by 4 t o g e t 16. Write "16 X [ ] X ( ) = 384" underneath "4 X 12 X 8 = 384." Ask t h e s t u d e n t s t o p r e d i c t what s h o u l d happen t o 12 o r t o 8 o r t o 12 and 8 i n o r d e r t o make t h e statement true. T e s t t h e p r e d i c t i o n s and l e a d t h e s t u d e n t s t o c o n c l u d e t h a t s i n c e one o f t h e f a c t o r s was m u l t i p l i e d by 4 any one o f t h e f o l l o w i n g c o u l d be done: a. b.  d i v i d e any o t h e r f a c t o r by 4 d i v i d e e a c h o f t h e f a c t o r s by 2.  Similarly, refer t o "4 X 12 X 8 = 384" and t e l l the s t u d e n t s t h a t t h e p r o d u c t 384 i s f i x e d and " t h e f a c t o r 12 i s divided by 6 t o g e t 2. W r i t e ] X 2 X ( ) = 384" u n d e r n e a t h "4 X 12 X 8 = 384." Ask t h e s t u d e n t s to predict what should happen t o 4 o r t o 8 o r t o 4 and 8 i n o r d e r t o make t h e s t a t e m e n t t r u e . T e s t t h e p r e d i c t i o n s and l e a d t h e s t u d e n t s t o c o n c l u d e t h a t s i n c e cne o f t h e f a c t o r s was d i v i d e d by 6 any one o f t h e f o l l o w i n g c o u l d be done a.  multiply  any o t h e r  f a c t o r by 6  b.  multiply  one o f t h e f a c t o r s  by 2 and t h e o t h e r by 3.  4. (10 min) Worksheet Give each o f t h e s t u d e n t s a copy o f the attached worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e . p e r i o d .  166  Name: L a s t :  , First: L e s s o n B3 -  Complete  Worksheet  the following:  1.  8 X 5 = 40 £ J X 10 = 40 The p r o d u c t 40 s t a y e d t h e same; 5 was r e p l a c e d by 10. What number s h o u l d go i n t h e I ]? ___  2.  9 X 15 = 135 27 X [ ] = 135 The product 135 What number s h o u l d  3.  4.  5.  6.  7.  stayed t h e same; 9 was r e p l a c e d by 27. go i n t h e £ J?  34 X 12 X 9 = 3672 34 X £ ] X 36 = 3672 The p r o d u c t 3672 s t a y e d t h e same; 34 s t a y e d was replaced by 36. What number s h o u l d  t h e same; 9 go i n t h e £ ]?  11 X 15 X 22 = 3630 33 X £ ] X 22 = 3630 The p r o d u c t 3630 s t a y e d t h e same; 22 s t a y e d was replaced by 3 3 . What number s h o u l d  t h e same; 11 go i n t h e £ ]?  26 X 45 = 1170 13 X £ j = 1170 I f t h e p r o d u c t 1170 s t a y s t h e same and 26 i s 13, what number s h o u l d r e p l a c e 45? _____  replaced  by  28 X 8 X 23 = 5152 £ ] X 32 X 23 = 5152 If t h e p r o d u c t 5152 s t a y s t h e same, and 23 s t a y s t h e same and 8 i s r e p l a c e d by 32, what number should replace 28?  167  Treatment B  Lesson  4  Behavioral Objectives: 1. G i v e n a c u b o i d o r a d i a g r a m o f a c u b o i d , w h i c h i s nonpartitioned, partially partitioned or completely partitioned i n t o u n i t c u b e s , t h e s t u d e n t s w i l l be a b l e t o use t h e volume algorithm "V = L X W X H" i n order to d e t e r m i n e t h e volume 2. Given partially the t o t a l  a diagram of a partitioned c u b o i d which i s c o v e r e d t h e s t u d e n t s w i l l be a b l e to determine volume o f t h e c u b o i d .  3. Given a d i a g r a m o r a word d e s c r i p t i o n o f a c u b o i d o f known dimensions and given a proposed additive or multiplicative dimensional transformation the students w i l l be a b l e t o s t a t e t h e volume o f t h e c u b o i d t h a t would r e s u l t a f t e r the transformation.  Outline: 1. F e l l o w  up o f t h e worksheet  2. C l a r i f i c a t i o n 3. A l g o r i t h m  from  o f the concept  the previous l e s s o n .  o f volume.  f o r t h e volume o f a c u b o i d  "V = L X W X  H."  4. Application of the volume a l g o r i t h m t o p a r t i t i o n e d , p a r t i a l l y p a r t i t i o n e d and p a r t i a l l y c o v e r e d c u b o i d s . . 5. A p p l i c a t i o n o f t h e volume algorithm proposed dimensional t r a n s f o r m a t i o n s . 6..Worksheet.  Materials: 1. C a r d b o a r d 2. D e c i m e t r e  boxes. cubes.  to  cuboids  with  168  3. A p o s t e r o f p o l y h e d r a l m o d e l s b u i l t f r o m u n i t c u b e s . 4. Posters of partitioned, p a r t i a l l y p a r t i t i o n e d , nonp a r t i t i o n e d and p a r t i a l l y c o v e r e d c u b o i d s .  Activities: 1. lessen:  (4  min)  Fellow  up  of the  Give each student his p r e v i o u s l e s s o n and e x p l a i n how #5 and #7.  worksheet  from  the  previous  corrected worksheet from one can o b t a i n t h e answers  the to  For #5, e x p l a i n t h a t 26 was d i v i d e d by 2 i n o r d e r t o g e t 13. T h e r e f o r e 45 s h o u l d be m u l t i p l i e d by 2 i n o r d e r t o g e t the same product, 1170. C h e c k by w r i t i n g "13 X [ 9 0 ] = _" and c o m p u t i n g t h e answer (1170). W r i t e "1170" i n t h e b l a n k . F o r #7, e x p l a i n t h a t 18 was d i v i d e d by 3 t o g e t 6, 32 was divided by 2 to g e t 16. T h e r e f o r e 5 s h o u l d be m u l t i p l i e d by 3 X 2 = 6 i n o r d e r t o g e t t h e same p r o d u c t , 2880. Thus 5 s h o u l d be m u l t i p l i e d by 30. Check by writing 6 X [ 3 0 ] X 16 = ______ and c o m p u t i n g t h e answer (2880). W r i t e "2880" i n t h e b l a n k . 2.  (5 min)  Clarification  of the concept  of  volume:  Display the twe c l o s e d c a r d b o a r d boxes A and B o f s i z e s 6 cm X 4 cm X 2 cm and 40 cm X 20 cm X 10 cm r e s p e c t i v e l y . Ask the students to guess which i s b i g g e r , w h i c h o c c u p i e s more s p a c e and w h i c h has t h e g r e a t e r volume. C o n c l u d e t h a t box B is bigger than box A and t h a t any one o f t h e f o l l o w i n g s e n t e n c e s describes this fact. a. Eox B i s b i g g e r t h a n box A b. Box B t a k e s up more room t h a n box A c. Eox B o c c u p i e s more s p a c e t h a n box A d. Box B has a l a r g e r volume t h a n box A., D i s p l a y a c l o s e d c a r d b o a r d box in each of two distant locations o f t h e c l a s s r o o m (use box #1 and box #2) and ask t h e s t u d e n t s t o compare t h e volumes o f the boxes without moving them. Lead t h e d i s c u s s i o n i n o r d e r t o c o n c l u d e t h a t p e r c e p t i o n may be d e c e i v i n g ; d e t e r m i n e and compare t h e v o l u m e s using the following activities: a. Use d e c i m e t r e c u b e s a s u n i t s t o b u i l d a c u b o i d n e x t t o e a c h box w i t h t h e same s h a p e and volume as that of the box. b . , Count t h e number o f u n i t s and w r i t e t h e volumes o f t h e b o x e s on t h e b o a r d . c . Compare t h e volumes o f t h e boxes u s i n g t h e numbers of u n i t s f o u n d i n b. Stress  that  in  this  p r o c e s s , any  u n i t s of the  same  size  169  (in t h i s case u n i t s must be 3. _ _ _ _ _ _ _ _  d e c i m e t r e cubes) c o u l d used throughout.  (5 J _ l  min)  be  used,  A l g o r i t h m f o r the ~ ~  but  the  volume, o f  a  same cuboid  r  D i s p l a y a p o s t e r (#4.1) of a partitioned cuboid whose d i m e n s i o n s a r e 6, 3, and 4. Remind t h e s t u d e n t s t h a t t h e h e i g h t is always t h e v e r t i c a l d i m e n s i o n , the l e n g t h i s the l o n g e r o f t h e two h o r i z o n t a l d i m e n s i o n s and t h e w i d t h i s t h e shorter of the two h o r i z o n t a l d i m e n s i o n s . D i s c u s s with t h e s t u d e n t s how one can d e t e r m i n e t h e volume o f t h e c u b o i d . L e a d t h e a c t i v i t i e s i n order to determine: a. The number o f c u b e s a l o n g t h e l e n g t h o f t h e t o p layer. Write on the hoard "L ( l e n g t h ) = 6 " i . e., 6 c u b e s f i t along the l e n g t h of the top l a y e r . b. The number o f c u b e s a l o n g W r i t e on t h e b o a r d "W (width)  the width = 3 "  of the  c. Conclude t h a t t h e number o f c u b e s t h a t t o p l a y e r i s g i v e n by L X W.  can  top  layer.  f i t in  the  d. The number o f c u b e s a l o n g t h e h e i g h t o f t h e box. Write on the board "H ( h e i g h t ) = 4 " . This i s t h e number o f layers. e. C o n c l u d e t h a t t h e t o t a l volume i s t h e volume of one l a y e r (L X W) m u l t i p l i e d by t h e number o f l a y e r s (H) i . e . , V = L X W X H. 4. following a.  (5 min) cases: Partially  A p p l i c a t i o n of the partitioned  volume, a l g o r i t h m  to  the  and, n o n - p a r t i t i o n e d , c u b o i d s :  R e f e r t o t h e p a r t i a l l y p a r t i t i o n e d and t h e n o n - p a r t i t i o n e d c u b o i d s on t h e p o s t e r (#4.2). I n e a c h c a s e d e t e r m i n e t h e volume o f e a c h l a y e r , t h e number o f l a y e r s and t h e t o t a l volume. Ose the algorithm "V = L X W X H" t o compute t h e volume. Compare t h e two a n s w e r s . , b. . P a r t i a l l y  covered  cuboids:  P r e s e n t t h e p o s t e r (#4.3) of a partially covered cuboid (4X3X6). Point o u t t h a t t h e b l o c k s o f t h e t o p l a y e r a r e shown and t h e r e i s a t o t a l o f 6 l a y e r s . D e t e r m i n e t h e length, width and height o f t h e c u b o i d and use t h e a l g o r i t h m t o compute t h e v o l u m e . C o u n t t h e number o f b l o c k s i n t h e t o p layer (12) and c o n c l u d e t h a t e a c h o f t h e 6 l a y e r s has 12 b l o c k s . D e t e r m i n e t h e t o t a l volume ( 6 x 1 2 ) . Compare t h e two answers.  170  5. (7 min) A p p l i c a t i o n o f the, volume a l g o r i t h m t o c u b o i d s proposed dimensional, t r a n s f o r m a t i o n s :  with  W r i t e on t h e b o a r d t h e s t a t e m e n t 6 X 4 X 10 = 240. Review with the students how some o f the changes i n the f a c t o r s (6,4,10) a f f e c t t h e p r o d u c t (240) by a s k i n g this series of questions and encouraging the students to make generalizations. i^.What h a p p e n s t o t h e p r o d u c t three of the factors? i i . What h a p p e n s t o t h e p r o d u c t three of the factors?  i f we i n c r e a s e one, two  i f we d e c r e a s e  or  one, two o r  i i i . a. What h a p p e n s t o t h e p r o d u c t i f we d o u b l e o n l y one factor? b. What happens t o t h e p r o d u c t i f we double two factors? c. What h a p p e n s t o t h e p r o d u c t i f we d o u b l e a l l t h r e e factors? iv. and  What happens t o t h e product halve another f a c t o r ?  i f we d o u b l e  one f a c t o r  Display a poster (#4.4) of a partitioned cuboid of dimensions 6, 4, and 10. Ask t h e s t u d e n t s to state the a l g o r i t h m f o r f i n d i n g t h e volume o f t h e c u b o i d and w r i t e on t h e b o a r d V = L X W X H. R e p l a c e L, W, and H by 6, 4 and 10 respectively and compute t h e volume ( 2 4 0 ) . . W r i t e on t h e b o a r d 6 X 4 X 10 = 240 u n d e r n e a t h V = L X W X H. Ask t h e s t u d e n t s t o p r e d i c t t h e changes i n the volume (240) when L ( 6 ) , W ( 4 ) , and H (10) v a r y . H e l p t h e s t u d e n t s make t h e g e n e r a l i z a t i o n s by asking them this series of questions.. i. What h a p p e n s t o t h e volume i f we i n c r e a s e one, two o r t h r e e o f t h e dimensions? i i . What h a p p e n s t o t h e . v o l u m e i f we d e c r e a s e o n e , two t h r e e of the dimensions?  or  i i i . What h a p p e n s t o t h e volume i f we i n c r e a s e one o f t h e d i m e n s i o n s and d e c r e a s e a n o t h e r ? i v . a. What h a p p e n s t o t h e volume i f we double dimension? b. What happens t o t h e volume i f we dimensions? c . What h a p p e n s t o t h e volume i f we d o u b l e dimensions? v.  What  h a p p e n s t o t h e volume i f we d o u b l e  one  only  one  double  two  a l l three dimension  171  and  halve  6.  (9 min)  another dimension? W o r k s h e e t:  G i v e each o f t h e s t u d e n t s a copy of the a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e p e r i o d .  172  E c s t e r s of Lessen E-t  173  3.  A pile  of  cubes p a r t i a l l y  T o t a l volume  =  covered.  4. i p l a s t i c .ox. I f we i n c r e a s e d t i e width c f t h i s box by t h r e e u n i t s and the l e n g t h and t h e h e i g h t s t a y e d the sane, then i n the new box what would be t h e Length? lidth? . Height? Volute?  5. 1 wooden l e x . I f we decreased tfce l e n g t h by 1 u n i t and we i n c r e a s e d tfce h e i g h t by 2 u n i t s but tfce w i d t h s t a y e d t h e sane, then i n the new bcx what would be the length?  lidth? Height? Vcluae?  6. A r e c t a n g u l a r I f we doubled the widtfc but same, then i n would be t h e length? lidth? Height? Vcluae?  p i l e o f cubes. tfce l e n g t h and h a l v e d tfce h e i g h t s t a y e d t h e t h e new p i l e what  175  Treatment, C Lesson 1  Behavioral Objectives: 1. G i v e n students notation a. b.  a n u m e r a l i n Base 10 f o r a number n ( n < 1 0 ) , the will be able t o w r i t e t h e numeral i n expanded u s i n g e i t h e r of the forms i l l u s t r a t e d below: s  324 = 3 h u n d r e d s • 2 t e n s • 4 o n e s 324 = 3 X 100 + 2 X 10 • 4  2. G i v e n a n u m e r a l i n Base 10 f o r a number n (n<124) , t h e students w i l l be a b l e , w i t h m a n i p u l a t i v e a i d s , t o c o n v e r t t h e n u m e r a l t o t h e e q u i v a l e n t n u m e r a l i n Base 5.  