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Student difficulties with volumetric analysis Anamuah-Mensah, Jophus 1981

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STUDENT D I F F I C U L T I E S WITH VOLUMETRIC  ANALYSIS  by  JOPHUS ANAMUAH-MENSAH B . S c . ( E d . ) / U n i v e r s i t y o f Cape C o a s t , Ghana, 1971 B.Sc.(Chem.), U n i v e r s i t y o f Cape C o a s t , Ghana, 1972 M.Sc.(Chem.), U n i v e r s i t y o f Cape C o a s t , Ghana, 1974 M.A., The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in THE FACULTY  OF GRADUATE  DEPARTMENT OF MATHEMATICS AND  We  accept t h i s to  thesis  the required  THE UNIVERSITY  SCIENCE EDUCATION  as  conforming  standard  OF BRITISH COLUMBIA  August, (c^  STUDIES  1981  J o p h u s Anamuah-Mensah,  1981  In  presenting  requirements  this for  thesis  an a d v a n c e d  of  B r i t i s h Columbia, I  it  freely  agree for  available  that  or  understood  that  financial  by h i s  or her  the  copying of  granted  shall  I  make  further this  thesis  by t h e h e a d o f my  representatives.  n o t be  the  University  and s t u d y .  copying or p u b l i c a t i o n of  gain  of  the L i b r a r y s h a l l  extensive  p u r p o s e s may be  department for  reference  for  fulfilment  degree at  agree that  for  permission  scholarly  in partial  this  It  is  thesis  a l l o w e d w i t h o u t my w r i t t e n  permission.  Department  of  M/V-rKE MAT\C<L A^£>  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date  2.1  AMGrMJT  1  \  JOSUCEL  v3 C f y r \ ^  Abstract The that In  s t u d y was d e s i g n e d  g r a d e 12 c h e m i s t r y  the f i r s t  volumetric tegration  part  to investigate  s t u d e n t s have w i t h v o l u m e t r i c  o f the study,  analysis  This  c a l c u l a t i o n s was d e v e l o p e d  o f two t h e o r e t i c a l p e r s p e c t i v e s  t h e o r y and t h e c u m m u l a t i v e  integrated  model h y p o t h e s i z e d  among t h e v a r i a b l e s : proportional on  specific  calculations.  proportional  in a pilot inverse  c o n c e p t s and p e r f o r m a n c e The s e c o n d p a r t that  The f i r s t  s t u d y were u s e d t o m e a s u r e d i r e c t proportional  reasoning,  teaching  of the volumetric  analysis  i n t h e g r a d e 12 c h e m i s t r y c o u r s e p r i o r t o t h e analysis  unit.  After  analysis  unit,  p e r f o r m a n c e on v o l u m e t r i c  analysis  c a l c u l a t i o n s was  on  The f i n a l  a test  the teachers  taught the volumetric  istration  measuring  sample s i z e was 328.  o f t h e t e s t s , a s u b s a m p l e o f 47 s u b j e c t s  a titration  knowledge  t h r e e m e a s u r e s were a d m i n i s t e r e d t o  enrolled  the students.  examined  the students  subsumed c o n c e p t s and p e r f o r m a n c e on v o l u m e t r i c  subjects  to  inverse  analysis.  reasoning,  calculations.  had  t h e o r y o f Gagne.  reasoning,  c o n c e p t u a l and manual d i f f i c u l t i e s  Tests developed  schema i n  relationships  direct proportional  analysis  have w i t h v o l u m e t r i c  of  reasoning  some  prerequisite  from t h e i n -  of i n t e l l e c t u a l  learning  reasoning,  volumetric  analysis.  a model o f p e r f o r m a n c e on  p e r f o r m a n c e , namely, t h e p r o p o r t i o n a l Piaget's  the d i f f i c u l t i e s  task.  administered  After was  admininterviewed  The The  test  d a t a were a n a l y z e d  psychometric  internal  p r o p e r t i e s of the  consistency estimate  t e s t s was The  above  qualitative  procedures  number o f t i t r a t i o n s  that  and  The  p e r f o r m e d and  understanding  adequately  as u s i n g  r e v e a l e d t h a t a trimmed  direct  p r o p o r t i o n a l r e a s o n i n g was  effect  on  provided The  p r e r e q u i s i t e concepts a reasonable  trimmed and  in-  negligible  volumetric analysis the  proposed  as u s i n g a l g o r i t h m s  with  i n t e g r a t e d model i n w h i c h  volumetric  e x p l a n a t i o n of t h e i r  a n a l y s i s of the  a  assumed t o have n e g l i g i b l e  and  performance. on  the  r e v e a l e d a number o f  errors  (e.g. i n d i s c r i m i n a t e assumption of  by  subjects.  direct  analysis calculations  subjects' calculations  the v o l u m e t r i c a n a l y s i s instrument  the  and  e v a l u a t i o n of  i n t e g r a t e d model f o r s u b j e c t s i d e n t i f i e d understanding  analysis.  algorithms  proportional reasoning  The  prior  volumetric  e x p l a i n e d by  p r e r e q u i s i t e concepts  respectively.  response  i n t e g r a t e d model r e v e a l e d  v e r s e p r o p o r t i o n a l r e a s o n i n g were assumed t o have i n f l u e n c e on  of  r e l a t i o n s h i p between  subjects i d e n t i f i e d  c o u l d be  on  using  using c o r r e l a t i o n a l  e v a l u a t i o n of the proposed  calculations,  the  subjects  frequency  performance or  analyzed  i n t e g r a t e d model i n w h i c h d i r e c t  direct  The  f o r each o f  t h e w r i t t e n work o f t h e  errors).  the performance of  without  of r e l i a b i l i t y  (e.g. c a t e g o r i z i n g the  c a l c u l a t i o n s was  The  techniques.  t e s t s were a s s e s s e d .  a n a l y s i s c a l c u l a t i o n s were a n a l y z e d  patterns or conceptual  analysis  analysis  0.75.  i n t e r v i e w data  the v o l u m e t r i c  using path  1:1  items  in  conceptual  mole r a t i o s )  made  iv The during  a n a l y s i s of the manipulatory  the interview revealed  be a d e q u a t e l y  skills  that while  developed, other  important  of the subjects  some s k i l l s skills  seemed t o  seemed t o be  lacking. The involved  a n a l y s i s of the students i n an a c i d - b a s e  1  titration  understanding revealed  of concepts  that the concepts  o f pH and i n d i c a t o r b e h a v i o u r as w e l l as t h e u s e o f c e r t a i n scientific The calculate data, the by was  terminologies  s u c h as e n d p o i n t  were n o t w e l l  a n a l y s i s o f t h e a p p r o a c h e s u s e d by t h e s u b j e c t s t o the concentration  during  the interview,  o f the a c i d s o l u t i o n from t h e i r revealed  the subjects  i n their  u s e d by a g r e a t e r  solution.  However, t h e F o r m u l a  education  from the i n t e r p r e t a t i o n o f the r e s u l t s .  strategies reflecting  use a l g o r i t h m s  Approach  i n the secondary  Such i m p l i c a t i o n s i n c l u d e t h e d e v e l o p m e n t o f d i f f e r e n t tional  used  proportion of the subjects.  i m p l i c a t i o n s f o r chemical  s c h o o l were i n f e r r e d  own  t h a t two m a i n a p p r o a c h e s -  P r o p o r t i o n a l A p p r o a c h and t h e F o r m u l a A p p r o a c h - were  Several  who  understood.  with  the d i f f e r e n t  and w i t h o u t  path  instruc-  models f o r students  understanding.  V  TABLE OF CONTENTS Page CHAPTER I  II  III  INTRODUCTION  1  Background o f the Problem Analytic Orientations D e f i n i t i o n o f Terms Statement o f the Problem Research Questions Overview o f the Study  1 4 7 9 10 14  REVIEW OF LITERATURE  16  Introduction The P r o p o r t i o n a l i t y Schema i n P i a g e t ' s Model o f Adolescent Reasoning Gagne's Theory of Learning Comparison of Gagne's and P i a g e t ' s Theories Related S t u d i e s : P r o p o r t i o n a l Reasoning and Content P r o p o r t i o n a l Reasoning S t u d i e s P r o p o r t i o n a l Reasoning as an Explanatory Construct Content as an Explanatory C o n s t r u c t I n f l u e n c e of Content on L o g i c a l Reasoning P a t h - A n a l y t i c Model o f Performance S t u d i e s Related t o the Secondary Questions i n t h i s Study  16 16 33 38 41 41 42 48 51 54 60  PILOT TESTING OF INSTRUMENTS AND PROCEDURES  66  Introduction Instruments Classroom P r o p o r t i o n a l i t y Test Subconcepts T e s t ,  66 66 67 69  vi CHAPTER III  IV  Page ..Volumetrie A n a l y s i s Test Laboratory S k i l l s i n T i t r a t i o n ...... The I n t e r v i e w Task . . . . . .•„•-.•. .... P i l o t Study ..... ................. ..... Pu.3rpo S S • • •>••«-. • > • • • • « , • *•>••••-••«>-*'•-• * • • • • • • The Sample Data C o l l e c t i o n Procedure Scoring Data A n a l y s i s R e s u l t s and D i s c u s s i o n R e v i s i o n s o f the Instrument Volumetric A n a l y s i s T e s t Classroom P r o p o r t i o n a l i t y Test Subconcepts T e s t Laboratory S k i l l s i n T i t r a t i o n The Interview Task METHODS OF THE STUDY Introduction Instruments Volumetric A n a l y s i s Test Classroom P r o p o r t i o n a l i t y Test Subconcepts Test Population Sample Data C o l l e c t i o n Procedures Test A d m i n i s t r a t i o n Interview Sample Interview Data C o l l e c t i o n Procedure ... Data A n a l y s i s Data P r e p a r a t i o n Scoring P r e l i m i n a r y Analyses o f Data f o r Model T e s t i n g R e l i a b i l i t y o f Instruments Differences of Classes I Path A n a l y s i s o f Data A n a l y s i s o f Data f o r the S p e c i f i c Student D i f f i c u l t i e s A n a l y s i s of Conceptual E r r o r s A n a l y s i s o f Interview Data Manual S k i l l s i n T i t r a t i o n Ideas About the Concepts Involved i n Titration Problem S o l v i n g Approach and P r e d i c t i v e Behaviour Performance on VAT and Number o f T i t r a t i o n Experiments  73 74 76 78 -"78 78 79 83 83 85 97 97 98 98 99 99 101 101 101 10 2 102 102 102 103 104 104 105 106 108 108 109 109 109 l l 114 117 117 120 120 120 122 124  vii CHAPTER V  Page RESULTS A.ND DISCUSSION OF MODEL TESTING ..... Introduction E v a l u a t i o n o f the I n t e g r a t e d Model S t a t i s t i c a l Hypotheses Model E v a l u a t i o n f o r S u b j e c t s Using Algorithms without Understanding Model E v a l u a t i o n f o r S u b j e c t s Using Algorithms with Understanding  VI  RESULTS  AND  STUDENT  DIFFICULTIES  DISCUSSION  OF  DATA  ON  126 127 127 128 141  SPECIFIC  Introduction Conceptual E r r o r s on V o l u m e t r i c A n a l y s i s Test Manual S k i l l s i n T i t r a t i o n Concepts i n T i t r a t i o n Student D i f f i c u l t i e s w i t h pH Student D i f f i c u l t i e s w i t h C o n c e n t r a t i o n and i t s R e l a t i o n s h i p t o Moles and pH.. Student D i f f i c u l t i e s w i t h I n d i c a t o r Behaviour Student D i f f i c u l t i e s w i t h S c i e n t i f i c Terminology Approaches used i n the Computation of S o l u t i o n C o n c e n t r a t i o n R e l a t i o n s h i p Between P r i o r Number of T i t r a t i o n s and Performance on VAT R e l a t i o n o f Student D i f f i c u l t y Data to the V a l i d a t e d Model VII  126  SUMMARY AND IMPLICATIONS Summary Purpose Procedure Analysis Results D e l i m i t a t i o n s of the Study Implications Future Research REFERENCES  152  152 153 162 17 2 172 180 18 6 200 204 231 233 237 237 237 238 239 241 248 249 253 255  APPENDIX A B  VOLUMETRIC CLASSROOM B.l B.2  ANALYSIS TEST PROPORTIONALITY TEST Answer B o o k l e t T e s t i n g Procedures f o r each Item  266 276 277 285  viii APPENDIX  Page  C  S U B C O N C E P T S T E S T ...... . , . . . ...... . . ....  D  LABORATORY S K I L L S LIST  I N TITRATION  CHECK300  E  I N T E R V I E W PROCEDURE  303  F  SUMMARY I T E M S T A T I S T I C S FOR D I R E C T PROPORTIONALITY SUBTEST  306  SUMMARY I T E M S T A T I S T I C S FOR I N V E R S E PORTIONALITY SUBTEST  308  G H  I J  K L  M  PRO-  SUMMARY I T E M S T A T I S T I C S FOR T H E T O T A L T E S T AND S U B T E S T S OF THE S U B C O N C E P T S TEST  310  SUMMARY I T E M S T A T I S T I C S FOR THE VOLUMETRIC A N A L Y S I S TEST  315  MEAN AND STANDARD D E V I A T I O N FOR THE D I R E C T P R O P O R T I O N A L I T Y , I N V E R S E PROP O R T I O N A L I T Y , S U B C O N C E P T S AND V O L U M E T R I C A N A L Y S I S T E S T SCORES BY C L A S S , S E P A R A T E L Y AND COMBINED  317  DATA FOR THE L A B O R A T O R Y S K I L L S I N T I T R A T I O N INSTRUMENT  320  I N T E R V I E W DATA FOR E A C H I N T E R V I E W S U B J E C T ON C O N C E P T S I N V O L V E D I N A TITRATION  322  I N T E R V I E W DATA FOR E A C H S U B J E C T ON THE C O M P U T A T I O N P R O B L E M  408  290  ix  LIST OF TABLES TABLE 1  2  Page The C o m b i n a t o r i a l o f the P r o p o s i t i o n a l Elements of P i a g e t ' s Model o f Formal Thought  ,  23  R e l a t i o n s h i p between P r o p o s i t i o n s : The I,N,R, and C Operations  25  3  T e s t S t a t i s t i c s of the Classroom Prop o r t i o n a l i t y Test  86  4  T e s t S t a t i s t i c s o f the Subtests Subconcepts T e s t  88  5  T e s t S t a t i s t i c s o f the Volumetric Analysis Test  6 7 8  9  10  11  12  i n the  Test A d m i n i s t r a t i o n and I n t e r - t e s t Breaks Summary T e s t S t a t i s t i c s Tests  91 108  fora l l 4 I l l  T e s t o f D i f f e r e n c e s i n Means and Homog e n e i t y of D i s p e r s i o n among D i r e c t P r o p o r t i o n a l i t y , Inverse P r o p o r t i o n a l i t y , Subconcepts and Volumetric A n a l y s i s T e s t Scores o f C l a s s e s  112  Test o f D i f f e r e n c e s i n Means and Homog e n e i t y o f D i s p e r s i o n among D i r e c t P r o p o r t i o n a l i t y , Inverse P r o p o r t i o n a l i t y , Subconcepts and Volumetric A n a l y s i s T e s t Scores of C l a s s e s  113  Covariance M a t r i x f o r S u b j e c t s Using Algorithms Without Understanding  128  Parameter E s t i m a t e s , Standard E r r o r s and C r i t i c a l R a t i o s f o r Subjects Using Algorithms Without Understanding  129  R e s i d u a l M a t r i x R e s u l t i n g from the D i f f e r e n c e Between the Sample Covariance M a t r i x and t h e Reproduced Covariance M a t r i x f o r Subjects Using Algorithms Without Understanding  132  X  TABLE. 13  14  15  Page Parameter E s t i m a t e s , Standard E r r o r s and C r i t i c a l R a t i o s o f Trimmed I n t e grated Model f o r Subjects Using Algorithms Without Understanding  133  E f f e c t s A n a l y s i s i n the I n t e g r a t e d Model f o r S u b j e c t s Using Algorithms Without Understanding  135  Parameter E s t i m a t e s , Standard E r r o r s C r i t i c a l R a t i o s f o r Subjects Using Algorithms With Understanding  and 14 2  16  Covariance M a t r i x f o r Subjects Using Algorithms With Understanding  14 3  17  R e s i d u a l M a t r i x R e s u l t i n g from the D i f f e r e n c e Between the Sample Covariance M a t r i x and the Reproduced Covariance M a t r i x f o r S u b j e c t s Using Algorithms With Understanding  144  Parameter E s t i m a t e s , Standard E r r o r s and C r i t i c a l R a t i o s of Trimmed I n t e g r a t e d Model f o r Subjects Using Algorithms With Understanding  147  E f f e c t s A n a l y s i s i n the I n t e g r a t e d Model f o r S u b j e c t s Using Algorithms With Understanding  147  Parameter E s t i m a t e s , Standard E r r o r s and C r i t i c a l R a t i o s f o r S u b j e c t s Using Algorithms w i t h Understanding  148  R e s i d u a l M a t r i x f o r the Next Trimmed Model  149  Conceptual E r r o r s on the V o l u m e t r i c Analysis Test  154  23  Manual S k i l l s  163  24  Manual S k i l l s D i s p l a y e d by Subjects C l a s s i f i e d According to the Number of T i t r a t i o n s Performed Since Grade 11  18  19  20  21 22  in Titration  16 6  xi  TABLE 25  Page Questions and Response P a t t e r n s Derived from Interview Data i n the area of pH  173  Questions and Response P a t t e r n s D e r i v e d from Interview Data i n the Area of C o n c e n t r a t i o n  181  Questions and Response P a t t e r n s Derived from Interview Data i n the Area of I n d i c a t o r Behaviour  187  Questions and Response P a t t e r n s Derived from Interview Data i n the Area of S c i e n t i f i c Terminology  201  29  Approaches to Problem S o l v i n g ployed by S u b j e c t s  Em-  206  30  Approaches to Problem S o l v i n g ployed by Subjects  Em-  31  D i s t r i b u t i o n of Students to Approach  26  27  28  2 07  According 214  32  D i s t r i b u t i o n of Interview Sample by S t r a t e g y and Achievement  215  33  D i s t r i b u t i o n of S u b j e c t s According to Approach and Number of C o r r e c t Responses on the 3 P r e d i c t i o n Questions  219  Student P r e d i c t i o n s of the Concent r a t i o n of A c i d from T i t r a t i o n Results  223  34  xii  LIST OF FIGURES FIGURE 1  Page Proposed I n t e g r a t e d Model o f P e r formance on V o l u m e t r i c A n a l y s i s Calculations  11  H i e r a r c h i c a l Arrangement Types  of  35  3  Hierarchical Calculations  Titration  4  Proposed I n t e g r a t e d Model o f Performance on V o l u m e t r i c A n a l y s i s C a l c u l a t i o n s  58  5  H i e r a r c h i c a l Analysis of Calculations (Modified)  70  6  O v e r a l l Design cedure  7  I n t e g r a t e d Model f o r S u b j e c t s U s i n g Algorithms Without Understanding  8  Trimmed I n t e g r a t e d M o d e l f o r S u b j e c t s Using Algorithms Without Understanding  2  9 10 11 12 13 14 15 16  Analysis  of  of  Learning  50  Titration  Experimental  Pro-  107 130 ...  133  Proposed I n t e g r a t e d Model for Subjects Using Algorithms With Understanding  143  Trimmed M o d e l f o r S u b j e c t s U s i n g Algorithms With Understanding  146  The F u r t h e r Trimmed M o d e l f o r S u b j e c t s Using Algorithms With Understanding  14 9  The F o r m u l a A p p r o a c h t o S o l u t i o n - the b a s i c  Problem 208  The F o r m u l a A p p r o a c h t o - Variant  Problem  Solution 209  The P r o p o r t i o n a l A p p r o a c h t o Solution - Basic  Problem  The p r o p o r t i o n a l A p p r o a c h t o Solution - Variant  Problem  Diagramatic d i c t i o n s of Approaches  211  R e p r e s e n t a t i o n of the P r e S u b j e c t s U s i n g the D i f f e r e n t  212  220  xiii  ACKNOWLEDGEMENTS  The  author would l i k e t o express h i s s i n c e r e  gratitude  to Dr. Gaalen E r i c k s o n and Dr. P. James G a s k e l l , h i s t h e s i s a d v i s o r s , f o r t h e i r i n v a l u a b l e guidance, encouragement and generous c o n t r i b u t i o n o f t h e i r time and e x p e r t i s e the p e r i o d o f r e s e a r c h The  and i n the p r e p a r a t i o n  throughout  o f t h i s manuscript.  author would a l s o l i k e t o thank Dr. Todd Rogers f o r  the generous c o n t r i b u t i o n of h i s e x p e r t i s e throughout the period of research t h i s manuscript.  and h i s comments t o the i n i t i a l d r a f t o f The other members of my Committee, D r s .  Robert Thompson, Pat A r l i n and Reginald acknowledged f o r t h e i r v a l u a b l e the i n c e p t i o n o f t h i s r e s e a r c h . thank Dr. Stanley  Wild a r e g r a t e f u l l y  comments and suggestions  since  The author would a l s o l i k e t o  J . T a y l o r f o r h i s i n v a l u a b l e help  throughout  the p e r i o d of the author's study a t U.B.C. Finally,  I would l i k e t o express my s i n c e r e g r a t i t u d e t o  my w i f e , B e a t r i c e Anamuah-Mensah and my c h i l d r e n , Ebow, Ekua, Maame and S t a y l o r f o r t h e i r p a t i e n c e throughout my f o u r year sojourn  and emotional support  a t U.B.C.  1  CHAPTER I INTRODUCTION  Background to the Problem In recent years s c i e n c e education  researchers  been c a l l e d upon to i n c r e a s e the amount of r e s e a r c h difficulties  students have w i t h s p e c i f i c s c i e n c e  (Shulman and  Tamir, 1973;  L i n n , 1977).  D r i v e r and  Kuhn (1979) has  Easely,  i n the t h e o r i e s themselves.  the  psychological  ambiguities  psychological  remove some of the problems encountered when they  are a p p l i e d s e p a r a t e l y  ( S t i c h t , 19.71; G r i f f i t h s ,  i n v e s t i g a t i n g c h i l d r e n ' s d i f f i c u l t i e s with school As  and  I t has been noted t h a t  the simultaneous a p p l i c a t i o n of more than one theory may  Levine  suggested t h a t some of  t h e o r i e s to t h e i r s p e c i f i c areas r e f l e c t the  the  concepts  1978;  problems s c i e n c e educators encounter i n a p p l y i n g  inherent  on  have  such, there e x i s t s a need f o r r e s e a r c h  concept s p e c i f i c and psychological  theory.  The  present  1979)  in  concepts.  that i s science  t h a t attempts to i n t e g r a t e more than  one  study, i n keeping with the p o s i t i o n s  s t a t e d above, examined a p e r s i s t e n t problem i n h i g h chemistry, namely, the d i f f i c u l t i e s  school  t h a t students experience  2 with volumetric analysis. grew from t h e w r i t e r ' s the  The c h o i c e o f  experience  introductory chemistry  course  c o n c e r n e x p r e s s e d by o t h e r W h e e l e r and K a s s ,  1977).  and t h e d i f f i c u l t i e s analysis  done t o in  identify  i n high school  Both chemistry the  1976).  content)  or  structure  inherent  to the  those task  components  difficulties c o u l d be ing  students  identified,  encounter  then i t  Also,  isolation  understanding  collectively  Consider  sources of  sources  processing  solve the  alleviate  problems  abilities  20 mL o f  sodium h y d r o x i d e s o l u t i o n .  the  sodium h y d r o x i d e  (Duncan and  of  solution?  If  1977). to  such  the  tasks  design  these  teach-  difficul-  d i f f i c u l t y may h e l p  of  students  following t y p i c a l volumetric  1 M solution  components  involving volumetric  25 mL o f  been  analysis,  i n d e p e n d e n t l y and  problem:  of  to  has  difficulties  task  in solving  may be p o s s i b l e  the  how t h e s e  affect  s t u d e n t s come t o  of  study  which c o n t r i b u t e  s t r a t e g i e s which would h e l p to  ties. in  that  volumetric  task  1973; H e r r o n 1975; W h e e l e r and K a s s , structure  have  1973; J o h n s t o n e ,  Johnstone,  or  the  students  students1  i n the  of  and  in volumetric  have b e e n a t t r i b u t e d  aspect  and by  teachers  A l t h o u g h no r i g o r o u s  c o n t r i b u t i n g to  analysis  ( H e r r o n , 1975;  apprehensions  performing computation problems  (i.e.  educators  (Duncan and J o h n s t o n e ,  factors  these d i f f i c u l t i e s  this  t h e y e n c o u n t e r when d o i n g  calculations  M o r r i s o n and S h a r p ,  i n teaching  chemistry  r e s e a r c h e r s have b e e n aware o f  volumetric  hydrochloric acid What i s  the  '.  '.  and how analysis. analysis neutralizes  concentration  3 In s o l v i n g  s u c h a p r o b l e m a s t u d e n t may  reason:  " F i r s t , I need t o f i n d the s t o i c h i o m e t r i c r e l a t i o n between HC1 and NaOH. T h i s i s 1:1. T h e r e f o r e , t h e number o f m o l e s o f HC1 i n t h e r e a c t i o n w o u l d be e q u a l t o t h e number o f m o l e s o f NaOH i n t h e r e a c t i o n . Since I c a n c a l c u l a t e t h e number o f m o l e s o f HC1 from i t s c o n c e n t r a t i o n and v o l u m e , I c a n d e t e r m i n e t h e c o n c e n t r a t i o n o f t h e NaOH s o l u t i o n from t h e e q u a l i t y . " Such r e a s o n i n g a)  the  involves:  recognition  stoichiometric  b)  c)  reactants;  the  recognition  the  tural  of  understanding  inverse  of  s u c h as  equations,  and c a l c u l a t i o n  and r u l e s  of  proportionality the  the mole,  balancing  of  and  (i.e.  of  amount o f  proportionality  the  Although a l l  content  three  may be n e c e s s a r y  for  t h e y may n o t a l l  be r e q u i r e d  solve the problem.  concepts  chemical  reactants  constitute  is  the  from  for  V  a  together, the  problem  successfully  b e c a u s e s u b j e c t s may p e r f o r m  formulas  _  taken  a subject to  the  HC1 HC1  concepts  of  resorting  of  subordinate  struc-  understanding  showing t h e u n d e r s t a n d i n g use  the  problem.  requirements,  a subject's  This  of  task without  C  content  of  between  reactants;  the problem w h i l e the  constitute  to  the  equations.  and i n v e r s e  components  between  and t h e number o f m o l e s  subordinate  and r u l e s )  direct  proportionality  and volume o f  stoichiometric The  direct  coefficients  the  concentration  of  C  NaOH  such  V  l a i d out as  NaOH,  b  above by  the  4 where C and V r e p r e s e n t solutions,  and a and b a r e  (or mole r a t i o s ) This  alternate  for  the  of  coefficients.  s t u d y were using  seen  Analytic  this  adequate  Such l a c k  of  stoichiometric  issue,  subjects  in  volumetric  the c o r r e c t  and t h o s e who  coefficients  the c o e f f i c i e n t s  problems  of  regardless  of  the  analysis  the  i n terms  Mealings,  1969)  of  the problem.  growing concern over structures  sole  reliance  to  1977;  theory,  and  without  variables  such  seems t o be  performance  Smedslund,  as a  operational  performance  requiring equivalent  (Brown and D e f o r g e s ,  Ingle  i n problem  t h e s e c o n c e r n s were e x p r e s s e d  the d i f f e r e n t i a l  on t a s k s  1975;  there on  have  operational  Piaget's  Currently,  Some o f  the  to contextual  in analyzing student's  situations.  a reaction  the  of  (Herron,  using  adequate c o n s i d e r a t i o n  the content  t h e t y p e o u t l i n e d above  demanded by t h e p r o b l e m  1971;  subjects  1978).  i n the c h o i c e of  been a n a l y z e d o n l y  structures  solving  Smith,  without  have  Orientations:  generally  giving  i n chemistry which  formulas  clarify  has  coefficients.  Chemical  Shayer,  respectively.  i d e n t i f i e d who p e r f o r m e d  assumed u n i t v a l u e s f o r reacting  of  the  coefficients  a memorized a l g o r i t h m  areas  1974;  To f u r t h e r  calculations  actual  using  other  (Chappetta,  u n d e r s t a n d i n g may be  stoichiometric  a c i d and t h e b a s e ,  the predominant use  understanding  present  the  solution of  been n o t e d i n s t u d i e s reported  t h e c o n c e n t r a t i o n and volume o f  of  logical 1977).  experimental structures As a  result  as  5 many r e s e a r c h e r s of  the task  i n a d d i t i o n to  (Abramowitz, 1975) . the  have c a l l e d f o r  1975; D r i v e r  O t h e r s have  subject's  Berzonsky,  B u t as  In the present  Piaget's  theory of  the b a s i s  an  i d e n t i f y the  upon p e r f o r m a n c e The and  the content  constituted  — the  on  a set  inverse solve  three  t h e c o n t e n t on  (Inhelder  (1977) c u m m u l a t i v e l e a r n i n g  for  influences  of  structure  and i n v e r s e  i n the  s t u d y was t h e  employed.  theory—  concepts  study.  —  The performance  A s u b j e c t who  namely d i r e c t p r o p o r t i o n a l i t y ,  problems  s u b j e c t s m i s s i n g any o f  and c o n t e n t  proportionality —  student's  problems.  requirements,  the v o l u m e t r i c a n a l y s i s  interrelationships  and  problems.  subsumed p r e r e q u i s i t e  t o have some d i f f i c u l t i e s  —  d e v e l o p i n g an i n t e g r a t e d m o d e l i n  — direct  i n the  cummulative.  frameworks  p r o p o r t i o n a l i t y , and subsumed c o n c e p t s ,  contrast,  Legrenzi's  have b e e n done  two t h e o r e t i c a l  of volumetric analysis  possessed a l l  to  1971a; 1973;  1979) u s i n g G a g n e ' s  the p r e d i c t o r v a r i a b l e s  dependent v a r i a b l e  (Lovell,  of  on v o l u m e t r i c a n a l y s i s  structure  involved  1978; L u n z e r , 1965;  i n t e l l e c t u a l development  were t a k e n as to  structures  end,studies  the e f f e c t  study,  1958) and G a g n e ' s  attempt  content  more a t t e n t i o n be p a i d  To t h i s  Griffiths,  theory.  the  f a m i l i a r i t y may be n o t h i n g more t h a n  t o examine  (e.g.  of  J o h n s o n - L a i r d , L e g r e n z i and  the content.  which attempted  Piaget,  and E a s e l y ,  suggested that  (1972) s t u d y , i n d i c a t e d ,  performance  the o p e r a t i o n a l  f a m i l i a r i t y w i t h the task  1971).  knowledge o f  an a n a l y s i s  was e x p e c t e d  successfully.  these s k i l l s  w i t h the problems.  among t h e s e v a r i a b l e s ,  were  to  In expected  In examining  path a n a l y s i s  was  the  6 Apart tion  of  other  t h e above  areas  students'  relationships,  relevant  errors  analysis  three  of  One o f  of  the  these  concep-  made by s u b j e c t s when s o l v i n g t h e v o l u m e t r i c  calculation  problems. was an e x a m i n a t i o n o f  the  and c o n c e p t u a l u n d e r s t a n d i n g w h i c h s t u d e n t s t i t r a t i o n task.  analysis  was made o f  students  when t h e y a p p l i e d t h e r e s u l t s  the types  of volumetric analysis The t h i r d  area  of  calculations  examining these concerns. examination of frequency of  students'  e r r o r s , made,  a t i t r a t i o n experiment  displayed  F u r t h e r m o r e , an  strategies of  u s e d by  these  their titration  i n v o l v e d an i n v e s t i g a t i o n  and t h e number o f  Three d i f f e r e n t  laboratory  on v o l u m e t r i c  laboratory  analytic  of  analysis  titrations  techniques  ( 2 ) the use and  of  ( 3 ) t h e use  the type clinical of  performed.  were u s e d  T h e s e were r e s p e c t i v e l y , w r i t t e n work f o r  the  in  ( 1 ) the and  interviews  correlational  analysis. These i n v e s t i g a t i o n s insight  i n t o the d i f f i c u l t i e s  analysis, results  and  ( 2 ) data  from the p a t h  to  problems.  r e l a t i o n s h i p between p e r f o r m a n c e  in  understanding  i d e n t i f i c a t i o n and c l a s s i f i c a t i o n  when d o i n g a p r a c t i c a l  a set  valida-  study a l s o examined  with volumetric analysis.  The s e c o n d a r e a skills  the  to gaining further  difficulties  i n v o l v e d the tual  from t h e d e v e l o p m e n t and s t a t i s t i c a l  provided: students  to enrich analysis.  the  (1) additional  have i n v o l u m e t r i c  i n t e r p r e t a t i o n of  the  7 D e f i n i t i o n of Terms V o l u m e t r i c A n a l y s i s : the process whereby the amount of a chemical substance  i n a g i v e n sample i n v o l v e d i n a q u a n t i t a t i v e  chemical r e a c t i o n i s determined  with the a i d of a chemical  i n d i c a t o r i n a process c a l l e d t i t r a t i o n . sis,  the amount of one of the r e a c t a n t s i s unknown.  o b j e c t i v e t h e r e f o r e i s to determine of  In v o l u m e t r i c analy-.'  the d e s i r e d substance  composition  The  the mass or c o n c e n t r a t i o n  i n a sample whose q u a n t i t a t i v e  i s unknown (Toon and E l l i s ,  1973).  V o l u m e t r i c A n a l y s i s C a l c u l a t i o n or Problem: the d e t e r m i n a t i o n of  the c o n c e n t r a t i o n or mass of one chemical substance  the known c o n c e n t r a t i o n of the other substance reaction. was  i n an  from  acid-base  In the present study, performance on t h i s v a r i a b l e  measured by the  'Volumetric A n a l y s i s T e s t '  D i r e c t P r o p o r t i o n a l i t y Reasoning:  (see Appendix A ) .  two v a r i a b l e s so r e l a t e d  t h a t t h e i r r a t i o i s constant, are d i r e c t l y p r o p o r t i o n a l t o each other was  (Vance,  1962).  In t h i s study, d i r e c t  measured with the f i r s t  t i o n a l i t y Test'  s u b t e s t of the  proportionality 'Classroom  Propor-  (see Appendix B).  Inverse P r o p o r t i o n a l i t y Reasoning:  two v a r i a b l e s so r e l a t e d  t h a t t h e i r product i s a c o n s t a n t , are i n v e r s e l y p r o p o r t i o n a l to was  each o t h e r .  In the present study, i n v e r s e p r o p o r t i o n a l i t y  d e f i n e d i n terms of scores on the second  'Classroom  Proportionality Test'  Subsumed or P r e r e q u i s i t e Concepts:  s u b t e s t of the  (see Appendix B). concepts hypothesized t o  8 be n e c e s s a r y  for  understanding volumetric analysis  T h e s e subsumed c o n c e p t s volumetric  analysis  such problems are  identified  problems  therefore,  integrally related  in that  is  c a n n o t be u n d e r s t o o d  not a v a i l a b l e  prerequisite  are,  to the  concepts  are  the mole,  to  the  this  as  will  study, s t u d e n t s '  subsumed p r e r e q u i s i t e  concepts  Test  necessary  These  concepts  interpret-  reactants  and  (contained  Using  be  Gagne's  in a  shown i n C h a p t e r I I  in this  study.  knowledge about  For  the  was d e r i v e d t h r o u g h c o n t e n t  of volumetric analysis  concepts  subsumed  t o be  were a r r a n g e d  t h e o r d e r was n o t p a r t i c u l a r l y r e l e v a n t  analysis  the  b a l a n c i n g and  substances.  o r d e r even though,  of  concepts  concepts  (1977a)  mass o f  that  and c a l c u l a t i n g t h e c o n c e n t r a t i o n o f  cummulative model, these concepts  the purposes  the  to  c o n v e r t i n g a s o l u t i o n from one c o n c e n -  from t h e mass o f  hierarchical  study,  calculations.  calculating relative  another,  subsumed  In t h i s  gram/mole c o n v e r s i o n s ,  from e q u a t i o n s ,  solutions  the  hypothesized  included essentially  solving volumetric analysis  tration  if  subject.  for  products  it  by Gower, D a n i e l s and L l o y d  ing equations,  problems.  calculations  i n A p p e n d i x C)  u s i n g the  c o m p l e t e d by  Sub-  the  students. Reasoning Strategy: relationships problems  out of  a plan,  or  experience,  and g e n e r a t i n g  i n p r o b l e m s o l v i n g as  has,  its  aim,  for  generating  i n other words,  new k n o w l e d g e .  operate as  strategy  part  of  the o r g a n i z a t i o n of  for  Reasoning one b a s i c  experience  orderly  solving  strategies  process  which  i n t o meaning-  ;  9 ful  systems o f  objects,  a sense e x p l i c i t The r e a s o n i n g  their  solutions  view  situation.  to  interest  an e r r o r  i n reasoning  in his/her  a volumetric analysis rule  or  s o l v i n g problems study,  formulas  and r u l e s as  The p r e s e n t  As w i l l that  tional  chemical  and t h a t  influence  proportional  the  of  as  he o r  reasoning  she  aids In  in  this  calcula-  Problem  of  aspects of  intellectual  t h e o r y and t h e  volumetric analysis II,  Piaget's  reasoning  theory  precedes  both inverse  further  and d i r e c t  on c e r t a i n  performance  namely,  cummulative  for  an  calculations. suggested  inverse  suggested that  influences  two  performance,  a model t o a c c o u n t  achievement It  as  data meaningful.  integrated  Gagne,into  i n Chapter  concepts.  calculation  in Piaget's  proportional  reasoning  reasoning  of  performance  be a r g u e d  direct  study  schema  inter-  algorithms.  perspectives  theory of  individual's  verbalized  misconception  involved in stoichiometric  Statement of  learning  or a  were  problem.  or making a s e t  the p r o p o r t i o n a l  study  they  f o r m u l a e m p l o y e d by i n d i v i d u a l s  were c l a s s i f i e d  theoretical  i n the present as  in  1979).  a v o l u m e t r i c a n a l y s i s p r o b l e m i n an  by a s u b j e c t  Algorithm:  tions  of  They a r e  (Lawson,  o r a p p r o a c h e s u s e d by s u b j e c t s  Conceptual E r r o r :  solved  and s i t u a t i o n s .  guides to problem s o l v i n g  strategies  the procedures  displayed  events,  propor-  proportional  prerequisite inverse on v o l u m e t r i c  10  analysis  calculations.  Gagne s u g g e s t e d t h a t  The c u m m u l a t i v e l e a r n i n g t h e o r y o f  the p r e r e q u i s i t e  i n f l u e n c e .' p e r f o r m a n c e Taken t o g e t h e r ,  these  on v o l u m e t r i c a n a l y s i s interrelationships  i n t e g r a t e d model r e p r e s e n t e d  the v a l i d i t y of  the  thereby gaining greater of  following  issues  volumetric a)  of  the  suggested  understanding of inverse  purpose  related  calculations.  s t u d y was,  the  therefore,  to  of  the  to  in Figure 1 ,  relative  influence  p r o p o r t i o n a l i t y , and  on v o l u m e t r i c a n a l y s i s  A secondary  content  i n t e g r a t e d model p r o p o s e d  direct proportionality,  sumed c o n c e p t s  or  i n Figure 1 .  The p r i m a r y p u r p o s e test  concepts  sub-  performance.  t h e s t u d y was t o examine  students'  difficulties  the  with  analysis:  the c o n c e p t u a l e r r o r s  made by s t u d e n t s  in their  calculations, b)  the  subjects'  the concepts c)  laboratory involved  and u n d e r s t a n d i n g  i n a t i t r a t i o n experiment,  t h e r e l a t i o n s h i p between t h e number o f p r i o r titration  p e r f o r m e d by t h e  on v o l u m e t r i c a n a l y s i s Exploratory investigations difficulties  was  subjects  and  laboratory  and t h e i r  the main aim o f  to p r o v i d e a d e t a i l e d view of  with volumetric  The g e n e r a l  of  performance  calculations.  