Outline: 1. Review  o f Base  2. B u n d l i n g  10 p l a c e v a l u e  concepts  i n f i v e s and e x p r e s s i n g numbers i n Base 5  Materials: 1. P o p s i c l e s t i c k s 2. T w i s t t i e s 3. Base 10/Base 5 t a b l e 4. E a s e 10 S B a s e 5 mats: a p a p e r on one s i d e o f w h i c h t h e r e are t h r e e l a r g e columns with headings: ten-tens, t e n s and o n e s ; t h e o t h e r s i d e has t h r e e l a r g e columns w i t h h e a d i n g s : f i v e - f i v e s , f i v e s and o n e s .  176  Activities: G i v e e a c h s t u d e n t 45 p o p s i c l e s t i c k s , 5 t w i s t t i e s , a Base 10 and a Base 5 mat and a c o p y o f t h e Base 10/Base 5 t a b l e i l l u s t r a t e d below. Ask t h e s t u d e n t s t o l a y a s i d e t h e handouts, the sticks and t h e t w i s t t i e s b e c a u s e t h e y w i l l be used l a t e r i n the p e r i o d . expanded form | e x p a n d e d form I short| short, | [ f o r m | form | _ | f i v e - f i v e s f i v e s I onesj f t e n - t e n s l t e n s l o n e s l , ,- - I I  1.  (15 min)  Review o f Base  10  place value  concepts:  W r i t e "111" on the b o a r d t h e n ask the students to read this number. G i v e p u p i l s o p p o r t u n i t y t o r e s p o n d t h e n ask a b o u t t h e r e l a t i o n s h i p o f t h e "1" i n t h e t e n s p l a c e t o t h e "1" i n t h e ones p l a c e and t h e " 1 " i n the hundreds place. Ose popsicle sticks (1 stick, 1 b u n d l e o f 10, and 1 b u n d l e o f 10 t e n s ) i n o r d e r t o i l l u s t r a t e t h a t * 1* i n t h e t e n s p l a c e means 10 times as much as the •1• i n t h e o n e s p l a c e ; t h e •1* i n t h e 100's p l a c e means 10 t i m e s as much a s t h e '1* i n t h e 10«s place and 100 times as much as t h e '1' i n t h e ones p ^ a c e . C o n c l u d e t h a t "111 = (100) + (10) +1". W r i t e "444 = hundreds + . tens + and t h e n ask t h e c h i l d r e n t o s u g g e s t numbers t h a t a s s e r t i o n t r u e . Review w i t h t h e c h i l d r e n t h a t : 444 = 4 h u n d r e d s + 4 t e n s • 4 o n e s . 444 = (4 x 100) + (4 x 10) + 4. R e p e a t u s i n g "2795" i n o r d e r t o c o n c l u d e t h a t 2795 = (2 x 1000) + (7 x 100) + (9 x 10) • 5. Base  2. 5:  (20  min)  Bundling, i n ^ f i y e s  and  will  ones" make t h e  :  e x p r e s s i n g numbers i n  Tell t h e s t u d e n t s t h a t a f t e r what t h e y have j u s t done t h e f o l l o w i n g may seem t o be r i d i c u l o u s l y e a s y . G i v e the students the following instructions and make s u r e t h a t each s t u d e n t c o m p l e t e s t h e work. a.  Count o u t  14  sticks.  b.  Group t h e  sticks  i n t e n s and  o n e s . Use  the  twist  ties  177  to  bundle each  group  c. P l a c e t h e b u n d l e s s e c t i o n s on t h e Base  of t e n . (1) and s t i c k s 10 mat.  (4) i n t h e p r o p e r  d. E n t e r i n t h e t a b l e  on t h e l e f t  what t h e b u n d l e s show,  e. E n t e r i n t h e t a b l e t h e b u n d l e s show.  on t h e l e f t  the short  f.  of what  Unbundle t h e s t i c k s .  g. E e g r o u p ties.  the s t i c k s i n f i v e s  and ones u s i n g  h. P l a c e t h e b u n d l e s (2) and t h e s t i c k s s e c t i o n s on t h e Base 5 mat. i.  form  E n t e r i n t h e t a b l e on t h e r i g h t  j . Enter i n the table t h e b u n d l e s show.  on t h e r i g h t  the t w i s t  (4) i n t h e p r o p e r  what t h e b u n d l e s the s h o r t form  show.  o f what  E e p e a t t h e above s t e p s o f i n s t r u c t i o n f o r " t h i r t y - t w o " and " t w e n t y - s i x " ; i n s t e p g s a y t h a t a s soon as 5 b u n d l e s a r e made t h e y s h o u l d be b u n d l e d i n t o a l a r g e r b u n d l e o f f i v e - f i v e s . T e l l t h e s t u d e n t s t h a t when they bundled in fives the written numbers were i n Base 5, j u s t a s when t h e y b u n d l e d i n t e n s t h e numbers were i n Base 10. Show t h e e q u i v a l e n c e o f t h e n u m e r a l s by u s i n g t h e n u m e r a l s w r i t t e n on t h e t a b l e and w r i t i n g the following: 14  (Base  10) =  24  (Base 5)  22  (Base  10) =  42  (Base 5)  32  (Base  10) =  112  26  (Base  10) = 101  (Base  5)  (Base 5)  178  Treatment^ C Lesson  2  Behavioral,Objectives: 1. G i v e n a n u m e r a l i n Base 10 f o r a number n (n<124), t h e s t u d e n t s w i l l be a b l e , w i t h a n d w i t h o u t m a n i p u l a t i v e a i d s , to convert t h e numeral to the e q u i v a l e n t numeral i n Base 5. 2. Given a number no g r e a t e r t h a n 124 (Base 10) w h i c h i s s u g g e s t e d by a g i v e n r e a l life situation, the students will be able t o w r i t e t h e e q u i v a l e n t Base 5 n u m e r a l f o r t h a t g i v e n number.  Outline: 1. Beview  of section  2. C o u n t i n g  i n Base 5  3. C o n v e r t i n g 4.  2 of the previous l e s s o n  n u m e r a l s f r o m Base  10 t o Base 5  Worksheet  Materials: 1. P o p s i c l e s t i c k s 2. T w i s t t i e s 3. Base 10/Base 5 t a b l e 4. Ease 10 S Base 5 mats: a p a p e r on one s i d e o f which t h e r e a r e three l a r g e columns with headings: ten-tens, t e n s and o n e s ; t h e o t h e r s i d e has t h r e e l a r g e c o l u m n s w i t h h e a d i n g s : f i v e - f i v e s , f i v e s and o n e s .  Activities: 1.  (7 min) Beview o f s e c t i o n  Hold  up  22 s t i c k s .  2 of the previous l e s s o n :  Show t h e s t u d e n t s  that 4 bundles  of 5  179  sticks  e a c h c a n be made and t h e r e w i l l  be 2 s t i c k s  left  over.  Draw t h e Base 5 t a b l e on t h e b o a r d and w r i t e "4" under t h e " f i v e s " and " 2 " under t h e " o n e s " . Then w r i t e on t h e b o a r d t h e s t a t e m e n t "22 (Base 10) = 42 (Base 5 ) " . 2.  (8 min) C o u n t i n g i n B a s e 5:  Ask t h e s t u d e n t s t o e n t e r " 1 " , "2", "12" v e r t i c a l l y i n t h e l e f t s h o r t form s e c t i o n o f t h e t a b l e . F o r e a c h o f t h e s e n u m e r a l s w r i t e t h e e q u i v a l e n t Base 5 n u m e r a l . Emphasize t h a t i n Base 5, f i v e s must be g r o u p e d ( b u n d l e d ) . Make a c o r r e s p o n d e n c e t h a t i n Base 5 t h e r e a r e o n e s , f i v e s , f i v e - f i v e s , ... j u s t as in Base 10 t h e r e a r e o n e s , t e n s , ten^-tens, . . . A l s o p o i n t o u t t h a t i n Base 5 t h e o n l y d i g i t s needed a r e t h e f i v e digits "0, 1, 2, 3 and 4." C o n t r a s t w i t h B a s e 10 i n which t h e t e n d i g i t s "0, 1, 9" a r e n e e d e d . 3.  (10 min) C o n v e r t i n g n u m e r a l s  from  Base  10 t o Base 5:  T e l l t h e s t u d e n t s t o e n t e r "38" i n t h e l e f t short form section o f t h e t a b l e . Ask t h e s t u d e n t s t o t h i n k o f 38 s t i c k s and t h e number o f b u n d l e s t h a t c o u l d be made. Ask i f 1 bundle of f i v e c o u l d be made ( Y e s ) , 2 b u n d l e s ( Y e s ) , 3 b u n d l e s ( Y e s ) , 4 b u n d l e s (Yes) and 5 b u n d l e s ( Y e s ) . Ask what we would do with 5 b u n d l e s o f f i v e (Make 1 b u n d l e o f f i v e - f i v e s o r twenty f i v e ) . Ask t h e s t u d e n t s t o e n t e r t h e number o f b u n d l e s o f twenty f i v e (1) i n t h e t a b l e . C o n t i n u e by a s k i n g t h e s t u d e n t s t o c a l c u l a t e t h e number o f s t i c k s t h a t would be l e f t o v e r (13) , t h e number o f b u n d l e s o f five t h a t c o u l d be made (2) and t h e number o f s t i c k s l e f t o v e r ( 3 ) . Ask t h e s t u d e n t s t o e n t e r t h e numbers (2,3) r e p r e s e n t i n g t h e f i v e s and t h e o n e s i n t h e t a b l e and t o a l s o e n t e r t h e s h o r t form (123). W r i t e on t h e b o a r d "38 (Base 10) = 123 (Base 5 ) . " E e p e a t f o r "56" and c o n c l u d e t h a t : 56  (Base  10) = 211  (Base 5)  4.  (10 min) W o r k s h e e t :  G i v e each o f t h e s t u d e n t s a copy o f t h e a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . At t h e e n d o f t h e p e r i o d c o l l e c t t h e w o r k s h e e t s .  180  Name: L a s t :  •  , First:.  :  L e s s o n C2 Do e a c h c a l c u l a t i o n for i t .  and w r i t e  1. 938 = 9 h u n d r e d s  +  2. 4207= ( 3. 279 =  ) +  ( ___  (Base  |  provided  ones  x 100) +  (7 x  5's 4. 18  your answer i n t h e s p a c e  tens +  x 1000) • (2 x  Worksheet  (0 x 10) t 7  ) + ones  10) =  = _____  (Base 5)  I  5. 30  (Base  25«s 10) = —  |  5»s  25»s  |  (Base  • (Base 5)  I  5«s  |  ones  (Base 10) =  = _______ I  7. 57  ones =  I  6. 86  |  10) = _________  I  (Base 5)  8. The B a s e week i s  5 number f o r t h e number o f days i n a (Base 5)  9. The Base year i s  5 number f o r t h e number o f months i n a (Base 5)  10. The Ease 5 number f o r t h e number o f d a y s i n January i s . (Base 5)  (Base 5)  181  Treatment  Lesson  Behavioral  C  3  Objectives:  1. G i v e n a n u m e r a l i n Base 5 f o r a number 5) J , t h e s t u d e n t s w i l l be a b l e t o c o n v e r t t h e e q u i v a l e n t n u m e r a l i n Base 1 0 .  n £n<444(Base the numeral t o  Outline: 1. F o l l o w  up o f t h e w o r k s h e e t  2. C o n v e r t i n g 3.  from the p r e v i o u s  n u m e r a l s f r o m Base  5 t o Base  lesson  10  Worksheet  Materials: number mainly of (the a c t i v i t i e s o f t h i s l e s s o n c o n s i s t m a n i p u l a t i o n and do n o t r e q u i r e p h y s i c a l m a t e r i a l s )  182  Activities: 1. lesson:  (5  min)  Give each previous lesson and #10.  F e l l o w up o f t h e worksheet  from  the, p r e v i o u s  student his corrected worksheet from the and e x p l a i n how one c a n o b t a i n t h e answer t o #6  For #6 explain that from 86 sticks o n e can b u n d l e 3 bundles of 25's, 2 bundles o f 5's and have 1 stick left. Therefore " 3 " , "2" and " 1 " s h o u l d be w r i t t e n under " 2 5 « s " , "5's" and " o n e s " and t h u s "321" s h o u l d be w r i t t e n i n t h e s p a c e p r o v i d e d on t h e r i g h t .  allow under  Similarly, explain that i n #10 t h e 31 d a y s i n J a n u a r y f o r w r i t i n g " 1 " under " 2 5 « s " , " 1 " under " 5 « s " and " 1 " " o n e s " . The answer i s t h e r e f o r e 111 (Base 5 ) . 2.  (15 min) C o n v e r t i n g n u m e r a l s  Write the place 1 I i I I  v a l u e s f o r Base  f r o m Base  5 t o Base  10:  5 on t h e b o a r d a s f o l l o w s :  25»s  W r i t e a l s o "123 (Base 5 ) " on t h e b o a r d and ask t h e s t u d e n t s what "123 (Base 5 ) " means (1 t w e n t y - f i v e , 2 f i v e s and 3 o n e s ) . W r i t e " 1 " , "2" and " 3 " i n t h e p r o p e r sections i n the place value t a b l e on t h e b o a r d . Then w r i t e on t h e b o a r d 123 (Base 5) = (1 X ) + (2 X ) + 3 and a s k t h e s t u d e n t s t o s t a t e numbers t h a t w i l l make t h e a s s e r t i o n t r u e . C o n c l u d e t h a t 123 (Base 5) = (1 X 25) • ( 2 X 5 ) * 3 = 25 + 10 • 3 123 Base  (Base 5) = 38  (Base 1 0 ) .  L i k e w i s e ask t h e s t u d e n t s t o s t a t e the usual way ( i n 10) o f w r i t i n g t h e numbers i n t h e f o l l o w i n g e x e r c i s e s : 203 334  (Base 5) = (Base 5) =  3 . , ( 1 5 min)  (2 X (3 X  ) + (0 X ) + (3 X  ) + 3 = ) • 4 =  -  (Base 10) (Base 10)  Worksheet:  Give each o f t h e s t u d e n t s a copy of t h e a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e p e r i o d .  183  Name: L a s t :  First:  ;  L e s s o n C3  Do e a c h c a l c u l a t i o n for i t .  and  write  |  ones  5»s 1. 23  (Base 5) =  - Worksheet  your  answer i n t h e s p a c e  =  (Base  provided  10)  I  25*s 2.  401  |  5's  34  4.  332  I  (Base 5) =  (Base  (Base 5) =  5. 304  (Base  6.  (Base 5) =  432  ones =  I 3.  |  (Base 5) =  5) =  .  10)  (Base  10)  (Base  10)  (Base  10)  (Base  10)  184  Treatment C Lesson  4  Behavioral Objectives: 1. Given two numerals i n Base 5 w h i c h r e p r e s e n t two numbers t h e sum o f which i s n o t g r e a t e r t h a n 4 4 4 ( B a s e 5), the students will be able t o c a l c u l a t e . t h e sum o f t h e numbers and e x p r e s s i t by a numeral i n Base 5 without t r a n s l a t i n g t h e n u m e r a l s i n t o Base 10. 2. Given two numbers i n Base 5, n e i t h e r o f w h i c h i s g r e a t e r than 444(Base 5 ) , t h e s t u d e n t s will be able to c a l c u l a t e t h e d i f f e r e n c e o f t h e two numbers and e x p r e s s i t by a numeral i n Base 5 w i t h o u t t r a n s l a t i n g t h e n u m e r a l s i n t o E a s e 10.  Outline: 1. F o l l o w  up o f t h e w o r k s h e e t  2. A d d i t i o n i n Base  from t h e p r e v i o u s  5 w i t h and w i t h o u t  renaming  3. S u b t r a c t i o n i n Base 5 w i t h and w i t h o u t 4.  Worksheet  Materials; 1. 2. 3. 4.  Popsicle sticks Twist t i e s Base 5 t a b l e Ease 5 mat  lesson  renaming  185  Activities: 1. lesson:  (3  min)  Follow  up, o f t h e worksheet  Give each student his p r e v i o u s l e s s o n and e x p l a i n how answer t o #6.  