i n nature,  Research  into  techniques  these the  subjects'  analysis.  Questions  c o n c e r n s o u t l i n e d above were  two d i s t i n c t b u t r e l a t e d  groups  of  questions  grouped according  11  Subsumed  Figure  *  1:  3  Concepts  Proposed I n t e g r a t e d Model of Performance on V o l u m e t r i c A n a l y s i s C a l c u l a t i o n s *  A f u l l e r e x p l a n a t i o n and j u s t i f i c a t i o n i n c l u d i n g a d e s c r i p t i o n o f the symbols C h a p t e r Two.  of t h i s used i s  model given i n  12 t o whether they f e l l  under the primary purpose  (i.e.  or the  of  model t e s t i n g )  the  the  the model  and e x p l o r a t o r y  Is  q u e s t i o n which r e l a t e d  the arrangement  of  of  variables  Specifically,  model c l o s e l y m i r r o r  of  the  the  does t h e  subjects'  in  the  application integrated  performance  on  tests?  This question t h e m o d e l as techniques  related  hypothesized  to  the d e t e r m i n a t i o n of  a whole t h r o u g h :  and  the paths  (ii)  connecting pairs i n f l u e n c e of  algorithms  (i)  t h e use  examination of  the  in their  on  literature,  solutions  the v a l i d i t y of  o f model c o m p a r i s o n  significance  of v a r i a b l e s  one v a r i a b l e  As s u g g e s t e d i n t h e certain  the t e s t i n g  represented  model v a l i d a c c o r d i n g t o  path a n a l y s i s ?  the  to  1:  integrated  (that  (Chiappetta,  1974;  made i n t h i s  study  through the  form o f  Students  t h e y u s e d a 1:1  to  Smith,  1978).  who u s e d a l g o r i t h m s mole r a t i o  when i t  to  students  the  algo-  of  was  students  the V o l u m e t r i c A n a l y s i s  indiscriminately, was n o t a p p l i c a b l e )  (e.g.  mole r a t i o  comprised the  who use  T h u s , an a t t e m p t  who u s e d a l g o r i t h m s  when t o a p p l y i t )  the  to c h e m i c a l problems  one g r o u p w h i l e t h o s e t h e y u s e d a 1:1  each  another).  i d e n t i f y t h e s e two g r o u p s  t h e i r responses  of  is,  may do so e i t h e r w i t h o r w i t h o u t u n d e r s t a n d i n g o f  Test.  section  was:  Question  rithm  study  study. The f i r s t  of  secondary  of  (e.g. formed  discriminately  w i t h an u n d e r s t a n d i n g second group.  It  was  of  13 hypothesized that differently might f i t other.  the data  Hence,  for  of  subjects might respond  used i n the study,  the second major q u e s t i o n addressed  Question  2:  Does t h e  i n t e g r a t e d model a c c o u n t f o r  data  t h e two g r o u p s  for  hypotheses  be s t a t e d  of  the  for  to the  research  questions  addressed  i n the  secondary  on t h e  4:  experiment?  a titration  skills  do s u b j e c t s  display  5:  What d i f f i c u l t i e s an a c t u a l  do;subjects  titration  have w i t h c o n c e p t s  6:  What a r e  the d i f f e r e n t  subjects  i n solving a volumetric analysis  Question  7:  t h e number o f to  Test?  approaches  u s e d by  laboratory titrations  students1  involved  experiment?  Question  Analysis  extent?  Test?  in  related  same  a r e made by s t u d e n t s  laboratory  Is  observed  3:  What p r a c t i c a l  in  t h e above  was:  were:  Volumetric Analysis  Question  subjects  questions  What c o n c e p t u a l e r r o r s  Question  of  the  the  i n Chapter IV.  study  Question  the p r o p o s e d model  one g r o u p more c l o s e l y t h a n f o r  The r e s e a r c h part  these groups  to the t e s t s  The s t a t i s t i c a l will  since  p e r f o r m a n c e on t h e  the problem?  performed Volumetric  14  The t h i r d q u e s t i o n sought to i d e n t i f y the e r r o r s made by the s u b j e c t s as they s o l v e d the problems on the Volumetric A n a l y s i s Test.  The next three q u e s t i o n s e x p l o r e d  the type of understanding t h a t s u b j e c t s b r i n g i n t o a c t u a l titration situations.  The manipulatory behaviour  t i c a l s k i l l s ) of the s u b j e c t s as w e l l as t h e i r of  the concepts i n v o l v e d i n t i t r a t i o n  ( i . e . prac-  understanding  (e.g. pH,  indicators,  terminology) were the s u b j e c t s of the f o u r t h and f i f t h q u e s t i o n s respectively.  The  s i x t h q u e s t i o n examined the approaches  used  by the s u b j e c t s while doing a V o l u m e t r i c A n a l y s i s problem. In  the seventh q u e s t i o n , the r e l a t i o n s h i p between  p r i o r involvement  i n performing t i t r a t i o n s and performance  v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s was involvement was  examined.  on  The degree  of  d e f i n e d as the number of l a b o r a t o r y t i t r a t i o n s  done s i n c e grade e l e v e n .  Overview of the In  t h i s chapter, the background, d e f i n i t i o n of terms  and the r e s e a r c h q u e s t i o n s posed In  Chapter  Study  i n the study were d e s c r i b e d .  I I , r e l e v a n t l i t e r a t u r e r e l a t e d to the p r e s e n t study  are reviewed.  In p a r t i c u l a r , l i t e r a t u r e r e l a t e d t o the  content of a task and the i n t e l l e c t u a l s t r u c t u r e t i o n a l reasoning) demand by a task are reviewed.  ( i . e . proporA l s o the  e m p i r i c a l and t h e o r e t i c a l underpinnings of the hypothesized model are d i s c u s s e d . In  Chapter  I I I , the data c o l l e c t i o n instruments  the p i l o t study are d e s c r i b e d .  Following t h i s ,  the  and  15  procedures  employed i n the main study f o r data c o l l e c t i o n and  analyses are presented i n Chapter  IV.  In Chapter V, r e s u l t s  are r e p o r t e d and d i s c u s s e d f o r the r e s e a r c h q u e s t i o n s r e l a t e d to the t e s t i n g of the model. posed i n the secondary  The r e s u l t s f o r the q u e s t i o n s  p a r t of the study are then  and d i s c u s s e d i n Chapter V I .  Finally,  i n the l a s t  presented chapter,  Chapter V I I , a summary of the study together with the t h e o r e t i c a l and e d u c a t i o n a l i m p l i c a t i o n s are presented.  16  CHAPTER I I REVIEW OF  THE  LITERATURE  Introduction  structure  The  focus of t h i s  and  content  literature  related  proportional theory  are  Following Studies  the  to these  reasoning  this,  two  areas.  the  related  to the  are  discussed.  then  studies related  the  operational hand.  content  as  studies  the  reported  integrated  analysis calculations.  secondary p a r t of the  study  discussed.  The  P r o p o r t i o n a l i t y Schema I n P i a g e t ' s M o d e l Adolescent Reasoning P i a g e t p o s t u l a t e d two  general  t i o n s which c h a r a c t e r i z e a d o l e s c e n t operations  i n Piaget's  terminology).  of  summarized.  u s e d t o c o n s t r u c t an  to the  review  chapter,  problem a t  The  the v o l u m e t r i c  underlying  the  formal  s t r u c t u r e and/or the  c o n s t r u c t are  literature  of  In t h i s  schema i n P i a g e t ' s  model o f p e r f o r m a n c e on  are  effect  Gagne's c u m m u l a t i v e model a r e  employing  Finally,  - the  performance - suggests  d e s c r i b e d and  explanatory in  on  study  of  conceptual  thinking These  (or are:  acquisi-  formal  17 (a)  propositional  This  is  clearly  logic,  and  (b)  formal o p e r a t i o n a l  s t a t e d by I n h e l d e r and P i a g e t  schemata. as  (1958)  follows: formal t h i n k i n g ( i . e . adolescent reasoning) makes i t s p r e s e n c e known n o t o n l y by t h e c o n s t a n t u t i l i z a t i o n of the s i x t e e n b i n a r y p r o p o s i t i o n a l o p e r a t i o n s and some t e r n a r y o r s u p e r i o r c o m b i n a t i o n s w h i c h d e r i v e from them b u t a l s o by t h e s p o r a d i c e l a b o r a t i o n o f some c o n c e p t s o r schemata which are i n a c c e s s i b l e at the c o n c r e t e l e v e l (p. Since  308)  a comprehensive review of  intended here,  attention w i l l  operational  schemata  be r e f e r r e d  t o where i t  tion  former  of  the  manifest  the  schemata at  that in  otherwise  they reappear  t h e most d i v e r s e  themselves  data,  concepts  but are  They a r e  are  "show some r e l a t i o n s h i p t o of  them t o  sense  applicable  Inhelder in  the the  and t h a t lattice  all or  of  subject these  group  INRC g r o u p o f  inver-  s i o n s and r e c i p r o c i t i e s " ' (pp. 3 0 9 - 3 1 0 ) . These combinatorial  schemata  :reasoning.,.  or general  concepts  proportions,  double  and  the  o r d e d u c e d by t h e  structures',  is  manifest  i n the  not d i s c o v e r e d  abstracted  concepts  which  problems or are  In the view o f  own ' o p e r a t i o n a l  and s e v e r a l  general  308-309),  subject  b u t w h i c h do n o t  from h i s  structures,  concepts  f o r m a l l e v e l when t h e  situations.  objects  only  explana-  ( 1 9 5 8 , pp.  general  i n many d i f f e r e n t  these general  logic will  schemata).  (p.308).  Piaget,  not  formal  to c l a r i f y P i a g e t ' s  constitute  the  is  on t h e  I n h e l d e r and P i a g e t  faced with c e r t a i n kinds of themselves  be f o c u s s e d  serves  (i.e.  themselves  formal stage  w h i l e the p r o p o s i t i o n a l  According to the o p e r a t i o n a l  the  include systems  of  18  r e f e r e n c e , mechanical e q u i l i b r i u m , p r o b a b i l i t i e s , c o r r e l a t i o n s , m u l t i p l i c a t i v e compensations, and forms of c o n s e r v a t i o n which go beyond d i r e c t e m p i r i c a l v e r i f i c a t i o n . The  f o l l o w i n g s e c t i o n w i l l present a summary o f  P i a g e t ' s e x p l a n a t i o n o f the p r o p o r t i o n a l i t y schema. P r o p o r t i o n i s a r e l a t i o n o f one v a r i a b l e t o another; it  i s the equivalence  s o l u t i o n chemistry,  o f two r e l a t i o n s o r r a t i o s .  Thus i n  the r e l a t i o n : c o n c e n t r a t i o n = moles/  volume, i s a p r o p o r t i o n a l r e l a t i o n s h i p .  Proportional  reasoning  t h e r e f o r e i n v o l v e s the a b i l i t y t o use p r o p o r t i o n s and r a t i o s . Inhelder and P i a g e t t i o n a l schema as having metrical proportions.  two aspects  p.  317)  view the propor-  - l o g i c a l p r o p o r t i o n s and  They p o i n t out t h a t l o g i c a l  precedes m e t r i c a l p r o p o r t i o n . which expresses  (1958,  proportion  That i s , l o g i c a l p r o p o r t i o n ,  the compensation between two heterogeneous  f a c t o r s such t h a t an i n c r e a s e i n the value of one g i v e s the same r e s u l t as an i n c r e a s e o r decrease i n the value o f the other  ( i n q u a l i t a t i v e terms),  i s achieved  (or numerical) p r o p o r t i o n which expresses t i o n s i n q u a n t i t a t i v e terms.  e a r l i e r than m e t r i c a l these  same compensa-  They noted t h a t whereas with  l o g i c a l p r o p o r t i o n , compensation may be a d d i t i v e o r m u l t i p l i c a t i v e , m e t r i c p r o p o r t i o n a l i t y i n v o l v e s only m u l t i p l i c a t i v e compensations.  Inhelder and P i a g e t p o i n t out t h a t t h i s  accounts f o r the " i n i t i a l tendency o f the c h i l d t o look f o r p r o p o r t i o n a l i t y i n the e q u a l i t y o f a d d i t i v e d i f f e r e n c e s " p. 2 2 3 ) .  ^(1958,  They a l s o s t a t e d t h a t t h i s compensation, which forms  an important  aspect  i n the a c q u i s i t i o n o f p r o p o r t i o n , d e r i v e s  19  d i r e c t l y from the i d e a of The  reciprocity.  p o s i t i o n t h a t p r o p o r t i o n i s a second-order  o p e r a t i o n i s c l e a r l y expressed  by Inhelder and P i a g e t ,  t h i s sense p r o p o r t i o n s presuppose second degree and the same may  operations  be s a i d of p r o p o s i t i o n a l l o g i c i t s e l f ,  i n t e r p r o p o s i t i o n a l o p e r a t i o n s are performed on whose i n t r a p r o p o s i t i o n a l content r e l a t i o n a l operations"  (1958,  "In  since  statements  c o n s i s t s of c l a s s and  p...254).  Inhelder and  Piaget  s t a t e f u r t h e r t h a t the a c q u i s i t i o n of the o p e r a t i o n a l schema of i n v e r s e p r o p o r t i o n presupposes an understanding  of both  reciprocity  and  (that i s , compensation by equivalence)  p r o p o r t i o n s , a f e a t which i s achieved, only d u r i n g adolescence.  late  However, P i a g e t p o i n t s out t h a t problems of  d i r e c t p r o p o r t i o n a l i t y as i n the experiments on the n o t i o n of s i m i l a r t r i a n g l e s (Piaget, 1 9 5 7 ) and problems of  simple  r e c i p r o c i t y as i n the problem of communicating v e s s e l s (Inhelder and P i a g e t ,  1958,  d u r i n g e a r l y adolescence  pp.  133-147)  tend to be  solved  (that i s , around ages 1 1 and 1 2 ) .  T h i s means t h a t r e l a t i o n s of i n v e r s e p r o p o r t i o n a l i t y come l a t e r than those of d i r e c t p r o p o r t i o n a l i t y (Rogers, Piaget's  1977).  (Inhelder and P i a g e t , 1 9 5 8 ) i n t e r p r e t a t i o n of  e s s e n t i a l mechanism u n d e r l y i n g the understanding of r e c i p r o c i t y and p r o p o r t i o n s can be presented  the  of problems i n two  stages.  F i r s t , P i a g e t showed t h a t wherever two p h y s i c a l systems of a c t i o n s are r e c i p r o c a l l y r e l a t e d to one any  a c t i o n may  be c a n c e l l e d e i t h e r by executing  a c t i o n , i . e . performing  the  another, opposite  the i n v e r s e a c t i o n w i t h i n the same  20 action  system - a p r o c e s s  executing process  an a c t i o n w i t h i n t h e  called  vessels  as  ing the  level  leaving  the  they  other vessel  the other v e s s e l , that  vessel.  Since  produces  four  further  lities  i.e.  represent  of  same a c t i o n  each v e s s e l  Negation  actions out  (or  integrated  starting  point  for  of  system of  four  be a p o i n t on a v e c t o r particular  type  of  magnitude,  on i t ) .  1965)  of  in  raising  related this  as  a whole.  reversibi-  preadolescent  system at  the  taken the  which  as  'formal the  interdependence  to h o l d .  Thus,  (or t r a n s f o r m a t i o n s )  the  constitute four  (disregarding  by  action  of  d u r i n g the  by a s s u m i n g  one o f  an  two v e s s e l s  where t h e v e c t o r  action If  while  and R e c i p r o c i t y ,  continue  communicating v e s s e l s ,  to Lunzer,  lower-  and l o w e r e d  the a c t i o n  actions  The g r o u p n a t u r e (according  the  these transformations,  i n v e r s i o n and r e c i p r o c i t y the  a c t i o n but  into a single  irrespective  sense,  of  s y s t e m - o r by  t h e s e two forms  Inversion)  Thus,  logical  for  that  stage'.  of  action  - performing the .  c a n be r a i s e d  separate transformations  s t a g e become  a  communicating  the v e s s e l  Negation  the o r i g i n a l  points  or  result.  Reciprocity - executing  possible  Piaget  by r a i s i n g  i.e.  the  i.e.  system -  similar  the  -  u n d i s t u r b e d may be a n n u l l e d e i t h e r  same v e s s e l ,  to  action  showed t h a t  i n one v e s s e l  action within  reciprocal  related  or Negation  t h e e q u i l i b r i u m between  an e x a m p l e ,  lowering the  Inversion  R e c i p r o c i t y - to obtain  Thus u s i n g  inverse  called  a  actions any o f  in a in  the  group. c a n be the  represents  actions  is  to  the  the d i s t a n c e ,  these four vectors  shown  i.e.  chosen  the as  21 the s t a r t i n g p o i n t and c a l l e d I ( i d e n t i t y o p e r a t o r ) , then the v e c t o r r e p r e s e n t i n g i t s exact o p p o s i t e can be c a l l e d  N(negation,  r a i s i n g or lowering the v e s s e l ) , R ( r e c i p r o c i t y ) which has s i m i l a r e f f e c t t o N can stand f o r the t h i r d v e c t o r and the f o u r t h v e c t o r which i s the exact o p p o s i t e of R can be c a l l e d C(correlative).  The I,N,R, and C o p e r a t i o n s t h e r e f o r e  c o n s t i t u t e a group of t r a n s f o r m a t i o n s which d e s c r i b e the r e l a t i o n s between the f o u r v e c t o r s and has the f o l l o w i n g multiplicative  table.  I  2  2  2  2  =  N  IR  =  RI = R; IN = NI = N; IC = CI = C  NR  =  RN = C  CR  =  RC = N  CN  =  NC = R  IN  =  RC  = R  = C  = I  NRC = . I T h i s group - INRC - i s isomorphic  t o a well-known mathematical  group c a l l e d the f o u r group. Although,  the INRC group can be seen t o be  connected  to the p h y s i c a l system, i n the second stage o f h i s i n t e r p r e t a t i o n , P i a g e t argued t h a t t h i s occurs as a r e s u l t of the a d o l e s c e n t ' s use of p r o p o s i t i o n a l l o g i c .  Thus, he t r i e d t o  show the i n t e r c o n n e c t i o n between the INRC connected p h y s i c a l system and the isomorphic propositional logic.  group connected  w i t h the with the  I t i s t h i s aspect of h i s theory which  has r a i s e d a number of c r i t i c i s m s .  22 According  t o P i a g e t , f o r two elementary p r o p o s i t i o n s ,  P and q, p r o p o s i t i o n a l reasoning  c o n s t i t u t e s an i n t e g r a t e d  system of the c o m b i n a t o r i a l of the two p r o p o s i t i o n s - thus producing  16 b i n a r y p r o p o s i t i o n s  four t r a n s f o r m a t i o n s another  (Table 1) - and a group o f  which transform  the p r o p o s i t i o n s i n t o one  (Table 2 ) . Table 1 g i v e s a l t e r n a t e r e p r e s e n t a t i o n s of the 16  l o g i c a l p r o p o s i t i o n s which can be obtained  from the s e t of  four base elements, p.q, P.q, P.q and p.q (obtained from the i n i t i a l two p r o p o s i t i o n s ) by c o n s i d e r i n g them one-by-one, twoby-two, t h r e e - b y - t h r e e ,  a l l f o u r and i n c l u d i n g the n u l l s e t .  As i n the p h y s i c a l system, P i a g e t argued t h a t the 16 b i n a r y p r o p o s i t i o n s can be r e l a t e d t o one another by the group o f i n v e r s i o n s and r e c i p r o c i t i e s - the INRC (Table 2 ) . T h u s , i f we l e t P stand f o r "he i s hungry" and q f o r "he  i s t i r e d " , then t h e i r c a n c e l l a t i o n s P and q r e p r e s e n t  "he  i s not hungry" and "he i s not t i r e d " r e s p e c t i v e l y .  Now,  i f we choose t h e i r c o n j u n c t i o n , i . e . p.q as the i d e n t i c a l o p e r a t i o n , then the f o l l o w i n g f o u r r e l a t i o n s r e s u l t . I  (P.q) = c h e i s hungry and t i r e d  N(P.q) = P v q = he i s not hungry o r he i s not t i r e d . T h i s statement i s a strong d e n i a l o f the o r i g i n a l and hence i t s i n v e r s i o n . R(P.q) = P.q = he i s not hungry and he i s not t i r e d . T h i s statement i s a weak d e n i a l o f the o r i g i n a l p o s i t i o n and hence i t s r e c i p r o c a l  pro-  The  Combinatorial  Verbal  of the Propositional  Label  Class  1.  Complete  Affirmation  2.  Complete  Negation  3.  Conjunct ion  A  D isjunct ion  6.  Conjunctive  7.  Negation  Implication  A  A  1 2  A  1 2  A  1 2  A  A  A  1 2  A  1 2 A  1 2  9.  Reciprocal  A  1 2  A  ! 2  A  1 2  Reciprocal  11. 12.  Reciprocal  Exclusion  13.  Affirmation  of P  ]k.  Negation  15.  Affirmation  16.  Negation  A  1 2 A  I t  + Aj  Aj A^  o f Formal  '(P.q)v(P.q-)v(P.q)v(P.q~)  P.q  +  A  +  A  i 2 A  1 2 A  + +  A  A  A|A  2  +  (P.q)v(P.q)v(F.q)  Pvq  1 2 A  A'A A  1  1 2 A  (P.q)v(P.q^v(F.'ql F.q  A  A  (P.q)v(F.q)v(P\q) P.q"  + A A' +  +  A'A'  P.q  1 2  A  A  0  P/q  A  (P.q)v(F.q)  Notation  P*q  (P.qXF.q)v(P\q}  A  A'A' A  Piaget  ! 2  P.q"  +  Thought  Notation  0  A  A  Negation o f Implication Equ i v a 1 e n c e  !-  +  Model  Logic  A'A'  Non-lmplication  10.  Notation  A  8.  Implication  of Piaget's  0  k. 1 n c o m p a t i b i 1 i t y 5.  1 2  Elements  F.q PDq P.q qDP F.q P=q;or P^.q  Key  on next  of P of q  of q  A  1 2  A  1 2  A  A  A;A  A  1 2  A  1 2  A  A  2  +  A  1 2  +  A  1 2  +  A  A  A  1 2 A  +  A  +  A'A'  1 2 A  12  (P.qlv(F.q)  Pv  (P.q)v(P."ql  P[q]  (F.q)v(F.q)  F[q]  (P.q)v(F.q)  q[P]  (P.i)v(F.q)  ?[P]  vq  page. to  00  24  KEY VERBAL  TO  TABLE ' J  LABELS:  These are P i a g e t ' s labels. Some s y n o n y m o u s l a b e l s a r e : Tautology = Complete A f f i r m a t i o n ; C o n d i t i o n a l = Implication; Biconditional = Equivalence; "the inclusive or" = Disjunction; "the exclusive or" = Reciprocal Exclusion. CLASS  NOTATION:  Aj  =  occurrence  of  Class  1  A^  = occurrence  of  Class  2  There  are  k  possible  Each o f t h e s e p a i r s these pairs are the LOGIC  i  pairs: in t u r n product  A^  =  non-occurrence  of  Class  1  A^  =  non-occurrence  of  Class  2  A^2>  ^]^2'  represents of logical  ^1^2'  ^1^2'  a n e l e m e n t o f a new c a t e g o r y ; m u l t i p l i c a t i o n of classes.  NOTATION:  P  is the g e n e r a l i z e d a b s t r a c t i o n a s s e r t i o n P is t r u e  corresponding  to  A^;  q  is the g e n e r a l i z e d a b s t r a c t i o n a s s e r t i o n q is true  corresponding  to  A^;  P  is the g e n e r a l i z e d a b s t r a c t i o n a s s e r t i o n P is false  corresponding  to  AJ;  q  is the generalized a b s t r a c t i o n a s s e r t i o n q is f a l s e .  corresponding  to  A^;  Combinations  are  represented  by  the  logical  P and  q are  • =  and/both (e.g., P . q - both logical multiplication.  v  e i t h e r o r b o t h and c o r r e s p o n d s it stands for logical addition both are t r u e ) .  =  Parentheses  ( ),  Bound  or  set  off  conjunction true);  t o + in Class (e.g. P v q =  components.  it  • and  v  represents  Notation; e i t h e r P or  q  or  25  Table  2  R e l a t i o n s h i p Between T h e 1, N , R, a n d C  IDENTITY  : Application of  a  same of  of  the  proposition  truth  tables  an  l{(P.q)  Operator  has v  a  to  (P.q)  v =  is  the  unique  I(Disjunction)  to  equivalent  Equivalence  according  proposition  Example:  Identity  yields  proposition.  Every  Propositions: Operations.  a  RECIPROCITY:  The  C  (P.q)  C  (Conjunction)  Reciprocal  =  P v  NEGATION  conventions  of  standard  truth  table.  identifying  (F.q)'} =  P v  the  construction logic.  q  Disjunction conjunctions in by s u b s t i t u t i n g  the "v"  q =  Operator  Disjunction.  changes  R  (P.q)  the  signs  R  (Conjunction)  in  the  by a d d i t i o n o f t h e and i t s d e l e t i o n  = ~P.q =  : The N e g a t i o n O p e r a t o r changes in the l o g i c a l r e p r e s e n t a t i o n Example:  of  by  l o g i c a l representation of a p r o p o s i t i o n n e g a t i o n symbol (-), where i t is absent where present. Example:  representation  determined  CORRELATION: The c o r r e l a t i o n O p e r a t o r c h a n g e s t h e logical representation of a proposition f o r " . " and v i c e v e r s a . Example:  logical  representation  N  (P.q)  =  P v  N  (Conjunction)  Conjunctiveboth of a  Negation  s i g n s and conjunctions proposition:  q =  Incompatibility  C(P.q) = P v q = he i s hungry or t i r e d T h i s statement i s nearer t o the o r i g i n a l than the o t h e r s and hence may be c a l l e d i t s c o r r e l a t i v e . Thus,the f o u r i n t e r p r o p o s i t i o n s , P.q, P v q, P.q and P v q, form a group o f four t r a n s f o r m a t i o n s  isomorphic t o the INRC  group j u s t as problems o f r e c i p r o c i t y i n p h y s i c a l systems form a s i m i l a r group.  That i s , the p r o p o s i t i o n a l r e c i p r o c i t i e s and  i n v e r s i o n s express the r e c i p r o c i t i e s and i n v e r s i o n s  operating  i n the e q u i l i b r a t e d system under study. Inhelder  and P i a g e t  (1958)  explained  t h i s psycho-  l o g i c a l u n i t y between p r o p o s i t i o n a l l o g i c and r e c i p r o c i t y i n p h y s i c a l systems by saying two  levels -  t h a t the INRC group f u n c t i o n s a t  initially,  " . . . . i t governs the p r o p o s i t i o n a l o p e r a t i o n s which the s u b j e c t uses t o d e s c r i b e and e x p l a i n r e a l i t y ; as such i t c o n s t i t u t e s an i n t e g r a t e d s t r u c t u r e a t the i n t e r i o r of h i s thought, ... (and then) as a r e s u l t o f t h i s f i r s t f u n c t i o n i t i s p r o j e c t e d outs i d e i n t o the phenomena under study (since i n the given data, these c o n s i s t o f a p h y s i c a l system whose e q u i l i b r i u m r e p r e s e n t s the very problem t o be r e s o l v e d ) . " (Inhelder and P i a g e t , 1958, p. 32) A c c o r d i n g t o Inhelder  and P i a g e t ,  the INRC group i s t h e r e f o r e  not r e s t r i c t e d t o mechanical systems alone but i s a p p l i c a b l e i n a l l s i t u a t i o n s i n v o l v i n g the c o o r d i n a t i o n reference  variables l i k e concentration  o f two d i s t i n c t  and s o l u t i o n volume i n  t i t r a t i o n i n chemistry. The  r e l a t i o n s h i p o f the INRC group t o the propor-  t i o n a l schema, i . e . t o concepts i n v o l v i n g both r e c i p r o c i t y and proportion,  i s expressed by Inhelder  and P i a g e t  as f o l l o w s :  27 The p o s s i b i l i t y . - o f . r e a s o n i n g i n t e r m s o f a g r o u p s t r u c t u r e - INRC - i n d i c a t e s o u r u n d e r s t a n d i n g o f t h e e q u a l i t i e s NR = I C , RC = I N , NC = I R , e t c . , t h e e q u a l i t i e s between p r o d u c t s o f two t r a n s f o r m a t i o n s . The r e s u l t i s t h a t t h e INRC g r o u p i s i t s e l f e q u i v a l e n t t o a s y s t e m o f l o g i c a l proportions:  Ix _ Rx Cx Nx  Rx _ Cx Ix Nx  S i n c e IN = RC (where x = t h e o p e r a t i o n t r a n s f o r m e d by I , N , R, o r C ) . ( I n h e l d e r and P i a g e t , 1958, p . 1 7 7 ) .  That discovery of the Balance  is,  proportionality is  logical reciprocity. Task,  as  an e x a m p l e ,  proportion corresponds  to  a consequence  of  the  Using the E q u i l i b r i u m they argued  that  the  in  above  the numerical p r o p o r t i o n :  nx _ n : y ny n :x where x and y c o r r e s p o n d t o fulcrum,respectively; i n x or y .  Since  to p r o p o s i t i o n a l to  15 y e a r s  t h e w e i g h t and d i s t a n c e  and n = c o e f f i c i e n t  i n t h e i r view, logic,  subjects  increase  fundamental  from 12  years  d i s c o v e r p r o p o r t i o n a l i t y because they can t h i n k  of  propositional logic  stand  and t r a n s f o r m t h e e q u a l i t y One o f (1958)  Equilibrium  an  INRC g r o u p i s  they argue t h a t  terms  Piaget  the  given to  from the  to  the experiments illustrate  i n the Balance.  application of  his  chosen because i t  and t h e r e f o r e  analysis  of  two  are  able  by I n h e l d e r  proportional reasoning  of  this  under-  products.  designed  This w i l l  to  be u s e d t o  schema.  This  involves reciprocal proportion  is  and the  show t h e problem (i.e.  in  is  28 r e c i p r o c i t y and p r o p o r t i o n ) i n c o n t r a s t to the problem of the communicating it  v e s s e l s which i n v o l v e s only r e c i p r o c i t y ;  also,  i s isomorphic to some of the c a l c u l a t i o n s i n v o l u m e t r i c  a n a l y s i s as w i l l be shown l a t e r . The balance task concerns the d i s c o v e r y and explanat i o n of the r e l a t i o n between the weights p l a c e d on the balance and t h e i r d i s t a n c e s from the f u l c r u m . W/W where W and W  That i s ,  = L'/L are two unequal weights  L and L  1  are t h e i r c o r r e s p o n d i n g unequal d i s t a n c e s ,  The balance s c a l e used i s a simple seesaw type balance with v a r y i n g weights which can be hung a t d i f f e r e n t p o i n t s along i t s two arms. The s u c c e s s f u l s o l u t i o n of t h i s problem i s assumed to depend on two c o n c e p t u a l systems: the r e c i p r o c i t y  that  o b t a i n s between weight and d i s t a n c e and which a c c o r d i n g to P i a g e t i s r e f l e c t e d i n the p r o p o s i t i o n a l r e a s o n i n g of the s u b j e c t , and the n o t i o n of p r o p o r t i o n i t s e l f as opposed to a d d i t i v e d i f f e r e n c e s o r s e r i a t i o n and correspondences. is,  That  i n s t e a d of s u b t r a c t i n g from the d i s t a n c e along the arm,  L, whatever  i s added to the weight W,  i s s u b t r a c t e d from W,  or adding to L whatever  the s u c c e s s f u l a p p l i c a t i o n of the p r o -  p o r t i o n a l i t y schema i n v o l v e s the a b s t r a c t i o n of a second o r d e r r e l a t i o n from the elementary r e l a t i o n s . Thus g i v e n :  W  L  W  L'  the d i s c o v e r y of p r o p o r t i o n l i e s i n the a b i l i t y t o a b s t r a c t the second order W:W*  relation:  1  = L:L' i n s t e a d of r e l y i n g on the a d d i t i v e differences,  W-W  = L'-L.  1  In s h o r t , s u c c e s s f u l s o l u t i o n of the balance the use o f t r u e p r o p o r t i o n s or r a t i o s .  problem i n v o l v e s  P i a g e t analyzes the  p r o p o r t i o n a l i t y i n v o l v e d i n t h i s task as f o l l o w s : I f the i n c r e a s e i n weight i s represented decrease or negation  can be denoted by P.  by P i t s  However,  i n weight can be compensated by a corresponding  the i n c r e a s e  decrease i n  d i s t a n c e from the fulcrum and t h i s can be denoted by q w h i l e the negation  of t h i s , q,  i . e . i n c r e a s i n g the d i s t a n c e from  the fulcrum,  compensates f o r the decrease i n weight.  f o l l o w i n g f o r m u l a t i o n then r e s u l t s : I(P) ; N(P) ; R(q) ; C(q) T h i s i s e q u i v a l e n t t o the p r o p o r t i o n a l i t y : P —— q Also,  P q  =  q ,, — f — thus P  =  P R—— q  ,  Ix — Rx  =  Cx — Nx  and as such  , _ where x = P  (P.q)  =  R(P.q)  The  30 That i s , understanding the system o f i n v e r s i o n s and r e c i p r o c i t i e s f o l l o w s from an understanding o f the above p r o p o r t i o n a l relation.  In other words, i n c r e a s i n g the weight and reducing  the d i s t a n c e on one arm o f the balance i s s i m i l a r t o r e d u c i n g the weight and i n c r e a s i n g the d i s t a n c e on the other arm. A s i m i l a r a n a l y s i s c o u l d be a p p l i e d t o a t y p i c a l problem i n volumetric  a n a l y s i s such a s :  I f 20 mL of 2M H & n e u t r a l i z e s 100 mL o f NaOH s o l u t i o n , what w i l l be the m o l a r i t y o f the NaOH s o l u t i o n ? S0  2  S u c c e s s f u l s o l u t i o n o f t h i s problem may i n v o l v e the r e c o g n i t i o n of the two types o f p r o p o r t i o n s  - i n v e r s e p r o p o r t i o n and d i r e c t  p r o p o r t i o n i n the problem. In the f i r s t case, as i n the problem o f E q u i l i b r i u m i n the Balance, the s u b j e c t must be able t o r e c o g n i z e the r e c i p r o c i t y t h a t o b t a i n s between the c o n c e n t r a t i o n  and the  volume o f the s o l u t i o n s and a t the same time be able t o use r a t i o s and p r o p o r t i o n s  instead of additive d i f f e r e n c e s .  That  i s , the f i r s t p a r t o f the problem i s solved i f the s u b j e c t i s able t o i d e n t i f y the i n v e r s e p r o p o r t i o n a l i t y t h a t between the c o n c e n t r a t i o n  obtains  and the volume o f the s o l u t i o n s .  The s o l u t i o n t o the problem i s f i n a l l y completed by r e c o g n i z i n g the d i r e c t p r o p o r t i o n t h a t o b t a i n s between the product o f the c o n c e n t r a t i o n  and volume of the s o l u t i o n s  ( i . e . the number o f moles o f the r e a c t a n t s ) c o e f f i c i e n t o f the r e a c t a n t s m e t r i c equation.  (H S0  Symbolically,  2  4  and the  and NaOH) i n the s t o i c h i o -  i t can be s t a t e d t h a t :  31 reciprocal proportion  or and,  1/2  direct proportion  where C. J  stands  f o r the c o n c e n t r a t i o n of a c i d or base  V. J  stands  f o r the volume of a c i d or base  1 and metric  2 stand f o r the c o e f f i c i e n t i n the  stoichio-  equation.  C o n s i d e r i n g the r e l a t i o n between volume and c o n c e n t r a t i o n , i f the i n c r e a s e i n c o n c e n t r a t i o n i s s e t f o r t h as the i d e n t i c a l o p e r a t o r and represented  by P, then i t s decrease or  i s the i n v e r s e t r a n s f o r m a t i o n P.  negation  But the decrease i n the  volume of the s o l u t i o n compensates f o r the i n c r e a s e i n concent r a t i o n without  cancelling  i t . Hence i t p l a y s the p a r t of  r e c i p r o c a l t r a n s f o r m a t i o n , q.  The  i n v e r s e of the  reciprocal  produces a s i m i l a r e f f e c t as the decrease i n c o n c e n t r a t i o n and can be regarded  as p l a y i n g the r o l e of the c o r r e l a t e ,  Thus, the f o l l o w i n g t r a n s f o r m a t i o n s : I ( P ) ; N(P);  R(q); C(q)  result.  T h i s y i e l d s the f o l l o w i n g p r o p o r t i o n , P  P  q  q  or  (P.q)  = R  (P.q)  q.  Considering  the r e l a t i o n between the c o e f f i c i e n t i n the  balanced equation and number o f moles o f r e a c t a n t s  used i n  the t i t r a t i o n ,  the f o l l o w i n g s i t u a t i o n e x i s t s :  I f an i n c r e a s e  i n the s t o i c h i o m e t r i c c o e f f i c i e n t i s denoted as  the  i d e n t i c a l operator,  operation,  P.  P, then i t s decrease i s the i n v e r s e  The decrease i n the number o f moles o f r e a c t a n t s  compensates f o r the decrease i n s t o i c h i o m e t r i c thus a c t i n g as the r e c i p r o c a l t r a n s f o r m a t i o n , of the r e c i p r o c a l , q, i . e . i n c r e a s e  coefficient, q.  The i n v e r s e  i n the number o f moles,  produces an e f f e c t s i m i l a r t o the i n c r e a s e  i n the s t o i c h i o m e t r i c  c o e f f i c i e n t and i s thus i t s c o r r e l a t i v e . The p r o p o r t i o n a l r e l a t i o n r e s u l t i n g from the above transformation i s : P  =  q  —  q2 P  M  —  or  I — C x  Rx — N x  x=  In other words, the i n c r e a s e o f s t o i c h i o m e t r i c c o e f f i c i e n t i s to the i n c r e a s e o f number of moles as the decrease o f number of moles i s t o the decrease o f s t o i c h i o m e t r i c I t should  coefficient.  be noted t h a t the above r e l a t i o n r e q u i r e s  knowledge or a b i l i t y t o w r i t e and i n t e r p r e t a balanced equation of the r e a c t i o n between the r e a c t a n t s , v i z . , H S0 2  4  + 2 NaOH = N a S 0 2  4  + 2 H 0 2  That i s , knowledge o f content may p l a y an important p a r t i n obtaining has  a s u c c e s s f u l s o l u t i o n t o the problem.  Piaget  (1972)  acknowledged t h a t the content i n a p a r t i c u l a r area may  help  33 a subject  t o use  formal  reasoning  reasoning  in situations  requiring  It  has  theory o f f e r s content  been argued  theory  and a t t e m p t t o  Gagne1s  theory  and P i a g e t ' s  Gagne's  subordinate  to  skills  a variety  of  number o f  higher-order  ordered  and a d d i t i v e  complex  skill  which i s until  is  traditional  position  any p a r t i c u l a r insight,  summarize between  development  of  is  certain  concepts  intel-  and r u l e s a  is  seen  i n the cummulative e f f e c t  of  learn-  acquire  That i s ,  by t h e  the  capabilities l e a r n i n g of  learning of  the  less  complex skill  skill  is  that  all  l e a r n i n g c a n be a c c o u n t e d  there  s u c h as  m i g h t be as  one on,  encountered.  distinguishes  learning  a more  and so  20)  of  i n an  complex  p.  by b e l i e v i n g t h a t  to  development  by a much l e s s  prototype  as a  and i n l e a r n i n g  As s u c h ,  component o f  Gagne' (1970,  transfer  individuals  preceded  t h e most b a s i c  Gagne v i e w s  situations  manner.  a l s o preceded  briefly  learning.  belief  Gagne' a r g u e s t h a t  the  Learning  the  skills. of  of  1979).  According to him, l e a r n i n g  problem s o l v i n g  From t h i s  influence  differences  discriminations,  l o n g term e f f e c t  learning  theory.  involving  s u c h as  the  therefore  Piaget,  learning.  cummulative p r o c e s s lectual  to  cummulative  (Griffith,  identify  Theory of  In c o n t r a s t  ing,  t o examine  will  Gagne1s  the  Gagne's  i n l e a r n i n g or problem s o l v i n g section  proportional  it.  that  the o p p o r t u n i t y  The n e x t  as  which i n c l u d e s  himself  from  association many  the for or  different  by  types of l e a r n i n g as there are c o n d i t i o n s under which l e a r n i n g can occur.  