corrected one can  from  the  previous  worksheet from the obtain the correct  Explain that "4" means f o u r 25's o r 100, " 3 " means t h r e e f i v e s o r 15 and "2" means two o n e s . T h e r e f o r e "432" i n Base 5 i s e q u i v a l e n t t o "117" i n Ease 10 (100 + 1 5 + 2 ) . 2.  (12 min) A d d i t i o n  2.1. A d d i t i o n  i n Base  5 w i t h and w i t h o u t  without renaming  (using  renaming:  the sticks) :  G i v e e a c h s t u d e n t 40 p o p s i c l e s t i c k s , 5 t w i s t t i e s , 5 mat and a copy o f t h e Base 5 t a b l e i l l u s t r a t e d below.  a Base  short| expanded f o r m I form | j — - _ f i v e - f i v e s l f i v e s I ones I I I I I I I  I  I I I I I I  I I I I I  I I I I I I  H o l d up i n one hand one b u n d l e o f f i v e p o p s i c l e s t i c k s and 3 sticks and ask t h e s t u d e n t s t o use s t i c k s and t w i s t t i e s i n o r d e r t o i s o l a t e a s i m i l a r amount and p l a c e t h e b u n d l e and t h e sticks i n the proper p l a c e s on t h e mat. Ask t h e s t u d e n t s t o r e c o r d i n t h e Ease 5 t a b l e t h e expanded form of the numeral (13) r e p r e s e n t i n g a l l t h e s t i c k s on t h e mat. S i m i l a r l y , h o l d up i n t h e o t h e r hand two b u n d l e s o f f i v e s t i c k s and one s t i c k . Ask the students t o i s o l a t e a s i m i l a r amount, p l a c e t h e s t i c k s on t h e mat and r e c o r d i n t h e same table underneath "13" t h e numeral (21) r e p r e s e n t i n g a l l t h e s t i c k s j u s t p l a c e d on t h e mat. Ask t h e s t u d e n t s t o draw a h o r i z o n t a l l i n e u n d e r n e a t h "21" and add t h e two numbers s t e p by s t e p a s f o l l o w s : a. J o i n t h e s t i c k s (3 and 1) on t h e mat i n o r d e r t o get "4." Becord t h e number o f s t i c k s (4) i n t h e p r o p e r p l a c e i n the t a b l e . b. J o i n t h e b u n d l e s (1 and 2) and record the b u n d l e s (3) i n t h e p r o p e r p l a c e i n t h e t a b l e . 2.2. A d d i t i o n  with  renaming  (using  t h e sticks),.,:  number' o f  1 86  Repeat activities similar to the above s e c t i o n (2.1) u s i n g 2 b u n d l e s and 4 hand and 1 b u n d l e and 3 s t i c k s (13) i n s t u d e n t s t h a t i n Base 5 we have t o b u n d l e Ask t h e s t u d e n t s t o add 24 and 13 s t e p by  ones d e s c r i b e d i n t h e sticks (24) i n one t h e o t h e r . Remind t h e by f i v e s i f we c a n . step as f o l l o w s :  a. J o i n t h e s t i c k s (4 and 3) on t h e mat, make a b u n d l e o f f i v e s t i c k s and p l a c e i t with t h e bundles on t h e mat. R e c o r d i n t h e t a b l e t h e r e s u l t i n g number o f s t i c k s ( 2 ) . b. the  J o i n t h e b u n d l e s (1, 2 and 1) on t h e mat and r e c o r d i n t a b l e t h e r e s u l t i n g number o f b u n d l e s .  S i m i l a r l y , c o n s i d e r 3 b u n d l e s o f f i v e s and 2 s t i c k s (32) and 3 b u n d l e s o f f i v e s and 4 s t i c k s ( 3 4 ) . a s k t h e s t u d e n t s t o p l a c e t h e s t i c k s on t h e mat, r e c o r d t h e n u m e r a l s and add s t e p by s t e p as f o l l o w s : a. J o i n t h e s t i c k s (2 a n d 4) on t h e mat, make a b u n d l e o f f i v e s t i c k s and p l a c e i t with the bundles on t h e mat. R e c o r d i n t h e t a b l e t h e r e s u l t i n g number o f s t i c k s ( 1 ) . , b. Join t h e b u n d l e s ( 1 , 3 a n d 3) on t h e mat, b u n d l e f i v e o f them i n t o a b u n d l e o f f i v e - f i v e s and r e c o r d i n the t a b l e t h e r e s u l t i n g number o f f i v e s ( 2 ) . c. Record table. 2.3.  t h e number o f b u n d l e s o f f i v e - f i v e s  addition  Write  without using  (1) on t h e  the^sticks;  on t h e b o a r d +  31 (Base 5) 2 (Base 5) (Base 5)  Ask t h e s t u d e n t s t o copy t h e n u m e r a l s t o t h e s h o r t form s e c t i o n on t h e Base 5 t a b l e and add. H e l p t h e s t u d e n t s t o c a l c u l a t e t h e number o f o n e s and f i v e s resulting from addition without actually manipulating the s t i c k s . the  Similarly, h e l p t h e s t u d e n t s perform the a d d i t i o n s t i c k s f o rboth of the following exercises: +  23 (Base 5) 4 (Base 5) (Base 5)  3  (12  min)  without  124 (Base 5) + 133 (Base 5) _____  (Base 5)  Subtraction  i n Base 5 w i t h and w i t h o u t  187  renaming; 3.1.  Subtraction  without  renaming  (using the s t i c k s ) :  Hold up t h r e e b u n d l e s o f f i v e s and two s t i c k s and ask t h e s t u d e n t s t o i s o l a t e a s i m i l a r amount, p l a c e t h e s t i c k s on t h e mat and record i n t h e Base 5 t a b l e t h e e x p a n d e d form o f t h e n u m e r a l (32) r e p r e s e n t i n g a l l t h e s t i c k s . Ask t h e s t u d e n t s to f o l l o w e a c h o f t h e s t e p s below i n o r d e r t o s u b t r a c t ( t a k e away, remove) 22 f r o m t h e number r e p r e s e n t e d i n t h e t a b l e . a. R e c o r d "22" i n t h e t a b l e u n d e r n e a t h a h o r i z o n t a l l i n e u n d e r n e a t h "22." b. Remove 2 s t i c k s and resulting sticks (0).  record  the  c. Remove two b u n d l e s and r e c o r d r e s u l t i n g bundles ( 1 ) . 3.2. S u b t r a c t i o n  with  renaming  "32" and draw  number  of  the  t h e number o f t h e  (using the s t i c k s ) :  R e p e a t a c t i v i t i e s s i m i l a r t o t h e one described i n the above section (3.1) u s i n g 4 b u n d l e s , 2 s t i c k s and s u b t r a c t 2 bundles, 4 sticks. Ask the students to carry out the s u b t r a c t i o n ( t a k e away) by f o l l o w i n g t h e s t e p s b e l o w : a. P l a c e 4 b u n d l e s and two s t i c k s i n t h e p r o p e r c o l u m n s on the mat. R e c o r d "42" i n t h e e x p a n d e d form s e c t i o n o f t h e t a b l e . W r i t e "24" u n d e r n e a t h "42" and draw a horizontal l i n e u n d e r n e a t h "24." b. In do. "  the  o n e s column  on t h e mat, "2 t a k e  away 4:  can't  c. Regroup 1 bundle into sticks i n order to have s u f f i c i e n t amount o f s t i c k s t o a l l o w t h e " t a k e away" o f 4. P l a c e t h e s t i c k s i n t h e p r o p e r c o l u m n on t h e mat. d. the  Remove 4 s t i c k s table.  and r e c o r d t h e r e s u l t i n g  number  (3) on  e. Remove 2 b u n d l e s the t a b l e .  