He d i s t i n g u i s h e s e i g h t d i f f e r e n t types of l e a r n i n g  which i n h i s view form a h i e r a r c h i c a l s t r u c t u r e i n which the subordinate types are p r e r e q u i s i t e to the l e a r n i n g of the s u p e r o r d i n a t e ones.  The developmental  nature of t h i s cummula-  t i v e model i s represented i n F i g u r e 2 (Gagne, 1970, According to Gagne, the h i e r a r c h y can be generated  p.  70).  by a s k i n g  the q u e s t i o n "What must the student f i r s t be able to do i f he i s to achieve a p a r t i c u l a r c a p a b i l i t y ? " any new  c a p a b i l i t y generated  c a p a b i l i t i e s are a r r i v e d a t .  T h i s i s repeated f o r  u n t i l simple and l e s s demanding A p p l i e d to the I d e a l Gas  Law,  the complex s u p e r o r d i n a t e r u l e to be l e a r n e d can be mathematic a l l y expressed  as f o l l o w s :  PV = nRT  where P = p r e s s u r e , V = volume, n = moles,  R = u n i v e r s a l gas constant, and T = temperature.  I t may  be  hypothesized t h a t t h i s complex r u l e r e q u i r e s the l e a r n i n g of r u l e s governing- temperature and volume ( i . e . C h a r l e s  Law)  and pressure and volume when temperature i s h e l d constant ( i . e . Boyle's Law).  The  l e a r n i n g of these two  r u l e s can  f u r t h e r hypothesized to r e q u i r e the f o l l o w i n g concepts p r e r e q u i s i t e s - temperature,  volume^pressure,  be  as  mass and mole.  I t i s noteworthy t h a t Gagne has changed h i s t h e o r e t i c a l stance to some extent s i n c e the p u b l i c a t i o n of h i s book, C o n d i t i o n s of L e a r n i n g  (Gagne', 1970) .  In h i s e a r l i e r  w r i t i n g s Gagne (1970) b e l i e v e d t h a t the h i e r a r c h i c a l s t r u c t u r e c o u l d be a p p l i e d to any content a r e a . h i e r a r c h i e s are the b e s t way  He  stated, "learning  t o d e s c r i b e the s t r u c t u r e of  any  Problem S o l v i n g  (Type 8 )  which r e q u i r e s as Rules  prerequisites:  (Type 7)  I which r e q u i r e s as Concepts  prerequisites:  j l y p e 6)  which r e q u i r e s as  prerequisites:  o n sp r e (Type whichD i srcerqiumi ri n eas t ias r e q u i5) sites  Verbal A s s o c i a t i o n s or chains  (Type h)  (Type 3)  which r e q u i r e as  prerequisites  Stimulus-Response Connections Signal  Learning  Figure  or  (Type 2)  (Type 1)  2  H i e r a r c h i c a l Arrangement of Learning Types (Gagne', 1 9 7 0 , p . 6 6 )  36 topic,  course,  or d i s c i p l i n e "  i n h i s more r e c e n t suggesting that to c e r t a i n Briggs, five  domains  information, attitudes. it  is  of  learning  writings,  (1972)  skills,  skills  subordinate  eight  The i n t e l l e c t u a l  types of  the tive  intellectual  strategies will  o n l y domain w h i c h has t h e o r y and a l s o task;  thus  the  the  only to  and c o g n i t i v e  any s u b j e c t individual strategy  of  (Gagne  domains,  That  is,  intellectual of  the  by Gagne'  (1970)  skills  and i t s This  been a n a l y z e d  applicable  strategies  rules  domain may be  i n the other  be c o n s i d e r e d .  to  domains,  relation  is  to  1974,  p.  is  the  of  a  study.  intellectual  i n a meaningful  49).  cogni-  Gagne's  f o l l o w i n g way.  intellectual  to  the content  skills;  a n o v e l problem c o n s t i t u t e s  and B r i g g s ,  in  the proposed  i n the  only  because i t  in detail  or concepts  area represents attacks  and  require  the  i n any p a r t i c u l a r  t h e domain w h i c h r e l a t e d  making i t  application  five  skills.  Gagne makes a d i s t i n c t i o n between skills  strategies,  domain c o n s i s t s  i n some way by l e a r n i n g  learning of  verbal  2.  Although learning affected  skills,  learning o r i g i n a l l y postulated  and shown i n F i g u r e  only  distinguishes  motor  prerequisite  skills  by  Gagne' and  l e a r n i n g which  t h e h i e r a r c h i c a l m o d e l c a n be a p p l i e d skills.  he  of  However,'  position  1977;  cognitive  suggests that  intellectual  l e a r n i n g of  245).  c a n be a p p l i e d  1972;  l e a r n i n g w h i c h he c a l l s  intellectual  the  structure  (Gagne,  In t h e s e r e c e n t  Gagne  only  prior  of  p.  works he has moved from t h i s  the h i e r a r c h i c a l  kinds  1974).  (Gagne',- 1970,  Whereas way i n  t h e way an  his  cognitive  Gagne' and  Briggs  37 (1974)  consider  skill.  They r e l a t e  cognitive skill tual  cognitive  s t r a t e g y as  in that skills  1974,  p.  it  and a r e  situations.  between  cognitive the  their objects  that  i n novel or  a major  individual's  the  strategies  are  solving,  particular  laws,  if  an i n d i v i d u a l i s  a particular specifically,  problem,  In the  possible  the  cognitive  subject matter  to next  the  arrive  at  and r u l e s  the  oriented  processes he  thinks.  behaviour the  free'.  cognitive  a specific skills  as  strategies  studying or  'content  or  strategies he  maintains  solution  to  more  required for  its  solution  learner. a comparison  t h e o r y and P i a g e t ' s  c o m b i n a t i o n from a t h e o r e t i c a l  considered.  the  oriented  i n problem s o l v i n g ,  section,  namely G a g n e ' s  such  intellectual  the concepts  must be a v a i l a b l e  theories,  to  is  are  own t h o u g h t  suggests that  importance  skills  and how f l u e n t l y  as  problem  difference  skills  p e r t a i n more t o  and a r e  Briggs,  environment such  individual's  skills  intellec-  (Gagne' and  and i n t e l l e c t u a l  and s t a t e m e n t o f  A l t h o u g h Gagne of  important  to  intellectual learned  i n d i v i d u a l , r e g a r d l e s s o f what he i s  p r o b l e m he i s  are  "specially  organized  by r e f e r r i n g  kind of  Whereas i n t e l l e c t u a l  intellectual  whereas c o g n i t i v e  skills  generalization"  how c r i t i c a l l y he t h i n k s  is,  the  of  i n the  have  of  out of  therefore  skill.  formulas,  That  an i n t e r n a l l y  They s u g g e s t t h a t  graphs,  s u c h as  a very special  strategies  toward the o b j e c t s  as  intellectual  develops  solving  as  to  by a p r o c e s s  48)  object of  it  strategy  of  the  two  t h e o r y and  viewpoint w i l l  ..  .  their be  38 Comparison of It p.  48)  theory  posited  to  to  relate  the  study  that  the  Gagne  to  This  is  examine  Theories  (Gagne  domain o f  intellectual  by P i a g e t .  the p r e s e n t  and P i a g e t ' s  interesting  attempts  in his as  is  Gagne's  cognitive  structure  important  using  structure  to Piaget  sets  to  the  limits  to  capabilities  set  Gagne  p.  readiness  are  ties  as  of  and  that  (3)  argues  that  tions  are  entirely  (1972)  to  in  (2)  the  that  He, however,  misleading different  since  in  of  the  developmental  (1)  that  intellectual  such c a p a b i l i t i e s  that  these apparent  two t h e o r i e s  philosophical  positions.  in  the  skills".  an  two  indivi-  capabiliare  the  environment,  these c a p a b i l i t i e s  notes  Thus  s u r f ace,, t h e  i n t e r a c t i o n with his  acquisition  time  prerequisite  intellectual  following:  solving  same  learned.  on t h e  intellec-  contrast,  to differences  suggests that  number o f  the  In  "differences  learned  a g r e e on t h e  older,  problem  subordinate  290)  an i n d i v i d u a l ' s  sequential.  the  f o r what c a n be  an i n c r e a s i n g  he grows  product  suggests that  previously  of  stimulus.  primarily attributable  appear  develops  schema  identified  perform while at  limits  Strauss  dual  as  in  on a c h e m i c a l  individual's  the  number and k i n d o f  theories  the  the kinds  the environmental  t h e Gagnean t h e o r y  (1970,  attempt  reasoning  requirements  (1964),  an i n d i v i d u a l c a n s u c c e s s f u l l y accommodating  individual  theory.  According tual  the  students performance  t h e o r y and t h e c o n t e n t  Gagne's  of  1974,  strategies  given the  problem i n terms o f b o t h the p r o p o r t i o n a l Piaget's  and B r i g g s ,  is generaliza-  entertain  39 Strauss  suggested that  Gagne v i e w i n t e l l e c t u a l c a n be a p p l i e d differences Gagne's  structures concepts, archy; tions  capabilities  to d i f f e r e n t  i n the nature  theory the  forms  s u c h as rules  the  although both Piaget as  contents,  of of  the  there  forms.  thought  specific  forms  of  thought  are  He a r g u e s t h a t  are  narrowly  responses,  of mental  theory,  structures  they are  and a r e  which  theoretical in  defined  discriminations,  and p r o b l e m s o l v i n g w h i c h c o n s t i t u t e  i n the P i a g e t i a n  and  potential  therefore  a  hier-v.  organiza-  much more  comprehensive. A second d i f f e r e n c e t h e Gagnean t h e o r y , constructs  reality.  passive recipient Piaget's his  theory,  intellectual  t h e o r y may be since  between P i a g e t extent  this  that  only it  and Gagne  is  is  learner  control is  follow a single  potential  of  suggests.  is  as  the  in  suggests,  controlled.  the  of  suggest,  learning  (1979)  of  higher-  around  the  individuals  of  a  or whether  s h o u l d be a v a i l a b l e  and Dawson  skills  An i m p o r t a n t  hierarchic organization  routes  Gagne's  difference  centre  whether the m a j o r i t y  theory  of  acquisition  to  than  constructing  domain t h a t  a i d the  in  w h i l e i n .......  intellectual  in this  that  relatively  stimuli,  more l i k e l y  Gagne1s  Rowell  to  is  rather  interpretation  (1979)  is  a  very active  This  As G r i f f i t h s  variety  reality  learner  is  c o n t r o l l e d to  concept  theory  learner  structures.  particular  copies  the environmental  the  to which the  aspect of need t o  Hence t h e  of  Gagne c l a i m s  skills.  learner  justifiable  c a n be c a r e f u l l y order  the  i d e n t i f i e d by S t r a u s s  as  suggests  a  Piaget's that  40 Gagne d o e s n o t c o n s i d e r he a p p r o a c h e s instruction  rather  is  particular  Some e v i d e n c e  concept  since  processes  in different  (Schaeffer,  c h i l d r e n of  and d i f f e r e n t orders,  Eggleston  The d i f f e r e n c e s n o t e d by S t r a u s s the  which t h e i r  (1972)  theory  both theories  of  the  intellectual  structures  the other hand, Gagne's d e s c r i p t i o n of  the  of  The c o m b i n a t i o n o f of  the problems  alone  This  the the  need i s  which i n d i c a t e d the  when he p o s s e s s e s  all  elements"  (White,  1972).  reasoning  of  the  needed f o r  the  as  need  in situations these a  of  in  differences  Piaget's  a task,  detailed its  not  subject's  (1971a),  a analysis  solution.  a detailed  On  hierarchical  problem. two t h e o r i e s  e n c o u n t e r e d when e i t h e r  in research.  finding  with his  theory gives  content  (1979)  understanding  the content  w h i c h c a n be c o n t r a s t e d  same  positions  In f a c t ,  As n o t e d by L o v e l l  say much a b o u t  use  developmental  and Dawson  use may be a d v a n t a g e o u s .  on a t a s k .  any-  1974).  theoretical  and R o w e l l  'unforced'  hierarchy of  ages  or the  in reaching a  i n the  or  a single  different  and S c o t t ,  application of  does n o t  situation  because  the development of  processes,  when c o m b i n e d may h e l p i n b e t t e r performance  of n a t u r a l  suggests that  u n l i k e l y to characterize  methods  preclude  to development  than a c o n s i d e r a t i o n  different  goal  'multi-path'  l e a r n i n g p r i n c i p a l l y from the v i e w p o i n t o f  development. subskills  a  may remove  theory i s  some  employed  e x e m p l i f i e d by W h i t e ' s  "subject's  apparently  failure  necessary  The e x a m i n a t i o n  s u b j e c t may e x p l a i n t h i s  of  to  learn  even  subordinate  the  intellectual  finding.  On t h e  other hand S t i c h t (1971) suggests t h a t the problem of hori-... z o n t a l decalege i n P i a g e t i a n r e s e a r c h may ing  the a p p r o p r i a t e  students'  be  provid-  p r e r e q u i s i t e content.  Thus i t can be t h e o r i e s may  be removed by  seen t h a t the combination of the  f r u i t f u l i n a f f o r d i n g a better explanation  d i f f i c u l t i e s w i t h c e r t a i n tasks  i n the  two of  school  curriculum. The  next s e c t i o n w i l l be concerned w i t h the  s i o n of s t u d i e s r e l a t e d t o p r o p o r t i o n a l reasoning and  the content o f  science  discus-  structures  t a s k s , e s p e c i a l l y those concerned  w i t h chemical problems.  Related  Study; P r o p o r t i o n a l Reasoning and The  Piaget's  Content  f o l l o w i n g s e c t i o n w i l l attempt to show t h a t  theory  and  Gagne's cummulative theory  have been used  independently i n a n a l y s i n g a s u b j e c t ' s performance on a  task.  A p o s s i b l e combination of both t h e o r i e s i n a s i n g l e study i s suggested a f t e r reviewing  s t u d i e s which demonstrate the  i n f l u e n c e of content on l o g i c a l reasoning Piaget's  theory.  P r o p o r t i o n a l Reasoning The ing  as proposed i n  importance of studying  Studies the p r o p o r t i o n a l reason-  of students cannot be underestimated s i n c e i t forms a v i t a l  aspect  of the understanding of q u a n t i t a t i v e r e l a t i o n s i n  s c i e n c e and formal  a l s o because i t forms an e s s e n t i a l p a r t of  operations.  Piaget's  42 During the l a s t two the s c i e n c e education Piaget's  theory  Karplus,  196 5).  decades, a s u b s t a n t i a l p o r t i o n of  l i t e r a t u r e has been geared to  to s c i e n c e curriculunr.and i n s t r u c t i o n  (e.g.  A f a i r p r o p o r t i o n of t h i s l i t e r a t u r e has  concerned w i t h the p r o p o r t i o n a l reasoning the elementary and Chiappetta,  applying  1974).  high schools I t should  of students i n both  (e.g. Karplus and  Peterson,  per  se  only as p a r t of the t o t a l a b i l i t i e s which c h a r a c t e r i z e s Examples of such s t u d i e s i n c l u d e those by  (1961) and Lawson and t i o n a l reasoning ables or other  Renner  The  Lovell  schema i n d e c i d i n g on the c o g n i t i v e developmental Since.the' aim of such s t u d i e s was or i t s r e l a t i o n  not  here.  here d e a l s p e c i f i c a l l y with p r o p o r t i o n a l  order of p r e s e n t a t i o n  include references  a n a l y s i s of chemical concepts i n terms of P i a g e t ' s the presence of p r o p o r t i o n a l reasoning  in  theory  to  The rea-  the  theory,  and  adolescents.  P r o p o r t i o n a l Reasoning as an Explanatory From P i a g e t ' s  to  to some other  se, these s t u d i e s w i l l not be d i s c u s s e d  studies discussed soning.  formal  (1975) where tasks r e q u i r i n g propor-  p r o p o r t i o n a l reasoning  c o n s t r u c t per  but  were used w i t h tasks r e q u i r i n g c o n t r o l of v a r i -  l e v e l of the s u b j e c t . identify  1970;  be mentioned t h a t a number of  s t u d i e s have not d e a l t with p r o p o r t i o n a l reasoning  operations.  been  Construct  i t appears t h a t the  intellectual  s t r u c t u r e of the s u b j e c t should  d i c t a t e h i s performance on a  task r e q u i r i n g t h i s s t r u c t u r e .  To t h i s end  a number of  s t u d i e s have appeared i n the l i t e r a t u r e which attempt to e x p l a i n students d i f f i c u l t i e s i n terms of the  intellectual  structure.  Mealings  and  1971;  Shayer,  1976;  Karplus,  school  underlying a study in  Shayer,  1977;  science  (196 3)  and a number o f  1972;  Herron,  C h a p p e t a 1978)  solely  intellectual structure.  difficult  students  m e t r i c p r o p o r t i o n s u c h as Law. will  He a l s o  demands o f  chemistry course their  analyses,  difficulties  Ingle  most s e c o n d a r y  such as  In a s i m i l a r a n a l y s i s  s u c h as  constant  source of  a  it  analyzed the  stages.  '0'  and Good  Level  According will  its are  r e l a t e d to the t o p i c s  for (1976)  to  encounter  t h e m o l e c o n c e p t and  physics course,  difficulty  student  sulphate,  s i n c e p r o p o r t i o n s and r a t i o s  demands o f c e r t a i n t o p i c s formal  age,  (1971)  i n the  i t was c o n c l u d e d t h a t  k i n e t i c t h e o r y and e n e r g y w i l l  Hartford  Piaget's  of  school students  involved.  topics  find  i n the N u f f i e l d  i n terms o f P i a g e t ' s  Level  he  and magnesium  and S h a y e r  the t o p i c s  with topics  students  tests.  calculations  '0'  age w i l l  u n d e r 14 y e a r s  associated  Nuffield  of  magnesium s u l p h a t e  lead,  conducted  r e q u i r i n g an u n d e r s t a n d i n g o f  c a r b o n a t e b y means o f p r a c t i c a l From t h i s  their  From t h e r e s u l t s ,  t o d i s t i n g u i s h between p o t a s s i u m  potassium carbonate,  intellectual  (196 3)  in  v a l e n c y , e q u i v a l e n c y and B o y l e ' s  suggested that  n o t be a b l e  and G o o d ,  t h e r e a s o n i n g u s e d by  u n d e r 16 y e a r s  to undertake t o p i c s  Harford  i n terms o f  Mealings  s o l v i n g c e r t a i n science problems.  suggested that  1975;  (Ingle  have a n a l y z e d c o n c e p t s  curricular materials  i n w h i c h he i n v e s t i g a t e d  researchers  students  present  (Shayer,  a  1972).  analyzed the conceptual  i n t h e CHEM S t u d y t e x t s  o p e r a t i o n t h e o r y and n o t e d t h a t  i n terms  chemical  of  bond-  44  i n g and e q u i l i b r i u m r e q u i r e a h i g h l e v e l of c o g n i t i v e development.  Herron  competencies  (1975)  presented a l i s t of  which chemistry students who  t i o n a l c o u l d be expected to e x h i b i t .  16  commonly expected  are c o n c r e t e opera-  Each of these competen-  c i e s were compared w i t h the i n t e l l e c t u a l demands judged to be r e q u i r e d by the s c i e n c e c u r r i c u l u m .  He argued t h a t a number of  areas r e q u i r e d formal o p e r a t i o n a l a b i l i t i e s t o s u c c e s s f u l l y understand the.concepts and problems i n v o l v e d .  For example,  given the volume of base needed to n e u t r a l i z e 1 . 0 g of a c i d , a c o n c r e t e o p e r a t i o n a l student can c a l c u l a t e the volume of base needed t o n e u t r a l i z e any amount of a c i d .  However,  a c c o r d i n g to Herron, the same student cannot c a l c u l a t e the c o n c e n t r a t i o n of an a c i d i f he i s given the c o n c e n t r a t i o n and volume of base needed t o n e u t r a l i z e a c e r t a i n volume of the acid.  Herron s 1  a n a l y s i s suggests t h a t students  difficulties  i n v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s i s a t t r i b u t a b l e t o l a c k of understanding o f the i n v e r s e p r o p o r t i o n a l i t i e s i n v o l v e d . Karplus  (1977)  has a l s o suggested t h a t  certain  concepts such as chemical bonding, p e r i o d i c system and gas which are u s u a l l y i n c l u d e d i n secondary courses may  be more d i f f i c u l t  demand formal r e a s o n i n g .  ideal  school s c i e n c e  f o r students i n t h a t they  There i s much\recent evidence  t h a t suggests t h a t " a s many as 5 0 % of some samples of h i g h s c h o o l and c o l l e g e students have f a i l e d t o a c q u i r e a working understanding of the p r o p o r t i o n a l i t y schema (e.g. Karplus and Peterson, 1 9 7 0 ; Lunzer and Pumfrey, 1 9 6 6 ; L o v e l l , 1961;  Lunzer,  1965;  Wollman and K a r p l u s ,  1974).  In Karplus  and P e t e r s o n ' s  (1970)  direct proportion) to  12.  high  Their  school  requiring  administered  s t u d e n t s use of  by L o v e l l  Chiappetta 50% o f  was  a Paper C l i p s Task  (1974)  the a d u l t s  additive  studies  A similar  used  as  additive  This  subjects  curriculum materials development o f  lack  strategies  used  students  Griffiths mental  (1979)  l e v e l to  evidence study  reasoning  is  of  reviewed studies science  scarce.  proportional achievement  is  reasoning  are  and K a s s  (1977) .  relating  female  the  intellectual  the q u e s t i o n  is  interest  the evidence  of  high school The o n l y  achievement.  to  the  develop-  (1974)  in-service  investigated  reported  teachers'  the  ability  answer  between  and  (1974)  such  present  needed t o  students  studies  of  the  cognitive  t h o s e c o n d u c t e d by C h i a p p e t t a  Chiapetta between  to  an e m p i r i c a l r e l a t i o n s h i p  i n science?  question  science- e d u c a t i o n .  and t h e i r  Of p a r t i c u l a r  there  Piagetian  and t h e  leads  about  (1976)  a c h i e v e m e n t and s u g g e s t e d t h a t  and e v e n more s c a r c e  the question:  Chiappetta  e x i s t s between  students  by  solubility  reviewing  schools  w h e t h e r an e m p i r i c a l r e l a t i o n s h i p intellectual  A study  c o n g r u e n c e between  i n the  high school  c o n c l u s i o n was  on a  proportion.  of  situations  showed t h a t  t o h i g h s c h o o l and c o l l e g e apparent  in  (1966).  a similar conclusion after  related  from Grade 4  a l a r g e p r o p o r t i o n of  and B u t t e r w o r t h  using adults  (which m e a s u r e s  students  strategies  proportion.  problem which r e q u i r e d d i r e c t arrived at  to  finding indicated that  the use  arrived at  study  their on  this  and W h e e l e r  relationship to  reason  in  terms  o f p r o p o r t i o n and t h e i r s u b j e c t s were f i r s t to  assess  their  participated tory course.  achievement i n p h y s i c a l s c i e n c e .  given  the E q u i l i b r i u m i n the Balance  p r o p o r t i o n a l t h o u g h t d e v e l o p m e n t . They  i n a s e l f - p a c e d 12-unit p h y s i c a l science One o f t h e u n i t s was c h e m i c a l  o b j e c t i v e s were c o n c e r n e d w i t h t i o n s and r a t i o s .  s u b j e c t s was a s s e s s e d test at the  the completion  solubility  whose propor-  a p a p e r and p e n c i l  One a c h i e v e m e n t t e s t was  o f t h e u n i t on s o l u b i l i t y  end o f t h e c o u r s e .  labora-  achievement o f the  a t two p o i n t s u s i n g  f o l l o w e d by an i n t e r v i e w .  task  then  calculations involving  The p h y s i c a l s c i e n c e  The  given  and t h e o t h e r a t  A c o r r e l a t i o n o f 0.60  was  found .  between p e r f o r m a n c e on t h e E q u i l i b r i u m i n t h e B a l a n c e T a s k and achievement i n s o l u b i l i t y  while  a c o r r e l a t i o n o f 0.80  was  f o u n d b e t w e e n t h e E q u i l i b r i u m i n t h e B a l a n c e T a s k and ment on a l l t h e p h y s i c a l s c i e n c e concluded  that the p r o p o r t i o n a l reasoning  a p p e a r e d t o be d i r e c t l y achievement.  logical  Chiappetta  of the subjects  related to their physical  science  However, no a t t e m p t was made t o r e l a t e  achievement t o t h e i r the  laboratory units.  understanding  of the content  s t r u c t u r e was c o n s i d e r e d  achieve-  their  area.  Only  as r e l e v a n t i n e x p l a i n i n g  achievement. W h e e l e r and. K a s s the  proportional reasoning  of chemistry chemical metry.  - chemical  (1977) s t u d i e d t h e r e l a t i o n s h i p o f schema t o a c h i e v e m e n t i n f o u r  n o m e n c l a t u r e and w r i t i n g o f f o r m u l a e ,  r e a c t i o n s , t h e mole c o n c e p t and g r a v i m e t r i c The s u b j e c t s  students.  areas  i n v o l v e d were 168  The s u b j e c t s were i n i t i a l l y  high  school  administered  stoichio-  chemistry three  p r o p o r t i o n a l reasoning tasks and a g e n e r a l p r o p o r t i o n a l reasoning t e s t . Balance,  The  tasks used were E q u i l i b r i u m i n the  the R a t i o Task which i s a m o d i f i c a t i o n of the Paper  C l i p s Task  (Wollman and K a r p l u s , 1974)  and the M e t r i c P u z z l e  which i n v o l v e d c o n v e r s i o n from inches to c e n t i m e t e r s .  They  a l s o employed the I s l a n d Puzzle  1970)  which was  (Karplus and Karplus,  employed to assess deductive r e a s o n i n g .  At the  c o n c l u s i o n of each of the f o u r i n t r o d u c t o r y chemistry u n i t s which were taught by the r e g u l a r t e a c h e r s , a s u b t e s t measuring content i n t h a t u n i t was  administered.  At the end of the  course a chemistry achievement t e s t which had the items i n the chemistry  items s i m i l a r to  s u b t e s t s were administered.  The  s i m i l a r i t y of the chemistry achievement t e s t and the s u b t e s t s i s shown by a h i g h p o s i t i v e c o r r e l a t i o n c o e f f i c i e n t of  0.79.  The authors r e p o r t e d a s i g n i f i c a n t r e l a t i o n s h i p between the students' a b i l i t y to apply p r o p o r t i o n a l reasoning and ment i n chemistry.  achieve-  However, i n t h i s case the authors a l s o  examined the e f f e c t of content but t h i s i s d i s c u s s e d i n the next two  sections.  Even though the tasks on p r o p o r t i o n a l  reasoning i n c l u d e d both i n v e r s e and d i r e c t t a s k s , no attempt was  made to examine how  proportionality these two  aspects  of p r o p o r t i o n a l i t y s e p a r a t e l y and i n t e r a c t i v e l y a f f e c t s u b j e c t s achievement i n chemistry. Thus with the e x c e p t i o n of the study by Wheeler Kass, the above s t u d i e s have mainly examined the demands of chemical concepts  and  structural  i n c u r r i c u l a r m a t e r i a l s or have  examined only the i n f l u e n c e of p r o p o r t i o n a l reasoning  on  48 achievement.  Generally,  as  variable  a relevant  C o n t e n t as Apart ing  these  have  same  content  i n the  studies  focussed  to  of  to  students  that  ing of  identify  i n understanding -  react  students  (1973)  reason-  difficulties,  other  the  tasks  that  and some o f  that  the r e a c t i o n ,  another, (ii)  using a structured students'  its  to  of  solutions.  A study  difficulties  students'  p r i o r to  t h e i r use (1976)  i n terms  of  (i)  always  stoichioand  (iii)  by K a s s  in  (1977) in  stoichio-  conducted a t h e mole  concept  They d i s c o v e r e d t h i n k i n g of  number;  they  overcoming  s t u d e n t s have  perception of  involved  understand-  (i)  the  A  found  Specifically,  regardless  interview procedure.  o f mass and n o t  their  a compound w i l l  N o v i c k and M e n i s  difficulties  They  balancing equations,  balancing equations  high school  to  seem t o r e l a t e  one mole o f  reported  calculations. of  were r e l a t e d  the d i f f i c u l t i e s  on t h e mole c o n c e p t  metric  i n an  e n c o u n t e r e d by h i g h s c h o o l  subsumed by t h e m o l e .  manipulation of m o l a r i t y of  correctly  in explaining  conducted a study  t h e mole c o n c e p t  difficulties  w i t h one mole o f  metry ,of  terms  of  proportional  s o l u t i o n c o n c e n t r a t i o n and v o l u m e t r i c w o r k .  the misapprehension  the  their  on t h e c o n t e n t  the d i f f i c u l t i e s  the concepts  reported  study  subject's  and p e n c i l m u l t i p l e c h o i c e t e s t was u s e d . the  considered  difficulties.  applications paper  n o t been  Construct  explain  Duncan and J o h n s t o n e attempt  has  studies.  Explanatory  f r o m t h e use  i n t h e above  studies  the  this,  they  that  t h e mole believed  in  49 o r i g i n a t e s from q u a n t i t a t i v e o p e r a t i o n s based on mass measurements, ( i i ) r e s t r i c t i n g the mole concept to a c e r t a i n number of p a r t i c l e s of gas, and  (iii)  t h i n k i n g of the mole as a  p r o p e r t y of a molecule. The above s t u d i e s demonstrate  the importance  p r e r e q u i s i t e content i n chemistry achievement.  To  of  anticipate  students' d i f f i c u l t i e s i n terms of the subsumed p r e r e q u i s i t e concepts demanded by a p a r t i c u l a r task, a content a n a l y s i s can be performed  on the t a s k .  Gagne's h i e r a r c h i c a l theory p r o v i d e s  such a means f o r g e t t i n g a t the subsumed concepts.  A number of  s t u d i e s have analyzed c e r t a i n chemical concepts u s i n g t h i s approach.' Gower, D a n i e l s and L l o y d  (1977b) analyzed the mole  concept u s i n g Gagne'-type of h i e r a r c h i c a l a n a l y s i s and  found  t h a t i t subsumes other lower l e v e l concepts which students have to master b e f o r e understanding the mole.  T h e i r study i n v o l v e d  the w r i t t e n responses of 24 h i g h s c h o o l students t o a s e t of w r i t t e n items r e p r e s e n t i n g the elements of the hypothesized hierarchy. responded  They found t h a t the connections e s t a b l i s h e d c o r c l o s e l y t o those p r e d i c t e d i n the hypothesized  hierarchy.  However, the authors noted t h a t the s m a l l number  of s u b j e c t s i n v o l v e d i n the study p r e c l u d e s g e n e r a l i z a t i o n of the f i n d i n g s .  I n another work^Gower, D a n i e l s and L l o y d ,  (1977a), analyzed v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s u s i n g Gagne's theory and found i t t o subsume a number of important lower-order concepts  (Figure 3 ) .  A s i m i l a r l i n e of approach was and Kass  f o l l o w e d by  Griffiths  (1979) i n t h e i r attempt t o i d e n t i f y a h i e r a r c h y  50  T i t r a t ion Calculations  S o l u t ion Concent r a t ion C a l c u l a t ions  Concentrat ion  Molarity  Calculations from Equations  Conventions  I nterpret Equat ions  g/mole Convers ion  Mole  Figure  3  H i e r a r c h i c a l A n a l y s i s of T i t r a t i o n ( G o w e r , D a n i e l s and L l o y d ,  Calculations  1977a)  Write Equations  51  representing chemistry  the mole.  students,  Using a large  t h e y were a b l e  sample  o,f h i g h  to obtain a  school  relationship  between t h e e m p i r i c a l c o n n e c t i o n s and t h e h y p o t h e s i z e d o n e . That i s ,  achievement  d e p e n d e n t on t h e  i n t h e mole c o n c e p t was  subsumed c o n c e p t s .  o b t a i n e d by Okey and Gagne used  s o l u b i l i t y product problems A general  that  the  on a t a s k .  c o n t e n t was u s e d structural  students'  oh  logical  the concept of  However, a g a i n  of  above  t h e t a s k s was n o t  discussions  dealt  the probable reasoning  The s t u d i e s  a better  of  the  the  subjects.  considered.  studies  or content  the nature  and h e n c e d e m o n s t r a t e s  both i n order to achieve  is  subject's  which in explaining  t o be d i s c u s s e d  influence of  they  Reasoning  with  proportional reasoning  was  interest.  studies a  be  o n l y one v a r i a b l e ,  C o n t e n t on L o g i c a l  difficulties.  demonstrate  as  i n which  may be c r i t i c a l t o  component o f  employed e i t h e r  i n a study  i n e x p l a i n i n g the performance  Influence The  A similar result  c o n c l u s i o n f r o m t h e above  subsumed c o n c e p t s  performance  The  (1970)  found t o  of  below the  content  the need t o  examine  understanding of  students'  difficulties. A number o f  studies  a t a s k may have a s i g n i f i c a n t underlying of  this  find  logical  structure  was d e m o n s t r a t e d  the e f f e c t  of  have shown t h a t effect of  familiar  (1928)  A classic  example  i n her attempt  everyday content,  c o n t e n t and s y m b o l i c c o n t e n t on t h e  of  on t h e r e c o g n i t i o n o f  the t a s k .  by W i l k i n s  the content  to  unfamiliar  s o l u t i o n of  syllogisms.  52 She d i s c o v e r e d familiar  that  her subjects  content easier  or u n f a m i l i a r content.  In a s i m i l a r  and P a u l u s  (1971)  content  effect  found.  Another c l e a r is  g i v e n by a s e r i e s  Card Problem  (e.g.  Wason and S h a p i r o ,  of  studies,  demonstration  it.  Johnson-Laird, 1971;  proportional performance  s t u d i e s w h i c h examine tional  reasoning  and  and L i n n ,  has 1977;  Griffiths, search  structure in  the  searchers 1975)  that  of  a  in a  reasoning  the content  and  subject's  the need t h e r e f o r e , among c o n t e n t ,  the content  1971a;  for  propor-  1979;  similar  solving  by a number o f w r i t e r s 1973;  L e v i n e and L i n n reasoning  i n problem  Lunzer, (1977)  1965;  n o t e d the need f o r  school  subjects,  experience  Lovell,  1971a;  suggestions.  It  (Levine  Nagy and  i n a review of  the  1973; is  re-  research  logical  with the  i n e x p l a i n i n g t h e i r performance.  have made  1972;  achievement.  the content of  (Keating,  Four  1972).  the content  that  the r e l a t i o n s h i p  and s t u d e n t s p r e v i o u s  study  content  was c o n s i d e r e d  to a f f e c t  There e x i s t  Lovell,  on a d o l e s c e n t  d i r e c t e d to  reasoning  implication is  been r e i t e r a t e d  1979).  of  (1966)  than the p r o p o r t i o n a l  The i m p o r t a n c e o f situations  the e f f e c t  L u n z e r , H a r r i s o n and D a v e y ,  r e a s o n i n g may i n t e r a c t on a t a s k .  significant  L e g r e n z i and L e g r e n z i ,  one i m p l i c a t i o n i s  Another  and h i g h l y  of  symbolic  c o n d u c t e d by  s t u d i e s b a s e d on W a s o n ' s  t a s k may be more i m p o r t a n t underlying  study  a substantial  Although only deductive t h e above  with  to handle than problems w i t h  Roberge  was  found the problems  Other  variables rer-  Lunzer,  1965;  noteworthy t h a t  in  53 h i s more r e c e n t w r i t i n g s P i a g e t this  (1972)  has come t o acknowledge  position. The r e c o g n i t i o n of the need t o c o n s i d e r not o n l y  the s t r u c t u r e but a l s o the content of a task i n i d e n t i f y i n g a student's d i f f i c u l t y study conducted  i s r e l e v a n t t o the present study.  by Wheeler and Kass  e a r l i e r seems to f a l l  (1977)  i n t h i s domain.  The  and d i s c u s s e d  The authors c o n s i d e r e d  the p r o p o r t i o n a l reasoning s t r u c t u r e s and the content of c e r t a i n i n t r o d u c t o r y chemistry concepts on the performance of h i g h school students on a chemistry achievement t e s t .  Their analysis  showed t h a t both content and p r o p o r t i o n a l r e a s o n i n g were s i g n i f i c a n t l y r e l a t e d t o achievement. accounted  f o r a h i g h e r percentage  However, content  (63.4%) of the t o t a l v a r i a n c e  of the chemistry achievement t e s t s c o r e s .  Further, a  stepwise  r e g r e s s i o n a n a l y s i s f o r the p r e d i c t i o n of chemistry achievement a l s o r e v e a l e d t h a t the p r o p o r t i o n a l reasoning tasks d i d not add s i g n i f i c a n t l y to the r e g r e s s i o n equation once the scores from the chemistry content s u b t e s t s had been entered. s i m i l a r f i n d i n g was  r e p o r t e d by G r i f f i t h s  (1979)  A  when develop-  mental l e v e l scores were entered i n t o a r e g r e s s i o n equation i n which concepts  subsumed by the mole had f i r s t been entered.  These s t u d i e s c o n s i d e r e d o n l y the i n f l u e n c e of s t r u c t u r e and content on achievement but d i d not examine the s h i p s among them.  interrelation-  To examine the r e l a t i o n s h i p s among a l l the  r e l e v a n t v a r i a b l e s one approach i s to develop a model of performance i n which a l l the t h e o r e t i c a l r e l a t i o n s h i p s among the v a r i a b l e s are s p e c i f i e d . next  section.  Such a model i s developed  2  i n the  54 P a t h - A n a l y t i c Model of So f a r the need t o  individual  results cal  of  studies  study the e f f e c t  s o n i n g on a c h i e v e m e n t studies  assumptions  of  Performance  of  have b e e n c i t e d t o  c o n t e n t and p r o p o r t i o n a l  i n chemistry.  reported the  In the present  i n the l i t e r a t u r e  structural  theory of  c u m m u l a t i v e l e a r n i n g t h e o r y o f Gagne w i l l a causal inverse metric  model i n v o l v i n g  the  that  experiments  problems  of  on t h e n o t i o n o f  w h i l e problems of  the E q u i l i b r i u m 1958)  are  there  is  portion  solved at  was g i v e n b y R o g e r s  tional  a later  (1977)  i n 16 y e a r  The P a p e r C l i p s T a s k direct Task other  Task  and h i s  and v o l u -  (a f i r s t - o r d e r  (Inhelder  inverse  in  the  (Piaget,  1957)  around ages  and  1971b).  Stronger  That  is,  sequence o f old high  pro-  e v i d e n c e for'  i n an e x p e r i m e n t a l  as  Piaget,  d e v e l o p m e n t between d i r e c t  this  s t u d y aimed  a c q u i s i t i o n of  school physics  (Wollman and K a r p l u s ,  tasks measuring  associates  (i.e.  1974)  p r o p o r t i o n a l i t y task), the E q u i l i b r i u m  indirect  construct  p r o p o r t i o n a l i t y such  (Lovell,  proportion.  a developmental  reasoning  concepts  adolescence  inverse  age  a temporal order of  for  the  - direct proportion,  similar triangles  i n the Balance  and i n v e r s e  testing  and  d i r e c t p r o p o r t i o n a l i t y as  t e n d t o be s o l v e d d u r i n g e a r l y 11 and 12)  theoreti-  calculations.  S t u d i e s c o n d u c t e d by P i a g e t indicate  and t h e  Piaget  four v a r i a b l e s  rea-  section,  be u s e d t o  p r o p o r t i o n , subsumed p r e r e q u i s i t e analysis  justify  (a  propor-  students. first-order  i n the  Balance  p r o p o r t i o n a l i t y t a s k ) , and two  second-order  direct  and  p r o p o r t i o n a l i t y were u s e d  i n the  study.  at  second-order U s i n g an  ordering-theoretic sequence e x i s t and the  first  approach,  between order  L u n z e r and Pumfrey findings that  of  that  the  of  further  direct  subsumed  concepts. that  (1977),  (1971)  analyses of  concepts  I n g l e and S h a y e r  W h e e l e r and K a s s and S h a y e r  Their  interpretation  and H e r r o n  analyses,  of  stoichiometric  calculation  reasoning  already  testing  discussed,  using the  both d i r e c t  achievement  Both  theoretical  theory,  also  tested  final  propor-  can  test  and c h e m i c a l  ( I n g l e and study  proportionality  was  tasks.  Shayer, measured But  proportion  p r o p o r t i o n a l i t y was n e c e s s a r y  test.  and  between  in a criterion  i n the  be  (1977)  relationship  proportion  for  that  (direct)  involved only d i r e c t  direct  indicated  indicated  stoichiometry  reasoning  chemical  proportionality  require  and a c h i e v e m e n t  Ingle  conversions^and  direct  has  problems  and i n v e r s e  that  i n the  gram/mole  (1979)  direct  and  indicated  s t o i c h i o m e t r y ^ and o t h e r  require  gravimetric  concepts  (1979).  provided a  showed a s i g n i f i c a n t  which r e q u i r e  could conclude  above  precedes  is  an a d e q u a t e u n d e r s t a n d i n g  The p r o p o r t i o n a l  final  the  c o n d u c t e d by W h e e l e r and K a s s  reasoning  the mole,  reactions 1971).  before  The s t u d y  proportional  to  proportionality  b a s e d on P i a g e t ' s  equations  Chiappetta  achieved.  support  by  (1971), H e r r o n . ( 1 9 7 5 ) ,  (1975)  stoichiometry,  understanding.  tional  A study  (or c o n t e n t )  and C h i a p p e t t a  the mole c o n c e p t ,  the mole,  proportionality  proportionality.  prerequisite  by t h e work o f  prerequisite  proportionality  The c o n n e c t i o n between d i r e c t the  a  order d i r e c t  proportionality.  gives  acquisition  inverse  first  inverse (1966)  Rogers found t h a t  as one for  56 By  l o g i c a l e x t e n s i o n of the preceding  discussion,  a r e l a t i o n s h i p between d i r e c t p r o p o r t i o n a l i t y and a n a l y s i s c a l c u l a t i o n s can be e s t a b l i s h e d . P i a g e t ' s theory suggests t h a t d i r e c t and a l i t y c o n s t i t u t e the  (Herron, 1975;  assumptions i n v o l v e d  proportional  inverse  Wheeler and  reasoning.  (measured u s i n g  inverse proportional the E q u i l i b r i u m  found which may  of achievement i n chemical concepts on One  of the  inverse  Chiappetta  r e a s o n i n g of  i n the Balance Task) A moderate  i n d i c a t e the inverse  dependence  proportionality.  subsumed p r e r e q u i s i t e concepts i n the  present study i s the c a l c u l a t i o n s i n v o l v e d concentrated s o l u t i o n s .  Analysis  i n the d i l u t i o n  of such problems  (Herron, 1975).  The  of  using  P i a g e t ' s theory suggests t h a t i n v e r s e p r o p o r t i o n a l i t y such problems  the  volumetric  study by  performance on a chemical s o l u b i l i t y t e s t . (0.60) was  the  I n d i r e c t evidence i n favour of  (1974) t h a t looked at the  correlation  Kass, 1977).  the p r i o r presence of d i r e c t and  a n a l y s i s c a l c u l a t i o n s i s p r o v i d e d i n the  and  volumetric  ( i . e . volumetric analysis c a l c u l a -  connection between i n v e r s e p r o p o r t i o n a l i t y and  subjects  proportion-  i n the theory suggests t h a t  s o l u t i o n to such a problem tion) w i l l require  a p p l i c a t i o n of  formal s t r u c t u r e s u n d e r l y i n g  analysis calculations The  The  volumetric  underlies  implication i s that  s o l u t i o n s t o these problems depend on the understanding of inverse  proportionality. No  show the  empirical  study was  found i n the  l i t e r a t u r e to .„!••...  i n f l u e n c e of subsumed p r e r e q u i s i t e concepts on  v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s per  se.  However, a number  57 of  studies  have r e p o r t e d on t h e  on o t h e r c h e m i c a l c o n c e p t s . the e f f e c t that  the  of  concepts  Gagne was  i n the  (1979)  subsumed by t h e mole c o n c e p t  showed  theory.  and L l o y d  (1970)  to the mole.  s t u d y were g e n e r a t e d A similar  (1977b)  The subsumed by u s i n g  s t u d y were due t o hypothesized  the  i n the study.  1977)  indicate that  prior  mastery  success  concept  difficulties  other  i n a chemical task concepts.  although not s u f f i c i e n t calculations.  volumetric analysis)  the lower o r d e r concepts  1976;  for  subsumed by i t ,  Kass, on t h e  studies,  prerequisite successful  Also, the b a s i c  the l e a r n i n g of depends  the  concepts  depends  subsumed  in  studies  From t h e s e  knowledge o f  t h e Gagnean t h e o r y t h a t  (e.g.  of  N o v i c k and M e n i s ,  of p r e r e q u i s i t e  necessary  the  subsumed p r e r e q u i s i t e  f o r m a n c e on v o l u m e t r i c a n a l y s i s sumption of  Using the h i e r a r c h i c a l  The r e s u l t s  1971;  c a n be h y p o t h e s i z e d t h a t is  variable  s c h o o l chemistry s u b j e c t s used  lack of  (Duncan and J o h n s t o n e ,  Okey and  i n which the c r i t e r i o n  by G a g n e , t h e y f o u n d t h a t  e n c o u n t e r e d by t h e h i g h  Gagne1s  s t u d y c o n d u c t e d by Gower,  s o l u b i l i t y product c a l c u l a t i o n s .  concepts  on  prerequisite  gave i d e n t i c a l r e s u l t s .  conducted a study  theory postulated  it  concepts  subsumed c o n t e n t were r e q u i r e d f o r optimum p e r f o r m a n c e  hierarchical Daniels  subsumed  The s t u d y by G r i f f i t h s  on t h e p r o b l e m s r e l a t e d concepts  influence of  peras-  a higher order  on t h e l e a r n i n g o f  lends  support  to  this  hypothesis. A as  diagramatic  postulated  referred  t o as  above  is  representation  of  g i v e n i n F i g u r e 4.  the proposed  the  i n t e g r a t e d model-  T h i s model w i l l  integrated model.  The o m i s s i o n  be  58  Subsumed  3  Figure  4:  Concepts  Proposed I n t e g r a t e d Model of Performance Volumetric Analysis Calculations  on  59 of  t h e l i n k a g e between d i r e c t p r o p o r t i o n a l r e a s o n i n g  mance on v o l u m e t r i c a n a l y s i s model  is  essentially  calculations  a theoretical  assertion;  subsumed by a l l  the other v a r i a b l e s  these v a r i a b l e s  should mediate  portional  reasoning  shown i n t h i s exogeneous variables  are  variables.  are  Since  variance  of  of v a r i a b l e s  methodological  is  integrated  is model,  direct  pro-  calculations.  proportionality,  proposed,  is  a variable,  As  of  of  an  sub-  are to  account  residual variables  integrated  model t o  R^  indicate  i n t e g r a t e d model  convention i n  representing  i n the  form o f  observed a  the c a u s a l  and y i e l d i n g t h e v a r i a b l e s  i n path analysis  1973, P . 3 0 8 ) . is  the or  measured  recursive  u n i d i r e c t i o n a l arrows  flow o r i g i n a t i n g i n the v a r i a b l e s  ( K e r l i n g e r and P e d h a z u r ,  influence taken taken  The use  only c o l l o q u i a l ; i t  as as of "is  meant t o p r o v i d e no p h i l o s o p h i c a l m e a n i n g b e y o n d a  shorthand  designation  .(Bentler,  1980) . used  for  for  a hypothesized unobserved process"'  Thus o t h e r words l i k e e x p l a n a t o r y the  same e n d .  R3  subjects.  the d i r e c t i o n of  "causes" or explanatory  t h e word " c a u s e "  never p o s s i b l e  A l s o by c o n v e n t i o n s i n c e  paths  indicate  calculations  not i n c l u d e d i n the  literature  with the causal  effects  it  uses a standard  by s q u a r e s .  drawn t o  it  n o t i n f l u e n c e d by o t h e r  and v o l u m e t r i c a n a l y s i s  The f i g u r e  model  is  while inverse  p r i o r experience  variables  it  i n t r o d u c e d i n t o the  the e f f e c t s u c h as  (i.e.  i n t h e model)  the t o t a l  and R 2  since  influence of  perfor-  integrated  r e c u r s i v e model, d i r e c t p r o p o r t i o n a l i t y i s  sumed c o n c e p t s  for  the  i n the  on v o l u m e t r i c a n a l y s i s  variable  endogeneous  i n the  and  or process  c o u l d be  60  Studies Related to the Secondary Questions i n t h i s  Study  Questions posed i n t h e secondary p a r t o f the study r e l a t e t o the i d e n t i f i c a t i o n o f s p e c i f i c student For convenience,  these d i f f i c u l t i e s  which r e l a t e t o a misunderstanding conceptual d i f f i c u l t i e s ) l a b o r a t o r y work.  difficulties.  can be grouped  into  those  of chemical concepts ( i . e .  and those r e l a t e d t o the p l a c e o f  S t u d i e s w i l l be reviewed which:;relate t o  these broad a r e a s . A number of s t u d i e s have been r e p o r t e d i n the l i t e r a t u r e which examined the c o n c e p t u a l d i f f i c u l t i e s have on chemical concepts.  t h a t students  These s t u d i e s can be grouped  those which assess the c o n c e p t u a l d i f f i c u l t i e s  into  o f the students  from t h e i r w r i t t e n work and those which employ c l i n i c a l interviews. The former i n c l u d e t h e s t u d i e s by Duncan and Johnstone  (1973),  Johnstone, (1980).  Doran  (1972),  MacDonald and Webb  Wheeler and Kass (1977)  (1978),  and Rowell and Dawson  The study by Duncan and Johnstone  (1972)  which was  reviewed e a r l i e r employed m u l t i p l e c h o i c e techniques t o assess secondary  school students  1  d i f f i c u l t i e s with concepts  the mole, s t o i c h i o m e t r y and v o l u m e t r i c a n a l y s i s The  s t u d i e s by Johnstone, MacDonald and Webb  and Kass  (1978)  including  calculations.  (1977)  and Wheeler  a l s o examined the d i f f i c u l t i e s o f secondary  s c h o o l students on chemical e q u i l i b r i u m problems u s i n g m u l t i p l e choice t e s t s .  Doran  (1972)  a l s o a p p l i e d a m u l t i p l e response  format t o i d e n t i f y the misconceptions o f h i s s u b j e c t s on the p a r t i c u l a t e nature o f matter.  61 The above conceptual tors  studies  difficulties  of  have t h e r e f o r e  the  s u b j e c t s by d e v e l o p i n g i t e m  distrac-  according to predetermined misconception c a t e g o r i e s . (1978),  was n o t e d by W h e e l e r and K a s s the responses to presents  c e r t a i n problems  i n that  subject  can a l s o o b t a i n a p a r t i c u l a r of  choice tests  by g u e s s i n g o r by a r g u i n g  different to  p a t h ways..  therefore,  ties  of  the  and Dawson  (1980)  identified  the conceptual  students  subjects'  n e e d e d w h i c h examine  students using other  have w i t h  falls  in this  difficulties  by t h e  students  to  made by t h e m .  and Dawson a l l o w e d them t o  errors  made by t h e  c h e m i c a l problems  calculations  since  stoichiometric analysis  Rowell  problems  calculations.  difficul-  authors  secondary  involving  students.  calculations.  Rowell  to other  that  in their written calculations  students.  Studies  school  the  mole  They d i d n o t  The s t u d e n t s were g i v e n t h e  show t h e i r  i d e n t i f y the e r r o r s  students  The s t u d y by R o w e l l These  s t o i c h i o m e t r y problems  employ m u l t i p l e - c h o i c e i t e m s .  of  through  multiple-  the conceptual  domain.  the  The  difficulties.  approaches.  by e x a m i n i n g t h e w r i t t e n work o f  and were r e q u i r e d  errors  of  of  a  incorrectly.  T h u s , t h e use  As  tests  i n c o r r e c t response  i d e n t i f y the conceptual  may n o t r e v e a l much a b o u t are,  interpretation  a s u b j e c t may o b t a i n  answer  a variety  the  such m u l t i p l e - c h o i c e - t y p e d i a g n o s t i c  correct  to  assessed  This  s u c h as  which are  were  approach  the  (1980)  applied  analysis  used  prerequisite  by  conceptual  s h o u l d be  volumetric  used  examined  The a p p r o a c h u s e d  identify,  and Dawson  The s t e p s  question  to  it  only  for  volumetric  62 The s t u d i e s w h i c h e m p l o y e d c l i n i c a l assess are  the conceptual  difficulties  e x a m p l i f i e d by t h e  colleagues Clinical  interviews of  have  Erickson,  to  the  discussed  study  A n d e r s o n , 1965;  difficulties  secondary nature  that  subjects  responses of emphasized  the  since  it  the need f o r  research  situations  secondary  is  difficulties  Sullivan  a p p l i c a t i o n of  specific  science  the  The n e e d ,  the  approach to  the  the  has  clinical  in  the  paucity  interview  exists  s u c h as  this  initial  (1967)  investigate  with chemical concept  diffi-  difficulties  concepts,  therefore,  subjects  procedure  noted that  the conceptual  pre-  identify  school  t o examine  u t i l i z i n g chemical concepts  notable.  to  A similar  The a u t h o r s  Although,  the  investigate studies  w h i c h was  allowed the probing of  s t u d i e s w h i c h employ t h i s tual  fields  interviews  (1978)  in revealing  the^subjects.  interview to of  the  s t u d e n t s had i n u n d e r s t a n d i n g  of matter.  a p p r o a c h was u s e f u l , the  1978).  Hobbs and  (1976)  had w i t h t h e mole c o n c e p t .  that  particulate  of  identify  Kargbo,  employed s t r u c t u r e d  was u s e d by N o v i c k and Nussbaum culties  his  1980).  the conceptual in  chemistry  phenomena i n o t h e r  The s t u d y by N o v i c k and M e n i s viously  in  to  N o v i c k and Nussbaum,  a l s o been used  c h i l d r e n about  E r i c k s o n , 1975;  subjects  done by N o v i c k and  ( N o v i c k and M e n i s , 1976;  conceptions (e.g.  studies  of  interviews  for  the  concep-  volumetric  analysis. The above  s t u d i e s by N o v i c k and h i s  employing a question-and-answer  colleagues  technique d u r i n g the  clinical  i n t e r v i e w c a n be d i f f e r e n t i a t e d employ t h e (Ericsson Ericsson  'talking aloud1 and S i m o n ,  and Simon  information  about  stated  the  that  processing  The fully  to  Simon,  the  thinking  problems. type  strategies  and S i m o n ,  interested  of  has  problems  applied  authors  concepts  to  (e.g.  used  investigating  analysis  laboratory  work t o  tual  difficulties  Infact, that  studies  students the  the  i n general  are  subjects  were  interested subjects.  by t h e  subjects  indicate  difficulties use  are  by D o r a n  of  by  with  the concep-  similarly (1978)  techniques  The o n l y  w i t h the  that  used  i n overcoming t h e i r  laboratory  lacking.  certain  problems.  work and t h e  literature  which d e a l t  used  of  More  strategies  students'  students  to  results  the  i n volumetric analysis  examining the  literature  chemistry  laboratory assist  a review of  the  and  1974).  by t h e i r  strategies  assess  it  Paige  were more  s t u d e n t s when s o l v i n g v o l u m e t r i c a n a l y s i s  volumetric  also  success-  processes  Flaherty,  However, t h e i r  c o u l d be u s e d  Studies  They  been u s e d  or c o g n i t i v e  (1980)  i n the  solutions.  approach  processes.  reliable  aloud d i d not a f f e c t  1972;  However, t h e  and f r e q u e n c y  at  can produce  to  subjects.  G o r o d e t s k y and Hoz  arrive  this  the  Newell  They were n o t to  of  cognitive  on c e r t a i n m a t h e m a t i c a l  recently,  in  a subject's  which  verbalization"  both techniques  (1980)  interviews  However, a c c o r d i n g  t a l k i n g aloud techniques  1966;  science  technique.  demand f o r  identify  subjects  or ("concurrent  1980)  abilities  from c l i n i c a l  of  studies  laboratory  indicates chemistry reported  skills  t h o s e by E g l e n and Kempa  lacking.  of  (1974)  c..^ and  in  Johnstone 1979).  and McCallum  (1972; c i t e d i n Johnstone  The study by Eglen and Kempa  and Sharp,  (1974) sought  t o examine  the degree o f concordance among teachers ,judgments  of a  1  student's m a n i p u l a t i v e l a b o r a t o r y s k i l l s , as presented on a video-tape.  D i f f e r e n t assessment techniques such as c h e c k l i s t s  andiopen-ended schedules were used by the teachers t o compare the r e l a t i v e e f f e c t i v e n e s s o f each o f these techniques.  This  study d i d not seek t o assess o r i n v e s t i g a t e the l a b o r a t o r y s k i l l s per se o f the s t u d e n t s . conducted  i n other s c i e n c e areas t o assess the a c q u i s i t i o n o f  l a b o r a t o r y techniques 1970) .  However, s t u d i e s have been  (e.g. Kruglak, 1954; Tamir and Glassman,  In some o f these s t u d i e s (e.g. T y l e r , 1942) c h e c k l i s t s  were used t o assess the s k i l l s o f the students w h i l e i n o t h e r s (e.g.  Kruglak, 1954) l a b o r a t o r y p r a c t i c a l examinations  employed i n a group  were  situation.  The r e s u l t s o f these s t u d i e s i n d i c a t e t h a t while c e r t a i n laboratory s k i l l s are learned quite w e l l , others are p o o r l y l e a r n e d (Johnstone 1979; to  Doran, 1978).  and Sharp, 1979; Johnstone  Johnstone  and Wham,  and Wham (1979) a t t r i b u t e  the i n s u f f i c i e n t emphasis on the mastery o f s k i l l s .  this Doran  (1978) suggests t h a t t h i s might be due t o the emphasis each teacher g i v e s t o students' equipment m a n i p u l a t i o n and laboratory  techniques. Although  the above r e s u l t may be g e n e r a l i z e d t o  i n c l u d e chemistry students, the p a u c i t y o f s t u d i e s d e a l i n g with laboratory s k i l l s s t u d i e s which attempt  i n chemistry suggest the need f o r t o assess l a b o r a t o r y s k i l l s  those found i n t i t r a t i o n .  such as  65  Lacking t h e use However,  of  i n the l i t e r a t u r e  studies  related  to  l a b o r a t o r y work t o r e i n f o r c e t h e o r e t i c a l k n o w l e d g e .  since  this  important o b j e c t i v e studies  are  is  r e g a r d e d by c h e m i s t r y t e a c h e r s  for  l a b o r a t o r y work i t  t o be c o n d u c t e d t o a s s e s s In the next c h a p t e r ,  and t h e p i l o t  study w i l l  be  the  an  seems i m p o r t a n t  for  claim.  the data  discussed.  as  c o l l e c t i o n instruments  66  CHAPTER I I I PILOT TESTING OF INSTRUMENTS AND PROCEDURES  Introduction The p r e v i o u s two chapters d e a l t with the s p e c i f i c a t i o n o f the q u e s t i o n s posed i n t h i s study and the r a t i o n a l e drawn from the l i t e r a t u r e t o support the study o f these questions.  In t h i s chapter, the instruments and the p i l o t  are d i s c u s s e d .  study  The i n s t r u m e n t a t i o n and p i l o t t e s t i n g i n v o l v e d  s e v e r a l stages: p r e p a r a t i o n o f the instruments, p r e t e s t i n g the instruments and other a n a l y t i c a l techniques i n the p i l o t  study,  a n a l y z i n g the r e s u l t s o f the study, and refinement o f the instrument and techniques f o r use i n the main study.  Instruments To o b t a i n data f o r the v a r i a b l e s i n the proposed i n t e g r a t e d model, t h r e e separate t e s t instruments were developed.  Two o f these instruments - Subconcepts T e s t and  V o l u m e t r i c A n a l y s i s T e s t were used t o measure p r e r e q u i s i t e concepts and achievement i n . v o l u m e t r i c a n a l y s i s respectively.  calculations  The t h i r d t e s t , the Classroom P r o p o r t i o n a l i t y  T e s t was used t o measure d i r e c t and i n v e r s e p r o p o r t i o n a l i t y .  67 On each t e s t , the f o l l o w i n g i n f o r m a t i o n was c o l l e c t e d : ( 1 ) gender,  ( 2 ) date of b i r t h ,  ( 3 ) name, and ( 4 ) s c h o o l .  In  a d d i t i o n t o the above t e s t s , a c h e c k l i s t and p r o t o c o l s were developed t o assess the q u a l i t a t i v e questions study.  posed i n the  A b r i e f d e s c r i p t i o n of each of the above  instruments  follows.  Classroom P r o p o r t i o n a l i t y T e s t T h i s i s a 1 4 - i t e m group administered a s s e s s i n g the p r o p o r t i o n a l reasoning  t e s t used i n  of s u b j e c t s .  I t consists  of two p a r t s - a d i r e c t p r o p o r t i o n a l i t y s u b t e s t and an i n v e r s e proportionality  subtest.  Since the s u b t e s t s assess two. v a r i a b l e s of i n t e r e s t i n the study The  they are d e s c r i b e d  separately.  items i n the i n v e r s e p r o p o r t i o n a l i t y subtest  were d e r i v e d from a classroom developed by Lawson ( 1 9 7 7 ,  t e s t of formal  1978).  operations  Lawson s t e s t was chosen 1  as a source o f items because i t i s "at present the most appropriately validated test" f a i r l y high r e l i a b i l i t y  (Nagy and G r i f f i t h s , 1 9 7 9 ) with  (KR-20 i s . 7 8 ) .  In Lawson s t e s t only 1  two items were used t o i n v e s t i g a t e i n v e r s e p r o p o r t i o n a l i t y s i n c e h i s o b j e c t i v e was t o assess a wider range of formal o p e r a t i o n a l thought.  However, i n the present  study, i n  a d d i t i o n t o the 1 : 2 and 2 : 3 s i t u a t i o n s i n v e s t i g a t e d by Lawson, the i n v e r s e p r o p o r t i o n a l i t y items were extended t o i n c l u d e the f o l l o w i n g r a t i o s i t u a t i o n s - 1 : 3 , 3 : 4 , 3 : 5 , 1 : 4 and 1 : 5 . That is,  seven  items were c o n s t r u c t e d f o r the i n v e r s e  proportions  68 a l i t y subtest.  F o l l o w i n g the procedure used by Lawson  and Shayer and Wharry  (1975),  each item i n v o l v e d a  t i o n u s i n g a balance beam and hanging weights Piaget, 1 9 5 8 ) .  (1978)  demonstra-  (Inhelder and  For each item, the demonstration was  used t o  pose a q u e s t i o n t o the c l a s s r a s a whole or c a l l f o r a prediction. i n h i s own  Each student responded to the q u e s t i o n s i n w r i t i n g test booklet.  The t e s t b o o k l e t s c o n t a i n e d o n l y the  q u e s t i o n s f o l l o w e d by a number of p o s s i b l e answers.  Students  were i n s t r u c t e d t o respond by checking the box next t o the best answer and then t o e x p l a i n why had f o r each q u e s t i o n .  they chose the answer they  As noted by Lawson ( 1 9 7 7 ) ,  this  demonstration procedure r e t a i n s some important aspects o f the clinical (e.g.  i n t e r v i e w employed by Inhelder and P i a g e t  (1958)  a s k i n g the-.students to e x p l a i n t h e i r responses)_ w h i l e  a l l o w i n g f o r a l a r g e number of students to be t e s t e d i n a r e l a t i v e l y s h o r t time u s i n g a s u b s t a n t i a l number of  problems.  The seven items i n the d i r e c t p r o p o r t i o n a l i t y s u b t e s t i n v o l v e d seven demonstrations u s i n g four p l a s t i c  graduated  c y l i n d e r s of d i f f e r e n t diameters and a j a r o f c o l o u r e d water. These items were a l s o o b t a i n e d from Lawson's t e s t 1977,  1978).  Apart from the 2 : 3 s i t u a t i o n i n v e s t i g a t e d by the  two items i n Lawson s t e s t , the f o l l o w i n g r a t i o 1  were i n v e s t i g a t e d — the  (Lawson,  2 : 5 , 1 : 3 and 1 : 2 .  With the e x c e p t i o n of  l a s t s i t u a t i o n , each s i t u a t i o n i n v o l v e d two  and hence two t e s t items.  demonstrations  For each s i t u a t i o n , s u b j e c t s were  allowed enough time t o complete demonstration was  situations  performed.  the item b e f o r e the next  69  The v a l i d i t y of the items was e s t a b l i s h e d by Lawson  (1977).  He compared s u b j e c t ' s performance on the  w r i t t e n task with t h e i r performance on an i n t e r v i e w task, the E q u i l i b r i u m i n the Balance Task. Performance on the Classroom P r o p o r t i o n a l i t y Test was scored as f o l l o w s : Those who s e l e c t e d the c o r r e c t answer and reasoned u s i n g r a t i o o r p r o p o r t i o n were awarded three p o i n t s . Those who used the c o r r e c t reasoning i . e . , r a t i o o r p r o p o r t i o n but s e l e c t e d the wrong answer were awarded two p o i n t s .  Subjects  who s e l e c t e d the c o r r e c t answer but used a d d i t i v e o r any other reasoning o r p r o v i d e d no reason were given one p o i n t while s u b j e c t s who used wrong reasoning and s e l e c t e d a wrong a l t e r n a t i v e were scored zero. used by Lawson  T h i s was the same s c o r i n g procedure  (Lawson, Nordland and DeVito,  1974).  Subconcepts Test T h i s was a t e s t designed content  to t e s t the knowledge o f the  represented by the subsumed p r e r e q u i s i t e concepts and  hypothesized  t o be r e q u i r e d f o r doing c a l c u l a t i o n s i n v o l u -  metric a n a l y s i s .  I t c o n s i s t e d of a s e r i e s of s u b t e s t s  r e p r e s e n t i n g each of the seven subordinate represented  i n F i g u r e 5.  The subordinate  p r e r e q u i s i t e concepts concepts  i n the  f i g u r e were d e r i v e d from the one proposed by Gower, D a n i e l s and Lloyd  (1977a).  The only important  d i f f e r e n c e between t h i s  f i g u r e and the one proposed by Gower, D a n i e l s and L l o y d (see F i g u r e 3), i s t h a t subordinate  concepts  requiring verbaliza-  t i o n of d e f i n i t i o n s such as ' c o n c e n t r a t i o n ' and 'molarity conventions'  were not i n c l u d e d i n F i g u r e 5.  T h i s was done f o r  70 D E T E R M I N E AMOUNT ACTANT OF  CALCULATE C O N C E N T R A T I O N OF S O L U T I O N S FROM MASSj OF S U B S T A N C E S  FROM  OF  RE-  CONCENTRATION  ANOTHER  CONVERT FROM ONE T I O N TO  SOLUTIONS  CALCULATIONS FROM E Q U A T I O N S  CONCENTRAANOTHER  WRITE EQUATIONS  g/mo 1e CONVERSION  Figure  INTERPRET EQUATIONS  5-  Hierarchical Analysis of T i t r a t i o n (Gower, D a n i e l s , and L l o y d ,  Calculations  1977a)  (Modified)  71 two r e a s o n s : because of that only tion  of  (1)  time  to  decrease the  limitations  concepts  of  definitions  the  test  schools,  and  (2)  i n the  higher  order  are  does n o t e n s u r e  A description subordinate  length of  concepts  A to  to  included since  ensure verbaliza-  understanding.  and i l l u s t r a t i v e  labelled  constructed  example o f  H in Figure  6 is  each of  the  presented  below: A.  Determine the c o n c e n t r a t i o n involved  in a titration  reactant  and t h e  Example:  "What i s  B.  Calculate of  the  the  mL o f  a hydrochloric  which i s  M sodium  concentration  'Calculate ing  5.0  of  another  acid  neutralized  by  hydroxide?'  solutions  from the  Calculate  the  mass  products  from b a l a n c e d  Example:  (a)  a 0.20  solution.' to  another.  M hydrocholoric to  M solution?1'  mass o r m o l e s  of  reactants  and  equations.  'How many grams o f g of  11.70  contain-  t h a t must be m e a s u r e d o u t  250 mL o f  relative  50.0  a solution  N a C l p e r mL o f  t h e volume o f  solution  prepare  of  f r o m one c o n c e n t r a t i o n  'Calculate acid  the m o l a r i t y  mg o f  Convert a s o l u t i o n Example:  D.  of  stoichiometry.  mL o f  0.10  reactant  concentration  the m o l a r i t y of 30.0  one  substances.  Example:  C.  from the  reaction  solution 48.0  o r mass o f  CaCC>3 i s  CC^ a r e  p r o d u c e d when  decomposed?'  72 (b)  ' C a l c u l a t e t h e grams o f A g C l 0.500  liter  of  0.10  an e x c e s s o f N a C l E.  C o n v e r t t h e mass o f of  moles,  Example:  and  M AgN0  f o r m e d when reacts  3  with  solution.1  an e l e m e n t o r compound t o  its  number  vice-versa.  'How many m o l e s  of NaCl  are  present  i n 29.Og o f  NaCl?' F.  Calculate  t h e masses o f  different  compounds c o n t a i n i n g t h e atoms o r  molecules.  Example:  'Calculate  same o r p r o p o r t i o n a t e  t h e mass o f  same number o f m o l e s G.  'Balance  molecules  o r atoms o f  'What d o e s t h e  Each subordinate multiple choice for  symbols.  equation:7  i n terms  o f number  chemical species  involved.  f o l l o w i n g e q u a t i o n mean: —>  2 C0  2  + 3 H 0' 2  c o n c e p t was r e p r e s e n t e d  (five-response)  the e n t i r e  carbon.'  2  of  C2H5OH + 3 0 2  questions,  by  four  t h u s y i e l d i n g 28  test.  The i t e m s  i n the t e s t  o b t a i n e d f r o m two s o u r c e s .  The f i r s t  s o u r c e was a t e s t  by W h e e l e r and K a s s  (1977)  the  co '  balanced chemical equations  moles,  grams o f  using chemical  Interpret  Example:  items  24.0  the f o l l o w i n g chemical  CO + o 2 -» H.  as  or  numbers o f  copper r e p r e s e n t i n g  Write balanced chemical equations Example:  chemical elements  to  a s s e s s knowledge o f  were used  introductory  73  chemical concepts such as the mole and s t o i c h i o m e t r y . other source was  a t e s t used by Johnstone  The  and Duncan  (1973)  to i d e n t i f y students d i f f i c u l t i e s w i t h c e r t a i n chemical c a l c u l a t i o n s i n c l u d i n g v o l u m e t r i c a n a l y s i s , s t o i c h i o m e t r y and gram mole c o n v e r s i o n s , The score of each student  was  i n d i c a t e d by the number of c o r r e c t answers g i v e n f o r the items i n the t e s t .  Volumetric A n a l y s i s Test T h i s was  a w r i t t e n t e s t designed t o t e s t s u b j e c t ' s  a b i l i t y t o do v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s .  Eight  i n the t e s t were s e l e c t e d from Duncan and Johnston's test.  items  (1973)  S i m i l a r items were b u i l t t o i n c r e a s e the t o t a l number  of items t o f i f t e e n .  The t e s t items r e q u i r e d s u b j e c t s t o  c a l c u l a t e the volume, c o n c e n t r a t i o n or mass of one of the reactants involved i n a t i t r a t i o n .  The s u b j e c t s were asked t o  respond t o the items by showing t h e i r answers as w e l l as the steps used i n t h e i r c a l c u l a t i o n s . was  Achievement i n t h i s  test  assessed by the number of c o r r e c t answers o b t a i n e d i n the  t e s t independent of the procedures used.  However, the  c a l c u l a t i o n s were examined f o r any c o n c e p t u a l e r r o r s t h a t the students may who  have made.  I t was  a l s o used t o i d e n t i f y those  used a l g o r i t h m s w i t h understanding and those who  used  a l g o r i t h m s without understanding.  In the study, u s i n g  a l g o r i t h m s w i t h understanding was  d e f i n e d as employing  the  c o r r e c t s t o i c h i o m e t r i c or mole r a t i o s i n computations. S p e c i f i c a l l y , i t r e f e r r e d t o the use of 1 : 1 and 1 : 2  stoichio-  74 metric  ratios  correctly  items  employed t h i s  algorithms standing  in test  with understanding.  was  defined  as  use  mole r a t i o s  o t h e r mole r a t i o s . considered  above i n s t r u m e n t s ,  in  the  of  the  algorithms  lab  referred  to  or the  requiring were  understanding.  psychometric was  a number o f  other  analytical  Skills  d a t a on  a l s o used  to  the  assess  techniques  the  used  in Titration  a p p r o a c h e s have b e e n u t i l i z e d i n  skills.  These are  checklists  T y l e r . . , 1 9 . 4 2 ; l A l l e h y ;197.2)  nations  (e.g.  checklist  Kruglak,  approach  techniques  The c h e c k l i s t The t a s k the next  is  1954).  According  appropriate  a p p r o a c h was  to which the  for  was  assessing scales  practical  to Doran  (1978),  i d e n t i f y i n g the  i n an i n t e r v i e w  therefore  checklist  or r a t i n g  and l a b o r a t o r y  u s e d by s t u d e n t s  is  the  manipula-  situation.  employed i n t h i s applied  exami-  study.  described  in  section. The i t e m s  of  gathering  under-  study.  student  Skills  it  study  Two m a j o r  tive  without  such s t r a t e g i e s without  using  stoichiometric  pilot  Laboratory  (e.g.  as  i n more t h a n one c o m p u t a t i o n  In a d d i t i o n to  effectiveness  Specifically,  Subjects using  t o be u s i n g  to  Using algorithms  employing i n c o r r e c t  i n computations.  1:1  T h o s e who  a l g o r i t h m were r e f e r r e d  mole r a t i o of  r e q u i r i n g them.  category  i n the  referred  the e v a l u a t i o n  of  checklist  were b a s e d on t h e  t o by K l o p f e r  learning  (1971)  in science.  two s u b c a t e g o r i e s u n d e r t h e M a n u a l S k i l l s .  in his  Klopfer These  Manual  discussion  identified are:  (a)  development of  and  (b)  performance  and s a f e t y . student  skills of  While the  manipulates  common l a b o r a t o r y former i s  various  w i t h the c a r r y i n g out of desired  goal.  Examples  skills  under the  react  to detect  safety  Some o f  performance  good r e s u l t s  these  c a u t i o n near  also  funnels  of  checking off overall  skill  summarized as  the  so t h a t  An sub-  are  setting  the  (1971)  good r e s u l t s  sufficient  added).  This  calls  are  : •,  attention  statement  student,  t h e end p o i n t o f  i d e n t i f y the  up  to  the equipment or  in transferring  for  about  example,  the r e a c t i o n  and  solutions.  this  c h e c k l i s t was u s e d by  students'  In s e c t i o n A of  adequate or  manual s k i l l s  the c h e c k l i s t ,  i n h a n d l i n g the b u r e t t e  inadequate,  three behaviours  a  include  and b u r e t t e s .  to Klopfer  r e q u i r e d the  items.  used  to reach  t h e s e and o t h e r t e c h n i q u e s  During the t i t r a t i o n , to  concerned  and u s i n g a w h i t e b a c k g r o u n d  According  (italics  and s a f e t y  investigator  is  subcategory  subskills  injuring either  to exercise t o use  latter  would i n c l u d e the  and t o be c a r r i e d o u t w i t h  the experimenter"  the  first  pipettes  a d d i n g an i n d i c a t o r ,  to prevent  the  when d e t e r m i n i n g t h e volume o f  them t o be done c a r e f u l l y ,  obtained,  care  c o m p l e t e l y w i t h a c e r t a i n volume o f an  c o l o u r changes.  "successful  techniques with  of manipulations  second subcategory  in titration.  experiment,  for  a series  r e q u i r e d by a s t u d e n t  base r e q u i r e d to acid  equipment,  s u c h as  equipment  c o n c e r n e d w i t h how t h e  under the  manipulating glassware, example  i n u s i n g common l a b o r a t o r y  or p i p e t t e  superior.  under:: a g i v e n s e c t i o n ,  say  by the  was  I f each of  the b u r e t t e ,  the  were  76 checked "yes", skills.  the s u b j e c t was regarded  as d i s p l a y i n g s u p e r i o r  I f only two of the i n c l u d e d behaviours  the s u b j e c t was regarded the b u r e t t e .  as having  were present  adequate s k i l l s i n h a n d l i n g  However, i f o n l y one o r none o f t h e behaviours  was checked "yes" the s u b j e c t was regarded  as having  inadequate s k i l l s i n h a n d l i n g the b u r e t t e .  The  Interview  Task  To assess the s u b j e c t s understanding i n v o l v e d i n an acid-base approach was favoured  t i t r a t i o n an i n d i v i d u a l  task —  Following Piaget's  pro-  the t i t r a t i o n o f HC1 a g a i n s t NaOH  was used as the b a s i s f o r e l i c i t i n g the s u b j e c t ' s The  interview  s i n c e i t allows one t o probe f u r t h e r  i n t o a s u b j e c t ' s i n i t i a l responses. cedure, a concrete  o f the concepts  —  responses.  task r e q u i r e d the s u b j e c t t o determine the c o n c e n t r a t i o n  of the HC1 given "the"concentration o f NaOH. T h i s task was broken down i n t o two p a r t s .  