and r e c o r d t h e r e s u l t i n g  number  (1) on  S i m i l a r l y , consider 1 bundle of f i v e - f i v e s , 1 bundle of f i v e s and 2 s t i c k s (112). Ask t h e s t u d e n t s t o p l a c e t h e b u n d l e s and s t i c k s on t h e mat and r e c o r d "112" i n t h e e x p a n d e d f o r m o f t h e t a b l e . Ask the students to record "34" on the t a b l e underneath "112", draw a h o r i z o n t a l l i n e u n d e r n e a t h "34" and s u b t r a c t ( t a k e away) by f o l l o w i n g t h e s t e p s b e l o w : a. I n t h e ones column on t h e mat, "2 t a k e do. "  away 4:  can't  188  b. Regroup 1 bundle into sticks i n order to have s u f f i c i e n t amount o f s t i c k s t o a l l o w t h e " t a k e away" o f 4,. P l a c e t h e s t i c k s i n t h e p r o p e r c o l u m n on t h e mat. c . Remove 4 s t i c k s the table. d. In do. "  the f i v e s  and r e c o r d  column  the r e s u l t i n g  number  (3) on  on t h e mat, "0 t a k e away 3: c a n ' t  e. Regroup t h e b u n d l e o f f i v e - f i v e s i n o r d e r t o g e t s u f f i c i e n t amount o f f i v e s t o a l l o w t h e " t a k e away.'! P l a c e t h e f i v e s i n t h e p r o p e r c o l u m n on t h e mat. f . Remove 3 f i v e s and r e c o r d (2) on t h e t a b l e . 3.3. S u b t r a c t i o n w i t h o u t Write  the r e s u l t i n g  using  number o f f i v e s  the s t i c k s :  on t h e b o a r d  -  44 32  (Base 5) (Base 5) (Base 5)  Ask t h e s t u d e n t s t o copy t h e n u m e r a l s t o t h e s h o r t form s e c t i o n on t h e Base 5 t a b l e and subtract. Help the students to calculate the number of ones and fives resulting from s u b t r a c t i o n without a c t u a l l y manipulating the s t i c k s . Similarly, help the students perform the subtraction without the s t i c k s f o r both o f the f o l l o w i n g e x e r c i s e s : 23 4  (Base 5) (Base 5)  6.  (Base 5) 4.  340 - 243  (Base 5) (Base 5) (Base 5)  (8 min) W o r k s h e e t :  G i v e each o f t h e s t u d e n t s a copy o f the a t t a c h e d worksheet and e x p l a i n i t t o them. Go a r o u n d and h e l p them t o c o m p l e t e i t . C o l l e c t t h e w o r k s h e e t s a t t h e end o f t h e p e r i o d .  189  Name: L a s t :  .  ._,  First:  L e s s o n C4  Do e a c h c a l c u l a t i o n for i t .  1. +  21 4  and  (Base 5) (Base 5)  write  2.  34 - 32  y o u r answer i n t h e s p a c e  34 + 3  ._,_(Base 5)  4.  - Worksheet  (Base 5) (Base 5)  .__(Base  (Base 5) (Base 5)  5. -  23 4  (Base 5)  7.  143  (Base 5)  • 31  (Base  8. .  201  (Base 5)  - 32  (Base 5)  9.  244  (Ease 5)  +'40  (Base 5) =  10,.  300  (Base  5) - 223  =  .__  (Base (Base (Base  6. -  5)  5) =  (Base 5) =  213 + 144  5)  (Base 5) (Base 5) (Base  3.  provided  340 143  5)  (Base  5)  (Base  5)  (Base  5)  5)  (Base 5) (Base 5)  __(Base  (Base  5) 5)  5)  Appendix B  ITEM  STATISTICS  191  Table Bl Volume A c h i e v e m e n t P r e t e s t Item Percent Number C o r r e c t 1 2 3 4 5 6 7 8 9 10 11 12 13 14  80.7 73. 1 52.0 28. 1 8.8 22.8 51. 5 25. 1 13. 5 27.5 24.0 32.7 8. 2 14.6  Point-biserial Coefficient 0. 35 0.51 0.58 0.66 0. 49 0.68 0.53 0.72 0.69 0.66 0.67 0.65 0.65 0.63  1 2 3 4 5 6  72.5 76.6 35.7 56. 1 58.5 53.8  Point-biserial Coefficient 0.67 0.75 0.52 0.62 0.65 0.73  Statistics  Item Percent Number C o r r e c t 15 16 17 18 19 20 21 22 23 24 25 26 27  T a b l e B2 Volume C o n s e r v a t i o n P r e t e s t Item Percent Number C o r r e c t  Item  4.1 19.9 11.7 35. 1 18. 1 11. 1 13.5 19.3 19.3 9.9 11.7 21.6 45.6  Item  0.47 0.69 0.49 0.64 0.74 0.74 0.74 0.68 0.68 0.67 0.68 0. 58 0.54  Statistics  Item Percent Number C o r r e c t 7 8 9 10 11  Point-biserial Coefficient  57.3 66. 1 56. 1 56. 1 42. 1  Point-biserial Coefficient 0.78 0.76 0.72 0.71 0.46  192  T a b l e B3 Volume A c h i e v e m e n t P o s t t e s t Item Percent Number C o r r e c t 1 2 3 4 5 6 7 8 9 10 11 12 13 14  78.4 74. 9 62.6 53. 2 27.9 41. 5 60.8 57. 9 48*5 54. 4 4 3-3 51.5 23.4 43.3  Point-biserial Coefficient 0.50 0.61 0.66 0.72 0.63 0.64 0.60 0. 76 0.73 0. 73 0.70 0.75 0.64 0.74  1 2 3 4 5 6  86.0 87. 1 59.6 66.7 74.3 6 7.3  Point-biserial Coefficient 0.65 0.75 0.60 0. 47 0.56 0.70  Statistics  Item Percent Number C o r r e c t 15 16 17 18 19 20 21 22 23 24 25 26 27  T a b l e B4 Volume C o n s e r v a t i o n P o s t t e s t Item Percent Number C o r r e c t  Item  8.2 42.1 17.0 58.5 39.8 32.7 36.3 40.9 33.9 32.2 30.4 41.5 56.7  Item  0.38 0.77 0. 35 0.79 0.65 0.73 0.66 0.67 0.62 0.66 0. 65 0.62 0.61  '  Statistics  Item Percent Number C o r r e c t 7 8 9 10 1 1  Point-biserial Coefficient  63.7 76.0 7 3. 1 71.9 56.1  Point-biserial Coefficient 0.72 0.75 0.72 0.70 0.46  193  Multiplication Item Percent Number C o r r e c t 1 2 3 4 5 6 7 8 9 10  85.4 80.7 79.5 49.7 50. 9 33-3 8.2 43.3 36.6 84.8  Point-biserial Coefficient  1 2 3. 4 5 6 7 8 9 10 11 12 13 14  87. 1 82.5 74.3 59.6 29.2 46.8 63. 2 59. 1 49. 1 59. 1 45.6 56.1 24. 6 46.2  Point-biserial Coefficient  71.3 63.7 33.9 22.8 19.9 67.3 71.3 23.4 22.8 23.2  0.58 0. 58 0.53 0.37 0.42 0. 59 0.58 0.46 0.42 •0. 36  T a b l e B6 B e t e n t i o n T e s t Item  Statistics  Point-biserial Coefficient 0.39 0.54 0.57 0.62 0.59 0.65 0. 58 0.73 0.78 0.68 0.70 0.67 0.58 0.69  Statistics  Item Percent Number C o r r e c t 11 12 13 14 15 16 17 18 19 20  0-58 0.59 0.47 0.60 0.65 0.62 0.34 0.56 0.67 0.58  Volume A c h i e v e m e n t Item Percent Number C o r r e c t  T a b l e B5 P o s t t e s t Item  Item Percent Number C o r r e c t 15 16 17 18 19 20 21 22 23 24 25 26 27  12.9 54.4 22.8 60.2 46.2 36.8 42. 1 45.6 40.4 34.5 37.4 38.6 6 3. 2  Point-biserial Coefficient 0.41 0.71 0.41 0.71 0.60 0.65 0.70 0.73 0.70 0. 68 0.69 0.51 0.55  194  Volume C o n s e r v a t i o n Item Percent Number C o r r e c t 1 2 3 4 5 6  90.1 91.2 68.4 81.9 79.5 71.9  Point-biserial Coefficient  1 2 3 4 5 6 7 8 9 10  93.0 91. 2 80.1 47.4 56. 1 30.4 7.6 50. 9 36.6 88.9  71.9 82.5 76.0 75. 4 59.1  T a b l e B8 R e t e n t i o n T e s t Item  Point-biserial Coefficient 0.50 0. 46 0.35 0.56 0.55 0. 55 0.34 0.55 0.53 0.44  Item Percent Number C o r r e c t 7 8 9 10 11  0.54 0, 59 0.60 0.54 0.60 0.65  Multiplication Item Percent Number C o r r e c t  T a b l e B7 R e t e n t i o n T e s t Item  Point-biserial Coefficient 0.77 0.70 0.72 0.60 0. 33  Statistics  Item Percent Number C o r r e c t 11 12 13 14 15 16 17 18 19 20  Statistics  77.8 66.7 28.7 19.3 22.8 71.3 73.7 22.2 26.3 14.6  Point-biserial Coefficient 0. 40 0. 44 0.35 0.35 0. 35 0.50 0.47 0.50 0. 