The f i r s t  p a r t was t o f i n d the volume o f base r e q u i r e d t o n e u t r a l i z e 25 ml of a given a c i d while the second p a r t r e q u i r e d the computation of the c o n c e n t r a t i o n o f the a c i d from the data obtained  from  the f i r s t p a r t . The  s u b j e c t s were s u p p l i e d with a l l the s o l u t i o n s  needed f o r t h e t i t r a t i o n i n c l u d i n g one i n d i c a t o r and a white sheet o f paper.  The s o l u t i o n s were HC1 (approximately  and a d i l u t e NaOH s o l u t i o n . blue.  0.1 M),  The i n d i c a t o r used was bromthymol  The apparatus c o n s i s t e d o f a 50 ml b u r e t t e , a 25 ml  p i p e t t e , a f u n n e l , a b u r e t t e stand,  f o u r 250 ml beakers and  f o u r 250  ml c o n i c a l f l a s k s . The  concepts i n v o l v e d i n an acid-base  were of i n t e r e s t i n the present mole concept and  study were: pH,  i n d i c a t o r behaviour.  titration  concentration,  A l s o of i n t e r e s t was  the s u b j e c t s ' understanding of some of the chemical o l o g i e s used i n acid-base  titrations.  s t o i c h i o m e t r i c p o i n t , equivalence  that  termi-  These i n c l u d e d endpoint,  p o i n t , and n e u t r a l i z a t i o n  point. To assess  the above concepts and  s c i e n t i f i c terms i n  an i n t e r v i e w s i t u a t i o n , a number of questions These questions  were developed.  were based upon the procedures used i n doing  t i t r a t i o n and  the o b s e r v a t i o n s  made by the s u b j e c t s .  such as: "Why  d i d you choose t h i s i n d i c a t o r " and  Questions  "Why  does  the i n d i c a t o r change c o l o u r " were asked, f o r example. Two  chemistry  teachers who  expressed i n t e r e s t i n  p a r t i c i p a t i n g i n the p i l o t study were asked to comment on questions.  T h e i r major suggestion  was  to i n c l u d e a  the  question  on the pH a t the s t o i c h i o m e t r i c p o i n t f o r the t i t r a t i o n of a weak a c i d a g a i n s t a strong base. To i d e n t i f y the s t r a t e g i e s used by the students t h e i r s o l u t i o n to the v o l u m e t r i c  a n a l y s i s c a l c u l a t i o n s during  the i n t e r v i e w , the f o l l o w i n g q u e s t i o n was  asked: "How  o b t a i n the c o n c e n t r a t i o n of the HCl from the obtained  can  the r e a c t i n g mole r a t i o i s  1:1.  you  data?"  Since the i n t e r v i e w task i n v o l v e d the t i t r a t i o n of NaOH a g a i n s t HCl,  in  78 Furthermore,  i n o r d e r t o examine how t h e  used  such t i t r a t i o n data  acid  i n a problem s i t u a t i o n  1:1,  three  1.  If  questions  If  you used  3.  If  than  are:  w i t h t h e b a s e i n a mole r a t i o  what w i l l  acid,  other  the  be t h e c o n c e n t r a t i o n o f  of  the  instead  2:1 acid? of  what w o u l d be t h e c o n c e n t r a t i o n o f  the  acid?  instead  carbonate  These  s u l p h u r i c a c i d i n the t i t r a t i o n  hydrochloric sulphuric  the c o n c e n t r a t i o n of  r e q u i r i n g a mole r a t i o  were u s e d .  the a c i d reacted  respectively, 2.  to p r e d i c t  subjects  of to  s o d i u m h y d r o x i d e , y o u had u s e d titrate  concentration of  hydrochloric acid,  sodium  what w o u l d be  the  the h y d r o c h l o r i c a c i d ?  Pilot  Study  Purpose The  purposes  of  the p i l o t  mine t h e p s y c h o m e t r i c p r o p e r t i e s ality  Test  Analysis  (CPT),  Test  Subconcepts  (VAT);  (2)  the procedure used to t h e V A T ; and of The  the  (3)  study were:  of  Test  (1)  to  deter-  the Classroom P r o p o r t i o n (SCT)  and t h e  Volumetric  to determine the appropriateness  i d e n t i f y the conceptual e r r o r s  to determine  the appropriateness  i n t e r v i e w on t h e t i t r a t i o n  of  of  made on the  format  task.  Sample The  secondary first  year  pilot  schools  sample  consisted  of  158  students  i n the Vancouver m e t r o p o l i t a n area  u n i v e r s i t y chemistry students.  The  i n two and 5  secondary  school  subjects belonged  schools.  Forty subjects  Grade 11 c h e m i s t r y  to  six  intact  classes  i n the  i n two c l a s s e s were e n r o l l e d i n  course w h i l e the  remaining subjects  c l a s s e s were e n r o l l e d i n t h e Grade 12 c h e m i s t r y Grade 11 s u b j e c t s were u s e d b e c a u s e o f o b t a i n i n g G r a d e 12 c l a s s e s The u n i v e r s i t y the of  study. the  been  They were  i n t e r v i e w most  formally  analysis tions.  the  i n the p i l o t  in  the d i f f i c u l t y  four The  in  schools.  s u b j e c t s were a s k e d t o v o l u n t e e r  i n c l u d e d i n the  study  G r a d e 12 c h e m i s t r y laboratory  because at  students  in that  for  the  s t u d e n t s had n o t  techniques  s t u d e n t s were t h o u g h t  study,  the  course.  p r i o r to  time yet  in volumetric  a l t h o u g h t h e y had done v o l u m e t r i c a n a l y s i s  G r a d e 12 c h e m i s t r y tion  i n other  introduced to  The u n i v e r s i t y  two  calcula-  t o be  similar  their  participa-  t h e y had n o t done any t i t r a t i o n s  to  at  university.  Data C o l l e c t i o n Procedure I n May 1980, Subconcepts students.  Test  of  the  format  tests of  were a d m i n i s t e r e d  The f o u r  the purposes of  participating  the p i l o t .  and t o  the  was  teachers  t h e G r a d e 12  t e a c h e r s were  any s o u r c e s o f  and  the  chemistry  informed of t o be  confusion  critical in  the  tests.  administered for  to  They were e n c o u r a g e d  indicate  I n September Test  the V o l u m e t r i c A n a l y s i s Test  1980, to  the  the  Classroom  Proportionality  two Grade 11 c l a s s e s .  The two  t h e s e c l a s s e s were a s k e d t o be p r e s e n t d u r i n g  administration  of  the t e s t  so  that  they c o u l d provide  their  the  80 comments on the i n s t r u c t i o n s .  They were a l s o asked t o comment  on the format. Immediately a f t e r the a d m i n i s t r a t i o n o f t h i s t e s t , 8 students were randomly s e l e c t e d from the 4 0 students who took t h i s t e s t and i n t e r v i e w e d Balance Task  on the E q u i l i b r i u m i n the )•  (Inhelder and P i a g e t ,  1958).  T h i s was done t o  assess the v a l i d i t y o f the Classroom P r o p o r t i o n a l i t y T e s t . The  i n t e r v i e w task i n v o l v e d the use o f a balance beam and hang-  ing  weights t o assess the p r o p o r t i o n a l reasoning  The  task r e q u i r e d the s u b j e c t s t o simply  of subjects.  r e s t o r e the e q u i l i -  brium o f the balance by p u t t i n g a s p e c i f i e d weight a t the c o r r e c t p o i n t where these would counter-balance those p l a c e d by the i n t e r v i e w e r .  A f t e r p l a c i n g the weights, the s u b j e c t s  were asked t o e x p l a i n t h e i r answers. s i t u a t i o n s were used.  Three d i f f e r e n t r a t i o  Performance on the Balance Task was  assessed on the b a s i s o f the q u a l i t y o f the e x p l a n a t i o n s  and  the a b i l i t y t o hang weights i n the c o r r e c t l o c a t i o n s on the beam of the b a l a n c e .  The v a l i d i t y and r e l i a b i l i t y of the  E q u i l i b r i u m i n the Balance Task have been e s t a b l i s h e d by numerous i n v e s t i g a t o r s (e.g. B a r t , 1 9 7 1 ; DeVries, 1 9 7 4 ; Pumfrey, 1 9 6 8 ; Lawson, Nordland and Devito, Following  1974).  the i n t e r v i e w on the E q u i l i b r i u m i n the  Balance Task, f i v e u n i v e r s i t y chemistry students were i n t e r viewed on the t i t r a t i o n t a s k . one  The i n t e r v i e w was conducted i n  of the l e c t u r e rooms i n the F a c u l t y o f E d u c a t i o n of the  U n i v e r s i t y o f B r i t i s h Columbia. The  interview  was  d i v i d e d i n t o two major sect-ions.  81 In the f i r s t p a r t of the i n t e r v i e w a type of P i a g e t i a n c l i n i c a l i n t e r v i e w technique student's The  was  used t o e l i c i t  the  understanding of the concepts i n v o l v e d i n t i t r a t i o n .  s u b j e c t was  titration.  asked q u e s t i o n s  He was  could write. list,  (Piaget, 1929)  as he was  performing  the  a l s o given a sheet of paper on which he  A l s o d u r i n g t h i s p a r t of the i n t e r v i e w a check-  Laboratory  S k i l l s i n T i t r a t i o n , was  used to i d e n t i f y  manipulatory s k i l l s used by the s u b j e c t as he performed  the  the  titration. In the second p a r t of the i n t e r v i e w , the was  subject  asked t o v e r b a l i z e h i s t h i n k i n g as he performed c a l c u l a ^  t i o n s and to p r e d i c t the c o n c e n t r a t i o n  of the a c i d i n three  problem s i t u a t i o n s . The tion  t a l k i n g aloud technique or concurrent v e r b a l i z a -  ( E r i c s s o n and  Simon, 1980)  employed i n t h i s study,  been used s u c c e s s f u l l y by a number of i n v e s t i g a t o r s F l a h e r t y , 1974;  Gorodetsky and  Hoz,  1980)  has  (e.g.  as an a l t e r n a t i v e  approach to the question-and-answer technique i n i d e n t i f y i n g the problem s o l v i n g processes  of s u b j e c t s .  I t must be noted  t h a t although a review a r t i c l e by N i s b e t t and Wilson sought t o c r i t i z e the use of t h i s technique, r e a n a l y s i s of t h e i r paper by E r i c s s o n and  a recent  Simon (1980), u s i n g  a model of v e r b a l i z a t i o n which p r e d i c t s the k i n d of t h a t can r e l i a b l y be r e p o r t e d ,  (1977)  showed t h a t the  content  experimental  c o n d i t i o n s and procedures used i n most of the s t u d i e s reviewed by N i s b e t t and Wilson were such t h a t " v e r d i c a l r e p o r t s  could  h a r d l y be e x p e c t e d " ( E r i c s s o n  their  and  Simon, 1980)  .  Using  82 model, E r i c s s o n  and Simon concluded t h a t v e r b a l  reports  e l i c i t e d by a s k i n g s u b j e c t s t o t h i n k aloud produce a v a l u a b l e and  thoroughly r e l i a b l e source o f data about  cognitive  processes. In the present study, a f t e r the s u b j e c t s had completed the  t i t r a t i o n they were asked t o t h i n k aloud as they used the  o b t a i n e d data t o c a l c u l a t e the c o n c e n t r a t i o n o f the a c i d .  The  s u b j e c t s were g i v e n sheets o f paper with t h e i r names on so t h a t they c o u l d verbalized  their  show t h e i r c a l c u l a t i o n s on the sheets as they thinking.  A f t e r c a l c u l a t i n g the c o n c e n t r a t i o n o f the a c i d used i n the t i t r a t i o n , the s u b j e c t  was asked t o use the data o b t a i n e d  i n the t i t r a t i o n t o p r e d i c t the c o n c e n t r a t i o n of the a c i d i n three s i t u a t i o n s r e q u i r i n g a mole r a t i o o f 2:1.  T h i s was done  i n order t o f i n d out i f problem s i t u a t i o n s r e q u i r i n g the use of 2:1 r e a c t i o n mole r a t i o s would present any d i f f i c u l t i e s t o the subjects — use  that  the c o r r e c t  t h e i r use.  i s , whether the s u b j e c t stoichiometric  1.  ratios i n situations  requiring  I t was a l s o done t o f i n d out i f the same approach  used i n the i n i t i a l applied  r e a l i z e d the need t o  s i t u a t i o n r e q u i r i n g a 1:1 mole r a t i o was  i n these s i t u a t i o n s . In these p r e d i c t i v e s i t u a t i o n s , the s u b j e c t  was  asked t o use the volumes o b t a i n e d i n the t i t r a t i o n and the known c o n c e n t r a t i o n o f the base. subject  was asked t o e x p l a i n  A f t e r each p r e d i c t i o n , the  his prediction  i r r e s p e c t i v e of  whether he showed h i s c a l c u l a t i o n s . The  above procedure, t h a t i s , a s k i n g f o r p r e d i c t i o n s  83 i n s t e a d o f asking s u b j e c t s t o perform more t i t r a t i o n s was employed because o f the l e n g t h o f the i n t e r v i e w . i n t e r v i e w was audio-taped  The e n t i r e  f o r subsequent t r a n s c r i p t i o n .  i n t e r v i e w took from 45 minutes t o one hour f o r  Each  completion.  Scoring In s c o r i n g the s u b t e s t s o f the CPT, 3 p o i n t s were given f o r those  s u b j e c t s who s e l e c t e d the c o r r e c t answer and  reasoned i n terms o f r a t i o o r p r o p o r t i o n ; 2 p o i n t s were awarded to those who s e l e c t e d the wrong answer but used r a t i o o r porportion i n t h e i r explanation.  Subjects who s e l e c t e d the c o r r e c t  answer but employed a d d i t i v e o r any other e x p l a n a t i o n s  o r no  e x p l a n a t i o n were given a score o f one. A zero score was awarded to those who s e l e c t e d the wrong a l t e r n a t i v e and gave the wrong reasons. CPT  The maximum score o b t a i n a b l e on each subtest o f the  was 21. The  s c o r i n g procedure used f o r the E q u i l i b r i u m i n  the Balance Task was s i m i l a r t o t h a t f o r the CPT.  The maximum  score f o r the E q u i l i b r i u m i n the Balance Task was 9. The was  s c o r i n g procedure used f o r the VAT and the SCT  t o g i v e one p o i n t f o r each c o r r e c t answer.  The maximum  score o b t a i n a b l e on the VAT was 15 while the maximum score f o r the SCT was 28. Data A n a l y s i s A n a l y s i s was done s e p a r a t e l y f o r the t e s t data and the i n t e r v i e w data.  In p r e p a r a t i o n of the t e s t data f o r  a n a l y s i s by computer, the i n v e s t i g a t o r coded the scores on f o r e r a n statement forms.  Each student was given an i d e n t i f i c a -  84 t i o n number t h a t score  on t h e  identified his  individual  s t u d e n t was a l s o verified at  items  coded.  of  of B r i t i s h  Item a n a l y s i s  reliabilities  and s c h o o l . by e a c h  of  Columbia. by t h e  t h e Computer C e n t r e The d a t a  were h a n d -  investigator.  was p e r f o r m e d u s i n g t h e LERTAP  computer program i n o r d e r t o d e t e r m i n e of  performance of  t h e V A T , S C T , a n d CPT and t o examine  the  items  i n each t e s t .  The two s u b t e s t s since  as  The S t a t i s t i c a l  for  variables  the S o c i a l  i n the  Sciences  study.  program  (Nie, et.  t o o b t a i n t h e c o r r e l a t i o n between t h e on t h e CPT and t h e E q u i l i b r i u m The i n t e r v i e w d a t a a complete t r a n s c r i p t the  subjects  of  called  Response  to  the audio tapes.  together.  Package used  subjects  -  Task. obtaining  The r e s p o n s e s i n the study  of  were  These c a t e g o r i e s  were  Patterns.  volumetric analysis  obtained  from t h e i r t i t r a t i o n )  approach  (or  think  treated  was  the  of  r e s p o n s e s w h i c h were  and t h e w r i t t e n work  s u b j e c t s when t h e y were a s k e d  several  1975)  were a n a l y z e d by f i r s t  The i n t e r v i e w p r o t o c o l s from t h e  al.  i n the Balance  For each q u e s t i o n ,  s i m i l a r were c a t e g o r i z e d  t h e y were  scores of  t o each main q u e s t i o n asked  grouped t o g e t h e r .  the the  t h e CPT were i t e m - a n a l y z e d s e p a r a t e l y different  The  The c o d e d d a t a were k e y - p u n c h e d and  checked f o r key-punching e r r o r s  1974)  class  the t e s t s n t a k e n  by t h e K e y - p u n c h i n g S e r v i c e s  the U n i v e r s i t y  (Nelson,  grade,  strategy)  used.  t o work o u t  problems u s i n g the were a n a l y z e d t o As t h e  students  a l o u d when d o i n g t h e s e p r o b l e m s ,  (obtained  solutions data  identify  were a s k e d  the v e r b a l  the to  protocols  85 along with t h e i r w r i t t e n responses  were used t o c a t e g o r i z e  the s t r a t e g y they used i n terms o f a formula approach or a p r o p o r t i o n a l reasoning approach. The  responses  o b t a i n e d when the s u b j e c t s were asked  to p r e d i c t the c o n c e n t r a t i o n o f the a c i d i n three problem s i t u a t i o n s r e q u i r i n g 2:1 mole r a t i o s were analyzed t o see whether c o r r e c t p r e d i c t i o n s were made. The manual s k i l l s used by the s u b j e c t s were examined t o f i n d the frequency skill  o f occurrence o f each s p e c i f i c  i n the i n t e r v i e w sample. F i n a l l y , the w r i t t e n work o f the s u b j e c t s on the  VAT was analyzed i n order t o i d e n t i f y the conceptual e r r o r s made i n t h e i r s o l u t i o n s .  F o r each s u b j e c t , the i n d i v i d u a l  steps used t o s o l v e each q u e s t i o n on the VAT was examined f o r any e r r o r s i n r e a s o n i n g .  Each conceptual e r r o r was recorded  only once f o r any p a r t i c u l a r student r e g a r d l e s s o f whether he had made the same e r r o r on two or more other q u e s t i o n s i n the test. R e s u l t s and D i s c u s s i o n Classroom  P r o p o r t i o n a l i t y Test  Item A n a l y s i s : of  The item c h a r a c t e r i s t i c s f o r the two s u b t e s t s  the CPT are presented  i n Table 3.. The item-subtest  c o r r e l a t i o n s r e v e a l e d t h a t a l l the i t e m - t e s t v a l u e s f o r both the D i r e c t P r o p o r t i o n a l i t y Subtest and the Inverse P r o p o r t i o n a l i t y Subtest were p o s i t i v e and h i g h .  This indicated that  those who scored h i g h on the s u b t e s t got the i n d i v i d u a l correct.  items  86 Table 3 Test S t a t i s t i c s o f the Classroom P r o p o r t i o n a l i t y T e s t  SUBTEST A - DIRECT PROPORTION Test mean = 1 7 . 1 0 Standard d e v i a t i o n = 5 . 0 0 Highest sicore = 2 1 . 0 0 Lowest eicore = 4 . 0 0 Hoyt estimate o f r e l i a b i l i t y = 0 . 7 9 Standard e r r o r of measurement = 2 1 1 0  Test Item  1 2 3 4 5 6 7  Difficulty (%)  Index  62, 60, 87. 82. 87. 87. 82.  D i s c r i m i n a t i o n Index (Point B i s e r i a l )  0.74 0.80 0.53 0.69 0.80 0.59 0.61  SUBTEST B - INVERSE PROPORTION Test mean = 1 0 . 2 0 Standard d e v i a t i o n = 5. 9 5 Highest score = 2 1 . 0 0 Lowest score = 1 . 0 0 Hoyt estimate o f r e l i a b i l i t y = 0 . 8 4 Standard e r r o r of measurement = 2 . 2 1 T e s t Item  8 9 10 11 12 13 14  Difficulty (%)  42.5 47.5 32.5 32.5 53.0 30.0 25.0  Index  D i s c r i m i n a t i o n Index (Point B i s e r i a l )  0.49 0.62 0.78 0.78 0.80 0.82 0.83  87 The  difficulty  i n d i c e s of the items i n the two  sub-  t e s t s r e v e a l e d t h a t the items i n the Inverse P r o p o r t i o n a l i t y Subtest were more d i f f i c u l t than those tionality  Subtest. The  and  i n the D i r e c t Propor-  Inverse  reliabilities  f o r the D i r e c t P r o p o r t i o n a l i t y  P r o p o r t i o n a l i t y Subtests were high  respectively).  T h i s i m p l i e s t h a t there i s an  (.79  and  .84,  acceptable  i n t e r n a l c o n s i s t e n c y among the items i n each s u b t e s t . Validity:  The  Pearson product  moment c o r r e l a t i o n between the  t o t a l scores of the s u b j e c t s on the i n t e r v i e w task - the brium i n the Balance Task - and Test was  c a l c u l a t e d to be 0.91.  between the two a l i t y Test has  Equili-  the Classroom P r o p o r t i o n a l i t y The  c l o s e correspondence  measures a f f i r m s t h a t the Classroom P r o p o r t i o n convergent v a l i d i t y .  T h i s i m p l i e s t h a t the  Classroom P r o p o r t i o n a l i t y Test can be used i n p l a c e of i n d i v i d u a l i n t e r v i e w with reasonable  confidence  the  to measure  p r o p o r t i o n a l reasoning of s u b j e c t s . In a d m i n i s t e r i n g the t e s t one  hour was  found to be  sufficient. The  Subconcepts Test The  r e s u l t s of the item a n a l y s i s of t h i s t e s t  given i n Table  4.  The  item-subtest  t h a t almost a l l the items possessed properties. was  Only one  c o r r e l a t i o n s revealed acceptable  psychometric  item i n the s u b t e s t , D i l u t i o n C a l c u l a t i o n ,  found to have a c o r r e l a t i o n as low as 0.28.  Examination  of the c a l c u l a t i o n s done by the s u b j e c t s revealed t h a t did  are  not read t h i s q u e s t i o n  carefully.  they  88 Table 4 Test  Item  Statistics  of  the  Subtests  i n the  D i f f i c u l t y Index (%)  Subconcepts  D i s c r i m i n a t i o n Index (Point-Biserial)  SUBTEST A - CONCENTRATION CALCULATION T e s t iiiean = 3.20 S t a n d a r d d e v i a t i o n = 0.95 Hoyt e s t i m a t e o f r e l i a b i l i t y = 1 2 3 4  0.49  82.7 93.8 85.2 58.0  .65 .43 .64 .76  SUBTEST B - DILUTION CALCULATION T e s t mean = 2.02 Standard d e v i a t i o n =1.04 Hoyt e s t i m a t e o f r e l i a b i l i t y 5 6 7 8  =  0.30  53.1 71.6 61.7 76.0  .67 .71 .56 .28  SUBTEST C - STOICHIOMETRY T e s t mean = 2.49 S t a n d a r d d e v i a t i o n = 1.10 Hoyt e s t i m a t e o f r e l i a b i l i t y = 9 10 11 12  0.41  53.1 81.5 39.5 75.3  .65 .42 .68 .63  SUBTEST D - GRAM/MOLE CONVERSION T e s t mean =3.10 S t a n d a r d d e v i a t i o n = .98 Hoyt e s t i m a t e o f r e l i a b i l i t y = 13 14 15 16  71.6 87.7 61.7 88.9  Test  .43 .68 .61 .68 .44  89 Table 4 continued  Item  Difficulty (%)  Index  D i s c r i m i n a t i o n Index (Point-Biserial)  SUBTEST E - MOLE Test mean = 2.62 Standard d e v i a t i o n = 1.12 Hoyt estimate o f r e l i a b i l i t y 17 18 19 20  = .40  65.4 58.0 58.0 80.2  .52 .60 .67 .61  SUBTEST F - BALANCING EQUATIONS T e s t mean = 2.04 Standard d e v i a t i o n = 1.02 Hoyt estimate o f r e l i a b i l i t y 21 22 23 24  = .43  71.6 46.9 77.8 7.4  .75 .63 .64 .36  SUBTEST G - INTERPRETATION OF REACTIONS Test mean - 2.41 Standard d e v i a t i o n = 1.12 Hoyt estimate o f r e l i a b i l i t y 25 26 27 28 Reliability  = 0.40  65.4 49.4 79.0 46.9 (Cronbachs  alpha)  .59 .59 .54 .61 f o r composite  = 0.80  90 The r e l i a b i l i t i e s 0.5.  T h i s was a t t r i b u t e d  Also,  t h e low r e l i a b i l i t i e s  of  the  subtest content.  to  (Cronbach's composite  that  all  domain.  the  the  s u b t e s t s were a l l  few i t e m s  i n each  may be p a r t i a l l y  alpha)  was  due t o  0 . 8 0 which  the  to t h i s  nature  the  total  indicated  s u b t e s t s may have b e e n a s s e s s i n g t h e  The one h o u r a l l o c a t e d  below  subtest.  However, the r e l i a b i l i t y o f  test  the  for  t e s t was  same c o n t e n t found to  be  sufficient. The V o l u m e t r i c A n a l y s i s In t h i s tion  of  that  the  of  the  by a l l  one i t e m was  format  of  students.  t h e q u e s t i o n was  s u b j e c t s with the r e s u l t i t e m between t h e  and t h o s e  terms  culty  of  who s c o r e d  their  item  coefficients.  f o u n d t o be 0 . 8 7 . the  items  for  this  item  that  suggested  confusing  low on t h e t e s t .  This  reflects  the  One h o u r was  some  correctly  h i g h on t h e  (Table 5 )  i n t h e t e s t were test  to  t h e r e was no d i s c r i m i n a -  s u b j e c t s who s c o r e d  — total  correla-  acceptable  correlations  and  diffi^  the o v e r a l l t e s t  was  i n t e r n a l homogeneity o f f o u n d t o be  sufficient  test. Errors Six  Test.  likely  The r e l i a b i l i t y o f  i n the t e s t .  Conceptual  subjects'  this  One i t e m i n t h e t e s t was a n s w e r e d  The r e m a i n i n g i t e m s in  f o u n d t o have a  Further examination of  0.20.  t i o n on t h i s test  test,  Test  types  of  conceptual errors  solutions  to  the  These were:  items  were f o u n d i n  i n the V o l u m e t r i c  ( 1 ) the assumption  of  1:1 ratio  the  Analysis in  all  91 Table 5 Test  Statistics  of  the Volumetric A n a l y s i s  Test  T e s t mean = 1 0 . 2 6 Standard d e v i a t i o n = H i g h e s t s c o r e = 15 Lowest score = 2 Hoyt e s t i m a t e o f r e l i a b i l i t y = 0.87 S t a n d a r d e r r o r o f measurement = 1.33  Test  Item  Difficulty (%)  Index  3.79  D i s c r i m i n a t i o n Index (Point-Biserial)  1  77.8  0.35  2  77.8  0.44  3  6.3  0.20  4  70.4  0.63  5  100.0  0.00  6  48.1  0.87  7  48.1  0.88  8  33.3  0.77  9  66.7  0.45  10  63.0  0.75  11  81.5  0.49  12  48.1  0.79  13  96.3  0.43  14  44.4  0.84  15  74.1  0.56  92 reactions;  (2) the misconception  that sulphuric acid w i l l  r e a c t with any base i n a r a t i o o f 1:2; (3) the n o t i o n t h a t the c o n c e n t r a t i o n o f a stock s o l u t i o n o f a c i d i s d i f f e r e n t from the c o n c e n t r a t i o n o f the p o r t i o n used i n t i t r a t i o n ; r e v e r s a l o f the s t o i c h i o m e t r i c mole r a t i o s ; i n c o r r e c t formulas; chemical  equations The  (4) the  (5) the w r i t i n g o f  and (6) the f a i l u r e t o w r i t e  balanced  i n the problem s o l u t i o n s .  f i r s t conceptual  e r r o r may be a t t r i b u t e d t o a  number o f p o s s i b l e f a c t o r s i n c l u d i n g the use o f i n c o r r e c t formulas and the f a i l u r e t o w r i t e balanced  chemical  equations  f o r the r e a c t a n t s . The  second conceptual  e r r o r r e v e a l e d t h a t some of the  s u b j e c t s - f a i l e d t o d i s c r i m i n a t e between r e a c t i o n s i n which H2SO4 reacted, i n a 1:2 and 1:1 mole r a t i o s , r e s p e c t i v e l y , with substances. without  They only c o n s i d e r e d  t h e two protons  g i v i n g equal a t t e n t i o n t o the nature  r e a c t i n g substances.  other  i n H2SO4  o f the other  Once again t h i s d i f f i c u l t y might be  a t t r i b u t e d t o the i n a b i l i t y o f some o f the s u b j e c t s t o w r i t e c o r r e c t formulas and c o r r e c t balanced reacting  equations  f o r the  substances. The  t h i r d conceptual  e r r o r r e v e a l e d t h a t the s u b j e c t s  were a p p l y i n g a d i r e c t p r o p o r t i o n a l r e l a t i o n t o o b t a i n a d i f f e r e n t c o n c e n t r a t i o n f o r the t o t a l s o l u t i o n . students  d i d not see the i n v a r i a n c e o f the c o n c e n t r a t i o n when  o n l y p a r t o f i t has been removed.  T h i s i m p l i e d t h a t the sub-  j e c t s may not have had a good understanding was.  These  o f what a s o l u t i o n  However, t h e i r d i f f i c u l t y c o u l d a l s o have been viewed  as being  s t r u c t u r a l i n nature.  r e l a t i o n s h i p here.  But t h i s  That i s , there i s a part-whole i s ..notutheccommonsense one which  i m p l i e d t h a t the whole i s g r e a t e r than one o f i t s parts:, s i n c e i n t h i s case the c o n c e n t r a t i o n o f the t o t a l s o l u t i o n was the same as the c o n c e n t r a t i o n i n a p o r t i o n of i t . The ratios —  f o u r t h conceptual  error —  r e v e r s a l o f mole  suggested t h a t the s u b j e c t s who made t h i s e r r o r d i d  not recognize the nature  o f the r e l a t i o n s h i p between the  s t o i c h i o m e t r i c c o e f f i c i e n t s and the moles o f substances used i n the r e a c t i o n . The molecular  f i f t h conceptual  formulas —  t h i s e r r o r probably  error —  writing incorrect  suggested t h a t the s u b j e c t s who made  d i d not understand the concept o f v a l e n c y .  The w r i t t e n work o f the s u b j e c t s who made t h i s e r r o r suggested t h a t some o f them, even though they c o u l d w r i t e the formulas f o r t h e compounds i n v o l v e d i n t h e r e a c t i o n , c o u l d n o t balance the equation.  Other s u b j e c t s f a i l e d t o w r i t e equations i n  their solutions. Laboratory  Skills in Titration The  a n a l y s i s o f the l a b o r a t o r y techniques  o f the  f i v e u n i v e r s i t y s u b j e c t s r e v e a l e d t h a t three o f the s u b j e c t s who had done more than one t i t r a t i o n i n Grade 1 2 d i d not have any  d i f f i c u l t i e s i n proceeding  d i s p l a y e d reasonably  accurate  with the experiment.  They  s k i l l s i n the h a n d l i n g and  r e a d i n g o f the b u r e t t e and p i p e t t e , used a f u n n e l i n t r a n s f e r i n g s o l u t i o n s from the reagent i n adding the base.  b o t t l e s and e x e r c i s e d c a u t i o n  The other two students who had done only  one t i t r a t i o n l a b Despite  i n G r a d e 1 2 had t o be g i v e n some  the proper  skills  displayed  i n some a r e a s by most  them o n l y one s u b j e c t r e a l i z e d t h e n e e d f o r to a i d  i n observing  the c o l o u r changes  down t h e a c i d s o l u t i o n a d h e r i n g t o ensure  that  all  the a c i d  for  an e x t r a  solution, familiar for  some o f  burette the  sides of  made w e r e :  t o be u s e d  students  three of  the  washed  flask  w i t h the bromthymol b l u e  the  to  subjects  i n measuring the  indicated that  p h e n o l p h t h a l e i n , two s t u d e n t s  obtained  a white background  and no s t u d e n t  the  of  reacted.  Other observations asked  help.  acid  t h e y were  i n d i c a t o r and h e n c e w r o t e down t h e  not  asked  volumes  i n t h e t i t r a t i o n o n l y when t h e y were p r o m p t e d by  the  investigator. The  Interview The  Task analysis  of  the  protocols  which e x p l o r e d the  titration  concepts  did  revealed  n o t seem t o have c l e a r  found t h a t about  although,  the  first  part  subjects'  of  In g e n e r a l ,  had a good  used  it  was  t h e y d i d n o t have a  clear  scientific  in titration. With respect  indicated that  to moles,  the moles  of  l o w e r when d i s t i l l e d w a t e r on m o l e s  subjects  understanding  u n d e r s t a n d i n g o f p H , i n d i c a t o r b e h a v i o u r and t h e terms  of  a r e a s where t h e  understanding.  c o n c e n t r a t i o n and m o l e s ,  interview  understanding  a number o f  subjects  the  o n l y one o f  the  five  the a c i d i n a s o l u t i o n w i l l is  added.  A l l the other  and c o n c e n t r a t i o n were c o r r e c t l y  answered.  students be  questions  95 Most o f the s u b j e c t s r e a l i z e d t h a t the pH a t the s t o i c h i o m e t r i c p o i n t f o r the r e a c t i o n between HC1 and NaOH was  7.  They a l s o r e a l i z e d t h a t the pH o f the a c i d  with the a d d i t i o n of the base.  However, only two s u b j e c t s  c o u l d c a l c u l a t e the pH o f .1 M NaOH o r .1 M HC1. of the s u b j e c t s  increased  Also  four  i n d i c a t e d t h a t the pH f o r the r e a c t i o n  between a c e t i c a c i d and NaOH was 7 w h i l e one student d i d n ' t know.  Even though, they knew t h a t the c o n c e n t r a t i o n  of a  s o l u t i o n w i l l change w i t h a d d i t i o n o f d i s t i l l e d water, none o f them r e a l i z e d t h a t the pH w i l l be h i g h e r . Although they seemed t o have a c l e a r n o t i o n o f why i n d i c a t o r s are used i n t i t r a t i o n , they d i d not seem t o understand the mechanism o f i n d i c a t o r a c t i o n . seem t o know why bromthymol blue  They a l s o d i d not  i s used i n the t i t r a t i o n .  Here, i t must be noted, t h a t s i n c e some o f the students i n d i c a t e d t h a t they were more f a m i l i a r with  phenolphthalein,  the r e s u l t c o u l d have been d i f f e r e n t i f p h e n o l p h t h a l e i n had been used. The  terminology questions  i n d i c a t e d t h a t only one  s u b j e c t c o u l d e x p l a i n what endpoint and s t o i c h i o m e t r i c p o i n t meant.  However, a l l o f them i n d i c a t e d t h a t they had heard  the term s t o i c h i o m e t r i c p o i n t . The d i f f i c u l t i e s  of the s u b j e c t s  c o u l d be a t t r i b u t e d t o the f a c t t h a t such questions  a r e not  u s u a l l y asked d u r i n g experiments, as was i n d i c a t e d by two students. The and  a n a l y s i s o f the t r a n s c r i b e d v e r b a l  the w r i t t e n work obtained  protocols  from the s u b j e c t s as they used  96  t h e i r own data t o c a l c u l a t e the c o n c e n t r a t i o n i n d i c a t e d t h a t two b a s i c approaches — and the P r o p o r t i o n a l Approach — i n the study.  o f the HC1  the Formula  were employed  Approach  by the s u b j e c t s  The b a s i c form o f the Formula Approach  involved  the use o f the formula: M o l a r i t y x Volume = Moles, t o c a l c u l a t e the moles o f base i n i t i a l l y and then use the v a l u e o b t a i n e d and the mole r a t i o t o f i n d the moles o f a c i d .  The c o n c e n t r a -  t i o n o f the a c i d was then found by a second a p p l i c a t i o n o f the above formula but i n t h i s case u s i n g v a l u e s f o r the a c i d .  In  a v a r i a n t form o f the Formula Approach, one s u b j e c t used a formula which a v o i d s a double a p p l i c a t i o n o f the formula used i n the b a s i c form o f the Formula Approach, namely, m o l a r i t y o f base x volume o f base = m o l a r i t y of a c i d x volume o f a c i d . Thus, t h i s v a r i a n t form a c u t a l l y combines the b a s i c  some o f the steps i n  form. In the b a s i c form o f the P r o p o r t i o n a l Approach, the  subject  d,i-.d not use an e x p l i c i t formula but i n s t e a d uses the  p o r p o r t i o n a l r e l a t i o n between moles and volume t o o b t a i n the moles o f base.  Using the r e a c t i o n mole r a t i o s , the moles o f  a c i d was then o b t a i n e d .  Thereafter,  the same p r o p o r t i o n a l  r e l a t i o n between moles and volume was a p p l i e d t o the a c i d t o o b t a i n the c o n c e n t r a t i o n . of t h i s approach employed  The student who used a v a r i a n t form a p r o p o r t i o n a l r e l a t i o n s h i p between  the volumes and the c o n c e n t r a t i o n s o f the a c i d and the base, namely, Volume o f a c i d Volume o f base  _  M o l a r i t y o f base Molarity of acid  97 T h i s , i n a s e n s e ^ i s a combination  o f the two p r o p o r t i o n s used  i n the b a s i c form of the P r o p o r t i o n a l Approach.  The a n a l y s i s  showed t h a t two students used the b a s i c form o f the Formula Approach while one student used the v a r i a n t form o f the Formula Approach.  F o r the remaining two students, one used  the b a s i c form o f the P r o p o r t i o n a l Approach while the other used the v a r i a n t form. The a n a l y s i s o f the p r o t o c o l s on the s u b j e c t s ' p r e d i c t i o n o f the c o n c e n t r a t i o n o f the a c i d from t h e i r  data  r e v e a l e d t h a t they had problems with r e a c t i o n s i n v o l v i n g 2:1 mole r a t i o s .  I t was found t h a t while the i n i t i a l  problem  i n v o l v i n g NaOH and HCl i n a 1:1 r a t i o was answered c o r r e c t l y by a l l s u b j e c t s , three o f the s u b j e c t s f a i l e d t o make c o r r e c t p r e d i c t i o n s i n the t h r e e s i t u a t i o n s although they used the same approach which gave them the c o r r e c t answer i n the i n i t i a l problem. to  I t was c o n j e c t u r e d t h a t t h i s d i f f i c u l t y might be due  the i n a b i l i t y t o c o o r d i n a t e the s t o i c h i o m e t r i c mole r a t i o  with other sources o f i n f o r m a t i o n such as the a c t u a l moles o f base used i n the t i t r a t i o n .  However, t h i s i s not c l e a r s i n c e  the s u b j e c t s were not asked t o e x p l a i n t h e i r p r e d i c t i o n s .  R e v i s i o n s o f the Instruments Volumetric A n a l y s i s Test D i s c u s s i o n s w i t h the t e a c h e r s i n v o l v e d i n the p i l o t study confirmed the ambiguity  o f the item i n t h i s t e s t which had  an item - t o t a l t e s t c o r r e l a t i o n o f 0.20. T h i s item was m o d i f i e d to  remove the ambiguity. A l s o , the one item with-an  item-subtest  98 c o r r e l a t i o n of high  z e r o was  difficulty  from the  test  index.  u s e d by  revised Volumetric items  by and to the  the  I t was  Duncan and  s u b j e c t s might  i n d i c a t e the  calculations.  number o f t i t r a t i o n s  a result  attention  item final  a l s o c o n s i s t e d of interview  15  results,  laboratory such, they  Analysis  techniques  s u b j e c t s were had  done  asked  p r e v i o u s l y done  Test.  (CPT) participat-  For  example,  to t h e i r  and  c e r t a i n aspects  of the  testing  s u b j e c t s were a s k e d t o pay  explanations.  the  the  on  particular  They were a l s o a s k e d t o  i n v e s t i g a t o r was  performing  t e s t i n g procedures are  pay  demonstrations.  presented  i n Appendix  B.  The  Subconcepts For  Test  the  a c o r r e l a t i o n of administration questions  subtest, 0.28  of the  carefully.  i n Appendix  on  some w o r d i n g c h a n g e s were made i n t h e i n s t r u c t i o n s .  