44 0.40  appendix C  EAW  DATa  Codes: 1 = Pre Vol SAT V =  Nonconserver, 2 = P a r t i a l Conserver, 3 = Conserver = P r e t e s t , P o s t = Post T e s t , Ret = R e t e n t i o n T e s t = Volume, Mul = M u l t i p l i c a t i o n , &ch = A c h i e v e m e n t = S t a n f o r d Achievement T e s t , T r = Treatment Volume, M = M u l t i p l i c a t i o n , C = C o n t r o l  Subject Number  Conservation Level Ret Pre P o s t  002 004 006 007 008 014 015 016 022 023 024 027 031 032 033 034 036 037 03 8 039 041 042 043 044 045 047 051 052 053 056 057 058 059 061 06 2 063 06 5 066 067 06 9 070 071  3 2 1 1 3 3 2 1 1 1 3 1 1 1 3 2 1 1 3 2 2 1 3 1 1 3 3 1 1 1 2 1 3 1 1 1 2 1 2 3 1 1  2 1 1 2 3 3 3 3 1 1 3 2 1 3 2 2 2 1 1 2 1 1 3 1 1 3 3 1 2 1 2 1 3 1 1 3 2 1 1 3 1 3  2 1 1 1 3 3 3 1 3 1 3 1 1 3 2 3 3 1 2 2 2 1 3 1 3 3 3 1 2 1 3 2 3 1 1 1 2 3 1 3 1 3  Tr  Vol Vol Ach Ach SAT P r e P o s t  V V M C M C c M C V V M M V M V c c V M c V M c M V c V V c c M V V V V M c V c c M  30 33 29 25 31 40 37 31 32 26 30 35 43 34 39 25 28 32 38 42 31 21 39 31 21 29 35 39 34 40 18 26 40 28 35 43 40 37 33 39 28 30  4 4 4 1 3 16 3 5 5 17 8 4 4 14 12 7 3 3 1 11 6 3 11 3 6 4 20 9 7 2 2 4 9 7 2 18 3 2 5 2 3 2  18 12 2 4 0 23 1 6 6 18 15 10 8 21 21 15 0 2 12 17 11 10 12 2 3 18 25 21 25 1 0 19 19 13 5 22 3 2 6 5 2 2  Mul Ach Post 10 5 10 7 15 15 9 9 8 13 8 8 14 4 14 3 7 11 10 14 6 8 12 4 6 11 9. 13 6 8 0 10 12 11 3 12 14 5 8 13 6 8  V o l Mul Ach Ach Ret Ret 17 8 2 5 1 23 3 15 6 18 7 7 8 21 24 19 3 ii 12 16 12 9 11 5 4 21 22 25 23 1 0 18 23 23 10 25 1 3 5 4 4 6  14 9 8 7 11 13 7 •8 7 14 12 11 9 13 8 6 7 13 8 15 10 8 11 10 5 10 15 12 9 10 5 11 10 12 9 13 8 13 9 12 12 8  197  074 075 076 077 07 8 079 080 082 083 084 085 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 104 10 5 109 110 111 113 114 119 120 121 122 124 126 127 128 129 130 131 133 136 137 140 142 143 145 148 150  3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 3 3 2 3 1 3 3 3 2 3 3  3 1 3 1 3 1 1 3 1 3 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 3 3 3 1 1 3 3 3 2 1 3 1 3 3 2 1 3 3 2 3 3 3 3 3 2 3 1  3 1 3 3 3 1 1 3 3 3 1 2 1 1 1 1 3 1 1 3 2 1 2 1 2 1 1 3 3 3 1 1 3 3 3 1 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 2 3 3  M C M C V M C V V V c c c h  V M M V B C V c V M V V H M M C C V H M M C V c V M V M V  &  M _ V c M C M M M C  39 35 35 38 36 23 36 35 33 20 43 44 37 27 30 33 38 38 42 41 28 41 44 36 36 41 39 44 39 27 37 42 39 28 18 20 23 44 30 29 31 31 28 35 42 31 35 38 39 34 35 36 45 28  13 7 9 3 18 3 3 19 4 5 3 12 1 10 3 2 7 5 23 3 2 4 23 6 7 5 7 7 24 0 16 14 21 9 10 5 3 26 11 11 4 5 1 4 18 8 18 10 20 14 13 4 8 6  25 7 6 4 23 1 7 26 24 12 3 26 13 12 16 10 14 17 25 3 20 8 25 9 22 22 24 20 24 0 20 24 21 22 12 7 17 26 18 3 22 14 9 19 22 3 16 14 24 14 18 1 23 11  14 10 6 5 11 13 14 16 4 7 11 19 9 6 9 12 15 11 19 8 12 15 16 14 10 12 14 18 19 9 12 18 14 16 9 15 9 19 8 9 8 11 13 15 18 10 4 15 18 12 15 11 18 11  25 12 15 7 22 2 3 23 19 8 1 26 10 1f* 9 14 12 21 26 11 17 15 26 20 20 24 23 22 25 7 20 23 22 17 15 1 6 26 23 10 15 23 24 13 23 10 20 15 23 17 23 2 6 7  11 12 12 4 13 6 10 15 8 8 14 15 7 5 7 13 15 9 19 13 12 14 18 16 9 13 14 18 9 7 12 17 17 15 9 9 9 19 10 6 6 12 10 11 14 11 12 9 13 12 16 14 15 8  198  152 •154 157 160 161 164 165 169 170  3 3 3 3 3 2 3 1 3  Number o f  3 3 3 3 3 3 3 2 3  3 3 3 3 3 3 3 3 3  subjects =  v V V V M C M M M 105  36 33 33 33 29 36 36 36 36  15 4 2 17 6 12 6 16 9  17 16 14 25 18 14 10 20 4  9 10 7 11 15 12 17 11 11  20 18 13 21 9 19 10 22 17  9 10 5 11 11 11 13 15 11  Appendix  D  ONAJDSTED DESCRIPTIVE STATISTICS  200  T a b l e D1 U n a d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and Group S i z e s o f Volume A c h i e v e m e n t P o s t t e s t S c o r e s f o r T r e a t m e n t s by C o n s e r v a t i o n L e v e l s (Maximum S c o r e = 27) Conservation Level  Volume  Treatments Multiplication  Control  Non-conservers  17.68 (5.63) 19  12 .28 (7 .27) 18  6.55 (6.87) 20  11.75 (6.58) 57  P a r t i a l conservers  17. 17 (7.41) 6  8 .33 (10. 17) 6  6.50 (7.05) 4  11. 19 (8. 36) 16  Conservers  17.82 (4.29) 11  16 .93 (7 .54) 15  17.33 (8.59) 6  17.31 (6.62) 32  Total  17.64 (5.52) 36  13 .00 (7 .82) 39  8.70 (7.24) 30  13. 36 (6.86) 105  Regression  coefficient  =  Total  0.66  T a b l e D2 U n a d j u s t e d Means, S t a n d a r d D e v i a t i o n s , and Group S i z e s o f Volume A c h i e v e m e n t R e t e n t i o n T e s t S c o r e s f o r T r e a t m e n t s by C o n s e r v a t i o n L e v e l s (Maximum S c o r e = 27) Conservation Level  Volume  .Treatments Multiplication  Control  Total  Non-conservers  18.32 (6.51) 19  13 .56 (6 .34) 18  7.81 (6.90) 20  13. 13 (6.59) 57  Partial-conservers  15.83 (8.13) 6  9 -50 (10.00) 6  8.50 (8.66) 4  11.62 (8. 96) 16  Conservers  17.73 (5.08) 11  17 .00 (7 .96) 15  16.50 (9-0 5) 6  17. 16 (7.17) 32  Total  17.72 (6.34) 36  14 .26 (7 .53) 39  9.63 (7.56) 30  14. 12 (7. 13) 105  Regression  coefficient  =  0.67  Appendix  TESTS  Volume Conservation Test  Page 1  (on vhite)  Page 2  (on pink)  203  204  Page 5  (on green)  Page 6  (on pink)  Before  Before  After  After  206  Page 9  "1  (on goldenrod)  . j  After  Before  Page 10 (on green)  .  '  • '-  " * * '  sip Before  After  ' 'I  207  208  Name: L a s t :  School:  Volume Achievement Test  First:  Height= LengthWidth =  210  A rectangular  p i l e o f cubes.  Volume of the top l a y e r = Number of l a y e r s =  A rectangular  p i l e o f cubes.  Volume o f a l a y e r = T o t a l volume =  A g l a s s box w i t h some cubes i n i t . Volume o f the box =  A cardboard box f u l l o f cubes. Volume of the bottom l a y e r = flumber o f l a y e r s =  A box i s 10 u n i t s l o n g ,  5 u n i t s wide and 2 u n i t s  high.  What i s the volume o f the box?  A rectangular Volume =  b l o c k of wood. .  211  212  A r e c t a n g u l a r p i l e of cubes p a r t i a l l y covered. T o t a l volume =  A p i l e of cubes. 9 l a y e r s a r e covered. T o t a l volume =  A p i l e of cubes. I f we removed the top l a y e r , what would be the volume of the p a r t l e f t ?  A p i l e of cubes. I f we removed both of the shaded p o r t i o n s , what would be the. volume of the p a r t l e f t ?  A plastic  box.  I f we i n c r e a s e d the width of t h i s box by 1 u n i t but the l e n g t h and h e i g h t stayed the same, what would be the volume of the new box?  A metal box. If but  we doubled the l e n g t h of t h i s  box  the width and the h e i g h t stayed the  same, what would be the volume of the new box?  213  19. A r e c t a n g u l a r p i l e of cubes. I f we halved the l e n g t h and doubled the width but the h e i g h t stayed the same, what would be the volume of the new shape?  