attention while This test  The  o f comments r e c e i v e d f r o m t h e  Emphases were a l s o p l a c e d procedure.  As  (1973).  an  laboratory titrations  influence their  Classroom P r o p o r t i o n a l i t y Test  ing teachers  (VAT)  number o f  f r o n t page o f t h e V o l u m e t r i c  As  Johnstone  I t appeared from the  above, t h a t the  possibly their  t e s t because of i t s  however, r e p l a c e d w i t h  Analysis Test  (Appendix A ) .  discussed  removed f r o m t h e  C.  was  Dilution  C a l c u l a t i o n , the  retained.  However, d u r i n g  test, The  students  item the  were a s k e d t o r e a d  Subconcepts T e s t  (SCT)  can  with  be  the found  99 Laboratory  Skills  in Titration  As a r e s u l t supplied with three  of  the o b s e r v a t i o n s  indicators  t h a l e i n ^ a n d methyl orange — for  the  titration.  burette was  Also,  t o measure o u t  i n c l u d e d i n the  included  i n the  recorded  t h e i r volumes  final 20  items  of  the  additional was  choice of the  for  included there  was  Another find  out  to the  Skills  the  i t e m was if  the  subjects  in T i t r a t i o n , consisted  interview protocols  indicators  for  the  indicate  given to the  of  and n e a r  indicators Other  the  change  p o i n t was  heard of  e v e n t h o u g h t h e y had h e a r d o f  it  No r e v i s i o n s to  use  their  dropped  since  were made t o titration  A his  asked  not  of  explain  if  t h e pH a t  the  endpoint.  W i t h t h e q u e s t i o n s on t e r m i n o l o g y , equivalence  in  questions  subject to  i n the r a t e of  titration  subjects.  Another question  c o u l d be u s e d .  i n t e r v i e w r e q u i r e d the  any d i f f e r e n c e the  titration.  reflected  subject to explain  whether the o t h e r  titration  i n the  beginning of  subjects  burette  t h e y were o b t a i n e d . \ The  i n c l u d e d which r e q u i r e d the  indicator  subject  chosen  an e x t r a  a  Task  The r e v i s i o n o f  question  acid,  to  one  them wanted t o use  apparatus.  s o o n as  Laboratory  phenolph-  (Appendix D ) .  The I n t e r v i e w  part,  some o f  i n order  as  were  from which t h e y c o u l d choose  since  checklist  checklist,  — bromthymol b l u e ,  t h e volume o f  list  made, s t u d e n t s  the q u e s t i o n  no s u b j e c t  seemed t o  stoichiometric  the q u e s t i o n  data to c a l c u l a t e  on have  point.  requiring the  concentra-  100 t i o n o f t h e HC1.  However, w i t h r e g a r d s t o t h e p r e d i c t i o n  questions,  a further question  to explain  their predictions.  was  prediction;jquestions,  and  s o d i u m c a r b o n a t e were i n c l u d e d  nity  some i d e a s bring  i n the l i s t were a l s o  interview  included.  format o f the i n t e r v i e w  situation.  that  an o p p o r t u and  provided  students  I t also allowed the inves-  t o i d e n t i f y t h e a p p r o a c h e s u s e d by t h e s u b j e c t s  calculations.  acid  of solutions.  afforded  a b o u t t h e u n d e r s t a n d i n g and s k i l l s  into a titration  tigator their  the p i l o t  to assess the general  subjects  l a b e l l e d solutions of sulphuric  o f a c e t i c a c i d a n d ammonia In g e n e r a l  t o ask t h e  To i m p r o v e t h e c o n c r e t n e s s o f  the  Solutions  included  in  101  CHAPTER IV METHODS OF THE STUDY  Introduction In  this  of  data  c o l l e c t i o n and a n a l y s e s the procedures  students the  sample,  for  the  who p a r t i c i p a t e d  study  related  to  the  sub-sample  task,  procedures  for  chapter.  instruments  used,  and t h e p r o c e d u r e s i n the  study are  selection  only  i n the  of  the  model  (completed a l l w r i t t e n instruments)  titration  data  the  the  tion  Since  interest,  chapter,  for  the  the use  of path analysis  employed i n  described. sample testing was  and d a t a  study  the p r e l i m i n a r y r e s u l t s  popula-  of phase o f  closely  i n t e r v i e w i n g on t h e  sampling procedures  both aspects of  Also,  p r i o r to  the  selected  the  are  collection  described  in  c o n d u c t e d on t h e are  included in  this test  this  chapter.  Instruments Student reasoning,  data  knowledge o f  on v o l u m e t r i c a n a l y s i s  on d i r e c t and i n v e r s e prerequisite calculations  concepts  proportional and  performance  were c o l l e c t e d by a d m i n i s -  102  tering and  the  Classroom P r o p o r t i o n a l i t y Test,  Volumetric  Volumetric  the  Analysis  Test  test,  previous  pretested  chapter,  reactant  from the  t e s t measured the  i n the  c o n s i s t e d of  were a s k e d t o c a l c u l a t e t h e one  Test,  Analysis Test, r e s p e c t i v e l y .  This in  Subconcepts  pilot 15  items.  concentration,  known v a l u e s  study  and The  described subjects  volume o r mass o f  of another reactant.  s u b j e c t s ' p e r f o r m a n c e on  volumetric  The  analysis  calculations.  Classroom P r o p o r t i o n a l i t y Test The tested  i n the  seven items. direct  pilot The  of  study.  this  instrument  Each of the  were a l s o  subtests  D i r e c t P r o p o r t i o n a l i t y subtest  measured  Subconcepts  while  the  inverse proportional  subtest,  pre-  contained measured Inverse  Propor-  reasoning.  Test  The  by  subtests  p r o p o r t i o n a l reasoning  tionality,  study,  two  subconcepts t e s t ,  contained  f o u r items.  also pretested  seven s u b t e s t s , Thus, t h e  test  i n the  e a c h o f w h i c h was  contained  a total  pilot  represented  of  28  items.  T h i s t e s t measured the  s u b j e c t s ' knowledge o f t h e p r e r e q u i s i t e  concepts i n volumetric  analysis calculations.  Population The designed  to  present  study  i n v e s t i g a t e the  was  an  analytical  difficulties  that  study senior  primarily secondary  103 school  students  calculations. taught  f a c e when p e r f o r m i n g v o l u m e t r i c Since volumetric analysis  i n t h e G r a d e 12 c h e m i s t r y . c o u r s e  Columbia,  it  was d e c i d e d t h a t  12 c h e m i s t r y c o u r s e general  However,  the  was p r a c t i c a l  students  normally only these  in British  e n r o l l e d i n the  and a v a i l a b l e Thus f o r  of  Grade the  British.  to c a r r y out  the purpose  the  of  B r i t i s h Columbia c o n s t i t u t e d  S i n c e C h e m i s t r y 12 i s  selected  subjects  study  formally  this  e n r o l l e d i n G r a d e 12 c h e m i s t r y c o u r s e s  t h e Lower M a i n l a n d o f target population.  of  the only area  s t u d y was t h e Lower M a i n l a n d . study,  was o n l y  i n B r i t i s h Columbia would c o n s t i t u t e  population.  Columbia t h a t  students  analysis  the  an o p t i o n a l  by more a c a d e m i c a l l y - o r i e n t e d  constituted  a self  selected  in  course, students,  group.  Sample I n May 1979, the in  study  and r e q u e s t i n g  their district  Lower M a i n l a n d . the  study.  Boards the  d e s c r i b i n g the o b j e c t i v e s  p e r m i s s i o n t o use  were s e n t t o  four  Two S c h o o l B o a r d s  Schools  w i t h i n the  contacted,  The l o c a t i o n s  of  the  schools  the interest  Furthermore,  the catchment areas of  were s u c h t h a t  socio-economic  these levels  The s t u d e n t s  i n the  Of in  the  a wide  sample.  schools as  in  these School  representatives.  was o b t a i n e d  of  schools  10 v o l u n t e e r e d t o p a r t i c i p a t e  representation  School Board o f f i c i a l s .  in  showed i m m e d i a t e  j u r i s d i c t i o n of  geographical  a wide d i v e r s i t y o f  selected  School Boards  were c o n t a c t e d by S c h o o l B o a r d  21 s c h o o l s  study.  letters  represented  a s s e s s e d by  i n the i n t a c t  the  classes  in  104 these  schools  constituted  were i n i t i a l l y  the  first  due t o  reasons  logistical  number o f  (a d e l a y  participated  subjects of  fully  selection  of  tests  students  used to  enrollment of  i n t e a c h i n g the  i n the  There  (CPT and S C T ) . H o w e v e r ,  schools study,  3 2 8 . ( 2 0 3 b o y s and 125  The p r o c e d u r e s  study.  student  only eight  Data C o l l e c t i o n  administration of  the  two i n s t r u m e n t s  u n i t on v o l u m e t r i c a n a l y s i s ) 14 c l a s s e s  for  17 c l a s s e s w i t h a t o t a l  402 who w r o t e t h e  of  sample  required  with a  total  leaving a  total  girls).  Procedures i n the  study  related  the p a r t i c i p a t i n g  to  the  students  and  the  for interviewing.  Test Administration I n O c t o b e r and November, 19 8 0", t h e C l a s s r o o m P r o p o r t i o n ality in ing  T e s t was a d m i n i s t e r e d by t h e  each s e l e c t e d this  test  class.  intact  to  to assess  for  students  administer-  this  test  to  teaching volumetric After  t h e above students  t h e i r knowledge o f  Chemistry teachers  administer  before  the a d m i n i s t r a t i o n of  T e s t was a d m i n i s t e r e d t o t h e  classes  concepts.  The d e t a i l e d p r o c e d u r e  to a l l  c a n be f o u n d i n A p p e n d i x B .  Following Subconcepts  investigator  students  i n the  as  study  a review  -. agreed  test  analysis.  t e a c h i n g t h e u n i t on a c i d s  and b a s e s  which  i n c l u d e d v o l u m e t r i c a n a l y s i s , the V o l u m e t r i c A n a l y s i s Test a d m i n i s t e r e d by t h e  the  in their  prerequisite  participating  their  test,  investigator  to  the  students  as  an  was  end-of-  105  unit test.  In a d m i n i s t e r i n g t h i s t e s t , the students  were  asked t o show the steps they were u s i n g i n t h e i r s o l u t i o n t o the problems.  Interview  Sample One week a f t e r a d m i n i s t e r i n g the f i n a l t e s t —  Volumetric  A n a l y s i s Test —  the students  the  i n each school were  grouped i n t o h i g h , medium and low achievement groups based on t h e i r scores on t h i s t e s t .  Those who scored from zero t o  f i v e were c l a s s i f i e d as low a c h i e v e r s , s i x t o t e n were c l a s s i f i e d as medium a c h i e v e r s and those to f o u r t e e n as high a c h i e v e r s .  s c o r i n g from eleven  In each s c h o o l , two  students  were randomly s e l e c t e d from each group f o r the i n t e r v i e w . Thus, s i x students were s e l e c t e d from each s c h o o l , g i v i n g a t o t a l subsample of 48 s u b j e c t s f o r the i n t e r v i e w t a s k .  This  was done t o ensure t h a t the s u b j e c t s from the o r i g i n a l  sample  were adequately  represented.  i n d i v i d u a l l y i n a separate t i o n room.  The students were  interviewed  room, u s u a l l y the s c i e n c e  However, one student  prepara-  i n the low achievement group  i n one school c o u l d not be i n t e r v i e w e d because he was not a v a i l a b l e when o t h e r students  i n t h a t school were  interviewed;  t h i s l e f t a t o t a l i n t e r v i e w sample of 47.  Interview Data C o l l e c t i o n Procedure The  f i n a l phase i n the c o l l e c t i o n of data, the  i n t e r v i e w i n g of the subsample, was scheduled  about s i x to  106 seven days a f t e r the a d m i n i s t r a t i o n of the Volumetric Test.  Analysis  In c o n s u l t a t i o n with the c l a s s teachers and the students,  appointments were arranged  f o r i n t e r v i e w i n g a c c o r d i n g t o the  convenience of the students. The  i n t e r v i e w s were conducted by the i n v e s t i g a t o r .  At the beginning  of the i n t e r v i e w , the i n t e r v i e w e r  identified  h i m s e l f , i n d i c a t e d the i n t e n t of the i n t e r v i e w and h i s affiliation  t o the U n i v e r s i t y of B r i t i s h Columbia.  c o n f i d e n t i a l i t y o f t h e i r responses was emphasized. of each student was obtained before u s i n g the recorder.  The The consent  audio-tape  The i n t e r v i e w e r then proceeded with the i n t e r v i e w  a c c o r d i n g t o the e s t a b l i s h e d p r o t o c o l s . The  i n t e r v i e w e r showed the student  the r e q u i r e d  apparatus and s o l u t i o n s f o r the t i t r a t i o n t a s k . and  the stand were l e f t unassembled.  doing the t i t r a t i o n ,  The b u r e t t e  W h i l s t the s u b j e c t was  the i n t e r v i e w e r observed h i s l a b o r a t o r y  techniques  and completed the l a b o r a t o r y s k i l l s i n t i t r a t i o n  checklist.  The procedure employed d u r i n g the i n t e r v i e w can  be found i n Appendix E. The  sequence of events i n v o l v e d i n the data  collec-  t i o n f o r the e n t i r e study has been summarized i n F i g u r e 6 and Table  6.  107 SELECTION OF CHEMISTRY IN  12 STUDENTS  INTACT CLASSES  ADMINISTRATION OF THE CLASSROOM PROPORTIONALITY TEST TO ALL STUDENTS  TESTING ALL STUDENTS ON THE SUBSUMED CONCEPTS  ADMINISTRATION OF THE VOLUMETRIC ANALYSIS TEST  SELECTION OF '3 ..GROUPS OF INDIVIDUALS FOR INTERVIEWS ON'A'TITRATION TASK  APPLYING STATISTICAL AND QUALITATIVE ANALYSIS TO THE DATA : ANALYSIS,  PATH  INTERVIEW DATA ANALYSIS  QF WRITTEN WORK  Figure Overall  ;  :  ;  & ANALYSIS  6  Design o f Experimental  Procedure  108 Table 6 Test A d m i n i s t r a t i o n and I n t e r - t e s t Breaks  Tests administered p r i o r t o the t e a c h i n g of v o l u m e t r i c a n a l y s i s unit CPT: DP Subtest 3 weeks SCT IP Subtest Interval]  Interval during which volumetric Interval after analysis teaching (VA) i s taught VA 2 months5 months  1 week  Test administe red  Interval before Interinterview period view  VAT  7 days  3 days1 week  Data A n a l y s i s Data P r e p a r a t i o n For each student,  the f o l l o w i n g data s e t was  c o l l e c t e d : number o f l a b o r a t o r y t i t r a t i o n s completed, d i r e c t p r o p o r t i o n a l i t y t e s t score, i n v e r s e p r o p o r t i o n a l i t y t e s t score, subconcepts t e s t score, and v o l u m e t r i c score.  analysis test  The number of l a b o r a t o r y t i t r a t i o n s completed was  obtained by a s k i n g the s u b j e c t s t o i n d i c a t e the number o f t i t r a t i o n s they had performed s i n c e Grade 11-.on the f r o n t page of the Volumetric  A n a l y s i s Test b o o k l e t .  The v a l u e s given by the s t u -  dents were checked with t h e i r teachers  t o ensure t h a t no one  o v e r s t a t e d the t i t r a t i o n l a b o r a t o r i e s completed. Other background data c o l l e c t e d i n c l u d e d s c h o o l , c l a s s , and gender. A f t e r marking the Volumetric  A n a l y s i s Test the i n v e s t i g a t o r a l s o c a t e g o r i z e d  the s u b j e c t s i n t o a group who used the 1:1 s t o i c h i o m e t r i c r a t i o with understanding  i n s o l v i n g the v o l u m e t r i c  a n a l y s i s problems  109 and a group which d i d not use the s t o i c h i o m e t r i c r a t i o with understanding.*  T h i s i n f o r m a t i o n was  converted t o numerical  codes and recorded on f o r t r a n statement  forms.  Data were key-punched and v e r i f i e d by the s t a f f a t the U n i v e r s i t y of B r i t i s h Columbia Computer Centre. data were hand-checked f o r key-punching gator.  E r r o r s were double-checked  A l l the  e r r o r s by the  and c o r r e c t e d .  In  investiall,  only f i v e e r r o r s were d e t e c t e d .  Scoring The (see Chapter  same s c o r i n g procedure  used i n the p i l o t  I I I ) f o r the p r o p o r t i o n a l i t y t e s t s , subsumed  ( p r e r e q u i s i t e ) concepts t e s t , and v o l u m e t r i c a n a l y s i s t i o n t e s t were employed i n the main study. two  study  s u b t e s t s of the Classroom  The  calcula-  scores f o r the  P r o p o r t i o n a l i t y T e s t , namely,  d i r e c t p r o p o r t i o n a l i t y and i n v e r s e p r o p o r t i o n a l i t y , were kept separate s i n c e each was the  c o n s i d e r e d as a d i f f e r e n t v a r i a b l e i n  study.  P r e l i m i n a r y Analyses of Data f o r Model T e s t i n g R e l i a b i l i t y of  Instruments  Item analyses and performed  r e l i a b i l i t y e s t i m a t i o n were  on the Inverse P r o p o r t i o n a l i t y and D i r e c t  Propor-  t i o n a l i t y s u b t e s t s u s i n g the computer program LERTAP * Those s u b j e c t s who wrongly a p p l i e d a 1:1 mole r a t i o t o more than two problems i n the Volumetric A n a l y s i s T e s t were assumed to have used a l g o r i t h m s without understanding.  110 (Nelson, 1974).  Subjects who  were absent d u r i n g the  t r a t i o n of any of the remaining  t e s t s but who  e l i m i n a t e d b e f o r e the item a n a l y s i s was procedure  was  adminis-  took the CPT  done.  A  c a r r i e d out on the Subconcept and  were  similar Volumetric  Analysis Test scores. Hoyt r e l i a b i l i t y  index was  c a l c u l a t e d f o r the  p r o p o r t i o n a l i t y , i n v e r s e p r o p o r t i o n a l i t y and A n a l y s i s instruments.  This r e l i a b i l i t y  direct  Volumetric  coefficient  reflects  on the i n t e r n a l c o n s i s t e n c y or homogeneity of the groups of items i n each t e s t or s u b t e s t . The  scores f o r the e n t i r e sample on the D i r e c t  P r o p o r t i o n a l i t y Subtest of the Classroom ranged from 0 t o 21. t h i s s u b t e s t was  The  0.80.  P r o p o r t i o n a l i t y Test  Hoyt estimate of r e l i a b i l i t y  The d i f f i c u l t y  for  index and d i s c r i m i n a t i o n  index f o r each of the items i n t h i s s u b t e s t i s r e p o r t e d i n Appendix F.  Table 7 g i v e s the mean and standard d e v i a t i o n .  The ality  scores f o r the sample on the Inverse P r o p o r t i o n -  s u b t e s t a l s o ranged from 0 to 21.  reliability  f o r t h i s t e s t was  0.93.  t h i s t e s t are g i v e n i n Appendix The 0 t o 28.  1  item s t a t i s t i c s f o r  G.  scores f o r the Subconcepts Test ranged from  Cronbach's composite alpha was  t e s t i n order to determine how together.  The  H o y t s estimate of  calculated for this  the seven s u b t e s t s hung  The value f o r Cronbach's alpha was  0.78.  value r e f l e c t s on the homogeneity of the items i n the The  item d i f f i c u l t y  This test.  and the item-subtest c o r r e l a t i o n s f o r the  items i n t h i s t e s t are r e p o r t e d i n Appendix  H.  Ill The s c o r e s from 0 t o Appendix  15.  for  the V o l u m e t r i c A n a l y s i s Test  The H o y t ' s r e l i a b i l i t y e s t i m a t e was  comparable  Chapter  0.88  (see  I). The r e l i a b i l i t i e s  are  ranged  to  those obtained  Table  III).  reported  above f o r  i n the p i l o t  7 summarizes  the  the  instruments  study  statistics  (see  for  the  instruments.  Table 7 Summary T e s t  Number o f  items  Mean  For A l l 4 Tests  DP  IPI/.  SCT  VAT  7  7  28  15  18.75  13.92  19.04  7.08  deviation  4.07  7.29  4.91  4.23  reliability  0.80  0.93  0.78*  0.88  1.68  1.84  2.04  1.39  Standard Hoyt's  Statistics  Standard  error  * Cronbach's  Differences  composite  of  study  all  four  different were  were tests.  for  screened  all to  This left  classes,  employed.  (Cronbach,  1951)  Classes  The d a t a the  alpha  328  s u b j e c t s who  eliminate a final  participated'in  t h o s e who d i d n o t  sample o f  265  upon w h i c h t h e p a t h a n a l y t i c a l  write  i n 14 procedures  112 The 1975)  SPSS s u b p r o g r a m CONDESCRIPTIVE  was u s e d t o o b t a i n  standard  deviations)  proportionality, test  scores  Since  to t e s t direct  (means and  proportionality, inverse and v o l u m e t r i c  f o r e a c h o f t h e 14 c l a s s e s  proportion,  analysis  (see A p p e n d i x J ) .  b e l o n g e d t o c e r t a i n c l a s s e s i t was  inverse  a n a l y s i s scores  necessary  proportion, before  subsumed c o n c e p t s and  continuing  with  the a n a l y s i s .  t h e c o m p u t e r p r o g r a m OWMAR m a i n t a i n e d by t h e D e p a r t m e n t  of Psychology the  f o r the d i r e c t  a l .  f o r p o s s i b l e d i f f e r e n c e s among t h e c l a s s e s f o r t h e  volumetric Using  descriptive statistics  subsumed c o n c e p t s ,  the subjects  (Nie, e t .  (U.B.C),  a m u l t i v a r i a t e t e s t was p e r f o r m e d on  d i f f e r e n c e s i n means among t h e c l a s s e s and t o t e s t t h e  tenability  o f t h e assumption o f homogeneity o f v a r i a n c e -  covariance.  Table  8 shows t h e r e s u l t s o f t h i s  Table  analysis.  8  T e s t o f D i f f e r e n c e s i n Means and H o m o g e n e i t y o f D i s p e r s i o n among D i r e c t P r o p o r t i o n a l i t y , I n v e r s e P r o p o r t i o n a l i t y , Subc o n c e p t s and V o l u m e t r i c A n a l y s i s T e s t S c o r e s o f C l a s s e s  Test  DF1  DF2  among Means  52  963  3.8690  0.001*  B a r t l e t t - B o x Homogeneity o f Dispersion Test  130  17348.1  1.4749  0.000*  MANOVA L i k e l i h o o d R a t i o for Differences  P < -05  F-Ratio  PROB  Test  113 These r e s u l t s i n d i c a t e t h a t the d i f f e r e n c e s among the c l a s s means on a l l four v a r i a b l e s i s probably i n d i c a t e s t h a t the n u l l hypothesis covariance matrix  real.  I t also  t h a t the v a r i a n c e -  i s the same f o r a l l c l a s s e s i s untenable.  However, s i n c e t h i s h e t e r o g e n e i t y may be due t o d i f f e r e n c e s i n v a r i a n c e s , a l l the scores were s t a n d a r d i z e d w i t h i n c l a s s e s to a mean o f zero and standard standardized The  d e v i a t i o n o f one.  scores were then analyzed  r e s u l t s as shown i n Table  The  u s i n g the OWMAR PROGRAM.  9 i n d i c a t e t h a t the transforma-  t i o n was s u c c e s s f u l i n removing c l a s s d i f f e r e n c e s . Consequently, all  subsequent analyses were conducted u s i n g students'  standard-  i z e d scores and d i s r e g a r d i n g c l a s s as a f a c t o r , and the scores f o r a l l c l a s s e s pooled  together. Table 9  Test o f D i f f e r e n c e s i n Means and Homogeneity of D i s p e r s i o n among D i r e c t P r o p o r t i o n a l i t y , Inverse P r o p o r t i o n a l i t y , subconcepts and Volumetric A n a l y s i s Test Scores o f C l a s s e s . 3  Test  DF1  DF2  F-Ratio  PROB  52  963  .0.0019  1.000*  17348.1  0.5969  0.999*  MANOVA L i k e l i h o o d R a t i o Test f o r D i f f e r e n c e s among Means B a r t l e t t - B o x Homogeneity of D i s p e r s i o n T e s t p >  130  .05  The scores f o r a l l the v a r i a b l e s were transformed c l a s s e s t o a mean o f zero and a v a r i a n c e o f one.  within  114  Before t e s t i n g the model f o r the two groups of s u b j e c t s , that the  i s those who use a l g o r i t h m s with and without understanding, scores were s t a n d a r d i z e d  separately.  Path A n a l y s i s o f Data Path models and o t h e r s t r u c t u r a l equation, models have been e v a l u a t e d i n the past by u s i n g d i f f e r e n t procedures. The most popular o f these procedures i n v o l v e s e s t i m a t i n g and t e s t i n g f o r the s i g n i f i c a n c e parameters.  of the i n d i v i d u a l  structural  In t h i s procedure, the f i t of the model as a whole  to the sample data i s not e v a l u a t e d - only the i n d i v i d u a l s t r u c t u r a l parameter estimates are evaluated 1974).  the  The n o n - s i g n i f i c a n t  r e s u l t i n g modified  paths are u s u a l l y  (e.g. Gimmel, e l i m i n a t e d and  (or trimmed) model i s r e - e v a l u a t e d .  In t h i s procedure, the f i t of the model as a whole i s not evaluated.  As Land  t u r a l models  (1973)  has noted, the e v a l u a t i o n of s t r u c -  ( i . e . models i n which the c a u s a l d i r e c t i o n s are  specified)  based s o l e l y on the e v a l u a t i o n of the i n d i v i d u a l  structural  (path) c o e f f i c i e n t s may not always be  applicable  to the model as a whole e s p e c i a l l y when t h e r e are more than one o v e r - i d e n t i f y i n g has  been e l i m i n a t e d ) .  restrictions  ( i . e . more than one path  T h i s i s because i t may be p o s s i b l e f o r  some of the i n d i v i d u a l c o e f f i c i e n t s to be  statistically  s i g n i f i c a n t while the t e s t o f the model as a whole may  either  115 produce a s i g n i f i c a n t an  investigator  statistically essential  to  paths.  h a s " been s u g g e s t e d  chi-square  proposed model a g a i n s t general  alternative  extent.  difference  i n that  is  Bentler,  the  the  be z e r o w h i l e t h e  this  model a p p r a i s a l ,  if  compared w i t h t h e d e g r e e s o f  suggestions.  provides  a test  are  that  the  system  population. the c a u s a l  processes  procedures the  influences  The c o n v e r s e  In  for  of  is  a  will  freedom,  it  265  that  the present  is  was u s e d  suggest that  analysis  the  there  freedom  a l s o be is  zero. large  concluded that  the  representation in  the  the model m i r r o r s  data.  a combination of  to evaluate the proposed  subjects.  since  statistic  among t h e v a r i a b l e s  generated  the  correlated  the degrees of  variate  the  chi-square  saturated model,  the chi-square  will  In  matrices  of  is,  simply  h y p o t h e s i z e d model d o e s n o t p r o v i d e a p l a u s i b l e of  fit  1980;  covariance  i n a sense,  chi-square  a  1980).  the v a r i a b l e s  for  t h e m o d e l as  (Bentler,  saturated model,  This,  chi-square  to evaluate the  the r e s i d u a l  no o v e r - i d e n t i f y i n g r e s t r i c t i o n ,  will In  tests  statistic  that  t o an a r b i t r a r y test  of  have made s i m i l a r  (e.g.  is  p a t h may be  likelihood ratio  Other r e s e a r c h e r s  1980)  path which  times,  (1973), t h e r e f o r e ,  goodness-of-fit  the examination of  The  Land  i n d i v i d u a l Z (or t - )  and B o n e t t ,  addition,  a whole.  the  Atcother  even though t h i s  both the  which t e s t s  structural  Bentler  of  result.  a particular  non-significant t h e m o d e l as  whole and t h e of  insignificant  may e l i m i n a t e  recommends t h e u s e statistic  or  The SPSS s u b p r o g r a m ,  the  integrated REGRESSION  above model (Nie,  et.  a l . 1 9 7 5 ) was used f o r t h i s a n a l y s i s .  T h i s computer pro-  gram estimates the s t r u c t u r a l parameters through equation l e a s t squares procedure. LISREL  equation-by-  However, programs l i k e  (Joreskog and Sorbom, 1 9 7 8 ) used the f u l l  information  maximum l i k e l i h o o d method t o estimate the parameters simultaneously.  However, f o r r e c u r s i v e  dent d i s t u r b a n c e s and no  models  with  indepen-  a p r i o r i cross-equation c o n s t r a i n t s ,  as i s the case i n the proposed model the equation-by-equation  l e a s t squares  t h i s method reduces t o solution  (Land,  1973).  The chi-square t e s t s t a t i s t i c used t o e v a l u a t e the theory i s g i v e n by Land  (1973)  for recursive  models  with  independent e r r o r s and no c r o s s - e q u a t i o n c o n s t r a i n t s as G —  A .  N ^ > g ='i  K  l o g 0" /0"° gg gg y  th £r i s the estimate o f the g disturbance variance gg ( i . e . , the mean squared r e s i d u a l ) o b t a i n e d when y ^ i s r e g r e s s e d th on a l l antecedent v a r i a b l e s and i s the estimate o f the g gg  where  0  d i s t u r b a n c e v a r i a n c e o b t a i n e d when y^ i s r e g r e s s e d on a l l antecedent  v a r i a b l e s except those which are excluded a  p r i o r i from the g^  1  equation.  N i s , the sample  size.  In e v a l u a t i n g the ^"model, the proposed i n t e g r a t e d model  was f i r s t  t e s t e d a g a i n s t the s a t u r a t e d  model f o r the  sub samples. . The path c o e f f i c i e n t s , which r e f l e c t thes t r e n g t h o f the i n f l u e n c e o f one v a r i a b l e on another, were a l s o o b t a i n e d and t e s t e d f o r s i g n i f i c a n c e .  T h i s proce-  117 d u r e was u s e d  for  the  standing  (N = 105)  standing  (N = 1 6 0 ) .  in  and t h o s e Details  the next  chapter.  Analysis  of Data f o r  Analysis  of  Conceptual  Initially was  subjects using algorithms  scored  1 for  under-  using algorithms without underof  this  analysis  the S p e c i f i c  Student  are  presented  Difficulties  Errors  each item i n the V o l u m e t r i c A n a l y s i s  correct  The s t e p s u s e d by t h e examined f o r  with  answer  subject  and 0 f o r  in his  conceptual e r r o r s .  an i n c o r r e c t  calculations  answer.  were t h e n  The p r o c e d u r e u s e d t o  identify  the conceptual e r r o r s  involved a c r i t i c a l examination of  the  for  items  asked of  to  i n the t e s t show t h e  steps used  the conceptual e r r o r s  the  subject  incorrect  response.; the  from t h e a n a l y s i s .  showed t h e  students  final  step(s)  which produced  answers  i n only for  a few i t e m s . at  steps  least  every  two i t e m s  on t h e  the c o n c e p t u a l  analysis  of  * All  names u s e d  i n the  thesis  were  In a l l ,  to  type of  the  steps  by B e n *  this  analysis  the  The f o l l o w i n g p r o b l e m s o l u t i o n o f f e r e d illustrate  were  were shown were e l i m i n a t e d  Some s u b j e c t s made e x p l i c i t  steps  the  all  i n v o l v e d e x a m i n i n g e a c h s t e p u s e d by  Items f o r w h i c h no e x p l i c i t  t h e i r computations  subject  Since  i n each c a l c u l a t i o n ,  and i d e n t i f y i n g t h e  shown b u t o n l y  in  each student.  Test  are  fictitious.  used  test.  .serves errors.  118 Question:  A 500  class of  mL s o l u t i o n  experiment.  0.10  solution  S t e p 1:  N a 2 C 0 3 + HCl  S t e p 2:  25.0  S t e p 3:  of  mL  mL o f what  this is  the  for  use  neutralizes  in a 2 5 . 0 mL  concentration  of  HCl?  25.0  S t e p 4:  2.5  x 10~3 x 10~3  6.25 =  7.8 x  x  is  However,  m e t r i c mole r e l a t i o n 6,  m o l N a 2 C 0 3 x 1 mol H C l 1 mol N a 2 C 0 3 mol H C l  x 10~2mol L  the  m o l . H C l x 1000 mL L  x  1 x 500 mL 40 mL  10_1mol/L Ben's  written,  between  calculation o n l y the  i n S t e p 4,  Ben assumes t h a t t h e  40 mL o f  3  10=?mol/L  Examination of equation  L 1000 mL  Na2C03  C o n e . = 2 . 5 x 10 40 mL  S t e p 6:  (Step 1 ) .  mol/L x  x 10~3mol  = 6.25  balanced  x mol/L  mL x 0 . 1  = 2.5 S t e p 5:  4 0 . 0 mL  mol/L  = 2.5  the  40.0  prepared  response:  0.10  step  HCl i s  M Na2C03 solution,  the o r i g i n a l Ben's  If  of  shows t h a t  reactants  Ben assumes t h a t  the  two r e a c t a n t s  calculated  a c i d used i n the  are the  is  reaction  indicated stoichio-  1:1.  concentration and hence  no  In was  for  calculates  119  the m o l a r i t y f o r the 500 ml stock s o l u t i o n .  Three conceptual  e r r o r s can t h e r e f o r e be i d e n t i f i e d i n Ben's s o l u t i o n .  They  are: 1)  assumption o f 1:1 mole r a t i o ,  2)  t h a t the c o n c e n t r a t i o n o f stock s o l u t i o n i s d i f f e r e n t from the c o n c e n t r a t i o n o f the volume used i n the t i t r a t i o n ,  3)  use o f balanced chemical equations not c o n s i d e r e d i n problem s o l u t i o n . The r e s u l t s o f the p i l o t study i n d i c a t e d the presence  of these and other k i n d s o f conceptual e r r o r s which c o u l d account  f o r the i n c o r r e c t answers on the items i n the Volumetric  Analysis Test.  To check t o see whether other people knowledge-  a b l e i n t h i s area would a r r i v e a t s i m i l a r conceptual  errors,  twelve randomly s e l e c t e d answer b o o k l e t s were given t o a graduate  student i n chemistry t o analyze i n terms o f the e r r o r s  o c c u r r i n g i n the s o l u t i o n s .  Before doing the a n a l y s e s , f o u r  students' answer b o o k l e t s were used t o i n s t r u c t the other a n a l y s t about the i n t e n t o f the a n a l y s e s .  I t was emphasized  t h a t o n l y the i n c o r r e c t s o l u t i o n s should be analysed.  I t was  a l s o s t r e s s e d t h a t s i n c e f o r each s u b j e c t a l l the items which had  i n c o r r e c t responses  were t o be analyzed, any one p a r t i c u l a r  e r r o r i d e n t i f i e d f o r each s u b j e c t should be i n d i c a t e d o n l y once even though i t may occur many times.  That i s , i f a  s u b j e c t assumed 1:1 mole r a t i o s i n 3 problems r e q u i r i n g mole r a t i o s , t h i s e r r o r was i n d i c a t e d only once.  2:1  F o r the  analyses o f the 12 b o o k l e t s an agreement r a n g i n g from 75.4 t o 100%  was o b t a i n e d with a mean percent agreement of 9 1 . 5 .  120 D i s c u s s i o n on t h e d i s c r e p a n c i e s while all  for  any one i t e m on t h e  the p o s s i b l e  analyst the  errors  that  i n the a n a l y s i s  test  the  c o u l d be  i d e n t i f i e d o n l y one c o n c e p t u a l  test.  booklets  To t r y  to resolve  were a n a l y z e d .  this  This  revealed  investigator identified, error  for  discrepancy  analyzed  the each  six  produced a percent  that  other i t e m on  more  answer  agreement  of  95.4. Analyses  of  Interview  Data  The i n t e r v i e w s type  titration  task,  aspects  potential  difficulties  the  of  about  understanding  several of  with volumetric analysis.  w h i l e the  i n t e r v i e w data are results  of  and The  different  presented  these analyses  i n f o r m a t i o n from the c h e c k l i s t  i n T i t r a t i o n — the m a n i p u l a t o r y t e c h n i q u e s  s u b j e c t s were t a b u l a t e d  absence of as  address  laboratory-  in  are  in Titration  Using the  the  students'  using a  i n Chapter V I .  Manual S k i l l s  Skills  data to  a n a l y s e s u s e d on t h e s e  following sections  given  4 7 students,  generated  different  types  with  each  (Appendix K ) .  i t e m on t h e c h e c k l i s t  a w h o l e was t h e n r e c o r d e d  i n terms  sample u s i n g o r n o t u s i n g a p a r t i c u l a r Ideas About the Concepts  u s e d by  interview  all  or sample  the percentage of  the  technique.  Involved i n T i t r a t i o n  The t a p e s c o n t a i n i n g t h e s t u d e n t s were t r a n s c r i b e d  Laboratory  The p r e s e n c e  i n the of  —  interview data  v e r b a t i m by t h e  for  investigator  the 4 7 and  the  121 graduate, student who  a s s i s t e d i n the a n a l y s i s of the  e r r o r s i n the Volumetric A n a l y s i s T e s t .  conceptual  In t r y i n g to i d e n t i f y  the ideas t h a t s u b j e c t s have about c e r t a i n phenomena - o b j e c t s and processes  of nature - from p r o t o c o l s obtained i n an  inter-  view s e t t i n g , d i f f e r e n t a n a l y t i c techniques  have been a p p l i e d  by d i f f e r e n t r e s e a r c h e r s  These a n a l y t i c  techniques  ( E r i c k s o n , 1975).  range from the s e l e c t i o n and c l a s s i f i c a t i o n of  s p e c i f i c p e r t i n e n t s e c t i o n s of the t r a n s c r i p t Inhelder and P i a g e t , 1958)  (Piaget,  1929;  t o more e l a b o r a t e procedures which  employ the e n t i r e t r a n s c r i p t of the i n t e r v i e w (e.g. Witz, The  1970).  analyses adopted i n the present study i s c l o s e l y  r e l a t e d to t h a t of P i a g e t and o t h e r s  (e.g. Anderson, 1979) .  I t i n v o l v e d i d e n t i f y i n g the p a t t e r n of responses expressed  by  a group of s u b j e c t s to account f o r each major q u e s t i o n posed by the i n v e s t i g a t o r i n the course of the i n t e r v i e w .  Questions  which were used as  (Archenhold,  1979)  ' s t a r t e r s ' or  were not analyzed.  'linking questions'  T h i s analyses,was a p p l i e d to the  first  p a r t of the i n t e r v i e w . Before  i s o l a t i n g the response p a t t e r n s , each sub-  j e c t ' s response f o r each q u e s t i o n posed i n the i n t e r v i e w (Appendix L) was A subject's idea  examined to i d e n t i f y the i d e a being  expressed.  (or notion) as used here i s a statement or  e x p l a n a t i o n o f f e r e d by the s u b j e c t to account f o r a problem s i t u a t i o n i n i t i a t e d by the i n v e s t i g a t o r . responses given v e r b a l l y and  Thus they are  the  i n w r i t i n g by the s u b j e c t i n answer  to questions posed by the i n v e s t i g a t o r . The  ideas expressed  by each s u b j e c t f o r each problem  122 situation  (or q u e s t i o n )  subjects'  ideas  to  about the  go t o g e t h e r o r  together  as  and T a b l e s 25, t o patterns  are  t h e n examined  same q u e s t i o n .  expressed  u n d e r what  I n most c a s e s ,  was  is  here  c o u l d be  2 8'  similar  to  Ideas which  seemed  i n t e n t were  referred  to  as  i n Chapter V I ) ,  reflections of  the  then  of  other  grouped  a Response  s e e n by c o m p a r i s o n  (presented  direct  in relation  Pattern.  Appendix L  these  response  l a n g u a g e u s e d by  the  subjects. P r o b l e m S o l v i n g A p p r o a c h and P r e d i c t i v e In the the  s u b j e c t was  second  of  the the  late the  approach  was  titration. the  interview, his  Thus,  during  solutions,  of  the  i n the  transcribed  the main study  study.  was  acid  the  from the  analyses of  verbal  Essentially,  were e x a m i n e d  to  subject  and h i s  to  the  the  w r i t t e n work w i l l  calcufrom  of  the  supported  by  the  The  one u s e d i n t h e  and t h e w r i t t e n  analyses  pilot calculations  subject  this::.analysis, be  to  part  section  s t e p s u s e d by t h e  To i l l u s t r a t e  identify-  data obtained  this  protocols  the p r o t o c o l s  isolate  solve the problem. report  similar  of  the problem posed i n t h i s  s u b j e c t ' s w r i t t e n work p r o v i d e d t h e p r i m a r y d a t a . in  which  each  analyzed w i t h the o b j e c t i v e  The p r o b l e m r e q u i r e d  concentration  interview,  the  used i n s o l v i n g  interview.  the  of  asked to v e r b a l i z e  subjectSs protocol ing  part  Behaviour  to  Harry's  verbal  used.  Verbalization: "We have .1M o f NaOH, we added 2 5 . 4 m i l l i l i t e r s o f i t t o t h e a c i d ; and m o l e s i s c o n c e n t r a t i o n t i m e s v o l u m e . T h e r e f o r e 2 5 . 4 t i m e s .1 i s 2.54 m i l l i m o l e s o f NaOH  123 w h i c h i s t h e same as 2.54 m i l l i m o l e s o f OH i o n s . And 1 mole o f t h a t ( s t u d e n t p o i n t s t o t h e NaOH and H C l i n a b a l a n c e d e q u a t i o n he has w r i t t e n p r e v i o u s l y on t h e s u p p l i e d p a p e r ) r e a c t s w i t h 1 mole o f t h i s . We must have 2.54 m i l l i m o l e s o f a c i d o r H C l r e a c t i n g . We had 25 m i l l i l i t e r s so t h e r e f o r e i f we had 2.54 m i l l i m o l e s r e a c t i n g . . . (pause) . . . We had 25 m i l l i l i t e r s , we d i v i d e t h e m o l e s i n t o m i l l i l i t e r s , n o , by t h e m i l l i l i t e r s ; t h e m i l l i c a n c e l o u t and y o u have m o l e s p e r l i t e r w h i c h i s c o n c e n t r a t i o n and y o u have a b o u t 2.54 d i v i d e d by 2 5 , t h a t g i v e s y o u c o n c e n t r a t i o n o f . 1 0 1 . C o n c e n t r a t i o n e q u a l s t o 0.101 m o l a r . " W r i t t e n work: 0.1 2  'li  NaOH + H C l = N a +  m o l / L NaOH x 2 5 . 4  rcf31 25 mL  =  0.101 M  =  0.10  From t h e by H a r r y a r e : substitute formula  (i)  of  base used  into  Approach.  it  the to  of  (iii)  their  t o o b t a i n the moles  t h e r e l a t i o n between  the r e a c t a n t s  i n the.^equation  t o o b t a i n moles  o b t a i n the m o l a r i t y of  the  employing the b a s i c  i n v o l v e d i n the  solutions asked  in this  part  for  three  of  of the  and  the  acid,  substitute  acid.  form o f  a  From  this,  the Formula  the data  for  all  interview.  of  the  subjects  the  interview,  t o make p r e d i c t i o n s  predictions  (ii)  the base i n t o  f o r m u l a a p p l i e d p r e v i o u s l y and  as  followed  chemical equation,  use  In a d d i t i o n t o a s k i n g  were a l s o  steps  S i m i l a r a n a l y s e s were a p p l i e d t o  subjects  their  mmol NaOH  the r e l e v a n t  i n the r e a c t i o n  H a r r y was d e s c r i b e d  the  data,  write balanced  ratio  H20  M HCl  above  in titration,  rearrange  values  mL = 2.54  +  t h e volume and c o n c e n t r a t i o n o f  stoichiometric  (iv)  CI"  ( M o l a r i t y x volume = moles)  base used  moles  +  problems  to  and o f f e r involving  verbalize the  subjects  reasons a 2:1  for mole  r a t i o between r e a c t a n t s . the  subjects  The  protocols  o b t a i n e d from a s k i n g  t o p r e d i c t the c o n c e n t r a t i o n  three r e l a t e d problem s i t u a t i o n s  of the a c i d i n these  (using the data they  had  o b t a i n e d i n the t i t r a t i o n ) were analyzed i n terms of the of p r e d i c t i o n and Analysis  the e x p l a n a t i o n  showed t h a t the  obtaining  the m o l a r i t y  same approach t h a t was  of t h e ^ h y d r o c h l o r i c  s i t u a t i o n i n v o l v i n g 1:1  initially  HC1  i n the 1:1  using  the NC  had  subjects. used i n  a c i d i n the  initial  r e a c t i n g r a t i o s were used i n respond-  i n g to the p r e d i c t i o n q u e s t i o n s . who  o f f e r e d by the  type  difficulty  I t was  a l s o found t h a t those  i n c a l c u l a t i n g the m o l a r i t y  s i t u a t i o n , e s p e c i a l l y those c l a s s i f i e d  of  as  (not comprehensible) Approach made the same  e r r o r s i n t h e i r c a l c u l a t i o n s with the p r e d i c t i o n q u e s t i o n s even though they o c c a s i o n a l l y made the c o r r e c t p r e d i c t i o n s o f f e r e d appropriate  e x p l a n a t i o n s i n some cases.  A f u r t h e r a n a l y s i s was predictions  by the  subjects  Performance on VAT  and  i n making t h e i r p r e d i c t i o n s .  Number of T i t r a t i o n -  the S o c i a l Sciences  Experiments  the S t a t i s t i c a l Package  (Nie, e t . a l . , 1975)  was  the number of t i t r a t i o n experiments done by the t o the (VAT).  study and  „/  of the p o s s i b l e conceptual paths  C o r r e l a t i o n a n a l y s i s using for  to the  T h i s a n a l y s i s attempted to p r o v i d e a c l e a r  diagramatic r e p r e s e n t a t i o n followed  done to show the changes i n  i n the three s i t u a t i o n s with r e s p e c t  approach employed.  and  performed  students p r i o r  t h e i r scores on the V o l u m e t r i c A n a l y s i s  T h i s a n a l y s i s was  performed f o r the  265  on  subjects  Test who  completed a l l done t o two  find  the t e s t s  administered  out the nature  variables.  of  i n the  study.  This  t h e r e l a t i o n s h i p between  was these  126  CHAPTER V  RESULTS AND DISCUSSION OF MODEL TESTING  Introduction In  t h i s chapter, the r e s u l t s and i n t e r p r e t a t i o n of  the analyses conducted  t o answer the r e s e a r c h q u e s t i o n s .  r e l a t e d t o the v a l i d a t i o n of the proposed are presented.  The s t a t i s t i c a l  hypotheses  i n t e g r a t e d model corresponding t o  these r e s e a r c h q u e s t i o n s are presented f i r s t .  Since the  p r e l i m i n a r y a n a l y s i s of the data showed t h a t there might be a d i f f e r e n c e among the students i n terms of t h e i r use of s t o i c h i o m e t r i c r a t i o s i n c a l c u l a t i o n s , the t e s t of the proposed i n t e g r a t e d model was performed s u b j e c t s i d e n t i f i e d i n the study. d i f f e r e n c e s were observed  on the two groups of  As w i l l be shown,  among these two groups.  Therefore  the t e s t of the model f o r the t o t a l sample was d e l e t e d . For reasons of b r e v i t y , the v a r i a b l e s ,  direct  p r o p o r t i o n a l r e a s o n i n g , i n v e r s e p r o p o r t i o n a l reasoning, knowledge o f subsumed o r p r e r e q u i s i t e concepts,and  performance on  v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s w i l l be designated by DP, IP, SC,and VAC r e s p e c t i v e l y , i n t h i s  chapter.  127 As will  refer  used  i n this  c h a p t e r , t h e term  t o a model s i m i l a r  s a t u r a t e d model  t o the proposed  integrated  b u t w i t h an assumed c a u s a l c o n n e c t i o n between d i r e c t tional  r e a s o n i n g and p e r f o r m a n c e  calculations. refer of  Trimmed  to the f i n a l  model  propor-  on v o l u m e t r i c a n a l y s i s  (or modified)  model r e s u l t i n g  i n t e g r a t e d model  from  will  t h e d e l e t i o n o f some  the c a u s a l connections i n the proposed  integrated  model.  E v a l u a t i o n o f t h e I n t e g r a t e d Model Statistical 1.  Model E v a l u a t i o n f o r T o t a l The total  sample  hypothesis, were f o u n d  performed  f o r the  data. be shown i n t e s t i n g  differences  on t h e t o t a l  results.  coefficients  Because o f these  t h e p o o l i n g o f t h e s c o r e s from  t o confounded  t h e second  i n some o f t h e p a t h  among t h e two g r o u p s .  differences may l e a d  Sample  h y p o t h e s i z e d i n t e g r a t e d model a c c o u n t s  However, a s w i l l  2.  Hypotheses  t h e two  Hence no a n a l y s i s  groups was  sample.  M o d e l E v a l u a t i o n f o r t h e Two Groups The  two g r o u p s  of subjects considered i n this  were t h o s e s u b j e c t s who u s e d and  t h o s e who u s e d  (a)  The p r o p o s e d tion  algorithms with  algorithms without  study  understanding  understanding.  i n t e g r a t e d model e x p l a i n s t h e o b s e r v a -  d a t a f o r t h e s u b j e c t s who u s e a l g o r i t h m s w i t h  understanding. (b)  The p r o p o s e d tion  i n t e g r a t e d model e x p l a i n s t h e o b s e r v a -  d a t a f o r t h e s u b j e c t s who u s e a l g o r i t h m s w i t h o u t  understanding.  128 T h e s e h y p o t h e s e s f o r t h e two g r o u p s were t e s t e d at  the  .05 l e v e l  questions  of  i n Chapter  significance  and c o r r e s p o n d t o r e s e a r c h  I.  Model E v a l u a t i o n  for Subjects  Algorithms Without  Using  Understanding  The a n a l y s i s o f t h e sample d a t a the s u b j e c t s  who a p p l i e d  t h e 1:1  y i e l d e d the parameter estimates  mole; r a t i o  reported  ( T a b l e 10)  for  indiscriminately  i n Table  11.  T a b l e 10 Covariance Matrix f o r Subjects Using Algorithms Without standing (N = 160)  VAC  VAC SC IP DP  Under-  SC  IP  DP  (0.405)  (0.122)  (-0.053)  0.403  (1.000) 0.994  (0.378)  (0.196)  0.122  0.376  (1.000) 0.994  (0.386)  -0.053  0.195  0.383  (1.000)* 0.994  Correlations  i n parentheses  (1.000) 0.994  129 Table  11  P a r a m e t e r E s t i m a t e s , S t a n d a r d E r r o r s (SE) and C r i t i c a l R a t i o s (CR) f o r S u b j e c t s u s i n g A l g o r i t h m s W i t h o u t U n d e r s t a n d i n g (N = 160)  P  43  P  42  P  32  P  31  P  21  P <  2  0.078  -0.036  0.078  0.355  0.079  4.494*  0.059  0.079  0.747  0.386  0.073  5.288*  Model  =  2.3668  The e s t i m a t e s by P i j  which r e p r e s e n t s  dent v a r i a b l e , model. ratios  (df  = 1) , P >  presented  -0.462  on v a r i a b l e  i ,  f o r each parameter  value of  zero to  the  its  i n the proposed  the c r i t i c a l  influence  of  ratios  standard  estimate. of  j,  symbolized the indepen-  e r r o r and  Each c r i t i c a l  the parameter  standard  p r o v i d e an i n d i c a t i o n o f  meter e s t i m a t e  of v a r i a b l e  are  the dependent v a r i a b l e  r e p r e s e n t i n g the r a t i o  the n u l l  .05  i n T a b l e 11,  the e f f e c t  T a b l e 11 a l s o g i v e s  t-ratio  for  5.372*  0.419  .05  Note:  ratios  CR  SE  Estimate  Parameter  the  error.  critical ratio  The  i n d i r e c t p r o p o r t i o n on p e r f o r m a n c e  is  a  minus  critical  importance of each  two p a t h s ,  the  estimate  integrated model.  suggest that  in  para-  The v a l u e s namely,  the  i n volumetric  130 analysis  calculations  on knowledge o f cally model.  and t h e  subsumed p r e r e q u i s i t e  the  A l l remaining estimates are  concepts  which i s  essential  are  statisti-  indicated  are  i n d i c a t e d on t h e  i n parentheses  i n the  c o r r e s p o n d i n g to each path The c h i - s q u a r e  freedom  (see  of  the path  paths.  figure  the  Also  standard  errors  test  of  the  Table 11).  fit  of  the  value of  This value  proposed 2.3668 w i t h  is  not  signi-  * Standard e r r o r s Figure  7:  in  co-  coefficient.  i n t e g r a t e d model y i e l d e d a c h i - s q u a r e of  representation  individual are  the  to the model.  a diagrammatic  i n t e g r a t e d model, the numerical values  efficients  degree  direct proportion  i n s i g n i f i c a n t and h e n c e may n o t be i m p o r t a n t t o  I n F i g u r e 7, of  influence of  parentheses  I n t e g r a t e d Model f o r S u b j e c t s U s i n g A l g o r i t h m s Without Understanding.  1  131  f i c a n t a t the 5% l e v e l of s i g n i f i c a n c e .  This i n d i c a t e s that  the model cannot be d i s t i n g u i s h e d s t a t i s t i c a l l y  from the  s a t u r a t e d model i n which there i s no o v e r - i d e n t i f y i n g tion  ( i . e . a l l the c a u s a l connections  sent) .  T h i s suggests  restric-  are assumed to be  pre-  t h a t the model o f f e r s a p l a u s i b l e  e x p l a n a t i o n of the v a r i a n c e - c o v a r i a n c e matrix i n t h i s subsample. of  The r e s i d u a l matrix r e s u l t i n g from the d e v i a t i o n  the reproduced  matrix The  covariance matrix from the sample covariance  ( f o r the proposed i n t e g r a t e d model) i s given i n Table  t a b l e a l s o g i v e s the reproduced  i n t e g r a t e d model. almost  Examination  12.  covariance matrix f o r the  of t h i s matrix r e v e a l s t h a t  a l l the sample c o v a r i a n c e s have been e x p l a i n e d by  the  proposed i n t e g r a t e d model. The  r e s u l t of the chi-square t e s t suggests  the c a u s a l paths, taken together, may model as a whole.  be important  that a l l  to the  However, as the t e s t of the i n d i v i d u a l  path  c o e f f i c i e n t s r e v e a l e d , the paths between DP and SC and between IP and VAC  may  not be very c r u c i a l i n e x p l a i n i n g the  scores f o r the s u b j e c t s who ing  use algorithms without  understand-  even though when c o n s i d e r e d together with the other  they appear to be  pp.  317  paths,  important.  As o t h e r r e s e a r c h e r s 1973,  test  - 318;  (e.g. K e r l i n g e r and  Heise, 1969,  pp.  the goals of s t r u c t u r a l m o d e l l i n g may  59  whether a more parsimonious  have  suggested,  i n c l u d e model trimming  t h a t i s , the removal of paths which are i n s i g n i f i c a n t from the model.  - 60)  Pedhazur,  -  statistically  T h i s i s done i n o r d e r to assess  model c o u l d account  reasonably  132 T a b l e 12 R e s i d u a l M a t r i x R e s u l t i n g f r o m t h e D i f f e r e n c e between t h e Sample C o v a r i a n c e M a t r i x and t h e R e p r o d u c e d C o v a r i a n c e M a t r i x for Subjects using Algorithms Without Understanding  (0.994)* 0.000 [0.000]+  VAC  DP  IP  SC  VAC  (0.403)  (0.122)  (0.068)  (0.376)  (0.195)  (0.383)  SC  0.000 [0.000]  (0.994) 0.000 [0.000]  IP  - 0.000 [-0.031]  0.000 "[0.000]  (0.994) 0.000 [0.000]  DP  -0.121 [-0.112]  0.000 [0.050]  0.000 [0.000]  Reproduced c o v a r i a n c e s Residual  +  well  matrix  i n parentheses  f o r trimmed model i n square  f o r the data.  indirect  Following this  p r o p o r t i o n and p e r f o r m a n c e  calculations  and t h e one between  l e d g e o f subsumed  prerequisite  direct  concepts  referred  t o as t h e t r i m m e d i n t e g r a t e d i n Figure  Analysis integrated Figure  between  analysis  were d e l e t e d  from t h e  model, which  model  is  (or t h e Z - m o d e l ) ,  8.  of the data  13.  the path  p r o p o r t i o n a n d know-  The r e s u l t i n g  i n terms o f t h e trimmed  model y i e l d e d t h e parameter  8 and T a b l e  brackets  on v o l u m e t r i c  integrated  represented  model.  procedure,  proposed  is  (0.994) 0.000 [0.000]  estimates  The c r i t i c a l r a t i o s  given i n  show t h a t  a l l the  133 Table  13  P a r a m e t e r E s t i m a t e s , S t a n d a r d E r r o r s (SE) and C r i t i c a l R a t i o s (CR) o f Trimmed I n t e g r a t e d M o d e l f o r S u b j e c t s u s i n g A l g o r i t h m s without Understanding  Parameter  Estimate  SE  CR  P  4 3  0.406  0.072  5.639*  P  3 2  0.378  0.073  5.178*  P  2 1  0.386  0.073  5.288*  P < Note:  .05 M o d e l yC  2  = 2.0215  (df = 3 ) ,  P >  .05  * "Standard e r r o r s i n parentheses F i g u r e -8V  Trimmed I n t e g r a t e d M o d e l f o r S u b j e c t s Algorithms without Understanding  using  134  paths a r e e s s e n t i a l t o the model.., The chi-square fit  t e s t f o r t h i s modified  goodness-of-  model gave a value o f 2.0215  which was not s t a t i s t i c a l l y  (df = 3)  s i g n i f i c a n t a t the 5% l e v e l .  This  i n d i c a t e s t h a t the trimmed i n t e g r a t e d model a l s o p r o v i d e s a plausible representation  o f the observed data.  examination o f i t s r e s i d u a l s  (Table 12) r e v e a l e d  Further t h a t the  trimmed i n t e g r a t e d model accounts f o r a s u b s t a n t i a l p r o p o r t i o n of the covariances  i n the m a t r i x .  Residuals  c l o s e t o zero are  not unexpected s i n c e the trimmed i n t e g r a t e d model assumes t h a t these c o v a r i a n c e s two  are zero.  Thus, i t would appear from the  s e t s o f r e s u l t s t h a t both models are p l a u s i b l e  representa-  t i o n of the data. To.further was  c l a r i f y this situation, a detailed analysis  performed t o examine the t o t a l a s s o c i a t i o n s among the  v a r i a b l e s i n the proposed i n t e g r a t e d model. t h i s a n a l y s i s are presented i n Table 1 4 .  The r e s u l t s o f  In t h i s t a b l e , the  t o t a l a s s o c i a t i o n between any two v a r i a b l e s has been decomposed i n t o i n d i r e c t and d i r e c t e f f e c t s , t o t a l e f f e c t s , and spurious effects.  Spurious e f f e c t s between two v a r i a b l e s r e f e r t o the  compound paths t h a t a r e mathematically p a r t of the decomposit i o n but may be due t o the presence o f a common cause o r the presence o f causes which are c o r r e l a t e d 1975) .  (Alwin and Hauser,  The t o t a l e f f e c t i s t h a t p a r t o f the t o t a l a s s o c i a t i o n  between two v a r i a b l e s which i s not due t o spurious e f f e c t s . The  d i r e c t e f f e c t i s t h a t p a r t o f the t o t a l e f f e c t which i s  not t r a n s m i t t e d  v i a i n t e r v e n i n g v a r i a b l e s , w h i l e the i n d i r e c t  e f f e c t i s t h a t p a r t o f a v a r i a b l e ' s t o t a l e f f e c t which i s mediated by i n t e r v e n i n g v a r i a b l e s .  135 Table  14  E f f e c t s A n a l y s i s i n the I n t e g r a t e d Model Algorithms without Understanding  PreDependent T o t a l determined V a r i a b l e Effect Variables  Subjects  -  0.419  -  IP  0.113  -0.036  -  0.149  DP  0.068  -  -0.014  0.025  SC  IP DP  IP  DP  0.355  0.355  0.196  0.059  0.386  0.386  The e f f e c t s using Wright's  (1934)  path analysis  these procedures  (Duncan, 1 9 6 6 ) .  The r e s u l t s of  DP on SC  effect  (0.059).  of  total  the  integrated vealed that  effect.  model for  understanding, sing  the e f f e c t  i n Table  and t h e  the  of  is  far  _ —  14 was  —  obtained  fundamental  the  simpler  14 show t h a t  greater  indirect effect is  Figure  8 ).  proportional  direct  for  The e f f e c t s  theorem from procedure  reasoning  indirect  direct  for  about  the  trimmed  analysis  algorithms  proportional  the  than i t s  accounts  even g r e a t e r  s u b j e c t s who use  inverse  0.023  The v a l u e s o b t a i n e d  shown i n T a b l e  This  (see  0.057  (1975).  (0.137)  This  0.009  —  —  Spurious Effect  -0.014  -  0.137  were c o n f i r m e d by u s i n g  due t o A l w i n and H a u s e r  effect  -  decomposition procedures  using  Indirect Effect Via SC/IP IP SC  0.419  VAC  SC  of  Direct Effect  for  also  re-  without  may be  reasoning  70%  suppres-  on v o l u m e t r i c  136 analysis for  calculations.  the mediating e f f e c t  However, to  suppress the e f f e c t  choice of  of  Applied  obtained  This for  included  to  is  1975,  problem,  it  influence analysis  effect  of  of  inverse  reasoning  interview to  relation  89-90. c o u l d be  IP i s  this  of  since  some o f  path  (IP)  is this  on t h e  on v o l u m e t r i c  of  IP on VAC  of d i r e c t p r o p o r t i o n a l this  suppression. found  i n c o r r e c t l y apply a d i r e c t p r o p o r t i o n a l  problems,  inverse  s o l u t i o n when s o l v i n g  even though they c o u l d  proportion.  i n s o l v i n g the problems  t h e mole r a t i o s  reason-  t h e s e s u b j e c t s were  g r o u p who o b t a i n e d t h e c o r r e c t  difficulty  (-0.143)  Because o f  because of  between m o l a r i t y and volume o f  i n terms  value  the d i r e c t e f f e c t  negative  argued  suppressor  when t h i s  model.  indirect effect  the v o l u m e t r i c a n a l y s i s reason  integrated  (VAC),  A l s o the  T h i s may be i m p o r t a n t the  calculations  The  t h e n one  direct proportional  proportional reasoning  on VAC v i a  89-90).  d i r e c t proportional reasoning  calculations  negative.  of  found  "fully  becomes pp.  is  situation  pp.  s u g g e s t e d by t h e n e g a t i v e  the d i r e c t e f f e c t  reasoning.  is  1975,  suppression  the present  i n the proposed  suppressing  in  suppression  (Cohen and C o h e n ,  i n g on v o l u m e t r i c a n a l y s i s  in  the converse  (Cohen and C o h e n , the  -0.014  when one v a r i a b l e  another, since  proportional  d i r e c t p r o p o r t i o n a l r e a s o n i n g may be t h e  variable.  is  shown by t h e v a l u e o f  system,  the d i r e c t i o n of  meaningfulness  is  inverse  to e x i s t  symmetric or m u t u a l "  that  of  i n an i n t e r a c t i v e  c o u l d a l s o be s a i d  of  This  Also,  those  reacting  because they  in their calculations.  That i s ,  subjects  ratios  had  'reversed1 these  subjects  137  i n c o r r e c t l y assumed an i n v e r s e r e l a t i o n between s t o i c h i o m e t r i c c o e f f i c i e n t and moles o f the r e a c t i n g substances. The  substantive  meaning o f the d e l e t e d paths i n the  trimmed i n t e g r a t e d model i s t h a t d i r e c t p r o p o r t i o n a l  reasoning  does not i n f l u e n c e knowledge o f p r e r e q u i s i t e concepts d i r e c t l y and  t h a t i n v e r s e p r o p o r t i o n a l reasoning  i n f l u e n c e on performance i n v o l u m e t r i c T h i s seems t o be c o n t r a r y , t h e o r e t i c a l analyses  does not have a d i r e c t analysis calculations.  i n some sense, t o some o f the  performed on some chemical concepts.  example Herron  (1975) analyzed  ing  a n a l y s i s c a l c u l a t i o n s and concluded t h a t  volumetric  students w i l l have d i f f i c u l t i e s  For  some chemical concepts i n c l u d -  i n doing v o l u m e t r i c  analysis-  type c a l c u l a t i o n s because o f the i n v e r s e p r o p o r t i o n a l i t y involved.  He i m p l i e d by h i s a n a l y s i s t h a t s u c c e s s f u l p e r f o r -  mance on such a problem w i l l r e q u i r e the a b i l i t y t o reason using  indirect proportion.  The e m p i r i c a l r e s u l t obtained f o r  the trimmed i n t e g r a t e d model suggests that i n d i r e c t reasoning volumetric  proportional  does not have any d i r e c t e f f e c t on performance i n analysis calculations.  I t s i n f l u e n c e i s mainly  through the knowledge o f the subsumed concepts.  Even the  r e s u l t f o r the proposed i n t e g r a t e d model r e v e a l s , as a l r e a d y noted above, t h a t the mediating e f f e c t o f subsumed concepts on v o l u m e t r i c  a n a l y s i s c a l c u l a t i o n s i s very  substantial.  r e s u l t i s understandable i n view o f the suggestion  This  i n the  l i t e r a t u r e t h a t the content o f a task may i n t e r a c t with reasoning 1975)  .  a b i l i t y o f the s u b j e c t s on the task  (e.g. Lunzer,  138 Ingle and Shayer  (19 71)  a l s o analyzed  the mole  concept and i t s a s s o c i a t e d c a l c u l a t i o n s and concluded t h a t s i n c e the mole concept i n v o l v e s p r o p o r t i o n a l proportional)  reasoning,  high  school  (really, direct  students who can't  reason  i n t h i s way w i l l f i n d i t d i f f i c u l t t o comprehend these concepts. T h i s a n a l y s i s i m p l i e s a d i r e c t e f f e c t of d i r e c t reasoning.  The above e m p i r i c a l r e s u l t f o r the trimmed i n t e -  grated model i n d i c a t e s , reasoning concepts  however, t h a t d i r e c t  (which i n c l u d e s the mole concept).  inverse proportional  I t treats i t s transmitted v i a  reasoning.  I t c o u l d be t h a t , s i n c e d i r e c t  reasoning,  proportional  does not have any d i r e c t e f f e c t on the p r e r e q u i s i t e  e f f e c t on subsumed concepts s o l e l y as being  reasoning  proportional  proportional  precedes and i s subsumed by i n v e r s e  proportional  most of the e f f e c t of d i r e c t p r o p o r t i o n a l  or subsumed concepts i s t r a n s m i t t e d t i o n a l reasoning i s developed. t h a t with tasks  reasoning  through i n v e r s e propor-  as soon as t h i s l a t t e r reasoning  ability  A l s o , i t seems reasonable t o hypothesize involving s o l e l y d i r e c t proportions  or simple  r a t i o s , the unmediated i n f l u e n c e of d i r e c t  proportional  reasoning  may be c o n s i d e r a b l e  proportional  reasoning  w i l l be d i s p l a y e d  where formulas or algorithms e f f e c t may become s m a l l .  (i.e. direct  i n the s o l u t i o n  c o u l d be a p p l i e d t h i s  Similarly,  i n v e r s e p r o p o r t i o n a l reasoning r e q u i r i n g i t s use.  t o the t a s k ) . But direct  the d i r e c t i n f l u e n c e of  may become important i n tasks  However, where there  i s a formula which  139 c o u l d be a p p l i e d as i n the case of v o l u m e t r i c t i o n s , t h i s d i r e c t e f f e c t may  analysis calcula-  d i m i n i s h while i t s e f f e c t through  the a p p l i c a t i o n of the formula may  become s u b s t a n t i a l .  A l s o the above r e s u l t s i n d i c a t e t h a t i n both o r i g i n a l model and reasoning  has  reasoning. others  the trimmed model, d i r e c t p r o p o r t i o n a l  a d i r e c t i n f l u e n c e on i n v e r s e  proportional  T h i s i s . c o n s i s t e n t with the r e s u l t s obtained  (Rogers, 1977).  t i o n a l reasoning acquisition. reasoning  precedes i n v e r s e p r o p o r t i o n a l reasoning  I t suggests f u r t h e r t h a t i n v e r s e  The  proportional  volumetric  s i g n i f i c a n t e f f e c t of subsumed  p r e r e q u i s i t e concepts compared with i n v e r s e on performance i n v o l u m e t r i c  proportional  analysis calculations,  i s not unexpected s i n c e i n the s t u d i e s by Wheeler and Griffiths  (1979) subsumed concepts  Kass  (or content)  were found to be more important than the reasoning the  in  i n f l u e n c e s knowledge of subsumed p r e r e q u i s i t e  analysis calculations.  (1977) and  by  I t a l s o suggests t h a t d i r e c t propor-  concepts which i n t u r n a f f e c t s performance on  reasoning  the  ability  of  subjects. I t has  been suggested t h a t i n the e v a l u a t i o n  of  s t r u c t u r a l models, apart from the computation of  statistics  and  meaningful  the examination of r e s i d u a l s , the theory  i n t e r p r e t a t i o n should  and  d i c t a t e the choice of model.  From the  above d i s c u s s i o n of the r e s u l t s , i t seems the trimmed grated model would be p r e f e r a b l e t o the proposed model on the grounds of meaningfulness and been argued t h a t ,  inte-  integrated  parsimony.  It  has  140 " t h e p o t e n t i a l f o r r e f i n i n g o r t r i m m i n g a t h e o r y , and t h u s m a k i n g t h e t h e o r y more p a r s i m o n i o u s , c l e a r l y i s o f c o n s i d e r a b l e s i g n i f i c a n c e and c o u l d be l i s t e d a l o n g w i t h t h e i s s u e s o f e x p l a n a t i o n . . . as a b a s i c g a i n t o be a c q u i r e d f r o m t h e c o n s t r u c t i o n o f l i n e a r m o d e l s . "  (Heise, 1 9 6 9 , pp. 5 9 - 6 0 ) . It  is  is  consistent  the science variables  seen  is  the  fact  r e a s o n i n g may be  proportional  reasoning.  The r e s u l t s  ability  to  true  data  of  their  these  discussed  instead  of  negative  subjects  problems  the e f f e c t  of  It  inverse  use  subjects  seem  a direct proportional  is  to  the the  ability  c o u l d be a r g u e d  b o r n e o u t by t h e  direct  in  that  interview relation  one between m o l a r i t y and volume e v e n  inverse  s u b j e c t s may r e f l e c t analysis  (as  later).assume  It  subjects  proportional reasoning  using  inverse  proportion, this  p r o p o r t i o n tends  i n f l u e n c e on t h e i r p e r f o r m a n c e  calculations.  However,  in their calculations  calculations.  an i n v e r s e  using  students'  1977;  1977).  these  t h a t , w i t h these  inverse  though they can reason to reason  for  in  explanatory  u s i n g d i r e c t p r o p o r t i o n may be h i d i n g  analysis  some o f  the  t h e " s u b j e c t s who t e n d e d t o  e x p l a i n e d above) reason  effect  volumetric since  for  u s u a l l y made  (Wheeler and K a s s ,  that  model  to r a t i o n a l i z e  suppressing  stoichiometric ratios  (as  used  of  M a c D o n a l d and Webb,  proportional  suggest  i n terms  school concepts  taken of  trimmed i n t e g r a t e d  formulations  and c o n t e n t )  Johnstone,  1975;  incorrect  the  education l i t e r a t u r e  (structure  cognisance  that  with the kinds of  understanding of Herron,  also  suggested that on t h e  — that  is,  the  fact  slight  on v o l u m e t r i c  this  structural  t o have a  behaviour of  complexity of that  it  ability  analysis the volumetric  involves  both  141 d i r e c t and i n v e r s e r e l a t i o n s h i p s .  As such the students  probably become confused over which r e l a t i o n i s a p p l i c a b l e between which two v a r i a b l e s . Again, i t i s suggested here t h a t the s t r o n g i n f l u e n c e of p r e r e q u i s i t e concepts on v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s f o r these s u b j e c t s may mean t h a t the mediating i n f l u e n c e o f p r e r e q u i s i t e concepts may be e s s e n t i a l t o t h e i r performance  on  the c a l c u l a t i o n s s i n c e they tend t o get confused over which s t r u c t u r a l r e l a t i o n they have t o use.  I t c o u l d be t h a t some  of these s u b j e c t s employ formulas i n t h e i r s o l u t i o n s with an understanding o f the b a s i c concepts i n v o l v e d thereby the i n f l u e n c e o f the subconcepts  on performance  increasing  on v o l u m e t r i c  analysis calculations.  Model E v a l u a t i o n f o r S u b j e c t s Using Algorithms With The v a l i d a t i o n of the proposed  Understanding  i n t e g r a t e d model f o r  s u b j e c t s u s i n g a l g o r i t h m s w i t h understanding gave the parameter e s t i m a t e s , standard e r r o r s , and c r i t i c a l r a t i o s presented i n Table 15.  These v a l u e s were o b t a i n e d from the analyses o f the  sample v a r i a n c e - c o v a r i a n c e matrix f o r t h i s subsample  (Table 16).  As the v a l u e s i n d i c a t e , w i t h the e x c e p t i o n o f the path coe f f i c i e n t between d i r e c t p r o p o r t i o n a l r e a s o n i n g and knowledge of subsumed p r e r e q u i s i t e concepts, a l l other parameters are important t o the proposed  i n t e g r a t e d model.  That i s , o n l y  d i r e c t p r o p o r t i o n a l r e a s o n i n g has s t a t i s t i c a l l y  insignificant  d i r e c t i n f l u e n c e on knowledge o f subsumed p r e r e q u i s i t e  concepts.  142 Table  15  P a r a m e t e r E s t i m a t e s , S t a n d a r d E r r o r s (SE) and C r i t i c a l (CR) f o r S u b j e c t s u s i n g A l g o r i t h m s w i t h U n d e r s t a n d i n g .  Parameter  P  43  P  42  P  32  P  31  P  21  P < Note:  '  Estimate  SE  Ratios (N' —105)  CR  0.214  0.104  2.058*  0.208  0.104  2.000*  0.429  0.095  4.515*  0.123  0.095  1.295  0.439  0.088  4.989*  .05 Model  Figure  %  = 0.9183  9 gives  the  (df  =  integrated  1), P >  0.05  model and t h e  parameter  estimates.. A test  of  t h e m o d e l as  using  the c h i - s q u a r e  value  of  0.9183 w i t h 1 degree o f  value  is  not  underlying from t h e the  the  left  the  sample d a t a . of  sample c o v a r i a n c e of  at  the model i s  deviation  examination are  goodness-of-fit  significant  suggesting that  this  behind.  a whole f o r  representative  the reproduced  matrix  (Table  l e v e l of  The r e s i d u a l  matrix  is  shows  subjects  test yielded a  freedom ^05  these  given that  chi-square  15).  This  significance, of  the  matrix  covariance i n Table  processes  resulting matrix  17.  v i r t u a l l y no  from  The residuals  143 T a b l e 16 Covariance Matrix for Subjects standing (N = 105)  VAC  (1.000) 0.990  VAC SC IP DP  *  9:  SC  IP  DP  (0.315)  (0.312)  (0.184)  0.311  (1.000) 0.990  (0.483)  (0.311)  0.308  0.478  (1.000) 0.990  (0.439)  0.182  0.308  Correlations  Figure  using Algorithms w i t h Under-  are  0.435  (1.000) 0.990  i n parentheses  Proposed I n t e g r a t e d Model f o r S u b j e c t s Algorithms with Understanding.  Using  144 Table  17  R e s i d u a l M a t r i x R e s u l t i n g f r o m t h e D i f f e r e n c e between t h e Sample C o v a r i a n c e M a t r i x and t h e R e p r o d u c e d C o v a r i a n c e M a t r i x for Subjects Using Algorithms with Understanding  VAC  SC  VAC  (0.990)* 0.000 [0.000]+  SC  IP  DP  (0.311)  (0.308)  (0.156)  0.000 [0.000 [  (0.990) 0.000 [0.000]  (0.478)  (0.308)  IP  0.000 [0.000]  0.000 [0.000]  (0.990) 0.000 [0.000]  (0.435)  DP  0.026 [0.047]  0.000 [0.098]  0.000 [0.000]  Reproduced c o v a r i a n c e s +  Residual matrix  i n parentheses  f o r t h e trimmed model i n square  Although the r e s i d u a l s all  and t h e c h i - s q u a r e  t h e p a t h s among t h e v a r i a b l e s  covariances  i n t h e sample d a t a ,  paths  that  shows  much i n t h i s  test  may h e l p  the t e s t  brackets  indicate  i n explaining  that the  of the i n d i v i d u a l  t h e p a t h between DP and SC may n o t c o u n t  very  explanation. The n e x t  stage  i n the a n a l y s i s  a p p l y model t r i m m i n g p r o c e d u r e s c o u l d be o b t a i n e d data  (0.990) 0.000 [0.000]  to find  if  therefore,  a trimmed model  w h i c h i s more r e p r e s e n t a t i v e  than the proposed  integrated  model.  was t o  Since  o f t h e sample the path  •  145 between d i r e c t  proportional reasoning  sumed c o n c e p t s ; w a s n o t model r e - a n a l y z e d . Table  significant,  it  and knowledge o f was d e l e t e d  The e s t i m a t e s o b t a i n e d  are  and  subthe  presented  in  18.  Table  18  P a r a m e t e r E s t i m a t e s , S t a n d a r d E r r o r s (SE) and C r i t i c a l R a t i o s (CR) o f Trimmed I n t e g r a t e d M o d e l f o r S u b j e c t s u s i n g A l g o r i t h m s with Understanding  Parameter  P  43  P  42  P  32  P  21  Note: P <  Estimate  Model %  2  SE  CR  0.214  0.104  2.058*  0.207  0.104  2.000*  0.483  0.086  5.616*  0.439  0.088  4.989*  = 1.5859  (df  = 2),  P>  .05  .05  A diagrammatic  representation  in Figure  10.  As i n d i c a t e d  estimates  are  significant  the paths are  at  that  all  this  t r i m m e d model s u g g e s t s  reasonably residuals  well for  for  the  of  the  trimmed model i s  i n T a b l e 18, the  five  probably r e a l . that  all  percent  the  given  parameter  level,  The c h i - s q u a r e  suggesting test  the. t r i m m e d model a l s o  sample d a t a .  the trimmed i n t e g r a t e d  The e x a m i n a t i o n o f model  (given  for  accounts the  i n Table  17)  146  * Standard e r r o r s i n parentheses Figure  1 0 : Trimmed Model f o r S u b j e c t s Using Algorithms with Understanding  shows t h a t p a r t o f the c o v a r i a n c e between SC and DP i s l e f t unexplained.  However, t h i s remaining c o v a r i a n c e  constitutes  only about one t h i r d o f the t o t a l a s s o c i a t i o n between these two v a r i a b l e s .  Thus, i t c o u l d be argued  t h a t the trimmed  i n t e g r a t e d model a l s o p r o v i d e s a reasonable e x p l a n a t i o n  o f the  c o v a r i a n c e s i n the sample data. A d e t a i l e d a n a l y s i s o f the e f f e c t s o f each v a r i a b l e through the decomposition  of the t o t a l a s s o c i a t i o n s  v a r i a b l e s y i e l d e d the v a l u e s i n Table 1 9 .  