20. A r e c t a n g u l a r p i l e of cubes. I f we doubled the l e n g t h , t r i p l e d the height and halved the width, what would be the volume of the new shape?  21. A wooden  box.  If we decreased the width by 1 u n i t and we i n c r e a s e d the height by 2 u n i t s but the l e n g t h stayed the same, what would be the volume of the new shape? 22. A metal  box.  I f we decreased the width by 2 u n i t s , decreased the l e n g t h by 6 u n i t s and i n c r e a s e d the height by 8 u n i t s , what would be the volume of the new shape?  23.  A b r i c k has  a h e i g h t of 5 u n i t s , a l e n g t h of 6 u n i t s and  a  volume of 120 u n i t s . What i s i t s width?  24.  A r e c t a n g u l a r p i l e of cubes covered. T o t a l volume = 42 How  partially  cubes.  many l a y e r s are t h e r e  altogether?  , First:  Naae: L a s t :  5-lt^p_,j,cati on y e s t 1  A. |n each of ,.the_follpwjnq w r i t e your answer i n the__ox__t_the r i g h t . w  1.  (23 x 17) x 36 = 17 x ([ ] x 23) Ihat nuaber s h o u l d go i n t h e £ j ?  2.  13 x (5 x 19) = (£ J x ) x 5 i h a t two numbers s h o u l d go i n the [ ] and i n t h e ?  3.  6 X 6 = 9 X £ ] What number s h o u l d go i n t h e £ J?  4.  15 X [ ] = 30 X 25 i h a t number s h o u l d go i n t h e [ 3-  5.  24 X 11 = £ 3 X 33 i h a t nuaber s h o u l d go i n t h e £ J?  6.  12 X 15 X 22 = 36 X £ 3 X 22 i h a t nuaber s h o u l d go i n t h e £ }?  7.  41 X 50 X 7 = [ 3 X 5 X 35 What number s h o u l d go i n the £ 3?  6.  18 X 24 = 432 I f 18 i s r e p l a c e d by 9 and i f t h e p r o d u c t , 432, s t a y s t h e same, t h e n 24 Bust be r e p l a c e d by what number?  9.  9 X 13 X 21 = 2457 I f 9 i s r e p l a c e d oy 27 and i f t h e p r o d u c t , 2457, s t a y s t h e same, then 21 a u s t be r e p l a c e d by what nuaber?  215  B. JJL_each of. the f o l l o w i n g w r i t e t h e l e t t e r c o r r a s p p n d i n q to the c o r r e c t answer i n _ t h e box, ,aj fehe r i g h t * 10-  18 X 27 = 486 I f 27 i s r e p l a c e d by a l a r g e r number and i f 18 s t a y s t h e saae, what happens to the p r o d u c t , 486? a. I t becomes s m a l l e r b. I t becomes l a r g e r c. I t s t a y s t h e same  11-  £ } X 14  (  9X  14  i h a t number (or numbers) c o u l d go i n the £ J? a. Any number g r e a t e r than 9 b. Any number l e s s t h a n 9 c. Any nuober g r e a t e r t h a n 14 d. Any number l e s s than 14 e. 9 12.  £  3X  27  ^>  41 X 27  What number (or numbers) c o u l d go i n the [ ]? a. Any number l e s s t h a n 27 b. Any number g r e a t e r than 27 c. 27 d. Any number l e s s t h a n 41 e. Any number g r e a t e r than 41 13.  38 X 53 = 2014 I f 38 i s r e p l a c e d by a number t w i c e a s b i g and i f 53 i s r e p l a c e d by a number t h r e e t i m e s a s b i g , what happens t o t h e product, 2014? a. I t i n c r e a s e s t o f i v e t i m e s a s much b. I t i n c r e a s e s t o s i x t i a e s as much c. I t d e c r e a s e s t o one h a l f as much d. I t d e c r e a s e s t o one f i f t h as much e. I t changes b u t i t i s i m p o s s i b l e t o know how ouch i t i n c r e a s e s o r decreases.  •  14.  72  X 35 = 2520  I f 72 i s r e p l a c e d by a number h a l f as b i q and i f 35 i s r e p l a c e d by a nuaber one f i f t h as b i q , what happens t o the p r o d u c t , 2520? a. I t d e c r e a s e s t o one seventh as much b. I t i n c r e a s e s t o t e n t i m e s a s much c. I t decreases t o one t e n t h as much d. I t i n c r e a s e s t o seven t i m e s as auch e. I t changes b u t i t i s i a p o s s i b l e t o know how •uch i t i n c r e a s e s o r decreases. 15.  40  X 24 =  960  I f 40 i s r e p l a c e d by a number one f o u r t h as b i q and i f 24 i s r e p l a c e d by a number t w i c e as b i q , what happens t o the p r o d u c t , 960? a. I t i n c r e a s e s t o s i x t i m e s as auch b. I t d e c r e a s e s t o one s i x t h as much c. I t i n c r e a s e s t o t w i c e as much d. I t decreases t o one h a l f a s auch e. I t changes but i t i s i a p o s s i b l e t o know how auch i t i n c r e a s e s o r d e c r e a s e s . 16.  12 X 25  X 18  <^  12 X [  J X  18  Vhat number (or numbers) c o u l d go i n the [ ]? a. any number l e s s t h a n 25 b. Any nuaber g r e a t e r than 25 c.  12  d. Any nuaber g r e a t e r than 12 e. Any nuaber l e s s than 12  17.  14 X 9 1 1 3  =  1638  I f 9 i s r e p l a c e d by a s m a l l e r nuaber and i f 13 i s r e p l a c e d by a s m a l l e r number and i f 14 s t a y s t h e saae, what happens t o t h e p r o d u c t , 1638? a. I t decreases b. I t i n c r e a s e s c. I t s t a y s t h e saae d. I t changes b u t i t i s i m p o s s i b l e t o know whether i t i n c r e a s e s o r decreases e. I t c o u l d i n c r e a s e o r i t c o u l d decrease o r i t c o u l d s t a y t h e saae  18.  21 X 18 X 25 = 9450 I f 18 i s r e p l a c e d by a nuaber h a l f as b i g and i f 21 i s r e p l a c e d by a number one t h i r d as b i g and i f 25 s t a y s t h e same, what happens t o the p r o d u c t , 9450? a. I t i n c r e a s e s t o s i x times as auch b. I t i n c r e a s e s t o f i v e t i m e s a s much c. I t d e c r e a s e s t o one f i f t h a s much d. I t d e c r e a s e s t o one s i x t h as much e. I t changes but i t i s i m p o s s i b l e t o know hov much i t i n c r e a s e s o r decreases.  19.  4 X 27 X 17 = 1836 I f 4 i s r e p l a c e d by a number t w i c e as b i g and i f 17 i s r e p l a c e d by a number t h r e e times a s b i g and i f 27 i s r e p l a c e d by a number one t h i r d as b i g , what happens t o t h e p r o d u c t , 1836? a. I t i n c r e a s e s t o t h r e e t i m e s as much b. I t decreases t o one t h i r d as ouch c. I t i n c r e a s e s t o t w i c e as much d. I t decreases t o one h a l f a s much e. I t changes but i t i s i m p o s s i b l e t o know how much i t i n c r e a s e s o r decreases.  20.  16 X 154 X 2 = 4928 I f 16 i s r e p l a c e d by a number t w i c e a s b i q and i f 2 i s r e p l a c e d by a number seven times as b i q and i f t h e product, 4928, s t a y s t h e same, vhat happens t o 154? a. I t i n c r e a s e s t o nine t i m e s a s much b. I t i n c r e a s e s t o f o u r t e e n t i m e s a s much c. I t d e c r e a s e s t o one n i n t h as much d. I t d e c r e a s e s t o one f o u r t e e n t h as much e. I t changes but i t i s i m p o s s i b l e t o know how auch i t i n c r e a s e s o r decreases. ***************** End o f Test  ****************  

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