among the  T h i s t a b l e shows  t h a t both the d i r e c t and i n d i r e c t e f f e c t o f d i r e c t r e a s o n i n g on subsumed concepts a r e s u b s t a n t i a l  proportional  (0.123 and 0.188  147 Table 19 E f f e c t s A n a l y s i s i n the I n t e g r a t e d Model f o r Subjects Using Algorithms With Understanding  Predetermined Variables SC  Dependent T o t a l Variable Effect  Indirect Effect SpuVia rious IP SC SC/IP " E f f e c t  0.214  0.214  -  IP  0.300  0.208  -  DP  0.112  IP  VAC  Direct Effect  SC  DP DP  IP  respectively).  -  .092  .046 0.026  0.429  0.429  0.311  0.123  0.439  0.439  0.040  0.188 -  0.012 0.054  -  -  -  However, the i n d i r e c t e f f e c t through i n v e r s e  p r o p o r t i o n a l reasoning accounts effect  0.101  f o r about 60% o f the t o t a l  (0.311). Using the c r i t e r i a o f parsimony and the f a c t t h a t  the i n d i r e c t e f f e c t accounts  f o r a g r e a t e r p r o p o r t i o n o f the  c o v a r i a n c e , the trimmed i n t e g r a t e d model was r e t a i n e d . However, because the path between IP and VAC  (Table 18) i s r e l a t i v e l y  s m a l l , i t was decided t o d e l e t e t h i s path i n a d d i t i o n t o the path between DP and SC i n order t o determine more o v e r - i d e n t i f i e d  whether a much  model ( s i m i l a r t o the one f o r the s u b j e c t s  u s i n g a l g o r i t h m s without understanding  i . e . a Z-model) c o u l d  be o b t a i n e d which c o u l d a l s o p r o v i d e a reasonable e x p l a n a t i o n of the data f o r t h i s  subsample.  148  A n a l y s i s o f the data f o r these s u b j e c t s i n terms o f the Z-model gave the r e s u l t s presented i n Table 20 and Figure 11.  The r e s u l t shows t h a t a l l the three paths a r e  e s s e n t i a l t o the Z-model.  The c h i - s q u a r e t e s t o f the goodness-  o f - f i t gave a v a l u e o f 2.6655 i n s i g n i f i c a n t a t the 5% l e v e l , may  (df = 3 ) .  T h i s value was  suggesting t h a t the Z-model  a l s o p r o v i d e a reasonable r e p r e s e n t a t i o n o f the model.  However, examination matrix  o f the r e s i d u a l v a r i a n c e - c o v a r i a n c e  (Table 21) r e v e a l e d t h a t q u i t e a s u b s t a n t i a l p r o p o r t i o n  of the c o v a r i a n c e between VAC and IP was unexplained.  Further-  more, from Table 1 9 , i t c o u l d be seen t h a t the d i r e c t i n f l u e n c e of i n v e r s e p r o p o r t i o n a l r e a s o n i n g on v o l u m e t r i c a n a l y s i s c a l c u l a t i o n s i s g r e a t e r than i t s i n f l u e n c e through concepts.  As such,'the  prerequisite  Z-model was r e j e c t e d i n . f a v o u r o f the  o r i g i n a l trimmed model ( i . e . the non-Z-model) as being untenable. Table 20 Parameter Estimates, Standard E r r o r s (SE) and C r i t i c a l R a t i o s (CR) f o r Subjects u s i n g Algorithms With Understanding (N = 105) Parameter P  Estimate  43  P *32 P  21  * P< Note:  SE  CR  0.314  0.093  3 .376*  0.483  0.086  5 .616*  0.439  0.088  4 .989*  .05  Model X  2  =  2.6655  (df = 3 ) ,  P>  .05  149  * Standard e r r o r s i n parentheses F i g u r e 11:  The F u r t h e r Trimmed Model f o r Subjects Using Algorithms With Understanding  Table 21 . . .Residual M a t r i x f o r the Next Trimmed  VAC  SC  IP  VAC  0.000  SC  0.000  0.000  IP  0.158  0.000  0.000  DP  0.116  0.098  0.000  Model  DP  0.000  150 Substantively, algorithms reasoning  the r e s u l t s  with understanding has  a strong  These r e s u l t s  indicate that  performance influence ability direct  is  using  the  inverse  who use tions,  result  is  section).  for  It  the c o r r e c t ability more,  that  proportional reasonable metric lations  that  ratio,  using  inverse identify  chemical equations.  involved.  ratio  is  As s u c h ,  the  Further-  correctly into  it  subjects using the appropriate  t h e i r performance  depends  concepts.  for  subjects  calculations  may i n v o l v e some i n s i g h t  relationships  (see  calcula-  the a b i l i t y to  i d e n t i f y how t h i s  in calculations  moderate  reported  these  in their  the  analysis  t h e one  for  this  i n a r e a c t i o n may i n v o l v e  and i n t e r p r e t e  the a b i l i t y to  applied  that  stoichiometric ratio  influences  although  on v o l u m e t r i c a n a l y s i s  c o u l d be a r g u e d  to balance  concepts  the  results  without understanding  may be i n f l u e n c e d by t h e i r a b i l i t y t o r e a s o n proportion.  The  on v o l u m e t r i c  different  for  these subjects,  stoichiometric ratios  t h e i r performance  reported  calculations  The i m p l i c a t i o n i s  appropriate  prerequisite  p r o p o r t i o n may have a  subjects using algorithms  previous  proportional  those  Furthermore, for  using  direct proportional  without understanding.  i n f l u e n c e on t h e i r p e r f o r m a n c e This  subjects  i n f l u e n c e on  knowledge o f p r e r e q u i s i t e  moderate.  to reason  that  s i m i l a r to  on v o l u m e t r i c a n a l y s i s  calculations. for  a strong  are  subjects using algorithms also  indicate  the  i n f l u e n c e on i n v e r s e  r e a s o n i n g w h i c h i n t u r n has concepts.  for  seems stoichio-  on v o l u m e t r i c a n a l y s i s  on i n v e r s e p r o p o r t i o n a l r e a s o n i n g  the  and  calcusubsumed  151 The  r e s u l t of the model v a l i d a t i o n  f o r these  a l s o i n d i c a t e t h a t d i r e c t p r o p o r t i o n a l reasoning  subjects  influences  knowledge of subsumed concepts mainly through i t s e f f e c t inverse p r o p o r t i o n a l reasoning. obtained ing  f o r the s u b j e c t s who  (see the p r e v i o u s  use  T h i s i s s i m i l a r to the algorithms without  to the r e s e a r c h hypotheses s t a t e d at the beginning  1.,  result  understand-  s e c t i o n f o r the d i s c u s s i o n of t h i s r e s u l t ) .  F i n a l l y , r e l a t i n g the above r e s u l t s and  chapter,  on  discussions of  the  the f o l l o w i n g c o n c l u s i o n s can be drawn.  For the s u b j e c t s u s i n g algorithms with understanding,  even  though both the proposed i n t e g r a t e d model and the trimmed i n t e g r a t e d model o f f e r e d reasonable  e x p l a n a t i o n of  sample data, the trimmed i n t e g r a t e d model was tenable on the grounds of parsimony and and  their  r e t a i n e d as  meaningfulness  the f a c t t h a t the path between d i r e c t p r o p o r t i o n a l  reasoning  and  subsumed concepts c o n t r i b u t e d l i t t l e  to the  e x p l a n a t i o n of t h e i r performance. 2.  For the s u b j e c t s u s i n g algorithms without  understanding,  the trimmed i n t e g r a t e d model (or Z-model) was in  accepted  favour of the proposed i n t e g r a t e d model even though  both appeared to account f o r the o b s e r v a t i o n data f o r these  subjects.  model was  The  acceptance of the trimmed i n t e g r a t e d  made on the grounds of parsimony, meaningfulness  and the f a c t t h a t the d i r e c t e f f e c t of d i r e c t p r o p o r t i o n a l reasoning  on subsumed concepts and t h a t of i n v e r s e  p r o p o r t i o n a l reasoning  on performance on v o l u m e t r i c  ses c a l c u l a t i o n s were found to c o n t r i b u t e l i t t l e e x p l a n a t i o n of the sample data f o r these  analy-  to the  subjects.  152  CHAPTER V I RESULTS AND DISCUSSION OF DATA ON S P E C I F I C STUDENT D I F F I C U L T I E S  Introduction In t h i s on s p e c i f i c the  study  chapter,  student  are  the r e s u l t s  difficulties  presented  posed  and d i s c u s s e d .  for  the  i n the  four  questions  second p a r t  In a d d i t i o n ,  the  c o r r e l a t i o n between p e r f o r m a n c e  on v o l u m e t r i c a n a l y s i s  lations  titrations  subjects  and t h e p r i o r number o f is  r e p o r t e d and d i s c u s s e d .  with a discussion  of  the  interview results  the  two g r o u p s  of  how t h e c o n c e p t u a l e r r o r s bear  on t h e v a l i d a t e d  and where a p p r o p r i a t e  research Chapter  I.  the  statistical  and i n t e r p r e t a t i o n .  questions  calcuthe  concludes  and some  p a t h models  of for  subjects.  In r e p o r t i n g t h e s e r e s u l t s ,  the r e s u l t s  p e r f o r m e d by  The c h a p t e r  of  corresponds  the research hypothesis,  The n u m b e r i n g o f to  question, precede the  the numbering used  in  153 Conceptual  E r r o r s on V o l u m e t r i c A n a l y s i s T e s t  To b e t t e r u n d e r s t a n d encountered  by  the  students while doing  calculations  i n t h e VAT,  s t u d e n t s was  examined.  The 3.  some o f t h e  specific  difficulties  the v o l u m e t r i c  t h e w r i t t e n work d i s p l a y e d by  research question addressed  What c o n c e p t u a l e r r o r s  are. made by  analysis the  was:  s t u d e n t s on  the  Volumetric Analysis Test? A n a l y s i s of the in  that  h e l d by  7 students  one  error.  students  lists  difficulties  least  three percent of the  of The  be  280  the  indicated  Volumetric  s u b j e c t s made a t  s u b j e c t s who  s u b j e c t s who  table  also  made e a c h  inferred  reaction  that these  reacting  ratio  Only by  test  one  the  at are and  subjects i s  mole o f one  w i t h one  the  substances.  ratio  substance  mole o f t h e  s u b j e c t s performed  this  that  error.  indiscriminately  s t u d e n t s used  this  least  shows t h e number  t a k i n g i n t o account  r e l a t i o n s between the r e a c t i n g  Some o f t h e s e  took  m a j o r c o n c e p t u a l e r r o r made by  c a l c u l a t i o n s without  1:1  e r r o r on  needed t o r e a c t c o m p l e t e l y  c o u l d be  analysis  the conceptual d i f f i c u l t i e s  The  assuming t h a t i n every  will  The  problems  fundamental  o r e r r o r s w h i c h were e n c o u n t e r e d  i n the t a b l e .  percentage  the  the  with volumetric analysis c a l c u l a t i o n s .  those  reported  students.  the remaining  T a b l e -22  had  the  d i d n o t make any  Analysis Test while  It  in solving  the V o l u m e t r i c A n a l y s i s T e s t r e v e a l e d c e r t a i n  misconceptions  in  steps used  other.  their  stoichiometric Thus, they  in their  used  calculations.  consistently  in a l l  154  T a b l e 22 Conceptual Errors Error No,  on t h e V o l u m e t r i c A n a l y s i s  ,.„_-, „ Conceptual E r r o r  1.  Assumption of  2.  3. 4.  5. 6.  7.  1:1  Test*  Number o f Subjects 201  70.0  Concentration of stock s o l u t i o n d i f f e r e n t from c o n c e n t r a t i o n o f volume u s e d i n t i t r a t i o n  99  34.5  Incorrect weight  85  2 9.6  Assumption o f 2:1 mole r a t i o s i n p r o b l e m s r e q u i r i n g 1:1 mole r a t i o s ( e s p e c i a l l y assuming I^SO^ a l w a y s r e a c t s i n a 1:2 r a t i o )  12  4.2  Reversal ratios  45  15.7  37  12.9  29  10.1  formula  of  mole r a t i o  or  formula  stoichiometric  Assume d i r e c t concentration solutions  proportion and volume  Equating concentration (moles) o f s u b s t a n c e  mole  between of  with  amount  8.  C a l c u l a t i o n o f moles of a substance irrespective o f • i t s purity  55  19.2  9.  A d d i n g v o l u m e s o f a c i d and b a s e t o g e t h e r i n c a l c u l a t i n g unknown c o n c e n t r a t i o n or v i c e versa  17  5.9  D i f f i c u l t i e s i n u s i n g mass c a l c u l a t e concentration or versa  45  15.7  173  60.3  10.  11.  Total  to vice  Use o f b a l a n c e d c h e m i c a l e q u a t i o n s not considered i n problem solution  number o f  s u b j e c t s who w r o t e  this  test  =287  155 problems w h i l e others used i t i n a few problems. predominance o f t h i s conceptual  The  e r r o r c o u l d be a t t r i b u t e d  to the f a i l u r e on the p a r t o f the students t o w r i t e balanced chemical equations o r c o r r e c t chemical formulas. n i t i o n and the use o f the a p p r o p r i a t e volumetric  The recog-  stoichiometric ratio i n  a n a l y s i s c a l c u l a t i o n s r e q u i r e s knowledge of the  c o r r e c t l y balanced chemical equation between the r e a c t a n t s ; t h i s i n t u r n r e q u i r e s knowledge o f the formulas f o r the reactants.  However, s i n c e Table 22 shows t h a t q u i t e a l a r g e  number o f students d i d not w r i t e chemical equations or c o r r e c t formulas  (60.3%)  (29.6%), i t i s not s u r p r i s i n g t h a t the  i n d i s c r i m i n a t e use of a l : l : ; r a t i o was p r e v a l e n t . In the second conceptual  error, the  subjects  viewed  the m o l a r i t y c a l c u l a t e d from the volumes used i n the r e a c t i o n to be a f r a c t i o n o f the c o n c e n t r a t i o n This i s exemplified  o f the o r i g i n a l s o l u t i o n .  by Pat's s o l u t i o n t o the f o l l o w i n g  problem: I f 25.0 mL o f 0.5 M HCl n e u t r a l i z e d 30.0 mL o f a 250.0 mL stock s o l u t i o n o f NaOH, what i s the m o l a r i t y o f the 250.0 mL stock s o l u t i o n ? Pat's s o l u t i o n : 0.5  mol —=— L  .0125 X  0.42  X  L 7" 1000 mL  TrTrTn  X  1 mol NaOH 1 mol HCl  mol L  X  1 30  X  X  25 mL L 30 X  =  i  m->r  .0125  1000 mL L  mol  =  T I  m  HCl  0.42 mol NaOH L  250 mL _ 3.5 mol L  From t h i s s o l u t i o n one can see t h a t Pat and other  students who  made s i m i l a r e r r o r s d i d not see the c o n c e n t r a t i o n  o f the s o l u -  156  t i o n as remaining these  students  i n v a r i a n t i n t h i s problem.  I t seems t h a t  employed a d i r e c t p r o p o r t i o n a l r e l a t i o n s h i p  between the c o n c e n t r a t i o n of the volume used i n t i t r a t i o n the c o n c e n t r a t i o n of the stock s o l u t i o n .  This error implied  t h a t thesecstudents  may  what a s o l u t i o n was  s i n c e they a p p l i e d a p r o p o r t i o n a l r e l a t i o n  where i t was  have had  and  not needed.  The  some misunderstanding of  a p p l i c a t i o n of t h i s  relation  may  be a t t r i b u t e d to the f a c t t h a t a part-whole r e l a t i o n s h i p  is  i n v o l v e d here.  However, t h i s r e l a t i o n i s not the common-  sense one which suggests t h a t the whole i s g r e a t e r than  one  of i t s p a r t s and which allows the use of mathematical operations.  However, the i n v a r i a n c e of the c o n c e n t r a t i o n seemed ;to  be more problematic  when the s u b j e c t s were;.asked to perform  c a l c u l a t i o n s than when asked t o g i v e q u a l i t a t i v e p r e d i c t i o n s . During  the interview, almost a l l the i n t e r v i e w s u b j e c t s  i n d i c a t e d t h a t the c o n c e n t r a t i o n of the stock s o l u t i o n would remain i n v a r i a n t when you  remove a p o r t i o n of i t (see  Table  2 5 , Question 3 ) . E r r o r number 3 , i n c o r r e c t formula  and or  formula  mass, c o u l d be a t t r i b u t e d to a number of b a s i c misunderstandings.  F i r s t l y , a b i l i t y to c a l c u l a t e the c o r r e c t molar or  formula mass of a chemical ledge of t h e i r formulas.  substance seemed to r e q u i r e knowSecondly, knowledge of the  symbols  f o r the atoms and the formulas f o r i o n i c s p e c i e s such as carbonate i o n i s needed together with a knowledge of v a l e n c i e s b e f o r e one  can w r i t e an a c c u r a t e  the students who knowledge.  formula.  I t .is p o s s i b l e t h a t  made t h i s e r r o r l a c k e d some of t h i s b a s i c  157 Error ratio  was  number 4,  required,  was  u s i n g a 2:1  detected  and was p a r t i c u l a r l y e v i d e n t reaction  of  i n about  familiar  It  this  would r e a c t  write balanced example, carbonate  to  the  i o n t o be  their  (error  any p r o b l e m s  in calculating  stance  the  used  tration  of  t h e ...  1:2,  the  t h e y assumed  mole r a t i o .  failure  of  the  and c o r r e c t  the  o r volume was  students  number 5 ) , the moles  of  d i d not the  seem t o  standard  substance t o be  appears  that  in  have  sub-  calculating  i n t h e r e a c t i o n whose c o n c e n -  found,  t h e y became  confused  That  real  For  ratios  standard  thus,  to  the  of  s u b s t a n c e by t h e  stoichiometric  d i d nbt..seem  to  recognize  between t h e c o e f f i c i e n t s  and t h e a c t u a l  again,  formulas.  w h e t h e r t h e y had t o d i v i d e o r m u l t i p l y t h e m o l e s  relationship  that  Once  s t o i c h i o m e t r i c mole  However, i t  the other  these subjects  H2S04  s u b s t a n c e w i t h known m o l a r i t y and volume)  i n the r e a c t i o n .  the moles  the  univalent.  calculations  (i.e.  students  since,  s t u d e n t s were noted.".to assume  T h o s e who r e v e r s e d in  is  chemical equations  some o f  seems t h a t  w i t h N a 2 C 0 3 i n a 1:2  c o u l d be a t t r i b u t e d  the  1:1  w i t h t h e r e a c t i o n between  and NaOH i n w h i c h t h e mole r a t i o H2S04  4% o f  where a  i n the problem i n v o l v i n g  H2S04 with N a 2 C 0 3 .  s t u d e n t s were a l r e a d y  mole r a t i o  moles  of  r e l a t i o n even though they  the nature  the is,  of  i n the balanced  substances used  p r o b a b l y a p p l y an i n v e r s e  ratio.  over  the equation  i n the r e a c t i o n .  They  r e l a t i o n w i t h o u t knowing the seem t o know how t o o b t a i n  the  coefficients. For  t h o s e who made e r r o r number 6,  that  is  assumed  158 d i r e c t proportion  between c o n c e n t r a t i o n  and volume of r e a c t i n g  s o l u t i o n s , i t c o u l d be argued t h a t they attempted t o e s t a b l i s h an e q u a l i t y between the r a t i o s o f the volumes and m o l a r i t i e s of the a c i d and base i n v o l v e d  i n the r e a c t i o n .  doing, they assumed t h a t the c o n c e n t r a t i o n the .volume o f the s o l u t i o n .  than u s i n g  upon as a more c o n c e p t u a l l y  v a r i e d d i r e c t l y with  Having t o decide whether there i s  a d i r e c t or i n d i r e c t proportional the c a l c u l a t i o n s r a t h e r  However, i n so  r e l a t i o n s h i p b e f o r e doing the formula, can be looked  demanding approach and, t h e r e f o r e ,  l i k e l y t o l e a d t o more e r r o r s . In the next c o n c e p t u a l e r r o r which was d e t e c t e d i n the s o l u t i o n s of about 10% o f the s u b j e c t s the  subjects  substance.  seemed t o confuse m o l a r i t y P a r t o f Sid's  ( e r r o r number 7 ) ,  w i t h moles o f  solution illustrates this error.  25 mL o f a c i d x .1 M = 2.5 M NaOH I t ' s not c l e a r whether t h i s d i f f i c u l t y was due t o improper assignments o f u n i t s t o the v a r i a b l e s  involved  by Rowell and Dawson (1980) on s t o i c h i o m e t r i c  as was found calculations.  I f i t was, then probably the use o f dimensional  analysis  ( f a c t o r l a b e l l i n g method) c o u l d help i n removing t h i s difficulty.  However, i t c o u l d be t h a t the students have  l e a r n t c e r t a i n formulas which have no proper meanings f o r them and  hence makes i t d i f f i c u l t f o r them t o c o r r e c t l y r e c a l l and  apply these a l g o r i t h m s . E r r o r number 8 made by about 19% o f the s u b j e c t s who c a l c u l a t e d the moles o f a substance i r r e s p e c t i v e o f i t s p u r i t y , was d e t e c t e d only  i n s o l u t i o n s t o the t h i r d problem  159 i n the V o l u m e t r i c A n a l y s i s who  Test  (see Appendix A ) .  made t h i s e r r o r attempted i n the  initial  The  subjects  stages of t h e i r  s o l u t i o n to c a l c u l a t e the number of moles of B a f O H ^ . taking  i n t o account t h a t i t was  impure, the  the given weight of the impure substance by These s u b j e c t s  d i d not,  therefore,  subjects  i t s pure s t a t e .  divided  i t s molar mass.  seem to r e a l i z e t h a t  moles of a substance i n a g i v e n weight of an can be c a l c u l a t e d only when the  Without  impure substance  substance can be  Thus, they d i d not  the  isolated into  appear to show much under-  standing of when a p a r t i c u l a r memorized a l g o r i t h m  was  applicable. The of a c i d and  next conceptual e r r o r  base together i n c a l c u l a t i n g the unknown c o n c e n t r a -  t i o n or v i c e v e r s a , of the  (number 9), adding volumes  subjects.  was  d e t e c t e d i n the  I t c o u l d be t h a t these s u b j e c t s  d i f f e r e n t i a t e the problems i n c l u d e d Test  t i t r a t i o n graph.  6%  not Analysis  involved  of an a c i d s o l u t i o n from a  In t h i s l a t t e r problem, students may  asked to c a l c u l a t e the c o n c e n t r a t i o n a f t e r a c e r t a i n amount of base has student has  did  i n the V o l u m e t r i c  from problems they've done i n c l a s s which  c a l c u l a t i n g the c o n c e n t r a t i o n  the  s o l u t i o n s of about  be  of the remaining a c i d  been added.  To do  this,  to c a l c u l a t e the moles of a c i d remaining a f t e r  the a d d i t i o n of a s p e c i f i e d volume of base and t h i s by the t o t a l volume of s o l u t i o n  then d i v i d e  ( i . e . volume of a c i d +  volume of added base) i n order to f i n d the c o n c e n t r a t i o n the remaining a c i d .  I t seemed the  students who  e r r o r employed t h i s method i n c a l c u l a t i n g the  of  made t h i s concentration  160 at  the  stoichiometric  point  (which i s  the V o l u m e t r i c A n a l y s i s Test  are  in  test  solving question Molarity of  Also,John (see  a c i d = 0.50  in calculating  A p p e n d i x A) 20 x  John  2 on t h e  .02 .06  = 6.7  the volume.  to obtain  a composite  of  therefore,  learned  wrote:  acid  i n question  1  to  Thus,  situations here  molarity  illustrates  for  i n v o l v i n g the  he c o m b i n e d t h e the  total  calcu-  molarities  solution.  an i n d i s c r i m i n a t e  This  application  algorithms.  difficulties  number o f  w i t h problems  the  subjects'  to  students also  calculation  substance  c o u l d be a t t r i b u t e d  the  r e q u i r i n g the  a substance or the  g i v e n mass o f  the  A p p e n d i x A)  George,  M x 36 = 0 . 3 2 M 56  t h e volume o f  this  A substantial  mass o f  (see  F o r example,  mL H „ S O . ^  of  error,  about).  in  wrote:  seemed t o e x t e n d  lation  what t h e p r o b l e m s  four  (error  encountered  calculation  of  of  concentration  number 1 0 ) .  This  basic misunderstandings  the from a  error detected  in  solutions.  a)  d e f i n i n g m o l a r i t y as  grams p e r m i l l i l i t r e ,  b)  m u l t i p l y i n g grams by m o l a r mass t o o b t a i n m o l e s  of  the  substance, c)  d i v i d i n g m o l a r mass by grams o f  substance to o b t a i n  moles,  and, d) It  using is  e q u i l i b r i u m constant  evident  illustrate formulas  from the  incorrect  while  (d)  equations.  above t h a t  recall  (a),  (b)  and a p p l i c a t i o n  illustrates  the  and of  (c)  again,  memorized  indiscriminate  application  161 of  learned  formulas.  It  s t u d e n t s were a b l e t o use  of,  solve  problems  or the c a l c u l a t i o n o f ,  However, they  should',.be n o t e d t h a t  as  s o o n as  the r e a c t i o n  balanced  formulas leading to  equations.  the equations  first  the  conceptual  the  following:  o b t a i n volume,  Again,  required for It  two c o n c e p t u a l  22)  which r e l a t e  the o t h e r content tiate  to  area.  these errors  the  (1975)  may be v e r y d e m a n d i n g , culty  is  due t o  proportions  the  the  write  above  relate the  to  by volume t o  (error  to  the  the  that  numbers  structure  correct  of  apart  5 and 6,  analysis  from Table  the problems,  a misunderstanding  of  all  the  seems t o  substan-  volumetric analysis  problems  does n o t  i n a b i l i t y of  However, i t  25  obtain  illustrated-that  discussions  above  claim that it  the  calculations.  logical  Although,  Herron's  correct  were n o t i n c l u d e d i n T a b l e  difficulties  difficulties  involved  d i v i d i n g c o n c e n t r a t i o n by m o l e s  seems f r o m t h e  the  inability  t h a n 3% o f  s u b j e c t s who made them p r o b a b l y c o u l d n o t r e c a l l formulas  the  them c o r r e c t l y .  and m u l t i p l y i n g m o l e s  concentration.  to  s t u d e n t s were a b l e t o  w h i c h were made by l e s s  therefore,  error,  chemical substances  but c o u l d not balance  s u b j e c t s and w h i c h ,  masses,  algorithms.  an i n a b i l i t y t o w r i t e  Some o f  Other e r r o r s  i n c l u d e d the  for  the  substance.  involving  e r r o r number 11 c o u l d be a t t r i b u t e d  to w r i t e c o r r e c t in  the  they c o n f r o n t e d problems  As e x p l a i n e d u n d e r t h e conceptual  these  which d i d not r e q u i r e  t h e mass o f  started applying incorrect  generally  indicate  the  that  subjects  c o u l d be a r g u e d  that  this  diffi-  to handle part  of  inverse  the  162 difficulty  relates  and i n v e r s e  is  the d i f f i c u l t i e s to  the  lack of  question  discussions  stated  direct  basic  s u c h as  formulas,  To i d e n t i f y  the  question  specific  the  c o u l d be  subjects  concluded  include  the problems  equations  errors  (i.e.  and  the  the  and w r i t i n g  difficulties  titrations, used  to  that  students  an a n a l y s i s identify  The s p e c i f i c  was  these  research  was:  Question  4:  What p r a c t i c a l  laboratory  in  experiment?  a titration  The r e s u l t s u s e d by t h e  it  to  in Titration  d u r i n g the t i t r a t i o n .  Research  an  problems.  from the c h e c k l i s t  addressed  to  themselves,  proportionality)  balancing of  have when p e r f o r m i n g l a b o r a t o r y  difficulties  s t u d e n t s c o u l d be  discussions  of  1981),  concepts.  section,  structure  Manual S k i l l s  done on t h e d a t a  the  these  t h e above  and i n v e r s e  r e q u i r e d by t h e  concep-  on t h e i r own ( K a s s ,  made by t h e  logical  requirement of  of  in this  errors  to both the  direct  a p p l i c a t i o n of. t h e c o n c e p t s  relating  the conceptual  concepts  rarely  i n which these i d e a s manifest  Finally,  related  of both  students  e n c o u n t e r e d by some o f  during the classroom  that  since  i n c h e m i c a l terms  e v e r y day c o n t e x t  research  presence  same p r o b l e m .  suggested that  the world  attributed  the p o t e n t i a l  p r o p o r t i o n i n the  It tualize  to  subjects  of are  skills  the a n a l y s i s presented  do s u b j e c t s  of  the  display  laboratory  i n Table 23.  skills  The t a b l e  also  163 Table 2 3 Manual S k i l l s i n T i t r a t i o n  Type of S k i l l s  Total no. of students  Percent  A.  S k i l l s i n Using Lab Equipment  1.  S k i l l s displayed i n the handling and reading of burette a) Inadequate b) Adequate c) Superior  17 12 18  36.2 25.5 38.3  S k i l l s displayed i n the handling and reading of pipette a) Inadequate b) Adequate c) Superior  3 6 8  17.6 35.3 47.1  2.  B.  S k i l l s i n Performing Lab Techniques  3.  Does the student need help i n setting up the experiment? a) Yes b) No  9 38  19.1 80.9  Is an indicator added to the solution before titration? a) Yes b) No  46 1  97.9 2.1  Is caution exercised near the endpoint? a) Yes b) No  27 20  57.4 42.6  Does the student overshoot the endpoint? a) Yes b) No  23 24  48.9 51.1  Is the f l a s k constantly shaken during the titration? a) Yes b) No  39 8  83.0 17.0  Is a white background used to detect colour changes? a) Yes b) No  18 29  38.3 61.7  4.  5.  6.  7.  8.  164 T a b l e 2 3 Continued ...  T o t a l no. of students  Percent  9 38  19.1 80.9  32 15  68.1 31.9  26 21  55.3 44.7  1 34 12  2.1 72.3 25.5  17 6 24  36.2 12.8 51.0  14. The i n d i c a t o r chosen f o r t i t r a t i o n i s : a) P h e n o l p h t h a l e i n b) Bromthymol b l u e c) Methyl orange d) A l l 3 i n d i c a t o r s  31 13 2 1  66.0 27.7 4.3 2.1  15. C o n t a i n e r used f o r t i t r a t i o n : a) C o n i c a l f l a s k b) Beaker  28 19  59.6 40.4  25 22  53.2 46.8  Type o f S k i l l s 9. Is the i n s i d e o f the f l a s k o r beaker r i n s e d w i t h d i s t i l l e d water d u r i n g the t i t r a t i o n ? a) Yes b) No 10. Is the f i r s t a) Yes b) No  t i t r a t i o n regarded as f i n a l ?  11. Are d a t a r e c o r d e d immediately are made? a) Yes b) No 12. To how read? a) b) c)  a f t e r readings  many decimal p l a c e s i s the b u r e t t e 2 1 0  13. To o b t a i n volume o f a c i d student a) P i p e t t e b) B u r e t t e c) Measuring c y l i n d e r  16. I s f u n n e l used f o r t r a n s f e r r i n g into burette a) Yes b) No  uses:  solutions  165 g i v e s the t o t a l number and the percentage s t r a t i n g each s p e c i f i c s k i l l .  of s u b j e c t s demon-  For example, about 38% of the  s u b j e c t s demonstrated a c c u r a t e s k i l l s i n the h a n d l i n g of a burette.  Table 24 g i v e s a s i m i l a r i n f o r m a t i o n f o r the  inter-  view s u b j e c t s when they were grouped a c c o r d i n g t o the p r i o r number of l a b o r a t o r y t i t r a t i o n s performed. are those who laboratory  have completed  three groups  to 5 and 6 t o 12  titrations. The  l a r g e percentage  an adequate s k i l l to  0 to 2,3  The  of s u b j e c t s showing a t l e a s t  i n the use of the b u r e t t e may  be a s c r i b e d  the f a c t t h a t i n v a r i a b l y i n any t i t r a t i o n conducted  s c h o o l s , a b u r e t t e i s used.  As such, even.those who  i n the  have done  l e s s than f i v e t i t r a t i o n s seemed-to know to-some extent t h a t they have t o  (i) r i n s e the b u r e t t e w i t h d i s t i l l e d water and  the base t o be used i n it., ( i i ) read the bottom of the meniscus, and  ( i i i ) remove any trapped bubbles of a i r from the b u r e t t e .  However, i t seems from Table 24, t h a t doing more t i t r a t i o n s may  a l l o w these ideas t o become more entrenched  as p a r t of  the students' l a b o r a t o r y t e c h n i q u e s . The  few students s e l e c t i n g the p i p e t t e f o r use i n  measuring the volume of the base, imply t h a t , i n g e n e r a l , students may pipette. t o r y work.  not be conversant with the o p e r a t i o n of the  I t c o u l d be t h a t they do not use i t i n t h e i r l a b o r a Thus they tend t o use the measuring c y l i n d e r with  which they are f a m i l a r even though they were t o l d t o be a c c u r a t e i n t h e i r work as p o s s i b l e . those who  as  I t seems (Table 24) t h a t  have done l e s s than 5 t i t r a t i o n s tend t o a v o i d the  use of the p i p e t t e i n t h e i r work.  166  Table  24 -  Manual S k i l l s D i s p l a y e d by Subjects C l a s s i f i e d A c c o r d i n g t o the number o f Lab T i t r a t i o n s performed s i n c e Grade 1 1 * " Number o f Lab T i t r a t i o n s 0-2 (N=17)  3-5 (N=13)  6-12 (N=17)  8(47.1) 3(17.6) 6(35.3)  7(53.8) 5(38.5) 1(7.7)  3(17.6) 3(17.6) 11(64.7)  1(5.9) 1(5.9) 1(5.9)  1(7.7) 2(15.4) 0  1(5.9) 3(17.6) 7(41.2)  5(29.4) 12 ( 7 0 . 6 ) :  3(23.1) 10 ( 7 6 . 9 )  1(5.9) 16 (94 .1)  16 (94.1)  13 ( 1 0 0 . 0 ) .  17 (100. 0)  SKILLS 1. S k i l l s d i s p l a y e d i n . the h a n d l i n g and reading of burette a) Inadequate b) Adequate c) S u p e r i o r  2 . . . S k i l l s displayed i n h a n d l i n g and reading o f p i p e t t e a) Inadequate b) Adequate c) S u p e r i o r  3.  Does t h e student need h e l p i n s e t t i n g up the experiment? a) Yes b) NO  4.  I s an i n d i c a t o r added to the s o l u t i o n before titration? a) Yes  b) No 5.  Is caution exercised near the endpoint?  %  *"  a) Yes b) No  6.  0  1.(5.9)  0  „ .  10(59.9) 7(41.2)  6(46.2) 7(53.8)  11(64.7) 6(35.3)  8 (47.1) 9(52.9)  9 (69.2) 4(30.8)  6 (35 3) 11(64, 7)  Does t h e student overshoot the endpoint? a) Yes b) No  7. I s t h e f l a s k c o n s t a n t l y shaken d u r i n g t h e titration? a) Yes b) NO  . 12(70.6) 5(29.4)  '.  12(92.3) 1(7.7)  15(88.2) 2(11.8)  167  SKILLS  Number o f Lab T i t r a t i o n s . 0-2 3-5.. 6-12 (N=17) (N=17) (N=13)  8. Is a white background used t o d e t e c t c o l o u r changes? a) Yes b) No  6(35.3) 11(64.7)  3(23.1) 10(76.9)  9(52.9) 8(47.1)  9. I s the i n s i d e o f the f l a s k o r beaker r i n s e d with d i s t i l l e d water d u r i n g the titration? a) Yes b) No  6 (35.3) 11(64.7)  1 (7.7) 12(92.3)  4 (23.5) 13(17.6)  10. Is the f i r s t t i t r a t i o n regarded as final? a) Yes b) No  13(76.5) 4(23.5)  10(76.9) 3(23.1)  9(52.9) 8(47.1)  11. Are data recorded immediately a f t e r readings are made? a) Yes b) No  11(64.7) 6(35.3)  9(69.2) 4(30.8)  6(35.3) 11(64.7)  12. To how many decimal p l a c e s i s the b u r e t t e read? a) 2 b) 1 c) 0  0 13(76.5) 4 (23.5)  0 9(69.2) 4 (30.8)  1 (5.9) 12(70.6) 4 (23.5)  13. To o b t a i n volume o f a c i d , student uses a) P i p e t t e b) B u r e t t e c) Measuring c y l i n d e r  3(17.6) 4(23.5) 10(58.8);  3(23.1) 0 10 (76.9)  11(64.7) 2(11.8) 1 4 (23.5)  14. The i n d i c a t o r chosen for t i t r a t i o n i s a) P h e n o l p h t h a l e i n b) Bromothymol blue c) Methyl orange d) A l l 3 i n d i c a t o r s  9(52.9) 7(41.2) 1(5.9) 0  10(76.9) 3(23.1) 0 0  12(70.6) 3(17.6) 1(5.9) 1(5.9)  168  SKILLS  Number 0-2 (N=17)  o f Lab T i t r a t i o n s 3-5 6-12 (N=13) (N=17)  15. C o n t a i n e r u s e d f o r titration a) C o n i c a l f l a s k b) B e a k e r  6(35.3) 11(64.7)  8(61.5) 5(38.5)  14(82.4) 3(17.6)  16. F u n n e l u s e d f o r transferring solution into buret a) Y e s b) No  6(35.3) 11(64.7)  4(30.8) 9(69.2)  15(88.2) 2(11.8)  * Percentage of t o t a l  subjects i n parenthesis  169 This  seems t o be  also  t r u e f o r the  t h e y were g i v e n t h e o p t i o n t o use f u n n e l and  conical  flask)  t h e many s t u d e n t s who difficulties beaker.  n o t mix  solution well.  the t i t r a t i o n  the  seems as fact  swirling run  titration  what t h e y  of  funnel for transfering  by  the experience  of the  titration, experiment,  2 3 and  almost  shook t h e t i t r a t i n g half  been u s i n g i n  titrations)  flask.  of the  of the  endpoint. by  their with  the  choice  seemed t o be i n f l u e n c e d laboratory titrations.  suggest  subjects  that  these  observation of  r e p r e s e n t the  24 also  repeat  i n f l u e n c e s the  Similarly,  t i t r a t i o n s may  flask  solu-  the  to  i t seems t h a t e x p e r i e n c e  solutions  f l a s k ; . b e f o r e commencing t h e  almost  had  a  phenomenon.  show t h a t i n p e r f o r m i n g  a l l s t u d e n t s were a b l e t o s e t up  added d r o p s  had  dictated  t h e p a r t o f most o f t h e  adequately  Tables  p a r t of  t h e b e a k e r was  have t h e n o t i o n t h a t one  phenomenon may  spilled  student with  on  (68.1%) t o r e p e a t t h e i r  some o f t h e  t o o b t a i n a more a c c u r a t e  c h o i c e of the beaker or c o n i c a l  s t u d e n t s may  example,  i n the  the beaker d i d not attempt  Here t o o ,  failure  solution  i t spilled  ( i . e . the-vnumber o f p r i o r  The  For  s t i r r e d with a glass rod could  i f the c h o i c e t o use  t h a t t h i s was  the  T h o s e who  i n order  l a b o r a t o r y work.  (e.g.  chose the beaker f o r the t i t r a t i o n  some o f t h o s e who  the  equipment  experiment.  i n s w i r l i n g or s t i r r i n g  solution while  It  in their  T h o s e who t r i e d -to ; s w i r l  tion while  certain  subjects.when  indicator  the  the  to the a c i d  i n the  a d d i t i o n o f b a s e and c o n s t a n t l y  d u r i n g the t i t r a t i o n .  s u b j e c t s overan  the endpoint  However, because  they  170  d i d not exercise  c a u t i o n i n t h e a d d i t i o n o f b a s e as  a p p r o a c h e d t h e e n d p o i n t and d i d n o t a l s o use to detect of  the  asked that to  the c o l o u r changes  i n the  solution.  d i d not r e c o r d t h e i r readings  t o do s o .  In r e c o r d i n g t h e i r r e a d i n g s ,  a quarter  of  the  subjects  e v e n one d e c i m a l p l a c e .  burette  is  graduated  This  c o u l d be e s t i m a t e d most o f  the  is  despite  w h i l e the  (72.3%)  c o u l d a l s o be t h a t A large  need t o use  this  is  number o f  the d i s t i l l e d water  i n t o t h e m a i n body o f  the teachers  students use  it  to  are  involved  squirt  water  use  the next  skills well  the  place  place It  seemed  uncertainty place.  i n the  d i d not  schools.  see  the  solution.  study  the This  or  suggests  that  in their titrations.  Some  indicated that  t h e wash b o